LMFDB - Embedded newform 378.2.d.a.377.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,2,Mod(377,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, 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("378.377");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	378.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:	$3.01834519640$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			377.4
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	378.377
Dual form			378.2.d.a.377.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +1.73205 q^{5} +(-2.50000 + 0.866025i) q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{4} +1.73205 q^{5} +(-2.50000 + 0.866025i) q^{7} -1.00000i q^{8} +1.73205i q^{10} +3.00000i q^{11} +6.92820i q^{13} +(-0.866025 - 2.50000i) q^{14} +1.00000 q^{16} +6.92820 q^{17} +3.46410i q^{19} -1.73205 q^{20} -3.00000 q^{22} -6.00000i q^{23} -2.00000 q^{25} -6.92820 q^{26} +(2.50000 - 0.866025i) q^{28} +6.00000i q^{29} -5.19615i q^{31} +1.00000i q^{32} +6.92820i q^{34} +(-4.33013 + 1.50000i) q^{35} -2.00000 q^{37} -3.46410 q^{38} -1.73205i q^{40} +3.46410 q^{41} -2.00000 q^{43} -3.00000i q^{44} +6.00000 q^{46} -3.46410 q^{47} +(5.50000 - 4.33013i) q^{49} -2.00000i q^{50} -6.92820i q^{52} -3.00000i q^{53} +5.19615i q^{55} +(0.866025 + 2.50000i) q^{56} -6.00000 q^{58} +3.46410 q^{59} -6.92820i q^{61} +5.19615 q^{62} -1.00000 q^{64} +12.0000i q^{65} +2.00000 q^{67} -6.92820 q^{68} +(-1.50000 - 4.33013i) q^{70} -12.0000i q^{71} -12.1244i q^{73} -2.00000i q^{74} -3.46410i q^{76} +(-2.59808 - 7.50000i) q^{77} +8.00000 q^{79} +1.73205 q^{80} +3.46410i q^{82} -1.73205 q^{83} +12.0000 q^{85} -2.00000i q^{86} +3.00000 q^{88} +10.3923 q^{89} +(-6.00000 - 17.3205i) q^{91} +6.00000i q^{92} -3.46410i q^{94} +6.00000i q^{95} +12.1244i q^{97} +(4.33013 + 5.50000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 10 q^{7}+O(q^{10}) $ 4 * q - 4 * q^4 - 10 * q^7	$ 4 q - 4 q^{4} - 10 q^{7} + 4 q^{16} - 12 q^{22} - 8 q^{25} + 10 q^{28} - 8 q^{37} - 8 q^{43} + 24 q^{46} + 22 q^{49} - 24 q^{58} - 4 q^{64} + 8 q^{67} - 6 q^{70} + 32 q^{79} + 48 q^{85} + 12 q^{88} - 24 q^{91}+O(q^{100}) $ 4 * q - 4 * q^4 - 10 * q^7 + 4 * q^16 - 12 * q^22 - 8 * q^25 + 10 * q^28 - 8 * q^37 - 8 * q^43 + 24 * q^46 + 22 * q^49 - 24 * q^58 - 4 * q^64 + 8 * q^67 - 6 * q^70 + 32 * q^79 + 48 * q^85 + 12 * q^88 - 24 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$-1$	$-1$

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
$5$	1.73205			0.774597			0.387298	−	0.921954i	$-0.373408\pi$
$5$	1.73205			0.774597			0.387298	+	0.921954i	$0.373408\pi$
$6$		0			0
$7$	−2.50000	+	0.866025i	−0.944911	+	0.327327i
$7$	−2.50000	+	0.866025i	−0.944911	+	0.327327i
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$			1.73205i			0.547723i
$11$			3.00000i			0.904534i	0.891883	+	0.452267i	$0.149385\pi$
$11$			3.00000i			0.904534i	−0.891883	+	0.452267i	$0.850615\pi$
$12$		0			0
$13$			6.92820i			1.92154i	0.277350	+	0.960769i	$0.410544\pi$
$13$			6.92820i			1.92154i	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−0.866025	−	2.50000i	−0.231455	−	0.668153i
$15$		0			0
$16$	1.00000			0.250000
$17$	6.92820			1.68034			0.840168	−	0.542326i	$-0.182456\pi$
$17$	6.92820			1.68034			0.840168	+	0.542326i	$0.182456\pi$
$18$		0			0
$19$			3.46410i			0.794719i	0.917663	+	0.397360i	$0.130073\pi$
$19$			3.46410i			0.794719i	−0.917663	+	0.397360i	$0.869927\pi$
$20$	−1.73205			−0.387298
$21$		0			0
$22$	−3.00000			−0.639602
$23$		−	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
$25$	−2.00000			−0.400000
$26$	−6.92820			−1.35873
$27$		0			0
$28$	2.50000	−	0.866025i	0.472456	−	0.163663i
$29$			6.00000i			1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$			6.00000i			1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$		0			0
$31$		−	5.19615i		−	0.933257i	−0.884454	−	0.466628i	$-0.845469\pi$
$31$		−	5.19615i		−	0.933257i	0.884454	−	0.466628i	$-0.154531\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$			6.92820i			1.18818i
$35$	−4.33013	+	1.50000i	−0.731925	+	0.253546i
$36$		0			0
$37$	−2.00000			−0.328798			−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000			−0.328798			−0.164399	+	0.986394i	$0.552568\pi$
$38$	−3.46410			−0.561951
$39$		0			0
$40$		−	1.73205i		−	0.273861i
$41$	3.46410			0.541002			0.270501	−	0.962720i	$-0.412811\pi$
$41$	3.46410			0.541002			0.270501	+	0.962720i	$0.412811\pi$
$42$		0			0
$43$	−2.00000			−0.304997			−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000			−0.304997			−0.152499	+	0.988304i	$0.548732\pi$
$44$		−	3.00000i		−	0.452267i
$45$		0			0
$46$	6.00000			0.884652
$47$	−3.46410			−0.505291			−0.252646	−	0.967559i	$-0.581301\pi$
$47$	−3.46410			−0.505291			−0.252646	+	0.967559i	$0.581301\pi$
$48$		0			0
$49$	5.50000	−	4.33013i	0.785714	−	0.618590i
$50$		−	2.00000i		−	0.282843i
$51$		0			0
$52$		−	6.92820i		−	0.960769i
$53$		−	3.00000i		−	0.412082i	−0.978543	−	0.206041i	$-0.933942\pi$
$53$		−	3.00000i		−	0.412082i	0.978543	−	0.206041i	$-0.0660580\pi$
$54$		0			0
$55$			5.19615i			0.700649i
$56$	0.866025	+	2.50000i	0.115728	+	0.334077i
$57$		0			0
$58$	−6.00000			−0.787839
$59$	3.46410			0.450988			0.225494	−	0.974245i	$-0.427600\pi$
$59$	3.46410			0.450988			0.225494	+	0.974245i	$0.427600\pi$
$60$		0			0
$61$		−	6.92820i		−	0.887066i	−0.896258	−	0.443533i	$-0.853725\pi$
$61$		−	6.92820i		−	0.887066i	0.896258	−	0.443533i	$-0.146275\pi$
$62$	5.19615			0.659912
$63$		0			0
$64$	−1.00000			−0.125000
$65$			12.0000i			1.48842i
$66$		0			0
$67$	2.00000			0.244339			0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000			0.244339			0.122169	+	0.992509i	$0.461015\pi$
$68$	−6.92820			−0.840168
$69$		0			0
$70$	−1.50000	−	4.33013i	−0.179284	−	0.517549i
$71$		−	12.0000i		−	1.42414i	−0.702109	−	0.712069i	$-0.747758\pi$
$71$		−	12.0000i		−	1.42414i	0.702109	−	0.712069i	$-0.252242\pi$
$72$		0			0
$73$		−	12.1244i		−	1.41905i	−0.704681	−	0.709524i	$-0.748910\pi$
$73$		−	12.1244i		−	1.41905i	0.704681	−	0.709524i	$-0.251090\pi$
$74$		−	2.00000i		−	0.232495i
$75$		0			0
$76$		−	3.46410i		−	0.397360i
$77$	−2.59808	−	7.50000i	−0.296078	−	0.854704i
$78$		0			0
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$	1.73205			0.193649
$81$		0			0
$82$			3.46410i			0.382546i
$83$	−1.73205			−0.190117			−0.0950586	−	0.995472i	$-0.530304\pi$
$83$	−1.73205			−0.190117			−0.0950586	+	0.995472i	$0.530304\pi$
$84$		0			0
$85$	12.0000			1.30158
$86$		−	2.00000i		−	0.215666i
$87$		0			0
$88$	3.00000			0.319801
$89$	10.3923			1.10158			0.550791	−	0.834643i	$-0.314326\pi$
$89$	10.3923			1.10158			0.550791	+	0.834643i	$0.314326\pi$
$90$		0			0
$91$	−6.00000	−	17.3205i	−0.628971	−	1.81568i
$92$			6.00000i			0.625543i
$93$		0			0
$94$		−	3.46410i		−	0.357295i
$95$			6.00000i			0.615587i
$96$		0			0
$97$			12.1244i			1.23104i	0.788121	+	0.615521i	$0.211054\pi$
$97$			12.1244i			1.23104i	−0.788121	+	0.615521i	$0.788946\pi$
$98$	4.33013	+	5.50000i	0.437409	+	0.555584i
$99$		0			0
$100$	2.00000			0.200000
$101$	15.5885			1.55111			0.775555	−	0.631280i	$-0.217470\pi$
$101$	15.5885			1.55111			0.775555	+	0.631280i	$0.217470\pi$
$102$		0			0
$103$			3.46410i			0.341328i	0.985329	+	0.170664i	$0.0545913\pi$
$103$			3.46410i			0.341328i	−0.985329	+	0.170664i	$0.945409\pi$
$104$	6.92820			0.679366
$105$		0			0
$106$	3.00000			0.291386
$107$			9.00000i			0.870063i	0.900415	+	0.435031i	$0.143263\pi$
$107$			9.00000i			0.870063i	−0.900415	+	0.435031i	$0.856737\pi$
$108$		0			0
$109$	−14.0000			−1.34096			−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000			−1.34096			−0.670478	+	0.741929i	$0.733911\pi$
$110$	−5.19615			−0.495434
$111$		0			0
$112$	−2.50000	+	0.866025i	−0.236228	+	0.0818317i
$113$			6.00000i			0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$			6.00000i			0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$		0			0
$115$		−	10.3923i		−	0.969087i
$116$		−	6.00000i		−	0.557086i
$117$		0			0
$118$			3.46410i			0.318896i
$119$	−17.3205	+	6.00000i	−1.58777	+	0.550019i
$120$		0			0
$121$	2.00000			0.181818
$122$	6.92820			0.627250
$123$		0			0
$124$			5.19615i			0.466628i
$125$	−12.1244			−1.08444
$126$		0			0
$127$	5.00000			0.443678			0.221839	−	0.975083i	$-0.428794\pi$
$127$	5.00000			0.443678			0.221839	+	0.975083i	$0.428794\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	−12.0000			−1.05247
$131$	5.19615			0.453990			0.226995	−	0.973896i	$-0.427110\pi$
$131$	5.19615			0.453990			0.226995	+	0.973896i	$0.427110\pi$
$132$		0			0
$133$	−3.00000	−	8.66025i	−0.260133	−	0.750939i
$134$			2.00000i			0.172774i
$135$		0			0
$136$		−	6.92820i		−	0.594089i
$137$		−	18.0000i		−	1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$		−	18.0000i		−	1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$		0			0
$139$			13.8564i			1.17529i	0.809121	+	0.587643i	$0.199944\pi$
$139$			13.8564i			1.17529i	−0.809121	+	0.587643i	$0.800056\pi$
$140$	4.33013	−	1.50000i	0.365963	−	0.126773i
$141$		0			0
$142$	12.0000			1.00702
$143$	−20.7846			−1.73810
$144$		0			0
$145$			10.3923i			0.863034i
$146$	12.1244			1.00342
$147$		0			0
$148$	2.00000			0.164399
$149$		−	15.0000i		−	1.22885i	−0.788976	−	0.614424i	$-0.789388\pi$
$149$		−	15.0000i		−	1.22885i	0.788976	−	0.614424i	$-0.210612\pi$
$150$		0			0
$151$	19.0000			1.54620			0.773099	−	0.634285i	$-0.218706\pi$
$151$	19.0000			1.54620			0.773099	+	0.634285i	$0.218706\pi$
$152$	3.46410			0.280976
$153$		0			0
$154$	7.50000	−	2.59808i	0.604367	−	0.209359i
$155$		−	9.00000i		−	0.722897i
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$			8.00000i			0.636446i
$159$		0			0
$160$			1.73205i			0.136931i
$161$	5.19615	+	15.0000i	0.409514	+	1.18217i
$162$		0			0
$163$	−16.0000			−1.25322			−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000			−1.25322			−0.626608	+	0.779334i	$0.715557\pi$
$164$	−3.46410			−0.270501
$165$		0			0
$166$		−	1.73205i		−	0.134433i
$167$	−24.2487			−1.87642			−0.938211	−	0.346064i	$-0.887518\pi$
$167$	−24.2487			−1.87642			−0.938211	+	0.346064i	$0.887518\pi$
$168$		0			0
$169$	−35.0000			−2.69231
$170$			12.0000i			0.920358i
$171$		0			0
$172$	2.00000			0.152499
$173$	5.19615			0.395056			0.197528	−	0.980297i	$-0.436709\pi$
$173$	5.19615			0.395056			0.197528	+	0.980297i	$0.436709\pi$
$174$		0			0
$175$	5.00000	−	1.73205i	0.377964	−	0.130931i
$176$			3.00000i			0.226134i
$177$		0			0
$178$			10.3923i			0.778936i
$179$		−	15.0000i		−	1.12115i	−0.828103	−	0.560576i	$-0.810580\pi$
$179$		−	15.0000i		−	1.12115i	0.828103	−	0.560576i	$-0.189420\pi$
$180$		0			0
$181$			6.92820i			0.514969i	0.966282	+	0.257485i	$0.0828937\pi$
$181$			6.92820i			0.514969i	−0.966282	+	0.257485i	$0.917106\pi$
$182$	17.3205	−	6.00000i	1.28388	−	0.444750i
$183$		0			0
$184$	−6.00000			−0.442326
$185$	−3.46410			−0.254686
$186$		0			0
$187$			20.7846i			1.51992i
$188$	3.46410			0.252646
$189$		0			0
$190$	−6.00000			−0.435286
$191$		−	12.0000i		−	0.868290i	−0.900843	−	0.434145i	$-0.857051\pi$
$191$		−	12.0000i		−	0.868290i	0.900843	−	0.434145i	$-0.142949\pi$
$192$		0			0
$193$	13.0000			0.935760			0.467880	−	0.883792i	$-0.345018\pi$
$193$	13.0000			0.935760			0.467880	+	0.883792i	$0.345018\pi$
$194$	−12.1244			−0.870478
$195$		0			0
$196$	−5.50000	+	4.33013i	−0.392857	+	0.309295i
$197$			3.00000i			0.213741i	0.994273	+	0.106871i	$0.0340831\pi$
$197$			3.00000i			0.213741i	−0.994273	+	0.106871i	$0.965917\pi$
$198$		0			0
$199$			12.1244i			0.859473i	0.902954	+	0.429736i	$0.141394\pi$
$199$			12.1244i			0.859473i	−0.902954	+	0.429736i	$0.858606\pi$
$200$			2.00000i			0.141421i
$201$		0			0
$202$			15.5885i			1.09680i
$203$	−5.19615	−	15.0000i	−0.364698	−	1.05279i
$204$		0			0
$205$	6.00000			0.419058
$206$	−3.46410			−0.241355
$207$		0			0
$208$			6.92820i			0.480384i
$209$	−10.3923			−0.718851
$210$		0			0
$211$	−2.00000			−0.137686			−0.0688428	−	0.997628i	$-0.521931\pi$
$211$	−2.00000			−0.137686			−0.0688428	+	0.997628i	$0.521931\pi$
$212$			3.00000i			0.206041i
$213$		0			0
$214$	−9.00000			−0.615227
$215$	−3.46410			−0.236250
$216$		0			0
$217$	4.50000	+	12.9904i	0.305480	+	0.881845i
$218$		−	14.0000i		−	0.948200i
$219$		0			0
$220$		−	5.19615i		−	0.350325i
$221$			48.0000i			3.22883i
$222$		0			0
$223$			10.3923i			0.695920i	0.937509	+	0.347960i	$0.113126\pi$
$223$			10.3923i			0.695920i	−0.937509	+	0.347960i	$0.886874\pi$
$224$	−0.866025	−	2.50000i	−0.0578638	−	0.167038i
$225$		0			0
$226$	−6.00000			−0.399114
$227$	−10.3923			−0.689761			−0.344881	−	0.938647i	$-0.612081\pi$
$227$	−10.3923			−0.689761			−0.344881	+	0.938647i	$0.612081\pi$
$228$		0			0
$229$			10.3923i			0.686743i	0.939200	+	0.343371i	$0.111569\pi$
$229$			10.3923i			0.686743i	−0.939200	+	0.343371i	$0.888431\pi$
$230$	10.3923			0.685248
$231$		0			0
$232$	6.00000			0.393919
$233$			6.00000i			0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$			6.00000i			0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$		0			0
$235$	−6.00000			−0.391397
$236$	−3.46410			−0.225494
$237$		0			0
$238$	−6.00000	−	17.3205i	−0.388922	−	1.12272i
$239$			18.0000i			1.16432i	0.813073	+	0.582162i	$0.197793\pi$
$239$			18.0000i			1.16432i	−0.813073	+	0.582162i	$0.802207\pi$
$240$		0			0
$241$		−	6.92820i		−	0.446285i	−0.974786	−	0.223142i	$-0.928369\pi$
$241$		−	6.92820i		−	0.446285i	0.974786	−	0.223142i	$-0.0716315\pi$
$242$			2.00000i			0.128565i
$243$		0			0
$244$			6.92820i			0.443533i
$245$	9.52628	−	7.50000i	0.608612	−	0.479157i
$246$		0			0
$247$	−24.0000			−1.52708
$248$	−5.19615			−0.329956
$249$		0			0
$250$		−	12.1244i		−	0.766812i
$251$	17.3205			1.09326			0.546630	−	0.837374i	$-0.315910\pi$
$251$	17.3205			1.09326			0.546630	+	0.837374i	$0.315910\pi$
$252$		0			0
$253$	18.0000			1.13165
$254$			5.00000i			0.313728i
$255$		0			0
$256$	1.00000			0.0625000
$257$	20.7846			1.29651			0.648254	−	0.761424i	$-0.275499\pi$
$257$	20.7846			1.29651			0.648254	+	0.761424i	$0.275499\pi$
$258$		0			0
$259$	5.00000	−	1.73205i	0.310685	−	0.107624i
$260$		−	12.0000i		−	0.744208i
$261$		0			0
$262$			5.19615i			0.321019i
$263$		−	18.0000i		−	1.10993i	−0.831875	−	0.554964i	$-0.812732\pi$
$263$		−	18.0000i		−	1.10993i	0.831875	−	0.554964i	$-0.187268\pi$
$264$		0			0
$265$		−	5.19615i		−	0.319197i
$266$	8.66025	−	3.00000i	0.530994	−	0.183942i
$267$		0			0
$268$	−2.00000			−0.122169
$269$	13.8564			0.844840			0.422420	−	0.906400i	$-0.361181\pi$
$269$	13.8564			0.844840			0.422420	+	0.906400i	$0.361181\pi$
$270$		0			0
$271$		−	29.4449i		−	1.78865i	−0.447420	−	0.894324i	$-0.647657\pi$
$271$		−	29.4449i		−	1.78865i	0.447420	−	0.894324i	$-0.352343\pi$
$272$	6.92820			0.420084
$273$		0			0
$274$	18.0000			1.08742
$275$		−	6.00000i		−	0.361814i
$276$		0			0
$277$	−16.0000			−0.961347			−0.480673	−	0.876900i	$-0.659608\pi$
$277$	−16.0000			−0.961347			−0.480673	+	0.876900i	$0.659608\pi$
$278$	−13.8564			−0.831052
$279$		0			0
$280$	1.50000	+	4.33013i	0.0896421	+	0.258775i
$281$		−	12.0000i		−	0.715860i	−0.933748	−	0.357930i	$-0.883483\pi$
$281$		−	12.0000i		−	0.715860i	0.933748	−	0.357930i	$-0.116517\pi$
$282$		0			0
$283$		−	10.3923i		−	0.617758i	−0.951101	−	0.308879i	$-0.900046\pi$
$283$		−	10.3923i		−	0.617758i	0.951101	−	0.308879i	$-0.0999539\pi$
$284$			12.0000i			0.712069i
$285$		0			0
$286$		−	20.7846i		−	1.22902i
$287$	−8.66025	+	3.00000i	−0.511199	+	0.177084i
$288$		0			0
$289$	31.0000			1.82353
$290$	−10.3923			−0.610257
$291$		0			0
$292$			12.1244i			0.709524i
$293$	−6.92820			−0.404750			−0.202375	−	0.979308i	$-0.564866\pi$
$293$	−6.92820			−0.404750			−0.202375	+	0.979308i	$0.564866\pi$
$294$		0			0
$295$	6.00000			0.349334
$296$			2.00000i			0.116248i
$297$		0			0
$298$	15.0000			0.868927
$299$	41.5692			2.40401
$300$		0			0
$301$	5.00000	−	1.73205i	0.288195	−	0.0998337i
$302$			19.0000i			1.09333i
$303$		0			0
$304$			3.46410i			0.198680i
$305$		−	12.0000i		−	0.687118i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$	2.59808	+	7.50000i	0.148039	+	0.427352i
$309$		0			0
$310$	9.00000			0.511166
$311$	−24.2487			−1.37502			−0.687509	−	0.726176i	$-0.741296\pi$
$311$	−24.2487			−1.37502			−0.687509	+	0.726176i	$0.741296\pi$
$312$		0			0
$313$		−	8.66025i		−	0.489506i	−0.969585	−	0.244753i	$-0.921293\pi$
$313$		−	8.66025i		−	0.489506i	0.969585	−	0.244753i	$-0.0787070\pi$
$314$		0			0
$315$		0			0
$316$	−8.00000			−0.450035
$317$		−	9.00000i		−	0.505490i	−0.967533	−	0.252745i	$-0.918667\pi$
$317$		−	9.00000i		−	0.505490i	0.967533	−	0.252745i	$-0.0813334\pi$
$318$		0			0
$319$	−18.0000			−1.00781
$320$	−1.73205			−0.0968246
$321$		0			0
$322$	−15.0000	+	5.19615i	−0.835917	+	0.289570i
$323$			24.0000i			1.33540i
$324$		0			0
$325$		−	13.8564i		−	0.768615i
$326$		−	16.0000i		−	0.886158i
$327$		0			0
$328$		−	3.46410i		−	0.191273i
$329$	8.66025	−	3.00000i	0.477455	−	0.165395i
$330$		0			0
$331$	10.0000			0.549650			0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000			0.549650			0.274825	+	0.961494i	$0.411380\pi$
$332$	1.73205			0.0950586
$333$		0			0
$334$		−	24.2487i		−	1.32683i
$335$	3.46410			0.189264
$336$		0			0
$337$	−14.0000			−0.762629			−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000			−0.762629			−0.381314	+	0.924445i	$0.624528\pi$
$338$		−	35.0000i		−	1.90375i
$339$		0			0
$340$	−12.0000			−0.650791
$341$	15.5885			0.844162
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$			2.00000i			0.107833i
$345$		0			0
$346$			5.19615i			0.279347i
$347$			27.0000i			1.44944i	0.689046	+	0.724718i	$0.258030\pi$
$347$			27.0000i			1.44944i	−0.689046	+	0.724718i	$0.741970\pi$
$348$		0			0
$349$		−	17.3205i		−	0.927146i	−0.886059	−	0.463573i	$-0.846567\pi$
$349$		−	17.3205i		−	0.927146i	0.886059	−	0.463573i	$-0.153433\pi$
$350$	1.73205	+	5.00000i	0.0925820	+	0.267261i
$351$		0			0
$352$	−3.00000			−0.159901
$353$	−3.46410			−0.184376			−0.0921878	−	0.995742i	$-0.529386\pi$
$353$	−3.46410			−0.184376			−0.0921878	+	0.995742i	$0.529386\pi$
$354$		0			0
$355$		−	20.7846i		−	1.10313i
$356$	−10.3923			−0.550791
$357$		0			0
$358$	15.0000			0.792775
$359$			18.0000i			0.950004i	0.879985	+	0.475002i	$0.157553\pi$
$359$			18.0000i			0.950004i	−0.879985	+	0.475002i	$0.842447\pi$
$360$		0			0
$361$	7.00000			0.368421
$362$	−6.92820			−0.364138
$363$		0			0
$364$	6.00000	+	17.3205i	0.314485	+	0.907841i
$365$		−	21.0000i		−	1.09919i
$366$		0			0
$367$			1.73205i			0.0904123i	0.998978	+	0.0452062i	$0.0143945\pi$
$367$			1.73205i			0.0904123i	−0.998978	+	0.0452062i	$0.985606\pi$
$368$		−	6.00000i		−	0.312772i
$369$		0			0
$370$		−	3.46410i		−	0.180090i
$371$	2.59808	+	7.50000i	0.134885	+	0.389381i
$372$		0			0
$373$	−4.00000			−0.207112			−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000			−0.207112			−0.103556	+	0.994624i	$0.533022\pi$
$374$	−20.7846			−1.07475
$375$		0			0
$376$			3.46410i			0.178647i
$377$	−41.5692			−2.14092
$378$		0			0
$379$	16.0000			0.821865			0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000			0.821865			0.410932	+	0.911666i	$0.365203\pi$
$380$		−	6.00000i		−	0.307794i
$381$		0			0
$382$	12.0000			0.613973
$383$	13.8564			0.708029			0.354015	−	0.935240i	$-0.384816\pi$
$383$	13.8564			0.708029			0.354015	+	0.935240i	$0.384816\pi$
$384$		0			0
$385$	−4.50000	−	12.9904i	−0.229341	−	0.662051i
$386$			13.0000i			0.661683i
$387$		0			0
$388$		−	12.1244i		−	0.615521i
$389$			15.0000i			0.760530i	0.924878	+	0.380265i	$0.124167\pi$
$389$			15.0000i			0.760530i	−0.924878	+	0.380265i	$0.875833\pi$
$390$		0			0
$391$		−	41.5692i		−	2.10225i
$392$	−4.33013	−	5.50000i	−0.218704	−	0.277792i
$393$		0			0
$394$	−3.00000			−0.151138
$395$	13.8564			0.697191
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	−12.1244			−0.607739
$399$		0			0
$400$	−2.00000			−0.100000
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	36.0000			1.79329
$404$	−15.5885			−0.775555
$405$		0			0
$406$	15.0000	−	5.19615i	0.744438	−	0.257881i
$407$		−	6.00000i		−	0.297409i
$408$		0			0
$409$			29.4449i			1.45595i	0.685601	+	0.727977i	$0.259539\pi$
$409$			29.4449i			1.45595i	−0.685601	+	0.727977i	$0.740461\pi$
$410$			6.00000i			0.296319i
$411$		0			0
$412$		−	3.46410i		−	0.170664i
$413$	−8.66025	+	3.00000i	−0.426143	+	0.147620i
$414$		0			0
$415$	−3.00000			−0.147264
$416$	−6.92820			−0.339683
$417$		0			0
$418$		−	10.3923i		−	0.508304i
$419$	38.1051			1.86156			0.930778	−	0.365584i	$-0.119131\pi$
$419$	38.1051			1.86156			0.930778	+	0.365584i	$0.119131\pi$
$420$		0			0
$421$	−4.00000			−0.194948			−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000			−0.194948			−0.0974740	+	0.995238i	$0.531076\pi$
$422$		−	2.00000i		−	0.0973585i
$423$		0			0
$424$	−3.00000			−0.145693
$425$	−13.8564			−0.672134
$426$		0			0
$427$	6.00000	+	17.3205i	0.290360	+	0.838198i
$428$		−	9.00000i		−	0.435031i
$429$		0			0
$430$		−	3.46410i		−	0.167054i
$431$		−	30.0000i		−	1.44505i	−0.691345	−	0.722525i	$-0.742982\pi$
$431$		−	30.0000i		−	1.44505i	0.691345	−	0.722525i	$-0.257018\pi$
$432$		0			0
$433$		−	5.19615i		−	0.249711i	−0.992175	−	0.124856i	$-0.960153\pi$
$433$		−	5.19615i		−	0.249711i	0.992175	−	0.124856i	$-0.0398468\pi$
$434$	−12.9904	+	4.50000i	−0.623558	+	0.216007i
$435$		0			0
$436$	14.0000			0.670478
$437$	20.7846			0.994263
$438$		0			0
$439$		−	5.19615i		−	0.247999i	−0.992282	−	0.123999i	$-0.960428\pi$
$439$		−	5.19615i		−	0.247999i	0.992282	−	0.123999i	$-0.0395721\pi$
$440$	5.19615			0.247717
$441$		0			0
$442$	−48.0000			−2.28313
$443$			36.0000i			1.71041i	0.518289	+	0.855206i	$0.326569\pi$
$443$			36.0000i			1.71041i	−0.518289	+	0.855206i	$0.673431\pi$
$444$		0			0
$445$	18.0000			0.853282
$446$	−10.3923			−0.492090
$447$		0			0
$448$	2.50000	−	0.866025i	0.118114	−	0.0409159i
$449$		−	30.0000i		−	1.41579i	−0.706319	−	0.707894i	$-0.749646\pi$
$449$		−	30.0000i		−	1.41579i	0.706319	−	0.707894i	$-0.250354\pi$
$450$		0			0
$451$			10.3923i			0.489355i
$452$		−	6.00000i		−	0.282216i
$453$		0			0
$454$		−	10.3923i		−	0.487735i
$455$	−10.3923	−	30.0000i	−0.487199	−	1.40642i
$456$		0			0
$457$	−17.0000			−0.795226			−0.397613	−	0.917553i	$-0.630161\pi$
$457$	−17.0000			−0.795226			−0.397613	+	0.917553i	$0.630161\pi$
$458$	−10.3923			−0.485601
$459$		0			0
$460$			10.3923i			0.484544i
$461$	−12.1244			−0.564688			−0.282344	−	0.959313i	$-0.591112\pi$
$461$	−12.1244			−0.564688			−0.282344	+	0.959313i	$0.591112\pi$
$462$		0			0
$463$	−31.0000			−1.44069			−0.720346	−	0.693615i	$-0.756017\pi$
$463$	−31.0000			−1.44069			−0.720346	+	0.693615i	$0.756017\pi$
$464$			6.00000i			0.278543i
$465$		0			0
$466$	−6.00000			−0.277945
$467$	15.5885			0.721348			0.360674	−	0.932692i	$-0.382547\pi$
$467$	15.5885			0.721348			0.360674	+	0.932692i	$0.382547\pi$
$468$		0			0
$469$	−5.00000	+	1.73205i	−0.230879	+	0.0799787i
$470$		−	6.00000i		−	0.276759i
$471$		0			0
$472$		−	3.46410i		−	0.159448i
$473$		−	6.00000i		−	0.275880i
$474$		0			0
$475$		−	6.92820i		−	0.317888i
$476$	17.3205	−	6.00000i	0.793884	−	0.275010i
$477$		0			0
$478$	−18.0000			−0.823301
$479$	3.46410			0.158279			0.0791394	−	0.996864i	$-0.474783\pi$
$479$	3.46410			0.158279			0.0791394	+	0.996864i	$0.474783\pi$
$480$		0			0
$481$		−	13.8564i		−	0.631798i
$482$	6.92820			0.315571
$483$		0			0
$484$	−2.00000			−0.0909091
$485$			21.0000i			0.953561i
$486$		0			0
$487$	−8.00000			−0.362515			−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000			−0.362515			−0.181257	+	0.983436i	$0.558017\pi$
$488$	−6.92820			−0.313625
$489$		0			0
$490$	7.50000	+	9.52628i	0.338815	+	0.430353i
$491$			9.00000i			0.406164i	0.979162	+	0.203082i	$0.0650959\pi$
$491$			9.00000i			0.406164i	−0.979162	+	0.203082i	$0.934904\pi$
$492$		0			0
$493$			41.5692i			1.87218i
$494$		−	24.0000i		−	1.07981i
$495$		0			0
$496$		−	5.19615i		−	0.233314i
$497$	10.3923	+	30.0000i	0.466159	+	1.34568i
$498$		0			0
$499$	10.0000			0.447661			0.223831	−	0.974628i	$-0.428144\pi$
$499$	10.0000			0.447661			0.223831	+	0.974628i	$0.428144\pi$
$500$	12.1244			0.542218
$501$		0			0
$502$			17.3205i			0.773052i
$503$	24.2487			1.08120			0.540598	−	0.841281i	$-0.318198\pi$
$503$	24.2487			1.08120			0.540598	+	0.841281i	$0.318198\pi$
$504$		0			0
$505$	27.0000			1.20148
$506$			18.0000i			0.800198i
$507$		0			0
$508$	−5.00000			−0.221839
$509$	−32.9090			−1.45866			−0.729332	−	0.684160i	$-0.760169\pi$
$509$	−32.9090			−1.45866			−0.729332	+	0.684160i	$0.760169\pi$
$510$		0			0
$511$	10.5000	+	30.3109i	0.464493	+	1.34087i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$			20.7846i			0.916770i
$515$			6.00000i			0.264392i
$516$		0			0
$517$		−	10.3923i		−	0.457053i
$518$	1.73205	+	5.00000i	0.0761019	+	0.219687i
$519$		0			0
$520$	12.0000			0.526235
$521$	−6.92820			−0.303530			−0.151765	−	0.988417i	$-0.548496\pi$
$521$	−6.92820			−0.303530			−0.151765	+	0.988417i	$0.548496\pi$
$522$		0			0
$523$			6.92820i			0.302949i	0.988461	+	0.151475i	$0.0484022\pi$
$523$			6.92820i			0.302949i	−0.988461	+	0.151475i	$0.951598\pi$
$524$	−5.19615			−0.226995
$525$		0			0
$526$	18.0000			0.784837
$527$		−	36.0000i		−	1.56818i
$528$		0			0
$529$	−13.0000			−0.565217
$530$	5.19615			0.225706
$531$		0			0
$532$	3.00000	+	8.66025i	0.130066	+	0.375470i
$533$			24.0000i			1.03956i
$534$		0			0
$535$			15.5885i			0.673948i
$536$		−	2.00000i		−	0.0863868i
$537$		0			0
$538$			13.8564i			0.597392i
$539$	12.9904	+	16.5000i	0.559535	+	0.710705i
$540$		0			0
$541$	−16.0000			−0.687894			−0.343947	−	0.938989i	$-0.611764\pi$
$541$	−16.0000			−0.687894			−0.343947	+	0.938989i	$0.611764\pi$
$542$	29.4449			1.26477
$543$		0			0
$544$			6.92820i			0.297044i
$545$	−24.2487			−1.03870
$546$		0			0
$547$	8.00000			0.342055			0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000			0.342055			0.171028	+	0.985266i	$0.445291\pi$
$548$			18.0000i			0.768922i
$549$		0			0
$550$	6.00000			0.255841
$551$	−20.7846			−0.885454
$552$		0			0
$553$	−20.0000	+	6.92820i	−0.850487	+	0.294617i
$554$		−	16.0000i		−	0.679775i
$555$		0			0
$556$		−	13.8564i		−	0.587643i
$557$		−	33.0000i		−	1.39825i	−0.714997	−	0.699127i	$-0.753572\pi$
$557$		−	33.0000i		−	1.39825i	0.714997	−	0.699127i	$-0.246428\pi$
$558$		0			0
$559$		−	13.8564i		−	0.586064i
$560$	−4.33013	+	1.50000i	−0.182981	+	0.0633866i
$561$		0			0
$562$	12.0000			0.506189
$563$	25.9808			1.09496			0.547479	−	0.836819i	$-0.315587\pi$
$563$	25.9808			1.09496			0.547479	+	0.836819i	$0.315587\pi$
$564$		0			0
$565$			10.3923i			0.437208i
$566$	10.3923			0.436821
$567$		0			0
$568$	−12.0000			−0.503509
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	−40.0000			−1.67395			−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000			−1.67395			−0.836974	+	0.547243i	$0.815677\pi$
$572$	20.7846			0.869048
$573$		0			0
$574$	−3.00000	−	8.66025i	−0.125218	−	0.361472i
$575$			12.0000i			0.500435i
$576$		0			0
$577$			34.6410i			1.44212i	0.692870	+	0.721062i	$0.256346\pi$
$577$			34.6410i			1.44212i	−0.692870	+	0.721062i	$0.743654\pi$
$578$			31.0000i			1.28943i
$579$		0			0
$580$		−	10.3923i		−	0.431517i
$581$	4.33013	−	1.50000i	0.179644	−	0.0622305i
$582$		0			0
$583$	9.00000			0.372742
$584$	−12.1244			−0.501709
$585$		0			0
$586$		−	6.92820i		−	0.286201i
$587$	25.9808			1.07234			0.536170	−	0.844110i	$-0.319870\pi$
$587$	25.9808			1.07234			0.536170	+	0.844110i	$0.319870\pi$
$588$		0			0
$589$	18.0000			0.741677
$590$			6.00000i			0.247016i
$591$		0			0
$592$	−2.00000			−0.0821995
$593$	3.46410			0.142254			0.0711268	−	0.997467i	$-0.477341\pi$
$593$	3.46410			0.142254			0.0711268	+	0.997467i	$0.477341\pi$
$594$		0			0
$595$	−30.0000	+	10.3923i	−1.22988	+	0.426043i
$596$			15.0000i			0.614424i
$597$		0			0
$598$			41.5692i			1.69989i
$599$			18.0000i			0.735460i	0.929933	+	0.367730i	$0.119865\pi$
$599$			18.0000i			0.735460i	−0.929933	+	0.367730i	$0.880135\pi$
$600$		0			0
$601$		−	46.7654i		−	1.90760i	−0.300443	−	0.953800i	$-0.597135\pi$
$601$		−	46.7654i		−	1.90760i	0.300443	−	0.953800i	$-0.402865\pi$
$602$	1.73205	+	5.00000i	0.0705931	+	0.203785i
$603$		0			0
$604$	−19.0000			−0.773099
$605$	3.46410			0.140836
$606$		0			0
$607$			38.1051i			1.54664i	0.634017	+	0.773320i	$0.281405\pi$
$607$			38.1051i			1.54664i	−0.634017	+	0.773320i	$0.718595\pi$
$608$	−3.46410			−0.140488
$609$		0			0
$610$	12.0000			0.485866
$611$		−	24.0000i		−	0.970936i
$612$		0			0
$613$	−46.0000			−1.85792			−0.928961	−	0.370177i	$-0.879297\pi$
$613$	−46.0000			−1.85792			−0.928961	+	0.370177i	$0.879297\pi$
$614$		0			0
$615$		0			0
$616$	−7.50000	+	2.59808i	−0.302184	+	0.104679i
$617$			18.0000i			0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$			18.0000i			0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$		0			0
$619$			27.7128i			1.11387i	0.830555	+	0.556936i	$0.188023\pi$
$619$			27.7128i			1.11387i	−0.830555	+	0.556936i	$0.811977\pi$
$620$			9.00000i			0.361449i
$621$		0			0
$622$		−	24.2487i		−	0.972285i
$623$	−25.9808	+	9.00000i	−1.04090	+	0.360577i
$624$		0			0
$625$	−11.0000			−0.440000
$626$	8.66025			0.346133
$627$		0			0
$628$		0			0
$629$	−13.8564			−0.552491
$630$		0			0
$631$	29.0000			1.15447			0.577236	−	0.816577i	$-0.304131\pi$
$631$	29.0000			1.15447			0.577236	+	0.816577i	$0.304131\pi$
$632$		−	8.00000i		−	0.318223i
$633$		0			0
$634$	9.00000			0.357436
$635$	8.66025			0.343672
$636$		0			0
$637$	30.0000	+	38.1051i	1.18864	+	1.50978i
$638$		−	18.0000i		−	0.712627i
$639$		0			0
$640$		−	1.73205i		−	0.0684653i
$641$		−	18.0000i		−	0.710957i	−0.934684	−	0.355479i	$-0.884318\pi$
$641$		−	18.0000i		−	0.710957i	0.934684	−	0.355479i	$-0.115682\pi$
$642$		0			0
$643$			20.7846i			0.819665i	0.912161	+	0.409832i	$0.134413\pi$
$643$			20.7846i			0.819665i	−0.912161	+	0.409832i	$0.865587\pi$
$644$	−5.19615	−	15.0000i	−0.204757	−	0.591083i
$645$		0			0
$646$	−24.0000			−0.944267
$647$	−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
$650$	13.8564			0.543493
$651$		0			0
$652$	16.0000			0.626608
$653$		−	45.0000i		−	1.76099i	−0.474059	−	0.880493i	$-0.657212\pi$
$653$		−	45.0000i		−	1.76099i	0.474059	−	0.880493i	$-0.342788\pi$
$654$		0			0
$655$	9.00000			0.351659
$656$	3.46410			0.135250
$657$		0			0
$658$	3.00000	+	8.66025i	0.116952	+	0.337612i
$659$		−	3.00000i		−	0.116863i	−0.998291	−	0.0584317i	$-0.981390\pi$
$659$		−	3.00000i		−	0.116863i	0.998291	−	0.0584317i	$-0.0186100\pi$
$660$		0			0
$661$		−	24.2487i		−	0.943166i	−0.881822	−	0.471583i	$-0.843683\pi$
$661$		−	24.2487i		−	0.943166i	0.881822	−	0.471583i	$-0.156317\pi$
$662$			10.0000i			0.388661i
$663$		0			0
$664$			1.73205i			0.0672166i
$665$	−5.19615	−	15.0000i	−0.201498	−	0.581675i
$666$		0			0
$667$	36.0000			1.39393
$668$	24.2487			0.938211
$669$		0			0
$670$			3.46410i			0.133830i
$671$	20.7846			0.802381
$672$		0			0
$673$	−5.00000			−0.192736			−0.0963679	−	0.995346i	$-0.530723\pi$
$673$	−5.00000			−0.192736			−0.0963679	+	0.995346i	$0.530723\pi$
$674$		−	14.0000i		−	0.539260i
$675$		0			0
$676$	35.0000			1.34615
$677$	−41.5692			−1.59763			−0.798817	−	0.601574i	$-0.794541\pi$
$677$	−41.5692			−1.59763			−0.798817	+	0.601574i	$0.794541\pi$
$678$		0			0
$679$	−10.5000	−	30.3109i	−0.402953	−	1.16323i
$680$		−	12.0000i		−	0.460179i
$681$		0			0
$682$			15.5885i			0.596913i
$683$			12.0000i			0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$			12.0000i			0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$		0			0
$685$		−	31.1769i		−	1.19121i
$686$	−15.5885	−	10.0000i	−0.595170	−	0.381802i
$687$		0			0
$688$	−2.00000			−0.0762493
$689$	20.7846			0.791831
$690$		0			0
$691$		−	20.7846i		−	0.790684i	−0.918534	−	0.395342i	$-0.870626\pi$
$691$		−	20.7846i		−	0.790684i	0.918534	−	0.395342i	$-0.129374\pi$
$692$	−5.19615			−0.197528
$693$		0			0
$694$	−27.0000			−1.02491
$695$			24.0000i			0.910372i
$696$		0			0
$697$	24.0000			0.909065
$698$	17.3205			0.655591
$699$		0			0
$700$	−5.00000	+	1.73205i	−0.188982	+	0.0654654i
$701$			3.00000i			0.113308i	0.998394	+	0.0566542i	$0.0180433\pi$
$701$			3.00000i			0.113308i	−0.998394	+	0.0566542i	$0.981957\pi$
$702$		0			0
$703$		−	6.92820i		−	0.261302i
$704$		−	3.00000i		−	0.113067i
$705$		0			0
$706$		−	3.46410i		−	0.130373i
$707$	−38.9711	+	13.5000i	−1.46566	+	0.507720i
$708$		0			0
$709$	−44.0000			−1.65245			−0.826227	−	0.563337i	$-0.809517\pi$
$709$	−44.0000			−1.65245			−0.826227	+	0.563337i	$0.809517\pi$
$710$	20.7846			0.780033
$711$		0			0
$712$		−	10.3923i		−	0.389468i
$713$	−31.1769			−1.16758
$714$		0			0
$715$	−36.0000			−1.34632
$716$			15.0000i			0.560576i
$717$		0			0
$718$	−18.0000			−0.671754
$719$	−27.7128			−1.03351			−0.516757	−	0.856132i	$-0.672861\pi$
$719$	−27.7128			−1.03351			−0.516757	+	0.856132i	$0.672861\pi$
$720$		0			0
$721$	−3.00000	−	8.66025i	−0.111726	−	0.322525i
$722$			7.00000i			0.260513i
$723$		0			0
$724$		−	6.92820i		−	0.257485i
$725$		−	12.0000i		−	0.445669i
$726$		0			0
$727$		−	25.9808i		−	0.963573i	−0.876289	−	0.481787i	$-0.839988\pi$
$727$		−	25.9808i		−	0.963573i	0.876289	−	0.481787i	$-0.160012\pi$
$728$	−17.3205	+	6.00000i	−0.641941	+	0.222375i
$729$		0			0
$730$	21.0000			0.777245
$731$	−13.8564			−0.512498
$732$		0			0
$733$		−	10.3923i		−	0.383849i	−0.981410	−	0.191924i	$-0.938527\pi$
$733$		−	10.3923i		−	0.383849i	0.981410	−	0.191924i	$-0.0614728\pi$
$734$	−1.73205			−0.0639312
$735$		0			0
$736$	6.00000			0.221163
$737$			6.00000i			0.221013i
$738$		0			0
$739$	20.0000			0.735712			0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000			0.735712			0.367856	+	0.929883i	$0.380092\pi$
$740$	3.46410			0.127343
$741$		0			0
$742$	−7.50000	+	2.59808i	−0.275334	+	0.0953784i
$743$		−	36.0000i		−	1.32071i	−0.750953	−	0.660356i	$-0.770405\pi$
$743$		−	36.0000i		−	1.32071i	0.750953	−	0.660356i	$-0.229595\pi$
$744$		0			0
$745$		−	25.9808i		−	0.951861i
$746$		−	4.00000i		−	0.146450i
$747$		0			0
$748$		−	20.7846i		−	0.759961i
$749$	−7.79423	−	22.5000i	−0.284795	−	0.822132i
$750$		0			0
$751$	43.0000			1.56909			0.784546	−	0.620070i	$-0.212896\pi$
$751$	43.0000			1.56909			0.784546	+	0.620070i	$0.212896\pi$
$752$	−3.46410			−0.126323
$753$		0			0
$754$		−	41.5692i		−	1.51386i
$755$	32.9090			1.19768
$756$		0			0
$757$	−2.00000			−0.0726912			−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000			−0.0726912			−0.0363456	+	0.999339i	$0.511572\pi$
$758$			16.0000i			0.581146i
$759$		0			0
$760$	6.00000			0.217643
$761$	−48.4974			−1.75803			−0.879015	−	0.476794i	$-0.841799\pi$
$761$	−48.4974			−1.75803			−0.879015	+	0.476794i	$0.841799\pi$
$762$		0			0
$763$	35.0000	−	12.1244i	1.26709	−	0.438931i
$764$			12.0000i			0.434145i
$765$		0			0
$766$			13.8564i			0.500652i
$767$			24.0000i			0.866590i
$768$		0			0
$769$		−	1.73205i		−	0.0624593i	−0.999512	−	0.0312297i	$-0.990058\pi$
$769$		−	1.73205i		−	0.0624593i	0.999512	−	0.0312297i	$-0.00994233\pi$
$770$	12.9904	−	4.50000i	0.468141	−	0.162169i
$771$		0			0
$772$	−13.0000			−0.467880
$773$	27.7128			0.996761			0.498380	−	0.866959i	$-0.333928\pi$
$773$	27.7128			0.996761			0.498380	+	0.866959i	$0.333928\pi$
$774$		0			0
$775$			10.3923i			0.373303i
$776$	12.1244			0.435239
$777$		0			0
$778$	−15.0000			−0.537776
$779$			12.0000i			0.429945i
$780$		0			0
$781$	36.0000			1.28818
$782$	41.5692			1.48651
$783$		0			0
$784$	5.50000	−	4.33013i	0.196429	−	0.154647i
$785$		0			0
$786$		0			0
$787$		−	13.8564i		−	0.493928i	−0.969025	−	0.246964i	$-0.920567\pi$
$787$		−	13.8564i		−	0.493928i	0.969025	−	0.246964i	$-0.0794329\pi$
$788$		−	3.00000i		−	0.106871i
$789$		0			0
$790$			13.8564i			0.492989i
$791$	−5.19615	−	15.0000i	−0.184754	−	0.533339i
$792$		0			0
$793$	48.0000			1.70453
$794$		0			0
$795$		0			0
$796$		−	12.1244i		−	0.429736i
$797$	−46.7654			−1.65651			−0.828257	−	0.560348i	$-0.810667\pi$
$797$	−46.7654			−1.65651			−0.828257	+	0.560348i	$0.810667\pi$
$798$		0			0
$799$	−24.0000			−0.849059
$800$		−	2.00000i		−	0.0707107i
$801$		0			0
$802$		0			0
$803$	36.3731			1.28358
$804$		0			0
$805$	9.00000	+	25.9808i	0.317208	+	0.915702i
$806$			36.0000i			1.26805i
$807$		0			0
$808$		−	15.5885i		−	0.548400i
$809$			30.0000i			1.05474i	0.849635	+	0.527372i	$0.176823\pi$
$809$			30.0000i			1.05474i	−0.849635	+	0.527372i	$0.823177\pi$
$810$		0			0
$811$		−	31.1769i		−	1.09477i	−0.836881	−	0.547385i	$-0.815623\pi$
$811$		−	31.1769i		−	1.09477i	0.836881	−	0.547385i	$-0.184377\pi$
$812$	5.19615	+	15.0000i	0.182349	+	0.526397i
$813$		0			0
$814$	6.00000			0.210300
$815$	−27.7128			−0.970737
$816$		0			0
$817$		−	6.92820i		−	0.242387i
$818$	−29.4449			−1.02952
$819$		0			0
$820$	−6.00000			−0.209529
$821$			6.00000i			0.209401i	0.994504	+	0.104701i	$0.0333885\pi$
$821$			6.00000i			0.209401i	−0.994504	+	0.104701i	$0.966612\pi$
$822$		0			0
$823$	−13.0000			−0.453152			−0.226576	−	0.973994i	$-0.572753\pi$
$823$	−13.0000			−0.453152			−0.226576	+	0.973994i	$0.572753\pi$
$824$	3.46410			0.120678
$825$		0			0
$826$	−3.00000	−	8.66025i	−0.104383	−	0.301329i
$827$		−	48.0000i		−	1.66912i	−0.550914	−	0.834562i	$-0.685721\pi$
$827$		−	48.0000i		−	1.66912i	0.550914	−	0.834562i	$-0.314279\pi$
$828$		0			0
$829$			17.3205i			0.601566i	0.953693	+	0.300783i	$0.0972480\pi$
$829$			17.3205i			0.601566i	−0.953693	+	0.300783i	$0.902752\pi$
$830$		−	3.00000i		−	0.104132i
$831$		0			0
$832$		−	6.92820i		−	0.240192i
$833$	38.1051	−	30.0000i	1.32026	−	1.03944i
$834$		0			0
$835$	−42.0000			−1.45347
$836$	10.3923			0.359425
$837$		0			0
$838$			38.1051i			1.31632i
$839$	6.92820			0.239188			0.119594	−	0.992823i	$-0.461841\pi$
$839$	6.92820			0.239188			0.119594	+	0.992823i	$0.461841\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$		−	4.00000i		−	0.137849i
$843$		0			0
$844$	2.00000			0.0688428
$845$	−60.6218			−2.08545
$846$		0			0
$847$	−5.00000	+	1.73205i	−0.171802	+	0.0595140i
$848$		−	3.00000i		−	0.103020i
$849$		0			0
$850$		−	13.8564i		−	0.475271i
$851$			12.0000i			0.411355i
$852$		0			0
$853$			17.3205i			0.593043i	0.955026	+	0.296521i	$0.0958266\pi$
$853$			17.3205i			0.593043i	−0.955026	+	0.296521i	$0.904173\pi$
$854$	−17.3205	+	6.00000i	−0.592696	+	0.205316i
$855$		0			0
$856$	9.00000			0.307614
$857$	−34.6410			−1.18331			−0.591657	−	0.806190i	$-0.701526\pi$
$857$	−34.6410			−1.18331			−0.591657	+	0.806190i	$0.701526\pi$
$858$		0			0
$859$		−	55.4256i		−	1.89110i	−0.325480	−	0.945549i	$-0.605526\pi$
$859$		−	55.4256i		−	1.89110i	0.325480	−	0.945549i	$-0.394474\pi$
$860$	3.46410			0.118125
$861$		0			0
$862$	30.0000			1.02180
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$	9.00000			0.306009
$866$	5.19615			0.176572
$867$		0			0
$868$	−4.50000	−	12.9904i	−0.152740	−	0.440922i
$869$			24.0000i			0.814144i
$870$		0			0
$871$			13.8564i			0.469506i
$872$			14.0000i			0.474100i
$873$		0			0
$874$			20.7846i			0.703050i
$875$	30.3109	−	10.5000i	1.02470	−	0.354965i
$876$		0			0
$877$	−22.0000			−0.742887			−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000			−0.742887			−0.371444	+	0.928456i	$0.621137\pi$
$878$	5.19615			0.175362
$879$		0			0
$880$			5.19615i			0.175162i
$881$	20.7846			0.700251			0.350126	−	0.936703i	$-0.386139\pi$
$881$	20.7846			0.700251			0.350126	+	0.936703i	$0.386139\pi$
$882$		0			0
$883$	−2.00000			−0.0673054			−0.0336527	−	0.999434i	$-0.510714\pi$
$883$	−2.00000			−0.0673054			−0.0336527	+	0.999434i	$0.510714\pi$
$884$		−	48.0000i		−	1.61441i
$885$		0			0
$886$	−36.0000			−1.20944
$887$	6.92820			0.232626			0.116313	−	0.993213i	$-0.462892\pi$
$887$	6.92820			0.232626			0.116313	+	0.993213i	$0.462892\pi$
$888$		0			0
$889$	−12.5000	+	4.33013i	−0.419237	+	0.145228i
$890$			18.0000i			0.603361i
$891$		0			0
$892$		−	10.3923i		−	0.347960i
$893$		−	12.0000i		−	0.401565i
$894$		0			0
$895$		−	25.9808i		−	0.868441i
$896$	0.866025	+	2.50000i	0.0289319	+	0.0835191i
$897$		0			0
$898$	30.0000			1.00111
$899$	31.1769			1.03981
$900$		0			0
$901$		−	20.7846i		−	0.692436i
$902$	−10.3923			−0.346026
$903$		0			0
$904$	6.00000			0.199557
$905$			12.0000i			0.398893i
$906$		0			0
$907$	−26.0000			−0.863316			−0.431658	−	0.902037i	$-0.642071\pi$
$907$	−26.0000			−0.863316			−0.431658	+	0.902037i	$0.642071\pi$
$908$	10.3923			0.344881
$909$		0			0
$910$	30.0000	−	10.3923i	0.994490	−	0.344502i
$911$			6.00000i			0.198789i	0.995048	+	0.0993944i	$0.0316906\pi$
$911$			6.00000i			0.198789i	−0.995048	+	0.0993944i	$0.968309\pi$
$912$		0			0
$913$		−	5.19615i		−	0.171968i
$914$		−	17.0000i		−	0.562310i
$915$		0			0
$916$		−	10.3923i		−	0.343371i
$917$	−12.9904	+	4.50000i	−0.428980	+	0.148603i
$918$		0			0
$919$	−23.0000			−0.758700			−0.379350	−	0.925253i	$-0.623852\pi$
$919$	−23.0000			−0.758700			−0.379350	+	0.925253i	$0.623852\pi$
$920$	−10.3923			−0.342624
$921$		0			0
$922$		−	12.1244i		−	0.399294i
$923$	83.1384			2.73654
$924$		0			0
$925$	4.00000			0.131519
$926$		−	31.0000i		−	1.01872i
$927$		0			0
$928$	−6.00000			−0.196960
$929$	27.7128			0.909228			0.454614	−	0.890689i	$-0.349777\pi$
$929$	27.7128			0.909228			0.454614	+	0.890689i	$0.349777\pi$
$930$		0			0
$931$	15.0000	+	19.0526i	0.491605	+	0.624422i
$932$		−	6.00000i		−	0.196537i
$933$		0			0
$934$			15.5885i			0.510070i
$935$			36.0000i			1.17733i
$936$		0			0
$937$			19.0526i			0.622420i	0.950341	+	0.311210i	$0.100734\pi$
$937$			19.0526i			0.622420i	−0.950341	+	0.311210i	$0.899266\pi$
$938$	−1.73205	−	5.00000i	−0.0565535	−	0.163256i
$939$		0			0
$940$	6.00000			0.195698
$941$	19.0526			0.621096			0.310548	−	0.950558i	$-0.399488\pi$
$941$	19.0526			0.621096			0.310548	+	0.950558i	$0.399488\pi$
$942$		0			0
$943$		−	20.7846i		−	0.676840i
$944$	3.46410			0.112747
$945$		0			0
$946$	6.00000			0.195077
$947$		−	51.0000i		−	1.65728i	−0.559784	−	0.828639i	$-0.689116\pi$
$947$		−	51.0000i		−	1.65728i	0.559784	−	0.828639i	$-0.310884\pi$
$948$		0			0
$949$	84.0000			2.72676
$950$	6.92820			0.224781
$951$		0			0
$952$	6.00000	+	17.3205i	0.194461	+	0.561361i
$953$			36.0000i			1.16615i	0.812417	+	0.583077i	$0.198151\pi$
$953$			36.0000i			1.16615i	−0.812417	+	0.583077i	$0.801849\pi$
$954$		0			0
$955$		−	20.7846i		−	0.672574i
$956$		−	18.0000i		−	0.582162i
$957$		0			0
$958$			3.46410i			0.111920i
$959$	15.5885	+	45.0000i	0.503378	+	1.45313i
$960$		0			0
$961$	4.00000			0.129032
$962$	13.8564			0.446748
$963$		0			0
$964$			6.92820i			0.223142i
$965$	22.5167			0.724837
$966$		0			0
$967$	11.0000			0.353736			0.176868	−	0.984235i	$-0.443403\pi$
$967$	11.0000			0.353736			0.176868	+	0.984235i	$0.443403\pi$
$968$		−	2.00000i		−	0.0642824i
$969$		0			0
$970$	−21.0000			−0.674269
$971$	−43.3013			−1.38960			−0.694802	−	0.719201i	$-0.744508\pi$
$971$	−43.3013			−1.38960			−0.694802	+	0.719201i	$0.744508\pi$
$972$		0			0
$973$	−12.0000	−	34.6410i	−0.384702	−	1.11054i
$974$		−	8.00000i		−	0.256337i
$975$		0			0
$976$		−	6.92820i		−	0.221766i
$977$		−	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
$979$			31.1769i			0.996419i
$980$	−9.52628	+	7.50000i	−0.304306	+	0.239579i
$981$		0			0
$982$	−9.00000			−0.287202
$983$	3.46410			0.110488			0.0552438	−	0.998473i	$-0.482406\pi$
$983$	3.46410			0.110488			0.0552438	+	0.998473i	$0.482406\pi$
$984$		0			0
$985$			5.19615i			0.165563i
$986$	−41.5692			−1.32383
$987$		0			0
$988$	24.0000			0.763542
$989$			12.0000i			0.381578i
$990$		0			0
$991$	−29.0000			−0.921215			−0.460608	−	0.887604i	$-0.652368\pi$
$991$	−29.0000			−0.921215			−0.460608	+	0.887604i	$0.652368\pi$
$992$	5.19615			0.164978
$993$		0			0
$994$	−30.0000	+	10.3923i	−0.951542	+	0.329624i
$995$			21.0000i			0.665745i
$996$		0			0
$997$		−	62.3538i		−	1.97477i	−0.158352	−	0.987383i	$-0.550618\pi$
$997$		−	62.3538i		−	1.97477i	0.158352	−	0.987383i	$-0.449382\pi$
$998$			10.0000i			0.316544i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.2.d.a.377.4	yes	4
3.2	odd	2	inner	378.2.d.a.377.1	✓	4
4.3	odd	2		3024.2.k.j.1889.3		4
7.6	odd	2	inner	378.2.d.a.377.3	yes	4
9.2	odd	6		1134.2.m.f.377.1		4
9.4	even	3		1134.2.m.e.755.1		4
9.5	odd	6		1134.2.m.e.755.2		4
9.7	even	3		1134.2.m.f.377.2		4
12.11	even	2		3024.2.k.j.1889.1		4
21.20	even	2	inner	378.2.d.a.377.2	yes	4
28.27	even	2		3024.2.k.j.1889.2		4
63.13	odd	6		1134.2.m.f.755.1		4
63.20	even	6		1134.2.m.e.377.1		4
63.34	odd	6		1134.2.m.e.377.2		4
63.41	even	6		1134.2.m.f.755.2		4
84.83	odd	2		3024.2.k.j.1889.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.d.a.377.1	✓	4	3.2	odd	2	inner
378.2.d.a.377.2	yes	4	21.20	even	2	inner
378.2.d.a.377.3	yes	4	7.6	odd	2	inner
378.2.d.a.377.4	yes	4	1.1	even	1	trivial
1134.2.m.e.377.1		4	63.20	even	6
1134.2.m.e.377.2		4	63.34	odd	6
1134.2.m.e.755.1		4	9.4	even	3
1134.2.m.e.755.2		4	9.5	odd	6
1134.2.m.f.377.1		4	9.2	odd	6
1134.2.m.f.377.2		4	9.7	even	3
1134.2.m.f.755.1		4	63.13	odd	6
1134.2.m.f.755.2		4	63.41	even	6
3024.2.k.j.1889.1		4	12.11	even	2
3024.2.k.j.1889.2		4	28.27	even	2
3024.2.k.j.1889.3		4	4.3	odd	2
3024.2.k.j.1889.4		4	84.83	odd	2

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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	378.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +1.73205 q^{5} +(-2.50000 + 0.866025i) q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +1.73205 q^{5} +(-2.50000 + 0.866025i) q^{7} -1.00000i q^{8} +1.73205i q^{10} +3.00000i q^{11} +6.92820i q^{13} +(-0.866025 - 2.50000i) q^{14} +1.00000 q^{16} +6.92820 q^{17} +3.46410i q^{19} -1.73205 q^{20} -3.00000 q^{22} -6.00000i q^{23} -2.00000 q^{25} -6.92820 q^{26} +(2.50000 - 0.866025i) q^{28} +6.00000i q^{29} -5.19615i q^{31} +1.00000i q^{32} +6.92820i q^{34} +(-4.33013 + 1.50000i) q^{35} -2.00000 q^{37} -3.46410 q^{38} -1.73205i q^{40} +3.46410 q^{41} -2.00000 q^{43} -3.00000i q^{44} +6.00000 q^{46} -3.46410 q^{47} +(5.50000 - 4.33013i) q^{49} -2.00000i q^{50} -6.92820i q^{52} -3.00000i q^{53} +5.19615i q^{55} +(0.866025 + 2.50000i) q^{56} -6.00000 q^{58} +3.46410 q^{59} -6.92820i q^{61} +5.19615 q^{62} -1.00000 q^{64} +12.0000i q^{65} +2.00000 q^{67} -6.92820 q^{68} +(-1.50000 - 4.33013i) q^{70} -12.0000i q^{71} -12.1244i q^{73} -2.00000i q^{74} -3.46410i q^{76} +(-2.59808 - 7.50000i) q^{77} +8.00000 q^{79} +1.73205 q^{80} +3.46410i q^{82} -1.73205 q^{83} +12.0000 q^{85} -2.00000i q^{86} +3.00000 q^{88} +10.3923 q^{89} +(-6.00000 - 17.3205i) q^{91} +6.00000i q^{92} -3.46410i q^{94} +6.00000i q^{95} +12.1244i q^{97} +(4.33013 + 5.50000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 10 q^{7}+O(q^{10}) \) 4 * q - 4 * q^4 - 10 * q^7	\( 4 q - 4 q^{4} - 10 q^{7} + 4 q^{16} - 12 q^{22} - 8 q^{25} + 10 q^{28} - 8 q^{37} - 8 q^{43} + 24 q^{46} + 22 q^{49} - 24 q^{58} - 4 q^{64} + 8 q^{67} - 6 q^{70} + 32 q^{79} + 48 q^{85} + 12 q^{88} - 24 q^{91}+O(q^{100}) \) 4 * q - 4 * q^4 - 10 * q^7 + 4 * q^16 - 12 * q^22 - 8 * q^25 + 10 * q^28 - 8 * q^37 - 8 * q^43 + 24 * q^46 + 22 * q^49 - 24 * q^58 - 4 * q^64 + 8 * q^67 - 6 * q^70 + 32 * q^79 + 48 * q^85 + 12 * q^88 - 24 * q^91

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