LMFDB - Embedded newform 7440.2.a.bp.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7440,2,Mod(1,7440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7440, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7440.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7440.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:	$59.4086991038$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 465)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	7440.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -0.688892 q^{7} +1.00000 q^{9} -2.00000 q^{11} +3.73975 q^{13} -1.00000 q^{15} +2.28100 q^{17} +0.688892 q^{21} -2.90321 q^{23} +1.00000 q^{25} -1.00000 q^{27} -4.02074 q^{29} -1.00000 q^{31} +2.00000 q^{33} -0.688892 q^{35} -8.79060 q^{37} -3.73975 q^{39} +2.75557 q^{41} -1.05086 q^{43} +1.00000 q^{45} -4.70964 q^{47} -6.52543 q^{49} -2.28100 q^{51} +2.14764 q^{53} -2.00000 q^{55} +6.44938 q^{59} -5.05086 q^{61} -0.688892 q^{63} +3.73975 q^{65} +5.44446 q^{67} +2.90321 q^{69} -8.96989 q^{71} -8.92396 q^{73} -1.00000 q^{75} +1.37778 q^{77} +8.38271 q^{79} +1.00000 q^{81} +10.5161 q^{83} +2.28100 q^{85} +4.02074 q^{87} +13.5002 q^{89} -2.57628 q^{91} +1.00000 q^{93} -12.0000 q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9} - 6 q^{11} - 2 q^{13} - 3 q^{15} + 2 q^{21} - 2 q^{23} + 3 q^{25} - 3 q^{27} + 8 q^{29} - 3 q^{31} + 6 q^{33} - 2 q^{35} + 2 q^{39} + 8 q^{41} + 10 q^{43} + 3 q^{45}+ \cdots - 6 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^5 - 2 * q^7 + 3 * q^9 - 6 * q^11 - 2 * q^13 - 3 * q^15 + 2 * q^21 - 2 * q^23 + 3 * q^25 - 3 * q^27 + 8 * q^29 - 3 * q^31 + 6 * q^33 - 2 * q^35 + 2 * q^39 + 8 * q^41 + 10 * q^43 + 3 * q^45 + 6 * q^47 - 13 * q^49 - 6 * q^55 - 14 * q^59 - 2 * q^61 - 2 * q^63 - 2 * q^65 + 16 * q^67 + 2 * q^69 - 20 * q^71 - 3 * q^75 + 4 * q^77 - 8 * q^79 + 3 * q^81 - 2 * q^83 - 8 * q^87 - 6 * q^89 + 12 * q^91 + 3 * q^93 - 36 * q^97 - 6 * 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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.688892	−0.260377	−0.130188	−	0.991489i	$-0.541558\pi$
$7$	−0.688892	−0.260377	−0.130188	+	0.991489i	$0.541558\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	3.73975	1.03722	0.518610	−	0.855011i	$-0.326450\pi$
$13$	3.73975	1.03722	0.518610	+	0.855011i	$0.326450\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	2.28100	0.553223	0.276611	−	0.960982i	$-0.410789\pi$
$17$	2.28100	0.553223	0.276611	+	0.960982i	$0.410789\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0.688892	0.150329
$22$	0	0
$23$	−2.90321	−0.605362	−0.302681	−	0.953092i	$-0.597882\pi$
$23$	−2.90321	−0.605362	−0.302681	+	0.953092i	$0.597882\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.02074	−0.746633	−0.373317	−	0.927704i	$-0.621780\pi$
$29$	−4.02074	−0.746633	−0.373317	+	0.927704i	$0.621780\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	−0.688892	−0.116444
$36$	0	0
$37$	−8.79060	−1.44517	−0.722583	−	0.691284i	$-0.757045\pi$
$37$	−8.79060	−1.44517	−0.722583	+	0.691284i	$0.757045\pi$
$38$	0	0
$39$	−3.73975	−0.598839
$40$	0	0
$41$	2.75557	0.430348	0.215174	−	0.976576i	$-0.430968\pi$
$41$	2.75557	0.430348	0.215174	+	0.976576i	$0.430968\pi$
$42$	0	0
$43$	−1.05086	−0.160254	−0.0801270	−	0.996785i	$-0.525533\pi$
$43$	−1.05086	−0.160254	−0.0801270	+	0.996785i	$0.525533\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−4.70964	−0.686971	−0.343485	−	0.939158i	$-0.611608\pi$
$47$	−4.70964	−0.686971	−0.343485	+	0.939158i	$0.611608\pi$
$48$	0	0
$49$	−6.52543	−0.932204
$50$	0	0
$51$	−2.28100	−0.319403
$52$	0	0
$53$	2.14764	0.295001	0.147501	−	0.989062i	$-0.452877\pi$
$53$	2.14764	0.295001	0.147501	+	0.989062i	$0.452877\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.44938	0.839638	0.419819	−	0.907608i	$-0.362094\pi$
$59$	6.44938	0.839638	0.419819	+	0.907608i	$0.362094\pi$
$60$	0	0
$61$	−5.05086	−0.646696	−0.323348	−	0.946280i	$-0.604808\pi$
$61$	−5.05086	−0.646696	−0.323348	+	0.946280i	$0.604808\pi$
$62$	0	0
$63$	−0.688892	−0.0867923
$64$	0	0
$65$	3.73975	0.463859
$66$	0	0
$67$	5.44446	0.665147	0.332573	−	0.943077i	$-0.392083\pi$
$67$	5.44446	0.665147	0.332573	+	0.943077i	$0.392083\pi$
$68$	0	0
$69$	2.90321	0.349506
$70$	0	0
$71$	−8.96989	−1.06453	−0.532265	−	0.846578i	$-0.678659\pi$
$71$	−8.96989	−1.06453	−0.532265	+	0.846578i	$0.678659\pi$
$72$	0	0
$73$	−8.92396	−1.04447	−0.522235	−	0.852802i	$-0.674902\pi$
$73$	−8.92396	−1.04447	−0.522235	+	0.852802i	$0.674902\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	1.37778	0.157013
$78$	0	0
$79$	8.38271	0.943128	0.471564	−	0.881832i	$-0.343690\pi$
$79$	8.38271	0.943128	0.471564	+	0.881832i	$0.343690\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.5161	1.15429	0.577144	−	0.816643i	$-0.304167\pi$
$83$	10.5161	1.15429	0.577144	+	0.816643i	$0.304167\pi$
$84$	0	0
$85$	2.28100	0.247409
$86$	0	0
$87$	4.02074	0.431069
$88$	0	0
$89$	13.5002	1.43102	0.715511	−	0.698601i	$-0.246194\pi$
$89$	13.5002	1.43102	0.715511	+	0.698601i	$0.246194\pi$
$90$	0	0
$91$	−2.57628	−0.270068
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	8.29529	0.825412	0.412706	−	0.910864i	$-0.364584\pi$
$101$	8.29529	0.825412	0.412706	+	0.910864i	$0.364584\pi$
$102$	0	0
$103$	−2.98418	−0.294040	−0.147020	−	0.989134i	$-0.546968\pi$
$103$	−2.98418	−0.294040	−0.147020	+	0.989134i	$0.546968\pi$
$104$	0	0
$105$	0.688892	0.0672290
$106$	0	0
$107$	−1.52543	−0.147469	−0.0737343	−	0.997278i	$-0.523492\pi$
$107$	−1.52543	−0.147469	−0.0737343	+	0.997278i	$0.523492\pi$
$108$	0	0
$109$	3.76049	0.360190	0.180095	−	0.983649i	$-0.442360\pi$
$109$	3.76049	0.360190	0.180095	+	0.983649i	$0.442360\pi$
$110$	0	0
$111$	8.79060	0.834367
$112$	0	0
$113$	−9.18421	−0.863978	−0.431989	−	0.901879i	$-0.642188\pi$
$113$	−9.18421	−0.863978	−0.431989	+	0.901879i	$0.642188\pi$
$114$	0	0
$115$	−2.90321	−0.270726
$116$	0	0
$117$	3.73975	0.345740
$118$	0	0
$119$	−1.57136	−0.144046
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−2.75557	−0.248461
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	11.4795	1.01864	0.509320	−	0.860577i	$-0.329897\pi$
$127$	11.4795	1.01864	0.509320	+	0.860577i	$0.329897\pi$
$128$	0	0
$129$	1.05086	0.0925226
$130$	0	0
$131$	−13.0716	−1.14207	−0.571035	−	0.820925i	$-0.693458\pi$
$131$	−13.0716	−1.14207	−0.571035	+	0.820925i	$0.693458\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−11.1383	−0.951607	−0.475804	−	0.879552i	$-0.657843\pi$
$137$	−11.1383	−0.951607	−0.475804	+	0.879552i	$0.657843\pi$
$138$	0	0
$139$	−10.2351	−0.868127	−0.434063	−	0.900882i	$-0.642921\pi$
$139$	−10.2351	−0.868127	−0.434063	+	0.900882i	$0.642921\pi$
$140$	0	0
$141$	4.70964	0.396623
$142$	0	0
$143$	−7.47949	−0.625467
$144$	0	0
$145$	−4.02074	−0.333905
$146$	0	0
$147$	6.52543	0.538208
$148$	0	0
$149$	−1.51114	−0.123797	−0.0618986	−	0.998082i	$-0.519716\pi$
$149$	−1.51114	−0.123797	−0.0618986	+	0.998082i	$0.519716\pi$
$150$	0	0
$151$	1.76049	0.143267	0.0716334	−	0.997431i	$-0.477179\pi$
$151$	1.76049	0.143267	0.0716334	+	0.997431i	$0.477179\pi$
$152$	0	0
$153$	2.28100	0.184408
$154$	0	0
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	−3.43801	−0.274383	−0.137191	−	0.990545i	$-0.543808\pi$
$157$	−3.43801	−0.274383	−0.137191	+	0.990545i	$0.543808\pi$
$158$	0	0
$159$	−2.14764	−0.170319
$160$	0	0
$161$	2.00000	0.157622
$162$	0	0
$163$	10.5970	0.830023	0.415012	−	0.909816i	$-0.363778\pi$
$163$	10.5970	0.830023	0.415012	+	0.909816i	$0.363778\pi$
$164$	0	0
$165$	2.00000	0.155700
$166$	0	0
$167$	20.0415	1.55086	0.775428	−	0.631435i	$-0.217534\pi$
$167$	20.0415	1.55086	0.775428	+	0.631435i	$0.217534\pi$
$168$	0	0
$169$	0.985710	0.0758238
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−18.8988	−1.43685	−0.718423	−	0.695606i	$-0.755136\pi$
$173$	−18.8988	−1.43685	−0.718423	+	0.695606i	$0.755136\pi$
$174$	0	0
$175$	−0.688892	−0.0520754
$176$	0	0
$177$	−6.44938	−0.484765
$178$	0	0
$179$	−16.2953	−1.21797	−0.608983	−	0.793183i	$-0.708422\pi$
$179$	−16.2953	−1.21797	−0.608983	+	0.793183i	$0.708422\pi$
$180$	0	0
$181$	−1.70471	−0.126710	−0.0633552	−	0.997991i	$-0.520180\pi$
$181$	−1.70471	−0.126710	−0.0633552	+	0.997991i	$0.520180\pi$
$182$	0	0
$183$	5.05086	0.373370
$184$	0	0
$185$	−8.79060	−0.646298
$186$	0	0
$187$	−4.56199	−0.333606
$188$	0	0
$189$	0.688892	0.0501095
$190$	0	0
$191$	3.53188	0.255558	0.127779	−	0.991803i	$-0.459215\pi$
$191$	3.53188	0.255558	0.127779	+	0.991803i	$0.459215\pi$
$192$	0	0
$193$	17.4193	1.25387	0.626933	−	0.779073i	$-0.284310\pi$
$193$	17.4193	1.25387	0.626933	+	0.779073i	$0.284310\pi$
$194$	0	0
$195$	−3.73975	−0.267809
$196$	0	0
$197$	10.0143	0.713489	0.356744	−	0.934202i	$-0.383887\pi$
$197$	10.0143	0.713489	0.356744	+	0.934202i	$0.383887\pi$
$198$	0	0
$199$	−27.1798	−1.92672	−0.963361	−	0.268208i	$-0.913569\pi$
$199$	−27.1798	−1.92672	−0.963361	+	0.268208i	$0.913569\pi$
$200$	0	0
$201$	−5.44446	−0.384023
$202$	0	0
$203$	2.76986	0.194406
$204$	0	0
$205$	2.75557	0.192457
$206$	0	0
$207$	−2.90321	−0.201787
$208$	0	0
$209$	0	0
$210$	0	0
$211$	1.93978	0.133540	0.0667699	−	0.997768i	$-0.478731\pi$
$211$	1.93978	0.133540	0.0667699	+	0.997768i	$0.478731\pi$
$212$	0	0
$213$	8.96989	0.614607
$214$	0	0
$215$	−1.05086	−0.0716677
$216$	0	0
$217$	0.688892	0.0467650
$218$	0	0
$219$	8.92396	0.603025
$220$	0	0
$221$	8.53035	0.573813
$222$	0	0
$223$	−23.5526	−1.57720	−0.788600	−	0.614906i	$-0.789194\pi$
$223$	−23.5526	−1.57720	−0.788600	+	0.614906i	$0.789194\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−19.8020	−1.31430	−0.657152	−	0.753758i	$-0.728239\pi$
$227$	−19.8020	−1.31430	−0.657152	+	0.753758i	$0.728239\pi$
$228$	0	0
$229$	−24.4099	−1.61305	−0.806526	−	0.591199i	$-0.798655\pi$
$229$	−24.4099	−1.61305	−0.806526	+	0.591199i	$0.798655\pi$
$230$	0	0
$231$	−1.37778	−0.0906516
$232$	0	0
$233$	3.13828	0.205595	0.102798	−	0.994702i	$-0.467221\pi$
$233$	3.13828	0.205595	0.102798	+	0.994702i	$0.467221\pi$
$234$	0	0
$235$	−4.70964	−0.307223
$236$	0	0
$237$	−8.38271	−0.544515
$238$	0	0
$239$	−6.81579	−0.440877	−0.220438	−	0.975401i	$-0.570749\pi$
$239$	−6.81579	−0.440877	−0.220438	+	0.975401i	$0.570749\pi$
$240$	0	0
$241$	17.0923	1.10101	0.550507	−	0.834830i	$-0.314434\pi$
$241$	17.0923	1.10101	0.550507	+	0.834830i	$0.314434\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−6.52543	−0.416894
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−10.5161	−0.666428
$250$	0	0
$251$	−16.1748	−1.02095	−0.510473	−	0.859894i	$-0.670530\pi$
$251$	−16.1748	−1.02095	−0.510473	+	0.859894i	$0.670530\pi$
$252$	0	0
$253$	5.80642	0.365047
$254$	0	0
$255$	−2.28100	−0.142842
$256$	0	0
$257$	19.0464	1.18808	0.594041	−	0.804435i	$-0.297532\pi$
$257$	19.0464	1.18808	0.594041	+	0.804435i	$0.297532\pi$
$258$	0	0
$259$	6.05578	0.376288
$260$	0	0
$261$	−4.02074	−0.248878
$262$	0	0
$263$	6.23506	0.384470	0.192235	−	0.981349i	$-0.438426\pi$
$263$	6.23506	0.384470	0.192235	+	0.981349i	$0.438426\pi$
$264$	0	0
$265$	2.14764	0.131929
$266$	0	0
$267$	−13.5002	−0.826201
$268$	0	0
$269$	20.8365	1.27043	0.635213	−	0.772337i	$-0.280912\pi$
$269$	20.8365	1.27043	0.635213	+	0.772337i	$0.280912\pi$
$270$	0	0
$271$	−6.62222	−0.402271	−0.201135	−	0.979563i	$-0.564463\pi$
$271$	−6.62222	−0.402271	−0.201135	+	0.979563i	$0.564463\pi$
$272$	0	0
$273$	2.57628	0.155924
$274$	0	0
$275$	−2.00000	−0.120605
$276$	0	0
$277$	4.79060	0.287839	0.143920	−	0.989589i	$-0.454029\pi$
$277$	4.79060	0.287839	0.143920	+	0.989589i	$0.454029\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−20.8256	−1.24235	−0.621177	−	0.783671i	$-0.713345\pi$
$281$	−20.8256	−1.24235	−0.621177	+	0.783671i	$0.713345\pi$
$282$	0	0
$283$	11.8731	0.705783	0.352891	−	0.935664i	$-0.385199\pi$
$283$	11.8731	0.705783	0.352891	+	0.935664i	$0.385199\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.89829	−0.112053
$288$	0	0
$289$	−11.7971	−0.693944
$290$	0	0
$291$	12.0000	0.703452
$292$	0	0
$293$	−18.0143	−1.05241	−0.526203	−	0.850359i	$-0.676385\pi$
$293$	−18.0143	−1.05241	−0.526203	+	0.850359i	$0.676385\pi$
$294$	0	0
$295$	6.44938	0.375498
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	−10.8573	−0.627893
$300$	0	0
$301$	0.723926	0.0417264
$302$	0	0
$303$	−8.29529	−0.476552
$304$	0	0
$305$	−5.05086	−0.289211
$306$	0	0
$307$	−10.9240	−0.623463	−0.311732	−	0.950170i	$-0.600909\pi$
$307$	−10.9240	−0.623463	−0.311732	+	0.950170i	$0.600909\pi$
$308$	0	0
$309$	2.98418	0.169764
$310$	0	0
$311$	−5.10324	−0.289378	−0.144689	−	0.989477i	$-0.546218\pi$
$311$	−5.10324	−0.289378	−0.144689	+	0.989477i	$0.546218\pi$
$312$	0	0
$313$	−22.1082	−1.24963	−0.624814	−	0.780774i	$-0.714825\pi$
$313$	−22.1082	−1.24963	−0.624814	+	0.780774i	$0.714825\pi$
$314$	0	0
$315$	−0.688892	−0.0388147
$316$	0	0
$317$	−1.06515	−0.0598245	−0.0299123	−	0.999553i	$-0.509523\pi$
$317$	−1.06515	−0.0598245	−0.0299123	+	0.999553i	$0.509523\pi$
$318$	0	0
$319$	8.04149	0.450237
$320$	0	0
$321$	1.52543	0.0851411
$322$	0	0
$323$	0	0
$324$	0	0
$325$	3.73975	0.207444
$326$	0	0
$327$	−3.76049	−0.207956
$328$	0	0
$329$	3.24443	0.178871
$330$	0	0
$331$	24.9447	1.37108	0.685542	−	0.728033i	$-0.259565\pi$
$331$	24.9447	1.37108	0.685542	+	0.728033i	$0.259565\pi$
$332$	0	0
$333$	−8.79060	−0.481722
$334$	0	0
$335$	5.44446	0.297463
$336$	0	0
$337$	0.453829	0.0247216	0.0123608	−	0.999924i	$-0.496065\pi$
$337$	0.453829	0.0247216	0.0123608	+	0.999924i	$0.496065\pi$
$338$	0	0
$339$	9.18421	0.498818
$340$	0	0
$341$	2.00000	0.108306
$342$	0	0
$343$	9.31756	0.503101
$344$	0	0
$345$	2.90321	0.156304
$346$	0	0
$347$	29.7560	1.59739	0.798694	−	0.601737i	$-0.205525\pi$
$347$	29.7560	1.59739	0.798694	+	0.601737i	$0.205525\pi$
$348$	0	0
$349$	−9.65878	−0.517023	−0.258511	−	0.966008i	$-0.583232\pi$
$349$	−9.65878	−0.517023	−0.258511	+	0.966008i	$0.583232\pi$
$350$	0	0
$351$	−3.73975	−0.199613
$352$	0	0
$353$	−16.8617	−0.897459	−0.448730	−	0.893668i	$-0.648123\pi$
$353$	−16.8617	−0.897459	−0.448730	+	0.893668i	$0.648123\pi$
$354$	0	0
$355$	−8.96989	−0.476072
$356$	0	0
$357$	1.57136	0.0831652
$358$	0	0
$359$	−8.64296	−0.456158	−0.228079	−	0.973643i	$-0.573244\pi$
$359$	−8.64296	−0.456158	−0.228079	+	0.973643i	$0.573244\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	−8.92396	−0.467101
$366$	0	0
$367$	−6.95899	−0.363256	−0.181628	−	0.983367i	$-0.558137\pi$
$367$	−6.95899	−0.363256	−0.181628	+	0.983367i	$0.558137\pi$
$368$	0	0
$369$	2.75557	0.143449
$370$	0	0
$371$	−1.47949	−0.0768115
$372$	0	0
$373$	21.0005	1.08736	0.543682	−	0.839291i	$-0.317030\pi$
$373$	21.0005	1.08736	0.543682	+	0.839291i	$0.317030\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−15.0366	−0.774422
$378$	0	0
$379$	−29.7146	−1.52633	−0.763167	−	0.646201i	$-0.776357\pi$
$379$	−29.7146	−1.52633	−0.763167	+	0.646201i	$0.776357\pi$
$380$	0	0
$381$	−11.4795	−0.588112
$382$	0	0
$383$	−4.05578	−0.207241	−0.103620	−	0.994617i	$-0.533043\pi$
$383$	−4.05578	−0.207241	−0.103620	+	0.994617i	$0.533043\pi$
$384$	0	0
$385$	1.37778	0.0702184
$386$	0	0
$387$	−1.05086	−0.0534180
$388$	0	0
$389$	−10.9985	−0.557644	−0.278822	−	0.960343i	$-0.589944\pi$
$389$	−10.9985	−0.557644	−0.278822	+	0.960343i	$0.589944\pi$
$390$	0	0
$391$	−6.62222	−0.334900
$392$	0	0
$393$	13.0716	0.659375
$394$	0	0
$395$	8.38271	0.421780
$396$	0	0
$397$	0.193576	0.00971531	0.00485765	−	0.999988i	$-0.498454\pi$
$397$	0.193576	0.00971531	0.00485765	+	0.999988i	$0.498454\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−38.6242	−1.92880	−0.964401	−	0.264445i	$-0.914811\pi$
$401$	−38.6242	−1.92880	−0.964401	+	0.264445i	$0.914811\pi$
$402$	0	0
$403$	−3.73975	−0.186290
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	17.5812	0.871468
$408$	0	0
$409$	2.94914	0.145826	0.0729129	−	0.997338i	$-0.476770\pi$
$409$	2.94914	0.145826	0.0729129	+	0.997338i	$0.476770\pi$
$410$	0	0
$411$	11.1383	0.549411
$412$	0	0
$413$	−4.44293	−0.218622
$414$	0	0
$415$	10.5161	0.516213
$416$	0	0
$417$	10.2351	0.501213
$418$	0	0
$419$	26.0020	1.27028	0.635141	−	0.772397i	$-0.280942\pi$
$419$	26.0020	1.27028	0.635141	+	0.772397i	$0.280942\pi$
$420$	0	0
$421$	−20.0143	−0.975437	−0.487718	−	0.873001i	$-0.662171\pi$
$421$	−20.0143	−0.975437	−0.487718	+	0.873001i	$0.662171\pi$
$422$	0	0
$423$	−4.70964	−0.228990
$424$	0	0
$425$	2.28100	0.110645
$426$	0	0
$427$	3.47949	0.168385
$428$	0	0
$429$	7.47949	0.361113
$430$	0	0
$431$	2.77631	0.133730	0.0668651	−	0.997762i	$-0.478700\pi$
$431$	2.77631	0.133730	0.0668651	+	0.997762i	$0.478700\pi$
$432$	0	0
$433$	−30.7590	−1.47818	−0.739091	−	0.673606i	$-0.764744\pi$
$433$	−30.7590	−1.47818	−0.739091	+	0.673606i	$0.764744\pi$
$434$	0	0
$435$	4.02074	0.192780
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−26.8385	−1.28093	−0.640467	−	0.767986i	$-0.721259\pi$
$439$	−26.8385	−1.28093	−0.640467	+	0.767986i	$0.721259\pi$
$440$	0	0
$441$	−6.52543	−0.310735
$442$	0	0
$443$	11.7190	0.556787	0.278393	−	0.960467i	$-0.410198\pi$
$443$	11.7190	0.556787	0.278393	+	0.960467i	$0.410198\pi$
$444$	0	0
$445$	13.5002	0.639973
$446$	0	0
$447$	1.51114	0.0714743
$448$	0	0
$449$	33.8084	1.59552	0.797759	−	0.602976i	$-0.206019\pi$
$449$	33.8084	1.59552	0.797759	+	0.602976i	$0.206019\pi$
$450$	0	0
$451$	−5.51114	−0.259509
$452$	0	0
$453$	−1.76049	−0.0827151
$454$	0	0
$455$	−2.57628	−0.120778
$456$	0	0
$457$	5.61930	0.262860	0.131430	−	0.991325i	$-0.458043\pi$
$457$	5.61930	0.262860	0.131430	+	0.991325i	$0.458043\pi$
$458$	0	0
$459$	−2.28100	−0.106468
$460$	0	0
$461$	−2.93825	−0.136848	−0.0684239	−	0.997656i	$-0.521797\pi$
$461$	−2.93825	−0.136848	−0.0684239	+	0.997656i	$0.521797\pi$
$462$	0	0
$463$	1.61285	0.0749554	0.0374777	−	0.999297i	$-0.488068\pi$
$463$	1.61285	0.0749554	0.0374777	+	0.999297i	$0.488068\pi$
$464$	0	0
$465$	1.00000	0.0463739
$466$	0	0
$467$	11.5857	0.536120	0.268060	−	0.963402i	$-0.413617\pi$
$467$	11.5857	0.536120	0.268060	+	0.963402i	$0.413617\pi$
$468$	0	0
$469$	−3.75065	−0.173189
$470$	0	0
$471$	3.43801	0.158415
$472$	0	0
$473$	2.10171	0.0966367
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.14764	0.0983338
$478$	0	0
$479$	−26.0020	−1.18806	−0.594031	−	0.804442i	$-0.702464\pi$
$479$	−26.0020	−1.18806	−0.594031	+	0.804442i	$0.702464\pi$
$480$	0	0
$481$	−32.8746	−1.49895
$482$	0	0
$483$	−2.00000	−0.0910032
$484$	0	0
$485$	−12.0000	−0.544892
$486$	0	0
$487$	−40.8988	−1.85330	−0.926650	−	0.375925i	$-0.877325\pi$
$487$	−40.8988	−1.85330	−0.926650	+	0.375925i	$0.877325\pi$
$488$	0	0
$489$	−10.5970	−0.479214
$490$	0	0
$491$	−16.6953	−0.753450	−0.376725	−	0.926325i	$-0.622950\pi$
$491$	−16.6953	−0.753450	−0.376725	+	0.926325i	$0.622950\pi$
$492$	0	0
$493$	−9.17130	−0.413055
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	0	0
$497$	6.17929	0.277179
$498$	0	0
$499$	26.3684	1.18041	0.590206	−	0.807253i	$-0.299046\pi$
$499$	26.3684	1.18041	0.590206	+	0.807253i	$0.299046\pi$
$500$	0	0
$501$	−20.0415	−0.895388
$502$	0	0
$503$	9.47505	0.422472	0.211236	−	0.977435i	$-0.432251\pi$
$503$	9.47505	0.422472	0.211236	+	0.977435i	$0.432251\pi$
$504$	0	0
$505$	8.29529	0.369135
$506$	0	0
$507$	−0.985710	−0.0437769
$508$	0	0
$509$	21.9003	0.970714	0.485357	−	0.874316i	$-0.338690\pi$
$509$	21.9003	0.970714	0.485357	+	0.874316i	$0.338690\pi$
$510$	0	0
$511$	6.14764	0.271956
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2.98418	−0.131499
$516$	0	0
$517$	9.41927	0.414259
$518$	0	0
$519$	18.8988	0.829564
$520$	0	0
$521$	−36.2034	−1.58610	−0.793050	−	0.609156i	$-0.791508\pi$
$521$	−36.2034	−1.58610	−0.793050	+	0.609156i	$0.791508\pi$
$522$	0	0
$523$	14.7971	0.647030	0.323515	−	0.946223i	$-0.395135\pi$
$523$	14.7971	0.647030	0.323515	+	0.946223i	$0.395135\pi$
$524$	0	0
$525$	0.688892	0.0300657
$526$	0	0
$527$	−2.28100	−0.0993618
$528$	0	0
$529$	−14.5714	−0.633537
$530$	0	0
$531$	6.44938	0.279879
$532$	0	0
$533$	10.3051	0.446365
$534$	0	0
$535$	−1.52543	−0.0659500
$536$	0	0
$537$	16.2953	0.703194
$538$	0	0
$539$	13.0509	0.562140
$540$	0	0
$541$	−41.9956	−1.80553	−0.902765	−	0.430134i	$-0.858466\pi$
$541$	−41.9956	−1.80553	−0.902765	+	0.430134i	$0.858466\pi$
$542$	0	0
$543$	1.70471	0.0731563
$544$	0	0
$545$	3.76049	0.161082
$546$	0	0
$547$	34.8004	1.48796	0.743980	−	0.668202i	$-0.232936\pi$
$547$	34.8004	1.48796	0.743980	+	0.668202i	$0.232936\pi$
$548$	0	0
$549$	−5.05086	−0.215565
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−5.77478	−0.245569
$554$	0	0
$555$	8.79060	0.373140
$556$	0	0
$557$	26.3096	1.11477	0.557386	−	0.830253i	$-0.311804\pi$
$557$	26.3096	1.11477	0.557386	+	0.830253i	$0.311804\pi$
$558$	0	0
$559$	−3.92993	−0.166218
$560$	0	0
$561$	4.56199	0.192607
$562$	0	0
$563$	31.8306	1.34150	0.670749	−	0.741684i	$-0.265973\pi$
$563$	31.8306	1.34150	0.670749	+	0.741684i	$0.265973\pi$
$564$	0	0
$565$	−9.18421	−0.386383
$566$	0	0
$567$	−0.688892	−0.0289308
$568$	0	0
$569$	23.6622	0.991970	0.495985	−	0.868331i	$-0.334807\pi$
$569$	23.6622	0.991970	0.495985	+	0.868331i	$0.334807\pi$
$570$	0	0
$571$	−18.8889	−0.790477	−0.395238	−	0.918579i	$-0.629338\pi$
$571$	−18.8889	−0.790477	−0.395238	+	0.918579i	$0.629338\pi$
$572$	0	0
$573$	−3.53188	−0.147546
$574$	0	0
$575$	−2.90321	−0.121072
$576$	0	0
$577$	1.11108	0.0462548	0.0231274	−	0.999733i	$-0.492638\pi$
$577$	1.11108	0.0462548	0.0231274	+	0.999733i	$0.492638\pi$
$578$	0	0
$579$	−17.4193	−0.723920
$580$	0	0
$581$	−7.24443	−0.300550
$582$	0	0
$583$	−4.29529	−0.177893
$584$	0	0
$585$	3.73975	0.154620
$586$	0	0
$587$	17.3778	0.717258	0.358629	−	0.933480i	$-0.383244\pi$
$587$	17.3778	0.717258	0.358629	+	0.933480i	$0.383244\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−10.0143	−0.411933
$592$	0	0
$593$	−37.1753	−1.52661	−0.763304	−	0.646040i	$-0.776424\pi$
$593$	−37.1753	−1.52661	−0.763304	+	0.646040i	$0.776424\pi$
$594$	0	0
$595$	−1.57136	−0.0644195
$596$	0	0
$597$	27.1798	1.11239
$598$	0	0
$599$	−15.6207	−0.638244	−0.319122	−	0.947714i	$-0.603388\pi$
$599$	−15.6207	−0.638244	−0.319122	+	0.947714i	$0.603388\pi$
$600$	0	0
$601$	−26.7239	−1.09009	−0.545046	−	0.838406i	$-0.683488\pi$
$601$	−26.7239	−1.09009	−0.545046	+	0.838406i	$0.683488\pi$
$602$	0	0
$603$	5.44446	0.221716
$604$	0	0
$605$	−7.00000	−0.284590
$606$	0	0
$607$	−14.3936	−0.584218	−0.292109	−	0.956385i	$-0.594357\pi$
$607$	−14.3936	−0.584218	−0.292109	+	0.956385i	$0.594357\pi$
$608$	0	0
$609$	−2.76986	−0.112240
$610$	0	0
$611$	−17.6128	−0.712540
$612$	0	0
$613$	6.09526	0.246185	0.123093	−	0.992395i	$-0.460719\pi$
$613$	6.09526	0.246185	0.123093	+	0.992395i	$0.460719\pi$
$614$	0	0
$615$	−2.75557	−0.111115
$616$	0	0
$617$	1.34614	0.0541936	0.0270968	−	0.999633i	$-0.491374\pi$
$617$	1.34614	0.0541936	0.0270968	+	0.999633i	$0.491374\pi$
$618$	0	0
$619$	−5.71900	−0.229866	−0.114933	−	0.993373i	$-0.536665\pi$
$619$	−5.71900	−0.229866	−0.114933	+	0.993373i	$0.536665\pi$
$620$	0	0
$621$	2.90321	0.116502
$622$	0	0
$623$	−9.30021	−0.372605
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−20.0513	−0.799499
$630$	0	0
$631$	−37.3274	−1.48598	−0.742990	−	0.669302i	$-0.766593\pi$
$631$	−37.3274	−1.48598	−0.742990	+	0.669302i	$0.766593\pi$
$632$	0	0
$633$	−1.93978	−0.0770992
$634$	0	0
$635$	11.4795	0.455550
$636$	0	0
$637$	−24.4035	−0.966900
$638$	0	0
$639$	−8.96989	−0.354843
$640$	0	0
$641$	36.1323	1.42714	0.713570	−	0.700584i	$-0.247077\pi$
$641$	36.1323	1.42714	0.713570	+	0.700584i	$0.247077\pi$
$642$	0	0
$643$	3.10123	0.122301	0.0611504	−	0.998129i	$-0.480523\pi$
$643$	3.10123	0.122301	0.0611504	+	0.998129i	$0.480523\pi$
$644$	0	0
$645$	1.05086	0.0413774
$646$	0	0
$647$	−29.6227	−1.16459	−0.582294	−	0.812978i	$-0.697845\pi$
$647$	−29.6227	−1.16459	−0.582294	+	0.812978i	$0.697845\pi$
$648$	0	0
$649$	−12.8988	−0.506321
$650$	0	0
$651$	−0.688892	−0.0269998
$652$	0	0
$653$	−25.2114	−0.986599	−0.493299	−	0.869860i	$-0.664209\pi$
$653$	−25.2114	−0.986599	−0.493299	+	0.869860i	$0.664209\pi$
$654$	0	0
$655$	−13.0716	−0.510750
$656$	0	0
$657$	−8.92396	−0.348157
$658$	0	0
$659$	−40.6242	−1.58250	−0.791248	−	0.611496i	$-0.790568\pi$
$659$	−40.6242	−1.58250	−0.791248	+	0.611496i	$0.790568\pi$
$660$	0	0
$661$	−38.7783	−1.50830	−0.754151	−	0.656701i	$-0.771951\pi$
$661$	−38.7783	−1.50830	−0.754151	+	0.656701i	$0.771951\pi$
$662$	0	0
$663$	−8.53035	−0.331291
$664$	0	0
$665$	0	0
$666$	0	0
$667$	11.6731	0.451983
$668$	0	0
$669$	23.5526	0.910597
$670$	0	0
$671$	10.1017	0.389972
$672$	0	0
$673$	40.1813	1.54888	0.774438	−	0.632650i	$-0.218033\pi$
$673$	40.1813	1.54888	0.774438	+	0.632650i	$0.218033\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	10.6811	0.410506	0.205253	−	0.978709i	$-0.434198\pi$
$677$	10.6811	0.410506	0.205253	+	0.978709i	$0.434198\pi$
$678$	0	0
$679$	8.26671	0.317247
$680$	0	0
$681$	19.8020	0.758813
$682$	0	0
$683$	−31.6271	−1.21018	−0.605089	−	0.796158i	$-0.706863\pi$
$683$	−31.6271	−1.21018	−0.605089	+	0.796158i	$0.706863\pi$
$684$	0	0
$685$	−11.1383	−0.425572
$686$	0	0
$687$	24.4099	0.931296
$688$	0	0
$689$	8.03164	0.305981
$690$	0	0
$691$	−10.6508	−0.405175	−0.202588	−	0.979264i	$-0.564935\pi$
$691$	−10.6508	−0.405175	−0.202588	+	0.979264i	$0.564935\pi$
$692$	0	0
$693$	1.37778	0.0523377
$694$	0	0
$695$	−10.2351	−0.388238
$696$	0	0
$697$	6.28544	0.238078
$698$	0	0
$699$	−3.13828	−0.118700
$700$	0	0
$701$	39.3590	1.48657	0.743285	−	0.668974i	$-0.233266\pi$
$701$	39.3590	1.48657	0.743285	+	0.668974i	$0.233266\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	4.70964	0.177375
$706$	0	0
$707$	−5.71456	−0.214918
$708$	0	0
$709$	20.1847	0.758052	0.379026	−	0.925386i	$-0.376259\pi$
$709$	20.1847	0.758052	0.379026	+	0.925386i	$0.376259\pi$
$710$	0	0
$711$	8.38271	0.314376
$712$	0	0
$713$	2.90321	0.108726
$714$	0	0
$715$	−7.47949	−0.279717
$716$	0	0
$717$	6.81579	0.254540
$718$	0	0
$719$	−24.1017	−0.898842	−0.449421	−	0.893320i	$-0.648370\pi$
$719$	−24.1017	−0.898842	−0.449421	+	0.893320i	$0.648370\pi$
$720$	0	0
$721$	2.05578	0.0765611
$722$	0	0
$723$	−17.0923	−0.635671
$724$	0	0
$725$	−4.02074	−0.149327
$726$	0	0
$727$	14.1180	0.523608	0.261804	−	0.965121i	$-0.415683\pi$
$727$	14.1180	0.523608	0.261804	+	0.965121i	$0.415683\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−2.39700	−0.0886561
$732$	0	0
$733$	−23.1240	−0.854104	−0.427052	−	0.904227i	$-0.640448\pi$
$733$	−23.1240	−0.854104	−0.427052	+	0.904227i	$0.640448\pi$
$734$	0	0
$735$	6.52543	0.240694
$736$	0	0
$737$	−10.8889	−0.401099
$738$	0	0
$739$	0.128907	0.00474193	0.00237097	−	0.999997i	$-0.499245\pi$
$739$	0.128907	0.00474193	0.00237097	+	0.999997i	$0.499245\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−28.0129	−1.02769	−0.513847	−	0.857882i	$-0.671780\pi$
$743$	−28.0129	−1.02769	−0.513847	+	0.857882i	$0.671780\pi$
$744$	0	0
$745$	−1.51114	−0.0553638
$746$	0	0
$747$	10.5161	0.384763
$748$	0	0
$749$	1.05086	0.0383974
$750$	0	0
$751$	3.56247	0.129996	0.0649982	−	0.997885i	$-0.479296\pi$
$751$	3.56247	0.129996	0.0649982	+	0.997885i	$0.479296\pi$
$752$	0	0
$753$	16.1748	0.589444
$754$	0	0
$755$	1.76049	0.0640708
$756$	0	0
$757$	27.0573	0.983415	0.491707	−	0.870761i	$-0.336373\pi$
$757$	27.0573	0.983415	0.491707	+	0.870761i	$0.336373\pi$
$758$	0	0
$759$	−5.80642	−0.210760
$760$	0	0
$761$	13.1447	0.476496	0.238248	−	0.971204i	$-0.423427\pi$
$761$	13.1447	0.476496	0.238248	+	0.971204i	$0.423427\pi$
$762$	0	0
$763$	−2.59057	−0.0937850
$764$	0	0
$765$	2.28100	0.0824696
$766$	0	0
$767$	24.1191	0.870889
$768$	0	0
$769$	25.7418	0.928271	0.464136	−	0.885764i	$-0.346365\pi$
$769$	25.7418	0.928271	0.464136	+	0.885764i	$0.346365\pi$
$770$	0	0
$771$	−19.0464	−0.685940
$772$	0	0
$773$	50.3007	1.80919	0.904595	−	0.426272i	$-0.140173\pi$
$773$	50.3007	1.80919	0.904595	+	0.426272i	$0.140173\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	−6.05578	−0.217250
$778$	0	0
$779$	0	0
$780$	0	0
$781$	17.9398	0.641936
$782$	0	0
$783$	4.02074	0.143690
$784$	0	0
$785$	−3.43801	−0.122708
$786$	0	0
$787$	−8.70519	−0.310307	−0.155153	−	0.987890i	$-0.549587\pi$
$787$	−8.70519	−0.310307	−0.155153	+	0.987890i	$0.549587\pi$
$788$	0	0
$789$	−6.23506	−0.221974
$790$	0	0
$791$	6.32693	0.224960
$792$	0	0
$793$	−18.8889	−0.670765
$794$	0	0
$795$	−2.14764	−0.0761691
$796$	0	0
$797$	39.0464	1.38309	0.691547	−	0.722331i	$-0.256929\pi$
$797$	39.0464	1.38309	0.691547	+	0.722331i	$0.256929\pi$
$798$	0	0
$799$	−10.7427	−0.380048
$800$	0	0
$801$	13.5002	0.477007
$802$	0	0
$803$	17.8479	0.629839
$804$	0	0
$805$	2.00000	0.0704907
$806$	0	0
$807$	−20.8365	−0.733481
$808$	0	0
$809$	−24.0435	−0.845324	−0.422662	−	0.906287i	$-0.638904\pi$
$809$	−24.0435	−0.845324	−0.422662	+	0.906287i	$0.638904\pi$
$810$	0	0
$811$	−45.5812	−1.60057	−0.800286	−	0.599618i	$-0.795319\pi$
$811$	−45.5812	−1.60057	−0.800286	+	0.599618i	$0.795319\pi$
$812$	0	0
$813$	6.62222	0.232251
$814$	0	0
$815$	10.5970	0.371198
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−2.57628	−0.0900226
$820$	0	0
$821$	−32.2054	−1.12398	−0.561989	−	0.827145i	$-0.689964\pi$
$821$	−32.2054	−1.12398	−0.561989	+	0.827145i	$0.689964\pi$
$822$	0	0
$823$	−33.1052	−1.15398	−0.576988	−	0.816752i	$-0.695772\pi$
$823$	−33.1052	−1.15398	−0.576988	+	0.816752i	$0.695772\pi$
$824$	0	0
$825$	2.00000	0.0696311
$826$	0	0
$827$	0.147643	0.00513406	0.00256703	−	0.999997i	$-0.499183\pi$
$827$	0.147643	0.00513406	0.00256703	+	0.999997i	$0.499183\pi$
$828$	0	0
$829$	−20.9175	−0.726495	−0.363247	−	0.931693i	$-0.618332\pi$
$829$	−20.9175	−0.726495	−0.363247	+	0.931693i	$0.618332\pi$
$830$	0	0
$831$	−4.79060	−0.166184
$832$	0	0
$833$	−14.8845	−0.515717
$834$	0	0
$835$	20.0415	0.693564
$836$	0	0
$837$	1.00000	0.0345651
$838$	0	0
$839$	42.9066	1.48130	0.740650	−	0.671891i	$-0.234518\pi$
$839$	42.9066	1.48130	0.740650	+	0.671891i	$0.234518\pi$
$840$	0	0
$841$	−12.8336	−0.442539
$842$	0	0
$843$	20.8256	0.717273
$844$	0	0
$845$	0.985710	0.0339095
$846$	0	0
$847$	4.82225	0.165694
$848$	0	0
$849$	−11.8731	−0.407484
$850$	0	0
$851$	25.5210	0.874848
$852$	0	0
$853$	−49.2226	−1.68535	−0.842675	−	0.538422i	$-0.819021\pi$
$853$	−49.2226	−1.68535	−0.842675	+	0.538422i	$0.819021\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	34.2449	1.16978	0.584892	−	0.811111i	$-0.301137\pi$
$857$	34.2449	1.16978	0.584892	+	0.811111i	$0.301137\pi$
$858$	0	0
$859$	−14.5718	−0.497185	−0.248592	−	0.968608i	$-0.579968\pi$
$859$	−14.5718	−0.497185	−0.248592	+	0.968608i	$0.579968\pi$
$860$	0	0
$861$	1.89829	0.0646935
$862$	0	0
$863$	−44.6164	−1.51876	−0.759380	−	0.650648i	$-0.774497\pi$
$863$	−44.6164	−1.51876	−0.759380	+	0.650648i	$0.774497\pi$
$864$	0	0
$865$	−18.8988	−0.642577
$866$	0	0
$867$	11.7971	0.400649
$868$	0	0
$869$	−16.7654	−0.568728
$870$	0	0
$871$	20.3609	0.689903
$872$	0	0
$873$	−12.0000	−0.406138
$874$	0	0
$875$	−0.688892	−0.0232888
$876$	0	0
$877$	8.05038	0.271842	0.135921	−	0.990720i	$-0.456601\pi$
$877$	8.05038	0.271842	0.135921	+	0.990720i	$0.456601\pi$
$878$	0	0
$879$	18.0143	0.607607
$880$	0	0
$881$	−28.7130	−0.967366	−0.483683	−	0.875243i	$-0.660701\pi$
$881$	−28.7130	−0.967366	−0.483683	+	0.875243i	$0.660701\pi$
$882$	0	0
$883$	26.4197	0.889095	0.444548	−	0.895755i	$-0.353364\pi$
$883$	26.4197	0.889095	0.444548	+	0.895755i	$0.353364\pi$
$884$	0	0
$885$	−6.44938	−0.216794
$886$	0	0
$887$	44.7926	1.50399	0.751994	−	0.659170i	$-0.229092\pi$
$887$	44.7926	1.50399	0.751994	+	0.659170i	$0.229092\pi$
$888$	0	0
$889$	−7.90813	−0.265230
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−16.2953	−0.544691
$896$	0	0
$897$	10.8573	0.362514
$898$	0	0
$899$	4.02074	0.134099
$900$	0	0
$901$	4.89877	0.163202
$902$	0	0
$903$	−0.723926	−0.0240907
$904$	0	0
$905$	−1.70471	−0.0566666
$906$	0	0
$907$	−7.21924	−0.239711	−0.119855	−	0.992791i	$-0.538243\pi$
$907$	−7.21924	−0.239711	−0.119855	+	0.992791i	$0.538243\pi$
$908$	0	0
$909$	8.29529	0.275137
$910$	0	0
$911$	−21.1052	−0.699248	−0.349624	−	0.936890i	$-0.613691\pi$
$911$	−21.1052	−0.699248	−0.349624	+	0.936890i	$0.613691\pi$
$912$	0	0
$913$	−21.0321	−0.696062
$914$	0	0
$915$	5.05086	0.166976
$916$	0	0
$917$	9.00492	0.297369
$918$	0	0
$919$	−16.4415	−0.542357	−0.271178	−	0.962529i	$-0.587413\pi$
$919$	−16.4415	−0.542357	−0.271178	+	0.962529i	$0.587413\pi$
$920$	0	0
$921$	10.9240	0.359957
$922$	0	0
$923$	−33.5451	−1.10415
$924$	0	0
$925$	−8.79060	−0.289033
$926$	0	0
$927$	−2.98418	−0.0980133
$928$	0	0
$929$	6.26470	0.205538	0.102769	−	0.994705i	$-0.467230\pi$
$929$	6.26470	0.205538	0.102769	+	0.994705i	$0.467230\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	5.10324	0.167073
$934$	0	0
$935$	−4.56199	−0.149193
$936$	0	0
$937$	37.1437	1.21343	0.606715	−	0.794919i	$-0.292487\pi$
$937$	37.1437	1.21343	0.606715	+	0.794919i	$0.292487\pi$
$938$	0	0
$939$	22.1082	0.721473
$940$	0	0
$941$	39.1318	1.27566	0.637830	−	0.770177i	$-0.279832\pi$
$941$	39.1318	1.27566	0.637830	+	0.770177i	$0.279832\pi$
$942$	0	0
$943$	−8.00000	−0.260516
$944$	0	0
$945$	0.688892	0.0224097
$946$	0	0
$947$	53.0968	1.72541	0.862707	−	0.505704i	$-0.168767\pi$
$947$	53.0968	1.72541	0.862707	+	0.505704i	$0.168767\pi$
$948$	0	0
$949$	−33.3733	−1.08334
$950$	0	0
$951$	1.06515	0.0345397
$952$	0	0
$953$	−6.57628	−0.213027	−0.106513	−	0.994311i	$-0.533969\pi$
$953$	−6.57628	−0.213027	−0.106513	+	0.994311i	$0.533969\pi$
$954$	0	0
$955$	3.53188	0.114289
$956$	0	0
$957$	−8.04149	−0.259944
$958$	0	0
$959$	7.67307	0.247776
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−1.52543	−0.0491562
$964$	0	0
$965$	17.4193	0.560746
$966$	0	0
$967$	−54.2449	−1.74440	−0.872199	−	0.489151i	$-0.837307\pi$
$967$	−54.2449	−1.74440	−0.872199	+	0.489151i	$0.837307\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	42.8879	1.37634	0.688169	−	0.725551i	$-0.258415\pi$
$971$	42.8879	1.37634	0.688169	+	0.725551i	$0.258415\pi$
$972$	0	0
$973$	7.05086	0.226040
$974$	0	0
$975$	−3.73975	−0.119768
$976$	0	0
$977$	11.6316	0.372127	0.186064	−	0.982538i	$-0.440427\pi$
$977$	11.6316	0.372127	0.186064	+	0.982538i	$0.440427\pi$
$978$	0	0
$979$	−27.0005	−0.862939
$980$	0	0
$981$	3.76049	0.120063
$982$	0	0
$983$	30.3595	0.968318	0.484159	−	0.874980i	$-0.339126\pi$
$983$	30.3595	0.968318	0.484159	+	0.874980i	$0.339126\pi$
$984$	0	0
$985$	10.0143	0.319082
$986$	0	0
$987$	−3.24443	−0.103271
$988$	0	0
$989$	3.05086	0.0970115
$990$	0	0
$991$	−42.5589	−1.35193	−0.675964	−	0.736934i	$-0.736273\pi$
$991$	−42.5589	−1.35193	−0.675964	+	0.736934i	$0.736273\pi$
$992$	0	0
$993$	−24.9447	−0.791596
$994$	0	0
$995$	−27.1798	−0.861656
$996$	0	0
$997$	−33.5339	−1.06203	−0.531014	−	0.847363i	$-0.678189\pi$
$997$	−33.5339	−1.06203	−0.531014	+	0.847363i	$0.678189\pi$
$998$	0	0
$999$	8.79060	0.278122

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7440.2.a.bp.1.2		3
4.3	odd	2		465.2.a.f.1.2	✓	3
12.11	even	2		1395.2.a.i.1.2		3
20.3	even	4		2325.2.c.o.1024.3		6
20.7	even	4		2325.2.c.o.1024.4		6
20.19	odd	2		2325.2.a.q.1.2		3
60.59	even	2		6975.2.a.be.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.f.1.2	✓	3	4.3	odd	2
1395.2.a.i.1.2		3	12.11	even	2
2325.2.a.q.1.2		3	20.19	odd	2
2325.2.c.o.1024.3		6	20.3	even	4
2325.2.c.o.1024.4		6	20.7	even	4
6975.2.a.be.1.2		3	60.59	even	2
7440.2.a.bp.1.2		3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7440.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -0.688892 q^{7} +1.00000 q^{9} -2.00000 q^{11} +3.73975 q^{13} -1.00000 q^{15} +2.28100 q^{17} +0.688892 q^{21} -2.90321 q^{23} +1.00000 q^{25} -1.00000 q^{27} -4.02074 q^{29} -1.00000 q^{31} +2.00000 q^{33} -0.688892 q^{35} -8.79060 q^{37} -3.73975 q^{39} +2.75557 q^{41} -1.05086 q^{43} +1.00000 q^{45} -4.70964 q^{47} -6.52543 q^{49} -2.28100 q^{51} +2.14764 q^{53} -2.00000 q^{55} +6.44938 q^{59} -5.05086 q^{61} -0.688892 q^{63} +3.73975 q^{65} +5.44446 q^{67} +2.90321 q^{69} -8.96989 q^{71} -8.92396 q^{73} -1.00000 q^{75} +1.37778 q^{77} +8.38271 q^{79} +1.00000 q^{81} +10.5161 q^{83} +2.28100 q^{85} +4.02074 q^{87} +13.5002 q^{89} -2.57628 q^{91} +1.00000 q^{93} -12.0000 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9} - 6 q^{11} - 2 q^{13} - 3 q^{15} + 2 q^{21} - 2 q^{23} + 3 q^{25} - 3 q^{27} + 8 q^{29} - 3 q^{31} + 6 q^{33} - 2 q^{35} + 2 q^{39} + 8 q^{41} + 10 q^{43} + 3 q^{45}+ \cdots - 6 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^5 - 2 * q^7 + 3 * q^9 - 6 * q^11 - 2 * q^13 - 3 * q^15 + 2 * q^21 - 2 * q^23 + 3 * q^25 - 3 * q^27 + 8 * q^29 - 3 * q^31 + 6 * q^33 - 2 * q^35 + 2 * q^39 + 8 * q^41 + 10 * q^43 + 3 * q^45 + 6 * q^47 - 13 * q^49 - 6 * q^55 - 14 * q^59 - 2 * q^61 - 2 * q^63 - 2 * q^65 + 16 * q^67 + 2 * q^69 - 20 * q^71 - 3 * q^75 + 4 * q^77 - 8 * q^79 + 3 * q^81 - 2 * q^83 - 8 * q^87 - 6 * q^89 + 12 * q^91 + 3 * q^93 - 36 * q^97 - 6 * q^99

Introduction

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