LMFDB - Embedded newform 9.20.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9,20,Mod(1,9)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 20, names="a")

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

chi := DirichletCharacter("9.1");

S:= CuspForms(chi, 20);

N := Newforms(S);

Level:	$ N $	$=$	$ 9 = 3^{2} $
Weight:	$ k $	$=$	$ 20 $
Character orbit:	$[\chi]$	$=$	9.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:	$20.5935026901$
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} - 40190x^{2} + 83776000 $ x^4 - 40190*x^2 + 83776000
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{10}\cdot 5^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$46.9631$ of defining polynomial
Character	$\chi$	$=$	9.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+281.779 q^{2} -444889. q^{4} +8.20822e6 q^{5} -9.23883e7 q^{7} -2.73093e8 q^{8} +O(q^{10})$	\(q+281.779 q^{2} -444889. q^{4} +8.20822e6 q^{5} -9.23883e7 q^{7} -2.73093e8 q^{8} +2.31290e9 q^{10} +9.91745e9 q^{11} -1.56864e10 q^{13} -2.60331e10 q^{14} +1.56298e11 q^{16} +4.97295e11 q^{17} +1.50719e12 q^{19} -3.65174e12 q^{20} +2.79453e12 q^{22} +7.64436e12 q^{23} +4.83013e13 q^{25} -4.42008e12 q^{26} +4.11025e13 q^{28} -5.91555e12 q^{29} +1.11767e14 q^{31} +1.87221e14 q^{32} +1.40127e14 q^{34} -7.58343e14 q^{35} -8.39465e14 q^{37} +4.24693e14 q^{38} -2.24161e15 q^{40} +2.42907e15 q^{41} -3.11030e15 q^{43} -4.41216e15 q^{44} +2.15402e15 q^{46} +1.18055e15 q^{47} -2.86330e15 q^{49} +1.36103e16 q^{50} +6.97869e15 q^{52} -4.30508e16 q^{53} +8.14046e16 q^{55} +2.52306e16 q^{56} -1.66687e15 q^{58} +4.74797e16 q^{59} -2.24946e16 q^{61} +3.14935e16 q^{62} -2.91903e16 q^{64} -1.28757e17 q^{65} -1.96208e17 q^{67} -2.21241e17 q^{68} -2.13685e17 q^{70} +6.00058e17 q^{71} -2.97653e17 q^{73} -2.36543e17 q^{74} -6.70531e17 q^{76} -9.16257e17 q^{77} +1.40502e18 q^{79} +1.28293e18 q^{80} +6.84459e17 q^{82} +1.15626e18 q^{83} +4.08191e18 q^{85} -8.76416e17 q^{86} -2.70839e18 q^{88} -3.48883e18 q^{89} +1.44924e18 q^{91} -3.40089e18 q^{92} +3.32654e17 q^{94} +1.23713e19 q^{95} +2.73147e18 q^{97} -8.06816e17 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 796528 q^{4} + 166272080 q^{7}+O(q^{10}) $ 4 * q + 796528 * q^4 + 166272080 * q^7	$ 4 q + 796528 q^{4} + 166272080 q^{7} - 3773076480 q^{10} + 53157676520 q^{13} + 300543715456 q^{16} + 3768641418464 q^{19} + 36676792627200 q^{22} + 84248826573580 q^{25} + 378192741767360 q^{28} + 471591909029072 q^{31} - 399874435983360 q^{34} - 22\!\cdots\!60 q^{37}+ \cdots - 41\!\cdots\!20 q^{97}+O(q^{100}) $ 4 * q + 796528 * q^4 + 166272080 * q^7 - 3773076480 * q^10 + 53157676520 * q^13 + 300543715456 * q^16 + 3768641418464 * q^19 + 36676792627200 * q^22 + 84248826573580 * q^25 + 378192741767360 * q^28 + 471591909029072 * q^31 - 399874435983360 * q^34 - 2228516688441160 * q^37 - 7161324419358720 * q^40 - 12262326274792480 * q^43 - 17070303547514880 * q^46 + 33093211591411428 * q^49 + 85229426103167840 * q^52 + 67338095392972800 * q^55 + 187691248170316800 * q^58 + 160046271305607128 * q^61 - 525751810435539968 * q^64 - 486553865858307520 * q^67 - 1901577196642406400 * q^70 - 130672866884308840 * q^73 - 705101514474977152 * q^76 + 3246789793721214416 * q^79 + 1277162815720089600 * q^82 + 10252506464133672960 * q^85 + 4496007782235340800 * q^88 + 17735617946513428000 * q^91 - 34218764107889172480 * q^94 - 417886879084958920 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	281.779	0.389155	0.194578	−	0.980887i	$-0.437666\pi$
$2$	281.779	0.389155	0.194578	+	0.980887i	$0.437666\pi$
$3$	0	0
$3$	0	0
$4$	−444889.	−0.848558
$5$	8.20822e6	1.87946	0.939732	−	0.341913i	$-0.111075\pi$
$5$	8.20822e6	1.87946	0.939732	+	0.341913i	$0.111075\pi$
$6$	0	0
$7$	−9.23883e7	−0.865338	−0.432669	−	0.901553i	$-0.642428\pi$
$7$	−9.23883e7	−0.865338	−0.432669	+	0.901553i	$0.642428\pi$
$8$	−2.73093e8	−0.719376
$9$	0	0
$10$	2.31290e9	0.731403
$11$	9.91745e9	1.26815	0.634073	−	0.773273i	$-0.281382\pi$
$11$	9.91745e9	1.26815	0.634073	+	0.773273i	$0.281382\pi$
$12$	0	0
$13$	−1.56864e10	−0.410261	−0.205131	−	0.978735i	$-0.565762\pi$
$13$	−1.56864e10	−0.410261	−0.205131	+	0.978735i	$0.565762\pi$
$14$	−2.60331e10	−0.336751
$15$	0	0
$16$	1.56298e11	0.568609
$17$	4.97295e11	1.01707	0.508534	−	0.861042i	$-0.330188\pi$
$17$	4.97295e11	1.01707	0.508534	+	0.861042i	$0.330188\pi$
$18$	0	0
$19$	1.50719e12	1.07154	0.535770	−	0.844364i	$-0.320021\pi$
$19$	1.50719e12	1.07154	0.535770	+	0.844364i	$0.320021\pi$
$20$	−3.65174e12	−1.59483
$21$	0	0
$22$	2.79453e12	0.493506
$23$	7.64436e12	0.884966	0.442483	−	0.896777i	$-0.354098\pi$
$23$	7.64436e12	0.884966	0.442483	+	0.896777i	$0.354098\pi$
$24$	0	0
$25$	4.83013e13	2.53238
$26$	−4.42008e12	−0.159655
$27$	0	0
$28$	4.11025e13	0.734289
$29$	−5.91555e12	−0.0757206	−0.0378603	−	0.999283i	$-0.512054\pi$
$29$	−5.91555e12	−0.0757206	−0.0378603	+	0.999283i	$0.512054\pi$
$30$	0	0
$31$	1.11767e14	0.759237	0.379618	−	0.925143i	$-0.376055\pi$
$31$	1.11767e14	0.759237	0.379618	+	0.925143i	$0.376055\pi$
$32$	1.87221e14	0.940654
$33$	0	0
$34$	1.40127e14	0.395797
$35$	−7.58343e14	−1.62637
$36$	0	0
$37$	−8.39465e14	−1.06191	−0.530953	−	0.847401i	$-0.678166\pi$
$37$	−8.39465e14	−1.06191	−0.530953	+	0.847401i	$0.678166\pi$
$38$	4.24693e14	0.416995
$39$	0	0
$40$	−2.24161e15	−1.35204
$41$	2.42907e15	1.15876	0.579379	−	0.815058i	$-0.303295\pi$
$41$	2.42907e15	1.15876	0.579379	+	0.815058i	$0.303295\pi$
$42$	0	0
$43$	−3.11030e15	−0.943739	−0.471869	−	0.881669i	$-0.656421\pi$
$43$	−3.11030e15	−0.943739	−0.471869	+	0.881669i	$0.656421\pi$
$44$	−4.41216e15	−1.07610
$45$	0	0
$46$	2.15402e15	0.344389
$47$	1.18055e15	0.153870	0.0769351	−	0.997036i	$-0.475487\pi$
$47$	1.18055e15	0.153870	0.0769351	+	0.997036i	$0.475487\pi$
$48$	0	0
$49$	−2.86330e15	−0.251191
$50$	1.36103e16	0.985490
$51$	0	0
$52$	6.97869e15	0.348130
$53$	−4.30508e16	−1.79209	−0.896047	−	0.443959i	$-0.853573\pi$
$53$	−4.30508e16	−1.79209	−0.896047	+	0.443959i	$0.853573\pi$
$54$	0	0
$55$	8.14046e16	2.38344
$56$	2.52306e16	0.622504
$57$	0	0
$58$	−1.66687e15	−0.0294671
$59$	4.74797e16	0.713532	0.356766	−	0.934194i	$-0.383879\pi$
$59$	4.74797e16	0.713532	0.356766	+	0.934194i	$0.383879\pi$
$60$	0	0
$61$	−2.24946e16	−0.246289	−0.123144	−	0.992389i	$-0.539298\pi$
$61$	−2.24946e16	−0.246289	−0.123144	+	0.992389i	$0.539298\pi$
$62$	3.14935e16	0.295461
$63$	0	0
$64$	−2.91903e16	−0.202548
$65$	−1.28757e17	−0.771071
$66$	0	0
$67$	−1.96208e17	−0.881060	−0.440530	−	0.897738i	$-0.645210\pi$
$67$	−1.96208e17	−0.881060	−0.440530	+	0.897738i	$0.645210\pi$
$68$	−2.21241e17	−0.863041
$69$	0	0
$70$	−2.13685e17	−0.632911
$71$	6.00058e17	1.55324	0.776621	−	0.629968i	$-0.216932\pi$
$71$	6.00058e17	1.55324	0.776621	+	0.629968i	$0.216932\pi$
$72$	0	0
$73$	−2.97653e17	−0.591757	−0.295879	−	0.955226i	$-0.595612\pi$
$73$	−2.97653e17	−0.591757	−0.295879	+	0.955226i	$0.595612\pi$
$74$	−2.36543e17	−0.413246
$75$	0	0
$76$	−6.70531e17	−0.909264
$77$	−9.16257e17	−1.09738
$78$	0	0
$79$	1.40502e18	1.31894	0.659470	−	0.751731i	$-0.270781\pi$
$79$	1.40502e18	1.31894	0.659470	+	0.751731i	$0.270781\pi$
$80$	1.28293e18	1.06868
$81$	0	0
$82$	6.84459e17	0.450937
$83$	1.15626e18	0.678914	0.339457	−	0.940621i	$-0.389757\pi$
$83$	1.15626e18	0.678914	0.339457	+	0.940621i	$0.389757\pi$
$84$	0	0
$85$	4.08191e18	1.91154
$86$	−8.76416e17	−0.367261
$87$	0	0
$88$	−2.70839e18	−0.912275
$89$	−3.48883e18	−1.05554	−0.527771	−	0.849387i	$-0.676972\pi$
$89$	−3.48883e18	−1.05554	−0.527771	+	0.849387i	$0.676972\pi$
$90$	0	0
$91$	1.44924e18	0.355015
$92$	−3.40089e18	−0.750945
$93$	0	0
$94$	3.32654e17	0.0598795
$95$	1.23713e19	2.01392
$96$	0	0
$97$	2.73147e18	0.364808	0.182404	−	0.983224i	$-0.441612\pi$
$97$	2.73147e18	0.364808	0.182404	+	0.983224i	$0.441612\pi$
$98$	−8.06816e17	−0.0977522
$99$	0	0
$100$	−2.14887e19	−2.14887
$101$	−8.79291e18	−0.799981	−0.399991	−	0.916519i	$-0.630987\pi$
$101$	−8.79291e18	−0.799981	−0.399991	+	0.916519i	$0.630987\pi$
$102$	0	0
$103$	−6.44538e18	−0.486738	−0.243369	−	0.969934i	$-0.578253\pi$
$103$	−6.44538e18	−0.486738	−0.243369	+	0.969934i	$0.578253\pi$
$104$	4.28384e18	0.295132
$105$	0	0
$106$	−1.21308e19	−0.697403
$107$	−7.48175e18	−0.393421	−0.196710	−	0.980462i	$-0.563026\pi$
$107$	−7.48175e18	−0.393421	−0.196710	+	0.980462i	$0.563026\pi$
$108$	0	0
$109$	2.09184e19	0.922521	0.461260	−	0.887265i	$-0.347397\pi$
$109$	2.09184e19	0.922521	0.461260	+	0.887265i	$0.347397\pi$
$110$	2.29381e19	0.927527
$111$	0	0
$112$	−1.44401e19	−0.492039
$113$	−2.26811e19	−0.710263	−0.355132	−	0.934816i	$-0.615564\pi$
$113$	−2.26811e19	−0.710263	−0.355132	+	0.934816i	$0.615564\pi$
$114$	0	0
$115$	6.27465e19	1.66326
$116$	2.63176e18	0.0642533
$117$	0	0
$118$	1.33788e19	0.277675
$119$	−4.59443e19	−0.880107
$120$	0	0
$121$	3.71967e19	0.608196
$122$	−6.33849e18	−0.0958446
$123$	0	0
$124$	−4.97239e19	−0.644256
$125$	2.39909e20	2.88005
$126$	0	0
$127$	1.31998e20	1.36280	0.681402	−	0.731909i	$-0.261370\pi$
$127$	1.31998e20	1.36280	0.681402	+	0.731909i	$0.261370\pi$
$128$	−1.06383e20	−1.01948
$129$	0	0
$130$	−3.62810e19	−0.300066
$131$	−1.62027e20	−1.24598	−0.622989	−	0.782230i	$-0.714082\pi$
$131$	−1.62027e20	−1.24598	−0.622989	+	0.782230i	$0.714082\pi$
$132$	0	0
$133$	−1.39247e20	−0.927244
$134$	−5.52872e19	−0.342869
$135$	0	0
$136$	−1.35808e20	−0.731654
$137$	2.49355e18	0.0125306	0.00626531	−	0.999980i	$-0.498006\pi$
$137$	2.49355e18	0.0125306	0.00626531	+	0.999980i	$0.498006\pi$
$138$	0	0
$139$	−4.30038e20	−1.88307	−0.941535	−	0.336916i	$-0.890616\pi$
$139$	−4.30038e20	−1.88307	−0.941535	+	0.336916i	$0.890616\pi$
$140$	3.37378e20	1.38007
$141$	0	0
$142$	1.69084e20	0.604453
$143$	−1.55569e20	−0.520271
$144$	0	0
$145$	−4.85561e19	−0.142314
$146$	−8.38724e19	−0.230286
$147$	0	0
$148$	3.73468e20	0.901089
$149$	1.92753e19	0.0436247	0.0218124	−	0.999762i	$-0.493056\pi$
$149$	1.92753e19	0.0436247	0.0218124	+	0.999762i	$0.493056\pi$
$150$	0	0
$151$	2.20822e20	0.440313	0.220156	−	0.975465i	$-0.429343\pi$
$151$	2.20822e20	0.440313	0.220156	+	0.975465i	$0.429343\pi$
$152$	−4.11603e20	−0.770840
$153$	0	0
$154$	−2.58181e20	−0.427050
$155$	9.17407e20	1.42696
$156$	0	0
$157$	4.03486e20	0.555625	0.277812	−	0.960635i	$-0.410391\pi$
$157$	4.03486e20	0.555625	0.277812	+	0.960635i	$0.410391\pi$
$158$	3.95905e20	0.513273
$159$	0	0
$160$	1.53675e21	1.76792
$161$	−7.06249e20	−0.765794
$162$	0	0
$163$	−2.70445e20	−0.260794	−0.130397	−	0.991462i	$-0.541625\pi$
$163$	−2.70445e20	−0.260794	−0.130397	+	0.991462i	$0.541625\pi$
$164$	−1.08066e21	−0.983273
$165$	0	0
$166$	3.25810e20	0.264203
$167$	−6.18664e20	−0.473858	−0.236929	−	0.971527i	$-0.576141\pi$
$167$	−6.18664e20	−0.473858	−0.236929	+	0.971527i	$0.576141\pi$
$168$	0	0
$169$	−1.21586e21	−0.831686
$170$	1.15019e21	0.743886
$171$	0	0
$172$	1.38374e21	0.800817
$173$	3.37328e20	0.184763	0.0923814	−	0.995724i	$-0.470552\pi$
$173$	3.37328e20	0.184763	0.0923814	+	0.995724i	$0.470552\pi$
$174$	0	0
$175$	−4.46248e21	−2.19137
$176$	1.55008e21	0.721080
$177$	0	0
$178$	−9.83079e20	−0.410770
$179$	8.34660e20	0.330678	0.165339	−	0.986237i	$-0.447128\pi$
$179$	8.34660e20	0.330678	0.165339	+	0.986237i	$0.447128\pi$
$180$	0	0
$181$	−1.69837e21	−0.605462	−0.302731	−	0.953076i	$-0.597898\pi$
$181$	−1.69837e21	−0.605462	−0.302731	+	0.953076i	$0.597898\pi$
$182$	4.08364e20	0.138156
$183$	0	0
$184$	−2.08762e21	−0.636623
$185$	−6.89051e21	−1.99581
$186$	0	0
$187$	4.93190e21	1.28979
$188$	−5.25213e20	−0.130568
$189$	0	0
$190$	3.48598e21	0.783728
$191$	6.85996e21	1.46725	0.733626	−	0.679554i	$-0.237827\pi$
$191$	6.85996e21	1.46725	0.733626	+	0.679554i	$0.237827\pi$
$192$	0	0
$193$	−4.74127e21	−0.918545	−0.459272	−	0.888295i	$-0.651890\pi$
$193$	−4.74127e21	−0.918545	−0.459272	+	0.888295i	$0.651890\pi$
$194$	7.69669e20	0.141967
$195$	0	0
$196$	1.27385e21	0.213150
$197$	2.74849e21	0.438192	0.219096	−	0.975703i	$-0.429689\pi$
$197$	2.74849e21	0.438192	0.219096	+	0.975703i	$0.429689\pi$
$198$	0	0
$199$	8.65318e21	1.25335	0.626674	−	0.779282i	$-0.284416\pi$
$199$	8.65318e21	1.25335	0.626674	+	0.779282i	$0.284416\pi$
$200$	−1.31908e22	−1.82174
$201$	0	0
$202$	−2.47766e21	−0.311317
$203$	5.46527e20	0.0655239
$204$	0	0
$205$	1.99383e22	2.17784
$206$	−1.81617e21	−0.189417
$207$	0	0
$208$	−2.45175e21	−0.233278
$209$	1.49475e22	1.35887
$210$	0	0
$211$	−7.15229e21	−0.593966	−0.296983	−	0.954883i	$-0.595980\pi$
$211$	−7.15229e21	−0.593966	−0.296983	+	0.954883i	$0.595980\pi$
$212$	1.91528e22	1.52070
$213$	0	0
$214$	−2.10820e21	−0.153102
$215$	−2.55300e22	−1.77372
$216$	0	0
$217$	−1.03260e22	−0.656996
$218$	5.89435e21	0.359004
$219$	0	0
$220$	−3.62160e22	−2.02248
$221$	−7.80076e21	−0.417263
$222$	0	0
$223$	1.36002e22	0.667807	0.333903	−	0.942607i	$-0.391634\pi$
$223$	1.36002e22	0.667807	0.333903	+	0.942607i	$0.391634\pi$
$224$	−1.72970e22	−0.813983
$225$	0	0
$226$	−6.39105e21	−0.276403
$227$	−1.39594e22	−0.578924	−0.289462	−	0.957190i	$-0.593476\pi$
$227$	−1.39594e22	−0.578924	−0.289462	+	0.957190i	$0.593476\pi$
$228$	0	0
$229$	−1.37017e22	−0.522801	−0.261400	−	0.965230i	$-0.584184\pi$
$229$	−1.37017e22	−0.522801	−0.261400	+	0.965230i	$0.584184\pi$
$230$	1.76806e22	0.647267
$231$	0	0
$232$	1.61550e21	0.0544716
$233$	5.81916e22	1.88356	0.941780	−	0.336231i	$-0.109152\pi$
$233$	5.81916e22	1.88356	0.941780	+	0.336231i	$0.109152\pi$
$234$	0	0
$235$	9.69021e21	0.289193
$236$	−2.11232e22	−0.605474
$237$	0	0
$238$	−1.29461e22	−0.342498
$239$	−2.27842e22	−0.579234	−0.289617	−	0.957143i	$-0.593528\pi$
$239$	−2.27842e22	−0.579234	−0.289617	+	0.957143i	$0.593528\pi$
$240$	0	0
$241$	6.13221e22	1.44031	0.720154	−	0.693814i	$-0.244071\pi$
$241$	6.13221e22	1.44031	0.720154	+	0.693814i	$0.244071\pi$
$242$	1.04812e22	0.236683
$243$	0	0
$244$	1.00076e22	0.208990
$245$	−2.35026e22	−0.472104
$246$	0	0
$247$	−2.36423e22	−0.439611
$248$	−3.05228e22	−0.546177
$249$	0	0
$250$	6.76011e22	1.12079
$251$	7.61994e22	1.21633	0.608165	−	0.793811i	$-0.291906\pi$
$251$	7.61994e22	1.21633	0.608165	+	0.793811i	$0.291906\pi$
$252$	0	0
$253$	7.58125e22	1.12227
$254$	3.71943e22	0.530343
$255$	0	0
$256$	−1.46723e22	−0.194186
$257$	−8.58510e22	−1.09492	−0.547458	−	0.836833i	$-0.684404\pi$
$257$	−8.58510e22	−1.09492	−0.547458	+	0.836833i	$0.684404\pi$
$258$	0	0
$259$	7.75567e22	0.918907
$260$	5.72826e22	0.654298
$261$	0	0
$262$	−4.56558e22	−0.484879
$263$	−8.44602e22	−0.865113	−0.432556	−	0.901607i	$-0.642388\pi$
$263$	−8.44602e22	−0.865113	−0.432556	+	0.901607i	$0.642388\pi$
$264$	0	0
$265$	−3.53371e23	−3.36817
$266$	−3.92367e22	−0.360842
$267$	0	0
$268$	8.72906e22	0.747631
$269$	−1.26750e23	−1.04786	−0.523929	−	0.851762i	$-0.675534\pi$
$269$	−1.26750e23	−1.04786	−0.523929	+	0.851762i	$0.675534\pi$
$270$	0	0
$271$	−4.11104e22	−0.316769	−0.158385	−	0.987377i	$-0.550629\pi$
$271$	−4.11104e22	−0.316769	−0.158385	+	0.987377i	$0.550629\pi$
$272$	7.77263e22	0.578313
$273$	0	0
$274$	7.02628e20	0.00487636
$275$	4.79026e23	3.21143
$276$	0	0
$277$	−1.55361e23	−0.972264	−0.486132	−	0.873885i	$-0.661593\pi$
$277$	−1.55361e23	−0.972264	−0.486132	+	0.873885i	$0.661593\pi$
$278$	−1.21176e23	−0.732807
$279$	0	0
$280$	2.07098e23	1.16997
$281$	2.35762e21	0.0128755	0.00643774	−	0.999979i	$-0.497951\pi$
$281$	2.35762e21	0.0128755	0.00643774	+	0.999979i	$0.497951\pi$
$282$	0	0
$283$	−1.68070e23	−0.858062	−0.429031	−	0.903290i	$-0.641145\pi$
$283$	−1.68070e23	−0.858062	−0.429031	+	0.903290i	$0.641145\pi$
$284$	−2.66959e23	−1.31802
$285$	0	0
$286$	−4.38359e22	−0.202466
$287$	−2.24417e23	−1.00272
$288$	0	0
$289$	8.23027e21	0.0344259
$290$	−1.36821e22	−0.0553823
$291$	0	0
$292$	1.32423e23	0.502140
$293$	1.91367e23	0.702465	0.351232	−	0.936288i	$-0.385763\pi$
$293$	1.91367e23	0.702465	0.351232	+	0.936288i	$0.385763\pi$
$294$	0	0
$295$	3.89723e23	1.34106
$296$	2.29252e23	0.763910
$297$	0	0
$298$	5.43138e21	0.0169768
$299$	−1.19912e23	−0.363067
$300$	0	0
$301$	2.87355e23	0.816653
$302$	6.22229e22	0.171350
$303$	0	0
$304$	2.35571e23	0.609287
$305$	−1.84640e23	−0.462890
$306$	0	0
$307$	5.62254e22	0.132470	0.0662351	−	0.997804i	$-0.478901\pi$
$307$	5.62254e22	0.132470	0.0662351	+	0.997804i	$0.478901\pi$
$308$	4.07632e23	0.931187
$309$	0	0
$310$	2.58506e23	0.555308
$311$	−7.13579e23	−1.48668	−0.743341	−	0.668912i	$-0.766760\pi$
$311$	−7.13579e23	−1.48668	−0.743341	+	0.668912i	$0.766760\pi$
$312$	0	0
$313$	−4.16338e23	−0.816160	−0.408080	−	0.912946i	$-0.633802\pi$
$313$	−4.16338e23	−0.816160	−0.408080	+	0.912946i	$0.633802\pi$
$314$	1.13694e23	0.216224
$315$	0	0
$316$	−6.25078e23	−1.11920
$317$	5.09416e23	0.885136	0.442568	−	0.896735i	$-0.354068\pi$
$317$	5.09416e23	0.885136	0.442568	+	0.896735i	$0.354068\pi$
$318$	0	0
$319$	−5.86671e22	−0.0960248
$320$	−2.39600e23	−0.380682
$321$	0	0
$322$	−1.99006e23	−0.298013
$323$	7.49518e23	1.08983
$324$	0	0
$325$	−7.57672e23	−1.03894
$326$	−7.62057e22	−0.101489
$327$	0	0
$328$	−6.63362e23	−0.833583
$329$	−1.09069e23	−0.133150
$330$	0	0
$331$	4.97482e23	0.573339	0.286669	−	0.958030i	$-0.407452\pi$
$331$	4.97482e23	0.573339	0.286669	+	0.958030i	$0.407452\pi$
$332$	−5.14409e23	−0.576098
$333$	0	0
$334$	−1.74326e23	−0.184404
$335$	−1.61052e24	−1.65592
$336$	0	0
$337$	1.06814e24	1.03788	0.518939	−	0.854811i	$-0.326327\pi$
$337$	1.06814e24	1.03788	0.518939	+	0.854811i	$0.326327\pi$
$338$	−3.42603e23	−0.323655
$339$	0	0
$340$	−1.81600e24	−1.62205
$341$	1.10844e24	0.962824
$342$	0	0
$343$	1.31766e24	1.08270
$344$	8.49402e23	0.678903
$345$	0	0
$346$	9.50518e22	0.0719014
$347$	1.81575e24	1.33637	0.668186	−	0.743994i	$-0.267071\pi$
$347$	1.81575e24	1.33637	0.668186	+	0.743994i	$0.267071\pi$
$348$	0	0
$349$	−1.90241e24	−1.32575	−0.662876	−	0.748729i	$-0.730664\pi$
$349$	−1.90241e24	−1.32575	−0.662876	+	0.748729i	$0.730664\pi$
$350$	−1.25743e24	−0.852782
$351$	0	0
$352$	1.85675e24	1.19289
$353$	4.69409e23	0.293557	0.146778	−	0.989169i	$-0.453110\pi$
$353$	4.69409e23	0.293557	0.146778	+	0.989169i	$0.453110\pi$
$354$	0	0
$355$	4.92541e24	2.91926
$356$	1.55214e24	0.895688
$357$	0	0
$358$	2.35189e23	0.128685
$359$	−3.08622e24	−1.64448	−0.822242	−	0.569138i	$-0.807277\pi$
$359$	−3.08622e24	−1.64448	−0.822242	+	0.569138i	$0.807277\pi$
$360$	0	0
$361$	2.93196e23	0.148197
$362$	−4.78565e23	−0.235619
$363$	0	0
$364$	−6.44749e23	−0.301250
$365$	−2.44320e24	−1.11219
$366$	0	0
$367$	6.74399e22	0.0291467	0.0145733	−	0.999894i	$-0.495361\pi$
$367$	6.74399e22	0.0291467	0.0145733	+	0.999894i	$0.495361\pi$
$368$	1.19480e24	0.503199
$369$	0	0
$370$	−1.94160e24	−0.776681
$371$	3.97739e24	1.55077
$372$	0	0
$373$	−1.87501e24	−0.694655	−0.347328	−	0.937744i	$-0.612911\pi$
$373$	−1.87501e24	−0.694655	−0.347328	+	0.937744i	$0.612911\pi$
$374$	1.38970e24	0.501929
$375$	0	0
$376$	−3.22400e23	−0.110691
$377$	9.27934e22	0.0310652
$378$	0	0
$379$	−1.31545e24	−0.418796	−0.209398	−	0.977830i	$-0.567150\pi$
$379$	−1.31545e24	−0.418796	−0.209398	+	0.977830i	$0.567150\pi$
$380$	−5.50387e24	−1.70893
$381$	0	0
$382$	1.93299e24	0.570989
$383$	4.59128e23	0.132296	0.0661478	−	0.997810i	$-0.478929\pi$
$383$	4.59128e23	0.132296	0.0661478	+	0.997810i	$0.478929\pi$
$384$	0	0
$385$	−7.52083e24	−2.06248
$386$	−1.33599e24	−0.357457
$387$	0	0
$388$	−1.21520e24	−0.309561
$389$	−2.20351e24	−0.547766	−0.273883	−	0.961763i	$-0.588308\pi$
$389$	−2.20351e24	−0.547766	−0.273883	+	0.961763i	$0.588308\pi$
$390$	0	0
$391$	3.80150e24	0.900070
$392$	7.81947e23	0.180701
$393$	0	0
$394$	7.74465e23	0.170525
$395$	1.15327e25	2.47890
$396$	0	0
$397$	−8.45884e24	−1.73301	−0.866505	−	0.499168i	$-0.833639\pi$
$397$	−8.45884e24	−1.73301	−0.866505	+	0.499168i	$0.833639\pi$
$398$	2.43828e24	0.487747
$399$	0	0
$400$	7.54940e24	1.43993
$401$	−7.80627e24	−1.45403	−0.727013	−	0.686624i	$-0.759092\pi$
$401$	−7.80627e24	−1.45403	−0.727013	+	0.686624i	$0.759092\pi$
$402$	0	0
$403$	−1.75322e24	−0.311485
$404$	3.91187e24	0.678831
$405$	0	0
$406$	1.54000e23	0.0254990
$407$	−8.32535e24	−1.34665
$408$	0	0
$409$	3.60139e24	0.556030	0.278015	−	0.960577i	$-0.410323\pi$
$409$	3.60139e24	0.556030	0.278015	+	0.960577i	$0.410323\pi$
$410$	5.61819e24	0.847519
$411$	0	0
$412$	2.86748e24	0.413025
$413$	−4.38657e24	−0.617446
$414$	0	0
$415$	9.49086e24	1.27599
$416$	−2.93682e24	−0.385914
$417$	0	0
$418$	4.21188e24	0.528812
$419$	−1.04250e25	−1.27951	−0.639755	−	0.768579i	$-0.720964\pi$
$419$	−1.04250e25	−1.27951	−0.639755	+	0.768579i	$0.720964\pi$
$420$	0	0
$421$	−4.79218e23	−0.0562152	−0.0281076	−	0.999605i	$-0.508948\pi$
$421$	−4.79218e23	−0.0562152	−0.0281076	+	0.999605i	$0.508948\pi$
$422$	−2.01536e24	−0.231145
$423$	0	0
$424$	1.17569e25	1.28919
$425$	2.40200e25	2.57560
$426$	0	0
$427$	2.07824e24	0.213123
$428$	3.32855e24	0.333840
$429$	0	0
$430$	−7.19381e24	−0.690253
$431$	−2.31998e24	−0.217746	−0.108873	−	0.994056i	$-0.534724\pi$
$431$	−2.31998e24	−0.217746	−0.108873	+	0.994056i	$0.534724\pi$
$432$	0	0
$433$	−1.68443e25	−1.51293	−0.756463	−	0.654037i	$-0.773074\pi$
$433$	−1.68443e25	−1.51293	−0.756463	+	0.654037i	$0.773074\pi$
$434$	−2.90963e24	−0.255674
$435$	0	0
$436$	−9.30635e24	−0.782813
$437$	1.15215e25	0.948276
$438$	0	0
$439$	7.88255e24	0.621232	0.310616	−	0.950536i	$-0.399465\pi$
$439$	7.88255e24	0.621232	0.310616	+	0.950536i	$0.399465\pi$
$440$	−2.22310e25	−1.71459
$441$	0	0
$442$	−2.19809e24	−0.162380
$443$	−9.63334e24	−0.696533	−0.348266	−	0.937396i	$-0.613230\pi$
$443$	−9.63334e24	−0.696533	−0.348266	+	0.937396i	$0.613230\pi$
$444$	0	0
$445$	−2.86371e25	−1.98385
$446$	3.83226e24	0.259881
$447$	0	0
$448$	2.69684e24	0.175273
$449$	8.23900e24	0.524245	0.262122	−	0.965035i	$-0.415578\pi$
$449$	8.23900e24	0.524245	0.262122	+	0.965035i	$0.415578\pi$
$450$	0	0
$451$	2.40901e25	1.46947
$452$	1.00906e25	0.602699
$453$	0	0
$454$	−3.93346e24	−0.225291
$455$	1.18956e25	0.667237
$456$	0	0
$457$	−1.43352e25	−0.771256	−0.385628	−	0.922654i	$-0.626015\pi$
$457$	−1.43352e25	−0.771256	−0.385628	+	0.922654i	$0.626015\pi$
$458$	−3.86084e24	−0.203451
$459$	0	0
$460$	−2.79152e25	−1.41137
$461$	−9.92208e24	−0.491410	−0.245705	−	0.969345i	$-0.579019\pi$
$461$	−9.92208e24	−0.491410	−0.245705	+	0.969345i	$0.579019\pi$
$462$	0	0
$463$	−6.84033e24	−0.325131	−0.162565	−	0.986698i	$-0.551977\pi$
$463$	−6.84033e24	−0.325131	−0.162565	+	0.986698i	$0.551977\pi$
$464$	−9.24588e23	−0.0430554
$465$	0	0
$466$	1.63972e25	0.732997
$467$	−2.50815e25	−1.09861	−0.549304	−	0.835623i	$-0.685107\pi$
$467$	−2.50815e25	−1.09861	−0.549304	+	0.835623i	$0.685107\pi$
$468$	0	0
$469$	1.81273e25	0.762415
$470$	2.73049e24	0.112541
$471$	0	0
$472$	−1.29664e25	−0.513298
$473$	−3.08462e25	−1.19680
$474$	0	0
$475$	7.27992e25	2.71355
$476$	2.04401e25	0.746822
$477$	0	0
$478$	−6.42010e24	−0.225412
$479$	3.09246e25	1.06443	0.532215	−	0.846610i	$-0.321360\pi$
$479$	3.09246e25	1.06443	0.532215	+	0.846610i	$0.321360\pi$
$480$	0	0
$481$	1.31681e25	0.435659
$482$	1.72793e25	0.560504
$483$	0	0
$484$	−1.65484e25	−0.516090
$485$	2.24205e25	0.685643
$486$	0	0
$487$	5.27932e25	1.55258	0.776288	−	0.630378i	$-0.217100\pi$
$487$	5.27932e25	1.55258	0.776288	+	0.630378i	$0.217100\pi$
$488$	6.14312e24	0.177174
$489$	0	0
$490$	−6.62252e24	−0.183722
$491$	6.92908e24	0.188539	0.0942695	−	0.995547i	$-0.469948\pi$
$491$	6.92908e24	0.188539	0.0942695	+	0.995547i	$0.469948\pi$
$492$	0	0
$493$	−2.94177e24	−0.0770129
$494$	−6.66190e24	−0.171077
$495$	0	0
$496$	1.74689e25	0.431709
$497$	−5.54384e25	−1.34408
$498$	0	0
$499$	−5.31984e25	−1.24149	−0.620745	−	0.784013i	$-0.713170\pi$
$499$	−5.31984e25	−1.24149	−0.620745	+	0.784013i	$0.713170\pi$
$500$	−1.06733e26	−2.44389
$501$	0	0
$502$	2.14714e25	0.473341
$503$	6.39529e25	1.38345	0.691727	−	0.722159i	$-0.256850\pi$
$503$	6.39529e25	1.38345	0.691727	+	0.722159i	$0.256850\pi$
$504$	0	0
$505$	−7.21742e25	−1.50354
$506$	2.13623e25	0.436736
$507$	0	0
$508$	−5.87246e25	−1.15642
$509$	2.57084e25	0.496885	0.248442	−	0.968647i	$-0.420081\pi$
$509$	2.57084e25	0.496885	0.248442	+	0.968647i	$0.420081\pi$
$510$	0	0
$511$	2.74997e25	0.512070
$512$	5.16409e25	0.943908
$513$	0	0
$514$	−2.41910e25	−0.426092
$515$	−5.29051e25	−0.914805
$516$	0	0
$517$	1.17080e25	0.195130
$518$	2.18538e25	0.357598
$519$	0	0
$520$	3.51627e25	0.554690
$521$	−6.87494e25	−1.06490	−0.532452	−	0.846460i	$-0.678729\pi$
$521$	−6.87494e25	−1.06490	−0.532452	+	0.846460i	$0.678729\pi$
$522$	0	0
$523$	−8.28427e25	−1.23734	−0.618669	−	0.785652i	$-0.712328\pi$
$523$	−8.28427e25	−1.23734	−0.618669	+	0.785652i	$0.712328\pi$
$524$	7.20841e25	1.05729
$525$	0	0
$526$	−2.37991e25	−0.336663
$527$	5.55812e25	0.772195
$528$	0	0
$529$	−1.61793e25	−0.216836
$530$	−9.95723e25	−1.31074
$531$	0	0
$532$	6.19492e25	0.786820
$533$	−3.81032e25	−0.475393
$534$	0	0
$535$	−6.14118e25	−0.739420
$536$	5.35830e25	0.633814
$537$	0	0
$538$	−3.57155e25	−0.407779
$539$	−2.83966e25	−0.318547
$540$	0	0
$541$	7.89322e25	0.854831	0.427416	−	0.904055i	$-0.359424\pi$
$541$	7.89322e25	0.854831	0.427416	+	0.904055i	$0.359424\pi$
$542$	−1.15840e25	−0.123272
$543$	0	0
$544$	9.31041e25	0.956708
$545$	1.71703e26	1.73384
$546$	0	0
$547$	1.30213e26	1.26991	0.634957	−	0.772547i	$-0.281018\pi$
$547$	1.30213e26	1.26991	0.634957	+	0.772547i	$0.281018\pi$
$548$	−1.10935e24	−0.0106330
$549$	0	0
$550$	1.34979e26	1.24975
$551$	−8.91584e24	−0.0811376
$552$	0	0
$553$	−1.29807e26	−1.14133
$554$	−4.37774e25	−0.378362
$555$	0	0
$556$	1.91319e26	1.59789
$557$	−1.10085e25	−0.0903865	−0.0451932	−	0.998978i	$-0.514390\pi$
$557$	−1.10085e25	−0.0903865	−0.0451932	+	0.998978i	$0.514390\pi$
$558$	0	0
$559$	4.87893e25	0.387179
$560$	−1.18528e26	−0.924768
$561$	0	0
$562$	6.64326e23	0.00501056
$563$	−2.56014e26	−1.89860	−0.949301	−	0.314369i	$-0.898207\pi$
$563$	−2.56014e26	−1.89860	−0.949301	+	0.314369i	$0.898207\pi$
$564$	0	0
$565$	−1.86172e26	−1.33491
$566$	−4.73585e25	−0.333919
$567$	0	0
$568$	−1.63872e26	−1.11737
$569$	1.90233e26	1.27561	0.637806	−	0.770197i	$-0.279842\pi$
$569$	1.90233e26	1.27561	0.637806	+	0.770197i	$0.279842\pi$
$570$	0	0
$571$	5.07228e25	0.328973	0.164486	−	0.986379i	$-0.447403\pi$
$571$	5.07228e25	0.328973	0.164486	+	0.986379i	$0.447403\pi$
$572$	6.92108e25	0.441481
$573$	0	0
$574$	−6.32360e25	−0.390213
$575$	3.69233e26	2.24107
$576$	0	0
$577$	1.12271e26	0.659324	0.329662	−	0.944099i	$-0.393065\pi$
$577$	1.12271e26	0.659324	0.329662	+	0.944099i	$0.393065\pi$
$578$	2.31912e24	0.0133970
$579$	0	0
$580$	2.16021e25	0.120762
$581$	−1.06825e26	−0.587490
$582$	0	0
$583$	−4.26954e26	−2.27264
$584$	8.12872e25	0.425696
$585$	0	0
$586$	5.39231e25	0.273368
$587$	−2.34247e26	−1.16845	−0.584227	−	0.811591i	$-0.698602\pi$
$587$	−2.34247e26	−1.16845	−0.584227	+	0.811591i	$0.698602\pi$
$588$	0	0
$589$	1.68454e26	0.813552
$590$	1.09816e26	0.521880
$591$	0	0
$592$	−1.31207e26	−0.603809
$593$	−9.90623e25	−0.448631	−0.224315	−	0.974517i	$-0.572015\pi$
$593$	−9.90623e25	−0.448631	−0.224315	+	0.974517i	$0.572015\pi$
$594$	0	0
$595$	−3.77121e26	−1.65413
$596$	−8.57539e24	−0.0370181
$597$	0	0
$598$	−3.37887e25	−0.141290
$599$	−4.50491e26	−1.85409	−0.927046	−	0.374947i	$-0.877661\pi$
$599$	−4.50491e26	−1.85409	−0.927046	+	0.374947i	$0.877661\pi$
$600$	0	0
$601$	1.99011e26	0.793540	0.396770	−	0.917918i	$-0.370131\pi$
$601$	1.99011e26	0.793540	0.396770	+	0.917918i	$0.370131\pi$
$602$	8.09705e25	0.317805
$603$	0	0
$604$	−9.82412e25	−0.373631
$605$	3.05319e26	1.14308
$606$	0	0
$607$	2.39279e26	0.868184	0.434092	−	0.900868i	$-0.357069\pi$
$607$	2.39279e26	0.868184	0.434092	+	0.900868i	$0.357069\pi$
$608$	2.82177e26	1.00795
$609$	0	0
$610$	−5.20277e25	−0.180136
$611$	−1.85185e25	−0.0631270
$612$	0	0
$613$	8.73354e24	0.0288613	0.0144306	−	0.999896i	$-0.495406\pi$
$613$	8.73354e24	0.0288613	0.0144306	+	0.999896i	$0.495406\pi$
$614$	1.58431e25	0.0515515
$615$	0	0
$616$	2.50224e26	0.789426
$617$	−9.96451e25	−0.309561	−0.154781	−	0.987949i	$-0.549467\pi$
$617$	−9.96451e25	−0.309561	−0.154781	+	0.987949i	$0.549467\pi$
$618$	0	0
$619$	6.60071e25	0.198852	0.0994259	−	0.995045i	$-0.468299\pi$
$619$	6.60071e25	0.198852	0.0994259	+	0.995045i	$0.468299\pi$
$620$	−4.08144e26	−1.21086
$621$	0	0
$622$	−2.01071e26	−0.578551
$623$	3.22327e26	0.913400
$624$	0	0
$625$	1.04795e27	2.88057
$626$	−1.17315e26	−0.317613
$627$	0	0
$628$	−1.79506e26	−0.471480
$629$	−4.17462e26	−1.08003
$630$	0	0
$631$	5.92478e26	1.48728	0.743641	−	0.668579i	$-0.233097\pi$
$631$	5.92478e26	1.48728	0.743641	+	0.668579i	$0.233097\pi$
$632$	−3.83701e26	−0.948814
$633$	0	0
$634$	1.43543e26	0.344455
$635$	1.08347e27	2.56134
$636$	0	0
$637$	4.49147e25	0.103054
$638$	−1.65311e25	−0.0373686
$639$	0	0
$640$	−8.73214e26	−1.91607
$641$	2.24611e26	0.485601	0.242801	−	0.970076i	$-0.421934\pi$
$641$	2.24611e26	0.485601	0.242801	+	0.970076i	$0.421934\pi$
$642$	0	0
$643$	−7.66343e26	−1.60849	−0.804246	−	0.594296i	$-0.797431\pi$
$643$	−7.66343e26	−1.60849	−0.804246	+	0.594296i	$0.797431\pi$
$644$	3.14202e26	0.649821
$645$	0	0
$646$	2.11198e26	0.424112
$647$	3.32213e25	0.0657395	0.0328698	−	0.999460i	$-0.489535\pi$
$647$	3.32213e25	0.0657395	0.0328698	+	0.999460i	$0.489535\pi$
$648$	0	0
$649$	4.70877e26	0.904864
$650$	−2.13496e26	−0.404308
$651$	0	0
$652$	1.20318e26	0.221299
$653$	−8.34424e26	−1.51256	−0.756278	−	0.654250i	$-0.772984\pi$
$653$	−8.34424e26	−1.51256	−0.756278	+	0.654250i	$0.772984\pi$
$654$	0	0
$655$	−1.32995e27	−2.34177
$656$	3.79658e26	0.658880
$657$	0	0
$658$	−3.07333e25	−0.0518159
$659$	−2.07574e26	−0.344955	−0.172477	−	0.985013i	$-0.555177\pi$
$659$	−2.07574e26	−0.344955	−0.172477	+	0.985013i	$0.555177\pi$
$660$	0	0
$661$	7.82027e26	1.26272	0.631361	−	0.775489i	$-0.282496\pi$
$661$	7.82027e26	1.26272	0.631361	+	0.775489i	$0.282496\pi$
$662$	1.40180e26	0.223118
$663$	0	0
$664$	−3.15768e26	−0.488395
$665$	−1.14297e27	−1.74272
$666$	0	0
$667$	−4.52205e25	−0.0670101
$668$	2.75237e26	0.402096
$669$	0	0
$670$	−4.53809e26	−0.644410
$671$	−2.23089e26	−0.312330
$672$	0	0
$673$	−1.32693e27	−1.80594	−0.902970	−	0.429703i	$-0.858618\pi$
$673$	−1.32693e27	−1.80594	−0.902970	+	0.429703i	$0.858618\pi$
$674$	3.00980e26	0.403896
$675$	0	0
$676$	5.40922e26	0.705734
$677$	1.32647e27	1.70650	0.853250	−	0.521502i	$-0.174628\pi$
$677$	1.32647e27	1.70650	0.853250	+	0.521502i	$0.174628\pi$
$678$	0	0
$679$	−2.52356e26	−0.315682
$680$	−1.11474e27	−1.37512
$681$	0	0
$682$	3.12336e26	0.374688
$683$	1.57682e27	1.86546	0.932729	−	0.360577i	$-0.117420\pi$
$683$	1.57682e27	1.86546	0.932729	+	0.360577i	$0.117420\pi$
$684$	0	0
$685$	2.04676e25	0.0235508
$686$	3.71288e26	0.421340
$687$	0	0
$688$	−4.86133e26	−0.536618
$689$	6.75311e26	0.735227
$690$	0	0
$691$	1.02619e27	1.08690	0.543448	−	0.839443i	$-0.317118\pi$
$691$	1.02619e27	1.08690	0.543448	+	0.839443i	$0.317118\pi$
$692$	−1.50073e26	−0.156782
$693$	0	0
$694$	5.11641e26	0.520056
$695$	−3.52985e27	−3.53916
$696$	0	0
$697$	1.20796e27	1.17853
$698$	−5.36058e26	−0.515924
$699$	0	0
$700$	1.98531e27	1.85950
$701$	1.32278e27	1.22226	0.611132	−	0.791529i	$-0.290714\pi$
$701$	1.32278e27	1.22226	0.611132	+	0.791529i	$0.290714\pi$
$702$	0	0
$703$	−1.26523e27	−1.13787
$704$	−2.89493e26	−0.256861
$705$	0	0
$706$	1.32269e26	0.114239
$707$	8.12363e26	0.692254
$708$	0	0
$709$	1.06497e27	0.883479	0.441739	−	0.897143i	$-0.354362\pi$
$709$	1.06497e27	0.883479	0.441739	+	0.897143i	$0.354362\pi$
$710$	1.38788e27	1.13605
$711$	0	0
$712$	9.52777e26	0.759332
$713$	8.54386e26	0.671899
$714$	0	0
$715$	−1.27694e27	−0.977831
$716$	−3.71331e26	−0.280600
$717$	0	0
$718$	−8.69631e26	−0.639960
$719$	−1.47670e27	−1.07243	−0.536214	−	0.844082i	$-0.680146\pi$
$719$	−1.47670e27	−1.07243	−0.536214	+	0.844082i	$0.680146\pi$
$720$	0	0
$721$	5.95478e26	0.421192
$722$	8.26165e25	0.0576718
$723$	0	0
$724$	7.55587e26	0.513770
$725$	−2.85729e26	−0.191753
$726$	0	0
$727$	−7.42596e26	−0.485485	−0.242742	−	0.970091i	$-0.578047\pi$
$727$	−7.42596e26	−0.485485	−0.242742	+	0.970091i	$0.578047\pi$
$728$	−3.95777e26	−0.255389
$729$	0	0
$730$	−6.88443e26	−0.432813
$731$	−1.54674e27	−0.959846
$732$	0	0
$733$	−3.03125e27	−1.83288	−0.916439	−	0.400174i	$-0.868950\pi$
$733$	−3.03125e27	−1.83288	−0.916439	+	0.400174i	$0.868950\pi$
$734$	1.90031e25	0.0113426
$735$	0	0
$736$	1.43118e27	0.832446
$737$	−1.94588e27	−1.11731
$738$	0	0
$739$	−2.19719e27	−1.22955	−0.614775	−	0.788703i	$-0.710753\pi$
$739$	−2.19719e27	−1.22955	−0.614775	+	0.788703i	$0.710753\pi$
$740$	3.06551e27	1.69356
$741$	0	0
$742$	1.12074e27	0.603489
$743$	2.23128e27	1.18621	0.593104	−	0.805126i	$-0.297902\pi$
$743$	2.23128e27	1.18621	0.593104	+	0.805126i	$0.297902\pi$
$744$	0	0
$745$	1.58216e26	0.0819911
$746$	−5.28337e26	−0.270329
$747$	0	0
$748$	−2.19415e27	−1.09446
$749$	6.91226e26	0.340442
$750$	0	0
$751$	−1.66044e26	−0.0797341	−0.0398670	−	0.999205i	$-0.512693\pi$
$751$	−1.66044e26	−0.0797341	−0.0398670	+	0.999205i	$0.512693\pi$
$752$	1.84518e26	0.0874920
$753$	0	0
$754$	2.61472e25	0.0120892
$755$	1.81255e27	0.827552
$756$	0	0
$757$	2.64496e27	1.17763	0.588813	−	0.808269i	$-0.299595\pi$
$757$	2.64496e27	1.17763	0.588813	+	0.808269i	$0.299595\pi$
$758$	−3.70667e26	−0.162977
$759$	0	0
$760$	−3.37853e27	−1.44877
$761$	4.97849e26	0.210835	0.105418	−	0.994428i	$-0.466382\pi$
$761$	4.97849e26	0.210835	0.105418	+	0.994428i	$0.466382\pi$
$762$	0	0
$763$	−1.93261e27	−0.798292
$764$	−3.05192e27	−1.24505
$765$	0	0
$766$	1.29372e26	0.0514835
$767$	−7.44784e26	−0.292735
$768$	0	0
$769$	−1.84040e27	−0.705688	−0.352844	−	0.935682i	$-0.614785\pi$
$769$	−1.84040e27	−0.705688	−0.352844	+	0.935682i	$0.614785\pi$
$770$	−2.11921e27	−0.802624
$771$	0	0
$772$	2.10934e27	0.779439
$773$	−2.77092e27	−1.01139	−0.505695	−	0.862712i	$-0.668764\pi$
$773$	−2.77092e27	−1.01139	−0.505695	+	0.862712i	$0.668764\pi$
$774$	0	0
$775$	5.39849e27	1.92268
$776$	−7.45945e26	−0.262434
$777$	0	0
$778$	−6.20903e26	−0.213166
$779$	3.66106e27	1.24165
$780$	0	0
$781$	5.95105e27	1.96974
$782$	1.07118e27	0.350267
$783$	0	0
$784$	−4.47527e26	−0.142829
$785$	3.31190e27	1.04428
$786$	0	0
$787$	2.52061e27	0.775793	0.387896	−	0.921703i	$-0.373202\pi$
$787$	2.52061e27	0.775793	0.387896	+	0.921703i	$0.373202\pi$
$788$	−1.22277e27	−0.371831
$789$	0	0
$790$	3.24967e27	0.964677
$791$	2.09547e27	0.614617
$792$	0	0
$793$	3.52858e26	0.101043
$794$	−2.38352e27	−0.674410
$795$	0	0
$796$	−3.84970e27	−1.06354
$797$	−4.82823e27	−1.31806	−0.659028	−	0.752119i	$-0.729032\pi$
$797$	−4.82823e27	−1.31806	−0.659028	+	0.752119i	$0.729032\pi$
$798$	0	0
$799$	5.87082e26	0.156496
$800$	9.04302e27	2.38209
$801$	0	0
$802$	−2.19964e27	−0.565842
$803$	−2.95196e27	−0.750435
$804$	0	0
$805$	−5.79705e27	−1.43928
$806$	−4.94019e26	−0.121216
$807$	0	0
$808$	2.40129e27	0.575488
$809$	−3.38607e27	−0.802019	−0.401009	−	0.916074i	$-0.631341\pi$
$809$	−3.38607e27	−0.802019	−0.401009	+	0.916074i	$0.631341\pi$
$810$	0	0
$811$	8.33208e27	1.92777	0.963885	−	0.266318i	$-0.0858072\pi$
$811$	8.33208e27	1.92777	0.963885	+	0.266318i	$0.0858072\pi$
$812$	−2.43144e26	−0.0556008
$813$	0	0
$814$	−2.34591e27	−0.524057
$815$	−2.21987e27	−0.490152
$816$	0	0
$817$	−4.68780e27	−1.01125
$818$	1.01479e27	0.216382
$819$	0	0
$820$	−8.87033e27	−1.84803
$821$	4.64993e27	0.957604	0.478802	−	0.877923i	$-0.341071\pi$
$821$	4.64993e27	0.957604	0.478802	+	0.877923i	$0.341071\pi$
$822$	0	0
$823$	2.70296e27	0.543929	0.271964	−	0.962307i	$-0.412327\pi$
$823$	2.70296e27	0.543929	0.271964	+	0.962307i	$0.412327\pi$
$824$	1.76019e27	0.350148
$825$	0	0
$826$	−1.23604e27	−0.240283
$827$	2.17329e27	0.417653	0.208826	−	0.977953i	$-0.433036\pi$
$827$	2.17329e27	0.417653	0.208826	+	0.977953i	$0.433036\pi$
$828$	0	0
$829$	1.31671e27	0.247300	0.123650	−	0.992326i	$-0.460540\pi$
$829$	1.31671e27	0.247300	0.123650	+	0.992326i	$0.460540\pi$
$830$	2.67432e27	0.496560
$831$	0	0
$832$	4.57890e26	0.0830978
$833$	−1.42390e27	−0.255478
$834$	0	0
$835$	−5.07813e27	−0.890599
$836$	−6.64996e27	−1.15308
$837$	0	0
$838$	−2.93755e27	−0.497929
$839$	3.61025e27	0.605060	0.302530	−	0.953140i	$-0.402169\pi$
$839$	3.61025e27	0.605060	0.302530	+	0.953140i	$0.402169\pi$
$840$	0	0
$841$	−6.06827e27	−0.994266
$842$	−1.35034e26	−0.0218764
$843$	0	0
$844$	3.18197e27	0.504015
$845$	−9.98003e27	−1.56312
$846$	0	0
$847$	−3.43654e27	−0.526295
$848$	−6.72876e27	−1.01900
$849$	0	0
$850$	6.76833e27	1.00231
$851$	−6.41717e27	−0.939750
$852$	0	0
$853$	−7.13542e27	−1.02189	−0.510945	−	0.859614i	$-0.670704\pi$
$853$	−7.13542e27	−1.02189	−0.510945	+	0.859614i	$0.670704\pi$
$854$	5.85603e26	0.0829379
$855$	0	0
$856$	2.04322e27	0.283018
$857$	1.23314e28	1.68926	0.844629	−	0.535352i	$-0.179821\pi$
$857$	1.23314e28	1.68926	0.844629	+	0.535352i	$0.179821\pi$
$858$	0	0
$859$	1.28154e28	1.71710	0.858552	−	0.512726i	$-0.171364\pi$
$859$	1.28154e28	1.71710	0.858552	+	0.512726i	$0.171364\pi$
$860$	1.13580e28	1.50511
$861$	0	0
$862$	−6.53721e26	−0.0847371
$863$	1.02326e28	1.31184	0.655922	−	0.754829i	$-0.272280\pi$
$863$	1.02326e28	1.31184	0.655922	+	0.754829i	$0.272280\pi$
$864$	0	0
$865$	2.76886e27	0.347255
$866$	−4.74636e27	−0.588763
$867$	0	0
$868$	4.59390e27	0.557499
$869$	1.39342e28	1.67261
$870$	0	0
$871$	3.07779e27	0.361465
$872$	−5.71267e27	−0.663640
$873$	0	0
$874$	3.24651e27	0.369027
$875$	−2.21647e28	−2.49222
$876$	0	0
$877$	6.16918e27	0.678783	0.339392	−	0.940645i	$-0.389779\pi$
$877$	6.16918e27	0.678783	0.339392	+	0.940645i	$0.389779\pi$
$878$	2.22113e27	0.241756
$879$	0	0
$880$	1.27234e28	1.35524
$881$	7.92860e26	0.0835460	0.0417730	−	0.999127i	$-0.486699\pi$
$881$	7.92860e26	0.0835460	0.0417730	+	0.999127i	$0.486699\pi$
$882$	0	0
$883$	2.78247e27	0.286949	0.143474	−	0.989654i	$-0.454173\pi$
$883$	2.78247e27	0.286949	0.143474	+	0.989654i	$0.454173\pi$
$884$	3.47047e27	0.354072
$885$	0	0
$886$	−2.71447e27	−0.271059
$887$	−7.02289e27	−0.693812	−0.346906	−	0.937900i	$-0.612768\pi$
$887$	−7.02289e27	−0.693812	−0.346906	+	0.937900i	$0.612768\pi$
$888$	0	0
$889$	−1.21951e28	−1.17929
$890$	−8.06932e27	−0.772026
$891$	0	0
$892$	−6.05060e27	−0.566673
$893$	1.77931e27	0.164878
$894$	0	0
$895$	6.85107e27	0.621498
$896$	9.82854e27	0.882191
$897$	0	0
$898$	2.32157e27	0.204013
$899$	−6.61163e26	−0.0574899
$900$	0	0
$901$	−2.14090e28	−1.82268
$902$	6.78809e27	0.571854
$903$	0	0
$904$	6.19406e27	0.510946
$905$	−1.39406e28	−1.13794
$906$	0	0
$907$	−2.18256e28	−1.74461	−0.872303	−	0.488966i	$-0.837374\pi$
$907$	−2.18256e28	−1.74461	−0.872303	+	0.488966i	$0.837374\pi$
$908$	6.21039e27	0.491251
$909$	0	0
$910$	3.35194e27	0.259659
$911$	1.02445e28	0.785356	0.392678	−	0.919676i	$-0.371549\pi$
$911$	1.02445e28	0.785356	0.392678	+	0.919676i	$0.371549\pi$
$912$	0	0
$913$	1.14672e28	0.860963
$914$	−4.03934e27	−0.300138
$915$	0	0
$916$	6.09572e27	0.443627
$917$	1.49694e28	1.07819
$918$	0	0
$919$	6.02455e27	0.425038	0.212519	−	0.977157i	$-0.431833\pi$
$919$	6.02455e27	0.425038	0.212519	+	0.977157i	$0.431833\pi$
$920$	−1.71357e28	−1.19651
$921$	0	0
$922$	−2.79583e27	−0.191235
$923$	−9.41274e27	−0.637235
$924$	0	0
$925$	−4.05473e28	−2.68915
$926$	−1.92746e27	−0.126526
$927$	0	0
$928$	−1.10751e27	−0.0712268
$929$	−4.16217e27	−0.264954	−0.132477	−	0.991186i	$-0.542293\pi$
$929$	−4.16217e27	−0.264954	−0.132477	+	0.991186i	$0.542293\pi$
$930$	0	0
$931$	−4.31553e27	−0.269161
$932$	−2.58888e28	−1.59831
$933$	0	0
$934$	−7.06742e27	−0.427529
$935$	4.04821e28	2.42411
$936$	0	0
$937$	−2.21976e27	−0.130251	−0.0651253	−	0.997877i	$-0.520745\pi$
$937$	−2.21976e27	−0.130251	−0.0651253	+	0.997877i	$0.520745\pi$
$938$	5.10789e27	0.296698
$939$	0	0
$940$	−4.31107e27	−0.245397
$941$	8.50104e27	0.479039	0.239519	−	0.970892i	$-0.423010\pi$
$941$	8.50104e27	0.479039	0.239519	+	0.970892i	$0.423010\pi$
$942$	0	0
$943$	1.85686e28	1.02546
$944$	7.42098e27	0.405721
$945$	0	0
$946$	−8.69181e27	−0.465741
$947$	−1.99860e27	−0.106023	−0.0530116	−	0.998594i	$-0.516882\pi$
$947$	−1.99860e27	−0.106023	−0.0530116	+	0.998594i	$0.516882\pi$
$948$	0	0
$949$	4.66910e27	0.242775
$950$	2.05133e28	1.05599
$951$	0	0
$952$	1.25471e28	0.633128
$953$	−3.50965e28	−1.75340	−0.876701	−	0.481035i	$-0.840261\pi$
$953$	−3.50965e28	−1.75340	−0.876701	+	0.481035i	$0.840261\pi$
$954$	0	0
$955$	5.63080e28	2.75764
$956$	1.01364e28	0.491514
$957$	0	0
$958$	8.71389e27	0.414228
$959$	−2.30375e26	−0.0108432
$960$	0	0
$961$	−9.17882e27	−0.423560
$962$	3.71050e27	0.169539
$963$	0	0
$964$	−2.72815e28	−1.22218
$965$	−3.89174e28	−1.72637
$966$	0	0
$967$	2.65188e28	1.15346	0.576730	−	0.816935i	$-0.304329\pi$
$967$	2.65188e28	1.15346	0.576730	+	0.816935i	$0.304329\pi$
$968$	−1.01582e28	−0.437522
$969$	0	0
$970$	6.31761e27	0.266822
$971$	−2.71399e28	−1.13508	−0.567539	−	0.823346i	$-0.692104\pi$
$971$	−2.71399e28	−1.13508	−0.567539	+	0.823346i	$0.692104\pi$
$972$	0	0
$973$	3.97305e28	1.62949
$974$	1.48760e28	0.604194
$975$	0	0
$976$	−3.51586e27	−0.140042
$977$	1.15123e28	0.454111	0.227056	−	0.973882i	$-0.427090\pi$
$977$	1.15123e28	0.454111	0.227056	+	0.973882i	$0.427090\pi$
$978$	0	0
$979$	−3.46003e28	−1.33858
$980$	1.04560e28	0.400607
$981$	0	0
$982$	1.95247e27	0.0733710
$983$	2.79303e28	1.03948	0.519741	−	0.854324i	$-0.326028\pi$
$983$	2.79303e28	1.03948	0.519741	+	0.854324i	$0.326028\pi$
$984$	0	0
$985$	2.25602e28	0.823566
$986$	−8.28929e26	−0.0299700
$987$	0	0
$988$	1.05182e28	0.373036
$989$	−2.37762e28	−0.835176
$990$	0	0
$991$	−3.99823e28	−1.37774	−0.688872	−	0.724883i	$-0.741894\pi$
$991$	−3.99823e28	−1.37774	−0.688872	+	0.724883i	$0.741894\pi$
$992$	2.09251e28	0.714179
$993$	0	0
$994$	−1.56214e28	−0.523056
$995$	7.10272e28	2.35562
$996$	0	0
$997$	1.13500e28	0.369310	0.184655	−	0.982803i	$-0.440883\pi$
$997$	1.13500e28	0.369310	0.184655	+	0.982803i	$0.440883\pi$
$998$	−1.49902e28	−0.483132
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.20.a.d.1.3	yes	4
3.2	odd	2	inner	9.20.a.d.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.20.a.d.1.2	✓	4	3.2	odd	2	inner
9.20.a.d.1.3	yes	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	9.a (trivial)

\(f(q)\)	\(=\)	\(q+281.779 q^{2} -444889. q^{4} +8.20822e6 q^{5} -9.23883e7 q^{7} -2.73093e8 q^{8} +O(q^{10})\)	\(q+281.779 q^{2} -444889. q^{4} +8.20822e6 q^{5} -9.23883e7 q^{7} -2.73093e8 q^{8} +2.31290e9 q^{10} +9.91745e9 q^{11} -1.56864e10 q^{13} -2.60331e10 q^{14} +1.56298e11 q^{16} +4.97295e11 q^{17} +1.50719e12 q^{19} -3.65174e12 q^{20} +2.79453e12 q^{22} +7.64436e12 q^{23} +4.83013e13 q^{25} -4.42008e12 q^{26} +4.11025e13 q^{28} -5.91555e12 q^{29} +1.11767e14 q^{31} +1.87221e14 q^{32} +1.40127e14 q^{34} -7.58343e14 q^{35} -8.39465e14 q^{37} +4.24693e14 q^{38} -2.24161e15 q^{40} +2.42907e15 q^{41} -3.11030e15 q^{43} -4.41216e15 q^{44} +2.15402e15 q^{46} +1.18055e15 q^{47} -2.86330e15 q^{49} +1.36103e16 q^{50} +6.97869e15 q^{52} -4.30508e16 q^{53} +8.14046e16 q^{55} +2.52306e16 q^{56} -1.66687e15 q^{58} +4.74797e16 q^{59} -2.24946e16 q^{61} +3.14935e16 q^{62} -2.91903e16 q^{64} -1.28757e17 q^{65} -1.96208e17 q^{67} -2.21241e17 q^{68} -2.13685e17 q^{70} +6.00058e17 q^{71} -2.97653e17 q^{73} -2.36543e17 q^{74} -6.70531e17 q^{76} -9.16257e17 q^{77} +1.40502e18 q^{79} +1.28293e18 q^{80} +6.84459e17 q^{82} +1.15626e18 q^{83} +4.08191e18 q^{85} -8.76416e17 q^{86} -2.70839e18 q^{88} -3.48883e18 q^{89} +1.44924e18 q^{91} -3.40089e18 q^{92} +3.32654e17 q^{94} +1.23713e19 q^{95} +2.73147e18 q^{97} -8.06816e17 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 796528 q^{4} + 166272080 q^{7}+O(q^{10}) \) 4 * q + 796528 * q^4 + 166272080 * q^7	\( 4 q + 796528 q^{4} + 166272080 q^{7} - 3773076480 q^{10} + 53157676520 q^{13} + 300543715456 q^{16} + 3768641418464 q^{19} + 36676792627200 q^{22} + 84248826573580 q^{25} + 378192741767360 q^{28} + 471591909029072 q^{31} - 399874435983360 q^{34} - 22\!\cdots\!60 q^{37}+ \cdots - 41\!\cdots\!20 q^{97}+O(q^{100}) \) 4 * q + 796528 * q^4 + 166272080 * q^7 - 3773076480 * q^10 + 53157676520 * q^13 + 300543715456 * q^16 + 3768641418464 * q^19 + 36676792627200 * q^22 + 84248826573580 * q^25 + 378192741767360 * q^28 + 471591909029072 * q^31 - 399874435983360 * q^34 - 2228516688441160 * q^37 - 7161324419358720 * q^40 - 12262326274792480 * q^43 - 17070303547514880 * q^46 + 33093211591411428 * q^49 + 85229426103167840 * q^52 + 67338095392972800 * q^55 + 187691248170316800 * q^58 + 160046271305607128 * q^61 - 525751810435539968 * q^64 - 486553865858307520 * q^67 - 1901577196642406400 * q^70 - 130672866884308840 * q^73 - 705101514474977152 * q^76 + 3246789793721214416 * q^79 + 1277162815720089600 * q^82 + 10252506464133672960 * q^85 + 4496007782235340800 * q^88 + 17735617946513428000 * q^91 - 34218764107889172480 * q^94 - 417886879084958920 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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