LMFDB - Embedded newform 2646.2.d.d.2645.8

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			2645.8
Root			$0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	2646.2645
Dual form			2646.2.d.d.2645.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +2.44949 q^{5} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +2.44949 q^{5} -1.00000i q^{8} +2.44949i q^{10} +4.24264i q^{11} +4.18154i q^{13} +1.00000 q^{16} +2.44949 q^{17} +4.89898i q^{19} -2.44949 q^{20} -4.24264 q^{22} +6.00000i q^{23} +1.00000 q^{25} -4.18154 q^{26} -10.2426i q^{29} -5.61642i q^{31} +1.00000i q^{32} +2.44949i q^{34} -3.24264 q^{37} -4.89898 q^{38} -2.44949i q^{40} -2.44949 q^{41} +7.00000 q^{43} -4.24264i q^{44} -6.00000 q^{46} -7.94282 q^{47} +1.00000i q^{50} -4.18154i q^{52} -2.48528i q^{53} +10.3923i q^{55} +10.2426 q^{58} -2.44949 q^{59} -0.717439i q^{61} +5.61642 q^{62} -1.00000 q^{64} +10.2426i q^{65} -3.48528 q^{67} -2.44949 q^{68} +12.7279i q^{71} +15.2913i q^{73} -3.24264i q^{74} -4.89898i q^{76} +9.24264 q^{79} +2.44949 q^{80} -2.44949i q^{82} -5.49333 q^{83} +6.00000 q^{85} +7.00000i q^{86} +4.24264 q^{88} -17.7408 q^{89} -6.00000i q^{92} -7.94282i q^{94} +12.0000i q^{95} -6.63103i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$13$	4.18154i	1.15975i	0.814705	+	0.579875i	$0.196899\pi$
$13$	4.18154i	1.15975i	−0.814705	+	0.579875i	$0.803101\pi$
$14$	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	$0.189949\pi$
$19$	4.89898i	1.12390i	−0.827170	+	0.561951i	$0.810051\pi$
$20$	−2.44949	−0.547723
$21$	0	0
$22$	−4.24264	−0.904534
$23$	6.00000i	1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$	6.00000i	1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−4.18154	−0.820068
$27$	0	0
$28$	0	0
$29$	− 10.2426i	− 1.90201i	−0.309175	−	0.951005i	$-0.600053\pi$
$29$	− 10.2426i	− 1.90201i	0.309175	−	0.951005i	$-0.399947\pi$
$30$	0	0
$31$	− 5.61642i	− 1.00874i	−0.863488	−	0.504369i	$-0.831725\pi$
$31$	− 5.61642i	− 1.00874i	0.863488	−	0.504369i	$-0.168275\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	2.44949i	0.420084i
$35$	0	0
$36$	0	0
$37$	−3.24264	−0.533087	−0.266543	−	0.963823i	$-0.585882\pi$
$37$	−3.24264	−0.533087	−0.266543	+	0.963823i	$0.585882\pi$
$38$	−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
$47$	−7.94282	−1.15858	−0.579289	−	0.815122i	$-0.696670\pi$
$47$	−7.94282	−1.15858	−0.579289	+	0.815122i	$0.696670\pi$
$48$	0	0
$49$	0	0
$50$	1.00000i	0.141421i
$51$	0	0
$52$	− 4.18154i	− 0.579875i
$53$	− 2.48528i	− 0.341380i	−0.985325	−	0.170690i	$-0.945400\pi$
$53$	− 2.48528i	− 0.341380i	0.985325	−	0.170690i	$-0.0545996\pi$
$54$	0	0
$55$	10.3923i	1.40130i
$56$	0	0
$57$	0	0
$58$	10.2426	1.34492
$59$	−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
$61$	− 0.717439i	− 0.0918586i	−0.998945	−	0.0459293i	$-0.985375\pi$
$61$	− 0.717439i	− 0.0918586i	0.998945	−	0.0459293i	$-0.0146249\pi$
$62$	5.61642	0.713286
$63$	0	0
$64$	−1.00000	−0.125000
$65$	10.2426i	1.27044i
$66$	0	0
$67$	−3.48528	−0.425795	−0.212897	−	0.977075i	$-0.568290\pi$
$67$	−3.48528	−0.425795	−0.212897	+	0.977075i	$0.568290\pi$
$68$	−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
$73$	15.2913i	1.78971i	0.446357	+	0.894855i	$0.352721\pi$
$73$	15.2913i	1.78971i	−0.446357	+	0.894855i	$0.647279\pi$
$74$	− 3.24264i	− 0.376949i
$75$	0	0
$76$	− 4.89898i	− 0.561951i
$77$	0	0
$78$	0	0
$79$	9.24264	1.03988	0.519939	−	0.854203i	$-0.325955\pi$
$79$	9.24264	1.03988	0.519939	+	0.854203i	$0.325955\pi$
$80$	2.44949	0.273861
$81$	0	0
$82$	− 2.44949i	− 0.270501i
$83$	−5.49333	−0.602971	−0.301485	−	0.953471i	$-0.597482\pi$
$83$	−5.49333	−0.602971	−0.301485	+	0.953471i	$0.597482\pi$
$84$	0	0
$85$	6.00000	0.650791
$86$	7.00000i	0.754829i
$87$	0	0
$88$	4.24264	0.452267
$89$	−17.7408	−1.88052	−0.940259	−	0.340460i	$-0.889417\pi$
$89$	−17.7408	−1.88052	−0.940259	+	0.340460i	$0.889417\pi$
$90$	0	0
$91$	0	0
$92$	− 6.00000i	− 0.625543i
$93$	0	0
$94$	− 7.94282i	− 0.819239i
$95$	12.0000i	1.23117i
$96$	0	0
$97$	− 6.63103i	− 0.673279i	−0.941634	−	0.336640i	$-0.890710\pi$
$97$	− 6.63103i	− 0.673279i	0.941634	−	0.336640i	$-0.109290\pi$
$98$	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
$103$	6.21076i	0.611965i	0.952037	+	0.305982i	$0.0989849\pi$
$103$	6.21076i	0.611965i	−0.952037	+	0.305982i	$0.901015\pi$
$104$	4.18154	0.410034
$105$	0	0
$106$	2.48528	0.241392
$107$	14.4853i	1.40035i	0.713974	+	0.700173i	$0.246894\pi$
$107$	14.4853i	1.40035i	−0.713974	+	0.700173i	$0.753106\pi$
$108$	0	0
$109$	7.72792	0.740201	0.370100	−	0.928992i	$-0.379323\pi$
$109$	7.72792	0.740201	0.370100	+	0.928992i	$0.379323\pi$
$110$	−10.3923	−0.990867
$111$	0	0
$112$	0	0
$113$	1.75736i	0.165318i	0.996578	+	0.0826592i	$0.0263413\pi$
$113$	1.75736i	0.165318i	−0.996578	+	0.0826592i	$0.973659\pi$
$114$	0	0
$115$	14.6969i	1.37050i
$116$	10.2426i	0.951005i
$117$	0	0
$118$	− 2.44949i	− 0.225494i
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0.717439	0.0649539
$123$	0	0
$124$	5.61642i	0.504369i
$125$	−9.79796	−0.876356
$126$	0	0
$127$	17.7279	1.57310	0.786549	−	0.617527i	$-0.211866\pi$
$127$	17.7279	1.57310	0.786549	+	0.617527i	$0.211866\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	−10.2426	−0.898339
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	− 3.48528i	− 0.301082i
$135$	0	0
$136$	− 2.44949i	− 0.210042i
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	0	0
$139$	− 11.5300i	− 0.977963i	−0.872294	−	0.488981i	$-0.837369\pi$
$139$	− 11.5300i	− 0.977963i	0.872294	−	0.488981i	$-0.162631\pi$
$140$	0	0
$141$	0	0
$142$	−12.7279	−1.06810
$143$	−17.7408	−1.48356
$144$	0	0
$145$	− 25.0892i	− 2.08355i
$146$	−15.2913	−1.26552
$147$	0	0
$148$	3.24264	0.266543
$149$	7.75736i	0.635508i	0.948173	+	0.317754i	$0.102929\pi$
$149$	7.75736i	0.635508i	−0.948173	+	0.317754i	$0.897071\pi$
$150$	0	0
$151$	17.2426	1.40319	0.701593	−	0.712578i	$-0.252472\pi$
$151$	17.2426	1.40319	0.701593	+	0.712578i	$0.252472\pi$
$152$	4.89898	0.397360
$153$	0	0
$154$	0	0
$155$	− 13.7574i	− 1.10502i
$156$	0	0
$157$	10.3923i	0.829396i	0.909959	+	0.414698i	$0.136113\pi$
$157$	10.3923i	0.829396i	−0.909959	+	0.414698i	$0.863887\pi$
$158$	9.24264i	0.735305i
$159$	0	0
$160$	2.44949i	0.193649i
$161$	0	0
$162$	0	0
$163$	7.48528	0.586292	0.293146	−	0.956068i	$-0.405298\pi$
$163$	7.48528	0.586292	0.293146	+	0.956068i	$0.405298\pi$
$164$	2.44949	0.191273
$165$	0	0
$166$	− 5.49333i	− 0.426365i
$167$	20.1903	1.56237	0.781185	−	0.624300i	$-0.214616\pi$
$167$	20.1903	1.56237	0.781185	+	0.624300i	$0.214616\pi$
$168$	0	0
$169$	−4.48528	−0.345022
$170$	6.00000i	0.460179i
$171$	0	0
$172$	−7.00000	−0.533745
$173$	−20.7846	−1.58022	−0.790112	−	0.612962i	$-0.789978\pi$
$173$	−20.7846	−1.58022	−0.790112	+	0.612962i	$0.789978\pi$
$174$	0	0
$175$	0	0
$176$	4.24264i	0.319801i
$177$	0	0
$178$	− 17.7408i	− 1.32973i
$179$	18.7279i	1.39979i	0.714245	+	0.699895i	$0.246770\pi$
$179$	18.7279i	1.39979i	−0.714245	+	0.699895i	$0.753230\pi$
$180$	0	0
$181$	9.79796i	0.728277i	0.931345	+	0.364138i	$0.118636\pi$
$181$	9.79796i	0.728277i	−0.931345	+	0.364138i	$0.881364\pi$
$182$	0	0
$183$	0	0
$184$	6.00000	0.442326
$185$	−7.94282	−0.583967
$186$	0	0
$187$	10.3923i	0.759961i
$188$	7.94282	0.579289
$189$	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
$193$	15.4853	1.11465	0.557327	−	0.830293i	$-0.311827\pi$
$193$	15.4853	1.11465	0.557327	+	0.830293i	$0.311827\pi$
$194$	6.63103	0.476080
$195$	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
$199$	− 3.58719i	− 0.254289i	−0.991884	−	0.127145i	$-0.959419\pi$
$199$	− 3.58719i	− 0.254289i	0.991884	−	0.127145i	$-0.0405813\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	0	0
$202$	7.34847i	0.517036i
$203$	0	0
$204$	0	0
$205$	−6.00000	−0.419058
$206$	−6.21076	−0.432724
$207$	0	0
$208$	4.18154i	0.289938i
$209$	−20.7846	−1.43770
$210$	0	0
$211$	−13.4853	−0.928365	−0.464183	−	0.885740i	$-0.653652\pi$
$211$	−13.4853	−0.928365	−0.464183	+	0.885740i	$0.653652\pi$
$212$	2.48528i	0.170690i
$213$	0	0
$214$	−14.4853	−0.990193
$215$	17.1464	1.16938
$216$	0	0
$217$	0	0
$218$	7.72792i	0.523401i
$219$	0	0
$220$	− 10.3923i	− 0.700649i
$221$	10.2426i	0.688995i
$222$	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
$226$	−1.75736	−0.116898
$227$	−4.30463	−0.285709	−0.142854	−	0.989744i	$-0.545628\pi$
$227$	−4.30463	−0.285709	−0.142854	+	0.989744i	$0.545628\pi$
$228$	0	0
$229$	9.08052i	0.600058i	0.953930	+	0.300029i	$0.0969963\pi$
$229$	9.08052i	0.600058i	−0.953930	+	0.300029i	$0.903004\pi$
$230$	−14.6969	−0.969087
$231$	0	0
$232$	−10.2426	−0.672462
$233$	3.51472i	0.230257i	0.993351	+	0.115128i	$0.0367280\pi$
$233$	3.51472i	0.230257i	−0.993351	+	0.115128i	$0.963272\pi$
$234$	0	0
$235$	−19.4558	−1.26916
$236$	2.44949	0.159448
$237$	0	0
$238$	0	0
$239$	7.75736i	0.501782i	0.968015	+	0.250891i	$0.0807236\pi$
$239$	7.75736i	0.501782i	−0.968015	+	0.250891i	$0.919276\pi$
$240$	0	0
$241$	− 18.4582i	− 1.18900i	−0.804096	−	0.594499i	$-0.797350\pi$
$241$	− 18.4582i	− 1.18900i	0.804096	−	0.594499i	$-0.202650\pi$
$242$	− 7.00000i	− 0.449977i
$243$	0	0
$244$	0.717439i	0.0459293i
$245$	0	0
$246$	0	0
$247$	−20.4853	−1.30345
$248$	−5.61642	−0.356643
$249$	0	0
$250$	− 9.79796i	− 0.619677i
$251$	5.49333	0.346736	0.173368	−	0.984857i	$-0.444535\pi$
$251$	5.49333	0.346736	0.173368	+	0.984857i	$0.444535\pi$
$252$	0	0
$253$	−25.4558	−1.60040
$254$	17.7279i	1.11235i
$255$	0	0
$256$	1.00000	0.0625000
$257$	−4.30463	−0.268516	−0.134258	−	0.990946i	$-0.542865\pi$
$257$	−4.30463	−0.268516	−0.134258	+	0.990946i	$0.542865\pi$
$258$	0	0
$259$	0	0
$260$	− 10.2426i	− 0.635222i
$261$	0	0
$262$	0	0
$263$	− 18.7279i	− 1.15481i	−0.816457	−	0.577407i	$-0.804065\pi$
$263$	− 18.7279i	− 1.15481i	0.816457	−	0.577407i	$-0.195935\pi$
$264$	0	0
$265$	− 6.08767i	− 0.373963i
$266$	0	0
$267$	0	0
$268$	3.48528	0.212897
$269$	−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
$271$	15.4144i	0.936357i	0.883634	+	0.468178i	$0.155090\pi$
$271$	15.4144i	0.936357i	−0.883634	+	0.468178i	$0.844910\pi$
$272$	2.44949	0.148522
$273$	0	0
$274$	−6.00000	−0.362473
$275$	4.24264i	0.255841i
$276$	0	0
$277$	−1.72792	−0.103821	−0.0519104	−	0.998652i	$-0.516531\pi$
$277$	−1.72792	−0.103821	−0.0519104	+	0.998652i	$0.516531\pi$
$278$	11.5300	0.691524
$279$	0	0
$280$	0	0
$281$	− 22.2426i	− 1.32688i	−0.748227	−	0.663442i	$-0.769095\pi$
$281$	− 22.2426i	− 1.32688i	0.748227	−	0.663442i	$-0.230905\pi$
$282$	0	0
$283$	23.3572i	1.38844i	0.719762	+	0.694220i	$0.244251\pi$
$283$	23.3572i	1.38844i	−0.719762	+	0.694220i	$0.755749\pi$
$284$	− 12.7279i	− 0.755263i
$285$	0	0
$286$	− 17.7408i	− 1.04903i
$287$	0	0
$288$	0	0
$289$	−11.0000	−0.647059
$290$	25.0892	1.47329
$291$	0	0
$292$	− 15.2913i	− 0.894855i
$293$	7.94282	0.464024	0.232012	−	0.972713i	$-0.425469\pi$
$293$	7.94282	0.464024	0.232012	+	0.972713i	$0.425469\pi$
$294$	0	0
$295$	−6.00000	−0.349334
$296$	3.24264i	0.188475i
$297$	0	0
$298$	−7.75736	−0.449372
$299$	−25.0892	−1.45095
$300$	0	0
$301$	0	0
$302$	17.2426i	0.992202i
$303$	0	0
$304$	4.89898i	0.280976i
$305$	− 1.75736i	− 0.100626i
$306$	0	0
$307$	− 2.57258i	− 0.146825i	−0.997302	−	0.0734125i	$-0.976611\pi$
$307$	− 2.57258i	− 0.146825i	0.997302	−	0.0734125i	$-0.0233890\pi$
$308$	0	0
$309$	0	0
$310$	13.7574	0.781366
$311$	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
$313$	− 0.594346i	− 0.0335944i	−0.999859	−	0.0167972i	$-0.994653\pi$
$313$	− 0.594346i	− 0.0335944i	0.999859	−	0.0167972i	$-0.00534697\pi$
$314$	−10.3923	−0.586472
$315$	0	0
$316$	−9.24264	−0.519939
$317$	− 13.7574i	− 0.772690i	−0.922354	−	0.386345i	$-0.873737\pi$
$317$	− 13.7574i	− 0.772690i	0.922354	−	0.386345i	$-0.126263\pi$
$318$	0	0
$319$	43.4558	2.43306
$320$	−2.44949	−0.136931
$321$	0	0
$322$	0	0
$323$	12.0000i	0.667698i
$324$	0	0
$325$	4.18154i	0.231950i
$326$	7.48528i	0.414571i
$327$	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	$0.411380\pi$
$332$	5.49333	0.301485
$333$	0	0
$334$	20.1903i	1.10476i
$335$	−8.53716	−0.466435
$336$	0	0
$337$	29.4558	1.60456	0.802281	−	0.596947i	$-0.203620\pi$
$337$	29.4558	1.60456	0.802281	+	0.596947i	$0.203620\pi$
$338$	− 4.48528i	− 0.243967i
$339$	0	0
$340$	−6.00000	−0.325396
$341$	23.8284	1.29038
$342$	0	0
$343$	0	0
$344$	− 7.00000i	− 0.377415i
$345$	0	0
$346$	− 20.7846i	− 1.11739i
$347$	− 28.9706i	− 1.55522i	−0.628746	−	0.777611i	$-0.716432\pi$
$347$	− 28.9706i	− 1.55522i	0.628746	−	0.777611i	$-0.283568\pi$
$348$	0	0
$349$	24.3718i	1.30459i	0.757964	+	0.652296i	$0.226194\pi$
$349$	24.3718i	1.30459i	−0.757964	+	0.652296i	$0.773806\pi$
$350$	0	0
$351$	0	0
$352$	−4.24264	−0.226134
$353$	30.5826	1.62775	0.813873	−	0.581043i	$-0.197355\pi$
$353$	30.5826	1.62775	0.813873	+	0.581043i	$0.197355\pi$
$354$	0	0
$355$	31.1769i	1.65470i
$356$	17.7408	0.940259
$357$	0	0
$358$	−18.7279	−0.989801
$359$	2.48528i	0.131168i	0.997847	+	0.0655841i	$0.0208911\pi$
$359$	2.48528i	0.131168i	−0.997847	+	0.0655841i	$0.979109\pi$
$360$	0	0
$361$	−5.00000	−0.263158
$362$	−9.79796	−0.514969
$363$	0	0
$364$	0	0
$365$	37.4558i	1.96053i
$366$	0	0
$367$	− 9.20361i	− 0.480425i	−0.970720	−	0.240212i	$-0.922783\pi$
$367$	− 9.20361i	− 0.480425i	0.970720	−	0.240212i	$-0.0772171\pi$
$368$	6.00000i	0.312772i
$369$	0	0
$370$	− 7.94282i	− 0.412927i
$371$	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
$376$	7.94282i	0.409619i
$377$	42.8300	2.20586
$378$	0	0
$379$	9.48528	0.487226	0.243613	−	0.969872i	$-0.421667\pi$
$379$	9.48528	0.487226	0.243613	+	0.969872i	$0.421667\pi$
$380$	− 12.0000i	− 0.615587i
$381$	0	0
$382$	21.2132	1.08536
$383$	15.2913	0.781348	0.390674	−	0.920529i	$-0.372242\pi$
$383$	15.2913	0.781348	0.390674	+	0.920529i	$0.372242\pi$
$384$	0	0
$385$	0	0
$386$	15.4853i	0.788180i
$387$	0	0
$388$	6.63103i	0.336640i
$389$	− 32.4853i	− 1.64707i	−0.567266	−	0.823535i	$-0.691999\pi$
$389$	− 32.4853i	− 1.64707i	0.567266	−	0.823535i	$-0.308001\pi$
$390$	0	0
$391$	14.6969i	0.743256i
$392$	0	0
$393$	0	0
$394$	16.9706	0.854965
$395$	22.6398	1.13913
$396$	0	0
$397$	− 29.8651i	− 1.49889i	−0.662068	−	0.749444i	$-0.730321\pi$
$397$	− 29.8651i	− 1.49889i	0.662068	−	0.749444i	$-0.269679\pi$
$398$	3.58719	0.179810
$399$	0	0
$400$	1.00000	0.0500000
$401$	− 28.2426i	− 1.41037i	−0.709023	−	0.705185i	$-0.750864\pi$
$401$	− 28.2426i	− 1.41037i	0.709023	−	0.705185i	$-0.249136\pi$
$402$	0	0
$403$	23.4853	1.16989
$404$	−7.34847	−0.365600
$405$	0	0
$406$	0	0
$407$	− 13.7574i	− 0.681927i
$408$	0	0
$409$	13.5592i	0.670461i	0.942136	+	0.335230i	$0.108814\pi$
$409$	13.5592i	0.670461i	−0.942136	+	0.335230i	$0.891186\pi$
$410$	− 6.00000i	− 0.296319i
$411$	0	0
$412$	− 6.21076i	− 0.305982i
$413$	0	0
$414$	0	0
$415$	−13.4558	−0.660521
$416$	−4.18154	−0.205017
$417$	0	0
$418$	− 20.7846i	− 1.01661i
$419$	12.8418	0.627363	0.313681	−	0.949528i	$-0.398438\pi$
$419$	12.8418	0.627363	0.313681	+	0.949528i	$0.398438\pi$
$420$	0	0
$421$	27.4558	1.33812	0.669058	−	0.743210i	$-0.266698\pi$
$421$	27.4558	1.33812	0.669058	+	0.743210i	$0.266698\pi$
$422$	− 13.4853i	− 0.656453i
$423$	0	0
$424$	−2.48528	−0.120696
$425$	2.44949	0.118818
$426$	0	0
$427$	0	0
$428$	− 14.4853i	− 0.700173i
$429$	0	0
$430$	17.1464i	0.826874i
$431$	− 10.2426i	− 0.493371i	−0.969096	−	0.246685i	$-0.920659\pi$
$431$	− 10.2426i	− 0.493371i	0.969096	−	0.246685i	$-0.0793414\pi$
$432$	0	0
$433$	− 26.8213i	− 1.28895i	−0.764626	−	0.644475i	$-0.777076\pi$
$433$	− 26.8213i	− 1.28895i	0.764626	−	0.644475i	$-0.222924\pi$
$434$	0	0
$435$	0	0
$436$	−7.72792	−0.370100
$437$	−29.3939	−1.40610
$438$	0	0
$439$	− 25.0892i	− 1.19744i	−0.800957	−	0.598722i	$-0.795675\pi$
$439$	− 25.0892i	− 1.19744i	0.800957	−	0.598722i	$-0.204325\pi$
$440$	10.3923	0.495434
$441$	0	0
$442$	−10.2426	−0.487193
$443$	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
$445$	−43.4558	−2.06000
$446$	10.3923	0.492090
$447$	0	0
$448$	0	0
$449$	30.7279i	1.45014i	0.688675	+	0.725070i	$0.258193\pi$
$449$	30.7279i	1.45014i	−0.688675	+	0.725070i	$0.741807\pi$
$450$	0	0
$451$	− 10.3923i	− 0.489355i
$452$	− 1.75736i	− 0.0826592i
$453$	0	0
$454$	− 4.30463i	− 0.202026i
$455$	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	$0.680800\pi$
$458$	−9.08052	−0.424305
$459$	0	0
$460$	− 14.6969i	− 0.685248i
$461$	−42.2357	−1.96711	−0.983556	−	0.180605i	$-0.942194\pi$
$461$	−42.2357	−1.96711	−0.983556	+	0.180605i	$0.942194\pi$
$462$	0	0
$463$	−22.0000	−1.02243	−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000	−1.02243	−0.511213	+	0.859454i	$0.670804\pi$
$464$	− 10.2426i	− 0.475503i
$465$	0	0
$466$	−3.51472	−0.162816
$467$	3.04384	0.140852	0.0704260	−	0.997517i	$-0.477564\pi$
$467$	3.04384	0.140852	0.0704260	+	0.997517i	$0.477564\pi$
$468$	0	0
$469$	0	0
$470$	− 19.4558i	− 0.897431i
$471$	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
$478$	−7.75736	−0.354813
$479$	−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
$481$	− 13.5592i	− 0.618248i
$482$	18.4582	0.840749
$483$	0	0
$484$	7.00000	0.318182
$485$	− 16.2426i	− 0.737540i
$486$	0	0
$487$	22.0000	0.996915	0.498458	−	0.866914i	$-0.333900\pi$
$487$	22.0000	0.996915	0.498458	+	0.866914i	$0.333900\pi$
$488$	−0.717439	−0.0324769
$489$	0	0
$490$	0	0
$491$	1.02944i	0.0464579i	0.999730	+	0.0232289i	$0.00739466\pi$
$491$	1.02944i	0.0464579i	−0.999730	+	0.0232289i	$0.992605\pi$
$492$	0	0
$493$	− 25.0892i	− 1.12996i
$494$	− 20.4853i	− 0.921676i
$495$	0	0
$496$	− 5.61642i	− 0.252185i
$497$	0	0
$498$	0	0
$499$	26.4558	1.18433	0.592163	−	0.805818i	$-0.298274\pi$
$499$	26.4558	1.18433	0.592163	+	0.805818i	$0.298274\pi$
$500$	9.79796	0.438178
$501$	0	0
$502$	5.49333i	0.245179i
$503$	−20.1903	−0.900239	−0.450120	−	0.892968i	$-0.648619\pi$
$503$	−20.1903	−0.900239	−0.450120	+	0.892968i	$0.648619\pi$
$504$	0	0
$505$	18.0000	0.800989
$506$	− 25.4558i	− 1.13165i
$507$	0	0
$508$	−17.7279	−0.786549
$509$	29.9882	1.32920	0.664602	−	0.747197i	$-0.268601\pi$
$509$	29.9882	1.32920	0.664602	+	0.747197i	$0.268601\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	− 4.30463i	− 0.189869i
$515$	15.2132i	0.670374i
$516$	0	0
$517$	− 33.6985i	− 1.48206i
$518$	0	0
$519$	0	0
$520$	10.2426	0.449170
$521$	9.20361	0.403218	0.201609	−	0.979466i	$-0.435383\pi$
$521$	9.20361	0.403218	0.201609	+	0.979466i	$0.435383\pi$
$522$	0	0
$523$	17.6177i	0.770367i	0.922840	+	0.385184i	$0.125862\pi$
$523$	17.6177i	0.770367i	−0.922840	+	0.385184i	$0.874138\pi$
$524$	0	0
$525$	0	0
$526$	18.7279	0.816576
$527$	− 13.7574i	− 0.599280i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	6.08767	0.264432
$531$	0	0
$532$	0	0
$533$	− 10.2426i	− 0.443658i
$534$	0	0
$535$	35.4815i	1.53400i
$536$	3.48528i	0.150541i
$537$	0	0
$538$	− 9.79796i	− 0.422420i
$539$	0	0
$540$	0	0
$541$	45.4558	1.95430	0.977150	−	0.212552i	$-0.0681776\pi$
$541$	45.4558	1.95430	0.977150	+	0.212552i	$0.0681776\pi$
$542$	−15.4144	−0.662104
$543$	0	0
$544$	2.44949i	0.105021i
$545$	18.9295	0.810849
$546$	0	0
$547$	6.02944	0.257800	0.128900	−	0.991658i	$-0.458855\pi$
$547$	6.02944	0.257800	0.128900	+	0.991658i	$0.458855\pi$
$548$	− 6.00000i	− 0.256307i
$549$	0	0
$550$	−4.24264	−0.180907
$551$	50.1785	2.13768
$552$	0	0
$553$	0	0
$554$	− 1.72792i	− 0.0734124i
$555$	0	0
$556$	11.5300i	0.488981i
$557$	− 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
$559$	29.2708i	1.23802i
$560$	0	0
$561$	0	0
$562$	22.2426	0.938249
$563$	−16.4800	−0.694548	−0.347274	−	0.937764i	$-0.612893\pi$
$563$	−16.4800	−0.694548	−0.347274	+	0.937764i	$0.612893\pi$
$564$	0	0
$565$	4.30463i	0.181097i
$566$	−23.3572	−0.981776
$567$	0	0
$568$	12.7279	0.534052
$569$	− 1.75736i	− 0.0736723i	−0.999321	−	0.0368362i	$-0.988272\pi$
$569$	− 1.75736i	− 0.0736723i	0.999321	−	0.0368362i	$-0.0117280\pi$
$570$	0	0
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	17.7408	0.741779
$573$	0	0
$574$	0	0
$575$	6.00000i	0.250217i
$576$	0	0
$577$	− 17.6177i	− 0.733434i	−0.930333	−	0.366717i	$-0.880482\pi$
$577$	− 17.6177i	− 0.733434i	0.930333	−	0.366717i	$-0.119518\pi$
$578$	− 11.0000i	− 0.457540i
$579$	0	0
$580$	25.0892i	1.04177i
$581$	0	0
$582$	0	0
$583$	10.5442	0.436694
$584$	15.2913	0.632758
$585$	0	0
$586$	7.94282i	0.328115i
$587$	−11.6531	−0.480975	−0.240488	−	0.970652i	$-0.577307\pi$
$587$	−11.6531	−0.480975	−0.240488	+	0.970652i	$0.577307\pi$
$588$	0	0
$589$	27.5147	1.13372
$590$	− 6.00000i	− 0.247016i
$591$	0	0
$592$	−3.24264	−0.133272
$593$	22.6398	0.929703	0.464852	−	0.885389i	$-0.346108\pi$
$593$	22.6398	0.929703	0.464852	+	0.885389i	$0.346108\pi$
$594$	0	0
$595$	0	0
$596$	− 7.75736i	− 0.317754i
$597$	0	0
$598$	− 25.0892i	− 1.02598i
$599$	− 20.4853i	− 0.837006i	−0.908215	−	0.418503i	$-0.862555\pi$
$599$	− 20.4853i	− 0.837006i	0.908215	−	0.418503i	$-0.137445\pi$
$600$	0	0
$601$	− 17.2695i	− 0.704438i	−0.935918	−	0.352219i	$-0.885427\pi$
$601$	− 17.2695i	− 0.704438i	0.935918	−	0.352219i	$-0.114573\pi$
$602$	0	0
$603$	0	0
$604$	−17.2426	−0.701593
$605$	−17.1464	−0.697101
$606$	0	0
$607$	26.2779i	1.06659i	0.845930	+	0.533294i	$0.179046\pi$
$607$	26.2779i	1.06659i	−0.845930	+	0.533294i	$0.820954\pi$
$608$	−4.89898	−0.198680
$609$	0	0
$610$	1.75736	0.0711534
$611$	− 33.2132i	− 1.34366i
$612$	0	0
$613$	14.2132	0.574066	0.287033	−	0.957921i	$-0.407331\pi$
$613$	14.2132	0.574066	0.287033	+	0.957921i	$0.407331\pi$
$614$	2.57258	0.103821
$615$	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
$619$	32.3146i	1.29883i	0.760432	+	0.649417i	$0.224987\pi$
$619$	32.3146i	1.29883i	−0.760432	+	0.649417i	$0.775013\pi$
$620$	13.7574i	0.552509i
$621$	0	0
$622$	17.1464i	0.687509i
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0.594346	0.0237548
$627$	0	0
$628$	− 10.3923i	− 0.414698i
$629$	−7.94282	−0.316701
$630$	0	0
$631$	−23.2426	−0.925275	−0.462637	−	0.886548i	$-0.653097\pi$
$631$	−23.2426	−0.925275	−0.462637	+	0.886548i	$0.653097\pi$
$632$	− 9.24264i	− 0.367653i
$633$	0	0
$634$	13.7574	0.546375
$635$	43.4244	1.72324
$636$	0	0
$637$	0	0
$638$	43.4558i	1.72043i
$639$	0	0
$640$	− 2.44949i	− 0.0968246i
$641$	9.21320i	0.363900i	0.983308	+	0.181950i	$0.0582408\pi$
$641$	9.21320i	0.363900i	−0.983308	+	0.181950i	$0.941759\pi$
$642$	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	$-0.366030\pi$
$647$	20.7846	0.817127	0.408564	+	0.912730i	$0.366030\pi$
$648$	0	0
$649$	− 10.3923i	− 0.407934i
$650$	−4.18154	−0.164014
$651$	0	0
$652$	−7.48528	−0.293146
$653$	14.4853i	0.566853i	0.958994	+	0.283426i	$0.0914712\pi$
$653$	14.4853i	0.566853i	−0.958994	+	0.283426i	$0.908529\pi$
$654$	0	0
$655$	0	0
$656$	−2.44949	−0.0956365
$657$	0	0
$658$	0	0
$659$	13.7574i	0.535911i	0.963431	+	0.267955i	$0.0863480\pi$
$659$	13.7574i	0.535911i	−0.963431	+	0.267955i	$0.913652\pi$
$660$	0	0
$661$	− 4.89898i	− 0.190548i	−0.995451	−	0.0952741i	$-0.969627\pi$
$661$	− 4.89898i	− 0.190548i	0.995451	−	0.0952741i	$-0.0303728\pi$
$662$	10.0000i	0.388661i
$663$	0	0
$664$	5.49333i	0.213182i
$665$	0	0
$666$	0	0
$667$	61.4558	2.37958
$668$	−20.1903	−0.781185
$669$	0	0
$670$	− 8.53716i	− 0.329819i
$671$	3.04384	0.117506
$672$	0	0
$673$	5.45584	0.210307	0.105154	−	0.994456i	$-0.466467\pi$
$673$	5.45584	0.210307	0.105154	+	0.994456i	$0.466467\pi$
$674$	29.4558i	1.13460i
$675$	0	0
$676$	4.48528	0.172511
$677$	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
$682$	23.8284i	0.912438i
$683$	− 1.75736i	− 0.0672435i	−0.999435	−	0.0336217i	$-0.989296\pi$
$683$	− 1.75736i	− 0.0672435i	0.999435	−	0.0336217i	$-0.0107042\pi$
$684$	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
$691$	− 26.8213i	− 1.02033i	−0.860077	−	0.510165i	$-0.829584\pi$
$691$	− 26.8213i	− 1.02033i	0.860077	−	0.510165i	$-0.170416\pi$
$692$	20.7846	0.790112
$693$	0	0
$694$	28.9706	1.09971
$695$	− 28.2426i	− 1.07130i
$696$	0	0
$697$	−6.00000	−0.227266
$698$	−24.3718	−0.922486
$699$	0	0
$700$	0	0
$701$	3.51472i	0.132749i	0.997795	+	0.0663745i	$0.0211432\pi$
$701$	3.51472i	0.132749i	−0.997795	+	0.0663745i	$0.978857\pi$
$702$	0	0
$703$	− 15.8856i	− 0.599138i
$704$	− 4.24264i	− 0.159901i
$705$	0	0
$706$	30.5826i	1.15099i
$707$	0	0
$708$	0	0
$709$	−26.2132	−0.984458	−0.492229	−	0.870466i	$-0.663818\pi$
$709$	−26.2132	−0.984458	−0.492229	+	0.870466i	$0.663818\pi$
$710$	−31.1769	−1.17005
$711$	0	0
$712$	17.7408i	0.664864i
$713$	33.6985	1.26202
$714$	0	0
$715$	−43.4558	−1.62516
$716$	− 18.7279i	− 0.699895i
$717$	0	0
$718$	−2.48528	−0.0927499
$719$	−11.0588	−0.412422	−0.206211	−	0.978508i	$-0.566113\pi$
$719$	−11.0588	−0.412422	−0.206211	+	0.978508i	$0.566113\pi$
$720$	0	0
$721$	0	0
$722$	− 5.00000i	− 0.186081i
$723$	0	0
$724$	− 9.79796i	− 0.364138i
$725$	− 10.2426i	− 0.380402i
$726$	0	0
$727$	45.4026i	1.68389i	0.539564	+	0.841945i	$0.318589\pi$
$727$	45.4026i	1.68389i	−0.539564	+	0.841945i	$0.681411\pi$
$728$	0	0
$729$	0	0
$730$	−37.4558	−1.38630
$731$	17.1464	0.634184
$732$	0	0
$733$	16.0087i	0.591296i	0.955297	+	0.295648i	$0.0955355\pi$
$733$	16.0087i	0.591296i	−0.955297	+	0.295648i	$0.904465\pi$
$734$	9.20361	0.339712
$735$	0	0
$736$	−6.00000	−0.221163
$737$	− 14.7868i	− 0.544679i
$738$	0	0
$739$	42.4558	1.56176	0.780882	−	0.624679i	$-0.214770\pi$
$739$	42.4558	1.56176	0.780882	+	0.624679i	$0.214770\pi$
$740$	7.94282	0.291984
$741$	0	0
$742$	0	0
$743$	− 6.72792i	− 0.246824i	−0.992356	−	0.123412i	$-0.960616\pi$
$743$	− 6.72792i	− 0.246824i	0.992356	−	0.123412i	$-0.0393836\pi$
$744$	0	0
$745$	19.0016i	0.696164i
$746$	− 22.0000i	− 0.805477i
$747$	0	0
$748$	− 10.3923i	− 0.379980i
$749$	0	0
$750$	0	0
$751$	2.54416	0.0928376	0.0464188	−	0.998922i	$-0.485219\pi$
$751$	2.54416	0.0928376	0.0464188	+	0.998922i	$0.485219\pi$
$752$	−7.94282	−0.289645
$753$	0	0
$754$	42.8300i	1.55978i
$755$	42.2357	1.53711
$756$	0	0
$757$	41.2426	1.49899	0.749495	−	0.662010i	$-0.230297\pi$
$757$	41.2426	1.49899	0.749495	+	0.662010i	$0.230297\pi$
$758$	9.48528i	0.344521i
$759$	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
$766$	15.2913i	0.552497i
$767$	− 10.2426i	− 0.369840i
$768$	0	0
$769$	− 1.18869i	− 0.0428653i	−0.999770	−	0.0214327i	$-0.993177\pi$
$769$	− 1.18869i	− 0.0428653i	0.999770	−	0.0214327i	$-0.00682275\pi$
$770$	0	0
$771$	0	0
$772$	−15.4853	−0.557327
$773$	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
$775$	− 5.61642i	− 0.201748i
$776$	−6.63103	−0.238040
$777$	0	0
$778$	32.4853	1.16465
$779$	− 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
$787$	− 10.9357i	− 0.389814i	−0.980822	−	0.194907i	$-0.937559\pi$
$787$	− 10.9357i	− 0.389814i	0.980822	−	0.194907i	$-0.0624406\pi$
$788$	16.9706i	0.604551i
$789$	0	0
$790$	22.6398i	0.805486i
$791$	0	0
$792$	0	0
$793$	3.00000	0.106533
$794$	29.8651	1.05987
$795$	0	0
$796$	3.58719i	0.127145i
$797$	−3.04384	−0.107818	−0.0539091	−	0.998546i	$-0.517168\pi$
$797$	−3.04384	−0.107818	−0.0539091	+	0.998546i	$0.517168\pi$
$798$	0	0
$799$	−19.4558	−0.688298
$800$	1.00000i	0.0353553i
$801$	0	0
$802$	28.2426	0.997282
$803$	−64.8754	−2.28940
$804$	0	0
$805$	0	0
$806$	23.4853i	0.827234i
$807$	0	0
$808$	− 7.34847i	− 0.258518i
$809$	2.48528i	0.0873778i	0.999045	+	0.0436889i	$0.0139110\pi$
$809$	2.48528i	0.0873778i	−0.999045	+	0.0436889i	$0.986089\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	13.7574	0.482195
$815$	18.3351	0.642251
$816$	0	0
$817$	34.2929i	1.19976i
$818$	−13.5592	−0.474087
$819$	0	0
$820$	6.00000	0.209529
$821$	− 10.9706i	− 0.382875i	−0.981505	−	0.191438i	$-0.938685\pi$
$821$	− 10.9706i	− 0.382875i	0.981505	−	0.191438i	$-0.0613150\pi$
$822$	0	0
$823$	−11.2426	−0.391894	−0.195947	−	0.980615i	$-0.562778\pi$
$823$	−11.2426	−0.391894	−0.195947	+	0.980615i	$0.562778\pi$
$824$	6.21076	0.216362
$825$	0	0
$826$	0	0
$827$	− 20.4853i	− 0.712343i	−0.934421	−	0.356172i	$-0.884082\pi$
$827$	− 20.4853i	− 0.712343i	0.934421	−	0.356172i	$-0.115918\pi$
$828$	0	0
$829$	9.79796i	0.340297i	0.985418	+	0.170149i	$0.0544248\pi$
$829$	9.79796i	0.340297i	−0.985418	+	0.170149i	$0.945575\pi$
$830$	− 13.4558i	− 0.467059i
$831$	0	0
$832$	− 4.18154i	− 0.144969i
$833$	0	0
$834$	0	0
$835$	49.4558	1.71149
$836$	20.7846	0.718851
$837$	0	0
$838$	12.8418i	0.443612i
$839$	5.49333	0.189651	0.0948253	−	0.995494i	$-0.469771\pi$
$839$	5.49333	0.189651	0.0948253	+	0.995494i	$0.469771\pi$
$840$	0	0
$841$	−75.9117	−2.61764
$842$	27.4558i	0.946191i
$843$	0	0
$844$	13.4853	0.464183
$845$	−10.9867	−0.377952
$846$	0	0
$847$	0	0
$848$	− 2.48528i	− 0.0853449i
$849$	0	0
$850$	2.44949i	0.0840168i
$851$	− 19.4558i	− 0.666938i
$852$	0	0
$853$	51.3672i	1.75878i	0.476103	+	0.879389i	$0.342049\pi$
$853$	51.3672i	1.75878i	−0.476103	+	0.879389i	$0.657951\pi$
$854$	0	0
$855$	0	0
$856$	14.4853	0.495097
$857$	6.75412	0.230716	0.115358	−	0.993324i	$-0.463198\pi$
$857$	6.75412	0.230716	0.115358	+	0.993324i	$0.463198\pi$
$858$	0	0
$859$	− 10.9357i	− 0.373120i	−0.982444	−	0.186560i	$-0.940266\pi$
$859$	− 10.9357i	− 0.373120i	0.982444	−	0.186560i	$-0.0597339\pi$
$860$	−17.1464	−0.584688
$861$	0	0
$862$	10.2426	0.348866
$863$	− 13.4558i	− 0.458042i	−0.973421	−	0.229021i	$-0.926448\pi$
$863$	− 13.4558i	− 0.458042i	0.973421	−	0.229021i	$-0.0735525\pi$
$864$	0	0
$865$	−50.9117	−1.73105
$866$	26.8213	0.911425
$867$	0	0
$868$	0	0
$869$	39.2132i	1.33022i
$870$	0	0
$871$	− 14.5738i	− 0.493816i
$872$	− 7.72792i	− 0.261700i
$873$	0	0
$874$	− 29.3939i	− 0.994263i
$875$	0	0
$876$	0	0
$877$	−3.78680	−0.127871	−0.0639355	−	0.997954i	$-0.520365\pi$
$877$	−3.78680	−0.127871	−0.0639355	+	0.997954i	$0.520365\pi$
$878$	25.0892	0.846721
$879$	0	0
$880$	10.3923i	0.350325i
$881$	4.30463	0.145027	0.0725134	−	0.997367i	$-0.476898\pi$
$881$	4.30463	0.145027	0.0725134	+	0.997367i	$0.476898\pi$
$882$	0	0
$883$	−2.00000	−0.0673054	−0.0336527	−	0.999434i	$-0.510714\pi$
$883$	−2.00000	−0.0673054	−0.0336527	+	0.999434i	$0.510714\pi$
$884$	− 10.2426i	− 0.344497i
$885$	0	0
$886$	−25.4558	−0.855206
$887$	−29.9882	−1.00691	−0.503453	−	0.864023i	$-0.667937\pi$
$887$	−29.9882	−1.00691	−0.503453	+	0.864023i	$0.667937\pi$
$888$	0	0
$889$	0	0
$890$	− 43.4558i	− 1.45664i
$891$	0	0
$892$	10.3923i	0.347960i
$893$	− 38.9117i	− 1.30213i
$894$	0	0
$895$	45.8739i	1.53339i
$896$	0	0
$897$	0	0
$898$	−30.7279	−1.02540
$899$	−57.5270	−1.91863
$900$	0	0
$901$	− 6.08767i	− 0.202810i
$902$	10.3923	0.346026
$903$	0	0
$904$	1.75736	0.0584489
$905$	24.0000i	0.797787i
$906$	0	0
$907$	3.48528	0.115727	0.0578634	−	0.998325i	$-0.481571\pi$
$907$	3.48528	0.115727	0.0578634	+	0.998325i	$0.481571\pi$
$908$	4.30463	0.142854
$909$	0	0
$910$	0	0
$911$	− 1.75736i	− 0.0582239i	−0.999576	−	0.0291120i	$-0.990732\pi$
$911$	− 1.75736i	− 0.0582239i	0.999576	−	0.0291120i	$-0.00926793\pi$
$912$	0	0
$913$	− 23.3062i	− 0.771323i
$914$	− 23.0000i	− 0.760772i
$915$	0	0
$916$	− 9.08052i	− 0.300029i
$917$	0	0
$918$	0	0
$919$	0.272078	0.00897502	0.00448751	−	0.999990i	$-0.498572\pi$
$919$	0.272078	0.00897502	0.00448751	+	0.999990i	$0.498572\pi$
$920$	14.6969	0.484544
$921$	0	0
$922$	− 42.2357i	− 1.39096i
$923$	−53.2223	−1.75183
$924$	0	0
$925$	−3.24264	−0.106617
$926$	− 22.0000i	− 0.722965i
$927$	0	0
$928$	10.2426	0.336231
$929$	33.6264	1.10325	0.551623	−	0.834093i	$-0.314009\pi$
$929$	33.6264	1.10325	0.551623	+	0.834093i	$0.314009\pi$
$930$	0	0
$931$	0	0
$932$	− 3.51472i	− 0.115128i
$933$	0	0
$934$	3.04384i	0.0995973i
$935$	25.4558i	0.832495i
$936$	0	0
$937$	− 53.0992i	− 1.73468i	−0.497719	−	0.867338i	$-0.665829\pi$
$937$	− 53.0992i	− 1.73468i	0.497719	−	0.867338i	$-0.334171\pi$
$938$	0	0
$939$	0	0
$940$	19.4558	0.634580
$941$	−21.4511	−0.699285	−0.349642	−	0.936883i	$-0.613697\pi$
$941$	−21.4511	−0.699285	−0.349642	+	0.936883i	$0.613697\pi$
$942$	0	0
$943$	− 14.6969i	− 0.478598i
$944$	−2.44949	−0.0797241
$945$	0	0
$946$	−29.6985	−0.965581
$947$	− 7.75736i	− 0.252080i	−0.992025	−	0.126040i	$-0.959773\pi$
$947$	− 7.75736i	− 0.252080i	0.992025	−	0.126040i	$-0.0402268\pi$
$948$	0	0
$949$	−63.9411	−2.07562
$950$	−4.89898	−0.158944
$951$	0	0
$952$	0	0
$953$	− 34.9706i	− 1.13281i	−0.824128	−	0.566404i	$-0.808334\pi$
$953$	− 34.9706i	− 1.13281i	0.824128	−	0.566404i	$-0.191666\pi$
$954$	0	0
$955$	− 51.9615i	− 1.68144i
$956$	− 7.75736i	− 0.250891i
$957$	0	0
$958$	− 2.44949i	− 0.0791394i
$959$	0	0
$960$	0	0
$961$	−0.544156	−0.0175534
$962$	13.5592	0.437167
$963$	0	0
$964$	18.4582i	0.594499i
$965$	37.9310	1.22104
$966$	0	0
$967$	16.6985	0.536987	0.268494	−	0.963281i	$-0.413474\pi$
$967$	16.6985	0.536987	0.268494	+	0.963281i	$0.413474\pi$
$968$	7.00000i	0.224989i
$969$	0	0
$970$	16.2426	0.521520
$971$	−1.18869	−0.0381469	−0.0190735	−	0.999818i	$-0.506072\pi$
$971$	−1.18869	−0.0381469	−0.0190735	+	0.999818i	$0.506072\pi$
$972$	0	0
$973$	0	0
$974$	22.0000i	0.704925i
$975$	0	0
$976$	− 0.717439i	− 0.0229647i
$977$	6.00000i	0.191957i	0.995383	+	0.0959785i	$0.0305980\pi$
$977$	6.00000i	0.191957i	−0.995383	+	0.0959785i	$0.969402\pi$
$978$	0	0
$979$	− 75.2677i	− 2.40557i
$980$	0	0
$981$	0	0
$982$	−1.02944	−0.0328507
$983$	18.3351	0.584800	0.292400	−	0.956296i	$-0.405546\pi$
$983$	18.3351	0.584800	0.292400	+	0.956296i	$0.405546\pi$
$984$	0	0
$985$	− 41.5692i	− 1.32451i
$986$	25.0892	0.799004
$987$	0	0
$988$	20.4853	0.651724
$989$	42.0000i	1.33552i
$990$	0	0
$991$	22.2132	0.705626	0.352813	−	0.935694i	$-0.385225\pi$
$991$	22.2132	0.705626	0.352813	+	0.935694i	$0.385225\pi$
$992$	5.61642	0.178321
$993$	0	0
$994$	0	0
$995$	− 8.78680i	− 0.278560i
$996$	0	0
$997$	− 4.77589i	− 0.151254i	−0.997136	−	0.0756269i	$-0.975904\pi$
$997$	− 4.77589i	− 0.151254i	0.997136	−	0.0756269i	$-0.0240958\pi$
$998$	26.4558i	0.837445i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2646.2.d.d.2645.8		8
3.2	odd	2	inner	2646.2.d.d.2645.1		8
7.2	even	3		378.2.k.d.269.3	yes	8
7.3	odd	6		378.2.k.d.215.2	✓	8
7.6	odd	2	inner	2646.2.d.d.2645.6		8
21.2	odd	6		378.2.k.d.269.2	yes	8
21.17	even	6		378.2.k.d.215.3	yes	8
21.20	even	2	inner	2646.2.d.d.2645.3		8
63.2	odd	6		1134.2.t.f.1025.3		8
63.16	even	3		1134.2.t.f.1025.2		8
63.23	odd	6		1134.2.l.e.269.2		8
63.31	odd	6		1134.2.t.f.593.3		8
63.38	even	6		1134.2.l.e.215.1		8
63.52	odd	6		1134.2.l.e.215.4		8
63.58	even	3		1134.2.l.e.269.3		8
63.59	even	6		1134.2.t.f.593.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.k.d.215.2	✓	8	7.3	odd	6
378.2.k.d.215.3	yes	8	21.17	even	6
378.2.k.d.269.2	yes	8	21.2	odd	6
378.2.k.d.269.3	yes	8	7.2	even	3
1134.2.l.e.215.1		8	63.38	even	6
1134.2.l.e.215.4		8	63.52	odd	6
1134.2.l.e.269.2		8	63.23	odd	6
1134.2.l.e.269.3		8	63.58	even	3
1134.2.t.f.593.2		8	63.59	even	6
1134.2.t.f.593.3		8	63.31	odd	6
1134.2.t.f.1025.2		8	63.16	even	3
1134.2.t.f.1025.3		8	63.2	odd	6
2646.2.d.d.2645.1		8	3.2	odd	2	inner
2646.2.d.d.2645.3		8	21.20	even	2	inner
2646.2.d.d.2645.6		8	7.6	odd	2	inner
2646.2.d.d.2645.8		8	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:	\( N \)	\(=\)	\( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2646.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +2.44949 q^{5} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +2.44949 q^{5} -1.00000i q^{8} +2.44949i q^{10} +4.24264i q^{11} +4.18154i q^{13} +1.00000 q^{16} +2.44949 q^{17} +4.89898i q^{19} -2.44949 q^{20} -4.24264 q^{22} +6.00000i q^{23} +1.00000 q^{25} -4.18154 q^{26} -10.2426i q^{29} -5.61642i q^{31} +1.00000i q^{32} +2.44949i q^{34} -3.24264 q^{37} -4.89898 q^{38} -2.44949i q^{40} -2.44949 q^{41} +7.00000 q^{43} -4.24264i q^{44} -6.00000 q^{46} -7.94282 q^{47} +1.00000i q^{50} -4.18154i q^{52} -2.48528i q^{53} +10.3923i q^{55} +10.2426 q^{58} -2.44949 q^{59} -0.717439i q^{61} +5.61642 q^{62} -1.00000 q^{64} +10.2426i q^{65} -3.48528 q^{67} -2.44949 q^{68} +12.7279i q^{71} +15.2913i q^{73} -3.24264i q^{74} -4.89898i q^{76} +9.24264 q^{79} +2.44949 q^{80} -2.44949i q^{82} -5.49333 q^{83} +6.00000 q^{85} +7.00000i q^{86} +4.24264 q^{88} -17.7408 q^{89} -6.00000i q^{92} -7.94282i q^{94} +12.0000i q^{95} -6.63103i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4}+O(q^{10}) \) 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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