LMFDB - Embedded newform 245.10.a.m.1.13

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			1.13
Root			$-45.0506$ 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)

$f(q)$	$=$	$q+45.0506 q^{2} +165.739 q^{3} +1517.55 q^{4} +625.000 q^{5} +7466.64 q^{6} +45300.8 q^{8} +7786.42 q^{9} +28156.6 q^{10} +20286.8 q^{11} +251518. q^{12} -22464.9 q^{13} +103587. q^{15} +1.26384e6 q^{16} +396961. q^{17} +350783. q^{18} -439499. q^{19} +948471. q^{20} +913934. q^{22} -1.20400e6 q^{23} +7.50811e6 q^{24} +390625. q^{25} -1.01206e6 q^{26} -1.97173e6 q^{27} +148802. q^{29} +4.66665e6 q^{30} -6.69581e6 q^{31} +3.37427e7 q^{32} +3.36232e6 q^{33} +1.78833e7 q^{34} +1.18163e7 q^{36} +9.49510e6 q^{37} -1.97997e7 q^{38} -3.72332e6 q^{39} +2.83130e7 q^{40} -7.55290e6 q^{41} -3.35061e7 q^{43} +3.07864e7 q^{44} +4.86651e6 q^{45} -5.42410e7 q^{46} +2.49724e7 q^{47} +2.09467e8 q^{48} +1.75979e7 q^{50} +6.57920e7 q^{51} -3.40918e7 q^{52} +6.69659e7 q^{53} -8.88275e7 q^{54} +1.26793e7 q^{55} -7.28422e7 q^{57} +6.70361e6 q^{58} -1.13339e8 q^{59} +1.57199e8 q^{60} -2.61045e7 q^{61} -3.01650e8 q^{62} +8.73041e8 q^{64} -1.40406e7 q^{65} +1.51474e8 q^{66} -4.31792e7 q^{67} +6.02410e8 q^{68} -1.99550e8 q^{69} +3.69341e8 q^{71} +3.52731e8 q^{72} -2.93142e8 q^{73} +4.27760e8 q^{74} +6.47418e7 q^{75} -6.66964e8 q^{76} -1.67738e8 q^{78} +4.61623e8 q^{79} +7.89899e8 q^{80} -4.80052e8 q^{81} -3.40263e8 q^{82} +7.40278e7 q^{83} +2.48101e8 q^{85} -1.50947e9 q^{86} +2.46623e7 q^{87} +9.19010e8 q^{88} -2.43277e8 q^{89} +2.19239e8 q^{90} -1.82714e9 q^{92} -1.10976e9 q^{93} +1.12502e9 q^{94} -2.74687e8 q^{95} +5.59247e9 q^{96} -8.76163e8 q^{97} +1.57962e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$13 q - q^{2} + 268 q^{3} + 3563 q^{4} + 8125 q^{5} + 3040 q^{6} - 4695 q^{8} + 82107 q^{9} - 625 q^{10} + 129087 q^{11} + 356068 q^{12} + 35889 q^{13} + 167500 q^{15} + 1379187 q^{16} + 251650 q^{17} + 391089 q^{18}+ \cdots + 5266142099 q^{99}+O(q^{100})$ 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

)

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

$2$	45.0506	1.99097	0.995486	−	0.0949041i	$-0.0302545\pi$
$2$	45.0506	1.99097	0.995486	+	0.0949041i	$0.0302545\pi$
$3$	165.739	1.18135	0.590676	−	0.806909i	$-0.298861\pi$
$3$	165.739	1.18135	0.590676	+	0.806909i	$0.298861\pi$
$4$	1517.55	2.96397
$5$	625.000	0.447214
$5$	625.000	0.447214
$6$	7466.64	2.35204
$7$	0	0
$7$	0	0
$8$	45300.8	3.91022
$9$	7786.42	0.395591
$10$	28156.6	0.890390
$11$	20286.8	0.417780	0.208890	−	0.977939i	$-0.433015\pi$
$11$	20286.8	0.417780	0.208890	+	0.977939i	$0.433015\pi$
$12$	251518.	3.50149
$13$	−22464.9	−0.218153	−0.109076	−	0.994033i	$-0.534789\pi$
$13$	−22464.9	−0.218153	−0.109076	+	0.994033i	$0.534789\pi$
$14$	0	0
$15$	103587.	0.528316
$16$	1.26384e6	4.82116
$17$	396961.	1.15273	0.576366	−	0.817192i	$-0.304470\pi$
$17$	396961.	1.15273	0.576366	+	0.817192i	$0.304470\pi$
$18$	350783.	0.787611
$19$	−439499.	−0.773690	−0.386845	−	0.922145i	$-0.626435\pi$
$19$	−439499.	−0.773690	−0.386845	+	0.922145i	$0.626435\pi$
$20$	948471.	1.32553
$21$	0	0
$22$	913934.	0.831788
$23$	−1.20400e6	−0.897123	−0.448561	−	0.893752i	$-0.648063\pi$
$23$	−1.20400e6	−0.897123	−0.448561	+	0.893752i	$0.648063\pi$
$24$	7.50811e6	4.61934
$25$	390625.	0.200000
$26$	−1.01206e6	−0.434336
$27$	−1.97173e6	−0.714019
$28$	0	0
$29$	148802.	0.0390677	0.0195338	−	0.999809i	$-0.493782\pi$
$29$	148802.	0.0390677	0.0195338	+	0.999809i	$0.493782\pi$
$30$	4.66665e6	1.05186
$31$	−6.69581e6	−1.30219	−0.651097	−	0.758995i	$-0.725691\pi$
$31$	−6.69581e6	−1.30219	−0.651097	+	0.758995i	$0.725691\pi$
$32$	3.37427e7	5.68859
$33$	3.36232e6	0.493545
$34$	1.78833e7	2.29506
$35$	0	0
$36$	1.18163e7	1.17252
$37$	9.49510e6	0.832898	0.416449	−	0.909159i	$-0.363275\pi$
$37$	9.49510e6	0.832898	0.416449	+	0.909159i	$0.363275\pi$
$38$	−1.97997e7	−1.54040
$39$	−3.72332e6	−0.257715
$40$	2.83130e7	1.74870
$41$	−7.55290e6	−0.417433	−0.208716	−	0.977976i	$-0.566929\pi$
$41$	−7.55290e6	−0.417433	−0.208716	+	0.977976i	$0.566929\pi$
$42$	0	0
$43$	−3.35061e7	−1.49457	−0.747284	−	0.664505i	$-0.768643\pi$
$43$	−3.35061e7	−1.49457	−0.747284	+	0.664505i	$0.768643\pi$
$44$	3.07864e7	1.23829
$45$	4.86651e6	0.176914
$46$	−5.42410e7	−1.78615
$47$	2.49724e7	0.746482	0.373241	−	0.927734i	$-0.378246\pi$
$47$	2.49724e7	0.746482	0.373241	+	0.927734i	$0.378246\pi$
$48$	2.09467e8	5.69549
$49$	0	0
$50$	1.75979e7	0.398195
$51$	6.57920e7	1.36178
$52$	−3.40918e7	−0.646598
$53$	6.69659e7	1.16577	0.582884	−	0.812555i	$-0.301924\pi$
$53$	6.69659e7	1.16577	0.582884	+	0.812555i	$0.301924\pi$
$54$	−8.88275e7	−1.42159
$55$	1.26793e7	0.186837
$56$	0	0
$57$	−7.28422e7	−0.913999
$58$	6.70361e6	0.0777827
$59$	−1.13339e8	−1.21772	−0.608858	−	0.793279i	$-0.708372\pi$
$59$	−1.13339e8	−1.21772	−0.608858	+	0.793279i	$0.708372\pi$
$60$	1.57199e8	1.56592
$61$	−2.61045e7	−0.241397	−0.120699	−	0.992689i	$-0.538513\pi$
$61$	−2.61045e7	−0.241397	−0.120699	+	0.992689i	$0.538513\pi$
$62$	−3.01650e8	−2.59263
$63$	0	0
$64$	8.73041e8	6.50466
$65$	−1.40406e7	−0.0975608
$66$	1.51474e8	0.982634
$67$	−4.31792e7	−0.261781	−0.130891	−	0.991397i	$-0.541784\pi$
$67$	−4.31792e7	−0.261781	−0.130891	+	0.991397i	$0.541784\pi$
$68$	6.02410e8	3.41666
$69$	−1.99550e8	−1.05982
$70$	0	0
$71$	3.69341e8	1.72491	0.862453	−	0.506138i	$-0.168927\pi$
$71$	3.69341e8	1.72491	0.862453	+	0.506138i	$0.168927\pi$
$72$	3.52731e8	1.54685
$73$	−2.93142e8	−1.20816	−0.604082	−	0.796922i	$-0.706460\pi$
$73$	−2.93142e8	−1.20816	−0.604082	+	0.796922i	$0.706460\pi$
$74$	4.27760e8	1.65828
$75$	6.47418e7	0.236270
$76$	−6.66964e8	−2.29320
$77$	0	0
$78$	−1.67738e8	−0.513103
$79$	4.61623e8	1.33342	0.666708	−	0.745319i	$-0.267703\pi$
$79$	4.61623e8	1.33342	0.666708	+	0.745319i	$0.267703\pi$
$80$	7.89899e8	2.15609
$81$	−4.80052e8	−1.23910
$82$	−3.40263e8	−0.831098
$83$	7.40278e7	0.171216	0.0856078	−	0.996329i	$-0.472717\pi$
$83$	7.40278e7	0.171216	0.0856078	+	0.996329i	$0.472717\pi$
$84$	0	0
$85$	2.48101e8	0.515517
$86$	−1.50947e9	−2.97564
$87$	2.46623e7	0.0461527
$88$	9.19010e8	1.63361
$89$	−2.43277e8	−0.411004	−0.205502	−	0.978657i	$-0.565883\pi$
$89$	−2.43277e8	−0.411004	−0.205502	+	0.978657i	$0.565883\pi$
$90$	2.19239e8	0.352230
$91$	0	0
$92$	−1.82714e9	−2.65905
$93$	−1.10976e9	−1.53835
$94$	1.12502e9	1.48623
$95$	−2.74687e8	−0.346005
$96$	5.59247e9	6.72022
$97$	−8.76163e8	−1.00488	−0.502438	−	0.864613i	$-0.667563\pi$
$97$	−8.76163e8	−1.00488	−0.502438	+	0.864613i	$0.667563\pi$
$98$	0	0
$99$	1.57962e8	0.165270
$100$	5.92795e8	0.592795
$101$	−5.66571e8	−0.541762	−0.270881	−	0.962613i	$-0.587315\pi$
$101$	−5.66571e8	−0.541762	−0.270881	+	0.962613i	$0.587315\pi$
$102$	2.96397e9	2.71127
$103$	−1.07819e8	−0.0943900	−0.0471950	−	0.998886i	$-0.515028\pi$
$103$	−1.07819e8	−0.0943900	−0.0471950	+	0.998886i	$0.515028\pi$
$104$	−1.01768e9	−0.853024
$105$	0	0
$106$	3.01685e9	2.32101
$107$	−1.01951e9	−0.751905	−0.375952	−	0.926639i	$-0.622684\pi$
$107$	−1.01951e9	−0.751905	−0.375952	+	0.926639i	$0.622684\pi$
$108$	−2.99220e9	−2.11633
$109$	1.13937e9	0.773120	0.386560	−	0.922264i	$-0.373663\pi$
$109$	1.13937e9	0.773120	0.386560	+	0.922264i	$0.373663\pi$
$110$	5.71209e8	0.371987
$111$	1.57371e9	0.983945
$112$	0	0
$113$	−8.79575e8	−0.507481	−0.253740	−	0.967272i	$-0.581661\pi$
$113$	−8.79575e8	−0.507481	−0.253740	+	0.967272i	$0.581661\pi$
$114$	−3.28158e9	−1.81975
$115$	−7.52501e8	−0.401205
$116$	2.25815e8	0.115796
$117$	−1.74921e8	−0.0862992
$118$	−5.10599e9	−2.42444
$119$	0	0
$120$	4.69257e9	2.06583
$121$	−1.94639e9	−0.825460
$122$	−1.17602e9	−0.480615
$123$	−1.25181e9	−0.493135
$124$	−1.01613e10	−3.85967
$125$	2.44141e8	0.0894427
$126$	0	0
$127$	1.96913e9	0.671673	0.335837	−	0.941920i	$-0.390981\pi$
$127$	1.96913e9	0.671673	0.335837	+	0.941920i	$0.390981\pi$
$128$	2.20547e10	7.26201
$129$	−5.55327e9	−1.76561
$130$	−6.32537e8	−0.194241
$131$	4.37181e9	1.29700	0.648500	−	0.761214i	$-0.275397\pi$
$131$	4.37181e9	1.29700	0.648500	+	0.761214i	$0.275397\pi$
$132$	5.10250e9	1.46285
$133$	0	0
$134$	−1.94525e9	−0.521199
$135$	−1.23233e9	−0.319319
$136$	1.79827e10	4.50743
$137$	2.15436e9	0.522488	0.261244	−	0.965273i	$-0.415867\pi$
$137$	2.15436e9	0.522488	0.261244	+	0.965273i	$0.415867\pi$
$138$	−8.98984e9	−2.11007
$139$	−2.17041e9	−0.493147	−0.246573	−	0.969124i	$-0.579305\pi$
$139$	−2.17041e9	−0.493147	−0.246573	+	0.969124i	$0.579305\pi$
$140$	0	0
$141$	4.13890e9	0.881858
$142$	1.66390e10	3.43424
$143$	−4.55743e8	−0.0911397
$144$	9.84077e9	1.90721
$145$	9.30012e7	0.0174716
$146$	−1.32062e10	−2.40542
$147$	0	0
$148$	1.44093e10	2.46869
$149$	−8.79035e8	−0.146106	−0.0730530	−	0.997328i	$-0.523274\pi$
$149$	−8.79035e8	−0.146106	−0.0730530	+	0.997328i	$0.523274\pi$
$150$	2.91666e9	0.470408
$151$	−1.35164e9	−0.211575	−0.105787	−	0.994389i	$-0.533736\pi$
$151$	−1.35164e9	−0.211575	−0.105787	+	0.994389i	$0.533736\pi$
$152$	−1.99097e10	−3.02529
$153$	3.09091e9	0.456010
$154$	0	0
$155$	−4.18488e9	−0.582359
$156$	−5.65034e9	−0.763860
$157$	−4.72740e9	−0.620975	−0.310487	−	0.950577i	$-0.600492\pi$
$157$	−4.72740e9	−0.620975	−0.310487	+	0.950577i	$0.600492\pi$
$158$	2.07964e10	2.65480
$159$	1.10989e10	1.37718
$160$	2.10892e10	2.54401
$161$	0	0
$162$	−2.16266e10	−2.46701
$163$	−4.66247e9	−0.517335	−0.258667	−	0.965966i	$-0.583283\pi$
$163$	−4.66247e9	−0.517335	−0.258667	+	0.965966i	$0.583283\pi$
$164$	−1.14619e10	−1.23726
$165$	2.10145e9	0.220720
$166$	3.33499e9	0.340886
$167$	5.99996e9	0.596932	0.298466	−	0.954420i	$-0.403525\pi$
$167$	5.99996e9	0.596932	0.298466	+	0.954420i	$0.403525\pi$
$168$	0	0
$169$	−1.00998e10	−0.952409
$170$	1.11771e10	1.02638
$171$	−3.42212e9	−0.306065
$172$	−5.08473e10	−4.42986
$173$	3.11408e9	0.264315	0.132158	−	0.991229i	$-0.457809\pi$
$173$	3.11408e9	0.264315	0.132158	+	0.991229i	$0.457809\pi$
$174$	1.11105e9	0.0918887
$175$	0	0
$176$	2.56393e10	2.01418
$177$	−1.87847e10	−1.43855
$178$	−1.09598e10	−0.818299
$179$	−2.78648e9	−0.202870	−0.101435	−	0.994842i	$-0.532343\pi$
$179$	−2.78648e9	−0.202870	−0.101435	+	0.994842i	$0.532343\pi$
$180$	7.38519e9	0.524367
$181$	−1.26160e10	−0.873714	−0.436857	−	0.899531i	$-0.643908\pi$
$181$	−1.26160e10	−0.873714	−0.436857	+	0.899531i	$0.643908\pi$
$182$	0	0
$183$	−4.32654e9	−0.285175
$184$	−5.45422e10	−3.50794
$185$	5.93444e9	0.372483
$186$	−4.99952e10	−3.06281
$187$	8.05309e9	0.481588
$188$	3.78969e10	2.21255
$189$	0	0
$190$	−1.23748e10	−0.688886
$191$	1.40201e10	0.762256	0.381128	−	0.924522i	$-0.375536\pi$
$191$	1.40201e10	0.762256	0.381128	+	0.924522i	$0.375536\pi$
$192$	1.44697e11	7.68429
$193$	1.48730e10	0.771598	0.385799	−	0.922583i	$-0.373926\pi$
$193$	1.48730e10	0.771598	0.385799	+	0.922583i	$0.373926\pi$
$194$	−3.94717e10	−2.00068
$195$	−2.32707e9	−0.115254
$196$	0	0
$197$	2.76254e9	0.130680	0.0653401	−	0.997863i	$-0.479187\pi$
$197$	2.76254e9	0.130680	0.0653401	+	0.997863i	$0.479187\pi$
$198$	7.11627e9	0.329048
$199$	−9.38188e9	−0.424083	−0.212042	−	0.977261i	$-0.568011\pi$
$199$	−9.38188e9	−0.424083	−0.212042	+	0.977261i	$0.568011\pi$
$200$	1.76956e10	0.782043
$201$	−7.15648e9	−0.309255
$202$	−2.55243e10	−1.07863
$203$	0	0
$204$	9.98429e10	4.03628
$205$	−4.72057e9	−0.186682
$206$	−4.85729e9	−0.187928
$207$	−9.37486e9	−0.354894
$208$	−2.83921e10	−1.05175
$209$	−8.91605e9	−0.323232
$210$	0	0
$211$	−4.23543e10	−1.47105	−0.735523	−	0.677500i	$-0.763063\pi$
$211$	−4.23543e10	−1.47105	−0.735523	+	0.677500i	$0.763063\pi$
$212$	1.01624e11	3.45531
$213$	6.12142e10	2.03772
$214$	−4.59293e10	−1.49702
$215$	−2.09413e10	−0.668391
$216$	−8.93208e10	−2.79197
$217$	0	0
$218$	5.13294e10	1.53926
$219$	−4.85851e10	−1.42727
$220$	1.92415e10	0.553779
$221$	−8.91772e9	−0.251471
$222$	7.08964e10	1.95901
$223$	−1.86261e10	−0.504372	−0.252186	−	0.967679i	$-0.581149\pi$
$223$	−1.86261e10	−0.504372	−0.252186	+	0.967679i	$0.581149\pi$
$224$	0	0
$225$	3.04157e9	0.0791182
$226$	−3.96253e10	−1.01038
$227$	4.92185e10	1.23030	0.615152	−	0.788409i	$-0.289095\pi$
$227$	4.92185e10	1.23030	0.615152	+	0.788409i	$0.289095\pi$
$228$	−1.10542e11	−2.70907
$229$	−1.26986e9	−0.0305139	−0.0152570	−	0.999884i	$-0.504857\pi$
$229$	−1.26986e9	−0.0305139	−0.0152570	+	0.999884i	$0.504857\pi$
$230$	−3.39006e10	−0.798789
$231$	0	0
$232$	6.74085e9	0.152763
$233$	−4.60301e10	−1.02315	−0.511576	−	0.859238i	$-0.670938\pi$
$233$	−4.60301e10	−1.02315	−0.511576	+	0.859238i	$0.670938\pi$
$234$	−7.88031e9	−0.171819
$235$	1.56077e10	0.333837
$236$	−1.71998e11	−3.60928
$237$	7.65090e10	1.57523
$238$	0	0
$239$	−3.22863e9	−0.0640071	−0.0320035	−	0.999488i	$-0.510189\pi$
$239$	−3.22863e9	−0.0640071	−0.0320035	+	0.999488i	$0.510189\pi$
$240$	1.30917e11	2.54710
$241$	−6.81778e10	−1.30187	−0.650933	−	0.759135i	$-0.725622\pi$
$241$	−6.81778e10	−1.30187	−0.650933	+	0.759135i	$0.725622\pi$
$242$	−8.76861e10	−1.64347
$243$	−4.07539e10	−0.749791
$244$	−3.96151e10	−0.715494
$245$	0	0
$246$	−5.63948e10	−0.981818
$247$	9.87332e9	0.168782
$248$	−3.03326e11	−5.09186
$249$	1.22693e10	0.202266
$250$	1.09987e10	0.178078
$251$	1.07251e11	1.70557	0.852787	−	0.522259i	$-0.174910\pi$
$251$	1.07251e11	1.70557	0.852787	+	0.522259i	$0.174910\pi$
$252$	0	0
$253$	−2.44254e10	−0.374800
$254$	8.87105e10	1.33728
$255$	4.11200e10	0.609007
$256$	5.46582e11	7.95381
$257$	1.29031e11	1.84499	0.922494	−	0.386010i	$-0.126147\pi$
$257$	1.29031e11	1.84499	0.922494	+	0.386010i	$0.126147\pi$
$258$	−2.50178e11	−3.51528
$259$	0	0
$260$	−2.13074e10	−0.289168
$261$	1.15863e9	0.0154548
$262$	1.96952e11	2.58229
$263$	6.04752e10	0.779429	0.389714	−	0.920936i	$-0.372574\pi$
$263$	6.04752e10	0.779429	0.389714	+	0.920936i	$0.372574\pi$
$264$	1.52316e11	1.92987
$265$	4.18537e10	0.521347
$266$	0	0
$267$	−4.03205e10	−0.485541
$268$	−6.55268e10	−0.775912
$269$	4.88251e10	0.568536	0.284268	−	0.958745i	$-0.408250\pi$
$269$	4.88251e10	0.568536	0.284268	+	0.958745i	$0.408250\pi$
$270$	−5.55172e10	−0.635756
$271$	1.36973e11	1.54267	0.771335	−	0.636429i	$-0.219589\pi$
$271$	1.36973e11	1.54267	0.771335	+	0.636429i	$0.219589\pi$
$272$	5.01695e11	5.55750
$273$	0	0
$274$	9.70553e10	1.04026
$275$	7.92455e9	0.0835559
$276$	−3.02828e11	−3.14127
$277$	−4.92371e10	−0.502497	−0.251248	−	0.967923i	$-0.580841\pi$
$277$	−4.92371e10	−0.502497	−0.251248	+	0.967923i	$0.580841\pi$
$278$	−9.77784e10	−0.981842
$279$	−5.21364e10	−0.515136
$280$	0	0
$281$	4.52296e10	0.432757	0.216378	−	0.976310i	$-0.430576\pi$
$281$	4.52296e10	0.432757	0.216378	+	0.976310i	$0.430576\pi$
$282$	1.86460e11	1.75575
$283$	1.07440e11	0.995697	0.497848	−	0.867264i	$-0.334124\pi$
$283$	1.07440e11	0.995697	0.497848	+	0.867264i	$0.334124\pi$
$284$	5.60495e11	5.11257
$285$	−4.55263e10	−0.408753
$286$	−2.05315e10	−0.181457
$287$	0	0
$288$	2.62734e11	2.25035
$289$	3.89904e10	0.328789
$290$	4.18976e9	0.0347855
$291$	−1.45214e11	−1.18711
$292$	−4.44859e11	−3.58096
$293$	−4.03110e10	−0.319535	−0.159768	−	0.987155i	$-0.551074\pi$
$293$	−4.03110e10	−0.319535	−0.159768	+	0.987155i	$0.551074\pi$
$294$	0	0
$295$	−7.08370e10	−0.544579
$296$	4.30135e11	3.25681
$297$	−4.00001e10	−0.298303
$298$	−3.96010e10	−0.290893
$299$	2.70478e10	0.195710
$300$	9.82492e10	0.700299
$301$	0	0
$302$	−6.08920e10	−0.421239
$303$	−9.39029e10	−0.640011
$304$	−5.55456e11	−3.73008
$305$	−1.63153e10	−0.107956
$306$	1.39247e11	0.907904
$307$	−2.24789e11	−1.44428	−0.722142	−	0.691745i	$-0.756842\pi$
$307$	−2.24789e11	−1.44428	−0.722142	+	0.691745i	$0.756842\pi$
$308$	0	0
$309$	−1.78697e10	−0.111508
$310$	−1.88531e11	−1.15946
$311$	1.03200e11	0.625544	0.312772	−	0.949828i	$-0.398743\pi$
$311$	1.03200e11	0.625544	0.312772	+	0.949828i	$0.398743\pi$
$312$	−1.68669e11	−1.00772
$313$	1.41481e11	0.833196	0.416598	−	0.909091i	$-0.363222\pi$
$313$	1.41481e11	0.833196	0.416598	+	0.909091i	$0.363222\pi$
$314$	−2.12972e11	−1.23634
$315$	0	0
$316$	7.00539e11	3.95221
$317$	−9.69129e10	−0.539032	−0.269516	−	0.962996i	$-0.586864\pi$
$317$	−9.69129e10	−0.539032	−0.269516	+	0.962996i	$0.586864\pi$
$318$	5.00010e11	2.74193
$319$	3.01872e9	0.0163217
$320$	5.45650e11	2.90897
$321$	−1.68972e11	−0.888264
$322$	0	0
$323$	−1.74464e11	−0.891856
$324$	−7.28505e11	−3.67266
$325$	−8.77537e9	−0.0436305
$326$	−2.10047e11	−1.03000
$327$	1.88839e11	0.913326
$328$	−3.42153e11	−1.63225
$329$	0	0
$330$	9.46716e10	0.439447
$331$	5.83011e10	0.266963	0.133481	−	0.991051i	$-0.457384\pi$
$331$	5.83011e10	0.266963	0.133481	+	0.991051i	$0.457384\pi$
$332$	1.12341e11	0.507478
$333$	7.39328e10	0.329487
$334$	2.70302e11	1.18847
$335$	−2.69870e10	−0.117072
$336$	0	0
$337$	4.50513e10	0.190271	0.0951356	−	0.995464i	$-0.469672\pi$
$337$	4.50513e10	0.190271	0.0951356	+	0.995464i	$0.469672\pi$
$338$	−4.55003e11	−1.89622
$339$	−1.45780e11	−0.599513
$340$	3.76506e11	1.52798
$341$	−1.35837e11	−0.544030
$342$	−1.54169e11	−0.609366
$343$	0	0
$344$	−1.51785e12	−5.84409
$345$	−1.24719e11	−0.473965
$346$	1.40291e11	0.526245
$347$	−1.58277e11	−0.586050	−0.293025	−	0.956105i	$-0.594662\pi$
$347$	−1.58277e11	−0.586050	−0.293025	+	0.956105i	$0.594662\pi$
$348$	3.74264e10	0.136795
$349$	1.97806e11	0.713715	0.356858	−	0.934159i	$-0.383848\pi$
$349$	1.97806e11	0.713715	0.356858	+	0.934159i	$0.383848\pi$
$350$	0	0
$351$	4.42948e10	0.155765
$352$	6.84532e11	2.37658
$353$	3.16675e10	0.108550	0.0542748	−	0.998526i	$-0.482715\pi$
$353$	3.16675e10	0.108550	0.0542748	+	0.998526i	$0.482715\pi$
$354$	−8.46262e11	−2.86411
$355$	2.30838e11	0.771401
$356$	−3.69186e11	−1.21821
$357$	0	0
$358$	−1.25532e11	−0.403908
$359$	−2.51531e11	−0.799220	−0.399610	−	0.916685i	$-0.630855\pi$
$359$	−2.51531e11	−0.799220	−0.399610	+	0.916685i	$0.630855\pi$
$360$	2.20457e11	0.691771
$361$	−1.29528e11	−0.401404
$362$	−5.68360e11	−1.73954
$363$	−3.22593e11	−0.975158
$364$	0	0
$365$	−1.83214e11	−0.540307
$366$	−1.94913e11	−0.567775
$367$	−6.72529e11	−1.93514	−0.967572	−	0.252594i	$-0.918716\pi$
$367$	−6.72529e11	−1.93514	−0.967572	+	0.252594i	$0.918716\pi$
$368$	−1.52166e12	−4.32517
$369$	−5.88101e10	−0.165133
$370$	2.67350e11	0.741604
$371$	0	0
$372$	−1.68412e12	−4.55962
$373$	−5.99806e11	−1.60443	−0.802216	−	0.597034i	$-0.796346\pi$
$373$	−5.99806e11	−1.60443	−0.802216	+	0.597034i	$0.796346\pi$
$374$	3.62796e11	0.958828
$375$	4.04636e10	0.105663
$376$	1.13127e12	2.91891
$377$	−3.34283e9	−0.00852271
$378$	0	0
$379$	6.38588e11	1.58981	0.794904	−	0.606735i	$-0.207521\pi$
$379$	6.38588e11	1.58981	0.794904	+	0.606735i	$0.207521\pi$
$380$	−4.16852e11	−1.02555
$381$	3.26362e11	0.793482
$382$	6.31613e11	1.51763
$383$	−5.10218e11	−1.21161	−0.605803	−	0.795615i	$-0.707148\pi$
$383$	−5.10218e11	−1.21161	−0.605803	+	0.795615i	$0.707148\pi$
$384$	3.65533e12	8.57899
$385$	0	0
$386$	6.70038e11	1.53623
$387$	−2.60892e11	−0.591238
$388$	−1.32963e12	−2.97842
$389$	1.18918e11	0.263315	0.131658	−	0.991295i	$-0.457970\pi$
$389$	1.18918e11	0.263315	0.131658	+	0.991295i	$0.457970\pi$
$390$	−1.04836e11	−0.229467
$391$	−4.77942e11	−1.03414
$392$	0	0
$393$	7.24579e11	1.53221
$394$	1.24454e11	0.260181
$395$	2.88515e11	0.596322
$396$	2.39716e11	0.489855
$397$	−2.80888e11	−0.567513	−0.283756	−	0.958896i	$-0.591581\pi$
$397$	−2.80888e11	−0.567513	−0.283756	+	0.958896i	$0.591581\pi$
$398$	−4.22659e11	−0.844338
$399$	0	0
$400$	4.93687e11	0.964232
$401$	−8.32505e11	−1.60782	−0.803910	−	0.594752i	$-0.797250\pi$
$401$	−8.32505e11	−1.60782	−0.803910	+	0.594752i	$0.797250\pi$
$402$	−3.22404e11	−0.615719
$403$	1.50421e11	0.284077
$404$	−8.59802e11	−1.60577
$405$	−3.00033e11	−0.554142
$406$	0	0
$407$	1.92626e11	0.347968
$408$	2.98043e12	5.32486
$409$	4.62186e11	0.816698	0.408349	−	0.912826i	$-0.366105\pi$
$409$	4.62186e11	0.816698	0.408349	+	0.912826i	$0.366105\pi$
$410$	−2.12664e11	−0.371678
$411$	3.57062e11	0.617242
$412$	−1.63621e11	−0.279770
$413$	0	0
$414$	−4.22343e11	−0.706583
$415$	4.62674e10	0.0765699
$416$	−7.58027e11	−1.24098
$417$	−3.59722e11	−0.582579
$418$	−4.01673e11	−0.643546
$419$	−8.42911e11	−1.33604	−0.668019	−	0.744144i	$-0.732858\pi$
$419$	−8.42911e11	−1.33604	−0.668019	+	0.744144i	$0.732858\pi$
$420$	0	0
$421$	−5.73267e11	−0.889380	−0.444690	−	0.895684i	$-0.646686\pi$
$421$	−5.73267e11	−0.889380	−0.444690	+	0.895684i	$0.646686\pi$
$422$	−1.90808e12	−2.92881
$423$	1.94445e11	0.295302
$424$	3.03361e12	4.55841
$425$	1.55063e11	0.230546
$426$	2.75774e12	4.05704
$427$	0	0
$428$	−1.54716e12	−2.22863
$429$	−7.55344e10	−0.107668
$430$	−9.43418e11	−1.33075
$431$	−4.81918e9	−0.00672707	−0.00336353	−	0.999994i	$-0.501071\pi$
$431$	−4.81918e9	−0.00672707	−0.00336353	+	0.999994i	$0.501071\pi$
$432$	−2.49195e12	−3.44240
$433$	−1.62755e11	−0.222505	−0.111252	−	0.993792i	$-0.535486\pi$
$433$	−1.62755e11	−0.222505	−0.111252	+	0.993792i	$0.535486\pi$
$434$	0	0
$435$	1.54139e10	0.0206401
$436$	1.72906e12	2.29151
$437$	5.29158e11	0.694095
$438$	−2.18879e12	−2.84165
$439$	2.96307e11	0.380760	0.190380	−	0.981710i	$-0.439028\pi$
$439$	2.96307e11	0.380760	0.190380	+	0.981710i	$0.439028\pi$
$440$	5.74381e11	0.730572
$441$	0	0
$442$	−4.01748e11	−0.500672
$443$	1.18123e12	1.45720	0.728600	−	0.684940i	$-0.240172\pi$
$443$	1.18123e12	1.45720	0.728600	+	0.684940i	$0.240172\pi$
$444$	2.38819e12	2.91639
$445$	−1.52048e11	−0.183807
$446$	−8.39118e11	−1.00419
$447$	−1.45690e11	−0.172602
$448$	0	0
$449$	4.11866e11	0.478242	0.239121	−	0.970990i	$-0.423141\pi$
$449$	4.11866e11	0.478242	0.239121	+	0.970990i	$0.423141\pi$
$450$	1.37024e11	0.157522
$451$	−1.53225e11	−0.174395
$452$	−1.33480e12	−1.50416
$453$	−2.24019e11	−0.249944
$454$	2.21732e12	2.44950
$455$	0	0
$456$	−3.29981e12	−3.57394
$457$	−8.72606e11	−0.935826	−0.467913	−	0.883774i	$-0.654994\pi$
$457$	−8.72606e11	−0.935826	−0.467913	+	0.883774i	$0.654994\pi$
$458$	−5.72081e10	−0.0607524
$459$	−7.82700e11	−0.823073
$460$	−1.14196e12	−1.18916
$461$	−1.60405e12	−1.65411	−0.827055	−	0.562121i	$-0.809985\pi$
$461$	−1.60405e12	−1.65411	−0.827055	+	0.562121i	$0.809985\pi$
$462$	0	0
$463$	−3.46408e10	−0.0350327	−0.0175163	−	0.999847i	$-0.505576\pi$
$463$	−3.46408e10	−0.0350327	−0.0175163	+	0.999847i	$0.505576\pi$
$464$	1.88062e11	0.188352
$465$	−6.93598e11	−0.687970
$466$	−2.07368e12	−2.03707
$467$	1.62367e12	1.57969	0.789847	−	0.613304i	$-0.210160\pi$
$467$	1.62367e12	1.57969	0.789847	+	0.613304i	$0.210160\pi$
$468$	−2.65453e11	−0.255788
$469$	0	0
$470$	7.03137e11	0.664660
$471$	−7.83515e11	−0.733589
$472$	−5.13435e12	−4.76153
$473$	−6.79733e11	−0.624400
$474$	3.44677e12	3.13625
$475$	−1.71679e11	−0.154738
$476$	0	0
$477$	5.21424e11	0.461167
$478$	−1.45452e11	−0.127436
$479$	1.35039e12	1.17206	0.586030	−	0.810289i	$-0.300690\pi$
$479$	1.35039e12	1.17206	0.586030	+	0.810289i	$0.300690\pi$
$480$	3.49530e12	3.00537
$481$	−2.13307e11	−0.181699
$482$	−3.07145e12	−2.59198
$483$	0	0
$484$	−2.95375e12	−2.44664
$485$	−5.47602e11	−0.449394
$486$	−1.83598e12	−1.49281
$487$	4.90765e11	0.395361	0.197680	−	0.980267i	$-0.436659\pi$
$487$	4.90765e11	0.395361	0.197680	+	0.980267i	$0.436659\pi$
$488$	−1.18256e12	−0.943915
$489$	−7.72753e11	−0.611154
$490$	0	0
$491$	−9.73443e11	−0.755865	−0.377932	−	0.925833i	$-0.623365\pi$
$491$	−9.73443e11	−0.755865	−0.377932	+	0.925833i	$0.623365\pi$
$492$	−1.89969e12	−1.46164
$493$	5.90686e10	0.0450345
$494$	4.44799e11	0.336041
$495$	9.87261e10	0.0739109
$496$	−8.46243e12	−6.27809
$497$	0	0
$498$	5.52739e11	0.402706
$499$	1.54629e12	1.11645	0.558223	−	0.829691i	$-0.311484\pi$
$499$	1.54629e12	1.11645	0.558223	+	0.829691i	$0.311484\pi$
$500$	3.70497e11	0.265106
$501$	9.94428e11	0.705186
$502$	4.83173e12	3.39575
$503$	−8.19908e11	−0.571096	−0.285548	−	0.958364i	$-0.592176\pi$
$503$	−8.19908e11	−0.571096	−0.285548	+	0.958364i	$0.592176\pi$
$504$	0	0
$505$	−3.54107e11	−0.242283
$506$	−1.10038e12	−0.746216
$507$	−1.67393e12	−1.12513
$508$	2.98826e12	1.99082
$509$	−1.49021e12	−0.984048	−0.492024	−	0.870582i	$-0.663743\pi$
$509$	−1.49021e12	−0.984048	−0.492024	+	0.870582i	$0.663743\pi$
$510$	1.85248e12	1.21252
$511$	0	0
$512$	1.33318e13	8.57381
$513$	8.66573e11	0.552430
$514$	5.81290e12	3.67332
$515$	−6.73866e10	−0.0422125
$516$	−8.42738e12	−5.23322
$517$	5.06611e11	0.311865
$518$	0	0
$519$	5.16125e11	0.312249
$520$	−6.36050e11	−0.381484
$521$	2.71301e12	1.61318	0.806588	−	0.591114i	$-0.201312\pi$
$521$	2.71301e12	1.61318	0.806588	+	0.591114i	$0.201312\pi$
$522$	5.21971e10	0.0307701
$523$	2.77389e12	1.62118	0.810590	−	0.585613i	$-0.199146\pi$
$523$	2.77389e12	1.62118	0.810590	+	0.585613i	$0.199146\pi$
$524$	6.63445e12	3.84427
$525$	0	0
$526$	2.72444e12	1.55182
$527$	−2.65798e12	−1.50108
$528$	4.24943e12	2.37946
$529$	−3.51533e11	−0.195171
$530$	1.88553e12	1.03799
$531$	−8.82506e11	−0.481717
$532$	0	0
$533$	1.69676e11	0.0910640
$534$	−1.81646e12	−0.966698
$535$	−6.37191e11	−0.336262
$536$	−1.95605e12	−1.02362
$537$	−4.61828e11	−0.239660
$538$	2.19960e12	1.13194
$539$	0	0
$540$	−1.87013e12	−0.946453
$541$	1.38439e12	0.694816	0.347408	−	0.937714i	$-0.387062\pi$
$541$	1.38439e12	0.694816	0.347408	+	0.937714i	$0.387062\pi$
$542$	6.17071e12	3.07141
$543$	−2.09097e12	−1.03216
$544$	1.33945e13	6.55741
$545$	7.12108e11	0.345750
$546$	0	0
$547$	−3.12767e12	−1.49375	−0.746876	−	0.664964i	$-0.768447\pi$
$547$	−3.12767e12	−1.49375	−0.746876	+	0.664964i	$0.768447\pi$
$548$	3.26936e12	1.54864
$549$	−2.03261e11	−0.0954945
$550$	3.57005e11	0.166358
$551$	−6.53983e10	−0.0302263
$552$	−9.03977e12	−4.14411
$553$	0	0
$554$	−2.21816e12	−1.00046
$555$	9.83567e11	0.440033
$556$	−3.29372e12	−1.46167
$557$	−6.70472e11	−0.295143	−0.147571	−	0.989051i	$-0.547146\pi$
$557$	−6.70472e11	−0.295143	−0.147571	+	0.989051i	$0.547146\pi$
$558$	−2.34877e12	−1.02562
$559$	7.52713e11	0.326044
$560$	0	0
$561$	1.33471e12	0.568924
$562$	2.03762e12	0.861607
$563$	8.15317e11	0.342010	0.171005	−	0.985270i	$-0.445299\pi$
$563$	8.15317e11	0.342010	0.171005	+	0.985270i	$0.445299\pi$
$564$	6.28100e12	2.61380
$565$	−5.49734e11	−0.226952
$566$	4.84023e12	1.98240
$567$	0	0
$568$	1.67314e13	6.74475
$569$	2.60733e11	0.104278	0.0521388	−	0.998640i	$-0.483396\pi$
$569$	2.60733e11	0.104278	0.0521388	+	0.998640i	$0.483396\pi$
$570$	−2.05099e12	−0.813816
$571$	1.20991e12	0.476312	0.238156	−	0.971227i	$-0.423457\pi$
$571$	1.20991e12	0.476312	0.238156	+	0.971227i	$0.423457\pi$
$572$	−6.91614e11	−0.270136
$573$	2.32368e12	0.900492
$574$	0	0
$575$	−4.70313e11	−0.179425
$576$	6.79786e12	2.57318
$577$	−1.46209e12	−0.549139	−0.274570	−	0.961567i	$-0.588535\pi$
$577$	−1.46209e12	−0.549139	−0.274570	+	0.961567i	$0.588535\pi$
$578$	1.75654e12	0.654611
$579$	2.46504e12	0.911529
$580$	1.41134e11	0.0517853
$581$	0	0
$582$	−6.54199e12	−2.36351
$583$	1.35853e12	0.487034
$584$	−1.32796e13	−4.72418
$585$	−1.09326e11	−0.0385942
$586$	−1.81603e12	−0.636186
$587$	5.40641e12	1.87948	0.939740	−	0.341889i	$-0.111067\pi$
$587$	5.40641e12	1.87948	0.939740	+	0.341889i	$0.111067\pi$
$588$	0	0
$589$	2.94280e12	1.00749
$590$	−3.19125e12	−1.08424
$591$	4.57860e11	0.154379
$592$	1.20003e13	4.01553
$593$	1.61738e11	0.0537114	0.0268557	−	0.999639i	$-0.491451\pi$
$593$	1.61738e11	0.0537114	0.0268557	+	0.999639i	$0.491451\pi$
$594$	−1.80203e12	−0.593913
$595$	0	0
$596$	−1.33398e12	−0.433054
$597$	−1.55494e12	−0.500991
$598$	1.21852e12	0.389652
$599$	4.70677e12	1.49383	0.746917	−	0.664917i	$-0.231533\pi$
$599$	4.70677e12	1.49383	0.746917	+	0.664917i	$0.231533\pi$
$600$	2.93285e12	0.923868
$601$	−4.13046e12	−1.29141	−0.645704	−	0.763588i	$-0.723436\pi$
$601$	−4.13046e12	−1.29141	−0.645704	+	0.763588i	$0.723436\pi$
$602$	0	0
$603$	−3.36211e11	−0.103558
$604$	−2.05118e12	−0.627101
$605$	−1.21649e12	−0.369157
$606$	−4.23038e12	−1.27424
$607$	−3.69599e12	−1.10505	−0.552524	−	0.833497i	$-0.686335\pi$
$607$	−3.69599e12	−1.10505	−0.552524	+	0.833497i	$0.686335\pi$
$608$	−1.48299e13	−4.40120
$609$	0	0
$610$	−7.35015e11	−0.214938
$611$	−5.61003e11	−0.162847
$612$	4.69062e12	1.35160
$613$	−3.11060e12	−0.889758	−0.444879	−	0.895591i	$-0.646753\pi$
$613$	−3.11060e12	−0.889758	−0.444879	+	0.895591i	$0.646753\pi$
$614$	−1.01269e13	−2.87553
$615$	−7.82382e11	−0.220537
$616$	0	0
$617$	2.55933e11	0.0710956	0.0355478	−	0.999368i	$-0.488682\pi$
$617$	2.55933e11	0.0710956	0.0355478	+	0.999368i	$0.488682\pi$
$618$	−8.05042e11	−0.222009
$619$	8.95234e11	0.245092	0.122546	−	0.992463i	$-0.460894\pi$
$619$	8.95234e11	0.245092	0.122546	+	0.992463i	$0.460894\pi$
$620$	−6.35079e12	−1.72610
$621$	2.37396e12	0.640563
$622$	4.64921e12	1.24544
$623$	0	0
$624$	−4.70567e12	−1.24248
$625$	1.52588e11	0.0400000
$626$	6.37378e12	1.65887
$627$	−1.47774e12	−0.381850
$628$	−7.17409e12	−1.84055
$629$	3.76919e12	0.960107
$630$	0	0
$631$	7.24421e11	0.181911	0.0909554	−	0.995855i	$-0.471008\pi$
$631$	7.24421e11	0.181911	0.0909554	+	0.995855i	$0.471008\pi$
$632$	2.09119e13	5.21395
$633$	−7.01975e12	−1.73782
$634$	−4.36598e12	−1.07320
$635$	1.23071e12	0.300381
$636$	1.68431e13	4.08193
$637$	0	0
$638$	1.35995e11	0.0324960
$639$	2.87584e12	0.682357
$640$	1.37842e13	3.24767
$641$	8.35278e12	1.95420	0.977102	−	0.212771i	$-0.0682487\pi$
$641$	8.35278e12	1.95420	0.977102	+	0.212771i	$0.0682487\pi$
$642$	−7.61228e12	−1.76851
$643$	−6.96916e12	−1.60780	−0.803898	−	0.594768i	$-0.797244\pi$
$643$	−6.96916e12	−1.60780	−0.803898	+	0.594768i	$0.797244\pi$
$644$	0	0
$645$	−3.47079e12	−0.789605
$646$	−7.85971e12	−1.77566
$647$	−6.66337e12	−1.49494	−0.747472	−	0.664294i	$-0.768732\pi$
$647$	−6.66337e12	−1.49494	−0.747472	+	0.664294i	$0.768732\pi$
$648$	−2.17467e13	−4.84514
$649$	−2.29929e12	−0.508737
$650$	−3.95335e11	−0.0868672
$651$	0	0
$652$	−7.07555e12	−1.53337
$653$	5.38394e12	1.15875	0.579377	−	0.815060i	$-0.303296\pi$
$653$	5.38394e12	1.15875	0.579377	+	0.815060i	$0.303296\pi$
$654$	8.50728e12	1.81841
$655$	2.73238e12	0.580036
$656$	−9.54565e12	−2.01251
$657$	−2.28253e12	−0.477938
$658$	0	0
$659$	7.46401e12	1.54166	0.770829	−	0.637043i	$-0.219842\pi$
$659$	7.46401e12	1.54166	0.770829	+	0.637043i	$0.219842\pi$
$660$	3.18906e12	0.654208
$661$	3.41567e12	0.695937	0.347968	−	0.937506i	$-0.386872\pi$
$661$	3.41567e12	0.695937	0.347968	+	0.937506i	$0.386872\pi$
$662$	2.62650e12	0.531515
$663$	−1.47801e12	−0.297076
$664$	3.35352e12	0.669490
$665$	0	0
$666$	3.33071e12	0.655999
$667$	−1.79158e11	−0.0350485
$668$	9.10527e12	1.76929
$669$	−3.08708e12	−0.595840
$670$	−1.21578e12	−0.233087
$671$	−5.29579e11	−0.100851
$672$	0	0
$673$	4.37280e12	0.821659	0.410830	−	0.911712i	$-0.365239\pi$
$673$	4.37280e12	0.821659	0.410830	+	0.911712i	$0.365239\pi$
$674$	2.02959e12	0.378825
$675$	−7.70206e11	−0.142804
$676$	−1.53270e13	−2.82292
$677$	9.39052e12	1.71807	0.859035	−	0.511917i	$-0.171065\pi$
$677$	9.39052e12	1.71807	0.859035	+	0.511917i	$0.171065\pi$
$678$	−6.56746e12	−1.19361
$679$	0	0
$680$	1.12392e13	2.01578
$681$	8.15743e12	1.45342
$682$	−6.11953e12	−1.08315
$683$	7.07488e12	1.24402	0.622008	−	0.783011i	$-0.286317\pi$
$683$	7.07488e12	1.24402	0.622008	+	0.783011i	$0.286317\pi$
$684$	−5.19326e12	−0.907167
$685$	1.34648e12	0.233664
$686$	0	0
$687$	−2.10466e11	−0.0360476
$688$	−4.23463e13	−7.20556
$689$	−1.50439e12	−0.254315
$690$	−5.61865e12	−0.943651
$691$	1.06781e13	1.78174	0.890869	−	0.454261i	$-0.150097\pi$
$691$	1.06781e13	1.78174	0.890869	+	0.454261i	$0.150097\pi$
$692$	4.72579e12	0.783424
$693$	0	0
$694$	−7.13046e12	−1.16681
$695$	−1.35651e12	−0.220542
$696$	1.11722e12	0.180467
$697$	−2.99821e12	−0.481188
$698$	8.91128e12	1.42099
$699$	−7.62899e12	−1.20870
$700$	0	0
$701$	7.11262e11	0.111250	0.0556248	−	0.998452i	$-0.482285\pi$
$701$	7.11262e11	0.111250	0.0556248	+	0.998452i	$0.482285\pi$
$702$	1.99550e12	0.310124
$703$	−4.17309e12	−0.644404
$704$	1.77112e13	2.71751
$705$	2.58681e12	0.394379
$706$	1.42664e12	0.216119
$707$	0	0
$708$	−2.85068e13	−4.26382
$709$	8.03357e12	1.19399	0.596994	−	0.802245i	$-0.296361\pi$
$709$	8.03357e12	1.19399	0.596994	+	0.802245i	$0.296361\pi$
$710$	1.03994e13	1.53584
$711$	3.59439e12	0.527488
$712$	−1.10207e13	−1.60712
$713$	8.06177e12	1.16823
$714$	0	0
$715$	−2.84839e11	−0.0407589
$716$	−4.22863e12	−0.601300
$717$	−5.35110e11	−0.0756149
$718$	−1.13316e13	−1.59123
$719$	−1.03105e13	−1.43879	−0.719396	−	0.694600i	$-0.755581\pi$
$719$	−1.03105e13	−1.43879	−0.719396	+	0.694600i	$0.755581\pi$
$720$	6.15048e12	0.852929
$721$	0	0
$722$	−5.83532e12	−0.799185
$723$	−1.12997e13	−1.53796
$724$	−1.91455e13	−2.58967
$725$	5.81258e10	0.00781354
$726$	−1.45330e13	−1.94151
$727$	8.01194e12	1.06373	0.531867	−	0.846828i	$-0.321491\pi$
$727$	8.01194e12	1.06373	0.531867	+	0.846828i	$0.321491\pi$
$728$	0	0
$729$	2.69436e12	0.353332
$730$	−8.25389e12	−1.07574
$731$	−1.33006e13	−1.72284
$732$	−6.56576e12	−0.845250
$733$	−8.34520e12	−1.06775	−0.533874	−	0.845564i	$-0.679264\pi$
$733$	−8.34520e12	−1.06775	−0.533874	+	0.845564i	$0.679264\pi$
$734$	−3.02978e13	−3.85282
$735$	0	0
$736$	−4.06262e13	−5.10336
$737$	−8.75970e11	−0.109367
$738$	−2.64943e12	−0.328775
$739$	1.29964e13	1.60296	0.801479	−	0.598022i	$-0.204047\pi$
$739$	1.29964e13	1.60296	0.801479	+	0.598022i	$0.204047\pi$
$740$	9.00583e12	1.10403
$741$	1.63639e12	0.199391
$742$	0	0
$743$	−2.04689e12	−0.246403	−0.123201	−	0.992382i	$-0.539316\pi$
$743$	−2.04689e12	−0.246403	−0.123201	+	0.992382i	$0.539316\pi$
$744$	−5.02729e13	−6.01527
$745$	−5.49397e11	−0.0653406
$746$	−2.70216e13	−3.19438
$747$	5.76411e11	0.0677313
$748$	1.22210e13	1.42741
$749$	0	0
$750$	1.82291e12	0.210373
$751$	−9.14165e12	−1.04868	−0.524342	−	0.851508i	$-0.675689\pi$
$751$	−9.14165e12	−1.04868	−0.524342	+	0.851508i	$0.675689\pi$
$752$	3.15610e13	3.59891
$753$	1.77757e13	2.01488
$754$	−1.50596e11	−0.0169685
$755$	−8.44772e11	−0.0946190
$756$	0	0
$757$	−1.60644e13	−1.77801	−0.889005	−	0.457897i	$-0.848603\pi$
$757$	−1.60644e13	−1.77801	−0.889005	+	0.457897i	$0.848603\pi$
$758$	2.87688e13	3.16526
$759$	−4.04824e12	−0.442770
$760$	−1.24435e13	−1.35295
$761$	−3.42399e11	−0.0370085	−0.0185043	−	0.999829i	$-0.505890\pi$
$761$	−3.42399e11	−0.0370085	−0.0185043	+	0.999829i	$0.505890\pi$
$762$	1.47028e13	1.57980
$763$	0	0
$764$	2.12762e13	2.25930
$765$	1.93182e12	0.203934
$766$	−2.29856e13	−2.41227
$767$	2.54616e12	0.265648
$768$	9.05899e13	9.39624
$769$	5.75725e12	0.593671	0.296836	−	0.954929i	$-0.404069\pi$
$769$	5.75725e12	0.593671	0.296836	+	0.954929i	$0.404069\pi$
$770$	0	0
$771$	2.13854e13	2.17958
$772$	2.25706e13	2.28700
$773$	1.73573e13	1.74853	0.874267	−	0.485445i	$-0.161342\pi$
$773$	1.73573e13	1.74853	0.874267	+	0.485445i	$0.161342\pi$
$774$	−1.17534e13	−1.17714
$775$	−2.61555e12	−0.260439
$776$	−3.96909e13	−3.92928
$777$	0	0
$778$	5.35734e12	0.524253
$779$	3.31950e12	0.322964
$780$	−3.53146e12	−0.341608
$781$	7.49277e12	0.720630
$782$	−2.15316e13	−2.05895
$783$	−2.93397e11	−0.0278951
$784$	0	0
$785$	−2.95463e12	−0.277708
$786$	3.26427e13	3.05060
$787$	−5.96110e12	−0.553911	−0.276956	−	0.960883i	$-0.589326\pi$
$787$	−5.96110e12	−0.553911	−0.276956	+	0.960883i	$0.589326\pi$
$788$	4.19230e12	0.387333
$789$	1.00231e13	0.920779
$790$	1.29978e13	1.18726
$791$	0	0
$792$	7.15579e12	0.646241
$793$	5.86437e11	0.0526614
$794$	−1.26542e13	−1.12990
$795$	6.93679e12	0.615894
$796$	−1.42375e13	−1.25697
$797$	8.87849e11	0.0779430	0.0389715	−	0.999240i	$-0.487592\pi$
$797$	8.87849e11	0.0779430	0.0389715	+	0.999240i	$0.487592\pi$
$798$	0	0
$799$	9.91307e12	0.860493
$800$	1.31807e13	1.13772
$801$	−1.89426e12	−0.162590
$802$	−3.75048e13	−3.20112
$803$	−5.94693e12	−0.504746
$804$	−1.08603e13	−0.916625
$805$	0	0
$806$	6.77655e12	0.565589
$807$	8.09222e12	0.671640
$808$	−2.56661e13	−2.11840
$809$	1.23901e13	1.01696	0.508481	−	0.861073i	$-0.330207\pi$
$809$	1.23901e13	1.01696	0.508481	+	0.861073i	$0.330207\pi$
$810$	−1.35166e13	−1.10328
$811$	2.04283e13	1.65820	0.829102	−	0.559097i	$-0.188852\pi$
$811$	2.04283e13	1.65820	0.829102	+	0.559097i	$0.188852\pi$
$812$	0	0
$813$	2.27018e13	1.82244
$814$	8.67789e12	0.692794
$815$	−2.91404e12	−0.231359
$816$	8.31504e13	6.56537
$817$	1.47259e13	1.15633
$818$	2.08217e13	1.62602
$819$	0	0
$820$	−7.16371e12	−0.553319
$821$	1.06298e13	0.816548	0.408274	−	0.912859i	$-0.366131\pi$
$821$	1.06298e13	0.816548	0.408274	+	0.912859i	$0.366131\pi$
$822$	1.60858e13	1.22891
$823$	2.52356e13	1.91741	0.958703	−	0.284410i	$-0.0917978\pi$
$823$	2.52356e13	1.91741	0.958703	+	0.284410i	$0.0917978\pi$
$824$	−4.88427e12	−0.369085
$825$	1.31341e12	0.0987089
$826$	0	0
$827$	9.72410e12	0.722894	0.361447	−	0.932393i	$-0.382283\pi$
$827$	9.72410e12	0.722894	0.361447	+	0.932393i	$0.382283\pi$
$828$	−1.42269e13	−1.05189
$829$	1.93523e13	1.42310	0.711552	−	0.702633i	$-0.247993\pi$
$829$	1.93523e13	1.42310	0.711552	+	0.702633i	$0.247993\pi$
$830$	2.08437e12	0.152449
$831$	−8.16050e12	−0.593625
$832$	−1.96128e13	−1.41901
$833$	0	0
$834$	−1.62057e13	−1.15990
$835$	3.74998e12	0.266956
$836$	−1.35306e13	−0.958051
$837$	1.32023e13	0.929792
$838$	−3.79736e13	−2.66002
$839$	1.81349e13	1.26353	0.631767	−	0.775158i	$-0.282330\pi$
$839$	1.81349e13	1.26353	0.631767	+	0.775158i	$0.282330\pi$
$840$	0	0
$841$	−1.44850e13	−0.998474
$842$	−2.58260e13	−1.77073
$843$	7.49630e12	0.511238
$844$	−6.42749e13	−4.36014
$845$	−6.31239e12	−0.425930
$846$	8.75987e12	0.587937
$847$	0	0
$848$	8.46341e13	5.62036
$849$	1.78070e13	1.17627
$850$	6.98568e12	0.459011
$851$	−1.14321e13	−0.747211
$852$	9.28959e13	6.03974
$853$	−2.30395e13	−1.49005	−0.745027	−	0.667035i	$-0.767563\pi$
$853$	−2.30395e13	−1.49005	−0.745027	+	0.667035i	$0.767563\pi$
$854$	0	0
$855$	−2.13883e12	−0.136876
$856$	−4.61844e13	−2.94011
$857$	2.28436e12	0.144660	0.0723302	−	0.997381i	$-0.476956\pi$
$857$	2.28436e12	0.144660	0.0723302	+	0.997381i	$0.476956\pi$
$858$	−3.40287e12	−0.214364
$859$	−5.10406e12	−0.319850	−0.159925	−	0.987129i	$-0.551125\pi$
$859$	−5.10406e12	−0.319850	−0.159925	+	0.987129i	$0.551125\pi$
$860$	−3.17796e13	−1.98109
$861$	0	0
$862$	−2.17107e11	−0.0133934
$863$	1.02192e13	0.627144	0.313572	−	0.949564i	$-0.398474\pi$
$863$	1.02192e13	0.627144	0.313572	+	0.949564i	$0.398474\pi$
$864$	−6.65313e13	−4.06176
$865$	1.94630e12	0.118205
$866$	−7.33222e12	−0.443001
$867$	6.46224e12	0.388416
$868$	0	0
$869$	9.36488e12	0.557075
$870$	6.94406e11	0.0410939
$871$	9.70019e11	0.0571082
$872$	5.16145e13	3.02306
$873$	−6.82217e12	−0.397520
$874$	2.38389e13	1.38192
$875$	0	0
$876$	−7.37305e13	−4.23038
$877$	1.62906e13	0.929908	0.464954	−	0.885335i	$-0.346071\pi$
$877$	1.62906e13	0.929908	0.464954	+	0.885335i	$0.346071\pi$
$878$	1.33488e13	0.758083
$879$	−6.68110e12	−0.377483
$880$	1.60246e13	0.900770
$881$	−1.93634e12	−0.108291	−0.0541453	−	0.998533i	$-0.517243\pi$
$881$	−1.93634e12	−0.108291	−0.0541453	+	0.998533i	$0.517243\pi$
$882$	0	0
$883$	−2.44524e13	−1.35362	−0.676811	−	0.736157i	$-0.736639\pi$
$883$	−2.44524e13	−1.35362	−0.676811	+	0.736157i	$0.736639\pi$
$884$	−1.35331e13	−0.745354
$885$	−1.17404e13	−0.643339
$886$	5.32152e13	2.90124
$887$	2.00412e13	1.08709	0.543547	−	0.839379i	$-0.317081\pi$
$887$	2.00412e13	1.08709	0.543547	+	0.839379i	$0.317081\pi$
$888$	7.12902e13	3.84744
$889$	0	0
$890$	−6.84986e12	−0.365954
$891$	−9.73874e12	−0.517670
$892$	−2.82662e13	−1.49494
$893$	−1.09753e13	−0.577546
$894$	−6.56343e12	−0.343647
$895$	−1.74155e12	−0.0907260
$896$	0	0
$897$	4.48288e12	0.231202
$898$	1.85548e13	0.952166
$899$	−9.96350e11	−0.0508737
$900$	4.61575e12	0.234504
$901$	2.65829e13	1.34382
$902$	−6.90286e12	−0.347216
$903$	0	0
$904$	−3.98454e13	−1.98436
$905$	−7.88502e12	−0.390737
$906$	−1.00922e13	−0.497631
$907$	1.67489e13	0.821777	0.410889	−	0.911686i	$-0.365219\pi$
$907$	1.67489e13	0.821777	0.410889	+	0.911686i	$0.365219\pi$
$908$	7.46918e13	3.64659
$909$	−4.41156e12	−0.214316
$910$	0	0
$911$	9.01719e12	0.433749	0.216875	−	0.976199i	$-0.430414\pi$
$911$	9.01719e12	0.433749	0.216875	+	0.976199i	$0.430414\pi$
$912$	−9.20607e13	−4.40654
$913$	1.50179e12	0.0715304
$914$	−3.93114e13	−1.86320
$915$	−2.70409e12	−0.127534
$916$	−1.92709e12	−0.0904424
$917$	0	0
$918$	−3.52611e13	−1.63872
$919$	−4.86323e12	−0.224908	−0.112454	−	0.993657i	$-0.535871\pi$
$919$	−4.86323e12	−0.224908	−0.112454	+	0.993657i	$0.535871\pi$
$920$	−3.40889e13	−1.56880
$921$	−3.72563e13	−1.70621
$922$	−7.22635e13	−3.29329
$923$	−8.29723e12	−0.376292
$924$	0	0
$925$	3.70902e12	0.166580
$926$	−1.56059e12	−0.0697491
$927$	−8.39520e11	−0.0373398
$928$	5.02097e12	0.222240
$929$	3.42631e13	1.50923	0.754617	−	0.656166i	$-0.227823\pi$
$929$	3.42631e13	1.50923	0.754617	+	0.656166i	$0.227823\pi$
$930$	−3.12470e13	−1.36973
$931$	0	0
$932$	−6.98532e13	−3.03260
$933$	1.71042e13	0.738987
$934$	7.31474e13	3.14513
$935$	5.03318e12	0.215373
$936$	−7.92408e12	−0.337448
$937$	1.80465e13	0.764828	0.382414	−	0.923991i	$-0.375093\pi$
$937$	1.80465e13	0.764828	0.382414	+	0.923991i	$0.375093\pi$
$938$	0	0
$939$	2.34488e13	0.984297
$940$	2.36856e13	0.989484
$941$	8.08587e12	0.336181	0.168091	−	0.985772i	$-0.446240\pi$
$941$	8.08587e12	0.336181	0.168091	+	0.985772i	$0.446240\pi$
$942$	−3.52978e13	−1.46056
$943$	9.09371e12	0.374488
$944$	−1.43242e14	−5.87081
$945$	0	0
$946$	−3.06224e13	−1.24316
$947$	7.17527e12	0.289910	0.144955	−	0.989438i	$-0.453696\pi$
$947$	7.17527e12	0.289910	0.144955	+	0.989438i	$0.453696\pi$
$948$	1.16107e14	4.66895
$949$	6.58543e12	0.263564
$950$	−7.73425e12	−0.308079
$951$	−1.60622e13	−0.636786
$952$	0	0
$953$	−1.63556e13	−0.642316	−0.321158	−	0.947026i	$-0.604072\pi$
$953$	−1.63556e13	−0.642316	−0.321158	+	0.947026i	$0.604072\pi$
$954$	2.34905e13	0.918172
$955$	8.76256e12	0.340891
$956$	−4.89962e12	−0.189715
$957$	5.00320e11	0.0192816
$958$	6.08359e13	2.33354
$959$	0	0
$960$	9.04355e13	3.43652
$961$	1.83943e13	0.695708
$962$	−9.60959e12	−0.361757
$963$	−7.93830e12	−0.297447
$964$	−1.03463e14	−3.85869
$965$	9.29564e12	0.345069
$966$	0	0
$967$	1.76906e13	0.650614	0.325307	−	0.945609i	$-0.394532\pi$
$967$	1.76906e13	0.650614	0.325307	+	0.945609i	$0.394532\pi$
$968$	−8.81731e13	−3.22773
$969$	−2.89155e13	−1.05360
$970$	−2.46698e13	−0.894731
$971$	1.71469e13	0.619013	0.309507	−	0.950897i	$-0.399836\pi$
$971$	1.71469e13	0.619013	0.309507	+	0.950897i	$0.399836\pi$
$972$	−6.18462e13	−2.22236
$973$	0	0
$974$	2.21092e13	0.787152
$975$	−1.45442e12	−0.0515430
$976$	−3.29919e13	−1.16381
$977$	−4.65378e13	−1.63411	−0.817054	−	0.576561i	$-0.804394\pi$
$977$	−4.65378e13	−1.63411	−0.817054	+	0.576561i	$0.804394\pi$
$978$	−3.48130e13	−1.21679
$979$	−4.93533e12	−0.171709
$980$	0	0
$981$	8.87163e12	0.305839
$982$	−4.38542e13	−1.50491
$983$	−3.89989e13	−1.33218	−0.666088	−	0.745873i	$-0.732033\pi$
$983$	−3.89989e13	−1.33218	−0.666088	+	0.745873i	$0.732033\pi$
$984$	−5.67080e13	−1.92826
$985$	1.72658e12	0.0584420
$986$	2.66108e12	0.0896625
$987$	0	0
$988$	1.49833e13	0.500266
$989$	4.03414e13	1.34081
$990$	4.44767e12	0.147155
$991$	−1.44456e13	−0.475779	−0.237889	−	0.971292i	$-0.576456\pi$
$991$	−1.44456e13	−0.475779	−0.237889	+	0.971292i	$0.576456\pi$
$992$	−2.25934e14	−7.40764
$993$	9.66276e12	0.315377
$994$	0	0
$995$	−5.86367e12	−0.189656
$996$	1.86193e13	0.599510
$997$	3.12265e13	1.00091	0.500455	−	0.865763i	$-0.333166\pi$
$997$	3.12265e13	1.00091	0.500455	+	0.865763i	$0.333166\pi$
$998$	6.96611e13	2.22281
$999$	−1.87217e13	−0.594705

Display

a_p

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p

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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

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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.1.13		13
7.2	even	3		35.10.e.b.11.1	✓	26
7.4	even	3		35.10.e.b.16.1	yes	26
7.6	odd	2		245.10.a.l.1.13		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.10.e.b.11.1	✓	26	7.2	even	3
35.10.e.b.16.1	yes	26	7.4	even	3
245.10.a.l.1.13		13	7.6	odd	2
245.10.a.m.1.13		13	1.1	even	1	trivial

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Classical	Maass
Hilbert	Bianchi

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

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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.1.13

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