LMFDB - Embedded newform 1323.2.c.e.1322.10

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			1322.10
Root			$0.441700 + 0.255016i$ of defining polynomial
Character	$\chi$	$=$	1323.1322
Dual form			1323.2.c.e.1322.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.510032i q^{2} +1.73987 q^{4} +4.01755 q^{5} +1.90745i q^{8} +O(q^{10})$	$q+0.510032i q^{2} +1.73987 q^{4} +4.01755 q^{5} +1.90745i q^{8} +2.04908i q^{10} -4.55982i q^{11} -1.36710i q^{13} +2.50688 q^{16} -2.15716 q^{17} -2.66105i q^{19} +6.99000 q^{20} +2.32565 q^{22} +1.27649i q^{23} +11.1407 q^{25} +0.697263 q^{26} -8.75700i q^{29} +8.72803i q^{31} +5.09349i q^{32} -1.10022i q^{34} +7.85727 q^{37} +1.35722 q^{38} +7.66327i q^{40} -8.92372 q^{41} -7.31781 q^{43} -7.93349i q^{44} -0.651052 q^{46} -8.97751 q^{47} +5.68210i q^{50} -2.37857i q^{52} +7.89115i q^{53} -18.3193i q^{55} +4.46635 q^{58} -7.72819 q^{59} +6.02394i q^{61} -4.45157 q^{62} +2.41591 q^{64} -5.49238i q^{65} +8.15818 q^{67} -3.75318 q^{68} +0.301598i q^{71} +11.6457i q^{73} +4.00745i q^{74} -4.62988i q^{76} -4.86511 q^{79} +10.0715 q^{80} -4.55138i q^{82} -4.38346 q^{83} -8.66650 q^{85} -3.73231i q^{86} +8.69764 q^{88} +1.90928 q^{89} +2.22093i q^{92} -4.57881i q^{94} -10.6909i q^{95} +0.231417i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.510032i	0.360647i	0.983607	+	0.180323i	$0.0577144\pi$
$2$	0.510032i	0.360647i	−0.983607	+	0.180323i	$0.942286\pi$
$3$	0	0
$3$	0	0
$4$	1.73987	0.869934
$5$	4.01755	1.79670	0.898351	−	0.439278i	$-0.144766\pi$
$5$	4.01755	1.79670	0.898351	+	0.439278i	$0.144766\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.90745i	0.674386i
$9$	0	0
$10$	2.04908i	0.647975i
$11$	− 4.55982i	− 1.37484i	−0.726261	−	0.687419i	$-0.758743\pi$
$11$	− 4.55982i	− 1.37484i	0.726261	−	0.687419i	$-0.241257\pi$
$12$	0	0
$13$	− 1.36710i	− 0.379165i	−0.981865	−	0.189582i	$-0.939287\pi$
$13$	− 1.36710i	− 0.379165i	0.981865	−	0.189582i	$-0.0607134\pi$
$14$	0	0
$15$	0	0
$16$	2.50688	0.626719
$17$	−2.15716	−0.523189	−0.261594	−	0.965178i	$-0.584248\pi$
$17$	−2.15716	−0.523189	−0.261594	+	0.965178i	$0.584248\pi$
$18$	0	0
$19$	− 2.66105i	− 0.610488i	−0.952274	−	0.305244i	$-0.901262\pi$
$19$	− 2.66105i	− 0.610488i	0.952274	−	0.305244i	$-0.0987380\pi$
$20$	6.99000	1.56301
$21$	0	0
$22$	2.32565	0.495831
$23$	1.27649i	0.266167i	0.991105	+	0.133084i	$0.0424879\pi$
$23$	1.27649i	0.266167i	−0.991105	+	0.133084i	$0.957512\pi$
$24$	0	0
$25$	11.1407	2.22814
$26$	0.697263	0.136744
$27$	0	0
$28$	0	0
$29$	− 8.75700i	− 1.62613i	−0.582170	−	0.813067i	$-0.697796\pi$
$29$	− 8.75700i	− 1.62613i	0.582170	−	0.813067i	$-0.302204\pi$
$30$	0	0
$31$	8.72803i	1.56760i	0.621013	+	0.783801i	$0.286722\pi$
$31$	8.72803i	1.56760i	−0.621013	+	0.783801i	$0.713278\pi$
$32$	5.09349i	0.900410i
$33$	0	0
$34$	− 1.10022i	− 0.188686i
$35$	0	0
$36$	0	0
$37$	7.85727	1.29173	0.645863	−	0.763453i	$-0.276498\pi$
$37$	7.85727	1.29173	0.645863	+	0.763453i	$0.276498\pi$
$38$	1.35722	0.220170
$39$	0	0
$40$	7.66327i	1.21167i
$41$	−8.92372	−1.39365	−0.696826	−	0.717241i	$-0.745405\pi$
$41$	−8.92372	−1.39365	−0.696826	+	0.717241i	$0.745405\pi$
$42$	0	0
$43$	−7.31781	−1.11596	−0.557978	−	0.829856i	$-0.688423\pi$
$43$	−7.31781	−1.11596	−0.557978	+	0.829856i	$0.688423\pi$
$44$	− 7.93349i	− 1.19602i
$45$	0	0
$46$	−0.651052	−0.0959923
$47$	−8.97751	−1.30950	−0.654752	−	0.755843i	$-0.727227\pi$
$47$	−8.97751	−1.30950	−0.654752	+	0.755843i	$0.727227\pi$
$48$	0	0
$49$	0	0
$50$	5.68210i	0.803571i
$51$	0	0
$52$	− 2.37857i	− 0.329848i
$53$	7.89115i	1.08393i	0.840400	+	0.541967i	$0.182320\pi$
$53$	7.89115i	1.08393i	−0.840400	+	0.541967i	$0.817680\pi$
$54$	0	0
$55$	− 18.3193i	− 2.47018i
$56$	0	0
$57$	0	0
$58$	4.46635	0.586460
$59$	−7.72819	−1.00613	−0.503063	−	0.864250i	$-0.667793\pi$
$59$	−7.72819	−1.00613	−0.503063	+	0.864250i	$0.667793\pi$
$60$	0	0
$61$	6.02394i	0.771286i	0.922648	+	0.385643i	$0.126020\pi$
$61$	6.02394i	0.771286i	−0.922648	+	0.385643i	$0.873980\pi$
$62$	−4.45157	−0.565350
$63$	0	0
$64$	2.41591	0.301989
$65$	− 5.49238i	− 0.681246i
$66$	0	0
$67$	8.15818	0.996681	0.498340	−	0.866981i	$-0.333943\pi$
$67$	8.15818	0.996681	0.498340	+	0.866981i	$0.333943\pi$
$68$	−3.75318	−0.455140
$69$	0	0
$70$	0	0
$71$	0.301598i	0.0357931i	0.999840	+	0.0178965i	$0.00569695\pi$
$71$	0.301598i	0.0357931i	−0.999840	+	0.0178965i	$0.994303\pi$
$72$	0	0
$73$	11.6457i	1.36303i	0.731805	+	0.681514i	$0.238678\pi$
$73$	11.6457i	1.36303i	−0.731805	+	0.681514i	$0.761322\pi$
$74$	4.00745i	0.465857i
$75$	0	0
$76$	− 4.62988i	− 0.531084i
$77$	0	0
$78$	0	0
$79$	−4.86511	−0.547368	−0.273684	−	0.961820i	$-0.588242\pi$
$79$	−4.86511	−0.547368	−0.273684	+	0.961820i	$0.588242\pi$
$80$	10.0715	1.12603
$81$	0	0
$82$	− 4.55138i	− 0.502616i
$83$	−4.38346	−0.481148	−0.240574	−	0.970631i	$-0.577336\pi$
$83$	−4.38346	−0.481148	−0.240574	+	0.970631i	$0.577336\pi$
$84$	0	0
$85$	−8.66650	−0.940014
$86$	− 3.73231i	− 0.402466i
$87$	0	0
$88$	8.69764	0.927172
$89$	1.90928	0.202383	0.101191	−	0.994867i	$-0.467735\pi$
$89$	1.90928	0.202383	0.101191	+	0.994867i	$0.467735\pi$
$90$	0	0
$91$	0	0
$92$	2.22093i	0.231548i
$93$	0	0
$94$	− 4.57881i	− 0.472269i
$95$	− 10.6909i	− 1.09686i
$96$	0	0
$97$	0.231417i	0.0234968i	0.999931	+	0.0117484i	$0.00373972\pi$
$97$	0.231417i	0.0234968i	−0.999931	+	0.0117484i	$0.996260\pi$
$98$	0	0
$99$	0	0
$100$	19.3833	1.93833
$101$	7.04379	0.700883	0.350442	−	0.936585i	$-0.386031\pi$
$101$	7.04379	0.700883	0.350442	+	0.936585i	$0.386031\pi$
$102$	0	0
$103$	6.67360i	0.657569i	0.944405	+	0.328785i	$0.106639\pi$
$103$	6.67360i	0.657569i	−0.944405	+	0.328785i	$0.893361\pi$
$104$	2.60767	0.255703
$105$	0	0
$106$	−4.02474	−0.390917
$107$	16.4154i	1.58693i	0.608614	+	0.793466i	$0.291726\pi$
$107$	16.4154i	1.58693i	−0.608614	+	0.793466i	$0.708274\pi$
$108$	0	0
$109$	−6.07891	−0.582254	−0.291127	−	0.956684i	$-0.594030\pi$
$109$	−6.07891	−0.582254	−0.291127	+	0.956684i	$0.594030\pi$
$110$	9.34343	0.890861
$111$	0	0
$112$	0	0
$113$	− 12.0500i	− 1.13357i	−0.823866	−	0.566784i	$-0.808187\pi$
$113$	− 12.0500i	− 1.13357i	0.823866	−	0.566784i	$-0.191813\pi$
$114$	0	0
$115$	5.12837i	0.478223i
$116$	− 15.2360i	− 1.41463i
$117$	0	0
$118$	− 3.94162i	− 0.362856i
$119$	0	0
$120$	0	0
$121$	−9.79200	−0.890182
$122$	−3.07240	−0.278162
$123$	0	0
$124$	15.1856i	1.36371i
$125$	24.6705	2.20660
$126$	0	0
$127$	3.30791	0.293529	0.146765	−	0.989171i	$-0.453114\pi$
$127$	3.30791	0.293529	0.146765	+	0.989171i	$0.453114\pi$
$128$	11.4192i	1.00932i
$129$	0	0
$130$	2.80129	0.245689
$131$	−3.28845	−0.287313	−0.143657	−	0.989628i	$-0.545886\pi$
$131$	−3.28845	−0.287313	−0.143657	+	0.989628i	$0.545886\pi$
$132$	0	0
$133$	0	0
$134$	4.16093i	0.359450i
$135$	0	0
$136$	− 4.11468i	− 0.352831i
$137$	2.01367i	0.172039i	0.996293	+	0.0860196i	$0.0274148\pi$
$137$	2.01367i	0.172039i	−0.996293	+	0.0860196i	$0.972585\pi$
$138$	0	0
$139$	10.1678i	0.862421i	0.902251	+	0.431211i	$0.141913\pi$
$139$	10.1678i	0.862421i	−0.902251	+	0.431211i	$0.858087\pi$
$140$	0	0
$141$	0	0
$142$	−0.153824	−0.0129087
$143$	−6.23373	−0.521290
$144$	0	0
$145$	− 35.1817i	− 2.92168i
$146$	−5.93968	−0.491572
$147$	0	0
$148$	13.6706	1.12372
$149$	15.9456i	1.30631i	0.757223	+	0.653156i	$0.226555\pi$
$149$	15.9456i	1.30631i	−0.757223	+	0.653156i	$0.773445\pi$
$150$	0	0
$151$	4.63356	0.377074	0.188537	−	0.982066i	$-0.439625\pi$
$151$	4.63356	0.377074	0.188537	+	0.982066i	$0.439625\pi$
$152$	5.07583	0.411704
$153$	0	0
$154$	0	0
$155$	35.0653i	2.81651i
$156$	0	0
$157$	7.39104i	0.589869i	0.955517	+	0.294934i	$0.0952978\pi$
$157$	7.39104i	0.589869i	−0.955517	+	0.294934i	$0.904702\pi$
$158$	− 2.48136i	− 0.197406i
$159$	0	0
$160$	20.4633i	1.61777i
$161$	0	0
$162$	0	0
$163$	−1.00026	−0.0783462	−0.0391731	−	0.999232i	$-0.512472\pi$
$163$	−1.00026	−0.0783462	−0.0391731	+	0.999232i	$0.512472\pi$
$164$	−15.5261	−1.21238
$165$	0	0
$166$	− 2.23571i	− 0.173524i
$167$	9.28225	0.718282	0.359141	−	0.933283i	$-0.383070\pi$
$167$	9.28225	0.718282	0.359141	+	0.933283i	$0.383070\pi$
$168$	0	0
$169$	11.1310	0.856234
$170$	− 4.42019i	− 0.339013i
$171$	0	0
$172$	−12.7320	−0.970808
$173$	17.8848	1.35975	0.679876	−	0.733327i	$-0.262034\pi$
$173$	17.8848	1.35975	0.679876	+	0.733327i	$0.262034\pi$
$174$	0	0
$175$	0	0
$176$	− 11.4309i	− 0.861638i
$177$	0	0
$178$	0.973791i	0.0729887i
$179$	2.84001i	0.212272i	0.994352	+	0.106136i	$0.0338479\pi$
$179$	2.84001i	0.212272i	−0.994352	+	0.106136i	$0.966152\pi$
$180$	0	0
$181$	− 7.95711i	− 0.591447i	−0.955274	−	0.295724i	$-0.904439\pi$
$181$	− 7.95711i	− 0.591447i	0.955274	−	0.295724i	$-0.0955608\pi$
$182$	0	0
$183$	0	0
$184$	−2.43485	−0.179499
$185$	31.5669	2.32085
$186$	0	0
$187$	9.83628i	0.719300i
$188$	−15.6197	−1.13918
$189$	0	0
$190$	5.45270	0.395581
$191$	− 13.9123i	− 1.00666i	−0.864095	−	0.503328i	$-0.832109\pi$
$191$	− 13.9123i	− 1.00666i	0.864095	−	0.503328i	$-0.167891\pi$
$192$	0	0
$193$	8.05188	0.579587	0.289793	−	0.957089i	$-0.406413\pi$
$193$	8.05188	0.579587	0.289793	+	0.957089i	$0.406413\pi$
$194$	−0.118030	−0.00847404
$195$	0	0
$196$	0	0
$197$	− 7.00094i	− 0.498796i	−0.968401	−	0.249398i	$-0.919767\pi$
$197$	− 7.00094i	− 0.498796i	0.968401	−	0.249398i	$-0.0802328\pi$
$198$	0	0
$199$	− 14.7197i	− 1.04345i	−0.853114	−	0.521724i	$-0.825289\pi$
$199$	− 14.7197i	− 1.04345i	0.853114	−	0.521724i	$-0.174711\pi$
$200$	21.2503i	1.50262i
$201$	0	0
$202$	3.59256i	0.252771i
$203$	0	0
$204$	0	0
$205$	−35.8515	−2.50398
$206$	−3.40374	−0.237150
$207$	0	0
$208$	− 3.42714i	− 0.237630i
$209$	−12.1339	−0.839322
$210$	0	0
$211$	−22.3102	−1.53590	−0.767950	−	0.640510i	$-0.778723\pi$
$211$	−22.3102	−1.53590	−0.767950	+	0.640510i	$0.778723\pi$
$212$	13.7296i	0.942950i
$213$	0	0
$214$	−8.37235	−0.572322
$215$	−29.3997	−2.00504
$216$	0	0
$217$	0	0
$218$	− 3.10044i	− 0.209988i
$219$	0	0
$220$	− 31.8732i	− 2.14889i
$221$	2.94905i	0.198375i
$222$	0	0
$223$	6.40672i	0.429026i	0.976721	+	0.214513i	$0.0688164\pi$
$223$	6.40672i	0.429026i	−0.976721	+	0.214513i	$0.931184\pi$
$224$	0	0
$225$	0	0
$226$	6.14588	0.408818
$227$	−3.09048	−0.205123	−0.102561	−	0.994727i	$-0.532704\pi$
$227$	−3.09048	−0.205123	−0.102561	+	0.994727i	$0.532704\pi$
$228$	0	0
$229$	− 13.5657i	− 0.896447i	−0.893922	−	0.448223i	$-0.852057\pi$
$229$	− 13.5657i	− 0.896447i	0.893922	−	0.448223i	$-0.147943\pi$
$230$	−2.61563	−0.172470
$231$	0	0
$232$	16.7035	1.09664
$233$	− 24.3065i	− 1.59237i	−0.605052	−	0.796186i	$-0.706848\pi$
$233$	− 24.3065i	− 1.59237i	0.605052	−	0.796186i	$-0.293152\pi$
$234$	0	0
$235$	−36.0676	−2.35279
$236$	−13.4460	−0.875262
$237$	0	0
$238$	0	0
$239$	− 3.58210i	− 0.231707i	−0.993266	−	0.115853i	$-0.963040\pi$
$239$	− 3.58210i	− 0.231707i	0.993266	−	0.115853i	$-0.0369603\pi$
$240$	0	0
$241$	7.01695i	0.452002i	0.974127	+	0.226001i	$0.0725652\pi$
$241$	7.01695i	0.452002i	−0.974127	+	0.226001i	$0.927435\pi$
$242$	− 4.99423i	− 0.321041i
$243$	0	0
$244$	10.4809i	0.670968i
$245$	0	0
$246$	0	0
$247$	−3.63792	−0.231475
$248$	−16.6483	−1.05717
$249$	0	0
$250$	12.5827i	0.795803i
$251$	−5.29654	−0.334314	−0.167157	−	0.985930i	$-0.553459\pi$
$251$	−5.29654	−0.334314	−0.167157	+	0.985930i	$0.553459\pi$
$252$	0	0
$253$	5.82058	0.365937
$254$	1.68714i	0.105860i
$255$	0	0
$256$	−0.992308	−0.0620192
$257$	−14.8634	−0.927154	−0.463577	−	0.886057i	$-0.653434\pi$
$257$	−14.8634	−0.927154	−0.463577	+	0.886057i	$0.653434\pi$
$258$	0	0
$259$	0	0
$260$	− 9.55602i	− 0.592639i
$261$	0	0
$262$	− 1.67721i	− 0.103619i
$263$	− 15.2311i	− 0.939190i	−0.882882	−	0.469595i	$-0.844400\pi$
$263$	− 15.2311i	− 0.939190i	0.882882	−	0.469595i	$-0.155600\pi$
$264$	0	0
$265$	31.7031i	1.94751i
$266$	0	0
$267$	0	0
$268$	14.1942	0.867047
$269$	13.3418	0.813461	0.406731	−	0.913548i	$-0.366669\pi$
$269$	13.3418	0.813461	0.406731	+	0.913548i	$0.366669\pi$
$270$	0	0
$271$	− 24.2808i	− 1.47496i	−0.675371	−	0.737478i	$-0.736017\pi$
$271$	− 24.2808i	− 1.47496i	0.675371	−	0.737478i	$-0.263983\pi$
$272$	−5.40774	−0.327892
$273$	0	0
$274$	−1.02703	−0.0620454
$275$	− 50.7996i	− 3.06333i
$276$	0	0
$277$	−16.3970	−0.985199	−0.492599	−	0.870256i	$-0.663953\pi$
$277$	−16.3970	−0.985199	−0.492599	+	0.870256i	$0.663953\pi$
$278$	−5.18590	−0.311029
$279$	0	0
$280$	0	0
$281$	− 14.6475i	− 0.873799i	−0.899510	−	0.436900i	$-0.856077\pi$
$281$	− 14.6475i	− 0.873799i	0.899510	−	0.436900i	$-0.143923\pi$
$282$	0	0
$283$	− 8.38829i	− 0.498632i	−0.968422	−	0.249316i	$-0.919794\pi$
$283$	− 8.38829i	− 0.498632i	0.968422	−	0.249316i	$-0.0802058\pi$
$284$	0.524741i	0.0311376i
$285$	0	0
$286$	− 3.17940i	− 0.188002i
$287$	0	0
$288$	0	0
$289$	−12.3467	−0.726274
$290$	17.9438	1.05369
$291$	0	0
$292$	20.2620i	1.18574i
$293$	1.73053	0.101099	0.0505493	−	0.998722i	$-0.483903\pi$
$293$	1.73053	0.101099	0.0505493	+	0.998722i	$0.483903\pi$
$294$	0	0
$295$	−31.0484	−1.80771
$296$	14.9873i	0.871122i
$297$	0	0
$298$	−8.13274	−0.471117
$299$	1.74509	0.100921
$300$	0	0
$301$	0	0
$302$	2.36326i	0.135990i
$303$	0	0
$304$	− 6.67093i	− 0.382604i
$305$	24.2015i	1.38577i
$306$	0	0
$307$	− 14.0499i	− 0.801873i	−0.916106	−	0.400936i	$-0.868685\pi$
$307$	− 14.0499i	− 0.801873i	0.916106	−	0.400936i	$-0.131315\pi$
$308$	0	0
$309$	0	0
$310$	−17.8844	−1.01577
$311$	−16.2569	−0.921844	−0.460922	−	0.887441i	$-0.652481\pi$
$311$	−16.2569	−0.921844	−0.460922	+	0.887441i	$0.652481\pi$
$312$	0	0
$313$	6.86733i	0.388165i	0.980985	+	0.194082i	$0.0621729\pi$
$313$	6.86733i	0.388165i	−0.980985	+	0.194082i	$0.937827\pi$
$314$	−3.76966	−0.212734
$315$	0	0
$316$	−8.46465	−0.476174
$317$	16.2748i	0.914084i	0.889445	+	0.457042i	$0.151091\pi$
$317$	16.2748i	0.914084i	−0.889445	+	0.457042i	$0.848909\pi$
$318$	0	0
$319$	−39.9304	−2.23567
$320$	9.70605	0.542585
$321$	0	0
$322$	0	0
$323$	5.74032i	0.319400i
$324$	0	0
$325$	− 15.2304i	− 0.844831i
$326$	− 0.510163i	− 0.0282553i
$327$	0	0
$328$	− 17.0216i	− 0.939858i
$329$	0	0
$330$	0	0
$331$	−33.9193	−1.86437	−0.932187	−	0.361977i	$-0.882102\pi$
$331$	−33.9193	−1.86437	−0.932187	+	0.361977i	$0.882102\pi$
$332$	−7.62665	−0.418567
$333$	0	0
$334$	4.73424i	0.259046i
$335$	32.7759	1.79074
$336$	0	0
$337$	−0.427345	−0.0232790	−0.0116395	−	0.999932i	$-0.503705\pi$
$337$	−0.427345	−0.0232790	−0.0116395	+	0.999932i	$0.503705\pi$
$338$	5.67718i	0.308798i
$339$	0	0
$340$	−15.0786	−0.817750
$341$	39.7983	2.15520
$342$	0	0
$343$	0	0
$344$	− 13.9584i	− 0.752584i
$345$	0	0
$346$	9.12179i	0.490390i
$347$	9.03989i	0.485287i	0.970116	+	0.242643i	$0.0780145\pi$
$347$	9.03989i	0.485287i	−0.970116	+	0.242643i	$0.921986\pi$
$348$	0	0
$349$	22.9568i	1.22885i	0.788976	+	0.614424i	$0.210612\pi$
$349$	22.9568i	1.22885i	−0.788976	+	0.614424i	$0.789388\pi$
$350$	0	0
$351$	0	0
$352$	23.2254	1.23792
$353$	−11.9601	−0.636571	−0.318286	−	0.947995i	$-0.603107\pi$
$353$	−11.9601	−0.636571	−0.318286	+	0.947995i	$0.603107\pi$
$354$	0	0
$355$	1.21168i	0.0643095i
$356$	3.32189	0.176060
$357$	0	0
$358$	−1.44849	−0.0765552
$359$	18.7765i	0.990987i	0.868612	+	0.495494i	$0.165013\pi$
$359$	18.7765i	0.990987i	−0.868612	+	0.495494i	$0.834987\pi$
$360$	0	0
$361$	11.9188	0.627305
$362$	4.05838	0.213304
$363$	0	0
$364$	0	0
$365$	46.7872i	2.44896i
$366$	0	0
$367$	− 27.1985i	− 1.41975i	−0.704327	−	0.709876i	$-0.748751\pi$
$367$	− 27.1985i	− 1.41975i	0.704327	−	0.709876i	$-0.251249\pi$
$368$	3.20001i	0.166812i
$369$	0	0
$370$	16.1001i	0.837006i
$371$	0	0
$372$	0	0
$373$	0.697276	0.0361036	0.0180518	−	0.999837i	$-0.494254\pi$
$373$	0.697276	0.0361036	0.0180518	+	0.999837i	$0.494254\pi$
$374$	−5.01681	−0.259413
$375$	0	0
$376$	− 17.1242i	− 0.883111i
$377$	−11.9717	−0.616573
$378$	0	0
$379$	−0.998557	−0.0512924	−0.0256462	−	0.999671i	$-0.508164\pi$
$379$	−0.998557	−0.0512924	−0.0256462	+	0.999671i	$0.508164\pi$
$380$	− 18.6008i	− 0.954200i
$381$	0	0
$382$	7.09570	0.363047
$383$	35.0709	1.79204	0.896019	−	0.444016i	$-0.146446\pi$
$383$	35.0709	1.79204	0.896019	+	0.444016i	$0.146446\pi$
$384$	0	0
$385$	0	0
$386$	4.10671i	0.209026i
$387$	0	0
$388$	0.402634i	0.0204407i
$389$	− 27.7273i	− 1.40583i	−0.711275	−	0.702914i	$-0.751882\pi$
$389$	− 27.7273i	− 1.40583i	0.711275	−	0.702914i	$-0.248118\pi$
$390$	0	0
$391$	− 2.75360i	− 0.139256i
$392$	0	0
$393$	0	0
$394$	3.57070	0.179889
$395$	−19.5458	−0.983457
$396$	0	0
$397$	1.10713i	0.0555653i	0.999614	+	0.0277826i	$0.00884463\pi$
$397$	1.10713i	0.0555653i	−0.999614	+	0.0277826i	$0.991155\pi$
$398$	7.50749	0.376316
$399$	0	0
$400$	27.9283	1.39642
$401$	32.4305i	1.61950i	0.586775	+	0.809750i	$0.300397\pi$
$401$	32.4305i	1.61950i	−0.586775	+	0.809750i	$0.699603\pi$
$402$	0	0
$403$	11.9321	0.594379
$404$	12.2553	0.609722
$405$	0	0
$406$	0	0
$407$	− 35.8278i	− 1.77592i
$408$	0	0
$409$	− 0.201959i	− 0.00998622i	−0.999988	−	0.00499311i	$-0.998411\pi$
$409$	− 0.201959i	− 0.00998622i	0.999988	−	0.00499311i	$-0.00158936\pi$
$410$	− 18.2854i	− 0.903051i
$411$	0	0
$412$	11.6112i	0.572042i
$413$	0	0
$414$	0	0
$415$	−17.6108	−0.864479
$416$	6.96329	0.341404
$417$	0	0
$418$	− 6.18869i	− 0.302699i
$419$	2.89945	0.141647	0.0708236	−	0.997489i	$-0.477437\pi$
$419$	2.89945	0.141647	0.0708236	+	0.997489i	$0.477437\pi$
$420$	0	0
$421$	21.9825	1.07136	0.535681	−	0.844420i	$-0.320055\pi$
$421$	21.9825	1.07136	0.535681	+	0.844420i	$0.320055\pi$
$422$	− 11.3789i	− 0.553917i
$423$	0	0
$424$	−15.0520	−0.730989
$425$	−24.0323	−1.16574
$426$	0	0
$427$	0	0
$428$	28.5606i	1.38053i
$429$	0	0
$430$	− 14.9948i	− 0.723111i
$431$	− 22.1190i	− 1.06543i	−0.846293	−	0.532717i	$-0.821171\pi$
$431$	− 22.1190i	− 1.06543i	0.846293	−	0.532717i	$-0.178829\pi$
$432$	0	0
$433$	− 17.7613i	− 0.853552i	−0.904357	−	0.426776i	$-0.859649\pi$
$433$	− 17.7613i	− 0.853552i	0.904357	−	0.426776i	$-0.140351\pi$
$434$	0	0
$435$	0	0
$436$	−10.5765	−0.506523
$437$	3.39682	0.162492
$438$	0	0
$439$	− 8.96748i	− 0.427994i	−0.976834	−	0.213997i	$-0.931352\pi$
$439$	− 8.96748i	− 0.427994i	0.976834	−	0.213997i	$-0.0686483\pi$
$440$	34.9432	1.66585
$441$	0	0
$442$	−1.50411	−0.0715432
$443$	16.5048i	0.784165i	0.919930	+	0.392083i	$0.128245\pi$
$443$	16.5048i	0.784165i	−0.919930	+	0.392083i	$0.871755\pi$
$444$	0	0
$445$	7.67061	0.363622
$446$	−3.26763	−0.154727
$447$	0	0
$448$	0	0
$449$	− 8.93393i	− 0.421618i	−0.977527	−	0.210809i	$-0.932390\pi$
$449$	− 8.93393i	− 0.421618i	0.977527	−	0.210809i	$-0.0676098\pi$
$450$	0	0
$451$	40.6906i	1.91605i
$452$	− 20.9654i	− 0.986129i
$453$	0	0
$454$	− 1.57624i	− 0.0739768i
$455$	0	0
$456$	0	0
$457$	26.7557	1.25158	0.625789	−	0.779992i	$-0.284777\pi$
$457$	26.7557	1.25158	0.625789	+	0.779992i	$0.284777\pi$
$458$	6.91893	0.323301
$459$	0	0
$460$	8.92269i	0.416023i
$461$	11.6692	0.543489	0.271745	−	0.962369i	$-0.412399\pi$
$461$	11.6692	0.543489	0.271745	+	0.962369i	$0.412399\pi$
$462$	0	0
$463$	−17.8823	−0.831059	−0.415530	−	0.909580i	$-0.636404\pi$
$463$	−17.8823	−0.831059	−0.415530	+	0.909580i	$0.636404\pi$
$464$	− 21.9527i	− 1.01913i
$465$	0	0
$466$	12.3971	0.574284
$467$	12.7961	0.592131	0.296066	−	0.955168i	$-0.404325\pi$
$467$	12.7961	0.592131	0.296066	+	0.955168i	$0.404325\pi$
$468$	0	0
$469$	0	0
$470$	− 18.3956i	− 0.848526i
$471$	0	0
$472$	− 14.7411i	− 0.678516i
$473$	33.3679i	1.53426i
$474$	0	0
$475$	− 29.6460i	− 1.36025i
$476$	0	0
$477$	0	0
$478$	1.82698	0.0835643
$479$	2.03391	0.0929315	0.0464658	−	0.998920i	$-0.485204\pi$
$479$	2.03391	0.0929315	0.0464658	+	0.998920i	$0.485204\pi$
$480$	0	0
$481$	− 10.7417i	− 0.489777i
$482$	−3.57887	−0.163013
$483$	0	0
$484$	−17.0368	−0.774399
$485$	0.929727i	0.0422167i
$486$	0	0
$487$	14.0614	0.637184	0.318592	−	0.947892i	$-0.396790\pi$
$487$	14.0614	0.637184	0.318592	+	0.947892i	$0.396790\pi$
$488$	−11.4904	−0.520144
$489$	0	0
$490$	0	0
$491$	− 32.1988i	− 1.45311i	−0.687107	−	0.726557i	$-0.741120\pi$
$491$	− 32.1988i	− 1.45311i	0.687107	−	0.726557i	$-0.258880\pi$
$492$	0	0
$493$	18.8903i	0.850775i
$494$	− 1.85545i	− 0.0834808i
$495$	0	0
$496$	21.8801i	0.982445i
$497$	0	0
$498$	0	0
$499$	17.0426	0.762931	0.381466	−	0.924383i	$-0.375419\pi$
$499$	17.0426	0.762931	0.381466	+	0.924383i	$0.375419\pi$
$500$	42.9235	1.91960
$501$	0	0
$502$	− 2.70140i	− 0.120569i
$503$	−43.7250	−1.94960	−0.974801	−	0.223077i	$-0.928390\pi$
$503$	−43.7250	−1.94960	−0.974801	+	0.223077i	$0.928390\pi$
$504$	0	0
$505$	28.2988	1.25928
$506$	2.96868i	0.131974i
$507$	0	0
$508$	5.75532	0.255351
$509$	−13.8222	−0.612657	−0.306329	−	0.951926i	$-0.599101\pi$
$509$	−13.8222	−0.612657	−0.306329	+	0.951926i	$0.599101\pi$
$510$	0	0
$511$	0	0
$512$	22.3322i	0.986954i
$513$	0	0
$514$	− 7.58081i	− 0.334375i
$515$	26.8115i	1.18146i
$516$	0	0
$517$	40.9359i	1.80036i
$518$	0	0
$519$	0	0
$520$	10.4764	0.459422
$521$	−4.28818	−0.187868	−0.0939342	−	0.995578i	$-0.529944\pi$
$521$	−4.28818	−0.187868	−0.0939342	+	0.995578i	$0.529944\pi$
$522$	0	0
$523$	31.9987i	1.39921i	0.714532	+	0.699603i	$0.246640\pi$
$523$	31.9987i	1.39921i	−0.714532	+	0.699603i	$0.753360\pi$
$524$	−5.72147	−0.249943
$525$	0	0
$526$	7.76834	0.338716
$527$	− 18.8278i	− 0.820151i
$528$	0	0
$529$	21.3706	0.929155
$530$	−16.1696	−0.702361
$531$	0	0
$532$	0	0
$533$	12.1996i	0.528423i
$534$	0	0
$535$	65.9495i	2.85124i
$536$	15.5613i	0.672147i
$537$	0	0
$538$	6.80472i	0.293372i
$539$	0	0
$540$	0	0
$541$	−32.6342	−1.40305	−0.701527	−	0.712643i	$-0.747498\pi$
$541$	−32.6342	−1.40305	−0.701527	+	0.712643i	$0.747498\pi$
$542$	12.3840	0.531938
$543$	0	0
$544$	− 10.9875i	− 0.471084i
$545$	−24.4223	−1.04614
$546$	0	0
$547$	2.09795	0.0897019	0.0448510	−	0.998994i	$-0.485719\pi$
$547$	2.09795	0.0897019	0.0448510	+	0.998994i	$0.485719\pi$
$548$	3.50351i	0.149663i
$549$	0	0
$550$	25.9094	1.10478
$551$	−23.3029	−0.992735
$552$	0	0
$553$	0	0
$554$	− 8.36298i	− 0.355309i
$555$	0	0
$556$	17.6906i	0.750250i
$557$	− 30.3756i	− 1.28706i	−0.765422	−	0.643528i	$-0.777470\pi$
$557$	− 30.3756i	− 1.28706i	0.765422	−	0.643528i	$-0.222530\pi$
$558$	0	0
$559$	10.0042i	0.423131i
$560$	0	0
$561$	0	0
$562$	7.47071	0.315133
$563$	−7.47998	−0.315244	−0.157622	−	0.987500i	$-0.550383\pi$
$563$	−7.47998	−0.315244	−0.157622	+	0.987500i	$0.550383\pi$
$564$	0	0
$565$	− 48.4114i	− 2.03668i
$566$	4.27829	0.179830
$567$	0	0
$568$	−0.575283	−0.0241383
$569$	− 18.0817i	− 0.758024i	−0.925392	−	0.379012i	$-0.876264\pi$
$569$	− 18.0817i	− 0.758024i	0.925392	−	0.379012i	$-0.123736\pi$
$570$	0	0
$571$	0.137188	0.00574112	0.00287056	−	0.999996i	$-0.499086\pi$
$571$	0.137188	0.00574112	0.00287056	+	0.999996i	$0.499086\pi$
$572$	−10.8459	−0.453488
$573$	0	0
$574$	0	0
$575$	14.2210i	0.593057i
$576$	0	0
$577$	− 27.6089i	− 1.14937i	−0.818373	−	0.574687i	$-0.805124\pi$
$577$	− 27.6089i	− 1.14937i	0.818373	−	0.574687i	$-0.194876\pi$
$578$	− 6.29718i	− 0.261928i
$579$	0	0
$580$	− 61.2115i	− 2.54167i
$581$	0	0
$582$	0	0
$583$	35.9823	1.49023
$584$	−22.2136	−0.919207
$585$	0	0
$586$	0.882624i	0.0364609i
$587$	−46.4304	−1.91639	−0.958193	−	0.286123i	$-0.907633\pi$
$587$	−46.4304	−1.91639	−0.958193	+	0.286123i	$0.907633\pi$
$588$	0	0
$589$	23.2258	0.957001
$590$	− 15.8357i	− 0.651944i
$591$	0	0
$592$	19.6972	0.809550
$593$	−38.7090	−1.58959	−0.794794	−	0.606880i	$-0.792421\pi$
$593$	−38.7090	−1.58959	−0.794794	+	0.606880i	$0.792421\pi$
$594$	0	0
$595$	0	0
$596$	27.7432i	1.13641i
$597$	0	0
$598$	0.890051i	0.0363969i
$599$	24.1896i	0.988360i	0.869360	+	0.494180i	$0.164532\pi$
$599$	24.1896i	0.988360i	−0.869360	+	0.494180i	$0.835468\pi$
$600$	0	0
$601$	− 43.4070i	− 1.77061i	−0.465014	−	0.885304i	$-0.653951\pi$
$601$	− 43.4070i	− 1.77061i	0.465014	−	0.885304i	$-0.346049\pi$
$602$	0	0
$603$	0	0
$604$	8.06178	0.328029
$605$	−39.3398	−1.59939
$606$	0	0
$607$	41.4806i	1.68364i	0.539756	+	0.841822i	$0.318517\pi$
$607$	41.4806i	1.68364i	−0.539756	+	0.841822i	$0.681483\pi$
$608$	13.5540	0.549689
$609$	0	0
$610$	−12.3435	−0.499774
$611$	12.2731i	0.496518i
$612$	0	0
$613$	34.5432	1.39519	0.697594	−	0.716494i	$-0.254254\pi$
$613$	34.5432	1.39519	0.697594	+	0.716494i	$0.254254\pi$
$614$	7.16591	0.289193
$615$	0	0
$616$	0	0
$617$	− 5.87374i	− 0.236468i	−0.992986	−	0.118234i	$-0.962277\pi$
$617$	− 5.87374i	− 0.236468i	0.992986	−	0.118234i	$-0.0377232\pi$
$618$	0	0
$619$	− 41.8903i	− 1.68371i	−0.539702	−	0.841856i	$-0.681463\pi$
$619$	− 41.8903i	− 1.68371i	0.539702	−	0.841856i	$-0.318537\pi$
$620$	61.0090i	2.45018i
$621$	0	0
$622$	− 8.29153i	− 0.332460i
$623$	0	0
$624$	0	0
$625$	43.4116	1.73646
$626$	−3.50256	−0.139990
$627$	0	0
$628$	12.8594i	0.513147i
$629$	−16.9494	−0.675817
$630$	0	0
$631$	−43.9572	−1.74991	−0.874954	−	0.484206i	$-0.839109\pi$
$631$	−43.9572	−1.74991	−0.874954	+	0.484206i	$0.839109\pi$
$632$	− 9.27996i	− 0.369137i
$633$	0	0
$634$	−8.30066	−0.329661
$635$	13.2897	0.527385
$636$	0	0
$637$	0	0
$638$	− 20.3658i	− 0.806288i
$639$	0	0
$640$	45.8770i	1.81345i
$641$	41.1337i	1.62468i	0.583182	+	0.812341i	$0.301807\pi$
$641$	41.1337i	1.62468i	−0.583182	+	0.812341i	$0.698193\pi$
$642$	0	0
$643$	− 32.6357i	− 1.28703i	−0.765435	−	0.643513i	$-0.777476\pi$
$643$	− 32.6357i	− 1.28703i	0.765435	−	0.643513i	$-0.222524\pi$
$644$	0	0
$645$	0	0
$646$	−2.92775	−0.115191
$647$	43.2586	1.70067	0.850335	−	0.526242i	$-0.176399\pi$
$647$	43.2586	1.70067	0.850335	+	0.526242i	$0.176399\pi$
$648$	0	0
$649$	35.2392i	1.38326i
$650$	7.76799	0.304686
$651$	0	0
$652$	−1.74031	−0.0681560
$653$	38.0805i	1.49020i	0.666951	+	0.745102i	$0.267599\pi$
$653$	38.0805i	1.49020i	−0.666951	+	0.745102i	$0.732401\pi$
$654$	0	0
$655$	−13.2115	−0.516216
$656$	−22.3707	−0.873428
$657$	0	0
$658$	0	0
$659$	35.2754i	1.37413i	0.726594	+	0.687067i	$0.241102\pi$
$659$	35.2754i	1.37413i	−0.726594	+	0.687067i	$0.758898\pi$
$660$	0	0
$661$	19.2800i	0.749905i	0.927044	+	0.374952i	$0.122341\pi$
$661$	19.2800i	0.749905i	−0.927044	+	0.374952i	$0.877659\pi$
$662$	− 17.2999i	− 0.672380i
$663$	0	0
$664$	− 8.36124i	− 0.324479i
$665$	0	0
$666$	0	0
$667$	11.1783	0.432824
$668$	16.1499	0.624858
$669$	0	0
$670$	16.7167i	0.645824i
$671$	27.4681	1.06039
$672$	0	0
$673$	−3.95341	−0.152393	−0.0761964	−	0.997093i	$-0.524278\pi$
$673$	−3.95341	−0.152393	−0.0761964	+	0.997093i	$0.524278\pi$
$674$	− 0.217959i	− 0.00839548i
$675$	0	0
$676$	19.3665	0.744867
$677$	22.2397	0.854740	0.427370	−	0.904077i	$-0.359440\pi$
$677$	22.2397	0.854740	0.427370	+	0.904077i	$0.359440\pi$
$678$	0	0
$679$	0	0
$680$	− 16.5309i	− 0.633932i
$681$	0	0
$682$	20.2984i	0.777266i
$683$	− 16.7827i	− 0.642174i	−0.947050	−	0.321087i	$-0.895952\pi$
$683$	− 16.7827i	− 0.642174i	0.947050	−	0.321087i	$-0.104048\pi$
$684$	0	0
$685$	8.09001i	0.309103i
$686$	0	0
$687$	0	0
$688$	−18.3448	−0.699391
$689$	10.7880	0.410989
$690$	0	0
$691$	− 3.01862i	− 0.114834i	−0.998350	−	0.0574169i	$-0.981714\pi$
$691$	− 3.01862i	− 0.114834i	0.998350	−	0.0574169i	$-0.0182864\pi$
$692$	31.1171	1.18289
$693$	0	0
$694$	−4.61063	−0.175017
$695$	40.8496i	1.54951i
$696$	0	0
$697$	19.2499	0.729143
$698$	−11.7087	−0.443180
$699$	0	0
$700$	0	0
$701$	− 22.7605i	− 0.859653i	−0.902911	−	0.429827i	$-0.858575\pi$
$701$	− 22.7605i	− 0.859653i	0.902911	−	0.429827i	$-0.141425\pi$
$702$	0	0
$703$	− 20.9086i	− 0.788583i
$704$	− 11.0161i	− 0.415186i
$705$	0	0
$706$	− 6.10002i	− 0.229577i
$707$	0	0
$708$	0	0
$709$	16.9079	0.634988	0.317494	−	0.948260i	$-0.397159\pi$
$709$	16.9079	0.634988	0.317494	+	0.948260i	$0.397159\pi$
$710$	−0.617997	−0.0231930
$711$	0	0
$712$	3.64185i	0.136484i
$713$	−11.1413	−0.417244
$714$	0	0
$715$	−25.0443	−0.936603
$716$	4.94123i	0.184663i
$717$	0	0
$718$	−9.57662	−0.357396
$719$	18.4522	0.688149	0.344075	−	0.938942i	$-0.388193\pi$
$719$	18.4522	0.688149	0.344075	+	0.938942i	$0.388193\pi$
$720$	0	0
$721$	0	0
$722$	6.07896i	0.226235i
$723$	0	0
$724$	− 13.8443i	− 0.514520i
$725$	− 97.5591i	− 3.62325i
$726$	0	0
$727$	− 46.9647i	− 1.74182i	−0.491438	−	0.870912i	$-0.663529\pi$
$727$	− 46.9647i	− 1.74182i	0.491438	−	0.870912i	$-0.336471\pi$
$728$	0	0
$729$	0	0
$730$	−23.8630	−0.883208
$731$	15.7857	0.583855
$732$	0	0
$733$	− 49.7678i	− 1.83822i	−0.394006	−	0.919108i	$-0.628911\pi$
$733$	− 49.7678i	− 1.83822i	0.394006	−	0.919108i	$-0.371089\pi$
$734$	13.8721	0.512029
$735$	0	0
$736$	−6.50180	−0.239660
$737$	− 37.1999i	− 1.37028i
$738$	0	0
$739$	32.4025	1.19195	0.595973	−	0.803004i	$-0.296766\pi$
$739$	32.4025	1.19195	0.595973	+	0.803004i	$0.296766\pi$
$740$	54.9223	2.01898
$741$	0	0
$742$	0	0
$743$	− 2.49994i	− 0.0917138i	−0.998948	−	0.0458569i	$-0.985398\pi$
$743$	− 2.49994i	− 0.0917138i	0.998948	−	0.0458569i	$-0.0146018\pi$
$744$	0	0
$745$	64.0621i	2.34705i
$746$	0.355633i	0.0130206i
$747$	0	0
$748$	17.1138i	0.625744i
$749$	0	0
$750$	0	0
$751$	17.5546	0.640578	0.320289	−	0.947320i	$-0.396220\pi$
$751$	17.5546	0.640578	0.320289	+	0.947320i	$0.396220\pi$
$752$	−22.5055	−0.820691
$753$	0	0
$754$	− 6.10593i	− 0.222365i
$755$	18.6156	0.677489
$756$	0	0
$757$	20.3609	0.740030	0.370015	−	0.929026i	$-0.379353\pi$
$757$	20.3609	0.740030	0.370015	+	0.929026i	$0.379353\pi$
$758$	− 0.509296i	− 0.0184985i
$759$	0	0
$760$	20.3924	0.739710
$761$	−41.9777	−1.52169	−0.760846	−	0.648932i	$-0.775216\pi$
$761$	−41.9777	−1.52169	−0.760846	+	0.648932i	$0.775216\pi$
$762$	0	0
$763$	0	0
$764$	− 24.2055i	− 0.875725i
$765$	0	0
$766$	17.8873i	0.646293i
$767$	10.5652i	0.381487i
$768$	0	0
$769$	− 12.0753i	− 0.435447i	−0.976010	−	0.217724i	$-0.930137\pi$
$769$	− 12.0753i	− 0.435447i	0.976010	−	0.217724i	$-0.0698632\pi$
$770$	0	0
$771$	0	0
$772$	14.0092	0.504202
$773$	−5.30144	−0.190679	−0.0953397	−	0.995445i	$-0.530394\pi$
$773$	−5.30144	−0.190679	−0.0953397	+	0.995445i	$0.530394\pi$
$774$	0	0
$775$	97.2364i	3.49283i
$776$	−0.441416	−0.0158459
$777$	0	0
$778$	14.1418	0.507007
$779$	23.7465i	0.850807i
$780$	0	0
$781$	1.37523	0.0492097
$782$	1.40442	0.0502221
$783$	0	0
$784$	0	0
$785$	29.6938i	1.05982i
$786$	0	0
$787$	− 28.2573i	− 1.00726i	−0.863918	−	0.503632i	$-0.831997\pi$
$787$	− 28.2573i	− 1.00726i	0.863918	−	0.503632i	$-0.168003\pi$
$788$	− 12.1807i	− 0.433920i
$789$	0	0
$790$	− 9.96898i	− 0.354680i
$791$	0	0
$792$	0	0
$793$	8.23531	0.292444
$794$	−0.564672	−0.0200394
$795$	0	0
$796$	− 25.6103i	− 0.907731i
$797$	32.1596	1.13915	0.569576	−	0.821939i	$-0.307107\pi$
$797$	32.1596	1.13915	0.569576	+	0.821939i	$0.307107\pi$
$798$	0	0
$799$	19.3659	0.685118
$800$	56.7450i	2.00624i
$801$	0	0
$802$	−16.5406	−0.584067
$803$	53.1024	1.87394
$804$	0	0
$805$	0	0
$806$	6.08573i	0.214361i
$807$	0	0
$808$	13.4357i	0.472666i
$809$	− 29.9525i	− 1.05307i	−0.850152	−	0.526537i	$-0.823490\pi$
$809$	− 29.9525i	− 1.05307i	0.850152	−	0.526537i	$-0.176510\pi$
$810$	0	0
$811$	− 14.6461i	− 0.514293i	−0.966372	−	0.257146i	$-0.917218\pi$
$811$	− 14.6461i	− 0.514293i	0.966372	−	0.257146i	$-0.0827823\pi$
$812$	0	0
$813$	0	0
$814$	18.2733	0.640478
$815$	−4.01858	−0.140765
$816$	0	0
$817$	19.4731i	0.681277i
$818$	0.103005	0.00360150
$819$	0	0
$820$	−62.3768	−2.17829
$821$	− 19.7174i	− 0.688142i	−0.938944	−	0.344071i	$-0.888194\pi$
$821$	− 19.7174i	− 0.688142i	0.938944	−	0.344071i	$-0.111806\pi$
$822$	0	0
$823$	−11.5526	−0.402699	−0.201349	−	0.979519i	$-0.564533\pi$
$823$	−11.5526	−0.402699	−0.201349	+	0.979519i	$0.564533\pi$
$824$	−12.7296	−0.443455
$825$	0	0
$826$	0	0
$827$	4.65207i	0.161768i	0.996724	+	0.0808841i	$0.0257744\pi$
$827$	4.65207i	0.161768i	−0.996724	+	0.0808841i	$0.974226\pi$
$828$	0	0
$829$	9.41157i	0.326877i	0.986553	+	0.163439i	$0.0522586\pi$
$829$	9.41157i	0.326877i	−0.986553	+	0.163439i	$0.947741\pi$
$830$	− 8.98205i	− 0.311772i
$831$	0	0
$832$	− 3.30279i	− 0.114504i
$833$	0	0
$834$	0	0
$835$	37.2919	1.29054
$836$	−21.1115	−0.730155
$837$	0	0
$838$	1.47881i	0.0510846i
$839$	3.85859	0.133213	0.0666067	−	0.997779i	$-0.478783\pi$
$839$	3.85859	0.133213	0.0666067	+	0.997779i	$0.478783\pi$
$840$	0	0
$841$	−47.6851	−1.64431
$842$	11.2118i	0.386383i
$843$	0	0
$844$	−38.8168	−1.33613
$845$	44.7195	1.53840
$846$	0	0
$847$	0	0
$848$	19.7821i	0.679322i
$849$	0	0
$850$	− 12.2572i	− 0.420419i
$851$	10.0297i	0.343815i
$852$	0	0
$853$	31.6208i	1.08268i	0.840805	+	0.541338i	$0.182082\pi$
$853$	31.6208i	1.08268i	−0.840805	+	0.541338i	$0.817918\pi$
$854$	0	0
$855$	0	0
$856$	−31.3115	−1.07020
$857$	−34.3673	−1.17396	−0.586982	−	0.809600i	$-0.699684\pi$
$857$	−34.3673	−1.17396	−0.586982	+	0.809600i	$0.699684\pi$
$858$	0	0
$859$	− 3.52561i	− 0.120292i	−0.998190	−	0.0601462i	$-0.980843\pi$
$859$	− 3.52561i	− 0.120292i	0.998190	−	0.0601462i	$-0.0191567\pi$
$860$	−51.1515	−1.74425
$861$	0	0
$862$	11.2814	0.384246
$863$	− 35.7972i	− 1.21855i	−0.792959	−	0.609275i	$-0.791461\pi$
$863$	− 35.7972i	− 1.21855i	0.792959	−	0.609275i	$-0.208539\pi$
$864$	0	0
$865$	71.8528	2.44307
$866$	9.05881	0.307831
$867$	0	0
$868$	0	0
$869$	22.1840i	0.752542i
$870$	0	0
$871$	− 11.1530i	− 0.377906i
$872$	− 11.5952i	− 0.392664i
$873$	0	0
$874$	1.73248i	0.0586021i
$875$	0	0
$876$	0	0
$877$	40.2024	1.35754	0.678769	−	0.734352i	$-0.262514\pi$
$877$	40.2024	1.35754	0.678769	+	0.734352i	$0.262514\pi$
$878$	4.57370	0.154355
$879$	0	0
$880$	− 45.9242i	− 1.54811i
$881$	9.64863	0.325071	0.162535	−	0.986703i	$-0.448033\pi$
$881$	9.64863	0.325071	0.162535	+	0.986703i	$0.448033\pi$
$882$	0	0
$883$	17.1497	0.577134	0.288567	−	0.957460i	$-0.406821\pi$
$883$	17.1497	0.577134	0.288567	+	0.957460i	$0.406821\pi$
$884$	5.13096i	0.172573i
$885$	0	0
$886$	−8.41795	−0.282807
$887$	32.1275	1.07874	0.539368	−	0.842070i	$-0.318663\pi$
$887$	32.1275	1.07874	0.539368	+	0.842070i	$0.318663\pi$
$888$	0	0
$889$	0	0
$890$	3.91225i	0.131139i
$891$	0	0
$892$	11.1468i	0.373224i
$893$	23.8896i	0.799436i
$894$	0	0
$895$	11.4099i	0.381390i
$896$	0	0
$897$	0	0
$898$	4.55658	0.152055
$899$	76.4314	2.54913
$900$	0	0
$901$	− 17.0225i	− 0.567102i
$902$	−20.7535	−0.691016
$903$	0	0
$904$	22.9848	0.764462
$905$	− 31.9681i	− 1.06265i
$906$	0	0
$907$	−27.6267	−0.917330	−0.458665	−	0.888609i	$-0.651672\pi$
$907$	−27.6267	−0.917330	−0.458665	+	0.888609i	$0.651672\pi$
$908$	−5.37703	−0.178443
$909$	0	0
$910$	0	0
$911$	− 26.1620i	− 0.866785i	−0.901205	−	0.433392i	$-0.857316\pi$
$911$	− 26.1620i	− 0.866785i	0.901205	−	0.433392i	$-0.142684\pi$
$912$	0	0
$913$	19.9878i	0.661501i
$914$	13.6462i	0.451378i
$915$	0	0
$916$	− 23.6025i	− 0.779849i
$917$	0	0
$918$	0	0
$919$	−16.0437	−0.529232	−0.264616	−	0.964354i	$-0.585245\pi$
$919$	−16.0437	−0.529232	−0.264616	+	0.964354i	$0.585245\pi$
$920$	−9.78212	−0.322507
$921$	0	0
$922$	5.95167i	0.196008i
$923$	0.412314	0.0135715
$924$	0	0
$925$	87.5354	2.87815
$926$	− 9.12052i	− 0.299719i
$927$	0	0
$928$	44.6037	1.46419
$929$	−34.0763	−1.11801	−0.559003	−	0.829166i	$-0.688816\pi$
$929$	−34.0763	−1.11801	−0.559003	+	0.829166i	$0.688816\pi$
$930$	0	0
$931$	0	0
$932$	− 42.2901i	− 1.38526i
$933$	0	0
$934$	6.52640i	0.213550i
$935$	39.5177i	1.29237i
$936$	0	0
$937$	39.6263i	1.29454i	0.762263	+	0.647268i	$0.224088\pi$
$937$	39.6263i	1.29454i	−0.762263	+	0.647268i	$0.775912\pi$
$938$	0	0
$939$	0	0
$940$	−62.7528	−2.04677
$941$	−27.4892	−0.896124	−0.448062	−	0.894003i	$-0.647886\pi$
$941$	−27.4892	−0.896124	−0.448062	+	0.894003i	$0.647886\pi$
$942$	0	0
$943$	− 11.3911i	− 0.370944i
$944$	−19.3736	−0.630558
$945$	0	0
$946$	−17.0187	−0.553326
$947$	− 12.9435i	− 0.420606i	−0.977636	−	0.210303i	$-0.932555\pi$
$947$	− 12.9435i	− 0.420606i	0.977636	−	0.210303i	$-0.0674451\pi$
$948$	0	0
$949$	15.9208	0.516812
$950$	15.1204	0.490570
$951$	0	0
$952$	0	0
$953$	30.0287i	0.972726i	0.873757	+	0.486363i	$0.161677\pi$
$953$	30.0287i	0.972726i	−0.873757	+	0.486363i	$0.838323\pi$
$954$	0	0
$955$	− 55.8932i	− 1.80866i
$956$	− 6.23238i	− 0.201570i
$957$	0	0
$958$	1.03736i	0.0335155i
$959$	0	0
$960$	0	0
$961$	−45.1786	−1.45737
$962$	5.47858	0.176637
$963$	0	0
$964$	12.2086i	0.393212i
$965$	32.3488	1.04134
$966$	0	0
$967$	33.1357	1.06557	0.532787	−	0.846250i	$-0.321145\pi$
$967$	33.1357	1.06557	0.532787	+	0.846250i	$0.321145\pi$
$968$	− 18.6778i	− 0.600326i
$969$	0	0
$970$	−0.474190	−0.0152253
$971$	−34.4484	−1.10550	−0.552752	−	0.833346i	$-0.686422\pi$
$971$	−34.4484	−1.10550	−0.552752	+	0.833346i	$0.686422\pi$
$972$	0	0
$973$	0	0
$974$	7.17177i	0.229798i
$975$	0	0
$976$	15.1013i	0.483380i
$977$	52.6834i	1.68549i	0.538311	+	0.842746i	$0.319062\pi$
$977$	52.6834i	1.68549i	−0.538311	+	0.842746i	$0.680938\pi$
$978$	0	0
$979$	− 8.70596i	− 0.278244i
$980$	0	0
$981$	0	0
$982$	16.4224	0.524060
$983$	35.5076	1.13252	0.566259	−	0.824228i	$-0.308390\pi$
$983$	35.5076	1.13252	0.566259	+	0.824228i	$0.308390\pi$
$984$	0	0
$985$	− 28.1266i	− 0.896188i
$986$	−9.63464	−0.306829
$987$	0	0
$988$	−6.32950	−0.201368
$989$	− 9.34114i	− 0.297031i
$990$	0	0
$991$	−8.36244	−0.265642	−0.132821	−	0.991140i	$-0.542404\pi$
$991$	−8.36244	−0.265642	−0.132821	+	0.991140i	$0.542404\pi$
$992$	−44.4561	−1.41148
$993$	0	0
$994$	0	0
$995$	− 59.1369i	− 1.87477i
$996$	0	0
$997$	− 39.8996i	− 1.26363i	−0.775118	−	0.631816i	$-0.782310\pi$
$997$	− 39.8996i	− 1.26363i	0.775118	−	0.631816i	$-0.217690\pi$
$998$	8.69226i	0.275149i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.2.c.e.1322.10	yes	16
3.2	odd	2	inner	1323.2.c.e.1322.7	✓	16
7.6	odd	2	inner	1323.2.c.e.1322.9	yes	16
21.20	even	2	inner	1323.2.c.e.1322.8	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1323.2.c.e.1322.7	✓	16	3.2	odd	2	inner
1323.2.c.e.1322.8	yes	16	21.20	even	2	inner
1323.2.c.e.1322.9	yes	16	7.6	odd	2	inner
1323.2.c.e.1322.10	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)

\(f(q)\)	\(=\)	\(q+0.510032i q^{2} +1.73987 q^{4} +4.01755 q^{5} +1.90745i q^{8} +O(q^{10})\)	\(q+0.510032i q^{2} +1.73987 q^{4} +4.01755 q^{5} +1.90745i q^{8} +2.04908i q^{10} -4.55982i q^{11} -1.36710i q^{13} +2.50688 q^{16} -2.15716 q^{17} -2.66105i q^{19} +6.99000 q^{20} +2.32565 q^{22} +1.27649i q^{23} +11.1407 q^{25} +0.697263 q^{26} -8.75700i q^{29} +8.72803i q^{31} +5.09349i q^{32} -1.10022i q^{34} +7.85727 q^{37} +1.35722 q^{38} +7.66327i q^{40} -8.92372 q^{41} -7.31781 q^{43} -7.93349i q^{44} -0.651052 q^{46} -8.97751 q^{47} +5.68210i q^{50} -2.37857i q^{52} +7.89115i q^{53} -18.3193i q^{55} +4.46635 q^{58} -7.72819 q^{59} +6.02394i q^{61} -4.45157 q^{62} +2.41591 q^{64} -5.49238i q^{65} +8.15818 q^{67} -3.75318 q^{68} +0.301598i q^{71} +11.6457i q^{73} +4.00745i q^{74} -4.62988i q^{76} -4.86511 q^{79} +10.0715 q^{80} -4.55138i q^{82} -4.38346 q^{83} -8.66650 q^{85} -3.73231i q^{86} +8.69764 q^{88} +1.90928 q^{89} +2.22093i q^{92} -4.57881i q^{94} -10.6909i q^{95} +0.231417i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4}+O(q^{10}) \) 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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