LMFDB - Embedded newform 3920.2.a.bv.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3920,2,Mod(1,3920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3920, 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("3920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3920 = 2^{4} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3920.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:	$31.3013575923$
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 35)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	3920.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.414214 q^{3} -1.00000 q^{5} -2.82843 q^{9} +O(q^{10})$	$q-0.414214 q^{3} -1.00000 q^{5} -2.82843 q^{9} +0.828427 q^{11} +4.82843 q^{13} +0.414214 q^{15} -4.82843 q^{17} +2.82843 q^{19} -0.414214 q^{23} +1.00000 q^{25} +2.41421 q^{27} -1.00000 q^{29} -6.00000 q^{31} -0.343146 q^{33} -2.00000 q^{39} +7.82843 q^{41} -3.58579 q^{43} +2.82843 q^{45} +2.00000 q^{47} +2.00000 q^{51} -1.17157 q^{53} -0.828427 q^{55} -1.17157 q^{57} +4.48528 q^{59} -5.48528 q^{61} -4.82843 q^{65} -9.58579 q^{67} +0.171573 q^{69} -4.48528 q^{71} +0.828427 q^{73} -0.414214 q^{75} -14.8284 q^{79} +7.48528 q^{81} +13.7279 q^{83} +4.82843 q^{85} +0.414214 q^{87} +8.65685 q^{89} +2.48528 q^{93} -2.82843 q^{95} -11.6569 q^{97} -2.34315 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 2 q^{23} + 2 q^{25} + 2 q^{27} - 2 q^{29} - 12 q^{31} - 12 q^{33} - 4 q^{39} + 10 q^{41} - 10 q^{43} + 4 q^{47} + 4 q^{51} - 8 q^{53} + 4 q^{55} - 8 q^{57} - 8 q^{59} + 6 q^{61} - 4 q^{65} - 22 q^{67} + 6 q^{69} + 8 q^{71} - 4 q^{73} + 2 q^{75} - 24 q^{79} - 2 q^{81} + 2 q^{83} + 4 q^{85} - 2 q^{87} + 6 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 + 2 * q^23 + 2 * q^25 + 2 * q^27 - 2 * q^29 - 12 * q^31 - 12 * q^33 - 4 * q^39 + 10 * q^41 - 10 * q^43 + 4 * q^47 + 4 * q^51 - 8 * q^53 + 4 * q^55 - 8 * q^57 - 8 * q^59 + 6 * q^61 - 4 * q^65 - 22 * q^67 + 6 * q^69 + 8 * q^71 - 4 * q^73 + 2 * q^75 - 24 * q^79 - 2 * q^81 + 2 * q^83 + 4 * q^85 - 2 * q^87 + 6 * 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	$-0.538153\pi$
$3$	−0.414214	−0.239146	−0.119573	+	0.992825i	$0.538153\pi$
$4$	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
$13$	4.82843	1.33916	0.669582	−	0.742738i	$-0.266473\pi$
$13$	4.82843	1.33916	0.669582	+	0.742738i	$0.266473\pi$
$14$	0	0
$15$	0.414214	0.106949
$16$	0	0
$17$	−4.82843	−1.17107	−0.585533	−	0.810649i	$-0.699115\pi$
$17$	−4.82843	−1.17107	−0.585533	+	0.810649i	$0.699115\pi$
$18$	0	0
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.414214	−0.0863695	−0.0431847	−	0.999067i	$-0.513750\pi$
$23$	−0.414214	−0.0863695	−0.0431847	+	0.999067i	$0.513750\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	2.41421	0.464616
$28$	0	0
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	0	0
$33$	−0.343146	−0.0597340
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	7.82843	1.22259	0.611297	−	0.791401i	$-0.290648\pi$
$41$	7.82843	1.22259	0.611297	+	0.791401i	$0.290648\pi$
$42$	0	0
$43$	−3.58579	−0.546827	−0.273414	−	0.961897i	$-0.588153\pi$
$43$	−3.58579	−0.546827	−0.273414	+	0.961897i	$0.588153\pi$
$44$	0	0
$45$	2.82843	0.421637
$46$	0	0
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	2.00000	0.280056
$52$	0	0
$53$	−1.17157	−0.160928	−0.0804640	−	0.996758i	$-0.525640\pi$
$53$	−1.17157	−0.160928	−0.0804640	+	0.996758i	$0.525640\pi$
$54$	0	0
$55$	−0.828427	−0.111705
$56$	0	0
$57$	−1.17157	−0.155179
$58$	0	0
$59$	4.48528	0.583934	0.291967	−	0.956428i	$-0.405690\pi$
$59$	4.48528	0.583934	0.291967	+	0.956428i	$0.405690\pi$
$60$	0	0
$61$	−5.48528	−0.702318	−0.351159	−	0.936316i	$-0.614212\pi$
$61$	−5.48528	−0.702318	−0.351159	+	0.936316i	$0.614212\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.82843	−0.598893
$66$	0	0
$67$	−9.58579	−1.17109	−0.585545	−	0.810640i	$-0.699119\pi$
$67$	−9.58579	−1.17109	−0.585545	+	0.810640i	$0.699119\pi$
$68$	0	0
$69$	0.171573	0.0206549
$70$	0	0
$71$	−4.48528	−0.532305	−0.266152	−	0.963931i	$-0.585752\pi$
$71$	−4.48528	−0.532305	−0.266152	+	0.963931i	$0.585752\pi$
$72$	0	0
$73$	0.828427	0.0969601	0.0484800	−	0.998824i	$-0.484562\pi$
$73$	0.828427	0.0969601	0.0484800	+	0.998824i	$0.484562\pi$
$74$	0	0
$75$	−0.414214	−0.0478293
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−14.8284	−1.66833	−0.834164	−	0.551516i	$-0.814049\pi$
$79$	−14.8284	−1.66833	−0.834164	+	0.551516i	$0.814049\pi$
$80$	0	0
$81$	7.48528	0.831698
$82$	0	0
$83$	13.7279	1.50684	0.753418	−	0.657542i	$-0.228404\pi$
$83$	13.7279	1.50684	0.753418	+	0.657542i	$0.228404\pi$
$84$	0	0
$85$	4.82843	0.523716
$86$	0	0
$87$	0.414214	0.0444084
$88$	0	0
$89$	8.65685	0.917625	0.458812	−	0.888533i	$-0.348275\pi$
$89$	8.65685	0.917625	0.458812	+	0.888533i	$0.348275\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2.48528	0.257712
$94$	0	0
$95$	−2.82843	−0.290191
$96$	0	0
$97$	−11.6569	−1.18357	−0.591787	−	0.806094i	$-0.701577\pi$
$97$	−11.6569	−1.18357	−0.591787	+	0.806094i	$0.701577\pi$
$98$	0	0
$99$	−2.34315	−0.235495
$100$	0	0
$101$	10.3137	1.02625	0.513126	−	0.858313i	$-0.328487\pi$
$101$	10.3137	1.02625	0.513126	+	0.858313i	$0.328487\pi$
$102$	0	0
$103$	−2.41421	−0.237880	−0.118940	−	0.992901i	$-0.537950\pi$
$103$	−2.41421	−0.237880	−0.118940	+	0.992901i	$0.537950\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.2426	−1.08687	−0.543434	−	0.839452i	$-0.682876\pi$
$107$	−11.2426	−1.08687	−0.543434	+	0.839452i	$0.682876\pi$
$108$	0	0
$109$	−13.4853	−1.29166	−0.645828	−	0.763483i	$-0.723488\pi$
$109$	−13.4853	−1.29166	−0.645828	+	0.763483i	$0.723488\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−4.48528	−0.421940	−0.210970	−	0.977493i	$-0.567662\pi$
$113$	−4.48528	−0.421940	−0.210970	+	0.977493i	$0.567662\pi$
$114$	0	0
$115$	0.414214	0.0386256
$116$	0	0
$117$	−13.6569	−1.26258
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.3137	−0.937610
$122$	0	0
$123$	−3.24264	−0.292379
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	9.31371	0.826458	0.413229	−	0.910627i	$-0.364401\pi$
$127$	9.31371	0.826458	0.413229	+	0.910627i	$0.364401\pi$
$128$	0	0
$129$	1.48528	0.130772
$130$	0	0
$131$	−19.3137	−1.68745	−0.843723	−	0.536778i	$-0.819641\pi$
$131$	−19.3137	−1.68745	−0.843723	+	0.536778i	$0.819641\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−2.41421	−0.207782
$136$	0	0
$137$	−9.65685	−0.825041	−0.412520	−	0.910948i	$-0.635351\pi$
$137$	−9.65685	−0.825041	−0.412520	+	0.910948i	$0.635351\pi$
$138$	0	0
$139$	−16.1421	−1.36916	−0.684579	−	0.728939i	$-0.740014\pi$
$139$	−16.1421	−1.36916	−0.684579	+	0.728939i	$0.740014\pi$
$140$	0	0
$141$	−0.828427	−0.0697661
$142$	0	0
$143$	4.00000	0.334497
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.17157	−0.177902	−0.0889511	−	0.996036i	$-0.528351\pi$
$149$	−2.17157	−0.177902	−0.0889511	+	0.996036i	$0.528351\pi$
$150$	0	0
$151$	−11.6569	−0.948621	−0.474311	−	0.880358i	$-0.657303\pi$
$151$	−11.6569	−0.948621	−0.474311	+	0.880358i	$0.657303\pi$
$152$	0	0
$153$	13.6569	1.10409
$154$	0	0
$155$	6.00000	0.481932
$156$	0	0
$157$	−17.3137	−1.38178	−0.690892	−	0.722958i	$-0.742782\pi$
$157$	−17.3137	−1.38178	−0.690892	+	0.722958i	$0.742782\pi$
$158$	0	0
$159$	0.485281	0.0384853
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−12.3431	−0.966790	−0.483395	−	0.875402i	$-0.660596\pi$
$163$	−12.3431	−0.966790	−0.483395	+	0.875402i	$0.660596\pi$
$164$	0	0
$165$	0.343146	0.0267139
$166$	0	0
$167$	22.4142	1.73446	0.867232	−	0.497904i	$-0.165897\pi$
$167$	22.4142	1.73446	0.867232	+	0.497904i	$0.165897\pi$
$168$	0	0
$169$	10.3137	0.793362
$170$	0	0
$171$	−8.00000	−0.611775
$172$	0	0
$173$	−3.31371	−0.251937	−0.125968	−	0.992034i	$-0.540204\pi$
$173$	−3.31371	−0.251937	−0.125968	+	0.992034i	$0.540204\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1.85786	−0.139646
$178$	0	0
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	−2.65685	−0.197482	−0.0987412	−	0.995113i	$-0.531482\pi$
$181$	−2.65685	−0.197482	−0.0987412	+	0.995113i	$0.531482\pi$
$182$	0	0
$183$	2.27208	0.167957
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.00000	−0.292509
$188$	0	0
$189$	0	0
$190$	0	0
$191$	12.8284	0.928232	0.464116	−	0.885774i	$-0.346372\pi$
$191$	12.8284	0.928232	0.464116	+	0.885774i	$0.346372\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	0	0
$195$	2.00000	0.143223
$196$	0	0
$197$	−12.3431	−0.879413	−0.439706	−	0.898142i	$-0.644917\pi$
$197$	−12.3431	−0.879413	−0.439706	+	0.898142i	$0.644917\pi$
$198$	0	0
$199$	−9.65685	−0.684556	−0.342278	−	0.939599i	$-0.611199\pi$
$199$	−9.65685	−0.684556	−0.342278	+	0.939599i	$0.611199\pi$
$200$	0	0
$201$	3.97056	0.280062
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−7.82843	−0.546761
$206$	0	0
$207$	1.17157	0.0814299
$208$	0	0
$209$	2.34315	0.162079
$210$	0	0
$211$	−20.4853	−1.41026	−0.705132	−	0.709076i	$-0.749112\pi$
$211$	−20.4853	−1.41026	−0.705132	+	0.709076i	$0.749112\pi$
$212$	0	0
$213$	1.85786	0.127299
$214$	0	0
$215$	3.58579	0.244549
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−0.343146	−0.0231876
$220$	0	0
$221$	−23.3137	−1.56825
$222$	0	0
$223$	0.343146	0.0229787	0.0114894	−	0.999934i	$-0.496343\pi$
$223$	0.343146	0.0229787	0.0114894	+	0.999934i	$0.496343\pi$
$224$	0	0
$225$	−2.82843	−0.188562
$226$	0	0
$227$	−6.97056	−0.462652	−0.231326	−	0.972876i	$-0.574306\pi$
$227$	−6.97056	−0.462652	−0.231326	+	0.972876i	$0.574306\pi$
$228$	0	0
$229$	11.6569	0.770307	0.385153	−	0.922853i	$-0.374149\pi$
$229$	11.6569	0.770307	0.385153	+	0.922853i	$0.374149\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−16.8284	−1.10247	−0.551233	−	0.834351i	$-0.685843\pi$
$233$	−16.8284	−1.10247	−0.551233	+	0.834351i	$0.685843\pi$
$234$	0	0
$235$	−2.00000	−0.130466
$236$	0	0
$237$	6.14214	0.398975
$238$	0	0
$239$	21.3137	1.37867	0.689335	−	0.724443i	$-0.257903\pi$
$239$	21.3137	1.37867	0.689335	+	0.724443i	$0.257903\pi$
$240$	0	0
$241$	−27.6569	−1.78153	−0.890767	−	0.454460i	$-0.849832\pi$
$241$	−27.6569	−1.78153	−0.890767	+	0.454460i	$0.849832\pi$
$242$	0	0
$243$	−10.3431	−0.663513
$244$	0	0
$245$	0	0
$246$	0	0
$247$	13.6569	0.868965
$248$	0	0
$249$	−5.68629	−0.360354
$250$	0	0
$251$	−9.31371	−0.587876	−0.293938	−	0.955824i	$-0.594966\pi$
$251$	−9.31371	−0.587876	−0.293938	+	0.955824i	$0.594966\pi$
$252$	0	0
$253$	−0.343146	−0.0215734
$254$	0	0
$255$	−2.00000	−0.125245
$256$	0	0
$257$	−6.34315	−0.395675	−0.197837	−	0.980235i	$-0.563392\pi$
$257$	−6.34315	−0.395675	−0.197837	+	0.980235i	$0.563392\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2.82843	0.175075
$262$	0	0
$263$	29.0416	1.79078	0.895392	−	0.445279i	$-0.146896\pi$
$263$	29.0416	1.79078	0.895392	+	0.445279i	$0.146896\pi$
$264$	0	0
$265$	1.17157	0.0719691
$266$	0	0
$267$	−3.58579	−0.219447
$268$	0	0
$269$	−20.4558	−1.24721	−0.623607	−	0.781738i	$-0.714334\pi$
$269$	−20.4558	−1.24721	−0.623607	+	0.781738i	$0.714334\pi$
$270$	0	0
$271$	16.4853	1.00141	0.500705	−	0.865618i	$-0.333074\pi$
$271$	16.4853	1.00141	0.500705	+	0.865618i	$0.333074\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.828427	0.0499560
$276$	0	0
$277$	16.1421	0.969887	0.484943	−	0.874546i	$-0.338840\pi$
$277$	16.1421	0.969887	0.484943	+	0.874546i	$0.338840\pi$
$278$	0	0
$279$	16.9706	1.01600
$280$	0	0
$281$	−30.2843	−1.80661	−0.903304	−	0.429001i	$-0.858866\pi$
$281$	−30.2843	−1.80661	−0.903304	+	0.429001i	$0.858866\pi$
$282$	0	0
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	0	0
$285$	1.17157	0.0693980
$286$	0	0
$287$	0	0
$288$	0	0
$289$	6.31371	0.371395
$290$	0	0
$291$	4.82843	0.283047
$292$	0	0
$293$	16.0000	0.934730	0.467365	−	0.884064i	$-0.345203\pi$
$293$	16.0000	0.934730	0.467365	+	0.884064i	$0.345203\pi$
$294$	0	0
$295$	−4.48528	−0.261143
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	−2.00000	−0.115663
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−4.27208	−0.245424
$304$	0	0
$305$	5.48528	0.314086
$306$	0	0
$307$	−4.75736	−0.271517	−0.135758	−	0.990742i	$-0.543347\pi$
$307$	−4.75736	−0.271517	−0.135758	+	0.990742i	$0.543347\pi$
$308$	0	0
$309$	1.00000	0.0568880
$310$	0	0
$311$	−13.1716	−0.746891	−0.373446	−	0.927652i	$-0.621824\pi$
$311$	−13.1716	−0.746891	−0.373446	+	0.927652i	$0.621824\pi$
$312$	0	0
$313$	6.34315	0.358536	0.179268	−	0.983800i	$-0.442627\pi$
$313$	6.34315	0.358536	0.179268	+	0.983800i	$0.442627\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−13.7990	−0.775028	−0.387514	−	0.921864i	$-0.626666\pi$
$317$	−13.7990	−0.775028	−0.387514	+	0.921864i	$0.626666\pi$
$318$	0	0
$319$	−0.828427	−0.0463830
$320$	0	0
$321$	4.65685	0.259920
$322$	0	0
$323$	−13.6569	−0.759888
$324$	0	0
$325$	4.82843	0.267833
$326$	0	0
$327$	5.58579	0.308895
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−22.9706	−1.26258	−0.631288	−	0.775548i	$-0.717473\pi$
$331$	−22.9706	−1.26258	−0.631288	+	0.775548i	$0.717473\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	9.58579	0.523727
$336$	0	0
$337$	9.17157	0.499607	0.249804	−	0.968296i	$-0.419634\pi$
$337$	9.17157	0.499607	0.249804	+	0.968296i	$0.419634\pi$
$338$	0	0
$339$	1.85786	0.100905
$340$	0	0
$341$	−4.97056	−0.269171
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−0.171573	−0.00923717
$346$	0	0
$347$	−7.92893	−0.425647	−0.212824	−	0.977091i	$-0.568266\pi$
$347$	−7.92893	−0.425647	−0.212824	+	0.977091i	$0.568266\pi$
$348$	0	0
$349$	15.3431	0.821300	0.410650	−	0.911793i	$-0.365302\pi$
$349$	15.3431	0.821300	0.410650	+	0.911793i	$0.365302\pi$
$350$	0	0
$351$	11.6569	0.622197
$352$	0	0
$353$	26.8284	1.42793	0.713967	−	0.700180i	$-0.246897\pi$
$353$	26.8284	1.42793	0.713967	+	0.700180i	$0.246897\pi$
$354$	0	0
$355$	4.48528	0.238054
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.0000	0.527780	0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000	0.527780	0.263890	+	0.964553i	$0.414994\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	4.27208	0.224226
$364$	0	0
$365$	−0.828427	−0.0433619
$366$	0	0
$367$	−2.75736	−0.143933	−0.0719665	−	0.997407i	$-0.522927\pi$
$367$	−2.75736	−0.143933	−0.0719665	+	0.997407i	$0.522927\pi$
$368$	0	0
$369$	−22.1421	−1.15267
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−20.9706	−1.08581	−0.542907	−	0.839793i	$-0.682677\pi$
$373$	−20.9706	−1.08581	−0.542907	+	0.839793i	$0.682677\pi$
$374$	0	0
$375$	0.414214	0.0213899
$376$	0	0
$377$	−4.82843	−0.248677
$378$	0	0
$379$	−26.8284	−1.37808	−0.689042	−	0.724722i	$-0.741968\pi$
$379$	−26.8284	−1.37808	−0.689042	+	0.724722i	$0.741968\pi$
$380$	0	0
$381$	−3.85786	−0.197644
$382$	0	0
$383$	−2.89949	−0.148157	−0.0740786	−	0.997252i	$-0.523602\pi$
$383$	−2.89949	−0.148157	−0.0740786	+	0.997252i	$0.523602\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	10.1421	0.515554
$388$	0	0
$389$	23.6569	1.19945	0.599725	−	0.800206i	$-0.295277\pi$
$389$	23.6569	1.19945	0.599725	+	0.800206i	$0.295277\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	0	0
$393$	8.00000	0.403547
$394$	0	0
$395$	14.8284	0.746099
$396$	0	0
$397$	16.6274	0.834506	0.417253	−	0.908790i	$-0.362993\pi$
$397$	16.6274	0.834506	0.417253	+	0.908790i	$0.362993\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.3137	1.51379	0.756897	−	0.653534i	$-0.226714\pi$
$401$	30.3137	1.51379	0.756897	+	0.653534i	$0.226714\pi$
$402$	0	0
$403$	−28.9706	−1.44313
$404$	0	0
$405$	−7.48528	−0.371947
$406$	0	0
$407$	0	0
$408$	0	0
$409$	14.7990	0.731763	0.365881	−	0.930661i	$-0.380768\pi$
$409$	14.7990	0.731763	0.365881	+	0.930661i	$0.380768\pi$
$410$	0	0
$411$	4.00000	0.197305
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−13.7279	−0.673877
$416$	0	0
$417$	6.68629	0.327429
$418$	0	0
$419$	−0.686292	−0.0335275	−0.0167638	−	0.999859i	$-0.505336\pi$
$419$	−0.686292	−0.0335275	−0.0167638	+	0.999859i	$0.505336\pi$
$420$	0	0
$421$	13.4853	0.657232	0.328616	−	0.944464i	$-0.393418\pi$
$421$	13.4853	0.657232	0.328616	+	0.944464i	$0.393418\pi$
$422$	0	0
$423$	−5.65685	−0.275046
$424$	0	0
$425$	−4.82843	−0.234213
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1.65685	−0.0799937
$430$	0	0
$431$	−17.7990	−0.857347	−0.428674	−	0.903459i	$-0.641019\pi$
$431$	−17.7990	−0.857347	−0.428674	+	0.903459i	$0.641019\pi$
$432$	0	0
$433$	−7.79899	−0.374796	−0.187398	−	0.982284i	$-0.560005\pi$
$433$	−7.79899	−0.374796	−0.187398	+	0.982284i	$0.560005\pi$
$434$	0	0
$435$	−0.414214	−0.0198600
$436$	0	0
$437$	−1.17157	−0.0560439
$438$	0	0
$439$	−33.9411	−1.61992	−0.809961	−	0.586484i	$-0.800512\pi$
$439$	−33.9411	−1.61992	−0.809961	+	0.586484i	$0.800512\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−30.2132	−1.43547	−0.717736	−	0.696315i	$-0.754822\pi$
$443$	−30.2132	−1.43547	−0.717736	+	0.696315i	$0.754822\pi$
$444$	0	0
$445$	−8.65685	−0.410374
$446$	0	0
$447$	0.899495	0.0425447
$448$	0	0
$449$	3.82843	0.180675	0.0903373	−	0.995911i	$-0.471205\pi$
$449$	3.82843	0.180675	0.0903373	+	0.995911i	$0.471205\pi$
$450$	0	0
$451$	6.48528	0.305380
$452$	0	0
$453$	4.82843	0.226859
$454$	0	0
$455$	0	0
$456$	0	0
$457$	24.2843	1.13597	0.567985	−	0.823039i	$-0.307723\pi$
$457$	24.2843	1.13597	0.567985	+	0.823039i	$0.307723\pi$
$458$	0	0
$459$	−11.6569	−0.544095
$460$	0	0
$461$	−41.3137	−1.92417	−0.962086	−	0.272748i	$-0.912068\pi$
$461$	−41.3137	−1.92417	−0.962086	+	0.272748i	$0.912068\pi$
$462$	0	0
$463$	37.0416	1.72147	0.860735	−	0.509053i	$-0.170004\pi$
$463$	37.0416	1.72147	0.860735	+	0.509053i	$0.170004\pi$
$464$	0	0
$465$	−2.48528	−0.115252
$466$	0	0
$467$	−3.10051	−0.143474	−0.0717371	−	0.997424i	$-0.522854\pi$
$467$	−3.10051	−0.143474	−0.0717371	+	0.997424i	$0.522854\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	7.17157	0.330449
$472$	0	0
$473$	−2.97056	−0.136587
$474$	0	0
$475$	2.82843	0.129777
$476$	0	0
$477$	3.31371	0.151724
$478$	0	0
$479$	−35.6569	−1.62920	−0.814602	−	0.580021i	$-0.803044\pi$
$479$	−35.6569	−1.62920	−0.814602	+	0.580021i	$0.803044\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	11.6569	0.529310
$486$	0	0
$487$	−4.34315	−0.196807	−0.0984034	−	0.995147i	$-0.531374\pi$
$487$	−4.34315	−0.196807	−0.0984034	+	0.995147i	$0.531374\pi$
$488$	0	0
$489$	5.11270	0.231204
$490$	0	0
$491$	−9.31371	−0.420322	−0.210161	−	0.977667i	$-0.567399\pi$
$491$	−9.31371	−0.420322	−0.210161	+	0.977667i	$0.567399\pi$
$492$	0	0
$493$	4.82843	0.217461
$494$	0	0
$495$	2.34315	0.105317
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0.828427	0.0370855	0.0185427	−	0.999828i	$-0.494097\pi$
$499$	0.828427	0.0370855	0.0185427	+	0.999828i	$0.494097\pi$
$500$	0	0
$501$	−9.28427	−0.414791
$502$	0	0
$503$	−15.8701	−0.707611	−0.353805	−	0.935319i	$-0.615113\pi$
$503$	−15.8701	−0.707611	−0.353805	+	0.935319i	$0.615113\pi$
$504$	0	0
$505$	−10.3137	−0.458954
$506$	0	0
$507$	−4.27208	−0.189730
$508$	0	0
$509$	−13.3431	−0.591425	−0.295712	−	0.955277i	$-0.595557\pi$
$509$	−13.3431	−0.591425	−0.295712	+	0.955277i	$0.595557\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	6.82843	0.301482
$514$	0	0
$515$	2.41421	0.106383
$516$	0	0
$517$	1.65685	0.0728684
$518$	0	0
$519$	1.37258	0.0602497
$520$	0	0
$521$	−14.9706	−0.655872	−0.327936	−	0.944700i	$-0.606353\pi$
$521$	−14.9706	−0.655872	−0.327936	+	0.944700i	$0.606353\pi$
$522$	0	0
$523$	35.6569	1.55917	0.779583	−	0.626299i	$-0.215431\pi$
$523$	35.6569	1.55917	0.779583	+	0.626299i	$0.215431\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	28.9706	1.26198
$528$	0	0
$529$	−22.8284	−0.992540
$530$	0	0
$531$	−12.6863	−0.550538
$532$	0	0
$533$	37.7990	1.63726
$534$	0	0
$535$	11.2426	0.486062
$536$	0	0
$537$	−4.14214	−0.178746
$538$	0	0
$539$	0	0
$540$	0	0
$541$	7.34315	0.315706	0.157853	−	0.987463i	$-0.449543\pi$
$541$	7.34315	0.315706	0.157853	+	0.987463i	$0.449543\pi$
$542$	0	0
$543$	1.10051	0.0472272
$544$	0	0
$545$	13.4853	0.577646
$546$	0	0
$547$	−24.8995	−1.06463	−0.532313	−	0.846548i	$-0.678677\pi$
$547$	−24.8995	−1.06463	−0.532313	+	0.846548i	$0.678677\pi$
$548$	0	0
$549$	15.5147	0.662152
$550$	0	0
$551$	−2.82843	−0.120495
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	22.2843	0.944215	0.472107	−	0.881541i	$-0.343493\pi$
$557$	22.2843	0.944215	0.472107	+	0.881541i	$0.343493\pi$
$558$	0	0
$559$	−17.3137	−0.732292
$560$	0	0
$561$	1.65685	0.0699524
$562$	0	0
$563$	41.7279	1.75862	0.879311	−	0.476248i	$-0.158003\pi$
$563$	41.7279	1.75862	0.879311	+	0.476248i	$0.158003\pi$
$564$	0	0
$565$	4.48528	0.188697
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7.65685	0.320992	0.160496	−	0.987036i	$-0.448691\pi$
$569$	7.65685	0.320992	0.160496	+	0.987036i	$0.448691\pi$
$570$	0	0
$571$	−9.17157	−0.383818	−0.191909	−	0.981413i	$-0.561468\pi$
$571$	−9.17157	−0.383818	−0.191909	+	0.981413i	$0.561468\pi$
$572$	0	0
$573$	−5.31371	−0.221983
$574$	0	0
$575$	−0.414214	−0.0172739
$576$	0	0
$577$	43.9411	1.82929	0.914646	−	0.404255i	$-0.132469\pi$
$577$	43.9411	1.82929	0.914646	+	0.404255i	$0.132469\pi$
$578$	0	0
$579$	0.828427	0.0344283
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−0.970563	−0.0401966
$584$	0	0
$585$	13.6569	0.564641
$586$	0	0
$587$	−34.2843	−1.41506	−0.707532	−	0.706682i	$-0.750191\pi$
$587$	−34.2843	−1.41506	−0.707532	+	0.706682i	$0.750191\pi$
$588$	0	0
$589$	−16.9706	−0.699260
$590$	0	0
$591$	5.11270	0.210308
$592$	0	0
$593$	−4.20101	−0.172515	−0.0862574	−	0.996273i	$-0.527491\pi$
$593$	−4.20101	−0.172515	−0.0862574	+	0.996273i	$0.527491\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.00000	0.163709
$598$	0	0
$599$	−6.34315	−0.259174	−0.129587	−	0.991568i	$-0.541365\pi$
$599$	−6.34315	−0.259174	−0.129587	+	0.991568i	$0.541365\pi$
$600$	0	0
$601$	−19.6569	−0.801820	−0.400910	−	0.916117i	$-0.631306\pi$
$601$	−19.6569	−0.801820	−0.400910	+	0.916117i	$0.631306\pi$
$602$	0	0
$603$	27.1127	1.10411
$604$	0	0
$605$	10.3137	0.419312
$606$	0	0
$607$	38.2132	1.55103	0.775513	−	0.631332i	$-0.217491\pi$
$607$	38.2132	1.55103	0.775513	+	0.631332i	$0.217491\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	9.65685	0.390675
$612$	0	0
$613$	35.4558	1.43205	0.716024	−	0.698076i	$-0.245960\pi$
$613$	35.4558	1.43205	0.716024	+	0.698076i	$0.245960\pi$
$614$	0	0
$615$	3.24264	0.130756
$616$	0	0
$617$	11.3137	0.455473	0.227736	−	0.973723i	$-0.426868\pi$
$617$	11.3137	0.455473	0.227736	+	0.973723i	$0.426868\pi$
$618$	0	0
$619$	25.5147	1.02552	0.512762	−	0.858531i	$-0.328622\pi$
$619$	25.5147	1.02552	0.512762	+	0.858531i	$0.328622\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−0.970563	−0.0387605
$628$	0	0
$629$	0	0
$630$	0	0
$631$	20.1421	0.801846	0.400923	−	0.916112i	$-0.368690\pi$
$631$	20.1421	0.801846	0.400923	+	0.916112i	$0.368690\pi$
$632$	0	0
$633$	8.48528	0.337260
$634$	0	0
$635$	−9.31371	−0.369603
$636$	0	0
$637$	0	0
$638$	0	0
$639$	12.6863	0.501862
$640$	0	0
$641$	31.4853	1.24359	0.621797	−	0.783179i	$-0.286403\pi$
$641$	31.4853	1.24359	0.621797	+	0.783179i	$0.286403\pi$
$642$	0	0
$643$	26.2843	1.03655	0.518275	−	0.855214i	$-0.326574\pi$
$643$	26.2843	1.03655	0.518275	+	0.855214i	$0.326574\pi$
$644$	0	0
$645$	−1.48528	−0.0584829
$646$	0	0
$647$	−31.0416	−1.22037	−0.610186	−	0.792258i	$-0.708905\pi$
$647$	−31.0416	−1.22037	−0.610186	+	0.792258i	$0.708905\pi$
$648$	0	0
$649$	3.71573	0.145855
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−19.1716	−0.750242	−0.375121	−	0.926976i	$-0.622399\pi$
$653$	−19.1716	−0.750242	−0.375121	+	0.926976i	$0.622399\pi$
$654$	0	0
$655$	19.3137	0.754649
$656$	0	0
$657$	−2.34315	−0.0914148
$658$	0	0
$659$	−21.1716	−0.824727	−0.412364	−	0.911019i	$-0.635297\pi$
$659$	−21.1716	−0.824727	−0.412364	+	0.911019i	$0.635297\pi$
$660$	0	0
$661$	−31.8284	−1.23798	−0.618991	−	0.785398i	$-0.712458\pi$
$661$	−31.8284	−1.23798	−0.618991	+	0.785398i	$0.712458\pi$
$662$	0	0
$663$	9.65685	0.375041
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.414214	0.0160384
$668$	0	0
$669$	−0.142136	−0.00549528
$670$	0	0
$671$	−4.54416	−0.175425
$672$	0	0
$673$	29.6569	1.14319	0.571594	−	0.820537i	$-0.306325\pi$
$673$	29.6569	1.14319	0.571594	+	0.820537i	$0.306325\pi$
$674$	0	0
$675$	2.41421	0.0929231
$676$	0	0
$677$	28.1421	1.08159	0.540795	−	0.841154i	$-0.318123\pi$
$677$	28.1421	1.08159	0.540795	+	0.841154i	$0.318123\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	2.88730	0.110642
$682$	0	0
$683$	34.7574	1.32995	0.664977	−	0.746864i	$-0.268441\pi$
$683$	34.7574	1.32995	0.664977	+	0.746864i	$0.268441\pi$
$684$	0	0
$685$	9.65685	0.368969
$686$	0	0
$687$	−4.82843	−0.184216
$688$	0	0
$689$	−5.65685	−0.215509
$690$	0	0
$691$	0.828427	0.0315149	0.0157574	−	0.999876i	$-0.494984\pi$
$691$	0.828427	0.0315149	0.0157574	+	0.999876i	$0.494984\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	16.1421	0.612306
$696$	0	0
$697$	−37.7990	−1.43174
$698$	0	0
$699$	6.97056	0.263651
$700$	0	0
$701$	−3.20101	−0.120900	−0.0604502	−	0.998171i	$-0.519254\pi$
$701$	−3.20101	−0.120900	−0.0604502	+	0.998171i	$0.519254\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0.828427	0.0312004
$706$	0	0
$707$	0	0
$708$	0	0
$709$	15.6863	0.589111	0.294556	−	0.955634i	$-0.404828\pi$
$709$	15.6863	0.589111	0.294556	+	0.955634i	$0.404828\pi$
$710$	0	0
$711$	41.9411	1.57292
$712$	0	0
$713$	2.48528	0.0930745
$714$	0	0
$715$	−4.00000	−0.149592
$716$	0	0
$717$	−8.82843	−0.329704
$718$	0	0
$719$	21.1127	0.787371	0.393685	−	0.919245i	$-0.371200\pi$
$719$	21.1127	0.787371	0.393685	+	0.919245i	$0.371200\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	11.4558	0.426047
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	−37.5858	−1.39398	−0.696990	−	0.717081i	$-0.745478\pi$
$727$	−37.5858	−1.39398	−0.696990	+	0.717081i	$0.745478\pi$
$728$	0	0
$729$	−18.1716	−0.673021
$730$	0	0
$731$	17.3137	0.640371
$732$	0	0
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7.94113	−0.292515
$738$	0	0
$739$	21.1127	0.776643	0.388322	−	0.921524i	$-0.373055\pi$
$739$	21.1127	0.776643	0.388322	+	0.921524i	$0.373055\pi$
$740$	0	0
$741$	−5.65685	−0.207810
$742$	0	0
$743$	16.0711	0.589590	0.294795	−	0.955560i	$-0.404749\pi$
$743$	16.0711	0.589590	0.294795	+	0.955560i	$0.404749\pi$
$744$	0	0
$745$	2.17157	0.0795603
$746$	0	0
$747$	−38.8284	−1.42066
$748$	0	0
$749$	0	0
$750$	0	0
$751$	30.3431	1.10724	0.553619	−	0.832770i	$-0.313247\pi$
$751$	30.3431	1.10724	0.553619	+	0.832770i	$0.313247\pi$
$752$	0	0
$753$	3.85786	0.140588
$754$	0	0
$755$	11.6569	0.424236
$756$	0	0
$757$	−31.4558	−1.14328	−0.571641	−	0.820504i	$-0.693693\pi$
$757$	−31.4558	−1.14328	−0.571641	+	0.820504i	$0.693693\pi$
$758$	0	0
$759$	0.142136	0.00515920
$760$	0	0
$761$	−9.31371	−0.337622	−0.168811	−	0.985648i	$-0.553993\pi$
$761$	−9.31371	−0.337622	−0.168811	+	0.985648i	$0.553993\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−13.6569	−0.493765
$766$	0	0
$767$	21.6569	0.781984
$768$	0	0
$769$	0.627417	0.0226252	0.0113126	−	0.999936i	$-0.496399\pi$
$769$	0.627417	0.0226252	0.0113126	+	0.999936i	$0.496399\pi$
$770$	0	0
$771$	2.62742	0.0946241
$772$	0	0
$773$	37.1127	1.33485	0.667425	−	0.744677i	$-0.267396\pi$
$773$	37.1127	1.33485	0.667425	+	0.744677i	$0.267396\pi$
$774$	0	0
$775$	−6.00000	−0.215526
$776$	0	0
$777$	0	0
$778$	0	0
$779$	22.1421	0.793324
$780$	0	0
$781$	−3.71573	−0.132959
$782$	0	0
$783$	−2.41421	−0.0862770
$784$	0	0
$785$	17.3137	0.617953
$786$	0	0
$787$	2.55635	0.0911240	0.0455620	−	0.998962i	$-0.485492\pi$
$787$	2.55635	0.0911240	0.0455620	+	0.998962i	$0.485492\pi$
$788$	0	0
$789$	−12.0294	−0.428259
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−26.4853	−0.940520
$794$	0	0
$795$	−0.485281	−0.0172112
$796$	0	0
$797$	8.00000	0.283375	0.141687	−	0.989911i	$-0.454747\pi$
$797$	8.00000	0.283375	0.141687	+	0.989911i	$0.454747\pi$
$798$	0	0
$799$	−9.65685	−0.341635
$800$	0	0
$801$	−24.4853	−0.865145
$802$	0	0
$803$	0.686292	0.0242187
$804$	0	0
$805$	0	0
$806$	0	0
$807$	8.47309	0.298267
$808$	0	0
$809$	35.6274	1.25259	0.626297	−	0.779585i	$-0.284570\pi$
$809$	35.6274	1.25259	0.626297	+	0.779585i	$0.284570\pi$
$810$	0	0
$811$	−20.6274	−0.724327	−0.362163	−	0.932115i	$-0.617962\pi$
$811$	−20.6274	−0.724327	−0.362163	+	0.932115i	$0.617962\pi$
$812$	0	0
$813$	−6.82843	−0.239483
$814$	0	0
$815$	12.3431	0.432362
$816$	0	0
$817$	−10.1421	−0.354828
$818$	0	0
$819$	0	0
$820$	0	0
$821$	47.9411	1.67316	0.836578	−	0.547847i	$-0.184552\pi$
$821$	47.9411	1.67316	0.836578	+	0.547847i	$0.184552\pi$
$822$	0	0
$823$	−2.07107	−0.0721929	−0.0360964	−	0.999348i	$-0.511492\pi$
$823$	−2.07107	−0.0721929	−0.0360964	+	0.999348i	$0.511492\pi$
$824$	0	0
$825$	−0.343146	−0.0119468
$826$	0	0
$827$	26.2132	0.911522	0.455761	−	0.890102i	$-0.349367\pi$
$827$	26.2132	0.911522	0.455761	+	0.890102i	$0.349367\pi$
$828$	0	0
$829$	−29.3137	−1.01811	−0.509054	−	0.860735i	$-0.670005\pi$
$829$	−29.3137	−1.01811	−0.509054	+	0.860735i	$0.670005\pi$
$830$	0	0
$831$	−6.68629	−0.231945
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−22.4142	−0.775676
$836$	0	0
$837$	−14.4853	−0.500685
$838$	0	0
$839$	15.1716	0.523781	0.261890	−	0.965098i	$-0.415654\pi$
$839$	15.1716	0.523781	0.261890	+	0.965098i	$0.415654\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	12.5442	0.432044
$844$	0	0
$845$	−10.3137	−0.354802
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−5.79899	−0.199021
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−2.54416	−0.0871102	−0.0435551	−	0.999051i	$-0.513868\pi$
$853$	−2.54416	−0.0871102	−0.0435551	+	0.999051i	$0.513868\pi$
$854$	0	0
$855$	8.00000	0.273594
$856$	0	0
$857$	−34.2843	−1.17113	−0.585564	−	0.810626i	$-0.699127\pi$
$857$	−34.2843	−1.17113	−0.585564	+	0.810626i	$0.699127\pi$
$858$	0	0
$859$	1.37258	0.0468319	0.0234160	−	0.999726i	$-0.492546\pi$
$859$	1.37258	0.0468319	0.0234160	+	0.999726i	$0.492546\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−14.5563	−0.495504	−0.247752	−	0.968823i	$-0.579692\pi$
$863$	−14.5563	−0.495504	−0.247752	+	0.968823i	$0.579692\pi$
$864$	0	0
$865$	3.31371	0.112669
$866$	0	0
$867$	−2.61522	−0.0888177
$868$	0	0
$869$	−12.2843	−0.416715
$870$	0	0
$871$	−46.2843	−1.56828
$872$	0	0
$873$	32.9706	1.11588
$874$	0	0
$875$	0	0
$876$	0	0
$877$	25.1716	0.849984	0.424992	−	0.905197i	$-0.360277\pi$
$877$	25.1716	0.849984	0.424992	+	0.905197i	$0.360277\pi$
$878$	0	0
$879$	−6.62742	−0.223537
$880$	0	0
$881$	−1.82843	−0.0616013	−0.0308006	−	0.999526i	$-0.509806\pi$
$881$	−1.82843	−0.0616013	−0.0308006	+	0.999526i	$0.509806\pi$
$882$	0	0
$883$	−18.2843	−0.615315	−0.307657	−	0.951497i	$-0.599545\pi$
$883$	−18.2843	−0.615315	−0.307657	+	0.951497i	$0.599545\pi$
$884$	0	0
$885$	1.85786	0.0624514
$886$	0	0
$887$	29.9289	1.00492	0.502458	−	0.864602i	$-0.332429\pi$
$887$	29.9289	1.00492	0.502458	+	0.864602i	$0.332429\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	6.20101	0.207742
$892$	0	0
$893$	5.65685	0.189299
$894$	0	0
$895$	−10.0000	−0.334263
$896$	0	0
$897$	0.828427	0.0276604
$898$	0	0
$899$	6.00000	0.200111
$900$	0	0
$901$	5.65685	0.188457
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.65685	0.0883168
$906$	0	0
$907$	14.2132	0.471942	0.235971	−	0.971760i	$-0.424173\pi$
$907$	14.2132	0.471942	0.235971	+	0.971760i	$0.424173\pi$
$908$	0	0
$909$	−29.1716	−0.967560
$910$	0	0
$911$	10.2010	0.337975	0.168987	−	0.985618i	$-0.445950\pi$
$911$	10.2010	0.337975	0.168987	+	0.985618i	$0.445950\pi$
$912$	0	0
$913$	11.3726	0.376378
$914$	0	0
$915$	−2.27208	−0.0751126
$916$	0	0
$917$	0	0
$918$	0	0
$919$	43.1127	1.42216	0.711078	−	0.703113i	$-0.248207\pi$
$919$	43.1127	1.42216	0.711078	+	0.703113i	$0.248207\pi$
$920$	0	0
$921$	1.97056	0.0649323
$922$	0	0
$923$	−21.6569	−0.712844
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6.82843	0.224275
$928$	0	0
$929$	−5.48528	−0.179966	−0.0899831	−	0.995943i	$-0.528681\pi$
$929$	−5.48528	−0.179966	−0.0899831	+	0.995943i	$0.528681\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	5.45584	0.178616
$934$	0	0
$935$	4.00000	0.130814
$936$	0	0
$937$	−34.6274	−1.13123	−0.565614	−	0.824670i	$-0.691361\pi$
$937$	−34.6274	−1.13123	−0.565614	+	0.824670i	$0.691361\pi$
$938$	0	0
$939$	−2.62742	−0.0857425
$940$	0	0
$941$	46.2843	1.50882	0.754412	−	0.656401i	$-0.227922\pi$
$941$	46.2843	1.50882	0.754412	+	0.656401i	$0.227922\pi$
$942$	0	0
$943$	−3.24264	−0.105595
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−33.1838	−1.07833	−0.539164	−	0.842201i	$-0.681260\pi$
$947$	−33.1838	−1.07833	−0.539164	+	0.842201i	$0.681260\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	0	0
$951$	5.71573	0.185345
$952$	0	0
$953$	−13.6569	−0.442389	−0.221194	−	0.975230i	$-0.570996\pi$
$953$	−13.6569	−0.442389	−0.221194	+	0.975230i	$0.570996\pi$
$954$	0	0
$955$	−12.8284	−0.415118
$956$	0	0
$957$	0.343146	0.0110923
$958$	0	0
$959$	0	0
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	31.7990	1.02471
$964$	0	0
$965$	2.00000	0.0643823
$966$	0	0
$967$	−37.5269	−1.20678	−0.603392	−	0.797445i	$-0.706185\pi$
$967$	−37.5269	−1.20678	−0.603392	+	0.797445i	$0.706185\pi$
$968$	0	0
$969$	5.65685	0.181724
$970$	0	0
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−2.00000	−0.0640513
$976$	0	0
$977$	−1.31371	−0.0420293	−0.0210146	−	0.999779i	$-0.506690\pi$
$977$	−1.31371	−0.0420293	−0.0210146	+	0.999779i	$0.506690\pi$
$978$	0	0
$979$	7.17157	0.229204
$980$	0	0
$981$	38.1421	1.21778
$982$	0	0
$983$	28.2132	0.899861	0.449931	−	0.893063i	$-0.351449\pi$
$983$	28.2132	0.899861	0.449931	+	0.893063i	$0.351449\pi$
$984$	0	0
$985$	12.3431	0.393285
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.48528	0.0472292
$990$	0	0
$991$	4.34315	0.137965	0.0689823	−	0.997618i	$-0.478025\pi$
$991$	4.34315	0.137965	0.0689823	+	0.997618i	$0.478025\pi$
$992$	0	0
$993$	9.51472	0.301940
$994$	0	0
$995$	9.65685	0.306143
$996$	0	0
$997$	−33.4558	−1.05956	−0.529779	−	0.848136i	$-0.677725\pi$
$997$	−33.4558	−1.05956	−0.529779	+	0.848136i	$0.677725\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3920.2.a.bv.1.1		2
4.3	odd	2		245.2.a.g.1.2		2
7.3	odd	6		560.2.q.k.401.1		4
7.5	odd	6		560.2.q.k.81.1		4
7.6	odd	2		3920.2.a.bq.1.2		2
12.11	even	2		2205.2.a.q.1.1		2
20.3	even	4		1225.2.b.h.99.1		4
20.7	even	4		1225.2.b.h.99.4		4
20.19	odd	2		1225.2.a.m.1.1		2
28.3	even	6		35.2.e.a.16.1	yes	4
28.11	odd	6		245.2.e.e.226.1		4
28.19	even	6		35.2.e.a.11.1	✓	4
28.23	odd	6		245.2.e.e.116.1		4
28.27	even	2		245.2.a.h.1.2		2
84.47	odd	6		315.2.j.e.46.2		4
84.59	odd	6		315.2.j.e.226.2		4
84.83	odd	2		2205.2.a.n.1.1		2
140.3	odd	12		175.2.k.a.149.4		8
140.19	even	6		175.2.e.c.151.2		4
140.27	odd	4		1225.2.b.g.99.4		4
140.47	odd	12		175.2.k.a.74.4		8
140.59	even	6		175.2.e.c.51.2		4
140.83	odd	4		1225.2.b.g.99.1		4
140.87	odd	12		175.2.k.a.149.1		8
140.103	odd	12		175.2.k.a.74.1		8
140.139	even	2		1225.2.a.k.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.2.e.a.11.1	✓	4	28.19	even	6
35.2.e.a.16.1	yes	4	28.3	even	6
175.2.e.c.51.2		4	140.59	even	6
175.2.e.c.151.2		4	140.19	even	6
175.2.k.a.74.1		8	140.103	odd	12
175.2.k.a.74.4		8	140.47	odd	12
175.2.k.a.149.1		8	140.87	odd	12
175.2.k.a.149.4		8	140.3	odd	12
245.2.a.g.1.2		2	4.3	odd	2
245.2.a.h.1.2		2	28.27	even	2
245.2.e.e.116.1		4	28.23	odd	6
245.2.e.e.226.1		4	28.11	odd	6
315.2.j.e.46.2		4	84.47	odd	6
315.2.j.e.226.2		4	84.59	odd	6
560.2.q.k.81.1		4	7.5	odd	6
560.2.q.k.401.1		4	7.3	odd	6
1225.2.a.k.1.1		2	140.139	even	2
1225.2.a.m.1.1		2	20.19	odd	2
1225.2.b.g.99.1		4	140.83	odd	4
1225.2.b.g.99.4		4	140.27	odd	4
1225.2.b.h.99.1		4	20.3	even	4
1225.2.b.h.99.4		4	20.7	even	4
2205.2.a.n.1.1		2	84.83	odd	2
2205.2.a.q.1.1		2	12.11	even	2
3920.2.a.bq.1.2		2	7.6	odd	2
3920.2.a.bv.1.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3920.a (trivial)

\(f(q)\)	\(=\)	\(q-0.414214 q^{3} -1.00000 q^{5} -2.82843 q^{9} +O(q^{10})\)	\(q-0.414214 q^{3} -1.00000 q^{5} -2.82843 q^{9} +0.828427 q^{11} +4.82843 q^{13} +0.414214 q^{15} -4.82843 q^{17} +2.82843 q^{19} -0.414214 q^{23} +1.00000 q^{25} +2.41421 q^{27} -1.00000 q^{29} -6.00000 q^{31} -0.343146 q^{33} -2.00000 q^{39} +7.82843 q^{41} -3.58579 q^{43} +2.82843 q^{45} +2.00000 q^{47} +2.00000 q^{51} -1.17157 q^{53} -0.828427 q^{55} -1.17157 q^{57} +4.48528 q^{59} -5.48528 q^{61} -4.82843 q^{65} -9.58579 q^{67} +0.171573 q^{69} -4.48528 q^{71} +0.828427 q^{73} -0.414214 q^{75} -14.8284 q^{79} +7.48528 q^{81} +13.7279 q^{83} +4.82843 q^{85} +0.414214 q^{87} +8.65685 q^{89} +2.48528 q^{93} -2.82843 q^{95} -11.6569 q^{97} -2.34315 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5	\( 2 q + 2 q^{3} - 2 q^{5} - 4 q^{11} + 4 q^{13} - 2 q^{15} - 4 q^{17} + 2 q^{23} + 2 q^{25} + 2 q^{27} - 2 q^{29} - 12 q^{31} - 12 q^{33} - 4 q^{39} + 10 q^{41} - 10 q^{43} + 4 q^{47} + 4 q^{51} - 8 q^{53} + 4 q^{55} - 8 q^{57} - 8 q^{59} + 6 q^{61} - 4 q^{65} - 22 q^{67} + 6 q^{69} + 8 q^{71} - 4 q^{73} + 2 q^{75} - 24 q^{79} - 2 q^{81} + 2 q^{83} + 4 q^{85} - 2 q^{87} + 6 q^{89} - 12 q^{93} - 12 q^{97} - 16 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 - 4 * q^11 + 4 * q^13 - 2 * q^15 - 4 * q^17 + 2 * q^23 + 2 * q^25 + 2 * q^27 - 2 * q^29 - 12 * q^31 - 12 * q^33 - 4 * q^39 + 10 * q^41 - 10 * q^43 + 4 * q^47 + 4 * q^51 - 8 * q^53 + 4 * q^55 - 8 * q^57 - 8 * q^59 + 6 * q^61 - 4 * q^65 - 22 * q^67 + 6 * q^69 + 8 * q^71 - 4 * q^73 + 2 * q^75 - 24 * q^79 - 2 * q^81 + 2 * q^83 + 4 * q^85 - 2 * q^87 + 6 * q^89 - 12 * q^93 - 12 * q^97 - 16 * q^99

Introduction

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