LMFDB - Embedded newform 1323.2.c.c.1322.3

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:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 189)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1322.3
Root			$-1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1323.1322
Dual form			1323.2.c.c.1322.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.44949 q^{5} +2.82843i q^{8} +O(q^{10})$	$q+1.41421i q^{2} -2.44949 q^{5} +2.82843i q^{8} -3.46410i q^{10} +5.65685i q^{11} -3.46410i q^{13} -4.00000 q^{16} -2.44949 q^{17} +1.73205i q^{19} -8.00000 q^{22} -7.07107i q^{23} +1.00000 q^{25} +4.89898 q^{26} +1.41421i q^{29} +5.19615i q^{31} -3.46410i q^{34} -4.00000 q^{37} -2.44949 q^{38} -6.92820i q^{40} +2.44949 q^{41} -7.00000 q^{43} +10.0000 q^{46} -7.34847 q^{47} +1.41421i q^{50} +1.41421i q^{53} -13.8564i q^{55} -2.00000 q^{58} -14.6969 q^{59} -5.19615i q^{61} -7.34847 q^{62} -8.00000 q^{64} +8.48528i q^{65} +2.00000 q^{67} -2.82843i q^{71} -12.1244i q^{73} -5.65685i q^{74} -4.00000 q^{79} +9.79796 q^{80} +3.46410i q^{82} -9.79796 q^{83} +6.00000 q^{85} -9.89949i q^{86} -16.0000 q^{88} +2.44949 q^{89} -10.3923i q^{94} -4.24264i q^{95} +8.66025i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 16 q^{16} - 32 q^{22} + 4 q^{25} - 16 q^{37} - 28 q^{43} + 40 q^{46} - 8 q^{58} - 32 q^{64} + 8 q^{67} - 16 q^{79} + 24 q^{85} - 64 q^{88}+O(q^{100}) $ 4 * q - 16 * q^16 - 32 * q^22 + 4 * q^25 - 16 * q^37 - 28 * q^43 + 40 * q^46 - 8 * q^58 - 32 * q^64 + 8 * q^67 - 16 * q^79 + 24 * q^85 - 64 * 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 $

$2$	1.41421i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$2$	1.41421i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−2.44949	−1.09545	−0.547723	−	0.836660i	$-0.684505\pi$
$5$	−2.44949	−1.09545	−0.547723	+	0.836660i	$0.684505\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.82843i	1.00000i
$9$	0	0
$10$	− 3.46410i	− 1.09545i
$11$	5.65685i	1.70561i	0.522233	+	0.852803i	$0.325099\pi$
$11$	5.65685i	1.70561i	−0.522233	+	0.852803i	$0.674901\pi$
$12$	0	0
$13$	− 3.46410i	− 0.960769i	−0.877058	−	0.480384i	$-0.840497\pi$
$13$	− 3.46410i	− 0.960769i	0.877058	−	0.480384i	$-0.159503\pi$
$14$	0	0
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−2.44949	−0.594089	−0.297044	−	0.954864i	$-0.596001\pi$
$17$	−2.44949	−0.594089	−0.297044	+	0.954864i	$0.596001\pi$
$18$	0	0
$19$	1.73205i	0.397360i	0.980064	+	0.198680i	$0.0636654\pi$
$19$	1.73205i	0.397360i	−0.980064	+	0.198680i	$0.936335\pi$
$20$	0	0
$21$	0	0
$22$	−8.00000	−1.70561
$23$	− 7.07107i	− 1.47442i	−0.675664	−	0.737210i	$-0.736143\pi$
$23$	− 7.07107i	− 1.47442i	0.675664	−	0.737210i	$-0.263857\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	4.89898	0.960769
$27$	0	0
$28$	0	0
$29$	1.41421i	0.262613i	0.991342	+	0.131306i	$0.0419172\pi$
$29$	1.41421i	0.262613i	−0.991342	+	0.131306i	$0.958083\pi$
$30$	0	0
$31$	5.19615i	0.933257i	0.884454	+	0.466628i	$0.154531\pi$
$31$	5.19615i	0.933257i	−0.884454	+	0.466628i	$0.845469\pi$
$32$	0	0
$33$	0	0
$34$	− 3.46410i	− 0.594089i
$35$	0	0
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	−2.44949	−0.397360
$39$	0	0
$40$	− 6.92820i	− 1.09545i
$41$	2.44949	0.382546	0.191273	−	0.981537i	$-0.438738\pi$
$41$	2.44949	0.382546	0.191273	+	0.981537i	$0.438738\pi$
$42$	0	0
$43$	−7.00000	−1.06749	−0.533745	−	0.845645i	$-0.679216\pi$
$43$	−7.00000	−1.06749	−0.533745	+	0.845645i	$0.679216\pi$
$44$	0	0
$45$	0	0
$46$	10.0000	1.47442
$47$	−7.34847	−1.07188	−0.535942	−	0.844255i	$-0.680044\pi$
$47$	−7.34847	−1.07188	−0.535942	+	0.844255i	$0.680044\pi$
$48$	0	0
$49$	0	0
$50$	1.41421i	0.200000i
$51$	0	0
$52$	0	0
$53$	1.41421i	0.194257i	0.995272	+	0.0971286i	$0.0309658\pi$
$53$	1.41421i	0.194257i	−0.995272	+	0.0971286i	$0.969034\pi$
$54$	0	0
$55$	− 13.8564i	− 1.86840i
$56$	0	0
$57$	0	0
$58$	−2.00000	−0.262613
$59$	−14.6969	−1.91338	−0.956689	−	0.291111i	$-0.905975\pi$
$59$	−14.6969	−1.91338	−0.956689	+	0.291111i	$0.905975\pi$
$60$	0	0
$61$	− 5.19615i	− 0.665299i	−0.943051	−	0.332650i	$-0.892057\pi$
$61$	− 5.19615i	− 0.665299i	0.943051	−	0.332650i	$-0.107943\pi$
$62$	−7.34847	−0.933257
$63$	0	0
$64$	−8.00000	−1.00000
$65$	8.48528i	1.05247i
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 2.82843i	− 0.335673i	−0.985815	−	0.167836i	$-0.946322\pi$
$71$	− 2.82843i	− 0.335673i	0.985815	−	0.167836i	$-0.0536780\pi$
$72$	0	0
$73$	− 12.1244i	− 1.41905i	−0.704681	−	0.709524i	$-0.748910\pi$
$73$	− 12.1244i	− 1.41905i	0.704681	−	0.709524i	$-0.251090\pi$
$74$	− 5.65685i	− 0.657596i
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	9.79796	1.09545
$81$	0	0
$82$	3.46410i	0.382546i
$83$	−9.79796	−1.07547	−0.537733	−	0.843115i	$-0.680719\pi$
$83$	−9.79796	−1.07547	−0.537733	+	0.843115i	$0.680719\pi$
$84$	0	0
$85$	6.00000	0.650791
$86$	− 9.89949i	− 1.06749i
$87$	0	0
$88$	−16.0000	−1.70561
$89$	2.44949	0.259645	0.129823	−	0.991537i	$-0.458559\pi$
$89$	2.44949	0.259645	0.129823	+	0.991537i	$0.458559\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	− 10.3923i	− 1.07188i
$95$	− 4.24264i	− 0.435286i
$96$	0	0
$97$	8.66025i	0.879316i	0.898165	+	0.439658i	$0.144900\pi$
$97$	8.66025i	0.879316i	−0.898165	+	0.439658i	$0.855100\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	9.79796	0.974933	0.487467	−	0.873142i	$-0.337921\pi$
$101$	9.79796	0.974933	0.487467	+	0.873142i	$0.337921\pi$
$102$	0	0
$103$	13.8564i	1.36531i	0.730740	+	0.682656i	$0.239175\pi$
$103$	13.8564i	1.36531i	−0.730740	+	0.682656i	$0.760825\pi$
$104$	9.79796	0.960769
$105$	0	0
$106$	−2.00000	−0.194257
$107$	14.1421i	1.36717i	0.729870	+	0.683586i	$0.239581\pi$
$107$	14.1421i	1.36717i	−0.729870	+	0.683586i	$0.760419\pi$
$108$	0	0
$109$	17.0000	1.62830	0.814152	−	0.580651i	$-0.197202\pi$
$109$	17.0000	1.62830	0.814152	+	0.580651i	$0.197202\pi$
$110$	19.5959	1.86840
$111$	0	0
$112$	0	0
$113$	− 2.82843i	− 0.266076i	−0.991111	−	0.133038i	$-0.957527\pi$
$113$	− 2.82843i	− 0.266076i	0.991111	−	0.133038i	$-0.0424732\pi$
$114$	0	0
$115$	17.3205i	1.61515i
$116$	0	0
$117$	0	0
$118$	− 20.7846i	− 1.91338i
$119$	0	0
$120$	0	0
$121$	−21.0000	−1.90909
$122$	7.34847	0.665299
$123$	0	0
$124$	0	0
$125$	9.79796	0.876356
$126$	0	0
$127$	−1.00000	−0.0887357	−0.0443678	−	0.999015i	$-0.514127\pi$
$127$	−1.00000	−0.0887357	−0.0443678	+	0.999015i	$0.514127\pi$
$128$	− 11.3137i	− 1.00000i
$129$	0	0
$130$	−12.0000	−1.05247
$131$	−17.1464	−1.49809	−0.749045	−	0.662519i	$-0.769487\pi$
$131$	−17.1464	−1.49809	−0.749045	+	0.662519i	$0.769487\pi$
$132$	0	0
$133$	0	0
$134$	2.82843i	0.244339i
$135$	0	0
$136$	− 6.92820i	− 0.594089i
$137$	− 7.07107i	− 0.604122i	−0.953289	−	0.302061i	$-0.902325\pi$
$137$	− 7.07107i	− 0.604122i	0.953289	−	0.302061i	$-0.0976746\pi$
$138$	0	0
$139$	6.92820i	0.587643i	0.955860	+	0.293821i	$0.0949270\pi$
$139$	6.92820i	0.587643i	−0.955860	+	0.293821i	$0.905073\pi$
$140$	0	0
$141$	0	0
$142$	4.00000	0.335673
$143$	19.5959	1.63869
$144$	0	0
$145$	− 3.46410i	− 0.287678i
$146$	17.1464	1.41905
$147$	0	0
$148$	0	0
$149$	9.89949i	0.810998i	0.914095	+	0.405499i	$0.132902\pi$
$149$	9.89949i	0.810998i	−0.914095	+	0.405499i	$0.867098\pi$
$150$	0	0
$151$	11.0000	0.895167	0.447584	−	0.894242i	$-0.352285\pi$
$151$	11.0000	0.895167	0.447584	+	0.894242i	$0.352285\pi$
$152$	−4.89898	−0.397360
$153$	0	0
$154$	0	0
$155$	− 12.7279i	− 1.02233i
$156$	0	0
$157$	10.3923i	0.829396i	0.909959	+	0.414698i	$0.136113\pi$
$157$	10.3923i	0.829396i	−0.909959	+	0.414698i	$0.863887\pi$
$158$	− 5.65685i	− 0.450035i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	23.0000	1.80150	0.900750	−	0.434339i	$-0.143018\pi$
$163$	23.0000	1.80150	0.900750	+	0.434339i	$0.143018\pi$
$164$	0	0
$165$	0	0
$166$	− 13.8564i	− 1.07547i
$167$	−4.89898	−0.379094	−0.189547	−	0.981872i	$-0.560702\pi$
$167$	−4.89898	−0.379094	−0.189547	+	0.981872i	$0.560702\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	8.48528i	0.650791i
$171$	0	0
$172$	0	0
$173$	7.34847	0.558694	0.279347	−	0.960190i	$-0.409882\pi$
$173$	7.34847	0.558694	0.279347	+	0.960190i	$0.409882\pi$
$174$	0	0
$175$	0	0
$176$	− 22.6274i	− 1.70561i
$177$	0	0
$178$	3.46410i	0.259645i
$179$	9.89949i	0.739923i	0.929047	+	0.369961i	$0.120629\pi$
$179$	9.89949i	0.739923i	−0.929047	+	0.369961i	$0.879371\pi$
$180$	0	0
$181$	15.5885i	1.15868i	0.815086	+	0.579340i	$0.196690\pi$
$181$	15.5885i	1.15868i	−0.815086	+	0.579340i	$0.803310\pi$
$182$	0	0
$183$	0	0
$184$	20.0000	1.47442
$185$	9.79796	0.720360
$186$	0	0
$187$	− 13.8564i	− 1.01328i
$188$	0	0
$189$	0	0
$190$	6.00000	0.435286
$191$	1.41421i	0.102329i	0.998690	+	0.0511645i	$0.0162933\pi$
$191$	1.41421i	0.102329i	−0.998690	+	0.0511645i	$0.983707\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	−12.2474	−0.879316
$195$	0	0
$196$	0	0
$197$	− 2.82843i	− 0.201517i	−0.994911	−	0.100759i	$-0.967873\pi$
$197$	− 2.82843i	− 0.201517i	0.994911	−	0.100759i	$-0.0321270\pi$
$198$	0	0
$199$	− 1.73205i	− 0.122782i	−0.998114	−	0.0613909i	$-0.980446\pi$
$199$	− 1.73205i	− 0.122782i	0.998114	−	0.0613909i	$-0.0195536\pi$
$200$	2.82843i	0.200000i
$201$	0	0
$202$	13.8564i	0.974933i
$203$	0	0
$204$	0	0
$205$	−6.00000	−0.419058
$206$	−19.5959	−1.36531
$207$	0	0
$208$	13.8564i	0.960769i
$209$	−9.79796	−0.677739
$210$	0	0
$211$	−1.00000	−0.0688428	−0.0344214	−	0.999407i	$-0.510959\pi$
$211$	−1.00000	−0.0688428	−0.0344214	+	0.999407i	$0.510959\pi$
$212$	0	0
$213$	0	0
$214$	−20.0000	−1.36717
$215$	17.1464	1.16938
$216$	0	0
$217$	0	0
$218$	24.0416i	1.62830i
$219$	0	0
$220$	0	0
$221$	8.48528i	0.570782i
$222$	0	0
$223$	13.8564i	0.927894i	0.885863	+	0.463947i	$0.153567\pi$
$223$	13.8564i	0.927894i	−0.885863	+	0.463947i	$0.846433\pi$
$224$	0	0
$225$	0	0
$226$	4.00000	0.266076
$227$	2.44949	0.162578	0.0812892	−	0.996691i	$-0.474096\pi$
$227$	2.44949	0.162578	0.0812892	+	0.996691i	$0.474096\pi$
$228$	0	0
$229$	− 1.73205i	− 0.114457i	−0.998361	−	0.0572286i	$-0.981774\pi$
$229$	− 1.73205i	− 0.114457i	0.998361	−	0.0572286i	$-0.0182264\pi$
$230$	−24.4949	−1.61515
$231$	0	0
$232$	−4.00000	−0.262613
$233$	− 15.5563i	− 1.01913i	−0.860432	−	0.509565i	$-0.829806\pi$
$233$	− 15.5563i	− 1.01913i	0.860432	−	0.509565i	$-0.170194\pi$
$234$	0	0
$235$	18.0000	1.17419
$236$	0	0
$237$	0	0
$238$	0	0
$239$	9.89949i	0.640345i	0.947359	+	0.320173i	$0.103741\pi$
$239$	9.89949i	0.640345i	−0.947359	+	0.320173i	$0.896259\pi$
$240$	0	0
$241$	12.1244i	0.780998i	0.920603	+	0.390499i	$0.127698\pi$
$241$	12.1244i	0.780998i	−0.920603	+	0.390499i	$0.872302\pi$
$242$	− 29.6985i	− 1.90909i
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.00000	0.381771
$248$	−14.6969	−0.933257
$249$	0	0
$250$	13.8564i	0.876356i
$251$	−7.34847	−0.463831	−0.231916	−	0.972736i	$-0.574499\pi$
$251$	−7.34847	−0.463831	−0.231916	+	0.972736i	$0.574499\pi$
$252$	0	0
$253$	40.0000	2.51478
$254$	− 1.41421i	− 0.0887357i
$255$	0	0
$256$	0	0
$257$	19.5959	1.22236	0.611180	−	0.791492i	$-0.290695\pi$
$257$	19.5959	1.22236	0.611180	+	0.791492i	$0.290695\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	− 24.2487i	− 1.49809i
$263$	− 19.7990i	− 1.22086i	−0.792071	−	0.610429i	$-0.790997\pi$
$263$	− 19.7990i	− 1.22086i	0.792071	−	0.610429i	$-0.209003\pi$
$264$	0	0
$265$	− 3.46410i	− 0.212798i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−24.4949	−1.49348	−0.746740	−	0.665116i	$-0.768382\pi$
$269$	−24.4949	−1.49348	−0.746740	+	0.665116i	$0.768382\pi$
$270$	0	0
$271$	22.5167i	1.36779i	0.729581	+	0.683895i	$0.239715\pi$
$271$	22.5167i	1.36779i	−0.729581	+	0.683895i	$0.760285\pi$
$272$	9.79796	0.594089
$273$	0	0
$274$	10.0000	0.604122
$275$	5.65685i	0.341121i
$276$	0	0
$277$	−25.0000	−1.50210	−0.751052	−	0.660243i	$-0.770453\pi$
$277$	−25.0000	−1.50210	−0.751052	+	0.660243i	$0.770453\pi$
$278$	−9.79796	−0.587643
$279$	0	0
$280$	0	0
$281$	5.65685i	0.337460i	0.985662	+	0.168730i	$0.0539665\pi$
$281$	5.65685i	0.337460i	−0.985662	+	0.168730i	$0.946033\pi$
$282$	0	0
$283$	25.9808i	1.54440i	0.635382	+	0.772198i	$0.280843\pi$
$283$	25.9808i	1.54440i	−0.635382	+	0.772198i	$0.719157\pi$
$284$	0	0
$285$	0	0
$286$	27.7128i	1.63869i
$287$	0	0
$288$	0	0
$289$	−11.0000	−0.647059
$290$	4.89898	0.287678
$291$	0	0
$292$	0	0
$293$	24.4949	1.43101	0.715504	−	0.698609i	$-0.246197\pi$
$293$	24.4949	1.43101	0.715504	+	0.698609i	$0.246197\pi$
$294$	0	0
$295$	36.0000	2.09600
$296$	− 11.3137i	− 0.657596i
$297$	0	0
$298$	−14.0000	−0.810998
$299$	−24.4949	−1.41658
$300$	0	0
$301$	0	0
$302$	15.5563i	0.895167i
$303$	0	0
$304$	− 6.92820i	− 0.397360i
$305$	12.7279i	0.728799i
$306$	0	0
$307$	− 5.19615i	− 0.296560i	−0.988945	−	0.148280i	$-0.952626\pi$
$307$	− 5.19615i	− 0.296560i	0.988945	−	0.148280i	$-0.0473737\pi$
$308$	0	0
$309$	0	0
$310$	18.0000	1.02233
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	5.19615i	0.293704i	0.989158	+	0.146852i	$0.0469141\pi$
$313$	5.19615i	0.293704i	−0.989158	+	0.146852i	$0.953086\pi$
$314$	−14.6969	−0.829396
$315$	0	0
$316$	0	0
$317$	− 2.82843i	− 0.158860i	−0.996840	−	0.0794301i	$-0.974690\pi$
$317$	− 2.82843i	− 0.158860i	0.996840	−	0.0794301i	$-0.0253101\pi$
$318$	0	0
$319$	−8.00000	−0.447914
$320$	19.5959	1.09545
$321$	0	0
$322$	0	0
$323$	− 4.24264i	− 0.236067i
$324$	0	0
$325$	− 3.46410i	− 0.192154i
$326$	32.5269i	1.80150i
$327$	0	0
$328$	6.92820i	0.382546i
$329$	0	0
$330$	0	0
$331$	17.0000	0.934405	0.467202	−	0.884150i	$-0.345262\pi$
$331$	17.0000	0.934405	0.467202	+	0.884150i	$0.345262\pi$
$332$	0	0
$333$	0	0
$334$	− 6.92820i	− 0.379094i
$335$	−4.89898	−0.267660
$336$	0	0
$337$	−1.00000	−0.0544735	−0.0272367	−	0.999629i	$-0.508671\pi$
$337$	−1.00000	−0.0544735	−0.0272367	+	0.999629i	$0.508671\pi$
$338$	1.41421i	0.0769231i
$339$	0	0
$340$	0	0
$341$	−29.3939	−1.59177
$342$	0	0
$343$	0	0
$344$	− 19.7990i	− 1.06749i
$345$	0	0
$346$	10.3923i	0.558694i
$347$	31.1127i	1.67022i	0.550085	+	0.835109i	$0.314595\pi$
$347$	31.1127i	1.67022i	−0.550085	+	0.835109i	$0.685405\pi$
$348$	0	0
$349$	− 1.73205i	− 0.0927146i	−0.998925	−	0.0463573i	$-0.985239\pi$
$349$	− 1.73205i	− 0.0927146i	0.998925	−	0.0463573i	$-0.0147613\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−34.2929	−1.82522	−0.912612	−	0.408826i	$-0.865938\pi$
$353$	−34.2929	−1.82522	−0.912612	+	0.408826i	$0.865938\pi$
$354$	0	0
$355$	6.92820i	0.367711i
$356$	0	0
$357$	0	0
$358$	−14.0000	−0.739923
$359$	− 11.3137i	− 0.597115i	−0.954392	−	0.298557i	$-0.903495\pi$
$359$	− 11.3137i	− 0.597115i	0.954392	−	0.298557i	$-0.0965054\pi$
$360$	0	0
$361$	16.0000	0.842105
$362$	−22.0454	−1.15868
$363$	0	0
$364$	0	0
$365$	29.6985i	1.55449i
$366$	0	0
$367$	− 8.66025i	− 0.452062i	−0.974120	−	0.226031i	$-0.927425\pi$
$367$	− 8.66025i	− 0.452062i	0.974120	−	0.226031i	$-0.0725750\pi$
$368$	28.2843i	1.47442i
$369$	0	0
$370$	13.8564i	0.720360i
$371$	0	0
$372$	0	0
$373$	−1.00000	−0.0517780	−0.0258890	−	0.999665i	$-0.508242\pi$
$373$	−1.00000	−0.0517780	−0.0258890	+	0.999665i	$0.508242\pi$
$374$	19.5959	1.01328
$375$	0	0
$376$	− 20.7846i	− 1.07188i
$377$	4.89898	0.252310
$378$	0	0
$379$	−10.0000	−0.513665	−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000	−0.513665	−0.256833	+	0.966456i	$0.582679\pi$
$380$	0	0
$381$	0	0
$382$	−2.00000	−0.102329
$383$	34.2929	1.75228	0.876142	−	0.482054i	$-0.160109\pi$
$383$	34.2929	1.75228	0.876142	+	0.482054i	$0.160109\pi$
$384$	0	0
$385$	0	0
$386$	− 5.65685i	− 0.287926i
$387$	0	0
$388$	0	0
$389$	26.8701i	1.36237i	0.732113	+	0.681183i	$0.238534\pi$
$389$	26.8701i	1.36237i	−0.732113	+	0.681183i	$0.761466\pi$
$390$	0	0
$391$	17.3205i	0.875936i
$392$	0	0
$393$	0	0
$394$	4.00000	0.201517
$395$	9.79796	0.492989
$396$	0	0
$397$	− 19.0526i	− 0.956221i	−0.878300	−	0.478110i	$-0.841322\pi$
$397$	− 19.0526i	− 0.956221i	0.878300	−	0.478110i	$-0.158678\pi$
$398$	2.44949	0.122782
$399$	0	0
$400$	−4.00000	−0.200000
$401$	22.6274i	1.12996i	0.825105	+	0.564980i	$0.191116\pi$
$401$	22.6274i	1.12996i	−0.825105	+	0.564980i	$0.808884\pi$
$402$	0	0
$403$	18.0000	0.896644
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 22.6274i	− 1.12160i
$408$	0	0
$409$	10.3923i	0.513866i	0.966429	+	0.256933i	$0.0827120\pi$
$409$	10.3923i	0.513866i	−0.966429	+	0.256933i	$0.917288\pi$
$410$	− 8.48528i	− 0.419058i
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	24.0000	1.17811
$416$	0	0
$417$	0	0
$418$	− 13.8564i	− 0.677739i
$419$	2.44949	0.119665	0.0598327	−	0.998208i	$-0.480943\pi$
$419$	2.44949	0.119665	0.0598327	+	0.998208i	$0.480943\pi$
$420$	0	0
$421$	11.0000	0.536107	0.268054	−	0.963404i	$-0.413620\pi$
$421$	11.0000	0.536107	0.268054	+	0.963404i	$0.413620\pi$
$422$	− 1.41421i	− 0.0688428i
$423$	0	0
$424$	−4.00000	−0.194257
$425$	−2.44949	−0.118818
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	24.2487i	1.16938i
$431$	− 28.2843i	− 1.36241i	−0.732095	−	0.681203i	$-0.761457\pi$
$431$	− 28.2843i	− 1.36241i	0.732095	−	0.681203i	$-0.238543\pi$
$432$	0	0
$433$	− 5.19615i	− 0.249711i	−0.992175	−	0.124856i	$-0.960153\pi$
$433$	− 5.19615i	− 0.249711i	0.992175	−	0.124856i	$-0.0398468\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	12.2474	0.585875
$438$	0	0
$439$	20.7846i	0.991995i	0.868324	+	0.495998i	$0.165198\pi$
$439$	20.7846i	0.991995i	−0.868324	+	0.495998i	$0.834802\pi$
$440$	39.1918	1.86840
$441$	0	0
$442$	−12.0000	−0.570782
$443$	− 15.5563i	− 0.739104i	−0.929210	−	0.369552i	$-0.879511\pi$
$443$	− 15.5563i	− 0.739104i	0.929210	−	0.369552i	$-0.120489\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	−19.5959	−0.927894
$447$	0	0
$448$	0	0
$449$	35.3553i	1.66852i	0.551370	+	0.834261i	$0.314105\pi$
$449$	35.3553i	1.66852i	−0.551370	+	0.834261i	$0.685895\pi$
$450$	0	0
$451$	13.8564i	0.652473i
$452$	0	0
$453$	0	0
$454$	3.46410i	0.162578i
$455$	0	0
$456$	0	0
$457$	5.00000	0.233890	0.116945	−	0.993138i	$-0.462690\pi$
$457$	5.00000	0.233890	0.116945	+	0.993138i	$0.462690\pi$
$458$	2.44949	0.114457
$459$	0	0
$460$	0	0
$461$	−24.4949	−1.14084	−0.570421	−	0.821353i	$-0.693220\pi$
$461$	−24.4949	−1.14084	−0.570421	+	0.821353i	$0.693220\pi$
$462$	0	0
$463$	−1.00000	−0.0464739	−0.0232370	−	0.999730i	$-0.507397\pi$
$463$	−1.00000	−0.0464739	−0.0232370	+	0.999730i	$0.507397\pi$
$464$	− 5.65685i	− 0.262613i
$465$	0	0
$466$	22.0000	1.01913
$467$	−4.89898	−0.226698	−0.113349	−	0.993555i	$-0.536158\pi$
$467$	−4.89898	−0.226698	−0.113349	+	0.993555i	$0.536158\pi$
$468$	0	0
$469$	0	0
$470$	25.4558i	1.17419i
$471$	0	0
$472$	− 41.5692i	− 1.91338i
$473$	− 39.5980i	− 1.82072i
$474$	0	0
$475$	1.73205i	0.0794719i
$476$	0	0
$477$	0	0
$478$	−14.0000	−0.640345
$479$	2.44949	0.111920	0.0559600	−	0.998433i	$-0.482178\pi$
$479$	2.44949	0.111920	0.0559600	+	0.998433i	$0.482178\pi$
$480$	0	0
$481$	13.8564i	0.631798i
$482$	−17.1464	−0.780998
$483$	0	0
$484$	0	0
$485$	− 21.2132i	− 0.963242i
$486$	0	0
$487$	−13.0000	−0.589086	−0.294543	−	0.955638i	$-0.595167\pi$
$487$	−13.0000	−0.589086	−0.294543	+	0.955638i	$0.595167\pi$
$488$	14.6969	0.665299
$489$	0	0
$490$	0	0
$491$	5.65685i	0.255290i	0.991820	+	0.127645i	$0.0407419\pi$
$491$	5.65685i	0.255290i	−0.991820	+	0.127645i	$0.959258\pi$
$492$	0	0
$493$	− 3.46410i	− 0.156015i
$494$	8.48528i	0.381771i
$495$	0	0
$496$	− 20.7846i	− 0.933257i
$497$	0	0
$498$	0	0
$499$	−25.0000	−1.11915	−0.559577	−	0.828778i	$-0.689036\pi$
$499$	−25.0000	−1.11915	−0.559577	+	0.828778i	$0.689036\pi$
$500$	0	0
$501$	0	0
$502$	− 10.3923i	− 0.463831i
$503$	22.0454	0.982956	0.491478	−	0.870890i	$-0.336457\pi$
$503$	22.0454	0.982956	0.491478	+	0.870890i	$0.336457\pi$
$504$	0	0
$505$	−24.0000	−1.06799
$506$	56.5685i	2.51478i
$507$	0	0
$508$	0	0
$509$	19.5959	0.868574	0.434287	−	0.900775i	$-0.357000\pi$
$509$	19.5959	0.868574	0.434287	+	0.900775i	$0.357000\pi$
$510$	0	0
$511$	0	0
$512$	− 22.6274i	− 1.00000i
$513$	0	0
$514$	27.7128i	1.22236i
$515$	− 33.9411i	− 1.49562i
$516$	0	0
$517$	− 41.5692i	− 1.82821i
$518$	0	0
$519$	0	0
$520$	−24.0000	−1.05247
$521$	−24.4949	−1.07314	−0.536570	−	0.843856i	$-0.680280\pi$
$521$	−24.4949	−1.07314	−0.536570	+	0.843856i	$0.680280\pi$
$522$	0	0
$523$	− 3.46410i	− 0.151475i	−0.997128	−	0.0757373i	$-0.975869\pi$
$523$	− 3.46410i	− 0.151475i	0.997128	−	0.0757373i	$-0.0241310\pi$
$524$	0	0
$525$	0	0
$526$	28.0000	1.22086
$527$	− 12.7279i	− 0.554437i
$528$	0	0
$529$	−27.0000	−1.17391
$530$	4.89898	0.212798
$531$	0	0
$532$	0	0
$533$	− 8.48528i	− 0.367538i
$534$	0	0
$535$	− 34.6410i	− 1.49766i
$536$	5.65685i	0.244339i
$537$	0	0
$538$	− 34.6410i	− 1.49348i
$539$	0	0
$540$	0	0
$541$	−13.0000	−0.558914	−0.279457	−	0.960158i	$-0.590154\pi$
$541$	−13.0000	−0.558914	−0.279457	+	0.960158i	$0.590154\pi$
$542$	−31.8434	−1.36779
$543$	0	0
$544$	0	0
$545$	−41.6413	−1.78372
$546$	0	0
$547$	−25.0000	−1.06892	−0.534461	−	0.845193i	$-0.679486\pi$
$547$	−25.0000	−1.06892	−0.534461	+	0.845193i	$0.679486\pi$
$548$	0	0
$549$	0	0
$550$	−8.00000	−0.341121
$551$	−2.44949	−0.104352
$552$	0	0
$553$	0	0
$554$	− 35.3553i	− 1.50210i
$555$	0	0
$556$	0	0
$557$	1.41421i	0.0599222i	0.999551	+	0.0299611i	$0.00953833\pi$
$557$	1.41421i	0.0599222i	−0.999551	+	0.0299611i	$0.990462\pi$
$558$	0	0
$559$	24.2487i	1.02561i
$560$	0	0
$561$	0	0
$562$	−8.00000	−0.337460
$563$	22.0454	0.929103	0.464552	−	0.885546i	$-0.346216\pi$
$563$	22.0454	0.929103	0.464552	+	0.885546i	$0.346216\pi$
$564$	0	0
$565$	6.92820i	0.291472i
$566$	−36.7423	−1.54440
$567$	0	0
$568$	8.00000	0.335673
$569$	− 36.7696i	− 1.54146i	−0.637162	−	0.770730i	$-0.719892\pi$
$569$	− 36.7696i	− 1.54146i	0.637162	−	0.770730i	$-0.280108\pi$
$570$	0	0
$571$	41.0000	1.71580	0.857898	−	0.513820i	$-0.171770\pi$
$571$	41.0000	1.71580	0.857898	+	0.513820i	$0.171770\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 7.07107i	− 0.294884i
$576$	0	0
$577$	3.46410i	0.144212i	0.997397	+	0.0721062i	$0.0229721\pi$
$577$	3.46410i	0.144212i	−0.997397	+	0.0721062i	$0.977028\pi$
$578$	− 15.5563i	− 0.647059i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−8.00000	−0.331326
$584$	34.2929	1.41905
$585$	0	0
$586$	34.6410i	1.43101i
$587$	26.9444	1.11211	0.556057	−	0.831144i	$-0.312314\pi$
$587$	26.9444	1.11211	0.556057	+	0.831144i	$0.312314\pi$
$588$	0	0
$589$	−9.00000	−0.370839
$590$	50.9117i	2.09600i
$591$	0	0
$592$	16.0000	0.657596
$593$	9.79796	0.402354	0.201177	−	0.979555i	$-0.435523\pi$
$593$	9.79796	0.402354	0.201177	+	0.979555i	$0.435523\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	− 34.6410i	− 1.41658i
$599$	− 7.07107i	− 0.288916i	−0.989511	−	0.144458i	$-0.953856\pi$
$599$	− 7.07107i	− 0.288916i	0.989511	−	0.144458i	$-0.0461439\pi$
$600$	0	0
$601$	− 22.5167i	− 0.918474i	−0.888314	−	0.459237i	$-0.848123\pi$
$601$	− 22.5167i	− 0.918474i	0.888314	−	0.459237i	$-0.151877\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	51.4393	2.09130
$606$	0	0
$607$	29.4449i	1.19513i	0.801820	+	0.597565i	$0.203865\pi$
$607$	29.4449i	1.19513i	−0.801820	+	0.597565i	$0.796135\pi$
$608$	0	0
$609$	0	0
$610$	−18.0000	−0.728799
$611$	25.4558i	1.02983i
$612$	0	0
$613$	−31.0000	−1.25208	−0.626039	−	0.779792i	$-0.715325\pi$
$613$	−31.0000	−1.25208	−0.626039	+	0.779792i	$0.715325\pi$
$614$	7.34847	0.296560
$615$	0	0
$616$	0	0
$617$	− 45.2548i	− 1.82189i	−0.412527	−	0.910946i	$-0.635354\pi$
$617$	− 45.2548i	− 1.82189i	0.412527	−	0.910946i	$-0.364646\pi$
$618$	0	0
$619$	− 34.6410i	− 1.39234i	−0.717877	−	0.696170i	$-0.754886\pi$
$619$	− 34.6410i	− 1.39234i	0.717877	−	0.696170i	$-0.245114\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	−7.34847	−0.293704
$627$	0	0
$628$	0	0
$629$	9.79796	0.390670
$630$	0	0
$631$	11.0000	0.437903	0.218952	−	0.975736i	$-0.429736\pi$
$631$	11.0000	0.437903	0.218952	+	0.975736i	$0.429736\pi$
$632$	− 11.3137i	− 0.450035i
$633$	0	0
$634$	4.00000	0.158860
$635$	2.44949	0.0972050
$636$	0	0
$637$	0	0
$638$	− 11.3137i	− 0.447914i
$639$	0	0
$640$	27.7128i	1.09545i
$641$	− 7.07107i	− 0.279290i	−0.990202	−	0.139645i	$-0.955404\pi$
$641$	− 7.07107i	− 0.279290i	0.990202	−	0.139645i	$-0.0445962\pi$
$642$	0	0
$643$	22.5167i	0.887970i	0.896034	+	0.443985i	$0.146436\pi$
$643$	22.5167i	0.887970i	−0.896034	+	0.443985i	$0.853564\pi$
$644$	0	0
$645$	0	0
$646$	6.00000	0.236067
$647$	−24.4949	−0.962994	−0.481497	−	0.876448i	$-0.659907\pi$
$647$	−24.4949	−0.962994	−0.481497	+	0.876448i	$0.659907\pi$
$648$	0	0
$649$	− 83.1384i	− 3.26347i
$650$	4.89898	0.192154
$651$	0	0
$652$	0	0
$653$	− 15.5563i	− 0.608767i	−0.952550	−	0.304383i	$-0.901550\pi$
$653$	− 15.5563i	− 0.608767i	0.952550	−	0.304383i	$-0.0984504\pi$
$654$	0	0
$655$	42.0000	1.64108
$656$	−9.79796	−0.382546
$657$	0	0
$658$	0	0
$659$	− 7.07107i	− 0.275450i	−0.990471	−	0.137725i	$-0.956021\pi$
$659$	− 7.07107i	− 0.275450i	0.990471	−	0.137725i	$-0.0439790\pi$
$660$	0	0
$661$	46.7654i	1.81896i	0.415745	+	0.909481i	$0.363521\pi$
$661$	46.7654i	1.81896i	−0.415745	+	0.909481i	$0.636479\pi$
$662$	24.0416i	0.934405i
$663$	0	0
$664$	− 27.7128i	− 1.07547i
$665$	0	0
$666$	0	0
$667$	10.0000	0.387202
$668$	0	0
$669$	0	0
$670$	− 6.92820i	− 0.267660i
$671$	29.3939	1.13474
$672$	0	0
$673$	−25.0000	−0.963679	−0.481840	−	0.876259i	$-0.660031\pi$
$673$	−25.0000	−0.963679	−0.481840	+	0.876259i	$0.660031\pi$
$674$	− 1.41421i	− 0.0544735i
$675$	0	0
$676$	0	0
$677$	−14.6969	−0.564849	−0.282425	−	0.959289i	$-0.591139\pi$
$677$	−14.6969	−0.564849	−0.282425	+	0.959289i	$0.591139\pi$
$678$	0	0
$679$	0	0
$680$	16.9706i	0.650791i
$681$	0	0
$682$	− 41.5692i	− 1.59177i
$683$	− 36.7696i	− 1.40695i	−0.710721	−	0.703474i	$-0.751631\pi$
$683$	− 36.7696i	− 1.40695i	0.710721	−	0.703474i	$-0.248369\pi$
$684$	0	0
$685$	17.3205i	0.661783i
$686$	0	0
$687$	0	0
$688$	28.0000	1.06749
$689$	4.89898	0.186636
$690$	0	0
$691$	− 5.19615i	− 0.197671i	−0.995104	−	0.0988355i	$-0.968488\pi$
$691$	− 5.19615i	− 0.197671i	0.995104	−	0.0988355i	$-0.0315118\pi$
$692$	0	0
$693$	0	0
$694$	−44.0000	−1.67022
$695$	− 16.9706i	− 0.643730i
$696$	0	0
$697$	−6.00000	−0.227266
$698$	2.44949	0.0927146
$699$	0	0
$700$	0	0
$701$	− 15.5563i	− 0.587555i	−0.955874	−	0.293778i	$-0.905087\pi$
$701$	− 15.5563i	− 0.587555i	0.955874	−	0.293778i	$-0.0949125\pi$
$702$	0	0
$703$	− 6.92820i	− 0.261302i
$704$	− 45.2548i	− 1.70561i
$705$	0	0
$706$	− 48.4974i	− 1.82522i
$707$	0	0
$708$	0	0
$709$	5.00000	0.187779	0.0938895	−	0.995583i	$-0.470070\pi$
$709$	5.00000	0.187779	0.0938895	+	0.995583i	$0.470070\pi$
$710$	−9.79796	−0.367711
$711$	0	0
$712$	6.92820i	0.259645i
$713$	36.7423	1.37601
$714$	0	0
$715$	−48.0000	−1.79510
$716$	0	0
$717$	0	0
$718$	16.0000	0.597115
$719$	24.4949	0.913506	0.456753	−	0.889594i	$-0.349012\pi$
$719$	24.4949	0.913506	0.456753	+	0.889594i	$0.349012\pi$
$720$	0	0
$721$	0	0
$722$	22.6274i	0.842105i
$723$	0	0
$724$	0	0
$725$	1.41421i	0.0525226i
$726$	0	0
$727$	− 32.9090i	− 1.22053i	−0.792199	−	0.610263i	$-0.791064\pi$
$727$	− 32.9090i	− 1.22053i	0.792199	−	0.610263i	$-0.208936\pi$
$728$	0	0
$729$	0	0
$730$	−42.0000	−1.55449
$731$	17.1464	0.634184
$732$	0	0
$733$	− 43.3013i	− 1.59937i	−0.600420	−	0.799684i	$-0.705000\pi$
$733$	− 43.3013i	− 1.59937i	0.600420	−	0.799684i	$-0.295000\pi$
$734$	12.2474	0.452062
$735$	0	0
$736$	0	0
$737$	11.3137i	0.416746i
$738$	0	0
$739$	5.00000	0.183928	0.0919640	−	0.995762i	$-0.470686\pi$
$739$	5.00000	0.183928	0.0919640	+	0.995762i	$0.470686\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	35.3553i	1.29706i	0.761188	+	0.648531i	$0.224616\pi$
$743$	35.3553i	1.29706i	−0.761188	+	0.648531i	$0.775384\pi$
$744$	0	0
$745$	− 24.2487i	− 0.888404i
$746$	− 1.41421i	− 0.0517780i
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−37.0000	−1.35015	−0.675075	−	0.737749i	$-0.735889\pi$
$751$	−37.0000	−1.35015	−0.675075	+	0.737749i	$0.735889\pi$
$752$	29.3939	1.07188
$753$	0	0
$754$	6.92820i	0.252310i
$755$	−26.9444	−0.980607
$756$	0	0
$757$	−7.00000	−0.254419	−0.127210	−	0.991876i	$-0.540602\pi$
$757$	−7.00000	−0.254419	−0.127210	+	0.991876i	$0.540602\pi$
$758$	− 14.1421i	− 0.513665i
$759$	0	0
$760$	12.0000	0.435286
$761$	−9.79796	−0.355176	−0.177588	−	0.984105i	$-0.556829\pi$
$761$	−9.79796	−0.355176	−0.177588	+	0.984105i	$0.556829\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	48.4974i	1.75228i
$767$	50.9117i	1.83831i
$768$	0	0
$769$	− 50.2295i	− 1.81132i	−0.424003	−	0.905661i	$-0.639376\pi$
$769$	− 50.2295i	− 1.81132i	0.424003	−	0.905661i	$-0.360624\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	41.6413	1.49773	0.748867	−	0.662720i	$-0.230598\pi$
$773$	41.6413	1.49773	0.748867	+	0.662720i	$0.230598\pi$
$774$	0	0
$775$	5.19615i	0.186651i
$776$	−24.4949	−0.879316
$777$	0	0
$778$	−38.0000	−1.36237
$779$	4.24264i	0.152008i
$780$	0	0
$781$	16.0000	0.572525
$782$	−24.4949	−0.875936
$783$	0	0
$784$	0	0
$785$	− 25.4558i	− 0.908558i
$786$	0	0
$787$	25.9808i	0.926114i	0.886328	+	0.463057i	$0.153248\pi$
$787$	25.9808i	0.926114i	−0.886328	+	0.463057i	$0.846752\pi$
$788$	0	0
$789$	0	0
$790$	13.8564i	0.492989i
$791$	0	0
$792$	0	0
$793$	−18.0000	−0.639199
$794$	26.9444	0.956221
$795$	0	0
$796$	0	0
$797$	39.1918	1.38825	0.694123	−	0.719856i	$-0.255792\pi$
$797$	39.1918	1.38825	0.694123	+	0.719856i	$0.255792\pi$
$798$	0	0
$799$	18.0000	0.636794
$800$	0	0
$801$	0	0
$802$	−32.0000	−1.12996
$803$	68.5857	2.42034
$804$	0	0
$805$	0	0
$806$	25.4558i	0.896644i
$807$	0	0
$808$	27.7128i	0.974933i
$809$	− 11.3137i	− 0.397769i	−0.980023	−	0.198884i	$-0.936268\pi$
$809$	− 11.3137i	− 0.397769i	0.980023	−	0.198884i	$-0.0637318\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	32.0000	1.12160
$815$	−56.3383	−1.97344
$816$	0	0
$817$	− 12.1244i	− 0.424178i
$818$	−14.6969	−0.513866
$819$	0	0
$820$	0	0
$821$	39.5980i	1.38198i	0.722865	+	0.690990i	$0.242825\pi$
$821$	39.5980i	1.38198i	−0.722865	+	0.690990i	$0.757175\pi$
$822$	0	0
$823$	23.0000	0.801730	0.400865	−	0.916137i	$-0.368710\pi$
$823$	23.0000	0.801730	0.400865	+	0.916137i	$0.368710\pi$
$824$	−39.1918	−1.36531
$825$	0	0
$826$	0	0
$827$	22.6274i	0.786832i	0.919360	+	0.393416i	$0.128707\pi$
$827$	22.6274i	0.786832i	−0.919360	+	0.393416i	$0.871293\pi$
$828$	0	0
$829$	− 32.9090i	− 1.14298i	−0.820611	−	0.571488i	$-0.806366\pi$
$829$	− 32.9090i	− 1.14298i	0.820611	−	0.571488i	$-0.193634\pi$
$830$	33.9411i	1.17811i
$831$	0	0
$832$	27.7128i	0.960769i
$833$	0	0
$834$	0	0
$835$	12.0000	0.415277
$836$	0	0
$837$	0	0
$838$	3.46410i	0.119665i
$839$	−24.4949	−0.845658	−0.422829	−	0.906210i	$-0.638963\pi$
$839$	−24.4949	−0.845658	−0.422829	+	0.906210i	$0.638963\pi$
$840$	0	0
$841$	27.0000	0.931034
$842$	15.5563i	0.536107i
$843$	0	0
$844$	0	0
$845$	−2.44949	−0.0842650
$846$	0	0
$847$	0	0
$848$	− 5.65685i	− 0.194257i
$849$	0	0
$850$	− 3.46410i	− 0.118818i
$851$	28.2843i	0.969572i
$852$	0	0
$853$	8.66025i	0.296521i	0.988948	+	0.148261i	$0.0473675\pi$
$853$	8.66025i	0.296521i	−0.988948	+	0.148261i	$0.952633\pi$
$854$	0	0
$855$	0	0
$856$	−40.0000	−1.36717
$857$	−4.89898	−0.167346	−0.0836730	−	0.996493i	$-0.526665\pi$
$857$	−4.89898	−0.167346	−0.0836730	+	0.996493i	$0.526665\pi$
$858$	0	0
$859$	50.2295i	1.71381i	0.515476	+	0.856904i	$0.327615\pi$
$859$	50.2295i	1.71381i	−0.515476	+	0.856904i	$0.672385\pi$
$860$	0	0
$861$	0	0
$862$	40.0000	1.36241
$863$	1.41421i	0.0481404i	0.999710	+	0.0240702i	$0.00766252\pi$
$863$	1.41421i	0.0481404i	−0.999710	+	0.0240702i	$0.992337\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	7.34847	0.249711
$867$	0	0
$868$	0	0
$869$	− 22.6274i	− 0.767583i
$870$	0	0
$871$	− 6.92820i	− 0.234753i
$872$	48.0833i	1.62830i
$873$	0	0
$874$	17.3205i	0.585875i
$875$	0	0
$876$	0	0
$877$	35.0000	1.18187	0.590933	−	0.806721i	$-0.298760\pi$
$877$	35.0000	1.18187	0.590933	+	0.806721i	$0.298760\pi$
$878$	−29.3939	−0.991995
$879$	0	0
$880$	55.4256i	1.86840i
$881$	−36.7423	−1.23788	−0.618941	−	0.785438i	$-0.712438\pi$
$881$	−36.7423	−1.23788	−0.618941	+	0.785438i	$0.712438\pi$
$882$	0	0
$883$	−37.0000	−1.24515	−0.622575	−	0.782560i	$-0.713913\pi$
$883$	−37.0000	−1.24515	−0.622575	+	0.782560i	$0.713913\pi$
$884$	0	0
$885$	0	0
$886$	22.0000	0.739104
$887$	−53.8888	−1.80941	−0.904704	−	0.426041i	$-0.859908\pi$
$887$	−53.8888	−1.80941	−0.904704	+	0.426041i	$0.859908\pi$
$888$	0	0
$889$	0	0
$890$	− 8.48528i	− 0.284427i
$891$	0	0
$892$	0	0
$893$	− 12.7279i	− 0.425924i
$894$	0	0
$895$	− 24.2487i	− 0.810545i
$896$	0	0
$897$	0	0
$898$	−50.0000	−1.66852
$899$	−7.34847	−0.245085
$900$	0	0
$901$	− 3.46410i	− 0.115406i
$902$	−19.5959	−0.652473
$903$	0	0
$904$	8.00000	0.266076
$905$	− 38.1838i	− 1.26927i
$906$	0	0
$907$	−16.0000	−0.531271	−0.265636	−	0.964073i	$-0.585582\pi$
$907$	−16.0000	−0.531271	−0.265636	+	0.964073i	$0.585582\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	52.3259i	1.73363i	0.498626	+	0.866817i	$0.333838\pi$
$911$	52.3259i	1.73363i	−0.498626	+	0.866817i	$0.666162\pi$
$912$	0	0
$913$	− 55.4256i	− 1.83432i
$914$	7.07107i	0.233890i
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	5.00000	0.164935	0.0824674	−	0.996594i	$-0.473720\pi$
$919$	5.00000	0.164935	0.0824674	+	0.996594i	$0.473720\pi$
$920$	−48.9898	−1.61515
$921$	0	0
$922$	− 34.6410i	− 1.14084i
$923$	−9.79796	−0.322504
$924$	0	0
$925$	−4.00000	−0.131519
$926$	− 1.41421i	− 0.0464739i
$927$	0	0
$928$	0	0
$929$	−22.0454	−0.723286	−0.361643	−	0.932317i	$-0.617784\pi$
$929$	−22.0454	−0.723286	−0.361643	+	0.932317i	$0.617784\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	− 6.92820i	− 0.226698i
$935$	33.9411i	1.10999i
$936$	0	0
$937$	− 10.3923i	− 0.339502i	−0.985487	−	0.169751i	$-0.945704\pi$
$937$	− 10.3923i	− 0.339502i	0.985487	−	0.169751i	$-0.0542963\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−14.6969	−0.479107	−0.239553	−	0.970883i	$-0.577001\pi$
$941$	−14.6969	−0.479107	−0.239553	+	0.970883i	$0.577001\pi$
$942$	0	0
$943$	− 17.3205i	− 0.564033i
$944$	58.7878	1.91338
$945$	0	0
$946$	56.0000	1.82072
$947$	9.89949i	0.321690i	0.986980	+	0.160845i	$0.0514220\pi$
$947$	9.89949i	0.321690i	−0.986980	+	0.160845i	$0.948578\pi$
$948$	0	0
$949$	−42.0000	−1.36338
$950$	−2.44949	−0.0794719
$951$	0	0
$952$	0	0
$953$	18.3848i	0.595541i	0.954637	+	0.297771i	$0.0962431\pi$
$953$	18.3848i	0.595541i	−0.954637	+	0.297771i	$0.903757\pi$
$954$	0	0
$955$	− 3.46410i	− 0.112096i
$956$	0	0
$957$	0	0
$958$	3.46410i	0.111920i
$959$	0	0
$960$	0	0
$961$	4.00000	0.129032
$962$	−19.5959	−0.631798
$963$	0	0
$964$	0	0
$965$	9.79796	0.315407
$966$	0	0
$967$	−34.0000	−1.09337	−0.546683	−	0.837340i	$-0.684110\pi$
$967$	−34.0000	−1.09337	−0.546683	+	0.837340i	$0.684110\pi$
$968$	− 59.3970i	− 1.90909i
$969$	0	0
$970$	30.0000	0.963242
$971$	−12.2474	−0.393039	−0.196520	−	0.980500i	$-0.562964\pi$
$971$	−12.2474	−0.393039	−0.196520	+	0.980500i	$0.562964\pi$
$972$	0	0
$973$	0	0
$974$	− 18.3848i	− 0.589086i
$975$	0	0
$976$	20.7846i	0.665299i
$977$	− 36.7696i	− 1.17636i	−0.808729	−	0.588181i	$-0.799844\pi$
$977$	− 36.7696i	− 1.17636i	0.808729	−	0.588181i	$-0.200156\pi$
$978$	0	0
$979$	13.8564i	0.442853i
$980$	0	0
$981$	0	0
$982$	−8.00000	−0.255290
$983$	−4.89898	−0.156253	−0.0781266	−	0.996943i	$-0.524894\pi$
$983$	−4.89898	−0.156253	−0.0781266	+	0.996943i	$0.524894\pi$
$984$	0	0
$985$	6.92820i	0.220751i
$986$	4.89898	0.156015
$987$	0	0
$988$	0	0
$989$	49.4975i	1.57393i
$990$	0	0
$991$	−58.0000	−1.84243	−0.921215	−	0.389053i	$-0.872802\pi$
$991$	−58.0000	−1.84243	−0.921215	+	0.389053i	$0.872802\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4.24264i	0.134501i
$996$	0	0
$997$	− 39.8372i	− 1.26166i	−0.775923	−	0.630828i	$-0.782715\pi$
$997$	− 39.8372i	− 1.26166i	0.775923	−	0.630828i	$-0.217285\pi$
$998$	− 35.3553i	− 1.11915i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.2.c.c.1322.3		4
3.2	odd	2	inner	1323.2.c.c.1322.2		4
7.2	even	3		189.2.p.b.80.2	yes	4
7.3	odd	6		189.2.p.b.26.1	✓	4
7.6	odd	2	inner	1323.2.c.c.1322.4		4
21.2	odd	6		189.2.p.b.80.1	yes	4
21.17	even	6		189.2.p.b.26.2	yes	4
21.20	even	2	inner	1323.2.c.c.1322.1		4
63.2	odd	6		567.2.s.c.458.2		4
63.16	even	3		567.2.s.c.458.1		4
63.23	odd	6		567.2.i.e.269.1		4
63.31	odd	6		567.2.s.c.26.2		4
63.38	even	6		567.2.i.e.215.1		4
63.52	odd	6		567.2.i.e.215.2		4
63.58	even	3		567.2.i.e.269.2		4
63.59	even	6		567.2.s.c.26.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.p.b.26.1	✓	4	7.3	odd	6
189.2.p.b.26.2	yes	4	21.17	even	6
189.2.p.b.80.1	yes	4	21.2	odd	6
189.2.p.b.80.2	yes	4	7.2	even	3
567.2.i.e.215.1		4	63.38	even	6
567.2.i.e.215.2		4	63.52	odd	6
567.2.i.e.269.1		4	63.23	odd	6
567.2.i.e.269.2		4	63.58	even	3
567.2.s.c.26.1		4	63.59	even	6
567.2.s.c.26.2		4	63.31	odd	6
567.2.s.c.458.1		4	63.16	even	3
567.2.s.c.458.2		4	63.2	odd	6
1323.2.c.c.1322.1		4	21.20	even	2	inner
1323.2.c.c.1322.2		4	3.2	odd	2	inner
1323.2.c.c.1322.3		4	1.1	even	1	trivial
1323.2.c.c.1322.4		4	7.6	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.44949 q^{5} +2.82843i q^{8} +O(q^{10})\)	\(q+1.41421i q^{2} -2.44949 q^{5} +2.82843i q^{8} -3.46410i q^{10} +5.65685i q^{11} -3.46410i q^{13} -4.00000 q^{16} -2.44949 q^{17} +1.73205i q^{19} -8.00000 q^{22} -7.07107i q^{23} +1.00000 q^{25} +4.89898 q^{26} +1.41421i q^{29} +5.19615i q^{31} -3.46410i q^{34} -4.00000 q^{37} -2.44949 q^{38} -6.92820i q^{40} +2.44949 q^{41} -7.00000 q^{43} +10.0000 q^{46} -7.34847 q^{47} +1.41421i q^{50} +1.41421i q^{53} -13.8564i q^{55} -2.00000 q^{58} -14.6969 q^{59} -5.19615i q^{61} -7.34847 q^{62} -8.00000 q^{64} +8.48528i q^{65} +2.00000 q^{67} -2.82843i q^{71} -12.1244i q^{73} -5.65685i q^{74} -4.00000 q^{79} +9.79796 q^{80} +3.46410i q^{82} -9.79796 q^{83} +6.00000 q^{85} -9.89949i q^{86} -16.0000 q^{88} +2.44949 q^{89} -10.3923i q^{94} -4.24264i q^{95} +8.66025i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 16 q^{16} - 32 q^{22} + 4 q^{25} - 16 q^{37} - 28 q^{43} + 40 q^{46} - 8 q^{58} - 32 q^{64} + 8 q^{67} - 16 q^{79} + 24 q^{85} - 64 q^{88}+O(q^{100}) \) 4 * q - 16 * q^16 - 32 * q^22 + 4 * q^25 - 16 * q^37 - 28 * q^43 + 40 * q^46 - 8 * q^58 - 32 * q^64 + 8 * q^67 - 16 * q^79 + 24 * q^85 - 64 * q^88

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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