LMFDB - Embedded newform 3640.2.a.n.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
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			1.2
Root			$1.61803$ 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)

$f(q)$	$=$	$q+1.61803 q^{3} +1.00000 q^{5} +1.00000 q^{7} -0.381966 q^{9} +O(q^{10})$	$q+1.61803 q^{3} +1.00000 q^{5} +1.00000 q^{7} -0.381966 q^{9} -6.47214 q^{11} +1.00000 q^{13} +1.61803 q^{15} -4.61803 q^{17} -2.61803 q^{19} +1.61803 q^{21} +0.472136 q^{23} +1.00000 q^{25} -5.47214 q^{27} -2.38197 q^{29} +8.09017 q^{31} -10.4721 q^{33} +1.00000 q^{35} -4.09017 q^{37} +1.61803 q^{39} -8.61803 q^{41} -0.763932 q^{43} -0.381966 q^{45} +4.47214 q^{47} +1.00000 q^{49} -7.47214 q^{51} -4.47214 q^{53} -6.47214 q^{55} -4.23607 q^{57} -6.56231 q^{59} -3.52786 q^{61} -0.381966 q^{63} +1.00000 q^{65} -2.61803 q^{67} +0.763932 q^{69} -2.29180 q^{71} +2.47214 q^{73} +1.61803 q^{75} -6.47214 q^{77} -3.56231 q^{79} -7.70820 q^{81} -6.00000 q^{83} -4.61803 q^{85} -3.85410 q^{87} +7.32624 q^{89} +1.00000 q^{91} +13.0902 q^{93} -2.61803 q^{95} -15.7082 q^{97} +2.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + 2 q^{5} + 2 q^{7} - 3 q^{9}+O(q^{10}) $ 2 * q + q^3 + 2 * q^5 + 2 * q^7 - 3 * q^9	$ 2 q + q^{3} + 2 q^{5} + 2 q^{7} - 3 q^{9} - 4 q^{11} + 2 q^{13} + q^{15} - 7 q^{17} - 3 q^{19} + q^{21} - 8 q^{23} + 2 q^{25} - 2 q^{27} - 7 q^{29} + 5 q^{31} - 12 q^{33} + 2 q^{35} + 3 q^{37} + q^{39} - 15 q^{41} - 6 q^{43} - 3 q^{45} + 2 q^{49} - 6 q^{51} - 4 q^{55} - 4 q^{57} + 7 q^{59} - 16 q^{61} - 3 q^{63} + 2 q^{65} - 3 q^{67} + 6 q^{69} - 18 q^{71} - 4 q^{73} + q^{75} - 4 q^{77} + 13 q^{79} - 2 q^{81} - 12 q^{83} - 7 q^{85} - q^{87} - q^{89} + 2 q^{91} + 15 q^{93} - 3 q^{95} - 18 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q + q^3 + 2 * q^5 + 2 * q^7 - 3 * q^9 - 4 * q^11 + 2 * q^13 + q^15 - 7 * q^17 - 3 * q^19 + q^21 - 8 * q^23 + 2 * q^25 - 2 * q^27 - 7 * q^29 + 5 * q^31 - 12 * q^33 + 2 * q^35 + 3 * q^37 + q^39 - 15 * q^41 - 6 * q^43 - 3 * q^45 + 2 * q^49 - 6 * q^51 - 4 * q^55 - 4 * q^57 + 7 * q^59 - 16 * q^61 - 3 * q^63 + 2 * q^65 - 3 * q^67 + 6 * q^69 - 18 * q^71 - 4 * q^73 + q^75 - 4 * q^77 + 13 * q^79 - 2 * q^81 - 12 * q^83 - 7 * q^85 - q^87 - q^89 + 2 * q^91 + 15 * q^93 - 3 * q^95 - 18 * q^97 - 4 * 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
$3$	1.61803	0.934172	0.467086	−	0.884212i	$-0.345304\pi$
$3$	1.61803	0.934172	0.467086	+	0.884212i	$0.345304\pi$
$4$	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
$9$	−0.381966	−0.127322
$10$	0	0
$11$	−6.47214	−1.95142	−0.975711	−	0.219061i	$-0.929701\pi$
$11$	−6.47214	−1.95142	−0.975711	+	0.219061i	$0.929701\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	1.61803	0.417775
$16$	0	0
$17$	−4.61803	−1.12004	−0.560019	−	0.828480i	$-0.689206\pi$
$17$	−4.61803	−1.12004	−0.560019	+	0.828480i	$0.689206\pi$
$18$	0	0
$19$	−2.61803	−0.600618	−0.300309	−	0.953842i	$-0.597090\pi$
$19$	−2.61803	−0.600618	−0.300309	+	0.953842i	$0.597090\pi$
$20$	0	0
$21$	1.61803	0.353084
$22$	0	0
$23$	0.472136	0.0984472	0.0492236	−	0.998788i	$-0.484325\pi$
$23$	0.472136	0.0984472	0.0492236	+	0.998788i	$0.484325\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.47214	−1.05311
$28$	0	0
$29$	−2.38197	−0.442320	−0.221160	−	0.975238i	$-0.570984\pi$
$29$	−2.38197	−0.442320	−0.221160	+	0.975238i	$0.570984\pi$
$30$	0	0
$31$	8.09017	1.45304	0.726519	−	0.687147i	$-0.241137\pi$
$31$	8.09017	1.45304	0.726519	+	0.687147i	$0.241137\pi$
$32$	0	0
$33$	−10.4721	−1.82296
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−4.09017	−0.672420	−0.336210	−	0.941787i	$-0.609145\pi$
$37$	−4.09017	−0.672420	−0.336210	+	0.941787i	$0.609145\pi$
$38$	0	0
$39$	1.61803	0.259093
$40$	0	0
$41$	−8.61803	−1.34591	−0.672955	−	0.739683i	$-0.734975\pi$
$41$	−8.61803	−1.34591	−0.672955	+	0.739683i	$0.734975\pi$
$42$	0	0
$43$	−0.763932	−0.116499	−0.0582493	−	0.998302i	$-0.518552\pi$
$43$	−0.763932	−0.116499	−0.0582493	+	0.998302i	$0.518552\pi$
$44$	0	0
$45$	−0.381966	−0.0569401
$46$	0	0
$47$	4.47214	0.652328	0.326164	−	0.945313i	$-0.394244\pi$
$47$	4.47214	0.652328	0.326164	+	0.945313i	$0.394244\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−7.47214	−1.04631
$52$	0	0
$53$	−4.47214	−0.614295	−0.307148	−	0.951662i	$-0.599375\pi$
$53$	−4.47214	−0.614295	−0.307148	+	0.951662i	$0.599375\pi$
$54$	0	0
$55$	−6.47214	−0.872703
$56$	0	0
$57$	−4.23607	−0.561081
$58$	0	0
$59$	−6.56231	−0.854339	−0.427170	−	0.904171i	$-0.640489\pi$
$59$	−6.56231	−0.854339	−0.427170	+	0.904171i	$0.640489\pi$
$60$	0	0
$61$	−3.52786	−0.451697	−0.225848	−	0.974162i	$-0.572515\pi$
$61$	−3.52786	−0.451697	−0.225848	+	0.974162i	$0.572515\pi$
$62$	0	0
$63$	−0.381966	−0.0481232
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−2.61803	−0.319844	−0.159922	−	0.987130i	$-0.551124\pi$
$67$	−2.61803	−0.319844	−0.159922	+	0.987130i	$0.551124\pi$
$68$	0	0
$69$	0.763932	0.0919666
$70$	0	0
$71$	−2.29180	−0.271986	−0.135993	−	0.990710i	$-0.543422\pi$
$71$	−2.29180	−0.271986	−0.135993	+	0.990710i	$0.543422\pi$
$72$	0	0
$73$	2.47214	0.289342	0.144671	−	0.989480i	$-0.453788\pi$
$73$	2.47214	0.289342	0.144671	+	0.989480i	$0.453788\pi$
$74$	0	0
$75$	1.61803	0.186834
$76$	0	0
$77$	−6.47214	−0.737568
$78$	0	0
$79$	−3.56231	−0.400791	−0.200395	−	0.979715i	$-0.564223\pi$
$79$	−3.56231	−0.400791	−0.200395	+	0.979715i	$0.564223\pi$
$80$	0	0
$81$	−7.70820	−0.856467
$82$	0	0
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	−4.61803	−0.500896
$86$	0	0
$87$	−3.85410	−0.413203
$88$	0	0
$89$	7.32624	0.776580	0.388290	−	0.921537i	$-0.373066\pi$
$89$	7.32624	0.776580	0.388290	+	0.921537i	$0.373066\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	13.0902	1.35739
$94$	0	0
$95$	−2.61803	−0.268605
$96$	0	0
$97$	−15.7082	−1.59493	−0.797463	−	0.603368i	$-0.793825\pi$
$97$	−15.7082	−1.59493	−0.797463	+	0.603368i	$0.793825\pi$
$98$	0	0
$99$	2.47214	0.248459
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	9.38197	0.924433	0.462216	−	0.886767i	$-0.347054\pi$
$103$	9.38197	0.924433	0.462216	+	0.886767i	$0.347054\pi$
$104$	0	0
$105$	1.61803	0.157904
$106$	0	0
$107$	−16.6525	−1.60986	−0.804928	−	0.593373i	$-0.797796\pi$
$107$	−16.6525	−1.60986	−0.804928	+	0.593373i	$0.797796\pi$
$108$	0	0
$109$	6.94427	0.665141	0.332570	−	0.943078i	$-0.392084\pi$
$109$	6.94427	0.665141	0.332570	+	0.943078i	$0.392084\pi$
$110$	0	0
$111$	−6.61803	−0.628156
$112$	0	0
$113$	−6.76393	−0.636297	−0.318149	−	0.948041i	$-0.603061\pi$
$113$	−6.76393	−0.636297	−0.318149	+	0.948041i	$0.603061\pi$
$114$	0	0
$115$	0.472136	0.0440269
$116$	0	0
$117$	−0.381966	−0.0353128
$118$	0	0
$119$	−4.61803	−0.423334
$120$	0	0
$121$	30.8885	2.80805
$122$	0	0
$123$	−13.9443	−1.25731
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	12.4721	1.10672	0.553362	−	0.832941i	$-0.313345\pi$
$127$	12.4721	1.10672	0.553362	+	0.832941i	$0.313345\pi$
$128$	0	0
$129$	−1.23607	−0.108830
$130$	0	0
$131$	0.472136	0.0412507	0.0206254	−	0.999787i	$-0.493434\pi$
$131$	0.472136	0.0412507	0.0206254	+	0.999787i	$0.493434\pi$
$132$	0	0
$133$	−2.61803	−0.227012
$134$	0	0
$135$	−5.47214	−0.470966
$136$	0	0
$137$	−20.8541	−1.78169	−0.890843	−	0.454311i	$-0.849885\pi$
$137$	−20.8541	−1.78169	−0.890843	+	0.454311i	$0.849885\pi$
$138$	0	0
$139$	−3.70820	−0.314526	−0.157263	−	0.987557i	$-0.550267\pi$
$139$	−3.70820	−0.314526	−0.157263	+	0.987557i	$0.550267\pi$
$140$	0	0
$141$	7.23607	0.609387
$142$	0	0
$143$	−6.47214	−0.541227
$144$	0	0
$145$	−2.38197	−0.197812
$146$	0	0
$147$	1.61803	0.133453
$148$	0	0
$149$	19.8885	1.62933	0.814666	−	0.579930i	$-0.196920\pi$
$149$	19.8885	1.62933	0.814666	+	0.579930i	$0.196920\pi$
$150$	0	0
$151$	15.2361	1.23989	0.619947	−	0.784644i	$-0.287154\pi$
$151$	15.2361	1.23989	0.619947	+	0.784644i	$0.287154\pi$
$152$	0	0
$153$	1.76393	0.142605
$154$	0	0
$155$	8.09017	0.649818
$156$	0	0
$157$	0.0901699	0.00719634	0.00359817	−	0.999994i	$-0.498855\pi$
$157$	0.0901699	0.00719634	0.00359817	+	0.999994i	$0.498855\pi$
$158$	0	0
$159$	−7.23607	−0.573858
$160$	0	0
$161$	0.472136	0.0372095
$162$	0	0
$163$	12.0902	0.946975	0.473488	−	0.880800i	$-0.342995\pi$
$163$	12.0902	0.946975	0.473488	+	0.880800i	$0.342995\pi$
$164$	0	0
$165$	−10.4721	−0.815255
$166$	0	0
$167$	3.23607	0.250414	0.125207	−	0.992131i	$-0.460040\pi$
$167$	3.23607	0.250414	0.125207	+	0.992131i	$0.460040\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−21.0344	−1.59922	−0.799610	−	0.600520i	$-0.794960\pi$
$173$	−21.0344	−1.59922	−0.799610	+	0.600520i	$0.794960\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−10.6180	−0.798100
$178$	0	0
$179$	7.56231	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$179$	7.56231	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$180$	0	0
$181$	−10.7639	−0.800077	−0.400038	−	0.916498i	$-0.631003\pi$
$181$	−10.7639	−0.800077	−0.400038	+	0.916498i	$0.631003\pi$
$182$	0	0
$183$	−5.70820	−0.421963
$184$	0	0
$185$	−4.09017	−0.300715
$186$	0	0
$187$	29.8885	2.18567
$188$	0	0
$189$	−5.47214	−0.398039
$190$	0	0
$191$	18.5623	1.34312	0.671561	−	0.740950i	$-0.265624\pi$
$191$	18.5623	1.34312	0.671561	+	0.740950i	$0.265624\pi$
$192$	0	0
$193$	19.7426	1.42111	0.710553	−	0.703643i	$-0.248445\pi$
$193$	19.7426	1.42111	0.710553	+	0.703643i	$0.248445\pi$
$194$	0	0
$195$	1.61803	0.115870
$196$	0	0
$197$	4.09017	0.291413	0.145706	−	0.989328i	$-0.453455\pi$
$197$	4.09017	0.291413	0.145706	+	0.989328i	$0.453455\pi$
$198$	0	0
$199$	−11.2361	−0.796504	−0.398252	−	0.917276i	$-0.630383\pi$
$199$	−11.2361	−0.796504	−0.398252	+	0.917276i	$0.630383\pi$
$200$	0	0
$201$	−4.23607	−0.298789
$202$	0	0
$203$	−2.38197	−0.167181
$204$	0	0
$205$	−8.61803	−0.601910
$206$	0	0
$207$	−0.180340	−0.0125345
$208$	0	0
$209$	16.9443	1.17206
$210$	0	0
$211$	−21.0902	−1.45191	−0.725954	−	0.687744i	$-0.758601\pi$
$211$	−21.0902	−1.45191	−0.725954	+	0.687744i	$0.758601\pi$
$212$	0	0
$213$	−3.70820	−0.254082
$214$	0	0
$215$	−0.763932	−0.0520997
$216$	0	0
$217$	8.09017	0.549197
$218$	0	0
$219$	4.00000	0.270295
$220$	0	0
$221$	−4.61803	−0.310643
$222$	0	0
$223$	−19.7082	−1.31976	−0.659879	−	0.751371i	$-0.729393\pi$
$223$	−19.7082	−1.31976	−0.659879	+	0.751371i	$0.729393\pi$
$224$	0	0
$225$	−0.381966	−0.0254644
$226$	0	0
$227$	26.0000	1.72568	0.862840	−	0.505477i	$-0.168683\pi$
$227$	26.0000	1.72568	0.862840	+	0.505477i	$0.168683\pi$
$228$	0	0
$229$	−13.5623	−0.896222	−0.448111	−	0.893978i	$-0.647903\pi$
$229$	−13.5623	−0.896222	−0.448111	+	0.893978i	$0.647903\pi$
$230$	0	0
$231$	−10.4721	−0.689016
$232$	0	0
$233$	10.4721	0.686052	0.343026	−	0.939326i	$-0.388548\pi$
$233$	10.4721	0.686052	0.343026	+	0.939326i	$0.388548\pi$
$234$	0	0
$235$	4.47214	0.291730
$236$	0	0
$237$	−5.76393	−0.374408
$238$	0	0
$239$	14.0000	0.905585	0.452792	−	0.891616i	$-0.350428\pi$
$239$	14.0000	0.905585	0.452792	+	0.891616i	$0.350428\pi$
$240$	0	0
$241$	−5.85410	−0.377096	−0.188548	−	0.982064i	$-0.560378\pi$
$241$	−5.85410	−0.377096	−0.188548	+	0.982064i	$0.560378\pi$
$242$	0	0
$243$	3.94427	0.253025
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−2.61803	−0.166582
$248$	0	0
$249$	−9.70820	−0.615232
$250$	0	0
$251$	−5.70820	−0.360299	−0.180149	−	0.983639i	$-0.557658\pi$
$251$	−5.70820	−0.360299	−0.180149	+	0.983639i	$0.557658\pi$
$252$	0	0
$253$	−3.05573	−0.192112
$254$	0	0
$255$	−7.47214	−0.467923
$256$	0	0
$257$	−19.8885	−1.24061	−0.620307	−	0.784359i	$-0.712992\pi$
$257$	−19.8885	−1.24061	−0.620307	+	0.784359i	$0.712992\pi$
$258$	0	0
$259$	−4.09017	−0.254151
$260$	0	0
$261$	0.909830	0.0563171
$262$	0	0
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	0	0
$265$	−4.47214	−0.274721
$266$	0	0
$267$	11.8541	0.725459
$268$	0	0
$269$	−2.47214	−0.150729	−0.0753644	−	0.997156i	$-0.524012\pi$
$269$	−2.47214	−0.150729	−0.0753644	+	0.997156i	$0.524012\pi$
$270$	0	0
$271$	−0.944272	−0.0573604	−0.0286802	−	0.999589i	$-0.509130\pi$
$271$	−0.944272	−0.0573604	−0.0286802	+	0.999589i	$0.509130\pi$
$272$	0	0
$273$	1.61803	0.0979279
$274$	0	0
$275$	−6.47214	−0.390284
$276$	0	0
$277$	−1.52786	−0.0918005	−0.0459002	−	0.998946i	$-0.514616\pi$
$277$	−1.52786	−0.0918005	−0.0459002	+	0.998946i	$0.514616\pi$
$278$	0	0
$279$	−3.09017	−0.185004
$280$	0	0
$281$	4.18034	0.249378	0.124689	−	0.992196i	$-0.460207\pi$
$281$	4.18034	0.249378	0.124689	+	0.992196i	$0.460207\pi$
$282$	0	0
$283$	30.9787	1.84149	0.920747	−	0.390161i	$-0.127581\pi$
$283$	30.9787	1.84149	0.920747	+	0.390161i	$0.127581\pi$
$284$	0	0
$285$	−4.23607	−0.250923
$286$	0	0
$287$	−8.61803	−0.508706
$288$	0	0
$289$	4.32624	0.254485
$290$	0	0
$291$	−25.4164	−1.48994
$292$	0	0
$293$	−2.29180	−0.133888	−0.0669441	−	0.997757i	$-0.521325\pi$
$293$	−2.29180	−0.133888	−0.0669441	+	0.997757i	$0.521325\pi$
$294$	0	0
$295$	−6.56231	−0.382072
$296$	0	0
$297$	35.4164	2.05507
$298$	0	0
$299$	0.472136	0.0273043
$300$	0	0
$301$	−0.763932	−0.0440323
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3.52786	−0.202005
$306$	0	0
$307$	−6.00000	−0.342438	−0.171219	−	0.985233i	$-0.554771\pi$
$307$	−6.00000	−0.342438	−0.171219	+	0.985233i	$0.554771\pi$
$308$	0	0
$309$	15.1803	0.863579
$310$	0	0
$311$	6.18034	0.350455	0.175227	−	0.984528i	$-0.443934\pi$
$311$	6.18034	0.350455	0.175227	+	0.984528i	$0.443934\pi$
$312$	0	0
$313$	−0.854102	−0.0482767	−0.0241383	−	0.999709i	$-0.507684\pi$
$313$	−0.854102	−0.0482767	−0.0241383	+	0.999709i	$0.507684\pi$
$314$	0	0
$315$	−0.381966	−0.0215213
$316$	0	0
$317$	−5.41641	−0.304216	−0.152108	−	0.988364i	$-0.548606\pi$
$317$	−5.41641	−0.304216	−0.152108	+	0.988364i	$0.548606\pi$
$318$	0	0
$319$	15.4164	0.863153
$320$	0	0
$321$	−26.9443	−1.50388
$322$	0	0
$323$	12.0902	0.672715
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	11.2361	0.621356
$328$	0	0
$329$	4.47214	0.246557
$330$	0	0
$331$	−24.0000	−1.31916	−0.659580	−	0.751635i	$-0.729266\pi$
$331$	−24.0000	−1.31916	−0.659580	+	0.751635i	$0.729266\pi$
$332$	0	0
$333$	1.56231	0.0856138
$334$	0	0
$335$	−2.61803	−0.143038
$336$	0	0
$337$	−5.41641	−0.295051	−0.147525	−	0.989058i	$-0.547131\pi$
$337$	−5.41641	−0.295051	−0.147525	+	0.989058i	$0.547131\pi$
$338$	0	0
$339$	−10.9443	−0.594411
$340$	0	0
$341$	−52.3607	−2.83549
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0.763932	0.0411287
$346$	0	0
$347$	31.8885	1.71187	0.855933	−	0.517086i	$-0.172983\pi$
$347$	31.8885	1.71187	0.855933	+	0.517086i	$0.172983\pi$
$348$	0	0
$349$	29.7984	1.59507	0.797535	−	0.603272i	$-0.206137\pi$
$349$	29.7984	1.59507	0.797535	+	0.603272i	$0.206137\pi$
$350$	0	0
$351$	−5.47214	−0.292081
$352$	0	0
$353$	−10.9443	−0.582505	−0.291252	−	0.956646i	$-0.594072\pi$
$353$	−10.9443	−0.582505	−0.291252	+	0.956646i	$0.594072\pi$
$354$	0	0
$355$	−2.29180	−0.121636
$356$	0	0
$357$	−7.47214	−0.395467
$358$	0	0
$359$	7.12461	0.376023	0.188011	−	0.982167i	$-0.439796\pi$
$359$	7.12461	0.376023	0.188011	+	0.982167i	$0.439796\pi$
$360$	0	0
$361$	−12.1459	−0.639258
$362$	0	0
$363$	49.9787	2.62320
$364$	0	0
$365$	2.47214	0.129398
$366$	0	0
$367$	4.94427	0.258089	0.129044	−	0.991639i	$-0.458809\pi$
$367$	4.94427	0.258089	0.129044	+	0.991639i	$0.458809\pi$
$368$	0	0
$369$	3.29180	0.171364
$370$	0	0
$371$	−4.47214	−0.232182
$372$	0	0
$373$	−11.1246	−0.576011	−0.288005	−	0.957629i	$-0.592992\pi$
$373$	−11.1246	−0.576011	−0.288005	+	0.957629i	$0.592992\pi$
$374$	0	0
$375$	1.61803	0.0835549
$376$	0	0
$377$	−2.38197	−0.122677
$378$	0	0
$379$	3.88854	0.199741	0.0998705	−	0.995000i	$-0.468157\pi$
$379$	3.88854	0.199741	0.0998705	+	0.995000i	$0.468157\pi$
$380$	0	0
$381$	20.1803	1.03387
$382$	0	0
$383$	−26.1803	−1.33775	−0.668876	−	0.743374i	$-0.733224\pi$
$383$	−26.1803	−1.33775	−0.668876	+	0.743374i	$0.733224\pi$
$384$	0	0
$385$	−6.47214	−0.329851
$386$	0	0
$387$	0.291796	0.0148328
$388$	0	0
$389$	−32.6180	−1.65380	−0.826900	−	0.562349i	$-0.809898\pi$
$389$	−32.6180	−1.65380	−0.826900	+	0.562349i	$0.809898\pi$
$390$	0	0
$391$	−2.18034	−0.110265
$392$	0	0
$393$	0.763932	0.0385353
$394$	0	0
$395$	−3.56231	−0.179239
$396$	0	0
$397$	−4.18034	−0.209805	−0.104903	−	0.994482i	$-0.533453\pi$
$397$	−4.18034	−0.209805	−0.104903	+	0.994482i	$0.533453\pi$
$398$	0	0
$399$	−4.23607	−0.212069
$400$	0	0
$401$	−26.0689	−1.30182	−0.650909	−	0.759156i	$-0.725612\pi$
$401$	−26.0689	−1.30182	−0.650909	+	0.759156i	$0.725612\pi$
$402$	0	0
$403$	8.09017	0.403000
$404$	0	0
$405$	−7.70820	−0.383024
$406$	0	0
$407$	26.4721	1.31218
$408$	0	0
$409$	26.3607	1.30345	0.651726	−	0.758455i	$-0.274045\pi$
$409$	26.3607	1.30345	0.651726	+	0.758455i	$0.274045\pi$
$410$	0	0
$411$	−33.7426	−1.66440
$412$	0	0
$413$	−6.56231	−0.322910
$414$	0	0
$415$	−6.00000	−0.294528
$416$	0	0
$417$	−6.00000	−0.293821
$418$	0	0
$419$	22.1803	1.08358	0.541790	−	0.840514i	$-0.317747\pi$
$419$	22.1803	1.08358	0.541790	+	0.840514i	$0.317747\pi$
$420$	0	0
$421$	−35.8885	−1.74910	−0.874550	−	0.484935i	$-0.838843\pi$
$421$	−35.8885	−1.74910	−0.874550	+	0.484935i	$0.838843\pi$
$422$	0	0
$423$	−1.70820	−0.0830557
$424$	0	0
$425$	−4.61803	−0.224008
$426$	0	0
$427$	−3.52786	−0.170725
$428$	0	0
$429$	−10.4721	−0.505599
$430$	0	0
$431$	2.76393	0.133134	0.0665670	−	0.997782i	$-0.478795\pi$
$431$	2.76393	0.133134	0.0665670	+	0.997782i	$0.478795\pi$
$432$	0	0
$433$	−34.5066	−1.65828	−0.829140	−	0.559041i	$-0.811170\pi$
$433$	−34.5066	−1.65828	−0.829140	+	0.559041i	$0.811170\pi$
$434$	0	0
$435$	−3.85410	−0.184790
$436$	0	0
$437$	−1.23607	−0.0591292
$438$	0	0
$439$	−28.3607	−1.35358	−0.676791	−	0.736175i	$-0.736630\pi$
$439$	−28.3607	−1.35358	−0.676791	+	0.736175i	$0.736630\pi$
$440$	0	0
$441$	−0.381966	−0.0181889
$442$	0	0
$443$	−10.9443	−0.519978	−0.259989	−	0.965612i	$-0.583719\pi$
$443$	−10.9443	−0.519978	−0.259989	+	0.965612i	$0.583719\pi$
$444$	0	0
$445$	7.32624	0.347297
$446$	0	0
$447$	32.1803	1.52208
$448$	0	0
$449$	39.1246	1.84640	0.923202	−	0.384314i	$-0.125562\pi$
$449$	39.1246	1.84640	0.923202	+	0.384314i	$0.125562\pi$
$450$	0	0
$451$	55.7771	2.62644
$452$	0	0
$453$	24.6525	1.15827
$454$	0	0
$455$	1.00000	0.0468807
$456$	0	0
$457$	−4.85410	−0.227065	−0.113533	−	0.993534i	$-0.536217\pi$
$457$	−4.85410	−0.227065	−0.113533	+	0.993534i	$0.536217\pi$
$458$	0	0
$459$	25.2705	1.17953
$460$	0	0
$461$	−23.9098	−1.11359	−0.556796	−	0.830649i	$-0.687969\pi$
$461$	−23.9098	−1.11359	−0.556796	+	0.830649i	$0.687969\pi$
$462$	0	0
$463$	33.7426	1.56815	0.784077	−	0.620664i	$-0.213137\pi$
$463$	33.7426	1.56815	0.784077	+	0.620664i	$0.213137\pi$
$464$	0	0
$465$	13.0902	0.607042
$466$	0	0
$467$	36.5066	1.68932	0.844661	−	0.535301i	$-0.179802\pi$
$467$	36.5066	1.68932	0.844661	+	0.535301i	$0.179802\pi$
$468$	0	0
$469$	−2.61803	−0.120890
$470$	0	0
$471$	0.145898	0.00672263
$472$	0	0
$473$	4.94427	0.227338
$474$	0	0
$475$	−2.61803	−0.120124
$476$	0	0
$477$	1.70820	0.0782133
$478$	0	0
$479$	29.9787	1.36976	0.684881	−	0.728655i	$-0.259854\pi$
$479$	29.9787	1.36976	0.684881	+	0.728655i	$0.259854\pi$
$480$	0	0
$481$	−4.09017	−0.186496
$482$	0	0
$483$	0.763932	0.0347601
$484$	0	0
$485$	−15.7082	−0.713273
$486$	0	0
$487$	−5.79837	−0.262749	−0.131375	−	0.991333i	$-0.541939\pi$
$487$	−5.79837	−0.262749	−0.131375	+	0.991333i	$0.541939\pi$
$488$	0	0
$489$	19.5623	0.884638
$490$	0	0
$491$	−12.5836	−0.567890	−0.283945	−	0.958841i	$-0.591643\pi$
$491$	−12.5836	−0.567890	−0.283945	+	0.958841i	$0.591643\pi$
$492$	0	0
$493$	11.0000	0.495415
$494$	0	0
$495$	2.47214	0.111114
$496$	0	0
$497$	−2.29180	−0.102801
$498$	0	0
$499$	22.4721	1.00599	0.502995	−	0.864289i	$-0.332231\pi$
$499$	22.4721	1.00599	0.502995	+	0.864289i	$0.332231\pi$
$500$	0	0
$501$	5.23607	0.233930
$502$	0	0
$503$	−1.88854	−0.0842060	−0.0421030	−	0.999113i	$-0.513406\pi$
$503$	−1.88854	−0.0842060	−0.0421030	+	0.999113i	$0.513406\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.61803	0.0718594
$508$	0	0
$509$	29.2705	1.29739	0.648696	−	0.761047i	$-0.275315\pi$
$509$	29.2705	1.29739	0.648696	+	0.761047i	$0.275315\pi$
$510$	0	0
$511$	2.47214	0.109361
$512$	0	0
$513$	14.3262	0.632519
$514$	0	0
$515$	9.38197	0.413419
$516$	0	0
$517$	−28.9443	−1.27297
$518$	0	0
$519$	−34.0344	−1.49395
$520$	0	0
$521$	−44.3607	−1.94348	−0.971738	−	0.236061i	$-0.924144\pi$
$521$	−44.3607	−1.94348	−0.971738	+	0.236061i	$0.924144\pi$
$522$	0	0
$523$	24.0000	1.04945	0.524723	−	0.851273i	$-0.324169\pi$
$523$	24.0000	1.04945	0.524723	+	0.851273i	$0.324169\pi$
$524$	0	0
$525$	1.61803	0.0706168
$526$	0	0
$527$	−37.3607	−1.62746
$528$	0	0
$529$	−22.7771	−0.990308
$530$	0	0
$531$	2.50658	0.108776
$532$	0	0
$533$	−8.61803	−0.373288
$534$	0	0
$535$	−16.6525	−0.719949
$536$	0	0
$537$	12.2361	0.528025
$538$	0	0
$539$	−6.47214	−0.278775
$540$	0	0
$541$	−2.29180	−0.0985320	−0.0492660	−	0.998786i	$-0.515688\pi$
$541$	−2.29180	−0.0985320	−0.0492660	+	0.998786i	$0.515688\pi$
$542$	0	0
$543$	−17.4164	−0.747410
$544$	0	0
$545$	6.94427	0.297460
$546$	0	0
$547$	−40.2492	−1.72093	−0.860466	−	0.509507i	$-0.829828\pi$
$547$	−40.2492	−1.72093	−0.860466	+	0.509507i	$0.829828\pi$
$548$	0	0
$549$	1.34752	0.0575109
$550$	0	0
$551$	6.23607	0.265665
$552$	0	0
$553$	−3.56231	−0.151485
$554$	0	0
$555$	−6.61803	−0.280920
$556$	0	0
$557$	31.7426	1.34498	0.672490	−	0.740107i	$-0.265225\pi$
$557$	31.7426	1.34498	0.672490	+	0.740107i	$0.265225\pi$
$558$	0	0
$559$	−0.763932	−0.0323109
$560$	0	0
$561$	48.3607	2.04179
$562$	0	0
$563$	30.3262	1.27810	0.639049	−	0.769166i	$-0.279328\pi$
$563$	30.3262	1.27810	0.639049	+	0.769166i	$0.279328\pi$
$564$	0	0
$565$	−6.76393	−0.284561
$566$	0	0
$567$	−7.70820	−0.323714
$568$	0	0
$569$	2.96556	0.124323	0.0621613	−	0.998066i	$-0.480201\pi$
$569$	2.96556	0.124323	0.0621613	+	0.998066i	$0.480201\pi$
$570$	0	0
$571$	17.4377	0.729745	0.364872	−	0.931058i	$-0.381113\pi$
$571$	17.4377	0.729745	0.364872	+	0.931058i	$0.381113\pi$
$572$	0	0
$573$	30.0344	1.25471
$574$	0	0
$575$	0.472136	0.0196894
$576$	0	0
$577$	46.4721	1.93466	0.967330	−	0.253520i	$-0.0815883\pi$
$577$	46.4721	1.93466	0.967330	+	0.253520i	$0.0815883\pi$
$578$	0	0
$579$	31.9443	1.32756
$580$	0	0
$581$	−6.00000	−0.248922
$582$	0	0
$583$	28.9443	1.19875
$584$	0	0
$585$	−0.381966	−0.0157924
$586$	0	0
$587$	−25.5967	−1.05649	−0.528245	−	0.849092i	$-0.677150\pi$
$587$	−25.5967	−1.05649	−0.528245	+	0.849092i	$0.677150\pi$
$588$	0	0
$589$	−21.1803	−0.872721
$590$	0	0
$591$	6.61803	0.272230
$592$	0	0
$593$	−18.8328	−0.773371	−0.386686	−	0.922212i	$-0.626380\pi$
$593$	−18.8328	−0.773371	−0.386686	+	0.922212i	$0.626380\pi$
$594$	0	0
$595$	−4.61803	−0.189321
$596$	0	0
$597$	−18.1803	−0.744072
$598$	0	0
$599$	−24.9443	−1.01920	−0.509598	−	0.860413i	$-0.670206\pi$
$599$	−24.9443	−1.01920	−0.509598	+	0.860413i	$0.670206\pi$
$600$	0	0
$601$	11.7082	0.477588	0.238794	−	0.971070i	$-0.423248\pi$
$601$	11.7082	0.477588	0.238794	+	0.971070i	$0.423248\pi$
$602$	0	0
$603$	1.00000	0.0407231
$604$	0	0
$605$	30.8885	1.25580
$606$	0	0
$607$	−40.2705	−1.63453	−0.817265	−	0.576262i	$-0.804511\pi$
$607$	−40.2705	−1.63453	−0.817265	+	0.576262i	$0.804511\pi$
$608$	0	0
$609$	−3.85410	−0.156176
$610$	0	0
$611$	4.47214	0.180923
$612$	0	0
$613$	2.58359	0.104350	0.0521752	−	0.998638i	$-0.483385\pi$
$613$	2.58359	0.104350	0.0521752	+	0.998638i	$0.483385\pi$
$614$	0	0
$615$	−13.9443	−0.562287
$616$	0	0
$617$	7.09017	0.285439	0.142720	−	0.989763i	$-0.454415\pi$
$617$	7.09017	0.285439	0.142720	+	0.989763i	$0.454415\pi$
$618$	0	0
$619$	5.43769	0.218559	0.109280	−	0.994011i	$-0.465146\pi$
$619$	5.43769	0.218559	0.109280	+	0.994011i	$0.465146\pi$
$620$	0	0
$621$	−2.58359	−0.103676
$622$	0	0
$623$	7.32624	0.293520
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	27.4164	1.09491
$628$	0	0
$629$	18.8885	0.753136
$630$	0	0
$631$	19.1246	0.761339	0.380669	−	0.924711i	$-0.375694\pi$
$631$	19.1246	0.761339	0.380669	+	0.924711i	$0.375694\pi$
$632$	0	0
$633$	−34.1246	−1.35633
$634$	0	0
$635$	12.4721	0.494942
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	0.875388	0.0346298
$640$	0	0
$641$	−4.67376	−0.184603	−0.0923013	−	0.995731i	$-0.529422\pi$
$641$	−4.67376	−0.184603	−0.0923013	+	0.995731i	$0.529422\pi$
$642$	0	0
$643$	9.41641	0.371347	0.185673	−	0.982612i	$-0.440553\pi$
$643$	9.41641	0.371347	0.185673	+	0.982612i	$0.440553\pi$
$644$	0	0
$645$	−1.23607	−0.0486701
$646$	0	0
$647$	22.5623	0.887016	0.443508	−	0.896270i	$-0.353734\pi$
$647$	22.5623	0.887016	0.443508	+	0.896270i	$0.353734\pi$
$648$	0	0
$649$	42.4721	1.66718
$650$	0	0
$651$	13.0902	0.513044
$652$	0	0
$653$	26.6525	1.04299	0.521496	−	0.853254i	$-0.325374\pi$
$653$	26.6525	1.04299	0.521496	+	0.853254i	$0.325374\pi$
$654$	0	0
$655$	0.472136	0.0184479
$656$	0	0
$657$	−0.944272	−0.0368396
$658$	0	0
$659$	−37.0902	−1.44483	−0.722414	−	0.691461i	$-0.756967\pi$
$659$	−37.0902	−1.44483	−0.722414	+	0.691461i	$0.756967\pi$
$660$	0	0
$661$	−49.6312	−1.93043	−0.965215	−	0.261458i	$-0.915797\pi$
$661$	−49.6312	−1.93043	−0.965215	+	0.261458i	$0.915797\pi$
$662$	0	0
$663$	−7.47214	−0.290194
$664$	0	0
$665$	−2.61803	−0.101523
$666$	0	0
$667$	−1.12461	−0.0435451
$668$	0	0
$669$	−31.8885	−1.23288
$670$	0	0
$671$	22.8328	0.881451
$672$	0	0
$673$	6.11146	0.235579	0.117790	−	0.993039i	$-0.462419\pi$
$673$	6.11146	0.235579	0.117790	+	0.993039i	$0.462419\pi$
$674$	0	0
$675$	−5.47214	−0.210623
$676$	0	0
$677$	13.4164	0.515634	0.257817	−	0.966194i	$-0.416997\pi$
$677$	13.4164	0.515634	0.257817	+	0.966194i	$0.416997\pi$
$678$	0	0
$679$	−15.7082	−0.602826
$680$	0	0
$681$	42.0689	1.61208
$682$	0	0
$683$	−22.0344	−0.843124	−0.421562	−	0.906799i	$-0.638518\pi$
$683$	−22.0344	−0.843124	−0.421562	+	0.906799i	$0.638518\pi$
$684$	0	0
$685$	−20.8541	−0.796794
$686$	0	0
$687$	−21.9443	−0.837226
$688$	0	0
$689$	−4.47214	−0.170375
$690$	0	0
$691$	−21.2148	−0.807048	−0.403524	−	0.914969i	$-0.632215\pi$
$691$	−21.2148	−0.807048	−0.403524	+	0.914969i	$0.632215\pi$
$692$	0	0
$693$	2.47214	0.0939087
$694$	0	0
$695$	−3.70820	−0.140660
$696$	0	0
$697$	39.7984	1.50747
$698$	0	0
$699$	16.9443	0.640891
$700$	0	0
$701$	−36.6869	−1.38565	−0.692823	−	0.721108i	$-0.743633\pi$
$701$	−36.6869	−1.38565	−0.692823	+	0.721108i	$0.743633\pi$
$702$	0	0
$703$	10.7082	0.403868
$704$	0	0
$705$	7.23607	0.272526
$706$	0	0
$707$	0	0
$708$	0	0
$709$	23.4164	0.879422	0.439711	−	0.898139i	$-0.355081\pi$
$709$	23.4164	0.879422	0.439711	+	0.898139i	$0.355081\pi$
$710$	0	0
$711$	1.36068	0.0510295
$712$	0	0
$713$	3.81966	0.143047
$714$	0	0
$715$	−6.47214	−0.242044
$716$	0	0
$717$	22.6525	0.845972
$718$	0	0
$719$	21.5967	0.805423	0.402711	−	0.915327i	$-0.368068\pi$
$719$	21.5967	0.805423	0.402711	+	0.915327i	$0.368068\pi$
$720$	0	0
$721$	9.38197	0.349403
$722$	0	0
$723$	−9.47214	−0.352273
$724$	0	0
$725$	−2.38197	−0.0884640
$726$	0	0
$727$	14.9656	0.555042	0.277521	−	0.960720i	$-0.410487\pi$
$727$	14.9656	0.555042	0.277521	+	0.960720i	$0.410487\pi$
$728$	0	0
$729$	29.5066	1.09284
$730$	0	0
$731$	3.52786	0.130483
$732$	0	0
$733$	−30.9443	−1.14295	−0.571476	−	0.820619i	$-0.693629\pi$
$733$	−30.9443	−1.14295	−0.571476	+	0.820619i	$0.693629\pi$
$734$	0	0
$735$	1.61803	0.0596821
$736$	0	0
$737$	16.9443	0.624150
$738$	0	0
$739$	−37.5967	−1.38302	−0.691509	−	0.722367i	$-0.743054\pi$
$739$	−37.5967	−1.38302	−0.691509	+	0.722367i	$0.743054\pi$
$740$	0	0
$741$	−4.23607	−0.155616
$742$	0	0
$743$	−30.9787	−1.13650	−0.568249	−	0.822856i	$-0.692379\pi$
$743$	−30.9787	−1.13650	−0.568249	+	0.822856i	$0.692379\pi$
$744$	0	0
$745$	19.8885	0.728660
$746$	0	0
$747$	2.29180	0.0838524
$748$	0	0
$749$	−16.6525	−0.608468
$750$	0	0
$751$	−41.3951	−1.51053	−0.755265	−	0.655420i	$-0.772492\pi$
$751$	−41.3951	−1.51053	−0.755265	+	0.655420i	$0.772492\pi$
$752$	0	0
$753$	−9.23607	−0.336581
$754$	0	0
$755$	15.2361	0.554497
$756$	0	0
$757$	−42.0000	−1.52652	−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000	−1.52652	−0.763258	+	0.646094i	$0.776401\pi$
$758$	0	0
$759$	−4.94427	−0.179466
$760$	0	0
$761$	17.4377	0.632116	0.316058	−	0.948740i	$-0.397641\pi$
$761$	17.4377	0.632116	0.316058	+	0.948740i	$0.397641\pi$
$762$	0	0
$763$	6.94427	0.251400
$764$	0	0
$765$	1.76393	0.0637751
$766$	0	0
$767$	−6.56231	−0.236951
$768$	0	0
$769$	−48.4721	−1.74795	−0.873975	−	0.485971i	$-0.838466\pi$
$769$	−48.4721	−1.74795	−0.873975	+	0.485971i	$0.838466\pi$
$770$	0	0
$771$	−32.1803	−1.15895
$772$	0	0
$773$	−11.8885	−0.427601	−0.213801	−	0.976877i	$-0.568584\pi$
$773$	−11.8885	−0.427601	−0.213801	+	0.976877i	$0.568584\pi$
$774$	0	0
$775$	8.09017	0.290607
$776$	0	0
$777$	−6.61803	−0.237421
$778$	0	0
$779$	22.5623	0.808378
$780$	0	0
$781$	14.8328	0.530760
$782$	0	0
$783$	13.0344	0.465813
$784$	0	0
$785$	0.0901699	0.00321830
$786$	0	0
$787$	46.4721	1.65655	0.828276	−	0.560320i	$-0.189322\pi$
$787$	46.4721	1.65655	0.828276	+	0.560320i	$0.189322\pi$
$788$	0	0
$789$	−9.70820	−0.345621
$790$	0	0
$791$	−6.76393	−0.240498
$792$	0	0
$793$	−3.52786	−0.125278
$794$	0	0
$795$	−7.23607	−0.256637
$796$	0	0
$797$	8.45085	0.299344	0.149672	−	0.988736i	$-0.452178\pi$
$797$	8.45085	0.299344	0.149672	+	0.988736i	$0.452178\pi$
$798$	0	0
$799$	−20.6525	−0.730632
$800$	0	0
$801$	−2.79837	−0.0988757
$802$	0	0
$803$	−16.0000	−0.564628
$804$	0	0
$805$	0.472136	0.0166406
$806$	0	0
$807$	−4.00000	−0.140807
$808$	0	0
$809$	−52.1033	−1.83186	−0.915928	−	0.401343i	$-0.868543\pi$
$809$	−52.1033	−1.83186	−0.915928	+	0.401343i	$0.868543\pi$
$810$	0	0
$811$	23.0557	0.809596	0.404798	−	0.914406i	$-0.367342\pi$
$811$	23.0557	0.809596	0.404798	+	0.914406i	$0.367342\pi$
$812$	0	0
$813$	−1.52786	−0.0535845
$814$	0	0
$815$	12.0902	0.423500
$816$	0	0
$817$	2.00000	0.0699711
$818$	0	0
$819$	−0.381966	−0.0133470
$820$	0	0
$821$	−4.76393	−0.166262	−0.0831312	−	0.996539i	$-0.526492\pi$
$821$	−4.76393	−0.166262	−0.0831312	+	0.996539i	$0.526492\pi$
$822$	0	0
$823$	39.7771	1.38654	0.693271	−	0.720677i	$-0.256169\pi$
$823$	39.7771	1.38654	0.693271	+	0.720677i	$0.256169\pi$
$824$	0	0
$825$	−10.4721	−0.364593
$826$	0	0
$827$	4.94427	0.171929	0.0859646	−	0.996298i	$-0.472603\pi$
$827$	4.94427	0.171929	0.0859646	+	0.996298i	$0.472603\pi$
$828$	0	0
$829$	6.11146	0.212260	0.106130	−	0.994352i	$-0.466154\pi$
$829$	6.11146	0.212260	0.106130	+	0.994352i	$0.466154\pi$
$830$	0	0
$831$	−2.47214	−0.0857574
$832$	0	0
$833$	−4.61803	−0.160005
$834$	0	0
$835$	3.23607	0.111989
$836$	0	0
$837$	−44.2705	−1.53021
$838$	0	0
$839$	−23.7771	−0.820876	−0.410438	−	0.911888i	$-0.634624\pi$
$839$	−23.7771	−0.820876	−0.410438	+	0.911888i	$0.634624\pi$
$840$	0	0
$841$	−23.3262	−0.804353
$842$	0	0
$843$	6.76393	0.232962
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	30.8885	1.06134
$848$	0	0
$849$	50.1246	1.72027
$850$	0	0
$851$	−1.93112	−0.0661978
$852$	0	0
$853$	10.4721	0.358559	0.179280	−	0.983798i	$-0.442623\pi$
$853$	10.4721	0.358559	0.179280	+	0.983798i	$0.442623\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	0	0
$857$	3.72949	0.127397	0.0636985	−	0.997969i	$-0.479710\pi$
$857$	3.72949	0.127397	0.0636985	+	0.997969i	$0.479710\pi$
$858$	0	0
$859$	21.2361	0.724565	0.362283	−	0.932068i	$-0.381998\pi$
$859$	21.2361	0.724565	0.362283	+	0.932068i	$0.381998\pi$
$860$	0	0
$861$	−13.9443	−0.475220
$862$	0	0
$863$	−16.7984	−0.571823	−0.285912	−	0.958256i	$-0.592296\pi$
$863$	−16.7984	−0.571823	−0.285912	+	0.958256i	$0.592296\pi$
$864$	0	0
$865$	−21.0344	−0.715192
$866$	0	0
$867$	7.00000	0.237732
$868$	0	0
$869$	23.0557	0.782112
$870$	0	0
$871$	−2.61803	−0.0887087
$872$	0	0
$873$	6.00000	0.203069
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	23.4508	0.791879	0.395939	−	0.918277i	$-0.370419\pi$
$877$	23.4508	0.791879	0.395939	+	0.918277i	$0.370419\pi$
$878$	0	0
$879$	−3.70820	−0.125075
$880$	0	0
$881$	−41.9574	−1.41358	−0.706791	−	0.707423i	$-0.749858\pi$
$881$	−41.9574	−1.41358	−0.706791	+	0.707423i	$0.749858\pi$
$882$	0	0
$883$	29.8885	1.00583	0.502915	−	0.864336i	$-0.332261\pi$
$883$	29.8885	1.00583	0.502915	+	0.864336i	$0.332261\pi$
$884$	0	0
$885$	−10.6180	−0.356921
$886$	0	0
$887$	24.6738	0.828464	0.414232	−	0.910171i	$-0.364050\pi$
$887$	24.6738	0.828464	0.414232	+	0.910171i	$0.364050\pi$
$888$	0	0
$889$	12.4721	0.418302
$890$	0	0
$891$	49.8885	1.67133
$892$	0	0
$893$	−11.7082	−0.391800
$894$	0	0
$895$	7.56231	0.252780
$896$	0	0
$897$	0.763932	0.0255069
$898$	0	0
$899$	−19.2705	−0.642707
$900$	0	0
$901$	20.6525	0.688034
$902$	0	0
$903$	−1.23607	−0.0411338
$904$	0	0
$905$	−10.7639	−0.357805
$906$	0	0
$907$	16.0000	0.531271	0.265636	−	0.964073i	$-0.414418\pi$
$907$	16.0000	0.531271	0.265636	+	0.964073i	$0.414418\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−16.4377	−0.544605	−0.272303	−	0.962212i	$-0.587785\pi$
$911$	−16.4377	−0.544605	−0.272303	+	0.962212i	$0.587785\pi$
$912$	0	0
$913$	38.8328	1.28518
$914$	0	0
$915$	−5.70820	−0.188707
$916$	0	0
$917$	0.472136	0.0155913
$918$	0	0
$919$	32.4508	1.07045	0.535227	−	0.844708i	$-0.320226\pi$
$919$	32.4508	1.07045	0.535227	+	0.844708i	$0.320226\pi$
$920$	0	0
$921$	−9.70820	−0.319896
$922$	0	0
$923$	−2.29180	−0.0754354
$924$	0	0
$925$	−4.09017	−0.134484
$926$	0	0
$927$	−3.58359	−0.117701
$928$	0	0
$929$	35.3262	1.15902	0.579508	−	0.814966i	$-0.303245\pi$
$929$	35.3262	1.15902	0.579508	+	0.814966i	$0.303245\pi$
$930$	0	0
$931$	−2.61803	−0.0858026
$932$	0	0
$933$	10.0000	0.327385
$934$	0	0
$935$	29.8885	0.977460
$936$	0	0
$937$	−37.9230	−1.23889	−0.619445	−	0.785040i	$-0.712642\pi$
$937$	−37.9230	−1.23889	−0.619445	+	0.785040i	$0.712642\pi$
$938$	0	0
$939$	−1.38197	−0.0450988
$940$	0	0
$941$	12.6869	0.413582	0.206791	−	0.978385i	$-0.433698\pi$
$941$	12.6869	0.413582	0.206791	+	0.978385i	$0.433698\pi$
$942$	0	0
$943$	−4.06888	−0.132501
$944$	0	0
$945$	−5.47214	−0.178009
$946$	0	0
$947$	8.50658	0.276427	0.138213	−	0.990402i	$-0.455864\pi$
$947$	8.50658	0.276427	0.138213	+	0.990402i	$0.455864\pi$
$948$	0	0
$949$	2.47214	0.0802489
$950$	0	0
$951$	−8.76393	−0.284190
$952$	0	0
$953$	15.7771	0.511070	0.255535	−	0.966800i	$-0.417748\pi$
$953$	15.7771	0.511070	0.255535	+	0.966800i	$0.417748\pi$
$954$	0	0
$955$	18.5623	0.600662
$956$	0	0
$957$	24.9443	0.806334
$958$	0	0
$959$	−20.8541	−0.673414
$960$	0	0
$961$	34.4508	1.11132
$962$	0	0
$963$	6.36068	0.204970
$964$	0	0
$965$	19.7426	0.635538
$966$	0	0
$967$	52.2705	1.68091	0.840453	−	0.541884i	$-0.182289\pi$
$967$	52.2705	1.68091	0.840453	+	0.541884i	$0.182289\pi$
$968$	0	0
$969$	19.5623	0.628432
$970$	0	0
$971$	10.8328	0.347642	0.173821	−	0.984777i	$-0.444389\pi$
$971$	10.8328	0.347642	0.173821	+	0.984777i	$0.444389\pi$
$972$	0	0
$973$	−3.70820	−0.118880
$974$	0	0
$975$	1.61803	0.0518186
$976$	0	0
$977$	18.7984	0.601413	0.300707	−	0.953717i	$-0.402778\pi$
$977$	18.7984	0.601413	0.300707	+	0.953717i	$0.402778\pi$
$978$	0	0
$979$	−47.4164	−1.51543
$980$	0	0
$981$	−2.65248	−0.0846870
$982$	0	0
$983$	54.9443	1.75245	0.876225	−	0.481902i	$-0.160054\pi$
$983$	54.9443	1.75245	0.876225	+	0.481902i	$0.160054\pi$
$984$	0	0
$985$	4.09017	0.130324
$986$	0	0
$987$	7.23607	0.230327
$988$	0	0
$989$	−0.360680	−0.0114689
$990$	0	0
$991$	20.2148	0.642144	0.321072	−	0.947055i	$-0.395957\pi$
$991$	20.2148	0.642144	0.321072	+	0.947055i	$0.395957\pi$
$992$	0	0
$993$	−38.8328	−1.23232
$994$	0	0
$995$	−11.2361	−0.356207
$996$	0	0
$997$	−7.67376	−0.243030	−0.121515	−	0.992590i	$-0.538775\pi$
$997$	−7.67376	−0.243030	−0.121515	+	0.992590i	$0.538775\pi$
$998$	0	0
$999$	22.3820	0.708134

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.n.1.2	✓	2	1.1	even	1	trivial
7280.2.a.z.1.1		2	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)

\(f(q)\)	\(=\)	\(q+1.61803 q^{3} +1.00000 q^{5} +1.00000 q^{7} -0.381966 q^{9} +O(q^{10})\)	\(q+1.61803 q^{3} +1.00000 q^{5} +1.00000 q^{7} -0.381966 q^{9} -6.47214 q^{11} +1.00000 q^{13} +1.61803 q^{15} -4.61803 q^{17} -2.61803 q^{19} +1.61803 q^{21} +0.472136 q^{23} +1.00000 q^{25} -5.47214 q^{27} -2.38197 q^{29} +8.09017 q^{31} -10.4721 q^{33} +1.00000 q^{35} -4.09017 q^{37} +1.61803 q^{39} -8.61803 q^{41} -0.763932 q^{43} -0.381966 q^{45} +4.47214 q^{47} +1.00000 q^{49} -7.47214 q^{51} -4.47214 q^{53} -6.47214 q^{55} -4.23607 q^{57} -6.56231 q^{59} -3.52786 q^{61} -0.381966 q^{63} +1.00000 q^{65} -2.61803 q^{67} +0.763932 q^{69} -2.29180 q^{71} +2.47214 q^{73} +1.61803 q^{75} -6.47214 q^{77} -3.56231 q^{79} -7.70820 q^{81} -6.00000 q^{83} -4.61803 q^{85} -3.85410 q^{87} +7.32624 q^{89} +1.00000 q^{91} +13.0902 q^{93} -2.61803 q^{95} -15.7082 q^{97} +2.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 2 q^{5} + 2 q^{7} - 3 q^{9}+O(q^{10}) \) 2 * q + q^3 + 2 * q^5 + 2 * q^7 - 3 * q^9	\( 2 q + q^{3} + 2 q^{5} + 2 q^{7} - 3 q^{9} - 4 q^{11} + 2 q^{13} + q^{15} - 7 q^{17} - 3 q^{19} + q^{21} - 8 q^{23} + 2 q^{25} - 2 q^{27} - 7 q^{29} + 5 q^{31} - 12 q^{33} + 2 q^{35} + 3 q^{37} + q^{39} - 15 q^{41} - 6 q^{43} - 3 q^{45} + 2 q^{49} - 6 q^{51} - 4 q^{55} - 4 q^{57} + 7 q^{59} - 16 q^{61} - 3 q^{63} + 2 q^{65} - 3 q^{67} + 6 q^{69} - 18 q^{71} - 4 q^{73} + q^{75} - 4 q^{77} + 13 q^{79} - 2 q^{81} - 12 q^{83} - 7 q^{85} - q^{87} - q^{89} + 2 q^{91} + 15 q^{93} - 3 q^{95} - 18 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q + q^3 + 2 * q^5 + 2 * q^7 - 3 * q^9 - 4 * q^11 + 2 * q^13 + q^15 - 7 * q^17 - 3 * q^19 + q^21 - 8 * q^23 + 2 * q^25 - 2 * q^27 - 7 * q^29 + 5 * q^31 - 12 * q^33 + 2 * q^35 + 3 * q^37 + q^39 - 15 * q^41 - 6 * q^43 - 3 * q^45 + 2 * q^49 - 6 * q^51 - 4 * q^55 - 4 * q^57 + 7 * q^59 - 16 * q^61 - 3 * q^63 + 2 * q^65 - 3 * q^67 + 6 * q^69 - 18 * q^71 - 4 * q^73 + q^75 - 4 * q^77 + 13 * q^79 - 2 * q^81 - 12 * q^83 - 7 * q^85 - q^87 - q^89 + 2 * q^91 + 15 * q^93 - 3 * q^95 - 18 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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