LMFDB - Embedded newform 8046.2.a.m.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.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:	$64.2476334663$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 21 x^{10} + 116 x^{9} + 106 x^{8} - 774 x^{7} - 63 x^{6} + 2013 x^{5} - 417 x^{4} + \cdots - 375 $ x^12 - 5x^11 - 21x^10 + 116x^9 + 106x^8 - 774x^7 - 63x^6 + 2013x^5 - 417x^4 - 2249x^3 + 761x^2 + 910*x - 375
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-2.26764$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +2.26764 q^{5} -0.397853 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +2.26764 q^{5} -0.397853 q^{7} +1.00000 q^{8} +2.26764 q^{10} -3.96969 q^{11} +0.823528 q^{13} -0.397853 q^{14} +1.00000 q^{16} +6.85300 q^{17} -8.38692 q^{19} +2.26764 q^{20} -3.96969 q^{22} -8.28167 q^{23} +0.142177 q^{25} +0.823528 q^{26} -0.397853 q^{28} -8.78572 q^{29} +4.00807 q^{31} +1.00000 q^{32} +6.85300 q^{34} -0.902187 q^{35} +0.694576 q^{37} -8.38692 q^{38} +2.26764 q^{40} -9.11675 q^{41} -8.24158 q^{43} -3.96969 q^{44} -8.28167 q^{46} -1.58117 q^{47} -6.84171 q^{49} +0.142177 q^{50} +0.823528 q^{52} +7.80173 q^{53} -9.00181 q^{55} -0.397853 q^{56} -8.78572 q^{58} +10.9286 q^{59} -8.40891 q^{61} +4.00807 q^{62} +1.00000 q^{64} +1.86746 q^{65} -11.4002 q^{67} +6.85300 q^{68} -0.902187 q^{70} -14.7645 q^{71} +6.27672 q^{73} +0.694576 q^{74} -8.38692 q^{76} +1.57935 q^{77} +9.64978 q^{79} +2.26764 q^{80} -9.11675 q^{82} -8.76097 q^{83} +15.5401 q^{85} -8.24158 q^{86} -3.96969 q^{88} +9.24953 q^{89} -0.327643 q^{91} -8.28167 q^{92} -1.58117 q^{94} -19.0185 q^{95} -8.05434 q^{97} -6.84171 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} + 12 q^{4} - 5 q^{5} - 6 q^{7} + 12 q^{8}+O(q^{10}) $ 12 * q + 12 * q^2 + 12 * q^4 - 5 * q^5 - 6 * q^7 + 12 * q^8	$ 12 q + 12 q^{2} + 12 q^{4} - 5 q^{5} - 6 q^{7} + 12 q^{8} - 5 q^{10} - 10 q^{11} - q^{13} - 6 q^{14} + 12 q^{16} - 6 q^{17} - 10 q^{19} - 5 q^{20} - 10 q^{22} - 15 q^{23} + 7 q^{25} - q^{26} - 6 q^{28} - 33 q^{29} - 6 q^{31} + 12 q^{32} - 6 q^{34} - 16 q^{35} - 13 q^{37} - 10 q^{38} - 5 q^{40} - 20 q^{41} - 11 q^{43} - 10 q^{44} - 15 q^{46} - 15 q^{47} + 2 q^{49} + 7 q^{50} - q^{52} - 4 q^{53} - 17 q^{55} - 6 q^{56} - 33 q^{58} - 10 q^{59} - 12 q^{61} - 6 q^{62} + 12 q^{64} - 40 q^{65} - 19 q^{67} - 6 q^{68} - 16 q^{70} - 47 q^{71} - 2 q^{73} - 13 q^{74} - 10 q^{76} + 6 q^{77} - 15 q^{79} - 5 q^{80} - 20 q^{82} - 18 q^{83} - 25 q^{85} - 11 q^{86} - 10 q^{88} - 24 q^{89} - 3 q^{91} - 15 q^{92} - 15 q^{94} + 3 q^{95} - 25 q^{97} + 2 q^{98}+O(q^{100}) $ 12 * q + 12 * q^2 + 12 * q^4 - 5 * q^5 - 6 * q^7 + 12 * q^8 - 5 * q^10 - 10 * q^11 - q^13 - 6 * q^14 + 12 * q^16 - 6 * q^17 - 10 * q^19 - 5 * q^20 - 10 * q^22 - 15 * q^23 + 7 * q^25 - q^26 - 6 * q^28 - 33 * q^29 - 6 * q^31 + 12 * q^32 - 6 * q^34 - 16 * q^35 - 13 * q^37 - 10 * q^38 - 5 * q^40 - 20 * q^41 - 11 * q^43 - 10 * q^44 - 15 * q^46 - 15 * q^47 + 2 * q^49 + 7 * q^50 - q^52 - 4 * q^53 - 17 * q^55 - 6 * q^56 - 33 * q^58 - 10 * q^59 - 12 * q^61 - 6 * q^62 + 12 * q^64 - 40 * q^65 - 19 * q^67 - 6 * q^68 - 16 * q^70 - 47 * q^71 - 2 * q^73 - 13 * q^74 - 10 * q^76 + 6 * q^77 - 15 * q^79 - 5 * q^80 - 20 * q^82 - 18 * q^83 - 25 * q^85 - 11 * q^86 - 10 * q^88 - 24 * q^89 - 3 * q^91 - 15 * q^92 - 15 * q^94 + 3 * q^95 - 25 * q^97 + 2 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	2.26764	1.01412	0.507059	−	0.861911i	$-0.330733\pi$
$5$	2.26764	1.01412	0.507059	+	0.861911i	$0.330733\pi$
$6$	0	0
$7$	−0.397853	−0.150374	−0.0751872	−	0.997169i	$-0.523955\pi$
$7$	−0.397853	−0.150374	−0.0751872	+	0.997169i	$0.523955\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	2.26764	0.717090
$11$	−3.96969	−1.19691	−0.598453	−	0.801158i	$-0.704218\pi$
$11$	−3.96969	−1.19691	−0.598453	+	0.801158i	$0.704218\pi$
$12$	0	0
$13$	0.823528	0.228406	0.114203	−	0.993457i	$-0.463569\pi$
$13$	0.823528	0.228406	0.114203	+	0.993457i	$0.463569\pi$
$14$	−0.397853	−0.106331
$15$	0	0
$16$	1.00000	0.250000
$17$	6.85300	1.66210	0.831049	−	0.556200i	$-0.187741\pi$
$17$	6.85300	1.66210	0.831049	+	0.556200i	$0.187741\pi$
$18$	0	0
$19$	−8.38692	−1.92409	−0.962045	−	0.272890i	$-0.912020\pi$
$19$	−8.38692	−1.92409	−0.962045	+	0.272890i	$0.912020\pi$
$20$	2.26764	0.507059
$21$	0	0
$22$	−3.96969	−0.846340
$23$	−8.28167	−1.72685	−0.863424	−	0.504479i	$-0.831685\pi$
$23$	−8.28167	−1.72685	−0.863424	+	0.504479i	$0.831685\pi$
$24$	0	0
$25$	0.142177	0.0284355
$26$	0.823528	0.161507
$27$	0	0
$28$	−0.397853	−0.0751872
$29$	−8.78572	−1.63147	−0.815734	−	0.578428i	$-0.803667\pi$
$29$	−8.78572	−1.63147	−0.815734	+	0.578428i	$0.803667\pi$
$30$	0	0
$31$	4.00807	0.719871	0.359936	−	0.932977i	$-0.382799\pi$
$31$	4.00807	0.719871	0.359936	+	0.932977i	$0.382799\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	6.85300	1.17528
$35$	−0.902187	−0.152497
$36$	0	0
$37$	0.694576	0.114188	0.0570938	−	0.998369i	$-0.481817\pi$
$37$	0.694576	0.114188	0.0570938	+	0.998369i	$0.481817\pi$
$38$	−8.38692	−1.36054
$39$	0	0
$40$	2.26764	0.358545
$41$	−9.11675	−1.42380	−0.711899	−	0.702282i	$-0.752164\pi$
$41$	−9.11675	−1.42380	−0.711899	+	0.702282i	$0.752164\pi$
$42$	0	0
$43$	−8.24158	−1.25683	−0.628414	−	0.777879i	$-0.716296\pi$
$43$	−8.24158	−1.25683	−0.628414	+	0.777879i	$0.716296\pi$
$44$	−3.96969	−0.598453
$45$	0	0
$46$	−8.28167	−1.22107
$47$	−1.58117	−0.230638	−0.115319	−	0.993329i	$-0.536789\pi$
$47$	−1.58117	−0.230638	−0.115319	+	0.993329i	$0.536789\pi$
$48$	0	0
$49$	−6.84171	−0.977388
$50$	0.142177	0.0201069
$51$	0	0
$52$	0.823528	0.114203
$53$	7.80173	1.07165	0.535825	−	0.844329i	$-0.320001\pi$
$53$	7.80173	1.07165	0.535825	+	0.844329i	$0.320001\pi$
$54$	0	0
$55$	−9.00181	−1.21380
$56$	−0.397853	−0.0531654
$57$	0	0
$58$	−8.78572	−1.15362
$59$	10.9286	1.42278	0.711390	−	0.702797i	$-0.248066\pi$
$59$	10.9286	1.42278	0.711390	+	0.702797i	$0.248066\pi$
$60$	0	0
$61$	−8.40891	−1.07665	−0.538325	−	0.842737i	$-0.680943\pi$
$61$	−8.40891	−1.07665	−0.538325	+	0.842737i	$0.680943\pi$
$62$	4.00807	0.509026
$63$	0	0
$64$	1.00000	0.125000
$65$	1.86746	0.231630
$66$	0	0
$67$	−11.4002	−1.39276	−0.696379	−	0.717674i	$-0.745207\pi$
$67$	−11.4002	−1.39276	−0.696379	+	0.717674i	$0.745207\pi$
$68$	6.85300	0.831049
$69$	0	0
$70$	−0.902187	−0.107832
$71$	−14.7645	−1.75223	−0.876113	−	0.482105i	$-0.839872\pi$
$71$	−14.7645	−1.75223	−0.876113	+	0.482105i	$0.839872\pi$
$72$	0	0
$73$	6.27672	0.734635	0.367317	−	0.930096i	$-0.380276\pi$
$73$	6.27672	0.734635	0.367317	+	0.930096i	$0.380276\pi$
$74$	0.694576	0.0807428
$75$	0	0
$76$	−8.38692	−0.962045
$77$	1.57935	0.179984
$78$	0	0
$79$	9.64978	1.08569	0.542843	−	0.839834i	$-0.317348\pi$
$79$	9.64978	1.08569	0.542843	+	0.839834i	$0.317348\pi$
$80$	2.26764	0.253530
$81$	0	0
$82$	−9.11675	−1.00678
$83$	−8.76097	−0.961641	−0.480820	−	0.876819i	$-0.659661\pi$
$83$	−8.76097	−0.961641	−0.480820	+	0.876819i	$0.659661\pi$
$84$	0	0
$85$	15.5401	1.68556
$86$	−8.24158	−0.888712
$87$	0	0
$88$	−3.96969	−0.423170
$89$	9.24953	0.980448	0.490224	−	0.871596i	$-0.336915\pi$
$89$	9.24953	0.980448	0.490224	+	0.871596i	$0.336915\pi$
$90$	0	0
$91$	−0.327643	−0.0343464
$92$	−8.28167	−0.863424
$93$	0	0
$94$	−1.58117	−0.163086
$95$	−19.0185	−1.95125
$96$	0	0
$97$	−8.05434	−0.817794	−0.408897	−	0.912580i	$-0.634087\pi$
$97$	−8.05434	−0.817794	−0.408897	+	0.912580i	$0.634087\pi$
$98$	−6.84171	−0.691117
$99$	0	0
$100$	0.142177	0.0142177
$101$	9.77230	0.972380	0.486190	−	0.873853i	$-0.338386\pi$
$101$	9.77230	0.972380	0.486190	+	0.873853i	$0.338386\pi$
$102$	0	0
$103$	2.13814	0.210678	0.105339	−	0.994436i	$-0.466407\pi$
$103$	2.13814	0.210678	0.105339	+	0.994436i	$0.466407\pi$
$104$	0.823528	0.0807536
$105$	0	0
$106$	7.80173	0.757771
$107$	9.52464	0.920782	0.460391	−	0.887716i	$-0.347709\pi$
$107$	9.52464	0.920782	0.460391	+	0.887716i	$0.347709\pi$
$108$	0	0
$109$	17.2092	1.64834	0.824170	−	0.566343i	$-0.191642\pi$
$109$	17.2092	1.64834	0.824170	+	0.566343i	$0.191642\pi$
$110$	−9.00181	−0.858288
$111$	0	0
$112$	−0.397853	−0.0375936
$113$	0.853149	0.0802575	0.0401288	−	0.999195i	$-0.487223\pi$
$113$	0.853149	0.0802575	0.0401288	+	0.999195i	$0.487223\pi$
$114$	0	0
$115$	−18.7798	−1.75123
$116$	−8.78572	−0.815734
$117$	0	0
$118$	10.9286	1.00606
$119$	−2.72649	−0.249937
$120$	0	0
$121$	4.75840	0.432582
$122$	−8.40891	−0.761307
$123$	0	0
$124$	4.00807	0.359936
$125$	−11.0158	−0.985281
$126$	0	0
$127$	10.3267	0.916349	0.458175	−	0.888862i	$-0.348503\pi$
$127$	10.3267	0.916349	0.458175	+	0.888862i	$0.348503\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	1.86746	0.163787
$131$	−7.41831	−0.648141	−0.324070	−	0.946033i	$-0.605051\pi$
$131$	−7.41831	−0.648141	−0.324070	+	0.946033i	$0.605051\pi$
$132$	0	0
$133$	3.33676	0.289334
$134$	−11.4002	−0.984829
$135$	0	0
$136$	6.85300	0.587640
$137$	−15.1282	−1.29249	−0.646245	−	0.763130i	$-0.723662\pi$
$137$	−15.1282	−1.29249	−0.646245	+	0.763130i	$0.723662\pi$
$138$	0	0
$139$	−13.7814	−1.16892	−0.584460	−	0.811423i	$-0.698694\pi$
$139$	−13.7814	−1.16892	−0.584460	+	0.811423i	$0.698694\pi$
$140$	−0.902187	−0.0762487
$141$	0	0
$142$	−14.7645	−1.23901
$143$	−3.26915	−0.273380
$144$	0	0
$145$	−19.9228	−1.65450
$146$	6.27672	0.519465
$147$	0	0
$148$	0.694576	0.0570938
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	16.9871	1.38239	0.691197	−	0.722666i	$-0.257084\pi$
$151$	16.9871	1.38239	0.691197	+	0.722666i	$0.257084\pi$
$152$	−8.38692	−0.680269
$153$	0	0
$154$	1.57935	0.127268
$155$	9.08885	0.730034
$156$	0	0
$157$	1.17392	0.0936891	0.0468445	−	0.998902i	$-0.485083\pi$
$157$	1.17392	0.0936891	0.0468445	+	0.998902i	$0.485083\pi$
$158$	9.64978	0.767695
$159$	0	0
$160$	2.26764	0.179272
$161$	3.29489	0.259674
$162$	0	0
$163$	17.5592	1.37534	0.687670	−	0.726023i	$-0.258633\pi$
$163$	17.5592	1.37534	0.687670	+	0.726023i	$0.258633\pi$
$164$	−9.11675	−0.711899
$165$	0	0
$166$	−8.76097	−0.679983
$167$	−4.50119	−0.348312	−0.174156	−	0.984718i	$-0.555720\pi$
$167$	−4.50119	−0.348312	−0.174156	+	0.984718i	$0.555720\pi$
$168$	0	0
$169$	−12.3218	−0.947831
$170$	15.5401	1.19187
$171$	0	0
$172$	−8.24158	−0.628414
$173$	6.31389	0.480036	0.240018	−	0.970768i	$-0.422847\pi$
$173$	6.31389	0.480036	0.240018	+	0.970768i	$0.422847\pi$
$174$	0	0
$175$	−0.0565658	−0.00427597
$176$	−3.96969	−0.299226
$177$	0	0
$178$	9.24953	0.693282
$179$	−7.46421	−0.557901	−0.278951	−	0.960305i	$-0.589987\pi$
$179$	−7.46421	−0.557901	−0.278951	+	0.960305i	$0.589987\pi$
$180$	0	0
$181$	15.8449	1.17774	0.588872	−	0.808226i	$-0.299572\pi$
$181$	15.8449	1.17774	0.588872	+	0.808226i	$0.299572\pi$
$182$	−0.327643	−0.0242865
$183$	0	0
$184$	−8.28167	−0.610533
$185$	1.57505	0.115800
$186$	0	0
$187$	−27.2043	−1.98937
$188$	−1.58117	−0.115319
$189$	0	0
$190$	−19.0185	−1.37975
$191$	16.6457	1.20444	0.602221	−	0.798330i	$-0.294283\pi$
$191$	16.6457	1.20444	0.602221	+	0.798330i	$0.294283\pi$
$192$	0	0
$193$	−9.73110	−0.700460	−0.350230	−	0.936664i	$-0.613897\pi$
$193$	−9.73110	−0.700460	−0.350230	+	0.936664i	$0.613897\pi$
$194$	−8.05434	−0.578268
$195$	0	0
$196$	−6.84171	−0.488694
$197$	−2.03744	−0.145161	−0.0725806	−	0.997363i	$-0.523123\pi$
$197$	−2.03744	−0.145161	−0.0725806	+	0.997363i	$0.523123\pi$
$198$	0	0
$199$	−20.4304	−1.44827	−0.724137	−	0.689656i	$-0.757762\pi$
$199$	−20.4304	−1.44827	−0.724137	+	0.689656i	$0.757762\pi$
$200$	0.142177	0.0100535
$201$	0	0
$202$	9.77230	0.687577
$203$	3.49543	0.245331
$204$	0	0
$205$	−20.6735	−1.44390
$206$	2.13814	0.148972
$207$	0	0
$208$	0.823528	0.0571014
$209$	33.2934	2.30295
$210$	0	0
$211$	14.9296	1.02780	0.513898	−	0.857851i	$-0.328201\pi$
$211$	14.9296	1.02780	0.513898	+	0.857851i	$0.328201\pi$
$212$	7.80173	0.535825
$213$	0	0
$214$	9.52464	0.651091
$215$	−18.6889	−1.27457
$216$	0	0
$217$	−1.59463	−0.108250
$218$	17.2092	1.16555
$219$	0	0
$220$	−9.00181	−0.606902
$221$	5.64364	0.379632
$222$	0	0
$223$	−16.1349	−1.08047	−0.540235	−	0.841514i	$-0.681665\pi$
$223$	−16.1349	−1.08047	−0.540235	+	0.841514i	$0.681665\pi$
$224$	−0.397853	−0.0265827
$225$	0	0
$226$	0.853149	0.0567506
$227$	19.4983	1.29415	0.647074	−	0.762427i	$-0.275992\pi$
$227$	19.4983	1.29415	0.647074	+	0.762427i	$0.275992\pi$
$228$	0	0
$229$	−11.0086	−0.727467	−0.363734	−	0.931503i	$-0.618498\pi$
$229$	−11.0086	−0.727467	−0.363734	+	0.931503i	$0.618498\pi$
$230$	−18.7798	−1.23830
$231$	0	0
$232$	−8.78572	−0.576811
$233$	28.0188	1.83558	0.917788	−	0.397071i	$-0.129973\pi$
$233$	28.0188	1.83558	0.917788	+	0.397071i	$0.129973\pi$
$234$	0	0
$235$	−3.58553	−0.233894
$236$	10.9286	0.711390
$237$	0	0
$238$	−2.72649	−0.176732
$239$	−23.4730	−1.51834	−0.759172	−	0.650890i	$-0.774396\pi$
$239$	−23.4730	−1.51834	−0.759172	+	0.650890i	$0.774396\pi$
$240$	0	0
$241$	16.6148	1.07025	0.535126	−	0.844772i	$-0.320264\pi$
$241$	16.6148	1.07025	0.535126	+	0.844772i	$0.320264\pi$
$242$	4.75840	0.305882
$243$	0	0
$244$	−8.40891	−0.538325
$245$	−15.5145	−0.991186
$246$	0	0
$247$	−6.90686	−0.439473
$248$	4.00807	0.254513
$249$	0	0
$250$	−11.0158	−0.696699
$251$	−21.7503	−1.37287	−0.686434	−	0.727193i	$-0.740825\pi$
$251$	−21.7503	−1.37287	−0.686434	+	0.727193i	$0.740825\pi$
$252$	0	0
$253$	32.8756	2.06687
$254$	10.3267	0.647957
$255$	0	0
$256$	1.00000	0.0625000
$257$	−25.0548	−1.56288	−0.781438	−	0.623983i	$-0.785514\pi$
$257$	−25.0548	−1.56288	−0.781438	+	0.623983i	$0.785514\pi$
$258$	0	0
$259$	−0.276339	−0.0171709
$260$	1.86746	0.115815
$261$	0	0
$262$	−7.41831	−0.458305
$263$	1.10598	0.0681979	0.0340989	−	0.999418i	$-0.489144\pi$
$263$	1.10598	0.0681979	0.0340989	+	0.999418i	$0.489144\pi$
$264$	0	0
$265$	17.6915	1.08678
$266$	3.33676	0.204590
$267$	0	0
$268$	−11.4002	−0.696379
$269$	14.9971	0.914389	0.457195	−	0.889367i	$-0.348854\pi$
$269$	14.9971	0.914389	0.457195	+	0.889367i	$0.348854\pi$
$270$	0	0
$271$	25.6135	1.55591	0.777955	−	0.628320i	$-0.216257\pi$
$271$	25.6135	1.55591	0.777955	+	0.628320i	$0.216257\pi$
$272$	6.85300	0.415524
$273$	0	0
$274$	−15.1282	−0.913929
$275$	−0.564399	−0.0340346
$276$	0	0
$277$	−17.2735	−1.03786	−0.518932	−	0.854815i	$-0.673670\pi$
$277$	−17.2735	−1.03786	−0.518932	+	0.854815i	$0.673670\pi$
$278$	−13.7814	−0.826551
$279$	0	0
$280$	−0.902187	−0.0539160
$281$	−3.74260	−0.223265	−0.111632	−	0.993750i	$-0.535608\pi$
$281$	−3.74260	−0.223265	−0.111632	+	0.993750i	$0.535608\pi$
$282$	0	0
$283$	−7.91599	−0.470557	−0.235278	−	0.971928i	$-0.575600\pi$
$283$	−7.91599	−0.470557	−0.235278	+	0.971928i	$0.575600\pi$
$284$	−14.7645	−0.876113
$285$	0	0
$286$	−3.26915	−0.193309
$287$	3.62713	0.214103
$288$	0	0
$289$	29.9636	1.76257
$290$	−19.9228	−1.16991
$291$	0	0
$292$	6.27672	0.367317
$293$	26.0968	1.52459	0.762295	−	0.647230i	$-0.224073\pi$
$293$	26.0968	1.52459	0.762295	+	0.647230i	$0.224073\pi$
$294$	0	0
$295$	24.7821	1.44287
$296$	0.694576	0.0403714
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	−6.82019	−0.394422
$300$	0	0
$301$	3.27894	0.188995
$302$	16.9871	0.977500
$303$	0	0
$304$	−8.38692	−0.481023
$305$	−19.0684	−1.09185
$306$	0	0
$307$	8.74645	0.499186	0.249593	−	0.968351i	$-0.419703\pi$
$307$	8.74645	0.499186	0.249593	+	0.968351i	$0.419703\pi$
$308$	1.57935	0.0899920
$309$	0	0
$310$	9.08885	0.516212
$311$	−11.6816	−0.662403	−0.331201	−	0.943560i	$-0.607454\pi$
$311$	−11.6816	−0.662403	−0.331201	+	0.943560i	$0.607454\pi$
$312$	0	0
$313$	23.9981	1.35645	0.678226	−	0.734854i	$-0.262749\pi$
$313$	23.9981	1.35645	0.678226	+	0.734854i	$0.262749\pi$
$314$	1.17392	0.0662482
$315$	0	0
$316$	9.64978	0.542843
$317$	−30.2788	−1.70063	−0.850314	−	0.526276i	$-0.823588\pi$
$317$	−30.2788	−1.70063	−0.850314	+	0.526276i	$0.823588\pi$
$318$	0	0
$319$	34.8765	1.95271
$320$	2.26764	0.126765
$321$	0	0
$322$	3.29489	0.183617
$323$	−57.4755	−3.19803
$324$	0	0
$325$	0.117087	0.00649482
$326$	17.5592	0.972513
$327$	0	0
$328$	−9.11675	−0.503388
$329$	0.629075	0.0346820
$330$	0	0
$331$	4.27876	0.235182	0.117591	−	0.993062i	$-0.462483\pi$
$331$	4.27876	0.235182	0.117591	+	0.993062i	$0.462483\pi$
$332$	−8.76097	−0.480820
$333$	0	0
$334$	−4.50119	−0.246294
$335$	−25.8516	−1.41242
$336$	0	0
$337$	−22.8901	−1.24690	−0.623450	−	0.781863i	$-0.714270\pi$
$337$	−22.8901	−1.24690	−0.623450	+	0.781863i	$0.714270\pi$
$338$	−12.3218	−0.670218
$339$	0	0
$340$	15.5401	0.842781
$341$	−15.9108	−0.861618
$342$	0	0
$343$	5.50697	0.297349
$344$	−8.24158	−0.444356
$345$	0	0
$346$	6.31389	0.339437
$347$	−17.5781	−0.943645	−0.471822	−	0.881694i	$-0.656404\pi$
$347$	−17.5781	−0.943645	−0.471822	+	0.881694i	$0.656404\pi$
$348$	0	0
$349$	−6.75698	−0.361693	−0.180846	−	0.983511i	$-0.557884\pi$
$349$	−6.75698	−0.361693	−0.180846	+	0.983511i	$0.557884\pi$
$350$	−0.0565658	−0.00302357
$351$	0	0
$352$	−3.96969	−0.211585
$353$	−3.34841	−0.178218	−0.0891090	−	0.996022i	$-0.528402\pi$
$353$	−3.34841	−0.178218	−0.0891090	+	0.996022i	$0.528402\pi$
$354$	0	0
$355$	−33.4806	−1.77696
$356$	9.24953	0.490224
$357$	0	0
$358$	−7.46421	−0.394496
$359$	−19.7588	−1.04283	−0.521415	−	0.853304i	$-0.674595\pi$
$359$	−19.7588	−1.04283	−0.521415	+	0.853304i	$0.674595\pi$
$360$	0	0
$361$	51.3403	2.70212
$362$	15.8449	0.832791
$363$	0	0
$364$	−0.327643	−0.0171732
$365$	14.2333	0.745006
$366$	0	0
$367$	5.61191	0.292939	0.146470	−	0.989215i	$-0.453209\pi$
$367$	5.61191	0.292939	0.146470	+	0.989215i	$0.453209\pi$
$368$	−8.28167	−0.431712
$369$	0	0
$370$	1.57505	0.0818827
$371$	−3.10394	−0.161149
$372$	0	0
$373$	6.40480	0.331628	0.165814	−	0.986157i	$-0.446975\pi$
$373$	6.40480	0.331628	0.165814	+	0.986157i	$0.446975\pi$
$374$	−27.2043	−1.40670
$375$	0	0
$376$	−1.58117	−0.0815428
$377$	−7.23529	−0.372636
$378$	0	0
$379$	−17.1296	−0.879891	−0.439945	−	0.898025i	$-0.645002\pi$
$379$	−17.1296	−0.879891	−0.439945	+	0.898025i	$0.645002\pi$
$380$	−19.0185	−0.975627
$381$	0	0
$382$	16.6457	0.851669
$383$	−22.0618	−1.12730	−0.563652	−	0.826012i	$-0.690604\pi$
$383$	−22.0618	−1.12730	−0.563652	+	0.826012i	$0.690604\pi$
$384$	0	0
$385$	3.58140	0.182525
$386$	−9.73110	−0.495300
$387$	0	0
$388$	−8.05434	−0.408897
$389$	15.0190	0.761494	0.380747	−	0.924679i	$-0.375667\pi$
$389$	15.0190	0.761494	0.380747	+	0.924679i	$0.375667\pi$
$390$	0	0
$391$	−56.7543	−2.87019
$392$	−6.84171	−0.345559
$393$	0	0
$394$	−2.03744	−0.102645
$395$	21.8822	1.10101
$396$	0	0
$397$	−9.26395	−0.464944	−0.232472	−	0.972603i	$-0.574681\pi$
$397$	−9.26395	−0.464944	−0.232472	+	0.972603i	$0.574681\pi$
$398$	−20.4304	−1.02408
$399$	0	0
$400$	0.142177	0.00710887
$401$	12.8908	0.643733	0.321867	−	0.946785i	$-0.395690\pi$
$401$	12.8908	0.643733	0.321867	+	0.946785i	$0.395690\pi$
$402$	0	0
$403$	3.30076	0.164423
$404$	9.77230	0.486190
$405$	0	0
$406$	3.49543	0.173475
$407$	−2.75725	−0.136672
$408$	0	0
$409$	−9.86517	−0.487801	−0.243901	−	0.969800i	$-0.578427\pi$
$409$	−9.86517	−0.487801	−0.243901	+	0.969800i	$0.578427\pi$
$410$	−20.6735	−1.02099
$411$	0	0
$412$	2.13814	0.105339
$413$	−4.34797	−0.213950
$414$	0	0
$415$	−19.8667	−0.975218
$416$	0.823528	0.0403768
$417$	0	0
$418$	33.2934	1.62843
$419$	−16.8632	−0.823823	−0.411912	−	0.911224i	$-0.635139\pi$
$419$	−16.8632	−0.823823	−0.411912	+	0.911224i	$0.635139\pi$
$420$	0	0
$421$	16.7035	0.814078	0.407039	−	0.913411i	$-0.366561\pi$
$421$	16.7035	0.814078	0.407039	+	0.913411i	$0.366561\pi$
$422$	14.9296	0.726762
$423$	0	0
$424$	7.80173	0.378885
$425$	0.974342	0.0472625
$426$	0	0
$427$	3.34551	0.161901
$428$	9.52464	0.460391
$429$	0	0
$430$	−18.6889	−0.901259
$431$	3.42714	0.165079	0.0825397	−	0.996588i	$-0.473697\pi$
$431$	3.42714	0.165079	0.0825397	+	0.996588i	$0.473697\pi$
$432$	0	0
$433$	9.04647	0.434746	0.217373	−	0.976089i	$-0.430251\pi$
$433$	9.04647	0.434746	0.217373	+	0.976089i	$0.430251\pi$
$434$	−1.59463	−0.0765445
$435$	0	0
$436$	17.2092	0.824170
$437$	69.4577	3.32261
$438$	0	0
$439$	15.9907	0.763193	0.381597	−	0.924329i	$-0.375374\pi$
$439$	15.9907	0.763193	0.381597	+	0.924329i	$0.375374\pi$
$440$	−9.00181	−0.429144
$441$	0	0
$442$	5.64364	0.268441
$443$	2.78120	0.132139	0.0660693	−	0.997815i	$-0.478954\pi$
$443$	2.78120	0.132139	0.0660693	+	0.997815i	$0.478954\pi$
$444$	0	0
$445$	20.9746	0.994290
$446$	−16.1349	−0.764008
$447$	0	0
$448$	−0.397853	−0.0187968
$449$	22.6866	1.07065	0.535324	−	0.844647i	$-0.320190\pi$
$449$	22.6866	1.07065	0.535324	+	0.844647i	$0.320190\pi$
$450$	0	0
$451$	36.1906	1.70415
$452$	0.853149	0.0401288
$453$	0	0
$454$	19.4983	0.915101
$455$	−0.742976	−0.0348313
$456$	0	0
$457$	6.54589	0.306204	0.153102	−	0.988210i	$-0.451074\pi$
$457$	6.54589	0.306204	0.153102	+	0.988210i	$0.451074\pi$
$458$	−11.0086	−0.514397
$459$	0	0
$460$	−18.7798	−0.875614
$461$	−8.49455	−0.395631	−0.197815	−	0.980239i	$-0.563385\pi$
$461$	−8.49455	−0.395631	−0.197815	+	0.980239i	$0.563385\pi$
$462$	0	0
$463$	29.4676	1.36947	0.684737	−	0.728790i	$-0.259917\pi$
$463$	29.4676	1.36947	0.684737	+	0.728790i	$0.259917\pi$
$464$	−8.78572	−0.407867
$465$	0	0
$466$	28.0188	1.29795
$467$	−26.8493	−1.24244	−0.621218	−	0.783638i	$-0.713362\pi$
$467$	−26.8493	−1.24244	−0.621218	+	0.783638i	$0.713362\pi$
$468$	0	0
$469$	4.53562	0.209435
$470$	−3.58553	−0.165388
$471$	0	0
$472$	10.9286	0.503029
$473$	32.7165	1.50430
$474$	0	0
$475$	−1.19243	−0.0547124
$476$	−2.72649	−0.124968
$477$	0	0
$478$	−23.4730	−1.07363
$479$	−17.1551	−0.783837	−0.391919	−	0.920000i	$-0.628189\pi$
$479$	−17.1551	−0.783837	−0.391919	+	0.920000i	$0.628189\pi$
$480$	0	0
$481$	0.572003	0.0260811
$482$	16.6148	0.756782
$483$	0	0
$484$	4.75840	0.216291
$485$	−18.2643	−0.829340
$486$	0	0
$487$	14.0684	0.637501	0.318750	−	0.947839i	$-0.396737\pi$
$487$	14.0684	0.637501	0.318750	+	0.947839i	$0.396737\pi$
$488$	−8.40891	−0.380654
$489$	0	0
$490$	−15.5145	−0.700875
$491$	6.74375	0.304341	0.152171	−	0.988354i	$-0.451374\pi$
$491$	6.74375	0.304341	0.152171	+	0.988354i	$0.451374\pi$
$492$	0	0
$493$	−60.2086	−2.71166
$494$	−6.90686	−0.310754
$495$	0	0
$496$	4.00807	0.179968
$497$	5.87412	0.263490
$498$	0	0
$499$	4.81728	0.215651	0.107825	−	0.994170i	$-0.465611\pi$
$499$	4.81728	0.215651	0.107825	+	0.994170i	$0.465611\pi$
$500$	−11.0158	−0.492641
$501$	0	0
$502$	−21.7503	−0.970764
$503$	−36.1200	−1.61051	−0.805255	−	0.592929i	$-0.797972\pi$
$503$	−36.1200	−1.61051	−0.805255	+	0.592929i	$0.797972\pi$
$504$	0	0
$505$	22.1600	0.986108
$506$	32.8756	1.46150
$507$	0	0
$508$	10.3267	0.458175
$509$	−27.1181	−1.20199	−0.600994	−	0.799254i	$-0.705228\pi$
$509$	−27.1181	−1.20199	−0.600994	+	0.799254i	$0.705228\pi$
$510$	0	0
$511$	−2.49721	−0.110470
$512$	1.00000	0.0441942
$513$	0	0
$514$	−25.0548	−1.10512
$515$	4.84853	0.213652
$516$	0	0
$517$	6.27676	0.276052
$518$	−0.276339	−0.0121417
$519$	0	0
$520$	1.86746	0.0818937
$521$	−3.14188	−0.137648	−0.0688240	−	0.997629i	$-0.521925\pi$
$521$	−3.14188	−0.137648	−0.0688240	+	0.997629i	$0.521925\pi$
$522$	0	0
$523$	−27.3922	−1.19778	−0.598888	−	0.800833i	$-0.704390\pi$
$523$	−27.3922	−1.19778	−0.598888	+	0.800833i	$0.704390\pi$
$524$	−7.41831	−0.324070
$525$	0	0
$526$	1.10598	0.0482232
$527$	27.4673	1.19650
$528$	0	0
$529$	45.5861	1.98200
$530$	17.6915	0.768469
$531$	0	0
$532$	3.33676	0.144667
$533$	−7.50790	−0.325203
$534$	0	0
$535$	21.5984	0.933781
$536$	−11.4002	−0.492415
$537$	0	0
$538$	14.9971	0.646571
$539$	27.1594	1.16984
$540$	0	0
$541$	13.0959	0.563036	0.281518	−	0.959556i	$-0.409162\pi$
$541$	13.0959	0.563036	0.281518	+	0.959556i	$0.409162\pi$
$542$	25.6135	1.10019
$543$	0	0
$544$	6.85300	0.293820
$545$	39.0241	1.67161
$546$	0	0
$547$	−10.4468	−0.446674	−0.223337	−	0.974741i	$-0.571695\pi$
$547$	−10.4468	−0.446674	−0.223337	+	0.974741i	$0.571695\pi$
$548$	−15.1282	−0.646245
$549$	0	0
$550$	−0.564399	−0.0240661
$551$	73.6851	3.13909
$552$	0	0
$553$	−3.83920	−0.163259
$554$	−17.2735	−0.733881
$555$	0	0
$556$	−13.7814	−0.584460
$557$	−23.7259	−1.00530	−0.502650	−	0.864490i	$-0.667642\pi$
$557$	−23.7259	−1.00530	−0.502650	+	0.864490i	$0.667642\pi$
$558$	0	0
$559$	−6.78717	−0.287067
$560$	−0.902187	−0.0381244
$561$	0	0
$562$	−3.74260	−0.157872
$563$	−15.4966	−0.653106	−0.326553	−	0.945179i	$-0.605887\pi$
$563$	−15.4966	−0.653106	−0.326553	+	0.945179i	$0.605887\pi$
$564$	0	0
$565$	1.93463	0.0813906
$566$	−7.91599	−0.332734
$567$	0	0
$568$	−14.7645	−0.619506
$569$	−9.61882	−0.403242	−0.201621	−	0.979464i	$-0.564621\pi$
$569$	−9.61882	−0.403242	−0.201621	+	0.979464i	$0.564621\pi$
$570$	0	0
$571$	34.8896	1.46008	0.730042	−	0.683402i	$-0.239500\pi$
$571$	34.8896	1.46008	0.730042	+	0.683402i	$0.239500\pi$
$572$	−3.26915	−0.136690
$573$	0	0
$574$	3.62713	0.151393
$575$	−1.17747	−0.0491037
$576$	0	0
$577$	5.66430	0.235808	0.117904	−	0.993025i	$-0.462382\pi$
$577$	5.66430	0.235808	0.117904	+	0.993025i	$0.462382\pi$
$578$	29.9636	1.24632
$579$	0	0
$580$	−19.9228	−0.827250
$581$	3.48558	0.144606
$582$	0	0
$583$	−30.9704	−1.28266
$584$	6.27672	0.259733
$585$	0	0
$586$	26.0968	1.07805
$587$	−13.8612	−0.572112	−0.286056	−	0.958213i	$-0.592344\pi$
$587$	−13.8612	−0.572112	−0.286056	+	0.958213i	$0.592344\pi$
$588$	0	0
$589$	−33.6154	−1.38510
$590$	24.7821	1.02026
$591$	0	0
$592$	0.694576	0.0285469
$593$	−42.2000	−1.73295	−0.866474	−	0.499222i	$-0.833619\pi$
$593$	−42.2000	−1.73295	−0.866474	+	0.499222i	$0.833619\pi$
$594$	0	0
$595$	−6.18269	−0.253466
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	−6.82019	−0.278898
$599$	40.0156	1.63499	0.817497	−	0.575933i	$-0.195361\pi$
$599$	40.0156	1.63499	0.817497	+	0.575933i	$0.195361\pi$
$600$	0	0
$601$	12.4610	0.508296	0.254148	−	0.967165i	$-0.418205\pi$
$601$	12.4610	0.508296	0.254148	+	0.967165i	$0.418205\pi$
$602$	3.27894	0.133640
$603$	0	0
$604$	16.9871	0.691197
$605$	10.7903	0.438689
$606$	0	0
$607$	28.2278	1.14573	0.572865	−	0.819650i	$-0.305832\pi$
$607$	28.2278	1.14573	0.572865	+	0.819650i	$0.305832\pi$
$608$	−8.38692	−0.340134
$609$	0	0
$610$	−19.0684	−0.772055
$611$	−1.30214	−0.0526790
$612$	0	0
$613$	9.70098	0.391819	0.195909	−	0.980622i	$-0.437234\pi$
$613$	9.70098	0.391819	0.195909	+	0.980622i	$0.437234\pi$
$614$	8.74645	0.352978
$615$	0	0
$616$	1.57935	0.0636339
$617$	20.3190	0.818012	0.409006	−	0.912532i	$-0.365876\pi$
$617$	20.3190	0.818012	0.409006	+	0.912532i	$0.365876\pi$
$618$	0	0
$619$	15.6405	0.628645	0.314323	−	0.949316i	$-0.398223\pi$
$619$	15.6405	0.628645	0.314323	+	0.949316i	$0.398223\pi$
$620$	9.08885	0.365017
$621$	0	0
$622$	−11.6816	−0.468389
$623$	−3.67996	−0.147434
$624$	0	0
$625$	−25.6907	−1.02763
$626$	23.9981	0.959156
$627$	0	0
$628$	1.17392	0.0468445
$629$	4.75993	0.189791
$630$	0	0
$631$	4.45340	0.177287	0.0886436	−	0.996063i	$-0.471747\pi$
$631$	4.45340	0.177287	0.0886436	+	0.996063i	$0.471747\pi$
$632$	9.64978	0.383848
$633$	0	0
$634$	−30.2788	−1.20253
$635$	23.4173	0.929287
$636$	0	0
$637$	−5.63434	−0.223241
$638$	34.8765	1.38078
$639$	0	0
$640$	2.26764	0.0896362
$641$	−42.1227	−1.66375	−0.831874	−	0.554964i	$-0.812732\pi$
$641$	−42.1227	−1.66375	−0.831874	+	0.554964i	$0.812732\pi$
$642$	0	0
$643$	10.4065	0.410392	0.205196	−	0.978721i	$-0.434217\pi$
$643$	10.4065	0.410392	0.205196	+	0.978721i	$0.434217\pi$
$644$	3.29489	0.129837
$645$	0	0
$646$	−57.4755	−2.26135
$647$	32.4987	1.27765	0.638827	−	0.769350i	$-0.279420\pi$
$647$	32.4987	1.27765	0.638827	+	0.769350i	$0.279420\pi$
$648$	0	0
$649$	−43.3830	−1.70293
$650$	0.117087	0.00459253
$651$	0	0
$652$	17.5592	0.687670
$653$	−42.1198	−1.64827	−0.824137	−	0.566391i	$-0.808339\pi$
$653$	−42.1198	−1.64827	−0.824137	+	0.566391i	$0.808339\pi$
$654$	0	0
$655$	−16.8220	−0.657291
$656$	−9.11675	−0.355949
$657$	0	0
$658$	0.629075	0.0245239
$659$	−38.1964	−1.48792	−0.743960	−	0.668224i	$-0.767055\pi$
$659$	−38.1964	−1.48792	−0.743960	+	0.668224i	$0.767055\pi$
$660$	0	0
$661$	34.9691	1.36014	0.680070	−	0.733147i	$-0.261949\pi$
$661$	34.9691	1.36014	0.680070	+	0.733147i	$0.261949\pi$
$662$	4.27876	0.166299
$663$	0	0
$664$	−8.76097	−0.339991
$665$	7.56657	0.293419
$666$	0	0
$667$	72.7605	2.81730
$668$	−4.50119	−0.174156
$669$	0	0
$670$	−25.8516	−0.998733
$671$	33.3807	1.28865
$672$	0	0
$673$	9.28107	0.357759	0.178880	−	0.983871i	$-0.442753\pi$
$673$	9.28107	0.357759	0.178880	+	0.983871i	$0.442753\pi$
$674$	−22.8901	−0.881692
$675$	0	0
$676$	−12.3218	−0.473915
$677$	−43.5582	−1.67408	−0.837039	−	0.547143i	$-0.815715\pi$
$677$	−43.5582	−1.67408	−0.837039	+	0.547143i	$0.815715\pi$
$678$	0	0
$679$	3.20445	0.122975
$680$	15.5401	0.595936
$681$	0	0
$682$	−15.9108	−0.609256
$683$	−10.2891	−0.393702	−0.196851	−	0.980433i	$-0.563071\pi$
$683$	−10.2891	−0.393702	−0.196851	+	0.980433i	$0.563071\pi$
$684$	0	0
$685$	−34.3053	−1.31074
$686$	5.50697	0.210257
$687$	0	0
$688$	−8.24158	−0.314207
$689$	6.42494	0.244771
$690$	0	0
$691$	−19.7310	−0.750603	−0.375302	−	0.926903i	$-0.622461\pi$
$691$	−19.7310	−0.750603	−0.375302	+	0.926903i	$0.622461\pi$
$692$	6.31389	0.240018
$693$	0	0
$694$	−17.5781	−0.667257
$695$	−31.2511	−1.18542
$696$	0	0
$697$	−62.4771	−2.36649
$698$	−6.75698	−0.255755
$699$	0	0
$700$	−0.0565658	−0.00213798
$701$	−44.6961	−1.68815	−0.844073	−	0.536228i	$-0.819849\pi$
$701$	−44.6961	−1.68815	−0.844073	+	0.536228i	$0.819849\pi$
$702$	0	0
$703$	−5.82535	−0.219707
$704$	−3.96969	−0.149613
$705$	0	0
$706$	−3.34841	−0.126019
$707$	−3.88794	−0.146221
$708$	0	0
$709$	6.66685	0.250379	0.125189	−	0.992133i	$-0.460046\pi$
$709$	6.66685	0.250379	0.125189	+	0.992133i	$0.460046\pi$
$710$	−33.4806	−1.25650
$711$	0	0
$712$	9.24953	0.346641
$713$	−33.1935	−1.24311
$714$	0	0
$715$	−7.41324	−0.277239
$716$	−7.46421	−0.278951
$717$	0	0
$718$	−19.7588	−0.737391
$719$	−21.3307	−0.795502	−0.397751	−	0.917493i	$-0.630209\pi$
$719$	−21.3307	−0.795502	−0.397751	+	0.917493i	$0.630209\pi$
$720$	0	0
$721$	−0.850668	−0.0316805
$722$	51.3403	1.91069
$723$	0	0
$724$	15.8449	0.588872
$725$	−1.24913	−0.0463916
$726$	0	0
$727$	−28.2804	−1.04886	−0.524430	−	0.851453i	$-0.675722\pi$
$727$	−28.2804	−1.04886	−0.524430	+	0.851453i	$0.675722\pi$
$728$	−0.327643	−0.0121433
$729$	0	0
$730$	14.2333	0.526799
$731$	−56.4795	−2.08897
$732$	0	0
$733$	6.79861	0.251112	0.125556	−	0.992087i	$-0.459928\pi$
$733$	6.79861	0.251112	0.125556	+	0.992087i	$0.459928\pi$
$734$	5.61191	0.207139
$735$	0	0
$736$	−8.28167	−0.305266
$737$	45.2553	1.66700
$738$	0	0
$739$	−18.8897	−0.694870	−0.347435	−	0.937704i	$-0.612947\pi$
$739$	−18.8897	−0.694870	−0.347435	+	0.937704i	$0.612947\pi$
$740$	1.57505	0.0578998
$741$	0	0
$742$	−3.10394	−0.113949
$743$	19.5087	0.715704	0.357852	−	0.933778i	$-0.383509\pi$
$743$	19.5087	0.715704	0.357852	+	0.933778i	$0.383509\pi$
$744$	0	0
$745$	−2.26764	−0.0830798
$746$	6.40480	0.234496
$747$	0	0
$748$	−27.2043	−0.994686
$749$	−3.78941	−0.138462
$750$	0	0
$751$	13.1392	0.479458	0.239729	−	0.970840i	$-0.422941\pi$
$751$	13.1392	0.479458	0.239729	+	0.970840i	$0.422941\pi$
$752$	−1.58117	−0.0576594
$753$	0	0
$754$	−7.23529	−0.263494
$755$	38.5207	1.40191
$756$	0	0
$757$	−22.5014	−0.817827	−0.408913	−	0.912573i	$-0.634092\pi$
$757$	−22.5014	−0.817827	−0.408913	+	0.912573i	$0.634092\pi$
$758$	−17.1296	−0.622177
$759$	0	0
$760$	−19.0185	−0.689873
$761$	4.15691	0.150688	0.0753440	−	0.997158i	$-0.475995\pi$
$761$	4.15691	0.150688	0.0753440	+	0.997158i	$0.475995\pi$
$762$	0	0
$763$	−6.84672	−0.247868
$764$	16.6457	0.602221
$765$	0	0
$766$	−22.0618	−0.797125
$767$	8.99999	0.324971
$768$	0	0
$769$	−17.5274	−0.632055	−0.316028	−	0.948750i	$-0.602349\pi$
$769$	−17.5274	−0.632055	−0.316028	+	0.948750i	$0.602349\pi$
$770$	3.58140	0.129065
$771$	0	0
$772$	−9.73110	−0.350230
$773$	−5.94560	−0.213849	−0.106924	−	0.994267i	$-0.534100\pi$
$773$	−5.94560	−0.213849	−0.106924	+	0.994267i	$0.534100\pi$
$774$	0	0
$775$	0.569857	0.0204699
$776$	−8.05434	−0.289134
$777$	0	0
$778$	15.0190	0.538458
$779$	76.4614	2.73951
$780$	0	0
$781$	58.6105	2.09725
$782$	−56.7543	−2.02953
$783$	0	0
$784$	−6.84171	−0.244347
$785$	2.66203	0.0950118
$786$	0	0
$787$	−31.7311	−1.13109	−0.565545	−	0.824717i	$-0.691334\pi$
$787$	−31.7311	−1.13109	−0.565545	+	0.824717i	$0.691334\pi$
$788$	−2.03744	−0.0725806
$789$	0	0
$790$	21.8822	0.778534
$791$	−0.339428	−0.0120687
$792$	0	0
$793$	−6.92498	−0.245913
$794$	−9.26395	−0.328765
$795$	0	0
$796$	−20.4304	−0.724137
$797$	−21.1583	−0.749466	−0.374733	−	0.927133i	$-0.622266\pi$
$797$	−21.1583	−0.749466	−0.374733	+	0.927133i	$0.622266\pi$
$798$	0	0
$799$	−10.8358	−0.383342
$800$	0.142177	0.00502673
$801$	0	0
$802$	12.8908	0.455188
$803$	−24.9166	−0.879288
$804$	0	0
$805$	7.47162	0.263340
$806$	3.30076	0.116264
$807$	0	0
$808$	9.77230	0.343788
$809$	−18.8833	−0.663900	−0.331950	−	0.943297i	$-0.607707\pi$
$809$	−18.8833	−0.663900	−0.331950	+	0.943297i	$0.607707\pi$
$810$	0	0
$811$	−21.7651	−0.764275	−0.382137	−	0.924105i	$-0.624812\pi$
$811$	−21.7651	−0.764275	−0.382137	+	0.924105i	$0.624812\pi$
$812$	3.49543	0.122665
$813$	0	0
$814$	−2.75725	−0.0966415
$815$	39.8178	1.39476
$816$	0	0
$817$	69.1214	2.41825
$818$	−9.86517	−0.344928
$819$	0	0
$820$	−20.6735	−0.721949
$821$	−12.5288	−0.437259	−0.218630	−	0.975808i	$-0.570159\pi$
$821$	−12.5288	−0.437259	−0.218630	+	0.975808i	$0.570159\pi$
$822$	0	0
$823$	2.12788	0.0741733	0.0370867	−	0.999312i	$-0.488192\pi$
$823$	2.12788	0.0741733	0.0370867	+	0.999312i	$0.488192\pi$
$824$	2.13814	0.0744858
$825$	0	0
$826$	−4.34797	−0.151285
$827$	14.0345	0.488026	0.244013	−	0.969772i	$-0.421536\pi$
$827$	14.0345	0.488026	0.244013	+	0.969772i	$0.421536\pi$
$828$	0	0
$829$	−4.43826	−0.154147	−0.0770735	−	0.997025i	$-0.524558\pi$
$829$	−4.43826	−0.154147	−0.0770735	+	0.997025i	$0.524558\pi$
$830$	−19.8667	−0.689583
$831$	0	0
$832$	0.823528	0.0285507
$833$	−46.8863	−1.62451
$834$	0	0
$835$	−10.2071	−0.353230
$836$	33.2934	1.15148
$837$	0	0
$838$	−16.8632	−0.582531
$839$	−6.15626	−0.212538	−0.106269	−	0.994337i	$-0.533890\pi$
$839$	−6.15626	−0.212538	−0.106269	+	0.994337i	$0.533890\pi$
$840$	0	0
$841$	48.1889	1.66169
$842$	16.7035	0.575640
$843$	0	0
$844$	14.9296	0.513898
$845$	−27.9414	−0.961212
$846$	0	0
$847$	−1.89315	−0.0650493
$848$	7.80173	0.267912
$849$	0	0
$850$	0.974342	0.0334197
$851$	−5.75225	−0.197185
$852$	0	0
$853$	9.14473	0.313110	0.156555	−	0.987669i	$-0.449961\pi$
$853$	9.14473	0.313110	0.156555	+	0.987669i	$0.449961\pi$
$854$	3.34551	0.114481
$855$	0	0
$856$	9.52464	0.325546
$857$	−25.6375	−0.875762	−0.437881	−	0.899033i	$-0.644271\pi$
$857$	−25.6375	−0.875762	−0.437881	+	0.899033i	$0.644271\pi$
$858$	0	0
$859$	−5.45085	−0.185981	−0.0929903	−	0.995667i	$-0.529643\pi$
$859$	−5.45085	−0.185981	−0.0929903	+	0.995667i	$0.529643\pi$
$860$	−18.6889	−0.637286
$861$	0	0
$862$	3.42714	0.116729
$863$	15.4896	0.527272	0.263636	−	0.964622i	$-0.415078\pi$
$863$	15.4896	0.527272	0.263636	+	0.964622i	$0.415078\pi$
$864$	0	0
$865$	14.3176	0.486814
$866$	9.04647	0.307412
$867$	0	0
$868$	−1.59463	−0.0541251
$869$	−38.3066	−1.29946
$870$	0	0
$871$	−9.38840	−0.318114
$872$	17.2092	0.582776
$873$	0	0
$874$	69.4577	2.34944
$875$	4.38266	0.148161
$876$	0	0
$877$	−2.33464	−0.0788353	−0.0394177	−	0.999223i	$-0.512550\pi$
$877$	−2.33464	−0.0788353	−0.0394177	+	0.999223i	$0.512550\pi$
$878$	15.9907	0.539659
$879$	0	0
$880$	−9.00181	−0.303451
$881$	−57.6543	−1.94242	−0.971211	−	0.238220i	$-0.923436\pi$
$881$	−57.6543	−1.94242	−0.971211	+	0.238220i	$0.923436\pi$
$882$	0	0
$883$	25.7235	0.865663	0.432832	−	0.901475i	$-0.357514\pi$
$883$	25.7235	0.865663	0.432832	+	0.901475i	$0.357514\pi$
$884$	5.64364	0.189816
$885$	0	0
$886$	2.78120	0.0934361
$887$	−46.8090	−1.57169	−0.785846	−	0.618422i	$-0.787772\pi$
$887$	−46.8090	−1.57169	−0.785846	+	0.618422i	$0.787772\pi$
$888$	0	0
$889$	−4.10853	−0.137796
$890$	20.9746	0.703069
$891$	0	0
$892$	−16.1349	−0.540235
$893$	13.2612	0.443768
$894$	0	0
$895$	−16.9261	−0.565778
$896$	−0.397853	−0.0132913
$897$	0	0
$898$	22.6866	0.757062
$899$	−35.2138	−1.17445
$900$	0	0
$901$	53.4653	1.78119
$902$	36.1906	1.20502
$903$	0	0
$904$	0.853149	0.0283753
$905$	35.9305	1.19437
$906$	0	0
$907$	−34.4229	−1.14299	−0.571497	−	0.820604i	$-0.693637\pi$
$907$	−34.4229	−1.14299	−0.571497	+	0.820604i	$0.693637\pi$
$908$	19.4983	0.647074
$909$	0	0
$910$	−0.742976	−0.0246294
$911$	6.77761	0.224552	0.112276	−	0.993677i	$-0.464186\pi$
$911$	6.77761	0.224552	0.112276	+	0.993677i	$0.464186\pi$
$912$	0	0
$913$	34.7783	1.15099
$914$	6.54589	0.216519
$915$	0	0
$916$	−11.0086	−0.363734
$917$	2.95140	0.0974638
$918$	0	0
$919$	36.3820	1.20013	0.600066	−	0.799950i	$-0.295141\pi$
$919$	36.3820	1.20013	0.600066	+	0.799950i	$0.295141\pi$
$920$	−18.7798	−0.619152
$921$	0	0
$922$	−8.49455	−0.279753
$923$	−12.1590	−0.400218
$924$	0	0
$925$	0.0987530	0.00324698
$926$	29.4676	0.968365
$927$	0	0
$928$	−8.78572	−0.288405
$929$	49.3214	1.61818	0.809091	−	0.587683i	$-0.199960\pi$
$929$	49.3214	1.61818	0.809091	+	0.587683i	$0.199960\pi$
$930$	0	0
$931$	57.3809	1.88058
$932$	28.0188	0.917788
$933$	0	0
$934$	−26.8493	−0.878535
$935$	−61.6894	−2.01746
$936$	0	0
$937$	22.8573	0.746717	0.373358	−	0.927687i	$-0.378206\pi$
$937$	22.8573	0.746717	0.373358	+	0.927687i	$0.378206\pi$
$938$	4.53562	0.148093
$939$	0	0
$940$	−3.58553	−0.116947
$941$	33.8585	1.10375	0.551877	−	0.833925i	$-0.313912\pi$
$941$	33.8585	1.10375	0.551877	+	0.833925i	$0.313912\pi$
$942$	0	0
$943$	75.5019	2.45868
$944$	10.9286	0.355695
$945$	0	0
$946$	32.7165	1.06370
$947$	−0.515622	−0.0167555	−0.00837774	−	0.999965i	$-0.502667\pi$
$947$	−0.515622	−0.0167555	−0.00837774	+	0.999965i	$0.502667\pi$
$948$	0	0
$949$	5.16906	0.167795
$950$	−1.19243	−0.0386875
$951$	0	0
$952$	−2.72649	−0.0883661
$953$	−5.49709	−0.178068	−0.0890341	−	0.996029i	$-0.528378\pi$
$953$	−5.49709	−0.178068	−0.0890341	+	0.996029i	$0.528378\pi$
$954$	0	0
$955$	37.7464	1.22145
$956$	−23.4730	−0.759172
$957$	0	0
$958$	−17.1551	−0.554257
$959$	6.01881	0.194357
$960$	0	0
$961$	−14.9353	−0.481785
$962$	0.572003	0.0184421
$963$	0	0
$964$	16.6148	0.535126
$965$	−22.0666	−0.710349
$966$	0	0
$967$	10.6176	0.341440	0.170720	−	0.985320i	$-0.445391\pi$
$967$	10.6176	0.341440	0.170720	+	0.985320i	$0.445391\pi$
$968$	4.75840	0.152941
$969$	0	0
$970$	−18.2643	−0.586432
$971$	51.8122	1.66273	0.831366	−	0.555725i	$-0.187559\pi$
$971$	51.8122	1.66273	0.831366	+	0.555725i	$0.187559\pi$
$972$	0	0
$973$	5.48296	0.175776
$974$	14.0684	0.450781
$975$	0	0
$976$	−8.40891	−0.269163
$977$	33.6732	1.07730	0.538651	−	0.842529i	$-0.318934\pi$
$977$	33.6732	1.07730	0.538651	+	0.842529i	$0.318934\pi$
$978$	0	0
$979$	−36.7177	−1.17350
$980$	−15.5145	−0.495593
$981$	0	0
$982$	6.74375	0.215202
$983$	51.2822	1.63565	0.817823	−	0.575469i	$-0.195181\pi$
$983$	51.2822	1.63565	0.817823	+	0.575469i	$0.195181\pi$
$984$	0	0
$985$	−4.62016	−0.147211
$986$	−60.2086	−1.91743
$987$	0	0
$988$	−6.90686	−0.219737
$989$	68.2540	2.17035
$990$	0	0
$991$	6.69020	0.212521	0.106261	−	0.994338i	$-0.466112\pi$
$991$	6.69020	0.212521	0.106261	+	0.994338i	$0.466112\pi$
$992$	4.00807	0.127256
$993$	0	0
$994$	5.87412	0.186316
$995$	−46.3288	−1.46872
$996$	0	0
$997$	10.3841	0.328869	0.164435	−	0.986388i	$-0.447420\pi$
$997$	10.3841	0.328869	0.164435	+	0.986388i	$0.447420\pi$
$998$	4.81728	0.152488
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.m.1.11	yes	12
3.2	odd	2		8046.2.a.l.1.2	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.l.1.2	✓	12	3.2	odd	2
8046.2.a.m.1.11	yes	12	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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.26764 q^{5} -0.397853 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.26764 q^{5} -0.397853 q^{7} +1.00000 q^{8} +2.26764 q^{10} -3.96969 q^{11} +0.823528 q^{13} -0.397853 q^{14} +1.00000 q^{16} +6.85300 q^{17} -8.38692 q^{19} +2.26764 q^{20} -3.96969 q^{22} -8.28167 q^{23} +0.142177 q^{25} +0.823528 q^{26} -0.397853 q^{28} -8.78572 q^{29} +4.00807 q^{31} +1.00000 q^{32} +6.85300 q^{34} -0.902187 q^{35} +0.694576 q^{37} -8.38692 q^{38} +2.26764 q^{40} -9.11675 q^{41} -8.24158 q^{43} -3.96969 q^{44} -8.28167 q^{46} -1.58117 q^{47} -6.84171 q^{49} +0.142177 q^{50} +0.823528 q^{52} +7.80173 q^{53} -9.00181 q^{55} -0.397853 q^{56} -8.78572 q^{58} +10.9286 q^{59} -8.40891 q^{61} +4.00807 q^{62} +1.00000 q^{64} +1.86746 q^{65} -11.4002 q^{67} +6.85300 q^{68} -0.902187 q^{70} -14.7645 q^{71} +6.27672 q^{73} +0.694576 q^{74} -8.38692 q^{76} +1.57935 q^{77} +9.64978 q^{79} +2.26764 q^{80} -9.11675 q^{82} -8.76097 q^{83} +15.5401 q^{85} -8.24158 q^{86} -3.96969 q^{88} +9.24953 q^{89} -0.327643 q^{91} -8.28167 q^{92} -1.58117 q^{94} -19.0185 q^{95} -8.05434 q^{97} -6.84171 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 12 q^{4} - 5 q^{5} - 6 q^{7} + 12 q^{8}+O(q^{10}) \) 12 * q + 12 * q^2 + 12 * q^4 - 5 * q^5 - 6 * q^7 + 12 * q^8	\( 12 q + 12 q^{2} + 12 q^{4} - 5 q^{5} - 6 q^{7} + 12 q^{8} - 5 q^{10} - 10 q^{11} - q^{13} - 6 q^{14} + 12 q^{16} - 6 q^{17} - 10 q^{19} - 5 q^{20} - 10 q^{22} - 15 q^{23} + 7 q^{25} - q^{26} - 6 q^{28} - 33 q^{29} - 6 q^{31} + 12 q^{32} - 6 q^{34} - 16 q^{35} - 13 q^{37} - 10 q^{38} - 5 q^{40} - 20 q^{41} - 11 q^{43} - 10 q^{44} - 15 q^{46} - 15 q^{47} + 2 q^{49} + 7 q^{50} - q^{52} - 4 q^{53} - 17 q^{55} - 6 q^{56} - 33 q^{58} - 10 q^{59} - 12 q^{61} - 6 q^{62} + 12 q^{64} - 40 q^{65} - 19 q^{67} - 6 q^{68} - 16 q^{70} - 47 q^{71} - 2 q^{73} - 13 q^{74} - 10 q^{76} + 6 q^{77} - 15 q^{79} - 5 q^{80} - 20 q^{82} - 18 q^{83} - 25 q^{85} - 11 q^{86} - 10 q^{88} - 24 q^{89} - 3 q^{91} - 15 q^{92} - 15 q^{94} + 3 q^{95} - 25 q^{97} + 2 q^{98}+O(q^{100}) \) 12 * q + 12 * q^2 + 12 * q^4 - 5 * q^5 - 6 * q^7 + 12 * q^8 - 5 * q^10 - 10 * q^11 - q^13 - 6 * q^14 + 12 * q^16 - 6 * q^17 - 10 * q^19 - 5 * q^20 - 10 * q^22 - 15 * q^23 + 7 * q^25 - q^26 - 6 * q^28 - 33 * q^29 - 6 * q^31 + 12 * q^32 - 6 * q^34 - 16 * q^35 - 13 * q^37 - 10 * q^38 - 5 * q^40 - 20 * q^41 - 11 * q^43 - 10 * q^44 - 15 * q^46 - 15 * q^47 + 2 * q^49 + 7 * q^50 - q^52 - 4 * q^53 - 17 * q^55 - 6 * q^56 - 33 * q^58 - 10 * q^59 - 12 * q^61 - 6 * q^62 + 12 * q^64 - 40 * q^65 - 19 * q^67 - 6 * q^68 - 16 * q^70 - 47 * q^71 - 2 * q^73 - 13 * q^74 - 10 * q^76 + 6 * q^77 - 15 * q^79 - 5 * q^80 - 20 * q^82 - 18 * q^83 - 25 * q^85 - 11 * q^86 - 10 * q^88 - 24 * q^89 - 3 * q^91 - 15 * q^92 - 15 * q^94 + 3 * q^95 - 25 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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