LMFDB - Embedded newform 2205.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,2,Mod(1,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2205.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2205.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:	$17.6070136457$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	2205.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} -1.00000 q^{5} +2.82843 q^{8} +O(q^{10})$	$q-1.41421 q^{2} -1.00000 q^{5} +2.82843 q^{8} +1.41421 q^{10} -3.41421 q^{11} +1.58579 q^{13} -4.00000 q^{16} -6.24264 q^{17} +6.65685 q^{19} +4.82843 q^{22} +6.24264 q^{23} +1.00000 q^{25} -2.24264 q^{26} +0.242641 q^{29} +0.171573 q^{31} +8.82843 q^{34} -5.58579 q^{37} -9.41421 q^{38} -2.82843 q^{40} +2.24264 q^{41} -10.4142 q^{43} -8.82843 q^{46} +9.31371 q^{47} -1.41421 q^{50} -1.17157 q^{53} +3.41421 q^{55} -0.343146 q^{58} +1.41421 q^{59} -12.4853 q^{61} -0.242641 q^{62} +8.00000 q^{64} -1.58579 q^{65} +11.7279 q^{67} +3.41421 q^{71} -2.07107 q^{73} +7.89949 q^{74} -4.65685 q^{79} +4.00000 q^{80} -3.17157 q^{82} +5.41421 q^{83} +6.24264 q^{85} +14.7279 q^{86} -9.65685 q^{88} -3.75736 q^{89} -13.1716 q^{94} -6.65685 q^{95} +10.8284 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5}+O(q^{10}) $ 2 * q - 2 * q^5	$ 2 q - 2 q^{5} - 4 q^{11} + 6 q^{13} - 8 q^{16} - 4 q^{17} + 2 q^{19} + 4 q^{22} + 4 q^{23} + 2 q^{25} + 4 q^{26} - 8 q^{29} + 6 q^{31} + 12 q^{34} - 14 q^{37} - 16 q^{38} - 4 q^{41} - 18 q^{43} - 12 q^{46} - 4 q^{47} - 8 q^{53} + 4 q^{55} - 12 q^{58} - 8 q^{61} + 8 q^{62} + 16 q^{64} - 6 q^{65} - 2 q^{67} + 4 q^{71} + 10 q^{73} - 4 q^{74} + 2 q^{79} + 8 q^{80} - 12 q^{82} + 8 q^{83} + 4 q^{85} + 4 q^{86} - 8 q^{88} - 16 q^{89} - 32 q^{94} - 2 q^{95} + 16 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 4 * q^11 + 6 * q^13 - 8 * q^16 - 4 * q^17 + 2 * q^19 + 4 * q^22 + 4 * q^23 + 2 * q^25 + 4 * q^26 - 8 * q^29 + 6 * q^31 + 12 * q^34 - 14 * q^37 - 16 * q^38 - 4 * q^41 - 18 * q^43 - 12 * q^46 - 4 * q^47 - 8 * q^53 + 4 * q^55 - 12 * q^58 - 8 * q^61 + 8 * q^62 + 16 * q^64 - 6 * q^65 - 2 * q^67 + 4 * q^71 + 10 * q^73 - 4 * q^74 + 2 * q^79 + 8 * q^80 - 12 * q^82 + 8 * q^83 + 4 * q^85 + 4 * q^86 - 8 * q^88 - 16 * q^89 - 32 * q^94 - 2 * q^95 + 16 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.41421	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$2$	−1.41421	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.82843	1.00000
$9$	0	0
$10$	1.41421	0.447214
$11$	−3.41421	−1.02942	−0.514712	−	0.857363i	$-0.672101\pi$
$11$	−3.41421	−1.02942	−0.514712	+	0.857363i	$0.672101\pi$
$12$	0	0
$13$	1.58579	0.439818	0.219909	−	0.975520i	$-0.429424\pi$
$13$	1.58579	0.439818	0.219909	+	0.975520i	$0.429424\pi$
$14$	0	0
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−6.24264	−1.51406	−0.757031	−	0.653379i	$-0.773351\pi$
$17$	−6.24264	−1.51406	−0.757031	+	0.653379i	$0.773351\pi$
$18$	0	0
$19$	6.65685	1.52719	0.763594	−	0.645697i	$-0.223433\pi$
$19$	6.65685	1.52719	0.763594	+	0.645697i	$0.223433\pi$
$20$	0	0
$21$	0	0
$22$	4.82843	1.02942
$23$	6.24264	1.30168	0.650840	−	0.759215i	$-0.274417\pi$
$23$	6.24264	1.30168	0.650840	+	0.759215i	$0.274417\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.24264	−0.439818
$27$	0	0
$28$	0	0
$29$	0.242641	0.0450572	0.0225286	−	0.999746i	$-0.492828\pi$
$29$	0.242641	0.0450572	0.0225286	+	0.999746i	$0.492828\pi$
$30$	0	0
$31$	0.171573	0.0308154	0.0154077	−	0.999881i	$-0.495095\pi$
$31$	0.171573	0.0308154	0.0154077	+	0.999881i	$0.495095\pi$
$32$	0	0
$33$	0	0
$34$	8.82843	1.51406
$35$	0	0
$36$	0	0
$37$	−5.58579	−0.918298	−0.459149	−	0.888359i	$-0.651846\pi$
$37$	−5.58579	−0.918298	−0.459149	+	0.888359i	$0.651846\pi$
$38$	−9.41421	−1.52719
$39$	0	0
$40$	−2.82843	−0.447214
$41$	2.24264	0.350242	0.175121	−	0.984547i	$-0.443968\pi$
$41$	2.24264	0.350242	0.175121	+	0.984547i	$0.443968\pi$
$42$	0	0
$43$	−10.4142	−1.58815	−0.794076	−	0.607818i	$-0.792045\pi$
$43$	−10.4142	−1.58815	−0.794076	+	0.607818i	$0.792045\pi$
$44$	0	0
$45$	0	0
$46$	−8.82843	−1.30168
$47$	9.31371	1.35854	0.679272	−	0.733887i	$-0.262296\pi$
$47$	9.31371	1.35854	0.679272	+	0.733887i	$0.262296\pi$
$48$	0	0
$49$	0	0
$50$	−1.41421	−0.200000
$51$	0	0
$52$	0	0
$53$	−1.17157	−0.160928	−0.0804640	−	0.996758i	$-0.525640\pi$
$53$	−1.17157	−0.160928	−0.0804640	+	0.996758i	$0.525640\pi$
$54$	0	0
$55$	3.41421	0.460372
$56$	0	0
$57$	0	0
$58$	−0.343146	−0.0450572
$59$	1.41421	0.184115	0.0920575	−	0.995754i	$-0.470656\pi$
$59$	1.41421	0.184115	0.0920575	+	0.995754i	$0.470656\pi$
$60$	0	0
$61$	−12.4853	−1.59858	−0.799288	−	0.600948i	$-0.794790\pi$
$61$	−12.4853	−1.59858	−0.799288	+	0.600948i	$0.794790\pi$
$62$	−0.242641	−0.0308154
$63$	0	0
$64$	8.00000	1.00000
$65$	−1.58579	−0.196693
$66$	0	0
$67$	11.7279	1.43279	0.716397	−	0.697693i	$-0.245790\pi$
$67$	11.7279	1.43279	0.716397	+	0.697693i	$0.245790\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.41421	0.405193	0.202596	−	0.979262i	$-0.435062\pi$
$71$	3.41421	0.405193	0.202596	+	0.979262i	$0.435062\pi$
$72$	0	0
$73$	−2.07107	−0.242400	−0.121200	−	0.992628i	$-0.538674\pi$
$73$	−2.07107	−0.242400	−0.121200	+	0.992628i	$0.538674\pi$
$74$	7.89949	0.918298
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4.65685	−0.523937	−0.261969	−	0.965076i	$-0.584372\pi$
$79$	−4.65685	−0.523937	−0.261969	+	0.965076i	$0.584372\pi$
$80$	4.00000	0.447214
$81$	0	0
$82$	−3.17157	−0.350242
$83$	5.41421	0.594287	0.297144	−	0.954833i	$-0.403966\pi$
$83$	5.41421	0.594287	0.297144	+	0.954833i	$0.403966\pi$
$84$	0	0
$85$	6.24264	0.677109
$86$	14.7279	1.58815
$87$	0	0
$88$	−9.65685	−1.02942
$89$	−3.75736	−0.398279	−0.199140	−	0.979971i	$-0.563815\pi$
$89$	−3.75736	−0.398279	−0.199140	+	0.979971i	$0.563815\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−13.1716	−1.35854
$95$	−6.65685	−0.682979
$96$	0	0
$97$	10.8284	1.09946	0.549730	−	0.835342i	$-0.314731\pi$
$97$	10.8284	1.09946	0.549730	+	0.835342i	$0.314731\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.24264	−0.621166	−0.310583	−	0.950546i	$-0.600524\pi$
$101$	−6.24264	−0.621166	−0.310583	+	0.950546i	$0.600524\pi$
$102$	0	0
$103$	1.24264	0.122441	0.0612205	−	0.998124i	$-0.480501\pi$
$103$	1.24264	0.122441	0.0612205	+	0.998124i	$0.480501\pi$
$104$	4.48528	0.439818
$105$	0	0
$106$	1.65685	0.160928
$107$	−15.4142	−1.49015	−0.745074	−	0.666982i	$-0.767586\pi$
$107$	−15.4142	−1.49015	−0.745074	+	0.666982i	$0.767586\pi$
$108$	0	0
$109$	−13.4853	−1.29166	−0.645828	−	0.763483i	$-0.723488\pi$
$109$	−13.4853	−1.29166	−0.645828	+	0.763483i	$0.723488\pi$
$110$	−4.82843	−0.460372
$111$	0	0
$112$	0	0
$113$	4.34315	0.408569	0.204284	−	0.978912i	$-0.434513\pi$
$113$	4.34315	0.408569	0.204284	+	0.978912i	$0.434513\pi$
$114$	0	0
$115$	−6.24264	−0.582129
$116$	0	0
$117$	0	0
$118$	−2.00000	−0.184115
$119$	0	0
$120$	0	0
$121$	0.656854	0.0597140
$122$	17.6569	1.59858
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−18.0711	−1.60355	−0.801774	−	0.597627i	$-0.796110\pi$
$127$	−18.0711	−1.60355	−0.801774	+	0.597627i	$0.796110\pi$
$128$	−11.3137	−1.00000
$129$	0	0
$130$	2.24264	0.196693
$131$	−10.4853	−0.916103	−0.458052	−	0.888926i	$-0.651453\pi$
$131$	−10.4853	−0.916103	−0.458052	+	0.888926i	$0.651453\pi$
$132$	0	0
$133$	0	0
$134$	−16.5858	−1.43279
$135$	0	0
$136$	−17.6569	−1.51406
$137$	2.92893	0.250236	0.125118	−	0.992142i	$-0.460069\pi$
$137$	2.92893	0.250236	0.125118	+	0.992142i	$0.460069\pi$
$138$	0	0
$139$	−11.4853	−0.974169	−0.487084	−	0.873355i	$-0.661940\pi$
$139$	−11.4853	−0.974169	−0.487084	+	0.873355i	$0.661940\pi$
$140$	0	0
$141$	0	0
$142$	−4.82843	−0.405193
$143$	−5.41421	−0.452759
$144$	0	0
$145$	−0.242641	−0.0201502
$146$	2.92893	0.242400
$147$	0	0
$148$	0	0
$149$	−17.6569	−1.44651	−0.723253	−	0.690583i	$-0.757354\pi$
$149$	−17.6569	−1.44651	−0.723253	+	0.690583i	$0.757354\pi$
$150$	0	0
$151$	−6.48528	−0.527765	−0.263882	−	0.964555i	$-0.585003\pi$
$151$	−6.48528	−0.527765	−0.263882	+	0.964555i	$0.585003\pi$
$152$	18.8284	1.52719
$153$	0	0
$154$	0	0
$155$	−0.171573	−0.0137811
$156$	0	0
$157$	16.1421	1.28828	0.644141	−	0.764906i	$-0.277215\pi$
$157$	16.1421	1.28828	0.644141	+	0.764906i	$0.277215\pi$
$158$	6.58579	0.523937
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−19.3137	−1.51277	−0.756383	−	0.654129i	$-0.773035\pi$
$163$	−19.3137	−1.51277	−0.756383	+	0.654129i	$0.773035\pi$
$164$	0	0
$165$	0	0
$166$	−7.65685	−0.594287
$167$	11.7574	0.909812	0.454906	−	0.890540i	$-0.349673\pi$
$167$	11.7574	0.909812	0.454906	+	0.890540i	$0.349673\pi$
$168$	0	0
$169$	−10.4853	−0.806560
$170$	−8.82843	−0.677109
$171$	0	0
$172$	0	0
$173$	−2.82843	−0.215041	−0.107521	−	0.994203i	$-0.534291\pi$
$173$	−2.82843	−0.215041	−0.107521	+	0.994203i	$0.534291\pi$
$174$	0	0
$175$	0	0
$176$	13.6569	1.02942
$177$	0	0
$178$	5.31371	0.398279
$179$	−19.6569	−1.46922	−0.734611	−	0.678488i	$-0.762635\pi$
$179$	−19.6569	−1.46922	−0.734611	+	0.678488i	$0.762635\pi$
$180$	0	0
$181$	−0.656854	−0.0488236	−0.0244118	−	0.999702i	$-0.507771\pi$
$181$	−0.656854	−0.0488236	−0.0244118	+	0.999702i	$0.507771\pi$
$182$	0	0
$183$	0	0
$184$	17.6569	1.30168
$185$	5.58579	0.410675
$186$	0	0
$187$	21.3137	1.55861
$188$	0	0
$189$	0	0
$190$	9.41421	0.682979
$191$	−18.9706	−1.37266	−0.686331	−	0.727289i	$-0.740780\pi$
$191$	−18.9706	−1.37266	−0.686331	+	0.727289i	$0.740780\pi$
$192$	0	0
$193$	−6.75736	−0.486405	−0.243203	−	0.969975i	$-0.578198\pi$
$193$	−6.75736	−0.486405	−0.243203	+	0.969975i	$0.578198\pi$
$194$	−15.3137	−1.09946
$195$	0	0
$196$	0	0
$197$	−5.55635	−0.395873	−0.197937	−	0.980215i	$-0.563424\pi$
$197$	−5.55635	−0.395873	−0.197937	+	0.980215i	$0.563424\pi$
$198$	0	0
$199$	−5.51472	−0.390928	−0.195464	−	0.980711i	$-0.562621\pi$
$199$	−5.51472	−0.390928	−0.195464	+	0.980711i	$0.562621\pi$
$200$	2.82843	0.200000
$201$	0	0
$202$	8.82843	0.621166
$203$	0	0
$204$	0	0
$205$	−2.24264	−0.156633
$206$	−1.75736	−0.122441
$207$	0	0
$208$	−6.34315	−0.439818
$209$	−22.7279	−1.57212
$210$	0	0
$211$	0.142136	0.00978502	0.00489251	−	0.999988i	$-0.498443\pi$
$211$	0.142136	0.00978502	0.00489251	+	0.999988i	$0.498443\pi$
$212$	0	0
$213$	0	0
$214$	21.7990	1.49015
$215$	10.4142	0.710243
$216$	0	0
$217$	0	0
$218$	19.0711	1.29166
$219$	0	0
$220$	0	0
$221$	−9.89949	−0.665912
$222$	0	0
$223$	−17.1716	−1.14989	−0.574947	−	0.818191i	$-0.694977\pi$
$223$	−17.1716	−1.14989	−0.574947	+	0.818191i	$0.694977\pi$
$224$	0	0
$225$	0	0
$226$	−6.14214	−0.408569
$227$	23.0711	1.53128	0.765640	−	0.643269i	$-0.222422\pi$
$227$	23.0711	1.53128	0.765640	+	0.643269i	$0.222422\pi$
$228$	0	0
$229$	14.3137	0.945876	0.472938	−	0.881096i	$-0.343193\pi$
$229$	14.3137	0.945876	0.472938	+	0.881096i	$0.343193\pi$
$230$	8.82843	0.582129
$231$	0	0
$232$	0.686292	0.0450572
$233$	22.4853	1.47306	0.736530	−	0.676405i	$-0.236463\pi$
$233$	22.4853	1.47306	0.736530	+	0.676405i	$0.236463\pi$
$234$	0	0
$235$	−9.31371	−0.607559
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−17.3137	−1.11993	−0.559965	−	0.828516i	$-0.689186\pi$
$239$	−17.3137	−1.11993	−0.559965	+	0.828516i	$0.689186\pi$
$240$	0	0
$241$	10.3431	0.666261	0.333130	−	0.942881i	$-0.391895\pi$
$241$	10.3431	0.666261	0.333130	+	0.942881i	$0.391895\pi$
$242$	−0.928932	−0.0597140
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	10.5563	0.671684
$248$	0.485281	0.0308154
$249$	0	0
$250$	1.41421	0.0894427
$251$	−5.41421	−0.341742	−0.170871	−	0.985293i	$-0.554658\pi$
$251$	−5.41421	−0.341742	−0.170871	+	0.985293i	$0.554658\pi$
$252$	0	0
$253$	−21.3137	−1.33998
$254$	25.5563	1.60355
$255$	0	0
$256$	0	0
$257$	−29.8995	−1.86508	−0.932540	−	0.361068i	$-0.882412\pi$
$257$	−29.8995	−1.86508	−0.932540	+	0.361068i	$0.882412\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	14.8284	0.916103
$263$	−1.17157	−0.0722423	−0.0361211	−	0.999347i	$-0.511500\pi$
$263$	−1.17157	−0.0722423	−0.0361211	+	0.999347i	$0.511500\pi$
$264$	0	0
$265$	1.17157	0.0719691
$266$	0	0
$267$	0	0
$268$	0	0
$269$	8.14214	0.496435	0.248217	−	0.968704i	$-0.420155\pi$
$269$	8.14214	0.496435	0.248217	+	0.968704i	$0.420155\pi$
$270$	0	0
$271$	−20.1421	−1.22355	−0.611774	−	0.791033i	$-0.709544\pi$
$271$	−20.1421	−1.22355	−0.611774	+	0.791033i	$0.709544\pi$
$272$	24.9706	1.51406
$273$	0	0
$274$	−4.14214	−0.250236
$275$	−3.41421	−0.205885
$276$	0	0
$277$	−23.3848	−1.40506	−0.702528	−	0.711657i	$-0.747945\pi$
$277$	−23.3848	−1.40506	−0.702528	+	0.711657i	$0.747945\pi$
$278$	16.2426	0.974169
$279$	0	0
$280$	0	0
$281$	−9.65685	−0.576080	−0.288040	−	0.957618i	$-0.593004\pi$
$281$	−9.65685	−0.576080	−0.288040	+	0.957618i	$0.593004\pi$
$282$	0	0
$283$	13.2426	0.787193	0.393597	−	0.919283i	$-0.371231\pi$
$283$	13.2426	0.787193	0.393597	+	0.919283i	$0.371231\pi$
$284$	0	0
$285$	0	0
$286$	7.65685	0.452759
$287$	0	0
$288$	0	0
$289$	21.9706	1.29239
$290$	0.343146	0.0201502
$291$	0	0
$292$	0	0
$293$	−15.3137	−0.894636	−0.447318	−	0.894375i	$-0.647621\pi$
$293$	−15.3137	−0.894636	−0.447318	+	0.894375i	$0.647621\pi$
$294$	0	0
$295$	−1.41421	−0.0823387
$296$	−15.7990	−0.918298
$297$	0	0
$298$	24.9706	1.44651
$299$	9.89949	0.572503
$300$	0	0
$301$	0	0
$302$	9.17157	0.527765
$303$	0	0
$304$	−26.6274	−1.52719
$305$	12.4853	0.714905
$306$	0	0
$307$	−1.58579	−0.0905056	−0.0452528	−	0.998976i	$-0.514409\pi$
$307$	−1.58579	−0.0905056	−0.0452528	+	0.998976i	$0.514409\pi$
$308$	0	0
$309$	0	0
$310$	0.242641	0.0137811
$311$	−17.0711	−0.968011	−0.484006	−	0.875065i	$-0.660819\pi$
$311$	−17.0711	−0.968011	−0.484006	+	0.875065i	$0.660819\pi$
$312$	0	0
$313$	−26.2132	−1.48166	−0.740829	−	0.671694i	$-0.765567\pi$
$313$	−26.2132	−1.48166	−0.740829	+	0.671694i	$0.765567\pi$
$314$	−22.8284	−1.28828
$315$	0	0
$316$	0	0
$317$	−0.100505	−0.00564493	−0.00282246	−	0.999996i	$-0.500898\pi$
$317$	−0.100505	−0.00564493	−0.00282246	+	0.999996i	$0.500898\pi$
$318$	0	0
$319$	−0.828427	−0.0463830
$320$	−8.00000	−0.447214
$321$	0	0
$322$	0	0
$323$	−41.5563	−2.31226
$324$	0	0
$325$	1.58579	0.0879636
$326$	27.3137	1.51277
$327$	0	0
$328$	6.34315	0.350242
$329$	0	0
$330$	0	0
$331$	−27.6274	−1.51854	−0.759270	−	0.650776i	$-0.774444\pi$
$331$	−27.6274	−1.51854	−0.759270	+	0.650776i	$0.774444\pi$
$332$	0	0
$333$	0	0
$334$	−16.6274	−0.909812
$335$	−11.7279	−0.640765
$336$	0	0
$337$	−0.899495	−0.0489986	−0.0244993	−	0.999700i	$-0.507799\pi$
$337$	−0.899495	−0.0489986	−0.0244993	+	0.999700i	$0.507799\pi$
$338$	14.8284	0.806560
$339$	0	0
$340$	0	0
$341$	−0.585786	−0.0317221
$342$	0	0
$343$	0	0
$344$	−29.4558	−1.58815
$345$	0	0
$346$	4.00000	0.215041
$347$	24.9706	1.34049	0.670245	−	0.742140i	$-0.266189\pi$
$347$	24.9706	1.34049	0.670245	+	0.742140i	$0.266189\pi$
$348$	0	0
$349$	16.6274	0.890045	0.445023	−	0.895519i	$-0.353196\pi$
$349$	16.6274	0.890045	0.445023	+	0.895519i	$0.353196\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5.89949	0.313998	0.156999	−	0.987599i	$-0.449818\pi$
$353$	5.89949	0.313998	0.156999	+	0.987599i	$0.449818\pi$
$354$	0	0
$355$	−3.41421	−0.181208
$356$	0	0
$357$	0	0
$358$	27.7990	1.46922
$359$	26.5858	1.40314	0.701572	−	0.712599i	$-0.252482\pi$
$359$	26.5858	1.40314	0.701572	+	0.712599i	$0.252482\pi$
$360$	0	0
$361$	25.3137	1.33230
$362$	0.928932	0.0488236
$363$	0	0
$364$	0	0
$365$	2.07107	0.108405
$366$	0	0
$367$	19.5858	1.02237	0.511185	−	0.859471i	$-0.329207\pi$
$367$	19.5858	1.02237	0.511185	+	0.859471i	$0.329207\pi$
$368$	−24.9706	−1.30168
$369$	0	0
$370$	−7.89949	−0.410675
$371$	0	0
$372$	0	0
$373$	19.2426	0.996346	0.498173	−	0.867078i	$-0.334004\pi$
$373$	19.2426	0.996346	0.498173	+	0.867078i	$0.334004\pi$
$374$	−30.1421	−1.55861
$375$	0	0
$376$	26.3431	1.35854
$377$	0.384776	0.0198170
$378$	0	0
$379$	14.7990	0.760173	0.380087	−	0.924951i	$-0.375894\pi$
$379$	14.7990	0.760173	0.380087	+	0.924951i	$0.375894\pi$
$380$	0	0
$381$	0	0
$382$	26.8284	1.37266
$383$	12.4853	0.637968	0.318984	−	0.947760i	$-0.396658\pi$
$383$	12.4853	0.637968	0.318984	+	0.947760i	$0.396658\pi$
$384$	0	0
$385$	0	0
$386$	9.55635	0.486405
$387$	0	0
$388$	0	0
$389$	−16.8701	−0.855346	−0.427673	−	0.903934i	$-0.640666\pi$
$389$	−16.8701	−0.855346	−0.427673	+	0.903934i	$0.640666\pi$
$390$	0	0
$391$	−38.9706	−1.97083
$392$	0	0
$393$	0	0
$394$	7.85786	0.395873
$395$	4.65685	0.234312
$396$	0	0
$397$	24.2132	1.21523	0.607613	−	0.794233i	$-0.292127\pi$
$397$	24.2132	1.21523	0.607613	+	0.794233i	$0.292127\pi$
$398$	7.79899	0.390928
$399$	0	0
$400$	−4.00000	−0.200000
$401$	−0.485281	−0.0242338	−0.0121169	−	0.999927i	$-0.503857\pi$
$401$	−0.485281	−0.0242338	−0.0121169	+	0.999927i	$0.503857\pi$
$402$	0	0
$403$	0.272078	0.0135532
$404$	0	0
$405$	0	0
$406$	0	0
$407$	19.0711	0.945318
$408$	0	0
$409$	26.3137	1.30113	0.650565	−	0.759451i	$-0.274532\pi$
$409$	26.3137	1.30113	0.650565	+	0.759451i	$0.274532\pi$
$410$	3.17157	0.156633
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−5.41421	−0.265773
$416$	0	0
$417$	0	0
$418$	32.1421	1.57212
$419$	−3.17157	−0.154941	−0.0774707	−	0.996995i	$-0.524684\pi$
$419$	−3.17157	−0.154941	−0.0774707	+	0.996995i	$0.524684\pi$
$420$	0	0
$421$	27.4853	1.33955	0.669775	−	0.742564i	$-0.266390\pi$
$421$	27.4853	1.33955	0.669775	+	0.742564i	$0.266390\pi$
$422$	−0.201010	−0.00978502
$423$	0	0
$424$	−3.31371	−0.160928
$425$	−6.24264	−0.302813
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	−14.7279	−0.710243
$431$	−36.8284	−1.77396	−0.886981	−	0.461805i	$-0.847202\pi$
$431$	−36.8284	−1.77396	−0.886981	+	0.461805i	$0.847202\pi$
$432$	0	0
$433$	24.5563	1.18010	0.590051	−	0.807366i	$-0.299107\pi$
$433$	24.5563	1.18010	0.590051	+	0.807366i	$0.299107\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	41.5563	1.98791
$438$	0	0
$439$	−0.343146	−0.0163775	−0.00818873	−	0.999966i	$-0.502607\pi$
$439$	−0.343146	−0.0163775	−0.00818873	+	0.999966i	$0.502607\pi$
$440$	9.65685	0.460372
$441$	0	0
$442$	14.0000	0.665912
$443$	−0.485281	−0.0230564	−0.0115282	−	0.999934i	$-0.503670\pi$
$443$	−0.485281	−0.0230564	−0.0115282	+	0.999934i	$0.503670\pi$
$444$	0	0
$445$	3.75736	0.178116
$446$	24.2843	1.14989
$447$	0	0
$448$	0	0
$449$	6.97056	0.328961	0.164481	−	0.986380i	$-0.447405\pi$
$449$	6.97056	0.328961	0.164481	+	0.986380i	$0.447405\pi$
$450$	0	0
$451$	−7.65685	−0.360547
$452$	0	0
$453$	0	0
$454$	−32.6274	−1.53128
$455$	0	0
$456$	0	0
$457$	−20.8995	−0.977637	−0.488819	−	0.872385i	$-0.662572\pi$
$457$	−20.8995	−0.977637	−0.488819	+	0.872385i	$0.662572\pi$
$458$	−20.2426	−0.945876
$459$	0	0
$460$	0	0
$461$	−27.0711	−1.26083	−0.630413	−	0.776260i	$-0.717114\pi$
$461$	−27.0711	−1.26083	−0.630413	+	0.776260i	$0.717114\pi$
$462$	0	0
$463$	−10.4142	−0.483990	−0.241995	−	0.970278i	$-0.577802\pi$
$463$	−10.4142	−0.483990	−0.241995	+	0.970278i	$0.577802\pi$
$464$	−0.970563	−0.0450572
$465$	0	0
$466$	−31.7990	−1.47306
$467$	6.68629	0.309405	0.154702	−	0.987961i	$-0.450558\pi$
$467$	6.68629	0.309405	0.154702	+	0.987961i	$0.450558\pi$
$468$	0	0
$469$	0	0
$470$	13.1716	0.607559
$471$	0	0
$472$	4.00000	0.184115
$473$	35.5563	1.63488
$474$	0	0
$475$	6.65685	0.305437
$476$	0	0
$477$	0	0
$478$	24.4853	1.11993
$479$	−34.6274	−1.58217	−0.791084	−	0.611708i	$-0.790483\pi$
$479$	−34.6274	−1.58217	−0.791084	+	0.611708i	$0.790483\pi$
$480$	0	0
$481$	−8.85786	−0.403884
$482$	−14.6274	−0.666261
$483$	0	0
$484$	0	0
$485$	−10.8284	−0.491694
$486$	0	0
$487$	0.899495	0.0407600	0.0203800	−	0.999792i	$-0.493512\pi$
$487$	0.899495	0.0407600	0.0203800	+	0.999792i	$0.493512\pi$
$488$	−35.3137	−1.59858
$489$	0	0
$490$	0	0
$491$	10.1421	0.457708	0.228854	−	0.973461i	$-0.426502\pi$
$491$	10.1421	0.457708	0.228854	+	0.973461i	$0.426502\pi$
$492$	0	0
$493$	−1.51472	−0.0682195
$494$	−14.9289	−0.671684
$495$	0	0
$496$	−0.686292	−0.0308154
$497$	0	0
$498$	0	0
$499$	8.79899	0.393897	0.196948	−	0.980414i	$-0.436897\pi$
$499$	8.79899	0.393897	0.196948	+	0.980414i	$0.436897\pi$
$500$	0	0
$501$	0	0
$502$	7.65685	0.341742
$503$	−18.4853	−0.824218	−0.412109	−	0.911135i	$-0.635208\pi$
$503$	−18.4853	−0.824218	−0.412109	+	0.911135i	$0.635208\pi$
$504$	0	0
$505$	6.24264	0.277794
$506$	30.1421	1.33998
$507$	0	0
$508$	0	0
$509$	13.7990	0.611629	0.305815	−	0.952091i	$-0.401071\pi$
$509$	13.7990	0.611629	0.305815	+	0.952091i	$0.401071\pi$
$510$	0	0
$511$	0	0
$512$	22.6274	1.00000
$513$	0	0
$514$	42.2843	1.86508
$515$	−1.24264	−0.0547573
$516$	0	0
$517$	−31.7990	−1.39852
$518$	0	0
$519$	0	0
$520$	−4.48528	−0.196693
$521$	−39.1127	−1.71356	−0.856779	−	0.515683i	$-0.827538\pi$
$521$	−39.1127	−1.71356	−0.856779	+	0.515683i	$0.827538\pi$
$522$	0	0
$523$	7.58579	0.331703	0.165852	−	0.986151i	$-0.446963\pi$
$523$	7.58579	0.331703	0.165852	+	0.986151i	$0.446963\pi$
$524$	0	0
$525$	0	0
$526$	1.65685	0.0722423
$527$	−1.07107	−0.0466564
$528$	0	0
$529$	15.9706	0.694372
$530$	−1.65685	−0.0719691
$531$	0	0
$532$	0	0
$533$	3.55635	0.154043
$534$	0	0
$535$	15.4142	0.666415
$536$	33.1716	1.43279
$537$	0	0
$538$	−11.5147	−0.496435
$539$	0	0
$540$	0	0
$541$	32.3137	1.38927	0.694637	−	0.719360i	$-0.255565\pi$
$541$	32.3137	1.38927	0.694637	+	0.719360i	$0.255565\pi$
$542$	28.4853	1.22355
$543$	0	0
$544$	0	0
$545$	13.4853	0.577646
$546$	0	0
$547$	−31.7990	−1.35963	−0.679813	−	0.733385i	$-0.737939\pi$
$547$	−31.7990	−1.35963	−0.679813	+	0.733385i	$0.737939\pi$
$548$	0	0
$549$	0	0
$550$	4.82843	0.205885
$551$	1.61522	0.0688108
$552$	0	0
$553$	0	0
$554$	33.0711	1.40506
$555$	0	0
$556$	0	0
$557$	8.34315	0.353510	0.176755	−	0.984255i	$-0.443440\pi$
$557$	8.34315	0.353510	0.176755	+	0.984255i	$0.443440\pi$
$558$	0	0
$559$	−16.5147	−0.698498
$560$	0	0
$561$	0	0
$562$	13.6569	0.576080
$563$	−14.2843	−0.602010	−0.301005	−	0.953623i	$-0.597322\pi$
$563$	−14.2843	−0.602010	−0.301005	+	0.953623i	$0.597322\pi$
$564$	0	0
$565$	−4.34315	−0.182718
$566$	−18.7279	−0.787193
$567$	0	0
$568$	9.65685	0.405193
$569$	36.8701	1.54567	0.772837	−	0.634605i	$-0.218837\pi$
$569$	36.8701	1.54567	0.772837	+	0.634605i	$0.218837\pi$
$570$	0	0
$571$	27.9706	1.17053	0.585266	−	0.810841i	$-0.300990\pi$
$571$	27.9706	1.17053	0.585266	+	0.810841i	$0.300990\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6.24264	0.260336
$576$	0	0
$577$	−6.55635	−0.272944	−0.136472	−	0.990644i	$-0.543576\pi$
$577$	−6.55635	−0.272944	−0.136472	+	0.990644i	$0.543576\pi$
$578$	−31.0711	−1.29239
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	4.00000	0.165663
$584$	−5.85786	−0.242400
$585$	0	0
$586$	21.6569	0.894636
$587$	30.2426	1.24825	0.624124	−	0.781326i	$-0.285456\pi$
$587$	30.2426	1.24825	0.624124	+	0.781326i	$0.285456\pi$
$588$	0	0
$589$	1.14214	0.0470609
$590$	2.00000	0.0823387
$591$	0	0
$592$	22.3431	0.918298
$593$	13.4142	0.550856	0.275428	−	0.961322i	$-0.411180\pi$
$593$	13.4142	0.550856	0.275428	+	0.961322i	$0.411180\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	−14.0000	−0.572503
$599$	42.1421	1.72188	0.860940	−	0.508706i	$-0.169876\pi$
$599$	42.1421	1.72188	0.860940	+	0.508706i	$0.169876\pi$
$600$	0	0
$601$	24.1716	0.985979	0.492990	−	0.870035i	$-0.335904\pi$
$601$	24.1716	0.985979	0.492990	+	0.870035i	$0.335904\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−0.656854	−0.0267049
$606$	0	0
$607$	1.24264	0.0504372	0.0252186	−	0.999682i	$-0.491972\pi$
$607$	1.24264	0.0504372	0.0252186	+	0.999682i	$0.491972\pi$
$608$	0	0
$609$	0	0
$610$	−17.6569	−0.714905
$611$	14.7696	0.597512
$612$	0	0
$613$	10.1421	0.409637	0.204818	−	0.978800i	$-0.434340\pi$
$613$	10.1421	0.409637	0.204818	+	0.978800i	$0.434340\pi$
$614$	2.24264	0.0905056
$615$	0	0
$616$	0	0
$617$	8.82843	0.355419	0.177710	−	0.984083i	$-0.443131\pi$
$617$	8.82843	0.355419	0.177710	+	0.984083i	$0.443131\pi$
$618$	0	0
$619$	−39.9706	−1.60655	−0.803276	−	0.595607i	$-0.796912\pi$
$619$	−39.9706	−1.60655	−0.803276	+	0.595607i	$0.796912\pi$
$620$	0	0
$621$	0	0
$622$	24.1421	0.968011
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	37.0711	1.48166
$627$	0	0
$628$	0	0
$629$	34.8701	1.39036
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	−13.1716	−0.523937
$633$	0	0
$634$	0.142136	0.00564493
$635$	18.0711	0.717128
$636$	0	0
$637$	0	0
$638$	1.17157	0.0463830
$639$	0	0
$640$	11.3137	0.447214
$641$	13.2132	0.521890	0.260945	−	0.965354i	$-0.415966\pi$
$641$	13.2132	0.521890	0.260945	+	0.965354i	$0.415966\pi$
$642$	0	0
$643$	−24.5563	−0.968408	−0.484204	−	0.874955i	$-0.660891\pi$
$643$	−24.5563	−0.968408	−0.484204	+	0.874955i	$0.660891\pi$
$644$	0	0
$645$	0	0
$646$	58.7696	2.31226
$647$	9.61522	0.378013	0.189007	−	0.981976i	$-0.439473\pi$
$647$	9.61522	0.378013	0.189007	+	0.981976i	$0.439473\pi$
$648$	0	0
$649$	−4.82843	−0.189532
$650$	−2.24264	−0.0879636
$651$	0	0
$652$	0	0
$653$	−7.75736	−0.303569	−0.151784	−	0.988414i	$-0.548502\pi$
$653$	−7.75736	−0.303569	−0.151784	+	0.988414i	$0.548502\pi$
$654$	0	0
$655$	10.4853	0.409694
$656$	−8.97056	−0.350242
$657$	0	0
$658$	0	0
$659$	27.3137	1.06399	0.531996	−	0.846747i	$-0.321442\pi$
$659$	27.3137	1.06399	0.531996	+	0.846747i	$0.321442\pi$
$660$	0	0
$661$	−32.3137	−1.25686	−0.628429	−	0.777867i	$-0.716302\pi$
$661$	−32.3137	−1.25686	−0.628429	+	0.777867i	$0.716302\pi$
$662$	39.0711	1.51854
$663$	0	0
$664$	15.3137	0.594287
$665$	0	0
$666$	0	0
$667$	1.51472	0.0586501
$668$	0	0
$669$	0	0
$670$	16.5858	0.640765
$671$	42.6274	1.64561
$672$	0	0
$673$	2.27208	0.0875822	0.0437911	−	0.999041i	$-0.486056\pi$
$673$	2.27208	0.0875822	0.0437911	+	0.999041i	$0.486056\pi$
$674$	1.27208	0.0489986
$675$	0	0
$676$	0	0
$677$	3.89949	0.149870	0.0749349	−	0.997188i	$-0.476125\pi$
$677$	3.89949	0.149870	0.0749349	+	0.997188i	$0.476125\pi$
$678$	0	0
$679$	0	0
$680$	17.6569	0.677109
$681$	0	0
$682$	0.828427	0.0317221
$683$	−20.5858	−0.787693	−0.393847	−	0.919176i	$-0.628856\pi$
$683$	−20.5858	−0.787693	−0.393847	+	0.919176i	$0.628856\pi$
$684$	0	0
$685$	−2.92893	−0.111909
$686$	0	0
$687$	0	0
$688$	41.6569	1.58815
$689$	−1.85786	−0.0707790
$690$	0	0
$691$	−30.3137	−1.15319	−0.576594	−	0.817031i	$-0.695619\pi$
$691$	−30.3137	−1.15319	−0.576594	+	0.817031i	$0.695619\pi$
$692$	0	0
$693$	0	0
$694$	−35.3137	−1.34049
$695$	11.4853	0.435662
$696$	0	0
$697$	−14.0000	−0.530288
$698$	−23.5147	−0.890045
$699$	0	0
$700$	0	0
$701$	−41.2132	−1.55660	−0.778301	−	0.627892i	$-0.783918\pi$
$701$	−41.2132	−1.55660	−0.778301	+	0.627892i	$0.783918\pi$
$702$	0	0
$703$	−37.1838	−1.40241
$704$	−27.3137	−1.02942
$705$	0	0
$706$	−8.34315	−0.313998
$707$	0	0
$708$	0	0
$709$	−11.1127	−0.417346	−0.208673	−	0.977985i	$-0.566914\pi$
$709$	−11.1127	−0.417346	−0.208673	+	0.977985i	$0.566914\pi$
$710$	4.82843	0.181208
$711$	0	0
$712$	−10.6274	−0.398279
$713$	1.07107	0.0401118
$714$	0	0
$715$	5.41421	0.202480
$716$	0	0
$717$	0	0
$718$	−37.5980	−1.40314
$719$	17.5147	0.653189	0.326594	−	0.945165i	$-0.394099\pi$
$719$	17.5147	0.653189	0.326594	+	0.945165i	$0.394099\pi$
$720$	0	0
$721$	0	0
$722$	−35.7990	−1.33230
$723$	0	0
$724$	0	0
$725$	0.242641	0.00901145
$726$	0	0
$727$	−4.75736	−0.176441	−0.0882203	−	0.996101i	$-0.528118\pi$
$727$	−4.75736	−0.176441	−0.0882203	+	0.996101i	$0.528118\pi$
$728$	0	0
$729$	0	0
$730$	−2.92893	−0.108405
$731$	65.0122	2.40456
$732$	0	0
$733$	10.7574	0.397332	0.198666	−	0.980067i	$-0.436339\pi$
$733$	10.7574	0.397332	0.198666	+	0.980067i	$0.436339\pi$
$734$	−27.6985	−1.02237
$735$	0	0
$736$	0	0
$737$	−40.0416	−1.47495
$738$	0	0
$739$	−35.1421	−1.29272	−0.646362	−	0.763031i	$-0.723710\pi$
$739$	−35.1421	−1.29272	−0.646362	+	0.763031i	$0.723710\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	4.72792	0.173451	0.0867253	−	0.996232i	$-0.472360\pi$
$743$	4.72792	0.173451	0.0867253	+	0.996232i	$0.472360\pi$
$744$	0	0
$745$	17.6569	0.646897
$746$	−27.2132	−0.996346
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	19.6274	0.716215	0.358107	−	0.933680i	$-0.383422\pi$
$751$	19.6274	0.716215	0.358107	+	0.933680i	$0.383422\pi$
$752$	−37.2548	−1.35854
$753$	0	0
$754$	−0.544156	−0.0198170
$755$	6.48528	0.236024
$756$	0	0
$757$	41.5980	1.51190	0.755952	−	0.654627i	$-0.227174\pi$
$757$	41.5980	1.51190	0.755952	+	0.654627i	$0.227174\pi$
$758$	−20.9289	−0.760173
$759$	0	0
$760$	−18.8284	−0.682979
$761$	−20.9289	−0.758673	−0.379337	−	0.925259i	$-0.623848\pi$
$761$	−20.9289	−0.758673	−0.379337	+	0.925259i	$0.623848\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−17.6569	−0.637968
$767$	2.24264	0.0809771
$768$	0	0
$769$	15.9706	0.575913	0.287957	−	0.957643i	$-0.407024\pi$
$769$	15.9706	0.575913	0.287957	+	0.957643i	$0.407024\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	24.9289	0.896631	0.448316	−	0.893875i	$-0.352024\pi$
$773$	24.9289	0.896631	0.448316	+	0.893875i	$0.352024\pi$
$774$	0	0
$775$	0.171573	0.00616308
$776$	30.6274	1.09946
$777$	0	0
$778$	23.8579	0.855346
$779$	14.9289	0.534885
$780$	0	0
$781$	−11.6569	−0.417115
$782$	55.1127	1.97083
$783$	0	0
$784$	0	0
$785$	−16.1421	−0.576138
$786$	0	0
$787$	−13.1716	−0.469516	−0.234758	−	0.972054i	$-0.575430\pi$
$787$	−13.1716	−0.469516	−0.234758	+	0.972054i	$0.575430\pi$
$788$	0	0
$789$	0	0
$790$	−6.58579	−0.234312
$791$	0	0
$792$	0	0
$793$	−19.7990	−0.703083
$794$	−34.2426	−1.21523
$795$	0	0
$796$	0	0
$797$	10.4437	0.369933	0.184967	−	0.982745i	$-0.440782\pi$
$797$	10.4437	0.369933	0.184967	+	0.982745i	$0.440782\pi$
$798$	0	0
$799$	−58.1421	−2.05692
$800$	0	0
$801$	0	0
$802$	0.686292	0.0242338
$803$	7.07107	0.249533
$804$	0	0
$805$	0	0
$806$	−0.384776	−0.0135532
$807$	0	0
$808$	−17.6569	−0.621166
$809$	4.62742	0.162691	0.0813457	−	0.996686i	$-0.474078\pi$
$809$	4.62742	0.162691	0.0813457	+	0.996686i	$0.474078\pi$
$810$	0	0
$811$	22.9706	0.806606	0.403303	−	0.915067i	$-0.367862\pi$
$811$	22.9706	0.806606	0.403303	+	0.915067i	$0.367862\pi$
$812$	0	0
$813$	0	0
$814$	−26.9706	−0.945318
$815$	19.3137	0.676530
$816$	0	0
$817$	−69.3259	−2.42541
$818$	−37.2132	−1.30113
$819$	0	0
$820$	0	0
$821$	−13.5563	−0.473120	−0.236560	−	0.971617i	$-0.576020\pi$
$821$	−13.5563	−0.473120	−0.236560	+	0.971617i	$0.576020\pi$
$822$	0	0
$823$	22.2843	0.776781	0.388390	−	0.921495i	$-0.373031\pi$
$823$	22.2843	0.776781	0.388390	+	0.921495i	$0.373031\pi$
$824$	3.51472	0.122441
$825$	0	0
$826$	0	0
$827$	−12.8284	−0.446088	−0.223044	−	0.974808i	$-0.571599\pi$
$827$	−12.8284	−0.446088	−0.223044	+	0.974808i	$0.571599\pi$
$828$	0	0
$829$	18.6569	0.647979	0.323990	−	0.946061i	$-0.394976\pi$
$829$	18.6569	0.647979	0.323990	+	0.946061i	$0.394976\pi$
$830$	7.65685	0.265773
$831$	0	0
$832$	12.6863	0.439818
$833$	0	0
$834$	0	0
$835$	−11.7574	−0.406880
$836$	0	0
$837$	0	0
$838$	4.48528	0.154941
$839$	−7.27208	−0.251060	−0.125530	−	0.992090i	$-0.540063\pi$
$839$	−7.27208	−0.251060	−0.125530	+	0.992090i	$0.540063\pi$
$840$	0	0
$841$	−28.9411	−0.997970
$842$	−38.8701	−1.33955
$843$	0	0
$844$	0	0
$845$	10.4853	0.360705
$846$	0	0
$847$	0	0
$848$	4.68629	0.160928
$849$	0	0
$850$	8.82843	0.302813
$851$	−34.8701	−1.19533
$852$	0	0
$853$	−38.0122	−1.30151	−0.650756	−	0.759287i	$-0.725548\pi$
$853$	−38.0122	−1.30151	−0.650756	+	0.759287i	$0.725548\pi$
$854$	0	0
$855$	0	0
$856$	−43.5980	−1.49015
$857$	−6.24264	−0.213245	−0.106622	−	0.994300i	$-0.534004\pi$
$857$	−6.24264	−0.213245	−0.106622	+	0.994300i	$0.534004\pi$
$858$	0	0
$859$	−15.6569	−0.534205	−0.267102	−	0.963668i	$-0.586066\pi$
$859$	−15.6569	−0.534205	−0.267102	+	0.963668i	$0.586066\pi$
$860$	0	0
$861$	0	0
$862$	52.0833	1.77396
$863$	9.79899	0.333561	0.166781	−	0.985994i	$-0.446663\pi$
$863$	9.79899	0.333561	0.166781	+	0.985994i	$0.446663\pi$
$864$	0	0
$865$	2.82843	0.0961694
$866$	−34.7279	−1.18010
$867$	0	0
$868$	0	0
$869$	15.8995	0.539353
$870$	0	0
$871$	18.5980	0.630169
$872$	−38.1421	−1.29166
$873$	0	0
$874$	−58.7696	−1.98791
$875$	0	0
$876$	0	0
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	0.485281	0.0163775
$879$	0	0
$880$	−13.6569	−0.460372
$881$	−21.6569	−0.729638	−0.364819	−	0.931078i	$-0.618869\pi$
$881$	−21.6569	−0.729638	−0.364819	+	0.931078i	$0.618869\pi$
$882$	0	0
$883$	8.07107	0.271613	0.135807	−	0.990735i	$-0.456637\pi$
$883$	8.07107	0.271613	0.135807	+	0.990735i	$0.456637\pi$
$884$	0	0
$885$	0	0
$886$	0.686292	0.0230564
$887$	5.21320	0.175042	0.0875211	−	0.996163i	$-0.472105\pi$
$887$	5.21320	0.175042	0.0875211	+	0.996163i	$0.472105\pi$
$888$	0	0
$889$	0	0
$890$	−5.31371	−0.178116
$891$	0	0
$892$	0	0
$893$	62.0000	2.07475
$894$	0	0
$895$	19.6569	0.657056
$896$	0	0
$897$	0	0
$898$	−9.85786	−0.328961
$899$	0.0416306	0.00138846
$900$	0	0
$901$	7.31371	0.243655
$902$	10.8284	0.360547
$903$	0	0
$904$	12.2843	0.408569
$905$	0.656854	0.0218346
$906$	0	0
$907$	−8.61522	−0.286064	−0.143032	−	0.989718i	$-0.545685\pi$
$907$	−8.61522	−0.286064	−0.143032	+	0.989718i	$0.545685\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−4.24264	−0.140565	−0.0702825	−	0.997527i	$-0.522390\pi$
$911$	−4.24264	−0.140565	−0.0702825	+	0.997527i	$0.522390\pi$
$912$	0	0
$913$	−18.4853	−0.611774
$914$	29.5563	0.977637
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−46.6569	−1.53907	−0.769534	−	0.638606i	$-0.779511\pi$
$919$	−46.6569	−1.53907	−0.769534	+	0.638606i	$0.779511\pi$
$920$	−17.6569	−0.582129
$921$	0	0
$922$	38.2843	1.26083
$923$	5.41421	0.178211
$924$	0	0
$925$	−5.58579	−0.183660
$926$	14.7279	0.483990
$927$	0	0
$928$	0	0
$929$	−51.0122	−1.67366	−0.836828	−	0.547466i	$-0.815593\pi$
$929$	−51.0122	−1.67366	−0.836828	+	0.547466i	$0.815593\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	−9.45584	−0.309405
$935$	−21.3137	−0.697033
$936$	0	0
$937$	7.92893	0.259027	0.129513	−	0.991578i	$-0.458658\pi$
$937$	7.92893	0.259027	0.129513	+	0.991578i	$0.458658\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−11.2721	−0.367459	−0.183730	−	0.982977i	$-0.558817\pi$
$941$	−11.2721	−0.367459	−0.183730	+	0.982977i	$0.558817\pi$
$942$	0	0
$943$	14.0000	0.455903
$944$	−5.65685	−0.184115
$945$	0	0
$946$	−50.2843	−1.63488
$947$	4.92893	0.160169	0.0800844	−	0.996788i	$-0.474481\pi$
$947$	4.92893	0.160169	0.0800844	+	0.996788i	$0.474481\pi$
$948$	0	0
$949$	−3.28427	−0.106612
$950$	−9.41421	−0.305437
$951$	0	0
$952$	0	0
$953$	−48.4853	−1.57059	−0.785296	−	0.619120i	$-0.787489\pi$
$953$	−48.4853	−1.57059	−0.785296	+	0.619120i	$0.787489\pi$
$954$	0	0
$955$	18.9706	0.613873
$956$	0	0
$957$	0	0
$958$	48.9706	1.58217
$959$	0	0
$960$	0	0
$961$	−30.9706	−0.999050
$962$	12.5269	0.403884
$963$	0	0
$964$	0	0
$965$	6.75736	0.217527
$966$	0	0
$967$	23.3848	0.752004	0.376002	−	0.926619i	$-0.377299\pi$
$967$	23.3848	0.752004	0.376002	+	0.926619i	$0.377299\pi$
$968$	1.85786	0.0597140
$969$	0	0
$970$	15.3137	0.491694
$971$	14.3431	0.460293	0.230147	−	0.973156i	$-0.426079\pi$
$971$	14.3431	0.460293	0.230147	+	0.973156i	$0.426079\pi$
$972$	0	0
$973$	0	0
$974$	−1.27208	−0.0407600
$975$	0	0
$976$	49.9411	1.59858
$977$	−13.6985	−0.438253	−0.219127	−	0.975696i	$-0.570321\pi$
$977$	−13.6985	−0.438253	−0.219127	+	0.975696i	$0.570321\pi$
$978$	0	0
$979$	12.8284	0.409998
$980$	0	0
$981$	0	0
$982$	−14.3431	−0.457708
$983$	53.1543	1.69536	0.847680	−	0.530508i	$-0.177999\pi$
$983$	53.1543	1.69536	0.847680	+	0.530508i	$0.177999\pi$
$984$	0	0
$985$	5.55635	0.177040
$986$	2.14214	0.0682195
$987$	0	0
$988$	0	0
$989$	−65.0122	−2.06727
$990$	0	0
$991$	−30.6569	−0.973847	−0.486924	−	0.873445i	$-0.661881\pi$
$991$	−30.6569	−0.973847	−0.486924	+	0.873445i	$0.661881\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	5.51472	0.174828
$996$	0	0
$997$	−0.757359	−0.0239858	−0.0119929	−	0.999928i	$-0.503818\pi$
$997$	−0.757359	−0.0239858	−0.0119929	+	0.999928i	$0.503818\pi$
$998$	−12.4437	−0.393897
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.2.a.s.1.1		2
3.2	odd	2		735.2.a.j.1.2		2
7.3	odd	6		315.2.j.d.226.2		4
7.5	odd	6		315.2.j.d.46.2		4
7.6	odd	2		2205.2.a.u.1.1		2
15.14	odd	2		3675.2.a.x.1.1		2
21.2	odd	6		735.2.i.j.361.1		4
21.5	even	6		105.2.i.c.46.1	yes	4
21.11	odd	6		735.2.i.j.226.1		4
21.17	even	6		105.2.i.c.16.1	✓	4
21.20	even	2		735.2.a.i.1.2		2
84.47	odd	6		1680.2.bg.p.1201.1		4
84.59	odd	6		1680.2.bg.p.961.1		4
105.17	odd	12		525.2.r.g.499.2		8
105.38	odd	12		525.2.r.g.499.3		8
105.47	odd	12		525.2.r.g.424.3		8
105.59	even	6		525.2.i.g.226.2		4
105.68	odd	12		525.2.r.g.424.2		8
105.89	even	6		525.2.i.g.151.2		4
105.104	even	2		3675.2.a.z.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.i.c.16.1	✓	4	21.17	even	6
105.2.i.c.46.1	yes	4	21.5	even	6
315.2.j.d.46.2		4	7.5	odd	6
315.2.j.d.226.2		4	7.3	odd	6
525.2.i.g.151.2		4	105.89	even	6
525.2.i.g.226.2		4	105.59	even	6
525.2.r.g.424.2		8	105.68	odd	12
525.2.r.g.424.3		8	105.47	odd	12
525.2.r.g.499.2		8	105.17	odd	12
525.2.r.g.499.3		8	105.38	odd	12
735.2.a.i.1.2		2	21.20	even	2
735.2.a.j.1.2		2	3.2	odd	2
735.2.i.j.226.1		4	21.11	odd	6
735.2.i.j.361.1		4	21.2	odd	6
1680.2.bg.p.961.1		4	84.59	odd	6
1680.2.bg.p.1201.1		4	84.47	odd	6
2205.2.a.s.1.1		2	1.1	even	1	trivial
2205.2.a.u.1.1		2	7.6	odd	2
3675.2.a.x.1.1		2	15.14	odd	2
3675.2.a.z.1.1		2	105.104	even	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:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} -1.00000 q^{5} +2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} -1.00000 q^{5} +2.82843 q^{8} +1.41421 q^{10} -3.41421 q^{11} +1.58579 q^{13} -4.00000 q^{16} -6.24264 q^{17} +6.65685 q^{19} +4.82843 q^{22} +6.24264 q^{23} +1.00000 q^{25} -2.24264 q^{26} +0.242641 q^{29} +0.171573 q^{31} +8.82843 q^{34} -5.58579 q^{37} -9.41421 q^{38} -2.82843 q^{40} +2.24264 q^{41} -10.4142 q^{43} -8.82843 q^{46} +9.31371 q^{47} -1.41421 q^{50} -1.17157 q^{53} +3.41421 q^{55} -0.343146 q^{58} +1.41421 q^{59} -12.4853 q^{61} -0.242641 q^{62} +8.00000 q^{64} -1.58579 q^{65} +11.7279 q^{67} +3.41421 q^{71} -2.07107 q^{73} +7.89949 q^{74} -4.65685 q^{79} +4.00000 q^{80} -3.17157 q^{82} +5.41421 q^{83} +6.24264 q^{85} +14.7279 q^{86} -9.65685 q^{88} -3.75736 q^{89} -13.1716 q^{94} -6.65685 q^{95} +10.8284 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5}+O(q^{10}) \) 2 * q - 2 * q^5	\( 2 q - 2 q^{5} - 4 q^{11} + 6 q^{13} - 8 q^{16} - 4 q^{17} + 2 q^{19} + 4 q^{22} + 4 q^{23} + 2 q^{25} + 4 q^{26} - 8 q^{29} + 6 q^{31} + 12 q^{34} - 14 q^{37} - 16 q^{38} - 4 q^{41} - 18 q^{43} - 12 q^{46} - 4 q^{47} - 8 q^{53} + 4 q^{55} - 12 q^{58} - 8 q^{61} + 8 q^{62} + 16 q^{64} - 6 q^{65} - 2 q^{67} + 4 q^{71} + 10 q^{73} - 4 q^{74} + 2 q^{79} + 8 q^{80} - 12 q^{82} + 8 q^{83} + 4 q^{85} + 4 q^{86} - 8 q^{88} - 16 q^{89} - 32 q^{94} - 2 q^{95} + 16 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 4 * q^11 + 6 * q^13 - 8 * q^16 - 4 * q^17 + 2 * q^19 + 4 * q^22 + 4 * q^23 + 2 * q^25 + 4 * q^26 - 8 * q^29 + 6 * q^31 + 12 * q^34 - 14 * q^37 - 16 * q^38 - 4 * q^41 - 18 * q^43 - 12 * q^46 - 4 * q^47 - 8 * q^53 + 4 * q^55 - 12 * q^58 - 8 * q^61 + 8 * q^62 + 16 * q^64 - 6 * q^65 - 2 * q^67 + 4 * q^71 + 10 * q^73 - 4 * q^74 + 2 * q^79 + 8 * q^80 - 12 * q^82 + 8 * q^83 + 4 * q^85 + 4 * q^86 - 8 * q^88 - 16 * q^89 - 32 * q^94 - 2 * q^95 + 16 * q^97

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