LMFDB - Embedded newform 369.6.a.e.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [369,6,Mod(1,369)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(369, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("369.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 369 = 3^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	369.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:	$59.1816295110$
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} - 3 x^{9} - 259 x^{8} + 639 x^{7} + 22422 x^{6} - 38356 x^{5} - 735592 x^{4} + 422608 x^{3} + \cdots - 24923264 $ x^10 - 3x^9 - 259x^8 + 639x^7 + 22422x^6 - 38356x^5 - 735592x^4 + 422608x^3 + 8253728x^2 - 28352*x - 24923264
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 41)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-9.84411$ of defining polynomial
Character	$\chi$	$=$	369.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.84411 q^{2} +64.9065 q^{4} -85.4409 q^{5} -100.276 q^{7} +323.935 q^{8} +O(q^{10})$	\(q+9.84411 q^{2} +64.9065 q^{4} -85.4409 q^{5} -100.276 q^{7} +323.935 q^{8} -841.090 q^{10} +243.976 q^{11} +475.395 q^{13} -987.130 q^{14} +1111.85 q^{16} +1589.02 q^{17} +1884.44 q^{19} -5545.67 q^{20} +2401.72 q^{22} +361.895 q^{23} +4175.15 q^{25} +4679.85 q^{26} -6508.58 q^{28} +5037.18 q^{29} +4280.76 q^{31} +579.204 q^{32} +15642.4 q^{34} +8567.69 q^{35} +176.563 q^{37} +18550.7 q^{38} -27677.3 q^{40} -1681.00 q^{41} -1246.74 q^{43} +15835.6 q^{44} +3562.53 q^{46} -2735.89 q^{47} -6751.67 q^{49} +41100.6 q^{50} +30856.3 q^{52} +29387.9 q^{53} -20845.5 q^{55} -32483.0 q^{56} +49586.6 q^{58} +27355.6 q^{59} -10930.0 q^{61} +42140.2 q^{62} -29877.3 q^{64} -40618.2 q^{65} -1624.42 q^{67} +103137. q^{68} +84341.3 q^{70} +6504.80 q^{71} -41601.7 q^{73} +1738.11 q^{74} +122313. q^{76} -24465.0 q^{77} +84767.6 q^{79} -94997.1 q^{80} -16547.9 q^{82} -77910.7 q^{83} -135767. q^{85} -12273.1 q^{86} +79032.3 q^{88} -38889.0 q^{89} -47670.9 q^{91} +23489.3 q^{92} -26932.4 q^{94} -161009. q^{95} -92130.2 q^{97} -66464.2 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 3 q^{2} + 207 q^{4} - 32 q^{5} + 342 q^{7} - 249 q^{8}+O(q^{10}) $ 10 * q - 3 * q^2 + 207 * q^4 - 32 * q^5 + 342 * q^7 - 249 * q^8	$ 10 q - 3 q^{2} + 207 q^{4} - 32 q^{5} + 342 q^{7} - 249 q^{8} + 102 q^{10} - 846 q^{11} + 1504 q^{13} - 3468 q^{14} + 5859 q^{16} - 560 q^{17} + 4240 q^{19} + 6182 q^{20} - 2628 q^{22} + 1508 q^{23} + 11734 q^{25} + 22014 q^{26} - 8662 q^{28} + 124 q^{29} + 10384 q^{31} + 6619 q^{32} + 802 q^{34} + 17890 q^{35} + 5524 q^{37} + 46098 q^{38} - 61738 q^{40} - 16810 q^{41} + 24160 q^{43} + 21594 q^{44} + 42404 q^{46} - 58984 q^{47} + 70326 q^{49} - 6817 q^{50} + 64374 q^{52} - 23456 q^{53} + 96426 q^{55} - 80184 q^{56} - 13378 q^{58} - 52428 q^{59} + 113540 q^{61} + 113008 q^{62} + 37363 q^{64} + 22340 q^{65} + 85506 q^{67} + 71406 q^{68} - 71946 q^{70} - 75236 q^{71} - 85148 q^{73} + 23462 q^{74} + 113376 q^{76} + 172896 q^{77} + 178200 q^{79} + 401850 q^{80} + 5043 q^{82} + 125412 q^{83} - 245912 q^{85} + 18848 q^{86} - 135952 q^{88} + 62696 q^{89} + 30056 q^{91} + 236372 q^{92} + 419014 q^{94} - 28002 q^{95} + 154548 q^{97} - 288367 q^{98}+O(q^{100}) $ 10 * q - 3 * q^2 + 207 * q^4 - 32 * q^5 + 342 * q^7 - 249 * q^8 + 102 * q^10 - 846 * q^11 + 1504 * q^13 - 3468 * q^14 + 5859 * q^16 - 560 * q^17 + 4240 * q^19 + 6182 * q^20 - 2628 * q^22 + 1508 * q^23 + 11734 * q^25 + 22014 * q^26 - 8662 * q^28 + 124 * q^29 + 10384 * q^31 + 6619 * q^32 + 802 * q^34 + 17890 * q^35 + 5524 * q^37 + 46098 * q^38 - 61738 * q^40 - 16810 * q^41 + 24160 * q^43 + 21594 * q^44 + 42404 * q^46 - 58984 * q^47 + 70326 * q^49 - 6817 * q^50 + 64374 * q^52 - 23456 * q^53 + 96426 * q^55 - 80184 * q^56 - 13378 * q^58 - 52428 * q^59 + 113540 * q^61 + 113008 * q^62 + 37363 * q^64 + 22340 * q^65 + 85506 * q^67 + 71406 * q^68 - 71946 * q^70 - 75236 * q^71 - 85148 * q^73 + 23462 * q^74 + 113376 * q^76 + 172896 * q^77 + 178200 * q^79 + 401850 * q^80 + 5043 * q^82 + 125412 * q^83 - 245912 * q^85 + 18848 * q^86 - 135952 * q^88 + 62696 * q^89 + 30056 * q^91 + 236372 * q^92 + 419014 * q^94 - 28002 * q^95 + 154548 * q^97 - 288367 * 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 $

$2$	9.84411	1.74021	0.870105	−	0.492867i	$-0.164051\pi$
$2$	9.84411	1.74021	0.870105	+	0.492867i	$0.164051\pi$
$3$	0	0
$3$	0	0
$4$	64.9065	2.02833
$5$	−85.4409	−1.52841	−0.764207	−	0.644971i	$-0.776869\pi$
$5$	−85.4409	−1.52841	−0.764207	+	0.644971i	$0.776869\pi$
$6$	0	0
$7$	−100.276	−0.773487	−0.386743	−	0.922187i	$-0.626400\pi$
$7$	−100.276	−0.773487	−0.386743	+	0.922187i	$0.626400\pi$
$8$	323.935	1.78951
$9$	0	0
$10$	−841.090	−2.65976
$11$	243.976	0.607946	0.303973	−	0.952681i	$-0.401687\pi$
$11$	243.976	0.607946	0.303973	+	0.952681i	$0.401687\pi$
$12$	0	0
$13$	475.395	0.780183	0.390092	−	0.920776i	$-0.372443\pi$
$13$	475.395	0.780183	0.390092	+	0.920776i	$0.372443\pi$
$14$	−987.130	−1.34603
$15$	0	0
$16$	1111.85	1.08579
$17$	1589.02	1.33354	0.666770	−	0.745264i	$-0.267676\pi$
$17$	1589.02	1.33354	0.666770	+	0.745264i	$0.267676\pi$
$18$	0	0
$19$	1884.44	1.19757	0.598783	−	0.800911i	$-0.295651\pi$
$19$	1884.44	1.19757	0.598783	+	0.800911i	$0.295651\pi$
$20$	−5545.67	−3.10012
$21$	0	0
$22$	2401.72	1.05795
$23$	361.895	0.142647	0.0713235	−	0.997453i	$-0.477278\pi$
$23$	361.895	0.142647	0.0713235	+	0.997453i	$0.477278\pi$
$24$	0	0
$25$	4175.15	1.33605
$26$	4679.85	1.35768
$27$	0	0
$28$	−6508.58	−1.56889
$29$	5037.18	1.11222	0.556112	−	0.831107i	$-0.312292\pi$
$29$	5037.18	1.11222	0.556112	+	0.831107i	$0.312292\pi$
$30$	0	0
$31$	4280.76	0.800048	0.400024	−	0.916505i	$-0.369002\pi$
$31$	4280.76	0.800048	0.400024	+	0.916505i	$0.369002\pi$
$32$	579.204	0.0999901
$33$	0	0
$34$	15642.4	2.32064
$35$	8567.69	1.18221
$36$	0	0
$37$	176.563	0.0212029	0.0106015	−	0.999944i	$-0.496625\pi$
$37$	176.563	0.0212029	0.0106015	+	0.999944i	$0.496625\pi$
$38$	18550.7	2.08402
$39$	0	0
$40$	−27677.3	−2.73510
$41$	−1681.00	−0.156174
$41$	−1681.00	−0.156174
$42$	0	0
$43$	−1246.74	−0.102827	−0.0514133	−	0.998677i	$-0.516373\pi$
$43$	−1246.74	−0.102827	−0.0514133	+	0.998677i	$0.516373\pi$
$44$	15835.6	1.23311
$45$	0	0
$46$	3562.53	0.248236
$47$	−2735.89	−0.180657	−0.0903284	−	0.995912i	$-0.528792\pi$
$47$	−2735.89	−0.180657	−0.0903284	+	0.995912i	$0.528792\pi$
$48$	0	0
$49$	−6751.67	−0.401718
$50$	41100.6	2.32500
$51$	0	0
$52$	30856.3	1.58247
$53$	29387.9	1.43708	0.718538	−	0.695488i	$-0.244812\pi$
$53$	29387.9	1.43708	0.718538	+	0.695488i	$0.244812\pi$
$54$	0	0
$55$	−20845.5	−0.929193
$56$	−32483.0	−1.38416
$57$	0	0
$58$	49586.6	1.93550
$59$	27355.6	1.02310	0.511548	−	0.859255i	$-0.329072\pi$
$59$	27355.6	1.02310	0.511548	+	0.859255i	$0.329072\pi$
$60$	0	0
$61$	−10930.0	−0.376093	−0.188046	−	0.982160i	$-0.560216\pi$
$61$	−10930.0	−0.376093	−0.188046	+	0.982160i	$0.560216\pi$
$62$	42140.2	1.39225
$63$	0	0
$64$	−29877.3	−0.911783
$65$	−40618.2	−1.19244
$66$	0	0
$67$	−1624.42	−0.0442091	−0.0221046	−	0.999756i	$-0.507037\pi$
$67$	−1624.42	−0.0442091	−0.0221046	+	0.999756i	$0.507037\pi$
$68$	103137.	2.70486
$69$	0	0
$70$	84341.3	2.05729
$71$	6504.80	0.153140	0.0765699	−	0.997064i	$-0.475603\pi$
$71$	6504.80	0.153140	0.0765699	+	0.997064i	$0.475603\pi$
$72$	0	0
$73$	−41601.7	−0.913702	−0.456851	−	0.889543i	$-0.651023\pi$
$73$	−41601.7	−0.913702	−0.456851	+	0.889543i	$0.651023\pi$
$74$	1738.11	0.0368975
$75$	0	0
$76$	122313.	2.42906
$77$	−24465.0	−0.470238
$78$	0	0
$79$	84767.6	1.52814	0.764068	−	0.645136i	$-0.223199\pi$
$79$	84767.6	1.52814	0.764068	+	0.645136i	$0.223199\pi$
$80$	−94997.1	−1.65953
$81$	0	0
$82$	−16547.9	−0.271775
$83$	−77910.7	−1.24137	−0.620686	−	0.784060i	$-0.713146\pi$
$83$	−77910.7	−1.24137	−0.620686	+	0.784060i	$0.713146\pi$
$84$	0	0
$85$	−135767.	−2.03820
$86$	−12273.1	−0.178940
$87$	0	0
$88$	79032.3	1.08792
$89$	−38889.0	−0.520418	−0.260209	−	0.965552i	$-0.583791\pi$
$89$	−38889.0	−0.520418	−0.260209	+	0.965552i	$0.583791\pi$
$90$	0	0
$91$	−47670.9	−0.603462
$92$	23489.3	0.289335
$93$	0	0
$94$	−26932.4	−0.314380
$95$	−161009.	−1.83038
$96$	0	0
$97$	−92130.2	−0.994197	−0.497099	−	0.867694i	$-0.665601\pi$
$97$	−92130.2	−0.994197	−0.497099	+	0.867694i	$0.665601\pi$
$98$	−66464.2	−0.699073
$99$	0	0
$100$	270994.	2.70994
$101$	91980.7	0.897209	0.448604	−	0.893730i	$-0.351921\pi$
$101$	91980.7	0.897209	0.448604	+	0.893730i	$0.351921\pi$
$102$	0	0
$103$	−160210.	−1.48798	−0.743990	−	0.668191i	$-0.767069\pi$
$103$	−160210.	−1.48798	−0.743990	+	0.668191i	$0.767069\pi$
$104$	153997.	1.39614
$105$	0	0
$106$	289298.	2.50081
$107$	98739.3	0.833740	0.416870	−	0.908966i	$-0.363127\pi$
$107$	98739.3	0.833740	0.416870	+	0.908966i	$0.363127\pi$
$108$	0	0
$109$	175448.	1.41443	0.707214	−	0.706999i	$-0.249952\pi$
$109$	175448.	1.41443	0.707214	+	0.706999i	$0.249952\pi$
$110$	−205206.	−1.61699
$111$	0	0
$112$	−111492.	−0.839842
$113$	68588.7	0.505309	0.252654	−	0.967557i	$-0.418696\pi$
$113$	68588.7	0.505309	0.252654	+	0.967557i	$0.418696\pi$
$114$	0	0
$115$	−30920.6	−0.218024
$116$	326946.	2.25596
$117$	0	0
$118$	269292.	1.78040
$119$	−159341.	−1.03148
$120$	0	0
$121$	−101527.	−0.630402
$122$	−107596.	−0.654480
$123$	0	0
$124$	277849.	1.62276
$125$	−89725.6	−0.513619
$126$	0	0
$127$	−112653.	−0.619774	−0.309887	−	0.950773i	$-0.600291\pi$
$127$	−112653.	−0.619774	−0.309887	+	0.950773i	$0.600291\pi$
$128$	−312650.	−1.68668
$129$	0	0
$130$	−399850.	−2.07510
$131$	260926.	1.32843	0.664216	−	0.747541i	$-0.268766\pi$
$131$	260926.	1.32843	0.664216	+	0.747541i	$0.268766\pi$
$132$	0	0
$133$	−188965.	−0.926301
$134$	−15991.0	−0.0769331
$135$	0	0
$136$	514738.	2.38638
$137$	337989.	1.53851	0.769257	−	0.638939i	$-0.220626\pi$
$137$	337989.	1.53851	0.769257	+	0.638939i	$0.220626\pi$
$138$	0	0
$139$	263322.	1.15598	0.577990	−	0.816044i	$-0.303837\pi$
$139$	263322.	1.15598	0.577990	+	0.816044i	$0.303837\pi$
$140$	556099.	2.39791
$141$	0	0
$142$	64034.0	0.266495
$143$	115985.	0.474309
$144$	0	0
$145$	−430381.	−1.69994
$146$	−409532.	−1.59003
$147$	0	0
$148$	11460.1	0.0430065
$149$	−268057.	−0.989149	−0.494575	−	0.869135i	$-0.664676\pi$
$149$	−268057.	−0.989149	−0.494575	+	0.869135i	$0.664676\pi$
$150$	0	0
$151$	498960.	1.78083	0.890417	−	0.455146i	$-0.150413\pi$
$151$	498960.	1.78083	0.890417	+	0.455146i	$0.150413\pi$
$152$	610438.	2.14305
$153$	0	0
$154$	−240836.	−0.818313
$155$	−365752.	−1.22280
$156$	0	0
$157$	187505.	0.607105	0.303552	−	0.952815i	$-0.401827\pi$
$157$	187505.	0.607105	0.303552	+	0.952815i	$0.401827\pi$
$158$	834461.	2.65928
$159$	0	0
$160$	−49487.8	−0.152826
$161$	−36289.4	−0.110336
$162$	0	0
$163$	1692.50	0.00498953	0.00249477	−	0.999997i	$-0.499206\pi$
$163$	1692.50	0.00498953	0.00249477	+	0.999997i	$0.499206\pi$
$164$	−109108.	−0.316772
$165$	0	0
$166$	−766961.	−2.16025
$167$	69275.8	0.192216	0.0961082	−	0.995371i	$-0.469361\pi$
$167$	69275.8	0.192216	0.0961082	+	0.995371i	$0.469361\pi$
$168$	0	0
$169$	−145292.	−0.391314
$170$	−1.33650e6	−3.54689
$171$	0	0
$172$	−80921.7	−0.208566
$173$	−618184.	−1.57037	−0.785186	−	0.619260i	$-0.787433\pi$
$173$	−618184.	−1.57037	−0.785186	+	0.619260i	$0.787433\pi$
$174$	0	0
$175$	−418668.	−1.03342
$176$	271263.	0.660100
$177$	0	0
$178$	−382828.	−0.905636
$179$	−627935.	−1.46481	−0.732406	−	0.680868i	$-0.761603\pi$
$179$	−627935.	−1.46481	−0.732406	+	0.680868i	$0.761603\pi$
$180$	0	0
$181$	245217.	0.556357	0.278179	−	0.960529i	$-0.410269\pi$
$181$	245217.	0.556357	0.278179	+	0.960529i	$0.410269\pi$
$182$	−469277.	−1.05015
$183$	0	0
$184$	117230.	0.255268
$185$	−15085.7	−0.0324069
$186$	0	0
$187$	387681.	0.810720
$188$	−177577.	−0.366431
$189$	0	0
$190$	−1.58499e6	−3.18524
$191$	−582954.	−1.15625	−0.578124	−	0.815949i	$-0.696215\pi$
$191$	−582954.	−1.15625	−0.578124	+	0.815949i	$0.696215\pi$
$192$	0	0
$193$	133051.	0.257113	0.128557	−	0.991702i	$-0.458966\pi$
$193$	133051.	0.257113	0.128557	+	0.991702i	$0.458966\pi$
$194$	−906940.	−1.73011
$195$	0	0
$196$	−438228.	−0.814816
$197$	769033.	1.41182	0.705910	−	0.708301i	$-0.250538\pi$
$197$	769033.	1.41182	0.705910	+	0.708301i	$0.250538\pi$
$198$	0	0
$199$	61690.6	0.110430	0.0552149	−	0.998474i	$-0.482416\pi$
$199$	61690.6	0.110430	0.0552149	+	0.998474i	$0.482416\pi$
$200$	1.35248e6	2.39086
$201$	0	0
$202$	905469.	1.56133
$203$	−505110.	−0.860291
$204$	0	0
$205$	143626.	0.238698
$206$	−1.57713e6	−2.58940
$207$	0	0
$208$	528566.	0.847113
$209$	459759.	0.728055
$210$	0	0
$211$	161101.	0.249111	0.124555	−	0.992213i	$-0.460250\pi$
$211$	161101.	0.249111	0.124555	+	0.992213i	$0.460250\pi$
$212$	1.90747e6	2.91486
$213$	0	0
$214$	972001.	1.45088
$215$	106523.	0.157162
$216$	0	0
$217$	−429258.	−0.618827
$218$	1.72713e6	2.46140
$219$	0	0
$220$	−1.35301e6	−1.88471
$221$	755411.	1.04041
$222$	0	0
$223$	−171575.	−0.231043	−0.115522	−	0.993305i	$-0.536854\pi$
$223$	−171575.	−0.231043	−0.115522	+	0.993305i	$0.536854\pi$
$224$	−58080.5	−0.0773410
$225$	0	0
$226$	675195.	0.879343
$227$	−290832.	−0.374609	−0.187304	−	0.982302i	$-0.559975\pi$
$227$	−290832.	−0.374609	−0.187304	+	0.982302i	$0.559975\pi$
$228$	0	0
$229$	1.37073e6	1.72728	0.863641	−	0.504108i	$-0.168179\pi$
$229$	1.37073e6	1.72728	0.863641	+	0.504108i	$0.168179\pi$
$230$	−304386.	−0.379407
$231$	0	0
$232$	1.63172e6	1.99033
$233$	1.43362e6	1.73000	0.864998	−	0.501776i	$-0.167320\pi$
$233$	1.43362e6	1.73000	0.864998	+	0.501776i	$0.167320\pi$
$234$	0	0
$235$	233757.	0.276118
$236$	1.77556e6	2.07518
$237$	0	0
$238$	−1.56857e6	−1.79498
$239$	699348.	0.791951	0.395976	−	0.918261i	$-0.370406\pi$
$239$	699348.	0.791951	0.395976	+	0.918261i	$0.370406\pi$
$240$	0	0
$241$	−301531.	−0.334418	−0.167209	−	0.985921i	$-0.553475\pi$
$241$	−301531.	−0.334418	−0.167209	+	0.985921i	$0.553475\pi$
$242$	−999441.	−1.09703
$243$	0	0
$244$	−709427.	−0.762840
$245$	576869.	0.613991
$246$	0	0
$247$	895856.	0.934321
$248$	1.38669e6	1.43169
$249$	0	0
$250$	−883269.	−0.893805
$251$	−1.25120e6	−1.25355	−0.626777	−	0.779198i	$-0.715626\pi$
$251$	−1.25120e6	−1.25355	−0.626777	+	0.779198i	$0.715626\pi$
$252$	0	0
$253$	88293.5	0.0867217
$254$	−1.10897e6	−1.07854
$255$	0	0
$256$	−2.12169e6	−2.02340
$257$	−781162.	−0.737749	−0.368874	−	0.929479i	$-0.620257\pi$
$257$	−781162.	−0.737749	−0.368874	+	0.929479i	$0.620257\pi$
$258$	0	0
$259$	−17705.1	−0.0164002
$260$	−2.63639e6	−2.41866
$261$	0	0
$262$	2.56859e6	2.31175
$263$	232806.	0.207542	0.103771	−	0.994601i	$-0.466909\pi$
$263$	232806.	0.207542	0.103771	+	0.994601i	$0.466909\pi$
$264$	0	0
$265$	−2.51093e6	−2.19644
$266$	−1.86019e6	−1.61196
$267$	0	0
$268$	−105436.	−0.0896706
$269$	−970689.	−0.817898	−0.408949	−	0.912557i	$-0.634105\pi$
$269$	−970689.	−0.817898	−0.408949	+	0.912557i	$0.634105\pi$
$270$	0	0
$271$	24897.4	0.0205935	0.0102968	−	0.999947i	$-0.496722\pi$
$271$	24897.4	0.0205935	0.0102968	+	0.999947i	$0.496722\pi$
$272$	1.76674e6	1.44794
$273$	0	0
$274$	3.32720e6	2.67734
$275$	1.01864e6	0.812245
$276$	0	0
$277$	1.10919e6	0.868572	0.434286	−	0.900775i	$-0.357001\pi$
$277$	1.10919e6	0.868572	0.434286	+	0.900775i	$0.357001\pi$
$278$	2.59217e6	2.01165
$279$	0	0
$280$	2.77538e6	2.11557
$281$	−1.44136e6	−1.08895	−0.544474	−	0.838778i	$-0.683271\pi$
$281$	−1.44136e6	−1.08895	−0.544474	+	0.838778i	$0.683271\pi$
$282$	0	0
$283$	−1.02505e6	−0.760815	−0.380407	−	0.924819i	$-0.624216\pi$
$283$	−1.02505e6	−0.760815	−0.380407	+	0.924819i	$0.624216\pi$
$284$	422204.	0.310618
$285$	0	0
$286$	1.14177e6	0.825398
$287$	168564.	0.120798
$288$	0	0
$289$	1.10511e6	0.778327
$290$	−4.23672e6	−2.95825
$291$	0	0
$292$	−2.70022e6	−1.85329
$293$	−1.41083e6	−0.960079	−0.480039	−	0.877247i	$-0.659378\pi$
$293$	−1.41083e6	−0.960079	−0.480039	+	0.877247i	$0.659378\pi$
$294$	0	0
$295$	−2.33729e6	−1.56371
$296$	57195.1	0.0379428
$297$	0	0
$298$	−2.63879e6	−1.72133
$299$	172043.	0.111291
$300$	0	0
$301$	125019.	0.0795351
$302$	4.91182e6	3.09902
$303$	0	0
$304$	2.09521e6	1.30030
$305$	933868.	0.574825
$306$	0	0
$307$	−1.09964e6	−0.665891	−0.332946	−	0.942946i	$-0.608043\pi$
$307$	−1.09964e6	−0.665891	−0.332946	+	0.942946i	$0.608043\pi$
$308$	−1.58794e6	−0.953798
$309$	0	0
$310$	−3.60050e6	−2.12794
$311$	−1.58841e6	−0.931240	−0.465620	−	0.884985i	$-0.654169\pi$
$311$	−1.58841e6	−0.931240	−0.465620	+	0.884985i	$0.654169\pi$
$312$	0	0
$313$	349995.	0.201930	0.100965	−	0.994890i	$-0.467807\pi$
$313$	349995.	0.201930	0.100965	+	0.994890i	$0.467807\pi$
$314$	1.84582e6	1.05649
$315$	0	0
$316$	5.50197e6	3.09956
$317$	−2.07203e6	−1.15810	−0.579052	−	0.815291i	$-0.696577\pi$
$317$	−2.07203e6	−1.15810	−0.579052	+	0.815291i	$0.696577\pi$
$318$	0	0
$319$	1.22895e6	0.676173
$320$	2.55274e6	1.39358
$321$	0	0
$322$	−357237.	−0.192007
$323$	2.99441e6	1.59700
$324$	0	0
$325$	1.98485e6	1.04236
$326$	16661.2	0.00868283
$327$	0	0
$328$	−544535.	−0.279474
$329$	274345.	0.139736
$330$	0	0
$331$	−3.43076e6	−1.72116	−0.860579	−	0.509317i	$-0.829898\pi$
$331$	−3.43076e6	−1.72116	−0.860579	+	0.509317i	$0.829898\pi$
$332$	−5.05691e6	−2.51791
$333$	0	0
$334$	681959.	0.334497
$335$	138792.	0.0675698
$336$	0	0
$337$	−2.90033e6	−1.39115	−0.695573	−	0.718455i	$-0.744850\pi$
$337$	−2.90033e6	−1.39115	−0.695573	+	0.718455i	$0.744850\pi$
$338$	−1.43027e6	−0.680968
$339$	0	0
$340$	−8.81216e6	−4.13414
$341$	1.04440e6	0.486386
$342$	0	0
$343$	2.36238e6	1.08421
$344$	−403864.	−0.184009
$345$	0	0
$346$	−6.08547e6	−2.73277
$347$	−1.80133e6	−0.803100	−0.401550	−	0.915837i	$-0.631528\pi$
$347$	−1.80133e6	−0.803100	−0.401550	+	0.915837i	$0.631528\pi$
$348$	0	0
$349$	−1.90333e6	−0.836469	−0.418235	−	0.908339i	$-0.637351\pi$
$349$	−1.90333e6	−0.836469	−0.418235	+	0.908339i	$0.637351\pi$
$350$	−4.12142e6	−1.79836
$351$	0	0
$352$	141312.	0.0607886
$353$	−1.39179e6	−0.594480	−0.297240	−	0.954803i	$-0.596066\pi$
$353$	−1.39179e6	−0.594480	−0.297240	+	0.954803i	$0.596066\pi$
$354$	0	0
$355$	−555776.	−0.234061
$356$	−2.52415e6	−1.05558
$357$	0	0
$358$	−6.18146e6	−2.54908
$359$	293957.	0.120378	0.0601891	−	0.998187i	$-0.480830\pi$
$359$	293957.	0.120378	0.0601891	+	0.998187i	$0.480830\pi$
$360$	0	0
$361$	1.07503e6	0.434164
$362$	2.41394e6	0.968178
$363$	0	0
$364$	−3.09415e6	−1.22402
$365$	3.55449e6	1.39651
$366$	0	0
$367$	3.02905e6	1.17393	0.586964	−	0.809613i	$-0.300323\pi$
$367$	3.02905e6	1.17393	0.586964	+	0.809613i	$0.300323\pi$
$368$	402371.	0.154884
$369$	0	0
$370$	−148506.	−0.0563947
$371$	−2.94691e6	−1.11156
$372$	0	0
$373$	−1.92513e6	−0.716454	−0.358227	−	0.933635i	$-0.616619\pi$
$373$	−1.92513e6	−0.716454	−0.358227	+	0.933635i	$0.616619\pi$
$374$	3.81638e6	1.41082
$375$	0	0
$376$	−886251.	−0.323286
$377$	2.39465e6	0.867739
$378$	0	0
$379$	2.42934e6	0.868742	0.434371	−	0.900734i	$-0.356971\pi$
$379$	2.42934e6	0.868742	0.434371	+	0.900734i	$0.356971\pi$
$380$	−1.04505e7	−3.71260
$381$	0	0
$382$	−5.73866e6	−2.01211
$383$	−4.62103e6	−1.60969	−0.804845	−	0.593486i	$-0.797751\pi$
$383$	−4.62103e6	−1.60969	−0.804845	+	0.593486i	$0.797751\pi$
$384$	0	0
$385$	2.09031e6	0.718718
$386$	1.30977e6	0.447431
$387$	0	0
$388$	−5.97985e6	−2.01656
$389$	1.35751e6	0.454851	0.227425	−	0.973796i	$-0.426969\pi$
$389$	1.35751e6	0.454851	0.227425	+	0.973796i	$0.426969\pi$
$390$	0	0
$391$	575056.	0.190225
$392$	−2.18711e6	−0.718877
$393$	0	0
$394$	7.57045e6	2.45686
$395$	−7.24262e6	−2.33562
$396$	0	0
$397$	3.15108e6	1.00342	0.501711	−	0.865035i	$-0.332704\pi$
$397$	3.15108e6	1.00342	0.501711	+	0.865035i	$0.332704\pi$
$398$	607289.	0.192171
$399$	0	0
$400$	4.64212e6	1.45066
$401$	−51106.8	−0.0158715	−0.00793575	−	0.999969i	$-0.502526\pi$
$401$	−51106.8	−0.0158715	−0.00793575	+	0.999969i	$0.502526\pi$
$402$	0	0
$403$	2.03505e6	0.624184
$404$	5.97015e6	1.81983
$405$	0	0
$406$	−4.97235e6	−1.49709
$407$	43077.2	0.0128902
$408$	0	0
$409$	−305664.	−0.0903516	−0.0451758	−	0.998979i	$-0.514385\pi$
$409$	−305664.	−0.0903516	−0.0451758	+	0.998979i	$0.514385\pi$
$410$	1.41387e6	0.415385
$411$	0	0
$412$	−1.03987e7	−3.01811
$413$	−2.74312e6	−0.791352
$414$	0	0
$415$	6.65676e6	1.89733
$416$	275351.	0.0780106
$417$	0	0
$418$	4.52592e6	1.26697
$419$	−3.67817e6	−1.02352	−0.511761	−	0.859128i	$-0.671007\pi$
$419$	−3.67817e6	−1.02352	−0.511761	+	0.859128i	$0.671007\pi$
$420$	0	0
$421$	−2.07572e6	−0.570774	−0.285387	−	0.958412i	$-0.592122\pi$
$421$	−2.07572e6	−0.570774	−0.285387	+	0.958412i	$0.592122\pi$
$422$	1.58590e6	0.433505
$423$	0	0
$424$	9.51979e6	2.57165
$425$	6.63438e6	1.78167
$426$	0	0
$427$	1.09602e6	0.290903
$428$	6.40882e6	1.69110
$429$	0	0
$430$	1.04862e6	0.273494
$431$	3.61678e6	0.937841	0.468921	−	0.883240i	$-0.344643\pi$
$431$	3.61678e6	0.937841	0.468921	+	0.883240i	$0.344643\pi$
$432$	0	0
$433$	6.25345e6	1.60288	0.801438	−	0.598078i	$-0.204069\pi$
$433$	6.25345e6	1.60288	0.801438	+	0.598078i	$0.204069\pi$
$434$	−4.22566e6	−1.07689
$435$	0	0
$436$	1.13877e7	2.86893
$437$	681971.	0.170829
$438$	0	0
$439$	−4.34921e6	−1.07708	−0.538541	−	0.842599i	$-0.681024\pi$
$439$	−4.34921e6	−1.07708	−0.538541	+	0.842599i	$0.681024\pi$
$440$	−6.75260e6	−1.66280
$441$	0	0
$442$	7.43635e6	1.81052
$443$	4.40583e6	1.06664	0.533321	−	0.845913i	$-0.320944\pi$
$443$	4.40583e6	1.06664	0.533321	+	0.845913i	$0.320944\pi$
$444$	0	0
$445$	3.32271e6	0.795414
$446$	−1.68901e6	−0.402063
$447$	0	0
$448$	2.99598e6	0.705252
$449$	5.97947e6	1.39974	0.699869	−	0.714271i	$-0.253242\pi$
$449$	5.97947e6	1.39974	0.699869	+	0.714271i	$0.253242\pi$
$450$	0	0
$451$	−410123.	−0.0949452
$452$	4.45185e6	1.02493
$453$	0	0
$454$	−2.86299e6	−0.651898
$455$	4.07304e6	0.922339
$456$	0	0
$457$	187085.	0.0419033	0.0209517	−	0.999780i	$-0.493330\pi$
$457$	187085.	0.0419033	0.0209517	+	0.999780i	$0.493330\pi$
$458$	1.34936e7	3.00583
$459$	0	0
$460$	−2.00695e6	−0.442223
$461$	−255073.	−0.0559001	−0.0279501	−	0.999609i	$-0.508898\pi$
$461$	−255073.	−0.0559001	−0.0279501	+	0.999609i	$0.508898\pi$
$462$	0	0
$463$	4.09433e6	0.887627	0.443813	−	0.896119i	$-0.353625\pi$
$463$	4.09433e6	0.887627	0.443813	+	0.896119i	$0.353625\pi$
$464$	5.60057e6	1.20764
$465$	0	0
$466$	1.41127e7	3.01055
$467$	−3.42511e6	−0.726746	−0.363373	−	0.931644i	$-0.618375\pi$
$467$	−3.42511e6	−0.726746	−0.363373	+	0.931644i	$0.618375\pi$
$468$	0	0
$469$	162891.	0.0341952
$470$	2.30113e6	0.480503
$471$	0	0
$472$	8.86145e6	1.83084
$473$	−304175.	−0.0625131
$474$	0	0
$475$	7.86784e6	1.60000
$476$	−1.03422e7	−2.09217
$477$	0	0
$478$	6.88446e6	1.37816
$479$	−3.78950e6	−0.754646	−0.377323	−	0.926082i	$-0.623155\pi$
$479$	−3.78950e6	−0.754646	−0.377323	+	0.926082i	$0.623155\pi$
$480$	0	0
$481$	83937.4	0.0165422
$482$	−2.96830e6	−0.581957
$483$	0	0
$484$	−6.58975e6	−1.27866
$485$	7.87169e6	1.51954
$486$	0	0
$487$	−4.02381e6	−0.768804	−0.384402	−	0.923166i	$-0.625592\pi$
$487$	−4.02381e6	−0.768804	−0.384402	+	0.923166i	$0.625592\pi$
$488$	−3.54061e6	−0.673021
$489$	0	0
$490$	5.67876e6	1.06847
$491$	−1.00818e7	−1.88727	−0.943637	−	0.330981i	$-0.892620\pi$
$491$	−1.00818e7	−1.88727	−0.943637	+	0.330981i	$0.892620\pi$
$492$	0	0
$493$	8.00416e6	1.48320
$494$	8.81891e6	1.62591
$495$	0	0
$496$	4.75954e6	0.868682
$497$	−652277.	−0.118452
$498$	0	0
$499$	5.79984e6	1.04271	0.521357	−	0.853339i	$-0.325426\pi$
$499$	5.79984e6	1.04271	0.521357	+	0.853339i	$0.325426\pi$
$500$	−5.82378e6	−1.04179
$501$	0	0
$502$	−1.23170e7	−2.18145
$503$	1.31021e6	0.230898	0.115449	−	0.993313i	$-0.463169\pi$
$503$	1.31021e6	0.230898	0.115449	+	0.993313i	$0.463169\pi$
$504$	0	0
$505$	−7.85892e6	−1.37131
$506$	869171.	0.150914
$507$	0	0
$508$	−7.31191e6	−1.25710
$509$	3.43754e6	0.588104	0.294052	−	0.955789i	$-0.404996\pi$
$509$	3.43754e6	0.588104	0.294052	+	0.955789i	$0.404996\pi$
$510$	0	0
$511$	4.17167e6	0.706736
$512$	−1.08813e7	−1.83445
$513$	0	0
$514$	−7.68985e6	−1.28384
$515$	1.36885e7	2.27425
$516$	0	0
$517$	−667491.	−0.109830
$518$	−174291.	−0.0285398
$519$	0	0
$520$	−1.31577e7	−2.13388
$521$	6.45185e6	1.04133	0.520667	−	0.853760i	$-0.325683\pi$
$521$	6.45185e6	1.04133	0.520667	+	0.853760i	$0.325683\pi$
$522$	0	0
$523$	−7.51573e6	−1.20148	−0.600741	−	0.799444i	$-0.705128\pi$
$523$	−7.51573e6	−1.20148	−0.600741	+	0.799444i	$0.705128\pi$
$524$	1.69358e7	2.69450
$525$	0	0
$526$	2.29177e6	0.361166
$527$	6.80219e6	1.06690
$528$	0	0
$529$	−6.30538e6	−0.979652
$530$	−2.47179e7	−3.82227
$531$	0	0
$532$	−1.22651e7	−1.87884
$533$	−799140.	−0.121844
$534$	0	0
$535$	−8.43638e6	−1.27430
$536$	−526207.	−0.0791125
$537$	0	0
$538$	−9.55557e6	−1.42331
$539$	−1.64725e6	−0.244223
$540$	0	0
$541$	1.06675e7	1.56700	0.783499	−	0.621393i	$-0.213433\pi$
$541$	1.06675e7	1.56700	0.783499	+	0.621393i	$0.213433\pi$
$542$	245093.	0.0358370
$543$	0	0
$544$	920365.	0.133341
$545$	−1.49904e7	−2.16183
$546$	0	0
$547$	5.60509e6	0.800966	0.400483	−	0.916304i	$-0.368842\pi$
$547$	5.60509e6	0.800966	0.400483	+	0.916304i	$0.368842\pi$
$548$	2.19377e7	3.12061
$549$	0	0
$550$	1.00276e7	1.41348
$551$	9.49229e6	1.33196
$552$	0	0
$553$	−8.50018e6	−1.18199
$554$	1.09190e7	1.51150
$555$	0	0
$556$	1.70913e7	2.34471
$557$	2.14933e6	0.293538	0.146769	−	0.989171i	$-0.453113\pi$
$557$	2.14933e6	0.293538	0.146769	+	0.989171i	$0.453113\pi$
$558$	0	0
$559$	−592696.	−0.0802237
$560$	9.52595e6	1.28363
$561$	0	0
$562$	−1.41889e7	−1.89500
$563$	−1.37709e7	−1.83102	−0.915509	−	0.402298i	$-0.868211\pi$
$563$	−1.37709e7	−1.83102	−0.915509	+	0.402298i	$0.868211\pi$
$564$	0	0
$565$	−5.86028e6	−0.772320
$566$	−1.00907e7	−1.32398
$567$	0	0
$568$	2.10713e6	0.274045
$569$	4.74696e6	0.614660	0.307330	−	0.951603i	$-0.400564\pi$
$569$	4.74696e6	0.614660	0.307330	+	0.951603i	$0.400564\pi$
$570$	0	0
$571$	3.85510e6	0.494818	0.247409	−	0.968911i	$-0.420421\pi$
$571$	3.85510e6	0.494818	0.247409	+	0.968911i	$0.420421\pi$
$572$	7.52818e6	0.962055
$573$	0	0
$574$	1.65937e6	0.210214
$575$	1.51096e6	0.190583
$576$	0	0
$577$	−6.11382e6	−0.764492	−0.382246	−	0.924061i	$-0.624849\pi$
$577$	−6.11382e6	−0.764492	−0.382246	+	0.924061i	$0.624849\pi$
$578$	1.08789e7	1.35445
$579$	0	0
$580$	−2.79345e7	−3.44803
$581$	7.81259e6	0.960184
$582$	0	0
$583$	7.16995e6	0.873664
$584$	−1.34763e7	−1.63507
$585$	0	0
$586$	−1.38884e7	−1.67074
$587$	5.81966e6	0.697111	0.348556	−	0.937288i	$-0.386672\pi$
$587$	5.81966e6	0.697111	0.348556	+	0.937288i	$0.386672\pi$
$588$	0	0
$589$	8.06685e6	0.958111
$590$	−2.30085e7	−2.72119
$591$	0	0
$592$	196311.	0.0230219
$593$	3.61783e6	0.422485	0.211243	−	0.977434i	$-0.432249\pi$
$593$	3.61783e6	0.422485	0.211243	+	0.977434i	$0.432249\pi$
$594$	0	0
$595$	1.36142e7	1.57652
$596$	−1.73987e7	−2.00632
$597$	0	0
$598$	1.69361e6	0.193669
$599$	−5.72850e6	−0.652340	−0.326170	−	0.945311i	$-0.605758\pi$
$599$	−5.72850e6	−0.652340	−0.326170	+	0.945311i	$0.605758\pi$
$600$	0	0
$601$	−6.64361e6	−0.750271	−0.375135	−	0.926970i	$-0.622404\pi$
$601$	−6.64361e6	−0.750271	−0.375135	+	0.926970i	$0.622404\pi$
$602$	1.23070e6	0.138408
$603$	0	0
$604$	3.23857e7	3.61211
$605$	8.67454e6	0.963514
$606$	0	0
$607$	2.15025e6	0.236874	0.118437	−	0.992962i	$-0.462212\pi$
$607$	2.15025e6	0.236874	0.118437	+	0.992962i	$0.462212\pi$
$608$	1.09148e6	0.119745
$609$	0	0
$610$	9.19310e6	1.00032
$611$	−1.30063e6	−0.140945
$612$	0	0
$613$	1.62680e7	1.74857	0.874286	−	0.485411i	$-0.161330\pi$
$613$	1.62680e7	1.74857	0.874286	+	0.485411i	$0.161330\pi$
$614$	−1.08249e7	−1.15879
$615$	0	0
$616$	−7.92507e6	−0.841494
$617$	−1.42355e7	−1.50543	−0.752715	−	0.658347i	$-0.771256\pi$
$617$	−1.42355e7	−1.50543	−0.752715	+	0.658347i	$0.771256\pi$
$618$	0	0
$619$	1.30018e7	1.36388	0.681940	−	0.731408i	$-0.261136\pi$
$619$	1.30018e7	1.36388	0.681940	+	0.731408i	$0.261136\pi$
$620$	−2.37397e7	−2.48025
$621$	0	0
$622$	−1.56365e7	−1.62055
$623$	3.89965e6	0.402536
$624$	0	0
$625$	−5.38110e6	−0.551025
$626$	3.44539e6	0.351400
$627$	0	0
$628$	1.21703e7	1.23141
$629$	280562.	0.0282750
$630$	0	0
$631$	4.14398e6	0.414328	0.207164	−	0.978306i	$-0.433577\pi$
$631$	4.14398e6	0.414328	0.207164	+	0.978306i	$0.433577\pi$
$632$	2.74592e7	2.73461
$633$	0	0
$634$	−2.03973e7	−2.01534
$635$	9.62517e6	0.947271
$636$	0	0
$637$	−3.20972e6	−0.313414
$638$	1.20979e7	1.17668
$639$	0	0
$640$	2.67131e7	2.57795
$641$	1.10799e6	0.106510	0.0532550	−	0.998581i	$-0.483040\pi$
$641$	1.10799e6	0.106510	0.0532550	+	0.998581i	$0.483040\pi$
$642$	0	0
$643$	−1.24431e7	−1.18686	−0.593432	−	0.804884i	$-0.702227\pi$
$643$	−1.24431e7	−1.18686	−0.593432	+	0.804884i	$0.702227\pi$
$644$	−2.35542e6	−0.223797
$645$	0	0
$646$	2.94773e7	2.77912
$647$	1.38477e7	1.30052	0.650258	−	0.759713i	$-0.274661\pi$
$647$	1.38477e7	1.30052	0.650258	+	0.759713i	$0.274661\pi$
$648$	0	0
$649$	6.67411e6	0.621987
$650$	1.95390e7	1.81393
$651$	0	0
$652$	109854.	0.0101204
$653$	1.06824e7	0.980365	0.490182	−	0.871620i	$-0.336930\pi$
$653$	1.06824e7	0.980365	0.490182	+	0.871620i	$0.336930\pi$
$654$	0	0
$655$	−2.22938e7	−2.03039
$656$	−1.86901e6	−0.169571
$657$	0	0
$658$	2.70068e6	0.243169
$659$	2.07210e7	1.85865	0.929325	−	0.369262i	$-0.120389\pi$
$659$	2.07210e7	1.85865	0.929325	+	0.369262i	$0.120389\pi$
$660$	0	0
$661$	1.22773e7	1.09295	0.546476	−	0.837475i	$-0.315969\pi$
$661$	1.22773e7	1.09295	0.546476	+	0.837475i	$0.315969\pi$
$662$	−3.37728e7	−2.99517
$663$	0	0
$664$	−2.52380e7	−2.22144
$665$	1.61453e7	1.41577
$666$	0	0
$667$	1.82293e6	0.158656
$668$	4.49645e6	0.389878
$669$	0	0
$670$	1.36628e6	0.117586
$671$	−2.66665e6	−0.228644
$672$	0	0
$673$	−2.03516e7	−1.73206	−0.866028	−	0.499996i	$-0.833335\pi$
$673$	−2.03516e7	−1.73206	−0.866028	+	0.499996i	$0.833335\pi$
$674$	−2.85512e7	−2.42089
$675$	0	0
$676$	−9.43040e6	−0.793713
$677$	−5.64504e6	−0.473364	−0.236682	−	0.971587i	$-0.576060\pi$
$677$	−5.64504e6	−0.473364	−0.236682	+	0.971587i	$0.576060\pi$
$678$	0	0
$679$	9.23847e6	0.768999
$680$	−4.39797e7	−3.64737
$681$	0	0
$682$	1.02812e7	0.846414
$683$	1.22298e7	1.00315	0.501576	−	0.865113i	$-0.332754\pi$
$683$	1.22298e7	1.00315	0.501576	+	0.865113i	$0.332754\pi$
$684$	0	0
$685$	−2.88781e7	−2.35149
$686$	2.32555e7	1.88675
$687$	0	0
$688$	−1.38619e6	−0.111648
$689$	1.39709e7	1.12118
$690$	0	0
$691$	54837.6	0.00436901	0.00218451	−	0.999998i	$-0.499305\pi$
$691$	54837.6	0.00436901	0.00218451	+	0.999998i	$0.499305\pi$
$692$	−4.01241e7	−3.18523
$693$	0	0
$694$	−1.77325e7	−1.39756
$695$	−2.24985e7	−1.76682
$696$	0	0
$697$	−2.67114e6	−0.208264
$698$	−1.87366e7	−1.45563
$699$	0	0
$700$	−2.71743e7	−2.09611
$701$	−4.75849e6	−0.365741	−0.182871	−	0.983137i	$-0.558539\pi$
$701$	−4.75849e6	−0.365741	−0.182871	+	0.983137i	$0.558539\pi$
$702$	0	0
$703$	332724.	0.0253919
$704$	−7.28934e6	−0.554315
$705$	0	0
$706$	−1.37009e7	−1.03452
$707$	−9.22348e6	−0.693979
$708$	0	0
$709$	−2.06998e7	−1.54650	−0.773252	−	0.634099i	$-0.781371\pi$
$709$	−2.06998e7	−1.54650	−0.773252	+	0.634099i	$0.781371\pi$
$710$	−5.47112e6	−0.407315
$711$	0	0
$712$	−1.25975e7	−0.931291
$713$	1.54918e6	0.114124
$714$	0	0
$715$	−9.90986e6	−0.724941
$716$	−4.07571e7	−2.97112
$717$	0	0
$718$	2.89375e6	0.209483
$719$	−6.54016e6	−0.471809	−0.235905	−	0.971776i	$-0.575805\pi$
$719$	−6.54016e6	−0.471809	−0.235905	+	0.971776i	$0.575805\pi$
$720$	0	0
$721$	1.60653e7	1.15093
$722$	1.05827e7	0.755536
$723$	0	0
$724$	1.59162e7	1.12847
$725$	2.10310e7	1.48599
$726$	0	0
$727$	−1.42484e6	−0.0999840	−0.0499920	−	0.998750i	$-0.515920\pi$
$727$	−1.42484e6	−0.0999840	−0.0499920	+	0.998750i	$0.515920\pi$
$728$	−1.54423e7	−1.07990
$729$	0	0
$730$	3.49908e7	2.43023
$731$	−1.98109e6	−0.137123
$732$	0	0
$733$	1.88388e7	1.29507	0.647535	−	0.762036i	$-0.275800\pi$
$733$	1.88388e7	1.29507	0.647535	+	0.762036i	$0.275800\pi$
$734$	2.98183e7	2.04288
$735$	0	0
$736$	209611.	0.0142633
$737$	−396320.	−0.0268768
$738$	0	0
$739$	1.71175e7	1.15300	0.576501	−	0.817097i	$-0.304418\pi$
$739$	1.71175e7	1.15300	0.576501	+	0.817097i	$0.304418\pi$
$740$	−979162.	−0.0657317
$741$	0	0
$742$	−2.90097e7	−1.93434
$743$	1.29716e7	0.862030	0.431015	−	0.902345i	$-0.358156\pi$
$743$	1.29716e7	0.862030	0.431015	+	0.902345i	$0.358156\pi$
$744$	0	0
$745$	2.29031e7	1.51183
$746$	−1.89512e7	−1.24678
$747$	0	0
$748$	2.51630e7	1.64441
$749$	−9.90121e6	−0.644887
$750$	0	0
$751$	2.09193e7	1.35347	0.676735	−	0.736227i	$-0.263394\pi$
$751$	2.09193e7	1.35347	0.676735	+	0.736227i	$0.263394\pi$
$752$	−3.04189e6	−0.196155
$753$	0	0
$754$	2.35732e7	1.51005
$755$	−4.26316e7	−2.72185
$756$	0	0
$757$	−1.23702e7	−0.784582	−0.392291	−	0.919841i	$-0.628317\pi$
$757$	−1.23702e7	−0.784582	−0.392291	+	0.919841i	$0.628317\pi$
$758$	2.39147e7	1.51179
$759$	0	0
$760$	−5.21564e7	−3.27547
$761$	1.71801e7	1.07539	0.537694	−	0.843140i	$-0.319296\pi$
$761$	1.71801e7	1.07539	0.537694	+	0.843140i	$0.319296\pi$
$762$	0	0
$763$	−1.75932e7	−1.09404
$764$	−3.78375e7	−2.34525
$765$	0	0
$766$	−4.54899e7	−2.80120
$767$	1.30047e7	0.798203
$768$	0	0
$769$	−2.19179e7	−1.33654	−0.668271	−	0.743918i	$-0.732965\pi$
$769$	−2.19179e7	−1.33654	−0.668271	+	0.743918i	$0.732965\pi$
$770$	2.05772e7	1.25072
$771$	0	0
$772$	8.63586e6	0.521510
$773$	−9.36560e6	−0.563751	−0.281875	−	0.959451i	$-0.590956\pi$
$773$	−9.36560e6	−0.563751	−0.281875	+	0.959451i	$0.590956\pi$
$774$	0	0
$775$	1.78728e7	1.06890
$776$	−2.98442e7	−1.77912
$777$	0	0
$778$	1.33635e7	0.791536
$779$	−3.16775e6	−0.187028
$780$	0	0
$781$	1.58701e6	0.0931007
$782$	5.66092e6	0.331032
$783$	0	0
$784$	−7.50682e6	−0.436180
$785$	−1.60206e7	−0.927907
$786$	0	0
$787$	−1.00361e7	−0.577603	−0.288802	−	0.957389i	$-0.593257\pi$
$787$	−1.00361e7	−0.577603	−0.288802	+	0.957389i	$0.593257\pi$
$788$	4.99153e7	2.86364
$789$	0	0
$790$	−7.12971e7	−4.06447
$791$	−6.87782e6	−0.390850
$792$	0	0
$793$	−5.19607e6	−0.293421
$794$	3.10196e7	1.74616
$795$	0	0
$796$	4.00412e6	0.223988
$797$	−1.36904e7	−0.763432	−0.381716	−	0.924280i	$-0.624667\pi$
$797$	−1.36904e7	−0.763432	−0.381716	+	0.924280i	$0.624667\pi$
$798$	0	0
$799$	−4.34737e6	−0.240913
$800$	2.41826e6	0.133592
$801$	0	0
$802$	−503101.	−0.0276197
$803$	−1.01498e7	−0.555481
$804$	0	0
$805$	3.10060e6	0.168638
$806$	2.00333e7	1.08621
$807$	0	0
$808$	2.97958e7	1.60556
$809$	−1.60241e7	−0.860798	−0.430399	−	0.902639i	$-0.641627\pi$
$809$	−1.60241e7	−0.860798	−0.430399	+	0.902639i	$0.641627\pi$
$810$	0	0
$811$	1.65069e7	0.881280	0.440640	−	0.897684i	$-0.354751\pi$
$811$	1.65069e7	0.881280	0.440640	+	0.897684i	$0.354751\pi$
$812$	−3.27849e7	−1.74495
$813$	0	0
$814$	424056.	0.0224317
$815$	−144609.	−0.00762607
$816$	0	0
$817$	−2.34942e6	−0.123142
$818$	−3.00899e6	−0.157231
$819$	0	0
$820$	9.32227e6	0.484158
$821$	1.03502e7	0.535909	0.267954	−	0.963432i	$-0.413652\pi$
$821$	1.03502e7	0.535909	0.267954	+	0.963432i	$0.413652\pi$
$822$	0	0
$823$	−6.59246e6	−0.339272	−0.169636	−	0.985507i	$-0.554259\pi$
$823$	−6.59246e6	−0.339272	−0.169636	+	0.985507i	$0.554259\pi$
$824$	−5.18977e7	−2.66275
$825$	0	0
$826$	−2.70036e7	−1.37712
$827$	2.22190e7	1.12969	0.564846	−	0.825196i	$-0.308936\pi$
$827$	2.22190e7	1.12969	0.564846	+	0.825196i	$0.308936\pi$
$828$	0	0
$829$	−3.02725e7	−1.52990	−0.764948	−	0.644092i	$-0.777235\pi$
$829$	−3.02725e7	−1.52990	−0.764948	+	0.644092i	$0.777235\pi$
$830$	6.55298e7	3.30175
$831$	0	0
$832$	−1.42035e7	−0.711358
$833$	−1.07285e7	−0.535707
$834$	0	0
$835$	−5.91899e6	−0.293786
$836$	2.98413e7	1.47674
$837$	0	0
$838$	−3.62083e7	−1.78114
$839$	−3.18956e7	−1.56432	−0.782161	−	0.623077i	$-0.785883\pi$
$839$	−3.18956e7	−1.56432	−0.782161	+	0.623077i	$0.785883\pi$
$840$	0	0
$841$	4.86204e6	0.237044
$842$	−2.04336e7	−0.993266
$843$	0	0
$844$	1.04565e7	0.505278
$845$	1.24139e7	0.598090
$846$	0	0
$847$	1.01807e7	0.487607
$848$	3.26749e7	1.56036
$849$	0	0
$850$	6.53095e7	3.10048
$851$	63897.3	0.00302454
$852$	0	0
$853$	−2.64293e7	−1.24369	−0.621847	−	0.783139i	$-0.713617\pi$
$853$	−2.64293e7	−1.24369	−0.621847	+	0.783139i	$0.713617\pi$
$854$	1.07893e7	0.506232
$855$	0	0
$856$	3.19851e7	1.49198
$857$	−4.11029e7	−1.91170	−0.955851	−	0.293853i	$-0.905062\pi$
$857$	−4.11029e7	−1.91170	−0.955851	+	0.293853i	$0.905062\pi$
$858$	0	0
$859$	−2.35423e7	−1.08860	−0.544298	−	0.838892i	$-0.683204\pi$
$859$	−2.35423e7	−1.08860	−0.544298	+	0.838892i	$0.683204\pi$
$860$	6.91403e6	0.318775
$861$	0	0
$862$	3.56040e7	1.63204
$863$	1.46120e7	0.667857	0.333928	−	0.942598i	$-0.391626\pi$
$863$	1.46120e7	0.667857	0.333928	+	0.942598i	$0.391626\pi$
$864$	0	0
$865$	5.28182e7	2.40018
$866$	6.15596e7	2.78934
$867$	0	0
$868$	−2.78616e7	−1.25518
$869$	2.06812e7	0.929024
$870$	0	0
$871$	−772243.	−0.0344912
$872$	5.68336e7	2.53113
$873$	0	0
$874$	6.71339e6	0.297278
$875$	8.99735e6	0.397278
$876$	0	0
$877$	−3.40898e7	−1.49667	−0.748333	−	0.663323i	$-0.769146\pi$
$877$	−3.40898e7	−1.49667	−0.748333	+	0.663323i	$0.769146\pi$
$878$	−4.28141e7	−1.87435
$879$	0	0
$880$	−2.31770e7	−1.00891
$881$	−6.44135e6	−0.279600	−0.139800	−	0.990180i	$-0.544646\pi$
$881$	−6.44135e6	−0.279600	−0.139800	+	0.990180i	$0.544646\pi$
$882$	0	0
$883$	−2.66000e7	−1.14810	−0.574051	−	0.818820i	$-0.694629\pi$
$883$	−2.66000e7	−1.14810	−0.574051	+	0.818820i	$0.694629\pi$
$884$	4.90311e7	2.11028
$885$	0	0
$886$	4.33715e7	1.85618
$887$	−3.70827e7	−1.58257	−0.791284	−	0.611449i	$-0.790587\pi$
$887$	−3.70827e7	−1.58257	−0.791284	+	0.611449i	$0.790587\pi$
$888$	0	0
$889$	1.12964e7	0.479387
$890$	3.27092e7	1.38419
$891$	0	0
$892$	−1.11364e7	−0.468631
$893$	−5.15563e6	−0.216348
$894$	0	0
$895$	5.36513e7	2.23884
$896$	3.13514e7	1.30463
$897$	0	0
$898$	5.88625e7	2.43584
$899$	2.15629e7	0.889834
$900$	0	0
$901$	4.66979e7	1.91640
$902$	−4.03730e6	−0.165225
$903$	0	0
$904$	2.22183e7	0.904253
$905$	−2.09515e7	−0.850344
$906$	0	0
$907$	−2.53650e7	−1.02380	−0.511901	−	0.859044i	$-0.671059\pi$
$907$	−2.53650e7	−1.02380	−0.511901	+	0.859044i	$0.671059\pi$
$908$	−1.88769e7	−0.759830
$909$	0	0
$910$	4.00955e7	1.60506
$911$	3.58728e7	1.43209	0.716043	−	0.698056i	$-0.245951\pi$
$911$	3.58728e7	1.43209	0.716043	+	0.698056i	$0.245951\pi$
$912$	0	0
$913$	−1.90083e7	−0.754687
$914$	1.84169e6	0.0729206
$915$	0	0
$916$	8.89693e7	3.50349
$917$	−2.61647e7	−1.02752
$918$	0	0
$919$	−1.28550e6	−0.0502094	−0.0251047	−	0.999685i	$-0.507992\pi$
$919$	−1.28550e6	−0.0502094	−0.0251047	+	0.999685i	$0.507992\pi$
$920$	−1.00163e7	−0.390155
$921$	0	0
$922$	−2.51097e6	−0.0972779
$923$	3.09235e6	0.119477
$924$	0	0
$925$	737178.	0.0283281
$926$	4.03050e7	1.54466
$927$	0	0
$928$	2.91756e6	0.111211
$929$	−4.11678e7	−1.56501	−0.782507	−	0.622641i	$-0.786060\pi$
$929$	−4.11678e7	−1.56501	−0.782507	+	0.622641i	$0.786060\pi$
$930$	0	0
$931$	−1.27232e7	−0.481084
$932$	9.30514e7	3.50900
$933$	0	0
$934$	−3.37172e7	−1.26469
$935$	−3.31238e7	−1.23912
$936$	0	0
$937$	4.53598e6	0.168780	0.0843901	−	0.996433i	$-0.473106\pi$
$937$	4.53598e6	0.168780	0.0843901	+	0.996433i	$0.473106\pi$
$938$	1.60352e6	0.0595068
$939$	0	0
$940$	1.51723e7	0.560058
$941$	−2.36926e7	−0.872246	−0.436123	−	0.899887i	$-0.643649\pi$
$941$	−2.36926e7	−0.872246	−0.436123	+	0.899887i	$0.643649\pi$
$942$	0	0
$943$	−608345.	−0.0222777
$944$	3.04152e7	1.11086
$945$	0	0
$946$	−2.99433e6	−0.108786
$947$	−2.52845e7	−0.916177	−0.458088	−	0.888907i	$-0.651466\pi$
$947$	−2.52845e7	−0.916177	−0.458088	+	0.888907i	$0.651466\pi$
$948$	0	0
$949$	−1.97773e7	−0.712855
$950$	7.74518e7	2.78434
$951$	0	0
$952$	−5.16160e7	−1.84583
$953$	−1.98011e7	−0.706249	−0.353124	−	0.935576i	$-0.614881\pi$
$953$	−1.98011e7	−0.706249	−0.353124	+	0.935576i	$0.614881\pi$
$954$	0	0
$955$	4.98081e7	1.76722
$956$	4.53922e7	1.60634
$957$	0	0
$958$	−3.73043e7	−1.31324
$959$	−3.38923e7	−1.19002
$960$	0	0
$961$	−1.03043e7	−0.359923
$962$	826289.	0.0287869
$963$	0	0
$964$	−1.95713e7	−0.678309
$965$	−1.13680e7	−0.392975
$966$	0	0
$967$	−1.24216e7	−0.427179	−0.213590	−	0.976923i	$-0.568516\pi$
$967$	−1.24216e7	−0.427179	−0.213590	+	0.976923i	$0.568516\pi$
$968$	−3.28881e7	−1.12811
$969$	0	0
$970$	7.74897e7	2.64433
$971$	−1.74492e7	−0.593918	−0.296959	−	0.954890i	$-0.595972\pi$
$971$	−1.74492e7	−0.593918	−0.296959	+	0.954890i	$0.595972\pi$
$972$	0	0
$973$	−2.64050e7	−0.894135
$974$	−3.96109e7	−1.33788
$975$	0	0
$976$	−1.21525e7	−0.408357
$977$	4.42537e7	1.48325	0.741623	−	0.670817i	$-0.234056\pi$
$977$	4.42537e7	1.48325	0.741623	+	0.670817i	$0.234056\pi$
$978$	0	0
$979$	−9.48798e6	−0.316386
$980$	3.74426e7	1.24538
$981$	0	0
$982$	−9.92465e7	−3.28425
$983$	1.38881e7	0.458415	0.229208	−	0.973378i	$-0.426387\pi$
$983$	1.38881e7	0.458415	0.229208	+	0.973378i	$0.426387\pi$
$984$	0	0
$985$	−6.57069e7	−2.15785
$986$	7.87938e7	2.58107
$987$	0	0
$988$	5.81469e7	1.89511
$989$	−451190.	−0.0146679
$990$	0	0
$991$	−5.37809e7	−1.73958	−0.869790	−	0.493423i	$-0.835746\pi$
$991$	−5.37809e7	−1.73958	−0.869790	+	0.493423i	$0.835746\pi$
$992$	2.47943e6	0.0799969
$993$	0	0
$994$	−6.42109e6	−0.206131
$995$	−5.27090e6	−0.168782
$996$	0	0
$997$	1.87136e7	0.596238	0.298119	−	0.954529i	$-0.403641\pi$
$997$	1.87136e7	0.596238	0.298119	+	0.954529i	$0.403641\pi$
$998$	5.70943e7	1.81454
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	369.6.a.e.1.9		10
3.2	odd	2		41.6.a.b.1.2	✓	10
12.11	even	2		656.6.a.g.1.1		10
15.14	odd	2		1025.6.a.b.1.9		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.6.a.b.1.2	✓	10	3.2	odd	2
369.6.a.e.1.9		10	1.1	even	1	trivial
656.6.a.g.1.1		10	12.11	even	2
1025.6.a.b.1.9		10	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 369 = 3^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	369.a (trivial)

\(f(q)\)	\(=\)	\(q+9.84411 q^{2} +64.9065 q^{4} -85.4409 q^{5} -100.276 q^{7} +323.935 q^{8} +O(q^{10})\)	\(q+9.84411 q^{2} +64.9065 q^{4} -85.4409 q^{5} -100.276 q^{7} +323.935 q^{8} -841.090 q^{10} +243.976 q^{11} +475.395 q^{13} -987.130 q^{14} +1111.85 q^{16} +1589.02 q^{17} +1884.44 q^{19} -5545.67 q^{20} +2401.72 q^{22} +361.895 q^{23} +4175.15 q^{25} +4679.85 q^{26} -6508.58 q^{28} +5037.18 q^{29} +4280.76 q^{31} +579.204 q^{32} +15642.4 q^{34} +8567.69 q^{35} +176.563 q^{37} +18550.7 q^{38} -27677.3 q^{40} -1681.00 q^{41} -1246.74 q^{43} +15835.6 q^{44} +3562.53 q^{46} -2735.89 q^{47} -6751.67 q^{49} +41100.6 q^{50} +30856.3 q^{52} +29387.9 q^{53} -20845.5 q^{55} -32483.0 q^{56} +49586.6 q^{58} +27355.6 q^{59} -10930.0 q^{61} +42140.2 q^{62} -29877.3 q^{64} -40618.2 q^{65} -1624.42 q^{67} +103137. q^{68} +84341.3 q^{70} +6504.80 q^{71} -41601.7 q^{73} +1738.11 q^{74} +122313. q^{76} -24465.0 q^{77} +84767.6 q^{79} -94997.1 q^{80} -16547.9 q^{82} -77910.7 q^{83} -135767. q^{85} -12273.1 q^{86} +79032.3 q^{88} -38889.0 q^{89} -47670.9 q^{91} +23489.3 q^{92} -26932.4 q^{94} -161009. q^{95} -92130.2 q^{97} -66464.2 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 3 q^{2} + 207 q^{4} - 32 q^{5} + 342 q^{7} - 249 q^{8}+O(q^{10}) \) 10 * q - 3 * q^2 + 207 * q^4 - 32 * q^5 + 342 * q^7 - 249 * q^8	\( 10 q - 3 q^{2} + 207 q^{4} - 32 q^{5} + 342 q^{7} - 249 q^{8} + 102 q^{10} - 846 q^{11} + 1504 q^{13} - 3468 q^{14} + 5859 q^{16} - 560 q^{17} + 4240 q^{19} + 6182 q^{20} - 2628 q^{22} + 1508 q^{23} + 11734 q^{25} + 22014 q^{26} - 8662 q^{28} + 124 q^{29} + 10384 q^{31} + 6619 q^{32} + 802 q^{34} + 17890 q^{35} + 5524 q^{37} + 46098 q^{38} - 61738 q^{40} - 16810 q^{41} + 24160 q^{43} + 21594 q^{44} + 42404 q^{46} - 58984 q^{47} + 70326 q^{49} - 6817 q^{50} + 64374 q^{52} - 23456 q^{53} + 96426 q^{55} - 80184 q^{56} - 13378 q^{58} - 52428 q^{59} + 113540 q^{61} + 113008 q^{62} + 37363 q^{64} + 22340 q^{65} + 85506 q^{67} + 71406 q^{68} - 71946 q^{70} - 75236 q^{71} - 85148 q^{73} + 23462 q^{74} + 113376 q^{76} + 172896 q^{77} + 178200 q^{79} + 401850 q^{80} + 5043 q^{82} + 125412 q^{83} - 245912 q^{85} + 18848 q^{86} - 135952 q^{88} + 62696 q^{89} + 30056 q^{91} + 236372 q^{92} + 419014 q^{94} - 28002 q^{95} + 154548 q^{97} - 288367 q^{98}+O(q^{100}) \) 10 * q - 3 * q^2 + 207 * q^4 - 32 * q^5 + 342 * q^7 - 249 * q^8 + 102 * q^10 - 846 * q^11 + 1504 * q^13 - 3468 * q^14 + 5859 * q^16 - 560 * q^17 + 4240 * q^19 + 6182 * q^20 - 2628 * q^22 + 1508 * q^23 + 11734 * q^25 + 22014 * q^26 - 8662 * q^28 + 124 * q^29 + 10384 * q^31 + 6619 * q^32 + 802 * q^34 + 17890 * q^35 + 5524 * q^37 + 46098 * q^38 - 61738 * q^40 - 16810 * q^41 + 24160 * q^43 + 21594 * q^44 + 42404 * q^46 - 58984 * q^47 + 70326 * q^49 - 6817 * q^50 + 64374 * q^52 - 23456 * q^53 + 96426 * q^55 - 80184 * q^56 - 13378 * q^58 - 52428 * q^59 + 113540 * q^61 + 113008 * q^62 + 37363 * q^64 + 22340 * q^65 + 85506 * q^67 + 71406 * q^68 - 71946 * q^70 - 75236 * q^71 - 85148 * q^73 + 23462 * q^74 + 113376 * q^76 + 172896 * q^77 + 178200 * q^79 + 401850 * q^80 + 5043 * q^82 + 125412 * q^83 - 245912 * q^85 + 18848 * q^86 - 135952 * q^88 + 62696 * q^89 + 30056 * q^91 + 236372 * q^92 + 419014 * q^94 - 28002 * q^95 + 154548 * q^97 - 288367 * q^98

Introduction

L-functions

Modular forms

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Database

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