LMFDB - Embedded newform 75.22.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,22,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, 22, names="a")

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

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 22 $
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:	$209.608008215$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
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)

$f(q)$

$=$

$q-544.000 q^{2} -59049.0 q^{3} -1.80122e6 q^{4} +3.21227e7 q^{6} -1.27770e9 q^{7} +2.12071e9 q^{8} +3.48678e9 q^{9} +O(q^{10})$

\(q-544.000 q^{2} -59049.0 q^{3} -1.80122e6 q^{4} +3.21227e7 q^{6} -1.27770e9 q^{7} +2.12071e9 q^{8} +3.48678e9 q^{9} -7.75859e10 q^{11} +1.06360e11 q^{12} +4.34111e11 q^{13} +6.95068e11 q^{14} +2.62376e12 q^{16} -1.28489e13 q^{17} -1.89681e12 q^{18} -2.86053e13 q^{19} +7.54468e13 q^{21} +4.22067e13 q^{22} -2.24022e14 q^{23} -1.25226e14 q^{24} -2.36156e14 q^{26} -2.05891e14 q^{27} +2.30141e15 q^{28} -5.16760e13 q^{29} +8.92111e15 q^{31} -5.87478e15 q^{32} +4.58137e15 q^{33} +6.98981e15 q^{34} -6.28045e15 q^{36} -4.39772e16 q^{37} +1.55613e16 q^{38} -2.56338e16 q^{39} +5.81681e16 q^{41} -4.10431e16 q^{42} +1.61438e17 q^{43} +1.39749e17 q^{44} +1.21868e17 q^{46} +1.60065e17 q^{47} -1.54930e17 q^{48} +1.07397e18 q^{49} +7.58716e17 q^{51} -7.81927e17 q^{52} -2.29953e18 q^{53} +1.12005e17 q^{54} -2.70963e18 q^{56} +1.68911e18 q^{57} +2.81118e16 q^{58} +5.15426e18 q^{59} +1.25169e18 q^{61} -4.85308e18 q^{62} -4.45506e18 q^{63} -2.30654e18 q^{64} -2.49227e18 q^{66} +5.40779e18 q^{67} +2.31437e19 q^{68} +1.32283e19 q^{69} -1.10432e19 q^{71} +7.39447e18 q^{72} +3.77012e19 q^{73} +2.39236e19 q^{74} +5.15242e19 q^{76} +9.91314e19 q^{77} +1.39448e19 q^{78} +6.31554e19 q^{79} +1.21577e19 q^{81} -3.16434e19 q^{82} +1.45158e20 q^{83} -1.35896e20 q^{84} -8.78222e19 q^{86} +3.05142e18 q^{87} -1.64537e20 q^{88} +1.37255e20 q^{89} -5.54663e20 q^{91} +4.03512e20 q^{92} -5.26783e20 q^{93} -8.70752e19 q^{94} +3.46900e20 q^{96} +3.24306e20 q^{97} -5.84238e20 q^{98} -2.70525e20 q^{99} +O(q^{100})\)

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 $

$2$	−544.000	−0.375650	−0.187825	−	0.982202i	$-0.560144\pi$
$2$	−544.000	−0.375650	−0.187825	+	0.982202i	$0.560144\pi$
$3$	−59049.0	−0.577350
$3$	−59049.0	−0.577350
$4$	−1.80122e6	−0.858887
$5$	0	0
$5$	0	0
$6$	3.21227e7	0.216882
$7$	−1.27770e9	−1.70962	−0.854809	−	0.518943i	$-0.826326\pi$
$7$	−1.27770e9	−1.70962	−0.854809	+	0.518943i	$0.826326\pi$
$8$	2.12071e9	0.698292
$9$	3.48678e9	0.333333
$10$	0	0
$11$	−7.75859e10	−0.901903	−0.450951	−	0.892549i	$-0.648915\pi$
$11$	−7.75859e10	−0.901903	−0.450951	+	0.892549i	$0.648915\pi$
$12$	1.06360e11	0.495878
$13$	4.34111e11	0.873364	0.436682	−	0.899616i	$-0.356153\pi$
$13$	4.34111e11	0.873364	0.436682	+	0.899616i	$0.356153\pi$
$14$	6.95068e11	0.642219
$15$	0	0
$16$	2.62376e12	0.596573
$17$	−1.28489e13	−1.54580	−0.772899	−	0.634529i	$-0.781194\pi$
$17$	−1.28489e13	−1.54580	−0.772899	+	0.634529i	$0.781194\pi$
$18$	−1.89681e12	−0.125217
$19$	−2.86053e13	−1.07037	−0.535184	−	0.844736i	$-0.679758\pi$
$19$	−2.86053e13	−1.07037	−0.535184	+	0.844736i	$0.679758\pi$
$20$	0	0
$21$	7.54468e13	0.987048
$22$	4.22067e13	0.338800
$23$	−2.24022e14	−1.12758	−0.563792	−	0.825917i	$-0.690658\pi$
$23$	−2.24022e14	−1.12758	−0.563792	+	0.825917i	$0.690658\pi$
$24$	−1.25226e14	−0.403159
$25$	0	0
$26$	−2.36156e14	−0.328080
$27$	−2.05891e14	−0.192450
$28$	2.30141e15	1.46837
$29$	−5.16760e13	−0.0228092	−0.0114046	−	0.999935i	$-0.503630\pi$
$29$	−5.16760e13	−0.0228092	−0.0114046	+	0.999935i	$0.503630\pi$
$30$	0	0
$31$	8.92111e15	1.95488	0.977442	−	0.211203i	$-0.0677382\pi$
$31$	8.92111e15	1.95488	0.977442	+	0.211203i	$0.0677382\pi$
$32$	−5.87478e15	−0.922395
$33$	4.58137e15	0.520714
$34$	6.98981e15	0.580680
$35$	0	0
$36$	−6.28045e15	−0.286296
$37$	−4.39772e16	−1.50352	−0.751760	−	0.659437i	$-0.770795\pi$
$37$	−4.39772e16	−1.50352	−0.751760	+	0.659437i	$0.770795\pi$
$38$	1.55613e16	0.402084
$39$	−2.56338e16	−0.504237
$40$	0	0
$41$	5.81681e16	0.676791	0.338395	−	0.941004i	$-0.390116\pi$
$41$	5.81681e16	0.676791	0.338395	+	0.941004i	$0.390116\pi$
$42$	−4.10431e16	−0.370785
$43$	1.61438e17	1.13916	0.569582	−	0.821934i	$-0.307105\pi$
$43$	1.61438e17	1.13916	0.569582	+	0.821934i	$0.307105\pi$
$44$	1.39749e17	0.774632
$45$	0	0
$46$	1.21868e17	0.423577
$47$	1.60065e17	0.443883	0.221941	−	0.975060i	$-0.428761\pi$
$47$	1.60065e17	0.443883	0.221941	+	0.975060i	$0.428761\pi$
$48$	−1.54930e17	−0.344432
$49$	1.07397e18	1.92279
$50$	0	0
$51$	7.58716e17	0.892467
$52$	−7.81927e17	−0.750121
$53$	−2.29953e18	−1.80610	−0.903050	−	0.429535i	$-0.858678\pi$
$53$	−2.29953e18	−1.80610	−0.903050	+	0.429535i	$0.858678\pi$
$54$	1.12005e17	0.0722940
$55$	0	0
$56$	−2.70963e18	−1.19381
$57$	1.68911e18	0.617977
$58$	2.81118e16	0.00856829
$59$	5.15426e18	1.31286	0.656432	−	0.754385i	$-0.272065\pi$
$59$	5.15426e18	1.31286	0.656432	+	0.754385i	$0.272065\pi$
$60$	0	0
$61$	1.25169e18	0.224663	0.112332	−	0.993671i	$-0.464168\pi$
$61$	1.25169e18	0.224663	0.112332	+	0.993671i	$0.464168\pi$
$62$	−4.85308e18	−0.734353
$63$	−4.45506e18	−0.569872
$64$	−2.30654e18	−0.250075
$65$	0	0
$66$	−2.49227e18	−0.195606
$67$	5.40779e18	0.362438	0.181219	−	0.983443i	$-0.441996\pi$
$67$	5.40779e18	0.362438	0.181219	+	0.983443i	$0.441996\pi$
$68$	2.31437e19	1.32767
$69$	1.32283e19	0.651011
$70$	0	0
$71$	−1.10432e19	−0.402609	−0.201305	−	0.979529i	$-0.564518\pi$
$71$	−1.10432e19	−0.402609	−0.201305	+	0.979529i	$0.564518\pi$
$72$	7.39447e18	0.232764
$73$	3.77012e19	1.02675	0.513375	−	0.858164i	$-0.328395\pi$
$73$	3.77012e19	1.02675	0.513375	+	0.858164i	$0.328395\pi$
$74$	2.39236e19	0.564798
$75$	0	0
$76$	5.15242e19	0.919325
$77$	9.91314e19	1.54191
$78$	1.39448e19	0.189417
$79$	6.31554e19	0.750457	0.375229	−	0.926932i	$-0.377564\pi$
$79$	6.31554e19	0.750457	0.375229	+	0.926932i	$0.377564\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	−3.16434e19	−0.254237
$83$	1.45158e20	1.02688	0.513442	−	0.858124i	$-0.328370\pi$
$83$	1.45158e20	1.02688	0.513442	+	0.858124i	$0.328370\pi$
$84$	−1.35896e20	−0.847762
$85$	0	0
$86$	−8.78222e19	−0.427928
$87$	3.05142e18	0.0131689
$88$	−1.64537e20	−0.629791
$89$	1.37255e20	0.466588	0.233294	−	0.972406i	$-0.425050\pi$
$89$	1.37255e20	0.466588	0.233294	+	0.972406i	$0.425050\pi$
$90$	0	0
$91$	−5.54663e20	−1.49312
$92$	4.03512e20	0.968467
$93$	−5.26783e20	−1.12865
$94$	−8.70752e19	−0.166745
$95$	0	0
$96$	3.46900e20	0.532545
$97$	3.24306e20	0.446532	0.223266	−	0.974758i	$-0.428328\pi$
$97$	3.24306e20	0.446532	0.223266	+	0.974758i	$0.428328\pi$
$98$	−5.84238e20	−0.722298
$99$	−2.70525e20	−0.300634
$100$	0	0
$101$	1.14557e21	1.03192	0.515962	−	0.856611i	$-0.327435\pi$
$101$	1.14557e21	1.03192	0.515962	+	0.856611i	$0.327435\pi$
$102$	−4.12741e20	−0.335256
$103$	−8.95787e20	−0.656771	−0.328385	−	0.944544i	$-0.606504\pi$
$103$	−8.95787e20	−0.656771	−0.328385	+	0.944544i	$0.606504\pi$
$104$	9.20624e20	0.609863
$105$	0	0
$106$	1.25094e21	0.678463
$107$	−1.25783e21	−0.618146	−0.309073	−	0.951038i	$-0.600019\pi$
$107$	−1.25783e21	−0.618146	−0.309073	+	0.951038i	$0.600019\pi$
$108$	3.70854e20	0.165293
$109$	−1.66725e21	−0.674562	−0.337281	−	0.941404i	$-0.609507\pi$
$109$	−1.66725e21	−0.674562	−0.337281	+	0.941404i	$0.609507\pi$
$110$	0	0
$111$	2.59681e21	0.868057
$112$	−3.35237e21	−1.01991
$113$	6.51198e21	1.80463	0.902317	−	0.431073i	$-0.141865\pi$
$113$	6.51198e21	1.80463	0.902317	+	0.431073i	$0.141865\pi$
$114$	−9.18877e20	−0.232143
$115$	0	0
$116$	9.30797e19	0.0195905
$117$	1.51365e21	0.291121
$118$	−2.80392e21	−0.493178
$119$	1.64170e22	2.64272
$120$	0	0
$121$	−1.38067e21	−0.186571
$122$	−6.80917e20	−0.0843949
$123$	−3.43477e21	−0.390745
$124$	−1.60688e22	−1.67902
$125$	0	0
$126$	2.42355e21	0.214073
$127$	7.99589e21	0.650022	0.325011	−	0.945710i	$-0.394632\pi$
$127$	7.99589e21	0.650022	0.325011	+	0.945710i	$0.394632\pi$
$128$	1.35751e22	1.01634
$129$	−9.53274e21	−0.657697
$130$	0	0
$131$	−5.53406e21	−0.324859	−0.162430	−	0.986720i	$-0.551933\pi$
$131$	−5.53406e21	−0.324859	−0.162430	+	0.986720i	$0.551933\pi$
$132$	−8.25204e21	−0.447234
$133$	3.65489e22	1.82992
$134$	−2.94184e21	−0.136150
$135$	0	0
$136$	−2.72489e22	−1.07942
$137$	−5.02831e21	−0.184440	−0.0922201	−	0.995739i	$-0.529396\pi$
$137$	−5.02831e21	−0.184440	−0.0922201	+	0.995739i	$0.529396\pi$
$138$	−7.19619e21	−0.244552
$139$	3.06514e21	0.0965594	0.0482797	−	0.998834i	$-0.484626\pi$
$139$	3.06514e21	0.0965594	0.0482797	+	0.998834i	$0.484626\pi$
$140$	0	0
$141$	−9.45166e21	−0.256276
$142$	6.00752e21	0.151240
$143$	−3.36809e22	−0.787690
$144$	9.14847e21	0.198858
$145$	0	0
$146$	−2.05094e22	−0.385699
$147$	−6.34167e22	−1.11012
$148$	7.92124e22	1.29135
$149$	3.18365e22	0.483581	0.241791	−	0.970328i	$-0.422265\pi$
$149$	3.18365e22	0.483581	0.241791	+	0.970328i	$0.422265\pi$
$150$	0	0
$151$	6.56248e22	0.866583	0.433292	−	0.901254i	$-0.357352\pi$
$151$	6.56248e22	0.866583	0.433292	+	0.901254i	$0.357352\pi$
$152$	−6.06635e22	−0.747429
$153$	−4.48014e22	−0.515266
$154$	−5.39275e22	−0.579219
$155$	0	0
$156$	4.61720e22	0.433083
$157$	6.59284e22	0.578264	0.289132	−	0.957289i	$-0.406633\pi$
$157$	6.59284e22	0.578264	0.289132	+	0.957289i	$0.406633\pi$
$158$	−3.43565e22	−0.281910
$159$	1.35785e23	1.04275
$160$	0	0
$161$	2.86233e23	1.92774
$162$	−6.61377e21	−0.0417389
$163$	8.79055e21	0.0520052	0.0260026	−	0.999662i	$-0.491722\pi$
$163$	8.79055e21	0.0520052	0.0260026	+	0.999662i	$0.491722\pi$
$164$	−1.04773e23	−0.581286
$165$	0	0
$166$	−7.89661e22	−0.385750
$167$	1.51028e23	0.692683	0.346341	−	0.938109i	$-0.387424\pi$
$167$	1.51028e23	0.692683	0.346341	+	0.938109i	$0.387424\pi$
$168$	1.60001e23	0.689247
$169$	−5.86123e22	−0.237235
$170$	0	0
$171$	−9.97404e22	−0.356789
$172$	−2.90784e23	−0.978414
$173$	−6.14868e23	−1.94669	−0.973347	−	0.229338i	$-0.926344\pi$
$173$	−6.14868e23	−1.94669	−0.973347	+	0.229338i	$0.926344\pi$
$174$	−1.65997e21	−0.00494690
$175$	0	0
$176$	−2.03567e23	−0.538051
$177$	−3.04354e23	−0.757983
$178$	−7.46667e22	−0.175274
$179$	4.83508e23	1.07016	0.535078	−	0.844803i	$-0.320282\pi$
$179$	4.83508e23	1.07016	0.535078	+	0.844803i	$0.320282\pi$
$180$	0	0
$181$	−9.28497e23	−1.82876	−0.914378	−	0.404862i	$-0.867320\pi$
$181$	−9.28497e23	−1.82876	−0.914378	+	0.404862i	$0.867320\pi$
$182$	3.01737e23	0.560891
$183$	−7.39108e22	−0.129709
$184$	−4.75087e23	−0.787382
$185$	0	0
$186$	2.86570e23	0.423979
$187$	9.96895e23	1.39416
$188$	−2.88311e23	−0.381245
$189$	2.63067e23	0.329016
$190$	0	0
$191$	−5.96634e23	−0.668125	−0.334062	−	0.942551i	$-0.608420\pi$
$191$	−5.96634e23	−0.668125	−0.334062	+	0.942551i	$0.608420\pi$
$192$	1.36199e23	0.144381
$193$	8.56373e23	0.859629	0.429815	−	0.902917i	$-0.358579\pi$
$193$	8.56373e23	0.859629	0.429815	+	0.902917i	$0.358579\pi$
$194$	−1.76423e23	−0.167740
$195$	0	0
$196$	−1.93445e24	−1.65146
$197$	−1.10520e24	−0.894430	−0.447215	−	0.894427i	$-0.647584\pi$
$197$	−1.10520e24	−0.894430	−0.447215	+	0.894427i	$0.647584\pi$
$198$	1.47166e23	0.112933
$199$	1.53187e24	1.11497	0.557487	−	0.830186i	$-0.311766\pi$
$199$	1.53187e24	1.11497	0.557487	+	0.830186i	$0.311766\pi$
$200$	0	0
$201$	−3.19324e23	−0.209254
$202$	−6.23191e23	−0.387643
$203$	6.60264e22	0.0389950
$204$	−1.36661e24	−0.766528
$205$	0	0
$206$	4.87308e23	0.246716
$207$	−7.81117e23	−0.375861
$208$	1.13900e24	0.521026
$209$	2.21937e24	0.965368
$210$	0	0
$211$	2.15918e24	0.849811	0.424906	−	0.905238i	$-0.360307\pi$
$211$	2.15918e24	0.849811	0.424906	+	0.905238i	$0.360307\pi$
$212$	4.14195e24	1.55124
$213$	6.52092e23	0.232446
$214$	6.84258e23	0.232207
$215$	0	0
$216$	−4.36636e23	−0.134386
$217$	−1.13985e25	−3.34210
$218$	9.06983e23	0.253400
$219$	−2.22622e24	−0.592795
$220$	0	0
$221$	−5.57785e24	−1.35005
$222$	−1.41266e24	−0.326086
$223$	−7.86309e24	−1.73138	−0.865689	−	0.500582i	$-0.833119\pi$
$223$	−7.86309e24	−1.73138	−0.865689	+	0.500582i	$0.833119\pi$
$224$	7.50620e24	1.57694
$225$	0	0
$226$	−3.54252e24	−0.677912
$227$	−5.06979e24	−0.926229	−0.463115	−	0.886298i	$-0.653268\pi$
$227$	−5.06979e24	−0.926229	−0.463115	+	0.886298i	$0.653268\pi$
$228$	−3.04246e24	−0.530772
$229$	−6.10932e24	−1.01793	−0.508967	−	0.860786i	$-0.669973\pi$
$229$	−6.10932e24	−1.01793	−0.508967	+	0.860786i	$0.669973\pi$
$230$	0	0
$231$	−5.85361e24	−0.890221
$232$	−1.09590e23	−0.0159275
$233$	−2.40865e24	−0.334609	−0.167304	−	0.985905i	$-0.553506\pi$
$233$	−2.40865e24	−0.334609	−0.167304	+	0.985905i	$0.553506\pi$
$234$	−8.23426e23	−0.109360
$235$	0	0
$236$	−9.28393e24	−1.12760
$237$	−3.72926e24	−0.433277
$238$	−8.93087e24	−0.992740
$239$	−8.12956e23	−0.0864748	−0.0432374	−	0.999065i	$-0.513767\pi$
$239$	−8.12956e23	−0.0864748	−0.0432374	+	0.999065i	$0.513767\pi$
$240$	0	0
$241$	1.98089e24	0.193055	0.0965277	−	0.995330i	$-0.469226\pi$
$241$	1.98089e24	0.193055	0.0965277	+	0.995330i	$0.469226\pi$
$242$	7.51087e23	0.0700856
$243$	−7.17898e23	−0.0641500
$244$	−2.25456e24	−0.192960
$245$	0	0
$246$	1.86851e24	0.146784
$247$	−1.24179e25	−0.934821
$248$	1.89191e25	1.36508
$249$	−8.57145e24	−0.592872
$250$	0	0
$251$	−1.71966e25	−1.09363	−0.546813	−	0.837255i	$-0.684159\pi$
$251$	−1.71966e25	−1.09363	−0.546813	+	0.837255i	$0.684159\pi$
$252$	8.02452e24	0.489456
$253$	1.73810e25	1.01697
$254$	−4.34976e24	−0.244181
$255$	0	0
$256$	−2.54768e24	−0.131712
$257$	−1.42771e25	−0.708503	−0.354251	−	0.935150i	$-0.615264\pi$
$257$	−1.42771e25	−0.708503	−0.354251	+	0.935150i	$0.615264\pi$
$258$	5.18581e24	0.247064
$259$	5.61895e25	2.57044
$260$	0	0
$261$	−1.80183e23	−0.00760307
$262$	3.01053e24	0.122034
$263$	−1.23443e25	−0.480765	−0.240382	−	0.970678i	$-0.577273\pi$
$263$	−1.23443e25	−0.480765	−0.240382	+	0.970678i	$0.577273\pi$
$264$	9.71577e24	0.363610
$265$	0	0
$266$	−1.98826e25	−0.687410
$267$	−8.10477e24	−0.269384
$268$	−9.74059e24	−0.311293
$269$	−4.23127e24	−0.130039	−0.0650193	−	0.997884i	$-0.520711\pi$
$269$	−4.23127e24	−0.130039	−0.0650193	+	0.997884i	$0.520711\pi$
$270$	0	0
$271$	−5.48006e25	−1.55815	−0.779073	−	0.626933i	$-0.784310\pi$
$271$	−5.48006e25	−1.55815	−0.779073	+	0.626933i	$0.784310\pi$
$272$	−3.37124e25	−0.922182
$273$	3.27523e25	0.862053
$274$	2.73540e24	0.0692851
$275$	0	0
$276$	−2.38270e25	−0.559144
$277$	−1.74992e25	−0.395349	−0.197675	−	0.980268i	$-0.563339\pi$
$277$	−1.74992e25	−0.395349	−0.197675	+	0.980268i	$0.563339\pi$
$278$	−1.66743e24	−0.0362726
$279$	3.11060e25	0.651628
$280$	0	0
$281$	2.52311e25	0.490365	0.245183	−	0.969477i	$-0.421152\pi$
$281$	2.52311e25	0.490365	0.245183	+	0.969477i	$0.421152\pi$
$282$	5.14171e24	0.0962701
$283$	1.19713e25	0.215966	0.107983	−	0.994153i	$-0.465561\pi$
$283$	1.19713e25	0.215966	0.107983	+	0.994153i	$0.465561\pi$
$284$	1.98912e25	0.345796
$285$	0	0
$286$	1.83224e25	0.295896
$287$	−7.43213e25	−1.15705
$288$	−2.04841e25	−0.307465
$289$	9.60027e25	1.38949
$290$	0	0
$291$	−1.91500e25	−0.257805
$292$	−6.79080e25	−0.881863
$293$	−6.89938e25	−0.864371	−0.432186	−	0.901785i	$-0.642257\pi$
$293$	−6.89938e25	−0.864371	−0.432186	+	0.901785i	$0.642257\pi$
$294$	3.44987e25	0.417019
$295$	0	0
$296$	−9.32629e25	−1.04990
$297$	1.59743e25	0.173571
$298$	−1.73191e25	−0.181657
$299$	−9.72505e25	−0.984791
$300$	0	0
$301$	−2.06269e26	−1.94754
$302$	−3.56999e25	−0.325532
$303$	−6.76449e25	−0.595782
$304$	−7.50532e25	−0.638553
$305$	0	0
$306$	2.43720e25	0.193560
$307$	6.78268e25	0.520533	0.260267	−	0.965537i	$-0.416190\pi$
$307$	6.78268e25	0.520533	0.260267	+	0.965537i	$0.416190\pi$
$308$	−1.78557e26	−1.32432
$309$	5.28953e25	0.379187
$310$	0	0
$311$	1.00215e26	0.671347	0.335673	−	0.941978i	$-0.391036\pi$
$311$	1.00215e26	0.671347	0.335673	+	0.941978i	$0.391036\pi$
$312$	−5.43619e25	−0.352105
$313$	2.29792e26	1.43920	0.719599	−	0.694390i	$-0.244326\pi$
$313$	2.29792e26	1.43920	0.719599	+	0.694390i	$0.244326\pi$
$314$	−3.58651e25	−0.217225
$315$	0	0
$316$	−1.13756e26	−0.644558
$317$	−2.06457e26	−1.13164	−0.565818	−	0.824530i	$-0.691439\pi$
$317$	−2.06457e26	−1.13164	−0.565818	+	0.824530i	$0.691439\pi$
$318$	−7.38669e25	−0.391711
$319$	4.00933e24	0.0205717
$320$	0	0
$321$	7.42734e25	0.356887
$322$	−1.55711e26	−0.724155
$323$	3.67547e26	1.65457
$324$	−2.18986e25	−0.0954319
$325$	0	0
$326$	−4.78206e24	−0.0195358
$327$	9.84493e25	0.389459
$328$	1.23358e26	0.472597
$329$	−2.04515e26	−0.758870
$330$	0	0
$331$	−1.09215e26	−0.380268	−0.190134	−	0.981758i	$-0.560892\pi$
$331$	−1.09215e26	−0.380268	−0.190134	+	0.981758i	$0.560892\pi$
$332$	−2.61461e26	−0.881978
$333$	−1.53339e26	−0.501173
$334$	−8.21592e25	−0.260207
$335$	0	0
$336$	1.97954e26	0.588846
$337$	4.76493e26	1.37386	0.686930	−	0.726724i	$-0.258958\pi$
$337$	4.76493e26	1.37386	0.686930	+	0.726724i	$0.258958\pi$
$338$	3.18851e25	0.0891173
$339$	−3.84526e26	−1.04191
$340$	0	0
$341$	−6.92152e26	−1.76312
$342$	5.42588e25	0.134028
$343$	−6.58553e26	−1.57762
$344$	3.42363e26	0.795469
$345$	0	0
$346$	3.34488e26	0.731276
$347$	5.67729e26	1.20415	0.602077	−	0.798438i	$-0.294340\pi$
$347$	5.67729e26	1.20415	0.602077	+	0.798438i	$0.294340\pi$
$348$	−5.49626e24	−0.0113106
$349$	−4.85816e26	−0.970075	−0.485037	−	0.874493i	$-0.661194\pi$
$349$	−4.85816e26	−0.970075	−0.485037	+	0.874493i	$0.661194\pi$
$350$	0	0
$351$	−8.93796e25	−0.168079
$352$	4.55800e26	0.831910
$353$	−1.71788e26	−0.304340	−0.152170	−	0.988354i	$-0.548626\pi$
$353$	−1.71788e26	−0.304340	−0.152170	+	0.988354i	$0.548626\pi$
$354$	1.65568e26	0.284737
$355$	0	0
$356$	−2.47226e26	−0.400746
$357$	−9.69410e26	−1.52578
$358$	−2.63028e26	−0.402004
$359$	1.33028e27	1.97448	0.987239	−	0.159247i	$-0.0509068\pi$
$359$	1.33028e27	1.97448	0.987239	+	0.159247i	$0.0509068\pi$
$360$	0	0
$361$	1.04051e26	0.145687
$362$	5.05103e26	0.686973
$363$	8.15275e25	0.107717
$364$	9.99067e26	1.28242
$365$	0	0
$366$	4.02075e25	0.0487254
$367$	−3.40629e26	−0.401132	−0.200566	−	0.979680i	$-0.564278\pi$
$367$	−3.40629e26	−0.401132	−0.200566	+	0.979680i	$0.564278\pi$
$368$	−5.87780e26	−0.672686
$369$	2.02820e26	0.225597
$370$	0	0
$371$	2.93810e27	3.08774
$372$	9.48849e26	0.969385
$373$	1.62912e27	1.61812	0.809058	−	0.587729i	$-0.199978\pi$
$373$	1.62912e27	1.61812	0.809058	+	0.587729i	$0.199978\pi$
$374$	−5.42311e26	−0.523717
$375$	0	0
$376$	3.39451e26	0.309960
$377$	−2.24331e25	−0.0199207
$378$	−1.43108e26	−0.123595
$379$	3.25550e26	0.273467	0.136734	−	0.990608i	$-0.456340\pi$
$379$	3.25550e26	0.273467	0.136734	+	0.990608i	$0.456340\pi$
$380$	0	0
$381$	−4.72149e26	−0.375290
$382$	3.24569e26	0.250981
$383$	1.61833e27	1.21753	0.608765	−	0.793350i	$-0.291665\pi$
$383$	1.61833e27	1.21753	0.608765	+	0.793350i	$0.291665\pi$
$384$	−8.01594e26	−0.586782
$385$	0	0
$386$	−4.65867e26	−0.322920
$387$	5.62899e26	0.379722
$388$	−5.84146e26	−0.383520
$389$	−2.02736e27	−1.29557	−0.647784	−	0.761824i	$-0.724304\pi$
$389$	−2.02736e27	−1.29557	−0.647784	+	0.761824i	$0.724304\pi$
$390$	0	0
$391$	2.87844e27	1.74302
$392$	2.27758e27	1.34267
$393$	3.26781e26	0.187558
$394$	6.01230e26	0.335993
$395$	0	0
$396$	4.87275e26	0.258211
$397$	−1.57647e27	−0.813551	−0.406775	−	0.913528i	$-0.633347\pi$
$397$	−1.57647e27	−0.813551	−0.406775	+	0.913528i	$0.633347\pi$
$398$	−8.33337e26	−0.418840
$399$	−2.15818e27	−1.05650
$400$	0	0
$401$	2.38019e27	1.10559	0.552797	−	0.833316i	$-0.313561\pi$
$401$	2.38019e27	1.10559	0.552797	+	0.833316i	$0.313561\pi$
$402$	1.73712e26	0.0786063
$403$	3.87275e27	1.70733
$404$	−2.06342e27	−0.886306
$405$	0	0
$406$	−3.59184e25	−0.0146485
$407$	3.41201e27	1.35603
$408$	1.60902e27	0.623202
$409$	1.28701e27	0.485832	0.242916	−	0.970047i	$-0.421896\pi$
$409$	1.28701e27	0.485832	0.242916	+	0.970047i	$0.421896\pi$
$410$	0	0
$411$	2.96916e26	0.106487
$412$	1.61351e27	0.564092
$413$	−6.58558e27	−2.24450
$414$	4.24928e26	0.141192
$415$	0	0
$416$	−2.55031e27	−0.805587
$417$	−1.80993e26	−0.0557486
$418$	−1.20733e27	−0.362641
$419$	−6.36510e27	−1.86448	−0.932241	−	0.361839i	$-0.882149\pi$
$419$	−6.36510e27	−1.86448	−0.932241	+	0.361839i	$0.882149\pi$
$420$	0	0
$421$	1.16839e27	0.325556	0.162778	−	0.986663i	$-0.447955\pi$
$421$	1.16839e27	0.325556	0.162778	+	0.986663i	$0.447955\pi$
$422$	−1.17459e27	−0.319232
$423$	5.58111e26	0.147961
$424$	−4.87664e27	−1.26119
$425$	0	0
$426$	−3.54738e26	−0.0873186
$427$	−1.59928e27	−0.384088
$428$	2.26562e27	0.530917
$429$	1.98882e27	0.454773
$430$	0	0
$431$	4.87470e27	1.06154	0.530770	−	0.847516i	$-0.321903\pi$
$431$	4.87470e27	1.06154	0.530770	+	0.847516i	$0.321903\pi$
$432$	−5.40208e26	−0.114811
$433$	−3.42952e27	−0.711395	−0.355698	−	0.934601i	$-0.615757\pi$
$433$	−3.42952e27	−0.711395	−0.355698	+	0.934601i	$0.615757\pi$
$434$	6.20078e27	1.25546
$435$	0	0
$436$	3.00307e27	0.579372
$437$	6.40821e27	1.20693
$438$	1.21106e27	0.222684
$439$	6.10587e27	1.09615	0.548075	−	0.836429i	$-0.315361\pi$
$439$	6.10587e27	1.09615	0.548075	+	0.836429i	$0.315361\pi$
$440$	0	0
$441$	3.74469e27	0.640931
$442$	3.03435e27	0.507145
$443$	6.52956e27	1.06572	0.532862	−	0.846202i	$-0.321116\pi$
$443$	6.52956e27	1.06572	0.532862	+	0.846202i	$0.321116\pi$
$444$	−4.67741e27	−0.745563
$445$	0	0
$446$	4.27752e27	0.650393
$447$	−1.87991e27	−0.279196
$448$	2.94706e27	0.427533
$449$	3.56364e27	0.505018	0.252509	−	0.967595i	$-0.418744\pi$
$449$	3.56364e27	0.505018	0.252509	+	0.967595i	$0.418744\pi$
$450$	0	0
$451$	−4.51302e27	−0.610399
$452$	−1.17295e28	−1.54998
$453$	−3.87508e27	−0.500322
$454$	2.75797e27	0.347939
$455$	0	0
$456$	3.58212e27	0.431528
$457$	5.64234e25	0.00664262	0.00332131	−	0.999994i	$-0.498943\pi$
$457$	5.64234e25	0.00664262	0.00332131	+	0.999994i	$0.498943\pi$
$458$	3.32347e27	0.382388
$459$	2.64548e27	0.297489
$460$	0	0
$461$	3.68168e27	0.395537	0.197768	−	0.980249i	$-0.436631\pi$
$461$	3.68168e27	0.395537	0.197768	+	0.980249i	$0.436631\pi$
$462$	3.18436e27	0.334412
$463$	6.25388e27	0.642020	0.321010	−	0.947076i	$-0.395978\pi$
$463$	6.25388e27	0.642020	0.321010	+	0.947076i	$0.395978\pi$
$464$	−1.35585e26	−0.0136074
$465$	0	0
$466$	1.31031e27	0.125696
$467$	1.81487e26	0.0170223	0.00851114	−	0.999964i	$-0.497291\pi$
$467$	1.81487e26	0.0170223	0.00851114	+	0.999964i	$0.497291\pi$
$468$	−2.72641e27	−0.250040
$469$	−6.90952e27	−0.619631
$470$	0	0
$471$	−3.89301e27	−0.333861
$472$	1.09307e28	0.916762
$473$	−1.25253e28	−1.02742
$474$	2.02872e27	0.162761
$475$	0	0
$476$	−2.95706e28	−2.26980
$477$	−8.01796e27	−0.602034
$478$	4.42248e26	0.0324843
$479$	4.83395e27	0.347360	0.173680	−	0.984802i	$-0.444434\pi$
$479$	4.83395e27	0.347360	0.173680	+	0.984802i	$0.444434\pi$
$480$	0	0
$481$	−1.90910e28	−1.31312
$482$	−1.07761e27	−0.0725214
$483$	−1.69018e28	−1.11298
$484$	2.48689e27	0.160244
$485$	0	0
$486$	3.90537e26	0.0240980
$487$	6.30377e27	0.380668	0.190334	−	0.981719i	$-0.439043\pi$
$487$	6.30377e27	0.380668	0.190334	+	0.981719i	$0.439043\pi$
$488$	2.65447e27	0.156881
$489$	−5.19073e26	−0.0300252
$490$	0	0
$491$	3.67372e27	0.203587	0.101794	−	0.994806i	$-0.467542\pi$
$491$	3.67372e27	0.203587	0.101794	+	0.994806i	$0.467542\pi$
$492$	6.18676e27	0.335606
$493$	6.63981e26	0.0352584
$494$	6.75531e27	0.351166
$495$	0	0
$496$	2.34068e28	1.16623
$497$	1.41099e28	0.688307
$498$	4.66287e27	0.222713
$499$	−2.00088e28	−0.935761	−0.467881	−	0.883792i	$-0.654982\pi$
$499$	−2.00088e28	−0.935761	−0.467881	+	0.883792i	$0.654982\pi$
$500$	0	0
$501$	−8.91805e27	−0.399921
$502$	9.35496e27	0.410821
$503$	−1.40332e28	−0.603521	−0.301761	−	0.953384i	$-0.597574\pi$
$503$	−1.40332e28	−0.603521	−0.301761	+	0.953384i	$0.597574\pi$
$504$	−9.44790e27	−0.397937
$505$	0	0
$506$	−9.45525e27	−0.382026
$507$	3.46100e27	0.136967
$508$	−1.44023e28	−0.558295
$509$	−6.58158e27	−0.249916	−0.124958	−	0.992162i	$-0.539880\pi$
$509$	−6.58158e27	−0.249916	−0.124958	+	0.992162i	$0.539880\pi$
$510$	0	0
$511$	−4.81708e28	−1.75535
$512$	−2.70830e28	−0.966858
$513$	5.88957e27	0.205992
$514$	7.76673e27	0.266149
$515$	0	0
$516$	1.71705e28	0.564887
$517$	−1.24188e28	−0.400339
$518$	−3.05671e28	−0.965588
$519$	3.63073e28	1.12392
$520$	0	0
$521$	−4.65750e28	−1.38470	−0.692351	−	0.721561i	$-0.743425\pi$
$521$	−4.65750e28	−1.38470	−0.692351	+	0.721561i	$0.743425\pi$
$522$	9.80196e25	0.00285610
$523$	3.89578e28	1.11257	0.556284	−	0.830992i	$-0.312227\pi$
$523$	3.89578e28	1.11257	0.556284	+	0.830992i	$0.312227\pi$
$524$	9.96804e27	0.279017
$525$	0	0
$526$	6.71532e27	0.180600
$527$	−1.14627e29	−3.02186
$528$	1.20204e28	0.310644
$529$	1.07144e28	0.271445
$530$	0	0
$531$	1.79718e28	0.437621
$532$	−6.58324e28	−1.57169
$533$	2.52514e28	0.591085
$534$	4.40900e27	0.101194
$535$	0	0
$536$	1.14684e28	0.253088
$537$	−2.85507e28	−0.617854
$538$	2.30181e27	0.0488491
$539$	−8.33247e28	−1.73417
$540$	0	0
$541$	7.65629e27	0.153267	0.0766333	−	0.997059i	$-0.475583\pi$
$541$	7.65629e27	0.153267	0.0766333	+	0.997059i	$0.475583\pi$
$542$	2.98116e28	0.585318
$543$	5.48268e28	1.05583
$544$	7.54846e28	1.42584
$545$	0	0
$546$	−1.78172e28	−0.323830
$547$	−6.77366e28	−1.20769	−0.603847	−	0.797100i	$-0.706366\pi$
$547$	−6.77366e28	−1.20769	−0.603847	+	0.797100i	$0.706366\pi$
$548$	9.05707e27	0.158413
$549$	4.36436e27	0.0748878
$550$	0	0
$551$	1.47821e27	0.0244142
$552$	2.80534e28	0.454595
$553$	−8.06935e28	−1.28299
$554$	9.51957e27	0.148513
$555$	0	0
$556$	−5.52097e27	−0.0829336
$557$	5.64828e28	0.832600	0.416300	−	0.909227i	$-0.363327\pi$
$557$	5.64828e28	0.832600	0.416300	+	0.909227i	$0.363327\pi$
$558$	−1.69217e28	−0.244784
$559$	7.00819e28	0.994906
$560$	0	0
$561$	−5.88657e28	−0.804919
$562$	−1.37257e28	−0.184206
$563$	−1.32120e28	−0.174033	−0.0870166	−	0.996207i	$-0.527733\pi$
$563$	−1.32120e28	−0.174033	−0.0870166	+	0.996207i	$0.527733\pi$
$564$	1.70245e28	0.220112
$565$	0	0
$566$	−6.51240e27	−0.0811275
$567$	−1.55338e28	−0.189957
$568$	−2.34195e28	−0.281139
$569$	3.77888e27	0.0445332	0.0222666	−	0.999752i	$-0.492912\pi$
$569$	3.77888e27	0.0445332	0.0222666	+	0.999752i	$0.492912\pi$
$570$	0	0
$571$	−9.70592e28	−1.10245	−0.551223	−	0.834358i	$-0.685839\pi$
$571$	−9.70592e28	−1.10245	−0.551223	+	0.834358i	$0.685839\pi$
$572$	6.06666e28	0.676536
$573$	3.52306e28	0.385742
$574$	4.04308e28	0.434647
$575$	0	0
$576$	−8.04239e27	−0.0833584
$577$	1.40057e27	0.0142547	0.00712737	−	0.999975i	$-0.497731\pi$
$577$	1.40057e27	0.0142547	0.00712737	+	0.999975i	$0.497731\pi$
$578$	−5.22255e28	−0.521964
$579$	−5.05680e28	−0.496307
$580$	0	0
$581$	−1.85468e29	−1.75558
$582$	1.04176e28	0.0968447
$583$	1.78411e29	1.62893
$584$	7.99534e28	0.716972
$585$	0	0
$586$	3.75326e28	0.324701
$587$	9.16040e28	0.778421	0.389211	−	0.921149i	$-0.372748\pi$
$587$	9.16040e28	0.778421	0.389211	+	0.921149i	$0.372748\pi$
$588$	1.14227e29	0.953471
$589$	−2.55191e29	−2.09245
$590$	0	0
$591$	6.52610e28	0.516399
$592$	−1.15385e29	−0.896959
$593$	−1.47495e29	−1.12643	−0.563214	−	0.826311i	$-0.690435\pi$
$593$	−1.47495e29	−1.12643	−0.563214	+	0.826311i	$0.690435\pi$
$594$	−8.68999e27	−0.0652021
$595$	0	0
$596$	−5.73444e28	−0.415341
$597$	−9.04554e28	−0.643730
$598$	5.29043e28	0.369937
$599$	1.19182e29	0.818899	0.409450	−	0.912333i	$-0.365721\pi$
$599$	1.19182e29	0.818899	0.409450	+	0.912333i	$0.365721\pi$
$600$	0	0
$601$	2.64125e29	1.75238	0.876190	−	0.481966i	$-0.160077\pi$
$601$	2.64125e29	1.75238	0.876190	+	0.481966i	$0.160077\pi$
$602$	1.12210e29	0.731593
$603$	1.88558e28	0.120813
$604$	−1.18205e29	−0.744297
$605$	0	0
$606$	3.67988e28	0.223806
$607$	8.82263e28	0.527371	0.263686	−	0.964609i	$-0.415062\pi$
$607$	8.82263e28	0.527371	0.263686	+	0.964609i	$0.415062\pi$
$608$	1.68050e29	0.987302
$609$	−3.89879e27	−0.0225138
$610$	0	0
$611$	6.94859e28	0.387671
$612$	8.06970e28	0.442555
$613$	−3.27173e29	−1.76377	−0.881885	−	0.471465i	$-0.843725\pi$
$613$	−3.27173e29	−1.76377	−0.881885	+	0.471465i	$0.843725\pi$
$614$	−3.68978e28	−0.195538
$615$	0	0
$616$	2.10229e29	1.07670
$617$	−7.46428e28	−0.375832	−0.187916	−	0.982185i	$-0.560173\pi$
$617$	−7.46428e28	−0.375832	−0.187916	+	0.982185i	$0.560173\pi$
$618$	−2.87751e28	−0.142442
$619$	−8.32478e28	−0.405155	−0.202577	−	0.979266i	$-0.564932\pi$
$619$	−8.32478e28	−0.405155	−0.202577	+	0.979266i	$0.564932\pi$
$620$	0	0
$621$	4.61242e28	0.217004
$622$	−5.45167e28	−0.252192
$623$	−1.75371e29	−0.797686
$624$	−6.72569e28	−0.300814
$625$	0	0
$626$	−1.25007e29	−0.540635
$627$	−1.31051e29	−0.557355
$628$	−1.18751e29	−0.496663
$629$	5.65059e29	2.32414
$630$	0	0
$631$	3.63173e29	1.44479	0.722396	−	0.691479i	$-0.243041\pi$
$631$	3.63173e29	1.44479	0.722396	+	0.691479i	$0.243041\pi$
$632$	1.33934e29	0.524038
$633$	−1.27497e29	−0.490639
$634$	1.12312e29	0.425099
$635$	0	0
$636$	−2.44578e29	−0.895606
$637$	4.66221e29	1.67930
$638$	−2.18108e27	−0.00772776
$639$	−3.85054e28	−0.134203
$640$	0	0
$641$	−4.09788e28	−0.138213	−0.0691067	−	0.997609i	$-0.522015\pi$
$641$	−4.09788e28	−0.138213	−0.0691067	+	0.997609i	$0.522015\pi$
$642$	−4.04047e28	−0.134065
$643$	3.44481e29	1.12448	0.562238	−	0.826976i	$-0.309941\pi$
$643$	3.44481e29	1.12448	0.562238	+	0.826976i	$0.309941\pi$
$644$	−5.15567e29	−1.65571
$645$	0	0
$646$	−1.99945e29	−0.621541
$647$	2.99052e29	0.914645	0.457322	−	0.889301i	$-0.348809\pi$
$647$	2.99052e29	0.914645	0.457322	+	0.889301i	$0.348809\pi$
$648$	2.57829e28	0.0775880
$649$	−3.99898e29	−1.18408
$650$	0	0
$651$	6.73069e29	1.92956
$652$	−1.58337e28	−0.0446665
$653$	−6.43185e29	−1.78545	−0.892724	−	0.450603i	$-0.851209\pi$
$653$	−6.43185e29	−1.78545	−0.892724	+	0.450603i	$0.851209\pi$
$654$	−5.35564e28	−0.146300
$655$	0	0
$656$	1.52619e29	0.403755
$657$	1.31456e29	0.342250
$658$	1.11256e29	0.285070
$659$	1.00447e29	0.253303	0.126652	−	0.991947i	$-0.459577\pi$
$659$	1.00447e29	0.253303	0.126652	+	0.991947i	$0.459577\pi$
$660$	0	0
$661$	−4.70181e29	−1.14855	−0.574274	−	0.818663i	$-0.694716\pi$
$661$	−4.70181e29	−1.14855	−0.574274	+	0.818663i	$0.694716\pi$
$662$	5.94131e28	0.142848
$663$	3.29367e29	0.779449
$664$	3.07839e29	0.717065
$665$	0	0
$666$	8.34163e28	0.188266
$667$	1.15766e28	0.0257193
$668$	−2.72034e29	−0.594936
$669$	4.64307e29	0.999612
$670$	0	0
$671$	−9.71132e28	−0.202624
$672$	−4.43233e29	−0.910448
$673$	3.92345e29	0.793432	0.396716	−	0.917941i	$-0.370150\pi$
$673$	3.92345e29	0.793432	0.396716	+	0.917941i	$0.370150\pi$
$674$	−2.59212e29	−0.516091
$675$	0	0
$676$	1.05573e29	0.203758
$677$	1.52718e29	0.290207	0.145104	−	0.989416i	$-0.453648\pi$
$677$	1.52718e29	0.290207	0.145104	+	0.989416i	$0.453648\pi$
$678$	2.09182e29	0.391392
$679$	−4.14366e29	−0.763399
$680$	0	0
$681$	2.99366e29	0.534759
$682$	3.76531e29	0.662315
$683$	4.36102e29	0.755389	0.377695	−	0.925930i	$-0.376717\pi$
$683$	4.36102e29	0.755389	0.377695	+	0.925930i	$0.376717\pi$
$684$	1.79654e29	0.306442
$685$	0	0
$686$	3.58253e29	0.592634
$687$	3.60749e29	0.587705
$688$	4.23574e29	0.679595
$689$	−9.98250e29	−1.57738
$690$	0	0
$691$	5.96732e29	0.914660	0.457330	−	0.889297i	$-0.348806\pi$
$691$	5.96732e29	0.914660	0.457330	+	0.889297i	$0.348806\pi$
$692$	1.10751e30	1.67199
$693$	3.45650e29	0.513970
$694$	−3.08845e29	−0.452341
$695$	0	0
$696$	6.47118e27	0.00919573
$697$	−7.47397e29	−1.04618
$698$	2.64284e29	0.364409
$699$	1.42229e29	0.193186
$700$	0	0
$701$	5.37209e29	0.708116	0.354058	−	0.935223i	$-0.384802\pi$
$701$	5.37209e29	0.708116	0.354058	+	0.935223i	$0.384802\pi$
$702$	4.86225e28	0.0631390
$703$	1.25798e30	1.60932
$704$	1.78955e29	0.225543
$705$	0	0
$706$	9.34527e28	0.114325
$707$	−1.46370e30	−1.76420
$708$	5.48207e29	0.651021
$709$	−1.27317e29	−0.148971	−0.0744854	−	0.997222i	$-0.523731\pi$
$709$	−1.27317e29	−0.148971	−0.0744854	+	0.997222i	$0.523731\pi$
$710$	0	0
$711$	2.20209e29	0.250152
$712$	2.91078e29	0.325814
$713$	−1.99853e30	−2.20430
$714$	5.27359e29	0.573159
$715$	0	0
$716$	−8.70902e29	−0.919142
$717$	4.80043e28	0.0499262
$718$	−7.23674e29	−0.741713
$719$	−1.86676e30	−1.88553	−0.942767	−	0.333451i	$-0.891787\pi$
$719$	−1.86676e30	−1.88553	−0.942767	+	0.333451i	$0.891787\pi$
$720$	0	0
$721$	1.14455e30	1.12283
$722$	−5.66038e28	−0.0547275
$723$	−1.16970e29	−0.111461
$724$	1.67242e30	1.57069
$725$	0	0
$726$	−4.43509e28	−0.0404640
$727$	1.86570e30	1.67776	0.838882	−	0.544314i	$-0.183210\pi$
$727$	1.86570e30	1.67776	0.838882	+	0.544314i	$0.183210\pi$
$728$	−1.17628e30	−1.04263
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	−2.07430e30	−1.76092
$732$	1.33129e29	0.111406
$733$	4.88128e29	0.402663	0.201331	−	0.979523i	$-0.435473\pi$
$733$	4.88128e29	0.402663	0.201331	+	0.979523i	$0.435473\pi$
$734$	1.85302e29	0.150686
$735$	0	0
$736$	1.31608e30	1.04008
$737$	−4.19568e29	−0.326884
$738$	−1.10334e29	−0.0847456
$739$	1.67069e30	1.26511	0.632556	−	0.774515i	$-0.282006\pi$
$739$	1.67069e30	1.26511	0.632556	+	0.774515i	$0.282006\pi$
$740$	0	0
$741$	7.33262e29	0.539719
$742$	−1.59833e30	−1.15991
$743$	−1.62834e30	−1.16509	−0.582547	−	0.812797i	$-0.697944\pi$
$743$	−1.62834e30	−1.16509	−0.582547	+	0.812797i	$0.697944\pi$
$744$	−1.11715e30	−0.788129
$745$	0	0
$746$	−8.86240e29	−0.607846
$747$	5.06136e29	0.342295
$748$	−1.79562e30	−1.19743
$749$	1.60712e30	1.05679
$750$	0	0
$751$	5.83073e29	0.372824	0.186412	−	0.982472i	$-0.440314\pi$
$751$	5.83073e29	0.372824	0.186412	+	0.982472i	$0.440314\pi$
$752$	4.19971e29	0.264809
$753$	1.01544e30	0.631406
$754$	1.22036e28	0.00748324
$755$	0	0
$756$	−4.73840e29	−0.282587
$757$	−1.39333e30	−0.819494	−0.409747	−	0.912199i	$-0.634383\pi$
$757$	−1.39333e30	−0.819494	−0.409747	+	0.912199i	$0.634383\pi$
$758$	−1.77099e29	−0.102728
$759$	−1.02633e30	−0.587148
$760$	0	0
$761$	−9.17379e29	−0.510517	−0.255258	−	0.966873i	$-0.582161\pi$
$761$	−9.17379e29	−0.510517	−0.255258	+	0.966873i	$0.582161\pi$
$762$	2.56849e29	0.140978
$763$	2.13024e30	1.15324
$764$	1.07467e30	0.573843
$765$	0	0
$766$	−8.80371e29	−0.457366
$767$	2.23752e30	1.14661
$768$	1.50438e29	0.0760438
$769$	−1.13366e30	−0.565273	−0.282636	−	0.959227i	$-0.591209\pi$
$769$	−1.13366e30	−0.565273	−0.282636	+	0.959227i	$0.591209\pi$
$770$	0	0
$771$	8.43047e29	0.409054
$772$	−1.54251e30	−0.738324
$773$	1.81526e29	0.0857143	0.0428572	−	0.999081i	$-0.486354\pi$
$773$	1.81526e29	0.0857143	0.0428572	+	0.999081i	$0.486354\pi$
$774$	−3.06217e29	−0.142643
$775$	0	0
$776$	6.87761e29	0.311809
$777$	−3.31794e30	−1.48405
$778$	1.10289e30	0.486681
$779$	−1.66391e30	−0.724415
$780$	0	0
$781$	8.56799e29	0.363114
$782$	−1.56587e30	−0.654765
$783$	1.06396e28	0.00438963
$784$	2.81783e30	1.14709
$785$	0	0
$786$	−1.77769e29	−0.0704561
$787$	−7.88071e29	−0.308199	−0.154099	−	0.988055i	$-0.549248\pi$
$787$	−7.88071e29	−0.308199	−0.154099	+	0.988055i	$0.549248\pi$
$788$	1.99071e30	0.768214
$789$	7.28921e29	0.277570
$790$	0	0
$791$	−8.32034e30	−3.08523
$792$	−5.73706e29	−0.209930
$793$	5.43371e29	0.196213
$794$	8.57598e29	0.305611
$795$	0	0
$796$	−2.75923e30	−0.957636
$797$	3.61162e30	1.23706	0.618528	−	0.785763i	$-0.287729\pi$
$797$	3.61162e30	1.23706	0.618528	+	0.785763i	$0.287729\pi$
$798$	1.17405e30	0.396876
$799$	−2.05666e30	−0.686153
$800$	0	0
$801$	4.78579e29	0.155529
$802$	−1.29482e30	−0.415317
$803$	−2.92508e30	−0.926029
$804$	5.75172e29	0.179725
$805$	0	0
$806$	−2.10678e30	−0.641358
$807$	2.49853e29	0.0750778
$808$	2.42943e30	0.720584
$809$	6.55296e30	1.91857	0.959286	−	0.282436i	$-0.0911424\pi$
$809$	6.55296e30	1.91857	0.959286	+	0.282436i	$0.0911424\pi$
$810$	0	0
$811$	−2.95834e30	−0.843975	−0.421988	−	0.906602i	$-0.638667\pi$
$811$	−2.95834e30	−0.843975	−0.421988	+	0.906602i	$0.638667\pi$
$812$	−1.18928e29	−0.0334923
$813$	3.23592e30	0.899596
$814$	−1.85613e30	−0.509393
$815$	0	0
$816$	1.99069e30	0.532422
$817$	−4.61797e30	−1.21933
$818$	−7.00132e29	−0.182503
$819$	−1.93399e30	−0.497706
$820$	0	0
$821$	−6.60712e30	−1.65733	−0.828665	−	0.559744i	$-0.810899\pi$
$821$	−6.60712e30	−1.65733	−0.828665	+	0.559744i	$0.810899\pi$
$822$	−1.61523e29	−0.0400018
$823$	−4.27938e30	−1.04636	−0.523182	−	0.852221i	$-0.675255\pi$
$823$	−4.27938e30	−1.04636	−0.523182	+	0.852221i	$0.675255\pi$
$824$	−1.89971e30	−0.458618
$825$	0	0
$826$	3.58256e30	0.843146
$827$	−8.08422e29	−0.187858	−0.0939291	−	0.995579i	$-0.529943\pi$
$827$	−8.08422e29	−0.187858	−0.0939291	+	0.995579i	$0.529943\pi$
$828$	1.40696e30	0.322822
$829$	6.58152e30	1.49109	0.745544	−	0.666456i	$-0.232190\pi$
$829$	6.58152e30	1.49109	0.745544	+	0.666456i	$0.232190\pi$
$830$	0	0
$831$	1.03331e30	0.228255
$832$	−1.00129e30	−0.218407
$833$	−1.37993e31	−2.97225
$834$	9.84603e28	0.0209420
$835$	0	0
$836$	−3.99756e30	−0.829141
$837$	−1.83678e30	−0.376218
$838$	3.46262e30	0.700393
$839$	−8.11970e30	−1.62196	−0.810979	−	0.585075i	$-0.801065\pi$
$839$	−8.11970e30	−1.62196	−0.810979	+	0.585075i	$0.801065\pi$
$840$	0	0
$841$	−5.13017e30	−0.999480
$842$	−6.35604e29	−0.122295
$843$	−1.48987e30	−0.283113
$844$	−3.88914e30	−0.729892
$845$	0	0
$846$	−3.03613e29	−0.0555816
$847$	1.76409e30	0.318966
$848$	−6.03340e30	−1.07747
$849$	−7.06894e29	−0.124688
$850$	0	0
$851$	9.85186e30	1.69534
$852$	−1.17456e30	−0.199645
$853$	−1.29820e30	−0.217960	−0.108980	−	0.994044i	$-0.534759\pi$
$853$	−1.29820e30	−0.217960	−0.108980	+	0.994044i	$0.534759\pi$
$854$	8.70007e29	0.144283
$855$	0	0
$856$	−2.66749e30	−0.431646
$857$	1.05399e30	0.168476	0.0842380	−	0.996446i	$-0.473154\pi$
$857$	1.05399e30	0.168476	0.0842380	+	0.996446i	$0.473154\pi$
$858$	−1.08192e30	−0.170836
$859$	6.98517e30	1.08955	0.544777	−	0.838581i	$-0.316614\pi$
$859$	6.98517e30	1.08955	0.544777	+	0.838581i	$0.316614\pi$
$860$	0	0
$861$	4.38860e30	0.668025
$862$	−2.65184e30	−0.398768
$863$	−1.03987e31	−1.54478	−0.772392	−	0.635146i	$-0.780940\pi$
$863$	−1.03987e31	−1.54478	−0.772392	+	0.635146i	$0.780940\pi$
$864$	1.20956e30	0.177515
$865$	0	0
$866$	1.86566e30	0.267236
$867$	−5.66887e30	−0.802224
$868$	2.05311e31	2.87049
$869$	−4.89997e30	−0.676839
$870$	0	0
$871$	2.34758e30	0.316541
$872$	−3.53575e30	−0.471041
$873$	1.13079e30	0.148844
$874$	−3.48607e30	−0.453384
$875$	0	0
$876$	4.00990e30	0.509144
$877$	−5.66163e30	−0.710306	−0.355153	−	0.934808i	$-0.615571\pi$
$877$	−5.66163e30	−0.710306	−0.355153	+	0.934808i	$0.615571\pi$
$878$	−3.32159e30	−0.411770
$879$	4.07402e30	0.499045
$880$	0	0
$881$	3.59833e30	0.430382	0.215191	−	0.976572i	$-0.430963\pi$
$881$	3.59833e30	0.430382	0.215191	+	0.976572i	$0.430963\pi$
$882$	−2.03711e30	−0.240766
$883$	1.24249e31	1.45112	0.725562	−	0.688157i	$-0.241580\pi$
$883$	1.24249e31	1.45112	0.725562	+	0.688157i	$0.241580\pi$
$884$	1.00469e31	1.15954
$885$	0	0
$886$	−3.55208e30	−0.400340
$887$	9.60349e30	1.06962	0.534812	−	0.844971i	$-0.320383\pi$
$887$	9.60349e30	1.06962	0.534812	+	0.844971i	$0.320383\pi$
$888$	5.50708e30	0.606157
$889$	−1.02163e31	−1.11129
$890$	0	0
$891$	−9.43264e29	−0.100211
$892$	1.41631e31	1.48706
$893$	−4.57869e30	−0.475118
$894$	1.02267e30	0.104880
$895$	0	0
$896$	−1.73448e31	−1.73754
$897$	5.74254e30	0.568570
$898$	−1.93862e30	−0.189710
$899$	−4.61007e29	−0.0445894
$900$	0	0
$901$	2.95464e31	2.79187
$902$	2.45509e30	0.229297
$903$	1.21800e31	1.12441
$904$	1.38100e31	1.26016
$905$	0	0
$906$	2.10804e30	0.187946
$907$	1.94141e31	1.71097	0.855483	−	0.517830i	$-0.173260\pi$
$907$	1.94141e31	1.71097	0.855483	+	0.517830i	$0.173260\pi$
$908$	9.13179e30	0.795526
$909$	3.99436e30	0.343975
$910$	0	0
$911$	−4.71847e30	−0.397062	−0.198531	−	0.980095i	$-0.563617\pi$
$911$	−4.71847e30	−0.397062	−0.198531	+	0.980095i	$0.563617\pi$
$912$	4.43182e30	0.368669
$913$	−1.12622e31	−0.926150
$914$	−3.06943e28	−0.00249530
$915$	0	0
$916$	1.10042e31	0.874291
$917$	7.07086e30	0.555385
$918$	−1.43914e30	−0.111752
$919$	−1.97911e31	−1.51935	−0.759673	−	0.650305i	$-0.774641\pi$
$919$	−1.97911e31	−1.51935	−0.759673	+	0.650305i	$0.774641\pi$
$920$	0	0
$921$	−4.00510e30	−0.300530
$922$	−2.00284e30	−0.148584
$923$	−4.79399e30	−0.351624
$924$	1.05436e31	0.764599
$925$	0	0
$926$	−3.40211e30	−0.241175
$927$	−3.12342e30	−0.218924
$928$	3.03585e29	0.0210391
$929$	−2.30269e31	−1.57786	−0.788932	−	0.614480i	$-0.789366\pi$
$929$	−2.30269e31	−1.57786	−0.788932	+	0.614480i	$0.789366\pi$
$930$	0	0
$931$	−3.07211e31	−2.05809
$932$	4.33851e30	0.287391
$933$	−5.91757e30	−0.387602
$934$	−9.87288e28	−0.00639443
$935$	0	0
$936$	3.21002e30	0.203288
$937$	−2.58396e31	−1.61816	−0.809078	−	0.587701i	$-0.800033\pi$
$937$	−2.58396e31	−1.61816	−0.809078	+	0.587701i	$0.800033\pi$
$938$	3.75878e30	0.232765
$939$	−1.35690e31	−0.830921
$940$	0	0
$941$	2.93678e31	1.75865	0.879326	−	0.476221i	$-0.157994\pi$
$941$	2.93678e31	1.75865	0.879326	+	0.476221i	$0.157994\pi$
$942$	2.11780e30	0.125415
$943$	−1.30309e31	−0.763138
$944$	1.35235e31	0.783220
$945$	0	0
$946$	6.81377e30	0.385949
$947$	8.23900e29	0.0461530	0.0230765	−	0.999734i	$-0.492654\pi$
$947$	8.23900e29	0.0461530	0.0230765	+	0.999734i	$0.492654\pi$
$948$	6.71721e30	0.372135
$949$	1.63665e31	0.896728
$950$	0	0
$951$	1.21911e31	0.653350
$952$	3.48158e31	1.84539
$953$	3.34266e31	1.75233	0.876167	−	0.482008i	$-0.160093\pi$
$953$	3.34266e31	1.75233	0.876167	+	0.482008i	$0.160093\pi$
$954$	4.36177e30	0.226154
$955$	0	0
$956$	1.46431e30	0.0742720
$957$	−2.36747e29	−0.0118771
$958$	−2.62967e30	−0.130486
$959$	6.42466e30	0.315322
$960$	0	0
$961$	5.87607e31	2.82157
$962$	1.03855e31	0.493274
$963$	−4.38577e30	−0.206049
$964$	−3.56802e30	−0.165813
$965$	0	0
$966$	9.19456e30	0.418091
$967$	−2.28891e31	−1.02956	−0.514779	−	0.857323i	$-0.672126\pi$
$967$	−2.28891e31	−1.02956	−0.514779	+	0.857323i	$0.672126\pi$
$968$	−2.92801e30	−0.130281
$969$	−2.17033e31	−0.955268
$970$	0	0
$971$	−4.60425e31	−1.98316	−0.991578	−	0.129508i	$-0.958660\pi$
$971$	−4.60425e31	−1.98316	−0.991578	+	0.129508i	$0.958660\pi$
$972$	1.29309e30	0.0550976
$973$	−3.91632e30	−0.165080
$974$	−3.42925e30	−0.142998
$975$	0	0
$976$	3.28412e30	0.134028
$977$	−7.20727e30	−0.290990	−0.145495	−	0.989359i	$-0.546477\pi$
$977$	−7.20727e30	−0.290990	−0.145495	+	0.989359i	$0.546477\pi$
$978$	2.82376e29	0.0112790
$979$	−1.06491e31	−0.420817
$980$	0	0
$981$	−5.81333e30	−0.224854
$982$	−1.99850e30	−0.0764776
$983$	−1.58951e31	−0.601800	−0.300900	−	0.953656i	$-0.597287\pi$
$983$	−1.58951e31	−0.601800	−0.300900	+	0.953656i	$0.597287\pi$
$984$	−7.28415e30	−0.272854
$985$	0	0
$986$	−3.61206e29	−0.0132448
$987$	1.20764e31	0.438134
$988$	2.23672e31	0.802905
$989$	−3.61657e31	−1.28450
$990$	0	0
$991$	4.62622e31	1.60862	0.804309	−	0.594211i	$-0.202536\pi$
$991$	4.62622e31	1.60862	0.804309	+	0.594211i	$0.202536\pi$
$992$	−5.24095e31	−1.80317
$993$	6.44905e30	0.219548
$994$	−7.67580e30	−0.258563
$995$	0	0
$996$	1.54390e31	0.509210
$997$	1.13237e31	0.369562	0.184781	−	0.982780i	$-0.440842\pi$
$997$	1.13237e31	0.369562	0.184781	+	0.982780i	$0.440842\pi$
$998$	1.08848e31	0.351519
$999$	9.05451e30	0.289352

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.22.a.b.1.1		1
5.2	odd	4		75.22.b.c.49.1		2
5.3	odd	4		75.22.b.c.49.2		2
5.4	even	2		15.22.a.a.1.1	✓	1
15.14	odd	2		45.22.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.22.a.a.1.1	✓	1	5.4	even	2
45.22.a.a.1.1		1	15.14	odd	2
75.22.a.b.1.1		1	1.1	even	1	trivial
75.22.b.c.49.1		2	5.2	odd	4
75.22.b.c.49.2		2	5.3	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:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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