LMFDB - Embedded newform 4140.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(1,4140)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4140.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$33.0580664368$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ of defining polynomial
Character	$\chi$	$=$	4140.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -3.44949 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -3.44949 q^{7} -2.44949 q^{11} +4.44949 q^{13} +5.44949 q^{17} -1.55051 q^{19} +1.00000 q^{23} +1.00000 q^{25} +1.89898 q^{29} -7.00000 q^{31} +3.44949 q^{35} +6.34847 q^{37} +7.89898 q^{41} -8.89898 q^{43} -2.44949 q^{47} +4.89898 q^{49} -4.34847 q^{53} +2.44949 q^{55} -1.89898 q^{59} -5.34847 q^{61} -4.44949 q^{65} +1.44949 q^{67} +3.00000 q^{71} +9.34847 q^{73} +8.44949 q^{77} -4.00000 q^{79} -10.3485 q^{83} -5.44949 q^{85} -16.8990 q^{89} -15.3485 q^{91} +1.55051 q^{95} -14.8990 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 - 2 * q^7	$ 2 q - 2 q^{5} - 2 q^{7} + 4 q^{13} + 6 q^{17} - 8 q^{19} + 2 q^{23} + 2 q^{25} - 6 q^{29} - 14 q^{31} + 2 q^{35} - 2 q^{37} + 6 q^{41} - 8 q^{43} + 6 q^{53} + 6 q^{59} + 4 q^{61} - 4 q^{65} - 2 q^{67} + 6 q^{71} + 4 q^{73} + 12 q^{77} - 8 q^{79} - 6 q^{83} - 6 q^{85} - 24 q^{89} - 16 q^{91} + 8 q^{95} - 20 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^7 + 4 * q^13 + 6 * q^17 - 8 * q^19 + 2 * q^23 + 2 * q^25 - 6 * q^29 - 14 * q^31 + 2 * q^35 - 2 * q^37 + 6 * q^41 - 8 * q^43 + 6 * q^53 + 6 * q^59 + 4 * q^61 - 4 * q^65 - 2 * q^67 + 6 * q^71 + 4 * q^73 + 12 * q^77 - 8 * q^79 - 6 * q^83 - 6 * q^85 - 24 * q^89 - 16 * q^91 + 8 * q^95 - 20 * q^97

Display 100 coefficients 10 coefficients

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
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.44949	−1.30378	−0.651892	−	0.758312i	$-0.726025\pi$
$7$	−3.44949	−1.30378	−0.651892	+	0.758312i	$0.726025\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.44949	−0.738549	−0.369274	−	0.929320i	$-0.620394\pi$
$11$	−2.44949	−0.738549	−0.369274	+	0.929320i	$0.620394\pi$
$12$	0	0
$13$	4.44949	1.23407	0.617033	−	0.786937i	$-0.288334\pi$
$13$	4.44949	1.23407	0.617033	+	0.786937i	$0.288334\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.44949	1.32170	0.660848	−	0.750520i	$-0.270197\pi$
$17$	5.44949	1.32170	0.660848	+	0.750520i	$0.270197\pi$
$18$	0	0
$19$	−1.55051	−0.355711	−0.177856	−	0.984057i	$-0.556916\pi$
$19$	−1.55051	−0.355711	−0.177856	+	0.984057i	$0.556916\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.89898	0.352632	0.176316	−	0.984334i	$-0.443582\pi$
$29$	1.89898	0.352632	0.176316	+	0.984334i	$0.443582\pi$
$30$	0	0
$31$	−7.00000	−1.25724	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000	−1.25724	−0.628619	+	0.777714i	$0.716379\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.44949	0.583070
$36$	0	0
$37$	6.34847	1.04368	0.521841	−	0.853043i	$-0.325245\pi$
$37$	6.34847	1.04368	0.521841	+	0.853043i	$0.325245\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.89898	1.23361	0.616807	−	0.787115i	$-0.288426\pi$
$41$	7.89898	1.23361	0.616807	+	0.787115i	$0.288426\pi$
$42$	0	0
$43$	−8.89898	−1.35708	−0.678541	−	0.734563i	$-0.737387\pi$
$43$	−8.89898	−1.35708	−0.678541	+	0.734563i	$0.737387\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.44949	−0.357295	−0.178647	−	0.983913i	$-0.557172\pi$
$47$	−2.44949	−0.357295	−0.178647	+	0.983913i	$0.557172\pi$
$48$	0	0
$49$	4.89898	0.699854
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.34847	−0.597308	−0.298654	−	0.954361i	$-0.596538\pi$
$53$	−4.34847	−0.597308	−0.298654	+	0.954361i	$0.596538\pi$
$54$	0	0
$55$	2.44949	0.330289
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.89898	−0.247226	−0.123613	−	0.992330i	$-0.539448\pi$
$59$	−1.89898	−0.247226	−0.123613	+	0.992330i	$0.539448\pi$
$60$	0	0
$61$	−5.34847	−0.684801	−0.342401	−	0.939554i	$-0.611240\pi$
$61$	−5.34847	−0.684801	−0.342401	+	0.939554i	$0.611240\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.44949	−0.551891
$66$	0	0
$67$	1.44949	0.177083	0.0885417	−	0.996072i	$-0.471779\pi$
$67$	1.44949	0.177083	0.0885417	+	0.996072i	$0.471779\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0	0
$73$	9.34847	1.09416	0.547078	−	0.837082i	$-0.315740\pi$
$73$	9.34847	1.09416	0.547078	+	0.837082i	$0.315740\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.44949	0.962909
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−10.3485	−1.13589	−0.567946	−	0.823066i	$-0.692262\pi$
$83$	−10.3485	−1.13589	−0.567946	+	0.823066i	$0.692262\pi$
$84$	0	0
$85$	−5.44949	−0.591080
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−16.8990	−1.79129	−0.895644	−	0.444771i	$-0.853285\pi$
$89$	−16.8990	−1.79129	−0.895644	+	0.444771i	$0.853285\pi$
$90$	0	0
$91$	−15.3485	−1.60896
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.55051	0.159079
$96$	0	0
$97$	−14.8990	−1.51276	−0.756381	−	0.654131i	$-0.773034\pi$
$97$	−14.8990	−1.51276	−0.756381	+	0.654131i	$0.773034\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	18.7980	1.87047	0.935233	−	0.354032i	$-0.115190\pi$
$101$	18.7980	1.87047	0.935233	+	0.354032i	$0.115190\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.1464	1.36759	0.683793	−	0.729676i	$-0.260329\pi$
$107$	14.1464	1.36759	0.683793	+	0.729676i	$0.260329\pi$
$108$	0	0
$109$	−11.3485	−1.08699	−0.543493	−	0.839414i	$-0.682899\pi$
$109$	−11.3485	−1.08699	−0.543493	+	0.839414i	$0.682899\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−16.3485	−1.53793	−0.768967	−	0.639288i	$-0.779229\pi$
$113$	−16.3485	−1.53793	−0.768967	+	0.639288i	$0.779229\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−18.7980	−1.72321
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	3.34847	0.297129	0.148564	−	0.988903i	$-0.452535\pi$
$127$	3.34847	0.297129	0.148564	+	0.988903i	$0.452535\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	10.8990	0.952248	0.476124	−	0.879378i	$-0.342041\pi$
$131$	10.8990	0.952248	0.476124	+	0.879378i	$0.342041\pi$
$132$	0	0
$133$	5.34847	0.463771
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−16.8990	−1.44378	−0.721889	−	0.692009i	$-0.756726\pi$
$137$	−16.8990	−1.44378	−0.721889	+	0.692009i	$0.756726\pi$
$138$	0	0
$139$	−21.6969	−1.84031	−0.920155	−	0.391554i	$-0.871938\pi$
$139$	−21.6969	−1.84031	−0.920155	+	0.391554i	$0.871938\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−10.8990	−0.911418
$144$	0	0
$145$	−1.89898	−0.157702
$146$	0	0
$147$	0	0
$148$	0	0
$149$	18.2474	1.49489	0.747445	−	0.664324i	$-0.231281\pi$
$149$	18.2474	1.49489	0.747445	+	0.664324i	$0.231281\pi$
$150$	0	0
$151$	−19.7980	−1.61114	−0.805568	−	0.592504i	$-0.798139\pi$
$151$	−19.7980	−1.61114	−0.805568	+	0.592504i	$0.798139\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	7.00000	0.562254
$156$	0	0
$157$	−14.3485	−1.14513	−0.572566	−	0.819858i	$-0.694052\pi$
$157$	−14.3485	−1.14513	−0.572566	+	0.819858i	$0.694052\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.44949	−0.271858
$162$	0	0
$163$	−24.6969	−1.93441	−0.967207	−	0.253990i	$-0.918257\pi$
$163$	−24.6969	−1.93441	−0.967207	+	0.253990i	$0.918257\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.247449	0.0191482	0.00957408	−	0.999954i	$-0.496952\pi$
$167$	0.247449	0.0191482	0.00957408	+	0.999954i	$0.496952\pi$
$168$	0	0
$169$	6.79796	0.522920
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−9.79796	−0.744925	−0.372463	−	0.928047i	$-0.621486\pi$
$173$	−9.79796	−0.744925	−0.372463	+	0.928047i	$0.621486\pi$
$174$	0	0
$175$	−3.44949	−0.260757
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	0.898979	0.0668206	0.0334103	−	0.999442i	$-0.489363\pi$
$181$	0.898979	0.0668206	0.0334103	+	0.999442i	$0.489363\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.34847	−0.466749
$186$	0	0
$187$	−13.3485	−0.976137
$188$	0	0
$189$	0	0
$190$	0	0
$191$	15.5505	1.12520	0.562598	−	0.826731i	$-0.309802\pi$
$191$	15.5505	1.12520	0.562598	+	0.826731i	$0.309802\pi$
$192$	0	0
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−15.7980	−1.12556	−0.562779	−	0.826607i	$-0.690268\pi$
$197$	−15.7980	−1.12556	−0.562779	+	0.826607i	$0.690268\pi$
$198$	0	0
$199$	11.7980	0.836335	0.418168	−	0.908370i	$-0.362672\pi$
$199$	11.7980	0.836335	0.418168	+	0.908370i	$0.362672\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−6.55051	−0.459756
$204$	0	0
$205$	−7.89898	−0.551689
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3.79796	0.262710
$210$	0	0
$211$	13.6969	0.942936	0.471468	−	0.881883i	$-0.343724\pi$
$211$	13.6969	0.942936	0.471468	+	0.881883i	$0.343724\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	8.89898	0.606905
$216$	0	0
$217$	24.1464	1.63917
$218$	0	0
$219$	0	0
$220$	0	0
$221$	24.2474	1.63106
$222$	0	0
$223$	−17.1010	−1.14517	−0.572585	−	0.819846i	$-0.694059\pi$
$223$	−17.1010	−1.14517	−0.572585	+	0.819846i	$0.694059\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	−1.79796	−0.118812	−0.0594062	−	0.998234i	$-0.518921\pi$
$229$	−1.79796	−0.118812	−0.0594062	+	0.998234i	$0.518921\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	24.4949	1.60471	0.802357	−	0.596844i	$-0.203579\pi$
$233$	24.4949	1.60471	0.802357	+	0.596844i	$0.203579\pi$
$234$	0	0
$235$	2.44949	0.159787
$236$	0	0
$237$	0	0
$238$	0	0
$239$	19.8990	1.28716	0.643579	−	0.765380i	$-0.277449\pi$
$239$	19.8990	1.28716	0.643579	+	0.765380i	$0.277449\pi$
$240$	0	0
$241$	3.34847	0.215694	0.107847	−	0.994168i	$-0.465604\pi$
$241$	3.34847	0.215694	0.107847	+	0.994168i	$0.465604\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−4.89898	−0.312984
$246$	0	0
$247$	−6.89898	−0.438972
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1.10102	−0.0694958	−0.0347479	−	0.999396i	$-0.511063\pi$
$251$	−1.10102	−0.0694958	−0.0347479	+	0.999396i	$0.511063\pi$
$252$	0	0
$253$	−2.44949	−0.153998
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−19.3485	−1.20692	−0.603462	−	0.797392i	$-0.706213\pi$
$257$	−19.3485	−1.20692	−0.603462	+	0.797392i	$0.706213\pi$
$258$	0	0
$259$	−21.8990	−1.36074
$260$	0	0
$261$	0	0
$262$	0	0
$263$	21.2474	1.31017	0.655087	−	0.755554i	$-0.272632\pi$
$263$	21.2474	1.31017	0.655087	+	0.755554i	$0.272632\pi$
$264$	0	0
$265$	4.34847	0.267124
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5.69694	−0.347348	−0.173674	−	0.984803i	$-0.555564\pi$
$269$	−5.69694	−0.347348	−0.173674	+	0.984803i	$0.555564\pi$
$270$	0	0
$271$	25.6969	1.56098	0.780489	−	0.625170i	$-0.214970\pi$
$271$	25.6969	1.56098	0.780489	+	0.625170i	$0.214970\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.44949	−0.147710
$276$	0	0
$277$	−14.8990	−0.895193	−0.447596	−	0.894236i	$-0.647720\pi$
$277$	−14.8990	−0.895193	−0.447596	+	0.894236i	$0.647720\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−7.34847	−0.438373	−0.219186	−	0.975683i	$-0.570340\pi$
$281$	−7.34847	−0.438373	−0.219186	+	0.975683i	$0.570340\pi$
$282$	0	0
$283$	8.55051	0.508275	0.254138	−	0.967168i	$-0.418208\pi$
$283$	8.55051	0.508275	0.254138	+	0.967168i	$0.418208\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−27.2474	−1.60837
$288$	0	0
$289$	12.6969	0.746879
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−22.3485	−1.30561	−0.652806	−	0.757525i	$-0.726408\pi$
$293$	−22.3485	−1.30561	−0.652806	+	0.757525i	$0.726408\pi$
$294$	0	0
$295$	1.89898	0.110563
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.44949	0.257321
$300$	0	0
$301$	30.6969	1.76934
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5.34847	0.306252
$306$	0	0
$307$	−12.4495	−0.710530	−0.355265	−	0.934766i	$-0.615609\pi$
$307$	−12.4495	−0.710530	−0.355265	+	0.934766i	$0.615609\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8.69694	−0.493158	−0.246579	−	0.969123i	$-0.579307\pi$
$311$	−8.69694	−0.493158	−0.246579	+	0.969123i	$0.579307\pi$
$312$	0	0
$313$	3.65153	0.206397	0.103198	−	0.994661i	$-0.467092\pi$
$313$	3.65153	0.206397	0.103198	+	0.994661i	$0.467092\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−24.2474	−1.36187	−0.680936	−	0.732343i	$-0.738427\pi$
$317$	−24.2474	−1.36187	−0.680936	+	0.732343i	$0.738427\pi$
$318$	0	0
$319$	−4.65153	−0.260436
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−8.44949	−0.470142
$324$	0	0
$325$	4.44949	0.246813
$326$	0	0
$327$	0	0
$328$	0	0
$329$	8.44949	0.465835
$330$	0	0
$331$	−1.00000	−0.0549650	−0.0274825	−	0.999622i	$-0.508749\pi$
$331$	−1.00000	−0.0549650	−0.0274825	+	0.999622i	$0.508749\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.44949	−0.0791941
$336$	0	0
$337$	15.1010	0.822605	0.411303	−	0.911499i	$-0.365074\pi$
$337$	15.1010	0.822605	0.411303	+	0.911499i	$0.365074\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	17.1464	0.928531
$342$	0	0
$343$	7.24745	0.391325
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.20204	0.118212	0.0591059	−	0.998252i	$-0.481175\pi$
$347$	2.20204	0.118212	0.0591059	+	0.998252i	$0.481175\pi$
$348$	0	0
$349$	−3.69694	−0.197893	−0.0989463	−	0.995093i	$-0.531547\pi$
$349$	−3.69694	−0.197893	−0.0989463	+	0.995093i	$0.531547\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.4495	0.769069	0.384534	−	0.923111i	$-0.374362\pi$
$353$	14.4495	0.769069	0.384534	+	0.923111i	$0.374362\pi$
$354$	0	0
$355$	−3.00000	−0.159223
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11.1464	0.588286	0.294143	−	0.955761i	$-0.404966\pi$
$359$	11.1464	0.588286	0.294143	+	0.955761i	$0.404966\pi$
$360$	0	0
$361$	−16.5959	−0.873469
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−9.34847	−0.489321
$366$	0	0
$367$	5.24745	0.273915	0.136957	−	0.990577i	$-0.456268\pi$
$367$	5.24745	0.273915	0.136957	+	0.990577i	$0.456268\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	15.0000	0.778761
$372$	0	0
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.44949	0.435171
$378$	0	0
$379$	25.3939	1.30440	0.652198	−	0.758049i	$-0.273847\pi$
$379$	25.3939	1.30440	0.652198	+	0.758049i	$0.273847\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−10.3485	−0.528782	−0.264391	−	0.964416i	$-0.585171\pi$
$383$	−10.3485	−0.528782	−0.264391	+	0.964416i	$0.585171\pi$
$384$	0	0
$385$	−8.44949	−0.430626
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−22.8990	−1.16102	−0.580512	−	0.814252i	$-0.697148\pi$
$389$	−22.8990	−1.16102	−0.580512	+	0.814252i	$0.697148\pi$
$390$	0	0
$391$	5.44949	0.275593
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.00000	0.201262
$396$	0	0
$397$	−1.79796	−0.0902370	−0.0451185	−	0.998982i	$-0.514367\pi$
$397$	−1.79796	−0.0902370	−0.0451185	+	0.998982i	$0.514367\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8.69694	−0.434304	−0.217152	−	0.976138i	$-0.569677\pi$
$401$	−8.69694	−0.434304	−0.217152	+	0.976138i	$0.569677\pi$
$402$	0	0
$403$	−31.1464	−1.55151
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−15.5505	−0.770810
$408$	0	0
$409$	−20.5959	−1.01840	−0.509201	−	0.860647i	$-0.670059\pi$
$409$	−20.5959	−1.01840	−0.509201	+	0.860647i	$0.670059\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	6.55051	0.322330
$414$	0	0
$415$	10.3485	0.507986
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−17.1464	−0.837658	−0.418829	−	0.908065i	$-0.637559\pi$
$419$	−17.1464	−0.837658	−0.418829	+	0.908065i	$0.637559\pi$
$420$	0	0
$421$	−20.6515	−1.00649	−0.503247	−	0.864143i	$-0.667861\pi$
$421$	−20.6515	−1.00649	−0.503247	+	0.864143i	$0.667861\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.44949	0.264339
$426$	0	0
$427$	18.4495	0.892833
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−19.5959	−0.943902	−0.471951	−	0.881625i	$-0.656450\pi$
$431$	−19.5959	−0.943902	−0.471951	+	0.881625i	$0.656450\pi$
$432$	0	0
$433$	−26.8434	−1.29001	−0.645005	−	0.764178i	$-0.723145\pi$
$433$	−26.8434	−1.29001	−0.645005	+	0.764178i	$0.723145\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.55051	−0.0741710
$438$	0	0
$439$	6.89898	0.329270	0.164635	−	0.986355i	$-0.447355\pi$
$439$	6.89898	0.329270	0.164635	+	0.986355i	$0.447355\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	38.9444	1.85030	0.925152	−	0.379597i	$-0.123937\pi$
$443$	38.9444	1.85030	0.925152	+	0.379597i	$0.123937\pi$
$444$	0	0
$445$	16.8990	0.801088
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−26.3939	−1.24560	−0.622802	−	0.782379i	$-0.714006\pi$
$449$	−26.3939	−1.24560	−0.622802	+	0.782379i	$0.714006\pi$
$450$	0	0
$451$	−19.3485	−0.911084
$452$	0	0
$453$	0	0
$454$	0	0
$455$	15.3485	0.719547
$456$	0	0
$457$	−26.3485	−1.23253	−0.616265	−	0.787539i	$-0.711355\pi$
$457$	−26.3485	−1.23253	−0.616265	+	0.787539i	$0.711355\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9.30306	0.433287	0.216643	−	0.976251i	$-0.430489\pi$
$461$	9.30306	0.433287	0.216643	+	0.976251i	$0.430489\pi$
$462$	0	0
$463$	24.0454	1.11748	0.558742	−	0.829341i	$-0.311284\pi$
$463$	24.0454	1.11748	0.558742	+	0.829341i	$0.311284\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17.9444	0.830367	0.415184	−	0.909738i	$-0.363717\pi$
$467$	17.9444	0.830367	0.415184	+	0.909738i	$0.363717\pi$
$468$	0	0
$469$	−5.00000	−0.230879
$470$	0	0
$471$	0	0
$472$	0	0
$473$	21.7980	1.00227
$474$	0	0
$475$	−1.55051	−0.0711423
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−10.6515	−0.486681	−0.243340	−	0.969941i	$-0.578243\pi$
$479$	−10.6515	−0.486681	−0.243340	+	0.969941i	$0.578243\pi$
$480$	0	0
$481$	28.2474	1.28797
$482$	0	0
$483$	0	0
$484$	0	0
$485$	14.8990	0.676528
$486$	0	0
$487$	1.14643	0.0519496	0.0259748	−	0.999663i	$-0.491731\pi$
$487$	1.14643	0.0519496	0.0259748	+	0.999663i	$0.491731\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	38.3939	1.73269	0.866346	−	0.499445i	$-0.166463\pi$
$491$	38.3939	1.73269	0.866346	+	0.499445i	$0.166463\pi$
$492$	0	0
$493$	10.3485	0.466072
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−10.3485	−0.464192
$498$	0	0
$499$	−4.79796	−0.214786	−0.107393	−	0.994217i	$-0.534250\pi$
$499$	−4.79796	−0.214786	−0.107393	+	0.994217i	$0.534250\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−14.1464	−0.630758	−0.315379	−	0.948966i	$-0.602132\pi$
$503$	−14.1464	−0.630758	−0.315379	+	0.948966i	$0.602132\pi$
$504$	0	0
$505$	−18.7980	−0.836498
$506$	0	0
$507$	0	0
$508$	0	0
$509$	23.3939	1.03692	0.518458	−	0.855103i	$-0.326506\pi$
$509$	23.3939	1.03692	0.518458	+	0.855103i	$0.326506\pi$
$510$	0	0
$511$	−32.2474	−1.42654
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.00000	0.176261
$516$	0	0
$517$	6.00000	0.263880
$518$	0	0
$519$	0	0
$520$	0	0
$521$	31.3485	1.37340	0.686701	−	0.726940i	$-0.259058\pi$
$521$	31.3485	1.37340	0.686701	+	0.726940i	$0.259058\pi$
$522$	0	0
$523$	−12.2020	−0.533558	−0.266779	−	0.963758i	$-0.585959\pi$
$523$	−12.2020	−0.533558	−0.266779	+	0.963758i	$0.585959\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−38.1464	−1.66168
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	35.1464	1.52236
$534$	0	0
$535$	−14.1464	−0.611603
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−12.0000	−0.516877
$540$	0	0
$541$	15.1010	0.649244	0.324622	−	0.945844i	$-0.394763\pi$
$541$	15.1010	0.649244	0.324622	+	0.945844i	$0.394763\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	11.3485	0.486115
$546$	0	0
$547$	−33.3939	−1.42782	−0.713910	−	0.700238i	$-0.753077\pi$
$547$	−33.3939	−1.42782	−0.713910	+	0.700238i	$0.753077\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.94439	−0.125435
$552$	0	0
$553$	13.7980	0.586749
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−24.5505	−1.04024	−0.520119	−	0.854094i	$-0.674113\pi$
$557$	−24.5505	−1.04024	−0.520119	+	0.854094i	$0.674113\pi$
$558$	0	0
$559$	−39.5959	−1.67473
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−38.1464	−1.60768	−0.803840	−	0.594845i	$-0.797213\pi$
$563$	−38.1464	−1.60768	−0.803840	+	0.594845i	$0.797213\pi$
$564$	0	0
$565$	16.3485	0.687785
$566$	0	0
$567$	0	0
$568$	0	0
$569$	11.3939	0.477656	0.238828	−	0.971062i	$-0.423237\pi$
$569$	11.3939	0.477656	0.238828	+	0.971062i	$0.423237\pi$
$570$	0	0
$571$	−32.0454	−1.34106	−0.670529	−	0.741883i	$-0.733933\pi$
$571$	−32.0454	−1.34106	−0.670529	+	0.741883i	$0.733933\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−26.2929	−1.09459	−0.547293	−	0.836941i	$-0.684342\pi$
$577$	−26.2929	−1.09459	−0.547293	+	0.836941i	$0.684342\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	35.6969	1.48096
$582$	0	0
$583$	10.6515	0.441141
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.59592	0.0658706	0.0329353	−	0.999457i	$-0.489514\pi$
$587$	1.59592	0.0658706	0.0329353	+	0.999457i	$0.489514\pi$
$588$	0	0
$589$	10.8536	0.447214
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−41.1464	−1.68968	−0.844841	−	0.535018i	$-0.820305\pi$
$593$	−41.1464	−1.68968	−0.844841	+	0.535018i	$0.820305\pi$
$594$	0	0
$595$	18.7980	0.770641
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	0	0
$601$	14.7980	0.603621	0.301811	−	0.953368i	$-0.402409\pi$
$601$	14.7980	0.603621	0.301811	+	0.953368i	$0.402409\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	5.00000	0.203279
$606$	0	0
$607$	24.6515	1.00057	0.500287	−	0.865859i	$-0.333228\pi$
$607$	24.6515	1.00057	0.500287	+	0.865859i	$0.333228\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−10.8990	−0.440926
$612$	0	0
$613$	22.6969	0.916721	0.458360	−	0.888766i	$-0.348437\pi$
$613$	22.6969	0.916721	0.458360	+	0.888766i	$0.348437\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−16.3485	−0.658165	−0.329082	−	0.944301i	$-0.606739\pi$
$617$	−16.3485	−0.658165	−0.329082	+	0.944301i	$0.606739\pi$
$618$	0	0
$619$	−1.79796	−0.0722661	−0.0361330	−	0.999347i	$-0.511504\pi$
$619$	−1.79796	−0.0722661	−0.0361330	+	0.999347i	$0.511504\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	58.2929	2.33545
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	34.5959	1.37943
$630$	0	0
$631$	−22.2474	−0.885657	−0.442828	−	0.896606i	$-0.646025\pi$
$631$	−22.2474	−0.885657	−0.442828	+	0.896606i	$0.646025\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3.34847	−0.132880
$636$	0	0
$637$	21.7980	0.863667
$638$	0	0
$639$	0	0
$640$	0	0
$641$	26.4495	1.04469	0.522346	−	0.852734i	$-0.325057\pi$
$641$	26.4495	1.04469	0.522346	+	0.852734i	$0.325057\pi$
$642$	0	0
$643$	24.3485	0.960210	0.480105	−	0.877211i	$-0.340599\pi$
$643$	24.3485	0.960210	0.480105	+	0.877211i	$0.340599\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	20.4495	0.803952	0.401976	−	0.915650i	$-0.368323\pi$
$647$	20.4495	0.803952	0.401976	+	0.915650i	$0.368323\pi$
$648$	0	0
$649$	4.65153	0.182589
$650$	0	0
$651$	0	0
$652$	0	0
$653$	27.5505	1.07813	0.539067	−	0.842263i	$-0.318777\pi$
$653$	27.5505	1.07813	0.539067	+	0.842263i	$0.318777\pi$
$654$	0	0
$655$	−10.8990	−0.425858
$656$	0	0
$657$	0	0
$658$	0	0
$659$	14.4495	0.562872	0.281436	−	0.959580i	$-0.409189\pi$
$659$	14.4495	0.562872	0.281436	+	0.959580i	$0.409189\pi$
$660$	0	0
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5.34847	−0.207405
$666$	0	0
$667$	1.89898	0.0735288
$668$	0	0
$669$	0	0
$670$	0	0
$671$	13.1010	0.505759
$672$	0	0
$673$	18.0454	0.695599	0.347800	−	0.937569i	$-0.386929\pi$
$673$	18.0454	0.695599	0.347800	+	0.937569i	$0.386929\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−39.2474	−1.50840	−0.754201	−	0.656644i	$-0.771976\pi$
$677$	−39.2474	−1.50840	−0.754201	+	0.656644i	$0.771976\pi$
$678$	0	0
$679$	51.3939	1.97232
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−19.3485	−0.740349	−0.370174	−	0.928962i	$-0.620702\pi$
$683$	−19.3485	−0.740349	−0.370174	+	0.928962i	$0.620702\pi$
$684$	0	0
$685$	16.8990	0.645677
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−19.3485	−0.737118
$690$	0	0
$691$	43.3939	1.65078	0.825390	−	0.564562i	$-0.190955\pi$
$691$	43.3939	1.65078	0.825390	+	0.564562i	$0.190955\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	21.6969	0.823012
$696$	0	0
$697$	43.0454	1.63046
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−16.0454	−0.606027	−0.303013	−	0.952986i	$-0.597993\pi$
$701$	−16.0454	−0.606027	−0.303013	+	0.952986i	$0.597993\pi$
$702$	0	0
$703$	−9.84337	−0.371250
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−64.8434	−2.43869
$708$	0	0
$709$	52.9444	1.98837	0.994184	−	0.107694i	$-0.0343466\pi$
$709$	52.9444	1.98837	0.994184	+	0.107694i	$0.0343466\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−7.00000	−0.262152
$714$	0	0
$715$	10.8990	0.407599
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−49.2929	−1.83831	−0.919157	−	0.393892i	$-0.871128\pi$
$719$	−49.2929	−1.83831	−0.919157	+	0.393892i	$0.871128\pi$
$720$	0	0
$721$	13.7980	0.513863
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.89898	0.0705263
$726$	0	0
$727$	−17.6515	−0.654659	−0.327330	−	0.944910i	$-0.606149\pi$
$727$	−17.6515	−0.654659	−0.327330	+	0.944910i	$0.606149\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−48.4949	−1.79365
$732$	0	0
$733$	21.6515	0.799718	0.399859	−	0.916577i	$-0.369059\pi$
$733$	21.6515	0.799718	0.399859	+	0.916577i	$0.369059\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.55051	−0.130785
$738$	0	0
$739$	−45.6969	−1.68099	−0.840495	−	0.541820i	$-0.817735\pi$
$739$	−45.6969	−1.68099	−0.840495	+	0.541820i	$0.817735\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−14.2020	−0.521022	−0.260511	−	0.965471i	$-0.583891\pi$
$743$	−14.2020	−0.521022	−0.260511	+	0.965471i	$0.583891\pi$
$744$	0	0
$745$	−18.2474	−0.668535
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−48.7980	−1.78304
$750$	0	0
$751$	27.3485	0.997960	0.498980	−	0.866614i	$-0.333708\pi$
$751$	27.3485	0.997960	0.498980	+	0.866614i	$0.333708\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	19.7980	0.720521
$756$	0	0
$757$	10.7526	0.390808	0.195404	−	0.980723i	$-0.437398\pi$
$757$	10.7526	0.390808	0.195404	+	0.980723i	$0.437398\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−3.49490	−0.126690	−0.0633450	−	0.997992i	$-0.520177\pi$
$761$	−3.49490	−0.126690	−0.0633450	+	0.997992i	$0.520177\pi$
$762$	0	0
$763$	39.1464	1.41720
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.44949	−0.305093
$768$	0	0
$769$	31.1464	1.12317	0.561584	−	0.827419i	$-0.310192\pi$
$769$	31.1464	1.12317	0.561584	+	0.827419i	$0.310192\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	20.6969	0.744417	0.372209	−	0.928149i	$-0.378601\pi$
$773$	20.6969	0.744417	0.372209	+	0.928149i	$0.378601\pi$
$774$	0	0
$775$	−7.00000	−0.251447
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−12.2474	−0.438810
$780$	0	0
$781$	−7.34847	−0.262949
$782$	0	0
$783$	0	0
$784$	0	0
$785$	14.3485	0.512119
$786$	0	0
$787$	−13.8536	−0.493827	−0.246913	−	0.969038i	$-0.579416\pi$
$787$	−13.8536	−0.493827	−0.246913	+	0.969038i	$0.579416\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	56.3939	2.00514
$792$	0	0
$793$	−23.7980	−0.845090
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−3.85357	−0.136501	−0.0682503	−	0.997668i	$-0.521742\pi$
$797$	−3.85357	−0.136501	−0.0682503	+	0.997668i	$0.521742\pi$
$798$	0	0
$799$	−13.3485	−0.472235
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−22.8990	−0.808087
$804$	0	0
$805$	3.44949	0.121579
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−18.7980	−0.660901	−0.330451	−	0.943823i	$-0.607201\pi$
$809$	−18.7980	−0.660901	−0.330451	+	0.943823i	$0.607201\pi$
$810$	0	0
$811$	2.79796	0.0982496	0.0491248	−	0.998793i	$-0.484357\pi$
$811$	2.79796	0.0982496	0.0491248	+	0.998793i	$0.484357\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	24.6969	0.865096
$816$	0	0
$817$	13.7980	0.482729
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−13.5959	−0.474501	−0.237250	−	0.971449i	$-0.576246\pi$
$821$	−13.5959	−0.474501	−0.237250	+	0.971449i	$0.576246\pi$
$822$	0	0
$823$	12.8990	0.449630	0.224815	−	0.974401i	$-0.427822\pi$
$823$	12.8990	0.449630	0.224815	+	0.974401i	$0.427822\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20.7526	0.721637	0.360818	−	0.932636i	$-0.382497\pi$
$827$	20.7526	0.721637	0.360818	+	0.932636i	$0.382497\pi$
$828$	0	0
$829$	−24.3939	−0.847234	−0.423617	−	0.905841i	$-0.639240\pi$
$829$	−24.3939	−0.847234	−0.423617	+	0.905841i	$0.639240\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	26.6969	0.924994
$834$	0	0
$835$	−0.247449	−0.00856332
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−25.5959	−0.883669	−0.441835	−	0.897097i	$-0.645672\pi$
$839$	−25.5959	−0.883669	−0.441835	+	0.897097i	$0.645672\pi$
$840$	0	0
$841$	−25.3939	−0.875651
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−6.79796	−0.233857
$846$	0	0
$847$	17.2474	0.592629
$848$	0	0
$849$	0	0
$850$	0	0
$851$	6.34847	0.217623
$852$	0	0
$853$	23.7980	0.814827	0.407413	−	0.913244i	$-0.366431\pi$
$853$	23.7980	0.814827	0.407413	+	0.913244i	$0.366431\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	16.2929	0.556553	0.278277	−	0.960501i	$-0.410237\pi$
$857$	16.2929	0.556553	0.278277	+	0.960501i	$0.410237\pi$
$858$	0	0
$859$	17.0000	0.580033	0.290016	−	0.957022i	$-0.406339\pi$
$859$	17.0000	0.580033	0.290016	+	0.957022i	$0.406339\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	25.5959	0.871295	0.435648	−	0.900117i	$-0.356519\pi$
$863$	25.5959	0.871295	0.435648	+	0.900117i	$0.356519\pi$
$864$	0	0
$865$	9.79796	0.333141
$866$	0	0
$867$	0	0
$868$	0	0
$869$	9.79796	0.332373
$870$	0	0
$871$	6.44949	0.218533
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.44949	0.116614
$876$	0	0
$877$	23.7980	0.803600	0.401800	−	0.915727i	$-0.368385\pi$
$877$	23.7980	0.803600	0.401800	+	0.915727i	$0.368385\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−35.1464	−1.18411	−0.592057	−	0.805896i	$-0.701684\pi$
$881$	−35.1464	−1.18411	−0.592057	+	0.805896i	$0.701684\pi$
$882$	0	0
$883$	−37.5505	−1.26368	−0.631838	−	0.775101i	$-0.717699\pi$
$883$	−37.5505	−1.26368	−0.631838	+	0.775101i	$0.717699\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−37.5959	−1.26235	−0.631174	−	0.775642i	$-0.717426\pi$
$887$	−37.5959	−1.26235	−0.631174	+	0.775642i	$0.717426\pi$
$888$	0	0
$889$	−11.5505	−0.387392
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3.79796	0.127094
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−13.2929	−0.443342
$900$	0	0
$901$	−23.6969	−0.789459
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−0.898979	−0.0298831
$906$	0	0
$907$	−2.34847	−0.0779796	−0.0389898	−	0.999240i	$-0.512414\pi$
$907$	−2.34847	−0.0779796	−0.0389898	+	0.999240i	$0.512414\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	24.4949	0.811552	0.405776	−	0.913973i	$-0.367001\pi$
$911$	24.4949	0.811552	0.405776	+	0.913973i	$0.367001\pi$
$912$	0	0
$913$	25.3485	0.838912
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−37.5959	−1.24153
$918$	0	0
$919$	36.8990	1.21719	0.608593	−	0.793483i	$-0.291734\pi$
$919$	36.8990	1.21719	0.608593	+	0.793483i	$0.291734\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	13.3485	0.439370
$924$	0	0
$925$	6.34847	0.208736
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−27.0000	−0.885841	−0.442921	−	0.896561i	$-0.646058\pi$
$929$	−27.0000	−0.885841	−0.442921	+	0.896561i	$0.646058\pi$
$930$	0	0
$931$	−7.59592	−0.248946
$932$	0	0
$933$	0	0
$934$	0	0
$935$	13.3485	0.436542
$936$	0	0
$937$	28.6969	0.937488	0.468744	−	0.883334i	$-0.344707\pi$
$937$	28.6969	0.937488	0.468744	+	0.883334i	$0.344707\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	32.4495	1.05782	0.528912	−	0.848677i	$-0.322600\pi$
$941$	32.4495	1.05782	0.528912	+	0.848677i	$0.322600\pi$
$942$	0	0
$943$	7.89898	0.257226
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36.4949	1.18592	0.592962	−	0.805230i	$-0.297958\pi$
$947$	36.4949	1.18592	0.592962	+	0.805230i	$0.297958\pi$
$948$	0	0
$949$	41.5959	1.35026
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−32.2020	−1.04313	−0.521563	−	0.853213i	$-0.674651\pi$
$953$	−32.2020	−1.04313	−0.521563	+	0.853213i	$0.674651\pi$
$954$	0	0
$955$	−15.5505	−0.503203
$956$	0	0
$957$	0	0
$958$	0	0
$959$	58.2929	1.88237
$960$	0	0
$961$	18.0000	0.580645
$962$	0	0
$963$	0	0
$964$	0	0
$965$	16.0000	0.515058
$966$	0	0
$967$	−8.65153	−0.278214	−0.139107	−	0.990277i	$-0.544423\pi$
$967$	−8.65153	−0.278214	−0.139107	+	0.990277i	$0.544423\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−12.4949	−0.400980	−0.200490	−	0.979696i	$-0.564253\pi$
$971$	−12.4949	−0.400980	−0.200490	+	0.979696i	$0.564253\pi$
$972$	0	0
$973$	74.8434	2.39937
$974$	0	0
$975$	0	0
$976$	0	0
$977$	40.3485	1.29086	0.645431	−	0.763819i	$-0.276678\pi$
$977$	40.3485	1.29086	0.645431	+	0.763819i	$0.276678\pi$
$978$	0	0
$979$	41.3939	1.32295
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−16.3485	−0.521435	−0.260718	−	0.965415i	$-0.583959\pi$
$983$	−16.3485	−0.521435	−0.260718	+	0.965415i	$0.583959\pi$
$984$	0	0
$985$	15.7980	0.503365
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.89898	−0.282971
$990$	0	0
$991$	30.1010	0.956190	0.478095	−	0.878308i	$-0.341327\pi$
$991$	30.1010	0.956190	0.478095	+	0.878308i	$0.341327\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−11.7980	−0.374020
$996$	0	0
$997$	32.0000	1.01345	0.506725	−	0.862108i	$-0.330856\pi$
$997$	32.0000	1.01345	0.506725	+	0.862108i	$0.330856\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.a.l.1.1	✓	2
3.2	odd	2		4140.2.a.q.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.a.l.1.1	✓	2	1.1	even	1	trivial
4140.2.a.q.1.1	yes	2	3.2	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -3.44949 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -3.44949 q^{7} -2.44949 q^{11} +4.44949 q^{13} +5.44949 q^{17} -1.55051 q^{19} +1.00000 q^{23} +1.00000 q^{25} +1.89898 q^{29} -7.00000 q^{31} +3.44949 q^{35} +6.34847 q^{37} +7.89898 q^{41} -8.89898 q^{43} -2.44949 q^{47} +4.89898 q^{49} -4.34847 q^{53} +2.44949 q^{55} -1.89898 q^{59} -5.34847 q^{61} -4.44949 q^{65} +1.44949 q^{67} +3.00000 q^{71} +9.34847 q^{73} +8.44949 q^{77} -4.00000 q^{79} -10.3485 q^{83} -5.44949 q^{85} -16.8990 q^{89} -15.3485 q^{91} +1.55051 q^{95} -14.8990 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 - 2 * q^7	\( 2 q - 2 q^{5} - 2 q^{7} + 4 q^{13} + 6 q^{17} - 8 q^{19} + 2 q^{23} + 2 q^{25} - 6 q^{29} - 14 q^{31} + 2 q^{35} - 2 q^{37} + 6 q^{41} - 8 q^{43} + 6 q^{53} + 6 q^{59} + 4 q^{61} - 4 q^{65} - 2 q^{67} + 6 q^{71} + 4 q^{73} + 12 q^{77} - 8 q^{79} - 6 q^{83} - 6 q^{85} - 24 q^{89} - 16 q^{91} + 8 q^{95} - 20 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^7 + 4 * q^13 + 6 * q^17 - 8 * q^19 + 2 * q^23 + 2 * q^25 - 6 * q^29 - 14 * q^31 + 2 * q^35 - 2 * q^37 + 6 * q^41 - 8 * q^43 + 6 * q^53 + 6 * q^59 + 4 * q^61 - 4 * q^65 - 2 * q^67 + 6 * q^71 + 4 * q^73 + 12 * q^77 - 8 * q^79 - 6 * q^83 - 6 * q^85 - 24 * q^89 - 16 * q^91 + 8 * q^95 - 20 * q^97

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