LMFDB - Embedded newform 416.2.i.e.321.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [416,2,Mod(289,416)] 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("416.289"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(416, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.i (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$3.32177672409$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label			321.2
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	416.321
Dual form			416.2.i.e.289.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+4.46410 q^{5} +(1.50000 + 2.59808i) q^{9} +(-3.23205 + 1.59808i) q^{13} +(-2.96410 - 5.13397i) q^{17} +14.9282 q^{25} +(-0.767949 + 1.33013i) q^{29} +(-5.69615 + 9.86603i) q^{37} +(5.96410 - 10.3301i) q^{41} +(6.69615 + 11.5981i) q^{45} +(3.50000 - 6.06218i) q^{49} -3.53590 q^{53} +(-7.69615 - 13.3301i) q^{61} +(-14.4282 + 7.13397i) q^{65} -10.8564 q^{73} +(-4.50000 + 7.79423i) q^{81} +(-13.2321 - 22.9186i) q^{85} +(-5.00000 + 8.66025i) q^{89} +(-9.00000 - 15.5885i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + 6 q^{9} - 6 q^{13} + 2 q^{17} + 32 q^{25} - 10 q^{29} - 2 q^{37} + 10 q^{41} + 6 q^{45} + 14 q^{49} - 28 q^{53} - 10 q^{61} - 30 q^{65} + 12 q^{73} - 18 q^{81} - 46 q^{85} - 20 q^{89} - 36 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 + 6 * q^9 - 6 * q^13 + 2 * q^17 + 32 * q^25 - 10 * q^29 - 2 * q^37 + 10 * q^41 + 6 * q^45 + 14 * q^49 - 28 * q^53 - 10 * q^61 - 30 * q^65 + 12 * q^73 - 18 * q^81 - 46 * q^85 - 20 * q^89 - 36 * q^97

Character values

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

$n$	$261$	$287$	$353$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$3$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$4$		0			0
$5$	4.46410			1.99641			0.998203	−	0.0599153i	$-0.0190830\pi$
$5$	4.46410			1.99641			0.998203	+	0.0599153i	$0.0190830\pi$
$6$		0			0
$7$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$7$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$8$		0			0
$9$	1.50000	+	2.59808i	0.500000	+	0.866025i
$10$		0			0
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$		0			0
$13$	−3.23205	+	1.59808i	−0.896410	+	0.443227i
$13$	−3.23205	+	1.59808i	−0.896410	+	0.443227i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.96410	−	5.13397i	−0.718900	−	1.24517i	−0.961436	−	0.275029i	$-0.911312\pi$
$17$	−2.96410	−	5.13397i	−0.718900	−	1.24517i	0.242536	−	0.970143i	$-0.422021\pi$
$18$		0			0
$19$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$19$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$	14.9282			2.98564
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.767949	+	1.33013i	−0.142605	+	0.246998i	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−0.767949	+	1.33013i	−0.142605	+	0.246998i	0.785872	+	0.618389i	$0.212214\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5.69615	+	9.86603i	−0.936442	+	1.62196i	−0.164399	+	0.986394i	$0.552568\pi$
$37$	−5.69615	+	9.86603i	−0.936442	+	1.62196i	−0.772043	+	0.635571i	$0.780765\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5.96410	−	10.3301i	0.931436	−	1.61329i	0.150567	−	0.988600i	$-0.451890\pi$
$41$	5.96410	−	10.3301i	0.931436	−	1.61329i	0.780869	−	0.624695i	$-0.214777\pi$
$42$		0			0
$43$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$43$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$44$		0			0
$45$	6.69615	+	11.5981i	0.998203	+	1.72894i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	3.50000	−	6.06218i	0.500000	−	0.866025i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3.53590			−0.485693			−0.242846	−	0.970065i	$-0.578081\pi$
$53$	−3.53590			−0.485693			−0.242846	+	0.970065i	$0.578081\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	−7.69615	−	13.3301i	−0.985391	−	1.70675i	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−7.69615	−	13.3301i	−0.985391	−	1.70675i	−0.345207	−	0.938527i	$-0.612191\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−14.4282	+	7.13397i	−1.78960	+	0.884861i
$66$		0			0
$67$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$67$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$71$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$72$		0			0
$73$	−10.8564			−1.27065			−0.635323	−	0.772246i	$-0.719133\pi$
$73$	−10.8564			−1.27065			−0.635323	+	0.772246i	$0.719133\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$	−13.2321	−	22.9186i	−1.43522	−	2.48587i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.00000	+	8.66025i	−0.529999	+	0.917985i	0.469389	+	0.882992i	$0.344474\pi$
$89$	−5.00000	+	8.66025i	−0.529999	+	0.917985i	−0.999388	+	0.0349934i	$0.988859\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−9.00000	−	15.5885i	−0.913812	−	1.58277i	−0.808632	−	0.588315i	$-0.799792\pi$
$97$	−9.00000	−	15.5885i	−0.913812	−	1.58277i	−0.105180	−	0.994453i	$-0.533542\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−9.16025	+	15.8660i	−0.911479	+	1.57873i	−0.0995037	+	0.995037i	$0.531726\pi$
$101$	−9.16025	+	15.8660i	−0.911479	+	1.57873i	−0.811976	+	0.583691i	$0.801608\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$107$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$108$		0			0
$109$	6.00000			0.574696			0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000			0.574696			0.287348	+	0.957826i	$0.407226\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.42820	+	5.93782i	0.322498	+	0.558583i	0.981003	−	0.193993i	$-0.0621440\pi$
$113$	3.42820	+	5.93782i	0.322498	+	0.558583i	−0.658505	+	0.752577i	$0.728811\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−9.00000	−	6.00000i	−0.832050	−	0.554700i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.50000	+	9.52628i	0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	44.3205			3.96415
$126$		0			0
$127$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$127$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.03590	−	3.52628i	−0.173939	−	0.301270i	0.765855	−	0.643013i	$-0.222316\pi$
$137$	−2.03590	−	3.52628i	−0.173939	−	0.301270i	−0.939793	+	0.341743i	$0.888983\pi$
$138$		0			0
$139$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$139$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−3.42820	+	5.93782i	−0.284697	+	0.493109i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−5.16025	−	8.93782i	−0.422744	−	0.732215i	0.573462	−	0.819232i	$-0.305600\pi$
$149$	−5.16025	−	8.93782i	−0.422744	−	0.732215i	−0.996207	+	0.0870170i	$0.972267\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$	8.89230	−	15.4019i	0.718900	−	1.24517i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−21.3923			−1.70729			−0.853646	−	0.520854i	$-0.825614\pi$
$157$	−21.3923			−1.70729			−0.853646	+	0.520854i	$0.825614\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$163$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$167$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$168$		0			0
$169$	7.89230	−	10.3301i	0.607100	−	0.794625i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	13.0000	+	22.5167i	0.988372	+	1.71191i	0.625871	+	0.779926i	$0.284744\pi$
$173$	13.0000	+	22.5167i	0.988372	+	1.71191i	0.362500	+	0.931984i	$0.381923\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$179$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$180$		0			0
$181$	−8.32051			−0.618458			−0.309229	−	0.950988i	$-0.600071\pi$
$181$	−8.32051			−0.618458			−0.309229	+	0.950988i	$0.600071\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−25.4282	+	44.0429i	−1.86952	+	3.23810i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$		0			0
$193$	6.89230	−	11.9378i	0.496119	−	0.859303i	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	6.89230	−	11.9378i	0.496119	−	0.859303i	0.999990	+	0.00447566i	$0.00142465\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1.00000	−	1.73205i	0.0712470	−	0.123404i	−0.828201	−	0.560431i	$-0.810635\pi$
$197$	1.00000	−	1.73205i	0.0712470	−	0.123404i	0.899448	+	0.437028i	$0.143969\pi$
$198$		0			0
$199$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$199$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	26.6244	−	46.1147i	1.85953	−	3.22079i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$211$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	17.7846	+	11.8564i	1.19632	+	0.797548i
$222$		0			0
$223$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$223$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$224$		0			0
$225$	22.3923	+	38.7846i	1.49282	+	2.58564i
$226$		0			0
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$		0			0
$229$	30.0000			1.98246			0.991228	−	0.132164i	$-0.0421925\pi$
$229$	30.0000			1.98246			0.991228	+	0.132164i	$0.0421925\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	26.0000			1.70332			0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000			1.70332			0.851658	+	0.524097i	$0.175597\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−10.9641	−	18.9904i	−0.706260	−	1.22328i	−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−10.9641	−	18.9904i	−0.706260	−	1.22328i	0.259975	−	0.965615i	$-0.416286\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	15.6244	−	27.0622i	0.998203	−	1.72894i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$251$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−13.3564	+	23.1340i	−0.833150	+	1.44306i	0.0623783	+	0.998053i	$0.480131\pi$
$257$	−13.3564	+	23.1340i	−0.833150	+	1.44306i	−0.895528	+	0.445005i	$0.853202\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−4.60770			−0.285209
$262$		0			0
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$	−15.7846			−0.969641
$266$		0			0
$267$		0			0
$268$		0			0
$269$	13.0000	+	22.5167i	0.792624	+	1.37287i	0.924337	+	0.381577i	$0.124619\pi$
$269$	13.0000	+	22.5167i	0.792624	+	1.37287i	−0.131713	+	0.991288i	$0.542048\pi$
$270$		0			0
$271$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$271$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	7.62436	+	13.2058i	0.458103	+	0.793458i	0.998861	−	0.0477206i	$-0.0151957\pi$
$277$	7.62436	+	13.2058i	0.458103	+	0.793458i	−0.540758	+	0.841178i	$0.681862\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	22.7128			1.35493			0.677466	−	0.735554i	$-0.263078\pi$
$281$	22.7128			1.35493			0.677466	+	0.735554i	$0.263078\pi$
$282$		0			0
$283$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$283$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−9.07180	+	15.7128i	−0.533635	+	0.924283i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−6.76795	−	11.7224i	−0.395388	−	0.684832i	0.597763	−	0.801673i	$-0.296056\pi$
$293$	−6.76795	−	11.7224i	−0.395388	−	0.684832i	−0.993151	+	0.116841i	$0.962723\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−34.3564	−	59.5070i	−1.96724	−	3.40736i
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	26.0000			1.46961			0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000			1.46961			0.734803	+	0.678280i	$0.237274\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−35.2487			−1.97976			−0.989882	−	0.141890i	$-0.954682\pi$
$317$	−35.2487			−1.97976			−0.989882	+	0.141890i	$0.954682\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−48.2487	+	23.8564i	−2.67636	+	1.32332i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$331$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$332$		0			0
$333$	−34.1769			−1.87288
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−36.7128			−1.99987			−0.999937	−	0.0112091i	$-0.996432\pi$
$337$	−36.7128			−1.99987			−0.999937	+	0.0112091i	$0.996432\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$347$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$348$		0			0
$349$	5.00000	−	8.66025i	0.267644	−	0.463573i	−0.700609	−	0.713545i	$-0.747088\pi$
$349$	5.00000	−	8.66025i	0.267644	−	0.463573i	0.968253	+	0.249973i	$0.0804216\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	15.4282	−	26.7224i	0.821160	−	1.42229i	−0.0836583	−	0.996495i	$-0.526660\pi$
$353$	15.4282	−	26.7224i	0.821160	−	1.42229i	0.904819	−	0.425797i	$-0.140006\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	9.50000	−	16.4545i	0.500000	−	0.866025i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−48.4641			−2.53673
$366$		0			0
$367$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$367$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$368$		0			0
$369$	35.7846			1.86287
$370$		0			0
$371$		0			0
$372$		0			0
$373$	19.0885	+	33.0622i	0.988363	+	1.71189i	0.625917	+	0.779890i	$0.284725\pi$
$373$	19.0885	+	33.0622i	0.988363	+	1.71189i	0.362446	+	0.932005i	$0.381942\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	0.356406	−	5.52628i	0.0183559	−	0.284618i
$378$		0			0
$379$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$379$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	34.3205			1.74012			0.870059	−	0.492947i	$-0.164080\pi$
$389$	34.3205			1.74012			0.870059	+	0.492947i	$0.164080\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−19.0000	−	32.9090i	−0.953583	−	1.65165i	−0.737579	−	0.675261i	$-0.764031\pi$
$397$	−19.0000	−	32.9090i	−0.953583	−	1.65165i	−0.216004	−	0.976392i	$-0.569302\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−16.8205	+	29.1340i	−0.839976	+	1.45488i	0.0499376	+	0.998752i	$0.484098\pi$
$401$	−16.8205	+	29.1340i	−0.839976	+	1.45488i	−0.889914	+	0.456129i	$0.849236\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−20.0885	+	34.7942i	−0.998203	+	1.72894i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	15.8205	+	27.4019i	0.782274	+	1.35494i	0.930614	+	0.366002i	$0.119274\pi$
$409$	15.8205	+	27.4019i	0.782274	+	1.35494i	−0.148340	+	0.988936i	$0.547393\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$419$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$420$		0			0
$421$	9.24871			0.450755			0.225377	−	0.974272i	$-0.427639\pi$
$421$	9.24871			0.450755			0.225377	+	0.974272i	$0.427639\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−44.2487	−	76.6410i	−2.14638	−	3.71764i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$431$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$432$		0			0
$433$	−1.89230	−	3.27757i	−0.0909384	−	0.157510i	0.816968	−	0.576683i	$-0.195653\pi$
$433$	−1.89230	−	3.27757i	−0.0909384	−	0.157510i	−0.907906	+	0.419173i	$0.862320\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$		0			0
$441$	21.0000			1.00000
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$	−22.3205	+	38.6603i	−1.05809	+	1.83267i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	7.00000	+	12.1244i	0.330350	+	0.572184i	0.982581	−	0.185837i	$-0.0594997\pi$
$449$	7.00000	+	12.1244i	0.330350	+	0.572184i	−0.652230	+	0.758021i	$0.726166\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	7.03590	−	12.1865i	0.329125	−	0.570062i	−0.653213	−	0.757174i	$-0.726579\pi$
$457$	7.03590	−	12.1865i	0.329125	−	0.570062i	0.982339	+	0.187112i	$0.0599128\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	18.1603	+	31.4545i	0.845807	+	1.46498i	0.884918	+	0.465746i	$0.154214\pi$
$461$	18.1603	+	31.4545i	0.845807	+	1.46498i	−0.0391109	+	0.999235i	$0.512453\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−5.30385	−	9.18653i	−0.242846	−	0.420622i
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$	2.64359	−	40.9904i	0.120537	−	1.86900i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−40.1769	−	69.5885i	−1.82434	−	3.15985i
$486$		0			0
$487$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$487$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$491$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$492$		0			0
$493$	9.10512			0.410074
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$503$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$504$		0			0
$505$	−40.8923	+	70.8275i	−1.81968	+	3.15178i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	16.5526	−	28.6699i	0.733679	−	1.27077i	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	16.5526	−	28.6699i	0.733679	−	1.27077i	0.955300	−	0.295637i	$-0.0955319\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−23.6410			−1.03573			−0.517866	−	0.855462i	$-0.673273\pi$
$521$	−23.6410			−1.03573			−0.517866	+	0.855462i	$0.673273\pi$
$522$		0			0
$523$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$523$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2.76795	+	42.9186i	−0.119893	+	1.85901i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	3.67949			0.158194			0.0790969	−	0.996867i	$-0.474796\pi$
$541$	3.67949			0.158194			0.0790969	+	0.996867i	$0.474796\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	26.7846			1.14733
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$		0			0
$549$	23.0885	−	39.9904i	0.985391	−	1.70675i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2.62436	+	4.54552i	−0.111198	+	0.192600i	−0.916253	−	0.400599i	$-0.868802\pi$
$557$	−2.62436	+	4.54552i	−0.111198	+	0.192600i	0.805056	+	0.593199i	$0.202135\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$	15.3038	+	26.5070i	0.643838	+	1.11516i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−13.0000	+	22.5167i	−0.544988	+	0.943948i	0.453619	+	0.891196i	$0.350133\pi$
$569$	−13.0000	+	22.5167i	−0.544988	+	0.943948i	−0.998608	+	0.0527519i	$0.983201\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	40.5692			1.68892			0.844459	−	0.535620i	$-0.179922\pi$
$577$	40.5692			1.68892			0.844459	+	0.535620i	$0.179922\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−40.1769	−	26.7846i	−1.66111	−	1.10741i
$586$		0			0
$587$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$587$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	36.8564			1.51351			0.756756	−	0.653698i	$-0.226783\pi$
$593$	36.8564			1.51351			0.756756	+	0.653698i	$0.226783\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	23.2846	−	40.3301i	0.949799	−	1.64510i	0.203954	−	0.978980i	$-0.434621\pi$
$601$	23.2846	−	40.3301i	0.949799	−	1.64510i	0.745845	−	0.666120i	$-0.232046\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	24.5526	+	42.5263i	0.998203	+	1.72894i
$606$		0			0
$607$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$607$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	7.08846	−	12.2776i	0.286300	−	0.495886i	−0.686624	−	0.727013i	$-0.740908\pi$
$613$	7.08846	−	12.2776i	0.286300	−	0.495886i	0.972924	+	0.231127i	$0.0742412\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	4.35641	+	7.54552i	0.175382	+	0.303771i	0.940294	−	0.340365i	$-0.110551\pi$
$617$	4.35641	+	7.54552i	0.175382	+	0.303771i	−0.764911	+	0.644136i	$0.777217\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	123.210			4.92841
$626$		0			0
$627$		0			0
$628$		0			0
$629$	67.5359			2.69283
$630$		0			0
$631$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$631$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−1.62436	+	25.1865i	−0.0643593	+	0.997927i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	15.9641	+	27.6506i	0.630544	+	1.09213i	0.987441	+	0.157991i	$0.0505015\pi$
$641$	15.9641	+	27.6506i	0.630544	+	1.09213i	−0.356897	+	0.934144i	$0.616165\pi$
$642$		0			0
$643$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$643$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	13.0000	−	22.5167i	0.508729	−	0.881145i	−0.491220	−	0.871036i	$-0.663449\pi$
$653$	13.0000	−	22.5167i	0.508729	−	0.881145i	0.999949	−	0.0101092i	$-0.00321793\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−16.2846	−	28.2058i	−0.635323	−	1.10041i
$658$		0			0
$659$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$659$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$660$		0			0
$661$	−7.30385	+	12.6506i	−0.284087	+	0.492053i	−0.972387	−	0.233373i	$-0.925024\pi$
$661$	−7.30385	+	12.6506i	−0.284087	+	0.492053i	0.688301	+	0.725426i	$0.258357\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−21.8923	+	37.9186i	−0.843886	+	1.46165i	0.0426985	+	0.999088i	$0.486405\pi$
$673$	−21.8923	+	37.9186i	−0.843886	+	1.46165i	−0.886585	+	0.462566i	$0.846929\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−2.00000			−0.0768662			−0.0384331	−	0.999261i	$-0.512237\pi$
$677$	−2.00000			−0.0768662			−0.0384331	+	0.999261i	$0.512237\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$683$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$684$		0			0
$685$	−9.08846	−	15.7417i	−0.347252	−	0.601458i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	11.4282	−	5.65064i	0.435380	−	0.215272i
$690$		0			0
$691$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$691$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−70.7128			−2.67844
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−10.0000			−0.377695			−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000			−0.377695			−0.188847	+	0.982006i	$0.560475\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−11.5526	−	20.0096i	−0.433865	−	0.751477i	0.563337	−	0.826227i	$-0.309517\pi$
$709$	−11.5526	−	20.0096i	−0.433865	−	0.751477i	−0.997202	+	0.0747503i	$0.976184\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−11.4641	+	19.8564i	−0.425766	+	0.737448i
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−30.4641			−1.12522			−0.562609	−	0.826723i	$-0.690202\pi$
$733$	−30.4641			−1.12522			−0.562609	+	0.826723i	$0.690202\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$739$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$743$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$744$		0			0
$745$	−23.0359	−	39.8993i	−0.843970	−	1.46180i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$751$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	9.00000	−	15.5885i	0.327111	−	0.566572i	−0.654827	−	0.755779i	$-0.727258\pi$
$757$	9.00000	−	15.5885i	0.327111	−	0.566572i	0.981937	+	0.189207i	$0.0605917\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	19.0000	+	32.9090i	0.688749	+	1.19295i	0.972243	+	0.233975i	$0.0751733\pi$
$761$	19.0000	+	32.9090i	0.688749	+	1.19295i	−0.283493	+	0.958974i	$0.591493\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	39.6962	−	68.7558i	1.43522	−	2.48587i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−25.0000	+	43.3013i	−0.901523	+	1.56148i	−0.0760054	+	0.997107i	$0.524217\pi$
$769$	−25.0000	+	43.3013i	−0.901523	+	1.56148i	−0.825518	+	0.564376i	$0.809117\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	17.0000	+	29.4449i	0.611448	+	1.05906i	0.990997	+	0.133887i	$0.0427458\pi$
$773$	17.0000	+	29.4449i	0.611448	+	1.05906i	−0.379549	+	0.925172i	$0.623921\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−95.4974			−3.40845
$786$		0			0
$787$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$787$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	46.1769	+	30.7846i	1.63979	+	1.09319i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−11.0000	−	19.0526i	−0.389640	−	0.674876i	0.602761	−	0.797922i	$-0.294067\pi$
$797$	−11.0000	−	19.0526i	−0.389640	−	0.674876i	−0.992401	+	0.123045i	$0.960734\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−30.0000			−1.06000
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−21.7487	+	37.6699i	−0.764644	+	1.32440i	0.175791	+	0.984428i	$0.443752\pi$
$809$	−21.7487	+	37.6699i	−0.764644	+	1.32440i	−0.940435	+	0.339975i	$0.889582\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	25.0000	−	43.3013i	0.872506	−	1.51122i	0.0131101	−	0.999914i	$-0.495827\pi$
$821$	25.0000	−	43.3013i	0.872506	−	1.51122i	0.859396	−	0.511311i	$-0.170840\pi$
$822$		0			0
$823$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$823$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$	22.1603	−	38.3827i	0.769657	−	1.33309i	−0.168091	−	0.985771i	$-0.553760\pi$
$829$	22.1603	−	38.3827i	0.769657	−	1.33309i	0.937749	−	0.347314i	$-0.112906\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−41.4974			−1.43780
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$839$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$840$		0			0
$841$	13.3205	+	23.0718i	0.459328	+	0.795579i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	35.2321	−	46.1147i	1.21202	−	1.58640i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	8.17691			0.279972			0.139986	−	0.990153i	$-0.455294\pi$
$853$	8.17691			0.279972			0.139986	+	0.990153i	$0.455294\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−35.9282			−1.22728			−0.613642	−	0.789584i	$-0.710296\pi$
$857$	−35.9282			−1.22728			−0.613642	+	0.789584i	$0.710296\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	58.0333	+	100.517i	1.97319	+	3.41767i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	27.0000	−	46.7654i	0.913812	−	1.58277i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−19.6962	−	34.1147i	−0.665092	−	1.15197i	−0.979260	−	0.202606i	$-0.935059\pi$
$877$	−19.6962	−	34.1147i	−0.665092	−	1.15197i	0.314169	−	0.949367i	$-0.398274\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	26.3564	−	45.6506i	0.887970	−	1.53801i	0.0456985	−	0.998955i	$-0.485449\pi$
$881$	26.3564	−	45.6506i	0.887970	−	1.53801i	0.842271	−	0.539054i	$-0.181218\pi$
$882$		0			0
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	10.4808	+	18.1532i	0.349165	+	0.604771i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−37.1436			−1.23469
$906$		0			0
$907$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$907$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$908$		0			0
$909$	−54.9615			−1.82296
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$919$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−85.0333	+	147.282i	−2.79588	+	4.84260i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−28.8205	−	49.9186i	−0.945570	−	1.63778i	−0.754606	−	0.656179i	$-0.772172\pi$
$929$	−28.8205	−	49.9186i	−0.945570	−	1.63778i	−0.190965	−	0.981597i	$-0.561162\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−22.5692			−0.737304			−0.368652	−	0.929567i	$-0.620181\pi$
$937$	−22.5692			−0.737304			−0.368652	+	0.929567i	$0.620181\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−58.0000			−1.89075			−0.945373	−	0.325991i	$-0.894302\pi$
$941$	−58.0000			−1.89075			−0.945373	+	0.325991i	$0.894302\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$947$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$948$		0			0
$949$	35.0885	−	17.3494i	1.13902	−	0.563184i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−13.0000	−	22.5167i	−0.421111	−	0.729386i	0.574937	−	0.818198i	$-0.305026\pi$
$953$	−13.0000	−	22.5167i	−0.421111	−	0.729386i	−0.996048	+	0.0888114i	$0.971693\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	30.7679	−	53.2917i	0.990455	−	1.71552i
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$971$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−18.9641	+	32.8468i	−0.606715	+	1.05086i	0.385063	+	0.922890i	$0.374180\pi$
$977$	−18.9641	+	32.8468i	−0.606715	+	1.05086i	−0.991778	+	0.127971i	$0.959153\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	9.00000	+	15.5885i	0.287348	+	0.497701i
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$	4.46410	−	7.73205i	0.142238	−	0.246364i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$991$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	20.6962	+	35.8468i	0.655454	+	1.13528i	0.981780	+	0.190022i	$0.0608559\pi$
$997$	20.6962	+	35.8468i	0.655454	+	1.13528i	−0.326326	+	0.945257i	$0.605811\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.2.i.e.321.2	yes	4
4.3	odd	2	CM	416.2.i.e.321.2	yes	4
8.3	odd	2		832.2.i.m.321.1		4
8.5	even	2		832.2.i.m.321.1		4
13.3	even	3	inner	416.2.i.e.289.2	✓	4
13.4	even	6		5408.2.a.s.1.1		2
13.9	even	3		5408.2.a.ba.1.2		2
52.3	odd	6	inner	416.2.i.e.289.2	✓	4
52.35	odd	6		5408.2.a.ba.1.2		2
52.43	odd	6		5408.2.a.s.1.1		2
104.3	odd	6		832.2.i.m.705.1		4
104.29	even	6		832.2.i.m.705.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.2.i.e.289.2	✓	4	13.3	even	3	inner
416.2.i.e.289.2	✓	4	52.3	odd	6	inner
416.2.i.e.321.2	yes	4	1.1	even	1	trivial
416.2.i.e.321.2	yes	4	4.3	odd	2	CM
832.2.i.m.321.1		4	8.3	odd	2
832.2.i.m.321.1		4	8.5	even	2
832.2.i.m.705.1		4	104.3	odd	6
832.2.i.m.705.1		4	104.29	even	6
5408.2.a.s.1.1		2	13.4	even	6
5408.2.a.s.1.1		2	52.43	odd	6
5408.2.a.ba.1.2		2	13.9	even	3
5408.2.a.ba.1.2		2	52.35	odd	6

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.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+4.46410 q^{5} +(1.50000 + 2.59808i) q^{9} +(-3.23205 + 1.59808i) q^{13} +(-2.96410 - 5.13397i) q^{17} +14.9282 q^{25} +(-0.767949 + 1.33013i) q^{29} +(-5.69615 + 9.86603i) q^{37} +(5.96410 - 10.3301i) q^{41} +(6.69615 + 11.5981i) q^{45} +(3.50000 - 6.06218i) q^{49} -3.53590 q^{53} +(-7.69615 - 13.3301i) q^{61} +(-14.4282 + 7.13397i) q^{65} -10.8564 q^{73} +(-4.50000 + 7.79423i) q^{81} +(-13.2321 - 22.9186i) q^{85} +(-5.00000 + 8.66025i) q^{89} +(-9.00000 - 15.5885i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + 6 q^{9} - 6 q^{13} + 2 q^{17} + 32 q^{25} - 10 q^{29} - 2 q^{37} + 10 q^{41} + 6 q^{45} + 14 q^{49} - 28 q^{53} - 10 q^{61} - 30 q^{65} + 12 q^{73} - 18 q^{81} - 46 q^{85} - 20 q^{89} - 36 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 + 6 * q^9 - 6 * q^13 + 2 * q^17 + 32 * q^25 - 10 * q^29 - 2 * q^37 + 10 * q^41 + 6 * q^45 + 14 * q^49 - 28 * q^53 - 10 * q^61 - 30 * q^65 + 12 * q^73 - 18 * q^81 - 46 * q^85 - 20 * q^89 - 36 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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