LMFDB - Embedded newform 45.22.a.g.1.1

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			1.1
Root			$-1711.97$ 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)

$f(q)$	$=$	\(q-1487.97 q^{2} +116903. q^{4} -9.76562e6 q^{5} -1.26833e9 q^{7} +2.94655e9 q^{8} +1.45310e10 q^{10} -4.25851e10 q^{11} +7.14056e11 q^{13} +1.88724e12 q^{14} -4.62954e12 q^{16} -9.08430e12 q^{17} -2.62762e13 q^{19} -1.14163e12 q^{20} +6.33654e13 q^{22} +2.96171e13 q^{23} +9.53674e13 q^{25} -1.06249e15 q^{26} -1.48271e14 q^{28} -4.45551e15 q^{29} +4.93427e14 q^{31} +7.09254e14 q^{32} +1.35172e16 q^{34} +1.23860e16 q^{35} +8.82960e15 q^{37} +3.90982e16 q^{38} -2.87749e16 q^{40} -1.17100e17 q^{41} -4.77156e16 q^{43} -4.97831e15 q^{44} -4.40694e16 q^{46} -3.81704e17 q^{47} +1.05012e18 q^{49} -1.41904e17 q^{50} +8.34750e16 q^{52} -1.01485e18 q^{53} +4.15870e17 q^{55} -3.73720e18 q^{56} +6.62967e18 q^{58} +8.26140e17 q^{59} -6.60518e18 q^{61} -7.34204e17 q^{62} +8.65351e18 q^{64} -6.97320e18 q^{65} -2.57407e19 q^{67} -1.06198e18 q^{68} -1.84301e19 q^{70} -3.32319e19 q^{71} -2.66123e19 q^{73} -1.31382e19 q^{74} -3.07176e18 q^{76} +5.40120e19 q^{77} -7.85851e19 q^{79} +4.52104e19 q^{80} +1.74241e20 q^{82} -1.68959e20 q^{83} +8.87139e19 q^{85} +7.09994e19 q^{86} -1.25479e20 q^{88} -6.58830e19 q^{89} -9.05659e20 q^{91} +3.46232e18 q^{92} +5.67964e20 q^{94} +2.56604e20 q^{95} -1.13619e21 q^{97} -1.56254e21 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1487.97	−1.02749	−0.513747	−	0.857942i	$-0.671743\pi$
$2$	−1487.97	−1.02749	−0.513747	+	0.857942i	$0.671743\pi$
$3$	0	0
$3$	0	0
$4$	116903.	0.0557435
$5$	−9.76562e6	−0.447214
$5$	−9.76562e6	−0.447214
$6$	0	0
$7$	−1.26833e9	−1.69708	−0.848541	−	0.529129i	$-0.822519\pi$
$7$	−1.26833e9	−1.69708	−0.848541	+	0.529129i	$0.822519\pi$
$8$	2.94655e9	0.970218
$9$	0	0
$10$	1.45310e10	0.459509
$11$	−4.25851e10	−0.495033	−0.247517	−	0.968884i	$-0.579615\pi$
$11$	−4.25851e10	−0.495033	−0.247517	+	0.968884i	$0.579615\pi$
$12$	0	0
$13$	7.14056e11	1.43657	0.718285	−	0.695749i	$-0.244927\pi$
$13$	7.14056e11	1.43657	0.718285	+	0.695749i	$0.244927\pi$
$14$	1.88724e12	1.74374
$15$	0	0
$16$	−4.62954e12	−1.05264
$17$	−9.08430e12	−1.09289	−0.546447	−	0.837494i	$-0.684020\pi$
$17$	−9.08430e12	−1.09289	−0.546447	+	0.837494i	$0.684020\pi$
$18$	0	0
$19$	−2.62762e13	−0.983219	−0.491609	−	0.870816i	$-0.663591\pi$
$19$	−2.62762e13	−0.983219	−0.491609	+	0.870816i	$0.663591\pi$
$20$	−1.14163e12	−0.0249293
$21$	0	0
$22$	6.33654e13	0.508644
$23$	2.96171e13	0.149074	0.0745368	−	0.997218i	$-0.476252\pi$
$23$	2.96171e13	0.149074	0.0745368	+	0.997218i	$0.476252\pi$
$24$	0	0
$25$	9.53674e13	0.200000
$26$	−1.06249e15	−1.47607
$27$	0	0
$28$	−1.48271e14	−0.0946014
$29$	−4.45551e15	−1.96661	−0.983305	−	0.181963i	$-0.941755\pi$
$29$	−4.45551e15	−1.96661	−0.983305	+	0.181963i	$0.941755\pi$
$30$	0	0
$31$	4.93427e14	0.108125	0.0540623	−	0.998538i	$-0.482783\pi$
$31$	4.93427e14	0.108125	0.0540623	+	0.998538i	$0.482783\pi$
$32$	7.09254e14	0.111359
$33$	0	0
$34$	1.35172e16	1.12294
$35$	1.23860e16	0.758958
$36$	0	0
$37$	8.82960e15	0.301872	0.150936	−	0.988544i	$-0.451771\pi$
$37$	8.82960e15	0.301872	0.150936	+	0.988544i	$0.451771\pi$
$38$	3.90982e16	1.01025
$39$	0	0
$40$	−2.87749e16	−0.433895
$41$	−1.17100e17	−1.36247	−0.681235	−	0.732065i	$-0.738557\pi$
$41$	−1.17100e17	−1.36247	−0.681235	+	0.732065i	$0.738557\pi$
$42$	0	0
$43$	−4.77156e16	−0.336699	−0.168350	−	0.985727i	$-0.553844\pi$
$43$	−4.77156e16	−0.336699	−0.168350	+	0.985727i	$0.553844\pi$
$44$	−4.97831e15	−0.0275949
$45$	0	0
$46$	−4.40694e16	−0.153172
$47$	−3.81704e17	−1.05852	−0.529260	−	0.848460i	$-0.677530\pi$
$47$	−3.81704e17	−1.05852	−0.529260	+	0.848460i	$0.677530\pi$
$48$	0	0
$49$	1.05012e18	1.88009
$50$	−1.41904e17	−0.205499
$51$	0	0
$52$	8.34750e16	0.0800795
$53$	−1.01485e18	−0.797086	−0.398543	−	0.917150i	$-0.630484\pi$
$53$	−1.01485e18	−0.797086	−0.398543	+	0.917150i	$0.630484\pi$
$54$	0	0
$55$	4.15870e17	0.221386
$56$	−3.73720e18	−1.64654
$57$	0	0
$58$	6.62967e18	2.02068
$59$	8.26140e17	0.210430	0.105215	−	0.994450i	$-0.466447\pi$
$59$	8.26140e17	0.210430	0.105215	+	0.994450i	$0.466447\pi$
$60$	0	0
$61$	−6.60518e18	−1.18555	−0.592777	−	0.805367i	$-0.701969\pi$
$61$	−6.60518e18	−1.18555	−0.592777	+	0.805367i	$0.701969\pi$
$62$	−7.34204e17	−0.111097
$63$	0	0
$64$	8.65351e18	0.938215
$65$	−6.97320e18	−0.642454
$66$	0	0
$67$	−2.57407e19	−1.72518	−0.862590	−	0.505904i	$-0.831159\pi$
$67$	−2.57407e19	−1.72518	−0.862590	+	0.505904i	$0.831159\pi$
$68$	−1.06198e18	−0.0609218
$69$	0	0
$70$	−1.84301e19	−0.779825
$71$	−3.32319e19	−1.21155	−0.605777	−	0.795635i	$-0.707137\pi$
$71$	−3.32319e19	−1.21155	−0.605777	+	0.795635i	$0.707137\pi$
$72$	0	0
$73$	−2.66123e19	−0.724757	−0.362378	−	0.932031i	$-0.618035\pi$
$73$	−2.66123e19	−0.724757	−0.362378	+	0.932031i	$0.618035\pi$
$74$	−1.31382e19	−0.310172
$75$	0	0
$76$	−3.07176e18	−0.0548081
$77$	5.40120e19	0.840112
$78$	0	0
$79$	−7.85851e19	−0.933804	−0.466902	−	0.884309i	$-0.654630\pi$
$79$	−7.85851e19	−0.933804	−0.466902	+	0.884309i	$0.654630\pi$
$80$	4.52104e19	0.470753
$81$	0	0
$82$	1.74241e20	1.39993
$83$	−1.68959e20	−1.19526	−0.597630	−	0.801772i	$-0.703891\pi$
$83$	−1.68959e20	−1.19526	−0.597630	+	0.801772i	$0.703891\pi$
$84$	0	0
$85$	8.87139e19	0.488757
$86$	7.09994e19	0.345956
$87$	0	0
$88$	−1.25479e20	−0.480290
$89$	−6.58830e19	−0.223964	−0.111982	−	0.993710i	$-0.535720\pi$
$89$	−6.58830e19	−0.223964	−0.111982	+	0.993710i	$0.535720\pi$
$90$	0	0
$91$	−9.05659e20	−2.43798
$92$	3.46232e18	0.00830989
$93$	0	0
$94$	5.67964e20	1.08762
$95$	2.56604e20	0.439709
$96$	0	0
$97$	−1.13619e21	−1.56440	−0.782201	−	0.623026i	$-0.785903\pi$
$97$	−1.13619e21	−1.56440	−0.782201	+	0.623026i	$0.785903\pi$
$98$	−1.56254e21	−1.93178
$99$	0	0
$100$	1.11487e19	0.0111487
$101$	−6.08234e20	−0.547894	−0.273947	−	0.961745i	$-0.588329\pi$
$101$	−6.08234e20	−0.547894	−0.273947	+	0.961745i	$0.588329\pi$
$102$	0	0
$103$	1.93831e21	1.42113	0.710564	−	0.703633i	$-0.248440\pi$
$103$	1.93831e21	1.42113	0.710564	+	0.703633i	$0.248440\pi$
$104$	2.10400e21	1.39379
$105$	0	0
$106$	1.51007e21	0.819001
$107$	1.21945e20	0.0599286	0.0299643	−	0.999551i	$-0.490461\pi$
$107$	1.21945e20	0.0599286	0.0299643	+	0.999551i	$0.490461\pi$
$108$	0	0
$109$	4.65871e21	1.88490	0.942448	−	0.334351i	$-0.108517\pi$
$109$	4.65871e21	1.88490	0.942448	+	0.334351i	$0.108517\pi$
$110$	−6.18802e20	−0.227472
$111$	0	0
$112$	5.87179e21	1.78641
$113$	−1.27385e21	−0.353017	−0.176509	−	0.984299i	$-0.556480\pi$
$113$	−1.27385e21	−0.353017	−0.176509	+	0.984299i	$0.556480\pi$
$114$	0	0
$115$	−2.89230e20	−0.0666677
$116$	−5.20861e20	−0.109626
$117$	0	0
$118$	−1.22927e21	−0.216215
$119$	1.15219e22	1.85473
$120$	0	0
$121$	−5.58676e21	−0.754942
$122$	9.82831e21	1.21815
$123$	0	0
$124$	5.76829e19	0.00602725
$125$	−9.31323e20	−0.0894427
$126$	0	0
$127$	−1.08103e22	−0.878821	−0.439411	−	0.898286i	$-0.644813\pi$
$127$	−1.08103e22	−0.878821	−0.439411	+	0.898286i	$0.644813\pi$
$128$	−1.43636e22	−1.07537
$129$	0	0
$130$	1.03759e22	0.660118
$131$	2.63526e22	1.54694	0.773472	−	0.633830i	$-0.218518\pi$
$131$	2.63526e22	1.54694	0.773472	+	0.633830i	$0.218518\pi$
$132$	0	0
$133$	3.33269e22	1.66860
$134$	3.83013e22	1.77261
$135$	0	0
$136$	−2.67674e22	−1.06034
$137$	1.96734e22	0.721629	0.360815	−	0.932638i	$-0.382499\pi$
$137$	1.96734e22	0.721629	0.360815	+	0.932638i	$0.382499\pi$
$138$	0	0
$139$	1.37361e22	0.432722	0.216361	−	0.976313i	$-0.430581\pi$
$139$	1.37361e22	0.432722	0.216361	+	0.976313i	$0.430581\pi$
$140$	1.44796e21	0.0423070
$141$	0	0
$142$	4.94481e22	1.24486
$143$	−3.04081e22	−0.711151
$144$	0	0
$145$	4.35108e22	0.879495
$146$	3.95983e22	0.744683
$147$	0	0
$148$	1.03220e21	0.0168274
$149$	−9.14642e22	−1.38930	−0.694649	−	0.719349i	$-0.744440\pi$
$149$	−9.14642e22	−1.38930	−0.694649	+	0.719349i	$0.744440\pi$
$150$	0	0
$151$	1.22303e23	1.61502	0.807511	−	0.589853i	$-0.200814\pi$
$151$	1.22303e23	1.61502	0.807511	+	0.589853i	$0.200814\pi$
$152$	−7.74243e22	−0.953936
$153$	0	0
$154$	−8.03682e22	−0.863210
$155$	−4.81862e21	−0.0483548
$156$	0	0
$157$	−1.10353e23	−0.967918	−0.483959	−	0.875091i	$-0.660802\pi$
$157$	−1.10353e23	−0.967918	−0.483959	+	0.875091i	$0.660802\pi$
$158$	1.16932e23	0.959477
$159$	0	0
$160$	−6.92631e21	−0.0498015
$161$	−3.75643e22	−0.252990
$162$	0	0
$163$	3.12086e23	1.84631	0.923155	−	0.384429i	$-0.125602\pi$
$163$	3.12086e23	1.84631	0.923155	+	0.384429i	$0.125602\pi$
$164$	−1.36893e22	−0.0759488
$165$	0	0
$166$	2.51406e23	1.22812
$167$	−5.58649e22	−0.256222	−0.128111	−	0.991760i	$-0.540891\pi$
$167$	−5.58649e22	−0.256222	−0.128111	+	0.991760i	$0.540891\pi$
$168$	0	0
$169$	2.62811e23	1.06374
$170$	−1.32004e23	−0.502195
$171$	0	0
$172$	−5.57809e21	−0.0187688
$173$	1.34399e23	0.425512	0.212756	−	0.977105i	$-0.431756\pi$
$173$	1.34399e23	0.425512	0.212756	+	0.977105i	$0.431756\pi$
$174$	0	0
$175$	−1.20957e23	−0.339416
$176$	1.97150e23	0.521090
$177$	0	0
$178$	9.80319e22	0.230122
$179$	−5.77287e23	−1.27772	−0.638859	−	0.769324i	$-0.720593\pi$
$179$	−5.77287e23	−1.27772	−0.638859	+	0.769324i	$0.720593\pi$
$180$	0	0
$181$	2.15236e23	0.423925	0.211963	−	0.977278i	$-0.432014\pi$
$181$	2.15236e23	0.423925	0.211963	+	0.977278i	$0.432014\pi$
$182$	1.34759e24	2.50501
$183$	0	0
$184$	8.72684e22	0.144634
$185$	−8.62266e22	−0.135001
$186$	0	0
$187$	3.86856e23	0.541019
$188$	−4.46222e22	−0.0590057
$189$	0	0
$190$	−3.81819e23	−0.451798
$191$	−4.41173e23	−0.494036	−0.247018	−	0.969011i	$-0.579451\pi$
$191$	−4.41173e23	−0.494036	−0.247018	+	0.969011i	$0.579451\pi$
$192$	0	0
$193$	−7.07886e23	−0.710578	−0.355289	−	0.934757i	$-0.615617\pi$
$193$	−7.07886e23	−0.710578	−0.355289	+	0.934757i	$0.615617\pi$
$194$	1.69062e24	1.60741
$195$	0	0
$196$	1.22761e23	0.104803
$197$	−2.25891e23	−0.182812	−0.0914058	−	0.995814i	$-0.529136\pi$
$197$	−2.25891e23	−0.182812	−0.0914058	+	0.995814i	$0.529136\pi$
$198$	0	0
$199$	−5.29674e22	−0.0385524	−0.0192762	−	0.999814i	$-0.506136\pi$
$199$	−5.29674e22	−0.0385524	−0.0192762	+	0.999814i	$0.506136\pi$
$200$	2.81005e23	0.194044
$201$	0	0
$202$	9.05034e23	0.562957
$203$	5.65106e24	3.33750
$204$	0	0
$205$	1.14356e24	0.609315
$206$	−2.88415e24	−1.46020
$207$	0	0
$208$	−3.30575e24	−1.51219
$209$	1.11898e24	0.486726
$210$	0	0
$211$	2.91715e24	1.14813	0.574067	−	0.818808i	$-0.305365\pi$
$211$	2.91715e24	1.14813	0.574067	+	0.818808i	$0.305365\pi$
$212$	−1.18639e23	−0.0444324
$213$	0	0
$214$	−1.81450e23	−0.0615763
$215$	4.65973e23	0.150576
$216$	0	0
$217$	−6.25828e23	−0.183496
$218$	−6.93202e24	−1.93672
$219$	0	0
$220$	4.86163e22	0.0123408
$221$	−6.48670e24	−1.57002
$222$	0	0
$223$	4.10996e24	0.904975	0.452487	−	0.891771i	$-0.350537\pi$
$223$	4.10996e24	0.904975	0.452487	+	0.891771i	$0.350537\pi$
$224$	−8.99569e23	−0.188986
$225$	0	0
$226$	1.89546e24	0.362723
$227$	−4.85462e24	−0.886919	−0.443460	−	0.896294i	$-0.646249\pi$
$227$	−4.85462e24	−0.886919	−0.443460	+	0.896294i	$0.646249\pi$
$228$	0	0
$229$	−3.91845e24	−0.652893	−0.326446	−	0.945216i	$-0.605851\pi$
$229$	−3.91845e24	−0.652893	−0.326446	+	0.945216i	$0.605851\pi$
$230$	4.30365e23	0.0685007
$231$	0	0
$232$	−1.31284e25	−1.90804
$233$	−1.12801e25	−1.56702	−0.783511	−	0.621378i	$-0.786573\pi$
$233$	−1.12801e25	−1.56702	−0.783511	+	0.621378i	$0.786573\pi$
$234$	0	0
$235$	3.72758e24	0.473385
$236$	9.65779e22	0.0117301
$237$	0	0
$238$	−1.71442e25	−1.90572
$239$	9.29369e24	0.988577	0.494288	−	0.869298i	$-0.335429\pi$
$239$	9.29369e24	0.988577	0.494288	+	0.869298i	$0.335429\pi$
$240$	0	0
$241$	−2.88908e24	−0.281567	−0.140783	−	0.990040i	$-0.544962\pi$
$241$	−2.88908e24	−0.281567	−0.140783	+	0.990040i	$0.544962\pi$
$242$	8.31293e24	0.775698
$243$	0	0
$244$	−7.72163e23	−0.0660870
$245$	−1.02550e25	−0.840801
$246$	0	0
$247$	−1.87627e25	−1.41246
$248$	1.45391e24	0.104904
$249$	0	0
$250$	1.38578e24	0.0919018
$251$	−1.09500e25	−0.696370	−0.348185	−	0.937426i	$-0.613202\pi$
$251$	−1.09500e25	−0.696370	−0.348185	+	0.937426i	$0.613202\pi$
$252$	0	0
$253$	−1.26125e24	−0.0737964
$254$	1.60855e25	0.902983
$255$	0	0
$256$	3.22485e24	0.166721
$257$	1.13821e25	0.564840	0.282420	−	0.959291i	$-0.408863\pi$
$257$	1.13821e25	0.564840	0.282420	+	0.959291i	$0.408863\pi$
$258$	0	0
$259$	−1.11988e25	−0.512302
$260$	−8.15186e23	−0.0358127
$261$	0	0
$262$	−3.92119e25	−1.58948
$263$	−3.72385e23	−0.0145030	−0.00725148	−	0.999974i	$-0.502308\pi$
$263$	−3.72385e23	−0.0145030	−0.00725148	+	0.999974i	$0.502308\pi$
$264$	0	0
$265$	9.91065e24	0.356468
$266$	−4.95895e25	−1.71448
$267$	0	0
$268$	−3.00915e24	−0.0961676
$269$	−3.72748e25	−1.14556	−0.572779	−	0.819710i	$-0.694135\pi$
$269$	−3.72748e25	−1.14556	−0.572779	+	0.819710i	$0.694135\pi$
$270$	0	0
$271$	2.21667e25	0.630267	0.315133	−	0.949047i	$-0.397951\pi$
$271$	2.21667e25	0.630267	0.315133	+	0.949047i	$0.397951\pi$
$272$	4.20562e25	1.15042
$273$	0	0
$274$	−2.92735e25	−0.741470
$275$	−4.06123e24	−0.0990067
$276$	0	0
$277$	1.52006e25	0.343419	0.171709	−	0.985148i	$-0.445071\pi$
$277$	1.52006e25	0.343419	0.171709	+	0.985148i	$0.445071\pi$
$278$	−2.04389e25	−0.444619
$279$	0	0
$280$	3.64961e25	0.736355
$281$	4.82436e25	0.937612	0.468806	−	0.883301i	$-0.344684\pi$
$281$	4.82436e25	0.937612	0.468806	+	0.883301i	$0.344684\pi$
$282$	0	0
$283$	3.95762e25	0.713965	0.356983	−	0.934111i	$-0.383806\pi$
$283$	3.95762e25	0.713965	0.356983	+	0.934111i	$0.383806\pi$
$284$	−3.88490e24	−0.0675362
$285$	0	0
$286$	4.52464e25	0.730703
$287$	1.48522e26	2.31222
$288$	0	0
$289$	1.34326e25	0.194416
$290$	−6.47428e25	−0.903676
$291$	0	0
$292$	−3.11105e24	−0.0404005
$293$	6.72372e25	0.842364	0.421182	−	0.906976i	$-0.361615\pi$
$293$	6.72372e25	0.842364	0.421182	+	0.906976i	$0.361615\pi$
$294$	0	0
$295$	−8.06777e24	−0.0941071
$296$	2.60169e25	0.292882
$297$	0	0
$298$	1.36096e26	1.42749
$299$	2.11483e25	0.214155
$300$	0	0
$301$	6.05192e25	0.571406
$302$	−1.81983e26	−1.65942
$303$	0	0
$304$	1.21647e26	1.03497
$305$	6.45037e25	0.530196
$306$	0	0
$307$	−1.82823e25	−0.140307	−0.0701533	−	0.997536i	$-0.522349\pi$
$307$	−1.82823e25	−0.140307	−0.0701533	+	0.997536i	$0.522349\pi$
$308$	6.31414e24	0.0468308
$309$	0	0
$310$	7.16996e24	0.0496843
$311$	−1.19682e26	−0.801762	−0.400881	−	0.916130i	$-0.631296\pi$
$311$	−1.19682e26	−0.801762	−0.400881	+	0.916130i	$0.631296\pi$
$312$	0	0
$313$	−2.72575e26	−1.70715	−0.853573	−	0.520974i	$-0.825569\pi$
$313$	−2.72575e26	−1.70715	−0.853573	+	0.520974i	$0.825569\pi$
$314$	1.64202e26	0.994530
$315$	0	0
$316$	−9.18680e24	−0.0520535
$317$	6.50489e25	0.356548	0.178274	−	0.983981i	$-0.442949\pi$
$317$	6.50489e25	0.356548	0.178274	+	0.983981i	$0.442949\pi$
$318$	0	0
$319$	1.89738e26	0.973538
$320$	−8.45069e25	−0.419583
$321$	0	0
$322$	5.58945e25	0.259946
$323$	2.38701e26	1.07455
$324$	0	0
$325$	6.80977e25	0.287314
$326$	−4.64375e26	−1.89707
$327$	0	0
$328$	−3.45041e26	−1.32189
$329$	4.84127e26	1.79640
$330$	0	0
$331$	2.10948e26	0.734482	0.367241	−	0.930126i	$-0.380302\pi$
$331$	2.10948e26	0.734482	0.367241	+	0.930126i	$0.380302\pi$
$332$	−1.97518e25	−0.0666280
$333$	0	0
$334$	8.31254e25	0.263267
$335$	2.51374e26	0.771524
$336$	0	0
$337$	1.58978e26	0.458377	0.229189	−	0.973382i	$-0.426393\pi$
$337$	1.58978e26	0.458377	0.229189	+	0.973382i	$0.426393\pi$
$338$	−3.91056e26	−1.09298
$339$	0	0
$340$	1.03709e25	0.0272450
$341$	−2.10126e25	−0.0535253
$342$	0	0
$343$	−6.23473e26	−1.49358
$344$	−1.40597e26	−0.326671
$345$	0	0
$346$	−1.99982e26	−0.437211
$347$	3.27291e26	0.694184	0.347092	−	0.937831i	$-0.387169\pi$
$347$	3.27291e26	0.694184	0.347092	+	0.937831i	$0.387169\pi$
$348$	0	0
$349$	6.28483e26	1.25495	0.627475	−	0.778637i	$-0.284088\pi$
$349$	6.28483e26	1.25495	0.627475	+	0.778637i	$0.284088\pi$
$350$	1.79981e26	0.348748
$351$	0	0
$352$	−3.02037e25	−0.0551267
$353$	−5.78370e26	−1.02464	−0.512320	−	0.858795i	$-0.671214\pi$
$353$	−5.78370e26	−1.02464	−0.512320	+	0.858795i	$0.671214\pi$
$354$	0	0
$355$	3.24530e26	0.541823
$356$	−7.70189e24	−0.0124845
$357$	0	0
$358$	8.58985e26	1.31285
$359$	5.85883e25	0.0869599	0.0434799	−	0.999054i	$-0.486156\pi$
$359$	5.85883e25	0.0869599	0.0434799	+	0.999054i	$0.486156\pi$
$360$	0	0
$361$	−2.37694e25	−0.0332807
$362$	−3.20264e26	−0.435581
$363$	0	0
$364$	−1.05874e26	−0.135902
$365$	2.59886e26	0.324121
$366$	0	0
$367$	−6.27704e25	−0.0739198	−0.0369599	−	0.999317i	$-0.511767\pi$
$367$	−6.27704e25	−0.0739198	−0.0369599	+	0.999317i	$0.511767\pi$
$368$	−1.37114e26	−0.156920
$369$	0	0
$370$	1.28303e26	0.138713
$371$	1.28717e27	1.35272
$372$	0	0
$373$	−1.70679e27	−1.69526	−0.847631	−	0.530587i	$-0.821972\pi$
$373$	−1.70679e27	−1.69526	−0.847631	+	0.530587i	$0.821972\pi$
$374$	−5.75630e26	−0.555893
$375$	0	0
$376$	−1.12471e27	−1.02700
$377$	−3.18148e27	−2.82518
$378$	0	0
$379$	2.35215e27	1.97585	0.987923	−	0.154946i	$-0.0495204\pi$
$379$	2.35215e27	1.97585	0.987923	+	0.154946i	$0.0495204\pi$
$380$	2.99977e25	0.0245109
$381$	0	0
$382$	6.56452e26	0.507619
$383$	−9.18944e26	−0.691356	−0.345678	−	0.938353i	$-0.612351\pi$
$383$	−9.18944e26	−0.691356	−0.345678	+	0.938353i	$0.612351\pi$
$384$	0	0
$385$	−5.27461e26	−0.375710
$386$	1.05331e27	0.730114
$387$	0	0
$388$	−1.32824e26	−0.0872053
$389$	1.24941e27	0.798428	0.399214	−	0.916858i	$-0.369283\pi$
$389$	1.24941e27	0.798428	0.399214	+	0.916858i	$0.369283\pi$
$390$	0	0
$391$	−2.69051e26	−0.162922
$392$	3.09422e27	1.82409
$393$	0	0
$394$	3.36119e26	0.187838
$395$	7.67432e26	0.417610
$396$	0	0
$397$	−3.17780e27	−1.63993	−0.819967	−	0.572410i	$-0.806008\pi$
$397$	−3.17780e27	−1.63993	−0.819967	+	0.572410i	$0.806008\pi$
$398$	7.88139e25	0.0396123
$399$	0	0
$400$	−4.41508e26	−0.210527
$401$	2.00476e27	0.931208	0.465604	−	0.884993i	$-0.345837\pi$
$401$	2.00476e27	0.931208	0.465604	+	0.884993i	$0.345837\pi$
$402$	0	0
$403$	3.52334e26	0.155329
$404$	−7.11042e25	−0.0305415
$405$	0	0
$406$	−8.40860e27	−3.42926
$407$	−3.76009e26	−0.149437
$408$	0	0
$409$	1.65077e27	0.623150	0.311575	−	0.950222i	$-0.399144\pi$
$409$	1.65077e27	0.623150	0.311575	+	0.950222i	$0.399144\pi$
$410$	−1.70158e27	−0.626067
$411$	0	0
$412$	2.26594e26	0.0792187
$413$	−1.04782e27	−0.357117
$414$	0	0
$415$	1.64999e27	0.534536
$416$	5.06447e26	0.159976
$417$	0	0
$418$	−1.66500e27	−0.500108
$419$	−2.41867e27	−0.708484	−0.354242	−	0.935154i	$-0.615261\pi$
$419$	−2.41867e27	−0.708484	−0.354242	+	0.935154i	$0.615261\pi$
$420$	0	0
$421$	−5.33902e27	−1.48764	−0.743822	−	0.668377i	$-0.766989\pi$
$421$	−5.33902e27	−1.48764	−0.743822	+	0.668377i	$0.766989\pi$
$422$	−4.34063e27	−1.17970
$423$	0	0
$424$	−2.99031e27	−0.773347
$425$	−8.66347e26	−0.218579
$426$	0	0
$427$	8.37755e27	2.01198
$428$	1.42557e25	0.00334063
$429$	0	0
$430$	−6.93354e26	−0.154716
$431$	−1.47687e27	−0.321610	−0.160805	−	0.986986i	$-0.551409\pi$
$431$	−1.47687e27	−0.321610	−0.160805	+	0.986986i	$0.551409\pi$
$432$	0	0
$433$	1.41696e27	0.293924	0.146962	−	0.989142i	$-0.453051\pi$
$433$	1.41696e27	0.293924	0.146962	+	0.989142i	$0.453051\pi$
$434$	9.31213e26	0.188541
$435$	0	0
$436$	5.44616e26	0.105071
$437$	−7.78226e26	−0.146572
$438$	0	0
$439$	1.76504e27	0.316867	0.158433	−	0.987370i	$-0.449356\pi$
$439$	1.76504e27	0.316867	0.158433	+	0.987370i	$0.449356\pi$
$440$	1.22538e27	0.214792
$441$	0	0
$442$	9.65202e27	1.61318
$443$	−1.08549e28	−1.77169	−0.885843	−	0.463984i	$-0.846419\pi$
$443$	−1.08549e28	−1.77169	−0.885843	+	0.463984i	$0.846419\pi$
$444$	0	0
$445$	6.43388e26	0.100160
$446$	−6.11550e27	−0.929856
$447$	0	0
$448$	−1.09755e28	−1.59223
$449$	−7.34487e27	−1.04087	−0.520436	−	0.853900i	$-0.674231\pi$
$449$	−7.34487e27	−1.04087	−0.520436	+	0.853900i	$0.674231\pi$
$450$	0	0
$451$	4.98672e27	0.674468
$452$	−1.48917e26	−0.0196784
$453$	0	0
$454$	7.22353e27	0.911304
$455$	8.84432e27	1.09030
$456$	0	0
$457$	−1.52263e28	−1.79256	−0.896282	−	0.443485i	$-0.853742\pi$
$457$	−1.52263e28	−1.79256	−0.896282	+	0.443485i	$0.853742\pi$
$458$	5.83054e27	0.670843
$459$	0	0
$460$	−3.38117e25	−0.00371629
$461$	1.36988e28	1.47171	0.735855	−	0.677140i	$-0.236781\pi$
$461$	1.36988e28	1.47171	0.735855	+	0.677140i	$0.236781\pi$
$462$	0	0
$463$	−8.82461e27	−0.905931	−0.452965	−	0.891528i	$-0.649634\pi$
$463$	−8.82461e27	−0.905931	−0.452965	+	0.891528i	$0.649634\pi$
$464$	2.06270e28	2.07013
$465$	0	0
$466$	1.67844e28	1.61011
$467$	−1.04648e28	−0.981532	−0.490766	−	0.871291i	$-0.663283\pi$
$467$	−1.04648e28	−0.981532	−0.490766	+	0.871291i	$0.663283\pi$
$468$	0	0
$469$	3.26477e28	2.92777
$470$	−5.54652e27	−0.486400
$471$	0	0
$472$	2.43426e27	0.204163
$473$	2.03198e27	0.166677
$474$	0	0
$475$	−2.50590e27	−0.196644
$476$	1.34694e27	0.103389
$477$	0	0
$478$	−1.38287e28	−1.01576
$479$	9.99111e27	0.717944	0.358972	−	0.933348i	$-0.383127\pi$
$479$	9.99111e27	0.717944	0.358972	+	0.933348i	$0.383127\pi$
$480$	0	0
$481$	6.30483e27	0.433661
$482$	4.29887e27	0.289308
$483$	0	0
$484$	−6.53107e26	−0.0420831
$485$	1.10956e28	0.699622
$486$	0	0
$487$	−2.33380e28	−1.40932	−0.704661	−	0.709544i	$-0.748901\pi$
$487$	−2.33380e28	−1.40932	−0.704661	+	0.709544i	$0.748901\pi$
$488$	−1.94625e28	−1.15025
$489$	0	0
$490$	1.52592e28	0.863918
$491$	−2.29620e28	−1.27249	−0.636245	−	0.771487i	$-0.719513\pi$
$491$	−2.29620e28	−1.27249	−0.636245	+	0.771487i	$0.719513\pi$
$492$	0	0
$493$	4.04752e28	2.14930
$494$	2.79183e28	1.45130
$495$	0	0
$496$	−2.28434e27	−0.113816
$497$	4.21490e28	2.05611
$498$	0	0
$499$	−1.58930e28	−0.743276	−0.371638	−	0.928378i	$-0.621204\pi$
$499$	−1.58930e28	−0.743276	−0.371638	+	0.928378i	$0.621204\pi$
$500$	−1.08874e26	−0.00498585
$501$	0	0
$502$	1.62933e28	0.715516
$503$	4.31616e28	1.85624	0.928120	−	0.372282i	$-0.121425\pi$
$503$	4.31616e28	1.85624	0.928120	+	0.372282i	$0.121425\pi$
$504$	0	0
$505$	5.93978e27	0.245025
$506$	1.87670e27	0.0758253
$507$	0	0
$508$	−1.26376e27	−0.0489886
$509$	−3.30343e28	−1.25438	−0.627189	−	0.778867i	$-0.715794\pi$
$509$	−3.30343e28	−1.25438	−0.627189	+	0.778867i	$0.715794\pi$
$510$	0	0
$511$	3.37532e28	1.22997
$512$	2.53241e28	0.904065
$513$	0	0
$514$	−1.69362e28	−0.580369
$515$	−1.89288e28	−0.635547
$516$	0	0
$517$	1.62549e28	0.524003
$518$	1.66635e28	0.526387
$519$	0	0
$520$	−2.05469e28	−0.623320
$521$	−4.42108e27	−0.131442	−0.0657208	−	0.997838i	$-0.520935\pi$
$521$	−4.42108e27	−0.131442	−0.0657208	+	0.997838i	$0.520935\pi$
$522$	0	0
$523$	−6.75490e28	−1.92909	−0.964543	−	0.263927i	$-0.914982\pi$
$523$	−6.75490e28	−1.92909	−0.964543	+	0.263927i	$0.914982\pi$
$524$	3.08069e27	0.0862321
$525$	0	0
$526$	5.54097e26	0.0149017
$527$	−4.48244e27	−0.118169
$528$	0	0
$529$	−3.85944e28	−0.977777
$530$	−1.47467e28	−0.366268
$531$	0	0
$532$	3.89601e27	0.0930138
$533$	−8.36160e28	−1.95728
$534$	0	0
$535$	−1.19087e27	−0.0268009
$536$	−7.58462e28	−1.67380
$537$	0	0
$538$	5.54638e28	1.17705
$539$	−4.47193e28	−0.930706
$540$	0	0
$541$	9.90502e27	0.198282	0.0991412	−	0.995073i	$-0.468390\pi$
$541$	9.90502e27	0.198282	0.0991412	+	0.995073i	$0.468390\pi$
$542$	−3.29834e28	−0.647595
$543$	0	0
$544$	−6.44308e27	−0.121704
$545$	−4.54952e28	−0.842952
$546$	0	0
$547$	−1.52806e28	−0.272442	−0.136221	−	0.990678i	$-0.543496\pi$
$547$	−1.52806e28	−0.272442	−0.136221	+	0.990678i	$0.543496\pi$
$548$	2.29988e27	0.0402262
$549$	0	0
$550$	6.04299e27	0.101729
$551$	1.17074e29	1.93361
$552$	0	0
$553$	9.96718e28	1.58474
$554$	−2.26181e28	−0.352861
$555$	0	0
$556$	1.60579e27	0.0241214
$557$	6.67397e28	0.983795	0.491897	−	0.870653i	$-0.336304\pi$
$557$	6.67397e28	0.983795	0.491897	+	0.870653i	$0.336304\pi$
$558$	0	0
$559$	−3.40716e28	−0.483692
$560$	−5.73417e28	−0.798907
$561$	0	0
$562$	−7.17850e28	−0.963390
$563$	−4.32189e28	−0.569293	−0.284646	−	0.958633i	$-0.591876\pi$
$563$	−4.32189e28	−0.569293	−0.284646	+	0.958633i	$0.591876\pi$
$564$	0	0
$565$	1.24400e28	0.157874
$566$	−5.88883e28	−0.733595
$567$	0	0
$568$	−9.79195e28	−1.17547
$569$	2.28226e28	0.268960	0.134480	−	0.990916i	$-0.457064\pi$
$569$	2.28226e28	0.268960	0.134480	+	0.990916i	$0.457064\pi$
$570$	0	0
$571$	1.97421e28	0.224240	0.112120	−	0.993695i	$-0.464236\pi$
$571$	1.97421e28	0.224240	0.112120	+	0.993695i	$0.464236\pi$
$572$	−3.55479e27	−0.0396420
$573$	0	0
$574$	−2.20996e29	−2.37579
$575$	2.82451e27	0.0298147
$576$	0	0
$577$	−1.13098e29	−1.15109	−0.575546	−	0.817770i	$-0.695210\pi$
$577$	−1.13098e29	−1.15109	−0.575546	+	0.817770i	$0.695210\pi$
$578$	−1.99873e28	−0.199762
$579$	0	0
$580$	5.08653e27	0.0490262
$581$	2.14296e29	2.02845
$582$	0	0
$583$	4.32175e28	0.394584
$584$	−7.84145e28	−0.703172
$585$	0	0
$586$	−1.00047e29	−0.865523
$587$	6.65281e28	0.565334	0.282667	−	0.959218i	$-0.408781\pi$
$587$	6.65281e28	0.565334	0.282667	+	0.959218i	$0.408781\pi$
$588$	0	0
$589$	−1.29654e28	−0.106310
$590$	1.20046e28	0.0966944
$591$	0	0
$592$	−4.08770e28	−0.317761
$593$	3.42927e28	0.261895	0.130948	−	0.991389i	$-0.458198\pi$
$593$	3.42927e28	0.261895	0.130948	+	0.991389i	$0.458198\pi$
$594$	0	0
$595$	−1.12518e29	−0.829461
$596$	−1.06924e28	−0.0774443
$597$	0	0
$598$	−3.14680e28	−0.220043
$599$	7.57014e28	0.520143	0.260071	−	0.965589i	$-0.416254\pi$
$599$	7.57014e28	0.520143	0.260071	+	0.965589i	$0.416254\pi$
$600$	0	0
$601$	2.14743e29	1.42475	0.712374	−	0.701800i	$-0.247620\pi$
$601$	2.14743e29	1.42475	0.712374	+	0.701800i	$0.247620\pi$
$602$	−9.00507e28	−0.587116
$603$	0	0
$604$	1.42975e28	0.0900270
$605$	5.45582e28	0.337620
$606$	0	0
$607$	3.33523e28	0.199363	0.0996815	−	0.995019i	$-0.468218\pi$
$607$	3.33523e28	0.199363	0.0996815	+	0.995019i	$0.468218\pi$
$608$	−1.86365e28	−0.109491
$609$	0	0
$610$	−9.59796e28	−0.544773
$611$	−2.72558e29	−1.52064
$612$	0	0
$613$	6.67007e28	0.359580	0.179790	−	0.983705i	$-0.442458\pi$
$613$	6.67007e28	0.359580	0.179790	+	0.983705i	$0.442458\pi$
$614$	2.72035e28	0.144164
$615$	0	0
$616$	1.59149e29	0.815092
$617$	2.22057e29	1.11807	0.559037	−	0.829143i	$-0.311171\pi$
$617$	2.22057e29	1.11807	0.559037	+	0.829143i	$0.311171\pi$
$618$	0	0
$619$	1.99692e29	0.971869	0.485935	−	0.873995i	$-0.338479\pi$
$619$	1.99692e29	0.971869	0.485935	+	0.873995i	$0.338479\pi$
$620$	−5.63309e26	−0.00269547
$621$	0	0
$622$	1.78084e29	0.823806
$623$	8.35614e28	0.380085
$624$	0	0
$625$	9.09495e27	0.0400000
$626$	4.05583e29	1.75408
$627$	0	0
$628$	−1.29006e28	−0.0539552
$629$	−8.02107e28	−0.329914
$630$	0	0
$631$	1.07700e29	0.428456	0.214228	−	0.976784i	$-0.431276\pi$
$631$	1.07700e29	0.428456	0.214228	+	0.976784i	$0.431276\pi$
$632$	−2.31555e29	−0.905993
$633$	0	0
$634$	−9.67908e28	−0.366351
$635$	1.05570e29	0.393021
$636$	0	0
$637$	7.49841e29	2.70088
$638$	−2.82325e29	−1.00030
$639$	0	0
$640$	1.40269e29	0.480920
$641$	4.82362e28	0.162691	0.0813455	−	0.996686i	$-0.474078\pi$
$641$	4.82362e28	0.162691	0.0813455	+	0.996686i	$0.474078\pi$
$642$	0	0
$643$	1.12338e29	0.366699	0.183350	−	0.983048i	$-0.441306\pi$
$643$	1.12338e29	0.366699	0.183350	+	0.983048i	$0.441306\pi$
$644$	−4.39136e27	−0.0141026
$645$	0	0
$646$	−3.55180e29	−1.10410
$647$	2.58352e28	0.0790162	0.0395081	−	0.999219i	$-0.487421\pi$
$647$	2.58352e28	0.0790162	0.0395081	+	0.999219i	$0.487421\pi$
$648$	0	0
$649$	−3.51812e28	−0.104170
$650$	−1.01327e29	−0.295214
$651$	0	0
$652$	3.64837e28	0.102920
$653$	−5.85458e29	−1.62520	−0.812601	−	0.582820i	$-0.801949\pi$
$653$	−5.85458e29	−1.62520	−0.812601	+	0.582820i	$0.801949\pi$
$654$	0	0
$655$	−2.57350e29	−0.691815
$656$	5.42120e29	1.43418
$657$	0	0
$658$	−7.20366e29	−1.84579
$659$	1.22801e29	0.309673	0.154837	−	0.987940i	$-0.450515\pi$
$659$	1.22801e29	0.309673	0.154837	+	0.987940i	$0.450515\pi$
$660$	0	0
$661$	4.50136e29	1.09958	0.549792	−	0.835301i	$-0.314707\pi$
$661$	4.50136e29	1.09958	0.549792	+	0.835301i	$0.314707\pi$
$662$	−3.13884e29	−0.754676
$663$	0	0
$664$	−4.97848e29	−1.15966
$665$	−3.25458e29	−0.746222
$666$	0	0
$667$	−1.31959e29	−0.293170
$668$	−6.53076e27	−0.0142827
$669$	0	0
$670$	−3.74036e29	−0.792736
$671$	2.81282e29	0.586889
$672$	0	0
$673$	6.19299e29	1.25240	0.626199	−	0.779663i	$-0.284610\pi$
$673$	6.19299e29	1.25240	0.626199	+	0.779663i	$0.284610\pi$
$674$	−2.36555e29	−0.470980
$675$	0	0
$676$	3.07234e28	0.0592964
$677$	−9.15907e28	−0.174049	−0.0870243	−	0.996206i	$-0.527736\pi$
$677$	−9.15907e28	−0.174049	−0.0870243	+	0.996206i	$0.527736\pi$
$678$	0	0
$679$	1.44107e30	2.65492
$680$	2.61400e29	0.474201
$681$	0	0
$682$	3.12661e28	0.0549969
$683$	1.08753e30	1.88375	0.941876	−	0.335961i	$-0.109061\pi$
$683$	1.08753e30	1.88375	0.941876	+	0.335961i	$0.109061\pi$
$684$	0	0
$685$	−1.92123e29	−0.322722
$686$	9.27708e29	1.53465
$687$	0	0
$688$	2.20902e29	0.354422
$689$	−7.24660e29	−1.14507
$690$	0	0
$691$	−4.53419e29	−0.694992	−0.347496	−	0.937681i	$-0.612968\pi$
$691$	−4.53419e29	−0.694992	−0.347496	+	0.937681i	$0.612968\pi$
$692$	1.57116e28	0.0237196
$693$	0	0
$694$	−4.86999e29	−0.713269
$695$	−1.34142e29	−0.193519
$696$	0	0
$697$	1.06377e30	1.48903
$698$	−9.35163e29	−1.28945
$699$	0	0
$700$	−1.41402e28	−0.0189203
$701$	9.03929e29	1.19150	0.595752	−	0.803169i	$-0.296854\pi$
$701$	9.03929e29	1.19150	0.595752	+	0.803169i	$0.296854\pi$
$702$	0	0
$703$	−2.32009e29	−0.296806
$704$	−3.68510e29	−0.464448
$705$	0	0
$706$	8.60597e29	1.05281
$707$	7.71441e29	0.929821
$708$	0	0
$709$	−9.52975e29	−1.11505	−0.557527	−	0.830159i	$-0.688250\pi$
$709$	−9.52975e29	−1.11505	−0.557527	+	0.830159i	$0.688250\pi$
$710$	−4.82891e29	−0.556720
$711$	0	0
$712$	−1.94128e29	−0.217294
$713$	1.46139e28	0.0161185
$714$	0	0
$715$	2.96955e29	0.318036
$716$	−6.74864e28	−0.0712245
$717$	0	0
$718$	−8.71776e28	−0.0893507
$719$	1.28953e30	1.30250	0.651249	−	0.758864i	$-0.274245\pi$
$719$	1.28953e30	1.30250	0.651249	+	0.758864i	$0.274245\pi$
$720$	0	0
$721$	−2.45842e30	−2.41177
$722$	3.53681e28	0.0341957
$723$	0	0
$724$	2.51616e28	0.0236311
$725$	−4.24911e29	−0.393322
$726$	0	0
$727$	1.09976e30	0.988981	0.494490	−	0.869183i	$-0.335355\pi$
$727$	1.09976e30	0.988981	0.494490	+	0.869183i	$0.335355\pi$
$728$	−2.66857e30	−2.36537
$729$	0	0
$730$	−3.86702e29	−0.333032
$731$	4.33463e29	0.367976
$732$	0	0
$733$	5.44126e29	0.448856	0.224428	−	0.974491i	$-0.427949\pi$
$733$	5.44126e29	0.448856	0.224428	+	0.974491i	$0.427949\pi$
$734$	9.34004e28	0.0759522
$735$	0	0
$736$	2.10061e28	0.0166008
$737$	1.09617e30	0.854021
$738$	0	0
$739$	−3.50055e29	−0.265076	−0.132538	−	0.991178i	$-0.542313\pi$
$739$	−3.50055e29	−0.265076	−0.132538	+	0.991178i	$0.542313\pi$
$740$	−1.00801e28	−0.00752545
$741$	0	0
$742$	−1.91526e30	−1.38991
$743$	−1.31687e30	−0.942239	−0.471120	−	0.882069i	$-0.656150\pi$
$743$	−1.31687e30	−0.942239	−0.471120	+	0.882069i	$0.656150\pi$
$744$	0	0
$745$	8.93205e29	0.621313
$746$	2.53965e30	1.74187
$747$	0	0
$748$	4.52245e28	0.0301583
$749$	−1.54666e29	−0.101704
$750$	0	0
$751$	2.67518e30	1.71054	0.855271	−	0.518181i	$-0.173391\pi$
$751$	2.67518e30	1.71054	0.855271	+	0.518181i	$0.173391\pi$
$752$	1.76712e30	1.11424
$753$	0	0
$754$	4.73395e30	2.90285
$755$	−1.19436e30	−0.722260
$756$	0	0
$757$	−1.57579e30	−0.926812	−0.463406	−	0.886146i	$-0.653373\pi$
$757$	−1.57579e30	−0.926812	−0.463406	+	0.886146i	$0.653373\pi$
$758$	−3.49993e30	−2.03017
$759$	0	0
$760$	7.56096e29	0.426613
$761$	−1.14447e30	−0.636893	−0.318447	−	0.947941i	$-0.603161\pi$
$761$	−1.14447e30	−0.636893	−0.318447	+	0.947941i	$0.603161\pi$
$762$	0	0
$763$	−5.90878e30	−3.19883
$764$	−5.15743e28	−0.0275393
$765$	0	0
$766$	1.36736e30	0.710364
$767$	5.89910e29	0.302297
$768$	0	0
$769$	−1.55372e29	−0.0774724	−0.0387362	−	0.999249i	$-0.512333\pi$
$769$	−1.55372e29	−0.0774724	−0.0387362	+	0.999249i	$0.512333\pi$
$770$	7.84846e29	0.386039
$771$	0	0
$772$	−8.27538e28	−0.0396101
$773$	−2.20959e30	−1.04334	−0.521671	−	0.853147i	$-0.674691\pi$
$773$	−2.20959e30	−1.04334	−0.521671	+	0.853147i	$0.674691\pi$
$774$	0	0
$775$	4.70568e28	0.0216249
$776$	−3.34785e30	−1.51781
$777$	0	0
$778$	−1.85909e30	−0.820380
$779$	3.07695e30	1.33961
$780$	0	0
$781$	1.41518e30	0.599759
$782$	4.00340e29	0.167401
$783$	0	0
$784$	−4.86155e30	−1.97905
$785$	1.07767e30	0.432866
$786$	0	0
$787$	−2.75620e30	−1.07789	−0.538947	−	0.842340i	$-0.681178\pi$
$787$	−2.75620e30	−1.07789	−0.538947	+	0.842340i	$0.681178\pi$
$788$	−2.64073e28	−0.0101906
$789$	0	0
$790$	−1.14192e30	−0.429091
$791$	1.61567e30	0.599099
$792$	0	0
$793$	−4.71647e30	−1.70313
$794$	4.72847e30	1.68502
$795$	0	0
$796$	−6.19203e27	−0.00214905
$797$	−4.23039e30	−1.44900	−0.724500	−	0.689275i	$-0.757929\pi$
$797$	−4.23039e30	−1.44900	−0.724500	+	0.689275i	$0.757929\pi$
$798$	0	0
$799$	3.46751e30	1.15685
$800$	6.76398e28	0.0222719
$801$	0	0
$802$	−2.98302e30	−0.956810
$803$	1.13329e30	0.358779
$804$	0	0
$805$	3.66839e29	0.113141
$806$	−5.24263e29	−0.159599
$807$	0	0
$808$	−1.79219e30	−0.531576
$809$	−3.93165e30	−1.15111	−0.575553	−	0.817764i	$-0.695213\pi$
$809$	−3.93165e30	−1.15111	−0.575553	+	0.817764i	$0.695213\pi$
$810$	0	0
$811$	−1.50931e30	−0.430585	−0.215293	−	0.976550i	$-0.569071\pi$
$811$	−1.50931e30	−0.430585	−0.215293	+	0.976550i	$0.569071\pi$
$812$	6.60624e29	0.186044
$813$	0	0
$814$	5.59491e29	0.153545
$815$	−3.04771e30	−0.825695
$816$	0	0
$817$	1.25379e30	0.331049
$818$	−2.45630e30	−0.640282
$819$	0	0
$820$	1.33685e29	0.0339654
$821$	1.87901e29	0.0471332	0.0235666	−	0.999722i	$-0.492498\pi$
$821$	1.87901e29	0.0471332	0.0235666	+	0.999722i	$0.492498\pi$
$822$	0	0
$823$	−5.28445e30	−1.29212	−0.646058	−	0.763289i	$-0.723583\pi$
$823$	−5.28445e30	−1.29212	−0.646058	+	0.763289i	$0.723583\pi$
$824$	5.71134e30	1.37880
$825$	0	0
$826$	1.55912e30	0.366935
$827$	2.20857e30	0.513220	0.256610	−	0.966515i	$-0.417394\pi$
$827$	2.20857e30	0.513220	0.256610	+	0.966515i	$0.417394\pi$
$828$	0	0
$829$	5.77035e30	1.30731	0.653657	−	0.756791i	$-0.273234\pi$
$829$	5.77035e30	1.30731	0.653657	+	0.756791i	$0.273234\pi$
$830$	−2.45514e30	−0.549233
$831$	0	0
$832$	6.17909e30	1.34781
$833$	−9.53957e30	−2.05474
$834$	0	0
$835$	5.45556e29	0.114586
$836$	1.30811e29	0.0271318
$837$	0	0
$838$	3.59892e30	0.727963
$839$	4.79329e30	0.957488	0.478744	−	0.877955i	$-0.341092\pi$
$839$	4.79329e30	0.957488	0.478744	+	0.877955i	$0.341092\pi$
$840$	0	0
$841$	1.47187e31	2.86756
$842$	7.94430e30	1.52855
$843$	0	0
$844$	3.41022e29	0.0640010
$845$	−2.56652e30	−0.475717
$846$	0	0
$847$	7.08586e30	1.28120
$848$	4.69829e30	0.839042
$849$	0	0
$850$	1.28910e30	0.224588
$851$	2.61507e29	0.0450011
$852$	0	0
$853$	−2.24317e30	−0.376615	−0.188308	−	0.982110i	$-0.560300\pi$
$853$	−2.24317e30	−0.376615	−0.188308	+	0.982110i	$0.560300\pi$
$854$	−1.24655e31	−2.06730
$855$	0	0
$856$	3.59317e29	0.0581438
$857$	1.67730e30	0.268109	0.134055	−	0.990974i	$-0.457200\pi$
$857$	1.67730e30	0.268109	0.134055	+	0.990974i	$0.457200\pi$
$858$	0	0
$859$	−2.75580e30	−0.429853	−0.214926	−	0.976630i	$-0.568951\pi$
$859$	−2.75580e30	−0.429853	−0.214926	+	0.976630i	$0.568951\pi$
$860$	5.44735e28	0.00839366
$861$	0	0
$862$	2.19753e30	0.330453
$863$	−4.97847e30	−0.739576	−0.369788	−	0.929116i	$-0.620570\pi$
$863$	−4.97847e30	−0.739576	−0.369788	+	0.929116i	$0.620570\pi$
$864$	0	0
$865$	−1.31249e30	−0.190295
$866$	−2.10839e30	−0.302005
$867$	0	0
$868$	−7.31609e28	−0.0102287
$869$	3.34655e30	0.462264
$870$	0	0
$871$	−1.83803e31	−2.47834
$872$	1.37271e31	1.82876
$873$	0	0
$874$	1.15798e30	0.150602
$875$	1.18122e30	0.151792
$876$	0	0
$877$	−5.93482e29	−0.0744581	−0.0372291	−	0.999307i	$-0.511853\pi$
$877$	−5.93482e29	−0.0744581	−0.0372291	+	0.999307i	$0.511853\pi$
$878$	−2.62632e30	−0.325579
$879$	0	0
$880$	−1.92529e30	−0.233039
$881$	1.27954e31	1.53040	0.765202	−	0.643791i	$-0.222639\pi$
$881$	1.27954e31	1.53040	0.765202	+	0.643791i	$0.222639\pi$
$882$	0	0
$883$	1.12787e30	0.131726	0.0658632	−	0.997829i	$-0.479020\pi$
$883$	1.12787e30	0.131726	0.0658632	+	0.997829i	$0.479020\pi$
$884$	−7.58313e29	−0.0875184
$885$	0	0
$886$	1.61518e31	1.82040
$887$	6.34061e30	0.706209	0.353104	−	0.935584i	$-0.385126\pi$
$887$	6.34061e30	0.706209	0.353104	+	0.935584i	$0.385126\pi$
$888$	0	0
$889$	1.37111e31	1.49143
$890$	−9.57343e29	−0.102913
$891$	0	0
$892$	4.80465e29	0.0504465
$893$	1.00297e31	1.04076
$894$	0	0
$895$	5.63757e30	0.571413
$896$	1.82178e31	1.82499
$897$	0	0
$898$	1.09289e31	1.06949
$899$	−2.19847e30	−0.212639
$900$	0	0
$901$	9.21921e30	0.871130
$902$	−7.42009e30	−0.693011
$903$	0	0
$904$	−3.75347e30	−0.342503
$905$	−2.10191e30	−0.189585
$906$	0	0
$907$	−4.56592e30	−0.402394	−0.201197	−	0.979551i	$-0.564483\pi$
$907$	−4.56592e30	−0.402394	−0.201197	+	0.979551i	$0.564483\pi$
$908$	−5.67518e29	−0.0494400
$909$	0	0
$910$	−1.31601e31	−1.12027
$911$	2.06100e31	1.73434	0.867172	−	0.498009i	$-0.165935\pi$
$911$	2.06100e31	1.73434	0.867172	+	0.498009i	$0.165935\pi$
$912$	0	0
$913$	7.19515e30	0.591694
$914$	2.26563e31	1.84185
$915$	0	0
$916$	−4.58077e29	−0.0363945
$917$	−3.34238e31	−2.62529
$918$	0	0
$919$	−1.86231e31	−1.42968	−0.714839	−	0.699289i	$-0.753500\pi$
$919$	−1.86231e31	−1.42968	−0.714839	+	0.699289i	$0.753500\pi$
$920$	−8.52230e29	−0.0646822
$921$	0	0
$922$	−2.03834e31	−1.51217
$923$	−2.37294e31	−1.74048
$924$	0	0
$925$	8.42056e29	0.0603744
$926$	1.31308e31	0.930838
$927$	0	0
$928$	−3.16009e30	−0.219001
$929$	−2.63294e31	−1.80417	−0.902083	−	0.431562i	$-0.857963\pi$
$929$	−2.63294e31	−1.80417	−0.902083	+	0.431562i	$0.857963\pi$
$930$	0	0
$931$	−2.75931e31	−1.84854
$932$	−1.31867e30	−0.0873513
$933$	0	0
$934$	1.55713e31	1.00852
$935$	−3.77789e30	−0.241951
$936$	0	0
$937$	4.07588e30	0.255244	0.127622	−	0.991823i	$-0.459266\pi$
$937$	4.07588e30	0.255244	0.127622	+	0.991823i	$0.459266\pi$
$938$	−4.85787e31	−3.00827
$939$	0	0
$940$	4.35764e29	0.0263881
$941$	3.48852e30	0.208906	0.104453	−	0.994530i	$-0.466691\pi$
$941$	3.48852e30	0.208906	0.104453	+	0.994530i	$0.466691\pi$
$942$	0	0
$943$	−3.46817e30	−0.203108
$944$	−3.82465e30	−0.221506
$945$	0	0
$946$	−3.02352e30	−0.171260
$947$	3.11529e31	1.74511	0.872557	−	0.488513i	$-0.162460\pi$
$947$	3.11529e31	1.74511	0.872557	+	0.488513i	$0.162460\pi$
$948$	0	0
$949$	−1.90027e31	−1.04116
$950$	3.72870e30	0.202050
$951$	0	0
$952$	3.39499e31	1.79949
$953$	1.98167e31	1.03886	0.519429	−	0.854514i	$-0.326145\pi$
$953$	1.98167e31	1.03886	0.519429	+	0.854514i	$0.326145\pi$
$954$	0	0
$955$	4.30833e30	0.220940
$956$	1.08646e30	0.0551068
$957$	0	0
$958$	−1.48665e31	−0.737683
$959$	−2.49524e31	−1.22466
$960$	0	0
$961$	−2.05820e31	−0.988309
$962$	−9.38140e30	−0.445584
$963$	0	0
$964$	−3.37742e29	−0.0156955
$965$	6.91295e30	0.317780
$966$	0	0
$967$	−8.26115e30	−0.371589	−0.185794	−	0.982589i	$-0.559486\pi$
$967$	−8.26115e30	−0.371589	−0.185794	+	0.982589i	$0.559486\pi$
$968$	−1.64617e31	−0.732458
$969$	0	0
$970$	−1.65099e31	−0.718857
$971$	−5.95729e30	−0.256594	−0.128297	−	0.991736i	$-0.540951\pi$
$971$	−5.95729e30	−0.256594	−0.128297	+	0.991736i	$0.540951\pi$
$972$	0	0
$973$	−1.74219e31	−0.734365
$974$	3.47263e31	1.44807
$975$	0	0
$976$	3.05790e31	1.24796
$977$	−3.99242e31	−1.61192	−0.805960	−	0.591970i	$-0.798351\pi$
$977$	−3.99242e31	−1.61192	−0.805960	+	0.591970i	$0.798351\pi$
$978$	0	0
$979$	2.80563e30	0.110870
$980$	−1.19884e30	−0.0468692
$981$	0	0
$982$	3.41668e31	1.30747
$983$	−2.59366e30	−0.0981977	−0.0490988	−	0.998794i	$-0.515635\pi$
$983$	−2.59366e30	−0.0981977	−0.0490988	+	0.998794i	$0.515635\pi$
$984$	0	0
$985$	2.20597e30	0.0817558
$986$	−6.02259e31	−2.20839
$987$	0	0
$988$	−2.19341e30	−0.0787357
$989$	−1.41320e30	−0.0501929
$990$	0	0
$991$	6.46664e30	0.224856	0.112428	−	0.993660i	$-0.464137\pi$
$991$	6.46664e30	0.224856	0.112428	+	0.993660i	$0.464137\pi$
$992$	3.49965e29	0.0120407
$993$	0	0
$994$	−6.27165e31	−2.11264
$995$	5.17260e29	0.0172411
$996$	0	0
$997$	3.67859e31	1.20056	0.600278	−	0.799791i	$-0.295057\pi$
$997$	3.67859e31	1.20056	0.600278	+	0.799791i	$0.295057\pi$
$998$	2.36483e31	0.763712
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.22.a.g.1.1		4
3.2	odd	2		15.22.a.e.1.4	✓	4
15.2	even	4		75.22.b.h.49.7		8
15.8	even	4		75.22.b.h.49.2		8
15.14	odd	2		75.22.a.h.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.22.a.e.1.4	✓	4	3.2	odd	2
45.22.a.g.1.1		4	1.1	even	1	trivial
75.22.a.h.1.1		4	15.14	odd	2
75.22.b.h.49.2		8	15.8	even	4
75.22.b.h.49.7		8	15.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	45.a (trivial)

\(f(q)\)	\(=\)	\(q-1487.97 q^{2} +116903. q^{4} -9.76562e6 q^{5} -1.26833e9 q^{7} +2.94655e9 q^{8} +1.45310e10 q^{10} -4.25851e10 q^{11} +7.14056e11 q^{13} +1.88724e12 q^{14} -4.62954e12 q^{16} -9.08430e12 q^{17} -2.62762e13 q^{19} -1.14163e12 q^{20} +6.33654e13 q^{22} +2.96171e13 q^{23} +9.53674e13 q^{25} -1.06249e15 q^{26} -1.48271e14 q^{28} -4.45551e15 q^{29} +4.93427e14 q^{31} +7.09254e14 q^{32} +1.35172e16 q^{34} +1.23860e16 q^{35} +8.82960e15 q^{37} +3.90982e16 q^{38} -2.87749e16 q^{40} -1.17100e17 q^{41} -4.77156e16 q^{43} -4.97831e15 q^{44} -4.40694e16 q^{46} -3.81704e17 q^{47} +1.05012e18 q^{49} -1.41904e17 q^{50} +8.34750e16 q^{52} -1.01485e18 q^{53} +4.15870e17 q^{55} -3.73720e18 q^{56} +6.62967e18 q^{58} +8.26140e17 q^{59} -6.60518e18 q^{61} -7.34204e17 q^{62} +8.65351e18 q^{64} -6.97320e18 q^{65} -2.57407e19 q^{67} -1.06198e18 q^{68} -1.84301e19 q^{70} -3.32319e19 q^{71} -2.66123e19 q^{73} -1.31382e19 q^{74} -3.07176e18 q^{76} +5.40120e19 q^{77} -7.85851e19 q^{79} +4.52104e19 q^{80} +1.74241e20 q^{82} -1.68959e20 q^{83} +8.87139e19 q^{85} +7.09994e19 q^{86} -1.25479e20 q^{88} -6.58830e19 q^{89} -9.05659e20 q^{91} +3.46232e18 q^{92} +5.67964e20 q^{94} +2.56604e20 q^{95} -1.13619e21 q^{97} -1.56254e21 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 897 q^{2} - 1163123 q^{4} - 39062500 q^{5} - 234577504 q^{7} - 76855629 q^{8} - 8759765625 q^{10} - 31491830256 q^{11} - 27017977768 q^{13} + 2233308126672 q^{14} - 11324681311775 q^{16} + 2946095028888 q^{17}+ \cdots - 19\!\cdots\!63 q^{98}+O(q^{100}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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