LMFDB - Embedded newform 75.20.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,20,Mod(1,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 20);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 20 $
Character orbit:	$[\chi]$	$=$	75.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:	$171.612522417$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	75.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1104.00 q^{2} -19683.0 q^{3} +694528. q^{4} -2.17300e7 q^{6} +1.95591e8 q^{7} +1.87945e8 q^{8} +3.87420e8 q^{9} +O(q^{10})$

\(q+1104.00 q^{2} -19683.0 q^{3} +694528. q^{4} -2.17300e7 q^{6} +1.95591e8 q^{7} +1.87945e8 q^{8} +3.87420e8 q^{9} -2.74686e9 q^{11} -1.36704e10 q^{12} +4.44004e10 q^{13} +2.15932e11 q^{14} -1.56641e11 q^{16} +7.85983e11 q^{17} +4.27712e11 q^{18} +3.15410e11 q^{19} -3.84981e12 q^{21} -3.03253e12 q^{22} -4.90056e12 q^{23} -3.69932e12 q^{24} +4.90181e13 q^{26} -7.62560e12 q^{27} +1.35843e14 q^{28} +1.21885e13 q^{29} -4.27137e13 q^{31} -2.71469e14 q^{32} +5.40664e13 q^{33} +8.67725e14 q^{34} +2.69074e14 q^{36} +4.23452e14 q^{37} +3.48213e14 q^{38} -8.73934e14 q^{39} -1.11392e15 q^{41} -4.25019e15 q^{42} -1.13610e15 q^{43} -1.90777e15 q^{44} -5.41022e15 q^{46} -1.53137e15 q^{47} +3.08317e15 q^{48} +2.68568e16 q^{49} -1.54705e16 q^{51} +3.08374e16 q^{52} +1.80593e16 q^{53} -8.41866e15 q^{54} +3.67603e16 q^{56} -6.20822e15 q^{57} +1.34561e16 q^{58} +9.27004e16 q^{59} +2.13530e16 q^{61} -4.71559e16 q^{62} +7.57758e16 q^{63} -2.17577e17 q^{64} +5.96893e16 q^{66} -2.68065e17 q^{67} +5.45887e17 q^{68} +9.64577e16 q^{69} -1.13274e17 q^{71} +7.28137e16 q^{72} +5.45956e17 q^{73} +4.67491e17 q^{74} +2.19061e17 q^{76} -5.37260e17 q^{77} -9.64823e17 q^{78} -1.80761e18 q^{79} +1.50095e17 q^{81} -1.22977e18 q^{82} -1.46996e18 q^{83} -2.67380e18 q^{84} -1.25425e18 q^{86} -2.39907e17 q^{87} -5.16258e17 q^{88} +2.97404e18 q^{89} +8.68431e18 q^{91} -3.40358e18 q^{92} +8.40733e17 q^{93} -1.69063e18 q^{94} +5.34333e18 q^{96} +6.92569e18 q^{97} +2.96499e19 q^{98} -1.06419e18 q^{99} +O(q^{100})\)

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 $

$2$	1104.00	1.52470	0.762349	−	0.647166i	$-0.224046\pi$
$2$	1104.00	1.52470	0.762349	+	0.647166i	$0.224046\pi$
$3$	−19683.0	−0.577350
$3$	−19683.0	−0.577350
$4$	694528.	1.32471
$5$	0	0
$5$	0	0
$6$	−2.17300e7	−0.880285
$7$	1.95591e8	1.83196	0.915981	−	0.401221i	$-0.131414\pi$
$7$	1.95591e8	1.83196	0.915981	+	0.401221i	$0.131414\pi$
$8$	1.87945e8	0.495080
$9$	3.87420e8	0.333333
$10$	0	0
$11$	−2.74686e9	−0.351241	−0.175621	−	0.984458i	$-0.556193\pi$
$11$	−2.74686e9	−0.351241	−0.175621	+	0.984458i	$0.556193\pi$
$12$	−1.36704e10	−0.764820
$13$	4.44004e10	1.16125	0.580625	−	0.814171i	$-0.302808\pi$
$13$	4.44004e10	1.16125	0.580625	+	0.814171i	$0.302808\pi$
$14$	2.15932e11	2.79319
$15$	0	0
$16$	−1.56641e11	−0.569858
$17$	7.85983e11	1.60749	0.803745	−	0.594974i	$-0.202838\pi$
$17$	7.85983e11	1.60749	0.803745	+	0.594974i	$0.202838\pi$
$18$	4.27712e11	0.508233
$19$	3.15410e11	0.224242	0.112121	−	0.993695i	$-0.464236\pi$
$19$	3.15410e11	0.224242	0.112121	+	0.993695i	$0.464236\pi$
$20$	0	0
$21$	−3.84981e12	−1.05768
$22$	−3.03253e12	−0.535537
$23$	−4.90056e12	−0.567324	−0.283662	−	0.958924i	$-0.591549\pi$
$23$	−4.90056e12	−0.567324	−0.283662	+	0.958924i	$0.591549\pi$
$24$	−3.69932e12	−0.285835
$25$	0	0
$26$	4.90181e13	1.77056
$27$	−7.62560e12	−0.192450
$28$	1.35843e14	2.42681
$29$	1.21885e13	0.156016	0.0780082	−	0.996953i	$-0.475144\pi$
$29$	1.21885e13	0.156016	0.0780082	+	0.996953i	$0.475144\pi$
$30$	0	0
$31$	−4.27137e13	−0.290155	−0.145078	−	0.989420i	$-0.546343\pi$
$31$	−4.27137e13	−0.290155	−0.145078	+	0.989420i	$0.546343\pi$
$32$	−2.71469e14	−1.36394
$33$	5.40664e13	0.202789
$34$	8.67725e14	2.45094
$35$	0	0
$36$	2.69074e14	0.441569
$37$	4.23452e14	0.535659	0.267829	−	0.963466i	$-0.413694\pi$
$37$	4.23452e14	0.535659	0.267829	+	0.963466i	$0.413694\pi$
$38$	3.48213e14	0.341901
$39$	−8.73934e14	−0.670448
$40$	0	0
$41$	−1.11392e15	−0.531383	−0.265691	−	0.964058i	$-0.585600\pi$
$41$	−1.11392e15	−0.531383	−0.265691	+	0.964058i	$0.585600\pi$
$42$	−4.25019e15	−1.61265
$43$	−1.13610e15	−0.344720	−0.172360	−	0.985034i	$-0.555139\pi$
$43$	−1.13610e15	−0.344720	−0.172360	+	0.985034i	$0.555139\pi$
$44$	−1.90777e15	−0.465292
$45$	0	0
$46$	−5.41022e15	−0.864999
$47$	−1.53137e15	−0.199596	−0.0997978	−	0.995008i	$-0.531820\pi$
$47$	−1.53137e15	−0.199596	−0.0997978	+	0.995008i	$0.531820\pi$
$48$	3.08317e15	0.329008
$49$	2.68568e16	2.35609
$50$	0	0
$51$	−1.54705e16	−0.928084
$52$	3.08374e16	1.53832
$53$	1.80593e16	0.751762	0.375881	−	0.926668i	$-0.377340\pi$
$53$	1.80593e16	0.751762	0.375881	+	0.926668i	$0.377340\pi$
$54$	−8.41866e15	−0.293428
$55$	0	0
$56$	3.67603e16	0.906969
$57$	−6.20822e15	−0.129466
$58$	1.34561e16	0.237878
$59$	9.27004e16	1.39312	0.696559	−	0.717500i	$-0.254714\pi$
$59$	9.27004e16	1.39312	0.696559	+	0.717500i	$0.254714\pi$
$60$	0	0
$61$	2.13530e16	0.233789	0.116895	−	0.993144i	$-0.462706\pi$
$61$	2.13530e16	0.233789	0.116895	+	0.993144i	$0.462706\pi$
$62$	−4.71559e16	−0.442400
$63$	7.57758e16	0.610654
$64$	−2.17577e17	−1.50974
$65$	0	0
$66$	5.96893e16	0.309193
$67$	−2.68065e17	−1.20373	−0.601866	−	0.798597i	$-0.705576\pi$
$67$	−2.68065e17	−1.20373	−0.601866	+	0.798597i	$0.705576\pi$
$68$	5.45887e17	2.12945
$69$	9.64577e16	0.327545
$70$	0	0
$71$	−1.13274e17	−0.293207	−0.146603	−	0.989195i	$-0.546834\pi$
$71$	−1.13274e17	−0.293207	−0.146603	+	0.989195i	$0.546834\pi$
$72$	7.28137e16	0.165027
$73$	5.45956e17	1.08540	0.542701	−	0.839926i	$-0.317402\pi$
$73$	5.45956e17	1.08540	0.542701	+	0.839926i	$0.317402\pi$
$74$	4.67491e17	0.816718
$75$	0	0
$76$	2.19061e17	0.297055
$77$	−5.37260e17	−0.643461
$78$	−9.64823e17	−1.02223
$79$	−1.80761e18	−1.69687	−0.848433	−	0.529303i	$-0.822453\pi$
$79$	−1.80761e18	−1.69687	−0.848433	+	0.529303i	$0.822453\pi$
$80$	0	0
$81$	1.50095e17	0.111111
$82$	−1.22977e18	−0.810199
$83$	−1.46996e18	−0.863104	−0.431552	−	0.902088i	$-0.642034\pi$
$83$	−1.46996e18	−0.863104	−0.431552	+	0.902088i	$0.642034\pi$
$84$	−2.67380e18	−1.40112
$85$	0	0
$86$	−1.25425e18	−0.525594
$87$	−2.39907e17	−0.0900761
$88$	−5.16258e17	−0.173893
$89$	2.97404e18	0.899791	0.449896	−	0.893081i	$-0.351461\pi$
$89$	2.97404e18	0.899791	0.449896	+	0.893081i	$0.351461\pi$
$90$	0	0
$91$	8.68431e18	2.12737
$92$	−3.40358e18	−0.751538
$93$	8.40733e17	0.167521
$94$	−1.69063e18	−0.304323
$95$	0	0
$96$	5.34333e18	0.787473
$97$	6.92569e18	0.924978	0.462489	−	0.886625i	$-0.346957\pi$
$97$	6.92569e18	0.924978	0.462489	+	0.886625i	$0.346957\pi$
$98$	2.96499e19	3.59232
$99$	−1.06419e18	−0.117080
$100$	0	0
$101$	1.78746e19	1.62623	0.813117	−	0.582100i	$-0.197769\pi$
$101$	1.78746e19	1.62623	0.813117	+	0.582100i	$0.197769\pi$
$102$	−1.70794e19	−1.41505
$103$	7.05681e18	0.532911	0.266456	−	0.963847i	$-0.414147\pi$
$103$	7.05681e18	0.532911	0.266456	+	0.963847i	$0.414147\pi$
$104$	8.34484e18	0.574912
$105$	0	0
$106$	1.99375e19	1.14621
$107$	1.77197e19	0.931772	0.465886	−	0.884845i	$-0.345736\pi$
$107$	1.77197e19	0.931772	0.465886	+	0.884845i	$0.345736\pi$
$108$	−5.29619e18	−0.254940
$109$	8.11766e18	0.357997	0.178998	−	0.983849i	$-0.442714\pi$
$109$	8.11766e18	0.357997	0.178998	+	0.983849i	$0.442714\pi$
$110$	0	0
$111$	−8.33481e18	−0.309263
$112$	−3.06376e19	−1.04396
$113$	4.38285e19	1.37250	0.686249	−	0.727367i	$-0.259256\pi$
$113$	4.38285e19	1.37250	0.686249	+	0.727367i	$0.259256\pi$
$114$	−6.85388e18	−0.197397
$115$	0	0
$116$	8.46527e18	0.206676
$117$	1.72016e19	0.387083
$118$	1.02341e20	2.12408
$119$	1.53731e20	2.94486
$120$	0	0
$121$	−5.36139e19	−0.876629
$122$	2.35737e19	0.356458
$123$	2.19253e19	0.306794
$124$	−2.96658e19	−0.384371
$125$	0	0
$126$	8.36565e19	0.931064
$127$	−2.94833e19	−0.304398	−0.152199	−	0.988350i	$-0.548635\pi$
$127$	−2.94833e19	−0.304398	−0.152199	+	0.988350i	$0.548635\pi$
$128$	−9.78769e19	−0.937962
$129$	2.23619e19	0.199024
$130$	0	0
$131$	1.29096e20	0.992738	0.496369	−	0.868112i	$-0.334666\pi$
$131$	1.29096e20	0.992738	0.496369	+	0.868112i	$0.334666\pi$
$132$	3.75506e19	0.268636
$133$	6.16913e19	0.410803
$134$	−2.95944e20	−1.83533
$135$	0	0
$136$	1.47721e20	0.795837
$137$	1.15732e20	0.581576	0.290788	−	0.956788i	$-0.406083\pi$
$137$	1.15732e20	0.581576	0.290788	+	0.956788i	$0.406083\pi$
$138$	1.06489e20	0.499407
$139$	−5.77640e19	−0.252939	−0.126470	−	0.991970i	$-0.540365\pi$
$139$	−5.77640e19	−0.252939	−0.126470	+	0.991970i	$0.540365\pi$
$140$	0	0
$141$	3.01420e19	0.115237
$142$	−1.25054e20	−0.447052
$143$	−1.21962e20	−0.407879
$144$	−6.06861e19	−0.189953
$145$	0	0
$146$	6.02736e20	1.65491
$147$	−5.28622e20	−1.36029
$148$	2.94100e20	0.709591
$149$	9.58388e19	0.216906	0.108453	−	0.994102i	$-0.465410\pi$
$149$	9.58388e19	0.216906	0.108453	+	0.994102i	$0.465410\pi$
$150$	0	0
$151$	1.32704e20	0.264608	0.132304	−	0.991209i	$-0.457763\pi$
$151$	1.32704e20	0.264608	0.132304	+	0.991209i	$0.457763\pi$
$152$	5.92798e19	0.111018
$153$	3.04506e20	0.535830
$154$	−5.93135e20	−0.981084
$155$	0	0
$156$	−6.06972e20	−0.888147
$157$	9.90999e20	1.36467	0.682333	−	0.731041i	$-0.260965\pi$
$157$	9.90999e20	1.36467	0.682333	+	0.731041i	$0.260965\pi$
$158$	−1.99560e21	−2.58721
$159$	−3.55462e20	−0.434030
$160$	0	0
$161$	−9.58503e20	−1.03932
$162$	1.65704e20	0.169411
$163$	−1.74144e21	−1.67929	−0.839647	−	0.543133i	$-0.817238\pi$
$163$	−1.74144e21	−1.67929	−0.839647	+	0.543133i	$0.817238\pi$
$164$	−7.73649e20	−0.703927
$165$	0	0
$166$	−1.62283e21	−1.31597
$167$	2.34167e21	1.79357	0.896785	−	0.442466i	$-0.145896\pi$
$167$	2.34167e21	1.79357	0.896785	+	0.442466i	$0.145896\pi$
$168$	−7.23552e20	−0.523639
$169$	5.09479e20	0.348500
$170$	0	0
$171$	1.22196e20	0.0747473
$172$	−7.89053e20	−0.456653
$173$	5.36374e20	0.293785	0.146892	−	0.989152i	$-0.453073\pi$
$173$	5.36374e20	0.293785	0.146892	+	0.989152i	$0.453073\pi$
$174$	−2.64857e20	−0.137339
$175$	0	0
$176$	4.30272e20	0.200158
$177$	−1.82462e21	−0.804317
$178$	3.28334e21	1.37191
$179$	3.33179e21	1.32000	0.660000	−	0.751266i	$-0.270556\pi$
$179$	3.33179e21	1.32000	0.660000	+	0.751266i	$0.270556\pi$
$180$	0	0
$181$	−3.97005e20	−0.141530	−0.0707652	−	0.997493i	$-0.522544\pi$
$181$	−3.97005e20	−0.141530	−0.0707652	+	0.997493i	$0.522544\pi$
$182$	9.58748e21	3.24359
$183$	−4.20290e20	−0.134978
$184$	−9.21036e20	−0.280871
$185$	0	0
$186$	9.28169e20	0.255420
$187$	−2.15898e21	−0.564617
$188$	−1.06358e21	−0.264406
$189$	−1.49150e21	−0.352561
$190$	0	0
$191$	1.18297e20	0.0253022	0.0126511	−	0.999920i	$-0.495973\pi$
$191$	1.18297e20	0.0253022	0.0126511	+	0.999920i	$0.495973\pi$
$192$	4.28257e21	0.871651
$193$	−3.76434e21	−0.729281	−0.364640	−	0.931148i	$-0.618808\pi$
$193$	−3.76434e21	−0.729281	−0.364640	+	0.931148i	$0.618808\pi$
$194$	7.64596e21	1.41031
$195$	0	0
$196$	1.86528e22	3.12112
$197$	−4.37987e21	−0.698283	−0.349142	−	0.937070i	$-0.613527\pi$
$197$	−4.37987e21	−0.698283	−0.349142	+	0.937070i	$0.613527\pi$
$198$	−1.17486e21	−0.178512
$199$	−1.32104e22	−1.91343	−0.956715	−	0.291025i	$-0.906004\pi$
$199$	−1.32104e22	−1.91343	−0.956715	+	0.291025i	$0.906004\pi$
$200$	0	0
$201$	5.27632e21	0.694975
$202$	1.97335e22	2.47952
$203$	2.38396e21	0.285816
$204$	−1.07447e22	−1.22944
$205$	0	0
$206$	7.79072e21	0.812530
$207$	−1.89858e21	−0.189108
$208$	−6.95495e21	−0.661748
$209$	−8.66388e20	−0.0787631
$210$	0	0
$211$	−1.55480e22	−1.29120	−0.645598	−	0.763678i	$-0.723392\pi$
$211$	−1.55480e22	−1.29120	−0.645598	+	0.763678i	$0.723392\pi$
$212$	1.25427e22	0.995865
$213$	2.22956e21	0.169283
$214$	1.95625e22	1.42067
$215$	0	0
$216$	−1.43319e21	−0.0952783
$217$	−8.35439e21	−0.531554
$218$	8.96190e21	0.545837
$219$	−1.07461e22	−0.626657
$220$	0	0
$221$	3.48980e22	1.86670
$222$	−9.20163e21	−0.471533
$223$	1.04802e21	0.0514602	0.0257301	−	0.999669i	$-0.491809\pi$
$223$	1.04802e21	0.0514602	0.0257301	+	0.999669i	$0.491809\pi$
$224$	−5.30969e22	−2.49869
$225$	0	0
$226$	4.83867e22	2.09265
$227$	4.90340e21	0.203354	0.101677	−	0.994817i	$-0.467579\pi$
$227$	4.90340e21	0.203354	0.101677	+	0.994817i	$0.467579\pi$
$228$	−4.31179e21	−0.171505
$229$	1.69013e22	0.644885	0.322443	−	0.946589i	$-0.395496\pi$
$229$	1.69013e22	0.644885	0.322443	+	0.946589i	$0.395496\pi$
$230$	0	0
$231$	1.05749e22	0.371502
$232$	2.29077e21	0.0772406
$233$	2.94863e22	0.954418	0.477209	−	0.878790i	$-0.341648\pi$
$233$	2.94863e22	0.954418	0.477209	+	0.878790i	$0.341648\pi$
$234$	1.89906e22	0.590185
$235$	0	0
$236$	6.43831e22	1.84547
$237$	3.55792e22	0.979686
$238$	1.69719e23	4.49003
$239$	−3.48672e22	−0.886415	−0.443208	−	0.896419i	$-0.646160\pi$
$239$	−3.48672e22	−0.886415	−0.443208	+	0.896419i	$0.646160\pi$
$240$	0	0
$241$	3.10412e22	0.729083	0.364542	−	0.931187i	$-0.381226\pi$
$241$	3.10412e22	0.729083	0.364542	+	0.931187i	$0.381226\pi$
$242$	−5.91897e22	−1.33660
$243$	−2.95431e21	−0.0641500
$244$	1.48302e22	0.309702
$245$	0	0
$246$	2.42055e22	0.467769
$247$	1.40044e22	0.260401
$248$	−8.02782e21	−0.143650
$249$	2.89332e22	0.498313
$250$	0	0
$251$	2.43352e21	0.0388449	0.0194224	−	0.999811i	$-0.493817\pi$
$251$	2.43352e21	0.0388449	0.0194224	+	0.999811i	$0.493817\pi$
$252$	5.26284e22	0.808938
$253$	1.34611e22	0.199268
$254$	−3.25496e22	−0.464115
$255$	0	0
$256$	6.01696e21	0.0796338
$257$	7.09300e22	0.904618	0.452309	−	0.891861i	$-0.350600\pi$
$257$	7.09300e22	0.904618	0.452309	+	0.891861i	$0.350600\pi$
$258$	2.46875e22	0.303452
$259$	8.28233e22	0.981307
$260$	0	0
$261$	4.72208e21	0.0520054
$262$	1.42522e23	1.51363
$263$	−8.10179e22	−0.829854	−0.414927	−	0.909855i	$-0.636193\pi$
$263$	−8.10179e22	−0.829854	−0.414927	+	0.909855i	$0.636193\pi$
$264$	1.01615e22	0.100397
$265$	0	0
$266$	6.81072e22	0.626351
$267$	−5.85380e22	−0.519495
$268$	−1.86179e23	−1.59459
$269$	−2.10267e23	−1.73830	−0.869150	−	0.494549i	$-0.835333\pi$
$269$	−2.10267e23	−1.73830	−0.869150	+	0.494549i	$0.835333\pi$
$270$	0	0
$271$	−9.67500e22	−0.745491	−0.372746	−	0.927934i	$-0.621584\pi$
$271$	−9.67500e22	−0.745491	−0.372746	+	0.927934i	$0.621584\pi$
$272$	−1.23117e23	−0.916041
$273$	−1.70933e23	−1.22823
$274$	1.27768e23	0.886728
$275$	0	0
$276$	6.69926e22	0.433901
$277$	1.50223e23	0.940112	0.470056	−	0.882637i	$-0.344234\pi$
$277$	1.50223e23	0.940112	0.470056	+	0.882637i	$0.344234\pi$
$278$	−6.37714e22	−0.385656
$279$	−1.65481e22	−0.0967185
$280$	0	0
$281$	−2.78843e23	−1.52282	−0.761412	−	0.648268i	$-0.775494\pi$
$281$	−2.78843e23	−1.52282	−0.761412	+	0.648268i	$0.775494\pi$
$282$	3.32768e22	0.175701
$283$	9.74687e22	0.497615	0.248808	−	0.968553i	$-0.419961\pi$
$283$	9.74687e22	0.497615	0.248808	+	0.968553i	$0.419961\pi$
$284$	−7.86716e22	−0.388413
$285$	0	0
$286$	−1.34646e23	−0.621892
$287$	−2.17872e23	−0.973474
$288$	−1.05173e23	−0.454648
$289$	3.78696e23	1.58402
$290$	0	0
$291$	−1.36318e23	−0.534036
$292$	3.79182e23	1.43784
$293$	−2.78548e23	−1.02249	−0.511243	−	0.859436i	$-0.670815\pi$
$293$	−2.78548e23	−1.02249	−0.511243	+	0.859436i	$0.670815\pi$
$294$	−5.83599e23	−2.07403
$295$	0	0
$296$	7.95857e22	0.265194
$297$	2.09464e22	0.0675964
$298$	1.05806e23	0.330716
$299$	−2.17587e23	−0.658805
$300$	0	0
$301$	−2.22211e23	−0.631514
$302$	1.46505e23	0.403447
$303$	−3.51826e23	−0.938907
$304$	−4.94064e22	−0.127786
$305$	0	0
$306$	3.36174e23	0.816979
$307$	−4.84526e23	−1.14157	−0.570785	−	0.821099i	$-0.693361\pi$
$307$	−4.84526e23	−1.14157	−0.570785	+	0.821099i	$0.693361\pi$
$308$	−3.73142e23	−0.852397
$309$	−1.38899e23	−0.307677
$310$	0	0
$311$	5.63561e23	1.17413	0.587067	−	0.809539i	$-0.300283\pi$
$311$	5.63561e23	1.17413	0.587067	+	0.809539i	$0.300283\pi$
$312$	−1.64251e23	−0.331926
$313$	5.79355e23	1.13573	0.567863	−	0.823123i	$-0.307770\pi$
$313$	5.79355e23	1.13573	0.567863	+	0.823123i	$0.307770\pi$
$314$	1.09406e24	2.08071
$315$	0	0
$316$	−1.25544e24	−2.24785
$317$	1.07037e24	1.85982	0.929909	−	0.367791i	$-0.119886\pi$
$317$	1.07037e24	1.85982	0.929909	+	0.367791i	$0.119886\pi$
$318$	−3.92430e23	−0.661766
$319$	−3.34801e22	−0.0547994
$320$	0	0
$321$	−3.48776e23	−0.537959
$322$	−1.05819e24	−1.58465
$323$	2.47907e23	0.360467
$324$	1.04245e23	0.147190
$325$	0	0
$326$	−1.92255e24	−2.56042
$327$	−1.59780e23	−0.206690
$328$	−2.09356e23	−0.263077
$329$	−2.99522e23	−0.365652
$330$	0	0
$331$	−3.07182e23	−0.354022	−0.177011	−	0.984209i	$-0.556643\pi$
$331$	−3.07182e23	−0.354022	−0.177011	+	0.984209i	$0.556643\pi$
$332$	−1.02093e24	−1.14336
$333$	1.64054e23	0.178553
$334$	2.58520e24	2.73465
$335$	0	0
$336$	6.03040e23	0.602730
$337$	−1.59649e24	−1.55125	−0.775625	−	0.631194i	$-0.782565\pi$
$337$	−1.59649e24	−1.55125	−0.775625	+	0.631194i	$0.782565\pi$
$338$	5.62465e23	0.531358
$339$	−8.62677e23	−0.792412
$340$	0	0
$341$	1.17328e23	0.101915
$342$	1.34905e23	0.113967
$343$	3.02342e24	2.48430
$344$	−2.13524e23	−0.170664
$345$	0	0
$346$	5.92157e23	0.447934
$347$	−1.89344e24	−1.39354	−0.696772	−	0.717292i	$-0.745381\pi$
$347$	−1.89344e24	−1.39354	−0.696772	+	0.717292i	$0.745381\pi$
$348$	−1.66622e23	−0.119324
$349$	3.19249e23	0.222478	0.111239	−	0.993794i	$-0.464518\pi$
$349$	3.19249e23	0.222478	0.111239	+	0.993794i	$0.464518\pi$
$350$	0	0
$351$	−3.38580e23	−0.223483
$352$	7.45688e23	0.479073
$353$	−2.10677e24	−1.31752	−0.658761	−	0.752352i	$-0.728919\pi$
$353$	−2.10677e24	−1.31752	−0.658761	+	0.752352i	$0.728919\pi$
$354$	−2.01438e24	−1.22634
$355$	0	0
$356$	2.06555e24	1.19196
$357$	−3.02588e24	−1.70022
$358$	3.67830e24	2.01260
$359$	−1.45227e24	−0.773836	−0.386918	−	0.922114i	$-0.626460\pi$
$359$	−1.45227e24	−0.773836	−0.386918	+	0.922114i	$0.626460\pi$
$360$	0	0
$361$	−1.87894e24	−0.949716
$362$	−4.38294e23	−0.215791
$363$	1.05528e24	0.506122
$364$	6.03150e24	2.81814
$365$	0	0
$366$	−4.64001e23	−0.205801
$367$	3.52842e23	0.152494	0.0762470	−	0.997089i	$-0.475706\pi$
$367$	3.52842e23	0.152494	0.0762470	+	0.997089i	$0.475706\pi$
$368$	7.67631e23	0.323294
$369$	−4.31556e23	−0.177128
$370$	0	0
$371$	3.53223e24	1.37720
$372$	5.83913e23	0.221917
$373$	−3.12604e24	−1.15814	−0.579069	−	0.815278i	$-0.696584\pi$
$373$	−3.12604e24	−1.15814	−0.579069	+	0.815278i	$0.696584\pi$
$374$	−2.38352e24	−0.860871
$375$	0	0
$376$	−2.87814e23	−0.0988159
$377$	5.41176e23	0.181174
$378$	−1.64661e24	−0.537550
$379$	5.18388e23	0.165037	0.0825187	−	0.996590i	$-0.473704\pi$
$379$	5.18388e23	0.165037	0.0825187	+	0.996590i	$0.473704\pi$
$380$	0	0
$381$	5.80321e23	0.175744
$382$	1.30600e23	0.0385782
$383$	−2.12429e23	−0.0612106	−0.0306053	−	0.999532i	$-0.509743\pi$
$383$	−2.12429e23	−0.0612106	−0.0306053	+	0.999532i	$0.509743\pi$
$384$	1.92651e24	0.541533
$385$	0	0
$386$	−4.15583e24	−1.11193
$387$	−4.40149e23	−0.114907
$388$	4.81008e24	1.22532
$389$	−4.43856e24	−1.10337	−0.551685	−	0.834053i	$-0.686015\pi$
$389$	−4.43856e24	−1.10337	−0.551685	+	0.834053i	$0.686015\pi$
$390$	0	0
$391$	−3.85175e24	−0.911968
$392$	5.04760e24	1.16645
$393$	−2.54099e24	−0.573158
$394$	−4.83537e24	−1.06467
$395$	0	0
$396$	−7.39109e23	−0.155097
$397$	−4.22183e24	−0.864951	−0.432475	−	0.901646i	$-0.642360\pi$
$397$	−4.22183e24	−0.864951	−0.432475	+	0.901646i	$0.642360\pi$
$398$	−1.45843e25	−2.91741
$399$	−1.21427e24	−0.237177
$400$	0	0
$401$	3.29680e24	0.614075	0.307037	−	0.951697i	$-0.400662\pi$
$401$	3.29680e24	0.614075	0.307037	+	0.951697i	$0.400662\pi$
$402$	5.82506e24	1.05963
$403$	−1.89651e24	−0.336943
$404$	1.24144e25	2.15428
$405$	0	0
$406$	2.63189e24	0.435783
$407$	−1.16316e24	−0.188146
$408$	−2.90760e24	−0.459477
$409$	3.40224e24	0.525284	0.262642	−	0.964893i	$-0.415406\pi$
$409$	3.40224e24	0.525284	0.262642	+	0.964893i	$0.415406\pi$
$410$	0	0
$411$	−2.27794e24	−0.335773
$412$	4.90116e24	0.705952
$413$	1.81313e25	2.55214
$414$	−2.09603e24	−0.288333
$415$	0	0
$416$	−1.20534e25	−1.58388
$417$	1.13697e24	0.146035
$418$	−9.56492e23	−0.120090
$419$	2.53386e24	0.310992	0.155496	−	0.987837i	$-0.450302\pi$
$419$	2.53386e24	0.310992	0.155496	+	0.987837i	$0.450302\pi$
$420$	0	0
$421$	−1.07229e25	−1.25786	−0.628930	−	0.777462i	$-0.716507\pi$
$421$	−1.07229e25	−1.25786	−0.628930	+	0.777462i	$0.716507\pi$
$422$	−1.71650e25	−1.96868
$423$	−5.93285e23	−0.0665319
$424$	3.39416e24	0.372183
$425$	0	0
$426$	2.46144e24	0.258106
$427$	4.17644e24	0.428293
$428$	1.23068e25	1.23433
$429$	2.40057e24	0.235489
$430$	0	0
$431$	1.57678e25	1.47992	0.739960	−	0.672651i	$-0.234845\pi$
$431$	1.57678e25	1.47992	0.739960	+	0.672651i	$0.234845\pi$
$432$	1.19448e24	0.109669
$433$	−7.64728e24	−0.686866	−0.343433	−	0.939177i	$-0.611590\pi$
$433$	−7.64728e24	−0.686866	−0.343433	+	0.939177i	$0.611590\pi$
$434$	−9.22325e24	−0.810460
$435$	0	0
$436$	5.63794e24	0.474241
$437$	−1.54569e24	−0.127218
$438$	−1.18636e25	−0.955464
$439$	−6.56625e24	−0.517493	−0.258747	−	0.965945i	$-0.583309\pi$
$439$	−6.56625e24	−0.517493	−0.258747	+	0.965945i	$0.583309\pi$
$440$	0	0
$441$	1.04049e25	0.785362
$442$	3.85274e25	2.84615
$443$	−7.45718e24	−0.539187	−0.269594	−	0.962974i	$-0.586889\pi$
$443$	−7.45718e24	−0.539187	−0.269594	+	0.962974i	$0.586889\pi$
$444$	−5.78876e24	−0.409683
$445$	0	0
$446$	1.15701e24	0.0784614
$447$	−1.88639e24	−0.125231
$448$	−4.25560e25	−2.76579
$449$	−2.64059e25	−1.68020	−0.840101	−	0.542430i	$-0.817504\pi$
$449$	−2.64059e25	−1.68020	−0.840101	+	0.542430i	$0.817504\pi$
$450$	0	0
$451$	3.05978e24	0.186644
$452$	3.04401e25	1.81816
$453$	−2.61201e24	−0.152771
$454$	5.41336e24	0.310053
$455$	0	0
$456$	−1.16680e24	−0.0640962
$457$	−1.31845e25	−0.709347	−0.354674	−	0.934990i	$-0.615408\pi$
$457$	−1.31845e25	−0.709347	−0.354674	+	0.934990i	$0.615408\pi$
$458$	1.86590e25	0.983256
$459$	−5.99359e24	−0.309361
$460$	0	0
$461$	−3.04421e25	−1.50770	−0.753851	−	0.657046i	$-0.771806\pi$
$461$	−3.04421e25	−1.50770	−0.753851	+	0.657046i	$0.771806\pi$
$462$	1.16747e25	0.566429
$463$	3.64943e25	1.73463	0.867313	−	0.497764i	$-0.165845\pi$
$463$	3.64943e25	1.73463	0.867313	+	0.497764i	$0.165845\pi$
$464$	−1.90923e24	−0.0889072
$465$	0	0
$466$	3.25529e25	1.45520
$467$	−3.33264e25	−1.45975	−0.729875	−	0.683581i	$-0.760422\pi$
$467$	−3.33264e25	−1.45975	−0.729875	+	0.683581i	$0.760422\pi$
$468$	1.19470e25	0.512772
$469$	−5.24310e25	−2.20519
$470$	0	0
$471$	−1.95058e25	−0.787890
$472$	1.74226e25	0.689705
$473$	3.12071e24	0.121080
$474$	3.92794e25	1.49373
$475$	0	0
$476$	1.06770e26	3.90108
$477$	6.99655e24	0.250587
$478$	−3.84934e25	−1.35152
$479$	8.10234e24	0.278884	0.139442	−	0.990230i	$-0.455469\pi$
$479$	8.10234e24	0.278884	0.139442	+	0.990230i	$0.455469\pi$
$480$	0	0
$481$	1.88015e25	0.622033
$482$	3.42695e25	1.11163
$483$	1.88662e25	0.600050
$484$	−3.72363e25	−1.16128
$485$	0	0
$486$	−3.26156e24	−0.0978095
$487$	−4.90469e25	−1.44240	−0.721202	−	0.692725i	$-0.756410\pi$
$487$	−4.90469e25	−1.44240	−0.721202	+	0.692725i	$0.756410\pi$
$488$	4.01318e24	0.115745
$489$	3.42768e25	0.969541
$490$	0	0
$491$	4.15501e25	1.13057	0.565285	−	0.824895i	$-0.308766\pi$
$491$	4.15501e25	1.13057	0.565285	+	0.824895i	$0.308766\pi$
$492$	1.52277e25	0.406412
$493$	9.57996e24	0.250795
$494$	1.54608e25	0.397033
$495$	0	0
$496$	6.69073e24	0.165347
$497$	−2.21552e25	−0.537144
$498$	3.19423e25	0.759778
$499$	−2.09237e25	−0.488297	−0.244148	−	0.969738i	$-0.578508\pi$
$499$	−2.09237e25	−0.488297	−0.244148	+	0.969738i	$0.578508\pi$
$500$	0	0
$501$	−4.60910e25	−1.03552
$502$	2.68660e24	0.0592267
$503$	8.92504e25	1.93070	0.965349	−	0.260963i	$-0.0840401\pi$
$503$	8.92504e25	1.93070	0.965349	+	0.260963i	$0.0840401\pi$
$504$	1.42417e25	0.302323
$505$	0	0
$506$	1.48611e25	0.303823
$507$	−1.00281e25	−0.201207
$508$	−2.04770e25	−0.403238
$509$	7.25766e25	1.40274	0.701371	−	0.712797i	$-0.252572\pi$
$509$	7.25766e25	1.40274	0.701371	+	0.712797i	$0.252572\pi$
$510$	0	0
$511$	1.06784e26	1.98842
$512$	5.79584e25	1.05938
$513$	−2.40519e24	−0.0431554
$514$	7.83067e25	1.37927
$515$	0	0
$516$	1.55309e25	0.263649
$517$	4.20646e24	0.0701063
$518$	9.14369e25	1.49620
$519$	−1.05574e25	−0.169617
$520$	0	0
$521$	−4.13657e25	−0.640741	−0.320370	−	0.947292i	$-0.603807\pi$
$521$	−4.13657e25	−0.640741	−0.320370	+	0.947292i	$0.603807\pi$
$522$	5.21318e24	0.0792926
$523$	1.17484e26	1.75475	0.877373	−	0.479810i	$-0.159294\pi$
$523$	1.17484e26	1.75475	0.877373	+	0.479810i	$0.159294\pi$
$524$	8.96607e25	1.31509
$525$	0	0
$526$	−8.94438e25	−1.26528
$527$	−3.35722e25	−0.466422
$528$	−8.46904e24	−0.115561
$529$	−5.06000e25	−0.678143
$530$	0	0
$531$	3.59140e25	0.464372
$532$	4.28463e25	0.544193
$533$	−4.94586e25	−0.617068
$534$	−6.46260e25	−0.792073
$535$	0	0
$536$	−5.03815e25	−0.595944
$537$	−6.55796e25	−0.762102
$538$	−2.32135e26	−2.65038
$539$	−7.37718e25	−0.827555
$540$	0	0
$541$	−4.86212e25	−0.526565	−0.263282	−	0.964719i	$-0.584805\pi$
$541$	−4.86212e25	−0.526565	−0.263282	+	0.964719i	$0.584805\pi$
$542$	−1.06812e26	−1.13665
$543$	7.81425e24	0.0817126
$544$	−2.13370e26	−2.19252
$545$	0	0
$546$	−1.88710e26	−1.87269
$547$	3.16162e25	0.308340	0.154170	−	0.988044i	$-0.450730\pi$
$547$	3.16162e25	0.308340	0.154170	+	0.988044i	$0.450730\pi$
$548$	8.03788e25	0.770417
$549$	8.27258e24	0.0779298
$550$	0	0
$551$	3.84439e24	0.0349854
$552$	1.81287e25	0.162161
$553$	−3.53551e26	−3.10859
$554$	1.65847e26	1.43339
$555$	0	0
$556$	−4.01187e25	−0.335071
$557$	1.82242e26	1.49632	0.748158	−	0.663520i	$-0.230938\pi$
$557$	1.82242e26	1.49632	0.748158	+	0.663520i	$0.230938\pi$
$558$	−1.82692e25	−0.147467
$559$	−5.04434e25	−0.400306
$560$	0	0
$561$	4.24952e25	0.325982
$562$	−3.07843e26	−2.32185
$563$	3.15854e25	0.234237	0.117119	−	0.993118i	$-0.462634\pi$
$563$	3.15854e25	0.234237	0.117119	+	0.993118i	$0.462634\pi$
$564$	2.09345e25	0.152655
$565$	0	0
$566$	1.07605e26	0.758714
$567$	2.93571e25	0.203551
$568$	−2.12892e25	−0.145161
$569$	1.47589e26	0.989666	0.494833	−	0.868988i	$-0.335229\pi$
$569$	1.47589e26	0.989666	0.494833	+	0.868988i	$0.335229\pi$
$570$	0	0
$571$	−1.69593e26	−1.09993	−0.549966	−	0.835187i	$-0.685359\pi$
$571$	−1.69593e26	−1.09993	−0.549966	+	0.835187i	$0.685359\pi$
$572$	−8.47058e25	−0.540320
$573$	−2.32844e24	−0.0146082
$574$	−2.40531e26	−1.48425
$575$	0	0
$576$	−8.42938e25	−0.503248
$577$	−7.03834e25	−0.413333	−0.206667	−	0.978411i	$-0.566262\pi$
$577$	−7.03834e25	−0.413333	−0.206667	+	0.978411i	$0.566262\pi$
$578$	4.18080e26	2.41516
$579$	7.40936e25	0.421050
$580$	0	0
$581$	−2.87510e26	−1.58117
$582$	−1.50495e26	−0.814244
$583$	−4.96064e25	−0.264050
$584$	1.02610e26	0.537361
$585$	0	0
$586$	−3.07517e26	−1.55898
$587$	−3.43137e25	−0.171162	−0.0855808	−	0.996331i	$-0.527275\pi$
$587$	−3.43137e25	−0.171162	−0.0855808	+	0.996331i	$0.527275\pi$
$588$	−3.67143e26	−1.80198
$589$	−1.34723e25	−0.0650650
$590$	0	0
$591$	8.62089e25	0.403154
$592$	−6.63302e25	−0.305250
$593$	−3.21922e26	−1.45791	−0.728955	−	0.684562i	$-0.759993\pi$
$593$	−3.21922e26	−1.45791	−0.728955	+	0.684562i	$0.759993\pi$
$594$	2.31249e25	0.103064
$595$	0	0
$596$	6.65627e25	0.287337
$597$	2.60021e26	1.10472
$598$	−2.40216e26	−1.00448
$599$	6.71773e25	0.276483	0.138241	−	0.990399i	$-0.455855\pi$
$599$	6.71773e25	0.276483	0.138241	+	0.990399i	$0.455855\pi$
$600$	0	0
$601$	−1.65737e26	−0.660863	−0.330432	−	0.943830i	$-0.607194\pi$
$601$	−1.65737e26	−0.660863	−0.330432	+	0.943830i	$0.607194\pi$
$602$	−2.45320e26	−0.962869
$603$	−1.03854e26	−0.401244
$604$	9.21666e25	0.350528
$605$	0	0
$606$	−3.88415e26	−1.43155
$607$	1.07010e26	0.388267	0.194133	−	0.980975i	$-0.437811\pi$
$607$	1.07010e26	0.388267	0.194133	+	0.980975i	$0.437811\pi$
$608$	−8.56243e25	−0.305853
$609$	−4.69235e25	−0.165016
$610$	0	0
$611$	−6.79936e25	−0.231780
$612$	2.11488e26	0.709818
$613$	−1.05520e26	−0.348708	−0.174354	−	0.984683i	$-0.555784\pi$
$613$	−1.05520e26	−0.348708	−0.174354	+	0.984683i	$0.555784\pi$
$614$	−5.34917e26	−1.74055
$615$	0	0
$616$	−1.00975e26	−0.318565
$617$	−5.70260e26	−1.77159	−0.885796	−	0.464074i	$-0.846387\pi$
$617$	−5.70260e26	−1.77159	−0.885796	+	0.464074i	$0.846387\pi$
$618$	−1.53345e26	−0.469114
$619$	5.68518e26	1.71271	0.856353	−	0.516390i	$-0.172725\pi$
$619$	5.68518e26	1.71271	0.856353	+	0.516390i	$0.172725\pi$
$620$	0	0
$621$	3.73697e25	0.109182
$622$	6.22172e26	1.79020
$623$	5.81694e26	1.64838
$624$	1.36894e26	0.382060
$625$	0	0
$626$	6.39608e26	1.73164
$627$	1.70531e25	0.0454739
$628$	6.88276e26	1.80778
$629$	3.32826e26	0.861066
$630$	0	0
$631$	−2.31534e26	−0.581214	−0.290607	−	0.956842i	$-0.593857\pi$
$631$	−2.31534e26	−0.581214	−0.290607	+	0.956842i	$0.593857\pi$
$632$	−3.39731e26	−0.840085
$633$	3.06032e26	0.745472
$634$	1.18169e27	2.83566
$635$	0	0
$636$	−2.46878e26	−0.574963
$637$	1.19245e27	2.73600
$638$	−3.69621e25	−0.0835526
$639$	−4.38845e25	−0.0977356
$640$	0	0
$641$	6.84136e26	1.47908	0.739539	−	0.673114i	$-0.235044\pi$
$641$	6.84136e26	1.47908	0.739539	+	0.673114i	$0.235044\pi$
$642$	−3.85049e26	−0.820225
$643$	4.41658e26	0.927004	0.463502	−	0.886096i	$-0.346593\pi$
$643$	4.41658e26	0.927004	0.463502	+	0.886096i	$0.346593\pi$
$644$	−6.65708e26	−1.37679
$645$	0	0
$646$	2.73689e26	0.549603
$647$	−2.91842e26	−0.577507	−0.288754	−	0.957403i	$-0.593241\pi$
$647$	−2.91842e26	−0.577507	−0.288754	+	0.957403i	$0.593241\pi$
$648$	2.82095e25	0.0550089
$649$	−2.54635e26	−0.489320
$650$	0	0
$651$	1.64439e26	0.306893
$652$	−1.20948e27	−2.22457
$653$	9.04573e26	1.63972	0.819858	−	0.572567i	$-0.194052\pi$
$653$	9.04573e26	1.63972	0.819858	+	0.572567i	$0.194052\pi$
$654$	−1.76397e26	−0.315139
$655$	0	0
$656$	1.74486e26	0.302813
$657$	2.11515e26	0.361801
$658$	−3.30672e26	−0.557509
$659$	−2.65595e26	−0.441375	−0.220688	−	0.975345i	$-0.570830\pi$
$659$	−2.65595e26	−0.441375	−0.220688	+	0.975345i	$0.570830\pi$
$660$	0	0
$661$	3.55026e26	0.573253	0.286627	−	0.958042i	$-0.407466\pi$
$661$	3.55026e26	0.573253	0.286627	+	0.958042i	$0.407466\pi$
$662$	−3.39129e26	−0.539777
$663$	−6.86897e26	−1.07774
$664$	−2.76271e26	−0.427306
$665$	0	0
$666$	1.81116e26	0.272239
$667$	−5.97306e25	−0.0885118
$668$	1.62635e27	2.37595
$669$	−2.06281e25	−0.0297106
$670$	0	0
$671$	−5.86536e25	−0.0821165
$672$	1.04511e27	1.44262
$673$	3.58165e26	0.487461	0.243731	−	0.969843i	$-0.421629\pi$
$673$	3.58165e26	0.487461	0.243731	+	0.969843i	$0.421629\pi$
$674$	−1.76252e27	−2.36519
$675$	0	0
$676$	3.53848e26	0.461660
$677$	−4.80665e26	−0.618373	−0.309186	−	0.951001i	$-0.600057\pi$
$677$	−4.80665e26	−0.618373	−0.309186	+	0.951001i	$0.600057\pi$
$678$	−9.52395e26	−1.20819
$679$	1.35460e27	1.69452
$680$	0	0
$681$	−9.65137e25	−0.117406
$682$	1.29531e26	0.155389
$683$	−9.71565e26	−1.14941	−0.574705	−	0.818360i	$-0.694883\pi$
$683$	−9.71565e26	−1.14941	−0.574705	+	0.818360i	$0.694883\pi$
$684$	8.48689e25	0.0990183
$685$	0	0
$686$	3.33785e27	3.78781
$687$	−3.32668e26	−0.372325
$688$	1.77960e26	0.196441
$689$	8.01842e26	0.872984
$690$	0	0
$691$	−6.78820e26	−0.718975	−0.359487	−	0.933150i	$-0.617048\pi$
$691$	−6.78820e26	−0.718975	−0.359487	+	0.933150i	$0.617048\pi$
$692$	3.72527e26	0.389179
$693$	−2.08145e26	−0.214487
$694$	−2.09035e27	−2.12474
$695$	0	0
$696$	−4.50892e25	−0.0445949
$697$	−8.75522e26	−0.854193
$698$	3.52451e26	0.339212
$699$	−5.80379e26	−0.551034
$700$	0	0
$701$	−1.64884e26	−0.152355	−0.0761776	−	0.997094i	$-0.524272\pi$
$701$	−1.64884e26	−0.152355	−0.0761776	+	0.997094i	$0.524272\pi$
$702$	−3.73792e26	−0.340744
$703$	1.33561e26	0.120117
$704$	5.97653e26	0.530285
$705$	0	0
$706$	−2.32588e27	−2.00882
$707$	3.49610e27	2.97920
$708$	−1.26725e27	−1.06548
$709$	1.52342e27	1.26381	0.631904	−	0.775047i	$-0.282274\pi$
$709$	1.52342e27	1.26381	0.631904	+	0.775047i	$0.282274\pi$
$710$	0	0
$711$	−7.00305e26	−0.565622
$712$	5.58956e26	0.445469
$713$	2.09321e26	0.164612
$714$	−3.34057e27	−2.59232
$715$	0	0
$716$	2.31402e27	1.74861
$717$	6.86291e26	0.511772
$718$	−1.60330e27	−1.17987
$719$	−2.54571e26	−0.184878	−0.0924390	−	0.995718i	$-0.529466\pi$
$719$	−2.54571e26	−0.184878	−0.0924390	+	0.995718i	$0.529466\pi$
$720$	0	0
$721$	1.38025e27	0.976274
$722$	−2.07435e27	−1.44803
$723$	−6.10985e26	−0.420936
$724$	−2.75731e26	−0.187486
$725$	0	0
$726$	1.16503e27	0.771684
$727$	2.12348e27	1.38826	0.694132	−	0.719847i	$-0.255788\pi$
$727$	2.12348e27	1.38826	0.694132	+	0.719847i	$0.255788\pi$
$728$	1.63217e27	1.05322
$729$	5.81497e25	0.0370370
$730$	0	0
$731$	−8.92955e26	−0.554134
$732$	−2.91903e26	−0.178807
$733$	1.37369e27	0.830619	0.415309	−	0.909680i	$-0.363673\pi$
$733$	1.37369e27	0.830619	0.415309	+	0.909680i	$0.363673\pi$
$734$	3.89538e26	0.232507
$735$	0	0
$736$	1.33035e27	0.773798
$737$	7.36336e26	0.422800
$738$	−4.76437e26	−0.270066
$739$	2.36197e27	1.32176	0.660880	−	0.750492i	$-0.270183\pi$
$739$	2.36197e27	1.32176	0.660880	+	0.750492i	$0.270183\pi$
$740$	0	0
$741$	−2.75648e26	−0.150342
$742$	3.89959e27	2.09982
$743$	2.96849e26	0.157813	0.0789063	−	0.996882i	$-0.474857\pi$
$743$	2.96849e26	0.157813	0.0789063	+	0.996882i	$0.474857\pi$
$744$	1.58012e26	0.0829365
$745$	0	0
$746$	−3.45115e27	−1.76581
$747$	−5.69492e26	−0.287701
$748$	−1.49947e27	−0.747952
$749$	3.46580e27	1.70697
$750$	0	0
$751$	−2.22426e27	−1.06808	−0.534042	−	0.845458i	$-0.679327\pi$
$751$	−2.22426e27	−1.06808	−0.534042	+	0.845458i	$0.679327\pi$
$752$	2.39876e26	0.113741
$753$	−4.78989e25	−0.0224271
$754$	5.97458e26	0.276236
$755$	0	0
$756$	−1.03589e27	−0.467040
$757$	1.92178e26	0.0855643	0.0427822	−	0.999084i	$-0.486378\pi$
$757$	1.92178e26	0.0855643	0.0427822	+	0.999084i	$0.486378\pi$
$758$	5.72300e26	0.251632
$759$	−2.64956e26	−0.115047
$760$	0	0
$761$	1.93340e27	0.818778	0.409389	−	0.912360i	$-0.365742\pi$
$761$	1.93340e27	0.818778	0.409389	+	0.912360i	$0.365742\pi$
$762$	6.40674e26	0.267957
$763$	1.58774e27	0.655837
$764$	8.21607e25	0.0335179
$765$	0	0
$766$	−2.34522e26	−0.0933277
$767$	4.11594e27	1.61776
$768$	−1.18432e26	−0.0459766
$769$	−1.55642e27	−0.596796	−0.298398	−	0.954442i	$-0.596452\pi$
$769$	−1.55642e27	−0.596796	−0.298398	+	0.954442i	$0.596452\pi$
$770$	0	0
$771$	−1.39611e27	−0.522281
$772$	−2.61444e27	−0.966083
$773$	−2.79822e27	−1.02136	−0.510678	−	0.859772i	$-0.670606\pi$
$773$	−2.79822e27	−1.02136	−0.510678	+	0.859772i	$0.670606\pi$
$774$	−4.85924e26	−0.175198
$775$	0	0
$776$	1.30165e27	0.457938
$777$	−1.63021e27	−0.566558
$778$	−4.90017e27	−1.68231
$779$	−3.51342e26	−0.119158
$780$	0	0
$781$	3.11146e26	0.102986
$782$	−4.25234e27	−1.39048
$783$	−9.29448e25	−0.0300254
$784$	−4.20689e27	−1.34264
$785$	0	0
$786$	−2.80526e27	−0.873893
$787$	−3.80897e27	−1.17232	−0.586161	−	0.810194i	$-0.699361\pi$
$787$	−3.80897e27	−1.17232	−0.586161	+	0.810194i	$0.699361\pi$
$788$	−3.04194e27	−0.925021
$789$	1.59468e27	0.479116
$790$	0	0
$791$	8.57244e27	2.51436
$792$	−2.00009e26	−0.0579643
$793$	9.48081e26	0.271488
$794$	−4.66090e27	−1.31879
$795$	0	0
$796$	−9.17501e27	−2.53474
$797$	−3.90140e26	−0.106504	−0.0532521	−	0.998581i	$-0.516959\pi$
$797$	−3.90140e26	−0.106504	−0.0532521	+	0.998581i	$0.516959\pi$
$798$	−1.34055e27	−0.361624
$799$	−1.20363e27	−0.320848
$800$	0	0
$801$	1.15220e27	0.299930
$802$	3.63967e27	0.936279
$803$	−1.49966e27	−0.381238
$804$	3.66455e27	0.920638
$805$	0	0
$806$	−2.09374e27	−0.513736
$807$	4.13869e27	1.00361
$808$	3.35944e27	0.805117
$809$	−8.85420e26	−0.209719	−0.104860	−	0.994487i	$-0.533439\pi$
$809$	−8.85420e26	−0.209719	−0.104860	+	0.994487i	$0.533439\pi$
$810$	0	0
$811$	−2.17253e27	−0.502651	−0.251326	−	0.967903i	$-0.580866\pi$
$811$	−2.17253e27	−0.502651	−0.251326	+	0.967903i	$0.580866\pi$
$812$	1.65573e27	0.378623
$813$	1.90433e27	0.430410
$814$	−1.28413e27	−0.286865
$815$	0	0
$816$	2.42332e27	0.528877
$817$	−3.58338e26	−0.0773007
$818$	3.75608e27	0.800900
$819$	3.36448e27	0.709122
$820$	0	0
$821$	−2.60443e27	−0.536355	−0.268177	−	0.963370i	$-0.586421\pi$
$821$	−2.60443e27	−0.536355	−0.268177	+	0.963370i	$0.586421\pi$
$822$	−2.51485e27	−0.511953
$823$	−3.71169e27	−0.746919	−0.373459	−	0.927647i	$-0.621828\pi$
$823$	−3.71169e27	−0.746919	−0.373459	+	0.927647i	$0.621828\pi$
$824$	1.32629e27	0.263834
$825$	0	0
$826$	2.00170e28	3.89124
$827$	−5.98762e27	−1.15067	−0.575336	−	0.817917i	$-0.695129\pi$
$827$	−5.98762e27	−1.15067	−0.575336	+	0.817917i	$0.695129\pi$
$828$	−1.31862e27	−0.250513
$829$	9.15991e27	1.72038	0.860188	−	0.509977i	$-0.170346\pi$
$829$	9.15991e27	1.72038	0.860188	+	0.509977i	$0.170346\pi$
$830$	0	0
$831$	−2.95685e27	−0.542774
$832$	−9.66052e27	−1.75319
$833$	2.11090e28	3.78738
$834$	1.25521e27	0.222659
$835$	0	0
$836$	−6.01731e26	−0.104338
$837$	3.25717e26	0.0558404
$838$	2.79738e27	0.474169
$839$	−1.90995e27	−0.320099	−0.160049	−	0.987109i	$-0.551165\pi$
$839$	−1.90995e27	−0.320099	−0.160049	+	0.987109i	$0.551165\pi$
$840$	0	0
$841$	−5.95470e27	−0.975659
$842$	−1.18381e28	−1.91786
$843$	5.48847e27	0.879203
$844$	−1.07985e28	−1.71046
$845$	0	0
$846$	−6.54987e26	−0.101441
$847$	−1.04864e28	−1.60595
$848$	−2.82884e27	−0.428398
$849$	−1.91848e27	−0.287298
$850$	0	0
$851$	−2.07515e27	−0.303892
$852$	1.54849e27	0.224250
$853$	−2.96003e27	−0.423917	−0.211958	−	0.977279i	$-0.567984\pi$
$853$	−2.96003e27	−0.423917	−0.211958	+	0.977279i	$0.567984\pi$
$854$	4.61079e27	0.653018
$855$	0	0
$856$	3.33032e27	0.461302
$857$	−6.39793e27	−0.876439	−0.438219	−	0.898868i	$-0.644391\pi$
$857$	−6.39793e27	−0.876439	−0.438219	+	0.898868i	$0.644391\pi$
$858$	2.65023e27	0.359050
$859$	1.49963e27	0.200932	0.100466	−	0.994941i	$-0.467967\pi$
$859$	1.49963e27	0.200932	0.100466	+	0.994941i	$0.467967\pi$
$860$	0	0
$861$	4.28838e27	0.562035
$862$	1.74077e28	2.25643
$863$	1.16811e28	1.49755	0.748774	−	0.662825i	$-0.230643\pi$
$863$	1.16811e28	1.49755	0.748774	+	0.662825i	$0.230643\pi$
$864$	2.07012e27	0.262491
$865$	0	0
$866$	−8.44260e27	−1.04726
$867$	−7.45387e27	−0.914536
$868$	−5.80236e27	−0.704153
$869$	4.96525e27	0.596009
$870$	0	0
$871$	−1.19022e28	−1.39783
$872$	1.52567e27	0.177237
$873$	2.68315e27	0.308326
$874$	−1.70644e27	−0.193969
$875$	0	0
$876$	−7.46344e27	−0.830137
$877$	−3.36367e27	−0.370098	−0.185049	−	0.982729i	$-0.559244\pi$
$877$	−3.36367e27	−0.370098	−0.185049	+	0.982729i	$0.559244\pi$
$878$	−7.24914e27	−0.789021
$879$	5.48265e27	0.590332
$880$	0	0
$881$	−8.62862e27	−0.909222	−0.454611	−	0.890690i	$-0.650222\pi$
$881$	−8.62862e27	−0.909222	−0.454611	+	0.890690i	$0.650222\pi$
$882$	1.14870e28	1.19744
$883$	−3.09180e27	−0.318849	−0.159425	−	0.987210i	$-0.550964\pi$
$883$	−3.09180e27	−0.318849	−0.159425	+	0.987210i	$0.550964\pi$
$884$	2.42376e28	2.47283
$885$	0	0
$886$	−8.23273e27	−0.822098
$887$	−1.06571e28	−1.05285	−0.526423	−	0.850223i	$-0.676467\pi$
$887$	−1.06571e28	−1.05285	−0.526423	+	0.850223i	$0.676467\pi$
$888$	−1.56649e27	−0.153110
$889$	−5.76666e27	−0.557645
$890$	0	0
$891$	−4.12289e26	−0.0390268
$892$	7.27876e26	0.0681697
$893$	−4.83011e26	−0.0447577
$894$	−2.08258e27	−0.190939
$895$	0	0
$896$	−1.91438e28	−1.71831
$897$	4.28277e27	0.380361
$898$	−2.91522e28	−2.56180
$899$	−5.20616e26	−0.0452690
$900$	0	0
$901$	1.41943e28	1.20845
$902$	3.37800e27	0.284575
$903$	4.37377e27	0.364605
$904$	8.23735e27	0.679497
$905$	0	0
$906$	−2.88366e27	−0.232930
$907$	1.19240e28	0.953133	0.476567	−	0.879138i	$-0.341881\pi$
$907$	1.19240e28	0.953133	0.476567	+	0.879138i	$0.341881\pi$
$908$	3.40555e27	0.269384
$909$	6.92498e27	0.542078
$910$	0	0
$911$	−1.44243e28	−1.10578	−0.552890	−	0.833254i	$-0.686475\pi$
$911$	−1.44243e28	−1.10578	−0.552890	+	0.833254i	$0.686475\pi$
$912$	9.72465e26	0.0737774
$913$	4.03777e27	0.303158
$914$	−1.45557e28	−1.08154
$915$	0	0
$916$	1.17384e28	0.854284
$917$	2.52499e28	1.81866
$918$	−6.61692e27	−0.471683
$919$	−2.45481e28	−1.73189	−0.865946	−	0.500138i	$-0.833283\pi$
$919$	−2.45481e28	−1.73189	−0.865946	+	0.500138i	$0.833283\pi$
$920$	0	0
$921$	9.53693e27	0.659086
$922$	−3.36080e28	−2.29879
$923$	−5.02940e27	−0.340486
$924$	7.34455e27	0.492132
$925$	0	0
$926$	4.02897e28	2.64478
$927$	2.73395e27	0.177637
$928$	−3.30881e27	−0.212797
$929$	2.24631e27	0.142995	0.0714974	−	0.997441i	$-0.477222\pi$
$929$	2.24631e27	0.142995	0.0714974	+	0.997441i	$0.477222\pi$
$930$	0	0
$931$	8.47091e27	0.528333
$932$	2.04790e28	1.26432
$933$	−1.10926e28	−0.677886
$934$	−3.67924e28	−2.22568
$935$	0	0
$936$	3.23296e27	0.191637
$937$	−5.69436e27	−0.334132	−0.167066	−	0.985946i	$-0.553429\pi$
$937$	−5.69436e27	−0.334132	−0.167066	+	0.985946i	$0.553429\pi$
$938$	−5.78838e28	−3.36225
$939$	−1.14034e28	−0.655712
$940$	0	0
$941$	−4.99691e27	−0.281579	−0.140790	−	0.990040i	$-0.544964\pi$
$941$	−4.99691e27	−0.281579	−0.140790	+	0.990040i	$0.544964\pi$
$942$	−2.15344e28	−1.20130
$943$	5.45884e27	0.301466
$944$	−1.45207e28	−0.793879
$945$	0	0
$946$	3.44526e27	0.184610
$947$	1.38781e28	0.736214	0.368107	−	0.929783i	$-0.380006\pi$
$947$	1.38781e28	0.736214	0.368107	+	0.929783i	$0.380006\pi$
$948$	2.47107e28	1.29780
$949$	2.42407e28	1.26042
$950$	0	0
$951$	−2.10681e28	−1.07377
$952$	2.88929e28	1.45794
$953$	3.71993e27	0.185846	0.0929228	−	0.995673i	$-0.470379\pi$
$953$	3.71993e27	0.185846	0.0929228	+	0.995673i	$0.470379\pi$
$954$	7.72419e27	0.382070
$955$	0	0
$956$	−2.42163e28	−1.17424
$957$	6.58989e26	0.0316384
$958$	8.94498e27	0.425213
$959$	2.26360e28	1.06542
$960$	0	0
$961$	−1.98462e28	−0.915810
$962$	2.07568e28	0.948414
$963$	6.86496e27	0.310591
$964$	2.15590e28	0.965822
$965$	0	0
$966$	2.08283e28	0.914895
$967$	−1.20501e28	−0.524132	−0.262066	−	0.965050i	$-0.584404\pi$
$967$	−1.20501e28	−0.524132	−0.262066	+	0.965050i	$0.584404\pi$
$968$	−1.00765e28	−0.434002
$969$	−4.87956e27	−0.208115
$970$	0	0
$971$	−8.01489e27	−0.335208	−0.167604	−	0.985854i	$-0.553603\pi$
$971$	−8.01489e27	−0.335208	−0.167604	+	0.985854i	$0.553603\pi$
$972$	−2.05185e27	−0.0849800
$973$	−1.12981e28	−0.463375
$974$	−5.41478e28	−2.19923
$975$	0	0
$976$	−3.34476e27	−0.133227
$977$	−3.97949e28	−1.56975	−0.784873	−	0.619656i	$-0.787272\pi$
$977$	−3.97949e28	−1.56975	−0.784873	+	0.619656i	$0.787272\pi$
$978$	3.78416e28	1.47826
$979$	−8.16927e27	−0.316044
$980$	0	0
$981$	3.14495e27	0.119332
$982$	4.58713e28	1.72378
$983$	−1.67921e28	−0.624950	−0.312475	−	0.949926i	$-0.601158\pi$
$983$	−1.67921e28	−0.624950	−0.312475	+	0.949926i	$0.601158\pi$
$984$	4.12075e27	0.151888
$985$	0	0
$986$	1.05763e28	0.382386
$987$	5.89549e27	0.211109
$988$	9.72642e27	0.344955
$989$	5.56753e27	0.195568
$990$	0	0
$991$	1.18105e28	0.406975	0.203488	−	0.979078i	$-0.434772\pi$
$991$	1.18105e28	0.406975	0.203488	+	0.979078i	$0.434772\pi$
$992$	1.15955e28	0.395755
$993$	6.04627e27	0.204395
$994$	−2.44594e28	−0.818983
$995$	0	0
$996$	2.00949e28	0.660119
$997$	4.34119e27	0.141255	0.0706277	−	0.997503i	$-0.477500\pi$
$997$	4.34119e27	0.141255	0.0706277	+	0.997503i	$0.477500\pi$
$998$	−2.30998e28	−0.744506
$999$	−3.22908e27	−0.103088

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.20.a.a.1.1		1
5.2	odd	4		75.20.b.a.49.2		2
5.3	odd	4		75.20.b.a.49.1		2
5.4	even	2		3.20.a.a.1.1	✓	1
15.14	odd	2		9.20.a.b.1.1		1
20.19	odd	2		48.20.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.20.a.a.1.1	✓	1	5.4	even	2
9.20.a.b.1.1		1	15.14	odd	2
48.20.a.c.1.1		1	20.19	odd	2
75.20.a.a.1.1		1	1.1	even	1	trivial
75.20.b.a.49.1		2	5.3	odd	4
75.20.b.a.49.2		2	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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