LMFDB - Embedded newform 1452.3.f.a.241.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1452,3,Mod(241,1452)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1452, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1452.241"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1452 = 2^{2} \cdot 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1452.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,8] 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:	$39.5641343851$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{3})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			241.1
Root			$-1.93185i$ of defining polynomial
Character	$\chi$	$=$	1452.241
Dual form			1452.3.f.a.241.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} +3.73205 q^{5} -2.44949i q^{7} +3.00000 q^{9} -14.5582i q^{13} -6.46410 q^{15} -8.52245i q^{17} -2.07055i q^{19} +4.24264i q^{21} +21.5167 q^{23} -11.0718 q^{25} -5.19615 q^{27} +18.4219i q^{29} -15.4641 q^{31} -9.14162i q^{35} -52.5167 q^{37} +25.2156i q^{39} -67.3374i q^{41} +4.24264i q^{43} +11.1962 q^{45} -31.4641 q^{47} +43.0000 q^{49} +14.7613i q^{51} -17.0000 q^{53} +3.58630i q^{57} -22.1962 q^{59} +17.7556i q^{61} -7.34847i q^{63} -54.3321i q^{65} +63.0859 q^{67} -37.2679 q^{69} +82.7321 q^{71} -86.4900i q^{73} +19.1769 q^{75} +37.1213i q^{79} +9.00000 q^{81} +146.117i q^{83} -31.8062i q^{85} -31.9077i q^{87} -87.8756 q^{89} -35.6603 q^{91} +26.7846 q^{93} -7.72741i q^{95} -116.660 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{5} + 12 q^{9} - 12 q^{15} - 4 q^{23} - 72 q^{25} - 48 q^{31} - 120 q^{37} + 24 q^{45} - 112 q^{47} + 172 q^{49} - 68 q^{53} - 68 q^{59} - 4 q^{67} - 156 q^{69} + 324 q^{71} - 48 q^{75} + 36 q^{81}+ \cdots - 432 q^{97}+O(q^{100}) $ 4 * q + 8 * q^5 + 12 * q^9 - 12 * q^15 - 4 * q^23 - 72 * q^25 - 48 * q^31 - 120 * q^37 + 24 * q^45 - 112 * q^47 + 172 * q^49 - 68 * q^53 - 68 * q^59 - 4 * q^67 - 156 * q^69 + 324 * q^71 - 48 * q^75 + 36 * q^81 - 400 * q^89 - 108 * q^91 + 24 * q^93 - 432 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1452\mathbb{Z}\right)^\times$.

$n$	$485$	$727$	$1333$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	−1.73205	−0.577350
$3$	−1.73205	−0.577350
$4$	0	0
$5$	3.73205	0.746410	0.373205	−	0.927749i	$-0.378259\pi$
$5$	3.73205	0.746410	0.373205	+	0.927749i	$0.378259\pi$
$6$	0	0
$7$	− 2.44949i	− 0.349927i	−0.984575	−	0.174964i	$-0.944019\pi$
$7$	− 2.44949i	− 0.349927i	0.984575	−	0.174964i	$-0.0559808\pi$
$8$	0	0
$9$	3.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	− 14.5582i	− 1.11986i	−0.828538	−	0.559932i	$-0.810827\pi$
$13$	− 14.5582i	− 1.11986i	0.828538	−	0.559932i	$-0.189173\pi$
$14$	0	0
$15$	−6.46410	−0.430940
$16$	0	0
$17$	− 8.52245i	− 0.501320i	−0.968075	−	0.250660i	$-0.919352\pi$
$17$	− 8.52245i	− 0.501320i	0.968075	−	0.250660i	$-0.0806477\pi$
$18$	0	0
$19$	− 2.07055i	− 0.108976i	−0.998514	−	0.0544882i	$-0.982647\pi$
$19$	− 2.07055i	− 0.108976i	0.998514	−	0.0544882i	$-0.0173527\pi$
$20$	0	0
$21$	4.24264i	0.202031i
$22$	0	0
$23$	21.5167	0.935507	0.467753	−	0.883859i	$-0.345064\pi$
$23$	21.5167	0.935507	0.467753	+	0.883859i	$0.345064\pi$
$24$	0	0
$25$	−11.0718	−0.442872
$26$	0	0
$27$	−5.19615	−0.192450
$28$	0	0
$29$	18.4219i	0.635239i	0.948218	+	0.317620i	$0.102884\pi$
$29$	18.4219i	0.635239i	−0.948218	+	0.317620i	$0.897116\pi$
$30$	0	0
$31$	−15.4641	−0.498842	−0.249421	−	0.968395i	$-0.580240\pi$
$31$	−15.4641	−0.498842	−0.249421	+	0.968395i	$0.580240\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 9.14162i	− 0.261189i
$36$	0	0
$37$	−52.5167	−1.41937	−0.709685	−	0.704520i	$-0.751163\pi$
$37$	−52.5167	−1.41937	−0.709685	+	0.704520i	$0.751163\pi$
$38$	0	0
$39$	25.2156i	0.646554i
$40$	0	0
$41$	− 67.3374i	− 1.64238i	−0.570658	−	0.821188i	$-0.693312\pi$
$41$	− 67.3374i	− 1.64238i	0.570658	−	0.821188i	$-0.306688\pi$
$42$	0	0
$43$	4.24264i	0.0986661i	0.998782	+	0.0493330i	$0.0157096\pi$
$43$	4.24264i	0.0986661i	−0.998782	+	0.0493330i	$0.984290\pi$
$44$	0	0
$45$	11.1962	0.248803
$46$	0	0
$47$	−31.4641	−0.669449	−0.334724	−	0.942316i	$-0.608643\pi$
$47$	−31.4641	−0.669449	−0.334724	+	0.942316i	$0.608643\pi$
$48$	0	0
$49$	43.0000	0.877551
$50$	0	0
$51$	14.7613i	0.289437i
$52$	0	0
$53$	−17.0000	−0.320755	−0.160377	−	0.987056i	$-0.551271\pi$
$53$	−17.0000	−0.320755	−0.160377	+	0.987056i	$0.551271\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.58630i	0.0629176i
$58$	0	0
$59$	−22.1962	−0.376206	−0.188103	−	0.982149i	$-0.560234\pi$
$59$	−22.1962	−0.376206	−0.188103	+	0.982149i	$0.560234\pi$
$60$	0	0
$61$	17.7556i	0.291076i	0.989353	+	0.145538i	$0.0464913\pi$
$61$	17.7556i	0.291076i	−0.989353	+	0.145538i	$0.953509\pi$
$62$	0	0
$63$	− 7.34847i	− 0.116642i
$64$	0	0
$65$	− 54.3321i	− 0.835878i
$66$	0	0
$67$	63.0859	0.941580	0.470790	−	0.882245i	$-0.343969\pi$
$67$	63.0859	0.941580	0.470790	+	0.882245i	$0.343969\pi$
$68$	0	0
$69$	−37.2679	−0.540115
$70$	0	0
$71$	82.7321	1.16524	0.582620	−	0.812745i	$-0.302028\pi$
$71$	82.7321	1.16524	0.582620	+	0.812745i	$0.302028\pi$
$72$	0	0
$73$	− 86.4900i	− 1.18479i	−0.805646	−	0.592397i	$-0.798182\pi$
$73$	− 86.4900i	− 1.18479i	0.805646	−	0.592397i	$-0.201818\pi$
$74$	0	0
$75$	19.1769	0.255692
$76$	0	0
$77$	0	0
$78$	0	0
$79$	37.1213i	0.469890i	0.972009	+	0.234945i	$0.0754909\pi$
$79$	37.1213i	0.469890i	−0.972009	+	0.234945i	$0.924509\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	146.117i	1.76045i	0.474558	+	0.880224i	$0.342608\pi$
$83$	146.117i	1.76045i	−0.474558	+	0.880224i	$0.657392\pi$
$84$	0	0
$85$	− 31.8062i	− 0.374191i
$86$	0	0
$87$	− 31.9077i	− 0.366756i
$88$	0	0
$89$	−87.8756	−0.987367	−0.493683	−	0.869642i	$-0.664350\pi$
$89$	−87.8756	−0.987367	−0.493683	+	0.869642i	$0.664350\pi$
$90$	0	0
$91$	−35.6603	−0.391871
$92$	0	0
$93$	26.7846	0.288007
$94$	0	0
$95$	− 7.72741i	− 0.0813411i
$96$	0	0
$97$	−116.660	−1.20268	−0.601342	−	0.798992i	$-0.705367\pi$
$97$	−116.660	−1.20268	−0.601342	+	0.798992i	$0.705367\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 50.9389i	− 0.504345i	−0.967682	−	0.252173i	$-0.918855\pi$
$101$	− 50.9389i	− 0.504345i	0.967682	−	0.252173i	$-0.0811451\pi$
$102$	0	0
$103$	−160.904	−1.56217	−0.781086	−	0.624423i	$-0.785334\pi$
$103$	−160.904	−1.56217	−0.781086	+	0.624423i	$0.785334\pi$
$104$	0	0
$105$	15.8338i	0.150798i
$106$	0	0
$107$	− 172.121i	− 1.60860i	−0.594221	−	0.804302i	$-0.702539\pi$
$107$	− 172.121i	− 1.60860i	0.594221	−	0.804302i	$-0.297461\pi$
$108$	0	0
$109$	− 215.775i	− 1.97959i	−0.142495	−	0.989795i	$-0.545513\pi$
$109$	− 215.775i	− 1.97959i	0.142495	−	0.989795i	$-0.454487\pi$
$110$	0	0
$111$	90.9615	0.819473
$112$	0	0
$113$	−20.0718	−0.177627	−0.0888133	−	0.996048i	$-0.528307\pi$
$113$	−20.0718	−0.177627	−0.0888133	+	0.996048i	$0.528307\pi$
$114$	0	0
$115$	80.3013	0.698272
$116$	0	0
$117$	− 43.6747i	− 0.373288i
$118$	0	0
$119$	−20.8756	−0.175426
$120$	0	0
$121$	0	0
$122$	0	0
$123$	116.632i	0.948226i
$124$	0	0
$125$	−134.622	−1.07697
$126$	0	0
$127$	11.2394i	0.0884990i	0.999021	+	0.0442495i	$0.0140897\pi$
$127$	11.2394i	0.0884990i	−0.999021	+	0.0442495i	$0.985910\pi$
$128$	0	0
$129$	− 7.34847i	− 0.0569649i
$130$	0	0
$131$	− 158.493i	− 1.20987i	−0.796273	−	0.604937i	$-0.793198\pi$
$131$	− 158.493i	− 1.20987i	0.796273	−	0.604937i	$-0.206802\pi$
$132$	0	0
$133$	−5.07180	−0.0381338
$134$	0	0
$135$	−19.3923	−0.143647
$136$	0	0
$137$	−131.205	−0.957701	−0.478851	−	0.877896i	$-0.658946\pi$
$137$	−131.205	−0.957701	−0.478851	+	0.877896i	$0.658946\pi$
$138$	0	0
$139$	− 71.9417i	− 0.517566i	−0.965935	−	0.258783i	$-0.916678\pi$
$139$	− 71.9417i	− 0.517566i	0.965935	−	0.258783i	$-0.0833215\pi$
$140$	0	0
$141$	54.4974	0.386507
$142$	0	0
$143$	0	0
$144$	0	0
$145$	68.7516i	0.474149i
$146$	0	0
$147$	−74.4782	−0.506654
$148$	0	0
$149$	− 14.5509i	− 0.0976574i	−0.998807	−	0.0488287i	$-0.984451\pi$
$149$	− 14.5509i	− 0.0976574i	0.998807	−	0.0488287i	$-0.0155488\pi$
$150$	0	0
$151$	− 126.413i	− 0.837169i	−0.908178	−	0.418585i	$-0.862526\pi$
$151$	− 126.413i	− 0.837169i	0.908178	−	0.418585i	$-0.137474\pi$
$152$	0	0
$153$	− 25.5673i	− 0.167107i
$154$	0	0
$155$	−57.7128	−0.372341
$156$	0	0
$157$	−124.967	−0.795966	−0.397983	−	0.917393i	$-0.630290\pi$
$157$	−124.967	−0.795966	−0.397983	+	0.917393i	$0.630290\pi$
$158$	0	0
$159$	29.4449	0.185188
$160$	0	0
$161$	− 52.7048i	− 0.327359i
$162$	0	0
$163$	203.899	1.25091	0.625456	−	0.780259i	$-0.284913\pi$
$163$	203.899	1.25091	0.625456	+	0.780259i	$0.284913\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 157.384i	− 0.942418i	−0.882022	−	0.471209i	$-0.843818\pi$
$167$	− 157.384i	− 0.942418i	0.882022	−	0.471209i	$-0.156182\pi$
$168$	0	0
$169$	−42.9423	−0.254096
$170$	0	0
$171$	− 6.21166i	− 0.0363255i
$172$	0	0
$173$	− 221.774i	− 1.28193i	−0.767570	−	0.640965i	$-0.778534\pi$
$173$	− 221.774i	− 1.28193i	0.767570	−	0.640965i	$-0.221466\pi$
$174$	0	0
$175$	27.1203i	0.154973i
$176$	0	0
$177$	38.4449	0.217203
$178$	0	0
$179$	−155.310	−0.867655	−0.433827	−	0.900996i	$-0.642837\pi$
$179$	−155.310	−0.867655	−0.433827	+	0.900996i	$0.642837\pi$
$180$	0	0
$181$	2.01924	0.0111560	0.00557801	−	0.999984i	$-0.498224\pi$
$181$	2.01924	0.0111560	0.00557801	+	0.999984i	$0.498224\pi$
$182$	0	0
$183$	− 30.7537i	− 0.168053i
$184$	0	0
$185$	−195.995	−1.05943
$186$	0	0
$187$	0	0
$188$	0	0
$189$	12.7279i	0.0673435i
$190$	0	0
$191$	95.4256	0.499611	0.249805	−	0.968296i	$-0.419633\pi$
$191$	95.4256	0.499611	0.249805	+	0.968296i	$0.419633\pi$
$192$	0	0
$193$	65.3684i	0.338696i	0.985556	+	0.169348i	$0.0541662\pi$
$193$	65.3684i	0.338696i	−0.985556	+	0.169348i	$0.945834\pi$
$194$	0	0
$195$	94.1059i	0.482595i
$196$	0	0
$197$	− 343.406i	− 1.74318i	−0.490236	−	0.871590i	$-0.663089\pi$
$197$	− 343.406i	− 1.74318i	0.490236	−	0.871590i	$-0.336911\pi$
$198$	0	0
$199$	258.081	1.29689	0.648444	−	0.761262i	$-0.275420\pi$
$199$	258.081	1.29689	0.648444	+	0.761262i	$0.275420\pi$
$200$	0	0
$201$	−109.268	−0.543622
$202$	0	0
$203$	45.1244	0.222287
$204$	0	0
$205$	− 251.307i	− 1.22589i
$206$	0	0
$207$	64.5500	0.311836
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 11.1923i	− 0.0530439i	−0.999648	−	0.0265219i	$-0.991557\pi$
$211$	− 11.1923i	− 0.0530439i	0.999648	−	0.0265219i	$-0.00844318\pi$
$212$	0	0
$213$	−143.296	−0.672752
$214$	0	0
$215$	15.8338i	0.0736454i
$216$	0	0
$217$	37.8792i	0.174558i
$218$	0	0
$219$	149.805i	0.684042i
$220$	0	0
$221$	−124.072	−0.561411
$222$	0	0
$223$	−312.956	−1.40339	−0.701696	−	0.712477i	$-0.747573\pi$
$223$	−312.956	−1.40339	−0.701696	+	0.712477i	$0.747573\pi$
$224$	0	0
$225$	−33.2154	−0.147624
$226$	0	0
$227$	240.984i	1.06160i	0.847496	+	0.530801i	$0.178109\pi$
$227$	240.984i	1.06160i	−0.847496	+	0.530801i	$0.821891\pi$
$228$	0	0
$229$	241.406	1.05418	0.527088	−	0.849811i	$-0.323284\pi$
$229$	241.406	1.05418	0.527088	+	0.849811i	$0.323284\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 267.391i	− 1.14760i	−0.818996	−	0.573800i	$-0.805469\pi$
$233$	− 267.391i	− 1.14760i	0.818996	−	0.573800i	$-0.194531\pi$
$234$	0	0
$235$	−117.426	−0.499684
$236$	0	0
$237$	− 64.2959i	− 0.271291i
$238$	0	0
$239$	− 301.849i	− 1.26297i	−0.775389	−	0.631484i	$-0.782446\pi$
$239$	− 301.849i	− 1.26297i	0.775389	−	0.631484i	$-0.217554\pi$
$240$	0	0
$241$	68.1125i	0.282625i	0.989965	+	0.141312i	$0.0451322\pi$
$241$	68.1125i	0.282625i	−0.989965	+	0.141312i	$0.954868\pi$
$242$	0	0
$243$	−15.5885	−0.0641500
$244$	0	0
$245$	160.478	0.655013
$246$	0	0
$247$	−30.1436	−0.122039
$248$	0	0
$249$	− 253.083i	− 1.01640i
$250$	0	0
$251$	−480.147	−1.91294	−0.956469	−	0.291834i	$-0.905734\pi$
$251$	−480.147	−1.91294	−0.956469	+	0.291834i	$0.905734\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	55.0900i	0.216039i
$256$	0	0
$257$	205.252	0.798648	0.399324	−	0.916810i	$-0.369245\pi$
$257$	205.252	0.798648	0.399324	+	0.916810i	$0.369245\pi$
$258$	0	0
$259$	128.639i	0.496676i
$260$	0	0
$261$	55.2658i	0.211746i
$262$	0	0
$263$	138.355i	0.526066i	0.964787	+	0.263033i	$0.0847228\pi$
$263$	138.355i	0.526066i	−0.964787	+	0.263033i	$0.915277\pi$
$264$	0	0
$265$	−63.4449	−0.239415
$266$	0	0
$267$	152.205	0.570056
$268$	0	0
$269$	−92.6640	−0.344476	−0.172238	−	0.985055i	$-0.555100\pi$
$269$	−92.6640	−0.344476	−0.172238	+	0.985055i	$0.555100\pi$
$270$	0	0
$271$	456.662i	1.68510i	0.538618	+	0.842550i	$0.318947\pi$
$271$	456.662i	1.68510i	−0.538618	+	0.842550i	$0.681053\pi$
$272$	0	0
$273$	61.7654	0.226247
$274$	0	0
$275$	0	0
$276$	0	0
$277$	517.511i	1.86827i	0.356921	+	0.934135i	$0.383827\pi$
$277$	517.511i	1.86827i	−0.356921	+	0.934135i	$0.616173\pi$
$278$	0	0
$279$	−46.3923	−0.166281
$280$	0	0
$281$	− 345.987i	− 1.23127i	−0.788031	−	0.615636i	$-0.788899\pi$
$281$	− 345.987i	− 1.23127i	0.788031	−	0.615636i	$-0.211101\pi$
$282$	0	0
$283$	153.689i	0.543070i	0.962429	+	0.271535i	$0.0875312\pi$
$283$	153.689i	0.543070i	−0.962429	+	0.271535i	$0.912469\pi$
$284$	0	0
$285$	13.3843i	0.0469623i
$286$	0	0
$287$	−164.942	−0.574712
$288$	0	0
$289$	216.368	0.748678
$290$	0	0
$291$	202.061	0.694369
$292$	0	0
$293$	8.66382i	0.0295693i	0.999891	+	0.0147847i	$0.00470628\pi$
$293$	8.66382i	0.0295693i	−0.999891	+	0.0147847i	$0.995294\pi$
$294$	0	0
$295$	−82.8372	−0.280804
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 313.245i	− 1.04764i
$300$	0	0
$301$	10.3923	0.0345259
$302$	0	0
$303$	88.2288i	0.291184i
$304$	0	0
$305$	66.2650i	0.217262i
$306$	0	0
$307$	335.263i	1.09206i	0.837765	+	0.546031i	$0.183862\pi$
$307$	335.263i	1.09206i	−0.837765	+	0.546031i	$0.816138\pi$
$308$	0	0
$309$	278.694	0.901921
$310$	0	0
$311$	376.105	1.20934	0.604671	−	0.796476i	$-0.293305\pi$
$311$	376.105	1.20934	0.604671	+	0.796476i	$0.293305\pi$
$312$	0	0
$313$	6.06149	0.0193658	0.00968289	−	0.999953i	$-0.496918\pi$
$313$	6.06149	0.0193658	0.00968289	+	0.999953i	$0.496918\pi$
$314$	0	0
$315$	− 27.4249i	− 0.0870630i
$316$	0	0
$317$	−3.98969	−0.0125858	−0.00629289	−	0.999980i	$-0.502003\pi$
$317$	−3.98969	−0.0125858	−0.00629289	+	0.999980i	$0.502003\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	298.122i	0.928728i
$322$	0	0
$323$	−17.6462	−0.0546321
$324$	0	0
$325$	161.186i	0.495956i
$326$	0	0
$327$	373.734i	1.14292i
$328$	0	0
$329$	77.0710i	0.234258i
$330$	0	0
$331$	310.105	0.936873	0.468437	−	0.883497i	$-0.344817\pi$
$331$	310.105	0.936873	0.468437	+	0.883497i	$0.344817\pi$
$332$	0	0
$333$	−157.550	−0.473123
$334$	0	0
$335$	235.440	0.702805
$336$	0	0
$337$	− 79.5849i	− 0.236157i	−0.993004	−	0.118078i	$-0.962327\pi$
$337$	− 79.5849i	− 0.236157i	0.993004	−	0.118078i	$-0.0376734\pi$
$338$	0	0
$339$	34.7654	0.102553
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 225.353i	− 0.657006i
$344$	0	0
$345$	−139.086	−0.403147
$346$	0	0
$347$	338.639i	0.975904i	0.872870	+	0.487952i	$0.162256\pi$
$347$	338.639i	0.975904i	−0.872870	+	0.487952i	$0.837744\pi$
$348$	0	0
$349$	228.618i	0.655064i	0.944840	+	0.327532i	$0.106217\pi$
$349$	228.618i	0.655064i	−0.944840	+	0.327532i	$0.893783\pi$
$350$	0	0
$351$	75.6468i	0.215518i
$352$	0	0
$353$	−699.932	−1.98281	−0.991405	−	0.130828i	$-0.958236\pi$
$353$	−699.932	−1.98281	−0.991405	+	0.130828i	$0.958236\pi$
$354$	0	0
$355$	308.760	0.869747
$356$	0	0
$357$	36.1577	0.101282
$358$	0	0
$359$	469.666i	1.30826i	0.756382	+	0.654130i	$0.226965\pi$
$359$	469.666i	1.30826i	−0.756382	+	0.654130i	$0.773035\pi$
$360$	0	0
$361$	356.713	0.988124
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 322.785i	− 0.884343i
$366$	0	0
$367$	457.373	1.24625	0.623124	−	0.782123i	$-0.285863\pi$
$367$	457.373	1.24625	0.623124	+	0.782123i	$0.285863\pi$
$368$	0	0
$369$	− 202.012i	− 0.547459i
$370$	0	0
$371$	41.6413i	0.112241i
$372$	0	0
$373$	− 70.3915i	− 0.188717i	−0.995538	−	0.0943586i	$-0.969920\pi$
$373$	− 70.3915i	− 0.188717i	0.995538	−	0.0943586i	$-0.0300800\pi$
$374$	0	0
$375$	233.172	0.621791
$376$	0	0
$377$	268.191	0.711382
$378$	0	0
$379$	195.611	0.516125	0.258063	−	0.966128i	$-0.416916\pi$
$379$	195.611	0.516125	0.258063	+	0.966128i	$0.416916\pi$
$380$	0	0
$381$	− 19.4672i	− 0.0510949i
$382$	0	0
$383$	−257.458	−0.672213	−0.336106	−	0.941824i	$-0.609110\pi$
$383$	−257.458	−0.672213	−0.336106	+	0.941824i	$0.609110\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	12.7279i	0.0328887i
$388$	0	0
$389$	734.878	1.88915	0.944573	−	0.328301i	$-0.106476\pi$
$389$	734.878	1.88915	0.944573	+	0.328301i	$0.106476\pi$
$390$	0	0
$391$	− 183.375i	− 0.468989i
$392$	0	0
$393$	274.519i	0.698521i
$394$	0	0
$395$	138.539i	0.350730i
$396$	0	0
$397$	154.138	0.388258	0.194129	−	0.980976i	$-0.437812\pi$
$397$	154.138	0.388258	0.194129	+	0.980976i	$0.437812\pi$
$398$	0	0
$399$	8.78461	0.0220166
$400$	0	0
$401$	−456.128	−1.13748	−0.568738	−	0.822519i	$-0.692568\pi$
$401$	−456.128	−1.13748	−0.568738	+	0.822519i	$0.692568\pi$
$402$	0	0
$403$	225.130i	0.558635i
$404$	0	0
$405$	33.5885	0.0829345
$406$	0	0
$407$	0	0
$408$	0	0
$409$	401.803i	0.982402i	0.871046	+	0.491201i	$0.163442\pi$
$409$	401.803i	0.982402i	−0.871046	+	0.491201i	$0.836558\pi$
$410$	0	0
$411$	227.254	0.552929
$412$	0	0
$413$	54.3692i	0.131645i
$414$	0	0
$415$	545.317i	1.31402i
$416$	0	0
$417$	124.607i	0.298817i
$418$	0	0
$419$	−502.922	−1.20029	−0.600145	−	0.799891i	$-0.704891\pi$
$419$	−502.922	−1.20029	−0.600145	+	0.799891i	$0.704891\pi$
$420$	0	0
$421$	−121.182	−0.287843	−0.143922	−	0.989589i	$-0.545971\pi$
$421$	−121.182	−0.287843	−0.143922	+	0.989589i	$0.545971\pi$
$422$	0	0
$423$	−94.3923	−0.223150
$424$	0	0
$425$	94.3588i	0.222021i
$426$	0	0
$427$	43.4923	0.101855
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 435.199i	− 1.00974i	−0.863195	−	0.504871i	$-0.831540\pi$
$431$	− 435.199i	− 1.00974i	0.863195	−	0.504871i	$-0.168460\pi$
$432$	0	0
$433$	256.142	0.591552	0.295776	−	0.955257i	$-0.404422\pi$
$433$	256.142	0.591552	0.295776	+	0.955257i	$0.404422\pi$
$434$	0	0
$435$	− 119.081i	− 0.273750i
$436$	0	0
$437$	− 44.5514i	− 0.101948i
$438$	0	0
$439$	755.864i	1.72179i	0.508785	+	0.860893i	$0.330095\pi$
$439$	755.864i	1.72179i	−0.508785	+	0.860893i	$0.669905\pi$
$440$	0	0
$441$	129.000	0.292517
$442$	0	0
$443$	678.771	1.53221	0.766107	−	0.642713i	$-0.222191\pi$
$443$	678.771	1.53221	0.766107	+	0.642713i	$0.222191\pi$
$444$	0	0
$445$	−327.956	−0.736981
$446$	0	0
$447$	25.2030i	0.0563825i
$448$	0	0
$449$	297.024	0.661524	0.330762	−	0.943714i	$-0.392694\pi$
$449$	297.024	0.661524	0.330762	+	0.943714i	$0.392694\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	218.953i	0.483340i
$454$	0	0
$455$	−133.086	−0.292496
$456$	0	0
$457$	− 541.146i	− 1.18413i	−0.805891	−	0.592064i	$-0.798313\pi$
$457$	− 541.146i	− 1.18413i	0.805891	−	0.592064i	$-0.201687\pi$
$458$	0	0
$459$	44.2839i	0.0964791i
$460$	0	0
$461$	393.607i	0.853812i	0.904296	+	0.426906i	$0.140396\pi$
$461$	393.607i	0.853812i	−0.904296	+	0.426906i	$0.859604\pi$
$462$	0	0
$463$	9.57576	0.0206820	0.0103410	−	0.999947i	$-0.496708\pi$
$463$	9.57576	0.0206820	0.0103410	+	0.999947i	$0.496708\pi$
$464$	0	0
$465$	99.9615	0.214971
$466$	0	0
$467$	−53.2013	−0.113921	−0.0569607	−	0.998376i	$-0.518141\pi$
$467$	−53.2013	−0.113921	−0.0569607	+	0.998376i	$0.518141\pi$
$468$	0	0
$469$	− 154.528i	− 0.329484i
$470$	0	0
$471$	216.449	0.459551
$472$	0	0
$473$	0	0
$474$	0	0
$475$	22.9247i	0.0482626i
$476$	0	0
$477$	−51.0000	−0.106918
$478$	0	0
$479$	− 594.200i	− 1.24050i	−0.784404	−	0.620250i	$-0.787031\pi$
$479$	− 594.200i	− 1.24050i	0.784404	−	0.620250i	$-0.212969\pi$
$480$	0	0
$481$	764.550i	1.58950i
$482$	0	0
$483$	91.2875i	0.189001i
$484$	0	0
$485$	−435.382	−0.897695
$486$	0	0
$487$	39.2398	0.0805745	0.0402873	−	0.999188i	$-0.487173\pi$
$487$	39.2398	0.0805745	0.0402873	+	0.999188i	$0.487173\pi$
$488$	0	0
$489$	−353.163	−0.722214
$490$	0	0
$491$	508.459i	1.03556i	0.855515	+	0.517779i	$0.173241\pi$
$491$	508.459i	1.03556i	−0.855515	+	0.517779i	$0.826759\pi$
$492$	0	0
$493$	157.000	0.318458
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 202.651i	− 0.407749i
$498$	0	0
$499$	−154.655	−0.309930	−0.154965	−	0.987920i	$-0.549526\pi$
$499$	−154.655	−0.309930	−0.154965	+	0.987920i	$0.549526\pi$
$500$	0	0
$501$	272.597i	0.544105i
$502$	0	0
$503$	− 987.208i	− 1.96264i	−0.192382	−	0.981320i	$-0.561621\pi$
$503$	− 987.208i	− 1.96264i	0.192382	−	0.981320i	$-0.438379\pi$
$504$	0	0
$505$	− 190.107i	− 0.376449i
$506$	0	0
$507$	74.3782	0.146703
$508$	0	0
$509$	−247.167	−0.485593	−0.242796	−	0.970077i	$-0.578065\pi$
$509$	−247.167	−0.485593	−0.242796	+	0.970077i	$0.578065\pi$
$510$	0	0
$511$	−211.856	−0.414592
$512$	0	0
$513$	10.7589i	0.0209725i
$514$	0	0
$515$	−600.501	−1.16602
$516$	0	0
$517$	0	0
$518$	0	0
$519$	384.124i	0.740123i
$520$	0	0
$521$	925.395	1.77619	0.888095	−	0.459660i	$-0.152029\pi$
$521$	925.395	1.77619	0.888095	+	0.459660i	$0.152029\pi$
$522$	0	0
$523$	− 322.049i	− 0.615773i	−0.951423	−	0.307886i	$-0.900378\pi$
$523$	− 322.049i	− 0.615773i	0.951423	−	0.307886i	$-0.0996217\pi$
$524$	0	0
$525$	− 46.9737i	− 0.0894736i
$526$	0	0
$527$	131.792i	0.250080i
$528$	0	0
$529$	−66.0333	−0.124827
$530$	0	0
$531$	−66.5885	−0.125402
$532$	0	0
$533$	−980.314	−1.83924
$534$	0	0
$535$	− 642.363i	− 1.20068i
$536$	0	0
$537$	269.005	0.500941
$538$	0	0
$539$	0	0
$540$	0	0
$541$	334.891i	0.619023i	0.950896	+	0.309511i	$0.100165\pi$
$541$	334.891i	0.619023i	−0.950896	+	0.309511i	$0.899835\pi$
$542$	0	0
$543$	−3.49742	−0.00644093
$544$	0	0
$545$	− 805.285i	− 1.47759i
$546$	0	0
$547$	678.016i	1.23952i	0.784793	+	0.619758i	$0.212769\pi$
$547$	678.016i	1.23952i	−0.784793	+	0.619758i	$0.787231\pi$
$548$	0	0
$549$	53.2669i	0.0970254i
$550$	0	0
$551$	38.1436	0.0692261
$552$	0	0
$553$	90.9282	0.164427
$554$	0	0
$555$	339.473	0.611663
$556$	0	0
$557$	73.3487i	0.131685i	0.997830	+	0.0658426i	$0.0209735\pi$
$557$	73.3487i	0.131685i	−0.997830	+	0.0658426i	$0.979026\pi$
$558$	0	0
$559$	61.7654	0.110493
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 271.001i	− 0.481352i	−0.970605	−	0.240676i	$-0.922631\pi$
$563$	− 271.001i	− 0.481352i	0.970605	−	0.240676i	$-0.0773692\pi$
$564$	0	0
$565$	−74.9090	−0.132582
$566$	0	0
$567$	− 22.0454i	− 0.0388808i
$568$	0	0
$569$	967.667i	1.70064i	0.526263	+	0.850322i	$0.323593\pi$
$569$	967.667i	1.70064i	−0.526263	+	0.850322i	$0.676407\pi$
$570$	0	0
$571$	852.736i	1.49341i	0.665156	+	0.746704i	$0.268365\pi$
$571$	852.736i	1.49341i	−0.665156	+	0.746704i	$0.731635\pi$
$572$	0	0
$573$	−165.282	−0.288450
$574$	0	0
$575$	−238.228	−0.414310
$576$	0	0
$577$	−628.058	−1.08849	−0.544244	−	0.838927i	$-0.683183\pi$
$577$	−628.058	−1.08849	−0.544244	+	0.838927i	$0.683183\pi$
$578$	0	0
$579$	− 113.221i	− 0.195546i
$580$	0	0
$581$	357.913	0.616029
$582$	0	0
$583$	0	0
$584$	0	0
$585$	− 162.996i	− 0.278626i
$586$	0	0
$587$	633.874	1.07985	0.539927	−	0.841712i	$-0.318452\pi$
$587$	633.874	1.07985	0.539927	+	0.841712i	$0.318452\pi$
$588$	0	0
$589$	32.0192i	0.0543620i
$590$	0	0
$591$	594.797i	1.00643i
$592$	0	0
$593$	− 615.166i	− 1.03738i	−0.854963	−	0.518689i	$-0.826420\pi$
$593$	− 615.166i	− 1.03738i	0.854963	−	0.518689i	$-0.173580\pi$
$594$	0	0
$595$	−77.9090	−0.130939
$596$	0	0
$597$	−447.009	−0.748759
$598$	0	0
$599$	347.503	0.580138	0.290069	−	0.957006i	$-0.406322\pi$
$599$	347.503	0.580138	0.290069	+	0.957006i	$0.406322\pi$
$600$	0	0
$601$	− 521.149i	− 0.867137i	−0.901121	−	0.433569i	$-0.857254\pi$
$601$	− 521.149i	− 0.867137i	0.901121	−	0.433569i	$-0.142746\pi$
$602$	0	0
$603$	189.258	0.313860
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 568.291i	− 0.936229i	−0.883668	−	0.468114i	$-0.844934\pi$
$607$	− 568.291i	− 0.936229i	0.883668	−	0.468114i	$-0.155066\pi$
$608$	0	0
$609$	−78.1577	−0.128338
$610$	0	0
$611$	458.062i	0.749692i
$612$	0	0
$613$	− 622.784i	− 1.01596i	−0.861369	−	0.507981i	$-0.830392\pi$
$613$	− 622.784i	− 1.01596i	0.861369	−	0.507981i	$-0.169608\pi$
$614$	0	0
$615$	435.276i	0.707766i
$616$	0	0
$617$	−533.249	−0.864260	−0.432130	−	0.901811i	$-0.642238\pi$
$617$	−533.249	−0.864260	−0.432130	+	0.901811i	$0.642238\pi$
$618$	0	0
$619$	696.291	1.12486	0.562432	−	0.826843i	$-0.309866\pi$
$619$	696.291	1.12486	0.562432	+	0.826843i	$0.309866\pi$
$620$	0	0
$621$	−111.804	−0.180038
$622$	0	0
$623$	215.250i	0.345506i
$624$	0	0
$625$	−225.620	−0.360993
$626$	0	0
$627$	0	0
$628$	0	0
$629$	447.570i	0.711559i
$630$	0	0
$631$	193.106	0.306032	0.153016	−	0.988224i	$-0.451101\pi$
$631$	193.106	0.306032	0.153016	+	0.988224i	$0.451101\pi$
$632$	0	0
$633$	19.3856i	0.0306249i
$634$	0	0
$635$	41.9459i	0.0660566i
$636$	0	0
$637$	− 626.004i	− 0.982738i
$638$	0	0
$639$	248.196	0.388413
$640$	0	0
$641$	695.788	1.08547	0.542737	−	0.839903i	$-0.317388\pi$
$641$	695.788	1.08547	0.542737	+	0.839903i	$0.317388\pi$
$642$	0	0
$643$	678.908	1.05584	0.527922	−	0.849293i	$-0.322971\pi$
$643$	678.908	1.05584	0.527922	+	0.849293i	$0.322971\pi$
$644$	0	0
$645$	− 27.4249i	− 0.0425192i
$646$	0	0
$647$	−78.8897	−0.121932	−0.0609658	−	0.998140i	$-0.519418\pi$
$647$	−78.8897	−0.121932	−0.0609658	+	0.998140i	$0.519418\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	− 65.6086i	− 0.100781i
$652$	0	0
$653$	909.549	1.39288	0.696438	−	0.717617i	$-0.254767\pi$
$653$	909.549	1.39288	0.696438	+	0.717617i	$0.254767\pi$
$654$	0	0
$655$	− 591.506i	− 0.903062i
$656$	0	0
$657$	− 259.470i	− 0.394932i
$658$	0	0
$659$	− 289.109i	− 0.438708i	−0.975645	−	0.219354i	$-0.929605\pi$
$659$	− 289.109i	− 0.438708i	0.975645	−	0.219354i	$-0.0703950\pi$
$660$	0	0
$661$	−678.682	−1.02675	−0.513375	−	0.858164i	$-0.671605\pi$
$661$	−678.682	−1.02675	−0.513375	+	0.858164i	$0.671605\pi$
$662$	0	0
$663$	214.899	0.324131
$664$	0	0
$665$	−18.9282	−0.0284635
$666$	0	0
$667$	396.379i	0.594271i
$668$	0	0
$669$	542.056	0.810249
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 98.1409i	− 0.145826i	−0.997338	−	0.0729130i	$-0.976770\pi$
$673$	− 98.1409i	− 0.145826i	0.997338	−	0.0729130i	$-0.0232295\pi$
$674$	0	0
$675$	57.5307	0.0852307
$676$	0	0
$677$	228.146i	0.336996i	0.985702	+	0.168498i	$0.0538917\pi$
$677$	228.146i	0.336996i	−0.985702	+	0.168498i	$0.946108\pi$
$678$	0	0
$679$	285.758i	0.420851i
$680$	0	0
$681$	− 417.396i	− 0.612916i
$682$	0	0
$683$	623.604	0.913036	0.456518	−	0.889714i	$-0.349096\pi$
$683$	623.604	0.913036	0.456518	+	0.889714i	$0.349096\pi$
$684$	0	0
$685$	−489.664	−0.714838
$686$	0	0
$687$	−418.128	−0.608629
$688$	0	0
$689$	247.490i	0.359202i
$690$	0	0
$691$	−63.4603	−0.0918384	−0.0459192	−	0.998945i	$-0.514622\pi$
$691$	−63.4603	−0.0918384	−0.0459192	+	0.998945i	$0.514622\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 268.490i	− 0.386317i
$696$	0	0
$697$	−573.879	−0.823356
$698$	0	0
$699$	463.134i	0.662567i
$700$	0	0
$701$	− 807.140i	− 1.15141i	−0.817657	−	0.575706i	$-0.804727\pi$
$701$	− 807.140i	− 1.15141i	0.817657	−	0.575706i	$-0.195273\pi$
$702$	0	0
$703$	108.738i	0.154678i
$704$	0	0
$705$	203.387	0.288492
$706$	0	0
$707$	−124.774	−0.176484
$708$	0	0
$709$	167.149	0.235753	0.117876	−	0.993028i	$-0.462391\pi$
$709$	167.149	0.235753	0.117876	+	0.993028i	$0.462391\pi$
$710$	0	0
$711$	111.364i	0.156630i
$712$	0	0
$713$	−332.736	−0.466670
$714$	0	0
$715$	0	0
$716$	0	0
$717$	522.818i	0.729175i
$718$	0	0
$719$	−1320.60	−1.83671	−0.918356	−	0.395755i	$-0.870483\pi$
$719$	−1320.60	−1.83671	−0.918356	+	0.395755i	$0.870483\pi$
$720$	0	0
$721$	394.132i	0.546647i
$722$	0	0
$723$	− 117.974i	− 0.163173i
$724$	0	0
$725$	− 203.964i	− 0.281330i
$726$	0	0
$727$	333.850	0.459216	0.229608	−	0.973283i	$-0.426256\pi$
$727$	333.850	0.459216	0.229608	+	0.973283i	$0.426256\pi$
$728$	0	0
$729$	27.0000	0.0370370
$730$	0	0
$731$	36.1577	0.0494633
$732$	0	0
$733$	156.175i	0.213063i	0.994309	+	0.106532i	$0.0339745\pi$
$733$	156.175i	0.213063i	−0.994309	+	0.106532i	$0.966025\pi$
$734$	0	0
$735$	−277.956	−0.378172
$736$	0	0
$737$	0	0
$738$	0	0
$739$	1292.40i	1.74885i	0.485162	+	0.874424i	$0.338760\pi$
$739$	1292.40i	1.74885i	−0.485162	+	0.874424i	$0.661240\pi$
$740$	0	0
$741$	52.2102	0.0704592
$742$	0	0
$743$	760.201i	1.02315i	0.859238	+	0.511576i	$0.170938\pi$
$743$	760.201i	1.02315i	−0.859238	+	0.511576i	$0.829062\pi$
$744$	0	0
$745$	− 54.3049i	− 0.0728925i
$746$	0	0
$747$	438.352i	0.586816i
$748$	0	0
$749$	−421.608	−0.562894
$750$	0	0
$751$	481.055	0.640553	0.320276	−	0.947324i	$-0.396224\pi$
$751$	481.055	0.640553	0.320276	+	0.947324i	$0.396224\pi$
$752$	0	0
$753$	831.640	1.10444
$754$	0	0
$755$	− 471.778i	− 0.624871i
$756$	0	0
$757$	509.344	0.672845	0.336422	−	0.941711i	$-0.390783\pi$
$757$	509.344	0.672845	0.336422	+	0.941711i	$0.390783\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	98.2417i	0.129096i	0.997915	+	0.0645478i	$0.0205605\pi$
$761$	98.2417i	0.129096i	−0.997915	+	0.0645478i	$0.979440\pi$
$762$	0	0
$763$	−528.540	−0.692713
$764$	0	0
$765$	− 95.4186i	− 0.124730i
$766$	0	0
$767$	323.137i	0.421300i
$768$	0	0
$769$	− 286.771i	− 0.372914i	−0.982463	−	0.186457i	$-0.940300\pi$
$769$	− 286.771i	− 0.372914i	0.982463	−	0.186457i	$-0.0597005\pi$
$770$	0	0
$771$	−355.508	−0.461100
$772$	0	0
$773$	31.6434	0.0409358	0.0204679	−	0.999791i	$-0.493484\pi$
$773$	31.6434	0.0409358	0.0204679	+	0.999791i	$0.493484\pi$
$774$	0	0
$775$	171.215	0.220923
$776$	0	0
$777$	− 222.809i	− 0.286756i
$778$	0	0
$779$	−139.426	−0.178980
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 95.7232i	− 0.122252i
$784$	0	0
$785$	−466.382	−0.594117
$786$	0	0
$787$	303.394i	0.385507i	0.981247	+	0.192754i	$0.0617418\pi$
$787$	303.394i	0.385507i	−0.981247	+	0.192754i	$0.938258\pi$
$788$	0	0
$789$	− 239.639i	− 0.303724i
$790$	0	0
$791$	49.1657i	0.0621563i
$792$	0	0
$793$	258.491	0.325966
$794$	0	0
$795$	109.890	0.138226
$796$	0	0
$797$	−107.836	−0.135302	−0.0676511	−	0.997709i	$-0.521550\pi$
$797$	−107.836	−0.135302	−0.0676511	+	0.997709i	$0.521550\pi$
$798$	0	0
$799$	268.151i	0.335608i
$800$	0	0
$801$	−263.627	−0.329122
$802$	0	0
$803$	0	0
$804$	0	0
$805$	− 196.697i	− 0.244344i
$806$	0	0
$807$	160.499	0.198883
$808$	0	0
$809$	276.782i	0.342128i	0.985260	+	0.171064i	$0.0547205\pi$
$809$	276.782i	0.342128i	−0.985260	+	0.171064i	$0.945279\pi$
$810$	0	0
$811$	− 63.0683i	− 0.0777661i	−0.999244	−	0.0388830i	$-0.987620\pi$
$811$	− 63.0683i	− 0.0777661i	0.999244	−	0.0388830i	$-0.0123800\pi$
$812$	0	0
$813$	− 790.962i	− 0.972893i
$814$	0	0
$815$	760.960	0.933693
$816$	0	0
$817$	8.78461	0.0107523
$818$	0	0
$819$	−106.981	−0.130624
$820$	0	0
$821$	596.231i	0.726225i	0.931745	+	0.363112i	$0.118286\pi$
$821$	596.231i	0.726225i	−0.931745	+	0.363112i	$0.881714\pi$
$822$	0	0
$823$	−1036.79	−1.25977	−0.629884	−	0.776690i	$-0.716897\pi$
$823$	−1036.79	−1.25977	−0.629884	+	0.776690i	$0.716897\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 460.026i	− 0.556258i	−0.960544	−	0.278129i	$-0.910286\pi$
$827$	− 460.026i	− 0.556258i	0.960544	−	0.278129i	$-0.0897143\pi$
$828$	0	0
$829$	782.069	0.943388	0.471694	−	0.881762i	$-0.343643\pi$
$829$	782.069	0.943388	0.471694	+	0.881762i	$0.343643\pi$
$830$	0	0
$831$	− 896.355i	− 1.07865i
$832$	0	0
$833$	− 366.465i	− 0.439934i
$834$	0	0
$835$	− 587.365i	− 0.703431i
$836$	0	0
$837$	80.3538	0.0960022
$838$	0	0
$839$	−1043.69	−1.24397	−0.621986	−	0.783029i	$-0.713674\pi$
$839$	−1043.69	−1.24397	−0.621986	+	0.783029i	$0.713674\pi$
$840$	0	0
$841$	501.632	0.596471
$842$	0	0
$843$	599.268i	0.710875i
$844$	0	0
$845$	−160.263	−0.189660
$846$	0	0
$847$	0	0
$848$	0	0
$849$	− 266.197i	− 0.313541i
$850$	0	0
$851$	−1129.98	−1.32783
$852$	0	0
$853$	598.911i	0.702123i	0.936352	+	0.351062i	$0.114179\pi$
$853$	598.911i	0.702123i	−0.936352	+	0.351062i	$0.885821\pi$
$854$	0	0
$855$	− 23.1822i	− 0.0271137i
$856$	0	0
$857$	1011.20i	1.17993i	0.807428	+	0.589966i	$0.200859\pi$
$857$	1011.20i	1.17993i	−0.807428	+	0.589966i	$0.799141\pi$
$858$	0	0
$859$	1066.19	1.24120	0.620602	−	0.784126i	$-0.286888\pi$
$859$	1066.19	1.24120	0.620602	+	0.784126i	$0.286888\pi$
$860$	0	0
$861$	285.688	0.331810
$862$	0	0
$863$	1013.93	1.17489	0.587447	−	0.809262i	$-0.300133\pi$
$863$	1013.93	1.17489	0.587447	+	0.809262i	$0.300133\pi$
$864$	0	0
$865$	− 827.672i	− 0.956846i
$866$	0	0
$867$	−374.760	−0.432249
$868$	0	0
$869$	0	0
$870$	0	0
$871$	− 918.419i	− 1.05444i
$872$	0	0
$873$	−349.981	−0.400894
$874$	0	0
$875$	329.755i	0.376862i
$876$	0	0
$877$	− 886.167i	− 1.01045i	−0.862987	−	0.505226i	$-0.831409\pi$
$877$	− 886.167i	− 1.01045i	0.862987	−	0.505226i	$-0.168591\pi$
$878$	0	0
$879$	− 15.0062i	− 0.0170719i
$880$	0	0
$881$	−267.175	−0.303263	−0.151631	−	0.988437i	$-0.548453\pi$
$881$	−267.175	−0.303263	−0.151631	+	0.988437i	$0.548453\pi$
$882$	0	0
$883$	−1265.37	−1.43303	−0.716516	−	0.697571i	$-0.754264\pi$
$883$	−1265.37	−1.43303	−0.716516	+	0.697571i	$0.754264\pi$
$884$	0	0
$885$	143.478	0.162122
$886$	0	0
$887$	− 747.020i	− 0.842187i	−0.907017	−	0.421094i	$-0.861646\pi$
$887$	− 747.020i	− 0.842187i	0.907017	−	0.421094i	$-0.138354\pi$
$888$	0	0
$889$	27.5307	0.0309682
$890$	0	0
$891$	0	0
$892$	0	0
$893$	65.1481i	0.0729542i
$894$	0	0
$895$	−579.626	−0.647626
$896$	0	0
$897$	542.556i	0.604856i
$898$	0	0
$899$	− 284.879i	− 0.316884i
$900$	0	0
$901$	144.882i	0.160801i
$902$	0	0
$903$	−18.0000	−0.0199336
$904$	0	0
$905$	7.53590	0.00832696
$906$	0	0
$907$	851.976	0.939334	0.469667	−	0.882844i	$-0.344374\pi$
$907$	851.976	0.939334	0.469667	+	0.882844i	$0.344374\pi$
$908$	0	0
$909$	− 152.817i	− 0.168115i
$910$	0	0
$911$	681.419	0.747990	0.373995	−	0.927431i	$-0.377988\pi$
$911$	681.419	0.747990	0.373995	+	0.927431i	$0.377988\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	− 114.774i	− 0.125436i
$916$	0	0
$917$	−388.228	−0.423368
$918$	0	0
$919$	1041.02i	1.13277i	0.824140	+	0.566386i	$0.191659\pi$
$919$	1041.02i	1.13277i	−0.824140	+	0.566386i	$0.808341\pi$
$920$	0	0
$921$	− 580.692i	− 0.630502i
$922$	0	0
$923$	− 1204.43i	− 1.30491i
$924$	0	0
$925$	581.454	0.628599
$926$	0	0
$927$	−482.711	−0.520724
$928$	0	0
$929$	80.7437	0.0869147	0.0434573	−	0.999055i	$-0.486163\pi$
$929$	80.7437	0.0869147	0.0434573	+	0.999055i	$0.486163\pi$
$930$	0	0
$931$	− 89.0338i	− 0.0956324i
$932$	0	0
$933$	−651.433	−0.698213
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 627.884i	− 0.670101i	−0.942200	−	0.335050i	$-0.891247\pi$
$937$	− 627.884i	− 0.670101i	0.942200	−	0.335050i	$-0.108753\pi$
$938$	0	0
$939$	−10.4988	−0.0111808
$940$	0	0
$941$	86.1721i	0.0915750i	0.998951	+	0.0457875i	$0.0145797\pi$
$941$	86.1721i	0.0915750i	−0.998951	+	0.0457875i	$0.985420\pi$
$942$	0	0
$943$	− 1448.88i	− 1.53645i
$944$	0	0
$945$	47.5013i	0.0502659i
$946$	0	0
$947$	202.876	0.214230	0.107115	−	0.994247i	$-0.465839\pi$
$947$	202.876	0.214230	0.107115	+	0.994247i	$0.465839\pi$
$948$	0	0
$949$	−1259.14	−1.32681
$950$	0	0
$951$	6.91035	0.00726640
$952$	0	0
$953$	− 1175.60i	− 1.23358i	−0.787128	−	0.616790i	$-0.788433\pi$
$953$	− 1175.60i	− 1.23358i	0.787128	−	0.616790i	$-0.211567\pi$
$954$	0	0
$955$	356.133	0.372914
$956$	0	0
$957$	0	0
$958$	0	0
$959$	321.385i	0.335126i
$960$	0	0
$961$	−721.862	−0.751157
$962$	0	0
$963$	− 516.362i	− 0.536201i
$964$	0	0
$965$	243.958i	0.252806i
$966$	0	0
$967$	− 89.7392i	− 0.0928016i	−0.998923	−	0.0464008i	$-0.985225\pi$
$967$	− 89.7392i	− 0.0928016i	0.998923	−	0.0464008i	$-0.0147751\pi$
$968$	0	0
$969$	30.5641	0.0315419
$970$	0	0
$971$	−906.683	−0.933762	−0.466881	−	0.884320i	$-0.654622\pi$
$971$	−906.683	−0.933762	−0.466881	+	0.884320i	$0.654622\pi$
$972$	0	0
$973$	−176.221	−0.181111
$974$	0	0
$975$	− 279.182i	− 0.286341i
$976$	0	0
$977$	−674.380	−0.690255	−0.345128	−	0.938556i	$-0.612164\pi$
$977$	−674.380	−0.690255	−0.345128	+	0.938556i	$0.612164\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	− 647.326i	− 0.659864i
$982$	0	0
$983$	−697.577	−0.709641	−0.354820	−	0.934935i	$-0.615458\pi$
$983$	−697.577	−0.709641	−0.354820	+	0.934935i	$0.615458\pi$
$984$	0	0
$985$	− 1281.61i	− 1.30113i
$986$	0	0
$987$	− 133.491i	− 0.135249i
$988$	0	0
$989$	91.2875i	0.0923028i
$990$	0	0
$991$	−995.751	−1.00479	−0.502397	−	0.864637i	$-0.667548\pi$
$991$	−995.751	−1.00479	−0.502397	+	0.864637i	$0.667548\pi$
$992$	0	0
$993$	−537.118	−0.540904
$994$	0	0
$995$	963.170	0.968010
$996$	0	0
$997$	1453.35i	1.45772i	0.684660	+	0.728862i	$0.259951\pi$
$997$	1453.35i	1.45772i	−0.684660	+	0.728862i	$0.740049\pi$
$998$	0	0
$999$	272.885	0.273158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1452.3.f.a.241.1	✓	4
3.2	odd	2		4356.3.f.d.1693.1		4
11.10	odd	2	inner	1452.3.f.a.241.2	yes	4
33.32	even	2		4356.3.f.d.1693.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1452.3.f.a.241.1	✓	4	1.1	even	1	trivial
1452.3.f.a.241.2	yes	4	11.10	odd	2	inner
4356.3.f.d.1693.1		4	3.2	odd	2
4356.3.f.d.1693.2		4	33.32	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:	\( N \)	\(=\)	\( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1452.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} +3.73205 q^{5} -2.44949i q^{7} +3.00000 q^{9} -14.5582i q^{13} -6.46410 q^{15} -8.52245i q^{17} -2.07055i q^{19} +4.24264i q^{21} +21.5167 q^{23} -11.0718 q^{25} -5.19615 q^{27} +18.4219i q^{29} -15.4641 q^{31} -9.14162i q^{35} -52.5167 q^{37} +25.2156i q^{39} -67.3374i q^{41} +4.24264i q^{43} +11.1962 q^{45} -31.4641 q^{47} +43.0000 q^{49} +14.7613i q^{51} -17.0000 q^{53} +3.58630i q^{57} -22.1962 q^{59} +17.7556i q^{61} -7.34847i q^{63} -54.3321i q^{65} +63.0859 q^{67} -37.2679 q^{69} +82.7321 q^{71} -86.4900i q^{73} +19.1769 q^{75} +37.1213i q^{79} +9.00000 q^{81} +146.117i q^{83} -31.8062i q^{85} -31.9077i q^{87} -87.8756 q^{89} -35.6603 q^{91} +26.7846 q^{93} -7.72741i q^{95} -116.660 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{5} + 12 q^{9} - 12 q^{15} - 4 q^{23} - 72 q^{25} - 48 q^{31} - 120 q^{37} + 24 q^{45} - 112 q^{47} + 172 q^{49} - 68 q^{53} - 68 q^{59} - 4 q^{67} - 156 q^{69} + 324 q^{71} - 48 q^{75} + 36 q^{81}+ \cdots - 432 q^{97}+O(q^{100}) \) 4 * q + 8 * q^5 + 12 * q^9 - 12 * q^15 - 4 * q^23 - 72 * q^25 - 48 * q^31 - 120 * q^37 + 24 * q^45 - 112 * q^47 + 172 * q^49 - 68 * q^53 - 68 * q^59 - 4 * q^67 - 156 * q^69 + 324 * q^71 - 48 * q^75 + 36 * q^81 - 400 * q^89 - 108 * q^91 + 24 * q^93 - 432 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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