LMFDB - Embedded newform 45.22.a.g.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,22,Mod(1,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("45.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$N$	$=$	$45 = 3^{2} \cdot 5$
Weight:	$k$	$=$	$22$
Character orbit:	$[\chi]$	$=$	45.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:	$125.764804929$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{3} - 3512166x^{2} + 363520480x + 2321089280000$ x^4 - x^3 - 3512166x^2 + 363520480x + 2321089280000
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{5}\cdot 3^{4}\cdot 5^{2}\cdot 7^{2}$
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1492.88$ of defining polynomial
Character	$\chi$	$=$	45.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1716.88 q^{2} +850541. q^{4} -9.76562e6 q^{5} -6.24477e8 q^{7} -2.14029e9 q^{8} -1.67665e10 q^{10} -1.47398e11 q^{11} -5.50754e11 q^{13} -1.07216e12 q^{14} -5.45834e12 q^{16} +4.04946e12 q^{17} +1.22679e12 q^{19} -8.30606e12 q^{20} -2.53065e14 q^{22} +2.50983e14 q^{23} +9.53674e13 q^{25} -9.45581e14 q^{26} -5.31144e14 q^{28} +3.49299e15 q^{29} -2.93209e15 q^{31} -4.88283e15 q^{32} +6.95245e15 q^{34} +6.09841e15 q^{35} +4.06107e16 q^{37} +2.10626e15 q^{38} +2.09012e16 q^{40} -1.45009e17 q^{41} +8.05941e16 q^{43} -1.25368e17 q^{44} +4.30909e17 q^{46} -4.95994e16 q^{47} -1.68574e17 q^{49} +1.63735e17 q^{50} -4.68439e17 q^{52} +1.24656e18 q^{53} +1.43943e18 q^{55} +1.33656e18 q^{56} +5.99706e18 q^{58} -7.20028e18 q^{59} -4.77418e18 q^{61} -5.03405e18 q^{62} +3.06371e18 q^{64} +5.37846e18 q^{65} +7.70841e18 q^{67} +3.44423e18 q^{68} +1.04703e19 q^{70} +1.63265e19 q^{71} -1.29930e19 q^{73} +6.97240e19 q^{74} +1.04344e18 q^{76} +9.20465e19 q^{77} +9.55293e19 q^{79} +5.33041e19 q^{80} -2.48964e20 q^{82} +2.20974e20 q^{83} -3.95455e19 q^{85} +1.38371e20 q^{86} +3.15473e20 q^{88} -3.15065e19 q^{89} +3.43933e20 q^{91} +2.13471e20 q^{92} -8.51565e19 q^{94} -1.19804e19 q^{95} +2.46548e20 q^{97} -2.89422e20 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 897 q^{2} - 1163123 q^{4} - 39062500 q^{5} - 234577504 q^{7} - 76855629 q^{8} - 8759765625 q^{10} - 31491830256 q^{11} - 27017977768 q^{13} + 2233308126672 q^{14} - 11324681311775 q^{16} + 2946095028888 q^{17}+ \cdots - 19\!\cdots\!63 q^{98}+O(q^{100})$ 4 * q + 897 * q^2 - 1163123 * q^4 - 39062500 * q^5 - 234577504 * q^7 - 76855629 * q^8 - 8759765625 * q^10 - 31491830256 * q^11 - 27017977768 * q^13 + 2233308126672 * q^14 - 11324681311775 * q^16 + 2946095028888 * q^17 - 24270353300752 * q^19 + 11358623046875 * q^20 - 56303932793676 * q^22 - 10350924920928 * q^23 + 381469726562500 * q^25 - 474751622871378 * q^26 - 1868468795909968 * q^28 - 4728924677079096 * q^29 - 1094923910405536 * q^31 - 7933327650814221 * q^32 + 3862795623387294 * q^34 + 2290795937500000 * q^35 + 4813435696247096 * q^37 - 21307487199674508 * q^38 + 750543251953125 * q^40 - 289731591445930344 * q^41 + 451091912658458000 * q^43 - 245434444420437276 * q^44 + 645145648102459824 * q^46 - 813883435638492480 * q^47 + 1545180753004755300 * q^49 + 85544586181640625 * q^50 + 884645329958480822 * q^52 - 697278335404085208 * q^53 + 307537404843750000 * q^55 - 5673275878853478000 * q^56 + 12277858736996428626 * q^58 - 6622888614569598192 * q^59 + 7390887218011683320 * q^61 - 11649094150323079392 * q^62 + 21525702776885072473 * q^64 + 263847439140625000 * q^65 - 24188188449376788688 * q^67 - 18995582935459304346 * q^68 - 21809649674531250000 * q^70 + 37390337803999713312 * q^71 - 37253672904265201432 * q^73 + 77875663243460100102 * q^74 - 44795472028972724668 * q^76 + 315245139112859776128 * q^77 - 109751291042259570400 * q^79 + 110592590935302734375 * q^80 - 210801413757052759338 * q^82 + 149023592088638482896 * q^83 - 28770459266484375000 * q^85 + 346200583098972939204 * q^86 - 116461091469581805156 * q^88 - 558502237151273959848 * q^89 + 30360075980445717184 * q^91 + 764380916144075558544 * q^92 + 1096939878232058257512 * q^94 + 237015168952656250000 * q^95 - 236244923748673777528 * q^97 - 197058383612642903463 * q^98

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

)

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

$2$	1716.88	1.18557	0.592784	−	0.805362i	$-0.298029\pi$
$2$	1716.88	1.18557	0.592784	+	0.805362i	$0.298029\pi$
$3$	0	0
$3$	0	0
$4$	850541.	0.405570
$5$	−9.76562e6	−0.447214
$5$	−9.76562e6	−0.447214
$6$	0	0
$7$	−6.24477e8	−0.835579	−0.417789	−	0.908544i	$-0.637195\pi$
$7$	−6.24477e8	−0.835579	−0.417789	+	0.908544i	$0.637195\pi$
$8$	−2.14029e9	−0.704737
$9$	0	0
$10$	−1.67665e10	−0.530202
$11$	−1.47398e11	−1.71343	−0.856717	−	0.515787i	$-0.827499\pi$
$11$	−1.47398e11	−1.71343	−0.856717	+	0.515787i	$0.827499\pi$
$12$	0	0
$13$	−5.50754e11	−1.10803	−0.554016	−	0.832506i	$-0.686905\pi$
$13$	−5.50754e11	−1.10803	−0.554016	+	0.832506i	$0.686905\pi$
$14$	−1.07216e12	−0.990635
$15$	0	0
$16$	−5.45834e12	−1.24108
$17$	4.04946e12	0.487173	0.243587	−	0.969879i	$-0.421676\pi$
$17$	4.04946e12	0.487173	0.243587	+	0.969879i	$0.421676\pi$
$18$	0	0
$19$	1.22679e12	0.0459048	0.0229524	−	0.999737i	$-0.492693\pi$
$19$	1.22679e12	0.0459048	0.0229524	+	0.999737i	$0.492693\pi$
$20$	−8.30606e12	−0.181376
$21$	0	0
$22$	−2.53065e14	−2.03139
$23$	2.50983e14	1.26329	0.631644	−	0.775259i	$-0.282380\pi$
$23$	2.50983e14	1.26329	0.631644	+	0.775259i	$0.282380\pi$
$24$	0	0
$25$	9.53674e13	0.200000
$26$	−9.45581e14	−1.31365
$27$	0	0
$28$	−5.31144e14	−0.338885
$29$	3.49299e15	1.54177	0.770883	−	0.636977i	$-0.219815\pi$
$29$	3.49299e15	1.54177	0.770883	+	0.636977i	$0.219815\pi$
$30$	0	0
$31$	−2.93209e15	−0.642508	−0.321254	−	0.946993i	$-0.604104\pi$
$31$	−2.93209e15	−0.642508	−0.321254	+	0.946993i	$0.604104\pi$
$32$	−4.88283e15	−0.766650
$33$	0	0
$34$	6.95245e15	0.577576
$35$	6.09841e15	0.373682
$36$	0	0
$37$	4.06107e16	1.38843	0.694213	−	0.719770i	$-0.255753\pi$
$37$	4.06107e16	1.38843	0.694213	+	0.719770i	$0.255753\pi$
$38$	2.10626e15	0.0544232
$39$	0	0
$40$	2.09012e16	0.315168
$41$	−1.45009e17	−1.68720	−0.843598	−	0.536976i	$-0.819567\pi$
$41$	−1.45009e17	−1.68720	−0.843598	+	0.536976i	$0.819567\pi$
$42$	0	0
$43$	8.05941e16	0.568702	0.284351	−	0.958720i	$-0.408222\pi$
$43$	8.05941e16	0.568702	0.284351	+	0.958720i	$0.408222\pi$
$44$	−1.25368e17	−0.694916
$45$	0	0
$46$	4.30909e17	1.49771
$47$	−4.95994e16	−0.137546	−0.0687732	−	0.997632i	$-0.521908\pi$
$47$	−4.95994e16	−0.137546	−0.0687732	+	0.997632i	$0.521908\pi$
$48$	0	0
$49$	−1.68574e17	−0.301808
$50$	1.63735e17	0.237113
$51$	0	0
$52$	−4.68439e17	−0.449384
$53$	1.24656e18	0.979079	0.489540	−	0.871981i	$-0.337165\pi$
$53$	1.24656e18	0.979079	0.489540	+	0.871981i	$0.337165\pi$
$54$	0	0
$55$	1.43943e18	0.766271
$56$	1.33656e18	0.588863
$57$	0	0
$58$	5.99706e18	1.82787
$59$	−7.20028e18	−1.83402	−0.917009	−	0.398867i	$-0.869404\pi$
$59$	−7.20028e18	−1.83402	−0.917009	+	0.398867i	$0.869404\pi$
$60$	0	0
$61$	−4.77418e18	−0.856910	−0.428455	−	0.903563i	$-0.640942\pi$
$61$	−4.77418e18	−0.856910	−0.428455	+	0.903563i	$0.640942\pi$
$62$	−5.03405e18	−0.761737
$63$	0	0
$64$	3.06371e18	0.332168
$65$	5.37846e18	0.495527
$66$	0	0
$67$	7.70841e18	0.516630	0.258315	−	0.966061i	$-0.416833\pi$
$67$	7.70841e18	0.516630	0.258315	+	0.966061i	$0.416833\pi$
$68$	3.44423e18	0.197583
$69$	0	0
$70$	1.04703e19	0.443025
$71$	1.63265e19	0.595223	0.297612	−	0.954687i	$-0.403810\pi$
$71$	1.63265e19	0.595223	0.297612	+	0.954687i	$0.403810\pi$
$72$	0	0
$73$	−1.29930e19	−0.353849	−0.176924	−	0.984224i	$-0.556615\pi$
$73$	−1.29930e19	−0.353849	−0.176924	+	0.984224i	$0.556615\pi$
$74$	6.97240e19	1.64607
$75$	0	0
$76$	1.04344e18	0.0186176
$77$	9.20465e19	1.43171
$78$	0	0
$79$	9.55293e19	1.13515	0.567574	−	0.823323i	$-0.307882\pi$
$79$	9.55293e19	1.13515	0.567574	+	0.823323i	$0.307882\pi$
$80$	5.33041e19	0.555029
$81$	0	0
$82$	−2.48964e20	−2.00028
$83$	2.20974e20	1.56323	0.781613	−	0.623764i	$-0.214397\pi$
$83$	2.20974e20	1.56323	0.781613	+	0.623764i	$0.214397\pi$
$84$	0	0
$85$	−3.95455e19	−0.217870
$86$	1.38371e20	0.674234
$87$	0	0
$88$	3.15473e20	1.20752
$89$	−3.15065e19	−0.107104	−0.0535519	−	0.998565i	$-0.517054\pi$
$89$	−3.15065e19	−0.107104	−0.0535519	+	0.998565i	$0.517054\pi$
$90$	0	0
$91$	3.43933e20	0.925848
$92$	2.13471e20	0.512351
$93$	0	0
$94$	−8.51565e19	−0.163070
$95$	−1.19804e19	−0.0205292
$96$	0	0
$97$	2.46548e20	0.339467	0.169733	−	0.985490i	$-0.445709\pi$
$97$	2.46548e20	0.339467	0.169733	+	0.985490i	$0.445709\pi$
$98$	−2.89422e20	−0.357814
$99$	0	0
$100$	8.11139e19	0.0811139
$101$	−2.15225e21	−1.93873	−0.969367	−	0.245616i	$-0.921010\pi$
$101$	−2.15225e21	−1.93873	−0.969367	+	0.245616i	$0.921010\pi$
$102$	0	0
$103$	−1.53340e21	−1.12426	−0.562128	−	0.827050i	$-0.690017\pi$
$103$	−1.53340e21	−1.12426	−0.562128	+	0.827050i	$0.690017\pi$
$104$	1.17877e21	0.780871
$105$	0	0
$106$	2.14021e21	1.16076
$107$	−1.54824e21	−0.760865	−0.380433	−	0.924809i	$-0.624225\pi$
$107$	−1.54824e21	−0.760865	−0.380433	+	0.924809i	$0.624225\pi$
$108$	0	0
$109$	2.85208e21	1.15394	0.576971	−	0.816764i	$-0.304234\pi$
$109$	2.85208e21	1.15394	0.576971	+	0.816764i	$0.304234\pi$
$110$	2.47133e21	0.908465
$111$	0	0
$112$	3.40861e21	1.03702
$113$	−3.30356e21	−0.915499	−0.457750	−	0.889081i	$-0.651344\pi$
$113$	−3.30356e21	−0.915499	−0.457750	+	0.889081i	$0.651344\pi$
$114$	0	0
$115$	−2.45101e21	−0.564959
$116$	2.97093e21	0.625293
$117$	0	0
$118$	−1.23621e22	−2.17435
$119$	−2.52880e21	−0.407071
$120$	0	0
$121$	1.43258e22	1.93585
$122$	−8.19671e21	−1.01592
$123$	0	0
$124$	−2.49386e21	−0.260582
$125$	−9.31323e20	−0.0894427
$126$	0	0
$127$	−8.37131e20	−0.0680541	−0.0340271	−	0.999421i	$-0.510833\pi$
$127$	−8.37131e20	−0.0680541	−0.0340271	+	0.999421i	$0.510833\pi$
$128$	1.55001e22	1.16046
$129$	0	0
$130$	9.23419e21	0.587481
$131$	−1.90600e22	−1.11886	−0.559429	−	0.828879i	$-0.688979\pi$
$131$	−1.90600e22	−1.11886	−0.559429	+	0.828879i	$0.688979\pi$
$132$	0	0
$133$	−7.66104e20	−0.0383571
$134$	1.32344e22	0.612499
$135$	0	0
$136$	−8.66701e21	−0.343329
$137$	8.63318e20	0.0316669	0.0158334	−	0.999875i	$-0.494960\pi$
$137$	8.63318e20	0.0316669	0.0158334	+	0.999875i	$0.494960\pi$
$138$	0	0
$139$	8.91438e21	0.280825	0.140412	−	0.990093i	$-0.455157\pi$
$139$	8.91438e21	0.280825	0.140412	+	0.990093i	$0.455157\pi$
$140$	5.18695e21	0.151554
$141$	0	0
$142$	2.80307e22	0.705677
$143$	8.11798e22	1.89854
$144$	0	0
$145$	−3.41112e22	−0.689498
$146$	−2.23074e22	−0.419512
$147$	0	0
$148$	3.45411e22	0.563103
$149$	9.42240e22	1.43122	0.715609	−	0.698501i	$-0.246149\pi$
$149$	9.42240e22	1.43122	0.715609	+	0.698501i	$0.246149\pi$
$150$	0	0
$151$	2.62382e22	0.346479	0.173239	−	0.984880i	$-0.444577\pi$
$151$	2.62382e22	0.346479	0.173239	+	0.984880i	$0.444577\pi$
$152$	−2.62569e21	−0.0323508
$153$	0	0
$154$	1.58033e23	1.69739
$155$	2.86336e22	0.287338
$156$	0	0
$157$	1.62221e23	1.42286	0.711428	−	0.702759i	$-0.248049\pi$
$157$	1.62221e23	1.42286	0.711428	+	0.702759i	$0.248049\pi$
$158$	1.64013e23	1.34579
$159$	0	0
$160$	4.76839e22	0.342856
$161$	−1.56733e23	−1.05558
$162$	0	0
$163$	2.32312e23	1.37436	0.687181	−	0.726486i	$-0.258848\pi$
$163$	2.32312e23	1.37436	0.687181	+	0.726486i	$0.258848\pi$
$164$	−1.23336e23	−0.684275
$165$	0	0
$166$	3.79387e23	1.85331
$167$	1.76825e23	0.811002	0.405501	−	0.914095i	$-0.367097\pi$
$167$	1.76825e23	0.811002	0.405501	+	0.914095i	$0.367097\pi$
$168$	0	0
$169$	5.62653e22	0.227735
$170$	−6.78951e22	−0.258300
$171$	0	0
$172$	6.85486e22	0.230648
$173$	−1.88916e23	−0.598113	−0.299057	−	0.954235i	$-0.596672\pi$
$173$	−1.88916e23	−0.598113	−0.299057	+	0.954235i	$0.596672\pi$
$174$	0	0
$175$	−5.95548e22	−0.167116
$176$	8.04546e23	2.12651
$177$	0	0
$178$	−5.40930e22	−0.126979
$179$	1.14808e23	0.254107	0.127053	−	0.991896i	$-0.459448\pi$
$179$	1.14808e23	0.254107	0.127053	+	0.991896i	$0.459448\pi$
$180$	0	0
$181$	3.15471e23	0.621347	0.310673	−	0.950517i	$-0.399445\pi$
$181$	3.15471e23	0.621347	0.310673	+	0.950517i	$0.399445\pi$
$182$	5.90494e23	1.09765
$183$	0	0
$184$	−5.37176e23	−0.890286
$185$	−3.96589e23	−0.620923
$186$	0	0
$187$	−5.96881e23	−0.834739
$188$	−4.21863e22	−0.0557846
$189$	0	0
$190$	−2.05689e22	−0.0243388
$191$	4.11188e23	0.460458	0.230229	−	0.973137i	$-0.426053\pi$
$191$	4.11188e23	0.460458	0.230229	+	0.973137i	$0.426053\pi$
$192$	0	0
$193$	−9.08221e22	−0.0911675	−0.0455837	−	0.998961i	$-0.514515\pi$
$193$	−9.08221e22	−0.0911675	−0.0455837	+	0.998961i	$0.514515\pi$
$194$	4.23294e23	0.402461
$195$	0	0
$196$	−1.43379e23	−0.122404
$197$	−1.22586e24	−0.992076	−0.496038	−	0.868301i	$-0.665212\pi$
$197$	−1.22586e24	−0.992076	−0.496038	+	0.868301i	$0.665212\pi$
$198$	0	0
$199$	1.05217e24	0.765824	0.382912	−	0.923785i	$-0.374921\pi$
$199$	1.05217e24	0.765824	0.382912	+	0.923785i	$0.374921\pi$
$200$	−2.04114e23	−0.140947
$201$	0	0
$202$	−3.69517e24	−2.29850
$203$	−2.18129e24	−1.28827
$204$	0	0
$205$	1.41611e24	0.754537
$206$	−2.63267e24	−1.33288
$207$	0	0
$208$	3.00620e24	1.37516
$209$	−1.80826e23	−0.0786548
$210$	0	0
$211$	5.34118e22	0.0210219	0.0105109	−	0.999945i	$-0.496654\pi$
$211$	5.34118e22	0.0210219	0.0105109	+	0.999945i	$0.496654\pi$
$212$	1.06025e24	0.397085
$213$	0	0
$214$	−2.65814e24	−0.902057
$215$	−7.87052e23	−0.254331
$216$	0	0
$217$	1.83102e24	0.536866
$218$	4.89670e24	1.36808
$219$	0	0
$220$	1.22429e24	0.310776
$221$	−2.23026e24	−0.539803
$222$	0	0
$223$	1.04158e24	0.229346	0.114673	−	0.993403i	$-0.463418\pi$
$223$	1.04158e24	0.229346	0.114673	+	0.993403i	$0.463418\pi$
$224$	3.04922e24	0.640596
$225$	0	0
$226$	−5.67182e24	−1.08539
$227$	−4.64270e24	−0.848201	−0.424101	−	0.905615i	$-0.639410\pi$
$227$	−4.64270e24	−0.848201	−0.424101	+	0.905615i	$0.639410\pi$
$228$	0	0
$229$	−3.77779e24	−0.629456	−0.314728	−	0.949182i	$-0.601913\pi$
$229$	−3.77779e24	−0.629456	−0.314728	+	0.949182i	$0.601913\pi$
$230$	−4.20810e24	−0.669797
$231$	0	0
$232$	−7.47600e24	−1.08654
$233$	−1.16866e25	−1.62350	−0.811750	−	0.584005i	$-0.801485\pi$
$233$	−1.16866e25	−1.62350	−0.811750	+	0.584005i	$0.801485\pi$
$234$	0	0
$235$	4.84369e23	0.0615126
$236$	−6.12414e24	−0.743822
$237$	0	0
$238$	−4.34165e24	−0.482611
$239$	1.39609e25	1.48503	0.742514	−	0.669831i	$-0.233633\pi$
$239$	1.39609e25	1.48503	0.742514	+	0.669831i	$0.233633\pi$
$240$	0	0
$241$	1.83127e25	1.78473	0.892366	−	0.451312i	$-0.149044\pi$
$241$	1.83127e25	1.78473	0.892366	+	0.451312i	$0.149044\pi$
$242$	2.45958e25	2.29508
$243$	0	0
$244$	−4.06063e24	−0.347537
$245$	1.64623e24	0.134973
$246$	0	0
$247$	−6.75660e23	−0.0508640
$248$	6.27550e24	0.452800
$249$	0	0
$250$	−1.59897e24	−0.106040
$251$	−2.08800e25	−1.32787	−0.663936	−	0.747789i	$-0.731115\pi$
$251$	−2.08800e25	−1.32787	−0.663936	+	0.747789i	$0.731115\pi$
$252$	0	0
$253$	−3.69943e25	−2.16456
$254$	−1.43726e24	−0.0806828
$255$	0	0
$256$	2.01868e25	1.04363
$257$	2.97816e24	0.147792	0.0738959	−	0.997266i	$-0.476457\pi$
$257$	2.97816e24	0.147792	0.0738959	+	0.997266i	$0.476457\pi$
$258$	0	0
$259$	−2.53605e25	−1.16014
$260$	4.57460e24	0.200971
$261$	0	0
$262$	−3.27238e25	−1.32648
$263$	4.48688e25	1.74747	0.873734	−	0.486404i	$-0.161692\pi$
$263$	4.48688e25	1.74747	0.873734	+	0.486404i	$0.161692\pi$
$264$	0	0
$265$	−1.21735e25	−0.437858
$266$	−1.31531e24	−0.0454749
$267$	0	0
$268$	6.55632e24	0.209529
$269$	−5.40734e25	−1.66182	−0.830912	−	0.556404i	$-0.812181\pi$
$269$	−5.40734e25	−1.66182	−0.830912	+	0.556404i	$0.812181\pi$
$270$	0	0
$271$	1.64805e25	0.468591	0.234295	−	0.972165i	$-0.424722\pi$
$271$	1.64805e25	0.468591	0.234295	+	0.972165i	$0.424722\pi$
$272$	−2.21033e25	−0.604622
$273$	0	0
$274$	1.48222e24	0.0375432
$275$	−1.40569e25	−0.342687
$276$	0	0
$277$	−2.02544e25	−0.457596	−0.228798	−	0.973474i	$-0.573480\pi$
$277$	−2.02544e25	−0.457596	−0.228798	+	0.973474i	$0.573480\pi$
$278$	1.53050e25	0.332937
$279$	0	0
$280$	−1.30524e25	−0.263348
$281$	−9.60686e25	−1.86709	−0.933545	−	0.358461i	$-0.883301\pi$
$281$	−9.60686e25	−1.86709	−0.933545	+	0.358461i	$0.883301\pi$
$282$	0	0
$283$	−9.26489e24	−0.167141	−0.0835704	−	0.996502i	$-0.526632\pi$
$283$	−9.26489e24	−0.167141	−0.0835704	+	0.996502i	$0.526632\pi$
$284$	1.38863e25	0.241404
$285$	0	0
$286$	1.39376e26	2.25085
$287$	9.05550e25	1.40978
$288$	0	0
$289$	−5.26938e25	−0.762662
$290$	−5.85650e25	−0.817447
$291$	0	0
$292$	−1.10510e25	−0.143510
$293$	−4.08300e25	−0.511528	−0.255764	−	0.966739i	$-0.582327\pi$
$293$	−4.08300e25	−0.511528	−0.255764	+	0.966739i	$0.582327\pi$
$294$	0	0
$295$	7.03153e25	0.820198
$296$	−8.69186e25	−0.978476
$297$	0	0
$298$	1.61772e26	1.69681
$299$	−1.38230e26	−1.39976
$300$	0	0
$301$	−5.03292e25	−0.475195
$302$	4.50480e25	0.410774
$303$	0	0
$304$	−6.69625e24	−0.0569716
$305$	4.66228e25	0.383222
$306$	0	0
$307$	1.09642e26	0.841445	0.420723	−	0.907189i	$-0.361777\pi$
$307$	1.09642e26	0.841445	0.420723	+	0.907189i	$0.361777\pi$
$308$	7.82893e25	0.580657
$309$	0	0
$310$	4.91607e25	0.340659
$311$	6.99628e25	0.468687	0.234343	−	0.972154i	$-0.424706\pi$
$311$	6.99628e25	0.468687	0.234343	+	0.972154i	$0.424706\pi$
$312$	0	0
$313$	6.95169e25	0.435386	0.217693	−	0.976017i	$-0.430147\pi$
$313$	6.95169e25	0.435386	0.217693	+	0.976017i	$0.430147\pi$
$314$	2.78515e26	1.68689
$315$	0	0
$316$	8.12516e25	0.460381
$317$	1.47035e26	0.805933	0.402967	−	0.915215i	$-0.367979\pi$
$317$	1.47035e26	0.805933	0.402967	+	0.915215i	$0.367979\pi$
$318$	0	0
$319$	−5.14858e26	−2.64171
$320$	−2.99190e25	−0.148550
$321$	0	0
$322$	−2.69093e26	−1.25146
$323$	4.96784e24	0.0223636
$324$	0	0
$325$	−5.25240e25	−0.221606
$326$	3.98853e26	1.62940
$327$	0	0
$328$	3.10362e26	1.18903
$329$	3.09737e25	0.114931
$330$	0	0
$331$	4.36548e26	1.51998	0.759990	−	0.649935i	$-0.225204\pi$
$331$	4.36548e26	1.51998	0.759990	+	0.649935i	$0.225204\pi$
$332$	1.87948e26	0.633997
$333$	0	0
$334$	3.03589e26	0.961497
$335$	−7.52774e25	−0.231044
$336$	0	0
$337$	4.15229e26	1.19722	0.598610	−	0.801041i	$-0.295720\pi$
$337$	4.15229e26	1.19722	0.598610	+	0.801041i	$0.295720\pi$
$338$	9.66010e25	0.269995
$339$	0	0
$340$	−3.36351e25	−0.0883616
$341$	4.32182e26	1.10090
$342$	0	0
$343$	4.54070e26	1.08776
$344$	−1.72495e26	−0.400785
$345$	0	0
$346$	−3.24346e26	−0.709104
$347$	−2.48113e26	−0.526246	−0.263123	−	0.964762i	$-0.584753\pi$
$347$	−2.48113e26	−0.526246	−0.263123	+	0.964762i	$0.584753\pi$
$348$	0	0
$349$	−8.30405e24	−0.0165815	−0.00829074	−	0.999966i	$-0.502639\pi$
$349$	−8.30405e24	−0.0165815	−0.00829074	+	0.999966i	$0.502639\pi$
$350$	−1.02249e26	−0.198127
$351$	0	0
$352$	7.19718e26	1.31360
$353$	−9.55014e25	−0.169190	−0.0845951	−	0.996415i	$-0.526960\pi$
$353$	−9.55014e25	−0.169190	−0.0845951	+	0.996415i	$0.526960\pi$
$354$	0	0
$355$	−1.59438e26	−0.266192
$356$	−2.67976e25	−0.0434381
$357$	0	0
$358$	1.97112e26	0.301260
$359$	−7.06762e25	−0.104901	−0.0524507	−	0.998624i	$-0.516703\pi$
$359$	−7.06762e25	−0.104901	−0.0524507	+	0.998624i	$0.516703\pi$
$360$	0	0
$361$	−7.12704e26	−0.997893
$362$	5.41627e26	0.736648
$363$	0	0
$364$	2.92529e26	0.375496
$365$	1.26884e26	0.158246
$366$	0	0
$367$	−1.33814e27	−1.57583	−0.787915	−	0.615785i	$-0.788839\pi$
$367$	−1.33814e27	−1.57583	−0.787915	+	0.615785i	$0.788839\pi$
$368$	−1.36995e27	−1.56784
$369$	0	0
$370$	−6.80898e26	−0.736146
$371$	−7.78451e26	−0.818098
$372$	0	0
$373$	−9.62756e24	−0.00956255	−0.00478127	−	0.999989i	$-0.501522\pi$
$373$	−9.62756e24	−0.00956255	−0.00478127	+	0.999989i	$0.501522\pi$
$374$	−1.02478e27	−0.989639
$375$	0	0
$376$	1.06157e26	0.0969340
$377$	−1.92378e27	−1.70833
$378$	0	0
$379$	−1.97813e27	−1.66166	−0.830830	−	0.556526i	$-0.812134\pi$
$379$	−1.97813e27	−1.66166	−0.830830	+	0.556526i	$0.812134\pi$
$380$	−1.01898e25	−0.00832604
$381$	0	0
$382$	7.05962e26	0.545904
$383$	2.14854e27	1.61643	0.808215	−	0.588888i	$-0.200434\pi$
$383$	2.14854e27	1.61643	0.808215	+	0.588888i	$0.200434\pi$
$384$	0	0
$385$	−8.98891e26	−0.640279
$386$	−1.55931e26	−0.108085
$387$	0	0
$388$	2.09699e26	0.137677
$389$	−2.40240e27	−1.53524	−0.767618	−	0.640907i	$-0.778558\pi$
$389$	−2.40240e27	−1.53524	−0.767618	+	0.640907i	$0.778558\pi$
$390$	0	0
$391$	1.01635e27	0.615440
$392$	3.60796e26	0.212696
$393$	0	0
$394$	−2.10466e27	−1.17617
$395$	−9.32903e26	−0.507653
$396$	0	0
$397$	2.20346e27	1.13711	0.568557	−	0.822644i	$-0.307502\pi$
$397$	2.20346e27	1.13711	0.568557	+	0.822644i	$0.307502\pi$
$398$	1.80646e27	0.907935
$399$	0	0
$400$	−5.20548e26	−0.248217
$401$	2.56251e27	1.19028	0.595141	−	0.803622i	$-0.297096\pi$
$401$	2.56251e27	1.19028	0.595141	+	0.803622i	$0.297096\pi$
$402$	0	0
$403$	1.61486e27	0.711920
$404$	−1.83058e27	−0.786292
$405$	0	0
$406$	−3.74503e27	−1.52733
$407$	−5.98593e27	−2.37898
$408$	0	0
$409$	−4.40538e27	−1.66299	−0.831494	−	0.555534i	$-0.812514\pi$
$409$	−4.40538e27	−1.66299	−0.831494	+	0.555534i	$0.812514\pi$
$410$	2.43129e27	0.894554
$411$	0	0
$412$	−1.30422e27	−0.455964
$413$	4.49641e27	1.53247
$414$	0	0
$415$	−2.15795e27	−0.699096
$416$	2.68924e27	0.849473
$417$	0	0
$418$	−3.10458e26	−0.0932505
$419$	−1.22792e27	−0.359686	−0.179843	−	0.983695i	$-0.557559\pi$
$419$	−1.22792e27	−0.359686	−0.179843	+	0.983695i	$0.557559\pi$
$420$	0	0
$421$	−2.61500e25	−0.00728633	−0.00364317	−	0.999993i	$-0.501160\pi$
$421$	−2.61500e25	−0.00728633	−0.00364317	+	0.999993i	$0.501160\pi$
$422$	9.17019e25	0.0249229
$423$	0	0
$424$	−2.66800e27	−0.689994
$425$	3.86187e26	0.0974346
$426$	0	0
$427$	2.98137e27	0.716016
$428$	−1.31684e27	−0.308584
$429$	0	0
$430$	−1.35128e27	−0.301527
$431$	3.68868e27	0.803265	0.401633	−	0.915801i	$-0.368443\pi$
$431$	3.68868e27	0.803265	0.401633	+	0.915801i	$0.368443\pi$
$432$	0	0
$433$	4.87388e27	1.01100	0.505501	−	0.862826i	$-0.331308\pi$
$433$	4.87388e27	1.01100	0.505501	+	0.862826i	$0.331308\pi$
$434$	3.14365e27	0.636491
$435$	0	0
$436$	2.42582e27	0.468004
$437$	3.07904e26	0.0579910
$438$	0	0
$439$	8.77061e27	1.57453	0.787267	−	0.616612i	$-0.211495\pi$
$439$	8.77061e27	1.57453	0.787267	+	0.616612i	$0.211495\pi$
$440$	−3.08079e27	−0.540019
$441$	0	0
$442$	−3.82909e27	−0.639973
$443$	−2.39913e26	−0.0391575	−0.0195787	−	0.999808i	$-0.506233\pi$
$443$	−2.39913e26	−0.0391575	−0.0195787	+	0.999808i	$0.506233\pi$
$444$	0	0
$445$	3.07681e26	0.0478983
$446$	1.78827e27	0.271905
$447$	0	0
$448$	−1.91322e27	−0.277552
$449$	1.18190e27	0.167491	0.0837457	−	0.996487i	$-0.473312\pi$
$449$	1.18190e27	0.167491	0.0837457	+	0.996487i	$0.473312\pi$
$450$	0	0
$451$	2.13740e28	2.89090
$452$	−2.80981e27	−0.371299
$453$	0	0
$454$	−7.97098e27	−1.00560
$455$	−3.35872e27	−0.414052
$456$	0	0
$457$	1.24883e28	1.47023	0.735114	−	0.677944i	$-0.237129\pi$
$457$	1.24883e28	1.47023	0.735114	+	0.677944i	$0.237129\pi$
$458$	−6.48603e27	−0.746262
$459$	0	0
$460$	−2.08468e27	−0.229130
$461$	4.81654e27	0.517459	0.258729	−	0.965950i	$-0.416696\pi$
$461$	4.81654e27	0.517459	0.258729	+	0.965950i	$0.416696\pi$
$462$	0	0
$463$	2.77535e27	0.284917	0.142458	−	0.989801i	$-0.454499\pi$
$463$	2.77535e27	0.284917	0.142458	+	0.989801i	$0.454499\pi$
$464$	−1.90659e28	−1.91346
$465$	0	0
$466$	−2.00646e28	−1.92477
$467$	1.30821e28	1.22702	0.613508	−	0.789688i	$-0.289758\pi$
$467$	1.30821e28	1.22702	0.613508	+	0.789688i	$0.289758\pi$
$468$	0	0
$469$	−4.81373e27	−0.431685
$470$	8.31606e26	0.0729273
$471$	0	0
$472$	1.54107e28	1.29250
$473$	−1.18794e28	−0.974433
$474$	0	0
$475$	1.16996e26	0.00918096
$476$	−2.15084e27	−0.165096
$477$	0	0
$478$	2.39692e28	1.76060
$479$	1.98117e28	1.42363	0.711817	−	0.702365i	$-0.247873\pi$
$479$	1.98117e28	1.42363	0.711817	+	0.702365i	$0.247873\pi$
$480$	0	0
$481$	−2.23665e28	−1.53842
$482$	3.14408e28	2.11592
$483$	0	0
$484$	1.21847e28	0.785123
$485$	−2.40769e27	−0.151814
$486$	0	0
$487$	−2.14424e28	−1.29485	−0.647425	−	0.762129i	$-0.724154\pi$
$487$	−2.14424e28	−1.29485	−0.647425	+	0.762129i	$0.724154\pi$
$488$	1.02181e28	0.603896
$489$	0	0
$490$	2.82639e27	0.160019
$491$	1.64770e28	0.913109	0.456554	−	0.889695i	$-0.349083\pi$
$491$	1.64770e28	0.913109	0.456554	+	0.889695i	$0.349083\pi$
$492$	0	0
$493$	1.41447e28	0.751107
$494$	−1.16003e27	−0.0603027
$495$	0	0
$496$	1.60043e28	0.797406
$497$	−1.01955e28	−0.497356
$498$	0	0
$499$	−2.46207e28	−1.15145	−0.575724	−	0.817644i	$-0.695280\pi$
$499$	−2.46207e28	−1.15145	−0.575724	+	0.817644i	$0.695280\pi$
$500$	−7.92128e26	−0.0362752
$501$	0	0
$502$	−3.58486e28	−1.57428
$503$	−9.01822e27	−0.387844	−0.193922	−	0.981017i	$-0.562121\pi$
$503$	−9.01822e27	−0.387844	−0.193922	+	0.981017i	$0.562121\pi$
$504$	0	0
$505$	2.10181e28	0.867028
$506$	−6.35150e28	−2.56623
$507$	0	0
$508$	−7.12015e26	−0.0276007
$509$	1.26550e28	0.480534	0.240267	−	0.970707i	$-0.422765\pi$
$509$	1.26550e28	0.480534	0.240267	+	0.970707i	$0.422765\pi$
$510$	0	0
$511$	8.11381e27	0.295669
$512$	2.15236e27	0.0768388
$513$	0	0
$514$	5.11316e27	0.175217
$515$	1.49746e28	0.502783
$516$	0	0
$517$	7.31084e27	0.235677
$518$	−4.35410e28	−1.37542
$519$	0	0
$520$	−1.15114e28	−0.349216
$521$	3.11290e28	0.925483	0.462742	−	0.886493i	$-0.346866\pi$
$521$	3.11290e28	0.925483	0.462742	+	0.886493i	$0.346866\pi$
$522$	0	0
$523$	−2.19095e28	−0.625699	−0.312850	−	0.949803i	$-0.601284\pi$
$523$	−2.19095e28	−0.625699	−0.312850	+	0.949803i	$0.601284\pi$
$524$	−1.62113e28	−0.453774
$525$	0	0
$526$	7.70346e28	2.07174
$527$	−1.18734e28	−0.313013
$528$	0	0
$529$	2.35210e28	0.595896
$530$	−2.09004e28	−0.519109
$531$	0	0
$532$	−6.51602e26	−0.0155565
$533$	7.98644e28	1.86947
$534$	0	0
$535$	1.51195e28	0.340269
$536$	−1.64982e28	−0.364088
$537$	0	0
$538$	−9.28378e28	−1.97020
$539$	2.48474e28	0.517129
$540$	0	0
$541$	8.64723e28	1.73104	0.865518	−	0.500879i	$-0.166990\pi$
$541$	8.64723e28	1.73104	0.865518	+	0.500879i	$0.166990\pi$
$542$	2.82952e28	0.555546
$543$	0	0
$544$	−1.97728e28	−0.373491
$545$	−2.78524e28	−0.516059
$546$	0	0
$547$	6.69174e28	1.19309	0.596544	−	0.802581i	$-0.296540\pi$
$547$	6.69174e28	1.19309	0.596544	+	0.802581i	$0.296540\pi$
$548$	7.34288e26	0.0128431
$549$	0	0
$550$	−2.41341e28	−0.406278
$551$	4.28517e27	0.0707744
$552$	0	0
$553$	−5.96559e28	−0.948505
$554$	−3.47745e28	−0.542511
$555$	0	0
$556$	7.58204e27	0.113894
$557$	6.14899e28	0.906409	0.453204	−	0.891407i	$-0.350281\pi$
$557$	6.14899e28	0.906409	0.453204	+	0.891407i	$0.350281\pi$
$558$	0	0
$559$	−4.43875e28	−0.630140
$560$	−3.32872e28	−0.463770
$561$	0	0
$562$	−1.64939e29	−2.21356
$563$	−1.12154e28	−0.147733	−0.0738663	−	0.997268i	$-0.523534\pi$
$563$	−1.12154e28	−0.147733	−0.0738663	+	0.997268i	$0.523534\pi$
$564$	0	0
$565$	3.22613e28	0.409424
$566$	−1.59067e28	−0.198157
$567$	0	0
$568$	−3.49434e28	−0.419476
$569$	1.24550e29	1.46779	0.733895	−	0.679263i	$-0.237700\pi$
$569$	1.24550e29	1.46779	0.733895	+	0.679263i	$0.237700\pi$
$570$	0	0
$571$	−6.70182e28	−0.761226	−0.380613	−	0.924734i	$-0.624287\pi$
$571$	−6.70182e28	−0.761226	−0.380613	+	0.924734i	$0.624287\pi$
$572$	6.90467e28	0.769990
$573$	0	0
$574$	1.55473e29	1.67139
$575$	2.39356e28	0.252658
$576$	0	0
$577$	1.60912e29	1.63773	0.818864	−	0.573987i	$-0.194604\pi$
$577$	1.60912e29	1.63773	0.818864	+	0.573987i	$0.194604\pi$
$578$	−9.04692e28	−0.904187
$579$	0	0
$580$	−2.90130e28	−0.279640
$581$	−1.37993e29	−1.30620
$582$	0	0
$583$	−1.83740e29	−1.67759
$584$	2.78087e28	0.249370
$585$	0	0
$586$	−7.01004e28	−0.606451
$587$	−1.88288e29	−1.60001	−0.800006	−	0.599992i	$-0.795170\pi$
$587$	−1.88288e29	−1.60001	−0.800006	+	0.599992i	$0.795170\pi$
$588$	0	0
$589$	−3.59706e27	−0.0294942
$590$	1.20723e29	0.972399
$591$	0	0
$592$	−2.21667e29	−1.72315
$593$	−2.04137e29	−1.55901	−0.779503	−	0.626398i	$-0.784529\pi$
$593$	−2.04137e29	−1.55901	−0.779503	+	0.626398i	$0.784529\pi$
$594$	0	0
$595$	2.46953e28	0.182048
$596$	8.01414e28	0.580459
$597$	0	0
$598$	−2.37325e29	−1.65951
$599$	−9.66401e28	−0.664012	−0.332006	−	0.943277i	$-0.607725\pi$
$599$	−9.66401e28	−0.664012	−0.332006	+	0.943277i	$0.607725\pi$
$600$	0	0
$601$	−1.22230e29	−0.810951	−0.405476	−	0.914106i	$-0.632894\pi$
$601$	−1.22230e29	−0.810951	−0.405476	+	0.914106i	$0.632894\pi$
$602$	−8.64095e28	−0.563376
$603$	0	0
$604$	2.23167e28	0.140521
$605$	−1.39900e29	−0.865740
$606$	0	0
$607$	−1.06271e29	−0.635235	−0.317617	−	0.948219i	$-0.602883\pi$
$607$	−1.06271e29	−0.635235	−0.317617	+	0.948219i	$0.602883\pi$
$608$	−5.99022e27	−0.0351929
$609$	0	0
$610$	8.00460e28	0.454335
$611$	2.73171e28	0.152406
$612$	0	0
$613$	3.11548e29	1.67954	0.839768	−	0.542945i	$-0.182691\pi$
$613$	3.11548e29	1.67954	0.839768	+	0.542945i	$0.182691\pi$
$614$	1.88244e29	0.997590
$615$	0	0
$616$	−1.97006e29	−1.00898
$617$	2.29331e29	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$617$	2.29331e29	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$618$	0	0
$619$	1.75961e29	0.856376	0.428188	−	0.903690i	$-0.359152\pi$
$619$	1.75961e29	0.856376	0.428188	+	0.903690i	$0.359152\pi$
$620$	2.43541e28	0.116536
$621$	0	0
$622$	1.20118e29	0.555660
$623$	1.96751e28	0.0894937
$624$	0	0
$625$	9.09495e27	0.0400000
$626$	1.19352e29	0.516180
$627$	0	0
$628$	1.37976e29	0.577067
$629$	1.64452e29	0.676404
$630$	0	0
$631$	−1.30487e28	−0.0519108	−0.0259554	−	0.999663i	$-0.508263\pi$
$631$	−1.30487e28	−0.0519108	−0.0259554	+	0.999663i	$0.508263\pi$
$632$	−2.04460e29	−0.799981
$633$	0	0
$634$	2.52442e29	0.955488
$635$	8.17511e27	0.0304347
$636$	0	0
$637$	9.28427e28	0.334413
$638$	−8.83952e29	−3.13193
$639$	0	0
$640$	−1.51368e29	−0.518972
$641$	3.70997e29	1.25130	0.625650	−	0.780104i	$-0.284834\pi$
$641$	3.70997e29	1.25130	0.625650	+	0.780104i	$0.284834\pi$
$642$	0	0
$643$	−1.15073e29	−0.375627	−0.187814	−	0.982205i	$-0.560140\pi$
$643$	−1.15073e29	−0.375627	−0.187814	+	0.982205i	$0.560140\pi$
$644$	−1.33308e29	−0.428110
$645$	0	0
$646$	8.52921e27	0.0265135
$647$	−2.45274e29	−0.750165	−0.375082	−	0.926991i	$-0.622386\pi$
$647$	−2.45274e29	−0.750165	−0.375082	+	0.926991i	$0.622386\pi$
$648$	0	0
$649$	1.06130e30	3.14247
$650$	−9.01776e28	−0.262729
$651$	0	0
$652$	1.97591e29	0.557400
$653$	−6.64107e29	−1.84353	−0.921764	−	0.387751i	$-0.873252\pi$
$653$	−6.64107e29	−1.84353	−0.921764	+	0.387751i	$0.873252\pi$
$654$	0	0
$655$	1.86133e29	0.500368
$656$	7.91510e29	2.09395
$657$	0	0
$658$	5.31783e28	0.136258
$659$	1.71049e29	0.431344	0.215672	−	0.976466i	$-0.430806\pi$
$659$	1.71049e29	0.431344	0.215672	+	0.976466i	$0.430806\pi$
$660$	0	0
$661$	−1.36074e29	−0.332399	−0.166200	−	0.986092i	$-0.553150\pi$
$661$	−1.36074e29	−0.332399	−0.166200	+	0.986092i	$0.553150\pi$
$662$	7.49502e29	1.80204
$663$	0	0
$664$	−4.72948e29	−1.10166
$665$	7.48148e27	0.0171538
$666$	0	0
$667$	8.76681e29	1.94769
$668$	1.50397e29	0.328918
$669$	0	0
$670$	−1.29243e29	−0.273918
$671$	7.03702e29	1.46826
$672$	0	0
$673$	−8.46772e29	−1.71241	−0.856206	−	0.516634i	$-0.827185\pi$
$673$	−8.46772e29	−1.71241	−0.856206	+	0.516634i	$0.827185\pi$
$674$	7.12900e29	1.41938
$675$	0	0
$676$	4.78559e28	0.0923625
$677$	−2.36669e29	−0.449738	−0.224869	−	0.974389i	$-0.572195\pi$
$677$	−2.36669e29	−0.449738	−0.224869	+	0.974389i	$0.572195\pi$
$678$	0	0
$679$	−1.53963e29	−0.283651
$680$	8.46387e28	0.153541
$681$	0	0
$682$	7.42007e29	1.30519
$683$	1.02867e30	1.78180	0.890898	−	0.454203i	$-0.150076\pi$
$683$	1.02867e30	1.78180	0.890898	+	0.454203i	$0.150076\pi$
$684$	0	0
$685$	−8.43084e27	−0.0141619
$686$	7.79586e29	1.28962
$687$	0	0
$688$	−4.39910e29	−0.705806
$689$	−6.86550e29	−1.08485
$690$	0	0
$691$	8.33280e29	1.27724	0.638619	−	0.769523i	$-0.279506\pi$
$691$	8.33280e29	1.27724	0.638619	+	0.769523i	$0.279506\pi$
$692$	−1.60680e29	−0.242577
$693$	0	0
$694$	−4.25981e29	−0.623901
$695$	−8.70545e28	−0.125589
$696$	0	0
$697$	−5.87209e29	−0.821956
$698$	−1.42571e28	−0.0196584
$699$	0	0
$700$	−5.06538e28	−0.0677770
$701$	−6.99649e29	−0.922234	−0.461117	−	0.887339i	$-0.652551\pi$
$701$	−6.99649e29	−0.922234	−0.461117	+	0.887339i	$0.652551\pi$
$702$	0	0
$703$	4.98209e28	0.0637354
$704$	−4.51583e29	−0.569148
$705$	0	0
$706$	−1.63965e29	−0.200586
$707$	1.34403e30	1.61997
$708$	0	0
$709$	2.89141e29	0.338318	0.169159	−	0.985589i	$-0.445895\pi$
$709$	2.89141e29	0.338318	0.169159	+	0.985589i	$0.445895\pi$
$710$	−2.73737e29	−0.315588
$711$	0	0
$712$	6.74330e28	0.0754801
$713$	−7.35904e29	−0.811673
$714$	0	0
$715$	−7.92771e29	−0.849053
$716$	9.76491e28	0.103058
$717$	0	0
$718$	−1.21343e29	−0.124368
$719$	−1.03572e29	−0.104614	−0.0523070	−	0.998631i	$-0.516657\pi$
$719$	−1.03572e29	−0.104614	−0.0523070	+	0.998631i	$0.516657\pi$
$720$	0	0
$721$	9.57575e29	0.939404
$722$	−1.22363e30	−1.18307
$723$	0	0
$724$	2.68321e29	0.251999
$725$	3.33117e29	0.308353
$726$	0	0
$727$	3.99248e29	0.359030	0.179515	−	0.983755i	$-0.442547\pi$
$727$	3.99248e29	0.359030	0.179515	+	0.983755i	$0.442547\pi$
$728$	−7.36116e29	−0.652479
$729$	0	0
$730$	2.17846e29	0.187611
$731$	3.26363e29	0.277056
$732$	0	0
$733$	1.02595e30	0.846317	0.423158	−	0.906056i	$-0.360921\pi$
$733$	1.02595e30	0.846317	0.423158	+	0.906056i	$0.360921\pi$
$734$	−2.29744e30	−1.86825
$735$	0	0
$736$	−1.22551e30	−0.968500
$737$	−1.13620e30	−0.885210
$738$	0	0
$739$	−5.23998e28	−0.0396792	−0.0198396	−	0.999803i	$-0.506316\pi$
$739$	−5.23998e28	−0.0396792	−0.0198396	+	0.999803i	$0.506316\pi$
$740$	−3.37315e29	−0.251827
$741$	0	0
$742$	−1.33651e30	−0.969910
$743$	−1.53956e30	−1.10158	−0.550788	−	0.834645i	$-0.685673\pi$
$743$	−1.53956e30	−1.10158	−0.550788	+	0.834645i	$0.685673\pi$
$744$	0	0
$745$	−9.20156e29	−0.640060
$746$	−1.65294e28	−0.0113370
$747$	0	0
$748$	−5.07671e29	−0.338545
$749$	9.66839e29	0.635763
$750$	0	0
$751$	1.28343e30	0.820639	0.410320	−	0.911942i	$-0.365417\pi$
$751$	1.28343e30	0.820639	0.410320	+	0.911942i	$0.365417\pi$
$752$	2.70731e29	0.170706
$753$	0	0
$754$	−3.30290e30	−2.02533
$755$	−2.56233e29	−0.154950
$756$	0	0
$757$	7.23137e29	0.425318	0.212659	−	0.977126i	$-0.431788\pi$
$757$	7.23137e29	0.425318	0.212659	+	0.977126i	$0.431788\pi$
$758$	−3.39622e30	−1.97001
$759$	0	0
$760$	2.56415e28	0.0144677
$761$	2.50610e29	0.139463	0.0697316	−	0.997566i	$-0.477786\pi$
$761$	2.50610e29	0.139463	0.0697316	+	0.997566i	$0.477786\pi$
$762$	0	0
$763$	−1.78106e30	−0.964210
$764$	3.49732e29	0.186748
$765$	0	0
$766$	3.68880e30	1.91639
$767$	3.96558e30	2.03215
$768$	0	0
$769$	−1.40362e30	−0.699880	−0.349940	−	0.936772i	$-0.613798\pi$
$769$	−1.40362e30	−0.699880	−0.349940	+	0.936772i	$0.613798\pi$
$770$	−1.54329e30	−0.759094
$771$	0	0
$772$	−7.72479e28	−0.0369747
$773$	1.31390e30	0.620410	0.310205	−	0.950670i	$-0.399602\pi$
$773$	1.31390e30	0.620410	0.310205	+	0.950670i	$0.399602\pi$
$774$	0	0
$775$	−2.79625e29	−0.128502
$776$	−5.27682e29	−0.239235
$777$	0	0
$778$	−4.12465e30	−1.82013
$779$	−1.77896e29	−0.0774503
$780$	0	0
$781$	−2.40648e30	−1.01988
$782$	1.74495e30	0.729645
$783$	0	0
$784$	9.20133e29	0.374569
$785$	−1.58419e30	−0.636321
$786$	0	0
$787$	−3.99554e30	−1.56257	−0.781287	−	0.624171i	$-0.785437\pi$
$787$	−3.99554e30	−1.56257	−0.781287	+	0.624171i	$0.785437\pi$
$788$	−1.04264e30	−0.402356
$789$	0	0
$790$	−1.60169e30	−0.601857
$791$	2.06300e30	0.764972
$792$	0	0
$793$	2.62940e30	0.949484
$794$	3.78308e30	1.34813
$795$	0	0
$796$	8.94914e29	0.310595
$797$	3.18052e30	1.08940	0.544698	−	0.838633i	$-0.316644\pi$
$797$	3.18052e30	1.08940	0.544698	+	0.838633i	$0.316644\pi$
$798$	0	0
$799$	−2.00851e29	−0.0670089
$800$	−4.65663e29	−0.153330
$801$	0	0
$802$	4.39953e30	1.41116
$803$	1.91513e30	0.606296
$804$	0	0
$805$	1.53060e30	0.472068
$806$	2.77252e30	0.844029
$807$	0	0
$808$	4.60643e30	1.36630
$809$	−2.04768e30	−0.599519	−0.299759	−	0.954015i	$-0.596906\pi$
$809$	−2.04768e30	−0.599519	−0.299759	+	0.954015i	$0.596906\pi$
$810$	0	0
$811$	2.02072e30	0.576485	0.288242	−	0.957557i	$-0.406929\pi$
$811$	2.02072e30	0.576485	0.288242	+	0.957557i	$0.406929\pi$
$812$	−1.85528e30	−0.522481
$813$	0	0
$814$	−1.02771e31	−2.82044
$815$	−2.26867e30	−0.614634
$816$	0	0
$817$	9.88722e28	0.0261061
$818$	−7.56354e30	−1.97158
$819$	0	0
$820$	1.20446e30	0.306017
$821$	3.31897e30	0.832531	0.416265	−	0.909243i	$-0.363339\pi$
$821$	3.31897e30	0.832531	0.416265	+	0.909243i	$0.363339\pi$
$822$	0	0
$823$	−8.46852e29	−0.207066	−0.103533	−	0.994626i	$-0.533015\pi$
$823$	−8.46852e29	−0.207066	−0.103533	+	0.994626i	$0.533015\pi$
$824$	3.28192e30	0.792305
$825$	0	0
$826$	7.71983e30	1.81684
$827$	7.77607e30	1.80697	0.903487	−	0.428614i	$-0.140998\pi$
$827$	7.77607e30	1.80697	0.903487	+	0.428614i	$0.140998\pi$
$828$	0	0
$829$	4.65905e30	1.05554	0.527770	−	0.849387i	$-0.323028\pi$
$829$	4.65905e30	1.05554	0.527770	+	0.849387i	$0.323028\pi$
$830$	−3.70495e30	−0.828825
$831$	0	0
$832$	−1.68735e30	−0.368053
$833$	−6.82633e29	−0.147033
$834$	0	0
$835$	−1.72681e30	−0.362691
$836$	−1.53800e29	−0.0319000
$837$	0	0
$838$	−2.10820e30	−0.426431
$839$	2.71344e29	0.0542025	0.0271013	−	0.999633i	$-0.491372\pi$
$839$	2.71344e29	0.0542025	0.0271013	+	0.999633i	$0.491372\pi$
$840$	0	0
$841$	7.06813e30	1.37704
$842$	−4.48965e28	−0.00863843
$843$	0	0
$844$	4.54289e28	0.00852584
$845$	−5.49466e29	−0.101846
$846$	0	0
$847$	−8.94614e30	−1.61756
$848$	−6.80417e30	−1.21512
$849$	0	0
$850$	6.63038e29	0.115515
$851$	1.01926e31	1.75398
$852$	0	0
$853$	9.28049e30	1.55814	0.779070	−	0.626937i	$-0.215692\pi$
$853$	9.28049e30	1.55814	0.779070	+	0.626937i	$0.215692\pi$
$854$	5.11866e30	0.848885
$855$	0	0
$856$	3.31367e30	0.536210
$857$	−4.26068e30	−0.681053	−0.340526	−	0.940235i	$-0.610605\pi$
$857$	−4.26068e30	−0.681053	−0.340526	+	0.940235i	$0.610605\pi$
$858$	0	0
$859$	−1.38058e30	−0.215344	−0.107672	−	0.994186i	$-0.534340\pi$
$859$	−1.38058e30	−0.215344	−0.107672	+	0.994186i	$0.534340\pi$
$860$	−6.69420e29	−0.103149
$861$	0	0
$862$	6.33303e30	0.952325
$863$	−6.77274e30	−1.00612	−0.503061	−	0.864251i	$-0.667793\pi$
$863$	−6.77274e30	−1.00612	−0.503061	+	0.864251i	$0.667793\pi$
$864$	0	0
$865$	1.84488e30	0.267484
$866$	8.36789e30	1.19861
$867$	0	0
$868$	1.55736e30	0.217737
$869$	−1.40808e31	−1.94500
$870$	0	0
$871$	−4.24544e30	−0.572442
$872$	−6.10428e30	−0.813226
$873$	0	0
$874$	5.28636e29	0.0687522
$875$	5.81590e29	0.0747364
$876$	0	0
$877$	7.75502e30	0.972943	0.486471	−	0.873697i	$-0.338284\pi$
$877$	7.75502e30	0.972943	0.486471	+	0.873697i	$0.338284\pi$
$878$	1.50581e31	1.86672
$879$	0	0
$880$	−7.85690e30	−0.951005
$881$	−1.13987e31	−1.36335	−0.681677	−	0.731653i	$-0.738749\pi$
$881$	−1.13987e31	−1.36335	−0.681677	+	0.731653i	$0.738749\pi$
$882$	0	0
$883$	−5.95571e28	−0.00695579	−0.00347789	−	0.999994i	$-0.501107\pi$
$883$	−5.95571e28	−0.00695579	−0.00347789	+	0.999994i	$0.501107\pi$
$884$	−1.89692e30	−0.218928
$885$	0	0
$886$	−4.11903e29	−0.0464238
$887$	−1.42633e31	−1.58863	−0.794315	−	0.607506i	$-0.792170\pi$
$887$	−1.42633e31	−1.58863	−0.794315	+	0.607506i	$0.792170\pi$
$888$	0	0
$889$	5.22770e29	0.0568646
$890$	5.28252e29	0.0567867
$891$	0	0
$892$	8.85904e29	0.0930156
$893$	−6.08481e28	−0.00631404
$894$	0	0
$895$	−1.12117e30	−0.113640
$896$	−9.67945e30	−0.969653
$897$	0	0
$898$	2.02918e30	0.198572
$899$	−1.02417e31	−0.990597
$900$	0	0
$901$	5.04791e30	0.476981
$902$	3.66967e31	3.42735
$903$	0	0
$904$	7.07056e30	0.645186
$905$	−3.08077e30	−0.277875
$906$	0	0
$907$	4.56820e30	0.402595	0.201297	−	0.979530i	$-0.435484\pi$
$907$	4.56820e30	0.402595	0.201297	+	0.979530i	$0.435484\pi$
$908$	−3.94880e30	−0.344005
$909$	0	0
$910$	−5.76654e30	−0.490886
$911$	1.83971e31	1.54813	0.774065	−	0.633107i	$-0.218220\pi$
$911$	1.83971e31	1.54813	0.774065	+	0.633107i	$0.218220\pi$
$912$	0	0
$913$	−3.25711e31	−2.67848
$914$	2.14410e31	1.74305
$915$	0	0
$916$	−3.21317e30	−0.255288
$917$	1.19025e31	0.934893
$918$	0	0
$919$	−6.31613e29	−0.0484884	−0.0242442	−	0.999706i	$-0.507718\pi$
$919$	−6.31613e29	−0.0484884	−0.0242442	+	0.999706i	$0.507718\pi$
$920$	5.24586e30	0.398148
$921$	0	0
$922$	8.26945e30	0.613482
$923$	−8.99187e30	−0.659527
$924$	0	0
$925$	3.87294e30	0.277685
$926$	4.76496e30	0.337788
$927$	0	0
$928$	−1.70557e31	−1.18199
$929$	−1.48189e31	−1.01543	−0.507717	−	0.861524i	$-0.669511\pi$
$929$	−1.48189e31	−1.01543	−0.507717	+	0.861524i	$0.669511\pi$
$930$	0	0
$931$	−2.06805e29	−0.0138544
$932$	−9.93996e30	−0.658442
$933$	0	0
$934$	2.24605e31	1.45471
$935$	5.82891e30	0.373306
$936$	0	0
$937$	1.44221e31	0.903154	0.451577	−	0.892232i	$-0.350862\pi$
$937$	1.44221e31	0.903154	0.451577	+	0.892232i	$0.350862\pi$
$938$	−8.26461e30	−0.511791
$939$	0	0
$940$	4.11976e29	0.0249476
$941$	3.60517e30	0.215891	0.107945	−	0.994157i	$-0.465573\pi$
$941$	3.60517e30	0.215891	0.107945	+	0.994157i	$0.465573\pi$
$942$	0	0
$943$	−3.63949e31	−2.13141
$944$	3.93016e31	2.27617
$945$	0	0
$946$	−2.03955e31	−1.15526
$947$	−9.57878e30	−0.536581	−0.268291	−	0.963338i	$-0.586459\pi$
$947$	−9.57878e30	−0.536581	−0.268291	+	0.963338i	$0.586459\pi$
$948$	0	0
$949$	7.15592e30	0.392076
$950$	2.00869e29	0.0108846
$951$	0	0
$952$	5.41235e30	0.286878
$953$	−1.99764e31	−1.04723	−0.523614	−	0.851956i	$-0.675417\pi$
$953$	−1.99764e31	−1.04723	−0.523614	+	0.851956i	$0.675417\pi$
$954$	0	0
$955$	−4.01551e30	−0.205923
$956$	1.18743e31	0.602282
$957$	0	0
$958$	3.40143e31	1.68781
$959$	−5.39123e29	−0.0264602
$960$	0	0
$961$	−1.22284e31	−0.587183
$962$	−3.84007e31	−1.82390
$963$	0	0
$964$	1.55757e31	0.723833
$965$	8.86935e29	0.0407713
$966$	0	0
$967$	−1.46532e30	−0.0659105	−0.0329552	−	0.999457i	$-0.510492\pi$
$967$	−1.46532e30	−0.0659105	−0.0329552	+	0.999457i	$0.510492\pi$
$968$	−3.06613e31	−1.36427
$969$	0	0
$970$	−4.13373e30	−0.179986
$971$	2.87520e31	1.23841	0.619207	−	0.785228i	$-0.287454\pi$
$971$	2.87520e31	1.23841	0.619207	+	0.785228i	$0.287454\pi$
$972$	0	0
$973$	−5.56683e30	−0.234651
$974$	−3.68141e31	−1.53513
$975$	0	0
$976$	2.60591e31	1.06350
$977$	−1.39361e31	−0.562662	−0.281331	−	0.959611i	$-0.590776\pi$
$977$	−1.39361e31	−0.562662	−0.281331	+	0.959611i	$0.590776\pi$
$978$	0	0
$979$	4.64398e30	0.183515
$980$	1.40019e30	0.0547409
$981$	0	0
$982$	2.82891e31	1.08255
$983$	1.40200e31	0.530805	0.265403	−	0.964138i	$-0.414495\pi$
$983$	1.40200e31	0.530805	0.265403	+	0.964138i	$0.414495\pi$
$984$	0	0
$985$	1.19713e31	0.443670
$986$	2.42848e31	0.890487
$987$	0	0
$988$	−5.74677e29	−0.0206289
$989$	2.02278e31	0.718434
$990$	0	0
$991$	−5.54876e31	−1.92940	−0.964701	−	0.263349i	$-0.915173\pi$
$991$	−5.54876e31	−1.92940	−0.964701	+	0.263349i	$0.915173\pi$
$992$	1.43169e31	0.492579
$993$	0	0
$994$	−1.75045e31	−0.589649
$995$	−1.02751e31	−0.342487
$996$	0	0
$997$	−5.26811e31	−1.71932	−0.859659	−	0.510869i	$-0.829324\pi$
$997$	−5.26811e31	−1.71932	−0.859659	+	0.510869i	$0.829324\pi$
$998$	−4.22709e31	−1.36512
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.22.a.g.1.4		4
3.2	odd	2		15.22.a.e.1.1	✓	4
15.2	even	4		75.22.b.h.49.1		8
15.8	even	4		75.22.b.h.49.8		8
15.14	odd	2		75.22.a.h.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.22.a.e.1.1	✓	4	3.2	odd	2
45.22.a.g.1.4		4	1.1	even	1	trivial
75.22.a.h.1.4		4	15.14	odd	2
75.22.b.h.49.1		8	15.2	even	4
75.22.b.h.49.8		8	15.8	even	4

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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	45.22.a.g.1.4

Level	$45$
Weight	$22$
Character	45.1
Self dual	yes
Analytic conductor	$125.765$
Analytic rank	$0$
Dimension	$4$
CM	no
Inner twists	$1$