LMFDB - Embedded newform 75.14.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,14,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, 14, names="a")

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

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 14);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 14 $
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:	$80.4231967139$
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)

$f(q)$

$=$

\(q+12.0000 q^{2} +729.000 q^{3} -8048.00 q^{4} +8748.00 q^{6} -235088. q^{7} -194880. q^{8} +531441. q^{9} -1.11829e7 q^{11} -5.86699e6 q^{12} -8.04961e6 q^{13} -2.82106e6 q^{14} +6.35907e7 q^{16} +1.17495e8 q^{17} +6.37729e6 q^{18} -2.14061e8 q^{19} -1.71379e8 q^{21} -1.34195e8 q^{22} -8.30556e8 q^{23} -1.42068e8 q^{24} -9.65954e7 q^{26} +3.87420e8 q^{27} +1.89199e9 q^{28} -1.25240e9 q^{29} +6.15935e9 q^{31} +2.35954e9 q^{32} -8.15234e9 q^{33} +1.40994e9 q^{34} -4.27704e9 q^{36} +5.49819e9 q^{37} -2.56874e9 q^{38} -5.86817e9 q^{39} -4.67869e9 q^{41} -2.05655e9 q^{42} -7.11501e9 q^{43} +9.00000e10 q^{44} -9.96667e9 q^{46} +2.95288e10 q^{47} +4.63576e10 q^{48} -4.16226e10 q^{49} +8.56536e10 q^{51} +6.47833e10 q^{52} +2.04125e11 q^{53} +4.64905e9 q^{54} +4.58139e10 q^{56} -1.56051e11 q^{57} -1.50288e10 q^{58} -2.99098e10 q^{59} -1.34392e11 q^{61} +7.39122e10 q^{62} -1.24935e11 q^{63} -4.92620e11 q^{64} -9.78281e10 q^{66} -3.48519e11 q^{67} -9.45597e11 q^{68} -6.05475e11 q^{69} +1.31434e12 q^{71} -1.03567e11 q^{72} +1.17888e12 q^{73} +6.59783e10 q^{74} +1.72277e12 q^{76} +2.62897e12 q^{77} -7.04180e10 q^{78} -1.07242e12 q^{79} +2.82430e11 q^{81} -5.61443e10 q^{82} -1.12403e12 q^{83} +1.37926e12 q^{84} -8.53802e10 q^{86} -9.13000e11 q^{87} +2.17933e12 q^{88} +2.23561e12 q^{89} +1.89237e12 q^{91} +6.68431e12 q^{92} +4.49017e12 q^{93} +3.54345e11 q^{94} +1.72011e12 q^{96} +1.42153e13 q^{97} -4.99472e11 q^{98} -5.94306e12 q^{99} +O(q^{100})\)

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 $

$2$	12.0000	0.132583	0.0662913	−	0.997800i	$-0.478883\pi$
$2$	12.0000	0.132583	0.0662913	+	0.997800i	$0.478883\pi$
$3$	729.000	0.577350
$3$	729.000	0.577350
$4$	−8048.00	−0.982422
$5$	0	0
$5$	0	0
$6$	8748.00	0.0765466
$7$	−235088.	−0.755254	−0.377627	−	0.925958i	$-0.623260\pi$
$7$	−235088.	−0.755254	−0.377627	+	0.925958i	$0.623260\pi$
$8$	−194880.	−0.262834
$9$	531441.	0.333333
$10$	0	0
$11$	−1.11829e7	−1.90328	−0.951639	−	0.307218i	$-0.900602\pi$
$11$	−1.11829e7	−1.90328	−0.951639	+	0.307218i	$0.900602\pi$
$12$	−5.86699e6	−0.567202
$13$	−8.04961e6	−0.462534	−0.231267	−	0.972890i	$-0.574287\pi$
$13$	−8.04961e6	−0.462534	−0.231267	+	0.972890i	$0.574287\pi$
$14$	−2.82106e6	−0.100134
$15$	0	0
$16$	6.35907e7	0.947575
$17$	1.17495e8	1.18059	0.590296	−	0.807187i	$-0.299011\pi$
$17$	1.17495e8	1.18059	0.590296	+	0.807187i	$0.299011\pi$
$18$	6.37729e6	0.0441942
$19$	−2.14061e8	−1.04385	−0.521927	−	0.852990i	$-0.674787\pi$
$19$	−2.14061e8	−1.04385	−0.521927	+	0.852990i	$0.674787\pi$
$20$	0	0
$21$	−1.71379e8	−0.436046
$22$	−1.34195e8	−0.252341
$23$	−8.30556e8	−1.16987	−0.584935	−	0.811080i	$-0.698880\pi$
$23$	−8.30556e8	−1.16987	−0.584935	+	0.811080i	$0.698880\pi$
$24$	−1.42068e8	−0.151748
$25$	0	0
$26$	−9.65954e7	−0.0613239
$27$	3.87420e8	0.192450
$28$	1.89199e9	0.741978
$29$	−1.25240e9	−0.390981	−0.195491	−	0.980706i	$-0.562630\pi$
$29$	−1.25240e9	−0.390981	−0.195491	+	0.980706i	$0.562630\pi$
$30$	0	0
$31$	6.15935e9	1.24648	0.623238	−	0.782032i	$-0.285817\pi$
$31$	6.15935e9	1.24648	0.623238	+	0.782032i	$0.285817\pi$
$32$	2.35954e9	0.388466
$33$	−8.15234e9	−1.09886
$34$	1.40994e9	0.156526
$35$	0	0
$36$	−4.27704e9	−0.327474
$37$	5.49819e9	0.352297	0.176148	−	0.984364i	$-0.443636\pi$
$37$	5.49819e9	0.352297	0.176148	+	0.984364i	$0.443636\pi$
$38$	−2.56874e9	−0.138397
$39$	−5.86817e9	−0.267044
$40$	0	0
$41$	−4.67869e9	−0.153826	−0.0769129	−	0.997038i	$-0.524506\pi$
$41$	−4.67869e9	−0.153826	−0.0769129	+	0.997038i	$0.524506\pi$
$42$	−2.05655e9	−0.0578121
$43$	−7.11501e9	−0.171645	−0.0858224	−	0.996310i	$-0.527352\pi$
$43$	−7.11501e9	−0.171645	−0.0858224	+	0.996310i	$0.527352\pi$
$44$	9.00000e10	1.86982
$45$	0	0
$46$	−9.96667e9	−0.155104
$47$	2.95288e10	0.399585	0.199793	−	0.979838i	$-0.435973\pi$
$47$	2.95288e10	0.399585	0.199793	+	0.979838i	$0.435973\pi$
$48$	4.63576e10	0.547082
$49$	−4.16226e10	−0.429591
$50$	0	0
$51$	8.56536e10	0.681615
$52$	6.47833e10	0.454403
$53$	2.04125e11	1.26504	0.632518	−	0.774545i	$-0.282021\pi$
$53$	2.04125e11	1.26504	0.632518	+	0.774545i	$0.282021\pi$
$54$	4.64905e9	0.0255155
$55$	0	0
$56$	4.58139e10	0.198507
$57$	−1.56051e11	−0.602670
$58$	−1.50288e10	−0.0518373
$59$	−2.99098e10	−0.0923157	−0.0461579	−	0.998934i	$-0.514698\pi$
$59$	−2.99098e10	−0.0923157	−0.0461579	+	0.998934i	$0.514698\pi$
$60$	0	0
$61$	−1.34392e11	−0.333987	−0.166993	−	0.985958i	$-0.553406\pi$
$61$	−1.34392e11	−0.333987	−0.166993	+	0.985958i	$0.553406\pi$
$62$	7.39122e10	0.165261
$63$	−1.24935e11	−0.251751
$64$	−4.92620e11	−0.896071
$65$	0	0
$66$	−9.78281e10	−0.145689
$67$	−3.48519e11	−0.470695	−0.235348	−	0.971911i	$-0.575623\pi$
$67$	−3.48519e11	−0.470695	−0.235348	+	0.971911i	$0.575623\pi$
$68$	−9.45597e11	−1.15984
$69$	−6.05475e11	−0.675425
$70$	0	0
$71$	1.31434e12	1.21766	0.608831	−	0.793300i	$-0.291639\pi$
$71$	1.31434e12	1.21766	0.608831	+	0.793300i	$0.291639\pi$
$72$	−1.03567e11	−0.0876115
$73$	1.17888e12	0.911737	0.455868	−	0.890047i	$-0.349329\pi$
$73$	1.17888e12	0.911737	0.455868	+	0.890047i	$0.349329\pi$
$74$	6.59783e10	0.0467084
$75$	0	0
$76$	1.72277e12	1.02551
$77$	2.62897e12	1.43746
$78$	−7.04180e10	−0.0354054
$79$	−1.07242e12	−0.496351	−0.248176	−	0.968715i	$-0.579831\pi$
$79$	−1.07242e12	−0.496351	−0.248176	+	0.968715i	$0.579831\pi$
$80$	0	0
$81$	2.82430e11	0.111111
$82$	−5.61443e10	−0.0203946
$83$	−1.12403e12	−0.377371	−0.188685	−	0.982038i	$-0.560423\pi$
$83$	−1.12403e12	−0.377371	−0.188685	+	0.982038i	$0.560423\pi$
$84$	1.37926e12	0.428381
$85$	0	0
$86$	−8.53802e10	−0.0227571
$87$	−9.13000e11	−0.225733
$88$	2.17933e12	0.500247
$89$	2.23561e12	0.476827	0.238414	−	0.971164i	$-0.423373\pi$
$89$	2.23561e12	0.476827	0.238414	+	0.971164i	$0.423373\pi$
$90$	0	0
$91$	1.89237e12	0.349330
$92$	6.68431e12	1.14931
$93$	4.49017e12	0.719653
$94$	3.54345e11	0.0529780
$95$	0	0
$96$	1.72011e12	0.224281
$97$	1.42153e13	1.73276	0.866380	−	0.499385i	$-0.166441\pi$
$97$	1.42153e13	1.73276	0.866380	+	0.499385i	$0.166441\pi$
$98$	−4.99472e11	−0.0569563
$99$	−5.94306e12	−0.634426
$100$	0	0
$101$	1.70194e13	1.59535	0.797675	−	0.603088i	$-0.206063\pi$
$101$	1.70194e13	1.59535	0.797675	+	0.603088i	$0.206063\pi$
$102$	1.02784e12	0.0903703
$103$	−1.09904e13	−0.906928	−0.453464	−	0.891275i	$-0.649812\pi$
$103$	−1.09904e13	−0.906928	−0.453464	+	0.891275i	$0.649812\pi$
$104$	1.56871e12	0.121570
$105$	0	0
$106$	2.44950e12	0.167722
$107$	1.96403e13	1.26519	0.632593	−	0.774485i	$-0.281991\pi$
$107$	1.96403e13	1.26519	0.632593	+	0.774485i	$0.281991\pi$
$108$	−3.11796e12	−0.189067
$109$	−9.82099e12	−0.560897	−0.280448	−	0.959869i	$-0.590483\pi$
$109$	−9.82099e12	−0.560897	−0.280448	+	0.959869i	$0.590483\pi$
$110$	0	0
$111$	4.00818e12	0.203399
$112$	−1.49494e13	−0.715660
$113$	1.70267e13	0.769344	0.384672	−	0.923053i	$-0.374315\pi$
$113$	1.70267e13	0.769344	0.384672	+	0.923053i	$0.374315\pi$
$114$	−1.87261e12	−0.0799035
$115$	0	0
$116$	1.00793e13	0.384109
$117$	−4.27789e12	−0.154178
$118$	−3.58918e11	−0.0122395
$119$	−2.76216e13	−0.891648
$120$	0	0
$121$	9.05347e13	2.62247
$122$	−1.61270e12	−0.0442808
$123$	−3.41076e12	−0.0888113
$124$	−4.95705e13	−1.22457
$125$	0	0
$126$	−1.49922e12	−0.0333778
$127$	4.49347e13	0.950292	0.475146	−	0.879907i	$-0.342395\pi$
$127$	4.49347e13	0.950292	0.475146	+	0.879907i	$0.342395\pi$
$128$	−2.52408e13	−0.507270
$129$	−5.18685e12	−0.0990992
$130$	0	0
$131$	−1.20182e12	−0.0207768	−0.0103884	−	0.999946i	$-0.503307\pi$
$131$	−1.20182e12	−0.0207768	−0.0103884	+	0.999946i	$0.503307\pi$
$132$	6.56100e13	1.07954
$133$	5.03233e13	0.788375
$134$	−4.18223e12	−0.0624060
$135$	0	0
$136$	−2.28974e13	−0.310300
$137$	−1.71562e13	−0.221685	−0.110842	−	0.993838i	$-0.535355\pi$
$137$	−1.71562e13	−0.221685	−0.110842	+	0.993838i	$0.535355\pi$
$138$	−7.26570e12	−0.0895495
$139$	1.05644e14	1.24236	0.621182	−	0.783666i	$-0.286653\pi$
$139$	1.05644e14	1.24236	0.621182	+	0.783666i	$0.286653\pi$
$140$	0	0
$141$	2.15265e13	0.230701
$142$	1.57720e13	0.161441
$143$	9.00181e13	0.880330
$144$	3.37947e13	0.315858
$145$	0	0
$146$	1.41465e13	0.120880
$147$	−3.03429e13	−0.248024
$148$	−4.42494e13	−0.346104
$149$	−8.53533e13	−0.639012	−0.319506	−	0.947584i	$-0.603517\pi$
$149$	−8.53533e13	−0.639012	−0.319506	+	0.947584i	$0.603517\pi$
$150$	0	0
$151$	−6.16414e13	−0.423177	−0.211589	−	0.977359i	$-0.567864\pi$
$151$	−6.16414e13	−0.423177	−0.211589	+	0.977359i	$0.567864\pi$
$152$	4.17163e13	0.274361
$153$	6.24415e13	0.393531
$154$	3.15476e13	0.190582
$155$	0	0
$156$	4.72270e13	0.262350
$157$	1.18021e14	0.628942	0.314471	−	0.949267i	$-0.398173\pi$
$157$	1.18021e14	0.628942	0.314471	+	0.949267i	$0.398173\pi$
$158$	−1.28690e13	−0.0658075
$159$	1.48807e14	0.730369
$160$	0	0
$161$	1.95254e14	0.883550
$162$	3.38915e12	0.0147314
$163$	−1.54710e14	−0.646099	−0.323050	−	0.946382i	$-0.604708\pi$
$163$	−1.54710e14	−0.646099	−0.323050	+	0.946382i	$0.604708\pi$
$164$	3.76541e13	0.151122
$165$	0	0
$166$	−1.34883e13	−0.0500328
$167$	−3.76012e14	−1.34136	−0.670679	−	0.741748i	$-0.733997\pi$
$167$	−3.76012e14	−1.34136	−0.670679	+	0.741748i	$0.733997\pi$
$168$	3.33984e13	0.114608
$169$	−2.38079e14	−0.786063
$170$	0	0
$171$	−1.13761e14	−0.347951
$172$	5.72616e13	0.168628
$173$	−3.73562e14	−1.05941	−0.529704	−	0.848182i	$-0.677697\pi$
$173$	−3.73562e14	−1.05941	−0.529704	+	0.848182i	$0.677697\pi$
$174$	−1.09560e13	−0.0299283
$175$	0	0
$176$	−7.11128e14	−1.80350
$177$	−2.18043e13	−0.0532985
$178$	2.68273e13	0.0632190
$179$	4.23349e13	0.0961952	0.0480976	−	0.998843i	$-0.484684\pi$
$179$	4.23349e13	0.0961952	0.0480976	+	0.998843i	$0.484684\pi$
$180$	0	0
$181$	−3.10447e14	−0.656261	−0.328130	−	0.944632i	$-0.606419\pi$
$181$	−3.10447e14	−0.656261	−0.328130	+	0.944632i	$0.606419\pi$
$182$	2.27084e13	0.0463151
$183$	−9.79718e13	−0.192827
$184$	1.61859e14	0.307482
$185$	0	0
$186$	5.38820e13	0.0954134
$187$	−1.31393e15	−2.24700
$188$	−2.37648e14	−0.392561
$189$	−9.10779e13	−0.145349
$190$	0	0
$191$	−8.62273e14	−1.28507	−0.642537	−	0.766255i	$-0.722118\pi$
$191$	−8.62273e14	−1.28507	−0.642537	+	0.766255i	$0.722118\pi$
$192$	−3.59120e14	−0.517347
$193$	9.37837e14	1.30618	0.653092	−	0.757278i	$-0.273471\pi$
$193$	9.37837e14	1.30618	0.653092	+	0.757278i	$0.273471\pi$
$194$	1.70583e14	0.229734
$195$	0	0
$196$	3.34979e14	0.422040
$197$	6.71715e14	0.818756	0.409378	−	0.912365i	$-0.365746\pi$
$197$	6.71715e14	0.818756	0.409378	+	0.912365i	$0.365746\pi$
$198$	−7.13167e13	−0.0841138
$199$	−4.36451e13	−0.0498185	−0.0249093	−	0.999690i	$-0.507930\pi$
$199$	−4.36451e13	−0.0498185	−0.0249093	+	0.999690i	$0.507930\pi$
$200$	0	0
$201$	−2.54070e14	−0.271756
$202$	2.04233e14	0.211515
$203$	2.94424e14	0.295290
$204$	−6.89340e14	−0.669634
$205$	0	0
$206$	−1.31885e14	−0.120243
$207$	−4.41391e14	−0.389957
$208$	−5.11880e14	−0.438285
$209$	2.39383e15	1.98675
$210$	0	0
$211$	−1.62162e15	−1.26507	−0.632534	−	0.774533i	$-0.717985\pi$
$211$	−1.62162e15	−1.26507	−0.632534	+	0.774533i	$0.717985\pi$
$212$	−1.64280e15	−1.24280
$213$	9.58151e14	0.703018
$214$	2.35684e14	0.167742
$215$	0	0
$216$	−7.55005e13	−0.0505825
$217$	−1.44799e15	−0.941406
$218$	−1.17852e14	−0.0743651
$219$	8.59401e14	0.526392
$220$	0	0
$221$	−9.45786e14	−0.546064
$222$	4.80982e13	0.0269671
$223$	−1.47333e15	−0.802266	−0.401133	−	0.916020i	$-0.631383\pi$
$223$	−1.47333e15	−0.802266	−0.401133	+	0.916020i	$0.631383\pi$
$224$	−5.54701e14	−0.293391
$225$	0	0
$226$	2.04320e14	0.102002
$227$	3.74889e15	1.81859	0.909294	−	0.416153i	$-0.136622\pi$
$227$	3.74889e15	1.81859	0.909294	+	0.416153i	$0.136622\pi$
$228$	1.25590e15	0.592076
$229$	−1.47993e13	−0.00678126	−0.00339063	−	0.999994i	$-0.501079\pi$
$229$	−1.47993e13	−0.00678126	−0.00339063	+	0.999994i	$0.501079\pi$
$230$	0	0
$231$	1.91652e15	0.829917
$232$	2.44068e14	0.102763
$233$	−3.63053e15	−1.48647	−0.743236	−	0.669030i	$-0.766710\pi$
$233$	−3.63053e15	−1.48647	−0.743236	+	0.669030i	$0.766710\pi$
$234$	−5.13347e13	−0.0204413
$235$	0	0
$236$	2.40714e14	0.0906930
$237$	−7.81795e14	−0.286569
$238$	−3.31459e14	−0.118217
$239$	−4.33900e15	−1.50592	−0.752962	−	0.658063i	$-0.771376\pi$
$239$	−4.33900e15	−1.50592	−0.752962	+	0.658063i	$0.771376\pi$
$240$	0	0
$241$	3.02372e15	0.994103	0.497051	−	0.867721i	$-0.334416\pi$
$241$	3.02372e15	0.994103	0.497051	+	0.867721i	$0.334416\pi$
$242$	1.08642e15	0.347693
$243$	2.05891e14	0.0641500
$244$	1.08159e15	0.328116
$245$	0	0
$246$	−4.09292e13	−0.0117748
$247$	1.72311e15	0.482818
$248$	−1.20033e15	−0.327617
$249$	−8.19414e14	−0.217875
$250$	0	0
$251$	−1.75146e15	−0.442099	−0.221050	−	0.975263i	$-0.570948\pi$
$251$	−1.75146e15	−0.442099	−0.221050	+	0.975263i	$0.570948\pi$
$252$	1.00548e15	0.247326
$253$	9.28803e15	2.22659
$254$	5.39216e14	0.125992
$255$	0	0
$256$	3.73265e15	0.828816
$257$	−4.87604e15	−1.05561	−0.527803	−	0.849367i	$-0.676984\pi$
$257$	−4.87604e15	−1.05561	−0.527803	+	0.849367i	$0.676984\pi$
$258$	−6.22421e13	−0.0131388
$259$	−1.29256e15	−0.266074
$260$	0	0
$261$	−6.65577e14	−0.130327
$262$	−1.44219e13	−0.00275463
$263$	−4.67882e15	−0.871815	−0.435907	−	0.899992i	$-0.643572\pi$
$263$	−4.67882e15	−0.871815	−0.435907	+	0.899992i	$0.643572\pi$
$264$	1.58873e15	0.288818
$265$	0	0
$266$	6.03879e14	0.104525
$267$	1.62976e15	0.275296
$268$	2.80488e15	0.462422
$269$	−1.80262e15	−0.290078	−0.145039	−	0.989426i	$-0.546331\pi$
$269$	−1.80262e15	−0.290078	−0.145039	+	0.989426i	$0.546331\pi$
$270$	0	0
$271$	6.10016e15	0.935494	0.467747	−	0.883862i	$-0.345066\pi$
$271$	6.10016e15	0.935494	0.467747	+	0.883862i	$0.345066\pi$
$272$	7.47156e15	1.11870
$273$	1.37954e15	0.201686
$274$	−2.05874e14	−0.0293915
$275$	0	0
$276$	4.87286e15	0.663552
$277$	1.07023e16	1.42351	0.711754	−	0.702428i	$-0.247901\pi$
$277$	1.07023e16	1.42351	0.711754	+	0.702428i	$0.247901\pi$
$278$	1.26773e15	0.164716
$279$	3.27333e15	0.415492
$280$	0	0
$281$	−2.45460e15	−0.297433	−0.148717	−	0.988880i	$-0.547514\pi$
$281$	−2.45460e15	−0.297433	−0.148717	+	0.988880i	$0.547514\pi$
$282$	2.58318e14	0.0305869
$283$	−4.01155e15	−0.464195	−0.232098	−	0.972692i	$-0.574559\pi$
$283$	−4.01155e15	−0.464195	−0.232098	+	0.972692i	$0.574559\pi$
$284$	−1.05778e16	−1.19626
$285$	0	0
$286$	1.08022e15	0.116716
$287$	1.09990e15	0.116178
$288$	1.25396e15	0.129489
$289$	3.90041e15	0.393799
$290$	0	0
$291$	1.03629e16	1.00041
$292$	−9.48759e15	−0.895710
$293$	−2.08187e15	−0.192227	−0.0961133	−	0.995370i	$-0.530641\pi$
$293$	−2.08187e15	−0.192227	−0.0961133	+	0.995370i	$0.530641\pi$
$294$	−3.64115e14	−0.0328837
$295$	0	0
$296$	−1.07149e15	−0.0925957
$297$	−4.33249e15	−0.366286
$298$	−1.02424e15	−0.0847219
$299$	6.68565e15	0.541104
$300$	0	0
$301$	1.67265e15	0.129636
$302$	−7.39697e14	−0.0561059
$303$	1.24072e16	0.921075
$304$	−1.36123e16	−0.989130
$305$	0	0
$306$	7.49298e14	0.0521753
$307$	1.32352e16	0.902260	0.451130	−	0.892458i	$-0.351021\pi$
$307$	1.32352e16	0.902260	0.451130	+	0.892458i	$0.351021\pi$
$308$	−2.11579e16	−1.41219
$309$	−8.01202e15	−0.523615
$310$	0	0
$311$	−8.09301e15	−0.507187	−0.253593	−	0.967311i	$-0.581612\pi$
$311$	−8.09301e15	−0.507187	−0.253593	+	0.967311i	$0.581612\pi$
$312$	1.14359e15	0.0701883
$313$	1.48181e16	0.890748	0.445374	−	0.895345i	$-0.353071\pi$
$313$	1.48181e16	0.890748	0.445374	+	0.895345i	$0.353071\pi$
$314$	1.41625e15	0.0833868
$315$	0	0
$316$	8.63084e15	0.487626
$317$	2.43171e16	1.34594	0.672970	−	0.739670i	$-0.265018\pi$
$317$	2.43171e16	1.34594	0.672970	+	0.739670i	$0.265018\pi$
$318$	1.78569e15	0.0968342
$319$	1.40055e16	0.744147
$320$	0	0
$321$	1.43178e16	0.730455
$322$	2.34304e15	0.117143
$323$	−2.51511e16	−1.23237
$324$	−2.27299e15	−0.109158
$325$	0	0
$326$	−1.85652e15	−0.0856615
$327$	−7.15950e15	−0.323834
$328$	9.11783e14	0.0404307
$329$	−6.94186e15	−0.301788
$330$	0	0
$331$	1.16232e16	0.485783	0.242892	−	0.970053i	$-0.421904\pi$
$331$	1.16232e16	0.485783	0.242892	+	0.970053i	$0.421904\pi$
$332$	9.04615e15	0.370737
$333$	2.92196e15	0.117432
$334$	−4.51214e15	−0.177841
$335$	0	0
$336$	−1.08981e16	−0.413186
$337$	−4.62652e16	−1.72052	−0.860262	−	0.509853i	$-0.829700\pi$
$337$	−4.62652e16	−1.72052	−0.860262	+	0.509853i	$0.829700\pi$
$338$	−2.85695e15	−0.104218
$339$	1.24125e16	0.444181
$340$	0	0
$341$	−6.88795e16	−2.37239
$342$	−1.36513e15	−0.0461323
$343$	3.25624e16	1.07970
$344$	1.38657e15	0.0451142
$345$	0	0
$346$	−4.48275e15	−0.140459
$347$	−4.79404e15	−0.147421	−0.0737106	−	0.997280i	$-0.523484\pi$
$347$	−4.79404e15	−0.147421	−0.0737106	+	0.997280i	$0.523484\pi$
$348$	7.34782e15	0.221765
$349$	3.76900e16	1.11651	0.558253	−	0.829671i	$-0.311472\pi$
$349$	3.76900e16	1.11651	0.558253	+	0.829671i	$0.311472\pi$
$350$	0	0
$351$	−3.11859e15	−0.0890146
$352$	−2.63866e16	−0.739360
$353$	−4.80179e16	−1.32089	−0.660446	−	0.750873i	$-0.729633\pi$
$353$	−4.80179e16	−1.32089	−0.660446	+	0.750873i	$0.729633\pi$
$354$	−2.61651e14	−0.00706645
$355$	0	0
$356$	−1.79922e16	−0.468446
$357$	−2.01361e16	−0.514793
$358$	5.08018e14	0.0127538
$359$	4.06616e16	1.00247	0.501234	−	0.865312i	$-0.332880\pi$
$359$	4.06616e16	1.00247	0.501234	+	0.865312i	$0.332880\pi$
$360$	0	0
$361$	3.76929e15	0.0896320
$362$	−3.72536e15	−0.0870087
$363$	6.59998e16	1.51408
$364$	−1.52298e16	−0.343190
$365$	0	0
$366$	−1.17566e15	−0.0255656
$367$	−2.96733e16	−0.633923	−0.316961	−	0.948438i	$-0.602663\pi$
$367$	−2.96733e16	−0.633923	−0.316961	+	0.948438i	$0.602663\pi$
$368$	−5.28156e16	−1.10854
$369$	−2.48645e15	−0.0512752
$370$	0	0
$371$	−4.79873e16	−0.955424
$372$	−3.61369e16	−0.707003
$373$	9.01346e16	1.73294	0.866471	−	0.499227i	$-0.166383\pi$
$373$	9.01346e16	1.73294	0.866471	+	0.499227i	$0.166383\pi$
$374$	−1.57672e16	−0.297912
$375$	0	0
$376$	−5.75457e15	−0.105025
$377$	1.00813e16	0.180842
$378$	−1.09293e15	−0.0192707
$379$	−1.54841e16	−0.268369	−0.134184	−	0.990956i	$-0.542841\pi$
$379$	−1.54841e16	−0.268369	−0.134184	+	0.990956i	$0.542841\pi$
$380$	0	0
$381$	3.27574e16	0.548652
$382$	−1.03473e16	−0.170378
$383$	−9.37088e15	−0.151701	−0.0758505	−	0.997119i	$-0.524167\pi$
$383$	−9.37088e15	−0.151701	−0.0758505	+	0.997119i	$0.524167\pi$
$384$	−1.84006e16	−0.292872
$385$	0	0
$386$	1.12540e16	0.173177
$387$	−3.78121e15	−0.0572150
$388$	−1.14404e17	−1.70230
$389$	2.95806e16	0.432847	0.216423	−	0.976300i	$-0.430561\pi$
$389$	2.95806e16	0.432847	0.216423	+	0.976300i	$0.430561\pi$
$390$	0	0
$391$	−9.75858e16	−1.38114
$392$	8.11142e15	0.112911
$393$	−8.76130e14	−0.0119955
$394$	8.06058e15	0.108553
$395$	0	0
$396$	4.78297e16	0.623274
$397$	−1.80617e16	−0.231538	−0.115769	−	0.993276i	$-0.536933\pi$
$397$	−1.80617e16	−0.231538	−0.115769	+	0.993276i	$0.536933\pi$
$398$	−5.23742e14	−0.00660507
$399$	3.66857e16	0.455169
$400$	0	0
$401$	−1.20412e17	−1.44621	−0.723107	−	0.690736i	$-0.757287\pi$
$401$	−1.20412e17	−1.44621	−0.723107	+	0.690736i	$0.757287\pi$
$402$	−3.04884e15	−0.0360301
$403$	−4.95804e16	−0.576537
$404$	−1.36972e17	−1.56731
$405$	0	0
$406$	3.53309e15	0.0391503
$407$	−6.14858e16	−0.670519
$408$	−1.66922e16	−0.179152
$409$	−1.77522e16	−0.187521	−0.0937606	−	0.995595i	$-0.529889\pi$
$409$	−1.77522e16	−0.187521	−0.0937606	+	0.995595i	$0.529889\pi$
$410$	0	0
$411$	−1.25068e16	−0.127990
$412$	8.84510e16	0.890986
$413$	7.03144e15	0.0697219
$414$	−5.29670e15	−0.0517015
$415$	0	0
$416$	−1.89934e16	−0.179679
$417$	7.70145e16	0.717279
$418$	2.87259e16	0.263408
$419$	1.75670e17	1.58602	0.793008	−	0.609212i	$-0.208514\pi$
$419$	1.75670e17	1.58602	0.793008	+	0.609212i	$0.208514\pi$
$420$	0	0
$421$	1.84473e17	1.61473	0.807365	−	0.590052i	$-0.200893\pi$
$421$	1.84473e17	1.61473	0.807365	+	0.590052i	$0.200893\pi$
$422$	−1.94595e16	−0.167726
$423$	1.56928e16	0.133195
$424$	−3.97799e16	−0.332495
$425$	0	0
$426$	1.14978e16	0.0932079
$427$	3.15939e16	0.252245
$428$	−1.58065e17	−1.24295
$429$	6.56232e16	0.508259
$430$	0	0
$431$	8.05532e16	0.605314	0.302657	−	0.953100i	$-0.402126\pi$
$431$	8.05532e16	0.605314	0.302657	+	0.953100i	$0.402126\pi$
$432$	2.46363e16	0.182361
$433$	1.97092e17	1.43714	0.718568	−	0.695457i	$-0.244798\pi$
$433$	1.97092e17	1.43714	0.718568	+	0.695457i	$0.244798\pi$
$434$	−1.73759e16	−0.124814
$435$	0	0
$436$	7.90393e16	0.551037
$437$	1.77790e17	1.22117
$438$	1.03128e16	0.0697903
$439$	9.89007e16	0.659447	0.329724	−	0.944078i	$-0.393044\pi$
$439$	9.89007e16	0.659447	0.329724	+	0.944078i	$0.393044\pi$
$440$	0	0
$441$	−2.21200e16	−0.143197
$442$	−1.13494e16	−0.0723985
$443$	1.25104e17	0.786404	0.393202	−	0.919452i	$-0.371367\pi$
$443$	1.25104e17	0.786404	0.393202	+	0.919452i	$0.371367\pi$
$444$	−3.22578e16	−0.199823
$445$	0	0
$446$	−1.76800e16	−0.106366
$447$	−6.22225e16	−0.368934
$448$	1.15809e17	0.676761
$449$	−1.80095e17	−1.03729	−0.518645	−	0.854990i	$-0.673563\pi$
$449$	−1.80095e17	−1.03729	−0.518645	+	0.854990i	$0.673563\pi$
$450$	0	0
$451$	5.23213e16	0.292773
$452$	−1.37031e17	−0.755820
$453$	−4.49366e16	−0.244322
$454$	4.49867e16	0.241113
$455$	0	0
$456$	3.04112e16	0.158402
$457$	9.43597e16	0.484542	0.242271	−	0.970209i	$-0.422108\pi$
$457$	9.43597e16	0.484542	0.242271	+	0.970209i	$0.422108\pi$
$458$	−1.77592e14	−0.000899076 0
$459$	4.55198e16	0.227205
$460$	0	0
$461$	8.00500e16	0.388423	0.194212	−	0.980960i	$-0.437785\pi$
$461$	8.00500e16	0.388423	0.194212	+	0.980960i	$0.437785\pi$
$462$	2.29982e16	0.110033
$463$	−2.14174e17	−1.01039	−0.505196	−	0.863004i	$-0.668580\pi$
$463$	−2.14174e17	−1.01039	−0.505196	+	0.863004i	$0.668580\pi$
$464$	−7.96410e16	−0.370484
$465$	0	0
$466$	−4.35663e16	−0.197080
$467$	1.80681e17	0.806031	0.403015	−	0.915193i	$-0.367962\pi$
$467$	1.80681e17	0.806031	0.403015	+	0.915193i	$0.367962\pi$
$468$	3.44285e16	0.151468
$469$	8.19326e16	0.355495
$470$	0	0
$471$	8.60372e16	0.363120
$472$	5.82883e15	0.0242638
$473$	7.95665e16	0.326688
$474$	−9.38154e15	−0.0379940
$475$	0	0
$476$	2.22298e17	0.875974
$477$	1.08480e17	0.421679
$478$	−5.20680e16	−0.199659
$479$	−2.66712e17	−1.00893	−0.504466	−	0.863431i	$-0.668311\pi$
$479$	−2.66712e17	−1.00893	−0.504466	+	0.863431i	$0.668311\pi$
$480$	0	0
$481$	−4.42583e16	−0.162949
$482$	3.62847e16	0.131801
$483$	1.42340e17	0.510118
$484$	−7.28623e17	−2.57637
$485$	0	0
$486$	2.47069e15	0.00850517
$487$	2.63552e17	0.895216	0.447608	−	0.894230i	$-0.352276\pi$
$487$	2.63552e17	0.895216	0.447608	+	0.894230i	$0.352276\pi$
$488$	2.61903e16	0.0877833
$489$	−1.12784e17	−0.373026
$490$	0	0
$491$	4.11733e17	1.32613	0.663065	−	0.748562i	$-0.269255\pi$
$491$	4.11733e17	1.32613	0.663065	+	0.748562i	$0.269255\pi$
$492$	2.74498e16	0.0872502
$493$	−1.47150e17	−0.461590
$494$	2.06773e16	0.0640132
$495$	0	0
$496$	3.91677e17	1.18113
$497$	−3.08984e17	−0.919645
$498$	−9.83297e15	−0.0288864
$499$	3.99658e17	1.15887	0.579435	−	0.815018i	$-0.303273\pi$
$499$	3.99658e17	1.15887	0.579435	+	0.815018i	$0.303273\pi$
$500$	0	0
$501$	−2.74113e17	−0.774433
$502$	−2.10175e16	−0.0586146
$503$	2.83581e17	0.780702	0.390351	−	0.920666i	$-0.372354\pi$
$503$	2.83581e17	0.780702	0.390351	+	0.920666i	$0.372354\pi$
$504$	2.43474e16	0.0661690
$505$	0	0
$506$	1.11456e17	0.295207
$507$	−1.73559e17	−0.453834
$508$	−3.61634e17	−0.933588
$509$	6.40327e17	1.63206	0.816030	−	0.578009i	$-0.196170\pi$
$509$	6.40327e17	1.63206	0.816030	+	0.578009i	$0.196170\pi$
$510$	0	0
$511$	−2.77140e17	−0.688593
$512$	2.51565e17	0.617156
$513$	−8.29318e16	−0.200890
$514$	−5.85124e16	−0.139955
$515$	0	0
$516$	4.17437e16	0.0973572
$517$	−3.30218e17	−0.760522
$518$	−1.55107e16	−0.0352767
$519$	−2.72327e17	−0.611650
$520$	0	0
$521$	−4.01348e17	−0.879175	−0.439588	−	0.898200i	$-0.644875\pi$
$521$	−4.01348e17	−0.879175	−0.439588	+	0.898200i	$0.644875\pi$
$522$	−7.98692e15	−0.0172791
$523$	5.05985e17	1.08113	0.540564	−	0.841303i	$-0.318211\pi$
$523$	5.05985e17	1.08113	0.540564	+	0.841303i	$0.318211\pi$
$524$	9.67228e15	0.0204115
$525$	0	0
$526$	−5.61459e16	−0.115587
$527$	7.23691e17	1.47158
$528$	−5.18413e17	−1.04125
$529$	1.85786e17	0.368597
$530$	0	0
$531$	−1.58953e16	−0.0307719
$532$	−4.05002e17	−0.774517
$533$	3.76616e16	0.0711496
$534$	1.95571e16	0.0364995
$535$	0	0
$536$	6.79193e16	0.123715
$537$	3.08621e16	0.0555383
$538$	−2.16314e16	−0.0384592
$539$	4.65462e17	0.817631
$540$	0	0
$541$	−1.69124e17	−0.290017	−0.145009	−	0.989430i	$-0.546321\pi$
$541$	−1.69124e17	−0.290017	−0.145009	+	0.989430i	$0.546321\pi$
$542$	7.32020e16	0.124030
$543$	−2.26316e17	−0.378892
$544$	2.77234e17	0.458620
$545$	0	0
$546$	1.65544e16	0.0267400
$547$	4.32104e17	0.689717	0.344858	−	0.938655i	$-0.387927\pi$
$547$	4.32104e17	0.689717	0.344858	+	0.938655i	$0.387927\pi$
$548$	1.38073e17	0.217788
$549$	−7.14214e16	−0.111329
$550$	0	0
$551$	2.68091e17	0.408128
$552$	1.17995e17	0.177525
$553$	2.52113e17	0.374871
$554$	1.28428e17	0.188732
$555$	0	0
$556$	−8.50224e17	−1.22053
$557$	−1.36804e18	−1.94107	−0.970534	−	0.240966i	$-0.922536\pi$
$557$	−1.36804e18	−1.94107	−0.970534	+	0.240966i	$0.922536\pi$
$558$	3.92800e16	0.0550870
$559$	5.72731e16	0.0793915
$560$	0	0
$561$	−9.57856e17	−1.29730
$562$	−2.94552e16	−0.0394345
$563$	9.52405e17	1.26043	0.630213	−	0.776422i	$-0.282968\pi$
$563$	9.52405e17	1.26043	0.630213	+	0.776422i	$0.282968\pi$
$564$	−1.73245e17	−0.226645
$565$	0	0
$566$	−4.81386e16	−0.0615442
$567$	−6.63958e16	−0.0839171
$568$	−2.56138e17	−0.320044
$569$	1.53632e17	0.189780	0.0948902	−	0.995488i	$-0.469750\pi$
$569$	1.53632e17	0.189780	0.0948902	+	0.995488i	$0.469750\pi$
$570$	0	0
$571$	−1.27956e18	−1.54500	−0.772498	−	0.635017i	$-0.780993\pi$
$571$	−1.27956e18	−1.54500	−0.772498	+	0.635017i	$0.780993\pi$
$572$	−7.24466e17	−0.864856
$573$	−6.28597e17	−0.741938
$574$	1.31988e16	0.0154031
$575$	0	0
$576$	−2.61799e17	−0.298690
$577$	−3.56770e17	−0.402481	−0.201241	−	0.979542i	$-0.564497\pi$
$577$	−3.56770e17	−0.402481	−0.201241	+	0.979542i	$0.564497\pi$
$578$	4.68049e16	0.0522108
$579$	6.83683e17	0.754126
$580$	0	0
$581$	2.64245e17	0.285011
$582$	1.24355e17	0.132637
$583$	−2.28271e18	−2.40772
$584$	−2.29739e17	−0.239636
$585$	0	0
$586$	−2.49824e16	−0.0254859
$587$	−1.28968e18	−1.30118	−0.650588	−	0.759431i	$-0.725477\pi$
$587$	−1.28968e18	−1.30118	−0.650588	+	0.759431i	$0.725477\pi$
$588$	2.44200e17	0.243665
$589$	−1.31848e18	−1.30114
$590$	0	0
$591$	4.89680e17	0.472709
$592$	3.49634e17	0.333827
$593$	1.88640e18	1.78147	0.890735	−	0.454523i	$-0.150190\pi$
$593$	1.88640e18	1.78147	0.890735	+	0.454523i	$0.150190\pi$
$594$	−5.19899e16	−0.0485631
$595$	0	0
$596$	6.86923e17	0.627780
$597$	−3.18173e16	−0.0287627
$598$	8.02278e16	0.0717410
$599$	−1.44668e18	−1.27967	−0.639834	−	0.768513i	$-0.720997\pi$
$599$	−1.44668e18	−1.27967	−0.639834	+	0.768513i	$0.720997\pi$
$600$	0	0
$601$	−4.44358e16	−0.0384635	−0.0192317	−	0.999815i	$-0.506122\pi$
$601$	−4.44358e16	−0.0384635	−0.0192317	+	0.999815i	$0.506122\pi$
$602$	2.00719e16	0.0171874
$603$	−1.85217e17	−0.156898
$604$	4.96090e17	0.415739
$605$	0	0
$606$	1.48886e17	0.122118
$607$	−2.98050e16	−0.0241860	−0.0120930	−	0.999927i	$-0.503849\pi$
$607$	−2.98050e16	−0.0241860	−0.0120930	+	0.999927i	$0.503849\pi$
$608$	−5.05087e17	−0.405502
$609$	2.14635e17	0.170486
$610$	0	0
$611$	−2.37695e17	−0.184822
$612$	−5.02529e17	−0.386613
$613$	8.84082e17	0.672976	0.336488	−	0.941688i	$-0.390761\pi$
$613$	8.84082e17	0.672976	0.336488	+	0.941688i	$0.390761\pi$
$614$	1.58823e17	0.119624
$615$	0	0
$616$	−5.12333e17	−0.377814
$617$	−1.43684e18	−1.04846	−0.524232	−	0.851575i	$-0.675648\pi$
$617$	−1.43684e18	−1.04846	−0.524232	+	0.851575i	$0.675648\pi$
$618$	−9.61443e16	−0.0694222
$619$	1.68862e18	1.20654	0.603272	−	0.797535i	$-0.293863\pi$
$619$	1.68862e18	1.20654	0.603272	+	0.797535i	$0.293863\pi$
$620$	0	0
$621$	−3.21774e17	−0.225142
$622$	−9.71162e16	−0.0672441
$623$	−5.25565e17	−0.360126
$624$	−3.73161e17	−0.253044
$625$	0	0
$626$	1.77817e17	0.118098
$627$	1.74510e18	1.14705
$628$	−9.49831e17	−0.617887
$629$	6.46008e17	0.415919
$630$	0	0
$631$	−3.53490e17	−0.222939	−0.111470	−	0.993768i	$-0.535556\pi$
$631$	−3.53490e17	−0.222939	−0.111470	+	0.993768i	$0.535556\pi$
$632$	2.08993e17	0.130458
$633$	−1.18216e18	−0.730387
$634$	2.91805e17	0.178448
$635$	0	0
$636$	−1.19760e18	−0.717531
$637$	3.35046e17	0.198700
$638$	1.68066e17	0.0986608
$639$	6.98492e17	0.405888
$640$	0	0
$641$	1.61802e18	0.921313	0.460656	−	0.887579i	$-0.347614\pi$
$641$	1.61802e18	0.921313	0.460656	+	0.887579i	$0.347614\pi$
$642$	1.71814e17	0.0968456
$643$	1.96065e18	1.09403	0.547015	−	0.837123i	$-0.315764\pi$
$643$	1.96065e18	1.09403	0.547015	+	0.837123i	$0.315764\pi$
$644$	−1.57140e18	−0.868018
$645$	0	0
$646$	−3.01813e17	−0.163390
$647$	5.96114e17	0.319486	0.159743	−	0.987159i	$-0.448933\pi$
$647$	5.96114e17	0.319486	0.159743	+	0.987159i	$0.448933\pi$
$648$	−5.50399e16	−0.0292038
$649$	3.34479e17	0.175703
$650$	0	0
$651$	−1.05558e18	−0.543521
$652$	1.24511e18	0.634742
$653$	2.58318e18	1.30382	0.651912	−	0.758295i	$-0.273967\pi$
$653$	2.58318e18	1.30382	0.651912	+	0.758295i	$0.273967\pi$
$654$	−8.59140e16	−0.0429347
$655$	0	0
$656$	−2.97521e17	−0.145761
$657$	6.26503e17	0.303912
$658$	−8.33023e16	−0.0400119
$659$	2.64137e18	1.25624	0.628121	−	0.778116i	$-0.283824\pi$
$659$	2.64137e18	1.25624	0.628121	+	0.778116i	$0.283824\pi$
$660$	0	0
$661$	4.12451e18	1.92337	0.961685	−	0.274156i	$-0.0883983\pi$
$661$	4.12451e18	1.92337	0.961685	+	0.274156i	$0.0883983\pi$
$662$	1.39478e17	0.0644064
$663$	−6.89478e17	−0.315270
$664$	2.19050e17	0.0991861
$665$	0	0
$666$	3.50636e16	0.0155695
$667$	1.04019e18	0.457398
$668$	3.02614e18	1.31778
$669$	−1.07406e18	−0.463189
$670$	0	0
$671$	1.50289e18	0.635670
$672$	−4.04377e17	−0.169389
$673$	−2.79726e18	−1.16047	−0.580236	−	0.814449i	$-0.697039\pi$
$673$	−2.79726e18	−1.16047	−0.580236	+	0.814449i	$0.697039\pi$
$674$	−5.55183e17	−0.228111
$675$	0	0
$676$	1.91606e18	0.772245
$677$	4.25553e18	1.69874	0.849372	−	0.527795i	$-0.176981\pi$
$677$	4.25553e18	1.69874	0.849372	+	0.527795i	$0.176981\pi$
$678$	1.48950e17	0.0588906
$679$	−3.34184e18	−1.30867
$680$	0	0
$681$	2.73294e18	1.04996
$682$	−8.26553e17	−0.314538
$683$	−1.60893e18	−0.606461	−0.303230	−	0.952917i	$-0.598065\pi$
$683$	−1.60893e18	−0.606461	−0.303230	+	0.952917i	$0.598065\pi$
$684$	9.15548e17	0.341835
$685$	0	0
$686$	3.90749e17	0.143150
$687$	−1.07887e16	−0.00391516
$688$	−4.52448e17	−0.162646
$689$	−1.64313e18	−0.585122
$690$	0	0
$691$	−3.06331e18	−1.07049	−0.535247	−	0.844696i	$-0.679782\pi$
$691$	−3.06331e18	−1.07049	−0.535247	+	0.844696i	$0.679782\pi$
$692$	3.00643e18	1.04079
$693$	1.39714e18	0.479153
$694$	−5.75285e16	−0.0195455
$695$	0	0
$696$	1.77925e17	0.0593305
$697$	−5.49721e17	−0.181606
$698$	4.52280e17	0.148029
$699$	−2.64666e18	−0.858214
$700$	0	0
$701$	2.99144e18	0.952166	0.476083	−	0.879400i	$-0.342056\pi$
$701$	2.99144e18	0.952166	0.476083	+	0.879400i	$0.342056\pi$
$702$	−3.74230e16	−0.0118018
$703$	−1.17695e18	−0.367747
$704$	5.50893e18	1.70547
$705$	0	0
$706$	−5.76215e17	−0.175127
$707$	−4.00106e18	−1.20489
$708$	1.75481e17	0.0523616
$709$	−5.31694e18	−1.57203	−0.786015	−	0.618207i	$-0.787859\pi$
$709$	−5.31694e18	−1.57203	−0.786015	+	0.618207i	$0.787859\pi$
$710$	0	0
$711$	−5.69928e17	−0.165450
$712$	−4.35676e17	−0.125327
$713$	−5.11568e18	−1.45822
$714$	−2.41634e17	−0.0682525
$715$	0	0
$716$	−3.40711e17	−0.0945043
$717$	−3.16313e18	−0.869446
$718$	4.87939e17	0.132910
$719$	−4.03153e18	−1.08826	−0.544129	−	0.839001i	$-0.683140\pi$
$719$	−4.03153e18	−1.08826	−0.544129	+	0.839001i	$0.683140\pi$
$720$	0	0
$721$	2.58372e18	0.684961
$722$	4.52315e16	0.0118836
$723$	2.20429e18	0.573945
$724$	2.49848e18	0.644725
$725$	0	0
$726$	7.91998e17	0.200741
$727$	4.77643e18	1.19986	0.599928	−	0.800054i	$-0.295196\pi$
$727$	4.77643e18	1.19986	0.599928	+	0.800054i	$0.295196\pi$
$728$	−3.68785e17	−0.0918161
$729$	1.50095e17	0.0370370
$730$	0	0
$731$	−8.35976e17	−0.202643
$732$	7.88477e17	0.189438
$733$	−1.71668e18	−0.408803	−0.204401	−	0.978887i	$-0.565525\pi$
$733$	−1.71668e18	−0.408803	−0.204401	+	0.978887i	$0.565525\pi$
$734$	−3.56080e17	−0.0840471
$735$	0	0
$736$	−1.95973e18	−0.454455
$737$	3.89745e18	0.895864
$738$	−2.98374e16	−0.00679820
$739$	−8.69723e17	−0.196423	−0.0982114	−	0.995166i	$-0.531312\pi$
$739$	−8.69723e17	−0.196423	−0.0982114	+	0.995166i	$0.531312\pi$
$740$	0	0
$741$	1.25615e18	0.278755
$742$	−5.75848e17	−0.126673
$743$	−2.40272e18	−0.523933	−0.261966	−	0.965077i	$-0.584371\pi$
$743$	−2.40272e18	−0.523933	−0.261966	+	0.965077i	$0.584371\pi$
$744$	−8.75044e17	−0.189150
$745$	0	0
$746$	1.08161e18	0.229758
$747$	−5.97353e17	−0.125790
$748$	1.05745e19	2.20750
$749$	−4.61721e18	−0.955537
$750$	0	0
$751$	9.37175e18	1.90617	0.953084	−	0.302706i	$-0.0978899\pi$
$751$	9.37175e18	1.90617	0.953084	+	0.302706i	$0.0978899\pi$
$752$	1.87775e18	0.378637
$753$	−1.27681e18	−0.255246
$754$	1.20976e17	0.0239765
$755$	0	0
$756$	7.32995e17	0.142794
$757$	−3.09120e18	−0.597040	−0.298520	−	0.954403i	$-0.596493\pi$
$757$	−3.09120e18	−0.597040	−0.298520	+	0.954403i	$0.596493\pi$
$758$	−1.85810e17	−0.0355810
$759$	6.77097e18	1.28552
$760$	0	0
$761$	−7.97787e18	−1.48897	−0.744486	−	0.667638i	$-0.767306\pi$
$761$	−7.97787e18	−1.48897	−0.744486	+	0.667638i	$0.767306\pi$
$762$	3.93088e17	0.0727416
$763$	2.30880e18	0.423620
$764$	6.93957e18	1.26248
$765$	0	0
$766$	−1.12451e17	−0.0201129
$767$	2.40763e17	0.0426991
$768$	2.72110e18	0.478517
$769$	7.37344e18	1.28573	0.642863	−	0.765981i	$-0.277746\pi$
$769$	7.37344e18	1.28573	0.642863	+	0.765981i	$0.277746\pi$
$770$	0	0
$771$	−3.55463e18	−0.609454
$772$	−7.54771e18	−1.28322
$773$	1.67335e18	0.282111	0.141056	−	0.990002i	$-0.454950\pi$
$773$	1.67335e18	0.282111	0.141056	+	0.990002i	$0.454950\pi$
$774$	−4.53745e16	−0.00758570
$775$	0	0
$776$	−2.77027e18	−0.455429
$777$	−9.42275e17	−0.153618
$778$	3.54967e17	0.0573879
$779$	1.00153e18	0.160572
$780$	0	0
$781$	−1.46981e19	−2.31755
$782$	−1.17103e18	−0.183115
$783$	−4.85206e17	−0.0752444
$784$	−2.64681e18	−0.407069
$785$	0	0
$786$	−1.05136e16	−0.00159039
$787$	3.75359e18	0.563133	0.281566	−	0.959542i	$-0.409146\pi$
$787$	3.75359e18	0.563133	0.281566	+	0.959542i	$0.409146\pi$
$788$	−5.40596e18	−0.804363
$789$	−3.41086e18	−0.503342
$790$	0	0
$791$	−4.00277e18	−0.581050
$792$	1.15818e18	0.166749
$793$	1.08180e18	0.154480
$794$	−2.16741e17	−0.0306978
$795$	0	0
$796$	3.51256e17	0.0489428
$797$	−3.38853e18	−0.468309	−0.234154	−	0.972199i	$-0.575232\pi$
$797$	−3.38853e18	−0.468309	−0.234154	+	0.972199i	$0.575232\pi$
$798$	4.40228e17	0.0603474
$799$	3.46947e18	0.471747
$800$	0	0
$801$	1.18810e18	0.158942
$802$	−1.44495e18	−0.191743
$803$	−1.31833e19	−1.73529
$804$	2.04476e18	0.266979
$805$	0	0
$806$	−5.94965e17	−0.0764387
$807$	−1.31411e18	−0.167476
$808$	−3.31674e18	−0.419313
$809$	−3.13119e18	−0.392685	−0.196343	−	0.980535i	$-0.562906\pi$
$809$	−3.13119e18	−0.392685	−0.196343	+	0.980535i	$0.562906\pi$
$810$	0	0
$811$	1.04731e19	1.29253	0.646264	−	0.763114i	$-0.276331\pi$
$811$	1.04731e19	1.29253	0.646264	+	0.763114i	$0.276331\pi$
$812$	−2.36953e18	−0.290100
$813$	4.44702e18	0.540108
$814$	−7.37829e17	−0.0888991
$815$	0	0
$816$	5.44677e18	0.645881
$817$	1.52305e18	0.179172
$818$	−2.13026e17	−0.0248620
$819$	1.00568e18	0.116443
$820$	0	0
$821$	−6.85162e18	−0.780841	−0.390421	−	0.920637i	$-0.627670\pi$
$821$	−6.85162e18	−0.780841	−0.390421	+	0.920637i	$0.627670\pi$
$822$	−1.50082e17	−0.0169692
$823$	−3.06934e17	−0.0344308	−0.0172154	−	0.999852i	$-0.505480\pi$
$823$	−3.06934e17	−0.0344308	−0.0172154	+	0.999852i	$0.505480\pi$
$824$	2.14181e18	0.238372
$825$	0	0
$826$	8.43773e16	0.00924390
$827$	−7.75365e18	−0.842792	−0.421396	−	0.906877i	$-0.638460\pi$
$827$	−7.75365e18	−0.842792	−0.421396	+	0.906877i	$0.638460\pi$
$828$	3.55232e18	0.383102
$829$	2.34336e18	0.250747	0.125373	−	0.992110i	$-0.459987\pi$
$829$	2.34336e18	0.250747	0.125373	+	0.992110i	$0.459987\pi$
$830$	0	0
$831$	7.80200e18	0.821863
$832$	3.96540e18	0.414463
$833$	−4.89044e18	−0.507172
$834$	9.24174e17	0.0950987
$835$	0	0
$836$	−1.92655e19	−1.95182
$837$	2.38626e18	0.239884
$838$	2.10804e18	0.210278
$839$	−1.63297e19	−1.61632	−0.808158	−	0.588965i	$-0.799535\pi$
$839$	−1.63297e19	−1.61632	−0.808158	+	0.588965i	$0.799535\pi$
$840$	0	0
$841$	−8.69212e18	−0.847134
$842$	2.21368e18	0.214085
$843$	−1.78940e18	−0.171723
$844$	1.30508e19	1.24283
$845$	0	0
$846$	1.88314e17	0.0176593
$847$	−2.12836e19	−1.98063
$848$	1.29804e19	1.19872
$849$	−2.92442e18	−0.268003
$850$	0	0
$851$	−4.56655e18	−0.412142
$852$	−7.71120e18	−0.690660
$853$	−1.93794e19	−1.72255	−0.861276	−	0.508137i	$-0.830334\pi$
$853$	−1.93794e19	−1.72255	−0.861276	+	0.508137i	$0.830334\pi$
$854$	3.79127e17	0.0334433
$855$	0	0
$856$	−3.82751e18	−0.332534
$857$	−1.20537e19	−1.03931	−0.519656	−	0.854376i	$-0.673940\pi$
$857$	−1.20537e19	−1.03931	−0.519656	+	0.854376i	$0.673940\pi$
$858$	7.87478e17	0.0673862
$859$	−1.00612e19	−0.854465	−0.427232	−	0.904142i	$-0.640511\pi$
$859$	−1.00612e19	−0.854465	−0.427232	+	0.904142i	$0.640511\pi$
$860$	0	0
$861$	8.01830e17	0.0670751
$862$	9.66639e17	0.0802541
$863$	−8.01407e18	−0.660363	−0.330182	−	0.943917i	$-0.607110\pi$
$863$	−8.01407e18	−0.660363	−0.330182	+	0.943917i	$0.607110\pi$
$864$	9.14136e17	0.0747604
$865$	0	0
$866$	2.36510e18	0.190539
$867$	2.84340e18	0.227360
$868$	1.16534e19	0.924858
$869$	1.19928e19	0.944694
$870$	0	0
$871$	2.80544e18	0.217712
$872$	1.91391e18	0.147423
$873$	7.55457e18	0.577587
$874$	2.13348e18	0.161906
$875$	0	0
$876$	−6.91646e18	−0.517139
$877$	−8.87791e17	−0.0658891	−0.0329445	−	0.999457i	$-0.510488\pi$
$877$	−8.87791e17	−0.0658891	−0.0329445	+	0.999457i	$0.510488\pi$
$878$	1.18681e18	0.0874312
$879$	−1.51768e18	−0.110982
$880$	0	0
$881$	3.78774e18	0.272921	0.136460	−	0.990646i	$-0.456427\pi$
$881$	3.78774e18	0.272921	0.136460	+	0.990646i	$0.456427\pi$
$882$	−2.65440e17	−0.0189854
$883$	2.75428e19	1.95553	0.977763	−	0.209712i	$-0.0672526\pi$
$883$	2.75428e19	1.95553	0.977763	+	0.209712i	$0.0672526\pi$
$884$	7.61169e18	0.536465
$885$	0	0
$886$	1.50124e18	0.104263
$887$	2.84165e19	1.95914	0.979572	−	0.201092i	$-0.0644489\pi$
$887$	2.84165e19	1.95914	0.979572	+	0.201092i	$0.0644489\pi$
$888$	−7.81114e17	−0.0534602
$889$	−1.05636e19	−0.717712
$890$	0	0
$891$	−3.15838e18	−0.211475
$892$	1.18574e19	0.788164
$893$	−6.32097e18	−0.417109
$894$	−7.46670e17	−0.0489142
$895$	0	0
$896$	5.93382e18	0.383118
$897$	4.87384e18	0.312407
$898$	−2.16114e18	−0.137527
$899$	−7.71397e18	−0.487349
$900$	0	0
$901$	2.39836e19	1.49349
$902$	6.27856e17	0.0388166
$903$	1.21937e18	0.0748451
$904$	−3.31816e18	−0.202210
$905$	0	0
$906$	−5.39239e17	−0.0323928
$907$	−7.51023e18	−0.447926	−0.223963	−	0.974598i	$-0.571899\pi$
$907$	−7.51023e18	−0.447926	−0.223963	+	0.974598i	$0.571899\pi$
$908$	−3.01711e19	−1.78662
$909$	9.04482e18	0.531783
$910$	0	0
$911$	−2.34028e19	−1.35643	−0.678216	−	0.734863i	$-0.737247\pi$
$911$	−2.34028e19	−1.35643	−0.678216	+	0.734863i	$0.737247\pi$
$912$	−9.92337e18	−0.571074
$913$	1.25699e19	0.718242
$914$	1.13232e18	0.0642418
$915$	0	0
$916$	1.19105e17	0.00666206
$917$	2.82535e17	0.0156917
$918$	5.46238e17	0.0301234
$919$	2.20881e19	1.20950	0.604752	−	0.796414i	$-0.293272\pi$
$919$	2.20881e19	1.20950	0.604752	+	0.796414i	$0.293272\pi$
$920$	0	0
$921$	9.64848e18	0.520920
$922$	9.60600e17	0.0514982
$923$	−1.05799e19	−0.563210
$924$	−1.54241e19	−0.815329
$925$	0	0
$926$	−2.57009e18	−0.133960
$927$	−5.84077e18	−0.302309
$928$	−2.95509e18	−0.151883
$929$	−1.05632e19	−0.539132	−0.269566	−	0.962982i	$-0.586880\pi$
$929$	−1.05632e19	−0.539132	−0.269566	+	0.962982i	$0.586880\pi$
$930$	0	0
$931$	8.90980e18	0.448430
$932$	2.92185e19	1.46034
$933$	−5.89981e18	−0.292824
$934$	2.16817e18	0.106866
$935$	0	0
$936$	8.33676e17	0.0405233
$937$	1.72833e19	0.834294	0.417147	−	0.908839i	$-0.363030\pi$
$937$	1.72833e19	0.834294	0.417147	+	0.908839i	$0.363030\pi$
$938$	9.83191e17	0.0471324
$939$	1.08024e19	0.514273
$940$	0	0
$941$	2.14038e19	1.00498	0.502490	−	0.864583i	$-0.332417\pi$
$941$	2.14038e19	1.00498	0.502490	+	0.864583i	$0.332417\pi$
$942$	1.03245e18	0.0481434
$943$	3.88591e18	0.179956
$944$	−1.90199e18	−0.0874760
$945$	0	0
$946$	9.54799e17	0.0433131
$947$	5.91308e17	0.0266403	0.0133202	−	0.999911i	$-0.495760\pi$
$947$	5.91308e17	0.0266403	0.0133202	+	0.999911i	$0.495760\pi$
$948$	6.29188e18	0.281531
$949$	−9.48950e18	−0.421709
$950$	0	0
$951$	1.77271e19	0.777079
$952$	5.38289e18	0.234356
$953$	2.39396e19	1.03517	0.517587	−	0.855631i	$-0.326830\pi$
$953$	2.39396e19	1.03517	0.517587	+	0.855631i	$0.326830\pi$
$954$	1.30177e18	0.0559073
$955$	0	0
$956$	3.49203e19	1.47945
$957$	1.02100e19	0.429633
$958$	−3.20055e18	−0.133767
$959$	4.03321e18	0.167429
$960$	0	0
$961$	1.35201e19	0.553702
$962$	−5.31100e17	−0.0216042
$963$	1.04377e19	0.421729
$964$	−2.43349e19	−0.976628
$965$	0	0
$966$	1.70808e18	0.0676327
$967$	−1.99045e18	−0.0782853	−0.0391427	−	0.999234i	$-0.512463\pi$
$967$	−1.99045e18	−0.0782853	−0.0391427	+	0.999234i	$0.512463\pi$
$968$	−1.76434e19	−0.689275
$969$	−1.83351e19	−0.711507
$970$	0	0
$971$	4.12385e19	1.57898	0.789492	−	0.613761i	$-0.210344\pi$
$971$	4.12385e19	1.57898	0.789492	+	0.613761i	$0.210344\pi$
$972$	−1.65701e18	−0.0630224
$973$	−2.48357e19	−0.938301
$974$	3.16262e18	0.118690
$975$	0	0
$976$	−8.54608e18	−0.316478
$977$	1.25374e19	0.461202	0.230601	−	0.973048i	$-0.425931\pi$
$977$	1.25374e19	0.461202	0.230601	+	0.973048i	$0.425931\pi$
$978$	−1.35340e18	−0.0494567
$979$	−2.50006e19	−0.907535
$980$	0	0
$981$	−5.21928e18	−0.186966
$982$	4.94080e18	0.175822
$983$	1.83966e19	0.650340	0.325170	−	0.945656i	$-0.394578\pi$
$983$	1.83966e19	0.650340	0.325170	+	0.945656i	$0.394578\pi$
$984$	6.64690e17	0.0233427
$985$	0	0
$986$	−1.76580e18	−0.0611987
$987$	−5.06062e18	−0.174238
$988$	−1.38676e19	−0.474331
$989$	5.90941e18	0.200802
$990$	0	0
$991$	−3.43064e19	−1.15053	−0.575263	−	0.817969i	$-0.695100\pi$
$991$	−3.43064e19	−1.15053	−0.575263	+	0.817969i	$0.695100\pi$
$992$	1.45333e19	0.484214
$993$	8.47329e18	0.280467
$994$	−3.70781e18	−0.121929
$995$	0	0
$996$	6.59465e18	0.214045
$997$	5.72458e19	1.84597	0.922985	−	0.384835i	$-0.125742\pi$
$997$	5.72458e19	1.84597	0.922985	+	0.384835i	$0.125742\pi$
$998$	4.79589e18	0.153646
$999$	2.13011e18	0.0677995

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.14.a.a.1.1		1
5.2	odd	4		75.14.b.b.49.2		2
5.3	odd	4		75.14.b.b.49.1		2
5.4	even	2		3.14.a.a.1.1	✓	1
15.14	odd	2		9.14.a.a.1.1		1
20.19	odd	2		48.14.a.c.1.1		1
35.34	odd	2		147.14.a.a.1.1		1
40.19	odd	2		192.14.a.e.1.1		1
40.29	even	2		192.14.a.j.1.1		1
60.59	even	2		144.14.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.14.a.a.1.1	✓	1	5.4	even	2
9.14.a.a.1.1		1	15.14	odd	2
48.14.a.c.1.1		1	20.19	odd	2
75.14.a.a.1.1		1	1.1	even	1	trivial
75.14.b.b.49.1		2	5.3	odd	4
75.14.b.b.49.2		2	5.2	odd	4
144.14.a.k.1.1		1	60.59	even	2
147.14.a.a.1.1		1	35.34	odd	2
192.14.a.e.1.1		1	40.19	odd	2
192.14.a.j.1.1		1	40.29	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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