LMFDB - Embedded newform 245.6.a.h.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [245,6,Mod(1,245)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(245, 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("245.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 245 = 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	245.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:	$39.2940358542$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 109x^{4} + 41x^{3} + 2208x^{2} - 3204x + 560 $ x^6 - x^5 - 109x^4 + 41x^3 + 2208x^2 - 3204x + 560
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$9.72556$ of defining polynomial
Character	$\chi$	$=$	245.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.72556 q^{2} -4.73965 q^{3} +44.1354 q^{4} +25.0000 q^{5} -41.3561 q^{6} +105.888 q^{8} -220.536 q^{9} +O(q^{10})$	\(q+8.72556 q^{2} -4.73965 q^{3} +44.1354 q^{4} +25.0000 q^{5} -41.3561 q^{6} +105.888 q^{8} -220.536 q^{9} +218.139 q^{10} -679.156 q^{11} -209.186 q^{12} -58.8790 q^{13} -118.491 q^{15} -488.399 q^{16} -1427.56 q^{17} -1924.30 q^{18} +2197.34 q^{19} +1103.39 q^{20} -5926.02 q^{22} -2223.33 q^{23} -501.873 q^{24} +625.000 q^{25} -513.753 q^{26} +2197.00 q^{27} +2079.59 q^{29} -1033.90 q^{30} -3599.58 q^{31} -7649.98 q^{32} +3218.96 q^{33} -12456.2 q^{34} -9733.43 q^{36} +9181.95 q^{37} +19173.0 q^{38} +279.066 q^{39} +2647.21 q^{40} -10061.1 q^{41} -13956.6 q^{43} -29974.8 q^{44} -5513.39 q^{45} -19399.8 q^{46} +16272.5 q^{47} +2314.84 q^{48} +5453.48 q^{50} +6766.12 q^{51} -2598.65 q^{52} +1267.14 q^{53} +19170.0 q^{54} -16978.9 q^{55} -10414.6 q^{57} +18145.6 q^{58} +13465.7 q^{59} -5229.66 q^{60} -35031.5 q^{61} -31408.4 q^{62} -51121.6 q^{64} -1471.98 q^{65} +28087.2 q^{66} -36919.0 q^{67} -63005.9 q^{68} +10537.8 q^{69} +4118.92 q^{71} -23352.1 q^{72} +3517.37 q^{73} +80117.7 q^{74} -2962.28 q^{75} +96980.4 q^{76} +2435.01 q^{78} -72440.8 q^{79} -12210.0 q^{80} +43177.2 q^{81} -87789.0 q^{82} +42252.6 q^{83} -35688.9 q^{85} -121779. q^{86} -9856.51 q^{87} -71914.7 q^{88} -94517.0 q^{89} -48107.5 q^{90} -98127.7 q^{92} +17060.8 q^{93} +141987. q^{94} +54933.4 q^{95} +36258.2 q^{96} -85271.1 q^{97} +149778. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 5 q^{2} - 20 q^{3} + 31 q^{4} + 150 q^{5} - 96 q^{6} - 135 q^{8} + 378 q^{9}+O(q^{10}) $ 6 * q - 5 * q^2 - 20 * q^3 + 31 * q^4 + 150 * q^5 - 96 * q^6 - 135 * q^8 + 378 * q^9	\( 6 q - 5 q^{2} - 20 q^{3} + 31 q^{4} + 150 q^{5} - 96 q^{6} - 135 q^{8} + 378 q^{9} - 125 q^{10} - 924 q^{11} - 370 q^{12} + 150 q^{13} - 500 q^{15} + 435 q^{16} - 1540 q^{17} + 195 q^{18} - 92 q^{19} + 775 q^{20} - 6855 q^{22} - 3920 q^{23} + 7200 q^{24} + 3750 q^{25} - 2635 q^{26} - 2060 q^{27} + 1264 q^{29} - 2400 q^{30} + 7160 q^{31} + 9105 q^{32} - 4460 q^{33} - 2166 q^{34} - 26375 q^{36} - 14170 q^{37} + 46215 q^{38} - 15376 q^{39} - 3375 q^{40} - 4098 q^{41} - 24460 q^{43} - 27873 q^{44} + 9450 q^{45} + 6815 q^{46} - 42940 q^{47} - 11610 q^{48} - 3125 q^{50} - 42008 q^{51} + 36115 q^{52} - 2450 q^{53} - 19566 q^{54} - 23100 q^{55} - 97100 q^{57} - 36110 q^{58} + 64600 q^{59} - 9250 q^{60} - 73620 q^{61} - 111440 q^{62} - 157997 q^{64} + 3750 q^{65} + 139138 q^{66} - 142620 q^{67} - 124330 q^{68} - 17344 q^{69} - 154256 q^{71} - 117495 q^{72} + 5120 q^{73} + 2785 q^{74} - 12500 q^{75} - 7775 q^{76} - 214090 q^{78} - 222504 q^{79} + 10875 q^{80} - 43986 q^{81} - 31665 q^{82} - 179580 q^{83} - 38500 q^{85} - 207160 q^{86} + 209300 q^{87} - 45145 q^{88} - 41648 q^{89} + 4875 q^{90} - 292185 q^{92} - 198520 q^{93} + 333699 q^{94} - 2300 q^{95} - 61824 q^{96} + 73980 q^{97} - 190772 q^{99}+O(q^{100}) \) 6 * q - 5 * q^2 - 20 * q^3 + 31 * q^4 + 150 * q^5 - 96 * q^6 - 135 * q^8 + 378 * q^9 - 125 * q^10 - 924 * q^11 - 370 * q^12 + 150 * q^13 - 500 * q^15 + 435 * q^16 - 1540 * q^17 + 195 * q^18 - 92 * q^19 + 775 * q^20 - 6855 * q^22 - 3920 * q^23 + 7200 * q^24 + 3750 * q^25 - 2635 * q^26 - 2060 * q^27 + 1264 * q^29 - 2400 * q^30 + 7160 * q^31 + 9105 * q^32 - 4460 * q^33 - 2166 * q^34 - 26375 * q^36 - 14170 * q^37 + 46215 * q^38 - 15376 * q^39 - 3375 * q^40 - 4098 * q^41 - 24460 * q^43 - 27873 * q^44 + 9450 * q^45 + 6815 * q^46 - 42940 * q^47 - 11610 * q^48 - 3125 * q^50 - 42008 * q^51 + 36115 * q^52 - 2450 * q^53 - 19566 * q^54 - 23100 * q^55 - 97100 * q^57 - 36110 * q^58 + 64600 * q^59 - 9250 * q^60 - 73620 * q^61 - 111440 * q^62 - 157997 * q^64 + 3750 * q^65 + 139138 * q^66 - 142620 * q^67 - 124330 * q^68 - 17344 * q^69 - 154256 * q^71 - 117495 * q^72 + 5120 * q^73 + 2785 * q^74 - 12500 * q^75 - 7775 * q^76 - 214090 * q^78 - 222504 * q^79 + 10875 * q^80 - 43986 * q^81 - 31665 * q^82 - 179580 * q^83 - 38500 * q^85 - 207160 * q^86 + 209300 * q^87 - 45145 * q^88 - 41648 * q^89 + 4875 * q^90 - 292185 * q^92 - 198520 * q^93 + 333699 * q^94 - 2300 * q^95 - 61824 * q^96 + 73980 * q^97 - 190772 * 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 $

$2$	8.72556	1.54248	0.771238	−	0.636547i	$-0.219638\pi$
$2$	8.72556	1.54248	0.771238	+	0.636547i	$0.219638\pi$
$3$	−4.73965	−0.304048	−0.152024	−	0.988377i	$-0.548579\pi$
$3$	−4.73965	−0.304048	−0.152024	+	0.988377i	$0.548579\pi$
$4$	44.1354	1.37923
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	−41.3561	−0.468987
$7$	0	0
$7$	0	0
$8$	105.888	0.584955
$9$	−220.536	−0.907555
$10$	218.139	0.689816
$11$	−679.156	−1.69234	−0.846171	−	0.532912i	$-0.821098\pi$
$11$	−679.156	−1.69234	−0.846171	+	0.532912i	$0.821098\pi$
$12$	−209.186	−0.419353
$13$	−58.8790	−0.0966279	−0.0483139	−	0.998832i	$-0.515385\pi$
$13$	−58.8790	−0.0966279	−0.0483139	+	0.998832i	$0.515385\pi$
$14$	0	0
$15$	−118.491	−0.135975
$16$	−488.399	−0.476952
$17$	−1427.56	−1.19804	−0.599020	−	0.800734i	$-0.704443\pi$
$17$	−1427.56	−1.19804	−0.599020	+	0.800734i	$0.704443\pi$
$18$	−1924.30	−1.39988
$19$	2197.34	1.39641	0.698205	−	0.715898i	$-0.253983\pi$
$19$	2197.34	1.39641	0.698205	+	0.715898i	$0.253983\pi$
$20$	1103.39	0.616811
$21$	0	0
$22$	−5926.02	−2.61040
$23$	−2223.33	−0.876365	−0.438182	−	0.898886i	$-0.644378\pi$
$23$	−2223.33	−0.876365	−0.438182	+	0.898886i	$0.644378\pi$
$24$	−501.873	−0.177855
$25$	625.000	0.200000
$26$	−513.753	−0.149046
$27$	2197.00	0.579989
$28$	0	0
$29$	2079.59	0.459179	0.229590	−	0.973288i	$-0.426262\pi$
$29$	2079.59	0.459179	0.229590	+	0.973288i	$0.426262\pi$
$30$	−1033.90	−0.209738
$31$	−3599.58	−0.672741	−0.336371	−	0.941730i	$-0.609199\pi$
$31$	−3599.58	−0.672741	−0.336371	+	0.941730i	$0.609199\pi$
$32$	−7649.98	−1.32064
$33$	3218.96	0.514554
$34$	−12456.2	−1.84795
$35$	0	0
$36$	−9733.43	−1.25173
$37$	9181.95	1.10263	0.551316	−	0.834296i	$-0.314126\pi$
$37$	9181.95	1.10263	0.551316	+	0.834296i	$0.314126\pi$
$38$	19173.0	2.15393
$39$	279.066	0.0293796
$40$	2647.21	0.261600
$41$	−10061.1	−0.934732	−0.467366	−	0.884064i	$-0.654797\pi$
$41$	−10061.1	−0.934732	−0.467366	+	0.884064i	$0.654797\pi$
$42$	0	0
$43$	−13956.6	−1.15108	−0.575542	−	0.817772i	$-0.695209\pi$
$43$	−13956.6	−1.15108	−0.575542	+	0.817772i	$0.695209\pi$
$44$	−29974.8	−2.33413
$45$	−5513.39	−0.405871
$46$	−19399.8	−1.35177
$47$	16272.5	1.07451	0.537254	−	0.843420i	$-0.319462\pi$
$47$	16272.5	1.07451	0.537254	+	0.843420i	$0.319462\pi$
$48$	2314.84	0.145017
$49$	0	0
$50$	5453.48	0.308495
$51$	6766.12	0.364262
$52$	−2598.65	−0.133272
$53$	1267.14	0.0619634	0.0309817	−	0.999520i	$-0.490137\pi$
$53$	1267.14	0.0619634	0.0309817	+	0.999520i	$0.490137\pi$
$54$	19170.0	0.894619
$55$	−16978.9	−0.756838
$56$	0	0
$57$	−10414.6	−0.424576
$58$	18145.6	0.708273
$59$	13465.7	0.503615	0.251807	−	0.967777i	$-0.418975\pi$
$59$	13465.7	0.503615	0.251807	+	0.967777i	$0.418975\pi$
$60$	−5229.66	−0.187540
$61$	−35031.5	−1.20541	−0.602703	−	0.797965i	$-0.705910\pi$
$61$	−35031.5	−1.20541	−0.602703	+	0.797965i	$0.705910\pi$
$62$	−31408.4	−1.03769
$63$	0	0
$64$	−51121.6	−1.56011
$65$	−1471.98	−0.0432133
$66$	28087.2	0.793687
$67$	−36919.0	−1.00476	−0.502380	−	0.864647i	$-0.667542\pi$
$67$	−36919.0	−1.00476	−0.502380	+	0.864647i	$0.667542\pi$
$68$	−63005.9	−1.65237
$69$	10537.8	0.266457
$70$	0	0
$71$	4118.92	0.0969700	0.0484850	−	0.998824i	$-0.484561\pi$
$71$	4118.92	0.0969700	0.0484850	+	0.998824i	$0.484561\pi$
$72$	−23352.1	−0.530879
$73$	3517.37	0.0772521	0.0386261	−	0.999254i	$-0.487702\pi$
$73$	3517.37	0.0772521	0.0386261	+	0.999254i	$0.487702\pi$
$74$	80117.7	1.70078
$75$	−2962.28	−0.0608097
$76$	96980.4	1.92597
$77$	0	0
$78$	2435.01	0.0453173
$79$	−72440.8	−1.30592	−0.652959	−	0.757394i	$-0.726472\pi$
$79$	−72440.8	−1.30592	−0.652959	+	0.757394i	$0.726472\pi$
$80$	−12210.0	−0.213299
$81$	43177.2	0.731210
$82$	−87789.0	−1.44180
$83$	42252.6	0.673223	0.336611	−	0.941644i	$-0.390719\pi$
$83$	42252.6	0.673223	0.336611	+	0.941644i	$0.390719\pi$
$84$	0	0
$85$	−35688.9	−0.535780
$86$	−121779.	−1.77552
$87$	−9856.51	−0.139613
$88$	−71914.7	−0.989944
$89$	−94517.0	−1.26484	−0.632419	−	0.774627i	$-0.717938\pi$
$89$	−94517.0	−1.26484	−0.632419	+	0.774627i	$0.717938\pi$
$90$	−48107.5	−0.626046
$91$	0	0
$92$	−98127.7	−1.20871
$93$	17060.8	0.204546
$94$	141987.	1.65740
$95$	54933.4	0.624493
$96$	36258.2	0.401539
$97$	−85271.1	−0.920180	−0.460090	−	0.887872i	$-0.652183\pi$
$97$	−85271.1	−0.920180	−0.460090	+	0.887872i	$0.652183\pi$
$98$	0	0
$99$	149778.	1.53589
$100$	27584.6	0.275846
$101$	180306.	1.75876	0.879380	−	0.476121i	$-0.157958\pi$
$101$	180306.	1.75876	0.879380	+	0.476121i	$0.157958\pi$
$102$	59038.2	0.561866
$103$	173368.	1.61018	0.805091	−	0.593152i	$-0.202117\pi$
$103$	173368.	1.61018	0.805091	+	0.593152i	$0.202117\pi$
$104$	−6234.60	−0.0565230
$105$	0	0
$106$	11056.5	0.0955770
$107$	155442.	1.31253	0.656264	−	0.754532i	$-0.272136\pi$
$107$	155442.	1.31253	0.656264	+	0.754532i	$0.272136\pi$
$108$	96965.3	0.799939
$109$	19088.6	0.153889	0.0769444	−	0.997035i	$-0.475484\pi$
$109$	19088.6	0.153889	0.0769444	+	0.997035i	$0.475484\pi$
$110$	−148151.	−1.16740
$111$	−43519.2	−0.335254
$112$	0	0
$113$	101150.	0.745197	0.372599	−	0.927993i	$-0.378467\pi$
$113$	101150.	0.745197	0.372599	+	0.927993i	$0.378467\pi$
$114$	−90873.3	−0.654899
$115$	−55583.3	−0.391922
$116$	91783.5	0.633315
$117$	12984.9	0.0876951
$118$	117496.	0.776814
$119$	0	0
$120$	−12546.8	−0.0795391
$121$	300202.	1.86402
$122$	−305669.	−1.85931
$123$	47686.2	0.284204
$124$	−158869.	−0.927866
$125$	15625.0	0.0894427
$126$	0	0
$127$	−112788.	−0.620514	−0.310257	−	0.950653i	$-0.600415\pi$
$127$	−112788.	−0.620514	−0.310257	+	0.950653i	$0.600415\pi$
$128$	−201265.	−1.08578
$129$	66149.1	0.349985
$130$	−12843.8	−0.0666555
$131$	183577.	0.934632	0.467316	−	0.884090i	$-0.345221\pi$
$131$	183577.	0.934632	0.467316	+	0.884090i	$0.345221\pi$
$132$	142070.	0.709689
$133$	0	0
$134$	−322139.	−1.54982
$135$	54924.9	0.259379
$136$	−151162.	−0.700800
$137$	−92212.1	−0.419746	−0.209873	−	0.977729i	$-0.567305\pi$
$137$	−92212.1	−0.419746	−0.209873	+	0.977729i	$0.567305\pi$
$138$	91948.3	0.411004
$139$	−424349.	−1.86288	−0.931442	−	0.363889i	$-0.881449\pi$
$139$	−424349.	−1.86288	−0.931442	+	0.363889i	$0.881449\pi$
$140$	0	0
$141$	−77125.9	−0.326703
$142$	35939.9	0.149574
$143$	39988.1	0.163527
$144$	107709.	0.432860
$145$	51989.7	0.205351
$146$	30691.0	0.119160
$147$	0	0
$148$	405249.	1.52079
$149$	137894.	0.508838	0.254419	−	0.967094i	$-0.418116\pi$
$149$	137894.	0.508838	0.254419	+	0.967094i	$0.418116\pi$
$150$	−25847.5	−0.0937975
$151$	−308590.	−1.10139	−0.550693	−	0.834708i	$-0.685636\pi$
$151$	−308590.	−1.10139	−0.550693	+	0.834708i	$0.685636\pi$
$152$	232672.	0.816837
$153$	314828.	1.08729
$154$	0	0
$155$	−89989.6	−0.300859
$156$	12316.7	0.0405212
$157$	209003.	0.676710	0.338355	−	0.941018i	$-0.390129\pi$
$157$	209003.	0.676710	0.338355	+	0.941018i	$0.390129\pi$
$158$	−632087.	−2.01435
$159$	−6005.80	−0.0188399
$160$	−191249.	−0.590609
$161$	0	0
$162$	376745.	1.12787
$163$	−249015.	−0.734103	−0.367051	−	0.930201i	$-0.619633\pi$
$163$	−249015.	−0.734103	−0.367051	+	0.930201i	$0.619633\pi$
$164$	−444052.	−1.28921
$165$	80474.0	0.230116
$166$	368678.	1.03843
$167$	−550497.	−1.52744	−0.763720	−	0.645548i	$-0.776629\pi$
$167$	−550497.	−1.52744	−0.763720	+	0.645548i	$0.776629\pi$
$168$	0	0
$169$	−367826.	−0.990663
$170$	−311406.	−0.826428
$171$	−484592.	−1.26732
$172$	−615978.	−1.58761
$173$	103026.	0.261717	0.130859	−	0.991401i	$-0.458227\pi$
$173$	103026.	0.261717	0.130859	+	0.991401i	$0.458227\pi$
$174$	−86003.6	−0.215349
$175$	0	0
$176$	331699.	0.807166
$177$	−63822.6	−0.153123
$178$	−824714.	−1.95098
$179$	470596.	1.09778	0.548891	−	0.835894i	$-0.315050\pi$
$179$	470596.	1.09778	0.548891	+	0.835894i	$0.315050\pi$
$180$	−243336.	−0.559790
$181$	529125.	1.20050	0.600250	−	0.799812i	$-0.295068\pi$
$181$	529125.	1.20050	0.600250	+	0.799812i	$0.295068\pi$
$182$	0	0
$183$	166037.	0.366502
$184$	−235425.	−0.512634
$185$	229549.	0.493112
$186$	148865.	0.315507
$187$	969535.	2.02749
$188$	718194.	1.48200
$189$	0	0
$190$	479325.	0.963266
$191$	180604.	0.358215	0.179108	−	0.983829i	$-0.442679\pi$
$191$	180604.	0.358215	0.179108	+	0.983829i	$0.442679\pi$
$192$	242298.	0.474348
$193$	−512865.	−0.991083	−0.495541	−	0.868584i	$-0.665030\pi$
$193$	−512865.	−0.991083	−0.495541	+	0.868584i	$0.665030\pi$
$194$	−744038.	−1.41935
$195$	6976.65	0.0131389
$196$	0	0
$197$	−28199.0	−0.0517687	−0.0258844	−	0.999665i	$-0.508240\pi$
$197$	−28199.0	−0.0517687	−0.0258844	+	0.999665i	$0.508240\pi$
$198$	1.30690e6	2.36908
$199$	−934000.	−1.67192	−0.835958	−	0.548794i	$-0.815087\pi$
$199$	−934000.	−1.67192	−0.835958	+	0.548794i	$0.815087\pi$
$200$	66180.1	0.116991
$201$	174983.	0.305496
$202$	1.57327e6	2.71284
$203$	0	0
$204$	298625.	0.502402
$205$	−251528.	−0.418025
$206$	1.51273e6	2.48367
$207$	490324.	0.795349
$208$	28756.5	0.0460869
$209$	−1.49234e6	−2.36320
$210$	0	0
$211$	−206876.	−0.319893	−0.159946	−	0.987126i	$-0.551132\pi$
$211$	−206876.	−0.319893	−0.159946	+	0.987126i	$0.551132\pi$
$212$	55925.8	0.0854619
$213$	−19522.2	−0.0294836
$214$	1.35632e6	2.02454
$215$	−348914.	−0.514780
$216$	232636.	0.339268
$217$	0	0
$218$	166559.	0.237370
$219$	−16671.1	−0.0234884
$220$	−749371.	−1.04386
$221$	84053.3	0.115764
$222$	−379730.	−0.517121
$223$	−347934.	−0.468527	−0.234263	−	0.972173i	$-0.575268\pi$
$223$	−347934.	−0.468527	−0.234263	+	0.972173i	$0.575268\pi$
$224$	0	0
$225$	−137835.	−0.181511
$226$	882594.	1.14945
$227$	792997.	1.02143	0.510713	−	0.859751i	$-0.329381\pi$
$227$	792997.	1.02143	0.510713	+	0.859751i	$0.329381\pi$
$228$	−459653.	−0.585589
$229$	275845.	0.347598	0.173799	−	0.984781i	$-0.444396\pi$
$229$	275845.	0.347598	0.173799	+	0.984781i	$0.444396\pi$
$230$	−484996.	−0.604531
$231$	0	0
$232$	220204.	0.268599
$233$	1.43590e6	1.73275	0.866373	−	0.499398i	$-0.166446\pi$
$233$	1.43590e6	1.73275	0.866373	+	0.499398i	$0.166446\pi$
$234$	113301.	0.135267
$235$	406813.	0.480535
$236$	594314.	0.694602
$237$	343344.	0.397062
$238$	0	0
$239$	127902.	0.144838	0.0724191	−	0.997374i	$-0.476928\pi$
$239$	127902.	0.144838	0.0724191	+	0.997374i	$0.476928\pi$
$240$	57871.0	0.0648534
$241$	−602677.	−0.668409	−0.334204	−	0.942501i	$-0.608468\pi$
$241$	−602677.	−0.668409	−0.334204	+	0.942501i	$0.608468\pi$
$242$	2.61943e6	2.87521
$243$	−738515.	−0.802312
$244$	−1.54613e6	−1.66253
$245$	0	0
$246$	416089.	0.438377
$247$	−129377.	−0.134932
$248$	−381154.	−0.393524
$249$	−200263.	−0.204692
$250$	136337.	0.137963
$251$	756978.	0.758401	0.379201	−	0.925314i	$-0.376199\pi$
$251$	756978.	0.758401	0.379201	+	0.925314i	$0.376199\pi$
$252$	0	0
$253$	1.50999e6	1.48311
$254$	−984135.	−0.957128
$255$	169153.	0.162903
$256$	−120261.	−0.114689
$257$	−606610.	−0.572898	−0.286449	−	0.958096i	$-0.592475\pi$
$257$	−606610.	−0.572898	−0.286449	+	0.958096i	$0.592475\pi$
$258$	577188.	0.539844
$259$	0	0
$260$	−64966.3	−0.0596011
$261$	−458624.	−0.416730
$262$	1.60181e6	1.44165
$263$	−1.41962e6	−1.26556	−0.632780	−	0.774331i	$-0.718086\pi$
$263$	−1.41962e6	−1.26556	−0.632780	+	0.774331i	$0.718086\pi$
$264$	340850.	0.300991
$265$	31678.5	0.0277109
$266$	0	0
$267$	447977.	0.384572
$268$	−1.62943e6	−1.38580
$269$	496984.	0.418756	0.209378	−	0.977835i	$-0.432856\pi$
$269$	496984.	0.418756	0.209378	+	0.977835i	$0.432856\pi$
$270$	479250.	0.400086
$271$	−155568.	−0.128676	−0.0643378	−	0.997928i	$-0.520494\pi$
$271$	−155568.	−0.128676	−0.0643378	+	0.997928i	$0.520494\pi$
$272$	697218.	0.571408
$273$	0	0
$274$	−804602.	−0.647448
$275$	−424473.	−0.338468
$276$	465091.	0.367506
$277$	−1.77069e6	−1.38657	−0.693285	−	0.720663i	$-0.743837\pi$
$277$	−1.77069e6	−1.38657	−0.693285	+	0.720663i	$0.743837\pi$
$278$	−3.70268e6	−2.87345
$279$	793837.	0.610549
$280$	0	0
$281$	−2.38775e6	−1.80394	−0.901972	−	0.431794i	$-0.857881\pi$
$281$	−2.38775e6	−1.80394	−0.901972	+	0.431794i	$0.857881\pi$
$282$	−672967.	−0.503931
$283$	193515.	0.143631	0.0718156	−	0.997418i	$-0.477121\pi$
$283$	193515.	0.143631	0.0718156	+	0.997418i	$0.477121\pi$
$284$	181790.	0.133744
$285$	−260365.	−0.189876
$286$	348918.	0.252237
$287$	0	0
$288$	1.68709e6	1.19855
$289$	618065.	0.435301
$290$	453639.	0.316749
$291$	404155.	0.279779
$292$	155240.	0.106549
$293$	66529.2	0.0452734	0.0226367	−	0.999744i	$-0.492794\pi$
$293$	66529.2	0.0452734	0.0226367	+	0.999744i	$0.492794\pi$
$294$	0	0
$295$	336642.	0.225223
$296$	972261.	0.644991
$297$	−1.49210e6	−0.981540
$298$	1.20320e6	0.784870
$299$	130908.	0.0846813
$300$	−130741.	−0.0838706
$301$	0	0
$302$	−2.69262e6	−1.69886
$303$	−854586.	−0.534748
$304$	−1.07318e6	−0.666020
$305$	−875786.	−0.539074
$306$	2.74705e6	1.67711
$307$	−1.08295e6	−0.655787	−0.327893	−	0.944715i	$-0.606339\pi$
$307$	−1.08295e6	−0.655787	−0.327893	+	0.944715i	$0.606339\pi$
$308$	0	0
$309$	−821701.	−0.489573
$310$	−785210.	−0.464068
$311$	1.48395e6	0.869999	0.435000	−	0.900431i	$-0.356749\pi$
$311$	1.48395e6	0.869999	0.435000	+	0.900431i	$0.356749\pi$
$312$	29549.8	0.0171857
$313$	2.59381e6	1.49650	0.748251	−	0.663416i	$-0.230894\pi$
$313$	2.59381e6	1.49650	0.748251	+	0.663416i	$0.230894\pi$
$314$	1.82367e6	1.04381
$315$	0	0
$316$	−3.19721e6	−1.80116
$317$	−2.77640e6	−1.55180	−0.775898	−	0.630858i	$-0.782703\pi$
$317$	−2.77640e6	−1.55180	−0.775898	+	0.630858i	$0.782703\pi$
$318$	−52404.0	−0.0290601
$319$	−1.41237e6	−0.777088
$320$	−1.27804e6	−0.697701
$321$	−736739.	−0.399072
$322$	0	0
$323$	−3.13683e6	−1.67295
$324$	1.90564e6	1.00851
$325$	−36799.4	−0.0193256
$326$	−2.17280e6	−1.13234
$327$	−90473.1	−0.0467897
$328$	−1.06535e6	−0.546776
$329$	0	0
$330$	702181.	0.354948
$331$	−1.99398e6	−1.00035	−0.500174	−	0.865925i	$-0.666731\pi$
$331$	−1.99398e6	−1.00035	−0.500174	+	0.865925i	$0.666731\pi$
$332$	1.86484e6	0.928530
$333$	−2.02495e6	−1.00070
$334$	−4.80340e6	−2.35604
$335$	−922974.	−0.449343
$336$	0	0
$337$	−19232.2	−0.00922473	−0.00461236	−	0.999989i	$-0.501468\pi$
$337$	−19232.2	−0.00922473	−0.00461236	+	0.999989i	$0.501468\pi$
$338$	−3.20949e6	−1.52807
$339$	−479417.	−0.226576
$340$	−1.57515e6	−0.738965
$341$	2.44468e6	1.13851
$342$	−4.22833e6	−1.95481
$343$	0	0
$344$	−1.47783e6	−0.673333
$345$	263445.	0.119163
$346$	898961.	0.403692
$347$	−2.23061e6	−0.994489	−0.497245	−	0.867610i	$-0.665655\pi$
$347$	−2.23061e6	−0.994489	−0.497245	+	0.867610i	$0.665655\pi$
$348$	−435021.	−0.192558
$349$	313791.	0.137904	0.0689520	−	0.997620i	$-0.478034\pi$
$349$	313791.	0.137904	0.0689520	+	0.997620i	$0.478034\pi$
$350$	0	0
$351$	−129357.	−0.0560431
$352$	5.19553e6	2.23498
$353$	−1.47348e6	−0.629370	−0.314685	−	0.949196i	$-0.601899\pi$
$353$	−1.47348e6	−0.629370	−0.314685	+	0.949196i	$0.601899\pi$
$354$	−556888.	−0.236189
$355$	102973.	0.0433663
$356$	−4.17155e6	−1.74450
$357$	0	0
$358$	4.10622e6	1.69330
$359$	−1.21115e6	−0.495977	−0.247988	−	0.968763i	$-0.579769\pi$
$359$	−1.21115e6	−0.495977	−0.247988	+	0.968763i	$0.579769\pi$
$360$	−583803.	−0.237416
$361$	2.35219e6	0.949960
$362$	4.61692e6	1.85174
$363$	−1.42285e6	−0.566753
$364$	0	0
$365$	87934.1	0.0345482
$366$	1.44876e6	0.565321
$367$	−2.43854e6	−0.945073	−0.472536	−	0.881311i	$-0.656661\pi$
$367$	−2.43854e6	−0.945073	−0.472536	+	0.881311i	$0.656661\pi$
$368$	1.08587e6	0.417984
$369$	2.21884e6	0.848320
$370$	2.00294e6	0.760614
$371$	0	0
$372$	752983.	0.282116
$373$	−3.48803e6	−1.29810	−0.649050	−	0.760746i	$-0.724833\pi$
$373$	−3.48803e6	−1.29810	−0.649050	+	0.760746i	$0.724833\pi$
$374$	8.45974e6	3.12736
$375$	−74057.0	−0.0271949
$376$	1.72307e6	0.628539
$377$	−122444.	−0.0443695
$378$	0	0
$379$	−964503.	−0.344910	−0.172455	−	0.985017i	$-0.555170\pi$
$379$	−964503.	−0.344910	−0.172455	+	0.985017i	$0.555170\pi$
$380$	2.42451e6	0.861321
$381$	534573.	0.188666
$382$	1.57587e6	0.552539
$383$	−3.36433e6	−1.17193	−0.585965	−	0.810336i	$-0.699284\pi$
$383$	−3.36433e6	−1.17193	−0.585965	+	0.810336i	$0.699284\pi$
$384$	953926.	0.330131
$385$	0	0
$386$	−4.47504e6	−1.52872
$387$	3.07792e6	1.04467
$388$	−3.76348e6	−1.26914
$389$	−1.18786e6	−0.398007	−0.199003	−	0.979999i	$-0.563770\pi$
$389$	−1.18786e6	−0.398007	−0.199003	+	0.979999i	$0.563770\pi$
$390$	60875.2	0.0202665
$391$	3.17394e6	1.04992
$392$	0	0
$393$	−870091.	−0.284173
$394$	−246052.	−0.0798520
$395$	−1.81102e6	−0.584024
$396$	6.61052e6	2.11835
$397$	−1.63665e6	−0.521171	−0.260585	−	0.965451i	$-0.583916\pi$
$397$	−1.63665e6	−0.521171	−0.260585	+	0.965451i	$0.583916\pi$
$398$	−8.14967e6	−2.57889
$399$	0	0
$400$	−305249.	−0.0953904
$401$	−892182.	−0.277072	−0.138536	−	0.990357i	$-0.544240\pi$
$401$	−892182.	−0.277072	−0.138536	+	0.990357i	$0.544240\pi$
$402$	1.52682e6	0.471220
$403$	211940.	0.0650056
$404$	7.95787e6	2.42574
$405$	1.07943e6	0.327007
$406$	0	0
$407$	−6.23598e6	−1.86603
$408$	716453.	0.213077
$409$	3.38800e6	1.00146	0.500732	−	0.865603i	$-0.333064\pi$
$409$	3.38800e6	1.00146	0.500732	+	0.865603i	$0.333064\pi$
$410$	−2.19472e6	−0.644793
$411$	437053.	0.127623
$412$	7.65165e6	2.22081
$413$	0	0
$414$	4.27836e6	1.22681
$415$	1.05632e6	0.301074
$416$	450423.	0.127611
$417$	2.01126e6	0.566407
$418$	−1.30215e7	−3.64518
$419$	5.29618e6	1.47376	0.736882	−	0.676022i	$-0.236297\pi$
$419$	5.29618e6	1.47376	0.736882	+	0.676022i	$0.236297\pi$
$420$	0	0
$421$	5.38110e6	1.47967	0.739836	−	0.672787i	$-0.234903\pi$
$421$	5.38110e6	1.47967	0.739836	+	0.672787i	$0.234903\pi$
$422$	−1.80511e6	−0.493427
$423$	−3.58867e6	−0.975175
$424$	134175.	0.0362458
$425$	−892224.	−0.239608
$426$	−170342.	−0.0454777
$427$	0	0
$428$	6.86049e6	1.81028
$429$	−189529.	−0.0497203
$430$	−3.04447e6	−0.794036
$431$	−6.29797e6	−1.63308	−0.816540	−	0.577290i	$-0.804110\pi$
$431$	−6.29797e6	−1.63308	−0.816540	+	0.577290i	$0.804110\pi$
$432$	−1.07301e6	−0.276627
$433$	4.54572e6	1.16515	0.582577	−	0.812775i	$-0.302044\pi$
$433$	4.54572e6	1.16515	0.582577	+	0.812775i	$0.302044\pi$
$434$	0	0
$435$	−246413.	−0.0624367
$436$	842482.	0.212248
$437$	−4.88541e6	−1.22376
$438$	−145464.	−0.0362303
$439$	2.69494e6	0.667402	0.333701	−	0.942679i	$-0.391702\pi$
$439$	2.69494e6	0.667402	0.333701	+	0.942679i	$0.391702\pi$
$440$	−1.79787e6	−0.442717
$441$	0	0
$442$	733412.	0.178563
$443$	−3.00924e6	−0.728530	−0.364265	−	0.931295i	$-0.618680\pi$
$443$	−3.00924e6	−0.728530	−0.364265	+	0.931295i	$0.618680\pi$
$444$	−1.92074e6	−0.462393
$445$	−2.36292e6	−0.565653
$446$	−3.03592e6	−0.722691
$447$	−653569.	−0.154711
$448$	0	0
$449$	3.52092e6	0.824214	0.412107	−	0.911135i	$-0.364793\pi$
$449$	3.52092e6	0.824214	0.412107	+	0.911135i	$0.364793\pi$
$450$	−1.20269e6	−0.279976
$451$	6.83308e6	1.58189
$452$	4.46431e6	1.02780
$453$	1.46261e6	0.334875
$454$	6.91935e6	1.57553
$455$	0	0
$456$	−1.10278e6	−0.248358
$457$	2.51620e6	0.563578	0.281789	−	0.959476i	$-0.409072\pi$
$457$	2.51620e6	0.563578	0.281789	+	0.959476i	$0.409072\pi$
$458$	2.40691e6	0.536161
$459$	−3.13634e6	−0.694850
$460$	−2.45319e6	−0.540552
$461$	200800.	0.0440060	0.0220030	−	0.999758i	$-0.492996\pi$
$461$	200800.	0.0440060	0.0220030	+	0.999758i	$0.492996\pi$
$462$	0	0
$463$	192190.	0.0416656	0.0208328	−	0.999783i	$-0.493368\pi$
$463$	192190.	0.0416656	0.0208328	+	0.999783i	$0.493368\pi$
$464$	−1.01567e6	−0.219007
$465$	426519.	0.0914757
$466$	1.25290e7	2.67272
$467$	−1.49891e6	−0.318040	−0.159020	−	0.987275i	$-0.550833\pi$
$467$	−1.49891e6	−0.318040	−0.159020	+	0.987275i	$0.550833\pi$
$468$	573095.	0.120952
$469$	0	0
$470$	3.54967e6	0.741213
$471$	−990599.	−0.205753
$472$	1.42586e6	0.294592
$473$	9.47868e6	1.94803
$474$	2.99587e6	0.612459
$475$	1.37334e6	0.279282
$476$	0	0
$477$	−279450.	−0.0562352
$478$	1.11602e6	0.223409
$479$	−7.48400e6	−1.49037	−0.745186	−	0.666857i	$-0.767639\pi$
$479$	−7.48400e6	−1.49037	−0.745186	+	0.666857i	$0.767639\pi$
$480$	906455.	0.179574
$481$	−540625.	−0.106545
$482$	−5.25870e6	−1.03100
$483$	0	0
$484$	1.32496e7	2.57092
$485$	−2.13178e6	−0.411517
$486$	−6.44395e6	−1.23755
$487$	−2.19160e6	−0.418734	−0.209367	−	0.977837i	$-0.567140\pi$
$487$	−2.19160e6	−0.418734	−0.209367	+	0.977837i	$0.567140\pi$
$488$	−3.70942e6	−0.705109
$489$	1.18024e6	0.223203
$490$	0	0
$491$	3.71570e6	0.695564	0.347782	−	0.937575i	$-0.386935\pi$
$491$	3.71570e6	0.695564	0.347782	+	0.937575i	$0.386935\pi$
$492$	2.10465e6	0.391983
$493$	−2.96873e6	−0.550115
$494$	−1.12889e6	−0.208129
$495$	3.74446e6	0.686872
$496$	1.75803e6	0.320865
$497$	0	0
$498$	−1.74740e6	−0.315733
$499$	5.74147e6	1.03222	0.516109	−	0.856523i	$-0.327380\pi$
$499$	5.74147e6	1.03222	0.516109	+	0.856523i	$0.327380\pi$
$500$	689616.	0.123362
$501$	2.60916e6	0.464416
$502$	6.60506e6	1.16982
$503$	−8.85276e6	−1.56012	−0.780061	−	0.625703i	$-0.784812\pi$
$503$	−8.85276e6	−1.56012	−0.780061	+	0.625703i	$0.784812\pi$
$504$	0	0
$505$	4.50765e6	0.786541
$506$	1.31755e7	2.28766
$507$	1.74337e6	0.301210
$508$	−4.97792e6	−0.855833
$509$	−478571.	−0.0818751	−0.0409376	−	0.999162i	$-0.513034\pi$
$509$	−478571.	−0.0818751	−0.0409376	+	0.999162i	$0.513034\pi$
$510$	1.47595e6	0.251274
$511$	0	0
$512$	5.39114e6	0.908879
$513$	4.82754e6	0.809902
$514$	−5.29302e6	−0.883681
$515$	4.33419e6	0.720095
$516$	2.91952e6	0.482711
$517$	−1.10516e7	−1.81844
$518$	0	0
$519$	−488307.	−0.0795747
$520$	−155865.	−0.0252778
$521$	8.96169e6	1.44642	0.723212	−	0.690626i	$-0.242665\pi$
$521$	8.96169e6	1.44642	0.723212	+	0.690626i	$0.242665\pi$
$522$	−4.00175e6	−0.642796
$523$	−1.08551e7	−1.73531	−0.867656	−	0.497164i	$-0.834375\pi$
$523$	−1.08551e7	−1.73531	−0.867656	+	0.497164i	$0.834375\pi$
$524$	8.10226e6	1.28907
$525$	0	0
$526$	−1.23870e7	−1.95210
$527$	5.13861e6	0.805971
$528$	−1.57214e6	−0.245418
$529$	−1.49313e6	−0.231984
$530$	276413.	0.0427434
$531$	−2.96967e6	−0.457058
$532$	0	0
$533$	592389.	0.0903211
$534$	3.90885e6	0.593193
$535$	3.88604e6	0.586980
$536$	−3.90929e6	−0.587740
$537$	−2.23046e6	−0.333779
$538$	4.33646e6	0.645922
$539$	0	0
$540$	2.42413e6	0.357744
$541$	−1.15845e7	−1.70170	−0.850852	−	0.525405i	$-0.823914\pi$
$541$	−1.15845e7	−1.70170	−0.850852	+	0.525405i	$0.823914\pi$
$542$	−1.35742e6	−0.198479
$543$	−2.50787e6	−0.365010
$544$	1.09208e7	1.58218
$545$	477214.	0.0688212
$546$	0	0
$547$	594715.	0.0849846	0.0424923	−	0.999097i	$-0.486470\pi$
$547$	594715.	0.0849846	0.0424923	+	0.999097i	$0.486470\pi$
$548$	−4.06982e6	−0.578927
$549$	7.72569e6	1.09397
$550$	−3.70376e6	−0.522079
$551$	4.56956e6	0.641202
$552$	1.11583e6	0.155866
$553$	0	0
$554$	−1.54502e7	−2.13875
$555$	−1.08798e6	−0.149930
$556$	−1.87288e7	−2.56935
$557$	4.57366e6	0.624634	0.312317	−	0.949978i	$-0.398895\pi$
$557$	4.57366e6	0.624634	0.312317	+	0.949978i	$0.398895\pi$
$558$	6.92667e6	0.941758
$559$	821748.	0.111227
$560$	0	0
$561$	−4.59525e6	−0.616456
$562$	−2.08345e7	−2.78254
$563$	−2.88071e6	−0.383027	−0.191513	−	0.981490i	$-0.561340\pi$
$563$	−2.88071e6	−0.383027	−0.191513	+	0.981490i	$0.561340\pi$
$564$	−3.40398e6	−0.450599
$565$	2.52876e6	0.333262
$566$	1.68853e6	0.221548
$567$	0	0
$568$	436145.	0.0567231
$569$	1.18609e7	1.53581	0.767903	−	0.640566i	$-0.221300\pi$
$569$	1.18609e7	1.53581	0.767903	+	0.640566i	$0.221300\pi$
$570$	−2.27183e6	−0.292880
$571$	−3.94838e6	−0.506791	−0.253396	−	0.967363i	$-0.581547\pi$
$571$	−3.94838e6	−0.506791	−0.253396	+	0.967363i	$0.581547\pi$
$572$	1.76489e6	0.225542
$573$	−856000.	−0.108915
$574$	0	0
$575$	−1.38958e6	−0.175273
$576$	1.12741e7	1.41588
$577$	−1.33953e6	−0.167500	−0.0837498	−	0.996487i	$-0.526690\pi$
$577$	−1.33953e6	−0.167500	−0.0837498	+	0.996487i	$0.526690\pi$
$578$	5.39296e6	0.671441
$579$	2.43080e6	0.301337
$580$	2.29459e6	0.283227
$581$	0	0
$582$	3.52648e6	0.431553
$583$	−860587.	−0.104863
$584$	372448.	0.0451890
$585$	324623.	0.0392184
$586$	580504.	0.0698331
$587$	1.53600e7	1.83990	0.919951	−	0.392034i	$-0.128228\pi$
$587$	1.53600e7	1.83990	0.919951	+	0.392034i	$0.128228\pi$
$588$	0	0
$589$	−7.90950e6	−0.939422
$590$	2.93739e6	0.347402
$591$	133653.	0.0157402
$592$	−4.48446e6	−0.525903
$593$	963371.	0.112501	0.0562506	−	0.998417i	$-0.482085\pi$
$593$	963371.	0.112501	0.0562506	+	0.998417i	$0.482085\pi$
$594$	−1.30194e7	−1.51400
$595$	0	0
$596$	6.08600e6	0.701805
$597$	4.42683e6	0.508343
$598$	1.14224e6	0.130619
$599$	−1.39812e6	−0.159212	−0.0796062	−	0.996826i	$-0.525366\pi$
$599$	−1.39812e6	−0.159212	−0.0796062	+	0.996826i	$0.525366\pi$
$600$	−313670.	−0.0355710
$601$	−9.79835e6	−1.10654	−0.553270	−	0.833002i	$-0.686620\pi$
$601$	−9.79835e6	−1.10654	−0.553270	+	0.833002i	$0.686620\pi$
$602$	0	0
$603$	8.14195e6	0.911875
$604$	−1.36197e7	−1.51907
$605$	7.50506e6	0.833616
$606$	−7.45674e6	−0.824836
$607$	1.05240e7	1.15933	0.579666	−	0.814854i	$-0.303183\pi$
$607$	1.05240e7	1.15933	0.579666	+	0.814854i	$0.303183\pi$
$608$	−1.68096e7	−1.84416
$609$	0	0
$610$	−7.64173e6	−0.831509
$611$	−958110.	−0.103827
$612$	1.38950e7	1.49962
$613$	−1.17388e7	−1.26175	−0.630874	−	0.775886i	$-0.717303\pi$
$613$	−1.17388e7	−1.26175	−0.630874	+	0.775886i	$0.717303\pi$
$614$	−9.44935e6	−1.01154
$615$	1.19215e6	0.127100
$616$	0	0
$617$	−6.83389e6	−0.722695	−0.361348	−	0.932431i	$-0.617683\pi$
$617$	−6.83389e6	−0.722695	−0.361348	+	0.932431i	$0.617683\pi$
$618$	−7.16980e6	−0.755155
$619$	−1.06954e7	−1.12194	−0.560969	−	0.827837i	$-0.689571\pi$
$619$	−1.06954e7	−1.12194	−0.560969	+	0.827837i	$0.689571\pi$
$620$	−3.97173e6	−0.414954
$621$	−4.88465e6	−0.508282
$622$	1.29483e7	1.34195
$623$	0	0
$624$	−136295.	−0.0140126
$625$	390625.	0.0400000
$626$	2.26324e7	2.30832
$627$	7.07315e6	0.718528
$628$	9.22442e6	0.933340
$629$	−1.31078e7	−1.32100
$630$	0	0
$631$	5.17826e6	0.517739	0.258869	−	0.965912i	$-0.416650\pi$
$631$	5.17826e6	0.517739	0.258869	+	0.965912i	$0.416650\pi$
$632$	−7.67063e6	−0.763903
$633$	980521.	0.0972630
$634$	−2.42257e7	−2.39361
$635$	−2.81969e6	−0.277502
$636$	−265068.	−0.0259846
$637$	0	0
$638$	−1.23237e7	−1.19864
$639$	−908369.	−0.0880055
$640$	−5.03163e6	−0.485578
$641$	2.00548e7	1.92785	0.963927	−	0.266165i	$-0.0857568\pi$
$641$	2.00548e7	1.92785	0.963927	+	0.266165i	$0.0857568\pi$
$642$	−6.42846e6	−0.615559
$643$	1.00964e6	0.0963032	0.0481516	−	0.998840i	$-0.484667\pi$
$643$	1.00964e6	0.0963032	0.0481516	+	0.998840i	$0.484667\pi$
$644$	0	0
$645$	1.65373e6	0.156518
$646$	−2.73706e7	−2.58049
$647$	−2.66309e6	−0.250107	−0.125053	−	0.992150i	$-0.539910\pi$
$647$	−2.66309e6	−0.250107	−0.125053	+	0.992150i	$0.539910\pi$
$648$	4.57196e6	0.427725
$649$	−9.14531e6	−0.852289
$650$	−321095.	−0.0298092
$651$	0	0
$652$	−1.09904e7	−1.01250
$653$	6.98872e6	0.641379	0.320689	−	0.947184i	$-0.396085\pi$
$653$	6.98872e6	0.641379	0.320689	+	0.947184i	$0.396085\pi$
$654$	−789429.	−0.0721720
$655$	4.58943e6	0.417980
$656$	4.91384e6	0.445822
$657$	−775705.	−0.0701105
$658$	0	0
$659$	1.22406e7	1.09797	0.548985	−	0.835833i	$-0.315015\pi$
$659$	1.22406e7	1.09797	0.548985	+	0.835833i	$0.315015\pi$
$660$	3.55175e6	0.317383
$661$	−8.94566e6	−0.796359	−0.398180	−	0.917307i	$-0.630358\pi$
$661$	−8.94566e6	−0.796359	−0.398180	+	0.917307i	$0.630358\pi$
$662$	−1.73986e7	−1.54301
$663$	−398383.	−0.0351979
$664$	4.47406e6	0.393805
$665$	0	0
$666$	−1.76688e7	−1.54355
$667$	−4.62362e6	−0.402409
$668$	−2.42964e7	−2.10669
$669$	1.64908e6	0.142455
$670$	−8.05347e6	−0.693100
$671$	2.37918e7	2.03996
$672$	0	0
$673$	−8.40438e6	−0.715267	−0.357633	−	0.933862i	$-0.616416\pi$
$673$	−8.40438e6	−0.715267	−0.357633	+	0.933862i	$0.616416\pi$
$674$	−167811.	−0.0142289
$675$	1.37312e6	0.115998
$676$	−1.62342e7	−1.36635
$677$	−2.67443e6	−0.224264	−0.112132	−	0.993693i	$-0.535768\pi$
$677$	−2.67443e6	−0.224264	−0.112132	+	0.993693i	$0.535768\pi$
$678$	−4.18318e6	−0.349488
$679$	0	0
$680$	−3.77904e6	−0.313407
$681$	−3.75853e6	−0.310563
$682$	2.13312e7	1.75612
$683$	8.95973e6	0.734925	0.367463	−	0.930038i	$-0.380227\pi$
$683$	8.95973e6	0.734925	0.367463	+	0.930038i	$0.380227\pi$
$684$	−2.13876e7	−1.74792
$685$	−2.30530e6	−0.187716
$686$	0	0
$687$	−1.30741e6	−0.105687
$688$	6.81636e6	0.549012
$689$	−74608.0	−0.00598739
$690$	2.29871e6	0.183807
$691$	22616.0	0.00180186	0.000900928	−	1.00000i	$-0.499713\pi$
$691$	22616.0	0.00180186	0.000900928		1.00000i	$0.499713\pi$
$692$	4.54710e6	0.360968
$693$	0	0
$694$	−1.94633e7	−1.53398
$695$	−1.06087e7	−0.833107
$696$	−1.04369e6	−0.0816672
$697$	1.43628e7	1.11985
$698$	2.73800e6	0.212714
$699$	−6.80566e6	−0.526839
$700$	0	0
$701$	1.80427e7	1.38677	0.693386	−	0.720566i	$-0.256118\pi$
$701$	1.80427e7	1.38677	0.693386	+	0.720566i	$0.256118\pi$
$702$	−1.12871e6	−0.0864451
$703$	2.01759e7	1.53973
$704$	3.47195e7	2.64023
$705$	−1.92815e6	−0.146106
$706$	−1.28569e7	−0.970789
$707$	0	0
$708$	−2.81684e6	−0.211193
$709$	1.52807e7	1.14164	0.570819	−	0.821076i	$-0.306626\pi$
$709$	1.52807e7	1.14164	0.570819	+	0.821076i	$0.306626\pi$
$710$	898497.	0.0668915
$711$	1.59758e7	1.18519
$712$	−1.00082e7	−0.739874
$713$	8.00308e6	0.589567
$714$	0	0
$715$	999702.	0.0731317
$716$	2.07700e7	1.51409
$717$	−606211.	−0.0440378
$718$	−1.05679e7	−0.765032
$719$	−1.45192e7	−1.04742	−0.523710	−	0.851897i	$-0.675452\pi$
$719$	−1.45192e7	−1.04742	−0.523710	+	0.851897i	$0.675452\pi$
$720$	2.69274e6	0.193581
$721$	0	0
$722$	2.05242e7	1.46529
$723$	2.85648e6	0.203229
$724$	2.33532e7	1.65577
$725$	1.29974e6	0.0918359
$726$	−1.24152e7	−0.874202
$727$	−1.98706e7	−1.39436	−0.697179	−	0.716897i	$-0.745562\pi$
$727$	−1.98706e7	−1.39436	−0.697179	+	0.716897i	$0.745562\pi$
$728$	0	0
$729$	−6.99176e6	−0.487268
$730$	767275.	0.0532898
$731$	1.99238e7	1.37904
$732$	7.32810e6	0.505491
$733$	1.58619e7	1.09042	0.545211	−	0.838299i	$-0.316450\pi$
$733$	1.58619e7	1.09042	0.545211	+	0.838299i	$0.316450\pi$
$734$	−2.12777e7	−1.45775
$735$	0	0
$736$	1.70085e7	1.15736
$737$	2.50738e7	1.70040
$738$	1.93606e7	1.30851
$739$	2.51524e7	1.69421	0.847107	−	0.531422i	$-0.178342\pi$
$739$	2.51524e7	1.69421	0.847107	+	0.531422i	$0.178342\pi$
$740$	1.01312e7	0.680116
$741$	613202.	0.0410259
$742$	0	0
$743$	7.30561e6	0.485495	0.242747	−	0.970090i	$-0.421951\pi$
$743$	7.30561e6	0.485495	0.242747	+	0.970090i	$0.421951\pi$
$744$	1.80653e6	0.119650
$745$	3.44735e6	0.227559
$746$	−3.04350e7	−2.00229
$747$	−9.31822e6	−0.610986
$748$	4.27908e7	2.79638
$749$	0	0
$750$	−646189.	−0.0419475
$751$	−826829.	−0.0534953	−0.0267477	−	0.999642i	$-0.508515\pi$
$751$	−826829.	−0.0534953	−0.0267477	+	0.999642i	$0.508515\pi$
$752$	−7.94747e6	−0.512489
$753$	−3.58781e6	−0.230591
$754$	−1.06839e6	−0.0684389
$755$	−7.71475e6	−0.492554
$756$	0	0
$757$	4.70431e6	0.298371	0.149185	−	0.988809i	$-0.452335\pi$
$757$	4.70431e6	0.298371	0.149185	+	0.988809i	$0.452335\pi$
$758$	−8.41583e6	−0.532015
$759$	−7.15683e6	−0.450937
$760$	5.81680e6	0.365301
$761$	1.42681e6	0.0893107	0.0446553	−	0.999002i	$-0.485781\pi$
$761$	1.42681e6	0.0893107	0.0446553	+	0.999002i	$0.485781\pi$
$762$	4.66445e6	0.291013
$763$	0	0
$764$	7.97104e6	0.494062
$765$	7.87069e6	0.486250
$766$	−2.93557e7	−1.80767
$767$	−792847.	−0.0486632
$768$	569993.	0.0348712
$769$	9.32444e6	0.568600	0.284300	−	0.958735i	$-0.408239\pi$
$769$	9.32444e6	0.568600	0.284300	+	0.958735i	$0.408239\pi$
$770$	0	0
$771$	2.87512e6	0.174189
$772$	−2.26355e7	−1.36693
$773$	1.52132e7	0.915741	0.457870	−	0.889019i	$-0.348612\pi$
$773$	1.52132e7	0.915741	0.457870	+	0.889019i	$0.348612\pi$
$774$	2.68566e7	1.61138
$775$	−2.24974e6	−0.134548
$776$	−9.02921e6	−0.538264
$777$	0	0
$778$	−1.03647e7	−0.613916
$779$	−2.21077e7	−1.30527
$780$	307917.	0.0181216
$781$	−2.79739e6	−0.164106
$782$	2.76944e7	1.61948
$783$	4.56885e6	0.266319
$784$	0	0
$785$	5.22507e6	0.302634
$786$	−7.59203e6	−0.438331
$787$	−2.20016e7	−1.26624	−0.633122	−	0.774052i	$-0.718227\pi$
$787$	−2.20016e7	−1.26624	−0.633122	+	0.774052i	$0.718227\pi$
$788$	−1.24457e6	−0.0714011
$789$	6.72850e6	0.384792
$790$	−1.58022e7	−0.900843
$791$	0	0
$792$	1.58598e7	0.898428
$793$	2.06262e6	0.116476
$794$	−1.42807e7	−0.803893
$795$	−150145.	−0.00842545
$796$	−4.12225e7	−2.30596
$797$	−3.02905e7	−1.68912	−0.844559	−	0.535462i	$-0.820138\pi$
$797$	−3.02905e7	−1.68912	−0.844559	+	0.535462i	$0.820138\pi$
$798$	0	0
$799$	−2.32299e7	−1.28730
$800$	−4.78124e6	−0.264128
$801$	2.08444e7	1.14791
$802$	−7.78479e6	−0.427377
$803$	−2.38884e6	−0.130737
$804$	7.72294e6	0.421350
$805$	0	0
$806$	1.84930e6	0.100269
$807$	−2.35553e6	−0.127322
$808$	1.90923e7	1.02880
$809$	−1.29572e7	−0.696050	−0.348025	−	0.937485i	$-0.613148\pi$
$809$	−1.29572e7	−0.696050	−0.348025	+	0.937485i	$0.613148\pi$
$810$	9.41863e6	0.504400
$811$	1.87248e7	0.999688	0.499844	−	0.866115i	$-0.333391\pi$
$811$	1.87248e7	0.999688	0.499844	+	0.866115i	$0.333391\pi$
$812$	0	0
$813$	737336.	0.0391236
$814$	−5.44124e7	−2.87831
$815$	−6.22538e6	−0.328301
$816$	−3.30457e6	−0.173736
$817$	−3.06673e7	−1.60738
$818$	2.95622e7	1.54473
$819$	0	0
$820$	−1.11013e7	−0.576553
$821$	2.07863e7	1.07626	0.538132	−	0.842861i	$-0.319130\pi$
$821$	2.07863e7	1.07626	0.538132	+	0.842861i	$0.319130\pi$
$822$	3.81353e6	0.196856
$823$	−2.40815e7	−1.23932	−0.619660	−	0.784870i	$-0.712730\pi$
$823$	−2.40815e7	−1.23932	−0.619660	+	0.784870i	$0.712730\pi$
$824$	1.83576e7	0.941884
$825$	2.01185e6	0.102911
$826$	0	0
$827$	−1.53428e6	−0.0780081	−0.0390041	−	0.999239i	$-0.512419\pi$
$827$	−1.53428e6	−0.0780081	−0.0390041	+	0.999239i	$0.512419\pi$
$828$	2.16407e7	1.09697
$829$	1.47802e7	0.746952	0.373476	−	0.927640i	$-0.378166\pi$
$829$	1.47802e7	0.746952	0.373476	+	0.927640i	$0.378166\pi$
$830$	9.21695e6	0.464400
$831$	8.39242e6	0.421585
$832$	3.00999e6	0.150750
$833$	0	0
$834$	1.75494e7	0.873669
$835$	−1.37624e7	−0.683092
$836$	−6.58649e7	−3.25940
$837$	−7.90827e6	−0.390183
$838$	4.62121e7	2.27324
$839$	−2.41688e6	−0.118536	−0.0592681	−	0.998242i	$-0.518877\pi$
$839$	−2.41688e6	−0.118536	−0.0592681	+	0.998242i	$0.518877\pi$
$840$	0	0
$841$	−1.61865e7	−0.789154
$842$	4.69531e7	2.28236
$843$	1.13171e7	0.548486
$844$	−9.13057e6	−0.441206
$845$	−9.19566e6	−0.443038
$846$	−3.13132e7	−1.50418
$847$	0	0
$848$	−618870.	−0.0295536
$849$	−917194.	−0.0436709
$850$	−7.78515e6	−0.369590
$851$	−2.04145e7	−0.966309
$852$	−861621.	−0.0406647
$853$	3.98657e7	1.87598	0.937988	−	0.346669i	$-0.112687\pi$
$853$	3.98657e7	1.87598	0.937988	+	0.346669i	$0.112687\pi$
$854$	0	0
$855$	−1.21148e7	−0.566762
$856$	1.64595e7	0.767770
$857$	−3.36474e7	−1.56495	−0.782473	−	0.622684i	$-0.786042\pi$
$857$	−3.36474e7	−1.56495	−0.782473	+	0.622684i	$0.786042\pi$
$858$	−1.65375e6	−0.0766923
$859$	−8.27879e6	−0.382811	−0.191405	−	0.981511i	$-0.561305\pi$
$859$	−8.27879e6	−0.382811	−0.191405	+	0.981511i	$0.561305\pi$
$860$	−1.53995e7	−0.710001
$861$	0	0
$862$	−5.49533e7	−2.51898
$863$	1.06111e7	0.484990	0.242495	−	0.970153i	$-0.422034\pi$
$863$	1.06111e7	0.484990	0.242495	+	0.970153i	$0.422034\pi$
$864$	−1.68070e7	−0.765958
$865$	2.57565e6	0.117043
$866$	3.96640e7	1.79722
$867$	−2.92941e6	−0.132353
$868$	0	0
$869$	4.91987e7	2.21006
$870$	−2.15009e6	−0.0963072
$871$	2.17375e6	0.0970879
$872$	2.02126e6	0.0900181
$873$	1.88053e7	0.835113
$874$	−4.26280e7	−1.88763
$875$	0	0
$876$	−735784.	−0.0323959
$877$	−1.80051e6	−0.0790492	−0.0395246	−	0.999219i	$-0.512584\pi$
$877$	−1.80051e6	−0.0790492	−0.0395246	+	0.999219i	$0.512584\pi$
$878$	2.35148e7	1.02945
$879$	−315325.	−0.0137653
$880$	8.29248e6	0.360976
$881$	3.86161e7	1.67621	0.838105	−	0.545508i	$-0.183663\pi$
$881$	3.86161e7	1.67621	0.838105	+	0.545508i	$0.183663\pi$
$882$	0	0
$883$	8.80155e6	0.379890	0.189945	−	0.981795i	$-0.439169\pi$
$883$	8.80155e6	0.379890	0.189945	+	0.981795i	$0.439169\pi$
$884$	3.70972e6	0.159665
$885$	−1.59557e6	−0.0684789
$886$	−2.62573e7	−1.12374
$887$	−2.76706e7	−1.18089	−0.590444	−	0.807078i	$-0.701047\pi$
$887$	−2.76706e7	−1.18089	−0.590444	+	0.807078i	$0.701047\pi$
$888$	−4.60817e6	−0.196108
$889$	0	0
$890$	−2.06178e7	−0.872505
$891$	−2.93241e7	−1.23746
$892$	−1.53562e7	−0.646207
$893$	3.57562e7	1.50045
$894$	−5.70275e6	−0.238639
$895$	1.17649e7	0.490943
$896$	0	0
$897$	−620457.	−0.0257472
$898$	3.07220e7	1.27133
$899$	−7.48565e6	−0.308909
$900$	−6.08340e6	−0.250346
$901$	−1.80892e6	−0.0742347
$902$	5.96224e7	2.44002
$903$	0	0
$904$	1.07106e7	0.435907
$905$	1.32281e7	0.536880
$906$	1.27621e7	0.516536
$907$	−3.47961e7	−1.40447	−0.702235	−	0.711945i	$-0.747815\pi$
$907$	−3.47961e7	−1.40447	−0.702235	+	0.711945i	$0.747815\pi$
$908$	3.49993e7	1.40878
$909$	−3.97639e7	−1.59617
$910$	0	0
$911$	2.48752e7	0.993049	0.496524	−	0.868023i	$-0.334609\pi$
$911$	2.48752e7	0.993049	0.496524	+	0.868023i	$0.334609\pi$
$912$	5.08648e6	0.202502
$913$	−2.86962e7	−1.13932
$914$	2.19552e7	0.869306
$915$	4.15092e6	0.163905
$916$	1.21745e7	0.479418
$917$	0	0
$918$	−2.73663e7	−1.07179
$919$	−1.88971e7	−0.738086	−0.369043	−	0.929412i	$-0.620314\pi$
$919$	−1.88971e7	−0.738086	−0.369043	+	0.929412i	$0.620314\pi$
$920$	−5.88562e6	−0.229257
$921$	5.13280e6	0.199391
$922$	1.75209e6	0.0678782
$923$	−242518.	−0.00937000
$924$	0	0
$925$	5.73872e6	0.220527
$926$	1.67696e6	0.0642681
$927$	−3.82337e7	−1.46133
$928$	−1.59088e7	−0.606412
$929$	2.29202e7	0.871324	0.435662	−	0.900110i	$-0.356514\pi$
$929$	2.29202e7	0.871324	0.435662	+	0.900110i	$0.356514\pi$
$930$	3.72162e6	0.141099
$931$	0	0
$932$	6.33741e7	2.38986
$933$	−7.03341e6	−0.264522
$934$	−1.30788e7	−0.490570
$935$	2.42384e7	0.906723
$936$	1.37495e6	0.0512977
$937$	−3.70307e7	−1.37789	−0.688943	−	0.724815i	$-0.741925\pi$
$937$	−3.70307e7	−1.37789	−0.688943	+	0.724815i	$0.741925\pi$
$938$	0	0
$939$	−1.22937e7	−0.455009
$940$	1.79548e7	0.662769
$941$	1.25128e7	0.460660	0.230330	−	0.973113i	$-0.426019\pi$
$941$	1.25128e7	0.460660	0.230330	+	0.973113i	$0.426019\pi$
$942$	−8.64353e6	−0.317369
$943$	2.23692e7	0.819166
$944$	−6.57663e6	−0.240200
$945$	0	0
$946$	8.27068e7	3.00479
$947$	−1.22987e7	−0.445640	−0.222820	−	0.974860i	$-0.571526\pi$
$947$	−1.22987e7	−0.445640	−0.222820	+	0.974860i	$0.571526\pi$
$948$	1.51536e7	0.547641
$949$	−207099.	−0.00746471
$950$	1.19831e7	0.430786
$951$	1.31592e7	0.471821
$952$	0	0
$953$	3.53421e6	0.126055	0.0630275	−	0.998012i	$-0.479924\pi$
$953$	3.53421e6	0.126055	0.0630275	+	0.998012i	$0.479924\pi$
$954$	−2.43836e6	−0.0867414
$955$	4.51510e6	0.160199
$956$	5.64501e6	0.199765
$957$	6.69411e6	0.236273
$958$	−6.53021e7	−2.29886
$959$	0	0
$960$	6.05746e6	0.212135
$961$	−1.56721e7	−0.547419
$962$	−4.71725e6	−0.164343
$963$	−3.42805e7	−1.19119
$964$	−2.65994e7	−0.921890
$965$	−1.28216e7	−0.443226
$966$	0	0
$967$	−9.47189e6	−0.325740	−0.162870	−	0.986648i	$-0.552075\pi$
$967$	−9.47189e6	−0.325740	−0.162870	+	0.986648i	$0.552075\pi$
$968$	3.17879e7	1.09037
$969$	1.48675e7	0.508659
$970$	−1.86010e7	−0.634755
$971$	−1.18431e7	−0.403106	−0.201553	−	0.979478i	$-0.564599\pi$
$971$	−1.18431e7	−0.403106	−0.201553	+	0.979478i	$0.564599\pi$
$972$	−3.25946e7	−1.10657
$973$	0	0
$974$	−1.91229e7	−0.645887
$975$	174416.	0.00587591
$976$	1.71093e7	0.574921
$977$	−7.17342e6	−0.240431	−0.120215	−	0.992748i	$-0.538358\pi$
$977$	−7.17342e6	−0.240431	−0.120215	+	0.992748i	$0.538358\pi$
$978$	1.02983e7	0.344285
$979$	6.41918e7	2.14054
$980$	0	0
$981$	−4.20971e6	−0.139663
$982$	3.24216e7	1.07289
$983$	−3.00611e7	−0.992251	−0.496126	−	0.868251i	$-0.665245\pi$
$983$	−3.00611e7	−0.992251	−0.496126	+	0.868251i	$0.665245\pi$
$984$	5.04940e6	0.166246
$985$	−704974.	−0.0231517
$986$	−2.59039e7	−0.848540
$987$	0	0
$988$	−5.71011e6	−0.186103
$989$	3.10301e7	1.00877
$990$	3.26725e7	1.05948
$991$	−2.55378e7	−0.826035	−0.413018	−	0.910723i	$-0.635525\pi$
$991$	−2.55378e7	−0.826035	−0.413018	+	0.910723i	$0.635525\pi$
$992$	2.75367e7	0.888451
$993$	9.45077e6	0.304154
$994$	0	0
$995$	−2.33500e7	−0.747703
$996$	−8.83867e6	−0.282318
$997$	2.42489e7	0.772598	0.386299	−	0.922374i	$-0.373753\pi$
$997$	2.42489e7	0.772598	0.386299	+	0.922374i	$0.373753\pi$
$998$	5.00975e7	1.59217
$999$	2.01727e7	0.639515

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.6.a.h.1.6		6
7.3	odd	6		35.6.e.a.16.1	yes	12
7.5	odd	6		35.6.e.a.11.1	✓	12
7.6	odd	2		245.6.a.i.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.6.e.a.11.1	✓	12	7.5	odd	6
35.6.e.a.16.1	yes	12	7.3	odd	6
245.6.a.h.1.6		6	1.1	even	1	trivial
245.6.a.i.1.6		6	7.6	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:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	245.a (trivial)

\(f(q)\)	\(=\)	\(q+8.72556 q^{2} -4.73965 q^{3} +44.1354 q^{4} +25.0000 q^{5} -41.3561 q^{6} +105.888 q^{8} -220.536 q^{9} +O(q^{10})\)	\(q+8.72556 q^{2} -4.73965 q^{3} +44.1354 q^{4} +25.0000 q^{5} -41.3561 q^{6} +105.888 q^{8} -220.536 q^{9} +218.139 q^{10} -679.156 q^{11} -209.186 q^{12} -58.8790 q^{13} -118.491 q^{15} -488.399 q^{16} -1427.56 q^{17} -1924.30 q^{18} +2197.34 q^{19} +1103.39 q^{20} -5926.02 q^{22} -2223.33 q^{23} -501.873 q^{24} +625.000 q^{25} -513.753 q^{26} +2197.00 q^{27} +2079.59 q^{29} -1033.90 q^{30} -3599.58 q^{31} -7649.98 q^{32} +3218.96 q^{33} -12456.2 q^{34} -9733.43 q^{36} +9181.95 q^{37} +19173.0 q^{38} +279.066 q^{39} +2647.21 q^{40} -10061.1 q^{41} -13956.6 q^{43} -29974.8 q^{44} -5513.39 q^{45} -19399.8 q^{46} +16272.5 q^{47} +2314.84 q^{48} +5453.48 q^{50} +6766.12 q^{51} -2598.65 q^{52} +1267.14 q^{53} +19170.0 q^{54} -16978.9 q^{55} -10414.6 q^{57} +18145.6 q^{58} +13465.7 q^{59} -5229.66 q^{60} -35031.5 q^{61} -31408.4 q^{62} -51121.6 q^{64} -1471.98 q^{65} +28087.2 q^{66} -36919.0 q^{67} -63005.9 q^{68} +10537.8 q^{69} +4118.92 q^{71} -23352.1 q^{72} +3517.37 q^{73} +80117.7 q^{74} -2962.28 q^{75} +96980.4 q^{76} +2435.01 q^{78} -72440.8 q^{79} -12210.0 q^{80} +43177.2 q^{81} -87789.0 q^{82} +42252.6 q^{83} -35688.9 q^{85} -121779. q^{86} -9856.51 q^{87} -71914.7 q^{88} -94517.0 q^{89} -48107.5 q^{90} -98127.7 q^{92} +17060.8 q^{93} +141987. q^{94} +54933.4 q^{95} +36258.2 q^{96} -85271.1 q^{97} +149778. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 5 q^{2} - 20 q^{3} + 31 q^{4} + 150 q^{5} - 96 q^{6} - 135 q^{8} + 378 q^{9}+O(q^{10}) \) 6 * q - 5 * q^2 - 20 * q^3 + 31 * q^4 + 150 * q^5 - 96 * q^6 - 135 * q^8 + 378 * q^9	\( 6 q - 5 q^{2} - 20 q^{3} + 31 q^{4} + 150 q^{5} - 96 q^{6} - 135 q^{8} + 378 q^{9} - 125 q^{10} - 924 q^{11} - 370 q^{12} + 150 q^{13} - 500 q^{15} + 435 q^{16} - 1540 q^{17} + 195 q^{18} - 92 q^{19} + 775 q^{20} - 6855 q^{22} - 3920 q^{23} + 7200 q^{24} + 3750 q^{25} - 2635 q^{26} - 2060 q^{27} + 1264 q^{29} - 2400 q^{30} + 7160 q^{31} + 9105 q^{32} - 4460 q^{33} - 2166 q^{34} - 26375 q^{36} - 14170 q^{37} + 46215 q^{38} - 15376 q^{39} - 3375 q^{40} - 4098 q^{41} - 24460 q^{43} - 27873 q^{44} + 9450 q^{45} + 6815 q^{46} - 42940 q^{47} - 11610 q^{48} - 3125 q^{50} - 42008 q^{51} + 36115 q^{52} - 2450 q^{53} - 19566 q^{54} - 23100 q^{55} - 97100 q^{57} - 36110 q^{58} + 64600 q^{59} - 9250 q^{60} - 73620 q^{61} - 111440 q^{62} - 157997 q^{64} + 3750 q^{65} + 139138 q^{66} - 142620 q^{67} - 124330 q^{68} - 17344 q^{69} - 154256 q^{71} - 117495 q^{72} + 5120 q^{73} + 2785 q^{74} - 12500 q^{75} - 7775 q^{76} - 214090 q^{78} - 222504 q^{79} + 10875 q^{80} - 43986 q^{81} - 31665 q^{82} - 179580 q^{83} - 38500 q^{85} - 207160 q^{86} + 209300 q^{87} - 45145 q^{88} - 41648 q^{89} + 4875 q^{90} - 292185 q^{92} - 198520 q^{93} + 333699 q^{94} - 2300 q^{95} - 61824 q^{96} + 73980 q^{97} - 190772 q^{99}+O(q^{100}) \) 6 * q - 5 * q^2 - 20 * q^3 + 31 * q^4 + 150 * q^5 - 96 * q^6 - 135 * q^8 + 378 * q^9 - 125 * q^10 - 924 * q^11 - 370 * q^12 + 150 * q^13 - 500 * q^15 + 435 * q^16 - 1540 * q^17 + 195 * q^18 - 92 * q^19 + 775 * q^20 - 6855 * q^22 - 3920 * q^23 + 7200 * q^24 + 3750 * q^25 - 2635 * q^26 - 2060 * q^27 + 1264 * q^29 - 2400 * q^30 + 7160 * q^31 + 9105 * q^32 - 4460 * q^33 - 2166 * q^34 - 26375 * q^36 - 14170 * q^37 + 46215 * q^38 - 15376 * q^39 - 3375 * q^40 - 4098 * q^41 - 24460 * q^43 - 27873 * q^44 + 9450 * q^45 + 6815 * q^46 - 42940 * q^47 - 11610 * q^48 - 3125 * q^50 - 42008 * q^51 + 36115 * q^52 - 2450 * q^53 - 19566 * q^54 - 23100 * q^55 - 97100 * q^57 - 36110 * q^58 + 64600 * q^59 - 9250 * q^60 - 73620 * q^61 - 111440 * q^62 - 157997 * q^64 + 3750 * q^65 + 139138 * q^66 - 142620 * q^67 - 124330 * q^68 - 17344 * q^69 - 154256 * q^71 - 117495 * q^72 + 5120 * q^73 + 2785 * q^74 - 12500 * q^75 - 7775 * q^76 - 214090 * q^78 - 222504 * q^79 + 10875 * q^80 - 43986 * q^81 - 31665 * q^82 - 179580 * q^83 - 38500 * q^85 - 207160 * q^86 + 209300 * q^87 - 45145 * q^88 - 41648 * q^89 + 4875 * q^90 - 292185 * q^92 - 198520 * q^93 + 333699 * q^94 - 2300 * q^95 - 61824 * q^96 + 73980 * q^97 - 190772 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more