LMFDB - Embedded newform 3432.2.a.x.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3432,2,Mod(1,3432)] 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("3432.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3432, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3432.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,5,0,1,0,-3,0,5,0,5,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$27.4046579737$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.46437524.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 20x^{3} + 8x^{2} + 70x + 28 $ x^5 - x^4 - 20x^3 + 8x^2 + 70*x + 28
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.447720$ of defining polynomial
Character	$\chi$	$=$	3432.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -0.447720 q^{5} +3.31638 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} -0.447720 q^{15} +4.86866 q^{17} +3.83157 q^{19} +3.31638 q^{21} -5.31638 q^{23} -4.79955 q^{25} +1.00000 q^{27} +8.83664 q^{29} +1.93253 q^{31} +1.00000 q^{33} -1.48481 q^{35} -6.63276 q^{37} +1.00000 q^{39} -1.31638 q^{41} +7.24891 q^{43} -0.447720 q^{45} +4.48317 q^{47} +3.99836 q^{49} +4.86866 q^{51} -11.1159 q^{53} -0.447720 q^{55} +3.83157 q^{57} -5.63112 q^{59} +0.380248 q^{61} +3.31638 q^{63} -0.447720 q^{65} +2.44772 q^{67} -5.31638 q^{69} -7.25087 q^{71} +11.7995 q^{73} -4.79955 q^{75} +3.31638 q^{77} +13.9104 q^{79} +1.00000 q^{81} +0.895440 q^{83} -2.17979 q^{85} +8.83664 q^{87} -9.46072 q^{89} +3.31638 q^{91} +1.93253 q^{93} -1.71547 q^{95} -4.22048 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} + q^{5} - 3 q^{7} + 5 q^{9} + 5 q^{11} + 5 q^{13} + q^{15} + 8 q^{17} + 10 q^{19} - 3 q^{21} - 7 q^{23} + 16 q^{25} + 5 q^{27} - 3 q^{29} - 4 q^{31} + 5 q^{33} + 3 q^{35} + 6 q^{37} + 5 q^{39}+ \cdots + 5 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 + q^5 - 3 * q^7 + 5 * q^9 + 5 * q^11 + 5 * q^13 + q^15 + 8 * q^17 + 10 * q^19 - 3 * q^21 - 7 * q^23 + 16 * q^25 + 5 * q^27 - 3 * q^29 - 4 * q^31 + 5 * q^33 + 3 * q^35 + 6 * q^37 + 5 * q^39 + 13 * q^41 + 3 * q^43 + q^45 + 2 * q^47 + 10 * q^49 + 8 * q^51 + 4 * q^53 + q^55 + 10 * q^57 + 21 * q^59 - 15 * q^61 - 3 * q^63 + q^65 + 9 * q^67 - 7 * q^69 - 4 * q^71 + 19 * q^73 + 16 * q^75 - 3 * q^77 + 8 * q^79 + 5 * q^81 - 2 * q^83 + 46 * q^85 - 3 * q^87 + 12 * q^89 - 3 * q^91 - 4 * q^93 + 32 * q^97 + 5 * q^99

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−0.447720	−0.200226	−0.100113	−	0.994976i	$-0.531920\pi$
$5$	−0.447720	−0.200226	−0.100113	+	0.994976i	$0.531920\pi$
$6$	0	0
$7$	3.31638	1.25347	0.626736	−	0.779231i	$-0.284390\pi$
$7$	3.31638	1.25347	0.626736	+	0.779231i	$0.284390\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−0.447720	−0.115601
$16$	0	0
$17$	4.86866	1.18082	0.590411	−	0.807102i	$-0.298965\pi$
$17$	4.86866	1.18082	0.590411	+	0.807102i	$0.298965\pi$
$18$	0	0
$19$	3.83157	0.879022	0.439511	−	0.898237i	$-0.355152\pi$
$19$	3.83157	0.879022	0.439511	+	0.898237i	$0.355152\pi$
$20$	0	0
$21$	3.31638	0.723693
$22$	0	0
$23$	−5.31638	−1.10854	−0.554271	−	0.832336i	$-0.687003\pi$
$23$	−5.31638	−1.10854	−0.554271	+	0.832336i	$0.687003\pi$
$24$	0	0
$25$	−4.79955	−0.959909
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	8.83664	1.64092	0.820461	−	0.571703i	$-0.193717\pi$
$29$	8.83664	1.64092	0.820461	+	0.571703i	$0.193717\pi$
$30$	0	0
$31$	1.93253	0.347092	0.173546	−	0.984826i	$-0.444477\pi$
$31$	1.93253	0.347092	0.173546	+	0.984826i	$0.444477\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−1.48481	−0.250978
$36$	0	0
$37$	−6.63276	−1.09042	−0.545209	−	0.838300i	$-0.683550\pi$
$37$	−6.63276	−1.09042	−0.545209	+	0.838300i	$0.683550\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−1.31638	−0.205584	−0.102792	−	0.994703i	$-0.532778\pi$
$41$	−1.31638	−0.205584	−0.102792	+	0.994703i	$0.532778\pi$
$42$	0	0
$43$	7.24891	1.10545	0.552724	−	0.833364i	$-0.313588\pi$
$43$	7.24891	1.10545	0.552724	+	0.833364i	$0.313588\pi$
$44$	0	0
$45$	−0.447720	−0.0667421
$46$	0	0
$47$	4.48317	0.653937	0.326969	−	0.945035i	$-0.393973\pi$
$47$	4.48317	0.653937	0.326969	+	0.945035i	$0.393973\pi$
$48$	0	0
$49$	3.99836	0.571194
$50$	0	0
$51$	4.86866	0.681748
$52$	0	0
$53$	−11.1159	−1.52689	−0.763445	−	0.645873i	$-0.776494\pi$
$53$	−11.1159	−1.52689	−0.763445	+	0.645873i	$0.776494\pi$
$54$	0	0
$55$	−0.447720	−0.0603705
$56$	0	0
$57$	3.83157	0.507504
$58$	0	0
$59$	−5.63112	−0.733109	−0.366554	−	0.930397i	$-0.619463\pi$
$59$	−5.63112	−0.733109	−0.366554	+	0.930397i	$0.619463\pi$
$60$	0	0
$61$	0.380248	0.0486857	0.0243429	−	0.999704i	$-0.492251\pi$
$61$	0.380248	0.0486857	0.0243429	+	0.999704i	$0.492251\pi$
$62$	0	0
$63$	3.31638	0.417824
$64$	0	0
$65$	−0.447720	−0.0555328
$66$	0	0
$67$	2.44772	0.299037	0.149518	−	0.988759i	$-0.452228\pi$
$67$	2.44772	0.299037	0.149518	+	0.988759i	$0.452228\pi$
$68$	0	0
$69$	−5.31638	−0.640017
$70$	0	0
$71$	−7.25087	−0.860520	−0.430260	−	0.902705i	$-0.641578\pi$
$71$	−7.25087	−0.860520	−0.430260	+	0.902705i	$0.641578\pi$
$72$	0	0
$73$	11.7995	1.38103	0.690516	−	0.723317i	$-0.257383\pi$
$73$	11.7995	1.38103	0.690516	+	0.723317i	$0.257383\pi$
$74$	0	0
$75$	−4.79955	−0.554204
$76$	0	0
$77$	3.31638	0.377936
$78$	0	0
$79$	13.9104	1.56504	0.782521	−	0.622624i	$-0.213933\pi$
$79$	13.9104	1.56504	0.782521	+	0.622624i	$0.213933\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.895440	0.0982873	0.0491436	−	0.998792i	$-0.484351\pi$
$83$	0.895440	0.0982873	0.0491436	+	0.998792i	$0.484351\pi$
$84$	0	0
$85$	−2.17979	−0.236432
$86$	0	0
$87$	8.83664	0.947387
$88$	0	0
$89$	−9.46072	−1.00283	−0.501417	−	0.865206i	$-0.667188\pi$
$89$	−9.46072	−1.00283	−0.501417	+	0.865206i	$0.667188\pi$
$90$	0	0
$91$	3.31638	0.347651
$92$	0	0
$93$	1.93253	0.200394
$94$	0	0
$95$	−1.71547	−0.176003
$96$	0	0
$97$	−4.22048	−0.428525	−0.214263	−	0.976776i	$-0.568735\pi$
$97$	−4.22048	−0.428525	−0.214263	+	0.976776i	$0.568735\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−14.8800	−1.48062	−0.740309	−	0.672267i	$-0.765321\pi$
$101$	−14.8800	−1.48062	−0.740309	+	0.672267i	$0.765321\pi$
$102$	0	0
$103$	8.73568	0.860752	0.430376	−	0.902650i	$-0.358381\pi$
$103$	8.73568	0.860752	0.430376	+	0.902650i	$0.358381\pi$
$104$	0	0
$105$	−1.48481	−0.144902
$106$	0	0
$107$	−2.25251	−0.217758	−0.108879	−	0.994055i	$-0.534726\pi$
$107$	−2.25251	−0.217758	−0.108879	+	0.994055i	$0.534726\pi$
$108$	0	0
$109$	11.2844	1.08085	0.540423	−	0.841394i	$-0.318264\pi$
$109$	11.2844	1.08085	0.540423	+	0.841394i	$0.318264\pi$
$110$	0	0
$111$	−6.63276	−0.629553
$112$	0	0
$113$	15.6311	1.47045	0.735226	−	0.677822i	$-0.237076\pi$
$113$	15.6311	1.47045	0.735226	+	0.677822i	$0.237076\pi$
$114$	0	0
$115$	2.38025	0.221959
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	16.1463	1.48013
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−1.31638	−0.118694
$124$	0	0
$125$	4.38745	0.392426
$126$	0	0
$127$	−15.5014	−1.37553	−0.687764	−	0.725934i	$-0.741408\pi$
$127$	−15.5014	−1.37553	−0.687764	+	0.725934i	$0.741408\pi$
$128$	0	0
$129$	7.24891	0.638231
$130$	0	0
$131$	2.30607	0.201482	0.100741	−	0.994913i	$-0.467879\pi$
$131$	2.30607	0.201482	0.100741	+	0.994913i	$0.467879\pi$
$132$	0	0
$133$	12.7069	1.10183
$134$	0	0
$135$	−0.447720	−0.0385336
$136$	0	0
$137$	−0.700227	−0.0598245	−0.0299122	−	0.999553i	$-0.509523\pi$
$137$	−0.700227	−0.0598245	−0.0299122	+	0.999553i	$0.509523\pi$
$138$	0	0
$139$	−13.8350	−1.17347	−0.586735	−	0.809779i	$-0.699587\pi$
$139$	−13.8350	−1.17347	−0.586735	+	0.809779i	$0.699587\pi$
$140$	0	0
$141$	4.48317	0.377551
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−3.95634	−0.328556
$146$	0	0
$147$	3.99836	0.329779
$148$	0	0
$149$	18.4757	1.51359	0.756794	−	0.653654i	$-0.226765\pi$
$149$	18.4757	1.51359	0.756794	+	0.653654i	$0.226765\pi$
$150$	0	0
$151$	−8.05205	−0.655267	−0.327633	−	0.944805i	$-0.606251\pi$
$151$	−8.05205	−0.655267	−0.327633	+	0.944805i	$0.606251\pi$
$152$	0	0
$153$	4.86866	0.393608
$154$	0	0
$155$	−0.865231	−0.0694970
$156$	0	0
$157$	19.3598	1.54508	0.772539	−	0.634968i	$-0.218987\pi$
$157$	19.3598	1.54508	0.772539	+	0.634968i	$0.218987\pi$
$158$	0	0
$159$	−11.1159	−0.881550
$160$	0	0
$161$	−17.6311	−1.38953
$162$	0	0
$163$	14.0354	1.09934	0.549671	−	0.835381i	$-0.314753\pi$
$163$	14.0354	1.09934	0.549671	+	0.835381i	$0.314753\pi$
$164$	0	0
$165$	−0.447720	−0.0348549
$166$	0	0
$167$	−16.3988	−1.26898	−0.634489	−	0.772932i	$-0.718790\pi$
$167$	−16.3988	−1.26898	−0.634489	+	0.772932i	$0.718790\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	3.83157	0.293007
$172$	0	0
$173$	−2.75109	−0.209162	−0.104581	−	0.994516i	$-0.533350\pi$
$173$	−2.75109	−0.209162	−0.104581	+	0.994516i	$0.533350\pi$
$174$	0	0
$175$	−15.9171	−1.20322
$176$	0	0
$177$	−5.63112	−0.423261
$178$	0	0
$179$	−13.3598	−0.998556	−0.499278	−	0.866442i	$-0.666401\pi$
$179$	−13.3598	−0.998556	−0.499278	+	0.866442i	$0.666401\pi$
$180$	0	0
$181$	23.0229	1.71128	0.855639	−	0.517572i	$-0.173164\pi$
$181$	23.0229	1.71128	0.855639	+	0.517572i	$0.173164\pi$
$182$	0	0
$183$	0.380248	0.0281087
$184$	0	0
$185$	2.96962	0.218331
$186$	0	0
$187$	4.86866	0.356032
$188$	0	0
$189$	3.31638	0.241231
$190$	0	0
$191$	5.22212	0.377860	0.188930	−	0.981991i	$-0.439498\pi$
$191$	5.22212	0.377860	0.188930	+	0.981991i	$0.439498\pi$
$192$	0	0
$193$	14.7270	1.06007	0.530037	−	0.847975i	$-0.322178\pi$
$193$	14.7270	1.06007	0.530037	+	0.847975i	$0.322178\pi$
$194$	0	0
$195$	−0.447720	−0.0320619
$196$	0	0
$197$	−13.4935	−0.961370	−0.480685	−	0.876893i	$-0.659612\pi$
$197$	−13.4935	−0.961370	−0.480685	+	0.876893i	$0.659612\pi$
$198$	0	0
$199$	−14.5266	−1.02976	−0.514880	−	0.857262i	$-0.672164\pi$
$199$	−14.5266	−1.02976	−0.514880	+	0.857262i	$0.672164\pi$
$200$	0	0
$201$	2.44772	0.172649
$202$	0	0
$203$	29.3056	2.05685
$204$	0	0
$205$	0.589368	0.0411633
$206$	0	0
$207$	−5.31638	−0.369514
$208$	0	0
$209$	3.83157	0.265035
$210$	0	0
$211$	9.48677	0.653096	0.326548	−	0.945181i	$-0.394114\pi$
$211$	9.48677	0.653096	0.326548	+	0.945181i	$0.394114\pi$
$212$	0	0
$213$	−7.25087	−0.496821
$214$	0	0
$215$	−3.24548	−0.221340
$216$	0	0
$217$	6.40899	0.435071
$218$	0	0
$219$	11.7995	0.797339
$220$	0	0
$221$	4.86866	0.327501
$222$	0	0
$223$	−14.8989	−0.997702	−0.498851	−	0.866688i	$-0.666244\pi$
$223$	−14.8989	−0.997702	−0.498851	+	0.866688i	$0.666244\pi$
$224$	0	0
$225$	−4.79955	−0.319970
$226$	0	0
$227$	21.6544	1.43725	0.718627	−	0.695395i	$-0.244771\pi$
$227$	21.6544	1.43725	0.718627	+	0.695395i	$0.244771\pi$
$228$	0	0
$229$	−21.1057	−1.39471	−0.697354	−	0.716727i	$-0.745639\pi$
$229$	−21.1057	−1.39471	−0.697354	+	0.716727i	$0.745639\pi$
$230$	0	0
$231$	3.31638	0.218202
$232$	0	0
$233$	12.1195	0.793977	0.396988	−	0.917824i	$-0.370055\pi$
$233$	12.1195	0.793977	0.396988	+	0.917824i	$0.370055\pi$
$234$	0	0
$235$	−2.00720	−0.130936
$236$	0	0
$237$	13.9104	0.903578
$238$	0	0
$239$	9.61227	0.621766	0.310883	−	0.950448i	$-0.399375\pi$
$239$	9.61227	0.621766	0.310883	+	0.950448i	$0.399375\pi$
$240$	0	0
$241$	19.1159	1.23136	0.615682	−	0.787994i	$-0.288880\pi$
$241$	19.1159	1.23136	0.615682	+	0.787994i	$0.288880\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.79015	−0.114368
$246$	0	0
$247$	3.83157	0.243797
$248$	0	0
$249$	0.895440	0.0567462
$250$	0	0
$251$	−20.3701	−1.28575	−0.642874	−	0.765972i	$-0.722258\pi$
$251$	−20.3701	−1.28575	−0.642874	+	0.765972i	$0.722258\pi$
$252$	0	0
$253$	−5.31638	−0.334238
$254$	0	0
$255$	−2.17979	−0.136504
$256$	0	0
$257$	−19.1593	−1.19513	−0.597563	−	0.801822i	$-0.703864\pi$
$257$	−19.1593	−1.19513	−0.597563	+	0.801822i	$0.703864\pi$
$258$	0	0
$259$	−21.9967	−1.36681
$260$	0	0
$261$	8.83664	0.546974
$262$	0	0
$263$	−15.0405	−0.927438	−0.463719	−	0.885982i	$-0.653485\pi$
$263$	−15.0405	−0.927438	−0.463719	+	0.885982i	$0.653485\pi$
$264$	0	0
$265$	4.97682	0.305724
$266$	0	0
$267$	−9.46072	−0.578987
$268$	0	0
$269$	17.1192	1.04378	0.521888	−	0.853014i	$-0.325228\pi$
$269$	17.1192	1.04378	0.521888	+	0.853014i	$0.325228\pi$
$270$	0	0
$271$	4.18700	0.254342	0.127171	−	0.991881i	$-0.459410\pi$
$271$	4.18700	0.254342	0.127171	+	0.991881i	$0.459410\pi$
$272$	0	0
$273$	3.31638	0.200716
$274$	0	0
$275$	−4.79955	−0.289424
$276$	0	0
$277$	14.3988	0.865141	0.432571	−	0.901600i	$-0.357607\pi$
$277$	14.3988	0.865141	0.432571	+	0.901600i	$0.357607\pi$
$278$	0	0
$279$	1.93253	0.115697
$280$	0	0
$281$	−20.7136	−1.23567	−0.617834	−	0.786309i	$-0.711989\pi$
$281$	−20.7136	−1.23567	−0.617834	+	0.786309i	$0.711989\pi$
$282$	0	0
$283$	16.9976	1.01040	0.505201	−	0.863002i	$-0.331418\pi$
$283$	16.9976	1.01040	0.505201	+	0.863002i	$0.331418\pi$
$284$	0	0
$285$	−1.71547	−0.101616
$286$	0	0
$287$	−4.36561	−0.257694
$288$	0	0
$289$	6.70383	0.394343
$290$	0	0
$291$	−4.22048	−0.247409
$292$	0	0
$293$	2.18700	0.127766	0.0638829	−	0.997957i	$-0.479652\pi$
$293$	2.18700	0.127766	0.0638829	+	0.997957i	$0.479652\pi$
$294$	0	0
$295$	2.52116	0.146788
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−5.31638	−0.307454
$300$	0	0
$301$	24.0401	1.38565
$302$	0	0
$303$	−14.8800	−0.854835
$304$	0	0
$305$	−0.170244	−0.00974817
$306$	0	0
$307$	−7.30620	−0.416987	−0.208493	−	0.978024i	$-0.566856\pi$
$307$	−7.30620	−0.416987	−0.208493	+	0.978024i	$0.566856\pi$
$308$	0	0
$309$	8.73568	0.496955
$310$	0	0
$311$	−19.0011	−1.07745	−0.538726	−	0.842481i	$-0.681094\pi$
$311$	−19.0011	−1.07745	−0.538726	+	0.842481i	$0.681094\pi$
$312$	0	0
$313$	24.2974	1.37337	0.686684	−	0.726956i	$-0.259066\pi$
$313$	24.2974	1.37337	0.686684	+	0.726956i	$0.259066\pi$
$314$	0	0
$315$	−1.48481	−0.0836595
$316$	0	0
$317$	23.2977	1.30853	0.654264	−	0.756266i	$-0.272978\pi$
$317$	23.2977	1.30853	0.654264	+	0.756266i	$0.272978\pi$
$318$	0	0
$319$	8.83664	0.494757
$320$	0	0
$321$	−2.25251	−0.125723
$322$	0	0
$323$	18.6546	1.03797
$324$	0	0
$325$	−4.79955	−0.266231
$326$	0	0
$327$	11.2844	0.624026
$328$	0	0
$329$	14.8679	0.819693
$330$	0	0
$331$	8.66820	0.476448	0.238224	−	0.971210i	$-0.423435\pi$
$331$	8.66820	0.476448	0.238224	+	0.971210i	$0.423435\pi$
$332$	0	0
$333$	−6.63276	−0.363473
$334$	0	0
$335$	−1.09589	−0.0598750
$336$	0	0
$337$	−8.36679	−0.455768	−0.227884	−	0.973688i	$-0.573181\pi$
$337$	−8.36679	−0.455768	−0.227884	+	0.973688i	$0.573181\pi$
$338$	0	0
$339$	15.6311	0.848966
$340$	0	0
$341$	1.93253	0.104652
$342$	0	0
$343$	−9.95457	−0.537496
$344$	0	0
$345$	2.38025	0.128148
$346$	0	0
$347$	8.29589	0.445347	0.222673	−	0.974893i	$-0.428522\pi$
$347$	8.29589	0.445347	0.222673	+	0.974893i	$0.428522\pi$
$348$	0	0
$349$	−18.9995	−1.01702	−0.508511	−	0.861055i	$-0.669804\pi$
$349$	−18.9995	−1.01702	−0.508511	+	0.861055i	$0.669804\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−22.3140	−1.18765	−0.593826	−	0.804593i	$-0.702383\pi$
$353$	−22.3140	−1.18765	−0.593826	+	0.804593i	$0.702383\pi$
$354$	0	0
$355$	3.24636	0.172299
$356$	0	0
$357$	16.1463	0.854553
$358$	0	0
$359$	−22.1376	−1.16838	−0.584190	−	0.811617i	$-0.698588\pi$
$359$	−22.1376	−1.16838	−0.584190	+	0.811617i	$0.698588\pi$
$360$	0	0
$361$	−4.31907	−0.227320
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−5.28289	−0.276519
$366$	0	0
$367$	−30.5846	−1.59650	−0.798251	−	0.602325i	$-0.794241\pi$
$367$	−30.5846	−1.59650	−0.798251	+	0.602325i	$0.794241\pi$
$368$	0	0
$369$	−1.31638	−0.0685279
$370$	0	0
$371$	−36.8646	−1.91391
$372$	0	0
$373$	−9.42200	−0.487852	−0.243926	−	0.969794i	$-0.578435\pi$
$373$	−9.42200	−0.487852	−0.243926	+	0.969794i	$0.578435\pi$
$374$	0	0
$375$	4.38745	0.226567
$376$	0	0
$377$	8.83664	0.455110
$378$	0	0
$379$	23.9657	1.23104	0.615519	−	0.788122i	$-0.288947\pi$
$379$	23.9657	1.23104	0.615519	+	0.788122i	$0.288947\pi$
$380$	0	0
$381$	−15.5014	−0.794161
$382$	0	0
$383$	3.02318	0.154477	0.0772386	−	0.997013i	$-0.475390\pi$
$383$	3.02318	0.154477	0.0772386	+	0.997013i	$0.475390\pi$
$384$	0	0
$385$	−1.48481	−0.0756728
$386$	0	0
$387$	7.24891	0.368483
$388$	0	0
$389$	22.5092	1.14126	0.570630	−	0.821207i	$-0.306699\pi$
$389$	22.5092	1.14126	0.570630	+	0.821207i	$0.306699\pi$
$390$	0	0
$391$	−25.8836	−1.30899
$392$	0	0
$393$	2.30607	0.116326
$394$	0	0
$395$	−6.22796	−0.313363
$396$	0	0
$397$	−18.9502	−0.951083	−0.475541	−	0.879693i	$-0.657748\pi$
$397$	−18.9502	−0.951083	−0.475541	+	0.879693i	$0.657748\pi$
$398$	0	0
$399$	12.7069	0.636142
$400$	0	0
$401$	31.8162	1.58882	0.794411	−	0.607380i	$-0.207779\pi$
$401$	31.8162	1.58882	0.794411	+	0.607380i	$0.207779\pi$
$402$	0	0
$403$	1.93253	0.0962661
$404$	0	0
$405$	−0.447720	−0.0222474
$406$	0	0
$407$	−6.63276	−0.328773
$408$	0	0
$409$	18.7692	0.928075	0.464038	−	0.885815i	$-0.346400\pi$
$409$	18.7692	0.928075	0.464038	+	0.885815i	$0.346400\pi$
$410$	0	0
$411$	−0.700227	−0.0345397
$412$	0	0
$413$	−18.6749	−0.918932
$414$	0	0
$415$	−0.400906	−0.0196797
$416$	0	0
$417$	−13.8350	−0.677503
$418$	0	0
$419$	−25.8007	−1.26045	−0.630224	−	0.776413i	$-0.717037\pi$
$419$	−25.8007	−1.26045	−0.630224	+	0.776413i	$0.717037\pi$
$420$	0	0
$421$	−3.27569	−0.159647	−0.0798236	−	0.996809i	$-0.525436\pi$
$421$	−3.27569	−0.159647	−0.0798236	+	0.996809i	$0.525436\pi$
$422$	0	0
$423$	4.48317	0.217979
$424$	0	0
$425$	−23.3674	−1.13348
$426$	0	0
$427$	1.26105	0.0610263
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	16.7778	0.808160	0.404080	−	0.914724i	$-0.367592\pi$
$431$	16.7778	0.808160	0.404080	+	0.914724i	$0.367592\pi$
$432$	0	0
$433$	−41.2667	−1.98315	−0.991575	−	0.129530i	$-0.958653\pi$
$433$	−41.2667	−1.98315	−0.991575	+	0.129530i	$0.958653\pi$
$434$	0	0
$435$	−3.95634	−0.189692
$436$	0	0
$437$	−20.3701	−0.974433
$438$	0	0
$439$	−33.5161	−1.59963	−0.799817	−	0.600244i	$-0.795070\pi$
$439$	−33.5161	−1.59963	−0.799817	+	0.600244i	$0.795070\pi$
$440$	0	0
$441$	3.99836	0.190398
$442$	0	0
$443$	−36.4410	−1.73136	−0.865681	−	0.500595i	$-0.833114\pi$
$443$	−36.4410	−1.73136	−0.865681	+	0.500595i	$0.833114\pi$
$444$	0	0
$445$	4.23575	0.200794
$446$	0	0
$447$	18.4757	0.873870
$448$	0	0
$449$	−1.01524	−0.0479122	−0.0239561	−	0.999713i	$-0.507626\pi$
$449$	−1.01524	−0.0479122	−0.0239561	+	0.999713i	$0.507626\pi$
$450$	0	0
$451$	−1.31638	−0.0619858
$452$	0	0
$453$	−8.05205	−0.378319
$454$	0	0
$455$	−1.48481	−0.0696089
$456$	0	0
$457$	−3.69393	−0.172795	−0.0863973	−	0.996261i	$-0.527535\pi$
$457$	−3.69393	−0.172795	−0.0863973	+	0.996261i	$0.527535\pi$
$458$	0	0
$459$	4.86866	0.227249
$460$	0	0
$461$	−19.2308	−0.895667	−0.447834	−	0.894117i	$-0.647804\pi$
$461$	−19.2308	−0.895667	−0.447834	+	0.894117i	$0.647804\pi$
$462$	0	0
$463$	−7.59443	−0.352943	−0.176472	−	0.984306i	$-0.556468\pi$
$463$	−7.59443	−0.352943	−0.176472	+	0.984306i	$0.556468\pi$
$464$	0	0
$465$	−0.865231	−0.0401241
$466$	0	0
$467$	−24.4003	−1.12911	−0.564555	−	0.825395i	$-0.690952\pi$
$467$	−24.4003	−1.12911	−0.564555	+	0.825395i	$0.690952\pi$
$468$	0	0
$469$	8.11756	0.374834
$470$	0	0
$471$	19.3598	0.892051
$472$	0	0
$473$	7.24891	0.333305
$474$	0	0
$475$	−18.3898	−0.843782
$476$	0	0
$477$	−11.1159	−0.508963
$478$	0	0
$479$	18.0359	0.824082	0.412041	−	0.911165i	$-0.364816\pi$
$479$	18.0359	0.824082	0.412041	+	0.911165i	$0.364816\pi$
$480$	0	0
$481$	−6.63276	−0.302428
$482$	0	0
$483$	−17.6311	−0.802244
$484$	0	0
$485$	1.88959	0.0858021
$486$	0	0
$487$	−32.3881	−1.46765	−0.733823	−	0.679340i	$-0.762266\pi$
$487$	−32.3881	−1.46765	−0.733823	+	0.679340i	$0.762266\pi$
$488$	0	0
$489$	14.0354	0.634705
$490$	0	0
$491$	17.3044	0.780936	0.390468	−	0.920616i	$-0.372313\pi$
$491$	17.3044	0.780936	0.390468	+	0.920616i	$0.372313\pi$
$492$	0	0
$493$	43.0226	1.93764
$494$	0	0
$495$	−0.447720	−0.0201235
$496$	0	0
$497$	−24.0466	−1.07864
$498$	0	0
$499$	−18.1243	−0.811354	−0.405677	−	0.914017i	$-0.632964\pi$
$499$	−18.1243	−0.811354	−0.405677	+	0.914017i	$0.632964\pi$
$500$	0	0
$501$	−16.3988	−0.732645
$502$	0	0
$503$	−9.65594	−0.430537	−0.215268	−	0.976555i	$-0.569063\pi$
$503$	−9.65594	−0.430537	−0.215268	+	0.976555i	$0.569063\pi$
$504$	0	0
$505$	6.66208	0.296459
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	10.0729	0.446472	0.223236	−	0.974764i	$-0.428338\pi$
$509$	10.0729	0.446472	0.223236	+	0.974764i	$0.428338\pi$
$510$	0	0
$511$	39.1318	1.73109
$512$	0	0
$513$	3.83157	0.169168
$514$	0	0
$515$	−3.91114	−0.172345
$516$	0	0
$517$	4.48317	0.197170
$518$	0	0
$519$	−2.75109	−0.120760
$520$	0	0
$521$	15.7020	0.687918	0.343959	−	0.938985i	$-0.388232\pi$
$521$	15.7020	0.687918	0.343959	+	0.938985i	$0.388232\pi$
$522$	0	0
$523$	3.42600	0.149809	0.0749043	−	0.997191i	$-0.476135\pi$
$523$	3.42600	0.149809	0.0749043	+	0.997191i	$0.476135\pi$
$524$	0	0
$525$	−15.9171	−0.694680
$526$	0	0
$527$	9.40882	0.409855
$528$	0	0
$529$	5.26387	0.228864
$530$	0	0
$531$	−5.63112	−0.244370
$532$	0	0
$533$	−1.31638	−0.0570186
$534$	0	0
$535$	1.00849	0.0436009
$536$	0	0
$537$	−13.3598	−0.576516
$538$	0	0
$539$	3.99836	0.172222
$540$	0	0
$541$	0.734489	0.0315782	0.0157891	−	0.999875i	$-0.494974\pi$
$541$	0.734489	0.0315782	0.0157891	+	0.999875i	$0.494974\pi$
$542$	0	0
$543$	23.0229	0.988007
$544$	0	0
$545$	−5.05223	−0.216414
$546$	0	0
$547$	13.1861	0.563797	0.281898	−	0.959444i	$-0.409036\pi$
$547$	13.1861	0.563797	0.281898	+	0.959444i	$0.409036\pi$
$548$	0	0
$549$	0.380248	0.0162286
$550$	0	0
$551$	33.8582	1.44241
$552$	0	0
$553$	46.1322	1.96174
$554$	0	0
$555$	2.96962	0.126053
$556$	0	0
$557$	18.3889	0.779163	0.389582	−	0.920992i	$-0.372620\pi$
$557$	18.3889	0.779163	0.389582	+	0.920992i	$0.372620\pi$
$558$	0	0
$559$	7.24891	0.306596
$560$	0	0
$561$	4.86866	0.205555
$562$	0	0
$563$	3.45999	0.145821	0.0729106	−	0.997338i	$-0.476771\pi$
$563$	3.45999	0.145821	0.0729106	+	0.997338i	$0.476771\pi$
$564$	0	0
$565$	−6.99836	−0.294423
$566$	0	0
$567$	3.31638	0.139275
$568$	0	0
$569$	−30.5955	−1.28263	−0.641315	−	0.767278i	$-0.721611\pi$
$569$	−30.5955	−1.28263	−0.641315	+	0.767278i	$0.721611\pi$
$570$	0	0
$571$	0.208391	0.00872088	0.00436044	−	0.999990i	$-0.498612\pi$
$571$	0.208391	0.00872088	0.00436044	+	0.999990i	$0.498612\pi$
$572$	0	0
$573$	5.22212	0.218157
$574$	0	0
$575$	25.5162	1.06410
$576$	0	0
$577$	7.36724	0.306702	0.153351	−	0.988172i	$-0.450993\pi$
$577$	7.36724	0.306702	0.153351	+	0.988172i	$0.450993\pi$
$578$	0	0
$579$	14.7270	0.612034
$580$	0	0
$581$	2.96962	0.123200
$582$	0	0
$583$	−11.1159	−0.460375
$584$	0	0
$585$	−0.447720	−0.0185109
$586$	0	0
$587$	22.3048	0.920619	0.460310	−	0.887758i	$-0.347738\pi$
$587$	22.3048	0.920619	0.460310	+	0.887758i	$0.347738\pi$
$588$	0	0
$589$	7.40462	0.305102
$590$	0	0
$591$	−13.4935	−0.555047
$592$	0	0
$593$	28.1490	1.15594	0.577970	−	0.816058i	$-0.303845\pi$
$593$	28.1490	1.15594	0.577970	+	0.816058i	$0.303845\pi$
$594$	0	0
$595$	−7.22902	−0.296361
$596$	0	0
$597$	−14.5266	−0.594532
$598$	0	0
$599$	−34.4876	−1.40913	−0.704563	−	0.709641i	$-0.748857\pi$
$599$	−34.4876	−1.40913	−0.704563	+	0.709641i	$0.748857\pi$
$600$	0	0
$601$	16.2959	0.664723	0.332362	−	0.943152i	$-0.392155\pi$
$601$	16.2959	0.664723	0.332362	+	0.943152i	$0.392155\pi$
$602$	0	0
$603$	2.44772	0.0996789
$604$	0	0
$605$	−0.447720	−0.0182024
$606$	0	0
$607$	6.43782	0.261303	0.130652	−	0.991428i	$-0.458293\pi$
$607$	6.43782	0.261303	0.130652	+	0.991428i	$0.458293\pi$
$608$	0	0
$609$	29.3056	1.18752
$610$	0	0
$611$	4.48317	0.181370
$612$	0	0
$613$	−1.07523	−0.0434283	−0.0217142	−	0.999764i	$-0.506912\pi$
$613$	−1.07523	−0.0434283	−0.0217142	+	0.999764i	$0.506912\pi$
$614$	0	0
$615$	0.589368	0.0237656
$616$	0	0
$617$	−25.9612	−1.04516	−0.522580	−	0.852590i	$-0.675030\pi$
$617$	−25.9612	−1.04516	−0.522580	+	0.852590i	$0.675030\pi$
$618$	0	0
$619$	−4.16319	−0.167333	−0.0836664	−	0.996494i	$-0.526663\pi$
$619$	−4.16319	−0.167333	−0.0836664	+	0.996494i	$0.526663\pi$
$620$	0	0
$621$	−5.31638	−0.213339
$622$	0	0
$623$	−31.3753	−1.25703
$624$	0	0
$625$	22.0334	0.881335
$626$	0	0
$627$	3.83157	0.153018
$628$	0	0
$629$	−32.2926	−1.28759
$630$	0	0
$631$	−17.8263	−0.709653	−0.354827	−	0.934932i	$-0.615460\pi$
$631$	−17.8263	−0.709653	−0.354827	+	0.934932i	$0.615460\pi$
$632$	0	0
$633$	9.48677	0.377065
$634$	0	0
$635$	6.94029	0.275417
$636$	0	0
$637$	3.99836	0.158421
$638$	0	0
$639$	−7.25087	−0.286840
$640$	0	0
$641$	14.9535	0.590627	0.295313	−	0.955400i	$-0.404576\pi$
$641$	14.9535	0.590627	0.295313	+	0.955400i	$0.404576\pi$
$642$	0	0
$643$	43.6402	1.72100	0.860502	−	0.509448i	$-0.170150\pi$
$643$	43.6402	1.72100	0.860502	+	0.509448i	$0.170150\pi$
$644$	0	0
$645$	−3.24548	−0.127791
$646$	0	0
$647$	−35.0109	−1.37642	−0.688211	−	0.725511i	$-0.741603\pi$
$647$	−35.0109	−1.37642	−0.688211	+	0.725511i	$0.741603\pi$
$648$	0	0
$649$	−5.63112	−0.221041
$650$	0	0
$651$	6.40899	0.251188
$652$	0	0
$653$	−30.7255	−1.20238	−0.601191	−	0.799105i	$-0.705307\pi$
$653$	−30.7255	−1.20238	−0.601191	+	0.799105i	$0.705307\pi$
$654$	0	0
$655$	−1.03247	−0.0403421
$656$	0	0
$657$	11.7995	0.460344
$658$	0	0
$659$	−4.76770	−0.185723	−0.0928616	−	0.995679i	$-0.529601\pi$
$659$	−4.76770	−0.185723	−0.0928616	+	0.995679i	$0.529601\pi$
$660$	0	0
$661$	26.2914	1.02262	0.511308	−	0.859397i	$-0.329161\pi$
$661$	26.2914	1.02262	0.511308	+	0.859397i	$0.329161\pi$
$662$	0	0
$663$	4.86866	0.189083
$664$	0	0
$665$	−5.68915	−0.220616
$666$	0	0
$667$	−46.9789	−1.81903
$668$	0	0
$669$	−14.8989	−0.576023
$670$	0	0
$671$	0.380248	0.0146793
$672$	0	0
$673$	−34.0287	−1.31171	−0.655855	−	0.754887i	$-0.727692\pi$
$673$	−34.0287	−1.31171	−0.655855	+	0.754887i	$0.727692\pi$
$674$	0	0
$675$	−4.79955	−0.184735
$676$	0	0
$677$	−49.6016	−1.90634	−0.953172	−	0.302428i	$-0.902203\pi$
$677$	−49.6016	−1.90634	−0.953172	+	0.302428i	$0.902203\pi$
$678$	0	0
$679$	−13.9967	−0.537145
$680$	0	0
$681$	21.6544	0.829799
$682$	0	0
$683$	−12.8111	−0.490203	−0.245101	−	0.969497i	$-0.578821\pi$
$683$	−12.8111	−0.490203	−0.245101	+	0.969497i	$0.578821\pi$
$684$	0	0
$685$	0.313506	0.0119784
$686$	0	0
$687$	−21.1057	−0.805235
$688$	0	0
$689$	−11.1159	−0.423483
$690$	0	0
$691$	−20.2960	−0.772098	−0.386049	−	0.922478i	$-0.626160\pi$
$691$	−20.2960	−0.772098	−0.386049	+	0.922478i	$0.626160\pi$
$692$	0	0
$693$	3.31638	0.125979
$694$	0	0
$695$	6.19420	0.234959
$696$	0	0
$697$	−6.40899	−0.242758
$698$	0	0
$699$	12.1195	0.458403
$700$	0	0
$701$	−3.02391	−0.114211	−0.0571057	−	0.998368i	$-0.518187\pi$
$701$	−3.02391	−0.114211	−0.0571057	+	0.998368i	$0.518187\pi$
$702$	0	0
$703$	−25.4139	−0.958502
$704$	0	0
$705$	−2.00720	−0.0755957
$706$	0	0
$707$	−49.3478	−1.85591
$708$	0	0
$709$	16.3829	0.615274	0.307637	−	0.951504i	$-0.400462\pi$
$709$	16.3829	0.615274	0.307637	+	0.951504i	$0.400462\pi$
$710$	0	0
$711$	13.9104	0.521681
$712$	0	0
$713$	−10.2740	−0.384766
$714$	0	0
$715$	−0.447720	−0.0167438
$716$	0	0
$717$	9.61227	0.358977
$718$	0	0
$719$	−10.3875	−0.387387	−0.193693	−	0.981062i	$-0.562047\pi$
$719$	−10.3875	−0.387387	−0.193693	+	0.981062i	$0.562047\pi$
$720$	0	0
$721$	28.9708	1.07893
$722$	0	0
$723$	19.1159	0.710929
$724$	0	0
$725$	−42.4118	−1.57514
$726$	0	0
$727$	−13.4447	−0.498636	−0.249318	−	0.968422i	$-0.580206\pi$
$727$	−13.4447	−0.498636	−0.249318	+	0.968422i	$0.580206\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	35.2924	1.30534
$732$	0	0
$733$	−17.3104	−0.639373	−0.319686	−	0.947523i	$-0.603578\pi$
$733$	−17.3104	−0.639373	−0.319686	+	0.947523i	$0.603578\pi$
$734$	0	0
$735$	−1.79015	−0.0660305
$736$	0	0
$737$	2.44772	0.0901629
$738$	0	0
$739$	43.0938	1.58523	0.792616	−	0.609722i	$-0.208719\pi$
$739$	43.0938	1.58523	0.792616	+	0.609722i	$0.208719\pi$
$740$	0	0
$741$	3.83157	0.140756
$742$	0	0
$743$	−34.6213	−1.27013	−0.635067	−	0.772457i	$-0.719028\pi$
$743$	−34.6213	−1.27013	−0.635067	+	0.772457i	$0.719028\pi$
$744$	0	0
$745$	−8.27193	−0.303060
$746$	0	0
$747$	0.895440	0.0327624
$748$	0	0
$749$	−7.47017	−0.272954
$750$	0	0
$751$	−25.5001	−0.930512	−0.465256	−	0.885176i	$-0.654038\pi$
$751$	−25.5001	−0.930512	−0.465256	+	0.885176i	$0.654038\pi$
$752$	0	0
$753$	−20.3701	−0.742327
$754$	0	0
$755$	3.60506	0.131202
$756$	0	0
$757$	45.3361	1.64777	0.823885	−	0.566757i	$-0.191802\pi$
$757$	45.3361	1.64777	0.823885	+	0.566757i	$0.191802\pi$
$758$	0	0
$759$	−5.31638	−0.192972
$760$	0	0
$761$	22.3800	0.811273	0.405637	−	0.914034i	$-0.367050\pi$
$761$	22.3800	0.811273	0.405637	+	0.914034i	$0.367050\pi$
$762$	0	0
$763$	37.4232	1.35481
$764$	0	0
$765$	−2.17979	−0.0788106
$766$	0	0
$767$	−5.63112	−0.203328
$768$	0	0
$769$	17.1707	0.619190	0.309595	−	0.950868i	$-0.399806\pi$
$769$	17.1707	0.619190	0.309595	+	0.950868i	$0.399806\pi$
$770$	0	0
$771$	−19.1593	−0.690006
$772$	0	0
$773$	−40.0468	−1.44038	−0.720191	−	0.693776i	$-0.755946\pi$
$773$	−40.0468	−1.44038	−0.720191	+	0.693776i	$0.755946\pi$
$774$	0	0
$775$	−9.27526	−0.333177
$776$	0	0
$777$	−21.9967	−0.789128
$778$	0	0
$779$	−5.04379	−0.180713
$780$	0	0
$781$	−7.25087	−0.259456
$782$	0	0
$783$	8.83664	0.315796
$784$	0	0
$785$	−8.66775	−0.309365
$786$	0	0
$787$	−1.19881	−0.0427331	−0.0213666	−	0.999772i	$-0.506802\pi$
$787$	−1.19881	−0.0427331	−0.0213666	+	0.999772i	$0.506802\pi$
$788$	0	0
$789$	−15.0405	−0.535457
$790$	0	0
$791$	51.8387	1.84317
$792$	0	0
$793$	0.380248	0.0135030
$794$	0	0
$795$	4.97682	0.176510
$796$	0	0
$797$	−17.0591	−0.604264	−0.302132	−	0.953266i	$-0.597698\pi$
$797$	−17.0591	−0.604264	−0.302132	+	0.953266i	$0.597698\pi$
$798$	0	0
$799$	21.8270	0.772184
$800$	0	0
$801$	−9.46072	−0.334278
$802$	0	0
$803$	11.7995	0.416397
$804$	0	0
$805$	7.89380	0.278220
$806$	0	0
$807$	17.1192	0.602624
$808$	0	0
$809$	−7.68992	−0.270363	−0.135182	−	0.990821i	$-0.543162\pi$
$809$	−7.68992	−0.270363	−0.135182	+	0.990821i	$0.543162\pi$
$810$	0	0
$811$	−26.8012	−0.941117	−0.470558	−	0.882369i	$-0.655948\pi$
$811$	−26.8012	−0.941117	−0.470558	+	0.882369i	$0.655948\pi$
$812$	0	0
$813$	4.18700	0.146844
$814$	0	0
$815$	−6.28395	−0.220117
$816$	0	0
$817$	27.7747	0.971713
$818$	0	0
$819$	3.31638	0.115884
$820$	0	0
$821$	−3.01154	−0.105103	−0.0525517	−	0.998618i	$-0.516735\pi$
$821$	−3.01154	−0.105103	−0.0525517	+	0.998618i	$0.516735\pi$
$822$	0	0
$823$	−7.41660	−0.258527	−0.129263	−	0.991610i	$-0.541261\pi$
$823$	−7.41660	−0.258527	−0.129263	+	0.991610i	$0.541261\pi$
$824$	0	0
$825$	−4.79955	−0.167099
$826$	0	0
$827$	21.3618	0.742823	0.371411	−	0.928468i	$-0.378874\pi$
$827$	21.3618	0.742823	0.371411	+	0.928468i	$0.378874\pi$
$828$	0	0
$829$	−47.4073	−1.64652	−0.823262	−	0.567662i	$-0.807848\pi$
$829$	−47.4073	−1.64652	−0.823262	+	0.567662i	$0.807848\pi$
$830$	0	0
$831$	14.3988	0.499489
$832$	0	0
$833$	19.4667	0.674480
$834$	0	0
$835$	7.34208	0.254083
$836$	0	0
$837$	1.93253	0.0667979
$838$	0	0
$839$	−17.8148	−0.615034	−0.307517	−	0.951543i	$-0.599498\pi$
$839$	−17.8148	−0.615034	−0.307517	+	0.951543i	$0.599498\pi$
$840$	0	0
$841$	49.0861	1.69262
$842$	0	0
$843$	−20.7136	−0.713413
$844$	0	0
$845$	−0.447720	−0.0154020
$846$	0	0
$847$	3.31638	0.113952
$848$	0	0
$849$	16.9976	0.583356
$850$	0	0
$851$	35.2622	1.20877
$852$	0	0
$853$	48.4227	1.65796	0.828981	−	0.559276i	$-0.188921\pi$
$853$	48.4227	1.65796	0.828981	+	0.559276i	$0.188921\pi$
$854$	0	0
$855$	−1.71547	−0.0586678
$856$	0	0
$857$	28.4459	0.971694	0.485847	−	0.874044i	$-0.338511\pi$
$857$	28.4459	0.971694	0.485847	+	0.874044i	$0.338511\pi$
$858$	0	0
$859$	−13.7373	−0.468711	−0.234356	−	0.972151i	$-0.575298\pi$
$859$	−13.7373	−0.468711	−0.234356	+	0.972151i	$0.575298\pi$
$860$	0	0
$861$	−4.36561	−0.148779
$862$	0	0
$863$	−29.7600	−1.01304	−0.506522	−	0.862227i	$-0.669069\pi$
$863$	−29.7600	−1.01304	−0.506522	+	0.862227i	$0.669069\pi$
$864$	0	0
$865$	1.23172	0.0418797
$866$	0	0
$867$	6.70383	0.227674
$868$	0	0
$869$	13.9104	0.471878
$870$	0	0
$871$	2.44772	0.0829378
$872$	0	0
$873$	−4.22048	−0.142842
$874$	0	0
$875$	14.5504	0.491895
$876$	0	0
$877$	−23.4058	−0.790359	−0.395180	−	0.918604i	$-0.629318\pi$
$877$	−23.4058	−0.790359	−0.395180	+	0.918604i	$0.629318\pi$
$878$	0	0
$879$	2.18700	0.0737656
$880$	0	0
$881$	−25.4702	−0.858112	−0.429056	−	0.903278i	$-0.641154\pi$
$881$	−25.4702	−0.858112	−0.429056	+	0.903278i	$0.641154\pi$
$882$	0	0
$883$	−6.15410	−0.207102	−0.103551	−	0.994624i	$-0.533020\pi$
$883$	−6.15410	−0.207102	−0.103551	+	0.994624i	$0.533020\pi$
$884$	0	0
$885$	2.52116	0.0847480
$886$	0	0
$887$	−45.5829	−1.53052	−0.765262	−	0.643719i	$-0.777391\pi$
$887$	−45.5829	−1.53052	−0.765262	+	0.643719i	$0.777391\pi$
$888$	0	0
$889$	−51.4085	−1.72419
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	17.1776	0.574826
$894$	0	0
$895$	5.98143	0.199937
$896$	0	0
$897$	−5.31638	−0.177509
$898$	0	0
$899$	17.0770	0.569551
$900$	0	0
$901$	−54.1196	−1.80299
$902$	0	0
$903$	24.0401	0.800005
$904$	0	0
$905$	−10.3078	−0.342643
$906$	0	0
$907$	10.0365	0.333256	0.166628	−	0.986020i	$-0.446712\pi$
$907$	10.0365	0.333256	0.166628	+	0.986020i	$0.446712\pi$
$908$	0	0
$909$	−14.8800	−0.493539
$910$	0	0
$911$	−34.7183	−1.15027	−0.575134	−	0.818059i	$-0.695050\pi$
$911$	−34.7183	−1.15027	−0.575134	+	0.818059i	$0.695050\pi$
$912$	0	0
$913$	0.895440	0.0296347
$914$	0	0
$915$	−0.170244	−0.00562811
$916$	0	0
$917$	7.64780	0.252553
$918$	0	0
$919$	−29.9020	−0.986375	−0.493187	−	0.869923i	$-0.664168\pi$
$919$	−29.9020	−0.986375	−0.493187	+	0.869923i	$0.664168\pi$
$920$	0	0
$921$	−7.30620	−0.240748
$922$	0	0
$923$	−7.25087	−0.238665
$924$	0	0
$925$	31.8342	1.04670
$926$	0	0
$927$	8.73568	0.286917
$928$	0	0
$929$	−6.50844	−0.213535	−0.106768	−	0.994284i	$-0.534050\pi$
$929$	−6.50844	−0.213535	−0.106768	+	0.994284i	$0.534050\pi$
$930$	0	0
$931$	15.3200	0.502093
$932$	0	0
$933$	−19.0011	−0.622067
$934$	0	0
$935$	−2.17979	−0.0712869
$936$	0	0
$937$	22.9227	0.748851	0.374426	−	0.927257i	$-0.377840\pi$
$937$	22.9227	0.748851	0.374426	+	0.927257i	$0.377840\pi$
$938$	0	0
$939$	24.2974	0.792914
$940$	0	0
$941$	−42.2872	−1.37852	−0.689261	−	0.724513i	$-0.742065\pi$
$941$	−42.2872	−1.37852	−0.689261	+	0.724513i	$0.742065\pi$
$942$	0	0
$943$	6.99836	0.227898
$944$	0	0
$945$	−1.48481	−0.0483008
$946$	0	0
$947$	−3.87642	−0.125967	−0.0629834	−	0.998015i	$-0.520062\pi$
$947$	−3.87642	−0.125967	−0.0629834	+	0.998015i	$0.520062\pi$
$948$	0	0
$949$	11.7995	0.383030
$950$	0	0
$951$	23.2977	0.755479
$952$	0	0
$953$	−21.8529	−0.707885	−0.353943	−	0.935267i	$-0.615159\pi$
$953$	−21.8529	−0.707885	−0.353943	+	0.935267i	$0.615159\pi$
$954$	0	0
$955$	−2.33805	−0.0756575
$956$	0	0
$957$	8.83664	0.285648
$958$	0	0
$959$	−2.32222	−0.0749884
$960$	0	0
$961$	−27.2653	−0.879527
$962$	0	0
$963$	−2.25251	−0.0725860
$964$	0	0
$965$	−6.59357	−0.212255
$966$	0	0
$967$	−55.7960	−1.79428	−0.897138	−	0.441750i	$-0.854358\pi$
$967$	−55.7960	−1.79428	−0.897138	+	0.441750i	$0.854358\pi$
$968$	0	0
$969$	18.6546	0.599272
$970$	0	0
$971$	21.4307	0.687743	0.343871	−	0.939017i	$-0.388262\pi$
$971$	21.4307	0.687743	0.343871	+	0.939017i	$0.388262\pi$
$972$	0	0
$973$	−45.8821	−1.47091
$974$	0	0
$975$	−4.79955	−0.153709
$976$	0	0
$977$	47.1672	1.50901	0.754506	−	0.656293i	$-0.227876\pi$
$977$	47.1672	1.50901	0.754506	+	0.656293i	$0.227876\pi$
$978$	0	0
$979$	−9.46072	−0.302366
$980$	0	0
$981$	11.2844	0.360282
$982$	0	0
$983$	36.3165	1.15832	0.579158	−	0.815215i	$-0.303381\pi$
$983$	36.3165	1.15832	0.579158	+	0.815215i	$0.303381\pi$
$984$	0	0
$985$	6.04130	0.192492
$986$	0	0
$987$	14.8679	0.473250
$988$	0	0
$989$	−38.5379	−1.22543
$990$	0	0
$991$	10.1361	0.321985	0.160992	−	0.986956i	$-0.448531\pi$
$991$	10.1361	0.321985	0.160992	+	0.986956i	$0.448531\pi$
$992$	0	0
$993$	8.66820	0.275077
$994$	0	0
$995$	6.50383	0.206185
$996$	0	0
$997$	42.6875	1.35193	0.675963	−	0.736935i	$-0.263728\pi$
$997$	42.6875	1.35193	0.675963	+	0.736935i	$0.263728\pi$
$998$	0	0
$999$	−6.63276	−0.209851

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3432.2.a.x.1.3	✓	5
4.3	odd	2		6864.2.a.ce.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.x.1.3	✓	5	1.1	even	1	trivial
6864.2.a.ce.1.3		5	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3432.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.447720 q^{5} +3.31638 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} -0.447720 q^{15} +4.86866 q^{17} +3.83157 q^{19} +3.31638 q^{21} -5.31638 q^{23} -4.79955 q^{25} +1.00000 q^{27} +8.83664 q^{29} +1.93253 q^{31} +1.00000 q^{33} -1.48481 q^{35} -6.63276 q^{37} +1.00000 q^{39} -1.31638 q^{41} +7.24891 q^{43} -0.447720 q^{45} +4.48317 q^{47} +3.99836 q^{49} +4.86866 q^{51} -11.1159 q^{53} -0.447720 q^{55} +3.83157 q^{57} -5.63112 q^{59} +0.380248 q^{61} +3.31638 q^{63} -0.447720 q^{65} +2.44772 q^{67} -5.31638 q^{69} -7.25087 q^{71} +11.7995 q^{73} -4.79955 q^{75} +3.31638 q^{77} +13.9104 q^{79} +1.00000 q^{81} +0.895440 q^{83} -2.17979 q^{85} +8.83664 q^{87} -9.46072 q^{89} +3.31638 q^{91} +1.93253 q^{93} -1.71547 q^{95} -4.22048 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} + q^{5} - 3 q^{7} + 5 q^{9} + 5 q^{11} + 5 q^{13} + q^{15} + 8 q^{17} + 10 q^{19} - 3 q^{21} - 7 q^{23} + 16 q^{25} + 5 q^{27} - 3 q^{29} - 4 q^{31} + 5 q^{33} + 3 q^{35} + 6 q^{37} + 5 q^{39}+ \cdots + 5 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 + q^5 - 3 * q^7 + 5 * q^9 + 5 * q^11 + 5 * q^13 + q^15 + 8 * q^17 + 10 * q^19 - 3 * q^21 - 7 * q^23 + 16 * q^25 + 5 * q^27 - 3 * q^29 - 4 * q^31 + 5 * q^33 + 3 * q^35 + 6 * q^37 + 5 * q^39 + 13 * q^41 + 3 * q^43 + q^45 + 2 * q^47 + 10 * q^49 + 8 * q^51 + 4 * q^53 + q^55 + 10 * q^57 + 21 * q^59 - 15 * q^61 - 3 * q^63 + q^65 + 9 * q^67 - 7 * q^69 - 4 * q^71 + 19 * q^73 + 16 * q^75 - 3 * q^77 + 8 * q^79 + 5 * q^81 - 2 * q^83 + 46 * q^85 - 3 * q^87 + 12 * q^89 - 3 * q^91 - 4 * q^93 + 32 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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