LMFDB - Embedded newform 912.2.k.h.607.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,2,Mod(607,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("912.607");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	912.k (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:	$7.28235666434$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			607.1
Root			$1.68614 + 0.396143i$ of defining polynomial
Character	$\chi$	$=$	912.607
Dual form			912.2.k.h.607.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -3.37228 q^{5} -0.792287i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.37228 q^{5} -0.792287i q^{7} +1.00000 q^{9} -0.792287i q^{11} +5.04868i q^{13} -3.37228 q^{15} +5.37228 q^{17} +(4.00000 + 1.73205i) q^{19} -0.792287i q^{21} +8.51278i q^{23} +6.37228 q^{25} +1.00000 q^{27} -10.0974i q^{29} +8.74456 q^{31} -0.792287i q^{33} +2.67181i q^{35} +5.04868i q^{37} +5.04868i q^{39} -6.92820i q^{41} +9.30506i q^{43} -3.37228 q^{45} +4.25639i q^{47} +6.37228 q^{49} +5.37228 q^{51} +3.16915i q^{53} +2.67181i q^{55} +(4.00000 + 1.73205i) q^{57} -4.00000 q^{59} +5.37228 q^{61} -0.792287i q^{63} -17.0256i q^{65} -9.48913 q^{67} +8.51278i q^{69} -4.00000 q^{71} -4.11684 q^{73} +6.37228 q^{75} -0.627719 q^{77} +4.74456 q^{79} +1.00000 q^{81} +3.46410i q^{83} -18.1168 q^{85} -10.0974i q^{87} +13.2665i q^{89} +4.00000 q^{91} +8.74456 q^{93} +(-13.4891 - 5.84096i) q^{95} -13.2665i q^{97} -0.792287i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 2 q^{5} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 2 * q^5 + 4 * q^9	$ 4 q + 4 q^{3} - 2 q^{5} + 4 q^{9} - 2 q^{15} + 10 q^{17} + 16 q^{19} + 14 q^{25} + 4 q^{27} + 12 q^{31} - 2 q^{45} + 14 q^{49} + 10 q^{51} + 16 q^{57} - 16 q^{59} + 10 q^{61} + 8 q^{67} - 16 q^{71} + 18 q^{73} + 14 q^{75} - 14 q^{77} - 4 q^{79} + 4 q^{81} - 38 q^{85} + 16 q^{91} + 12 q^{93} - 8 q^{95}+O(q^{100}) $ 4 * q + 4 * q^3 - 2 * q^5 + 4 * q^9 - 2 * q^15 + 10 * q^17 + 16 * q^19 + 14 * q^25 + 4 * q^27 + 12 * q^31 - 2 * q^45 + 14 * q^49 + 10 * q^51 + 16 * q^57 - 16 * q^59 + 10 * q^61 + 8 * q^67 - 16 * q^71 + 18 * q^73 + 14 * q^75 - 14 * q^77 - 4 * q^79 + 4 * q^81 - 38 * q^85 + 16 * q^91 + 12 * q^93 - 8 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$-1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$	−3.37228			−1.50813			−0.754065	−	0.656800i	$-0.771910\pi$
$5$	−3.37228			−1.50813			−0.754065	+	0.656800i	$0.771910\pi$
$6$		0			0
$7$		−	0.792287i		−	0.299456i	−0.988727	−	0.149728i	$-0.952160\pi$
$7$		−	0.792287i		−	0.299456i	0.988727	−	0.149728i	$-0.0478399\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	0.792287i		−	0.238884i	−0.992841	−	0.119442i	$-0.961890\pi$
$11$		−	0.792287i		−	0.238884i	0.992841	−	0.119442i	$-0.0381105\pi$
$12$		0			0
$13$			5.04868i			1.40025i	0.714020	+	0.700125i	$0.246873\pi$
$13$			5.04868i			1.40025i	−0.714020	+	0.700125i	$0.753127\pi$
$14$		0			0
$15$	−3.37228			−0.870719
$16$		0			0
$17$	5.37228			1.30297			0.651485	−	0.758662i	$-0.274146\pi$
$17$	5.37228			1.30297			0.651485	+	0.758662i	$0.274146\pi$
$18$		0			0
$19$	4.00000	+	1.73205i	0.917663	+	0.397360i
$19$	4.00000	+	1.73205i	0.917663	+	0.397360i
$20$		0			0
$21$		−	0.792287i		−	0.172891i
$22$		0			0
$23$			8.51278i			1.77504i	0.460772	+	0.887518i	$0.347572\pi$
$23$			8.51278i			1.77504i	−0.460772	+	0.887518i	$0.652428\pi$
$24$		0			0
$25$	6.37228			1.27446
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$		−	10.0974i		−	1.87503i	−0.347943	−	0.937516i	$-0.613120\pi$
$29$		−	10.0974i		−	1.87503i	0.347943	−	0.937516i	$-0.386880\pi$
$30$		0			0
$31$	8.74456			1.57057			0.785285	−	0.619135i	$-0.212516\pi$
$31$	8.74456			1.57057			0.785285	+	0.619135i	$0.212516\pi$
$32$		0			0
$33$		−	0.792287i		−	0.137919i
$34$		0			0
$35$			2.67181i			0.451619i
$36$		0			0
$37$			5.04868i			0.829997i	0.909822	+	0.414999i	$0.136218\pi$
$37$			5.04868i			0.829997i	−0.909822	+	0.414999i	$0.863782\pi$
$38$		0			0
$39$			5.04868i			0.808435i
$40$		0			0
$41$		−	6.92820i		−	1.08200i	−0.841021	−	0.541002i	$-0.818045\pi$
$41$		−	6.92820i		−	1.08200i	0.841021	−	0.541002i	$-0.181955\pi$
$42$		0			0
$43$			9.30506i			1.41901i	0.704701	+	0.709504i	$0.251081\pi$
$43$			9.30506i			1.41901i	−0.704701	+	0.709504i	$0.748919\pi$
$44$		0			0
$45$	−3.37228			−0.502710
$46$		0			0
$47$			4.25639i			0.620858i	0.950597	+	0.310429i	$0.100473\pi$
$47$			4.25639i			0.620858i	−0.950597	+	0.310429i	$0.899527\pi$
$48$		0			0
$49$	6.37228			0.910326
$50$		0			0
$51$	5.37228			0.752270
$52$		0			0
$53$			3.16915i			0.435316i	0.976025	+	0.217658i	$0.0698417\pi$
$53$			3.16915i			0.435316i	−0.976025	+	0.217658i	$0.930158\pi$
$54$		0			0
$55$			2.67181i			0.360267i
$56$		0			0
$57$	4.00000	+	1.73205i	0.529813	+	0.229416i
$58$		0			0
$59$	−4.00000			−0.520756			−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000			−0.520756			−0.260378	+	0.965507i	$0.583847\pi$
$60$		0			0
$61$	5.37228			0.687850			0.343925	−	0.938997i	$-0.388243\pi$
$61$	5.37228			0.687850			0.343925	+	0.938997i	$0.388243\pi$
$62$		0			0
$63$		−	0.792287i		−	0.0998188i
$64$		0			0
$65$		−	17.0256i		−	2.11176i
$66$		0			0
$67$	−9.48913			−1.15928			−0.579641	−	0.814872i	$-0.696807\pi$
$67$	−9.48913			−1.15928			−0.579641	+	0.814872i	$0.696807\pi$
$68$		0			0
$69$			8.51278i			1.02482i
$70$		0			0
$71$	−4.00000			−0.474713			−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000			−0.474713			−0.237356	+	0.971423i	$0.576281\pi$
$72$		0			0
$73$	−4.11684			−0.481840			−0.240920	−	0.970545i	$-0.577449\pi$
$73$	−4.11684			−0.481840			−0.240920	+	0.970545i	$0.577449\pi$
$74$		0			0
$75$	6.37228			0.735808
$76$		0			0
$77$	−0.627719			−0.0715352
$78$		0			0
$79$	4.74456			0.533805			0.266903	−	0.963724i	$-0.414000\pi$
$79$	4.74456			0.533805			0.266903	+	0.963724i	$0.414000\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			3.46410i			0.380235i	0.981761	+	0.190117i	$0.0608868\pi$
$83$			3.46410i			0.380235i	−0.981761	+	0.190117i	$0.939113\pi$
$84$		0			0
$85$	−18.1168			−1.96505
$86$		0			0
$87$		−	10.0974i		−	1.08255i
$88$		0			0
$89$			13.2665i			1.40625i	0.711068	+	0.703123i	$0.248212\pi$
$89$			13.2665i			1.40625i	−0.711068	+	0.703123i	$0.751788\pi$
$90$		0			0
$91$	4.00000			0.419314
$92$		0			0
$93$	8.74456			0.906769
$94$		0			0
$95$	−13.4891	−	5.84096i	−1.38396	−	0.599270i
$96$		0			0
$97$		−	13.2665i		−	1.34701i	−0.739183	−	0.673504i	$-0.764788\pi$
$97$		−	13.2665i		−	1.34701i	0.739183	−	0.673504i	$-0.235212\pi$
$98$		0			0
$99$		−	0.792287i		−	0.0796278i
$100$		0			0
$101$	−14.7446			−1.46714			−0.733569	−	0.679615i	$-0.762147\pi$
$101$	−14.7446			−1.46714			−0.733569	+	0.679615i	$0.762147\pi$
$102$		0			0
$103$	−12.7446			−1.25576			−0.627880	−	0.778311i	$-0.716077\pi$
$103$	−12.7446			−1.25576			−0.627880	+	0.778311i	$0.716077\pi$
$104$		0			0
$105$			2.67181i			0.260742i
$106$		0			0
$107$	9.48913			0.917348			0.458674	−	0.888605i	$-0.348325\pi$
$107$	9.48913			0.917348			0.458674	+	0.888605i	$0.348325\pi$
$108$		0			0
$109$		−	8.21782i		−	0.787125i	−0.919298	−	0.393562i	$-0.871243\pi$
$109$		−	8.21782i		−	0.787125i	0.919298	−	0.393562i	$-0.128757\pi$
$110$		0			0
$111$			5.04868i			0.479199i
$112$		0			0
$113$		−	3.75906i		−	0.353622i	−0.984245	−	0.176811i	$-0.943422\pi$
$113$		−	3.75906i		−	0.353622i	0.984245	−	0.176811i	$-0.0565782\pi$
$114$		0			0
$115$		−	28.7075i		−	2.67699i
$116$		0			0
$117$			5.04868i			0.466750i
$118$		0			0
$119$		−	4.25639i		−	0.390183i
$120$		0			0
$121$	10.3723			0.942935
$122$		0			0
$123$		−	6.92820i		−	0.624695i
$124$		0			0
$125$	−4.62772			−0.413916
$126$		0			0
$127$	12.7446			1.13090			0.565449	−	0.824784i	$-0.308703\pi$
$127$	12.7446			1.13090			0.565449	+	0.824784i	$0.308703\pi$
$128$		0			0
$129$			9.30506i			0.819265i
$130$		0			0
$131$			2.37686i			0.207667i	0.994595	+	0.103834i	$0.0331110\pi$
$131$			2.37686i			0.207667i	−0.994595	+	0.103834i	$0.966889\pi$
$132$		0			0
$133$	1.37228	−	3.16915i	0.118992	−	0.274800i
$134$		0			0
$135$	−3.37228			−0.290240
$136$		0			0
$137$	−4.11684			−0.351726			−0.175863	−	0.984415i	$-0.556272\pi$
$137$	−4.11684			−0.351726			−0.175863	+	0.984415i	$0.556272\pi$
$138$		0			0
$139$			5.54601i			0.470406i	0.971946	+	0.235203i	$0.0755756\pi$
$139$			5.54601i			0.470406i	−0.971946	+	0.235203i	$0.924424\pi$
$140$		0			0
$141$			4.25639i			0.358453i
$142$		0			0
$143$	4.00000			0.334497
$144$		0			0
$145$			34.0511i			2.82779i
$146$		0			0
$147$	6.37228			0.525577
$148$		0			0
$149$	2.11684			0.173419			0.0867093	−	0.996234i	$-0.472365\pi$
$149$	2.11684			0.173419			0.0867093	+	0.996234i	$0.472365\pi$
$150$		0			0
$151$	−8.74456			−0.711622			−0.355811	−	0.934558i	$-0.615795\pi$
$151$	−8.74456			−0.711622			−0.355811	+	0.934558i	$0.615795\pi$
$152$		0			0
$153$	5.37228			0.434323
$154$		0			0
$155$	−29.4891			−2.36862
$156$		0			0
$157$	7.48913			0.597697			0.298849	−	0.954301i	$-0.403397\pi$
$157$	7.48913			0.597697			0.298849	+	0.954301i	$0.403397\pi$
$158$		0			0
$159$			3.16915i			0.251330i
$160$		0			0
$161$	6.74456			0.531546
$162$		0			0
$163$			19.8997i			1.55867i	0.626608	+	0.779334i	$0.284443\pi$
$163$			19.8997i			1.55867i	−0.626608	+	0.779334i	$0.715557\pi$
$164$		0			0
$165$			2.67181i			0.208000i
$166$		0			0
$167$	17.4891			1.35335			0.676675	−	0.736282i	$-0.263420\pi$
$167$	17.4891			1.35335			0.676675	+	0.736282i	$0.263420\pi$
$168$		0			0
$169$	−12.4891			−0.960702
$170$		0			0
$171$	4.00000	+	1.73205i	0.305888	+	0.132453i
$172$		0			0
$173$		−	20.1947i		−	1.53537i	−0.640825	−	0.767687i	$-0.721407\pi$
$173$		−	20.1947i		−	1.53537i	0.640825	−	0.767687i	$-0.278593\pi$
$174$		0			0
$175$		−	5.04868i		−	0.381644i
$176$		0			0
$177$	−4.00000			−0.300658
$178$		0			0
$179$	1.48913			0.111302			0.0556512	−	0.998450i	$-0.482277\pi$
$179$	1.48913			0.111302			0.0556512	+	0.998450i	$0.482277\pi$
$180$		0			0
$181$			5.63858i			0.419113i	0.977797	+	0.209556i	$0.0672019\pi$
$181$			5.63858i			0.419113i	−0.977797	+	0.209556i	$0.932798\pi$
$182$		0			0
$183$	5.37228			0.397130
$184$		0			0
$185$		−	17.0256i		−	1.25174i
$186$		0			0
$187$		−	4.25639i		−	0.311258i
$188$		0			0
$189$		−	0.792287i		−	0.0576304i
$190$		0			0
$191$			3.66648i			0.265297i	0.991163	+	0.132649i	$0.0423482\pi$
$191$			3.66648i			0.265297i	−0.991163	+	0.132649i	$0.957652\pi$
$192$		0			0
$193$		−	13.8564i		−	0.997406i	−0.866773	−	0.498703i	$-0.833810\pi$
$193$		−	13.8564i		−	0.997406i	0.866773	−	0.498703i	$-0.166190\pi$
$194$		0			0
$195$		−	17.0256i		−	1.21923i
$196$		0			0
$197$	−5.25544			−0.374434			−0.187217	−	0.982319i	$-0.559947\pi$
$197$	−5.25544			−0.374434			−0.187217	+	0.982319i	$0.559947\pi$
$198$		0			0
$199$		−	21.5769i		−	1.52955i	−0.644300	−	0.764773i	$-0.722851\pi$
$199$		−	21.5769i		−	1.52955i	0.644300	−	0.764773i	$-0.277149\pi$
$200$		0			0
$201$	−9.48913			−0.669311
$202$		0			0
$203$	−8.00000			−0.561490
$204$		0			0
$205$			23.3639i			1.63180i
$206$		0			0
$207$			8.51278i			0.591679i
$208$		0			0
$209$	1.37228	−	3.16915i	0.0949227	−	0.219215i
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$		0			0
$213$	−4.00000			−0.274075
$214$		0			0
$215$		−	31.3793i		−	2.14005i
$216$		0			0
$217$		−	6.92820i		−	0.470317i
$218$		0			0
$219$	−4.11684			−0.278191
$220$		0			0
$221$			27.1229i			1.82448i
$222$		0			0
$223$	0.744563			0.0498596			0.0249298	−	0.999689i	$-0.492064\pi$
$223$	0.744563			0.0498596			0.0249298	+	0.999689i	$0.492064\pi$
$224$		0			0
$225$	6.37228			0.424819
$226$		0			0
$227$	−12.0000			−0.796468			−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000			−0.796468			−0.398234	+	0.917284i	$0.630377\pi$
$228$		0			0
$229$	−24.1168			−1.59369			−0.796843	−	0.604186i	$-0.793498\pi$
$229$	−24.1168			−1.59369			−0.796843	+	0.604186i	$0.793498\pi$
$230$		0			0
$231$	−0.627719			−0.0413009
$232$		0			0
$233$	−4.11684			−0.269703			−0.134852	−	0.990866i	$-0.543056\pi$
$233$	−4.11684			−0.269703			−0.134852	+	0.990866i	$0.543056\pi$
$234$		0			0
$235$		−	14.3537i		−	0.936335i
$236$		0			0
$237$	4.74456			0.308192
$238$		0			0
$239$		−	29.7947i		−	1.92726i	−0.267240	−	0.963630i	$-0.586112\pi$
$239$		−	29.7947i		−	1.92726i	0.267240	−	0.963630i	$-0.413888\pi$
$240$		0			0
$241$			17.0256i			1.09671i	0.836245	+	0.548356i	$0.184746\pi$
$241$			17.0256i			1.09671i	−0.836245	+	0.548356i	$0.815254\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$	−21.4891			−1.37289
$246$		0			0
$247$	−8.74456	+	20.1947i	−0.556403	+	1.28496i
$248$		0			0
$249$			3.46410i			0.219529i
$250$		0			0
$251$			6.13592i			0.387296i	0.981071	+	0.193648i	$0.0620319\pi$
$251$			6.13592i			0.387296i	−0.981071	+	0.193648i	$0.937968\pi$
$252$		0			0
$253$	6.74456			0.424027
$254$		0			0
$255$	−18.1168			−1.13452
$256$		0			0
$257$			3.75906i			0.234483i	0.993103	+	0.117242i	$0.0374052\pi$
$257$			3.75906i			0.234483i	−0.993103	+	0.117242i	$0.962595\pi$
$258$		0			0
$259$	4.00000			0.248548
$260$		0			0
$261$		−	10.0974i		−	0.625010i
$262$		0			0
$263$			18.1128i			1.11688i	0.829544	+	0.558441i	$0.188600\pi$
$263$			18.1128i			1.11688i	−0.829544	+	0.558441i	$0.811400\pi$
$264$		0			0
$265$		−	10.6873i		−	0.656513i
$266$		0			0
$267$			13.2665i			0.811897i
$268$		0			0
$269$		−	10.6873i		−	0.651614i	−0.945436	−	0.325807i	$-0.894364\pi$
$269$		−	10.6873i		−	0.651614i	0.945436	−	0.325807i	$-0.105636\pi$
$270$		0			0
$271$		−	13.5615i		−	0.823800i	−0.911229	−	0.411900i	$-0.864865\pi$
$271$		−	13.5615i		−	0.823800i	0.911229	−	0.411900i	$-0.135135\pi$
$272$		0			0
$273$	4.00000			0.242091
$274$		0			0
$275$		−	5.04868i		−	0.304447i
$276$		0			0
$277$	−2.62772			−0.157884			−0.0789422	−	0.996879i	$-0.525154\pi$
$277$	−2.62772			−0.157884			−0.0789422	+	0.996879i	$0.525154\pi$
$278$		0			0
$279$	8.74456			0.523523
$280$		0			0
$281$		−	10.0974i		−	0.602357i	−0.953568	−	0.301179i	$-0.902620\pi$
$281$		−	10.0974i		−	0.602357i	0.953568	−	0.301179i	$-0.0973801\pi$
$282$		0			0
$283$		−	27.3253i		−	1.62432i	−0.583435	−	0.812160i	$-0.698292\pi$
$283$		−	27.3253i		−	1.62432i	0.583435	−	0.812160i	$-0.301708\pi$
$284$		0			0
$285$	−13.4891	−	5.84096i	−0.799027	−	0.345989i
$286$		0			0
$287$	−5.48913			−0.324013
$288$		0			0
$289$	11.8614			0.697730
$290$		0			0
$291$		−	13.2665i		−	0.777696i
$292$		0			0
$293$		−	13.8564i		−	0.809500i	−0.914427	−	0.404750i	$-0.867359\pi$
$293$		−	13.8564i		−	0.809500i	0.914427	−	0.404750i	$-0.132641\pi$
$294$		0			0
$295$	13.4891			0.785367
$296$		0			0
$297$		−	0.792287i		−	0.0459732i
$298$		0			0
$299$	−42.9783			−2.48550
$300$		0			0
$301$	7.37228			0.424931
$302$		0			0
$303$	−14.7446			−0.847053
$304$		0			0
$305$	−18.1168			−1.03737
$306$		0			0
$307$	−17.4891			−0.998157			−0.499079	−	0.866557i	$-0.666328\pi$
$307$	−17.4891			−0.998157			−0.499079	+	0.866557i	$0.666328\pi$
$308$		0			0
$309$	−12.7446			−0.725013
$310$		0			0
$311$			14.3537i			0.813926i	0.913445	+	0.406963i	$0.133412\pi$
$311$			14.3537i			0.813926i	−0.913445	+	0.406963i	$0.866588\pi$
$312$		0			0
$313$	19.4891			1.10159			0.550795	−	0.834640i	$-0.314325\pi$
$313$	19.4891			1.10159			0.550795	+	0.834640i	$0.314325\pi$
$314$		0			0
$315$			2.67181i			0.150540i
$316$		0			0
$317$			10.6873i			0.600256i	0.953899	+	0.300128i	$0.0970294\pi$
$317$			10.6873i			0.600256i	−0.953899	+	0.300128i	$0.902971\pi$
$318$		0			0
$319$	−8.00000			−0.447914
$320$		0			0
$321$	9.48913			0.529631
$322$		0			0
$323$	21.4891	+	9.30506i	1.19569	+	0.517748i
$324$		0			0
$325$			32.1716i			1.78456i
$326$		0			0
$327$		−	8.21782i		−	0.454447i
$328$		0			0
$329$	3.37228			0.185920
$330$		0			0
$331$	9.48913			0.521569			0.260785	−	0.965397i	$-0.416019\pi$
$331$	9.48913			0.521569			0.260785	+	0.965397i	$0.416019\pi$
$332$		0			0
$333$			5.04868i			0.276666i
$334$		0			0
$335$	32.0000			1.74835
$336$		0			0
$337$		−	20.7846i		−	1.13221i	−0.824333	−	0.566105i	$-0.808450\pi$
$337$		−	20.7846i		−	1.13221i	0.824333	−	0.566105i	$-0.191550\pi$
$338$		0			0
$339$		−	3.75906i		−	0.204164i
$340$		0			0
$341$		−	6.92820i		−	0.375183i
$342$		0			0
$343$		−	10.5947i		−	0.572059i
$344$		0			0
$345$		−	28.7075i		−	1.54556i
$346$		0			0
$347$			13.0641i			0.701319i	0.936503	+	0.350659i	$0.114042\pi$
$347$			13.0641i			0.701319i	−0.936503	+	0.350659i	$0.885958\pi$
$348$		0			0
$349$	−25.6060			−1.37066			−0.685328	−	0.728234i	$-0.740341\pi$
$349$	−25.6060			−1.37066			−0.685328	+	0.728234i	$0.740341\pi$
$350$		0			0
$351$			5.04868i			0.269478i
$352$		0			0
$353$	−23.4891			−1.25020			−0.625100	−	0.780545i	$-0.714942\pi$
$353$	−23.4891			−1.25020			−0.625100	+	0.780545i	$0.714942\pi$
$354$		0			0
$355$	13.4891			0.715928
$356$		0			0
$357$		−	4.25639i		−	0.225272i
$358$		0			0
$359$			13.7638i			0.726427i	0.931706	+	0.363214i	$0.118320\pi$
$359$			13.7638i			0.726427i	−0.931706	+	0.363214i	$0.881680\pi$
$360$		0			0
$361$	13.0000	+	13.8564i	0.684211	+	0.729285i
$362$		0			0
$363$	10.3723			0.544404
$364$		0			0
$365$	13.8832			0.726678
$366$		0			0
$367$			26.8280i			1.40041i	0.713943	+	0.700204i	$0.246908\pi$
$367$			26.8280i			1.40041i	−0.713943	+	0.700204i	$0.753092\pi$
$368$		0			0
$369$		−	6.92820i		−	0.360668i
$370$		0			0
$371$	2.51087			0.130358
$372$		0			0
$373$			1.87953i			0.0973183i	0.998815	+	0.0486591i	$0.0154948\pi$
$373$			1.87953i			0.0973183i	−0.998815	+	0.0486591i	$0.984505\pi$
$374$		0			0
$375$	−4.62772			−0.238974
$376$		0			0
$377$	50.9783			2.62551
$378$		0			0
$379$	26.9783			1.38578			0.692890	−	0.721043i	$-0.256337\pi$
$379$	26.9783			1.38578			0.692890	+	0.721043i	$0.256337\pi$
$380$		0			0
$381$	12.7446			0.652924
$382$		0			0
$383$	30.9783			1.58291			0.791457	−	0.611224i	$-0.209323\pi$
$383$	30.9783			1.58291			0.791457	+	0.611224i	$0.209323\pi$
$384$		0			0
$385$	2.11684			0.107884
$386$		0			0
$387$			9.30506i			0.473003i
$388$		0			0
$389$	6.11684			0.310136			0.155068	−	0.987904i	$-0.450440\pi$
$389$	6.11684			0.310136			0.155068	+	0.987904i	$0.450440\pi$
$390$		0			0
$391$			45.7330i			2.31282i
$392$		0			0
$393$			2.37686i			0.119897i
$394$		0			0
$395$	−16.0000			−0.805047
$396$		0			0
$397$	11.8832			0.596399			0.298199	−	0.954504i	$-0.403614\pi$
$397$	11.8832			0.596399			0.298199	+	0.954504i	$0.403614\pi$
$398$		0			0
$399$	1.37228	−	3.16915i	0.0687000	−	0.158656i
$400$		0			0
$401$			23.9538i			1.19619i	0.801424	+	0.598097i	$0.204076\pi$
$401$			23.9538i			1.19619i	−0.801424	+	0.598097i	$0.795924\pi$
$402$		0			0
$403$			44.1485i			2.19919i
$404$		0			0
$405$	−3.37228			−0.167570
$406$		0			0
$407$	4.00000			0.198273
$408$		0			0
$409$			6.92820i			0.342578i	0.985221	+	0.171289i	$0.0547931\pi$
$409$			6.92820i			0.342578i	−0.985221	+	0.171289i	$0.945207\pi$
$410$		0			0
$411$	−4.11684			−0.203069
$412$		0			0
$413$			3.16915i			0.155944i
$414$		0			0
$415$		−	11.6819i		−	0.573443i
$416$		0			0
$417$			5.54601i			0.271589i
$418$		0			0
$419$		−	0.294954i		−	0.0144094i	−0.999974	−	0.00720471i	$-0.997707\pi$
$419$		−	0.294954i		−	0.0144094i	0.999974	−	0.00720471i	$-0.00229335\pi$
$420$		0			0
$421$		−	32.7615i		−	1.59670i	−0.602196	−	0.798349i	$-0.705707\pi$
$421$		−	32.7615i		−	1.59670i	0.602196	−	0.798349i	$-0.294293\pi$
$422$		0			0
$423$			4.25639i			0.206953i
$424$		0			0
$425$	34.2337			1.66058
$426$		0			0
$427$		−	4.25639i		−	0.205981i
$428$		0			0
$429$	4.00000			0.193122
$430$		0			0
$431$	18.5109			0.891637			0.445819	−	0.895123i	$-0.352913\pi$
$431$	18.5109			0.891637			0.445819	+	0.895123i	$0.352913\pi$
$432$		0			0
$433$		−	16.4356i		−	0.789847i	−0.918714	−	0.394923i	$-0.870771\pi$
$433$		−	16.4356i		−	0.789847i	0.918714	−	0.394923i	$-0.129229\pi$
$434$		0			0
$435$			34.0511i			1.63263i
$436$		0			0
$437$	−14.7446	+	34.0511i	−0.705328	+	1.62889i
$438$		0			0
$439$	−39.7228			−1.89587			−0.947933	−	0.318469i	$-0.896831\pi$
$439$	−39.7228			−1.89587			−0.947933	+	0.318469i	$0.896831\pi$
$440$		0			0
$441$	6.37228			0.303442
$442$		0			0
$443$			19.4024i			0.921837i	0.887443	+	0.460918i	$0.152480\pi$
$443$			19.4024i			0.921837i	−0.887443	+	0.460918i	$0.847520\pi$
$444$		0			0
$445$		−	44.7384i		−	2.12080i
$446$		0			0
$447$	2.11684			0.100123
$448$		0			0
$449$			17.6155i			0.831325i	0.909519	+	0.415663i	$0.136450\pi$
$449$			17.6155i			0.831325i	−0.909519	+	0.415663i	$0.863550\pi$
$450$		0			0
$451$	−5.48913			−0.258473
$452$		0			0
$453$	−8.74456			−0.410855
$454$		0			0
$455$	−13.4891			−0.632380
$456$		0			0
$457$	−8.11684			−0.379690			−0.189845	−	0.981814i	$-0.560799\pi$
$457$	−8.11684			−0.379690			−0.189845	+	0.981814i	$0.560799\pi$
$458$		0			0
$459$	5.37228			0.250757
$460$		0			0
$461$	32.6277			1.51962			0.759812	−	0.650143i	$-0.225291\pi$
$461$	32.6277			1.51962			0.759812	+	0.650143i	$0.225291\pi$
$462$		0			0
$463$		−	0.202380i		−	0.00940538i	−0.999989	−	0.00470269i	$-0.998503\pi$
$463$		−	0.202380i		−	0.00940538i	0.999989	−	0.00470269i	$-0.00149692\pi$
$464$		0			0
$465$	−29.4891			−1.36753
$466$		0			0
$467$		−	31.0843i		−	1.43841i	−0.694797	−	0.719206i	$-0.744506\pi$
$467$		−	31.0843i		−	1.43841i	0.694797	−	0.719206i	$-0.255494\pi$
$468$		0			0
$469$			7.51811i			0.347154i
$470$		0			0
$471$	7.48913			0.345081
$472$		0			0
$473$	7.37228			0.338978
$474$		0			0
$475$	25.4891	+	11.0371i	1.16952	+	0.506418i
$476$		0			0
$477$			3.16915i			0.145105i
$478$		0			0
$479$		−	21.7793i		−	0.995121i	−0.867429	−	0.497560i	$-0.834229\pi$
$479$		−	21.7793i		−	0.995121i	0.867429	−	0.497560i	$-0.165771\pi$
$480$		0			0
$481$	−25.4891			−1.16220
$482$		0			0
$483$	6.74456			0.306888
$484$		0			0
$485$			44.7384i			2.03146i
$486$		0			0
$487$	5.76631			0.261297			0.130648	−	0.991429i	$-0.458294\pi$
$487$	5.76631			0.261297			0.130648	+	0.991429i	$0.458294\pi$
$488$		0			0
$489$			19.8997i			0.899898i
$490$		0			0
$491$			21.0796i			0.951307i	0.879633	+	0.475654i	$0.157788\pi$
$491$			21.0796i			0.951307i	−0.879633	+	0.475654i	$0.842212\pi$
$492$		0			0
$493$		−	54.2458i		−	2.44311i
$494$		0			0
$495$			2.67181i			0.120089i
$496$		0			0
$497$			3.16915i			0.142156i
$498$		0			0
$499$		−	4.55134i		−	0.203746i	−0.994797	−	0.101873i	$-0.967516\pi$
$499$		−	4.55134i		−	0.203746i	0.994797	−	0.101873i	$-0.0324835\pi$
$500$		0			0
$501$	17.4891			0.781356
$502$		0			0
$503$			16.0309i			0.714782i	0.933955	+	0.357391i	$0.116334\pi$
$503$			16.0309i			0.714782i	−0.933955	+	0.357391i	$0.883666\pi$
$504$		0			0
$505$	49.7228			2.21264
$506$		0			0
$507$	−12.4891			−0.554662
$508$		0			0
$509$		−	13.2665i		−	0.588027i	−0.955801	−	0.294014i	$-0.905009\pi$
$509$		−	13.2665i		−	0.588027i	0.955801	−	0.294014i	$-0.0949911\pi$
$510$		0			0
$511$			3.26172i			0.144290i
$512$		0			0
$513$	4.00000	+	1.73205i	0.176604	+	0.0764719i
$514$		0			0
$515$	42.9783			1.89385
$516$		0			0
$517$	3.37228			0.148313
$518$		0			0
$519$		−	20.1947i		−	0.886449i
$520$		0			0
$521$		−	3.75906i		−	0.164687i	−0.996604	−	0.0823436i	$-0.973760\pi$
$521$		−	3.75906i		−	0.164687i	0.996604	−	0.0823436i	$-0.0262405\pi$
$522$		0			0
$523$	32.0000			1.39926			0.699631	−	0.714504i	$-0.253348\pi$
$523$	32.0000			1.39926			0.699631	+	0.714504i	$0.253348\pi$
$524$		0			0
$525$		−	5.04868i		−	0.220342i
$526$		0			0
$527$	46.9783			2.04640
$528$		0			0
$529$	−49.4674			−2.15076
$530$		0			0
$531$	−4.00000			−0.173585
$532$		0			0
$533$	34.9783			1.51508
$534$		0			0
$535$	−32.0000			−1.38348
$536$		0			0
$537$	1.48913			0.0642605
$538$		0			0
$539$		−	5.04868i		−	0.217462i
$540$		0			0
$541$	−4.11684			−0.176997			−0.0884985	−	0.996076i	$-0.528207\pi$
$541$	−4.11684			−0.176997			−0.0884985	+	0.996076i	$0.528207\pi$
$542$		0			0
$543$			5.63858i			0.241975i
$544$		0			0
$545$			27.7128i			1.18709i
$546$		0			0
$547$	−29.4891			−1.26086			−0.630432	−	0.776245i	$-0.717122\pi$
$547$	−29.4891			−1.26086			−0.630432	+	0.776245i	$0.717122\pi$
$548$		0			0
$549$	5.37228			0.229283
$550$		0			0
$551$	17.4891	−	40.3894i	0.745062	−	1.72065i
$552$		0			0
$553$		−	3.75906i		−	0.159851i
$554$		0			0
$555$		−	17.0256i		−	0.722695i
$556$		0			0
$557$	11.1386			0.471957			0.235979	−	0.971758i	$-0.424170\pi$
$557$	11.1386			0.471957			0.235979	+	0.971758i	$0.424170\pi$
$558$		0			0
$559$	−46.9783			−1.98697
$560$		0			0
$561$		−	4.25639i		−	0.179705i
$562$		0			0
$563$	−22.9783			−0.968418			−0.484209	−	0.874952i	$-0.660893\pi$
$563$	−22.9783			−0.968418			−0.484209	+	0.874952i	$0.660893\pi$
$564$		0			0
$565$			12.6766i			0.533308i
$566$		0			0
$567$		−	0.792287i		−	0.0332729i
$568$		0			0
$569$		−	27.1229i		−	1.13705i	−0.822666	−	0.568526i	$-0.807514\pi$
$569$		−	27.1229i		−	1.13705i	0.822666	−	0.568526i	$-0.192486\pi$
$570$		0			0
$571$		−	2.87419i		−	0.120281i	−0.998190	−	0.0601406i	$-0.980845\pi$
$571$		−	2.87419i		−	0.120281i	0.998190	−	0.0601406i	$-0.0191549\pi$
$572$		0			0
$573$			3.66648i			0.153169i
$574$		0			0
$575$			54.2458i			2.26221i
$576$		0			0
$577$	−37.6060			−1.56556			−0.782778	−	0.622300i	$-0.786198\pi$
$577$	−37.6060			−1.56556			−0.782778	+	0.622300i	$0.786198\pi$
$578$		0			0
$579$		−	13.8564i		−	0.575853i
$580$		0			0
$581$	2.74456			0.113864
$582$		0			0
$583$	2.51087			0.103990
$584$		0			0
$585$		−	17.0256i		−	0.703920i
$586$		0			0
$587$		−	1.38219i		−	0.0570493i	−0.999593	−	0.0285246i	$-0.990919\pi$
$587$		−	1.38219i		−	0.0570493i	0.999593	−	0.0285246i	$-0.00908090\pi$
$588$		0			0
$589$	34.9783	+	15.1460i	1.44125	+	0.624081i
$590$		0			0
$591$	−5.25544			−0.216180
$592$		0			0
$593$	−23.4891			−0.964583			−0.482291	−	0.876011i	$-0.660195\pi$
$593$	−23.4891			−0.964583			−0.482291	+	0.876011i	$0.660195\pi$
$594$		0			0
$595$			14.3537i			0.588446i
$596$		0			0
$597$		−	21.5769i		−	0.883084i
$598$		0			0
$599$	44.4674			1.81689			0.908444	−	0.418007i	$-0.137271\pi$
$599$	44.4674			1.81689			0.908444	+	0.418007i	$0.137271\pi$
$600$		0			0
$601$		−	19.6048i		−	0.799696i	−0.916581	−	0.399848i	$-0.869063\pi$
$601$		−	19.6048i		−	0.799696i	0.916581	−	0.399848i	$-0.130937\pi$
$602$		0			0
$603$	−9.48913			−0.386427
$604$		0			0
$605$	−34.9783			−1.42207
$606$		0			0
$607$	−40.7446			−1.65377			−0.826885	−	0.562370i	$-0.809889\pi$
$607$	−40.7446			−1.65377			−0.826885	+	0.562370i	$0.809889\pi$
$608$		0			0
$609$	−8.00000			−0.324176
$610$		0			0
$611$	−21.4891			−0.869357
$612$		0			0
$613$	5.37228			0.216984			0.108492	−	0.994097i	$-0.465398\pi$
$613$	5.37228			0.216984			0.108492	+	0.994097i	$0.465398\pi$
$614$		0			0
$615$			23.3639i			0.942121i
$616$		0			0
$617$	−19.0951			−0.768740			−0.384370	−	0.923179i	$-0.625581\pi$
$617$	−19.0951			−0.768740			−0.384370	+	0.923179i	$0.625581\pi$
$618$		0			0
$619$		−	10.3923i		−	0.417702i	−0.977947	−	0.208851i	$-0.933028\pi$
$619$		−	10.3923i		−	0.417702i	0.977947	−	0.208851i	$-0.0669724\pi$
$620$		0			0
$621$			8.51278i			0.341606i
$622$		0			0
$623$	10.5109			0.421109
$624$		0			0
$625$	−16.2554			−0.650217
$626$		0			0
$627$	1.37228	−	3.16915i	0.0548036	−	0.126564i
$628$		0			0
$629$			27.1229i			1.08146i
$630$		0			0
$631$			42.7663i			1.70250i	0.524762	+	0.851249i	$0.324154\pi$
$631$			42.7663i			1.70250i	−0.524762	+	0.851249i	$0.675846\pi$
$632$		0			0
$633$	4.00000			0.158986
$634$		0			0
$635$	−42.9783			−1.70554
$636$		0			0
$637$			32.1716i			1.27468i
$638$		0			0
$639$	−4.00000			−0.158238
$640$		0			0
$641$			13.2665i			0.523995i	0.965069	+	0.261998i	$0.0843813\pi$
$641$			13.2665i			0.523995i	−0.965069	+	0.261998i	$0.915619\pi$
$642$		0			0
$643$		−	0.792287i		−	0.0312447i	−0.999878	−	0.0156224i	$-0.995027\pi$
$643$		−	0.792287i		−	0.0312447i	0.999878	−	0.0156224i	$-0.00497296\pi$
$644$		0			0
$645$		−	31.3793i		−	1.23556i
$646$		0			0
$647$		−	36.7229i		−	1.44373i	−0.692035	−	0.721864i	$-0.743286\pi$
$647$		−	36.7229i		−	1.44373i	0.692035	−	0.721864i	$-0.256714\pi$
$648$		0			0
$649$			3.16915i			0.124400i
$650$		0			0
$651$		−	6.92820i		−	0.271538i
$652$		0			0
$653$	22.1168			0.865499			0.432749	−	0.901514i	$-0.357543\pi$
$653$	22.1168			0.865499			0.432749	+	0.901514i	$0.357543\pi$
$654$		0			0
$655$		−	8.01544i		−	0.313189i
$656$		0			0
$657$	−4.11684			−0.160613
$658$		0			0
$659$	9.48913			0.369644			0.184822	−	0.982772i	$-0.440829\pi$
$659$	9.48913			0.369644			0.184822	+	0.982772i	$0.440829\pi$
$660$		0			0
$661$		−	46.0280i		−	1.79028i	−0.445784	−	0.895141i	$-0.647075\pi$
$661$		−	46.0280i		−	1.79028i	0.445784	−	0.895141i	$-0.352925\pi$
$662$		0			0
$663$			27.1229i			1.05337i
$664$		0			0
$665$	−4.62772	+	10.6873i	−0.179455	+	0.414434i
$666$		0			0
$667$	85.9565			3.32825
$668$		0			0
$669$	0.744563			0.0287865
$670$		0			0
$671$		−	4.25639i		−	0.164316i
$672$		0			0
$673$		−	14.4463i		−	0.556864i	−0.960456	−	0.278432i	$-0.910185\pi$
$673$		−	14.4463i		−	0.556864i	0.960456	−	0.278432i	$-0.0898148\pi$
$674$		0			0
$675$	6.37228			0.245269
$676$		0			0
$677$			21.3745i			0.821489i	0.911750	+	0.410745i	$0.134731\pi$
$677$			21.3745i			0.821489i	−0.911750	+	0.410745i	$0.865269\pi$
$678$		0			0
$679$	−10.5109			−0.403370
$680$		0			0
$681$	−12.0000			−0.459841
$682$		0			0
$683$	6.97825			0.267015			0.133508	−	0.991048i	$-0.457376\pi$
$683$	6.97825			0.267015			0.133508	+	0.991048i	$0.457376\pi$
$684$		0			0
$685$	13.8832			0.530448
$686$		0			0
$687$	−24.1168			−0.920115
$688$		0			0
$689$	−16.0000			−0.609551
$690$		0			0
$691$		−	27.9152i		−	1.06194i	−0.847389	−	0.530972i	$-0.821827\pi$
$691$		−	27.9152i		−	1.06194i	0.847389	−	0.530972i	$-0.178173\pi$
$692$		0			0
$693$	−0.627719			−0.0238451
$694$		0			0
$695$		−	18.7027i		−	0.709434i
$696$		0			0
$697$		−	37.2203i		−	1.40982i
$698$		0			0
$699$	−4.11684			−0.155713
$700$		0			0
$701$	37.7228			1.42477			0.712385	−	0.701788i	$-0.247615\pi$
$701$	37.7228			1.42477			0.712385	+	0.701788i	$0.247615\pi$
$702$		0			0
$703$	−8.74456	+	20.1947i	−0.329807	+	0.761658i
$704$		0			0
$705$		−	14.3537i		−	0.540593i
$706$		0			0
$707$			11.6819i			0.439344i
$708$		0			0
$709$	43.9565			1.65082			0.825411	−	0.564533i	$-0.190944\pi$
$709$	43.9565			1.65082			0.825411	+	0.564533i	$0.190944\pi$
$710$		0			0
$711$	4.74456			0.177935
$712$		0			0
$713$			74.4405i			2.78782i
$714$		0			0
$715$	−13.4891			−0.504465
$716$		0			0
$717$		−	29.7947i		−	1.11270i
$718$		0			0
$719$			13.7638i			0.513304i	0.966504	+	0.256652i	$0.0826195\pi$
$719$			13.7638i			0.513304i	−0.966504	+	0.256652i	$0.917381\pi$
$720$		0			0
$721$			10.0974i			0.376045i
$722$		0			0
$723$			17.0256i			0.633187i
$724$		0			0
$725$		−	64.3432i		−	2.38965i
$726$		0			0
$727$		−	24.7460i		−	0.917780i	−0.888493	−	0.458890i	$-0.848247\pi$
$727$		−	24.7460i		−	0.917780i	0.888493	−	0.458890i	$-0.151753\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$			49.9894i			1.84893i
$732$		0			0
$733$	−20.9783			−0.774849			−0.387425	−	0.921901i	$-0.626635\pi$
$733$	−20.9783			−0.774849			−0.387425	+	0.921901i	$0.626635\pi$
$734$		0			0
$735$	−21.4891			−0.792638
$736$		0			0
$737$			7.51811i			0.276933i
$738$		0			0
$739$		−	38.0125i		−	1.39831i	−0.714968	−	0.699157i	$-0.753559\pi$
$739$		−	38.0125i		−	1.39831i	0.714968	−	0.699157i	$-0.246441\pi$
$740$		0			0
$741$	−8.74456	+	20.1947i	−0.321240	+	0.741871i
$742$		0			0
$743$	−34.9783			−1.28323			−0.641614	−	0.767028i	$-0.721735\pi$
$743$	−34.9783			−1.28323			−0.641614	+	0.767028i	$0.721735\pi$
$744$		0			0
$745$	−7.13859			−0.261538
$746$		0			0
$747$			3.46410i			0.126745i
$748$		0			0
$749$		−	7.51811i		−	0.274706i
$750$		0			0
$751$	15.7228			0.573734			0.286867	−	0.957970i	$-0.407386\pi$
$751$	15.7228			0.573734			0.286867	+	0.957970i	$0.407386\pi$
$752$		0			0
$753$			6.13592i			0.223605i
$754$		0			0
$755$	29.4891			1.07322
$756$		0			0
$757$	10.8614			0.394765			0.197382	−	0.980327i	$-0.436756\pi$
$757$	10.8614			0.394765			0.197382	+	0.980327i	$0.436756\pi$
$758$		0			0
$759$	6.74456			0.244812
$760$		0			0
$761$	6.86141			0.248726			0.124363	−	0.992237i	$-0.460311\pi$
$761$	6.86141			0.248726			0.124363	+	0.992237i	$0.460311\pi$
$762$		0			0
$763$	−6.51087			−0.235709
$764$		0			0
$765$	−18.1168			−0.655016
$766$		0			0
$767$		−	20.1947i		−	0.729188i
$768$		0			0
$769$	−9.13859			−0.329546			−0.164773	−	0.986332i	$-0.552689\pi$
$769$	−9.13859			−0.329546			−0.164773	+	0.986332i	$0.552689\pi$
$770$		0			0
$771$			3.75906i			0.135379i
$772$		0			0
$773$		−	39.7995i		−	1.43149i	−0.698363	−	0.715744i	$-0.746088\pi$
$773$		−	39.7995i		−	1.43149i	0.698363	−	0.715744i	$-0.253912\pi$
$774$		0			0
$775$	55.7228			2.00162
$776$		0			0
$777$	4.00000			0.143499
$778$		0			0
$779$	12.0000	−	27.7128i	0.429945	−	0.992915i
$780$		0			0
$781$			3.16915i			0.113401i
$782$		0			0
$783$		−	10.0974i		−	0.360850i
$784$		0			0
$785$	−25.2554			−0.901405
$786$		0			0
$787$	10.9783			0.391332			0.195666	−	0.980671i	$-0.437313\pi$
$787$	10.9783			0.391332			0.195666	+	0.980671i	$0.437313\pi$
$788$		0			0
$789$			18.1128i			0.644833i
$790$		0			0
$791$	−2.97825			−0.105894
$792$		0			0
$793$			27.1229i			0.963163i
$794$		0			0
$795$		−	10.6873i		−	0.379038i
$796$		0			0
$797$		−	10.0974i		−	0.357667i	−0.983879	−	0.178833i	$-0.942768\pi$
$797$		−	10.0974i		−	0.357667i	0.983879	−	0.178833i	$-0.0572323\pi$
$798$		0			0
$799$			22.8665i			0.808959i
$800$		0			0
$801$			13.2665i			0.468749i
$802$		0			0
$803$			3.26172i			0.115104i
$804$		0			0
$805$	−22.7446			−0.801640
$806$		0			0
$807$		−	10.6873i		−	0.376209i
$808$		0			0
$809$	−32.1168			−1.12917			−0.564584	−	0.825376i	$-0.690963\pi$
$809$	−32.1168			−1.12917			−0.564584	+	0.825376i	$0.690963\pi$
$810$		0			0
$811$	−32.4674			−1.14008			−0.570042	−	0.821616i	$-0.693073\pi$
$811$	−32.4674			−1.14008			−0.570042	+	0.821616i	$0.693073\pi$
$812$		0			0
$813$		−	13.5615i		−	0.475621i
$814$		0			0
$815$		−	67.1076i		−	2.35067i
$816$		0			0
$817$	−16.1168	+	37.2203i	−0.563857	+	1.30217i
$818$		0			0
$819$	4.00000			0.139771
$820$		0			0
$821$	16.6277			0.580311			0.290156	−	0.956979i	$-0.406293\pi$
$821$	16.6277			0.580311			0.290156	+	0.956979i	$0.406293\pi$
$822$		0			0
$823$			16.2333i			0.565856i	0.959141	+	0.282928i	$0.0913058\pi$
$823$			16.2333i			0.565856i	−0.959141	+	0.282928i	$0.908694\pi$
$824$		0			0
$825$		−	5.04868i		−	0.175772i
$826$		0			0
$827$	−1.48913			−0.0517820			−0.0258910	−	0.999665i	$-0.508242\pi$
$827$	−1.48913			−0.0517820			−0.0258910	+	0.999665i	$0.508242\pi$
$828$		0			0
$829$			43.4487i			1.50904i	0.656279	+	0.754519i	$0.272130\pi$
$829$			43.4487i			1.50904i	−0.656279	+	0.754519i	$0.727870\pi$
$830$		0			0
$831$	−2.62772			−0.0911546
$832$		0			0
$833$	34.2337			1.18613
$834$		0			0
$835$	−58.9783			−2.04103
$836$		0			0
$837$	8.74456			0.302256
$838$		0			0
$839$	−24.0000			−0.828572			−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000			−0.828572			−0.414286	+	0.910147i	$0.635969\pi$
$840$		0			0
$841$	−72.9565			−2.51574
$842$		0			0
$843$		−	10.0974i		−	0.347771i
$844$		0			0
$845$	42.1168			1.44886
$846$		0			0
$847$		−	8.21782i		−	0.282368i
$848$		0			0
$849$		−	27.3253i		−	0.937801i
$850$		0			0
$851$	−42.9783			−1.47328
$852$		0			0
$853$	22.0000			0.753266			0.376633	−	0.926363i	$-0.377082\pi$
$853$	22.0000			0.753266			0.376633	+	0.926363i	$0.377082\pi$
$854$		0			0
$855$	−13.4891	−	5.84096i	−0.461318	−	0.199757i
$856$		0			0
$857$			47.3176i			1.61634i	0.588950	+	0.808169i	$0.299541\pi$
$857$			47.3176i			1.61634i	−0.588950	+	0.808169i	$0.700459\pi$
$858$		0			0
$859$		−	52.4589i		−	1.78987i	−0.446193	−	0.894937i	$-0.647221\pi$
$859$		−	52.4589i		−	1.78987i	0.446193	−	0.894937i	$-0.352779\pi$
$860$		0			0
$861$	−5.48913			−0.187069
$862$		0			0
$863$	−56.4674			−1.92217			−0.961086	−	0.276250i	$-0.910908\pi$
$863$	−56.4674			−1.92217			−0.961086	+	0.276250i	$0.910908\pi$
$864$		0			0
$865$			68.1022i			2.31554i
$866$		0			0
$867$	11.8614			0.402834
$868$		0			0
$869$		−	3.75906i		−	0.127517i
$870$		0			0
$871$		−	47.9075i		−	1.62328i
$872$		0			0
$873$		−	13.2665i		−	0.449003i
$874$		0			0
$875$			3.66648i			0.123950i
$876$		0			0
$877$		−	33.3514i		−	1.12620i	−0.826390	−	0.563098i	$-0.809609\pi$
$877$		−	33.3514i		−	1.12620i	0.826390	−	0.563098i	$-0.190391\pi$
$878$		0			0
$879$		−	13.8564i		−	0.467365i
$880$		0			0
$881$	−33.6060			−1.13221			−0.566107	−	0.824332i	$-0.691551\pi$
$881$	−33.6060			−1.13221			−0.566107	+	0.824332i	$0.691551\pi$
$882$		0			0
$883$			6.13592i			0.206490i	0.994656	+	0.103245i	$0.0329226\pi$
$883$			6.13592i			0.206490i	−0.994656	+	0.103245i	$0.967077\pi$
$884$		0			0
$885$	13.4891			0.453432
$886$		0			0
$887$	28.0000			0.940148			0.470074	−	0.882627i	$-0.344227\pi$
$887$	28.0000			0.940148			0.470074	+	0.882627i	$0.344227\pi$
$888$		0			0
$889$		−	10.0974i		−	0.338654i
$890$		0			0
$891$		−	0.792287i		−	0.0265426i
$892$		0			0
$893$	−7.37228	+	17.0256i	−0.246704	+	0.569738i
$894$		0			0
$895$	−5.02175			−0.167859
$896$		0			0
$897$	−42.9783			−1.43500
$898$		0			0
$899$		−	88.2969i		−	2.94487i
$900$		0			0
$901$			17.0256i			0.567203i
$902$		0			0
$903$	7.37228			0.245334
$904$		0			0
$905$		−	19.0149i		−	0.632076i
$906$		0			0
$907$	−48.4674			−1.60933			−0.804666	−	0.593727i	$-0.797656\pi$
$907$	−48.4674			−1.60933			−0.804666	+	0.593727i	$0.797656\pi$
$908$		0			0
$909$	−14.7446			−0.489046
$910$		0			0
$911$	25.4891			0.844492			0.422246	−	0.906481i	$-0.361242\pi$
$911$	25.4891			0.844492			0.422246	+	0.906481i	$0.361242\pi$
$912$		0			0
$913$	2.74456			0.0908318
$914$		0			0
$915$	−18.1168			−0.598924
$916$		0			0
$917$	1.88316			0.0621873
$918$		0			0
$919$		−	51.3716i		−	1.69459i	−0.531120	−	0.847297i	$-0.678229\pi$
$919$		−	51.3716i		−	1.69459i	0.531120	−	0.847297i	$-0.321771\pi$
$920$		0			0
$921$	−17.4891			−0.576286
$922$		0			0
$923$		−	20.1947i		−	0.664717i
$924$		0			0
$925$			32.1716i			1.05780i
$926$		0			0
$927$	−12.7446			−0.418586
$928$		0			0
$929$	−55.4891			−1.82054			−0.910270	−	0.414016i	$-0.864126\pi$
$929$	−55.4891			−1.82054			−0.910270	+	0.414016i	$0.864126\pi$
$930$		0			0
$931$	25.4891	+	11.0371i	0.835372	+	0.361727i
$932$		0			0
$933$			14.3537i			0.469920i
$934$		0			0
$935$			14.3537i			0.469417i
$936$		0			0
$937$	17.3723			0.567528			0.283764	−	0.958894i	$-0.408417\pi$
$937$	17.3723			0.567528			0.283764	+	0.958894i	$0.408417\pi$
$938$		0			0
$939$	19.4891			0.636004
$940$		0			0
$941$			3.75906i			0.122542i	0.998121	+	0.0612708i	$0.0195153\pi$
$941$			3.75906i			0.122542i	−0.998121	+	0.0612708i	$0.980485\pi$
$942$		0			0
$943$	58.9783			1.92060
$944$		0			0
$945$			2.67181i			0.0869141i
$946$		0			0
$947$		−	12.9715i		−	0.421519i	−0.977538	−	0.210759i	$-0.932406\pi$
$947$		−	12.9715i		−	0.421519i	0.977538	−	0.210759i	$-0.0675936\pi$
$948$		0			0
$949$		−	20.7846i		−	0.674697i
$950$		0			0
$951$			10.6873i			0.346558i
$952$		0			0
$953$		−	37.8102i		−	1.22479i	−0.790551	−	0.612396i	$-0.790206\pi$
$953$		−	37.8102i		−	1.22479i	0.790551	−	0.612396i	$-0.209794\pi$
$954$		0			0
$955$		−	12.3644i		−	0.400103i
$956$		0			0
$957$	−8.00000			−0.258603
$958$		0			0
$959$			3.26172i			0.105326i
$960$		0			0
$961$	45.4674			1.46669
$962$		0			0
$963$	9.48913			0.305783
$964$		0			0
$965$			46.7277i			1.50422i
$966$		0			0
$967$		−	16.1407i		−	0.519050i	−0.965736	−	0.259525i	$-0.916434\pi$
$967$		−	16.1407i		−	0.519050i	0.965736	−	0.259525i	$-0.0835659\pi$
$968$		0			0
$969$	21.4891	+	9.30506i	0.690330	+	0.298922i
$970$		0			0
$971$	44.4674			1.42703			0.713513	−	0.700642i	$-0.247103\pi$
$971$	44.4674			1.42703			0.713513	+	0.700642i	$0.247103\pi$
$972$		0			0
$973$	4.39403			0.140866
$974$		0			0
$975$			32.1716i			1.03032i
$976$		0			0
$977$			19.6048i			0.627213i	0.949553	+	0.313606i	$0.101537\pi$
$977$			19.6048i			0.627213i	−0.949553	+	0.313606i	$0.898463\pi$
$978$		0			0
$979$	10.5109			0.335929
$980$		0			0
$981$		−	8.21782i		−	0.262375i
$982$		0			0
$983$	−5.02175			−0.160169			−0.0800845	−	0.996788i	$-0.525519\pi$
$983$	−5.02175			−0.160169			−0.0800845	+	0.996788i	$0.525519\pi$
$984$		0			0
$985$	17.7228			0.564696
$986$		0			0
$987$	3.37228			0.107341
$988$		0			0
$989$	−79.2119			−2.51879
$990$		0			0
$991$	8.74456			0.277780			0.138890	−	0.990308i	$-0.455647\pi$
$991$	8.74456			0.277780			0.138890	+	0.990308i	$0.455647\pi$
$992$		0			0
$993$	9.48913			0.301128
$994$		0			0
$995$			72.7634i			2.30675i
$996$		0			0
$997$	−53.6060			−1.69772			−0.848859	−	0.528619i	$-0.822710\pi$
$997$	−53.6060			−1.69772			−0.848859	+	0.528619i	$0.822710\pi$
$998$		0			0
$999$			5.04868i			0.159733i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.k.h.607.1	yes	4
3.2	odd	2		2736.2.k.o.2431.3		4
4.3	odd	2		912.2.k.g.607.2	yes	4
8.3	odd	2		3648.2.k.h.2431.4		4
8.5	even	2		3648.2.k.g.2431.3		4
12.11	even	2		2736.2.k.n.2431.4		4
19.18	odd	2		912.2.k.g.607.1	✓	4
57.56	even	2		2736.2.k.n.2431.3		4
76.75	even	2	inner	912.2.k.h.607.2	yes	4
152.37	odd	2		3648.2.k.h.2431.3		4
152.75	even	2		3648.2.k.g.2431.4		4
228.227	odd	2		2736.2.k.o.2431.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.2.k.g.607.1	✓	4	19.18	odd	2
912.2.k.g.607.2	yes	4	4.3	odd	2
912.2.k.h.607.1	yes	4	1.1	even	1	trivial
912.2.k.h.607.2	yes	4	76.75	even	2	inner
2736.2.k.n.2431.3		4	57.56	even	2
2736.2.k.n.2431.4		4	12.11	even	2
2736.2.k.o.2431.3		4	3.2	odd	2
2736.2.k.o.2431.4		4	228.227	odd	2
3648.2.k.g.2431.3		4	8.5	even	2
3648.2.k.g.2431.4		4	152.75	even	2
3648.2.k.h.2431.3		4	152.37	odd	2
3648.2.k.h.2431.4		4	8.3	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:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	912.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.37228 q^{5} -0.792287i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.37228 q^{5} -0.792287i q^{7} +1.00000 q^{9} -0.792287i q^{11} +5.04868i q^{13} -3.37228 q^{15} +5.37228 q^{17} +(4.00000 + 1.73205i) q^{19} -0.792287i q^{21} +8.51278i q^{23} +6.37228 q^{25} +1.00000 q^{27} -10.0974i q^{29} +8.74456 q^{31} -0.792287i q^{33} +2.67181i q^{35} +5.04868i q^{37} +5.04868i q^{39} -6.92820i q^{41} +9.30506i q^{43} -3.37228 q^{45} +4.25639i q^{47} +6.37228 q^{49} +5.37228 q^{51} +3.16915i q^{53} +2.67181i q^{55} +(4.00000 + 1.73205i) q^{57} -4.00000 q^{59} +5.37228 q^{61} -0.792287i q^{63} -17.0256i q^{65} -9.48913 q^{67} +8.51278i q^{69} -4.00000 q^{71} -4.11684 q^{73} +6.37228 q^{75} -0.627719 q^{77} +4.74456 q^{79} +1.00000 q^{81} +3.46410i q^{83} -18.1168 q^{85} -10.0974i q^{87} +13.2665i q^{89} +4.00000 q^{91} +8.74456 q^{93} +(-13.4891 - 5.84096i) q^{95} -13.2665i q^{97} -0.792287i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 2 q^{5} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 2 * q^5 + 4 * q^9	\( 4 q + 4 q^{3} - 2 q^{5} + 4 q^{9} - 2 q^{15} + 10 q^{17} + 16 q^{19} + 14 q^{25} + 4 q^{27} + 12 q^{31} - 2 q^{45} + 14 q^{49} + 10 q^{51} + 16 q^{57} - 16 q^{59} + 10 q^{61} + 8 q^{67} - 16 q^{71} + 18 q^{73} + 14 q^{75} - 14 q^{77} - 4 q^{79} + 4 q^{81} - 38 q^{85} + 16 q^{91} + 12 q^{93} - 8 q^{95}+O(q^{100}) \) 4 * q + 4 * q^3 - 2 * q^5 + 4 * q^9 - 2 * q^15 + 10 * q^17 + 16 * q^19 + 14 * q^25 + 4 * q^27 + 12 * q^31 - 2 * q^45 + 14 * q^49 + 10 * q^51 + 16 * q^57 - 16 * q^59 + 10 * q^61 + 8 * q^67 - 16 * q^71 + 18 * q^73 + 14 * q^75 - 14 * q^77 - 4 * q^79 + 4 * q^81 - 38 * q^85 + 16 * q^91 + 12 * q^93 - 8 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more