LMFDB - Embedded newform 912.2.d.b.191.6

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			191.6
Character	$\chi$	$=$	912.191
Dual form			912.2.d.b.191.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.50956 + 0.849252i) q^{3} -0.771899i q^{5} +1.48496i q^{7} +(1.55754 - 2.56399i) q^{9} +O(q^{10})$	\(q+(-1.50956 + 0.849252i) q^{3} -0.771899i q^{5} +1.48496i q^{7} +(1.55754 - 2.56399i) q^{9} +4.34856 q^{11} -5.42004 q^{13} +(0.655537 + 1.16523i) q^{15} -7.36768i q^{17} +1.00000i q^{19} +(-1.26110 - 2.24163i) q^{21} +1.65261 q^{23} +4.40417 q^{25} +(-0.173726 + 5.19325i) q^{27} +5.57672i q^{29} +1.97544i q^{31} +(-6.56441 + 3.69302i) q^{33} +1.14624 q^{35} +10.2727 q^{37} +(8.18188 - 4.60298i) q^{39} -3.92937i q^{41} -8.99665i q^{43} +(-1.97914 - 1.20227i) q^{45} +3.62677 q^{47} +4.79490 q^{49} +(6.25702 + 11.1220i) q^{51} -2.72185i q^{53} -3.35665i q^{55} +(-0.849252 - 1.50956i) q^{57} +1.94462 q^{59} +12.2553 q^{61} +(3.80742 + 2.31288i) q^{63} +4.18372i q^{65} +10.8690i q^{67} +(-2.49471 + 1.40348i) q^{69} +16.0982 q^{71} +4.95000 q^{73} +(-6.64836 + 3.74025i) q^{75} +6.45743i q^{77} +3.32493i q^{79} +(-4.14813 - 7.98706i) q^{81} -17.0770 q^{83} -5.68711 q^{85} +(-4.73604 - 8.41839i) q^{87} -4.94478i q^{89} -8.04853i q^{91} +(-1.67765 - 2.98205i) q^{93} +0.771899 q^{95} +5.51162 q^{97} +(6.77307 - 11.1497i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.50956	+	0.849252i	−0.871545	+	0.490316i
$3$	−1.50956	+	0.849252i	−0.871545	+	0.490316i
$4$		0			0
$5$		−	0.771899i		−	0.345204i	−0.984992	−	0.172602i	$-0.944783\pi$
$5$		−	0.771899i		−	0.345204i	0.984992	−	0.172602i	$-0.0552174\pi$
$6$		0			0
$7$			1.48496i			0.561261i	0.959816	+	0.280631i	$0.0905436\pi$
$7$			1.48496i			0.561261i	−0.959816	+	0.280631i	$0.909456\pi$
$8$		0			0
$9$	1.55754	−	2.56399i	0.519181	−	0.854665i
$10$		0			0
$11$	4.34856			1.31114			0.655570	−	0.755134i	$-0.272428\pi$
$11$	4.34856			1.31114			0.655570	+	0.755134i	$0.272428\pi$
$12$		0			0
$13$	−5.42004			−1.50325			−0.751625	−	0.659591i	$-0.770729\pi$
$13$	−5.42004			−1.50325			−0.751625	+	0.659591i	$0.770729\pi$
$14$		0			0
$15$	0.655537	+	1.16523i	0.169259	+	0.300861i
$16$		0			0
$17$		−	7.36768i		−	1.78693i	−0.449138	−	0.893463i	$-0.648269\pi$
$17$		−	7.36768i		−	1.78693i	0.449138	−	0.893463i	$-0.351731\pi$
$18$		0			0
$19$			1.00000i			0.229416i
$19$			1.00000i			0.229416i
$20$		0			0
$21$	−1.26110	−	2.24163i	−0.275195	−	0.489164i
$22$		0			0
$23$	1.65261			0.344592			0.172296	−	0.985045i	$-0.444881\pi$
$23$	1.65261			0.344592			0.172296	+	0.985045i	$0.444881\pi$
$24$		0			0
$25$	4.40417			0.880834
$26$		0			0
$27$	−0.173726	+	5.19325i	−0.0334336	+	0.999441i
$28$		0			0
$29$			5.57672i			1.03557i	0.855510	+	0.517786i	$0.173243\pi$
$29$			5.57672i			1.03557i	−0.855510	+	0.517786i	$0.826757\pi$
$30$		0			0
$31$			1.97544i			0.354800i	0.984139	+	0.177400i	$0.0567686\pi$
$31$			1.97544i			0.354800i	−0.984139	+	0.177400i	$0.943231\pi$
$32$		0			0
$33$	−6.56441	+	3.69302i	−1.14272	+	0.642873i
$34$		0			0
$35$	1.14624			0.193749
$36$		0			0
$37$	10.2727			1.68882			0.844411	−	0.535696i	$-0.179951\pi$
$37$	10.2727			1.68882			0.844411	+	0.535696i	$0.179951\pi$
$38$		0			0
$39$	8.18188	−	4.60298i	1.31015	−	0.737067i
$40$		0			0
$41$		−	3.92937i		−	0.613665i	−0.951764	−	0.306832i	$-0.900731\pi$
$41$		−	3.92937i		−	0.613665i	0.951764	−	0.306832i	$-0.0992691\pi$
$42$		0			0
$43$		−	8.99665i		−	1.37198i	−0.727613	−	0.685988i	$-0.759370\pi$
$43$		−	8.99665i		−	1.37198i	0.727613	−	0.685988i	$-0.240630\pi$
$44$		0			0
$45$	−1.97914	−	1.20227i	−0.295033	−	0.179223i
$46$		0			0
$47$	3.62677			0.529019			0.264509	−	0.964383i	$-0.414790\pi$
$47$	3.62677			0.529019			0.264509	+	0.964383i	$0.414790\pi$
$48$		0			0
$49$	4.79490			0.684986
$50$		0			0
$51$	6.25702	+	11.1220i	0.876158	+	1.55739i
$52$		0			0
$53$		−	2.72185i		−	0.373875i	−0.982372	−	0.186937i	$-0.940144\pi$
$53$		−	2.72185i		−	0.373875i	0.982372	−	0.186937i	$-0.0598562\pi$
$54$		0			0
$55$		−	3.35665i		−	0.452611i
$56$		0			0
$57$	−0.849252	−	1.50956i	−0.112486	−	0.199946i
$58$		0			0
$59$	1.94462			0.253168			0.126584	−	0.991956i	$-0.459599\pi$
$59$	1.94462			0.253168			0.126584	+	0.991956i	$0.459599\pi$
$60$		0			0
$61$	12.2553			1.56913			0.784566	−	0.620046i	$-0.212886\pi$
$61$	12.2553			1.56913			0.784566	+	0.620046i	$0.212886\pi$
$62$		0			0
$63$	3.80742	+	2.31288i	0.479690	+	0.291396i
$64$		0			0
$65$			4.18372i			0.518927i
$66$		0			0
$67$			10.8690i			1.32786i	0.747793	+	0.663932i	$0.231114\pi$
$67$			10.8690i			1.32786i	−0.747793	+	0.663932i	$0.768886\pi$
$68$		0			0
$69$	−2.49471	+	1.40348i	−0.300328	+	0.168959i
$70$		0			0
$71$	16.0982			1.91051			0.955254	−	0.295787i	$-0.0955819\pi$
$71$	16.0982			1.91051			0.955254	+	0.295787i	$0.0955819\pi$
$72$		0			0
$73$	4.95000			0.579354			0.289677	−	0.957125i	$-0.406452\pi$
$73$	4.95000			0.579354			0.289677	+	0.957125i	$0.406452\pi$
$74$		0			0
$75$	−6.64836	+	3.74025i	−0.767687	+	0.431887i
$76$		0			0
$77$			6.45743i			0.735892i
$78$		0			0
$79$			3.32493i			0.374084i	0.982352	+	0.187042i	$0.0598900\pi$
$79$			3.32493i			0.374084i	−0.982352	+	0.187042i	$0.940110\pi$
$80$		0			0
$81$	−4.14813	−	7.98706i	−0.460903	−	0.887451i
$82$		0			0
$83$	−17.0770			−1.87444			−0.937220	−	0.348740i	$-0.886610\pi$
$83$	−17.0770			−1.87444			−0.937220	+	0.348740i	$0.886610\pi$
$84$		0			0
$85$	−5.68711			−0.616853
$86$		0			0
$87$	−4.73604	−	8.41839i	−0.507757	−	0.902547i
$88$		0			0
$89$		−	4.94478i		−	0.524146i	−0.965048	−	0.262073i	$-0.915594\pi$
$89$		−	4.94478i		−	0.524146i	0.965048	−	0.262073i	$-0.0844061\pi$
$90$		0			0
$91$		−	8.04853i		−	0.843715i
$92$		0			0
$93$	−1.67765	−	2.98205i	−0.173964	−	0.309224i
$94$		0			0
$95$	0.771899			0.0791952
$96$		0			0
$97$	5.51162			0.559621			0.279810	−	0.960055i	$-0.409728\pi$
$97$	5.51162			0.559621			0.279810	+	0.960055i	$0.409728\pi$
$98$		0			0
$99$	6.77307	−	11.1497i	0.680719	−	1.12059i
$100$		0			0
$101$			16.8495i			1.67659i	0.545220	+	0.838293i	$0.316446\pi$
$101$			16.8495i			1.67659i	−0.545220	+	0.838293i	$0.683554\pi$
$102$		0			0
$103$		−	0.908890i		−	0.0895555i	−0.998997	−	0.0447778i	$-0.985742\pi$
$103$		−	0.908890i		−	0.0895555i	0.998997	−	0.0447778i	$-0.0142580\pi$
$104$		0			0
$105$	−1.73031	+	0.973444i	−0.168861	+	0.0949984i
$106$		0			0
$107$	3.64328			0.352209			0.176105	−	0.984371i	$-0.443650\pi$
$107$	3.64328			0.352209			0.176105	+	0.984371i	$0.443650\pi$
$108$		0			0
$109$	−7.75050			−0.742363			−0.371182	−	0.928560i	$-0.621047\pi$
$109$	−7.75050			−0.742363			−0.371182	+	0.928560i	$0.621047\pi$
$110$		0			0
$111$	−15.5073	+	8.72411i	−1.47188	+	0.828056i
$112$		0			0
$113$		−	7.03860i		−	0.662135i	−0.943607	−	0.331068i	$-0.892591\pi$
$113$		−	7.03860i		−	0.662135i	0.943607	−	0.331068i	$-0.107409\pi$
$114$		0			0
$115$		−	1.27565i		−	0.118955i
$116$		0			0
$117$	−8.44194	+	13.8970i	−0.780458	+	1.28477i
$118$		0			0
$119$	10.9407			1.00293
$120$		0			0
$121$	7.90999			0.719090
$122$		0			0
$123$	3.33703	+	5.93162i	0.300890	+	0.534836i
$124$		0			0
$125$		−	7.25907i		−	0.649271i
$126$		0			0
$127$		−	17.5203i		−	1.55468i	−0.629083	−	0.777338i	$-0.716570\pi$
$127$		−	17.5203i		−	1.55468i	0.629083	−	0.777338i	$-0.283430\pi$
$128$		0			0
$129$	7.64042	+	13.5810i	0.672702	+	1.19574i
$130$		0			0
$131$	−12.1482			−1.06139			−0.530697	−	0.847561i	$-0.678070\pi$
$131$	−12.1482			−1.06139			−0.530697	+	0.847561i	$0.678070\pi$
$132$		0			0
$133$	−1.48496			−0.128762
$134$		0			0
$135$	4.00866	+	0.134099i	0.345011	+	0.0115414i
$136$		0			0
$137$		−	13.4577i		−	1.14977i	−0.818234	−	0.574885i	$-0.805047\pi$
$137$		−	13.4577i		−	1.14977i	0.818234	−	0.574885i	$-0.194953\pi$
$138$		0			0
$139$			0.446425i			0.0378652i	0.999821	+	0.0189326i	$0.00602680\pi$
$139$			0.446425i			0.0378652i	−0.999821	+	0.0189326i	$0.993973\pi$
$140$		0			0
$141$	−5.47483	+	3.08004i	−0.461064	+	0.259386i
$142$		0			0
$143$	−23.5694			−1.97097
$144$		0			0
$145$	4.30467			0.357483
$146$		0			0
$147$	−7.23819	+	4.07208i	−0.596996	+	0.335859i
$148$		0			0
$149$		−	0.974462i		−	0.0798311i	−0.999203	−	0.0399155i	$-0.987291\pi$
$149$		−	0.974462i		−	0.0798311i	0.999203	−	0.0399155i	$-0.0127089\pi$
$150$		0			0
$151$			13.6741i			1.11278i	0.830921	+	0.556390i	$0.187814\pi$
$151$			13.6741i			1.11278i	−0.830921	+	0.556390i	$0.812186\pi$
$152$		0			0
$153$	−18.8907	−	11.4755i	−1.52722	−	0.927737i
$154$		0			0
$155$	1.52484			0.122478
$156$		0			0
$157$	−10.5971			−0.845740			−0.422870	−	0.906190i	$-0.638977\pi$
$157$	−10.5971			−0.845740			−0.422870	+	0.906190i	$0.638977\pi$
$158$		0			0
$159$	2.31154	+	4.10879i	0.183317	+	0.325849i
$160$		0			0
$161$			2.45405i			0.193406i
$162$		0			0
$163$		−	0.0136222i		−	0.00106697i	−1.00000		0.000533487i	$-0.999830\pi$
$163$		−	0.0136222i		−	0.00106697i	1.00000		0.000533487i	$-0.000169814\pi$
$164$		0			0
$165$	2.85064	+	5.06706i	0.221922	+	0.394470i
$166$		0			0
$167$	−6.87662			−0.532129			−0.266064	−	0.963955i	$-0.585723\pi$
$167$	−6.87662			−0.532129			−0.266064	+	0.963955i	$0.585723\pi$
$168$		0			0
$169$	16.3769			1.25976
$170$		0			0
$171$	2.56399	+	1.55754i	0.196073	+	0.119108i
$172$		0			0
$173$		−	16.2201i		−	1.23319i	−0.787281	−	0.616595i	$-0.788512\pi$
$173$		−	16.2201i		−	1.23319i	0.787281	−	0.616595i	$-0.211488\pi$
$174$		0			0
$175$			6.54001i			0.494378i
$176$		0			0
$177$	−2.93552	+	1.65147i	−0.220647	+	0.124132i
$178$		0			0
$179$	3.41994			0.255618			0.127809	−	0.991799i	$-0.459205\pi$
$179$	3.41994			0.255618			0.127809	+	0.991799i	$0.459205\pi$
$180$		0			0
$181$	−2.90781			−0.216136			−0.108068	−	0.994144i	$-0.534466\pi$
$181$	−2.90781			−0.216136			−0.108068	+	0.994144i	$0.534466\pi$
$182$		0			0
$183$	−18.5001	+	10.4078i	−1.36757	+	0.769370i
$184$		0			0
$185$		−	7.92949i		−	0.582988i
$186$		0			0
$187$		−	32.0388i		−	2.34291i
$188$		0			0
$189$	−7.71175	−	0.257976i	−0.560947	−	0.0187650i
$190$		0			0
$191$	−16.9029			−1.22305			−0.611527	−	0.791224i	$-0.709444\pi$
$191$	−16.9029			−1.22305			−0.611527	+	0.791224i	$0.709444\pi$
$192$		0			0
$193$	−4.36692			−0.314338			−0.157169	−	0.987572i	$-0.550237\pi$
$193$	−4.36692			−0.314338			−0.157169	+	0.987572i	$0.550237\pi$
$194$		0			0
$195$	−3.55304	−	6.31558i	−0.254438	−	0.452268i
$196$		0			0
$197$			15.3575i			1.09418i	0.837075	+	0.547088i	$0.184264\pi$
$197$			15.3575i			1.09418i	−0.837075	+	0.547088i	$0.815736\pi$
$198$		0			0
$199$		−	10.0160i		−	0.710018i	−0.934863	−	0.355009i	$-0.884478\pi$
$199$		−	10.0160i		−	0.710018i	0.934863	−	0.355009i	$-0.115522\pi$
$200$		0			0
$201$	−9.23056	−	16.4075i	−0.651073	−	1.15729i
$202$		0			0
$203$	−8.28119			−0.581226
$204$		0			0
$205$	−3.03308			−0.211839
$206$		0			0
$207$	2.57400	−	4.23727i	0.178906	−	0.294511i
$208$		0			0
$209$			4.34856i			0.300796i
$210$		0			0
$211$		−	27.8220i		−	1.91534i	−0.287862	−	0.957672i	$-0.592945\pi$
$211$		−	27.8220i		−	1.91534i	0.287862	−	0.957672i	$-0.407055\pi$
$212$		0			0
$213$	−24.3012	+	13.6715i	−1.66509	+	0.936752i
$214$		0			0
$215$	−6.94451			−0.473611
$216$		0			0
$217$	−2.93345			−0.199135
$218$		0			0
$219$	−7.47232	+	4.20380i	−0.504933	+	0.284066i
$220$		0			0
$221$			39.9331i			2.68619i
$222$		0			0
$223$		−	8.47655i		−	0.567632i	−0.958879	−	0.283816i	$-0.908400\pi$
$223$		−	8.47655i		−	0.567632i	0.958879	−	0.283816i	$-0.0916005\pi$
$224$		0			0
$225$	6.85968	−	11.2923i	0.457312	−	0.752818i
$226$		0			0
$227$	−14.1089			−0.936443			−0.468221	−	0.883611i	$-0.655105\pi$
$227$	−14.1089			−0.936443			−0.468221	+	0.883611i	$0.655105\pi$
$228$		0			0
$229$	3.07414			0.203145			0.101573	−	0.994828i	$-0.467613\pi$
$229$	3.07414			0.203145			0.101573	+	0.994828i	$0.467613\pi$
$230$		0			0
$231$	−5.48398	−	9.74788i	−0.360820	−	0.641363i
$232$		0			0
$233$			22.6364i			1.48296i	0.670976	+	0.741479i	$0.265875\pi$
$233$			22.6364i			1.48296i	−0.670976	+	0.741479i	$0.734125\pi$
$234$		0			0
$235$		−	2.79950i		−	0.182619i
$236$		0			0
$237$	−2.82370	−	5.01918i	−0.183419	−	0.326031i
$238$		0			0
$239$	−1.96500			−0.127106			−0.0635528	−	0.997978i	$-0.520243\pi$
$239$	−1.96500			−0.127106			−0.0635528	+	0.997978i	$0.520243\pi$
$240$		0			0
$241$	10.6457			0.685749			0.342875	−	0.939381i	$-0.388599\pi$
$241$	10.6457			0.685749			0.342875	+	0.939381i	$0.388599\pi$
$242$		0			0
$243$	13.0449	+	8.53413i	0.836829	+	0.547465i
$244$		0			0
$245$		−	3.70118i		−	0.236460i
$246$		0			0
$247$		−	5.42004i		−	0.344869i
$248$		0			0
$249$	25.7787	−	14.5026i	1.63366	−	0.919067i
$250$		0			0
$251$	20.3661			1.28549			0.642747	−	0.766078i	$-0.277794\pi$
$251$	20.3661			1.28549			0.642747	+	0.766078i	$0.277794\pi$
$252$		0			0
$253$	7.18646			0.451809
$254$		0			0
$255$	8.58503	−	4.82979i	0.537615	−	0.302453i
$256$		0			0
$257$			4.23279i			0.264034i	0.991247	+	0.132017i	$0.0421454\pi$
$257$			4.23279i			0.264034i	−0.991247	+	0.132017i	$0.957855\pi$
$258$		0			0
$259$			15.2545i			0.947870i
$260$		0			0
$261$	14.2987	+	8.68598i	0.885066	+	0.537649i
$262$		0			0
$263$	−11.9725			−0.738258			−0.369129	−	0.929378i	$-0.620344\pi$
$263$	−11.9725			−0.738258			−0.369129	+	0.929378i	$0.620344\pi$
$264$		0			0
$265$	−2.10099			−0.129063
$266$		0			0
$267$	4.19937	+	7.46445i	0.256997	+	0.456817i
$268$		0			0
$269$		−	12.2090i		−	0.744395i	−0.928154	−	0.372198i	$-0.878604\pi$
$269$		−	12.2090i		−	0.744395i	0.928154	−	0.372198i	$-0.121396\pi$
$270$		0			0
$271$			21.9421i			1.33289i	0.745556	+	0.666443i	$0.232184\pi$
$271$			21.9421i			1.33289i	−0.745556	+	0.666443i	$0.767816\pi$
$272$		0			0
$273$	6.83523	+	12.1497i	0.413687	+	0.735336i
$274$		0			0
$275$	19.1518			1.15490
$276$		0			0
$277$	14.8788			0.893983			0.446992	−	0.894538i	$-0.352495\pi$
$277$	14.8788			0.893983			0.446992	+	0.894538i	$0.352495\pi$
$278$		0			0
$279$	5.06502	+	3.07683i	0.303235	+	0.184205i
$280$		0			0
$281$			22.9820i			1.37099i	0.728078	+	0.685494i	$0.240414\pi$
$281$			22.9820i			1.37099i	−0.728078	+	0.685494i	$0.759586\pi$
$282$		0			0
$283$			7.85519i			0.466943i	0.972364	+	0.233471i	$0.0750085\pi$
$283$			7.85519i			0.466943i	−0.972364	+	0.233471i	$0.924991\pi$
$284$		0			0
$285$	−1.16523	+	0.655537i	−0.0690221	+	0.0388306i
$286$		0			0
$287$	5.83495			0.344426
$288$		0			0
$289$	−37.2827			−2.19310
$290$		0			0
$291$	−8.32013	+	4.68076i	−0.487734	+	0.274391i
$292$		0			0
$293$			17.8190i			1.04100i	0.853862	+	0.520499i	$0.174254\pi$
$293$			17.8190i			1.04100i	−0.853862	+	0.520499i	$0.825746\pi$
$294$		0			0
$295$		−	1.50105i		−	0.0873945i
$296$		0			0
$297$	−0.755459	+	22.5832i	−0.0438362	+	1.31041i
$298$		0			0
$299$	−8.95720			−0.518008
$300$		0			0
$301$	13.3596			0.770037
$302$		0			0
$303$	−14.3095	−	25.4353i	−0.822057	−	1.46122i
$304$		0			0
$305$		−	9.45986i		−	0.541670i
$306$		0			0
$307$			6.56643i			0.374766i	0.982287	+	0.187383i	$0.0600006\pi$
$307$			6.56643i			0.374766i	−0.982287	+	0.187383i	$0.939999\pi$
$308$		0			0
$309$	0.771876	+	1.37202i	0.0439105	+	0.0780517i
$310$		0			0
$311$	0.203589			0.0115445			0.00577224	−	0.999983i	$-0.498163\pi$
$311$	0.203589			0.0115445			0.00577224	+	0.999983i	$0.498163\pi$
$312$		0			0
$313$	9.87656			0.558256			0.279128	−	0.960254i	$-0.409955\pi$
$313$	9.87656			0.558256			0.279128	+	0.960254i	$0.409955\pi$
$314$		0			0
$315$	1.78531	−	2.93894i	0.100591	−	0.165591i
$316$		0			0
$317$		−	29.5186i		−	1.65793i	−0.559299	−	0.828966i	$-0.688930\pi$
$317$		−	29.5186i		−	1.65793i	0.559299	−	0.828966i	$-0.311070\pi$
$318$		0			0
$319$			24.2507i			1.35778i
$320$		0			0
$321$	−5.49975	+	3.09406i	−0.306966	+	0.172694i
$322$		0			0
$323$	7.36768			0.409949
$324$		0			0
$325$	−23.8708			−1.32411
$326$		0			0
$327$	11.6998	−	6.58213i	0.647003	−	0.363992i
$328$		0			0
$329$			5.38560i			0.296918i
$330$		0			0
$331$		−	26.9811i		−	1.48302i	−0.670944	−	0.741508i	$-0.734111\pi$
$331$		−	26.9811i		−	1.48302i	0.670944	−	0.741508i	$-0.265889\pi$
$332$		0			0
$333$	16.0002	−	26.3391i	0.876804	−	1.44338i
$334$		0			0
$335$	8.38980			0.458384
$336$		0			0
$337$	7.04071			0.383532			0.191766	−	0.981441i	$-0.438579\pi$
$337$	7.04071			0.383532			0.191766	+	0.981441i	$0.438579\pi$
$338$		0			0
$339$	5.97754	+	10.6252i	0.324656	+	0.577081i
$340$		0			0
$341$			8.59033i			0.465192i
$342$		0			0
$343$			17.5149i			0.945717i
$344$		0			0
$345$	1.08334	+	1.92566i	0.0583253	+	0.103674i
$346$		0			0
$347$	19.4535			1.04432			0.522161	−	0.852847i	$-0.325126\pi$
$347$	19.4535			1.04432			0.522161	+	0.852847i	$0.325126\pi$
$348$		0			0
$349$	0.983778			0.0526604			0.0263302	−	0.999653i	$-0.491618\pi$
$349$	0.983778			0.0526604			0.0263302	+	0.999653i	$0.491618\pi$
$350$		0			0
$351$	0.941603	−	28.1476i	0.0502591	−	1.50241i
$352$		0			0
$353$		−	14.2453i		−	0.758200i	−0.925356	−	0.379100i	$-0.876234\pi$
$353$		−	14.2453i		−	0.758200i	0.925356	−	0.379100i	$-0.123766\pi$
$354$		0			0
$355$		−	12.4262i		−	0.659514i
$356$		0			0
$357$	−16.5156	+	9.29141i	−0.874100	+	0.491753i
$358$		0			0
$359$	−31.2181			−1.64763			−0.823814	−	0.566860i	$-0.808158\pi$
$359$	−31.2181			−1.64763			−0.823814	+	0.566860i	$0.808158\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$		0			0
$363$	−11.9406	+	6.71757i	−0.626719	+	0.352581i
$364$		0			0
$365$		−	3.82090i		−	0.199995i
$366$		0			0
$367$			9.33319i			0.487188i	0.969877	+	0.243594i	$0.0783265\pi$
$367$			9.33319i			0.487188i	−0.969877	+	0.243594i	$0.921673\pi$
$368$		0			0
$369$	−10.0749	−	6.12016i	−0.524478	−	0.318603i
$370$		0			0
$371$	4.04183			0.209841
$372$		0			0
$373$	−24.6575			−1.27672			−0.638359	−	0.769738i	$-0.720387\pi$
$373$	−24.6575			−1.27672			−0.638359	+	0.769738i	$0.720387\pi$
$374$		0			0
$375$	6.16478	+	10.9580i	0.318348	+	0.565869i
$376$		0			0
$377$		−	30.2261i		−	1.55672i
$378$		0			0
$379$			17.7338i			0.910924i	0.890255	+	0.455462i	$0.150526\pi$
$379$			17.7338i			0.910924i	−0.890255	+	0.455462i	$0.849474\pi$
$380$		0			0
$381$	14.8792	+	26.4480i	0.762282	+	1.35497i
$382$		0			0
$383$	9.79058			0.500275			0.250138	−	0.968210i	$-0.419524\pi$
$383$	9.79058			0.500275			0.250138	+	0.968210i	$0.419524\pi$
$384$		0			0
$385$	4.98448			0.254033
$386$		0			0
$387$	−23.0674	−	14.0127i	−1.17258	−	0.712304i
$388$		0			0
$389$		−	4.73528i		−	0.240088i	−0.992769	−	0.120044i	$-0.961696\pi$
$389$		−	4.73528i		−	0.240088i	0.992769	−	0.120044i	$-0.0383036\pi$
$390$		0			0
$391$		−	12.1759i		−	0.615761i
$392$		0			0
$393$	18.3385	−	10.3169i	0.925053	−	0.520419i
$394$		0			0
$395$	2.56651			0.129135
$396$		0			0
$397$	25.2117			1.26534			0.632669	−	0.774422i	$-0.281959\pi$
$397$	25.2117			1.26534			0.632669	+	0.774422i	$0.281959\pi$
$398$		0			0
$399$	2.24163	−	1.26110i	0.112222	−	0.0631341i
$400$		0			0
$401$		−	8.47152i		−	0.423047i	−0.977373	−	0.211524i	$-0.932157\pi$
$401$		−	8.47152i		−	0.423047i	0.977373	−	0.211524i	$-0.0678425\pi$
$402$		0			0
$403$		−	10.7070i		−	0.533353i
$404$		0			0
$405$	−6.16520	+	3.20193i	−0.306351	+	0.159105i
$406$		0			0
$407$	44.6715			2.21428
$408$		0			0
$409$	5.39447			0.266739			0.133370	−	0.991066i	$-0.457420\pi$
$409$	5.39447			0.266739			0.133370	+	0.991066i	$0.457420\pi$
$410$		0			0
$411$	11.4290	+	20.3152i	0.563750	+	1.00208i
$412$		0			0
$413$			2.88768i			0.142093i
$414$		0			0
$415$			13.1817i			0.647063i
$416$		0			0
$417$	−0.379127	−	0.673905i	−0.0185659	−	0.0330013i
$418$		0			0
$419$	34.0530			1.66360			0.831799	−	0.555077i	$-0.187311\pi$
$419$	34.0530			1.66360			0.831799	+	0.555077i	$0.187311\pi$
$420$		0			0
$421$	−0.368706			−0.0179696			−0.00898481	−	0.999960i	$-0.502860\pi$
$421$	−0.368706			−0.0179696			−0.00898481	+	0.999960i	$0.502860\pi$
$422$		0			0
$423$	5.64885	−	9.29902i	0.274656	−	0.452134i
$424$		0			0
$425$		−	32.4485i		−	1.57399i
$426$		0			0
$427$			18.1986i			0.880693i
$428$		0			0
$429$	35.5794	−	20.0163i	1.71779	−	0.966398i
$430$		0			0
$431$	−13.0041			−0.626386			−0.313193	−	0.949690i	$-0.601399\pi$
$431$	−13.0041			−0.626386			−0.313193	+	0.949690i	$0.601399\pi$
$432$		0			0
$433$	−25.1441			−1.20835			−0.604175	−	0.796852i	$-0.706497\pi$
$433$	−25.1441			−1.20835			−0.604175	+	0.796852i	$0.706497\pi$
$434$		0			0
$435$	−6.49815	+	3.65575i	−0.311562	+	0.175280i
$436$		0			0
$437$			1.65261i			0.0790549i
$438$		0			0
$439$			7.55875i			0.360759i	0.983597	+	0.180380i	$0.0577326\pi$
$439$			7.55875i			0.360759i	−0.983597	+	0.180380i	$0.942267\pi$
$440$		0			0
$441$	7.46826	−	12.2941i	0.355631	−	0.585433i
$442$		0			0
$443$	4.71379			0.223959			0.111980	−	0.993711i	$-0.464281\pi$
$443$	4.71379			0.223959			0.111980	+	0.993711i	$0.464281\pi$
$444$		0			0
$445$	−3.81687			−0.180937
$446$		0			0
$447$	0.827564	+	1.47101i	0.0391424	+	0.0695764i
$448$		0			0
$449$			24.2223i			1.14312i	0.820560	+	0.571560i	$0.193662\pi$
$449$			24.2223i			1.14312i	−0.820560	+	0.571560i	$0.806338\pi$
$450$		0			0
$451$		−	17.0871i		−	0.804601i
$452$		0			0
$453$	−11.6127	−	20.6418i	−0.545614	−	0.969837i
$454$		0			0
$455$	−6.21265			−0.291254
$456$		0			0
$457$	5.86940			0.274559			0.137279	−	0.990532i	$-0.456164\pi$
$457$	5.86940			0.274559			0.137279	+	0.990532i	$0.456164\pi$
$458$		0			0
$459$	38.2622	+	1.27996i	1.78593	+	0.0597434i
$460$		0			0
$461$		−	19.5224i		−	0.909250i	−0.890683	−	0.454625i	$-0.849773\pi$
$461$		−	19.5224i		−	0.909250i	0.890683	−	0.454625i	$-0.150227\pi$
$462$		0			0
$463$			30.5441i			1.41950i	0.704452	+	0.709751i	$0.251193\pi$
$463$			30.5441i			1.41950i	−0.704452	+	0.709751i	$0.748807\pi$
$464$		0			0
$465$	−2.30184	+	1.29497i	−0.106745	+	0.0600530i
$466$		0			0
$467$	−34.1261			−1.57917			−0.789584	−	0.613643i	$-0.789704\pi$
$467$	−34.1261			−1.57917			−0.789584	+	0.613643i	$0.789704\pi$
$468$		0			0
$469$	−16.1401			−0.745279
$470$		0			0
$471$	15.9969	−	8.99960i	0.737100	−	0.414680i
$472$		0			0
$473$		−	39.1225i		−	1.79885i
$474$		0			0
$475$			4.40417i			0.202077i
$476$		0			0
$477$	−6.97880	−	4.23939i	−0.319538	−	0.194109i
$478$		0			0
$479$	23.7779			1.08644			0.543220	−	0.839590i	$-0.317205\pi$
$479$	23.7779			1.08644			0.543220	+	0.839590i	$0.317205\pi$
$480$		0			0
$481$	−55.6785			−2.53872
$482$		0			0
$483$	−2.08411	−	3.70454i	−0.0948302	−	0.168562i
$484$		0			0
$485$		−	4.25442i		−	0.193183i
$486$		0			0
$487$			22.7746i			1.03202i	0.856584	+	0.516008i	$0.172583\pi$
$487$			22.7746i			1.03202i	−0.856584	+	0.516008i	$0.827417\pi$
$488$		0			0
$489$	0.0115687	+	0.0205636i	0.000523155	+	0.000929916i
$490$		0			0
$491$	−12.4456			−0.561660			−0.280830	−	0.959757i	$-0.590610\pi$
$491$	−12.4456			−0.561660			−0.280830	+	0.959757i	$0.590610\pi$
$492$		0			0
$493$	41.0875			1.85049
$494$		0			0
$495$	−8.60643	−	5.22812i	−0.386830	−	0.234987i
$496$		0			0
$497$			23.9052i			1.07229i
$498$		0			0
$499$			27.4605i			1.22930i	0.788800	+	0.614650i	$0.210703\pi$
$499$			27.4605i			1.22930i	−0.788800	+	0.614650i	$0.789297\pi$
$500$		0			0
$501$	10.3807	−	5.83998i	0.463774	−	0.260911i
$502$		0			0
$503$	0.0570190			0.00254235			0.00127118	−	0.999999i	$-0.499595\pi$
$503$	0.0570190			0.00254235			0.00127118	+	0.999999i	$0.499595\pi$
$504$		0			0
$505$	13.0061			0.578764
$506$		0			0
$507$	−24.7218	+	13.9081i	−1.09794	+	0.617679i
$508$		0			0
$509$		−	1.80663i		−	0.0800775i	−0.999198	−	0.0400387i	$-0.987252\pi$
$509$		−	1.80663i		−	0.0800775i	0.999198	−	0.0400387i	$-0.0127481\pi$
$510$		0			0
$511$			7.35054i			0.325169i
$512$		0			0
$513$	−5.19325	−	0.173726i	−0.229287	−	0.00767020i
$514$		0			0
$515$	−0.701571			−0.0309149
$516$		0			0
$517$	15.7712			0.693618
$518$		0			0
$519$	13.7749	+	24.4852i	0.604652	+	1.07478i
$520$		0			0
$521$		−	0.439375i		−	0.0192494i	−0.999954	−	0.00962469i	$-0.996936\pi$
$521$		−	0.439375i		−	0.0192494i	0.999954	−	0.00962469i	$-0.00306368\pi$
$522$		0			0
$523$		−	30.1583i		−	1.31873i	−0.751822	−	0.659366i	$-0.770825\pi$
$523$		−	30.1583i		−	1.31873i	0.751822	−	0.659366i	$-0.229175\pi$
$524$		0			0
$525$	−5.55412	−	9.87253i	−0.242401	−	0.430873i
$526$		0			0
$527$	14.5544			0.634001
$528$		0			0
$529$	−20.2689			−0.881256
$530$		0			0
$531$	3.02883	−	4.98599i	0.131440	−	0.216374i
$532$		0			0
$533$			21.2974i			0.922491i
$534$		0			0
$535$		−	2.81224i		−	0.121584i
$536$		0			0
$537$	−5.16261	+	2.90439i	−0.222783	+	0.125334i
$538$		0			0
$539$	20.8509			0.898113
$540$		0			0
$541$	−34.9412			−1.50224			−0.751120	−	0.660165i	$-0.770486\pi$
$541$	−34.9412			−1.50224			−0.751120	+	0.660165i	$0.770486\pi$
$542$		0			0
$543$	4.38951	−	2.46946i	0.188372	−	0.105975i
$544$		0			0
$545$			5.98260i			0.256266i
$546$		0			0
$547$			31.7856i			1.35905i	0.733650	+	0.679527i	$0.237815\pi$
$547$			31.7856i			1.35905i	−0.733650	+	0.679527i	$0.762185\pi$
$548$		0			0
$549$	19.0882	−	31.4225i	0.814663	−	1.34108i
$550$		0			0
$551$	−5.57672			−0.237576
$552$		0			0
$553$	−4.93738			−0.209959
$554$		0			0
$555$	6.73414	+	11.9700i	0.285848	+	0.508100i
$556$		0			0
$557$			26.3099i			1.11478i	0.830249	+	0.557392i	$0.188198\pi$
$557$			26.3099i			1.11478i	−0.830249	+	0.557392i	$0.811802\pi$
$558$		0			0
$559$			48.7622i			2.06242i
$560$		0			0
$561$	27.2090	+	48.3645i	1.14877	+	2.04195i
$562$		0			0
$563$	−20.8169			−0.877328			−0.438664	−	0.898651i	$-0.644548\pi$
$563$	−20.8169			−0.877328			−0.438664	+	0.898651i	$0.644548\pi$
$564$		0			0
$565$	−5.43309			−0.228572
$566$		0			0
$567$	11.8604	−	6.15979i	0.498092	−	0.258687i
$568$		0			0
$569$		−	26.2506i		−	1.10048i	−0.835005	−	0.550242i	$-0.814536\pi$
$569$		−	26.2506i		−	1.10048i	0.835005	−	0.550242i	$-0.185464\pi$
$570$		0			0
$571$		−	20.9006i		−	0.874662i	−0.899301	−	0.437331i	$-0.855924\pi$
$571$		−	20.9006i		−	0.874662i	0.899301	−	0.437331i	$-0.144076\pi$
$572$		0			0
$573$	25.5160	−	14.3548i	1.06595	−	0.599682i
$574$		0			0
$575$	7.27836			0.303529
$576$		0			0
$577$	21.8267			0.908656			0.454328	−	0.890835i	$-0.349879\pi$
$577$	21.8267			0.908656			0.454328	+	0.890835i	$0.349879\pi$
$578$		0			0
$579$	6.59213	−	3.70862i	0.273960	−	0.154125i
$580$		0			0
$581$		−	25.3586i		−	1.05205i
$582$		0			0
$583$		−	11.8361i		−	0.490202i
$584$		0			0
$585$	10.7270	+	6.51633i	0.443509	+	0.269417i
$586$		0			0
$587$	−0.0435337			−0.00179683			−0.000898413	−	1.00000i	$-0.500286\pi$
$587$	−0.0435337			−0.00179683			−0.000898413		1.00000i	$0.500286\pi$
$588$		0			0
$589$	−1.97544			−0.0813967
$590$		0			0
$591$	−13.0424	−	23.1830i	−0.536491	−	0.953623i
$592$		0			0
$593$			34.7289i			1.42614i	0.701090	+	0.713072i	$0.252697\pi$
$593$			34.7289i			1.42614i	−0.701090	+	0.713072i	$0.747303\pi$
$594$		0			0
$595$		−	8.44511i		−	0.346216i
$596$		0			0
$597$	8.50614	+	15.1198i	0.348133	+	0.618813i
$598$		0			0
$599$	−5.23660			−0.213962			−0.106981	−	0.994261i	$-0.534118\pi$
$599$	−5.23660			−0.213962			−0.106981	+	0.994261i	$0.534118\pi$
$600$		0			0
$601$	−12.0980			−0.493489			−0.246745	−	0.969081i	$-0.579361\pi$
$601$	−12.0980			−0.493489			−0.246745	+	0.969081i	$0.579361\pi$
$602$		0			0
$603$	27.8682	+	16.9290i	1.13488	+	0.689402i
$604$		0			0
$605$		−	6.10571i		−	0.248232i
$606$		0			0
$607$		−	31.7542i		−	1.28886i	−0.764662	−	0.644431i	$-0.777094\pi$
$607$		−	31.7542i		−	1.28886i	0.764662	−	0.644431i	$-0.222906\pi$
$608$		0			0
$609$	12.5010	−	7.03282i	0.506564	−	0.284984i
$610$		0			0
$611$	−19.6572			−0.795247
$612$		0			0
$613$	−9.54539			−0.385535			−0.192767	−	0.981244i	$-0.561746\pi$
$613$	−9.54539			−0.385535			−0.192767	+	0.981244i	$0.561746\pi$
$614$		0			0
$615$	4.57861	−	2.57585i	0.184628	−	0.103868i
$616$		0			0
$617$			12.5959i			0.507091i	0.967323	+	0.253546i	$0.0815968\pi$
$617$			12.5959i			0.507091i	−0.967323	+	0.253546i	$0.918403\pi$
$618$		0			0
$619$		−	23.0595i		−	0.926841i	−0.886138	−	0.463421i	$-0.846622\pi$
$619$		−	23.0595i		−	0.926841i	0.886138	−	0.463421i	$-0.153378\pi$
$620$		0			0
$621$	−0.287101	+	8.58239i	−0.0115210	+	0.344400i
$622$		0			0
$623$	7.34279			0.294183
$624$		0			0
$625$	16.4176			0.656704
$626$		0			0
$627$	−3.69302	−	6.56441i	−0.147485	−	0.262157i
$628$		0			0
$629$		−	75.6860i		−	3.01780i
$630$		0			0
$631$		−	1.14798i		−	0.0457005i	−0.999739	−	0.0228502i	$-0.992726\pi$
$631$		−	1.14798i		−	0.0457005i	0.999739	−	0.0228502i	$-0.00727409\pi$
$632$		0			0
$633$	23.6279	+	41.9989i	0.939123	+	1.66931i
$634$		0			0
$635$	−13.5239			−0.536680
$636$		0			0
$637$	−25.9886			−1.02970
$638$		0			0
$639$	25.0737	−	41.2757i	0.991899	−	1.63284i
$640$		0			0
$641$			46.9017i			1.85250i	0.376905	+	0.926252i	$0.376988\pi$
$641$			46.9017i			1.85250i	−0.376905	+	0.926252i	$0.623012\pi$
$642$		0			0
$643$			19.3474i			0.762986i	0.924372	+	0.381493i	$0.124590\pi$
$643$			19.3474i			0.762986i	−0.924372	+	0.381493i	$0.875410\pi$
$644$		0			0
$645$	10.4831	−	5.89764i	0.412774	−	0.232219i
$646$		0			0
$647$	13.2766			0.521956			0.260978	−	0.965345i	$-0.415955\pi$
$647$	13.2766			0.521956			0.260978	+	0.965345i	$0.415955\pi$
$648$		0			0
$649$	8.45630			0.331939
$650$		0			0
$651$	4.42821	−	2.49124i	0.173555	−	0.0976392i
$652$		0			0
$653$		−	28.4844i		−	1.11468i	−0.830284	−	0.557341i	$-0.811822\pi$
$653$		−	28.4844i		−	1.11468i	0.830284	−	0.557341i	$-0.188178\pi$
$654$		0			0
$655$			9.37720i			0.366397i
$656$		0			0
$657$	7.70983	−	12.6918i	0.300789	−	0.495153i
$658$		0			0
$659$	48.4760			1.88836			0.944179	−	0.329434i	$-0.106858\pi$
$659$	48.4760			1.88836			0.944179	+	0.329434i	$0.106858\pi$
$660$		0			0
$661$	−25.2334			−0.981465			−0.490732	−	0.871310i	$-0.663271\pi$
$661$	−25.2334			−0.981465			−0.490732	+	0.871310i	$0.663271\pi$
$662$		0			0
$663$	−33.9133	−	60.2815i	−1.31708	−	2.34114i
$664$		0			0
$665$			1.14624i			0.0444492i
$666$		0			0
$667$			9.21613i			0.356850i
$668$		0			0
$669$	7.19873	+	12.7959i	0.278319	+	0.494717i
$670$		0			0
$671$	53.2930			2.05735
$672$		0			0
$673$	−3.96502			−0.152840			−0.0764201	−	0.997076i	$-0.524349\pi$
$673$	−3.96502			−0.152840			−0.0764201	+	0.997076i	$0.524349\pi$
$674$		0			0
$675$	−0.765120	+	22.8720i	−0.0294495	+	0.880342i
$676$		0			0
$677$			36.0398i			1.38512i	0.721359	+	0.692561i	$0.243518\pi$
$677$			36.0398i			1.38512i	−0.721359	+	0.692561i	$0.756482\pi$
$678$		0			0
$679$			8.18453i			0.314093i
$680$		0			0
$681$	21.2983	−	11.9820i	0.816152	−	0.459153i
$682$		0			0
$683$	−21.1211			−0.808175			−0.404087	−	0.914720i	$-0.632411\pi$
$683$	−21.1211			−0.808175			−0.404087	+	0.914720i	$0.632411\pi$
$684$		0			0
$685$	−10.3880			−0.396905
$686$		0			0
$687$	−4.64060	+	2.61072i	−0.177050	+	0.0996052i
$688$		0			0
$689$			14.7525i			0.562027i
$690$		0			0
$691$		−	8.72828i		−	0.332039i	−0.986122	−	0.166020i	$-0.946908\pi$
$691$		−	8.72828i		−	0.332039i	0.986122	−	0.166020i	$-0.0530915\pi$
$692$		0			0
$693$	16.5568	+	10.0577i	0.628941	+	0.382061i
$694$		0			0
$695$	0.344595			0.0130712
$696$		0			0
$697$	−28.9504			−1.09657
$698$		0			0
$699$	−19.2240	−	34.1710i	−0.727118	−	1.29246i
$700$		0			0
$701$		−	38.2944i		−	1.44636i	−0.690660	−	0.723180i	$-0.742680\pi$
$701$		−	38.2944i		−	1.44636i	0.690660	−	0.723180i	$-0.257320\pi$
$702$		0			0
$703$			10.2727i			0.387442i
$704$		0			0
$705$	2.37748	+	4.22601i	0.0895411	+	0.159161i
$706$		0			0
$707$	−25.0208			−0.941003
$708$		0			0
$709$	−15.4736			−0.581123			−0.290561	−	0.956856i	$-0.593842\pi$
$709$	−15.4736			−0.581123			−0.290561	+	0.956856i	$0.593842\pi$
$710$		0			0
$711$	8.52510	+	5.17872i	0.319716	+	0.194217i
$712$		0			0
$713$			3.26463i			0.122261i
$714$		0			0
$715$			18.1932i			0.680387i
$716$		0			0
$717$	2.96629	−	1.66878i	0.110778	−	0.0623219i
$718$		0			0
$719$	−10.1416			−0.378217			−0.189109	−	0.981956i	$-0.560560\pi$
$719$	−10.1416			−0.378217			−0.189109	+	0.981956i	$0.560560\pi$
$720$		0			0
$721$	1.34966			0.0502641
$722$		0			0
$723$	−16.0703	+	9.04088i	−0.597661	+	0.336234i
$724$		0			0
$725$			24.5608i			0.912167i
$726$		0			0
$727$		−	33.6882i		−	1.24943i	−0.780854	−	0.624713i	$-0.785216\pi$
$727$		−	33.6882i		−	1.24943i	0.780854	−	0.624713i	$-0.214784\pi$
$728$		0			0
$729$	−26.9396	−	1.80441i	−0.997764	−	0.0668299i
$730$		0			0
$731$	−66.2845			−2.45162
$732$		0			0
$733$	−43.8862			−1.62097			−0.810487	−	0.585756i	$-0.800798\pi$
$733$	−43.8862			−1.62097			−0.810487	+	0.585756i	$0.800798\pi$
$734$		0			0
$735$	3.14323	+	5.58715i	0.115940	+	0.206085i
$736$		0			0
$737$			47.2647i			1.74102i
$738$		0			0
$739$		−	16.4719i		−	0.605928i	−0.953002	−	0.302964i	$-0.902024\pi$
$739$		−	16.4719i		−	0.605928i	0.953002	−	0.302964i	$-0.0979762\pi$
$740$		0			0
$741$	4.60298	+	8.18188i	0.169095	+	0.300569i
$742$		0			0
$743$	−27.8678			−1.02237			−0.511186	−	0.859470i	$-0.670794\pi$
$743$	−27.8678			−1.02237			−0.511186	+	0.859470i	$0.670794\pi$
$744$		0			0
$745$	−0.752187			−0.0275580
$746$		0			0
$747$	−26.5981	+	43.7852i	−0.973173	+	1.60202i
$748$		0			0
$749$			5.41011i			0.197681i
$750$		0			0
$751$			8.58017i			0.313095i	0.987670	+	0.156547i	$0.0500364\pi$
$751$			8.58017i			0.313095i	−0.987670	+	0.156547i	$0.949964\pi$
$752$		0			0
$753$	−30.7438	+	17.2959i	−1.12037	+	0.630299i
$754$		0			0
$755$	10.5550			0.384136
$756$		0			0
$757$	46.1734			1.67820			0.839100	−	0.543978i	$-0.183082\pi$
$757$	46.1734			1.67820			0.839100	+	0.543978i	$0.183082\pi$
$758$		0			0
$759$	−10.8484	+	6.10312i	−0.393772	+	0.221529i
$760$		0			0
$761$		−	6.95402i		−	0.252083i	−0.992025	−	0.126042i	$-0.959773\pi$
$761$		−	6.95402i		−	0.252083i	0.992025	−	0.126042i	$-0.0402273\pi$
$762$		0			0
$763$		−	11.5092i		−	0.416660i
$764$		0			0
$765$	−8.85791	+	14.5817i	−0.320258	+	0.527203i
$766$		0			0
$767$	−10.5399			−0.380574
$768$		0			0
$769$	21.7792			0.785377			0.392688	−	0.919672i	$-0.371545\pi$
$769$	21.7792			0.785377			0.392688	+	0.919672i	$0.371545\pi$
$770$		0			0
$771$	−3.59470	−	6.38965i	−0.129460	−	0.230118i
$772$		0			0
$773$			53.0921i			1.90959i	0.297264	+	0.954795i	$0.403926\pi$
$773$			53.0921i			1.90959i	−0.297264	+	0.954795i	$0.596074\pi$
$774$		0			0
$775$			8.70019i			0.312520i
$776$		0			0
$777$	−12.9549	−	23.0276i	−0.464756	−	0.826111i
$778$		0			0
$779$	3.92937			0.140784
$780$		0			0
$781$	70.0041			2.50494
$782$		0			0
$783$	−28.9613	−	0.968822i	−1.03499	−	0.0346229i
$784$		0			0
$785$			8.17988i			0.291953i
$786$		0			0
$787$			29.2161i			1.04144i	0.853727	+	0.520721i	$0.174337\pi$
$787$			29.2161i			1.04144i	−0.853727	+	0.520721i	$0.825663\pi$
$788$		0			0
$789$	18.0733	−	10.1677i	0.643425	−	0.361980i
$790$		0			0
$791$	10.4520			0.371631
$792$		0			0
$793$	−66.4243			−2.35880
$794$		0			0
$795$	3.17157	−	1.78427i	0.112484	−	0.0632816i
$796$		0			0
$797$		−	32.7208i		−	1.15903i	−0.814962	−	0.579515i	$-0.803242\pi$
$797$		−	32.7208i		−	1.15903i	0.814962	−	0.579515i	$-0.196758\pi$
$798$		0			0
$799$		−	26.7209i		−	0.945317i
$800$		0			0
$801$	−12.6784	−	7.70171i	−0.447969	−	0.272126i
$802$		0			0
$803$	21.5254			0.759614
$804$		0			0
$805$	1.89428			0.0667646
$806$		0			0
$807$	10.3685	+	18.4302i	0.364989	+	0.648774i
$808$		0			0
$809$		−	3.02723i		−	0.106432i	−0.998583	−	0.0532158i	$-0.983053\pi$
$809$		−	3.02723i		−	0.106432i	0.998583	−	0.0532158i	$-0.0169471\pi$
$810$		0			0
$811$			10.2095i			0.358504i	0.983803	+	0.179252i	$0.0573677\pi$
$811$			10.2095i			0.358504i	−0.983803	+	0.179252i	$0.942632\pi$
$812$		0			0
$813$	−18.6343	−	33.1229i	−0.653535	−	1.16167i
$814$		0			0
$815$	−0.0105150			−0.000368324
$816$		0			0
$817$	8.99665			0.314753
$818$		0			0
$819$	−20.6364	−	12.5359i	−0.721094	−	0.438041i
$820$		0			0
$821$			11.3681i			0.396748i	0.980126	+	0.198374i	$0.0635661\pi$
$821$			11.3681i			0.396748i	−0.980126	+	0.198374i	$0.936434\pi$
$822$		0			0
$823$		−	38.6752i		−	1.34813i	−0.738671	−	0.674066i	$-0.764546\pi$
$823$		−	38.6752i		−	1.34813i	0.738671	−	0.674066i	$-0.235454\pi$
$824$		0			0
$825$	−28.9108	+	16.2647i	−1.00655	+	0.566265i
$826$		0			0
$827$	50.7631			1.76521			0.882604	−	0.470118i	$-0.155789\pi$
$827$	50.7631			1.76521			0.882604	+	0.470118i	$0.155789\pi$
$828$		0			0
$829$	4.93921			0.171546			0.0857729	−	0.996315i	$-0.472664\pi$
$829$	4.93921			0.171546			0.0857729	+	0.996315i	$0.472664\pi$
$830$		0			0
$831$	−22.4605	+	12.6359i	−0.779146	+	0.438334i
$832$		0			0
$833$		−	35.3273i		−	1.22402i
$834$		0			0
$835$			5.30805i			0.183693i
$836$		0			0
$837$	−10.2590	−	0.343186i	−0.354602	−	0.0118622i
$838$		0			0
$839$	−7.34156			−0.253459			−0.126729	−	0.991937i	$-0.540448\pi$
$839$	−7.34156			−0.253459			−0.126729	+	0.991937i	$0.540448\pi$
$840$		0			0
$841$	−2.09982			−0.0724075
$842$		0			0
$843$	−19.5175	−	34.6926i	−0.672218	−	1.19488i
$844$		0			0
$845$		−	12.6413i		−	0.434873i
$846$		0			0
$847$			11.7460i			0.403597i
$848$		0			0
$849$	−6.67104	−	11.8579i	−0.228949	−	0.406961i
$850$		0			0
$851$	16.9767			0.581955
$852$		0			0
$853$	31.5517			1.08031			0.540156	−	0.841565i	$-0.318365\pi$
$853$	31.5517			1.08031			0.540156	+	0.841565i	$0.318365\pi$
$854$		0			0
$855$	1.20227	−	1.97914i	0.0411166	−	0.0676853i
$856$		0			0
$857$			50.5708i			1.72747i	0.503950	+	0.863733i	$0.331879\pi$
$857$			50.5708i			1.72747i	−0.503950	+	0.863733i	$0.668121\pi$
$858$		0			0
$859$			51.3852i			1.75324i	0.481184	+	0.876620i	$0.340207\pi$
$859$			51.3852i			1.75324i	−0.481184	+	0.876620i	$0.659793\pi$
$860$		0			0
$861$	−8.80821	+	4.95534i	−0.300183	+	0.168878i
$862$		0			0
$863$	−32.0771			−1.09192			−0.545958	−	0.837812i	$-0.683834\pi$
$863$	−32.0771			−1.09192			−0.545958	+	0.837812i	$0.683834\pi$
$864$		0			0
$865$	−12.5203			−0.425701
$866$		0			0
$867$	56.2805	−	31.6624i	1.91139	−	1.07531i
$868$		0			0
$869$			14.4587i			0.490476i
$870$		0			0
$871$		−	58.9107i		−	1.99611i
$872$		0			0
$873$	8.58458	−	14.1318i	0.290544	−	0.478288i
$874$		0			0
$875$	10.7794			0.364411
$876$		0			0
$877$	−39.2697			−1.32604			−0.663022	−	0.748600i	$-0.730726\pi$
$877$	−39.2697			−1.32604			−0.663022	+	0.748600i	$0.730726\pi$
$878$		0			0
$879$	−15.1328	−	26.8989i	−0.510418	−	0.907277i
$880$		0			0
$881$			12.3038i			0.414524i	0.978285	+	0.207262i	$0.0664553\pi$
$881$			12.3038i			0.414524i	−0.978285	+	0.207262i	$0.933545\pi$
$882$		0			0
$883$		−	22.2964i		−	0.750332i	−0.926958	−	0.375166i	$-0.877586\pi$
$883$		−	22.2964i		−	0.750332i	0.926958	−	0.375166i	$-0.122414\pi$
$884$		0			0
$885$	1.27477	+	2.26592i	0.0428509	+	0.0761682i
$886$		0			0
$887$	−44.6596			−1.49952			−0.749762	−	0.661708i	$-0.769832\pi$
$887$	−44.6596			−1.49952			−0.749762	+	0.661708i	$0.769832\pi$
$888$		0			0
$889$	26.0169			0.872579
$890$		0			0
$891$	−18.0384	−	34.7322i	−0.604308	−	1.16357i
$892$		0			0
$893$			3.62677i			0.121365i
$894$		0			0
$895$		−	2.63985i		−	0.0882404i
$896$		0			0
$897$	13.5214	−	7.60692i	0.451467	−	0.253988i
$898$		0			0
$899$	−11.0165			−0.367420
$900$		0			0
$901$	−20.0537			−0.668086
$902$		0			0
$903$	−20.1672	+	11.3457i	−0.671122	+	0.377561i
$904$		0			0
$905$			2.24453i			0.0746108i
$906$		0			0
$907$		−	41.6382i		−	1.38257i	−0.722580	−	0.691287i	$-0.757044\pi$
$907$		−	41.6382i		−	1.38257i	0.722580	−	0.691287i	$-0.242956\pi$
$908$		0			0
$909$	43.2020	+	26.2438i	1.43292	+	0.870451i
$910$		0			0
$911$	−21.4962			−0.712201			−0.356101	−	0.934448i	$-0.615894\pi$
$911$	−21.4962			−0.712201			−0.356101	+	0.934448i	$0.615894\pi$
$912$		0			0
$913$	−74.2602			−2.45765
$914$		0			0
$915$	8.03381	+	14.2802i	0.265589	+	0.472090i
$916$		0			0
$917$		−	18.0396i		−	0.595720i
$918$		0			0
$919$			34.1951i			1.12799i	0.825777	+	0.563997i	$0.190737\pi$
$919$			34.1951i			1.12799i	−0.825777	+	0.563997i	$0.809263\pi$
$920$		0			0
$921$	−5.57656	−	9.91242i	−0.183754	−	0.326625i
$922$		0			0
$923$	−87.2531			−2.87197
$924$		0			0
$925$	45.2428			1.48757
$926$		0			0
$927$	−2.33039	−	1.41563i	−0.0765400	−	0.0464955i
$928$		0			0
$929$		−	32.7452i		−	1.07433i	−0.843476	−	0.537167i	$-0.819494\pi$
$929$		−	32.7452i		−	1.07433i	0.843476	−	0.537167i	$-0.180506\pi$
$930$		0			0
$931$			4.79490i			0.157147i
$932$		0			0
$933$	−0.307330	+	0.172898i	−0.0100615	+	0.00566044i
$934$		0			0
$935$	−24.7307			−0.808781
$936$		0			0
$937$	−25.1312			−0.821000			−0.410500	−	0.911861i	$-0.634646\pi$
$937$	−25.1312			−0.821000			−0.410500	+	0.911861i	$0.634646\pi$
$938$		0			0
$939$	−14.9093	+	8.38769i	−0.486545	+	0.273722i
$940$		0			0
$941$		−	26.4701i		−	0.862902i	−0.902136	−	0.431451i	$-0.858002\pi$
$941$		−	26.4701i		−	0.862902i	0.902136	−	0.431451i	$-0.141998\pi$
$942$		0			0
$943$		−	6.49371i		−	0.211464i
$944$		0			0
$945$	−0.199131	+	5.95269i	−0.00647775	+	0.193641i
$946$		0			0
$947$	−57.0169			−1.85280			−0.926400	−	0.376541i	$-0.877113\pi$
$947$	−57.0169			−1.85280			−0.926400	+	0.376541i	$0.877113\pi$
$948$		0			0
$949$	−26.8292			−0.870913
$950$		0			0
$951$	25.0688	+	44.5601i	0.812910	+	1.44496i
$952$		0			0
$953$			24.4039i			0.790519i	0.918570	+	0.395259i	$0.129345\pi$
$953$			24.4039i			0.790519i	−0.918570	+	0.395259i	$0.870655\pi$
$954$		0			0
$955$			13.0474i			0.422202i
$956$		0			0
$957$	−20.5950	−	36.6079i	−0.665741	−	1.18337i
$958$		0			0
$959$	19.9841			0.645321
$960$		0			0
$961$	27.0976			0.874117
$962$		0			0
$963$	5.67456	−	9.34134i	0.182860	−	0.301021i
$964$		0			0
$965$			3.37082i			0.108511i
$966$		0			0
$967$			47.3042i			1.52120i	0.649221	+	0.760600i	$0.275095\pi$
$967$			47.3042i			1.52120i	−0.649221	+	0.760600i	$0.724905\pi$
$968$		0			0
$969$	−11.1220	+	6.25702i	−0.357289	+	0.201004i
$970$		0			0
$971$	−19.7843			−0.634907			−0.317454	−	0.948274i	$-0.602828\pi$
$971$	−19.7843			−0.634907			−0.317454	+	0.948274i	$0.602828\pi$
$972$		0			0
$973$	−0.662922			−0.0212523
$974$		0			0
$975$	36.0344	−	20.2723i	1.15402	−	0.649234i
$976$		0			0
$977$			41.8422i			1.33865i	0.742970	+	0.669325i	$0.233417\pi$
$977$			41.8422i			1.33865i	−0.742970	+	0.669325i	$0.766583\pi$
$978$		0			0
$979$		−	21.5027i		−	0.687229i
$980$		0			0
$981$	−12.0717	+	19.8722i	−0.385421	+	0.634471i
$982$		0			0
$983$	−25.6659			−0.818614			−0.409307	−	0.912397i	$-0.634230\pi$
$983$	−25.6659			−0.818614			−0.409307	+	0.912397i	$0.634230\pi$
$984$		0			0
$985$	11.8544			0.377713
$986$		0			0
$987$	−4.57373	−	8.12989i	−0.145583	−	0.258777i
$988$		0			0
$989$		−	14.8679i		−	0.472772i
$990$		0			0
$991$			55.2997i			1.75665i	0.478061	+	0.878326i	$0.341340\pi$
$991$			55.2997i			1.75665i	−0.478061	+	0.878326i	$0.658660\pi$
$992$		0			0
$993$	22.9138	+	40.7296i	0.727147	+	1.29252i
$994$		0			0
$995$	−7.73137			−0.245101
$996$		0			0
$997$	0.146047			0.00462536			0.00231268	−	0.999997i	$-0.499264\pi$
$997$	0.146047			0.00462536			0.00231268	+	0.999997i	$0.499264\pi$
$998$		0			0
$999$	−1.78464	+	53.3487i	−0.0564634	+	1.68788i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.d.b.191.6	yes	24
3.2	odd	2	inner	912.2.d.b.191.20	yes	24
4.3	odd	2	inner	912.2.d.b.191.19	yes	24
12.11	even	2	inner	912.2.d.b.191.5	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.2.d.b.191.5	✓	24	12.11	even	2	inner
912.2.d.b.191.6	yes	24	1.1	even	1	trivial
912.2.d.b.191.19	yes	24	4.3	odd	2	inner
912.2.d.b.191.20	yes	24	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-1.50956 + 0.849252i) q^{3} -0.771899i q^{5} +1.48496i q^{7} +(1.55754 - 2.56399i) q^{9} +O(q^{10})\)	\(q+(-1.50956 + 0.849252i) q^{3} -0.771899i q^{5} +1.48496i q^{7} +(1.55754 - 2.56399i) q^{9} +4.34856 q^{11} -5.42004 q^{13} +(0.655537 + 1.16523i) q^{15} -7.36768i q^{17} +1.00000i q^{19} +(-1.26110 - 2.24163i) q^{21} +1.65261 q^{23} +4.40417 q^{25} +(-0.173726 + 5.19325i) q^{27} +5.57672i q^{29} +1.97544i q^{31} +(-6.56441 + 3.69302i) q^{33} +1.14624 q^{35} +10.2727 q^{37} +(8.18188 - 4.60298i) q^{39} -3.92937i q^{41} -8.99665i q^{43} +(-1.97914 - 1.20227i) q^{45} +3.62677 q^{47} +4.79490 q^{49} +(6.25702 + 11.1220i) q^{51} -2.72185i q^{53} -3.35665i q^{55} +(-0.849252 - 1.50956i) q^{57} +1.94462 q^{59} +12.2553 q^{61} +(3.80742 + 2.31288i) q^{63} +4.18372i q^{65} +10.8690i q^{67} +(-2.49471 + 1.40348i) q^{69} +16.0982 q^{71} +4.95000 q^{73} +(-6.64836 + 3.74025i) q^{75} +6.45743i q^{77} +3.32493i q^{79} +(-4.14813 - 7.98706i) q^{81} -17.0770 q^{83} -5.68711 q^{85} +(-4.73604 - 8.41839i) q^{87} -4.94478i q^{89} -8.04853i q^{91} +(-1.67765 - 2.98205i) q^{93} +0.771899 q^{95} +5.51162 q^{97} +(6.77307 - 11.1497i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{9}+O(q^{10}) \) 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

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