LMFDB - Embedded newform 3640.2.a.z.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:	$5$
Coefficient field:	5.5.2112217.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 10x^{3} - x^{2} + 10x + 4 $ x^5 - 10x^3 - x^2 + 10x + 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			1.4
Root			$-0.769609$ 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)

$f(q)$	$=$	$q+0.769609 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.40770 q^{9} +O(q^{10})$	$q+0.769609 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.40770 q^{9} -1.48346 q^{11} -1.00000 q^{13} +0.769609 q^{15} +3.27320 q^{17} +4.93143 q^{19} +0.769609 q^{21} +5.02268 q^{23} +1.00000 q^{25} -4.16182 q^{27} -3.79229 q^{29} +2.76961 q^{31} -1.14169 q^{33} +1.00000 q^{35} -0.427834 q^{37} -0.769609 q^{39} +1.86848 q^{41} -2.88818 q^{43} -2.40770 q^{45} -0.286018 q^{47} +1.00000 q^{49} +2.51909 q^{51} +12.5061 q^{53} -1.48346 q^{55} +3.79527 q^{57} +1.05563 q^{59} +10.2201 q^{61} -2.40770 q^{63} -1.00000 q^{65} +11.6448 q^{67} +3.86550 q^{69} -12.7108 q^{71} -12.7665 q^{73} +0.769609 q^{75} -1.48346 q^{77} -3.55922 q^{79} +4.02013 q^{81} +17.8679 q^{83} +3.27320 q^{85} -2.91858 q^{87} +9.52925 q^{89} -1.00000 q^{91} +2.13152 q^{93} +4.93143 q^{95} +19.2324 q^{97} +3.57173 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^5 + 5 * q^7 + 5 * q^9	$ 5 q + 5 q^{5} + 5 q^{7} + 5 q^{9} + 7 q^{11} - 5 q^{13} + 3 q^{17} + 3 q^{19} + 3 q^{23} + 5 q^{25} - 3 q^{27} + 7 q^{29} + 10 q^{31} + 17 q^{33} + 5 q^{35} + 10 q^{37} + 4 q^{43} + 5 q^{45} - 3 q^{47} + 5 q^{49} + 26 q^{53} + 7 q^{55} - 20 q^{57} + 3 q^{59} + 13 q^{61} + 5 q^{63} - 5 q^{65} + 12 q^{67} + 23 q^{69} + 8 q^{71} + q^{73} + 7 q^{77} - 6 q^{79} + 25 q^{81} - 8 q^{83} + 3 q^{85} - 43 q^{87} + 3 q^{89} - 5 q^{91} + 20 q^{93} + 3 q^{95} - 15 q^{97} + 16 q^{99}+O(q^{100}) $ 5 * q + 5 * q^5 + 5 * q^7 + 5 * q^9 + 7 * q^11 - 5 * q^13 + 3 * q^17 + 3 * q^19 + 3 * q^23 + 5 * q^25 - 3 * q^27 + 7 * q^29 + 10 * q^31 + 17 * q^33 + 5 * q^35 + 10 * q^37 + 4 * q^43 + 5 * q^45 - 3 * q^47 + 5 * q^49 + 26 * q^53 + 7 * q^55 - 20 * q^57 + 3 * q^59 + 13 * q^61 + 5 * q^63 - 5 * q^65 + 12 * q^67 + 23 * q^69 + 8 * q^71 + q^73 + 7 * q^77 - 6 * q^79 + 25 * q^81 - 8 * q^83 + 3 * q^85 - 43 * q^87 + 3 * q^89 - 5 * q^91 + 20 * q^93 + 3 * q^95 - 15 * q^97 + 16 * 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
$3$	0.769609	0.444334	0.222167	−	0.975009i	$-0.428687\pi$
$3$	0.769609	0.444334	0.222167	+	0.975009i	$0.428687\pi$
$4$	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
$9$	−2.40770	−0.802567
$10$	0	0
$11$	−1.48346	−0.447280	−0.223640	−	0.974672i	$-0.571794\pi$
$11$	−1.48346	−0.447280	−0.223640	+	0.974672i	$0.571794\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0.769609	0.198712
$16$	0	0
$17$	3.27320	0.793868	0.396934	−	0.917847i	$-0.370074\pi$
$17$	3.27320	0.793868	0.396934	+	0.917847i	$0.370074\pi$
$18$	0	0
$19$	4.93143	1.13135	0.565673	−	0.824629i	$-0.308616\pi$
$19$	4.93143	1.13135	0.565673	+	0.824629i	$0.308616\pi$
$20$	0	0
$21$	0.769609	0.167943
$22$	0	0
$23$	5.02268	1.04730	0.523651	−	0.851933i	$-0.324570\pi$
$23$	5.02268	1.04730	0.523651	+	0.851933i	$0.324570\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.16182	−0.800942
$28$	0	0
$29$	−3.79229	−0.704210	−0.352105	−	0.935960i	$-0.614534\pi$
$29$	−3.79229	−0.704210	−0.352105	+	0.935960i	$0.614534\pi$
$30$	0	0
$31$	2.76961	0.497437	0.248718	−	0.968576i	$-0.419991\pi$
$31$	2.76961	0.497437	0.248718	+	0.968576i	$0.419991\pi$
$32$	0	0
$33$	−1.14169	−0.198742
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−0.427834	−0.0703354	−0.0351677	−	0.999381i	$-0.511197\pi$
$37$	−0.427834	−0.0703354	−0.0351677	+	0.999381i	$0.511197\pi$
$38$	0	0
$39$	−0.769609	−0.123236
$40$	0	0
$41$	1.86848	0.291808	0.145904	−	0.989299i	$-0.453391\pi$
$41$	1.86848	0.291808	0.145904	+	0.989299i	$0.453391\pi$
$42$	0	0
$43$	−2.88818	−0.440443	−0.220222	−	0.975450i	$-0.570678\pi$
$43$	−2.88818	−0.440443	−0.220222	+	0.975450i	$0.570678\pi$
$44$	0	0
$45$	−2.40770	−0.358919
$46$	0	0
$47$	−0.286018	−0.0417200	−0.0208600	−	0.999782i	$-0.506640\pi$
$47$	−0.286018	−0.0417200	−0.0208600	+	0.999782i	$0.506640\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.51909	0.352743
$52$	0	0
$53$	12.5061	1.71785	0.858925	−	0.512101i	$-0.171133\pi$
$53$	12.5061	1.71785	0.858925	+	0.512101i	$0.171133\pi$
$54$	0	0
$55$	−1.48346	−0.200030
$56$	0	0
$57$	3.79527	0.502696
$58$	0	0
$59$	1.05563	0.137431	0.0687155	−	0.997636i	$-0.478110\pi$
$59$	1.05563	0.137431	0.0687155	+	0.997636i	$0.478110\pi$
$60$	0	0
$61$	10.2201	1.30855	0.654276	−	0.756256i	$-0.272973\pi$
$61$	10.2201	1.30855	0.654276	+	0.756256i	$0.272973\pi$
$62$	0	0
$63$	−2.40770	−0.303342
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	11.6448	1.42264	0.711322	−	0.702866i	$-0.248097\pi$
$67$	11.6448	1.42264	0.711322	+	0.702866i	$0.248097\pi$
$68$	0	0
$69$	3.86550	0.465352
$70$	0	0
$71$	−12.7108	−1.50849	−0.754245	−	0.656593i	$-0.771997\pi$
$71$	−12.7108	−1.50849	−0.754245	+	0.656593i	$0.771997\pi$
$72$	0	0
$73$	−12.7665	−1.49421	−0.747104	−	0.664707i	$-0.768557\pi$
$73$	−12.7665	−1.49421	−0.747104	+	0.664707i	$0.768557\pi$
$74$	0	0
$75$	0.769609	0.0888668
$76$	0	0
$77$	−1.48346	−0.169056
$78$	0	0
$79$	−3.55922	−0.400444	−0.200222	−	0.979751i	$-0.564166\pi$
$79$	−3.55922	−0.400444	−0.200222	+	0.979751i	$0.564166\pi$
$80$	0	0
$81$	4.02013	0.446681
$82$	0	0
$83$	17.8679	1.96126	0.980631	−	0.195864i	$-0.0627512\pi$
$83$	17.8679	1.96126	0.980631	+	0.195864i	$0.0627512\pi$
$84$	0	0
$85$	3.27320	0.355029
$86$	0	0
$87$	−2.91858	−0.312905
$88$	0	0
$89$	9.52925	1.01010	0.505049	−	0.863090i	$-0.331474\pi$
$89$	9.52925	1.01010	0.505049	+	0.863090i	$0.331474\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	2.13152	0.221028
$94$	0	0
$95$	4.93143	0.505954
$96$	0	0
$97$	19.2324	1.95275	0.976377	−	0.216073i	$-0.0693248\pi$
$97$	19.2324	1.95275	0.976377	+	0.216073i	$0.0693248\pi$
$98$	0	0
$99$	3.57173	0.358972
$100$	0	0
$101$	12.1489	1.20886	0.604429	−	0.796659i	$-0.293401\pi$
$101$	12.1489	1.20886	0.604429	+	0.796659i	$0.293401\pi$
$102$	0	0
$103$	9.74683	0.960384	0.480192	−	0.877164i	$-0.340567\pi$
$103$	9.74683	0.960384	0.480192	+	0.877164i	$0.340567\pi$
$104$	0	0
$105$	0.769609	0.0751062
$106$	0	0
$107$	2.20463	0.213129	0.106565	−	0.994306i	$-0.466015\pi$
$107$	2.20463	0.213129	0.106565	+	0.994306i	$0.466015\pi$
$108$	0	0
$109$	−5.36181	−0.513568	−0.256784	−	0.966469i	$-0.582663\pi$
$109$	−5.36181	−0.513568	−0.256784	+	0.966469i	$0.582663\pi$
$110$	0	0
$111$	−0.329265	−0.0312524
$112$	0	0
$113$	11.9974	1.12862	0.564308	−	0.825564i	$-0.309143\pi$
$113$	11.9974	1.12862	0.564308	+	0.825564i	$0.309143\pi$
$114$	0	0
$115$	5.02268	0.468367
$116$	0	0
$117$	2.40770	0.222592
$118$	0	0
$119$	3.27320	0.300054
$120$	0	0
$121$	−8.79934	−0.799940
$122$	0	0
$123$	1.43800	0.129660
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−11.8381	−1.05046	−0.525230	−	0.850960i	$-0.676021\pi$
$127$	−11.8381	−1.05046	−0.525230	+	0.850960i	$0.676021\pi$
$128$	0	0
$129$	−2.22277	−0.195704
$130$	0	0
$131$	−9.09281	−0.794442	−0.397221	−	0.917723i	$-0.630025\pi$
$131$	−9.09281	−0.794442	−0.397221	+	0.917723i	$0.630025\pi$
$132$	0	0
$133$	4.93143	0.427609
$134$	0	0
$135$	−4.16182	−0.358192
$136$	0	0
$137$	−14.1613	−1.20988	−0.604940	−	0.796271i	$-0.706803\pi$
$137$	−14.1613	−1.20988	−0.604940	+	0.796271i	$0.706803\pi$
$138$	0	0
$139$	−2.04026	−0.173053	−0.0865265	−	0.996250i	$-0.527577\pi$
$139$	−2.04026	−0.173053	−0.0865265	+	0.996250i	$0.527577\pi$
$140$	0	0
$141$	−0.220122	−0.0185376
$142$	0	0
$143$	1.48346	0.124053
$144$	0	0
$145$	−3.79229	−0.314932
$146$	0	0
$147$	0.769609	0.0634763
$148$	0	0
$149$	−10.1799	−0.833966	−0.416983	−	0.908914i	$-0.636913\pi$
$149$	−10.1799	−0.833966	−0.416983	+	0.908914i	$0.636913\pi$
$150$	0	0
$151$	9.89537	0.805273	0.402637	−	0.915360i	$-0.368094\pi$
$151$	9.89537	0.805273	0.402637	+	0.915360i	$0.368094\pi$
$152$	0	0
$153$	−7.88089	−0.637133
$154$	0	0
$155$	2.76961	0.222460
$156$	0	0
$157$	−9.44011	−0.753403	−0.376702	−	0.926335i	$-0.622942\pi$
$157$	−9.44011	−0.753403	−0.376702	+	0.926335i	$0.622942\pi$
$158$	0	0
$159$	9.62484	0.763300
$160$	0	0
$161$	5.02268	0.395843
$162$	0	0
$163$	−18.1210	−1.41935	−0.709674	−	0.704530i	$-0.751158\pi$
$163$	−18.1210	−1.41935	−0.709674	+	0.704530i	$0.751158\pi$
$164$	0	0
$165$	−1.14169	−0.0888801
$166$	0	0
$167$	−6.19775	−0.479596	−0.239798	−	0.970823i	$-0.577081\pi$
$167$	−6.19775	−0.479596	−0.239798	+	0.970823i	$0.577081\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−11.8734	−0.907982
$172$	0	0
$173$	1.24588	0.0947228	0.0473614	−	0.998878i	$-0.484919\pi$
$173$	1.24588	0.0947228	0.0473614	+	0.998878i	$0.484919\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	0.812421	0.0610653
$178$	0	0
$179$	−22.3463	−1.67024	−0.835122	−	0.550065i	$-0.814603\pi$
$179$	−22.3463	−1.67024	−0.835122	+	0.550065i	$0.814603\pi$
$180$	0	0
$181$	17.9571	1.33474	0.667370	−	0.744726i	$-0.267420\pi$
$181$	17.9571	1.33474	0.667370	+	0.744726i	$0.267420\pi$
$182$	0	0
$183$	7.86550	0.581435
$184$	0	0
$185$	−0.427834	−0.0314549
$186$	0	0
$187$	−4.85567	−0.355082
$188$	0	0
$189$	−4.16182	−0.302728
$190$	0	0
$191$	5.94381	0.430079	0.215039	−	0.976605i	$-0.431012\pi$
$191$	5.94381	0.430079	0.215039	+	0.976605i	$0.431012\pi$
$192$	0	0
$193$	17.3648	1.24994	0.624972	−	0.780647i	$-0.285110\pi$
$193$	17.3648	1.24994	0.624972	+	0.780647i	$0.285110\pi$
$194$	0	0
$195$	−0.769609	−0.0551129
$196$	0	0
$197$	27.7235	1.97522	0.987608	−	0.156939i	$-0.0501627\pi$
$197$	27.7235	1.97522	0.987608	+	0.156939i	$0.0501627\pi$
$198$	0	0
$199$	16.0597	1.13844	0.569222	−	0.822184i	$-0.307244\pi$
$199$	16.0597	1.13844	0.569222	+	0.822184i	$0.307244\pi$
$200$	0	0
$201$	8.96198	0.632129
$202$	0	0
$203$	−3.79229	−0.266166
$204$	0	0
$205$	1.86848	0.130501
$206$	0	0
$207$	−12.0931	−0.840529
$208$	0	0
$209$	−7.31558	−0.506029
$210$	0	0
$211$	1.49684	0.103047	0.0515234	−	0.998672i	$-0.483592\pi$
$211$	1.49684	0.103047	0.0515234	+	0.998672i	$0.483592\pi$
$212$	0	0
$213$	−9.78232	−0.670274
$214$	0	0
$215$	−2.88818	−0.196972
$216$	0	0
$217$	2.76961	0.188013
$218$	0	0
$219$	−9.82524	−0.663928
$220$	0	0
$221$	−3.27320	−0.220179
$222$	0	0
$223$	−23.9562	−1.60423	−0.802113	−	0.597172i	$-0.796291\pi$
$223$	−23.9562	−1.60423	−0.802113	+	0.597172i	$0.796291\pi$
$224$	0	0
$225$	−2.40770	−0.160513
$226$	0	0
$227$	−16.1566	−1.07235	−0.536176	−	0.844106i	$-0.680132\pi$
$227$	−16.1566	−1.07235	−0.536176	+	0.844106i	$0.680132\pi$
$228$	0	0
$229$	−27.4248	−1.81228	−0.906140	−	0.422977i	$-0.860985\pi$
$229$	−27.4248	−1.81228	−0.906140	+	0.422977i	$0.860985\pi$
$230$	0	0
$231$	−1.14169	−0.0751174
$232$	0	0
$233$	7.97676	0.522575	0.261287	−	0.965261i	$-0.415853\pi$
$233$	7.97676	0.522575	0.261287	+	0.965261i	$0.415853\pi$
$234$	0	0
$235$	−0.286018	−0.0186578
$236$	0	0
$237$	−2.73921	−0.177931
$238$	0	0
$239$	11.4355	0.739698	0.369849	−	0.929092i	$-0.379409\pi$
$239$	11.4355	0.739698	0.369849	+	0.929092i	$0.379409\pi$
$240$	0	0
$241$	8.39190	0.540570	0.270285	−	0.962780i	$-0.412882\pi$
$241$	8.39190	0.540570	0.270285	+	0.962780i	$0.412882\pi$
$242$	0	0
$243$	15.5794	0.999418
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−4.93143	−0.313779
$248$	0	0
$249$	13.7513	0.871456
$250$	0	0
$251$	17.2273	1.08738	0.543689	−	0.839287i	$-0.317027\pi$
$251$	17.2273	1.08738	0.543689	+	0.839287i	$0.317027\pi$
$252$	0	0
$253$	−7.45095	−0.468437
$254$	0	0
$255$	2.51909	0.157751
$256$	0	0
$257$	22.2115	1.38552	0.692758	−	0.721170i	$-0.256395\pi$
$257$	22.2115	1.38552	0.692758	+	0.721170i	$0.256395\pi$
$258$	0	0
$259$	−0.427834	−0.0265843
$260$	0	0
$261$	9.13070	0.565176
$262$	0	0
$263$	−1.34922	−0.0831966	−0.0415983	−	0.999134i	$-0.513245\pi$
$263$	−1.34922	−0.0831966	−0.0415983	+	0.999134i	$0.513245\pi$
$264$	0	0
$265$	12.5061	0.768246
$266$	0	0
$267$	7.33380	0.448821
$268$	0	0
$269$	−9.79960	−0.597492	−0.298746	−	0.954333i	$-0.596568\pi$
$269$	−9.79960	−0.597492	−0.298746	+	0.954333i	$0.596568\pi$
$270$	0	0
$271$	14.9968	0.910990	0.455495	−	0.890238i	$-0.349462\pi$
$271$	14.9968	0.910990	0.455495	+	0.890238i	$0.349462\pi$
$272$	0	0
$273$	−0.769609	−0.0465789
$274$	0	0
$275$	−1.48346	−0.0894561
$276$	0	0
$277$	1.06407	0.0639335	0.0319668	−	0.999489i	$-0.489823\pi$
$277$	1.06407	0.0639335	0.0319668	+	0.999489i	$0.489823\pi$
$278$	0	0
$279$	−6.66839	−0.399226
$280$	0	0
$281$	1.80286	0.107550	0.0537749	−	0.998553i	$-0.482875\pi$
$281$	1.80286	0.107550	0.0537749	+	0.998553i	$0.482875\pi$
$282$	0	0
$283$	−20.0785	−1.19354	−0.596770	−	0.802412i	$-0.703550\pi$
$283$	−20.0785	−1.19354	−0.596770	+	0.802412i	$0.703550\pi$
$284$	0	0
$285$	3.79527	0.224812
$286$	0	0
$287$	1.86848	0.110293
$288$	0	0
$289$	−6.28615	−0.369773
$290$	0	0
$291$	14.8014	0.867675
$292$	0	0
$293$	10.1662	0.593913	0.296956	−	0.954891i	$-0.404028\pi$
$293$	10.1662	0.593913	0.296956	+	0.954891i	$0.404028\pi$
$294$	0	0
$295$	1.05563	0.0614610
$296$	0	0
$297$	6.17389	0.358246
$298$	0	0
$299$	−5.02268	−0.290469
$300$	0	0
$301$	−2.88818	−0.166472
$302$	0	0
$303$	9.34988	0.537137
$304$	0	0
$305$	10.2201	0.585202
$306$	0	0
$307$	−1.07308	−0.0612440	−0.0306220	−	0.999531i	$-0.509749\pi$
$307$	−1.07308	−0.0612440	−0.0306220	+	0.999531i	$0.509749\pi$
$308$	0	0
$309$	7.50125	0.426731
$310$	0	0
$311$	8.08649	0.458543	0.229271	−	0.973363i	$-0.426366\pi$
$311$	8.08649	0.458543	0.229271	+	0.973363i	$0.426366\pi$
$312$	0	0
$313$	11.1541	0.630467	0.315233	−	0.949014i	$-0.397917\pi$
$313$	11.1541	0.630467	0.315233	+	0.949014i	$0.397917\pi$
$314$	0	0
$315$	−2.40770	−0.135659
$316$	0	0
$317$	21.5417	1.20990	0.604950	−	0.796263i	$-0.293193\pi$
$317$	21.5417	1.20990	0.604950	+	0.796263i	$0.293193\pi$
$318$	0	0
$319$	5.62571	0.314979
$320$	0	0
$321$	1.69670	0.0947007
$322$	0	0
$323$	16.1416	0.898140
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−4.12650	−0.228196
$328$	0	0
$329$	−0.286018	−0.0157687
$330$	0	0
$331$	22.8134	1.25394	0.626969	−	0.779044i	$-0.284295\pi$
$331$	22.8134	1.25394	0.626969	+	0.779044i	$0.284295\pi$
$332$	0	0
$333$	1.03010	0.0564489
$334$	0	0
$335$	11.6448	0.636226
$336$	0	0
$337$	−17.2727	−0.940902	−0.470451	−	0.882426i	$-0.655909\pi$
$337$	−17.2727	−0.940902	−0.470451	+	0.882426i	$0.655909\pi$
$338$	0	0
$339$	9.23327	0.501483
$340$	0	0
$341$	−4.10861	−0.222494
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	3.86550	0.208112
$346$	0	0
$347$	−25.3747	−1.36218	−0.681091	−	0.732198i	$-0.738494\pi$
$347$	−25.3747	−1.36218	−0.681091	+	0.732198i	$0.738494\pi$
$348$	0	0
$349$	34.0098	1.82050	0.910250	−	0.414058i	$-0.135889\pi$
$349$	34.0098	1.82050	0.910250	+	0.414058i	$0.135889\pi$
$350$	0	0
$351$	4.16182	0.222141
$352$	0	0
$353$	−14.8933	−0.792689	−0.396345	−	0.918102i	$-0.629721\pi$
$353$	−14.8933	−0.792689	−0.396345	+	0.918102i	$0.629721\pi$
$354$	0	0
$355$	−12.7108	−0.674618
$356$	0	0
$357$	2.51909	0.133324
$358$	0	0
$359$	−18.2044	−0.960790	−0.480395	−	0.877052i	$-0.659507\pi$
$359$	−18.2044	−0.960790	−0.480395	+	0.877052i	$0.659507\pi$
$360$	0	0
$361$	5.31897	0.279946
$362$	0	0
$363$	−6.77206	−0.355441
$364$	0	0
$365$	−12.7665	−0.668231
$366$	0	0
$367$	−30.2903	−1.58114	−0.790569	−	0.612373i	$-0.790215\pi$
$367$	−30.2903	−1.58114	−0.790569	+	0.612373i	$0.790215\pi$
$368$	0	0
$369$	−4.49875	−0.234196
$370$	0	0
$371$	12.5061	0.649286
$372$	0	0
$373$	−25.7815	−1.33491	−0.667457	−	0.744649i	$-0.732617\pi$
$373$	−25.7815	−1.33491	−0.667457	+	0.744649i	$0.732617\pi$
$374$	0	0
$375$	0.769609	0.0397425
$376$	0	0
$377$	3.79229	0.195313
$378$	0	0
$379$	−18.4069	−0.945497	−0.472748	−	0.881197i	$-0.656738\pi$
$379$	−18.4069	−0.945497	−0.472748	+	0.881197i	$0.656738\pi$
$380$	0	0
$381$	−9.11070	−0.466755
$382$	0	0
$383$	−35.7684	−1.82768	−0.913840	−	0.406074i	$-0.866898\pi$
$383$	−35.7684	−1.82768	−0.913840	+	0.406074i	$0.866898\pi$
$384$	0	0
$385$	−1.48346	−0.0756042
$386$	0	0
$387$	6.95388	0.353485
$388$	0	0
$389$	25.7134	1.30372	0.651862	−	0.758338i	$-0.273988\pi$
$389$	25.7134	1.30372	0.651862	+	0.758338i	$0.273988\pi$
$390$	0	0
$391$	16.4402	0.831419
$392$	0	0
$393$	−6.99791	−0.352998
$394$	0	0
$395$	−3.55922	−0.179084
$396$	0	0
$397$	13.9031	0.697777	0.348889	−	0.937164i	$-0.386559\pi$
$397$	13.9031	0.697777	0.348889	+	0.937164i	$0.386559\pi$
$398$	0	0
$399$	3.79527	0.190001
$400$	0	0
$401$	−12.2575	−0.612109	−0.306055	−	0.952014i	$-0.599009\pi$
$401$	−12.2575	−0.612109	−0.306055	+	0.952014i	$0.599009\pi$
$402$	0	0
$403$	−2.76961	−0.137964
$404$	0	0
$405$	4.02013	0.199762
$406$	0	0
$407$	0.634674	0.0314596
$408$	0	0
$409$	−4.59793	−0.227353	−0.113676	−	0.993518i	$-0.536263\pi$
$409$	−4.59793	−0.227353	−0.113676	+	0.993518i	$0.536263\pi$
$410$	0	0
$411$	−10.8987	−0.537591
$412$	0	0
$413$	1.05563	0.0519440
$414$	0	0
$415$	17.8679	0.877103
$416$	0	0
$417$	−1.57021	−0.0768933
$418$	0	0
$419$	−3.62749	−0.177214	−0.0886072	−	0.996067i	$-0.528242\pi$
$419$	−3.62749	−0.177214	−0.0886072	+	0.996067i	$0.528242\pi$
$420$	0	0
$421$	12.1396	0.591648	0.295824	−	0.955243i	$-0.404406\pi$
$421$	12.1396	0.591648	0.295824	+	0.955243i	$0.404406\pi$
$422$	0	0
$423$	0.688646	0.0334831
$424$	0	0
$425$	3.27320	0.158774
$426$	0	0
$427$	10.2201	0.494586
$428$	0	0
$429$	1.14169	0.0551211
$430$	0	0
$431$	−13.4140	−0.646132	−0.323066	−	0.946376i	$-0.604714\pi$
$431$	−13.4140	−0.646132	−0.323066	+	0.946376i	$0.604714\pi$
$432$	0	0
$433$	−8.28126	−0.397972	−0.198986	−	0.980002i	$-0.563765\pi$
$433$	−8.28126	−0.397972	−0.198986	+	0.980002i	$0.563765\pi$
$434$	0	0
$435$	−2.91858	−0.139935
$436$	0	0
$437$	24.7690	1.18486
$438$	0	0
$439$	−35.2548	−1.68262	−0.841309	−	0.540554i	$-0.818215\pi$
$439$	−35.2548	−1.68262	−0.841309	+	0.540554i	$0.818215\pi$
$440$	0	0
$441$	−2.40770	−0.114652
$442$	0	0
$443$	−13.1966	−0.626988	−0.313494	−	0.949590i	$-0.601500\pi$
$443$	−13.1966	−0.626988	−0.313494	+	0.949590i	$0.601500\pi$
$444$	0	0
$445$	9.52925	0.451730
$446$	0	0
$447$	−7.83451	−0.370560
$448$	0	0
$449$	−5.60517	−0.264525	−0.132262	−	0.991215i	$-0.542224\pi$
$449$	−5.60517	−0.264525	−0.132262	+	0.991215i	$0.542224\pi$
$450$	0	0
$451$	−2.77182	−0.130520
$452$	0	0
$453$	7.61556	0.357810
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	−37.2397	−1.74200	−0.871000	−	0.491283i	$-0.836528\pi$
$457$	−37.2397	−1.74200	−0.871000	+	0.491283i	$0.836528\pi$
$458$	0	0
$459$	−13.6225	−0.635842
$460$	0	0
$461$	−7.58389	−0.353217	−0.176608	−	0.984281i	$-0.556513\pi$
$461$	−7.58389	−0.353217	−0.176608	+	0.984281i	$0.556513\pi$
$462$	0	0
$463$	40.2483	1.87050	0.935249	−	0.353990i	$-0.115175\pi$
$463$	40.2483	1.87050	0.935249	+	0.353990i	$0.115175\pi$
$464$	0	0
$465$	2.13152	0.0988467
$466$	0	0
$467$	32.5817	1.50770	0.753851	−	0.657046i	$-0.228194\pi$
$467$	32.5817	1.50770	0.753851	+	0.657046i	$0.228194\pi$
$468$	0	0
$469$	11.6448	0.537709
$470$	0	0
$471$	−7.26520	−0.334763
$472$	0	0
$473$	4.28450	0.197002
$474$	0	0
$475$	4.93143	0.226269
$476$	0	0
$477$	−30.1111	−1.37869
$478$	0	0
$479$	8.94947	0.408912	0.204456	−	0.978876i	$-0.434458\pi$
$479$	8.94947	0.408912	0.204456	+	0.978876i	$0.434458\pi$
$480$	0	0
$481$	0.427834	0.0195075
$482$	0	0
$483$	3.86550	0.175886
$484$	0	0
$485$	19.2324	0.873298
$486$	0	0
$487$	−21.7923	−0.987503	−0.493751	−	0.869603i	$-0.664375\pi$
$487$	−21.7923	−0.987503	−0.493751	+	0.869603i	$0.664375\pi$
$488$	0	0
$489$	−13.9461	−0.630665
$490$	0	0
$491$	17.9338	0.809343	0.404672	−	0.914462i	$-0.367386\pi$
$491$	17.9338	0.809343	0.404672	+	0.914462i	$0.367386\pi$
$492$	0	0
$493$	−12.4129	−0.559050
$494$	0	0
$495$	3.57173	0.160537
$496$	0	0
$497$	−12.7108	−0.570156
$498$	0	0
$499$	27.3824	1.22580	0.612902	−	0.790159i	$-0.290002\pi$
$499$	27.3824	1.22580	0.612902	+	0.790159i	$0.290002\pi$
$500$	0	0
$501$	−4.76984	−0.213101
$502$	0	0
$503$	−25.4456	−1.13457	−0.567283	−	0.823523i	$-0.692005\pi$
$503$	−25.4456	−1.13457	−0.567283	+	0.823523i	$0.692005\pi$
$504$	0	0
$505$	12.1489	0.540618
$506$	0	0
$507$	0.769609	0.0341795
$508$	0	0
$509$	−19.8256	−0.878753	−0.439377	−	0.898303i	$-0.644801\pi$
$509$	−19.8256	−0.878753	−0.439377	+	0.898303i	$0.644801\pi$
$510$	0	0
$511$	−12.7665	−0.564758
$512$	0	0
$513$	−20.5237	−0.906143
$514$	0	0
$515$	9.74683	0.429497
$516$	0	0
$517$	0.424297	0.0186605
$518$	0	0
$519$	0.958844	0.0420886
$520$	0	0
$521$	32.6112	1.42872	0.714362	−	0.699776i	$-0.246717\pi$
$521$	32.6112	1.42872	0.714362	+	0.699776i	$0.246717\pi$
$522$	0	0
$523$	−4.01040	−0.175363	−0.0876813	−	0.996149i	$-0.527946\pi$
$523$	−4.01040	−0.175363	−0.0876813	+	0.996149i	$0.527946\pi$
$524$	0	0
$525$	0.769609	0.0335885
$526$	0	0
$527$	9.06549	0.394899
$528$	0	0
$529$	2.22731	0.0968395
$530$	0	0
$531$	−2.54164	−0.110298
$532$	0	0
$533$	−1.86848	−0.0809330
$534$	0	0
$535$	2.20463	0.0953144
$536$	0	0
$537$	−17.1980	−0.742146
$538$	0	0
$539$	−1.48346	−0.0638972
$540$	0	0
$541$	28.4690	1.22398	0.611989	−	0.790867i	$-0.290370\pi$
$541$	28.4690	1.22398	0.611989	+	0.790867i	$0.290370\pi$
$542$	0	0
$543$	13.8199	0.593070
$544$	0	0
$545$	−5.36181	−0.229675
$546$	0	0
$547$	−36.9592	−1.58026	−0.790131	−	0.612938i	$-0.789988\pi$
$547$	−36.9592	−1.58026	−0.790131	+	0.612938i	$0.789988\pi$
$548$	0	0
$549$	−24.6070	−1.05020
$550$	0	0
$551$	−18.7014	−0.796706
$552$	0	0
$553$	−3.55922	−0.151353
$554$	0	0
$555$	−0.329265	−0.0139765
$556$	0	0
$557$	38.1161	1.61503	0.807515	−	0.589847i	$-0.200812\pi$
$557$	38.1161	1.61503	0.807515	+	0.589847i	$0.200812\pi$
$558$	0	0
$559$	2.88818	0.122157
$560$	0	0
$561$	−3.73697	−0.157775
$562$	0	0
$563$	−44.4782	−1.87453	−0.937266	−	0.348615i	$-0.886652\pi$
$563$	−44.4782	−1.87453	−0.937266	+	0.348615i	$0.886652\pi$
$564$	0	0
$565$	11.9974	0.504732
$566$	0	0
$567$	4.02013	0.168830
$568$	0	0
$569$	−12.7360	−0.533919	−0.266960	−	0.963708i	$-0.586019\pi$
$569$	−12.7360	−0.533919	−0.266960	+	0.963708i	$0.586019\pi$
$570$	0	0
$571$	−18.0500	−0.755370	−0.377685	−	0.925934i	$-0.623280\pi$
$571$	−18.0500	−0.755370	−0.377685	+	0.925934i	$0.623280\pi$
$572$	0	0
$573$	4.57441	0.191099
$574$	0	0
$575$	5.02268	0.209460
$576$	0	0
$577$	−16.3493	−0.680629	−0.340314	−	0.940312i	$-0.610534\pi$
$577$	−16.3493	−0.680629	−0.340314	+	0.940312i	$0.610534\pi$
$578$	0	0
$579$	13.3641	0.555393
$580$	0	0
$581$	17.8679	0.741287
$582$	0	0
$583$	−18.5524	−0.768361
$584$	0	0
$585$	2.40770	0.0995462
$586$	0	0
$587$	−9.91973	−0.409431	−0.204716	−	0.978821i	$-0.565627\pi$
$587$	−9.91973	−0.409431	−0.204716	+	0.978821i	$0.565627\pi$
$588$	0	0
$589$	13.6581	0.562773
$590$	0	0
$591$	21.3362	0.877656
$592$	0	0
$593$	27.7961	1.14145	0.570724	−	0.821142i	$-0.306663\pi$
$593$	27.7961	1.14145	0.570724	+	0.821142i	$0.306663\pi$
$594$	0	0
$595$	3.27320	0.134188
$596$	0	0
$597$	12.3597	0.505850
$598$	0	0
$599$	19.6222	0.801741	0.400871	−	0.916135i	$-0.368708\pi$
$599$	19.6222	0.801741	0.400871	+	0.916135i	$0.368708\pi$
$600$	0	0
$601$	−28.9667	−1.18158	−0.590789	−	0.806826i	$-0.701183\pi$
$601$	−28.9667	−1.18158	−0.590789	+	0.806826i	$0.701183\pi$
$602$	0	0
$603$	−28.0373	−1.14177
$604$	0	0
$605$	−8.79934	−0.357744
$606$	0	0
$607$	−10.3123	−0.418565	−0.209283	−	0.977855i	$-0.567113\pi$
$607$	−10.3123	−0.418565	−0.209283	+	0.977855i	$0.567113\pi$
$608$	0	0
$609$	−2.91858	−0.118267
$610$	0	0
$611$	0.286018	0.0115711
$612$	0	0
$613$	21.8912	0.884177	0.442088	−	0.896971i	$-0.354238\pi$
$613$	21.8912	0.884177	0.442088	+	0.896971i	$0.354238\pi$
$614$	0	0
$615$	1.43800	0.0579858
$616$	0	0
$617$	−41.2670	−1.66135	−0.830674	−	0.556759i	$-0.812045\pi$
$617$	−41.2670	−1.66135	−0.830674	+	0.556759i	$0.812045\pi$
$618$	0	0
$619$	−45.6680	−1.83555	−0.917776	−	0.397099i	$-0.870017\pi$
$619$	−45.6680	−1.83555	−0.917776	+	0.397099i	$0.870017\pi$
$620$	0	0
$621$	−20.9035	−0.838827
$622$	0	0
$623$	9.52925	0.381782
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−5.63014	−0.224846
$628$	0	0
$629$	−1.40039	−0.0558370
$630$	0	0
$631$	17.8208	0.709435	0.354717	−	0.934974i	$-0.384577\pi$
$631$	17.8208	0.709435	0.354717	+	0.934974i	$0.384577\pi$
$632$	0	0
$633$	1.15198	0.0457872
$634$	0	0
$635$	−11.8381	−0.469780
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	30.6037	1.21067
$640$	0	0
$641$	29.8744	1.17997	0.589984	−	0.807415i	$-0.299134\pi$
$641$	29.8744	1.17997	0.589984	+	0.807415i	$0.299134\pi$
$642$	0	0
$643$	−21.8739	−0.862623	−0.431311	−	0.902203i	$-0.641949\pi$
$643$	−21.8739	−0.862623	−0.431311	+	0.902203i	$0.641949\pi$
$644$	0	0
$645$	−2.22277	−0.0875215
$646$	0	0
$647$	−22.4341	−0.881974	−0.440987	−	0.897514i	$-0.645371\pi$
$647$	−22.4341	−0.881974	−0.440987	+	0.897514i	$0.645371\pi$
$648$	0	0
$649$	−1.56598	−0.0614702
$650$	0	0
$651$	2.13152	0.0835407
$652$	0	0
$653$	−13.5848	−0.531616	−0.265808	−	0.964026i	$-0.585639\pi$
$653$	−13.5848	−0.531616	−0.265808	+	0.964026i	$0.585639\pi$
$654$	0	0
$655$	−9.09281	−0.355285
$656$	0	0
$657$	30.7380	1.19920
$658$	0	0
$659$	21.0155	0.818649	0.409324	−	0.912389i	$-0.365764\pi$
$659$	21.0155	0.818649	0.409324	+	0.912389i	$0.365764\pi$
$660$	0	0
$661$	−16.6894	−0.649142	−0.324571	−	0.945861i	$-0.605220\pi$
$661$	−16.6894	−0.649142	−0.324571	+	0.945861i	$0.605220\pi$
$662$	0	0
$663$	−2.51909	−0.0978332
$664$	0	0
$665$	4.93143	0.191233
$666$	0	0
$667$	−19.0474	−0.737520
$668$	0	0
$669$	−18.4369	−0.712813
$670$	0	0
$671$	−15.1612	−0.585290
$672$	0	0
$673$	−10.1181	−0.390026	−0.195013	−	0.980801i	$-0.562475\pi$
$673$	−10.1181	−0.390026	−0.195013	+	0.980801i	$0.562475\pi$
$674$	0	0
$675$	−4.16182	−0.160188
$676$	0	0
$677$	−28.3458	−1.08942	−0.544709	−	0.838625i	$-0.683360\pi$
$677$	−28.3458	−1.08942	−0.544709	+	0.838625i	$0.683360\pi$
$678$	0	0
$679$	19.2324	0.738072
$680$	0	0
$681$	−12.4343	−0.476483
$682$	0	0
$683$	7.72180	0.295466	0.147733	−	0.989027i	$-0.452802\pi$
$683$	7.72180	0.295466	0.147733	+	0.989027i	$0.452802\pi$
$684$	0	0
$685$	−14.1613	−0.541075
$686$	0	0
$687$	−21.1064	−0.805258
$688$	0	0
$689$	−12.5061	−0.476446
$690$	0	0
$691$	−0.675634	−0.0257023	−0.0128512	−	0.999917i	$-0.504091\pi$
$691$	−0.675634	−0.0257023	−0.0128512	+	0.999917i	$0.504091\pi$
$692$	0	0
$693$	3.57173	0.135679
$694$	0	0
$695$	−2.04026	−0.0773916
$696$	0	0
$697$	6.11592	0.231657
$698$	0	0
$699$	6.13898	0.232198
$700$	0	0
$701$	21.8858	0.826614	0.413307	−	0.910592i	$-0.364374\pi$
$701$	21.8858	0.826614	0.413307	+	0.910592i	$0.364374\pi$
$702$	0	0
$703$	−2.10983	−0.0795737
$704$	0	0
$705$	−0.220122	−0.00829028
$706$	0	0
$707$	12.1489	0.456905
$708$	0	0
$709$	−39.3479	−1.47774	−0.738871	−	0.673847i	$-0.764641\pi$
$709$	−39.3479	−1.47774	−0.738871	+	0.673847i	$0.764641\pi$
$710$	0	0
$711$	8.56954	0.321383
$712$	0	0
$713$	13.9109	0.520966
$714$	0	0
$715$	1.48346	0.0554783
$716$	0	0
$717$	8.80083	0.328673
$718$	0	0
$719$	22.1051	0.824381	0.412190	−	0.911098i	$-0.364764\pi$
$719$	22.1051	0.824381	0.412190	+	0.911098i	$0.364764\pi$
$720$	0	0
$721$	9.74683	0.362991
$722$	0	0
$723$	6.45849	0.240194
$724$	0	0
$725$	−3.79229	−0.140842
$726$	0	0
$727$	−1.10543	−0.0409981	−0.0204990	−	0.999790i	$-0.506526\pi$
$727$	−1.10543	−0.0409981	−0.0204990	+	0.999790i	$0.506526\pi$
$728$	0	0
$729$	−0.0703594	−0.00260591
$730$	0	0
$731$	−9.45360	−0.349654
$732$	0	0
$733$	9.87635	0.364791	0.182396	−	0.983225i	$-0.441615\pi$
$733$	9.87635	0.364791	0.182396	+	0.983225i	$0.441615\pi$
$734$	0	0
$735$	0.769609	0.0283875
$736$	0	0
$737$	−17.2747	−0.636321
$738$	0	0
$739$	−7.28877	−0.268122	−0.134061	−	0.990973i	$-0.542802\pi$
$739$	−7.28877	−0.268122	−0.134061	+	0.990973i	$0.542802\pi$
$740$	0	0
$741$	−3.79527	−0.139423
$742$	0	0
$743$	−36.2853	−1.33118	−0.665589	−	0.746319i	$-0.731820\pi$
$743$	−36.2853	−1.33118	−0.665589	+	0.746319i	$0.731820\pi$
$744$	0	0
$745$	−10.1799	−0.372961
$746$	0	0
$747$	−43.0207	−1.57404
$748$	0	0
$749$	2.20463	0.0805554
$750$	0	0
$751$	−31.2618	−1.14076	−0.570380	−	0.821381i	$-0.693204\pi$
$751$	−31.2618	−1.14076	−0.570380	+	0.821381i	$0.693204\pi$
$752$	0	0
$753$	13.2583	0.483159
$754$	0	0
$755$	9.89537	0.360129
$756$	0	0
$757$	43.2946	1.57357	0.786784	−	0.617228i	$-0.211744\pi$
$757$	43.2946	1.57357	0.786784	+	0.617228i	$0.211744\pi$
$758$	0	0
$759$	−5.73432	−0.208143
$760$	0	0
$761$	14.5650	0.527982	0.263991	−	0.964525i	$-0.414961\pi$
$761$	14.5650	0.527982	0.263991	+	0.964525i	$0.414961\pi$
$762$	0	0
$763$	−5.36181	−0.194110
$764$	0	0
$765$	−7.88089	−0.284934
$766$	0	0
$767$	−1.05563	−0.0381165
$768$	0	0
$769$	5.17572	0.186641	0.0933207	−	0.995636i	$-0.470252\pi$
$769$	5.17572	0.186641	0.0933207	+	0.995636i	$0.470252\pi$
$770$	0	0
$771$	17.0942	0.615632
$772$	0	0
$773$	41.6131	1.49672	0.748360	−	0.663293i	$-0.230842\pi$
$773$	41.6131	1.49672	0.748360	+	0.663293i	$0.230842\pi$
$774$	0	0
$775$	2.76961	0.0994873
$776$	0	0
$777$	−0.329265	−0.0118123
$778$	0	0
$779$	9.21429	0.330136
$780$	0	0
$781$	18.8559	0.674718
$782$	0	0
$783$	15.7828	0.564032
$784$	0	0
$785$	−9.44011	−0.336932
$786$	0	0
$787$	−40.9920	−1.46121	−0.730604	−	0.682802i	$-0.760761\pi$
$787$	−40.9920	−1.46121	−0.730604	+	0.682802i	$0.760761\pi$
$788$	0	0
$789$	−1.03837	−0.0369671
$790$	0	0
$791$	11.9974	0.426577
$792$	0	0
$793$	−10.2201	−0.362927
$794$	0	0
$795$	9.62484	0.341358
$796$	0	0
$797$	2.55459	0.0904882	0.0452441	−	0.998976i	$-0.485593\pi$
$797$	2.55459	0.0904882	0.0452441	+	0.998976i	$0.485593\pi$
$798$	0	0
$799$	−0.936195	−0.0331202
$800$	0	0
$801$	−22.9436	−0.810672
$802$	0	0
$803$	18.9386	0.668330
$804$	0	0
$805$	5.02268	0.177026
$806$	0	0
$807$	−7.54187	−0.265486
$808$	0	0
$809$	−4.01761	−0.141252	−0.0706259	−	0.997503i	$-0.522500\pi$
$809$	−4.01761	−0.141252	−0.0706259	+	0.997503i	$0.522500\pi$
$810$	0	0
$811$	20.6759	0.726028	0.363014	−	0.931784i	$-0.381748\pi$
$811$	20.6759	0.726028	0.363014	+	0.931784i	$0.381748\pi$
$812$	0	0
$813$	11.5417	0.404784
$814$	0	0
$815$	−18.1210	−0.634752
$816$	0	0
$817$	−14.2428	−0.498294
$818$	0	0
$819$	2.40770	0.0841319
$820$	0	0
$821$	28.9523	1.01044	0.505222	−	0.862989i	$-0.331411\pi$
$821$	28.9523	1.01044	0.505222	+	0.862989i	$0.331411\pi$
$822$	0	0
$823$	0.793683	0.0276661	0.0138330	−	0.999904i	$-0.495597\pi$
$823$	0.793683	0.0276661	0.0138330	+	0.999904i	$0.495597\pi$
$824$	0	0
$825$	−1.14169	−0.0397484
$826$	0	0
$827$	5.01228	0.174294	0.0871470	−	0.996195i	$-0.472225\pi$
$827$	5.01228	0.174294	0.0871470	+	0.996195i	$0.472225\pi$
$828$	0	0
$829$	13.8936	0.482544	0.241272	−	0.970458i	$-0.422435\pi$
$829$	13.8936	0.482544	0.241272	+	0.970458i	$0.422435\pi$
$830$	0	0
$831$	0.818915	0.0284078
$832$	0	0
$833$	3.27320	0.113410
$834$	0	0
$835$	−6.19775	−0.214482
$836$	0	0
$837$	−11.5266	−0.398418
$838$	0	0
$839$	4.82493	0.166575	0.0832876	−	0.996526i	$-0.473458\pi$
$839$	4.82493	0.166575	0.0832876	+	0.996526i	$0.473458\pi$
$840$	0	0
$841$	−14.6185	−0.504088
$842$	0	0
$843$	1.38750	0.0477880
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−8.79934	−0.302349
$848$	0	0
$849$	−15.4526	−0.530331
$850$	0	0
$851$	−2.14887	−0.0736623
$852$	0	0
$853$	8.05132	0.275672	0.137836	−	0.990455i	$-0.455985\pi$
$853$	8.05132	0.275672	0.137836	+	0.990455i	$0.455985\pi$
$854$	0	0
$855$	−11.8734	−0.406062
$856$	0	0
$857$	−50.5604	−1.72711	−0.863555	−	0.504255i	$-0.831767\pi$
$857$	−50.5604	−1.72711	−0.863555	+	0.504255i	$0.831767\pi$
$858$	0	0
$859$	−36.0027	−1.22840	−0.614198	−	0.789152i	$-0.710520\pi$
$859$	−36.0027	−1.22840	−0.614198	+	0.789152i	$0.710520\pi$
$860$	0	0
$861$	1.43800	0.0490070
$862$	0	0
$863$	32.9586	1.12193	0.560963	−	0.827841i	$-0.310431\pi$
$863$	32.9586	1.12193	0.560963	+	0.827841i	$0.310431\pi$
$864$	0	0
$865$	1.24588	0.0423613
$866$	0	0
$867$	−4.83788	−0.164303
$868$	0	0
$869$	5.27996	0.179111
$870$	0	0
$871$	−11.6448	−0.394570
$872$	0	0
$873$	−46.3059	−1.56722
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−5.94483	−0.200743	−0.100371	−	0.994950i	$-0.532003\pi$
$877$	−5.94483	−0.200743	−0.100371	+	0.994950i	$0.532003\pi$
$878$	0	0
$879$	7.82396	0.263896
$880$	0	0
$881$	16.7555	0.564506	0.282253	−	0.959340i	$-0.408918\pi$
$881$	16.7555	0.564506	0.282253	+	0.959340i	$0.408918\pi$
$882$	0	0
$883$	29.2344	0.983816	0.491908	−	0.870647i	$-0.336300\pi$
$883$	29.2344	0.983816	0.491908	+	0.870647i	$0.336300\pi$
$884$	0	0
$885$	0.812421	0.0273092
$886$	0	0
$887$	−23.9361	−0.803696	−0.401848	−	0.915706i	$-0.631632\pi$
$887$	−23.9361	−0.803696	−0.401848	+	0.915706i	$0.631632\pi$
$888$	0	0
$889$	−11.8381	−0.397037
$890$	0	0
$891$	−5.96371	−0.199792
$892$	0	0
$893$	−1.41048	−0.0471998
$894$	0	0
$895$	−22.3463	−0.746956
$896$	0	0
$897$	−3.86550	−0.129065
$898$	0	0
$899$	−10.5032	−0.350300
$900$	0	0
$901$	40.9351	1.36375
$902$	0	0
$903$	−2.22277	−0.0739691
$904$	0	0
$905$	17.9571	0.596914
$906$	0	0
$907$	−35.3087	−1.17241	−0.586203	−	0.810164i	$-0.699378\pi$
$907$	−35.3087	−1.17241	−0.586203	+	0.810164i	$0.699378\pi$
$908$	0	0
$909$	−29.2509	−0.970190
$910$	0	0
$911$	15.7092	0.520470	0.260235	−	0.965545i	$-0.416200\pi$
$911$	15.7092	0.520470	0.260235	+	0.965545i	$0.416200\pi$
$912$	0	0
$913$	−26.5064	−0.877234
$914$	0	0
$915$	7.86550	0.260025
$916$	0	0
$917$	−9.09281	−0.300271
$918$	0	0
$919$	−23.3945	−0.771713	−0.385857	−	0.922559i	$-0.626094\pi$
$919$	−23.3945	−0.771713	−0.385857	+	0.922559i	$0.626094\pi$
$920$	0	0
$921$	−0.825853	−0.0272128
$922$	0	0
$923$	12.7108	0.418380
$924$	0	0
$925$	−0.427834	−0.0140671
$926$	0	0
$927$	−23.4675	−0.770772
$928$	0	0
$929$	−51.7237	−1.69700	−0.848501	−	0.529195i	$-0.822494\pi$
$929$	−51.7237	−1.69700	−0.848501	+	0.529195i	$0.822494\pi$
$930$	0	0
$931$	4.93143	0.161621
$932$	0	0
$933$	6.22344	0.203746
$934$	0	0
$935$	−4.85567	−0.158797
$936$	0	0
$937$	24.3195	0.794484	0.397242	−	0.917714i	$-0.369967\pi$
$937$	24.3195	0.794484	0.397242	+	0.917714i	$0.369967\pi$
$938$	0	0
$939$	8.58430	0.280138
$940$	0	0
$941$	21.4845	0.700373	0.350187	−	0.936680i	$-0.386118\pi$
$941$	21.4845	0.700373	0.350187	+	0.936680i	$0.386118\pi$
$942$	0	0
$943$	9.38479	0.305611
$944$	0	0
$945$	−4.16182	−0.135384
$946$	0	0
$947$	−26.7720	−0.869972	−0.434986	−	0.900437i	$-0.643247\pi$
$947$	−26.7720	−0.869972	−0.434986	+	0.900437i	$0.643247\pi$
$948$	0	0
$949$	12.7665	0.414419
$950$	0	0
$951$	16.5787	0.537600
$952$	0	0
$953$	−14.8453	−0.480885	−0.240443	−	0.970663i	$-0.577293\pi$
$953$	−14.8453	−0.480885	−0.240443	+	0.970663i	$0.577293\pi$
$954$	0	0
$955$	5.94381	0.192337
$956$	0	0
$957$	4.32960	0.139956
$958$	0	0
$959$	−14.1613	−0.457292
$960$	0	0
$961$	−23.3293	−0.752557
$962$	0	0
$963$	−5.30809	−0.171051
$964$	0	0
$965$	17.3648	0.558992
$966$	0	0
$967$	49.6994	1.59823	0.799113	−	0.601181i	$-0.205303\pi$
$967$	49.6994	1.59823	0.799113	+	0.601181i	$0.205303\pi$
$968$	0	0
$969$	12.4227	0.399074
$970$	0	0
$971$	20.4365	0.655837	0.327919	−	0.944706i	$-0.393653\pi$
$971$	20.4365	0.655837	0.327919	+	0.944706i	$0.393653\pi$
$972$	0	0
$973$	−2.04026	−0.0654079
$974$	0	0
$975$	−0.769609	−0.0246472
$976$	0	0
$977$	−11.5927	−0.370882	−0.185441	−	0.982655i	$-0.559371\pi$
$977$	−11.5927	−0.370882	−0.185441	+	0.982655i	$0.559371\pi$
$978$	0	0
$979$	−14.1363	−0.451797
$980$	0	0
$981$	12.9096	0.412173
$982$	0	0
$983$	−8.83487	−0.281789	−0.140894	−	0.990025i	$-0.544998\pi$
$983$	−8.83487	−0.281789	−0.140894	+	0.990025i	$0.544998\pi$
$984$	0	0
$985$	27.7235	0.883344
$986$	0	0
$987$	−0.220122	−0.00700656
$988$	0	0
$989$	−14.5064	−0.461277
$990$	0	0
$991$	−14.4766	−0.459864	−0.229932	−	0.973207i	$-0.573850\pi$
$991$	−14.4766	−0.459864	−0.229932	+	0.973207i	$0.573850\pi$
$992$	0	0
$993$	17.5574	0.557168
$994$	0	0
$995$	16.0597	0.509128
$996$	0	0
$997$	26.8086	0.849038	0.424519	−	0.905419i	$-0.360443\pi$
$997$	26.8086	0.849038	0.424519	+	0.905419i	$0.360443\pi$
$998$	0	0
$999$	1.78057	0.0563346

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.z.1.4	✓	5
4.3	odd	2		7280.2.a.cd.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.z.1.4	✓	5	1.1	even	1	trivial
7280.2.a.cd.1.2		5	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)

\(f(q)\)	\(=\)	\(q+0.769609 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.40770 q^{9} +O(q^{10})\)	\(q+0.769609 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.40770 q^{9} -1.48346 q^{11} -1.00000 q^{13} +0.769609 q^{15} +3.27320 q^{17} +4.93143 q^{19} +0.769609 q^{21} +5.02268 q^{23} +1.00000 q^{25} -4.16182 q^{27} -3.79229 q^{29} +2.76961 q^{31} -1.14169 q^{33} +1.00000 q^{35} -0.427834 q^{37} -0.769609 q^{39} +1.86848 q^{41} -2.88818 q^{43} -2.40770 q^{45} -0.286018 q^{47} +1.00000 q^{49} +2.51909 q^{51} +12.5061 q^{53} -1.48346 q^{55} +3.79527 q^{57} +1.05563 q^{59} +10.2201 q^{61} -2.40770 q^{63} -1.00000 q^{65} +11.6448 q^{67} +3.86550 q^{69} -12.7108 q^{71} -12.7665 q^{73} +0.769609 q^{75} -1.48346 q^{77} -3.55922 q^{79} +4.02013 q^{81} +17.8679 q^{83} +3.27320 q^{85} -2.91858 q^{87} +9.52925 q^{89} -1.00000 q^{91} +2.13152 q^{93} +4.93143 q^{95} +19.2324 q^{97} +3.57173 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^5 + 5 * q^7 + 5 * q^9	\( 5 q + 5 q^{5} + 5 q^{7} + 5 q^{9} + 7 q^{11} - 5 q^{13} + 3 q^{17} + 3 q^{19} + 3 q^{23} + 5 q^{25} - 3 q^{27} + 7 q^{29} + 10 q^{31} + 17 q^{33} + 5 q^{35} + 10 q^{37} + 4 q^{43} + 5 q^{45} - 3 q^{47} + 5 q^{49} + 26 q^{53} + 7 q^{55} - 20 q^{57} + 3 q^{59} + 13 q^{61} + 5 q^{63} - 5 q^{65} + 12 q^{67} + 23 q^{69} + 8 q^{71} + q^{73} + 7 q^{77} - 6 q^{79} + 25 q^{81} - 8 q^{83} + 3 q^{85} - 43 q^{87} + 3 q^{89} - 5 q^{91} + 20 q^{93} + 3 q^{95} - 15 q^{97} + 16 q^{99}+O(q^{100}) \) 5 * q + 5 * q^5 + 5 * q^7 + 5 * q^9 + 7 * q^11 - 5 * q^13 + 3 * q^17 + 3 * q^19 + 3 * q^23 + 5 * q^25 - 3 * q^27 + 7 * q^29 + 10 * q^31 + 17 * q^33 + 5 * q^35 + 10 * q^37 + 4 * q^43 + 5 * q^45 - 3 * q^47 + 5 * q^49 + 26 * q^53 + 7 * q^55 - 20 * q^57 + 3 * q^59 + 13 * q^61 + 5 * q^63 - 5 * q^65 + 12 * q^67 + 23 * q^69 + 8 * q^71 + q^73 + 7 * q^77 - 6 * q^79 + 25 * q^81 - 8 * q^83 + 3 * q^85 - 43 * q^87 + 3 * q^89 - 5 * q^91 + 20 * q^93 + 3 * q^95 - 15 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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