LMFDB - Embedded newform 3640.2.a.p.1.3

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:	$3$
Coefficient field:	3.3.1101.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 12 $ x^3 - x^2 - 9*x + 12
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.11903$ 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)

$f(q)$	$=$	$q+3.11903 q^{3} -1.00000 q^{5} +1.00000 q^{7} +6.72833 q^{9} +O(q^{10})$	$q+3.11903 q^{3} -1.00000 q^{5} +1.00000 q^{7} +6.72833 q^{9} -1.00000 q^{13} -3.11903 q^{15} +3.72833 q^{17} +1.72833 q^{19} +3.11903 q^{21} +1.21860 q^{23} +1.00000 q^{25} +11.6288 q^{27} -6.33763 q^{29} +7.11903 q^{31} -1.00000 q^{35} +3.90043 q^{37} -3.11903 q^{39} +3.72833 q^{41} -10.2381 q^{43} -6.72833 q^{45} +2.78140 q^{47} +1.00000 q^{49} +11.6288 q^{51} +2.00000 q^{53} +5.39070 q^{57} +13.3571 q^{59} -10.0000 q^{61} +6.72833 q^{63} +1.00000 q^{65} -3.29112 q^{67} +3.80085 q^{69} -6.23805 q^{71} +15.6947 q^{73} +3.11903 q^{75} -1.72833 q^{79} +16.0854 q^{81} -1.21860 q^{83} -3.72833 q^{85} -19.7672 q^{87} +10.1385 q^{89} -1.00000 q^{91} +22.2044 q^{93} -1.72833 q^{95} -5.80085 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} - 3 q^{5} + 3 q^{7} + 10 q^{9}+O(q^{10}) $ 3 * q - q^3 - 3 * q^5 + 3 * q^7 + 10 * q^9	$ 3 q - q^{3} - 3 q^{5} + 3 q^{7} + 10 q^{9} - 3 q^{13} + q^{15} + q^{17} - 5 q^{19} - q^{21} + 4 q^{23} + 3 q^{25} + 14 q^{27} - 9 q^{29} + 11 q^{31} - 3 q^{35} + q^{37} + q^{39} + q^{41} - 10 q^{43} - 10 q^{45} + 8 q^{47} + 3 q^{49} + 14 q^{51} + 6 q^{53} + 16 q^{57} + 9 q^{59} - 30 q^{61} + 10 q^{63} + 3 q^{65} + q^{67} - 10 q^{69} + 2 q^{71} + 6 q^{73} - q^{75} + 5 q^{79} + 7 q^{81} - 4 q^{83} - q^{85} - 7 q^{87} - q^{89} - 3 q^{91} + 15 q^{93} + 5 q^{95} + 4 q^{97}+O(q^{100}) $ 3 * q - q^3 - 3 * q^5 + 3 * q^7 + 10 * q^9 - 3 * q^13 + q^15 + q^17 - 5 * q^19 - q^21 + 4 * q^23 + 3 * q^25 + 14 * q^27 - 9 * q^29 + 11 * q^31 - 3 * q^35 + q^37 + q^39 + q^41 - 10 * q^43 - 10 * q^45 + 8 * q^47 + 3 * q^49 + 14 * q^51 + 6 * q^53 + 16 * q^57 + 9 * q^59 - 30 * q^61 + 10 * q^63 + 3 * q^65 + q^67 - 10 * q^69 + 2 * q^71 + 6 * q^73 - q^75 + 5 * q^79 + 7 * q^81 - 4 * q^83 - q^85 - 7 * q^87 - q^89 - 3 * q^91 + 15 * q^93 + 5 * q^95 + 4 * q^97

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
$3$	3.11903	1.80077	0.900385	−	0.435093i	$-0.143285\pi$
$3$	3.11903	1.80077	0.900385	+	0.435093i	$0.143285\pi$
$4$	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
$9$	6.72833	2.24278
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−3.11903	−0.805329
$16$	0	0
$17$	3.72833	0.904252	0.452126	−	0.891954i	$-0.350666\pi$
$17$	3.72833	0.904252	0.452126	+	0.891954i	$0.350666\pi$
$18$	0	0
$19$	1.72833	0.396505	0.198253	−	0.980151i	$-0.436473\pi$
$19$	1.72833	0.396505	0.198253	+	0.980151i	$0.436473\pi$
$20$	0	0
$21$	3.11903	0.680627
$22$	0	0
$23$	1.21860	0.254096	0.127048	−	0.991897i	$-0.459450\pi$
$23$	1.21860	0.254096	0.127048	+	0.991897i	$0.459450\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	11.6288	2.23795
$28$	0	0
$29$	−6.33763	−1.17687	−0.588434	−	0.808545i	$-0.700255\pi$
$29$	−6.33763	−1.17687	−0.588434	+	0.808545i	$0.700255\pi$
$30$	0	0
$31$	7.11903	1.27861	0.639307	−	0.768951i	$-0.279221\pi$
$31$	7.11903	1.27861	0.639307	+	0.768951i	$0.279221\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	3.90043	0.641226	0.320613	−	0.947210i	$-0.396111\pi$
$37$	3.90043	0.641226	0.320613	+	0.947210i	$0.396111\pi$
$38$	0	0
$39$	−3.11903	−0.499444
$40$	0	0
$41$	3.72833	0.582267	0.291133	−	0.956682i	$-0.405968\pi$
$41$	3.72833	0.582267	0.291133	+	0.956682i	$0.405968\pi$
$42$	0	0
$43$	−10.2381	−1.56129	−0.780644	−	0.624976i	$-0.785109\pi$
$43$	−10.2381	−1.56129	−0.780644	+	0.624976i	$0.785109\pi$
$44$	0	0
$45$	−6.72833	−1.00300
$46$	0	0
$47$	2.78140	0.405709	0.202854	−	0.979209i	$-0.434978\pi$
$47$	2.78140	0.405709	0.202854	+	0.979209i	$0.434978\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	11.6288	1.62835
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.39070	0.714016
$58$	0	0
$59$	13.3571	1.73894	0.869472	−	0.493982i	$-0.164459\pi$
$59$	13.3571	1.73894	0.869472	+	0.493982i	$0.164459\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	6.72833	0.847690
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−3.29112	−0.402075	−0.201037	−	0.979584i	$-0.564431\pi$
$67$	−3.29112	−0.402075	−0.201037	+	0.979584i	$0.564431\pi$
$68$	0	0
$69$	3.80085	0.457569
$70$	0	0
$71$	−6.23805	−0.740321	−0.370160	−	0.928968i	$-0.620697\pi$
$71$	−6.23805	−0.740321	−0.370160	+	0.928968i	$0.620697\pi$
$72$	0	0
$73$	15.6947	1.83693	0.918463	−	0.395506i	$-0.129431\pi$
$73$	15.6947	1.83693	0.918463	+	0.395506i	$0.129431\pi$
$74$	0	0
$75$	3.11903	0.360154
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−1.72833	−0.194452	−0.0972260	−	0.995262i	$-0.530997\pi$
$79$	−1.72833	−0.194452	−0.0972260	+	0.995262i	$0.530997\pi$
$80$	0	0
$81$	16.0854	1.78727
$82$	0	0
$83$	−1.21860	−0.133759	−0.0668794	−	0.997761i	$-0.521304\pi$
$83$	−1.21860	−0.133759	−0.0668794	+	0.997761i	$0.521304\pi$
$84$	0	0
$85$	−3.72833	−0.404394
$86$	0	0
$87$	−19.7672	−2.11927
$88$	0	0
$89$	10.1385	1.07468	0.537338	−	0.843367i	$-0.319430\pi$
$89$	10.1385	1.07468	0.537338	+	0.843367i	$0.319430\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	22.2044	2.30249
$94$	0	0
$95$	−1.72833	−0.177323
$96$	0	0
$97$	−5.80085	−0.588987	−0.294494	−	0.955653i	$-0.595151\pi$
$97$	−5.80085	−0.588987	−0.294494	+	0.955653i	$0.595151\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.21860	0.320263	0.160131	−	0.987096i	$-0.448808\pi$
$101$	3.21860	0.320263	0.160131	+	0.987096i	$0.448808\pi$
$102$	0	0
$103$	3.29112	0.324284	0.162142	−	0.986767i	$-0.448160\pi$
$103$	3.29112	0.324284	0.162142	+	0.986767i	$0.448160\pi$
$104$	0	0
$105$	−3.11903	−0.304386
$106$	0	0
$107$	−5.01945	−0.485249	−0.242624	−	0.970120i	$-0.578008\pi$
$107$	−5.01945	−0.485249	−0.242624	+	0.970120i	$0.578008\pi$
$108$	0	0
$109$	−3.56280	−0.341254	−0.170627	−	0.985336i	$-0.554579\pi$
$109$	−3.56280	−0.341254	−0.170627	+	0.985336i	$0.554579\pi$
$110$	0	0
$111$	12.1655	1.15470
$112$	0	0
$113$	13.8009	1.29827	0.649137	−	0.760671i	$-0.275130\pi$
$113$	13.8009	1.29827	0.649137	+	0.760671i	$0.275130\pi$
$114$	0	0
$115$	−1.21860	−0.113635
$116$	0	0
$117$	−6.72833	−0.622034
$118$	0	0
$119$	3.72833	0.341775
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	11.6288	1.04853
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−5.69471	−0.505324	−0.252662	−	0.967555i	$-0.581306\pi$
$127$	−5.69471	−0.505324	−0.252662	+	0.967555i	$0.581306\pi$
$128$	0	0
$129$	−31.9328	−2.81152
$130$	0	0
$131$	−13.6947	−1.19651	−0.598256	−	0.801305i	$-0.704139\pi$
$131$	−13.6947	−1.19651	−0.598256	+	0.801305i	$0.704139\pi$
$132$	0	0
$133$	1.72833	0.149865
$134$	0	0
$135$	−11.6288	−1.00084
$136$	0	0
$137$	9.66237	0.825512	0.412756	−	0.910842i	$-0.364566\pi$
$137$	9.66237	0.825512	0.412756	+	0.910842i	$0.364566\pi$
$138$	0	0
$139$	−7.45665	−0.632465	−0.316233	−	0.948682i	$-0.602418\pi$
$139$	−7.45665	−0.632465	−0.316233	+	0.948682i	$0.602418\pi$
$140$	0	0
$141$	8.67526	0.730588
$142$	0	0
$143$	0	0
$144$	0	0
$145$	6.33763	0.526311
$146$	0	0
$147$	3.11903	0.257253
$148$	0	0
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	0	0
$151$	−0.675256	−0.0549516	−0.0274758	−	0.999622i	$-0.508747\pi$
$151$	−0.675256	−0.0549516	−0.0274758	+	0.999622i	$0.508747\pi$
$152$	0	0
$153$	25.0854	2.02803
$154$	0	0
$155$	−7.11903	−0.571814
$156$	0	0
$157$	−16.5757	−1.32288	−0.661442	−	0.749997i	$-0.730055\pi$
$157$	−16.5757	−1.32288	−0.661442	+	0.749997i	$0.730055\pi$
$158$	0	0
$159$	6.23805	0.494710
$160$	0	0
$161$	1.21860	0.0960392
$162$	0	0
$163$	4.88097	0.382307	0.191154	−	0.981560i	$-0.438777\pi$
$163$	4.88097	0.382307	0.191154	+	0.981560i	$0.438777\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−21.1514	−1.63674	−0.818371	−	0.574691i	$-0.805122\pi$
$167$	−21.1514	−1.63674	−0.818371	+	0.574691i	$0.805122\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	11.6288	0.889273
$172$	0	0
$173$	7.01288	0.533180	0.266590	−	0.963810i	$-0.414103\pi$
$173$	7.01288	0.533180	0.266590	+	0.963810i	$0.414103\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	41.6611	3.13144
$178$	0	0
$179$	13.7283	1.02610	0.513052	−	0.858358i	$-0.328515\pi$
$179$	13.7283	1.02610	0.513052	+	0.858358i	$0.328515\pi$
$180$	0	0
$181$	9.80085	0.728491	0.364246	−	0.931303i	$-0.381327\pi$
$181$	9.80085	0.728491	0.364246	+	0.931303i	$0.381327\pi$
$182$	0	0
$183$	−31.1903	−2.30565
$184$	0	0
$185$	−3.90043	−0.286765
$186$	0	0
$187$	0	0
$188$	0	0
$189$	11.6288	0.845867
$190$	0	0
$191$	8.88097	0.642605	0.321302	−	0.946977i	$-0.395879\pi$
$191$	8.88097	0.642605	0.321302	+	0.946977i	$0.395879\pi$
$192$	0	0
$193$	27.1850	1.95682	0.978409	−	0.206678i	$-0.0662654\pi$
$193$	27.1850	1.95682	0.978409	+	0.206678i	$0.0662654\pi$
$194$	0	0
$195$	3.11903	0.223358
$196$	0	0
$197$	−19.2898	−1.37434	−0.687172	−	0.726495i	$-0.741148\pi$
$197$	−19.2898	−1.37434	−0.687172	+	0.726495i	$0.741148\pi$
$198$	0	0
$199$	11.8009	0.836540	0.418270	−	0.908323i	$-0.362637\pi$
$199$	11.8009	0.836540	0.418270	+	0.908323i	$0.362637\pi$
$200$	0	0
$201$	−10.2651	−0.724045
$202$	0	0
$203$	−6.33763	−0.444814
$204$	0	0
$205$	−3.72833	−0.260398
$206$	0	0
$207$	8.19915	0.569880
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−15.7672	−1.08546	−0.542730	−	0.839907i	$-0.682609\pi$
$211$	−15.7672	−1.08546	−0.542730	+	0.839907i	$0.682609\pi$
$212$	0	0
$213$	−19.4567	−1.33315
$214$	0	0
$215$	10.2381	0.698229
$216$	0	0
$217$	7.11903	0.483271
$218$	0	0
$219$	48.9522	3.30788
$220$	0	0
$221$	−3.72833	−0.250794
$222$	0	0
$223$	−1.01945	−0.0682675	−0.0341338	−	0.999417i	$-0.510867\pi$
$223$	−1.01945	−0.0682675	−0.0341338	+	0.999417i	$0.510867\pi$
$224$	0	0
$225$	6.72833	0.448555
$226$	0	0
$227$	9.21860	0.611860	0.305930	−	0.952054i	$-0.401033\pi$
$227$	9.21860	0.611860	0.305930	+	0.952054i	$0.401033\pi$
$228$	0	0
$229$	−4.27167	−0.282280	−0.141140	−	0.989990i	$-0.545077\pi$
$229$	−4.27167	−0.282280	−0.141140	+	0.989990i	$0.545077\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.78140	−0.575289	−0.287644	−	0.957737i	$-0.592872\pi$
$233$	−8.78140	−0.575289	−0.287644	+	0.957737i	$0.592872\pi$
$234$	0	0
$235$	−2.78140	−0.181438
$236$	0	0
$237$	−5.39070	−0.350164
$238$	0	0
$239$	2.78140	0.179914	0.0899569	−	0.995946i	$-0.471327\pi$
$239$	2.78140	0.179914	0.0899569	+	0.995946i	$0.471327\pi$
$240$	0	0
$241$	−14.6416	−0.943151	−0.471575	−	0.881826i	$-0.656314\pi$
$241$	−14.6416	−0.943151	−0.471575	+	0.881826i	$0.656314\pi$
$242$	0	0
$243$	15.2846	0.980505
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−1.72833	−0.109971
$248$	0	0
$249$	−3.80085	−0.240869
$250$	0	0
$251$	25.1514	1.58754	0.793770	−	0.608218i	$-0.208115\pi$
$251$	25.1514	1.58754	0.793770	+	0.608218i	$0.208115\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−11.6288	−0.728221
$256$	0	0
$257$	−14.9522	−0.932693	−0.466347	−	0.884602i	$-0.654430\pi$
$257$	−14.9522	−0.932693	−0.466347	+	0.884602i	$0.654430\pi$
$258$	0	0
$259$	3.90043	0.242361
$260$	0	0
$261$	−42.6416	−2.63945
$262$	0	0
$263$	14.7814	0.911460	0.455730	−	0.890118i	$-0.349378\pi$
$263$	14.7814	0.911460	0.455730	+	0.890118i	$0.349378\pi$
$264$	0	0
$265$	−2.00000	−0.122859
$266$	0	0
$267$	31.6222	1.93525
$268$	0	0
$269$	−17.2575	−1.05221	−0.526104	−	0.850420i	$-0.676348\pi$
$269$	−17.2575	−1.05221	−0.526104	+	0.850420i	$0.676348\pi$
$270$	0	0
$271$	−5.56280	−0.337916	−0.168958	−	0.985623i	$-0.554040\pi$
$271$	−5.56280	−0.337916	−0.168958	+	0.985623i	$0.554040\pi$
$272$	0	0
$273$	−3.11903	−0.188772
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−21.2575	−1.27724	−0.638620	−	0.769522i	$-0.720494\pi$
$277$	−21.2575	−1.27724	−0.638620	+	0.769522i	$0.720494\pi$
$278$	0	0
$279$	47.8991	2.86765
$280$	0	0
$281$	10.3311	0.616299	0.308150	−	0.951338i	$-0.400290\pi$
$281$	10.3311	0.616299	0.308150	+	0.951338i	$0.400290\pi$
$282$	0	0
$283$	−31.7672	−1.88837	−0.944183	−	0.329422i	$-0.893146\pi$
$283$	−31.7672	−1.88837	−0.944183	+	0.329422i	$0.893146\pi$
$284$	0	0
$285$	−5.39070	−0.319317
$286$	0	0
$287$	3.72833	0.220076
$288$	0	0
$289$	−3.09957	−0.182328
$290$	0	0
$291$	−18.0930	−1.06063
$292$	0	0
$293$	−15.0195	−0.877446	−0.438723	−	0.898622i	$-0.644569\pi$
$293$	−15.0195	−0.877446	−0.438723	+	0.898622i	$0.644569\pi$
$294$	0	0
$295$	−13.3571	−0.777679
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.21860	−0.0704735
$300$	0	0
$301$	−10.2381	−0.590112
$302$	0	0
$303$	10.0389	0.576720
$304$	0	0
$305$	10.0000	0.572598
$306$	0	0
$307$	9.21860	0.526133	0.263067	−	0.964778i	$-0.415266\pi$
$307$	9.21860	0.526133	0.263067	+	0.964778i	$0.415266\pi$
$308$	0	0
$309$	10.2651	0.583961
$310$	0	0
$311$	−21.4956	−1.21890	−0.609451	−	0.792824i	$-0.708610\pi$
$311$	−21.4956	−1.21890	−0.609451	+	0.792824i	$0.708610\pi$
$312$	0	0
$313$	−3.01288	−0.170298	−0.0851491	−	0.996368i	$-0.527137\pi$
$313$	−3.01288	−0.170298	−0.0851491	+	0.996368i	$0.527137\pi$
$314$	0	0
$315$	−6.72833	−0.379098
$316$	0	0
$317$	−20.9133	−1.17461	−0.587304	−	0.809366i	$-0.699811\pi$
$317$	−20.9133	−1.17461	−0.587304	+	0.809366i	$0.699811\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−15.6558	−0.873822
$322$	0	0
$323$	6.44377	0.358541
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−11.1125	−0.614520
$328$	0	0
$329$	2.78140	0.153343
$330$	0	0
$331$	−5.56280	−0.305759	−0.152879	−	0.988245i	$-0.548855\pi$
$331$	−5.56280	−0.305759	−0.152879	+	0.988245i	$0.548855\pi$
$332$	0	0
$333$	26.2433	1.43813
$334$	0	0
$335$	3.29112	0.179813
$336$	0	0
$337$	−10.4761	−0.570670	−0.285335	−	0.958428i	$-0.592105\pi$
$337$	−10.4761	−0.570670	−0.285335	+	0.958428i	$0.592105\pi$
$338$	0	0
$339$	43.0452	2.33790
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−3.80085	−0.204631
$346$	0	0
$347$	−18.5691	−0.996842	−0.498421	−	0.866935i	$-0.666087\pi$
$347$	−18.5691	−0.996842	−0.498421	+	0.866935i	$0.666087\pi$
$348$	0	0
$349$	27.4890	1.47145	0.735726	−	0.677279i	$-0.236841\pi$
$349$	27.4890	1.47145	0.735726	+	0.677279i	$0.236841\pi$
$350$	0	0
$351$	−11.6288	−0.620697
$352$	0	0
$353$	−37.3894	−1.99004	−0.995019	−	0.0996863i	$-0.968216\pi$
$353$	−37.3894	−1.99004	−0.995019	+	0.0996863i	$0.968216\pi$
$354$	0	0
$355$	6.23805	0.331081
$356$	0	0
$357$	11.6288	0.615459
$358$	0	0
$359$	8.33106	0.439697	0.219848	−	0.975534i	$-0.429444\pi$
$359$	8.33106	0.439697	0.219848	+	0.975534i	$0.429444\pi$
$360$	0	0
$361$	−16.0129	−0.842783
$362$	0	0
$363$	−34.3093	−1.80077
$364$	0	0
$365$	−15.6947	−0.821499
$366$	0	0
$367$	17.5628	0.916771	0.458385	−	0.888754i	$-0.348428\pi$
$367$	17.5628	0.916771	0.458385	+	0.888754i	$0.348428\pi$
$368$	0	0
$369$	25.0854	1.30589
$370$	0	0
$371$	2.00000	0.103835
$372$	0	0
$373$	33.1903	1.71853	0.859263	−	0.511533i	$-0.170922\pi$
$373$	33.1903	1.71853	0.859263	+	0.511533i	$0.170922\pi$
$374$	0	0
$375$	−3.11903	−0.161066
$376$	0	0
$377$	6.33763	0.326404
$378$	0	0
$379$	0.344196	0.0176801	0.00884007	−	0.999961i	$-0.497186\pi$
$379$	0.344196	0.0176801	0.00884007	+	0.999961i	$0.497186\pi$
$380$	0	0
$381$	−17.7619	−0.909972
$382$	0	0
$383$	−31.9328	−1.63169	−0.815844	−	0.578272i	$-0.803727\pi$
$383$	−31.9328	−1.63169	−0.815844	+	0.578272i	$0.803727\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−68.8850	−3.50162
$388$	0	0
$389$	−23.1177	−1.17212	−0.586058	−	0.810269i	$-0.699321\pi$
$389$	−23.1177	−1.17212	−0.586058	+	0.810269i	$0.699321\pi$
$390$	0	0
$391$	4.54335	0.229767
$392$	0	0
$393$	−42.7142	−2.15464
$394$	0	0
$395$	1.72833	0.0869616
$396$	0	0
$397$	−29.9328	−1.50228	−0.751141	−	0.660142i	$-0.770496\pi$
$397$	−29.9328	−1.50228	−0.751141	+	0.660142i	$0.770496\pi$
$398$	0	0
$399$	5.39070	0.269873
$400$	0	0
$401$	21.0584	1.05160	0.525802	−	0.850607i	$-0.323765\pi$
$401$	21.0584	1.05160	0.525802	+	0.850607i	$0.323765\pi$
$402$	0	0
$403$	−7.11903	−0.354624
$404$	0	0
$405$	−16.0854	−0.799290
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−36.9133	−1.82525	−0.912623	−	0.408803i	$-0.865946\pi$
$409$	−36.9133	−1.82525	−0.912623	+	0.408803i	$0.865946\pi$
$410$	0	0
$411$	30.1372	1.48656
$412$	0	0
$413$	13.3571	0.657259
$414$	0	0
$415$	1.21860	0.0598188
$416$	0	0
$417$	−23.2575	−1.13892
$418$	0	0
$419$	0.543345	0.0265441	0.0132721	−	0.999912i	$-0.495775\pi$
$419$	0.543345	0.0265441	0.0132721	+	0.999912i	$0.495775\pi$
$420$	0	0
$421$	−18.4761	−0.900470	−0.450235	−	0.892910i	$-0.648660\pi$
$421$	−18.4761	−0.900470	−0.450235	+	0.892910i	$0.648660\pi$
$422$	0	0
$423$	18.7142	0.909914
$424$	0	0
$425$	3.72833	0.180850
$426$	0	0
$427$	−10.0000	−0.483934
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−14.5822	−0.702402	−0.351201	−	0.936300i	$-0.614227\pi$
$431$	−14.5822	−0.702402	−0.351201	+	0.936300i	$0.614227\pi$
$432$	0	0
$433$	−14.7089	−0.706863	−0.353432	−	0.935460i	$-0.614985\pi$
$433$	−14.7089	−0.706863	−0.353432	+	0.935460i	$0.614985\pi$
$434$	0	0
$435$	19.7672	0.947766
$436$	0	0
$437$	2.10614	0.100750
$438$	0	0
$439$	6.91331	0.329954	0.164977	−	0.986297i	$-0.447245\pi$
$439$	6.91331	0.329954	0.164977	+	0.986297i	$0.447245\pi$
$440$	0	0
$441$	6.72833	0.320397
$442$	0	0
$443$	−1.21860	−0.0578975	−0.0289488	−	0.999581i	$-0.509216\pi$
$443$	−1.21860	−0.0578975	−0.0289488	+	0.999581i	$0.509216\pi$
$444$	0	0
$445$	−10.1385	−0.480610
$446$	0	0
$447$	6.23805	0.295050
$448$	0	0
$449$	6.19915	0.292556	0.146278	−	0.989244i	$-0.453271\pi$
$449$	6.19915	0.292556	0.146278	+	0.989244i	$0.453271\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−2.10614	−0.0989552
$454$	0	0
$455$	1.00000	0.0468807
$456$	0	0
$457$	1.66237	0.0777625	0.0388812	−	0.999244i	$-0.487621\pi$
$457$	1.66237	0.0777625	0.0388812	+	0.999244i	$0.487621\pi$
$458$	0	0
$459$	43.3558	2.02368
$460$	0	0
$461$	31.2898	1.45731	0.728657	−	0.684879i	$-0.240145\pi$
$461$	31.2898	1.45731	0.728657	+	0.684879i	$0.240145\pi$
$462$	0	0
$463$	−16.6416	−0.773402	−0.386701	−	0.922205i	$-0.626386\pi$
$463$	−16.6416	−0.773402	−0.386701	+	0.922205i	$0.626386\pi$
$464$	0	0
$465$	−22.2044	−1.02971
$466$	0	0
$467$	32.0983	1.48533	0.742666	−	0.669662i	$-0.233561\pi$
$467$	32.0983	1.48533	0.742666	+	0.669662i	$0.233561\pi$
$468$	0	0
$469$	−3.29112	−0.151970
$470$	0	0
$471$	−51.7000	−2.38221
$472$	0	0
$473$	0	0
$474$	0	0
$475$	1.72833	0.0793011
$476$	0	0
$477$	13.4567	0.616138
$478$	0	0
$479$	17.5562	0.802165	0.401082	−	0.916042i	$-0.368634\pi$
$479$	17.5562	0.802165	0.401082	+	0.916042i	$0.368634\pi$
$480$	0	0
$481$	−3.90043	−0.177844
$482$	0	0
$483$	3.80085	0.172945
$484$	0	0
$485$	5.80085	0.263403
$486$	0	0
$487$	4.27039	0.193510	0.0967549	−	0.995308i	$-0.469154\pi$
$487$	4.27039	0.193510	0.0967549	+	0.995308i	$0.469154\pi$
$488$	0	0
$489$	15.2239	0.688448
$490$	0	0
$491$	14.0389	0.633567	0.316783	−	0.948498i	$-0.397397\pi$
$491$	14.0389	0.633567	0.316783	+	0.948498i	$0.397397\pi$
$492$	0	0
$493$	−23.6288	−1.06419
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.23805	−0.279815
$498$	0	0
$499$	−33.3505	−1.49297	−0.746487	−	0.665400i	$-0.768261\pi$
$499$	−33.3505	−1.49297	−0.746487	+	0.665400i	$0.768261\pi$
$500$	0	0
$501$	−65.9717	−2.94740
$502$	0	0
$503$	17.5628	0.783086	0.391543	−	0.920160i	$-0.371941\pi$
$503$	17.5628	0.783086	0.391543	+	0.920160i	$0.371941\pi$
$504$	0	0
$505$	−3.21860	−0.143226
$506$	0	0
$507$	3.11903	0.138521
$508$	0	0
$509$	25.2911	1.12101	0.560505	−	0.828151i	$-0.310607\pi$
$509$	25.2911	1.12101	0.560505	+	0.828151i	$0.310607\pi$
$510$	0	0
$511$	15.6947	0.694293
$512$	0	0
$513$	20.0983	0.887361
$514$	0	0
$515$	−3.29112	−0.145024
$516$	0	0
$517$	0	0
$518$	0	0
$519$	21.8734	0.960135
$520$	0	0
$521$	−11.9070	−0.521655	−0.260827	−	0.965385i	$-0.583995\pi$
$521$	−11.9070	−0.521655	−0.260827	+	0.965385i	$0.583995\pi$
$522$	0	0
$523$	39.3894	1.72238	0.861189	−	0.508284i	$-0.169720\pi$
$523$	39.3894	1.72238	0.861189	+	0.508284i	$0.169720\pi$
$524$	0	0
$525$	3.11903	0.136125
$526$	0	0
$527$	26.5421	1.15619
$528$	0	0
$529$	−21.5150	−0.935435
$530$	0	0
$531$	89.8708	3.90006
$532$	0	0
$533$	−3.72833	−0.161492
$534$	0	0
$535$	5.01945	0.217010
$536$	0	0
$537$	42.8190	1.84778
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−12.9805	−0.558077	−0.279039	−	0.960280i	$-0.590016\pi$
$541$	−12.9805	−0.558077	−0.279039	+	0.960280i	$0.590016\pi$
$542$	0	0
$543$	30.5691	1.31185
$544$	0	0
$545$	3.56280	0.152613
$546$	0	0
$547$	21.6947	0.927599	0.463799	−	0.885940i	$-0.346486\pi$
$547$	21.6947	0.927599	0.463799	+	0.885940i	$0.346486\pi$
$548$	0	0
$549$	−67.2833	−2.87158
$550$	0	0
$551$	−10.9535	−0.466635
$552$	0	0
$553$	−1.72833	−0.0734960
$554$	0	0
$555$	−12.1655	−0.516398
$556$	0	0
$557$	−14.6416	−0.620386	−0.310193	−	0.950674i	$-0.600394\pi$
$557$	−14.6416	−0.620386	−0.310193	+	0.950674i	$0.600394\pi$
$558$	0	0
$559$	10.2381	0.433024
$560$	0	0
$561$	0	0
$562$	0	0
$563$	25.1177	1.05859	0.529293	−	0.848439i	$-0.322457\pi$
$563$	25.1177	1.05859	0.529293	+	0.848439i	$0.322457\pi$
$564$	0	0
$565$	−13.8009	−0.580606
$566$	0	0
$567$	16.0854	0.675524
$568$	0	0
$569$	21.2509	0.890886	0.445443	−	0.895310i	$-0.353046\pi$
$569$	21.2509	0.890886	0.445443	+	0.895310i	$0.353046\pi$
$570$	0	0
$571$	−12.8810	−0.539052	−0.269526	−	0.962993i	$-0.586867\pi$
$571$	−12.8810	−0.539052	−0.269526	+	0.962993i	$0.586867\pi$
$572$	0	0
$573$	27.7000	1.15718
$574$	0	0
$575$	1.21860	0.0508192
$576$	0	0
$577$	−12.3831	−0.515515	−0.257758	−	0.966210i	$-0.582984\pi$
$577$	−12.3831	−0.515515	−0.257758	+	0.966210i	$0.582984\pi$
$578$	0	0
$579$	84.7907	3.52378
$580$	0	0
$581$	−1.21860	−0.0505561
$582$	0	0
$583$	0	0
$584$	0	0
$585$	6.72833	0.278182
$586$	0	0
$587$	−22.3700	−0.923307	−0.461654	−	0.887060i	$-0.652744\pi$
$587$	−22.3700	−0.923307	−0.461654	+	0.887060i	$0.652744\pi$
$588$	0	0
$589$	12.3040	0.506978
$590$	0	0
$591$	−60.1655	−2.47488
$592$	0	0
$593$	14.3442	0.589046	0.294523	−	0.955644i	$-0.404839\pi$
$593$	14.3442	0.589046	0.294523	+	0.955644i	$0.404839\pi$
$594$	0	0
$595$	−3.72833	−0.152847
$596$	0	0
$597$	36.8072	1.50642
$598$	0	0
$599$	13.5628	0.554161	0.277080	−	0.960847i	$-0.410633\pi$
$599$	13.5628	0.554161	0.277080	+	0.960847i	$0.410633\pi$
$600$	0	0
$601$	−45.1903	−1.84335	−0.921675	−	0.387964i	$-0.873179\pi$
$601$	−45.1903	−1.84335	−0.921675	+	0.387964i	$0.873179\pi$
$602$	0	0
$603$	−22.1438	−0.901764
$604$	0	0
$605$	11.0000	0.447214
$606$	0	0
$607$	−12.8810	−0.522823	−0.261411	−	0.965228i	$-0.584188\pi$
$607$	−12.8810	−0.522823	−0.261411	+	0.965228i	$0.584188\pi$
$608$	0	0
$609$	−19.7672	−0.801009
$610$	0	0
$611$	−2.78140	−0.112523
$612$	0	0
$613$	−8.03890	−0.324688	−0.162344	−	0.986734i	$-0.551905\pi$
$613$	−8.03890	−0.324688	−0.162344	+	0.986734i	$0.551905\pi$
$614$	0	0
$615$	−11.6288	−0.468917
$616$	0	0
$617$	24.8150	0.999015	0.499507	−	0.866310i	$-0.333514\pi$
$617$	24.8150	0.999015	0.499507	+	0.866310i	$0.333514\pi$
$618$	0	0
$619$	−13.0801	−0.525735	−0.262867	−	0.964832i	$-0.584668\pi$
$619$	−13.0801	−0.525735	−0.262867	+	0.964832i	$0.584668\pi$
$620$	0	0
$621$	14.1708	0.568655
$622$	0	0
$623$	10.1385	0.406190
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	14.5421	0.579830
$630$	0	0
$631$	−18.3700	−0.731297	−0.365648	−	0.930753i	$-0.619153\pi$
$631$	−18.3700	−0.731297	−0.365648	+	0.930753i	$0.619153\pi$
$632$	0	0
$633$	−49.1784	−1.95467
$634$	0	0
$635$	5.69471	0.225988
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−41.9717	−1.66037
$640$	0	0
$641$	−29.0518	−1.14748	−0.573738	−	0.819039i	$-0.694507\pi$
$641$	−29.0518	−1.14748	−0.573738	+	0.819039i	$0.694507\pi$
$642$	0	0
$643$	40.8203	1.60980	0.804898	−	0.593413i	$-0.202220\pi$
$643$	40.8203	1.60980	0.804898	+	0.593413i	$0.202220\pi$
$644$	0	0
$645$	31.9328	1.25735
$646$	0	0
$647$	11.1190	0.437134	0.218567	−	0.975822i	$-0.429862\pi$
$647$	11.1190	0.437134	0.218567	+	0.975822i	$0.429862\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	22.2044	0.870260
$652$	0	0
$653$	35.9717	1.40768	0.703840	−	0.710359i	$-0.251467\pi$
$653$	35.9717	1.40768	0.703840	+	0.710359i	$0.251467\pi$
$654$	0	0
$655$	13.6947	0.535097
$656$	0	0
$657$	105.599	4.11981
$658$	0	0
$659$	22.0594	0.859312	0.429656	−	0.902993i	$-0.358635\pi$
$659$	22.0594	0.859312	0.429656	+	0.902993i	$0.358635\pi$
$660$	0	0
$661$	4.05939	0.157892	0.0789459	−	0.996879i	$-0.474845\pi$
$661$	4.05939	0.157892	0.0789459	+	0.996879i	$0.474845\pi$
$662$	0	0
$663$	−11.6288	−0.451623
$664$	0	0
$665$	−1.72833	−0.0670217
$666$	0	0
$667$	−7.72304	−0.299037
$668$	0	0
$669$	−3.17970	−0.122934
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−36.1708	−1.39428	−0.697141	−	0.716934i	$-0.745545\pi$
$673$	−36.1708	−1.39428	−0.697141	+	0.716934i	$0.745545\pi$
$674$	0	0
$675$	11.6288	0.447591
$676$	0	0
$677$	−8.03890	−0.308960	−0.154480	−	0.987996i	$-0.549370\pi$
$677$	−8.03890	−0.308960	−0.154480	+	0.987996i	$0.549370\pi$
$678$	0	0
$679$	−5.80085	−0.222616
$680$	0	0
$681$	28.7531	1.10182
$682$	0	0
$683$	20.3106	0.777163	0.388581	−	0.921414i	$-0.372965\pi$
$683$	20.3106	0.777163	0.388581	+	0.921414i	$0.372965\pi$
$684$	0	0
$685$	−9.66237	−0.369180
$686$	0	0
$687$	−13.3235	−0.508322
$688$	0	0
$689$	−2.00000	−0.0761939
$690$	0	0
$691$	19.5951	0.745434	0.372717	−	0.927945i	$-0.378426\pi$
$691$	19.5951	0.745434	0.372717	+	0.927945i	$0.378426\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.45665	0.282847
$696$	0	0
$697$	13.9004	0.526516
$698$	0	0
$699$	−27.3894	−1.03596
$700$	0	0
$701$	44.2044	1.66958	0.834789	−	0.550570i	$-0.185589\pi$
$701$	44.2044	1.66958	0.834789	+	0.550570i	$0.185589\pi$
$702$	0	0
$703$	6.74121	0.254250
$704$	0	0
$705$	−8.67526	−0.326729
$706$	0	0
$707$	3.21860	0.121048
$708$	0	0
$709$	−12.1708	−0.457085	−0.228542	−	0.973534i	$-0.573396\pi$
$709$	−12.1708	−0.457085	−0.228542	+	0.973534i	$0.573396\pi$
$710$	0	0
$711$	−11.6288	−0.436112
$712$	0	0
$713$	8.67526	0.324891
$714$	0	0
$715$	0	0
$716$	0	0
$717$	8.67526	0.323983
$718$	0	0
$719$	32.8850	1.22640	0.613201	−	0.789927i	$-0.289881\pi$
$719$	32.8850	1.22640	0.613201	+	0.789927i	$0.289881\pi$
$720$	0	0
$721$	3.29112	0.122568
$722$	0	0
$723$	−45.6677	−1.69840
$724$	0	0
$725$	−6.33763	−0.235374
$726$	0	0
$727$	27.7943	1.03083	0.515416	−	0.856940i	$-0.327637\pi$
$727$	27.7943	1.03083	0.515416	+	0.856940i	$0.327637\pi$
$728$	0	0
$729$	−0.583281	−0.0216030
$730$	0	0
$731$	−38.1708	−1.41180
$732$	0	0
$733$	0.649487	0.0239894	0.0119947	−	0.999928i	$-0.496182\pi$
$733$	0.649487	0.0239894	0.0119947	+	0.999928i	$0.496182\pi$
$734$	0	0
$735$	−3.11903	−0.115047
$736$	0	0
$737$	0	0
$738$	0	0
$739$	25.0195	0.920355	0.460178	−	0.887827i	$-0.347786\pi$
$739$	25.0195	0.920355	0.460178	+	0.887827i	$0.347786\pi$
$740$	0	0
$741$	−5.39070	−0.198032
$742$	0	0
$743$	−14.8928	−0.546365	−0.273182	−	0.961962i	$-0.588076\pi$
$743$	−14.8928	−0.546365	−0.273182	+	0.961962i	$0.588076\pi$
$744$	0	0
$745$	−2.00000	−0.0732743
$746$	0	0
$747$	−8.19915	−0.299991
$748$	0	0
$749$	−5.01945	−0.183407
$750$	0	0
$751$	−46.7076	−1.70438	−0.852192	−	0.523229i	$-0.824727\pi$
$751$	−46.7076	−1.70438	−0.852192	+	0.523229i	$0.824727\pi$
$752$	0	0
$753$	78.4478	2.85880
$754$	0	0
$755$	0.675256	0.0245751
$756$	0	0
$757$	7.56280	0.274875	0.137437	−	0.990510i	$-0.456113\pi$
$757$	7.56280	0.274875	0.137437	+	0.990510i	$0.456113\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−45.3287	−1.64317	−0.821583	−	0.570089i	$-0.806909\pi$
$761$	−45.3287	−1.64317	−0.821583	+	0.570089i	$0.806909\pi$
$762$	0	0
$763$	−3.56280	−0.128982
$764$	0	0
$765$	−25.0854	−0.906965
$766$	0	0
$767$	−13.3571	−0.482296
$768$	0	0
$769$	−35.4283	−1.27758	−0.638789	−	0.769382i	$-0.720564\pi$
$769$	−35.4283	−1.27758	−0.638789	+	0.769382i	$0.720564\pi$
$770$	0	0
$771$	−46.6364	−1.67957
$772$	0	0
$773$	−48.9911	−1.76209	−0.881044	−	0.473034i	$-0.843159\pi$
$773$	−48.9911	−1.76209	−0.881044	+	0.473034i	$0.843159\pi$
$774$	0	0
$775$	7.11903	0.255723
$776$	0	0
$777$	12.1655	0.436436
$778$	0	0
$779$	6.44377	0.230872
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−73.6987	−2.63378
$784$	0	0
$785$	16.5757	0.591611
$786$	0	0
$787$	−52.9522	−1.88754	−0.943771	−	0.330599i	$-0.892749\pi$
$787$	−52.9522	−1.88754	−0.943771	+	0.330599i	$0.892749\pi$
$788$	0	0
$789$	46.1036	1.64133
$790$	0	0
$791$	13.8009	0.490702
$792$	0	0
$793$	10.0000	0.355110
$794$	0	0
$795$	−6.23805	−0.221241
$796$	0	0
$797$	−7.22517	−0.255929	−0.127964	−	0.991779i	$-0.540844\pi$
$797$	−7.22517	−0.255929	−0.127964	+	0.991779i	$0.540844\pi$
$798$	0	0
$799$	10.3700	0.366863
$800$	0	0
$801$	68.2150	2.41026
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−1.21860	−0.0429501
$806$	0	0
$807$	−53.8266	−1.89479
$808$	0	0
$809$	−37.2239	−1.30872	−0.654361	−	0.756182i	$-0.727062\pi$
$809$	−37.2239	−1.30872	−0.654361	+	0.756182i	$0.727062\pi$
$810$	0	0
$811$	−8.68839	−0.305091	−0.152545	−	0.988296i	$-0.548747\pi$
$811$	−8.68839	−0.305091	−0.152545	+	0.988296i	$0.548747\pi$
$812$	0	0
$813$	−17.3505	−0.608509
$814$	0	0
$815$	−4.88097	−0.170973
$816$	0	0
$817$	−17.6947	−0.619059
$818$	0	0
$819$	−6.72833	−0.235107
$820$	0	0
$821$	4.36996	0.152513	0.0762564	−	0.997088i	$-0.475703\pi$
$821$	4.36996	0.152513	0.0762564	+	0.997088i	$0.475703\pi$
$822$	0	0
$823$	52.9522	1.84580	0.922899	−	0.385042i	$-0.125813\pi$
$823$	52.9522	1.84580	0.922899	+	0.385042i	$0.125813\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−26.5150	−0.922017	−0.461009	−	0.887396i	$-0.652512\pi$
$827$	−26.5150	−0.922017	−0.461009	+	0.887396i	$0.652512\pi$
$828$	0	0
$829$	−35.2964	−1.22589	−0.612947	−	0.790124i	$-0.710016\pi$
$829$	−35.2964	−1.22589	−0.612947	+	0.790124i	$0.710016\pi$
$830$	0	0
$831$	−66.3027	−2.30002
$832$	0	0
$833$	3.72833	0.129179
$834$	0	0
$835$	21.1514	0.731973
$836$	0	0
$837$	82.7854	2.86148
$838$	0	0
$839$	−18.4372	−0.636523	−0.318261	−	0.948003i	$-0.603099\pi$
$839$	−18.4372	−0.636523	−0.318261	+	0.948003i	$0.603099\pi$
$840$	0	0
$841$	11.1655	0.385018
$842$	0	0
$843$	32.2229	1.10981
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−11.0000	−0.377964
$848$	0	0
$849$	−99.0828	−3.40051
$850$	0	0
$851$	4.75306	0.162933
$852$	0	0
$853$	−20.3053	−0.695240	−0.347620	−	0.937636i	$-0.613010\pi$
$853$	−20.3053	−0.695240	−0.347620	+	0.937636i	$0.613010\pi$
$854$	0	0
$855$	−11.6288	−0.397695
$856$	0	0
$857$	50.4154	1.72216	0.861079	−	0.508471i	$-0.169789\pi$
$857$	50.4154	1.72216	0.861079	+	0.508471i	$0.169789\pi$
$858$	0	0
$859$	17.8939	0.610531	0.305265	−	0.952267i	$-0.401255\pi$
$859$	17.8939	0.610531	0.305265	+	0.952267i	$0.401255\pi$
$860$	0	0
$861$	11.6288	0.396307
$862$	0	0
$863$	9.33003	0.317598	0.158799	−	0.987311i	$-0.449238\pi$
$863$	9.33003	0.317598	0.158799	+	0.987311i	$0.449238\pi$
$864$	0	0
$865$	−7.01288	−0.238445
$866$	0	0
$867$	−9.66766	−0.328331
$868$	0	0
$869$	0	0
$870$	0	0
$871$	3.29112	0.111515
$872$	0	0
$873$	−39.0300	−1.32097
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	25.5549	0.862929	0.431465	−	0.902130i	$-0.357997\pi$
$877$	25.5549	0.862929	0.431465	+	0.902130i	$0.357997\pi$
$878$	0	0
$879$	−46.8461	−1.58008
$880$	0	0
$881$	−41.4026	−1.39489	−0.697444	−	0.716640i	$-0.745679\pi$
$881$	−41.4026	−1.39489	−0.697444	+	0.716640i	$0.745679\pi$
$882$	0	0
$883$	29.3505	0.987723	0.493862	−	0.869540i	$-0.335585\pi$
$883$	29.3505	0.987723	0.493862	+	0.869540i	$0.335585\pi$
$884$	0	0
$885$	−41.6611	−1.40042
$886$	0	0
$887$	−26.4438	−0.887895	−0.443947	−	0.896053i	$-0.646422\pi$
$887$	−26.4438	−0.887895	−0.443947	+	0.896053i	$0.646422\pi$
$888$	0	0
$889$	−5.69471	−0.190994
$890$	0	0
$891$	0	0
$892$	0	0
$893$	4.80717	0.160866
$894$	0	0
$895$	−13.7283	−0.458887
$896$	0	0
$897$	−3.80085	−0.126907
$898$	0	0
$899$	−45.1177	−1.50476
$900$	0	0
$901$	7.45665	0.248417
$902$	0	0
$903$	−31.9328	−1.06266
$904$	0	0
$905$	−9.80085	−0.325791
$906$	0	0
$907$	0.610584	0.0202741	0.0101370	−	0.999949i	$-0.496773\pi$
$907$	0.610584	0.0202741	0.0101370	+	0.999949i	$0.496773\pi$
$908$	0	0
$909$	21.6558	0.718278
$910$	0	0
$911$	−6.53549	−0.216531	−0.108265	−	0.994122i	$-0.534530\pi$
$911$	−6.53549	−0.216531	−0.108265	+	0.994122i	$0.534530\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	31.1903	1.03112
$916$	0	0
$917$	−13.6947	−0.452239
$918$	0	0
$919$	−17.5562	−0.579127	−0.289563	−	0.957159i	$-0.593510\pi$
$919$	−17.5562	−0.579127	−0.289563	+	0.957159i	$0.593510\pi$
$920$	0	0
$921$	28.7531	0.947446
$922$	0	0
$923$	6.23805	0.205328
$924$	0	0
$925$	3.90043	0.128245
$926$	0	0
$927$	22.1438	0.727297
$928$	0	0
$929$	−33.2509	−1.09093	−0.545464	−	0.838134i	$-0.683647\pi$
$929$	−33.2509	−1.09093	−0.545464	+	0.838134i	$0.683647\pi$
$930$	0	0
$931$	1.72833	0.0566436
$932$	0	0
$933$	−67.0452	−2.19496
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−35.1850	−1.14944	−0.574722	−	0.818349i	$-0.694890\pi$
$937$	−35.1850	−1.14944	−0.574722	+	0.818349i	$0.694890\pi$
$938$	0	0
$939$	−9.39727	−0.306668
$940$	0	0
$941$	27.3300	0.890933	0.445467	−	0.895298i	$-0.353038\pi$
$941$	27.3300	0.890933	0.445467	+	0.895298i	$0.353038\pi$
$942$	0	0
$943$	4.54335	0.147952
$944$	0	0
$945$	−11.6288	−0.378283
$946$	0	0
$947$	0.853922	0.0277487	0.0138744	−	0.999904i	$-0.495584\pi$
$947$	0.853922	0.0277487	0.0138744	+	0.999904i	$0.495584\pi$
$948$	0	0
$949$	−15.6947	−0.509472
$950$	0	0
$951$	−65.2292	−2.11520
$952$	0	0
$953$	60.2486	1.95164	0.975822	−	0.218566i	$-0.0701379\pi$
$953$	60.2486	1.95164	0.975822	+	0.218566i	$0.0701379\pi$
$954$	0	0
$955$	−8.88097	−0.287382
$956$	0	0
$957$	0	0
$958$	0	0
$959$	9.66237	0.312014
$960$	0	0
$961$	19.6805	0.634856
$962$	0	0
$963$	−33.7725	−1.08830
$964$	0	0
$965$	−27.1850	−0.875116
$966$	0	0
$967$	46.0323	1.48030	0.740150	−	0.672442i	$-0.234754\pi$
$967$	46.0323	1.48030	0.740150	+	0.672442i	$0.234754\pi$
$968$	0	0
$969$	20.0983	0.645650
$970$	0	0
$971$	−15.3894	−0.493870	−0.246935	−	0.969032i	$-0.579423\pi$
$971$	−15.3894	−0.493870	−0.246935	+	0.969032i	$0.579423\pi$
$972$	0	0
$973$	−7.45665	−0.239049
$974$	0	0
$975$	−3.11903	−0.0998888
$976$	0	0
$977$	−30.0311	−0.960779	−0.480389	−	0.877055i	$-0.659505\pi$
$977$	−30.0311	−0.960779	−0.480389	+	0.877055i	$0.659505\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−23.9717	−0.765356
$982$	0	0
$983$	−22.3053	−0.711428	−0.355714	−	0.934595i	$-0.615762\pi$
$983$	−22.3053	−0.711428	−0.355714	+	0.934595i	$0.615762\pi$
$984$	0	0
$985$	19.2898	0.614625
$986$	0	0
$987$	8.67526	0.276136
$988$	0	0
$989$	−12.4761	−0.396717
$990$	0	0
$991$	−8.24334	−0.261858	−0.130929	−	0.991392i	$-0.541796\pi$
$991$	−8.24334	−0.261858	−0.130929	+	0.991392i	$0.541796\pi$
$992$	0	0
$993$	−17.3505	−0.550602
$994$	0	0
$995$	−11.8009	−0.374112
$996$	0	0
$997$	−33.1461	−1.04975	−0.524873	−	0.851180i	$-0.675887\pi$
$997$	−33.1461	−1.04975	−0.524873	+	0.851180i	$0.675887\pi$
$998$	0	0
$999$	45.3571	1.43503

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.p.1.3	✓	3
4.3	odd	2		7280.2.a.bn.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.p.1.3	✓	3	1.1	even	1	trivial
7280.2.a.bn.1.1		3	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)

\(f(q)\)	\(=\)	\(q+3.11903 q^{3} -1.00000 q^{5} +1.00000 q^{7} +6.72833 q^{9} +O(q^{10})\)	\(q+3.11903 q^{3} -1.00000 q^{5} +1.00000 q^{7} +6.72833 q^{9} -1.00000 q^{13} -3.11903 q^{15} +3.72833 q^{17} +1.72833 q^{19} +3.11903 q^{21} +1.21860 q^{23} +1.00000 q^{25} +11.6288 q^{27} -6.33763 q^{29} +7.11903 q^{31} -1.00000 q^{35} +3.90043 q^{37} -3.11903 q^{39} +3.72833 q^{41} -10.2381 q^{43} -6.72833 q^{45} +2.78140 q^{47} +1.00000 q^{49} +11.6288 q^{51} +2.00000 q^{53} +5.39070 q^{57} +13.3571 q^{59} -10.0000 q^{61} +6.72833 q^{63} +1.00000 q^{65} -3.29112 q^{67} +3.80085 q^{69} -6.23805 q^{71} +15.6947 q^{73} +3.11903 q^{75} -1.72833 q^{79} +16.0854 q^{81} -1.21860 q^{83} -3.72833 q^{85} -19.7672 q^{87} +10.1385 q^{89} -1.00000 q^{91} +22.2044 q^{93} -1.72833 q^{95} -5.80085 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} - 3 q^{5} + 3 q^{7} + 10 q^{9}+O(q^{10}) \) 3 * q - q^3 - 3 * q^5 + 3 * q^7 + 10 * q^9	\( 3 q - q^{3} - 3 q^{5} + 3 q^{7} + 10 q^{9} - 3 q^{13} + q^{15} + q^{17} - 5 q^{19} - q^{21} + 4 q^{23} + 3 q^{25} + 14 q^{27} - 9 q^{29} + 11 q^{31} - 3 q^{35} + q^{37} + q^{39} + q^{41} - 10 q^{43} - 10 q^{45} + 8 q^{47} + 3 q^{49} + 14 q^{51} + 6 q^{53} + 16 q^{57} + 9 q^{59} - 30 q^{61} + 10 q^{63} + 3 q^{65} + q^{67} - 10 q^{69} + 2 q^{71} + 6 q^{73} - q^{75} + 5 q^{79} + 7 q^{81} - 4 q^{83} - q^{85} - 7 q^{87} - q^{89} - 3 q^{91} + 15 q^{93} + 5 q^{95} + 4 q^{97}+O(q^{100}) \) 3 * q - q^3 - 3 * q^5 + 3 * q^7 + 10 * q^9 - 3 * q^13 + q^15 + q^17 - 5 * q^19 - q^21 + 4 * q^23 + 3 * q^25 + 14 * q^27 - 9 * q^29 + 11 * q^31 - 3 * q^35 + q^37 + q^39 + q^41 - 10 * q^43 - 10 * q^45 + 8 * q^47 + 3 * q^49 + 14 * q^51 + 6 * q^53 + 16 * q^57 + 9 * q^59 - 30 * q^61 + 10 * q^63 + 3 * q^65 + q^67 - 10 * q^69 + 2 * q^71 + 6 * q^73 - q^75 + 5 * q^79 + 7 * q^81 - 4 * q^83 - q^85 - 7 * q^87 - q^89 - 3 * q^91 + 15 * q^93 + 5 * q^95 + 4 * q^97

Introduction

L-functions

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