LMFDB - Embedded newform 7440.2.a.bs.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:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
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			$0.571993$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -2.42801 q^{7} +1.00000 q^{9} -1.14399 q^{11} +1.57199 q^{13} -1.00000 q^{15} +4.67282 q^{17} -5.34565 q^{19} -2.42801 q^{21} +1.81681 q^{23} +1.00000 q^{25} +1.00000 q^{27} +1.95684 q^{29} -1.00000 q^{31} -1.14399 q^{33} +2.42801 q^{35} +3.57199 q^{37} +1.57199 q^{39} -3.14399 q^{41} -2.85601 q^{43} -1.00000 q^{45} -7.16246 q^{47} -1.10478 q^{49} +4.67282 q^{51} -3.81681 q^{53} +1.14399 q^{55} -5.34565 q^{57} +6.24482 q^{59} -2.00000 q^{61} -2.42801 q^{63} -1.57199 q^{65} +0.917641 q^{67} +1.81681 q^{69} +6.44648 q^{71} +6.71598 q^{73} +1.00000 q^{75} +2.77761 q^{77} -9.52884 q^{79} +1.00000 q^{81} +3.16246 q^{83} -4.67282 q^{85} +1.95684 q^{87} -15.2241 q^{89} -3.81681 q^{91} -1.00000 q^{93} +5.34565 q^{95} +10.7776 q^{97} -1.14399 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{5} - 8 q^{7} + 3 q^{9} - 2 q^{11} + 4 q^{13} - 3 q^{15} + 4 q^{17} + 4 q^{19} - 8 q^{21} - 6 q^{23} + 3 q^{25} + 3 q^{27} - 2 q^{29} - 3 q^{31} - 2 q^{33} + 8 q^{35} + 10 q^{37} + 4 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 - 8 * q^7 + 3 * q^9 - 2 * q^11 + 4 * q^13 - 3 * q^15 + 4 * q^17 + 4 * q^19 - 8 * q^21 - 6 * q^23 + 3 * q^25 + 3 * q^27 - 2 * q^29 - 3 * q^31 - 2 * q^33 + 8 * q^35 + 10 * q^37 + 4 * q^39 - 8 * q^41 - 10 * q^43 - 3 * q^45 + 10 * q^47 + 11 * q^49 + 4 * q^51 + 2 * q^55 + 4 * q^57 + 8 * q^59 - 6 * q^61 - 8 * q^63 - 4 * q^65 - 18 * q^67 - 6 * q^69 - 10 * q^71 + 18 * q^73 + 3 * q^75 - 16 * q^77 - 20 * q^79 + 3 * q^81 - 22 * q^83 - 4 * q^85 - 2 * q^87 + 8 * q^89 - 3 * q^93 - 4 * q^95 + 8 * q^97 - 2 * 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
$7$	−2.42801	−0.917700	−0.458850	−	0.888514i	$-0.651739\pi$
$7$	−2.42801	−0.917700	−0.458850	+	0.888514i	$0.651739\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.14399	−0.344925	−0.172462	−	0.985016i	$-0.555172\pi$
$11$	−1.14399	−0.344925	−0.172462	+	0.985016i	$0.555172\pi$
$12$	0	0
$13$	1.57199	0.435992	0.217996	−	0.975950i	$-0.430048\pi$
$13$	1.57199	0.435992	0.217996	+	0.975950i	$0.430048\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	4.67282	1.13333	0.566663	−	0.823950i	$-0.308234\pi$
$17$	4.67282	1.13333	0.566663	+	0.823950i	$0.308234\pi$
$18$	0	0
$19$	−5.34565	−1.22638	−0.613188	−	0.789937i	$-0.710113\pi$
$19$	−5.34565	−1.22638	−0.613188	+	0.789937i	$0.710113\pi$
$20$	0	0
$21$	−2.42801	−0.529835
$22$	0	0
$23$	1.81681	0.378831	0.189416	−	0.981897i	$-0.439341\pi$
$23$	1.81681	0.378831	0.189416	+	0.981897i	$0.439341\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.95684	0.363377	0.181688	−	0.983356i	$-0.441844\pi$
$29$	1.95684	0.363377	0.181688	+	0.983356i	$0.441844\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	−1.14399	−0.199142
$34$	0	0
$35$	2.42801	0.410408
$36$	0	0
$37$	3.57199	0.587232	0.293616	−	0.955923i	$-0.405141\pi$
$37$	3.57199	0.587232	0.293616	+	0.955923i	$0.405141\pi$
$38$	0	0
$39$	1.57199	0.251720
$40$	0	0
$41$	−3.14399	−0.491008	−0.245504	−	0.969396i	$-0.578953\pi$
$41$	−3.14399	−0.491008	−0.245504	+	0.969396i	$0.578953\pi$
$42$	0	0
$43$	−2.85601	−0.435538	−0.217769	−	0.976000i	$-0.569878\pi$
$43$	−2.85601	−0.435538	−0.217769	+	0.976000i	$0.569878\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−7.16246	−1.04475	−0.522376	−	0.852715i	$-0.674954\pi$
$47$	−7.16246	−1.04475	−0.522376	+	0.852715i	$0.674954\pi$
$48$	0	0
$49$	−1.10478	−0.157826
$50$	0	0
$51$	4.67282	0.654326
$52$	0	0
$53$	−3.81681	−0.524279	−0.262140	−	0.965030i	$-0.584428\pi$
$53$	−3.81681	−0.524279	−0.262140	+	0.965030i	$0.584428\pi$
$54$	0	0
$55$	1.14399	0.154255
$56$	0	0
$57$	−5.34565	−0.708048
$58$	0	0
$59$	6.24482	0.813006	0.406503	−	0.913649i	$-0.366748\pi$
$59$	6.24482	0.813006	0.406503	+	0.913649i	$0.366748\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	−2.42801	−0.305900
$64$	0	0
$65$	−1.57199	−0.194982
$66$	0	0
$67$	0.917641	0.112108	0.0560538	−	0.998428i	$-0.482148\pi$
$67$	0.917641	0.112108	0.0560538	+	0.998428i	$0.482148\pi$
$68$	0	0
$69$	1.81681	0.218718
$70$	0	0
$71$	6.44648	0.765056	0.382528	−	0.923944i	$-0.375054\pi$
$71$	6.44648	0.765056	0.382528	+	0.923944i	$0.375054\pi$
$72$	0	0
$73$	6.71598	0.786046	0.393023	−	0.919529i	$-0.371429\pi$
$73$	6.71598	0.786046	0.393023	+	0.919529i	$0.371429\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	2.77761	0.316538
$78$	0	0
$79$	−9.52884	−1.07208	−0.536039	−	0.844193i	$-0.680080\pi$
$79$	−9.52884	−1.07208	−0.536039	+	0.844193i	$0.680080\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.16246	0.347125	0.173562	−	0.984823i	$-0.444472\pi$
$83$	3.16246	0.347125	0.173562	+	0.984823i	$0.444472\pi$
$84$	0	0
$85$	−4.67282	−0.506839
$86$	0	0
$87$	1.95684	0.209796
$88$	0	0
$89$	−15.2241	−1.61375	−0.806875	−	0.590722i	$-0.798843\pi$
$89$	−15.2241	−1.61375	−0.806875	+	0.590722i	$0.798843\pi$
$90$	0	0
$91$	−3.81681	−0.400110
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	5.34565	0.548452
$96$	0	0
$97$	10.7776	1.09430	0.547150	−	0.837035i	$-0.315713\pi$
$97$	10.7776	1.09430	0.547150	+	0.837035i	$0.315713\pi$
$98$	0	0
$99$	−1.14399	−0.114975
$100$	0	0
$101$	−3.51037	−0.349294	−0.174647	−	0.984631i	$-0.555879\pi$
$101$	−3.51037	−0.349294	−0.174647	+	0.984631i	$0.555879\pi$
$102$	0	0
$103$	−18.2633	−1.79954	−0.899768	−	0.436369i	$-0.856264\pi$
$103$	−18.2633	−1.79954	−0.899768	+	0.436369i	$0.856264\pi$
$104$	0	0
$105$	2.42801	0.236949
$106$	0	0
$107$	4.96080	0.479578	0.239789	−	0.970825i	$-0.422922\pi$
$107$	4.96080	0.479578	0.239789	+	0.970825i	$0.422922\pi$
$108$	0	0
$109$	6.30644	0.604048	0.302024	−	0.953300i	$-0.402338\pi$
$109$	6.30644	0.604048	0.302024	+	0.953300i	$0.402338\pi$
$110$	0	0
$111$	3.57199	0.339039
$112$	0	0
$113$	−3.43196	−0.322852	−0.161426	−	0.986885i	$-0.551609\pi$
$113$	−3.43196	−0.322852	−0.161426	+	0.986885i	$0.551609\pi$
$114$	0	0
$115$	−1.81681	−0.169418
$116$	0	0
$117$	1.57199	0.145331
$118$	0	0
$119$	−11.3456	−1.04005
$120$	0	0
$121$	−9.69129	−0.881027
$122$	0	0
$123$	−3.14399	−0.283484
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−13.8353	−1.22768	−0.613841	−	0.789429i	$-0.710377\pi$
$127$	−13.8353	−1.22768	−0.613841	+	0.789429i	$0.710377\pi$
$128$	0	0
$129$	−2.85601	−0.251458
$130$	0	0
$131$	−9.79213	−0.855542	−0.427771	−	0.903887i	$-0.640701\pi$
$131$	−9.79213	−0.855542	−0.427771	+	0.903887i	$0.640701\pi$
$132$	0	0
$133$	12.9793	1.12545
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	15.3641	1.31265	0.656323	−	0.754480i	$-0.272111\pi$
$137$	15.3641	1.31265	0.656323	+	0.754480i	$0.272111\pi$
$138$	0	0
$139$	19.1809	1.62691	0.813453	−	0.581631i	$-0.197585\pi$
$139$	19.1809	1.62691	0.813453	+	0.581631i	$0.197585\pi$
$140$	0	0
$141$	−7.16246	−0.603188
$142$	0	0
$143$	−1.79834	−0.150385
$144$	0	0
$145$	−1.95684	−0.162507
$146$	0	0
$147$	−1.10478	−0.0911210
$148$	0	0
$149$	−11.5473	−0.945992	−0.472996	−	0.881064i	$-0.656828\pi$
$149$	−11.5473	−0.945992	−0.472996	+	0.881064i	$0.656828\pi$
$150$	0	0
$151$	−5.61515	−0.456954	−0.228477	−	0.973549i	$-0.573375\pi$
$151$	−5.61515	−0.456954	−0.228477	+	0.973549i	$0.573375\pi$
$152$	0	0
$153$	4.67282	0.377775
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	−16.1233	−1.28678	−0.643388	−	0.765540i	$-0.722472\pi$
$157$	−16.1233	−1.28678	−0.643388	+	0.765540i	$0.722472\pi$
$158$	0	0
$159$	−3.81681	−0.302693
$160$	0	0
$161$	−4.41123	−0.347653
$162$	0	0
$163$	−6.71598	−0.526036	−0.263018	−	0.964791i	$-0.584718\pi$
$163$	−6.71598	−0.526036	−0.263018	+	0.964791i	$0.584718\pi$
$164$	0	0
$165$	1.14399	0.0890592
$166$	0	0
$167$	−2.16472	−0.167511	−0.0837555	−	0.996486i	$-0.526691\pi$
$167$	−2.16472	−0.167511	−0.0837555	+	0.996486i	$0.526691\pi$
$168$	0	0
$169$	−10.5288	−0.809911
$170$	0	0
$171$	−5.34565	−0.408792
$172$	0	0
$173$	−14.8560	−1.12948	−0.564741	−	0.825268i	$-0.691024\pi$
$173$	−14.8560	−1.12948	−0.564741	+	0.825268i	$0.691024\pi$
$174$	0	0
$175$	−2.42801	−0.183540
$176$	0	0
$177$	6.24482	0.469389
$178$	0	0
$179$	−20.0369	−1.49763	−0.748816	−	0.662778i	$-0.769377\pi$
$179$	−20.0369	−1.49763	−0.748816	+	0.662778i	$0.769377\pi$
$180$	0	0
$181$	−0.287973	−0.0214049	−0.0107024	−	0.999943i	$-0.503407\pi$
$181$	−0.287973	−0.0214049	−0.0107024	+	0.999943i	$0.503407\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	−3.57199	−0.262618
$186$	0	0
$187$	−5.34565	−0.390912
$188$	0	0
$189$	−2.42801	−0.176612
$190$	0	0
$191$	−10.2448	−0.741289	−0.370644	−	0.928775i	$-0.620863\pi$
$191$	−10.2448	−0.741289	−0.370644	+	0.928775i	$0.620863\pi$
$192$	0	0
$193$	17.8353	1.28381	0.641906	−	0.766783i	$-0.278144\pi$
$193$	17.8353	1.28381	0.641906	+	0.766783i	$0.278144\pi$
$194$	0	0
$195$	−1.57199	−0.112573
$196$	0	0
$197$	3.32718	0.237051	0.118526	−	0.992951i	$-0.462183\pi$
$197$	3.32718	0.237051	0.118526	+	0.992951i	$0.462183\pi$
$198$	0	0
$199$	5.65209	0.400666	0.200333	−	0.979728i	$-0.435798\pi$
$199$	5.65209	0.400666	0.200333	+	0.979728i	$0.435798\pi$
$200$	0	0
$201$	0.917641	0.0647254
$202$	0	0
$203$	−4.75123	−0.333471
$204$	0	0
$205$	3.14399	0.219586
$206$	0	0
$207$	1.81681	0.126277
$208$	0	0
$209$	6.11535	0.423008
$210$	0	0
$211$	22.6050	1.55619	0.778096	−	0.628146i	$-0.216186\pi$
$211$	22.6050	1.55619	0.778096	+	0.628146i	$0.216186\pi$
$212$	0	0
$213$	6.44648	0.441705
$214$	0	0
$215$	2.85601	0.194779
$216$	0	0
$217$	2.42801	0.164824
$218$	0	0
$219$	6.71598	0.453824
$220$	0	0
$221$	7.34565	0.494122
$222$	0	0
$223$	4.69129	0.314152	0.157076	−	0.987586i	$-0.449793\pi$
$223$	4.69129	0.314152	0.157076	+	0.987586i	$0.449793\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−1.90312	−0.126315	−0.0631573	−	0.998004i	$-0.520117\pi$
$227$	−1.90312	−0.126315	−0.0631573	+	0.998004i	$0.520117\pi$
$228$	0	0
$229$	−21.1440	−1.39723	−0.698617	−	0.715496i	$-0.746201\pi$
$229$	−21.1440	−1.39723	−0.698617	+	0.715496i	$0.746201\pi$
$230$	0	0
$231$	2.77761	0.182753
$232$	0	0
$233$	−26.7882	−1.75495	−0.877476	−	0.479621i	$-0.840774\pi$
$233$	−26.7882	−1.75495	−0.877476	+	0.479621i	$0.840774\pi$
$234$	0	0
$235$	7.16246	0.467227
$236$	0	0
$237$	−9.52884	−0.618964
$238$	0	0
$239$	−24.3249	−1.57345	−0.786724	−	0.617305i	$-0.788224\pi$
$239$	−24.3249	−1.57345	−0.786724	+	0.617305i	$0.788224\pi$
$240$	0	0
$241$	6.48963	0.418034	0.209017	−	0.977912i	$-0.432974\pi$
$241$	6.48963	0.418034	0.209017	+	0.977912i	$0.432974\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.10478	0.0705820
$246$	0	0
$247$	−8.40332	−0.534691
$248$	0	0
$249$	3.16246	0.200413
$250$	0	0
$251$	21.9216	1.38368	0.691839	−	0.722051i	$-0.256801\pi$
$251$	21.9216	1.38368	0.691839	+	0.722051i	$0.256801\pi$
$252$	0	0
$253$	−2.07841	−0.130668
$254$	0	0
$255$	−4.67282	−0.292624
$256$	0	0
$257$	−7.73050	−0.482215	−0.241108	−	0.970498i	$-0.577511\pi$
$257$	−7.73050	−0.482215	−0.241108	+	0.970498i	$0.577511\pi$
$258$	0	0
$259$	−8.67282	−0.538903
$260$	0	0
$261$	1.95684	0.121126
$262$	0	0
$263$	−25.4689	−1.57048	−0.785240	−	0.619192i	$-0.787460\pi$
$263$	−25.4689	−1.57048	−0.785240	+	0.619192i	$0.787460\pi$
$264$	0	0
$265$	3.81681	0.234465
$266$	0	0
$267$	−15.2241	−0.931699
$268$	0	0
$269$	−3.75518	−0.228958	−0.114479	−	0.993426i	$-0.536520\pi$
$269$	−3.75518	−0.228958	−0.114479	+	0.993426i	$0.536520\pi$
$270$	0	0
$271$	13.6257	0.827703	0.413852	−	0.910344i	$-0.364183\pi$
$271$	13.6257	0.827703	0.413852	+	0.910344i	$0.364183\pi$
$272$	0	0
$273$	−3.81681	−0.231004
$274$	0	0
$275$	−1.14399	−0.0689850
$276$	0	0
$277$	−19.3994	−1.16560	−0.582798	−	0.812617i	$-0.698042\pi$
$277$	−19.3994	−1.16560	−0.582798	+	0.812617i	$0.698042\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−9.14399	−0.545485	−0.272742	−	0.962087i	$-0.587931\pi$
$281$	−9.14399	−0.545485	−0.272742	+	0.962087i	$0.587931\pi$
$282$	0	0
$283$	−22.0616	−1.31143	−0.655714	−	0.755010i	$-0.727632\pi$
$283$	−22.0616	−1.31143	−0.655714	+	0.755010i	$0.727632\pi$
$284$	0	0
$285$	5.34565	0.316649
$286$	0	0
$287$	7.63362	0.450598
$288$	0	0
$289$	4.83528	0.284428
$290$	0	0
$291$	10.7776	0.631795
$292$	0	0
$293$	−7.90312	−0.461705	−0.230853	−	0.972989i	$-0.574152\pi$
$293$	−7.90312	−0.461705	−0.230853	+	0.972989i	$0.574152\pi$
$294$	0	0
$295$	−6.24482	−0.363587
$296$	0	0
$297$	−1.14399	−0.0663808
$298$	0	0
$299$	2.85601	0.165168
$300$	0	0
$301$	6.93442	0.399693
$302$	0	0
$303$	−3.51037	−0.201665
$304$	0	0
$305$	2.00000	0.114520
$306$	0	0
$307$	24.3865	1.39181	0.695907	−	0.718132i	$-0.255003\pi$
$307$	24.3865	1.39181	0.695907	+	0.718132i	$0.255003\pi$
$308$	0	0
$309$	−18.2633	−1.03896
$310$	0	0
$311$	−32.3681	−1.83542	−0.917712	−	0.397245i	$-0.869966\pi$
$311$	−32.3681	−1.83542	−0.917712	+	0.397245i	$0.869966\pi$
$312$	0	0
$313$	10.8313	0.612223	0.306111	−	0.951996i	$-0.400972\pi$
$313$	10.8313	0.612223	0.306111	+	0.951996i	$0.400972\pi$
$314$	0	0
$315$	2.42801	0.136803
$316$	0	0
$317$	−15.3641	−0.862935	−0.431467	−	0.902129i	$-0.642004\pi$
$317$	−15.3641	−0.862935	−0.431467	+	0.902129i	$0.642004\pi$
$318$	0	0
$319$	−2.23860	−0.125338
$320$	0	0
$321$	4.96080	0.276885
$322$	0	0
$323$	−24.9793	−1.38988
$324$	0	0
$325$	1.57199	0.0871985
$326$	0	0
$327$	6.30644	0.348747
$328$	0	0
$329$	17.3905	0.958769
$330$	0	0
$331$	−22.1417	−1.21702	−0.608510	−	0.793546i	$-0.708232\pi$
$331$	−22.1417	−1.21702	−0.608510	+	0.793546i	$0.708232\pi$
$332$	0	0
$333$	3.57199	0.195744
$334$	0	0
$335$	−0.917641	−0.0501361
$336$	0	0
$337$	25.6873	1.39928	0.699639	−	0.714496i	$-0.253344\pi$
$337$	25.6873	1.39928	0.699639	+	0.714496i	$0.253344\pi$
$338$	0	0
$339$	−3.43196	−0.186398
$340$	0	0
$341$	1.14399	0.0619503
$342$	0	0
$343$	19.6785	1.06254
$344$	0	0
$345$	−1.81681	−0.0978138
$346$	0	0
$347$	−17.2593	−0.926530	−0.463265	−	0.886220i	$-0.653322\pi$
$347$	−17.2593	−0.926530	−0.463265	+	0.886220i	$0.653322\pi$
$348$	0	0
$349$	−30.7098	−1.64386	−0.821928	−	0.569591i	$-0.807101\pi$
$349$	−30.7098	−1.64386	−0.821928	+	0.569591i	$0.807101\pi$
$350$	0	0
$351$	1.57199	0.0839068
$352$	0	0
$353$	−32.1338	−1.71031	−0.855155	−	0.518372i	$-0.826538\pi$
$353$	−32.1338	−1.71031	−0.855155	+	0.518372i	$0.826538\pi$
$354$	0	0
$355$	−6.44648	−0.342144
$356$	0	0
$357$	−11.3456	−0.600475
$358$	0	0
$359$	25.0594	1.32258	0.661291	−	0.750129i	$-0.270009\pi$
$359$	25.0594	1.32258	0.661291	+	0.750129i	$0.270009\pi$
$360$	0	0
$361$	9.57595	0.503997
$362$	0	0
$363$	−9.69129	−0.508661
$364$	0	0
$365$	−6.71598	−0.351530
$366$	0	0
$367$	1.71203	0.0893671	0.0446835	−	0.999001i	$-0.485772\pi$
$367$	1.71203	0.0893671	0.0446835	+	0.999001i	$0.485772\pi$
$368$	0	0
$369$	−3.14399	−0.163669
$370$	0	0
$371$	9.26724	0.481131
$372$	0	0
$373$	−32.4482	−1.68010	−0.840051	−	0.542507i	$-0.817475\pi$
$373$	−32.4482	−1.68010	−0.840051	+	0.542507i	$0.817475\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	3.07615	0.158430
$378$	0	0
$379$	1.51827	0.0779884	0.0389942	−	0.999239i	$-0.487585\pi$
$379$	1.51827	0.0779884	0.0389942	+	0.999239i	$0.487585\pi$
$380$	0	0
$381$	−13.8353	−0.708803
$382$	0	0
$383$	−30.2201	−1.54418	−0.772088	−	0.635515i	$-0.780788\pi$
$383$	−30.2201	−1.54418	−0.772088	+	0.635515i	$0.780788\pi$
$384$	0	0
$385$	−2.77761	−0.141560
$386$	0	0
$387$	−2.85601	−0.145179
$388$	0	0
$389$	7.67678	0.389228	0.194614	−	0.980880i	$-0.437655\pi$
$389$	7.67678	0.389228	0.194614	+	0.980880i	$0.437655\pi$
$390$	0	0
$391$	8.48963	0.429339
$392$	0	0
$393$	−9.79213	−0.493947
$394$	0	0
$395$	9.52884	0.479448
$396$	0	0
$397$	−14.3664	−0.721028	−0.360514	−	0.932754i	$-0.617399\pi$
$397$	−14.3664	−0.721028	−0.360514	+	0.932754i	$0.617399\pi$
$398$	0	0
$399$	12.9793	0.649776
$400$	0	0
$401$	10.2448	0.511602	0.255801	−	0.966729i	$-0.417661\pi$
$401$	10.2448	0.511602	0.255801	+	0.966729i	$0.417661\pi$
$402$	0	0
$403$	−1.57199	−0.0783066
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−4.08631	−0.202551
$408$	0	0
$409$	−24.2880	−1.20096	−0.600481	−	0.799639i	$-0.705024\pi$
$409$	−24.2880	−1.20096	−0.600481	+	0.799639i	$0.705024\pi$
$410$	0	0
$411$	15.3641	0.757856
$412$	0	0
$413$	−15.1625	−0.746096
$414$	0	0
$415$	−3.16246	−0.155239
$416$	0	0
$417$	19.1809	0.939294
$418$	0	0
$419$	37.9938	1.85612	0.928059	−	0.372433i	$-0.121476\pi$
$419$	37.9938	1.85612	0.928059	+	0.372433i	$0.121476\pi$
$420$	0	0
$421$	31.5658	1.53842	0.769211	−	0.638995i	$-0.220650\pi$
$421$	31.5658	1.53842	0.769211	+	0.638995i	$0.220650\pi$
$422$	0	0
$423$	−7.16246	−0.348251
$424$	0	0
$425$	4.67282	0.226665
$426$	0	0
$427$	4.85601	0.234999
$428$	0	0
$429$	−1.79834	−0.0868246
$430$	0	0
$431$	14.8992	0.717668	0.358834	−	0.933401i	$-0.383174\pi$
$431$	14.8992	0.717668	0.358834	+	0.933401i	$0.383174\pi$
$432$	0	0
$433$	−15.9753	−0.767725	−0.383862	−	0.923390i	$-0.625406\pi$
$433$	−15.9753	−0.767725	−0.383862	+	0.923390i	$0.625406\pi$
$434$	0	0
$435$	−1.95684	−0.0938235
$436$	0	0
$437$	−9.71203	−0.464589
$438$	0	0
$439$	32.0739	1.53080	0.765401	−	0.643553i	$-0.222540\pi$
$439$	32.0739	1.53080	0.765401	+	0.643553i	$0.222540\pi$
$440$	0	0
$441$	−1.10478	−0.0526087
$442$	0	0
$443$	12.5944	0.598379	0.299189	−	0.954194i	$-0.403284\pi$
$443$	12.5944	0.598379	0.299189	+	0.954194i	$0.403284\pi$
$444$	0	0
$445$	15.2241	0.721691
$446$	0	0
$447$	−11.5473	−0.546169
$448$	0	0
$449$	−7.46721	−0.352399	−0.176200	−	0.984354i	$-0.556380\pi$
$449$	−7.46721	−0.352399	−0.176200	+	0.984354i	$0.556380\pi$
$450$	0	0
$451$	3.59668	0.169361
$452$	0	0
$453$	−5.61515	−0.263823
$454$	0	0
$455$	3.81681	0.178935
$456$	0	0
$457$	−20.1849	−0.944209	−0.472104	−	0.881543i	$-0.656505\pi$
$457$	−20.1849	−0.944209	−0.472104	+	0.881543i	$0.656505\pi$
$458$	0	0
$459$	4.67282	0.218109
$460$	0	0
$461$	28.0017	1.30417	0.652084	−	0.758146i	$-0.273895\pi$
$461$	28.0017	1.30417	0.652084	+	0.758146i	$0.273895\pi$
$462$	0	0
$463$	38.4896	1.78876	0.894382	−	0.447303i	$-0.147615\pi$
$463$	38.4896	1.78876	0.894382	+	0.447303i	$0.147615\pi$
$464$	0	0
$465$	1.00000	0.0463739
$466$	0	0
$467$	20.2985	0.939304	0.469652	−	0.882852i	$-0.344379\pi$
$467$	20.2985	0.939304	0.469652	+	0.882852i	$0.344379\pi$
$468$	0	0
$469$	−2.22804	−0.102881
$470$	0	0
$471$	−16.1233	−0.742920
$472$	0	0
$473$	3.26724	0.150228
$474$	0	0
$475$	−5.34565	−0.245275
$476$	0	0
$477$	−3.81681	−0.174760
$478$	0	0
$479$	19.0594	0.870845	0.435422	−	0.900226i	$-0.356599\pi$
$479$	19.0594	0.870845	0.435422	+	0.900226i	$0.356599\pi$
$480$	0	0
$481$	5.61515	0.256029
$482$	0	0
$483$	−4.41123	−0.200718
$484$	0	0
$485$	−10.7776	−0.489386
$486$	0	0
$487$	−13.0656	−0.592058	−0.296029	−	0.955179i	$-0.595662\pi$
$487$	−13.0656	−0.592058	−0.296029	+	0.955179i	$0.595662\pi$
$488$	0	0
$489$	−6.71598	−0.303707
$490$	0	0
$491$	−24.3170	−1.09741	−0.548706	−	0.836016i	$-0.684879\pi$
$491$	−24.3170	−1.09741	−0.548706	+	0.836016i	$0.684879\pi$
$492$	0	0
$493$	9.14399	0.411824
$494$	0	0
$495$	1.14399	0.0514184
$496$	0	0
$497$	−15.6521	−0.702092
$498$	0	0
$499$	16.4033	0.734314	0.367157	−	0.930159i	$-0.380331\pi$
$499$	16.4033	0.734314	0.367157	+	0.930159i	$0.380331\pi$
$500$	0	0
$501$	−2.16472	−0.0967125
$502$	0	0
$503$	0.837542	0.0373442	0.0186721	−	0.999826i	$-0.494056\pi$
$503$	0.837542	0.0373442	0.0186721	+	0.999826i	$0.494056\pi$
$504$	0	0
$505$	3.51037	0.156209
$506$	0	0
$507$	−10.5288	−0.467602
$508$	0	0
$509$	−1.38880	−0.0615576	−0.0307788	−	0.999526i	$-0.509799\pi$
$509$	−1.38880	−0.0615576	−0.0307788	+	0.999526i	$0.509799\pi$
$510$	0	0
$511$	−16.3064	−0.721355
$512$	0	0
$513$	−5.34565	−0.236016
$514$	0	0
$515$	18.2633	0.804777
$516$	0	0
$517$	8.19376	0.360361
$518$	0	0
$519$	−14.8560	−0.652107
$520$	0	0
$521$	33.1025	1.45025	0.725124	−	0.688618i	$-0.241782\pi$
$521$	33.1025	1.45025	0.725124	+	0.688618i	$0.241782\pi$
$522$	0	0
$523$	−1.62571	−0.0710875	−0.0355438	−	0.999368i	$-0.511316\pi$
$523$	−1.62571	−0.0710875	−0.0355438	+	0.999368i	$0.511316\pi$
$524$	0	0
$525$	−2.42801	−0.105967
$526$	0	0
$527$	−4.67282	−0.203551
$528$	0	0
$529$	−19.6992	−0.856487
$530$	0	0
$531$	6.24482	0.271002
$532$	0	0
$533$	−4.94233	−0.214076
$534$	0	0
$535$	−4.96080	−0.214474
$536$	0	0
$537$	−20.0369	−0.864658
$538$	0	0
$539$	1.26386	0.0544382
$540$	0	0
$541$	26.5865	1.14304	0.571522	−	0.820587i	$-0.306353\pi$
$541$	26.5865	1.14304	0.571522	+	0.820587i	$0.306353\pi$
$542$	0	0
$543$	−0.287973	−0.0123581
$544$	0	0
$545$	−6.30644	−0.270138
$546$	0	0
$547$	19.9753	0.854083	0.427041	−	0.904232i	$-0.359556\pi$
$547$	19.9753	0.854083	0.427041	+	0.904232i	$0.359556\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−10.4606	−0.445636
$552$	0	0
$553$	23.1361	0.983846
$554$	0	0
$555$	−3.57199	−0.151623
$556$	0	0
$557$	−29.2488	−1.23931	−0.619655	−	0.784874i	$-0.712728\pi$
$557$	−29.2488	−1.23931	−0.619655	+	0.784874i	$0.712728\pi$
$558$	0	0
$559$	−4.48963	−0.189891
$560$	0	0
$561$	−5.34565	−0.225693
$562$	0	0
$563$	−0.960797	−0.0404928	−0.0202464	−	0.999795i	$-0.506445\pi$
$563$	−0.960797	−0.0404928	−0.0202464	+	0.999795i	$0.506445\pi$
$564$	0	0
$565$	3.43196	0.144384
$566$	0	0
$567$	−2.42801	−0.101967
$568$	0	0
$569$	−3.87844	−0.162593	−0.0812963	−	0.996690i	$-0.525906\pi$
$569$	−3.87844	−0.162593	−0.0812963	+	0.996690i	$0.525906\pi$
$570$	0	0
$571$	6.20166	0.259531	0.129766	−	0.991545i	$-0.458577\pi$
$571$	6.20166	0.259531	0.129766	+	0.991545i	$0.458577\pi$
$572$	0	0
$573$	−10.2448	−0.427983
$574$	0	0
$575$	1.81681	0.0757662
$576$	0	0
$577$	35.6336	1.48345	0.741724	−	0.670706i	$-0.234009\pi$
$577$	35.6336	1.48345	0.741724	+	0.670706i	$0.234009\pi$
$578$	0	0
$579$	17.8353	0.741209
$580$	0	0
$581$	−7.67847	−0.318557
$582$	0	0
$583$	4.36638	0.180837
$584$	0	0
$585$	−1.57199	−0.0649939
$586$	0	0
$587$	43.7569	1.80604	0.903020	−	0.429599i	$-0.141345\pi$
$587$	43.7569	1.80604	0.903020	+	0.429599i	$0.141345\pi$
$588$	0	0
$589$	5.34565	0.220264
$590$	0	0
$591$	3.32718	0.136862
$592$	0	0
$593$	17.3536	0.712625	0.356313	−	0.934367i	$-0.384034\pi$
$593$	17.3536	0.712625	0.356313	+	0.934367i	$0.384034\pi$
$594$	0	0
$595$	11.3456	0.465126
$596$	0	0
$597$	5.65209	0.231325
$598$	0	0
$599$	−23.7552	−0.970610	−0.485305	−	0.874345i	$-0.661292\pi$
$599$	−23.7552	−0.970610	−0.485305	+	0.874345i	$0.661292\pi$
$600$	0	0
$601$	−41.9506	−1.71120	−0.855601	−	0.517636i	$-0.826812\pi$
$601$	−41.9506	−1.71120	−0.855601	+	0.517636i	$0.826812\pi$
$602$	0	0
$603$	0.917641	0.0373692
$604$	0	0
$605$	9.69129	0.394007
$606$	0	0
$607$	4.25538	0.172721	0.0863603	−	0.996264i	$-0.472476\pi$
$607$	4.25538	0.172721	0.0863603	+	0.996264i	$0.472476\pi$
$608$	0	0
$609$	−4.75123	−0.192530
$610$	0	0
$611$	−11.2593	−0.455504
$612$	0	0
$613$	11.7367	0.474041	0.237021	−	0.971505i	$-0.423829\pi$
$613$	11.7367	0.474041	0.237021	+	0.971505i	$0.423829\pi$
$614$	0	0
$615$	3.14399	0.126778
$616$	0	0
$617$	−10.5759	−0.425772	−0.212886	−	0.977077i	$-0.568286\pi$
$617$	−10.5759	−0.425772	−0.212886	+	0.977077i	$0.568286\pi$
$618$	0	0
$619$	−29.1994	−1.17362	−0.586811	−	0.809724i	$-0.699617\pi$
$619$	−29.1994	−1.17362	−0.586811	+	0.809724i	$0.699617\pi$
$620$	0	0
$621$	1.81681	0.0729061
$622$	0	0
$623$	36.9642	1.48094
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	6.11535	0.244224
$628$	0	0
$629$	16.6913	0.665526
$630$	0	0
$631$	−19.6706	−0.783073	−0.391536	−	0.920163i	$-0.628056\pi$
$631$	−19.6706	−0.783073	−0.391536	+	0.920163i	$0.628056\pi$
$632$	0	0
$633$	22.6050	0.898467
$634$	0	0
$635$	13.8353	0.549036
$636$	0	0
$637$	−1.73671	−0.0688110
$638$	0	0
$639$	6.44648	0.255019
$640$	0	0
$641$	40.3681	1.59444	0.797221	−	0.603687i	$-0.206302\pi$
$641$	40.3681	1.59444	0.797221	+	0.603687i	$0.206302\pi$
$642$	0	0
$643$	−2.20166	−0.0868250	−0.0434125	−	0.999057i	$-0.513823\pi$
$643$	−2.20166	−0.0868250	−0.0434125	+	0.999057i	$0.513823\pi$
$644$	0	0
$645$	2.85601	0.112455
$646$	0	0
$647$	−41.0162	−1.61251	−0.806257	−	0.591566i	$-0.798510\pi$
$647$	−41.0162	−1.61251	−0.806257	+	0.591566i	$0.798510\pi$
$648$	0	0
$649$	−7.14399	−0.280426
$650$	0	0
$651$	2.42801	0.0951611
$652$	0	0
$653$	−5.00226	−0.195754	−0.0978768	−	0.995199i	$-0.531205\pi$
$653$	−5.00226	−0.195754	−0.0978768	+	0.995199i	$0.531205\pi$
$654$	0	0
$655$	9.79213	0.382610
$656$	0	0
$657$	6.71598	0.262015
$658$	0	0
$659$	−11.5826	−0.451192	−0.225596	−	0.974221i	$-0.572433\pi$
$659$	−11.5826	−0.451192	−0.225596	+	0.974221i	$0.572433\pi$
$660$	0	0
$661$	48.2386	1.87626	0.938132	−	0.346278i	$-0.112554\pi$
$661$	48.2386	1.87626	0.938132	+	0.346278i	$0.112554\pi$
$662$	0	0
$663$	7.34565	0.285281
$664$	0	0
$665$	−12.9793	−0.503314
$666$	0	0
$667$	3.55521	0.137658
$668$	0	0
$669$	4.69129	0.181376
$670$	0	0
$671$	2.28797	0.0883262
$672$	0	0
$673$	18.3417	0.707020	0.353510	−	0.935431i	$-0.384988\pi$
$673$	18.3417	0.707020	0.353510	+	0.935431i	$0.384988\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	5.33508	0.205044	0.102522	−	0.994731i	$-0.467309\pi$
$677$	5.33508	0.205044	0.102522	+	0.994731i	$0.467309\pi$
$678$	0	0
$679$	−26.1681	−1.00424
$680$	0	0
$681$	−1.90312	−0.0729278
$682$	0	0
$683$	−14.2695	−0.546007	−0.273004	−	0.962013i	$-0.588017\pi$
$683$	−14.2695	−0.546007	−0.273004	+	0.962013i	$0.588017\pi$
$684$	0	0
$685$	−15.3641	−0.587033
$686$	0	0
$687$	−21.1440	−0.806693
$688$	0	0
$689$	−6.00000	−0.228582
$690$	0	0
$691$	24.1233	0.917692	0.458846	−	0.888516i	$-0.348263\pi$
$691$	24.1233	0.917692	0.458846	+	0.888516i	$0.348263\pi$
$692$	0	0
$693$	2.77761	0.105513
$694$	0	0
$695$	−19.1809	−0.727574
$696$	0	0
$697$	−14.6913	−0.556472
$698$	0	0
$699$	−26.7882	−1.01322
$700$	0	0
$701$	−3.26724	−0.123402	−0.0617010	−	0.998095i	$-0.519653\pi$
$701$	−3.26724	−0.123402	−0.0617010	+	0.998095i	$0.519653\pi$
$702$	0	0
$703$	−19.0946	−0.720167
$704$	0	0
$705$	7.16246	0.269754
$706$	0	0
$707$	8.52319	0.320548
$708$	0	0
$709$	49.1104	1.84438	0.922190	−	0.386736i	$-0.126398\pi$
$709$	49.1104	1.84438	0.922190	+	0.386736i	$0.126398\pi$
$710$	0	0
$711$	−9.52884	−0.357359
$712$	0	0
$713$	−1.81681	−0.0680401
$714$	0	0
$715$	1.79834	0.0672541
$716$	0	0
$717$	−24.3249	−0.908431
$718$	0	0
$719$	−11.4320	−0.426340	−0.213170	−	0.977015i	$-0.568379\pi$
$719$	−11.4320	−0.426340	−0.213170	+	0.977015i	$0.568379\pi$
$720$	0	0
$721$	44.3434	1.65143
$722$	0	0
$723$	6.48963	0.241352
$724$	0	0
$725$	1.95684	0.0726754
$726$	0	0
$727$	0.514319	0.0190750	0.00953752	−	0.999955i	$-0.496964\pi$
$727$	0.514319	0.0190750	0.00953752	+	0.999955i	$0.496964\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−13.3456	−0.493607
$732$	0	0
$733$	35.7859	1.32178	0.660891	−	0.750482i	$-0.270178\pi$
$733$	35.7859	1.32178	0.660891	+	0.750482i	$0.270178\pi$
$734$	0	0
$735$	1.10478	0.0407505
$736$	0	0
$737$	−1.04977	−0.0386687
$738$	0	0
$739$	−33.2857	−1.22443	−0.612217	−	0.790690i	$-0.709722\pi$
$739$	−33.2857	−1.22443	−0.612217	+	0.790690i	$0.709722\pi$
$740$	0	0
$741$	−8.40332	−0.308704
$742$	0	0
$743$	−38.0448	−1.39573	−0.697865	−	0.716229i	$-0.745866\pi$
$743$	−38.0448	−1.39573	−0.697865	+	0.716229i	$0.745866\pi$
$744$	0	0
$745$	11.5473	0.423061
$746$	0	0
$747$	3.16246	0.115708
$748$	0	0
$749$	−12.0448	−0.440109
$750$	0	0
$751$	6.20166	0.226302	0.113151	−	0.993578i	$-0.463906\pi$
$751$	6.20166	0.226302	0.113151	+	0.993578i	$0.463906\pi$
$752$	0	0
$753$	21.9216	0.798867
$754$	0	0
$755$	5.61515	0.204356
$756$	0	0
$757$	−0.522225	−0.0189806	−0.00949029	−	0.999955i	$-0.503021\pi$
$757$	−0.522225	−0.0189806	−0.00949029	+	0.999955i	$0.503021\pi$
$758$	0	0
$759$	−2.07841	−0.0754414
$760$	0	0
$761$	−4.07219	−0.147617	−0.0738084	−	0.997272i	$-0.523515\pi$
$761$	−4.07219	−0.147617	−0.0738084	+	0.997272i	$0.523515\pi$
$762$	0	0
$763$	−15.3121	−0.554335
$764$	0	0
$765$	−4.67282	−0.168946
$766$	0	0
$767$	9.81681	0.354464
$768$	0	0
$769$	−19.8089	−0.714327	−0.357164	−	0.934042i	$-0.616256\pi$
$769$	−19.8089	−0.714327	−0.357164	+	0.934042i	$0.616256\pi$
$770$	0	0
$771$	−7.73050	−0.278407
$772$	0	0
$773$	−40.3803	−1.45238	−0.726190	−	0.687494i	$-0.758711\pi$
$773$	−40.3803	−1.45238	−0.726190	+	0.687494i	$0.758711\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	−8.67282	−0.311136
$778$	0	0
$779$	16.8066	0.602160
$780$	0	0
$781$	−7.37468	−0.263887
$782$	0	0
$783$	1.95684	0.0699319
$784$	0	0
$785$	16.1233	0.575464
$786$	0	0
$787$	−19.1730	−0.683444	−0.341722	−	0.939801i	$-0.611010\pi$
$787$	−19.1730	−0.683444	−0.341722	+	0.939801i	$0.611010\pi$
$788$	0	0
$789$	−25.4689	−0.906717
$790$	0	0
$791$	8.33282	0.296281
$792$	0	0
$793$	−3.14399	−0.111646
$794$	0	0
$795$	3.81681	0.135368
$796$	0	0
$797$	39.0347	1.38268	0.691340	−	0.722530i	$-0.257021\pi$
$797$	39.0347	1.38268	0.691340	+	0.722530i	$0.257021\pi$
$798$	0	0
$799$	−33.4689	−1.18404
$800$	0	0
$801$	−15.2241	−0.537917
$802$	0	0
$803$	−7.68299	−0.271127
$804$	0	0
$805$	4.41123	0.155475
$806$	0	0
$807$	−3.75518	−0.132189
$808$	0	0
$809$	−25.4627	−0.895220	−0.447610	−	0.894229i	$-0.647725\pi$
$809$	−25.4627	−0.895220	−0.447610	+	0.894229i	$0.647725\pi$
$810$	0	0
$811$	43.1440	1.51499	0.757495	−	0.652841i	$-0.226423\pi$
$811$	43.1440	1.51499	0.757495	+	0.652841i	$0.226423\pi$
$812$	0	0
$813$	13.6257	0.477875
$814$	0	0
$815$	6.71598	0.235251
$816$	0	0
$817$	15.2672	0.534133
$818$	0	0
$819$	−3.81681	−0.133370
$820$	0	0
$821$	12.2527	0.427623	0.213811	−	0.976875i	$-0.431412\pi$
$821$	12.2527	0.427623	0.213811	+	0.976875i	$0.431412\pi$
$822$	0	0
$823$	33.7569	1.17669	0.588345	−	0.808610i	$-0.299780\pi$
$823$	33.7569	1.17669	0.588345	+	0.808610i	$0.299780\pi$
$824$	0	0
$825$	−1.14399	−0.0398285
$826$	0	0
$827$	37.2778	1.29628	0.648138	−	0.761523i	$-0.275548\pi$
$827$	37.2778	1.29628	0.648138	+	0.761523i	$0.275548\pi$
$828$	0	0
$829$	26.3170	0.914028	0.457014	−	0.889460i	$-0.348919\pi$
$829$	26.3170	0.914028	0.457014	+	0.889460i	$0.348919\pi$
$830$	0	0
$831$	−19.3994	−0.672957
$832$	0	0
$833$	−5.16246	−0.178869
$834$	0	0
$835$	2.16472	0.0749132
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	−28.1664	−0.972412	−0.486206	−	0.873844i	$-0.661620\pi$
$839$	−28.1664	−0.972412	−0.486206	+	0.873844i	$0.661620\pi$
$840$	0	0
$841$	−25.1708	−0.867957
$842$	0	0
$843$	−9.14399	−0.314936
$844$	0	0
$845$	10.5288	0.362203
$846$	0	0
$847$	23.5305	0.808519
$848$	0	0
$849$	−22.0616	−0.757153
$850$	0	0
$851$	6.48963	0.222462
$852$	0	0
$853$	−7.67056	−0.262635	−0.131318	−	0.991340i	$-0.541921\pi$
$853$	−7.67056	−0.262635	−0.131318	+	0.991340i	$0.541921\pi$
$854$	0	0
$855$	5.34565	0.182817
$856$	0	0
$857$	−42.6992	−1.45858	−0.729288	−	0.684206i	$-0.760149\pi$
$857$	−42.6992	−1.45858	−0.729288	+	0.684206i	$0.760149\pi$
$858$	0	0
$859$	44.2465	1.50967	0.754836	−	0.655914i	$-0.227717\pi$
$859$	44.2465	1.50967	0.754836	+	0.655914i	$0.227717\pi$
$860$	0	0
$861$	7.63362	0.260153
$862$	0	0
$863$	−22.8639	−0.778297	−0.389148	−	0.921175i	$-0.627231\pi$
$863$	−22.8639	−0.778297	−0.389148	+	0.921175i	$0.627231\pi$
$864$	0	0
$865$	14.8560	0.505120
$866$	0	0
$867$	4.83528	0.164215
$868$	0	0
$869$	10.9009	0.369786
$870$	0	0
$871$	1.44252	0.0488781
$872$	0	0
$873$	10.7776	0.364767
$874$	0	0
$875$	2.42801	0.0820816
$876$	0	0
$877$	31.0577	1.04874	0.524372	−	0.851490i	$-0.324300\pi$
$877$	31.0577	1.04874	0.524372	+	0.851490i	$0.324300\pi$
$878$	0	0
$879$	−7.90312	−0.266566
$880$	0	0
$881$	−17.8784	−0.602340	−0.301170	−	0.953570i	$-0.597377\pi$
$881$	−17.8784	−0.602340	−0.301170	+	0.953570i	$0.597377\pi$
$882$	0	0
$883$	22.2096	0.747411	0.373706	−	0.927547i	$-0.378087\pi$
$883$	22.2096	0.747411	0.373706	+	0.927547i	$0.378087\pi$
$884$	0	0
$885$	−6.24482	−0.209917
$886$	0	0
$887$	4.10478	0.137825	0.0689126	−	0.997623i	$-0.478047\pi$
$887$	4.10478	0.137825	0.0689126	+	0.997623i	$0.478047\pi$
$888$	0	0
$889$	33.5922	1.12664
$890$	0	0
$891$	−1.14399	−0.0383250
$892$	0	0
$893$	38.2880	1.28126
$894$	0	0
$895$	20.0369	0.669761
$896$	0	0
$897$	2.85601	0.0953595
$898$	0	0
$899$	−1.95684	−0.0652644
$900$	0	0
$901$	−17.8353	−0.594179
$902$	0	0
$903$	6.93442	0.230763
$904$	0	0
$905$	0.287973	0.00957255
$906$	0	0
$907$	−8.62967	−0.286543	−0.143272	−	0.989683i	$-0.545762\pi$
$907$	−8.62967	−0.286543	−0.143272	+	0.989683i	$0.545762\pi$
$908$	0	0
$909$	−3.51037	−0.116431
$910$	0	0
$911$	−7.83528	−0.259594	−0.129797	−	0.991541i	$-0.541433\pi$
$911$	−7.83528	−0.259594	−0.129797	+	0.991541i	$0.541433\pi$
$912$	0	0
$913$	−3.61781	−0.119732
$914$	0	0
$915$	2.00000	0.0661180
$916$	0	0
$917$	23.7753	0.785131
$918$	0	0
$919$	−15.2514	−0.503098	−0.251549	−	0.967845i	$-0.580940\pi$
$919$	−15.2514	−0.503098	−0.251549	+	0.967845i	$0.580940\pi$
$920$	0	0
$921$	24.3865	0.803564
$922$	0	0
$923$	10.1338	0.333559
$924$	0	0
$925$	3.57199	0.117446
$926$	0	0
$927$	−18.2633	−0.599845
$928$	0	0
$929$	−4.15850	−0.136436	−0.0682181	−	0.997670i	$-0.521731\pi$
$929$	−4.15850	−0.136436	−0.0682181	+	0.997670i	$0.521731\pi$
$930$	0	0
$931$	5.90578	0.193554
$932$	0	0
$933$	−32.3681	−1.05968
$934$	0	0
$935$	5.34565	0.174821
$936$	0	0
$937$	−18.2175	−0.595139	−0.297569	−	0.954700i	$-0.596176\pi$
$937$	−18.2175	−0.595139	−0.297569	+	0.954700i	$0.596176\pi$
$938$	0	0
$939$	10.8313	0.353467
$940$	0	0
$941$	−5.84150	−0.190427	−0.0952137	−	0.995457i	$-0.530353\pi$
$941$	−5.84150	−0.190427	−0.0952137	+	0.995457i	$0.530353\pi$
$942$	0	0
$943$	−5.71203	−0.186009
$944$	0	0
$945$	2.42801	0.0789831
$946$	0	0
$947$	−57.9276	−1.88240	−0.941198	−	0.337856i	$-0.890298\pi$
$947$	−57.9276	−1.88240	−0.941198	+	0.337856i	$0.890298\pi$
$948$	0	0
$949$	10.5575	0.342710
$950$	0	0
$951$	−15.3641	−0.498216
$952$	0	0
$953$	22.9977	0.744970	0.372485	−	0.928038i	$-0.378506\pi$
$953$	22.9977	0.744970	0.372485	+	0.928038i	$0.378506\pi$
$954$	0	0
$955$	10.2448	0.331514
$956$	0	0
$957$	−2.23860	−0.0723638
$958$	0	0
$959$	−37.3042	−1.20461
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	4.96080	0.159859
$964$	0	0
$965$	−17.8353	−0.574138
$966$	0	0
$967$	−45.2672	−1.45570	−0.727848	−	0.685738i	$-0.759479\pi$
$967$	−45.2672	−1.45570	−0.727848	+	0.685738i	$0.759479\pi$
$968$	0	0
$969$	−24.9793	−0.802450
$970$	0	0
$971$	−7.59837	−0.243843	−0.121922	−	0.992540i	$-0.538906\pi$
$971$	−7.59837	−0.243843	−0.121922	+	0.992540i	$0.538906\pi$
$972$	0	0
$973$	−46.5714	−1.49301
$974$	0	0
$975$	1.57199	0.0503441
$976$	0	0
$977$	−19.1519	−0.612723	−0.306362	−	0.951915i	$-0.599112\pi$
$977$	−19.1519	−0.612723	−0.306362	+	0.951915i	$0.599112\pi$
$978$	0	0
$979$	17.4161	0.556623
$980$	0	0
$981$	6.30644	0.201349
$982$	0	0
$983$	−4.20957	−0.134264	−0.0671322	−	0.997744i	$-0.521385\pi$
$983$	−4.20957	−0.134264	−0.0671322	+	0.997744i	$0.521385\pi$
$984$	0	0
$985$	−3.32718	−0.106013
$986$	0	0
$987$	17.3905	0.553546
$988$	0	0
$989$	−5.18883	−0.164995
$990$	0	0
$991$	37.3121	1.18526	0.592629	−	0.805476i	$-0.298090\pi$
$991$	37.3121	1.18526	0.592629	+	0.805476i	$0.298090\pi$
$992$	0	0
$993$	−22.1417	−0.702646
$994$	0	0
$995$	−5.65209	−0.179183
$996$	0	0
$997$	−62.0034	−1.96367	−0.981833	−	0.189745i	$-0.939234\pi$
$997$	−62.0034	−1.96367	−0.981833	+	0.189745i	$0.939234\pi$
$998$	0	0
$999$	3.57199	0.113013

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7440.2.a.bs.1.2		3
4.3	odd	2		465.2.a.e.1.2	✓	3
12.11	even	2		1395.2.a.j.1.2		3
20.3	even	4		2325.2.c.k.1024.3		6
20.7	even	4		2325.2.c.k.1024.4		6
20.19	odd	2		2325.2.a.r.1.2		3
60.59	even	2		6975.2.a.bf.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.e.1.2	✓	3	4.3	odd	2
1395.2.a.j.1.2		3	12.11	even	2
2325.2.a.r.1.2		3	20.19	odd	2
2325.2.c.k.1024.3		6	20.3	even	4
2325.2.c.k.1024.4		6	20.7	even	4
6975.2.a.bf.1.2		3	60.59	even	2
7440.2.a.bs.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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -2.42801 q^{7} +1.00000 q^{9} -1.14399 q^{11} +1.57199 q^{13} -1.00000 q^{15} +4.67282 q^{17} -5.34565 q^{19} -2.42801 q^{21} +1.81681 q^{23} +1.00000 q^{25} +1.00000 q^{27} +1.95684 q^{29} -1.00000 q^{31} -1.14399 q^{33} +2.42801 q^{35} +3.57199 q^{37} +1.57199 q^{39} -3.14399 q^{41} -2.85601 q^{43} -1.00000 q^{45} -7.16246 q^{47} -1.10478 q^{49} +4.67282 q^{51} -3.81681 q^{53} +1.14399 q^{55} -5.34565 q^{57} +6.24482 q^{59} -2.00000 q^{61} -2.42801 q^{63} -1.57199 q^{65} +0.917641 q^{67} +1.81681 q^{69} +6.44648 q^{71} +6.71598 q^{73} +1.00000 q^{75} +2.77761 q^{77} -9.52884 q^{79} +1.00000 q^{81} +3.16246 q^{83} -4.67282 q^{85} +1.95684 q^{87} -15.2241 q^{89} -3.81681 q^{91} -1.00000 q^{93} +5.34565 q^{95} +10.7776 q^{97} -1.14399 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5} - 8 q^{7} + 3 q^{9} - 2 q^{11} + 4 q^{13} - 3 q^{15} + 4 q^{17} + 4 q^{19} - 8 q^{21} - 6 q^{23} + 3 q^{25} + 3 q^{27} - 2 q^{29} - 3 q^{31} - 2 q^{33} + 8 q^{35} + 10 q^{37} + 4 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 - 8 * q^7 + 3 * q^9 - 2 * q^11 + 4 * q^13 - 3 * q^15 + 4 * q^17 + 4 * q^19 - 8 * q^21 - 6 * q^23 + 3 * q^25 + 3 * q^27 - 2 * q^29 - 3 * q^31 - 2 * q^33 + 8 * q^35 + 10 * q^37 + 4 * q^39 - 8 * q^41 - 10 * q^43 - 3 * q^45 + 10 * q^47 + 11 * q^49 + 4 * q^51 + 2 * q^55 + 4 * q^57 + 8 * q^59 - 6 * q^61 - 8 * q^63 - 4 * q^65 - 18 * q^67 - 6 * q^69 - 10 * q^71 + 18 * q^73 + 3 * q^75 - 16 * q^77 - 20 * q^79 + 3 * q^81 - 22 * q^83 - 4 * q^85 - 2 * q^87 + 8 * q^89 - 3 * q^93 - 4 * q^95 + 8 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

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Database

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