LMFDB - Embedded newform 3762.2.a.bl.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3762,2,Mod(1,3762)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3762, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3762.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3762.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:	$30.0397212404$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.7578576.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 12x^{3} + 14x^{2} + 35x - 12 $ x^5 - 2x^4 - 12x^3 + 14x^2 + 35x - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.314438$ of defining polynomial
Character	$\chi$	$=$	3762.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +0.685562 q^{5} -0.816139 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +0.685562 q^{5} -0.816139 q^{7} +1.00000 q^{8} +0.685562 q^{10} +1.00000 q^{11} -4.21557 q^{13} -0.816139 q^{14} +1.00000 q^{16} -4.97049 q^{17} +1.00000 q^{19} +0.685562 q^{20} +1.00000 q^{22} +6.21557 q^{23} -4.53001 q^{25} -4.21557 q^{26} -0.816139 q^{28} +6.90113 q^{29} +9.10107 q^{31} +1.00000 q^{32} -4.97049 q^{34} -0.559513 q^{35} +10.9294 q^{37} +1.00000 q^{38} +0.685562 q^{40} +3.70164 q^{41} +7.53001 q^{43} +1.00000 q^{44} +6.21557 q^{46} +6.21557 q^{47} -6.33392 q^{49} -4.53001 q^{50} -4.21557 q^{52} +0.885502 q^{53} +0.685562 q^{55} -0.816139 q^{56} +6.90113 q^{58} +1.93064 q^{59} +14.0478 q^{61} +9.10107 q^{62} +1.00000 q^{64} -2.89003 q^{65} -4.34162 q^{67} -4.97049 q^{68} -0.559513 q^{70} +2.44842 q^{71} +9.96279 q^{73} +10.9294 q^{74} +1.00000 q^{76} -0.816139 q^{77} -5.21897 q^{79} +0.685562 q^{80} +3.70164 q^{82} +3.70164 q^{83} -3.40758 q^{85} +7.53001 q^{86} +1.00000 q^{88} -4.41098 q^{89} +3.44049 q^{91} +6.21557 q^{92} +6.21557 q^{94} +0.685562 q^{95} -7.12937 q^{97} -6.33392 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{2} + 5 q^{4} + 3 q^{5} + 6 q^{7} + 5 q^{8}+O(q^{10}) $ 5 * q + 5 * q^2 + 5 * q^4 + 3 * q^5 + 6 * q^7 + 5 * q^8	$ 5 q + 5 q^{2} + 5 q^{4} + 3 q^{5} + 6 q^{7} + 5 q^{8} + 3 q^{10} + 5 q^{11} + 6 q^{13} + 6 q^{14} + 5 q^{16} + 3 q^{17} + 5 q^{19} + 3 q^{20} + 5 q^{22} + 4 q^{23} + 4 q^{25} + 6 q^{26} + 6 q^{28} + 7 q^{29} + 8 q^{31} + 5 q^{32} + 3 q^{34} - 4 q^{35} + 11 q^{37} + 5 q^{38} + 3 q^{40} - 2 q^{41} + 11 q^{43} + 5 q^{44} + 4 q^{46} + 4 q^{47} + 9 q^{49} + 4 q^{50} + 6 q^{52} - 6 q^{53} + 3 q^{55} + 6 q^{56} + 7 q^{58} + 10 q^{59} + 13 q^{61} + 8 q^{62} + 5 q^{64} + 4 q^{65} + 7 q^{67} + 3 q^{68} - 4 q^{70} - 18 q^{71} + 10 q^{73} + 11 q^{74} + 5 q^{76} + 6 q^{77} + 22 q^{79} + 3 q^{80} - 2 q^{82} - 2 q^{83} - 9 q^{85} + 11 q^{86} + 5 q^{88} + 7 q^{89} + 16 q^{91} + 4 q^{92} + 4 q^{94} + 3 q^{95} + 18 q^{97} + 9 q^{98}+O(q^{100}) $ 5 * q + 5 * q^2 + 5 * q^4 + 3 * q^5 + 6 * q^7 + 5 * q^8 + 3 * q^10 + 5 * q^11 + 6 * q^13 + 6 * q^14 + 5 * q^16 + 3 * q^17 + 5 * q^19 + 3 * q^20 + 5 * q^22 + 4 * q^23 + 4 * q^25 + 6 * q^26 + 6 * q^28 + 7 * q^29 + 8 * q^31 + 5 * q^32 + 3 * q^34 - 4 * q^35 + 11 * q^37 + 5 * q^38 + 3 * q^40 - 2 * q^41 + 11 * q^43 + 5 * q^44 + 4 * q^46 + 4 * q^47 + 9 * q^49 + 4 * q^50 + 6 * q^52 - 6 * q^53 + 3 * q^55 + 6 * q^56 + 7 * q^58 + 10 * q^59 + 13 * q^61 + 8 * q^62 + 5 * q^64 + 4 * q^65 + 7 * q^67 + 3 * q^68 - 4 * q^70 - 18 * q^71 + 10 * q^73 + 11 * q^74 + 5 * q^76 + 6 * q^77 + 22 * q^79 + 3 * q^80 - 2 * q^82 - 2 * q^83 - 9 * q^85 + 11 * q^86 + 5 * q^88 + 7 * q^89 + 16 * q^91 + 4 * q^92 + 4 * q^94 + 3 * q^95 + 18 * q^97 + 9 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0.685562	0.306592	0.153296	−	0.988180i	$-0.451011\pi$
$5$	0.685562	0.306592	0.153296	+	0.988180i	$0.451011\pi$
$6$	0	0
$7$	−0.816139	−0.308471	−0.154236	−	0.988034i	$-0.549292\pi$
$7$	−0.816139	−0.308471	−0.154236	+	0.988034i	$0.549292\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0.685562	0.216794
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−4.21557	−1.16919	−0.584594	−	0.811326i	$-0.698746\pi$
$13$	−4.21557	−1.16919	−0.584594	+	0.811326i	$0.698746\pi$
$14$	−0.816139	−0.218122
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.97049	−1.20552	−0.602761	−	0.797922i	$-0.705933\pi$
$17$	−4.97049	−1.20552	−0.602761	+	0.797922i	$0.705933\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0.685562	0.153296
$21$	0	0
$22$	1.00000	0.213201
$23$	6.21557	1.29604	0.648018	−	0.761625i	$-0.275598\pi$
$23$	6.21557	1.29604	0.648018	+	0.761625i	$0.275598\pi$
$24$	0	0
$25$	−4.53001	−0.906001
$26$	−4.21557	−0.826741
$27$	0	0
$28$	−0.816139	−0.154236
$29$	6.90113	1.28151	0.640754	−	0.767747i	$-0.278622\pi$
$29$	6.90113	1.28151	0.640754	+	0.767747i	$0.278622\pi$
$30$	0	0
$31$	9.10107	1.63460	0.817300	−	0.576212i	$-0.195470\pi$
$31$	9.10107	1.63460	0.817300	+	0.576212i	$0.195470\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−4.97049	−0.852432
$35$	−0.559513	−0.0945750
$36$	0	0
$37$	10.9294	1.79679	0.898394	−	0.439191i	$-0.144735\pi$
$37$	10.9294	1.79679	0.898394	+	0.439191i	$0.144735\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	0.685562	0.108397
$41$	3.70164	0.578099	0.289050	−	0.957314i	$-0.406661\pi$
$41$	3.70164	0.578099	0.289050	+	0.957314i	$0.406661\pi$
$42$	0	0
$43$	7.53001	1.14832	0.574158	−	0.818745i	$-0.305330\pi$
$43$	7.53001	1.14832	0.574158	+	0.818745i	$0.305330\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	6.21557	0.916435
$47$	6.21557	0.906634	0.453317	−	0.891349i	$-0.350241\pi$
$47$	6.21557	0.906634	0.453317	+	0.891349i	$0.350241\pi$
$48$	0	0
$49$	−6.33392	−0.904845
$50$	−4.53001	−0.640639
$51$	0	0
$52$	−4.21557	−0.584594
$53$	0.885502	0.121633	0.0608165	−	0.998149i	$-0.480630\pi$
$53$	0.885502	0.121633	0.0608165	+	0.998149i	$0.480630\pi$
$54$	0	0
$55$	0.685562	0.0924411
$56$	−0.816139	−0.109061
$57$	0	0
$58$	6.90113	0.906163
$59$	1.93064	0.251347	0.125674	−	0.992072i	$-0.459891\pi$
$59$	1.93064	0.251347	0.125674	+	0.992072i	$0.459891\pi$
$60$	0	0
$61$	14.0478	1.79863	0.899317	−	0.437297i	$-0.144064\pi$
$61$	14.0478	1.79863	0.899317	+	0.437297i	$0.144064\pi$
$62$	9.10107	1.15584
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.89003	−0.358464
$66$	0	0
$67$	−4.34162	−0.530413	−0.265206	−	0.964192i	$-0.585440\pi$
$67$	−4.34162	−0.530413	−0.265206	+	0.964192i	$0.585440\pi$
$68$	−4.97049	−0.602761
$69$	0	0
$70$	−0.559513	−0.0668746
$71$	2.44842	0.290573	0.145287	−	0.989390i	$-0.453590\pi$
$71$	2.44842	0.290573	0.145287	+	0.989390i	$0.453590\pi$
$72$	0	0
$73$	9.96279	1.16606	0.583028	−	0.812452i	$-0.301868\pi$
$73$	9.96279	1.16606	0.583028	+	0.812452i	$0.301868\pi$
$74$	10.9294	1.27052
$75$	0	0
$76$	1.00000	0.114708
$77$	−0.816139	−0.0930076
$78$	0	0
$79$	−5.21897	−0.587180	−0.293590	−	0.955931i	$-0.594850\pi$
$79$	−5.21897	−0.587180	−0.293590	+	0.955931i	$0.594850\pi$
$80$	0.685562	0.0766481
$81$	0	0
$82$	3.70164	0.408778
$83$	3.70164	0.406308	0.203154	−	0.979147i	$-0.434881\pi$
$83$	3.70164	0.406308	0.203154	+	0.979147i	$0.434881\pi$
$84$	0	0
$85$	−3.40758	−0.369604
$86$	7.53001	0.811981
$87$	0	0
$88$	1.00000	0.106600
$89$	−4.41098	−0.467563	−0.233781	−	0.972289i	$-0.575110\pi$
$89$	−4.41098	−0.467563	−0.233781	+	0.972289i	$0.575110\pi$
$90$	0	0
$91$	3.44049	0.360661
$92$	6.21557	0.648018
$93$	0	0
$94$	6.21557	0.641087
$95$	0.685562	0.0703371
$96$	0	0
$97$	−7.12937	−0.723878	−0.361939	−	0.932202i	$-0.617885\pi$
$97$	−7.12937	−0.723878	−0.361939	+	0.932202i	$0.617885\pi$
$98$	−6.33392	−0.639822
$99$	0	0
$100$	−4.53001	−0.453001
$101$	−15.8306	−1.57520	−0.787600	−	0.616187i	$-0.788677\pi$
$101$	−15.8306	−1.57520	−0.787600	+	0.616187i	$0.788677\pi$
$102$	0	0
$103$	15.8999	1.56667	0.783333	−	0.621602i	$-0.213518\pi$
$103$	15.8999	1.56667	0.783333	+	0.621602i	$0.213518\pi$
$104$	−4.21557	−0.413370
$105$	0	0
$106$	0.885502	0.0860076
$107$	−16.2350	−1.56950	−0.784751	−	0.619812i	$-0.787209\pi$
$107$	−16.2350	−1.56950	−0.784751	+	0.619812i	$0.787209\pi$
$108$	0	0
$109$	−5.98657	−0.573410	−0.286705	−	0.958019i	$-0.592560\pi$
$109$	−5.98657	−0.573410	−0.286705	+	0.958019i	$0.592560\pi$
$110$	0.685562	0.0653657
$111$	0	0
$112$	−0.816139	−0.0771178
$113$	−7.50819	−0.706312	−0.353156	−	0.935565i	$-0.614891\pi$
$113$	−7.50819	−0.706312	−0.353156	+	0.935565i	$0.614891\pi$
$114$	0	0
$115$	4.26115	0.397355
$116$	6.90113	0.640754
$117$	0	0
$118$	1.93064	0.177729
$119$	4.05661	0.371869
$120$	0	0
$121$	1.00000	0.0909091
$122$	14.0478	1.27183
$123$	0	0
$124$	9.10107	0.817300
$125$	−6.53341	−0.584366
$126$	0	0
$127$	8.31784	0.738089	0.369044	−	0.929412i	$-0.379685\pi$
$127$	8.31784	0.738089	0.369044	+	0.929412i	$0.379685\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−2.89003	−0.253472
$131$	−6.44389	−0.563005	−0.281503	−	0.959560i	$-0.590833\pi$
$131$	−6.44389	−0.563005	−0.281503	+	0.959560i	$0.590833\pi$
$132$	0	0
$133$	−0.816139	−0.0707682
$134$	−4.34162	−0.375058
$135$	0	0
$136$	−4.97049	−0.426216
$137$	10.9444	0.935042	0.467521	−	0.883982i	$-0.345147\pi$
$137$	10.9444	0.935042	0.467521	+	0.883982i	$0.345147\pi$
$138$	0	0
$139$	−18.1439	−1.53894	−0.769472	−	0.638681i	$-0.779480\pi$
$139$	−18.1439	−1.53894	−0.769472	+	0.638681i	$0.779480\pi$
$140$	−0.559513	−0.0472875
$141$	0	0
$142$	2.44842	0.205466
$143$	−4.21557	−0.352523
$144$	0	0
$145$	4.73115	0.392901
$146$	9.96279	0.824527
$147$	0	0
$148$	10.9294	0.898394
$149$	7.66058	0.627579	0.313790	−	0.949493i	$-0.398401\pi$
$149$	7.66058	0.627579	0.313790	+	0.949493i	$0.398401\pi$
$150$	0	0
$151$	11.0033	0.895438	0.447719	−	0.894174i	$-0.352237\pi$
$151$	11.0033	0.895438	0.447719	+	0.894174i	$0.352237\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	−0.816139	−0.0657663
$155$	6.23934	0.501156
$156$	0	0
$157$	4.74225	0.378472	0.189236	−	0.981932i	$-0.439399\pi$
$157$	4.74225	0.378472	0.189236	+	0.981932i	$0.439399\pi$
$158$	−5.21897	−0.415199
$159$	0	0
$160$	0.685562	0.0541984
$161$	−5.07276	−0.399790
$162$	0	0
$163$	7.07276	0.553982	0.276991	−	0.960873i	$-0.410663\pi$
$163$	7.07276	0.553982	0.276991	+	0.960873i	$0.410663\pi$
$164$	3.70164	0.289050
$165$	0	0
$166$	3.70164	0.287303
$167$	−12.6357	−0.977778	−0.488889	−	0.872346i	$-0.662598\pi$
$167$	−12.6357	−0.977778	−0.488889	+	0.872346i	$0.662598\pi$
$168$	0	0
$169$	4.77100	0.367000
$170$	−3.40758	−0.261349
$171$	0	0
$172$	7.53001	0.574158
$173$	−19.7321	−1.50021	−0.750104	−	0.661320i	$-0.769996\pi$
$173$	−19.7321	−1.50021	−0.750104	+	0.661320i	$0.769996\pi$
$174$	0	0
$175$	3.69711	0.279475
$176$	1.00000	0.0753778
$177$	0	0
$178$	−4.41098	−0.330617
$179$	−2.89003	−0.216011	−0.108006	−	0.994150i	$-0.534446\pi$
$179$	−2.89003	−0.216011	−0.108006	+	0.994150i	$0.534446\pi$
$180$	0	0
$181$	−6.92943	−0.515061	−0.257530	−	0.966270i	$-0.582909\pi$
$181$	−6.92943	−0.515061	−0.257530	+	0.966270i	$0.582909\pi$
$182$	3.44049	0.255026
$183$	0	0
$184$	6.21557	0.458218
$185$	7.49280	0.550882
$186$	0	0
$187$	−4.97049	−0.363478
$188$	6.21557	0.453317
$189$	0	0
$190$	0.685562	0.0497359
$191$	7.35089	0.531892	0.265946	−	0.963988i	$-0.414316\pi$
$191$	7.35089	0.531892	0.265946	+	0.963988i	$0.414316\pi$
$192$	0	0
$193$	5.00770	0.360462	0.180231	−	0.983624i	$-0.442315\pi$
$193$	5.00770	0.360462	0.180231	+	0.983624i	$0.442315\pi$
$194$	−7.12937	−0.511859
$195$	0	0
$196$	−6.33392	−0.452423
$197$	17.6016	1.25406	0.627030	−	0.778995i	$-0.284270\pi$
$197$	17.6016	1.25406	0.627030	+	0.778995i	$0.284270\pi$
$198$	0	0
$199$	3.90453	0.276785	0.138392	−	0.990377i	$-0.455807\pi$
$199$	3.90453	0.276785	0.138392	+	0.990377i	$0.455807\pi$
$200$	−4.53001	−0.320320
$201$	0	0
$202$	−15.8306	−1.11383
$203$	−5.63228	−0.395308
$204$	0	0
$205$	2.53770	0.177241
$206$	15.8999	1.10780
$207$	0	0
$208$	−4.21557	−0.292297
$209$	1.00000	0.0691714
$210$	0	0
$211$	−9.69569	−0.667479	−0.333739	−	0.942665i	$-0.608311\pi$
$211$	−9.69569	−0.667479	−0.333739	+	0.942665i	$0.608311\pi$
$212$	0.885502	0.0608165
$213$	0	0
$214$	−16.2350	−1.10980
$215$	5.16228	0.352065
$216$	0	0
$217$	−7.42773	−0.504227
$218$	−5.98657	−0.405462
$219$	0	0
$220$	0.685562	0.0462206
$221$	20.9534	1.40948
$222$	0	0
$223$	17.4756	1.17025	0.585126	−	0.810942i	$-0.301045\pi$
$223$	17.4756	1.17025	0.585126	+	0.810942i	$0.301045\pi$
$224$	−0.816139	−0.0545306
$225$	0	0
$226$	−7.50819	−0.499438
$227$	−17.5622	−1.16564	−0.582821	−	0.812601i	$-0.698051\pi$
$227$	−17.5622	−1.16564	−0.582821	+	0.812601i	$0.698051\pi$
$228$	0	0
$229$	20.1110	1.32897	0.664485	−	0.747302i	$-0.268651\pi$
$229$	20.1110	1.32897	0.664485	+	0.747302i	$0.268651\pi$
$230$	4.26115	0.280972
$231$	0	0
$232$	6.90113	0.453081
$233$	16.7244	1.09565	0.547827	−	0.836591i	$-0.315455\pi$
$233$	16.7244	1.09565	0.547827	+	0.836591i	$0.315455\pi$
$234$	0	0
$235$	4.26115	0.277967
$236$	1.93064	0.125674
$237$	0	0
$238$	4.05661	0.262951
$239$	−16.7444	−1.08311	−0.541554	−	0.840666i	$-0.682164\pi$
$239$	−16.7444	−1.08311	−0.541554	+	0.840666i	$0.682164\pi$
$240$	0	0
$241$	−17.6846	−1.13916	−0.569582	−	0.821934i	$-0.692895\pi$
$241$	−17.6846	−1.13916	−0.569582	+	0.821934i	$0.692895\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	14.0478	0.899317
$245$	−4.34229	−0.277419
$246$	0	0
$247$	−4.21557	−0.268230
$248$	9.10107	0.577918
$249$	0	0
$250$	−6.53341	−0.413209
$251$	−15.4076	−0.972518	−0.486259	−	0.873815i	$-0.661639\pi$
$251$	−15.4076	−0.972518	−0.486259	+	0.873815i	$0.661639\pi$
$252$	0	0
$253$	6.21557	0.390769
$254$	8.31784	0.521908
$255$	0	0
$256$	1.00000	0.0625000
$257$	30.0087	1.87189	0.935945	−	0.352145i	$-0.114548\pi$
$257$	30.0087	1.87189	0.935945	+	0.352145i	$0.114548\pi$
$258$	0	0
$259$	−8.91993	−0.554258
$260$	−2.89003	−0.179232
$261$	0	0
$262$	−6.44389	−0.398105
$263$	−7.42124	−0.457613	−0.228807	−	0.973472i	$-0.573482\pi$
$263$	−7.42124	−0.457613	−0.228807	+	0.973472i	$0.573482\pi$
$264$	0	0
$265$	0.607066	0.0372918
$266$	−0.816139	−0.0500407
$267$	0	0
$268$	−4.34162	−0.265206
$269$	5.43001	0.331073	0.165537	−	0.986204i	$-0.447064\pi$
$269$	5.43001	0.331073	0.165537	+	0.986204i	$0.447064\pi$
$270$	0	0
$271$	8.72496	0.530004	0.265002	−	0.964248i	$-0.414627\pi$
$271$	8.72496	0.530004	0.265002	+	0.964248i	$0.414627\pi$
$272$	−4.97049	−0.301380
$273$	0	0
$274$	10.9444	0.661174
$275$	−4.53001	−0.273170
$276$	0	0
$277$	25.1078	1.50858	0.754291	−	0.656541i	$-0.227981\pi$
$277$	25.1078	1.50858	0.754291	+	0.656541i	$0.227981\pi$
$278$	−18.1439	−1.08820
$279$	0	0
$280$	−0.559513	−0.0334373
$281$	−28.7720	−1.71639	−0.858197	−	0.513321i	$-0.828415\pi$
$281$	−28.7720	−1.71639	−0.858197	+	0.513321i	$0.828415\pi$
$282$	0	0
$283$	13.6517	0.811508	0.405754	−	0.913982i	$-0.367009\pi$
$283$	13.6517	0.811508	0.405754	+	0.913982i	$0.367009\pi$
$284$	2.44842	0.145287
$285$	0	0
$286$	−4.21557	−0.249272
$287$	−3.02105	−0.178327
$288$	0	0
$289$	7.70579	0.453282
$290$	4.73115	0.277823
$291$	0	0
$292$	9.96279	0.583028
$293$	1.87328	0.109438	0.0547190	−	0.998502i	$-0.482574\pi$
$293$	1.87328	0.109438	0.0547190	+	0.998502i	$0.482574\pi$
$294$	0	0
$295$	1.32357	0.0770612
$296$	10.9294	0.635260
$297$	0	0
$298$	7.66058	0.443766
$299$	−26.2021	−1.51531
$300$	0	0
$301$	−6.14553	−0.354222
$302$	11.0033	0.633170
$303$	0	0
$304$	1.00000	0.0573539
$305$	9.63062	0.551448
$306$	0	0
$307$	−9.06001	−0.517082	−0.258541	−	0.966000i	$-0.583242\pi$
$307$	−9.06001	−0.517082	−0.258541	+	0.966000i	$0.583242\pi$
$308$	−0.816139	−0.0465038
$309$	0	0
$310$	6.23934	0.354371
$311$	−15.1378	−0.858383	−0.429192	−	0.903213i	$-0.641202\pi$
$311$	−15.1378	−0.858383	−0.429192	+	0.903213i	$0.641202\pi$
$312$	0	0
$313$	20.0849	1.13526	0.567632	−	0.823283i	$-0.307860\pi$
$313$	20.0849	1.13526	0.567632	+	0.823283i	$0.307860\pi$
$314$	4.74225	0.267620
$315$	0	0
$316$	−5.21897	−0.293590
$317$	−23.4146	−1.31510	−0.657548	−	0.753413i	$-0.728406\pi$
$317$	−23.4146	−1.31510	−0.657548	+	0.753413i	$0.728406\pi$
$318$	0	0
$319$	6.90113	0.386389
$320$	0.685562	0.0383241
$321$	0	0
$322$	−5.07276	−0.282694
$323$	−4.97049	−0.276566
$324$	0	0
$325$	19.0965	1.05929
$326$	7.07276	0.391724
$327$	0	0
$328$	3.70164	0.204389
$329$	−5.07276	−0.279671
$330$	0	0
$331$	−5.84112	−0.321057	−0.160528	−	0.987031i	$-0.551320\pi$
$331$	−5.84112	−0.321057	−0.160528	+	0.987031i	$0.551320\pi$
$332$	3.70164	0.203154
$333$	0	0
$334$	−12.6357	−0.691394
$335$	−2.97644	−0.162621
$336$	0	0
$337$	20.3135	1.10655	0.553273	−	0.833000i	$-0.313379\pi$
$337$	20.3135	1.10655	0.553273	+	0.833000i	$0.313379\pi$
$338$	4.77100	0.259508
$339$	0	0
$340$	−3.40758	−0.184802
$341$	9.10107	0.492851
$342$	0	0
$343$	10.8823	0.587590
$344$	7.53001	0.405991
$345$	0	0
$346$	−19.7321	−1.06081
$347$	15.1615	0.813914	0.406957	−	0.913447i	$-0.366590\pi$
$347$	15.1615	0.813914	0.406957	+	0.913447i	$0.366590\pi$
$348$	0	0
$349$	−27.1069	−1.45100	−0.725499	−	0.688223i	$-0.758391\pi$
$349$	−27.1069	−1.45100	−0.725499	+	0.688223i	$0.758391\pi$
$350$	3.69711	0.197619
$351$	0	0
$352$	1.00000	0.0533002
$353$	−21.7523	−1.15776	−0.578879	−	0.815413i	$-0.696510\pi$
$353$	−21.7523	−1.15776	−0.578879	+	0.815413i	$0.696510\pi$
$354$	0	0
$355$	1.67854	0.0890876
$356$	−4.41098	−0.233781
$357$	0	0
$358$	−2.89003	−0.152743
$359$	−36.1923	−1.91016	−0.955079	−	0.296351i	$-0.904230\pi$
$359$	−36.1923	−1.91016	−0.955079	+	0.296351i	$0.904230\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−6.92943	−0.364203
$363$	0	0
$364$	3.44049	0.180331
$365$	6.83011	0.357504
$366$	0	0
$367$	−29.9679	−1.56431	−0.782157	−	0.623082i	$-0.785880\pi$
$367$	−29.9679	−1.56431	−0.782157	+	0.623082i	$0.785880\pi$
$368$	6.21557	0.324009
$369$	0	0
$370$	7.49280	0.389532
$371$	−0.722692	−0.0375203
$372$	0	0
$373$	14.2477	0.737719	0.368860	−	0.929485i	$-0.379748\pi$
$373$	14.2477	0.737719	0.368860	+	0.929485i	$0.379748\pi$
$374$	−4.97049	−0.257018
$375$	0	0
$376$	6.21557	0.320543
$377$	−29.0922	−1.49832
$378$	0	0
$379$	−8.76240	−0.450094	−0.225047	−	0.974348i	$-0.572254\pi$
$379$	−8.76240	−0.450094	−0.225047	+	0.974348i	$0.572254\pi$
$380$	0.685562	0.0351686
$381$	0	0
$382$	7.35089	0.376104
$383$	−39.0615	−1.99595	−0.997976	−	0.0635990i	$-0.979742\pi$
$383$	−39.0615	−1.99595	−0.997976	+	0.0635990i	$0.979742\pi$
$384$	0	0
$385$	−0.559513	−0.0285154
$386$	5.00770	0.254885
$387$	0	0
$388$	−7.12937	−0.361939
$389$	9.07344	0.460042	0.230021	−	0.973186i	$-0.426121\pi$
$389$	9.07344	0.460042	0.230021	+	0.973186i	$0.426121\pi$
$390$	0	0
$391$	−30.8944	−1.56240
$392$	−6.33392	−0.319911
$393$	0	0
$394$	17.6016	0.886754
$395$	−3.57792	−0.180025
$396$	0	0
$397$	−25.5165	−1.28064	−0.640318	−	0.768110i	$-0.721198\pi$
$397$	−25.5165	−1.28064	−0.640318	+	0.768110i	$0.721198\pi$
$398$	3.90453	0.195716
$399$	0	0
$400$	−4.53001	−0.226500
$401$	−0.457241	−0.0228335	−0.0114168	−	0.999935i	$-0.503634\pi$
$401$	−0.457241	−0.0228335	−0.0114168	+	0.999935i	$0.503634\pi$
$402$	0	0
$403$	−38.3662	−1.91115
$404$	−15.8306	−0.787600
$405$	0	0
$406$	−5.63228	−0.279525
$407$	10.9294	0.541752
$408$	0	0
$409$	5.46230	0.270093	0.135047	−	0.990839i	$-0.456882\pi$
$409$	5.46230	0.270093	0.135047	+	0.990839i	$0.456882\pi$
$410$	2.53770	0.125328
$411$	0	0
$412$	15.8999	0.783333
$413$	−1.57567	−0.0775335
$414$	0	0
$415$	2.53770	0.124571
$416$	−4.21557	−0.206685
$417$	0	0
$418$	1.00000	0.0489116
$419$	26.2477	1.28228	0.641141	−	0.767423i	$-0.278461\pi$
$419$	26.2477	1.28228	0.641141	+	0.767423i	$0.278461\pi$
$420$	0	0
$421$	18.4504	0.899217	0.449608	−	0.893226i	$-0.351564\pi$
$421$	18.4504	0.899217	0.449608	+	0.893226i	$0.351564\pi$
$422$	−9.69569	−0.471979
$423$	0	0
$424$	0.885502	0.0430038
$425$	22.5164	1.09220
$426$	0	0
$427$	−11.4649	−0.554827
$428$	−16.2350	−0.784751
$429$	0	0
$430$	5.16228	0.248947
$431$	12.0956	0.582623	0.291312	−	0.956628i	$-0.405908\pi$
$431$	12.0956	0.582623	0.291312	+	0.956628i	$0.405908\pi$
$432$	0	0
$433$	17.5257	0.842232	0.421116	−	0.907007i	$-0.361639\pi$
$433$	17.5257	0.842232	0.421116	+	0.907007i	$0.361639\pi$
$434$	−7.42773	−0.356543
$435$	0	0
$436$	−5.98657	−0.286705
$437$	6.21557	0.297331
$438$	0	0
$439$	12.8623	0.613886	0.306943	−	0.951728i	$-0.400694\pi$
$439$	12.8623	0.613886	0.306943	+	0.951728i	$0.400694\pi$
$440$	0.685562	0.0326829
$441$	0	0
$442$	20.9534	0.996654
$443$	36.8648	1.75150	0.875749	−	0.482767i	$-0.160368\pi$
$443$	36.8648	1.75150	0.875749	+	0.482767i	$0.160368\pi$
$444$	0	0
$445$	−3.02400	−0.143351
$446$	17.4756	0.827493
$447$	0	0
$448$	−0.816139	−0.0385589
$449$	−9.86127	−0.465382	−0.232691	−	0.972551i	$-0.574753\pi$
$449$	−9.86127	−0.465382	−0.232691	+	0.972551i	$0.574753\pi$
$450$	0	0
$451$	3.70164	0.174303
$452$	−7.50819	−0.353156
$453$	0	0
$454$	−17.5622	−0.824233
$455$	2.35867	0.110576
$456$	0	0
$457$	3.97555	0.185968	0.0929841	−	0.995668i	$-0.470359\pi$
$457$	3.97555	0.185968	0.0929841	+	0.995668i	$0.470359\pi$
$458$	20.1110	0.939724
$459$	0	0
$460$	4.26115	0.198677
$461$	−37.6405	−1.75309	−0.876547	−	0.481316i	$-0.840159\pi$
$461$	−37.6405	−1.75309	−0.876547	+	0.481316i	$0.840159\pi$
$462$	0	0
$463$	−7.74361	−0.359876	−0.179938	−	0.983678i	$-0.557590\pi$
$463$	−7.74361	−0.359876	−0.179938	+	0.983678i	$0.557590\pi$
$464$	6.90113	0.320377
$465$	0	0
$466$	16.7244	0.774745
$467$	25.3256	1.17193	0.585964	−	0.810337i	$-0.300716\pi$
$467$	25.3256	1.17193	0.585964	+	0.810337i	$0.300716\pi$
$468$	0	0
$469$	3.54336	0.163617
$470$	4.26115	0.196552
$471$	0	0
$472$	1.93064	0.0888647
$473$	7.53001	0.346230
$474$	0	0
$475$	−4.53001	−0.207851
$476$	4.05661	0.185934
$477$	0	0
$478$	−16.7444	−0.765873
$479$	−17.6114	−0.804684	−0.402342	−	0.915489i	$-0.631804\pi$
$479$	−17.6114	−0.804684	−0.402342	+	0.915489i	$0.631804\pi$
$480$	0	0
$481$	−46.0738	−2.10078
$482$	−17.6846	−0.805511
$483$	0	0
$484$	1.00000	0.0454545
$485$	−4.88763	−0.221936
$486$	0	0
$487$	32.6177	1.47805	0.739025	−	0.673678i	$-0.235287\pi$
$487$	32.6177	1.47805	0.739025	+	0.673678i	$0.235287\pi$
$488$	14.0478	0.635913
$489$	0	0
$490$	−4.34229	−0.196165
$491$	−10.6601	−0.481085	−0.240542	−	0.970639i	$-0.577325\pi$
$491$	−10.6601	−0.481085	−0.240542	+	0.970639i	$0.577325\pi$
$492$	0	0
$493$	−34.3020	−1.54488
$494$	−4.21557	−0.189667
$495$	0	0
$496$	9.10107	0.408650
$497$	−1.99825	−0.0896336
$498$	0	0
$499$	−20.7398	−0.928443	−0.464221	−	0.885719i	$-0.653666\pi$
$499$	−20.7398	−0.928443	−0.464221	+	0.885719i	$0.653666\pi$
$500$	−6.53341	−0.292183
$501$	0	0
$502$	−15.4076	−0.687674
$503$	−41.5413	−1.85223	−0.926117	−	0.377237i	$-0.876874\pi$
$503$	−41.5413	−1.85223	−0.926117	+	0.377237i	$0.876874\pi$
$504$	0	0
$505$	−10.8528	−0.482944
$506$	6.21557	0.276316
$507$	0	0
$508$	8.31784	0.369044
$509$	34.0655	1.50993	0.754964	−	0.655766i	$-0.227654\pi$
$509$	34.0655	1.50993	0.754964	+	0.655766i	$0.227654\pi$
$510$	0	0
$511$	−8.13102	−0.359695
$512$	1.00000	0.0441942
$513$	0	0
$514$	30.0087	1.32363
$515$	10.9004	0.480328
$516$	0	0
$517$	6.21557	0.273360
$518$	−8.91993	−0.391919
$519$	0	0
$520$	−2.89003	−0.126736
$521$	5.58676	0.244761	0.122380	−	0.992483i	$-0.460947\pi$
$521$	5.58676	0.244761	0.122380	+	0.992483i	$0.460947\pi$
$522$	0	0
$523$	30.3721	1.32808	0.664040	−	0.747697i	$-0.268840\pi$
$523$	30.3721	1.32808	0.664040	+	0.747697i	$0.268840\pi$
$524$	−6.44389	−0.281503
$525$	0	0
$526$	−7.42124	−0.323581
$527$	−45.2368	−1.97055
$528$	0	0
$529$	15.6333	0.679707
$530$	0.607066	0.0263693
$531$	0	0
$532$	−0.816139	−0.0353841
$533$	−15.6045	−0.675907
$534$	0	0
$535$	−11.1301	−0.481197
$536$	−4.34162	−0.187529
$537$	0	0
$538$	5.43001	0.234104
$539$	−6.33392	−0.272821
$540$	0	0
$541$	15.2010	0.653542	0.326771	−	0.945103i	$-0.394039\pi$
$541$	15.2010	0.653542	0.326771	+	0.945103i	$0.394039\pi$
$542$	8.72496	0.374769
$543$	0	0
$544$	−4.97049	−0.213108
$545$	−4.10416	−0.175803
$546$	0	0
$547$	39.7245	1.69850	0.849249	−	0.527992i	$-0.177055\pi$
$547$	39.7245	1.69850	0.849249	+	0.527992i	$0.177055\pi$
$548$	10.9444	0.467521
$549$	0	0
$550$	−4.53001	−0.193160
$551$	6.90113	0.293998
$552$	0	0
$553$	4.25940	0.181128
$554$	25.1078	1.06673
$555$	0	0
$556$	−18.1439	−0.769472
$557$	35.0606	1.48556	0.742782	−	0.669534i	$-0.233506\pi$
$557$	35.0606	1.48556	0.742782	+	0.669534i	$0.233506\pi$
$558$	0	0
$559$	−31.7432	−1.34260
$560$	−0.559513	−0.0236438
$561$	0	0
$562$	−28.7720	−1.21367
$563$	10.4295	0.439550	0.219775	−	0.975551i	$-0.429468\pi$
$563$	10.4295	0.439550	0.219775	+	0.975551i	$0.429468\pi$
$564$	0	0
$565$	−5.14733	−0.216550
$566$	13.6517	0.573823
$567$	0	0
$568$	2.44842	0.102733
$569$	38.3571	1.60801	0.804007	−	0.594620i	$-0.202698\pi$
$569$	38.3571	1.60801	0.804007	+	0.594620i	$0.202698\pi$
$570$	0	0
$571$	−42.8465	−1.79307	−0.896535	−	0.442972i	$-0.853924\pi$
$571$	−42.8465	−1.79307	−0.896535	+	0.442972i	$0.853924\pi$
$572$	−4.21557	−0.176262
$573$	0	0
$574$	−3.02105	−0.126096
$575$	−28.1566	−1.17421
$576$	0	0
$577$	23.2061	0.966084	0.483042	−	0.875597i	$-0.339532\pi$
$577$	23.2061	0.966084	0.483042	+	0.875597i	$0.339532\pi$
$578$	7.70579	0.320519
$579$	0	0
$580$	4.73115	0.196450
$581$	−3.02105	−0.125334
$582$	0	0
$583$	0.885502	0.0366738
$584$	9.96279	0.412263
$585$	0	0
$586$	1.87328	0.0773843
$587$	−0.149825	−0.00618393	−0.00309197	−	0.999995i	$-0.500984\pi$
$587$	−0.149825	−0.00618393	−0.00309197	+	0.999995i	$0.500984\pi$
$588$	0	0
$589$	9.10107	0.375003
$590$	1.32357	0.0544905
$591$	0	0
$592$	10.9294	0.449197
$593$	−31.0858	−1.27654	−0.638269	−	0.769813i	$-0.720349\pi$
$593$	−31.0858	−1.27654	−0.638269	+	0.769813i	$0.720349\pi$
$594$	0	0
$595$	2.78106	0.114012
$596$	7.66058	0.313790
$597$	0	0
$598$	−26.2021	−1.07149
$599$	−35.4581	−1.44878	−0.724389	−	0.689391i	$-0.757878\pi$
$599$	−35.4581	−1.44878	−0.724389	+	0.689391i	$0.757878\pi$
$600$	0	0
$601$	3.15118	0.128540	0.0642698	−	0.997933i	$-0.479528\pi$
$601$	3.15118	0.128540	0.0642698	+	0.997933i	$0.479528\pi$
$602$	−6.14553	−0.250473
$603$	0	0
$604$	11.0033	0.447719
$605$	0.685562	0.0278720
$606$	0	0
$607$	25.2397	1.02445	0.512223	−	0.858852i	$-0.328822\pi$
$607$	25.2397	1.02445	0.512223	+	0.858852i	$0.328822\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	9.63062	0.389932
$611$	−26.2021	−1.06003
$612$	0	0
$613$	5.70746	0.230522	0.115261	−	0.993335i	$-0.463230\pi$
$613$	5.70746	0.230522	0.115261	+	0.993335i	$0.463230\pi$
$614$	−9.06001	−0.365632
$615$	0	0
$616$	−0.816139	−0.0328832
$617$	7.77100	0.312849	0.156424	−	0.987690i	$-0.450003\pi$
$617$	7.77100	0.312849	0.156424	+	0.987690i	$0.450003\pi$
$618$	0	0
$619$	−21.8716	−0.879095	−0.439547	−	0.898219i	$-0.644861\pi$
$619$	−21.8716	−0.879095	−0.439547	+	0.898219i	$0.644861\pi$
$620$	6.23934	0.250578
$621$	0	0
$622$	−15.1378	−0.606969
$623$	3.59997	0.144230
$624$	0	0
$625$	18.1710	0.726839
$626$	20.0849	0.802752
$627$	0	0
$628$	4.74225	0.189236
$629$	−54.3247	−2.16607
$630$	0	0
$631$	14.7379	0.586708	0.293354	−	0.956004i	$-0.405228\pi$
$631$	14.7379	0.586708	0.293354	+	0.956004i	$0.405228\pi$
$632$	−5.21897	−0.207599
$633$	0	0
$634$	−23.4146	−0.929913
$635$	5.70239	0.226292
$636$	0	0
$637$	26.7011	1.05793
$638$	6.90113	0.273218
$639$	0	0
$640$	0.685562	0.0270992
$641$	14.1132	0.557439	0.278719	−	0.960373i	$-0.410090\pi$
$641$	14.1132	0.557439	0.278719	+	0.960373i	$0.410090\pi$
$642$	0	0
$643$	0.980299	0.0386592	0.0193296	−	0.999813i	$-0.493847\pi$
$643$	0.980299	0.0386592	0.0193296	+	0.999813i	$0.493847\pi$
$644$	−5.07276	−0.199895
$645$	0	0
$646$	−4.97049	−0.195561
$647$	−42.0064	−1.65144	−0.825720	−	0.564080i	$-0.809231\pi$
$647$	−42.0064	−1.65144	−0.825720	+	0.564080i	$0.809231\pi$
$648$	0	0
$649$	1.93064	0.0757841
$650$	19.0965	0.749028
$651$	0	0
$652$	7.07276	0.276991
$653$	38.1911	1.49453	0.747267	−	0.664524i	$-0.231366\pi$
$653$	38.1911	1.49453	0.747267	+	0.664524i	$0.231366\pi$
$654$	0	0
$655$	−4.41768	−0.172613
$656$	3.70164	0.144525
$657$	0	0
$658$	−5.07276	−0.197757
$659$	14.2021	0.553237	0.276618	−	0.960980i	$-0.410786\pi$
$659$	14.2021	0.553237	0.276618	+	0.960980i	$0.410786\pi$
$660$	0	0
$661$	−15.5219	−0.603730	−0.301865	−	0.953351i	$-0.597609\pi$
$661$	−15.5219	−0.603730	−0.301865	+	0.953351i	$0.597609\pi$
$662$	−5.84112	−0.227021
$663$	0	0
$664$	3.70164	0.143652
$665$	−0.559513	−0.0216970
$666$	0	0
$667$	42.8944	1.66088
$668$	−12.6357	−0.488889
$669$	0	0
$670$	−2.97644	−0.114990
$671$	14.0478	0.542309
$672$	0	0
$673$	−17.3169	−0.667516	−0.333758	−	0.942659i	$-0.608317\pi$
$673$	−17.3169	−0.667516	−0.333758	+	0.942659i	$0.608317\pi$
$674$	20.3135	0.782446
$675$	0	0
$676$	4.77100	0.183500
$677$	−6.28335	−0.241489	−0.120744	−	0.992684i	$-0.538528\pi$
$677$	−6.28335	−0.241489	−0.120744	+	0.992684i	$0.538528\pi$
$678$	0	0
$679$	5.81856	0.223296
$680$	−3.40758	−0.130675
$681$	0	0
$682$	9.10107	0.348498
$683$	−17.5801	−0.672682	−0.336341	−	0.941740i	$-0.609190\pi$
$683$	−17.5801	−0.672682	−0.336341	+	0.941740i	$0.609190\pi$
$684$	0	0
$685$	7.50305	0.286677
$686$	10.8823	0.415489
$687$	0	0
$688$	7.53001	0.287079
$689$	−3.73289	−0.142212
$690$	0	0
$691$	−12.3468	−0.469695	−0.234848	−	0.972032i	$-0.575459\pi$
$691$	−12.3468	−0.469695	−0.234848	+	0.972032i	$0.575459\pi$
$692$	−19.7321	−0.750104
$693$	0	0
$694$	15.1615	0.575524
$695$	−12.4387	−0.471828
$696$	0	0
$697$	−18.3990	−0.696911
$698$	−27.1069	−1.02601
$699$	0	0
$700$	3.69711	0.139738
$701$	28.3782	1.07183	0.535915	−	0.844272i	$-0.319967\pi$
$701$	28.3782	1.07183	0.535915	+	0.844272i	$0.319967\pi$
$702$	0	0
$703$	10.9294	0.412211
$704$	1.00000	0.0376889
$705$	0	0
$706$	−21.7523	−0.818659
$707$	12.9199	0.485904
$708$	0	0
$709$	−39.3802	−1.47895	−0.739477	−	0.673182i	$-0.764927\pi$
$709$	−39.3802	−1.47895	−0.739477	+	0.673182i	$0.764927\pi$
$710$	1.67854	0.0629945
$711$	0	0
$712$	−4.41098	−0.165308
$713$	56.5683	2.11850
$714$	0	0
$715$	−2.89003	−0.108081
$716$	−2.89003	−0.108006
$717$	0	0
$718$	−36.1923	−1.35069
$719$	−47.6131	−1.77567	−0.887835	−	0.460162i	$-0.847791\pi$
$719$	−47.6131	−1.77567	−0.887835	+	0.460162i	$0.847791\pi$
$720$	0	0
$721$	−12.9765	−0.483272
$722$	1.00000	0.0372161
$723$	0	0
$724$	−6.92943	−0.257530
$725$	−31.2621	−1.16105
$726$	0	0
$727$	−36.3044	−1.34646	−0.673228	−	0.739435i	$-0.735093\pi$
$727$	−36.3044	−1.34646	−0.673228	+	0.739435i	$0.735093\pi$
$728$	3.44049	0.127513
$729$	0	0
$730$	6.83011	0.252794
$731$	−37.4278	−1.38432
$732$	0	0
$733$	−17.7703	−0.656362	−0.328181	−	0.944615i	$-0.606436\pi$
$733$	−17.7703	−0.656362	−0.328181	+	0.944615i	$0.606436\pi$
$734$	−29.9679	−1.10614
$735$	0	0
$736$	6.21557	0.229109
$737$	−4.34162	−0.159925
$738$	0	0
$739$	−20.6207	−0.758544	−0.379272	−	0.925285i	$-0.623825\pi$
$739$	−20.6207	−0.758544	−0.379272	+	0.925285i	$0.623825\pi$
$740$	7.49280	0.275441
$741$	0	0
$742$	−0.722692	−0.0265309
$743$	46.9823	1.72361	0.861807	−	0.507237i	$-0.169333\pi$
$743$	46.9823	1.72361	0.861807	+	0.507237i	$0.169333\pi$
$744$	0	0
$745$	5.25180	0.192411
$746$	14.2477	0.521646
$747$	0	0
$748$	−4.97049	−0.181739
$749$	13.2500	0.484146
$750$	0	0
$751$	3.27345	0.119450	0.0597250	−	0.998215i	$-0.480978\pi$
$751$	3.27345	0.119450	0.0597250	+	0.998215i	$0.480978\pi$
$752$	6.21557	0.226658
$753$	0	0
$754$	−29.0922	−1.05947
$755$	7.54346	0.274535
$756$	0	0
$757$	−31.0798	−1.12961	−0.564807	−	0.825223i	$-0.691049\pi$
$757$	−31.0798	−1.12961	−0.564807	+	0.825223i	$0.691049\pi$
$758$	−8.76240	−0.318265
$759$	0	0
$760$	0.685562	0.0248679
$761$	−18.2418	−0.661266	−0.330633	−	0.943759i	$-0.607262\pi$
$761$	−18.2418	−0.661266	−0.330633	+	0.943759i	$0.607262\pi$
$762$	0	0
$763$	4.88587	0.176880
$764$	7.35089	0.265946
$765$	0	0
$766$	−39.0615	−1.41135
$767$	−8.13873	−0.293872
$768$	0	0
$769$	−51.9350	−1.87283	−0.936413	−	0.350901i	$-0.885875\pi$
$769$	−51.9350	−1.87283	−0.936413	+	0.350901i	$0.885875\pi$
$770$	−0.559513	−0.0201635
$771$	0	0
$772$	5.00770	0.180231
$773$	25.9686	0.934026	0.467013	−	0.884250i	$-0.345330\pi$
$773$	25.9686	0.934026	0.467013	+	0.884250i	$0.345330\pi$
$774$	0	0
$775$	−41.2279	−1.48095
$776$	−7.12937	−0.255930
$777$	0	0
$778$	9.07344	0.325299
$779$	3.70164	0.132625
$780$	0	0
$781$	2.44842	0.0876112
$782$	−30.8944	−1.10478
$783$	0	0
$784$	−6.33392	−0.226211
$785$	3.25110	0.116037
$786$	0	0
$787$	17.7686	0.633382	0.316691	−	0.948529i	$-0.397428\pi$
$787$	17.7686	0.633382	0.316691	+	0.948529i	$0.397428\pi$
$788$	17.6016	0.627030
$789$	0	0
$790$	−3.57792	−0.127297
$791$	6.12773	0.217877
$792$	0	0
$793$	−59.2194	−2.10294
$794$	−25.5165	−0.905547
$795$	0	0
$796$	3.90453	0.138392
$797$	−14.1001	−0.499451	−0.249726	−	0.968317i	$-0.580340\pi$
$797$	−14.1001	−0.499451	−0.249726	+	0.968317i	$0.580340\pi$
$798$	0	0
$799$	−30.8944	−1.09297
$800$	−4.53001	−0.160160
$801$	0	0
$802$	−0.457241	−0.0161458
$803$	9.96279	0.351579
$804$	0	0
$805$	−3.47769	−0.122573
$806$	−38.3662	−1.35139
$807$	0	0
$808$	−15.8306	−0.556917
$809$	34.2519	1.20423	0.602117	−	0.798408i	$-0.294324\pi$
$809$	34.2519	1.20423	0.602117	+	0.798408i	$0.294324\pi$
$810$	0	0
$811$	−1.71439	−0.0602005	−0.0301003	−	0.999547i	$-0.509583\pi$
$811$	−1.71439	−0.0602005	−0.0301003	+	0.999547i	$0.509583\pi$
$812$	−5.63228	−0.197654
$813$	0	0
$814$	10.9294	0.383076
$815$	4.84882	0.169847
$816$	0	0
$817$	7.53001	0.263442
$818$	5.46230	0.190985
$819$	0	0
$820$	2.53770	0.0886204
$821$	−41.2497	−1.43962	−0.719812	−	0.694169i	$-0.755772\pi$
$821$	−41.2497	−1.43962	−0.719812	+	0.694169i	$0.755772\pi$
$822$	0	0
$823$	2.96454	0.103337	0.0516687	−	0.998664i	$-0.483546\pi$
$823$	2.96454	0.103337	0.0516687	+	0.998664i	$0.483546\pi$
$824$	15.8999	0.553900
$825$	0	0
$826$	−1.57567	−0.0548245
$827$	−4.06936	−0.141506	−0.0707528	−	0.997494i	$-0.522540\pi$
$827$	−4.06936	−0.141506	−0.0707528	+	0.997494i	$0.522540\pi$
$828$	0	0
$829$	−35.9218	−1.24762	−0.623808	−	0.781577i	$-0.714415\pi$
$829$	−35.9218	−1.24762	−0.623808	+	0.781577i	$0.714415\pi$
$830$	2.53770	0.0880849
$831$	0	0
$832$	−4.21557	−0.146148
$833$	31.4827	1.09081
$834$	0	0
$835$	−8.66254	−0.299779
$836$	1.00000	0.0345857
$837$	0	0
$838$	26.2477	0.906710
$839$	23.9982	0.828510	0.414255	−	0.910161i	$-0.364042\pi$
$839$	23.9982	0.828510	0.414255	+	0.910161i	$0.364042\pi$
$840$	0	0
$841$	18.6256	0.642261
$842$	18.4504	0.635842
$843$	0	0
$844$	−9.69569	−0.333739
$845$	3.27082	0.112520
$846$	0	0
$847$	−0.816139	−0.0280429
$848$	0.885502	0.0304083
$849$	0	0
$850$	22.5164	0.772305
$851$	67.9326	2.32870
$852$	0	0
$853$	−8.38109	−0.286963	−0.143482	−	0.989653i	$-0.545830\pi$
$853$	−8.38109	−0.286963	−0.143482	+	0.989653i	$0.545830\pi$
$854$	−11.4649	−0.392322
$855$	0	0
$856$	−16.2350	−0.554902
$857$	11.2492	0.384264	0.192132	−	0.981369i	$-0.438460\pi$
$857$	11.2492	0.384264	0.192132	+	0.981369i	$0.438460\pi$
$858$	0	0
$859$	21.9454	0.748767	0.374383	−	0.927274i	$-0.377854\pi$
$859$	21.9454	0.748767	0.374383	+	0.927274i	$0.377854\pi$
$860$	5.16228	0.176032
$861$	0	0
$862$	12.0956	0.411977
$863$	52.2205	1.77761	0.888803	−	0.458289i	$-0.151537\pi$
$863$	52.2205	1.77761	0.888803	+	0.458289i	$0.151537\pi$
$864$	0	0
$865$	−13.5276	−0.459952
$866$	17.5257	0.595548
$867$	0	0
$868$	−7.42773	−0.252114
$869$	−5.21897	−0.177041
$870$	0	0
$871$	18.3024	0.620152
$872$	−5.98657	−0.202731
$873$	0	0
$874$	6.21557	0.210245
$875$	5.33216	0.180260
$876$	0	0
$877$	−39.9588	−1.34931	−0.674657	−	0.738132i	$-0.735708\pi$
$877$	−39.9588	−1.34931	−0.674657	+	0.738132i	$0.735708\pi$
$878$	12.8623	0.434083
$879$	0	0
$880$	0.685562	0.0231103
$881$	−2.14447	−0.0722491	−0.0361246	−	0.999347i	$-0.511501\pi$
$881$	−2.14447	−0.0722491	−0.0361246	+	0.999347i	$0.511501\pi$
$882$	0	0
$883$	−12.5259	−0.421529	−0.210764	−	0.977537i	$-0.567595\pi$
$883$	−12.5259	−0.421529	−0.210764	+	0.977537i	$0.567595\pi$
$884$	20.9534	0.704741
$885$	0	0
$886$	36.8648	1.23850
$887$	−34.7557	−1.16698	−0.583491	−	0.812120i	$-0.698314\pi$
$887$	−34.7557	−1.16698	−0.583491	+	0.812120i	$0.698314\pi$
$888$	0	0
$889$	−6.78851	−0.227679
$890$	−3.02400	−0.101365
$891$	0	0
$892$	17.4756	0.585126
$893$	6.21557	0.207996
$894$	0	0
$895$	−1.98129	−0.0662274
$896$	−0.816139	−0.0272653
$897$	0	0
$898$	−9.86127	−0.329075
$899$	62.8076	2.09475
$900$	0	0
$901$	−4.40138	−0.146631
$902$	3.70164	0.123251
$903$	0	0
$904$	−7.50819	−0.249719
$905$	−4.75055	−0.157914
$906$	0	0
$907$	−37.0430	−1.22999	−0.614996	−	0.788530i	$-0.710842\pi$
$907$	−37.0430	−1.22999	−0.614996	+	0.788530i	$0.710842\pi$
$908$	−17.5622	−0.582821
$909$	0	0
$910$	2.35867	0.0781890
$911$	10.5483	0.349480	0.174740	−	0.984615i	$-0.444091\pi$
$911$	10.5483	0.349480	0.174740	+	0.984615i	$0.444091\pi$
$912$	0	0
$913$	3.70164	0.122506
$914$	3.97555	0.131499
$915$	0	0
$916$	20.1110	0.664485
$917$	5.25910	0.173671
$918$	0	0
$919$	29.3261	0.967378	0.483689	−	0.875240i	$-0.339297\pi$
$919$	29.3261	0.967378	0.483689	+	0.875240i	$0.339297\pi$
$920$	4.26115	0.140486
$921$	0	0
$922$	−37.6405	−1.23962
$923$	−10.3215	−0.339735
$924$	0	0
$925$	−49.5104	−1.62789
$926$	−7.74361	−0.254471
$927$	0	0
$928$	6.90113	0.226541
$929$	32.2021	1.05652	0.528259	−	0.849083i	$-0.322845\pi$
$929$	32.2021	1.05652	0.528259	+	0.849083i	$0.322845\pi$
$930$	0	0
$931$	−6.33392	−0.207586
$932$	16.7244	0.547827
$933$	0	0
$934$	25.3256	0.828678
$935$	−3.40758	−0.111440
$936$	0	0
$937$	−17.9628	−0.586819	−0.293409	−	0.955987i	$-0.594790\pi$
$937$	−17.9628	−0.586819	−0.293409	+	0.955987i	$0.594790\pi$
$938$	3.54336	0.115695
$939$	0	0
$940$	4.26115	0.138984
$941$	31.7844	1.03614	0.518070	−	0.855338i	$-0.326651\pi$
$941$	31.7844	1.03614	0.518070	+	0.855338i	$0.326651\pi$
$942$	0	0
$943$	23.0078	0.749237
$944$	1.93064	0.0628369
$945$	0	0
$946$	7.53001	0.244822
$947$	26.0836	0.847602	0.423801	−	0.905755i	$-0.360696\pi$
$947$	26.0836	0.847602	0.423801	+	0.905755i	$0.360696\pi$
$948$	0	0
$949$	−41.9988	−1.36334
$950$	−4.53001	−0.146973
$951$	0	0
$952$	4.05661	0.131475
$953$	−33.5768	−1.08766	−0.543830	−	0.839196i	$-0.683026\pi$
$953$	−33.5768	−1.08766	−0.543830	+	0.839196i	$0.683026\pi$
$954$	0	0
$955$	5.03949	0.163074
$956$	−16.7444	−0.541554
$957$	0	0
$958$	−17.6114	−0.568998
$959$	−8.93213	−0.288434
$960$	0	0
$961$	51.8295	1.67192
$962$	−46.0738	−1.48548
$963$	0	0
$964$	−17.6846	−0.569582
$965$	3.43309	0.110515
$966$	0	0
$967$	20.4044	0.656160	0.328080	−	0.944650i	$-0.393598\pi$
$967$	20.4044	0.656160	0.328080	+	0.944650i	$0.393598\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	−4.88763	−0.156932
$971$	10.7077	0.343626	0.171813	−	0.985130i	$-0.445038\pi$
$971$	10.7077	0.343626	0.171813	+	0.985130i	$0.445038\pi$
$972$	0	0
$973$	14.8079	0.474720
$974$	32.6177	1.04514
$975$	0	0
$976$	14.0478	0.449659
$977$	−24.0828	−0.770478	−0.385239	−	0.922817i	$-0.625881\pi$
$977$	−24.0828	−0.770478	−0.385239	+	0.922817i	$0.625881\pi$
$978$	0	0
$979$	−4.41098	−0.140975
$980$	−4.34229	−0.138709
$981$	0	0
$982$	−10.6601	−0.340178
$983$	−27.5895	−0.879968	−0.439984	−	0.898006i	$-0.645016\pi$
$983$	−27.5895	−0.879968	−0.439984	+	0.898006i	$0.645016\pi$
$984$	0	0
$985$	12.0670	0.384485
$986$	−34.3020	−1.09240
$987$	0	0
$988$	−4.21557	−0.134115
$989$	46.8033	1.48826
$990$	0	0
$991$	−46.1932	−1.46738	−0.733688	−	0.679486i	$-0.762203\pi$
$991$	−46.1932	−1.46738	−0.733688	+	0.679486i	$0.762203\pi$
$992$	9.10107	0.288959
$993$	0	0
$994$	−1.99825	−0.0633805
$995$	2.67680	0.0848601
$996$	0	0
$997$	−29.2854	−0.927477	−0.463739	−	0.885972i	$-0.653492\pi$
$997$	−29.2854	−0.927477	−0.463739	+	0.885972i	$0.653492\pi$
$998$	−20.7398	−0.656508
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3762.2.a.bl.1.3	yes	5
3.2	odd	2		3762.2.a.bi.1.3	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3762.2.a.bi.1.3	✓	5	3.2	odd	2
3762.2.a.bl.1.3	yes	5	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:	\( N \)	\(=\)	\( 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3762.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.685562 q^{5} -0.816139 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.685562 q^{5} -0.816139 q^{7} +1.00000 q^{8} +0.685562 q^{10} +1.00000 q^{11} -4.21557 q^{13} -0.816139 q^{14} +1.00000 q^{16} -4.97049 q^{17} +1.00000 q^{19} +0.685562 q^{20} +1.00000 q^{22} +6.21557 q^{23} -4.53001 q^{25} -4.21557 q^{26} -0.816139 q^{28} +6.90113 q^{29} +9.10107 q^{31} +1.00000 q^{32} -4.97049 q^{34} -0.559513 q^{35} +10.9294 q^{37} +1.00000 q^{38} +0.685562 q^{40} +3.70164 q^{41} +7.53001 q^{43} +1.00000 q^{44} +6.21557 q^{46} +6.21557 q^{47} -6.33392 q^{49} -4.53001 q^{50} -4.21557 q^{52} +0.885502 q^{53} +0.685562 q^{55} -0.816139 q^{56} +6.90113 q^{58} +1.93064 q^{59} +14.0478 q^{61} +9.10107 q^{62} +1.00000 q^{64} -2.89003 q^{65} -4.34162 q^{67} -4.97049 q^{68} -0.559513 q^{70} +2.44842 q^{71} +9.96279 q^{73} +10.9294 q^{74} +1.00000 q^{76} -0.816139 q^{77} -5.21897 q^{79} +0.685562 q^{80} +3.70164 q^{82} +3.70164 q^{83} -3.40758 q^{85} +7.53001 q^{86} +1.00000 q^{88} -4.41098 q^{89} +3.44049 q^{91} +6.21557 q^{92} +6.21557 q^{94} +0.685562 q^{95} -7.12937 q^{97} -6.33392 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{2} + 5 q^{4} + 3 q^{5} + 6 q^{7} + 5 q^{8}+O(q^{10}) \) 5 * q + 5 * q^2 + 5 * q^4 + 3 * q^5 + 6 * q^7 + 5 * q^8	\( 5 q + 5 q^{2} + 5 q^{4} + 3 q^{5} + 6 q^{7} + 5 q^{8} + 3 q^{10} + 5 q^{11} + 6 q^{13} + 6 q^{14} + 5 q^{16} + 3 q^{17} + 5 q^{19} + 3 q^{20} + 5 q^{22} + 4 q^{23} + 4 q^{25} + 6 q^{26} + 6 q^{28} + 7 q^{29} + 8 q^{31} + 5 q^{32} + 3 q^{34} - 4 q^{35} + 11 q^{37} + 5 q^{38} + 3 q^{40} - 2 q^{41} + 11 q^{43} + 5 q^{44} + 4 q^{46} + 4 q^{47} + 9 q^{49} + 4 q^{50} + 6 q^{52} - 6 q^{53} + 3 q^{55} + 6 q^{56} + 7 q^{58} + 10 q^{59} + 13 q^{61} + 8 q^{62} + 5 q^{64} + 4 q^{65} + 7 q^{67} + 3 q^{68} - 4 q^{70} - 18 q^{71} + 10 q^{73} + 11 q^{74} + 5 q^{76} + 6 q^{77} + 22 q^{79} + 3 q^{80} - 2 q^{82} - 2 q^{83} - 9 q^{85} + 11 q^{86} + 5 q^{88} + 7 q^{89} + 16 q^{91} + 4 q^{92} + 4 q^{94} + 3 q^{95} + 18 q^{97} + 9 q^{98}+O(q^{100}) \) 5 * q + 5 * q^2 + 5 * q^4 + 3 * q^5 + 6 * q^7 + 5 * q^8 + 3 * q^10 + 5 * q^11 + 6 * q^13 + 6 * q^14 + 5 * q^16 + 3 * q^17 + 5 * q^19 + 3 * q^20 + 5 * q^22 + 4 * q^23 + 4 * q^25 + 6 * q^26 + 6 * q^28 + 7 * q^29 + 8 * q^31 + 5 * q^32 + 3 * q^34 - 4 * q^35 + 11 * q^37 + 5 * q^38 + 3 * q^40 - 2 * q^41 + 11 * q^43 + 5 * q^44 + 4 * q^46 + 4 * q^47 + 9 * q^49 + 4 * q^50 + 6 * q^52 - 6 * q^53 + 3 * q^55 + 6 * q^56 + 7 * q^58 + 10 * q^59 + 13 * q^61 + 8 * q^62 + 5 * q^64 + 4 * q^65 + 7 * q^67 + 3 * q^68 - 4 * q^70 - 18 * q^71 + 10 * q^73 + 11 * q^74 + 5 * q^76 + 6 * q^77 + 22 * q^79 + 3 * q^80 - 2 * q^82 - 2 * q^83 - 9 * q^85 + 11 * q^86 + 5 * q^88 + 7 * q^89 + 16 * q^91 + 4 * q^92 + 4 * q^94 + 3 * q^95 + 18 * q^97 + 9 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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