LMFDB - Embedded newform 912.2.d.b.191.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,2,Mod(191,912)]

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, 1, 0]))

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

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

chi := DirichletCharacter("912.191");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	912.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:	$7.28235666434$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.18
Character	$\chi$	$=$	912.191
Dual form			912.2.d.b.191.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.909651 + 1.47395i) q^{3} +2.62895i q^{5} +3.58561i q^{7} +(-1.34507 + 2.68156i) q^{9} +O(q^{10})$	\(q+(0.909651 + 1.47395i) q^{3} +2.62895i q^{5} +3.58561i q^{7} +(-1.34507 + 2.68156i) q^{9} -1.56630 q^{11} +1.23349 q^{13} +(-3.87495 + 2.39143i) q^{15} -3.38560i q^{17} -1.00000i q^{19} +(-5.28501 + 3.26165i) q^{21} +1.04413 q^{23} -1.91139 q^{25} +(-5.17604 + 0.456722i) q^{27} -7.20096i q^{29} +1.42596i q^{31} +(-1.42479 - 2.30865i) q^{33} -9.42639 q^{35} +4.55367 q^{37} +(1.12205 + 1.81811i) q^{39} -10.8210i q^{41} +9.96678i q^{43} +(-7.04971 - 3.53612i) q^{45} +0.136958 q^{47} -5.85657 q^{49} +(4.99022 - 3.07972i) q^{51} +9.69295i q^{53} -4.11773i q^{55} +(1.47395 - 0.909651i) q^{57} +6.50974 q^{59} +6.11212 q^{61} +(-9.61503 - 4.82289i) q^{63} +3.24279i q^{65} +14.1977i q^{67} +(0.949792 + 1.53899i) q^{69} +10.0857 q^{71} -9.78698 q^{73} +(-1.73870 - 2.81730i) q^{75} -5.61613i q^{77} +10.5281i q^{79} +(-5.38158 - 7.21378i) q^{81} -13.4190 q^{83} +8.90059 q^{85} +(10.6139 - 6.55036i) q^{87} -4.31626i q^{89} +4.42281i q^{91} +(-2.10180 + 1.29713i) q^{93} +2.62895 q^{95} +6.67044 q^{97} +(2.10678 - 4.20013i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 4 q^{9}+O(q^{10}) $ 24 * q + 4 * q^9	$ 24 q + 4 q^{9} - 12 q^{21} - 64 q^{25} + 12 q^{33} + 64 q^{37} - 16 q^{45} - 8 q^{49} - 24 q^{61} - 8 q^{69} - 16 q^{73} - 4 q^{81} - 8 q^{85} + 32 q^{93} - 8 q^{97}+O(q^{100}) $ 24 * q + 4 * q^9 - 12 * q^21 - 64 * q^25 + 12 * q^33 + 64 * q^37 - 16 * q^45 - 8 * q^49 - 24 * q^61 - 8 * q^69 - 16 * q^73 - 4 * q^81 - 8 * q^85 + 32 * q^93 - 8 * q^97

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
$3$	0.909651	+	1.47395i	0.525187	+	0.850987i
$3$	0.909651	+	1.47395i	0.525187	+	0.850987i
$4$		0			0
$5$			2.62895i			1.17570i	0.808969	+	0.587852i	$0.200026\pi$
$5$			2.62895i			1.17570i	−0.808969	+	0.587852i	$0.799974\pi$
$6$		0			0
$7$			3.58561i			1.35523i	0.735416	+	0.677616i	$0.236987\pi$
$7$			3.58561i			1.35523i	−0.735416	+	0.677616i	$0.763013\pi$
$8$		0			0
$9$	−1.34507	+	2.68156i	−0.448356	+	0.893855i
$10$		0			0
$11$	−1.56630			−0.472257			−0.236129	−	0.971722i	$-0.575879\pi$
$11$	−1.56630			−0.472257			−0.236129	+	0.971722i	$0.575879\pi$
$12$		0			0
$13$	1.23349			0.342109			0.171054	−	0.985262i	$-0.445283\pi$
$13$	1.23349			0.342109			0.171054	+	0.985262i	$0.445283\pi$
$14$		0			0
$15$	−3.87495	+	2.39143i	−1.00051	+	0.617465i
$16$		0			0
$17$		−	3.38560i		−	0.821129i	−0.911832	−	0.410565i	$-0.865332\pi$
$17$		−	3.38560i		−	0.821129i	0.911832	−	0.410565i	$-0.134668\pi$
$18$		0			0
$19$		−	1.00000i		−	0.229416i
$19$		−	1.00000i		−	0.229416i
$20$		0			0
$21$	−5.28501	+	3.26165i	−1.15328	+	0.711751i
$22$		0			0
$23$	1.04413			0.217716			0.108858	−	0.994057i	$-0.465281\pi$
$23$	1.04413			0.217716			0.108858	+	0.994057i	$0.465281\pi$
$24$		0			0
$25$	−1.91139			−0.382279
$26$		0			0
$27$	−5.17604	+	0.456722i	−0.996130	+	0.0878963i
$28$		0			0
$29$		−	7.20096i		−	1.33718i	−0.743629	−	0.668592i	$-0.766897\pi$
$29$		−	7.20096i		−	1.33718i	0.743629	−	0.668592i	$-0.233103\pi$
$30$		0			0
$31$			1.42596i			0.256111i	0.991767	+	0.128055i	$0.0408735\pi$
$31$			1.42596i			0.256111i	−0.991767	+	0.128055i	$0.959126\pi$
$32$		0			0
$33$	−1.42479	−	2.30865i	−0.248024	−	0.401884i
$34$		0			0
$35$	−9.42639			−1.59335
$36$		0			0
$37$	4.55367			0.748619			0.374309	−	0.927304i	$-0.377880\pi$
$37$	4.55367			0.748619			0.374309	+	0.927304i	$0.377880\pi$
$38$		0			0
$39$	1.12205	+	1.81811i	0.179671	+	0.291130i
$40$		0			0
$41$		−	10.8210i		−	1.68996i	−0.534796	−	0.844981i	$-0.679612\pi$
$41$		−	10.8210i		−	1.68996i	0.534796	−	0.844981i	$-0.320388\pi$
$42$		0			0
$43$			9.96678i			1.51992i	0.649970	+	0.759960i	$0.274781\pi$
$43$			9.96678i			1.51992i	−0.649970	+	0.759960i	$0.725219\pi$
$44$		0			0
$45$	−7.04971	−	3.53612i	−1.05091	−	0.527134i
$46$		0			0
$47$	0.136958			0.0199774			0.00998870	−	0.999950i	$-0.496820\pi$
$47$	0.136958			0.0199774			0.00998870	+	0.999950i	$0.496820\pi$
$48$		0			0
$49$	−5.85657			−0.836653
$50$		0			0
$51$	4.99022	−	3.07972i	0.698770	−	0.431247i
$52$		0			0
$53$			9.69295i			1.33143i	0.746206	+	0.665715i	$0.231873\pi$
$53$			9.69295i			1.33143i	−0.746206	+	0.665715i	$0.768127\pi$
$54$		0			0
$55$		−	4.11773i		−	0.555234i
$56$		0			0
$57$	1.47395	−	0.909651i	0.195230	−	0.120486i
$58$		0			0
$59$	6.50974			0.847496			0.423748	−	0.905780i	$-0.360714\pi$
$59$	6.50974			0.847496			0.423748	+	0.905780i	$0.360714\pi$
$60$		0			0
$61$	6.11212			0.782577			0.391288	−	0.920268i	$-0.372029\pi$
$61$	6.11212			0.782577			0.391288	+	0.920268i	$0.372029\pi$
$62$		0			0
$63$	−9.61503	−	4.82289i	−1.21138	−	0.607627i
$64$		0			0
$65$			3.24279i			0.402218i
$66$		0			0
$67$			14.1977i			1.73453i	0.497846	+	0.867265i	$0.334124\pi$
$67$			14.1977i			1.73453i	−0.497846	+	0.867265i	$0.665876\pi$
$68$		0			0
$69$	0.949792	+	1.53899i	0.114342	+	0.185273i
$70$		0			0
$71$	10.0857			1.19695			0.598474	−	0.801143i	$-0.295774\pi$
$71$	10.0857			1.19695			0.598474	+	0.801143i	$0.295774\pi$
$72$		0			0
$73$	−9.78698			−1.14548			−0.572740	−	0.819737i	$-0.694119\pi$
$73$	−9.78698			−1.14548			−0.572740	+	0.819737i	$0.694119\pi$
$74$		0			0
$75$	−1.73870	−	2.81730i	−0.200768	−	0.325314i
$76$		0			0
$77$		−	5.61613i		−	0.640018i
$78$		0			0
$79$			10.5281i			1.18450i	0.805753	+	0.592252i	$0.201761\pi$
$79$			10.5281i			1.18450i	−0.805753	+	0.592252i	$0.798239\pi$
$80$		0			0
$81$	−5.38158	−	7.21378i	−0.597953	−	0.801531i
$82$		0			0
$83$	−13.4190			−1.47293			−0.736465	−	0.676476i	$-0.763506\pi$
$83$	−13.4190			−1.47293			−0.736465	+	0.676476i	$0.763506\pi$
$84$		0			0
$85$	8.90059			0.965404
$86$		0			0
$87$	10.6139	−	6.55036i	1.13793	−	0.702272i
$88$		0			0
$89$		−	4.31626i		−	0.457523i	−0.973483	−	0.228761i	$-0.926532\pi$
$89$		−	4.31626i		−	0.457523i	0.973483	−	0.228761i	$-0.0734676\pi$
$90$		0			0
$91$			4.42281i			0.463636i
$92$		0			0
$93$	−2.10180	+	1.29713i	−0.217947	+	0.134506i
$94$		0			0
$95$	2.62895			0.269725
$96$		0			0
$97$	6.67044			0.677280			0.338640	−	0.940916i	$-0.390033\pi$
$97$	6.67044			0.677280			0.338640	+	0.940916i	$0.390033\pi$
$98$		0			0
$99$	2.10678	−	4.20013i	0.211739	−	0.422129i
$100$		0			0
$101$			16.5666i			1.64844i	0.566268	+	0.824221i	$0.308387\pi$
$101$			16.5666i			1.64844i	−0.566268	+	0.824221i	$0.691613\pi$
$102$		0			0
$103$		−	0.313018i		−	0.0308426i	−0.999881	−	0.0154213i	$-0.995091\pi$
$103$		−	0.313018i		−	0.0308426i	0.999881	−	0.0154213i	$-0.00490894\pi$
$104$		0			0
$105$	−8.57473	−	13.8940i	−0.836808	−	1.35592i
$106$		0			0
$107$	−14.7286			−1.42387			−0.711934	−	0.702247i	$-0.752180\pi$
$107$	−14.7286			−1.42387			−0.711934	+	0.702247i	$0.752180\pi$
$108$		0			0
$109$	6.01635			0.576262			0.288131	−	0.957591i	$-0.406966\pi$
$109$	6.01635			0.576262			0.288131	+	0.957591i	$0.406966\pi$
$110$		0			0
$111$	4.14225	+	6.71189i	0.393165	+	0.637064i
$112$		0			0
$113$		−	5.73478i		−	0.539483i	−0.962933	−	0.269741i	$-0.913062\pi$
$113$		−	5.73478i		−	0.539483i	0.962933	−	0.269741i	$-0.0869382\pi$
$114$		0			0
$115$			2.74496i			0.255969i
$116$		0			0
$117$	−1.65913	+	3.30768i	−0.153387	+	0.305795i
$118$		0			0
$119$	12.1394			1.11282
$120$		0			0
$121$	−8.54671			−0.776973
$122$		0			0
$123$	15.9497	−	9.84337i	1.43814	−	0.887547i
$124$		0			0
$125$			8.11980i			0.726257i
$126$		0			0
$127$		−	8.79148i		−	0.780118i	−0.920790	−	0.390059i	$-0.872455\pi$
$127$		−	8.79148i		−	0.780118i	0.920790	−	0.390059i	$-0.127545\pi$
$128$		0			0
$129$	−14.6906	+	9.06629i	−1.29343	+	0.798243i
$130$		0			0
$131$	8.37456			0.731688			0.365844	−	0.930676i	$-0.380780\pi$
$131$	8.37456			0.731688			0.365844	+	0.930676i	$0.380780\pi$
$132$		0			0
$133$	3.58561			0.310911
$134$		0			0
$135$	−1.20070	−	13.6076i	−0.103340	−	1.17115i
$136$		0			0
$137$			17.5826i			1.50218i	0.660201	+	0.751089i	$0.270471\pi$
$137$			17.5826i			1.50218i	−0.660201	+	0.751089i	$0.729529\pi$
$138$		0			0
$139$			4.53795i			0.384904i	0.981306	+	0.192452i	$0.0616440\pi$
$139$			4.53795i			0.384904i	−0.981306	+	0.192452i	$0.938356\pi$
$140$		0			0
$141$	0.124584	+	0.201870i	0.0104919	+	0.0170005i
$142$		0			0
$143$	−1.93202			−0.161563
$144$		0			0
$145$	18.9310			1.57213
$146$		0			0
$147$	−5.32744	−	8.63230i	−0.439399	−	0.711980i
$148$		0			0
$149$		−	14.3996i		−	1.17966i	−0.807528	−	0.589829i	$-0.799195\pi$
$149$		−	14.3996i		−	1.17966i	0.807528	−	0.589829i	$-0.200805\pi$
$150$		0			0
$151$		−	14.0819i		−	1.14597i	−0.819566	−	0.572985i	$-0.805785\pi$
$151$		−	14.0819i		−	1.14597i	0.819566	−	0.572985i	$-0.194215\pi$
$152$		0			0
$153$	9.07871	+	4.55387i	0.733970	+	0.368158i
$154$		0			0
$155$	−3.74879			−0.301110
$156$		0			0
$157$	20.4473			1.63187			0.815936	−	0.578142i	$-0.196222\pi$
$157$	20.4473			1.63187			0.815936	+	0.578142i	$0.196222\pi$
$158$		0			0
$159$	−14.2869	+	8.81721i	−1.13303	+	0.699250i
$160$		0			0
$161$			3.74383i			0.295055i
$162$		0			0
$163$		−	12.9161i		−	1.01166i	−0.862632	−	0.505832i	$-0.831185\pi$
$163$		−	12.9161i		−	1.01166i	0.862632	−	0.505832i	$-0.168815\pi$
$164$		0			0
$165$	6.06933	−	3.74570i	0.472497	−	0.291602i
$166$		0			0
$167$	−14.8709			−1.15074			−0.575371	−	0.817892i	$-0.695142\pi$
$167$	−14.8709			−1.15074			−0.575371	+	0.817892i	$0.695142\pi$
$168$		0			0
$169$	−11.4785			−0.882962
$170$		0			0
$171$	2.68156	+	1.34507i	0.205064	+	0.102860i
$172$		0			0
$173$			13.3205i			1.01274i	0.862316	+	0.506370i	$0.169013\pi$
$173$			13.3205i			1.01274i	−0.862316	+	0.506370i	$0.830987\pi$
$174$		0			0
$175$		−	6.85350i		−	0.518076i
$176$		0			0
$177$	5.92159	+	9.59504i	0.445094	+	0.721208i
$178$		0			0
$179$	−0.567468			−0.0424145			−0.0212073	−	0.999775i	$-0.506751\pi$
$179$	−0.567468			−0.0424145			−0.0212073	+	0.999775i	$0.506751\pi$
$180$		0			0
$181$	21.2587			1.58015			0.790074	−	0.613011i	$-0.210042\pi$
$181$	21.2587			1.58015			0.790074	+	0.613011i	$0.210042\pi$
$182$		0			0
$183$	5.55990	+	9.00897i	0.411000	+	0.665962i
$184$		0			0
$185$			11.9714i			0.880154i
$186$		0			0
$187$			5.30287i			0.387784i
$188$		0			0
$189$	−1.63763	−	18.5592i	−0.119120	−	1.34999i
$190$		0			0
$191$	8.13476			0.588611			0.294305	−	0.955711i	$-0.404912\pi$
$191$	8.13476			0.588611			0.294305	+	0.955711i	$0.404912\pi$
$192$		0			0
$193$	15.0670			1.08455			0.542274	−	0.840202i	$-0.317564\pi$
$193$	15.0670			1.08455			0.542274	+	0.840202i	$0.317564\pi$
$194$		0			0
$195$	−4.77971	+	2.94981i	−0.342282	+	0.211240i
$196$		0			0
$197$		−	2.78221i		−	0.198224i	−0.995076	−	0.0991122i	$-0.968400\pi$
$197$		−	2.78221i		−	0.198224i	0.995076	−	0.0991122i	$-0.0316003\pi$
$198$		0			0
$199$		−	12.4471i		−	0.882351i	−0.897421	−	0.441176i	$-0.854562\pi$
$199$		−	12.4471i		−	0.882351i	0.897421	−	0.441176i	$-0.145438\pi$
$200$		0			0
$201$	−20.9268	+	12.9150i	−1.47606	+	0.910954i
$202$		0			0
$203$	25.8198			1.81219
$204$		0			0
$205$	28.4480			1.98689
$206$		0			0
$207$	−1.40442	+	2.79990i	−0.0976142	+	0.194606i
$208$		0			0
$209$			1.56630i			0.108343i
$210$		0			0
$211$		−	15.9686i		−	1.09932i	−0.835388	−	0.549661i	$-0.814757\pi$
$211$		−	15.9686i		−	1.09932i	0.835388	−	0.549661i	$-0.185243\pi$
$212$		0			0
$213$	9.17443	+	14.8658i	0.628622	+	1.01859i
$214$		0			0
$215$	−26.2022			−1.78697
$216$		0			0
$217$	−5.11295			−0.347089
$218$		0			0
$219$	−8.90274	−	14.4255i	−0.601591	−	0.974787i
$220$		0			0
$221$		−	4.17611i		−	0.280915i
$222$		0			0
$223$			4.22684i			0.283050i	0.989935	+	0.141525i	$0.0452005\pi$
$223$			4.22684i			0.283050i	−0.989935	+	0.141525i	$0.954799\pi$
$224$		0			0
$225$	2.57095	−	5.12552i	0.171397	−	0.341702i
$226$		0			0
$227$	15.1403			1.00489			0.502447	−	0.864608i	$-0.332433\pi$
$227$	15.1403			1.00489			0.502447	+	0.864608i	$0.332433\pi$
$228$		0			0
$229$	−11.3412			−0.749446			−0.374723	−	0.927137i	$-0.622262\pi$
$229$	−11.3412			−0.749446			−0.374723	+	0.927137i	$0.622262\pi$
$230$		0			0
$231$	8.27791	−	5.10872i	0.544646	−	0.336129i
$232$		0			0
$233$			4.56159i			0.298840i	0.988774	+	0.149420i	$0.0477406\pi$
$233$			4.56159i			0.298840i	−0.988774	+	0.149420i	$0.952259\pi$
$234$		0			0
$235$			0.360056i			0.0234875i
$236$		0			0
$237$	−15.5179	+	9.57691i	−1.00800	+	0.622087i
$238$		0			0
$239$	−15.6650			−1.01328			−0.506641	−	0.862157i	$-0.669113\pi$
$239$	−15.6650			−1.01328			−0.506641	+	0.862157i	$0.669113\pi$
$240$		0			0
$241$	3.70475			0.238644			0.119322	−	0.992856i	$-0.461928\pi$
$241$	3.70475			0.238644			0.119322	+	0.992856i	$0.461928\pi$
$242$		0			0
$243$	5.73740	−	14.4942i	0.368054	−	0.929804i
$244$		0			0
$245$		−	15.3966i		−	0.983655i
$246$		0			0
$247$		−	1.23349i		−	0.0784851i
$248$		0			0
$249$	−12.2066	−	19.7790i	−0.773565	−	1.25344i
$250$		0			0
$251$	28.7827			1.81675			0.908375	−	0.418157i	$-0.137324\pi$
$251$	28.7827			1.81675			0.908375	+	0.418157i	$0.137324\pi$
$252$		0			0
$253$	−1.63542			−0.102818
$254$		0			0
$255$	8.09643	+	13.1190i	0.507018	+	0.821546i
$256$		0			0
$257$		−	22.9658i		−	1.43257i	−0.697809	−	0.716284i	$-0.745841\pi$
$257$		−	22.9658i		−	1.43257i	0.697809	−	0.716284i	$-0.254159\pi$
$258$		0			0
$259$			16.3277i			1.01455i
$260$		0			0
$261$	19.3098	+	9.68578i	1.19525	+	0.599535i
$262$		0			0
$263$	13.7535			0.848075			0.424038	−	0.905645i	$-0.360612\pi$
$263$	13.7535			0.848075			0.424038	+	0.905645i	$0.360612\pi$
$264$		0			0
$265$	−25.4823			−1.56537
$266$		0			0
$267$	6.36196	−	3.92629i	0.389346	−	0.240285i
$268$		0			0
$269$		−	2.93383i		−	0.178879i	−0.995992	−	0.0894393i	$-0.971492\pi$
$269$		−	2.93383i		−	0.178879i	0.995992	−	0.0894393i	$-0.0285075\pi$
$270$		0			0
$271$		−	21.3061i		−	1.29425i	−0.762382	−	0.647127i	$-0.775970\pi$
$271$		−	21.3061i		−	1.29425i	0.762382	−	0.647127i	$-0.224030\pi$
$272$		0			0
$273$	−6.51901	+	4.02321i	−0.394548	+	0.243496i
$274$		0			0
$275$	2.99381			0.180534
$276$		0			0
$277$	−5.89065			−0.353935			−0.176968	−	0.984217i	$-0.556629\pi$
$277$	−5.89065			−0.353935			−0.176968	+	0.984217i	$0.556629\pi$
$278$		0			0
$279$	−3.82382	−	1.91802i	−0.228926	−	0.114829i
$280$		0			0
$281$		−	30.8471i		−	1.84018i	−0.391702	−	0.920092i	$-0.628113\pi$
$281$		−	30.8471i		−	1.84018i	0.391702	−	0.920092i	$-0.371887\pi$
$282$		0			0
$283$		−	25.9416i		−	1.54207i	−0.636793	−	0.771035i	$-0.719739\pi$
$283$		−	25.9416i		−	1.54207i	0.636793	−	0.771035i	$-0.280261\pi$
$284$		0			0
$285$	2.39143	+	3.87495i	0.141656	+	0.229532i
$286$		0			0
$287$	38.8000			2.29029
$288$		0			0
$289$	5.53770			0.325747
$290$		0			0
$291$	6.06777	+	9.83190i	0.355699	+	0.576356i
$292$		0			0
$293$			33.1106i			1.93434i	0.254130	+	0.967170i	$0.418211\pi$
$293$			33.1106i			1.93434i	−0.254130	+	0.967170i	$0.581789\pi$
$294$		0			0
$295$			17.1138i			0.996404i
$296$		0			0
$297$	8.10723	−	0.715364i	0.470429	−	0.0415096i
$298$		0			0
$299$	1.28792			0.0744824
$300$		0			0
$301$	−35.7369			−2.05984
$302$		0			0
$303$	−24.4184	+	15.0699i	−1.40280	+	0.865741i
$304$		0			0
$305$			16.0685i			0.920078i
$306$		0			0
$307$			0.857386i			0.0489336i	0.999701	+	0.0244668i	$0.00778880\pi$
$307$			0.857386i			0.0489336i	−0.999701	+	0.0244668i	$0.992211\pi$
$308$		0			0
$309$	0.461373	−	0.284737i	0.0262466	−	0.0161981i
$310$		0			0
$311$	18.8346			1.06801			0.534006	−	0.845481i	$-0.320686\pi$
$311$	18.8346			1.06801			0.534006	+	0.845481i	$0.320686\pi$
$312$		0			0
$313$	28.4391			1.60747			0.803737	−	0.594985i	$-0.202842\pi$
$313$	28.4391			1.60747			0.803737	+	0.594985i	$0.202842\pi$
$314$		0			0
$315$	12.6791	−	25.2775i	0.714389	−	1.42422i
$316$		0			0
$317$		−	12.9564i		−	0.727705i	−0.931457	−	0.363852i	$-0.881461\pi$
$317$		−	12.9564i		−	0.727705i	0.931457	−	0.363852i	$-0.118539\pi$
$318$		0			0
$319$			11.2789i			0.631495i
$320$		0			0
$321$	−13.3979	−	21.7092i	−0.747797	−	1.21169i
$322$		0			0
$323$	−3.38560			−0.188380
$324$		0			0
$325$	−2.35768			−0.130781
$326$		0			0
$327$	5.47278	+	8.86781i	0.302646	+	0.490391i
$328$		0			0
$329$			0.491078i			0.0270740i
$330$		0			0
$331$		−	6.45891i		−	0.355014i	−0.984120	−	0.177507i	$-0.943197\pi$
$331$		−	6.45891i		−	0.355014i	0.984120	−	0.177507i	$-0.0568032\pi$
$332$		0			0
$333$	−6.12500	+	12.2110i	−0.335648	+	0.669157i
$334$		0			0
$335$	−37.3252			−2.03929
$336$		0			0
$337$	−16.5456			−0.901297			−0.450648	−	0.892702i	$-0.648807\pi$
$337$	−16.5456			−0.901297			−0.450648	+	0.892702i	$0.648807\pi$
$338$		0			0
$339$	8.45279	−	5.21665i	0.459093	−	0.283330i
$340$		0			0
$341$		−	2.23349i		−	0.120950i
$342$		0			0
$343$			4.09990i			0.221374i
$344$		0			0
$345$	−4.04594	+	2.49696i	−0.217826	+	0.134432i
$346$		0			0
$347$	16.5061			0.886095			0.443047	−	0.896498i	$-0.353897\pi$
$347$	16.5061			0.886095			0.443047	+	0.896498i	$0.353897\pi$
$348$		0			0
$349$	−8.12035			−0.434673			−0.217336	−	0.976097i	$-0.569737\pi$
$349$	−8.12035			−0.434673			−0.217336	+	0.976097i	$0.569737\pi$
$350$		0			0
$351$	−6.38460	+	0.563363i	−0.340785	+	0.0300701i
$352$		0			0
$353$		−	23.2255i		−	1.23617i	−0.786111	−	0.618085i	$-0.787909\pi$
$353$		−	23.2255i		−	1.23617i	0.786111	−	0.618085i	$-0.212091\pi$
$354$		0			0
$355$			26.5147i			1.40725i
$356$		0			0
$357$	11.0427	+	17.8929i	0.584439	+	0.946995i
$358$		0			0
$359$	−10.5501			−0.556813			−0.278406	−	0.960463i	$-0.589806\pi$
$359$	−10.5501			−0.556813			−0.278406	+	0.960463i	$0.589806\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$		0			0
$363$	−7.77452	−	12.5974i	−0.408057	−	0.661194i
$364$		0			0
$365$		−	25.7295i		−	1.34674i
$366$		0			0
$367$			16.7774i			0.875772i	0.899030	+	0.437886i	$0.144273\pi$
$367$			16.7774i			0.875772i	−0.899030	+	0.437886i	$0.855727\pi$
$368$		0			0
$369$	29.0173	+	14.5550i	1.51058	+	0.757705i
$370$		0			0
$371$	−34.7551			−1.80440
$372$		0			0
$373$	2.88732			0.149500			0.0747498	−	0.997202i	$-0.476184\pi$
$373$	2.88732			0.149500			0.0747498	+	0.997202i	$0.476184\pi$
$374$		0			0
$375$	−11.9682	+	7.38619i	−0.618035	+	0.381421i
$376$		0			0
$377$		−	8.88231i		−	0.457462i
$378$		0			0
$379$			20.0903i			1.03197i	0.856598	+	0.515985i	$0.172574\pi$
$379$			20.0903i			1.03197i	−0.856598	+	0.515985i	$0.827426\pi$
$380$		0			0
$381$	12.9582	−	7.99718i	0.663870	−	0.409708i
$382$		0			0
$383$	−2.45494			−0.125442			−0.0627209	−	0.998031i	$-0.519978\pi$
$383$	−2.45494			−0.125442			−0.0627209	+	0.998031i	$0.519978\pi$
$384$		0			0
$385$	14.7645			0.752471
$386$		0			0
$387$	−26.7266	−	13.4060i	−1.35859	−	0.681465i
$388$		0			0
$389$			9.41557i			0.477388i	0.971095	+	0.238694i	$0.0767193\pi$
$389$			9.41557i			0.477388i	−0.971095	+	0.238694i	$0.923281\pi$
$390$		0			0
$391$		−	3.53500i		−	0.178773i
$392$		0			0
$393$	7.61793	+	12.3437i	0.384274	+	0.622657i
$394$		0			0
$395$	−27.6779			−1.39263
$396$		0			0
$397$	−39.3046			−1.97264			−0.986320	−	0.164841i	$-0.947289\pi$
$397$	−39.3046			−1.97264			−0.986320	+	0.164841i	$0.947289\pi$
$398$		0			0
$399$	3.26165	+	5.28501i	0.163287	+	0.264581i
$400$		0			0
$401$		−	10.2498i		−	0.511851i	−0.966696	−	0.255926i	$-0.917620\pi$
$401$		−	10.2498i		−	0.511851i	0.966696	−	0.255926i	$-0.0823802\pi$
$402$		0			0
$403$			1.75891i			0.0876177i
$404$		0			0
$405$	18.9647	−	14.1479i	0.942363	−	0.703016i
$406$		0			0
$407$	−7.13241			−0.353541
$408$		0			0
$409$	−22.7232			−1.12359			−0.561795	−	0.827276i	$-0.689889\pi$
$409$	−22.7232			−1.12359			−0.561795	+	0.827276i	$0.689889\pi$
$410$		0			0
$411$	−25.9158	+	15.9940i	−1.27833	+	0.788925i
$412$		0			0
$413$			23.3414i			1.14855i
$414$		0			0
$415$		−	35.2780i		−	1.73173i
$416$		0			0
$417$	−6.68872	+	4.12795i	−0.327548	+	0.202147i
$418$		0			0
$419$	−36.8117			−1.79837			−0.899185	−	0.437569i	$-0.855840\pi$
$419$	−36.8117			−1.79837			−0.899185	+	0.437569i	$0.855840\pi$
$420$		0			0
$421$	11.3910			0.555164			0.277582	−	0.960702i	$-0.410467\pi$
$421$	11.3910			0.555164			0.277582	+	0.960702i	$0.410467\pi$
$422$		0			0
$423$	−0.184218	+	0.367262i	−0.00895699	+	0.0178569i
$424$		0			0
$425$			6.47122i			0.313900i
$426$		0			0
$427$			21.9157i			1.06057i
$428$		0			0
$429$	−1.75746	−	2.84770i	−0.0848510	−	0.137488i
$430$		0			0
$431$	−26.4348			−1.27332			−0.636659	−	0.771146i	$-0.719684\pi$
$431$	−26.4348			−1.27332			−0.636659	+	0.771146i	$0.719684\pi$
$432$		0			0
$433$	11.2338			0.539860			0.269930	−	0.962880i	$-0.412999\pi$
$433$	11.2338			0.539860			0.269930	+	0.962880i	$0.412999\pi$
$434$		0			0
$435$	17.2206	+	27.9034i	0.825664	+	1.33786i
$436$		0			0
$437$		−	1.04413i		−	0.0499474i
$438$		0			0
$439$			21.3692i			1.01990i	0.860205	+	0.509949i	$0.170336\pi$
$439$			21.3692i			1.01990i	−0.860205	+	0.509949i	$0.829664\pi$
$440$		0			0
$441$	7.87749	−	15.7048i	0.375118	−	0.747846i
$442$		0			0
$443$	15.7849			0.749963			0.374981	−	0.927032i	$-0.377649\pi$
$443$	15.7849			0.749963			0.374981	+	0.927032i	$0.377649\pi$
$444$		0			0
$445$	11.3472			0.537911
$446$		0			0
$447$	21.2243	−	13.0986i	1.00387	−	0.619541i
$448$		0			0
$449$			10.5162i			0.496292i	0.968723	+	0.248146i	$0.0798213\pi$
$449$			10.5162i			0.496292i	−0.968723	+	0.248146i	$0.920179\pi$
$450$		0			0
$451$			16.9490i			0.798097i
$452$		0			0
$453$	20.7561	−	12.8096i	0.975205	−	0.601849i
$454$		0			0
$455$	−11.6274			−0.545099
$456$		0			0
$457$	21.4547			1.00361			0.501804	−	0.864981i	$-0.332670\pi$
$457$	21.4547			1.00361			0.501804	+	0.864981i	$0.332670\pi$
$458$		0			0
$459$	1.54628	+	17.5240i	0.0721742	+	0.817951i
$460$		0			0
$461$			27.2511i			1.26921i	0.772836	+	0.634606i	$0.218837\pi$
$461$			27.2511i			1.26921i	−0.772836	+	0.634606i	$0.781163\pi$
$462$		0			0
$463$		−	27.4287i		−	1.27472i	−0.770567	−	0.637359i	$-0.780027\pi$
$463$		−	27.4287i		−	1.27472i	0.770567	−	0.637359i	$-0.219973\pi$
$464$		0			0
$465$	−3.41010	−	5.52554i	−0.158139	−	0.256241i
$466$		0			0
$467$	21.1682			0.979546			0.489773	−	0.871850i	$-0.337080\pi$
$467$	21.1682			0.979546			0.489773	+	0.871850i	$0.337080\pi$
$468$		0			0
$469$	−50.9075			−2.35069
$470$		0			0
$471$	18.5999	+	30.1383i	0.857039	+	1.38870i
$472$		0			0
$473$		−	15.6110i		−	0.717793i
$474$		0			0
$475$			1.91139i			0.0877007i
$476$		0			0
$477$	−25.9923	−	13.0377i	−1.19010	−	0.596955i
$478$		0			0
$479$	22.9110			1.04683			0.523415	−	0.852078i	$-0.324658\pi$
$479$	22.9110			1.04683			0.523415	+	0.852078i	$0.324658\pi$
$480$		0			0
$481$	5.61691			0.256109
$482$		0			0
$483$	−5.51823	+	3.40558i	−0.251088	+	0.154959i
$484$		0			0
$485$			17.5363i			0.796281i
$486$		0			0
$487$			26.0702i			1.18135i	0.806908	+	0.590677i	$0.201139\pi$
$487$			26.0702i			1.18135i	−0.806908	+	0.590677i	$0.798861\pi$
$488$		0			0
$489$	19.0377	−	11.7491i	0.860913	−	0.531314i
$490$		0			0
$491$	2.35468			0.106265			0.0531327	−	0.998587i	$-0.483079\pi$
$491$	2.35468			0.106265			0.0531327	+	0.998587i	$0.483079\pi$
$492$		0			0
$493$	−24.3796			−1.09800
$494$		0			0
$495$	11.0420	+	5.53863i	0.496299	+	0.248943i
$496$		0			0
$497$			36.1632i			1.62214i
$498$		0			0
$499$			8.01852i			0.358958i	0.983762	+	0.179479i	$0.0574412\pi$
$499$			8.01852i			0.358958i	−0.983762	+	0.179479i	$0.942559\pi$
$500$		0			0
$501$	−13.5273	−	21.9189i	−0.604356	−	0.979266i
$502$		0			0
$503$	−28.8205			−1.28504			−0.642522	−	0.766267i	$-0.722112\pi$
$503$	−28.8205			−1.28504			−0.642522	+	0.766267i	$0.722112\pi$
$504$		0			0
$505$	−43.5529			−1.93808
$506$		0			0
$507$	−10.4414	−	16.9188i	−0.463720	−	0.751389i
$508$		0			0
$509$			35.4676i			1.57207i	0.618179	+	0.786037i	$0.287871\pi$
$509$			35.4676i			1.57207i	−0.618179	+	0.786037i	$0.712129\pi$
$510$		0			0
$511$		−	35.0922i		−	1.55239i
$512$		0			0
$513$	0.456722	+	5.17604i	0.0201648	+	0.228528i
$514$		0			0
$515$	0.822909			0.0362617
$516$		0			0
$517$	−0.214517			−0.00943447
$518$		0			0
$519$	−19.6338	+	12.1170i	−0.861829	+	0.531879i
$520$		0			0
$521$			11.1183i			0.487103i	0.969888	+	0.243551i	$0.0783124\pi$
$521$			11.1183i			0.487103i	−0.969888	+	0.243551i	$0.921688\pi$
$522$		0			0
$523$			19.5917i			0.856684i	0.903617	+	0.428342i	$0.140902\pi$
$523$			19.5917i			0.856684i	−0.903617	+	0.428342i	$0.859098\pi$
$524$		0			0
$525$	10.1017	−	6.23430i	0.440876	−	0.272087i
$526$		0			0
$527$	4.82775			0.210300
$528$		0			0
$529$	−21.9098			−0.952600
$530$		0			0
$531$	−8.75605	+	17.4563i	−0.379980	+	0.757539i
$532$		0			0
$533$		−	13.3476i		−	0.578151i
$534$		0			0
$535$		−	38.7208i		−	1.67405i
$536$		0			0
$537$	−0.516198	−	0.836420i	−0.0222756	−	0.0360942i
$538$		0			0
$539$	9.17314			0.395115
$540$		0			0
$541$	−29.9449			−1.28743			−0.643716	−	0.765264i	$-0.722608\pi$
$541$	−29.9449			−1.28743			−0.643716	+	0.765264i	$0.722608\pi$
$542$		0			0
$543$	19.3380	+	31.3343i	0.829874	+	1.34469i
$544$		0			0
$545$			15.8167i			0.677513i
$546$		0			0
$547$			17.1892i			0.734955i	0.930032	+	0.367478i	$0.119779\pi$
$547$			17.1892i			0.734955i	−0.930032	+	0.367478i	$0.880221\pi$
$548$		0			0
$549$	−8.22122	+	16.3900i	−0.350873	+	0.699510i
$550$		0			0
$551$	−7.20096			−0.306771
$552$		0			0
$553$	−37.7496			−1.60528
$554$		0			0
$555$	−17.6452	+	10.8898i	−0.748999	+	0.462246i
$556$		0			0
$557$		−	39.7539i		−	1.68443i	−0.539144	−	0.842214i	$-0.681252\pi$
$557$		−	39.7539i		−	1.68443i	0.539144	−	0.842214i	$-0.318748\pi$
$558$		0			0
$559$			12.2939i			0.519978i
$560$		0			0
$561$	−7.81617	+	4.82376i	−0.329999	+	0.203659i
$562$		0			0
$563$	8.43451			0.355472			0.177736	−	0.984078i	$-0.443123\pi$
$563$	8.43451			0.355472			0.177736	+	0.984078i	$0.443123\pi$
$564$		0			0
$565$	15.0765			0.634272
$566$		0			0
$567$	25.8658	−	19.2962i	1.08626	−	0.810365i
$568$		0			0
$569$		−	28.4016i		−	1.19066i	−0.803482	−	0.595329i	$-0.797022\pi$
$569$		−	28.4016i		−	1.19066i	0.803482	−	0.595329i	$-0.202978\pi$
$570$		0			0
$571$		−	16.3746i		−	0.685254i	−0.939472	−	0.342627i	$-0.888683\pi$
$571$		−	16.3746i		−	0.685254i	0.939472	−	0.342627i	$-0.111317\pi$
$572$		0			0
$573$	7.39980	+	11.9902i	0.309131	+	0.500900i
$574$		0			0
$575$	−1.99574			−0.0832280
$576$		0			0
$577$	36.4598			1.51784			0.758920	−	0.651184i	$-0.225727\pi$
$577$	36.4598			1.51784			0.758920	+	0.651184i	$0.225727\pi$
$578$		0			0
$579$	13.7057	+	22.2081i	0.569591	+	0.922936i
$580$		0			0
$581$		−	48.1154i		−	1.99616i
$582$		0			0
$583$		−	15.1821i		−	0.628777i
$584$		0			0
$585$	−8.69574	−	4.36177i	−0.359525	−	0.180337i
$586$		0			0
$587$	−18.3627			−0.757911			−0.378955	−	0.925415i	$-0.623717\pi$
$587$	−18.3627			−0.757911			−0.378955	+	0.925415i	$0.623717\pi$
$588$		0			0
$589$	1.42596			0.0587558
$590$		0			0
$591$	4.10085	−	2.53084i	0.168686	−	0.104105i
$592$		0			0
$593$			37.9486i			1.55836i	0.626798	+	0.779182i	$0.284365\pi$
$593$			37.9486i			1.55836i	−0.626798	+	0.779182i	$0.715635\pi$
$594$		0			0
$595$			31.9140i			1.30835i
$596$		0			0
$597$	18.3464	−	11.3225i	0.750869	−	0.463400i
$598$		0			0
$599$	6.14228			0.250967			0.125483	−	0.992096i	$-0.459952\pi$
$599$	6.14228			0.250967			0.125483	+	0.992096i	$0.459952\pi$
$600$		0			0
$601$	27.2065			1.10978			0.554889	−	0.831924i	$-0.312761\pi$
$601$	27.2065			1.10978			0.554889	+	0.831924i	$0.312761\pi$
$602$		0			0
$603$	−38.0722	−	19.0969i	−1.55042	−	0.777688i
$604$		0			0
$605$		−	22.4689i		−	0.913490i
$606$		0			0
$607$		−	28.9061i		−	1.17326i	−0.809855	−	0.586630i	$-0.800454\pi$
$607$		−	28.9061i		−	1.17326i	0.809855	−	0.586630i	$-0.199546\pi$
$608$		0			0
$609$	23.4870	+	38.0571i	0.951742	+	1.54215i
$610$		0			0
$611$	0.168937			0.00683444
$612$		0			0
$613$	24.3751			0.984502			0.492251	−	0.870453i	$-0.336174\pi$
$613$	24.3751			0.984502			0.492251	+	0.870453i	$0.336174\pi$
$614$		0			0
$615$	25.8778	+	41.9310i	1.04349	+	1.69082i
$616$		0			0
$617$		−	10.6419i		−	0.428429i	−0.976787	−	0.214214i	$-0.931281\pi$
$617$		−	10.6419i		−	0.428429i	0.976787	−	0.214214i	$-0.0687191\pi$
$618$		0			0
$619$			2.83221i			0.113836i	0.998379	+	0.0569181i	$0.0181274\pi$
$619$			2.83221i			0.113836i	−0.998379	+	0.0569181i	$0.981873\pi$
$620$		0			0
$621$	−5.40445	+	0.476877i	−0.216873	+	0.0191364i
$622$		0			0
$623$	15.4764			0.620049
$624$		0			0
$625$	−30.9035			−1.23614
$626$		0			0
$627$	−2.30865	+	1.42479i	−0.0921986	+	0.0569005i
$628$		0			0
$629$		−	15.4169i		−	0.614713i
$630$		0			0
$631$		−	25.1883i		−	1.00273i	−0.865236	−	0.501365i	$-0.832831\pi$
$631$		−	25.1883i		−	1.00273i	0.865236	−	0.501365i	$-0.167169\pi$
$632$		0			0
$633$	23.5369	−	14.5258i	0.935508	−	0.577350i
$634$		0			0
$635$	23.1124			0.917187
$636$		0			0
$637$	−7.22402			−0.286226
$638$		0			0
$639$	−13.5659	+	27.0453i	−0.536659	+	1.06990i
$640$		0			0
$641$			16.6362i			0.657091i	0.944488	+	0.328545i	$0.106558\pi$
$641$			16.6362i			0.657091i	−0.944488	+	0.328545i	$0.893442\pi$
$642$		0			0
$643$		−	40.2224i		−	1.58622i	−0.609081	−	0.793108i	$-0.708462\pi$
$643$		−	40.2224i		−	1.58622i	0.609081	−	0.793108i	$-0.291538\pi$
$644$		0			0
$645$	−23.8349	−	38.6208i	−0.938497	−	1.52069i
$646$		0			0
$647$	21.8899			0.860580			0.430290	−	0.902691i	$-0.358411\pi$
$647$	21.8899			0.860580			0.430290	+	0.902691i	$0.358411\pi$
$648$		0			0
$649$	−10.1962			−0.400236
$650$		0			0
$651$	−4.65100	−	7.53624i	−0.182287	−	0.295368i
$652$		0			0
$653$		−	2.25896i		−	0.0884001i	−0.999023	−	0.0442000i	$-0.985926\pi$
$653$		−	2.25896i		−	0.0884001i	0.999023	−	0.0442000i	$-0.0140739\pi$
$654$		0			0
$655$			22.0163i			0.860248i
$656$		0			0
$657$	13.1642	−	26.2444i	0.513583	−	1.02389i
$658$		0			0
$659$	−17.0566			−0.664430			−0.332215	−	0.943204i	$-0.607796\pi$
$659$	−17.0566			−0.664430			−0.332215	+	0.943204i	$0.607796\pi$
$660$		0			0
$661$	28.3209			1.10156			0.550778	−	0.834652i	$-0.314331\pi$
$661$	28.3209			1.10156			0.550778	+	0.834652i	$0.314331\pi$
$662$		0			0
$663$	6.15538	−	3.79880i	0.239055	−	0.147533i
$664$		0			0
$665$			9.42639i			0.365540i
$666$		0			0
$667$		−	7.51872i		−	0.291126i
$668$		0			0
$669$	−6.23015	+	3.84495i	−0.240872	+	0.148654i
$670$		0			0
$671$	−9.57341			−0.369577
$672$		0			0
$673$	33.0681			1.27468			0.637341	−	0.770582i	$-0.280034\pi$
$673$	33.0681			1.27468			0.637341	+	0.770582i	$0.280034\pi$
$674$		0			0
$675$	9.89345	−	0.872976i	0.380799	−	0.0336009i
$676$		0			0
$677$			23.9824i			0.921719i	0.887473	+	0.460859i	$0.152459\pi$
$677$			23.9824i			0.921719i	−0.887473	+	0.460859i	$0.847541\pi$
$678$		0			0
$679$			23.9176i			0.917872i
$680$		0			0
$681$	13.7724	+	22.3160i	0.527758	+	0.855151i
$682$		0			0
$683$	0.0577275			0.00220888			0.00110444	−	0.999999i	$-0.499648\pi$
$683$	0.0577275			0.00220888			0.00110444	+	0.999999i	$0.499648\pi$
$684$		0			0
$685$	−46.2237			−1.76612
$686$		0			0
$687$	−10.3165	−	16.7164i	−0.393600	−	0.637769i
$688$		0			0
$689$			11.9562i			0.455493i
$690$		0			0
$691$		−	27.4729i		−	1.04512i	−0.852603	−	0.522558i	$-0.824978\pi$
$691$		−	27.4729i		−	1.04512i	0.852603	−	0.522558i	$-0.175022\pi$
$692$		0			0
$693$	15.0600	+	7.55408i	0.572083	+	0.286956i
$694$		0			0
$695$	−11.9301			−0.452533
$696$		0			0
$697$	−36.6357			−1.38768
$698$		0			0
$699$	−6.72356	+	4.14945i	−0.254308	+	0.156947i
$700$		0			0
$701$		−	50.3391i		−	1.90128i	−0.310293	−	0.950641i	$-0.600427\pi$
$701$		−	50.3391i		−	1.90128i	0.310293	−	0.950641i	$-0.399573\pi$
$702$		0			0
$703$		−	4.55367i		−	0.171745i
$704$		0			0
$705$	−0.530706	+	0.327526i	−0.0199875	+	0.0123353i
$706$		0			0
$707$	−59.4014			−2.23402
$708$		0			0
$709$	21.9250			0.823410			0.411705	−	0.911317i	$-0.364933\pi$
$709$	21.9250			0.823410			0.411705	+	0.911317i	$0.364933\pi$
$710$		0			0
$711$	−28.2318	−	14.1610i	−1.05878	−	0.531080i
$712$		0			0
$713$			1.48889i			0.0557593i
$714$		0			0
$715$		−	5.07918i		−	0.189950i
$716$		0			0
$717$	−14.2496	−	23.0894i	−0.532163	−	0.862289i
$718$		0			0
$719$	5.74026			0.214075			0.107038	−	0.994255i	$-0.465863\pi$
$719$	5.74026			0.214075			0.107038	+	0.994255i	$0.465863\pi$
$720$		0			0
$721$	1.12236			0.0417988
$722$		0			0
$723$	3.37003	+	5.46063i	0.125333	+	0.203083i
$724$		0			0
$725$			13.7639i			0.511177i
$726$		0			0
$727$		−	27.3443i		−	1.01414i	−0.861904	−	0.507071i	$-0.830728\pi$
$727$		−	27.3443i		−	1.01414i	0.861904	−	0.507071i	$-0.169272\pi$
$728$		0			0
$729$	26.5828	−	4.72803i	0.984548	−	0.175112i
$730$		0			0
$731$	33.7435			1.24805
$732$		0			0
$733$	−14.2323			−0.525684			−0.262842	−	0.964839i	$-0.584660\pi$
$733$	−14.2323			−0.525684			−0.262842	+	0.964839i	$0.584660\pi$
$734$		0			0
$735$	22.6939	−	14.0056i	0.837077	−	0.516603i
$736$		0			0
$737$		−	22.2379i		−	0.819144i
$738$		0			0
$739$		−	33.5861i		−	1.23548i	−0.786381	−	0.617742i	$-0.788048\pi$
$739$		−	33.5861i		−	1.23548i	0.786381	−	0.617742i	$-0.211952\pi$
$740$		0			0
$741$	1.81811	−	1.12205i	0.0667898	−	0.0412194i
$742$		0			0
$743$	3.96649			0.145517			0.0727583	−	0.997350i	$-0.476820\pi$
$743$	3.96649			0.145517			0.0727583	+	0.997350i	$0.476820\pi$
$744$		0			0
$745$	37.8558			1.38693
$746$		0			0
$747$	18.0495	−	35.9840i	0.660397	−	1.31659i
$748$		0			0
$749$		−	52.8109i		−	1.92967i
$750$		0			0
$751$		−	29.8793i		−	1.09031i	−0.838335	−	0.545156i	$-0.816470\pi$
$751$		−	29.8793i		−	1.09031i	0.838335	−	0.545156i	$-0.183530\pi$
$752$		0			0
$753$	26.1823	+	42.4244i	0.954134	+	1.54603i
$754$		0			0
$755$	37.0207			1.34732
$756$		0			0
$757$	−51.4613			−1.87039			−0.935196	−	0.354130i	$-0.884777\pi$
$757$	−51.4613			−1.87039			−0.935196	+	0.354130i	$0.884777\pi$
$758$		0			0
$759$	−1.48766	−	2.41053i	−0.0539986	−	0.0874965i
$760$		0			0
$761$		−	4.79445i		−	0.173799i	−0.996217	−	0.0868994i	$-0.972304\pi$
$761$		−	4.79445i		−	0.173799i	0.996217	−	0.0868994i	$-0.0276959\pi$
$762$		0			0
$763$			21.5723i			0.780968i
$764$		0			0
$765$	−11.9719	+	23.8675i	−0.432845	+	0.862931i
$766$		0			0
$767$	8.02970			0.289936
$768$		0			0
$769$	−9.98056			−0.359908			−0.179954	−	0.983675i	$-0.557595\pi$
$769$	−9.98056			−0.359908			−0.179954	+	0.983675i	$0.557595\pi$
$770$		0			0
$771$	33.8505	−	20.8909i	1.21910	−	0.752367i
$772$		0			0
$773$		−	4.11463i		−	0.147993i	−0.997259	−	0.0739964i	$-0.976425\pi$
$773$		−	4.11463i		−	0.147993i	0.997259	−	0.0739964i	$-0.0235753\pi$
$774$		0			0
$775$		−	2.72558i		−	0.0979057i
$776$		0			0
$777$	−24.0662	+	14.8525i	−0.863370	+	0.532830i
$778$		0			0
$779$	−10.8210			−0.387704
$780$		0			0
$781$	−15.7972			−0.565267
$782$		0			0
$783$	3.28884	+	37.2725i	0.117534	+	1.33201i
$784$		0			0
$785$			53.7550i			1.91860i
$786$		0			0
$787$		−	27.5187i		−	0.980936i	−0.871459	−	0.490468i	$-0.836826\pi$
$787$		−	27.5187i		−	0.980936i	0.871459	−	0.490468i	$-0.163174\pi$
$788$		0			0
$789$	12.5109	+	20.2720i	0.445399	+	0.721701i
$790$		0			0
$791$	20.5627			0.731124
$792$		0			0
$793$	7.53924			0.267726
$794$		0			0
$795$	−23.1800	−	37.5597i	−0.822111	−	1.33211i
$796$		0			0
$797$			40.2348i			1.42519i	0.701575	+	0.712596i	$0.252481\pi$
$797$			40.2348i			1.42519i	−0.701575	+	0.712596i	$0.747519\pi$
$798$		0			0
$799$		−	0.463686i		−	0.0164040i
$800$		0			0
$801$	11.5743	+	5.80567i	0.408959	+	0.205133i
$802$		0			0
$803$	15.3293			0.540961
$804$		0			0
$805$	−9.84235			−0.346897
$806$		0			0
$807$	4.32432	−	2.66876i	0.152223	−	0.0939448i
$808$		0			0
$809$		−	30.6395i		−	1.07723i	−0.842553	−	0.538614i	$-0.818948\pi$
$809$		−	30.6395i		−	1.07723i	0.842553	−	0.538614i	$-0.181052\pi$
$810$		0			0
$811$			48.3574i			1.69806i	0.528345	+	0.849030i	$0.322813\pi$
$811$			48.3574i			1.69806i	−0.528345	+	0.849030i	$0.677187\pi$
$812$		0			0
$813$	31.4042	−	19.3811i	1.10139	−	0.679726i
$814$		0			0
$815$	33.9557			1.18942
$816$		0			0
$817$	9.96678			0.348693
$818$		0			0
$819$	−11.8601	−	5.94898i	−0.414424	−	0.207874i
$820$		0			0
$821$			17.2475i			0.601944i	0.953633	+	0.300972i	$0.0973110\pi$
$821$			17.2475i			0.601944i	−0.953633	+	0.300972i	$0.902689\pi$
$822$		0			0
$823$			28.4658i			0.992255i	0.868250	+	0.496128i	$0.165245\pi$
$823$			28.4658i			0.992255i	−0.868250	+	0.496128i	$0.834755\pi$
$824$		0			0
$825$	2.72333	+	4.41274i	0.0948141	+	0.153632i
$826$		0			0
$827$	−7.03686			−0.244695			−0.122348	−	0.992487i	$-0.539042\pi$
$827$	−7.03686			−0.244695			−0.122348	+	0.992487i	$0.539042\pi$
$828$		0			0
$829$	26.1522			0.908303			0.454151	−	0.890924i	$-0.349942\pi$
$829$	26.1522			0.908303			0.454151	+	0.890924i	$0.349942\pi$
$830$		0			0
$831$	−5.35844	−	8.68254i	−0.185882	−	0.301194i
$832$		0			0
$833$			19.8280i			0.687000i
$834$		0			0
$835$		−	39.0948i		−	1.35293i
$836$		0			0
$837$	−0.651270	−	7.38085i	−0.0225112	−	0.255120i
$838$		0			0
$839$	−0.634411			−0.0219023			−0.0109511	−	0.999940i	$-0.503486\pi$
$839$	−0.634411			−0.0219023			−0.0109511	+	0.999940i	$0.503486\pi$
$840$		0			0
$841$	−22.8538			−0.788062
$842$		0			0
$843$	45.4672	−	28.0601i	1.56597	−	0.966442i
$844$		0			0
$845$		−	30.1764i		−	1.03810i
$846$		0			0
$847$		−	30.6451i		−	1.05298i
$848$		0			0
$849$	38.2367	−	23.5978i	1.31228	−	0.809876i
$850$		0			0
$851$	4.75461			0.162986
$852$		0			0
$853$	16.9254			0.579516			0.289758	−	0.957100i	$-0.406425\pi$
$853$	16.9254			0.579516			0.289758	+	0.957100i	$0.406425\pi$
$854$		0			0
$855$	−3.53612	+	7.04971i	−0.120933	+	0.241095i
$856$		0			0
$857$		−	33.9804i		−	1.16075i	−0.814350	−	0.580374i	$-0.802907\pi$
$857$		−	33.9804i		−	1.16075i	0.814350	−	0.580374i	$-0.197093\pi$
$858$		0			0
$859$			16.3639i			0.558330i	0.960243	+	0.279165i	$0.0900576\pi$
$859$			16.3639i			0.558330i	−0.960243	+	0.279165i	$0.909942\pi$
$860$		0			0
$861$	35.2945	+	57.1893i	1.20283	+	1.94901i
$862$		0			0
$863$	12.2331			0.416419			0.208210	−	0.978084i	$-0.433236\pi$
$863$	12.2331			0.416419			0.208210	+	0.978084i	$0.433236\pi$
$864$		0			0
$865$	−35.0190			−1.19068
$866$		0			0
$867$	5.03737	+	8.16230i	0.171078	+	0.277206i
$868$		0			0
$869$		−	16.4902i		−	0.559391i
$870$		0			0
$871$			17.5128i			0.593398i
$872$		0			0
$873$	−8.97220	+	17.8872i	−0.303663	+	0.605390i
$874$		0			0
$875$	−29.1144			−0.984247
$876$		0			0
$877$	8.59003			0.290065			0.145032	−	0.989427i	$-0.453671\pi$
$877$	8.59003			0.290065			0.145032	+	0.989427i	$0.453671\pi$
$878$		0			0
$879$	−48.8034	+	30.1191i	−1.64610	+	1.01589i
$880$		0			0
$881$		−	20.1146i		−	0.677677i	−0.940845	−	0.338838i	$-0.889966\pi$
$881$		−	20.1146i		−	0.677677i	0.940845	−	0.338838i	$-0.110034\pi$
$882$		0			0
$883$		−	32.1457i		−	1.08179i	−0.841091	−	0.540894i	$-0.818086\pi$
$883$		−	32.1457i		−	1.08179i	0.841091	−	0.540894i	$-0.181914\pi$
$884$		0			0
$885$	−25.2249	+	15.5676i	−0.847926	+	0.523299i
$886$		0			0
$887$	10.2485			0.344112			0.172056	−	0.985087i	$-0.444959\pi$
$887$	10.2485			0.344112			0.172056	+	0.985087i	$0.444959\pi$
$888$		0			0
$889$	31.5228			1.05724
$890$		0			0
$891$	8.42917	+	11.2989i	0.282388	+	0.378529i
$892$		0			0
$893$		−	0.136958i		−	0.00458313i
$894$		0			0
$895$		−	1.49185i		−	0.0498669i
$896$		0			0
$897$	1.17156	+	1.89833i	0.0391172	+	0.0633835i
$898$		0			0
$899$	10.2683			0.342467
$900$		0			0
$901$	32.8165			1.09328
$902$		0			0
$903$	−32.5082	−	52.6745i	−1.08180	−	1.75290i
$904$		0			0
$905$			55.8882i			1.85779i
$906$		0			0
$907$			32.9063i			1.09264i	0.837578	+	0.546318i	$0.183971\pi$
$907$			32.9063i			1.09264i	−0.837578	+	0.546318i	$0.816029\pi$
$908$		0			0
$909$	−44.4245	−	22.2833i	−1.47347	−	0.739089i
$910$		0			0
$911$	49.4178			1.63728			0.818642	−	0.574304i	$-0.194727\pi$
$911$	49.4178			1.63728			0.818642	+	0.574304i	$0.194727\pi$
$912$		0			0
$913$	21.0182			0.695602
$914$		0			0
$915$	−23.6842	+	14.6167i	−0.782974	+	0.483214i
$916$		0			0
$917$			30.0279i			0.991607i
$918$		0			0
$919$		−	6.63018i		−	0.218709i	−0.994003	−	0.109355i	$-0.965122\pi$
$919$		−	6.63018i		−	0.218709i	0.994003	−	0.109355i	$-0.0348784\pi$
$920$		0			0
$921$	−1.26375	+	0.779923i	−0.0416418	+	0.0256993i
$922$		0			0
$923$	12.4406			0.409486
$924$		0			0
$925$	−8.70385			−0.286181
$926$		0			0
$927$	0.839377	+	0.421030i	0.0275688	+	0.0138285i
$928$		0			0
$929$			39.5553i			1.29777i	0.760888	+	0.648884i	$0.224764\pi$
$929$			39.5553i			1.29777i	−0.760888	+	0.648884i	$0.775236\pi$
$930$		0			0
$931$			5.85657i			0.191941i
$932$		0			0
$933$	17.1329	+	27.7613i	0.560907	+	0.908864i
$934$		0			0
$935$	−13.9410			−0.455919
$936$		0			0
$937$	24.2158			0.791096			0.395548	−	0.918445i	$-0.370555\pi$
$937$	24.2158			0.791096			0.395548	+	0.918445i	$0.370555\pi$
$938$		0			0
$939$	25.8697	+	41.9179i	0.844225	+	1.36794i
$940$		0			0
$941$			56.6132i			1.84554i	0.385354	+	0.922769i	$0.374079\pi$
$941$			56.6132i			1.84554i	−0.385354	+	0.922769i	$0.625921\pi$
$942$		0			0
$943$		−	11.2985i		−	0.367931i
$944$		0			0
$945$	48.7914	−	4.30524i	1.58718	−	0.140050i
$946$		0			0
$947$	5.31711			0.172783			0.0863914	−	0.996261i	$-0.472466\pi$
$947$	5.31711			0.172783			0.0863914	+	0.996261i	$0.472466\pi$
$948$		0			0
$949$	−12.0721			−0.391878
$950$		0			0
$951$	19.0971	−	11.7858i	0.619267	−	0.382181i
$952$		0			0
$953$		−	20.0357i		−	0.649021i	−0.945882	−	0.324511i	$-0.894800\pi$
$953$		−	20.0357i		−	0.649021i	0.945882	−	0.324511i	$-0.105200\pi$
$954$		0			0
$955$			21.3859i			0.692032i
$956$		0			0
$957$	−16.6245	+	10.2598i	−0.537394	+	0.331653i
$958$		0			0
$959$	−63.0441			−2.03580
$960$		0			0
$961$	28.9666			0.934407
$962$		0			0
$963$	19.8110	−	39.4957i	0.638400	−	1.27273i
$964$		0			0
$965$			39.6105i			1.27511i
$966$		0			0
$967$		−	14.4593i		−	0.464979i	−0.972599	−	0.232490i	$-0.925313\pi$
$967$		−	14.4593i		−	0.464979i	0.972599	−	0.232490i	$-0.0746872\pi$
$968$		0			0
$969$	−3.07972	−	4.99022i	−0.0989348	−	0.160309i
$970$		0			0
$971$	2.01730			0.0647382			0.0323691	−	0.999476i	$-0.489695\pi$
$971$	2.01730			0.0647382			0.0323691	+	0.999476i	$0.489695\pi$
$972$		0			0
$973$	−16.2713			−0.521634
$974$		0			0
$975$	−2.14467	−	3.47511i	−0.0686844	−	0.111293i
$976$		0			0
$977$		−	3.17018i		−	0.101423i	−0.998713	−	0.0507116i	$-0.983851\pi$
$977$		−	3.17018i		−	0.101423i	0.998713	−	0.0507116i	$-0.0161489\pi$
$978$		0			0
$979$			6.76056i			0.216068i
$980$		0			0
$981$	−8.09241	+	16.1332i	−0.258371	+	0.515095i
$982$		0			0
$983$	13.9228			0.444067			0.222034	−	0.975039i	$-0.428731\pi$
$983$	13.9228			0.444067			0.222034	+	0.975039i	$0.428731\pi$
$984$		0			0
$985$	7.31431			0.233053
$986$		0			0
$987$	−0.723825	+	0.446710i	−0.0230396	+	0.0142189i
$988$		0			0
$989$			10.4066i			0.330910i
$990$		0			0
$991$		−	9.88678i		−	0.314064i	−0.987594	−	0.157032i	$-0.949807\pi$
$991$		−	9.88678i		−	0.314064i	0.987594	−	0.157032i	$-0.0501926\pi$
$992$		0			0
$993$	9.52013	−	5.87536i	0.302112	−	0.186449i
$994$		0			0
$995$	32.7228			1.03738
$996$		0			0
$997$	55.2982			1.75131			0.875657	−	0.482934i	$-0.160429\pi$
$997$	55.2982			1.75131			0.875657	+	0.482934i	$0.160429\pi$
$998$		0			0
$999$	−23.5700	+	2.07976i	−0.745721	+	0.0658008i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.d.b.191.18	yes	24
3.2	odd	2	inner	912.2.d.b.191.8	yes	24
4.3	odd	2	inner	912.2.d.b.191.7	✓	24
12.11	even	2	inner	912.2.d.b.191.17	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.2.d.b.191.7	✓	24	4.3	odd	2	inner
912.2.d.b.191.8	yes	24	3.2	odd	2	inner
912.2.d.b.191.17	yes	24	12.11	even	2	inner
912.2.d.b.191.18	yes	24	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	912.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.909651 + 1.47395i) q^{3} +2.62895i q^{5} +3.58561i q^{7} +(-1.34507 + 2.68156i) q^{9} +O(q^{10})\)	\(q+(0.909651 + 1.47395i) q^{3} +2.62895i q^{5} +3.58561i q^{7} +(-1.34507 + 2.68156i) q^{9} -1.56630 q^{11} +1.23349 q^{13} +(-3.87495 + 2.39143i) q^{15} -3.38560i q^{17} -1.00000i q^{19} +(-5.28501 + 3.26165i) q^{21} +1.04413 q^{23} -1.91139 q^{25} +(-5.17604 + 0.456722i) q^{27} -7.20096i q^{29} +1.42596i q^{31} +(-1.42479 - 2.30865i) q^{33} -9.42639 q^{35} +4.55367 q^{37} +(1.12205 + 1.81811i) q^{39} -10.8210i q^{41} +9.96678i q^{43} +(-7.04971 - 3.53612i) q^{45} +0.136958 q^{47} -5.85657 q^{49} +(4.99022 - 3.07972i) q^{51} +9.69295i q^{53} -4.11773i q^{55} +(1.47395 - 0.909651i) q^{57} +6.50974 q^{59} +6.11212 q^{61} +(-9.61503 - 4.82289i) q^{63} +3.24279i q^{65} +14.1977i q^{67} +(0.949792 + 1.53899i) q^{69} +10.0857 q^{71} -9.78698 q^{73} +(-1.73870 - 2.81730i) q^{75} -5.61613i q^{77} +10.5281i q^{79} +(-5.38158 - 7.21378i) q^{81} -13.4190 q^{83} +8.90059 q^{85} +(10.6139 - 6.55036i) q^{87} -4.31626i q^{89} +4.42281i q^{91} +(-2.10180 + 1.29713i) q^{93} +2.62895 q^{95} +6.67044 q^{97} +(2.10678 - 4.20013i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{9}+O(q^{10}) \) 24 * q + 4 * q^9	\( 24 q + 4 q^{9} - 12 q^{21} - 64 q^{25} + 12 q^{33} + 64 q^{37} - 16 q^{45} - 8 q^{49} - 24 q^{61} - 8 q^{69} - 16 q^{73} - 4 q^{81} - 8 q^{85} + 32 q^{93} - 8 q^{97}+O(q^{100}) \) 24 * q + 4 * q^9 - 12 * q^21 - 64 * q^25 + 12 * q^33 + 64 * q^37 - 16 * q^45 - 8 * q^49 - 24 * q^61 - 8 * q^69 - 16 * q^73 - 4 * q^81 - 8 * q^85 + 32 * q^93 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more