LMFDB - Embedded newform 475.6.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,6,Mod(1,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, 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("475.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	475.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:	$76.1823144112$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{177}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 44 $ x^2 - x - 44
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 19)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.15207$ of defining polynomial
Character	$\chi$	$=$	475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.1521 q^{2} -16.4562 q^{3} +71.0645 q^{4} -167.064 q^{6} +57.1289 q^{7} +396.585 q^{8} +27.8066 q^{9} +O(q^{10})$	\(q+10.1521 q^{2} -16.4562 q^{3} +71.0645 q^{4} -167.064 q^{6} +57.1289 q^{7} +396.585 q^{8} +27.8066 q^{9} -438.977 q^{11} -1169.45 q^{12} +477.590 q^{13} +579.977 q^{14} +1752.10 q^{16} -1671.39 q^{17} +282.294 q^{18} -361.000 q^{19} -940.125 q^{21} -4456.52 q^{22} -459.514 q^{23} -6526.29 q^{24} +4848.53 q^{26} +3541.27 q^{27} +4059.84 q^{28} -2696.78 q^{29} -2314.43 q^{31} +5096.67 q^{32} +7223.89 q^{33} -16968.0 q^{34} +1976.06 q^{36} -3022.51 q^{37} -3664.90 q^{38} -7859.32 q^{39} +5483.05 q^{41} -9544.22 q^{42} -5101.80 q^{43} -31195.7 q^{44} -4665.02 q^{46} -5803.63 q^{47} -28832.8 q^{48} -13543.3 q^{49} +27504.7 q^{51} +33939.7 q^{52} +23207.5 q^{53} +35951.2 q^{54} +22656.5 q^{56} +5940.69 q^{57} -27377.9 q^{58} -32849.5 q^{59} -46911.5 q^{61} -23496.2 q^{62} +1588.56 q^{63} -4325.31 q^{64} +73337.4 q^{66} +3565.12 q^{67} -118776. q^{68} +7561.86 q^{69} +2346.44 q^{71} +11027.7 q^{72} -57370.0 q^{73} -30684.7 q^{74} -25654.3 q^{76} -25078.3 q^{77} -79788.3 q^{78} -29715.8 q^{79} -65032.8 q^{81} +55664.3 q^{82} -60526.0 q^{83} -66809.5 q^{84} -51793.8 q^{86} +44378.8 q^{87} -174092. q^{88} +19134.9 q^{89} +27284.2 q^{91} -32655.1 q^{92} +38086.7 q^{93} -58918.9 q^{94} -83871.9 q^{96} +43189.4 q^{97} -137492. q^{98} -12206.4 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 7 q^{2} + 7 q^{3} + 49 q^{4} - 241 q^{6} - 72 q^{7} + 567 q^{8} + 335 q^{9}+O(q^{10}) $ 2 * q + 7 * q^2 + 7 * q^3 + 49 * q^4 - 241 * q^6 - 72 * q^7 + 567 * q^8 + 335 * q^9	\( 2 q + 7 q^{2} + 7 q^{3} + 49 q^{4} - 241 q^{6} - 72 q^{7} + 567 q^{8} + 335 q^{9} - 705 q^{11} - 1687 q^{12} + 1341 q^{13} + 987 q^{14} + 1921 q^{16} - 2784 q^{17} - 686 q^{18} - 722 q^{19} - 3969 q^{21} - 3618 q^{22} + 2713 q^{23} - 2529 q^{24} + 2127 q^{26} + 5047 q^{27} + 6909 q^{28} - 7775 q^{29} + 7132 q^{31} - 889 q^{32} + 984 q^{33} - 13461 q^{34} - 4802 q^{36} + 6248 q^{37} - 2527 q^{38} + 12393 q^{39} - 4174 q^{41} + 3 q^{42} - 25357 q^{43} - 25326 q^{44} - 14665 q^{46} - 11727 q^{47} - 24871 q^{48} - 13676 q^{49} + 1407 q^{51} + 14889 q^{52} + 29133 q^{53} + 31205 q^{54} + 651 q^{56} - 2527 q^{57} - 11371 q^{58} - 64515 q^{59} - 40939 q^{61} - 53272 q^{62} - 38079 q^{63} + 9137 q^{64} + 93006 q^{66} - 19039 q^{67} - 94227 q^{68} + 81977 q^{69} - 70236 q^{71} + 63378 q^{72} - 67058 q^{73} - 59906 q^{74} - 17689 q^{76} + 9273 q^{77} - 143625 q^{78} - 32850 q^{79} - 104362 q^{81} + 86104 q^{82} - 71534 q^{83} + 21 q^{84} + 12052 q^{86} - 74737 q^{87} - 219426 q^{88} - 87268 q^{89} - 84207 q^{91} - 102655 q^{92} + 259664 q^{93} - 40248 q^{94} - 224273 q^{96} + 62458 q^{97} - 137074 q^{98} - 93927 q^{99}+O(q^{100}) \) 2 * q + 7 * q^2 + 7 * q^3 + 49 * q^4 - 241 * q^6 - 72 * q^7 + 567 * q^8 + 335 * q^9 - 705 * q^11 - 1687 * q^12 + 1341 * q^13 + 987 * q^14 + 1921 * q^16 - 2784 * q^17 - 686 * q^18 - 722 * q^19 - 3969 * q^21 - 3618 * q^22 + 2713 * q^23 - 2529 * q^24 + 2127 * q^26 + 5047 * q^27 + 6909 * q^28 - 7775 * q^29 + 7132 * q^31 - 889 * q^32 + 984 * q^33 - 13461 * q^34 - 4802 * q^36 + 6248 * q^37 - 2527 * q^38 + 12393 * q^39 - 4174 * q^41 + 3 * q^42 - 25357 * q^43 - 25326 * q^44 - 14665 * q^46 - 11727 * q^47 - 24871 * q^48 - 13676 * q^49 + 1407 * q^51 + 14889 * q^52 + 29133 * q^53 + 31205 * q^54 + 651 * q^56 - 2527 * q^57 - 11371 * q^58 - 64515 * q^59 - 40939 * q^61 - 53272 * q^62 - 38079 * q^63 + 9137 * q^64 + 93006 * q^66 - 19039 * q^67 - 94227 * q^68 + 81977 * q^69 - 70236 * q^71 + 63378 * q^72 - 67058 * q^73 - 59906 * q^74 - 17689 * q^76 + 9273 * q^77 - 143625 * q^78 - 32850 * q^79 - 104362 * q^81 + 86104 * q^82 - 71534 * q^83 + 21 * q^84 + 12052 * q^86 - 74737 * q^87 - 219426 * q^88 - 87268 * q^89 - 84207 * q^91 - 102655 * q^92 + 259664 * q^93 - 40248 * q^94 - 224273 * q^96 + 62458 * q^97 - 137074 * q^98 - 93927 * q^99

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 $

$2$	10.1521	1.79465	0.897324	−	0.441372i	$-0.145508\pi$
$2$	10.1521	1.79465	0.897324	+	0.441372i	$0.145508\pi$
$3$	−16.4562	−1.05567	−0.527833	−	0.849348i	$-0.676995\pi$
$3$	−16.4562	−1.05567	−0.527833	+	0.849348i	$0.676995\pi$
$4$	71.0645	2.22076
$5$	0	0
$5$	0	0
$6$	−167.064	−1.89455
$7$	57.1289	0.440668	0.220334	−	0.975425i	$-0.429285\pi$
$7$	57.1289	0.440668	0.220334	+	0.975425i	$0.429285\pi$
$8$	396.585	2.19084
$9$	27.8066	0.114430
$10$	0	0
$11$	−438.977	−1.09386	−0.546928	−	0.837180i	$-0.684203\pi$
$11$	−438.977	−1.09386	−0.546928	+	0.837180i	$0.684203\pi$
$12$	−1169.45	−2.34439
$13$	477.590	0.783785	0.391892	−	0.920011i	$-0.371821\pi$
$13$	477.590	0.783785	0.391892	+	0.920011i	$0.371821\pi$
$14$	579.977	0.790844
$15$	0	0
$16$	1752.10	1.71103
$17$	−1671.39	−1.40267	−0.701334	−	0.712833i	$-0.747412\pi$
$17$	−1671.39	−1.40267	−0.701334	+	0.712833i	$0.747412\pi$
$18$	282.294	0.205362
$19$	−361.000	−0.229416
$19$	−361.000	−0.229416
$20$	0	0
$21$	−940.125	−0.465198
$22$	−4456.52	−1.96309
$23$	−459.514	−0.181125	−0.0905627	−	0.995891i	$-0.528867\pi$
$23$	−459.514	−0.181125	−0.0905627	+	0.995891i	$0.528867\pi$
$24$	−6526.29	−2.31280
$25$	0	0
$26$	4848.53	1.40662
$27$	3541.27	0.934866
$28$	4059.84	0.978619
$29$	−2696.78	−0.595457	−0.297729	−	0.954651i	$-0.596229\pi$
$29$	−2696.78	−0.595457	−0.297729	+	0.954651i	$0.596229\pi$
$30$	0	0
$31$	−2314.43	−0.432553	−0.216277	−	0.976332i	$-0.569391\pi$
$31$	−2314.43	−0.432553	−0.216277	+	0.976332i	$0.569391\pi$
$32$	5096.67	0.879856
$33$	7223.89	1.15475
$34$	−16968.0	−2.51730
$35$	0	0
$36$	1976.06	0.254123
$37$	−3022.51	−0.362964	−0.181482	−	0.983394i	$-0.558089\pi$
$37$	−3022.51	−0.362964	−0.181482	+	0.983394i	$0.558089\pi$
$38$	−3664.90	−0.411721
$39$	−7859.32	−0.827415
$40$	0	0
$41$	5483.05	0.509404	0.254702	−	0.967020i	$-0.418023\pi$
$41$	5483.05	0.509404	0.254702	+	0.967020i	$0.418023\pi$
$42$	−9544.22	−0.834867
$43$	−5101.80	−0.420777	−0.210388	−	0.977618i	$-0.567473\pi$
$43$	−5101.80	−0.420777	−0.210388	+	0.977618i	$0.567473\pi$
$44$	−31195.7	−2.42920
$45$	0	0
$46$	−4665.02	−0.325057
$47$	−5803.63	−0.383226	−0.191613	−	0.981471i	$-0.561372\pi$
$47$	−5803.63	−0.383226	−0.191613	+	0.981471i	$0.561372\pi$
$48$	−28832.8	−1.80628
$49$	−13543.3	−0.805812
$50$	0	0
$51$	27504.7	1.48075
$52$	33939.7	1.74060
$53$	23207.5	1.13485	0.567426	−	0.823424i	$-0.307939\pi$
$53$	23207.5	1.13485	0.567426	+	0.823424i	$0.307939\pi$
$54$	35951.2	1.67776
$55$	0	0
$56$	22656.5	0.965434
$57$	5940.69	0.242186
$58$	−27377.9	−1.06864
$59$	−32849.5	−1.22857	−0.614284	−	0.789085i	$-0.710555\pi$
$59$	−32849.5	−1.22857	−0.614284	+	0.789085i	$0.710555\pi$
$60$	0	0
$61$	−46911.5	−1.61419	−0.807095	−	0.590422i	$-0.798961\pi$
$61$	−46911.5	−1.61419	−0.807095	+	0.590422i	$0.798961\pi$
$62$	−23496.2	−0.776281
$63$	1588.56	0.0504258
$64$	−4325.31	−0.131998
$65$	0	0
$66$	73337.4	2.07236
$67$	3565.12	0.0970257	0.0485128	−	0.998823i	$-0.484552\pi$
$67$	3565.12	0.0970257	0.0485128	+	0.998823i	$0.484552\pi$
$68$	−118776.	−3.11499
$69$	7561.86	0.191208
$70$	0	0
$71$	2346.44	0.0552413	0.0276207	−	0.999618i	$-0.491207\pi$
$71$	2346.44	0.0552413	0.0276207	+	0.999618i	$0.491207\pi$
$72$	11027.7	0.250699
$73$	−57370.0	−1.26002	−0.630010	−	0.776587i	$-0.716949\pi$
$73$	−57370.0	−1.26002	−0.630010	+	0.776587i	$0.716949\pi$
$74$	−30684.7	−0.651393
$75$	0	0
$76$	−25654.3	−0.509478
$77$	−25078.3	−0.482027
$78$	−79788.3	−1.48492
$79$	−29715.8	−0.535698	−0.267849	−	0.963461i	$-0.586313\pi$
$79$	−29715.8	−0.535698	−0.267849	+	0.963461i	$0.586313\pi$
$80$	0	0
$81$	−65032.8	−1.10134
$82$	55664.3	0.914202
$83$	−60526.0	−0.964377	−0.482188	−	0.876068i	$-0.660158\pi$
$83$	−60526.0	−0.964377	−0.482188	+	0.876068i	$0.660158\pi$
$84$	−66809.5	−1.03309
$85$	0	0
$86$	−51793.8	−0.755147
$87$	44378.8	0.628604
$88$	−174092.	−2.39647
$89$	19134.9	0.256066	0.128033	−	0.991770i	$-0.459134\pi$
$89$	19134.9	0.256066	0.128033	+	0.991770i	$0.459134\pi$
$90$	0	0
$91$	27284.2	0.345389
$92$	−32655.1	−0.402237
$93$	38086.7	0.456631
$94$	−58918.9	−0.687756
$95$	0	0
$96$	−83871.9	−0.928834
$97$	43189.4	0.466067	0.233033	−	0.972469i	$-0.425135\pi$
$97$	43189.4	0.466067	0.233033	+	0.972469i	$0.425135\pi$
$98$	−137492.	−1.44615
$99$	−12206.4	−0.125170
$100$	0	0
$101$	−55330.5	−0.539711	−0.269855	−	0.962901i	$-0.586976\pi$
$101$	−55330.5	−0.539711	−0.269855	+	0.962901i	$0.586976\pi$
$102$	279229.	2.65742
$103$	205641.	1.90993	0.954964	−	0.296722i	$-0.0958933\pi$
$103$	205641.	1.90993	0.954964	+	0.296722i	$0.0958933\pi$
$104$	189405.	1.71715
$105$	0	0
$106$	235604.	2.03666
$107$	−181889.	−1.53585	−0.767923	−	0.640543i	$-0.778709\pi$
$107$	−181889.	−1.53585	−0.767923	+	0.640543i	$0.778709\pi$
$108$	251658.	2.07612
$109$	24288.8	0.195813	0.0979063	−	0.995196i	$-0.468785\pi$
$109$	24288.8	0.195813	0.0979063	+	0.995196i	$0.468785\pi$
$110$	0	0
$111$	49739.0	0.383169
$112$	100095.	0.753996
$113$	229667.	1.69201	0.846006	−	0.533174i	$-0.179001\pi$
$113$	229667.	1.69201	0.846006	+	0.533174i	$0.179001\pi$
$114$	60310.3	0.434639
$115$	0	0
$116$	−191645.	−1.32237
$117$	13280.1	0.0896888
$118$	−333491.	−2.20485
$119$	−95484.6	−0.618110
$120$	0	0
$121$	31649.7	0.196520
$122$	−476248.	−2.89690
$123$	−90230.2	−0.537761
$124$	−164474.	−0.960599
$125$	0	0
$126$	16127.2	0.0904965
$127$	−8086.69	−0.0444899	−0.0222450	−	0.999753i	$-0.507081\pi$
$127$	−8086.69	−0.0444899	−0.0222450	+	0.999753i	$0.507081\pi$
$128$	−207004.	−1.11675
$129$	83956.2	0.444200
$130$	0	0
$131$	118712.	0.604390	0.302195	−	0.953246i	$-0.402281\pi$
$131$	118712.	0.604390	0.302195	+	0.953246i	$0.402281\pi$
$132$	513362.	2.56442
$133$	−20623.5	−0.101096
$134$	36193.3	0.174127
$135$	0	0
$136$	−662847.	−3.07303
$137$	−65348.7	−0.297465	−0.148732	−	0.988877i	$-0.547519\pi$
$137$	−65348.7	−0.297465	−0.148732	+	0.988877i	$0.547519\pi$
$138$	76768.5	0.343151
$139$	−28437.6	−0.124841	−0.0624203	−	0.998050i	$-0.519882\pi$
$139$	−28437.6	−0.124841	−0.0624203	+	0.998050i	$0.519882\pi$
$140$	0	0
$141$	95505.7	0.404559
$142$	23821.3	0.0991388
$143$	−209651.	−0.857347
$144$	48719.8	0.195794
$145$	0	0
$146$	−582424.	−2.26130
$147$	222871.	0.850668
$148$	−214793.	−0.806057
$149$	115852.	0.427502	0.213751	−	0.976888i	$-0.431432\pi$
$149$	115852.	0.427502	0.213751	+	0.976888i	$0.431432\pi$
$150$	0	0
$151$	35980.9	0.128419	0.0642096	−	0.997936i	$-0.479547\pi$
$151$	35980.9	0.128419	0.0642096	+	0.997936i	$0.479547\pi$
$152$	−143167.	−0.502614
$153$	−46475.6	−0.160508
$154$	−254596.	−0.865069
$155$	0	0
$156$	−558518.	−1.83749
$157$	−566234.	−1.83336	−0.916678	−	0.399627i	$-0.869140\pi$
$157$	−566234.	−1.83336	−0.916678	+	0.399627i	$0.869140\pi$
$158$	−301677.	−0.961390
$159$	−381908.	−1.19802
$160$	0	0
$161$	−26251.6	−0.0798161
$162$	−660217.	−1.97651
$163$	307841.	0.907523	0.453761	−	0.891123i	$-0.350082\pi$
$163$	307841.	0.907523	0.453761	+	0.891123i	$0.350082\pi$
$164$	389650.	1.13127
$165$	0	0
$166$	−614464.	−1.73072
$167$	−126991.	−0.352357	−0.176179	−	0.984358i	$-0.556374\pi$
$167$	−126991.	−0.352357	−0.176179	+	0.984358i	$0.556374\pi$
$168$	−372840.	−1.01918
$169$	−143201.	−0.385681
$170$	0	0
$171$	−10038.2	−0.0262521
$172$	−362556.	−0.934446
$173$	582515.	1.47976	0.739881	−	0.672738i	$-0.234882\pi$
$173$	582515.	1.47976	0.739881	+	0.672738i	$0.234882\pi$
$174$	450536.	1.12812
$175$	0	0
$176$	−769130.	−1.87162
$177$	540579.	1.29696
$178$	194259.	0.459548
$179$	−233819.	−0.545440	−0.272720	−	0.962093i	$-0.587923\pi$
$179$	−233819.	−0.545440	−0.272720	+	0.962093i	$0.587923\pi$
$180$	0	0
$181$	318180.	0.721899	0.360950	−	0.932585i	$-0.382453\pi$
$181$	318180.	0.721899	0.360950	+	0.932585i	$0.382453\pi$
$182$	276991.	0.619851
$183$	771985.	1.70404
$184$	−182237.	−0.396818
$185$	0	0
$186$	386659.	0.819493
$187$	733700.	1.53432
$188$	−412432.	−0.851055
$189$	202309.	0.411965
$190$	0	0
$191$	−182032.	−0.361047	−0.180523	−	0.983571i	$-0.557779\pi$
$191$	−182032.	−0.361047	−0.180523	+	0.983571i	$0.557779\pi$
$192$	71178.2	0.139346
$193$	−286026.	−0.552729	−0.276365	−	0.961053i	$-0.589130\pi$
$193$	−286026.	−0.552729	−0.276365	+	0.961053i	$0.589130\pi$
$194$	438462.	0.836426
$195$	0	0
$196$	−962446.	−1.78952
$197$	465880.	0.855281	0.427640	−	0.903949i	$-0.359345\pi$
$197$	465880.	0.855281	0.427640	+	0.903949i	$0.359345\pi$
$198$	−123921.	−0.224637
$199$	1.06447e6	1.90547	0.952736	−	0.303798i	$-0.0982548\pi$
$199$	1.06447e6	1.90547	0.952736	+	0.303798i	$0.0982548\pi$
$200$	0	0
$201$	−58668.3	−0.102427
$202$	−561719.	−0.968591
$203$	−154064.	−0.262399
$204$	1.95461e6	3.28839
$205$	0	0
$206$	2.08768e6	3.42765
$207$	−12777.5	−0.0207263
$208$	836784.	1.34108
$209$	158471.	0.250948
$210$	0	0
$211$	1.00586e6	1.55536	0.777679	−	0.628662i	$-0.216397\pi$
$211$	1.00586e6	1.55536	0.777679	+	0.628662i	$0.216397\pi$
$212$	1.64923e6	2.52024
$213$	−38613.5	−0.0583164
$214$	−1.84655e6	−2.75630
$215$	0	0
$216$	1.40441e6	2.04814
$217$	−132221.	−0.190612
$218$	246582.	0.351415
$219$	944092.	1.33016
$220$	0	0
$221$	−798238.	−1.09939
$222$	504954.	0.687653
$223$	377403.	0.508210	0.254105	−	0.967177i	$-0.418219\pi$
$223$	377403.	0.508210	0.254105	+	0.967177i	$0.418219\pi$
$224$	291167.	0.387724
$225$	0	0
$226$	2.33160e6	3.03657
$227$	1.28433e6	1.65429	0.827145	−	0.561989i	$-0.189964\pi$
$227$	1.28433e6	1.65429	0.827145	+	0.561989i	$0.189964\pi$
$228$	422172.	0.537839
$229$	299103.	0.376905	0.188452	−	0.982082i	$-0.439653\pi$
$229$	299103.	0.376905	0.188452	+	0.982082i	$0.439653\pi$
$230$	0	0
$231$	412693.	0.508859
$232$	−1.06950e6	−1.30455
$233$	−564942.	−0.681732	−0.340866	−	0.940112i	$-0.610720\pi$
$233$	−564942.	−0.681732	−0.340866	+	0.940112i	$0.610720\pi$
$234$	134821.	0.160960
$235$	0	0
$236$	−2.33443e6	−2.72836
$237$	489010.	0.565518
$238$	−969366.	−1.10929
$239$	−1.03635e6	−1.17358	−0.586789	−	0.809740i	$-0.699608\pi$
$239$	−1.03635e6	−1.17358	−0.586789	+	0.809740i	$0.699608\pi$
$240$	0	0
$241$	821274.	0.910847	0.455423	−	0.890275i	$-0.349488\pi$
$241$	821274.	0.910847	0.455423	+	0.890275i	$0.349488\pi$
$242$	321310.	0.352684
$243$	209665.	0.227777
$244$	−3.33374e6	−3.58473
$245$	0	0
$246$	−916023.	−0.965092
$247$	−172410.	−0.179813
$248$	−917868.	−0.947656
$249$	996028.	1.01806
$250$	0	0
$251$	1.23062e6	1.23293	0.616467	−	0.787380i	$-0.288563\pi$
$251$	1.23062e6	1.23293	0.616467	+	0.787380i	$0.288563\pi$
$252$	112890.	0.111984
$253$	201716.	0.198125
$254$	−82096.7	−0.0798438
$255$	0	0
$256$	−1.96311e6	−1.87217
$257$	1.05206e6	0.993592	0.496796	−	0.867867i	$-0.334510\pi$
$257$	1.05206e6	0.993592	0.496796	+	0.867867i	$0.334510\pi$
$258$	852329.	0.797183
$259$	−172673.	−0.159946
$260$	0	0
$261$	−74988.2	−0.0681384
$262$	1.20517e6	1.08467
$263$	−1.88157e6	−1.67738	−0.838688	−	0.544612i	$-0.816677\pi$
$263$	−1.88157e6	−1.67738	−0.838688	+	0.544612i	$0.816677\pi$
$264$	2.86489e6	2.52987
$265$	0	0
$266$	−209372.	−0.181432
$267$	−314888.	−0.270320
$268$	253353.	0.215471
$269$	−1.05389e6	−0.888002	−0.444001	−	0.896026i	$-0.646441\pi$
$269$	−1.05389e6	−0.888002	−0.444001	+	0.896026i	$0.646441\pi$
$270$	0	0
$271$	688686.	0.569637	0.284818	−	0.958581i	$-0.408067\pi$
$271$	688686.	0.569637	0.284818	+	0.958581i	$0.408067\pi$
$272$	−2.92843e6	−2.40001
$273$	−448995.	−0.364615
$274$	−663425.	−0.533845
$275$	0	0
$276$	537380.	0.424628
$277$	522720.	0.409326	0.204663	−	0.978832i	$-0.434390\pi$
$277$	522720.	0.409326	0.204663	+	0.978832i	$0.434390\pi$
$278$	−288701.	−0.224045
$279$	−64356.3	−0.0494972
$280$	0	0
$281$	−2.00251e6	−1.51290	−0.756448	−	0.654053i	$-0.773067\pi$
$281$	−2.00251e6	−1.51290	−0.756448	+	0.654053i	$0.773067\pi$
$282$	969581.	0.726041
$283$	2.25219e6	1.67163	0.835813	−	0.549014i	$-0.184997\pi$
$283$	2.25219e6	1.67163	0.835813	+	0.549014i	$0.184997\pi$
$284$	166749.	0.122678
$285$	0	0
$286$	−2.12839e6	−1.53864
$287$	313241.	0.224478
$288$	141721.	0.100682
$289$	1.37368e6	0.967476
$290$	0	0
$291$	−710734.	−0.492011
$292$	−4.07697e6	−2.79821
$293$	−1.76627e6	−1.20195	−0.600976	−	0.799267i	$-0.705221\pi$
$293$	−1.76627e6	−1.20195	−0.600976	+	0.799267i	$0.705221\pi$
$294$	2.26260e6	1.52665
$295$	0	0
$296$	−1.19868e6	−0.795197
$297$	−1.55453e6	−1.02261
$298$	1.17614e6	0.767216
$299$	−219459.	−0.141963
$300$	0	0
$301$	−291460.	−0.185423
$302$	365281.	0.230467
$303$	910530.	0.569754
$304$	−632507.	−0.392537
$305$	0	0
$306$	−471823.	−0.288055
$307$	1.71867e6	1.04075	0.520376	−	0.853937i	$-0.325792\pi$
$307$	1.71867e6	1.04075	0.520376	+	0.853937i	$0.325792\pi$
$308$	−1.78218e6	−1.07047
$309$	−3.38407e6	−2.01625
$310$	0	0
$311$	1.58149e6	0.927182	0.463591	−	0.886049i	$-0.346561\pi$
$311$	1.58149e6	0.927182	0.463591	+	0.886049i	$0.346561\pi$
$312$	−3.11689e6	−1.81274
$313$	−54617.0	−0.0315114	−0.0157557	−	0.999876i	$-0.505015\pi$
$313$	−54617.0	−0.0315114	−0.0157557	+	0.999876i	$0.505015\pi$
$314$	−5.74844e6	−3.29023
$315$	0	0
$316$	−2.11174e6	−1.18966
$317$	1.03467e6	0.578303	0.289151	−	0.957283i	$-0.406627\pi$
$317$	1.03467e6	0.578303	0.289151	+	0.957283i	$0.406627\pi$
$318$	−3.87715e6	−2.15003
$319$	1.18382e6	0.651344
$320$	0	0
$321$	2.99321e6	1.62134
$322$	−266508.	−0.143242
$323$	603371.	0.321794
$324$	−4.62152e6	−2.44581
$325$	0	0
$326$	3.12522e6	1.62868
$327$	−399702.	−0.206713
$328$	2.17450e6	1.11603
$329$	−331555.	−0.168875
$330$	0	0
$331$	2.03987e6	1.02337	0.511685	−	0.859173i	$-0.329021\pi$
$331$	2.03987e6	1.02337	0.511685	+	0.859173i	$0.329021\pi$
$332$	−4.30125e6	−2.14165
$333$	−84045.7	−0.0415341
$334$	−1.28923e6	−0.632358
$335$	0	0
$336$	−1.64719e6	−0.795968
$337$	−1.22774e6	−0.588887	−0.294444	−	0.955669i	$-0.595134\pi$
$337$	−1.22774e6	−0.588887	−0.294444	+	0.955669i	$0.595134\pi$
$338$	−1.45378e6	−0.692162
$339$	−3.77945e6	−1.78620
$340$	0	0
$341$	1.01598e6	0.473150
$342$	−101908.	−0.0471134
$343$	−1.73388e6	−0.795763
$344$	−2.02330e6	−0.921857
$345$	0	0
$346$	5.91373e6	2.65565
$347$	−3.93267e6	−1.75333	−0.876665	−	0.481101i	$-0.840237\pi$
$347$	−3.93267e6	−1.75333	−0.876665	+	0.481101i	$0.840237\pi$
$348$	3.15375e6	1.39598
$349$	−3.46176e6	−1.52136	−0.760682	−	0.649125i	$-0.775135\pi$
$349$	−3.46176e6	−1.52136	−0.760682	+	0.649125i	$0.775135\pi$
$350$	0	0
$351$	1.69127e6	0.732734
$352$	−2.23732e6	−0.962436
$353$	3.91227e6	1.67106	0.835531	−	0.549443i	$-0.185160\pi$
$353$	3.91227e6	1.67106	0.835531	+	0.549443i	$0.185160\pi$
$354$	5.48799e6	2.32758
$355$	0	0
$356$	1.35981e6	0.568662
$357$	1.57131e6	0.652518
$358$	−2.37375e6	−0.978874
$359$	−949856.	−0.388975	−0.194487	−	0.980905i	$-0.562304\pi$
$359$	−949856.	−0.388975	−0.194487	+	0.980905i	$0.562304\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	3.23018e6	1.29556
$363$	−520834.	−0.207459
$364$	1.93894e6	0.767027
$365$	0	0
$366$	7.83724e6	3.05816
$367$	−3.72584e6	−1.44397	−0.721986	−	0.691908i	$-0.756770\pi$
$367$	−3.72584e6	−1.44397	−0.721986	+	0.691908i	$0.756770\pi$
$368$	−805113.	−0.309911
$369$	152465.	0.0582914
$370$	0	0
$371$	1.32582e6	0.500093
$372$	2.70661e6	1.01407
$373$	4.15376e6	1.54586	0.772929	−	0.634493i	$-0.218791\pi$
$373$	4.15376e6	1.54586	0.772929	+	0.634493i	$0.218791\pi$
$374$	7.44857e6	2.75356
$375$	0	0
$376$	−2.30163e6	−0.839589
$377$	−1.28796e6	−0.466710
$378$	2.05385e6	0.739332
$379$	2.29264e6	0.819857	0.409928	−	0.912118i	$-0.365554\pi$
$379$	2.29264e6	0.819857	0.409928	+	0.912118i	$0.365554\pi$
$380$	0	0
$381$	133076.	0.0469665
$382$	−1.84800e6	−0.647952
$383$	−3.09365e6	−1.07764	−0.538821	−	0.842420i	$-0.681130\pi$
$383$	−3.09365e6	−1.07764	−0.538821	+	0.842420i	$0.681130\pi$
$384$	3.40651e6	1.17891
$385$	0	0
$386$	−2.90376e6	−0.991955
$387$	−141864.	−0.0481497
$388$	3.06923e6	1.03502
$389$	−4.53732e6	−1.52029	−0.760144	−	0.649755i	$-0.774871\pi$
$389$	−4.53732e6	−1.52029	−0.760144	+	0.649755i	$0.774871\pi$
$390$	0	0
$391$	768026.	0.254059
$392$	−5.37107e6	−1.76541
$393$	−1.95355e6	−0.638034
$394$	4.72965e6	1.53493
$395$	0	0
$396$	−867445.	−0.277974
$397$	404604.	0.128841	0.0644205	−	0.997923i	$-0.479480\pi$
$397$	404604.	0.128841	0.0644205	+	0.997923i	$0.479480\pi$
$398$	1.08066e7	3.41965
$399$	339385.	0.106724
$400$	0	0
$401$	−5.08752e6	−1.57996	−0.789978	−	0.613135i	$-0.789908\pi$
$401$	−5.08752e6	−1.57996	−0.789978	+	0.613135i	$0.789908\pi$
$402$	−595604.	−0.183820
$403$	−1.10535e6	−0.339029
$404$	−3.93203e6	−1.19857
$405$	0	0
$406$	−1.56407e6	−0.470913
$407$	1.32681e6	0.397030
$408$	1.09079e7	3.24409
$409$	367772.	0.108710	0.0543551	−	0.998522i	$-0.482690\pi$
$409$	367772.	0.108710	0.0543551	+	0.998522i	$0.482690\pi$
$410$	0	0
$411$	1.07539e6	0.314024
$412$	1.46138e7	4.24150
$413$	−1.87666e6	−0.541390
$414$	−129718.	−0.0371964
$415$	0	0
$416$	2.43412e6	0.689618
$417$	467975.	0.131790
$418$	1.60880e6	0.450363
$419$	−4.12612e6	−1.14817	−0.574086	−	0.818795i	$-0.694643\pi$
$419$	−4.12612e6	−1.14817	−0.574086	+	0.818795i	$0.694643\pi$
$420$	0	0
$421$	−2.05918e6	−0.566224	−0.283112	−	0.959087i	$-0.591367\pi$
$421$	−2.05918e6	−0.566224	−0.283112	+	0.959087i	$0.591367\pi$
$422$	1.02115e7	2.79132
$423$	−161379.	−0.0438527
$424$	9.20376e6	2.48628
$425$	0	0
$426$	−392007.	−0.104657
$427$	−2.68000e6	−0.711321
$428$	−1.29259e7	−3.41075
$429$	3.45006e6	0.905072
$430$	0	0
$431$	−3.23089e6	−0.837777	−0.418889	−	0.908038i	$-0.637580\pi$
$431$	−3.23089e6	−0.837777	−0.418889	+	0.908038i	$0.637580\pi$
$432$	6.20464e6	1.59958
$433$	4.46477e6	1.14440	0.572202	−	0.820113i	$-0.306089\pi$
$433$	4.46477e6	1.14440	0.572202	+	0.820113i	$0.306089\pi$
$434$	−1.34231e6	−0.342082
$435$	0	0
$436$	1.72607e6	0.434854
$437$	165885.	0.0415530
$438$	9.58449e6	2.38717
$439$	3.96204e6	0.981201	0.490601	−	0.871385i	$-0.336777\pi$
$439$	3.96204e6	0.981201	0.490601	+	0.871385i	$0.336777\pi$
$440$	0	0
$441$	−376592.	−0.0922094
$442$	−8.10376e6	−1.97302
$443$	−5.87689e6	−1.42278	−0.711391	−	0.702797i	$-0.751934\pi$
$443$	−5.87689e6	−1.42278	−0.711391	+	0.702797i	$0.751934\pi$
$444$	3.53468e6	0.850927
$445$	0	0
$446$	3.83142e6	0.912059
$447$	−1.90649e6	−0.451299
$448$	−247101.	−0.0581673
$449$	−2.62215e6	−0.613820	−0.306910	−	0.951739i	$-0.599295\pi$
$449$	−2.62215e6	−0.613820	−0.306910	+	0.951739i	$0.599295\pi$
$450$	0	0
$451$	−2.40693e6	−0.557215
$452$	1.63212e7	3.75756
$453$	−592109.	−0.135568
$454$	1.30386e7	2.96887
$455$	0	0
$456$	2.35599e6	0.530593
$457$	−5.12578e6	−1.14807	−0.574037	−	0.818830i	$-0.694623\pi$
$457$	−5.12578e6	−1.14807	−0.574037	+	0.818830i	$0.694623\pi$
$458$	3.03651e6	0.676411
$459$	−5.91883e6	−1.31131
$460$	0	0
$461$	2.65571e6	0.582008	0.291004	−	0.956722i	$-0.406011\pi$
$461$	2.65571e6	0.582008	0.291004	+	0.956722i	$0.406011\pi$
$462$	4.18969e6	0.913223
$463$	2.94632e6	0.638744	0.319372	−	0.947629i	$-0.396528\pi$
$463$	2.94632e6	0.638744	0.319372	+	0.947629i	$0.396528\pi$
$464$	−4.72502e6	−1.01885
$465$	0	0
$466$	−5.73533e6	−1.22347
$467$	−1.82563e6	−0.387364	−0.193682	−	0.981064i	$-0.562043\pi$
$467$	−1.82563e6	−0.387364	−0.193682	+	0.981064i	$0.562043\pi$
$468$	943747.	0.199178
$469$	203671.	0.0427561
$470$	0	0
$471$	9.31806e6	1.93541
$472$	−1.30276e7	−2.69160
$473$	2.23957e6	0.460269
$474$	4.96446e6	1.01491
$475$	0	0
$476$	−6.78556e6	−1.37268
$477$	645322.	0.129862
$478$	−1.05211e7	−2.10616
$479$	5.92850e6	1.18061	0.590304	−	0.807181i	$-0.299008\pi$
$479$	5.92850e6	1.18061	0.590304	+	0.807181i	$0.299008\pi$
$480$	0	0
$481$	−1.44352e6	−0.284486
$482$	8.33763e6	1.63465
$483$	432001.	0.0842592
$484$	2.24917e6	0.436424
$485$	0	0
$486$	2.12853e6	0.408780
$487$	3.61261e6	0.690237	0.345118	−	0.938559i	$-0.387839\pi$
$487$	3.61261e6	0.690237	0.345118	+	0.938559i	$0.387839\pi$
$488$	−1.86044e7	−3.53644
$489$	−5.06589e6	−0.958041
$490$	0	0
$491$	−4.32852e6	−0.810281	−0.405140	−	0.914255i	$-0.632777\pi$
$491$	−4.32852e6	−0.810281	−0.405140	+	0.914255i	$0.632777\pi$
$492$	−6.41216e6	−1.19424
$493$	4.50736e6	0.835228
$494$	−1.75032e6	−0.322700
$495$	0	0
$496$	−4.05510e6	−0.740112
$497$	134050.	0.0243431
$498$	1.01117e7	1.82706
$499$	−4.93615e6	−0.887435	−0.443718	−	0.896167i	$-0.646341\pi$
$499$	−4.93615e6	−0.887435	−0.443718	+	0.896167i	$0.646341\pi$
$500$	0	0
$501$	2.08980e6	0.371972
$502$	1.24933e7	2.21268
$503$	−1.03023e6	−0.181557	−0.0907786	−	0.995871i	$-0.528936\pi$
$503$	−1.03023e6	−0.181557	−0.0907786	+	0.995871i	$0.528936\pi$
$504$	630000.	0.110475
$505$	0	0
$506$	2.04784e6	0.355565
$507$	2.35654e6	0.407151
$508$	−574677.	−0.0988017
$509$	4.99120e6	0.853907	0.426954	−	0.904274i	$-0.359587\pi$
$509$	4.99120e6	0.853907	0.426954	+	0.904274i	$0.359587\pi$
$510$	0	0
$511$	−3.27749e6	−0.555250
$512$	−1.33055e7	−2.24314
$513$	−1.27840e6	−0.214473
$514$	1.06806e7	1.78315
$515$	0	0
$516$	5.96630e6	0.986463
$517$	2.54766e6	0.419194
$518$	−1.75299e6	−0.287048
$519$	−9.58598e6	−1.56213
$520$	0	0
$521$	9.78050e6	1.57858	0.789291	−	0.614020i	$-0.210449\pi$
$521$	9.78050e6	1.57858	0.789291	+	0.614020i	$0.210449\pi$
$522$	−761286.	−0.122284
$523$	6.73157e6	1.07612	0.538062	−	0.842905i	$-0.319157\pi$
$523$	6.73157e6	1.07612	0.538062	+	0.842905i	$0.319157\pi$
$524$	8.43622e6	1.34221
$525$	0	0
$526$	−1.91018e7	−3.01030
$527$	3.86830e6	0.606728
$528$	1.26570e7	1.97581
$529$	−6.22519e6	−0.967194
$530$	0	0
$531$	−913433.	−0.140586
$532$	−1.46560e6	−0.224511
$533$	2.61865e6	0.399264
$534$	−3.19676e6	−0.485129
$535$	0	0
$536$	1.41387e6	0.212568
$537$	3.84777e6	0.575803
$538$	−1.06991e7	−1.59365
$539$	5.94519e6	0.881442
$540$	0	0
$541$	−6.68215e6	−0.981574	−0.490787	−	0.871280i	$-0.663291\pi$
$541$	−6.68215e6	−0.981574	−0.490787	+	0.871280i	$0.663291\pi$
$542$	6.99159e6	1.02230
$543$	−5.23603e6	−0.762084
$544$	−8.51851e6	−1.23415
$545$	0	0
$546$	−4.55822e6	−0.654356
$547$	235187.	0.0336081	0.0168041	−	0.999859i	$-0.494651\pi$
$547$	235187.	0.0336081	0.0168041	+	0.999859i	$0.494651\pi$
$548$	−4.64397e6	−0.660600
$549$	−1.30445e6	−0.184712
$550$	0	0
$551$	973538.	0.136607
$552$	2.99892e6	0.418907
$553$	−1.69763e6	−0.236065
$554$	5.30668e6	0.734597
$555$	0	0
$556$	−2.02090e6	−0.277242
$557$	4.80203e6	0.655824	0.327912	−	0.944708i	$-0.393655\pi$
$557$	4.80203e6	0.655824	0.327912	+	0.944708i	$0.393655\pi$
$558$	−653350.	−0.0888301
$559$	−2.43657e6	−0.329799
$560$	0	0
$561$	−1.20739e7	−1.61972
$562$	−2.03296e7	−2.71512
$563$	8.37831e6	1.11400	0.557000	−	0.830512i	$-0.311952\pi$
$563$	8.37831e6	1.11400	0.557000	+	0.830512i	$0.311952\pi$
$564$	6.78706e6	0.898430
$565$	0	0
$566$	2.28644e7	2.99998
$567$	−3.71525e6	−0.485323
$568$	930565.	0.121025
$569$	−8.38130e6	−1.08525	−0.542626	−	0.839974i	$-0.682570\pi$
$569$	−8.38130e6	−1.08525	−0.542626	+	0.839974i	$0.682570\pi$
$570$	0	0
$571$	3.97400e6	0.510079	0.255040	−	0.966931i	$-0.417912\pi$
$571$	3.97400e6	0.510079	0.255040	+	0.966931i	$0.417912\pi$
$572$	−1.48987e7	−1.90397
$573$	2.99555e6	0.381145
$574$	3.18004e6	0.402859
$575$	0	0
$576$	−120272.	−0.0151046
$577$	−1.40884e7	−1.76166	−0.880830	−	0.473432i	$-0.843015\pi$
$577$	−1.40884e7	−1.76166	−0.880830	+	0.473432i	$0.843015\pi$
$578$	1.39457e7	1.73628
$579$	4.70690e6	0.583497
$580$	0	0
$581$	−3.45779e6	−0.424970
$582$	−7.21542e6	−0.882986
$583$	−1.01876e7	−1.24136
$584$	−2.27521e7	−2.76051
$585$	0	0
$586$	−1.79313e7	−2.15708
$587$	−3.14647e6	−0.376901	−0.188451	−	0.982083i	$-0.560347\pi$
$587$	−3.14647e6	−0.376901	−0.188451	+	0.982083i	$0.560347\pi$
$588$	1.58382e7	1.88913
$589$	835508.	0.0992345
$590$	0	0
$591$	−7.66662e6	−0.902891
$592$	−5.29573e6	−0.621043
$593$	4.29402e6	0.501450	0.250725	−	0.968058i	$-0.419331\pi$
$593$	4.29402e6	0.501450	0.250725	+	0.968058i	$0.419331\pi$
$594$	−1.57817e7	−1.83522
$595$	0	0
$596$	8.23297e6	0.949382
$597$	−1.75172e7	−2.01154
$598$	−2.22797e6	−0.254774
$599$	−6.82650e6	−0.777376	−0.388688	−	0.921370i	$-0.627072\pi$
$599$	−6.82650e6	−0.777376	−0.388688	+	0.921370i	$0.627072\pi$
$600$	0	0
$601$	1.82575e6	0.206184	0.103092	−	0.994672i	$-0.467126\pi$
$601$	1.82575e6	0.206184	0.103092	+	0.994672i	$0.467126\pi$
$602$	−2.95892e6	−0.332769
$603$	99133.7	0.0111027
$604$	2.55697e6	0.285189
$605$	0	0
$606$	9.24376e6	1.02251
$607$	2.44884e6	0.269767	0.134883	−	0.990861i	$-0.456934\pi$
$607$	2.44884e6	0.269767	0.134883	+	0.990861i	$0.456934\pi$
$608$	−1.83990e6	−0.201853
$609$	2.53531e6	0.277005
$610$	0	0
$611$	−2.77176e6	−0.300367
$612$	−3.30276e6	−0.356450
$613$	6.05994e6	0.651354	0.325677	−	0.945481i	$-0.394408\pi$
$613$	6.05994e6	0.651354	0.325677	+	0.945481i	$0.394408\pi$
$614$	1.74481e7	1.86778
$615$	0	0
$616$	−9.94568e6	−1.05605
$617$	−9.33155e6	−0.986827	−0.493414	−	0.869795i	$-0.664251\pi$
$617$	−9.33155e6	−0.986827	−0.493414	+	0.869795i	$0.664251\pi$
$618$	−3.43553e7	−3.61845
$619$	−9.73252e6	−1.02094	−0.510468	−	0.859897i	$-0.670528\pi$
$619$	−9.73252e6	−1.02094	−0.510468	+	0.859897i	$0.670528\pi$
$620$	0	0
$621$	−1.62726e6	−0.169328
$622$	1.60554e7	1.66397
$623$	1.09316e6	0.112840
$624$	−1.37703e7	−1.41573
$625$	0	0
$626$	−554475.	−0.0565518
$627$	−2.60783e6	−0.264917
$628$	−4.02391e7	−4.07145
$629$	5.05178e6	0.509118
$630$	0	0
$631$	8.09784e6	0.809647	0.404824	−	0.914395i	$-0.367333\pi$
$631$	8.09784e6	0.809647	0.404824	+	0.914395i	$0.367333\pi$
$632$	−1.17849e7	−1.17363
$633$	−1.65526e7	−1.64194
$634$	1.05041e7	1.03785
$635$	0	0
$636$	−2.71401e7	−2.66053
$637$	−6.46814e6	−0.631583
$638$	1.20183e7	1.16893
$639$	65246.6	0.00632129
$640$	0	0
$641$	−1.54883e7	−1.48887	−0.744436	−	0.667694i	$-0.767282\pi$
$641$	−1.54883e7	−1.48887	−0.744436	+	0.667694i	$0.767282\pi$
$642$	3.03872e7	2.90973
$643$	−1.43966e7	−1.37320	−0.686600	−	0.727036i	$-0.740897\pi$
$643$	−1.43966e7	−1.37320	−0.686600	+	0.727036i	$0.740897\pi$
$644$	−1.86555e6	−0.177253
$645$	0	0
$646$	6.12546e6	0.577507
$647$	6.47992e6	0.608568	0.304284	−	0.952581i	$-0.401583\pi$
$647$	6.47992e6	0.608568	0.304284	+	0.952581i	$0.401583\pi$
$648$	−2.57910e7	−2.41286
$649$	1.44202e7	1.34388
$650$	0	0
$651$	2.17585e6	0.201223
$652$	2.18766e7	2.01539
$653$	7.89619e6	0.724660	0.362330	−	0.932050i	$-0.381981\pi$
$653$	7.89619e6	0.724660	0.362330	+	0.932050i	$0.381981\pi$
$654$	−4.05780e6	−0.370977
$655$	0	0
$656$	9.60683e6	0.871607
$657$	−1.59526e6	−0.144185
$658$	−3.36597e6	−0.303072
$659$	−1.08756e7	−0.975530	−0.487765	−	0.872975i	$-0.662188\pi$
$659$	−1.08756e7	−0.975530	−0.487765	+	0.872975i	$0.662188\pi$
$660$	0	0
$661$	−8.09950e6	−0.721032	−0.360516	−	0.932753i	$-0.617399\pi$
$661$	−8.09950e6	−0.721032	−0.360516	+	0.932753i	$0.617399\pi$
$662$	2.07089e7	1.83659
$663$	1.31360e7	1.16059
$664$	−2.40037e7	−2.11280
$665$	0	0
$666$	−853238.	−0.0745391
$667$	1.23921e6	0.107852
$668$	−9.02458e6	−0.782503
$669$	−6.21062e6	−0.536500
$670$	0	0
$671$	2.05930e7	1.76569
$672$	−4.79151e6	−0.409307
$673$	−6.23497e6	−0.530636	−0.265318	−	0.964161i	$-0.585477\pi$
$673$	−6.23497e6	−0.530636	−0.265318	+	0.964161i	$0.585477\pi$
$674$	−1.24641e7	−1.05685
$675$	0	0
$676$	−1.01765e7	−0.856507
$677$	−1.16588e7	−0.977649	−0.488824	−	0.872382i	$-0.662574\pi$
$677$	−1.16588e7	−0.977649	−0.488824	+	0.872382i	$0.662574\pi$
$678$	−3.83693e7	−3.20560
$679$	2.46737e6	0.205380
$680$	0	0
$681$	−2.11352e7	−1.74638
$682$	1.03143e7	0.849139
$683$	1.22340e6	0.100350	0.0501748	−	0.998740i	$-0.484022\pi$
$683$	1.22340e6	0.100350	0.0501748	+	0.998740i	$0.484022\pi$
$684$	−713358.	−0.0582998
$685$	0	0
$686$	−1.76025e7	−1.42811
$687$	−4.92209e6	−0.397885
$688$	−8.93884e6	−0.719962
$689$	1.10837e7	0.889480
$690$	0	0
$691$	−5.82015e6	−0.463702	−0.231851	−	0.972751i	$-0.574478\pi$
$691$	−5.82015e6	−0.463702	−0.231851	+	0.972751i	$0.574478\pi$
$692$	4.13961e7	3.28620
$693$	−697341.	−0.0551585
$694$	−3.99247e7	−3.14661
$695$	0	0
$696$	1.76000e7	1.37717
$697$	−9.16430e6	−0.714525
$698$	−3.51440e7	−2.73031
$699$	9.29680e6	0.719682
$700$	0	0
$701$	9.44191e6	0.725713	0.362856	−	0.931845i	$-0.381802\pi$
$701$	9.44191e6	0.725713	0.362856	+	0.931845i	$0.381802\pi$
$702$	1.71699e7	1.31500
$703$	1.09113e6	0.0832696
$704$	1.89871e6	0.144387
$705$	0	0
$706$	3.97177e7	2.99897
$707$	−3.16097e6	−0.237833
$708$	3.84159e7	2.88024
$709$	7.88964e6	0.589443	0.294721	−	0.955583i	$-0.404773\pi$
$709$	7.88964e6	0.589443	0.294721	+	0.955583i	$0.404773\pi$
$710$	0	0
$711$	−826296.	−0.0613002
$712$	7.58862e6	0.561000
$713$	1.06351e6	0.0783464
$714$	1.59521e7	1.17104
$715$	0	0
$716$	−1.66162e7	−1.21129
$717$	1.70544e7	1.23891
$718$	−9.64300e6	−0.698073
$719$	1.89500e7	1.36706	0.683530	−	0.729922i	$-0.260444\pi$
$719$	1.89500e7	1.36706	0.683530	+	0.729922i	$0.260444\pi$
$720$	0	0
$721$	1.17481e7	0.841643
$722$	1.32303e6	0.0944552
$723$	−1.35150e7	−0.961550
$724$	2.26113e7	1.60317
$725$	0	0
$726$	−5.28754e6	−0.372316
$727$	−6.46947e6	−0.453976	−0.226988	−	0.973898i	$-0.572888\pi$
$727$	−6.46947e6	−0.453976	−0.226988	+	0.973898i	$0.572888\pi$
$728$	1.08205e7	0.756693
$729$	1.23527e7	0.860879
$730$	0	0
$731$	8.52707e6	0.590210
$732$	5.48607e7	3.78428
$733$	−6.39936e6	−0.439923	−0.219962	−	0.975509i	$-0.570593\pi$
$733$	−6.39936e6	−0.439923	−0.219962	+	0.975509i	$0.570593\pi$
$734$	−3.78249e7	−2.59142
$735$	0	0
$736$	−2.34199e6	−0.159364
$737$	−1.56500e6	−0.106132
$738$	1.54783e6	0.104613
$739$	−1.28565e7	−0.865987	−0.432993	−	0.901397i	$-0.642543\pi$
$739$	−1.28565e7	−0.865987	−0.432993	+	0.901397i	$0.642543\pi$
$740$	0	0
$741$	2.83721e6	0.189822
$742$	1.34598e7	0.897491
$743$	2.19966e7	1.46179	0.730893	−	0.682492i	$-0.239104\pi$
$743$	2.19966e7	1.46179	0.730893	+	0.682492i	$0.239104\pi$
$744$	1.51046e7	1.00041
$745$	0	0
$746$	4.21693e7	2.77427
$747$	−1.68302e6	−0.110354
$748$	5.21400e7	3.40735
$749$	−1.03911e7	−0.676797
$750$	0	0
$751$	4.73982e6	0.306663	0.153332	−	0.988175i	$-0.451000\pi$
$751$	4.73982e6	0.306663	0.153332	+	0.988175i	$0.451000\pi$
$752$	−1.01685e7	−0.655712
$753$	−2.02514e7	−1.30157
$754$	−1.30754e7	−0.837581
$755$	0	0
$756$	1.43770e7	0.914877
$757$	8.15849e6	0.517452	0.258726	−	0.965951i	$-0.416697\pi$
$757$	8.15849e6	0.517452	0.258726	+	0.965951i	$0.416697\pi$
$758$	2.32750e7	1.47135
$759$	−3.31948e6	−0.209154
$760$	0	0
$761$	−2.40805e7	−1.50731	−0.753657	−	0.657268i	$-0.771712\pi$
$761$	−2.40805e7	−1.50731	−0.753657	+	0.657268i	$0.771712\pi$
$762$	1.35100e6	0.0842884
$763$	1.38760e6	0.0862883
$764$	−1.29360e7	−0.801800
$765$	0	0
$766$	−3.14069e7	−1.93399
$767$	−1.56886e7	−0.962933
$768$	3.23054e7	1.97639
$769$	−3.57395e6	−0.217938	−0.108969	−	0.994045i	$-0.534755\pi$
$769$	−3.57395e6	−0.217938	−0.108969	+	0.994045i	$0.534755\pi$
$770$	0	0
$771$	−1.73129e7	−1.04890
$772$	−2.03263e7	−1.22748
$773$	−1.38262e7	−0.832251	−0.416126	−	0.909307i	$-0.636612\pi$
$773$	−1.38262e7	−0.832251	−0.416126	+	0.909307i	$0.636612\pi$
$774$	−1.44021e6	−0.0864117
$775$	0	0
$776$	1.71283e7	1.02108
$777$	2.84154e6	0.168850
$778$	−4.60632e7	−2.72838
$779$	−1.97938e6	−0.116865
$780$	0	0
$781$	−1.03003e6	−0.0604260
$782$	7.79705e6	0.455946
$783$	−9.55002e6	−0.556672
$784$	−2.37291e7	−1.37877
$785$	0	0
$786$	−1.98326e7	−1.14505
$787$	−2.52043e7	−1.45057	−0.725283	−	0.688451i	$-0.758291\pi$
$787$	−2.52043e7	−1.45057	−0.725283	+	0.688451i	$0.758291\pi$
$788$	3.31075e7	1.89938
$789$	3.09635e7	1.77075
$790$	0	0
$791$	1.31207e7	0.745615
$792$	−4.84090e6	−0.274229
$793$	−2.24045e7	−1.26518
$794$	4.10757e6	0.231224
$795$	0	0
$796$	7.56464e7	4.23161
$797$	−3.37562e7	−1.88238	−0.941190	−	0.337877i	$-0.890291\pi$
$797$	−3.37562e7	−1.88238	−0.941190	+	0.337877i	$0.890291\pi$
$798$	3.44546e6	0.191532
$799$	9.70011e6	0.537539
$800$	0	0
$801$	532076.	0.0293017
$802$	−5.16488e7	−2.83547
$803$	2.51841e7	1.37828
$804$	−4.16923e6	−0.227466
$805$	0	0
$806$	−1.12216e7	−0.608437
$807$	1.73430e7	0.937433
$808$	−2.19433e7	−1.18242
$809$	−2.10318e7	−1.12981	−0.564906	−	0.825155i	$-0.691088\pi$
$809$	−2.10318e7	−1.12981	−0.564906	+	0.825155i	$0.691088\pi$
$810$	0	0
$811$	−2.27692e7	−1.21561	−0.607807	−	0.794085i	$-0.707950\pi$
$811$	−2.27692e7	−1.21561	−0.607807	+	0.794085i	$0.707950\pi$
$812$	−1.09485e7	−0.582726
$813$	−1.13332e7	−0.601346
$814$	1.34699e7	0.712530
$815$	0	0
$816$	4.81908e7	2.53361
$817$	1.84175e6	0.0965328
$818$	3.73365e6	0.195097
$819$	758681.	0.0395230
$820$	0	0
$821$	−2.30273e7	−1.19230	−0.596149	−	0.802874i	$-0.703303\pi$
$821$	−2.30273e7	−1.19230	−0.596149	+	0.802874i	$0.703303\pi$
$822$	1.09175e7	0.563562
$823$	1.70408e7	0.876979	0.438490	−	0.898736i	$-0.355514\pi$
$823$	1.70408e7	0.876979	0.438490	+	0.898736i	$0.355514\pi$
$824$	8.15542e7	4.18435
$825$	0	0
$826$	−1.90520e7	−0.971605
$827$	−3.62225e6	−0.184168	−0.0920840	−	0.995751i	$-0.529353\pi$
$827$	−3.62225e6	−0.184168	−0.0920840	+	0.995751i	$0.529353\pi$
$828$	−908028.	−0.0460281
$829$	1.36980e7	0.692261	0.346131	−	0.938186i	$-0.387495\pi$
$829$	1.36980e7	0.692261	0.346131	+	0.938186i	$0.387495\pi$
$830$	0	0
$831$	−8.60198e6	−0.432112
$832$	−2.06573e6	−0.103458
$833$	2.26361e7	1.13029
$834$	4.75091e6	0.236517
$835$	0	0
$836$	1.12616e7	0.557296
$837$	−8.19600e6	−0.404379
$838$	−4.18887e7	−2.06057
$839$	−8.15272e6	−0.399850	−0.199925	−	0.979811i	$-0.564070\pi$
$839$	−8.15272e6	−0.399850	−0.199925	+	0.979811i	$0.564070\pi$
$840$	0	0
$841$	−1.32385e7	−0.645431
$842$	−2.09049e7	−1.01617
$843$	3.29537e7	1.59711
$844$	7.14807e7	3.45408
$845$	0	0
$846$	−1.63833e6	−0.0787002
$847$	1.80811e6	0.0865999
$848$	4.06618e7	1.94177
$849$	−3.70625e7	−1.76468
$850$	0	0
$851$	1.38889e6	0.0657420
$852$	−2.74405e6	−0.129507
$853$	3.76514e7	1.77178	0.885888	−	0.463899i	$-0.153550\pi$
$853$	3.76514e7	1.77178	0.885888	+	0.463899i	$0.153550\pi$
$854$	−2.72076e7	−1.27657
$855$	0	0
$856$	−7.21345e7	−3.36480
$857$	−1.18679e7	−0.551980	−0.275990	−	0.961160i	$-0.589006\pi$
$857$	−1.18679e7	−0.551980	−0.275990	+	0.961160i	$0.589006\pi$
$858$	3.50252e7	1.62429
$859$	−8.63985e6	−0.399506	−0.199753	−	0.979846i	$-0.564014\pi$
$859$	−8.63985e6	−0.399506	−0.199753	+	0.979846i	$0.564014\pi$
$860$	0	0
$861$	−5.15476e6	−0.236974
$862$	−3.28002e7	−1.50352
$863$	1.39409e7	0.637181	0.318591	−	0.947892i	$-0.396790\pi$
$863$	1.39409e7	0.637181	0.318591	+	0.947892i	$0.396790\pi$
$864$	1.80487e7	0.822547
$865$	0	0
$866$	4.53267e7	2.05380
$867$	−2.26055e7	−1.02133
$868$	−9.39620e6	−0.423305
$869$	1.30446e7	0.585976
$870$	0	0
$871$	1.70266e6	0.0760473
$872$	9.63260e6	0.428995
$873$	1.20095e6	0.0533322
$874$	1.68407e6	0.0745731
$875$	0	0
$876$	6.70914e7	2.95397
$877$	−2.92561e7	−1.28445	−0.642224	−	0.766517i	$-0.721988\pi$
$877$	−2.92561e7	−1.28445	−0.642224	+	0.766517i	$0.721988\pi$
$878$	4.02229e7	1.76091
$879$	2.90661e7	1.26886
$880$	0	0
$881$	−1.44235e7	−0.626081	−0.313041	−	0.949740i	$-0.601348\pi$
$881$	−1.44235e7	−0.626081	−0.313041	+	0.949740i	$0.601348\pi$
$882$	−3.82319e6	−0.165483
$883$	4.94360e6	0.213374	0.106687	−	0.994293i	$-0.465976\pi$
$883$	4.94360e6	0.213374	0.106687	+	0.994293i	$0.465976\pi$
$884$	−5.67263e7	−2.44149
$885$	0	0
$886$	−5.96626e7	−2.55339
$887$	1.32368e7	0.564903	0.282452	−	0.959282i	$-0.408852\pi$
$887$	1.32368e7	0.564903	0.282452	+	0.959282i	$0.408852\pi$
$888$	1.97258e7	0.839463
$889$	−461984.	−0.0196053
$890$	0	0
$891$	2.85479e7	1.20470
$892$	2.68200e7	1.12862
$893$	2.09511e6	0.0879181
$894$	−1.93548e7	−0.809924
$895$	0	0
$896$	−1.18259e7	−0.492114
$897$	3.61147e6	0.149866
$898$	−2.66202e7	−1.10159
$899$	6.24150e6	0.257567
$900$	0	0
$901$	−3.87888e7	−1.59182
$902$	−2.44353e7	−1.00001
$903$	4.79633e6	0.195744
$904$	9.10827e7	3.70693
$905$	0	0
$906$	−6.01113e6	−0.243297
$907$	−7.51989e6	−0.303524	−0.151762	−	0.988417i	$-0.548495\pi$
$907$	−7.51989e6	−0.303524	−0.151762	+	0.988417i	$0.548495\pi$
$908$	9.12701e7	3.67379
$909$	−1.53855e6	−0.0617593
$910$	0	0
$911$	4.53248e7	1.80942	0.904710	−	0.426027i	$-0.140087\pi$
$911$	4.53248e7	1.80942	0.904710	+	0.426027i	$0.140087\pi$
$912$	1.04087e7	0.414388
$913$	2.65695e7	1.05489
$914$	−5.20373e7	−2.06039
$915$	0	0
$916$	2.12556e7	0.837016
$917$	6.78190e6	0.266335
$918$	−6.00883e7	−2.35333
$919$	1.47088e7	0.574497	0.287249	−	0.957856i	$-0.407259\pi$
$919$	1.47088e7	0.574497	0.287249	+	0.957856i	$0.407259\pi$
$920$	0	0
$921$	−2.82828e7	−1.09869
$922$	2.69610e7	1.04450
$923$	1.12064e6	0.0432973
$924$	2.93278e7	1.13006
$925$	0	0
$926$	2.99112e7	1.14632
$927$	5.71818e6	0.218554
$928$	−1.37446e7	−0.523917
$929$	−1.35395e7	−0.514711	−0.257356	−	0.966317i	$-0.582851\pi$
$929$	−1.35395e7	−0.514711	−0.257356	+	0.966317i	$0.582851\pi$
$930$	0	0
$931$	4.88913e6	0.184866
$932$	−4.01473e7	−1.51397
$933$	−2.60253e7	−0.978795
$934$	−1.85339e7	−0.695183
$935$	0	0
$936$	5.26671e6	0.196494
$937$	1.21203e7	0.450988	0.225494	−	0.974245i	$-0.427600\pi$
$937$	1.21203e7	0.450988	0.225494	+	0.974245i	$0.427600\pi$
$938$	2.06769e6	0.0767321
$939$	898788.	0.0332655
$940$	0	0
$941$	2.61927e7	0.964288	0.482144	−	0.876092i	$-0.339858\pi$
$941$	2.61927e7	0.964288	0.482144	+	0.876092i	$0.339858\pi$
$942$	9.45976e7	3.47338
$943$	−2.51954e6	−0.0922661
$944$	−5.75555e7	−2.10212
$945$	0	0
$946$	2.27363e7	0.826021
$947$	3.47875e7	1.26051	0.630257	−	0.776386i	$-0.282949\pi$
$947$	3.47875e7	1.26051	0.630257	+	0.776386i	$0.282949\pi$
$948$	3.47512e7	1.25588
$949$	−2.73993e7	−0.987585
$950$	0	0
$951$	−1.70268e7	−0.610494
$952$	−3.78678e7	−1.35418
$953$	−1.16268e7	−0.414695	−0.207347	−	0.978267i	$-0.566483\pi$
$953$	−1.16268e7	−0.414695	−0.207347	+	0.978267i	$0.566483\pi$
$954$	6.55136e6	0.233056
$955$	0	0
$956$	−7.36477e7	−2.60624
$957$	−1.94812e7	−0.687602
$958$	6.01865e7	2.11878
$959$	−3.73330e6	−0.131083
$960$	0	0
$961$	−2.32726e7	−0.812898
$962$	−1.46547e7	−0.510552
$963$	−5.05772e6	−0.175747
$964$	5.83634e7	2.02278
$965$	0	0
$966$	4.38570e6	0.151216
$967$	2.70316e7	0.929622	0.464811	−	0.885410i	$-0.346122\pi$
$967$	2.70316e7	0.929622	0.464811	+	0.885410i	$0.346122\pi$
$968$	1.25518e7	0.430544
$969$	−9.92919e6	−0.339707
$970$	0	0
$971$	−5.13472e6	−0.174771	−0.0873854	−	0.996175i	$-0.527851\pi$
$971$	−5.13472e6	−0.174771	−0.0873854	+	0.996175i	$0.527851\pi$
$972$	1.48997e7	0.505840
$973$	−1.62461e6	−0.0550132
$974$	3.66754e7	1.23873
$975$	0	0
$976$	−8.21934e7	−2.76193
$977$	5.11922e6	0.171580	0.0857902	−	0.996313i	$-0.472659\pi$
$977$	5.11922e6	0.171580	0.0857902	+	0.996313i	$0.472659\pi$
$978$	−5.14293e7	−1.71935
$979$	−8.39978e6	−0.280099
$980$	0	0
$981$	675390.	0.0224069
$982$	−4.39434e7	−1.45417
$983$	−3.30087e7	−1.08954	−0.544771	−	0.838585i	$-0.683384\pi$
$983$	−3.30087e7	−1.08954	−0.544771	+	0.838585i	$0.683384\pi$
$984$	−3.57840e7	−1.17815
$985$	0	0
$986$	4.57590e7	1.49894
$987$	5.45614e6	0.178276
$988$	−1.22522e7	−0.399321
$989$	2.34435e6	0.0762134
$990$	0	0
$991$	−3.54462e7	−1.14653	−0.573265	−	0.819370i	$-0.694324\pi$
$991$	−3.54462e7	−1.14653	−0.573265	+	0.819370i	$0.694324\pi$
$992$	−1.17959e7	−0.380585
$993$	−3.35685e7	−1.08034
$994$	1.36088e6	0.0436873
$995$	0	0
$996$	7.07822e7	2.26087
$997$	−4.51197e7	−1.43757	−0.718784	−	0.695233i	$-0.755301\pi$
$997$	−4.51197e7	−1.43757	−0.718784	+	0.695233i	$0.755301\pi$
$998$	−5.01121e7	−1.59264
$999$	−1.07035e7	−0.339322

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.6.a.d.1.2		2
5.4	even	2		19.6.a.c.1.1	✓	2
15.14	odd	2		171.6.a.f.1.2		2
20.19	odd	2		304.6.a.g.1.1		2
35.34	odd	2		931.6.a.c.1.1		2
95.94	odd	2		361.6.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.6.a.c.1.1	✓	2	5.4	even	2
171.6.a.f.1.2		2	15.14	odd	2
304.6.a.g.1.1		2	20.19	odd	2
361.6.a.d.1.2		2	95.94	odd	2
475.6.a.d.1.2		2	1.1	even	1	trivial
931.6.a.c.1.1		2	35.34	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:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	475.a (trivial)

\(f(q)\)	\(=\)	\(q+10.1521 q^{2} -16.4562 q^{3} +71.0645 q^{4} -167.064 q^{6} +57.1289 q^{7} +396.585 q^{8} +27.8066 q^{9} +O(q^{10})\)	\(q+10.1521 q^{2} -16.4562 q^{3} +71.0645 q^{4} -167.064 q^{6} +57.1289 q^{7} +396.585 q^{8} +27.8066 q^{9} -438.977 q^{11} -1169.45 q^{12} +477.590 q^{13} +579.977 q^{14} +1752.10 q^{16} -1671.39 q^{17} +282.294 q^{18} -361.000 q^{19} -940.125 q^{21} -4456.52 q^{22} -459.514 q^{23} -6526.29 q^{24} +4848.53 q^{26} +3541.27 q^{27} +4059.84 q^{28} -2696.78 q^{29} -2314.43 q^{31} +5096.67 q^{32} +7223.89 q^{33} -16968.0 q^{34} +1976.06 q^{36} -3022.51 q^{37} -3664.90 q^{38} -7859.32 q^{39} +5483.05 q^{41} -9544.22 q^{42} -5101.80 q^{43} -31195.7 q^{44} -4665.02 q^{46} -5803.63 q^{47} -28832.8 q^{48} -13543.3 q^{49} +27504.7 q^{51} +33939.7 q^{52} +23207.5 q^{53} +35951.2 q^{54} +22656.5 q^{56} +5940.69 q^{57} -27377.9 q^{58} -32849.5 q^{59} -46911.5 q^{61} -23496.2 q^{62} +1588.56 q^{63} -4325.31 q^{64} +73337.4 q^{66} +3565.12 q^{67} -118776. q^{68} +7561.86 q^{69} +2346.44 q^{71} +11027.7 q^{72} -57370.0 q^{73} -30684.7 q^{74} -25654.3 q^{76} -25078.3 q^{77} -79788.3 q^{78} -29715.8 q^{79} -65032.8 q^{81} +55664.3 q^{82} -60526.0 q^{83} -66809.5 q^{84} -51793.8 q^{86} +44378.8 q^{87} -174092. q^{88} +19134.9 q^{89} +27284.2 q^{91} -32655.1 q^{92} +38086.7 q^{93} -58918.9 q^{94} -83871.9 q^{96} +43189.4 q^{97} -137492. q^{98} -12206.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 7 q^{2} + 7 q^{3} + 49 q^{4} - 241 q^{6} - 72 q^{7} + 567 q^{8} + 335 q^{9}+O(q^{10}) \) 2 * q + 7 * q^2 + 7 * q^3 + 49 * q^4 - 241 * q^6 - 72 * q^7 + 567 * q^8 + 335 * q^9	\( 2 q + 7 q^{2} + 7 q^{3} + 49 q^{4} - 241 q^{6} - 72 q^{7} + 567 q^{8} + 335 q^{9} - 705 q^{11} - 1687 q^{12} + 1341 q^{13} + 987 q^{14} + 1921 q^{16} - 2784 q^{17} - 686 q^{18} - 722 q^{19} - 3969 q^{21} - 3618 q^{22} + 2713 q^{23} - 2529 q^{24} + 2127 q^{26} + 5047 q^{27} + 6909 q^{28} - 7775 q^{29} + 7132 q^{31} - 889 q^{32} + 984 q^{33} - 13461 q^{34} - 4802 q^{36} + 6248 q^{37} - 2527 q^{38} + 12393 q^{39} - 4174 q^{41} + 3 q^{42} - 25357 q^{43} - 25326 q^{44} - 14665 q^{46} - 11727 q^{47} - 24871 q^{48} - 13676 q^{49} + 1407 q^{51} + 14889 q^{52} + 29133 q^{53} + 31205 q^{54} + 651 q^{56} - 2527 q^{57} - 11371 q^{58} - 64515 q^{59} - 40939 q^{61} - 53272 q^{62} - 38079 q^{63} + 9137 q^{64} + 93006 q^{66} - 19039 q^{67} - 94227 q^{68} + 81977 q^{69} - 70236 q^{71} + 63378 q^{72} - 67058 q^{73} - 59906 q^{74} - 17689 q^{76} + 9273 q^{77} - 143625 q^{78} - 32850 q^{79} - 104362 q^{81} + 86104 q^{82} - 71534 q^{83} + 21 q^{84} + 12052 q^{86} - 74737 q^{87} - 219426 q^{88} - 87268 q^{89} - 84207 q^{91} - 102655 q^{92} + 259664 q^{93} - 40248 q^{94} - 224273 q^{96} + 62458 q^{97} - 137074 q^{98} - 93927 q^{99}+O(q^{100}) \) 2 * q + 7 * q^2 + 7 * q^3 + 49 * q^4 - 241 * q^6 - 72 * q^7 + 567 * q^8 + 335 * q^9 - 705 * q^11 - 1687 * q^12 + 1341 * q^13 + 987 * q^14 + 1921 * q^16 - 2784 * q^17 - 686 * q^18 - 722 * q^19 - 3969 * q^21 - 3618 * q^22 + 2713 * q^23 - 2529 * q^24 + 2127 * q^26 + 5047 * q^27 + 6909 * q^28 - 7775 * q^29 + 7132 * q^31 - 889 * q^32 + 984 * q^33 - 13461 * q^34 - 4802 * q^36 + 6248 * q^37 - 2527 * q^38 + 12393 * q^39 - 4174 * q^41 + 3 * q^42 - 25357 * q^43 - 25326 * q^44 - 14665 * q^46 - 11727 * q^47 - 24871 * q^48 - 13676 * q^49 + 1407 * q^51 + 14889 * q^52 + 29133 * q^53 + 31205 * q^54 + 651 * q^56 - 2527 * q^57 - 11371 * q^58 - 64515 * q^59 - 40939 * q^61 - 53272 * q^62 - 38079 * q^63 + 9137 * q^64 + 93006 * q^66 - 19039 * q^67 - 94227 * q^68 + 81977 * q^69 - 70236 * q^71 + 63378 * q^72 - 67058 * q^73 - 59906 * q^74 - 17689 * q^76 + 9273 * q^77 - 143625 * q^78 - 32850 * q^79 - 104362 * q^81 + 86104 * q^82 - 71534 * q^83 + 21 * q^84 + 12052 * q^86 - 74737 * q^87 - 219426 * q^88 - 87268 * q^89 - 84207 * q^91 - 102655 * q^92 + 259664 * q^93 - 40248 * q^94 - 224273 * q^96 + 62458 * q^97 - 137074 * q^98 - 93927 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more