LMFDB - Embedded newform 1960.2.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1960,2,Mod(1,1960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1960, 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("1960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1960 = 2^{3} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1960.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:	$15.6506787962$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 280)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	1960.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214 q^{3} +1.00000 q^{5} -2.82843 q^{9} +O(q^{10})$	$q+0.414214 q^{3} +1.00000 q^{5} -2.82843 q^{9} +0.828427 q^{11} +2.00000 q^{13} +0.414214 q^{15} -7.65685 q^{17} -5.65685 q^{19} -5.58579 q^{23} +1.00000 q^{25} -2.41421 q^{27} -7.82843 q^{29} +0.828427 q^{31} +0.343146 q^{33} +5.65685 q^{37} +0.828427 q^{39} +5.82843 q^{41} -6.89949 q^{43} -2.82843 q^{45} +11.6569 q^{47} -3.17157 q^{51} -5.65685 q^{53} +0.828427 q^{55} -2.34315 q^{57} -4.00000 q^{59} +6.65685 q^{61} +2.00000 q^{65} -12.8995 q^{67} -2.31371 q^{69} -12.0000 q^{71} +3.65685 q^{73} +0.414214 q^{75} -4.00000 q^{79} +7.48528 q^{81} -4.75736 q^{83} -7.65685 q^{85} -3.24264 q^{87} -5.34315 q^{89} +0.343146 q^{93} -5.65685 q^{95} +6.00000 q^{97} -2.34315 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^5	$ 2 q - 2 q^{3} + 2 q^{5} - 4 q^{11} + 4 q^{13} - 2 q^{15} - 4 q^{17} - 14 q^{23} + 2 q^{25} - 2 q^{27} - 10 q^{29} - 4 q^{31} + 12 q^{33} - 4 q^{39} + 6 q^{41} + 6 q^{43} + 12 q^{47} - 12 q^{51} - 4 q^{55} - 16 q^{57} - 8 q^{59} + 2 q^{61} + 4 q^{65} - 6 q^{67} + 18 q^{69} - 24 q^{71} - 4 q^{73} - 2 q^{75} - 8 q^{79} - 2 q^{81} - 18 q^{83} - 4 q^{85} + 2 q^{87} - 22 q^{89} + 12 q^{93} + 12 q^{97} - 16 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 - 4 * q^11 + 4 * q^13 - 2 * q^15 - 4 * q^17 - 14 * q^23 + 2 * q^25 - 2 * q^27 - 10 * q^29 - 4 * q^31 + 12 * q^33 - 4 * q^39 + 6 * q^41 + 6 * q^43 + 12 * q^47 - 12 * q^51 - 4 * q^55 - 16 * q^57 - 8 * q^59 + 2 * q^61 + 4 * q^65 - 6 * q^67 + 18 * q^69 - 24 * q^71 - 4 * q^73 - 2 * q^75 - 8 * q^79 - 2 * q^81 - 18 * q^83 - 4 * q^85 + 2 * q^87 - 22 * q^89 + 12 * q^93 + 12 * q^97 - 16 * q^99

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
$3$	0.414214	0.239146	0.119573	−	0.992825i	$-0.461847\pi$
$3$	0.414214	0.239146	0.119573	+	0.992825i	$0.461847\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.82843	−0.942809
$10$	0	0
$11$	0.828427	0.249780	0.124890	−	0.992171i	$-0.460142\pi$
$11$	0.828427	0.249780	0.124890	+	0.992171i	$0.460142\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	0.414214	0.106949
$16$	0	0
$17$	−7.65685	−1.85706	−0.928530	−	0.371257i	$-0.878927\pi$
$17$	−7.65685	−1.85706	−0.928530	+	0.371257i	$0.878927\pi$
$18$	0	0
$19$	−5.65685	−1.29777	−0.648886	−	0.760886i	$-0.724765\pi$
$19$	−5.65685	−1.29777	−0.648886	+	0.760886i	$0.724765\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.58579	−1.16472	−0.582358	−	0.812932i	$-0.697870\pi$
$23$	−5.58579	−1.16472	−0.582358	+	0.812932i	$0.697870\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.41421	−0.464616
$28$	0	0
$29$	−7.82843	−1.45370	−0.726851	−	0.686795i	$-0.759017\pi$
$29$	−7.82843	−1.45370	−0.726851	+	0.686795i	$0.759017\pi$
$30$	0	0
$31$	0.828427	0.148790	0.0743950	−	0.997229i	$-0.476297\pi$
$31$	0.828427	0.148790	0.0743950	+	0.997229i	$0.476297\pi$
$32$	0	0
$33$	0.343146	0.0597340
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.65685	0.929981	0.464991	−	0.885316i	$-0.346058\pi$
$37$	5.65685	0.929981	0.464991	+	0.885316i	$0.346058\pi$
$38$	0	0
$39$	0.828427	0.132655
$40$	0	0
$41$	5.82843	0.910247	0.455124	−	0.890428i	$-0.349595\pi$
$41$	5.82843	0.910247	0.455124	+	0.890428i	$0.349595\pi$
$42$	0	0
$43$	−6.89949	−1.05216	−0.526082	−	0.850434i	$-0.676339\pi$
$43$	−6.89949	−1.05216	−0.526082	+	0.850434i	$0.676339\pi$
$44$	0	0
$45$	−2.82843	−0.421637
$46$	0	0
$47$	11.6569	1.70033	0.850163	−	0.526519i	$-0.176503\pi$
$47$	11.6569	1.70033	0.850163	+	0.526519i	$0.176503\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−3.17157	−0.444109
$52$	0	0
$53$	−5.65685	−0.777029	−0.388514	−	0.921443i	$-0.627012\pi$
$53$	−5.65685	−0.777029	−0.388514	+	0.921443i	$0.627012\pi$
$54$	0	0
$55$	0.828427	0.111705
$56$	0	0
$57$	−2.34315	−0.310357
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	6.65685	0.852323	0.426161	−	0.904647i	$-0.359866\pi$
$61$	6.65685	0.852323	0.426161	+	0.904647i	$0.359866\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	−12.8995	−1.57592	−0.787962	−	0.615724i	$-0.788864\pi$
$67$	−12.8995	−1.57592	−0.787962	+	0.615724i	$0.788864\pi$
$68$	0	0
$69$	−2.31371	−0.278538
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	3.65685	0.428002	0.214001	−	0.976833i	$-0.431350\pi$
$73$	3.65685	0.428002	0.214001	+	0.976833i	$0.431350\pi$
$74$	0	0
$75$	0.414214	0.0478293
$76$	0	0
$77$	0	0
$78$	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
$81$	7.48528	0.831698
$82$	0	0
$83$	−4.75736	−0.522188	−0.261094	−	0.965313i	$-0.584083\pi$
$83$	−4.75736	−0.522188	−0.261094	+	0.965313i	$0.584083\pi$
$84$	0	0
$85$	−7.65685	−0.830502
$86$	0	0
$87$	−3.24264	−0.347648
$88$	0	0
$89$	−5.34315	−0.566372	−0.283186	−	0.959065i	$-0.591391\pi$
$89$	−5.34315	−0.566372	−0.283186	+	0.959065i	$0.591391\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0.343146	0.0355826
$94$	0	0
$95$	−5.65685	−0.580381
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	0	0
$99$	−2.34315	−0.235495
$100$	0	0
$101$	−11.4853	−1.14283	−0.571414	−	0.820662i	$-0.693605\pi$
$101$	−11.4853	−1.14283	−0.571414	+	0.820662i	$0.693605\pi$
$102$	0	0
$103$	7.58579	0.747450	0.373725	−	0.927540i	$-0.378080\pi$
$103$	7.58579	0.747450	0.373725	+	0.927540i	$0.378080\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.58579	−0.539998	−0.269999	−	0.962861i	$-0.587023\pi$
$107$	−5.58579	−0.539998	−0.269999	+	0.962861i	$0.587023\pi$
$108$	0	0
$109$	18.3137	1.75414	0.877068	−	0.480367i	$-0.159497\pi$
$109$	18.3137	1.75414	0.877068	+	0.480367i	$0.159497\pi$
$110$	0	0
$111$	2.34315	0.222402
$112$	0	0
$113$	11.3137	1.06430	0.532152	−	0.846649i	$-0.321383\pi$
$113$	11.3137	1.06430	0.532152	+	0.846649i	$0.321383\pi$
$114$	0	0
$115$	−5.58579	−0.520877
$116$	0	0
$117$	−5.65685	−0.522976
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.3137	−0.937610
$122$	0	0
$123$	2.41421	0.217682
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.34315	−0.385392	−0.192696	−	0.981259i	$-0.561723\pi$
$127$	−4.34315	−0.385392	−0.192696	+	0.981259i	$0.561723\pi$
$128$	0	0
$129$	−2.85786	−0.251621
$130$	0	0
$131$	13.6569	1.19320	0.596602	−	0.802537i	$-0.296517\pi$
$131$	13.6569	1.19320	0.596602	+	0.802537i	$0.296517\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−2.41421	−0.207782
$136$	0	0
$137$	−4.00000	−0.341743	−0.170872	−	0.985293i	$-0.554658\pi$
$137$	−4.00000	−0.341743	−0.170872	+	0.985293i	$0.554658\pi$
$138$	0	0
$139$	2.48528	0.210799	0.105399	−	0.994430i	$-0.466388\pi$
$139$	2.48528	0.210799	0.105399	+	0.994430i	$0.466388\pi$
$140$	0	0
$141$	4.82843	0.406627
$142$	0	0
$143$	1.65685	0.138553
$144$	0	0
$145$	−7.82843	−0.650115
$146$	0	0
$147$	0	0
$148$	0	0
$149$	4.65685	0.381504	0.190752	−	0.981638i	$-0.438907\pi$
$149$	4.65685	0.381504	0.190752	+	0.981638i	$0.438907\pi$
$150$	0	0
$151$	−11.1716	−0.909130	−0.454565	−	0.890714i	$-0.650205\pi$
$151$	−11.1716	−0.909130	−0.454565	+	0.890714i	$0.650205\pi$
$152$	0	0
$153$	21.6569	1.75085
$154$	0	0
$155$	0.828427	0.0665409
$156$	0	0
$157$	1.31371	0.104845	0.0524227	−	0.998625i	$-0.483306\pi$
$157$	1.31371	0.104845	0.0524227	+	0.998625i	$0.483306\pi$
$158$	0	0
$159$	−2.34315	−0.185824
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−15.6569	−1.22634	−0.613170	−	0.789951i	$-0.710106\pi$
$163$	−15.6569	−1.22634	−0.613170	+	0.789951i	$0.710106\pi$
$164$	0	0
$165$	0.343146	0.0267139
$166$	0	0
$167$	2.07107	0.160264	0.0801320	−	0.996784i	$-0.474466\pi$
$167$	2.07107	0.160264	0.0801320	+	0.996784i	$0.474466\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	16.0000	1.22355
$172$	0	0
$173$	10.3431	0.786375	0.393187	−	0.919458i	$-0.371372\pi$
$173$	10.3431	0.786375	0.393187	+	0.919458i	$0.371372\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1.65685	−0.124537
$178$	0	0
$179$	−6.48528	−0.484733	−0.242366	−	0.970185i	$-0.577924\pi$
$179$	−6.48528	−0.484733	−0.242366	+	0.970185i	$0.577924\pi$
$180$	0	0
$181$	−4.17157	−0.310071	−0.155035	−	0.987909i	$-0.549549\pi$
$181$	−4.17157	−0.310071	−0.155035	+	0.987909i	$0.549549\pi$
$182$	0	0
$183$	2.75736	0.203830
$184$	0	0
$185$	5.65685	0.415900
$186$	0	0
$187$	−6.34315	−0.463857
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.51472	−0.399031	−0.199516	−	0.979895i	$-0.563937\pi$
$191$	−5.51472	−0.399031	−0.199516	+	0.979895i	$0.563937\pi$
$192$	0	0
$193$	−5.31371	−0.382489	−0.191245	−	0.981542i	$-0.561252\pi$
$193$	−5.31371	−0.382489	−0.191245	+	0.981542i	$0.561252\pi$
$194$	0	0
$195$	0.828427	0.0593249
$196$	0	0
$197$	0.343146	0.0244481	0.0122241	−	0.999925i	$-0.496109\pi$
$197$	0.343146	0.0244481	0.0122241	+	0.999925i	$0.496109\pi$
$198$	0	0
$199$	−23.3137	−1.65266	−0.826332	−	0.563183i	$-0.809577\pi$
$199$	−23.3137	−1.65266	−0.826332	+	0.563183i	$0.809577\pi$
$200$	0	0
$201$	−5.34315	−0.376876
$202$	0	0
$203$	0	0
$204$	0	0
$205$	5.82843	0.407075
$206$	0	0
$207$	15.7990	1.09811
$208$	0	0
$209$	−4.68629	−0.324158
$210$	0	0
$211$	26.6274	1.83311	0.916553	−	0.399912i	$-0.130959\pi$
$211$	26.6274	1.83311	0.916553	+	0.399912i	$0.130959\pi$
$212$	0	0
$213$	−4.97056	−0.340577
$214$	0	0
$215$	−6.89949	−0.470542
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1.51472	0.102355
$220$	0	0
$221$	−15.3137	−1.03011
$222$	0	0
$223$	−14.9706	−1.00250	−0.501252	−	0.865302i	$-0.667127\pi$
$223$	−14.9706	−1.00250	−0.501252	+	0.865302i	$0.667127\pi$
$224$	0	0
$225$	−2.82843	−0.188562
$226$	0	0
$227$	14.0000	0.929213	0.464606	−	0.885517i	$-0.346196\pi$
$227$	14.0000	0.929213	0.464606	+	0.885517i	$0.346196\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	11.6569	0.763666	0.381833	−	0.924231i	$-0.375293\pi$
$233$	11.6569	0.763666	0.381833	+	0.924231i	$0.375293\pi$
$234$	0	0
$235$	11.6569	0.760409
$236$	0	0
$237$	−1.65685	−0.107624
$238$	0	0
$239$	−30.4853	−1.97193	−0.985964	−	0.166955i	$-0.946606\pi$
$239$	−30.4853	−1.97193	−0.985964	+	0.166955i	$0.946606\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	10.3431	0.663513
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−11.3137	−0.719874
$248$	0	0
$249$	−1.97056	−0.124879
$250$	0	0
$251$	−27.4558	−1.73300	−0.866499	−	0.499179i	$-0.833635\pi$
$251$	−27.4558	−1.73300	−0.866499	+	0.499179i	$0.833635\pi$
$252$	0	0
$253$	−4.62742	−0.290923
$254$	0	0
$255$	−3.17157	−0.198612
$256$	0	0
$257$	15.3137	0.955243	0.477621	−	0.878566i	$-0.341499\pi$
$257$	15.3137	0.955243	0.477621	+	0.878566i	$0.341499\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	22.1421	1.37056
$262$	0	0
$263$	−15.7279	−0.969825	−0.484913	−	0.874563i	$-0.661149\pi$
$263$	−15.7279	−0.969825	−0.484913	+	0.874563i	$0.661149\pi$
$264$	0	0
$265$	−5.65685	−0.347498
$266$	0	0
$267$	−2.21320	−0.135446
$268$	0	0
$269$	−6.65685	−0.405876	−0.202938	−	0.979192i	$-0.565049\pi$
$269$	−6.65685	−0.405876	−0.202938	+	0.979192i	$0.565049\pi$
$270$	0	0
$271$	−3.31371	−0.201293	−0.100647	−	0.994922i	$-0.532091\pi$
$271$	−3.31371	−0.201293	−0.100647	+	0.994922i	$0.532091\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.828427	0.0499560
$276$	0	0
$277$	28.6274	1.72005	0.860027	−	0.510248i	$-0.170446\pi$
$277$	28.6274	1.72005	0.860027	+	0.510248i	$0.170446\pi$
$278$	0	0
$279$	−2.34315	−0.140280
$280$	0	0
$281$	2.68629	0.160251	0.0801254	−	0.996785i	$-0.474468\pi$
$281$	2.68629	0.160251	0.0801254	+	0.996785i	$0.474468\pi$
$282$	0	0
$283$	18.0000	1.06999	0.534994	−	0.844856i	$-0.320314\pi$
$283$	18.0000	1.06999	0.534994	+	0.844856i	$0.320314\pi$
$284$	0	0
$285$	−2.34315	−0.138796
$286$	0	0
$287$	0	0
$288$	0	0
$289$	41.6274	2.44867
$290$	0	0
$291$	2.48528	0.145690
$292$	0	0
$293$	16.9706	0.991431	0.495715	−	0.868485i	$-0.334906\pi$
$293$	16.9706	0.991431	0.495715	+	0.868485i	$0.334906\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	0	0
$297$	−2.00000	−0.116052
$298$	0	0
$299$	−11.1716	−0.646069
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−4.75736	−0.273303
$304$	0	0
$305$	6.65685	0.381170
$306$	0	0
$307$	4.75736	0.271517	0.135758	−	0.990742i	$-0.456653\pi$
$307$	4.75736	0.271517	0.135758	+	0.990742i	$0.456653\pi$
$308$	0	0
$309$	3.14214	0.178750
$310$	0	0
$311$	21.6569	1.22805	0.614024	−	0.789288i	$-0.289550\pi$
$311$	21.6569	1.22805	0.614024	+	0.789288i	$0.289550\pi$
$312$	0	0
$313$	20.9706	1.18533	0.592663	−	0.805450i	$-0.298077\pi$
$313$	20.9706	1.18533	0.592663	+	0.805450i	$0.298077\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	0	0
$319$	−6.48528	−0.363106
$320$	0	0
$321$	−2.31371	−0.129139
$322$	0	0
$323$	43.3137	2.41004
$324$	0	0
$325$	2.00000	0.110940
$326$	0	0
$327$	7.58579	0.419495
$328$	0	0
$329$	0	0
$330$	0	0
$331$	26.4853	1.45576	0.727881	−	0.685703i	$-0.240505\pi$
$331$	26.4853	1.45576	0.727881	+	0.685703i	$0.240505\pi$
$332$	0	0
$333$	−16.0000	−0.876795
$334$	0	0
$335$	−12.8995	−0.704775
$336$	0	0
$337$	24.9706	1.36023	0.680117	−	0.733104i	$-0.261929\pi$
$337$	24.9706	1.36023	0.680117	+	0.733104i	$0.261929\pi$
$338$	0	0
$339$	4.68629	0.254524
$340$	0	0
$341$	0.686292	0.0371648
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−2.31371	−0.124566
$346$	0	0
$347$	11.3848	0.611167	0.305583	−	0.952165i	$-0.401149\pi$
$347$	11.3848	0.611167	0.305583	+	0.952165i	$0.401149\pi$
$348$	0	0
$349$	9.82843	0.526104	0.263052	−	0.964782i	$-0.415271\pi$
$349$	9.82843	0.526104	0.263052	+	0.964782i	$0.415271\pi$
$350$	0	0
$351$	−4.82843	−0.257722
$352$	0	0
$353$	33.6569	1.79137	0.895687	−	0.444685i	$-0.146685\pi$
$353$	33.6569	1.79137	0.895687	+	0.444685i	$0.146685\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6.48528	−0.342280	−0.171140	−	0.985247i	$-0.554745\pi$
$359$	−6.48528	−0.342280	−0.171140	+	0.985247i	$0.554745\pi$
$360$	0	0
$361$	13.0000	0.684211
$362$	0	0
$363$	−4.27208	−0.224226
$364$	0	0
$365$	3.65685	0.191408
$366$	0	0
$367$	21.5858	1.12677	0.563384	−	0.826195i	$-0.309499\pi$
$367$	21.5858	1.12677	0.563384	+	0.826195i	$0.309499\pi$
$368$	0	0
$369$	−16.4853	−0.858189
$370$	0	0
$371$	0	0
$372$	0	0
$373$	12.0000	0.621336	0.310668	−	0.950518i	$-0.399447\pi$
$373$	12.0000	0.621336	0.310668	+	0.950518i	$0.399447\pi$
$374$	0	0
$375$	0.414214	0.0213899
$376$	0	0
$377$	−15.6569	−0.806369
$378$	0	0
$379$	4.68629	0.240719	0.120359	−	0.992730i	$-0.461595\pi$
$379$	4.68629	0.240719	0.120359	+	0.992730i	$0.461595\pi$
$380$	0	0
$381$	−1.79899	−0.0921650
$382$	0	0
$383$	−0.899495	−0.0459620	−0.0229810	−	0.999736i	$-0.507316\pi$
$383$	−0.899495	−0.0459620	−0.0229810	+	0.999736i	$0.507316\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	19.5147	0.991989
$388$	0	0
$389$	5.31371	0.269416	0.134708	−	0.990885i	$-0.456990\pi$
$389$	5.31371	0.269416	0.134708	+	0.990885i	$0.456990\pi$
$390$	0	0
$391$	42.7696	2.16295
$392$	0	0
$393$	5.65685	0.285351
$394$	0	0
$395$	−4.00000	−0.201262
$396$	0	0
$397$	−24.6274	−1.23601	−0.618007	−	0.786172i	$-0.712060\pi$
$397$	−24.6274	−1.23601	−0.618007	+	0.786172i	$0.712060\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−32.3137	−1.61367	−0.806835	−	0.590777i	$-0.798821\pi$
$401$	−32.3137	−1.61367	−0.806835	+	0.590777i	$0.798821\pi$
$402$	0	0
$403$	1.65685	0.0825338
$404$	0	0
$405$	7.48528	0.371947
$406$	0	0
$407$	4.68629	0.232291
$408$	0	0
$409$	−25.1421	−1.24320	−0.621599	−	0.783335i	$-0.713517\pi$
$409$	−25.1421	−1.24320	−0.621599	+	0.783335i	$0.713517\pi$
$410$	0	0
$411$	−1.65685	−0.0817266
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−4.75736	−0.233530
$416$	0	0
$417$	1.02944	0.0504118
$418$	0	0
$419$	−15.3137	−0.748124	−0.374062	−	0.927404i	$-0.622035\pi$
$419$	−15.3137	−0.748124	−0.374062	+	0.927404i	$0.622035\pi$
$420$	0	0
$421$	27.3431	1.33262	0.666312	−	0.745673i	$-0.267872\pi$
$421$	27.3431	1.33262	0.666312	+	0.745673i	$0.267872\pi$
$422$	0	0
$423$	−32.9706	−1.60308
$424$	0	0
$425$	−7.65685	−0.371412
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0.686292	0.0331345
$430$	0	0
$431$	−0.828427	−0.0399039	−0.0199520	−	0.999801i	$-0.506351\pi$
$431$	−0.828427	−0.0399039	−0.0199520	+	0.999801i	$0.506351\pi$
$432$	0	0
$433$	−19.3137	−0.928158	−0.464079	−	0.885794i	$-0.653615\pi$
$433$	−19.3137	−0.928158	−0.464079	+	0.885794i	$0.653615\pi$
$434$	0	0
$435$	−3.24264	−0.155473
$436$	0	0
$437$	31.5980	1.51154
$438$	0	0
$439$	−18.3431	−0.875471	−0.437735	−	0.899104i	$-0.644219\pi$
$439$	−18.3431	−0.875471	−0.437735	+	0.899104i	$0.644219\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−15.5858	−0.740503	−0.370252	−	0.928932i	$-0.620729\pi$
$443$	−15.5858	−0.740503	−0.370252	+	0.928932i	$0.620729\pi$
$444$	0	0
$445$	−5.34315	−0.253289
$446$	0	0
$447$	1.92893	0.0912354
$448$	0	0
$449$	−7.48528	−0.353252	−0.176626	−	0.984278i	$-0.556518\pi$
$449$	−7.48528	−0.353252	−0.176626	+	0.984278i	$0.556518\pi$
$450$	0	0
$451$	4.82843	0.227362
$452$	0	0
$453$	−4.62742	−0.217415
$454$	0	0
$455$	0	0
$456$	0	0
$457$	28.9706	1.35519	0.677593	−	0.735437i	$-0.263023\pi$
$457$	28.9706	1.35519	0.677593	+	0.735437i	$0.263023\pi$
$458$	0	0
$459$	18.4853	0.862819
$460$	0	0
$461$	1.31371	0.0611855	0.0305928	−	0.999532i	$-0.490261\pi$
$461$	1.31371	0.0611855	0.0305928	+	0.999532i	$0.490261\pi$
$462$	0	0
$463$	14.8995	0.692438	0.346219	−	0.938154i	$-0.387465\pi$
$463$	14.8995	0.692438	0.346219	+	0.938154i	$0.387465\pi$
$464$	0	0
$465$	0.343146	0.0159130
$466$	0	0
$467$	−37.8701	−1.75242	−0.876209	−	0.481932i	$-0.839935\pi$
$467$	−37.8701	−1.75242	−0.876209	+	0.481932i	$0.839935\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0.544156	0.0250734
$472$	0	0
$473$	−5.71573	−0.262809
$474$	0	0
$475$	−5.65685	−0.259554
$476$	0	0
$477$	16.0000	0.732590
$478$	0	0
$479$	−1.51472	−0.0692093	−0.0346046	−	0.999401i	$-0.511017\pi$
$479$	−1.51472	−0.0692093	−0.0346046	+	0.999401i	$0.511017\pi$
$480$	0	0
$481$	11.3137	0.515861
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6.00000	0.272446
$486$	0	0
$487$	−38.2843	−1.73483	−0.867413	−	0.497589i	$-0.834219\pi$
$487$	−38.2843	−1.73483	−0.867413	+	0.497589i	$0.834219\pi$
$488$	0	0
$489$	−6.48528	−0.293275
$490$	0	0
$491$	1.51472	0.0683583	0.0341791	−	0.999416i	$-0.489118\pi$
$491$	1.51472	0.0683583	0.0341791	+	0.999416i	$0.489118\pi$
$492$	0	0
$493$	59.9411	2.69961
$494$	0	0
$495$	−2.34315	−0.105317
$496$	0	0
$497$	0	0
$498$	0	0
$499$	44.1421	1.97607	0.988037	−	0.154219i	$-0.0492861\pi$
$499$	44.1421	1.97607	0.988037	+	0.154219i	$0.0492861\pi$
$500$	0	0
$501$	0.857864	0.0383266
$502$	0	0
$503$	−3.92893	−0.175182	−0.0875912	−	0.996157i	$-0.527917\pi$
$503$	−3.92893	−0.175182	−0.0875912	+	0.996157i	$0.527917\pi$
$504$	0	0
$505$	−11.4853	−0.511088
$506$	0	0
$507$	−3.72792	−0.165563
$508$	0	0
$509$	−33.4853	−1.48421	−0.742105	−	0.670284i	$-0.766172\pi$
$509$	−33.4853	−1.48421	−0.742105	+	0.670284i	$0.766172\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	13.6569	0.602965
$514$	0	0
$515$	7.58579	0.334270
$516$	0	0
$517$	9.65685	0.424708
$518$	0	0
$519$	4.28427	0.188059
$520$	0	0
$521$	36.6274	1.60468	0.802338	−	0.596870i	$-0.203589\pi$
$521$	36.6274	1.60468	0.802338	+	0.596870i	$0.203589\pi$
$522$	0	0
$523$	27.9411	1.22178	0.610890	−	0.791715i	$-0.290812\pi$
$523$	27.9411	1.22178	0.610890	+	0.791715i	$0.290812\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6.34315	−0.276312
$528$	0	0
$529$	8.20101	0.356566
$530$	0	0
$531$	11.3137	0.490973
$532$	0	0
$533$	11.6569	0.504914
$534$	0	0
$535$	−5.58579	−0.241495
$536$	0	0
$537$	−2.68629	−0.115922
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−23.4853	−1.00971	−0.504856	−	0.863204i	$-0.668454\pi$
$541$	−23.4853	−1.00971	−0.504856	+	0.863204i	$0.668454\pi$
$542$	0	0
$543$	−1.72792	−0.0741522
$544$	0	0
$545$	18.3137	0.784473
$546$	0	0
$547$	−2.27208	−0.0971470	−0.0485735	−	0.998820i	$-0.515468\pi$
$547$	−2.27208	−0.0971470	−0.0485735	+	0.998820i	$0.515468\pi$
$548$	0	0
$549$	−18.8284	−0.803578
$550$	0	0
$551$	44.2843	1.88657
$552$	0	0
$553$	0	0
$554$	0	0
$555$	2.34315	0.0994610
$556$	0	0
$557$	−17.3137	−0.733605	−0.366803	−	0.930299i	$-0.619548\pi$
$557$	−17.3137	−0.733605	−0.366803	+	0.930299i	$0.619548\pi$
$558$	0	0
$559$	−13.7990	−0.583635
$560$	0	0
$561$	−2.62742	−0.110930
$562$	0	0
$563$	−4.07107	−0.171575	−0.0857875	−	0.996313i	$-0.527341\pi$
$563$	−4.07107	−0.171575	−0.0857875	+	0.996313i	$0.527341\pi$
$564$	0	0
$565$	11.3137	0.475971
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−36.6274	−1.53550	−0.767751	−	0.640749i	$-0.778624\pi$
$569$	−36.6274	−1.53550	−0.767751	+	0.640749i	$0.778624\pi$
$570$	0	0
$571$	−20.9706	−0.877591	−0.438795	−	0.898587i	$-0.644595\pi$
$571$	−20.9706	−0.877591	−0.438795	+	0.898587i	$0.644595\pi$
$572$	0	0
$573$	−2.28427	−0.0954268
$574$	0	0
$575$	−5.58579	−0.232943
$576$	0	0
$577$	−22.2843	−0.927706	−0.463853	−	0.885912i	$-0.653533\pi$
$577$	−22.2843	−0.927706	−0.463853	+	0.885912i	$0.653533\pi$
$578$	0	0
$579$	−2.20101	−0.0914709
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−4.68629	−0.194086
$584$	0	0
$585$	−5.65685	−0.233882
$586$	0	0
$587$	−22.6863	−0.936363	−0.468182	−	0.883632i	$-0.655091\pi$
$587$	−22.6863	−0.936363	−0.468182	+	0.883632i	$0.655091\pi$
$588$	0	0
$589$	−4.68629	−0.193095
$590$	0	0
$591$	0.142136	0.00584668
$592$	0	0
$593$	29.9411	1.22953	0.614767	−	0.788709i	$-0.289250\pi$
$593$	29.9411	1.22953	0.614767	+	0.788709i	$0.289250\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−9.65685	−0.395229
$598$	0	0
$599$	42.6274	1.74171	0.870855	−	0.491541i	$-0.163566\pi$
$599$	42.6274	1.74171	0.870855	+	0.491541i	$0.163566\pi$
$600$	0	0
$601$	−34.0000	−1.38689	−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000	−1.38689	−0.693444	+	0.720510i	$0.743908\pi$
$602$	0	0
$603$	36.4853	1.48580
$604$	0	0
$605$	−10.3137	−0.419312
$606$	0	0
$607$	27.2426	1.10574	0.552872	−	0.833266i	$-0.313532\pi$
$607$	27.2426	1.10574	0.552872	+	0.833266i	$0.313532\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	23.3137	0.943172
$612$	0	0
$613$	38.9706	1.57401	0.787003	−	0.616949i	$-0.211632\pi$
$613$	38.9706	1.57401	0.787003	+	0.616949i	$0.211632\pi$
$614$	0	0
$615$	2.41421	0.0973505
$616$	0	0
$617$	−12.6863	−0.510731	−0.255365	−	0.966845i	$-0.582196\pi$
$617$	−12.6863	−0.510731	−0.255365	+	0.966845i	$0.582196\pi$
$618$	0	0
$619$	39.4558	1.58586	0.792932	−	0.609310i	$-0.208553\pi$
$619$	39.4558	1.58586	0.792932	+	0.609310i	$0.208553\pi$
$620$	0	0
$621$	13.4853	0.541146
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−1.94113	−0.0775211
$628$	0	0
$629$	−43.3137	−1.72703
$630$	0	0
$631$	−1.51472	−0.0603000	−0.0301500	−	0.999545i	$-0.509598\pi$
$631$	−1.51472	−0.0603000	−0.0301500	+	0.999545i	$0.509598\pi$
$632$	0	0
$633$	11.0294	0.438381
$634$	0	0
$635$	−4.34315	−0.172352
$636$	0	0
$637$	0	0
$638$	0	0
$639$	33.9411	1.34269
$640$	0	0
$641$	14.1127	0.557418	0.278709	−	0.960376i	$-0.410093\pi$
$641$	14.1127	0.557418	0.278709	+	0.960376i	$0.410093\pi$
$642$	0	0
$643$	26.0000	1.02534	0.512670	−	0.858586i	$-0.328656\pi$
$643$	26.0000	1.02534	0.512670	+	0.858586i	$0.328656\pi$
$644$	0	0
$645$	−2.85786	−0.112528
$646$	0	0
$647$	−20.7574	−0.816056	−0.408028	−	0.912969i	$-0.633783\pi$
$647$	−20.7574	−0.816056	−0.408028	+	0.912969i	$0.633783\pi$
$648$	0	0
$649$	−3.31371	−0.130074
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−15.6569	−0.612700	−0.306350	−	0.951919i	$-0.599108\pi$
$653$	−15.6569	−0.612700	−0.306350	+	0.951919i	$0.599108\pi$
$654$	0	0
$655$	13.6569	0.533617
$656$	0	0
$657$	−10.3431	−0.403525
$658$	0	0
$659$	12.6863	0.494188	0.247094	−	0.968992i	$-0.420524\pi$
$659$	12.6863	0.494188	0.247094	+	0.968992i	$0.420524\pi$
$660$	0	0
$661$	−50.3137	−1.95698	−0.978488	−	0.206303i	$-0.933857\pi$
$661$	−50.3137	−1.95698	−0.978488	+	0.206303i	$0.933857\pi$
$662$	0	0
$663$	−6.34315	−0.246347
$664$	0	0
$665$	0	0
$666$	0	0
$667$	43.7279	1.69315
$668$	0	0
$669$	−6.20101	−0.239745
$670$	0	0
$671$	5.51472	0.212893
$672$	0	0
$673$	−5.65685	−0.218056	−0.109028	−	0.994039i	$-0.534774\pi$
$673$	−5.65685	−0.218056	−0.109028	+	0.994039i	$0.534774\pi$
$674$	0	0
$675$	−2.41421	−0.0929231
$676$	0	0
$677$	22.9706	0.882830	0.441415	−	0.897303i	$-0.354477\pi$
$677$	22.9706	0.882830	0.441415	+	0.897303i	$0.354477\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	5.79899	0.222218
$682$	0	0
$683$	−1.92893	−0.0738085	−0.0369043	−	0.999319i	$-0.511750\pi$
$683$	−1.92893	−0.0738085	−0.0369043	+	0.999319i	$0.511750\pi$
$684$	0	0
$685$	−4.00000	−0.152832
$686$	0	0
$687$	−5.79899	−0.221245
$688$	0	0
$689$	−11.3137	−0.431018
$690$	0	0
$691$	−42.7696	−1.62703	−0.813515	−	0.581544i	$-0.802449\pi$
$691$	−42.7696	−1.62703	−0.813515	+	0.581544i	$0.802449\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.48528	0.0942721
$696$	0	0
$697$	−44.6274	−1.69038
$698$	0	0
$699$	4.82843	0.182628
$700$	0	0
$701$	−11.0000	−0.415464	−0.207732	−	0.978186i	$-0.566608\pi$
$701$	−11.0000	−0.415464	−0.207732	+	0.978186i	$0.566608\pi$
$702$	0	0
$703$	−32.0000	−1.20690
$704$	0	0
$705$	4.82843	0.181849
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−35.4264	−1.33047	−0.665233	−	0.746636i	$-0.731668\pi$
$709$	−35.4264	−1.33047	−0.665233	+	0.746636i	$0.731668\pi$
$710$	0	0
$711$	11.3137	0.424297
$712$	0	0
$713$	−4.62742	−0.173298
$714$	0	0
$715$	1.65685	0.0619628
$716$	0	0
$717$	−12.6274	−0.471580
$718$	0	0
$719$	−25.7990	−0.962140	−0.481070	−	0.876682i	$-0.659752\pi$
$719$	−25.7990	−0.962140	−0.481070	+	0.876682i	$0.659752\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−4.14214	−0.154048
$724$	0	0
$725$	−7.82843	−0.290740
$726$	0	0
$727$	38.0711	1.41198	0.705989	−	0.708223i	$-0.250503\pi$
$727$	38.0711	1.41198	0.705989	+	0.708223i	$0.250503\pi$
$728$	0	0
$729$	−18.1716	−0.673021
$730$	0	0
$731$	52.8284	1.95393
$732$	0	0
$733$	7.65685	0.282812	0.141406	−	0.989952i	$-0.454838\pi$
$733$	7.65685	0.282812	0.141406	+	0.989952i	$0.454838\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−10.6863	−0.393635
$738$	0	0
$739$	16.8284	0.619044	0.309522	−	0.950892i	$-0.399831\pi$
$739$	16.8284	0.619044	0.309522	+	0.950892i	$0.399831\pi$
$740$	0	0
$741$	−4.68629	−0.172155
$742$	0	0
$743$	−10.7574	−0.394649	−0.197325	−	0.980338i	$-0.563225\pi$
$743$	−10.7574	−0.394649	−0.197325	+	0.980338i	$0.563225\pi$
$744$	0	0
$745$	4.65685	0.170614
$746$	0	0
$747$	13.4558	0.492324
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−12.0000	−0.437886	−0.218943	−	0.975738i	$-0.570261\pi$
$751$	−12.0000	−0.437886	−0.218943	+	0.975738i	$0.570261\pi$
$752$	0	0
$753$	−11.3726	−0.414440
$754$	0	0
$755$	−11.1716	−0.406575
$756$	0	0
$757$	6.00000	0.218074	0.109037	−	0.994038i	$-0.465223\pi$
$757$	6.00000	0.218074	0.109037	+	0.994038i	$0.465223\pi$
$758$	0	0
$759$	−1.91674	−0.0695732
$760$	0	0
$761$	−43.9411	−1.59286	−0.796432	−	0.604728i	$-0.793282\pi$
$761$	−43.9411	−1.59286	−0.796432	+	0.604728i	$0.793282\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	21.6569	0.783005
$766$	0	0
$767$	−8.00000	−0.288863
$768$	0	0
$769$	43.2548	1.55981	0.779905	−	0.625898i	$-0.215268\pi$
$769$	43.2548	1.55981	0.779905	+	0.625898i	$0.215268\pi$
$770$	0	0
$771$	6.34315	0.228443
$772$	0	0
$773$	24.3431	0.875562	0.437781	−	0.899082i	$-0.355765\pi$
$773$	24.3431	0.875562	0.437781	+	0.899082i	$0.355765\pi$
$774$	0	0
$775$	0.828427	0.0297580
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−32.9706	−1.18129
$780$	0	0
$781$	−9.94113	−0.355721
$782$	0	0
$783$	18.8995	0.675413
$784$	0	0
$785$	1.31371	0.0468883
$786$	0	0
$787$	−1.58579	−0.0565272	−0.0282636	−	0.999601i	$-0.508998\pi$
$787$	−1.58579	−0.0565272	−0.0282636	+	0.999601i	$0.508998\pi$
$788$	0	0
$789$	−6.51472	−0.231930
$790$	0	0
$791$	0	0
$792$	0	0
$793$	13.3137	0.472784
$794$	0	0
$795$	−2.34315	−0.0831028
$796$	0	0
$797$	−35.3137	−1.25088	−0.625438	−	0.780274i	$-0.715080\pi$
$797$	−35.3137	−1.25088	−0.625438	+	0.780274i	$0.715080\pi$
$798$	0	0
$799$	−89.2548	−3.15761
$800$	0	0
$801$	15.1127	0.533981
$802$	0	0
$803$	3.02944	0.106907
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2.75736	−0.0970636
$808$	0	0
$809$	−35.9706	−1.26466	−0.632329	−	0.774700i	$-0.717901\pi$
$809$	−35.9706	−1.26466	−0.632329	+	0.774700i	$0.717901\pi$
$810$	0	0
$811$	52.1421	1.83096	0.915479	−	0.402366i	$-0.131812\pi$
$811$	52.1421	1.83096	0.915479	+	0.402366i	$0.131812\pi$
$812$	0	0
$813$	−1.37258	−0.0481386
$814$	0	0
$815$	−15.6569	−0.548436
$816$	0	0
$817$	39.0294	1.36547
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−32.6274	−1.13870	−0.569352	−	0.822094i	$-0.692806\pi$
$821$	−32.6274	−1.13870	−0.569352	+	0.822094i	$0.692806\pi$
$822$	0	0
$823$	30.0122	1.04616	0.523080	−	0.852284i	$-0.324783\pi$
$823$	30.0122	1.04616	0.523080	+	0.852284i	$0.324783\pi$
$824$	0	0
$825$	0.343146	0.0119468
$826$	0	0
$827$	13.5269	0.470377	0.235188	−	0.971950i	$-0.424429\pi$
$827$	13.5269	0.470377	0.235188	+	0.971950i	$0.424429\pi$
$828$	0	0
$829$	5.31371	0.184553	0.0922764	−	0.995733i	$-0.470586\pi$
$829$	5.31371	0.184553	0.0922764	+	0.995733i	$0.470586\pi$
$830$	0	0
$831$	11.8579	0.411345
$832$	0	0
$833$	0	0
$834$	0	0
$835$	2.07107	0.0716723
$836$	0	0
$837$	−2.00000	−0.0691301
$838$	0	0
$839$	20.1421	0.695384	0.347692	−	0.937609i	$-0.386966\pi$
$839$	20.1421	0.695384	0.347692	+	0.937609i	$0.386966\pi$
$840$	0	0
$841$	32.2843	1.11325
$842$	0	0
$843$	1.11270	0.0383234
$844$	0	0
$845$	−9.00000	−0.309609
$846$	0	0
$847$	0	0
$848$	0	0
$849$	7.45584	0.255884
$850$	0	0
$851$	−31.5980	−1.08316
$852$	0	0
$853$	12.6863	0.434370	0.217185	−	0.976130i	$-0.430312\pi$
$853$	12.6863	0.434370	0.217185	+	0.976130i	$0.430312\pi$
$854$	0	0
$855$	16.0000	0.547188
$856$	0	0
$857$	−18.9706	−0.648022	−0.324011	−	0.946053i	$-0.605031\pi$
$857$	−18.9706	−0.648022	−0.324011	+	0.946053i	$0.605031\pi$
$858$	0	0
$859$	−30.6274	−1.04499	−0.522497	−	0.852641i	$-0.674999\pi$
$859$	−30.6274	−1.04499	−0.522497	+	0.852641i	$0.674999\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−56.0122	−1.90668	−0.953339	−	0.301903i	$-0.902378\pi$
$863$	−56.0122	−1.90668	−0.953339	+	0.301903i	$0.902378\pi$
$864$	0	0
$865$	10.3431	0.351678
$866$	0	0
$867$	17.2426	0.585591
$868$	0	0
$869$	−3.31371	−0.112410
$870$	0	0
$871$	−25.7990	−0.874165
$872$	0	0
$873$	−16.9706	−0.574367
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−28.2843	−0.955092	−0.477546	−	0.878607i	$-0.658474\pi$
$877$	−28.2843	−0.955092	−0.477546	+	0.878607i	$0.658474\pi$
$878$	0	0
$879$	7.02944	0.237097
$880$	0	0
$881$	−26.4558	−0.891320	−0.445660	−	0.895202i	$-0.647031\pi$
$881$	−26.4558	−0.891320	−0.445660	+	0.895202i	$0.647031\pi$
$882$	0	0
$883$	6.68629	0.225012	0.112506	−	0.993651i	$-0.464112\pi$
$883$	6.68629	0.225012	0.112506	+	0.993651i	$0.464112\pi$
$884$	0	0
$885$	−1.65685	−0.0556945
$886$	0	0
$887$	17.5858	0.590473	0.295236	−	0.955424i	$-0.404602\pi$
$887$	17.5858	0.590473	0.295236	+	0.955424i	$0.404602\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	6.20101	0.207742
$892$	0	0
$893$	−65.9411	−2.20664
$894$	0	0
$895$	−6.48528	−0.216779
$896$	0	0
$897$	−4.62742	−0.154505
$898$	0	0
$899$	−6.48528	−0.216296
$900$	0	0
$901$	43.3137	1.44299
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4.17157	−0.138668
$906$	0	0
$907$	46.2132	1.53448	0.767242	−	0.641358i	$-0.221628\pi$
$907$	46.2132	1.53448	0.767242	+	0.641358i	$0.221628\pi$
$908$	0	0
$909$	32.4853	1.07747
$910$	0	0
$911$	−18.4853	−0.612445	−0.306222	−	0.951960i	$-0.599065\pi$
$911$	−18.4853	−0.612445	−0.306222	+	0.951960i	$0.599065\pi$
$912$	0	0
$913$	−3.94113	−0.130432
$914$	0	0
$915$	2.75736	0.0911555
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−33.2548	−1.09698	−0.548488	−	0.836159i	$-0.684796\pi$
$919$	−33.2548	−1.09698	−0.548488	+	0.836159i	$0.684796\pi$
$920$	0	0
$921$	1.97056	0.0649323
$922$	0	0
$923$	−24.0000	−0.789970
$924$	0	0
$925$	5.65685	0.185996
$926$	0	0
$927$	−21.4558	−0.704702
$928$	0	0
$929$	9.20101	0.301875	0.150938	−	0.988543i	$-0.451771\pi$
$929$	9.20101	0.301875	0.150938	+	0.988543i	$0.451771\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	8.97056	0.293683
$934$	0	0
$935$	−6.34315	−0.207443
$936$	0	0
$937$	18.6274	0.608531	0.304266	−	0.952587i	$-0.401589\pi$
$937$	18.6274	0.608531	0.304266	+	0.952587i	$0.401589\pi$
$938$	0	0
$939$	8.68629	0.283466
$940$	0	0
$941$	−10.0000	−0.325991	−0.162995	−	0.986627i	$-0.552116\pi$
$941$	−10.0000	−0.325991	−0.162995	+	0.986627i	$0.552116\pi$
$942$	0	0
$943$	−32.5563	−1.06018
$944$	0	0
$945$	0	0
$946$	0	0
$947$	11.1005	0.360718	0.180359	−	0.983601i	$-0.442274\pi$
$947$	11.1005	0.360718	0.180359	+	0.983601i	$0.442274\pi$
$948$	0	0
$949$	7.31371	0.237413
$950$	0	0
$951$	−9.11270	−0.295499
$952$	0	0
$953$	−54.6274	−1.76956	−0.884778	−	0.466013i	$-0.845690\pi$
$953$	−54.6274	−1.76956	−0.884778	+	0.466013i	$0.845690\pi$
$954$	0	0
$955$	−5.51472	−0.178452
$956$	0	0
$957$	−2.68629	−0.0868355
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.3137	−0.977862
$962$	0	0
$963$	15.7990	0.509115
$964$	0	0
$965$	−5.31371	−0.171054
$966$	0	0
$967$	−24.7574	−0.796143	−0.398072	−	0.917354i	$-0.630320\pi$
$967$	−24.7574	−0.796143	−0.398072	+	0.917354i	$0.630320\pi$
$968$	0	0
$969$	17.9411	0.576352
$970$	0	0
$971$	8.00000	0.256732	0.128366	−	0.991727i	$-0.459027\pi$
$971$	8.00000	0.256732	0.128366	+	0.991727i	$0.459027\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0.828427	0.0265309
$976$	0	0
$977$	30.2843	0.968880	0.484440	−	0.874825i	$-0.339023\pi$
$977$	30.2843	0.968880	0.484440	+	0.874825i	$0.339023\pi$
$978$	0	0
$979$	−4.42641	−0.141469
$980$	0	0
$981$	−51.7990	−1.65381
$982$	0	0
$983$	−41.3848	−1.31997	−0.659985	−	0.751279i	$-0.729437\pi$
$983$	−41.3848	−1.31997	−0.659985	+	0.751279i	$0.729437\pi$
$984$	0	0
$985$	0.343146	0.0109335
$986$	0	0
$987$	0	0
$988$	0	0
$989$	38.5391	1.22547
$990$	0	0
$991$	4.82843	0.153380	0.0766900	−	0.997055i	$-0.475565\pi$
$991$	4.82843	0.153380	0.0766900	+	0.997055i	$0.475565\pi$
$992$	0	0
$993$	10.9706	0.348140
$994$	0	0
$995$	−23.3137	−0.739094
$996$	0	0
$997$	−14.3431	−0.454252	−0.227126	−	0.973865i	$-0.572933\pi$
$997$	−14.3431	−0.454252	−0.227126	+	0.973865i	$0.572933\pi$
$998$	0	0
$999$	−13.6569	−0.432084

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1960.2.a.p.1.2		2
4.3	odd	2		3920.2.a.bz.1.1		2
5.4	even	2		9800.2.a.bz.1.1		2
7.2	even	3		280.2.q.d.81.1	✓	4
7.3	odd	6		1960.2.q.q.961.2		4
7.4	even	3		280.2.q.d.121.1	yes	4
7.5	odd	6		1960.2.q.q.361.2		4
7.6	odd	2		1960.2.a.t.1.1		2
21.2	odd	6		2520.2.bi.k.361.2		4
21.11	odd	6		2520.2.bi.k.1801.2		4
28.11	odd	6		560.2.q.j.401.2		4
28.23	odd	6		560.2.q.j.81.2		4
28.27	even	2		3920.2.a.bp.1.2		2
35.2	odd	12		1400.2.bh.g.249.3		8
35.4	even	6		1400.2.q.h.401.2		4
35.9	even	6		1400.2.q.h.1201.2		4
35.18	odd	12		1400.2.bh.g.849.3		8
35.23	odd	12		1400.2.bh.g.249.2		8
35.32	odd	12		1400.2.bh.g.849.2		8
35.34	odd	2		9800.2.a.br.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.q.d.81.1	✓	4	7.2	even	3
280.2.q.d.121.1	yes	4	7.4	even	3
560.2.q.j.81.2		4	28.23	odd	6
560.2.q.j.401.2		4	28.11	odd	6
1400.2.q.h.401.2		4	35.4	even	6
1400.2.q.h.1201.2		4	35.9	even	6
1400.2.bh.g.249.2		8	35.23	odd	12
1400.2.bh.g.249.3		8	35.2	odd	12
1400.2.bh.g.849.2		8	35.32	odd	12
1400.2.bh.g.849.3		8	35.18	odd	12
1960.2.a.p.1.2		2	1.1	even	1	trivial
1960.2.a.t.1.1		2	7.6	odd	2
1960.2.q.q.361.2		4	7.5	odd	6
1960.2.q.q.961.2		4	7.3	odd	6
2520.2.bi.k.361.2		4	21.2	odd	6
2520.2.bi.k.1801.2		4	21.11	odd	6
3920.2.a.bp.1.2		2	28.27	even	2
3920.2.a.bz.1.1		2	4.3	odd	2
9800.2.a.br.1.2		2	35.34	odd	2
9800.2.a.bz.1.1		2	5.4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1960.a (trivial)

\(f(q)\)	\(=\)	\(q+0.414214 q^{3} +1.00000 q^{5} -2.82843 q^{9} +O(q^{10})\)	\(q+0.414214 q^{3} +1.00000 q^{5} -2.82843 q^{9} +0.828427 q^{11} +2.00000 q^{13} +0.414214 q^{15} -7.65685 q^{17} -5.65685 q^{19} -5.58579 q^{23} +1.00000 q^{25} -2.41421 q^{27} -7.82843 q^{29} +0.828427 q^{31} +0.343146 q^{33} +5.65685 q^{37} +0.828427 q^{39} +5.82843 q^{41} -6.89949 q^{43} -2.82843 q^{45} +11.6569 q^{47} -3.17157 q^{51} -5.65685 q^{53} +0.828427 q^{55} -2.34315 q^{57} -4.00000 q^{59} +6.65685 q^{61} +2.00000 q^{65} -12.8995 q^{67} -2.31371 q^{69} -12.0000 q^{71} +3.65685 q^{73} +0.414214 q^{75} -4.00000 q^{79} +7.48528 q^{81} -4.75736 q^{83} -7.65685 q^{85} -3.24264 q^{87} -5.34315 q^{89} +0.343146 q^{93} -5.65685 q^{95} +6.00000 q^{97} -2.34315 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^5	\( 2 q - 2 q^{3} + 2 q^{5} - 4 q^{11} + 4 q^{13} - 2 q^{15} - 4 q^{17} - 14 q^{23} + 2 q^{25} - 2 q^{27} - 10 q^{29} - 4 q^{31} + 12 q^{33} - 4 q^{39} + 6 q^{41} + 6 q^{43} + 12 q^{47} - 12 q^{51} - 4 q^{55} - 16 q^{57} - 8 q^{59} + 2 q^{61} + 4 q^{65} - 6 q^{67} + 18 q^{69} - 24 q^{71} - 4 q^{73} - 2 q^{75} - 8 q^{79} - 2 q^{81} - 18 q^{83} - 4 q^{85} + 2 q^{87} - 22 q^{89} + 12 q^{93} + 12 q^{97} - 16 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 - 4 * q^11 + 4 * q^13 - 2 * q^15 - 4 * q^17 - 14 * q^23 + 2 * q^25 - 2 * q^27 - 10 * q^29 - 4 * q^31 + 12 * q^33 - 4 * q^39 + 6 * q^41 + 6 * q^43 + 12 * q^47 - 12 * q^51 - 4 * q^55 - 16 * q^57 - 8 * q^59 + 2 * q^61 + 4 * q^65 - 6 * q^67 + 18 * q^69 - 24 * q^71 - 4 * q^73 - 2 * q^75 - 8 * q^79 - 2 * q^81 - 18 * q^83 - 4 * q^85 + 2 * q^87 - 22 * q^89 + 12 * q^93 + 12 * q^97 - 16 * q^99

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