LMFDB - Embedded newform 245.10.a.m.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [245,10,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, 10, names="a")

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

chi := DirichletCharacter("245.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$N$	$=$	$245 = 5 \cdot 7^{2}$
Weight:	$k$	$=$	$10$
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:	$126.183779860$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{13} - x^{12} - 5109 x^{11} + 3203 x^{10} + 9635922 x^{9} + 242128 x^{8} - 8405086048 x^{7} + \cdots - 96\!\cdots\!52$ x^13 - x^12 - 5109x^11 + 3203x^10 + 9635922x^9 + 242128x^8 - 8405086048x^7 - 4775917264x^6 + 3475841466656x^5 + 1485626212224x^4 - 636895157285888x^3 + 323463590100992x^2 + 40992624146128896*x - 96856528223895552
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2^{14}\cdot 3^{3}\cdot 5^{3}\cdot 7^{7}$
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$9.52028$ 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)

$f(q)$	$=$	$q-9.52028 q^{2} -175.497 q^{3} -421.364 q^{4} +625.000 q^{5} +1670.78 q^{6} +8885.89 q^{8} +11116.2 q^{9} -5950.18 q^{10} -26269.4 q^{11} +73948.1 q^{12} -43764.6 q^{13} -109686. q^{15} +131142. q^{16} +313832. q^{17} -105829. q^{18} +232987. q^{19} -263353. q^{20} +250093. q^{22} +1.23092e6 q^{23} -1.55945e6 q^{24} +390625. q^{25} +416651. q^{26} +1.50345e6 q^{27} +1.65095e6 q^{29} +1.04424e6 q^{30} +1.09098e6 q^{31} -5.79809e6 q^{32} +4.61021e6 q^{33} -2.98777e6 q^{34} -4.68396e6 q^{36} -5.56473e6 q^{37} -2.21810e6 q^{38} +7.68055e6 q^{39} +5.55368e6 q^{40} +7.29660e6 q^{41} -4.17651e7 q^{43} +1.10690e7 q^{44} +6.94762e6 q^{45} -1.17187e7 q^{46} -6.13817e7 q^{47} -2.30151e7 q^{48} -3.71886e6 q^{50} -5.50765e7 q^{51} +1.84408e7 q^{52} -2.19202e7 q^{53} -1.43133e7 q^{54} -1.64184e7 q^{55} -4.08886e7 q^{57} -1.57175e7 q^{58} +1.65061e7 q^{59} +4.62176e7 q^{60} -1.22052e8 q^{61} -1.03864e7 q^{62} -1.19454e7 q^{64} -2.73529e7 q^{65} -4.38905e7 q^{66} +1.45742e8 q^{67} -1.32237e8 q^{68} -2.16023e8 q^{69} -5.74040e7 q^{71} +9.87773e7 q^{72} -3.90283e8 q^{73} +5.29778e7 q^{74} -6.85535e7 q^{75} -9.81725e7 q^{76} -7.31210e7 q^{78} +4.47245e8 q^{79} +8.19639e7 q^{80} -4.82651e8 q^{81} -6.94657e7 q^{82} +5.77741e8 q^{83} +1.96145e8 q^{85} +3.97615e8 q^{86} -2.89737e8 q^{87} -2.33427e8 q^{88} -7.75704e8 q^{89} -6.61433e7 q^{90} -5.18667e8 q^{92} -1.91463e8 q^{93} +5.84372e8 q^{94} +1.45617e8 q^{95} +1.01755e9 q^{96} +7.65402e8 q^{97} -2.92016e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$13 q - q^{2} + 268 q^{3} + 3563 q^{4} + 8125 q^{5} + 3040 q^{6} - 4695 q^{8} + 82107 q^{9} - 625 q^{10} + 129087 q^{11} + 356068 q^{12} + 35889 q^{13} + 167500 q^{15} + 1379187 q^{16} + 251650 q^{17} + 391089 q^{18}+ \cdots + 5266142099 q^{99}+O(q^{100})$ 13 * q - q^2 + 268 * q^3 + 3563 * q^4 + 8125 * q^5 + 3040 * q^6 - 4695 * q^8 + 82107 * q^9 - 625 * q^10 + 129087 * q^11 + 356068 * q^12 + 35889 * q^13 + 167500 * q^15 + 1379187 * q^16 + 251650 * q^17 + 391089 * q^18 - 22537 * q^19 + 2226875 * q^20 - 867693 * q^22 + 1640451 * q^23 + 3300104 * q^24 + 5078125 * q^25 - 215949 * q^26 + 4833346 * q^27 + 175192 * q^29 + 1900000 * q^30 + 3363236 * q^31 + 11618029 * q^32 + 1855346 * q^33 + 30084718 * q^34 + 2860133 * q^36 + 57134155 * q^37 + 28383363 * q^38 - 10216786 * q^39 - 2934375 * q^40 + 46043487 * q^41 - 23472776 * q^43 + 171806579 * q^44 + 51316875 * q^45 - 53075341 * q^46 - 1336787 * q^47 + 304545376 * q^48 - 390625 * q^50 + 234254040 * q^51 + 80069087 * q^52 + 122467755 * q^53 - 99983754 * q^54 + 80679375 * q^55 + 32652030 * q^57 + 37365152 * q^58 - 136056496 * q^59 + 222542500 * q^60 - 50448822 * q^61 - 109231036 * q^62 + 1109008839 * q^64 + 22430625 * q^65 - 500348610 * q^66 + 585418818 * q^67 + 480342846 * q^68 - 521416036 * q^69 + 497194080 * q^71 - 870751121 * q^72 - 346232592 * q^73 - 613439215 * q^74 + 104687500 * q^75 - 1436195213 * q^76 - 327612138 * q^78 + 136453862 * q^79 + 861991875 * q^80 - 344488883 * q^81 - 224863259 * q^82 - 523042314 * q^83 + 157281250 * q^85 - 2285275668 * q^86 + 2310323838 * q^87 - 1816859183 * q^88 - 843350780 * q^89 + 244430625 * q^90 - 172327913 * q^92 + 198172924 * q^93 + 2777934057 * q^94 - 14085625 * q^95 - 1583073484 * q^96 + 694973158 * q^97 + 5266142099 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	−9.52028	−0.420741	−0.210371	−	0.977622i	$-0.567467\pi$
$2$	−9.52028	−0.420741	−0.210371	+	0.977622i	$0.567467\pi$
$3$	−175.497	−1.25090	−0.625452	−	0.780263i	$-0.715085\pi$
$3$	−175.497	−1.25090	−0.625452	+	0.780263i	$0.715085\pi$
$4$	−421.364	−0.822977
$5$	625.000	0.447214
$5$	625.000	0.447214
$6$	1670.78	0.526307
$7$	0	0
$7$	0	0
$8$	8885.89	0.767001
$9$	11116.2	0.564761
$10$	−5950.18	−0.188161
$11$	−26269.4	−0.540983	−0.270492	−	0.962722i	$-0.587186\pi$
$11$	−26269.4	−0.540983	−0.270492	+	0.962722i	$0.587186\pi$
$12$	73948.1	1.02947
$13$	−43764.6	−0.424989	−0.212494	−	0.977162i	$-0.568159\pi$
$13$	−43764.6	−0.424989	−0.212494	+	0.977162i	$0.568159\pi$
$14$	0	0
$15$	−109686.	−0.559421
$16$	131142.	0.500268
$17$	313832.	0.911332	0.455666	−	0.890151i	$-0.349401\pi$
$17$	313832.	0.911332	0.455666	+	0.890151i	$0.349401\pi$
$18$	−105829.	−0.237618
$19$	232987.	0.410148	0.205074	−	0.978746i	$-0.434256\pi$
$19$	232987.	0.410148	0.205074	+	0.978746i	$0.434256\pi$
$20$	−263353.	−0.368046
$21$	0	0
$22$	250093.	0.227614
$23$	1.23092e6	0.917182	0.458591	−	0.888647i	$-0.348354\pi$
$23$	1.23092e6	0.917182	0.458591	+	0.888647i	$0.348354\pi$
$24$	−1.55945e6	−0.959445
$25$	390625.	0.200000
$26$	416651.	0.178810
$27$	1.50345e6	0.544442
$28$	0	0
$29$	1.65095e6	0.433454	0.216727	−	0.976232i	$-0.430462\pi$
$29$	1.65095e6	0.433454	0.216727	+	0.976232i	$0.430462\pi$
$30$	1.04424e6	0.235372
$31$	1.09098e6	0.212172	0.106086	−	0.994357i	$-0.466168\pi$
$31$	1.09098e6	0.212172	0.106086	+	0.994357i	$0.466168\pi$
$32$	−5.79809e6	−0.977485
$33$	4.61021e6	0.676718
$34$	−2.98777e6	−0.383435
$35$	0	0
$36$	−4.68396e6	−0.464785
$37$	−5.56473e6	−0.488131	−0.244065	−	0.969759i	$-0.578481\pi$
$37$	−5.56473e6	−0.488131	−0.244065	+	0.969759i	$0.578481\pi$
$38$	−2.21810e6	−0.172566
$39$	7.68055e6	0.531620
$40$	5.55368e6	0.343013
$41$	7.29660e6	0.403268	0.201634	−	0.979461i	$-0.435375\pi$
$41$	7.29660e6	0.403268	0.201634	+	0.979461i	$0.435375\pi$
$42$	0	0
$43$	−4.17651e7	−1.86297	−0.931483	−	0.363784i	$-0.881485\pi$
$43$	−4.17651e7	−1.86297	−0.931483	+	0.363784i	$0.881485\pi$
$44$	1.10690e7	0.445217
$45$	6.94762e6	0.252569
$46$	−1.17187e7	−0.385896
$47$	−6.13817e7	−1.83484	−0.917421	−	0.397917i	$-0.869733\pi$
$47$	−6.13817e7	−1.83484	−0.917421	+	0.397917i	$0.869733\pi$
$48$	−2.30151e7	−0.625787
$49$	0	0
$50$	−3.71886e6	−0.0841482
$51$	−5.50765e7	−1.13999
$52$	1.84408e7	0.349756
$53$	−2.19202e7	−0.381595	−0.190797	−	0.981629i	$-0.561107\pi$
$53$	−2.19202e7	−0.381595	−0.190797	+	0.981629i	$0.561107\pi$
$54$	−1.43133e7	−0.229069
$55$	−1.64184e7	−0.241935
$56$	0	0
$57$	−4.08886e7	−0.513056
$58$	−1.57175e7	−0.182372
$59$	1.65061e7	0.177341	0.0886705	−	0.996061i	$-0.471738\pi$
$59$	1.65061e7	0.177341	0.0886705	+	0.996061i	$0.471738\pi$
$60$	4.62176e7	0.460391
$61$	−1.22052e8	−1.12865	−0.564326	−	0.825552i	$-0.690864\pi$
$61$	−1.22052e8	−1.12865	−0.564326	+	0.825552i	$0.690864\pi$
$62$	−1.03864e7	−0.0892694
$63$	0	0
$64$	−1.19454e7	−0.0890000
$65$	−2.73529e7	−0.190061
$66$	−4.38905e7	−0.284723
$67$	1.45742e8	0.883584	0.441792	−	0.897118i	$-0.354343\pi$
$67$	1.45742e8	0.883584	0.441792	+	0.897118i	$0.354343\pi$
$68$	−1.32237e8	−0.750005
$69$	−2.16023e8	−1.14731
$70$	0	0
$71$	−5.74040e7	−0.268089	−0.134045	−	0.990975i	$-0.542797\pi$
$71$	−5.74040e7	−0.268089	−0.134045	+	0.990975i	$0.542797\pi$
$72$	9.87773e7	0.433172
$73$	−3.90283e8	−1.60852	−0.804260	−	0.594278i	$-0.797438\pi$
$73$	−3.90283e8	−1.60852	−0.804260	+	0.594278i	$0.797438\pi$
$74$	5.29778e7	0.205377
$75$	−6.85535e7	−0.250181
$76$	−9.81725e7	−0.337543
$77$	0	0
$78$	−7.31210e7	−0.223675
$79$	4.47245e8	1.29188	0.645942	−	0.763386i	$-0.276465\pi$
$79$	4.47245e8	1.29188	0.645942	+	0.763386i	$0.276465\pi$
$80$	8.19639e7	0.223727
$81$	−4.82651e8	−1.24581
$82$	−6.94657e7	−0.169671
$83$	5.77741e8	1.33623	0.668115	−	0.744058i	$-0.267101\pi$
$83$	5.77741e8	1.33623	0.668115	+	0.744058i	$0.267101\pi$
$84$	0	0
$85$	1.96145e8	0.407560
$86$	3.97615e8	0.783827
$87$	−2.89737e8	−0.542209
$88$	−2.33427e8	−0.414935
$89$	−7.75704e8	−1.31051	−0.655256	−	0.755407i	$-0.727439\pi$
$89$	−7.75704e8	−1.31051	−0.655256	+	0.755407i	$0.727439\pi$
$90$	−6.61433e7	−0.106266
$91$	0	0
$92$	−5.18667e8	−0.754820
$93$	−1.91463e8	−0.265407
$94$	5.84372e8	0.771994
$95$	1.45617e8	0.183424
$96$	1.01755e9	1.22274
$97$	7.65402e8	0.877843	0.438922	−	0.898525i	$-0.355361\pi$
$97$	7.65402e8	0.877843	0.438922	+	0.898525i	$0.355361\pi$
$98$	0	0
$99$	−2.92016e8	−0.305526
$100$	−1.64595e8	−0.164595
$101$	9.48735e8	0.907191	0.453596	−	0.891208i	$-0.350141\pi$
$101$	9.48735e8	0.907191	0.453596	+	0.891208i	$0.350141\pi$
$102$	5.24344e8	0.479640
$103$	−5.63675e8	−0.493471	−0.246735	−	0.969083i	$-0.579358\pi$
$103$	−5.63675e8	−0.493471	−0.246735	+	0.969083i	$0.579358\pi$
$104$	−3.88887e8	−0.325967
$105$	0	0
$106$	2.08686e8	0.160553
$107$	−4.28146e8	−0.315765	−0.157883	−	0.987458i	$-0.550467\pi$
$107$	−4.28146e8	−0.315765	−0.157883	+	0.987458i	$0.550467\pi$
$108$	−6.33500e8	−0.448063
$109$	7.20541e8	0.488922	0.244461	−	0.969659i	$-0.421389\pi$
$109$	7.20541e8	0.488922	0.244461	+	0.969659i	$0.421389\pi$
$110$	1.56308e8	0.101792
$111$	9.76593e8	0.610605
$112$	0	0
$113$	3.34613e9	1.93059	0.965295	−	0.261163i	$-0.0841059\pi$
$113$	3.34613e9	1.93059	0.965295	+	0.261163i	$0.0841059\pi$
$114$	3.89271e8	0.215864
$115$	7.69327e8	0.410176
$116$	−6.95651e8	−0.356722
$117$	−4.86495e8	−0.240017
$118$	−1.57142e8	−0.0746147
$119$	0	0
$120$	−9.74655e8	−0.429077
$121$	−1.66786e9	−0.707337
$122$	1.16197e9	0.474870
$123$	−1.28053e9	−0.504449
$124$	−4.59698e8	−0.174613
$125$	2.44141e8	0.0894427
$126$	0	0
$127$	2.86346e9	0.976728	0.488364	−	0.872640i	$-0.337594\pi$
$127$	2.86346e9	0.976728	0.488364	+	0.872640i	$0.337594\pi$
$128$	3.08234e9	1.01493
$129$	7.32964e9	2.33039
$130$	2.60407e8	0.0799664
$131$	1.80384e8	0.0535152	0.0267576	−	0.999642i	$-0.491482\pi$
$131$	1.80384e8	0.0535152	0.0267576	+	0.999642i	$0.491482\pi$
$132$	−1.94258e9	−0.556923
$133$	0	0
$134$	−1.38750e9	−0.371760
$135$	9.39656e8	0.243482
$136$	2.78867e9	0.698993
$137$	−2.97988e9	−0.722698	−0.361349	−	0.932431i	$-0.617684\pi$
$137$	−2.97988e9	−0.722698	−0.361349	+	0.932431i	$0.617684\pi$
$138$	2.05660e9	0.482719
$139$	−7.43567e9	−1.68948	−0.844741	−	0.535175i	$-0.820246\pi$
$139$	−7.43567e9	−1.68948	−0.844741	+	0.535175i	$0.820246\pi$
$140$	0	0
$141$	1.07723e10	2.29521
$142$	5.46502e8	0.112796
$143$	1.14967e9	0.229912
$144$	1.45780e9	0.282532
$145$	1.03184e9	0.193846
$146$	3.71560e9	0.676770
$147$	0	0
$148$	2.34478e9	0.401720
$149$	−1.39313e9	−0.231555	−0.115778	−	0.993275i	$-0.536936\pi$
$149$	−1.39313e9	−0.231555	−0.115778	+	0.993275i	$0.536936\pi$
$150$	6.52649e8	0.105261
$151$	−7.46459e9	−1.16845	−0.584224	−	0.811592i	$-0.698601\pi$
$151$	−7.46459e9	−1.16845	−0.584224	+	0.811592i	$0.698601\pi$
$152$	2.07030e9	0.314584
$153$	3.48861e9	0.514685
$154$	0	0
$155$	6.81860e8	0.0948861
$156$	−3.23631e9	−0.437511
$157$	−5.46781e9	−0.718233	−0.359116	−	0.933293i	$-0.616922\pi$
$157$	−5.46781e9	−0.718233	−0.359116	+	0.933293i	$0.616922\pi$
$158$	−4.25790e9	−0.543549
$159$	3.84692e9	0.477339
$160$	−3.62381e9	−0.437144
$161$	0	0
$162$	4.59497e9	0.524162
$163$	1.17928e10	1.30849	0.654246	−	0.756281i	$-0.272986\pi$
$163$	1.17928e10	1.30849	0.654246	+	0.756281i	$0.272986\pi$
$164$	−3.07453e9	−0.331880
$165$	2.88138e9	0.302637
$166$	−5.50025e9	−0.562207
$167$	−1.23448e10	−1.22818	−0.614088	−	0.789238i	$-0.710476\pi$
$167$	−1.23448e10	−1.22818	−0.614088	+	0.789238i	$0.710476\pi$
$168$	0	0
$169$	−8.68916e9	−0.819384
$170$	−1.86735e9	−0.171477
$171$	2.58993e9	0.231636
$172$	1.75983e10	1.53318
$173$	−1.20327e10	−1.02130	−0.510652	−	0.859787i	$-0.670596\pi$
$173$	−1.20327e10	−1.02130	−0.510652	+	0.859787i	$0.670596\pi$
$174$	2.75837e9	0.228130
$175$	0	0
$176$	−3.44503e9	−0.270637
$177$	−2.89676e9	−0.221837
$178$	7.38492e9	0.551386
$179$	1.93725e10	1.41041	0.705207	−	0.709002i	$-0.250854\pi$
$179$	1.93725e10	1.41041	0.705207	+	0.709002i	$0.250854\pi$
$180$	−2.92748e9	−0.207858
$181$	−2.20051e10	−1.52395	−0.761974	−	0.647608i	$-0.775769\pi$
$181$	−2.20051e10	−1.52395	−0.761974	+	0.647608i	$0.775769\pi$
$182$	0	0
$183$	2.14197e10	1.41184
$184$	1.09378e10	0.703480
$185$	−3.47796e9	−0.218299
$186$	1.82278e9	0.111667
$187$	−8.24418e9	−0.493015
$188$	2.58641e10	1.51003
$189$	0	0
$190$	−1.38632e9	−0.0771740
$191$	2.70149e10	1.46877	0.734385	−	0.678733i	$-0.237471\pi$
$191$	2.70149e10	1.46877	0.734385	+	0.678733i	$0.237471\pi$
$192$	2.09638e9	0.111331
$193$	1.92005e10	0.996103	0.498052	−	0.867147i	$-0.334049\pi$
$193$	1.92005e10	0.996103	0.498052	+	0.867147i	$0.334049\pi$
$194$	−7.28685e9	−0.369345
$195$	4.80034e9	0.237748
$196$	0	0
$197$	3.62794e10	1.71618	0.858088	−	0.513503i	$-0.171652\pi$
$197$	3.62794e10	1.71618	0.858088	+	0.513503i	$0.171652\pi$
$198$	2.78008e9	0.128547
$199$	3.21330e10	1.45249	0.726243	−	0.687438i	$-0.241265\pi$
$199$	3.21330e10	1.45249	0.726243	+	0.687438i	$0.241265\pi$
$200$	3.47105e9	0.153400
$201$	−2.55773e10	−1.10528
$202$	−9.03223e9	−0.381693
$203$	0	0
$204$	2.32073e10	0.938185
$205$	4.56038e9	0.180347
$206$	5.36635e9	0.207623
$207$	1.36832e10	0.517989
$208$	−5.73938e9	−0.212608
$209$	−6.12044e9	−0.221883
$210$	0	0
$211$	−2.68209e10	−0.931542	−0.465771	−	0.884905i	$-0.654223\pi$
$211$	−2.68209e10	−0.931542	−0.465771	+	0.884905i	$0.654223\pi$
$212$	9.23638e9	0.314044
$213$	1.00742e10	0.335354
$214$	4.07607e9	0.132855
$215$	−2.61032e10	−0.833144
$216$	1.33595e10	0.417588
$217$	0	0
$218$	−6.85976e9	−0.205710
$219$	6.84934e10	2.01210
$220$	6.91813e9	0.199107
$221$	−1.37347e10	−0.387306
$222$	−9.29745e9	−0.256907
$223$	−3.04272e10	−0.823929	−0.411965	−	0.911200i	$-0.635157\pi$
$223$	−3.04272e10	−0.823929	−0.411965	+	0.911200i	$0.635157\pi$
$224$	0	0
$225$	4.34226e9	0.112952
$226$	−3.18561e10	−0.812278
$227$	−2.92758e10	−0.731801	−0.365900	−	0.930654i	$-0.619239\pi$
$227$	−2.92758e10	−0.731801	−0.365900	+	0.930654i	$0.619239\pi$
$228$	1.72290e10	0.422233
$229$	1.45201e10	0.348906	0.174453	−	0.984666i	$-0.444184\pi$
$229$	1.45201e10	0.348906	0.174453	+	0.984666i	$0.444184\pi$
$230$	−7.32421e9	−0.172578
$231$	0	0
$232$	1.46702e10	0.332460
$233$	4.74755e10	1.05528	0.527640	−	0.849468i	$-0.323077\pi$
$233$	4.74755e10	1.05528	0.527640	+	0.849468i	$0.323077\pi$
$234$	4.63157e9	0.100985
$235$	−3.83636e10	−0.820567
$236$	−6.95506e9	−0.145948
$237$	−7.84902e10	−1.61602
$238$	0	0
$239$	7.94829e10	1.57574	0.787868	−	0.615845i	$-0.211185\pi$
$239$	7.94829e10	1.57574	0.787868	+	0.615845i	$0.211185\pi$
$240$	−1.43844e10	−0.279861
$241$	4.80731e10	0.917963	0.458982	−	0.888446i	$-0.348214\pi$
$241$	4.80731e10	0.917963	0.458982	+	0.888446i	$0.348214\pi$
$242$	1.58785e10	0.297606
$243$	5.51114e10	1.01394
$244$	5.14283e10	0.928855
$245$	0	0
$246$	1.21910e10	0.212243
$247$	−1.01966e10	−0.174308
$248$	9.69430e9	0.162736
$249$	−1.01392e11	−1.67150
$250$	−2.32429e9	−0.0376322
$251$	6.79354e10	1.08035	0.540175	−	0.841553i	$-0.318358\pi$
$251$	6.79354e10	1.08035	0.540175	+	0.841553i	$0.318358\pi$
$252$	0	0
$253$	−3.23356e10	−0.496180
$254$	−2.72609e10	−0.410950
$255$	−3.44228e10	−0.509819
$256$	−2.32288e10	−0.338023
$257$	8.20972e10	1.17390	0.586948	−	0.809625i	$-0.300329\pi$
$257$	8.20972e10	1.17390	0.586948	+	0.809625i	$0.300329\pi$
$258$	−6.97803e10	−0.980492
$259$	0	0
$260$	1.15255e10	0.156416
$261$	1.83523e10	0.244798
$262$	−1.71731e9	−0.0225160
$263$	1.39217e10	0.179429	0.0897146	−	0.995968i	$-0.471405\pi$
$263$	1.39217e10	0.179429	0.0897146	+	0.995968i	$0.471405\pi$
$264$	4.09658e10	0.519044
$265$	−1.37001e10	−0.170654
$266$	0	0
$267$	1.36134e11	1.63932
$268$	−6.14104e10	−0.727169
$269$	2.61399e10	0.304382	0.152191	−	0.988351i	$-0.451367\pi$
$269$	2.61399e10	0.304382	0.152191	+	0.988351i	$0.451367\pi$
$270$	−8.94579e9	−0.102443
$271$	−1.45817e11	−1.64228	−0.821140	−	0.570727i	$-0.806661\pi$
$271$	−1.45817e11	−1.64228	−0.821140	+	0.570727i	$0.806661\pi$
$272$	4.11566e10	0.455910
$273$	0	0
$274$	2.83693e10	0.304069
$275$	−1.02615e10	−0.108197
$276$	9.10244e10	0.944207
$277$	1.18191e11	1.20622	0.603108	−	0.797660i	$-0.293929\pi$
$277$	1.18191e11	1.20622	0.603108	+	0.797660i	$0.293929\pi$
$278$	7.07897e10	0.710835
$279$	1.21275e10	0.119826
$280$	0	0
$281$	−1.41444e11	−1.35334	−0.676670	−	0.736287i	$-0.736577\pi$
$281$	−1.41444e11	−1.35334	−0.676670	+	0.736287i	$0.736577\pi$
$282$	−1.02555e11	−0.965690
$283$	1.80216e11	1.67015	0.835073	−	0.550139i	$-0.185425\pi$
$283$	1.80216e11	1.67015	0.835073	+	0.550139i	$0.185425\pi$
$284$	2.41880e10	0.220631
$285$	−2.55553e10	−0.229446
$286$	−1.09452e10	−0.0967334
$287$	0	0
$288$	−6.44527e10	−0.552045
$289$	−2.00975e10	−0.169474
$290$	−9.82344e9	−0.0815591
$291$	−1.34326e11	−1.09810
$292$	1.64451e11	1.32377
$293$	1.16123e10	0.0920482	0.0460241	−	0.998940i	$-0.485345\pi$
$293$	1.16123e10	0.0920482	0.0460241	+	0.998940i	$0.485345\pi$
$294$	0	0
$295$	1.03163e10	0.0793093
$296$	−4.94476e10	−0.374397
$297$	−3.94948e10	−0.294534
$298$	1.32630e10	0.0974247
$299$	−5.38708e10	−0.389792
$300$	2.88860e10	0.205893
$301$	0	0
$302$	7.10650e10	0.491614
$303$	−1.66500e11	−1.13481
$304$	3.05545e10	0.205184
$305$	−7.62824e10	−0.504749
$306$	−3.32126e10	−0.216549
$307$	−7.59903e10	−0.488242	−0.244121	−	0.969745i	$-0.578499\pi$
$307$	−7.59903e10	−0.488242	−0.244121	+	0.969745i	$0.578499\pi$
$308$	0	0
$309$	9.89233e10	0.617284
$310$	−6.49150e9	−0.0399225
$311$	−6.05505e10	−0.367025	−0.183513	−	0.983017i	$-0.558747\pi$
$311$	−6.05505e10	−0.367025	−0.183513	+	0.983017i	$0.558747\pi$
$312$	6.82485e10	0.407754
$313$	−1.84759e11	−1.08807	−0.544033	−	0.839064i	$-0.683104\pi$
$313$	−1.84759e11	−1.08807	−0.544033	+	0.839064i	$0.683104\pi$
$314$	5.20551e10	0.302190
$315$	0	0
$316$	−1.88453e11	−1.06319
$317$	9.74554e10	0.542050	0.271025	−	0.962572i	$-0.412637\pi$
$317$	9.74554e10	0.542050	0.271025	+	0.962572i	$0.412637\pi$
$318$	−3.66238e10	−0.200836
$319$	−4.33695e10	−0.234491
$320$	−7.46586e9	−0.0398020
$321$	7.51383e10	0.394992
$322$	0	0
$323$	7.31188e10	0.373781
$324$	2.03372e11	1.02527
$325$	−1.70955e10	−0.0849978
$326$	−1.12270e11	−0.550537
$327$	−1.26453e11	−0.611595
$328$	6.48368e10	0.309307
$329$	0	0
$330$	−2.74316e10	−0.127332
$331$	2.09598e11	0.959755	0.479877	−	0.877336i	$-0.340681\pi$
$331$	2.09598e11	0.959755	0.479877	+	0.877336i	$0.340681\pi$
$332$	−2.43439e11	−1.09969
$333$	−6.18586e10	−0.275677
$334$	1.17526e11	0.516744
$335$	9.10887e10	0.395151
$336$	0	0
$337$	−3.91362e11	−1.65289	−0.826446	−	0.563016i	$-0.809641\pi$
$337$	−3.91362e11	−1.65289	−0.826446	+	0.563016i	$0.809641\pi$
$338$	8.27233e10	0.344749
$339$	−5.87236e11	−2.41498
$340$	−8.26484e10	−0.335413
$341$	−2.86593e10	−0.114781
$342$	−2.46569e10	−0.0974587
$343$	0	0
$344$	−3.71120e11	−1.42890
$345$	−1.35015e11	−0.513091
$346$	1.14555e11	0.429705
$347$	4.38929e11	1.62522	0.812610	−	0.582808i	$-0.198046\pi$
$347$	4.38929e11	1.62522	0.812610	+	0.582808i	$0.198046\pi$
$348$	1.22085e11	0.446226
$349$	−1.91496e11	−0.690948	−0.345474	−	0.938428i	$-0.612282\pi$
$349$	−1.91496e11	−0.690948	−0.345474	+	0.938428i	$0.612282\pi$
$350$	0	0
$351$	−6.57978e10	−0.231382
$352$	1.52313e11	0.528803
$353$	−6.04366e10	−0.207164	−0.103582	−	0.994621i	$-0.533030\pi$
$353$	−6.04366e10	−0.207164	−0.103582	+	0.994621i	$0.533030\pi$
$354$	2.75780e10	0.0933358
$355$	−3.58775e10	−0.119893
$356$	3.26854e11	1.07852
$357$	0	0
$358$	−1.84431e11	−0.593419
$359$	−2.91644e10	−0.0926676	−0.0463338	−	0.998926i	$-0.514754\pi$
$359$	−2.91644e10	−0.0926676	−0.0463338	+	0.998926i	$0.514754\pi$
$360$	6.17358e10	0.193721
$361$	−2.68405e11	−0.831778
$362$	2.09495e11	0.641187
$363$	2.92705e11	0.884811
$364$	0	0
$365$	−2.43927e11	−0.719352
$366$	−2.03922e11	−0.594017
$367$	−4.89761e11	−1.40925	−0.704623	−	0.709582i	$-0.748884\pi$
$367$	−4.89761e11	−1.40925	−0.704623	+	0.709582i	$0.748884\pi$
$368$	1.61426e11	0.458837
$369$	8.11105e10	0.227750
$370$	3.31111e10	0.0918473
$371$	0	0
$372$	8.06757e10	0.218424
$373$	1.99118e11	0.532624	0.266312	−	0.963887i	$-0.414195\pi$
$373$	1.99118e11	0.532624	0.266312	+	0.963887i	$0.414195\pi$
$374$	7.84870e10	0.207432
$375$	−4.28459e10	−0.111884
$376$	−5.45431e11	−1.40733
$377$	−7.22531e10	−0.184213
$378$	0	0
$379$	−1.32403e11	−0.329626	−0.164813	−	0.986325i	$-0.552702\pi$
$379$	−1.32403e11	−0.329626	−0.164813	+	0.986325i	$0.552702\pi$
$380$	−6.13578e10	−0.150954
$381$	−5.02528e11	−1.22179
$382$	−2.57190e11	−0.617972
$383$	6.04046e11	1.43442	0.717209	−	0.696858i	$-0.245419\pi$
$383$	6.04046e11	1.43442	0.717209	+	0.696858i	$0.245419\pi$
$384$	−5.40942e11	−1.26958
$385$	0	0
$386$	−1.82794e11	−0.419102
$387$	−4.64268e11	−1.05213
$388$	−3.22513e11	−0.722445
$389$	5.00639e11	1.10854	0.554270	−	0.832337i	$-0.312998\pi$
$389$	5.00639e11	1.10854	0.554270	+	0.832337i	$0.312998\pi$
$390$	−4.57006e10	−0.100030
$391$	3.86303e11	0.835857
$392$	0	0
$393$	−3.16568e10	−0.0669424
$394$	−3.45390e11	−0.722066
$395$	2.79528e11	0.577748
$396$	1.23045e11	0.251441
$397$	−2.63517e10	−0.0532416	−0.0266208	−	0.999646i	$-0.508475\pi$
$397$	−2.63517e10	−0.0532416	−0.0266208	+	0.999646i	$0.508475\pi$
$398$	−3.05915e11	−0.611121
$399$	0	0
$400$	5.12274e10	0.100054
$401$	−8.61548e11	−1.66391	−0.831955	−	0.554844i	$-0.812778\pi$
$401$	−8.61548e11	−1.66391	−0.831955	+	0.554844i	$0.812778\pi$
$402$	2.43503e11	0.465036
$403$	−4.77461e10	−0.0901707
$404$	−3.99763e11	−0.746597
$405$	−3.01657e11	−0.557141
$406$	0	0
$407$	1.46182e11	0.264071
$408$	−4.89404e11	−0.874373
$409$	3.38790e11	0.598653	0.299327	−	0.954151i	$-0.403238\pi$
$409$	3.38790e11	0.598653	0.299327	+	0.954151i	$0.403238\pi$
$410$	−4.34161e10	−0.0758793
$411$	5.22961e11	0.904026
$412$	2.37512e11	0.406115
$413$	0	0
$414$	−1.30268e11	−0.217939
$415$	3.61088e11	0.597580
$416$	2.53751e11	0.415420
$417$	1.30494e12	2.11338
$418$	5.82684e10	0.0933554
$419$	−3.72424e11	−0.590303	−0.295151	−	0.955451i	$-0.595370\pi$
$419$	−3.72424e11	−0.590303	−0.295151	+	0.955451i	$0.595370\pi$
$420$	0	0
$421$	−2.08072e11	−0.322808	−0.161404	−	0.986888i	$-0.551602\pi$
$421$	−2.08072e11	−0.322808	−0.161404	+	0.986888i	$0.551602\pi$
$422$	2.55343e11	0.391938
$423$	−6.82331e11	−1.03625
$424$	−1.94780e11	−0.292684
$425$	1.22591e11	0.182266
$426$	−9.59095e10	−0.141097
$427$	0	0
$428$	1.80405e11	0.259868
$429$	−2.01764e11	−0.287598
$430$	2.48510e11	0.350538
$431$	−8.40866e11	−1.17376	−0.586880	−	0.809674i	$-0.699644\pi$
$431$	−8.40866e11	−1.17376	−0.586880	+	0.809674i	$0.699644\pi$
$432$	1.97166e11	0.272367
$433$	1.06266e12	1.45278	0.726392	−	0.687281i	$-0.241196\pi$
$433$	1.06266e12	1.45278	0.726392	+	0.687281i	$0.241196\pi$
$434$	0	0
$435$	−1.81085e11	−0.242483
$436$	−3.03610e11	−0.402372
$437$	2.86789e11	0.376181
$438$	−6.52077e11	−0.846575
$439$	1.99978e11	0.256976	0.128488	−	0.991711i	$-0.458988\pi$
$439$	1.99978e11	0.256976	0.128488	+	0.991711i	$0.458988\pi$
$440$	−1.45892e11	−0.185564
$441$	0	0
$442$	1.30758e11	0.162956
$443$	−5.42567e11	−0.669324	−0.334662	−	0.942338i	$-0.608622\pi$
$443$	−5.42567e11	−0.669324	−0.334662	+	0.942338i	$0.608622\pi$
$444$	−4.11501e11	−0.502514
$445$	−4.84815e11	−0.586079
$446$	2.89675e11	0.346661
$447$	2.44491e11	0.289653
$448$	0	0
$449$	1.13888e11	0.132243	0.0661213	−	0.997812i	$-0.478938\pi$
$449$	1.13888e11	0.132243	0.0661213	+	0.997812i	$0.478938\pi$
$450$	−4.13396e10	−0.0475236
$451$	−1.91678e11	−0.218161
$452$	−1.40994e12	−1.58883
$453$	1.31001e12	1.46162
$454$	2.78714e11	0.307899
$455$	0	0
$456$	−3.63331e11	−0.393515
$457$	6.46996e11	0.693871	0.346936	−	0.937889i	$-0.387222\pi$
$457$	6.46996e11	0.693871	0.346936	+	0.937889i	$0.387222\pi$
$458$	−1.38235e11	−0.146799
$459$	4.71830e11	0.496168
$460$	−3.24167e11	−0.337566
$461$	−3.02404e11	−0.311841	−0.155921	−	0.987770i	$-0.549834\pi$
$461$	−3.02404e11	−0.311841	−0.155921	+	0.987770i	$0.549834\pi$
$462$	0	0
$463$	6.37663e11	0.644877	0.322439	−	0.946590i	$-0.395497\pi$
$463$	6.37663e11	0.644877	0.322439	+	0.946590i	$0.395497\pi$
$464$	2.16509e11	0.216843
$465$	−1.19664e11	−0.118693
$466$	−4.51980e11	−0.444000
$467$	1.62208e12	1.57814	0.789070	−	0.614304i	$-0.210563\pi$
$467$	1.62208e12	1.57814	0.789070	+	0.614304i	$0.210563\pi$
$468$	2.04992e11	0.197529
$469$	0	0
$470$	3.65232e11	0.345246
$471$	9.59585e11	0.898440
$472$	1.46671e11	0.136021
$473$	1.09714e12	1.00783
$474$	7.47249e11	0.679927
$475$	9.10106e10	0.0820297
$476$	0	0
$477$	−2.43669e11	−0.215510
$478$	−7.56700e11	−0.662977
$479$	7.61461e11	0.660903	0.330452	−	0.943823i	$-0.392799\pi$
$479$	7.61461e11	0.660903	0.330452	+	0.943823i	$0.392799\pi$
$480$	6.35967e11	0.546826
$481$	2.43538e11	0.207450
$482$	−4.57669e11	−0.386225
$483$	0	0
$484$	7.02778e11	0.582122
$485$	4.78376e11	0.392583
$486$	−5.24676e11	−0.426607
$487$	−1.66188e12	−1.33881	−0.669404	−	0.742898i	$-0.733451\pi$
$487$	−1.66188e12	−1.33881	−0.669404	+	0.742898i	$0.733451\pi$
$488$	−1.08454e12	−0.865678
$489$	−2.06959e12	−1.63680
$490$	0	0
$491$	−7.13111e11	−0.553720	−0.276860	−	0.960910i	$-0.589294\pi$
$491$	−7.13111e11	−0.553720	−0.276860	+	0.960910i	$0.589294\pi$
$492$	5.39570e11	0.415150
$493$	5.18120e11	0.395020
$494$	9.70744e10	0.0733387
$495$	−1.82510e11	−0.136635
$496$	1.43073e11	0.106143
$497$	0	0
$498$	9.65278e11	0.703267
$499$	3.42238e11	0.247102	0.123551	−	0.992338i	$-0.460572\pi$
$499$	3.42238e11	0.247102	0.123551	+	0.992338i	$0.460572\pi$
$500$	−1.02872e11	−0.0736093
$501$	2.16648e12	1.53633
$502$	−6.46765e11	−0.454548
$503$	1.72343e12	1.20043	0.600217	−	0.799837i	$-0.295081\pi$
$503$	1.72343e12	1.20043	0.600217	+	0.799837i	$0.295081\pi$
$504$	0	0
$505$	5.92959e11	0.405708
$506$	3.07845e11	0.208763
$507$	1.52492e12	1.02497
$508$	−1.20656e12	−0.803825
$509$	−1.18787e12	−0.784401	−0.392201	−	0.919880i	$-0.628286\pi$
$509$	−1.18787e12	−0.784401	−0.392201	+	0.919880i	$0.628286\pi$
$510$	3.27715e11	0.214502
$511$	0	0
$512$	−1.35702e12	−0.872710
$513$	3.50284e11	0.223302
$514$	−7.81589e11	−0.493906
$515$	−3.52297e11	−0.220687
$516$	−3.08845e12	−1.91786
$517$	1.61246e12	0.992619
$518$	0	0
$519$	2.11170e12	1.27755
$520$	−2.43055e11	−0.145777
$521$	−2.40085e11	−0.142757	−0.0713783	−	0.997449i	$-0.522740\pi$
$521$	−2.40085e11	−0.142757	−0.0713783	+	0.997449i	$0.522740\pi$
$522$	−1.74719e11	−0.102996
$523$	7.10285e9	0.00415121	0.00207561	−	0.999998i	$-0.499339\pi$
$523$	7.10285e9	0.00415121	0.00207561	+	0.999998i	$0.499339\pi$
$524$	−7.60073e10	−0.0440418
$525$	0	0
$526$	−1.32539e11	−0.0754932
$527$	3.42383e11	0.193359
$528$	6.04593e11	0.338540
$529$	−2.85982e11	−0.158777
$530$	1.30429e11	0.0718013
$531$	1.83485e11	0.100155
$532$	0	0
$533$	−3.19333e11	−0.171384
$534$	−1.29603e12	−0.689731
$535$	−2.67591e11	−0.141215
$536$	1.29505e12	0.677710
$537$	−3.39981e12	−1.76429
$538$	−2.48859e11	−0.128066
$539$	0	0
$540$	−3.95937e11	−0.200380
$541$	9.97063e11	0.500420	0.250210	−	0.968192i	$-0.419500\pi$
$541$	9.97063e11	0.500420	0.250210	+	0.968192i	$0.419500\pi$
$542$	1.38822e12	0.690974
$543$	3.86183e12	1.90631
$544$	−1.81962e12	−0.890813
$545$	4.50338e11	0.218653
$546$	0	0
$547$	−2.39224e11	−0.114251	−0.0571256	−	0.998367i	$-0.518194\pi$
$547$	−2.39224e11	−0.114251	−0.0571256	+	0.998367i	$0.518194\pi$
$548$	1.25562e12	0.594764
$549$	−1.35675e12	−0.637419
$550$	9.76924e10	0.0455228
$551$	3.84650e11	0.177780
$552$	−1.91956e12	−0.879986
$553$	0	0
$554$	−1.12521e12	−0.507505
$555$	6.10371e11	0.273071
$556$	3.13313e12	1.39041
$557$	1.28056e12	0.563704	0.281852	−	0.959458i	$-0.409051\pi$
$557$	1.28056e12	0.563704	0.281852	+	0.959458i	$0.409051\pi$
$558$	−1.15457e11	−0.0504159
$559$	1.82783e12	0.791740
$560$	0	0
$561$	1.44683e12	0.616715
$562$	1.34659e12	0.569405
$563$	4.17614e12	1.75181	0.875905	−	0.482484i	$-0.160265\pi$
$563$	4.17614e12	1.75181	0.875905	+	0.482484i	$0.160265\pi$
$564$	−4.53907e12	−1.88891
$565$	2.09133e12	0.863386
$566$	−1.71571e12	−0.702699
$567$	0	0
$568$	−5.10086e11	−0.205625
$569$	7.47920e11	0.299123	0.149562	−	0.988752i	$-0.452214\pi$
$569$	7.47920e11	0.299123	0.149562	+	0.988752i	$0.452214\pi$
$570$	2.43294e11	0.0965372
$571$	3.30415e12	1.30076	0.650380	−	0.759609i	$-0.274610\pi$
$571$	3.30415e12	1.30076	0.650380	+	0.759609i	$0.274610\pi$
$572$	−4.84430e11	−0.189212
$573$	−4.74104e12	−1.83729
$574$	0	0
$575$	4.80829e11	0.183436
$576$	−1.32787e11	−0.0502638
$577$	4.91162e11	0.184473	0.0922367	−	0.995737i	$-0.470598\pi$
$577$	4.91162e11	0.184473	0.0922367	+	0.995737i	$0.470598\pi$
$578$	1.91334e11	0.0713046
$579$	−3.36963e12	−1.24603
$580$	−4.34782e11	−0.159531
$581$	0	0
$582$	1.27882e12	0.462015
$583$	5.75831e11	0.206436
$584$	−3.46801e12	−1.23374
$585$	−3.04060e11	−0.107339
$586$	−1.10553e11	−0.0387285
$587$	3.13584e12	1.09014	0.545070	−	0.838390i	$-0.316503\pi$
$587$	3.13584e12	1.09014	0.545070	+	0.838390i	$0.316503\pi$
$588$	0	0
$589$	2.54184e11	0.0870219
$590$	−9.82140e10	−0.0333687
$591$	−6.36692e12	−2.14677
$592$	−7.29771e11	−0.244196
$593$	2.78743e12	0.925673	0.462837	−	0.886444i	$-0.346832\pi$
$593$	2.78743e12	0.925673	0.462837	+	0.886444i	$0.346832\pi$
$594$	3.76001e11	0.123923
$595$	0	0
$596$	5.87016e11	0.190564
$597$	−5.63924e12	−1.81692
$598$	5.12865e11	0.164002
$599$	−4.50849e12	−1.43090	−0.715452	−	0.698662i	$-0.753779\pi$
$599$	−4.50849e12	−1.43090	−0.715452	+	0.698662i	$0.753779\pi$
$600$	−6.09159e11	−0.191889
$601$	−4.42349e12	−1.38303	−0.691513	−	0.722364i	$-0.743056\pi$
$601$	−4.42349e12	−1.38303	−0.691513	+	0.722364i	$0.743056\pi$
$602$	0	0
$603$	1.62009e12	0.499014
$604$	3.14531e12	0.961606
$605$	−1.04242e12	−0.316331
$606$	1.58513e12	0.477461
$607$	−2.66914e12	−0.798037	−0.399018	−	0.916943i	$-0.630649\pi$
$607$	−2.66914e12	−0.798037	−0.399018	+	0.916943i	$0.630649\pi$
$608$	−1.35088e12	−0.400914
$609$	0	0
$610$	7.26230e11	0.212368
$611$	2.68635e12	0.779788
$612$	−1.46998e12	−0.423574
$613$	−2.71304e12	−0.776040	−0.388020	−	0.921651i	$-0.626841\pi$
$613$	−2.71304e12	−0.776040	−0.388020	+	0.921651i	$0.626841\pi$
$614$	7.23449e11	0.205424
$615$	−8.00333e11	−0.225597
$616$	0	0
$617$	−4.55367e11	−0.126497	−0.0632483	−	0.997998i	$-0.520146\pi$
$617$	−4.55367e11	−0.126497	−0.0632483	+	0.997998i	$0.520146\pi$
$618$	−9.41778e11	−0.259717
$619$	3.92204e12	1.07375	0.536876	−	0.843661i	$-0.319604\pi$
$619$	3.92204e12	1.07375	0.536876	+	0.843661i	$0.319604\pi$
$620$	−2.87311e11	−0.0780891
$621$	1.85063e12	0.499353
$622$	5.76458e11	0.154423
$623$	0	0
$624$	1.00724e12	0.265953
$625$	1.52588e11	0.0400000
$626$	1.75896e12	0.457794
$627$	1.07412e12	0.277555
$628$	2.30394e12	0.591089
$629$	−1.74639e12	−0.444849
$630$	0	0
$631$	2.74330e12	0.688876	0.344438	−	0.938809i	$-0.388070\pi$
$631$	2.74330e12	0.688876	0.344438	+	0.938809i	$0.388070\pi$
$632$	3.97417e12	0.990877
$633$	4.70699e12	1.16527
$634$	−9.27803e11	−0.228063
$635$	1.78966e12	0.436806
$636$	−1.62096e12	−0.392839
$637$	0	0
$638$	4.12890e11	0.0986601
$639$	−6.38114e11	−0.151406
$640$	1.92647e12	0.453891
$641$	4.11504e12	0.962749	0.481374	−	0.876515i	$-0.340138\pi$
$641$	4.11504e12	0.962749	0.481374	+	0.876515i	$0.340138\pi$
$642$	−7.15338e11	−0.166189
$643$	3.64691e12	0.841349	0.420675	−	0.907212i	$-0.361793\pi$
$643$	3.64691e12	0.841349	0.420675	+	0.907212i	$0.361793\pi$
$644$	0	0
$645$	4.58103e12	1.04218
$646$	−6.96112e11	−0.157265
$647$	2.34539e12	0.526193	0.263097	−	0.964769i	$-0.415256\pi$
$647$	2.34539e12	0.526193	0.263097	+	0.964769i	$0.415256\pi$
$648$	−4.28878e12	−0.955535
$649$	−4.33605e11	−0.0959385
$650$	1.62754e11	0.0357621
$651$	0	0
$652$	−4.96905e12	−1.07686
$653$	1.73236e11	0.0372846	0.0186423	−	0.999826i	$-0.494066\pi$
$653$	1.73236e11	0.0372846	0.0186423	+	0.999826i	$0.494066\pi$
$654$	1.20387e12	0.257323
$655$	1.12740e11	0.0239327
$656$	9.56893e11	0.201742
$657$	−4.33846e12	−0.908429
$658$	0	0
$659$	3.10137e12	0.640574	0.320287	−	0.947320i	$-0.396221\pi$
$659$	3.10137e12	0.640574	0.320287	+	0.947320i	$0.396221\pi$
$660$	−1.21411e12	−0.249064
$661$	2.62494e12	0.534826	0.267413	−	0.963582i	$-0.413831\pi$
$661$	2.62494e12	0.534826	0.267413	+	0.963582i	$0.413831\pi$
$662$	−1.99543e12	−0.403808
$663$	2.41040e12	0.484483
$664$	5.13374e12	1.02489
$665$	0	0
$666$	5.88911e11	0.115989
$667$	2.03219e12	0.397556
$668$	5.20166e12	1.01076
$669$	5.33988e12	1.03066
$670$	−8.67190e11	−0.166256
$671$	3.20623e12	0.610582
$672$	0	0
$673$	8.40711e12	1.57972	0.789858	−	0.613290i	$-0.210154\pi$
$673$	8.40711e12	1.57972	0.789858	+	0.613290i	$0.210154\pi$
$674$	3.72588e12	0.695439
$675$	5.87285e11	0.108888
$676$	3.66130e12	0.674334
$677$	−5.51141e12	−1.00836	−0.504178	−	0.863600i	$-0.668204\pi$
$677$	−5.51141e12	−1.00836	−0.504178	+	0.863600i	$0.668204\pi$
$678$	5.59065e12	1.01608
$679$	0	0
$680$	1.74292e12	0.312599
$681$	5.13782e12	0.915413
$682$	2.72845e11	0.0482932
$683$	5.28804e11	0.0929826	0.0464913	−	0.998919i	$-0.485196\pi$
$683$	5.28804e11	0.0929826	0.0464913	+	0.998919i	$0.485196\pi$
$684$	−1.09130e12	−0.190631
$685$	−1.86243e12	−0.323200
$686$	0	0
$687$	−2.54823e12	−0.436448
$688$	−5.47716e12	−0.931982
$689$	9.59327e11	0.162174
$690$	1.28538e12	0.215879
$691$	7.40471e12	1.23554	0.617770	−	0.786359i	$-0.288036\pi$
$691$	7.40471e12	1.23554	0.617770	+	0.786359i	$0.288036\pi$
$692$	5.07014e12	0.840510
$693$	0	0
$694$	−4.17873e12	−0.683797
$695$	−4.64730e12	−0.755560
$696$	−2.57457e12	−0.415875
$697$	2.28991e12	0.367511
$698$	1.82310e12	0.290710
$699$	−8.33180e12	−1.32005
$700$	0	0
$701$	3.15485e12	0.493455	0.246727	−	0.969085i	$-0.420645\pi$
$701$	3.15485e12	0.493455	0.246727	+	0.969085i	$0.420645\pi$
$702$	6.26414e11	0.0973519
$703$	−1.29651e12	−0.200206
$704$	3.13799e11	0.0481475
$705$	6.73269e12	1.02645
$706$	5.75374e11	0.0871623
$707$	0	0
$708$	1.22059e12	0.182566
$709$	1.22184e12	0.181596	0.0907982	−	0.995869i	$-0.471058\pi$
$709$	1.22184e12	0.181596	0.0907982	+	0.995869i	$0.471058\pi$
$710$	3.41564e11	0.0504440
$711$	4.97166e12	0.729606
$712$	−6.89282e12	−1.00516
$713$	1.34291e12	0.194600
$714$	0	0
$715$	7.18544e11	0.102820
$716$	−8.16286e12	−1.16074
$717$	−1.39490e13	−1.97109
$718$	2.77653e11	0.0389891
$719$	1.24183e12	0.173294	0.0866468	−	0.996239i	$-0.472385\pi$
$719$	1.24183e12	0.173294	0.0866468	+	0.996239i	$0.472385\pi$
$720$	9.11126e11	0.126352
$721$	0	0
$722$	2.55529e12	0.349963
$723$	−8.43668e12	−1.14828
$724$	9.27217e12	1.25417
$725$	6.44902e11	0.0866907
$726$	−2.78664e12	−0.372276
$727$	−6.23382e12	−0.827656	−0.413828	−	0.910355i	$-0.635808\pi$
$727$	−6.23382e12	−0.827656	−0.413828	+	0.910355i	$0.635808\pi$
$728$	0	0
$729$	−1.71863e11	−0.0225377
$730$	2.32225e12	0.302661
$731$	−1.31072e13	−1.69778
$732$	−9.02551e12	−1.16191
$733$	1.13604e13	1.45354	0.726769	−	0.686882i	$-0.241021\pi$
$733$	1.13604e13	1.45354	0.726769	+	0.686882i	$0.241021\pi$
$734$	4.66266e12	0.592927
$735$	0	0
$736$	−7.13700e12	−0.896531
$737$	−3.82856e12	−0.478004
$738$	−7.72195e11	−0.0958237
$739$	6.78685e12	0.837082	0.418541	−	0.908198i	$-0.362542\pi$
$739$	6.78685e12	0.837082	0.418541	+	0.908198i	$0.362542\pi$
$740$	1.46549e12	0.179655
$741$	1.78947e12	0.218043
$742$	0	0
$743$	−3.07367e12	−0.370005	−0.185002	−	0.982738i	$-0.559229\pi$
$743$	−3.07367e12	−0.370005	−0.185002	+	0.982738i	$0.559229\pi$
$744$	−1.70132e12	−0.203567
$745$	−8.70708e11	−0.103555
$746$	−1.89566e12	−0.224097
$747$	6.42227e12	0.754651
$748$	3.47380e12	0.405740
$749$	0	0
$750$	4.07906e11	0.0470743
$751$	1.55928e13	1.78873	0.894366	−	0.447337i	$-0.147627\pi$
$751$	1.55928e13	1.78873	0.894366	+	0.447337i	$0.147627\pi$
$752$	−8.04974e12	−0.917913
$753$	−1.19225e13	−1.35141
$754$	6.87870e11	0.0775060
$755$	−4.66537e12	−0.522546
$756$	0	0
$757$	−2.01197e11	−0.0222684	−0.0111342	−	0.999938i	$-0.503544\pi$
$757$	−2.01197e11	−0.0222684	−0.0111342	+	0.999938i	$0.503544\pi$
$758$	1.26051e12	0.138687
$759$	5.67481e12	0.620674
$760$	1.29394e12	0.140686
$761$	−1.00060e13	−1.08151	−0.540755	−	0.841180i	$-0.681861\pi$
$761$	−1.00060e13	−1.08151	−0.540755	+	0.841180i	$0.681861\pi$
$762$	4.78421e12	0.514059
$763$	0	0
$764$	−1.13831e13	−1.20876
$765$	2.18038e12	0.230174
$766$	−5.75069e12	−0.603519
$767$	−7.22381e11	−0.0753680
$768$	4.07658e12	0.422834
$769$	−6.54056e11	−0.0674445	−0.0337223	−	0.999431i	$-0.510736\pi$
$769$	−6.54056e11	−0.0674445	−0.0337223	+	0.999431i	$0.510736\pi$
$770$	0	0
$771$	−1.44078e13	−1.46843
$772$	−8.09040e12	−0.819770
$773$	−1.50342e13	−1.51451	−0.757255	−	0.653120i	$-0.773460\pi$
$773$	−1.50342e13	−1.51451	−0.757255	+	0.653120i	$0.773460\pi$
$774$	4.41997e12	0.442675
$775$	4.26163e11	0.0424344
$776$	6.80128e12	0.673307
$777$	0	0
$778$	−4.76622e12	−0.466408
$779$	1.70002e12	0.165400
$780$	−2.02269e12	−0.195661
$781$	1.50797e12	0.145032
$782$	−3.67771e12	−0.351680
$783$	2.48212e12	0.235991
$784$	0	0
$785$	−3.41738e12	−0.321203
$786$	3.01382e11	0.0281654
$787$	−2.10567e12	−0.195661	−0.0978305	−	0.995203i	$-0.531190\pi$
$787$	−2.10567e12	−0.195661	−0.0978305	+	0.995203i	$0.531190\pi$
$788$	−1.52868e13	−1.41237
$789$	−2.44323e12	−0.224449
$790$	−2.66119e12	−0.243082
$791$	0	0
$792$	−2.59482e12	−0.234339
$793$	5.34155e12	0.479665
$794$	2.50875e11	0.0224009
$795$	2.40433e12	0.213472
$796$	−1.35397e13	−1.19536
$797$	−1.24161e13	−1.08999	−0.544997	−	0.838438i	$-0.683469\pi$
$797$	−1.24161e13	−1.08999	−0.544997	+	0.838438i	$0.683469\pi$
$798$	0	0
$799$	−1.92635e13	−1.67215
$800$	−2.26488e12	−0.195497
$801$	−8.62287e12	−0.740126
$802$	8.20218e12	0.700075
$803$	1.02525e13	0.870182
$804$	1.07773e13	0.909619
$805$	0	0
$806$	4.54557e11	0.0379385
$807$	−4.58747e12	−0.380752
$808$	8.43036e12	0.695817
$809$	−1.17105e13	−0.961183	−0.480591	−	0.876945i	$-0.659578\pi$
$809$	−1.17105e13	−0.961183	−0.480591	+	0.876945i	$0.659578\pi$
$810$	2.87186e12	0.234412
$811$	−1.51560e13	−1.23024	−0.615122	−	0.788432i	$-0.710893\pi$
$811$	−1.51560e13	−1.23024	−0.615122	+	0.788432i	$0.710893\pi$
$812$	0	0
$813$	2.55905e13	2.05433
$814$	−1.39170e12	−0.111105
$815$	7.37048e12	0.585176
$816$	−7.22286e12	−0.570300
$817$	−9.73073e12	−0.764093
$818$	−3.22538e12	−0.251878
$819$	0	0
$820$	−1.92158e12	−0.148421
$821$	−7.80111e12	−0.599255	−0.299628	−	0.954056i	$-0.596862\pi$
$821$	−7.80111e12	−0.599255	−0.299628	+	0.954056i	$0.596862\pi$
$822$	−4.97873e12	−0.380361
$823$	6.32663e12	0.480699	0.240350	−	0.970686i	$-0.422738\pi$
$823$	6.32663e12	0.480699	0.240350	+	0.970686i	$0.422738\pi$
$824$	−5.00876e12	−0.378493
$825$	1.80086e12	0.135344
$826$	0	0
$827$	−2.24559e13	−1.66938	−0.834689	−	0.550721i	$-0.814353\pi$
$827$	−2.24559e13	−1.66938	−0.834689	+	0.550721i	$0.814353\pi$
$828$	−5.76560e12	−0.426293
$829$	−1.48421e13	−1.09144	−0.545718	−	0.837969i	$-0.683743\pi$
$829$	−1.48421e13	−1.09144	−0.545718	+	0.837969i	$0.683743\pi$
$830$	−3.43766e12	−0.251427
$831$	−2.07421e13	−1.50886
$832$	5.22785e11	0.0378240
$833$	0	0
$834$	−1.24234e13	−0.889186
$835$	−7.71551e12	−0.549257
$836$	2.57894e12	0.182605
$837$	1.64023e12	0.115515
$838$	3.54558e12	0.248365
$839$	−2.98994e12	−0.208321	−0.104160	−	0.994561i	$-0.533216\pi$
$839$	−2.98994e12	−0.208321	−0.104160	+	0.994561i	$0.533216\pi$
$840$	0	0
$841$	−1.17815e13	−0.812118
$842$	1.98090e12	0.135818
$843$	2.48230e13	1.69290
$844$	1.13014e13	0.766638
$845$	−5.43073e12	−0.366440
$846$	6.49599e12	0.435992
$847$	0	0
$848$	−2.87466e12	−0.190900
$849$	−3.16274e13	−2.08919
$850$	−1.16710e12	−0.0766870
$851$	−6.84975e12	−0.447705
$852$	−4.24492e12	−0.275989
$853$	3.18183e12	0.205781	0.102891	−	0.994693i	$-0.467191\pi$
$853$	3.18183e12	0.205781	0.102891	+	0.994693i	$0.467191\pi$
$854$	0	0
$855$	1.61871e12	0.103591
$856$	−3.80446e12	−0.242192
$857$	−4.67041e10	−0.00295761	−0.00147881	−	0.999999i	$-0.500471\pi$
$857$	−4.67041e10	−0.00295761	−0.00147881	+	0.999999i	$0.500471\pi$
$858$	1.92085e12	0.121004
$859$	2.45887e13	1.54087	0.770436	−	0.637518i	$-0.220039\pi$
$859$	2.45887e13	1.54087	0.770436	+	0.637518i	$0.220039\pi$
$860$	1.09989e13	0.685658
$861$	0	0
$862$	8.00529e12	0.493849
$863$	−4.91452e12	−0.301601	−0.150800	−	0.988564i	$-0.548185\pi$
$863$	−4.91452e12	−0.301601	−0.150800	+	0.988564i	$0.548185\pi$
$864$	−8.71713e12	−0.532184
$865$	−7.52043e12	−0.456741
$866$	−1.01169e13	−0.611246
$867$	3.52706e12	0.211996
$868$	0	0
$869$	−1.17489e13	−0.698888
$870$	1.72398e12	0.102023
$871$	−6.37833e12	−0.375513
$872$	6.40265e12	0.375004
$873$	8.50836e12	0.495772
$874$	−2.73032e12	−0.158275
$875$	0	0
$876$	−2.88607e13	−1.65591
$877$	6.49150e12	0.370550	0.185275	−	0.982687i	$-0.440682\pi$
$877$	6.49150e12	0.370550	0.185275	+	0.982687i	$0.440682\pi$
$878$	−1.90385e12	−0.108120
$879$	−2.03793e12	−0.115143
$880$	−2.15315e12	−0.121032
$881$	6.22021e11	0.0347867	0.0173934	−	0.999849i	$-0.494463\pi$
$881$	6.22021e11	0.0347867	0.0173934	+	0.999849i	$0.494463\pi$
$882$	0	0
$883$	2.83144e13	1.56741	0.783707	−	0.621131i	$-0.213326\pi$
$883$	2.83144e13	1.56741	0.783707	+	0.621131i	$0.213326\pi$
$884$	5.78732e12	0.318744
$885$	−1.81048e12	−0.0992084
$886$	5.16539e12	0.281612
$887$	2.76786e13	1.50137	0.750685	−	0.660660i	$-0.229724\pi$
$887$	2.76786e13	1.50137	0.750685	+	0.660660i	$0.229724\pi$
$888$	8.67790e12	0.468335
$889$	0	0
$890$	4.61558e12	0.246587
$891$	1.26790e13	0.673960
$892$	1.28209e13	0.678075
$893$	−1.43012e13	−0.752558
$894$	−2.32762e12	−0.121869
$895$	1.21078e13	0.630756
$896$	0	0
$897$	9.45416e12	0.487593
$898$	−1.08425e12	−0.0556399
$899$	1.80115e12	0.0919667
$900$	−1.82967e12	−0.0929571
$901$	−6.87924e12	−0.347760
$902$	1.82483e12	0.0917893
$903$	0	0
$904$	2.97334e13	1.48076
$905$	−1.37532e13	−0.681530
$906$	−1.24717e13	−0.614962
$907$	1.55835e13	0.764598	0.382299	−	0.924039i	$-0.375132\pi$
$907$	1.55835e13	0.764598	0.382299	+	0.924039i	$0.375132\pi$
$908$	1.23358e13	0.602255
$909$	1.05463e13	0.512346
$910$	0	0
$911$	−3.32939e13	−1.60152	−0.800760	−	0.598985i	$-0.795571\pi$
$911$	−3.32939e13	−1.60152	−0.800760	+	0.598985i	$0.795571\pi$
$912$	−5.36222e12	−0.256666
$913$	−1.51769e13	−0.722878
$914$	−6.15959e12	−0.291940
$915$	1.33873e13	0.631392
$916$	−6.11823e12	−0.287142
$917$	0	0
$918$	−4.49196e12	−0.208758
$919$	3.33563e13	1.54262	0.771309	−	0.636461i	$-0.219602\pi$
$919$	3.33563e13	1.54262	0.771309	+	0.636461i	$0.219602\pi$
$920$	6.83615e12	0.314606
$921$	1.33361e13	0.610744
$922$	2.87897e12	0.131204
$923$	2.51226e12	0.113935
$924$	0	0
$925$	−2.17372e12	−0.0976262
$926$	−6.07074e12	−0.271326
$927$	−6.26592e12	−0.278693
$928$	−9.57235e12	−0.423694
$929$	2.59300e13	1.14217	0.571086	−	0.820890i	$-0.306522\pi$
$929$	2.59300e13	1.14217	0.571086	+	0.820890i	$0.306522\pi$
$930$	1.13924e12	0.0499392
$931$	0	0
$932$	−2.00045e13	−0.868471
$933$	1.06264e13	0.459113
$934$	−1.54426e13	−0.663988
$935$	−5.15261e12	−0.220483
$936$	−4.32295e12	−0.184093
$937$	1.05742e13	0.448148	0.224074	−	0.974572i	$-0.428064\pi$
$937$	1.05742e13	0.448148	0.224074	+	0.974572i	$0.428064\pi$
$938$	0	0
$939$	3.24246e13	1.36107
$940$	1.61650e13	0.675307
$941$	−4.43349e11	−0.0184329	−0.00921643	−	0.999958i	$-0.502934\pi$
$941$	−4.43349e11	−0.0184329	−0.00921643	+	0.999958i	$0.502934\pi$
$942$	−9.13552e12	−0.378011
$943$	8.98156e12	0.369870
$944$	2.16464e12	0.0887181
$945$	0	0
$946$	−1.04451e13	−0.424037
$947$	2.08797e13	0.843625	0.421812	−	0.906683i	$-0.361394\pi$
$947$	2.08797e13	0.843625	0.421812	+	0.906683i	$0.361394\pi$
$948$	3.30729e13	1.32995
$949$	1.70806e13	0.683603
$950$	−8.66447e11	−0.0345132
$951$	−1.71031e13	−0.678052
$952$	0	0
$953$	2.68247e13	1.05346	0.526728	−	0.850034i	$-0.323419\pi$
$953$	2.68247e13	1.05346	0.526728	+	0.850034i	$0.323419\pi$
$954$	2.31980e12	0.0906739
$955$	1.68843e13	0.656854
$956$	−3.34913e13	−1.29679
$957$	7.61122e12	0.293326
$958$	−7.24933e12	−0.278069
$959$	0	0
$960$	1.31024e12	0.0497885
$961$	−2.52494e13	−0.954983
$962$	−2.31855e12	−0.0872828
$963$	−4.75935e12	−0.178332
$964$	−2.02563e13	−0.755463
$965$	1.20003e13	0.445471
$966$	0	0
$967$	4.80960e13	1.76885	0.884423	−	0.466686i	$-0.154552\pi$
$967$	4.80960e13	1.76885	0.884423	+	0.466686i	$0.154552\pi$
$968$	−1.48205e13	−0.542529
$969$	−1.28321e13	−0.467565
$970$	−4.55428e12	−0.165176
$971$	−4.95038e13	−1.78711	−0.893556	−	0.448952i	$-0.851797\pi$
$971$	−4.95038e13	−1.78711	−0.893556	+	0.448952i	$0.851797\pi$
$972$	−2.32220e13	−0.834451
$973$	0	0
$974$	1.58215e13	0.563292
$975$	3.00022e12	0.106324
$976$	−1.60062e13	−0.564629
$977$	3.89809e13	1.36876	0.684378	−	0.729127i	$-0.260074\pi$
$977$	3.89809e13	1.36876	0.684378	+	0.729127i	$0.260074\pi$
$978$	1.97031e13	0.688669
$979$	2.03773e13	0.708965
$980$	0	0
$981$	8.00968e12	0.276124
$982$	6.78901e12	0.232973
$983$	4.13652e13	1.41301	0.706504	−	0.707709i	$-0.250271\pi$
$983$	4.13652e13	1.41301	0.706504	+	0.707709i	$0.250271\pi$
$984$	−1.13787e13	−0.386913
$985$	2.26746e13	0.767497
$986$	−4.93265e12	−0.166201
$987$	0	0
$988$	4.29648e12	0.143452
$989$	−5.14096e13	−1.70868
$990$	1.73755e12	0.0574882
$991$	−2.23173e13	−0.735039	−0.367519	−	0.930016i	$-0.619793\pi$
$991$	−2.23173e13	−0.735039	−0.367519	+	0.930016i	$0.619793\pi$
$992$	−6.32558e12	−0.207395
$993$	−3.67837e13	−1.20056
$994$	0	0
$995$	2.00831e13	0.649572
$996$	4.27228e13	1.37560
$997$	−4.27668e13	−1.37081	−0.685406	−	0.728161i	$-0.740375\pi$
$997$	−4.27668e13	−1.37081	−0.685406	+	0.728161i	$0.740375\pi$
$998$	−3.25821e12	−0.103966
$999$	−8.36629e12	−0.265759

Display

a_p

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p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.10.a.m.1.6		13
7.2	even	3		35.10.e.b.11.8	✓	26
7.4	even	3		35.10.e.b.16.8	yes	26
7.6	odd	2		245.10.a.l.1.6		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.10.e.b.11.8	✓	26	7.2	even	3
35.10.e.b.16.8	yes	26	7.4	even	3
245.10.a.l.1.6		13	7.6	odd	2
245.10.a.m.1.6		13	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

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Fields

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Groups

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Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

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

Label	245.10.a.m.1.6

Level	$245$
Weight	$10$
Character	245.1
Self dual	yes
Analytic conductor	$126.184$
Analytic rank	$0$
Dimension	$13$
CM	no
Inner twists	$1$