LMFDB - Embedded newform 3640.2.a.z.1.1

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			1.1
Root			$3.01413$ 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)

$f(q)$	$=$	$q-3.01413 q^{3} +1.00000 q^{5} +1.00000 q^{7} +6.08498 q^{9} +O(q^{10})$	$q-3.01413 q^{3} +1.00000 q^{5} +1.00000 q^{7} +6.08498 q^{9} +0.174677 q^{11} -1.00000 q^{13} -3.01413 q^{15} +3.58322 q^{17} +6.28439 q^{19} -3.01413 q^{21} -4.20294 q^{23} +1.00000 q^{25} -9.29852 q^{27} +9.21707 q^{29} -1.01413 q^{31} -0.526500 q^{33} +1.00000 q^{35} +2.31295 q^{37} +3.01413 q^{39} +0.943281 q^{41} -2.46526 q^{43} +6.08498 q^{45} -5.15241 q^{47} +1.00000 q^{49} -10.8003 q^{51} +1.62239 q^{53} +0.174677 q^{55} -18.9420 q^{57} +2.13828 q^{59} -5.53002 q^{61} +6.08498 q^{63} -1.00000 q^{65} +9.87380 q^{67} +12.6682 q^{69} -3.49000 q^{71} +2.36358 q^{73} -3.01413 q^{75} +0.174677 q^{77} -8.73562 q^{79} +9.77203 q^{81} -9.38113 q^{83} +3.58322 q^{85} -27.7814 q^{87} -5.33050 q^{89} -1.00000 q^{91} +3.05672 q^{93} +6.28439 q^{95} -18.2852 q^{97} +1.06291 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.01413	−1.74021	−0.870104	−	0.492868i	$-0.835949\pi$
$3$	−3.01413	−1.74021	−0.870104	+	0.492868i	$0.835949\pi$
$4$	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
$9$	6.08498	2.02833
$10$	0	0
$11$	0.174677	0.0526671	0.0263336	−	0.999653i	$-0.491617\pi$
$11$	0.174677	0.0526671	0.0263336	+	0.999653i	$0.491617\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−3.01413	−0.778245
$16$	0	0
$17$	3.58322	0.869058	0.434529	−	0.900658i	$-0.356915\pi$
$17$	3.58322	0.869058	0.434529	+	0.900658i	$0.356915\pi$
$18$	0	0
$19$	6.28439	1.44174	0.720870	−	0.693071i	$-0.243743\pi$
$19$	6.28439	1.44174	0.720870	+	0.693071i	$0.243743\pi$
$20$	0	0
$21$	−3.01413	−0.657737
$22$	0	0
$23$	−4.20294	−0.876373	−0.438186	−	0.898884i	$-0.644379\pi$
$23$	−4.20294	−0.876373	−0.438186	+	0.898884i	$0.644379\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−9.29852	−1.78950
$28$	0	0
$29$	9.21707	1.71157	0.855783	−	0.517335i	$-0.173076\pi$
$29$	9.21707	1.71157	0.855783	+	0.517335i	$0.173076\pi$
$30$	0	0
$31$	−1.01413	−0.182143	−0.0910715	−	0.995844i	$-0.529029\pi$
$31$	−1.01413	−0.182143	−0.0910715	+	0.995844i	$0.529029\pi$
$32$	0	0
$33$	−0.526500	−0.0916518
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	2.31295	0.380247	0.190124	−	0.981760i	$-0.439111\pi$
$37$	2.31295	0.380247	0.190124	+	0.981760i	$0.439111\pi$
$38$	0	0
$39$	3.01413	0.482647
$40$	0	0
$41$	0.943281	0.147316	0.0736579	−	0.997284i	$-0.476533\pi$
$41$	0.943281	0.147316	0.0736579	+	0.997284i	$0.476533\pi$
$42$	0	0
$43$	−2.46526	−0.375949	−0.187974	−	0.982174i	$-0.560192\pi$
$43$	−2.46526	−0.375949	−0.187974	+	0.982174i	$0.560192\pi$
$44$	0	0
$45$	6.08498	0.907095
$46$	0	0
$47$	−5.15241	−0.751556	−0.375778	−	0.926710i	$-0.622624\pi$
$47$	−5.15241	−0.751556	−0.375778	+	0.926710i	$0.622624\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−10.8003	−1.51234
$52$	0	0
$53$	1.62239	0.222852	0.111426	−	0.993773i	$-0.464458\pi$
$53$	1.62239	0.222852	0.111426	+	0.993773i	$0.464458\pi$
$54$	0	0
$55$	0.174677	0.0235535
$56$	0	0
$57$	−18.9420	−2.50893
$58$	0	0
$59$	2.13828	0.278380	0.139190	−	0.990266i	$-0.455550\pi$
$59$	2.13828	0.278380	0.139190	+	0.990266i	$0.455550\pi$
$60$	0	0
$61$	−5.53002	−0.708046	−0.354023	−	0.935237i	$-0.615187\pi$
$61$	−5.53002	−0.708046	−0.354023	+	0.935237i	$0.615187\pi$
$62$	0	0
$63$	6.08498	0.766635
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	9.87380	1.20628	0.603138	−	0.797637i	$-0.293917\pi$
$67$	9.87380	1.20628	0.603138	+	0.797637i	$0.293917\pi$
$68$	0	0
$69$	12.6682	1.52507
$70$	0	0
$71$	−3.49000	−0.414187	−0.207093	−	0.978321i	$-0.566400\pi$
$71$	−3.49000	−0.414187	−0.207093	+	0.978321i	$0.566400\pi$
$72$	0	0
$73$	2.36358	0.276636	0.138318	−	0.990388i	$-0.455830\pi$
$73$	2.36358	0.276636	0.138318	+	0.990388i	$0.455830\pi$
$74$	0	0
$75$	−3.01413	−0.348042
$76$	0	0
$77$	0.174677	0.0199063
$78$	0	0
$79$	−8.73562	−0.982834	−0.491417	−	0.870924i	$-0.663521\pi$
$79$	−8.73562	−0.982834	−0.491417	+	0.870924i	$0.663521\pi$
$80$	0	0
$81$	9.77203	1.08578
$82$	0	0
$83$	−9.38113	−1.02971	−0.514857	−	0.857276i	$-0.672155\pi$
$83$	−9.38113	−1.02971	−0.514857	+	0.857276i	$0.672155\pi$
$84$	0	0
$85$	3.58322	0.388655
$86$	0	0
$87$	−27.7814	−2.97848
$88$	0	0
$89$	−5.33050	−0.565032	−0.282516	−	0.959263i	$-0.591169\pi$
$89$	−5.33050	−0.565032	−0.282516	+	0.959263i	$0.591169\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	3.05672	0.316967
$94$	0	0
$95$	6.28439	0.644765
$96$	0	0
$97$	−18.2852	−1.85659	−0.928293	−	0.371850i	$-0.878724\pi$
$97$	−18.2852	−1.85659	−0.928293	+	0.371850i	$0.878724\pi$
$98$	0	0
$99$	1.06291	0.106826
$100$	0	0
$101$	19.7212	1.96233	0.981166	−	0.193166i	$-0.0618755\pi$
$101$	19.7212	1.96233	0.981166	+	0.193166i	$0.0618755\pi$
$102$	0	0
$103$	−5.88556	−0.579922	−0.289961	−	0.957039i	$-0.593642\pi$
$103$	−5.88556	−0.579922	−0.289961	+	0.957039i	$0.593642\pi$
$104$	0	0
$105$	−3.01413	−0.294149
$106$	0	0
$107$	3.86761	0.373896	0.186948	−	0.982370i	$-0.440140\pi$
$107$	3.86761	0.373896	0.186948	+	0.982370i	$0.440140\pi$
$108$	0	0
$109$	11.0035	1.05395	0.526973	−	0.849882i	$-0.323327\pi$
$109$	11.0035	1.05395	0.526973	+	0.849882i	$0.323327\pi$
$110$	0	0
$111$	−6.97154	−0.661709
$112$	0	0
$113$	5.90059	0.555081	0.277541	−	0.960714i	$-0.410481\pi$
$113$	5.90059	0.555081	0.277541	+	0.960714i	$0.410481\pi$
$114$	0	0
$115$	−4.20294	−0.391926
$116$	0	0
$117$	−6.08498	−0.562556
$118$	0	0
$119$	3.58322	0.328473
$120$	0	0
$121$	−10.9695	−0.997226
$122$	0	0
$123$	−2.84317	−0.256360
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	14.3729	1.27539	0.637694	−	0.770290i	$-0.279888\pi$
$127$	14.3729	1.27539	0.637694	+	0.770290i	$0.279888\pi$
$128$	0	0
$129$	7.43061	0.654229
$130$	0	0
$131$	−10.3329	−0.902787	−0.451394	−	0.892325i	$-0.649073\pi$
$131$	−10.3329	−0.902787	−0.451394	+	0.892325i	$0.649073\pi$
$132$	0	0
$133$	6.28439	0.544926
$134$	0	0
$135$	−9.29852	−0.800290
$136$	0	0
$137$	7.02589	0.600262	0.300131	−	0.953898i	$-0.402970\pi$
$137$	7.02589	0.600262	0.300131	+	0.953898i	$0.402970\pi$
$138$	0	0
$139$	−13.5441	−1.14879	−0.574396	−	0.818578i	$-0.694763\pi$
$139$	−13.5441	−1.14879	−0.574396	+	0.818578i	$0.694763\pi$
$140$	0	0
$141$	15.5300	1.30786
$142$	0	0
$143$	−0.174677	−0.0146072
$144$	0	0
$145$	9.21707	0.765436
$146$	0	0
$147$	−3.01413	−0.248601
$148$	0	0
$149$	17.0741	1.39876	0.699381	−	0.714749i	$-0.253459\pi$
$149$	17.0741	1.39876	0.699381	+	0.714749i	$0.253459\pi$
$150$	0	0
$151$	17.6600	1.43715	0.718573	−	0.695451i	$-0.244795\pi$
$151$	17.6600	1.43715	0.718573	+	0.695451i	$0.244795\pi$
$152$	0	0
$153$	21.8038	1.76273
$154$	0	0
$155$	−1.01413	−0.0814569
$156$	0	0
$157$	15.0682	1.20257	0.601286	−	0.799034i	$-0.294655\pi$
$157$	15.0682	1.20257	0.601286	+	0.799034i	$0.294655\pi$
$158$	0	0
$159$	−4.89008	−0.387809
$160$	0	0
$161$	−4.20294	−0.331238
$162$	0	0
$163$	14.5699	1.14121	0.570603	−	0.821226i	$-0.306710\pi$
$163$	14.5699	1.14121	0.570603	+	0.821226i	$0.306710\pi$
$164$	0	0
$165$	−0.526500	−0.0409879
$166$	0	0
$167$	−11.9148	−0.921997	−0.460998	−	0.887401i	$-0.652509\pi$
$167$	−11.9148	−0.921997	−0.460998	+	0.887401i	$0.652509\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	38.2404	2.92432
$172$	0	0
$173$	−12.3835	−0.941500	−0.470750	−	0.882267i	$-0.656017\pi$
$173$	−12.3835	−0.941500	−0.470750	+	0.882267i	$0.656017\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−6.44504	−0.484439
$178$	0	0
$179$	4.67211	0.349210	0.174605	−	0.984639i	$-0.444135\pi$
$179$	4.67211	0.349210	0.174605	+	0.984639i	$0.444135\pi$
$180$	0	0
$181$	0.356543	0.0265016	0.0132508	−	0.999912i	$-0.495782\pi$
$181$	0.356543	0.0265016	0.0132508	+	0.999912i	$0.495782\pi$
$182$	0	0
$183$	16.6682	1.23215
$184$	0	0
$185$	2.31295	0.170052
$186$	0	0
$187$	0.625906	0.0457708
$188$	0	0
$189$	−9.29852	−0.676368
$190$	0	0
$191$	6.60354	0.477815	0.238908	−	0.971042i	$-0.423211\pi$
$191$	6.60354	0.477815	0.238908	+	0.971042i	$0.423211\pi$
$192$	0	0
$193$	−8.72844	−0.628287	−0.314143	−	0.949376i	$-0.601717\pi$
$193$	−8.72844	−0.628287	−0.314143	+	0.949376i	$0.601717\pi$
$194$	0	0
$195$	3.01413	0.215846
$196$	0	0
$197$	1.98482	0.141412	0.0707062	−	0.997497i	$-0.477475\pi$
$197$	1.98482	0.141412	0.0707062	+	0.997497i	$0.477475\pi$
$198$	0	0
$199$	13.9835	0.991265	0.495633	−	0.868532i	$-0.334936\pi$
$199$	13.9835	0.991265	0.495633	+	0.868532i	$0.334936\pi$
$200$	0	0
$201$	−29.7609	−2.09917
$202$	0	0
$203$	9.21707	0.646911
$204$	0	0
$205$	0.943281	0.0658816
$206$	0	0
$207$	−25.5748	−1.77757
$208$	0	0
$209$	1.09774	0.0759323
$210$	0	0
$211$	2.65270	0.182619	0.0913097	−	0.995823i	$-0.470895\pi$
$211$	2.65270	0.182619	0.0913097	+	0.995823i	$0.470895\pi$
$212$	0	0
$213$	10.5193	0.720771
$214$	0	0
$215$	−2.46526	−0.168129
$216$	0	0
$217$	−1.01413	−0.0688436
$218$	0	0
$219$	−7.12415	−0.481405
$220$	0	0
$221$	−3.58322	−0.241033
$222$	0	0
$223$	4.14355	0.277473	0.138736	−	0.990329i	$-0.455696\pi$
$223$	4.14355	0.277473	0.138736	+	0.990329i	$0.455696\pi$
$224$	0	0
$225$	6.08498	0.405665
$226$	0	0
$227$	0.129321	0.00858335	0.00429168	−	0.999991i	$-0.498634\pi$
$227$	0.129321	0.00858335	0.00429168	+	0.999991i	$0.498634\pi$
$228$	0	0
$229$	14.7286	0.973295	0.486648	−	0.873598i	$-0.338220\pi$
$229$	14.7286	0.973295	0.486648	+	0.873598i	$0.338220\pi$
$230$	0	0
$231$	−0.526500	−0.0346411
$232$	0	0
$233$	18.9447	1.24111	0.620556	−	0.784162i	$-0.286907\pi$
$233$	18.9447	1.24111	0.620556	+	0.784162i	$0.286907\pi$
$234$	0	0
$235$	−5.15241	−0.336106
$236$	0	0
$237$	26.3303	1.71034
$238$	0	0
$239$	22.1318	1.43159	0.715793	−	0.698312i	$-0.246065\pi$
$239$	22.1318	1.43159	0.715793	+	0.698312i	$0.246065\pi$
$240$	0	0
$241$	5.07075	0.326636	0.163318	−	0.986574i	$-0.447780\pi$
$241$	5.07075	0.326636	0.163318	+	0.986574i	$0.447780\pi$
$242$	0	0
$243$	−1.55858	−0.0999828
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−6.28439	−0.399866
$248$	0	0
$249$	28.2760	1.79192
$250$	0	0
$251$	9.66468	0.610029	0.305014	−	0.952348i	$-0.401339\pi$
$251$	9.66468	0.610029	0.305014	+	0.952348i	$0.401339\pi$
$252$	0	0
$253$	−0.734157	−0.0461560
$254$	0	0
$255$	−10.8003	−0.676340
$256$	0	0
$257$	19.8204	1.23636	0.618181	−	0.786035i	$-0.287870\pi$
$257$	19.8204	1.23636	0.618181	+	0.786035i	$0.287870\pi$
$258$	0	0
$259$	2.31295	0.143720
$260$	0	0
$261$	56.0856	3.47161
$262$	0	0
$263$	5.49020	0.338540	0.169270	−	0.985570i	$-0.445859\pi$
$263$	5.49020	0.338540	0.169270	+	0.985570i	$0.445859\pi$
$264$	0	0
$265$	1.62239	0.0996624
$266$	0	0
$267$	16.0668	0.983274
$268$	0	0
$269$	2.01423	0.122810	0.0614049	−	0.998113i	$-0.480442\pi$
$269$	2.01423	0.122810	0.0614049	+	0.998113i	$0.480442\pi$
$270$	0	0
$271$	10.6424	0.646480	0.323240	−	0.946317i	$-0.395228\pi$
$271$	10.6424	0.646480	0.323240	+	0.946317i	$0.395228\pi$
$272$	0	0
$273$	3.01413	0.182423
$274$	0	0
$275$	0.174677	0.0105334
$276$	0	0
$277$	−30.4459	−1.82932	−0.914659	−	0.404227i	$-0.867541\pi$
$277$	−30.4459	−1.82932	−0.914659	+	0.404227i	$0.867541\pi$
$278$	0	0
$279$	−6.17096	−0.369446
$280$	0	0
$281$	20.5690	1.22704	0.613521	−	0.789678i	$-0.289752\pi$
$281$	20.5690	1.22704	0.613521	+	0.789678i	$0.289752\pi$
$282$	0	0
$283$	23.1227	1.37450	0.687252	−	0.726419i	$-0.258817\pi$
$283$	23.1227	1.37450	0.687252	+	0.726419i	$0.258817\pi$
$284$	0	0
$285$	−18.9420	−1.12203
$286$	0	0
$287$	0.943281	0.0556801
$288$	0	0
$289$	−4.16055	−0.244738
$290$	0	0
$291$	55.1141	3.23085
$292$	0	0
$293$	26.2263	1.53216	0.766078	−	0.642748i	$-0.222206\pi$
$293$	26.2263	1.53216	0.766078	+	0.642748i	$0.222206\pi$
$294$	0	0
$295$	2.13828	0.124495
$296$	0	0
$297$	−1.62424	−0.0942479
$298$	0	0
$299$	4.20294	0.243062
$300$	0	0
$301$	−2.46526	−0.142095
$302$	0	0
$303$	−59.4422	−3.41487
$304$	0	0
$305$	−5.53002	−0.316648
$306$	0	0
$307$	−29.8771	−1.70518	−0.852589	−	0.522583i	$-0.824969\pi$
$307$	−29.8771	−1.70518	−0.852589	+	0.522583i	$0.824969\pi$
$308$	0	0
$309$	17.7398	1.00918
$310$	0	0
$311$	11.6383	0.659946	0.329973	−	0.943990i	$-0.392960\pi$
$311$	11.6383	0.659946	0.329973	+	0.943990i	$0.392960\pi$
$312$	0	0
$313$	−18.2206	−1.02989	−0.514944	−	0.857224i	$-0.672187\pi$
$313$	−18.2206	−1.02989	−0.514944	+	0.857224i	$0.672187\pi$
$314$	0	0
$315$	6.08498	0.342850
$316$	0	0
$317$	−22.0776	−1.24000	−0.620000	−	0.784601i	$-0.712868\pi$
$317$	−22.0776	−1.24000	−0.620000	+	0.784601i	$0.712868\pi$
$318$	0	0
$319$	1.61001	0.0901433
$320$	0	0
$321$	−11.6575	−0.650658
$322$	0	0
$323$	22.5184	1.25295
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−33.1660	−1.83409
$328$	0	0
$329$	−5.15241	−0.284061
$330$	0	0
$331$	29.1170	1.60041	0.800207	−	0.599724i	$-0.204723\pi$
$331$	29.1170	1.60041	0.800207	+	0.599724i	$0.204723\pi$
$332$	0	0
$333$	14.0743	0.771265
$334$	0	0
$335$	9.87380	0.539463
$336$	0	0
$337$	8.74120	0.476163	0.238082	−	0.971245i	$-0.423481\pi$
$337$	8.74120	0.476163	0.238082	+	0.971245i	$0.423481\pi$
$338$	0	0
$339$	−17.7852	−0.965957
$340$	0	0
$341$	−0.177145	−0.00959295
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	12.6682	0.682033
$346$	0	0
$347$	14.5006	0.778431	0.389215	−	0.921147i	$-0.372746\pi$
$347$	14.5006	0.778431	0.389215	+	0.921147i	$0.372746\pi$
$348$	0	0
$349$	−0.846493	−0.0453117	−0.0226559	−	0.999743i	$-0.507212\pi$
$349$	−0.846493	−0.0453117	−0.0226559	+	0.999743i	$0.507212\pi$
$350$	0	0
$351$	9.29852	0.496319
$352$	0	0
$353$	−14.8187	−0.788722	−0.394361	−	0.918956i	$-0.629034\pi$
$353$	−14.8187	−0.788722	−0.394361	+	0.918956i	$0.629034\pi$
$354$	0	0
$355$	−3.49000	−0.185230
$356$	0	0
$357$	−10.8003	−0.571612
$358$	0	0
$359$	−33.8513	−1.78660	−0.893302	−	0.449456i	$-0.851618\pi$
$359$	−33.8513	−1.78660	−0.893302	+	0.449456i	$0.851618\pi$
$360$	0	0
$361$	20.4936	1.07861
$362$	0	0
$363$	33.0635	1.73538
$364$	0	0
$365$	2.36358	0.123716
$366$	0	0
$367$	−25.0058	−1.30529	−0.652646	−	0.757663i	$-0.726341\pi$
$367$	−25.0058	−1.30529	−0.652646	+	0.757663i	$0.726341\pi$
$368$	0	0
$369$	5.73985	0.298804
$370$	0	0
$371$	1.62239	0.0842301
$372$	0	0
$373$	5.01940	0.259895	0.129947	−	0.991521i	$-0.458519\pi$
$373$	5.01940	0.259895	0.129947	+	0.991521i	$0.458519\pi$
$374$	0	0
$375$	−3.01413	−0.155649
$376$	0	0
$377$	−9.21707	−0.474703
$378$	0	0
$379$	28.6513	1.47172	0.735859	−	0.677135i	$-0.236779\pi$
$379$	28.6513	1.47172	0.735859	+	0.677135i	$0.236779\pi$
$380$	0	0
$381$	−43.3218	−2.21944
$382$	0	0
$383$	−16.6323	−0.849871	−0.424935	−	0.905224i	$-0.639703\pi$
$383$	−16.6323	−0.849871	−0.424935	+	0.905224i	$0.639703\pi$
$384$	0	0
$385$	0.174677	0.00890237
$386$	0	0
$387$	−15.0011	−0.762546
$388$	0	0
$389$	−5.47682	−0.277686	−0.138843	−	0.990314i	$-0.544338\pi$
$389$	−5.47682	−0.277686	−0.138843	+	0.990314i	$0.544338\pi$
$390$	0	0
$391$	−15.0600	−0.761619
$392$	0	0
$393$	31.1446	1.57104
$394$	0	0
$395$	−8.73562	−0.439537
$396$	0	0
$397$	28.1128	1.41094	0.705471	−	0.708738i	$-0.250735\pi$
$397$	28.1128	1.41094	0.705471	+	0.708738i	$0.250735\pi$
$398$	0	0
$399$	−18.9420	−0.948285
$400$	0	0
$401$	−15.8983	−0.793925	−0.396963	−	0.917835i	$-0.629936\pi$
$401$	−15.8983	−0.793925	−0.396963	+	0.917835i	$0.629936\pi$
$402$	0	0
$403$	1.01413	0.0505174
$404$	0	0
$405$	9.77203	0.485576
$406$	0	0
$407$	0.404020	0.0200265
$408$	0	0
$409$	−9.45947	−0.467741	−0.233870	−	0.972268i	$-0.575139\pi$
$409$	−9.45947	−0.467741	−0.233870	+	0.972268i	$0.575139\pi$
$410$	0	0
$411$	−21.1769	−1.04458
$412$	0	0
$413$	2.13828	0.105218
$414$	0	0
$415$	−9.38113	−0.460502
$416$	0	0
$417$	40.8235	1.99914
$418$	0	0
$419$	4.79068	0.234040	0.117020	−	0.993130i	$-0.462666\pi$
$419$	4.79068	0.234040	0.117020	+	0.993130i	$0.462666\pi$
$420$	0	0
$421$	−26.6181	−1.29729	−0.648644	−	0.761092i	$-0.724664\pi$
$421$	−26.6181	−1.29729	−0.648644	+	0.761092i	$0.724664\pi$
$422$	0	0
$423$	−31.3523	−1.52440
$424$	0	0
$425$	3.58322	0.173812
$426$	0	0
$427$	−5.53002	−0.267616
$428$	0	0
$429$	0.526500	0.0254196
$430$	0	0
$431$	25.4566	1.22620	0.613101	−	0.790005i	$-0.289922\pi$
$431$	25.4566	1.22620	0.613101	+	0.790005i	$0.289922\pi$
$432$	0	0
$433$	−27.2780	−1.31090	−0.655449	−	0.755240i	$-0.727520\pi$
$433$	−27.2780	−1.31090	−0.655449	+	0.755240i	$0.727520\pi$
$434$	0	0
$435$	−27.7814	−1.33202
$436$	0	0
$437$	−26.4129	−1.26350
$438$	0	0
$439$	23.7301	1.13258	0.566288	−	0.824208i	$-0.308379\pi$
$439$	23.7301	1.13258	0.566288	+	0.824208i	$0.308379\pi$
$440$	0	0
$441$	6.08498	0.289761
$442$	0	0
$443$	3.82718	0.181835	0.0909173	−	0.995858i	$-0.471020\pi$
$443$	3.82718	0.181835	0.0909173	+	0.995858i	$0.471020\pi$
$444$	0	0
$445$	−5.33050	−0.252690
$446$	0	0
$447$	−51.4635	−2.43414
$448$	0	0
$449$	37.4783	1.76871	0.884355	−	0.466815i	$-0.154599\pi$
$449$	37.4783	1.76871	0.884355	+	0.466815i	$0.154599\pi$
$450$	0	0
$451$	0.164770	0.00775870
$452$	0	0
$453$	−53.2294	−2.50094
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	−0.917590	−0.0429231	−0.0214615	−	0.999770i	$-0.506832\pi$
$457$	−0.917590	−0.0429231	−0.0214615	+	0.999770i	$0.506832\pi$
$458$	0	0
$459$	−33.3186	−1.55518
$460$	0	0
$461$	−20.6029	−0.959574	−0.479787	−	0.877385i	$-0.659286\pi$
$461$	−20.6029	−0.959574	−0.479787	+	0.877385i	$0.659286\pi$
$462$	0	0
$463$	30.9243	1.43718	0.718588	−	0.695436i	$-0.244789\pi$
$463$	30.9243	1.43718	0.718588	+	0.695436i	$0.244789\pi$
$464$	0	0
$465$	3.05672	0.141752
$466$	0	0
$467$	−13.6160	−0.630076	−0.315038	−	0.949079i	$-0.602017\pi$
$467$	−13.6160	−0.630076	−0.315038	+	0.949079i	$0.602017\pi$
$468$	0	0
$469$	9.87380	0.455930
$470$	0	0
$471$	−45.4175	−2.09273
$472$	0	0
$473$	−0.430624	−0.0198001
$474$	0	0
$475$	6.28439	0.288348
$476$	0	0
$477$	9.87218	0.452016
$478$	0	0
$479$	−22.0882	−1.00924	−0.504618	−	0.863343i	$-0.668367\pi$
$479$	−22.0882	−1.00924	−0.504618	+	0.863343i	$0.668367\pi$
$480$	0	0
$481$	−2.31295	−0.105462
$482$	0	0
$483$	12.6682	0.576423
$484$	0	0
$485$	−18.2852	−0.830290
$486$	0	0
$487$	−8.78293	−0.397993	−0.198996	−	0.980000i	$-0.563768\pi$
$487$	−8.78293	−0.397993	−0.198996	+	0.980000i	$0.563768\pi$
$488$	0	0
$489$	−43.9157	−1.98594
$490$	0	0
$491$	11.3013	0.510020	0.255010	−	0.966938i	$-0.417921\pi$
$491$	11.3013	0.510020	0.255010	+	0.966938i	$0.417921\pi$
$492$	0	0
$493$	33.0268	1.48745
$494$	0	0
$495$	1.06291	0.0477741
$496$	0	0
$497$	−3.49000	−0.156548
$498$	0	0
$499$	−36.3511	−1.62730	−0.813649	−	0.581357i	$-0.802522\pi$
$499$	−36.3511	−1.62730	−0.813649	+	0.581357i	$0.802522\pi$
$500$	0	0
$501$	35.9128	1.60447
$502$	0	0
$503$	23.7681	1.05977	0.529883	−	0.848071i	$-0.322236\pi$
$503$	23.7681	1.05977	0.529883	+	0.848071i	$0.322236\pi$
$504$	0	0
$505$	19.7212	0.877582
$506$	0	0
$507$	−3.01413	−0.133862
$508$	0	0
$509$	−1.29982	−0.0576137	−0.0288069	−	0.999585i	$-0.509171\pi$
$509$	−1.29982	−0.0576137	−0.0288069	+	0.999585i	$0.509171\pi$
$510$	0	0
$511$	2.36358	0.104559
$512$	0	0
$513$	−58.4356	−2.57999
$514$	0	0
$515$	−5.88556	−0.259349
$516$	0	0
$517$	−0.900007	−0.0395823
$518$	0	0
$519$	37.3255	1.63841
$520$	0	0
$521$	1.20003	0.0525743	0.0262872	−	0.999654i	$-0.491632\pi$
$521$	1.20003	0.0525743	0.0262872	+	0.999654i	$0.491632\pi$
$522$	0	0
$523$	−16.5523	−0.723781	−0.361891	−	0.932221i	$-0.617869\pi$
$523$	−16.5523	−0.723781	−0.361891	+	0.932221i	$0.617869\pi$
$524$	0	0
$525$	−3.01413	−0.131547
$526$	0	0
$527$	−3.63385	−0.158293
$528$	0	0
$529$	−5.33532	−0.231971
$530$	0	0
$531$	13.0114	0.564645
$532$	0	0
$533$	−0.943281	−0.0408580
$534$	0	0
$535$	3.86761	0.167211
$536$	0	0
$537$	−14.0824	−0.607699
$538$	0	0
$539$	0.174677	0.00752388
$540$	0	0
$541$	29.7187	1.27771	0.638854	−	0.769328i	$-0.279409\pi$
$541$	29.7187	1.27771	0.638854	+	0.769328i	$0.279409\pi$
$542$	0	0
$543$	−1.07467	−0.0461184
$544$	0	0
$545$	11.0035	0.471339
$546$	0	0
$547$	28.9347	1.23716	0.618579	−	0.785723i	$-0.287709\pi$
$547$	28.9347	1.23716	0.618579	+	0.785723i	$0.287709\pi$
$548$	0	0
$549$	−33.6500	−1.43615
$550$	0	0
$551$	57.9237	2.46763
$552$	0	0
$553$	−8.73562	−0.371477
$554$	0	0
$555$	−6.97154	−0.295925
$556$	0	0
$557$	−29.9815	−1.27036	−0.635179	−	0.772365i	$-0.719073\pi$
$557$	−29.9815	−1.27036	−0.635179	+	0.772365i	$0.719073\pi$
$558$	0	0
$559$	2.46526	0.104269
$560$	0	0
$561$	−1.88656	−0.0796507
$562$	0	0
$563$	27.7431	1.16923	0.584617	−	0.811310i	$-0.301245\pi$
$563$	27.7431	1.16923	0.584617	+	0.811310i	$0.301245\pi$
$564$	0	0
$565$	5.90059	0.248240
$566$	0	0
$567$	9.77203	0.410386
$568$	0	0
$569$	−7.37833	−0.309316	−0.154658	−	0.987968i	$-0.549428\pi$
$569$	−7.37833	−0.309316	−0.154658	+	0.987968i	$0.549428\pi$
$570$	0	0
$571$	5.30244	0.221900	0.110950	−	0.993826i	$-0.464611\pi$
$571$	5.30244	0.221900	0.110950	+	0.993826i	$0.464611\pi$
$572$	0	0
$573$	−19.9039	−0.831498
$574$	0	0
$575$	−4.20294	−0.175275
$576$	0	0
$577$	−35.7354	−1.48769	−0.743843	−	0.668355i	$-0.766999\pi$
$577$	−35.7354	−1.48769	−0.743843	+	0.668355i	$0.766999\pi$
$578$	0	0
$579$	26.3086	1.09335
$580$	0	0
$581$	−9.38113	−0.389195
$582$	0	0
$583$	0.283394	0.0117370
$584$	0	0
$585$	−6.08498	−0.251583
$586$	0	0
$587$	27.0718	1.11737	0.558687	−	0.829379i	$-0.311305\pi$
$587$	27.0718	1.11737	0.558687	+	0.829379i	$0.311305\pi$
$588$	0	0
$589$	−6.37319	−0.262603
$590$	0	0
$591$	−5.98250	−0.246087
$592$	0	0
$593$	−0.613729	−0.0252028	−0.0126014	−	0.999921i	$-0.504011\pi$
$593$	−0.613729	−0.0252028	−0.0126014	+	0.999921i	$0.504011\pi$
$594$	0	0
$595$	3.58322	0.146898
$596$	0	0
$597$	−42.1481	−1.72501
$598$	0	0
$599$	−0.989489	−0.0404294	−0.0202147	−	0.999796i	$-0.506435\pi$
$599$	−0.989489	−0.0404294	−0.0202147	+	0.999796i	$0.506435\pi$
$600$	0	0
$601$	16.4981	0.672971	0.336486	−	0.941689i	$-0.390762\pi$
$601$	16.4981	0.672971	0.336486	+	0.941689i	$0.390762\pi$
$602$	0	0
$603$	60.0819	2.44672
$604$	0	0
$605$	−10.9695	−0.445973
$606$	0	0
$607$	−15.5571	−0.631444	−0.315722	−	0.948852i	$-0.602247\pi$
$607$	−15.5571	−0.631444	−0.315722	+	0.948852i	$0.602247\pi$
$608$	0	0
$609$	−27.7814	−1.12576
$610$	0	0
$611$	5.15241	0.208444
$612$	0	0
$613$	−16.3259	−0.659396	−0.329698	−	0.944086i	$-0.606947\pi$
$613$	−16.3259	−0.659396	−0.329698	+	0.944086i	$0.606947\pi$
$614$	0	0
$615$	−2.84317	−0.114648
$616$	0	0
$617$	−18.8843	−0.760254	−0.380127	−	0.924934i	$-0.624120\pi$
$617$	−18.8843	−0.760254	−0.380127	+	0.924934i	$0.624120\pi$
$618$	0	0
$619$	−11.8547	−0.476480	−0.238240	−	0.971206i	$-0.576570\pi$
$619$	−11.8547	−0.476480	−0.238240	+	0.971206i	$0.576570\pi$
$620$	0	0
$621$	39.0811	1.56827
$622$	0	0
$623$	−5.33050	−0.213562
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−3.30873	−0.132138
$628$	0	0
$629$	8.28782	0.330457
$630$	0	0
$631$	−16.8780	−0.671902	−0.335951	−	0.941880i	$-0.609058\pi$
$631$	−16.8780	−0.671902	−0.335951	+	0.941880i	$0.609058\pi$
$632$	0	0
$633$	−7.99558	−0.317796
$634$	0	0
$635$	14.3729	0.570371
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−21.2366	−0.840106
$640$	0	0
$641$	−6.67839	−0.263781	−0.131890	−	0.991264i	$-0.542105\pi$
$641$	−6.67839	−0.263781	−0.131890	+	0.991264i	$0.542105\pi$
$642$	0	0
$643$	24.8310	0.979238	0.489619	−	0.871937i	$-0.337136\pi$
$643$	24.8310	0.979238	0.489619	+	0.871937i	$0.337136\pi$
$644$	0	0
$645$	7.43061	0.292580
$646$	0	0
$647$	44.0499	1.73178	0.865890	−	0.500234i	$-0.166753\pi$
$647$	44.0499	1.73178	0.865890	+	0.500234i	$0.166753\pi$
$648$	0	0
$649$	0.373508	0.0146615
$650$	0	0
$651$	3.05672	0.119802
$652$	0	0
$653$	26.4178	1.03381	0.516905	−	0.856043i	$-0.327084\pi$
$653$	26.4178	1.03381	0.516905	+	0.856043i	$0.327084\pi$
$654$	0	0
$655$	−10.3329	−0.403739
$656$	0	0
$657$	14.3824	0.561109
$658$	0	0
$659$	−24.4639	−0.952977	−0.476488	−	0.879181i	$-0.658091\pi$
$659$	−24.4639	−0.952977	−0.476488	+	0.879181i	$0.658091\pi$
$660$	0	0
$661$	−42.4969	−1.65294	−0.826468	−	0.562984i	$-0.809653\pi$
$661$	−42.4969	−1.65294	−0.826468	+	0.562984i	$0.809653\pi$
$662$	0	0
$663$	10.8003	0.419448
$664$	0	0
$665$	6.28439	0.243698
$666$	0	0
$667$	−38.7387	−1.49997
$668$	0	0
$669$	−12.4892	−0.482860
$670$	0	0
$671$	−0.965968	−0.0372908
$672$	0	0
$673$	−38.5327	−1.48533	−0.742664	−	0.669664i	$-0.766438\pi$
$673$	−38.5327	−1.48533	−0.742664	+	0.669664i	$0.766438\pi$
$674$	0	0
$675$	−9.29852	−0.357900
$676$	0	0
$677$	24.9965	0.960695	0.480347	−	0.877078i	$-0.340511\pi$
$677$	24.9965	0.960695	0.480347	+	0.877078i	$0.340511\pi$
$678$	0	0
$679$	−18.2852	−0.701723
$680$	0	0
$681$	−0.389791	−0.0149368
$682$	0	0
$683$	−22.8435	−0.874083	−0.437042	−	0.899441i	$-0.643974\pi$
$683$	−22.8435	−0.874083	−0.437042	+	0.899441i	$0.643974\pi$
$684$	0	0
$685$	7.02589	0.268445
$686$	0	0
$687$	−44.3940	−1.69374
$688$	0	0
$689$	−1.62239	−0.0618080
$690$	0	0
$691$	−3.21440	−0.122282	−0.0611408	−	0.998129i	$-0.519474\pi$
$691$	−3.21440	−0.122282	−0.0611408	+	0.998129i	$0.519474\pi$
$692$	0	0
$693$	1.06291	0.0403765
$694$	0	0
$695$	−13.5441	−0.513755
$696$	0	0
$697$	3.37998	0.128026
$698$	0	0
$699$	−57.1019	−2.15979
$700$	0	0
$701$	29.4484	1.11225	0.556125	−	0.831099i	$-0.312288\pi$
$701$	29.4484	1.11225	0.556125	+	0.831099i	$0.312288\pi$
$702$	0	0
$703$	14.5355	0.548217
$704$	0	0
$705$	15.5300	0.584894
$706$	0	0
$707$	19.7212	0.741692
$708$	0	0
$709$	−24.1481	−0.906901	−0.453450	−	0.891282i	$-0.649807\pi$
$709$	−24.1481	−0.906901	−0.453450	+	0.891282i	$0.649807\pi$
$710$	0	0
$711$	−53.1561	−1.99351
$712$	0	0
$713$	4.26232	0.159625
$714$	0	0
$715$	−0.174677	−0.00653255
$716$	0	0
$717$	−66.7081	−2.49126
$718$	0	0
$719$	1.57765	0.0588362	0.0294181	−	0.999567i	$-0.490635\pi$
$719$	1.57765	0.0588362	0.0294181	+	0.999567i	$0.490635\pi$
$720$	0	0
$721$	−5.88556	−0.219190
$722$	0	0
$723$	−15.2839	−0.568414
$724$	0	0
$725$	9.21707	0.342313
$726$	0	0
$727$	−46.0379	−1.70745	−0.853726	−	0.520722i	$-0.825663\pi$
$727$	−46.0379	−1.70745	−0.853726	+	0.520722i	$0.825663\pi$
$728$	0	0
$729$	−24.6183	−0.911790
$730$	0	0
$731$	−8.83356	−0.326721
$732$	0	0
$733$	18.4581	0.681765	0.340883	−	0.940106i	$-0.389274\pi$
$733$	18.4581	0.681765	0.340883	+	0.940106i	$0.389274\pi$
$734$	0	0
$735$	−3.01413	−0.111178
$736$	0	0
$737$	1.72473	0.0635311
$738$	0	0
$739$	11.6550	0.428737	0.214368	−	0.976753i	$-0.431231\pi$
$739$	11.6550	0.428737	0.214368	+	0.976753i	$0.431231\pi$
$740$	0	0
$741$	18.9420	0.695851
$742$	0	0
$743$	27.3207	1.00230	0.501150	−	0.865360i	$-0.332910\pi$
$743$	27.3207	1.00230	0.501150	+	0.865360i	$0.332910\pi$
$744$	0	0
$745$	17.0741	0.625546
$746$	0	0
$747$	−57.0840	−2.08859
$748$	0	0
$749$	3.86761	0.141320
$750$	0	0
$751$	12.5435	0.457720	0.228860	−	0.973459i	$-0.426500\pi$
$751$	12.5435	0.457720	0.228860	+	0.973459i	$0.426500\pi$
$752$	0	0
$753$	−29.1306	−1.06158
$754$	0	0
$755$	17.6600	0.642712
$756$	0	0
$757$	−32.3511	−1.17582	−0.587910	−	0.808927i	$-0.700049\pi$
$757$	−32.3511	−1.17582	−0.587910	+	0.808927i	$0.700049\pi$
$758$	0	0
$759$	2.21284	0.0803212
$760$	0	0
$761$	−47.0142	−1.70426	−0.852132	−	0.523327i	$-0.824690\pi$
$761$	−47.0142	−1.70426	−0.852132	+	0.523327i	$0.824690\pi$
$762$	0	0
$763$	11.0035	0.398354
$764$	0	0
$765$	21.8038	0.788318
$766$	0	0
$767$	−2.13828	−0.0772087
$768$	0	0
$769$	49.5041	1.78516	0.892582	−	0.450886i	$-0.148892\pi$
$769$	49.5041	1.78516	0.892582	+	0.450886i	$0.148892\pi$
$770$	0	0
$771$	−59.7413	−2.15153
$772$	0	0
$773$	6.19591	0.222852	0.111426	−	0.993773i	$-0.464458\pi$
$773$	6.19591	0.222852	0.111426	+	0.993773i	$0.464458\pi$
$774$	0	0
$775$	−1.01413	−0.0364286
$776$	0	0
$777$	−6.97154	−0.250103
$778$	0	0
$779$	5.92795	0.212391
$780$	0	0
$781$	−0.609623	−0.0218140
$782$	0	0
$783$	−85.7051	−3.06285
$784$	0	0
$785$	15.0682	0.537806
$786$	0	0
$787$	33.8272	1.20581	0.602905	−	0.797813i	$-0.294010\pi$
$787$	33.8272	1.20581	0.602905	+	0.797813i	$0.294010\pi$
$788$	0	0
$789$	−16.5482	−0.589130
$790$	0	0
$791$	5.90059	0.209801
$792$	0	0
$793$	5.53002	0.196377
$794$	0	0
$795$	−4.89008	−0.173433
$796$	0	0
$797$	14.8694	0.526700	0.263350	−	0.964700i	$-0.415173\pi$
$797$	14.8694	0.526700	0.263350	+	0.964700i	$0.415173\pi$
$798$	0	0
$799$	−18.4622	−0.653145
$800$	0	0
$801$	−32.4360	−1.14607
$802$	0	0
$803$	0.412864	0.0145697
$804$	0	0
$805$	−4.20294	−0.148134
$806$	0	0
$807$	−6.07115	−0.213715
$808$	0	0
$809$	3.31924	0.116698	0.0583491	−	0.998296i	$-0.481416\pi$
$809$	3.31924	0.116698	0.0583491	+	0.998296i	$0.481416\pi$
$810$	0	0
$811$	−43.9877	−1.54462	−0.772308	−	0.635248i	$-0.780898\pi$
$811$	−43.9877	−1.54462	−0.772308	+	0.635248i	$0.780898\pi$
$812$	0	0
$813$	−32.0776	−1.12501
$814$	0	0
$815$	14.5699	0.510363
$816$	0	0
$817$	−15.4927	−0.542020
$818$	0	0
$819$	−6.08498	−0.212626
$820$	0	0
$821$	−32.8875	−1.14778	−0.573891	−	0.818932i	$-0.694567\pi$
$821$	−32.8875	−1.14778	−0.573891	+	0.818932i	$0.694567\pi$
$822$	0	0
$823$	34.6612	1.20821	0.604107	−	0.796903i	$-0.293530\pi$
$823$	34.6612	1.20821	0.604107	+	0.796903i	$0.293530\pi$
$824$	0	0
$825$	−0.526500	−0.0183304
$826$	0	0
$827$	−16.7552	−0.582636	−0.291318	−	0.956626i	$-0.594094\pi$
$827$	−16.7552	−0.582636	−0.291318	+	0.956626i	$0.594094\pi$
$828$	0	0
$829$	−4.24276	−0.147357	−0.0736785	−	0.997282i	$-0.523474\pi$
$829$	−4.24276	−0.147357	−0.0736785	+	0.997282i	$0.523474\pi$
$830$	0	0
$831$	91.7679	3.18339
$832$	0	0
$833$	3.58322	0.124151
$834$	0	0
$835$	−11.9148	−0.412329
$836$	0	0
$837$	9.42991	0.325945
$838$	0	0
$839$	−10.1178	−0.349304	−0.174652	−	0.984630i	$-0.555880\pi$
$839$	−10.1178	−0.349304	−0.174652	+	0.984630i	$0.555880\pi$
$840$	0	0
$841$	55.9543	1.92946
$842$	0	0
$843$	−61.9976	−2.13531
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−10.9695	−0.376916
$848$	0	0
$849$	−69.6950	−2.39192
$850$	0	0
$851$	−9.72120	−0.333238
$852$	0	0
$853$	−29.8557	−1.02224	−0.511120	−	0.859509i	$-0.670769\pi$
$853$	−29.8557	−1.02224	−0.511120	+	0.859509i	$0.670769\pi$
$854$	0	0
$855$	38.2404	1.30779
$856$	0	0
$857$	24.1301	0.824268	0.412134	−	0.911123i	$-0.364784\pi$
$857$	24.1301	0.824268	0.412134	+	0.911123i	$0.364784\pi$
$858$	0	0
$859$	14.0330	0.478801	0.239401	−	0.970921i	$-0.423049\pi$
$859$	14.0330	0.478801	0.239401	+	0.970921i	$0.423049\pi$
$860$	0	0
$861$	−2.84317	−0.0968951
$862$	0	0
$863$	27.1767	0.925107	0.462554	−	0.886591i	$-0.346933\pi$
$863$	27.1767	0.925107	0.462554	+	0.886591i	$0.346933\pi$
$864$	0	0
$865$	−12.3835	−0.421052
$866$	0	0
$867$	12.5404	0.425895
$868$	0	0
$869$	−1.52591	−0.0517631
$870$	0	0
$871$	−9.87380	−0.334561
$872$	0	0
$873$	−111.265	−3.76576
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	48.2579	1.62955	0.814776	−	0.579776i	$-0.196860\pi$
$877$	48.2579	1.62955	0.814776	+	0.579776i	$0.196860\pi$
$878$	0	0
$879$	−79.0494	−2.66627
$880$	0	0
$881$	−40.3022	−1.35782	−0.678908	−	0.734223i	$-0.737546\pi$
$881$	−40.3022	−1.35782	−0.678908	+	0.734223i	$0.737546\pi$
$882$	0	0
$883$	−31.5722	−1.06249	−0.531244	−	0.847219i	$-0.678275\pi$
$883$	−31.5722	−1.06249	−0.531244	+	0.847219i	$0.678275\pi$
$884$	0	0
$885$	−6.44504	−0.216648
$886$	0	0
$887$	37.9818	1.27530	0.637652	−	0.770325i	$-0.279906\pi$
$887$	37.9818	1.27530	0.637652	+	0.770325i	$0.279906\pi$
$888$	0	0
$889$	14.3729	0.482051
$890$	0	0
$891$	1.70695	0.0571850
$892$	0	0
$893$	−32.3798	−1.08355
$894$	0	0
$895$	4.67211	0.156172
$896$	0	0
$897$	−12.6682	−0.422979
$898$	0	0
$899$	−9.34730	−0.311750
$900$	0	0
$901$	5.81336	0.193671
$902$	0	0
$903$	7.43061	0.247275
$904$	0	0
$905$	0.356543	0.0118519
$906$	0	0
$907$	−30.9495	−1.02766	−0.513830	−	0.857892i	$-0.671774\pi$
$907$	−30.9495	−1.02766	−0.513830	+	0.857892i	$0.671774\pi$
$908$	0	0
$909$	120.003	3.98025
$910$	0	0
$911$	24.9732	0.827399	0.413699	−	0.910414i	$-0.364236\pi$
$911$	24.9732	0.827399	0.413699	+	0.910414i	$0.364236\pi$
$912$	0	0
$913$	−1.63867	−0.0542320
$914$	0	0
$915$	16.6682	0.551034
$916$	0	0
$917$	−10.3329	−0.341222
$918$	0	0
$919$	−31.3214	−1.03320	−0.516599	−	0.856227i	$-0.672802\pi$
$919$	−31.3214	−1.03320	−0.516599	+	0.856227i	$0.672802\pi$
$920$	0	0
$921$	90.0535	2.96736
$922$	0	0
$923$	3.49000	0.114875
$924$	0	0
$925$	2.31295	0.0760494
$926$	0	0
$927$	−35.8135	−1.17627
$928$	0	0
$929$	−12.0011	−0.393743	−0.196872	−	0.980429i	$-0.563078\pi$
$929$	−12.0011	−0.393743	−0.196872	+	0.980429i	$0.563078\pi$
$930$	0	0
$931$	6.28439	0.205963
$932$	0	0
$933$	−35.0793	−1.14844
$934$	0	0
$935$	0.625906	0.0204693
$936$	0	0
$937$	47.8056	1.56174	0.780870	−	0.624693i	$-0.214776\pi$
$937$	47.8056	1.56174	0.780870	+	0.624693i	$0.214776\pi$
$938$	0	0
$939$	54.9192	1.79222
$940$	0	0
$941$	33.3873	1.08840	0.544198	−	0.838957i	$-0.316834\pi$
$941$	33.3873	1.08840	0.544198	+	0.838957i	$0.316834\pi$
$942$	0	0
$943$	−3.96455	−0.129104
$944$	0	0
$945$	−9.29852	−0.302481
$946$	0	0
$947$	34.1378	1.10933	0.554665	−	0.832074i	$-0.312846\pi$
$947$	34.1378	1.10933	0.554665	+	0.832074i	$0.312846\pi$
$948$	0	0
$949$	−2.36358	−0.0767252
$950$	0	0
$951$	66.5447	2.15786
$952$	0	0
$953$	33.4816	1.08458	0.542288	−	0.840193i	$-0.317558\pi$
$953$	33.4816	1.08458	0.542288	+	0.840193i	$0.317558\pi$
$954$	0	0
$955$	6.60354	0.213685
$956$	0	0
$957$	−4.85278	−0.156868
$958$	0	0
$959$	7.02589	0.226878
$960$	0	0
$961$	−29.9715	−0.966824
$962$	0	0
$963$	23.5343	0.758384
$964$	0	0
$965$	−8.72844	−0.280978
$966$	0	0
$967$	−31.7030	−1.01950	−0.509750	−	0.860323i	$-0.670262\pi$
$967$	−31.7030	−1.01950	−0.509750	+	0.860323i	$0.670262\pi$
$968$	0	0
$969$	−67.8733	−2.18040
$970$	0	0
$971$	−13.6758	−0.438878	−0.219439	−	0.975626i	$-0.570423\pi$
$971$	−13.6758	−0.438878	−0.219439	+	0.975626i	$0.570423\pi$
$972$	0	0
$973$	−13.5441	−0.434202
$974$	0	0
$975$	3.01413	0.0965294
$976$	0	0
$977$	54.1090	1.73110	0.865550	−	0.500823i	$-0.166969\pi$
$977$	54.1090	1.73110	0.865550	+	0.500823i	$0.166969\pi$
$978$	0	0
$979$	−0.931117	−0.0297586
$980$	0	0
$981$	66.9562	2.13775
$982$	0	0
$983$	21.7305	0.693095	0.346547	−	0.938032i	$-0.387354\pi$
$983$	21.7305	0.693095	0.346547	+	0.938032i	$0.387354\pi$
$984$	0	0
$985$	1.98482	0.0632416
$986$	0	0
$987$	15.5300	0.494326
$988$	0	0
$989$	10.3613	0.329471
$990$	0	0
$991$	41.4971	1.31820	0.659099	−	0.752056i	$-0.270938\pi$
$991$	41.4971	1.31820	0.659099	+	0.752056i	$0.270938\pi$
$992$	0	0
$993$	−87.7623	−2.78505
$994$	0	0
$995$	13.9835	0.443307
$996$	0	0
$997$	16.9392	0.536469	0.268235	−	0.963354i	$-0.413560\pi$
$997$	16.9392	0.536469	0.268235	+	0.963354i	$0.413560\pi$
$998$	0	0
$999$	−21.5071	−0.680453

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.z.1.1	✓	5
4.3	odd	2		7280.2.a.cd.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.z.1.1	✓	5	1.1	even	1	trivial
7280.2.a.cd.1.5		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)

\(f(q)\)	\(=\)	\(q-3.01413 q^{3} +1.00000 q^{5} +1.00000 q^{7} +6.08498 q^{9} +O(q^{10})\)	\(q-3.01413 q^{3} +1.00000 q^{5} +1.00000 q^{7} +6.08498 q^{9} +0.174677 q^{11} -1.00000 q^{13} -3.01413 q^{15} +3.58322 q^{17} +6.28439 q^{19} -3.01413 q^{21} -4.20294 q^{23} +1.00000 q^{25} -9.29852 q^{27} +9.21707 q^{29} -1.01413 q^{31} -0.526500 q^{33} +1.00000 q^{35} +2.31295 q^{37} +3.01413 q^{39} +0.943281 q^{41} -2.46526 q^{43} +6.08498 q^{45} -5.15241 q^{47} +1.00000 q^{49} -10.8003 q^{51} +1.62239 q^{53} +0.174677 q^{55} -18.9420 q^{57} +2.13828 q^{59} -5.53002 q^{61} +6.08498 q^{63} -1.00000 q^{65} +9.87380 q^{67} +12.6682 q^{69} -3.49000 q^{71} +2.36358 q^{73} -3.01413 q^{75} +0.174677 q^{77} -8.73562 q^{79} +9.77203 q^{81} -9.38113 q^{83} +3.58322 q^{85} -27.7814 q^{87} -5.33050 q^{89} -1.00000 q^{91} +3.05672 q^{93} +6.28439 q^{95} -18.2852 q^{97} +1.06291 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) \) 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

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Representations

Groups

Database

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