LMFDB - Embedded newform 75.22.a.c.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:	$0$
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)

$f(q)$

$=$

$q+2844.00 q^{2} +59049.0 q^{3} +5.99118e6 q^{4} +1.67935e8 q^{6} -3.63304e8 q^{7} +1.10746e10 q^{8} +3.48678e9 q^{9} +O(q^{10})$

\(q+2844.00 q^{2} +59049.0 q^{3} +5.99118e6 q^{4} +1.67935e8 q^{6} -3.63304e8 q^{7} +1.10746e10 q^{8} +3.48678e9 q^{9} +1.45818e10 q^{11} +3.53773e11 q^{12} -1.13351e11 q^{13} -1.03324e12 q^{14} +1.89318e13 q^{16} +8.58939e12 q^{17} +9.91641e12 q^{18} -2.92029e13 q^{19} -2.14527e13 q^{21} +4.14707e13 q^{22} +1.55899e14 q^{23} +6.53946e14 q^{24} -3.22370e14 q^{26} +2.05891e14 q^{27} -2.17662e15 q^{28} +2.40079e15 q^{29} +2.23982e15 q^{31} +3.06169e16 q^{32} +8.61043e14 q^{33} +2.44282e16 q^{34} +2.08900e16 q^{36} +3.07851e16 q^{37} -8.30532e16 q^{38} -6.69325e15 q^{39} -1.03208e17 q^{41} -6.10116e16 q^{42} +1.65557e17 q^{43} +8.73624e16 q^{44} +4.43377e17 q^{46} +6.65872e16 q^{47} +1.11790e18 q^{48} -4.26556e17 q^{49} +5.07195e17 q^{51} -6.79105e17 q^{52} -4.35423e17 q^{53} +5.85554e17 q^{54} -4.02346e18 q^{56} -1.72440e18 q^{57} +6.82784e18 q^{58} +5.53437e18 q^{59} -7.17621e18 q^{61} +6.37005e18 q^{62} -1.26676e18 q^{63} +4.73716e19 q^{64} +2.44881e18 q^{66} +1.57554e19 q^{67} +5.14606e19 q^{68} +9.20569e18 q^{69} +2.64579e19 q^{71} +3.86148e19 q^{72} -1.34712e19 q^{73} +8.75527e19 q^{74} -1.74960e20 q^{76} -5.29764e18 q^{77} -1.90356e19 q^{78} -1.68861e19 q^{79} +1.21577e19 q^{81} -2.93522e20 q^{82} +1.70688e20 q^{83} -1.28527e20 q^{84} +4.70845e20 q^{86} +1.41764e20 q^{87} +1.61488e20 q^{88} -3.12592e20 q^{89} +4.11808e19 q^{91} +9.34021e20 q^{92} +1.32259e20 q^{93} +1.89374e20 q^{94} +1.80790e21 q^{96} -9.49015e20 q^{97} -1.21313e21 q^{98} +5.08437e19 q^{99} +O(q^{100})\)

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 $

$2$	2844.00	1.96388	0.981939	−	0.189196i	$-0.0605882\pi$
$2$	2844.00	1.96388	0.981939	+	0.189196i	$0.0605882\pi$
$3$	59049.0	0.577350
$3$	59049.0	0.577350
$4$	5.99118e6	2.85682
$5$	0	0
$5$	0	0
$6$	1.67935e8	1.13385
$7$	−3.63304e8	−0.486117	−0.243058	−	0.970012i	$-0.578151\pi$
$7$	−3.63304e8	−0.486117	−0.243058	+	0.970012i	$0.578151\pi$
$8$	1.10746e10	3.64657
$9$	3.48678e9	0.333333
$10$	0	0
$11$	1.45818e10	0.169508	0.0847538	−	0.996402i	$-0.472990\pi$
$11$	1.45818e10	0.169508	0.0847538	+	0.996402i	$0.472990\pi$
$12$	3.53773e11	1.64939
$13$	−1.13351e11	−0.228044	−0.114022	−	0.993478i	$-0.536373\pi$
$13$	−1.13351e11	−0.228044	−0.114022	+	0.993478i	$0.536373\pi$
$14$	−1.03324e12	−0.954674
$15$	0	0
$16$	1.89318e13	4.30460
$17$	8.58939e12	1.03335	0.516676	−	0.856181i	$-0.327169\pi$
$17$	8.58939e12	1.03335	0.516676	+	0.856181i	$0.327169\pi$
$18$	9.91641e12	0.654626
$19$	−2.92029e13	−1.09273	−0.546366	−	0.837546i	$-0.683989\pi$
$19$	−2.92029e13	−1.09273	−0.546366	+	0.837546i	$0.683989\pi$
$20$	0	0
$21$	−2.14527e13	−0.280660
$22$	4.14707e13	0.332892
$23$	1.55899e14	0.784696	0.392348	−	0.919817i	$-0.371663\pi$
$23$	1.55899e14	0.784696	0.392348	+	0.919817i	$0.371663\pi$
$24$	6.53946e14	2.10535
$25$	0	0
$26$	−3.22370e14	−0.447851
$27$	2.05891e14	0.192450
$28$	−2.17662e15	−1.38875
$29$	2.40079e15	1.05968	0.529840	−	0.848097i	$-0.322252\pi$
$29$	2.40079e15	1.05968	0.529840	+	0.848097i	$0.322252\pi$
$30$	0	0
$31$	2.23982e15	0.490812	0.245406	−	0.969420i	$-0.421079\pi$
$31$	2.23982e15	0.490812	0.245406	+	0.969420i	$0.421079\pi$
$32$	3.06169e16	4.80714
$33$	8.61043e14	0.0978652
$34$	2.44282e16	2.02938
$35$	0	0
$36$	2.08900e16	0.952273
$37$	3.07851e16	1.05250	0.526250	−	0.850330i	$-0.323598\pi$
$37$	3.07851e16	1.05250	0.526250	+	0.850330i	$0.323598\pi$
$38$	−8.30532e16	−2.14599
$39$	−6.69325e15	−0.131661
$40$	0	0
$41$	−1.03208e17	−1.20083	−0.600414	−	0.799689i	$-0.704998\pi$
$41$	−1.03208e17	−1.20083	−0.600414	+	0.799689i	$0.704998\pi$
$42$	−6.10116e16	−0.551182
$43$	1.65557e17	1.16823	0.584117	−	0.811670i	$-0.301441\pi$
$43$	1.65557e17	1.16823	0.584117	+	0.811670i	$0.301441\pi$
$44$	8.73624e16	0.484252
$45$	0	0
$46$	4.43377e17	1.54105
$47$	6.65872e16	0.184656	0.0923280	−	0.995729i	$-0.470569\pi$
$47$	6.65872e16	0.184656	0.0923280	+	0.995729i	$0.470569\pi$
$48$	1.11790e18	2.48526
$49$	−4.26556e17	−0.763690
$50$	0	0
$51$	5.07195e17	0.596607
$52$	−6.79105e17	−0.651482
$53$	−4.35423e17	−0.341991	−0.170995	−	0.985272i	$-0.554698\pi$
$53$	−4.35423e17	−0.341991	−0.170995	+	0.985272i	$0.554698\pi$
$54$	5.85554e17	0.377949
$55$	0	0
$56$	−4.02346e18	−1.77266
$57$	−1.72440e18	−0.630889
$58$	6.82784e18	2.08108
$59$	5.53437e18	1.40968	0.704842	−	0.709364i	$-0.251018\pi$
$59$	5.53437e18	1.40968	0.704842	+	0.709364i	$0.251018\pi$
$60$	0	0
$61$	−7.17621e18	−1.28805	−0.644023	−	0.765006i	$-0.722736\pi$
$61$	−7.17621e18	−1.28805	−0.644023	+	0.765006i	$0.722736\pi$
$62$	6.37005e18	0.963896
$63$	−1.26676e18	−0.162039
$64$	4.73716e19	5.13604
$65$	0	0
$66$	2.44881e18	0.192195
$67$	1.57554e19	1.05595	0.527977	−	0.849258i	$-0.322951\pi$
$67$	1.57554e19	1.05595	0.527977	+	0.849258i	$0.322951\pi$
$68$	5.14606e19	2.95210
$69$	9.20569e18	0.453045
$70$	0	0
$71$	2.64579e19	0.964588	0.482294	−	0.876009i	$-0.339804\pi$
$71$	2.64579e19	0.964588	0.482294	+	0.876009i	$0.339804\pi$
$72$	3.86148e19	1.21552
$73$	−1.34712e19	−0.366875	−0.183437	−	0.983031i	$-0.558722\pi$
$73$	−1.34712e19	−0.366875	−0.183437	+	0.983031i	$0.558722\pi$
$74$	8.75527e19	2.06698
$75$	0	0
$76$	−1.74960e20	−3.12174
$77$	−5.29764e18	−0.0824005
$78$	−1.90356e19	−0.258567
$79$	−1.68861e19	−0.200653	−0.100326	−	0.994955i	$-0.531989\pi$
$79$	−1.68861e19	−0.200653	−0.100326	+	0.994955i	$0.531989\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	−2.93522e20	−2.35828
$83$	1.70688e20	1.20749	0.603744	−	0.797178i	$-0.293675\pi$
$83$	1.70688e20	1.20749	0.603744	+	0.797178i	$0.293675\pi$
$84$	−1.28527e20	−0.801794
$85$	0	0
$86$	4.70845e20	2.29427
$87$	1.41764e20	0.611807
$88$	1.61488e20	0.618121
$89$	−3.12592e20	−1.06263	−0.531317	−	0.847173i	$-0.678303\pi$
$89$	−3.12592e20	−1.06263	−0.531317	+	0.847173i	$0.678303\pi$
$90$	0	0
$91$	4.11808e19	0.110856
$92$	9.34021e20	2.24174
$93$	1.32259e20	0.283371
$94$	1.89374e20	0.362642
$95$	0	0
$96$	1.80790e21	2.77540
$97$	−9.49015e20	−1.30668	−0.653341	−	0.757064i	$-0.726633\pi$
$97$	−9.49015e20	−1.30668	−0.653341	+	0.757064i	$0.726633\pi$
$98$	−1.21313e21	−1.49980
$99$	5.08437e19	0.0565025
$100$	0	0
$101$	1.44798e20	0.130433	0.0652166	−	0.997871i	$-0.479226\pi$
$101$	1.44798e20	0.130433	0.0652166	+	0.997871i	$0.479226\pi$
$102$	1.44246e21	1.17166
$103$	−2.19627e21	−1.61025	−0.805127	−	0.593102i	$-0.797903\pi$
$103$	−2.19627e21	−1.61025	−0.805127	+	0.593102i	$0.797903\pi$
$104$	−1.25532e21	−0.831579
$105$	0	0
$106$	−1.23834e21	−0.671629
$107$	1.63087e20	0.0801473	0.0400737	−	0.999197i	$-0.487241\pi$
$107$	1.63087e20	0.0801473	0.0400737	+	0.999197i	$0.487241\pi$
$108$	1.23353e21	0.549795
$109$	2.24852e20	0.0909743	0.0454871	−	0.998965i	$-0.485516\pi$
$109$	2.24852e20	0.0909743	0.0454871	+	0.998965i	$0.485516\pi$
$110$	0	0
$111$	1.81783e21	0.607661
$112$	−6.87800e21	−2.09254
$113$	4.24118e21	1.17534	0.587670	−	0.809101i	$-0.300045\pi$
$113$	4.24118e21	1.17534	0.587670	+	0.809101i	$0.300045\pi$
$114$	−4.90421e21	−1.23899
$115$	0	0
$116$	1.43836e22	3.02732
$117$	−3.95230e20	−0.0760148
$118$	1.57397e22	2.76845
$119$	−3.12056e21	−0.502330
$120$	0	0
$121$	−7.18762e21	−0.971267
$122$	−2.04091e22	−2.52957
$123$	−6.09430e21	−0.693299
$124$	1.34192e22	1.40216
$125$	0	0
$126$	−3.60267e21	−0.318225
$127$	−1.66312e21	−0.135202	−0.0676012	−	0.997712i	$-0.521535\pi$
$127$	−1.66312e21	−0.135202	−0.0676012	+	0.997712i	$0.521535\pi$
$128$	7.05165e22	5.27942
$129$	9.77599e21	0.674480
$130$	0	0
$131$	6.40663e21	0.376081	0.188040	−	0.982161i	$-0.439786\pi$
$131$	6.40663e21	0.376081	0.188040	+	0.982161i	$0.439786\pi$
$132$	5.15867e21	0.279583
$133$	1.06095e22	0.531196
$134$	4.48085e22	2.07377
$135$	0	0
$136$	9.51243e22	3.76819
$137$	1.98314e22	0.727423	0.363711	−	0.931512i	$-0.381509\pi$
$137$	1.98314e22	0.727423	0.363711	+	0.931512i	$0.381509\pi$
$138$	2.61810e22	0.889725
$139$	−5.20143e21	−0.163858	−0.0819290	−	0.996638i	$-0.526108\pi$
$139$	−5.20143e21	−0.163858	−0.0819290	+	0.996638i	$0.526108\pi$
$140$	0	0
$141$	3.93191e21	0.106611
$142$	7.52461e22	1.89433
$143$	−1.65286e21	−0.0386552
$144$	6.60112e22	1.43487
$145$	0	0
$146$	−3.83122e22	−0.720498
$147$	−2.51877e22	−0.440917
$148$	1.84439e23	3.00680
$149$	−7.47631e22	−1.13562	−0.567808	−	0.823161i	$-0.692208\pi$
$149$	−7.47631e22	−1.13562	−0.567808	+	0.823161i	$0.692208\pi$
$150$	0	0
$151$	1.11044e23	1.46635	0.733174	−	0.680042i	$-0.238038\pi$
$151$	1.11044e23	1.46635	0.733174	+	0.680042i	$0.238038\pi$
$152$	−3.23412e23	−3.98472
$153$	2.99493e22	0.344451
$154$	−1.50665e22	−0.161824
$155$	0	0
$156$	−4.01005e22	−0.376133
$157$	−4.36563e22	−0.382913	−0.191457	−	0.981501i	$-0.561321\pi$
$157$	−4.36563e22	−0.382913	−0.191457	+	0.981501i	$0.561321\pi$
$158$	−4.80241e22	−0.394058
$159$	−2.57113e22	−0.197449
$160$	0	0
$161$	−5.66388e22	−0.381454
$162$	3.45764e22	0.218209
$163$	−2.85661e23	−1.68998	−0.844989	−	0.534783i	$-0.820393\pi$
$163$	−2.85661e23	−1.68998	−0.844989	+	0.534783i	$0.820393\pi$
$164$	−6.18336e23	−3.43055
$165$	0	0
$166$	4.85437e23	2.37136
$167$	−2.66950e23	−1.22435	−0.612177	−	0.790720i	$-0.709706\pi$
$167$	−2.66950e23	−1.22435	−0.612177	+	0.790720i	$0.709706\pi$
$168$	−2.37581e23	−1.02344
$169$	−2.34216e23	−0.947996
$170$	0	0
$171$	−1.01824e23	−0.364244
$172$	9.91884e23	3.33743
$173$	2.28496e23	0.723425	0.361713	−	0.932290i	$-0.382192\pi$
$173$	2.28496e23	0.723425	0.361713	+	0.932290i	$0.382192\pi$
$174$	4.03177e23	1.20151
$175$	0	0
$176$	2.76061e23	0.729661
$177$	3.26799e23	0.813881
$178$	−8.89013e23	−2.08688
$179$	−1.29151e22	−0.0285852	−0.0142926	−	0.999898i	$-0.504550\pi$
$179$	−1.29151e22	−0.0285852	−0.0142926	+	0.999898i	$0.504550\pi$
$180$	0	0
$181$	−8.75338e23	−1.72405	−0.862026	−	0.506863i	$-0.830805\pi$
$181$	−8.75338e23	−1.72405	−0.862026	+	0.506863i	$0.830805\pi$
$182$	1.17118e23	0.217708
$183$	−4.23748e23	−0.743654
$184$	1.72653e24	2.86145
$185$	0	0
$186$	3.76145e23	0.556506
$187$	1.25249e23	0.175161
$188$	3.98936e23	0.527529
$189$	−7.48011e22	−0.0935532
$190$	0	0
$191$	1.33961e24	1.50013	0.750066	−	0.661363i	$-0.230022\pi$
$191$	1.33961e24	1.50013	0.750066	+	0.661363i	$0.230022\pi$
$192$	2.79725e24	2.96529
$193$	2.13970e23	0.214783	0.107392	−	0.994217i	$-0.465750\pi$
$193$	2.13970e23	0.214783	0.107392	+	0.994217i	$0.465750\pi$
$194$	−2.69900e24	−2.56616
$195$	0	0
$196$	−2.55558e24	−2.18173
$197$	−1.42895e24	−1.15644	−0.578220	−	0.815881i	$-0.696252\pi$
$197$	−1.42895e24	−1.15644	−0.578220	+	0.815881i	$0.696252\pi$
$198$	1.44600e23	0.110964
$199$	7.45856e23	0.542872	0.271436	−	0.962456i	$-0.412501\pi$
$199$	7.45856e23	0.542872	0.271436	+	0.962456i	$0.412501\pi$
$200$	0	0
$201$	9.30344e23	0.609656
$202$	4.11805e23	0.256155
$203$	−8.72216e23	−0.515129
$204$	3.03870e24	1.70440
$205$	0	0
$206$	−6.24619e24	−3.16234
$207$	5.43587e23	0.261565
$208$	−2.14594e24	−0.981639
$209$	−4.25832e23	−0.185226
$210$	0	0
$211$	−2.65090e24	−1.04334	−0.521671	−	0.853147i	$-0.674691\pi$
$211$	−2.65090e24	−1.04334	−0.521671	+	0.853147i	$0.674691\pi$
$212$	−2.60870e24	−0.977006
$213$	1.56231e24	0.556905
$214$	4.63819e23	0.157400
$215$	0	0
$216$	2.28017e24	0.701782
$217$	−8.13736e23	−0.238592
$218$	6.39479e23	0.178662
$219$	−7.95464e23	−0.211815
$220$	0	0
$221$	−9.73614e23	−0.235650
$222$	5.16990e24	1.19337
$223$	−3.97174e24	−0.874539	−0.437270	−	0.899330i	$-0.644054\pi$
$223$	−3.97174e24	−0.874539	−0.437270	+	0.899330i	$0.644054\pi$
$224$	−1.11232e25	−2.33683
$225$	0	0
$226$	1.20619e25	2.30823
$227$	2.14690e24	0.392229	0.196114	−	0.980581i	$-0.437168\pi$
$227$	2.14690e24	0.392229	0.196114	+	0.980581i	$0.437168\pi$
$228$	−1.03312e25	−1.80234
$229$	3.66024e24	0.609869	0.304934	−	0.952373i	$-0.401365\pi$
$229$	3.66024e24	0.609869	0.304934	+	0.952373i	$0.401365\pi$
$230$	0	0
$231$	−3.12820e23	−0.0475739
$232$	2.65878e25	3.86420
$233$	2.90921e24	0.404145	0.202073	−	0.979371i	$-0.435232\pi$
$233$	2.90921e24	0.404145	0.202073	+	0.979371i	$0.435232\pi$
$234$	−1.12403e24	−0.149284
$235$	0	0
$236$	3.31574e25	4.02721
$237$	−9.97109e23	−0.115847
$238$	−8.87487e24	−0.986516
$239$	−4.80224e24	−0.510818	−0.255409	−	0.966833i	$-0.582210\pi$
$239$	−4.80224e24	−0.510818	−0.255409	+	0.966833i	$0.582210\pi$
$240$	0	0
$241$	7.86263e24	0.766282	0.383141	−	0.923690i	$-0.374842\pi$
$241$	7.86263e24	0.766282	0.383141	+	0.923690i	$0.374842\pi$
$242$	−2.04416e25	−1.90745
$243$	7.17898e23	0.0641500
$244$	−4.29940e25	−3.67972
$245$	0	0
$246$	−1.73322e25	−1.36155
$247$	3.31018e24	0.249191
$248$	2.48052e25	1.78978
$249$	1.00790e25	0.697144
$250$	0	0
$251$	−1.99525e25	−1.26889	−0.634443	−	0.772969i	$-0.718771\pi$
$251$	−1.99525e25	−1.26889	−0.634443	+	0.772969i	$0.718771\pi$
$252$	−7.58941e24	−0.462916
$253$	2.27330e24	0.133012
$254$	−4.72991e24	−0.265521
$255$	0	0
$256$	1.01203e26	5.23210
$257$	−1.33580e25	−0.662894	−0.331447	−	0.943474i	$-0.607537\pi$
$257$	−1.33580e25	−0.662894	−0.331447	+	0.943474i	$0.607537\pi$
$258$	2.78029e25	1.32460
$259$	−1.11843e25	−0.511638
$260$	0	0
$261$	8.37103e24	0.353227
$262$	1.82204e25	0.738576
$263$	5.83637e24	0.227304	0.113652	−	0.993521i	$-0.463745\pi$
$263$	5.83637e24	0.227304	0.113652	+	0.993521i	$0.463745\pi$
$264$	9.53573e24	0.356872
$265$	0	0
$266$	3.01735e25	1.04320
$267$	−1.84583e25	−0.613512
$268$	9.43938e25	3.01667
$269$	5.35297e25	1.64511	0.822557	−	0.568683i	$-0.192547\pi$
$269$	5.35297e25	1.64511	0.822557	+	0.568683i	$0.192547\pi$
$270$	0	0
$271$	−1.04403e25	−0.296849	−0.148425	−	0.988924i	$-0.547420\pi$
$271$	−1.04403e25	−0.296849	−0.148425	+	0.988924i	$0.547420\pi$
$272$	1.62613e26	4.44817
$273$	2.43168e24	0.0640029
$274$	5.64004e25	1.42857
$275$	0	0
$276$	5.51530e25	1.29427
$277$	−3.14884e25	−0.711399	−0.355699	−	0.934600i	$-0.615757\pi$
$277$	−3.14884e25	−0.711399	−0.355699	+	0.934600i	$0.615757\pi$
$278$	−1.47929e25	−0.321797
$279$	7.80977e24	0.163604
$280$	0	0
$281$	−1.15887e25	−0.225225	−0.112613	−	0.993639i	$-0.535922\pi$
$281$	−1.15887e25	−0.225225	−0.112613	+	0.993639i	$0.535922\pi$
$282$	1.11823e25	0.209371
$283$	4.80399e25	0.866652	0.433326	−	0.901237i	$-0.357340\pi$
$283$	4.80399e25	0.866652	0.433326	+	0.901237i	$0.357340\pi$
$284$	1.58514e26	2.75565
$285$	0	0
$286$	−4.70074e24	−0.0759142
$287$	3.74957e25	0.583743
$288$	1.06755e26	1.60238
$289$	4.68568e24	0.0678180
$290$	0	0
$291$	−5.60384e25	−0.754413
$292$	−8.07087e25	−1.04809
$293$	7.96714e25	0.998142	0.499071	−	0.866561i	$-0.333675\pi$
$293$	7.96714e25	0.998142	0.499071	+	0.866561i	$0.333675\pi$
$294$	−7.16339e25	−0.865907
$295$	0	0
$296$	3.40933e26	3.83801
$297$	3.00227e24	0.0326217
$298$	−2.12626e26	−2.23021
$299$	−1.76713e25	−0.178946
$300$	0	0
$301$	−6.01476e25	−0.567898
$302$	3.15809e26	2.87973
$303$	8.55018e24	0.0753056
$304$	−5.52865e26	−4.70377
$305$	0	0
$306$	8.51760e25	0.676460
$307$	−1.51498e26	−1.16266	−0.581332	−	0.813666i	$-0.697468\pi$
$307$	−1.51498e26	−1.16266	−0.581332	+	0.813666i	$0.697468\pi$
$308$	−3.17391e25	−0.235403
$309$	−1.29687e26	−0.929681
$310$	0	0
$311$	1.41406e26	0.947292	0.473646	−	0.880715i	$-0.342938\pi$
$311$	1.41406e26	0.947292	0.473646	+	0.880715i	$0.342938\pi$
$312$	−7.41253e25	−0.480112
$313$	−4.99598e25	−0.312899	−0.156450	−	0.987686i	$-0.550005\pi$
$313$	−4.99598e25	−0.312899	−0.156450	+	0.987686i	$0.550005\pi$
$314$	−1.24158e26	−0.751995
$315$	0	0
$316$	−1.01168e26	−0.573229
$317$	−1.81535e26	−0.995033	−0.497517	−	0.867454i	$-0.665755\pi$
$317$	−1.81535e26	−0.995033	−0.497517	+	0.867454i	$0.665755\pi$
$318$	−7.31229e25	−0.387765
$319$	3.50079e25	0.179624
$320$	0	0
$321$	9.63011e24	0.0462731
$322$	−1.61081e26	−0.749130
$323$	−2.50835e26	−1.12918
$324$	7.28388e25	0.317424
$325$	0	0
$326$	−8.12420e26	−3.31891
$327$	1.32773e25	0.0525240
$328$	−1.14299e27	−4.37890
$329$	−2.41914e25	−0.0897644
$330$	0	0
$331$	1.44090e26	0.501695	0.250848	−	0.968027i	$-0.419291\pi$
$331$	1.44090e26	0.501695	0.250848	+	0.968027i	$0.419291\pi$
$332$	1.02262e27	3.44958
$333$	1.07341e26	0.350833
$334$	−7.59206e26	−2.40448
$335$	0	0
$336$	−4.06139e26	−1.20813
$337$	3.63051e26	1.04678	0.523388	−	0.852095i	$-0.324668\pi$
$337$	3.63051e26	1.04678	0.523388	+	0.852095i	$0.324668\pi$
$338$	−6.66111e26	−1.86175
$339$	2.50438e26	0.678583
$340$	0	0
$341$	3.26607e25	0.0831964
$342$	−2.89588e26	−0.715331
$343$	3.57891e26	0.857360
$344$	1.83349e27	4.26004
$345$	0	0
$346$	6.49841e26	1.42072
$347$	7.09622e25	0.150511	0.0752554	−	0.997164i	$-0.476023\pi$
$347$	7.09622e25	0.150511	0.0752554	+	0.997164i	$0.476023\pi$
$348$	8.49335e26	1.74782
$349$	−7.03939e26	−1.40562	−0.702810	−	0.711378i	$-0.748072\pi$
$349$	−7.03939e26	−1.40562	−0.702810	+	0.711378i	$0.748072\pi$
$350$	0	0
$351$	−2.33379e25	−0.0438872
$352$	4.46451e26	0.814846
$353$	1.08085e26	0.191483	0.0957414	−	0.995406i	$-0.469478\pi$
$353$	1.08085e26	0.191483	0.0957414	+	0.995406i	$0.469478\pi$
$354$	9.29416e26	1.59836
$355$	0	0
$356$	−1.87280e27	−3.03575
$357$	−1.84266e26	−0.290020
$358$	−3.67305e25	−0.0561378
$359$	−1.67492e26	−0.248601	−0.124301	−	0.992245i	$-0.539669\pi$
$359$	−1.67492e26	−0.248601	−0.124301	+	0.992245i	$0.539669\pi$
$360$	0	0
$361$	1.38602e26	0.194064
$362$	−2.48946e27	−3.38583
$363$	−4.24422e26	−0.560761
$364$	2.46722e26	0.316696
$365$	0	0
$366$	−1.20514e27	−1.46045
$367$	−9.83667e26	−1.15839	−0.579194	−	0.815190i	$-0.696633\pi$
$367$	−9.83667e26	−1.15839	−0.579194	+	0.815190i	$0.696633\pi$
$368$	2.95146e27	3.37780
$369$	−3.59863e26	−0.400276
$370$	0	0
$371$	1.58191e26	0.166248
$372$	7.92389e26	0.809539
$373$	−1.00058e26	−0.0993824	−0.0496912	−	0.998765i	$-0.515824\pi$
$373$	−1.00058e26	−0.0993824	−0.0496912	+	0.998765i	$0.515824\pi$
$374$	3.56208e26	0.343995
$375$	0	0
$376$	7.37429e26	0.673360
$377$	−2.72131e26	−0.241654
$378$	−2.12734e26	−0.183727
$379$	9.23905e25	0.0776096	0.0388048	−	0.999247i	$-0.487645\pi$
$379$	9.23905e25	0.0776096	0.0388048	+	0.999247i	$0.487645\pi$
$380$	0	0
$381$	−9.82055e25	−0.0780591
$382$	3.80986e27	2.94608
$383$	2.13677e27	1.60757	0.803786	−	0.594919i	$-0.202816\pi$
$383$	2.13677e27	1.60757	0.803786	+	0.594919i	$0.202816\pi$
$384$	4.16393e27	3.04807
$385$	0	0
$386$	6.08530e26	0.421809
$387$	5.77263e26	0.389411
$388$	−5.68572e27	−3.73295
$389$	−1.25581e27	−0.802516	−0.401258	−	0.915965i	$-0.631427\pi$
$389$	−1.25581e27	−0.802516	−0.401258	+	0.915965i	$0.631427\pi$
$390$	0	0
$391$	1.33908e27	0.810868
$392$	−4.72395e27	−2.78485
$393$	3.78305e26	0.217130
$394$	−4.06394e27	−2.27111
$395$	0	0
$396$	3.04614e26	0.161417
$397$	−1.78165e27	−0.919439	−0.459719	−	0.888064i	$-0.652050\pi$
$397$	−1.78165e27	−0.919439	−0.459719	+	0.888064i	$0.652050\pi$
$398$	2.12121e27	1.06614
$399$	6.26483e26	0.306686
$400$	0	0
$401$	1.40181e27	0.651141	0.325570	−	0.945518i	$-0.394444\pi$
$401$	1.40181e27	0.651141	0.325570	+	0.945518i	$0.394444\pi$
$402$	2.64590e27	1.19729
$403$	−2.53885e26	−0.111927
$404$	8.67511e26	0.372624
$405$	0	0
$406$	−2.48058e27	−1.01165
$407$	4.48903e26	0.178407
$408$	5.61699e27	2.17557
$409$	2.17493e27	0.821015	0.410507	−	0.911857i	$-0.365352\pi$
$409$	2.17493e27	0.821015	0.410507	+	0.911857i	$0.365352\pi$
$410$	0	0
$411$	1.17102e27	0.419978
$412$	−1.31582e28	−4.60021
$413$	−2.01066e27	−0.685271
$414$	1.54596e27	0.513683
$415$	0	0
$416$	−3.47045e27	−1.09624
$417$	−3.07139e26	−0.0946034
$418$	−1.21107e27	−0.363762
$419$	−6.07636e27	−1.77990	−0.889952	−	0.456055i	$-0.849262\pi$
$419$	−6.07636e27	−1.77990	−0.889952	+	0.456055i	$0.849262\pi$
$420$	0	0
$421$	−1.89993e27	−0.529389	−0.264695	−	0.964332i	$-0.585271\pi$
$421$	−1.89993e27	−0.529389	−0.264695	+	0.964332i	$0.585271\pi$
$422$	−7.53915e27	−2.04900
$423$	2.32175e26	0.0615520
$424$	−4.82214e27	−1.24709
$425$	0	0
$426$	4.44321e27	1.09369
$427$	2.60714e27	0.626141
$428$	9.77083e26	0.228966
$429$	−9.75999e25	−0.0223176
$430$	0	0
$431$	−8.08572e24	−0.00176079	−0.000880395	−	1.00000i	$-0.500280\pi$
$431$	−8.08572e24	−0.00176079	−0.000880395		1.00000i	$0.500280\pi$
$432$	3.89789e27	0.828420
$433$	5.60439e27	1.16253	0.581267	−	0.813713i	$-0.302557\pi$
$433$	5.60439e27	1.16253	0.581267	+	0.813713i	$0.302557\pi$
$434$	−2.31426e27	−0.468566
$435$	0	0
$436$	1.34713e27	0.259897
$437$	−4.55272e27	−0.857463
$438$	−2.26230e27	−0.415979
$439$	−8.51110e27	−1.52795	−0.763973	−	0.645248i	$-0.776754\pi$
$439$	−8.51110e27	−1.52795	−0.763973	+	0.645248i	$0.776754\pi$
$440$	0	0
$441$	−1.48731e27	−0.254563
$442$	−2.76896e27	−0.462789
$443$	6.63134e27	1.08234	0.541168	−	0.840915i	$-0.317982\pi$
$443$	6.63134e27	1.08234	0.541168	+	0.840915i	$0.317982\pi$
$444$	1.08909e28	1.73598
$445$	0	0
$446$	−1.12956e28	−1.71749
$447$	−4.41468e27	−0.655648
$448$	−1.72103e28	−2.49671
$449$	1.30394e28	1.84787	0.923933	−	0.382555i	$-0.124956\pi$
$449$	1.30394e28	1.84787	0.923933	+	0.382555i	$0.124956\pi$
$450$	0	0
$451$	−1.50496e27	−0.203549
$452$	2.54097e28	3.35773
$453$	6.55703e27	0.846596
$454$	6.10577e27	0.770289
$455$	0	0
$456$	−1.90971e28	−2.30058
$457$	−4.72949e27	−0.556793	−0.278397	−	0.960466i	$-0.589803\pi$
$457$	−4.72949e27	−0.556793	−0.278397	+	0.960466i	$0.589803\pi$
$458$	1.04097e28	1.19771
$459$	1.76848e27	0.198869
$460$	0	0
$461$	4.92722e27	0.529349	0.264675	−	0.964338i	$-0.414735\pi$
$461$	4.92722e27	0.529349	0.264675	+	0.964338i	$0.414735\pi$
$462$	−8.89661e26	−0.0934294
$463$	−1.20207e28	−1.23404	−0.617021	−	0.786947i	$-0.711661\pi$
$463$	−1.20207e28	−1.23404	−0.617021	+	0.786947i	$0.711661\pi$
$464$	4.54513e28	4.56150
$465$	0	0
$466$	8.27379e27	0.793693
$467$	−1.09969e28	−1.03144	−0.515719	−	0.856758i	$-0.672475\pi$
$467$	−1.09969e28	−1.03144	−0.515719	+	0.856758i	$0.672475\pi$
$468$	−2.36789e27	−0.217161
$469$	−5.72402e27	−0.513317
$470$	0	0
$471$	−2.57786e27	−0.221075
$472$	6.12910e28	5.14051
$473$	2.41413e27	0.198024
$474$	−2.83578e27	−0.227510
$475$	0	0
$476$	−1.86958e28	−1.43507
$477$	−1.51823e27	−0.113997
$478$	−1.36576e28	−1.00318
$479$	−1.32717e28	−0.953684	−0.476842	−	0.878989i	$-0.658219\pi$
$479$	−1.32717e28	−0.953684	−0.476842	+	0.878989i	$0.658219\pi$
$480$	0	0
$481$	−3.48951e27	−0.240017
$482$	2.23613e28	1.50489
$483$	−3.34446e27	−0.220233
$484$	−4.30624e28	−2.77473
$485$	0	0
$486$	2.04170e27	0.125983
$487$	−2.62576e28	−1.58563	−0.792813	−	0.609466i	$-0.791384\pi$
$487$	−2.62576e28	−1.58563	−0.792813	+	0.609466i	$0.791384\pi$
$488$	−7.94738e28	−4.69695
$489$	−1.68680e28	−0.975710
$490$	0	0
$491$	−2.54066e28	−1.40796	−0.703982	−	0.710218i	$-0.748596\pi$
$491$	−2.54066e28	−1.40796	−0.703982	+	0.710218i	$0.748596\pi$
$492$	−3.65121e28	−1.98063
$493$	2.06213e28	1.09502
$494$	9.41414e27	0.489382
$495$	0	0
$496$	4.24039e28	2.11275
$497$	−9.61224e27	−0.468903
$498$	2.86645e28	1.36911
$499$	−1.30048e28	−0.608204	−0.304102	−	0.952640i	$-0.598356\pi$
$499$	−1.30048e28	−0.608204	−0.304102	+	0.952640i	$0.598356\pi$
$500$	0	0
$501$	−1.57631e28	−0.706882
$502$	−5.67449e28	−2.49194
$503$	−1.34993e27	−0.0580559	−0.0290280	−	0.999579i	$-0.509241\pi$
$503$	−1.34993e27	−0.0580559	−0.0290280	+	0.999579i	$0.509241\pi$
$504$	−1.40289e28	−0.590886
$505$	0	0
$506$	6.46525e27	0.261219
$507$	−1.38302e28	−0.547326
$508$	−9.96406e27	−0.386249
$509$	−4.04902e28	−1.53749	−0.768746	−	0.639554i	$-0.779119\pi$
$509$	−4.04902e28	−1.53749	−0.768746	+	0.639554i	$0.779119\pi$
$510$	0	0
$511$	4.89416e27	0.178344
$512$	1.39939e29	4.99579
$513$	−6.01263e27	−0.210296
$514$	−3.79902e28	−1.30184
$515$	0	0
$516$	5.85698e28	1.92687
$517$	9.70964e26	0.0313006
$518$	−3.18083e28	−1.00479
$519$	1.34924e28	0.417670
$520$	0	0
$521$	5.40378e28	1.60658	0.803289	−	0.595590i	$-0.203082\pi$
$521$	5.40378e28	1.60658	0.803289	+	0.595590i	$0.203082\pi$
$522$	2.38072e28	0.693695
$523$	−1.54066e28	−0.439988	−0.219994	−	0.975501i	$-0.570604\pi$
$523$	−1.54066e28	−0.439988	−0.219994	+	0.975501i	$0.570604\pi$
$524$	3.83833e28	1.07439
$525$	0	0
$526$	1.65986e28	0.446398
$527$	1.92387e28	0.507182
$528$	1.63011e28	0.421270
$529$	−1.51670e28	−0.384252
$530$	0	0
$531$	1.92971e28	0.469895
$532$	6.35637e28	1.51753
$533$	1.16987e28	0.273842
$534$	−5.24953e28	−1.20486
$535$	0	0
$536$	1.74486e29	3.85061
$537$	−7.62623e26	−0.0165037
$538$	1.52238e29	3.23080
$539$	−6.21997e27	−0.129451
$540$	0	0
$541$	−7.54478e28	−1.51034	−0.755171	−	0.655528i	$-0.772446\pi$
$541$	−7.54478e28	−1.51034	−0.755171	+	0.655528i	$0.772446\pi$
$542$	−2.96923e28	−0.582976
$543$	−5.16878e28	−0.995382
$544$	2.62981e29	4.96747
$545$	0	0
$546$	6.91571e27	0.125694
$547$	−7.90524e28	−1.40944	−0.704722	−	0.709483i	$-0.748928\pi$
$547$	−7.90524e28	−1.40944	−0.704722	+	0.709483i	$0.748928\pi$
$548$	1.18813e29	2.07812
$549$	−2.50219e28	−0.429349
$550$	0	0
$551$	−7.01101e28	−1.15795
$552$	1.01950e29	1.65206
$553$	6.13480e27	0.0975408
$554$	−8.95531e28	−1.39710
$555$	0	0
$556$	−3.11627e28	−0.468113
$557$	−1.17729e29	−1.73541	−0.867707	−	0.497075i	$-0.834407\pi$
$557$	−1.17729e29	−1.73541	−0.867707	+	0.497075i	$0.834407\pi$
$558$	2.22110e28	0.321299
$559$	−1.87660e28	−0.266409
$560$	0	0
$561$	7.39583e27	0.101129
$562$	−3.29582e28	−0.442315
$563$	9.20807e28	1.21291	0.606457	−	0.795116i	$-0.292590\pi$
$563$	9.20807e28	1.21291	0.606457	+	0.795116i	$0.292590\pi$
$564$	2.35568e28	0.304569
$565$	0	0
$566$	1.36626e29	1.70200
$567$	−4.41693e27	−0.0540130
$568$	2.93011e29	3.51744
$569$	−2.71795e27	−0.0320304	−0.0160152	−	0.999872i	$-0.505098\pi$
$569$	−2.71795e27	−0.0320304	−0.0160152	+	0.999872i	$0.505098\pi$
$570$	0	0
$571$	1.28086e28	0.145487	0.0727434	−	0.997351i	$-0.476825\pi$
$571$	1.28086e28	0.145487	0.0727434	+	0.997351i	$0.476825\pi$
$572$	−9.90260e27	−0.110431
$573$	7.91029e28	0.866102
$574$	1.06638e29	1.14640
$575$	0	0
$576$	1.65175e29	1.71201
$577$	−1.49329e29	−1.51984	−0.759922	−	0.650015i	$-0.774763\pi$
$577$	−1.49329e29	−1.51984	−0.759922	+	0.650015i	$0.774763\pi$
$578$	1.33261e28	0.133186
$579$	1.26347e28	0.124005
$580$	0	0
$581$	−6.20116e28	−0.586980
$582$	−1.59373e29	−1.48158
$583$	−6.34926e27	−0.0579700
$584$	−1.49189e29	−1.33783
$585$	0	0
$586$	2.26586e29	1.96023
$587$	−2.62975e28	−0.223468	−0.111734	−	0.993738i	$-0.535640\pi$
$587$	−2.62975e28	−0.223468	−0.111734	+	0.993738i	$0.535640\pi$
$588$	−1.50904e29	−1.25962
$589$	−6.54093e28	−0.536326
$590$	0	0
$591$	−8.43783e28	−0.667671
$592$	5.82817e29	4.53059
$593$	−1.92294e29	−1.46856	−0.734278	−	0.678849i	$-0.762479\pi$
$593$	−1.92294e29	−1.46856	−0.734278	+	0.678849i	$0.762479\pi$
$594$	8.53846e27	0.0640651
$595$	0	0
$596$	−4.47919e29	−3.24425
$597$	4.40421e28	0.313428
$598$	−5.02572e28	−0.351427
$599$	−2.45874e29	−1.68940	−0.844699	−	0.535242i	$-0.820220\pi$
$599$	−2.45874e29	−1.68940	−0.844699	+	0.535242i	$0.820220\pi$
$600$	0	0
$601$	7.47252e28	0.495776	0.247888	−	0.968789i	$-0.420264\pi$
$601$	7.47252e28	0.495776	0.247888	+	0.968789i	$0.420264\pi$
$602$	−1.71060e29	−1.11528
$603$	5.49359e28	0.351985
$604$	6.65285e29	4.18909
$605$	0	0
$606$	2.43167e28	0.147891
$607$	1.23466e29	0.738016	0.369008	−	0.929426i	$-0.379698\pi$
$607$	1.23466e29	0.738016	0.369008	+	0.929426i	$0.379698\pi$
$608$	−8.94104e29	−5.25291
$609$	−5.15035e28	−0.297410
$610$	0	0
$611$	−7.54771e27	−0.0421098
$612$	1.79432e29	0.984034
$613$	1.59244e29	0.858476	0.429238	−	0.903191i	$-0.358782\pi$
$613$	1.59244e29	0.858476	0.429238	+	0.903191i	$0.358782\pi$
$614$	−4.30861e29	−2.28333
$615$	0	0
$616$	−5.86694e28	−0.300479
$617$	−1.23256e29	−0.620602	−0.310301	−	0.950638i	$-0.600430\pi$
$617$	−1.23256e29	−0.620602	−0.310301	+	0.950638i	$0.600430\pi$
$618$	−3.68831e29	−1.82578
$619$	−5.18990e28	−0.252585	−0.126292	−	0.991993i	$-0.540308\pi$
$619$	−5.18990e28	−0.252585	−0.126292	+	0.991993i	$0.540308\pi$
$620$	0	0
$621$	3.20983e28	0.151015
$622$	4.02159e29	1.86037
$623$	1.13566e29	0.516564
$624$	−1.26715e29	−0.566750
$625$	0	0
$626$	−1.42086e29	−0.614497
$627$	−2.51450e28	−0.106940
$628$	−2.61553e29	−1.09391
$629$	2.64425e29	1.08760
$630$	0	0
$631$	−2.05208e28	−0.0816366	−0.0408183	−	0.999167i	$-0.512996\pi$
$631$	−2.05208e28	−0.0816366	−0.0408183	+	0.999167i	$0.512996\pi$
$632$	−1.87008e29	−0.731694
$633$	−1.56533e29	−0.602374
$634$	−5.16285e29	−1.95412
$635$	0	0
$636$	−1.54041e29	−0.564075
$637$	4.83505e28	0.174155
$638$	9.95625e28	0.352759
$639$	9.22528e28	0.321529
$640$	0	0
$641$	5.14342e29	1.73477	0.867386	−	0.497636i	$-0.165798\pi$
$641$	5.14342e29	1.73477	0.867386	+	0.497636i	$0.165798\pi$
$642$	2.73880e28	0.0908747
$643$	8.85766e28	0.289137	0.144569	−	0.989495i	$-0.453821\pi$
$643$	8.85766e28	0.289137	0.144569	+	0.989495i	$0.453821\pi$
$644$	−3.39333e29	−1.08975
$645$	0	0
$646$	−7.13376e29	−2.21757
$647$	−5.46916e29	−1.67273	−0.836364	−	0.548174i	$-0.815323\pi$
$647$	−5.46916e29	−1.67273	−0.836364	+	0.548174i	$0.815323\pi$
$648$	1.34642e29	0.405174
$649$	8.07012e28	0.238952
$650$	0	0
$651$	−4.80503e28	−0.137751
$652$	−1.71145e30	−4.82796
$653$	5.66153e29	1.57161	0.785806	−	0.618473i	$-0.212248\pi$
$653$	5.66153e29	1.57161	0.785806	+	0.618473i	$0.212248\pi$
$654$	3.77606e28	0.103151
$655$	0	0
$656$	−1.95391e30	−5.16908
$657$	−4.69713e28	−0.122292
$658$	−6.88003e28	−0.176286
$659$	−1.48653e29	−0.374867	−0.187434	−	0.982277i	$-0.560017\pi$
$659$	−1.48653e29	−0.374867	−0.187434	+	0.982277i	$0.560017\pi$
$660$	0	0
$661$	−4.04669e29	−0.988517	−0.494259	−	0.869315i	$-0.664560\pi$
$661$	−4.04669e29	−0.988517	−0.494259	+	0.869315i	$0.664560\pi$
$662$	4.09792e29	0.985268
$663$	−5.74909e28	−0.136053
$664$	1.89031e30	4.40319
$665$	0	0
$666$	3.05278e29	0.688994
$667$	3.74281e29	0.831527
$668$	−1.59935e30	−3.49776
$669$	−2.34527e29	−0.504916
$670$	0	0
$671$	−1.04642e29	−0.218334
$672$	−6.56816e29	−1.34917
$673$	1.54590e29	0.312625	0.156313	−	0.987708i	$-0.450039\pi$
$673$	1.54590e29	0.312625	0.156313	+	0.987708i	$0.450039\pi$
$674$	1.03252e30	2.05574
$675$	0	0
$676$	−1.40323e30	−2.70825
$677$	9.88421e29	1.87828	0.939142	−	0.343530i	$-0.111623\pi$
$677$	9.88421e29	1.87828	0.939142	+	0.343530i	$0.111623\pi$
$678$	7.12245e29	1.33265
$679$	3.44781e29	0.635200
$680$	0	0
$681$	1.26772e29	0.226453
$682$	9.28870e28	0.163388
$683$	1.41169e29	0.244524	0.122262	−	0.992498i	$-0.460985\pi$
$683$	1.41169e29	0.244524	0.122262	+	0.992498i	$0.460985\pi$
$684$	−6.10048e29	−1.04058
$685$	0	0
$686$	1.01784e30	1.68375
$687$	2.16133e29	0.352108
$688$	3.13430e30	5.02877
$689$	4.93555e28	0.0779891
$690$	0	0
$691$	7.11585e29	1.09070	0.545352	−	0.838207i	$-0.316396\pi$
$691$	7.11585e29	1.09070	0.545352	+	0.838207i	$0.316396\pi$
$692$	1.36896e30	2.06669
$693$	−1.84717e28	−0.0274668
$694$	2.01817e29	0.295585
$695$	0	0
$696$	1.56999e30	2.23099
$697$	−8.86490e29	−1.24088
$698$	−2.00200e30	−2.76047
$699$	1.71786e29	0.233334
$700$	0	0
$701$	−8.80754e29	−1.16096	−0.580478	−	0.814276i	$-0.697134\pi$
$701$	−8.80754e29	−1.16096	−0.580478	+	0.814276i	$0.697134\pi$
$702$	−6.63731e28	−0.0861891
$703$	−8.99015e29	−1.15010
$704$	6.90765e29	0.870597
$705$	0	0
$706$	3.07393e29	0.376049
$707$	−5.26057e28	−0.0634058
$708$	1.95791e30	2.32511
$709$	−2.79000e29	−0.326452	−0.163226	−	0.986589i	$-0.552190\pi$
$709$	−2.79000e29	−0.326452	−0.163226	+	0.986589i	$0.552190\pi$
$710$	0	0
$711$	−5.88783e28	−0.0668843
$712$	−3.46185e30	−3.87496
$713$	3.49186e29	0.385139
$714$	−5.24052e29	−0.569565
$715$	0	0
$716$	−7.73767e28	−0.0816626
$717$	−2.83568e29	−0.294921
$718$	−4.76349e29	−0.488223
$719$	1.21404e30	1.22625	0.613124	−	0.789987i	$-0.289913\pi$
$719$	1.21404e30	1.22625	0.613124	+	0.789987i	$0.289913\pi$
$720$	0	0
$721$	7.97913e29	0.782772
$722$	3.94185e29	0.381118
$723$	4.64280e29	0.442413
$724$	−5.24431e30	−4.92531
$725$	0	0
$726$	−1.20706e30	−1.10127
$727$	6.54831e29	0.588868	0.294434	−	0.955672i	$-0.404869\pi$
$727$	6.54831e29	0.588868	0.294434	+	0.955672i	$0.404869\pi$
$728$	4.56062e29	0.404245
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	1.42204e30	1.20720
$732$	−2.53875e30	−2.12449
$733$	−2.20665e29	−0.182029	−0.0910147	−	0.995850i	$-0.529011\pi$
$733$	−2.20665e29	−0.182029	−0.0910147	+	0.995850i	$0.529011\pi$
$734$	−2.79755e30	−2.27493
$735$	0	0
$736$	4.77315e30	3.77214
$737$	2.29743e29	0.178992
$738$	−1.02345e30	−0.786094
$739$	−4.07297e29	−0.308422	−0.154211	−	0.988038i	$-0.549283\pi$
$739$	−4.07297e29	−0.308422	−0.154211	+	0.988038i	$0.549283\pi$
$740$	0	0
$741$	1.95463e29	0.143871
$742$	4.49895e29	0.326490
$743$	−3.97218e29	−0.284214	−0.142107	−	0.989851i	$-0.545388\pi$
$743$	−3.97218e29	−0.284214	−0.142107	+	0.989851i	$0.545388\pi$
$744$	1.46472e30	1.03333
$745$	0	0
$746$	−2.84565e29	−0.195175
$747$	5.95152e29	0.402496
$748$	7.50390e29	0.500403
$749$	−5.92500e28	−0.0389610
$750$	0	0
$751$	−1.86890e30	−1.19500	−0.597498	−	0.801871i	$-0.703838\pi$
$751$	−1.86890e30	−1.19500	−0.597498	+	0.801871i	$0.703838\pi$
$752$	1.26062e30	0.794869
$753$	−1.17817e30	−0.732592
$754$	−7.73941e29	−0.474579
$755$	0	0
$756$	−4.48147e29	−0.267265
$757$	2.99103e30	1.75919	0.879597	−	0.475720i	$-0.157812\pi$
$757$	2.99103e30	1.75919	0.879597	+	0.475720i	$0.157812\pi$
$758$	2.62759e29	0.152416
$759$	1.34236e29	0.0767945
$760$	0	0
$761$	1.51341e30	0.842206	0.421103	−	0.907013i	$-0.361643\pi$
$761$	1.51341e30	0.842206	0.421103	+	0.907013i	$0.361643\pi$
$762$	−2.79297e29	−0.153299
$763$	−8.16896e28	−0.0442241
$764$	8.02588e30	4.28561
$765$	0	0
$766$	6.07697e30	3.15707
$767$	−6.27325e29	−0.321470
$768$	5.97596e30	3.02075
$769$	2.53401e30	1.26352	0.631759	−	0.775165i	$-0.282333\pi$
$769$	2.53401e30	1.26352	0.631759	+	0.775165i	$0.282333\pi$
$770$	0	0
$771$	−7.88777e29	−0.382722
$772$	1.28193e30	0.613598
$773$	1.69545e30	0.800571	0.400285	−	0.916390i	$-0.368911\pi$
$773$	1.69545e30	0.800571	0.400285	+	0.916390i	$0.368911\pi$
$774$	1.64173e30	0.764756
$775$	0	0
$776$	−1.05100e31	−4.76490
$777$	−6.60424e29	−0.295394
$778$	−3.57153e30	−1.57604
$779$	3.01396e30	1.31218
$780$	0	0
$781$	3.85804e29	0.163505
$782$	3.80834e30	1.59245
$783$	4.94301e29	0.203936
$784$	−8.07548e30	−3.28738
$785$	0	0
$786$	1.07590e30	0.426417
$787$	1.87332e30	0.732616	0.366308	−	0.930494i	$-0.380622\pi$
$787$	1.87332e30	0.732616	0.366308	+	0.930494i	$0.380622\pi$
$788$	−8.56113e30	−3.30374
$789$	3.44632e29	0.131234
$790$	0	0
$791$	−1.54084e30	−0.571353
$792$	5.63075e29	0.206040
$793$	8.13429e29	0.293732
$794$	−5.06702e30	−1.80567
$795$	0	0
$796$	4.46856e30	1.55089
$797$	7.79664e29	0.267051	0.133526	−	0.991045i	$-0.457370\pi$
$797$	7.79664e29	0.267051	0.133526	+	0.991045i	$0.457370\pi$
$798$	1.78172e30	0.602294
$799$	5.71944e29	0.190815
$800$	0	0
$801$	−1.08994e30	−0.354211
$802$	3.98676e30	1.27876
$803$	−1.96436e29	−0.0621880
$804$	5.57386e30	1.74168
$805$	0	0
$806$	−7.22050e29	−0.219811
$807$	3.16088e30	0.949807
$808$	1.60358e30	0.475633
$809$	8.82262e29	0.258308	0.129154	−	0.991625i	$-0.458774\pi$
$809$	8.82262e29	0.258308	0.129154	+	0.991625i	$0.458774\pi$
$810$	0	0
$811$	−2.06044e30	−0.587815	−0.293907	−	0.955834i	$-0.594956\pi$
$811$	−2.06044e30	−0.587815	−0.293907	+	0.955834i	$0.594956\pi$
$812$	−5.22561e30	−1.47163
$813$	−6.16490e29	−0.171386
$814$	1.27668e30	0.350369
$815$	0	0
$816$	9.60212e30	2.56815
$817$	−4.83476e30	−1.27657
$818$	6.18551e30	1.61237
$819$	1.43589e29	0.0369521
$820$	0	0
$821$	−1.83846e30	−0.461160	−0.230580	−	0.973053i	$-0.574062\pi$
$821$	−1.83846e30	−0.461160	−0.230580	+	0.973053i	$0.574062\pi$
$822$	3.33039e30	0.824785
$823$	−7.73766e29	−0.189196	−0.0945979	−	0.995516i	$-0.530157\pi$
$823$	−7.73766e29	−0.189196	−0.0945979	+	0.995516i	$0.530157\pi$
$824$	−2.43229e31	−5.87190
$825$	0	0
$826$	−5.71831e30	−1.34579
$827$	4.59989e30	1.06891	0.534453	−	0.845198i	$-0.320518\pi$
$827$	4.59989e30	1.06891	0.534453	+	0.845198i	$0.320518\pi$
$828$	3.25673e30	0.747245
$829$	7.93000e30	1.79660	0.898298	−	0.439386i	$-0.144804\pi$
$829$	7.93000e30	1.79660	0.898298	+	0.439386i	$0.144804\pi$
$830$	0	0
$831$	−1.85936e30	−0.410726
$832$	−5.36961e30	−1.17124
$833$	−3.66386e30	−0.789162
$834$	−8.73504e29	−0.185790
$835$	0	0
$836$	−2.55124e30	−0.529158
$837$	4.61159e29	0.0944569
$838$	−1.72812e31	−3.49551
$839$	4.84033e30	0.966884	0.483442	−	0.875376i	$-0.339386\pi$
$839$	4.84033e30	0.966884	0.483442	+	0.875376i	$0.339386\pi$
$840$	0	0
$841$	6.30944e29	0.122923
$842$	−5.40340e30	−1.03966
$843$	−6.84300e29	−0.130034
$844$	−1.58820e31	−2.98064
$845$	0	0
$846$	6.60306e29	0.120881
$847$	2.61129e30	0.472149
$848$	−8.24334e30	−1.47213
$849$	2.83671e30	0.500362
$850$	0	0
$851$	4.79937e30	0.825893
$852$	9.36009e30	1.59098
$853$	−2.96903e30	−0.498483	−0.249242	−	0.968441i	$-0.580181\pi$
$853$	−2.96903e30	−0.498483	−0.249242	+	0.968441i	$0.580181\pi$
$854$	7.41472e30	1.22967
$855$	0	0
$856$	1.80612e30	0.292263
$857$	3.70009e30	0.591444	0.295722	−	0.955274i	$-0.404440\pi$
$857$	3.70009e30	0.591444	0.295722	+	0.955274i	$0.404440\pi$
$858$	−2.77574e29	−0.0438291
$859$	7.75385e29	0.120945	0.0604726	−	0.998170i	$-0.480739\pi$
$859$	7.75385e29	0.120945	0.0604726	+	0.998170i	$0.480739\pi$
$860$	0	0
$861$	2.21408e30	0.337024
$862$	−2.29958e28	−0.00345798
$863$	1.29544e31	1.92443	0.962217	−	0.272283i	$-0.0877786\pi$
$863$	1.29544e31	1.92443	0.962217	+	0.272283i	$0.0877786\pi$
$864$	6.30375e30	0.925134
$865$	0	0
$866$	1.59389e31	2.28307
$867$	2.76685e29	0.0391548
$868$	−4.87524e30	−0.681615
$869$	−2.46231e29	−0.0340122
$870$	0	0
$871$	−1.78589e30	−0.240805
$872$	2.49015e30	0.331744
$873$	−3.30901e30	−0.435560
$874$	−1.29479e31	−1.68395
$875$	0	0
$876$	−4.76577e30	−0.605118
$877$	1.57355e31	1.97417	0.987083	−	0.160207i	$-0.0512162\pi$
$877$	1.57355e31	1.97417	0.987083	+	0.160207i	$0.0512162\pi$
$878$	−2.42056e31	−3.00070
$879$	4.70452e30	0.576278
$880$	0	0
$881$	−1.47526e31	−1.76450	−0.882252	−	0.470778i	$-0.843973\pi$
$881$	−1.47526e31	−1.76450	−0.882252	+	0.470778i	$0.843973\pi$
$882$	−4.22991e30	−0.499932
$883$	5.64453e30	0.659235	0.329617	−	0.944115i	$-0.393080\pi$
$883$	5.64453e30	0.659235	0.329617	+	0.944115i	$0.393080\pi$
$884$	−5.83310e30	−0.673210
$885$	0	0
$886$	1.88595e31	2.12558
$887$	−5.89300e30	−0.656354	−0.328177	−	0.944616i	$-0.606434\pi$
$887$	−5.89300e30	−0.656354	−0.328177	+	0.944616i	$0.606434\pi$
$888$	2.01318e31	2.21588
$889$	6.04218e29	0.0657242
$890$	0	0
$891$	1.77281e29	0.0188342
$892$	−2.37954e31	−2.49840
$893$	−1.94454e30	−0.201780
$894$	−1.25554e31	−1.28761
$895$	0	0
$896$	−2.56189e31	−2.56641
$897$	−1.04347e30	−0.103314
$898$	3.70840e31	3.62898
$899$	5.37734e30	0.520104
$900$	0	0
$901$	−3.74002e30	−0.353397
$902$	−4.28009e30	−0.399746
$903$	−3.55166e30	−0.327876
$904$	4.69695e31	4.28596
$905$	0	0
$906$	1.86482e31	1.66261
$907$	−8.32782e30	−0.733931	−0.366965	−	0.930235i	$-0.619603\pi$
$907$	−8.32782e30	−0.733931	−0.366965	+	0.930235i	$0.619603\pi$
$908$	1.28624e31	1.12053
$909$	5.04879e29	0.0434777
$910$	0	0
$911$	1.08086e31	0.909548	0.454774	−	0.890607i	$-0.349720\pi$
$911$	1.08086e31	0.909548	0.454774	+	0.890607i	$0.349720\pi$
$912$	−3.26461e31	−2.71572
$913$	2.48894e30	0.204678
$914$	−1.34507e31	−1.09347
$915$	0	0
$916$	2.19291e31	1.74229
$917$	−2.32755e30	−0.182819
$918$	5.02955e30	0.390554
$919$	6.55205e30	0.502996	0.251498	−	0.967858i	$-0.419077\pi$
$919$	6.55205e30	0.502996	0.251498	+	0.967858i	$0.419077\pi$
$920$	0	0
$921$	−8.94582e30	−0.671265
$922$	1.40130e31	1.03958
$923$	−2.99902e30	−0.219969
$924$	−1.87416e30	−0.135910
$925$	0	0
$926$	−3.41869e31	−2.42351
$927$	−7.65791e30	−0.536752
$928$	7.35047e31	5.09403
$929$	1.04036e31	0.712883	0.356442	−	0.934318i	$-0.383990\pi$
$929$	1.04036e31	0.712883	0.356442	+	0.934318i	$0.383990\pi$
$930$	0	0
$931$	1.24567e31	0.834509
$932$	1.74296e31	1.15457
$933$	8.34989e30	0.546919
$934$	−3.12752e31	−2.02562
$935$	0	0
$936$	−4.37702e30	−0.277193
$937$	2.09831e31	1.31402	0.657012	−	0.753880i	$-0.271820\pi$
$937$	2.09831e31	1.31402	0.657012	+	0.753880i	$0.271820\pi$
$938$	−1.62791e31	−1.00809
$939$	−2.95007e30	−0.180653
$940$	0	0
$941$	2.20967e31	1.32323	0.661617	−	0.749842i	$-0.269870\pi$
$941$	2.20967e31	1.32323	0.661617	+	0.749842i	$0.269870\pi$
$942$	−7.33143e30	−0.434165
$943$	−1.60900e31	−0.942286
$944$	1.04776e32	6.06812
$945$	0	0
$946$	6.86578e30	0.388896
$947$	2.59604e31	1.45424	0.727121	−	0.686509i	$-0.240858\pi$
$947$	2.59604e31	1.45424	0.727121	+	0.686509i	$0.240858\pi$
$948$	−5.97386e30	−0.330954
$949$	1.52698e30	0.0836637
$950$	0	0
$951$	−1.07194e31	−0.574483
$952$	−3.45590e31	−1.83178
$953$	2.94729e31	1.54507	0.772535	−	0.634973i	$-0.218989\pi$
$953$	2.94729e31	1.54507	0.772535	+	0.634973i	$0.218989\pi$
$954$	−4.31783e30	−0.223876
$955$	0	0
$956$	−2.87711e31	−1.45931
$957$	2.06718e30	0.103706
$958$	−3.77448e31	−1.87292
$959$	−7.20482e30	−0.353612
$960$	0	0
$961$	−1.58087e31	−0.759103
$962$	−9.92417e30	−0.471364
$963$	5.68648e29	0.0267158
$964$	4.71064e31	2.18913
$965$	0	0
$966$	−9.51166e30	−0.432510
$967$	−6.98077e30	−0.313997	−0.156998	−	0.987599i	$-0.550182\pi$
$967$	−6.98077e30	−0.313997	−0.156998	+	0.987599i	$0.550182\pi$
$968$	−7.96002e31	−3.54179
$969$	−1.48116e31	−0.651931
$970$	0	0
$971$	−1.30234e30	−0.0560946	−0.0280473	−	0.999607i	$-0.508929\pi$
$971$	−1.30234e30	−0.0560946	−0.0280473	+	0.999607i	$0.508929\pi$
$972$	4.30106e30	0.183265
$973$	1.88970e30	0.0796541
$974$	−7.46765e31	−3.11398
$975$	0	0
$976$	−1.35859e32	−5.54452
$977$	−3.06599e31	−1.23788	−0.618939	−	0.785439i	$-0.712437\pi$
$977$	−3.06599e31	−1.23788	−0.618939	+	0.785439i	$0.712437\pi$
$978$	−4.79726e31	−1.91618
$979$	−4.55817e30	−0.180124
$980$	0	0
$981$	7.84011e29	0.0303248
$982$	−7.22564e31	−2.76507
$983$	−8.31325e30	−0.314745	−0.157373	−	0.987539i	$-0.550302\pi$
$983$	−8.31325e30	−0.314745	−0.157373	+	0.987539i	$0.550302\pi$
$984$	−6.74921e31	−2.52816
$985$	0	0
$986$	5.86470e31	2.15049
$987$	−1.42848e30	−0.0518255
$988$	1.98319e31	0.711895
$989$	2.58102e31	0.916708
$990$	0	0
$991$	−1.32568e31	−0.460962	−0.230481	−	0.973077i	$-0.574030\pi$
$991$	−1.32568e31	−0.460962	−0.230481	+	0.973077i	$0.574030\pi$
$992$	6.85764e31	2.35940
$993$	8.50837e30	0.289654
$994$	−2.73372e31	−0.920868
$995$	0	0
$996$	6.03849e31	1.99161
$997$	−3.42048e31	−1.11632	−0.558159	−	0.829734i	$-0.688492\pi$
$997$	−3.42048e31	−1.11632	−0.558159	+	0.829734i	$0.688492\pi$
$998$	−3.69857e31	−1.19444
$999$	6.33837e30	0.202554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.22.a.c.1.1		1
5.2	odd	4		75.22.b.a.49.2		2
5.3	odd	4		75.22.b.a.49.1		2
5.4	even	2		3.22.a.a.1.1	✓	1
15.14	odd	2		9.22.a.d.1.1		1
20.19	odd	2		48.22.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.22.a.a.1.1	✓	1	5.4	even	2
9.22.a.d.1.1		1	15.14	odd	2
48.22.a.e.1.1		1	20.19	odd	2
75.22.a.c.1.1		1	1.1	even	1	trivial
75.22.b.a.49.1		2	5.3	odd	4
75.22.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:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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