LMFDB - Embedded newform 2646.2.a.bo.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2646,2,Mod(1,2646)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2646, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2646.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$2646 = 2 \cdot 3^{3} \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2646.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2,0,2,2,0,0,2,0,2,2,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$21.1284163748$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - 7$ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 378)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.64575$ of defining polynomial
Character	$\chi$	$=$	2646.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.64575 q^{5} +1.00000 q^{8} -1.64575 q^{10} -1.64575 q^{11} -0.645751 q^{13} +1.00000 q^{16} +1.64575 q^{17} -2.00000 q^{19} -1.64575 q^{20} -1.64575 q^{22} +9.29150 q^{23} -2.29150 q^{25} -0.645751 q^{26} +7.64575 q^{29} -0.645751 q^{31} +1.00000 q^{32} +1.64575 q^{34} +3.93725 q^{37} -2.00000 q^{38} -1.64575 q^{40} +4.93725 q^{41} +5.00000 q^{43} -1.64575 q^{44} +9.29150 q^{46} +10.9373 q^{47} -2.29150 q^{50} -0.645751 q^{52} +6.00000 q^{53} +2.70850 q^{55} +7.64575 q^{58} -13.6458 q^{59} -12.6458 q^{61} -0.645751 q^{62} +1.00000 q^{64} +1.06275 q^{65} +8.29150 q^{67} +1.64575 q^{68} +10.3542 q^{71} +10.5830 q^{73} +3.93725 q^{74} -2.00000 q^{76} -15.2288 q^{79} -1.64575 q^{80} +4.93725 q^{82} +2.70850 q^{83} -2.70850 q^{85} +5.00000 q^{86} -1.64575 q^{88} +10.9373 q^{89} +9.29150 q^{92} +10.9373 q^{94} +3.29150 q^{95} +7.58301 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8} + 2 q^{10} + 2 q^{11} + 4 q^{13} + 2 q^{16} - 2 q^{17} - 4 q^{19} + 2 q^{20} + 2 q^{22} + 8 q^{23} + 6 q^{25} + 4 q^{26} + 10 q^{29} + 4 q^{31} + 2 q^{32} - 2 q^{34}+ \cdots - 6 q^{97}+O(q^{100})$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^8 + 2 * q^10 + 2 * q^11 + 4 * q^13 + 2 * q^16 - 2 * q^17 - 4 * q^19 + 2 * q^20 + 2 * q^22 + 8 * q^23 + 6 * q^25 + 4 * q^26 + 10 * q^29 + 4 * q^31 + 2 * q^32 - 2 * q^34 - 8 * q^37 - 4 * q^38 + 2 * q^40 - 6 * q^41 + 10 * q^43 + 2 * q^44 + 8 * q^46 + 6 * q^47 + 6 * q^50 + 4 * q^52 + 12 * q^53 + 16 * q^55 + 10 * q^58 - 22 * q^59 - 20 * q^61 + 4 * q^62 + 2 * q^64 + 18 * q^65 + 6 * q^67 - 2 * q^68 + 26 * q^71 - 8 * q^74 - 4 * q^76 - 4 * q^79 + 2 * q^80 - 6 * q^82 + 16 * q^83 - 16 * q^85 + 10 * q^86 + 2 * q^88 + 6 * q^89 + 8 * q^92 + 6 * q^94 - 4 * q^95 - 6 * q^97

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
$5$	−1.64575	−0.736002	−0.368001	−	0.929825i	$-0.619958\pi$
$5$	−1.64575	−0.736002	−0.368001	+	0.929825i	$0.619958\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.64575	−0.520432
$11$	−1.64575	−0.496213	−0.248106	−	0.968733i	$-0.579808\pi$
$11$	−1.64575	−0.496213	−0.248106	+	0.968733i	$0.579808\pi$
$12$	0	0
$13$	−0.645751	−0.179099	−0.0895496	−	0.995982i	$-0.528543\pi$
$13$	−0.645751	−0.179099	−0.0895496	+	0.995982i	$0.528543\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	1.64575	0.399153	0.199577	−	0.979882i	$-0.436043\pi$
$17$	1.64575	0.399153	0.199577	+	0.979882i	$0.436043\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	−1.64575	−0.368001
$21$	0	0
$22$	−1.64575	−0.350875
$23$	9.29150	1.93741	0.968706	−	0.248211i	$-0.0798425\pi$
$23$	9.29150	1.93741	0.968706	+	0.248211i	$0.0798425\pi$
$24$	0	0
$25$	−2.29150	−0.458301
$26$	−0.645751	−0.126642
$27$	0	0
$28$	0	0
$29$	7.64575	1.41978	0.709890	−	0.704312i	$-0.248745\pi$
$29$	7.64575	1.41978	0.709890	+	0.704312i	$0.248745\pi$
$30$	0	0
$31$	−0.645751	−0.115980	−0.0579902	−	0.998317i	$-0.518469\pi$
$31$	−0.645751	−0.115980	−0.0579902	+	0.998317i	$0.518469\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.64575	0.282244
$35$	0	0
$36$	0	0
$37$	3.93725	0.647281	0.323640	−	0.946180i	$-0.395093\pi$
$37$	3.93725	0.647281	0.323640	+	0.946180i	$0.395093\pi$
$38$	−2.00000	−0.324443
$39$	0	0
$40$	−1.64575	−0.260216
$41$	4.93725	0.771070	0.385535	−	0.922693i	$-0.374017\pi$
$41$	4.93725	0.771070	0.385535	+	0.922693i	$0.374017\pi$
$42$	0	0
$43$	5.00000	0.762493	0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000	0.762493	0.381246	+	0.924473i	$0.375495\pi$
$44$	−1.64575	−0.248106
$45$	0	0
$46$	9.29150	1.36996
$47$	10.9373	1.59536	0.797681	−	0.603079i	$-0.206060\pi$
$47$	10.9373	1.59536	0.797681	+	0.603079i	$0.206060\pi$
$48$	0	0
$49$	0	0
$50$	−2.29150	−0.324067
$51$	0	0
$52$	−0.645751	−0.0895496
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	2.70850	0.365214
$56$	0	0
$57$	0	0
$58$	7.64575	1.00394
$59$	−13.6458	−1.77653	−0.888263	−	0.459336i	$-0.848088\pi$
$59$	−13.6458	−1.77653	−0.888263	+	0.459336i	$0.848088\pi$
$60$	0	0
$61$	−12.6458	−1.61912	−0.809561	−	0.587035i	$-0.800295\pi$
$61$	−12.6458	−1.61912	−0.809561	+	0.587035i	$0.800295\pi$
$62$	−0.645751	−0.0820105
$63$	0	0
$64$	1.00000	0.125000
$65$	1.06275	0.131817
$66$	0	0
$67$	8.29150	1.01297	0.506484	−	0.862249i	$-0.330945\pi$
$67$	8.29150	1.01297	0.506484	+	0.862249i	$0.330945\pi$
$68$	1.64575	0.199577
$69$	0	0
$70$	0	0
$71$	10.3542	1.22882	0.614412	−	0.788986i	$-0.289393\pi$
$71$	10.3542	1.22882	0.614412	+	0.788986i	$0.289393\pi$
$72$	0	0
$73$	10.5830	1.23865	0.619324	−	0.785136i	$-0.287407\pi$
$73$	10.5830	1.23865	0.619324	+	0.785136i	$0.287407\pi$
$74$	3.93725	0.457696
$75$	0	0
$76$	−2.00000	−0.229416
$77$	0	0
$78$	0	0
$79$	−15.2288	−1.71337	−0.856684	−	0.515841i	$-0.827480\pi$
$79$	−15.2288	−1.71337	−0.856684	+	0.515841i	$0.827480\pi$
$80$	−1.64575	−0.184001
$81$	0	0
$82$	4.93725	0.545228
$83$	2.70850	0.297296	0.148648	−	0.988890i	$-0.452508\pi$
$83$	2.70850	0.297296	0.148648	+	0.988890i	$0.452508\pi$
$84$	0	0
$85$	−2.70850	−0.293778
$86$	5.00000	0.539164
$87$	0	0
$88$	−1.64575	−0.175438
$89$	10.9373	1.15935	0.579673	−	0.814849i	$-0.303180\pi$
$89$	10.9373	1.15935	0.579673	+	0.814849i	$0.303180\pi$
$90$	0	0
$91$	0	0
$92$	9.29150	0.968706
$93$	0	0
$94$	10.9373	1.12809
$95$	3.29150	0.337701
$96$	0	0
$97$	7.58301	0.769938	0.384969	−	0.922930i	$-0.374212\pi$
$97$	7.58301	0.769938	0.384969	+	0.922930i	$0.374212\pi$
$98$	0	0
$99$	0	0
$100$	−2.29150	−0.229150
$101$	−13.6458	−1.35780	−0.678902	−	0.734229i	$-0.737544\pi$
$101$	−13.6458	−1.35780	−0.678902	+	0.734229i	$0.737544\pi$
$102$	0	0
$103$	11.9373	1.17621	0.588106	−	0.808784i	$-0.299874\pi$
$103$	11.9373	1.17621	0.588106	+	0.808784i	$0.299874\pi$
$104$	−0.645751	−0.0633211
$105$	0	0
$106$	6.00000	0.582772
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	−8.64575	−0.828113	−0.414056	−	0.910251i	$-0.635888\pi$
$109$	−8.64575	−0.828113	−0.414056	+	0.910251i	$0.635888\pi$
$110$	2.70850	0.258245
$111$	0	0
$112$	0	0
$113$	7.64575	0.719252	0.359626	−	0.933097i	$-0.382904\pi$
$113$	7.64575	0.719252	0.359626	+	0.933097i	$0.382904\pi$
$114$	0	0
$115$	−15.2915	−1.42594
$116$	7.64575	0.709890
$117$	0	0
$118$	−13.6458	−1.25619
$119$	0	0
$120$	0	0
$121$	−8.29150	−0.753773
$122$	−12.6458	−1.14489
$123$	0	0
$124$	−0.645751	−0.0579902
$125$	12.0000	1.07331
$126$	0	0
$127$	6.64575	0.589715	0.294858	−	0.955541i	$-0.404728\pi$
$127$	6.64575	0.589715	0.294858	+	0.955541i	$0.404728\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	1.06275	0.0932090
$131$	−6.58301	−0.575160	−0.287580	−	0.957757i	$-0.592851\pi$
$131$	−6.58301	−0.575160	−0.287580	+	0.957757i	$0.592851\pi$
$132$	0	0
$133$	0	0
$134$	8.29150	0.716277
$135$	0	0
$136$	1.64575	0.141122
$137$	−9.29150	−0.793827	−0.396913	−	0.917856i	$-0.629919\pi$
$137$	−9.29150	−0.793827	−0.396913	+	0.917856i	$0.629919\pi$
$138$	0	0
$139$	19.5830	1.66101	0.830504	−	0.557012i	$-0.188052\pi$
$139$	19.5830	1.66101	0.830504	+	0.557012i	$0.188052\pi$
$140$	0	0
$141$	0	0
$142$	10.3542	0.868909
$143$	1.06275	0.0888713
$144$	0	0
$145$	−12.5830	−1.04496
$146$	10.5830	0.875856
$147$	0	0
$148$	3.93725	0.323640
$149$	7.06275	0.578603	0.289301	−	0.957238i	$-0.406577\pi$
$149$	7.06275	0.578603	0.289301	+	0.957238i	$0.406577\pi$
$150$	0	0
$151$	7.22876	0.588268	0.294134	−	0.955764i	$-0.404969\pi$
$151$	7.22876	0.588268	0.294134	+	0.955764i	$0.404969\pi$
$152$	−2.00000	−0.162221
$153$	0	0
$154$	0	0
$155$	1.06275	0.0853618
$156$	0	0
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	−15.2288	−1.21153
$159$	0	0
$160$	−1.64575	−0.130108
$161$	0	0
$162$	0	0
$163$	−1.00000	−0.0783260	−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000	−0.0783260	−0.0391630	+	0.999233i	$0.512469\pi$
$164$	4.93725	0.385535
$165$	0	0
$166$	2.70850	0.210220
$167$	−12.5830	−0.973702	−0.486851	−	0.873485i	$-0.661855\pi$
$167$	−12.5830	−0.973702	−0.486851	+	0.873485i	$0.661855\pi$
$168$	0	0
$169$	−12.5830	−0.967923
$170$	−2.70850	−0.207732
$171$	0	0
$172$	5.00000	0.381246
$173$	−6.58301	−0.500497	−0.250248	−	0.968182i	$-0.580512\pi$
$173$	−6.58301	−0.500497	−0.250248	+	0.968182i	$0.580512\pi$
$174$	0	0
$175$	0	0
$176$	−1.64575	−0.124053
$177$	0	0
$178$	10.9373	0.819782
$179$	1.06275	0.0794334	0.0397167	−	0.999211i	$-0.487354\pi$
$179$	1.06275	0.0794334	0.0397167	+	0.999211i	$0.487354\pi$
$180$	0	0
$181$	13.2915	0.987950	0.493975	−	0.869476i	$-0.335543\pi$
$181$	13.2915	0.987950	0.493975	+	0.869476i	$0.335543\pi$
$182$	0	0
$183$	0	0
$184$	9.29150	0.684979
$185$	−6.47974	−0.476400
$186$	0	0
$187$	−2.70850	−0.198065
$188$	10.9373	0.797681
$189$	0	0
$190$	3.29150	0.238791
$191$	26.8118	1.94003	0.970015	−	0.243043i	$-0.0781456\pi$
$191$	26.8118	1.94003	0.970015	+	0.243043i	$0.0781456\pi$
$192$	0	0
$193$	−7.00000	−0.503871	−0.251936	−	0.967744i	$-0.581067\pi$
$193$	−7.00000	−0.503871	−0.251936	+	0.967744i	$0.581067\pi$
$194$	7.58301	0.544428
$195$	0	0
$196$	0	0
$197$	15.2915	1.08947	0.544737	−	0.838607i	$-0.316629\pi$
$197$	15.2915	1.08947	0.544737	+	0.838607i	$0.316629\pi$
$198$	0	0
$199$	−21.9373	−1.55509	−0.777545	−	0.628827i	$-0.783535\pi$
$199$	−21.9373	−1.55509	−0.777545	+	0.628827i	$0.783535\pi$
$200$	−2.29150	−0.162034
$201$	0	0
$202$	−13.6458	−0.960112
$203$	0	0
$204$	0	0
$205$	−8.12549	−0.567509
$206$	11.9373	0.831708
$207$	0	0
$208$	−0.645751	−0.0447748
$209$	3.29150	0.227678
$210$	0	0
$211$	−16.8745	−1.16169	−0.580845	−	0.814015i	$-0.697278\pi$
$211$	−16.8745	−1.16169	−0.580845	+	0.814015i	$0.697278\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	−6.00000	−0.410152
$215$	−8.22876	−0.561197
$216$	0	0
$217$	0	0
$218$	−8.64575	−0.585564
$219$	0	0
$220$	2.70850	0.182607
$221$	−1.06275	−0.0714880
$222$	0	0
$223$	−17.8745	−1.19697	−0.598483	−	0.801136i	$-0.704230\pi$
$223$	−17.8745	−1.19697	−0.598483	+	0.801136i	$0.704230\pi$
$224$	0	0
$225$	0	0
$226$	7.64575	0.508588
$227$	−6.00000	−0.398234	−0.199117	−	0.979976i	$-0.563807\pi$
$227$	−6.00000	−0.398234	−0.199117	+	0.979976i	$0.563807\pi$
$228$	0	0
$229$	14.6458	0.967818	0.483909	−	0.875118i	$-0.339216\pi$
$229$	14.6458	0.967818	0.483909	+	0.875118i	$0.339216\pi$
$230$	−15.2915	−1.00829
$231$	0	0
$232$	7.64575	0.501968
$233$	−8.70850	−0.570513	−0.285256	−	0.958451i	$-0.592079\pi$
$233$	−8.70850	−0.570513	−0.285256	+	0.958451i	$0.592079\pi$
$234$	0	0
$235$	−18.0000	−1.17419
$236$	−13.6458	−0.888263
$237$	0	0
$238$	0	0
$239$	4.93725	0.319364	0.159682	−	0.987168i	$-0.448953\pi$
$239$	4.93725	0.319364	0.159682	+	0.987168i	$0.448953\pi$
$240$	0	0
$241$	−5.00000	−0.322078	−0.161039	−	0.986948i	$-0.551485\pi$
$241$	−5.00000	−0.322078	−0.161039	+	0.986948i	$0.551485\pi$
$242$	−8.29150	−0.532998
$243$	0	0
$244$	−12.6458	−0.809561
$245$	0	0
$246$	0	0
$247$	1.29150	0.0821763
$248$	−0.645751	−0.0410052
$249$	0	0
$250$	12.0000	0.758947
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	0	0
$253$	−15.2915	−0.961369
$254$	6.64575	0.416992
$255$	0	0
$256$	1.00000	0.0625000
$257$	15.8745	0.990225	0.495112	−	0.868829i	$-0.335127\pi$
$257$	15.8745	0.990225	0.495112	+	0.868829i	$0.335127\pi$
$258$	0	0
$259$	0	0
$260$	1.06275	0.0659087
$261$	0	0
$262$	−6.58301	−0.406699
$263$	10.9373	0.674420	0.337210	−	0.941429i	$-0.390517\pi$
$263$	10.9373	0.674420	0.337210	+	0.941429i	$0.390517\pi$
$264$	0	0
$265$	−9.87451	−0.606586
$266$	0	0
$267$	0	0
$268$	8.29150	0.506484
$269$	27.2915	1.66399	0.831996	−	0.554781i	$-0.187198\pi$
$269$	27.2915	1.66399	0.831996	+	0.554781i	$0.187198\pi$
$270$	0	0
$271$	21.2288	1.28956	0.644778	−	0.764370i	$-0.276950\pi$
$271$	21.2288	1.28956	0.644778	+	0.764370i	$0.276950\pi$
$272$	1.64575	0.0997883
$273$	0	0
$274$	−9.29150	−0.561320
$275$	3.77124	0.227415
$276$	0	0
$277$	−17.9373	−1.07775	−0.538873	−	0.842387i	$-0.681150\pi$
$277$	−17.9373	−1.07775	−0.538873	+	0.842387i	$0.681150\pi$
$278$	19.5830	1.17451
$279$	0	0
$280$	0	0
$281$	−17.5203	−1.04517	−0.522586	−	0.852587i	$-0.675032\pi$
$281$	−17.5203	−1.04517	−0.522586	+	0.852587i	$0.675032\pi$
$282$	0	0
$283$	−8.29150	−0.492879	−0.246439	−	0.969158i	$-0.579261\pi$
$283$	−8.29150	−0.492879	−0.246439	+	0.969158i	$0.579261\pi$
$284$	10.3542	0.614412
$285$	0	0
$286$	1.06275	0.0628415
$287$	0	0
$288$	0	0
$289$	−14.2915	−0.840677
$290$	−12.5830	−0.738900
$291$	0	0
$292$	10.5830	0.619324
$293$	−22.9373	−1.34001	−0.670004	−	0.742357i	$-0.733708\pi$
$293$	−22.9373	−1.34001	−0.670004	+	0.742357i	$0.733708\pi$
$294$	0	0
$295$	22.4575	1.30753
$296$	3.93725	0.228848
$297$	0	0
$298$	7.06275	0.409134
$299$	−6.00000	−0.346989
$300$	0	0
$301$	0	0
$302$	7.22876	0.415968
$303$	0	0
$304$	−2.00000	−0.114708
$305$	20.8118	1.19168
$306$	0	0
$307$	13.5830	0.775223	0.387612	−	0.921823i	$-0.373300\pi$
$307$	13.5830	0.775223	0.387612	+	0.921823i	$0.373300\pi$
$308$	0	0
$309$	0	0
$310$	1.06275	0.0603599
$311$	−23.5203	−1.33371	−0.666856	−	0.745187i	$-0.732360\pi$
$311$	−23.5203	−1.33371	−0.666856	+	0.745187i	$0.732360\pi$
$312$	0	0
$313$	−23.2915	−1.31651	−0.658257	−	0.752793i	$-0.728706\pi$
$313$	−23.2915	−1.31651	−0.658257	+	0.752793i	$0.728706\pi$
$314$	4.00000	0.225733
$315$	0	0
$316$	−15.2288	−0.856684
$317$	−20.8118	−1.16890	−0.584452	−	0.811428i	$-0.698691\pi$
$317$	−20.8118	−1.16890	−0.584452	+	0.811428i	$0.698691\pi$
$318$	0	0
$319$	−12.5830	−0.704513
$320$	−1.64575	−0.0920003
$321$	0	0
$322$	0	0
$323$	−3.29150	−0.183144
$324$	0	0
$325$	1.47974	0.0820813
$326$	−1.00000	−0.0553849
$327$	0	0
$328$	4.93725	0.272614
$329$	0	0
$330$	0	0
$331$	−0.125492	−0.00689767	−0.00344884	−	0.999994i	$-0.501098\pi$
$331$	−0.125492	−0.00689767	−0.00344884	+	0.999994i	$0.501098\pi$
$332$	2.70850	0.148648
$333$	0	0
$334$	−12.5830	−0.688511
$335$	−13.6458	−0.745547
$336$	0	0
$337$	−9.41699	−0.512976	−0.256488	−	0.966547i	$-0.582565\pi$
$337$	−9.41699	−0.512976	−0.256488	+	0.966547i	$0.582565\pi$
$338$	−12.5830	−0.684425
$339$	0	0
$340$	−2.70850	−0.146889
$341$	1.06275	0.0575509
$342$	0	0
$343$	0	0
$344$	5.00000	0.269582
$345$	0	0
$346$	−6.58301	−0.353905
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	−25.2288	−1.35046	−0.675232	−	0.737605i	$-0.735957\pi$
$349$	−25.2288	−1.35046	−0.675232	+	0.737605i	$0.735957\pi$
$350$	0	0
$351$	0	0
$352$	−1.64575	−0.0877188
$353$	12.0000	0.638696	0.319348	−	0.947638i	$-0.396536\pi$
$353$	12.0000	0.638696	0.319348	+	0.947638i	$0.396536\pi$
$354$	0	0
$355$	−17.0405	−0.904417
$356$	10.9373	0.579673
$357$	0	0
$358$	1.06275	0.0561679
$359$	−31.1660	−1.64488	−0.822440	−	0.568853i	$-0.807388\pi$
$359$	−31.1660	−1.64488	−0.822440	+	0.568853i	$0.807388\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	13.2915	0.698586
$363$	0	0
$364$	0	0
$365$	−17.4170	−0.911647
$366$	0	0
$367$	1.87451	0.0978485	0.0489243	−	0.998802i	$-0.484421\pi$
$367$	1.87451	0.0978485	0.0489243	+	0.998802i	$0.484421\pi$
$368$	9.29150	0.484353
$369$	0	0
$370$	−6.47974	−0.336866
$371$	0	0
$372$	0	0
$373$	−16.5830	−0.858635	−0.429318	−	0.903154i	$-0.641246\pi$
$373$	−16.5830	−0.858635	−0.429318	+	0.903154i	$0.641246\pi$
$374$	−2.70850	−0.140053
$375$	0	0
$376$	10.9373	0.564046
$377$	−4.93725	−0.254282
$378$	0	0
$379$	4.41699	0.226886	0.113443	−	0.993545i	$-0.463812\pi$
$379$	4.41699	0.226886	0.113443	+	0.993545i	$0.463812\pi$
$380$	3.29150	0.168851
$381$	0	0
$382$	26.8118	1.37181
$383$	0.583005	0.0297902	0.0148951	−	0.999889i	$-0.495259\pi$
$383$	0.583005	0.0297902	0.0148951	+	0.999889i	$0.495259\pi$
$384$	0	0
$385$	0	0
$386$	−7.00000	−0.356291
$387$	0	0
$388$	7.58301	0.384969
$389$	8.70850	0.441538	0.220769	−	0.975326i	$-0.429143\pi$
$389$	8.70850	0.441538	0.220769	+	0.975326i	$0.429143\pi$
$390$	0	0
$391$	15.2915	0.773325
$392$	0	0
$393$	0	0
$394$	15.2915	0.770375
$395$	25.0627	1.26104
$396$	0	0
$397$	11.3542	0.569853	0.284927	−	0.958549i	$-0.408031\pi$
$397$	11.3542	0.569853	0.284927	+	0.958549i	$0.408031\pi$
$398$	−21.9373	−1.09962
$399$	0	0
$400$	−2.29150	−0.114575
$401$	26.8118	1.33892	0.669458	−	0.742850i	$-0.266527\pi$
$401$	26.8118	1.33892	0.669458	+	0.742850i	$0.266527\pi$
$402$	0	0
$403$	0.416995	0.0207720
$404$	−13.6458	−0.678902
$405$	0	0
$406$	0	0
$407$	−6.47974	−0.321189
$408$	0	0
$409$	16.8745	0.834391	0.417195	−	0.908817i	$-0.363013\pi$
$409$	16.8745	0.834391	0.417195	+	0.908817i	$0.363013\pi$
$410$	−8.12549	−0.401289
$411$	0	0
$412$	11.9373	0.588106
$413$	0	0
$414$	0	0
$415$	−4.45751	−0.218811
$416$	−0.645751	−0.0316606
$417$	0	0
$418$	3.29150	0.160993
$419$	−13.0627	−0.638157	−0.319078	−	0.947728i	$-0.603373\pi$
$419$	−13.0627	−0.638157	−0.319078	+	0.947728i	$0.603373\pi$
$420$	0	0
$421$	−25.2915	−1.23263	−0.616316	−	0.787499i	$-0.711376\pi$
$421$	−25.2915	−1.23263	−0.616316	+	0.787499i	$0.711376\pi$
$422$	−16.8745	−0.821438
$423$	0	0
$424$	6.00000	0.291386
$425$	−3.77124	−0.182932
$426$	0	0
$427$	0	0
$428$	−6.00000	−0.290021
$429$	0	0
$430$	−8.22876	−0.396826
$431$	−8.81176	−0.424448	−0.212224	−	0.977221i	$-0.568071\pi$
$431$	−8.81176	−0.424448	−0.212224	+	0.977221i	$0.568071\pi$
$432$	0	0
$433$	19.0000	0.913082	0.456541	−	0.889702i	$-0.349088\pi$
$433$	19.0000	0.913082	0.456541	+	0.889702i	$0.349088\pi$
$434$	0	0
$435$	0	0
$436$	−8.64575	−0.414056
$437$	−18.5830	−0.888946
$438$	0	0
$439$	17.1660	0.819289	0.409644	−	0.912245i	$-0.365653\pi$
$439$	17.1660	0.819289	0.409644	+	0.912245i	$0.365653\pi$
$440$	2.70850	0.129123
$441$	0	0
$442$	−1.06275	−0.0505497
$443$	−8.70850	−0.413753	−0.206877	−	0.978367i	$-0.566330\pi$
$443$	−8.70850	−0.413753	−0.206877	+	0.978367i	$0.566330\pi$
$444$	0	0
$445$	−18.0000	−0.853282
$446$	−17.8745	−0.846382
$447$	0	0
$448$	0	0
$449$	7.64575	0.360825	0.180413	−	0.983591i	$-0.442257\pi$
$449$	7.64575	0.360825	0.180413	+	0.983591i	$0.442257\pi$
$450$	0	0
$451$	−8.12549	−0.382614
$452$	7.64575	0.359626
$453$	0	0
$454$	−6.00000	−0.281594
$455$	0	0
$456$	0	0
$457$	2.29150	0.107192	0.0535960	−	0.998563i	$-0.482932\pi$
$457$	2.29150	0.107192	0.0535960	+	0.998563i	$0.482932\pi$
$458$	14.6458	0.684351
$459$	0	0
$460$	−15.2915	−0.712970
$461$	2.22876	0.103804	0.0519018	−	0.998652i	$-0.483472\pi$
$461$	2.22876	0.103804	0.0519018	+	0.998652i	$0.483472\pi$
$462$	0	0
$463$	29.2915	1.36129	0.680646	−	0.732613i	$-0.261699\pi$
$463$	29.2915	1.36129	0.680646	+	0.732613i	$0.261699\pi$
$464$	7.64575	0.354945
$465$	0	0
$466$	−8.70850	−0.403413
$467$	26.2288	1.21372	0.606861	−	0.794808i	$-0.292428\pi$
$467$	26.2288	1.21372	0.606861	+	0.794808i	$0.292428\pi$
$468$	0	0
$469$	0	0
$470$	−18.0000	−0.830278
$471$	0	0
$472$	−13.6458	−0.628097
$473$	−8.22876	−0.378359
$474$	0	0
$475$	4.58301	0.210283
$476$	0	0
$477$	0	0
$478$	4.93725	0.225825
$479$	1.64575	0.0751963	0.0375981	−	0.999293i	$-0.488029\pi$
$479$	1.64575	0.0751963	0.0375981	+	0.999293i	$0.488029\pi$
$480$	0	0
$481$	−2.54249	−0.115927
$482$	−5.00000	−0.227744
$483$	0	0
$484$	−8.29150	−0.376886
$485$	−12.4797	−0.566676
$486$	0	0
$487$	−7.87451	−0.356828	−0.178414	−	0.983956i	$-0.557097\pi$
$487$	−7.87451	−0.356828	−0.178414	+	0.983956i	$0.557097\pi$
$488$	−12.6458	−0.572446
$489$	0	0
$490$	0	0
$491$	−37.7490	−1.70359	−0.851795	−	0.523876i	$-0.824486\pi$
$491$	−37.7490	−1.70359	−0.851795	+	0.523876i	$0.824486\pi$
$492$	0	0
$493$	12.5830	0.566710
$494$	1.29150	0.0581075
$495$	0	0
$496$	−0.645751	−0.0289951
$497$	0	0
$498$	0	0
$499$	36.1660	1.61901	0.809506	−	0.587111i	$-0.199735\pi$
$499$	36.1660	1.61901	0.809506	+	0.587111i	$0.199735\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	18.0000	0.803379
$503$	27.8745	1.24286	0.621431	−	0.783469i	$-0.286551\pi$
$503$	27.8745	1.24286	0.621431	+	0.783469i	$0.286551\pi$
$504$	0	0
$505$	22.4575	0.999346
$506$	−15.2915	−0.679790
$507$	0	0
$508$	6.64575	0.294858
$509$	−39.8745	−1.76741	−0.883703	−	0.468048i	$-0.844958\pi$
$509$	−39.8745	−1.76741	−0.883703	+	0.468048i	$0.844958\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	15.8745	0.700195
$515$	−19.6458	−0.865695
$516$	0	0
$517$	−18.0000	−0.791639
$518$	0	0
$519$	0	0
$520$	1.06275	0.0466045
$521$	39.8745	1.74693	0.873467	−	0.486883i	$-0.161866\pi$
$521$	39.8745	1.74693	0.873467	+	0.486883i	$0.161866\pi$
$522$	0	0
$523$	1.00000	0.0437269	0.0218635	−	0.999761i	$-0.493040\pi$
$523$	1.00000	0.0437269	0.0218635	+	0.999761i	$0.493040\pi$
$524$	−6.58301	−0.287580
$525$	0	0
$526$	10.9373	0.476887
$527$	−1.06275	−0.0462939
$528$	0	0
$529$	63.3320	2.75357
$530$	−9.87451	−0.428921
$531$	0	0
$532$	0	0
$533$	−3.18824	−0.138098
$534$	0	0
$535$	9.87451	0.426912
$536$	8.29150	0.358138
$537$	0	0
$538$	27.2915	1.17662
$539$	0	0
$540$	0	0
$541$	−41.1660	−1.76987	−0.884933	−	0.465719i	$-0.845796\pi$
$541$	−41.1660	−1.76987	−0.884933	+	0.465719i	$0.845796\pi$
$542$	21.2288	0.911853
$543$	0	0
$544$	1.64575	0.0705610
$545$	14.2288	0.609493
$546$	0	0
$547$	7.70850	0.329592	0.164796	−	0.986328i	$-0.447303\pi$
$547$	7.70850	0.329592	0.164796	+	0.986328i	$0.447303\pi$
$548$	−9.29150	−0.396913
$549$	0	0
$550$	3.77124	0.160806
$551$	−15.2915	−0.651440
$552$	0	0
$553$	0	0
$554$	−17.9373	−0.762081
$555$	0	0
$556$	19.5830	0.830504
$557$	−32.2288	−1.36558	−0.682788	−	0.730616i	$-0.739233\pi$
$557$	−32.2288	−1.36558	−0.682788	+	0.730616i	$0.739233\pi$
$558$	0	0
$559$	−3.22876	−0.136562
$560$	0	0
$561$	0	0
$562$	−17.5203	−0.739048
$563$	34.4575	1.45221	0.726106	−	0.687583i	$-0.241328\pi$
$563$	34.4575	1.45221	0.726106	+	0.687583i	$0.241328\pi$
$564$	0	0
$565$	−12.5830	−0.529371
$566$	−8.29150	−0.348518
$567$	0	0
$568$	10.3542	0.434455
$569$	−1.06275	−0.0445526	−0.0222763	−	0.999752i	$-0.507091\pi$
$569$	−1.06275	−0.0445526	−0.0222763	+	0.999752i	$0.507091\pi$
$570$	0	0
$571$	−1.29150	−0.0540477	−0.0270239	−	0.999635i	$-0.508603\pi$
$571$	−1.29150	−0.0540477	−0.0270239	+	0.999635i	$0.508603\pi$
$572$	1.06275	0.0444356
$573$	0	0
$574$	0	0
$575$	−21.2915	−0.887917
$576$	0	0
$577$	21.7085	0.903737	0.451868	−	0.892085i	$-0.350758\pi$
$577$	21.7085	0.903737	0.451868	+	0.892085i	$0.350758\pi$
$578$	−14.2915	−0.594448
$579$	0	0
$580$	−12.5830	−0.522481
$581$	0	0
$582$	0	0
$583$	−9.87451	−0.408960
$584$	10.5830	0.437928
$585$	0	0
$586$	−22.9373	−0.947529
$587$	−38.2288	−1.57787	−0.788935	−	0.614477i	$-0.789367\pi$
$587$	−38.2288	−1.57787	−0.788935	+	0.614477i	$0.789367\pi$
$588$	0	0
$589$	1.29150	0.0532154
$590$	22.4575	0.924561
$591$	0	0
$592$	3.93725	0.161820
$593$	−25.0627	−1.02920	−0.514602	−	0.857429i	$-0.672060\pi$
$593$	−25.0627	−1.02920	−0.514602	+	0.857429i	$0.672060\pi$
$594$	0	0
$595$	0	0
$596$	7.06275	0.289301
$597$	0	0
$598$	−6.00000	−0.245358
$599$	−21.8745	−0.893768	−0.446884	−	0.894592i	$-0.647466\pi$
$599$	−21.8745	−0.893768	−0.446884	+	0.894592i	$0.647466\pi$
$600$	0	0
$601$	4.87451	0.198835	0.0994177	−	0.995046i	$-0.468302\pi$
$601$	4.87451	0.198835	0.0994177	+	0.995046i	$0.468302\pi$
$602$	0	0
$603$	0	0
$604$	7.22876	0.294134
$605$	13.6458	0.554779
$606$	0	0
$607$	−26.5830	−1.07897	−0.539485	−	0.841995i	$-0.681381\pi$
$607$	−26.5830	−1.07897	−0.539485	+	0.841995i	$0.681381\pi$
$608$	−2.00000	−0.0811107
$609$	0	0
$610$	20.8118	0.842644
$611$	−7.06275	−0.285728
$612$	0	0
$613$	26.3948	1.06607	0.533037	−	0.846092i	$-0.321051\pi$
$613$	26.3948	1.06607	0.533037	+	0.846092i	$0.321051\pi$
$614$	13.5830	0.548165
$615$	0	0
$616$	0	0
$617$	−35.5203	−1.42999	−0.714996	−	0.699129i	$-0.753571\pi$
$617$	−35.5203	−1.42999	−0.714996	+	0.699129i	$0.753571\pi$
$618$	0	0
$619$	−2.29150	−0.0921033	−0.0460516	−	0.998939i	$-0.514664\pi$
$619$	−2.29150	−0.0921033	−0.0460516	+	0.998939i	$0.514664\pi$
$620$	1.06275	0.0426809
$621$	0	0
$622$	−23.5203	−0.943076
$623$	0	0
$624$	0	0
$625$	−8.29150	−0.331660
$626$	−23.2915	−0.930916
$627$	0	0
$628$	4.00000	0.159617
$629$	6.47974	0.258364
$630$	0	0
$631$	21.9373	0.873308	0.436654	−	0.899629i	$-0.356163\pi$
$631$	21.9373	0.873308	0.436654	+	0.899629i	$0.356163\pi$
$632$	−15.2288	−0.605767
$633$	0	0
$634$	−20.8118	−0.826541
$635$	−10.9373	−0.434032
$636$	0	0
$637$	0	0
$638$	−12.5830	−0.498166
$639$	0	0
$640$	−1.64575	−0.0650540
$641$	8.22876	0.325016	0.162508	−	0.986707i	$-0.448042\pi$
$641$	8.22876	0.325016	0.162508	+	0.986707i	$0.448042\pi$
$642$	0	0
$643$	34.8745	1.37532	0.687658	−	0.726035i	$-0.258639\pi$
$643$	34.8745	1.37532	0.687658	+	0.726035i	$0.258639\pi$
$644$	0	0
$645$	0	0
$646$	−3.29150	−0.129502
$647$	21.8745	0.859976	0.429988	−	0.902835i	$-0.358518\pi$
$647$	21.8745	0.859976	0.429988	+	0.902835i	$0.358518\pi$
$648$	0	0
$649$	22.4575	0.881534
$650$	1.47974	0.0580402
$651$	0	0
$652$	−1.00000	−0.0391630
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	0	0
$655$	10.8340	0.423319
$656$	4.93725	0.192767
$657$	0	0
$658$	0	0
$659$	2.22876	0.0868200	0.0434100	−	0.999057i	$-0.486178\pi$
$659$	2.22876	0.0868200	0.0434100	+	0.999057i	$0.486178\pi$
$660$	0	0
$661$	−39.1660	−1.52338	−0.761691	−	0.647941i	$-0.775630\pi$
$661$	−39.1660	−1.52338	−0.761691	+	0.647941i	$0.775630\pi$
$662$	−0.125492	−0.00487739
$663$	0	0
$664$	2.70850	0.105110
$665$	0	0
$666$	0	0
$667$	71.0405	2.75070
$668$	−12.5830	−0.486851
$669$	0	0
$670$	−13.6458	−0.527181
$671$	20.8118	0.803429
$672$	0	0
$673$	−47.7490	−1.84059	−0.920295	−	0.391226i	$-0.872051\pi$
$673$	−47.7490	−1.84059	−0.920295	+	0.391226i	$0.872051\pi$
$674$	−9.41699	−0.362729
$675$	0	0
$676$	−12.5830	−0.483962
$677$	14.1255	0.542887	0.271443	−	0.962454i	$-0.412499\pi$
$677$	14.1255	0.542887	0.271443	+	0.962454i	$0.412499\pi$
$678$	0	0
$679$	0	0
$680$	−2.70850	−0.103866
$681$	0	0
$682$	1.06275	0.0406947
$683$	20.8118	0.796340	0.398170	−	0.917312i	$-0.369645\pi$
$683$	20.8118	0.796340	0.398170	+	0.917312i	$0.369645\pi$
$684$	0	0
$685$	15.2915	0.584258
$686$	0	0
$687$	0	0
$688$	5.00000	0.190623
$689$	−3.87451	−0.147607
$690$	0	0
$691$	−24.7490	−0.941497	−0.470748	−	0.882267i	$-0.656016\pi$
$691$	−24.7490	−0.941497	−0.470748	+	0.882267i	$0.656016\pi$
$692$	−6.58301	−0.250248
$693$	0	0
$694$	−12.0000	−0.455514
$695$	−32.2288	−1.22251
$696$	0	0
$697$	8.12549	0.307775
$698$	−25.2288	−0.954923
$699$	0	0
$700$	0	0
$701$	−5.41699	−0.204597	−0.102299	−	0.994754i	$-0.532620\pi$
$701$	−5.41699	−0.204597	−0.102299	+	0.994754i	$0.532620\pi$
$702$	0	0
$703$	−7.87451	−0.296993
$704$	−1.64575	−0.0620266
$705$	0	0
$706$	12.0000	0.451626
$707$	0	0
$708$	0	0
$709$	−9.81176	−0.368488	−0.184244	−	0.982880i	$-0.558984\pi$
$709$	−9.81176	−0.368488	−0.184244	+	0.982880i	$0.558984\pi$
$710$	−17.0405	−0.639519
$711$	0	0
$712$	10.9373	0.409891
$713$	−6.00000	−0.224702
$714$	0	0
$715$	−1.74902	−0.0654095
$716$	1.06275	0.0397167
$717$	0	0
$718$	−31.1660	−1.16311
$719$	−7.06275	−0.263396	−0.131698	−	0.991290i	$-0.542043\pi$
$719$	−7.06275	−0.263396	−0.131698	+	0.991290i	$0.542043\pi$
$720$	0	0
$721$	0	0
$722$	−15.0000	−0.558242
$723$	0	0
$724$	13.2915	0.493975
$725$	−17.5203	−0.650686
$726$	0	0
$727$	−25.2288	−0.935683	−0.467841	−	0.883812i	$-0.654968\pi$
$727$	−25.2288	−0.935683	−0.467841	+	0.883812i	$0.654968\pi$
$728$	0	0
$729$	0	0
$730$	−17.4170	−0.644632
$731$	8.22876	0.304352
$732$	0	0
$733$	−8.77124	−0.323973	−0.161987	−	0.986793i	$-0.551790\pi$
$733$	−8.77124	−0.323973	−0.161987	+	0.986793i	$0.551790\pi$
$734$	1.87451	0.0691893
$735$	0	0
$736$	9.29150	0.342489
$737$	−13.6458	−0.502648
$738$	0	0
$739$	21.4575	0.789327	0.394664	−	0.918826i	$-0.370861\pi$
$739$	21.4575	0.789327	0.394664	+	0.918826i	$0.370861\pi$
$740$	−6.47974	−0.238200
$741$	0	0
$742$	0	0
$743$	−13.0627	−0.479226	−0.239613	−	0.970869i	$-0.577021\pi$
$743$	−13.0627	−0.479226	−0.239613	+	0.970869i	$0.577021\pi$
$744$	0	0
$745$	−11.6235	−0.425853
$746$	−16.5830	−0.607147
$747$	0	0
$748$	−2.70850	−0.0990325
$749$	0	0
$750$	0	0
$751$	0.457513	0.0166949	0.00834745	−	0.999965i	$-0.497343\pi$
$751$	0.457513	0.0166949	0.00834745	+	0.999965i	$0.497343\pi$
$752$	10.9373	0.398841
$753$	0	0
$754$	−4.93725	−0.179804
$755$	−11.8967	−0.432967
$756$	0	0
$757$	−40.9778	−1.48936	−0.744681	−	0.667420i	$-0.767398\pi$
$757$	−40.9778	−1.48936	−0.744681	+	0.667420i	$0.767398\pi$
$758$	4.41699	0.160432
$759$	0	0
$760$	3.29150	0.119395
$761$	18.5830	0.673633	0.336817	−	0.941570i	$-0.390650\pi$
$761$	18.5830	0.673633	0.336817	+	0.941570i	$0.390650\pi$
$762$	0	0
$763$	0	0
$764$	26.8118	0.970015
$765$	0	0
$766$	0.583005	0.0210648
$767$	8.81176	0.318174
$768$	0	0
$769$	−19.4170	−0.700195	−0.350097	−	0.936713i	$-0.613852\pi$
$769$	−19.4170	−0.700195	−0.350097	+	0.936713i	$0.613852\pi$
$770$	0	0
$771$	0	0
$772$	−7.00000	−0.251936
$773$	−38.3320	−1.37871	−0.689353	−	0.724425i	$-0.742105\pi$
$773$	−38.3320	−1.37871	−0.689353	+	0.724425i	$0.742105\pi$
$774$	0	0
$775$	1.47974	0.0531539
$776$	7.58301	0.272214
$777$	0	0
$778$	8.70850	0.312215
$779$	−9.87451	−0.353791
$780$	0	0
$781$	−17.0405	−0.609758
$782$	15.2915	0.546823
$783$	0	0
$784$	0	0
$785$	−6.58301	−0.234958
$786$	0	0
$787$	22.2915	0.794606	0.397303	−	0.917687i	$-0.369946\pi$
$787$	22.2915	0.794606	0.397303	+	0.917687i	$0.369946\pi$
$788$	15.2915	0.544737
$789$	0	0
$790$	25.0627	0.891692
$791$	0	0
$792$	0	0
$793$	8.16601	0.289984
$794$	11.3542	0.402947
$795$	0	0
$796$	−21.9373	−0.777545
$797$	−32.8118	−1.16225	−0.581126	−	0.813814i	$-0.697388\pi$
$797$	−32.8118	−1.16225	−0.581126	+	0.813814i	$0.697388\pi$
$798$	0	0
$799$	18.0000	0.636794
$800$	−2.29150	−0.0810169
$801$	0	0
$802$	26.8118	0.946756
$803$	−17.4170	−0.614632
$804$	0	0
$805$	0	0
$806$	0.416995	0.0146880
$807$	0	0
$808$	−13.6458	−0.480056
$809$	36.5830	1.28619	0.643095	−	0.765786i	$-0.277650\pi$
$809$	36.5830	1.28619	0.643095	+	0.765786i	$0.277650\pi$
$810$	0	0
$811$	18.7085	0.656944	0.328472	−	0.944514i	$-0.393466\pi$
$811$	18.7085	0.656944	0.328472	+	0.944514i	$0.393466\pi$
$812$	0	0
$813$	0	0
$814$	−6.47974	−0.227115
$815$	1.64575	0.0576482
$816$	0	0
$817$	−10.0000	−0.349856
$818$	16.8745	0.590003
$819$	0	0
$820$	−8.12549	−0.283754
$821$	−0.583005	−0.0203470	−0.0101735	−	0.999948i	$-0.503238\pi$
$821$	−0.583005	−0.0203470	−0.0101735	+	0.999948i	$0.503238\pi$
$822$	0	0
$823$	23.1033	0.805329	0.402665	−	0.915348i	$-0.368084\pi$
$823$	23.1033	0.805329	0.402665	+	0.915348i	$0.368084\pi$
$824$	11.9373	0.415854
$825$	0	0
$826$	0	0
$827$	19.7490	0.686741	0.343370	−	0.939200i	$-0.388431\pi$
$827$	19.7490	0.686741	0.343370	+	0.939200i	$0.388431\pi$
$828$	0	0
$829$	18.7085	0.649773	0.324886	−	0.945753i	$-0.394674\pi$
$829$	18.7085	0.649773	0.324886	+	0.945753i	$0.394674\pi$
$830$	−4.45751	−0.154723
$831$	0	0
$832$	−0.645751	−0.0223874
$833$	0	0
$834$	0	0
$835$	20.7085	0.716647
$836$	3.29150	0.113839
$837$	0	0
$838$	−13.0627	−0.451245
$839$	38.7085	1.33637	0.668183	−	0.743997i	$-0.267072\pi$
$839$	38.7085	1.33637	0.668183	+	0.743997i	$0.267072\pi$
$840$	0	0
$841$	29.4575	1.01578
$842$	−25.2915	−0.871603
$843$	0	0
$844$	−16.8745	−0.580845
$845$	20.7085	0.712394
$846$	0	0
$847$	0	0
$848$	6.00000	0.206041
$849$	0	0
$850$	−3.77124	−0.129353
$851$	36.5830	1.25405
$852$	0	0
$853$	−16.1255	−0.552126	−0.276063	−	0.961139i	$-0.589030\pi$
$853$	−16.1255	−0.552126	−0.276063	+	0.961139i	$0.589030\pi$
$854$	0	0
$855$	0	0
$856$	−6.00000	−0.205076
$857$	−20.8118	−0.710916	−0.355458	−	0.934692i	$-0.615675\pi$
$857$	−20.8118	−0.710916	−0.355458	+	0.934692i	$0.615675\pi$
$858$	0	0
$859$	7.00000	0.238837	0.119418	−	0.992844i	$-0.461897\pi$
$859$	7.00000	0.238837	0.119418	+	0.992844i	$0.461897\pi$
$860$	−8.22876	−0.280598
$861$	0	0
$862$	−8.81176	−0.300130
$863$	−13.1660	−0.448176	−0.224088	−	0.974569i	$-0.571940\pi$
$863$	−13.1660	−0.448176	−0.224088	+	0.974569i	$0.571940\pi$
$864$	0	0
$865$	10.8340	0.368367
$866$	19.0000	0.645646
$867$	0	0
$868$	0	0
$869$	25.0627	0.850195
$870$	0	0
$871$	−5.35425	−0.181422
$872$	−8.64575	−0.292782
$873$	0	0
$874$	−18.5830	−0.628580
$875$	0	0
$876$	0	0
$877$	21.3542	0.721082	0.360541	−	0.932743i	$-0.382592\pi$
$877$	21.3542	0.721082	0.360541	+	0.932743i	$0.382592\pi$
$878$	17.1660	0.579325
$879$	0	0
$880$	2.70850	0.0913034
$881$	−27.8745	−0.939116	−0.469558	−	0.882902i	$-0.655587\pi$
$881$	−27.8745	−0.939116	−0.469558	+	0.882902i	$0.655587\pi$
$882$	0	0
$883$	11.8745	0.399609	0.199805	−	0.979836i	$-0.435969\pi$
$883$	11.8745	0.399609	0.199805	+	0.979836i	$0.435969\pi$
$884$	−1.06275	−0.0357440
$885$	0	0
$886$	−8.70850	−0.292568
$887$	15.8745	0.533014	0.266507	−	0.963833i	$-0.414130\pi$
$887$	15.8745	0.533014	0.266507	+	0.963833i	$0.414130\pi$
$888$	0	0
$889$	0	0
$890$	−18.0000	−0.603361
$891$	0	0
$892$	−17.8745	−0.598483
$893$	−21.8745	−0.732002
$894$	0	0
$895$	−1.74902	−0.0584631
$896$	0	0
$897$	0	0
$898$	7.64575	0.255142
$899$	−4.93725	−0.164667
$900$	0	0
$901$	9.87451	0.328968
$902$	−8.12549	−0.270549
$903$	0	0
$904$	7.64575	0.254294
$905$	−21.8745	−0.727133
$906$	0	0
$907$	25.7085	0.853637	0.426818	−	0.904337i	$-0.359634\pi$
$907$	25.7085	0.853637	0.426818	+	0.904337i	$0.359634\pi$
$908$	−6.00000	−0.199117
$909$	0	0
$910$	0	0
$911$	13.0627	0.432788	0.216394	−	0.976306i	$-0.430570\pi$
$911$	13.0627	0.432788	0.216394	+	0.976306i	$0.430570\pi$
$912$	0	0
$913$	−4.45751	−0.147522
$914$	2.29150	0.0757962
$915$	0	0
$916$	14.6458	0.483909
$917$	0	0
$918$	0	0
$919$	55.2288	1.82183	0.910914	−	0.412596i	$-0.135378\pi$
$919$	55.2288	1.82183	0.910914	+	0.412596i	$0.135378\pi$
$920$	−15.2915	−0.504146
$921$	0	0
$922$	2.22876	0.0734002
$923$	−6.68627	−0.220081
$924$	0	0
$925$	−9.02223	−0.296649
$926$	29.2915	0.962579
$927$	0	0
$928$	7.64575	0.250984
$929$	1.06275	0.0348676	0.0174338	−	0.999848i	$-0.494450\pi$
$929$	1.06275	0.0348676	0.0174338	+	0.999848i	$0.494450\pi$
$930$	0	0
$931$	0	0
$932$	−8.70850	−0.285256
$933$	0	0
$934$	26.2288	0.858231
$935$	4.45751	0.145776
$936$	0	0
$937$	−18.7490	−0.612504	−0.306252	−	0.951951i	$-0.599075\pi$
$937$	−18.7490	−0.612504	−0.306252	+	0.951951i	$0.599075\pi$
$938$	0	0
$939$	0	0
$940$	−18.0000	−0.587095
$941$	−15.3948	−0.501855	−0.250928	−	0.968006i	$-0.580736\pi$
$941$	−15.3948	−0.501855	−0.250928	+	0.968006i	$0.580736\pi$
$942$	0	0
$943$	45.8745	1.49388
$944$	−13.6458	−0.444131
$945$	0	0
$946$	−8.22876	−0.267540
$947$	47.5203	1.54420	0.772100	−	0.635500i	$-0.219206\pi$
$947$	47.5203	1.54420	0.772100	+	0.635500i	$0.219206\pi$
$948$	0	0
$949$	−6.83399	−0.221841
$950$	4.58301	0.148692
$951$	0	0
$952$	0	0
$953$	−32.3320	−1.04734	−0.523668	−	0.851922i	$-0.675437\pi$
$953$	−32.3320	−1.04734	−0.523668	+	0.851922i	$0.675437\pi$
$954$	0	0
$955$	−44.1255	−1.42787
$956$	4.93725	0.159682
$957$	0	0
$958$	1.64575	0.0531718
$959$	0	0
$960$	0	0
$961$	−30.5830	−0.986549
$962$	−2.54249	−0.0819731
$963$	0	0
$964$	−5.00000	−0.161039
$965$	11.5203	0.370850
$966$	0	0
$967$	51.9373	1.67019	0.835095	−	0.550106i	$-0.185413\pi$
$967$	51.9373	1.67019	0.835095	+	0.550106i	$0.185413\pi$
$968$	−8.29150	−0.266499
$969$	0	0
$970$	−12.4797	−0.400700
$971$	−21.8745	−0.701986	−0.350993	−	0.936378i	$-0.614156\pi$
$971$	−21.8745	−0.701986	−0.350993	+	0.936378i	$0.614156\pi$
$972$	0	0
$973$	0	0
$974$	−7.87451	−0.252316
$975$	0	0
$976$	−12.6458	−0.404781
$977$	−20.1255	−0.643872	−0.321936	−	0.946762i	$-0.604334\pi$
$977$	−20.1255	−0.643872	−0.321936	+	0.946762i	$0.604334\pi$
$978$	0	0
$979$	−18.0000	−0.575282
$980$	0	0
$981$	0	0
$982$	−37.7490	−1.20462
$983$	26.8118	0.855162	0.427581	−	0.903977i	$-0.359366\pi$
$983$	26.8118	0.855162	0.427581	+	0.903977i	$0.359366\pi$
$984$	0	0
$985$	−25.1660	−0.801856
$986$	12.5830	0.400725
$987$	0	0
$988$	1.29150	0.0410882
$989$	46.4575	1.47726
$990$	0	0
$991$	−41.9373	−1.33218	−0.666090	−	0.745871i	$-0.732034\pi$
$991$	−41.9373	−1.33218	−0.666090	+	0.745871i	$0.732034\pi$
$992$	−0.645751	−0.0205026
$993$	0	0
$994$	0	0
$995$	36.1033	1.14455
$996$	0	0
$997$	55.6863	1.76360	0.881801	−	0.471622i	$-0.156331\pi$
$997$	55.6863	1.76360	0.881801	+	0.471622i	$0.156331\pi$
$998$	36.1660	1.14482
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2646.2.a.bo.1.1		2
3.2	odd	2		2646.2.a.bf.1.2		2
7.3	odd	6		378.2.g.g.163.1	yes	4
7.5	odd	6		378.2.g.g.109.1	✓	4
7.6	odd	2		2646.2.a.bl.1.2		2
21.5	even	6		378.2.g.h.109.2	yes	4
21.17	even	6		378.2.g.h.163.2	yes	4
21.20	even	2		2646.2.a.bi.1.1		2
63.5	even	6		1134.2.e.q.865.2		4
63.31	odd	6		1134.2.h.q.541.2		4
63.38	even	6		1134.2.e.q.919.2		4
63.40	odd	6		1134.2.e.t.865.1		4
63.47	even	6		1134.2.h.t.109.1		4
63.52	odd	6		1134.2.e.t.919.1		4
63.59	even	6		1134.2.h.t.541.1		4
63.61	odd	6		1134.2.h.q.109.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.g.g.109.1	✓	4	7.5	odd	6
378.2.g.g.163.1	yes	4	7.3	odd	6
378.2.g.h.109.2	yes	4	21.5	even	6
378.2.g.h.163.2	yes	4	21.17	even	6
1134.2.e.q.865.2		4	63.5	even	6
1134.2.e.q.919.2		4	63.38	even	6
1134.2.e.t.865.1		4	63.40	odd	6
1134.2.e.t.919.1		4	63.52	odd	6
1134.2.h.q.109.2		4	63.61	odd	6
1134.2.h.q.541.2		4	63.31	odd	6
1134.2.h.t.109.1		4	63.47	even	6
1134.2.h.t.541.1		4	63.59	even	6
2646.2.a.bf.1.2		2	3.2	odd	2
2646.2.a.bi.1.1		2	21.20	even	2
2646.2.a.bl.1.2		2	7.6	odd	2
2646.2.a.bo.1.1		2	1.1	even	1	trivial

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2646.2.a.bo.1.1

Level	$2646$
Weight	$2$
Character	2646.1
Self dual	yes
Analytic conductor	$21.128$
Analytic rank	$0$
Dimension	$2$
CM	no
Inner twists	$1$