LMFDB - Embedded newform 3680.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3680,2,Mod(1,3680)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3680 = 2^{5} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3680.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:	$29.3849479438$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	3680.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -0.732051 q^{7} -2.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} -0.732051 q^{7} -2.00000 q^{9} +0.732051 q^{11} +1.73205 q^{13} -1.00000 q^{15} -1.46410 q^{17} +0.732051 q^{19} -0.732051 q^{21} -1.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} +7.92820 q^{29} -3.73205 q^{31} +0.732051 q^{33} +0.732051 q^{35} +0.732051 q^{37} +1.73205 q^{39} -4.46410 q^{41} -3.26795 q^{43} +2.00000 q^{45} +7.92820 q^{47} -6.46410 q^{49} -1.46410 q^{51} -7.46410 q^{53} -0.732051 q^{55} +0.732051 q^{57} -10.3923 q^{59} -9.66025 q^{61} +1.46410 q^{63} -1.73205 q^{65} +5.66025 q^{67} -1.00000 q^{69} -12.1244 q^{71} -15.7321 q^{73} +1.00000 q^{75} -0.535898 q^{77} +15.6603 q^{79} +1.00000 q^{81} -4.00000 q^{83} +1.46410 q^{85} +7.92820 q^{87} -0.339746 q^{89} -1.26795 q^{91} -3.73205 q^{93} -0.732051 q^{95} +9.66025 q^{97} -1.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} - 4 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 - 4 * q^9	$ 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} - 4 q^{9} - 2 q^{11} - 2 q^{15} + 4 q^{17} - 2 q^{19} + 2 q^{21} - 2 q^{23} + 2 q^{25} - 10 q^{27} + 2 q^{29} - 4 q^{31} - 2 q^{33} - 2 q^{35} - 2 q^{37} - 2 q^{41} - 10 q^{43} + 4 q^{45} + 2 q^{47} - 6 q^{49} + 4 q^{51} - 8 q^{53} + 2 q^{55} - 2 q^{57} - 2 q^{61} - 4 q^{63} - 6 q^{67} - 2 q^{69} - 28 q^{73} + 2 q^{75} - 8 q^{77} + 14 q^{79} + 2 q^{81} - 8 q^{83} - 4 q^{85} + 2 q^{87} - 18 q^{89} - 6 q^{91} - 4 q^{93} + 2 q^{95} + 2 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 - 4 * q^9 - 2 * q^11 - 2 * q^15 + 4 * q^17 - 2 * q^19 + 2 * q^21 - 2 * q^23 + 2 * q^25 - 10 * q^27 + 2 * q^29 - 4 * q^31 - 2 * q^33 - 2 * q^35 - 2 * q^37 - 2 * q^41 - 10 * q^43 + 4 * q^45 + 2 * q^47 - 6 * q^49 + 4 * q^51 - 8 * q^53 + 2 * q^55 - 2 * q^57 - 2 * q^61 - 4 * q^63 - 6 * q^67 - 2 * q^69 - 28 * q^73 + 2 * q^75 - 8 * q^77 + 14 * q^79 + 2 * q^81 - 8 * q^83 - 4 * q^85 + 2 * q^87 - 18 * q^89 - 6 * q^91 - 4 * q^93 + 2 * q^95 + 2 * 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
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.732051	−0.276689	−0.138345	−	0.990384i	$-0.544178\pi$
$7$	−0.732051	−0.276689	−0.138345	+	0.990384i	$0.544178\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	0.732051	0.220722	0.110361	−	0.993892i	$-0.464799\pi$
$11$	0.732051	0.220722	0.110361	+	0.993892i	$0.464799\pi$
$12$	0	0
$13$	1.73205	0.480384	0.240192	−	0.970725i	$-0.422790\pi$
$13$	1.73205	0.480384	0.240192	+	0.970725i	$0.422790\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−1.46410	−0.355097	−0.177548	−	0.984112i	$-0.556817\pi$
$17$	−1.46410	−0.355097	−0.177548	+	0.984112i	$0.556817\pi$
$18$	0	0
$19$	0.732051	0.167944	0.0839720	−	0.996468i	$-0.473239\pi$
$19$	0.732051	0.167944	0.0839720	+	0.996468i	$0.473239\pi$
$20$	0	0
$21$	−0.732051	−0.159747
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	7.92820	1.47223	0.736115	−	0.676856i	$-0.236658\pi$
$29$	7.92820	1.47223	0.736115	+	0.676856i	$0.236658\pi$
$30$	0	0
$31$	−3.73205	−0.670296	−0.335148	−	0.942165i	$-0.608786\pi$
$31$	−3.73205	−0.670296	−0.335148	+	0.942165i	$0.608786\pi$
$32$	0	0
$33$	0.732051	0.127434
$34$	0	0
$35$	0.732051	0.123739
$36$	0	0
$37$	0.732051	0.120348	0.0601742	−	0.998188i	$-0.480834\pi$
$37$	0.732051	0.120348	0.0601742	+	0.998188i	$0.480834\pi$
$38$	0	0
$39$	1.73205	0.277350
$40$	0	0
$41$	−4.46410	−0.697176	−0.348588	−	0.937276i	$-0.613339\pi$
$41$	−4.46410	−0.697176	−0.348588	+	0.937276i	$0.613339\pi$
$42$	0	0
$43$	−3.26795	−0.498358	−0.249179	−	0.968458i	$-0.580161\pi$
$43$	−3.26795	−0.498358	−0.249179	+	0.968458i	$0.580161\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	7.92820	1.15645	0.578224	−	0.815878i	$-0.303746\pi$
$47$	7.92820	1.15645	0.578224	+	0.815878i	$0.303746\pi$
$48$	0	0
$49$	−6.46410	−0.923443
$50$	0	0
$51$	−1.46410	−0.205015
$52$	0	0
$53$	−7.46410	−1.02527	−0.512637	−	0.858606i	$-0.671331\pi$
$53$	−7.46410	−1.02527	−0.512637	+	0.858606i	$0.671331\pi$
$54$	0	0
$55$	−0.732051	−0.0987097
$56$	0	0
$57$	0.732051	0.0969625
$58$	0	0
$59$	−10.3923	−1.35296	−0.676481	−	0.736460i	$-0.736496\pi$
$59$	−10.3923	−1.35296	−0.676481	+	0.736460i	$0.736496\pi$
$60$	0	0
$61$	−9.66025	−1.23687	−0.618434	−	0.785836i	$-0.712233\pi$
$61$	−9.66025	−1.23687	−0.618434	+	0.785836i	$0.712233\pi$
$62$	0	0
$63$	1.46410	0.184459
$64$	0	0
$65$	−1.73205	−0.214834
$66$	0	0
$67$	5.66025	0.691510	0.345755	−	0.938325i	$-0.387623\pi$
$67$	5.66025	0.691510	0.345755	+	0.938325i	$0.387623\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−12.1244	−1.43890	−0.719448	−	0.694546i	$-0.755605\pi$
$71$	−12.1244	−1.43890	−0.719448	+	0.694546i	$0.755605\pi$
$72$	0	0
$73$	−15.7321	−1.84130	−0.920649	−	0.390392i	$-0.872339\pi$
$73$	−15.7321	−1.84130	−0.920649	+	0.390392i	$0.872339\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−0.535898	−0.0610713
$78$	0	0
$79$	15.6603	1.76192	0.880958	−	0.473194i	$-0.156899\pi$
$79$	15.6603	1.76192	0.880958	+	0.473194i	$0.156899\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	1.46410	0.158804
$86$	0	0
$87$	7.92820	0.849993
$88$	0	0
$89$	−0.339746	−0.0360130	−0.0180065	−	0.999838i	$-0.505732\pi$
$89$	−0.339746	−0.0360130	−0.0180065	+	0.999838i	$0.505732\pi$
$90$	0	0
$91$	−1.26795	−0.132917
$92$	0	0
$93$	−3.73205	−0.386996
$94$	0	0
$95$	−0.732051	−0.0751068
$96$	0	0
$97$	9.66025	0.980850	0.490425	−	0.871483i	$-0.336842\pi$
$97$	9.66025	0.980850	0.490425	+	0.871483i	$0.336842\pi$
$98$	0	0
$99$	−1.46410	−0.147148
$100$	0	0
$101$	9.85641	0.980749	0.490375	−	0.871512i	$-0.336860\pi$
$101$	9.85641	0.980749	0.490375	+	0.871512i	$0.336860\pi$
$102$	0	0
$103$	−1.46410	−0.144262	−0.0721311	−	0.997395i	$-0.522980\pi$
$103$	−1.46410	−0.144262	−0.0721311	+	0.997395i	$0.522980\pi$
$104$	0	0
$105$	0.732051	0.0714408
$106$	0	0
$107$	−10.7321	−1.03751	−0.518753	−	0.854924i	$-0.673604\pi$
$107$	−10.7321	−1.03751	−0.518753	+	0.854924i	$0.673604\pi$
$108$	0	0
$109$	−19.6603	−1.88311	−0.941555	−	0.336858i	$-0.890636\pi$
$109$	−19.6603	−1.88311	−0.941555	+	0.336858i	$0.890636\pi$
$110$	0	0
$111$	0.732051	0.0694832
$112$	0	0
$113$	−10.7321	−1.00959	−0.504793	−	0.863240i	$-0.668431\pi$
$113$	−10.7321	−1.00959	−0.504793	+	0.863240i	$0.668431\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	−3.46410	−0.320256
$118$	0	0
$119$	1.07180	0.0982514
$120$	0	0
$121$	−10.4641	−0.951282
$122$	0	0
$123$	−4.46410	−0.402514
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−16.8564	−1.49576	−0.747882	−	0.663832i	$-0.768929\pi$
$127$	−16.8564	−1.49576	−0.747882	+	0.663832i	$0.768929\pi$
$128$	0	0
$129$	−3.26795	−0.287727
$130$	0	0
$131$	−4.66025	−0.407168	−0.203584	−	0.979057i	$-0.565259\pi$
$131$	−4.66025	−0.407168	−0.203584	+	0.979057i	$0.565259\pi$
$132$	0	0
$133$	−0.535898	−0.0464683
$134$	0	0
$135$	5.00000	0.430331
$136$	0	0
$137$	−1.46410	−0.125087	−0.0625433	−	0.998042i	$-0.519921\pi$
$137$	−1.46410	−0.125087	−0.0625433	+	0.998042i	$0.519921\pi$
$138$	0	0
$139$	−4.80385	−0.407457	−0.203729	−	0.979027i	$-0.565306\pi$
$139$	−4.80385	−0.407457	−0.203729	+	0.979027i	$0.565306\pi$
$140$	0	0
$141$	7.92820	0.667675
$142$	0	0
$143$	1.26795	0.106031
$144$	0	0
$145$	−7.92820	−0.658401
$146$	0	0
$147$	−6.46410	−0.533150
$148$	0	0
$149$	−8.92820	−0.731427	−0.365713	−	0.930727i	$-0.619175\pi$
$149$	−8.92820	−0.731427	−0.365713	+	0.930727i	$0.619175\pi$
$150$	0	0
$151$	20.1244	1.63770	0.818848	−	0.574010i	$-0.194613\pi$
$151$	20.1244	1.63770	0.818848	+	0.574010i	$0.194613\pi$
$152$	0	0
$153$	2.92820	0.236731
$154$	0	0
$155$	3.73205	0.299766
$156$	0	0
$157$	2.73205	0.218041	0.109021	−	0.994039i	$-0.465229\pi$
$157$	2.73205	0.218041	0.109021	+	0.994039i	$0.465229\pi$
$158$	0	0
$159$	−7.46410	−0.591942
$160$	0	0
$161$	0.732051	0.0576937
$162$	0	0
$163$	−6.46410	−0.506308	−0.253154	−	0.967426i	$-0.581468\pi$
$163$	−6.46410	−0.506308	−0.253154	+	0.967426i	$0.581468\pi$
$164$	0	0
$165$	−0.732051	−0.0569901
$166$	0	0
$167$	20.9282	1.61947	0.809736	−	0.586794i	$-0.199610\pi$
$167$	20.9282	1.61947	0.809736	+	0.586794i	$0.199610\pi$
$168$	0	0
$169$	−10.0000	−0.769231
$170$	0	0
$171$	−1.46410	−0.111963
$172$	0	0
$173$	−7.85641	−0.597312	−0.298656	−	0.954361i	$-0.596538\pi$
$173$	−7.85641	−0.597312	−0.298656	+	0.954361i	$0.596538\pi$
$174$	0	0
$175$	−0.732051	−0.0553378
$176$	0	0
$177$	−10.3923	−0.781133
$178$	0	0
$179$	14.6603	1.09576	0.547879	−	0.836557i	$-0.315435\pi$
$179$	14.6603	1.09576	0.547879	+	0.836557i	$0.315435\pi$
$180$	0	0
$181$	−17.6603	−1.31268	−0.656338	−	0.754467i	$-0.727896\pi$
$181$	−17.6603	−1.31268	−0.656338	+	0.754467i	$0.727896\pi$
$182$	0	0
$183$	−9.66025	−0.714107
$184$	0	0
$185$	−0.732051	−0.0538214
$186$	0	0
$187$	−1.07180	−0.0783775
$188$	0	0
$189$	3.66025	0.266244
$190$	0	0
$191$	18.3923	1.33082	0.665410	−	0.746478i	$-0.268257\pi$
$191$	18.3923	1.33082	0.665410	+	0.746478i	$0.268257\pi$
$192$	0	0
$193$	−24.6603	−1.77508	−0.887542	−	0.460727i	$-0.847589\pi$
$193$	−24.6603	−1.77508	−0.887542	+	0.460727i	$0.847589\pi$
$194$	0	0
$195$	−1.73205	−0.124035
$196$	0	0
$197$	−0.660254	−0.0470412	−0.0235206	−	0.999723i	$-0.507488\pi$
$197$	−0.660254	−0.0470412	−0.0235206	+	0.999723i	$0.507488\pi$
$198$	0	0
$199$	22.1962	1.57344	0.786722	−	0.617308i	$-0.211777\pi$
$199$	22.1962	1.57344	0.786722	+	0.617308i	$0.211777\pi$
$200$	0	0
$201$	5.66025	0.399244
$202$	0	0
$203$	−5.80385	−0.407350
$204$	0	0
$205$	4.46410	0.311786
$206$	0	0
$207$	2.00000	0.139010
$208$	0	0
$209$	0.535898	0.0370689
$210$	0	0
$211$	−12.3923	−0.853121	−0.426561	−	0.904459i	$-0.640275\pi$
$211$	−12.3923	−0.853121	−0.426561	+	0.904459i	$0.640275\pi$
$212$	0	0
$213$	−12.1244	−0.830747
$214$	0	0
$215$	3.26795	0.222872
$216$	0	0
$217$	2.73205	0.185464
$218$	0	0
$219$	−15.7321	−1.06307
$220$	0	0
$221$	−2.53590	−0.170583
$222$	0	0
$223$	−3.07180	−0.205703	−0.102851	−	0.994697i	$-0.532797\pi$
$223$	−3.07180	−0.205703	−0.102851	+	0.994697i	$0.532797\pi$
$224$	0	0
$225$	−2.00000	−0.133333
$226$	0	0
$227$	4.73205	0.314077	0.157039	−	0.987592i	$-0.449805\pi$
$227$	4.73205	0.314077	0.157039	+	0.987592i	$0.449805\pi$
$228$	0	0
$229$	−13.1244	−0.867282	−0.433641	−	0.901086i	$-0.642771\pi$
$229$	−13.1244	−0.867282	−0.433641	+	0.901086i	$0.642771\pi$
$230$	0	0
$231$	−0.535898	−0.0352595
$232$	0	0
$233$	15.0526	0.986126	0.493063	−	0.869994i	$-0.335877\pi$
$233$	15.0526	0.986126	0.493063	+	0.869994i	$0.335877\pi$
$234$	0	0
$235$	−7.92820	−0.517179
$236$	0	0
$237$	15.6603	1.01724
$238$	0	0
$239$	14.2679	0.922917	0.461458	−	0.887162i	$-0.347326\pi$
$239$	14.2679	0.922917	0.461458	+	0.887162i	$0.347326\pi$
$240$	0	0
$241$	24.2487	1.56200	0.780998	−	0.624533i	$-0.214711\pi$
$241$	24.2487	1.56200	0.780998	+	0.624533i	$0.214711\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	6.46410	0.412976
$246$	0	0
$247$	1.26795	0.0806777
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	−1.07180	−0.0676512	−0.0338256	−	0.999428i	$-0.510769\pi$
$251$	−1.07180	−0.0676512	−0.0338256	+	0.999428i	$0.510769\pi$
$252$	0	0
$253$	−0.732051	−0.0460236
$254$	0	0
$255$	1.46410	0.0916856
$256$	0	0
$257$	0.267949	0.0167142	0.00835711	−	0.999965i	$-0.497340\pi$
$257$	0.267949	0.0167142	0.00835711	+	0.999965i	$0.497340\pi$
$258$	0	0
$259$	−0.535898	−0.0332991
$260$	0	0
$261$	−15.8564	−0.981487
$262$	0	0
$263$	−14.0526	−0.866518	−0.433259	−	0.901269i	$-0.642636\pi$
$263$	−14.0526	−0.866518	−0.433259	+	0.901269i	$0.642636\pi$
$264$	0	0
$265$	7.46410	0.458516
$266$	0	0
$267$	−0.339746	−0.0207921
$268$	0	0
$269$	−8.07180	−0.492146	−0.246073	−	0.969251i	$-0.579140\pi$
$269$	−8.07180	−0.492146	−0.246073	+	0.969251i	$0.579140\pi$
$270$	0	0
$271$	−26.2487	−1.59450	−0.797248	−	0.603652i	$-0.793712\pi$
$271$	−26.2487	−1.59450	−0.797248	+	0.603652i	$0.793712\pi$
$272$	0	0
$273$	−1.26795	−0.0767398
$274$	0	0
$275$	0.732051	0.0441443
$276$	0	0
$277$	9.05256	0.543916	0.271958	−	0.962309i	$-0.412329\pi$
$277$	9.05256	0.543916	0.271958	+	0.962309i	$0.412329\pi$
$278$	0	0
$279$	7.46410	0.446864
$280$	0	0
$281$	−13.1244	−0.782933	−0.391467	−	0.920192i	$-0.628032\pi$
$281$	−13.1244	−0.782933	−0.391467	+	0.920192i	$0.628032\pi$
$282$	0	0
$283$	21.3205	1.26737	0.633686	−	0.773590i	$-0.281541\pi$
$283$	21.3205	1.26737	0.633686	+	0.773590i	$0.281541\pi$
$284$	0	0
$285$	−0.732051	−0.0433629
$286$	0	0
$287$	3.26795	0.192901
$288$	0	0
$289$	−14.8564	−0.873906
$290$	0	0
$291$	9.66025	0.566294
$292$	0	0
$293$	27.7128	1.61900	0.809500	−	0.587120i	$-0.199738\pi$
$293$	27.7128	1.61900	0.809500	+	0.587120i	$0.199738\pi$
$294$	0	0
$295$	10.3923	0.605063
$296$	0	0
$297$	−3.66025	−0.212389
$298$	0	0
$299$	−1.73205	−0.100167
$300$	0	0
$301$	2.39230	0.137890
$302$	0	0
$303$	9.85641	0.566236
$304$	0	0
$305$	9.66025	0.553145
$306$	0	0
$307$	4.53590	0.258877	0.129439	−	0.991587i	$-0.458682\pi$
$307$	4.53590	0.258877	0.129439	+	0.991587i	$0.458682\pi$
$308$	0	0
$309$	−1.46410	−0.0832898
$310$	0	0
$311$	5.05256	0.286504	0.143252	−	0.989686i	$-0.454244\pi$
$311$	5.05256	0.286504	0.143252	+	0.989686i	$0.454244\pi$
$312$	0	0
$313$	−5.07180	−0.286675	−0.143337	−	0.989674i	$-0.545783\pi$
$313$	−5.07180	−0.286675	−0.143337	+	0.989674i	$0.545783\pi$
$314$	0	0
$315$	−1.46410	−0.0824928
$316$	0	0
$317$	7.07180	0.397192	0.198596	−	0.980081i	$-0.436362\pi$
$317$	7.07180	0.397192	0.198596	+	0.980081i	$0.436362\pi$
$318$	0	0
$319$	5.80385	0.324953
$320$	0	0
$321$	−10.7321	−0.599005
$322$	0	0
$323$	−1.07180	−0.0596364
$324$	0	0
$325$	1.73205	0.0960769
$326$	0	0
$327$	−19.6603	−1.08721
$328$	0	0
$329$	−5.80385	−0.319976
$330$	0	0
$331$	−3.73205	−0.205132	−0.102566	−	0.994726i	$-0.532705\pi$
$331$	−3.73205	−0.205132	−0.102566	+	0.994726i	$0.532705\pi$
$332$	0	0
$333$	−1.46410	−0.0802323
$334$	0	0
$335$	−5.66025	−0.309253
$336$	0	0
$337$	3.41154	0.185839	0.0929193	−	0.995674i	$-0.470380\pi$
$337$	3.41154	0.185839	0.0929193	+	0.995674i	$0.470380\pi$
$338$	0	0
$339$	−10.7321	−0.582885
$340$	0	0
$341$	−2.73205	−0.147949
$342$	0	0
$343$	9.85641	0.532196
$344$	0	0
$345$	1.00000	0.0538382
$346$	0	0
$347$	−19.8564	−1.06595	−0.532974	−	0.846132i	$-0.678926\pi$
$347$	−19.8564	−1.06595	−0.532974	+	0.846132i	$0.678926\pi$
$348$	0	0
$349$	−11.0000	−0.588817	−0.294408	−	0.955680i	$-0.595123\pi$
$349$	−11.0000	−0.588817	−0.294408	+	0.955680i	$0.595123\pi$
$350$	0	0
$351$	−8.66025	−0.462250
$352$	0	0
$353$	16.2679	0.865856	0.432928	−	0.901429i	$-0.357481\pi$
$353$	16.2679	0.865856	0.432928	+	0.901429i	$0.357481\pi$
$354$	0	0
$355$	12.1244	0.643494
$356$	0	0
$357$	1.07180	0.0567255
$358$	0	0
$359$	27.6603	1.45985	0.729926	−	0.683526i	$-0.239554\pi$
$359$	27.6603	1.45985	0.729926	+	0.683526i	$0.239554\pi$
$360$	0	0
$361$	−18.4641	−0.971795
$362$	0	0
$363$	−10.4641	−0.549223
$364$	0	0
$365$	15.7321	0.823453
$366$	0	0
$367$	21.4641	1.12042	0.560208	−	0.828352i	$-0.310721\pi$
$367$	21.4641	1.12042	0.560208	+	0.828352i	$0.310721\pi$
$368$	0	0
$369$	8.92820	0.464784
$370$	0	0
$371$	5.46410	0.283682
$372$	0	0
$373$	−7.07180	−0.366164	−0.183082	−	0.983098i	$-0.558607\pi$
$373$	−7.07180	−0.366164	−0.183082	+	0.983098i	$0.558607\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	13.7321	0.707237
$378$	0	0
$379$	16.3923	0.842016	0.421008	−	0.907057i	$-0.361676\pi$
$379$	16.3923	0.842016	0.421008	+	0.907057i	$0.361676\pi$
$380$	0	0
$381$	−16.8564	−0.863580
$382$	0	0
$383$	6.19615	0.316609	0.158304	−	0.987390i	$-0.449397\pi$
$383$	6.19615	0.316609	0.158304	+	0.987390i	$0.449397\pi$
$384$	0	0
$385$	0.535898	0.0273119
$386$	0	0
$387$	6.53590	0.332238
$388$	0	0
$389$	−32.3923	−1.64236	−0.821178	−	0.570673i	$-0.806682\pi$
$389$	−32.3923	−1.64236	−0.821178	+	0.570673i	$0.806682\pi$
$390$	0	0
$391$	1.46410	0.0740428
$392$	0	0
$393$	−4.66025	−0.235079
$394$	0	0
$395$	−15.6603	−0.787953
$396$	0	0
$397$	−2.26795	−0.113825	−0.0569126	−	0.998379i	$-0.518126\pi$
$397$	−2.26795	−0.113825	−0.0569126	+	0.998379i	$0.518126\pi$
$398$	0	0
$399$	−0.535898	−0.0268285
$400$	0	0
$401$	4.14359	0.206921	0.103461	−	0.994634i	$-0.467008\pi$
$401$	4.14359	0.206921	0.103461	+	0.994634i	$0.467008\pi$
$402$	0	0
$403$	−6.46410	−0.322000
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	0.535898	0.0265635
$408$	0	0
$409$	−19.0000	−0.939490	−0.469745	−	0.882802i	$-0.655654\pi$
$409$	−19.0000	−0.939490	−0.469745	+	0.882802i	$0.655654\pi$
$410$	0	0
$411$	−1.46410	−0.0722188
$412$	0	0
$413$	7.60770	0.374350
$414$	0	0
$415$	4.00000	0.196352
$416$	0	0
$417$	−4.80385	−0.235245
$418$	0	0
$419$	1.26795	0.0619434	0.0309717	−	0.999520i	$-0.490140\pi$
$419$	1.26795	0.0619434	0.0309717	+	0.999520i	$0.490140\pi$
$420$	0	0
$421$	−12.7321	−0.620522	−0.310261	−	0.950651i	$-0.600416\pi$
$421$	−12.7321	−0.620522	−0.310261	+	0.950651i	$0.600416\pi$
$422$	0	0
$423$	−15.8564	−0.770965
$424$	0	0
$425$	−1.46410	−0.0710194
$426$	0	0
$427$	7.07180	0.342228
$428$	0	0
$429$	1.26795	0.0612172
$430$	0	0
$431$	−16.1962	−0.780141	−0.390071	−	0.920785i	$-0.627549\pi$
$431$	−16.1962	−0.780141	−0.390071	+	0.920785i	$0.627549\pi$
$432$	0	0
$433$	8.92820	0.429062	0.214531	−	0.976717i	$-0.431178\pi$
$433$	8.92820	0.429062	0.214531	+	0.976717i	$0.431178\pi$
$434$	0	0
$435$	−7.92820	−0.380128
$436$	0	0
$437$	−0.732051	−0.0350187
$438$	0	0
$439$	25.1962	1.20255	0.601273	−	0.799043i	$-0.294660\pi$
$439$	25.1962	1.20255	0.601273	+	0.799043i	$0.294660\pi$
$440$	0	0
$441$	12.9282	0.615629
$442$	0	0
$443$	−15.2487	−0.724488	−0.362244	−	0.932083i	$-0.617989\pi$
$443$	−15.2487	−0.724488	−0.362244	+	0.932083i	$0.617989\pi$
$444$	0	0
$445$	0.339746	0.0161055
$446$	0	0
$447$	−8.92820	−0.422290
$448$	0	0
$449$	17.8564	0.842696	0.421348	−	0.906899i	$-0.361557\pi$
$449$	17.8564	0.842696	0.421348	+	0.906899i	$0.361557\pi$
$450$	0	0
$451$	−3.26795	−0.153882
$452$	0	0
$453$	20.1244	0.945525
$454$	0	0
$455$	1.26795	0.0594424
$456$	0	0
$457$	30.9282	1.44676	0.723380	−	0.690450i	$-0.242587\pi$
$457$	30.9282	1.44676	0.723380	+	0.690450i	$0.242587\pi$
$458$	0	0
$459$	7.32051	0.341692
$460$	0	0
$461$	−11.5359	−0.537280	−0.268640	−	0.963241i	$-0.586574\pi$
$461$	−11.5359	−0.537280	−0.268640	+	0.963241i	$0.586574\pi$
$462$	0	0
$463$	7.46410	0.346886	0.173443	−	0.984844i	$-0.444511\pi$
$463$	7.46410	0.346886	0.173443	+	0.984844i	$0.444511\pi$
$464$	0	0
$465$	3.73205	0.173070
$466$	0	0
$467$	4.14359	0.191743	0.0958713	−	0.995394i	$-0.469436\pi$
$467$	4.14359	0.191743	0.0958713	+	0.995394i	$0.469436\pi$
$468$	0	0
$469$	−4.14359	−0.191333
$470$	0	0
$471$	2.73205	0.125886
$472$	0	0
$473$	−2.39230	−0.109998
$474$	0	0
$475$	0.732051	0.0335888
$476$	0	0
$477$	14.9282	0.683515
$478$	0	0
$479$	−37.1769	−1.69866	−0.849328	−	0.527865i	$-0.822993\pi$
$479$	−37.1769	−1.69866	−0.849328	+	0.527865i	$0.822993\pi$
$480$	0	0
$481$	1.26795	0.0578135
$482$	0	0
$483$	0.732051	0.0333095
$484$	0	0
$485$	−9.66025	−0.438650
$486$	0	0
$487$	−2.60770	−0.118166	−0.0590830	−	0.998253i	$-0.518818\pi$
$487$	−2.60770	−0.118166	−0.0590830	+	0.998253i	$0.518818\pi$
$488$	0	0
$489$	−6.46410	−0.292317
$490$	0	0
$491$	0.660254	0.0297968	0.0148984	−	0.999889i	$-0.495258\pi$
$491$	0.660254	0.0297968	0.0148984	+	0.999889i	$0.495258\pi$
$492$	0	0
$493$	−11.6077	−0.522784
$494$	0	0
$495$	1.46410	0.0658065
$496$	0	0
$497$	8.87564	0.398127
$498$	0	0
$499$	−11.7321	−0.525199	−0.262599	−	0.964905i	$-0.584580\pi$
$499$	−11.7321	−0.525199	−0.262599	+	0.964905i	$0.584580\pi$
$500$	0	0
$501$	20.9282	0.935003
$502$	0	0
$503$	2.39230	0.106668	0.0533338	−	0.998577i	$-0.483015\pi$
$503$	2.39230	0.106668	0.0533338	+	0.998577i	$0.483015\pi$
$504$	0	0
$505$	−9.85641	−0.438604
$506$	0	0
$507$	−10.0000	−0.444116
$508$	0	0
$509$	−2.32051	−0.102855	−0.0514274	−	0.998677i	$-0.516377\pi$
$509$	−2.32051	−0.102855	−0.0514274	+	0.998677i	$0.516377\pi$
$510$	0	0
$511$	11.5167	0.509467
$512$	0	0
$513$	−3.66025	−0.161604
$514$	0	0
$515$	1.46410	0.0645160
$516$	0	0
$517$	5.80385	0.255253
$518$	0	0
$519$	−7.85641	−0.344858
$520$	0	0
$521$	19.6077	0.859029	0.429514	−	0.903060i	$-0.358685\pi$
$521$	19.6077	0.859029	0.429514	+	0.903060i	$0.358685\pi$
$522$	0	0
$523$	−38.5885	−1.68736	−0.843678	−	0.536850i	$-0.819614\pi$
$523$	−38.5885	−1.68736	−0.843678	+	0.536850i	$0.819614\pi$
$524$	0	0
$525$	−0.732051	−0.0319493
$526$	0	0
$527$	5.46410	0.238020
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	20.7846	0.901975
$532$	0	0
$533$	−7.73205	−0.334912
$534$	0	0
$535$	10.7321	0.463987
$536$	0	0
$537$	14.6603	0.632637
$538$	0	0
$539$	−4.73205	−0.203824
$540$	0	0
$541$	21.3923	0.919727	0.459864	−	0.887990i	$-0.347898\pi$
$541$	21.3923	0.919727	0.459864	+	0.887990i	$0.347898\pi$
$542$	0	0
$543$	−17.6603	−0.757874
$544$	0	0
$545$	19.6603	0.842153
$546$	0	0
$547$	−13.7846	−0.589387	−0.294694	−	0.955592i	$-0.595218\pi$
$547$	−13.7846	−0.589387	−0.294694	+	0.955592i	$0.595218\pi$
$548$	0	0
$549$	19.3205	0.824579
$550$	0	0
$551$	5.80385	0.247252
$552$	0	0
$553$	−11.4641	−0.487503
$554$	0	0
$555$	−0.732051	−0.0310738
$556$	0	0
$557$	11.1244	0.471354	0.235677	−	0.971831i	$-0.424269\pi$
$557$	11.1244	0.471354	0.235677	+	0.971831i	$0.424269\pi$
$558$	0	0
$559$	−5.66025	−0.239403
$560$	0	0
$561$	−1.07180	−0.0452513
$562$	0	0
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	0	0
$565$	10.7321	0.451501
$566$	0	0
$567$	−0.732051	−0.0307432
$568$	0	0
$569$	−7.85641	−0.329358	−0.164679	−	0.986347i	$-0.552659\pi$
$569$	−7.85641	−0.329358	−0.164679	+	0.986347i	$0.552659\pi$
$570$	0	0
$571$	−19.3205	−0.808538	−0.404269	−	0.914640i	$-0.632474\pi$
$571$	−19.3205	−0.808538	−0.404269	+	0.914640i	$0.632474\pi$
$572$	0	0
$573$	18.3923	0.768350
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	33.4449	1.39233	0.696164	−	0.717883i	$-0.254889\pi$
$577$	33.4449	1.39233	0.696164	+	0.717883i	$0.254889\pi$
$578$	0	0
$579$	−24.6603	−1.02485
$580$	0	0
$581$	2.92820	0.121482
$582$	0	0
$583$	−5.46410	−0.226300
$584$	0	0
$585$	3.46410	0.143223
$586$	0	0
$587$	−25.5359	−1.05398	−0.526990	−	0.849872i	$-0.676679\pi$
$587$	−25.5359	−1.05398	−0.526990	+	0.849872i	$0.676679\pi$
$588$	0	0
$589$	−2.73205	−0.112572
$590$	0	0
$591$	−0.660254	−0.0271592
$592$	0	0
$593$	−39.7128	−1.63081	−0.815405	−	0.578891i	$-0.803486\pi$
$593$	−39.7128	−1.63081	−0.815405	+	0.578891i	$0.803486\pi$
$594$	0	0
$595$	−1.07180	−0.0439394
$596$	0	0
$597$	22.1962	0.908428
$598$	0	0
$599$	6.67949	0.272917	0.136458	−	0.990646i	$-0.456428\pi$
$599$	6.67949	0.272917	0.136458	+	0.990646i	$0.456428\pi$
$600$	0	0
$601$	−11.9282	−0.486562	−0.243281	−	0.969956i	$-0.578224\pi$
$601$	−11.9282	−0.486562	−0.243281	+	0.969956i	$0.578224\pi$
$602$	0	0
$603$	−11.3205	−0.461007
$604$	0	0
$605$	10.4641	0.425426
$606$	0	0
$607$	14.3923	0.584166	0.292083	−	0.956393i	$-0.405652\pi$
$607$	14.3923	0.584166	0.292083	+	0.956393i	$0.405652\pi$
$608$	0	0
$609$	−5.80385	−0.235184
$610$	0	0
$611$	13.7321	0.555539
$612$	0	0
$613$	−34.2487	−1.38329	−0.691646	−	0.722236i	$-0.743114\pi$
$613$	−34.2487	−1.38329	−0.691646	+	0.722236i	$0.743114\pi$
$614$	0	0
$615$	4.46410	0.180010
$616$	0	0
$617$	28.7321	1.15671	0.578354	−	0.815786i	$-0.303695\pi$
$617$	28.7321	1.15671	0.578354	+	0.815786i	$0.303695\pi$
$618$	0	0
$619$	−46.2487	−1.85889	−0.929446	−	0.368957i	$-0.879715\pi$
$619$	−46.2487	−1.85889	−0.929446	+	0.368957i	$0.879715\pi$
$620$	0	0
$621$	5.00000	0.200643
$622$	0	0
$623$	0.248711	0.00996441
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0.535898	0.0214017
$628$	0	0
$629$	−1.07180	−0.0427353
$630$	0	0
$631$	27.8038	1.10685	0.553427	−	0.832898i	$-0.313320\pi$
$631$	27.8038	1.10685	0.553427	+	0.832898i	$0.313320\pi$
$632$	0	0
$633$	−12.3923	−0.492550
$634$	0	0
$635$	16.8564	0.668926
$636$	0	0
$637$	−11.1962	−0.443608
$638$	0	0
$639$	24.2487	0.959264
$640$	0	0
$641$	25.5167	1.00785	0.503924	−	0.863748i	$-0.331889\pi$
$641$	25.5167	1.00785	0.503924	+	0.863748i	$0.331889\pi$
$642$	0	0
$643$	20.5359	0.809857	0.404928	−	0.914348i	$-0.367296\pi$
$643$	20.5359	0.809857	0.404928	+	0.914348i	$0.367296\pi$
$644$	0	0
$645$	3.26795	0.128675
$646$	0	0
$647$	6.46410	0.254130	0.127065	−	0.991894i	$-0.459444\pi$
$647$	6.46410	0.254130	0.127065	+	0.991894i	$0.459444\pi$
$648$	0	0
$649$	−7.60770	−0.298628
$650$	0	0
$651$	2.73205	0.107078
$652$	0	0
$653$	12.8038	0.501053	0.250527	−	0.968110i	$-0.419396\pi$
$653$	12.8038	0.501053	0.250527	+	0.968110i	$0.419396\pi$
$654$	0	0
$655$	4.66025	0.182091
$656$	0	0
$657$	31.4641	1.22753
$658$	0	0
$659$	−7.21539	−0.281072	−0.140536	−	0.990076i	$-0.544883\pi$
$659$	−7.21539	−0.281072	−0.140536	+	0.990076i	$0.544883\pi$
$660$	0	0
$661$	−29.7128	−1.15569	−0.577847	−	0.816145i	$-0.696107\pi$
$661$	−29.7128	−1.15569	−0.577847	+	0.816145i	$0.696107\pi$
$662$	0	0
$663$	−2.53590	−0.0984861
$664$	0	0
$665$	0.535898	0.0207812
$666$	0	0
$667$	−7.92820	−0.306981
$668$	0	0
$669$	−3.07180	−0.118763
$670$	0	0
$671$	−7.07180	−0.273004
$672$	0	0
$673$	36.9090	1.42274	0.711368	−	0.702820i	$-0.248076\pi$
$673$	36.9090	1.42274	0.711368	+	0.702820i	$0.248076\pi$
$674$	0	0
$675$	−5.00000	−0.192450
$676$	0	0
$677$	−4.92820	−0.189406	−0.0947031	−	0.995506i	$-0.530190\pi$
$677$	−4.92820	−0.189406	−0.0947031	+	0.995506i	$0.530190\pi$
$678$	0	0
$679$	−7.07180	−0.271391
$680$	0	0
$681$	4.73205	0.181333
$682$	0	0
$683$	37.1051	1.41979	0.709894	−	0.704309i	$-0.248743\pi$
$683$	37.1051	1.41979	0.709894	+	0.704309i	$0.248743\pi$
$684$	0	0
$685$	1.46410	0.0559404
$686$	0	0
$687$	−13.1244	−0.500725
$688$	0	0
$689$	−12.9282	−0.492525
$690$	0	0
$691$	−7.60770	−0.289410	−0.144705	−	0.989475i	$-0.546223\pi$
$691$	−7.60770	−0.289410	−0.144705	+	0.989475i	$0.546223\pi$
$692$	0	0
$693$	1.07180	0.0407142
$694$	0	0
$695$	4.80385	0.182220
$696$	0	0
$697$	6.53590	0.247565
$698$	0	0
$699$	15.0526	0.569340
$700$	0	0
$701$	7.26795	0.274507	0.137253	−	0.990536i	$-0.456173\pi$
$701$	7.26795	0.274507	0.137253	+	0.990536i	$0.456173\pi$
$702$	0	0
$703$	0.535898	0.0202118
$704$	0	0
$705$	−7.92820	−0.298593
$706$	0	0
$707$	−7.21539	−0.271363
$708$	0	0
$709$	26.5885	0.998550	0.499275	−	0.866443i	$-0.333600\pi$
$709$	26.5885	0.998550	0.499275	+	0.866443i	$0.333600\pi$
$710$	0	0
$711$	−31.3205	−1.17461
$712$	0	0
$713$	3.73205	0.139766
$714$	0	0
$715$	−1.26795	−0.0474186
$716$	0	0
$717$	14.2679	0.532846
$718$	0	0
$719$	36.3923	1.35720	0.678602	−	0.734506i	$-0.262586\pi$
$719$	36.3923	1.35720	0.678602	+	0.734506i	$0.262586\pi$
$720$	0	0
$721$	1.07180	0.0399158
$722$	0	0
$723$	24.2487	0.901819
$724$	0	0
$725$	7.92820	0.294446
$726$	0	0
$727$	−27.1769	−1.00794	−0.503968	−	0.863722i	$-0.668127\pi$
$727$	−27.1769	−1.00794	−0.503968	+	0.863722i	$0.668127\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	4.78461	0.176965
$732$	0	0
$733$	32.7846	1.21093	0.605464	−	0.795873i	$-0.292988\pi$
$733$	32.7846	1.21093	0.605464	+	0.795873i	$0.292988\pi$
$734$	0	0
$735$	6.46410	0.238432
$736$	0	0
$737$	4.14359	0.152631
$738$	0	0
$739$	−32.3731	−1.19086	−0.595431	−	0.803406i	$-0.703019\pi$
$739$	−32.3731	−1.19086	−0.595431	+	0.803406i	$0.703019\pi$
$740$	0	0
$741$	1.26795	0.0465793
$742$	0	0
$743$	49.1769	1.80413	0.902063	−	0.431604i	$-0.142052\pi$
$743$	49.1769	1.80413	0.902063	+	0.431604i	$0.142052\pi$
$744$	0	0
$745$	8.92820	0.327104
$746$	0	0
$747$	8.00000	0.292705
$748$	0	0
$749$	7.85641	0.287067
$750$	0	0
$751$	−24.9808	−0.911561	−0.455780	−	0.890092i	$-0.650640\pi$
$751$	−24.9808	−0.911561	−0.455780	+	0.890092i	$0.650640\pi$
$752$	0	0
$753$	−1.07180	−0.0390584
$754$	0	0
$755$	−20.1244	−0.732400
$756$	0	0
$757$	1.80385	0.0655620	0.0327810	−	0.999463i	$-0.489564\pi$
$757$	1.80385	0.0655620	0.0327810	+	0.999463i	$0.489564\pi$
$758$	0	0
$759$	−0.732051	−0.0265718
$760$	0	0
$761$	−1.78461	−0.0646921	−0.0323460	−	0.999477i	$-0.510298\pi$
$761$	−1.78461	−0.0646921	−0.0323460	+	0.999477i	$0.510298\pi$
$762$	0	0
$763$	14.3923	0.521036
$764$	0	0
$765$	−2.92820	−0.105869
$766$	0	0
$767$	−18.0000	−0.649942
$768$	0	0
$769$	0.928203	0.0334719	0.0167359	−	0.999860i	$-0.494673\pi$
$769$	0.928203	0.0334719	0.0167359	+	0.999860i	$0.494673\pi$
$770$	0	0
$771$	0.267949	0.00964995
$772$	0	0
$773$	−1.66025	−0.0597152	−0.0298576	−	0.999554i	$-0.509505\pi$
$773$	−1.66025	−0.0597152	−0.0298576	+	0.999554i	$0.509505\pi$
$774$	0	0
$775$	−3.73205	−0.134059
$776$	0	0
$777$	−0.535898	−0.0192252
$778$	0	0
$779$	−3.26795	−0.117086
$780$	0	0
$781$	−8.87564	−0.317596
$782$	0	0
$783$	−39.6410	−1.41665
$784$	0	0
$785$	−2.73205	−0.0975111
$786$	0	0
$787$	31.7128	1.13044	0.565220	−	0.824940i	$-0.308791\pi$
$787$	31.7128	1.13044	0.565220	+	0.824940i	$0.308791\pi$
$788$	0	0
$789$	−14.0526	−0.500284
$790$	0	0
$791$	7.85641	0.279342
$792$	0	0
$793$	−16.7321	−0.594173
$794$	0	0
$795$	7.46410	0.264724
$796$	0	0
$797$	52.0526	1.84380	0.921898	−	0.387432i	$-0.126638\pi$
$797$	52.0526	1.84380	0.921898	+	0.387432i	$0.126638\pi$
$798$	0	0
$799$	−11.6077	−0.410651
$800$	0	0
$801$	0.679492	0.0240087
$802$	0	0
$803$	−11.5167	−0.406414
$804$	0	0
$805$	−0.732051	−0.0258014
$806$	0	0
$807$	−8.07180	−0.284141
$808$	0	0
$809$	−17.0718	−0.600212	−0.300106	−	0.953906i	$-0.597022\pi$
$809$	−17.0718	−0.600212	−0.300106	+	0.953906i	$0.597022\pi$
$810$	0	0
$811$	11.0526	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$811$	11.0526	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$812$	0	0
$813$	−26.2487	−0.920582
$814$	0	0
$815$	6.46410	0.226428
$816$	0	0
$817$	−2.39230	−0.0836962
$818$	0	0
$819$	2.53590	0.0886115
$820$	0	0
$821$	−18.1436	−0.633216	−0.316608	−	0.948557i	$-0.602544\pi$
$821$	−18.1436	−0.633216	−0.316608	+	0.948557i	$0.602544\pi$
$822$	0	0
$823$	−35.7846	−1.24737	−0.623687	−	0.781675i	$-0.714366\pi$
$823$	−35.7846	−1.24737	−0.623687	+	0.781675i	$0.714366\pi$
$824$	0	0
$825$	0.732051	0.0254867
$826$	0	0
$827$	0.392305	0.0136418	0.00682089	−	0.999977i	$-0.497829\pi$
$827$	0.392305	0.0136418	0.00682089	+	0.999977i	$0.497829\pi$
$828$	0	0
$829$	−39.3205	−1.36566	−0.682829	−	0.730578i	$-0.739251\pi$
$829$	−39.3205	−1.36566	−0.682829	+	0.730578i	$0.739251\pi$
$830$	0	0
$831$	9.05256	0.314030
$832$	0	0
$833$	9.46410	0.327912
$834$	0	0
$835$	−20.9282	−0.724250
$836$	0	0
$837$	18.6603	0.644993
$838$	0	0
$839$	18.3397	0.633158	0.316579	−	0.948566i	$-0.397466\pi$
$839$	18.3397	0.633158	0.316579	+	0.948566i	$0.397466\pi$
$840$	0	0
$841$	33.8564	1.16746
$842$	0	0
$843$	−13.1244	−0.452027
$844$	0	0
$845$	10.0000	0.344010
$846$	0	0
$847$	7.66025	0.263209
$848$	0	0
$849$	21.3205	0.731718
$850$	0	0
$851$	−0.732051	−0.0250944
$852$	0	0
$853$	−22.7846	−0.780130	−0.390065	−	0.920787i	$-0.627547\pi$
$853$	−22.7846	−0.780130	−0.390065	+	0.920787i	$0.627547\pi$
$854$	0	0
$855$	1.46410	0.0500712
$856$	0	0
$857$	−25.0526	−0.855779	−0.427890	−	0.903831i	$-0.640743\pi$
$857$	−25.0526	−0.855779	−0.427890	+	0.903831i	$0.640743\pi$
$858$	0	0
$859$	10.6603	0.363723	0.181862	−	0.983324i	$-0.441788\pi$
$859$	10.6603	0.363723	0.181862	+	0.983324i	$0.441788\pi$
$860$	0	0
$861$	3.26795	0.111371
$862$	0	0
$863$	−43.2487	−1.47220	−0.736102	−	0.676871i	$-0.763335\pi$
$863$	−43.2487	−1.47220	−0.736102	+	0.676871i	$0.763335\pi$
$864$	0	0
$865$	7.85641	0.267126
$866$	0	0
$867$	−14.8564	−0.504550
$868$	0	0
$869$	11.4641	0.388893
$870$	0	0
$871$	9.80385	0.332191
$872$	0	0
$873$	−19.3205	−0.653900
$874$	0	0
$875$	0.732051	0.0247478
$876$	0	0
$877$	5.60770	0.189358	0.0946792	−	0.995508i	$-0.469817\pi$
$877$	5.60770	0.189358	0.0946792	+	0.995508i	$0.469817\pi$
$878$	0	0
$879$	27.7128	0.934730
$880$	0	0
$881$	12.3923	0.417507	0.208754	−	0.977968i	$-0.433059\pi$
$881$	12.3923	0.417507	0.208754	+	0.977968i	$0.433059\pi$
$882$	0	0
$883$	34.0000	1.14419	0.572096	−	0.820187i	$-0.306131\pi$
$883$	34.0000	1.14419	0.572096	+	0.820187i	$0.306131\pi$
$884$	0	0
$885$	10.3923	0.349334
$886$	0	0
$887$	39.1051	1.31302	0.656511	−	0.754317i	$-0.272032\pi$
$887$	39.1051	1.31302	0.656511	+	0.754317i	$0.272032\pi$
$888$	0	0
$889$	12.3397	0.413862
$890$	0	0
$891$	0.732051	0.0245246
$892$	0	0
$893$	5.80385	0.194218
$894$	0	0
$895$	−14.6603	−0.490038
$896$	0	0
$897$	−1.73205	−0.0578315
$898$	0	0
$899$	−29.5885	−0.986830
$900$	0	0
$901$	10.9282	0.364071
$902$	0	0
$903$	2.39230	0.0796109
$904$	0	0
$905$	17.6603	0.587047
$906$	0	0
$907$	21.2679	0.706191	0.353095	−	0.935587i	$-0.385129\pi$
$907$	21.2679	0.706191	0.353095	+	0.935587i	$0.385129\pi$
$908$	0	0
$909$	−19.7128	−0.653833
$910$	0	0
$911$	38.5885	1.27849	0.639246	−	0.769002i	$-0.279246\pi$
$911$	38.5885	1.27849	0.639246	+	0.769002i	$0.279246\pi$
$912$	0	0
$913$	−2.92820	−0.0969094
$914$	0	0
$915$	9.66025	0.319358
$916$	0	0
$917$	3.41154	0.112659
$918$	0	0
$919$	28.1962	0.930105	0.465053	−	0.885283i	$-0.346035\pi$
$919$	28.1962	0.930105	0.465053	+	0.885283i	$0.346035\pi$
$920$	0	0
$921$	4.53590	0.149463
$922$	0	0
$923$	−21.0000	−0.691223
$924$	0	0
$925$	0.732051	0.0240697
$926$	0	0
$927$	2.92820	0.0961748
$928$	0	0
$929$	1.39230	0.0456800	0.0228400	−	0.999739i	$-0.492729\pi$
$929$	1.39230	0.0456800	0.0228400	+	0.999739i	$0.492729\pi$
$930$	0	0
$931$	−4.73205	−0.155087
$932$	0	0
$933$	5.05256	0.165413
$934$	0	0
$935$	1.07180	0.0350515
$936$	0	0
$937$	28.3397	0.925819	0.462910	−	0.886406i	$-0.346805\pi$
$937$	28.3397	0.925819	0.462910	+	0.886406i	$0.346805\pi$
$938$	0	0
$939$	−5.07180	−0.165512
$940$	0	0
$941$	−45.5692	−1.48551	−0.742757	−	0.669561i	$-0.766482\pi$
$941$	−45.5692	−1.48551	−0.742757	+	0.669561i	$0.766482\pi$
$942$	0	0
$943$	4.46410	0.145371
$944$	0	0
$945$	−3.66025	−0.119068
$946$	0	0
$947$	33.5359	1.08977	0.544885	−	0.838511i	$-0.316573\pi$
$947$	33.5359	1.08977	0.544885	+	0.838511i	$0.316573\pi$
$948$	0	0
$949$	−27.2487	−0.884531
$950$	0	0
$951$	7.07180	0.229319
$952$	0	0
$953$	23.1769	0.750774	0.375387	−	0.926868i	$-0.377510\pi$
$953$	23.1769	0.750774	0.375387	+	0.926868i	$0.377510\pi$
$954$	0	0
$955$	−18.3923	−0.595161
$956$	0	0
$957$	5.80385	0.187612
$958$	0	0
$959$	1.07180	0.0346101
$960$	0	0
$961$	−17.0718	−0.550703
$962$	0	0
$963$	21.4641	0.691671
$964$	0	0
$965$	24.6603	0.793842
$966$	0	0
$967$	10.4641	0.336503	0.168251	−	0.985744i	$-0.446188\pi$
$967$	10.4641	0.336503	0.168251	+	0.985744i	$0.446188\pi$
$968$	0	0
$969$	−1.07180	−0.0344311
$970$	0	0
$971$	−18.0000	−0.577647	−0.288824	−	0.957382i	$-0.593264\pi$
$971$	−18.0000	−0.577647	−0.288824	+	0.957382i	$0.593264\pi$
$972$	0	0
$973$	3.51666	0.112739
$974$	0	0
$975$	1.73205	0.0554700
$976$	0	0
$977$	56.4449	1.80583	0.902916	−	0.429818i	$-0.141422\pi$
$977$	56.4449	1.80583	0.902916	+	0.429818i	$0.141422\pi$
$978$	0	0
$979$	−0.248711	−0.00794885
$980$	0	0
$981$	39.3205	1.25541
$982$	0	0
$983$	−23.1769	−0.739229	−0.369614	−	0.929185i	$-0.620510\pi$
$983$	−23.1769	−0.739229	−0.369614	+	0.929185i	$0.620510\pi$
$984$	0	0
$985$	0.660254	0.0210374
$986$	0	0
$987$	−5.80385	−0.184739
$988$	0	0
$989$	3.26795	0.103915
$990$	0	0
$991$	−44.0000	−1.39771	−0.698853	−	0.715265i	$-0.746306\pi$
$991$	−44.0000	−1.39771	−0.698853	+	0.715265i	$0.746306\pi$
$992$	0	0
$993$	−3.73205	−0.118433
$994$	0	0
$995$	−22.1962	−0.703665
$996$	0	0
$997$	−41.3205	−1.30863	−0.654317	−	0.756221i	$-0.727044\pi$
$997$	−41.3205	−1.30863	−0.654317	+	0.756221i	$0.727044\pi$
$998$	0	0
$999$	−3.66025	−0.115805

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3680.2.a.n.1.1	yes	2
4.3	odd	2		3680.2.a.l.1.2	✓	2
8.3	odd	2		7360.2.a.bs.1.2		2
8.5	even	2		7360.2.a.bg.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3680.2.a.l.1.2	✓	2	4.3	odd	2
3680.2.a.n.1.1	yes	2	1.1	even	1	trivial
7360.2.a.bg.1.1		2	8.5	even	2
7360.2.a.bs.1.2		2	8.3	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:	\( N \)	\(=\)	\( 3680 = 2^{5} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3680.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -0.732051 q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} -0.732051 q^{7} -2.00000 q^{9} +0.732051 q^{11} +1.73205 q^{13} -1.00000 q^{15} -1.46410 q^{17} +0.732051 q^{19} -0.732051 q^{21} -1.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} +7.92820 q^{29} -3.73205 q^{31} +0.732051 q^{33} +0.732051 q^{35} +0.732051 q^{37} +1.73205 q^{39} -4.46410 q^{41} -3.26795 q^{43} +2.00000 q^{45} +7.92820 q^{47} -6.46410 q^{49} -1.46410 q^{51} -7.46410 q^{53} -0.732051 q^{55} +0.732051 q^{57} -10.3923 q^{59} -9.66025 q^{61} +1.46410 q^{63} -1.73205 q^{65} +5.66025 q^{67} -1.00000 q^{69} -12.1244 q^{71} -15.7321 q^{73} +1.00000 q^{75} -0.535898 q^{77} +15.6603 q^{79} +1.00000 q^{81} -4.00000 q^{83} +1.46410 q^{85} +7.92820 q^{87} -0.339746 q^{89} -1.26795 q^{91} -3.73205 q^{93} -0.732051 q^{95} +9.66025 q^{97} -1.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} - 4 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 - 4 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} - 4 q^{9} - 2 q^{11} - 2 q^{15} + 4 q^{17} - 2 q^{19} + 2 q^{21} - 2 q^{23} + 2 q^{25} - 10 q^{27} + 2 q^{29} - 4 q^{31} - 2 q^{33} - 2 q^{35} - 2 q^{37} - 2 q^{41} - 10 q^{43} + 4 q^{45} + 2 q^{47} - 6 q^{49} + 4 q^{51} - 8 q^{53} + 2 q^{55} - 2 q^{57} - 2 q^{61} - 4 q^{63} - 6 q^{67} - 2 q^{69} - 28 q^{73} + 2 q^{75} - 8 q^{77} + 14 q^{79} + 2 q^{81} - 8 q^{83} - 4 q^{85} + 2 q^{87} - 18 q^{89} - 6 q^{91} - 4 q^{93} + 2 q^{95} + 2 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 - 4 * q^9 - 2 * q^11 - 2 * q^15 + 4 * q^17 - 2 * q^19 + 2 * q^21 - 2 * q^23 + 2 * q^25 - 10 * q^27 + 2 * q^29 - 4 * q^31 - 2 * q^33 - 2 * q^35 - 2 * q^37 - 2 * q^41 - 10 * q^43 + 4 * q^45 + 2 * q^47 - 6 * q^49 + 4 * q^51 - 8 * q^53 + 2 * q^55 - 2 * q^57 - 2 * q^61 - 4 * q^63 - 6 * q^67 - 2 * q^69 - 28 * q^73 + 2 * q^75 - 8 * q^77 + 14 * q^79 + 2 * q^81 - 8 * q^83 - 4 * q^85 + 2 * q^87 - 18 * q^89 - 6 * q^91 - 4 * q^93 + 2 * q^95 + 2 * q^97 + 4 * q^99

Introduction

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