LMFDB - Embedded newform 3640.2.a.v.1.2

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:	4.4.2225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 2x + 4 $ x^4 - x^3 - 5x^2 + 2x + 4
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			1.2
Root			$-0.820249$ 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)

$f(q)$	$=$	$q-0.820249 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.32719 q^{9} +O(q^{10})$	$q-0.820249 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.32719 q^{9} +1.23607 q^{11} -1.00000 q^{13} +0.820249 q^{15} +2.34107 q^{17} -4.56326 q^{19} +0.820249 q^{21} -1.41831 q^{23} +1.00000 q^{25} +4.36962 q^{27} +3.42970 q^{29} -1.44686 q^{31} -1.01388 q^{33} +1.00000 q^{35} -0.0563172 q^{37} +0.820249 q^{39} -10.4537 q^{41} -10.1576 q^{43} +2.32719 q^{45} +1.34562 q^{47} +1.00000 q^{49} -1.92027 q^{51} -9.59865 q^{53} -1.23607 q^{55} +3.74301 q^{57} +10.6658 q^{59} +9.93039 q^{61} +2.32719 q^{63} +1.00000 q^{65} +6.81321 q^{67} +1.16337 q^{69} -9.97532 q^{71} +16.5848 q^{73} -0.820249 q^{75} -1.23607 q^{77} +13.5802 q^{79} +3.39739 q^{81} +4.83164 q^{83} -2.34107 q^{85} -2.81321 q^{87} +0.125929 q^{89} +1.00000 q^{91} +1.18679 q^{93} +4.56326 q^{95} +8.76702 q^{97} -2.87657 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} - 4 q^{5} - 4 q^{7} - q^{9}+O(q^{10}) $ 4 * q + q^3 - 4 * q^5 - 4 * q^7 - q^9	$ 4 q + q^{3} - 4 q^{5} - 4 q^{7} - q^{9} - 4 q^{11} - 4 q^{13} - q^{15} - 7 q^{17} - q^{19} - q^{21} + 2 q^{23} + 4 q^{25} + 4 q^{27} + q^{29} - q^{31} + 4 q^{33} + 4 q^{35} + 13 q^{37} - q^{39} + q^{41} - 6 q^{43} + q^{45} + 22 q^{47} + 4 q^{49} - 2 q^{51} + 14 q^{53} + 4 q^{55} + 2 q^{57} + 21 q^{59} + 12 q^{61} + q^{63} + 4 q^{65} - 7 q^{67} + 20 q^{69} - 4 q^{71} + 22 q^{73} + q^{75} + 4 q^{77} - 23 q^{79} - 16 q^{81} + 10 q^{83} + 7 q^{85} + 23 q^{87} + 15 q^{89} + 4 q^{91} + 39 q^{93} + q^{95} - 8 q^{97} + 6 q^{99}+O(q^{100}) $ 4 * q + q^3 - 4 * q^5 - 4 * q^7 - q^9 - 4 * q^11 - 4 * q^13 - q^15 - 7 * q^17 - q^19 - q^21 + 2 * q^23 + 4 * q^25 + 4 * q^27 + q^29 - q^31 + 4 * q^33 + 4 * q^35 + 13 * q^37 - q^39 + q^41 - 6 * q^43 + q^45 + 22 * q^47 + 4 * q^49 - 2 * q^51 + 14 * q^53 + 4 * q^55 + 2 * q^57 + 21 * q^59 + 12 * q^61 + q^63 + 4 * q^65 - 7 * q^67 + 20 * q^69 - 4 * q^71 + 22 * q^73 + q^75 + 4 * q^77 - 23 * q^79 - 16 * q^81 + 10 * q^83 + 7 * q^85 + 23 * q^87 + 15 * q^89 + 4 * q^91 + 39 * q^93 + q^95 - 8 * q^97 + 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−0.820249	−0.473571	−0.236786	−	0.971562i	$-0.576094\pi$
$3$	−0.820249	−0.473571	−0.236786	+	0.971562i	$0.576094\pi$
$4$	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
$9$	−2.32719	−0.775730
$10$	0	0
$11$	1.23607	0.372689	0.186344	−	0.982485i	$-0.440336\pi$
$11$	1.23607	0.372689	0.186344	+	0.982485i	$0.440336\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0.820249	0.211787
$16$	0	0
$17$	2.34107	0.567794	0.283897	−	0.958855i	$-0.408373\pi$
$17$	2.34107	0.567794	0.283897	+	0.958855i	$0.408373\pi$
$18$	0	0
$19$	−4.56326	−1.04688	−0.523442	−	0.852061i	$-0.675352\pi$
$19$	−4.56326	−1.04688	−0.523442	+	0.852061i	$0.675352\pi$
$20$	0	0
$21$	0.820249	0.178993
$22$	0	0
$23$	−1.41831	−0.295739	−0.147869	−	0.989007i	$-0.547242\pi$
$23$	−1.41831	−0.295739	−0.147869	+	0.989007i	$0.547242\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.36962	0.840935
$28$	0	0
$29$	3.42970	0.636880	0.318440	−	0.947943i	$-0.396841\pi$
$29$	3.42970	0.636880	0.318440	+	0.947943i	$0.396841\pi$
$30$	0	0
$31$	−1.44686	−0.259864	−0.129932	−	0.991523i	$-0.541476\pi$
$31$	−1.44686	−0.259864	−0.129932	+	0.991523i	$0.541476\pi$
$32$	0	0
$33$	−1.01388	−0.176495
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−0.0563172	−0.00925850	−0.00462925	−	0.999989i	$-0.501474\pi$
$37$	−0.0563172	−0.00925850	−0.00462925	+	0.999989i	$0.501474\pi$
$38$	0	0
$39$	0.820249	0.131345
$40$	0	0
$41$	−10.4537	−1.63260	−0.816298	−	0.577632i	$-0.803977\pi$
$41$	−10.4537	−1.63260	−0.816298	+	0.577632i	$0.803977\pi$
$42$	0	0
$43$	−10.1576	−1.54901	−0.774507	−	0.632565i	$-0.782002\pi$
$43$	−10.1576	−1.54901	−0.774507	+	0.632565i	$0.782002\pi$
$44$	0	0
$45$	2.32719	0.346917
$46$	0	0
$47$	1.34562	0.196279	0.0981393	−	0.995173i	$-0.468711\pi$
$47$	1.34562	0.196279	0.0981393	+	0.995173i	$0.468711\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.92027	−0.268891
$52$	0	0
$53$	−9.59865	−1.31848	−0.659238	−	0.751934i	$-0.729121\pi$
$53$	−9.59865	−1.31848	−0.659238	+	0.751934i	$0.729121\pi$
$54$	0	0
$55$	−1.23607	−0.166671
$56$	0	0
$57$	3.74301	0.495774
$58$	0	0
$59$	10.6658	1.38857	0.694283	−	0.719703i	$-0.255722\pi$
$59$	10.6658	1.38857	0.694283	+	0.719703i	$0.255722\pi$
$60$	0	0
$61$	9.93039	1.27146	0.635728	−	0.771913i	$-0.280700\pi$
$61$	9.93039	1.27146	0.635728	+	0.771913i	$0.280700\pi$
$62$	0	0
$63$	2.32719	0.293199
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	6.81321	0.832366	0.416183	−	0.909281i	$-0.363368\pi$
$67$	6.81321	0.832366	0.416183	+	0.909281i	$0.363368\pi$
$68$	0	0
$69$	1.16337	0.140053
$70$	0	0
$71$	−9.97532	−1.18385	−0.591926	−	0.805992i	$-0.701632\pi$
$71$	−9.97532	−1.18385	−0.591926	+	0.805992i	$0.701632\pi$
$72$	0	0
$73$	16.5848	1.94110	0.970550	−	0.240899i	$-0.0774421\pi$
$73$	16.5848	1.94110	0.970550	+	0.240899i	$0.0774421\pi$
$74$	0	0
$75$	−0.820249	−0.0947142
$76$	0	0
$77$	−1.23607	−0.140863
$78$	0	0
$79$	13.5802	1.52789	0.763947	−	0.645278i	$-0.223259\pi$
$79$	13.5802	1.52789	0.763947	+	0.645278i	$0.223259\pi$
$80$	0	0
$81$	3.39739	0.377488
$82$	0	0
$83$	4.83164	0.530341	0.265171	−	0.964202i	$-0.414572\pi$
$83$	4.83164	0.530341	0.265171	+	0.964202i	$0.414572\pi$
$84$	0	0
$85$	−2.34107	−0.253925
$86$	0	0
$87$	−2.81321	−0.301608
$88$	0	0
$89$	0.125929	0.0133485	0.00667423	−	0.999978i	$-0.497876\pi$
$89$	0.125929	0.0133485	0.00667423	+	0.999978i	$0.497876\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	1.18679	0.123064
$94$	0	0
$95$	4.56326	0.468181
$96$	0	0
$97$	8.76702	0.890156	0.445078	−	0.895492i	$-0.353176\pi$
$97$	8.76702	0.890156	0.445078	+	0.895492i	$0.353176\pi$
$98$	0	0
$99$	−2.87657	−0.289106
$100$	0	0
$101$	−3.66327	−0.364509	−0.182255	−	0.983251i	$-0.558340\pi$
$101$	−3.66327	−0.364509	−0.182255	+	0.983251i	$0.558340\pi$
$102$	0	0
$103$	4.89499	0.482318	0.241159	−	0.970486i	$-0.422472\pi$
$103$	4.89499	0.482318	0.241159	+	0.970486i	$0.422472\pi$
$104$	0	0
$105$	−0.820249	−0.0800481
$106$	0	0
$107$	−8.53984	−0.825578	−0.412789	−	0.910827i	$-0.635445\pi$
$107$	−8.53984	−0.825578	−0.412789	+	0.910827i	$0.635445\pi$
$108$	0	0
$109$	9.85051	0.943508	0.471754	−	0.881730i	$-0.343621\pi$
$109$	9.85051	0.943508	0.471754	+	0.881730i	$0.343621\pi$
$110$	0	0
$111$	0.0461942	0.00438456
$112$	0	0
$113$	−4.28989	−0.403559	−0.201779	−	0.979431i	$-0.564672\pi$
$113$	−4.28989	−0.403559	−0.201779	+	0.979431i	$0.564672\pi$
$114$	0	0
$115$	1.41831	0.132258
$116$	0	0
$117$	2.32719	0.215149
$118$	0	0
$119$	−2.34107	−0.214606
$120$	0	0
$121$	−9.47214	−0.861103
$122$	0	0
$123$	8.57465	0.773150
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−5.23607	−0.464626	−0.232313	−	0.972641i	$-0.574629\pi$
$127$	−5.23607	−0.464626	−0.232313	+	0.972641i	$0.574629\pi$
$128$	0	0
$129$	8.33173	0.733568
$130$	0	0
$131$	−6.02777	−0.526649	−0.263324	−	0.964707i	$-0.584819\pi$
$131$	−6.02777	−0.526649	−0.263324	+	0.964707i	$0.584819\pi$
$132$	0	0
$133$	4.56326	0.395685
$134$	0	0
$135$	−4.36962	−0.376077
$136$	0	0
$137$	17.4450	1.49042	0.745212	−	0.666828i	$-0.232348\pi$
$137$	17.4450	1.49042	0.745212	+	0.666828i	$0.232348\pi$
$138$	0	0
$139$	−6.98612	−0.592555	−0.296277	−	0.955102i	$-0.595745\pi$
$139$	−6.98612	−0.592555	−0.296277	+	0.955102i	$0.595745\pi$
$140$	0	0
$141$	−1.10374	−0.0929518
$142$	0	0
$143$	−1.23607	−0.103365
$144$	0	0
$145$	−3.42970	−0.284821
$146$	0	0
$147$	−0.820249	−0.0676530
$148$	0	0
$149$	22.7948	1.86742	0.933711	−	0.358028i	$-0.116551\pi$
$149$	22.7948	1.86742	0.933711	+	0.358028i	$0.116551\pi$
$150$	0	0
$151$	−17.4386	−1.41913	−0.709565	−	0.704640i	$-0.751109\pi$
$151$	−17.4386	−1.41913	−0.709565	+	0.704640i	$0.751109\pi$
$152$	0	0
$153$	−5.44813	−0.440455
$154$	0	0
$155$	1.44686	0.116215
$156$	0	0
$157$	21.8095	1.74058	0.870292	−	0.492536i	$-0.163930\pi$
$157$	21.8095	1.74058	0.870292	+	0.492536i	$0.163930\pi$
$158$	0	0
$159$	7.87329	0.624392
$160$	0	0
$161$	1.41831	0.111779
$162$	0	0
$163$	7.87906	0.617136	0.308568	−	0.951202i	$-0.400150\pi$
$163$	7.87906	0.617136	0.308568	+	0.951202i	$0.400150\pi$
$164$	0	0
$165$	1.01388	0.0789307
$166$	0	0
$167$	18.7670	1.45224	0.726118	−	0.687570i	$-0.241323\pi$
$167$	18.7670	1.45224	0.726118	+	0.687570i	$0.241323\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	10.6196	0.812099
$172$	0	0
$173$	21.3879	1.62609	0.813044	−	0.582202i	$-0.197809\pi$
$173$	21.3879	1.62609	0.813044	+	0.582202i	$0.197809\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−8.74859	−0.657584
$178$	0	0
$179$	−16.7714	−1.25355	−0.626775	−	0.779200i	$-0.715626\pi$
$179$	−16.7714	−1.25355	−0.626775	+	0.779200i	$0.715626\pi$
$180$	0	0
$181$	21.8935	1.62733	0.813667	−	0.581331i	$-0.197468\pi$
$181$	21.8935	1.62733	0.813667	+	0.581331i	$0.197468\pi$
$182$	0	0
$183$	−8.14539	−0.602125
$184$	0	0
$185$	0.0563172	0.00414052
$186$	0	0
$187$	2.89373	0.211610
$188$	0	0
$189$	−4.36962	−0.317843
$190$	0	0
$191$	10.6133	0.767953	0.383976	−	0.923343i	$-0.374554\pi$
$191$	10.6133	0.767953	0.383976	+	0.923343i	$0.374554\pi$
$192$	0	0
$193$	9.62207	0.692612	0.346306	−	0.938122i	$-0.387436\pi$
$193$	9.62207	0.692612	0.346306	+	0.938122i	$0.387436\pi$
$194$	0	0
$195$	−0.820249	−0.0587393
$196$	0	0
$197$	10.5285	0.750121	0.375061	−	0.927000i	$-0.377622\pi$
$197$	10.5285	0.750121	0.375061	+	0.927000i	$0.377622\pi$
$198$	0	0
$199$	−23.5618	−1.67025	−0.835126	−	0.550059i	$-0.814605\pi$
$199$	−23.5618	−1.67025	−0.835126	+	0.550059i	$0.814605\pi$
$200$	0	0
$201$	−5.58853	−0.394185
$202$	0	0
$203$	−3.42970	−0.240718
$204$	0	0
$205$	10.4537	0.730119
$206$	0	0
$207$	3.30069	0.229414
$208$	0	0
$209$	−5.64050	−0.390161
$210$	0	0
$211$	23.3334	1.60633	0.803167	−	0.595753i	$-0.203146\pi$
$211$	23.3334	1.60633	0.803167	+	0.595753i	$0.203146\pi$
$212$	0	0
$213$	8.18225	0.560638
$214$	0	0
$215$	10.1576	0.692740
$216$	0	0
$217$	1.44686	0.0982195
$218$	0	0
$219$	−13.6036	−0.919249
$220$	0	0
$221$	−2.34107	−0.157478
$222$	0	0
$223$	−16.2253	−1.08652	−0.543262	−	0.839563i	$-0.682811\pi$
$223$	−16.2253	−1.08652	−0.543262	+	0.839563i	$0.682811\pi$
$224$	0	0
$225$	−2.32719	−0.155146
$226$	0	0
$227$	−1.69603	−0.112570	−0.0562849	−	0.998415i	$-0.517926\pi$
$227$	−1.69603	−0.112570	−0.0562849	+	0.998415i	$0.517926\pi$
$228$	0	0
$229$	−13.3702	−0.883529	−0.441764	−	0.897131i	$-0.645647\pi$
$229$	−13.3702	−0.883529	−0.441764	+	0.897131i	$0.645647\pi$
$230$	0	0
$231$	1.01388	0.0667087
$232$	0	0
$233$	26.3934	1.72909	0.864546	−	0.502554i	$-0.167606\pi$
$233$	26.3934	1.72909	0.864546	+	0.502554i	$0.167606\pi$
$234$	0	0
$235$	−1.34562	−0.0877784
$236$	0	0
$237$	−11.1392	−0.723567
$238$	0	0
$239$	18.6575	1.20685	0.603426	−	0.797419i	$-0.293802\pi$
$239$	18.6575	1.20685	0.603426	+	0.797419i	$0.293802\pi$
$240$	0	0
$241$	−18.3613	−1.18276	−0.591378	−	0.806394i	$-0.701416\pi$
$241$	−18.3613	−1.18276	−0.591378	+	0.806394i	$0.701416\pi$
$242$	0	0
$243$	−15.8956	−1.01970
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	4.56326	0.290353
$248$	0	0
$249$	−3.96315	−0.251154
$250$	0	0
$251$	−18.5277	−1.16946	−0.584728	−	0.811229i	$-0.698799\pi$
$251$	−18.5277	−1.16946	−0.584728	+	0.811229i	$0.698799\pi$
$252$	0	0
$253$	−1.75313	−0.110219
$254$	0	0
$255$	1.92027	0.120252
$256$	0	0
$257$	21.8708	1.36426	0.682130	−	0.731231i	$-0.261054\pi$
$257$	21.8708	1.36426	0.682130	+	0.731231i	$0.261054\pi$
$258$	0	0
$259$	0.0563172	0.00349938
$260$	0	0
$261$	−7.98157	−0.494047
$262$	0	0
$263$	−1.63741	−0.100967	−0.0504836	−	0.998725i	$-0.516076\pi$
$263$	−1.63741	−0.100967	−0.0504836	+	0.998725i	$0.516076\pi$
$264$	0	0
$265$	9.59865	0.589641
$266$	0	0
$267$	−0.103293	−0.00632145
$268$	0	0
$269$	28.1404	1.71575	0.857875	−	0.513858i	$-0.171784\pi$
$269$	28.1404	1.71575	0.857875	+	0.513858i	$0.171784\pi$
$270$	0	0
$271$	−26.5879	−1.61510	−0.807549	−	0.589801i	$-0.799206\pi$
$271$	−26.5879	−1.61510	−0.807549	+	0.589801i	$0.799206\pi$
$272$	0	0
$273$	−0.820249	−0.0496437
$274$	0	0
$275$	1.23607	0.0745377
$276$	0	0
$277$	−23.7391	−1.42634	−0.713171	−	0.700990i	$-0.752742\pi$
$277$	−23.7391	−1.42634	−0.713171	+	0.700990i	$0.752742\pi$
$278$	0	0
$279$	3.36713	0.201585
$280$	0	0
$281$	−4.59728	−0.274251	−0.137125	−	0.990554i	$-0.543786\pi$
$281$	−4.59728	−0.274251	−0.137125	+	0.990554i	$0.543786\pi$
$282$	0	0
$283$	−3.78716	−0.225123	−0.112562	−	0.993645i	$-0.535906\pi$
$283$	−3.78716	−0.225123	−0.112562	+	0.993645i	$0.535906\pi$
$284$	0	0
$285$	−3.74301	−0.221717
$286$	0	0
$287$	10.4537	0.617063
$288$	0	0
$289$	−11.5194	−0.677610
$290$	0	0
$291$	−7.19114	−0.421552
$292$	0	0
$293$	8.98612	0.524975	0.262487	−	0.964935i	$-0.415457\pi$
$293$	8.98612	0.524975	0.262487	+	0.964935i	$0.415457\pi$
$294$	0	0
$295$	−10.6658	−0.620985
$296$	0	0
$297$	5.40115	0.313407
$298$	0	0
$299$	1.41831	0.0820232
$300$	0	0
$301$	10.1576	0.585472
$302$	0	0
$303$	3.00480	0.172621
$304$	0	0
$305$	−9.93039	−0.568612
$306$	0	0
$307$	32.5529	1.85789	0.928946	−	0.370214i	$-0.120716\pi$
$307$	32.5529	1.85789	0.928946	+	0.370214i	$0.120716\pi$
$308$	0	0
$309$	−4.01511	−0.228412
$310$	0	0
$311$	19.1543	1.08614	0.543070	−	0.839687i	$-0.317262\pi$
$311$	19.1543	1.08614	0.543070	+	0.839687i	$0.317262\pi$
$312$	0	0
$313$	19.8928	1.12440	0.562202	−	0.827000i	$-0.309954\pi$
$313$	19.8928	1.12440	0.562202	+	0.827000i	$0.309954\pi$
$314$	0	0
$315$	−2.32719	−0.131122
$316$	0	0
$317$	4.78109	0.268533	0.134266	−	0.990945i	$-0.457132\pi$
$317$	4.78109	0.268533	0.134266	+	0.990945i	$0.457132\pi$
$318$	0	0
$319$	4.23935	0.237358
$320$	0	0
$321$	7.00480	0.390970
$322$	0	0
$323$	−10.6829	−0.594414
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−8.07988	−0.446818
$328$	0	0
$329$	−1.34562	−0.0741863
$330$	0	0
$331$	19.6916	1.08235	0.541174	−	0.840911i	$-0.317980\pi$
$331$	19.6916	1.08235	0.541174	+	0.840911i	$0.317980\pi$
$332$	0	0
$333$	0.131061	0.00718210
$334$	0	0
$335$	−6.81321	−0.372245
$336$	0	0
$337$	−32.1138	−1.74935	−0.874675	−	0.484709i	$-0.838925\pi$
$337$	−32.1138	−1.74935	−0.874675	+	0.484709i	$0.838925\pi$
$338$	0	0
$339$	3.51878	0.191114
$340$	0	0
$341$	−1.78842	−0.0968485
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−1.16337	−0.0626338
$346$	0	0
$347$	−6.93367	−0.372219	−0.186109	−	0.982529i	$-0.559588\pi$
$347$	−6.93367	−0.372219	−0.186109	+	0.982529i	$0.559588\pi$
$348$	0	0
$349$	5.73367	0.306916	0.153458	−	0.988155i	$-0.450959\pi$
$349$	5.73367	0.306916	0.153458	+	0.988155i	$0.450959\pi$
$350$	0	0
$351$	−4.36962	−0.233233
$352$	0	0
$353$	−6.66964	−0.354989	−0.177494	−	0.984122i	$-0.556799\pi$
$353$	−6.66964	−0.354989	−0.177494	+	0.984122i	$0.556799\pi$
$354$	0	0
$355$	9.97532	0.529435
$356$	0	0
$357$	1.92027	0.101631
$358$	0	0
$359$	−17.9753	−0.948701	−0.474350	−	0.880336i	$-0.657317\pi$
$359$	−17.9753	−0.948701	−0.474350	+	0.880336i	$0.657317\pi$
$360$	0	0
$361$	1.82333	0.0959649
$362$	0	0
$363$	7.76951	0.407794
$364$	0	0
$365$	−16.5848	−0.868087
$366$	0	0
$367$	8.24496	0.430383	0.215192	−	0.976572i	$-0.430962\pi$
$367$	8.24496	0.430383	0.215192	+	0.976572i	$0.430962\pi$
$368$	0	0
$369$	24.3278	1.26645
$370$	0	0
$371$	9.59865	0.498337
$372$	0	0
$373$	20.9165	1.08302	0.541508	−	0.840696i	$-0.317854\pi$
$373$	20.9165	1.08302	0.541508	+	0.840696i	$0.317854\pi$
$374$	0	0
$375$	0.820249	0.0423575
$376$	0	0
$377$	−3.42970	−0.176639
$378$	0	0
$379$	−14.3423	−0.736717	−0.368358	−	0.929684i	$-0.620080\pi$
$379$	−14.3423	−0.736717	−0.368358	+	0.929684i	$0.620080\pi$
$380$	0	0
$381$	4.29488	0.220033
$382$	0	0
$383$	7.42892	0.379600	0.189800	−	0.981823i	$-0.439216\pi$
$383$	7.42892	0.379600	0.189800	+	0.981823i	$0.439216\pi$
$384$	0	0
$385$	1.23607	0.0629959
$386$	0	0
$387$	23.6386	1.20162
$388$	0	0
$389$	4.34860	0.220483	0.110241	−	0.993905i	$-0.464838\pi$
$389$	4.34860	0.220483	0.110241	+	0.993905i	$0.464838\pi$
$390$	0	0
$391$	−3.32038	−0.167919
$392$	0	0
$393$	4.94427	0.249406
$394$	0	0
$395$	−13.5802	−0.683295
$396$	0	0
$397$	17.3038	0.868451	0.434226	−	0.900804i	$-0.357022\pi$
$397$	17.3038	0.868451	0.434226	+	0.900804i	$0.357022\pi$
$398$	0	0
$399$	−3.74301	−0.187385
$400$	0	0
$401$	4.95179	0.247281	0.123640	−	0.992327i	$-0.460543\pi$
$401$	4.95179	0.247281	0.123640	+	0.992327i	$0.460543\pi$
$402$	0	0
$403$	1.44686	0.0720734
$404$	0	0
$405$	−3.39739	−0.168818
$406$	0	0
$407$	−0.0696119	−0.00345053
$408$	0	0
$409$	−16.6175	−0.821684	−0.410842	−	0.911707i	$-0.634765\pi$
$409$	−16.6175	−0.821684	−0.410842	+	0.911707i	$0.634765\pi$
$410$	0	0
$411$	−14.3092	−0.705821
$412$	0	0
$413$	−10.6658	−0.524828
$414$	0	0
$415$	−4.83164	−0.237176
$416$	0	0
$417$	5.73036	0.280617
$418$	0	0
$419$	26.3429	1.28693	0.643467	−	0.765474i	$-0.277495\pi$
$419$	26.3429	1.28693	0.643467	+	0.765474i	$0.277495\pi$
$420$	0	0
$421$	−30.8150	−1.50183	−0.750916	−	0.660397i	$-0.770388\pi$
$421$	−30.8150	−1.50183	−0.750916	+	0.660397i	$0.770388\pi$
$422$	0	0
$423$	−3.13151	−0.152259
$424$	0	0
$425$	2.34107	0.113559
$426$	0	0
$427$	−9.93039	−0.480565
$428$	0	0
$429$	1.01388	0.0489508
$430$	0	0
$431$	13.1296	0.632431	0.316215	−	0.948687i	$-0.397588\pi$
$431$	13.1296	0.632431	0.316215	+	0.948687i	$0.397588\pi$
$432$	0	0
$433$	−27.5575	−1.32433	−0.662163	−	0.749360i	$-0.730361\pi$
$433$	−27.5575	−1.32433	−0.662163	+	0.749360i	$0.730361\pi$
$434$	0	0
$435$	2.81321	0.134883
$436$	0	0
$437$	6.47214	0.309604
$438$	0	0
$439$	17.5543	0.837820	0.418910	−	0.908028i	$-0.362412\pi$
$439$	17.5543	0.837820	0.418910	+	0.908028i	$0.362412\pi$
$440$	0	0
$441$	−2.32719	−0.110819
$442$	0	0
$443$	−3.83335	−0.182128	−0.0910640	−	0.995845i	$-0.529027\pi$
$443$	−3.83335	−0.182128	−0.0910640	+	0.995845i	$0.529027\pi$
$444$	0	0
$445$	−0.125929	−0.00596961
$446$	0	0
$447$	−18.6974	−0.884357
$448$	0	0
$449$	11.7557	0.554784	0.277392	−	0.960757i	$-0.410530\pi$
$449$	11.7557	0.554784	0.277392	+	0.960757i	$0.410530\pi$
$450$	0	0
$451$	−12.9215	−0.608449
$452$	0	0
$453$	14.3040	0.672059
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	23.4827	1.09847	0.549237	−	0.835666i	$-0.314918\pi$
$457$	23.4827	1.09847	0.549237	+	0.835666i	$0.314918\pi$
$458$	0	0
$459$	10.2296	0.477478
$460$	0	0
$461$	−29.2399	−1.36184	−0.680920	−	0.732358i	$-0.738420\pi$
$461$	−29.2399	−1.36184	−0.680920	+	0.732358i	$0.738420\pi$
$462$	0	0
$463$	−16.0296	−0.744958	−0.372479	−	0.928041i	$-0.621492\pi$
$463$	−16.0296	−0.744958	−0.372479	+	0.928041i	$0.621492\pi$
$464$	0	0
$465$	−1.18679	−0.0550360
$466$	0	0
$467$	34.9658	1.61802	0.809012	−	0.587792i	$-0.200003\pi$
$467$	34.9658	1.61802	0.809012	+	0.587792i	$0.200003\pi$
$468$	0	0
$469$	−6.81321	−0.314605
$470$	0	0
$471$	−17.8892	−0.824290
$472$	0	0
$473$	−12.5554	−0.577300
$474$	0	0
$475$	−4.56326	−0.209377
$476$	0	0
$477$	22.3379	1.02278
$478$	0	0
$479$	4.30010	0.196477	0.0982383	−	0.995163i	$-0.468679\pi$
$479$	4.30010	0.196477	0.0982383	+	0.995163i	$0.468679\pi$
$480$	0	0
$481$	0.0563172	0.00256784
$482$	0	0
$483$	−1.16337	−0.0529352
$484$	0	0
$485$	−8.76702	−0.398090
$486$	0	0
$487$	−12.2320	−0.554285	−0.277143	−	0.960829i	$-0.589387\pi$
$487$	−12.2320	−0.554285	−0.277143	+	0.960829i	$0.589387\pi$
$488$	0	0
$489$	−6.46279	−0.292258
$490$	0	0
$491$	−10.7811	−0.486544	−0.243272	−	0.969958i	$-0.578221\pi$
$491$	−10.7811	−0.486544	−0.243272	+	0.969958i	$0.578221\pi$
$492$	0	0
$493$	8.02919	0.361617
$494$	0	0
$495$	2.87657	0.129292
$496$	0	0
$497$	9.97532	0.447454
$498$	0	0
$499$	−25.6233	−1.14706	−0.573529	−	0.819185i	$-0.694426\pi$
$499$	−25.6233	−1.14706	−0.573529	+	0.819185i	$0.694426\pi$
$500$	0	0
$501$	−15.3936	−0.687737
$502$	0	0
$503$	8.39035	0.374107	0.187054	−	0.982350i	$-0.440106\pi$
$503$	8.39035	0.374107	0.187054	+	0.982350i	$0.440106\pi$
$504$	0	0
$505$	3.66327	0.163014
$506$	0	0
$507$	−0.820249	−0.0364285
$508$	0	0
$509$	28.9233	1.28200	0.641002	−	0.767539i	$-0.278519\pi$
$509$	28.9233	1.28200	0.641002	+	0.767539i	$0.278519\pi$
$510$	0	0
$511$	−16.5848	−0.733667
$512$	0	0
$513$	−19.9397	−0.880361
$514$	0	0
$515$	−4.89499	−0.215699
$516$	0	0
$517$	1.66327	0.0731507
$518$	0	0
$519$	−17.5434	−0.770069
$520$	0	0
$521$	30.6125	1.34116	0.670580	−	0.741837i	$-0.266045\pi$
$521$	30.6125	1.34116	0.670580	+	0.741837i	$0.266045\pi$
$522$	0	0
$523$	−39.3967	−1.72270	−0.861349	−	0.508014i	$-0.830380\pi$
$523$	−39.3967	−1.72270	−0.861349	+	0.508014i	$0.830380\pi$
$524$	0	0
$525$	0.820249	0.0357986
$526$	0	0
$527$	−3.38722	−0.147549
$528$	0	0
$529$	−20.9884	−0.912538
$530$	0	0
$531$	−24.8213	−1.07715
$532$	0	0
$533$	10.4537	0.452800
$534$	0	0
$535$	8.53984	0.369210
$536$	0	0
$537$	13.7567	0.593645
$538$	0	0
$539$	1.23607	0.0532412
$540$	0	0
$541$	23.1593	0.995695	0.497848	−	0.867265i	$-0.334124\pi$
$541$	23.1593	0.995695	0.497848	+	0.867265i	$0.334124\pi$
$542$	0	0
$543$	−17.9582	−0.770658
$544$	0	0
$545$	−9.85051	−0.421950
$546$	0	0
$547$	−20.4843	−0.875846	−0.437923	−	0.899013i	$-0.644286\pi$
$547$	−20.4843	−0.875846	−0.437923	+	0.899013i	$0.644286\pi$
$548$	0	0
$549$	−23.1099	−0.986307
$550$	0	0
$551$	−15.6506	−0.666739
$552$	0	0
$553$	−13.5802	−0.577490
$554$	0	0
$555$	−0.0461942	−0.00196083
$556$	0	0
$557$	−0.186596	−0.00790634	−0.00395317	−	0.999992i	$-0.501258\pi$
$557$	−0.186596	−0.00790634	−0.00395317	+	0.999992i	$0.501258\pi$
$558$	0	0
$559$	10.1576	0.429619
$560$	0	0
$561$	−2.37358	−0.100213
$562$	0	0
$563$	39.2591	1.65457	0.827287	−	0.561780i	$-0.189883\pi$
$563$	39.2591	1.65457	0.827287	+	0.561780i	$0.189883\pi$
$564$	0	0
$565$	4.28989	0.180477
$566$	0	0
$567$	−3.39739	−0.142677
$568$	0	0
$569$	−38.4978	−1.61391	−0.806955	−	0.590613i	$-0.798886\pi$
$569$	−38.4978	−1.61391	−0.806955	+	0.590613i	$0.798886\pi$
$570$	0	0
$571$	−23.6474	−0.989615	−0.494807	−	0.869003i	$-0.664761\pi$
$571$	−23.6474	−0.989615	−0.494807	+	0.869003i	$0.664761\pi$
$572$	0	0
$573$	−8.70557	−0.363680
$574$	0	0
$575$	−1.41831	−0.0591478
$576$	0	0
$577$	5.44300	0.226595	0.113297	−	0.993561i	$-0.463859\pi$
$577$	5.44300	0.226595	0.113297	+	0.993561i	$0.463859\pi$
$578$	0	0
$579$	−7.89250	−0.328001
$580$	0	0
$581$	−4.83164	−0.200450
$582$	0	0
$583$	−11.8646	−0.491381
$584$	0	0
$585$	−2.32719	−0.0962175
$586$	0	0
$587$	35.8380	1.47919	0.739596	−	0.673051i	$-0.235016\pi$
$587$	35.8380	1.47919	0.739596	+	0.673051i	$0.235016\pi$
$588$	0	0
$589$	6.60242	0.272048
$590$	0	0
$591$	−8.63596	−0.355236
$592$	0	0
$593$	−21.5987	−0.886950	−0.443475	−	0.896287i	$-0.646255\pi$
$593$	−21.5987	−0.886950	−0.443475	+	0.896287i	$0.646255\pi$
$594$	0	0
$595$	2.34107	0.0959747
$596$	0	0
$597$	19.3265	0.790983
$598$	0	0
$599$	−5.34309	−0.218313	−0.109156	−	0.994025i	$-0.534815\pi$
$599$	−5.34309	−0.218313	−0.109156	+	0.994025i	$0.534815\pi$
$600$	0	0
$601$	−5.92422	−0.241654	−0.120827	−	0.992674i	$-0.538555\pi$
$601$	−5.92422	−0.241654	−0.120827	+	0.992674i	$0.538555\pi$
$602$	0	0
$603$	−15.8556	−0.645692
$604$	0	0
$605$	9.47214	0.385097
$606$	0	0
$607$	23.1859	0.941087	0.470544	−	0.882377i	$-0.344058\pi$
$607$	23.1859	0.941087	0.470544	+	0.882377i	$0.344058\pi$
$608$	0	0
$609$	2.81321	0.113997
$610$	0	0
$611$	−1.34562	−0.0544379
$612$	0	0
$613$	39.6451	1.60125	0.800625	−	0.599165i	$-0.204501\pi$
$613$	39.6451	1.60125	0.800625	+	0.599165i	$0.204501\pi$
$614$	0	0
$615$	−8.57465	−0.345763
$616$	0	0
$617$	−34.7461	−1.39882	−0.699412	−	0.714719i	$-0.746555\pi$
$617$	−34.7461	−1.39882	−0.699412	+	0.714719i	$0.746555\pi$
$618$	0	0
$619$	24.7491	0.994749	0.497375	−	0.867536i	$-0.334297\pi$
$619$	24.7491	0.994749	0.497375	+	0.867536i	$0.334297\pi$
$620$	0	0
$621$	−6.19750	−0.248697
$622$	0	0
$623$	−0.125929	−0.00504524
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	4.62661	0.184769
$628$	0	0
$629$	−0.131843	−0.00525692
$630$	0	0
$631$	1.97532	0.0786361	0.0393181	−	0.999227i	$-0.487481\pi$
$631$	1.97532	0.0786361	0.0393181	+	0.999227i	$0.487481\pi$
$632$	0	0
$633$	−19.1392	−0.760714
$634$	0	0
$635$	5.23607	0.207787
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	23.2145	0.918350
$640$	0	0
$641$	−3.65878	−0.144513	−0.0722566	−	0.997386i	$-0.523020\pi$
$641$	−3.65878	−0.144513	−0.0722566	+	0.997386i	$0.523020\pi$
$642$	0	0
$643$	−47.9921	−1.89262	−0.946312	−	0.323256i	$-0.895223\pi$
$643$	−47.9921	−1.89262	−0.946312	+	0.323256i	$0.895223\pi$
$644$	0	0
$645$	−8.33173	−0.328062
$646$	0	0
$647$	23.9948	0.943332	0.471666	−	0.881777i	$-0.343653\pi$
$647$	23.9948	0.943332	0.471666	+	0.881777i	$0.343653\pi$
$648$	0	0
$649$	13.1836	0.517502
$650$	0	0
$651$	−1.18679	−0.0465139
$652$	0	0
$653$	18.3011	0.716175	0.358088	−	0.933688i	$-0.383429\pi$
$653$	18.3011	0.716175	0.358088	+	0.933688i	$0.383429\pi$
$654$	0	0
$655$	6.02777	0.235524
$656$	0	0
$657$	−38.5959	−1.50577
$658$	0	0
$659$	−35.8801	−1.39769	−0.698845	−	0.715273i	$-0.746302\pi$
$659$	−35.8801	−1.39769	−0.698845	+	0.715273i	$0.746302\pi$
$660$	0	0
$661$	−8.74360	−0.340087	−0.170043	−	0.985437i	$-0.554391\pi$
$661$	−8.74360	−0.340087	−0.170043	+	0.985437i	$0.554391\pi$
$662$	0	0
$663$	1.92027	0.0745769
$664$	0	0
$665$	−4.56326	−0.176956
$666$	0	0
$667$	−4.86440	−0.188350
$668$	0	0
$669$	13.3088	0.514547
$670$	0	0
$671$	12.2746	0.473857
$672$	0	0
$673$	−7.85960	−0.302965	−0.151483	−	0.988460i	$-0.548405\pi$
$673$	−7.85960	−0.302965	−0.151483	+	0.988460i	$0.548405\pi$
$674$	0	0
$675$	4.36962	0.168187
$676$	0	0
$677$	−31.4503	−1.20873	−0.604367	−	0.796706i	$-0.706574\pi$
$677$	−31.4503	−1.20873	−0.604367	+	0.796706i	$0.706574\pi$
$678$	0	0
$679$	−8.76702	−0.336447
$680$	0	0
$681$	1.39117	0.0533098
$682$	0	0
$683$	−1.45761	−0.0557739	−0.0278870	−	0.999611i	$-0.508878\pi$
$683$	−1.45761	−0.0557739	−0.0278870	+	0.999611i	$0.508878\pi$
$684$	0	0
$685$	−17.4450	−0.666538
$686$	0	0
$687$	10.9669	0.418414
$688$	0	0
$689$	9.59865	0.365680
$690$	0	0
$691$	−13.2771	−0.505086	−0.252543	−	0.967586i	$-0.581267\pi$
$691$	−13.2771	−0.505086	−0.252543	+	0.967586i	$0.581267\pi$
$692$	0	0
$693$	2.87657	0.109272
$694$	0	0
$695$	6.98612	0.264998
$696$	0	0
$697$	−24.4729	−0.926978
$698$	0	0
$699$	−21.6492	−0.818848
$700$	0	0
$701$	29.0712	1.09801	0.549003	−	0.835821i	$-0.315008\pi$
$701$	29.0712	1.09801	0.549003	+	0.835821i	$0.315008\pi$
$702$	0	0
$703$	0.256990	0.00969257
$704$	0	0
$705$	1.10374	0.0415693
$706$	0	0
$707$	3.66327	0.137772
$708$	0	0
$709$	37.9516	1.42530	0.712651	−	0.701519i	$-0.247494\pi$
$709$	37.9516	1.42530	0.712651	+	0.701519i	$0.247494\pi$
$710$	0	0
$711$	−31.6038	−1.18523
$712$	0	0
$713$	2.05211	0.0768520
$714$	0	0
$715$	1.23607	0.0462263
$716$	0	0
$717$	−15.3038	−0.571530
$718$	0	0
$719$	−25.5241	−0.951886	−0.475943	−	0.879476i	$-0.657893\pi$
$719$	−25.5241	−0.951886	−0.475943	+	0.879476i	$0.657893\pi$
$720$	0	0
$721$	−4.89499	−0.182299
$722$	0	0
$723$	15.0609	0.560119
$724$	0	0
$725$	3.42970	0.127376
$726$	0	0
$727$	−1.87716	−0.0696198	−0.0348099	−	0.999394i	$-0.511083\pi$
$727$	−1.87716	−0.0696198	−0.0348099	+	0.999394i	$0.511083\pi$
$728$	0	0
$729$	2.84616	0.105413
$730$	0	0
$731$	−23.7796	−0.879521
$732$	0	0
$733$	3.40272	0.125682	0.0628411	−	0.998024i	$-0.479984\pi$
$733$	3.40272	0.125682	0.0628411	+	0.998024i	$0.479984\pi$
$734$	0	0
$735$	0.820249	0.0302553
$736$	0	0
$737$	8.42159	0.310213
$738$	0	0
$739$	−17.2829	−0.635762	−0.317881	−	0.948131i	$-0.602971\pi$
$739$	−17.2829	−0.635762	−0.317881	+	0.948131i	$0.602971\pi$
$740$	0	0
$741$	−3.74301	−0.137503
$742$	0	0
$743$	9.04503	0.331830	0.165915	−	0.986140i	$-0.446942\pi$
$743$	9.04503	0.331830	0.165915	+	0.986140i	$0.446942\pi$
$744$	0	0
$745$	−22.7948	−0.835136
$746$	0	0
$747$	−11.2441	−0.411402
$748$	0	0
$749$	8.53984	0.312039
$750$	0	0
$751$	−46.8598	−1.70994	−0.854969	−	0.518679i	$-0.826424\pi$
$751$	−46.8598	−1.70994	−0.854969	+	0.518679i	$0.826424\pi$
$752$	0	0
$753$	15.1973	0.553821
$754$	0	0
$755$	17.4386	0.634654
$756$	0	0
$757$	−9.44917	−0.343436	−0.171718	−	0.985146i	$-0.554932\pi$
$757$	−9.44917	−0.343436	−0.171718	+	0.985146i	$0.554932\pi$
$758$	0	0
$759$	1.43801	0.0521963
$760$	0	0
$761$	7.94115	0.287867	0.143933	−	0.989587i	$-0.454025\pi$
$761$	7.94115	0.287867	0.143933	+	0.989587i	$0.454025\pi$
$762$	0	0
$763$	−9.85051	−0.356612
$764$	0	0
$765$	5.44813	0.196978
$766$	0	0
$767$	−10.6658	−0.385119
$768$	0	0
$769$	−24.6175	−0.887731	−0.443865	−	0.896093i	$-0.646393\pi$
$769$	−24.6175	−0.887731	−0.443865	+	0.896093i	$0.646393\pi$
$770$	0	0
$771$	−17.9395	−0.646074
$772$	0	0
$773$	16.4253	0.590777	0.295388	−	0.955377i	$-0.404551\pi$
$773$	16.4253	0.590777	0.295388	+	0.955377i	$0.404551\pi$
$774$	0	0
$775$	−1.44686	−0.0519729
$776$	0	0
$777$	−0.0461942	−0.00165721
$778$	0	0
$779$	47.7030	1.70914
$780$	0	0
$781$	−12.3302	−0.441208
$782$	0	0
$783$	14.9865	0.535574
$784$	0	0
$785$	−21.8095	−0.778413
$786$	0	0
$787$	38.9165	1.38722	0.693612	−	0.720349i	$-0.256018\pi$
$787$	38.9165	1.38722	0.693612	+	0.720349i	$0.256018\pi$
$788$	0	0
$789$	1.34309	0.0478152
$790$	0	0
$791$	4.28989	0.152531
$792$	0	0
$793$	−9.93039	−0.352638
$794$	0	0
$795$	−7.87329	−0.279237
$796$	0	0
$797$	−45.9626	−1.62808	−0.814039	−	0.580810i	$-0.802736\pi$
$797$	−45.9626	−1.62808	−0.814039	+	0.580810i	$0.802736\pi$
$798$	0	0
$799$	3.15019	0.111446
$800$	0	0
$801$	−0.293061	−0.0103548
$802$	0	0
$803$	20.4999	0.723426
$804$	0	0
$805$	−1.41831	−0.0499890
$806$	0	0
$807$	−23.0821	−0.812530
$808$	0	0
$809$	−33.9070	−1.19211	−0.596053	−	0.802945i	$-0.703265\pi$
$809$	−33.9070	−1.19211	−0.596053	+	0.802945i	$0.703265\pi$
$810$	0	0
$811$	29.1399	1.02324	0.511620	−	0.859212i	$-0.329046\pi$
$811$	29.1399	1.02324	0.511620	+	0.859212i	$0.329046\pi$
$812$	0	0
$813$	21.8087	0.764863
$814$	0	0
$815$	−7.87906	−0.275992
$816$	0	0
$817$	46.3516	1.62164
$818$	0	0
$819$	−2.32719	−0.0813186
$820$	0	0
$821$	−9.94446	−0.347064	−0.173532	−	0.984828i	$-0.555518\pi$
$821$	−9.94446	−0.347064	−0.173532	+	0.984828i	$0.555518\pi$
$822$	0	0
$823$	−49.4120	−1.72239	−0.861197	−	0.508272i	$-0.830284\pi$
$823$	−49.4120	−1.72239	−0.861197	+	0.508272i	$0.830284\pi$
$824$	0	0
$825$	−1.01388	−0.0352989
$826$	0	0
$827$	28.4961	0.990906	0.495453	−	0.868635i	$-0.335002\pi$
$827$	28.4961	0.990906	0.495453	+	0.868635i	$0.335002\pi$
$828$	0	0
$829$	38.0264	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$829$	38.0264	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$830$	0	0
$831$	19.4719	0.675474
$832$	0	0
$833$	2.34107	0.0811134
$834$	0	0
$835$	−18.7670	−0.649459
$836$	0	0
$837$	−6.32225	−0.218529
$838$	0	0
$839$	29.8233	1.02961	0.514807	−	0.857306i	$-0.327863\pi$
$839$	29.8233	1.02961	0.514807	+	0.857306i	$0.327863\pi$
$840$	0	0
$841$	−17.2371	−0.594384
$842$	0	0
$843$	3.77092	0.129877
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	9.47214	0.325466
$848$	0	0
$849$	3.10641	0.106612
$850$	0	0
$851$	0.0798755	0.00273810
$852$	0	0
$853$	−30.9318	−1.05908	−0.529542	−	0.848284i	$-0.677636\pi$
$853$	−30.9318	−1.05908	−0.529542	+	0.848284i	$0.677636\pi$
$854$	0	0
$855$	−10.6196	−0.363182
$856$	0	0
$857$	−13.1726	−0.449966	−0.224983	−	0.974363i	$-0.572233\pi$
$857$	−13.1726	−0.449966	−0.224983	+	0.974363i	$0.572233\pi$
$858$	0	0
$859$	−16.2746	−0.555283	−0.277642	−	0.960685i	$-0.589553\pi$
$859$	−16.2746	−0.555283	−0.277642	+	0.960685i	$0.589553\pi$
$860$	0	0
$861$	−8.57465	−0.292223
$862$	0	0
$863$	−7.31311	−0.248941	−0.124471	−	0.992223i	$-0.539723\pi$
$863$	−7.31311	−0.248941	−0.124471	+	0.992223i	$0.539723\pi$
$864$	0	0
$865$	−21.3879	−0.727209
$866$	0	0
$867$	9.44875	0.320896
$868$	0	0
$869$	16.7861	0.569429
$870$	0	0
$871$	−6.81321	−0.230857
$872$	0	0
$873$	−20.4025	−0.690521
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−25.9615	−0.876659	−0.438329	−	0.898814i	$-0.644430\pi$
$877$	−25.9615	−0.876659	−0.438329	+	0.898814i	$0.644430\pi$
$878$	0	0
$879$	−7.37086	−0.248613
$880$	0	0
$881$	0.529235	0.0178304	0.00891519	−	0.999960i	$-0.497162\pi$
$881$	0.529235	0.0178304	0.00891519	+	0.999960i	$0.497162\pi$
$882$	0	0
$883$	35.5446	1.19617	0.598086	−	0.801432i	$-0.295928\pi$
$883$	35.5446	1.19617	0.598086	+	0.801432i	$0.295928\pi$
$884$	0	0
$885$	8.74859	0.294081
$886$	0	0
$887$	18.4264	0.618699	0.309349	−	0.950948i	$-0.399889\pi$
$887$	18.4264	0.618699	0.309349	+	0.950948i	$0.399889\pi$
$888$	0	0
$889$	5.23607	0.175612
$890$	0	0
$891$	4.19941	0.140685
$892$	0	0
$893$	−6.14040	−0.205481
$894$	0	0
$895$	16.7714	0.560605
$896$	0	0
$897$	−1.16337	−0.0388438
$898$	0	0
$899$	−4.96231	−0.165502
$900$	0	0
$901$	−22.4712	−0.748623
$902$	0	0
$903$	−8.33173	−0.277263
$904$	0	0
$905$	−21.8935	−0.727766
$906$	0	0
$907$	9.22199	0.306211	0.153106	−	0.988210i	$-0.451073\pi$
$907$	9.22199	0.306211	0.153106	+	0.988210i	$0.451073\pi$
$908$	0	0
$909$	8.52514	0.282761
$910$	0	0
$911$	−56.2776	−1.86456	−0.932281	−	0.361736i	$-0.882184\pi$
$911$	−56.2776	−1.86456	−0.932281	+	0.361736i	$0.882184\pi$
$912$	0	0
$913$	5.97223	0.197652
$914$	0	0
$915$	8.14539	0.269278
$916$	0	0
$917$	6.02777	0.199054
$918$	0	0
$919$	−54.6916	−1.80411	−0.902055	−	0.431621i	$-0.857942\pi$
$919$	−54.6916	−1.80411	−0.902055	+	0.431621i	$0.857942\pi$
$920$	0	0
$921$	−26.7015	−0.879844
$922$	0	0
$923$	9.97532	0.328342
$924$	0	0
$925$	−0.0563172	−0.00185170
$926$	0	0
$927$	−11.3916	−0.374149
$928$	0	0
$929$	−31.3217	−1.02763	−0.513815	−	0.857901i	$-0.671768\pi$
$929$	−31.3217	−1.02763	−0.513815	+	0.857901i	$0.671768\pi$
$930$	0	0
$931$	−4.56326	−0.149555
$932$	0	0
$933$	−15.7113	−0.514364
$934$	0	0
$935$	−2.89373	−0.0946350
$936$	0	0
$937$	−10.6032	−0.346391	−0.173196	−	0.984887i	$-0.555409\pi$
$937$	−10.6032	−0.346391	−0.173196	+	0.984887i	$0.555409\pi$
$938$	0	0
$939$	−16.3170	−0.532486
$940$	0	0
$941$	31.8854	1.03943	0.519717	−	0.854339i	$-0.326037\pi$
$941$	31.8854	1.03943	0.519717	+	0.854339i	$0.326037\pi$
$942$	0	0
$943$	14.8266	0.482822
$944$	0	0
$945$	4.36962	0.142144
$946$	0	0
$947$	52.8384	1.71702	0.858509	−	0.512798i	$-0.171391\pi$
$947$	52.8384	1.71702	0.858509	+	0.512798i	$0.171391\pi$
$948$	0	0
$949$	−16.5848	−0.538364
$950$	0	0
$951$	−3.92169	−0.127169
$952$	0	0
$953$	−42.6390	−1.38121	−0.690606	−	0.723232i	$-0.742656\pi$
$953$	−42.6390	−1.38121	−0.690606	+	0.723232i	$0.742656\pi$
$954$	0	0
$955$	−10.6133	−0.343439
$956$	0	0
$957$	−3.47732	−0.112406
$958$	0	0
$959$	−17.4450	−0.563327
$960$	0	0
$961$	−28.9066	−0.932470
$962$	0	0
$963$	19.8738	0.640426
$964$	0	0
$965$	−9.62207	−0.309745
$966$	0	0
$967$	15.2627	0.490816	0.245408	−	0.969420i	$-0.421078\pi$
$967$	15.2627	0.490816	0.245408	+	0.969420i	$0.421078\pi$
$968$	0	0
$969$	8.76267	0.281497
$970$	0	0
$971$	−12.9356	−0.415122	−0.207561	−	0.978222i	$-0.566553\pi$
$971$	−12.9356	−0.415122	−0.207561	+	0.978222i	$0.566553\pi$
$972$	0	0
$973$	6.98612	0.223965
$974$	0	0
$975$	0.820249	0.0262690
$976$	0	0
$977$	−7.97131	−0.255025	−0.127512	−	0.991837i	$-0.540699\pi$
$977$	−7.97131	−0.255025	−0.127512	+	0.991837i	$0.540699\pi$
$978$	0	0
$979$	0.155657	0.00497482
$980$	0	0
$981$	−22.9240	−0.731908
$982$	0	0
$983$	−1.16494	−0.0371557	−0.0185778	−	0.999827i	$-0.505914\pi$
$983$	−1.16494	−0.0371557	−0.0185778	+	0.999827i	$0.505914\pi$
$984$	0	0
$985$	−10.5285	−0.335464
$986$	0	0
$987$	1.10374	0.0351325
$988$	0	0
$989$	14.4066	0.458104
$990$	0	0
$991$	−30.5925	−0.971804	−0.485902	−	0.874013i	$-0.661509\pi$
$991$	−30.5925	−0.971804	−0.485902	+	0.874013i	$0.661509\pi$
$992$	0	0
$993$	−16.1520	−0.512569
$994$	0	0
$995$	23.5618	0.746959
$996$	0	0
$997$	12.5486	0.397417	0.198708	−	0.980059i	$-0.436325\pi$
$997$	12.5486	0.397417	0.198708	+	0.980059i	$0.436325\pi$
$998$	0	0
$999$	−0.246085	−0.00778579

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.v.1.2	✓	4
4.3	odd	2		7280.2.a.bt.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.v.1.2	✓	4	1.1	even	1	trivial
7280.2.a.bt.1.3		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)

\(f(q)\)	\(=\)	\(q-0.820249 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.32719 q^{9} +O(q^{10})\)	\(q-0.820249 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.32719 q^{9} +1.23607 q^{11} -1.00000 q^{13} +0.820249 q^{15} +2.34107 q^{17} -4.56326 q^{19} +0.820249 q^{21} -1.41831 q^{23} +1.00000 q^{25} +4.36962 q^{27} +3.42970 q^{29} -1.44686 q^{31} -1.01388 q^{33} +1.00000 q^{35} -0.0563172 q^{37} +0.820249 q^{39} -10.4537 q^{41} -10.1576 q^{43} +2.32719 q^{45} +1.34562 q^{47} +1.00000 q^{49} -1.92027 q^{51} -9.59865 q^{53} -1.23607 q^{55} +3.74301 q^{57} +10.6658 q^{59} +9.93039 q^{61} +2.32719 q^{63} +1.00000 q^{65} +6.81321 q^{67} +1.16337 q^{69} -9.97532 q^{71} +16.5848 q^{73} -0.820249 q^{75} -1.23607 q^{77} +13.5802 q^{79} +3.39739 q^{81} +4.83164 q^{83} -2.34107 q^{85} -2.81321 q^{87} +0.125929 q^{89} +1.00000 q^{91} +1.18679 q^{93} +4.56326 q^{95} +8.76702 q^{97} -2.87657 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} - 4 q^{5} - 4 q^{7} - q^{9}+O(q^{10}) \) 4 * q + q^3 - 4 * q^5 - 4 * q^7 - q^9	\( 4 q + q^{3} - 4 q^{5} - 4 q^{7} - q^{9} - 4 q^{11} - 4 q^{13} - q^{15} - 7 q^{17} - q^{19} - q^{21} + 2 q^{23} + 4 q^{25} + 4 q^{27} + q^{29} - q^{31} + 4 q^{33} + 4 q^{35} + 13 q^{37} - q^{39} + q^{41} - 6 q^{43} + q^{45} + 22 q^{47} + 4 q^{49} - 2 q^{51} + 14 q^{53} + 4 q^{55} + 2 q^{57} + 21 q^{59} + 12 q^{61} + q^{63} + 4 q^{65} - 7 q^{67} + 20 q^{69} - 4 q^{71} + 22 q^{73} + q^{75} + 4 q^{77} - 23 q^{79} - 16 q^{81} + 10 q^{83} + 7 q^{85} + 23 q^{87} + 15 q^{89} + 4 q^{91} + 39 q^{93} + q^{95} - 8 q^{97} + 6 q^{99}+O(q^{100}) \) 4 * q + q^3 - 4 * q^5 - 4 * q^7 - q^9 - 4 * q^11 - 4 * q^13 - q^15 - 7 * q^17 - q^19 - q^21 + 2 * q^23 + 4 * q^25 + 4 * q^27 + q^29 - q^31 + 4 * q^33 + 4 * q^35 + 13 * q^37 - q^39 + q^41 - 6 * q^43 + q^45 + 22 * q^47 + 4 * q^49 - 2 * q^51 + 14 * q^53 + 4 * q^55 + 2 * q^57 + 21 * q^59 + 12 * q^61 + q^63 + 4 * q^65 - 7 * q^67 + 20 * q^69 - 4 * q^71 + 22 * q^73 + q^75 + 4 * q^77 - 23 * q^79 - 16 * q^81 + 10 * q^83 + 7 * q^85 + 23 * q^87 + 15 * q^89 + 4 * q^91 + 39 * q^93 + q^95 - 8 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

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