LMFDB - Embedded newform 18.12.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [18,12,Mod(1,18)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(18, 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("18.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 18 = 2 \cdot 3^{2} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	18.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:	$13.8301772501$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 6)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	18.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+32.0000 q^{2} +1024.00 q^{4} -5766.00 q^{5} +72464.0 q^{7} +32768.0 q^{8} +O(q^{10})$

\(q+32.0000 q^{2} +1024.00 q^{4} -5766.00 q^{5} +72464.0 q^{7} +32768.0 q^{8} -184512. q^{10} +408948. q^{11} +1.36756e6 q^{13} +2.31885e6 q^{14} +1.04858e6 q^{16} -5.42291e6 q^{17} +1.51661e7 q^{19} -5.90438e6 q^{20} +1.30863e7 q^{22} +5.21941e7 q^{23} -1.55814e7 q^{25} +4.37619e7 q^{26} +7.42031e7 q^{28} -1.18581e8 q^{29} -5.76524e7 q^{31} +3.35544e7 q^{32} -1.73533e8 q^{34} -4.17827e8 q^{35} -3.75985e8 q^{37} +4.85315e8 q^{38} -1.88940e8 q^{40} -8.56316e8 q^{41} -1.24519e9 q^{43} +4.18763e8 q^{44} +1.67021e9 q^{46} +1.30676e9 q^{47} +3.27370e9 q^{49} -4.98604e8 q^{50} +1.40038e9 q^{52} -4.09556e8 q^{53} -2.35799e9 q^{55} +2.37450e9 q^{56} -3.79460e9 q^{58} +2.88287e9 q^{59} +5.73177e9 q^{61} -1.84488e9 q^{62} +1.07374e9 q^{64} -7.88534e9 q^{65} +3.89327e9 q^{67} -5.55306e9 q^{68} -1.33705e10 q^{70} +9.07589e9 q^{71} -1.55718e10 q^{73} -1.20315e10 q^{74} +1.55301e10 q^{76} +2.96340e10 q^{77} -3.01968e10 q^{79} -6.04609e9 q^{80} -2.74021e10 q^{82} -2.31353e10 q^{83} +3.12685e10 q^{85} -3.98461e10 q^{86} +1.34004e10 q^{88} +2.56148e10 q^{89} +9.90987e10 q^{91} +5.34467e10 q^{92} +4.18164e10 q^{94} -8.74477e10 q^{95} -6.19376e10 q^{97} +1.04759e11 q^{98} +O(q^{100})\)

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 $

$2$	32.0000	0.707107
$2$	32.0000	0.707107
$3$	0	0
$3$	0	0
$4$	1024.00	0.500000
$5$	−5766.00	−0.825163	−0.412581	−	0.910921i	$-0.635373\pi$
$5$	−5766.00	−0.825163	−0.412581	+	0.910921i	$0.635373\pi$
$6$	0	0
$7$	72464.0	1.62961	0.814804	−	0.579737i	$-0.196845\pi$
$7$	72464.0	1.62961	0.814804	+	0.579737i	$0.196845\pi$
$8$	32768.0	0.353553
$9$	0	0
$10$	−184512.	−0.583478
$11$	408948.	0.765611	0.382806	−	0.923829i	$-0.374958\pi$
$11$	408948.	0.765611	0.382806	+	0.923829i	$0.374958\pi$
$12$	0	0
$13$	1.36756e6	1.02154	0.510772	−	0.859716i	$-0.329360\pi$
$13$	1.36756e6	1.02154	0.510772	+	0.859716i	$0.329360\pi$
$14$	2.31885e6	1.15231
$15$	0	0
$16$	1.04858e6	0.250000
$17$	−5.42291e6	−0.926326	−0.463163	−	0.886273i	$-0.653285\pi$
$17$	−5.42291e6	−0.926326	−0.463163	+	0.886273i	$0.653285\pi$
$18$	0	0
$19$	1.51661e7	1.40517	0.702585	−	0.711599i	$-0.252029\pi$
$19$	1.51661e7	1.40517	0.702585	+	0.711599i	$0.252029\pi$
$20$	−5.90438e6	−0.412581
$21$	0	0
$22$	1.30863e7	0.541369
$23$	5.21941e7	1.69090	0.845450	−	0.534054i	$-0.179332\pi$
$23$	5.21941e7	1.69090	0.845450	+	0.534054i	$0.179332\pi$
$24$	0	0
$25$	−1.55814e7	−0.319106
$26$	4.37619e7	0.722341
$27$	0	0
$28$	7.42031e7	0.814804
$29$	−1.18581e8	−1.07356	−0.536780	−	0.843722i	$-0.680360\pi$
$29$	−1.18581e8	−1.07356	−0.536780	+	0.843722i	$0.680360\pi$
$30$	0	0
$31$	−5.76524e7	−0.361683	−0.180842	−	0.983512i	$-0.557882\pi$
$31$	−5.76524e7	−0.361683	−0.180842	+	0.983512i	$0.557882\pi$
$32$	3.35544e7	0.176777
$33$	0	0
$34$	−1.73533e8	−0.655011
$35$	−4.17827e8	−1.34469
$36$	0	0
$37$	−3.75985e8	−0.891377	−0.445688	−	0.895188i	$-0.647041\pi$
$37$	−3.75985e8	−0.891377	−0.445688	+	0.895188i	$0.647041\pi$
$38$	4.85315e8	0.993606
$39$	0	0
$40$	−1.88940e8	−0.291739
$41$	−8.56316e8	−1.15431	−0.577156	−	0.816634i	$-0.695837\pi$
$41$	−8.56316e8	−1.15431	−0.577156	+	0.816634i	$0.695837\pi$
$42$	0	0
$43$	−1.24519e9	−1.29169	−0.645846	−	0.763468i	$-0.723495\pi$
$43$	−1.24519e9	−1.29169	−0.645846	+	0.763468i	$0.723495\pi$
$44$	4.18763e8	0.382806
$45$	0	0
$46$	1.67021e9	1.19565
$47$	1.30676e9	0.831110	0.415555	−	0.909568i	$-0.363587\pi$
$47$	1.30676e9	0.831110	0.415555	+	0.909568i	$0.363587\pi$
$48$	0	0
$49$	3.27370e9	1.65562
$50$	−4.98604e8	−0.225642
$51$	0	0
$52$	1.40038e9	0.510772
$53$	−4.09556e8	−0.134523	−0.0672615	−	0.997735i	$-0.521426\pi$
$53$	−4.09556e8	−0.134523	−0.0672615	+	0.997735i	$0.521426\pi$
$54$	0	0
$55$	−2.35799e9	−0.631754
$56$	2.37450e9	0.576153
$57$	0	0
$58$	−3.79460e9	−0.759122
$59$	2.88287e9	0.524975	0.262487	−	0.964935i	$-0.415457\pi$
$59$	2.88287e9	0.524975	0.262487	+	0.964935i	$0.415457\pi$
$60$	0	0
$61$	5.73177e9	0.868909	0.434455	−	0.900694i	$-0.356941\pi$
$61$	5.73177e9	0.868909	0.434455	+	0.900694i	$0.356941\pi$
$62$	−1.84488e9	−0.255749
$63$	0	0
$64$	1.07374e9	0.125000
$65$	−7.88534e9	−0.842941
$66$	0	0
$67$	3.89327e9	0.352292	0.176146	−	0.984364i	$-0.443637\pi$
$67$	3.89327e9	0.352292	0.176146	+	0.984364i	$0.443637\pi$
$68$	−5.55306e9	−0.463163
$69$	0	0
$70$	−1.33705e10	−0.950841
$71$	9.07589e9	0.596992	0.298496	−	0.954411i	$-0.403515\pi$
$71$	9.07589e9	0.596992	0.298496	+	0.954411i	$0.403515\pi$
$72$	0	0
$73$	−1.55718e10	−0.879152	−0.439576	−	0.898206i	$-0.644871\pi$
$73$	−1.55718e10	−0.879152	−0.439576	+	0.898206i	$0.644871\pi$
$74$	−1.20315e10	−0.630298
$75$	0	0
$76$	1.55301e10	0.702585
$77$	2.96340e10	1.24765
$78$	0	0
$79$	−3.01968e10	−1.10411	−0.552054	−	0.833809i	$-0.686156\pi$
$79$	−3.01968e10	−1.10411	−0.552054	+	0.833809i	$0.686156\pi$
$80$	−6.04609e9	−0.206291
$81$	0	0
$82$	−2.74021e10	−0.816221
$83$	−2.31353e10	−0.644681	−0.322340	−	0.946624i	$-0.604470\pi$
$83$	−2.31353e10	−0.644681	−0.322340	+	0.946624i	$0.604470\pi$
$84$	0	0
$85$	3.12685e10	0.764369
$86$	−3.98461e10	−0.913364
$87$	0	0
$88$	1.34004e10	0.270684
$89$	2.56148e10	0.486235	0.243118	−	0.969997i	$-0.421830\pi$
$89$	2.56148e10	0.486235	0.243118	+	0.969997i	$0.421830\pi$
$90$	0	0
$91$	9.90987e10	1.66472
$92$	5.34467e10	0.845450
$93$	0	0
$94$	4.18164e10	0.587684
$95$	−8.74477e10	−1.15949
$96$	0	0
$97$	−6.19376e10	−0.732335	−0.366167	−	0.930549i	$-0.619330\pi$
$97$	−6.19376e10	−0.732335	−0.366167	+	0.930549i	$0.619330\pi$
$98$	1.04759e11	1.17070
$99$	0	0
$100$	−1.59553e10	−0.159553
$101$	9.49642e10	0.899067	0.449534	−	0.893263i	$-0.351590\pi$
$101$	9.49642e10	0.899067	0.449534	+	0.893263i	$0.351590\pi$
$102$	0	0
$103$	5.43719e10	0.462136	0.231068	−	0.972938i	$-0.425778\pi$
$103$	5.43719e10	0.462136	0.231068	+	0.972938i	$0.425778\pi$
$104$	4.48121e10	0.361171
$105$	0	0
$106$	−1.31058e10	−0.0951221
$107$	−5.71348e9	−0.0393813	−0.0196906	−	0.999806i	$-0.506268\pi$
$107$	−5.71348e9	−0.0393813	−0.0196906	+	0.999806i	$0.506268\pi$
$108$	0	0
$109$	−4.96724e10	−0.309221	−0.154611	−	0.987975i	$-0.549412\pi$
$109$	−4.96724e10	−0.309221	−0.154611	+	0.987975i	$0.549412\pi$
$110$	−7.54558e10	−0.446717
$111$	0	0
$112$	7.59840e10	0.407402
$113$	−1.77803e11	−0.907834	−0.453917	−	0.891044i	$-0.649974\pi$
$113$	−1.77803e11	−0.907834	−0.453917	+	0.891044i	$0.649974\pi$
$114$	0	0
$115$	−3.00951e11	−1.39527
$116$	−1.21427e11	−0.536780
$117$	0	0
$118$	9.22517e10	0.371213
$119$	−3.92966e11	−1.50955
$120$	0	0
$121$	−1.18073e11	−0.413839
$122$	1.83417e11	0.614412
$123$	0	0
$124$	−5.90361e10	−0.180842
$125$	3.71385e11	1.08848
$126$	0	0
$127$	−4.73708e11	−1.27230	−0.636150	−	0.771565i	$-0.719474\pi$
$127$	−4.73708e11	−1.27230	−0.636150	+	0.771565i	$0.719474\pi$
$128$	3.43597e10	0.0883883
$129$	0	0
$130$	−2.52331e11	−0.596049
$131$	−1.68574e11	−0.381767	−0.190883	−	0.981613i	$-0.561135\pi$
$131$	−1.68574e11	−0.381767	−0.190883	+	0.981613i	$0.561135\pi$
$132$	0	0
$133$	1.09900e12	2.28988
$134$	1.24585e11	0.249108
$135$	0	0
$136$	−1.77698e11	−0.327506
$137$	8.01479e11	1.41883	0.709413	−	0.704793i	$-0.248960\pi$
$137$	8.01479e11	1.41883	0.709413	+	0.704793i	$0.248960\pi$
$138$	0	0
$139$	5.24839e11	0.857916	0.428958	−	0.903324i	$-0.358881\pi$
$139$	5.24839e11	0.857916	0.428958	+	0.903324i	$0.358881\pi$
$140$	−4.27855e11	−0.672346
$141$	0	0
$142$	2.90428e11	0.422137
$143$	5.59260e11	0.782106
$144$	0	0
$145$	6.83739e11	0.885862
$146$	−4.98298e11	−0.621654
$147$	0	0
$148$	−3.85009e11	−0.445688
$149$	−1.25702e12	−1.40223	−0.701113	−	0.713051i	$-0.747313\pi$
$149$	−1.25702e12	−1.40223	−0.701113	+	0.713051i	$0.747313\pi$
$150$	0	0
$151$	−4.92117e11	−0.510147	−0.255074	−	0.966922i	$-0.582100\pi$
$151$	−4.92117e11	−0.510147	−0.255074	+	0.966922i	$0.582100\pi$
$152$	4.96963e11	0.496803
$153$	0	0
$154$	9.48288e11	0.882219
$155$	3.32424e11	0.298447
$156$	0	0
$157$	−9.97092e11	−0.834232	−0.417116	−	0.908853i	$-0.636959\pi$
$157$	−9.97092e11	−0.834232	−0.417116	+	0.908853i	$0.636959\pi$
$158$	−9.66296e11	−0.780722
$159$	0	0
$160$	−1.93475e11	−0.145870
$161$	3.78219e12	2.75550
$162$	0	0
$163$	−4.84142e11	−0.329565	−0.164782	−	0.986330i	$-0.552692\pi$
$163$	−4.84142e11	−0.329565	−0.164782	+	0.986330i	$0.552692\pi$
$164$	−8.76868e11	−0.577156
$165$	0	0
$166$	−7.40328e11	−0.455858
$167$	−2.31537e12	−1.37937	−0.689685	−	0.724110i	$-0.742251\pi$
$167$	−2.31537e12	−1.37937	−0.689685	+	0.724110i	$0.742251\pi$
$168$	0	0
$169$	7.80545e10	0.0435533
$170$	1.00059e12	0.540491
$171$	0	0
$172$	−1.27507e12	−0.645846
$173$	−3.37157e12	−1.65416	−0.827081	−	0.562083i	$-0.810000\pi$
$173$	−3.37157e12	−1.65416	−0.827081	+	0.562083i	$0.810000\pi$
$174$	0	0
$175$	−1.12909e12	−0.520018
$176$	4.28813e11	0.191403
$177$	0	0
$178$	8.19674e11	0.343820
$179$	−1.64598e12	−0.669472	−0.334736	−	0.942312i	$-0.608647\pi$
$179$	−1.64598e12	−0.669472	−0.334736	+	0.942312i	$0.608647\pi$
$180$	0	0
$181$	2.74217e12	1.04921	0.524605	−	0.851346i	$-0.324213\pi$
$181$	2.74217e12	1.04921	0.524605	+	0.851346i	$0.324213\pi$
$182$	3.17116e12	1.17713
$183$	0	0
$184$	1.71030e12	0.597824
$185$	2.16793e12	0.735531
$186$	0	0
$187$	−2.21769e12	−0.709205
$188$	1.33812e12	0.415555
$189$	0	0
$190$	−2.79833e12	−0.819886
$191$	−4.08409e12	−1.16255	−0.581275	−	0.813707i	$-0.697446\pi$
$191$	−4.08409e12	−1.16255	−0.581275	+	0.813707i	$0.697446\pi$
$192$	0	0
$193$	−4.47239e12	−1.20219	−0.601097	−	0.799176i	$-0.705269\pi$
$193$	−4.47239e12	−1.20219	−0.601097	+	0.799176i	$0.705269\pi$
$194$	−1.98200e12	−0.517839
$195$	0	0
$196$	3.35227e12	0.827811
$197$	7.37025e12	1.76977	0.884887	−	0.465805i	$-0.154235\pi$
$197$	7.37025e12	1.76977	0.884887	+	0.465805i	$0.154235\pi$
$198$	0	0
$199$	2.69308e12	0.611728	0.305864	−	0.952075i	$-0.401055\pi$
$199$	2.69308e12	0.611728	0.305864	+	0.952075i	$0.401055\pi$
$200$	−5.10570e11	−0.112821
$201$	0	0
$202$	3.03885e12	0.635736
$203$	−8.59286e12	−1.74948
$204$	0	0
$205$	4.93752e12	0.952495
$206$	1.73990e12	0.326780
$207$	0	0
$208$	1.43399e12	0.255386
$209$	6.20215e12	1.07581
$210$	0	0
$211$	6.63458e12	1.09209	0.546047	−	0.837755i	$-0.316132\pi$
$211$	6.63458e12	1.09209	0.546047	+	0.837755i	$0.316132\pi$
$212$	−4.19386e11	−0.0672615
$213$	0	0
$214$	−1.82831e11	−0.0278468
$215$	7.17976e12	1.06586
$216$	0	0
$217$	−4.17772e12	−0.589401
$218$	−1.58952e12	−0.218652
$219$	0	0
$220$	−2.41459e12	−0.315877
$221$	−7.41615e12	−0.946283
$222$	0	0
$223$	−5.99559e12	−0.728040	−0.364020	−	0.931391i	$-0.618596\pi$
$223$	−5.99559e12	−0.728040	−0.364020	+	0.931391i	$0.618596\pi$
$224$	2.43149e12	0.288077
$225$	0	0
$226$	−5.68968e12	−0.641936
$227$	7.74930e12	0.853337	0.426668	−	0.904408i	$-0.359687\pi$
$227$	7.74930e12	0.853337	0.426668	+	0.904408i	$0.359687\pi$
$228$	0	0
$229$	3.75804e12	0.394336	0.197168	−	0.980370i	$-0.436826\pi$
$229$	3.75804e12	0.394336	0.197168	+	0.980370i	$0.436826\pi$
$230$	−9.63043e12	−0.986604
$231$	0	0
$232$	−3.88567e12	−0.379561
$233$	−9.02676e12	−0.861141	−0.430570	−	0.902557i	$-0.641688\pi$
$233$	−9.02676e12	−0.861141	−0.430570	+	0.902557i	$0.641688\pi$
$234$	0	0
$235$	−7.53479e12	−0.685801
$236$	2.95206e12	0.262487
$237$	0	0
$238$	−1.25749e13	−1.06741
$239$	4.02269e12	0.333679	0.166839	−	0.985984i	$-0.446644\pi$
$239$	4.02269e12	0.333679	0.166839	+	0.985984i	$0.446644\pi$
$240$	0	0
$241$	−1.49997e13	−1.18847	−0.594235	−	0.804292i	$-0.702545\pi$
$241$	−1.49997e13	−1.18847	−0.594235	+	0.804292i	$0.702545\pi$
$242$	−3.77834e12	−0.292629
$243$	0	0
$244$	5.86933e12	0.434455
$245$	−1.88762e13	−1.36616
$246$	0	0
$247$	2.07405e13	1.43544
$248$	−1.88915e12	−0.127874
$249$	0	0
$250$	1.18843e13	0.769670
$251$	1.46817e13	0.930187	0.465094	−	0.885262i	$-0.346021\pi$
$251$	1.46817e13	0.930187	0.465094	+	0.885262i	$0.346021\pi$
$252$	0	0
$253$	2.13447e13	1.29457
$254$	−1.51586e13	−0.899653
$255$	0	0
$256$	1.09951e12	0.0625000
$257$	1.64476e13	0.915106	0.457553	−	0.889182i	$-0.348726\pi$
$257$	1.64476e13	0.915106	0.457553	+	0.889182i	$0.348726\pi$
$258$	0	0
$259$	−2.72454e13	−1.45259
$260$	−8.07459e12	−0.421470
$261$	0	0
$262$	−5.39436e12	−0.269950
$263$	1.96818e13	0.964514	0.482257	−	0.876030i	$-0.339817\pi$
$263$	1.96818e13	0.964514	0.482257	+	0.876030i	$0.339817\pi$
$264$	0	0
$265$	2.36150e12	0.111003
$266$	3.51679e13	1.61919
$267$	0	0
$268$	3.98671e12	0.176146
$269$	1.16174e13	0.502889	0.251444	−	0.967872i	$-0.419094\pi$
$269$	1.16174e13	0.502889	0.251444	+	0.967872i	$0.419094\pi$
$270$	0	0
$271$	7.46788e12	0.310360	0.155180	−	0.987886i	$-0.450404\pi$
$271$	7.46788e12	0.310360	0.155180	+	0.987886i	$0.450404\pi$
$272$	−5.68634e12	−0.231581
$273$	0	0
$274$	2.56473e13	1.00326
$275$	−6.37197e12	−0.244311
$276$	0	0
$277$	1.59564e13	0.587892	0.293946	−	0.955822i	$-0.405031\pi$
$277$	1.59564e13	0.587892	0.293946	+	0.955822i	$0.405031\pi$
$278$	1.67949e13	0.606638
$279$	0	0
$280$	−1.36914e13	−0.475420
$281$	−3.96115e13	−1.34877	−0.674383	−	0.738382i	$-0.735590\pi$
$281$	−3.96115e13	−1.34877	−0.674383	+	0.738382i	$0.735590\pi$
$282$	0	0
$283$	1.50001e13	0.491211	0.245605	−	0.969370i	$-0.421013\pi$
$283$	1.50001e13	0.491211	0.245605	+	0.969370i	$0.421013\pi$
$284$	9.29371e12	0.298496
$285$	0	0
$286$	1.78963e13	0.553032
$287$	−6.20521e13	−1.88107
$288$	0	0
$289$	−4.86390e12	−0.141921
$290$	2.18796e13	0.626399
$291$	0	0
$292$	−1.59455e13	−0.439576
$293$	−3.88255e13	−1.05038	−0.525188	−	0.850986i	$-0.676005\pi$
$293$	−3.88255e13	−1.05038	−0.525188	+	0.850986i	$0.676005\pi$
$294$	0	0
$295$	−1.66226e13	−0.433190
$296$	−1.23203e13	−0.315149
$297$	0	0
$298$	−4.02246e13	−0.991523
$299$	7.13784e13	1.72733
$300$	0	0
$301$	−9.02314e13	−2.10495
$302$	−1.57478e13	−0.360729
$303$	0	0
$304$	1.59028e13	0.351293
$305$	−3.30494e13	−0.716992
$306$	0	0
$307$	7.58194e13	1.58679	0.793394	−	0.608708i	$-0.208312\pi$
$307$	7.58194e13	1.58679	0.793394	+	0.608708i	$0.208312\pi$
$308$	3.03452e13	0.623823
$309$	0	0
$310$	1.06376e13	0.211034
$311$	7.99907e13	1.55904	0.779520	−	0.626378i	$-0.215463\pi$
$311$	7.99907e13	1.55904	0.779520	+	0.626378i	$0.215463\pi$
$312$	0	0
$313$	−6.85614e13	−1.28999	−0.644994	−	0.764187i	$-0.723140\pi$
$313$	−6.85614e13	−1.28999	−0.644994	+	0.764187i	$0.723140\pi$
$314$	−3.19069e13	−0.589891
$315$	0	0
$316$	−3.09215e13	−0.552054
$317$	−6.37891e12	−0.111923	−0.0559617	−	0.998433i	$-0.517822\pi$
$317$	−6.37891e12	−0.111923	−0.0559617	+	0.998433i	$0.517822\pi$
$318$	0	0
$319$	−4.84935e13	−0.821930
$320$	−6.19120e12	−0.103145
$321$	0	0
$322$	1.21030e14	1.94844
$323$	−8.22445e13	−1.30165
$324$	0	0
$325$	−2.13084e13	−0.325981
$326$	−1.54925e13	−0.233038
$327$	0	0
$328$	−2.80598e13	−0.408111
$329$	9.46932e13	1.35438
$330$	0	0
$331$	8.90210e13	1.23151	0.615756	−	0.787937i	$-0.288851\pi$
$331$	8.90210e13	1.23151	0.615756	+	0.787937i	$0.288851\pi$
$332$	−2.36905e13	−0.322340
$333$	0	0
$334$	−7.40920e13	−0.975361
$335$	−2.24486e13	−0.290699
$336$	0	0
$337$	1.42601e14	1.78714	0.893568	−	0.448928i	$-0.148194\pi$
$337$	1.42601e14	1.78714	0.893568	+	0.448928i	$0.148194\pi$
$338$	2.49774e12	0.0307968
$339$	0	0
$340$	3.20190e13	0.382185
$341$	−2.35768e13	−0.276909
$342$	0	0
$343$	9.39407e13	1.06841
$344$	−4.08024e13	−0.456682
$345$	0	0
$346$	−1.07890e14	−1.16967
$347$	1.33726e14	1.42693	0.713467	−	0.700689i	$-0.247124\pi$
$347$	1.33726e14	1.42693	0.713467	+	0.700689i	$0.247124\pi$
$348$	0	0
$349$	−6.30474e13	−0.651820	−0.325910	−	0.945401i	$-0.605671\pi$
$349$	−6.30474e13	−0.651820	−0.325910	+	0.945401i	$0.605671\pi$
$350$	−3.61308e13	−0.367708
$351$	0	0
$352$	1.37220e13	0.135342
$353$	2.89749e13	0.281359	0.140680	−	0.990055i	$-0.455071\pi$
$353$	2.89749e13	0.281359	0.140680	+	0.990055i	$0.455071\pi$
$354$	0	0
$355$	−5.23316e13	−0.492615
$356$	2.62296e13	0.243118
$357$	0	0
$358$	−5.26713e13	−0.473388
$359$	−4.33108e13	−0.383334	−0.191667	−	0.981460i	$-0.561389\pi$
$359$	−4.33108e13	−0.383334	−0.191667	+	0.981460i	$0.561389\pi$
$360$	0	0
$361$	1.13520e14	0.974505
$362$	8.77495e13	0.741904
$363$	0	0
$364$	1.01477e14	0.832358
$365$	8.97871e13	0.725443
$366$	0	0
$367$	−1.12471e14	−0.881814	−0.440907	−	0.897553i	$-0.645343\pi$
$367$	−1.12471e14	−0.881814	−0.440907	+	0.897553i	$0.645343\pi$
$368$	5.47295e13	0.422725
$369$	0	0
$370$	6.93738e13	0.520099
$371$	−2.96781e13	−0.219220
$372$	0	0
$373$	1.29948e14	0.931900	0.465950	−	0.884811i	$-0.345713\pi$
$373$	1.29948e14	0.931900	0.465950	+	0.884811i	$0.345713\pi$
$374$	−7.09661e13	−0.501484
$375$	0	0
$376$	4.28200e13	0.293842
$377$	−1.62167e14	−1.09669
$378$	0	0
$379$	−9.45382e13	−0.621000	−0.310500	−	0.950573i	$-0.600496\pi$
$379$	−9.45382e13	−0.621000	−0.310500	+	0.950573i	$0.600496\pi$
$380$	−8.95465e13	−0.579747
$381$	0	0
$382$	−1.30691e14	−0.822047
$383$	−5.45117e13	−0.337985	−0.168992	−	0.985617i	$-0.554051\pi$
$383$	−5.45117e13	−0.337985	−0.168992	+	0.985617i	$0.554051\pi$
$384$	0	0
$385$	−1.70870e14	−1.02951
$386$	−1.43116e14	−0.850079
$387$	0	0
$388$	−6.34241e13	−0.366167
$389$	5.11379e13	0.291085	0.145543	−	0.989352i	$-0.453507\pi$
$389$	5.11379e13	0.291085	0.145543	+	0.989352i	$0.453507\pi$
$390$	0	0
$391$	−2.83044e14	−1.56632
$392$	1.07273e14	0.585351
$393$	0	0
$394$	2.35848e14	1.25142
$395$	1.74115e14	0.911068
$396$	0	0
$397$	−4.87234e13	−0.247965	−0.123982	−	0.992284i	$-0.539567\pi$
$397$	−4.87234e13	−0.247965	−0.123982	+	0.992284i	$0.539567\pi$
$398$	8.61787e13	0.432557
$399$	0	0
$400$	−1.63382e13	−0.0797766
$401$	−3.76407e14	−1.81286	−0.906430	−	0.422356i	$-0.861203\pi$
$401$	−3.76407e14	−1.81286	−0.906430	+	0.422356i	$0.861203\pi$
$402$	0	0
$403$	−7.88430e13	−0.369475
$404$	9.72433e13	0.449534
$405$	0	0
$406$	−2.74972e14	−1.23707
$407$	−1.53758e14	−0.682448
$408$	0	0
$409$	3.12467e14	1.34998	0.674988	−	0.737828i	$-0.264149\pi$
$409$	3.12467e14	1.34998	0.674988	+	0.737828i	$0.264149\pi$
$410$	1.58001e14	0.673515
$411$	0	0
$412$	5.56768e13	0.231068
$413$	2.08904e14	0.855503
$414$	0	0
$415$	1.33398e14	0.531967
$416$	4.58876e13	0.180585
$417$	0	0
$418$	1.98469e14	0.760716
$419$	−2.81729e14	−1.06575	−0.532874	−	0.846195i	$-0.678888\pi$
$419$	−2.81729e14	−1.06575	−0.532874	+	0.846195i	$0.678888\pi$
$420$	0	0
$421$	−3.01064e14	−1.10945	−0.554724	−	0.832034i	$-0.687176\pi$
$421$	−3.01064e14	−1.10945	−0.554724	+	0.832034i	$0.687176\pi$
$422$	2.12306e14	0.772227
$423$	0	0
$424$	−1.34203e13	−0.0475610
$425$	8.44964e13	0.295596
$426$	0	0
$427$	4.15347e14	1.41598
$428$	−5.85060e12	−0.0196906
$429$	0	0
$430$	2.29752e14	0.753674
$431$	−3.46694e14	−1.12285	−0.561425	−	0.827528i	$-0.689747\pi$
$431$	−3.46694e14	−1.12285	−0.561425	+	0.827528i	$0.689747\pi$
$432$	0	0
$433$	4.82345e14	1.52291	0.761456	−	0.648217i	$-0.224485\pi$
$433$	4.82345e14	1.52291	0.761456	+	0.648217i	$0.224485\pi$
$434$	−1.33687e14	−0.416770
$435$	0	0
$436$	−5.08645e13	−0.154611
$437$	7.91581e14	2.37600
$438$	0	0
$439$	1.18934e14	0.348138	0.174069	−	0.984733i	$-0.444308\pi$
$439$	1.18934e14	0.348138	0.174069	+	0.984733i	$0.444308\pi$
$440$	−7.72668e13	−0.223359
$441$	0	0
$442$	−2.37317e14	−0.669123
$443$	1.99914e14	0.556703	0.278351	−	0.960479i	$-0.410212\pi$
$443$	1.99914e14	0.556703	0.278351	+	0.960479i	$0.410212\pi$
$444$	0	0
$445$	−1.47695e14	−0.401223
$446$	−1.91859e14	−0.514802
$447$	0	0
$448$	7.78076e13	0.203701
$449$	5.06702e14	1.31038	0.655191	−	0.755464i	$-0.272588\pi$
$449$	5.06702e14	1.31038	0.655191	+	0.755464i	$0.272588\pi$
$450$	0	0
$451$	−3.50189e14	−0.883754
$452$	−1.82070e14	−0.453917
$453$	0	0
$454$	2.47978e14	0.603400
$455$	−5.71403e14	−1.37366
$456$	0	0
$457$	−3.62768e14	−0.851315	−0.425658	−	0.904884i	$-0.639957\pi$
$457$	−3.62768e14	−0.851315	−0.425658	+	0.904884i	$0.639957\pi$
$458$	1.20257e14	0.278838
$459$	0	0
$460$	−3.08174e14	−0.697634
$461$	−6.56466e13	−0.146844	−0.0734221	−	0.997301i	$-0.523392\pi$
$461$	−6.56466e13	−0.146844	−0.0734221	+	0.997301i	$0.523392\pi$
$462$	0	0
$463$	−2.06748e14	−0.451592	−0.225796	−	0.974175i	$-0.572498\pi$
$463$	−2.06748e14	−0.451592	−0.225796	+	0.974175i	$0.572498\pi$
$464$	−1.24341e14	−0.268390
$465$	0	0
$466$	−2.88856e14	−0.608919
$467$	3.79401e14	0.790416	0.395208	−	0.918592i	$-0.370673\pi$
$467$	3.79401e14	0.790416	0.395208	+	0.918592i	$0.370673\pi$
$468$	0	0
$469$	2.82122e14	0.574099
$470$	−2.41113e14	−0.484935
$471$	0	0
$472$	9.44658e13	0.185607
$473$	−5.09218e14	−0.988934
$474$	0	0
$475$	−2.36309e14	−0.448399
$476$	−4.02397e14	−0.754774
$477$	0	0
$478$	1.28726e14	0.235946
$479$	−3.67939e14	−0.666701	−0.333350	−	0.942803i	$-0.608179\pi$
$479$	−3.67939e14	−0.666701	−0.333350	+	0.942803i	$0.608179\pi$
$480$	0	0
$481$	−5.14182e14	−0.910581
$482$	−4.79989e14	−0.840375
$483$	0	0
$484$	−1.20907e14	−0.206920
$485$	3.57132e14	0.604295
$486$	0	0
$487$	6.31645e14	1.04487	0.522437	−	0.852678i	$-0.325023\pi$
$487$	6.31645e14	1.04487	0.522437	+	0.852678i	$0.325023\pi$
$488$	1.87819e14	0.307206
$489$	0	0
$490$	−6.04038e14	−0.966019
$491$	−2.78183e14	−0.439929	−0.219965	−	0.975508i	$-0.570594\pi$
$491$	−2.78183e14	−0.439929	−0.219965	+	0.975508i	$0.570594\pi$
$492$	0	0
$493$	6.43055e14	0.994467
$494$	6.63697e14	1.01501
$495$	0	0
$496$	−6.04529e13	−0.0904208
$497$	6.57675e14	0.972862
$498$	0	0
$499$	4.59050e14	0.664213	0.332106	−	0.943242i	$-0.392241\pi$
$499$	4.59050e14	0.664213	0.332106	+	0.943242i	$0.392241\pi$
$500$	3.80298e14	0.544239
$501$	0	0
$502$	4.69814e14	0.657742
$503$	9.15703e14	1.26803	0.634017	−	0.773319i	$-0.281405\pi$
$503$	9.15703e14	1.26803	0.634017	+	0.773319i	$0.281405\pi$
$504$	0	0
$505$	−5.47563e14	−0.741877
$506$	6.83029e14	0.915401
$507$	0	0
$508$	−4.85077e14	−0.636150
$509$	−1.08562e15	−1.40842	−0.704208	−	0.709993i	$-0.748698\pi$
$509$	−1.08562e15	−1.40842	−0.704208	+	0.709993i	$0.748698\pi$
$510$	0	0
$511$	−1.12840e15	−1.43267
$512$	3.51844e13	0.0441942
$513$	0	0
$514$	5.26324e14	0.647077
$515$	−3.13508e14	−0.381337
$516$	0	0
$517$	5.34398e14	0.636307
$518$	−8.71852e14	−1.02714
$519$	0	0
$520$	−2.58387e14	−0.298024
$521$	1.63925e14	0.187084	0.0935422	−	0.995615i	$-0.470181\pi$
$521$	1.63925e14	0.187084	0.0935422	+	0.995615i	$0.470181\pi$
$522$	0	0
$523$	4.69127e13	0.0524241	0.0262121	−	0.999656i	$-0.491655\pi$
$523$	4.69127e13	0.0524241	0.0262121	+	0.999656i	$0.491655\pi$
$524$	−1.72620e14	−0.190883
$525$	0	0
$526$	6.29818e14	0.682015
$527$	3.12644e14	0.335036
$528$	0	0
$529$	1.77141e15	1.85914
$530$	7.55681e13	0.0784912
$531$	0	0
$532$	1.12537e15	1.14494
$533$	−1.17106e15	−1.17918
$534$	0	0
$535$	3.29439e13	0.0324960
$536$	1.27575e14	0.124554
$537$	0	0
$538$	3.71757e14	0.355596
$539$	1.33877e15	1.26756
$540$	0	0
$541$	−1.28196e15	−1.18929	−0.594647	−	0.803987i	$-0.702708\pi$
$541$	−1.28196e15	−1.18929	−0.594647	+	0.803987i	$0.702708\pi$
$542$	2.38972e14	0.219458
$543$	0	0
$544$	−1.81963e14	−0.163753
$545$	2.86411e14	0.255158
$546$	0	0
$547$	1.10564e15	0.965347	0.482673	−	0.875801i	$-0.339666\pi$
$547$	1.10564e15	0.965347	0.482673	+	0.875801i	$0.339666\pi$
$548$	8.20715e14	0.709413
$549$	0	0
$550$	−2.03903e14	−0.172754
$551$	−1.79841e15	−1.50854
$552$	0	0
$553$	−2.18818e15	−1.79926
$554$	5.10606e14	0.415702
$555$	0	0
$556$	5.37435e14	0.428958
$557$	−2.12161e15	−1.67672	−0.838362	−	0.545113i	$-0.816487\pi$
$557$	−2.12161e15	−1.67672	−0.838362	+	0.545113i	$0.816487\pi$
$558$	0	0
$559$	−1.70287e15	−1.31952
$560$	−4.38124e14	−0.336173
$561$	0	0
$562$	−1.26757e15	−0.953721
$563$	2.44888e15	1.82462	0.912309	−	0.409503i	$-0.134298\pi$
$563$	2.44888e15	1.82462	0.912309	+	0.409503i	$0.134298\pi$
$564$	0	0
$565$	1.02521e15	0.749111
$566$	4.80002e14	0.347338
$567$	0	0
$568$	2.97399e14	0.211068
$569$	7.38095e14	0.518794	0.259397	−	0.965771i	$-0.416476\pi$
$569$	7.38095e14	0.518794	0.259397	+	0.965771i	$0.416476\pi$
$570$	0	0
$571$	−2.10103e13	−0.0144855	−0.00724274	−	0.999974i	$-0.502305\pi$
$571$	−2.10103e13	−0.0144855	−0.00724274	+	0.999974i	$0.502305\pi$
$572$	5.72682e14	0.391053
$573$	0	0
$574$	−1.98567e15	−1.33012
$575$	−8.13255e14	−0.539577
$576$	0	0
$577$	1.13249e15	0.737168	0.368584	−	0.929594i	$-0.379843\pi$
$577$	1.13249e15	0.737168	0.368584	+	0.929594i	$0.379843\pi$
$578$	−1.55645e14	−0.100353
$579$	0	0
$580$	7.00149e14	0.442931
$581$	−1.67647e15	−1.05058
$582$	0	0
$583$	−1.67487e14	−0.102992
$584$	−5.10257e14	−0.310827
$585$	0	0
$586$	−1.24242e15	−0.742728
$587$	2.91255e15	1.72490	0.862448	−	0.506145i	$-0.168930\pi$
$587$	2.91255e15	1.72490	0.862448	+	0.506145i	$0.168930\pi$
$588$	0	0
$589$	−8.74362e14	−0.508226
$590$	−5.31923e14	−0.306311
$591$	0	0
$592$	−3.94249e14	−0.222844
$593$	−1.52901e15	−0.856270	−0.428135	−	0.903715i	$-0.640829\pi$
$593$	−1.52901e15	−0.856270	−0.428135	+	0.903715i	$0.640829\pi$
$594$	0	0
$595$	2.26584e15	1.24562
$596$	−1.28719e15	−0.701113
$597$	0	0
$598$	2.28411e15	1.22141
$599$	−2.71517e15	−1.43863	−0.719315	−	0.694684i	$-0.755544\pi$
$599$	−2.71517e15	−1.43863	−0.719315	+	0.694684i	$0.755544\pi$
$600$	0	0
$601$	−2.59104e15	−1.34792	−0.673960	−	0.738768i	$-0.735408\pi$
$601$	−2.59104e15	−1.34792	−0.673960	+	0.738768i	$0.735408\pi$
$602$	−2.88740e15	−1.48842
$603$	0	0
$604$	−5.03928e14	−0.255074
$605$	6.80810e14	0.341485
$606$	0	0
$607$	1.68761e15	0.831255	0.415628	−	0.909535i	$-0.363562\pi$
$607$	1.68761e15	0.831255	0.415628	+	0.909535i	$0.363562\pi$
$608$	5.08890e14	0.248401
$609$	0	0
$610$	−1.05758e15	−0.506990
$611$	1.78707e15	0.849016
$612$	0	0
$613$	−5.35764e14	−0.250000	−0.125000	−	0.992157i	$-0.539893\pi$
$613$	−5.35764e14	−0.250000	−0.125000	+	0.992157i	$0.539893\pi$
$614$	2.42622e15	1.12203
$615$	0	0
$616$	9.71047e14	0.441109
$617$	2.84367e15	1.28030	0.640148	−	0.768251i	$-0.278873\pi$
$617$	2.84367e15	1.28030	0.640148	+	0.768251i	$0.278873\pi$
$618$	0	0
$619$	8.14487e14	0.360235	0.180117	−	0.983645i	$-0.442352\pi$
$619$	8.14487e14	0.360235	0.180117	+	0.983645i	$0.442352\pi$
$620$	3.40402e14	0.149224
$621$	0	0
$622$	2.55970e15	1.10241
$623$	1.85615e15	0.792372
$624$	0	0
$625$	−1.38060e15	−0.579065
$626$	−2.19397e15	−0.912160
$627$	0	0
$628$	−1.02102e15	−0.417116
$629$	2.03894e15	0.825705
$630$	0	0
$631$	−1.45868e15	−0.580497	−0.290248	−	0.956951i	$-0.593738\pi$
$631$	−1.45868e15	−0.580497	−0.290248	+	0.956951i	$0.593738\pi$
$632$	−9.89488e14	−0.390361
$633$	0	0
$634$	−2.04125e14	−0.0791417
$635$	2.73140e15	1.04986
$636$	0	0
$637$	4.47698e15	1.69129
$638$	−1.55179e15	−0.581192
$639$	0	0
$640$	−1.98118e14	−0.0729348
$641$	1.16297e15	0.424473	0.212236	−	0.977218i	$-0.431925\pi$
$641$	1.16297e15	0.424473	0.212236	+	0.977218i	$0.431925\pi$
$642$	0	0
$643$	4.73461e15	1.69873	0.849364	−	0.527807i	$-0.176985\pi$
$643$	4.73461e15	1.69873	0.849364	+	0.527807i	$0.176985\pi$
$644$	3.87296e15	1.37775
$645$	0	0
$646$	−2.63182e15	−0.920402
$647$	−5.47527e15	−1.89859	−0.949297	−	0.314380i	$-0.898203\pi$
$647$	−5.47527e15	−1.89859	−0.949297	+	0.314380i	$0.898203\pi$
$648$	0	0
$649$	1.17894e15	0.401927
$650$	−6.81870e14	−0.230504
$651$	0	0
$652$	−4.95761e14	−0.164782
$653$	8.87971e14	0.292669	0.146334	−	0.989235i	$-0.453252\pi$
$653$	8.87971e14	0.292669	0.146334	+	0.989235i	$0.453252\pi$
$654$	0	0
$655$	9.71997e14	0.315020
$656$	−8.97913e14	−0.288578
$657$	0	0
$658$	3.03018e15	0.957694
$659$	−3.14454e15	−0.985571	−0.492785	−	0.870151i	$-0.664021\pi$
$659$	−3.14454e15	−0.985571	−0.492785	+	0.870151i	$0.664021\pi$
$660$	0	0
$661$	1.32982e15	0.409907	0.204954	−	0.978772i	$-0.434296\pi$
$661$	1.32982e15	0.409907	0.204954	+	0.978772i	$0.434296\pi$
$662$	2.84867e15	0.870810
$663$	0	0
$664$	−7.58096e14	−0.227929
$665$	−6.33681e15	−1.88952
$666$	0	0
$667$	−6.18923e15	−1.81528
$668$	−2.37094e15	−0.689685
$669$	0	0
$670$	−7.18355e14	−0.205555
$671$	2.34399e15	0.665247
$672$	0	0
$673$	−6.84438e15	−1.91096	−0.955479	−	0.295058i	$-0.904661\pi$
$673$	−6.84438e15	−1.91096	−0.955479	+	0.295058i	$0.904661\pi$
$674$	4.56323e15	1.26370
$675$	0	0
$676$	7.99278e13	0.0217766
$677$	−8.88820e14	−0.240202	−0.120101	−	0.992762i	$-0.538322\pi$
$677$	−8.88820e14	−0.240202	−0.120101	+	0.992762i	$0.538322\pi$
$678$	0	0
$679$	−4.48824e15	−1.19342
$680$	1.02461e15	0.270245
$681$	0	0
$682$	−7.54459e14	−0.195804
$683$	−2.22402e15	−0.572565	−0.286282	−	0.958145i	$-0.592420\pi$
$683$	−2.22402e15	−0.572565	−0.286282	+	0.958145i	$0.592420\pi$
$684$	0	0
$685$	−4.62133e15	−1.17076
$686$	3.00610e15	0.755477
$687$	0	0
$688$	−1.30568e15	−0.322923
$689$	−5.60092e14	−0.137421
$690$	0	0
$691$	4.59721e15	1.11011	0.555053	−	0.831815i	$-0.312698\pi$
$691$	4.59721e15	1.11011	0.555053	+	0.831815i	$0.312698\pi$
$692$	−3.45248e15	−0.827081
$693$	0	0
$694$	4.27923e15	1.00899
$695$	−3.02622e15	−0.707921
$696$	0	0
$697$	4.64373e15	1.06927
$698$	−2.01752e15	−0.460906
$699$	0	0
$700$	−1.15619e15	−0.260009
$701$	2.58701e15	0.577231	0.288616	−	0.957445i	$-0.406805\pi$
$701$	2.58701e15	0.577231	0.288616	+	0.957445i	$0.406805\pi$
$702$	0	0
$703$	−5.70223e15	−1.25254
$704$	4.39105e14	0.0957014
$705$	0	0
$706$	9.27197e14	0.198951
$707$	6.88148e15	1.46513
$708$	0	0
$709$	2.52081e15	0.528427	0.264214	−	0.964464i	$-0.414888\pi$
$709$	2.52081e15	0.528427	0.264214	+	0.964464i	$0.414888\pi$
$710$	−1.67461e15	−0.348332
$711$	0	0
$712$	8.39346e14	0.171910
$713$	−3.00911e15	−0.611570
$714$	0	0
$715$	−3.22469e15	−0.645365
$716$	−1.68548e15	−0.334736
$717$	0	0
$718$	−1.38595e15	−0.271058
$719$	−3.03716e15	−0.589467	−0.294733	−	0.955580i	$-0.595231\pi$
$719$	−3.03716e15	−0.589467	−0.294733	+	0.955580i	$0.595231\pi$
$720$	0	0
$721$	3.94001e15	0.753100
$722$	3.63265e15	0.689079
$723$	0	0
$724$	2.80799e15	0.524605
$725$	1.84766e15	0.342580
$726$	0	0
$727$	6.52086e15	1.19087	0.595436	−	0.803402i	$-0.296979\pi$
$727$	6.52086e15	1.19087	0.595436	+	0.803402i	$0.296979\pi$
$728$	3.24727e15	0.588566
$729$	0	0
$730$	2.87319e15	0.512966
$731$	6.75255e15	1.19653
$732$	0	0
$733$	−1.48208e15	−0.258703	−0.129351	−	0.991599i	$-0.541290\pi$
$733$	−1.48208e15	−0.258703	−0.129351	+	0.991599i	$0.541290\pi$
$734$	−3.59907e15	−0.623536
$735$	0	0
$736$	1.75134e15	0.298912
$737$	1.59215e15	0.269719
$738$	0	0
$739$	5.39343e15	0.900161	0.450081	−	0.892988i	$-0.351395\pi$
$739$	5.39343e15	0.900161	0.450081	+	0.892988i	$0.351395\pi$
$740$	2.21996e15	0.367765
$741$	0	0
$742$	−9.49699e14	−0.155012
$743$	−8.96192e15	−1.45199	−0.725993	−	0.687702i	$-0.758620\pi$
$743$	−8.96192e15	−1.45199	−0.725993	+	0.687702i	$0.758620\pi$
$744$	0	0
$745$	7.24798e15	1.15706
$746$	4.15832e15	0.658953
$747$	0	0
$748$	−2.27091e15	−0.354603
$749$	−4.14021e14	−0.0641760
$750$	0	0
$751$	−7.78720e15	−1.18949	−0.594746	−	0.803913i	$-0.702747\pi$
$751$	−7.78720e15	−1.18949	−0.594746	+	0.803913i	$0.702747\pi$
$752$	1.37024e15	0.207777
$753$	0	0
$754$	−5.18933e15	−0.775477
$755$	2.83755e15	0.420955
$756$	0	0
$757$	−2.41922e15	−0.353711	−0.176856	−	0.984237i	$-0.556593\pi$
$757$	−2.41922e15	−0.353711	−0.176856	+	0.984237i	$0.556593\pi$
$758$	−3.02522e15	−0.439113
$759$	0	0
$760$	−2.86549e15	−0.409943
$761$	5.02552e15	0.713782	0.356891	−	0.934146i	$-0.383837\pi$
$761$	5.02552e15	0.713782	0.356891	+	0.934146i	$0.383837\pi$
$762$	0	0
$763$	−3.59946e15	−0.503909
$764$	−4.18211e15	−0.581275
$765$	0	0
$766$	−1.74438e15	−0.238991
$767$	3.94249e15	0.536285
$768$	0	0
$769$	1.21056e16	1.62327	0.811634	−	0.584166i	$-0.198578\pi$
$769$	1.21056e16	1.62327	0.811634	+	0.584166i	$0.198578\pi$
$770$	−5.46783e15	−0.727974
$771$	0	0
$772$	−4.57973e15	−0.601097
$773$	7.45630e15	0.971709	0.485854	−	0.874040i	$-0.338508\pi$
$773$	7.45630e15	0.971709	0.485854	+	0.874040i	$0.338508\pi$
$774$	0	0
$775$	8.98303e14	0.115415
$776$	−2.02957e15	−0.258919
$777$	0	0
$778$	1.63641e15	0.205828
$779$	−1.29870e16	−1.62200
$780$	0	0
$781$	3.71157e15	0.457064
$782$	−9.05741e15	−1.10756
$783$	0	0
$784$	3.43273e15	0.413905
$785$	5.74923e15	0.688377
$786$	0	0
$787$	−1.12580e16	−1.32923	−0.664615	−	0.747186i	$-0.731404\pi$
$787$	−1.12580e16	−1.32923	−0.664615	+	0.747186i	$0.731404\pi$
$788$	7.54713e15	0.884887
$789$	0	0
$790$	5.57167e15	0.644223
$791$	−1.28843e16	−1.47941
$792$	0	0
$793$	7.83852e15	0.887630
$794$	−1.55915e15	−0.175338
$795$	0	0
$796$	2.75772e15	0.305864
$797$	1.04669e16	1.15291	0.576457	−	0.817127i	$-0.304435\pi$
$797$	1.04669e16	1.15291	0.576457	+	0.817127i	$0.304435\pi$
$798$	0	0
$799$	−7.08646e15	−0.769878
$800$	−5.22824e14	−0.0564106
$801$	0	0
$802$	−1.20450e16	−1.28189
$803$	−6.36807e15	−0.673088
$804$	0	0
$805$	−2.18081e16	−2.27374
$806$	−2.52298e15	−0.261259
$807$	0	0
$808$	3.11179e15	0.317868
$809$	1.28180e16	1.30048	0.650238	−	0.759730i	$-0.274669\pi$
$809$	1.28180e16	1.30048	0.650238	+	0.759730i	$0.274669\pi$
$810$	0	0
$811$	9.79295e15	0.980164	0.490082	−	0.871676i	$-0.336967\pi$
$811$	9.79295e15	0.980164	0.490082	+	0.871676i	$0.336967\pi$
$812$	−8.79909e15	−0.874742
$813$	0	0
$814$	−4.92027e15	−0.482564
$815$	2.79156e15	0.271945
$816$	0	0
$817$	−1.88847e16	−1.81505
$818$	9.99895e15	0.954578
$819$	0	0
$820$	5.05602e15	0.476247
$821$	−7.08410e15	−0.662823	−0.331411	−	0.943486i	$-0.607525\pi$
$821$	−7.08410e15	−0.662823	−0.331411	+	0.943486i	$0.607525\pi$
$822$	0	0
$823$	1.98569e16	1.83321	0.916606	−	0.399792i	$-0.130918\pi$
$823$	1.98569e16	1.83321	0.916606	+	0.399792i	$0.130918\pi$
$824$	1.78166e15	0.163390
$825$	0	0
$826$	6.68493e15	0.604932
$827$	2.05537e16	1.84760	0.923802	−	0.382871i	$-0.125065\pi$
$827$	2.05537e16	1.84760	0.923802	+	0.382871i	$0.125065\pi$
$828$	0	0
$829$	−1.45920e16	−1.29439	−0.647197	−	0.762323i	$-0.724059\pi$
$829$	−1.45920e16	−1.29439	−0.647197	+	0.762323i	$0.724059\pi$
$830$	4.26873e15	0.376157
$831$	0	0
$832$	1.46840e15	0.127693
$833$	−1.77530e16	−1.53364
$834$	0	0
$835$	1.33504e16	1.13820
$836$	6.35100e15	0.537907
$837$	0	0
$838$	−9.01533e15	−0.753598
$839$	−6.54963e15	−0.543909	−0.271954	−	0.962310i	$-0.587670\pi$
$839$	−6.54963e15	−0.543909	−0.271954	+	0.962310i	$0.587670\pi$
$840$	0	0
$841$	1.86098e15	0.152533
$842$	−9.63404e15	−0.784498
$843$	0	0
$844$	6.79381e15	0.546047
$845$	−4.50062e14	−0.0359386
$846$	0	0
$847$	−8.55606e15	−0.674396
$848$	−4.29451e14	−0.0336307
$849$	0	0
$850$	2.70389e15	0.209018
$851$	−1.96242e16	−1.50723
$852$	0	0
$853$	1.99244e16	1.51066	0.755329	−	0.655346i	$-0.227477\pi$
$853$	1.99244e16	1.51066	0.755329	+	0.655346i	$0.227477\pi$
$854$	1.32911e16	1.00125
$855$	0	0
$856$	−1.87219e14	−0.0139234
$857$	−1.59149e16	−1.17600	−0.588002	−	0.808859i	$-0.700085\pi$
$857$	−1.59149e16	−1.17600	−0.588002	+	0.808859i	$0.700085\pi$
$858$	0	0
$859$	−1.19020e15	−0.0868272	−0.0434136	−	0.999057i	$-0.513823\pi$
$859$	−1.19020e15	−0.0868272	−0.0434136	+	0.999057i	$0.513823\pi$
$860$	7.35208e15	0.532928
$861$	0	0
$862$	−1.10942e16	−0.793975
$863$	1.73343e15	0.123267	0.0616336	−	0.998099i	$-0.480369\pi$
$863$	1.73343e15	0.123267	0.0616336	+	0.998099i	$0.480369\pi$
$864$	0	0
$865$	1.94405e16	1.36495
$866$	1.54351e16	1.07686
$867$	0	0
$868$	−4.27799e15	−0.294701
$869$	−1.23489e16	−0.845317
$870$	0	0
$871$	5.32428e15	0.359882
$872$	−1.62766e15	−0.109326
$873$	0	0
$874$	2.53306e16	1.68009
$875$	2.69121e16	1.77379
$876$	0	0
$877$	1.16602e16	0.758943	0.379471	−	0.925203i	$-0.376106\pi$
$877$	1.16602e16	0.758943	0.379471	+	0.925203i	$0.376106\pi$
$878$	3.80589e15	0.246171
$879$	0	0
$880$	−2.47254e15	−0.157938
$881$	1.66091e16	1.05434	0.527168	−	0.849761i	$-0.323254\pi$
$881$	1.66091e16	1.05434	0.527168	+	0.849761i	$0.323254\pi$
$882$	0	0
$883$	−1.28205e16	−0.803750	−0.401875	−	0.915695i	$-0.631641\pi$
$883$	−1.28205e16	−0.803750	−0.401875	+	0.915695i	$0.631641\pi$
$884$	−7.59414e15	−0.473141
$885$	0	0
$886$	6.39726e15	0.393648
$887$	2.21358e14	0.0135368	0.00676840	−	0.999977i	$-0.497846\pi$
$887$	2.21358e14	0.0135368	0.00676840	+	0.999977i	$0.497846\pi$
$888$	0	0
$889$	−3.43267e16	−2.07335
$890$	−4.72624e15	−0.283708
$891$	0	0
$892$	−6.13948e15	−0.364020
$893$	1.98185e16	1.16785
$894$	0	0
$895$	9.49070e15	0.552423
$896$	2.48984e15	0.144038
$897$	0	0
$898$	1.62145e16	0.926579
$899$	6.83649e15	0.388289
$900$	0	0
$901$	2.22099e15	0.124612
$902$	−1.12060e16	−0.624908
$903$	0	0
$904$	−5.82623e15	−0.320968
$905$	−1.58114e16	−0.865770
$906$	0	0
$907$	−2.48361e16	−1.34352	−0.671758	−	0.740770i	$-0.734461\pi$
$907$	−2.48361e16	−1.34352	−0.671758	+	0.740770i	$0.734461\pi$
$908$	7.93529e15	0.426668
$909$	0	0
$910$	−1.82849e16	−0.971326
$911$	−8.10295e15	−0.427851	−0.213925	−	0.976850i	$-0.568625\pi$
$911$	−8.10295e15	−0.427851	−0.213925	+	0.976850i	$0.568625\pi$
$912$	0	0
$913$	−9.46112e15	−0.493575
$914$	−1.16086e16	−0.601971
$915$	0	0
$916$	3.84823e15	0.197168
$917$	−1.22155e16	−0.622130
$918$	0	0
$919$	−3.33814e16	−1.67984	−0.839922	−	0.542707i	$-0.817399\pi$
$919$	−3.33814e16	−1.67984	−0.839922	+	0.542707i	$0.817399\pi$
$920$	−9.86156e15	−0.493302
$921$	0	0
$922$	−2.10069e15	−0.103835
$923$	1.24118e16	0.609854
$924$	0	0
$925$	5.85836e15	0.284444
$926$	−6.61595e15	−0.319324
$927$	0	0
$928$	−3.97892e15	−0.189781
$929$	−3.56713e16	−1.69134	−0.845672	−	0.533703i	$-0.820800\pi$
$929$	−3.56713e16	−1.69134	−0.845672	+	0.533703i	$0.820800\pi$
$930$	0	0
$931$	4.96493e16	2.32643
$932$	−9.24340e15	−0.430570
$933$	0	0
$934$	1.21408e16	0.558908
$935$	1.27872e16	0.585210
$936$	0	0
$937$	3.96248e16	1.79225	0.896126	−	0.443800i	$-0.146370\pi$
$937$	3.96248e16	1.79225	0.896126	+	0.443800i	$0.146370\pi$
$938$	9.02791e15	0.405949
$939$	0	0
$940$	−7.71563e15	−0.342900
$941$	−5.37193e15	−0.237349	−0.118675	−	0.992933i	$-0.537865\pi$
$941$	−5.37193e15	−0.237349	−0.118675	+	0.992933i	$0.537865\pi$
$942$	0	0
$943$	−4.46946e16	−1.95183
$944$	3.02290e15	0.131244
$945$	0	0
$946$	−1.62950e16	−0.699282
$947$	−2.22241e16	−0.948199	−0.474099	−	0.880471i	$-0.657226\pi$
$947$	−2.22241e16	−0.948199	−0.474099	+	0.880471i	$0.657226\pi$
$948$	0	0
$949$	−2.12954e16	−0.898092
$950$	−7.56188e15	−0.317066
$951$	0	0
$952$	−1.28767e16	−0.533706
$953$	−7.66867e15	−0.316016	−0.158008	−	0.987438i	$-0.550507\pi$
$953$	−7.66867e15	−0.316016	−0.158008	+	0.987438i	$0.550507\pi$
$954$	0	0
$955$	2.35489e16	0.959293
$956$	4.11924e15	0.166839
$957$	0	0
$958$	−1.17741e16	−0.471429
$959$	5.80784e16	2.31213
$960$	0	0
$961$	−2.20847e16	−0.869185
$962$	−1.64538e16	−0.643878
$963$	0	0
$964$	−1.53597e16	−0.594235
$965$	2.57878e16	0.992006
$966$	0	0
$967$	2.80474e16	1.06671	0.533355	−	0.845891i	$-0.320931\pi$
$967$	2.80474e16	1.06671	0.533355	+	0.845891i	$0.320931\pi$
$968$	−3.86902e15	−0.146314
$969$	0	0
$970$	1.14282e16	0.427301
$971$	2.43427e16	0.905031	0.452516	−	0.891757i	$-0.350527\pi$
$971$	2.43427e16	0.905031	0.452516	+	0.891757i	$0.350527\pi$
$972$	0	0
$973$	3.80320e16	1.39807
$974$	2.02126e16	0.738837
$975$	0	0
$976$	6.01019e15	0.217227
$977$	2.19144e16	0.787608	0.393804	−	0.919194i	$-0.371159\pi$
$977$	2.19144e16	0.787608	0.393804	+	0.919194i	$0.371159\pi$
$978$	0	0
$979$	1.04751e16	0.372267
$980$	−1.93292e16	−0.683079
$981$	0	0
$982$	−8.90186e15	−0.311077
$983$	7.04586e15	0.244844	0.122422	−	0.992478i	$-0.460934\pi$
$983$	7.04586e15	0.244844	0.122422	+	0.992478i	$0.460934\pi$
$984$	0	0
$985$	−4.24969e16	−1.46035
$986$	2.05778e16	0.703194
$987$	0	0
$988$	2.12383e16	0.717722
$989$	−6.49915e16	−2.18412
$990$	0	0
$991$	4.04873e16	1.34559	0.672796	−	0.739828i	$-0.265093\pi$
$991$	4.04873e16	1.34559	0.672796	+	0.739828i	$0.265093\pi$
$992$	−1.93449e15	−0.0639371
$993$	0	0
$994$	2.10456e16	0.687918
$995$	−1.55283e16	−0.504775
$996$	0	0
$997$	1.79607e16	0.577432	0.288716	−	0.957415i	$-0.406772\pi$
$997$	1.79607e16	0.577432	0.288716	+	0.957415i	$0.406772\pi$
$998$	1.46896e16	0.469669
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	18.12.a.c.1.1		1
3.2	odd	2		6.12.a.a.1.1	✓	1
4.3	odd	2		144.12.a.b.1.1		1
9.2	odd	6		162.12.c.g.109.1		2
9.4	even	3		162.12.c.d.55.1		2
9.5	odd	6		162.12.c.g.55.1		2
9.7	even	3		162.12.c.d.109.1		2
12.11	even	2		48.12.a.h.1.1		1
15.2	even	4		150.12.c.f.49.1		2
15.8	even	4		150.12.c.f.49.2		2
15.14	odd	2		150.12.a.g.1.1		1
24.5	odd	2		192.12.a.l.1.1		1
24.11	even	2		192.12.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.12.a.a.1.1	✓	1	3.2	odd	2
18.12.a.c.1.1		1	1.1	even	1	trivial
48.12.a.h.1.1		1	12.11	even	2
144.12.a.b.1.1		1	4.3	odd	2
150.12.a.g.1.1		1	15.14	odd	2
150.12.c.f.49.1		2	15.2	even	4
150.12.c.f.49.2		2	15.8	even	4
162.12.c.d.55.1		2	9.4	even	3
162.12.c.d.109.1		2	9.7	even	3
162.12.c.g.55.1		2	9.5	odd	6
162.12.c.g.109.1		2	9.2	odd	6
192.12.a.b.1.1		1	24.11	even	2
192.12.a.l.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:	\( N \)	\(=\)	\( 18 = 2 \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	18.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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