LMFDB - Embedded newform 416.2.bk.a.271.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [416,2,Mod(15,416)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("416.15"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(416, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 6, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.bk (of order $12$, degree $4$, not minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$3.32177672409$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$12$ over $\Q(\zeta_{12})$
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			271.11
Character	$\chi$	$=$	416.271
Dual form			416.2.bk.a.175.11

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.57612 + 2.72991i) q^{3} +(-2.13831 + 2.13831i) q^{5} +(1.43298 - 0.383965i) q^{7} +(-3.46828 + 6.00723i) q^{9} +(1.14540 + 0.306909i) q^{11} +(-1.50651 - 3.27573i) q^{13} +(-9.20762 - 2.46717i) q^{15} +(-0.917474 - 0.529704i) q^{17} +(3.01087 - 0.806761i) q^{19} +(3.30673 + 3.30673i) q^{21} +(2.44668 + 4.23777i) q^{23} -4.14473i q^{25} -12.4089 q^{27} +(-0.430121 + 0.248330i) q^{29} +(-0.180109 + 0.180109i) q^{31} +(0.967448 + 3.61056i) q^{33} +(-2.24311 + 3.88518i) q^{35} +(0.869322 - 3.24435i) q^{37} +(6.56803 - 9.27557i) q^{39} +(1.95534 - 7.29743i) q^{41} +(0.979452 + 0.565487i) q^{43} +(-5.42907 - 20.2616i) q^{45} +(5.29277 + 5.29277i) q^{47} +(-4.15619 + 2.39957i) q^{49} -3.33950i q^{51} +13.2050i q^{53} +(-3.10549 + 1.79295i) q^{55} +(6.94787 + 6.94787i) q^{57} +(1.58626 + 5.92000i) q^{59} +(9.32964 + 5.38647i) q^{61} +(-2.66339 + 9.93992i) q^{63} +(10.2259 + 3.78314i) q^{65} +(3.22308 - 12.0287i) q^{67} +(-7.71248 + 13.3584i) q^{69} +(-1.99632 - 7.45038i) q^{71} +(4.04472 - 4.04472i) q^{73} +(11.3148 - 6.53258i) q^{75} +1.75917 q^{77} +0.933621i q^{79} +(-9.15307 - 15.8536i) q^{81} +(6.19701 + 6.19701i) q^{83} +(3.09451 - 0.829172i) q^{85} +(-1.35584 - 0.782794i) q^{87} +(2.06105 + 0.552258i) q^{89} +(-3.41656 - 4.11560i) q^{91} +(-0.775556 - 0.207810i) q^{93} +(-4.71307 + 8.16328i) q^{95} +(-3.94895 + 1.05812i) q^{97} +(-5.81624 + 5.81624i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 4 q^{3} - 20 q^{9} + 8 q^{11} - 12 q^{17} + 8 q^{19} - 8 q^{27} + 4 q^{33} + 4 q^{35} + 12 q^{43} - 60 q^{49} + 36 q^{57} + 64 q^{59} - 16 q^{65} + 8 q^{67} - 12 q^{73} - 24 q^{75} - 8 q^{81} + 48 q^{83}+ \cdots - 168 q^{99}+O(q^{100}) $ 48 * q + 4 * q^3 - 20 * q^9 + 8 * q^11 - 12 * q^17 + 8 * q^19 - 8 * q^27 + 4 * q^33 + 4 * q^35 + 12 * q^43 - 60 * q^49 + 36 * q^57 + 64 * q^59 - 16 * q^65 + 8 * q^67 - 12 * q^73 - 24 * q^75 - 8 * q^81 + 48 * q^83 + 12 * q^89 - 104 * q^91 + 4 * q^97 - 168 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

$n$	$261$	$287$	$353$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{7}{12}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	1.57612	+	2.72991i	0.909970	+	1.57612i	0.814103	+	0.580721i	$0.197229\pi$
$3$	1.57612	+	2.72991i	0.909970	+	1.57612i	0.0958679	+	0.995394i	$0.469437\pi$
$4$		0			0
$5$	−2.13831	+	2.13831i	−0.956281	+	0.956281i	−0.999084	−	0.0428025i	$-0.986371\pi$
$5$	−2.13831	+	2.13831i	−0.956281	+	0.956281i	0.0428025	+	0.999084i	$0.486371\pi$
$6$		0			0
$7$	1.43298	−	0.383965i	0.541614	−	0.145125i	0.0223680	−	0.999750i	$-0.492879\pi$
$7$	1.43298	−	0.383965i	0.541614	−	0.145125i	0.519246	+	0.854625i	$0.326213\pi$
$8$		0			0
$9$	−3.46828	+	6.00723i	−1.15609	+	2.00241i
$10$		0			0
$11$	1.14540	+	0.306909i	0.345351	+	0.0925365i	0.427325	−	0.904098i	$-0.359456\pi$
$11$	1.14540	+	0.306909i	0.345351	+	0.0925365i	−0.0819743	+	0.996634i	$0.526123\pi$
$12$		0			0
$13$	−1.50651	−	3.27573i	−0.417831	−	0.908525i
$13$	−1.50651	−	3.27573i	−0.417831	−	0.908525i
$14$		0			0
$15$	−9.20762	−	2.46717i	−2.37740	−	0.637021i
$16$		0			0
$17$	−0.917474	−	0.529704i	−0.222520	−	0.128472i	0.384597	−	0.923085i	$-0.374341\pi$
$17$	−0.917474	−	0.529704i	−0.222520	−	0.128472i	−0.607117	+	0.794613i	$0.707674\pi$
$18$		0			0
$19$	3.01087	−	0.806761i	0.690742	−	0.185084i	0.103662	−	0.994613i	$-0.466944\pi$
$19$	3.01087	−	0.806761i	0.690742	−	0.185084i	0.587080	+	0.809529i	$0.300277\pi$
$20$		0			0
$21$	3.30673	+	3.30673i	0.721587	+	0.721587i
$22$		0			0
$23$	2.44668	+	4.23777i	0.510167	+	0.883635i	0.999931	+	0.0117799i	$0.00374976\pi$
$23$	2.44668	+	4.23777i	0.510167	+	0.883635i	−0.489764	+	0.871855i	$0.662917\pi$
$24$		0			0
$25$		−	4.14473i		−	0.828947i
$26$		0			0
$27$	−12.4089			−2.38810
$28$		0			0
$29$	−0.430121	+	0.248330i	−0.0798714	+	0.0461138i	−0.539404	−	0.842047i	$-0.681350\pi$
$29$	−0.430121	+	0.248330i	−0.0798714	+	0.0461138i	0.459532	+	0.888161i	$0.348017\pi$
$30$		0			0
$31$	−0.180109	+	0.180109i	−0.0323486	+	0.0323486i	−0.723096	−	0.690747i	$-0.757282\pi$
$31$	−0.180109	+	0.180109i	−0.0323486	+	0.0323486i	0.690747	+	0.723096i	$0.257282\pi$
$32$		0			0
$33$	0.967448	+	3.61056i	0.168411	+	0.628518i
$34$		0			0
$35$	−2.24311	+	3.88518i	−0.379155	+	0.656716i
$36$		0			0
$37$	0.869322	−	3.24435i	0.142916	−	0.533368i	−0.856924	−	0.515443i	$-0.827627\pi$
$37$	0.869322	−	3.24435i	0.142916	−	0.533368i	0.999839	−	0.0179251i	$-0.00570604\pi$
$38$		0			0
$39$	6.56803	−	9.27557i	1.05173	−	1.48528i
$40$		0			0
$41$	1.95534	−	7.29743i	0.305373	−	1.13967i	−0.627251	−	0.778817i	$-0.715820\pi$
$41$	1.95534	−	7.29743i	0.305373	−	1.13967i	0.932624	−	0.360850i	$-0.117513\pi$
$42$		0			0
$43$	0.979452	+	0.565487i	0.149365	+	0.0862359i	0.572820	−	0.819681i	$-0.305849\pi$
$43$	0.979452	+	0.565487i	0.149365	+	0.0862359i	−0.423455	+	0.905917i	$0.639183\pi$
$44$		0			0
$45$	−5.42907	−	20.2616i	−0.809318	−	3.02042i
$46$		0			0
$47$	5.29277	+	5.29277i	0.772030	+	0.772030i	0.978461	−	0.206431i	$-0.0661849\pi$
$47$	5.29277	+	5.29277i	0.772030	+	0.772030i	−0.206431	+	0.978461i	$0.566185\pi$
$48$		0			0
$49$	−4.15619	+	2.39957i	−0.593741	+	0.342796i
$50$		0			0
$51$		−	3.33950i		−	0.467623i
$52$		0			0
$53$			13.2050i			1.81385i	0.421292	+	0.906925i	$0.361577\pi$
$53$			13.2050i			1.81385i	−0.421292	+	0.906925i	$0.638423\pi$
$54$		0			0
$55$	−3.10549	+	1.79295i	−0.418744	+	0.241762i
$56$		0			0
$57$	6.94787	+	6.94787i	0.920268	+	0.920268i
$58$		0			0
$59$	1.58626	+	5.92000i	0.206513	+	0.770718i	0.988983	+	0.148029i	$0.0472930\pi$
$59$	1.58626	+	5.92000i	0.206513	+	0.770718i	−0.782470	+	0.622689i	$0.786040\pi$
$60$		0			0
$61$	9.32964	+	5.38647i	1.19454	+	0.689667i	0.959332	−	0.282279i	$-0.0910903\pi$
$61$	9.32964	+	5.38647i	1.19454	+	0.689667i	0.235206	+	0.971946i	$0.424424\pi$
$62$		0			0
$63$	−2.66339	+	9.93992i	−0.335556	+	1.25231i
$64$		0			0
$65$	10.2259	+	3.78314i	1.26837	+	0.469241i
$66$		0			0
$67$	3.22308	−	12.0287i	0.393762	−	1.46954i	−0.430118	−	0.902773i	$-0.641528\pi$
$67$	3.22308	−	12.0287i	0.393762	−	1.46954i	0.823879	−	0.566765i	$-0.191805\pi$
$68$		0			0
$69$	−7.71248	+	13.3584i	−0.928474	+	1.60816i
$70$		0			0
$71$	−1.99632	−	7.45038i	−0.236920	−	0.884197i	−0.977274	−	0.211980i	$-0.932009\pi$
$71$	−1.99632	−	7.45038i	−0.236920	−	0.884197i	0.740354	−	0.672217i	$-0.234658\pi$
$72$		0			0
$73$	4.04472	−	4.04472i	0.473399	−	0.473399i	−0.429614	−	0.903013i	$-0.641350\pi$
$73$	4.04472	−	4.04472i	0.473399	−	0.473399i	0.903013	+	0.429614i	$0.141350\pi$
$74$		0			0
$75$	11.3148	−	6.53258i	1.30652	−	0.754317i
$76$		0			0
$77$	1.75917			0.200476
$78$		0			0
$79$			0.933621i			0.105041i	0.998620	+	0.0525203i	$0.0167254\pi$
$79$			0.933621i			0.105041i	−0.998620	+	0.0525203i	$0.983275\pi$
$80$		0			0
$81$	−9.15307	−	15.8536i	−1.01701	−	1.76151i
$82$		0			0
$83$	6.19701	+	6.19701i	0.680210	+	0.680210i	0.960047	−	0.279837i	$-0.0902805\pi$
$83$	6.19701	+	6.19701i	0.680210	+	0.680210i	−0.279837	+	0.960047i	$0.590281\pi$
$84$		0			0
$85$	3.09451	−	0.829172i	0.335647	−	0.0899363i
$86$		0			0
$87$	−1.35584	−	0.782794i	−0.145361	−	0.0839244i
$88$		0			0
$89$	2.06105	+	0.552258i	0.218471	+	0.0585392i	0.366394	−	0.930460i	$-0.380592\pi$
$89$	2.06105	+	0.552258i	0.218471	+	0.0585392i	−0.147923	+	0.988999i	$0.547259\pi$
$90$		0			0
$91$	−3.41656	−	4.11560i	−0.358153	−	0.431432i
$92$		0			0
$93$	−0.775556	−	0.207810i	−0.0804214	−	0.0215488i
$94$		0			0
$95$	−4.71307	+	8.16328i	−0.483551	+	0.837535i
$96$		0			0
$97$	−3.94895	+	1.05812i	−0.400955	+	0.107436i	−0.453661	−	0.891174i	$-0.649882\pi$
$97$	−3.94895	+	1.05812i	−0.400955	+	0.107436i	0.0527054	+	0.998610i	$0.483216\pi$
$98$		0			0
$99$	−5.81624	+	5.81624i	−0.584554	+	0.584554i
$100$		0			0
$101$	−5.45375	−	9.44617i	−0.542668	−	0.939929i	−0.998750	−	0.0499909i	$-0.984081\pi$
$101$	−5.45375	−	9.44617i	−0.542668	−	0.939929i	0.456081	−	0.889938i	$-0.349253\pi$
$102$		0			0
$103$	5.93866			0.585154			0.292577	−	0.956242i	$-0.405487\pi$
$103$	5.93866			0.585154			0.292577	+	0.956242i	$0.405487\pi$
$104$		0			0
$105$	−14.1416			−1.38008
$106$		0			0
$107$	−3.86681	−	6.69752i	−0.373819	−	0.647474i	0.616330	−	0.787488i	$-0.288619\pi$
$107$	−3.86681	−	6.69752i	−0.373819	−	0.647474i	−0.990150	+	0.140014i	$0.955285\pi$
$108$		0			0
$109$	4.46618	−	4.46618i	0.427782	−	0.427782i	−0.460090	−	0.887872i	$-0.652183\pi$
$109$	4.46618	−	4.46618i	0.427782	−	0.427782i	0.887872	+	0.460090i	$0.152183\pi$
$110$		0			0
$111$	10.2269	−	2.74030i	0.970699	−	0.260098i
$112$		0			0
$113$	5.74089	−	9.94352i	0.540058	−	0.935408i	−0.458842	−	0.888518i	$-0.651736\pi$
$113$	5.74089	−	9.94352i	0.540058	−	0.935408i	0.998900	−	0.0468899i	$-0.0149310\pi$
$114$		0			0
$115$	−14.2934	−	3.82991i	−1.33287	−	0.357141i
$116$		0			0
$117$	24.9031	+	2.31119i	2.30229	+	0.213670i
$118$		0			0
$119$	−1.51811	−	0.406775i	−0.139165	−	0.0372890i
$120$		0			0
$121$	−8.30853	−	4.79693i	−0.755321	−	0.436085i
$122$		0			0
$123$	23.0032	−	6.16368i	2.07413	−	0.555761i
$124$		0			0
$125$	−1.82882	−	1.82882i	−0.163575	−	0.163575i
$126$		0			0
$127$	−4.40510	−	7.62986i	−0.390890	−	0.677041i	0.601678	−	0.798739i	$-0.294499\pi$
$127$	−4.40510	−	7.62986i	−0.390890	−	0.677041i	−0.992567	+	0.121698i	$0.961166\pi$
$128$		0			0
$129$			3.56509i			0.313889i
$130$		0			0
$131$	14.7822			1.29153			0.645765	−	0.763536i	$-0.276539\pi$
$131$	14.7822			1.29153			0.645765	+	0.763536i	$0.276539\pi$
$132$		0			0
$133$	4.00474	−	2.31214i	0.347255	−	0.200488i
$134$		0			0
$135$	26.5341	−	26.5341i	2.28369	−	2.28369i
$136$		0			0
$137$	0.0261039	+	0.0974210i	0.00223021	+	0.00832324i	0.967032	−	0.254655i	$-0.0819619\pi$
$137$	0.0261039	+	0.0974210i	0.00223021	+	0.00832324i	−0.964802	+	0.262978i	$0.915295\pi$
$138$		0			0
$139$	−3.39564	+	5.88143i	−0.288015	+	0.498856i	−0.973336	−	0.229385i	$-0.926329\pi$
$139$	−3.39564	+	5.88143i	−0.288015	+	0.498856i	0.685321	+	0.728241i	$0.259662\pi$
$140$		0			0
$141$	−6.10678	+	22.7908i	−0.514284	+	1.91933i
$142$		0			0
$143$	−0.720205	−	4.21439i	−0.0602266	−	0.352425i
$144$		0			0
$145$	0.388724	−	1.45074i	0.0322818	−	0.120477i
$146$		0			0
$147$	−13.1013	−	7.56401i	−1.08057	−	0.623869i
$148$		0			0
$149$	−3.37169	−	12.5833i	−0.276220	−	1.03087i	−0.955020	−	0.296543i	$-0.904166\pi$
$149$	−3.37169	−	12.5833i	−0.276220	−	1.03087i	0.678800	−	0.734323i	$-0.262500\pi$
$150$		0			0
$151$	8.53114	+	8.53114i	0.694254	+	0.694254i	0.963165	−	0.268911i	$-0.0866638\pi$
$151$	8.53114	+	8.53114i	0.694254	+	0.694254i	−0.268911	+	0.963165i	$0.586664\pi$
$152$		0			0
$153$	6.36411	−	3.67432i	0.514507	−	0.297051i
$154$		0			0
$155$		−	0.770259i		−	0.0618687i
$156$		0			0
$157$			0.522164i			0.0416732i	0.999783	+	0.0208366i	$0.00663298\pi$
$157$			0.522164i			0.0416732i	−0.999783	+	0.0208366i	$0.993367\pi$
$158$		0			0
$159$	−36.0486	+	20.8126i	−2.85884	+	1.65055i
$160$		0			0
$161$	5.13318	+	5.13318i	0.404551	+	0.404551i
$162$		0			0
$163$	−6.43789	−	24.0265i	−0.504255	−	1.88190i	−0.470353	−	0.882479i	$-0.655873\pi$
$163$	−6.43789	−	24.0265i	−0.504255	−	1.88190i	−0.0339021	−	0.999425i	$-0.510793\pi$
$164$		0			0
$165$	−9.78920	−	5.65180i	−0.762089	−	0.439992i
$166$		0			0
$167$	0.809102	−	3.01961i	0.0626102	−	0.233665i	−0.927529	−	0.373751i	$-0.878071\pi$
$167$	0.809102	−	3.01961i	0.0626102	−	0.233665i	0.990139	+	0.140087i	$0.0447381\pi$
$168$		0			0
$169$	−8.46085	+	9.86985i	−0.650835	+	0.759220i
$170$		0			0
$171$	−5.59614	+	20.8851i	−0.427948	+	1.59712i
$172$		0			0
$173$	3.28703	−	5.69330i	0.249908	−	0.432853i	−0.713592	−	0.700561i	$-0.752933\pi$
$173$	3.28703	−	5.69330i	0.249908	−	0.432853i	0.963500	+	0.267708i	$0.0862663\pi$
$174$		0			0
$175$	−1.59143	−	5.93931i	−0.120301	−	0.448969i
$176$		0			0
$177$	−13.6609	+	13.6609i	−1.02682	+	1.02682i
$178$		0			0
$179$	−19.1587	+	11.0613i	−1.43199	+	0.826760i	−0.997273	−	0.0738069i	$-0.976485\pi$
$179$	−19.1587	+	11.0613i	−1.43199	+	0.826760i	−0.434718	+	0.900567i	$0.643152\pi$
$180$		0			0
$181$	−21.7718			−1.61828			−0.809141	−	0.587614i	$-0.800067\pi$
$181$	−21.7718			−1.61828			−0.809141	+	0.587614i	$0.800067\pi$
$182$		0			0
$183$			33.9588i			2.51031i
$184$		0			0
$185$	5.07855	+	8.79631i	0.373383	+	0.646718i
$186$		0			0
$187$	−0.888303	−	0.888303i	−0.0649592	−	0.0649592i
$188$		0			0
$189$	−17.7817	+	4.76459i	−1.29343	+	0.346573i
$190$		0			0
$191$	−1.44924	−	0.836719i	−0.104863	−	0.0605429i	0.446651	−	0.894708i	$-0.352617\pi$
$191$	−1.44924	−	0.836719i	−0.104863	−	0.0605429i	−0.551514	+	0.834165i	$0.685950\pi$
$192$		0			0
$193$	−5.90121	−	1.58123i	−0.424779	−	0.113819i	0.0400956	−	0.999196i	$-0.487234\pi$
$193$	−5.90121	−	1.58123i	−0.424779	−	0.113819i	−0.464874	+	0.885377i	$0.653900\pi$
$194$		0			0
$195$	5.78957	+	33.8785i	0.414600	+	2.42609i
$196$		0			0
$197$	−8.91936	−	2.38994i	−0.635478	−	0.170276i	−0.0733239	−	0.997308i	$-0.523361\pi$
$197$	−8.91936	−	2.38994i	−0.635478	−	0.170276i	−0.562154	+	0.827032i	$0.690027\pi$
$198$		0			0
$199$	10.3397	−	17.9089i	0.732961	−	1.26953i	−0.222651	−	0.974898i	$-0.571471\pi$
$199$	10.3397	−	17.9089i	0.732961	−	1.26953i	0.955612	−	0.294628i	$-0.0951957\pi$
$200$		0			0
$201$	37.9172	−	10.1599i	2.67447	−	0.716623i
$202$		0			0
$203$	−0.521003	+	0.521003i	−0.0365672	+	0.0365672i
$204$		0			0
$205$	11.4230	+	19.7853i	0.797820	+	1.38186i
$206$		0			0
$207$	−33.9430			−2.35920
$208$		0			0
$209$	3.69625			0.255675
$210$		0			0
$211$	10.7064	+	18.5440i	0.737059	+	1.27662i	0.953814	+	0.300398i	$0.0971194\pi$
$211$	10.7064	+	18.5440i	0.737059	+	1.27662i	−0.216755	+	0.976226i	$0.569547\pi$
$212$		0			0
$213$	17.1924	−	17.1924i	1.17801	−	1.17801i
$214$		0			0
$215$	−3.30356	+	0.885185i	−0.225301	+	0.0603691i
$216$		0			0
$217$	−0.188937	+	0.327248i	−0.0128259	+	0.0222151i
$218$		0			0
$219$	17.4167	+	4.66678i	1.17691	+	0.315352i
$220$		0			0
$221$	−0.352984	+	3.80340i	−0.0237442	+	0.255845i
$222$		0			0
$223$	9.84578	+	2.63817i	0.659323	+	0.176665i	0.572940	−	0.819597i	$-0.305803\pi$
$223$	9.84578	+	2.63817i	0.659323	+	0.176665i	0.0863822	+	0.996262i	$0.472469\pi$
$224$		0			0
$225$	24.8984	+	14.3751i	1.65989	+	0.958339i
$226$		0			0
$227$	−3.62620	+	0.971637i	−0.240679	+	0.0644898i	−0.377142	−	0.926155i	$-0.623093\pi$
$227$	−3.62620	+	0.971637i	−0.240679	+	0.0644898i	0.136463	+	0.990645i	$0.456427\pi$
$228$		0			0
$229$	−5.43768	−	5.43768i	−0.359332	−	0.359332i	0.504235	−	0.863567i	$-0.331775\pi$
$229$	−5.43768	−	5.43768i	−0.359332	−	0.359332i	−0.863567	+	0.504235i	$0.831775\pi$
$230$		0			0
$231$	2.77266	+	4.80239i	0.182428	+	0.315974i
$232$		0			0
$233$			2.92911i			0.191892i	0.995387	+	0.0959461i	$0.0305876\pi$
$233$			2.92911i			0.191892i	−0.995387	+	0.0959461i	$0.969412\pi$
$234$		0			0
$235$	−22.6352			−1.47656
$236$		0			0
$237$	−2.54870	+	1.47149i	−0.165556	+	0.0955838i
$238$		0			0
$239$	−14.3258	+	14.3258i	−0.926659	+	0.926659i	−0.997488	−	0.0708299i	$-0.977435\pi$
$239$	−14.3258	+	14.3258i	−0.926659	+	0.926659i	0.0708299	+	0.997488i	$0.477435\pi$
$240$		0			0
$241$	3.51676	+	13.1247i	0.226534	+	0.845437i	0.981784	+	0.190000i	$0.0608488\pi$
$241$	3.51676	+	13.1247i	0.226534	+	0.845437i	−0.755250	+	0.655437i	$0.772484\pi$
$242$		0			0
$243$	10.2392	−	17.7348i	0.656844	−	1.13769i
$244$		0			0
$245$	3.75618	−	14.0182i	0.239973	−	0.895593i
$246$		0			0
$247$	−7.17865	−	8.64742i	−0.456766	−	0.550222i
$248$		0			0
$249$	−7.15009	+	26.6845i	−0.453118	+	1.69106i
$250$		0			0
$251$	9.51992	+	5.49633i	0.600892	+	0.346925i	0.769392	−	0.638777i	$-0.220559\pi$
$251$	9.51992	+	5.49633i	0.600892	+	0.346925i	−0.168501	+	0.985702i	$0.553893\pi$
$252$		0			0
$253$	1.50181	+	5.60484i	0.0944182	+	0.352373i
$254$		0			0
$255$	7.14087	+	7.14087i	0.447179	+	0.447179i
$256$		0			0
$257$	−11.6764	+	6.74138i	−0.728355	+	0.420516i	−0.817820	−	0.575474i	$-0.804818\pi$
$257$	−11.6764	+	6.74138i	−0.728355	+	0.420516i	0.0894651	+	0.995990i	$0.471484\pi$
$258$		0			0
$259$		−	4.98287i		−	0.309621i
$260$		0			0
$261$		−	3.44511i		−	0.213247i
$262$		0			0
$263$	18.3764	−	10.6096i	1.13313	−	0.654216i	0.188413	−	0.982090i	$-0.439666\pi$
$263$	18.3764	−	10.6096i	1.13313	−	0.654216i	0.944721	+	0.327874i	$0.106332\pi$
$264$		0			0
$265$	−28.2364	−	28.2364i	−1.73455	−	1.73455i
$266$		0			0
$267$	1.74084	+	6.49692i	0.106538	+	0.397605i
$268$		0			0
$269$	25.1549	+	14.5232i	1.53372	+	0.885495i	0.999186	+	0.0403505i	$0.0128475\pi$
$269$	25.1549	+	14.5232i	1.53372	+	0.885495i	0.534537	+	0.845145i	$0.320486\pi$
$270$		0			0
$271$	3.45071	−	12.8782i	0.209616	−	0.782296i	−0.778377	−	0.627797i	$-0.783957\pi$
$271$	3.45071	−	12.8782i	0.209616	−	0.782296i	0.987993	−	0.154500i	$-0.0493765\pi$
$272$		0			0
$273$	5.85033	−	15.8136i	0.354078	−	0.957081i
$274$		0			0
$275$	1.27206	−	4.74738i	0.0767079	−	0.286278i
$276$		0			0
$277$	12.3608	−	21.4095i	0.742686	−	1.28637i	−0.208583	−	0.978005i	$-0.566885\pi$
$277$	12.3608	−	21.4095i	0.742686	−	1.28637i	0.951268	−	0.308364i	$-0.0997816\pi$
$278$		0			0
$279$	−0.457290	−	1.70663i	−0.0273772	−	0.102173i
$280$		0			0
$281$	−19.1640	+	19.1640i	−1.14323	+	1.14323i	−0.155371	+	0.987856i	$0.549657\pi$
$281$	−19.1640	+	19.1640i	−1.14323	+	1.14323i	−0.987856	+	0.155371i	$0.950343\pi$
$282$		0			0
$283$	−16.9756	+	9.80085i	−1.00909	+	0.582600i	−0.910926	−	0.412570i	$-0.864631\pi$
$283$	−16.9756	+	9.80085i	−1.00909	+	0.582600i	−0.0981670	+	0.995170i	$0.531298\pi$
$284$		0			0
$285$	−29.7134			−1.76007
$286$		0			0
$287$		−	11.2078i		−	0.661577i
$288$		0			0
$289$	−7.93883	−	13.7505i	−0.466990	−	0.808850i
$290$		0			0
$291$	−9.11257	−	9.11257i	−0.534189	−	0.534189i
$292$		0			0
$293$	−10.7632	+	2.88398i	−0.628792	+	0.168484i	−0.559121	−	0.829086i	$-0.688861\pi$
$293$	−10.7632	+	2.88398i	−0.628792	+	0.168484i	−0.0696703	+	0.997570i	$0.522195\pi$
$294$		0			0
$295$	−16.0507	−	9.26687i	−0.934507	−	0.539538i
$296$		0			0
$297$	−14.2132	−	3.80841i	−0.824733	−	0.220986i
$298$		0			0
$299$	10.1958	−	14.3989i	0.589641	−	0.832710i
$300$		0			0
$301$	1.62066	+	0.434254i	0.0934132	+	0.0250300i
$302$		0			0
$303$	17.1915	−	29.7765i	0.987624	−	1.71062i
$304$		0			0
$305$	−31.4676	+	8.43172i	−1.80183	+	0.482799i
$306$		0			0
$307$	8.52938	−	8.52938i	0.486797	−	0.486797i	−0.420497	−	0.907294i	$-0.638144\pi$
$307$	8.52938	−	8.52938i	0.486797	−	0.486797i	0.907294	+	0.420497i	$0.138144\pi$
$308$		0			0
$309$	9.36001	+	16.2120i	0.532473	+	0.922270i
$310$		0			0
$311$	−8.81350			−0.499768			−0.249884	−	0.968276i	$-0.580392\pi$
$311$	−8.81350			−0.499768			−0.249884	+	0.968276i	$0.580392\pi$
$312$		0			0
$313$	13.8328			0.781877			0.390938	−	0.920417i	$-0.372151\pi$
$313$	13.8328			0.781877			0.390938	+	0.920417i	$0.372151\pi$
$314$		0			0
$315$	−15.5595	−	26.9498i	−0.876677	−	1.51845i
$316$		0			0
$317$	−11.7109	+	11.7109i	−0.657749	+	0.657749i	−0.954847	−	0.297098i	$-0.903981\pi$
$317$	−11.7109	+	11.7109i	−0.657749	+	0.657749i	0.297098	+	0.954847i	$0.403981\pi$
$318$		0			0
$319$	−0.568875	+	0.152430i	−0.0318509	+	0.00853442i
$320$		0			0
$321$	12.1891	−	21.1121i	0.680329	−	1.17836i
$322$		0			0
$323$	−3.18974	−	0.854688i	−0.177482	−	0.0475561i
$324$		0			0
$325$	−13.5770	+	6.24409i	−0.753119	+	0.346360i
$326$		0			0
$327$	19.2315	+	5.15306i	1.06350	+	0.284965i
$328$		0			0
$329$	9.61666	+	5.55218i	0.530184	+	0.306102i
$330$		0			0
$331$	−13.8576	+	3.71314i	−0.761683	+	0.204092i	−0.618694	−	0.785632i	$-0.712338\pi$
$331$	−13.8576	+	3.71314i	−0.761683	+	0.204092i	−0.142989	+	0.989724i	$0.545671\pi$
$332$		0			0
$333$	16.4745	+	16.4745i	0.902799	+	0.902799i
$334$		0			0
$335$	18.8291	+	32.6130i	1.02874	+	1.78184i
$336$		0			0
$337$		−	26.5479i		−	1.44616i	−0.690765	−	0.723079i	$-0.742726\pi$
$337$		−	26.5479i		−	1.44616i	0.690765	−	0.723079i	$-0.257274\pi$
$338$		0			0
$339$	36.1932			1.96575
$340$		0			0
$341$	−0.261574	+	0.151020i	−0.0141650	+	0.00817819i
$342$		0			0
$343$	−12.3774	+	12.3774i	−0.668319	+	0.668319i
$344$		0			0
$345$	−12.0727	−	45.0561i	−0.649975	−	2.42574i
$346$		0			0
$347$	0.588542	−	1.01939i	0.0315946	−	0.0547235i	−0.849796	−	0.527112i	$-0.823275\pi$
$347$	0.588542	−	1.01939i	0.0315946	−	0.0547235i	0.881390	+	0.472389i	$0.156608\pi$
$348$		0			0
$349$	7.72090	−	28.8148i	0.413291	−	1.54242i	−0.374945	−	0.927047i	$-0.622338\pi$
$349$	7.72090	−	28.8148i	0.413291	−	1.54242i	0.788235	−	0.615374i	$-0.210995\pi$
$350$		0			0
$351$	18.6942	+	40.6483i	0.997822	+	2.16965i
$352$		0			0
$353$	6.75802	−	25.2213i	0.359693	−	1.34239i	−0.514781	−	0.857322i	$-0.672127\pi$
$353$	6.75802	−	25.2213i	0.359693	−	1.34239i	0.874474	−	0.485072i	$-0.161207\pi$
$354$		0			0
$355$	20.2000	+	11.6625i	1.07210	+	0.618979i
$356$		0			0
$357$	−1.28225	−	4.78542i	−0.0678638	−	0.253271i
$358$		0			0
$359$	−21.1519	−	21.1519i	−1.11636	−	1.11636i	−0.992272	−	0.124086i	$-0.960400\pi$
$359$	−21.1519	−	21.1519i	−1.11636	−	1.11636i	−0.124086	−	0.992272i	$-0.539600\pi$
$360$		0			0
$361$	−8.03999	+	4.64189i	−0.423157	+	0.244310i
$362$		0			0
$363$		−	30.2421i		−	1.58730i
$364$		0			0
$365$			17.2977i			0.905405i
$366$		0			0
$367$	17.9908	−	10.3870i	0.939114	−	0.542198i	0.0494315	−	0.998778i	$-0.484259\pi$
$367$	17.9908	−	10.3870i	0.939114	−	0.542198i	0.889682	+	0.456580i	$0.150926\pi$
$368$		0			0
$369$	37.0557	+	37.0557i	1.92904	+	1.92904i
$370$		0			0
$371$	5.07027	+	18.9225i	0.263235	+	0.982407i
$372$		0			0
$373$	−12.6271	−	7.29024i	−0.653804	−	0.377474i	0.136108	−	0.990694i	$-0.456541\pi$
$373$	−12.6271	−	7.29024i	−0.653804	−	0.377474i	−0.789912	+	0.613220i	$0.789874\pi$
$374$		0			0
$375$	2.11009	−	7.87496i	0.108965	−	0.406661i
$376$		0			0
$377$	1.46145	+	1.03485i	0.0752683	+	0.0532974i
$378$		0			0
$379$	−3.47679	+	12.9756i	−0.178591	+	0.666509i	0.817321	+	0.576182i	$0.195458\pi$
$379$	−3.47679	+	12.9756i	−0.178591	+	0.666509i	−0.995912	+	0.0903274i	$0.971209\pi$
$380$		0			0
$381$	13.8859	−	24.0511i	0.711396	−	1.23217i
$382$		0			0
$383$	−3.31127	−	12.3578i	−0.169198	−	0.631456i	−0.997467	−	0.0711259i	$-0.977341\pi$
$383$	−3.31127	−	12.3578i	−0.169198	−	0.631456i	0.828269	−	0.560331i	$-0.189326\pi$
$384$		0			0
$385$	−3.76166	+	3.76166i	−0.191712	+	0.191712i
$386$		0			0
$387$	−6.79402	+	3.92253i	−0.345360	+	0.199393i
$388$		0			0
$389$	−9.99223			−0.506626			−0.253313	−	0.967384i	$-0.581520\pi$
$389$	−9.99223			−0.506626			−0.253313	+	0.967384i	$0.581520\pi$
$390$		0			0
$391$		−	5.18405i		−	0.262169i
$392$		0			0
$393$	23.2985	+	40.3542i	1.17525	+	2.03560i
$394$		0			0
$395$	−1.99637	−	1.99637i	−0.100448	−	0.100448i
$396$		0			0
$397$	4.54273	−	1.21722i	0.227993	−	0.0610906i	−0.143014	−	0.989721i	$-0.545679\pi$
$397$	4.54273	−	1.21722i	0.227993	−	0.0610906i	0.371007	+	0.928630i	$0.379013\pi$
$398$		0			0
$399$	12.6239	+	7.28839i	0.631984	+	0.364876i
$400$		0			0
$401$	−2.57682	−	0.690456i	−0.128680	−	0.0344797i	0.193904	−	0.981020i	$-0.437885\pi$
$401$	−2.57682	−	0.690456i	−0.128680	−	0.0344797i	−0.322584	+	0.946541i	$0.604552\pi$
$402$		0			0
$403$	0.861327	+	0.318653i	0.0429058	+	0.0158733i
$404$		0			0
$405$	53.4720	+	14.3278i	2.65704	+	0.711952i
$406$		0			0
$407$	1.99144	−	3.44928i	0.0987121	−	0.170974i
$408$		0			0
$409$	25.4794	−	6.82718i	1.25987	−	0.337582i	0.433731	−	0.901042i	$-0.357197\pi$
$409$	25.4794	−	6.82718i	1.25987	−	0.337582i	0.826143	+	0.563460i	$0.190530\pi$
$410$		0			0
$411$	−0.224808	+	0.224808i	−0.0110890	+	0.0110890i
$412$		0			0
$413$	4.54614	+	7.87415i	0.223701	+	0.387461i
$414$		0			0
$415$	−26.5022			−1.30094
$416$		0			0
$417$	−21.4077			−1.04834
$418$		0			0
$419$	−14.0047	−	24.2568i	−0.684174	−	1.18502i	−0.973696	−	0.227853i	$-0.926830\pi$
$419$	−14.0047	−	24.2568i	−0.684174	−	1.18502i	0.289522	−	0.957171i	$-0.406504\pi$
$420$		0			0
$421$	0.583969	−	0.583969i	0.0284609	−	0.0284609i	−0.692733	−	0.721194i	$-0.743594\pi$
$421$	0.583969	−	0.583969i	0.0284609	−	0.0284609i	0.721194	+	0.692733i	$0.243594\pi$
$422$		0			0
$423$	−50.1517	+	13.4381i	−2.43846	+	0.653384i
$424$		0			0
$425$	−2.19548	+	3.80268i	−0.106496	+	0.184457i
$426$		0			0
$427$	15.4374	+	4.13643i	0.747067	+	0.200176i
$428$		0			0
$429$	10.3698	−	8.60845i	0.500657	−	0.415620i
$430$		0			0
$431$	−16.3546	−	4.38219i	−0.787772	−	0.211083i	−0.157563	−	0.987509i	$-0.550364\pi$
$431$	−16.3546	−	4.38219i	−0.787772	−	0.211083i	−0.630208	+	0.776426i	$0.717031\pi$
$432$		0			0
$433$	19.1457	+	11.0538i	0.920082	+	0.531209i	0.883661	−	0.468127i	$-0.155071\pi$
$433$	19.1457	+	11.0538i	0.920082	+	0.531209i	0.0364205	+	0.999337i	$0.488404\pi$
$434$		0			0
$435$	4.57306	−	1.22535i	0.219261	−	0.0587509i
$436$		0			0
$437$	10.7855	+	10.7855i	0.515940	+	0.515940i
$438$		0			0
$439$	18.7931	+	32.5507i	0.896947	+	1.55356i	0.831376	+	0.555710i	$0.187553\pi$
$439$	18.7931	+	32.5507i	0.896947	+	1.55356i	0.0655708	+	0.997848i	$0.479113\pi$
$440$		0			0
$441$		−	33.2896i		−	1.58522i
$442$		0			0
$443$	−16.9312			−0.804428			−0.402214	−	0.915546i	$-0.631759\pi$
$443$	−16.9312			−0.804428			−0.402214	+	0.915546i	$0.631759\pi$
$444$		0			0
$445$	−5.58807	+	3.22627i	−0.264900	+	0.152940i
$446$		0			0
$447$	29.0372	−	29.0372i	1.37341	−	1.37341i
$448$		0			0
$449$	−0.0228713	−	0.0853569i	−0.00107936	−	0.00402824i	0.965384	−	0.260833i	$-0.0839971\pi$
$449$	−0.0228713	−	0.0853569i	−0.00107936	−	0.00402824i	−0.966463	+	0.256804i	$0.917330\pi$
$450$		0			0
$451$	4.47929	−	7.75836i	0.210922	−	0.365327i
$452$		0			0
$453$	−9.84319	+	36.7353i	−0.462474	+	1.72597i
$454$		0			0
$455$	16.1061	+	1.49476i	0.755065	+	0.0700756i
$456$		0			0
$457$	−6.35265	+	23.7084i	−0.297164	+	1.10903i	0.642319	+	0.766438i	$0.277973\pi$
$457$	−6.35265	+	23.7084i	−0.297164	+	1.10903i	−0.939483	+	0.342595i	$0.888694\pi$
$458$		0			0
$459$	11.3849	+	6.57305i	0.531400	+	0.306804i
$460$		0			0
$461$	1.14969	+	4.29072i	0.0535466	+	0.199839i	0.987517	−	0.157511i	$-0.0503470\pi$
$461$	1.14969	+	4.29072i	0.0535466	+	0.199839i	−0.933971	+	0.357350i	$0.883680\pi$
$462$		0			0
$463$	15.6374	+	15.6374i	0.726731	+	0.726731i	0.969967	−	0.243236i	$-0.0782089\pi$
$463$	15.6374	+	15.6374i	0.726731	+	0.726731i	−0.243236	+	0.969967i	$0.578209\pi$
$464$		0			0
$465$	2.10274	−	1.21402i	0.0975122	−	0.0562987i
$466$		0			0
$467$		−	8.82470i		−	0.408358i	−0.978934	−	0.204179i	$-0.934547\pi$
$467$		−	8.82470i		−	0.408358i	0.978934	−	0.204179i	$-0.0654525\pi$
$468$		0			0
$469$		−	18.4744i		−	0.853067i
$470$		0			0
$471$	−1.42546	+	0.822990i	−0.0656818	+	0.0379214i
$472$		0			0
$473$	0.948311	+	0.948311i	0.0436034	+	0.0436034i
$474$		0			0
$475$	−3.34381	−	12.4793i	−0.153424	−	0.572588i
$476$		0			0
$477$	−79.3257	−	45.7987i	−3.63207	−	2.09698i
$478$		0			0
$479$	−8.93153	+	33.3329i	−0.408092	+	1.52302i	0.390189	+	0.920735i	$0.372409\pi$
$479$	−8.93153	+	33.3329i	−0.408092	+	1.52302i	−0.798281	+	0.602285i	$0.794257\pi$
$480$		0			0
$481$	−11.9373	+	2.03999i	−0.544293	+	0.0930154i
$482$		0			0
$483$	−5.92265	+	22.1036i	−0.269490	+	1.00575i
$484$		0			0
$485$	6.18150	−	10.7067i	0.280687	−	0.486165i
$486$		0			0
$487$	6.09032	+	22.7294i	0.275979	+	1.02997i	0.955183	+	0.296017i	$0.0956586\pi$
$487$	6.09032	+	22.7294i	0.275979	+	1.02997i	−0.679204	+	0.733950i	$0.737675\pi$
$488$		0			0
$489$	55.4435	−	55.4435i	2.50724	−	2.50724i
$490$		0			0
$491$	28.1779	−	16.2685i	1.27165	−	0.734187i	0.296351	−	0.955079i	$-0.404230\pi$
$491$	28.1779	−	16.2685i	1.27165	−	0.734187i	0.975298	+	0.220892i	$0.0708967\pi$
$492$		0			0
$493$	0.526166			0.0236973
$494$		0			0
$495$		−	24.8738i		−	1.11800i
$496$		0			0
$497$	−5.72137	−	9.90970i	−0.256638	−	0.444511i
$498$		0			0
$499$	−0.938780	−	0.938780i	−0.0420256	−	0.0420256i	0.685782	−	0.727807i	$-0.259460\pi$
$499$	−0.938780	−	0.938780i	−0.0420256	−	0.0420256i	−0.727807	+	0.685782i	$0.759460\pi$
$500$		0			0
$501$	9.51851	−	2.55048i	0.425256	−	0.113947i
$502$		0			0
$503$	−3.62331	−	2.09192i	−0.161555	−	0.0932739i	0.417043	−	0.908887i	$-0.363067\pi$
$503$	−3.62331	−	2.09192i	−0.161555	−	0.0932739i	−0.578598	+	0.815613i	$0.696400\pi$
$504$		0			0
$505$	31.8606	+	8.53703i	1.41778	+	0.379893i
$506$		0			0
$507$	−40.2791	−	7.54135i	−1.78886	−	0.334923i
$508$		0			0
$509$	23.6241	+	6.33007i	1.04712	+	0.280575i	0.741062	−	0.671437i	$-0.234323\pi$
$509$	23.6241	+	6.33007i	1.04712	+	0.280575i	0.306060	+	0.952012i	$0.400989\pi$
$510$		0			0
$511$	4.24296	−	7.34902i	0.187697	−	0.325102i
$512$		0			0
$513$	−37.3617	+	10.0110i	−1.64956	+	0.441998i
$514$		0			0
$515$	−12.6987	+	12.6987i	−0.559571	+	0.559571i
$516$		0			0
$517$	4.43794	+	7.68674i	0.195180	+	0.338062i
$518$		0			0
$519$	20.7229			0.909635
$520$		0			0
$521$	−10.9083			−0.477900			−0.238950	−	0.971032i	$-0.576803\pi$
$521$	−10.9083			−0.477900			−0.238950	+	0.971032i	$0.576803\pi$
$522$		0			0
$523$	9.42167	+	16.3188i	0.411981	+	0.713572i	0.995106	−	0.0988105i	$-0.0315038\pi$
$523$	9.42167	+	16.3188i	0.411981	+	0.713572i	−0.583126	+	0.812382i	$0.698170\pi$
$524$		0			0
$525$	13.7055	−	13.7055i	0.598157	−	0.598157i
$526$		0			0
$527$	0.260650	−	0.0698410i	0.0113541	−	0.00304232i
$528$		0			0
$529$	−0.472439	+	0.818288i	−0.0205408	+	0.0355777i
$530$		0			0
$531$	−41.0644	−	11.0032i	−1.78204	−	0.477497i
$532$		0			0
$533$	−26.8502	+	4.58848i	−1.16301	+	0.198749i
$534$		0			0
$535$	22.5898	+	6.05292i	0.976643	+	0.261691i
$536$		0			0
$537$	−60.3927	−	34.8677i	−2.60614	−	1.50465i
$538$		0			0
$539$	−5.49694	+	1.47290i	−0.236770	+	0.0634424i
$540$		0			0
$541$	−1.32217	−	1.32217i	−0.0568444	−	0.0568444i	0.678113	−	0.734958i	$-0.262798\pi$
$541$	−1.32217	−	1.32217i	−0.0568444	−	0.0568444i	−0.734958	+	0.678113i	$0.762798\pi$
$542$		0			0
$543$	−34.3148	−	59.4350i	−1.47259	−	2.55060i
$544$		0			0
$545$			19.1001i			0.818160i
$546$		0			0
$547$	−17.9127			−0.765894			−0.382947	−	0.923770i	$-0.625091\pi$
$547$	−17.9127			−0.765894			−0.382947	+	0.923770i	$0.625091\pi$
$548$		0			0
$549$	−64.7156	+	37.3636i	−2.76199	+	1.59464i
$550$		0			0
$551$	−1.09470	+	1.09470i	−0.0466356	+	0.0466356i
$552$		0			0
$553$	0.358478	+	1.33786i	0.0152440	+	0.0568915i
$554$		0			0
$555$	−16.0088	+	27.7280i	−0.679534	+	1.17699i
$556$		0			0
$557$	10.4145	−	38.8674i	0.441276	−	1.64686i	−0.284309	−	0.958733i	$-0.591764\pi$
$557$	10.4145	−	38.8674i	0.441276	−	1.64686i	0.725585	−	0.688132i	$-0.241569\pi$
$558$		0			0
$559$	0.376829	−	4.06033i	0.0159382	−	0.171734i
$560$		0			0
$561$	1.02492	−	3.82506i	0.0432722	−	0.161494i
$562$		0			0
$563$	9.19054	+	5.30616i	0.387335	+	0.223628i	0.681005	−	0.732279i	$-0.261543\pi$
$563$	9.19054	+	5.30616i	0.387335	+	0.223628i	−0.293670	+	0.955907i	$0.594877\pi$
$564$		0			0
$565$	8.98651	+	33.5381i	0.378066	+	1.41096i
$566$		0			0
$567$	−19.2034	−	19.2034i	−0.806465	−	0.806465i
$568$		0			0
$569$	5.37353	−	3.10241i	0.225270	−	0.130060i	−0.383118	−	0.923699i	$-0.625150\pi$
$569$	5.37353	−	3.10241i	0.225270	−	0.130060i	0.608388	+	0.793640i	$0.291816\pi$
$570$		0			0
$571$			10.5785i			0.442697i	0.975195	+	0.221348i	$0.0710458\pi$
$571$			10.5785i			0.442697i	−0.975195	+	0.221348i	$0.928954\pi$
$572$		0			0
$573$		−	5.27506i		−	0.220369i
$574$		0			0
$575$	17.5644	−	10.1408i	0.732487	−	0.422901i
$576$		0			0
$577$	−8.26875	−	8.26875i	−0.344233	−	0.344233i	0.513723	−	0.857956i	$-0.328266\pi$
$577$	−8.26875	−	8.26875i	−0.344233	−	0.344233i	−0.857956	+	0.513723i	$0.828266\pi$
$578$		0			0
$579$	−4.98439	−	18.6020i	−0.207144	−	0.773072i
$580$		0			0
$581$	11.2596	+	6.50073i	0.467127	+	0.269696i
$582$		0			0
$583$	−4.05274	+	15.1250i	−0.167847	+	0.626415i
$584$		0			0
$585$	−58.1925	+	48.3085i	−2.40597	+	1.99731i
$586$		0			0
$587$	4.88481	−	18.2304i	0.201618	−	0.752447i	−0.788836	−	0.614603i	$-0.789316\pi$
$587$	4.88481	−	18.2304i	0.201618	−	0.752447i	0.990454	−	0.137844i	$-0.0440172\pi$
$588$		0			0
$589$	−0.396981	+	0.687592i	−0.0163573	+	0.0283317i
$590$		0			0
$591$	−7.53363	−	28.1159i	−0.309892	−	1.15653i
$592$		0			0
$593$	−8.56694	+	8.56694i	−0.351802	+	0.351802i	−0.860780	−	0.508978i	$-0.830024\pi$
$593$	−8.56694	+	8.56694i	−0.351802	+	0.351802i	0.508978	+	0.860780i	$0.330024\pi$
$594$		0			0
$595$	4.11599	−	2.37637i	0.168739	−	0.0974216i
$596$		0			0
$597$	65.1862			2.66789
$598$		0			0
$599$		−	22.4062i		−	0.915491i	−0.889083	−	0.457745i	$-0.848657\pi$
$599$		−	22.4062i		−	0.915491i	0.889083	−	0.457745i	$-0.151343\pi$
$600$		0			0
$601$	11.5194	+	19.9522i	0.469887	+	0.813867i	0.999407	−	0.0344297i	$-0.0109615\pi$
$601$	11.5194	+	19.9522i	0.469887	+	0.813867i	−0.529521	+	0.848297i	$0.677628\pi$
$602$		0			0
$603$	61.0806	+	61.0806i	2.48739	+	2.48739i
$604$		0			0
$605$	28.0235	−	7.50888i	1.13932	−	0.305280i
$606$		0			0
$607$	−15.0714	−	8.70147i	−0.611729	−	0.353182i	0.161913	−	0.986805i	$-0.448234\pi$
$607$	−15.0714	−	8.70147i	−0.611729	−	0.353182i	−0.773642	+	0.633623i	$0.781567\pi$
$608$		0			0
$609$	−2.24345	−	0.601131i	−0.0909092	−	0.0243591i
$610$		0			0
$611$	9.36409	−	25.3113i	0.378831	−	1.02399i
$612$		0			0
$613$	−23.9979	−	6.43022i	−0.969266	−	0.259714i	−0.260748	−	0.965407i	$-0.583969\pi$
$613$	−23.9979	−	6.43022i	−0.969266	−	0.259714i	−0.708518	+	0.705693i	$0.750636\pi$
$614$		0			0
$615$	−36.0081	+	62.3678i	−1.45198	+	2.51491i
$616$		0			0
$617$	16.8453	−	4.51370i	0.678168	−	0.181715i	0.0967368	−	0.995310i	$-0.469160\pi$
$617$	16.8453	−	4.51370i	0.678168	−	0.181715i	0.581431	+	0.813595i	$0.302493\pi$
$618$		0			0
$619$	22.2153	−	22.2153i	0.892907	−	0.892907i	−0.101889	−	0.994796i	$-0.532489\pi$
$619$	22.2153	−	22.2153i	0.892907	−	0.892907i	0.994796	+	0.101889i	$0.0324885\pi$
$620$		0			0
$621$	−30.3606	−	52.5861i	−1.21833	−	2.11021i
$622$		0			0
$623$	3.16549			0.126823
$624$		0			0
$625$	28.5448			1.14179
$626$		0			0
$627$	5.82572	+	10.0904i	0.232657	+	0.402974i
$628$		0			0
$629$	−2.51613	+	2.51613i	−0.100324	+	0.100324i
$630$		0			0
$631$	−23.0172	+	6.16745i	−0.916302	+	0.245522i	−0.686004	−	0.727598i	$-0.740637\pi$
$631$	−23.0172	+	6.16745i	−0.916302	+	0.245522i	−0.230298	+	0.973120i	$0.573970\pi$
$632$		0			0
$633$	−33.7490	+	58.4551i	−1.34140	+	2.32338i
$634$		0			0
$635$	25.7345	+	6.89553i	1.02124	+	0.273641i
$636$		0			0
$637$	14.1217	+	9.99957i	0.559522	+	0.396197i
$638$		0			0
$639$	51.6800	+	13.8476i	2.04443	+	0.547803i
$640$		0			0
$641$	19.3494	+	11.1714i	0.764257	+	0.441244i	0.830822	−	0.556538i	$-0.187871\pi$
$641$	19.3494	+	11.1714i	0.764257	+	0.441244i	−0.0665652	+	0.997782i	$0.521204\pi$
$642$		0			0
$643$	4.36483	−	1.16955i	0.172132	−	0.0461227i	−0.171724	−	0.985145i	$-0.554934\pi$
$643$	4.36483	−	1.16955i	0.172132	−	0.0461227i	0.343856	+	0.939023i	$0.388267\pi$
$644$		0			0
$645$	−7.62326	−	7.62326i	−0.300166	−	0.300166i
$646$		0			0
$647$	−4.37758	−	7.58218i	−0.172100	−	0.298086i	0.767054	−	0.641583i	$-0.221722\pi$
$647$	−4.37758	−	7.58218i	−0.172100	−	0.298086i	−0.939154	+	0.343497i	$0.888389\pi$
$648$		0			0
$649$			7.26760i			0.285278i
$650$		0			0
$651$	−1.19114			−0.0466846
$652$		0			0
$653$	5.94625	−	3.43307i	0.232695	−	0.134346i	−0.379120	−	0.925348i	$-0.623773\pi$
$653$	5.94625	−	3.43307i	0.232695	−	0.134346i	0.611815	+	0.791001i	$0.290440\pi$
$654$		0			0
$655$	−31.6090	+	31.6090i	−1.23507	+	1.23507i
$656$		0			0
$657$	10.2694	+	38.3258i	0.400646	+	1.49523i
$658$		0			0
$659$	7.77012	−	13.4582i	0.302681	−	0.524259i	−0.674061	−	0.738675i	$-0.735452\pi$
$659$	7.77012	−	13.4582i	0.302681	−	0.524259i	0.976742	+	0.214417i	$0.0687850\pi$
$660$		0			0
$661$	−10.9023	+	40.6880i	−0.424051	+	1.58258i	0.341936	+	0.939723i	$0.388918\pi$
$661$	−10.9023	+	40.6880i	−0.424051	+	1.58258i	−0.765987	+	0.642856i	$0.777749\pi$
$662$		0			0
$663$	−10.9393	+	5.03099i	−0.424847	+	0.195387i
$664$		0			0
$665$	−3.61931	+	13.5074i	−0.140351	+	0.523796i
$666$		0			0
$667$	−2.10473	−	1.21517i	−0.0814955	−	0.0470515i
$668$		0			0
$669$	8.31612	+	31.0362i	0.321520	+	1.19993i
$670$		0			0
$671$	9.03301	+	9.03301i	0.348716	+	0.348716i
$672$		0			0
$673$	5.42846	−	3.13412i	0.209252	−	0.120812i	−0.391712	−	0.920088i	$-0.628117\pi$
$673$	5.42846	−	3.13412i	0.209252	−	0.120812i	0.600964	+	0.799276i	$0.294784\pi$
$674$		0			0
$675$			51.4317i			1.97961i
$676$		0			0
$677$			21.1245i			0.811881i	0.913900	+	0.405940i	$0.133056\pi$
$677$			21.1245i			0.811881i	−0.913900	+	0.405940i	$0.866944\pi$
$678$		0			0
$679$	−5.25248	+	3.03252i	−0.201572	+	0.116377i
$680$		0			0
$681$	−8.36779	−	8.36779i	−0.320654	−	0.320654i
$682$		0			0
$683$	7.19116	+	26.8378i	0.275162	+	1.02692i	0.955733	+	0.294236i	$0.0950653\pi$
$683$	7.19116	+	26.8378i	0.275162	+	1.02692i	−0.680571	+	0.732683i	$0.738268\pi$
$684$		0			0
$685$	−0.264135	−	0.152498i	−0.0100921	−	0.00582665i
$686$		0			0
$687$	6.27397	−	23.4148i	0.239367	−	0.893329i
$688$		0			0
$689$	43.2561	−	19.8935i	1.64793	−	0.757883i
$690$		0			0
$691$	−4.73219	+	17.6608i	−0.180021	+	0.671848i	0.815621	+	0.578587i	$0.196396\pi$
$691$	−4.73219	+	17.6608i	−0.180021	+	0.671848i	−0.995642	+	0.0932606i	$0.970271\pi$
$692$		0			0
$693$	−6.10130	+	10.5678i	−0.231769	+	0.401436i
$694$		0			0
$695$	−5.31537	−	19.8372i	−0.201624	−	0.752470i
$696$		0			0
$697$	−5.65945	+	5.65945i	−0.214367	+	0.214367i
$698$		0			0
$699$	−7.99620	+	4.61661i	−0.302444	+	0.174616i
$700$		0			0
$701$	−17.2336			−0.650904			−0.325452	−	0.945559i	$-0.605516\pi$
$701$	−17.2336			−0.650904			−0.325452	+	0.945559i	$0.605516\pi$
$702$		0			0
$703$		−	10.4697i		−	0.394871i
$704$		0			0
$705$	−35.6756	−	61.7920i	−1.34362	−	2.32722i
$706$		0			0
$707$	−11.4421	−	11.4421i	−0.430324	−	0.430324i
$708$		0			0
$709$	−32.3352	+	8.66420i	−1.21437	+	0.325391i	−0.808477	−	0.588528i	$-0.799708\pi$
$709$	−32.3352	+	8.66420i	−1.21437	+	0.325391i	−0.405898	+	0.913919i	$0.633041\pi$
$710$		0			0
$711$	−5.60848	−	3.23806i	−0.210334	−	0.121437i
$712$		0			0
$713$	−1.20393	−	0.322592i	−0.0450876	−	0.0120812i
$714$		0			0
$715$	10.5517	+	7.47164i	0.394611	+	0.279423i
$716$		0			0
$717$	−61.6872	−	16.5290i	−2.30375	−	0.617289i
$718$		0			0
$719$	−8.93537	+	15.4765i	−0.333233	+	0.577177i	−0.983144	−	0.182834i	$-0.941473\pi$
$719$	−8.93537	+	15.4765i	−0.333233	+	0.577177i	0.649911	+	0.760011i	$0.274806\pi$
$720$		0			0
$721$	8.50996	−	2.28024i	0.316928	−	0.0849205i
$722$		0			0
$723$	−30.2865	+	30.2865i	−1.12637	+	1.12637i
$724$		0			0
$725$	1.02926	+	1.78274i	0.0382259	+	0.0662091i
$726$		0			0
$727$	−21.2697			−0.788848			−0.394424	−	0.918928i	$-0.629056\pi$
$727$	−21.2697			−0.788848			−0.394424	+	0.918928i	$0.629056\pi$
$728$		0			0
$729$	9.63413			0.356820
$730$		0			0
$731$	−0.599081	−	1.03764i	−0.0221578	−	0.0383784i
$732$		0			0
$733$	−4.74608	+	4.74608i	−0.175300	+	0.175300i	−0.789304	−	0.614003i	$-0.789558\pi$
$733$	−4.74608	+	4.74608i	−0.175300	+	0.175300i	0.614003	+	0.789304i	$0.289558\pi$
$734$		0			0
$735$	44.1887	−	11.8403i	1.62993	−	0.436737i
$736$		0			0
$737$	7.38342	−	12.7885i	0.271972	−	0.471069i
$738$		0			0
$739$	39.5252	+	10.5907i	1.45396	+	0.389587i	0.897398	−	0.441221i	$-0.145455\pi$
$739$	39.5252	+	10.5907i	1.45396	+	0.389587i	0.556559	+	0.830808i	$0.312121\pi$
$740$		0			0
$741$	12.2923	−	33.2264i	0.451570	−	1.22060i
$742$		0			0
$743$	−38.1054	−	10.2103i	−1.39795	−	0.374580i	−0.520343	−	0.853957i	$-0.674196\pi$
$743$	−38.1054	−	10.2103i	−1.39795	−	0.374580i	−0.877609	+	0.479377i	$0.840863\pi$
$744$		0			0
$745$	34.1167	+	19.6973i	1.24994	+	0.721654i
$746$		0			0
$747$	−58.7198	+	15.7339i	−2.14845	+	0.575674i
$748$		0			0
$749$	−8.11267	−	8.11267i	−0.296430	−	0.296430i
$750$		0			0
$751$	11.2537	+	19.4920i	0.410655	+	0.711275i	0.994961	−	0.100258i	$-0.0319668\pi$
$751$	11.2537	+	19.4920i	0.410655	+	0.711275i	−0.584307	+	0.811533i	$0.698633\pi$
$752$		0			0
$753$			34.6514i			1.26277i
$754$		0			0
$755$	−36.4844			−1.32780
$756$		0			0
$757$	40.5049	−	23.3855i	1.47217	−	0.849960i	0.472664	−	0.881243i	$-0.343293\pi$
$757$	40.5049	−	23.3855i	1.47217	−	0.849960i	0.999511	+	0.0312829i	$0.00995927\pi$
$758$		0			0
$759$	−12.9337	+	12.9337i	−0.469463	+	0.469463i
$760$		0			0
$761$	−5.99174	−	22.3615i	−0.217200	−	0.810603i	−0.985380	−	0.170369i	$-0.945504\pi$
$761$	−5.99174	−	22.3615i	−0.217200	−	0.810603i	0.768180	−	0.640234i	$-0.221163\pi$
$762$		0			0
$763$	4.68507	−	8.11478i	0.169611	−	0.293775i
$764$		0			0
$765$	−5.75160	+	21.4653i	−0.207949	+	0.776078i
$766$		0			0
$767$	17.0026	−	14.1147i	0.613929	−	0.509652i
$768$		0			0
$769$	1.46149	−	5.45437i	0.0527028	−	0.196689i	−0.934555	−	0.355819i	$-0.884202\pi$
$769$	1.46149	−	5.45437i	0.0527028	−	0.196689i	0.987258	+	0.159129i	$0.0508687\pi$
$770$		0			0
$771$	−36.8068	−	21.2504i	−1.32556	−	0.765314i
$772$		0			0
$773$	−13.1234	−	48.9774i	−0.472018	−	1.76159i	−0.632506	−	0.774556i	$-0.717973\pi$
$773$	−13.1234	−	48.9774i	−0.472018	−	1.76159i	0.160488	−	0.987038i	$-0.448693\pi$
$774$		0			0
$775$	0.746506	+	0.746506i	0.0268153	+	0.0268153i
$776$		0			0
$777$	13.6028	−	7.85358i	0.487998	−	0.281746i
$778$		0			0
$779$		−	23.5491i		−	0.843735i
$780$		0			0
$781$		−	9.14635i		−	0.327282i
$782$		0			0
$783$	5.33734	−	3.08151i	0.190741	−	0.110124i
$784$		0			0
$785$	−1.11655	−	1.11655i	−0.0398513	−	0.0398513i
$786$		0			0
$787$	1.63955	+	6.11890i	0.0584438	+	0.218115i	0.988971	−	0.148107i	$-0.0473179\pi$
$787$	1.63955	+	6.11890i	0.0584438	+	0.218115i	−0.930528	+	0.366222i	$0.880651\pi$
$788$		0			0
$789$	57.9265	+	33.4439i	2.06224	+	1.19063i
$790$		0			0
$791$	4.40860	−	16.4531i	0.156752	−	0.585006i
$792$		0			0
$793$	3.58943	−	38.6762i	0.127465	−	1.37343i
$794$		0			0
$795$	32.5791	−	121.587i	1.15546	−	4.31224i
$796$		0			0
$797$	−15.6429	+	27.0944i	−0.554101	+	0.959732i	0.443871	+	0.896090i	$0.353605\pi$
$797$	−15.6429	+	27.0944i	−0.554101	+	0.959732i	−0.997973	+	0.0636413i	$0.979729\pi$
$798$		0			0
$799$	−2.05238	−	7.65958i	−0.0726079	−	0.270976i
$800$		0			0
$801$	−10.4659	+	10.4659i	−0.369793	+	0.369793i
$802$		0			0
$803$	5.87418	−	3.39146i	0.207295	−	0.119682i
$804$		0			0
$805$	−21.9527			−0.773730
$806$		0			0
$807$			91.5610i			3.22310i
$808$		0			0
$809$	−7.70970	−	13.3536i	−0.271059	−	0.469487i	0.698075	−	0.716025i	$-0.254040\pi$
$809$	−7.70970	−	13.3536i	−0.271059	−	0.469487i	−0.969133	+	0.246538i	$0.920707\pi$
$810$		0			0
$811$	5.80801	+	5.80801i	0.203947	+	0.203947i	0.801689	−	0.597742i	$-0.203935\pi$
$811$	5.80801	+	5.80801i	0.203947	+	0.203947i	−0.597742	+	0.801689i	$0.703935\pi$
$812$		0			0
$813$	40.5951	−	10.8774i	1.42373	−	0.381488i
$814$		0			0
$815$	65.1424	+	37.6100i	2.28184	+	1.31742i
$816$		0			0
$817$	3.40522	+	0.912425i	0.119133	+	0.0319217i
$818$		0			0
$819$	36.5730	−	6.25003i	1.27796	−	0.218394i
$820$		0			0
$821$	−32.4496	−	8.69486i	−1.13250	−	0.303453i	−0.356568	−	0.934269i	$-0.616053\pi$
$821$	−32.4496	−	8.69486i	−1.13250	−	0.303453i	−0.775932	+	0.630817i	$0.782720\pi$
$822$		0			0
$823$	12.8798	−	22.3085i	0.448962	−	0.777625i	−0.549357	−	0.835588i	$-0.685127\pi$
$823$	12.8798	−	22.3085i	0.448962	−	0.777625i	0.998319	+	0.0579627i	$0.0184605\pi$
$824$		0			0
$825$	14.9648	−	4.00981i	0.521008	−	0.139604i
$826$		0			0
$827$	−22.6971	+	22.6971i	−0.789255	+	0.789255i	−0.981372	−	0.192117i	$-0.938465\pi$
$827$	−22.6971	+	22.6971i	−0.789255	+	0.789255i	0.192117	+	0.981372i	$0.438465\pi$
$828$		0			0
$829$	28.4583	+	49.2912i	0.988397	+	1.71195i	0.625743	+	0.780029i	$0.284796\pi$
$829$	28.4583	+	49.2912i	0.988397	+	1.71195i	0.362654	+	0.931924i	$0.381871\pi$
$830$		0			0
$831$	77.9279			2.70329
$832$		0			0
$833$	5.08425			0.176159
$834$		0			0
$835$	4.72675	+	8.18697i	0.163576	+	0.283322i
$836$		0			0
$837$	2.23496	−	2.23496i	0.0772517	−	0.0772517i
$838$		0			0
$839$	0.535224	−	0.143413i	0.0184780	−	0.00495116i	−0.249568	−	0.968357i	$-0.580289\pi$
$839$	0.535224	−	0.143413i	0.0184780	−	0.00495116i	0.268046	+	0.963406i	$0.413622\pi$
$840$		0			0
$841$	−14.3767	+	24.9011i	−0.495747	+	0.858659i
$842$		0			0
$843$	−82.5206	−	22.1113i	−2.84216	−	0.761554i
$844$		0			0
$845$	−3.01288	−	39.1967i	−0.103646	−	1.34841i
$846$		0			0
$847$	−13.7478	−	3.68371i	−0.472379	−	0.126574i
$848$		0			0
$849$	−53.5109	−	30.8945i	−1.83649	−	1.06030i
$850$		0			0
$851$	15.8758	−	4.25390i	0.544214	−	0.145822i
$852$		0			0
$853$	6.92158	+	6.92158i	0.236990	+	0.236990i	0.815603	−	0.578612i	$-0.196406\pi$
$853$	6.92158	+	6.92158i	0.236990	+	0.236990i	−0.578612	+	0.815603i	$0.696406\pi$
$854$		0			0
$855$	−32.6925	−	56.6251i	−1.11806	−	1.93654i
$856$		0			0
$857$		−	23.8544i		−	0.814849i	−0.913239	−	0.407425i	$-0.866427\pi$
$857$		−	23.8544i		−	0.814849i	0.913239	−	0.407425i	$-0.133573\pi$
$858$		0			0
$859$	47.4998			1.62067			0.810336	−	0.585966i	$-0.199285\pi$
$859$	47.4998			1.62067			0.810336	+	0.585966i	$0.199285\pi$
$860$		0			0
$861$	30.5964	−	17.6648i	1.04272	−	0.602016i
$862$		0			0
$863$	−6.61684	+	6.61684i	−0.225240	+	0.225240i	−0.810701	−	0.585461i	$-0.800914\pi$
$863$	−6.61684	+	6.61684i	−0.225240	+	0.225240i	0.585461	+	0.810701i	$0.300914\pi$
$864$		0			0
$865$	5.14535	+	19.2027i	0.174947	+	0.652912i
$866$		0			0
$867$	25.0250	−	43.3446i	0.849894	−	1.47206i
$868$		0			0
$869$	−0.286537	+	1.06937i	−0.00972009	+	0.0362759i
$870$		0			0
$871$	−44.2584	+	7.56341i	−1.49964	+	0.256276i
$872$		0			0
$873$	7.33970	−	27.3921i	0.248411	−	0.927083i
$874$		0			0
$875$	−3.32286	−	1.91846i	−0.112333	−	0.0648557i
$876$		0			0
$877$	5.43918	+	20.2993i	0.183668	+	0.685458i	0.994912	+	0.100750i	$0.0321243\pi$
$877$	5.43918	+	20.2993i	0.183668	+	0.685458i	−0.811244	+	0.584708i	$0.801209\pi$
$878$		0			0
$879$	−24.8370	−	24.8370i	−0.837732	−	0.837732i
$880$		0			0
$881$	−43.8036	+	25.2900i	−1.47578	+	0.852043i	−0.999627	−	0.0273185i	$-0.991303\pi$
$881$	−43.8036	+	25.2900i	−1.47578	+	0.852043i	−0.476155	+	0.879361i	$0.657970\pi$
$882$		0			0
$883$			8.15325i			0.274379i	0.990545	+	0.137189i	$0.0438069\pi$
$883$			8.15325i			0.274379i	−0.990545	+	0.137189i	$0.956193\pi$
$884$		0			0
$885$		−	58.4226i		−	1.96386i
$886$		0			0
$887$	35.3131	−	20.3880i	1.18570	−	0.684563i	0.228373	−	0.973574i	$-0.426660\pi$
$887$	35.3131	−	20.3880i	1.18570	−	0.684563i	0.957326	+	0.289010i	$0.0933262\pi$
$888$		0			0
$889$	−9.24201	−	9.24201i	−0.309967	−	0.309967i
$890$		0			0
$891$	−5.61832	−	20.9678i	−0.188221	−	0.702449i
$892$		0			0
$893$	20.2059	+	11.6659i	0.676164	+	0.390383i
$894$		0			0
$895$	17.3148	−	64.6197i	0.578770	−	2.16000i
$896$		0			0
$897$	55.3775	+	5.13944i	1.84900	+	0.171601i
$898$		0			0
$899$	0.0327422	−	0.122195i	0.00109201	−	0.00407544i
$900$		0			0
$901$	6.99475	−	12.1153i	0.233029	−	0.403618i
$902$		0			0
$903$	1.36887	+	5.10869i	0.0455531	+	0.170007i
$904$		0			0
$905$	46.5548	−	46.5548i	1.54753	−	1.54753i
$906$		0			0
$907$	−16.2086	+	9.35804i	−0.538198	+	0.310729i	−0.744348	−	0.667792i	$-0.767240\pi$
$907$	−16.2086	+	9.35804i	−0.538198	+	0.310729i	0.206151	+	0.978520i	$0.433906\pi$
$908$		0			0
$909$	75.6605			2.50950
$910$		0			0
$911$			18.7926i			0.622627i	0.950307	+	0.311313i	$0.100769\pi$
$911$			18.7926i			0.622627i	−0.950307	+	0.311313i	$0.899231\pi$
$912$		0			0
$913$	5.19613	+	8.99997i	0.171967	+	0.297856i
$914$		0			0
$915$	−72.6144	−	72.6144i	−2.40056	−	2.40056i
$916$		0			0
$917$	21.1826	−	5.67586i	0.699511	−	0.187433i
$918$		0			0
$919$	41.2891	+	23.8383i	1.36200	+	0.786352i	0.989890	−	0.141836i	$-0.0453005\pi$
$919$	41.2891	+	23.8383i	1.36200	+	0.786352i	0.372112	+	0.928188i	$0.378634\pi$
$920$		0			0
$921$	36.7277	+	9.84116i	1.21022	+	0.324277i
$922$		0			0
$923$	−21.3980	+	17.7635i	−0.704323	+	0.584693i
$924$		0			0
$925$	−13.4470	−	3.60311i	−0.442134	−	0.118469i
$926$		0			0
$927$	−20.5969	+	35.6749i	−0.676492	+	1.17172i
$928$		0			0
$929$	−8.04178	+	2.15479i	−0.263842	+	0.0706963i	−0.388315	−	0.921527i	$-0.626943\pi$
$929$	−8.04178	+	2.15479i	−0.263842	+	0.0706963i	0.124473	+	0.992223i	$0.460276\pi$
$930$		0			0
$931$	−10.5779	+	10.5779i	−0.346675	+	0.346675i
$932$		0			0
$933$	−13.8911	−	24.0601i	−0.454774	−	0.787692i
$934$		0			0
$935$	3.79893			0.124238
$936$		0			0
$937$	−46.0216			−1.50346			−0.751730	−	0.659471i	$-0.770780\pi$
$937$	−46.0216			−1.50346			−0.751730	+	0.659471i	$0.770780\pi$
$938$		0			0
$939$	21.8021	+	37.7624i	0.711485	+	1.23233i
$940$		0			0
$941$	−16.6053	+	16.6053i	−0.541316	+	0.541316i	−0.923915	−	0.382599i	$-0.875029\pi$
$941$	−16.6053	+	16.6053i	−0.541316	+	0.541316i	0.382599	+	0.923915i	$0.375029\pi$
$942$		0			0
$943$	35.7089	−	9.56817i	1.16284	−	0.311582i
$944$		0			0
$945$	27.8346	−	48.2110i	0.905460	−	1.56830i
$946$		0			0
$947$	16.2699	+	4.35951i	0.528701	+	0.141665i	0.513289	−	0.858216i	$-0.328427\pi$
$947$	16.2699	+	4.35951i	0.528701	+	0.141665i	0.0154129	+	0.999881i	$0.495094\pi$
$948$		0			0
$949$	−19.3428	−	7.15601i	−0.627895	−	0.232294i
$950$		0			0
$951$	−50.4274	−	13.5120i	−1.63522	−	0.438156i
$952$		0			0
$953$	−45.2296	−	26.1133i	−1.46513	−	0.845893i	−0.465888	−	0.884844i	$-0.654265\pi$
$953$	−45.2296	−	26.1133i	−1.46513	−	0.845893i	−0.999241	+	0.0389507i	$0.987598\pi$
$954$		0			0
$955$	4.88809	−	1.30976i	0.158175	−	0.0423828i
$956$		0			0
$957$	−1.31273	−	1.31273i	−0.0424346	−	0.0424346i
$958$		0			0
$959$	0.0748125	+	0.129579i	0.00241582	+	0.00418433i
$960$		0			0
$961$			30.9351i			0.997907i
$962$		0			0
$963$	53.6447			1.72868
$964$		0			0
$965$	15.9998	−	9.23747i	0.515051	−	0.297365i
$966$		0			0
$967$	−18.5489	+	18.5489i	−0.596493	+	0.596493i	−0.939377	−	0.342885i	$-0.888596\pi$
$967$	−18.5489	+	18.5489i	−0.596493	+	0.596493i	0.342885	+	0.939377i	$0.388596\pi$
$968$		0			0
$969$	−2.69417	−	10.0548i	−0.0865493	−	0.323007i
$970$		0			0
$971$	−25.3476	+	43.9034i	−0.813444	+	1.40893i	0.0969959	+	0.995285i	$0.469077\pi$
$971$	−25.3476	+	43.9034i	−0.813444	+	1.40893i	−0.910440	+	0.413642i	$0.864257\pi$
$972$		0			0
$973$	−2.60762	+	9.73176i	−0.0835963	+	0.311986i
$974$		0			0
$975$	−38.4448	−	27.2227i	−1.23122	−	0.871825i
$976$		0			0
$977$	5.21978	−	19.4805i	0.166996	−	0.623236i	−0.830782	−	0.556598i	$-0.812106\pi$
$977$	5.21978	−	19.4805i	0.166996	−	0.623236i	0.997777	−	0.0666376i	$-0.0212271\pi$
$978$		0			0
$979$	2.19124	+	1.26511i	0.0700323	+	0.0404332i
$980$		0			0
$981$	11.3394	+	42.3193i	0.362040	+	1.35115i
$982$		0			0
$983$	21.7716	+	21.7716i	0.694405	+	0.694405i	0.963198	−	0.268793i	$-0.0866248\pi$
$983$	21.7716	+	21.7716i	0.694405	+	0.694405i	−0.268793	+	0.963198i	$0.586625\pi$
$984$		0			0
$985$	24.1828	−	13.9619i	0.770528	−	0.444864i
$986$		0			0
$987$			35.0035i			1.11417i
$988$		0			0
$989$			5.53425i			0.175979i
$990$		0			0
$991$	−12.8654	+	7.42785i	−0.408683	+	0.235954i	−0.690224	−	0.723596i	$-0.742488\pi$
$991$	−12.8654	+	7.42785i	−0.408683	+	0.235954i	0.281540	+	0.959549i	$0.409155\pi$
$992$		0			0
$993$	−31.9777	−	31.9777i	−1.01478	−	1.01478i
$994$		0			0
$995$	16.1852	+	60.4042i	0.513107	+	1.91494i
$996$		0			0
$997$	−1.48602	−	0.857953i	−0.0470627	−	0.0271717i	0.476284	−	0.879291i	$-0.341983\pi$
$997$	−1.48602	−	0.857953i	−0.0470627	−	0.0271717i	−0.523347	+	0.852120i	$0.675317\pi$
$998$		0			0
$999$	−10.7874	+	40.2590i	−0.341297	+	1.27374i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.2.bk.a.271.11		48
4.3	odd	2		104.2.u.a.11.1	✓	48
8.3	odd	2	inner	416.2.bk.a.271.12		48
8.5	even	2		104.2.u.a.11.10	yes	48
12.11	even	2		936.2.ed.d.739.12		48
13.6	odd	12	inner	416.2.bk.a.175.12		48
24.5	odd	2		936.2.ed.d.739.3		48
52.19	even	12		104.2.u.a.19.10	yes	48
104.19	even	12	inner	416.2.bk.a.175.11		48
104.45	odd	12		104.2.u.a.19.1	yes	48
156.71	odd	12		936.2.ed.d.19.3		48
312.149	even	12		936.2.ed.d.19.12		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.u.a.11.1	✓	48	4.3	odd	2
104.2.u.a.11.10	yes	48	8.5	even	2
104.2.u.a.19.1	yes	48	104.45	odd	12
104.2.u.a.19.10	yes	48	52.19	even	12
416.2.bk.a.175.11		48	104.19	even	12	inner
416.2.bk.a.175.12		48	13.6	odd	12	inner
416.2.bk.a.271.11		48	1.1	even	1	trivial
416.2.bk.a.271.12		48	8.3	odd	2	inner
936.2.ed.d.19.3		48	156.71	odd	12
936.2.ed.d.19.12		48	312.149	even	12
936.2.ed.d.739.3		48	24.5	odd	2
936.2.ed.d.739.12		48	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.bk (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.57612 + 2.72991i) q^{3} +(-2.13831 + 2.13831i) q^{5} +(1.43298 - 0.383965i) q^{7} +(-3.46828 + 6.00723i) q^{9} +(1.14540 + 0.306909i) q^{11} +(-1.50651 - 3.27573i) q^{13} +(-9.20762 - 2.46717i) q^{15} +(-0.917474 - 0.529704i) q^{17} +(3.01087 - 0.806761i) q^{19} +(3.30673 + 3.30673i) q^{21} +(2.44668 + 4.23777i) q^{23} -4.14473i q^{25} -12.4089 q^{27} +(-0.430121 + 0.248330i) q^{29} +(-0.180109 + 0.180109i) q^{31} +(0.967448 + 3.61056i) q^{33} +(-2.24311 + 3.88518i) q^{35} +(0.869322 - 3.24435i) q^{37} +(6.56803 - 9.27557i) q^{39} +(1.95534 - 7.29743i) q^{41} +(0.979452 + 0.565487i) q^{43} +(-5.42907 - 20.2616i) q^{45} +(5.29277 + 5.29277i) q^{47} +(-4.15619 + 2.39957i) q^{49} -3.33950i q^{51} +13.2050i q^{53} +(-3.10549 + 1.79295i) q^{55} +(6.94787 + 6.94787i) q^{57} +(1.58626 + 5.92000i) q^{59} +(9.32964 + 5.38647i) q^{61} +(-2.66339 + 9.93992i) q^{63} +(10.2259 + 3.78314i) q^{65} +(3.22308 - 12.0287i) q^{67} +(-7.71248 + 13.3584i) q^{69} +(-1.99632 - 7.45038i) q^{71} +(4.04472 - 4.04472i) q^{73} +(11.3148 - 6.53258i) q^{75} +1.75917 q^{77} +0.933621i q^{79} +(-9.15307 - 15.8536i) q^{81} +(6.19701 + 6.19701i) q^{83} +(3.09451 - 0.829172i) q^{85} +(-1.35584 - 0.782794i) q^{87} +(2.06105 + 0.552258i) q^{89} +(-3.41656 - 4.11560i) q^{91} +(-0.775556 - 0.207810i) q^{93} +(-4.71307 + 8.16328i) q^{95} +(-3.94895 + 1.05812i) q^{97} +(-5.81624 + 5.81624i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 4 q^{3} - 20 q^{9} + 8 q^{11} - 12 q^{17} + 8 q^{19} - 8 q^{27} + 4 q^{33} + 4 q^{35} + 12 q^{43} - 60 q^{49} + 36 q^{57} + 64 q^{59} - 16 q^{65} + 8 q^{67} - 12 q^{73} - 24 q^{75} - 8 q^{81} + 48 q^{83}+ \cdots - 168 q^{99}+O(q^{100}) \) 48 * q + 4 * q^3 - 20 * q^9 + 8 * q^11 - 12 * q^17 + 8 * q^19 - 8 * q^27 + 4 * q^33 + 4 * q^35 + 12 * q^43 - 60 * q^49 + 36 * q^57 + 64 * q^59 - 16 * q^65 + 8 * q^67 - 12 * q^73 - 24 * q^75 - 8 * q^81 + 48 * q^83 + 12 * q^89 - 104 * q^91 + 4 * q^97 - 168 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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