LMFDB - Embedded newform 3640.2.a.w.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3640,2,Mod(1,3640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3640, 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("3640.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3640.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:	$29.0655463357$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.23724.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 7x^{2} + 4x + 10 $ x^4 - x^3 - 7x^2 + 4x + 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.09285$ of defining polynomial
Character	$\chi$	$=$	3640.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.80569 q^{3} -1.00000 q^{5} +1.00000 q^{7} +4.87187 q^{9} +O(q^{10})$	$q+2.80569 q^{3} -1.00000 q^{5} +1.00000 q^{7} +4.87187 q^{9} -2.18569 q^{11} +1.00000 q^{13} -2.80569 q^{15} +3.87187 q^{17} -2.44619 q^{19} +2.80569 q^{21} +5.42568 q^{23} +1.00000 q^{25} +5.25188 q^{27} +3.38001 q^{29} +0.805685 q^{31} -6.13237 q^{33} -1.00000 q^{35} +2.75236 q^{37} +2.80569 q^{39} +6.63189 q^{41} +0.574323 q^{43} -4.87187 q^{45} -7.74374 q^{47} +1.00000 q^{49} +10.8632 q^{51} +5.61137 q^{53} +2.18569 q^{55} -6.86325 q^{57} -4.93806 q^{59} +8.50376 q^{61} +4.87187 q^{63} -1.00000 q^{65} +1.73950 q^{67} +15.2227 q^{69} +7.03705 q^{71} -0.760017 q^{73} +2.80569 q^{75} -2.18569 q^{77} +7.35087 q^{79} +0.119509 q^{81} +5.61137 q^{83} -3.87187 q^{85} +9.48324 q^{87} -3.51238 q^{89} +1.00000 q^{91} +2.26050 q^{93} +2.44619 q^{95} -11.3551 q^{97} -10.6484 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} - 4 q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10}) $ 4 * q + q^3 - 4 * q^5 + 4 * q^7 + 3 * q^9	$ 4 q + q^{3} - 4 q^{5} + 4 q^{7} + 3 q^{9} + 2 q^{11} + 4 q^{13} - q^{15} - q^{17} - 3 q^{19} + q^{21} + 12 q^{23} + 4 q^{25} + 4 q^{27} + 13 q^{29} - 7 q^{31} - 12 q^{33} - 4 q^{35} - q^{37} + q^{39} + 9 q^{41} + 12 q^{43} - 3 q^{45} + 2 q^{47} + 4 q^{49} + 6 q^{51} + 2 q^{53} - 2 q^{55} + 10 q^{57} + 3 q^{59} + 3 q^{63} - 4 q^{65} + 3 q^{67} + 20 q^{69} - 2 q^{71} - 2 q^{73} + q^{75} + 2 q^{77} + 5 q^{79} - 4 q^{81} + 2 q^{83} + q^{85} + q^{87} - q^{89} + 4 q^{91} + 13 q^{93} + 3 q^{95} + 8 q^{97} + 8 q^{99}+O(q^{100}) $ 4 * q + q^3 - 4 * q^5 + 4 * q^7 + 3 * q^9 + 2 * q^11 + 4 * q^13 - q^15 - q^17 - 3 * q^19 + q^21 + 12 * q^23 + 4 * q^25 + 4 * q^27 + 13 * q^29 - 7 * q^31 - 12 * q^33 - 4 * q^35 - q^37 + q^39 + 9 * q^41 + 12 * q^43 - 3 * q^45 + 2 * q^47 + 4 * q^49 + 6 * q^51 + 2 * q^53 - 2 * q^55 + 10 * q^57 + 3 * q^59 + 3 * q^63 - 4 * q^65 + 3 * q^67 + 20 * q^69 - 2 * q^71 - 2 * q^73 + q^75 + 2 * q^77 + 5 * q^79 - 4 * q^81 + 2 * q^83 + q^85 + q^87 - q^89 + 4 * q^91 + 13 * q^93 + 3 * q^95 + 8 * q^97 + 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.80569	1.61986	0.809932	−	0.586524i	$-0.199504\pi$
$3$	2.80569	1.61986	0.809932	+	0.586524i	$0.199504\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	4.87187	1.62396
$10$	0	0
$11$	−2.18569	−0.659012	−0.329506	−	0.944154i	$-0.606882\pi$
$11$	−2.18569	−0.659012	−0.329506	+	0.944154i	$0.606882\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−2.80569	−0.724425
$16$	0	0
$17$	3.87187	0.939066	0.469533	−	0.882915i	$-0.344422\pi$
$17$	3.87187	0.939066	0.469533	+	0.882915i	$0.344422\pi$
$18$	0	0
$19$	−2.44619	−0.561195	−0.280598	−	0.959825i	$-0.590533\pi$
$19$	−2.44619	−0.561195	−0.280598	+	0.959825i	$0.590533\pi$
$20$	0	0
$21$	2.80569	0.612251
$22$	0	0
$23$	5.42568	1.13133	0.565666	−	0.824635i	$-0.308619\pi$
$23$	5.42568	1.13133	0.565666	+	0.824635i	$0.308619\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.25188	1.01072
$28$	0	0
$29$	3.38001	0.627652	0.313826	−	0.949481i	$-0.398389\pi$
$29$	3.38001	0.627652	0.313826	+	0.949481i	$0.398389\pi$
$30$	0	0
$31$	0.805685	0.144705	0.0723527	−	0.997379i	$-0.476949\pi$
$31$	0.805685	0.144705	0.0723527	+	0.997379i	$0.476949\pi$
$32$	0	0
$33$	−6.13237	−1.06751
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	2.75236	0.452485	0.226243	−	0.974071i	$-0.427356\pi$
$37$	2.75236	0.452485	0.226243	+	0.974071i	$0.427356\pi$
$38$	0	0
$39$	2.80569	0.449269
$40$	0	0
$41$	6.63189	1.03573	0.517863	−	0.855463i	$-0.326728\pi$
$41$	6.63189	1.03573	0.517863	+	0.855463i	$0.326728\pi$
$42$	0	0
$43$	0.574323	0.0875835	0.0437917	−	0.999041i	$-0.486056\pi$
$43$	0.574323	0.0875835	0.0437917	+	0.999041i	$0.486056\pi$
$44$	0	0
$45$	−4.87187	−0.726256
$46$	0	0
$47$	−7.74374	−1.12954	−0.564770	−	0.825248i	$-0.691035\pi$
$47$	−7.74374	−1.12954	−0.564770	+	0.825248i	$0.691035\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	10.8632	1.52116
$52$	0	0
$53$	5.61137	0.770781	0.385391	−	0.922754i	$-0.374067\pi$
$53$	5.61137	0.770781	0.385391	+	0.922754i	$0.374067\pi$
$54$	0	0
$55$	2.18569	0.294719
$56$	0	0
$57$	−6.86325	−0.909060
$58$	0	0
$59$	−4.93806	−0.642880	−0.321440	−	0.946930i	$-0.604167\pi$
$59$	−4.93806	−0.642880	−0.321440	+	0.946930i	$0.604167\pi$
$60$	0	0
$61$	8.50376	1.08879	0.544397	−	0.838827i	$-0.316758\pi$
$61$	8.50376	1.08879	0.544397	+	0.838827i	$0.316758\pi$
$62$	0	0
$63$	4.87187	0.613798
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	1.73950	0.212514	0.106257	−	0.994339i	$-0.466113\pi$
$67$	1.73950	0.212514	0.106257	+	0.994339i	$0.466113\pi$
$68$	0	0
$69$	15.2227	1.83260
$70$	0	0
$71$	7.03705	0.835144	0.417572	−	0.908644i	$-0.362881\pi$
$71$	7.03705	0.835144	0.417572	+	0.908644i	$0.362881\pi$
$72$	0	0
$73$	−0.760017	−0.0889533	−0.0444767	−	0.999010i	$-0.514162\pi$
$73$	−0.760017	−0.0889533	−0.0444767	+	0.999010i	$0.514162\pi$
$74$	0	0
$75$	2.80569	0.323973
$76$	0	0
$77$	−2.18569	−0.249083
$78$	0	0
$79$	7.35087	0.827038	0.413519	−	0.910496i	$-0.364300\pi$
$79$	7.35087	0.827038	0.413519	+	0.910496i	$0.364300\pi$
$80$	0	0
$81$	0.119509	0.0132788
$82$	0	0
$83$	5.61137	0.615928	0.307964	−	0.951398i	$-0.400352\pi$
$83$	5.61137	0.615928	0.307964	+	0.951398i	$0.400352\pi$
$84$	0	0
$85$	−3.87187	−0.419963
$86$	0	0
$87$	9.48324	1.01671
$88$	0	0
$89$	−3.51238	−0.372311	−0.186156	−	0.982520i	$-0.559603\pi$
$89$	−3.51238	−0.372311	−0.186156	+	0.982520i	$0.559603\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	2.26050	0.234403
$94$	0	0
$95$	2.44619	0.250974
$96$	0	0
$97$	−11.3551	−1.15294	−0.576468	−	0.817119i	$-0.695570\pi$
$97$	−11.3551	−1.15294	−0.576468	+	0.817119i	$0.695570\pi$
$98$	0	0
$99$	−10.6484	−1.07021
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	7.29755	0.719049	0.359524	−	0.933136i	$-0.382939\pi$
$103$	7.29755	0.719049	0.359524	+	0.933136i	$0.382939\pi$
$104$	0	0
$105$	−2.80569	−0.273807
$106$	0	0
$107$	8.57432	0.828911	0.414456	−	0.910070i	$-0.363972\pi$
$107$	8.57432	0.828911	0.414456	+	0.910070i	$0.363972\pi$
$108$	0	0
$109$	3.62861	0.347558	0.173779	−	0.984785i	$-0.444402\pi$
$109$	3.62861	0.347558	0.173779	+	0.984785i	$0.444402\pi$
$110$	0	0
$111$	7.72226	0.732964
$112$	0	0
$113$	−16.9665	−1.59607	−0.798036	−	0.602610i	$-0.794128\pi$
$113$	−16.9665	−1.59607	−0.798036	+	0.602610i	$0.794128\pi$
$114$	0	0
$115$	−5.42568	−0.505947
$116$	0	0
$117$	4.87187	0.450405
$118$	0	0
$119$	3.87187	0.354934
$120$	0	0
$121$	−6.22274	−0.565704
$122$	0	0
$123$	18.6070	1.67774
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−10.6895	−0.948535	−0.474268	−	0.880381i	$-0.657287\pi$
$127$	−10.6895	−0.948535	−0.474268	+	0.880381i	$0.657287\pi$
$128$	0	0
$129$	1.61137	0.141873
$130$	0	0
$131$	2.85135	0.249124	0.124562	−	0.992212i	$-0.460247\pi$
$131$	2.85135	0.249124	0.124562	+	0.992212i	$0.460247\pi$
$132$	0	0
$133$	−2.44619	−0.212112
$134$	0	0
$135$	−5.25188	−0.452010
$136$	0	0
$137$	−9.97510	−0.852231	−0.426115	−	0.904669i	$-0.640118\pi$
$137$	−9.97510	−0.852231	−0.426115	+	0.904669i	$0.640118\pi$
$138$	0	0
$139$	2.13237	0.180865	0.0904326	−	0.995903i	$-0.471175\pi$
$139$	2.13237	0.180865	0.0904326	+	0.995903i	$0.471175\pi$
$140$	0	0
$141$	−21.7265	−1.82970
$142$	0	0
$143$	−2.18569	−0.182777
$144$	0	0
$145$	−3.38001	−0.280694
$146$	0	0
$147$	2.80569	0.231409
$148$	0	0
$149$	12.3714	1.01350	0.506752	−	0.862092i	$-0.330846\pi$
$149$	12.3714	1.01350	0.506752	+	0.862092i	$0.330846\pi$
$150$	0	0
$151$	−3.29331	−0.268006	−0.134003	−	0.990981i	$-0.542783\pi$
$151$	−3.29331	−0.268006	−0.134003	+	0.990981i	$0.542783\pi$
$152$	0	0
$153$	18.8632	1.52500
$154$	0	0
$155$	−0.805685	−0.0647142
$156$	0	0
$157$	−1.71036	−0.136502	−0.0682509	−	0.997668i	$-0.521742\pi$
$157$	−1.71036	−0.136502	−0.0682509	+	0.997668i	$0.521742\pi$
$158$	0	0
$159$	15.7437	1.24856
$160$	0	0
$161$	5.42568	0.427603
$162$	0	0
$163$	20.0902	1.57359	0.786794	−	0.617215i	$-0.211739\pi$
$163$	20.0902	1.57359	0.786794	+	0.617215i	$0.211739\pi$
$164$	0	0
$165$	6.13237	0.477404
$166$	0	0
$167$	−6.87515	−0.532015	−0.266007	−	0.963971i	$-0.585705\pi$
$167$	−6.87515	−0.532015	−0.266007	+	0.963971i	$0.585705\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−11.9175	−0.911357
$172$	0	0
$173$	1.04965	0.0798038	0.0399019	−	0.999204i	$-0.487295\pi$
$173$	1.04965	0.0798038	0.0399019	+	0.999204i	$0.487295\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−13.8546	−1.04138
$178$	0	0
$179$	−7.61561	−0.569217	−0.284609	−	0.958644i	$-0.591864\pi$
$179$	−7.61561	−0.569217	−0.284609	+	0.958644i	$0.591864\pi$
$180$	0	0
$181$	7.23998	0.538144	0.269072	−	0.963120i	$-0.413283\pi$
$181$	7.23998	0.538144	0.269072	+	0.963120i	$0.413283\pi$
$182$	0	0
$183$	23.8589	1.76370
$184$	0	0
$185$	−2.75236	−0.202358
$186$	0	0
$187$	−8.46272	−0.618856
$188$	0	0
$189$	5.25188	0.382018
$190$	0	0
$191$	−13.3885	−0.968757	−0.484379	−	0.874859i	$-0.660954\pi$
$191$	−13.3885	−0.968757	−0.484379	+	0.874859i	$0.660954\pi$
$192$	0	0
$193$	−2.84711	−0.204940	−0.102470	−	0.994736i	$-0.532675\pi$
$193$	−2.84711	−0.204940	−0.102470	+	0.994736i	$0.532675\pi$
$194$	0	0
$195$	−2.80569	−0.200919
$196$	0	0
$197$	9.14099	0.651269	0.325634	−	0.945496i	$-0.394422\pi$
$197$	9.14099	0.651269	0.325634	+	0.945496i	$0.394422\pi$
$198$	0	0
$199$	−23.5941	−1.67254	−0.836272	−	0.548315i	$-0.815269\pi$
$199$	−23.5941	−1.67254	−0.836272	+	0.548315i	$0.815269\pi$
$200$	0	0
$201$	4.88049	0.344243
$202$	0	0
$203$	3.38001	0.237230
$204$	0	0
$205$	−6.63189	−0.463191
$206$	0	0
$207$	26.4332	1.83723
$208$	0	0
$209$	5.34663	0.369834
$210$	0	0
$211$	−12.2433	−0.842861	−0.421430	−	0.906861i	$-0.638472\pi$
$211$	−12.2433	−0.842861	−0.421430	+	0.906861i	$0.638472\pi$
$212$	0	0
$213$	19.7437	1.35282
$214$	0	0
$215$	−0.574323	−0.0391685
$216$	0	0
$217$	0.805685	0.0546935
$218$	0	0
$219$	−2.13237	−0.144092
$220$	0	0
$221$	3.87187	0.260450
$222$	0	0
$223$	−5.41339	−0.362507	−0.181254	−	0.983436i	$-0.558015\pi$
$223$	−5.41339	−0.362507	−0.181254	+	0.983436i	$0.558015\pi$
$224$	0	0
$225$	4.87187	0.324791
$226$	0	0
$227$	4.17340	0.276999	0.138499	−	0.990363i	$-0.455772\pi$
$227$	4.17340	0.276999	0.138499	+	0.990363i	$0.455772\pi$
$228$	0	0
$229$	9.08613	0.600428	0.300214	−	0.953872i	$-0.402942\pi$
$229$	9.08613	0.600428	0.300214	+	0.953872i	$0.402942\pi$
$230$	0	0
$231$	−6.13237	−0.403480
$232$	0	0
$233$	0.760017	0.0497904	0.0248952	−	0.999690i	$-0.492075\pi$
$233$	0.760017	0.0497904	0.0248952	+	0.999690i	$0.492075\pi$
$234$	0	0
$235$	7.74374	0.505146
$236$	0	0
$237$	20.6242	1.33969
$238$	0	0
$239$	15.9294	1.03039	0.515195	−	0.857073i	$-0.327720\pi$
$239$	15.9294	1.03039	0.515195	+	0.857073i	$0.327720\pi$
$240$	0	0
$241$	−0.847113	−0.0545674	−0.0272837	−	0.999628i	$-0.508686\pi$
$241$	−0.847113	−0.0545674	−0.0272837	+	0.999628i	$0.508686\pi$
$242$	0	0
$243$	−15.4203	−0.989215
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−2.44619	−0.155648
$248$	0	0
$249$	15.7437	0.997719
$250$	0	0
$251$	12.2647	0.774144	0.387072	−	0.922050i	$-0.373487\pi$
$251$	12.2647	0.774144	0.387072	+	0.922050i	$0.373487\pi$
$252$	0	0
$253$	−11.8589	−0.745561
$254$	0	0
$255$	−10.8632	−0.680283
$256$	0	0
$257$	6.23902	0.389179	0.194590	−	0.980885i	$-0.437662\pi$
$257$	6.23902	0.389179	0.194590	+	0.980885i	$0.437662\pi$
$258$	0	0
$259$	2.75236	0.171023
$260$	0	0
$261$	16.4670	1.01928
$262$	0	0
$263$	−1.42568	−0.0879110	−0.0439555	−	0.999033i	$-0.513996\pi$
$263$	−1.42568	−0.0879110	−0.0439555	+	0.999033i	$0.513996\pi$
$264$	0	0
$265$	−5.61137	−0.344704
$266$	0	0
$267$	−9.85463	−0.603094
$268$	0	0
$269$	23.4702	1.43101	0.715503	−	0.698610i	$-0.246198\pi$
$269$	23.4702	1.43101	0.715503	+	0.698610i	$0.246198\pi$
$270$	0	0
$271$	−28.7808	−1.74831	−0.874154	−	0.485649i	$-0.838584\pi$
$271$	−28.7808	−1.74831	−0.874154	+	0.485649i	$0.838584\pi$
$272$	0	0
$273$	2.80569	0.169808
$274$	0	0
$275$	−2.18569	−0.131802
$276$	0	0
$277$	−11.8589	−0.712530	−0.356265	−	0.934385i	$-0.615950\pi$
$277$	−11.8589	−0.712530	−0.356265	+	0.934385i	$0.615950\pi$
$278$	0	0
$279$	3.92519	0.234995
$280$	0	0
$281$	−11.4618	−0.683751	−0.341876	−	0.939745i	$-0.611062\pi$
$281$	−11.4618	−0.683751	−0.341876	+	0.939745i	$0.611062\pi$
$282$	0	0
$283$	−27.4299	−1.63054	−0.815270	−	0.579082i	$-0.803411\pi$
$283$	−27.4299	−1.63054	−0.815270	+	0.579082i	$0.803411\pi$
$284$	0	0
$285$	6.86325	0.406544
$286$	0	0
$287$	6.63189	0.391468
$288$	0	0
$289$	−2.00862	−0.118154
$290$	0	0
$291$	−31.8589	−1.86760
$292$	0	0
$293$	−14.2065	−0.829951	−0.414975	−	0.909833i	$-0.636210\pi$
$293$	−14.2065	−0.829951	−0.414975	+	0.909833i	$0.636210\pi$
$294$	0	0
$295$	4.93806	0.287505
$296$	0	0
$297$	−11.4790	−0.666079
$298$	0	0
$299$	5.42568	0.313775
$300$	0	0
$301$	0.574323	0.0331034
$302$	0	0
$303$	16.8341	0.967094
$304$	0	0
$305$	−8.50376	−0.486924
$306$	0	0
$307$	−26.9665	−1.53906	−0.769529	−	0.638612i	$-0.779509\pi$
$307$	−26.9665	−1.53906	−0.769529	+	0.638612i	$0.779509\pi$
$308$	0	0
$309$	20.4746	1.16476
$310$	0	0
$311$	4.00848	0.227300	0.113650	−	0.993521i	$-0.463746\pi$
$311$	4.00848	0.227300	0.113650	+	0.993521i	$0.463746\pi$
$312$	0	0
$313$	7.45410	0.421331	0.210665	−	0.977558i	$-0.432437\pi$
$313$	7.45410	0.421331	0.210665	+	0.977558i	$0.432437\pi$
$314$	0	0
$315$	−4.87187	−0.274499
$316$	0	0
$317$	18.2647	1.02585	0.512925	−	0.858433i	$-0.328562\pi$
$317$	18.2647	1.02585	0.512925	+	0.858433i	$0.328562\pi$
$318$	0	0
$319$	−7.38767	−0.413630
$320$	0	0
$321$	24.0569	1.34272
$322$	0	0
$323$	−9.47134	−0.527000
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	10.1807	0.562996
$328$	0	0
$329$	−7.74374	−0.426926
$330$	0	0
$331$	−14.2770	−0.784737	−0.392368	−	0.919808i	$-0.628344\pi$
$331$	−14.2770	−0.784737	−0.392368	+	0.919808i	$0.628344\pi$
$332$	0	0
$333$	13.4091	0.734817
$334$	0	0
$335$	−1.73950	−0.0950391
$336$	0	0
$337$	−15.7027	−0.855381	−0.427690	−	0.903925i	$-0.640673\pi$
$337$	−15.7027	−0.855381	−0.427690	+	0.903925i	$0.640673\pi$
$338$	0	0
$339$	−47.6026	−2.58542
$340$	0	0
$341$	−1.76098	−0.0953625
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−15.2227	−0.819565
$346$	0	0
$347$	−15.1111	−0.811209	−0.405604	−	0.914049i	$-0.632939\pi$
$347$	−15.1111	−0.811209	−0.405604	+	0.914049i	$0.632939\pi$
$348$	0	0
$349$	−13.7188	−0.734353	−0.367176	−	0.930151i	$-0.619675\pi$
$349$	−13.7188	−0.734353	−0.367176	+	0.930151i	$0.619675\pi$
$350$	0	0
$351$	5.25188	0.280325
$352$	0	0
$353$	−11.7848	−0.627240	−0.313620	−	0.949549i	$-0.601542\pi$
$353$	−11.7848	−0.627240	−0.313620	+	0.949549i	$0.601542\pi$
$354$	0	0
$355$	−7.03705	−0.373488
$356$	0	0
$357$	10.8632	0.574944
$358$	0	0
$359$	8.82182	0.465598	0.232799	−	0.972525i	$-0.425212\pi$
$359$	8.82182	0.465598	0.232799	+	0.972525i	$0.425212\pi$
$360$	0	0
$361$	−13.0161	−0.685060
$362$	0	0
$363$	−17.4591	−0.916363
$364$	0	0
$365$	0.760017	0.0397811
$366$	0	0
$367$	−24.7808	−1.29355	−0.646773	−	0.762682i	$-0.723882\pi$
$367$	−24.7808	−1.29355	−0.646773	+	0.762682i	$0.723882\pi$
$368$	0	0
$369$	32.3097	1.68198
$370$	0	0
$371$	5.61137	0.291328
$372$	0	0
$373$	1.60289	0.0829945	0.0414973	−	0.999139i	$-0.486787\pi$
$373$	1.60289	0.0829945	0.0414973	+	0.999139i	$0.486787\pi$
$374$	0	0
$375$	−2.80569	−0.144885
$376$	0	0
$377$	3.38001	0.174079
$378$	0	0
$379$	−8.94571	−0.459510	−0.229755	−	0.973248i	$-0.573793\pi$
$379$	−8.94571	−0.459510	−0.229755	+	0.973248i	$0.573793\pi$
$380$	0	0
$381$	−29.9912	−1.53650
$382$	0	0
$383$	0.149611	0.00764476	0.00382238	−	0.999993i	$-0.498783\pi$
$383$	0.149611	0.00764476	0.00382238	+	0.999993i	$0.498783\pi$
$384$	0	0
$385$	2.18569	0.111393
$386$	0	0
$387$	2.79803	0.142232
$388$	0	0
$389$	25.6156	1.29876	0.649382	−	0.760463i	$-0.275028\pi$
$389$	25.6156	1.29876	0.649382	+	0.760463i	$0.275028\pi$
$390$	0	0
$391$	21.0075	1.06240
$392$	0	0
$393$	8.00000	0.403547
$394$	0	0
$395$	−7.35087	−0.369862
$396$	0	0
$397$	−2.49721	−0.125331	−0.0626656	−	0.998035i	$-0.519960\pi$
$397$	−2.49721	−0.125331	−0.0626656	+	0.998035i	$0.519960\pi$
$398$	0	0
$399$	−6.86325	−0.343592
$400$	0	0
$401$	−23.4618	−1.17162	−0.585812	−	0.810447i	$-0.699224\pi$
$401$	−23.4618	−1.17162	−0.585812	+	0.810447i	$0.699224\pi$
$402$	0	0
$403$	0.805685	0.0401340
$404$	0	0
$405$	−0.119509	−0.00593846
$406$	0	0
$407$	−6.01582	−0.298193
$408$	0	0
$409$	11.5458	0.570901	0.285450	−	0.958393i	$-0.407857\pi$
$409$	11.5458	0.570901	0.285450	+	0.958393i	$0.407857\pi$
$410$	0	0
$411$	−27.9870	−1.38050
$412$	0	0
$413$	−4.93806	−0.242986
$414$	0	0
$415$	−5.61137	−0.275451
$416$	0	0
$417$	5.98276	0.292977
$418$	0	0
$419$	37.5616	1.83500	0.917502	−	0.397732i	$-0.130203\pi$
$419$	37.5616	1.83500	0.917502	+	0.397732i	$0.130203\pi$
$420$	0	0
$421$	12.3952	0.604104	0.302052	−	0.953291i	$-0.402328\pi$
$421$	12.3952	0.604104	0.302052	+	0.953291i	$0.402328\pi$
$422$	0	0
$423$	−37.7265	−1.83433
$424$	0	0
$425$	3.87187	0.187813
$426$	0	0
$427$	8.50376	0.411526
$428$	0	0
$429$	−6.13237	−0.296074
$430$	0	0
$431$	0.813342	0.0391773	0.0195886	−	0.999808i	$-0.493764\pi$
$431$	0.813342	0.0391773	0.0195886	+	0.999808i	$0.493764\pi$
$432$	0	0
$433$	0.649129	0.0311951	0.0155976	−	0.999878i	$-0.495035\pi$
$433$	0.649129	0.0311951	0.0155976	+	0.999878i	$0.495035\pi$
$434$	0	0
$435$	−9.48324	−0.454687
$436$	0	0
$437$	−13.2723	−0.634898
$438$	0	0
$439$	23.6179	1.12722	0.563611	−	0.826040i	$-0.309412\pi$
$439$	23.6179	1.12722	0.563611	+	0.826040i	$0.309412\pi$
$440$	0	0
$441$	4.87187	0.231994
$442$	0	0
$443$	22.1274	1.05131	0.525653	−	0.850699i	$-0.323821\pi$
$443$	22.1274	1.05131	0.525653	+	0.850699i	$0.323821\pi$
$444$	0	0
$445$	3.51238	0.166503
$446$	0	0
$447$	34.7102	1.64174
$448$	0	0
$449$	13.7437	0.648607	0.324304	−	0.945953i	$-0.394870\pi$
$449$	13.7437	0.648607	0.324304	+	0.945953i	$0.394870\pi$
$450$	0	0
$451$	−14.4953	−0.682556
$452$	0	0
$453$	−9.23998	−0.434132
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	−19.4951	−0.911944	−0.455972	−	0.889994i	$-0.650708\pi$
$457$	−19.4951	−0.911944	−0.455972	+	0.889994i	$0.650708\pi$
$458$	0	0
$459$	20.3346	0.949138
$460$	0	0
$461$	−30.7417	−1.43178	−0.715891	−	0.698212i	$-0.753979\pi$
$461$	−30.7417	−1.43178	−0.715891	+	0.698212i	$0.753979\pi$
$462$	0	0
$463$	−25.3421	−1.17775	−0.588874	−	0.808225i	$-0.700429\pi$
$463$	−25.3421	−1.17775	−0.588874	+	0.808225i	$0.700429\pi$
$464$	0	0
$465$	−2.26050	−0.104828
$466$	0	0
$467$	−27.0240	−1.25052	−0.625262	−	0.780415i	$-0.715008\pi$
$467$	−27.0240	−1.25052	−0.625262	+	0.780415i	$0.715008\pi$
$468$	0	0
$469$	1.73950	0.0803227
$470$	0	0
$471$	−4.79874	−0.221114
$472$	0	0
$473$	−1.25530	−0.0577185
$474$	0	0
$475$	−2.44619	−0.112239
$476$	0	0
$477$	27.3379	1.25172
$478$	0	0
$479$	−36.8625	−1.68429	−0.842146	−	0.539249i	$-0.818708\pi$
$479$	−36.8625	−1.68429	−0.842146	+	0.539249i	$0.818708\pi$
$480$	0	0
$481$	2.75236	0.125497
$482$	0	0
$483$	15.2227	0.692659
$484$	0	0
$485$	11.3551	0.515609
$486$	0	0
$487$	−35.4864	−1.60804	−0.804021	−	0.594601i	$-0.797310\pi$
$487$	−35.4864	−1.60804	−0.804021	+	0.594601i	$0.797310\pi$
$488$	0	0
$489$	56.3669	2.54900
$490$	0	0
$491$	−21.4856	−0.969629	−0.484815	−	0.874617i	$-0.661113\pi$
$491$	−21.4856	−0.969629	−0.484815	+	0.874617i	$0.661113\pi$
$492$	0	0
$493$	13.0870	0.589407
$494$	0	0
$495$	10.6484	0.478611
$496$	0	0
$497$	7.03705	0.315655
$498$	0	0
$499$	0.574323	0.0257102	0.0128551	−	0.999917i	$-0.495908\pi$
$499$	0.574323	0.0257102	0.0128551	+	0.999917i	$0.495908\pi$
$500$	0	0
$501$	−19.2895	−0.861791
$502$	0	0
$503$	15.2608	0.680443	0.340222	−	0.940345i	$-0.389498\pi$
$503$	15.2608	0.680443	0.340222	+	0.940345i	$0.389498\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	2.80569	0.124605
$508$	0	0
$509$	42.5411	1.88560	0.942800	−	0.333360i	$-0.108182\pi$
$509$	42.5411	1.88560	0.942800	+	0.333360i	$0.108182\pi$
$510$	0	0
$511$	−0.760017	−0.0336212
$512$	0	0
$513$	−12.8471	−0.567214
$514$	0	0
$515$	−7.29755	−0.321568
$516$	0	0
$517$	16.9254	0.744380
$518$	0	0
$519$	2.94500	0.129271
$520$	0	0
$521$	−4.12389	−0.180671	−0.0903354	−	0.995911i	$-0.528794\pi$
$521$	−4.12389	−0.180671	−0.0903354	+	0.995911i	$0.528794\pi$
$522$	0	0
$523$	14.0618	0.614880	0.307440	−	0.951568i	$-0.400528\pi$
$523$	14.0618	0.614880	0.307440	+	0.951568i	$0.400528\pi$
$524$	0	0
$525$	2.80569	0.122450
$526$	0	0
$527$	3.11951	0.135888
$528$	0	0
$529$	6.43797	0.279912
$530$	0	0
$531$	−24.0576	−1.04401
$532$	0	0
$533$	6.63189	0.287259
$534$	0	0
$535$	−8.57432	−0.370700
$536$	0	0
$537$	−21.3670	−0.922054
$538$	0	0
$539$	−2.18569	−0.0941445
$540$	0	0
$541$	−19.1161	−0.821865	−0.410933	−	0.911666i	$-0.634797\pi$
$541$	−19.1161	−0.821865	−0.410933	+	0.911666i	$0.634797\pi$
$542$	0	0
$543$	20.3131	0.871719
$544$	0	0
$545$	−3.62861	−0.155433
$546$	0	0
$547$	31.7971	1.35954	0.679772	−	0.733423i	$-0.262079\pi$
$547$	31.7971	1.35954	0.679772	+	0.733423i	$0.262079\pi$
$548$	0	0
$549$	41.4292	1.76816
$550$	0	0
$551$	−8.26816	−0.352235
$552$	0	0
$553$	7.35087	0.312591
$554$	0	0
$555$	−7.72226	−0.327792
$556$	0	0
$557$	15.8631	0.672142	0.336071	−	0.941837i	$-0.390902\pi$
$557$	15.8631	0.672142	0.336071	+	0.941837i	$0.390902\pi$
$558$	0	0
$559$	0.574323	0.0242913
$560$	0	0
$561$	−23.7437	−1.00246
$562$	0	0
$563$	−45.8839	−1.93377	−0.966887	−	0.255203i	$-0.917858\pi$
$563$	−45.8839	−1.93377	−0.966887	+	0.255203i	$0.917858\pi$
$564$	0	0
$565$	16.9665	0.713785
$566$	0	0
$567$	0.119509	0.00501891
$568$	0	0
$569$	41.2216	1.72810	0.864050	−	0.503405i	$-0.167920\pi$
$569$	41.2216	1.72810	0.864050	+	0.503405i	$0.167920\pi$
$570$	0	0
$571$	−42.9654	−1.79805	−0.899023	−	0.437902i	$-0.855722\pi$
$571$	−42.9654	−1.79805	−0.899023	+	0.437902i	$0.855722\pi$
$572$	0	0
$573$	−37.5639	−1.56925
$574$	0	0
$575$	5.42568	0.226266
$576$	0	0
$577$	−2.49721	−0.103960	−0.0519800	−	0.998648i	$-0.516553\pi$
$577$	−2.49721	−0.103960	−0.0519800	+	0.998648i	$0.516553\pi$
$578$	0	0
$579$	−7.98810	−0.331974
$580$	0	0
$581$	5.61137	0.232799
$582$	0	0
$583$	−12.2647	−0.507954
$584$	0	0
$585$	−4.87187	−0.201427
$586$	0	0
$587$	33.7596	1.39341	0.696703	−	0.717360i	$-0.254650\pi$
$587$	33.7596	1.39341	0.696703	+	0.717360i	$0.254650\pi$
$588$	0	0
$589$	−1.97086	−0.0812080
$590$	0	0
$591$	25.6467	1.05497
$592$	0	0
$593$	36.8254	1.51224	0.756118	−	0.654435i	$-0.227094\pi$
$593$	36.8254	1.51224	0.756118	+	0.654435i	$0.227094\pi$
$594$	0	0
$595$	−3.87187	−0.158731
$596$	0	0
$597$	−66.1977	−2.70929
$598$	0	0
$599$	−28.1482	−1.15010	−0.575052	−	0.818117i	$-0.695018\pi$
$599$	−28.1482	−1.15010	−0.575052	+	0.818117i	$0.695018\pi$
$600$	0	0
$601$	21.4292	0.874116	0.437058	−	0.899433i	$-0.356021\pi$
$601$	21.4292	0.874116	0.437058	+	0.899433i	$0.356021\pi$
$602$	0	0
$603$	8.47462	0.345113
$604$	0	0
$605$	6.22274	0.252990
$606$	0	0
$607$	28.9876	1.17657	0.588285	−	0.808654i	$-0.299803\pi$
$607$	28.9876	1.17657	0.588285	+	0.808654i	$0.299803\pi$
$608$	0	0
$609$	9.48324	0.384280
$610$	0	0
$611$	−7.74374	−0.313278
$612$	0	0
$613$	37.5941	1.51841	0.759206	−	0.650850i	$-0.225587\pi$
$613$	37.5941	1.51841	0.759206	+	0.650850i	$0.225587\pi$
$614$	0	0
$615$	−18.6070	−0.750306
$616$	0	0
$617$	−7.33363	−0.295241	−0.147620	−	0.989044i	$-0.547161\pi$
$617$	−7.33363	−0.295241	−0.147620	+	0.989044i	$0.547161\pi$
$618$	0	0
$619$	−30.8541	−1.24013	−0.620065	−	0.784551i	$-0.712894\pi$
$619$	−30.8541	−1.24013	−0.620065	+	0.784551i	$0.712894\pi$
$620$	0	0
$621$	28.4950	1.14346
$622$	0	0
$623$	−3.51238	−0.140720
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	15.0010	0.599081
$628$	0	0
$629$	10.6568	0.424914
$630$	0	0
$631$	43.1262	1.71683	0.858413	−	0.512959i	$-0.171451\pi$
$631$	43.1262	1.71683	0.858413	+	0.512959i	$0.171451\pi$
$632$	0	0
$633$	−34.3507	−1.36532
$634$	0	0
$635$	10.6895	0.424198
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	34.2836	1.35624
$640$	0	0
$641$	−12.9761	−0.512524	−0.256262	−	0.966607i	$-0.582491\pi$
$641$	−12.9761	−0.512524	−0.256262	+	0.966607i	$0.582491\pi$
$642$	0	0
$643$	8.39518	0.331074	0.165537	−	0.986204i	$-0.447064\pi$
$643$	8.39518	0.331074	0.165537	+	0.986204i	$0.447064\pi$
$644$	0	0
$645$	−1.61137	−0.0634477
$646$	0	0
$647$	3.04470	0.119700	0.0598498	−	0.998207i	$-0.480938\pi$
$647$	3.04470	0.119700	0.0598498	+	0.998207i	$0.480938\pi$
$648$	0	0
$649$	10.7931	0.423665
$650$	0	0
$651$	2.26050	0.0885960
$652$	0	0
$653$	13.8094	0.540402	0.270201	−	0.962804i	$-0.412910\pi$
$653$	13.8094	0.540402	0.270201	+	0.962804i	$0.412910\pi$
$654$	0	0
$655$	−2.85135	−0.111412
$656$	0	0
$657$	−3.70271	−0.144456
$658$	0	0
$659$	−10.9880	−0.428030	−0.214015	−	0.976830i	$-0.568654\pi$
$659$	−10.9880	−0.428030	−0.214015	+	0.976830i	$0.568654\pi$
$660$	0	0
$661$	−1.59865	−0.0621803	−0.0310901	−	0.999517i	$-0.509898\pi$
$661$	−1.59865	−0.0621803	−0.0310901	+	0.999517i	$0.509898\pi$
$662$	0	0
$663$	10.8632	0.421894
$664$	0	0
$665$	2.44619	0.0948593
$666$	0	0
$667$	18.3388	0.710083
$668$	0	0
$669$	−15.1883	−0.587212
$670$	0	0
$671$	−18.5866	−0.717528
$672$	0	0
$673$	10.0668	0.388045	0.194022	−	0.980997i	$-0.437847\pi$
$673$	10.0668	0.388045	0.194022	+	0.980997i	$0.437847\pi$
$674$	0	0
$675$	5.25188	0.202145
$676$	0	0
$677$	−10.7428	−0.412878	−0.206439	−	0.978459i	$-0.566188\pi$
$677$	−10.7428	−0.412878	−0.206439	+	0.978459i	$0.566188\pi$
$678$	0	0
$679$	−11.3551	−0.435769
$680$	0	0
$681$	11.7093	0.448700
$682$	0	0
$683$	29.3851	1.12439	0.562194	−	0.827005i	$-0.309957\pi$
$683$	29.3851	1.12439	0.562194	+	0.827005i	$0.309957\pi$
$684$	0	0
$685$	9.97510	0.381129
$686$	0	0
$687$	25.4928	0.972612
$688$	0	0
$689$	5.61137	0.213776
$690$	0	0
$691$	−8.39326	−0.319295	−0.159647	−	0.987174i	$-0.551036\pi$
$691$	−8.39326	−0.319295	−0.159647	+	0.987174i	$0.551036\pi$
$692$	0	0
$693$	−10.6484	−0.404500
$694$	0	0
$695$	−2.13237	−0.0808854
$696$	0	0
$697$	25.6778	0.972616
$698$	0	0
$699$	2.13237	0.0806536
$700$	0	0
$701$	−23.1356	−0.873821	−0.436911	−	0.899505i	$-0.643927\pi$
$701$	−23.1356	−0.873821	−0.436911	+	0.899505i	$0.643927\pi$
$702$	0	0
$703$	−6.73281	−0.253933
$704$	0	0
$705$	21.7265	0.818267
$706$	0	0
$707$	6.00000	0.225653
$708$	0	0
$709$	−39.0816	−1.46774	−0.733870	−	0.679290i	$-0.762288\pi$
$709$	−39.0816	−1.46774	−0.733870	+	0.679290i	$0.762288\pi$
$710$	0	0
$711$	35.8125	1.34307
$712$	0	0
$713$	4.37139	0.163710
$714$	0	0
$715$	2.18569	0.0817403
$716$	0	0
$717$	44.6930	1.66909
$718$	0	0
$719$	20.2647	0.755747	0.377874	−	0.925857i	$-0.376655\pi$
$719$	20.2647	0.755747	0.377874	+	0.925857i	$0.376655\pi$
$720$	0	0
$721$	7.29755	0.271775
$722$	0	0
$723$	−2.37673	−0.0883916
$724$	0	0
$725$	3.38001	0.125530
$726$	0	0
$727$	−13.2353	−0.490872	−0.245436	−	0.969413i	$-0.578931\pi$
$727$	−13.2353	−0.490872	−0.245436	+	0.969413i	$0.578931\pi$
$728$	0	0
$729$	−43.6231	−1.61567
$730$	0	0
$731$	2.22371	0.0822467
$732$	0	0
$733$	−40.7340	−1.50455	−0.752273	−	0.658852i	$-0.771042\pi$
$733$	−40.7340	−1.50455	−0.752273	+	0.658852i	$0.771042\pi$
$734$	0	0
$735$	−2.80569	−0.103489
$736$	0	0
$737$	−3.80202	−0.140049
$738$	0	0
$739$	−26.2188	−0.964472	−0.482236	−	0.876041i	$-0.660175\pi$
$739$	−26.2188	−0.964472	−0.482236	+	0.876041i	$0.660175\pi$
$740$	0	0
$741$	−6.86325	−0.252128
$742$	0	0
$743$	6.00424	0.220274	0.110137	−	0.993916i	$-0.464871\pi$
$743$	6.00424	0.220274	0.110137	+	0.993916i	$0.464871\pi$
$744$	0	0
$745$	−12.3714	−0.453253
$746$	0	0
$747$	27.3379	1.00024
$748$	0	0
$749$	8.57432	0.313299
$750$	0	0
$751$	34.4206	1.25602	0.628012	−	0.778203i	$-0.283869\pi$
$751$	34.4206	1.25602	0.628012	+	0.778203i	$0.283869\pi$
$752$	0	0
$753$	34.4110	1.25401
$754$	0	0
$755$	3.29331	0.119856
$756$	0	0
$757$	47.3379	1.72052	0.860262	−	0.509852i	$-0.170300\pi$
$757$	47.3379	1.72052	0.860262	+	0.509852i	$0.170300\pi$
$758$	0	0
$759$	−33.2723	−1.20771
$760$	0	0
$761$	−13.0324	−0.472425	−0.236212	−	0.971701i	$-0.575906\pi$
$761$	−13.0324	−0.472425	−0.236212	+	0.971701i	$0.575906\pi$
$762$	0	0
$763$	3.62861	0.131365
$764$	0	0
$765$	−18.8632	−0.682002
$766$	0	0
$767$	−4.93806	−0.178303
$768$	0	0
$769$	39.9072	1.43909	0.719546	−	0.694445i	$-0.244350\pi$
$769$	39.9072	1.43909	0.719546	+	0.694445i	$0.244350\pi$
$770$	0	0
$771$	17.5047	0.630417
$772$	0	0
$773$	20.1892	0.726156	0.363078	−	0.931759i	$-0.381726\pi$
$773$	20.1892	0.726156	0.363078	+	0.931759i	$0.381726\pi$
$774$	0	0
$775$	0.805685	0.0289411
$776$	0	0
$777$	7.72226	0.277035
$778$	0	0
$779$	−16.2229	−0.581245
$780$	0	0
$781$	−15.3808	−0.550369
$782$	0	0
$783$	17.7514	0.634383
$784$	0	0
$785$	1.71036	0.0610455
$786$	0	0
$787$	−6.26281	−0.223245	−0.111623	−	0.993751i	$-0.535605\pi$
$787$	−6.26281	−0.223245	−0.111623	+	0.993751i	$0.535605\pi$
$788$	0	0
$789$	−4.00000	−0.142404
$790$	0	0
$791$	−16.9665	−0.603259
$792$	0	0
$793$	8.50376	0.301977
$794$	0	0
$795$	−15.7437	−0.558373
$796$	0	0
$797$	−23.6433	−0.837490	−0.418745	−	0.908104i	$-0.637530\pi$
$797$	−23.6433	−0.837490	−0.418745	+	0.908104i	$0.637530\pi$
$798$	0	0
$799$	−29.9828	−1.06071
$800$	0	0
$801$	−17.1119	−0.604618
$802$	0	0
$803$	1.66117	0.0586213
$804$	0	0
$805$	−5.42568	−0.191230
$806$	0	0
$807$	65.8501	2.31803
$808$	0	0
$809$	−27.5831	−0.969769	−0.484884	−	0.874578i	$-0.661138\pi$
$809$	−27.5831	−0.969769	−0.484884	+	0.874578i	$0.661138\pi$
$810$	0	0
$811$	−52.6139	−1.84753	−0.923763	−	0.382966i	$-0.874903\pi$
$811$	−52.6139	−1.84753	−0.923763	+	0.382966i	$0.874903\pi$
$812$	0	0
$813$	−80.7498	−2.83202
$814$	0	0
$815$	−20.0902	−0.703730
$816$	0	0
$817$	−1.40491	−0.0491514
$818$	0	0
$819$	4.87187	0.170237
$820$	0	0
$821$	−12.3476	−0.430934	−0.215467	−	0.976511i	$-0.569127\pi$
$821$	−12.3476	−0.430934	−0.215467	+	0.976511i	$0.569127\pi$
$822$	0	0
$823$	0.161901	0.00564352	0.00282176	−	0.999996i	$-0.499102\pi$
$823$	0.161901	0.00564352	0.00282176	+	0.999996i	$0.499102\pi$
$824$	0	0
$825$	−6.13237	−0.213502
$826$	0	0
$827$	−23.7522	−0.825946	−0.412973	−	0.910743i	$-0.635510\pi$
$827$	−23.7522	−0.825946	−0.412973	+	0.910743i	$0.635510\pi$
$828$	0	0
$829$	−18.2885	−0.635187	−0.317593	−	0.948227i	$-0.602875\pi$
$829$	−18.2885	−0.635187	−0.317593	+	0.948227i	$0.602875\pi$
$830$	0	0
$831$	−33.2723	−1.15420
$832$	0	0
$833$	3.87187	0.134152
$834$	0	0
$835$	6.87515	0.237924
$836$	0	0
$837$	4.23136	0.146257
$838$	0	0
$839$	−42.1942	−1.45670	−0.728352	−	0.685203i	$-0.759714\pi$
$839$	−42.1942	−1.45670	−0.728352	+	0.685203i	$0.759714\pi$
$840$	0	0
$841$	−17.5755	−0.606053
$842$	0	0
$843$	−32.1581	−1.10758
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−6.22274	−0.213816
$848$	0	0
$849$	−76.9597	−2.64125
$850$	0	0
$851$	14.9334	0.511911
$852$	0	0
$853$	20.8183	0.712805	0.356402	−	0.934333i	$-0.384003\pi$
$853$	20.8183	0.712805	0.356402	+	0.934333i	$0.384003\pi$
$854$	0	0
$855$	11.9175	0.407571
$856$	0	0
$857$	−42.7436	−1.46009	−0.730047	−	0.683397i	$-0.760502\pi$
$857$	−42.7436	−1.46009	−0.730047	+	0.683397i	$0.760502\pi$
$858$	0	0
$859$	1.43797	0.0490628	0.0245314	−	0.999699i	$-0.492191\pi$
$859$	1.43797	0.0490628	0.0245314	+	0.999699i	$0.492191\pi$
$860$	0	0
$861$	18.6070	0.634125
$862$	0	0
$863$	24.5993	0.837371	0.418686	−	0.908131i	$-0.362491\pi$
$863$	24.5993	0.837371	0.418686	+	0.908131i	$0.362491\pi$
$864$	0	0
$865$	−1.04965	−0.0356893
$866$	0	0
$867$	−5.63556	−0.191394
$868$	0	0
$869$	−16.0668	−0.545027
$870$	0	0
$871$	1.73950	0.0589407
$872$	0	0
$873$	−55.3206	−1.87232
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−29.7069	−1.00313	−0.501566	−	0.865119i	$-0.667243\pi$
$877$	−29.7069	−1.00313	−0.501566	+	0.865119i	$0.667243\pi$
$878$	0	0
$879$	−39.8589	−1.34441
$880$	0	0
$881$	29.3961	0.990381	0.495191	−	0.868784i	$-0.335098\pi$
$881$	29.3961	0.990381	0.495191	+	0.868784i	$0.335098\pi$
$882$	0	0
$883$	−39.6406	−1.33401	−0.667007	−	0.745052i	$-0.732425\pi$
$883$	−39.6406	−1.33401	−0.667007	+	0.745052i	$0.732425\pi$
$884$	0	0
$885$	13.8546	0.465718
$886$	0	0
$887$	−12.2502	−0.411322	−0.205661	−	0.978623i	$-0.565934\pi$
$887$	−12.2502	−0.411322	−0.205661	+	0.978623i	$0.565934\pi$
$888$	0	0
$889$	−10.6895	−0.358513
$890$	0	0
$891$	−0.261210	−0.00875088
$892$	0	0
$893$	18.9427	0.633893
$894$	0	0
$895$	7.61561	0.254562
$896$	0	0
$897$	15.2227	0.508273
$898$	0	0
$899$	2.72322	0.0908246
$900$	0	0
$901$	21.7265	0.723815
$902$	0	0
$903$	1.61137	0.0536231
$904$	0	0
$905$	−7.23998	−0.240665
$906$	0	0
$907$	35.1694	1.16778	0.583891	−	0.811832i	$-0.301530\pi$
$907$	35.1694	1.16778	0.583891	+	0.811832i	$0.301530\pi$
$908$	0	0
$909$	29.2312	0.969538
$910$	0	0
$911$	43.5726	1.44363	0.721813	−	0.692088i	$-0.243309\pi$
$911$	43.5726	1.44363	0.721813	+	0.692088i	$0.243309\pi$
$912$	0	0
$913$	−12.2647	−0.405904
$914$	0	0
$915$	−23.8589	−0.788750
$916$	0	0
$917$	2.85135	0.0941600
$918$	0	0
$919$	−51.3215	−1.69294	−0.846469	−	0.532438i	$-0.821276\pi$
$919$	−51.3215	−1.69294	−0.846469	+	0.532438i	$0.821276\pi$
$920$	0	0
$921$	−75.6595	−2.49306
$922$	0	0
$923$	7.03705	0.231627
$924$	0	0
$925$	2.75236	0.0904971
$926$	0	0
$927$	35.5527	1.16770
$928$	0	0
$929$	54.4206	1.78548	0.892741	−	0.450571i	$-0.148779\pi$
$929$	54.4206	1.78548	0.892741	+	0.450571i	$0.148779\pi$
$930$	0	0
$931$	−2.44619	−0.0801708
$932$	0	0
$933$	11.2465	0.368195
$934$	0	0
$935$	8.46272	0.276761
$936$	0	0
$937$	−12.6062	−0.411826	−0.205913	−	0.978570i	$-0.566016\pi$
$937$	−12.6062	−0.411826	−0.205913	+	0.978570i	$0.566016\pi$
$938$	0	0
$939$	20.9139	0.682498
$940$	0	0
$941$	−52.7648	−1.72008	−0.860041	−	0.510225i	$-0.829562\pi$
$941$	−52.7648	−1.72008	−0.860041	+	0.510225i	$0.829562\pi$
$942$	0	0
$943$	35.9825	1.17175
$944$	0	0
$945$	−5.25188	−0.170844
$946$	0	0
$947$	30.0373	0.976081	0.488041	−	0.872821i	$-0.337712\pi$
$947$	30.0373	0.976081	0.488041	+	0.872821i	$0.337712\pi$
$948$	0	0
$949$	−0.760017	−0.0246712
$950$	0	0
$951$	51.2451	1.66174
$952$	0	0
$953$	−40.4025	−1.30877	−0.654383	−	0.756163i	$-0.727072\pi$
$953$	−40.4025	−1.30877	−0.654383	+	0.756163i	$0.727072\pi$
$954$	0	0
$955$	13.3885	0.433241
$956$	0	0
$957$	−20.7275	−0.670024
$958$	0	0
$959$	−9.97510	−0.322113
$960$	0	0
$961$	−30.3509	−0.979060
$962$	0	0
$963$	41.7730	1.34612
$964$	0	0
$965$	2.84711	0.0916518
$966$	0	0
$967$	10.7266	0.344945	0.172473	−	0.985014i	$-0.444824\pi$
$967$	10.7266	0.344945	0.172473	+	0.985014i	$0.444824\pi$
$968$	0	0
$969$	−26.5736	−0.853667
$970$	0	0
$971$	−31.9264	−1.02457	−0.512284	−	0.858816i	$-0.671200\pi$
$971$	−31.9264	−1.02457	−0.512284	+	0.858816i	$0.671200\pi$
$972$	0	0
$973$	2.13237	0.0683606
$974$	0	0
$975$	2.80569	0.0898538
$976$	0	0
$977$	13.6309	0.436092	0.218046	−	0.975939i	$-0.430032\pi$
$977$	13.6309	0.436092	0.218046	+	0.975939i	$0.430032\pi$
$978$	0	0
$979$	7.67699	0.245358
$980$	0	0
$981$	17.6781	0.564419
$982$	0	0
$983$	−30.8322	−0.983394	−0.491697	−	0.870766i	$-0.663623\pi$
$983$	−30.8322	−0.983394	−0.491697	+	0.870766i	$0.663623\pi$
$984$	0	0
$985$	−9.14099	−0.291256
$986$	0	0
$987$	−21.7265	−0.691562
$988$	0	0
$989$	3.11609	0.0990860
$990$	0	0
$991$	−37.3574	−1.18670	−0.593349	−	0.804945i	$-0.702195\pi$
$991$	−37.3574	−1.18670	−0.593349	+	0.804945i	$0.702195\pi$
$992$	0	0
$993$	−40.0569	−1.27117
$994$	0	0
$995$	23.5941	0.747984
$996$	0	0
$997$	41.3487	1.30953	0.654763	−	0.755835i	$-0.272769\pi$
$997$	41.3487	1.30953	0.654763	+	0.755835i	$0.272769\pi$
$998$	0	0
$999$	14.4551	0.457338

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.w.1.4	✓	4
4.3	odd	2		7280.2.a.bs.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.w.1.4	✓	4	1.1	even	1	trivial
7280.2.a.bs.1.1		4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3640.a (trivial)

\(f(q)\)	\(=\)	\(q+2.80569 q^{3} -1.00000 q^{5} +1.00000 q^{7} +4.87187 q^{9} +O(q^{10})\)	\(q+2.80569 q^{3} -1.00000 q^{5} +1.00000 q^{7} +4.87187 q^{9} -2.18569 q^{11} +1.00000 q^{13} -2.80569 q^{15} +3.87187 q^{17} -2.44619 q^{19} +2.80569 q^{21} +5.42568 q^{23} +1.00000 q^{25} +5.25188 q^{27} +3.38001 q^{29} +0.805685 q^{31} -6.13237 q^{33} -1.00000 q^{35} +2.75236 q^{37} +2.80569 q^{39} +6.63189 q^{41} +0.574323 q^{43} -4.87187 q^{45} -7.74374 q^{47} +1.00000 q^{49} +10.8632 q^{51} +5.61137 q^{53} +2.18569 q^{55} -6.86325 q^{57} -4.93806 q^{59} +8.50376 q^{61} +4.87187 q^{63} -1.00000 q^{65} +1.73950 q^{67} +15.2227 q^{69} +7.03705 q^{71} -0.760017 q^{73} +2.80569 q^{75} -2.18569 q^{77} +7.35087 q^{79} +0.119509 q^{81} +5.61137 q^{83} -3.87187 q^{85} +9.48324 q^{87} -3.51238 q^{89} +1.00000 q^{91} +2.26050 q^{93} +2.44619 q^{95} -11.3551 q^{97} -10.6484 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} - 4 q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10}) \) 4 * q + q^3 - 4 * q^5 + 4 * q^7 + 3 * q^9	\( 4 q + q^{3} - 4 q^{5} + 4 q^{7} + 3 q^{9} + 2 q^{11} + 4 q^{13} - q^{15} - q^{17} - 3 q^{19} + q^{21} + 12 q^{23} + 4 q^{25} + 4 q^{27} + 13 q^{29} - 7 q^{31} - 12 q^{33} - 4 q^{35} - q^{37} + q^{39} + 9 q^{41} + 12 q^{43} - 3 q^{45} + 2 q^{47} + 4 q^{49} + 6 q^{51} + 2 q^{53} - 2 q^{55} + 10 q^{57} + 3 q^{59} + 3 q^{63} - 4 q^{65} + 3 q^{67} + 20 q^{69} - 2 q^{71} - 2 q^{73} + q^{75} + 2 q^{77} + 5 q^{79} - 4 q^{81} + 2 q^{83} + q^{85} + q^{87} - q^{89} + 4 q^{91} + 13 q^{93} + 3 q^{95} + 8 q^{97} + 8 q^{99}+O(q^{100}) \) 4 * q + q^3 - 4 * q^5 + 4 * q^7 + 3 * q^9 + 2 * q^11 + 4 * q^13 - q^15 - q^17 - 3 * q^19 + q^21 + 12 * q^23 + 4 * q^25 + 4 * q^27 + 13 * q^29 - 7 * q^31 - 12 * q^33 - 4 * q^35 - q^37 + q^39 + 9 * q^41 + 12 * q^43 - 3 * q^45 + 2 * q^47 + 4 * q^49 + 6 * q^51 + 2 * q^53 - 2 * q^55 + 10 * q^57 + 3 * q^59 + 3 * q^63 - 4 * q^65 + 3 * q^67 + 20 * q^69 - 2 * q^71 - 2 * q^73 + q^75 + 2 * q^77 + 5 * q^79 - 4 * q^81 + 2 * q^83 + q^85 + q^87 - q^89 + 4 * q^91 + 13 * q^93 + 3 * q^95 + 8 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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