LMFDB - Embedded newform 6776.2.a.bp.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6776,2,Mod(1,6776)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6776, 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("6776.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6776 = 2^{3} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6776.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:	$54.1066324096$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 23x^{8} + 40x^{7} + 178x^{6} - 266x^{5} - 515x^{4} + 700x^{3} + 433x^{2} - 624x + 132 $ x^10 - 2x^9 - 23x^8 + 40x^7 + 178x^6 - 266x^5 - 515x^4 + 700x^3 + 433x^2 - 624*x + 132
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$3.33612$ of defining polynomial
Character	$\chi$	$=$	6776.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.33612 q^{3} +4.16244 q^{5} +1.00000 q^{7} +8.12971 q^{9} +O(q^{10})$	$q+3.33612 q^{3} +4.16244 q^{5} +1.00000 q^{7} +8.12971 q^{9} -3.85036 q^{13} +13.8864 q^{15} -0.171569 q^{17} +3.33657 q^{19} +3.33612 q^{21} -1.51024 q^{23} +12.3259 q^{25} +17.1134 q^{27} -9.49721 q^{29} -8.97027 q^{31} +4.16244 q^{35} -5.01901 q^{37} -12.8453 q^{39} +5.76810 q^{41} +5.29822 q^{43} +33.8395 q^{45} -0.0721270 q^{47} +1.00000 q^{49} -0.572376 q^{51} +1.25593 q^{53} +11.1312 q^{57} +4.15485 q^{59} +13.1909 q^{61} +8.12971 q^{63} -16.0269 q^{65} -12.0475 q^{67} -5.03836 q^{69} -3.95433 q^{71} -2.05848 q^{73} +41.1209 q^{75} +1.12520 q^{79} +32.7031 q^{81} +12.1809 q^{83} -0.714147 q^{85} -31.6839 q^{87} -6.53593 q^{89} -3.85036 q^{91} -29.9259 q^{93} +13.8883 q^{95} +4.25054 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{3} + 4 q^{5} + 10 q^{7} + 20 q^{9}+O(q^{10}) $ 10 * q + 2 * q^3 + 4 * q^5 + 10 * q^7 + 20 * q^9	$ 10 q + 2 q^{3} + 4 q^{5} + 10 q^{7} + 20 q^{9} - 12 q^{13} + 10 q^{15} + 2 q^{17} + 2 q^{19} + 2 q^{21} + 10 q^{23} + 8 q^{25} + 14 q^{27} - 16 q^{29} + 14 q^{31} + 4 q^{35} + 2 q^{37} - 4 q^{39} + 18 q^{41} + 16 q^{43} + 52 q^{45} - 4 q^{47} + 10 q^{49} + 16 q^{51} + 32 q^{53} - 24 q^{57} + 4 q^{59} + 16 q^{61} + 20 q^{63} - 4 q^{65} + 10 q^{67} - 14 q^{69} - 10 q^{71} - 22 q^{73} + 48 q^{75} + 36 q^{79} + 50 q^{81} + 2 q^{83} - 4 q^{85} - 2 q^{87} + 36 q^{89} - 12 q^{91} - 2 q^{93} + 52 q^{95} + 44 q^{97}+O(q^{100}) $ 10 * q + 2 * q^3 + 4 * q^5 + 10 * q^7 + 20 * q^9 - 12 * q^13 + 10 * q^15 + 2 * q^17 + 2 * q^19 + 2 * q^21 + 10 * q^23 + 8 * q^25 + 14 * q^27 - 16 * q^29 + 14 * q^31 + 4 * q^35 + 2 * q^37 - 4 * q^39 + 18 * q^41 + 16 * q^43 + 52 * q^45 - 4 * q^47 + 10 * q^49 + 16 * q^51 + 32 * q^53 - 24 * q^57 + 4 * q^59 + 16 * q^61 + 20 * q^63 - 4 * q^65 + 10 * q^67 - 14 * q^69 - 10 * q^71 - 22 * q^73 + 48 * q^75 + 36 * q^79 + 50 * q^81 + 2 * q^83 - 4 * q^85 - 2 * q^87 + 36 * q^89 - 12 * q^91 - 2 * q^93 + 52 * q^95 + 44 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.33612	1.92611	0.963056	−	0.269302i	$-0.0867931\pi$
$3$	3.33612	1.92611	0.963056	+	0.269302i	$0.0867931\pi$
$4$	0	0
$5$	4.16244	1.86150	0.930751	−	0.365654i	$-0.119155\pi$
$5$	4.16244	1.86150	0.930751	+	0.365654i	$0.119155\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	8.12971	2.70990
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−3.85036	−1.06790	−0.533949	−	0.845517i	$-0.679292\pi$
$13$	−3.85036	−1.06790	−0.533949	+	0.845517i	$0.679292\pi$
$14$	0	0
$15$	13.8864	3.58546
$16$	0	0
$17$	−0.171569	−0.0416117	−0.0208058	−	0.999784i	$-0.506623\pi$
$17$	−0.171569	−0.0416117	−0.0208058	+	0.999784i	$0.506623\pi$
$18$	0	0
$19$	3.33657	0.765461	0.382730	−	0.923860i	$-0.374984\pi$
$19$	3.33657	0.765461	0.382730	+	0.923860i	$0.374984\pi$
$20$	0	0
$21$	3.33612	0.728002
$22$	0	0
$23$	−1.51024	−0.314908	−0.157454	−	0.987526i	$-0.550329\pi$
$23$	−1.51024	−0.314908	−0.157454	+	0.987526i	$0.550329\pi$
$24$	0	0
$25$	12.3259	2.46519
$26$	0	0
$27$	17.1134	3.29347
$28$	0	0
$29$	−9.49721	−1.76359	−0.881794	−	0.471635i	$-0.843664\pi$
$29$	−9.49721	−1.76359	−0.881794	+	0.471635i	$0.843664\pi$
$30$	0	0
$31$	−8.97027	−1.61111	−0.805554	−	0.592523i	$-0.798132\pi$
$31$	−8.97027	−1.61111	−0.805554	+	0.592523i	$0.798132\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.16244	0.703582
$36$	0	0
$37$	−5.01901	−0.825119	−0.412560	−	0.910931i	$-0.635365\pi$
$37$	−5.01901	−0.825119	−0.412560	+	0.910931i	$0.635365\pi$
$38$	0	0
$39$	−12.8453	−2.05689
$40$	0	0
$41$	5.76810	0.900827	0.450413	−	0.892820i	$-0.351277\pi$
$41$	5.76810	0.900827	0.450413	+	0.892820i	$0.351277\pi$
$42$	0	0
$43$	5.29822	0.807971	0.403986	−	0.914765i	$-0.367625\pi$
$43$	5.29822	0.807971	0.403986	+	0.914765i	$0.367625\pi$
$44$	0	0
$45$	33.8395	5.04449
$46$	0	0
$47$	−0.0721270	−0.0105208	−0.00526040	−	0.999986i	$-0.501674\pi$
$47$	−0.0721270	−0.0105208	−0.00526040	+	0.999986i	$0.501674\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.572376	−0.0801487
$52$	0	0
$53$	1.25593	0.172516	0.0862578	−	0.996273i	$-0.472509\pi$
$53$	1.25593	0.172516	0.0862578	+	0.996273i	$0.472509\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	11.1312	1.47436
$58$	0	0
$59$	4.15485	0.540915	0.270457	−	0.962732i	$-0.412825\pi$
$59$	4.15485	0.540915	0.270457	+	0.962732i	$0.412825\pi$
$60$	0	0
$61$	13.1909	1.68892	0.844462	−	0.535616i	$-0.179920\pi$
$61$	13.1909	1.68892	0.844462	+	0.535616i	$0.179920\pi$
$62$	0	0
$63$	8.12971	1.02425
$64$	0	0
$65$	−16.0269	−1.98789
$66$	0	0
$67$	−12.0475	−1.47183	−0.735917	−	0.677072i	$-0.763249\pi$
$67$	−12.0475	−1.47183	−0.735917	+	0.677072i	$0.763249\pi$
$68$	0	0
$69$	−5.03836	−0.606547
$70$	0	0
$71$	−3.95433	−0.469293	−0.234646	−	0.972081i	$-0.575393\pi$
$71$	−3.95433	−0.469293	−0.234646	+	0.972081i	$0.575393\pi$
$72$	0	0
$73$	−2.05848	−0.240927	−0.120463	−	0.992718i	$-0.538438\pi$
$73$	−2.05848	−0.240927	−0.120463	+	0.992718i	$0.538438\pi$
$74$	0	0
$75$	41.1209	4.74823
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1.12520	0.126595	0.0632974	−	0.997995i	$-0.479838\pi$
$79$	1.12520	0.126595	0.0632974	+	0.997995i	$0.479838\pi$
$80$	0	0
$81$	32.7031	3.63368
$82$	0	0
$83$	12.1809	1.33703	0.668513	−	0.743700i	$-0.266931\pi$
$83$	12.1809	1.33703	0.668513	+	0.743700i	$0.266931\pi$
$84$	0	0
$85$	−0.714147	−0.0774602
$86$	0	0
$87$	−31.6839	−3.39687
$88$	0	0
$89$	−6.53593	−0.692807	−0.346404	−	0.938086i	$-0.612597\pi$
$89$	−6.53593	−0.692807	−0.346404	+	0.938086i	$0.612597\pi$
$90$	0	0
$91$	−3.85036	−0.403627
$92$	0	0
$93$	−29.9259	−3.10317
$94$	0	0
$95$	13.8883	1.42491
$96$	0	0
$97$	4.25054	0.431577	0.215789	−	0.976440i	$-0.430768\pi$
$97$	4.25054	0.431577	0.215789	+	0.976440i	$0.430768\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−16.3051	−1.62242	−0.811208	−	0.584758i	$-0.801189\pi$
$101$	−16.3051	−1.62242	−0.811208	+	0.584758i	$0.801189\pi$
$102$	0	0
$103$	−5.13285	−0.505755	−0.252877	−	0.967498i	$-0.581377\pi$
$103$	−5.13285	−0.505755	−0.252877	+	0.967498i	$0.581377\pi$
$104$	0	0
$105$	13.8864	1.35518
$106$	0	0
$107$	1.83467	0.177364	0.0886822	−	0.996060i	$-0.471734\pi$
$107$	1.83467	0.177364	0.0886822	+	0.996060i	$0.471734\pi$
$108$	0	0
$109$	0.247450	0.0237014	0.0118507	−	0.999930i	$-0.496228\pi$
$109$	0.247450	0.0237014	0.0118507	+	0.999930i	$0.496228\pi$
$110$	0	0
$111$	−16.7440	−1.58927
$112$	0	0
$113$	−14.9101	−1.40262	−0.701310	−	0.712856i	$-0.747401\pi$
$113$	−14.9101	−1.40262	−0.701310	+	0.712856i	$0.747401\pi$
$114$	0	0
$115$	−6.28631	−0.586201
$116$	0	0
$117$	−31.3023	−2.89390
$118$	0	0
$119$	−0.171569	−0.0157277
$120$	0	0
$121$	0	0
$122$	0	0
$123$	19.2431	1.73509
$124$	0	0
$125$	30.4938	2.72745
$126$	0	0
$127$	4.07013	0.361166	0.180583	−	0.983560i	$-0.442202\pi$
$127$	4.07013	0.361166	0.180583	+	0.983560i	$0.442202\pi$
$128$	0	0
$129$	17.6755	1.55624
$130$	0	0
$131$	−11.2835	−0.985841	−0.492921	−	0.870074i	$-0.664071\pi$
$131$	−11.2835	−0.985841	−0.492921	+	0.870074i	$0.664071\pi$
$132$	0	0
$133$	3.33657	0.289317
$134$	0	0
$135$	71.2334	6.13080
$136$	0	0
$137$	5.33965	0.456197	0.228099	−	0.973638i	$-0.426749\pi$
$137$	5.33965	0.456197	0.228099	+	0.973638i	$0.426749\pi$
$138$	0	0
$139$	2.49692	0.211786	0.105893	−	0.994378i	$-0.466230\pi$
$139$	2.49692	0.211786	0.105893	+	0.994378i	$0.466230\pi$
$140$	0	0
$141$	−0.240624	−0.0202642
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−39.5316	−3.28292
$146$	0	0
$147$	3.33612	0.275159
$148$	0	0
$149$	8.08160	0.662071	0.331035	−	0.943618i	$-0.392602\pi$
$149$	8.08160	0.662071	0.331035	+	0.943618i	$0.392602\pi$
$150$	0	0
$151$	5.63214	0.458337	0.229169	−	0.973387i	$-0.426399\pi$
$151$	5.63214	0.458337	0.229169	+	0.973387i	$0.426399\pi$
$152$	0	0
$153$	−1.39481	−0.112764
$154$	0	0
$155$	−37.3382	−2.99908
$156$	0	0
$157$	2.63163	0.210027	0.105014	−	0.994471i	$-0.466511\pi$
$157$	2.63163	0.210027	0.105014	+	0.994471i	$0.466511\pi$
$158$	0	0
$159$	4.18994	0.332284
$160$	0	0
$161$	−1.51024	−0.119024
$162$	0	0
$163$	7.89916	0.618710	0.309355	−	0.950947i	$-0.399887\pi$
$163$	7.89916	0.618710	0.309355	+	0.950947i	$0.399887\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	22.5236	1.74293	0.871466	−	0.490456i	$-0.163170\pi$
$167$	22.5236	1.74293	0.871466	+	0.490456i	$0.163170\pi$
$168$	0	0
$169$	1.82526	0.140405
$170$	0	0
$171$	27.1253	2.07433
$172$	0	0
$173$	−17.3001	−1.31530	−0.657652	−	0.753322i	$-0.728450\pi$
$173$	−17.3001	−1.31530	−0.657652	+	0.753322i	$0.728450\pi$
$174$	0	0
$175$	12.3259	0.931754
$176$	0	0
$177$	13.8611	1.04186
$178$	0	0
$179$	4.56290	0.341047	0.170523	−	0.985354i	$-0.445454\pi$
$179$	4.56290	0.341047	0.170523	+	0.985354i	$0.445454\pi$
$180$	0	0
$181$	−18.1826	−1.35150	−0.675750	−	0.737130i	$-0.736180\pi$
$181$	−18.1826	−1.35150	−0.675750	+	0.737130i	$0.736180\pi$
$182$	0	0
$183$	44.0065	3.25306
$184$	0	0
$185$	−20.8913	−1.53596
$186$	0	0
$187$	0	0
$188$	0	0
$189$	17.1134	1.24481
$190$	0	0
$191$	4.47216	0.323594	0.161797	−	0.986824i	$-0.448271\pi$
$191$	4.47216	0.323594	0.161797	+	0.986824i	$0.448271\pi$
$192$	0	0
$193$	14.7425	1.06118	0.530592	−	0.847627i	$-0.321970\pi$
$193$	14.7425	1.06118	0.530592	+	0.847627i	$0.321970\pi$
$194$	0	0
$195$	−53.4677	−3.82890
$196$	0	0
$197$	−23.5816	−1.68012	−0.840061	−	0.542492i	$-0.817481\pi$
$197$	−23.5816	−1.68012	−0.840061	+	0.542492i	$0.817481\pi$
$198$	0	0
$199$	20.5941	1.45987	0.729937	−	0.683514i	$-0.239549\pi$
$199$	20.5941	1.45987	0.729937	+	0.683514i	$0.239549\pi$
$200$	0	0
$201$	−40.1919	−2.83492
$202$	0	0
$203$	−9.49721	−0.666573
$204$	0	0
$205$	24.0094	1.67689
$206$	0	0
$207$	−12.2779	−0.853370
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−22.5505	−1.55244	−0.776220	−	0.630462i	$-0.782866\pi$
$211$	−22.5505	−1.55244	−0.776220	+	0.630462i	$0.782866\pi$
$212$	0	0
$213$	−13.1921	−0.903910
$214$	0	0
$215$	22.0536	1.50404
$216$	0	0
$217$	−8.97027	−0.608941
$218$	0	0
$219$	−6.86735	−0.464052
$220$	0	0
$221$	0.660603	0.0444370
$222$	0	0
$223$	−8.84382	−0.592226	−0.296113	−	0.955153i	$-0.595690\pi$
$223$	−8.84382	−0.592226	−0.296113	+	0.955153i	$0.595690\pi$
$224$	0	0
$225$	100.206	6.68043
$226$	0	0
$227$	−3.77152	−0.250325	−0.125162	−	0.992136i	$-0.539945\pi$
$227$	−3.77152	−0.250325	−0.125162	+	0.992136i	$0.539945\pi$
$228$	0	0
$229$	0.162265	0.0107228	0.00536139	−	0.999986i	$-0.498293\pi$
$229$	0.162265	0.0107228	0.00536139	+	0.999986i	$0.498293\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	11.1868	0.732870	0.366435	−	0.930444i	$-0.380578\pi$
$233$	11.1868	0.732870	0.366435	+	0.930444i	$0.380578\pi$
$234$	0	0
$235$	−0.300224	−0.0195845
$236$	0	0
$237$	3.75380	0.243836
$238$	0	0
$239$	−24.0430	−1.55521	−0.777606	−	0.628752i	$-0.783566\pi$
$239$	−24.0430	−1.55521	−0.777606	+	0.628752i	$0.783566\pi$
$240$	0	0
$241$	−18.0337	−1.16166	−0.580828	−	0.814027i	$-0.697271\pi$
$241$	−18.0337	−1.16166	−0.580828	+	0.814027i	$0.697271\pi$
$242$	0	0
$243$	57.7615	3.70540
$244$	0	0
$245$	4.16244	0.265929
$246$	0	0
$247$	−12.8470	−0.817433
$248$	0	0
$249$	40.6369	2.57526
$250$	0	0
$251$	12.4667	0.786894	0.393447	−	0.919347i	$-0.371283\pi$
$251$	12.4667	0.786894	0.393447	+	0.919347i	$0.371283\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−2.38248	−0.149197
$256$	0	0
$257$	−2.43093	−0.151637	−0.0758185	−	0.997122i	$-0.524157\pi$
$257$	−2.43093	−0.151637	−0.0758185	+	0.997122i	$0.524157\pi$
$258$	0	0
$259$	−5.01901	−0.311866
$260$	0	0
$261$	−77.2096	−4.77915
$262$	0	0
$263$	−7.02978	−0.433475	−0.216737	−	0.976230i	$-0.569541\pi$
$263$	−7.02978	−0.433475	−0.216737	+	0.976230i	$0.569541\pi$
$264$	0	0
$265$	5.22775	0.321138
$266$	0	0
$267$	−21.8047	−1.33442
$268$	0	0
$269$	3.12678	0.190643	0.0953215	−	0.995447i	$-0.469612\pi$
$269$	3.12678	0.190643	0.0953215	+	0.995447i	$0.469612\pi$
$270$	0	0
$271$	−18.8109	−1.14268	−0.571341	−	0.820713i	$-0.693577\pi$
$271$	−18.8109	−1.14268	−0.571341	+	0.820713i	$0.693577\pi$
$272$	0	0
$273$	−12.8453	−0.777431
$274$	0	0
$275$	0	0
$276$	0	0
$277$	15.3698	0.923480	0.461740	−	0.887015i	$-0.347225\pi$
$277$	15.3698	0.923480	0.461740	+	0.887015i	$0.347225\pi$
$278$	0	0
$279$	−72.9257	−4.36595
$280$	0	0
$281$	8.21188	0.489880	0.244940	−	0.969538i	$-0.421232\pi$
$281$	8.21188	0.489880	0.244940	+	0.969538i	$0.421232\pi$
$282$	0	0
$283$	−11.4336	−0.679659	−0.339829	−	0.940487i	$-0.610369\pi$
$283$	−11.4336	−0.679659	−0.339829	+	0.940487i	$0.610369\pi$
$284$	0	0
$285$	46.3330	2.74453
$286$	0	0
$287$	5.76810	0.340480
$288$	0	0
$289$	−16.9706	−0.998268
$290$	0	0
$291$	14.1803	0.831266
$292$	0	0
$293$	−14.7057	−0.859118	−0.429559	−	0.903039i	$-0.641331\pi$
$293$	−14.7057	−0.859118	−0.429559	+	0.903039i	$0.641331\pi$
$294$	0	0
$295$	17.2943	1.00691
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.81498	0.336289
$300$	0	0
$301$	5.29822	0.305384
$302$	0	0
$303$	−54.3957	−3.12495
$304$	0	0
$305$	54.9065	3.14393
$306$	0	0
$307$	−9.65550	−0.551068	−0.275534	−	0.961291i	$-0.588855\pi$
$307$	−9.65550	−0.551068	−0.275534	+	0.961291i	$0.588855\pi$
$308$	0	0
$309$	−17.1238	−0.974140
$310$	0	0
$311$	28.6430	1.62420	0.812098	−	0.583521i	$-0.198325\pi$
$311$	28.6430	1.62420	0.812098	+	0.583521i	$0.198325\pi$
$312$	0	0
$313$	−22.2278	−1.25639	−0.628196	−	0.778055i	$-0.716206\pi$
$313$	−22.2278	−1.25639	−0.628196	+	0.778055i	$0.716206\pi$
$314$	0	0
$315$	33.8395	1.90664
$316$	0	0
$317$	−1.01347	−0.0569222	−0.0284611	−	0.999595i	$-0.509061\pi$
$317$	−1.01347	−0.0569222	−0.0284611	+	0.999595i	$0.509061\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	6.12069	0.341623
$322$	0	0
$323$	−0.572452	−0.0318521
$324$	0	0
$325$	−47.4593	−2.63257
$326$	0	0
$327$	0.825524	0.0456516
$328$	0	0
$329$	−0.0721270	−0.00397649
$330$	0	0
$331$	−32.3266	−1.77683	−0.888416	−	0.459039i	$-0.848194\pi$
$331$	−32.3266	−1.77683	−0.888416	+	0.459039i	$0.848194\pi$
$332$	0	0
$333$	−40.8031	−2.23600
$334$	0	0
$335$	−50.1470	−2.73982
$336$	0	0
$337$	−13.6733	−0.744831	−0.372416	−	0.928066i	$-0.621470\pi$
$337$	−13.6733	−0.744831	−0.372416	+	0.928066i	$0.621470\pi$
$338$	0	0
$339$	−49.7418	−2.70160
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−20.9719	−1.12909
$346$	0	0
$347$	−0.667995	−0.0358598	−0.0179299	−	0.999839i	$-0.505708\pi$
$347$	−0.667995	−0.0358598	−0.0179299	+	0.999839i	$0.505708\pi$
$348$	0	0
$349$	9.47655	0.507268	0.253634	−	0.967300i	$-0.418374\pi$
$349$	9.47655	0.507268	0.253634	+	0.967300i	$0.418374\pi$
$350$	0	0
$351$	−65.8926	−3.51708
$352$	0	0
$353$	2.60510	0.138655	0.0693277	−	0.997594i	$-0.477915\pi$
$353$	2.60510	0.138655	0.0693277	+	0.997594i	$0.477915\pi$
$354$	0	0
$355$	−16.4597	−0.873590
$356$	0	0
$357$	−0.572376	−0.0302934
$358$	0	0
$359$	−32.5704	−1.71900	−0.859501	−	0.511134i	$-0.829226\pi$
$359$	−32.5704	−1.71900	−0.859501	+	0.511134i	$0.829226\pi$
$360$	0	0
$361$	−7.86733	−0.414070
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−8.56831	−0.448486
$366$	0	0
$367$	29.3387	1.53147	0.765735	−	0.643156i	$-0.222375\pi$
$367$	29.3387	1.53147	0.765735	+	0.643156i	$0.222375\pi$
$368$	0	0
$369$	46.8930	2.44115
$370$	0	0
$371$	1.25593	0.0652047
$372$	0	0
$373$	10.8933	0.564033	0.282016	−	0.959410i	$-0.408997\pi$
$373$	10.8933	0.564033	0.282016	+	0.959410i	$0.408997\pi$
$374$	0	0
$375$	101.731	5.25338
$376$	0	0
$377$	36.5677	1.88333
$378$	0	0
$379$	7.03536	0.361382	0.180691	−	0.983540i	$-0.442167\pi$
$379$	7.03536	0.361382	0.180691	+	0.983540i	$0.442167\pi$
$380$	0	0
$381$	13.5785	0.695645
$382$	0	0
$383$	9.75280	0.498345	0.249172	−	0.968459i	$-0.419841\pi$
$383$	9.75280	0.498345	0.249172	+	0.968459i	$0.419841\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	43.0730	2.18953
$388$	0	0
$389$	10.4742	0.531064	0.265532	−	0.964102i	$-0.414452\pi$
$389$	10.4742	0.531064	0.265532	+	0.964102i	$0.414452\pi$
$390$	0	0
$391$	0.259111	0.0131038
$392$	0	0
$393$	−37.6430	−1.89884
$394$	0	0
$395$	4.68358	0.235656
$396$	0	0
$397$	14.7138	0.738466	0.369233	−	0.929337i	$-0.379621\pi$
$397$	14.7138	0.738466	0.369233	+	0.929337i	$0.379621\pi$
$398$	0	0
$399$	11.1312	0.557257
$400$	0	0
$401$	12.4244	0.620444	0.310222	−	0.950664i	$-0.399597\pi$
$401$	12.4244	0.620444	0.310222	+	0.950664i	$0.399597\pi$
$402$	0	0
$403$	34.5387	1.72050
$404$	0	0
$405$	136.125	6.76410
$406$	0	0
$407$	0	0
$408$	0	0
$409$	25.3431	1.25313	0.626567	−	0.779367i	$-0.284459\pi$
$409$	25.3431	1.25313	0.626567	+	0.779367i	$0.284459\pi$
$410$	0	0
$411$	17.8137	0.878687
$412$	0	0
$413$	4.15485	0.204447
$414$	0	0
$415$	50.7023	2.48888
$416$	0	0
$417$	8.33003	0.407923
$418$	0	0
$419$	−19.9926	−0.976704	−0.488352	−	0.872647i	$-0.662402\pi$
$419$	−19.9926	−0.976704	−0.488352	+	0.872647i	$0.662402\pi$
$420$	0	0
$421$	−21.6898	−1.05709	−0.528547	−	0.848904i	$-0.677263\pi$
$421$	−21.6898	−1.05709	−0.528547	+	0.848904i	$0.677263\pi$
$422$	0	0
$423$	−0.586372	−0.0285104
$424$	0	0
$425$	−2.11475	−0.102581
$426$	0	0
$427$	13.1909	0.638353
$428$	0	0
$429$	0	0
$430$	0	0
$431$	16.8780	0.812982	0.406491	−	0.913655i	$-0.366752\pi$
$431$	16.8780	0.812982	0.406491	+	0.913655i	$0.366752\pi$
$432$	0	0
$433$	−30.2556	−1.45399	−0.726996	−	0.686642i	$-0.759084\pi$
$433$	−30.2556	−1.45399	−0.726996	+	0.686642i	$0.759084\pi$
$434$	0	0
$435$	−131.882	−6.32327
$436$	0	0
$437$	−5.03903	−0.241049
$438$	0	0
$439$	−0.329665	−0.0157341	−0.00786703	−	0.999969i	$-0.502504\pi$
$439$	−0.329665	−0.0157341	−0.00786703	+	0.999969i	$0.502504\pi$
$440$	0	0
$441$	8.12971	0.387129
$442$	0	0
$443$	−12.0446	−0.572258	−0.286129	−	0.958191i	$-0.592369\pi$
$443$	−12.0446	−0.572258	−0.286129	+	0.958191i	$0.592369\pi$
$444$	0	0
$445$	−27.2055	−1.28966
$446$	0	0
$447$	26.9612	1.27522
$448$	0	0
$449$	−11.6408	−0.549364	−0.274682	−	0.961535i	$-0.588573\pi$
$449$	−11.6408	−0.549364	−0.274682	+	0.961535i	$0.588573\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	18.7895	0.882809
$454$	0	0
$455$	−16.0269	−0.751353
$456$	0	0
$457$	−23.7011	−1.10869	−0.554345	−	0.832287i	$-0.687031\pi$
$457$	−23.7011	−1.10869	−0.554345	+	0.832287i	$0.687031\pi$
$458$	0	0
$459$	−2.93613	−0.137047
$460$	0	0
$461$	6.66466	0.310404	0.155202	−	0.987883i	$-0.450397\pi$
$461$	6.66466	0.310404	0.155202	+	0.987883i	$0.450397\pi$
$462$	0	0
$463$	33.3591	1.55033	0.775165	−	0.631759i	$-0.217667\pi$
$463$	33.3591	1.55033	0.775165	+	0.631759i	$0.217667\pi$
$464$	0	0
$465$	−124.565	−5.77656
$466$	0	0
$467$	31.9429	1.47814	0.739072	−	0.673627i	$-0.235264\pi$
$467$	31.9429	1.47814	0.739072	+	0.673627i	$0.235264\pi$
$468$	0	0
$469$	−12.0475	−0.556301
$470$	0	0
$471$	8.77945	0.404536
$472$	0	0
$473$	0	0
$474$	0	0
$475$	41.1263	1.88701
$476$	0	0
$477$	10.2104	0.467501
$478$	0	0
$479$	16.9719	0.775467	0.387734	−	0.921772i	$-0.373258\pi$
$479$	16.9719	0.775467	0.387734	+	0.921772i	$0.373258\pi$
$480$	0	0
$481$	19.3250	0.881143
$482$	0	0
$483$	−5.03836	−0.229253
$484$	0	0
$485$	17.6927	0.803382
$486$	0	0
$487$	−30.6364	−1.38827	−0.694133	−	0.719847i	$-0.744212\pi$
$487$	−30.6364	−1.38827	−0.694133	+	0.719847i	$0.744212\pi$
$488$	0	0
$489$	26.3526	1.19170
$490$	0	0
$491$	−18.3039	−0.826044	−0.413022	−	0.910721i	$-0.635527\pi$
$491$	−18.3039	−0.826044	−0.413022	+	0.910721i	$0.635527\pi$
$492$	0	0
$493$	1.62943	0.0733858
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.95433	−0.177376
$498$	0	0
$499$	−8.07843	−0.361640	−0.180820	−	0.983516i	$-0.557875\pi$
$499$	−8.07843	−0.361640	−0.180820	+	0.983516i	$0.557875\pi$
$500$	0	0
$501$	75.1416	3.35708
$502$	0	0
$503$	5.20936	0.232274	0.116137	−	0.993233i	$-0.462949\pi$
$503$	5.20936	0.232274	0.116137	+	0.993233i	$0.462949\pi$
$504$	0	0
$505$	−67.8690	−3.02013
$506$	0	0
$507$	6.08929	0.270435
$508$	0	0
$509$	−40.0781	−1.77643	−0.888214	−	0.459430i	$-0.848054\pi$
$509$	−40.0781	−1.77643	−0.888214	+	0.459430i	$0.848054\pi$
$510$	0	0
$511$	−2.05848	−0.0910618
$512$	0	0
$513$	57.0998	2.52102
$514$	0	0
$515$	−21.3652	−0.941463
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−57.7153	−2.53342
$520$	0	0
$521$	1.66669	0.0730191	0.0365095	−	0.999333i	$-0.488376\pi$
$521$	1.66669	0.0730191	0.0365095	+	0.999333i	$0.488376\pi$
$522$	0	0
$523$	1.83581	0.0802745	0.0401373	−	0.999194i	$-0.487220\pi$
$523$	1.83581	0.0802745	0.0401373	+	0.999194i	$0.487220\pi$
$524$	0	0
$525$	41.1209	1.79466
$526$	0	0
$527$	1.53902	0.0670408
$528$	0	0
$529$	−20.7192	−0.900833
$530$	0	0
$531$	33.7777	1.46583
$532$	0	0
$533$	−22.2093	−0.961990
$534$	0	0
$535$	7.63672	0.330164
$536$	0	0
$537$	15.2224	0.656894
$538$	0	0
$539$	0	0
$540$	0	0
$541$	20.4752	0.880297	0.440149	−	0.897925i	$-0.354926\pi$
$541$	20.4752	0.880297	0.440149	+	0.897925i	$0.354926\pi$
$542$	0	0
$543$	−60.6593	−2.60314
$544$	0	0
$545$	1.03000	0.0441202
$546$	0	0
$547$	13.8616	0.592678	0.296339	−	0.955083i	$-0.404234\pi$
$547$	13.8616	0.592678	0.296339	+	0.955083i	$0.404234\pi$
$548$	0	0
$549$	107.238	4.57682
$550$	0	0
$551$	−31.6881	−1.34996
$552$	0	0
$553$	1.12520	0.0478483
$554$	0	0
$555$	−69.6961	−2.95843
$556$	0	0
$557$	−17.5768	−0.744753	−0.372377	−	0.928082i	$-0.621457\pi$
$557$	−17.5768	−0.744753	−0.372377	+	0.928082i	$0.621457\pi$
$558$	0	0
$559$	−20.4001	−0.862830
$560$	0	0
$561$	0	0
$562$	0	0
$563$	13.6619	0.575781	0.287891	−	0.957663i	$-0.407046\pi$
$563$	13.6619	0.575781	0.287891	+	0.957663i	$0.407046\pi$
$564$	0	0
$565$	−62.0623	−2.61098
$566$	0	0
$567$	32.7031	1.37340
$568$	0	0
$569$	−32.4625	−1.36090	−0.680449	−	0.732795i	$-0.738215\pi$
$569$	−32.4625	−1.36090	−0.680449	+	0.732795i	$0.738215\pi$
$570$	0	0
$571$	24.9992	1.04618	0.523091	−	0.852277i	$-0.324779\pi$
$571$	24.9992	1.04618	0.523091	+	0.852277i	$0.324779\pi$
$572$	0	0
$573$	14.9197	0.623278
$574$	0	0
$575$	−18.6152	−0.776307
$576$	0	0
$577$	−25.3793	−1.05655	−0.528277	−	0.849072i	$-0.677162\pi$
$577$	−25.3793	−1.05655	−0.528277	+	0.849072i	$0.677162\pi$
$578$	0	0
$579$	49.1826	2.04396
$580$	0	0
$581$	12.1809	0.505348
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−130.294	−5.38700
$586$	0	0
$587$	−6.40928	−0.264539	−0.132270	−	0.991214i	$-0.542226\pi$
$587$	−6.40928	−0.264539	−0.132270	+	0.991214i	$0.542226\pi$
$588$	0	0
$589$	−29.9299	−1.23324
$590$	0	0
$591$	−78.6712	−3.23610
$592$	0	0
$593$	15.5921	0.640290	0.320145	−	0.947369i	$-0.396268\pi$
$593$	15.5921	0.640290	0.320145	+	0.947369i	$0.396268\pi$
$594$	0	0
$595$	−0.714147	−0.0292772
$596$	0	0
$597$	68.7043	2.81188
$598$	0	0
$599$	28.3587	1.15870	0.579352	−	0.815077i	$-0.303306\pi$
$599$	28.3587	1.15870	0.579352	+	0.815077i	$0.303306\pi$
$600$	0	0
$601$	26.4199	1.07769	0.538845	−	0.842405i	$-0.318861\pi$
$601$	26.4199	1.07769	0.538845	+	0.842405i	$0.318861\pi$
$602$	0	0
$603$	−97.9426	−3.98853
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−36.5781	−1.48466	−0.742330	−	0.670034i	$-0.766280\pi$
$607$	−36.5781	−1.48466	−0.742330	+	0.670034i	$0.766280\pi$
$608$	0	0
$609$	−31.6839	−1.28389
$610$	0	0
$611$	0.277715	0.0112351
$612$	0	0
$613$	−7.45409	−0.301068	−0.150534	−	0.988605i	$-0.548099\pi$
$613$	−7.45409	−0.301068	−0.150534	+	0.988605i	$0.548099\pi$
$614$	0	0
$615$	80.0984	3.22988
$616$	0	0
$617$	22.4820	0.905093	0.452546	−	0.891741i	$-0.350516\pi$
$617$	22.4820	0.905093	0.452546	+	0.891741i	$0.350516\pi$
$618$	0	0
$619$	−16.1961	−0.650977	−0.325488	−	0.945546i	$-0.605529\pi$
$619$	−16.1961	−0.650977	−0.325488	+	0.945546i	$0.605529\pi$
$620$	0	0
$621$	−25.8453	−1.03714
$622$	0	0
$623$	−6.53593	−0.261857
$624$	0	0
$625$	65.2992	2.61197
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.861107	0.0343346
$630$	0	0
$631$	19.9873	0.795682	0.397841	−	0.917454i	$-0.369760\pi$
$631$	19.9873	0.795682	0.397841	+	0.917454i	$0.369760\pi$
$632$	0	0
$633$	−75.2312	−2.99017
$634$	0	0
$635$	16.9417	0.672310
$636$	0	0
$637$	−3.85036	−0.152557
$638$	0	0
$639$	−32.1476	−1.27174
$640$	0	0
$641$	45.7414	1.80667	0.903337	−	0.428931i	$-0.141110\pi$
$641$	45.7414	1.80667	0.903337	+	0.428931i	$0.141110\pi$
$642$	0	0
$643$	−20.4660	−0.807100	−0.403550	−	0.914958i	$-0.632224\pi$
$643$	−20.4660	−0.807100	−0.403550	+	0.914958i	$0.632224\pi$
$644$	0	0
$645$	73.5734	2.89695
$646$	0	0
$647$	27.4482	1.07910	0.539549	−	0.841954i	$-0.318595\pi$
$647$	27.4482	1.07910	0.539549	+	0.841954i	$0.318595\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−29.9259	−1.17289
$652$	0	0
$653$	6.74625	0.264001	0.132001	−	0.991250i	$-0.457860\pi$
$653$	6.74625	0.264001	0.132001	+	0.991250i	$0.457860\pi$
$654$	0	0
$655$	−46.9668	−1.83515
$656$	0	0
$657$	−16.7349	−0.652889
$658$	0	0
$659$	15.4127	0.600392	0.300196	−	0.953878i	$-0.402948\pi$
$659$	15.4127	0.600392	0.300196	+	0.953878i	$0.402948\pi$
$660$	0	0
$661$	−30.1422	−1.17239	−0.586197	−	0.810168i	$-0.699376\pi$
$661$	−30.1422	−1.17239	−0.586197	+	0.810168i	$0.699376\pi$
$662$	0	0
$663$	2.20385	0.0855906
$664$	0	0
$665$	13.8883	0.538564
$666$	0	0
$667$	14.3431	0.555367
$668$	0	0
$669$	−29.5041	−1.14069
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−49.6267	−1.91297	−0.956484	−	0.291784i	$-0.905751\pi$
$673$	−49.6267	−1.91297	−0.956484	+	0.291784i	$0.905751\pi$
$674$	0	0
$675$	210.938	8.11902
$676$	0	0
$677$	25.1846	0.967924	0.483962	−	0.875089i	$-0.339197\pi$
$677$	25.1846	0.967924	0.483962	+	0.875089i	$0.339197\pi$
$678$	0	0
$679$	4.25054	0.163121
$680$	0	0
$681$	−12.5823	−0.482154
$682$	0	0
$683$	23.1143	0.884445	0.442222	−	0.896905i	$-0.354190\pi$
$683$	23.1143	0.884445	0.442222	+	0.896905i	$0.354190\pi$
$684$	0	0
$685$	22.2260	0.849212
$686$	0	0
$687$	0.541336	0.0206533
$688$	0	0
$689$	−4.83579	−0.184229
$690$	0	0
$691$	28.0820	1.06829	0.534145	−	0.845393i	$-0.320634\pi$
$691$	28.0820	1.06829	0.534145	+	0.845393i	$0.320634\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	10.3933	0.394240
$696$	0	0
$697$	−0.989629	−0.0374849
$698$	0	0
$699$	37.3205	1.41159
$700$	0	0
$701$	36.5936	1.38212	0.691060	−	0.722797i	$-0.257144\pi$
$701$	36.5936	1.38212	0.691060	+	0.722797i	$0.257144\pi$
$702$	0	0
$703$	−16.7462	−0.631597
$704$	0	0
$705$	−1.00159	−0.0377219
$706$	0	0
$707$	−16.3051	−0.613215
$708$	0	0
$709$	37.5296	1.40945	0.704726	−	0.709480i	$-0.251070\pi$
$709$	37.5296	1.40945	0.704726	+	0.709480i	$0.251070\pi$
$710$	0	0
$711$	9.14755	0.343060
$712$	0	0
$713$	13.5473	0.507350
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−80.2104	−2.99551
$718$	0	0
$719$	33.3325	1.24309	0.621547	−	0.783377i	$-0.286505\pi$
$719$	33.3325	1.24309	0.621547	+	0.783377i	$0.286505\pi$
$720$	0	0
$721$	−5.13285	−0.191157
$722$	0	0
$723$	−60.1628	−2.23748
$724$	0	0
$725$	−117.062	−4.34758
$726$	0	0
$727$	16.4013	0.608291	0.304146	−	0.952626i	$-0.401629\pi$
$727$	16.4013	0.608291	0.304146	+	0.952626i	$0.401629\pi$
$728$	0	0
$729$	94.5902	3.50334
$730$	0	0
$731$	−0.909012	−0.0336210
$732$	0	0
$733$	27.2901	1.00798	0.503992	−	0.863708i	$-0.331864\pi$
$733$	27.2901	1.00798	0.503992	+	0.863708i	$0.331864\pi$
$734$	0	0
$735$	13.8864	0.512209
$736$	0	0
$737$	0	0
$738$	0	0
$739$	19.8366	0.729701	0.364851	−	0.931066i	$-0.381120\pi$
$739$	19.8366	0.729701	0.364851	+	0.931066i	$0.381120\pi$
$740$	0	0
$741$	−42.8591	−1.57447
$742$	0	0
$743$	4.89503	0.179581	0.0897906	−	0.995961i	$-0.471380\pi$
$743$	4.89503	0.179581	0.0897906	+	0.995961i	$0.471380\pi$
$744$	0	0
$745$	33.6392	1.23245
$746$	0	0
$747$	99.0271	3.62321
$748$	0	0
$749$	1.83467	0.0670374
$750$	0	0
$751$	−25.9301	−0.946204	−0.473102	−	0.881008i	$-0.656866\pi$
$751$	−25.9301	−0.946204	−0.473102	+	0.881008i	$0.656866\pi$
$752$	0	0
$753$	41.5906	1.51565
$754$	0	0
$755$	23.4435	0.853196
$756$	0	0
$757$	5.56113	0.202123	0.101061	−	0.994880i	$-0.467776\pi$
$757$	5.56113	0.202123	0.101061	+	0.994880i	$0.467776\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	17.6867	0.641143	0.320571	−	0.947224i	$-0.396125\pi$
$761$	17.6867	0.641143	0.320571	+	0.947224i	$0.396125\pi$
$762$	0	0
$763$	0.247450	0.00895829
$764$	0	0
$765$	−5.80581	−0.209910
$766$	0	0
$767$	−15.9976	−0.577641
$768$	0	0
$769$	−29.9260	−1.07916	−0.539580	−	0.841934i	$-0.681417\pi$
$769$	−29.9260	−1.07916	−0.539580	+	0.841934i	$0.681417\pi$
$770$	0	0
$771$	−8.10987	−0.292070
$772$	0	0
$773$	−4.78449	−0.172086	−0.0860431	−	0.996291i	$-0.527422\pi$
$773$	−4.78449	−0.172086	−0.0860431	+	0.996291i	$0.527422\pi$
$774$	0	0
$775$	−110.567	−3.97168
$776$	0	0
$777$	−16.7440	−0.600688
$778$	0	0
$779$	19.2457	0.689547
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−162.529	−5.80832
$784$	0	0
$785$	10.9540	0.390966
$786$	0	0
$787$	31.2008	1.11219	0.556095	−	0.831119i	$-0.312299\pi$
$787$	31.2008	1.11219	0.556095	+	0.831119i	$0.312299\pi$
$788$	0	0
$789$	−23.4522	−0.834920
$790$	0	0
$791$	−14.9101	−0.530141
$792$	0	0
$793$	−50.7898	−1.80360
$794$	0	0
$795$	17.4404	0.618547
$796$	0	0
$797$	−2.02638	−0.0717782	−0.0358891	−	0.999356i	$-0.511426\pi$
$797$	−2.02638	−0.0717782	−0.0358891	+	0.999356i	$0.511426\pi$
$798$	0	0
$799$	0.0123748	0.000437788 0
$800$	0	0
$801$	−53.1353	−1.87744
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−6.28631	−0.221563
$806$	0	0
$807$	10.4313	0.367200
$808$	0	0
$809$	−44.7925	−1.57482	−0.787410	−	0.616430i	$-0.788578\pi$
$809$	−44.7925	−1.57482	−0.787410	+	0.616430i	$0.788578\pi$
$810$	0	0
$811$	51.8941	1.82225	0.911123	−	0.412134i	$-0.135216\pi$
$811$	51.8941	1.82225	0.911123	+	0.412134i	$0.135216\pi$
$812$	0	0
$813$	−62.7555	−2.20093
$814$	0	0
$815$	32.8798	1.15173
$816$	0	0
$817$	17.6779	0.618470
$818$	0	0
$819$	−31.3023	−1.09379
$820$	0	0
$821$	−44.3257	−1.54698	−0.773488	−	0.633810i	$-0.781490\pi$
$821$	−44.3257	−1.54698	−0.773488	+	0.633810i	$0.781490\pi$
$822$	0	0
$823$	28.3734	0.989034	0.494517	−	0.869168i	$-0.335345\pi$
$823$	28.3734	0.989034	0.494517	+	0.869168i	$0.335345\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	54.4341	1.89286	0.946429	−	0.322912i	$-0.104662\pi$
$827$	54.4341	1.89286	0.946429	+	0.322912i	$0.104662\pi$
$828$	0	0
$829$	48.9072	1.69862	0.849308	−	0.527898i	$-0.177020\pi$
$829$	48.9072	1.69862	0.849308	+	0.527898i	$0.177020\pi$
$830$	0	0
$831$	51.2754	1.77872
$832$	0	0
$833$	−0.171569	−0.00594452
$834$	0	0
$835$	93.7534	3.24447
$836$	0	0
$837$	−153.511	−5.30613
$838$	0	0
$839$	47.3689	1.63535	0.817677	−	0.575677i	$-0.195261\pi$
$839$	47.3689	1.63535	0.817677	+	0.575677i	$0.195261\pi$
$840$	0	0
$841$	61.1970	2.11024
$842$	0	0
$843$	27.3959	0.943563
$844$	0	0
$845$	7.59754	0.261363
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−38.1440	−1.30910
$850$	0	0
$851$	7.57992	0.259836
$852$	0	0
$853$	−45.1972	−1.54752	−0.773761	−	0.633478i	$-0.781627\pi$
$853$	−45.1972	−1.54752	−0.773761	+	0.633478i	$0.781627\pi$
$854$	0	0
$855$	112.908	3.86136
$856$	0	0
$857$	−22.1966	−0.758222	−0.379111	−	0.925351i	$-0.623770\pi$
$857$	−22.1966	−0.758222	−0.379111	+	0.925351i	$0.623770\pi$
$858$	0	0
$859$	43.6079	1.48788	0.743941	−	0.668245i	$-0.232954\pi$
$859$	43.6079	1.48788	0.743941	+	0.668245i	$0.232954\pi$
$860$	0	0
$861$	19.2431	0.655803
$862$	0	0
$863$	17.0049	0.578853	0.289426	−	0.957200i	$-0.406535\pi$
$863$	17.0049	0.578853	0.289426	+	0.957200i	$0.406535\pi$
$864$	0	0
$865$	−72.0108	−2.44844
$866$	0	0
$867$	−56.6159	−1.92278
$868$	0	0
$869$	0	0
$870$	0	0
$871$	46.3871	1.57177
$872$	0	0
$873$	34.5557	1.16953
$874$	0	0
$875$	30.4938	1.03088
$876$	0	0
$877$	−42.8265	−1.44615	−0.723075	−	0.690770i	$-0.757272\pi$
$877$	−42.8265	−1.44615	−0.723075	+	0.690770i	$0.757272\pi$
$878$	0	0
$879$	−49.0601	−1.65476
$880$	0	0
$881$	50.8518	1.71324	0.856620	−	0.515948i	$-0.172560\pi$
$881$	50.8518	1.71324	0.856620	+	0.515948i	$0.172560\pi$
$882$	0	0
$883$	6.53906	0.220057	0.110028	−	0.993928i	$-0.464906\pi$
$883$	6.53906	0.220057	0.110028	+	0.993928i	$0.464906\pi$
$884$	0	0
$885$	57.6960	1.93943
$886$	0	0
$887$	−20.8366	−0.699624	−0.349812	−	0.936820i	$-0.613755\pi$
$887$	−20.8366	−0.699624	−0.349812	+	0.936820i	$0.613755\pi$
$888$	0	0
$889$	4.07013	0.136508
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−0.240656	−0.00805326
$894$	0	0
$895$	18.9928	0.634859
$896$	0	0
$897$	19.3995	0.647730
$898$	0	0
$899$	85.1925	2.84133
$900$	0	0
$901$	−0.215479	−0.00717866
$902$	0	0
$903$	17.6755	0.588204
$904$	0	0
$905$	−75.6840	−2.51582
$906$	0	0
$907$	−5.13048	−0.170355	−0.0851773	−	0.996366i	$-0.527146\pi$
$907$	−5.13048	−0.170355	−0.0851773	+	0.996366i	$0.527146\pi$
$908$	0	0
$909$	−132.556	−4.39659
$910$	0	0
$911$	−47.3024	−1.56720	−0.783600	−	0.621266i	$-0.786619\pi$
$911$	−47.3024	−1.56720	−0.783600	+	0.621266i	$0.786619\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	183.175	6.05557
$916$	0	0
$917$	−11.2835	−0.372613
$918$	0	0
$919$	14.2389	0.469699	0.234850	−	0.972032i	$-0.424540\pi$
$919$	14.2389	0.469699	0.234850	+	0.972032i	$0.424540\pi$
$920$	0	0
$921$	−32.2119	−1.06142
$922$	0	0
$923$	15.2256	0.501157
$924$	0	0
$925$	−61.8640	−2.03408
$926$	0	0
$927$	−41.7286	−1.37055
$928$	0	0
$929$	−41.4007	−1.35831	−0.679156	−	0.733994i	$-0.737654\pi$
$929$	−41.4007	−1.35831	−0.679156	+	0.733994i	$0.737654\pi$
$930$	0	0
$931$	3.33657	0.109352
$932$	0	0
$933$	95.5566	3.12838
$934$	0	0
$935$	0	0
$936$	0	0
$937$	46.2961	1.51243	0.756214	−	0.654324i	$-0.227047\pi$
$937$	46.2961	1.51243	0.756214	+	0.654324i	$0.227047\pi$
$938$	0	0
$939$	−74.1548	−2.41995
$940$	0	0
$941$	18.9579	0.618010	0.309005	−	0.951060i	$-0.400004\pi$
$941$	18.9579	0.618010	0.309005	+	0.951060i	$0.400004\pi$
$942$	0	0
$943$	−8.71125	−0.283677
$944$	0	0
$945$	71.2334	2.31722
$946$	0	0
$947$	−39.6431	−1.28823	−0.644114	−	0.764930i	$-0.722774\pi$
$947$	−39.6431	−1.28823	−0.644114	+	0.764930i	$0.722774\pi$
$948$	0	0
$949$	7.92589	0.257285
$950$	0	0
$951$	−3.38106	−0.109639
$952$	0	0
$953$	15.1661	0.491279	0.245640	−	0.969361i	$-0.421002\pi$
$953$	15.1661	0.491279	0.245640	+	0.969361i	$0.421002\pi$
$954$	0	0
$955$	18.6151	0.602371
$956$	0	0
$957$	0	0
$958$	0	0
$959$	5.33965	0.172426
$960$	0	0
$961$	49.4657	1.59567
$962$	0	0
$963$	14.9154	0.480641
$964$	0	0
$965$	61.3646	1.97540
$966$	0	0
$967$	−13.8192	−0.444396	−0.222198	−	0.975002i	$-0.571323\pi$
$967$	−13.8192	−0.444396	−0.222198	+	0.975002i	$0.571323\pi$
$968$	0	0
$969$	−1.90977	−0.0613507
$970$	0	0
$971$	41.1757	1.32139	0.660696	−	0.750654i	$-0.270261\pi$
$971$	41.1757	1.32139	0.660696	+	0.750654i	$0.270261\pi$
$972$	0	0
$973$	2.49692	0.0800475
$974$	0	0
$975$	−158.330	−5.07062
$976$	0	0
$977$	−45.5257	−1.45650	−0.728248	−	0.685314i	$-0.759665\pi$
$977$	−45.5257	−1.45650	−0.728248	+	0.685314i	$0.759665\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	2.01170	0.0642286
$982$	0	0
$983$	−24.0323	−0.766510	−0.383255	−	0.923643i	$-0.625197\pi$
$983$	−24.0323	−0.766510	−0.383255	+	0.923643i	$0.625197\pi$
$984$	0	0
$985$	−98.1572	−3.12755
$986$	0	0
$987$	−0.240624	−0.00765916
$988$	0	0
$989$	−8.00161	−0.254436
$990$	0	0
$991$	26.6312	0.845968	0.422984	−	0.906137i	$-0.360983\pi$
$991$	26.6312	0.845968	0.422984	+	0.906137i	$0.360983\pi$
$992$	0	0
$993$	−107.846	−3.42238
$994$	0	0
$995$	85.7217	2.71756
$996$	0	0
$997$	−2.88291	−0.0913028	−0.0456514	−	0.998957i	$-0.514536\pi$
$997$	−2.88291	−0.0913028	−0.0456514	+	0.998957i	$0.514536\pi$
$998$	0	0
$999$	−85.8920	−2.71750

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6776.2.a.bp.1.10	yes	10
11.10	odd	2		6776.2.a.bo.1.10	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6776.2.a.bo.1.10	✓	10	11.10	odd	2
6776.2.a.bp.1.10	yes	10	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6776 = 2^{3} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6776.a (trivial)

\(f(q)\)	\(=\)	\(q+3.33612 q^{3} +4.16244 q^{5} +1.00000 q^{7} +8.12971 q^{9} +O(q^{10})\)	\(q+3.33612 q^{3} +4.16244 q^{5} +1.00000 q^{7} +8.12971 q^{9} -3.85036 q^{13} +13.8864 q^{15} -0.171569 q^{17} +3.33657 q^{19} +3.33612 q^{21} -1.51024 q^{23} +12.3259 q^{25} +17.1134 q^{27} -9.49721 q^{29} -8.97027 q^{31} +4.16244 q^{35} -5.01901 q^{37} -12.8453 q^{39} +5.76810 q^{41} +5.29822 q^{43} +33.8395 q^{45} -0.0721270 q^{47} +1.00000 q^{49} -0.572376 q^{51} +1.25593 q^{53} +11.1312 q^{57} +4.15485 q^{59} +13.1909 q^{61} +8.12971 q^{63} -16.0269 q^{65} -12.0475 q^{67} -5.03836 q^{69} -3.95433 q^{71} -2.05848 q^{73} +41.1209 q^{75} +1.12520 q^{79} +32.7031 q^{81} +12.1809 q^{83} -0.714147 q^{85} -31.6839 q^{87} -6.53593 q^{89} -3.85036 q^{91} -29.9259 q^{93} +13.8883 q^{95} +4.25054 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{3} + 4 q^{5} + 10 q^{7} + 20 q^{9}+O(q^{10}) \) 10 * q + 2 * q^3 + 4 * q^5 + 10 * q^7 + 20 * q^9	\( 10 q + 2 q^{3} + 4 q^{5} + 10 q^{7} + 20 q^{9} - 12 q^{13} + 10 q^{15} + 2 q^{17} + 2 q^{19} + 2 q^{21} + 10 q^{23} + 8 q^{25} + 14 q^{27} - 16 q^{29} + 14 q^{31} + 4 q^{35} + 2 q^{37} - 4 q^{39} + 18 q^{41} + 16 q^{43} + 52 q^{45} - 4 q^{47} + 10 q^{49} + 16 q^{51} + 32 q^{53} - 24 q^{57} + 4 q^{59} + 16 q^{61} + 20 q^{63} - 4 q^{65} + 10 q^{67} - 14 q^{69} - 10 q^{71} - 22 q^{73} + 48 q^{75} + 36 q^{79} + 50 q^{81} + 2 q^{83} - 4 q^{85} - 2 q^{87} + 36 q^{89} - 12 q^{91} - 2 q^{93} + 52 q^{95} + 44 q^{97}+O(q^{100}) \) 10 * q + 2 * q^3 + 4 * q^5 + 10 * q^7 + 20 * q^9 - 12 * q^13 + 10 * q^15 + 2 * q^17 + 2 * q^19 + 2 * q^21 + 10 * q^23 + 8 * q^25 + 14 * q^27 - 16 * q^29 + 14 * q^31 + 4 * q^35 + 2 * q^37 - 4 * q^39 + 18 * q^41 + 16 * q^43 + 52 * q^45 - 4 * q^47 + 10 * q^49 + 16 * q^51 + 32 * q^53 - 24 * q^57 + 4 * q^59 + 16 * q^61 + 20 * q^63 - 4 * q^65 + 10 * q^67 - 14 * q^69 - 10 * q^71 - 22 * q^73 + 48 * q^75 + 36 * q^79 + 50 * q^81 + 2 * q^83 - 4 * q^85 - 2 * q^87 + 36 * q^89 - 12 * q^91 - 2 * q^93 + 52 * q^95 + 44 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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