LMFDB - Embedded newform 9.14.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9,14,Mod(1,9)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

N = Newforms(chi, 14, names="a")

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

chi := DirichletCharacter("9.1");

S:= CuspForms(chi, 14);

N := Newforms(S);

Level:	$ N $	$=$	$ 9 = 3^{2} $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	9.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$9.65078360567$
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	$\chi$	$=$	9.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+12.0000 q^{2} -8048.00 q^{4} +30210.0 q^{5} +235088. q^{7} -194880. q^{8} +O(q^{10})$

\(q+12.0000 q^{2} -8048.00 q^{4} +30210.0 q^{5} +235088. q^{7} -194880. q^{8} +362520. q^{10} +1.11829e7 q^{11} +8.04961e6 q^{13} +2.82106e6 q^{14} +6.35907e7 q^{16} +1.17495e8 q^{17} -2.14061e8 q^{19} -2.43130e8 q^{20} +1.34195e8 q^{22} -8.30556e8 q^{23} -3.08059e8 q^{25} +9.65954e7 q^{26} -1.89199e9 q^{28} +1.25240e9 q^{29} +6.15935e9 q^{31} +2.35954e9 q^{32} +1.40994e9 q^{34} +7.10201e9 q^{35} -5.49819e9 q^{37} -2.56874e9 q^{38} -5.88732e9 q^{40} +4.67869e9 q^{41} +7.11501e9 q^{43} -9.00000e10 q^{44} -9.96667e9 q^{46} +2.95288e10 q^{47} -4.16226e10 q^{49} -3.69671e9 q^{50} -6.47833e10 q^{52} +2.04125e11 q^{53} +3.37836e11 q^{55} -4.58139e10 q^{56} +1.50288e10 q^{58} +2.99098e10 q^{59} -1.34392e11 q^{61} +7.39122e10 q^{62} -4.92620e11 q^{64} +2.43179e11 q^{65} +3.48519e11 q^{67} -9.45597e11 q^{68} +8.52241e10 q^{70} -1.31434e12 q^{71} -1.17888e12 q^{73} -6.59783e10 q^{74} +1.72277e12 q^{76} +2.62897e12 q^{77} -1.07242e12 q^{79} +1.92107e12 q^{80} +5.61443e10 q^{82} -1.12403e12 q^{83} +3.54951e12 q^{85} +8.53802e10 q^{86} -2.17933e12 q^{88} -2.23561e12 q^{89} +1.89237e12 q^{91} +6.68431e12 q^{92} +3.54345e11 q^{94} -6.46679e12 q^{95} -1.42153e13 q^{97} -4.99472e11 q^{98} +O(q^{100})\)

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 $

$2$	12.0000	0.132583	0.0662913	−	0.997800i	$-0.478883\pi$
$2$	12.0000	0.132583	0.0662913	+	0.997800i	$0.478883\pi$
$3$	0	0
$3$	0	0
$4$	−8048.00	−0.982422
$5$	30210.0	0.864661	0.432330	−	0.901715i	$-0.357691\pi$
$5$	30210.0	0.864661	0.432330	+	0.901715i	$0.357691\pi$
$6$	0	0
$7$	235088.	0.755254	0.377627	−	0.925958i	$-0.376740\pi$
$7$	235088.	0.755254	0.377627	+	0.925958i	$0.376740\pi$
$8$	−194880.	−0.262834
$9$	0	0
$10$	362520.	0.114639
$11$	1.11829e7	1.90328	0.951639	−	0.307218i	$-0.0993981\pi$
$11$	1.11829e7	1.90328	0.951639	+	0.307218i	$0.0993981\pi$
$12$	0	0
$13$	8.04961e6	0.462534	0.231267	−	0.972890i	$-0.425713\pi$
$13$	8.04961e6	0.462534	0.231267	+	0.972890i	$0.425713\pi$
$14$	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	$0.299011\pi$
$18$	0	0
$19$	−2.14061e8	−1.04385	−0.521927	−	0.852990i	$-0.674787\pi$
$19$	−2.14061e8	−1.04385	−0.521927	+	0.852990i	$0.674787\pi$
$20$	−2.43130e8	−0.849462
$21$	0	0
$22$	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	$0.698880\pi$
$24$	0	0
$25$	−3.08059e8	−0.252362
$26$	9.65954e7	0.0613239
$27$	0	0
$28$	−1.89199e9	−0.741978
$29$	1.25240e9	0.390981	0.195491	−	0.980706i	$-0.437370\pi$
$29$	1.25240e9	0.390981	0.195491	+	0.980706i	$0.437370\pi$
$30$	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
$33$	0	0
$34$	1.40994e9	0.156526
$35$	7.10201e9	0.653039
$36$	0	0
$37$	−5.49819e9	−0.352297	−0.176148	−	0.984364i	$-0.556364\pi$
$37$	−5.49819e9	−0.352297	−0.176148	+	0.984364i	$0.556364\pi$
$38$	−2.56874e9	−0.138397
$39$	0	0
$40$	−5.88732e9	−0.227263
$41$	4.67869e9	0.153826	0.0769129	−	0.997038i	$-0.475494\pi$
$41$	4.67869e9	0.153826	0.0769129	+	0.997038i	$0.475494\pi$
$42$	0	0
$43$	7.11501e9	0.171645	0.0858224	−	0.996310i	$-0.472648\pi$
$43$	7.11501e9	0.171645	0.0858224	+	0.996310i	$0.472648\pi$
$44$	−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	$0.435973\pi$
$48$	0	0
$49$	−4.16226e10	−0.429591
$50$	−3.69671e9	−0.0334588
$51$	0	0
$52$	−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	$0.282021\pi$
$54$	0	0
$55$	3.37836e11	1.64569
$56$	−4.58139e10	−0.198507
$57$	0	0
$58$	1.50288e10	0.0518373
$59$	2.99098e10	0.0923157	0.0461579	−	0.998934i	$-0.485302\pi$
$59$	2.99098e10	0.0923157	0.0461579	+	0.998934i	$0.485302\pi$
$60$	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
$63$	0	0
$64$	−4.92620e11	−0.896071
$65$	2.43179e11	0.399935
$66$	0	0
$67$	3.48519e11	0.470695	0.235348	−	0.971911i	$-0.424377\pi$
$67$	3.48519e11	0.470695	0.235348	+	0.971911i	$0.424377\pi$
$68$	−9.45597e11	−1.15984
$69$	0	0
$70$	8.52241e10	0.0865815
$71$	−1.31434e12	−1.21766	−0.608831	−	0.793300i	$-0.708361\pi$
$71$	−1.31434e12	−1.21766	−0.608831	+	0.793300i	$0.708361\pi$
$72$	0	0
$73$	−1.17888e12	−0.911737	−0.455868	−	0.890047i	$-0.650671\pi$
$73$	−1.17888e12	−0.911737	−0.455868	+	0.890047i	$0.650671\pi$
$74$	−6.59783e10	−0.0467084
$75$	0	0
$76$	1.72277e12	1.02551
$77$	2.62897e12	1.43746
$78$	0	0
$79$	−1.07242e12	−0.496351	−0.248176	−	0.968715i	$-0.579831\pi$
$79$	−1.07242e12	−0.496351	−0.248176	+	0.968715i	$0.579831\pi$
$80$	1.92107e12	0.819330
$81$	0	0
$82$	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	$0.560423\pi$
$84$	0	0
$85$	3.54951e12	1.02081
$86$	8.53802e10	0.0227571
$87$	0	0
$88$	−2.17933e12	−0.500247
$89$	−2.23561e12	−0.476827	−0.238414	−	0.971164i	$-0.576627\pi$
$89$	−2.23561e12	−0.476827	−0.238414	+	0.971164i	$0.576627\pi$
$90$	0	0
$91$	1.89237e12	0.349330
$92$	6.68431e12	1.14931
$93$	0	0
$94$	3.54345e11	0.0529780
$95$	−6.46679e12	−0.902580
$96$	0	0
$97$	−1.42153e13	−1.73276	−0.866380	−	0.499385i	$-0.833559\pi$
$97$	−1.42153e13	−1.73276	−0.866380	+	0.499385i	$0.833559\pi$
$98$	−4.99472e11	−0.0569563
$99$	0	0
$100$	2.47926e12	0.247926
$101$	−1.70194e13	−1.59535	−0.797675	−	0.603088i	$-0.793937\pi$
$101$	−1.70194e13	−1.59535	−0.797675	+	0.603088i	$0.793937\pi$
$102$	0	0
$103$	1.09904e13	0.906928	0.453464	−	0.891275i	$-0.350188\pi$
$103$	1.09904e13	0.906928	0.453464	+	0.891275i	$0.350188\pi$
$104$	−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	$0.281991\pi$
$108$	0	0
$109$	−9.82099e12	−0.560897	−0.280448	−	0.959869i	$-0.590483\pi$
$109$	−9.82099e12	−0.560897	−0.280448	+	0.959869i	$0.590483\pi$
$110$	4.05403e12	0.218190
$111$	0	0
$112$	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	$0.374315\pi$
$114$	0	0
$115$	−2.50911e13	−1.01154
$116$	−1.00793e13	−0.384109
$117$	0	0
$118$	3.58918e11	0.0122395
$119$	2.76216e13	0.891648
$120$	0	0
$121$	9.05347e13	2.62247
$122$	−1.61270e12	−0.0442808
$123$	0	0
$124$	−4.95705e13	−1.22457
$125$	−4.61839e13	−1.08287
$126$	0	0
$127$	−4.49347e13	−0.950292	−0.475146	−	0.879907i	$-0.657605\pi$
$127$	−4.49347e13	−0.950292	−0.475146	+	0.879907i	$0.657605\pi$
$128$	−2.52408e13	−0.507270
$129$	0	0
$130$	2.91815e12	0.0530243
$131$	1.20182e12	0.0207768	0.0103884	−	0.999946i	$-0.496693\pi$
$131$	1.20182e12	0.0207768	0.0103884	+	0.999946i	$0.496693\pi$
$132$	0	0
$133$	−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	$0.535355\pi$
$138$	0	0
$139$	1.05644e14	1.24236	0.621182	−	0.783666i	$-0.286653\pi$
$139$	1.05644e14	1.24236	0.621182	+	0.783666i	$0.286653\pi$
$140$	−5.71570e13	−0.641559
$141$	0	0
$142$	−1.57720e13	−0.161441
$143$	9.00181e13	0.880330
$144$	0	0
$145$	3.78350e13	0.338066
$146$	−1.41465e13	−0.120880
$147$	0	0
$148$	4.42494e13	0.346104
$149$	8.53533e13	0.639012	0.319506	−	0.947584i	$-0.396483\pi$
$149$	8.53533e13	0.639012	0.319506	+	0.947584i	$0.396483\pi$
$150$	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
$153$	0	0
$154$	3.15476e13	0.190582
$155$	1.86074e14	1.07778
$156$	0	0
$157$	−1.18021e14	−0.628942	−0.314471	−	0.949267i	$-0.601827\pi$
$157$	−1.18021e14	−0.628942	−0.314471	+	0.949267i	$0.601827\pi$
$158$	−1.28690e13	−0.0658075
$159$	0	0
$160$	7.12818e13	0.335892
$161$	−1.95254e14	−0.883550
$162$	0	0
$163$	1.54710e14	0.646099	0.323050	−	0.946382i	$-0.395292\pi$
$163$	1.54710e14	0.646099	0.323050	+	0.946382i	$0.395292\pi$
$164$	−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	$0.733997\pi$
$168$	0	0
$169$	−2.38079e14	−0.786063
$170$	4.25942e13	0.135342
$171$	0	0
$172$	−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	$0.677697\pi$
$174$	0	0
$175$	−7.24210e13	−0.190597
$176$	7.11128e14	1.80350
$177$	0	0
$178$	−2.68273e13	−0.0632190
$179$	−4.23349e13	−0.0961952	−0.0480976	−	0.998843i	$-0.515316\pi$
$179$	−4.23349e13	−0.0961952	−0.0480976	+	0.998843i	$0.515316\pi$
$180$	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
$183$	0	0
$184$	1.61859e14	0.307482
$185$	−1.66100e14	−0.304617
$186$	0	0
$187$	1.31393e15	2.24700
$188$	−2.37648e14	−0.392561
$189$	0	0
$190$	−7.76015e13	−0.119666
$191$	8.62273e14	1.28507	0.642537	−	0.766255i	$-0.277882\pi$
$191$	8.62273e14	1.28507	0.642537	+	0.766255i	$0.277882\pi$
$192$	0	0
$193$	−9.37837e14	−1.30618	−0.653092	−	0.757278i	$-0.726529\pi$
$193$	−9.37837e14	−1.30618	−0.653092	+	0.757278i	$0.726529\pi$
$194$	−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	$0.365746\pi$
$198$	0	0
$199$	−4.36451e13	−0.0498185	−0.0249093	−	0.999690i	$-0.507930\pi$
$199$	−4.36451e13	−0.0498185	−0.0249093	+	0.999690i	$0.507930\pi$
$200$	6.00345e13	0.0663294
$201$	0	0
$202$	−2.04233e14	−0.211515
$203$	2.94424e14	0.295290
$204$	0	0
$205$	1.41343e14	0.133007
$206$	1.31885e14	0.120243
$207$	0	0
$208$	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
$213$	0	0
$214$	2.35684e14	0.167742
$215$	2.14945e14	0.148415
$216$	0	0
$217$	1.44799e15	0.941406
$218$	−1.17852e14	−0.0743651
$219$	0	0
$220$	−2.71890e15	−1.61676
$221$	9.45786e14	0.546064
$222$	0	0
$223$	1.47333e15	0.802266	0.401133	−	0.916020i	$-0.368617\pi$
$223$	1.47333e15	0.802266	0.401133	+	0.916020i	$0.368617\pi$
$224$	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	$0.136622\pi$
$228$	0	0
$229$	−1.47993e13	−0.00678126	−0.00339063	−	0.999994i	$-0.501079\pi$
$229$	−1.47993e13	−0.00678126	−0.00339063	+	0.999994i	$0.501079\pi$
$230$	−3.01093e14	−0.134113
$231$	0	0
$232$	−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	$0.766710\pi$
$234$	0	0
$235$	8.92064e14	0.345506
$236$	−2.40714e14	−0.0906930
$237$	0	0
$238$	3.31459e14	0.118217
$239$	4.33900e15	1.50592	0.752962	−	0.658063i	$-0.228624\pi$
$239$	4.33900e15	1.50592	0.752962	+	0.658063i	$0.228624\pi$
$240$	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
$243$	0	0
$244$	1.08159e15	0.328116
$245$	−1.25742e15	−0.371450
$246$	0	0
$247$	−1.72311e15	−0.482818
$248$	−1.20033e15	−0.327617
$249$	0	0
$250$	−5.54207e14	−0.143569
$251$	1.75146e15	0.442099	0.221050	−	0.975263i	$-0.429052\pi$
$251$	1.75146e15	0.442099	0.221050	+	0.975263i	$0.429052\pi$
$252$	0	0
$253$	−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	$0.676984\pi$
$258$	0	0
$259$	−1.29256e15	−0.266074
$260$	−1.95710e15	−0.392904
$261$	0	0
$262$	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	$0.643572\pi$
$264$	0	0
$265$	6.16662e15	1.09383
$266$	−6.03879e14	−0.104525
$267$	0	0
$268$	−2.80488e15	−0.462422
$269$	1.80262e15	0.290078	0.145039	−	0.989426i	$-0.453669\pi$
$269$	1.80262e15	0.290078	0.145039	+	0.989426i	$0.453669\pi$
$270$	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
$273$	0	0
$274$	−2.05874e14	−0.0293915
$275$	−3.44500e15	−0.480315
$276$	0	0
$277$	−1.07023e16	−1.42351	−0.711754	−	0.702428i	$-0.752099\pi$
$277$	−1.07023e16	−1.42351	−0.711754	+	0.702428i	$0.752099\pi$
$278$	1.26773e15	0.164716
$279$	0	0
$280$	−1.38404e15	−0.171641
$281$	2.45460e15	0.297433	0.148717	−	0.988880i	$-0.452486\pi$
$281$	2.45460e15	0.297433	0.148717	+	0.988880i	$0.452486\pi$
$282$	0	0
$283$	4.01155e15	0.464195	0.232098	−	0.972692i	$-0.425441\pi$
$283$	4.01155e15	0.464195	0.232098	+	0.972692i	$0.425441\pi$
$284$	1.05778e16	1.19626
$285$	0	0
$286$	1.08022e15	0.116716
$287$	1.09990e15	0.116178
$288$	0	0
$289$	3.90041e15	0.393799
$290$	4.54020e14	0.0448217
$291$	0	0
$292$	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	$0.530641\pi$
$294$	0	0
$295$	9.03576e14	0.0798218
$296$	1.07149e15	0.0925957
$297$	0	0
$298$	1.02424e15	0.0847219
$299$	−6.68565e15	−0.541104
$300$	0	0
$301$	1.67265e15	0.129636
$302$	−7.39697e14	−0.0561059
$303$	0	0
$304$	−1.36123e16	−0.989130
$305$	−4.05998e15	−0.288785
$306$	0	0
$307$	−1.32352e16	−0.902260	−0.451130	−	0.892458i	$-0.648979\pi$
$307$	−1.32352e16	−0.902260	−0.451130	+	0.892458i	$0.648979\pi$
$308$	−2.11579e16	−1.41219
$309$	0	0
$310$	2.23289e15	0.142895
$311$	8.09301e15	0.507187	0.253593	−	0.967311i	$-0.418388\pi$
$311$	8.09301e15	0.507187	0.253593	+	0.967311i	$0.418388\pi$
$312$	0	0
$313$	−1.48181e16	−0.890748	−0.445374	−	0.895345i	$-0.646929\pi$
$313$	−1.48181e16	−0.890748	−0.445374	+	0.895345i	$0.646929\pi$
$314$	−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	$0.265018\pi$
$318$	0	0
$319$	1.40055e16	0.744147
$320$	−1.48821e16	−0.774797
$321$	0	0
$322$	−2.34304e15	−0.117143
$323$	−2.51511e16	−1.23237
$324$	0	0
$325$	−2.47976e15	−0.116726
$326$	1.85652e15	0.0856615
$327$	0	0
$328$	−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
$333$	0	0
$334$	−4.51214e15	−0.177841
$335$	1.05288e16	0.406992
$336$	0	0
$337$	4.62652e16	1.72052	0.860262	−	0.509853i	$-0.170300\pi$
$337$	4.62652e16	1.72052	0.860262	+	0.509853i	$0.170300\pi$
$338$	−2.85695e15	−0.104218
$339$	0	0
$340$	−2.85665e16	−1.00287
$341$	6.88795e16	2.37239
$342$	0	0
$343$	−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	$0.523484\pi$
$348$	0	0
$349$	3.76900e16	1.11651	0.558253	−	0.829671i	$-0.311472\pi$
$349$	3.76900e16	1.11651	0.558253	+	0.829671i	$0.311472\pi$
$350$	−8.69052e14	−0.0252699
$351$	0	0
$352$	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	$0.729633\pi$
$354$	0	0
$355$	−3.97061e16	−1.05286
$356$	1.79922e16	0.468446
$357$	0	0
$358$	−5.08018e14	−0.0127538
$359$	−4.06616e16	−1.00247	−0.501234	−	0.865312i	$-0.667120\pi$
$359$	−4.06616e16	−1.00247	−0.501234	+	0.865312i	$0.667120\pi$
$360$	0	0
$361$	3.76929e15	0.0896320
$362$	−3.72536e15	−0.0870087
$363$	0	0
$364$	−1.52298e16	−0.343190
$365$	−3.56138e16	−0.788343
$366$	0	0
$367$	2.96733e16	0.633923	0.316961	−	0.948438i	$-0.397337\pi$
$367$	2.96733e16	0.633923	0.316961	+	0.948438i	$0.397337\pi$
$368$	−5.28156e16	−1.10854
$369$	0	0
$370$	−1.99320e15	−0.0403869
$371$	4.79873e16	0.955424
$372$	0	0
$373$	−9.01346e16	−1.73294	−0.866471	−	0.499227i	$-0.833617\pi$
$373$	−9.01346e16	−1.73294	−0.866471	+	0.499227i	$0.833617\pi$
$374$	1.57672e16	0.297912
$375$	0	0
$376$	−5.75457e15	−0.105025
$377$	1.00813e16	0.180842
$378$	0	0
$379$	−1.54841e16	−0.268369	−0.134184	−	0.990956i	$-0.542841\pi$
$379$	−1.54841e16	−0.268369	−0.134184	+	0.990956i	$0.542841\pi$
$380$	5.20448e16	0.886714
$381$	0	0
$382$	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	$0.524167\pi$
$384$	0	0
$385$	7.94211e16	1.24291
$386$	−1.12540e16	−0.173177
$387$	0	0
$388$	1.14404e17	1.70230
$389$	−2.95806e16	−0.432847	−0.216423	−	0.976300i	$-0.569439\pi$
$389$	−2.95806e16	−0.432847	−0.216423	+	0.976300i	$0.569439\pi$
$390$	0	0
$391$	−9.75858e16	−1.38114
$392$	8.11142e15	0.112911
$393$	0	0
$394$	8.06058e15	0.108553
$395$	−3.23978e16	−0.429175
$396$	0	0
$397$	1.80617e16	0.231538	0.115769	−	0.993276i	$-0.463067\pi$
$397$	1.80617e16	0.231538	0.115769	+	0.993276i	$0.463067\pi$
$398$	−5.23742e14	−0.00660507
$399$	0	0
$400$	−1.95897e16	−0.239132
$401$	1.20412e17	1.44621	0.723107	−	0.690736i	$-0.242713\pi$
$401$	1.20412e17	1.44621	0.723107	+	0.690736i	$0.242713\pi$
$402$	0	0
$403$	4.95804e16	0.576537
$404$	1.36972e17	1.56731
$405$	0	0
$406$	3.53309e15	0.0391503
$407$	−6.14858e16	−0.670519
$408$	0	0
$409$	−1.77522e16	−0.187521	−0.0937606	−	0.995595i	$-0.529889\pi$
$409$	−1.77522e16	−0.187521	−0.0937606	+	0.995595i	$0.529889\pi$
$410$	1.69612e15	0.0176344
$411$	0	0
$412$	−8.84510e16	−0.890986
$413$	7.03144e15	0.0697219
$414$	0	0
$415$	−3.39568e16	−0.326298
$416$	1.89934e16	0.179679
$417$	0	0
$418$	−2.87259e16	−0.263408
$419$	−1.75670e17	−1.58602	−0.793008	−	0.609212i	$-0.791486\pi$
$419$	−1.75670e17	−1.58602	−0.793008	+	0.609212i	$0.791486\pi$
$420$	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
$423$	0	0
$424$	−3.97799e16	−0.332495
$425$	−3.61953e16	−0.297937
$426$	0	0
$427$	−3.15939e16	−0.252245
$428$	−1.58065e17	−1.24295
$429$	0	0
$430$	2.57933e15	0.0196772
$431$	−8.05532e16	−0.605314	−0.302657	−	0.953100i	$-0.597874\pi$
$431$	−8.05532e16	−0.605314	−0.302657	+	0.953100i	$0.597874\pi$
$432$	0	0
$433$	−1.97092e17	−1.43714	−0.718568	−	0.695457i	$-0.755202\pi$
$433$	−1.97092e17	−1.43714	−0.718568	+	0.695457i	$0.755202\pi$
$434$	1.73759e16	0.124814
$435$	0	0
$436$	7.90393e16	0.551037
$437$	1.77790e17	1.22117
$438$	0	0
$439$	9.89007e16	0.659447	0.329724	−	0.944078i	$-0.393044\pi$
$439$	9.89007e16	0.659447	0.329724	+	0.944078i	$0.393044\pi$
$440$	−6.58374e16	−0.432544
$441$	0	0
$442$	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	$0.371367\pi$
$444$	0	0
$445$	−6.75378e16	−0.412294
$446$	1.76800e16	0.106366
$447$	0	0
$448$	−1.15809e17	−0.676761
$449$	1.80095e17	1.03729	0.518645	−	0.854990i	$-0.326437\pi$
$449$	1.80095e17	1.03729	0.518645	+	0.854990i	$0.326437\pi$
$450$	0	0
$451$	5.23213e16	0.292773
$452$	−1.37031e17	−0.755820
$453$	0	0
$454$	4.49867e16	0.241113
$455$	5.71684e16	0.302052
$456$	0	0
$457$	−9.43597e16	−0.484542	−0.242271	−	0.970209i	$-0.577892\pi$
$457$	−9.43597e16	−0.484542	−0.242271	+	0.970209i	$0.577892\pi$
$458$	−1.77592e14	−0.000899076 0
$459$	0	0
$460$	2.01933e17	0.993760
$461$	−8.00500e16	−0.388423	−0.194212	−	0.980960i	$-0.562215\pi$
$461$	−8.00500e16	−0.388423	−0.194212	+	0.980960i	$0.562215\pi$
$462$	0	0
$463$	2.14174e17	1.01039	0.505196	−	0.863004i	$-0.331420\pi$
$463$	2.14174e17	1.01039	0.505196	+	0.863004i	$0.331420\pi$
$464$	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	$0.367962\pi$
$468$	0	0
$469$	8.19326e16	0.355495
$470$	1.07048e16	0.0458080
$471$	0	0
$472$	−5.82883e15	−0.0242638
$473$	7.95665e16	0.326688
$474$	0	0
$475$	6.59435e16	0.263429
$476$	−2.22298e17	−0.875974
$477$	0	0
$478$	5.20680e16	0.199659
$479$	2.66712e17	1.00893	0.504466	−	0.863431i	$-0.331689\pi$
$479$	2.66712e17	1.00893	0.504466	+	0.863431i	$0.331689\pi$
$480$	0	0
$481$	−4.42583e16	−0.162949
$482$	3.62847e16	0.131801
$483$	0	0
$484$	−7.28623e17	−2.57637
$485$	−4.29443e17	−1.49825
$486$	0	0
$487$	−2.63552e17	−0.895216	−0.447608	−	0.894230i	$-0.647724\pi$
$487$	−2.63552e17	−0.895216	−0.447608	+	0.894230i	$0.647724\pi$
$488$	2.61903e16	0.0877833
$489$	0	0
$490$	−1.50890e16	−0.0492478
$491$	−4.11733e17	−1.32613	−0.663065	−	0.748562i	$-0.730745\pi$
$491$	−4.11733e17	−1.32613	−0.663065	+	0.748562i	$0.730745\pi$
$492$	0	0
$493$	1.47150e17	0.461590
$494$	−2.06773e16	−0.0640132
$495$	0	0
$496$	3.91677e17	1.18113
$497$	−3.08984e17	−0.919645
$498$	0	0
$499$	3.99658e17	1.15887	0.579435	−	0.815018i	$-0.303273\pi$
$499$	3.99658e17	1.15887	0.579435	+	0.815018i	$0.303273\pi$
$500$	3.71688e17	1.06383
$501$	0	0
$502$	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	$0.372354\pi$
$504$	0	0
$505$	−5.14157e17	−1.37944
$506$	−1.11456e17	−0.295207
$507$	0	0
$508$	3.61634e17	0.933588
$509$	−6.40327e17	−1.63206	−0.816030	−	0.578009i	$-0.803830\pi$
$509$	−6.40327e17	−1.63206	−0.816030	+	0.578009i	$0.803830\pi$
$510$	0	0
$511$	−2.77140e17	−0.688593
$512$	2.51565e17	0.617156
$513$	0	0
$514$	−5.85124e16	−0.139955
$515$	3.32021e17	0.784185
$516$	0	0
$517$	3.30218e17	0.760522
$518$	−1.55107e16	−0.0352767
$519$	0	0
$520$	−4.73907e16	−0.105117
$521$	4.01348e17	0.879175	0.439588	−	0.898200i	$-0.355125\pi$
$521$	4.01348e17	0.879175	0.439588	+	0.898200i	$0.355125\pi$
$522$	0	0
$523$	−5.05985e17	−1.08113	−0.540564	−	0.841303i	$-0.681789\pi$
$523$	−5.05985e17	−1.08113	−0.540564	+	0.841303i	$0.681789\pi$
$524$	−9.67228e15	−0.0204115
$525$	0	0
$526$	−5.61459e16	−0.115587
$527$	7.23691e17	1.47158
$528$	0	0
$529$	1.85786e17	0.368597
$530$	7.39994e16	0.145022
$531$	0	0
$532$	4.05002e17	0.774517
$533$	3.76616e16	0.0711496
$534$	0	0
$535$	5.93334e17	1.09396
$536$	−6.79193e16	−0.123715
$537$	0	0
$538$	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
$543$	0	0
$544$	2.77234e17	0.458620
$545$	−2.96692e17	−0.484986
$546$	0	0
$547$	−4.32104e17	−0.689717	−0.344858	−	0.938655i	$-0.612073\pi$
$547$	−4.32104e17	−0.689717	−0.344858	+	0.938655i	$0.612073\pi$
$548$	1.38073e17	0.217788
$549$	0	0
$550$	−4.13399e16	−0.0636814
$551$	−2.68091e17	−0.408128
$552$	0	0
$553$	−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	$0.922536\pi$
$558$	0	0
$559$	5.72731e16	0.0793915
$560$	4.51621e17	0.618803
$561$	0	0
$562$	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	$0.282968\pi$
$564$	0	0
$565$	5.14377e17	0.665221
$566$	4.81386e16	0.0615442
$567$	0	0
$568$	2.56138e17	0.320044
$569$	−1.53632e17	−0.189780	−0.0948902	−	0.995488i	$-0.530250\pi$
$569$	−1.53632e17	−0.189780	−0.0948902	+	0.995488i	$0.530250\pi$
$570$	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
$573$	0	0
$574$	1.31988e16	0.0154031
$575$	2.55860e17	0.295231
$576$	0	0
$577$	3.56770e17	0.402481	0.201241	−	0.979542i	$-0.435503\pi$
$577$	3.56770e17	0.402481	0.201241	+	0.979542i	$0.435503\pi$
$578$	4.68049e16	0.0522108
$579$	0	0
$580$	−3.04496e17	−0.332124
$581$	−2.64245e17	−0.285011
$582$	0	0
$583$	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	$0.725477\pi$
$588$	0	0
$589$	−1.31848e18	−1.30114
$590$	1.08429e16	0.0105830
$591$	0	0
$592$	−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	$0.150190\pi$
$594$	0	0
$595$	8.34448e17	0.770973
$596$	−6.86923e17	−0.627780
$597$	0	0
$598$	−8.02278e16	−0.0717410
$599$	1.44668e18	1.27967	0.639834	−	0.768513i	$-0.279003\pi$
$599$	1.44668e18	1.27967	0.639834	+	0.768513i	$0.279003\pi$
$600$	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
$603$	0	0
$604$	4.96090e17	0.415739
$605$	2.73505e18	2.26755
$606$	0	0
$607$	2.98050e16	0.0241860	0.0120930	−	0.999927i	$-0.496151\pi$
$607$	2.98050e16	0.0241860	0.0120930	+	0.999927i	$0.496151\pi$
$608$	−5.05087e17	−0.405502
$609$	0	0
$610$	−4.87198e16	−0.0382879
$611$	2.37695e17	0.184822
$612$	0	0
$613$	−8.84082e17	−0.672976	−0.336488	−	0.941688i	$-0.609239\pi$
$613$	−8.84082e17	−0.672976	−0.336488	+	0.941688i	$0.609239\pi$
$614$	−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	$0.675648\pi$
$618$	0	0
$619$	1.68862e18	1.20654	0.603272	−	0.797535i	$-0.293863\pi$
$619$	1.68862e18	1.20654	0.603272	+	0.797535i	$0.293863\pi$
$620$	−1.49752e18	−1.05883
$621$	0	0
$622$	9.71162e16	0.0672441
$623$	−5.25565e17	−0.360126
$624$	0	0
$625$	−1.01917e18	−0.683951
$626$	−1.77817e17	−0.118098
$627$	0	0
$628$	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
$633$	0	0
$634$	2.91805e17	0.178448
$635$	−1.35748e18	−0.821680
$636$	0	0
$637$	−3.35046e17	−0.198700
$638$	1.68066e17	0.0986608
$639$	0	0
$640$	−7.62526e17	−0.438616
$641$	−1.61802e18	−0.921313	−0.460656	−	0.887579i	$-0.652386\pi$
$641$	−1.61802e18	−0.921313	−0.460656	+	0.887579i	$0.652386\pi$
$642$	0	0
$643$	−1.96065e18	−1.09403	−0.547015	−	0.837123i	$-0.684236\pi$
$643$	−1.96065e18	−1.09403	−0.547015	+	0.837123i	$0.684236\pi$
$644$	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	$0.448933\pi$
$648$	0	0
$649$	3.34479e17	0.175703
$650$	−2.97571e16	−0.0154758
$651$	0	0
$652$	−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	$0.273967\pi$
$654$	0	0
$655$	3.63071e16	0.0179648
$656$	2.97521e17	0.145761
$657$	0	0
$658$	8.33023e16	0.0400119
$659$	−2.64137e18	−1.25624	−0.628121	−	0.778116i	$-0.716176\pi$
$659$	−2.64137e18	−1.25624	−0.628121	+	0.778116i	$0.716176\pi$
$660$	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
$663$	0	0
$664$	2.19050e17	0.0991861
$665$	−1.52027e18	−0.681677
$666$	0	0
$667$	−1.04019e18	−0.457398
$668$	3.02614e18	1.31778
$669$	0	0
$670$	1.26345e17	0.0539600
$671$	−1.50289e18	−0.635670
$672$	0	0
$673$	2.79726e18	1.16047	0.580236	−	0.814449i	$-0.302961\pi$
$673$	2.79726e18	1.16047	0.580236	+	0.814449i	$0.302961\pi$
$674$	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	$0.176981\pi$
$678$	0	0
$679$	−3.34184e18	−1.30867
$680$	−6.91729e17	−0.268305
$681$	0	0
$682$	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	$0.598065\pi$
$684$	0	0
$685$	−5.18287e17	−0.191682
$686$	−3.90749e17	−0.143150
$687$	0	0
$688$	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
$693$	0	0
$694$	−5.75285e16	−0.0195455
$695$	3.19151e18	1.07422
$696$	0	0
$697$	5.49721e17	0.181606
$698$	4.52280e17	0.148029
$699$	0	0
$700$	5.82844e17	0.187247
$701$	−2.99144e18	−0.952166	−0.476083	−	0.879400i	$-0.657944\pi$
$701$	−2.99144e18	−0.952166	−0.476083	+	0.879400i	$0.657944\pi$
$702$	0	0
$703$	1.17695e18	0.367747
$704$	−5.50893e18	−1.70547
$705$	0	0
$706$	−5.76215e17	−0.175127
$707$	−4.00106e18	−1.20489
$708$	0	0
$709$	−5.31694e18	−1.57203	−0.786015	−	0.618207i	$-0.787859\pi$
$709$	−5.31694e18	−1.57203	−0.786015	+	0.618207i	$0.787859\pi$
$710$	−4.76473e17	−0.139591
$711$	0	0
$712$	4.35676e17	0.125327
$713$	−5.11568e18	−1.45822
$714$	0	0
$715$	2.71945e18	0.761187
$716$	3.40711e17	0.0945043
$717$	0	0
$718$	−4.87939e17	−0.132910
$719$	4.03153e18	1.08826	0.544129	−	0.839001i	$-0.316860\pi$
$719$	4.03153e18	1.08826	0.544129	+	0.839001i	$0.316860\pi$
$720$	0	0
$721$	2.58372e18	0.684961
$722$	4.52315e16	0.0118836
$723$	0	0
$724$	2.49848e18	0.644725
$725$	−3.85813e17	−0.0986688
$726$	0	0
$727$	−4.77643e18	−1.19986	−0.599928	−	0.800054i	$-0.704804\pi$
$727$	−4.77643e18	−1.19986	−0.599928	+	0.800054i	$0.704804\pi$
$728$	−3.68785e17	−0.0918161
$729$	0	0
$730$	−4.27366e17	−0.104520
$731$	8.35976e17	0.202643
$732$	0	0
$733$	1.71668e18	0.408803	0.204401	−	0.978887i	$-0.434475\pi$
$733$	1.71668e18	0.408803	0.204401	+	0.978887i	$0.434475\pi$
$734$	3.56080e17	0.0840471
$735$	0	0
$736$	−1.95973e18	−0.454455
$737$	3.89745e18	0.895864
$738$	0	0
$739$	−8.69723e17	−0.196423	−0.0982114	−	0.995166i	$-0.531312\pi$
$739$	−8.69723e17	−0.196423	−0.0982114	+	0.995166i	$0.531312\pi$
$740$	1.33678e18	0.299263
$741$	0	0
$742$	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	$0.584371\pi$
$744$	0	0
$745$	2.57852e18	0.552529
$746$	−1.08161e18	−0.229758
$747$	0	0
$748$	−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
$753$	0	0
$754$	1.20976e17	0.0239765
$755$	−1.86219e18	−0.365905
$756$	0	0
$757$	3.09120e18	0.597040	0.298520	−	0.954403i	$-0.403507\pi$
$757$	3.09120e18	0.597040	0.298520	+	0.954403i	$0.403507\pi$
$758$	−1.85810e17	−0.0355810
$759$	0	0
$760$	1.26025e18	0.237229
$761$	7.97787e18	1.48897	0.744486	−	0.667638i	$-0.232694\pi$
$761$	7.97787e18	1.48897	0.744486	+	0.667638i	$0.232694\pi$
$762$	0	0
$763$	−2.30880e18	−0.423620
$764$	−6.93957e18	−1.26248
$765$	0	0
$766$	−1.12451e17	−0.0201129
$767$	2.40763e17	0.0426991
$768$	0	0
$769$	7.37344e18	1.28573	0.642863	−	0.765981i	$-0.277746\pi$
$769$	7.37344e18	1.28573	0.642863	+	0.765981i	$0.277746\pi$
$770$	9.53053e17	0.164789
$771$	0	0
$772$	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	$0.454950\pi$
$774$	0	0
$775$	−1.89744e18	−0.314563
$776$	2.77027e18	0.455429
$777$	0	0
$778$	−3.54967e17	−0.0573879
$779$	−1.00153e18	−0.160572
$780$	0	0
$781$	−1.46981e19	−2.31755
$782$	−1.17103e18	−0.183115
$783$	0	0
$784$	−2.64681e18	−0.407069
$785$	−3.56541e18	−0.543822
$786$	0	0
$787$	−3.75359e18	−0.563133	−0.281566	−	0.959542i	$-0.590854\pi$
$787$	−3.75359e18	−0.563133	−0.281566	+	0.959542i	$0.590854\pi$
$788$	−5.40596e18	−0.804363
$789$	0	0
$790$	−3.88774e17	−0.0569012
$791$	4.00277e18	0.581050
$792$	0	0
$793$	−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	$0.575232\pi$
$798$	0	0
$799$	3.46947e18	0.471747
$800$	−7.26879e17	−0.0980341
$801$	0	0
$802$	1.44495e18	0.191743
$803$	−1.31833e19	−1.73529
$804$	0	0
$805$	−5.89861e18	−0.763971
$806$	5.94965e17	0.0764387
$807$	0	0
$808$	3.31674e18	0.419313
$809$	3.13119e18	0.392685	0.196343	−	0.980535i	$-0.437094\pi$
$809$	3.13119e18	0.392685	0.196343	+	0.980535i	$0.437094\pi$
$810$	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
$813$	0	0
$814$	−7.37829e17	−0.0888991
$815$	4.67379e18	0.558657
$816$	0	0
$817$	−1.52305e18	−0.179172
$818$	−2.13026e17	−0.0248620
$819$	0	0
$820$	−1.13753e18	−0.130669
$821$	6.85162e18	0.780841	0.390421	−	0.920637i	$-0.372330\pi$
$821$	6.85162e18	0.780841	0.390421	+	0.920637i	$0.372330\pi$
$822$	0	0
$823$	3.06934e17	0.0344308	0.0172154	−	0.999852i	$-0.494520\pi$
$823$	3.06934e17	0.0344308	0.0172154	+	0.999852i	$0.494520\pi$
$824$	−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	$0.638460\pi$
$828$	0	0
$829$	2.34336e18	0.250747	0.125373	−	0.992110i	$-0.459987\pi$
$829$	2.34336e18	0.250747	0.125373	+	0.992110i	$0.459987\pi$
$830$	−4.07482e17	−0.0432614
$831$	0	0
$832$	−3.96540e18	−0.414463
$833$	−4.89044e18	−0.507172
$834$	0	0
$835$	−1.13593e19	−1.15982
$836$	1.92655e19	1.95182
$837$	0	0
$838$	−2.10804e18	−0.210278
$839$	1.63297e19	1.61632	0.808158	−	0.588965i	$-0.200465\pi$
$839$	1.63297e19	1.61632	0.808158	+	0.588965i	$0.200465\pi$
$840$	0	0
$841$	−8.69212e18	−0.847134
$842$	2.21368e18	0.214085
$843$	0	0
$844$	1.30508e19	1.24283
$845$	−7.19236e18	−0.679677
$846$	0	0
$847$	2.12836e19	1.98063
$848$	1.29804e19	1.19872
$849$	0	0
$850$	−4.34343e17	−0.0395012
$851$	4.56655e18	0.412142
$852$	0	0
$853$	1.93794e19	1.72255	0.861276	−	0.508137i	$-0.169666\pi$
$853$	1.93794e19	1.72255	0.861276	+	0.508137i	$0.169666\pi$
$854$	−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	$0.673940\pi$
$858$	0	0
$859$	−1.00612e19	−0.854465	−0.427232	−	0.904142i	$-0.640511\pi$
$859$	−1.00612e19	−0.854465	−0.427232	+	0.904142i	$0.640511\pi$
$860$	−1.72987e18	−0.145806
$861$	0	0
$862$	−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	$0.607110\pi$
$864$	0	0
$865$	−1.12853e19	−0.916029
$866$	−2.36510e18	−0.190539
$867$	0	0
$868$	−1.16534e19	−0.924858
$869$	−1.19928e19	−0.944694
$870$	0	0
$871$	2.80544e18	0.217712
$872$	1.91391e18	0.147423
$873$	0	0
$874$	2.13348e18	0.161906
$875$	−1.08573e19	−0.817841
$876$	0	0
$877$	8.87791e17	0.0658891	0.0329445	−	0.999457i	$-0.489512\pi$
$877$	8.87791e17	0.0658891	0.0329445	+	0.999457i	$0.489512\pi$
$878$	1.18681e18	0.0874312
$879$	0	0
$880$	2.14832e19	1.55941
$881$	−3.78774e18	−0.272921	−0.136460	−	0.990646i	$-0.543573\pi$
$881$	−3.78774e18	−0.272921	−0.136460	+	0.990646i	$0.543573\pi$
$882$	0	0
$883$	−2.75428e19	−1.95553	−0.977763	−	0.209712i	$-0.932747\pi$
$883$	−2.75428e19	−1.95553	−0.977763	+	0.209712i	$0.932747\pi$
$884$	−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	$0.0644489\pi$
$888$	0	0
$889$	−1.05636e19	−0.717712
$890$	−8.10454e17	−0.0546630
$891$	0	0
$892$	−1.18574e19	−0.788164
$893$	−6.32097e18	−0.417109
$894$	0	0
$895$	−1.27894e18	−0.0831762
$896$	−5.93382e18	−0.383118
$897$	0	0
$898$	2.16114e18	0.137527
$899$	7.71397e18	0.487349
$900$	0	0
$901$	2.39836e19	1.49349
$902$	6.27856e17	0.0388166
$903$	0	0
$904$	−3.31816e18	−0.202210
$905$	−9.37860e18	−0.567443
$906$	0	0
$907$	7.51023e18	0.447926	0.223963	−	0.974598i	$-0.428101\pi$
$907$	7.51023e18	0.447926	0.223963	+	0.974598i	$0.428101\pi$
$908$	−3.01711e19	−1.78662
$909$	0	0
$910$	6.86021e17	0.0400469
$911$	2.34028e19	1.35643	0.678216	−	0.734863i	$-0.262753\pi$
$911$	2.34028e19	1.35643	0.678216	+	0.734863i	$0.262753\pi$
$912$	0	0
$913$	−1.25699e19	−0.718242
$914$	−1.13232e18	−0.0642418
$915$	0	0
$916$	1.19105e17	0.00666206
$917$	2.82535e17	0.0156917
$918$	0	0
$919$	2.20881e19	1.20950	0.604752	−	0.796414i	$-0.293272\pi$
$919$	2.20881e19	1.20950	0.604752	+	0.796414i	$0.293272\pi$
$920$	4.88975e18	0.265868
$921$	0	0
$922$	−9.60600e17	−0.0514982
$923$	−1.05799e19	−0.563210
$924$	0	0
$925$	1.69377e18	0.0889063
$926$	2.57009e18	0.133960
$927$	0	0
$928$	2.95509e18	0.151883
$929$	1.05632e19	0.539132	0.269566	−	0.962982i	$-0.413120\pi$
$929$	1.05632e19	0.539132	0.269566	+	0.962982i	$0.413120\pi$
$930$	0	0
$931$	8.90980e18	0.448430
$932$	2.92185e19	1.46034
$933$	0	0
$934$	2.16817e18	0.106866
$935$	3.96939e19	1.94289
$936$	0	0
$937$	−1.72833e19	−0.834294	−0.417147	−	0.908839i	$-0.636970\pi$
$937$	−1.72833e19	−0.834294	−0.417147	+	0.908839i	$0.636970\pi$
$938$	9.83191e17	0.0471324
$939$	0	0
$940$	−7.17933e18	−0.339432
$941$	−2.14038e19	−1.00498	−0.502490	−	0.864583i	$-0.667583\pi$
$941$	−2.14038e19	−1.00498	−0.502490	+	0.864583i	$0.667583\pi$
$942$	0	0
$943$	−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	$0.495760\pi$
$948$	0	0
$949$	−9.48950e18	−0.421709
$950$	7.91322e17	0.0349261
$951$	0	0
$952$	−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	$0.326830\pi$
$954$	0	0
$955$	2.60493e19	1.11115
$956$	−3.49203e19	−1.47945
$957$	0	0
$958$	3.20055e18	0.133767
$959$	−4.03321e18	−0.167429
$960$	0	0
$961$	1.35201e19	0.553702
$962$	−5.31100e17	−0.0216042
$963$	0	0
$964$	−2.43349e19	−0.976628
$965$	−2.83320e19	−1.12941
$966$	0	0
$967$	1.99045e18	0.0782853	0.0391427	−	0.999234i	$-0.487537\pi$
$967$	1.99045e18	0.0782853	0.0391427	+	0.999234i	$0.487537\pi$
$968$	−1.76434e19	−0.689275
$969$	0	0
$970$	−5.15332e18	−0.198642
$971$	−4.12385e19	−1.57898	−0.789492	−	0.613761i	$-0.789656\pi$
$971$	−4.12385e19	−1.57898	−0.789492	+	0.613761i	$0.789656\pi$
$972$	0	0
$973$	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	$0.425931\pi$
$978$	0	0
$979$	−2.50006e19	−0.907535
$980$	1.01197e19	0.364921
$981$	0	0
$982$	−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	$0.394578\pi$
$984$	0	0
$985$	2.02925e19	0.707946
$986$	1.76580e18	0.0611987
$987$	0	0
$988$	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
$993$	0	0
$994$	−3.70781e18	−0.121929
$995$	−1.31852e18	−0.0430761
$996$	0	0
$997$	−5.72458e19	−1.84597	−0.922985	−	0.384835i	$-0.874258\pi$
$997$	−5.72458e19	−1.84597	−0.922985	+	0.384835i	$0.874258\pi$
$998$	4.79589e18	0.153646
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.14.a.a.1.1		1
3.2	odd	2		3.14.a.a.1.1	✓	1
4.3	odd	2		144.14.a.k.1.1		1
12.11	even	2		48.14.a.c.1.1		1
15.2	even	4		75.14.b.b.49.1		2
15.8	even	4		75.14.b.b.49.2		2
15.14	odd	2		75.14.a.a.1.1		1
21.20	even	2		147.14.a.a.1.1		1
24.5	odd	2		192.14.a.j.1.1		1
24.11	even	2		192.14.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.14.a.a.1.1	✓	1	3.2	odd	2
9.14.a.a.1.1		1	1.1	even	1	trivial
48.14.a.c.1.1		1	12.11	even	2
75.14.a.a.1.1		1	15.14	odd	2
75.14.b.b.49.1		2	15.2	even	4
75.14.b.b.49.2		2	15.8	even	4
144.14.a.k.1.1		1	4.3	odd	2
147.14.a.a.1.1		1	21.20	even	2
192.14.a.e.1.1		1	24.11	even	2
192.14.a.j.1.1		1	24.5	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	9.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more