LMFDB - Embedded newform 912.2.d.b.191.15

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			191.15
Character	$\chi$	$=$	912.191
Dual form			912.2.d.b.191.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.881895 - 1.49073i) q^{3} -1.27921i q^{5} -0.100648i q^{7} +(-1.44452 - 2.62933i) q^{9} +O(q^{10})$	\(q+(0.881895 - 1.49073i) q^{3} -1.27921i q^{5} -0.100648i q^{7} +(-1.44452 - 2.62933i) q^{9} +3.88598 q^{11} +5.91860 q^{13} +(-1.90695 - 1.12813i) q^{15} +2.12219i q^{17} -1.00000i q^{19} +(-0.150039 - 0.0887611i) q^{21} -8.22016 q^{23} +3.36363 q^{25} +(-5.19352 - 0.165399i) q^{27} -7.33207i q^{29} -2.91462i q^{31} +(3.42702 - 5.79293i) q^{33} -0.128750 q^{35} -3.36227 q^{37} +(5.21958 - 8.82301i) q^{39} -0.0327171i q^{41} -7.23138i q^{43} +(-3.36345 + 1.84785i) q^{45} -7.65152 q^{47} +6.98987 q^{49} +(3.16360 + 1.87155i) q^{51} +13.7044i q^{53} -4.97097i q^{55} +(-1.49073 - 0.881895i) q^{57} -8.10338 q^{59} -4.04692 q^{61} +(-0.264637 + 0.145389i) q^{63} -7.57112i q^{65} +10.5995i q^{67} +(-7.24931 + 12.2540i) q^{69} +3.62697 q^{71} +9.76518 q^{73} +(2.96636 - 5.01424i) q^{75} -0.391117i q^{77} +0.457075i q^{79} +(-4.82670 + 7.59625i) q^{81} -2.00500 q^{83} +2.71472 q^{85} +(-10.9301 - 6.46611i) q^{87} -11.0372i q^{89} -0.595697i q^{91} +(-4.34490 - 2.57039i) q^{93} -1.27921 q^{95} +12.7987 q^{97} +(-5.61339 - 10.2175i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.881895	−	1.49073i	0.509162	−	0.860671i
$3$	0.881895	−	1.49073i	0.509162	−	0.860671i
$4$		0			0
$5$		−	1.27921i		−	0.572079i	−0.958218	−	0.286040i	$-0.907661\pi$
$5$		−	1.27921i		−	0.572079i	0.958218	−	0.286040i	$-0.0923389\pi$
$6$		0			0
$7$		−	0.100648i		−	0.0380414i	−0.999819	−	0.0190207i	$-0.993945\pi$
$7$		−	0.100648i		−	0.0380414i	0.999819	−	0.0190207i	$-0.00605485\pi$
$8$		0			0
$9$	−1.44452	−	2.62933i	−0.481508	−	0.876442i
$10$		0			0
$11$	3.88598			1.17167			0.585833	−	0.810432i	$-0.300767\pi$
$11$	3.88598			1.17167			0.585833	+	0.810432i	$0.300767\pi$
$12$		0			0
$13$	5.91860			1.64152			0.820762	−	0.571270i	$-0.193549\pi$
$13$	5.91860			1.64152			0.820762	+	0.571270i	$0.193549\pi$
$14$		0			0
$15$	−1.90695	−	1.12813i	−0.492372	−	0.291281i
$16$		0			0
$17$			2.12219i			0.514706i	0.966317	+	0.257353i	$0.0828504\pi$
$17$			2.12219i			0.514706i	−0.966317	+	0.257353i	$0.917150\pi$
$18$		0			0
$19$		−	1.00000i		−	0.229416i
$19$		−	1.00000i		−	0.229416i
$20$		0			0
$21$	−0.150039	−	0.0887611i	−0.0327411	−	0.0193693i
$22$		0			0
$23$	−8.22016			−1.71402			−0.857011	−	0.515299i	$-0.827681\pi$
$23$	−8.22016			−1.71402			−0.857011	+	0.515299i	$0.827681\pi$
$24$		0			0
$25$	3.36363			0.672725
$26$		0			0
$27$	−5.19352	−	0.165399i	−0.999493	−	0.0318310i
$28$		0			0
$29$		−	7.33207i		−	1.36153i	−0.732502	−	0.680765i	$-0.761647\pi$
$29$		−	7.33207i		−	1.36153i	0.732502	−	0.680765i	$-0.238353\pi$
$30$		0			0
$31$		−	2.91462i		−	0.523482i	−0.965138	−	0.261741i	$-0.915703\pi$
$31$		−	2.91462i		−	0.523482i	0.965138	−	0.261741i	$-0.0842966\pi$
$32$		0			0
$33$	3.42702	−	5.79293i	0.596568	−	1.00842i
$34$		0			0
$35$	−0.128750			−0.0217627
$36$		0			0
$37$	−3.36227			−0.552754			−0.276377	−	0.961049i	$-0.589134\pi$
$37$	−3.36227			−0.552754			−0.276377	+	0.961049i	$0.589134\pi$
$38$		0			0
$39$	5.21958	−	8.82301i	0.835802	−	1.41281i
$40$		0			0
$41$		−	0.0327171i		−	0.00510955i	−0.999997	−	0.00255478i	$-0.999187\pi$
$41$		−	0.0327171i		−	0.00510955i	0.999997	−	0.00255478i	$-0.000813211\pi$
$42$		0			0
$43$		−	7.23138i		−	1.10277i	−0.834249	−	0.551387i	$-0.814099\pi$
$43$		−	7.23138i		−	1.10277i	0.834249	−	0.551387i	$-0.185901\pi$
$44$		0			0
$45$	−3.36345	+	1.84785i	−0.501394	+	0.275461i
$46$		0			0
$47$	−7.65152			−1.11609			−0.558044	−	0.829811i	$-0.688448\pi$
$47$	−7.65152			−1.11609			−0.558044	+	0.829811i	$0.688448\pi$
$48$		0			0
$49$	6.98987			0.998553
$50$		0			0
$51$	3.16360	+	1.87155i	0.442993	+	0.262069i
$52$		0			0
$53$			13.7044i			1.88244i	0.337793	+	0.941221i	$0.390320\pi$
$53$			13.7044i			1.88244i	−0.337793	+	0.941221i	$0.609680\pi$
$54$		0			0
$55$		−	4.97097i		−	0.670286i
$56$		0			0
$57$	−1.49073	−	0.881895i	−0.197451	−	0.116810i
$58$		0			0
$59$	−8.10338			−1.05497			−0.527485	−	0.849564i	$-0.676865\pi$
$59$	−8.10338			−1.05497			−0.527485	+	0.849564i	$0.676865\pi$
$60$		0			0
$61$	−4.04692			−0.518155			−0.259078	−	0.965857i	$-0.583419\pi$
$61$	−4.04692			−0.518155			−0.259078	+	0.965857i	$0.583419\pi$
$62$		0			0
$63$	−0.264637	+	0.145389i	−0.0333411	+	0.0183173i
$64$		0			0
$65$		−	7.57112i		−	0.939082i
$66$		0			0
$67$			10.5995i			1.29494i	0.762093	+	0.647468i	$0.224172\pi$
$67$			10.5995i			1.29494i	−0.762093	+	0.647468i	$0.775828\pi$
$68$		0			0
$69$	−7.24931	+	12.2540i	−0.872715	+	1.47521i
$70$		0			0
$71$	3.62697			0.430442			0.215221	−	0.976565i	$-0.430953\pi$
$71$	3.62697			0.430442			0.215221	+	0.976565i	$0.430953\pi$
$72$		0			0
$73$	9.76518			1.14293			0.571464	−	0.820627i	$-0.306376\pi$
$73$	9.76518			1.14293			0.571464	+	0.820627i	$0.306376\pi$
$74$		0			0
$75$	2.96636	−	5.01424i	0.342526	−	0.578995i
$76$		0			0
$77$		−	0.391117i		−	0.0445719i
$78$		0			0
$79$			0.457075i			0.0514249i	0.999669	+	0.0257125i	$0.00818544\pi$
$79$			0.457075i			0.0514249i	−0.999669	+	0.0257125i	$0.991815\pi$
$80$		0			0
$81$	−4.82670	+	7.59625i	−0.536300	+	0.844027i
$82$		0			0
$83$	−2.00500			−0.220077			−0.110038	−	0.993927i	$-0.535097\pi$
$83$	−2.00500			−0.220077			−0.110038	+	0.993927i	$0.535097\pi$
$84$		0			0
$85$	2.71472			0.294453
$86$		0			0
$87$	−10.9301	−	6.46611i	−1.17183	−	0.693240i
$88$		0			0
$89$		−	11.0372i		−	1.16994i	−0.811056	−	0.584969i	$-0.801107\pi$
$89$		−	11.0372i		−	1.16994i	0.811056	−	0.584969i	$-0.198893\pi$
$90$		0			0
$91$		−	0.595697i		−	0.0624460i
$92$		0			0
$93$	−4.34490	−	2.57039i	−0.450546	−	0.266537i
$94$		0			0
$95$	−1.27921			−0.131244
$96$		0			0
$97$	12.7987			1.29951			0.649756	−	0.760143i	$-0.274871\pi$
$97$	12.7987			1.29951			0.649756	+	0.760143i	$0.274871\pi$
$98$		0			0
$99$	−5.61339	−	10.2175i	−0.564167	−	1.02690i
$100$		0			0
$101$			14.8354i			1.47618i	0.674702	+	0.738090i	$0.264272\pi$
$101$			14.8354i			1.47618i	−0.674702	+	0.738090i	$0.735728\pi$
$102$		0			0
$103$		−	0.792024i		−	0.0780405i	−0.999238	−	0.0390202i	$-0.987576\pi$
$103$		−	0.792024i		−	0.0780405i	0.999238	−	0.0390202i	$-0.0124237\pi$
$104$		0			0
$105$	−0.113544	+	0.191931i	−0.0110807	+	0.0187305i
$106$		0			0
$107$	14.6438			1.41567			0.707835	−	0.706378i	$-0.249672\pi$
$107$	14.6438			1.41567			0.707835	+	0.706378i	$0.249672\pi$
$108$		0			0
$109$	3.66235			0.350789			0.175395	−	0.984498i	$-0.443880\pi$
$109$	3.66235			0.350789			0.175395	+	0.984498i	$0.443880\pi$
$110$		0			0
$111$	−2.96517	+	5.01222i	−0.281441	+	0.475739i
$112$		0			0
$113$			1.19881i			0.112775i	0.998409	+	0.0563874i	$0.0179582\pi$
$113$			1.19881i			0.112775i	−0.998409	+	0.0563874i	$0.982042\pi$
$114$		0			0
$115$			10.5153i			0.980556i
$116$		0			0
$117$	−8.54956	−	15.5619i	−0.790407	−	1.43870i
$118$		0			0
$119$	0.213594			0.0195802
$120$		0			0
$121$	4.10082			0.372802
$122$		0			0
$123$	−0.0487722	−	0.0288530i	−0.00439764	−	0.00260159i
$124$		0			0
$125$		−	10.6988i		−	0.956931i
$126$		0			0
$127$		−	3.60447i		−	0.319845i	−0.987130	−	0.159923i	$-0.948876\pi$
$127$		−	3.60447i		−	0.319845i	0.987130	−	0.159923i	$-0.0511244\pi$
$128$		0			0
$129$	−10.7800	−	6.37731i	−0.949126	−	0.561491i
$130$		0			0
$131$	13.5235			1.18155			0.590777	−	0.806835i	$-0.298821\pi$
$131$	13.5235			1.18155			0.590777	+	0.806835i	$0.298821\pi$
$132$		0			0
$133$	−0.100648			−0.00872730
$134$		0			0
$135$	−0.211579	+	6.64359i	−0.0182099	+	0.571789i
$136$		0			0
$137$			1.79109i			0.153024i	0.997069	+	0.0765118i	$0.0243783\pi$
$137$			1.79109i			0.153024i	−0.997069	+	0.0765118i	$0.975622\pi$
$138$		0			0
$139$			8.56873i			0.726790i	0.931635	+	0.363395i	$0.118382\pi$
$139$			8.56873i			0.726790i	−0.931635	+	0.363395i	$0.881618\pi$
$140$		0			0
$141$	−6.74783	+	11.4063i	−0.568270	+	0.960585i
$142$		0			0
$143$	22.9996			1.92332
$144$		0			0
$145$	−9.37924			−0.778903
$146$		0			0
$147$	6.16433	−	10.4200i	0.508425	−	0.859425i
$148$		0			0
$149$			3.12815i			0.256268i	0.991757	+	0.128134i	$0.0408987\pi$
$149$			3.12815i			0.256268i	−0.991757	+	0.128134i	$0.959101\pi$
$150$		0			0
$151$			22.4355i			1.82577i	0.408214	+	0.912886i	$0.366152\pi$
$151$			22.4355i			1.82577i	−0.408214	+	0.912886i	$0.633848\pi$
$152$		0			0
$153$	5.57992	−	3.06555i	0.451110	−	0.247835i
$154$		0			0
$155$	−3.72841			−0.299473
$156$		0			0
$157$	1.61390			0.128803			0.0644016	−	0.997924i	$-0.479486\pi$
$157$	1.61390			0.128803			0.0644016	+	0.997924i	$0.479486\pi$
$158$		0			0
$159$	20.4295	+	12.0858i	1.62016	+	0.958468i
$160$		0			0
$161$			0.827344i			0.0652038i
$162$		0			0
$163$			24.3665i			1.90854i	0.298953	+	0.954268i	$0.403362\pi$
$163$			24.3665i			1.90854i	−0.298953	+	0.954268i	$0.596638\pi$
$164$		0			0
$165$	−7.41036	−	4.38387i	−0.576895	−	0.341284i
$166$		0			0
$167$	−4.30981			−0.333503			−0.166752	−	0.985999i	$-0.553328\pi$
$167$	−4.30981			−0.333503			−0.166752	+	0.985999i	$0.553328\pi$
$168$		0			0
$169$	22.0299			1.69460
$170$		0			0
$171$	−2.62933	+	1.44452i	−0.201070	+	0.110466i
$172$		0			0
$173$		−	15.6581i		−	1.19046i	−0.803555	−	0.595231i	$-0.797061\pi$
$173$		−	15.6581i		−	1.19046i	0.803555	−	0.595231i	$-0.202939\pi$
$174$		0			0
$175$		−	0.338543i		−	0.0255914i
$176$		0			0
$177$	−7.14633	+	12.0799i	−0.537151	+	0.907982i
$178$		0			0
$179$	−7.30876			−0.546282			−0.273141	−	0.961974i	$-0.588063\pi$
$179$	−7.30876			−0.546282			−0.273141	+	0.961974i	$0.588063\pi$
$180$		0			0
$181$	−11.6189			−0.863624			−0.431812	−	0.901964i	$-0.642126\pi$
$181$	−11.6189			−0.863624			−0.431812	+	0.901964i	$0.642126\pi$
$182$		0			0
$183$	−3.56896	+	6.03285i	−0.263825	+	0.445961i
$184$		0			0
$185$			4.30104i			0.316219i
$186$		0			0
$187$			8.24678i			0.603064i
$188$		0			0
$189$	−0.0166471	+	0.522718i	−0.00121090	+	0.0380222i
$190$		0			0
$191$	−7.90448			−0.571948			−0.285974	−	0.958237i	$-0.592317\pi$
$191$	−7.90448			−0.571948			−0.285974	+	0.958237i	$0.592317\pi$
$192$		0			0
$193$	−4.16420			−0.299746			−0.149873	−	0.988705i	$-0.547886\pi$
$193$	−4.16420			−0.299746			−0.149873	+	0.988705i	$0.547886\pi$
$194$		0			0
$195$	−11.2865	−	6.67693i	−0.808241	−	0.478145i
$196$		0			0
$197$			9.08053i			0.646961i	0.946235	+	0.323480i	$0.104853\pi$
$197$			9.08053i			0.646961i	−0.946235	+	0.323480i	$0.895147\pi$
$198$		0			0
$199$		−	11.4253i		−	0.809921i	−0.914334	−	0.404961i	$-0.867285\pi$
$199$		−	11.4253i		−	0.809921i	0.914334	−	0.404961i	$-0.132715\pi$
$200$		0			0
$201$	15.8009	+	9.34764i	1.11451	+	0.659332i
$202$		0			0
$203$	−0.737959			−0.0517946
$204$		0			0
$205$	−0.0418520			−0.00292307
$206$		0			0
$207$	11.8742	+	21.6135i	0.825315	+	1.50224i
$208$		0			0
$209$		−	3.88598i		−	0.268799i
$210$		0			0
$211$		−	5.72904i		−	0.394403i	−0.980363	−	0.197202i	$-0.936815\pi$
$211$		−	5.72904i		−	0.394403i	0.980363	−	0.197202i	$-0.0631854\pi$
$212$		0			0
$213$	3.19860	−	5.40681i	0.219165	−	0.370469i
$214$		0			0
$215$	−9.25044			−0.630875
$216$		0			0
$217$	−0.293352			−0.0199140
$218$		0			0
$219$	8.61186	−	14.5572i	0.581936	−	0.983685i
$220$		0			0
$221$			12.5604i			0.844903i
$222$		0			0
$223$			4.08122i			0.273299i	0.990619	+	0.136649i	$0.0436334\pi$
$223$			4.08122i			0.273299i	−0.990619	+	0.136649i	$0.956367\pi$
$224$		0			0
$225$	−4.85884	−	8.84407i	−0.323923	−	0.589605i
$226$		0			0
$227$	19.8827			1.31966			0.659832	−	0.751413i	$-0.270627\pi$
$227$	19.8827			1.31966			0.659832	+	0.751413i	$0.270627\pi$
$228$		0			0
$229$	−20.5894			−1.36058			−0.680292	−	0.732941i	$-0.738147\pi$
$229$	−20.5894			−1.36058			−0.680292	+	0.732941i	$0.738147\pi$
$230$		0			0
$231$	−0.583047	−	0.344924i	−0.0383617	−	0.0226943i
$232$		0			0
$233$		−	27.7105i		−	1.81537i	−0.419647	−	0.907687i	$-0.637846\pi$
$233$		−	27.7105i		−	1.81537i	0.419647	−	0.907687i	$-0.362154\pi$
$234$		0			0
$235$			9.78788i			0.638491i
$236$		0			0
$237$	0.681373	+	0.403092i	0.0442599	+	0.0261836i
$238$		0			0
$239$	0.747254			0.0483358			0.0241679	−	0.999708i	$-0.492306\pi$
$239$	0.747254			0.0483358			0.0241679	+	0.999708i	$0.492306\pi$
$240$		0			0
$241$	−4.69019			−0.302122			−0.151061	−	0.988524i	$-0.548269\pi$
$241$	−4.69019			−0.302122			−0.151061	+	0.988524i	$0.548269\pi$
$242$		0			0
$243$	7.06728	+	13.8944i	0.453366	+	0.891324i
$244$		0			0
$245$		−	8.94150i		−	0.571251i
$246$		0			0
$247$		−	5.91860i		−	0.376592i
$248$		0			0
$249$	−1.76819	+	2.98890i	−0.112055	+	0.189414i
$250$		0			0
$251$	−16.1795			−1.02124			−0.510619	−	0.859807i	$-0.670584\pi$
$251$	−16.1795			−1.02124			−0.510619	+	0.859807i	$0.670584\pi$
$252$		0			0
$253$	−31.9433			−2.00826
$254$		0			0
$255$	2.39410	−	4.04690i	0.149924	−	0.253427i
$256$		0			0
$257$			22.7206i			1.41727i	0.705575	+	0.708635i	$0.250689\pi$
$257$			22.7206i			1.41727i	−0.705575	+	0.708635i	$0.749311\pi$
$258$		0			0
$259$			0.338407i			0.0210276i
$260$		0			0
$261$	−19.2784	+	10.5913i	−1.19330	+	0.655588i
$262$		0			0
$263$	27.1674			1.67522			0.837608	−	0.546272i	$-0.183953\pi$
$263$	27.1674			1.67522			0.837608	+	0.546272i	$0.183953\pi$
$264$		0			0
$265$	17.5307			1.07691
$266$		0			0
$267$	−16.4534	−	9.73362i	−1.00693	−	0.595688i
$268$		0			0
$269$			19.8337i			1.20928i	0.796497	+	0.604642i	$0.206684\pi$
$269$			19.8337i			1.20928i	−0.796497	+	0.604642i	$0.793316\pi$
$270$		0			0
$271$			2.19563i			0.133375i	0.997774	+	0.0666874i	$0.0212430\pi$
$271$			2.19563i			0.133375i	−0.997774	+	0.0666874i	$0.978757\pi$
$272$		0			0
$273$	−0.888020	−	0.525342i	−0.0537454	−	0.0317951i
$274$		0			0
$275$	13.0710			0.788210
$276$		0			0
$277$	12.2605			0.736663			0.368332	−	0.929694i	$-0.379929\pi$
$277$	12.2605			0.736663			0.368332	+	0.929694i	$0.379929\pi$
$278$		0			0
$279$	−7.66349	+	4.21024i	−0.458801	+	0.252061i
$280$		0			0
$281$			20.4798i			1.22172i	0.791737	+	0.610862i	$0.209177\pi$
$281$			20.4798i			1.22172i	−0.791737	+	0.610862i	$0.790823\pi$
$282$		0			0
$283$			22.7443i			1.35201i	0.736899	+	0.676003i	$0.236289\pi$
$283$			22.7443i			1.35201i	−0.736899	+	0.676003i	$0.763711\pi$
$284$		0			0
$285$	−1.12813	+	1.90695i	−0.0668244	+	0.112958i
$286$		0			0
$287$	−0.00329292			−0.000194375
$288$		0			0
$289$	12.4963			0.735077
$290$		0			0
$291$	11.2871	−	19.0793i	0.661662	−	1.11845i
$292$		0			0
$293$			14.2431i			0.832088i	0.909344	+	0.416044i	$0.136584\pi$
$293$			14.2431i			0.832088i	−0.909344	+	0.416044i	$0.863416\pi$
$294$		0			0
$295$			10.3659i			0.603527i
$296$		0			0
$297$	−20.1819	−	0.642736i	−1.17107	−	0.0372953i
$298$		0			0
$299$	−48.6518			−2.81361
$300$		0			0
$301$	−0.727825			−0.0419511
$302$		0			0
$303$	22.1155	+	13.0833i	1.27050	+	0.751615i
$304$		0			0
$305$			5.17685i			0.296426i
$306$		0			0
$307$		−	4.53966i		−	0.259092i	−0.991573	−	0.129546i	$-0.958648\pi$
$307$		−	4.53966i		−	0.259092i	0.991573	−	0.129546i	$-0.0413520\pi$
$308$		0			0
$309$	−1.18069	−	0.698482i	−0.0671671	−	0.0397352i
$310$		0			0
$311$	13.1346			0.744797			0.372399	−	0.928073i	$-0.378535\pi$
$311$	13.1346			0.744797			0.372399	+	0.928073i	$0.378535\pi$
$312$		0			0
$313$	6.78945			0.383763			0.191881	−	0.981418i	$-0.438541\pi$
$313$	6.78945			0.383763			0.191881	+	0.981418i	$0.438541\pi$
$314$		0			0
$315$	0.185982	+	0.338525i	0.0104789	+	0.0190737i
$316$		0			0
$317$			26.6244i			1.49538i	0.664049	+	0.747689i	$0.268836\pi$
$317$			26.6244i			1.49538i	−0.664049	+	0.747689i	$0.731164\pi$
$318$		0			0
$319$		−	28.4922i		−	1.59526i
$320$		0			0
$321$	12.9143	−	21.8299i	0.720806	−	1.21843i
$322$		0			0
$323$	2.12219			0.118082
$324$		0			0
$325$	19.9080			1.10430
$326$		0			0
$327$	3.22981	−	5.45956i	0.178609	−	0.301914i
$328$		0			0
$329$			0.770111i			0.0424576i
$330$		0			0
$331$		−	14.4504i		−	0.794266i	−0.917761	−	0.397133i	$-0.870005\pi$
$331$		−	14.4504i		−	0.794266i	0.917761	−	0.397133i	$-0.129995\pi$
$332$		0			0
$333$	4.85688	+	8.84051i	0.266156	+	0.484457i
$334$		0			0
$335$	13.5590			0.740805
$336$		0			0
$337$	−35.6195			−1.94032			−0.970158	−	0.242475i	$-0.922041\pi$
$337$	−35.6195			−1.94032			−0.970158	+	0.242475i	$0.922041\pi$
$338$		0			0
$339$	1.78710	+	1.05723i	0.0970619	+	0.0574206i
$340$		0			0
$341$		−	11.3262i		−	0.613346i
$342$		0			0
$343$		−	1.40805i		−	0.0760278i
$344$		0			0
$345$	15.6754	+	9.27338i	0.843936	+	0.499262i
$346$		0			0
$347$	−25.6301			−1.37589			−0.687947	−	0.725761i	$-0.741488\pi$
$347$	−25.6301			−1.37589			−0.687947	+	0.725761i	$0.741488\pi$
$348$		0			0
$349$	27.0648			1.44874			0.724372	−	0.689409i	$-0.242130\pi$
$349$	27.0648			1.44874			0.724372	+	0.689409i	$0.242130\pi$
$350$		0			0
$351$	−30.7384	−	0.978930i	−1.64069	−	0.0522514i
$352$		0			0
$353$		−	4.40282i		−	0.234339i	−0.993112	−	0.117169i	$-0.962618\pi$
$353$		−	4.40282i		−	0.234339i	0.993112	−	0.117169i	$-0.0373820\pi$
$354$		0			0
$355$		−	4.63964i		−	0.246247i
$356$		0			0
$357$	0.188368	−	0.318411i	0.00996948	−	0.0168521i
$358$		0			0
$359$	−19.0650			−1.00621			−0.503106	−	0.864225i	$-0.667809\pi$
$359$	−19.0650			−1.00621			−0.503106	+	0.864225i	$0.667809\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$		0			0
$363$	3.61649	−	6.11320i	0.189817	−	0.320860i
$364$		0			0
$365$		−	12.4917i		−	0.653845i
$366$		0			0
$367$			0.163503i			0.00853479i	0.999991	+	0.00426740i	$0.00135836\pi$
$367$			0.163503i			0.00853479i	−0.999991	+	0.00426740i	$0.998642\pi$
$368$		0			0
$369$	−0.0860239	+	0.0472606i	−0.00447822	+	0.00246029i
$370$		0			0
$371$	1.37932			0.0716108
$372$		0			0
$373$	−32.2631			−1.67052			−0.835260	−	0.549855i	$-0.814683\pi$
$373$	−32.2631			−1.67052			−0.835260	+	0.549855i	$0.814683\pi$
$374$		0			0
$375$	−15.9490	−	9.43523i	−0.823603	−	0.487233i
$376$		0			0
$377$		−	43.3956i		−	2.23499i
$378$		0			0
$379$		−	5.35650i		−	0.275145i	−0.990492	−	0.137572i	$-0.956070\pi$
$379$		−	5.35650i		−	0.275145i	0.990492	−	0.137572i	$-0.0439300\pi$
$380$		0			0
$381$	−5.37328	−	3.17876i	−0.275281	−	0.162853i
$382$		0			0
$383$	15.2815			0.780850			0.390425	−	0.920635i	$-0.372328\pi$
$383$	15.2815			0.780850			0.390425	+	0.920635i	$0.372328\pi$
$384$		0			0
$385$	−0.500319			−0.0254986
$386$		0			0
$387$	−19.0136	+	10.4459i	−0.966518	+	0.530995i
$388$		0			0
$389$		−	15.8728i		−	0.804781i	−0.915468	−	0.402391i	$-0.868179\pi$
$389$		−	15.8728i		−	0.804781i	0.915468	−	0.402391i	$-0.131821\pi$
$390$		0			0
$391$		−	17.4447i		−	0.882218i
$392$		0			0
$393$	11.9263	−	20.1598i	0.601602	−	1.01693i
$394$		0			0
$395$	0.584694			0.0294191
$396$		0			0
$397$	36.1647			1.81505			0.907527	−	0.419994i	$-0.137968\pi$
$397$	36.1647			1.81505			0.907527	+	0.419994i	$0.137968\pi$
$398$		0			0
$399$	−0.0887611	+	0.150039i	−0.00444361	+	0.00751133i
$400$		0			0
$401$		−	21.9001i		−	1.09364i	−0.837250	−	0.546820i	$-0.815838\pi$
$401$		−	21.9001i		−	1.09364i	0.837250	−	0.546820i	$-0.184162\pi$
$402$		0			0
$403$		−	17.2505i		−	0.859309i
$404$		0			0
$405$	9.71718	+	6.17435i	0.482850	+	0.306806i
$406$		0			0
$407$	−13.0657			−0.647643
$408$		0			0
$409$	−16.7238			−0.826939			−0.413470	−	0.910518i	$-0.635683\pi$
$409$	−16.7238			−0.826939			−0.413470	+	0.910518i	$0.635683\pi$
$410$		0			0
$411$	2.67003	+	1.57956i	0.131703	+	0.0779138i
$412$		0			0
$413$			0.815591i			0.0401326i
$414$		0			0
$415$			2.56481i			0.125901i
$416$		0			0
$417$	12.7736	+	7.55671i	0.625527	+	0.370054i
$418$		0			0
$419$	2.79539			0.136564			0.0682818	−	0.997666i	$-0.478248\pi$
$419$	2.79539			0.136564			0.0682818	+	0.997666i	$0.478248\pi$
$420$		0			0
$421$	−1.57745			−0.0768802			−0.0384401	−	0.999261i	$-0.512239\pi$
$421$	−1.57745			−0.0768802			−0.0384401	+	0.999261i	$0.512239\pi$
$422$		0			0
$423$	11.0528	+	20.1183i	0.537406	+	0.978187i
$424$		0			0
$425$			7.13825i			0.346256i
$426$		0			0
$427$			0.407315i			0.0197114i
$428$		0			0
$429$	20.2832	−	34.2860i	0.979281	−	1.65534i
$430$		0			0
$431$	−14.9744			−0.721291			−0.360646	−	0.932703i	$-0.617444\pi$
$431$	−14.9744			−0.721291			−0.360646	+	0.932703i	$0.617444\pi$
$432$		0			0
$433$	−24.0511			−1.15582			−0.577912	−	0.816099i	$-0.696132\pi$
$433$	−24.0511			−1.15582			−0.577912	+	0.816099i	$0.696132\pi$
$434$		0			0
$435$	−8.27150	+	13.9819i	−0.396588	+	0.670379i
$436$		0			0
$437$			8.22016i			0.393223i
$438$		0			0
$439$			1.70617i			0.0814312i	0.999171	+	0.0407156i	$0.0129638\pi$
$439$			1.70617i			0.0814312i	−0.999171	+	0.0407156i	$0.987036\pi$
$440$		0			0
$441$	−10.0970	−	18.3786i	−0.480811	−	0.875173i
$442$		0			0
$443$	32.0303			1.52181			0.760904	−	0.648865i	$-0.224756\pi$
$443$	32.0303			1.52181			0.760904	+	0.648865i	$0.224756\pi$
$444$		0			0
$445$	−14.1188			−0.669297
$446$		0			0
$447$	4.66321	+	2.75870i	0.220562	+	0.130482i
$448$		0			0
$449$			28.6602i			1.35256i	0.736645	+	0.676279i	$0.236409\pi$
$449$			28.6602i			1.35256i	−0.736645	+	0.676279i	$0.763591\pi$
$450$		0			0
$451$		−	0.127138i		−	0.00598669i
$452$		0			0
$453$	33.4451	+	19.7857i	1.57139	+	0.929614i
$454$		0			0
$455$	−0.762020			−0.0357240
$456$		0			0
$457$	6.38873			0.298852			0.149426	−	0.988773i	$-0.452257\pi$
$457$	6.38873			0.298852			0.149426	+	0.988773i	$0.452257\pi$
$458$		0			0
$459$	0.351007	−	11.0216i	0.0163836	−	0.514445i
$460$		0			0
$461$		−	8.43191i		−	0.392713i	−0.980533	−	0.196357i	$-0.937089\pi$
$461$		−	8.43191i		−	0.392713i	0.980533	−	0.196357i	$-0.0629110\pi$
$462$		0			0
$463$		−	16.9362i		−	0.787094i	−0.919305	−	0.393547i	$-0.871248\pi$
$463$		−	16.9362i		−	0.787094i	0.919305	−	0.393547i	$-0.128752\pi$
$464$		0			0
$465$	−3.28806	+	5.55804i	−0.152480	+	0.257748i
$466$		0			0
$467$	13.0664			0.604640			0.302320	−	0.953206i	$-0.402239\pi$
$467$	13.0664			0.604640			0.302320	+	0.953206i	$0.402239\pi$
$468$		0			0
$469$	1.06682			0.0492612
$470$		0			0
$471$	1.42329	−	2.40588i	0.0655817	−	0.110857i
$472$		0			0
$473$		−	28.1010i		−	1.29208i
$474$		0			0
$475$		−	3.36363i		−	0.154334i
$476$		0			0
$477$	36.0333	−	19.7963i	1.64985	−	0.906410i
$478$		0			0
$479$	−7.32913			−0.334876			−0.167438	−	0.985883i	$-0.553549\pi$
$479$	−7.32913			−0.334876			−0.167438	+	0.985883i	$0.553549\pi$
$480$		0			0
$481$	−19.8999			−0.907360
$482$		0			0
$483$	1.23334	+	0.729630i	0.0561190	+	0.0331993i
$484$		0			0
$485$		−	16.3722i		−	0.743423i
$486$		0			0
$487$		−	26.6161i		−	1.20609i	−0.797707	−	0.603045i	$-0.793954\pi$
$487$		−	26.6161i		−	1.20609i	0.797707	−	0.603045i	$-0.206046\pi$
$488$		0			0
$489$	36.3238	+	21.4887i	1.64262	+	0.971754i
$490$		0			0
$491$	−9.68024			−0.436863			−0.218432	−	0.975852i	$-0.570094\pi$
$491$	−9.68024			−0.436863			−0.218432	+	0.975852i	$0.570094\pi$
$492$		0			0
$493$	15.5600			0.700788
$494$		0			0
$495$	−13.0703	+	7.18069i	−0.587467	+	0.322748i
$496$		0			0
$497$		−	0.365048i		−	0.0163746i
$498$		0			0
$499$		−	38.0029i		−	1.70124i	−0.525779	−	0.850621i	$-0.676226\pi$
$499$		−	38.0029i		−	1.70124i	0.525779	−	0.850621i	$-0.323774\pi$
$500$		0			0
$501$	−3.80080	+	6.42474i	−0.169807	+	0.287036i
$502$		0			0
$503$	−8.85823			−0.394969			−0.197484	−	0.980306i	$-0.563277\pi$
$503$	−8.85823			−0.394969			−0.197484	+	0.980306i	$0.563277\pi$
$504$		0			0
$505$	18.9776			0.844492
$506$		0			0
$507$	19.4280	−	32.8405i	0.862828	−	1.45850i
$508$		0			0
$509$		−	32.7646i		−	1.45227i	−0.687554	−	0.726133i	$-0.741316\pi$
$509$		−	32.7646i		−	1.45227i	0.687554	−	0.726133i	$-0.258684\pi$
$510$		0			0
$511$		−	0.982848i		−	0.0434786i
$512$		0			0
$513$	−0.165399	+	5.19352i	−0.00730253	+	0.229299i
$514$		0			0
$515$	−1.01316			−0.0446453
$516$		0			0
$517$	−29.7336			−1.30768
$518$		0			0
$519$	−23.3419	−	13.8088i	−1.02460	−	0.606138i
$520$		0			0
$521$			29.7576i			1.30370i	0.758347	+	0.651851i	$0.226007\pi$
$521$			29.7576i			1.30370i	−0.758347	+	0.651851i	$0.773993\pi$
$522$		0			0
$523$			6.17821i			0.270154i	0.990835	+	0.135077i	$0.0431282\pi$
$523$			6.17821i			0.270154i	−0.990835	+	0.135077i	$0.956872\pi$
$524$		0			0
$525$	−0.504675	−	0.298559i	−0.0220258	−	0.0130302i
$526$		0			0
$527$	6.18538			0.269439
$528$		0			0
$529$	44.5710			1.93787
$530$		0			0
$531$	11.7055	+	21.3064i	0.507977	+	0.924620i
$532$		0			0
$533$		−	0.193639i		−	0.00838746i
$534$		0			0
$535$		−	18.7325i		−	0.809876i
$536$		0			0
$537$	−6.44555	+	10.8953i	−0.278146	+	0.470169i
$538$		0			0
$539$	27.1625			1.16997
$540$		0			0
$541$	2.46051			0.105786			0.0528928	−	0.998600i	$-0.483156\pi$
$541$	2.46051			0.105786			0.0528928	+	0.998600i	$0.483156\pi$
$542$		0			0
$543$	−10.2466	+	17.3205i	−0.439725	+	0.743296i
$544$		0			0
$545$		−	4.68491i		−	0.200679i
$546$		0			0
$547$			3.86440i			0.165230i	0.996582	+	0.0826149i	$0.0263272\pi$
$547$			3.86440i			0.165230i	−0.996582	+	0.0826149i	$0.973673\pi$
$548$		0			0
$549$	5.84588	+	10.6407i	0.249496	+	0.454133i
$550$		0			0
$551$	−7.33207			−0.312357
$552$		0			0
$553$	0.0460038			0.00195628
$554$		0			0
$555$	6.41168	+	3.79307i	0.272161	+	0.161007i
$556$		0			0
$557$		−	9.99554i		−	0.423525i	−0.977321	−	0.211762i	$-0.932080\pi$
$557$		−	9.99554i		−	0.423525i	0.977321	−	0.211762i	$-0.0679203\pi$
$558$		0			0
$559$		−	42.7997i		−	1.81023i
$560$		0			0
$561$	12.2937	+	7.27279i	0.519040	+	0.307057i
$562$		0			0
$563$	−41.7233			−1.75843			−0.879214	−	0.476427i	$-0.841932\pi$
$563$	−41.7233			−1.75843			−0.879214	+	0.476427i	$0.841932\pi$
$564$		0			0
$565$	1.53353			0.0645161
$566$		0			0
$567$	0.764548	+	0.485799i	0.0321080	+	0.0204016i
$568$		0			0
$569$		−	13.2833i		−	0.556865i	−0.960456	−	0.278433i	$-0.910185\pi$
$569$		−	13.2833i		−	0.556865i	0.960456	−	0.278433i	$-0.0898149\pi$
$570$		0			0
$571$		−	18.5260i		−	0.775290i	−0.921809	−	0.387645i	$-0.873289\pi$
$571$		−	18.5260i		−	0.775290i	0.921809	−	0.387645i	$-0.126711\pi$
$572$		0			0
$573$	−6.97092	+	11.7834i	−0.291214	+	0.492259i
$574$		0			0
$575$	−27.6495			−1.15307
$576$		0			0
$577$	−32.6787			−1.36043			−0.680216	−	0.733011i	$-0.738114\pi$
$577$	−32.6787			−1.36043			−0.680216	+	0.733011i	$0.738114\pi$
$578$		0			0
$579$	−3.67238	+	6.20768i	−0.152619	+	0.257982i
$580$		0			0
$581$			0.201799i			0.00837204i
$582$		0			0
$583$			53.2549i			2.20559i
$584$		0			0
$585$	−19.9069	+	10.9367i	−0.823051	+	0.452176i
$586$		0			0
$587$	34.8959			1.44031			0.720154	−	0.693814i	$-0.244071\pi$
$587$	34.8959			1.44031			0.720154	+	0.693814i	$0.244071\pi$
$588$		0			0
$589$	−2.91462			−0.120095
$590$		0			0
$591$	13.5366	+	8.00807i	0.556820	+	0.329408i
$592$		0			0
$593$			5.91033i			0.242708i	0.992609	+	0.121354i	$0.0387237\pi$
$593$			5.91033i			0.242708i	−0.992609	+	0.121354i	$0.961276\pi$
$594$		0			0
$595$		−	0.273232i		−	0.0112014i
$596$		0			0
$597$	−17.0321	−	10.0760i	−0.697076	−	0.412381i
$598$		0			0
$599$	−7.56236			−0.308990			−0.154495	−	0.987994i	$-0.549375\pi$
$599$	−7.56236			−0.308990			−0.154495	+	0.987994i	$0.549375\pi$
$600$		0			0
$601$	−20.9123			−0.853033			−0.426516	−	0.904480i	$-0.640259\pi$
$601$	−20.9123			−0.853033			−0.426516	+	0.904480i	$0.640259\pi$
$602$		0			0
$603$	27.8695	−	15.3112i	1.13494	−	0.623522i
$604$		0			0
$605$		−	5.24580i		−	0.213272i
$606$		0			0
$607$		−	38.7045i		−	1.57097i	−0.618882	−	0.785484i	$-0.712414\pi$
$607$		−	38.7045i		−	1.57097i	0.618882	−	0.785484i	$-0.287586\pi$
$608$		0			0
$609$	−0.650802	+	1.10009i	−0.0263718	+	0.0445781i
$610$		0			0
$611$	−45.2863			−1.83209
$612$		0			0
$613$	−13.5425			−0.546978			−0.273489	−	0.961875i	$-0.588178\pi$
$613$	−13.5425			−0.546978			−0.273489	+	0.961875i	$0.588178\pi$
$614$		0			0
$615$	−0.0369090	+	0.0623898i	−0.00148832	+	0.00251580i
$616$		0			0
$617$			28.7699i			1.15823i	0.815245	+	0.579116i	$0.196602\pi$
$617$			28.7699i			1.15823i	−0.815245	+	0.579116i	$0.803398\pi$
$618$		0			0
$619$			34.2134i			1.37515i	0.726113	+	0.687576i	$0.241325\pi$
$619$			34.2134i			1.37515i	−0.726113	+	0.687576i	$0.758675\pi$
$620$		0			0
$621$	42.6915	+	1.35960i	1.71315	+	0.0545590i
$622$		0			0
$623$	−1.11087			−0.0445061
$624$		0			0
$625$	3.13212			0.125285
$626$		0			0
$627$	−5.79293	−	3.42702i	−0.231347	−	0.136862i
$628$		0			0
$629$		−	7.13537i		−	0.284506i
$630$		0			0
$631$		−	36.4764i		−	1.45210i	−0.687641	−	0.726051i	$-0.741354\pi$
$631$		−	36.4764i		−	1.45210i	0.687641	−	0.726051i	$-0.258646\pi$
$632$		0			0
$633$	−8.54042	−	5.05241i	−0.339451	−	0.200815i
$634$		0			0
$635$	−4.61087			−0.182977
$636$		0			0
$637$	41.3703			1.63915
$638$		0			0
$639$	−5.23924	−	9.53647i	−0.207261	−	0.377257i
$640$		0			0
$641$		−	14.6502i		−	0.578648i	−0.957231	−	0.289324i	$-0.906570\pi$
$641$		−	14.6502i		−	0.578648i	0.957231	−	0.289324i	$-0.0934304\pi$
$642$		0			0
$643$			38.3736i			1.51331i	0.653816	+	0.756654i	$0.273167\pi$
$643$			38.3736i			1.51331i	−0.653816	+	0.756654i	$0.726833\pi$
$644$		0			0
$645$	−8.15791	+	13.7899i	−0.321217	+	0.542975i
$646$		0			0
$647$	22.2148			0.873356			0.436678	−	0.899618i	$-0.356155\pi$
$647$	22.2148			0.873356			0.436678	+	0.899618i	$0.356155\pi$
$648$		0			0
$649$	−31.4896			−1.23607
$650$		0			0
$651$	−0.258705	+	0.437307i	−0.0101395	+	0.0171394i
$652$		0			0
$653$		−	24.2260i		−	0.948036i	−0.880515	−	0.474018i	$-0.842803\pi$
$653$		−	24.2260i		−	0.948036i	0.880515	−	0.474018i	$-0.157197\pi$
$654$		0			0
$655$		−	17.2994i		−	0.675942i
$656$		0			0
$657$	−14.1060	−	25.6758i	−0.550329	−	1.00171i
$658$		0			0
$659$	6.39874			0.249259			0.124630	−	0.992203i	$-0.460226\pi$
$659$	6.39874			0.249259			0.124630	+	0.992203i	$0.460226\pi$
$660$		0			0
$661$	−35.9440			−1.39806			−0.699029	−	0.715093i	$-0.746384\pi$
$661$	−35.9440			−1.39806			−0.699029	+	0.715093i	$0.746384\pi$
$662$		0			0
$663$	18.7241	+	11.0769i	0.727183	+	0.430193i
$664$		0			0
$665$			0.128750i			0.00499271i
$666$		0			0
$667$			60.2707i			2.33369i
$668$		0			0
$669$	6.08398	+	3.59921i	0.235220	+	0.139153i
$670$		0			0
$671$	−15.7262			−0.607105
$672$		0			0
$673$	−31.4428			−1.21203			−0.606016	−	0.795452i	$-0.707233\pi$
$673$	−31.4428			−1.21203			−0.606016	+	0.795452i	$0.707233\pi$
$674$		0			0
$675$	−17.4691	−	0.556340i	−0.672385	−	0.0214135i
$676$		0			0
$677$			20.4484i			0.785894i	0.919561	+	0.392947i	$0.128544\pi$
$677$			20.4484i			0.785894i	−0.919561	+	0.392947i	$0.871456\pi$
$678$		0			0
$679$		−	1.28817i		−	0.0494353i
$680$		0			0
$681$	17.5345	−	29.6397i	0.671923	−	1.13580i
$682$		0			0
$683$	−11.1038			−0.424877			−0.212438	−	0.977174i	$-0.568140\pi$
$683$	−11.1038			−0.424877			−0.212438	+	0.977174i	$0.568140\pi$
$684$		0			0
$685$	2.29118			0.0875416
$686$		0			0
$687$	−18.1577	+	30.6931i	−0.692758	+	1.17101i
$688$		0			0
$689$			81.1108i			3.09007i
$690$		0			0
$691$			2.76382i			0.105141i	0.998617	+	0.0525704i	$0.0167414\pi$
$691$			2.76382i			0.105141i	−0.998617	+	0.0525704i	$0.983259\pi$
$692$		0			0
$693$	−1.02837	+	0.564977i	−0.0390646	+	0.0214617i
$694$		0			0
$695$	10.9612			0.415781
$696$		0			0
$697$	0.0694319			0.00262992
$698$		0			0
$699$	−41.3087	−	24.4377i	−1.56244	−	0.924320i
$700$		0			0
$701$			31.8865i			1.20434i	0.798369	+	0.602169i	$0.205697\pi$
$701$			31.8865i			1.20434i	−0.798369	+	0.602169i	$0.794303\pi$
$702$		0			0
$703$			3.36227i			0.126810i
$704$		0			0
$705$	14.5910	+	8.63188i	0.549531	+	0.325095i
$706$		0			0
$707$	1.49316			0.0561560
$708$		0			0
$709$	−24.3078			−0.912898			−0.456449	−	0.889750i	$-0.650879\pi$
$709$	−24.3078			−0.912898			−0.456449	+	0.889750i	$0.650879\pi$
$710$		0			0
$711$	1.20180	−	0.660256i	0.0450710	−	0.0247615i
$712$		0			0
$713$			23.9587i			0.897259i
$714$		0			0
$715$		−	29.4212i		−	1.10029i
$716$		0			0
$717$	0.658999	−	1.11395i	0.0246108	−	0.0416012i
$718$		0			0
$719$	37.7696			1.40857			0.704284	−	0.709918i	$-0.251268\pi$
$719$	37.7696			1.40857			0.704284	+	0.709918i	$0.251268\pi$
$720$		0			0
$721$	−0.0797158			−0.00296877
$722$		0			0
$723$	−4.13626	+	6.99179i	−0.153829	+	0.260028i
$724$		0			0
$725$		−	24.6623i		−	0.915936i
$726$		0			0
$727$		−	14.5927i		−	0.541212i	−0.962690	−	0.270606i	$-0.912776\pi$
$727$		−	14.5927i		−	0.541212i	0.962690	−	0.270606i	$-0.0872240\pi$
$728$		0			0
$729$	26.9453	+	1.71800i	0.997974	+	0.0636298i
$730$		0			0
$731$	15.3463			0.567605
$732$		0			0
$733$	32.5494			1.20224			0.601119	−	0.799160i	$-0.294722\pi$
$733$	32.5494			1.20224			0.601119	+	0.799160i	$0.294722\pi$
$734$		0			0
$735$	−13.3293	−	7.88546i	−0.491659	−	0.290859i
$736$		0			0
$737$			41.1894i			1.51723i
$738$		0			0
$739$		−	43.9384i		−	1.61630i	−0.588977	−	0.808150i	$-0.700469\pi$
$739$		−	43.9384i		−	1.61630i	0.588977	−	0.808150i	$-0.299531\pi$
$740$		0			0
$741$	−8.82301	−	5.21958i	−0.324121	−	0.191746i
$742$		0			0
$743$	6.29687			0.231010			0.115505	−	0.993307i	$-0.463151\pi$
$743$	6.29687			0.231010			0.115505	+	0.993307i	$0.463151\pi$
$744$		0			0
$745$	4.00155			0.146605
$746$		0			0
$747$	2.89626	+	5.27178i	0.105969	+	0.192885i
$748$		0			0
$749$		−	1.47387i		−	0.0538541i
$750$		0			0
$751$			46.4736i			1.69585i	0.530120	+	0.847923i	$0.322147\pi$
$751$			46.4736i			1.69585i	−0.530120	+	0.847923i	$0.677853\pi$
$752$		0			0
$753$	−14.2686	+	24.1191i	−0.519976	+	0.878950i
$754$		0			0
$755$	28.6996			1.04449
$756$		0			0
$757$	−12.7839			−0.464638			−0.232319	−	0.972640i	$-0.574631\pi$
$757$	−12.7839			−0.464638			−0.232319	+	0.972640i	$0.574631\pi$
$758$		0			0
$759$	−28.1707	+	47.6188i	−1.02253	+	1.72845i
$760$		0			0
$761$			2.80771i			0.101780i	0.998704	+	0.0508898i	$0.0162057\pi$
$761$			2.80771i			0.101780i	−0.998704	+	0.0508898i	$0.983794\pi$
$762$		0			0
$763$		−	0.368609i		−	0.0133445i
$764$		0			0
$765$	−3.92148	−	7.13788i	−0.141781	−	0.258071i
$766$		0			0
$767$	−47.9607			−1.73176
$768$		0			0
$769$	−47.7601			−1.72227			−0.861137	−	0.508373i	$-0.830247\pi$
$769$	−47.7601			−1.72227			−0.861137	+	0.508373i	$0.830247\pi$
$770$		0			0
$771$	33.8701	+	20.0371i	1.21980	+	0.721620i
$772$		0			0
$773$		−	27.7196i		−	0.997005i	−0.866888	−	0.498503i	$-0.833883\pi$
$773$		−	27.7196i		−	0.997005i	0.866888	−	0.498503i	$-0.166117\pi$
$774$		0			0
$775$		−	9.80371i		−	0.352160i
$776$		0			0
$777$	0.504471	+	0.298439i	0.0180978	+	0.0107064i
$778$		0			0
$779$	−0.0327171			−0.00117221
$780$		0			0
$781$	14.0943			0.504334
$782$		0			0
$783$	−1.21271	+	38.0792i	−0.0433389	+	1.36084i
$784$		0			0
$785$		−	2.06451i		−	0.0736856i
$786$		0			0
$787$			40.2707i			1.43550i	0.696302	+	0.717748i	$0.254827\pi$
$787$			40.2707i			1.43550i	−0.696302	+	0.717748i	$0.745173\pi$
$788$		0			0
$789$	23.9588	−	40.4992i	0.852956	−	1.44181i
$790$		0			0
$791$	0.120658			0.00429011
$792$		0			0
$793$	−23.9521			−0.850565
$794$		0			0
$795$	15.4603	−	26.1335i	0.548319	−	0.926861i
$796$		0			0
$797$		−	11.9250i		−	0.422405i	−0.977442	−	0.211203i	$-0.932262\pi$
$797$		−	11.9250i		−	0.422405i	0.977442	−	0.211203i	$-0.0677380\pi$
$798$		0			0
$799$		−	16.2380i		−	0.574458i
$800$		0			0
$801$	−29.0203	+	15.9435i	−1.02538	+	0.563334i
$802$		0			0
$803$	37.9473			1.33913
$804$		0			0
$805$	1.05834			0.0373018
$806$		0			0
$807$	29.5666	+	17.4913i	1.04080	+	0.615722i
$808$		0			0
$809$		−	41.0152i		−	1.44202i	−0.692926	−	0.721009i	$-0.743679\pi$
$809$		−	41.0152i		−	1.44202i	0.692926	−	0.721009i	$-0.256321\pi$
$810$		0			0
$811$			55.2584i			1.94039i	0.242332	+	0.970193i	$0.422088\pi$
$811$			55.2584i			1.94039i	−0.242332	+	0.970193i	$0.577912\pi$
$812$		0			0
$813$	3.27307	+	1.93631i	0.114792	+	0.0679094i
$814$		0			0
$815$	31.1699			1.09183
$816$		0			0
$817$	−7.23138			−0.252994
$818$		0			0
$819$	−1.56628	+	0.860498i	−0.0547302	+	0.0300682i
$820$		0			0
$821$			50.3367i			1.75676i	0.477961	+	0.878381i	$0.341376\pi$
$821$			50.3367i			1.75676i	−0.477961	+	0.878381i	$0.658624\pi$
$822$		0			0
$823$		−	27.7153i		−	0.966096i	−0.875594	−	0.483048i	$-0.839530\pi$
$823$		−	27.7153i		−	0.966096i	0.875594	−	0.483048i	$-0.160470\pi$
$824$		0			0
$825$	11.5272	−	19.4852i	0.401326	−	0.678389i
$826$		0			0
$827$	−4.21549			−0.146587			−0.0732935	−	0.997310i	$-0.523351\pi$
$827$	−4.21549			−0.146587			−0.0732935	+	0.997310i	$0.523351\pi$
$828$		0			0
$829$	−42.4837			−1.47552			−0.737760	−	0.675063i	$-0.764116\pi$
$829$	−42.4837			−1.47552			−0.737760	+	0.675063i	$0.764116\pi$
$830$		0			0
$831$	10.8125	−	18.2771i	0.375081	−	0.634024i
$832$		0			0
$833$			14.8338i			0.513961i
$834$		0			0
$835$			5.51314i			0.190790i
$836$		0			0
$837$	−0.482075	+	15.1372i	−0.0166630	+	0.523217i
$838$		0			0
$839$	12.8333			0.443056			0.221528	−	0.975154i	$-0.428896\pi$
$839$	12.8333			0.443056			0.221528	+	0.975154i	$0.428896\pi$
$840$		0			0
$841$	−24.7592			−0.853765
$842$		0			0
$843$	30.5298	+	18.0611i	1.05150	+	0.622056i
$844$		0			0
$845$		−	28.1808i		−	0.969448i
$846$		0			0
$847$		−	0.412740i		−	0.0141819i
$848$		0			0
$849$	33.9055	+	20.0580i	1.16363	+	0.688390i
$850$		0			0
$851$	27.6384			0.947432
$852$		0			0
$853$	45.2356			1.54884			0.774419	−	0.632672i	$-0.218042\pi$
$853$	45.2356			1.54884			0.774419	+	0.632672i	$0.218042\pi$
$854$		0			0
$855$	1.84785	+	3.36345i	0.0631950	+	0.115028i
$856$		0			0
$857$		−	11.7824i		−	0.402480i	−0.979542	−	0.201240i	$-0.935503\pi$
$857$		−	11.7824i		−	0.402480i	0.979542	−	0.201240i	$-0.0644972\pi$
$858$		0			0
$859$			2.64817i			0.0903544i	0.998979	+	0.0451772i	$0.0143852\pi$
$859$			2.64817i			0.0903544i	−0.998979	+	0.0451772i	$0.985615\pi$
$860$		0			0
$861$	−0.00290400	+	0.00490883i	−9.89682e−5	+	0.000167293i
$862$		0			0
$863$	−24.5234			−0.834788			−0.417394	−	0.908726i	$-0.637056\pi$
$863$	−24.5234			−0.834788			−0.417394	+	0.908726i	$0.637056\pi$
$864$		0			0
$865$	−20.0299			−0.681038
$866$		0			0
$867$	11.0204	−	18.6286i	0.374274	−	0.632660i
$868$		0			0
$869$			1.77618i			0.0602529i
$870$		0			0
$871$			62.7342i			2.12567i
$872$		0			0
$873$	−18.4880	−	33.6519i	−0.625725	−	1.13895i
$874$		0			0
$875$	−1.07682			−0.0364030
$876$		0			0
$877$	36.2156			1.22291			0.611456	−	0.791278i	$-0.290584\pi$
$877$	36.2156			1.22291			0.611456	+	0.791278i	$0.290584\pi$
$878$		0			0
$879$	21.2325	+	12.5609i	0.716154	+	0.423668i
$880$		0			0
$881$		−	38.6301i		−	1.30148i	−0.759300	−	0.650741i	$-0.774458\pi$
$881$		−	38.6301i		−	1.30148i	0.759300	−	0.650741i	$-0.225542\pi$
$882$		0			0
$883$		−	9.05968i		−	0.304882i	−0.988313	−	0.152441i	$-0.951287\pi$
$883$		−	9.05968i		−	0.304882i	0.988313	−	0.152441i	$-0.0487135\pi$
$884$		0			0
$885$	15.4527	+	9.14164i	0.519438	+	0.307293i
$886$		0			0
$887$	42.6523			1.43212			0.716062	−	0.698036i	$-0.245943\pi$
$887$	42.6523			1.43212			0.716062	+	0.698036i	$0.245943\pi$
$888$		0			0
$889$	−0.362783			−0.0121674
$890$		0			0
$891$	−18.7564	+	29.5188i	−0.628365	+	0.988918i
$892$		0			0
$893$			7.65152i			0.256048i
$894$		0			0
$895$			9.34942i			0.312517i
$896$		0			0
$897$	−42.9058	+	72.5265i	−1.43258	+	2.42159i
$898$		0			0
$899$	−21.3702			−0.712737
$900$		0			0
$901$	−29.0833			−0.968904
$902$		0			0
$903$	−0.641865	+	1.08499i	−0.0213599	+	0.0361061i
$904$		0			0
$905$			14.8630i			0.494061i
$906$		0			0
$907$			45.5414i			1.51218i	0.654470	+	0.756088i	$0.272892\pi$
$907$			45.5414i			1.51218i	−0.654470	+	0.756088i	$0.727108\pi$
$908$		0			0
$909$	39.0072	−	21.4301i	1.29379	−	0.710793i
$910$		0			0
$911$	−30.2643			−1.00270			−0.501351	−	0.865244i	$-0.667163\pi$
$911$	−30.2643			−1.00270			−0.501351	+	0.865244i	$0.667163\pi$
$912$		0			0
$913$	−7.79137			−0.257857
$914$		0			0
$915$	7.71727	+	4.56544i	0.255125	+	0.150929i
$916$		0			0
$917$		−	1.36112i		−	0.0449480i
$918$		0			0
$919$		−	35.6913i		−	1.17735i	−0.808370	−	0.588674i	$-0.799650\pi$
$919$		−	35.6913i		−	1.17735i	0.808370	−	0.588674i	$-0.200350\pi$
$920$		0			0
$921$	−6.76739	−	4.00350i	−0.222993	−	0.131920i
$922$		0			0
$923$	21.4666			0.706581
$924$		0			0
$925$	−11.3094			−0.371852
$926$		0			0
$927$	−2.08249	+	1.14410i	−0.0683979	+	0.0375771i
$928$		0			0
$929$		−	44.8227i		−	1.47058i	−0.677750	−	0.735292i	$-0.737045\pi$
$929$		−	44.8227i		−	1.47058i	0.677750	−	0.735292i	$-0.262955\pi$
$930$		0			0
$931$		−	6.98987i		−	0.229084i
$932$		0			0
$933$	11.5834	−	19.5801i	0.379223	−	0.641025i
$934$		0			0
$935$	10.5493			0.345000
$936$		0			0
$937$	−11.6590			−0.380883			−0.190442	−	0.981699i	$-0.560992\pi$
$937$	−11.6590			−0.380883			−0.190442	+	0.981699i	$0.560992\pi$
$938$		0			0
$939$	5.98758	−	10.1212i	0.195397	−	0.330293i
$940$		0			0
$941$		−	17.8763i		−	0.582752i	−0.956609	−	0.291376i	$-0.905887\pi$
$941$		−	17.8763i		−	0.582752i	0.956609	−	0.291376i	$-0.0941131\pi$
$942$		0			0
$943$			0.268940i			0.00875788i
$944$		0			0
$945$	0.668665	+	0.0212951i	0.0217517	+	0.000692729i
$946$		0			0
$947$	−50.5577			−1.64291			−0.821453	−	0.570276i	$-0.806836\pi$
$947$	−50.5577			−1.64291			−0.821453	+	0.570276i	$0.806836\pi$
$948$		0			0
$949$	57.7962			1.87614
$950$		0			0
$951$	39.6897	+	23.4799i	1.28703	+	0.761389i
$952$		0			0
$953$			32.0630i			1.03862i	0.854585	+	0.519311i	$0.173811\pi$
$953$			32.0630i			1.03862i	−0.854585	+	0.519311i	$0.826189\pi$
$954$		0			0
$955$			10.1115i			0.327200i
$956$		0			0
$957$	−42.4741	−	25.1272i	−1.37299	−	0.812246i
$958$		0			0
$959$	0.180270			0.00582124
$960$		0			0
$961$	22.5050			0.725967
$962$		0			0
$963$	−21.1533	−	38.5033i	−0.681657	−	1.24075i
$964$		0			0
$965$			5.32687i			0.171478i
$966$		0			0
$967$			25.2660i			0.812501i	0.913762	+	0.406250i	$0.133164\pi$
$967$			25.2660i			0.812501i	−0.913762	+	0.406250i	$0.866836\pi$
$968$		0			0
$969$	1.87155	−	3.16360i	0.0601227	−	0.101629i
$970$		0			0
$971$	15.5254			0.498233			0.249116	−	0.968474i	$-0.419860\pi$
$971$	15.5254			0.498233			0.249116	+	0.968474i	$0.419860\pi$
$972$		0			0
$973$	0.862427			0.0276481
$974$		0			0
$975$	17.5567	−	29.6773i	0.562265	−	0.950435i
$976$		0			0
$977$		−	44.8656i		−	1.43538i	−0.696364	−	0.717689i	$-0.745200\pi$
$977$		−	44.8656i		−	1.43538i	0.696364	−	0.717689i	$-0.254800\pi$
$978$		0			0
$979$		−	42.8902i		−	1.37078i
$980$		0			0
$981$	−5.29035	−	9.62951i	−0.168908	−	0.307446i
$982$		0			0
$983$	5.58521			0.178141			0.0890703	−	0.996025i	$-0.471610\pi$
$983$	5.58521			0.178141			0.0890703	+	0.996025i	$0.471610\pi$
$984$		0			0
$985$	11.6159			0.370113
$986$		0			0
$987$	1.14802	+	0.679157i	0.0365420	+	0.0216178i
$988$		0			0
$989$			59.4431i			1.89018i
$990$		0			0
$991$			24.9378i			0.792174i	0.918213	+	0.396087i	$0.129632\pi$
$991$			24.9378i			0.792174i	−0.918213	+	0.396087i	$0.870368\pi$
$992$		0			0
$993$	−21.5416	−	12.7437i	−0.683602	−	0.404410i
$994$		0			0
$995$	−14.6154			−0.463339
$996$		0			0
$997$	19.6223			0.621446			0.310723	−	0.950501i	$-0.399429\pi$
$997$	19.6223			0.621446			0.310723	+	0.950501i	$0.399429\pi$
$998$		0			0
$999$	17.4620	+	0.556116i	0.552474	+	0.0175947i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.d.b.191.15	yes	24
3.2	odd	2	inner	912.2.d.b.191.9	✓	24
4.3	odd	2	inner	912.2.d.b.191.10	yes	24
12.11	even	2	inner	912.2.d.b.191.16	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.2.d.b.191.9	✓	24	3.2	odd	2	inner
912.2.d.b.191.10	yes	24	4.3	odd	2	inner
912.2.d.b.191.15	yes	24	1.1	even	1	trivial
912.2.d.b.191.16	yes	24	12.11	even	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)

\(f(q)\)	\(=\)	\(q+(0.881895 - 1.49073i) q^{3} -1.27921i q^{5} -0.100648i q^{7} +(-1.44452 - 2.62933i) q^{9} +O(q^{10})\)	\(q+(0.881895 - 1.49073i) q^{3} -1.27921i q^{5} -0.100648i q^{7} +(-1.44452 - 2.62933i) q^{9} +3.88598 q^{11} +5.91860 q^{13} +(-1.90695 - 1.12813i) q^{15} +2.12219i q^{17} -1.00000i q^{19} +(-0.150039 - 0.0887611i) q^{21} -8.22016 q^{23} +3.36363 q^{25} +(-5.19352 - 0.165399i) q^{27} -7.33207i q^{29} -2.91462i q^{31} +(3.42702 - 5.79293i) q^{33} -0.128750 q^{35} -3.36227 q^{37} +(5.21958 - 8.82301i) q^{39} -0.0327171i q^{41} -7.23138i q^{43} +(-3.36345 + 1.84785i) q^{45} -7.65152 q^{47} +6.98987 q^{49} +(3.16360 + 1.87155i) q^{51} +13.7044i q^{53} -4.97097i q^{55} +(-1.49073 - 0.881895i) q^{57} -8.10338 q^{59} -4.04692 q^{61} +(-0.264637 + 0.145389i) q^{63} -7.57112i q^{65} +10.5995i q^{67} +(-7.24931 + 12.2540i) q^{69} +3.62697 q^{71} +9.76518 q^{73} +(2.96636 - 5.01424i) q^{75} -0.391117i q^{77} +0.457075i q^{79} +(-4.82670 + 7.59625i) q^{81} -2.00500 q^{83} +2.71472 q^{85} +(-10.9301 - 6.46611i) q^{87} -11.0372i q^{89} -0.595697i q^{91} +(-4.34490 - 2.57039i) q^{93} -1.27921 q^{95} +12.7987 q^{97} +(-5.61339 - 10.2175i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{9}+O(q^{10}) \) 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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