LMFDB - Embedded newform 1323.2.c.e.1322.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,2,Mod(1322,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1323.1322");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1323.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$10.5642081874$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 12x^{14} + 106x^{12} - 384x^{10} + 1005x^{8} - 1200x^{6} + 1030x^{4} - 252x^{2} + 49 $ x^16 - 12x^14 + 106x^12 - 384x^10 + 1005x^8 - 1200x^6 + 1030x^4 - 252*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1322.14
Root			$1.47769 + 0.853147i$ of defining polynomial
Character	$\chi$	$=$	1323.1322
Dual form			1323.2.c.e.1322.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.70629i q^{2} -0.911441 q^{4} +0.829297 q^{5} +1.85740i q^{8} +O(q^{10})$	$q+1.70629i q^{2} -0.911441 q^{4} +0.829297 q^{5} +1.85740i q^{8} +1.41503i q^{10} +0.190855i q^{11} -3.53188i q^{13} -4.99216 q^{16} -4.60484 q^{17} +4.09593i q^{19} -0.755855 q^{20} -0.325654 q^{22} +8.79438i q^{23} -4.31227 q^{25} +6.02643 q^{26} +7.99271i q^{29} +6.56325i q^{31} -4.80328i q^{32} -7.85721i q^{34} +1.45644 q^{37} -6.98887 q^{38} +1.54034i q^{40} +12.1217 q^{41} +2.83253 q^{43} -0.173953i q^{44} -15.0058 q^{46} +1.49423 q^{47} -7.35800i q^{50} +3.21910i q^{52} +4.55622i q^{53} +0.158275i q^{55} -13.6379 q^{58} +5.67378 q^{59} -10.9229i q^{61} -11.1988 q^{62} -1.78850 q^{64} -2.92898i q^{65} -4.64347 q^{67} +4.19704 q^{68} +10.0568i q^{71} +7.95020i q^{73} +2.48512i q^{74} -3.73320i q^{76} -5.96332 q^{79} -4.13998 q^{80} +20.6832i q^{82} -2.05346 q^{83} -3.81878 q^{85} +4.83313i q^{86} -0.354494 q^{88} -8.83575 q^{89} -8.01556i q^{92} +2.54959i q^{94} +3.39675i q^{95} -7.15962i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4}+O(q^{10}) $ 16 * q - 16 * q^4	$ 16 q - 16 q^{4} + 48 q^{16} + 16 q^{22} + 32 q^{25} - 16 q^{37} + 32 q^{43} - 80 q^{46} - 96 q^{58} - 176 q^{64} + 96 q^{67} - 64 q^{79} - 32 q^{85} + 112 q^{88}+O(q^{100}) $ 16 * q - 16 * q^4 + 48 * q^16 + 16 * q^22 + 32 * q^25 - 16 * q^37 + 32 * q^43 - 80 * q^46 - 96 * q^58 - 176 * q^64 + 96 * q^67 - 64 * q^79 - 32 * q^85 + 112 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$
$\chi(n)$	$-1$	$-1$

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 $

$2$	1.70629i	1.20653i	0.797540	+	0.603266i	$0.206134\pi$
$2$	1.70629i	1.20653i	−0.797540	+	0.603266i	$0.793866\pi$
$3$	0	0
$3$	0	0
$4$	−0.911441	−0.455720
$5$	0.829297	0.370873	0.185437	−	0.982656i	$-0.440630\pi$
$5$	0.829297	0.370873	0.185437	+	0.982656i	$0.440630\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.85740i	0.656691i
$9$	0	0
$10$	1.41503i	0.447470i
$11$	0.190855i	0.0575449i	0.999586	+	0.0287724i	$0.00915981\pi$
$11$	0.190855i	0.0575449i	−0.999586	+	0.0287724i	$0.990840\pi$
$12$	0	0
$13$	− 3.53188i	− 0.979568i	−0.871844	−	0.489784i	$-0.837076\pi$
$13$	− 3.53188i	− 0.979568i	0.871844	−	0.489784i	$-0.162924\pi$
$14$	0	0
$15$	0	0
$16$	−4.99216	−1.24804
$17$	−4.60484	−1.11684	−0.558418	−	0.829559i	$-0.688592\pi$
$17$	−4.60484	−1.11684	−0.558418	+	0.829559i	$0.688592\pi$
$18$	0	0
$19$	4.09593i	0.939671i	0.882754	+	0.469836i	$0.155687\pi$
$19$	4.09593i	0.939671i	−0.882754	+	0.469836i	$0.844313\pi$
$20$	−0.755855	−0.169014
$21$	0	0
$22$	−0.325654	−0.0694297
$23$	8.79438i	1.83376i	0.399169	+	0.916878i	$0.369299\pi$
$23$	8.79438i	1.83376i	−0.399169	+	0.916878i	$0.630701\pi$
$24$	0	0
$25$	−4.31227	−0.862453
$26$	6.02643	1.18188
$27$	0	0
$28$	0	0
$29$	7.99271i	1.48421i	0.670284	+	0.742105i	$0.266172\pi$
$29$	7.99271i	1.48421i	−0.670284	+	0.742105i	$0.733828\pi$
$30$	0	0
$31$	6.56325i	1.17879i	0.807843	+	0.589397i	$0.200635\pi$
$31$	6.56325i	1.17879i	−0.807843	+	0.589397i	$0.799365\pi$
$32$	− 4.80328i	− 0.849109i
$33$	0	0
$34$	− 7.85721i	− 1.34750i
$35$	0	0
$36$	0	0
$37$	1.45644	0.239438	0.119719	−	0.992808i	$-0.461801\pi$
$37$	1.45644	0.239438	0.119719	+	0.992808i	$0.461801\pi$
$38$	−6.98887	−1.13374
$39$	0	0
$40$	1.54034i	0.243549i
$41$	12.1217	1.89309	0.946546	−	0.322569i	$-0.104546\pi$
$41$	12.1217	1.89309	0.946546	+	0.322569i	$0.104546\pi$
$42$	0	0
$43$	2.83253	0.431957	0.215978	−	0.976398i	$-0.430706\pi$
$43$	2.83253	0.431957	0.215978	+	0.976398i	$0.430706\pi$
$44$	− 0.173953i	− 0.0262244i
$45$	0	0
$46$	−15.0058	−2.21248
$47$	1.49423	0.217955	0.108978	−	0.994044i	$-0.465242\pi$
$47$	1.49423	0.217955	0.108978	+	0.994044i	$0.465242\pi$
$48$	0	0
$49$	0	0
$50$	− 7.35800i	− 1.04058i
$51$	0	0
$52$	3.21910i	0.446409i
$53$	4.55622i	0.625845i	0.949779	+	0.312922i	$0.101308\pi$
$53$	4.55622i	0.625845i	−0.949779	+	0.312922i	$0.898692\pi$
$54$	0	0
$55$	0.158275i	0.0213418i
$56$	0	0
$57$	0	0
$58$	−13.6379	−1.79075
$59$	5.67378	0.738663	0.369331	−	0.929298i	$-0.379587\pi$
$59$	5.67378	0.738663	0.369331	+	0.929298i	$0.379587\pi$
$60$	0	0
$61$	− 10.9229i	− 1.39854i	−0.714859	−	0.699268i	$-0.753509\pi$
$61$	− 10.9229i	− 1.39854i	0.714859	−	0.699268i	$-0.246491\pi$
$62$	−11.1988	−1.42225
$63$	0	0
$64$	−1.78850	−0.223562
$65$	− 2.92898i	− 0.363295i
$66$	0	0
$67$	−4.64347	−0.567290	−0.283645	−	0.958929i	$-0.591544\pi$
$67$	−4.64347	−0.567290	−0.283645	+	0.958929i	$0.591544\pi$
$68$	4.19704	0.508965
$69$	0	0
$70$	0	0
$71$	10.0568i	1.19352i	0.802418	+	0.596762i	$0.203546\pi$
$71$	10.0568i	1.19352i	−0.802418	+	0.596762i	$0.796454\pi$
$72$	0	0
$73$	7.95020i	0.930501i	0.885179	+	0.465250i	$0.154036\pi$
$73$	7.95020i	0.930501i	−0.885179	+	0.465250i	$0.845964\pi$
$74$	2.48512i	0.288889i
$75$	0	0
$76$	− 3.73320i	− 0.428227i
$77$	0	0
$78$	0	0
$79$	−5.96332	−0.670926	−0.335463	−	0.942053i	$-0.608893\pi$
$79$	−5.96332	−0.670926	−0.335463	+	0.942053i	$0.608893\pi$
$80$	−4.13998	−0.462864
$81$	0	0
$82$	20.6832i	2.28408i
$83$	−2.05346	−0.225396	−0.112698	−	0.993629i	$-0.535949\pi$
$83$	−2.05346	−0.225396	−0.112698	+	0.993629i	$0.535949\pi$
$84$	0	0
$85$	−3.81878	−0.414205
$86$	4.83313i	0.521170i
$87$	0	0
$88$	−0.354494	−0.0377892
$89$	−8.83575	−0.936587	−0.468294	−	0.883573i	$-0.655131\pi$
$89$	−8.83575	−0.936587	−0.468294	+	0.883573i	$0.655131\pi$
$90$	0	0
$91$	0	0
$92$	− 8.01556i	− 0.835680i
$93$	0	0
$94$	2.54959i	0.262970i
$95$	3.39675i	0.348499i
$96$	0	0
$97$	− 7.15962i	− 0.726949i	−0.931604	−	0.363475i	$-0.881590\pi$
$97$	− 7.15962i	− 0.726949i	0.931604	−	0.363475i	$-0.118410\pi$
$98$	0	0
$99$	0	0
$100$	3.93037	0.393037
$101$	9.87162	0.982263	0.491132	−	0.871085i	$-0.336583\pi$
$101$	9.87162	0.982263	0.491132	+	0.871085i	$0.336583\pi$
$102$	0	0
$103$	− 8.10848i	− 0.798952i	−0.916744	−	0.399476i	$-0.869192\pi$
$103$	− 8.10848i	− 0.798952i	0.916744	−	0.399476i	$-0.130808\pi$
$104$	6.56013	0.643273
$105$	0	0
$106$	−7.77425	−0.755102
$107$	− 15.1370i	− 1.46335i	−0.681654	−	0.731675i	$-0.738739\pi$
$107$	− 15.1370i	− 1.46335i	0.681654	−	0.731675i	$-0.261261\pi$
$108$	0	0
$109$	−13.5779	−1.30053	−0.650266	−	0.759707i	$-0.725342\pi$
$109$	−13.5779	−1.30053	−0.650266	+	0.759707i	$0.725342\pi$
$110$	−0.270064	−0.0257496
$111$	0	0
$112$	0	0
$113$	− 13.9456i	− 1.31189i	−0.754809	−	0.655944i	$-0.772271\pi$
$113$	− 13.9456i	− 1.31189i	0.754809	−	0.655944i	$-0.227729\pi$
$114$	0	0
$115$	7.29316i	0.680090i
$116$	− 7.28488i	− 0.676384i
$117$	0	0
$118$	9.68113i	0.891221i
$119$	0	0
$120$	0	0
$121$	10.9636	0.996689
$122$	18.6377	1.68738
$123$	0	0
$124$	− 5.98201i	− 0.537201i
$125$	−7.72264	−0.690734
$126$	0	0
$127$	17.6627	1.56731	0.783654	−	0.621198i	$-0.213354\pi$
$127$	17.6627	1.56731	0.783654	+	0.621198i	$0.213354\pi$
$128$	− 12.6583i	− 1.11884i
$129$	0	0
$130$	4.99770	0.438328
$131$	−21.0008	−1.83485	−0.917424	−	0.397910i	$-0.869736\pi$
$131$	−21.0008	−1.83485	−0.917424	+	0.397910i	$0.869736\pi$
$132$	0	0
$133$	0	0
$134$	− 7.92312i	− 0.684453i
$135$	0	0
$136$	− 8.55303i	− 0.733417i
$137$	4.08661i	0.349143i	0.984645	+	0.174571i	$0.0558540\pi$
$137$	4.08661i	0.349143i	−0.984645	+	0.174571i	$0.944146\pi$
$138$	0	0
$139$	− 15.0668i	− 1.27795i	−0.769229	−	0.638974i	$-0.779359\pi$
$139$	− 15.0668i	− 1.27795i	0.769229	−	0.638974i	$-0.220641\pi$
$140$	0	0
$141$	0	0
$142$	−17.1599	−1.44003
$143$	0.674076	0.0563691
$144$	0	0
$145$	6.62833i	0.550453i
$146$	−13.5654	−1.12268
$147$	0	0
$148$	−1.32746	−0.109117
$149$	− 3.91310i	− 0.320574i	−0.987070	−	0.160287i	$-0.948758\pi$
$149$	− 3.91310i	− 0.320574i	0.987070	−	0.160287i	$-0.0512419\pi$
$150$	0	0
$151$	16.3370	1.32949	0.664743	−	0.747072i	$-0.268541\pi$
$151$	16.3370	1.32949	0.664743	+	0.747072i	$0.268541\pi$
$152$	−7.60779	−0.617074
$153$	0	0
$154$	0	0
$155$	5.44289i	0.437183i
$156$	0	0
$157$	− 7.39104i	− 0.589869i	−0.955517	−	0.294934i	$-0.904702\pi$
$157$	− 7.39104i	− 0.589869i	0.955517	−	0.294934i	$-0.0952978\pi$
$158$	− 10.1752i	− 0.809494i
$159$	0	0
$160$	− 3.98335i	− 0.314912i
$161$	0	0
$162$	0	0
$163$	18.6571	1.46134	0.730669	−	0.682732i	$-0.239208\pi$
$163$	18.6571	1.46134	0.730669	+	0.682732i	$0.239208\pi$
$164$	−11.0482	−0.862721
$165$	0	0
$166$	− 3.50381i	− 0.271948i
$167$	12.7703	0.988195	0.494098	−	0.869406i	$-0.335499\pi$
$167$	12.7703	0.988195	0.494098	+	0.869406i	$0.335499\pi$
$168$	0	0
$169$	0.525810	0.0404469
$170$	− 6.51596i	− 0.499751i
$171$	0	0
$172$	−2.58168	−0.196852
$173$	−5.20168	−0.395477	−0.197738	−	0.980255i	$-0.563360\pi$
$173$	−5.20168	−0.395477	−0.197738	+	0.980255i	$0.563360\pi$
$174$	0	0
$175$	0	0
$176$	− 0.952777i	− 0.0718182i
$177$	0	0
$178$	− 15.0764i	− 1.13002i
$179$	4.97724i	0.372016i	0.982548	+	0.186008i	$0.0595551\pi$
$179$	4.97724i	0.372016i	−0.982548	+	0.186008i	$0.940445\pi$
$180$	0	0
$181$	20.9730i	1.55891i	0.626458	+	0.779455i	$0.284504\pi$
$181$	20.9730i	1.55891i	−0.626458	+	0.779455i	$0.715496\pi$
$182$	0	0
$183$	0	0
$184$	−16.3347	−1.20421
$185$	1.20782	0.0888009
$186$	0	0
$187$	− 0.878855i	− 0.0642682i
$188$	−1.36190	−0.0993267
$189$	0	0
$190$	−5.79585	−0.420475
$191$	− 11.1280i	− 0.805190i	−0.915378	−	0.402595i	$-0.868108\pi$
$191$	− 11.1280i	− 0.805190i	0.915378	−	0.402595i	$-0.131892\pi$
$192$	0	0
$193$	9.60498	0.691381	0.345691	−	0.938349i	$-0.387645\pi$
$193$	9.60498	0.691381	0.345691	+	0.938349i	$0.387645\pi$
$194$	12.2164	0.877088
$195$	0	0
$196$	0	0
$197$	2.94588i	0.209885i	0.994478	+	0.104943i	$0.0334659\pi$
$197$	2.94588i	0.209885i	−0.994478	+	0.104943i	$0.966534\pi$
$198$	0	0
$199$	− 0.571631i	− 0.0405219i	−0.999795	−	0.0202609i	$-0.993550\pi$
$199$	− 0.571631i	− 0.0405219i	0.999795	−	0.0202609i	$-0.00644970\pi$
$200$	− 8.00961i	− 0.566365i
$201$	0	0
$202$	16.8439i	1.18513i
$203$	0	0
$204$	0	0
$205$	10.0525	0.702097
$206$	13.8354	0.963961
$207$	0	0
$208$	17.6317i	1.22254i
$209$	−0.781728	−0.0540732
$210$	0	0
$211$	14.9965	1.03240	0.516201	−	0.856467i	$-0.327346\pi$
$211$	14.9965	1.03240	0.516201	+	0.856467i	$0.327346\pi$
$212$	− 4.15272i	− 0.285210i
$213$	0	0
$214$	25.8282	1.76558
$215$	2.34901	0.160201
$216$	0	0
$217$	0	0
$218$	− 23.1680i	− 1.56913i
$219$	0	0
$220$	− 0.144259i	− 0.00972591i
$221$	16.2637i	1.09402i
$222$	0	0
$223$	− 3.78315i	− 0.253339i	−0.991945	−	0.126669i	$-0.959571\pi$
$223$	− 3.78315i	− 0.253339i	0.991945	−	0.126669i	$-0.0404287\pi$
$224$	0	0
$225$	0	0
$226$	23.7953	1.58284
$227$	13.8016	0.916043	0.458021	−	0.888941i	$-0.348558\pi$
$227$	13.8016	0.916043	0.458021	+	0.888941i	$0.348558\pi$
$228$	0	0
$229$	− 14.9876i	− 0.990412i	−0.868776	−	0.495206i	$-0.835093\pi$
$229$	− 14.9876i	− 0.990412i	0.868776	−	0.495206i	$-0.164907\pi$
$230$	−12.4443	−0.820551
$231$	0	0
$232$	−14.8457	−0.974667
$233$	10.5808i	0.693170i	0.938018	+	0.346585i	$0.112659\pi$
$233$	10.5808i	0.693170i	−0.938018	+	0.346585i	$0.887341\pi$
$234$	0	0
$235$	1.23916	0.0808338
$236$	−5.17131	−0.336624
$237$	0	0
$238$	0	0
$239$	7.98571i	0.516553i	0.966071	+	0.258276i	$0.0831545\pi$
$239$	7.98571i	0.516553i	−0.966071	+	0.258276i	$0.916845\pi$
$240$	0	0
$241$	21.5364i	1.38728i	0.720321	+	0.693640i	$0.243994\pi$
$241$	21.5364i	1.38728i	−0.720321	+	0.693640i	$0.756006\pi$
$242$	18.7071i	1.20254i
$243$	0	0
$244$	9.95559i	0.637342i
$245$	0	0
$246$	0	0
$247$	14.4663	0.920472
$248$	−12.1906	−0.774104
$249$	0	0
$250$	− 13.1771i	− 0.833393i
$251$	14.2569	0.899890	0.449945	−	0.893056i	$-0.351444\pi$
$251$	14.2569	0.899890	0.449945	+	0.893056i	$0.351444\pi$
$252$	0	0
$253$	−1.67845	−0.105523
$254$	30.1377i	1.89101i
$255$	0	0
$256$	18.0217	1.12636
$257$	−1.44368	−0.0900542	−0.0450271	−	0.998986i	$-0.514337\pi$
$257$	−1.44368	−0.0900542	−0.0450271	+	0.998986i	$0.514337\pi$
$258$	0	0
$259$	0	0
$260$	2.66959i	0.165561i
$261$	0	0
$262$	− 35.8336i	− 2.21380i
$263$	− 15.5401i	− 0.958241i	−0.877749	−	0.479121i	$-0.840956\pi$
$263$	− 15.5401i	− 0.958241i	0.877749	−	0.479121i	$-0.159044\pi$
$264$	0	0
$265$	3.77846i	0.232109i
$266$	0	0
$267$	0	0
$268$	4.23224	0.258525
$269$	−13.6464	−0.832033	−0.416016	−	0.909357i	$-0.636574\pi$
$269$	−13.6464	−0.832033	−0.416016	+	0.909357i	$0.636574\pi$
$270$	0	0
$271$	− 4.27251i	− 0.259537i	−0.991544	−	0.129768i	$-0.958577\pi$
$271$	− 4.27251i	− 0.259537i	0.991544	−	0.129768i	$-0.0414233\pi$
$272$	22.9881	1.39386
$273$	0	0
$274$	−6.97297	−0.421252
$275$	− 0.823016i	− 0.0496297i
$276$	0	0
$277$	5.91170	0.355199	0.177600	−	0.984103i	$-0.443167\pi$
$277$	5.91170	0.355199	0.177600	+	0.984103i	$0.443167\pi$
$278$	25.7084	1.54188
$279$	0	0
$280$	0	0
$281$	23.7012i	1.41390i	0.707265	+	0.706948i	$0.249929\pi$
$281$	23.7012i	1.41390i	−0.707265	+	0.706948i	$0.750071\pi$
$282$	0	0
$283$	− 21.0056i	− 1.24865i	−0.781164	−	0.624326i	$-0.785374\pi$
$283$	− 21.0056i	− 1.24865i	0.781164	−	0.624326i	$-0.214626\pi$
$284$	− 9.16619i	− 0.543913i
$285$	0	0
$286$	1.15017i	0.0680111i
$287$	0	0
$288$	0	0
$289$	4.20452	0.247324
$290$	−11.3099	−0.664140
$291$	0	0
$292$	− 7.24614i	− 0.424048i
$293$	−29.4325	−1.71947	−0.859734	−	0.510743i	$-0.829371\pi$
$293$	−29.4325	−1.71947	−0.859734	+	0.510743i	$0.829371\pi$
$294$	0	0
$295$	4.70525	0.273950
$296$	2.70520i	0.157236i
$297$	0	0
$298$	6.67690	0.386782
$299$	31.0607	1.79629
$300$	0	0
$301$	0	0
$302$	27.8757i	1.60407i
$303$	0	0
$304$	− 20.4475i	− 1.17275i
$305$	− 9.05835i	− 0.518679i
$306$	0	0
$307$	16.6735i	0.951608i	0.879551	+	0.475804i	$0.157843\pi$
$307$	16.6735i	0.951608i	−0.879551	+	0.475804i	$0.842157\pi$
$308$	0	0
$309$	0	0
$310$	−9.28717	−0.527476
$311$	−29.7981	−1.68970	−0.844848	−	0.535006i	$-0.820309\pi$
$311$	−29.7981	−1.68970	−0.844848	+	0.535006i	$0.820309\pi$
$312$	0	0
$313$	− 23.5935i	− 1.33358i	−0.745244	−	0.666792i	$-0.767667\pi$
$313$	− 23.5935i	− 1.33358i	0.745244	−	0.666792i	$-0.232333\pi$
$314$	12.6113	0.711696
$315$	0	0
$316$	5.43521	0.305754
$317$	12.6339i	0.709592i	0.934944	+	0.354796i	$0.115450\pi$
$317$	12.6339i	0.709592i	−0.934944	+	0.354796i	$0.884550\pi$
$318$	0	0
$319$	−1.52545	−0.0854086
$320$	−1.48320	−0.0829131
$321$	0	0
$322$	0	0
$323$	− 18.8611i	− 1.04946i
$324$	0	0
$325$	15.2304i	0.844831i
$326$	31.8345i	1.76315i
$327$	0	0
$328$	22.5149i	1.24318i
$329$	0	0
$330$	0	0
$331$	15.0909	0.829470	0.414735	−	0.909942i	$-0.363874\pi$
$331$	15.0909	0.829470	0.414735	+	0.909942i	$0.363874\pi$
$332$	1.87161	0.102718
$333$	0	0
$334$	21.7899i	1.19229i
$335$	−3.85081	−0.210392
$336$	0	0
$337$	−32.8864	−1.79143	−0.895717	−	0.444624i	$-0.853337\pi$
$337$	−32.8864	−1.79143	−0.895717	+	0.444624i	$0.853337\pi$
$338$	0.897187i	0.0488005i
$339$	0	0
$340$	3.48059	0.188762
$341$	−1.25263	−0.0678336
$342$	0	0
$343$	0	0
$344$	5.26115i	0.283662i
$345$	0	0
$346$	− 8.87560i	− 0.477155i
$347$	0.127644i	0.00685227i	0.999994	+	0.00342613i	$0.00109057\pi$
$347$	0.127644i	0.00685227i	−0.999994	+	0.00342613i	$0.998909\pi$
$348$	0	0
$349$	22.3227i	1.19491i	0.801903	+	0.597454i	$0.203821\pi$
$349$	22.3227i	1.19491i	−0.801903	+	0.597454i	$0.796179\pi$
$350$	0	0
$351$	0	0
$352$	0.916729	0.0488618
$353$	4.12738	0.219678	0.109839	−	0.993949i	$-0.464966\pi$
$353$	4.12738	0.219678	0.109839	+	0.993949i	$0.464966\pi$
$354$	0	0
$355$	8.34009i	0.442646i
$356$	8.05326	0.426822
$357$	0	0
$358$	−8.49263	−0.448850
$359$	30.0512i	1.58604i	0.609195	+	0.793021i	$0.291493\pi$
$359$	30.0512i	1.58604i	−0.609195	+	0.793021i	$0.708507\pi$
$360$	0	0
$361$	2.22334	0.117018
$362$	−35.7861	−1.88087
$363$	0	0
$364$	0	0
$365$	6.59308i	0.345098i
$366$	0	0
$367$	− 5.65946i	− 0.295421i	−0.989031	−	0.147711i	$-0.952810\pi$
$367$	− 5.65946i	− 0.295421i	0.989031	−	0.147711i	$-0.0471905\pi$
$368$	− 43.9029i	− 2.28860i
$369$	0	0
$370$	2.06090i	0.107141i
$371$	0	0
$372$	0	0
$373$	19.4449	1.00682	0.503408	−	0.864049i	$-0.332079\pi$
$373$	19.4449	1.00682	0.503408	+	0.864049i	$0.332079\pi$
$374$	1.49958	0.0775417
$375$	0	0
$376$	2.77538i	0.143129i
$377$	28.2293	1.45388
$378$	0	0
$379$	14.4544	0.742473	0.371236	−	0.928538i	$-0.378934\pi$
$379$	14.4544	0.742473	0.371236	+	0.928538i	$0.378934\pi$
$380$	− 3.09593i	− 0.158818i
$381$	0	0
$382$	18.9876	0.971488
$383$	32.2952	1.65021	0.825103	−	0.564982i	$-0.191117\pi$
$383$	32.2952	1.65021	0.825103	+	0.564982i	$0.191117\pi$
$384$	0	0
$385$	0	0
$386$	16.3889i	0.834174i
$387$	0	0
$388$	6.52557i	0.331286i
$389$	− 24.5805i	− 1.24628i	−0.782111	−	0.623139i	$-0.785857\pi$
$389$	− 24.5805i	− 1.24628i	0.782111	−	0.623139i	$-0.214143\pi$
$390$	0	0
$391$	− 40.4967i	− 2.04801i
$392$	0	0
$393$	0	0
$394$	−5.02654	−0.253234
$395$	−4.94536	−0.248828
$396$	0	0
$397$	− 24.7615i	− 1.24274i	−0.783516	−	0.621372i	$-0.786576\pi$
$397$	− 24.7615i	− 1.24274i	0.783516	−	0.621372i	$-0.213424\pi$
$398$	0.975371	0.0488909
$399$	0	0
$400$	21.5275	1.07638
$401$	5.67595i	0.283443i	0.989907	+	0.141722i	$0.0452638\pi$
$401$	5.67595i	0.283443i	−0.989907	+	0.141722i	$0.954736\pi$
$402$	0	0
$403$	23.1806	1.15471
$404$	−8.99740	−0.447637
$405$	0	0
$406$	0	0
$407$	0.277969i	0.0137784i
$408$	0	0
$409$	15.7394i	0.778265i	0.921182	+	0.389132i	$0.127225\pi$
$409$	15.7394i	0.778265i	−0.921182	+	0.389132i	$0.872775\pi$
$410$	17.1525i	0.847103i
$411$	0	0
$412$	7.39039i	0.364099i
$413$	0	0
$414$	0	0
$415$	−1.70293	−0.0835935
$416$	−16.9646	−0.831760
$417$	0	0
$418$	− 1.33386i	− 0.0652411i
$419$	−26.4182	−1.29061	−0.645307	−	0.763923i	$-0.723271\pi$
$419$	−26.4182	−1.29061	−0.645307	+	0.763923i	$0.723271\pi$
$420$	0	0
$421$	19.3312	0.942145	0.471073	−	0.882094i	$-0.343867\pi$
$421$	19.3312	0.942145	0.471073	+	0.882094i	$0.343867\pi$
$422$	25.5885i	1.24563i
$423$	0	0
$424$	−8.46274	−0.410987
$425$	19.8573	0.963219
$426$	0	0
$427$	0	0
$428$	13.7965i	0.666878i
$429$	0	0
$430$	4.00810i	0.193288i
$431$	11.5013i	0.553996i	0.960870	+	0.276998i	$0.0893396\pi$
$431$	11.5013i	0.553996i	−0.960870	+	0.276998i	$0.910660\pi$
$432$	0	0
$433$	1.87564i	0.0901374i	0.998984	+	0.0450687i	$0.0143507\pi$
$433$	1.87564i	0.0901374i	−0.998984	+	0.0450687i	$0.985649\pi$
$434$	0	0
$435$	0	0
$436$	12.3755	0.592679
$437$	−36.0212	−1.72313
$438$	0	0
$439$	15.8957i	0.758659i	0.925262	+	0.379330i	$0.123845\pi$
$439$	15.8957i	0.758659i	−0.925262	+	0.379330i	$0.876155\pi$
$440$	−0.293981	−0.0140150
$441$	0	0
$442$	−27.7507	−1.31997
$443$	15.2772i	0.725840i	0.931820	+	0.362920i	$0.118220\pi$
$443$	15.2772i	0.725840i	−0.931820	+	0.362920i	$0.881780\pi$
$444$	0	0
$445$	−7.32746	−0.347355
$446$	6.45517	0.305661
$447$	0	0
$448$	0	0
$449$	− 22.7142i	− 1.07195i	−0.844234	−	0.535975i	$-0.819944\pi$
$449$	− 22.7142i	− 1.07195i	0.844234	−	0.535975i	$-0.180056\pi$
$450$	0	0
$451$	2.31348i	0.108938i
$452$	12.7106i	0.597855i
$453$	0	0
$454$	23.5495i	1.10523i
$455$	0	0
$456$	0	0
$457$	−22.8978	−1.07111	−0.535557	−	0.844499i	$-0.679898\pi$
$457$	−22.8978	−1.07111	−0.535557	+	0.844499i	$0.679898\pi$
$458$	25.5733	1.19496
$459$	0	0
$460$	− 6.64728i	− 0.309931i
$461$	25.9796	1.20999	0.604995	−	0.796230i	$-0.293175\pi$
$461$	25.9796	1.20999	0.604995	+	0.796230i	$0.293175\pi$
$462$	0	0
$463$	4.42642	0.205713	0.102857	−	0.994696i	$-0.467202\pi$
$463$	4.42642	0.205713	0.102857	+	0.994696i	$0.467202\pi$
$464$	− 39.9009i	− 1.85235i
$465$	0	0
$466$	−18.0539	−0.836333
$467$	−19.1657	−0.886884	−0.443442	−	0.896303i	$-0.646243\pi$
$467$	−19.1657	−0.886884	−0.443442	+	0.896303i	$0.646243\pi$
$468$	0	0
$469$	0	0
$470$	2.11437i	0.0975285i
$471$	0	0
$472$	10.5385i	0.485073i
$473$	0.540602i	0.0248569i
$474$	0	0
$475$	− 17.6627i	− 0.810422i
$476$	0	0
$477$	0	0
$478$	−13.6260	−0.623238
$479$	33.5569	1.53326	0.766628	−	0.642092i	$-0.221933\pi$
$479$	33.5569	1.53326	0.766628	+	0.642092i	$0.221933\pi$
$480$	0	0
$481$	− 5.14398i	− 0.234545i
$482$	−36.7474	−1.67380
$483$	0	0
$484$	−9.99265	−0.454211
$485$	− 5.93745i	− 0.269606i
$486$	0	0
$487$	18.9091	0.856855	0.428428	−	0.903576i	$-0.359068\pi$
$487$	18.9091	0.856855	0.428428	+	0.903576i	$0.359068\pi$
$488$	20.2883	0.918406
$489$	0	0
$490$	0	0
$491$	2.27292i	0.102575i	0.998684	+	0.0512876i	$0.0163325\pi$
$491$	2.27292i	0.102575i	−0.998684	+	0.0512876i	$0.983667\pi$
$492$	0	0
$493$	− 36.8051i	− 1.65762i
$494$	24.6838i	1.11058i
$495$	0	0
$496$	− 32.7648i	− 1.47118i
$497$	0	0
$498$	0	0
$499$	−4.35630	−0.195015	−0.0975074	−	0.995235i	$-0.531087\pi$
$499$	−4.35630	−0.195015	−0.0975074	+	0.995235i	$0.531087\pi$
$500$	7.03873	0.314781
$501$	0	0
$502$	24.3265i	1.08575i
$503$	−31.8828	−1.42158	−0.710792	−	0.703402i	$-0.751663\pi$
$503$	−31.8828	−1.42158	−0.710792	+	0.703402i	$0.751663\pi$
$504$	0	0
$505$	8.18651	0.364295
$506$	− 2.86393i	− 0.127317i
$507$	0	0
$508$	−16.0985	−0.714254
$509$	−31.0185	−1.37487	−0.687435	−	0.726245i	$-0.741264\pi$
$509$	−31.0185	−1.37487	−0.687435	+	0.726245i	$0.741264\pi$
$510$	0	0
$511$	0	0
$512$	5.43386i	0.240145i
$513$	0	0
$514$	− 2.46334i	− 0.108653i
$515$	− 6.72434i	− 0.296310i
$516$	0	0
$517$	0.285180i	0.0125422i
$518$	0	0
$519$	0	0
$520$	5.44030	0.238573
$521$	27.1816	1.19085	0.595425	−	0.803411i	$-0.296984\pi$
$521$	27.1816	1.19085	0.595425	+	0.803411i	$0.296984\pi$
$522$	0	0
$523$	20.8034i	0.909667i	0.890576	+	0.454834i	$0.150301\pi$
$523$	20.8034i	0.909667i	−0.890576	+	0.454834i	$0.849699\pi$
$524$	19.1410	0.836178
$525$	0	0
$526$	26.5159	1.15615
$527$	− 30.2227i	− 1.31652i
$528$	0	0
$529$	−54.3411	−2.36266
$530$	−6.44717	−0.280047
$531$	0	0
$532$	0	0
$533$	− 42.8124i	− 1.85441i
$534$	0	0
$535$	− 12.5531i	− 0.542717i
$536$	− 8.62479i	− 0.372534i
$537$	0	0
$538$	− 23.2847i	− 1.00387i
$539$	0	0
$540$	0	0
$541$	3.11946	0.134116	0.0670581	−	0.997749i	$-0.478639\pi$
$541$	3.11946	0.134116	0.0670581	+	0.997749i	$0.478639\pi$
$542$	7.29016	0.313139
$543$	0	0
$544$	22.1183i	0.948316i
$545$	−11.2602	−0.482332
$546$	0	0
$547$	19.5589	0.836278	0.418139	−	0.908383i	$-0.362682\pi$
$547$	19.5589	0.836278	0.418139	+	0.908383i	$0.362682\pi$
$548$	− 3.72471i	− 0.159112i
$549$	0	0
$550$	1.40431	0.0598799
$551$	−32.7376	−1.39467
$552$	0	0
$553$	0	0
$554$	10.0871i	0.428560i
$555$	0	0
$556$	13.7325i	0.582387i
$557$	− 9.72325i	− 0.411988i	−0.978553	−	0.205994i	$-0.933957\pi$
$557$	− 9.72325i	− 0.411988i	0.978553	−	0.205994i	$-0.0660427\pi$
$558$	0	0
$559$	− 10.0042i	− 0.423131i
$560$	0	0
$561$	0	0
$562$	−40.4413	−1.70591
$563$	−13.3090	−0.560908	−0.280454	−	0.959867i	$-0.590485\pi$
$563$	−13.3090	−0.560908	−0.280454	+	0.959867i	$0.590485\pi$
$564$	0	0
$565$	− 11.5650i	− 0.486544i
$566$	35.8417	1.50654
$567$	0	0
$568$	−18.6796	−0.783777
$569$	− 30.3767i	− 1.27346i	−0.771088	−	0.636729i	$-0.780287\pi$
$569$	− 30.3767i	− 1.27346i	0.771088	−	0.636729i	$-0.219713\pi$
$570$	0	0
$571$	1.23540	0.0516997	0.0258498	−	0.999666i	$-0.491771\pi$
$571$	1.23540	0.0516997	0.0258498	+	0.999666i	$0.491771\pi$
$572$	−0.614381	−0.0256885
$573$	0	0
$574$	0	0
$575$	− 37.9237i	− 1.58153i
$576$	0	0
$577$	24.3910i	1.01541i	0.861531	+	0.507705i	$0.169506\pi$
$577$	24.3910i	1.01541i	−0.861531	+	0.507705i	$0.830494\pi$
$578$	7.17414i	0.298405i
$579$	0	0
$580$	− 6.04133i	− 0.250853i
$581$	0	0
$582$	0	0
$583$	−0.869576	−0.0360142
$584$	−14.7667	−0.611051
$585$	0	0
$586$	− 50.2206i	− 2.07459i
$587$	−2.65150	−0.109439	−0.0547196	−	0.998502i	$-0.517426\pi$
$587$	−2.65150	−0.109439	−0.0547196	+	0.998502i	$0.517426\pi$
$588$	0	0
$589$	−26.8826	−1.10768
$590$	8.02854i	0.330530i
$591$	0	0
$592$	−7.27079	−0.298827
$593$	44.8945	1.84360	0.921799	−	0.387668i	$-0.126719\pi$
$593$	44.8945	1.84360	0.921799	+	0.387668i	$0.126719\pi$
$594$	0	0
$595$	0	0
$596$	3.56656i	0.146092i
$597$	0	0
$598$	52.9987i	2.16728i
$599$	10.8714i	0.444194i	0.975025	+	0.222097i	$0.0712902\pi$
$599$	10.8714i	0.444194i	−0.975025	+	0.222097i	$0.928710\pi$
$600$	0	0
$601$	28.7100i	1.17111i	0.810634	+	0.585553i	$0.199123\pi$
$601$	28.7100i	1.17111i	−0.810634	+	0.585553i	$0.800877\pi$
$602$	0	0
$603$	0	0
$604$	−14.8902	−0.605874
$605$	9.09206	0.369645
$606$	0	0
$607$	33.1928i	1.34726i	0.739071	+	0.673628i	$0.235265\pi$
$607$	33.1928i	1.34726i	−0.739071	+	0.673628i	$0.764735\pi$
$608$	19.6739	0.797883
$609$	0	0
$610$	15.4562	0.625804
$611$	− 5.27743i	− 0.213502i
$612$	0	0
$613$	2.08420	0.0841800	0.0420900	−	0.999114i	$-0.486598\pi$
$613$	2.08420	0.0841800	0.0420900	+	0.999114i	$0.486598\pi$
$614$	−28.4499	−1.14815
$615$	0	0
$616$	0	0
$617$	− 12.4764i	− 0.502283i	−0.967950	−	0.251141i	$-0.919194\pi$
$617$	− 12.4764i	− 0.502283i	0.967950	−	0.251141i	$-0.0808059\pi$
$618$	0	0
$619$	48.2241i	1.93829i	0.246489	+	0.969146i	$0.420723\pi$
$619$	48.2241i	1.93829i	−0.246489	+	0.969146i	$0.579277\pi$
$620$	− 4.96087i	− 0.199233i
$621$	0	0
$622$	− 50.8444i	− 2.03867i
$623$	0	0
$624$	0	0
$625$	15.1570	0.606279
$626$	40.2574	1.60901
$627$	0	0
$628$	6.73649i	0.268815i
$629$	−6.70668	−0.267413
$630$	0	0
$631$	22.7024	0.903767	0.451883	−	0.892077i	$-0.350752\pi$
$631$	22.7024	0.903767	0.451883	+	0.892077i	$0.350752\pi$
$632$	− 11.0763i	− 0.440591i
$633$	0	0
$634$	−21.5572	−0.856146
$635$	14.6476	0.581272
$636$	0	0
$637$	0	0
$638$	− 2.60286i	− 0.103048i
$639$	0	0
$640$	− 10.4975i	− 0.414949i
$641$	− 0.622832i	− 0.0246004i	−0.999924	−	0.0123002i	$-0.996085\pi$
$641$	− 0.622832i	− 0.0246004i	0.999924	−	0.0123002i	$-0.00391537\pi$
$642$	0	0
$643$	− 12.8900i	− 0.508332i	−0.967161	−	0.254166i	$-0.918199\pi$
$643$	− 12.8900i	− 0.508332i	0.967161	−	0.254166i	$-0.0818010\pi$
$644$	0	0
$645$	0	0
$646$	32.1826	1.26621
$647$	18.1592	0.713913	0.356957	−	0.934121i	$-0.383814\pi$
$647$	18.1592	0.713913	0.356957	+	0.934121i	$0.383814\pi$
$648$	0	0
$649$	1.08287i	0.0425062i
$650$	−25.9876	−1.01932
$651$	0	0
$652$	−17.0049	−0.665961
$653$	− 22.5352i	− 0.881870i	−0.897539	−	0.440935i	$-0.854647\pi$
$653$	− 22.5352i	− 0.881870i	0.897539	−	0.440935i	$-0.145353\pi$
$654$	0	0
$655$	−17.4159	−0.680496
$656$	−60.5135	−2.36265
$657$	0	0
$658$	0	0
$659$	2.39800i	0.0934130i	0.998909	+	0.0467065i	$0.0148725\pi$
$659$	2.39800i	0.0934130i	−0.998909	+	0.0467065i	$0.985127\pi$
$660$	0	0
$661$	8.08466i	0.314457i	0.987562	+	0.157228i	$0.0502559\pi$
$661$	8.08466i	0.314457i	−0.987562	+	0.157228i	$0.949744\pi$
$662$	25.7495i	1.00078i
$663$	0	0
$664$	− 3.81410i	− 0.148016i
$665$	0	0
$666$	0	0
$667$	−70.2910	−2.72168
$668$	−11.6394	−0.450341
$669$	0	0
$670$	− 6.57062i	− 0.253845i
$671$	2.08469	0.0804786
$672$	0	0
$673$	−27.3603	−1.05466	−0.527331	−	0.849660i	$-0.676807\pi$
$673$	−27.3603	−1.05466	−0.527331	+	0.849660i	$0.676807\pi$
$674$	− 56.1138i	− 2.16142i
$675$	0	0
$676$	−0.479245	−0.0184325
$677$	25.4870	0.979545	0.489773	−	0.871850i	$-0.337080\pi$
$677$	25.4870	0.979545	0.489773	+	0.871850i	$0.337080\pi$
$678$	0	0
$679$	0	0
$680$	− 7.09301i	− 0.272004i
$681$	0	0
$682$	− 2.13735i	− 0.0818434i
$683$	− 31.6527i	− 1.21116i	−0.795786	−	0.605578i	$-0.792942\pi$
$683$	− 31.6527i	− 1.21116i	0.795786	−	0.605578i	$-0.207058\pi$
$684$	0	0
$685$	3.38902i	0.129488i
$686$	0	0
$687$	0	0
$688$	−14.1404	−0.539099
$689$	16.0920	0.613058
$690$	0	0
$691$	9.59867i	0.365150i	0.983192	+	0.182575i	$0.0584433\pi$
$691$	9.59867i	0.365150i	−0.983192	+	0.182575i	$0.941557\pi$
$692$	4.74102	0.180227
$693$	0	0
$694$	−0.217798	−0.00826748
$695$	− 12.4948i	− 0.473956i
$696$	0	0
$697$	−55.8185	−2.11427
$698$	−38.0891	−1.44170
$699$	0	0
$700$	0	0
$701$	8.57884i	0.324018i	0.986789	+	0.162009i	$0.0517974\pi$
$701$	8.57884i	0.324018i	−0.986789	+	0.162009i	$0.948203\pi$
$702$	0	0
$703$	5.96549i	0.224993i
$704$	− 0.341343i	− 0.0128648i
$705$	0	0
$706$	7.04253i	0.265049i
$707$	0	0
$708$	0	0
$709$	−14.9079	−0.559876	−0.279938	−	0.960018i	$-0.590314\pi$
$709$	−14.9079	−0.559876	−0.279938	+	0.960018i	$0.590314\pi$
$710$	−14.2306	−0.534067
$711$	0	0
$712$	− 16.4115i	− 0.615048i
$713$	−57.7197	−2.16162
$714$	0	0
$715$	0.559010	0.0209058
$716$	− 4.53646i	− 0.169535i
$717$	0	0
$718$	−51.2762	−1.91361
$719$	−24.8323	−0.926087	−0.463043	−	0.886336i	$-0.653243\pi$
$719$	−24.8323	−0.926087	−0.463043	+	0.886336i	$0.653243\pi$
$720$	0	0
$721$	0	0
$722$	3.79368i	0.141186i
$723$	0	0
$724$	− 19.1156i	− 0.710427i
$725$	− 34.4667i	− 1.28006i
$726$	0	0
$727$	4.55497i	0.168934i	0.996426	+	0.0844672i	$0.0269188\pi$
$727$	4.55497i	0.168934i	−0.996426	+	0.0844672i	$0.973081\pi$
$728$	0	0
$729$	0	0
$730$	−11.2497	−0.416371
$731$	−13.0433	−0.482425
$732$	0	0
$733$	11.4165i	0.421678i	0.977521	+	0.210839i	$0.0676195\pi$
$733$	11.4165i	0.421678i	−0.977521	+	0.210839i	$0.932380\pi$
$734$	9.65671	0.356436
$735$	0	0
$736$	42.2419	1.55706
$737$	− 0.886227i	− 0.0326446i
$738$	0	0
$739$	15.3965	0.566368	0.283184	−	0.959066i	$-0.408609\pi$
$739$	15.3965	0.566368	0.283184	+	0.959066i	$0.408609\pi$
$740$	−1.10086	−0.0404684
$741$	0	0
$742$	0	0
$743$	− 45.6525i	− 1.67483i	−0.546571	−	0.837413i	$-0.684067\pi$
$743$	− 45.6525i	− 1.67483i	0.546571	−	0.837413i	$-0.315933\pi$
$744$	0	0
$745$	− 3.24512i	− 0.118892i
$746$	33.1787i	1.21476i
$747$	0	0
$748$	0.801024i	0.0292883i
$749$	0	0
$750$	0	0
$751$	21.7591	0.793999	0.397000	−	0.917819i	$-0.370051\pi$
$751$	21.7591	0.793999	0.397000	+	0.917819i	$0.370051\pi$
$752$	−7.45941	−0.272017
$753$	0	0
$754$	48.1675i	1.75416i
$755$	13.5482	0.493071
$756$	0	0
$757$	−50.5031	−1.83556	−0.917782	−	0.397084i	$-0.870022\pi$
$757$	−50.5031	−1.83556	−0.917782	+	0.397084i	$0.870022\pi$
$758$	24.6635i	0.895817i
$759$	0	0
$760$	−6.30912	−0.228856
$761$	−7.75425	−0.281092	−0.140546	−	0.990074i	$-0.544886\pi$
$761$	−7.75425	−0.281092	−0.140546	+	0.990074i	$0.544886\pi$
$762$	0	0
$763$	0	0
$764$	10.1425i	0.366942i
$765$	0	0
$766$	55.1051i	1.99103i
$767$	− 20.0391i	− 0.723570i
$768$	0	0
$769$	− 25.4354i	− 0.917226i	−0.888636	−	0.458613i	$-0.848346\pi$
$769$	− 25.4354i	− 0.917226i	0.888636	−	0.458613i	$-0.151654\pi$
$770$	0	0
$771$	0	0
$772$	−8.75437	−0.315077
$773$	−1.43074	−0.0514602	−0.0257301	−	0.999669i	$-0.508191\pi$
$773$	−1.43074	−0.0514602	−0.0257301	+	0.999669i	$0.508191\pi$
$774$	0	0
$775$	− 28.3025i	− 1.01666i
$776$	13.2983	0.477381
$777$	0	0
$778$	41.9415	1.50368
$779$	49.6497i	1.77888i
$780$	0	0
$781$	−1.91939	−0.0686812
$782$	69.0993	2.47098
$783$	0	0
$784$	0	0
$785$	− 6.12937i	− 0.218766i
$786$	0	0
$787$	− 45.5756i	− 1.62459i	−0.583244	−	0.812297i	$-0.698217\pi$
$787$	− 45.5756i	− 1.62459i	0.583244	−	0.812297i	$-0.301783\pi$
$788$	− 2.68500i	− 0.0956491i
$789$	0	0
$790$	− 8.43825i	− 0.300219i
$791$	0	0
$792$	0	0
$793$	−38.5785	−1.36996
$794$	42.2504	1.49941
$795$	0	0
$796$	0.521008i	0.0184666i
$797$	20.8400	0.738189	0.369095	−	0.929392i	$-0.379668\pi$
$797$	20.8400	0.738189	0.369095	+	0.929392i	$0.379668\pi$
$798$	0	0
$799$	−6.88067	−0.243421
$800$	20.7130i	0.732317i
$801$	0	0
$802$	−9.68484	−0.341984
$803$	−1.51733	−0.0535455
$804$	0	0
$805$	0	0
$806$	39.5530i	1.39319i
$807$	0	0
$808$	18.3356i	0.645043i
$809$	51.6009i	1.81419i	0.420926	+	0.907095i	$0.361705\pi$
$809$	51.6009i	1.81419i	−0.420926	+	0.907095i	$0.638295\pi$
$810$	0	0
$811$	− 23.4591i	− 0.823759i	−0.911238	−	0.411879i	$-0.864873\pi$
$811$	− 23.4591i	− 0.823759i	0.911238	−	0.411879i	$-0.135127\pi$
$812$	0	0
$813$	0	0
$814$	−0.474296	−0.0166241
$815$	15.4723	0.541971
$816$	0	0
$817$	11.6019i	0.405897i
$818$	−26.8561	−0.939002
$819$	0	0
$820$	−9.16226	−0.319960
$821$	22.1858i	0.774289i	0.922019	+	0.387145i	$0.126539\pi$
$821$	22.1858i	0.774289i	−0.922019	+	0.387145i	$0.873461\pi$
$822$	0	0
$823$	6.09677	0.212520	0.106260	−	0.994338i	$-0.466112\pi$
$823$	6.09677	0.212520	0.106260	+	0.994338i	$0.466112\pi$
$824$	15.0607	0.524664
$825$	0	0
$826$	0	0
$827$	39.2806i	1.36592i	0.730456	+	0.682960i	$0.239308\pi$
$827$	39.2806i	1.36592i	−0.730456	+	0.682960i	$0.760692\pi$
$828$	0	0
$829$	− 12.2813i	− 0.426548i	−0.976992	−	0.213274i	$-0.931587\pi$
$829$	− 12.2813i	− 0.426548i	0.976992	−	0.213274i	$-0.0684127\pi$
$830$	− 2.90570i	− 0.100858i
$831$	0	0
$832$	6.31676i	0.218994i
$833$	0	0
$834$	0	0
$835$	10.5904	0.366495
$836$	0.712499	0.0246423
$837$	0	0
$838$	− 45.0773i	− 1.55717i
$839$	23.9901	0.828229	0.414115	−	0.910225i	$-0.364091\pi$
$839$	23.9901	0.828229	0.414115	+	0.910225i	$0.364091\pi$
$840$	0	0
$841$	−34.8834	−1.20288
$842$	32.9847i	1.13673i
$843$	0	0
$844$	−13.6684	−0.470487
$845$	0.436053	0.0150007
$846$	0	0
$847$	0	0
$848$	− 22.7454i	− 0.781079i
$849$	0	0
$850$	33.8824i	1.16216i
$851$	12.8085i	0.439070i
$852$	0	0
$853$	19.7464i	0.676103i	0.941128	+	0.338051i	$0.109768\pi$
$853$	19.7464i	0.676103i	−0.941128	+	0.338051i	$0.890232\pi$
$854$	0	0
$855$	0	0
$856$	28.1155	0.960969
$857$	−33.6350	−1.14895	−0.574475	−	0.818522i	$-0.694794\pi$
$857$	−33.6350	−1.14895	−0.574475	+	0.818522i	$0.694794\pi$
$858$	0	0
$859$	− 14.9836i	− 0.511233i	−0.966778	−	0.255617i	$-0.917722\pi$
$859$	− 14.9836i	− 0.511233i	0.966778	−	0.255617i	$-0.0822785\pi$
$860$	−2.14098	−0.0730069
$861$	0	0
$862$	−19.6245	−0.668414
$863$	− 17.7153i	− 0.603037i	−0.953460	−	0.301519i	$-0.902506\pi$
$863$	− 17.7153i	− 0.603037i	0.953460	−	0.301519i	$-0.0974936\pi$
$864$	0	0
$865$	−4.31374	−0.146672
$866$	−3.20039	−0.108754
$867$	0	0
$868$	0	0
$869$	− 1.13813i	− 0.0386083i
$870$	0	0
$871$	16.4002i	0.555699i
$872$	− 25.2197i	− 0.854047i
$873$	0	0
$874$	− 61.4627i	− 2.07901i
$875$	0	0
$876$	0	0
$877$	9.93977	0.335642	0.167821	−	0.985817i	$-0.446327\pi$
$877$	9.93977	0.335642	0.167821	+	0.985817i	$0.446327\pi$
$878$	−27.1227	−0.915347
$879$	0	0
$880$	− 0.790135i	− 0.0266354i
$881$	−31.8591	−1.07336	−0.536680	−	0.843786i	$-0.680322\pi$
$881$	−31.8591	−1.07336	−0.536680	+	0.843786i	$0.680322\pi$
$882$	0	0
$883$	44.3061	1.49102	0.745510	−	0.666495i	$-0.232206\pi$
$883$	44.3061	1.49102	0.745510	+	0.666495i	$0.232206\pi$
$884$	− 14.8234i	− 0.498566i
$885$	0	0
$886$	−26.0673	−0.875749
$887$	−14.9375	−0.501551	−0.250775	−	0.968045i	$-0.580686\pi$
$887$	−14.9375	−0.501551	−0.250775	+	0.968045i	$0.580686\pi$
$888$	0	0
$889$	0	0
$890$	− 12.5028i	− 0.419095i
$891$	0	0
$892$	3.44812i	0.115452i
$893$	6.12025i	0.204806i
$894$	0	0
$895$	4.12761i	0.137971i
$896$	0	0
$897$	0	0
$898$	38.7571	1.29334
$899$	−52.4582	−1.74958
$900$	0	0
$901$	− 20.9806i	− 0.698967i
$902$	−3.94748	−0.131437
$903$	0	0
$904$	25.9025	0.861506
$905$	17.3928i	0.578158i
$906$	0	0
$907$	43.4257	1.44193	0.720963	−	0.692973i	$-0.243700\pi$
$907$	43.4257	1.44193	0.720963	+	0.692973i	$0.243700\pi$
$908$	−12.5793	−0.417459
$909$	0	0
$910$	0	0
$911$	35.2646i	1.16837i	0.811621	+	0.584185i	$0.198586\pi$
$911$	35.2646i	1.16837i	−0.811621	+	0.584185i	$0.801414\pi$
$912$	0	0
$913$	− 0.391912i	− 0.0129704i
$914$	− 39.0704i	− 1.29233i
$915$	0	0
$916$	13.6604i	0.451351i
$917$	0	0
$918$	0	0
$919$	−37.8975	−1.25012	−0.625061	−	0.780576i	$-0.714926\pi$
$919$	−37.8975	−1.25012	−0.625061	+	0.780576i	$0.714926\pi$
$920$	−13.5463	−0.446609
$921$	0	0
$922$	44.3288i	1.45989i
$923$	35.5195	1.16914
$924$	0	0
$925$	−6.28056	−0.206504
$926$	7.55277i	0.248199i
$927$	0	0
$928$	38.3913	1.26026
$929$	11.6636	0.382671	0.191335	−	0.981525i	$-0.438718\pi$
$929$	11.6636	0.382671	0.191335	+	0.981525i	$0.438718\pi$
$930$	0	0
$931$	0	0
$932$	− 9.64376i	− 0.315892i
$933$	0	0
$934$	− 32.7024i	− 1.07005i
$935$	− 0.728832i	− 0.0238353i
$936$	0	0
$937$	4.46450i	0.145849i	0.997337	+	0.0729244i	$0.0232332\pi$
$937$	4.46450i	0.145849i	−0.997337	+	0.0729244i	$0.976767\pi$
$938$	0	0
$939$	0	0
$940$	−1.12942	−0.0368376
$941$	52.6172	1.71527	0.857636	−	0.514257i	$-0.171932\pi$
$941$	52.6172	1.71527	0.857636	+	0.514257i	$0.171932\pi$
$942$	0	0
$943$	106.603i	3.47147i
$944$	−28.3244	−0.921880
$945$	0	0
$946$	−0.922426	−0.0299906
$947$	− 7.88685i	− 0.256288i	−0.991756	−	0.128144i	$-0.959098\pi$
$947$	− 7.88685i	− 0.256288i	0.991756	−	0.128144i	$-0.0409020\pi$
$948$	0	0
$949$	28.0792	0.911488
$950$	30.1378	0.977801
$951$	0	0
$952$	0	0
$953$	28.5322i	0.924250i	0.886815	+	0.462125i	$0.152913\pi$
$953$	28.5322i	0.924250i	−0.886815	+	0.462125i	$0.847087\pi$
$954$	0	0
$955$	− 9.22838i	− 0.298623i
$956$	− 7.27850i	− 0.235404i
$957$	0	0
$958$	57.2580i	1.84992i
$959$	0	0
$960$	0	0
$961$	−12.0763	−0.389556
$962$	8.77714	0.282986
$963$	0	0
$964$	− 19.6291i	− 0.632212i
$965$	7.96538	0.256415
$966$	0	0
$967$	41.0897	1.32135	0.660677	−	0.750670i	$-0.270269\pi$
$967$	41.0897	1.32135	0.660677	+	0.750670i	$0.270269\pi$
$968$	20.3638i	0.654516i
$969$	0	0
$970$	10.1310	0.325288
$971$	43.4767	1.39523	0.697617	−	0.716470i	$-0.254244\pi$
$971$	43.4767	1.39523	0.697617	+	0.716470i	$0.254244\pi$
$972$	0	0
$973$	0	0
$974$	32.2646i	1.03382i
$975$	0	0
$976$	54.5289i	1.74543i
$977$	8.99914i	0.287908i	0.989584	+	0.143954i	$0.0459818\pi$
$977$	8.99914i	0.287908i	−0.989584	+	0.143954i	$0.954018\pi$
$978$	0	0
$979$	− 1.68634i	− 0.0538958i
$980$	0	0
$981$	0	0
$982$	−3.87826	−0.123760
$983$	−54.7852	−1.74738	−0.873689	−	0.486486i	$-0.838279\pi$
$983$	−54.7852	−1.74738	−0.873689	+	0.486486i	$0.838279\pi$
$984$	0	0
$985$	2.44301i	0.0778408i
$986$	62.8004	1.99997
$987$	0	0
$988$	−13.1852	−0.419478
$989$	24.9103i	0.792103i
$990$	0	0
$991$	1.33301	0.0423443	0.0211722	−	0.999776i	$-0.493260\pi$
$991$	1.33301	0.0423443	0.0211722	+	0.999776i	$0.493260\pi$
$992$	31.5252	1.00092
$993$	0	0
$994$	0	0
$995$	− 0.474052i	− 0.0150285i
$996$	0	0
$997$	− 24.4835i	− 0.775399i	−0.921786	−	0.387700i	$-0.873270\pi$
$997$	− 24.4835i	− 0.775399i	0.921786	−	0.387700i	$-0.126730\pi$
$998$	− 7.43313i	− 0.235292i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.2.c.e.1322.14	yes	16
3.2	odd	2	inner	1323.2.c.e.1322.3	✓	16
7.6	odd	2	inner	1323.2.c.e.1322.13	yes	16
21.20	even	2	inner	1323.2.c.e.1322.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1323.2.c.e.1322.3	✓	16	3.2	odd	2	inner
1323.2.c.e.1322.4	yes	16	21.20	even	2	inner
1323.2.c.e.1322.13	yes	16	7.6	odd	2	inner
1323.2.c.e.1322.14	yes	16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1323.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.70629i q^{2} -0.911441 q^{4} +0.829297 q^{5} +1.85740i q^{8} +O(q^{10})\)	\(q+1.70629i q^{2} -0.911441 q^{4} +0.829297 q^{5} +1.85740i q^{8} +1.41503i q^{10} +0.190855i q^{11} -3.53188i q^{13} -4.99216 q^{16} -4.60484 q^{17} +4.09593i q^{19} -0.755855 q^{20} -0.325654 q^{22} +8.79438i q^{23} -4.31227 q^{25} +6.02643 q^{26} +7.99271i q^{29} +6.56325i q^{31} -4.80328i q^{32} -7.85721i q^{34} +1.45644 q^{37} -6.98887 q^{38} +1.54034i q^{40} +12.1217 q^{41} +2.83253 q^{43} -0.173953i q^{44} -15.0058 q^{46} +1.49423 q^{47} -7.35800i q^{50} +3.21910i q^{52} +4.55622i q^{53} +0.158275i q^{55} -13.6379 q^{58} +5.67378 q^{59} -10.9229i q^{61} -11.1988 q^{62} -1.78850 q^{64} -2.92898i q^{65} -4.64347 q^{67} +4.19704 q^{68} +10.0568i q^{71} +7.95020i q^{73} +2.48512i q^{74} -3.73320i q^{76} -5.96332 q^{79} -4.13998 q^{80} +20.6832i q^{82} -2.05346 q^{83} -3.81878 q^{85} +4.83313i q^{86} -0.354494 q^{88} -8.83575 q^{89} -8.01556i q^{92} +2.54959i q^{94} +3.39675i q^{95} -7.15962i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4}+O(q^{10}) \) 16 * q - 16 * q^4	\( 16 q - 16 q^{4} + 48 q^{16} + 16 q^{22} + 32 q^{25} - 16 q^{37} + 32 q^{43} - 80 q^{46} - 96 q^{58} - 176 q^{64} + 96 q^{67} - 64 q^{79} - 32 q^{85} + 112 q^{88}+O(q^{100}) \) 16 * q - 16 * q^4 + 48 * q^16 + 16 * q^22 + 32 * q^25 - 16 * q^37 + 32 * q^43 - 80 * q^46 - 96 * q^58 - 176 * q^64 + 96 * q^67 - 64 * q^79 - 32 * q^85 + 112 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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