LMFDB - Embedded newform 1600.2.o.i.607.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1600,2,Mod(543,1600)] 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("1600.543"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1600, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 3])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.o (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(39)] == traces)

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

Self dual:	no
Analytic conductor:	$12.7760643234$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{24})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			607.2
Root			$-0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1600.607
Dual form			1600.2.o.i.543.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 0.707107i) q^{3} +2.00000i q^{9} +1.73205 q^{11} +(-2.82843 - 2.82843i) q^{13} +(-1.22474 - 1.22474i) q^{17} +5.19615i q^{19} +(4.89898 + 4.89898i) q^{23} +(-3.53553 - 3.53553i) q^{27} -6.92820 q^{29} +4.00000i q^{31} +(-1.22474 + 1.22474i) q^{33} +(5.65685 - 5.65685i) q^{37} +4.00000 q^{39} +3.00000 q^{41} +(-5.65685 + 5.65685i) q^{43} +(-4.89898 + 4.89898i) q^{47} +7.00000i q^{49} +1.73205 q^{51} +(-8.48528 - 8.48528i) q^{53} +(-3.67423 - 3.67423i) q^{57} +10.3923i q^{59} +(-4.94975 - 4.94975i) q^{67} -6.92820 q^{69} -12.0000i q^{71} +(-6.12372 + 6.12372i) q^{73} +8.00000 q^{79} -1.00000 q^{81} +(-10.6066 + 10.6066i) q^{83} +(4.89898 - 4.89898i) q^{87} +3.00000i q^{89} +(-2.82843 - 2.82843i) q^{93} +3.46410i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 32 q^{39} + 24 q^{41} + 64 q^{79} - 8 q^{81}+O(q^{100}) $ 8 * q + 32 * q^39 + 24 * q^41 + 64 * q^79 - 8 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$-1$

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
$3$	−0.707107	+	0.707107i	−0.408248	+	0.408248i	−0.881127	−	0.472879i	$-0.843215\pi$
$3$	−0.707107	+	0.707107i	−0.408248	+	0.408248i	0.472879	+	0.881127i	$0.343215\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$		0			0
$9$			2.00000i			0.666667i
$10$		0			0
$11$	1.73205			0.522233			0.261116	−	0.965307i	$-0.415909\pi$
$11$	1.73205			0.522233			0.261116	+	0.965307i	$0.415909\pi$
$12$		0			0
$13$	−2.82843	−	2.82843i	−0.784465	−	0.784465i	0.196116	−	0.980581i	$-0.437167\pi$
$13$	−2.82843	−	2.82843i	−0.784465	−	0.784465i	−0.980581	+	0.196116i	$0.937167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.22474	−	1.22474i	−0.297044	−	0.297044i	0.542811	−	0.839855i	$-0.317360\pi$
$17$	−1.22474	−	1.22474i	−0.297044	−	0.297044i	−0.839855	+	0.542811i	$0.817360\pi$
$18$		0			0
$19$			5.19615i			1.19208i	0.802955	+	0.596040i	$0.203260\pi$
$19$			5.19615i			1.19208i	−0.802955	+	0.596040i	$0.796740\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.89898	+	4.89898i	1.02151	+	1.02151i	0.999764	+	0.0217443i	$0.00692196\pi$
$23$	4.89898	+	4.89898i	1.02151	+	1.02151i	0.0217443	+	0.999764i	$0.493078\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−3.53553	−	3.53553i	−0.680414	−	0.680414i
$28$		0			0
$29$	−6.92820			−1.28654			−0.643268	−	0.765641i	$-0.722422\pi$
$29$	−6.92820			−1.28654			−0.643268	+	0.765641i	$0.722422\pi$
$30$		0			0
$31$			4.00000i			0.718421i	0.933257	+	0.359211i	$0.116954\pi$
$31$			4.00000i			0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$		0			0
$33$	−1.22474	+	1.22474i	−0.213201	+	0.213201i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	5.65685	−	5.65685i	0.929981	−	0.929981i	−0.0677230	−	0.997704i	$-0.521573\pi$
$37$	5.65685	−	5.65685i	0.929981	−	0.929981i	0.997704	+	0.0677230i	$0.0215734\pi$
$38$		0			0
$39$	4.00000			0.640513
$40$		0			0
$41$	3.00000			0.468521			0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000			0.468521			0.234261	+	0.972174i	$0.424733\pi$
$42$		0			0
$43$	−5.65685	+	5.65685i	−0.862662	+	0.862662i	−0.991647	−	0.128984i	$-0.958828\pi$
$43$	−5.65685	+	5.65685i	−0.862662	+	0.862662i	0.128984	+	0.991647i	$0.458828\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.89898	+	4.89898i	−0.714590	+	0.714590i	−0.967492	−	0.252902i	$-0.918615\pi$
$47$	−4.89898	+	4.89898i	−0.714590	+	0.714590i	0.252902	+	0.967492i	$0.418615\pi$
$48$		0			0
$49$			7.00000i			1.00000i
$50$		0			0
$51$	1.73205			0.242536
$52$		0			0
$53$	−8.48528	−	8.48528i	−1.16554	−	1.16554i	−0.983243	−	0.182300i	$-0.941646\pi$
$53$	−8.48528	−	8.48528i	−1.16554	−	1.16554i	−0.182300	−	0.983243i	$-0.558354\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−3.67423	−	3.67423i	−0.486664	−	0.486664i
$58$		0			0
$59$			10.3923i			1.35296i	0.736460	+	0.676481i	$0.236496\pi$
$59$			10.3923i			1.35296i	−0.736460	+	0.676481i	$0.763504\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−4.94975	−	4.94975i	−0.604708	−	0.604708i	0.336850	−	0.941558i	$-0.390638\pi$
$67$	−4.94975	−	4.94975i	−0.604708	−	0.604708i	−0.941558	+	0.336850i	$0.890638\pi$
$68$		0			0
$69$	−6.92820			−0.834058
$70$		0			0
$71$		−	12.0000i		−	1.42414i	−0.702109	−	0.712069i	$-0.747758\pi$
$71$		−	12.0000i		−	1.42414i	0.702109	−	0.712069i	$-0.252242\pi$
$72$		0			0
$73$	−6.12372	+	6.12372i	−0.716728	+	0.716728i	−0.967934	−	0.251206i	$-0.919173\pi$
$73$	−6.12372	+	6.12372i	−0.716728	+	0.716728i	0.251206	+	0.967934i	$0.419173\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$	−10.6066	+	10.6066i	−1.16423	+	1.16423i	−0.180685	+	0.983541i	$0.557831\pi$
$83$	−10.6066	+	10.6066i	−1.16423	+	1.16423i	−0.983541	+	0.180685i	$0.942169\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	4.89898	−	4.89898i	0.525226	−	0.525226i
$88$		0			0
$89$			3.00000i			0.317999i	0.987279	+	0.159000i	$0.0508269\pi$
$89$			3.00000i			0.317999i	−0.987279	+	0.159000i	$0.949173\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−2.82843	−	2.82843i	−0.293294	−	0.293294i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$97$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$98$		0			0
$99$			3.46410i			0.348155i
$100$		0			0
$101$			13.8564i			1.37876i	0.724398	+	0.689382i	$0.242118\pi$
$101$			13.8564i			1.37876i	−0.724398	+	0.689382i	$0.757882\pi$
$102$		0			0
$103$	4.89898	+	4.89898i	0.482711	+	0.482711i	0.905996	−	0.423286i	$-0.139123\pi$
$103$	4.89898	+	4.89898i	0.482711	+	0.482711i	−0.423286	+	0.905996i	$0.639123\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.12132	+	2.12132i	0.205076	+	0.205076i	0.802171	−	0.597095i	$-0.203678\pi$
$107$	2.12132	+	2.12132i	0.205076	+	0.205076i	−0.597095	+	0.802171i	$0.703678\pi$
$108$		0			0
$109$	−13.8564			−1.32720			−0.663602	−	0.748086i	$-0.730973\pi$
$109$	−13.8564			−1.32720			−0.663602	+	0.748086i	$0.730973\pi$
$110$		0			0
$111$			8.00000i			0.759326i
$112$		0			0
$113$	−8.57321	+	8.57321i	−0.806500	+	0.806500i	−0.984102	−	0.177602i	$-0.943166\pi$
$113$	−8.57321	+	8.57321i	−0.806500	+	0.806500i	0.177602	+	0.984102i	$0.443166\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	5.65685	−	5.65685i	0.522976	−	0.522976i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−8.00000			−0.727273
$122$		0			0
$123$	−2.12132	+	2.12132i	−0.191273	+	0.191273i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−14.6969	+	14.6969i	−1.30414	+	1.30414i	−0.378570	+	0.925573i	$0.623584\pi$
$127$	−14.6969	+	14.6969i	−1.30414	+	1.30414i	−0.925573	+	0.378570i	$0.876416\pi$
$128$		0			0
$129$		−	8.00000i		−	0.704361i
$130$		0			0
$131$	3.46410			0.302660			0.151330	−	0.988483i	$-0.451644\pi$
$131$	3.46410			0.302660			0.151330	+	0.988483i	$0.451644\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3.67423	+	3.67423i	0.313911	+	0.313911i	0.846423	−	0.532512i	$-0.178752\pi$
$137$	3.67423	+	3.67423i	0.313911	+	0.313911i	−0.532512	+	0.846423i	$0.678752\pi$
$138$		0			0
$139$			1.73205i			0.146911i	0.997299	+	0.0734553i	$0.0234026\pi$
$139$			1.73205i			0.146911i	−0.997299	+	0.0734553i	$0.976597\pi$
$140$		0			0
$141$		−	6.92820i		−	0.583460i
$142$		0			0
$143$	−4.89898	−	4.89898i	−0.409673	−	0.409673i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−4.94975	−	4.94975i	−0.408248	−	0.408248i
$148$		0			0
$149$	6.92820			0.567581			0.283790	−	0.958886i	$-0.408408\pi$
$149$	6.92820			0.567581			0.283790	+	0.958886i	$0.408408\pi$
$150$		0			0
$151$		−	16.0000i		−	1.30206i	−0.759051	−	0.651031i	$-0.774337\pi$
$151$		−	16.0000i		−	1.30206i	0.759051	−	0.651031i	$-0.225663\pi$
$152$		0			0
$153$	2.44949	−	2.44949i	0.198030	−	0.198030i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	2.82843	−	2.82843i	0.225733	−	0.225733i	−0.585174	−	0.810907i	$-0.698974\pi$
$157$	2.82843	−	2.82843i	0.225733	−	0.225733i	0.810907	+	0.585174i	$0.198974\pi$
$158$		0			0
$159$	12.0000			0.951662
$160$		0			0
$161$		0			0
$162$		0			0
$163$	4.94975	−	4.94975i	0.387694	−	0.387694i	−0.486170	−	0.873864i	$-0.661606\pi$
$163$	4.94975	−	4.94975i	0.387694	−	0.387694i	0.873864	+	0.486170i	$0.161606\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	14.6969	−	14.6969i	1.13728	−	1.13728i	0.148348	−	0.988935i	$-0.452604\pi$
$167$	14.6969	−	14.6969i	1.13728	−	1.13728i	0.988935	−	0.148348i	$-0.0473956\pi$
$168$		0			0
$169$			3.00000i			0.230769i
$170$		0			0
$171$	−10.3923			−0.794719
$172$		0			0
$173$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$173$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−7.34847	−	7.34847i	−0.552345	−	0.552345i
$178$		0			0
$179$		−	5.19615i		−	0.388379i	−0.980964	−	0.194189i	$-0.937792\pi$
$179$		−	5.19615i		−	0.388379i	0.980964	−	0.194189i	$-0.0622076\pi$
$180$		0			0
$181$		−	6.92820i		−	0.514969i	−0.966282	−	0.257485i	$-0.917106\pi$
$181$		−	6.92820i		−	0.514969i	0.966282	−	0.257485i	$-0.0828937\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−2.12132	−	2.12132i	−0.155126	−	0.155126i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			12.0000i			0.868290i	0.900843	+	0.434145i	$0.142949\pi$
$191$			12.0000i			0.868290i	−0.900843	+	0.434145i	$0.857051\pi$
$192$		0			0
$193$	11.0227	−	11.0227i	0.793432	−	0.793432i	−0.188619	−	0.982050i	$-0.560401\pi$
$193$	11.0227	−	11.0227i	0.793432	−	0.793432i	0.982050	+	0.188619i	$0.0604011\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−8.48528	+	8.48528i	−0.604551	+	0.604551i	−0.941517	−	0.336966i	$-0.890599\pi$
$197$	−8.48528	+	8.48528i	−0.604551	+	0.604551i	0.336966	+	0.941517i	$0.390599\pi$
$198$		0			0
$199$	−16.0000			−1.13421			−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000			−1.13421			−0.567105	+	0.823646i	$0.691937\pi$
$200$		0			0
$201$	7.00000			0.493742
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−9.79796	+	9.79796i	−0.681005	+	0.681005i
$208$		0			0
$209$			9.00000i			0.622543i
$210$		0			0
$211$	−8.66025			−0.596196			−0.298098	−	0.954535i	$-0.596352\pi$
$211$	−8.66025			−0.596196			−0.298098	+	0.954535i	$0.596352\pi$
$212$		0			0
$213$	8.48528	+	8.48528i	0.581402	+	0.581402i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		−	8.66025i		−	0.585206i
$220$		0			0
$221$			6.92820i			0.466041i
$222$		0			0
$223$	−9.79796	−	9.79796i	−0.656120	−	0.656120i	0.298340	−	0.954460i	$-0.403567\pi$
$223$	−9.79796	−	9.79796i	−0.656120	−	0.656120i	−0.954460	+	0.298340i	$0.903567\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	16.9706	+	16.9706i	1.12638	+	1.12638i	0.990762	+	0.135614i	$0.0433007\pi$
$227$	16.9706	+	16.9706i	1.12638	+	1.12638i	0.135614	+	0.990762i	$0.456699\pi$
$228$		0			0
$229$	13.8564			0.915657			0.457829	−	0.889041i	$-0.348627\pi$
$229$	13.8564			0.915657			0.457829	+	0.889041i	$0.348627\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$233$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−5.65685	+	5.65685i	−0.367452	+	0.367452i
$238$		0			0
$239$	−12.0000			−0.776215			−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000			−0.776215			−0.388108	+	0.921614i	$0.626871\pi$
$240$		0			0
$241$	1.00000			0.0644157			0.0322078	−	0.999481i	$-0.489746\pi$
$241$	1.00000			0.0644157			0.0322078	+	0.999481i	$0.489746\pi$
$242$		0			0
$243$	11.3137	−	11.3137i	0.725775	−	0.725775i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	14.6969	−	14.6969i	0.935144	−	0.935144i
$248$		0			0
$249$		−	15.0000i		−	0.950586i
$250$		0			0
$251$	29.4449			1.85854			0.929272	−	0.369397i	$-0.120436\pi$
$251$	29.4449			1.85854			0.929272	+	0.369397i	$0.120436\pi$
$252$		0			0
$253$	8.48528	+	8.48528i	0.533465	+	0.533465i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	19.5959	+	19.5959i	1.22236	+	1.22236i	0.966791	+	0.255569i	$0.0822628\pi$
$257$	19.5959	+	19.5959i	1.22236	+	1.22236i	0.255569	+	0.966791i	$0.417737\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		−	13.8564i		−	0.857690i
$262$		0			0
$263$	14.6969	+	14.6969i	0.906252	+	0.906252i	0.995967	−	0.0897154i	$-0.0285957\pi$
$263$	14.6969	+	14.6969i	0.906252	+	0.906252i	−0.0897154	+	0.995967i	$0.528596\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−2.12132	−	2.12132i	−0.129823	−	0.129823i
$268$		0			0
$269$	27.7128			1.68968			0.844840	−	0.535019i	$-0.179696\pi$
$269$	27.7128			1.68968			0.844840	+	0.535019i	$0.179696\pi$
$270$		0			0
$271$			28.0000i			1.70088i	0.526073	+	0.850439i	$0.323664\pi$
$271$			28.0000i			1.70088i	−0.526073	+	0.850439i	$0.676336\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−2.82843	+	2.82843i	−0.169944	+	0.169944i	−0.786955	−	0.617011i	$-0.788343\pi$
$277$	−2.82843	+	2.82843i	−0.169944	+	0.169944i	0.617011	+	0.786955i	$0.288343\pi$
$278$		0			0
$279$	−8.00000			−0.478947
$280$		0			0
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		0			0
$283$	20.5061	−	20.5061i	1.21896	−	1.21896i	0.250965	−	0.967996i	$-0.419252\pi$
$283$	20.5061	−	20.5061i	1.21896	−	1.21896i	0.967996	−	0.250965i	$-0.0807478\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$		−	14.0000i		−	0.823529i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−6.12372	−	6.12372i	−0.355335	−	0.355335i
$298$		0			0
$299$		−	27.7128i		−	1.60267i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−9.79796	−	9.79796i	−0.562878	−	0.562878i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	4.94975	+	4.94975i	0.282497	+	0.282497i	0.834104	−	0.551607i	$-0.185985\pi$
$307$	4.94975	+	4.94975i	0.282497	+	0.282497i	−0.551607	+	0.834104i	$0.685985\pi$
$308$		0			0
$309$	−6.92820			−0.394132
$310$		0			0
$311$		−	24.0000i		−	1.36092i	−0.732787	−	0.680458i	$-0.761781\pi$
$311$		−	24.0000i		−	1.36092i	0.732787	−	0.680458i	$-0.238219\pi$
$312$		0			0
$313$	19.5959	−	19.5959i	1.10763	−	1.10763i	0.114165	−	0.993462i	$-0.463581\pi$
$313$	19.5959	−	19.5959i	1.10763	−	1.10763i	0.993462	−	0.114165i	$-0.0364192\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−16.9706	+	16.9706i	−0.953162	+	0.953162i	−0.998951	−	0.0457894i	$-0.985420\pi$
$317$	−16.9706	+	16.9706i	−0.953162	+	0.953162i	0.0457894	+	0.998951i	$0.485420\pi$
$318$		0			0
$319$	−12.0000			−0.671871
$320$		0			0
$321$	−3.00000			−0.167444
$322$		0			0
$323$	6.36396	−	6.36396i	0.354100	−	0.354100i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	9.79796	−	9.79796i	0.541828	−	0.541828i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−29.4449			−1.61844			−0.809218	−	0.587508i	$-0.800109\pi$
$331$	−29.4449			−1.61844			−0.809218	+	0.587508i	$0.800109\pi$
$332$		0			0
$333$	11.3137	+	11.3137i	0.619987	+	0.619987i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−8.57321	−	8.57321i	−0.467013	−	0.467013i	0.433933	−	0.900945i	$-0.357126\pi$
$337$	−8.57321	−	8.57321i	−0.467013	−	0.467013i	−0.900945	+	0.433933i	$0.857126\pi$
$338$		0			0
$339$		−	12.1244i		−	0.658505i
$340$		0			0
$341$			6.92820i			0.375183i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	19.0919	+	19.0919i	1.02491	+	1.02491i	0.999682	+	0.0252242i	$0.00802995\pi$
$347$	19.0919	+	19.0919i	1.02491	+	1.02491i	0.0252242	+	0.999682i	$0.491970\pi$
$348$		0			0
$349$	27.7128			1.48343			0.741716	−	0.670714i	$-0.234012\pi$
$349$	27.7128			1.48343			0.741716	+	0.670714i	$0.234012\pi$
$350$		0			0
$351$			20.0000i			1.06752i
$352$		0			0
$353$	19.5959	−	19.5959i	1.04299	−	1.04299i	0.0439518	−	0.999034i	$-0.486005\pi$
$353$	19.5959	−	19.5959i	1.04299	−	1.04299i	0.999034	−	0.0439518i	$-0.0139948\pi$
$354$		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
$361$	−8.00000			−0.421053
$362$		0			0
$363$	5.65685	−	5.65685i	0.296908	−	0.296908i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−9.79796	+	9.79796i	−0.511449	+	0.511449i	−0.914970	−	0.403521i	$-0.867786\pi$
$367$	−9.79796	+	9.79796i	−0.511449	+	0.511449i	0.403521	+	0.914970i	$0.367786\pi$
$368$		0			0
$369$			6.00000i			0.312348i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	19.7990	+	19.7990i	1.02515	+	1.02515i	0.999675	+	0.0254774i	$0.00811060\pi$
$373$	19.7990	+	19.7990i	1.02515	+	1.02515i	0.0254774	+	0.999675i	$0.491889\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	19.5959	+	19.5959i	1.00924	+	1.00924i
$378$		0			0
$379$		−	1.73205i		−	0.0889695i	−0.999010	−	0.0444847i	$-0.985835\pi$
$379$		−	1.73205i		−	0.0889695i	0.999010	−	0.0444847i	$-0.0141646\pi$
$380$		0			0
$381$		−	20.7846i		−	1.06483i
$382$		0			0
$383$	−19.5959	−	19.5959i	−1.00130	−	1.00130i	−0.999999	−	0.00130548i	$-0.999584\pi$
$383$	−19.5959	−	19.5959i	−1.00130	−	1.00130i	−0.00130548	−	0.999999i	$-0.500416\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−11.3137	−	11.3137i	−0.575108	−	0.575108i
$388$		0			0
$389$	6.92820			0.351274			0.175637	−	0.984455i	$-0.443802\pi$
$389$	6.92820			0.351274			0.175637	+	0.984455i	$0.443802\pi$
$390$		0			0
$391$		−	12.0000i		−	0.606866i
$392$		0			0
$393$	−2.44949	+	2.44949i	−0.123560	+	0.123560i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	11.3137	−	11.3137i	0.567819	−	0.567819i	−0.363698	−	0.931517i	$-0.618486\pi$
$397$	11.3137	−	11.3137i	0.567819	−	0.567819i	0.931517	+	0.363698i	$0.118486\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	15.0000			0.749064			0.374532	−	0.927214i	$-0.377803\pi$
$401$	15.0000			0.749064			0.374532	+	0.927214i	$0.377803\pi$
$402$		0			0
$403$	11.3137	−	11.3137i	0.563576	−	0.563576i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	9.79796	−	9.79796i	0.485667	−	0.485667i
$408$		0			0
$409$		−	5.00000i		−	0.247234i	−0.992330	−	0.123617i	$-0.960551\pi$
$409$		−	5.00000i		−	0.247234i	0.992330	−	0.123617i	$-0.0394494\pi$
$410$		0			0
$411$	−5.19615			−0.256307
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−1.22474	−	1.22474i	−0.0599760	−	0.0599760i
$418$		0			0
$419$		−	19.0526i		−	0.930778i	−0.885106	−	0.465389i	$-0.845914\pi$
$419$		−	19.0526i		−	0.930778i	0.885106	−	0.465389i	$-0.154086\pi$
$420$		0			0
$421$		−	27.7128i		−	1.35064i	−0.737525	−	0.675320i	$-0.764006\pi$
$421$		−	27.7128i		−	1.35064i	0.737525	−	0.675320i	$-0.235994\pi$
$422$		0			0
$423$	−9.79796	−	9.79796i	−0.476393	−	0.476393i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	6.92820			0.334497
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	−20.8207	+	20.8207i	−1.00058	+	1.00058i	−0.000577367		1.00000i	$0.500184\pi$
$433$	−20.8207	+	20.8207i	−1.00058	+	1.00058i	−1.00000		0.000577367i	$0.999816\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−25.4558	+	25.4558i	−1.21772	+	1.21772i
$438$		0			0
$439$	−4.00000			−0.190910			−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000			−0.190910			−0.0954548	+	0.995434i	$0.530431\pi$
$440$		0			0
$441$	−14.0000			−0.666667
$442$		0			0
$443$	2.12132	−	2.12132i	0.100787	−	0.100787i	−0.654915	−	0.755702i	$-0.727296\pi$
$443$	2.12132	−	2.12132i	0.100787	−	0.100787i	0.755702	+	0.654915i	$0.227296\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−4.89898	+	4.89898i	−0.231714	+	0.231714i
$448$		0			0
$449$		−	39.0000i		−	1.84052i	−0.391303	−	0.920262i	$-0.627976\pi$
$449$		−	39.0000i		−	1.84052i	0.391303	−	0.920262i	$-0.372024\pi$
$450$		0			0
$451$	5.19615			0.244677
$452$		0			0
$453$	11.3137	+	11.3137i	0.531564	+	0.531564i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−3.67423	−	3.67423i	−0.171873	−	0.171873i	0.615929	−	0.787802i	$-0.288781\pi$
$457$	−3.67423	−	3.67423i	−0.171873	−	0.171873i	−0.787802	+	0.615929i	$0.788781\pi$
$458$		0			0
$459$			8.66025i			0.404226i
$460$		0			0
$461$			41.5692i			1.93607i	0.250812	+	0.968036i	$0.419302\pi$
$461$			41.5692i			1.93607i	−0.250812	+	0.968036i	$0.580698\pi$
$462$		0			0
$463$	−9.79796	−	9.79796i	−0.455350	−	0.455350i	0.441776	−	0.897126i	$-0.354349\pi$
$463$	−9.79796	−	9.79796i	−0.455350	−	0.455350i	−0.897126	+	0.441776i	$0.854349\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	16.9706	+	16.9706i	0.785304	+	0.785304i	0.980720	−	0.195416i	$-0.0626058\pi$
$467$	16.9706	+	16.9706i	0.785304	+	0.785304i	−0.195416	+	0.980720i	$0.562606\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$			4.00000i			0.184310i
$472$		0			0
$473$	−9.79796	+	9.79796i	−0.450511	+	0.450511i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	16.9706	−	16.9706i	0.777029	−	0.777029i
$478$		0			0
$479$	−12.0000			−0.548294			−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000			−0.548294			−0.274147	+	0.961688i	$0.588395\pi$
$480$		0			0
$481$	−32.0000			−1.45907
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	4.89898	−	4.89898i	0.221994	−	0.221994i	−0.587344	−	0.809338i	$-0.699826\pi$
$487$	4.89898	−	4.89898i	0.221994	−	0.221994i	0.809338	+	0.587344i	$0.199826\pi$
$488$		0			0
$489$			7.00000i			0.316551i
$490$		0			0
$491$	10.3923			0.468998			0.234499	−	0.972116i	$-0.424655\pi$
$491$	10.3923			0.468998			0.234499	+	0.972116i	$0.424655\pi$
$492$		0			0
$493$	8.48528	+	8.48528i	0.382158	+	0.382158i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	24.2487i		−	1.08552i	−0.839887	−	0.542761i	$-0.817379\pi$
$499$		−	24.2487i		−	1.08552i	0.839887	−	0.542761i	$-0.182621\pi$
$500$		0			0
$501$			20.7846i			0.928588i
$502$		0			0
$503$	19.5959	+	19.5959i	0.873739	+	0.873739i	0.992878	−	0.119139i	$-0.0380134\pi$
$503$	19.5959	+	19.5959i	0.873739	+	0.873739i	−0.119139	+	0.992878i	$0.538013\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−2.12132	−	2.12132i	−0.0942111	−	0.0942111i
$508$		0			0
$509$	−6.92820			−0.307087			−0.153544	−	0.988142i	$-0.549069\pi$
$509$	−6.92820			−0.307087			−0.153544	+	0.988142i	$0.549069\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	18.3712	−	18.3712i	0.811107	−	0.811107i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−8.48528	+	8.48528i	−0.373182	+	0.373182i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3.00000			0.131432			0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000			0.131432			0.0657162	+	0.997838i	$0.479067\pi$
$522$		0			0
$523$	−9.19239	+	9.19239i	−0.401955	+	0.401955i	−0.878922	−	0.476966i	$-0.841736\pi$
$523$	−9.19239	+	9.19239i	−0.401955	+	0.401955i	0.476966	+	0.878922i	$0.341736\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	4.89898	−	4.89898i	0.213403	−	0.213403i
$528$		0			0
$529$			25.0000i			1.08696i
$530$		0			0
$531$	−20.7846			−0.901975
$532$		0			0
$533$	−8.48528	−	8.48528i	−0.367538	−	0.367538i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	3.67423	+	3.67423i	0.158555	+	0.158555i
$538$		0			0
$539$			12.1244i			0.522233i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	4.89898	+	4.89898i	0.210235	+	0.210235i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−12.0208	−	12.0208i	−0.513973	−	0.513973i	0.401768	−	0.915741i	$-0.368396\pi$
$547$	−12.0208	−	12.0208i	−0.513973	−	0.513973i	−0.915741	+	0.401768i	$0.868396\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	36.0000i		−	1.53365i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−8.48528	+	8.48528i	−0.359533	+	0.359533i	−0.863641	−	0.504108i	$-0.831821\pi$
$557$	−8.48528	+	8.48528i	−0.359533	+	0.359533i	0.504108	+	0.863641i	$0.331821\pi$
$558$		0			0
$559$	32.0000			1.35346
$560$		0			0
$561$	3.00000			0.126660
$562$		0			0
$563$	−16.9706	+	16.9706i	−0.715224	+	0.715224i	−0.967623	−	0.252399i	$-0.918780\pi$
$563$	−16.9706	+	16.9706i	−0.715224	+	0.715224i	0.252399	+	0.967623i	$0.418780\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	27.0000i		−	1.13190i	−0.824440	−	0.565949i	$-0.808510\pi$
$569$		−	27.0000i		−	1.13190i	0.824440	−	0.565949i	$-0.191490\pi$
$570$		0			0
$571$	−10.3923			−0.434904			−0.217452	−	0.976071i	$-0.569775\pi$
$571$	−10.3923			−0.434904			−0.217452	+	0.976071i	$0.569775\pi$
$572$		0			0
$573$	−8.48528	−	8.48528i	−0.354478	−	0.354478i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	18.3712	+	18.3712i	0.764802	+	0.764802i	0.977186	−	0.212384i	$-0.0681229\pi$
$577$	18.3712	+	18.3712i	0.764802	+	0.764802i	−0.212384	+	0.977186i	$0.568123\pi$
$578$		0			0
$579$			15.5885i			0.647834i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−14.6969	−	14.6969i	−0.608685	−	0.608685i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	19.0919	+	19.0919i	0.788006	+	0.788006i	0.981167	−	0.193161i	$-0.0618739\pi$
$587$	19.0919	+	19.0919i	0.788006	+	0.788006i	−0.193161	+	0.981167i	$0.561874\pi$
$588$		0			0
$589$	−20.7846			−0.856415
$590$		0			0
$591$		−	12.0000i		−	0.493614i
$592$		0			0
$593$	−28.1691	+	28.1691i	−1.15677	+	1.15677i	−0.171601	+	0.985167i	$0.554894\pi$
$593$	−28.1691	+	28.1691i	−1.15677	+	1.15677i	−0.985167	+	0.171601i	$0.945106\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	11.3137	−	11.3137i	0.463039	−	0.463039i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	19.0000			0.775026			0.387513	−	0.921864i	$-0.373334\pi$
$601$	19.0000			0.775026			0.387513	+	0.921864i	$0.373334\pi$
$602$		0			0
$603$	9.89949	−	9.89949i	0.403139	−	0.403139i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−19.5959	+	19.5959i	−0.795374	+	0.795374i	−0.982362	−	0.186988i	$-0.940127\pi$
$607$	−19.5959	+	19.5959i	−0.795374	+	0.795374i	0.186988	+	0.982362i	$0.440127\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	27.7128			1.12114
$612$		0			0
$613$	−14.1421	−	14.1421i	−0.571195	−	0.571195i	0.361267	−	0.932462i	$-0.382344\pi$
$613$	−14.1421	−	14.1421i	−0.571195	−	0.571195i	−0.932462	+	0.361267i	$0.882344\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−19.5959	−	19.5959i	−0.788902	−	0.788902i	0.192412	−	0.981314i	$-0.438369\pi$
$617$	−19.5959	−	19.5959i	−0.788902	−	0.788902i	−0.981314	+	0.192412i	$0.938369\pi$
$618$		0			0
$619$		−	10.3923i		−	0.417702i	−0.977947	−	0.208851i	$-0.933028\pi$
$619$		−	10.3923i		−	0.417702i	0.977947	−	0.208851i	$-0.0669724\pi$
$620$		0			0
$621$		−	34.6410i		−	1.39010i
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−6.36396	−	6.36396i	−0.254152	−	0.254152i
$628$		0			0
$629$	−13.8564			−0.552491
$630$		0			0
$631$			28.0000i			1.11466i	0.830290	+	0.557331i	$0.188175\pi$
$631$			28.0000i			1.11466i	−0.830290	+	0.557331i	$0.811825\pi$
$632$		0			0
$633$	6.12372	−	6.12372i	0.243396	−	0.243396i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	19.7990	−	19.7990i	0.784465	−	0.784465i
$638$		0			0
$639$	24.0000			0.949425
$640$		0			0
$641$	18.0000			0.710957			0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000			0.710957			0.355479	+	0.934684i	$0.384318\pi$
$642$		0			0
$643$	−5.65685	+	5.65685i	−0.223085	+	0.223085i	−0.809796	−	0.586711i	$-0.800422\pi$
$643$	−5.65685	+	5.65685i	−0.223085	+	0.223085i	0.586711	+	0.809796i	$0.300422\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$647$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$648$		0			0
$649$			18.0000i			0.706562i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−16.9706	−	16.9706i	−0.664109	−	0.664109i	0.292237	−	0.956346i	$-0.405601\pi$
$653$	−16.9706	−	16.9706i	−0.664109	−	0.664109i	−0.956346	+	0.292237i	$0.905601\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−12.2474	−	12.2474i	−0.477818	−	0.477818i
$658$		0			0
$659$			36.3731i			1.41689i	0.705764	+	0.708447i	$0.250604\pi$
$659$			36.3731i			1.41689i	−0.705764	+	0.708447i	$0.749396\pi$
$660$		0			0
$661$		−	6.92820i		−	0.269476i	−0.990881	−	0.134738i	$-0.956981\pi$
$661$		−	6.92820i		−	0.269476i	0.990881	−	0.134738i	$-0.0430193\pi$
$662$		0			0
$663$	−4.89898	−	4.89898i	−0.190261	−	0.190261i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−33.9411	−	33.9411i	−1.31421	−	1.31421i
$668$		0			0
$669$	13.8564			0.535720
$670$		0			0
$671$		0			0
$672$		0			0
$673$	19.5959	−	19.5959i	0.755367	−	0.755367i	−0.220108	−	0.975475i	$-0.570641\pi$
$673$	19.5959	−	19.5959i	0.755367	−	0.755367i	0.975475	+	0.220108i	$0.0706411\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−24.0000			−0.919682
$682$		0			0
$683$	−31.8198	+	31.8198i	−1.21755	+	1.21755i	−0.249064	+	0.968487i	$0.580123\pi$
$683$	−31.8198	+	31.8198i	−1.21755	+	1.21755i	−0.968487	+	0.249064i	$0.919877\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−9.79796	+	9.79796i	−0.373815	+	0.373815i
$688$		0			0
$689$			48.0000i			1.82865i
$690$		0			0
$691$	−5.19615			−0.197671			−0.0988355	−	0.995104i	$-0.531512\pi$
$691$	−5.19615			−0.197671			−0.0988355	+	0.995104i	$0.531512\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−3.67423	−	3.67423i	−0.139172	−	0.139172i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			34.6410i			1.30837i	0.756333	+	0.654187i	$0.226989\pi$
$701$			34.6410i			1.30837i	−0.756333	+	0.654187i	$0.773011\pi$
$702$		0			0
$703$	29.3939	+	29.3939i	1.10861	+	1.10861i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−41.5692			−1.56116			−0.780582	−	0.625053i	$-0.785077\pi$
$709$	−41.5692			−1.56116			−0.780582	+	0.625053i	$0.785077\pi$
$710$		0			0
$711$			16.0000i			0.600047i
$712$		0			0
$713$	−19.5959	+	19.5959i	−0.733873	+	0.733873i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	8.48528	−	8.48528i	0.316889	−	0.316889i
$718$		0			0
$719$	−24.0000			−0.895049			−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000			−0.895049			−0.447524	+	0.894272i	$0.647694\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−0.707107	+	0.707107i	−0.0262976	+	0.0262976i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−4.89898	+	4.89898i	−0.181693	+	0.181693i	−0.792093	−	0.610400i	$-0.791009\pi$
$727$	−4.89898	+	4.89898i	−0.181693	+	0.181693i	0.610400	+	0.792093i	$0.291009\pi$
$728$		0			0
$729$			13.0000i			0.481481i
$730$		0			0
$731$	13.8564			0.512498
$732$		0			0
$733$	2.82843	+	2.82843i	0.104470	+	0.104470i	0.757410	−	0.652940i	$-0.226464\pi$
$733$	2.82843	+	2.82843i	0.104470	+	0.104470i	−0.652940	+	0.757410i	$0.726464\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−8.57321	−	8.57321i	−0.315798	−	0.315798i
$738$		0			0
$739$			51.9615i			1.91144i	0.294285	+	0.955718i	$0.404919\pi$
$739$			51.9615i			1.91144i	−0.294285	+	0.955718i	$0.595081\pi$
$740$		0			0
$741$			20.7846i			0.763542i
$742$		0			0
$743$	24.4949	+	24.4949i	0.898631	+	0.898631i	0.995315	−	0.0966845i	$-0.0308238\pi$
$743$	24.4949	+	24.4949i	0.898631	+	0.898631i	−0.0966845	+	0.995315i	$0.530824\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−21.2132	−	21.2132i	−0.776151	−	0.776151i
$748$		0			0
$749$		0			0
$750$		0			0
$751$			4.00000i			0.145962i	0.997333	+	0.0729810i	$0.0232513\pi$
$751$			4.00000i			0.145962i	−0.997333	+	0.0729810i	$0.976749\pi$
$752$		0			0
$753$	−20.8207	+	20.8207i	−0.758747	+	0.758747i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	28.2843	−	28.2843i	1.02801	−	1.02801i	0.0284131	−	0.999596i	$-0.490955\pi$
$757$	28.2843	−	28.2843i	1.02801	−	1.02801i	0.999596	−	0.0284131i	$-0.00904537\pi$
$758$		0			0
$759$	−12.0000			−0.435572
$760$		0			0
$761$	21.0000			0.761249			0.380625	−	0.924730i	$-0.375709\pi$
$761$	21.0000			0.761249			0.380625	+	0.924730i	$0.375709\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	29.3939	−	29.3939i	1.06135	−	1.06135i
$768$		0			0
$769$		−	41.0000i		−	1.47850i	−0.673432	−	0.739249i	$-0.735181\pi$
$769$		−	41.0000i		−	1.47850i	0.673432	−	0.739249i	$-0.264819\pi$
$770$		0			0
$771$	−27.7128			−0.998053
$772$		0			0
$773$	−25.4558	−	25.4558i	−0.915583	−	0.915583i	0.0811212	−	0.996704i	$-0.474150\pi$
$773$	−25.4558	−	25.4558i	−0.915583	−	0.915583i	−0.996704	+	0.0811212i	$0.974150\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$			15.5885i			0.558514i
$780$		0			0
$781$		−	20.7846i		−	0.743732i
$782$		0			0
$783$	24.4949	+	24.4949i	0.875376	+	0.875376i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−5.65685	−	5.65685i	−0.201645	−	0.201645i	0.599059	−	0.800705i	$-0.295541\pi$
$787$	−5.65685	−	5.65685i	−0.201645	−	0.201645i	−0.800705	+	0.599059i	$0.795541\pi$
$788$		0			0
$789$	−20.7846			−0.739952
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−25.4558	+	25.4558i	−0.901692	+	0.901692i	−0.995583	−	0.0938903i	$-0.970070\pi$
$797$	−25.4558	+	25.4558i	−0.901692	+	0.901692i	0.0938903	+	0.995583i	$0.470070\pi$
$798$		0			0
$799$	12.0000			0.424529
$800$		0			0
$801$	−6.00000			−0.212000
$802$		0			0
$803$	−10.6066	+	10.6066i	−0.374299	+	0.374299i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−19.5959	+	19.5959i	−0.689809	+	0.689809i
$808$		0			0
$809$			42.0000i			1.47664i	0.674450	+	0.738321i	$0.264381\pi$
$809$			42.0000i			1.47664i	−0.674450	+	0.738321i	$0.735619\pi$
$810$		0			0
$811$	38.1051			1.33805			0.669026	−	0.743239i	$-0.266712\pi$
$811$	38.1051			1.33805			0.669026	+	0.743239i	$0.266712\pi$
$812$		0			0
$813$	−19.7990	−	19.7990i	−0.694381	−	0.694381i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−29.3939	−	29.3939i	−1.02836	−	1.02836i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	20.7846i		−	0.725388i	−0.931908	−	0.362694i	$-0.881857\pi$
$821$		−	20.7846i		−	0.725388i	0.931908	−	0.362694i	$-0.118143\pi$
$822$		0			0
$823$	−9.79796	−	9.79796i	−0.341535	−	0.341535i	0.515409	−	0.856944i	$-0.327640\pi$
$823$	−9.79796	−	9.79796i	−0.341535	−	0.341535i	−0.856944	+	0.515409i	$0.827640\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−2.12132	−	2.12132i	−0.0737655	−	0.0737655i	0.669261	−	0.743027i	$-0.266611\pi$
$827$	−2.12132	−	2.12132i	−0.0737655	−	0.0737655i	−0.743027	+	0.669261i	$0.766611\pi$
$828$		0			0
$829$	−34.6410			−1.20313			−0.601566	−	0.798823i	$-0.705456\pi$
$829$	−34.6410			−1.20313			−0.601566	+	0.798823i	$0.705456\pi$
$830$		0			0
$831$		−	4.00000i		−	0.138758i
$832$		0			0
$833$	8.57321	−	8.57321i	0.297044	−	0.297044i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	14.1421	−	14.1421i	0.488824	−	0.488824i
$838$		0			0
$839$	48.0000			1.65714			0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000			1.65714			0.828572	+	0.559883i	$0.189154\pi$
$840$		0			0
$841$	19.0000			0.655172
$842$		0			0
$843$	−4.24264	+	4.24264i	−0.146124	+	0.146124i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$			29.0000i			0.995277i
$850$		0			0
$851$	55.4256			1.89997
$852$		0			0
$853$	−31.1127	−	31.1127i	−1.06528	−	1.06528i	−0.997715	−	0.0675635i	$-0.978477\pi$
$853$	−31.1127	−	31.1127i	−1.06528	−	1.06528i	−0.0675635	−	0.997715i	$-0.521523\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	25.7196	+	25.7196i	0.878566	+	0.878566i	0.993386	−	0.114820i	$-0.0366291\pi$
$857$	25.7196	+	25.7196i	0.878566	+	0.878566i	−0.114820	+	0.993386i	$0.536629\pi$
$858$		0			0
$859$			43.3013i			1.47742i	0.674023	+	0.738710i	$0.264565\pi$
$859$			43.3013i			1.47742i	−0.674023	+	0.738710i	$0.735435\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	4.89898	+	4.89898i	0.166763	+	0.166763i	0.785555	−	0.618792i	$-0.212378\pi$
$863$	4.89898	+	4.89898i	0.166763	+	0.166763i	−0.618792	+	0.785555i	$0.712378\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	9.89949	+	9.89949i	0.336204	+	0.336204i
$868$		0			0
$869$	13.8564			0.470046
$870$		0			0
$871$			28.0000i			0.948744i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	5.65685	−	5.65685i	0.191018	−	0.191018i	−0.605118	−	0.796136i	$-0.706874\pi$
$877$	5.65685	−	5.65685i	0.191018	−	0.191018i	0.796136	+	0.605118i	$0.206874\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	30.0000			1.01073			0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000			1.01073			0.505363	+	0.862907i	$0.331359\pi$
$882$		0			0
$883$	−17.6777	+	17.6777i	−0.594901	+	0.594901i	−0.938951	−	0.344050i	$-0.888201\pi$
$883$	−17.6777	+	17.6777i	−0.594901	+	0.594901i	0.344050	+	0.938951i	$0.388201\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−19.5959	+	19.5959i	−0.657967	+	0.657967i	−0.954899	−	0.296932i	$-0.904037\pi$
$887$	−19.5959	+	19.5959i	−0.657967	+	0.657967i	0.296932	+	0.954899i	$0.404037\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−1.73205			−0.0580259
$892$		0			0
$893$	−25.4558	−	25.4558i	−0.851847	−	0.851847i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	19.5959	+	19.5959i	0.654289	+	0.654289i
$898$		0			0
$899$		−	27.7128i		−	0.924274i
$900$		0			0
$901$			20.7846i			0.692436i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	5.65685	+	5.65685i	0.187833	+	0.187833i	0.794759	−	0.606926i	$-0.207597\pi$
$907$	5.65685	+	5.65685i	0.187833	+	0.187833i	−0.606926	+	0.794759i	$0.707597\pi$
$908$		0			0
$909$	−27.7128			−0.919176
$910$		0			0
$911$		−	24.0000i		−	0.795155i	−0.917568	−	0.397578i	$-0.869851\pi$
$911$		−	24.0000i		−	0.795155i	0.917568	−	0.397578i	$-0.130149\pi$
$912$		0			0
$913$	−18.3712	+	18.3712i	−0.607997	+	0.607997i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	20.0000			0.659739			0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000			0.659739			0.329870	+	0.944027i	$0.392995\pi$
$920$		0			0
$921$	−7.00000			−0.230658
$922$		0			0
$923$	−33.9411	+	33.9411i	−1.11719	+	1.11719i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−9.79796	+	9.79796i	−0.321807	+	0.321807i
$928$		0			0
$929$		−	18.0000i		−	0.590561i	−0.955411	−	0.295280i	$-0.904587\pi$
$929$		−	18.0000i		−	0.590561i	0.955411	−	0.295280i	$-0.0954131\pi$
$930$		0			0
$931$	−36.3731			−1.19208
$932$		0			0
$933$	16.9706	+	16.9706i	0.555591	+	0.555591i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−25.7196	−	25.7196i	−0.840224	−	0.840224i	0.148664	−	0.988888i	$-0.452503\pi$
$937$	−25.7196	−	25.7196i	−0.840224	−	0.840224i	−0.988888	+	0.148664i	$0.952503\pi$
$938$		0			0
$939$			27.7128i			0.904373i
$940$		0			0
$941$			34.6410i			1.12926i	0.825342	+	0.564632i	$0.190982\pi$
$941$			34.6410i			1.12926i	−0.825342	+	0.564632i	$0.809018\pi$
$942$		0			0
$943$	14.6969	+	14.6969i	0.478598	+	0.478598i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−16.9706	−	16.9706i	−0.551469	−	0.551469i	0.375396	−	0.926865i	$-0.377507\pi$
$947$	−16.9706	−	16.9706i	−0.551469	−	0.551469i	−0.926865	+	0.375396i	$0.877507\pi$
$948$		0			0
$949$	34.6410			1.12449
$950$		0			0
$951$		−	24.0000i		−	0.778253i
$952$		0			0
$953$	13.4722	−	13.4722i	0.436407	−	0.436407i	−0.454394	−	0.890801i	$-0.650144\pi$
$953$	13.4722	−	13.4722i	0.436407	−	0.436407i	0.890801	+	0.454394i	$0.150144\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	8.48528	−	8.48528i	0.274290	−	0.274290i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	15.0000			0.483871
$962$		0			0
$963$	−4.24264	+	4.24264i	−0.136717	+	0.136717i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−9.79796	+	9.79796i	−0.315081	+	0.315081i	−0.846874	−	0.531793i	$-0.821518\pi$
$967$	−9.79796	+	9.79796i	−0.315081	+	0.315081i	0.531793	+	0.846874i	$0.321518\pi$
$968$		0			0
$969$			9.00000i			0.289122i
$970$		0			0
$971$	−29.4449			−0.944931			−0.472465	−	0.881349i	$-0.656636\pi$
$971$	−29.4449			−0.944931			−0.472465	+	0.881349i	$0.656636\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.22474	−	1.22474i	−0.0391831	−	0.0391831i	0.687244	−	0.726427i	$-0.258820\pi$
$977$	−1.22474	−	1.22474i	−0.0391831	−	0.0391831i	−0.726427	+	0.687244i	$0.758820\pi$
$978$		0			0
$979$			5.19615i			0.166070i
$980$		0			0
$981$		−	27.7128i		−	0.884802i
$982$		0			0
$983$	9.79796	+	9.79796i	0.312506	+	0.312506i	0.845880	−	0.533373i	$-0.179076\pi$
$983$	9.79796	+	9.79796i	0.312506	+	0.312506i	−0.533373	+	0.845880i	$0.679076\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−55.4256			−1.76243
$990$		0			0
$991$		−	40.0000i		−	1.27064i	−0.772248	−	0.635321i	$-0.780868\pi$
$991$		−	40.0000i		−	1.27064i	0.772248	−	0.635321i	$-0.219132\pi$
$992$		0			0
$993$	20.8207	−	20.8207i	0.660724	−	0.660724i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	11.3137	−	11.3137i	0.358309	−	0.358309i	−0.504881	−	0.863189i	$-0.668463\pi$
$997$	11.3137	−	11.3137i	0.358309	−	0.358309i	0.863189	+	0.504881i	$0.168463\pi$
$998$		0			0
$999$	−40.0000			−1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.o.i.607.2	yes	8
4.3	odd	2		1600.2.o.h.607.3	yes	8
5.2	odd	4		1600.2.o.h.543.4	yes	8
5.3	odd	4		1600.2.o.h.543.2	yes	8
5.4	even	2	inner	1600.2.o.i.607.4	yes	8
8.3	odd	2		1600.2.o.h.607.2	yes	8
8.5	even	2	inner	1600.2.o.i.607.3	yes	8
20.3	even	4	inner	1600.2.o.i.543.3	yes	8
20.7	even	4	inner	1600.2.o.i.543.1	yes	8
20.19	odd	2		1600.2.o.h.607.1	yes	8
40.3	even	4	inner	1600.2.o.i.543.2	yes	8
40.13	odd	4		1600.2.o.h.543.3	yes	8
40.19	odd	2		1600.2.o.h.607.4	yes	8
40.27	even	4	inner	1600.2.o.i.543.4	yes	8
40.29	even	2	inner	1600.2.o.i.607.1	yes	8
40.37	odd	4		1600.2.o.h.543.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1600.2.o.h.543.1	✓	8	40.37	odd	4
1600.2.o.h.543.2	yes	8	5.3	odd	4
1600.2.o.h.543.3	yes	8	40.13	odd	4
1600.2.o.h.543.4	yes	8	5.2	odd	4
1600.2.o.h.607.1	yes	8	20.19	odd	2
1600.2.o.h.607.2	yes	8	8.3	odd	2
1600.2.o.h.607.3	yes	8	4.3	odd	2
1600.2.o.h.607.4	yes	8	40.19	odd	2
1600.2.o.i.543.1	yes	8	20.7	even	4	inner
1600.2.o.i.543.2	yes	8	40.3	even	4	inner
1600.2.o.i.543.3	yes	8	20.3	even	4	inner
1600.2.o.i.543.4	yes	8	40.27	even	4	inner
1600.2.o.i.607.1	yes	8	40.29	even	2	inner
1600.2.o.i.607.2	yes	8	1.1	even	1	trivial
1600.2.o.i.607.3	yes	8	8.5	even	2	inner
1600.2.o.i.607.4	yes	8	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.o (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 0.707107i) q^{3} +2.00000i q^{9} +1.73205 q^{11} +(-2.82843 - 2.82843i) q^{13} +(-1.22474 - 1.22474i) q^{17} +5.19615i q^{19} +(4.89898 + 4.89898i) q^{23} +(-3.53553 - 3.53553i) q^{27} -6.92820 q^{29} +4.00000i q^{31} +(-1.22474 + 1.22474i) q^{33} +(5.65685 - 5.65685i) q^{37} +4.00000 q^{39} +3.00000 q^{41} +(-5.65685 + 5.65685i) q^{43} +(-4.89898 + 4.89898i) q^{47} +7.00000i q^{49} +1.73205 q^{51} +(-8.48528 - 8.48528i) q^{53} +(-3.67423 - 3.67423i) q^{57} +10.3923i q^{59} +(-4.94975 - 4.94975i) q^{67} -6.92820 q^{69} -12.0000i q^{71} +(-6.12372 + 6.12372i) q^{73} +8.00000 q^{79} -1.00000 q^{81} +(-10.6066 + 10.6066i) q^{83} +(4.89898 - 4.89898i) q^{87} +3.00000i q^{89} +(-2.82843 - 2.82843i) q^{93} +3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 32 q^{39} + 24 q^{41} + 64 q^{79} - 8 q^{81}+O(q^{100}) \) 8 * q + 32 * q^39 + 24 * q^41 + 64 * q^79 - 8 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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