LMFDB - Embedded newform 2646.2.d.d.2645.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2646,2,Mod(2645,2646)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2646, 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("2646.2645");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2646 = 2 \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2646.d (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:	$21.1284163748$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 378)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2645.4
Root			$0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	2646.2645
Dual form			2646.2.d.d.2645.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +2.44949 q^{5} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} +2.44949 q^{5} +1.00000i q^{8} -2.44949i q^{10} +4.24264i q^{11} -0.717439i q^{13} +1.00000 q^{16} +2.44949 q^{17} -4.89898i q^{19} -2.44949 q^{20} +4.24264 q^{22} -6.00000i q^{23} +1.00000 q^{25} -0.717439 q^{26} +1.75736i q^{29} +9.08052i q^{31} -1.00000i q^{32} -2.44949i q^{34} +5.24264 q^{37} -4.89898 q^{38} +2.44949i q^{40} -2.44949 q^{41} +7.00000 q^{43} -4.24264i q^{44} -6.00000 q^{46} +12.8418 q^{47} -1.00000i q^{50} +0.717439i q^{52} -14.4853i q^{53} +10.3923i q^{55} +1.75736 q^{58} -2.44949 q^{59} +4.18154i q^{61} +9.08052 q^{62} -1.00000 q^{64} -1.75736i q^{65} +13.4853 q^{67} -2.44949 q^{68} +12.7279i q^{71} +5.49333i q^{73} -5.24264i q^{74} +4.89898i q^{76} +0.757359 q^{79} +2.44949 q^{80} +2.44949i q^{82} +15.2913 q^{83} +6.00000 q^{85} -7.00000i q^{86} -4.24264 q^{88} +3.04384 q^{89} +6.00000i q^{92} -12.8418i q^{94} -12.0000i q^{95} +3.16693i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4}+O(q^{10}) $ 8 * q - 8 * q^4	$ 8 q - 8 q^{4} + 8 q^{16} + 8 q^{25} + 8 q^{37} + 56 q^{43} - 48 q^{46} + 48 q^{58} - 8 q^{64} + 40 q^{67} + 40 q^{79} + 48 q^{85}+O(q^{100}) $ 8 * q - 8 * q^4 + 8 * q^16 + 8 * q^25 + 8 * q^37 + 56 * q^43 - 48 * q^46 + 48 * q^58 - 8 * q^64 + 40 * q^67 + 40 * q^79 + 48 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times$.

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

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	2.44949	1.09545	0.547723	−	0.836660i	$-0.315495\pi$
$5$	2.44949	1.09545	0.547723	+	0.836660i	$0.315495\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000i	0.353553i
$9$	0	0
$10$	− 2.44949i	− 0.774597i
$11$	4.24264i	1.27920i	0.768706	+	0.639602i	$0.220901\pi$
$11$	4.24264i	1.27920i	−0.768706	+	0.639602i	$0.779099\pi$
$12$	0	0
$13$	− 0.717439i	− 0.198982i	−0.995038	−	0.0994909i	$-0.968279\pi$
$13$	− 0.717439i	− 0.198982i	0.995038	−	0.0994909i	$-0.0317214\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	2.44949	0.594089	0.297044	−	0.954864i	$-0.403999\pi$
$17$	2.44949	0.594089	0.297044	+	0.954864i	$0.403999\pi$
$18$	0	0
$19$	− 4.89898i	− 1.12390i	−0.827170	−	0.561951i	$-0.810051\pi$
$19$	− 4.89898i	− 1.12390i	0.827170	−	0.561951i	$-0.189949\pi$
$20$	−2.44949	−0.547723
$21$	0	0
$22$	4.24264	0.904534
$23$	− 6.00000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 6.00000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.717439	−0.140701
$27$	0	0
$28$	0	0
$29$	1.75736i	0.326333i	0.986599	+	0.163167i	$0.0521708\pi$
$29$	1.75736i	0.326333i	−0.986599	+	0.163167i	$0.947829\pi$
$30$	0	0
$31$	9.08052i	1.63091i	0.578821	+	0.815455i	$0.303513\pi$
$31$	9.08052i	1.63091i	−0.578821	+	0.815455i	$0.696487\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	− 2.44949i	− 0.420084i
$35$	0	0
$36$	0	0
$37$	5.24264	0.861885	0.430942	−	0.902379i	$-0.358181\pi$
$37$	5.24264	0.861885	0.430942	+	0.902379i	$0.358181\pi$
$38$	−4.89898	−0.794719
$39$	0	0
$40$	2.44949i	0.387298i
$41$	−2.44949	−0.382546	−0.191273	−	0.981537i	$-0.561262\pi$
$41$	−2.44949	−0.382546	−0.191273	+	0.981537i	$0.561262\pi$
$42$	0	0
$43$	7.00000	1.06749	0.533745	−	0.845645i	$-0.320784\pi$
$43$	7.00000	1.06749	0.533745	+	0.845645i	$0.320784\pi$
$44$	− 4.24264i	− 0.639602i
$45$	0	0
$46$	−6.00000	−0.884652
$47$	12.8418	1.87317	0.936584	−	0.350443i	$-0.113969\pi$
$47$	12.8418	1.87317	0.936584	+	0.350443i	$0.113969\pi$
$48$	0	0
$49$	0	0
$50$	− 1.00000i	− 0.141421i
$51$	0	0
$52$	0.717439i	0.0994909i
$53$	− 14.4853i	− 1.98971i	−0.101327	−	0.994853i	$-0.532309\pi$
$53$	− 14.4853i	− 1.98971i	0.101327	−	0.994853i	$-0.467691\pi$
$54$	0	0
$55$	10.3923i	1.40130i
$56$	0	0
$57$	0	0
$58$	1.75736	0.230753
$59$	−2.44949	−0.318896	−0.159448	−	0.987206i	$-0.550971\pi$
$59$	−2.44949	−0.318896	−0.159448	+	0.987206i	$0.550971\pi$
$60$	0	0
$61$	4.18154i	0.535391i	0.963504	+	0.267696i	$0.0862622\pi$
$61$	4.18154i	0.535391i	−0.963504	+	0.267696i	$0.913738\pi$
$62$	9.08052	1.15323
$63$	0	0
$64$	−1.00000	−0.125000
$65$	− 1.75736i	− 0.217974i
$66$	0	0
$67$	13.4853	1.64749	0.823745	−	0.566961i	$-0.191881\pi$
$67$	13.4853	1.64749	0.823745	+	0.566961i	$0.191881\pi$
$68$	−2.44949	−0.297044
$69$	0	0
$70$	0	0
$71$	12.7279i	1.51053i	0.655422	+	0.755263i	$0.272491\pi$
$71$	12.7279i	1.51053i	−0.655422	+	0.755263i	$0.727509\pi$
$72$	0	0
$73$	5.49333i	0.642945i	0.946919	+	0.321473i	$0.104178\pi$
$73$	5.49333i	0.642945i	−0.946919	+	0.321473i	$0.895822\pi$
$74$	− 5.24264i	− 0.609445i
$75$	0	0
$76$	4.89898i	0.561951i
$77$	0	0
$78$	0	0
$79$	0.757359	0.0852096	0.0426048	−	0.999092i	$-0.486434\pi$
$79$	0.757359	0.0852096	0.0426048	+	0.999092i	$0.486434\pi$
$80$	2.44949	0.273861
$81$	0	0
$82$	2.44949i	0.270501i
$83$	15.2913	1.67844	0.839218	−	0.543795i	$-0.183013\pi$
$83$	15.2913	1.67844	0.839218	+	0.543795i	$0.183013\pi$
$84$	0	0
$85$	6.00000	0.650791
$86$	− 7.00000i	− 0.754829i
$87$	0	0
$88$	−4.24264	−0.452267
$89$	3.04384	0.322646	0.161323	−	0.986902i	$-0.448424\pi$
$89$	3.04384	0.322646	0.161323	+	0.986902i	$0.448424\pi$
$90$	0	0
$91$	0	0
$92$	6.00000i	0.625543i
$93$	0	0
$94$	− 12.8418i	− 1.32453i
$95$	− 12.0000i	− 1.23117i
$96$	0	0
$97$	3.16693i	0.321553i	0.986991	+	0.160776i	$0.0513998\pi$
$97$	3.16693i	0.321553i	−0.986991	+	0.160776i	$0.948600\pi$
$98$	0	0
$99$	0	0
$100$	−1.00000	−0.100000
$101$	7.34847	0.731200	0.365600	−	0.930772i	$-0.380864\pi$
$101$	7.34847	0.731200	0.365600	+	0.930772i	$0.380864\pi$
$102$	0	0
$103$	11.1097i	1.09468i	0.836912	+	0.547338i	$0.184359\pi$
$103$	11.1097i	1.09468i	−0.836912	+	0.547338i	$0.815641\pi$
$104$	0.717439	0.0703507
$105$	0	0
$106$	−14.4853	−1.40693
$107$	2.48528i	0.240261i	0.992758	+	0.120131i	$0.0383313\pi$
$107$	2.48528i	0.240261i	−0.992758	+	0.120131i	$0.961669\pi$
$108$	0	0
$109$	−17.7279	−1.69803	−0.849013	−	0.528371i	$-0.822803\pi$
$109$	−17.7279	−1.69803	−0.849013	+	0.528371i	$0.822803\pi$
$110$	10.3923	0.990867
$111$	0	0
$112$	0	0
$113$	− 10.2426i	− 0.963547i	−0.876296	−	0.481773i	$-0.839993\pi$
$113$	− 10.2426i	− 0.963547i	0.876296	−	0.481773i	$-0.160007\pi$
$114$	0	0
$115$	− 14.6969i	− 1.37050i
$116$	− 1.75736i	− 0.163167i
$117$	0	0
$118$	2.44949i	0.225494i
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	4.18154	0.378579
$123$	0	0
$124$	− 9.08052i	− 0.815455i
$125$	−9.79796	−0.876356
$126$	0	0
$127$	−7.72792	−0.685742	−0.342871	−	0.939382i	$-0.611399\pi$
$127$	−7.72792	−0.685742	−0.342871	+	0.939382i	$0.611399\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	−1.75736	−0.154131
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	− 13.4853i	− 1.16495i
$135$	0	0
$136$	2.44949i	0.210042i
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	0	0
$139$	8.06591i	0.684141i	0.939674	+	0.342071i	$0.111128\pi$
$139$	8.06591i	0.684141i	−0.939674	+	0.342071i	$0.888872\pi$
$140$	0	0
$141$	0	0
$142$	12.7279	1.06810
$143$	3.04384	0.254538
$144$	0	0
$145$	4.30463i	0.357480i
$146$	5.49333	0.454631
$147$	0	0
$148$	−5.24264	−0.430942
$149$	− 16.2426i	− 1.33065i	−0.746554	−	0.665324i	$-0.768293\pi$
$149$	− 16.2426i	− 1.33065i	0.746554	−	0.665324i	$-0.231707\pi$
$150$	0	0
$151$	8.75736	0.712664	0.356332	−	0.934359i	$-0.384027\pi$
$151$	8.75736	0.712664	0.356332	+	0.934359i	$0.384027\pi$
$152$	4.89898	0.397360
$153$	0	0
$154$	0	0
$155$	22.2426i	1.78657i
$156$	0	0
$157$	10.3923i	0.829396i	0.909959	+	0.414698i	$0.136113\pi$
$157$	10.3923i	0.829396i	−0.909959	+	0.414698i	$0.863887\pi$
$158$	− 0.757359i	− 0.0602523i
$159$	0	0
$160$	− 2.44949i	− 0.193649i
$161$	0	0
$162$	0	0
$163$	−9.48528	−0.742945	−0.371472	−	0.928444i	$-0.621147\pi$
$163$	−9.48528	−0.742945	−0.371472	+	0.928444i	$0.621147\pi$
$164$	2.44949	0.191273
$165$	0	0
$166$	− 15.2913i	− 1.18683i
$167$	−0.594346	−0.0459919	−0.0229959	−	0.999736i	$-0.507320\pi$
$167$	−0.594346	−0.0459919	−0.0229959	+	0.999736i	$0.507320\pi$
$168$	0	0
$169$	12.4853	0.960406
$170$	− 6.00000i	− 0.460179i
$171$	0	0
$172$	−7.00000	−0.533745
$173$	20.7846	1.58022	0.790112	−	0.612962i	$-0.210022\pi$
$173$	20.7846	1.58022	0.790112	+	0.612962i	$0.210022\pi$
$174$	0	0
$175$	0	0
$176$	4.24264i	0.319801i
$177$	0	0
$178$	− 3.04384i	− 0.228145i
$179$	6.72792i	0.502869i	0.967874	+	0.251434i	$0.0809022\pi$
$179$	6.72792i	0.502869i	−0.967874	+	0.251434i	$0.919098\pi$
$180$	0	0
$181$	− 9.79796i	− 0.728277i	−0.931345	−	0.364138i	$-0.881364\pi$
$181$	− 9.79796i	− 0.728277i	0.931345	−	0.364138i	$-0.118636\pi$
$182$	0	0
$183$	0	0
$184$	6.00000	0.442326
$185$	12.8418	0.944148
$186$	0	0
$187$	10.3923i	0.759961i
$188$	−12.8418	−0.936584
$189$	0	0
$190$	−12.0000	−0.870572
$191$	− 21.2132i	− 1.53493i	−0.641089	−	0.767467i	$-0.721517\pi$
$191$	− 21.2132i	− 1.53493i	0.641089	−	0.767467i	$-0.278483\pi$
$192$	0	0
$193$	−1.48528	−0.106913	−0.0534564	−	0.998570i	$-0.517024\pi$
$193$	−1.48528	−0.106913	−0.0534564	+	0.998570i	$0.517024\pi$
$194$	3.16693	0.227372
$195$	0	0
$196$	0	0
$197$	− 16.9706i	− 1.20910i	−0.796566	−	0.604551i	$-0.793352\pi$
$197$	− 16.9706i	− 1.20910i	0.796566	−	0.604551i	$-0.206648\pi$
$198$	0	0
$199$	20.9077i	1.48211i	0.671446	+	0.741054i	$0.265674\pi$
$199$	20.9077i	1.48211i	−0.671446	+	0.741054i	$0.734326\pi$
$200$	1.00000i	0.0707107i
$201$	0	0
$202$	− 7.34847i	− 0.517036i
$203$	0	0
$204$	0	0
$205$	−6.00000	−0.419058
$206$	11.1097	0.774053
$207$	0	0
$208$	− 0.717439i	− 0.0497454i
$209$	20.7846	1.43770
$210$	0	0
$211$	3.48528	0.239937	0.119968	−	0.992778i	$-0.461721\pi$
$211$	3.48528	0.239937	0.119968	+	0.992778i	$0.461721\pi$
$212$	14.4853i	0.994853i
$213$	0	0
$214$	2.48528	0.169890
$215$	17.1464	1.16938
$216$	0	0
$217$	0	0
$218$	17.7279i	1.20069i
$219$	0	0
$220$	− 10.3923i	− 0.700649i
$221$	− 1.75736i	− 0.118213i
$222$	0	0
$223$	− 10.3923i	− 0.695920i	−0.937509	−	0.347960i	$-0.886874\pi$
$223$	− 10.3923i	− 0.695920i	0.937509	−	0.347960i	$-0.113126\pi$
$224$	0	0
$225$	0	0
$226$	−10.2426	−0.681330
$227$	−25.0892	−1.66523	−0.832616	−	0.553851i	$-0.813158\pi$
$227$	−25.0892	−1.66523	−0.832616	+	0.553851i	$0.813158\pi$
$228$	0	0
$229$	− 5.61642i	− 0.371143i	−0.982631	−	0.185572i	$-0.940586\pi$
$229$	− 5.61642i	− 0.371143i	0.982631	−	0.185572i	$-0.0594137\pi$
$230$	−14.6969	−0.969087
$231$	0	0
$232$	−1.75736	−0.115376
$233$	− 20.4853i	− 1.34204i	−0.741441	−	0.671018i	$-0.765857\pi$
$233$	− 20.4853i	− 1.34204i	0.741441	−	0.671018i	$-0.234143\pi$
$234$	0	0
$235$	31.4558	2.05195
$236$	2.44949	0.159448
$237$	0	0
$238$	0	0
$239$	− 16.2426i	− 1.05065i	−0.850902	−	0.525325i	$-0.823944\pi$
$239$	− 16.2426i	− 1.05065i	0.850902	−	0.525325i	$-0.176056\pi$
$240$	0	0
$241$	1.13770i	0.0732860i	0.999328	+	0.0366430i	$0.0116664\pi$
$241$	1.13770i	0.0732860i	−0.999328	+	0.0366430i	$0.988334\pi$
$242$	7.00000i	0.449977i
$243$	0	0
$244$	− 4.18154i	− 0.267696i
$245$	0	0
$246$	0	0
$247$	−3.51472	−0.223636
$248$	−9.08052	−0.576614
$249$	0	0
$250$	9.79796i	0.619677i
$251$	−15.2913	−0.965177	−0.482589	−	0.875847i	$-0.660303\pi$
$251$	−15.2913	−0.965177	−0.482589	+	0.875847i	$0.660303\pi$
$252$	0	0
$253$	25.4558	1.60040
$254$	7.72792i	0.484893i
$255$	0	0
$256$	1.00000	0.0625000
$257$	−25.0892	−1.56502	−0.782512	−	0.622636i	$-0.786062\pi$
$257$	−25.0892	−1.56502	−0.782512	+	0.622636i	$0.786062\pi$
$258$	0	0
$259$	0	0
$260$	1.75736i	0.108987i
$261$	0	0
$262$	0	0
$263$	− 6.72792i	− 0.414861i	−0.978250	−	0.207431i	$-0.933490\pi$
$263$	− 6.72792i	− 0.414861i	0.978250	−	0.207431i	$-0.0665102\pi$
$264$	0	0
$265$	− 35.4815i	− 2.17961i
$266$	0	0
$267$	0	0
$268$	−13.4853	−0.823745
$269$	−9.79796	−0.597392	−0.298696	−	0.954348i	$-0.596552\pi$
$269$	−9.79796	−0.597392	−0.298696	+	0.954348i	$0.596552\pi$
$270$	0	0
$271$	− 18.8785i	− 1.14679i	−0.819280	−	0.573393i	$-0.805627\pi$
$271$	− 18.8785i	− 1.14679i	0.819280	−	0.573393i	$-0.194373\pi$
$272$	2.44949	0.148522
$273$	0	0
$274$	−6.00000	−0.362473
$275$	4.24264i	0.255841i
$276$	0	0
$277$	23.7279	1.42567	0.712836	−	0.701330i	$-0.247410\pi$
$277$	23.7279	1.42567	0.712836	+	0.701330i	$0.247410\pi$
$278$	8.06591	0.483761
$279$	0	0
$280$	0	0
$281$	13.7574i	0.820695i	0.911929	+	0.410348i	$0.134593\pi$
$281$	13.7574i	0.820695i	−0.911929	+	0.410348i	$0.865407\pi$
$282$	0	0
$283$	− 6.03668i	− 0.358844i	−0.983772	−	0.179422i	$-0.942577\pi$
$283$	− 6.03668i	− 0.358844i	0.983772	−	0.179422i	$-0.0574227\pi$
$284$	− 12.7279i	− 0.755263i
$285$	0	0
$286$	− 3.04384i	− 0.179986i
$287$	0	0
$288$	0	0
$289$	−11.0000	−0.647059
$290$	4.30463	0.252777
$291$	0	0
$292$	− 5.49333i	− 0.321473i
$293$	−12.8418	−0.750226	−0.375113	−	0.926979i	$-0.622396\pi$
$293$	−12.8418	−0.750226	−0.375113	+	0.926979i	$0.622396\pi$
$294$	0	0
$295$	−6.00000	−0.349334
$296$	5.24264i	0.304722i
$297$	0	0
$298$	−16.2426	−0.940911
$299$	−4.30463	−0.248943
$300$	0	0
$301$	0	0
$302$	− 8.75736i	− 0.503929i
$303$	0	0
$304$	− 4.89898i	− 0.280976i
$305$	10.2426i	0.586492i
$306$	0	0
$307$	26.8213i	1.53077i	0.643571	+	0.765386i	$0.277452\pi$
$307$	26.8213i	1.53077i	−0.643571	+	0.765386i	$0.722548\pi$
$308$	0	0
$309$	0	0
$310$	22.2426	1.26330
$311$	17.1464	0.972285	0.486142	−	0.873880i	$-0.338404\pi$
$311$	17.1464	0.972285	0.486142	+	0.873880i	$0.338404\pi$
$312$	0	0
$313$	− 20.1903i	− 1.14122i	−0.821221	−	0.570611i	$-0.806707\pi$
$313$	− 20.1903i	− 1.14122i	0.821221	−	0.570611i	$-0.193293\pi$
$314$	10.3923	0.586472
$315$	0	0
$316$	−0.757359	−0.0426048
$317$	22.2426i	1.24927i	0.780916	+	0.624636i	$0.214752\pi$
$317$	22.2426i	1.24927i	−0.780916	+	0.624636i	$0.785248\pi$
$318$	0	0
$319$	−7.45584	−0.417447
$320$	−2.44949	−0.136931
$321$	0	0
$322$	0	0
$323$	− 12.0000i	− 0.667698i
$324$	0	0
$325$	− 0.717439i	− 0.0397964i
$326$	9.48528i	0.525341i
$327$	0	0
$328$	− 2.44949i	− 0.135250i
$329$	0	0
$330$	0	0
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	−15.2913	−0.839218
$333$	0	0
$334$	0.594346i	0.0325212i
$335$	33.0321	1.80473
$336$	0	0
$337$	−21.4558	−1.16877	−0.584387	−	0.811475i	$-0.698665\pi$
$337$	−21.4558	−1.16877	−0.584387	+	0.811475i	$0.698665\pi$
$338$	− 12.4853i	− 0.679110i
$339$	0	0
$340$	−6.00000	−0.325396
$341$	−38.5254	−2.08627
$342$	0	0
$343$	0	0
$344$	7.00000i	0.377415i
$345$	0	0
$346$	− 20.7846i	− 1.11739i
$347$	− 4.97056i	− 0.266834i	−0.991060	−	0.133417i	$-0.957405\pi$
$347$	− 4.97056i	− 0.266834i	0.991060	−	0.133417i	$-0.0425949\pi$
$348$	0	0
$349$	− 0.123093i	− 0.00658902i	−0.999995	−	0.00329451i	$-0.998951\pi$
$349$	− 0.123093i	− 0.00658902i	0.999995	−	0.00329451i	$-0.00104868\pi$
$350$	0	0
$351$	0	0
$352$	4.24264	0.226134
$353$	−10.9867	−0.584760	−0.292380	−	0.956302i	$-0.594447\pi$
$353$	−10.9867	−0.584760	−0.292380	+	0.956302i	$0.594447\pi$
$354$	0	0
$355$	31.1769i	1.65470i
$356$	−3.04384	−0.161323
$357$	0	0
$358$	6.72792	0.355582
$359$	14.4853i	0.764504i	0.924058	+	0.382252i	$0.124851\pi$
$359$	14.4853i	0.764504i	−0.924058	+	0.382252i	$0.875149\pi$
$360$	0	0
$361$	−5.00000	−0.263158
$362$	−9.79796	−0.514969
$363$	0	0
$364$	0	0
$365$	13.4558i	0.704311i
$366$	0	0
$367$	29.9882i	1.56537i	0.622416	+	0.782686i	$0.286151\pi$
$367$	29.9882i	1.56537i	−0.622416	+	0.782686i	$0.713849\pi$
$368$	− 6.00000i	− 0.312772i
$369$	0	0
$370$	− 12.8418i	− 0.667613i
$371$	0	0
$372$	0	0
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	10.3923	0.537373
$375$	0	0
$376$	12.8418i	0.662265i
$377$	1.26080	0.0649344
$378$	0	0
$379$	−7.48528	−0.384493	−0.192247	−	0.981347i	$-0.561577\pi$
$379$	−7.48528	−0.384493	−0.192247	+	0.981347i	$0.561577\pi$
$380$	12.0000i	0.615587i
$381$	0	0
$382$	−21.2132	−1.08536
$383$	−5.49333	−0.280696	−0.140348	−	0.990102i	$-0.544822\pi$
$383$	−5.49333	−0.280696	−0.140348	+	0.990102i	$0.544822\pi$
$384$	0	0
$385$	0	0
$386$	1.48528i	0.0755988i
$387$	0	0
$388$	− 3.16693i	− 0.160776i
$389$	15.5147i	0.786627i	0.919404	+	0.393314i	$0.128671\pi$
$389$	15.5147i	0.786627i	−0.919404	+	0.393314i	$0.871329\pi$
$390$	0	0
$391$	− 14.6969i	− 0.743256i
$392$	0	0
$393$	0	0
$394$	−16.9706	−0.854965
$395$	1.85514	0.0933424
$396$	0	0
$397$	− 15.1682i	− 0.761270i	−0.924725	−	0.380635i	$-0.875705\pi$
$397$	− 15.1682i	− 0.761270i	0.924725	−	0.380635i	$-0.124295\pi$
$398$	20.9077	1.04801
$399$	0	0
$400$	1.00000	0.0500000
$401$	19.7574i	0.986635i	0.869849	+	0.493318i	$0.164216\pi$
$401$	19.7574i	0.986635i	−0.869849	+	0.493318i	$0.835784\pi$
$402$	0	0
$403$	6.51472	0.324521
$404$	−7.34847	−0.365600
$405$	0	0
$406$	0	0
$407$	22.2426i	1.10253i
$408$	0	0
$409$	3.76127i	0.185983i	0.995667	+	0.0929915i	$0.0296430\pi$
$409$	3.76127i	0.185983i	−0.995667	+	0.0929915i	$0.970357\pi$
$410$	6.00000i	0.296319i
$411$	0	0
$412$	− 11.1097i	− 0.547338i
$413$	0	0
$414$	0	0
$415$	37.4558	1.83864
$416$	−0.717439	−0.0351753
$417$	0	0
$418$	− 20.7846i	− 1.01661i
$419$	−7.94282	−0.388032	−0.194016	−	0.980998i	$-0.562151\pi$
$419$	−7.94282	−0.388032	−0.194016	+	0.980998i	$0.562151\pi$
$420$	0	0
$421$	−23.4558	−1.14317	−0.571584	−	0.820544i	$-0.693671\pi$
$421$	−23.4558	−1.14317	−0.571584	+	0.820544i	$0.693671\pi$
$422$	− 3.48528i	− 0.169661i
$423$	0	0
$424$	14.4853	0.703467
$425$	2.44949	0.118818
$426$	0	0
$427$	0	0
$428$	− 2.48528i	− 0.120131i
$429$	0	0
$430$	− 17.1464i	− 0.826874i
$431$	1.75736i	0.0846490i	0.999104	+	0.0423245i	$0.0134763\pi$
$431$	1.75736i	0.0846490i	−0.999104	+	0.0423245i	$0.986524\pi$
$432$	0	0
$433$	2.57258i	0.123630i	0.998088	+	0.0618152i	$0.0196889\pi$
$433$	2.57258i	0.123630i	−0.998088	+	0.0618152i	$0.980311\pi$
$434$	0	0
$435$	0	0
$436$	17.7279	0.849013
$437$	−29.3939	−1.40610
$438$	0	0
$439$	4.30463i	0.205449i	0.994710	+	0.102724i	$0.0327560\pi$
$439$	4.30463i	0.205449i	−0.994710	+	0.102724i	$0.967244\pi$
$440$	−10.3923	−0.495434
$441$	0	0
$442$	−1.75736	−0.0835891
$443$	25.4558i	1.20944i	0.796437	+	0.604722i	$0.206716\pi$
$443$	25.4558i	1.20944i	−0.796437	+	0.604722i	$0.793284\pi$
$444$	0	0
$445$	7.45584	0.353441
$446$	−10.3923	−0.492090
$447$	0	0
$448$	0	0
$449$	− 5.27208i	− 0.248805i	−0.992232	−	0.124402i	$-0.960299\pi$
$449$	− 5.27208i	− 0.248805i	0.992232	−	0.124402i	$-0.0397014\pi$
$450$	0	0
$451$	− 10.3923i	− 0.489355i
$452$	10.2426i	0.481773i
$453$	0	0
$454$	25.0892i	1.17750i
$455$	0	0
$456$	0	0
$457$	−23.0000	−1.07589	−0.537947	−	0.842978i	$-0.680800\pi$
$457$	−23.0000	−1.07589	−0.537947	+	0.842978i	$0.680800\pi$
$458$	−5.61642	−0.262438
$459$	0	0
$460$	14.6969i	0.685248i
$461$	−21.4511	−0.999076	−0.499538	−	0.866292i	$-0.666497\pi$
$461$	−21.4511	−0.999076	−0.499538	+	0.866292i	$0.666497\pi$
$462$	0	0
$463$	−22.0000	−1.02243	−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000	−1.02243	−0.511213	+	0.859454i	$0.670804\pi$
$464$	1.75736i	0.0815834i
$465$	0	0
$466$	−20.4853	−0.948962
$467$	−17.7408	−0.820945	−0.410473	−	0.911873i	$-0.634636\pi$
$467$	−17.7408	−0.820945	−0.410473	+	0.911873i	$0.634636\pi$
$468$	0	0
$469$	0	0
$470$	− 31.4558i	− 1.45095i
$471$	0	0
$472$	− 2.44949i	− 0.112747i
$473$	29.6985i	1.36554i
$474$	0	0
$475$	− 4.89898i	− 0.224781i
$476$	0	0
$477$	0	0
$478$	−16.2426	−0.742921
$479$	−2.44949	−0.111920	−0.0559600	−	0.998433i	$-0.517822\pi$
$479$	−2.44949	−0.111920	−0.0559600	+	0.998433i	$0.517822\pi$
$480$	0	0
$481$	− 3.76127i	− 0.171499i
$482$	1.13770	0.0518210
$483$	0	0
$484$	7.00000	0.318182
$485$	7.75736i	0.352244i
$486$	0	0
$487$	22.0000	0.996915	0.498458	−	0.866914i	$-0.333900\pi$
$487$	22.0000	0.996915	0.498458	+	0.866914i	$0.333900\pi$
$488$	−4.18154	−0.189289
$489$	0	0
$490$	0	0
$491$	− 34.9706i	− 1.57820i	−0.614265	−	0.789100i	$-0.710547\pi$
$491$	− 34.9706i	− 1.57820i	0.614265	−	0.789100i	$-0.289453\pi$
$492$	0	0
$493$	4.30463i	0.193871i
$494$	3.51472i	0.158135i
$495$	0	0
$496$	9.08052i	0.407727i
$497$	0	0
$498$	0	0
$499$	−24.4558	−1.09479	−0.547397	−	0.836873i	$-0.684381\pi$
$499$	−24.4558	−1.09479	−0.547397	+	0.836873i	$0.684381\pi$
$500$	9.79796	0.438178
$501$	0	0
$502$	15.2913i	0.682483i
$503$	0.594346	0.0265006	0.0132503	−	0.999912i	$-0.495782\pi$
$503$	0.594346	0.0265006	0.0132503	+	0.999912i	$0.495782\pi$
$504$	0	0
$505$	18.0000	0.800989
$506$	− 25.4558i	− 1.13165i
$507$	0	0
$508$	7.72792	0.342871
$509$	9.20361	0.407943	0.203971	−	0.978977i	$-0.434615\pi$
$509$	9.20361	0.407943	0.203971	+	0.978977i	$0.434615\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	25.0892i	1.10664i
$515$	27.2132i	1.19916i
$516$	0	0
$517$	54.4831i	2.39616i
$518$	0	0
$519$	0	0
$520$	1.75736	0.0770653
$521$	29.9882	1.31381	0.656904	−	0.753974i	$-0.271866\pi$
$521$	29.9882	1.31381	0.656904	+	0.753974i	$0.271866\pi$
$522$	0	0
$523$	27.4156i	1.19880i	0.800449	+	0.599401i	$0.204595\pi$
$523$	27.4156i	1.19880i	−0.800449	+	0.599401i	$0.795405\pi$
$524$	0	0
$525$	0	0
$526$	−6.72792	−0.293351
$527$	22.2426i	0.968905i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	−35.4815	−1.54122
$531$	0	0
$532$	0	0
$533$	1.75736i	0.0761197i
$534$	0	0
$535$	6.08767i	0.263193i
$536$	13.4853i	0.582475i
$537$	0	0
$538$	9.79796i	0.422420i
$539$	0	0
$540$	0	0
$541$	−5.45584	−0.234565	−0.117283	−	0.993099i	$-0.537418\pi$
$541$	−5.45584	−0.234565	−0.117283	+	0.993099i	$0.537418\pi$
$542$	−18.8785	−0.810900
$543$	0	0
$544$	− 2.44949i	− 0.105021i
$545$	−43.4244	−1.86010
$546$	0	0
$547$	39.9706	1.70902	0.854509	−	0.519437i	$-0.173858\pi$
$547$	39.9706	1.70902	0.854509	+	0.519437i	$0.173858\pi$
$548$	6.00000i	0.256307i
$549$	0	0
$550$	4.24264	0.180907
$551$	8.60927	0.366767
$552$	0	0
$553$	0	0
$554$	− 23.7279i	− 1.00810i
$555$	0	0
$556$	− 8.06591i	− 0.342071i
$557$	− 21.2132i	− 0.898832i	−0.893323	−	0.449416i	$-0.851632\pi$
$557$	− 21.2132i	− 0.898832i	0.893323	−	0.449416i	$-0.148368\pi$
$558$	0	0
$559$	− 5.02207i	− 0.212411i
$560$	0	0
$561$	0	0
$562$	13.7574	0.580319
$563$	45.8739	1.93335	0.966676	−	0.256002i	$-0.0824054\pi$
$563$	45.8739	1.93335	0.966676	+	0.256002i	$0.0824054\pi$
$564$	0	0
$565$	− 25.0892i	− 1.05551i
$566$	−6.03668	−0.253741
$567$	0	0
$568$	−12.7279	−0.534052
$569$	10.2426i	0.429394i	0.976681	+	0.214697i	$0.0688764\pi$
$569$	10.2426i	0.429394i	−0.976681	+	0.214697i	$0.931124\pi$
$570$	0	0
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	−3.04384	−0.127269
$573$	0	0
$574$	0	0
$575$	− 6.00000i	− 0.250217i
$576$	0	0
$577$	− 27.4156i	− 1.14133i	−0.821184	−	0.570664i	$-0.806686\pi$
$577$	− 27.4156i	− 1.14133i	0.821184	−	0.570664i	$-0.193314\pi$
$578$	11.0000i	0.457540i
$579$	0	0
$580$	− 4.30463i	− 0.178740i
$581$	0	0
$582$	0	0
$583$	61.4558	2.54524
$584$	−5.49333	−0.227315
$585$	0	0
$586$	12.8418i	0.530490i
$587$	−32.4377	−1.33885	−0.669424	−	0.742881i	$-0.733459\pi$
$587$	−32.4377	−1.33885	−0.669424	+	0.742881i	$0.733459\pi$
$588$	0	0
$589$	44.4853	1.83298
$590$	6.00000i	0.247016i
$591$	0	0
$592$	5.24264	0.215471
$593$	1.85514	0.0761816	0.0380908	−	0.999274i	$-0.487872\pi$
$593$	1.85514	0.0761816	0.0380908	+	0.999274i	$0.487872\pi$
$594$	0	0
$595$	0	0
$596$	16.2426i	0.665324i
$597$	0	0
$598$	4.30463i	0.176030i
$599$	3.51472i	0.143608i	0.997419	+	0.0718038i	$0.0228755\pi$
$599$	3.51472i	0.143608i	−0.997419	+	0.0718038i	$0.977124\pi$
$600$	0	0
$601$	41.5182i	1.69356i	0.531940	+	0.846782i	$0.321463\pi$
$601$	41.5182i	1.69356i	−0.531940	+	0.846782i	$0.678537\pi$
$602$	0	0
$603$	0	0
$604$	−8.75736	−0.356332
$605$	−17.1464	−0.697101
$606$	0	0
$607$	36.0759i	1.46428i	0.681157	+	0.732138i	$0.261477\pi$
$607$	36.0759i	1.46428i	−0.681157	+	0.732138i	$0.738523\pi$
$608$	−4.89898	−0.198680
$609$	0	0
$610$	10.2426	0.414712
$611$	− 9.21320i	− 0.372726i
$612$	0	0
$613$	−28.2132	−1.13952	−0.569760	−	0.821811i	$-0.692964\pi$
$613$	−28.2132	−1.13952	−0.569760	+	0.821811i	$0.692964\pi$
$614$	26.8213	1.08242
$615$	0	0
$616$	0	0
$617$	− 4.24264i	− 0.170802i	−0.996347	−	0.0854011i	$-0.972783\pi$
$617$	− 4.24264i	− 0.170802i	0.996347	−	0.0854011i	$-0.0272172\pi$
$618$	0	0
$619$	12.7187i	0.511208i	0.966782	+	0.255604i	$0.0822743\pi$
$619$	12.7187i	0.511208i	−0.966782	+	0.255604i	$0.917726\pi$
$620$	− 22.2426i	− 0.893286i
$621$	0	0
$622$	− 17.1464i	− 0.687509i
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	−20.1903	−0.806965
$627$	0	0
$628$	− 10.3923i	− 0.414698i
$629$	12.8418	0.512036
$630$	0	0
$631$	−14.7574	−0.587481	−0.293741	−	0.955885i	$-0.594900\pi$
$631$	−14.7574	−0.587481	−0.293741	+	0.955885i	$0.594900\pi$
$632$	0.757359i	0.0301261i
$633$	0	0
$634$	22.2426	0.883368
$635$	−18.9295	−0.751193
$636$	0	0
$637$	0	0
$638$	7.45584i	0.295180i
$639$	0	0
$640$	2.44949i	0.0968246i
$641$	33.2132i	1.31184i	0.754829	+	0.655921i	$0.227720\pi$
$641$	33.2132i	1.31184i	−0.754829	+	0.655921i	$0.772280\pi$
$642$	0	0
$643$	− 1.73205i	− 0.0683054i	−0.999417	−	0.0341527i	$-0.989127\pi$
$643$	− 1.73205i	− 0.0683054i	0.999417	−	0.0341527i	$-0.0108733\pi$
$644$	0	0
$645$	0	0
$646$	−12.0000	−0.472134
$647$	−20.7846	−0.817127	−0.408564	−	0.912730i	$-0.633970\pi$
$647$	−20.7846	−0.817127	−0.408564	+	0.912730i	$0.633970\pi$
$648$	0	0
$649$	− 10.3923i	− 0.407934i
$650$	−0.717439	−0.0281403
$651$	0	0
$652$	9.48528	0.371472
$653$	2.48528i	0.0972566i	0.998817	+	0.0486283i	$0.0154850\pi$
$653$	2.48528i	0.0972566i	−0.998817	+	0.0486283i	$0.984515\pi$
$654$	0	0
$655$	0	0
$656$	−2.44949	−0.0956365
$657$	0	0
$658$	0	0
$659$	− 22.2426i	− 0.866450i	−0.901286	−	0.433225i	$-0.857376\pi$
$659$	− 22.2426i	− 0.866450i	0.901286	−	0.433225i	$-0.142624\pi$
$660$	0	0
$661$	4.89898i	0.190548i	0.995451	+	0.0952741i	$0.0303728\pi$
$661$	4.89898i	0.190548i	−0.995451	+	0.0952741i	$0.969627\pi$
$662$	− 10.0000i	− 0.388661i
$663$	0	0
$664$	15.2913i	0.593417i
$665$	0	0
$666$	0	0
$667$	10.5442	0.408271
$668$	0.594346	0.0229959
$669$	0	0
$670$	− 33.0321i	− 1.27614i
$671$	−17.7408	−0.684875
$672$	0	0
$673$	−45.4558	−1.75219	−0.876097	−	0.482135i	$-0.839862\pi$
$673$	−45.4558	−1.75219	−0.876097	+	0.482135i	$0.839862\pi$
$674$	21.4558i	0.826448i
$675$	0	0
$676$	−12.4853	−0.480203
$677$	14.6969	0.564849	0.282425	−	0.959289i	$-0.408861\pi$
$677$	14.6969	0.564849	0.282425	+	0.959289i	$0.408861\pi$
$678$	0	0
$679$	0	0
$680$	6.00000i	0.230089i
$681$	0	0
$682$	38.5254i	1.47521i
$683$	10.2426i	0.391924i	0.980612	+	0.195962i	$0.0627829\pi$
$683$	10.2426i	0.391924i	−0.980612	+	0.195962i	$0.937217\pi$
$684$	0	0
$685$	− 14.6969i	− 0.561541i
$686$	0	0
$687$	0	0
$688$	7.00000	0.266872
$689$	−10.3923	−0.395915
$690$	0	0
$691$	2.57258i	0.0978657i	0.998802	+	0.0489328i	$0.0155820\pi$
$691$	2.57258i	0.0978657i	−0.998802	+	0.0489328i	$0.984418\pi$
$692$	−20.7846	−0.790112
$693$	0	0
$694$	−4.97056	−0.188680
$695$	19.7574i	0.749439i
$696$	0	0
$697$	−6.00000	−0.227266
$698$	−0.123093	−0.00465914
$699$	0	0
$700$	0	0
$701$	− 20.4853i	− 0.773718i	−0.922139	−	0.386859i	$-0.873560\pi$
$701$	− 20.4853i	− 0.773718i	0.922139	−	0.386859i	$-0.126440\pi$
$702$	0	0
$703$	− 25.6836i	− 0.968675i
$704$	− 4.24264i	− 0.159901i
$705$	0	0
$706$	10.9867i	0.413488i
$707$	0	0
$708$	0	0
$709$	16.2132	0.608900	0.304450	−	0.952528i	$-0.401527\pi$
$709$	16.2132	0.608900	0.304450	+	0.952528i	$0.401527\pi$
$710$	31.1769	1.17005
$711$	0	0
$712$	3.04384i	0.114073i
$713$	54.4831	2.04041
$714$	0	0
$715$	7.45584	0.278833
$716$	− 6.72792i	− 0.251434i
$717$	0	0
$718$	14.4853	0.540586
$719$	−52.6280	−1.96269	−0.981346	−	0.192249i	$-0.938422\pi$
$719$	−52.6280	−1.96269	−0.981346	+	0.192249i	$0.938422\pi$
$720$	0	0
$721$	0	0
$722$	5.00000i	0.186081i
$723$	0	0
$724$	9.79796i	0.364138i
$725$	1.75736i	0.0652667i
$726$	0	0
$727$	− 28.0821i	− 1.04151i	−0.853707	−	0.520754i	$-0.825651\pi$
$727$	− 28.0821i	− 1.04151i	0.853707	−	0.520754i	$-0.174349\pi$
$728$	0	0
$729$	0	0
$730$	13.4558	0.498023
$731$	17.1464	0.634184
$732$	0	0
$733$	1.31178i	0.0484519i	0.999707	+	0.0242259i	$0.00771211\pi$
$733$	1.31178i	0.0484519i	−0.999707	+	0.0242259i	$0.992288\pi$
$734$	29.9882	1.10689
$735$	0	0
$736$	−6.00000	−0.221163
$737$	57.2132i	2.10748i
$738$	0	0
$739$	−8.45584	−0.311053	−0.155527	−	0.987832i	$-0.549707\pi$
$739$	−8.45584	−0.311053	−0.155527	+	0.987832i	$0.549707\pi$
$740$	−12.8418	−0.472074
$741$	0	0
$742$	0	0
$743$	− 18.7279i	− 0.687061i	−0.939142	−	0.343530i	$-0.888377\pi$
$743$	− 18.7279i	− 0.687061i	0.939142	−	0.343530i	$-0.111623\pi$
$744$	0	0
$745$	− 39.7862i	− 1.45765i
$746$	22.0000i	0.805477i
$747$	0	0
$748$	− 10.3923i	− 0.379980i
$749$	0	0
$750$	0	0
$751$	53.4558	1.95063	0.975316	−	0.220815i	$-0.0708717\pi$
$751$	53.4558	1.95063	0.975316	+	0.220815i	$0.0708717\pi$
$752$	12.8418	0.468292
$753$	0	0
$754$	− 1.26080i	− 0.0459156i
$755$	21.4511	0.780684
$756$	0	0
$757$	32.7574	1.19059	0.595293	−	0.803509i	$-0.297036\pi$
$757$	32.7574	1.19059	0.595293	+	0.803509i	$0.297036\pi$
$758$	7.48528i	0.271878i
$759$	0	0
$760$	12.0000	0.435286
$761$	−24.4949	−0.887939	−0.443970	−	0.896042i	$-0.646430\pi$
$761$	−24.4949	−0.887939	−0.443970	+	0.896042i	$0.646430\pi$
$762$	0	0
$763$	0	0
$764$	21.2132i	0.767467i
$765$	0	0
$766$	5.49333i	0.198482i
$767$	1.75736i	0.0634546i
$768$	0	0
$769$	− 40.3805i	− 1.45616i	−0.685493	−	0.728080i	$-0.740413\pi$
$769$	− 40.3805i	− 1.45616i	0.685493	−	0.728080i	$-0.259587\pi$
$770$	0	0
$771$	0	0
$772$	1.48528	0.0534564
$773$	9.79796	0.352408	0.176204	−	0.984354i	$-0.443618\pi$
$773$	9.79796	0.352408	0.176204	+	0.984354i	$0.443618\pi$
$774$	0	0
$775$	9.08052i	0.326182i
$776$	−3.16693	−0.113686
$777$	0	0
$778$	15.5147	0.556230
$779$	12.0000i	0.429945i
$780$	0	0
$781$	−54.0000	−1.93227
$782$	−14.6969	−0.525561
$783$	0	0
$784$	0	0
$785$	25.4558i	0.908558i
$786$	0	0
$787$	28.2562i	1.00722i	0.863930	+	0.503612i	$0.167996\pi$
$787$	28.2562i	1.00722i	−0.863930	+	0.503612i	$0.832004\pi$
$788$	16.9706i	0.604551i
$789$	0	0
$790$	− 1.85514i	− 0.0660031i
$791$	0	0
$792$	0	0
$793$	3.00000	0.106533
$794$	−15.1682	−0.538299
$795$	0	0
$796$	− 20.9077i	− 0.741054i
$797$	17.7408	0.628410	0.314205	−	0.949355i	$-0.398262\pi$
$797$	17.7408	0.628410	0.314205	+	0.949355i	$0.398262\pi$
$798$	0	0
$799$	31.4558	1.11283
$800$	− 1.00000i	− 0.0353553i
$801$	0	0
$802$	19.7574	0.697657
$803$	−23.3062	−0.822458
$804$	0	0
$805$	0	0
$806$	− 6.51472i	− 0.229471i
$807$	0	0
$808$	7.34847i	0.258518i
$809$	14.4853i	0.509275i	0.967037	+	0.254638i	$0.0819562\pi$
$809$	14.4853i	0.509275i	−0.967037	+	0.254638i	$0.918044\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	22.2426	0.779604
$815$	−23.2341	−0.813855
$816$	0	0
$817$	− 34.2929i	− 1.19976i
$818$	3.76127	0.131510
$819$	0	0
$820$	6.00000	0.209529
$821$	− 22.9706i	− 0.801678i	−0.916148	−	0.400839i	$-0.868719\pi$
$821$	− 22.9706i	− 0.801678i	0.916148	−	0.400839i	$-0.131281\pi$
$822$	0	0
$823$	−2.75736	−0.0961155	−0.0480578	−	0.998845i	$-0.515303\pi$
$823$	−2.75736	−0.0961155	−0.0480578	+	0.998845i	$0.515303\pi$
$824$	−11.1097	−0.387026
$825$	0	0
$826$	0	0
$827$	3.51472i	0.122219i	0.998131	+	0.0611094i	$0.0194638\pi$
$827$	3.51472i	0.122219i	−0.998131	+	0.0611094i	$0.980536\pi$
$828$	0	0
$829$	− 9.79796i	− 0.340297i	−0.985418	−	0.170149i	$-0.945575\pi$
$829$	− 9.79796i	− 0.340297i	0.985418	−	0.170149i	$-0.0544248\pi$
$830$	− 37.4558i	− 1.30011i
$831$	0	0
$832$	0.717439i	0.0248727i
$833$	0	0
$834$	0	0
$835$	−1.45584	−0.0503816
$836$	−20.7846	−0.718851
$837$	0	0
$838$	7.94282i	0.274380i
$839$	−15.2913	−0.527914	−0.263957	−	0.964534i	$-0.585028\pi$
$839$	−15.2913	−0.527914	−0.263957	+	0.964534i	$0.585028\pi$
$840$	0	0
$841$	25.9117	0.893506
$842$	23.4558i	0.808342i
$843$	0	0
$844$	−3.48528	−0.119968
$845$	30.5826	1.05207
$846$	0	0
$847$	0	0
$848$	− 14.4853i	− 0.497427i
$849$	0	0
$850$	− 2.44949i	− 0.0840168i
$851$	− 31.4558i	− 1.07829i
$852$	0	0
$853$	31.7713i	1.08783i	0.839141	+	0.543914i	$0.183058\pi$
$853$	31.7713i	1.08783i	−0.839141	+	0.543914i	$0.816942\pi$
$854$	0	0
$855$	0	0
$856$	−2.48528	−0.0849452
$857$	27.5387	0.940705	0.470353	−	0.882479i	$-0.344127\pi$
$857$	27.5387	0.940705	0.470353	+	0.882479i	$0.344127\pi$
$858$	0	0
$859$	28.2562i	0.964088i	0.876147	+	0.482044i	$0.160105\pi$
$859$	28.2562i	0.964088i	−0.876147	+	0.482044i	$0.839895\pi$
$860$	−17.1464	−0.584688
$861$	0	0
$862$	1.75736	0.0598559
$863$	− 37.4558i	− 1.27501i	−0.770445	−	0.637506i	$-0.779966\pi$
$863$	− 37.4558i	− 1.27501i	0.770445	−	0.637506i	$-0.220034\pi$
$864$	0	0
$865$	50.9117	1.73105
$866$	2.57258	0.0874199
$867$	0	0
$868$	0	0
$869$	3.21320i	0.109000i
$870$	0	0
$871$	− 9.67487i	− 0.327820i
$872$	− 17.7279i	− 0.600343i
$873$	0	0
$874$	29.3939i	0.994263i
$875$	0	0
$876$	0	0
$877$	−46.2132	−1.56051	−0.780254	−	0.625462i	$-0.784910\pi$
$877$	−46.2132	−1.56051	−0.780254	+	0.625462i	$0.784910\pi$
$878$	4.30463	0.145274
$879$	0	0
$880$	10.3923i	0.350325i
$881$	25.0892	0.845278	0.422639	−	0.906298i	$-0.361104\pi$
$881$	25.0892	0.845278	0.422639	+	0.906298i	$0.361104\pi$
$882$	0	0
$883$	−2.00000	−0.0673054	−0.0336527	−	0.999434i	$-0.510714\pi$
$883$	−2.00000	−0.0673054	−0.0336527	+	0.999434i	$0.510714\pi$
$884$	1.75736i	0.0591064i
$885$	0	0
$886$	25.4558	0.855206
$887$	−9.20361	−0.309027	−0.154514	−	0.987991i	$-0.549381\pi$
$887$	−9.20361	−0.309027	−0.154514	+	0.987991i	$0.549381\pi$
$888$	0	0
$889$	0	0
$890$	− 7.45584i	− 0.249920i
$891$	0	0
$892$	10.3923i	0.347960i
$893$	− 62.9117i	− 2.10526i
$894$	0	0
$895$	16.4800i	0.550865i
$896$	0	0
$897$	0	0
$898$	−5.27208	−0.175932
$899$	−15.9577	−0.532220
$900$	0	0
$901$	− 35.4815i	− 1.18206i
$902$	−10.3923	−0.346026
$903$	0	0
$904$	10.2426	0.340665
$905$	− 24.0000i	− 0.797787i
$906$	0	0
$907$	−13.4853	−0.447771	−0.223886	−	0.974615i	$-0.571874\pi$
$907$	−13.4853	−0.447771	−0.223886	+	0.974615i	$0.571874\pi$
$908$	25.0892	0.832616
$909$	0	0
$910$	0	0
$911$	10.2426i	0.339354i	0.985500	+	0.169677i	$0.0542724\pi$
$911$	10.2426i	0.339354i	−0.985500	+	0.169677i	$0.945728\pi$
$912$	0	0
$913$	64.8754i	2.14706i
$914$	23.0000i	0.760772i
$915$	0	0
$916$	5.61642i	0.185572i
$917$	0	0
$918$	0	0
$919$	25.7279	0.848686	0.424343	−	0.905502i	$-0.360505\pi$
$919$	25.7279	0.848686	0.424343	+	0.905502i	$0.360505\pi$
$920$	14.6969	0.484544
$921$	0	0
$922$	21.4511i	0.706453i
$923$	9.13151	0.300567
$924$	0	0
$925$	5.24264	0.172377
$926$	22.0000i	0.722965i
$927$	0	0
$928$	1.75736	0.0576881
$929$	−28.7274	−0.942516	−0.471258	−	0.881995i	$-0.656200\pi$
$929$	−28.7274	−0.942516	−0.471258	+	0.881995i	$0.656200\pi$
$930$	0	0
$931$	0	0
$932$	20.4853i	0.671018i
$933$	0	0
$934$	17.7408i	0.580496i
$935$	25.4558i	0.832495i
$936$	0	0
$937$	− 33.5033i	− 1.09451i	−0.836967	−	0.547253i	$-0.815674\pi$
$937$	− 33.5033i	− 1.09451i	0.836967	−	0.547253i	$-0.184326\pi$
$938$	0	0
$939$	0	0
$940$	−31.4558	−1.02598
$941$	−42.2357	−1.37684	−0.688422	−	0.725311i	$-0.741696\pi$
$941$	−42.2357	−1.37684	−0.688422	+	0.725311i	$0.741696\pi$
$942$	0	0
$943$	14.6969i	0.478598i
$944$	−2.44949	−0.0797241
$945$	0	0
$946$	29.6985	0.965581
$947$	16.2426i	0.527815i	0.964548	+	0.263907i	$0.0850114\pi$
$947$	16.2426i	0.527815i	−0.964548	+	0.263907i	$0.914989\pi$
$948$	0	0
$949$	3.94113	0.127934
$950$	−4.89898	−0.158944
$951$	0	0
$952$	0	0
$953$	1.02944i	0.0333467i	0.999861	+	0.0166734i	$0.00530755\pi$
$953$	1.02944i	0.0333467i	−0.999861	+	0.0166734i	$0.994692\pi$
$954$	0	0
$955$	− 51.9615i	− 1.68144i
$956$	16.2426i	0.525325i
$957$	0	0
$958$	2.44949i	0.0791394i
$959$	0	0
$960$	0	0
$961$	−51.4558	−1.65987
$962$	−3.76127	−0.121268
$963$	0	0
$964$	− 1.13770i	− 0.0366430i
$965$	−3.63818	−0.117117
$966$	0	0
$967$	−42.6985	−1.37309	−0.686545	−	0.727087i	$-0.740874\pi$
$967$	−42.6985	−1.37309	−0.686545	+	0.727087i	$0.740874\pi$
$968$	− 7.00000i	− 0.224989i
$969$	0	0
$970$	7.75736	0.249074
$971$	40.3805	1.29587	0.647936	−	0.761694i	$-0.275632\pi$
$971$	40.3805	1.29587	0.647936	+	0.761694i	$0.275632\pi$
$972$	0	0
$973$	0	0
$974$	− 22.0000i	− 0.704925i
$975$	0	0
$976$	4.18154i	0.133848i
$977$	− 6.00000i	− 0.191957i	−0.995383	−	0.0959785i	$-0.969402\pi$
$977$	− 6.00000i	− 0.191957i	0.995383	−	0.0959785i	$-0.0305980\pi$
$978$	0	0
$979$	12.9139i	0.412730i
$980$	0	0
$981$	0	0
$982$	−34.9706	−1.11596
$983$	−23.2341	−0.741053	−0.370526	−	0.928822i	$-0.620823\pi$
$983$	−23.2341	−0.741053	−0.370526	+	0.928822i	$0.620823\pi$
$984$	0	0
$985$	− 41.5692i	− 1.32451i
$986$	4.30463	0.137087
$987$	0	0
$988$	3.51472	0.111818
$989$	− 42.0000i	− 1.33552i
$990$	0	0
$991$	−20.2132	−0.642094	−0.321047	−	0.947063i	$-0.604035\pi$
$991$	−20.2132	−0.642094	−0.321047	+	0.947063i	$0.604035\pi$
$992$	9.08052	0.288307
$993$	0	0
$994$	0	0
$995$	51.2132i	1.62357i
$996$	0	0
$997$	− 19.4728i	− 0.616711i	−0.951271	−	0.308355i	$-0.900221\pi$
$997$	− 19.4728i	− 0.616711i	0.951271	−	0.308355i	$-0.0997786\pi$
$998$	24.4558i	0.774136i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2646.2.d.d.2645.4		8
3.2	odd	2	inner	2646.2.d.d.2645.5		8
7.2	even	3		378.2.k.d.269.1	yes	8
7.3	odd	6		378.2.k.d.215.4	yes	8
7.6	odd	2	inner	2646.2.d.d.2645.2		8
21.2	odd	6		378.2.k.d.269.4	yes	8
21.17	even	6		378.2.k.d.215.1	✓	8
21.20	even	2	inner	2646.2.d.d.2645.7		8
63.2	odd	6		1134.2.t.f.1025.1		8
63.16	even	3		1134.2.t.f.1025.4		8
63.23	odd	6		1134.2.l.e.269.4		8
63.31	odd	6		1134.2.t.f.593.1		8
63.38	even	6		1134.2.l.e.215.3		8
63.52	odd	6		1134.2.l.e.215.2		8
63.58	even	3		1134.2.l.e.269.1		8
63.59	even	6		1134.2.t.f.593.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.k.d.215.1	✓	8	21.17	even	6
378.2.k.d.215.4	yes	8	7.3	odd	6
378.2.k.d.269.1	yes	8	7.2	even	3
378.2.k.d.269.4	yes	8	21.2	odd	6
1134.2.l.e.215.2		8	63.52	odd	6
1134.2.l.e.215.3		8	63.38	even	6
1134.2.l.e.269.1		8	63.58	even	3
1134.2.l.e.269.4		8	63.23	odd	6
1134.2.t.f.593.1		8	63.31	odd	6
1134.2.t.f.593.4		8	63.59	even	6
1134.2.t.f.1025.1		8	63.2	odd	6
1134.2.t.f.1025.4		8	63.16	even	3
2646.2.d.d.2645.2		8	7.6	odd	2	inner
2646.2.d.d.2645.4		8	1.1	even	1	trivial
2646.2.d.d.2645.5		8	3.2	odd	2	inner
2646.2.d.d.2645.7		8	21.20	even	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:	\( N \)	\(=\)	\( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2646.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +2.44949 q^{5} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +2.44949 q^{5} +1.00000i q^{8} -2.44949i q^{10} +4.24264i q^{11} -0.717439i q^{13} +1.00000 q^{16} +2.44949 q^{17} -4.89898i q^{19} -2.44949 q^{20} +4.24264 q^{22} -6.00000i q^{23} +1.00000 q^{25} -0.717439 q^{26} +1.75736i q^{29} +9.08052i q^{31} -1.00000i q^{32} -2.44949i q^{34} +5.24264 q^{37} -4.89898 q^{38} +2.44949i q^{40} -2.44949 q^{41} +7.00000 q^{43} -4.24264i q^{44} -6.00000 q^{46} +12.8418 q^{47} -1.00000i q^{50} +0.717439i q^{52} -14.4853i q^{53} +10.3923i q^{55} +1.75736 q^{58} -2.44949 q^{59} +4.18154i q^{61} +9.08052 q^{62} -1.00000 q^{64} -1.75736i q^{65} +13.4853 q^{67} -2.44949 q^{68} +12.7279i q^{71} +5.49333i q^{73} -5.24264i q^{74} +4.89898i q^{76} +0.757359 q^{79} +2.44949 q^{80} +2.44949i q^{82} +15.2913 q^{83} +6.00000 q^{85} -7.00000i q^{86} -4.24264 q^{88} +3.04384 q^{89} +6.00000i q^{92} -12.8418i q^{94} -12.0000i q^{95} +3.16693i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4}+O(q^{10}) \) 8 * q - 8 * q^4	\( 8 q - 8 q^{4} + 8 q^{16} + 8 q^{25} + 8 q^{37} + 56 q^{43} - 48 q^{46} + 48 q^{58} - 8 q^{64} + 40 q^{67} + 40 q^{79} + 48 q^{85}+O(q^{100}) \) 8 * q - 8 * q^4 + 8 * q^16 + 8 * q^25 + 8 * q^37 + 56 * q^43 - 48 * q^46 + 48 * q^58 - 8 * q^64 + 40 * q^67 + 40 * q^79 + 48 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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