LMFDB - Embedded newform 9.12.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9,12,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, 12, names="a")

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

chi := DirichletCharacter("9.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 9 = 3^{2} $
Weight:	$ k $	$=$	$ 12 $
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:	$6.91508862504$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1)
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)

$f(q)$

$=$

$q+24.0000 q^{2} -1472.00 q^{4} -4830.00 q^{5} -16744.0 q^{7} -84480.0 q^{8} +O(q^{10})$

\(q+24.0000 q^{2} -1472.00 q^{4} -4830.00 q^{5} -16744.0 q^{7} -84480.0 q^{8} -115920. q^{10} -534612. q^{11} -577738. q^{13} -401856. q^{14} +987136. q^{16} +6.90593e6 q^{17} +1.06614e7 q^{19} +7.10976e6 q^{20} -1.28307e7 q^{22} -1.86433e7 q^{23} -2.54992e7 q^{25} -1.38657e7 q^{26} +2.46472e7 q^{28} -1.28407e8 q^{29} -5.28432e7 q^{31} +1.96706e8 q^{32} +1.65742e8 q^{34} +8.08735e7 q^{35} -1.82213e8 q^{37} +2.55874e8 q^{38} +4.08038e8 q^{40} -3.08120e8 q^{41} -1.71257e7 q^{43} +7.86949e8 q^{44} -4.47439e8 q^{46} -2.68735e9 q^{47} -1.69697e9 q^{49} -6.11981e8 q^{50} +8.50430e8 q^{52} +1.59606e9 q^{53} +2.58218e9 q^{55} +1.41453e9 q^{56} -3.08176e9 q^{58} +5.18920e9 q^{59} +6.95648e9 q^{61} -1.26824e9 q^{62} +2.69930e9 q^{64} +2.79047e9 q^{65} -1.54818e10 q^{67} -1.01655e10 q^{68} +1.94096e9 q^{70} -9.79149e9 q^{71} +1.46379e9 q^{73} -4.37312e9 q^{74} -1.56936e10 q^{76} +8.95154e9 q^{77} +3.81168e10 q^{79} -4.76787e9 q^{80} -7.39489e9 q^{82} +2.93351e10 q^{83} -3.33557e10 q^{85} -4.11017e8 q^{86} +4.51640e10 q^{88} +2.49929e10 q^{89} +9.67365e9 q^{91} +2.74429e10 q^{92} -6.44964e10 q^{94} -5.14947e10 q^{95} +7.50136e10 q^{97} -4.07272e10 q^{98} +O(q^{100})\)

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 $

$2$	24.0000	0.530330	0.265165	−	0.964203i	$-0.414574\pi$
$2$	24.0000	0.530330	0.265165	+	0.964203i	$0.414574\pi$
$3$	0	0
$3$	0	0
$4$	−1472.00	−0.718750
$5$	−4830.00	−0.691213	−0.345607	−	0.938379i	$-0.612327\pi$
$5$	−4830.00	−0.691213	−0.345607	+	0.938379i	$0.612327\pi$
$6$	0	0
$7$	−16744.0	−0.376548	−0.188274	−	0.982117i	$-0.560289\pi$
$7$	−16744.0	−0.376548	−0.188274	+	0.982117i	$0.560289\pi$
$8$	−84480.0	−0.911505
$9$	0	0
$10$	−115920.	−0.366571
$11$	−534612.	−1.00087	−0.500436	−	0.865773i	$-0.666827\pi$
$11$	−534612.	−1.00087	−0.500436	+	0.865773i	$0.666827\pi$
$12$	0	0
$13$	−577738.	−0.431561	−0.215781	−	0.976442i	$-0.569230\pi$
$13$	−577738.	−0.431561	−0.215781	+	0.976442i	$0.569230\pi$
$14$	−401856.	−0.199695
$15$	0	0
$16$	987136.	0.235352
$17$	6.90593e6	1.17965	0.589825	−	0.807531i	$-0.299197\pi$
$17$	6.90593e6	1.17965	0.589825	+	0.807531i	$0.299197\pi$
$18$	0	0
$19$	1.06614e7	0.987803	0.493901	−	0.869518i	$-0.335570\pi$
$19$	1.06614e7	0.987803	0.493901	+	0.869518i	$0.335570\pi$
$20$	7.10976e6	0.496810
$21$	0	0
$22$	−1.28307e7	−0.530793
$23$	−1.86433e7	−0.603975	−0.301988	−	0.953312i	$-0.597650\pi$
$23$	−1.86433e7	−0.603975	−0.301988	+	0.953312i	$0.597650\pi$
$24$	0	0
$25$	−2.54992e7	−0.522224
$26$	−1.38657e7	−0.228870
$27$	0	0
$28$	2.46472e7	0.270644
$29$	−1.28407e8	−1.16251	−0.581257	−	0.813720i	$-0.697439\pi$
$29$	−1.28407e8	−1.16251	−0.581257	+	0.813720i	$0.697439\pi$
$30$	0	0
$31$	−5.28432e7	−0.331512	−0.165756	−	0.986167i	$-0.553006\pi$
$31$	−5.28432e7	−0.331512	−0.165756	+	0.986167i	$0.553006\pi$
$32$	1.96706e8	1.03632
$33$	0	0
$34$	1.65742e8	0.625604
$35$	8.08735e7	0.260275
$36$	0	0
$37$	−1.82213e8	−0.431987	−0.215993	−	0.976395i	$-0.569299\pi$
$37$	−1.82213e8	−0.431987	−0.215993	+	0.976395i	$0.569299\pi$
$38$	2.55874e8	0.523862
$39$	0	0
$40$	4.08038e8	0.630044
$41$	−3.08120e8	−0.415345	−0.207673	−	0.978198i	$-0.566589\pi$
$41$	−3.08120e8	−0.415345	−0.207673	+	0.978198i	$0.566589\pi$
$42$	0	0
$43$	−1.71257e7	−0.0177653	−0.00888264	−	0.999961i	$-0.502827\pi$
$43$	−1.71257e7	−0.0177653	−0.00888264	+	0.999961i	$0.502827\pi$
$44$	7.86949e8	0.719377
$45$	0	0
$46$	−4.47439e8	−0.320306
$47$	−2.68735e9	−1.70917	−0.854586	−	0.519310i	$-0.826189\pi$
$47$	−2.68735e9	−1.70917	−0.854586	+	0.519310i	$0.826189\pi$
$48$	0	0
$49$	−1.69697e9	−0.858212
$50$	−6.11981e8	−0.276951
$51$	0	0
$52$	8.50430e8	0.310185
$53$	1.59606e9	0.524241	0.262120	−	0.965035i	$-0.415578\pi$
$53$	1.59606e9	0.524241	0.262120	+	0.965035i	$0.415578\pi$
$54$	0	0
$55$	2.58218e9	0.691817
$56$	1.41453e9	0.343225
$57$	0	0
$58$	−3.08176e9	−0.616517
$59$	5.18920e9	0.944963	0.472481	−	0.881341i	$-0.343358\pi$
$59$	5.18920e9	0.944963	0.472481	+	0.881341i	$0.343358\pi$
$60$	0	0
$61$	6.95648e9	1.05457	0.527285	−	0.849689i	$-0.323210\pi$
$61$	6.95648e9	1.05457	0.527285	+	0.849689i	$0.323210\pi$
$62$	−1.26824e9	−0.175811
$63$	0	0
$64$	2.69930e9	0.314240
$65$	2.79047e9	0.298301
$66$	0	0
$67$	−1.54818e10	−1.40091	−0.700456	−	0.713696i	$-0.747020\pi$
$67$	−1.54818e10	−1.40091	−0.700456	+	0.713696i	$0.747020\pi$
$68$	−1.01655e10	−0.847874
$69$	0	0
$70$	1.94096e9	0.138032
$71$	−9.79149e9	−0.644062	−0.322031	−	0.946729i	$-0.604366\pi$
$71$	−9.79149e9	−0.644062	−0.322031	+	0.946729i	$0.604366\pi$
$72$	0	0
$73$	1.46379e9	0.0826425	0.0413212	−	0.999146i	$-0.486843\pi$
$73$	1.46379e9	0.0826425	0.0413212	+	0.999146i	$0.486843\pi$
$74$	−4.37312e9	−0.229096
$75$	0	0
$76$	−1.56936e10	−0.709983
$77$	8.95154e9	0.376876
$78$	0	0
$79$	3.81168e10	1.39370	0.696848	−	0.717219i	$-0.254585\pi$
$79$	3.81168e10	1.39370	0.696848	+	0.717219i	$0.254585\pi$
$80$	−4.76787e9	−0.162678
$81$	0	0
$82$	−7.39489e9	−0.220270
$83$	2.93351e10	0.817444	0.408722	−	0.912659i	$-0.365975\pi$
$83$	2.93351e10	0.817444	0.408722	+	0.912659i	$0.365975\pi$
$84$	0	0
$85$	−3.33557e10	−0.815390
$86$	−4.11017e8	−0.00942146
$87$	0	0
$88$	4.51640e10	0.912300
$89$	2.49929e10	0.474430	0.237215	−	0.971457i	$-0.423765\pi$
$89$	2.49929e10	0.474430	0.237215	+	0.971457i	$0.423765\pi$
$90$	0	0
$91$	9.67365e9	0.162503
$92$	2.74429e10	0.434107
$93$	0	0
$94$	−6.44964e10	−0.906425
$95$	−5.14947e10	−0.682782
$96$	0	0
$97$	7.50136e10	0.886942	0.443471	−	0.896289i	$-0.353747\pi$
$97$	7.50136e10	0.886942	0.443471	+	0.896289i	$0.353747\pi$
$98$	−4.07272e10	−0.455136
$99$	0	0
$100$	3.75349e10	0.375349
$101$	−8.17430e10	−0.773896	−0.386948	−	0.922101i	$-0.626471\pi$
$101$	−8.17430e10	−0.773896	−0.386948	+	0.922101i	$0.626471\pi$
$102$	0	0
$103$	−2.25755e11	−1.91881	−0.959407	−	0.282025i	$-0.908994\pi$
$103$	−2.25755e11	−1.91881	−0.959407	+	0.282025i	$0.908994\pi$
$104$	4.88073e10	0.393370
$105$	0	0
$106$	3.83053e10	0.278021
$107$	−9.02413e10	−0.622006	−0.311003	−	0.950409i	$-0.600665\pi$
$107$	−9.02413e10	−0.622006	−0.311003	+	0.950409i	$0.600665\pi$
$108$	0	0
$109$	7.34827e10	0.457445	0.228723	−	0.973492i	$-0.426545\pi$
$109$	7.34827e10	0.457445	0.228723	+	0.973492i	$0.426545\pi$
$110$	6.19722e10	0.366891
$111$	0	0
$112$	−1.65286e10	−0.0886211
$113$	8.51469e10	0.434748	0.217374	−	0.976088i	$-0.430251\pi$
$113$	8.51469e10	0.434748	0.217374	+	0.976088i	$0.430251\pi$
$114$	0	0
$115$	9.00470e10	0.417476
$116$	1.89015e11	0.835557
$117$	0	0
$118$	1.24541e11	0.501142
$119$	−1.15633e11	−0.444195
$120$	0	0
$121$	4.98320e8	0.00174658
$122$	1.66955e11	0.559270
$123$	0	0
$124$	7.77851e10	0.238274
$125$	3.59001e11	1.05218
$126$	0	0
$127$	−2.62717e11	−0.705615	−0.352808	−	0.935696i	$-0.614773\pi$
$127$	−2.62717e11	−0.705615	−0.352808	+	0.935696i	$0.614773\pi$
$128$	−3.38071e11	−0.869668
$129$	0	0
$130$	6.69714e10	0.158198
$131$	−6.31529e11	−1.43021	−0.715107	−	0.699015i	$-0.753622\pi$
$131$	−6.31529e11	−1.43021	−0.715107	+	0.699015i	$0.753622\pi$
$132$	0	0
$133$	−1.78515e11	−0.371955
$134$	−3.71564e11	−0.742946
$135$	0	0
$136$	−5.83413e11	−1.07526
$137$	2.97199e11	0.526119	0.263059	−	0.964780i	$-0.415268\pi$
$137$	2.97199e11	0.526119	0.263059	+	0.964780i	$0.415268\pi$
$138$	0	0
$139$	5.96794e11	0.975535	0.487767	−	0.872974i	$-0.337811\pi$
$139$	5.96794e11	0.975535	0.487767	+	0.872974i	$0.337811\pi$
$140$	−1.19046e11	−0.187073
$141$	0	0
$142$	−2.34996e11	−0.341565
$143$	3.08866e11	0.431938
$144$	0	0
$145$	6.20204e11	0.803546
$146$	3.51310e10	0.0438278
$147$	0	0
$148$	2.68218e11	0.310491
$149$	1.11543e12	1.24428	0.622142	−	0.782905i	$-0.286263\pi$
$149$	1.11543e12	1.24428	0.622142	+	0.782905i	$0.286263\pi$
$150$	0	0
$151$	−8.24447e11	−0.854653	−0.427326	−	0.904097i	$-0.640544\pi$
$151$	−8.24447e11	−0.854653	−0.427326	+	0.904097i	$0.640544\pi$
$152$	−9.00677e11	−0.900387
$153$	0	0
$154$	2.14837e11	0.199869
$155$	2.55233e11	0.229146
$156$	0	0
$157$	1.31512e12	1.10031	0.550156	−	0.835062i	$-0.314568\pi$
$157$	1.31512e12	1.10031	0.550156	+	0.835062i	$0.314568\pi$
$158$	9.14804e11	0.739119
$159$	0	0
$160$	−9.50091e11	−0.716317
$161$	3.12163e11	0.227425
$162$	0	0
$163$	−3.57833e11	−0.243584	−0.121792	−	0.992556i	$-0.538864\pi$
$163$	−3.57833e11	−0.243584	−0.121792	+	0.992556i	$0.538864\pi$
$164$	4.53553e11	0.298529
$165$	0	0
$166$	7.04042e11	0.433515
$167$	−2.75483e12	−1.64117	−0.820587	−	0.571521i	$-0.806354\pi$
$167$	−2.75483e12	−1.64117	−0.820587	+	0.571521i	$0.806354\pi$
$168$	0	0
$169$	−1.45838e12	−0.813755
$170$	−8.00536e11	−0.432426
$171$	0	0
$172$	2.52090e10	0.0127688
$173$	9.50387e11	0.466280	0.233140	−	0.972443i	$-0.425100\pi$
$173$	9.50387e11	0.466280	0.233140	+	0.972443i	$0.425100\pi$
$174$	0	0
$175$	4.26959e11	0.196642
$176$	−5.27735e11	−0.235557
$177$	0	0
$178$	5.99830e11	0.251604
$179$	−1.68138e12	−0.683873	−0.341936	−	0.939723i	$-0.611083\pi$
$179$	−1.68138e12	−0.683873	−0.341936	+	0.939723i	$0.611083\pi$
$180$	0	0
$181$	−9.96774e11	−0.381386	−0.190693	−	0.981650i	$-0.561073\pi$
$181$	−9.96774e11	−0.381386	−0.190693	+	0.981650i	$0.561073\pi$
$182$	2.32167e11	0.0861804
$183$	0	0
$184$	1.57498e12	0.550526
$185$	8.80090e11	0.298595
$186$	0	0
$187$	−3.69200e12	−1.18068
$188$	3.95578e12	1.22847
$189$	0	0
$190$	−1.23587e12	−0.362100
$191$	−2.76240e12	−0.786328	−0.393164	−	0.919468i	$-0.628619\pi$
$191$	−2.76240e12	−0.786328	−0.393164	+	0.919468i	$0.628619\pi$
$192$	0	0
$193$	5.44239e12	1.46293	0.731466	−	0.681878i	$-0.238836\pi$
$193$	5.44239e12	1.46293	0.731466	+	0.681878i	$0.238836\pi$
$194$	1.80033e12	0.470372
$195$	0	0
$196$	2.49793e12	0.616840
$197$	2.87609e12	0.690619	0.345309	−	0.938489i	$-0.387774\pi$
$197$	2.87609e12	0.690619	0.345309	+	0.938489i	$0.387774\pi$
$198$	0	0
$199$	7.28391e11	0.165452	0.0827262	−	0.996572i	$-0.473637\pi$
$199$	7.28391e11	0.165452	0.0827262	+	0.996572i	$0.473637\pi$
$200$	2.15417e12	0.476010
$201$	0	0
$202$	−1.96183e12	−0.410421
$203$	2.15004e12	0.437742
$204$	0	0
$205$	1.48822e12	0.287092
$206$	−5.41812e12	−1.01760
$207$	0	0
$208$	−5.70306e11	−0.101569
$209$	−5.69972e12	−0.988665
$210$	0	0
$211$	−6.79317e12	−1.11820	−0.559099	−	0.829101i	$-0.688853\pi$
$211$	−6.79317e12	−1.11820	−0.559099	+	0.829101i	$0.688853\pi$
$212$	−2.34939e12	−0.376798
$213$	0	0
$214$	−2.16579e12	−0.329868
$215$	8.27172e10	0.0122796
$216$	0	0
$217$	8.84806e11	0.124830
$218$	1.76358e12	0.242597
$219$	0	0
$220$	−3.80096e12	−0.497243
$221$	−3.98982e12	−0.509092
$222$	0	0
$223$	7.33486e12	0.890667	0.445333	−	0.895365i	$-0.353085\pi$
$223$	7.33486e12	0.890667	0.445333	+	0.895365i	$0.353085\pi$
$224$	−3.29365e12	−0.390223
$225$	0	0
$226$	2.04352e12	0.230560
$227$	1.35984e12	0.149743	0.0748713	−	0.997193i	$-0.476145\pi$
$227$	1.35984e12	0.149743	0.0748713	+	0.997193i	$0.476145\pi$
$228$	0	0
$229$	−1.18244e13	−1.24075	−0.620375	−	0.784305i	$-0.713020\pi$
$229$	−1.18244e13	−1.24075	−0.620375	+	0.784305i	$0.713020\pi$
$230$	2.16113e12	0.221400
$231$	0	0
$232$	1.08478e13	1.05964
$233$	1.75634e13	1.67552	0.837761	−	0.546038i	$-0.183865\pi$
$233$	1.75634e13	1.67552	0.837761	+	0.546038i	$0.183865\pi$
$234$	0	0
$235$	1.29799e13	1.18140
$236$	−7.63851e12	−0.679192
$237$	0	0
$238$	−2.77519e12	−0.235570
$239$	7.13958e12	0.592221	0.296111	−	0.955154i	$-0.404310\pi$
$239$	7.13958e12	0.592221	0.296111	+	0.955154i	$0.404310\pi$
$240$	0	0
$241$	−2.31307e11	−0.0183271	−0.00916357	−	0.999958i	$-0.502917\pi$
$241$	−2.31307e11	−0.0183271	−0.00916357	+	0.999958i	$0.502917\pi$
$242$	1.19597e10	0.000926264 0
$243$	0	0
$244$	−1.02399e13	−0.757972
$245$	8.19634e12	0.593207
$246$	0	0
$247$	−6.15951e12	−0.426297
$248$	4.46419e12	0.302175
$249$	0	0
$250$	8.61603e12	0.558004
$251$	−1.29831e13	−0.822567	−0.411284	−	0.911507i	$-0.634919\pi$
$251$	−1.29831e13	−0.822567	−0.411284	+	0.911507i	$0.634919\pi$
$252$	0	0
$253$	9.96692e12	0.604502
$254$	−6.30521e12	−0.374209
$255$	0	0
$256$	−1.36419e13	−0.775451
$257$	−2.39612e13	−1.33314	−0.666571	−	0.745442i	$-0.732239\pi$
$257$	−2.39612e13	−1.33314	−0.666571	+	0.745442i	$0.732239\pi$
$258$	0	0
$259$	3.05098e12	0.162664
$260$	−4.10758e12	−0.214404
$261$	0	0
$262$	−1.51567e13	−0.758485
$263$	2.42737e13	1.18954	0.594771	−	0.803895i	$-0.297243\pi$
$263$	2.42737e13	1.18954	0.594771	+	0.803895i	$0.297243\pi$
$264$	0	0
$265$	−7.70895e12	−0.362362
$266$	−4.28436e12	−0.197259
$267$	0	0
$268$	2.27892e13	1.00691
$269$	−2.58377e13	−1.11845	−0.559225	−	0.829016i	$-0.688901\pi$
$269$	−2.58377e13	−1.11845	−0.559225	+	0.829016i	$0.688901\pi$
$270$	0	0
$271$	−3.76793e12	−0.156593	−0.0782964	−	0.996930i	$-0.524948\pi$
$271$	−3.76793e12	−0.156593	−0.0782964	+	0.996930i	$0.524948\pi$
$272$	6.81710e12	0.277633
$273$	0	0
$274$	7.13277e12	0.279017
$275$	1.36322e13	0.522680
$276$	0	0
$277$	−1.64189e13	−0.604931	−0.302466	−	0.953160i	$-0.597810\pi$
$277$	−1.64189e13	−0.604931	−0.302466	+	0.953160i	$0.597810\pi$
$278$	1.43230e13	0.517355
$279$	0	0
$280$	−6.83219e12	−0.237242
$281$	−2.10357e13	−0.716263	−0.358132	−	0.933671i	$-0.616586\pi$
$281$	−2.10357e13	−0.716263	−0.358132	+	0.933671i	$0.616586\pi$
$282$	0	0
$283$	1.67132e13	0.547310	0.273655	−	0.961828i	$-0.411767\pi$
$283$	1.67132e13	0.547310	0.273655	+	0.961828i	$0.411767\pi$
$284$	1.44131e13	0.462920
$285$	0	0
$286$	7.41278e12	0.229070
$287$	5.15917e12	0.156397
$288$	0	0
$289$	1.34200e13	0.391575
$290$	1.48849e13	0.426144
$291$	0	0
$292$	−2.15470e12	−0.0593993
$293$	2.39269e13	0.647312	0.323656	−	0.946175i	$-0.395088\pi$
$293$	2.39269e13	0.647312	0.323656	+	0.946175i	$0.395088\pi$
$294$	0	0
$295$	−2.50639e13	−0.653171
$296$	1.53934e13	0.393758
$297$	0	0
$298$	2.67704e13	0.659881
$299$	1.07709e13	0.260652
$300$	0	0
$301$	2.86753e11	0.00668947
$302$	−1.97867e13	−0.453248
$303$	0	0
$304$	1.05243e13	0.232481
$305$	−3.35998e13	−0.728933
$306$	0	0
$307$	1.53111e13	0.320439	0.160219	−	0.987081i	$-0.448780\pi$
$307$	1.53111e13	0.320439	0.160219	+	0.987081i	$0.448780\pi$
$308$	−1.31767e13	−0.270880
$309$	0	0
$310$	6.12558e12	0.121523
$311$	−4.98752e13	−0.972080	−0.486040	−	0.873936i	$-0.661559\pi$
$311$	−4.98752e13	−0.972080	−0.486040	+	0.873936i	$0.661559\pi$
$312$	0	0
$313$	−9.94808e13	−1.87174	−0.935870	−	0.352345i	$-0.885384\pi$
$313$	−9.94808e13	−1.87174	−0.935870	+	0.352345i	$0.885384\pi$
$314$	3.15628e13	0.583529
$315$	0	0
$316$	−5.61080e13	−1.00172
$317$	−8.33692e13	−1.46278	−0.731392	−	0.681958i	$-0.761129\pi$
$317$	−8.33692e13	−1.46278	−0.731392	+	0.681958i	$0.761129\pi$
$318$	0	0
$319$	6.86477e13	1.16353
$320$	−1.30376e13	−0.217207
$321$	0	0
$322$	7.49191e12	0.120611
$323$	7.36271e13	1.16526
$324$	0	0
$325$	1.47319e13	0.225372
$326$	−8.58799e12	−0.129180
$327$	0	0
$328$	2.60300e13	0.378589
$329$	4.49970e13	0.643585
$330$	0	0
$331$	−6.35840e13	−0.879618	−0.439809	−	0.898091i	$-0.644954\pi$
$331$	−6.35840e13	−0.879618	−0.439809	+	0.898091i	$0.644954\pi$
$332$	−4.31813e13	−0.587538
$333$	0	0
$334$	−6.61160e13	−0.870364
$335$	7.47772e13	0.968329
$336$	0	0
$337$	1.21001e14	1.51644	0.758221	−	0.651997i	$-0.226069\pi$
$337$	1.21001e14	1.51644	0.758221	+	0.651997i	$0.226069\pi$
$338$	−3.50011e13	−0.431559
$339$	0	0
$340$	4.90995e13	0.586062
$341$	2.82506e13	0.331802
$342$	0	0
$343$	6.15223e13	0.699705
$344$	1.44678e12	0.0161931
$345$	0	0
$346$	2.28093e13	0.247283
$347$	1.55662e14	1.66100	0.830499	−	0.557020i	$-0.188055\pi$
$347$	1.55662e14	1.66100	0.830499	+	0.557020i	$0.188055\pi$
$348$	0	0
$349$	−2.56430e13	−0.265112	−0.132556	−	0.991176i	$-0.542318\pi$
$349$	−2.56430e13	−0.265112	−0.132556	+	0.991176i	$0.542318\pi$
$350$	1.02470e13	0.104285
$351$	0	0
$352$	−1.05162e14	−1.03722
$353$	−2.49098e13	−0.241885	−0.120943	−	0.992659i	$-0.538592\pi$
$353$	−2.49098e13	−0.241885	−0.120943	+	0.992659i	$0.538592\pi$
$354$	0	0
$355$	4.72929e13	0.445184
$356$	−3.67896e13	−0.340996
$357$	0	0
$358$	−4.03532e13	−0.362678
$359$	−1.57584e14	−1.39474	−0.697370	−	0.716712i	$-0.745646\pi$
$359$	−1.57584e14	−1.39474	−0.697370	+	0.716712i	$0.745646\pi$
$360$	0	0
$361$	−2.82438e12	−0.0242457
$362$	−2.39226e13	−0.202260
$363$	0	0
$364$	−1.42396e13	−0.116799
$365$	−7.07011e12	−0.0571236
$366$	0	0
$367$	−1.77901e14	−1.39481	−0.697406	−	0.716676i	$-0.745662\pi$
$367$	−1.77901e14	−1.39481	−0.697406	+	0.716676i	$0.745662\pi$
$368$	−1.84034e13	−0.142146
$369$	0	0
$370$	2.11222e13	0.158354
$371$	−2.67244e13	−0.197402
$372$	0	0
$373$	−5.51617e13	−0.395585	−0.197792	−	0.980244i	$-0.563377\pi$
$373$	−5.51617e13	−0.395585	−0.197792	+	0.980244i	$0.563377\pi$
$374$	−8.86079e13	−0.626150
$375$	0	0
$376$	2.27027e14	1.55792
$377$	7.41854e13	0.501696
$378$	0	0
$379$	1.46463e14	0.962083	0.481042	−	0.876698i	$-0.340259\pi$
$379$	1.46463e14	0.962083	0.481042	+	0.876698i	$0.340259\pi$
$380$	7.58001e13	0.490750
$381$	0	0
$382$	−6.62977e13	−0.417013
$383$	−2.31450e14	−1.43504	−0.717519	−	0.696539i	$-0.754722\pi$
$383$	−2.31450e14	−1.43504	−0.717519	+	0.696539i	$0.754722\pi$
$384$	0	0
$385$	−4.32360e13	−0.260502
$386$	1.30617e14	0.775837
$387$	0	0
$388$	−1.10420e14	−0.637490
$389$	1.49872e14	0.853093	0.426547	−	0.904466i	$-0.359730\pi$
$389$	1.49872e14	0.853093	0.426547	+	0.904466i	$0.359730\pi$
$390$	0	0
$391$	−1.28749e14	−0.712480
$392$	1.43360e14	0.782264
$393$	0	0
$394$	6.90262e13	0.366256
$395$	−1.84104e14	−0.963341
$396$	0	0
$397$	2.08111e14	1.05912	0.529562	−	0.848271i	$-0.322356\pi$
$397$	2.08111e14	1.05912	0.529562	+	0.848271i	$0.322356\pi$
$398$	1.74814e13	0.0877443
$399$	0	0
$400$	−2.51712e13	−0.122906
$401$	1.33408e14	0.642521	0.321261	−	0.946991i	$-0.395893\pi$
$401$	1.33408e14	0.642521	0.321261	+	0.946991i	$0.395893\pi$
$402$	0	0
$403$	3.05295e13	0.143068
$404$	1.20326e14	0.556238
$405$	0	0
$406$	5.16010e13	0.232148
$407$	9.74134e13	0.432364
$408$	0	0
$409$	−2.06168e14	−0.890722	−0.445361	−	0.895351i	$-0.646925\pi$
$409$	−2.06168e14	−0.890722	−0.445361	+	0.895351i	$0.646925\pi$
$410$	3.57173e13	0.152254
$411$	0	0
$412$	3.32312e14	1.37915
$413$	−8.68880e13	−0.355824
$414$	0	0
$415$	−1.41689e14	−0.565028
$416$	−1.13645e14	−0.447235
$417$	0	0
$418$	−1.36793e14	−0.524319
$419$	−7.34035e13	−0.277677	−0.138838	−	0.990315i	$-0.544337\pi$
$419$	−7.34035e13	−0.277677	−0.138838	+	0.990315i	$0.544337\pi$
$420$	0	0
$421$	1.71112e14	0.630563	0.315282	−	0.948998i	$-0.397901\pi$
$421$	1.71112e14	0.630563	0.315282	+	0.948998i	$0.397901\pi$
$422$	−1.63036e14	−0.593014
$423$	0	0
$424$	−1.34835e14	−0.477848
$425$	−1.76096e14	−0.616042
$426$	0	0
$427$	−1.16479e14	−0.397096
$428$	1.32835e14	0.447067
$429$	0	0
$430$	1.98521e12	0.00651224
$431$	7.17758e13	0.232463	0.116231	−	0.993222i	$-0.462919\pi$
$431$	7.17758e13	0.232463	0.116231	+	0.993222i	$0.462919\pi$
$432$	0	0
$433$	9.98812e13	0.315356	0.157678	−	0.987491i	$-0.449599\pi$
$433$	9.98812e13	0.315356	0.157678	+	0.987491i	$0.449599\pi$
$434$	2.12353e13	0.0662012
$435$	0	0
$436$	−1.08166e14	−0.328789
$437$	−1.98764e14	−0.596608
$438$	0	0
$439$	−2.90312e13	−0.0849788	−0.0424894	−	0.999097i	$-0.513529\pi$
$439$	−2.90312e13	−0.0849788	−0.0424894	+	0.999097i	$0.513529\pi$
$440$	−2.18142e14	−0.630594
$441$	0	0
$442$	−9.57557e13	−0.269987
$443$	−3.28370e14	−0.914414	−0.457207	−	0.889360i	$-0.651150\pi$
$443$	−3.28370e14	−0.914414	−0.457207	+	0.889360i	$0.651150\pi$
$444$	0	0
$445$	−1.20716e14	−0.327932
$446$	1.76037e14	0.472347
$447$	0	0
$448$	−4.51970e13	−0.118326
$449$	6.12368e14	1.58364	0.791822	−	0.610752i	$-0.209133\pi$
$449$	6.12368e14	1.58364	0.791822	+	0.610752i	$0.209133\pi$
$450$	0	0
$451$	1.64725e14	0.415708
$452$	−1.25336e14	−0.312475
$453$	0	0
$454$	3.26361e13	0.0794130
$455$	−4.67237e13	−0.112325
$456$	0	0
$457$	3.03483e14	0.712189	0.356095	−	0.934450i	$-0.384108\pi$
$457$	3.03483e14	0.712189	0.356095	+	0.934450i	$0.384108\pi$
$458$	−2.83786e14	−0.658007
$459$	0	0
$460$	−1.32549e14	−0.300061
$461$	7.29308e14	1.63138	0.815691	−	0.578487i	$-0.196357\pi$
$461$	7.29308e14	1.63138	0.815691	+	0.578487i	$0.196357\pi$
$462$	0	0
$463$	1.22188e14	0.266891	0.133445	−	0.991056i	$-0.457396\pi$
$463$	1.22188e14	0.266891	0.133445	+	0.991056i	$0.457396\pi$
$464$	−1.26755e14	−0.273600
$465$	0	0
$466$	4.21520e14	0.888579
$467$	6.17381e14	1.28621	0.643103	−	0.765780i	$-0.277647\pi$
$467$	6.17381e14	1.28621	0.643103	+	0.765780i	$0.277647\pi$
$468$	0	0
$469$	2.59228e14	0.527510
$470$	3.11517e14	0.626533
$471$	0	0
$472$	−4.38384e14	−0.861338
$473$	9.15561e12	0.0177808
$474$	0	0
$475$	−2.71858e14	−0.515854
$476$	1.70212e14	0.319265
$477$	0	0
$478$	1.71350e14	0.314073
$479$	−1.05084e15	−1.90410	−0.952052	−	0.305938i	$-0.901030\pi$
$479$	−1.05084e15	−1.90410	−0.952052	+	0.305938i	$0.901030\pi$
$480$	0	0
$481$	1.05272e14	0.186429
$482$	−5.55137e12	−0.00971944
$483$	0	0
$484$	−7.33527e11	−0.00125536
$485$	−3.62316e14	−0.613066
$486$	0	0
$487$	−2.19910e14	−0.363777	−0.181889	−	0.983319i	$-0.558221\pi$
$487$	−2.19910e14	−0.363777	−0.181889	+	0.983319i	$0.558221\pi$
$488$	−5.87683e14	−0.961246
$489$	0	0
$490$	1.96712e14	0.314596
$491$	4.83863e14	0.765199	0.382599	−	0.923914i	$-0.375029\pi$
$491$	4.83863e14	0.765199	0.382599	+	0.923914i	$0.375029\pi$
$492$	0	0
$493$	−8.86768e14	−1.37136
$494$	−1.47828e14	−0.226078
$495$	0	0
$496$	−5.21634e13	−0.0780219
$497$	1.63949e14	0.242520
$498$	0	0
$499$	−1.08878e14	−0.157538	−0.0787691	−	0.996893i	$-0.525099\pi$
$499$	−1.08878e14	−0.157538	−0.0787691	+	0.996893i	$0.525099\pi$
$500$	−5.28450e14	−0.756256
$501$	0	0
$502$	−3.11593e14	−0.436232
$503$	−5.06588e14	−0.701506	−0.350753	−	0.936468i	$-0.614074\pi$
$503$	−5.06588e14	−0.701506	−0.350753	+	0.936468i	$0.614074\pi$
$504$	0	0
$505$	3.94818e14	0.534927
$506$	2.39206e14	0.320586
$507$	0	0
$508$	3.86720e14	0.507161
$509$	−8.57534e13	−0.111251	−0.0556254	−	0.998452i	$-0.517715\pi$
$509$	−8.57534e13	−0.111251	−0.0556254	+	0.998452i	$0.517715\pi$
$510$	0	0
$511$	−2.45097e13	−0.0311188
$512$	3.64965e14	0.458423
$513$	0	0
$514$	−5.75069e14	−0.707005
$515$	1.09040e15	1.32631
$516$	0	0
$517$	1.43669e15	1.71066
$518$	7.32235e13	0.0862654
$519$	0	0
$520$	−2.35739e14	−0.271903
$521$	−9.27575e14	−1.05862	−0.529312	−	0.848428i	$-0.677550\pi$
$521$	−9.27575e14	−1.05862	−0.529312	+	0.848428i	$0.677550\pi$
$522$	0	0
$523$	−2.18187e13	−0.0243820	−0.0121910	−	0.999926i	$-0.503881\pi$
$523$	−2.18187e13	−0.0243820	−0.0121910	+	0.999926i	$0.503881\pi$
$524$	9.29610e14	1.02797
$525$	0	0
$526$	5.82569e14	0.630850
$527$	−3.64931e14	−0.391069
$528$	0	0
$529$	−6.05238e14	−0.635214
$530$	−1.85015e14	−0.192172
$531$	0	0
$532$	2.62774e14	0.267343
$533$	1.78013e14	0.179247
$534$	0	0
$535$	4.35865e14	0.429939
$536$	1.30790e15	1.27694
$537$	0	0
$538$	−6.20105e14	−0.593147
$539$	9.07218e14	0.858961
$540$	0	0
$541$	−1.69527e15	−1.57273	−0.786363	−	0.617765i	$-0.788038\pi$
$541$	−1.69527e15	−1.57273	−0.786363	+	0.617765i	$0.788038\pi$
$542$	−9.04304e13	−0.0830459
$543$	0	0
$544$	1.35844e15	1.22249
$545$	−3.54921e14	−0.316192
$546$	0	0
$547$	7.52145e14	0.656706	0.328353	−	0.944555i	$-0.393506\pi$
$547$	7.52145e14	0.656706	0.328353	+	0.944555i	$0.393506\pi$
$548$	−4.37477e14	−0.378148
$549$	0	0
$550$	3.27173e14	0.277193
$551$	−1.36900e15	−1.14834
$552$	0	0
$553$	−6.38228e14	−0.524793
$554$	−3.94054e14	−0.320813
$555$	0	0
$556$	−8.78480e14	−0.701166
$557$	−1.87489e14	−0.148174	−0.0740870	−	0.997252i	$-0.523604\pi$
$557$	−1.87489e14	−0.148174	−0.0740870	+	0.997252i	$0.523604\pi$
$558$	0	0
$559$	9.89417e12	0.00766681
$560$	7.98332e13	0.0612561
$561$	0	0
$562$	−5.04857e14	−0.379856
$563$	−2.44971e14	−0.182524	−0.0912618	−	0.995827i	$-0.529090\pi$
$563$	−2.44971e14	−0.182524	−0.0912618	+	0.995827i	$0.529090\pi$
$564$	0	0
$565$	−4.11259e14	−0.300503
$566$	4.01116e14	0.290255
$567$	0	0
$568$	8.27185e14	0.587066
$569$	−1.35243e15	−0.950596	−0.475298	−	0.879825i	$-0.657660\pi$
$569$	−1.35243e15	−0.950596	−0.475298	+	0.879825i	$0.657660\pi$
$570$	0	0
$571$	1.43223e15	0.987447	0.493723	−	0.869619i	$-0.335636\pi$
$571$	1.43223e15	0.987447	0.493723	+	0.869619i	$0.335636\pi$
$572$	−4.54650e14	−0.310455
$573$	0	0
$574$	1.23820e14	0.0829422
$575$	4.75389e14	0.315410
$576$	0	0
$577$	−8.77659e14	−0.571293	−0.285647	−	0.958335i	$-0.592208\pi$
$577$	−8.77659e14	−0.571293	−0.285647	+	0.958335i	$0.592208\pi$
$578$	3.22081e14	0.207664
$579$	0	0
$580$	−9.12940e14	−0.577548
$581$	−4.91187e14	−0.307807
$582$	0	0
$583$	−8.53271e14	−0.524698
$584$	−1.23661e14	−0.0753290
$585$	0	0
$586$	5.74245e14	0.343289
$587$	2.43425e15	1.44164	0.720818	−	0.693124i	$-0.243766\pi$
$587$	2.43425e15	1.44164	0.720818	+	0.693124i	$0.243766\pi$
$588$	0	0
$589$	−5.63383e14	−0.327469
$590$	−6.01532e14	−0.346396
$591$	0	0
$592$	−1.79869e14	−0.101669
$593$	3.03318e14	0.169863	0.0849313	−	0.996387i	$-0.472933\pi$
$593$	3.03318e14	0.169863	0.0849313	+	0.996387i	$0.472933\pi$
$594$	0	0
$595$	5.58507e14	0.307033
$596$	−1.64192e15	−0.894329
$597$	0	0
$598$	2.58502e14	0.138232
$599$	1.70198e15	0.901795	0.450898	−	0.892576i	$-0.351104\pi$
$599$	1.70198e15	0.901795	0.450898	+	0.892576i	$0.351104\pi$
$600$	0	0
$601$	2.33922e15	1.21692	0.608458	−	0.793586i	$-0.291788\pi$
$601$	2.33922e15	1.21692	0.608458	+	0.793586i	$0.291788\pi$
$602$	6.88207e12	0.00354763
$603$	0	0
$604$	1.21359e15	0.614282
$605$	−2.40689e12	−0.00120726
$606$	0	0
$607$	−2.49607e15	−1.22947	−0.614737	−	0.788732i	$-0.710738\pi$
$607$	−2.49607e15	−1.22947	−0.614737	+	0.788732i	$0.710738\pi$
$608$	2.09717e15	1.02368
$609$	0	0
$610$	−8.06395e14	−0.386575
$611$	1.55258e15	0.737612
$612$	0	0
$613$	2.47301e15	1.15397	0.576983	−	0.816756i	$-0.304230\pi$
$613$	2.47301e15	1.15397	0.576983	+	0.816756i	$0.304230\pi$
$614$	3.67466e14	0.169938
$615$	0	0
$616$	−7.56226e14	−0.343525
$617$	−2.43368e13	−0.0109571	−0.00547854	−	0.999985i	$-0.501744\pi$
$617$	−2.43368e13	−0.0109571	−0.00547854	+	0.999985i	$0.501744\pi$
$618$	0	0
$619$	4.22545e15	1.86885	0.934425	−	0.356160i	$-0.115914\pi$
$619$	4.22545e15	1.86885	0.934425	+	0.356160i	$0.115914\pi$
$620$	−3.75702e14	−0.164698
$621$	0	0
$622$	−1.19700e15	−0.515523
$623$	−4.18481e14	−0.178645
$624$	0	0
$625$	−4.88896e14	−0.205058
$626$	−2.38754e15	−0.992640
$627$	0	0
$628$	−1.93585e15	−0.790850
$629$	−1.25835e15	−0.509594
$630$	0	0
$631$	−4.26326e15	−1.69660	−0.848302	−	0.529513i	$-0.822375\pi$
$631$	−4.26326e15	−1.69660	−0.848302	+	0.529513i	$0.822375\pi$
$632$	−3.22011e15	−1.27036
$633$	0	0
$634$	−2.00086e15	−0.775758
$635$	1.26892e15	0.487731
$636$	0	0
$637$	9.80401e14	0.370371
$638$	1.64755e15	0.617055
$639$	0	0
$640$	1.63288e15	0.601126
$641$	−1.00830e15	−0.368018	−0.184009	−	0.982925i	$-0.558908\pi$
$641$	−1.00830e15	−0.368018	−0.184009	+	0.982925i	$0.558908\pi$
$642$	0	0
$643$	3.03982e14	0.109066	0.0545328	−	0.998512i	$-0.482633\pi$
$643$	3.03982e14	0.109066	0.0545328	+	0.998512i	$0.482633\pi$
$644$	−4.59504e14	−0.163462
$645$	0	0
$646$	1.76705e15	0.617974
$647$	−3.43583e15	−1.19140	−0.595700	−	0.803207i	$-0.703125\pi$
$647$	−3.43583e15	−1.19140	−0.595700	+	0.803207i	$0.703125\pi$
$648$	0	0
$649$	−2.77421e15	−0.945788
$650$	3.53565e14	0.119521
$651$	0	0
$652$	5.26730e14	0.175076
$653$	1.18539e15	0.390695	0.195347	−	0.980734i	$-0.437417\pi$
$653$	1.18539e15	0.390695	0.195347	+	0.980734i	$0.437417\pi$
$654$	0	0
$655$	3.05028e15	0.988583
$656$	−3.04157e14	−0.0977522
$657$	0	0
$658$	1.07993e15	0.341312
$659$	2.26510e15	0.709934	0.354967	−	0.934879i	$-0.384492\pi$
$659$	2.26510e15	0.709934	0.354967	+	0.934879i	$0.384492\pi$
$660$	0	0
$661$	−5.33012e15	−1.64297	−0.821484	−	0.570232i	$-0.806853\pi$
$661$	−5.33012e15	−1.64297	−0.821484	+	0.570232i	$0.806853\pi$
$662$	−1.52602e15	−0.466488
$663$	0	0
$664$	−2.47823e15	−0.745104
$665$	8.62227e14	0.257100
$666$	0	0
$667$	2.39392e15	0.702130
$668$	4.05512e15	1.17959
$669$	0	0
$670$	1.79465e15	0.513534
$671$	−3.71902e15	−1.05549
$672$	0	0
$673$	4.74120e15	1.32375	0.661874	−	0.749615i	$-0.269761\pi$
$673$	4.74120e15	1.32375	0.661874	+	0.749615i	$0.269761\pi$
$674$	2.90403e15	0.804215
$675$	0	0
$676$	2.14673e15	0.584886
$677$	1.41307e15	0.381880	0.190940	−	0.981602i	$-0.438846\pi$
$677$	1.41307e15	0.381880	0.190940	+	0.981602i	$0.438846\pi$
$678$	0	0
$679$	−1.25603e15	−0.333976
$680$	2.81789e15	0.743232
$681$	0	0
$682$	6.78014e14	0.175964
$683$	3.03116e15	0.780359	0.390180	−	0.920739i	$-0.372413\pi$
$683$	3.03116e15	0.780359	0.390180	+	0.920739i	$0.372413\pi$
$684$	0	0
$685$	−1.43547e15	−0.363660
$686$	1.47654e15	0.371075
$687$	0	0
$688$	−1.69054e13	−0.00418109
$689$	−9.22102e14	−0.226242
$690$	0	0
$691$	−2.74731e15	−0.663405	−0.331703	−	0.943384i	$-0.607623\pi$
$691$	−2.74731e15	−0.663405	−0.331703	+	0.943384i	$0.607623\pi$
$692$	−1.39897e15	−0.335139
$693$	0	0
$694$	3.73588e15	0.880878
$695$	−2.88251e15	−0.674303
$696$	0	0
$697$	−2.12786e15	−0.489962
$698$	−6.15433e14	−0.140597
$699$	0	0
$700$	−6.28484e14	−0.141337
$701$	−5.72747e15	−1.27795	−0.638974	−	0.769228i	$-0.720641\pi$
$701$	−5.72747e15	−1.27795	−0.638974	+	0.769228i	$0.720641\pi$
$702$	0	0
$703$	−1.94265e15	−0.426718
$704$	−1.44308e15	−0.314514
$705$	0	0
$706$	−5.97836e14	−0.128279
$707$	1.36870e15	0.291409
$708$	0	0
$709$	6.98326e14	0.146388	0.0731938	−	0.997318i	$-0.476681\pi$
$709$	6.98326e14	0.146388	0.0731938	+	0.997318i	$0.476681\pi$
$710$	1.13503e15	0.236095
$711$	0	0
$712$	−2.11140e15	−0.432445
$713$	9.85170e14	0.200225
$714$	0	0
$715$	−1.49182e15	−0.298561
$716$	2.47500e15	0.491534
$717$	0	0
$718$	−3.78202e15	−0.739672
$719$	−9.70979e15	−1.88452	−0.942260	−	0.334882i	$-0.891304\pi$
$719$	−9.70979e15	−1.88452	−0.942260	+	0.334882i	$0.891304\pi$
$720$	0	0
$721$	3.78004e15	0.722525
$722$	−6.77852e13	−0.0128582
$723$	0	0
$724$	1.46725e15	0.274121
$725$	3.27427e15	0.607093
$726$	0	0
$727$	2.46469e15	0.450114	0.225057	−	0.974346i	$-0.427743\pi$
$727$	2.46469e15	0.450114	0.225057	+	0.974346i	$0.427743\pi$
$728$	−8.17230e14	−0.148123
$729$	0	0
$730$	−1.69683e14	−0.0302944
$731$	−1.18269e14	−0.0209568
$732$	0	0
$733$	7.91285e15	1.38121	0.690607	−	0.723230i	$-0.257343\pi$
$733$	7.91285e15	1.38121	0.690607	+	0.723230i	$0.257343\pi$
$734$	−4.26963e15	−0.739711
$735$	0	0
$736$	−3.66725e15	−0.625911
$737$	8.27677e15	1.40213
$738$	0	0
$739$	−8.40694e15	−1.40312	−0.701558	−	0.712613i	$-0.747512\pi$
$739$	−8.40694e15	−1.40312	−0.701558	+	0.712613i	$0.747512\pi$
$740$	−1.29549e15	−0.214615
$741$	0	0
$742$	−6.41385e14	−0.104688
$743$	−1.36287e15	−0.220809	−0.110404	−	0.993887i	$-0.535215\pi$
$743$	−1.36287e15	−0.220809	−0.110404	+	0.993887i	$0.535215\pi$
$744$	0	0
$745$	−5.38754e15	−0.860065
$746$	−1.32388e15	−0.209790
$747$	0	0
$748$	5.43462e15	0.848614
$749$	1.51100e15	0.234215
$750$	0	0
$751$	6.81722e15	1.04133	0.520664	−	0.853762i	$-0.325684\pi$
$751$	6.81722e15	1.04133	0.520664	+	0.853762i	$0.325684\pi$
$752$	−2.65278e15	−0.402256
$753$	0	0
$754$	1.78045e15	0.266065
$755$	3.98208e15	0.590747
$756$	0	0
$757$	−6.67049e14	−0.0975282	−0.0487641	−	0.998810i	$-0.515528\pi$
$757$	−6.67049e14	−0.0975282	−0.0487641	+	0.998810i	$0.515528\pi$
$758$	3.51511e15	0.510222
$759$	0	0
$760$	4.35027e15	0.622360
$761$	7.74408e15	1.09990	0.549951	−	0.835197i	$-0.314646\pi$
$761$	7.74408e15	1.09990	0.549951	+	0.835197i	$0.314646\pi$
$762$	0	0
$763$	−1.23039e15	−0.172250
$764$	4.06626e15	0.565173
$765$	0	0
$766$	−5.55479e15	−0.761043
$767$	−2.99800e15	−0.407809
$768$	0	0
$769$	2.52411e15	0.338465	0.169232	−	0.985576i	$-0.445871\pi$
$769$	2.52411e15	0.338465	0.169232	+	0.985576i	$0.445871\pi$
$770$	−1.03766e15	−0.138152
$771$	0	0
$772$	−8.01119e15	−1.05148
$773$	1.11453e16	1.45246	0.726229	−	0.687453i	$-0.241271\pi$
$773$	1.11453e16	1.45246	0.726229	+	0.687453i	$0.241271\pi$
$774$	0	0
$775$	1.34746e15	0.173124
$776$	−6.33715e15	−0.808452
$777$	0	0
$778$	3.59692e15	0.452421
$779$	−3.28500e15	−0.410279
$780$	0	0
$781$	5.23465e15	0.644624
$782$	−3.08998e15	−0.377849
$783$	0	0
$784$	−1.67514e15	−0.201981
$785$	−6.35201e15	−0.760551
$786$	0	0
$787$	1.32271e16	1.56172	0.780861	−	0.624705i	$-0.214781\pi$
$787$	1.32271e16	1.56172	0.780861	+	0.624705i	$0.214781\pi$
$788$	−4.23361e15	−0.496382
$789$	0	0
$790$	−4.41850e15	−0.510889
$791$	−1.42570e15	−0.163703
$792$	0	0
$793$	−4.01902e15	−0.455112
$794$	4.99466e15	0.561685
$795$	0	0
$796$	−1.07219e15	−0.118919
$797$	−2.30248e15	−0.253615	−0.126807	−	0.991927i	$-0.540473\pi$
$797$	−2.30248e15	−0.253615	−0.126807	+	0.991927i	$0.540473\pi$
$798$	0	0
$799$	−1.85587e16	−2.01623
$800$	−5.01586e15	−0.541191
$801$	0	0
$802$	3.20179e15	0.340748
$803$	−7.82560e14	−0.0827146
$804$	0	0
$805$	−1.50775e15	−0.157199
$806$	7.32708e14	0.0758732
$807$	0	0
$808$	6.90565e15	0.705410
$809$	−5.60472e15	−0.568639	−0.284320	−	0.958730i	$-0.591768\pi$
$809$	−5.60472e15	−0.568639	−0.284320	+	0.958730i	$0.591768\pi$
$810$	0	0
$811$	−5.08516e15	−0.508968	−0.254484	−	0.967077i	$-0.581906\pi$
$811$	−5.08516e15	−0.508968	−0.254484	+	0.967077i	$0.581906\pi$
$812$	−3.16486e15	−0.314627
$813$	0	0
$814$	2.33792e15	0.229296
$815$	1.72833e15	0.168368
$816$	0	0
$817$	−1.82584e14	−0.0175486
$818$	−4.94802e15	−0.472377
$819$	0	0
$820$	−2.19066e15	−0.206348
$821$	−2.79111e14	−0.0261150	−0.0130575	−	0.999915i	$-0.504156\pi$
$821$	−2.79111e14	−0.0261150	−0.0130575	+	0.999915i	$0.504156\pi$
$822$	0	0
$823$	−1.35265e16	−1.24878	−0.624391	−	0.781112i	$-0.714653\pi$
$823$	−1.35265e16	−1.24878	−0.624391	+	0.781112i	$0.714653\pi$
$824$	1.90718e16	1.74901
$825$	0	0
$826$	−2.08531e15	−0.188704
$827$	−2.72544e14	−0.0244994	−0.0122497	−	0.999925i	$-0.503899\pi$
$827$	−2.72544e14	−0.0244994	−0.0122497	+	0.999925i	$0.503899\pi$
$828$	0	0
$829$	1.80459e16	1.60077	0.800385	−	0.599486i	$-0.204628\pi$
$829$	1.80459e16	1.60077	0.800385	+	0.599486i	$0.204628\pi$
$830$	−3.40052e15	−0.299651
$831$	0	0
$832$	−1.55949e15	−0.135614
$833$	−1.17191e16	−1.01239
$834$	0	0
$835$	1.33058e16	1.13440
$836$	8.38999e15	0.710603
$837$	0	0
$838$	−1.76168e15	−0.147260
$839$	7.96183e15	0.661184	0.330592	−	0.943774i	$-0.392752\pi$
$839$	7.96183e15	0.661184	0.330592	+	0.943774i	$0.392752\pi$
$840$	0	0
$841$	4.28775e15	0.351440
$842$	4.10669e15	0.334407
$843$	0	0
$844$	9.99954e15	0.803705
$845$	7.04397e15	0.562478
$846$	0	0
$847$	−8.34387e12	−0.000657671 0
$848$	1.57552e15	0.123381
$849$	0	0
$850$	−4.22630e15	−0.326706
$851$	3.39705e15	0.260909
$852$	0	0
$853$	−1.49826e16	−1.13598	−0.567988	−	0.823037i	$-0.692278\pi$
$853$	−1.49826e16	−1.13598	−0.567988	+	0.823037i	$0.692278\pi$
$854$	−2.79550e15	−0.210592
$855$	0	0
$856$	7.62358e15	0.566961
$857$	2.22561e16	1.64458	0.822290	−	0.569068i	$-0.192696\pi$
$857$	2.22561e16	1.64458	0.822290	+	0.569068i	$0.192696\pi$
$858$	0	0
$859$	5.44237e15	0.397032	0.198516	−	0.980098i	$-0.436388\pi$
$859$	5.44237e15	0.397032	0.198516	+	0.980098i	$0.436388\pi$
$860$	−1.21760e14	−0.00882596
$861$	0	0
$862$	1.72262e15	0.123282
$863$	−1.08110e16	−0.768787	−0.384393	−	0.923169i	$-0.625589\pi$
$863$	−1.08110e16	−0.768787	−0.384393	+	0.923169i	$0.625589\pi$
$864$	0	0
$865$	−4.59037e15	−0.322299
$866$	2.39715e15	0.167243
$867$	0	0
$868$	−1.30243e15	−0.0897217
$869$	−2.03777e16	−1.39491
$870$	0	0
$871$	8.94444e15	0.604579
$872$	−6.20782e15	−0.416964
$873$	0	0
$874$	−4.77033e15	−0.316399
$875$	−6.01111e15	−0.396197
$876$	0	0
$877$	−2.81024e16	−1.82914	−0.914568	−	0.404431i	$-0.867470\pi$
$877$	−2.81024e16	−1.82914	−0.914568	+	0.404431i	$0.867470\pi$
$878$	−6.96749e14	−0.0450668
$879$	0	0
$880$	2.54896e15	0.162820
$881$	−4.22209e15	−0.268016	−0.134008	−	0.990980i	$-0.542785\pi$
$881$	−4.22209e15	−0.268016	−0.134008	+	0.990980i	$0.542785\pi$
$882$	0	0
$883$	5.16092e14	0.0323551	0.0161776	−	0.999869i	$-0.494850\pi$
$883$	5.16092e14	0.0323551	0.0161776	+	0.999869i	$0.494850\pi$
$884$	5.87302e15	0.365910
$885$	0	0
$886$	−7.88088e15	−0.484941
$887$	−5.71906e15	−0.349740	−0.174870	−	0.984592i	$-0.555950\pi$
$887$	−5.71906e15	−0.349740	−0.174870	+	0.984592i	$0.555950\pi$
$888$	0	0
$889$	4.39894e15	0.265698
$890$	−2.89718e15	−0.173912
$891$	0	0
$892$	−1.07969e16	−0.640167
$893$	−2.86510e16	−1.68832
$894$	0	0
$895$	8.12109e15	0.472702
$896$	5.66067e15	0.327472
$897$	0	0
$898$	1.46968e16	0.839854
$899$	6.78541e15	0.385388
$900$	0	0
$901$	1.10223e16	0.618421
$902$	3.95340e15	0.220462
$903$	0	0
$904$	−7.19321e15	−0.396275
$905$	4.81442e15	0.263619
$906$	0	0
$907$	−8.43778e13	−0.00456445	−0.00228222	−	0.999997i	$-0.500726\pi$
$907$	−8.43778e13	−0.00456445	−0.00228222	+	0.999997i	$0.500726\pi$
$908$	−2.00168e15	−0.107628
$909$	0	0
$910$	−1.12137e15	−0.0595691
$911$	1.10091e16	0.581298	0.290649	−	0.956830i	$-0.406129\pi$
$911$	1.10091e16	0.581298	0.290649	+	0.956830i	$0.406129\pi$
$912$	0	0
$913$	−1.56829e16	−0.818158
$914$	7.28359e15	0.377695
$915$	0	0
$916$	1.74055e16	0.891789
$917$	1.05743e16	0.538544
$918$	0	0
$919$	−4.86351e15	−0.244746	−0.122373	−	0.992484i	$-0.539050\pi$
$919$	−4.86351e15	−0.244746	−0.122373	+	0.992484i	$0.539050\pi$
$920$	−7.60717e15	−0.380531
$921$	0	0
$922$	1.75034e16	0.865171
$923$	5.65691e15	0.277952
$924$	0	0
$925$	4.64630e15	0.225594
$926$	2.93252e15	0.141540
$927$	0	0
$928$	−2.52584e16	−1.20474
$929$	−3.57534e15	−0.169524	−0.0847620	−	0.996401i	$-0.527013\pi$
$929$	−3.57534e15	−0.169524	−0.0847620	+	0.996401i	$0.527013\pi$
$930$	0	0
$931$	−1.80921e16	−0.847744
$932$	−2.58533e16	−1.20428
$933$	0	0
$934$	1.48171e16	0.682113
$935$	1.78323e16	0.816102
$936$	0	0
$937$	3.86373e16	1.74759	0.873795	−	0.486295i	$-0.161652\pi$
$937$	3.86373e16	1.74759	0.873795	+	0.486295i	$0.161652\pi$
$938$	6.22147e15	0.279754
$939$	0	0
$940$	−1.91064e16	−0.849133
$941$	3.48997e16	1.54198	0.770991	−	0.636846i	$-0.219761\pi$
$941$	3.48997e16	1.54198	0.770991	+	0.636846i	$0.219761\pi$
$942$	0	0
$943$	5.74437e15	0.250858
$944$	5.12245e15	0.222398
$945$	0	0
$946$	2.19735e14	0.00942969
$947$	2.85123e16	1.21649	0.608243	−	0.793751i	$-0.291875\pi$
$947$	2.85123e16	1.21649	0.608243	+	0.793751i	$0.291875\pi$
$948$	0	0
$949$	−8.45688e14	−0.0356653
$950$	−6.52459e15	−0.273573
$951$	0	0
$952$	9.76867e15	0.404886
$953$	−4.00334e16	−1.64973	−0.824863	−	0.565332i	$-0.808748\pi$
$953$	−4.00334e16	−1.64973	−0.824863	+	0.565332i	$0.808748\pi$
$954$	0	0
$955$	1.33424e16	0.543520
$956$	−1.05095e16	−0.425659
$957$	0	0
$958$	−2.52201e16	−1.00980
$959$	−4.97630e15	−0.198109
$960$	0	0
$961$	−2.26161e16	−0.890100
$962$	2.52652e15	0.0988688
$963$	0	0
$964$	3.40484e14	0.0131726
$965$	−2.62867e16	−1.01120
$966$	0	0
$967$	1.84953e16	0.703422	0.351711	−	0.936109i	$-0.385600\pi$
$967$	1.84953e16	0.703422	0.351711	+	0.936109i	$0.385600\pi$
$968$	−4.20981e13	−0.00159202
$969$	0	0
$970$	−8.69557e15	−0.325127
$971$	2.14877e16	0.798884	0.399442	−	0.916759i	$-0.369204\pi$
$971$	2.14877e16	0.798884	0.399442	+	0.916759i	$0.369204\pi$
$972$	0	0
$973$	−9.99271e15	−0.367335
$974$	−5.27784e15	−0.192922
$975$	0	0
$976$	6.86699e15	0.248195
$977$	8.73880e15	0.314074	0.157037	−	0.987593i	$-0.449806\pi$
$977$	8.73880e15	0.314074	0.157037	+	0.987593i	$0.449806\pi$
$978$	0	0
$979$	−1.33615e16	−0.474844
$980$	−1.20650e16	−0.426368
$981$	0	0
$982$	1.16127e16	0.405808
$983$	−1.18924e16	−0.413263	−0.206631	−	0.978419i	$-0.566250\pi$
$983$	−1.18924e16	−0.413263	−0.206631	+	0.978419i	$0.566250\pi$
$984$	0	0
$985$	−1.38915e16	−0.477365
$986$	−2.12824e16	−0.727274
$987$	0	0
$988$	9.06679e15	0.306401
$989$	3.19279e14	0.0107298
$990$	0	0
$991$	2.34409e16	0.779056	0.389528	−	0.921015i	$-0.372638\pi$
$991$	2.34409e16	0.779056	0.389528	+	0.921015i	$0.372638\pi$
$992$	−1.03946e16	−0.343552
$993$	0	0
$994$	3.93477e15	0.128616
$995$	−3.51813e15	−0.114363
$996$	0	0
$997$	−2.14004e16	−0.688016	−0.344008	−	0.938967i	$-0.611785\pi$
$997$	−2.14004e16	−0.688016	−0.344008	+	0.938967i	$0.611785\pi$
$998$	−2.61307e15	−0.0835473
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9.12.a.b.1.1		1
3.2	odd	2		1.12.a.a.1.1	✓	1
4.3	odd	2		144.12.a.d.1.1		1
5.2	odd	4		225.12.b.d.199.2		2
5.3	odd	4		225.12.b.d.199.1		2
5.4	even	2		225.12.a.b.1.1		1
9.2	odd	6		81.12.c.d.28.1		2
9.4	even	3		81.12.c.b.55.1		2
9.5	odd	6		81.12.c.d.55.1		2
9.7	even	3		81.12.c.b.28.1		2
12.11	even	2		16.12.a.a.1.1		1
15.2	even	4		25.12.b.b.24.1		2
15.8	even	4		25.12.b.b.24.2		2
15.14	odd	2		25.12.a.b.1.1		1
21.2	odd	6		49.12.c.b.18.1		2
21.5	even	6		49.12.c.c.18.1		2
21.11	odd	6		49.12.c.b.30.1		2
21.17	even	6		49.12.c.c.30.1		2
21.20	even	2		49.12.a.a.1.1		1
24.5	odd	2		64.12.a.b.1.1		1
24.11	even	2		64.12.a.f.1.1		1
33.32	even	2		121.12.a.b.1.1		1
39.38	odd	2		169.12.a.a.1.1		1
48.5	odd	4		256.12.b.e.129.1		2
48.11	even	4		256.12.b.c.129.2		2
48.29	odd	4		256.12.b.e.129.2		2
48.35	even	4		256.12.b.c.129.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.12.a.a.1.1	✓	1	3.2	odd	2
9.12.a.b.1.1		1	1.1	even	1	trivial
16.12.a.a.1.1		1	12.11	even	2
25.12.a.b.1.1		1	15.14	odd	2
25.12.b.b.24.1		2	15.2	even	4
25.12.b.b.24.2		2	15.8	even	4
49.12.a.a.1.1		1	21.20	even	2
49.12.c.b.18.1		2	21.2	odd	6
49.12.c.b.30.1		2	21.11	odd	6
49.12.c.c.18.1		2	21.5	even	6
49.12.c.c.30.1		2	21.17	even	6
64.12.a.b.1.1		1	24.5	odd	2
64.12.a.f.1.1		1	24.11	even	2
81.12.c.b.28.1		2	9.7	even	3
81.12.c.b.55.1		2	9.4	even	3
81.12.c.d.28.1		2	9.2	odd	6
81.12.c.d.55.1		2	9.5	odd	6
121.12.a.b.1.1		1	33.32	even	2
144.12.a.d.1.1		1	4.3	odd	2
169.12.a.a.1.1		1	39.38	odd	2
225.12.a.b.1.1		1	5.4	even	2
225.12.b.d.199.1		2	5.3	odd	4
225.12.b.d.199.2		2	5.2	odd	4
256.12.b.c.129.1		2	48.35	even	4
256.12.b.c.129.2		2	48.11	even	4
256.12.b.e.129.1		2	48.5	odd	4
256.12.b.e.129.2		2	48.29	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:	\( N \)	\(=\)	\( 9 = 3^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	9.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more