LMFDB - Embedded newform 495.6.a.p.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,6,Mod(1,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("495.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	495.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:	$79.3899908074$
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} - 246 x^{8} + 640 x^{7} + 20433 x^{6} - 44595 x^{5} - 667026 x^{4} + 1173648 x^{3} + \cdots - 30445728 $ x^10 - 3x^9 - 246x^8 + 640x^7 + 20433x^6 - 44595x^5 - 667026x^4 + 1173648x^3 + 7949136x^2 - 8226800*x - 30445728
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{6} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-6.41346$ of defining polynomial
Character	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.41346 q^{2} +9.13253 q^{4} +25.0000 q^{5} +14.3353 q^{7} +146.660 q^{8} +O(q^{10})$	\(q-6.41346 q^{2} +9.13253 q^{4} +25.0000 q^{5} +14.3353 q^{7} +146.660 q^{8} -160.337 q^{10} -121.000 q^{11} -434.702 q^{13} -91.9389 q^{14} -1232.84 q^{16} +92.0787 q^{17} +1969.29 q^{19} +228.313 q^{20} +776.029 q^{22} +1123.87 q^{23} +625.000 q^{25} +2787.95 q^{26} +130.917 q^{28} -1311.10 q^{29} +8137.51 q^{31} +3213.65 q^{32} -590.543 q^{34} +358.382 q^{35} -5371.33 q^{37} -12630.0 q^{38} +3666.49 q^{40} -19026.6 q^{41} +11767.5 q^{43} -1105.04 q^{44} -7207.93 q^{46} +27941.3 q^{47} -16601.5 q^{49} -4008.42 q^{50} -3969.93 q^{52} -33963.1 q^{53} -3025.00 q^{55} +2102.41 q^{56} +8408.70 q^{58} -32534.8 q^{59} +47721.9 q^{61} -52189.6 q^{62} +18840.2 q^{64} -10867.5 q^{65} -8869.10 q^{67} +840.911 q^{68} -2298.47 q^{70} +26131.7 q^{71} -70330.2 q^{73} +34448.8 q^{74} +17984.6 q^{76} -1734.57 q^{77} +96260.0 q^{79} -30820.9 q^{80} +122027. q^{82} -71581.3 q^{83} +2301.97 q^{85} -75470.3 q^{86} -17745.8 q^{88} -40176.7 q^{89} -6231.58 q^{91} +10263.8 q^{92} -179201. q^{94} +49232.3 q^{95} -124148. q^{97} +106473. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 3 q^{2} + 181 q^{4} + 250 q^{5} + 116 q^{7} + 129 q^{8}+O(q^{10}) $ 10 * q + 3 * q^2 + 181 * q^4 + 250 * q^5 + 116 * q^7 + 129 * q^8	$ 10 q + 3 q^{2} + 181 q^{4} + 250 q^{5} + 116 q^{7} + 129 q^{8} + 75 q^{10} - 1210 q^{11} + 932 q^{13} - 1332 q^{14} + 2701 q^{16} + 96 q^{17} + 1664 q^{19} + 4525 q^{20} - 363 q^{22} + 6288 q^{23} + 6250 q^{25} + 13380 q^{26} + 13868 q^{28} + 11208 q^{29} + 9032 q^{31} + 9801 q^{32} + 14610 q^{34} + 2900 q^{35} + 21572 q^{37} + 15870 q^{38} + 3225 q^{40} + 10800 q^{41} + 21128 q^{43} - 21901 q^{44} + 83982 q^{46} - 17400 q^{47} + 71610 q^{49} + 1875 q^{50} + 40640 q^{52} + 5004 q^{53} - 30250 q^{55} - 54012 q^{56} - 9786 q^{58} - 25272 q^{59} + 52004 q^{61} + 34740 q^{62} + 56953 q^{64} + 23300 q^{65} + 4160 q^{67} - 87978 q^{68} - 33300 q^{70} - 65232 q^{71} + 44252 q^{73} - 49842 q^{74} + 233246 q^{76} - 14036 q^{77} + 112604 q^{79} + 67525 q^{80} + 167910 q^{82} + 70032 q^{83} + 2400 q^{85} - 72978 q^{86} - 15609 q^{88} - 46848 q^{89} + 130672 q^{91} + 121302 q^{92} + 252294 q^{94} + 41600 q^{95} + 129932 q^{97} - 316137 q^{98}+O(q^{100}) $ 10 * q + 3 * q^2 + 181 * q^4 + 250 * q^5 + 116 * q^7 + 129 * q^8 + 75 * q^10 - 1210 * q^11 + 932 * q^13 - 1332 * q^14 + 2701 * q^16 + 96 * q^17 + 1664 * q^19 + 4525 * q^20 - 363 * q^22 + 6288 * q^23 + 6250 * q^25 + 13380 * q^26 + 13868 * q^28 + 11208 * q^29 + 9032 * q^31 + 9801 * q^32 + 14610 * q^34 + 2900 * q^35 + 21572 * q^37 + 15870 * q^38 + 3225 * q^40 + 10800 * q^41 + 21128 * q^43 - 21901 * q^44 + 83982 * q^46 - 17400 * q^47 + 71610 * q^49 + 1875 * q^50 + 40640 * q^52 + 5004 * q^53 - 30250 * q^55 - 54012 * q^56 - 9786 * q^58 - 25272 * q^59 + 52004 * q^61 + 34740 * q^62 + 56953 * q^64 + 23300 * q^65 + 4160 * q^67 - 87978 * q^68 - 33300 * q^70 - 65232 * q^71 + 44252 * q^73 - 49842 * q^74 + 233246 * q^76 - 14036 * q^77 + 112604 * q^79 + 67525 * q^80 + 167910 * q^82 + 70032 * q^83 + 2400 * q^85 - 72978 * q^86 - 15609 * q^88 - 46848 * q^89 + 130672 * q^91 + 121302 * q^92 + 252294 * q^94 + 41600 * q^95 + 129932 * q^97 - 316137 * 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 $

$2$	−6.41346	−1.13375	−0.566876	−	0.823803i	$-0.691848\pi$
$2$	−6.41346	−1.13375	−0.566876	+	0.823803i	$0.691848\pi$
$3$	0	0
$3$	0	0
$4$	9.13253	0.285391
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	14.3353	0.110576	0.0552881	−	0.998470i	$-0.482392\pi$
$7$	14.3353	0.110576	0.0552881	+	0.998470i	$0.482392\pi$
$8$	146.660	0.810188
$9$	0	0
$10$	−160.337	−0.507029
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	−434.702	−0.713400	−0.356700	−	0.934219i	$-0.616098\pi$
$13$	−434.702	−0.713400	−0.356700	+	0.934219i	$0.616098\pi$
$14$	−91.9389	−0.125366
$15$	0	0
$16$	−1232.84	−1.20394
$17$	92.0787	0.0772746	0.0386373	−	0.999253i	$-0.487698\pi$
$17$	92.0787	0.0772746	0.0386373	+	0.999253i	$0.487698\pi$
$18$	0	0
$19$	1969.29	1.25149	0.625743	−	0.780029i	$-0.284796\pi$
$19$	1969.29	1.25149	0.625743	+	0.780029i	$0.284796\pi$
$20$	228.313	0.127631
$21$	0	0
$22$	776.029	0.341839
$23$	1123.87	0.442994	0.221497	−	0.975161i	$-0.428906\pi$
$23$	1123.87	0.442994	0.221497	+	0.975161i	$0.428906\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	2787.95	0.808818
$27$	0	0
$28$	130.917	0.0315575
$29$	−1311.10	−0.289495	−0.144748	−	0.989469i	$-0.546237\pi$
$29$	−1311.10	−0.289495	−0.144748	+	0.989469i	$0.546237\pi$
$30$	0	0
$31$	8137.51	1.52085	0.760427	−	0.649424i	$-0.224990\pi$
$31$	8137.51	1.52085	0.760427	+	0.649424i	$0.224990\pi$
$32$	3213.65	0.554784
$33$	0	0
$34$	−590.543	−0.0876102
$35$	358.382	0.0494511
$36$	0	0
$37$	−5371.33	−0.645026	−0.322513	−	0.946565i	$-0.604528\pi$
$37$	−5371.33	−0.645026	−0.322513	+	0.946565i	$0.604528\pi$
$38$	−12630.0	−1.41887
$39$	0	0
$40$	3666.49	0.362327
$41$	−19026.6	−1.76767	−0.883836	−	0.467796i	$-0.845048\pi$
$41$	−19026.6	−1.76767	−0.883836	+	0.467796i	$0.845048\pi$
$42$	0	0
$43$	11767.5	0.970537	0.485269	−	0.874365i	$-0.338722\pi$
$43$	11767.5	0.970537	0.485269	+	0.874365i	$0.338722\pi$
$44$	−1105.04	−0.0860487
$45$	0	0
$46$	−7207.93	−0.502245
$47$	27941.3	1.84503	0.922513	−	0.385965i	$-0.126131\pi$
$47$	27941.3	1.84503	0.922513	+	0.385965i	$0.126131\pi$
$48$	0	0
$49$	−16601.5	−0.987773
$50$	−4008.42	−0.226750
$51$	0	0
$52$	−3969.93	−0.203598
$53$	−33963.1	−1.66080	−0.830401	−	0.557166i	$-0.811889\pi$
$53$	−33963.1	−1.66080	−0.830401	+	0.557166i	$0.811889\pi$
$54$	0	0
$55$	−3025.00	−0.134840
$56$	2102.41	0.0895875
$57$	0	0
$58$	8408.70	0.328215
$59$	−32534.8	−1.21680	−0.608398	−	0.793632i	$-0.708188\pi$
$59$	−32534.8	−1.21680	−0.608398	+	0.793632i	$0.708188\pi$
$60$	0	0
$61$	47721.9	1.64207	0.821037	−	0.570875i	$-0.193396\pi$
$61$	47721.9	1.64207	0.821037	+	0.570875i	$0.193396\pi$
$62$	−52189.6	−1.72427
$63$	0	0
$64$	18840.2	0.574957
$65$	−10867.5	−0.319042
$66$	0	0
$67$	−8869.10	−0.241375	−0.120688	−	0.992691i	$-0.538510\pi$
$67$	−8869.10	−0.241375	−0.120688	+	0.992691i	$0.538510\pi$
$68$	840.911	0.0220535
$69$	0	0
$70$	−2298.47	−0.0560653
$71$	26131.7	0.615208	0.307604	−	0.951514i	$-0.400473\pi$
$71$	26131.7	0.615208	0.307604	+	0.951514i	$0.400473\pi$
$72$	0	0
$73$	−70330.2	−1.54467	−0.772333	−	0.635218i	$-0.780910\pi$
$73$	−70330.2	−1.54467	−0.772333	+	0.635218i	$0.780910\pi$
$74$	34448.8	0.731299
$75$	0	0
$76$	17984.6	0.357163
$77$	−1734.57	−0.0333400
$78$	0	0
$79$	96260.0	1.73531	0.867657	−	0.497163i	$-0.165625\pi$
$79$	96260.0	1.73531	0.867657	+	0.497163i	$0.165625\pi$
$80$	−30820.9	−0.538420
$81$	0	0
$82$	122027.	2.00410
$83$	−71581.3	−1.14052	−0.570262	−	0.821463i	$-0.693158\pi$
$83$	−71581.3	−1.14052	−0.570262	+	0.821463i	$0.693158\pi$
$84$	0	0
$85$	2301.97	0.0345583
$86$	−75470.3	−1.10035
$87$	0	0
$88$	−17745.8	−0.244281
$89$	−40176.7	−0.537649	−0.268825	−	0.963189i	$-0.586635\pi$
$89$	−40176.7	−0.537649	−0.268825	+	0.963189i	$0.586635\pi$
$90$	0	0
$91$	−6231.58	−0.0788850
$92$	10263.8	0.126427
$93$	0	0
$94$	−179201.	−2.09180
$95$	49232.3	0.559682
$96$	0	0
$97$	−124148.	−1.33971	−0.669853	−	0.742494i	$-0.733643\pi$
$97$	−124148.	−1.33971	−0.669853	+	0.742494i	$0.733643\pi$
$98$	106473.	1.11989
$99$	0	0
$100$	5707.83	0.0570783
$101$	−132784.	−1.29522	−0.647609	−	0.761973i	$-0.724231\pi$
$101$	−132784.	−1.29522	−0.647609	+	0.761973i	$0.724231\pi$
$102$	0	0
$103$	192036.	1.78357	0.891783	−	0.452462i	$-0.149454\pi$
$103$	192036.	1.78357	0.891783	+	0.452462i	$0.149454\pi$
$104$	−63753.3	−0.577988
$105$	0	0
$106$	217821.	1.88294
$107$	−83635.0	−0.706202	−0.353101	−	0.935585i	$-0.614873\pi$
$107$	−83635.0	−0.706202	−0.353101	+	0.935585i	$0.614873\pi$
$108$	0	0
$109$	62912.2	0.507188	0.253594	−	0.967311i	$-0.418387\pi$
$109$	62912.2	0.507188	0.253594	+	0.967311i	$0.418387\pi$
$110$	19400.7	0.152875
$111$	0	0
$112$	−17673.1	−0.133127
$113$	253548.	1.86795	0.933974	−	0.357342i	$-0.116317\pi$
$113$	253548.	1.86795	0.933974	+	0.357342i	$0.116317\pi$
$114$	0	0
$115$	28096.8	0.198113
$116$	−11973.7	−0.0826194
$117$	0	0
$118$	208661.	1.37954
$119$	1319.97	0.00854473
$120$	0	0
$121$	14641.0	0.0909091
$122$	−306062.	−1.86170
$123$	0	0
$124$	74316.0	0.434038
$125$	15625.0	0.0894427
$126$	0	0
$127$	94022.7	0.517277	0.258639	−	0.965974i	$-0.416726\pi$
$127$	94022.7	0.517277	0.258639	+	0.965974i	$0.416726\pi$
$128$	−223668.	−1.20664
$129$	0	0
$130$	69698.6	0.361714
$131$	−80165.5	−0.408140	−0.204070	−	0.978956i	$-0.565417\pi$
$131$	−80165.5	−0.408140	−0.204070	+	0.978956i	$0.565417\pi$
$132$	0	0
$133$	28230.4	0.138384
$134$	56881.7	0.273659
$135$	0	0
$136$	13504.2	0.0626070
$137$	259034.	1.17911	0.589557	−	0.807727i	$-0.299302\pi$
$137$	259034.	1.17911	0.589557	+	0.807727i	$0.299302\pi$
$138$	0	0
$139$	52425.5	0.230147	0.115074	−	0.993357i	$-0.463290\pi$
$139$	52425.5	0.230147	0.115074	+	0.993357i	$0.463290\pi$
$140$	3272.93	0.0141129
$141$	0	0
$142$	−167595.	−0.697493
$143$	52598.9	0.215098
$144$	0	0
$145$	−32777.5	−0.129466
$146$	451060.	1.75127
$147$	0	0
$148$	−49053.8	−0.184085
$149$	259235.	0.956596	0.478298	−	0.878197i	$-0.341254\pi$
$149$	259235.	0.956596	0.478298	+	0.878197i	$0.341254\pi$
$150$	0	0
$151$	48469.3	0.172991	0.0864956	−	0.996252i	$-0.472433\pi$
$151$	48469.3	0.172991	0.0864956	+	0.996252i	$0.472433\pi$
$152$	288816.	1.01394
$153$	0	0
$154$	11124.6	0.0377992
$155$	203438.	0.680146
$156$	0	0
$157$	580025.	1.87801	0.939004	−	0.343905i	$-0.111750\pi$
$157$	580025.	1.87801	0.939004	+	0.343905i	$0.111750\pi$
$158$	−617360.	−1.96741
$159$	0	0
$160$	80341.2	0.248107
$161$	16111.1	0.0489846
$162$	0	0
$163$	179159.	0.528164	0.264082	−	0.964500i	$-0.414931\pi$
$163$	179159.	0.528164	0.264082	+	0.964500i	$0.414931\pi$
$164$	−173761.	−0.504479
$165$	0	0
$166$	459084.	1.29307
$167$	355532.	0.986479	0.493239	−	0.869894i	$-0.335813\pi$
$167$	355532.	0.986479	0.493239	+	0.869894i	$0.335813\pi$
$168$	0	0
$169$	−182327.	−0.491060
$170$	−14763.6	−0.0391805
$171$	0	0
$172$	107467.	0.276983
$173$	663926.	1.68657	0.843285	−	0.537466i	$-0.180619\pi$
$173$	663926.	1.68657	0.843285	+	0.537466i	$0.180619\pi$
$174$	0	0
$175$	8959.56	0.0221152
$176$	149173.	0.363003
$177$	0	0
$178$	257672.	0.609561
$179$	−737106.	−1.71948	−0.859741	−	0.510731i	$-0.829375\pi$
$179$	−737106.	−1.71948	−0.859741	+	0.510731i	$0.829375\pi$
$180$	0	0
$181$	352875.	0.800617	0.400309	−	0.916380i	$-0.368903\pi$
$181$	352875.	0.800617	0.400309	+	0.916380i	$0.368903\pi$
$182$	39966.0	0.0894360
$183$	0	0
$184$	164827.	0.358909
$185$	−134283.	−0.288464
$186$	0	0
$187$	−11141.5	−0.0232992
$188$	255175.	0.526555
$189$	0	0
$190$	−315750.	−0.634540
$191$	901648.	1.78836	0.894178	−	0.447712i	$-0.147761\pi$
$191$	901648.	1.78836	0.894178	+	0.447712i	$0.147761\pi$
$192$	0	0
$193$	574940.	1.11104	0.555519	−	0.831504i	$-0.312520\pi$
$193$	574940.	1.11104	0.555519	+	0.831504i	$0.312520\pi$
$194$	796217.	1.51889
$195$	0	0
$196$	−151614.	−0.281902
$197$	445669.	0.818177	0.409088	−	0.912495i	$-0.365847\pi$
$197$	445669.	0.818177	0.409088	+	0.912495i	$0.365847\pi$
$198$	0	0
$199$	−123668.	−0.221372	−0.110686	−	0.993855i	$-0.535305\pi$
$199$	−123668.	−0.221372	−0.110686	+	0.993855i	$0.535305\pi$
$200$	91662.3	0.162038
$201$	0	0
$202$	851606.	1.46845
$203$	−18795.0	−0.0320112
$204$	0	0
$205$	−475665.	−0.790527
$206$	−1.23162e6	−2.02212
$207$	0	0
$208$	535917.	0.858893
$209$	−238284.	−0.377337
$210$	0	0
$211$	133049.	0.205733	0.102867	−	0.994695i	$-0.467199\pi$
$211$	133049.	0.205733	0.102867	+	0.994695i	$0.467199\pi$
$212$	−310169.	−0.473979
$213$	0	0
$214$	536390.	0.800657
$215$	294187.	0.434037
$216$	0	0
$217$	116654.	0.168170
$218$	−403485.	−0.575025
$219$	0	0
$220$	−27625.9	−0.0384822
$221$	−40026.8	−0.0551277
$222$	0	0
$223$	232071.	0.312507	0.156253	−	0.987717i	$-0.450058\pi$
$223$	232071.	0.312507	0.156253	+	0.987717i	$0.450058\pi$
$224$	46068.6	0.0613458
$225$	0	0
$226$	−1.62612e6	−2.11779
$227$	638865.	0.822895	0.411448	−	0.911433i	$-0.365023\pi$
$227$	638865.	0.822895	0.411448	+	0.911433i	$0.365023\pi$
$228$	0	0
$229$	−1.17104e6	−1.47565	−0.737823	−	0.674994i	$-0.764146\pi$
$229$	−1.17104e6	−1.47565	−0.737823	+	0.674994i	$0.764146\pi$
$230$	−180198.	−0.224611
$231$	0	0
$232$	−192286.	−0.234546
$233$	101098.	0.121998	0.0609990	−	0.998138i	$-0.480571\pi$
$233$	101098.	0.121998	0.0609990	+	0.998138i	$0.480571\pi$
$234$	0	0
$235$	698534.	0.825121
$236$	−297125.	−0.347263
$237$	0	0
$238$	−8465.61	−0.00968759
$239$	372908.	0.422286	0.211143	−	0.977455i	$-0.432281\pi$
$239$	372908.	0.422286	0.211143	+	0.977455i	$0.432281\pi$
$240$	0	0
$241$	67018.5	0.0743279	0.0371640	−	0.999309i	$-0.488168\pi$
$241$	67018.5	0.0743279	0.0371640	+	0.999309i	$0.488168\pi$
$242$	−93899.5	−0.103068
$243$	0	0
$244$	435821.	0.468634
$245$	−415037.	−0.441745
$246$	0	0
$247$	−856055.	−0.892810
$248$	1.19345e6	1.23218
$249$	0	0
$250$	−100210.	−0.101406
$251$	−173151.	−0.173476	−0.0867381	−	0.996231i	$-0.527644\pi$
$251$	−173151.	−0.173476	−0.0867381	+	0.996231i	$0.527644\pi$
$252$	0	0
$253$	−135989.	−0.133568
$254$	−603011.	−0.586463
$255$	0	0
$256$	831599.	0.793074
$257$	−1.14577e6	−1.08209	−0.541047	−	0.840992i	$-0.681972\pi$
$257$	−1.14577e6	−1.08209	−0.541047	+	0.840992i	$0.681972\pi$
$258$	0	0
$259$	−76999.5	−0.0713245
$260$	−99248.2	−0.0910519
$261$	0	0
$262$	514138.	0.462729
$263$	312523.	0.278607	0.139304	−	0.990250i	$-0.455514\pi$
$263$	312523.	0.278607	0.139304	+	0.990250i	$0.455514\pi$
$264$	0	0
$265$	−849078.	−0.742733
$266$	−181054.	−0.156894
$267$	0	0
$268$	−80997.3	−0.0688864
$269$	1.24870e6	1.05215	0.526074	−	0.850439i	$-0.323664\pi$
$269$	1.24870e6	1.05215	0.526074	+	0.850439i	$0.323664\pi$
$270$	0	0
$271$	1.10002e6	0.909863	0.454931	−	0.890526i	$-0.349664\pi$
$271$	1.10002e6	0.909863	0.454931	+	0.890526i	$0.349664\pi$
$272$	−113518.	−0.0930342
$273$	0	0
$274$	−1.66131e6	−1.33682
$275$	−75625.0	−0.0603023
$276$	0	0
$277$	934855.	0.732057	0.366029	−	0.930604i	$-0.380717\pi$
$277$	934855.	0.732057	0.366029	+	0.930604i	$0.380717\pi$
$278$	−336229.	−0.260930
$279$	0	0
$280$	52560.2	0.0400647
$281$	795677.	0.601133	0.300567	−	0.953761i	$-0.402824\pi$
$281$	795677.	0.601133	0.300567	+	0.953761i	$0.402824\pi$
$282$	0	0
$283$	2.52040e6	1.87070	0.935350	−	0.353723i	$-0.115085\pi$
$283$	2.52040e6	1.87070	0.935350	+	0.353723i	$0.115085\pi$
$284$	238649.	0.175575
$285$	0	0
$286$	−337341.	−0.243868
$287$	−272752.	−0.195462
$288$	0	0
$289$	−1.41138e6	−0.994029
$290$	210218.	0.146782
$291$	0	0
$292$	−642292.	−0.440835
$293$	326185.	0.221971	0.110985	−	0.993822i	$-0.464599\pi$
$293$	326185.	0.221971	0.110985	+	0.993822i	$0.464599\pi$
$294$	0	0
$295$	−813370.	−0.544168
$296$	−787757.	−0.522593
$297$	0	0
$298$	−1.66260e6	−1.08454
$299$	−488550.	−0.316032
$300$	0	0
$301$	168690.	0.107318
$302$	−310856.	−0.196129
$303$	0	0
$304$	−2.42782e6	−1.50672
$305$	1.19305e6	0.734358
$306$	0	0
$307$	−1.23434e6	−0.747462	−0.373731	−	0.927537i	$-0.621922\pi$
$307$	−1.23434e6	−0.747462	−0.373731	+	0.927537i	$0.621922\pi$
$308$	−15841.0	−0.00951494
$309$	0	0
$310$	−1.30474e6	−0.771117
$311$	−2.55370e6	−1.49716	−0.748581	−	0.663043i	$-0.769265\pi$
$311$	−2.55370e6	−1.49716	−0.748581	+	0.663043i	$0.769265\pi$
$312$	0	0
$313$	2.78210e6	1.60514	0.802569	−	0.596559i	$-0.203466\pi$
$313$	2.78210e6	1.60514	0.802569	+	0.596559i	$0.203466\pi$
$314$	−3.71997e6	−2.12919
$315$	0	0
$316$	879097.	0.495244
$317$	594498.	0.332278	0.166139	−	0.986102i	$-0.446870\pi$
$317$	594498.	0.332278	0.166139	+	0.986102i	$0.446870\pi$
$318$	0	0
$319$	158643.	0.0872861
$320$	471005.	0.257128
$321$	0	0
$322$	−103328.	−0.0555363
$323$	181330.	0.0967081
$324$	0	0
$325$	−271689.	−0.142680
$326$	−1.14903e6	−0.598806
$327$	0	0
$328$	−2.79044e6	−1.43215
$329$	400547.	0.204016
$330$	0	0
$331$	236027.	0.118411	0.0592055	−	0.998246i	$-0.481143\pi$
$331$	236027.	0.118411	0.0592055	+	0.998246i	$0.481143\pi$
$332$	−653718.	−0.325496
$333$	0	0
$334$	−2.28019e6	−1.11842
$335$	−221728.	−0.107946
$336$	0	0
$337$	3.22953e6	1.54905	0.774524	−	0.632545i	$-0.217990\pi$
$337$	3.22953e6	1.54905	0.774524	+	0.632545i	$0.217990\pi$
$338$	1.16935e6	0.556740
$339$	0	0
$340$	21022.8	0.00986263
$341$	−984639.	−0.458555
$342$	0	0
$343$	−478921.	−0.219800
$344$	1.72581e6	0.786318
$345$	0	0
$346$	−4.25807e6	−1.91215
$347$	−1.63430e6	−0.728633	−0.364316	−	0.931275i	$-0.618697\pi$
$347$	−1.63430e6	−0.728633	−0.364316	+	0.931275i	$0.618697\pi$
$348$	0	0
$349$	−985103.	−0.432930	−0.216465	−	0.976290i	$-0.569453\pi$
$349$	−985103.	−0.432930	−0.216465	+	0.976290i	$0.569453\pi$
$350$	−57461.8	−0.0250732
$351$	0	0
$352$	−388852.	−0.167274
$353$	−218596.	−0.0933695	−0.0466848	−	0.998910i	$-0.514866\pi$
$353$	−218596.	−0.0933695	−0.0466848	+	0.998910i	$0.514866\pi$
$354$	0	0
$355$	653293.	0.275129
$356$	−366915.	−0.153441
$357$	0	0
$358$	4.72740e6	1.94946
$359$	−3.97704e6	−1.62864	−0.814318	−	0.580419i	$-0.802889\pi$
$359$	−3.97704e6	−1.62864	−0.814318	+	0.580419i	$0.802889\pi$
$360$	0	0
$361$	1.40201e6	0.566218
$362$	−2.26315e6	−0.907701
$363$	0	0
$364$	−56910.0	−0.0225131
$365$	−1.75826e6	−0.690796
$366$	0	0
$367$	−1.44478e6	−0.559933	−0.279967	−	0.960010i	$-0.590323\pi$
$367$	−1.44478e6	−0.559933	−0.279967	+	0.960010i	$0.590323\pi$
$368$	−1.38555e6	−0.533340
$369$	0	0
$370$	861220.	0.327047
$371$	−486871.	−0.183645
$372$	0	0
$373$	−1.28919e6	−0.479784	−0.239892	−	0.970800i	$-0.577112\pi$
$373$	−1.28919e6	−0.479784	−0.239892	+	0.970800i	$0.577112\pi$
$374$	71455.8	0.0264155
$375$	0	0
$376$	4.09787e6	1.49482
$377$	569938.	0.206526
$378$	0	0
$379$	−4.37935e6	−1.56607	−0.783036	−	0.621976i	$-0.786330\pi$
$379$	−4.37935e6	−1.56607	−0.783036	+	0.621976i	$0.786330\pi$
$380$	449615.	0.159728
$381$	0	0
$382$	−5.78269e6	−2.02755
$383$	2.63073e6	0.916389	0.458194	−	0.888852i	$-0.348496\pi$
$383$	2.63073e6	0.916389	0.458194	+	0.888852i	$0.348496\pi$
$384$	0	0
$385$	−43364.3	−0.0149101
$386$	−3.68736e6	−1.25964
$387$	0	0
$388$	−1.13378e6	−0.382341
$389$	17933.8	0.00600896	0.00300448	−	0.999995i	$-0.499044\pi$
$389$	17933.8	0.00600896	0.00300448	+	0.999995i	$0.499044\pi$
$390$	0	0
$391$	103485.	0.0342322
$392$	−2.43477e6	−0.800282
$393$	0	0
$394$	−2.85828e6	−0.927608
$395$	2.40650e6	0.776056
$396$	0	0
$397$	−3.76201e6	−1.19796	−0.598981	−	0.800763i	$-0.704428\pi$
$397$	−3.76201e6	−1.19796	−0.598981	+	0.800763i	$0.704428\pi$
$398$	793138.	0.250981
$399$	0	0
$400$	−770524.	−0.240789
$401$	4.23902e6	1.31645	0.658225	−	0.752821i	$-0.271308\pi$
$401$	4.23902e6	1.31645	0.658225	+	0.752821i	$0.271308\pi$
$402$	0	0
$403$	−3.53739e6	−1.08498
$404$	−1.21265e6	−0.369644
$405$	0	0
$406$	120541.	0.0362928
$407$	649931.	0.194483
$408$	0	0
$409$	839682.	0.248203	0.124101	−	0.992270i	$-0.460395\pi$
$409$	839682.	0.248203	0.124101	+	0.992270i	$0.460395\pi$
$410$	3.05066e6	0.896261
$411$	0	0
$412$	1.75377e6	0.509015
$413$	−466396.	−0.134549
$414$	0	0
$415$	−1.78953e6	−0.510058
$416$	−1.39698e6	−0.395783
$417$	0	0
$418$	1.52823e6	0.427807
$419$	3.85828e6	1.07364	0.536820	−	0.843697i	$-0.319625\pi$
$419$	3.85828e6	1.07364	0.536820	+	0.843697i	$0.319625\pi$
$420$	0	0
$421$	−335468.	−0.0922457	−0.0461229	−	0.998936i	$-0.514687\pi$
$421$	−335468.	−0.0922457	−0.0461229	+	0.998936i	$0.514687\pi$
$422$	−853303.	−0.233250
$423$	0	0
$424$	−4.98102e6	−1.34556
$425$	57549.2	0.0154549
$426$	0	0
$427$	684107.	0.181574
$428$	−763799.	−0.201544
$429$	0	0
$430$	−1.88676e6	−0.492090
$431$	2.34221e6	0.607341	0.303670	−	0.952777i	$-0.401788\pi$
$431$	2.34221e6	0.607341	0.303670	+	0.952777i	$0.401788\pi$
$432$	0	0
$433$	6.28743e6	1.61159	0.805793	−	0.592197i	$-0.201739\pi$
$433$	6.28743e6	1.61159	0.805793	+	0.592197i	$0.201739\pi$
$434$	−748154.	−0.190663
$435$	0	0
$436$	574548.	0.144747
$437$	2.21324e6	0.554401
$438$	0	0
$439$	−3.29141e6	−0.815119	−0.407560	−	0.913179i	$-0.633620\pi$
$439$	−3.29141e6	−0.815119	−0.407560	+	0.913179i	$0.633620\pi$
$440$	−443646.	−0.109246
$441$	0	0
$442$	256710.	0.0625011
$443$	2.88842e6	0.699281	0.349640	−	0.936884i	$-0.386304\pi$
$443$	2.88842e6	0.699281	0.349640	+	0.936884i	$0.386304\pi$
$444$	0	0
$445$	−1.00442e6	−0.240444
$446$	−1.48838e6	−0.354305
$447$	0	0
$448$	270079.	0.0635765
$449$	201254.	0.0471116	0.0235558	−	0.999723i	$-0.492501\pi$
$449$	201254.	0.0471116	0.0235558	+	0.999723i	$0.492501\pi$
$450$	0	0
$451$	2.30222e6	0.532973
$452$	2.31554e6	0.533096
$453$	0	0
$454$	−4.09734e6	−0.932958
$455$	−155789.	−0.0352785
$456$	0	0
$457$	−176976.	−0.0396390	−0.0198195	−	0.999804i	$-0.506309\pi$
$457$	−176976.	−0.0396390	−0.0198195	+	0.999804i	$0.506309\pi$
$458$	7.51041e6	1.67302
$459$	0	0
$460$	256595.	0.0565398
$461$	6.58406e6	1.44292	0.721459	−	0.692458i	$-0.243472\pi$
$461$	6.58406e6	1.44292	0.721459	+	0.692458i	$0.243472\pi$
$462$	0	0
$463$	7.30509e6	1.58370	0.791850	−	0.610716i	$-0.209118\pi$
$463$	7.30509e6	1.58370	0.791850	+	0.610716i	$0.209118\pi$
$464$	1.61638e6	0.348536
$465$	0	0
$466$	−648389.	−0.138315
$467$	−3.36268e6	−0.713498	−0.356749	−	0.934200i	$-0.616115\pi$
$467$	−3.36268e6	−0.713498	−0.356749	+	0.934200i	$0.616115\pi$
$468$	0	0
$469$	−127141.	−0.0266903
$470$	−4.48002e6	−0.935482
$471$	0	0
$472$	−4.77154e6	−0.985834
$473$	−1.42386e6	−0.292628
$474$	0	0
$475$	1.23081e6	0.250297
$476$	12054.7	0.00243859
$477$	0	0
$478$	−2.39163e6	−0.478768
$479$	7.00089e6	1.39417	0.697083	−	0.716991i	$-0.254481\pi$
$479$	7.00089e6	1.39417	0.697083	+	0.716991i	$0.254481\pi$
$480$	0	0
$481$	2.33493e6	0.460162
$482$	−429821.	−0.0842693
$483$	0	0
$484$	133709.	0.0259447
$485$	−3.10369e6	−0.599135
$486$	0	0
$487$	8.58719e6	1.64070	0.820349	−	0.571863i	$-0.193779\pi$
$487$	8.58719e6	1.64070	0.820349	+	0.571863i	$0.193779\pi$
$488$	6.99888e6	1.33039
$489$	0	0
$490$	2.66183e6	0.500829
$491$	−5.26590e6	−0.985755	−0.492878	−	0.870099i	$-0.664055\pi$
$491$	−5.26590e6	−0.985755	−0.492878	+	0.870099i	$0.664055\pi$
$492$	0	0
$493$	−120724.	−0.0223706
$494$	5.49028e6	1.01222
$495$	0	0
$496$	−1.00322e7	−1.83102
$497$	374606.	0.0680273
$498$	0	0
$499$	−5.15093e6	−0.926050	−0.463025	−	0.886345i	$-0.653236\pi$
$499$	−5.15093e6	−0.926050	−0.463025	+	0.886345i	$0.653236\pi$
$500$	142696.	0.0255262
$501$	0	0
$502$	1.11050e6	0.196679
$503$	1.80665e6	0.318385	0.159193	−	0.987248i	$-0.449111\pi$
$503$	1.80665e6	0.318385	0.159193	+	0.987248i	$0.449111\pi$
$504$	0	0
$505$	−3.31960e6	−0.579239
$506$	872159.	0.151433
$507$	0	0
$508$	858664.	0.147626
$509$	1.70125e6	0.291055	0.145527	−	0.989354i	$-0.453512\pi$
$509$	1.70125e6	0.291055	0.145527	+	0.989354i	$0.453512\pi$
$510$	0	0
$511$	−1.00820e6	−0.170803
$512$	1.82394e6	0.307493
$513$	0	0
$514$	7.34837e6	1.22683
$515$	4.80090e6	0.797635
$516$	0	0
$517$	−3.38090e6	−0.556297
$518$	493834.	0.0808642
$519$	0	0
$520$	−1.59383e6	−0.258484
$521$	−9.85677e6	−1.59089	−0.795445	−	0.606026i	$-0.792763\pi$
$521$	−9.85677e6	−1.59089	−0.795445	+	0.606026i	$0.792763\pi$
$522$	0	0
$523$	7.95753e6	1.27211	0.636054	−	0.771645i	$-0.280566\pi$
$523$	7.95753e6	1.27211	0.636054	+	0.771645i	$0.280566\pi$
$524$	−732113.	−0.116480
$525$	0	0
$526$	−2.00435e6	−0.315871
$527$	749291.	0.117523
$528$	0	0
$529$	−5.17325e6	−0.803756
$530$	5.44553e6	0.842074
$531$	0	0
$532$	257815.	0.0394937
$533$	8.27091e6	1.26106
$534$	0	0
$535$	−2.09088e6	−0.315823
$536$	−1.30074e6	−0.195559
$537$	0	0
$538$	−8.00848e6	−1.19287
$539$	2.00878e6	0.297825
$540$	0	0
$541$	1.35980e7	1.99747	0.998735	−	0.0502761i	$-0.0160101\pi$
$541$	1.35980e7	1.99747	0.998735	+	0.0502761i	$0.0160101\pi$
$542$	−7.05492e6	−1.03156
$543$	0	0
$544$	295909.	0.0428707
$545$	1.57281e6	0.226821
$546$	0	0
$547$	2.25258e6	0.321893	0.160947	−	0.986963i	$-0.448545\pi$
$547$	2.25258e6	0.321893	0.160947	+	0.986963i	$0.448545\pi$
$548$	2.36564e6	0.336509
$549$	0	0
$550$	485018.	0.0683678
$551$	−2.58194e6	−0.362299
$552$	0	0
$553$	1.37992e6	0.191884
$554$	−5.99566e6	−0.829971
$555$	0	0
$556$	478778.	0.0656821
$557$	−5.71996e6	−0.781188	−0.390594	−	0.920563i	$-0.627730\pi$
$557$	−5.71996e6	−0.781188	−0.390594	+	0.920563i	$0.627730\pi$
$558$	0	0
$559$	−5.11534e6	−0.692381
$560$	−441827.	−0.0595364
$561$	0	0
$562$	−5.10304e6	−0.681536
$563$	5.88397e6	0.782347	0.391173	−	0.920317i	$-0.372069\pi$
$563$	5.88397e6	0.782347	0.391173	+	0.920317i	$0.372069\pi$
$564$	0	0
$565$	6.33871e6	0.835371
$566$	−1.61645e7	−2.12091
$567$	0	0
$568$	3.83247e6	0.498434
$569$	4.80355e6	0.621988	0.310994	−	0.950412i	$-0.399338\pi$
$569$	4.80355e6	0.621988	0.310994	+	0.950412i	$0.399338\pi$
$570$	0	0
$571$	1.04711e7	1.34401	0.672004	−	0.740548i	$-0.265434\pi$
$571$	1.04711e7	1.34401	0.672004	+	0.740548i	$0.265434\pi$
$572$	480361.	0.0613872
$573$	0	0
$574$	1.74929e6	0.221606
$575$	702421.	0.0885988
$576$	0	0
$577$	−1.20038e6	−0.150099	−0.0750497	−	0.997180i	$-0.523912\pi$
$577$	−1.20038e6	−0.150099	−0.0750497	+	0.997180i	$0.523912\pi$
$578$	9.05183e6	1.12698
$579$	0	0
$580$	−299342.	−0.0369485
$581$	−1.02614e6	−0.126115
$582$	0	0
$583$	4.10954e6	0.500751
$584$	−1.03146e7	−1.25147
$585$	0	0
$586$	−2.09198e6	−0.251659
$587$	−231939.	−0.0277830	−0.0138915	−	0.999904i	$-0.504422\pi$
$587$	−231939.	−0.0277830	−0.0138915	+	0.999904i	$0.504422\pi$
$588$	0	0
$589$	1.60251e7	1.90333
$590$	5.21652e6	0.616951
$591$	0	0
$592$	6.62197e6	0.776575
$593$	6.15905e6	0.719246	0.359623	−	0.933098i	$-0.382905\pi$
$593$	6.15905e6	0.719246	0.359623	+	0.933098i	$0.382905\pi$
$594$	0	0
$595$	32999.4	0.00382132
$596$	2.36747e6	0.273004
$597$	0	0
$598$	3.13330e6	0.358302
$599$	−9.08345e6	−1.03439	−0.517195	−	0.855868i	$-0.673024\pi$
$599$	−9.08345e6	−1.03439	−0.517195	+	0.855868i	$0.673024\pi$
$600$	0	0
$601$	8.78606e6	0.992220	0.496110	−	0.868260i	$-0.334761\pi$
$601$	8.78606e6	0.992220	0.496110	+	0.868260i	$0.334761\pi$
$602$	−1.08189e6	−0.121672
$603$	0	0
$604$	442647.	0.0493702
$605$	366025.	0.0406558
$606$	0	0
$607$	1.13475e7	1.25005	0.625025	−	0.780605i	$-0.285089\pi$
$607$	1.13475e7	1.25005	0.625025	+	0.780605i	$0.285089\pi$
$608$	6.32861e6	0.694304
$609$	0	0
$610$	−7.65156e6	−0.832579
$611$	−1.21462e7	−1.31624
$612$	0	0
$613$	−3.69201e6	−0.396836	−0.198418	−	0.980117i	$-0.563580\pi$
$613$	−3.69201e6	−0.396836	−0.198418	+	0.980117i	$0.563580\pi$
$614$	7.91640e6	0.847435
$615$	0	0
$616$	−254392.	−0.0270116
$617$	−5.36991e6	−0.567877	−0.283938	−	0.958843i	$-0.591641\pi$
$617$	−5.36991e6	−0.567877	−0.283938	+	0.958843i	$0.591641\pi$
$618$	0	0
$619$	1.31328e7	1.37762	0.688809	−	0.724942i	$-0.258134\pi$
$619$	1.31328e7	1.37762	0.688809	+	0.724942i	$0.258134\pi$
$620$	1.85790e6	0.194108
$621$	0	0
$622$	1.63781e7	1.69741
$623$	−575944.	−0.0594512
$624$	0	0
$625$	390625.	0.0400000
$626$	−1.78429e7	−1.81983
$627$	0	0
$628$	5.29709e6	0.535968
$629$	−494585.	−0.0498441
$630$	0	0
$631$	−6.24016e6	−0.623911	−0.311955	−	0.950097i	$-0.600984\pi$
$631$	−6.24016e6	−0.623911	−0.311955	+	0.950097i	$0.600984\pi$
$632$	1.41175e7	1.40593
$633$	0	0
$634$	−3.81279e6	−0.376721
$635$	2.35057e6	0.231333
$636$	0	0
$637$	7.21670e6	0.704677
$638$	−1.01745e6	−0.0989607
$639$	0	0
$640$	−5.59169e6	−0.539626
$641$	−3.28646e6	−0.315924	−0.157962	−	0.987445i	$-0.550492\pi$
$641$	−3.28646e6	−0.315924	−0.157962	+	0.987445i	$0.550492\pi$
$642$	0	0
$643$	−3.62449e6	−0.345716	−0.172858	−	0.984947i	$-0.555300\pi$
$643$	−3.62449e6	−0.345716	−0.172858	+	0.984947i	$0.555300\pi$
$644$	147135.	0.0139798
$645$	0	0
$646$	−1.16295e6	−0.109643
$647$	−1.26440e7	−1.18747	−0.593737	−	0.804659i	$-0.702348\pi$
$647$	−1.26440e7	−1.18747	−0.593737	+	0.804659i	$0.702348\pi$
$648$	0	0
$649$	3.93671e6	0.366878
$650$	1.74247e6	0.161764
$651$	0	0
$652$	1.63617e6	0.150733
$653$	−3.56673e6	−0.327331	−0.163666	−	0.986516i	$-0.552332\pi$
$653$	−3.56673e6	−0.327331	−0.163666	+	0.986516i	$0.552332\pi$
$654$	0	0
$655$	−2.00414e6	−0.182526
$656$	2.34567e7	2.12818
$657$	0	0
$658$	−2.56890e6	−0.231303
$659$	−2.01187e7	−1.80462	−0.902309	−	0.431089i	$-0.858129\pi$
$659$	−2.01187e7	−1.80462	−0.902309	+	0.431089i	$0.858129\pi$
$660$	0	0
$661$	−2.12952e7	−1.89574	−0.947870	−	0.318656i	$-0.896768\pi$
$661$	−2.12952e7	−1.89574	−0.947870	+	0.318656i	$0.896768\pi$
$662$	−1.51375e6	−0.134249
$663$	0	0
$664$	−1.04981e7	−0.924039
$665$	705759.	0.0618874
$666$	0	0
$667$	−1.47351e6	−0.128245
$668$	3.24691e6	0.281533
$669$	0	0
$670$	1.42204e6	0.122384
$671$	−5.77435e6	−0.495104
$672$	0	0
$673$	2.13853e7	1.82003	0.910013	−	0.414581i	$-0.136072\pi$
$673$	2.13853e7	1.82003	0.910013	+	0.414581i	$0.136072\pi$
$674$	−2.07125e7	−1.75623
$675$	0	0
$676$	−1.66511e6	−0.140144
$677$	−8.93251e6	−0.749034	−0.374517	−	0.927220i	$-0.622192\pi$
$677$	−8.93251e6	−0.749034	−0.374517	+	0.927220i	$0.622192\pi$
$678$	0	0
$679$	−1.77969e6	−0.148140
$680$	337606.	0.0279987
$681$	0	0
$682$	6.31495e6	0.519887
$683$	−2.46420e6	−0.202127	−0.101063	−	0.994880i	$-0.532224\pi$
$683$	−2.46420e6	−0.202127	−0.101063	+	0.994880i	$0.532224\pi$
$684$	0	0
$685$	6.47585e6	0.527316
$686$	3.07154e6	0.249199
$687$	0	0
$688$	−1.45074e7	−1.16847
$689$	1.47638e7	1.18482
$690$	0	0
$691$	5.04891e6	0.402256	0.201128	−	0.979565i	$-0.435539\pi$
$691$	5.04891e6	0.402256	0.201128	+	0.979565i	$0.435539\pi$
$692$	6.06332e6	0.481333
$693$	0	0
$694$	1.04815e7	0.826088
$695$	1.31064e6	0.102925
$696$	0	0
$697$	−1.75195e6	−0.136596
$698$	6.31792e6	0.490835
$699$	0	0
$700$	81823.4	0.00631150
$701$	−1.43357e7	−1.10186	−0.550928	−	0.834552i	$-0.685726\pi$
$701$	−1.43357e7	−1.10186	−0.550928	+	0.834552i	$0.685726\pi$
$702$	0	0
$703$	−1.05777e7	−0.807241
$704$	−2.27966e6	−0.173356
$705$	0	0
$706$	1.40196e6	0.105858
$707$	−1.90350e6	−0.143220
$708$	0	0
$709$	7.20312e6	0.538152	0.269076	−	0.963119i	$-0.413282\pi$
$709$	7.20312e6	0.538152	0.269076	+	0.963119i	$0.413282\pi$
$710$	−4.18987e6	−0.311928
$711$	0	0
$712$	−5.89230e6	−0.435597
$713$	9.14554e6	0.673729
$714$	0	0
$715$	1.31497e6	0.0961949
$716$	−6.73164e6	−0.490725
$717$	0	0
$718$	2.55066e7	1.84647
$719$	−2.46592e6	−0.177892	−0.0889461	−	0.996036i	$-0.528350\pi$
$719$	−2.46592e6	−0.177892	−0.0889461	+	0.996036i	$0.528350\pi$
$720$	0	0
$721$	2.75289e6	0.197220
$722$	−8.99175e6	−0.641950
$723$	0	0
$724$	3.22264e6	0.228489
$725$	−819438.	−0.0578990
$726$	0	0
$727$	−1.19300e7	−0.837149	−0.418574	−	0.908182i	$-0.637470\pi$
$727$	−1.19300e7	−0.837149	−0.418574	+	0.908182i	$0.637470\pi$
$728$	−913922.	−0.0639117
$729$	0	0
$730$	1.12765e7	0.783191
$731$	1.08353e6	0.0749979
$732$	0	0
$733$	−8.70541e6	−0.598452	−0.299226	−	0.954182i	$-0.596728\pi$
$733$	−8.70541e6	−0.598452	−0.299226	+	0.954182i	$0.596728\pi$
$734$	9.26604e6	0.634825
$735$	0	0
$736$	3.61174e6	0.245766
$737$	1.07316e6	0.0727774
$738$	0	0
$739$	2.66072e7	1.79221	0.896105	−	0.443843i	$-0.146385\pi$
$739$	2.66072e7	1.79221	0.896105	+	0.443843i	$0.146385\pi$
$740$	−1.22634e6	−0.0823253
$741$	0	0
$742$	3.12253e6	0.208208
$743$	−1.34181e7	−0.891702	−0.445851	−	0.895107i	$-0.647099\pi$
$743$	−1.34181e7	−0.891702	−0.445851	+	0.895107i	$0.647099\pi$
$744$	0	0
$745$	6.48089e6	0.427803
$746$	8.26819e6	0.543956
$747$	0	0
$748$	−101750.	−0.00664938
$749$	−1.19893e6	−0.0780890
$750$	0	0
$751$	2.00693e7	1.29847	0.649236	−	0.760587i	$-0.275089\pi$
$751$	2.00693e7	1.29847	0.649236	+	0.760587i	$0.275089\pi$
$752$	−3.44472e7	−2.22131
$753$	0	0
$754$	−3.65528e6	−0.234149
$755$	1.21173e6	0.0773641
$756$	0	0
$757$	−1.27259e7	−0.807137	−0.403568	−	0.914949i	$-0.632230\pi$
$757$	−1.27259e7	−0.807137	−0.403568	+	0.914949i	$0.632230\pi$
$758$	2.80868e7	1.77554
$759$	0	0
$760$	7.22040e6	0.453447
$761$	−2.45362e7	−1.53584	−0.767919	−	0.640548i	$-0.778707\pi$
$761$	−2.45362e7	−1.53584	−0.767919	+	0.640548i	$0.778707\pi$
$762$	0	0
$763$	901865.	0.0560829
$764$	8.23433e6	0.510381
$765$	0	0
$766$	−1.68721e7	−1.03896
$767$	1.41429e7	0.868063
$768$	0	0
$769$	−1.51811e7	−0.925734	−0.462867	−	0.886428i	$-0.653179\pi$
$769$	−1.51811e7	−0.925734	−0.462867	+	0.886428i	$0.653179\pi$
$770$	278115.	0.0169043
$771$	0	0
$772$	5.25065e6	0.317081
$773$	−2.65426e6	−0.159770	−0.0798849	−	0.996804i	$-0.525455\pi$
$773$	−2.65426e6	−0.159770	−0.0798849	+	0.996804i	$0.525455\pi$
$774$	0	0
$775$	5.08594e6	0.304171
$776$	−1.82075e7	−1.08541
$777$	0	0
$778$	−115018.	−0.00681266
$779$	−3.74690e7	−2.21222
$780$	0	0
$781$	−3.16194e6	−0.185492
$782$	−663696.	−0.0388108
$783$	0	0
$784$	2.04670e7	1.18922
$785$	1.45006e7	0.839871
$786$	0	0
$787$	4.08789e6	0.235268	0.117634	−	0.993057i	$-0.462469\pi$
$787$	4.08789e6	0.235268	0.117634	+	0.993057i	$0.462469\pi$
$788$	4.07008e6	0.233501
$789$	0	0
$790$	−1.54340e7	−0.879855
$791$	3.63469e6	0.206550
$792$	0	0
$793$	−2.07448e7	−1.17146
$794$	2.41275e7	1.35819
$795$	0	0
$796$	−1.12940e6	−0.0631778
$797$	−2.46193e7	−1.37287	−0.686435	−	0.727191i	$-0.740825\pi$
$797$	−2.46193e7	−1.37287	−0.686435	+	0.727191i	$0.740825\pi$
$798$	0	0
$799$	2.57280e6	0.142574
$800$	2.00853e6	0.110957
$801$	0	0
$802$	−2.71868e7	−1.49253
$803$	8.50995e6	0.465734
$804$	0	0
$805$	402776.	0.0219066
$806$	2.26869e7	1.23009
$807$	0	0
$808$	−1.94741e7	−1.04937
$809$	−1.60895e7	−0.864311	−0.432156	−	0.901799i	$-0.642247\pi$
$809$	−1.60895e7	−0.864311	−0.432156	+	0.901799i	$0.642247\pi$
$810$	0	0
$811$	3.15111e7	1.68233	0.841166	−	0.540777i	$-0.181870\pi$
$811$	3.15111e7	1.68233	0.841166	+	0.540777i	$0.181870\pi$
$812$	−171646.	−0.00913573
$813$	0	0
$814$	−4.16831e6	−0.220495
$815$	4.47896e6	0.236202
$816$	0	0
$817$	2.31736e7	1.21461
$818$	−5.38527e6	−0.281400
$819$	0	0
$820$	−4.34403e6	−0.225610
$821$	−1.23166e7	−0.637724	−0.318862	−	0.947801i	$-0.603301\pi$
$821$	−1.23166e7	−0.637724	−0.318862	+	0.947801i	$0.603301\pi$
$822$	0	0
$823$	−2.87086e7	−1.47745	−0.738723	−	0.674009i	$-0.764571\pi$
$823$	−2.87086e7	−1.47745	−0.738723	+	0.674009i	$0.764571\pi$
$824$	2.81639e7	1.44502
$825$	0	0
$826$	2.99121e6	0.152545
$827$	1.14498e7	0.582149	0.291075	−	0.956700i	$-0.405987\pi$
$827$	1.14498e7	0.582149	0.291075	+	0.956700i	$0.405987\pi$
$828$	0	0
$829$	3.04591e7	1.53933	0.769664	−	0.638449i	$-0.220424\pi$
$829$	3.04591e7	1.53933	0.769664	+	0.638449i	$0.220424\pi$
$830$	1.14771e7	0.578279
$831$	0	0
$832$	−8.18986e6	−0.410174
$833$	−1.52864e6	−0.0763298
$834$	0	0
$835$	8.88831e6	0.441167
$836$	−2.17614e6	−0.107689
$837$	0	0
$838$	−2.47450e7	−1.21724
$839$	1.15222e7	0.565106	0.282553	−	0.959252i	$-0.408819\pi$
$839$	1.15222e7	0.565106	0.282553	+	0.959252i	$0.408819\pi$
$840$	0	0
$841$	−1.87922e7	−0.916193
$842$	2.15151e6	0.104584
$843$	0	0
$844$	1.21507e6	0.0587145
$845$	−4.55818e6	−0.219609
$846$	0	0
$847$	209883.	0.0100524
$848$	4.18710e7	1.99951
$849$	0	0
$850$	−369090.	−0.0175220
$851$	−6.03669e6	−0.285743
$852$	0	0
$853$	2.13285e7	1.00366	0.501832	−	0.864965i	$-0.332660\pi$
$853$	2.13285e7	1.00366	0.501832	+	0.864965i	$0.332660\pi$
$854$	−4.38749e6	−0.205860
$855$	0	0
$856$	−1.22659e7	−0.572156
$857$	8.24400e6	0.383430	0.191715	−	0.981451i	$-0.438595\pi$
$857$	8.24400e6	0.383430	0.191715	+	0.981451i	$0.438595\pi$
$858$	0	0
$859$	−3.21430e7	−1.48629	−0.743145	−	0.669131i	$-0.766667\pi$
$859$	−3.21430e7	−1.48629	−0.743145	+	0.669131i	$0.766667\pi$
$860$	2.68667e6	0.123871
$861$	0	0
$862$	−1.50217e7	−0.688573
$863$	2.33824e7	1.06872	0.534359	−	0.845258i	$-0.320553\pi$
$863$	2.33824e7	1.06872	0.534359	+	0.845258i	$0.320553\pi$
$864$	0	0
$865$	1.65982e7	0.754257
$866$	−4.03242e7	−1.82714
$867$	0	0
$868$	1.06534e6	0.0479943
$869$	−1.16475e7	−0.523217
$870$	0	0
$871$	3.85542e6	0.172197
$872$	9.22669e6	0.410918
$873$	0	0
$874$	−1.41945e7	−0.628553
$875$	223989.	0.00989023
$876$	0	0
$877$	6.11225e6	0.268350	0.134175	−	0.990958i	$-0.457162\pi$
$877$	6.11225e6	0.268350	0.134175	+	0.990958i	$0.457162\pi$
$878$	2.11094e7	0.924142
$879$	0	0
$880$	3.72933e6	0.162340
$881$	−3.83167e7	−1.66322	−0.831608	−	0.555363i	$-0.812580\pi$
$881$	−3.83167e7	−1.66322	−0.831608	+	0.555363i	$0.812580\pi$
$882$	0	0
$883$	−6.45643e6	−0.278670	−0.139335	−	0.990245i	$-0.544497\pi$
$883$	−6.45643e6	−0.278670	−0.139335	+	0.990245i	$0.544497\pi$
$884$	−365546.	−0.0157330
$885$	0	0
$886$	−1.85248e7	−0.792810
$887$	2.63767e7	1.12567	0.562835	−	0.826569i	$-0.309711\pi$
$887$	2.63767e7	1.12567	0.562835	+	0.826569i	$0.309711\pi$
$888$	0	0
$889$	1.34784e6	0.0571985
$890$	6.44179e6	0.272604
$891$	0	0
$892$	2.11940e6	0.0891868
$893$	5.50247e7	2.30903
$894$	0	0
$895$	−1.84277e7	−0.768976
$896$	−3.20634e6	−0.133426
$897$	0	0
$898$	−1.29073e6	−0.0534128
$899$	−1.06691e7	−0.440280
$900$	0	0
$901$	−3.12728e6	−0.128338
$902$	−1.47652e7	−0.604259
$903$	0	0
$904$	3.71853e7	1.51339
$905$	8.82188e6	0.358047
$906$	0	0
$907$	−2.58593e7	−1.04376	−0.521878	−	0.853020i	$-0.674768\pi$
$907$	−2.58593e7	−1.04376	−0.521878	+	0.853020i	$0.674768\pi$
$908$	5.83445e6	0.234847
$909$	0	0
$910$	999150.	0.0399970
$911$	1.51122e7	0.603299	0.301649	−	0.953419i	$-0.402463\pi$
$911$	1.51122e7	0.603299	0.301649	+	0.953419i	$0.402463\pi$
$912$	0	0
$913$	8.66134e6	0.343881
$914$	1.13503e6	0.0449408
$915$	0	0
$916$	−1.06945e7	−0.421137
$917$	−1.14920e6	−0.0451305
$918$	0	0
$919$	−952884.	−0.0372178	−0.0186089	−	0.999827i	$-0.505924\pi$
$919$	−952884.	−0.0372178	−0.0186089	+	0.999827i	$0.505924\pi$
$920$	4.12068e6	0.160509
$921$	0	0
$922$	−4.22266e7	−1.63591
$923$	−1.13595e7	−0.438890
$924$	0	0
$925$	−3.35708e6	−0.129005
$926$	−4.68509e7	−1.79552
$927$	0	0
$928$	−4.21342e6	−0.160607
$929$	−5.37858e6	−0.204469	−0.102235	−	0.994760i	$-0.532599\pi$
$929$	−5.37858e6	−0.204469	−0.102235	+	0.994760i	$0.532599\pi$
$930$	0	0
$931$	−3.26932e7	−1.23618
$932$	923280.	0.0348172
$933$	0	0
$934$	2.15664e7	0.808930
$935$	−278538.	−0.0104197
$936$	0	0
$937$	2.23159e7	0.830360	0.415180	−	0.909739i	$-0.363719\pi$
$937$	2.23159e7	0.830360	0.415180	+	0.909739i	$0.363719\pi$
$938$	815415.	0.0302602
$939$	0	0
$940$	6.37938e6	0.235482
$941$	3.96205e7	1.45863	0.729317	−	0.684176i	$-0.239838\pi$
$941$	3.96205e7	1.45863	0.729317	+	0.684176i	$0.239838\pi$
$942$	0	0
$943$	−2.13835e7	−0.783069
$944$	4.01101e7	1.46495
$945$	0	0
$946$	9.13190e6	0.331767
$947$	2.18979e7	0.793464	0.396732	−	0.917935i	$-0.370144\pi$
$947$	2.18979e7	0.793464	0.396732	+	0.917935i	$0.370144\pi$
$948$	0	0
$949$	3.05727e7	1.10197
$950$	−7.89374e6	−0.283775
$951$	0	0
$952$	193587.	0.00692284
$953$	6.33415e6	0.225921	0.112960	−	0.993599i	$-0.463967\pi$
$953$	6.33415e6	0.225921	0.112960	+	0.993599i	$0.463967\pi$
$954$	0	0
$955$	2.25412e7	0.799777
$956$	3.40559e6	0.120517
$957$	0	0
$958$	−4.49000e7	−1.58064
$959$	3.71333e6	0.130382
$960$	0	0
$961$	3.75899e7	1.31300
$962$	−1.49750e7	−0.521709
$963$	0	0
$964$	612048.	0.0212125
$965$	1.43735e7	0.496872
$966$	0	0
$967$	−3.47493e7	−1.19503	−0.597517	−	0.801857i	$-0.703846\pi$
$967$	−3.47493e7	−1.19503	−0.597517	+	0.801857i	$0.703846\pi$
$968$	2.14725e6	0.0736535
$969$	0	0
$970$	1.99054e7	0.679270
$971$	3.15169e6	0.107274	0.0536371	−	0.998560i	$-0.482919\pi$
$971$	3.15169e6	0.107274	0.0536371	+	0.998560i	$0.482919\pi$
$972$	0	0
$973$	751535.	0.0254488
$974$	−5.50737e7	−1.86014
$975$	0	0
$976$	−5.88333e7	−1.97696
$977$	4.07493e7	1.36579	0.682896	−	0.730516i	$-0.260720\pi$
$977$	4.07493e7	1.36579	0.682896	+	0.730516i	$0.260720\pi$
$978$	0	0
$979$	4.86138e6	0.162107
$980$	−3.79034e6	−0.126070
$981$	0	0
$982$	3.37727e7	1.11760
$983$	−4.36952e7	−1.44228	−0.721141	−	0.692789i	$-0.756382\pi$
$983$	−4.36952e7	−1.44228	−0.721141	+	0.692789i	$0.756382\pi$
$984$	0	0
$985$	1.11417e7	0.365900
$986$	774262.	0.0253627
$987$	0	0
$988$	−7.81794e6	−0.254800
$989$	1.32252e7	0.429942
$990$	0	0
$991$	−1.56678e7	−0.506784	−0.253392	−	0.967364i	$-0.581546\pi$
$991$	−1.56678e7	−0.506784	−0.253392	+	0.967364i	$0.581546\pi$
$992$	2.61511e7	0.843744
$993$	0	0
$994$	−2.40252e6	−0.0771261
$995$	−3.09169e6	−0.0990008
$996$	0	0
$997$	−3.86408e7	−1.23114	−0.615571	−	0.788081i	$-0.711075\pi$
$997$	−3.86408e7	−1.23114	−0.615571	+	0.788081i	$0.711075\pi$
$998$	3.30353e7	1.04991
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.6.a.p.1.3	yes	10
3.2	odd	2		495.6.a.o.1.8	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.6.a.o.1.8	✓	10	3.2	odd	2
495.6.a.p.1.3	yes	10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q-6.41346 q^{2} +9.13253 q^{4} +25.0000 q^{5} +14.3353 q^{7} +146.660 q^{8} +O(q^{10})\)	\(q-6.41346 q^{2} +9.13253 q^{4} +25.0000 q^{5} +14.3353 q^{7} +146.660 q^{8} -160.337 q^{10} -121.000 q^{11} -434.702 q^{13} -91.9389 q^{14} -1232.84 q^{16} +92.0787 q^{17} +1969.29 q^{19} +228.313 q^{20} +776.029 q^{22} +1123.87 q^{23} +625.000 q^{25} +2787.95 q^{26} +130.917 q^{28} -1311.10 q^{29} +8137.51 q^{31} +3213.65 q^{32} -590.543 q^{34} +358.382 q^{35} -5371.33 q^{37} -12630.0 q^{38} +3666.49 q^{40} -19026.6 q^{41} +11767.5 q^{43} -1105.04 q^{44} -7207.93 q^{46} +27941.3 q^{47} -16601.5 q^{49} -4008.42 q^{50} -3969.93 q^{52} -33963.1 q^{53} -3025.00 q^{55} +2102.41 q^{56} +8408.70 q^{58} -32534.8 q^{59} +47721.9 q^{61} -52189.6 q^{62} +18840.2 q^{64} -10867.5 q^{65} -8869.10 q^{67} +840.911 q^{68} -2298.47 q^{70} +26131.7 q^{71} -70330.2 q^{73} +34448.8 q^{74} +17984.6 q^{76} -1734.57 q^{77} +96260.0 q^{79} -30820.9 q^{80} +122027. q^{82} -71581.3 q^{83} +2301.97 q^{85} -75470.3 q^{86} -17745.8 q^{88} -40176.7 q^{89} -6231.58 q^{91} +10263.8 q^{92} -179201. q^{94} +49232.3 q^{95} -124148. q^{97} +106473. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 3 q^{2} + 181 q^{4} + 250 q^{5} + 116 q^{7} + 129 q^{8}+O(q^{10}) \) 10 * q + 3 * q^2 + 181 * q^4 + 250 * q^5 + 116 * q^7 + 129 * q^8	\( 10 q + 3 q^{2} + 181 q^{4} + 250 q^{5} + 116 q^{7} + 129 q^{8} + 75 q^{10} - 1210 q^{11} + 932 q^{13} - 1332 q^{14} + 2701 q^{16} + 96 q^{17} + 1664 q^{19} + 4525 q^{20} - 363 q^{22} + 6288 q^{23} + 6250 q^{25} + 13380 q^{26} + 13868 q^{28} + 11208 q^{29} + 9032 q^{31} + 9801 q^{32} + 14610 q^{34} + 2900 q^{35} + 21572 q^{37} + 15870 q^{38} + 3225 q^{40} + 10800 q^{41} + 21128 q^{43} - 21901 q^{44} + 83982 q^{46} - 17400 q^{47} + 71610 q^{49} + 1875 q^{50} + 40640 q^{52} + 5004 q^{53} - 30250 q^{55} - 54012 q^{56} - 9786 q^{58} - 25272 q^{59} + 52004 q^{61} + 34740 q^{62} + 56953 q^{64} + 23300 q^{65} + 4160 q^{67} - 87978 q^{68} - 33300 q^{70} - 65232 q^{71} + 44252 q^{73} - 49842 q^{74} + 233246 q^{76} - 14036 q^{77} + 112604 q^{79} + 67525 q^{80} + 167910 q^{82} + 70032 q^{83} + 2400 q^{85} - 72978 q^{86} - 15609 q^{88} - 46848 q^{89} + 130672 q^{91} + 121302 q^{92} + 252294 q^{94} + 41600 q^{95} + 129932 q^{97} - 316137 q^{98}+O(q^{100}) \) 10 * q + 3 * q^2 + 181 * q^4 + 250 * q^5 + 116 * q^7 + 129 * q^8 + 75 * q^10 - 1210 * q^11 + 932 * q^13 - 1332 * q^14 + 2701 * q^16 + 96 * q^17 + 1664 * q^19 + 4525 * q^20 - 363 * q^22 + 6288 * q^23 + 6250 * q^25 + 13380 * q^26 + 13868 * q^28 + 11208 * q^29 + 9032 * q^31 + 9801 * q^32 + 14610 * q^34 + 2900 * q^35 + 21572 * q^37 + 15870 * q^38 + 3225 * q^40 + 10800 * q^41 + 21128 * q^43 - 21901 * q^44 + 83982 * q^46 - 17400 * q^47 + 71610 * q^49 + 1875 * q^50 + 40640 * q^52 + 5004 * q^53 - 30250 * q^55 - 54012 * q^56 - 9786 * q^58 - 25272 * q^59 + 52004 * q^61 + 34740 * q^62 + 56953 * q^64 + 23300 * q^65 + 4160 * q^67 - 87978 * q^68 - 33300 * q^70 - 65232 * q^71 + 44252 * q^73 - 49842 * q^74 + 233246 * q^76 - 14036 * q^77 + 112604 * q^79 + 67525 * q^80 + 167910 * q^82 + 70032 * q^83 + 2400 * q^85 - 72978 * q^86 - 15609 * q^88 - 46848 * q^89 + 130672 * q^91 + 121302 * q^92 + 252294 * q^94 + 41600 * q^95 + 129932 * q^97 - 316137 * q^98

Introduction

L-functions

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Database

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