LMFDB - Embedded newform 45.22.a.g.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [45,22,Mod(1,45)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [4,897] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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			1.3
Root			$1069.36$ 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)

$f(q)$	$=$	\(q+1293.36 q^{2} -424379. q^{4} -9.76562e6 q^{5} +1.27960e9 q^{7} -3.26124e9 q^{8} -1.26304e10 q^{10} +1.21178e11 q^{11} +7.37152e11 q^{13} +1.65498e12 q^{14} -3.32796e12 q^{16} -6.05462e12 q^{17} -3.23277e13 q^{19} +4.14433e12 q^{20} +1.56726e14 q^{22} +3.98109e13 q^{23} +9.53674e13 q^{25} +9.53401e14 q^{26} -5.43035e14 q^{28} -1.40928e15 q^{29} -2.62721e15 q^{31} +2.53508e15 q^{32} -7.83079e15 q^{34} -1.24961e16 q^{35} -3.44728e15 q^{37} -4.18113e16 q^{38} +3.18481e16 q^{40} -7.99269e16 q^{41} +2.07610e17 q^{43} -5.14254e16 q^{44} +5.14897e16 q^{46} +1.95408e17 q^{47} +1.07882e18 q^{49} +1.23344e17 q^{50} -3.12832e17 q^{52} +2.33305e17 q^{53} -1.18338e18 q^{55} -4.17307e18 q^{56} -1.82271e18 q^{58} +3.93099e18 q^{59} +8.99462e18 q^{61} -3.39792e18 q^{62} +1.02580e19 q^{64} -7.19875e18 q^{65} +9.21439e18 q^{67} +2.56946e18 q^{68} -1.61619e19 q^{70} +4.75444e19 q^{71} +1.81566e19 q^{73} -4.45857e18 q^{74} +1.37192e19 q^{76} +1.55059e20 q^{77} -1.11814e20 q^{79} +3.24996e19 q^{80} -1.03374e20 q^{82} +3.39283e19 q^{83} +5.91272e19 q^{85} +2.68514e20 q^{86} -3.95190e20 q^{88} -3.38998e20 q^{89} +9.43257e20 q^{91} -1.68949e19 q^{92} +2.52733e20 q^{94} +3.15700e20 q^{95} +1.12006e20 q^{97} +1.39530e21 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1293.36	0.893107	0.446553	−	0.894757i	$-0.352651\pi$
$2$	1293.36	0.893107	0.446553	+	0.894757i	$0.352651\pi$
$3$	0	0
$3$	0	0
$4$	−424379.	−0.202360
$5$	−9.76562e6	−0.447214
$5$	−9.76562e6	−0.447214
$6$	0	0
$7$	1.27960e9	1.71216	0.856079	−	0.516846i	$-0.172894\pi$
$7$	1.27960e9	1.71216	0.856079	+	0.516846i	$0.172894\pi$
$8$	−3.26124e9	−1.07384
$9$	0	0
$10$	−1.26304e10	−0.399410
$11$	1.21178e11	1.40864	0.704320	−	0.709883i	$-0.251252\pi$
$11$	1.21178e11	1.40864	0.704320	+	0.709883i	$0.251252\pi$
$12$	0	0
$13$	7.37152e11	1.48304	0.741518	−	0.670933i	$-0.234106\pi$
$13$	7.37152e11	1.48304	0.741518	+	0.670933i	$0.234106\pi$
$14$	1.65498e12	1.52914
$15$	0	0
$16$	−3.32796e12	−0.756691
$17$	−6.05462e12	−0.728406	−0.364203	−	0.931320i	$-0.618659\pi$
$17$	−6.05462e12	−0.728406	−0.364203	+	0.931320i	$0.618659\pi$
$18$	0	0
$19$	−3.23277e13	−1.20966	−0.604828	−	0.796356i	$-0.706758\pi$
$19$	−3.23277e13	−1.20966	−0.604828	+	0.796356i	$0.706758\pi$
$20$	4.14433e12	0.0904981
$21$	0	0
$22$	1.56726e14	1.25807
$23$	3.98109e13	0.200382	0.100191	−	0.994968i	$-0.468055\pi$
$23$	3.98109e13	0.200382	0.100191	+	0.994968i	$0.468055\pi$
$24$	0	0
$25$	9.53674e13	0.200000
$26$	9.53401e14	1.32451
$27$	0	0
$28$	−5.43035e14	−0.346472
$29$	−1.40928e15	−0.622041	−0.311021	−	0.950403i	$-0.600671\pi$
$29$	−1.40928e15	−0.622041	−0.311021	+	0.950403i	$0.600671\pi$
$30$	0	0
$31$	−2.62721e15	−0.575701	−0.287850	−	0.957675i	$-0.592941\pi$
$31$	−2.62721e15	−0.575701	−0.287850	+	0.957675i	$0.592941\pi$
$32$	2.53508e15	0.398030
$33$	0	0
$34$	−7.83079e15	−0.650544
$35$	−1.24961e16	−0.765700
$36$	0	0
$37$	−3.44728e15	−0.117858	−0.0589290	−	0.998262i	$-0.518769\pi$
$37$	−3.44728e15	−0.117858	−0.0589290	+	0.998262i	$0.518769\pi$
$38$	−4.18113e16	−1.08035
$39$	0	0
$40$	3.18481e16	0.480234
$41$	−7.99269e16	−0.929956	−0.464978	−	0.885322i	$-0.653938\pi$
$41$	−7.99269e16	−0.929956	−0.464978	+	0.885322i	$0.653938\pi$
$42$	0	0
$43$	2.07610e17	1.46498	0.732488	−	0.680780i	$-0.238359\pi$
$43$	2.07610e17	1.46498	0.732488	+	0.680780i	$0.238359\pi$
$44$	−5.14254e16	−0.285052
$45$	0	0
$46$	5.14897e16	0.178963
$47$	1.95408e17	0.541895	0.270948	−	0.962594i	$-0.412663\pi$
$47$	1.95408e17	0.541895	0.270948	+	0.962594i	$0.412663\pi$
$48$	0	0
$49$	1.07882e18	1.93148
$50$	1.23344e17	0.178621
$51$	0	0
$52$	−3.12832e17	−0.300107
$53$	2.33305e17	0.183243	0.0916214	−	0.995794i	$-0.470795\pi$
$53$	2.33305e17	0.183243	0.0916214	+	0.995794i	$0.470795\pi$
$54$	0	0
$55$	−1.18338e18	−0.629963
$56$	−4.17307e18	−1.83858
$57$	0	0
$58$	−1.82271e18	−0.555550
$59$	3.93099e18	1.00128	0.500640	−	0.865656i	$-0.333098\pi$
$59$	3.93099e18	1.00128	0.500640	+	0.865656i	$0.333098\pi$
$60$	0	0
$61$	8.99462e18	1.61443	0.807216	−	0.590256i	$-0.200973\pi$
$61$	8.99462e18	1.61443	0.807216	+	0.590256i	$0.200973\pi$
$62$	−3.39792e18	−0.514162
$63$	0	0
$64$	1.02580e19	1.11217
$65$	−7.19875e18	−0.663234
$66$	0	0
$67$	9.21439e18	0.617563	0.308781	−	0.951133i	$-0.400079\pi$
$67$	9.21439e18	0.617563	0.308781	+	0.951133i	$0.400079\pi$
$68$	2.56946e18	0.147400
$69$	0	0
$70$	−1.61619e19	−0.683852
$71$	4.75444e19	1.73335	0.866676	−	0.498871i	$-0.166252\pi$
$71$	4.75444e19	1.73335	0.866676	+	0.498871i	$0.166252\pi$
$72$	0	0
$73$	1.81566e19	0.494474	0.247237	−	0.968955i	$-0.420477\pi$
$73$	1.81566e19	0.494474	0.247237	+	0.968955i	$0.420477\pi$
$74$	−4.45857e18	−0.105260
$75$	0	0
$76$	1.37192e19	0.244786
$77$	1.55059e20	2.41181
$78$	0	0
$79$	−1.11814e20	−1.32865	−0.664327	−	0.747442i	$-0.731282\pi$
$79$	−1.11814e20	−1.32865	−0.664327	+	0.747442i	$0.731282\pi$
$80$	3.24996e19	0.338402
$81$	0	0
$82$	−1.03374e20	−0.830550
$83$	3.39283e19	0.240017	0.120009	−	0.992773i	$-0.461708\pi$
$83$	3.39283e19	0.240017	0.120009	+	0.992773i	$0.461708\pi$
$84$	0	0
$85$	5.91272e19	0.325753
$86$	2.68514e20	1.30838
$87$	0	0
$88$	−3.95190e20	−1.51265
$89$	−3.38998e20	−1.15240	−0.576199	−	0.817310i	$-0.695465\pi$
$89$	−3.38998e20	−1.15240	−0.576199	+	0.817310i	$0.695465\pi$
$90$	0	0
$91$	9.43257e20	2.53919
$92$	−1.68949e19	−0.0405494
$93$	0	0
$94$	2.52733e20	0.483970
$95$	3.15700e20	0.540975
$96$	0	0
$97$	1.12006e20	0.154219	0.0771097	−	0.997023i	$-0.475431\pi$
$97$	1.12006e20	0.154219	0.0771097	+	0.997023i	$0.475431\pi$
$98$	1.39530e21	1.72502
$99$	0	0
$100$	−4.04720e19	−0.0404720
$101$	−2.51635e20	−0.226672	−0.113336	−	0.993557i	$-0.536154\pi$
$101$	−2.51635e20	−0.226672	−0.113336	+	0.993557i	$0.536154\pi$
$102$	0	0
$103$	2.29653e21	1.68377	0.841883	−	0.539660i	$-0.181447\pi$
$103$	2.29653e21	1.68377	0.841883	+	0.539660i	$0.181447\pi$
$104$	−2.40403e21	−1.59254
$105$	0	0
$106$	3.01746e20	0.163655
$107$	−3.75913e21	−1.84739	−0.923694	−	0.383130i	$-0.874846\pi$
$107$	−3.75913e21	−1.84739	−0.923694	+	0.383130i	$0.874846\pi$
$108$	0	0
$109$	−1.44930e21	−0.586380	−0.293190	−	0.956054i	$-0.594717\pi$
$109$	−1.44930e21	−0.586380	−0.293190	+	0.956054i	$0.594717\pi$
$110$	−1.53053e21	−0.562624
$111$	0	0
$112$	−4.25845e21	−1.29557
$113$	5.37147e21	1.48857	0.744285	−	0.667862i	$-0.232790\pi$
$113$	5.37147e21	1.48857	0.744285	+	0.667862i	$0.232790\pi$
$114$	0	0
$115$	−3.88778e20	−0.0896137
$116$	5.98071e20	0.125876
$117$	0	0
$118$	5.08417e21	0.894250
$119$	−7.74747e21	−1.24715
$120$	0	0
$121$	7.28381e21	0.984266
$122$	1.16333e22	1.44186
$123$	0	0
$124$	1.11493e21	0.116499
$125$	−9.31323e20	−0.0894427
$126$	0	0
$127$	1.99248e22	1.61978	0.809888	−	0.586585i	$-0.199528\pi$
$127$	1.99248e22	1.61978	0.809888	+	0.586585i	$0.199528\pi$
$128$	7.95081e21	0.595260
$129$	0	0
$130$	−9.31056e21	−0.592339
$131$	−1.95301e22	−1.14645	−0.573225	−	0.819398i	$-0.694308\pi$
$131$	−1.95301e22	−1.14645	−0.573225	+	0.819398i	$0.694308\pi$
$132$	0	0
$133$	−4.13664e22	−2.07112
$134$	1.19175e22	0.551549
$135$	0	0
$136$	1.97456e22	0.782188
$137$	−2.54688e22	−0.934206	−0.467103	−	0.884203i	$-0.654702\pi$
$137$	−2.54688e22	−0.934206	−0.467103	+	0.884203i	$0.654702\pi$
$138$	0	0
$139$	−3.28445e22	−1.03468	−0.517341	−	0.855780i	$-0.673078\pi$
$139$	−3.28445e22	−1.03468	−0.517341	+	0.855780i	$0.673078\pi$
$140$	5.30307e21	0.154947
$141$	0	0
$142$	6.14919e22	1.54807
$143$	8.93265e22	2.08906
$144$	0	0
$145$	1.37625e22	0.278185
$146$	2.34829e22	0.441618
$147$	0	0
$148$	1.46296e21	0.0238497
$149$	4.10815e22	0.624009	0.312004	−	0.950081i	$-0.399000\pi$
$149$	4.10815e22	0.624009	0.312004	+	0.950081i	$0.399000\pi$
$150$	0	0
$151$	8.76777e22	1.15779	0.578897	−	0.815401i	$-0.303483\pi$
$151$	8.76777e22	1.15779	0.578897	+	0.815401i	$0.303483\pi$
$152$	1.05428e23	1.29897
$153$	0	0
$154$	2.00546e23	2.15401
$155$	2.56563e22	0.257461
$156$	0	0
$157$	1.06605e23	0.935040	0.467520	−	0.883982i	$-0.345148\pi$
$157$	1.06605e23	0.935040	0.467520	+	0.883982i	$0.345148\pi$
$158$	−1.44615e23	−1.18663
$159$	0	0
$160$	−2.47566e22	−0.178005
$161$	5.09419e22	0.343086
$162$	0	0
$163$	2.93390e23	1.73571	0.867853	−	0.496822i	$-0.165500\pi$
$163$	2.93390e23	1.73571	0.867853	+	0.496822i	$0.165500\pi$
$164$	3.39193e22	0.188186
$165$	0	0
$166$	4.38814e22	0.214361
$167$	−1.40812e23	−0.645830	−0.322915	−	0.946428i	$-0.604663\pi$
$167$	−1.40812e23	−0.645830	−0.322915	+	0.946428i	$0.604663\pi$
$168$	0	0
$169$	2.96329e23	1.19940
$170$	7.64725e22	0.290932
$171$	0	0
$172$	−8.81056e22	−0.296452
$173$	−3.14336e21	−0.00995198	−0.00497599	−	0.999988i	$-0.501584\pi$
$173$	−3.14336e21	−0.00995198	−0.00497599	+	0.999988i	$0.501584\pi$
$174$	0	0
$175$	1.22032e23	0.342431
$176$	−4.03275e23	−1.06590
$177$	0	0
$178$	−4.38446e23	−1.02921
$179$	7.66967e23	1.69754	0.848770	−	0.528763i	$-0.177344\pi$
$179$	7.66967e23	1.69754	0.848770	+	0.528763i	$0.177344\pi$
$180$	0	0
$181$	1.51630e23	0.298649	0.149325	−	0.988788i	$-0.452290\pi$
$181$	1.51630e23	0.298649	0.149325	+	0.988788i	$0.452290\pi$
$182$	1.21997e24	2.26777
$183$	0	0
$184$	−1.29833e23	−0.215178
$185$	3.36649e22	0.0527077
$186$	0	0
$187$	−7.33686e23	−1.02606
$188$	−8.29272e22	−0.109658
$189$	0	0
$190$	4.08313e23	0.483148
$191$	−6.06790e23	−0.679498	−0.339749	−	0.940516i	$-0.610342\pi$
$191$	−6.06790e23	−0.679498	−0.339749	+	0.940516i	$0.610342\pi$
$192$	0	0
$193$	1.87596e24	1.88309	0.941544	−	0.336889i	$-0.109375\pi$
$193$	1.87596e24	1.88309	0.941544	+	0.336889i	$0.109375\pi$
$194$	1.44864e23	0.137734
$195$	0	0
$196$	−4.57830e23	−0.390855
$197$	−2.26924e23	−0.183647	−0.0918237	−	0.995775i	$-0.529270\pi$
$197$	−2.26924e23	−0.183647	−0.0918237	+	0.995775i	$0.529270\pi$
$198$	0	0
$199$	−6.13967e23	−0.446876	−0.223438	−	0.974718i	$-0.571728\pi$
$199$	−6.13967e23	−0.446876	−0.223438	+	0.974718i	$0.571728\pi$
$200$	−3.11016e23	−0.214767
$201$	0	0
$202$	−3.25454e23	−0.202442
$203$	−1.80331e24	−1.06503
$204$	0	0
$205$	7.80536e23	0.415889
$206$	2.97024e24	1.50378
$207$	0	0
$208$	−2.45321e24	−1.12220
$209$	−3.91740e24	−1.70397
$210$	0	0
$211$	1.99189e24	0.783970	0.391985	−	0.919972i	$-0.371788\pi$
$211$	1.99189e24	0.783970	0.391985	+	0.919972i	$0.371788\pi$
$212$	−9.90097e22	−0.0370810
$213$	0	0
$214$	−4.86190e24	−1.64992
$215$	−2.02745e24	−0.655157
$216$	0	0
$217$	−3.36177e24	−0.985690
$218$	−1.87446e24	−0.523700
$219$	0	0
$220$	5.02201e23	0.127479
$221$	−4.46318e24	−1.08025
$222$	0	0
$223$	3.29187e24	0.724839	0.362420	−	0.932015i	$-0.381951\pi$
$223$	3.29187e24	0.724839	0.362420	+	0.932015i	$0.381951\pi$
$224$	3.24388e24	0.681491
$225$	0	0
$226$	6.94722e24	1.32945
$227$	1.53623e24	0.280663	0.140332	−	0.990105i	$-0.455183\pi$
$227$	1.53623e24	0.280663	0.140332	+	0.990105i	$0.455183\pi$
$228$	0	0
$229$	−3.70932e24	−0.618048	−0.309024	−	0.951054i	$-0.600002\pi$
$229$	−3.70932e24	−0.618048	−0.309024	+	0.951054i	$0.600002\pi$
$230$	−5.02829e23	−0.0800346
$231$	0	0
$232$	4.59601e24	0.667971
$233$	1.12524e25	1.56317	0.781587	−	0.623796i	$-0.214411\pi$
$233$	1.12524e25	1.56317	0.781587	+	0.623796i	$0.214411\pi$
$234$	0	0
$235$	−1.90828e24	−0.242343
$236$	−1.66823e24	−0.202619
$237$	0	0
$238$	−1.00202e25	−1.11383
$239$	−1.31053e25	−1.39402	−0.697008	−	0.717063i	$-0.745486\pi$
$239$	−1.31053e25	−1.39402	−0.697008	+	0.717063i	$0.745486\pi$
$240$	0	0
$241$	1.66470e24	0.162240	0.0811200	−	0.996704i	$-0.474150\pi$
$241$	1.66470e24	0.162240	0.0811200	+	0.996704i	$0.474150\pi$
$242$	9.42057e24	0.879055
$243$	0	0
$244$	−3.81713e24	−0.326696
$245$	−1.05354e25	−0.863785
$246$	0	0
$247$	−2.38304e25	−1.79396
$248$	8.56796e24	0.618208
$249$	0	0
$250$	−1.20453e24	−0.0798819
$251$	−2.32403e25	−1.47798	−0.738990	−	0.673717i	$-0.764697\pi$
$251$	−2.32403e25	−1.47798	−0.738990	+	0.673717i	$0.764697\pi$
$252$	0	0
$253$	4.82420e24	0.282267
$254$	2.57699e25	1.44663
$255$	0	0
$256$	−1.12293e25	−0.580543
$257$	2.65650e25	1.31829	0.659147	−	0.752014i	$-0.270918\pi$
$257$	2.65650e25	1.31829	0.659147	+	0.752014i	$0.270918\pi$
$258$	0	0
$259$	−4.41113e24	−0.201791
$260$	3.05500e24	0.134212
$261$	0	0
$262$	−2.52593e25	−1.02390
$263$	9.75846e24	0.380055	0.190027	−	0.981779i	$-0.439142\pi$
$263$	9.75846e24	0.380055	0.190027	+	0.981779i	$0.439142\pi$
$264$	0	0
$265$	−2.27837e24	−0.0819487
$266$	−5.35015e25	−1.84973
$267$	0	0
$268$	−3.91040e24	−0.124970
$269$	1.15143e25	0.353866	0.176933	−	0.984223i	$-0.443382\pi$
$269$	1.15143e25	0.353866	0.176933	+	0.984223i	$0.443382\pi$
$270$	0	0
$271$	1.57414e25	0.447575	0.223788	−	0.974638i	$-0.428158\pi$
$271$	1.57414e25	0.447575	0.223788	+	0.974638i	$0.428158\pi$
$272$	2.01495e25	0.551178
$273$	0	0
$274$	−3.29403e25	−0.834346
$275$	1.15564e25	0.281728
$276$	0	0
$277$	1.64872e25	0.372486	0.186243	−	0.982504i	$-0.440369\pi$
$277$	1.64872e25	0.372486	0.186243	+	0.982504i	$0.440369\pi$
$278$	−4.24796e25	−0.924081
$279$	0	0
$280$	4.07527e25	0.822236
$281$	−6.63562e25	−1.28963	−0.644815	−	0.764339i	$-0.723066\pi$
$281$	−6.63562e25	−1.28963	−0.644815	+	0.764339i	$0.723066\pi$
$282$	0	0
$283$	−6.72915e25	−1.21395	−0.606977	−	0.794719i	$-0.707618\pi$
$283$	−6.72915e25	−1.21395	−0.606977	+	0.794719i	$0.707618\pi$
$284$	−2.01769e25	−0.350761
$285$	0	0
$286$	1.15531e26	1.86576
$287$	−1.02274e26	−1.59223
$288$	0	0
$289$	−3.24335e25	−0.469425
$290$	1.77999e25	0.248449
$291$	0	0
$292$	−7.70527e24	−0.100062
$293$	1.51857e25	0.190250	0.0951248	−	0.995465i	$-0.469675\pi$
$293$	1.51857e25	0.190250	0.0951248	+	0.995465i	$0.469675\pi$
$294$	0	0
$295$	−3.83885e25	−0.447786
$296$	1.12424e25	0.126560
$297$	0	0
$298$	5.31331e25	0.557307
$299$	2.93467e25	0.297174
$300$	0	0
$301$	2.65658e26	2.50827
$302$	1.13399e26	1.03403
$303$	0	0
$304$	1.07585e26	0.915335
$305$	−8.78381e25	−0.721996
$306$	0	0
$307$	−2.16909e26	−1.66465	−0.832327	−	0.554285i	$-0.812992\pi$
$307$	−2.16909e26	−1.66465	−0.832327	+	0.554285i	$0.812992\pi$
$308$	−6.58037e25	−0.488054
$309$	0	0
$310$	3.31828e25	0.229940
$311$	9.86475e25	0.660848	0.330424	−	0.943833i	$-0.392808\pi$
$311$	9.86475e25	0.660848	0.330424	+	0.943833i	$0.392808\pi$
$312$	0	0
$313$	1.28416e26	0.804273	0.402137	−	0.915580i	$-0.368268\pi$
$313$	1.28416e26	0.804273	0.402137	+	0.915580i	$0.368268\pi$
$314$	1.37878e26	0.835091
$315$	0	0
$316$	4.74516e25	0.268866
$317$	1.45513e26	0.797588	0.398794	−	0.917041i	$-0.369429\pi$
$317$	1.45513e26	0.797588	0.398794	+	0.917041i	$0.369429\pi$
$318$	0	0
$319$	−1.70774e26	−0.876232
$320$	−1.00176e26	−0.497379
$321$	0	0
$322$	6.58860e25	0.306413
$323$	1.95732e26	0.881120
$324$	0	0
$325$	7.03003e25	0.296607
$326$	3.79458e26	1.55017
$327$	0	0
$328$	2.60661e26	0.998620
$329$	2.50044e26	0.927810
$330$	0	0
$331$	−4.40991e26	−1.53545	−0.767725	−	0.640780i	$-0.778611\pi$
$331$	−4.40991e26	−1.53545	−0.767725	+	0.640780i	$0.778611\pi$
$332$	−1.43985e25	−0.0485699
$333$	0	0
$334$	−1.82121e26	−0.576795
$335$	−8.99842e25	−0.276182
$336$	0	0
$337$	4.81798e26	1.38915	0.694577	−	0.719418i	$-0.255591\pi$
$337$	4.81798e26	1.38915	0.694577	+	0.719418i	$0.255591\pi$
$338$	3.83259e26	1.07119
$339$	0	0
$340$	−2.50924e25	−0.0659193
$341$	−3.18359e26	−0.810955
$342$	0	0
$343$	6.65743e26	1.59485
$344$	−6.77068e26	−1.57314
$345$	0	0
$346$	−4.06548e24	−0.00888819
$347$	3.64414e26	0.772922	0.386461	−	0.922306i	$-0.373697\pi$
$347$	3.64414e26	0.772922	0.386461	+	0.922306i	$0.373697\pi$
$348$	0	0
$349$	−3.60335e26	−0.719515	−0.359758	−	0.933046i	$-0.617141\pi$
$349$	−3.60335e26	−0.719515	−0.359758	+	0.933046i	$0.617141\pi$
$350$	1.57831e26	0.305828
$351$	0	0
$352$	3.07195e26	0.560681
$353$	3.52364e26	0.624247	0.312124	−	0.950042i	$-0.398960\pi$
$353$	3.52364e26	0.624247	0.312124	+	0.950042i	$0.398960\pi$
$354$	0	0
$355$	−4.64301e26	−0.775179
$356$	1.43864e26	0.233199
$357$	0	0
$358$	9.91962e26	1.51608
$359$	−1.17616e27	−1.74571	−0.872857	−	0.487977i	$-0.837735\pi$
$359$	−1.17616e27	−1.74571	−0.872857	+	0.487977i	$0.837735\pi$
$360$	0	0
$361$	3.30870e26	0.463268
$362$	1.96112e26	0.266726
$363$	0	0
$364$	−4.00299e26	−0.513831
$365$	−1.77310e26	−0.221136
$366$	0	0
$367$	1.52241e26	0.179282	0.0896411	−	0.995974i	$-0.471428\pi$
$367$	1.52241e26	0.179282	0.0896411	+	0.995974i	$0.471428\pi$
$368$	−1.32489e26	−0.151627
$369$	0	0
$370$	4.35407e25	0.0470736
$371$	2.98536e26	0.313740
$372$	0	0
$373$	−8.84632e26	−0.878658	−0.439329	−	0.898326i	$-0.644784\pi$
$373$	−8.84632e26	−0.878658	−0.439329	+	0.898326i	$0.644784\pi$
$374$	−9.48918e26	−0.916382
$375$	0	0
$376$	−6.37273e26	−0.581907
$377$	−1.03886e27	−0.922510
$378$	0	0
$379$	−3.83294e26	−0.321974	−0.160987	−	0.986957i	$-0.551468\pi$
$379$	−3.83294e26	−0.321974	−0.160987	+	0.986957i	$0.551468\pi$
$380$	−1.33977e26	−0.109472
$381$	0	0
$382$	−7.84796e26	−0.606864
$383$	−1.86870e27	−1.40589	−0.702947	−	0.711243i	$-0.748133\pi$
$383$	−1.86870e27	−1.40589	−0.702947	+	0.711243i	$0.748133\pi$
$384$	0	0
$385$	−1.51425e27	−1.07860
$386$	2.42628e27	1.68180
$387$	0	0
$388$	−4.75332e25	−0.0312078
$389$	7.58738e26	0.484865	0.242432	−	0.970168i	$-0.422055\pi$
$389$	7.58738e26	0.484865	0.242432	+	0.970168i	$0.422055\pi$
$390$	0	0
$391$	−2.41040e26	−0.145960
$392$	−3.51830e27	−2.07410
$393$	0	0
$394$	−2.93494e26	−0.164017
$395$	1.09193e27	0.594192
$396$	0	0
$397$	5.16201e26	0.266391	0.133195	−	0.991090i	$-0.457476\pi$
$397$	5.16201e26	0.266391	0.133195	+	0.991090i	$0.457476\pi$
$398$	−7.94078e26	−0.399108
$399$	0	0
$400$	−3.17379e26	−0.151338
$401$	−2.28825e27	−1.06289	−0.531443	−	0.847094i	$-0.678350\pi$
$401$	−2.28825e27	−1.06289	−0.531443	+	0.847094i	$0.678350\pi$
$402$	0	0
$403$	−1.93665e27	−0.853785
$404$	1.06789e26	0.0458692
$405$	0	0
$406$	−2.33233e27	−0.951188
$407$	−4.17734e26	−0.166019
$408$	0	0
$409$	−1.71019e27	−0.645581	−0.322791	−	0.946470i	$-0.604621\pi$
$409$	−1.71019e27	−0.645581	−0.322791	+	0.946470i	$0.604621\pi$
$410$	1.00951e27	0.371433
$411$	0	0
$412$	−9.74601e26	−0.340727
$413$	5.03008e27	1.71435
$414$	0	0
$415$	−3.31331e26	−0.107339
$416$	1.86874e27	0.590294
$417$	0	0
$418$	−5.06660e27	−1.52183
$419$	7.77024e25	0.0227608	0.0113804	−	0.999935i	$-0.496377\pi$
$419$	7.77024e25	0.0227608	0.0113804	+	0.999935i	$0.496377\pi$
$420$	0	0
$421$	2.53023e27	0.705015	0.352508	−	0.935809i	$-0.385329\pi$
$421$	2.53023e27	0.705015	0.352508	+	0.935809i	$0.385329\pi$
$422$	2.57622e27	0.700169
$423$	0	0
$424$	−7.60863e26	−0.196773
$425$	−5.77414e26	−0.145681
$426$	0	0
$427$	1.15095e28	2.76416
$428$	1.59530e27	0.373837
$429$	0	0
$430$	−2.62221e27	−0.585125
$431$	4.79626e26	0.104446	0.0522229	−	0.998635i	$-0.483369\pi$
$431$	4.79626e26	0.104446	0.0522229	+	0.998635i	$0.483369\pi$
$432$	0	0
$433$	8.00695e27	1.66090	0.830451	−	0.557092i	$-0.188083\pi$
$433$	8.00695e27	1.66090	0.830451	+	0.557092i	$0.188083\pi$
$434$	−4.34797e27	−0.880327
$435$	0	0
$436$	6.15052e26	0.118660
$437$	−1.28699e27	−0.242394
$438$	0	0
$439$	−9.55381e27	−1.71514	−0.857569	−	0.514368i	$-0.828026\pi$
$439$	−9.55381e27	−1.71514	−0.857569	+	0.514368i	$0.828026\pi$
$440$	3.85928e27	0.676477
$441$	0	0
$442$	−5.77248e27	−0.964781
$443$	−5.53824e27	−0.903925	−0.451962	−	0.892037i	$-0.649276\pi$
$443$	−5.53824e27	−0.903925	−0.451962	+	0.892037i	$0.649276\pi$
$444$	0	0
$445$	3.31053e27	0.515368
$446$	4.25756e27	0.647359
$447$	0	0
$448$	1.31261e28	1.90422
$449$	−5.56138e27	−0.788127	−0.394063	−	0.919083i	$-0.628931\pi$
$449$	−5.56138e27	−0.788127	−0.394063	+	0.919083i	$0.628931\pi$
$450$	0	0
$451$	−9.68536e27	−1.30997
$452$	−2.27954e27	−0.301227
$453$	0	0
$454$	1.98690e27	0.250662
$455$	−9.21150e27	−1.13556
$456$	0	0
$457$	5.87968e27	0.692203	0.346101	−	0.938197i	$-0.387505\pi$
$457$	5.87968e27	0.692203	0.346101	+	0.938197i	$0.387505\pi$
$458$	−4.79748e27	−0.551983
$459$	0	0
$460$	1.64989e26	0.0181342
$461$	1.13415e28	1.21846	0.609231	−	0.792993i	$-0.291478\pi$
$461$	1.13415e28	1.21846	0.609231	+	0.792993i	$0.291478\pi$
$462$	0	0
$463$	1.74305e28	1.78941	0.894705	−	0.446658i	$-0.147386\pi$
$463$	1.74305e28	1.78941	0.894705	+	0.446658i	$0.147386\pi$
$464$	4.69004e27	0.470693
$465$	0	0
$466$	1.45533e28	1.39608
$467$	−1.60704e27	−0.150730	−0.0753651	−	0.997156i	$-0.524012\pi$
$467$	−1.60704e27	−0.150730	−0.0753651	+	0.997156i	$0.524012\pi$
$468$	0	0
$469$	1.17907e28	1.05736
$470$	−2.46809e27	−0.216438
$471$	0	0
$472$	−1.28199e28	−1.07521
$473$	2.51578e28	2.06362
$474$	0	0
$475$	−3.08301e27	−0.241931
$476$	3.28787e27	0.252372
$477$	0	0
$478$	−1.69498e28	−1.24501
$479$	5.39349e27	0.387567	0.193783	−	0.981044i	$-0.437924\pi$
$479$	5.39349e27	0.387567	0.193783	+	0.981044i	$0.437924\pi$
$480$	0	0
$481$	−2.54117e27	−0.174788
$482$	2.15306e27	0.144898
$483$	0	0
$484$	−3.09110e27	−0.199176
$485$	−1.09381e27	−0.0689690
$486$	0	0
$487$	1.39994e28	0.845386	0.422693	−	0.906273i	$-0.361085\pi$
$487$	1.39994e28	0.845386	0.422693	+	0.906273i	$0.361085\pi$
$488$	−2.93336e28	−1.73363
$489$	0	0
$490$	−1.36260e28	−0.771453
$491$	−1.66296e27	−0.0921567	−0.0460783	−	0.998938i	$-0.514672\pi$
$491$	−1.66296e27	−0.0921567	−0.0460783	+	0.998938i	$0.514672\pi$
$492$	0	0
$493$	8.53268e27	0.453099
$494$	−3.08213e28	−1.60220
$495$	0	0
$496$	8.74325e27	0.435627
$497$	6.08377e28	2.96777
$498$	0	0
$499$	−2.24968e28	−1.05212	−0.526059	−	0.850448i	$-0.676331\pi$
$499$	−2.24968e28	−1.05212	−0.526059	+	0.850448i	$0.676331\pi$
$500$	3.95234e26	0.0180996
$501$	0	0
$502$	−3.00581e28	−1.31999
$503$	2.16964e28	0.933090	0.466545	−	0.884497i	$-0.345499\pi$
$503$	2.16964e28	0.933090	0.466545	+	0.884497i	$0.345499\pi$
$504$	0	0
$505$	2.45738e27	0.101371
$506$	6.23941e27	0.252094
$507$	0	0
$508$	−8.45567e27	−0.327778
$509$	1.86305e28	0.707436	0.353718	−	0.935352i	$-0.384917\pi$
$509$	1.86305e28	0.707436	0.353718	+	0.935352i	$0.384917\pi$
$510$	0	0
$511$	2.32331e28	0.846618
$512$	−3.11976e28	−1.11375
$513$	0	0
$514$	3.43580e28	1.17738
$515$	−2.24271e28	−0.753003
$516$	0	0
$517$	2.36791e28	0.763335
$518$	−5.70517e27	−0.180221
$519$	0	0
$520$	2.34769e28	0.712205
$521$	−6.39183e28	−1.90033	−0.950165	−	0.311747i	$-0.899086\pi$
$521$	−6.39183e28	−1.90033	−0.950165	+	0.311747i	$0.899086\pi$
$522$	0	0
$523$	−9.84531e27	−0.281165	−0.140583	−	0.990069i	$-0.544898\pi$
$523$	−9.84531e27	−0.281165	−0.140583	+	0.990069i	$0.544898\pi$
$524$	8.28815e27	0.231995
$525$	0	0
$526$	1.26212e28	0.339429
$527$	1.59068e28	0.419344
$528$	0	0
$529$	−3.78867e28	−0.959847
$530$	−2.94674e27	−0.0731889
$531$	0	0
$532$	1.75551e28	0.419112
$533$	−5.89183e28	−1.37916
$534$	0	0
$535$	3.67103e28	0.826177
$536$	−3.00503e28	−0.663161
$537$	0	0
$538$	1.48921e28	0.316040
$539$	1.30729e29	2.72076
$540$	0	0
$541$	6.36576e28	1.27432	0.637161	−	0.770731i	$-0.280109\pi$
$541$	6.36576e28	1.27432	0.637161	+	0.770731i	$0.280109\pi$
$542$	2.03593e28	0.399732
$543$	0	0
$544$	−1.53489e28	−0.289928
$545$	1.41533e28	0.262237
$546$	0	0
$547$	−7.99629e28	−1.42568	−0.712839	−	0.701328i	$-0.752591\pi$
$547$	−7.99629e28	−1.42568	−0.712839	+	0.701328i	$0.752591\pi$
$548$	1.08084e28	0.189046
$549$	0	0
$550$	1.49466e28	0.251613
$551$	4.55589e28	0.752456
$552$	0	0
$553$	−1.43077e29	−2.27486
$554$	2.13238e28	0.332669
$555$	0	0
$556$	1.39385e28	0.209378
$557$	−6.15885e28	−0.907862	−0.453931	−	0.891037i	$-0.649979\pi$
$557$	−6.15885e28	−0.907862	−0.453931	+	0.891037i	$0.649979\pi$
$558$	0	0
$559$	1.53040e29	2.17261
$560$	4.15864e28	0.579398
$561$	0	0
$562$	−8.58222e28	−1.15178
$563$	−2.00390e28	−0.263960	−0.131980	−	0.991252i	$-0.542134\pi$
$563$	−2.00390e28	−0.263960	−0.131980	+	0.991252i	$0.542134\pi$
$564$	0	0
$565$	−5.24557e28	−0.665709
$566$	−8.70319e28	−1.08419
$567$	0	0
$568$	−1.55054e29	−1.86134
$569$	−6.14459e28	−0.724126	−0.362063	−	0.932154i	$-0.617927\pi$
$569$	−6.14459e28	−0.724126	−0.362063	+	0.932154i	$0.617927\pi$
$570$	0	0
$571$	−4.65211e28	−0.528410	−0.264205	−	0.964467i	$-0.585110\pi$
$571$	−4.65211e28	−0.528410	−0.264205	+	0.964467i	$0.585110\pi$
$572$	−3.79083e28	−0.422743
$573$	0	0
$574$	−1.32277e29	−1.42203
$575$	3.79666e27	0.0400765
$576$	0	0
$577$	1.04152e28	0.106004	0.0530020	−	0.998594i	$-0.483121\pi$
$577$	1.04152e28	0.106004	0.0530020	+	0.998594i	$0.483121\pi$
$578$	−4.19481e28	−0.419247
$579$	0	0
$580$	−5.84054e27	−0.0562936
$581$	4.34146e28	0.410947
$582$	0	0
$583$	2.82714e28	0.258123
$584$	−5.92129e28	−0.530984
$585$	0	0
$586$	1.96405e28	0.169913
$587$	4.54423e28	0.386154	0.193077	−	0.981184i	$-0.438153\pi$
$587$	4.54423e28	0.386154	0.193077	+	0.981184i	$0.438153\pi$
$588$	0	0
$589$	8.49316e28	0.696400
$590$	−4.96501e28	−0.399921
$591$	0	0
$592$	1.14724e28	0.0891820
$593$	−3.29422e28	−0.251581	−0.125791	−	0.992057i	$-0.540147\pi$
$593$	−3.29422e28	−0.251581	−0.125791	+	0.992057i	$0.540147\pi$
$594$	0	0
$595$	7.56589e28	0.557740
$596$	−1.74342e28	−0.126274
$597$	0	0
$598$	3.79557e28	0.265409
$599$	1.44151e29	0.990458	0.495229	−	0.868763i	$-0.335084\pi$
$599$	1.44151e29	0.990458	0.495229	+	0.868763i	$0.335084\pi$
$600$	0	0
$601$	1.75962e28	0.116745	0.0583723	−	0.998295i	$-0.481409\pi$
$601$	1.75962e28	0.116745	0.0583723	+	0.998295i	$0.481409\pi$
$602$	3.43590e29	2.24015
$603$	0	0
$604$	−3.72086e28	−0.234291
$605$	−7.11310e28	−0.440177
$606$	0	0
$607$	−7.40539e28	−0.442656	−0.221328	−	0.975199i	$-0.571039\pi$
$607$	−7.40539e28	−0.442656	−0.221328	+	0.975199i	$0.571039\pi$
$608$	−8.19532e28	−0.481480
$609$	0	0
$610$	−1.13606e29	−0.644819
$611$	1.44046e29	0.803650
$612$	0	0
$613$	−2.04497e29	−1.10243	−0.551216	−	0.834363i	$-0.685836\pi$
$613$	−2.04497e29	−1.10243	−0.551216	+	0.834363i	$0.685836\pi$
$614$	−2.80540e29	−1.48671
$615$	0	0
$616$	−5.05684e29	−2.58989
$617$	−3.28969e29	−1.65638	−0.828192	−	0.560445i	$-0.810630\pi$
$617$	−3.28969e29	−1.65638	−0.828192	+	0.560445i	$0.810630\pi$
$618$	0	0
$619$	1.51436e29	0.737018	0.368509	−	0.929624i	$-0.379868\pi$
$619$	1.51436e29	0.737018	0.368509	+	0.929624i	$0.379868\pi$
$620$	−1.08880e28	−0.0520998
$621$	0	0
$622$	1.27586e29	0.590208
$623$	−4.33781e29	−1.97309
$624$	0	0
$625$	9.09495e27	0.0400000
$626$	1.66088e29	0.718302
$627$	0	0
$628$	−4.52409e28	−0.189215
$629$	2.08720e28	0.0858484
$630$	0	0
$631$	1.00288e29	0.398969	0.199485	−	0.979901i	$-0.436073\pi$
$631$	1.00288e29	0.398969	0.199485	+	0.979901i	$0.436073\pi$
$632$	3.64652e29	1.42676
$633$	0	0
$634$	1.88200e29	0.712331
$635$	−1.94578e29	−0.724386
$636$	0	0
$637$	7.95256e29	2.86446
$638$	−2.20872e29	−0.782569
$639$	0	0
$640$	−7.76447e28	−0.266208
$641$	2.60873e29	0.879871	0.439936	−	0.898029i	$-0.355001\pi$
$641$	2.60873e29	0.879871	0.439936	+	0.898029i	$0.355001\pi$
$642$	0	0
$643$	5.30533e28	0.173180	0.0865898	−	0.996244i	$-0.472403\pi$
$643$	5.30533e28	0.173180	0.0865898	+	0.996244i	$0.472403\pi$
$644$	−2.16187e28	−0.0694269
$645$	0	0
$646$	2.53151e29	0.786935
$647$	7.13885e27	0.0218340	0.0109170	−	0.999940i	$-0.496525\pi$
$647$	7.13885e27	0.0218340	0.0109170	+	0.999940i	$0.496525\pi$
$648$	0	0
$649$	4.76348e29	1.41044
$650$	9.09234e28	0.264902
$651$	0	0
$652$	−1.24509e29	−0.351237
$653$	8.18691e28	0.227264	0.113632	−	0.993523i	$-0.463751\pi$
$653$	8.18691e28	0.227264	0.113632	+	0.993523i	$0.463751\pi$
$654$	0	0
$655$	1.90723e29	0.512708
$656$	2.65993e29	0.703689
$657$	0	0
$658$	3.23396e29	0.828633
$659$	3.25306e28	0.0820342	0.0410171	−	0.999158i	$-0.486940\pi$
$659$	3.25306e28	0.0820342	0.0410171	+	0.999158i	$0.486940\pi$
$660$	0	0
$661$	8.22623e28	0.200949	0.100474	−	0.994940i	$-0.467964\pi$
$661$	8.22623e28	0.200949	0.100474	+	0.994940i	$0.467964\pi$
$662$	−5.70359e29	−1.37132
$663$	0	0
$664$	−1.10648e29	−0.257739
$665$	4.03969e29	0.926234
$666$	0	0
$667$	−5.61048e28	−0.124646
$668$	5.97579e28	0.130690
$669$	0	0
$670$	−1.16382e29	−0.246660
$671$	1.08995e30	2.27415
$672$	0	0
$673$	1.16937e29	0.236479	0.118239	−	0.992985i	$-0.462275\pi$
$673$	1.16937e29	0.236479	0.118239	+	0.992985i	$0.462275\pi$
$674$	6.23136e29	1.24066
$675$	0	0
$676$	−1.25756e29	−0.242710
$677$	−4.28660e28	−0.0814577	−0.0407289	−	0.999170i	$-0.512968\pi$
$677$	−4.28660e28	−0.0814577	−0.0407289	+	0.999170i	$0.512968\pi$
$678$	0	0
$679$	1.43323e29	0.264048
$680$	−1.92828e29	−0.349805
$681$	0	0
$682$	−4.11752e29	−0.724270
$683$	−5.45367e29	−0.944651	−0.472325	−	0.881424i	$-0.656585\pi$
$683$	−5.45367e29	−0.944651	−0.472325	+	0.881424i	$0.656585\pi$
$684$	0	0
$685$	2.48719e29	0.417790
$686$	8.61044e29	1.42437
$687$	0	0
$688$	−6.90919e29	−1.10853
$689$	1.71981e29	0.271756
$690$	0	0
$691$	−9.02075e29	−1.38269	−0.691343	−	0.722527i	$-0.742980\pi$
$691$	−9.02075e29	−1.38269	−0.691343	+	0.722527i	$0.742980\pi$
$692$	1.33398e27	0.00201388
$693$	0	0
$694$	4.71318e29	0.690302
$695$	3.20747e29	0.462724
$696$	0	0
$697$	4.83927e29	0.677385
$698$	−4.66042e29	−0.642604
$699$	0	0
$700$	−5.17878e28	−0.0692944
$701$	1.36429e29	0.179833	0.0899164	−	0.995949i	$-0.471340\pi$
$701$	1.36429e29	0.179833	0.0899164	+	0.995949i	$0.471340\pi$
$702$	0	0
$703$	1.11443e29	0.142568
$704$	1.24304e30	1.56665
$705$	0	0
$706$	4.55732e29	0.557520
$707$	−3.21992e29	−0.388097
$708$	0	0
$709$	−1.96075e29	−0.229423	−0.114712	−	0.993399i	$-0.536594\pi$
$709$	−1.96075e29	−0.229423	−0.114712	+	0.993399i	$0.536594\pi$
$710$	−6.00507e29	−0.692317
$711$	0	0
$712$	1.10555e30	1.23749
$713$	−1.04591e29	−0.115360
$714$	0	0
$715$	−8.72329e29	−0.934258
$716$	−3.25485e29	−0.343514
$717$	0	0
$718$	−1.52119e30	−1.55911
$719$	−6.61361e29	−0.668013	−0.334007	−	0.942571i	$-0.608401\pi$
$719$	−6.61361e29	−0.668013	−0.334007	+	0.942571i	$0.608401\pi$
$720$	0	0
$721$	2.93864e30	2.88287
$722$	4.27933e29	0.413748
$723$	0	0
$724$	−6.43488e28	−0.0604346
$725$	−1.34400e29	−0.124408
$726$	0	0
$727$	−5.62124e29	−0.505500	−0.252750	−	0.967532i	$-0.581335\pi$
$727$	−5.62124e29	−0.505500	−0.252750	+	0.967532i	$0.581335\pi$
$728$	−3.07619e30	−2.72668
$729$	0	0
$730$	−2.29325e29	−0.197498
$731$	−1.25700e30	−1.06710
$732$	0	0
$733$	−1.44462e30	−1.19169	−0.595843	−	0.803101i	$-0.703182\pi$
$733$	−1.44462e30	−1.19169	−0.595843	+	0.803101i	$0.703182\pi$
$734$	1.96902e29	0.160118
$735$	0	0
$736$	1.00924e29	0.0797583
$737$	1.11658e30	0.869923
$738$	0	0
$739$	4.75852e29	0.360334	0.180167	−	0.983636i	$-0.442336\pi$
$739$	4.75852e29	0.360334	0.180167	+	0.983636i	$0.442336\pi$
$740$	−1.42867e28	−0.0106659
$741$	0	0
$742$	3.86114e29	0.280204
$743$	1.64704e30	1.17848	0.589239	−	0.807959i	$-0.299428\pi$
$743$	1.64704e30	1.17848	0.589239	+	0.807959i	$0.299428\pi$
$744$	0	0
$745$	−4.01187e29	−0.279065
$746$	−1.14415e30	−0.784736
$747$	0	0
$748$	3.11361e29	0.207634
$749$	−4.81018e30	−3.16302
$750$	0	0
$751$	−1.15249e30	−0.736914	−0.368457	−	0.929645i	$-0.620114\pi$
$751$	−1.15249e30	−0.736914	−0.368457	+	0.929645i	$0.620114\pi$
$752$	−6.50311e29	−0.410047
$753$	0	0
$754$	−1.34361e30	−0.823900
$755$	−8.56228e29	−0.517781
$756$	0	0
$757$	6.06438e29	0.356681	0.178340	−	0.983969i	$-0.442927\pi$
$757$	6.06438e29	0.356681	0.178340	+	0.983969i	$0.442927\pi$
$758$	−4.95736e29	−0.287557
$759$	0	0
$760$	−1.02957e30	−0.580918
$761$	3.30053e30	1.83673	0.918364	−	0.395737i	$-0.129511\pi$
$761$	3.30053e30	1.83673	0.918364	+	0.395737i	$0.129511\pi$
$762$	0	0
$763$	−1.85452e30	−1.00398
$764$	2.57509e29	0.137503
$765$	0	0
$766$	−2.41690e30	−1.25561
$767$	2.89773e30	1.48493
$768$	0	0
$769$	−2.35995e30	−1.17673	−0.588363	−	0.808597i	$-0.700228\pi$
$769$	−2.35995e30	−1.17673	−0.588363	+	0.808597i	$0.700228\pi$
$770$	−1.95846e30	−0.963301
$771$	0	0
$772$	−7.96117e29	−0.381062
$773$	−2.75286e30	−1.29987	−0.649935	−	0.759990i	$-0.725204\pi$
$773$	−2.75286e30	−1.29987	−0.649935	+	0.759990i	$0.725204\pi$
$774$	0	0
$775$	−2.50550e29	−0.115140
$776$	−3.65279e29	−0.165606
$777$	0	0
$778$	9.81319e29	0.433036
$779$	2.58385e30	1.12493
$780$	0	0
$781$	5.76133e30	2.44167
$782$	−3.11751e29	−0.130358
$783$	0	0
$784$	−3.59028e30	−1.46153
$785$	−1.04106e30	−0.418163
$786$	0	0
$787$	−1.96178e30	−0.767213	−0.383606	−	0.923497i	$-0.625318\pi$
$787$	−1.96178e30	−0.767213	−0.383606	+	0.923497i	$0.625318\pi$
$788$	9.63019e28	0.0371629
$789$	0	0
$790$	1.41226e30	0.530677
$791$	6.87331e30	2.54867
$792$	0	0
$793$	6.63040e30	2.39426
$794$	6.67633e29	0.237915
$795$	0	0
$796$	2.60555e29	0.0904299
$797$	3.68558e30	1.26239	0.631195	−	0.775624i	$-0.282565\pi$
$797$	3.68558e30	1.26239	0.631195	+	0.775624i	$0.282565\pi$
$798$	0	0
$799$	−1.18312e30	−0.394720
$800$	2.41764e29	0.0796061
$801$	0	0
$802$	−2.95952e30	−0.949271
$803$	2.20017e30	0.696536
$804$	0	0
$805$	−4.97479e29	−0.153433
$806$	−2.50478e30	−0.762522
$807$	0	0
$808$	8.20643e29	0.243408
$809$	1.59133e30	0.465909	0.232954	−	0.972488i	$-0.425161\pi$
$809$	1.59133e30	0.465909	0.232954	+	0.972488i	$0.425161\pi$
$810$	0	0
$811$	2.03075e30	0.579345	0.289673	−	0.957126i	$-0.406454\pi$
$811$	2.03075e30	0.579345	0.289673	+	0.957126i	$0.406454\pi$
$812$	7.65290e29	0.215520
$813$	0	0
$814$	−5.40280e29	−0.148273
$815$	−2.86514e30	−0.776231
$816$	0	0
$817$	−6.71157e30	−1.77212
$818$	−2.21189e30	−0.576573
$819$	0	0
$820$	−3.31243e29	−0.0841592
$821$	1.16660e30	0.292631	0.146315	−	0.989238i	$-0.453259\pi$
$821$	1.16660e30	0.292631	0.146315	+	0.989238i	$0.453259\pi$
$822$	0	0
$823$	−4.40906e30	−1.07807	−0.539036	−	0.842283i	$-0.681211\pi$
$823$	−4.40906e30	−1.07807	−0.539036	+	0.842283i	$0.681211\pi$
$824$	−7.48954e30	−1.80809
$825$	0	0
$826$	6.50568e30	1.53110
$827$	2.06510e30	0.479881	0.239941	−	0.970788i	$-0.422872\pi$
$827$	2.06510e30	0.479881	0.239941	+	0.970788i	$0.422872\pi$
$828$	0	0
$829$	−3.07768e30	−0.697271	−0.348635	−	0.937258i	$-0.613355\pi$
$829$	−3.07768e30	−0.697271	−0.348635	+	0.937258i	$0.613355\pi$
$830$	−4.28530e29	−0.0958652
$831$	0	0
$832$	7.56170e30	1.64940
$833$	−6.53186e30	−1.40690
$834$	0	0
$835$	1.37512e30	0.288824
$836$	1.66246e30	0.344815
$837$	0	0
$838$	1.00497e29	0.0203278
$839$	4.30426e30	0.859800	0.429900	−	0.902876i	$-0.358549\pi$
$839$	4.30426e30	0.859800	0.429900	+	0.902876i	$0.358549\pi$
$840$	0	0
$841$	−3.14676e30	−0.613064
$842$	3.27249e30	0.629654
$843$	0	0
$844$	−8.45317e29	−0.158644
$845$	−2.89384e30	−0.536387
$846$	0	0
$847$	9.32035e30	1.68522
$848$	−7.76429e29	−0.138658
$849$	0	0
$850$	−7.46802e29	−0.130109
$851$	−1.37239e29	−0.0236167
$852$	0	0
$853$	−5.29542e30	−0.889069	−0.444534	−	0.895762i	$-0.646631\pi$
$853$	−5.29542e30	−0.889069	−0.444534	+	0.895762i	$0.646631\pi$
$854$	1.48859e31	2.46869
$855$	0	0
$856$	1.22594e31	1.98379
$857$	−8.07412e29	−0.129062	−0.0645308	−	0.997916i	$-0.520555\pi$
$857$	−8.07412e29	−0.129062	−0.0645308	+	0.997916i	$0.520555\pi$
$858$	0	0
$859$	4.76822e30	0.743751	0.371876	−	0.928283i	$-0.378715\pi$
$859$	4.76822e30	0.743751	0.371876	+	0.928283i	$0.378715\pi$
$860$	8.60406e29	0.132578
$861$	0	0
$862$	6.20328e29	0.0932814
$863$	−1.13640e31	−1.68817	−0.844085	−	0.536209i	$-0.819856\pi$
$863$	−1.13640e31	−1.68817	−0.844085	+	0.536209i	$0.819856\pi$
$864$	0	0
$865$	3.06969e28	0.00445066
$866$	1.03558e31	1.48336
$867$	0	0
$868$	1.42667e30	0.199464
$869$	−1.35494e31	−1.87159
$870$	0	0
$871$	6.79240e30	0.915868
$872$	4.72651e30	0.629676
$873$	0	0
$874$	−1.66454e30	−0.216484
$875$	−1.19172e30	−0.153140
$876$	0	0
$877$	6.52492e30	0.818615	0.409307	−	0.912397i	$-0.365770\pi$
$877$	6.52492e30	0.818615	0.409307	+	0.912397i	$0.365770\pi$
$878$	−1.23565e31	−1.53180
$879$	0	0
$880$	3.93823e30	0.476687
$881$	−6.24200e30	−0.746581	−0.373290	−	0.927715i	$-0.621770\pi$
$881$	−6.24200e30	−0.746581	−0.373290	+	0.927715i	$0.621770\pi$
$882$	0	0
$883$	−5.35499e30	−0.625419	−0.312709	−	0.949849i	$-0.601237\pi$
$883$	−5.35499e30	−0.625419	−0.312709	+	0.949849i	$0.601237\pi$
$884$	1.89408e30	0.218600
$885$	0	0
$886$	−7.16292e30	−0.807302
$887$	1.58739e31	1.76802	0.884009	−	0.467470i	$-0.154834\pi$
$887$	1.58739e31	1.76802	0.884009	+	0.467470i	$0.154834\pi$
$888$	0	0
$889$	2.54957e31	2.77331
$890$	4.28170e30	0.460279
$891$	0	0
$892$	−1.39700e30	−0.146678
$893$	−6.31710e30	−0.655507
$894$	0	0
$895$	−7.48991e30	−0.759163
$896$	1.01738e31	1.01918
$897$	0	0
$898$	−7.19285e30	−0.703881
$899$	3.70248e30	0.358110
$900$	0	0
$901$	−1.41257e30	−0.133475
$902$	−1.25266e31	−1.16995
$903$	0	0
$904$	−1.75176e31	−1.59848
$905$	−1.48077e30	−0.133560
$906$	0	0
$907$	−2.03878e31	−1.79677	−0.898386	−	0.439206i	$-0.855260\pi$
$907$	−2.03878e31	−1.79677	−0.898386	+	0.439206i	$0.855260\pi$
$908$	−6.51945e29	−0.0567950
$909$	0	0
$910$	−1.19138e31	−1.01418
$911$	−1.16077e31	−0.976792	−0.488396	−	0.872622i	$-0.662418\pi$
$911$	−1.16077e31	−0.976792	−0.488396	+	0.872622i	$0.662418\pi$
$912$	0	0
$913$	4.11136e30	0.338098
$914$	7.60452e30	0.618211
$915$	0	0
$916$	1.57416e30	0.125068
$917$	−2.49906e31	−1.96290
$918$	0	0
$919$	1.11455e31	0.855631	0.427816	−	0.903866i	$-0.359283\pi$
$919$	1.11455e31	0.855631	0.427816	+	0.903866i	$0.359283\pi$
$920$	1.26790e30	0.0962304
$921$	0	0
$922$	1.46687e31	1.08822
$923$	3.50475e31	2.57062
$924$	0	0
$925$	−3.28759e29	−0.0235716
$926$	2.25439e31	1.59813
$927$	0	0
$928$	−3.57264e30	−0.247591
$929$	1.58810e30	0.108821	0.0544105	−	0.998519i	$-0.482672\pi$
$929$	1.58810e30	0.108821	0.0544105	+	0.998519i	$0.482672\pi$
$930$	0	0
$931$	−3.48758e31	−2.33643
$932$	−4.77528e30	−0.316324
$933$	0	0
$934$	−2.07848e30	−0.134618
$935$	7.16490e30	0.458869
$936$	0	0
$937$	−3.78856e29	−0.0237251	−0.0118626	−	0.999930i	$-0.503776\pi$
$937$	−3.78856e29	−0.0237251	−0.0118626	+	0.999930i	$0.503776\pi$
$938$	1.52496e31	0.944339
$939$	0	0
$940$	8.09836e29	0.0490405
$941$	2.33225e31	1.39664	0.698320	−	0.715786i	$-0.253931\pi$
$941$	2.33225e31	1.39664	0.698320	+	0.715786i	$0.253931\pi$
$942$	0	0
$943$	−3.18196e30	−0.186347
$944$	−1.30822e31	−0.757659
$945$	0	0
$946$	3.25380e31	1.84304
$947$	1.89428e31	1.06113	0.530566	−	0.847643i	$-0.321979\pi$
$947$	1.89428e31	1.06113	0.530566	+	0.847643i	$0.321979\pi$
$948$	0	0
$949$	1.33841e31	0.733323
$950$	−3.98743e30	−0.216070
$951$	0	0
$952$	2.52664e31	1.33923
$953$	−2.74511e31	−1.43908	−0.719538	−	0.694453i	$-0.755646\pi$
$953$	−2.74511e31	−1.43908	−0.719538	+	0.694453i	$0.755646\pi$
$954$	0	0
$955$	5.92569e30	0.303881
$956$	5.56161e30	0.282093
$957$	0	0
$958$	6.97570e30	0.346139
$959$	−3.25898e31	−1.59951
$960$	0	0
$961$	−1.39233e31	−0.668569
$962$	−3.28664e30	−0.156104
$963$	0	0
$964$	−7.06466e29	−0.0328309
$965$	−1.83199e31	−0.842143
$966$	0	0
$967$	3.03551e31	1.36538	0.682690	−	0.730708i	$-0.260810\pi$
$967$	3.03551e31	1.36538	0.682690	+	0.730708i	$0.260810\pi$
$968$	−2.37543e31	−1.05694
$969$	0	0
$970$	−1.41469e30	−0.0615967
$971$	−3.36946e31	−1.45130	−0.725651	−	0.688063i	$-0.758461\pi$
$971$	−3.36946e31	−1.45130	−0.725651	+	0.688063i	$0.758461\pi$
$972$	0	0
$973$	−4.20277e31	−1.77154
$974$	1.81062e31	0.755020
$975$	0	0
$976$	−2.99337e31	−1.22163
$977$	−1.47911e31	−0.597184	−0.298592	−	0.954381i	$-0.596517\pi$
$977$	−1.47911e31	−0.597184	−0.298592	+	0.954381i	$0.596517\pi$
$978$	0	0
$979$	−4.10791e31	−1.62331
$980$	4.47099e30	0.174796
$981$	0	0
$982$	−2.15080e30	−0.0823057
$983$	−3.37648e31	−1.27836	−0.639180	−	0.769058i	$-0.720726\pi$
$983$	−3.37648e31	−1.27836	−0.639180	+	0.769058i	$0.720726\pi$
$984$	0	0
$985$	2.21605e30	0.0821297
$986$	1.10358e31	0.404665
$987$	0	0
$988$	1.01131e31	0.363026
$989$	8.26515e30	0.293555
$990$	0	0
$991$	−5.48368e31	−1.90677	−0.953387	−	0.301751i	$-0.902429\pi$
$991$	−5.48368e31	−1.90677	−0.953387	+	0.301751i	$0.902429\pi$
$992$	−6.66017e30	−0.229146
$993$	0	0
$994$	7.86848e31	2.65054
$995$	5.99577e30	0.199849
$996$	0	0
$997$	−2.82839e31	−0.923082	−0.461541	−	0.887119i	$-0.652703\pi$
$997$	−2.82839e31	−0.923082	−0.461541	+	0.887119i	$0.652703\pi$
$998$	−2.90964e31	−0.939654
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.22.a.g.1.3		4
3.2	odd	2		15.22.a.e.1.2	✓	4
15.2	even	4		75.22.b.h.49.3		8
15.8	even	4		75.22.b.h.49.6		8
15.14	odd	2		75.22.a.h.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.22.a.e.1.2	✓	4	3.2	odd	2
45.22.a.g.1.3		4	1.1	even	1	trivial
75.22.a.h.1.3		4	15.14	odd	2
75.22.b.h.49.3		8	15.2	even	4
75.22.b.h.49.6		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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+1293.36 q^{2} -424379. q^{4} -9.76562e6 q^{5} +1.27960e9 q^{7} -3.26124e9 q^{8} -1.26304e10 q^{10} +1.21178e11 q^{11} +7.37152e11 q^{13} +1.65498e12 q^{14} -3.32796e12 q^{16} -6.05462e12 q^{17} -3.23277e13 q^{19} +4.14433e12 q^{20} +1.56726e14 q^{22} +3.98109e13 q^{23} +9.53674e13 q^{25} +9.53401e14 q^{26} -5.43035e14 q^{28} -1.40928e15 q^{29} -2.62721e15 q^{31} +2.53508e15 q^{32} -7.83079e15 q^{34} -1.24961e16 q^{35} -3.44728e15 q^{37} -4.18113e16 q^{38} +3.18481e16 q^{40} -7.99269e16 q^{41} +2.07610e17 q^{43} -5.14254e16 q^{44} +5.14897e16 q^{46} +1.95408e17 q^{47} +1.07882e18 q^{49} +1.23344e17 q^{50} -3.12832e17 q^{52} +2.33305e17 q^{53} -1.18338e18 q^{55} -4.17307e18 q^{56} -1.82271e18 q^{58} +3.93099e18 q^{59} +8.99462e18 q^{61} -3.39792e18 q^{62} +1.02580e19 q^{64} -7.19875e18 q^{65} +9.21439e18 q^{67} +2.56946e18 q^{68} -1.61619e19 q^{70} +4.75444e19 q^{71} +1.81566e19 q^{73} -4.45857e18 q^{74} +1.37192e19 q^{76} +1.55059e20 q^{77} -1.11814e20 q^{79} +3.24996e19 q^{80} -1.03374e20 q^{82} +3.39283e19 q^{83} +5.91272e19 q^{85} +2.68514e20 q^{86} -3.95190e20 q^{88} -3.38998e20 q^{89} +9.43257e20 q^{91} -1.68949e19 q^{92} +2.52733e20 q^{94} +3.15700e20 q^{95} +1.12006e20 q^{97} +1.39530e21 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 897 q^{2} - 1163123 q^{4} - 39062500 q^{5} - 234577504 q^{7} - 76855629 q^{8} - 8759765625 q^{10} - 31491830256 q^{11} - 27017977768 q^{13} + 2233308126672 q^{14} - 11324681311775 q^{16} + 2946095028888 q^{17}+ \cdots - 19\!\cdots\!63 q^{98}+O(q^{100}) \) 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

Related objects

Downloads

Learn more