LMFDB - Embedded newform 471.2.b.a.313.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [471,2,Mod(313,471)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(471, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("471.313");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 471 = 3 \cdot 157 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	471.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$3.76095393520$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 15x^{10} + 77x^{8} + 158x^{6} + 111x^{4} + 21x^{2} + 1 $ x^12 + 15x^10 + 77x^8 + 158x^6 + 111x^4 + 21*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			313.5
Root			$-0.430002i$ of defining polynomial
Character	$\chi$	$=$	471.313
Dual form			471.2.b.a.313.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.430002i q^{2} -1.00000 q^{3} +1.81510 q^{4} -1.89557i q^{5} +0.430002i q^{6} +3.40343i q^{7} -1.64050i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.430002i q^{2} -1.00000 q^{3} +1.81510 q^{4} -1.89557i q^{5} +0.430002i q^{6} +3.40343i q^{7} -1.64050i q^{8} +1.00000 q^{9} -0.815098 q^{10} +2.33159 q^{11} -1.81510 q^{12} +1.46348 q^{13} +1.46348 q^{14} +1.89557i q^{15} +2.92478 q^{16} +5.58332 q^{17} -0.430002i q^{18} -6.04462 q^{19} -3.44065i q^{20} -3.40343i q^{21} -1.00259i q^{22} -8.56557i q^{23} +1.64050i q^{24} +1.40682 q^{25} -0.629299i q^{26} -1.00000 q^{27} +6.17756i q^{28} +8.19556i q^{29} +0.815098 q^{30} -4.52666 q^{31} -4.53866i q^{32} -2.33159 q^{33} -2.40084i q^{34} +6.45143 q^{35} +1.81510 q^{36} +9.83969 q^{37} +2.59920i q^{38} -1.46348 q^{39} -3.10968 q^{40} -3.26571i q^{41} -1.46348 q^{42} -10.3363i q^{43} +4.23207 q^{44} -1.89557i q^{45} -3.68321 q^{46} +1.67601 q^{47} -2.92478 q^{48} -4.58332 q^{49} -0.604933i q^{50} -5.58332 q^{51} +2.65636 q^{52} +0.511004i q^{53} +0.430002i q^{54} -4.41970i q^{55} +5.58332 q^{56} +6.04462 q^{57} +3.52410 q^{58} +9.95014i q^{59} +3.44065i q^{60} +1.38457i q^{61} +1.94647i q^{62} +3.40343i q^{63} +3.89793 q^{64} -2.77413i q^{65} +1.00259i q^{66} -11.0544 q^{67} +10.1343 q^{68} +8.56557i q^{69} -2.77413i q^{70} -12.7906 q^{71} -1.64050i q^{72} +16.3878i q^{73} -4.23109i q^{74} -1.40682 q^{75} -10.9716 q^{76} +7.93541i q^{77} +0.629299i q^{78} +1.85932i q^{79} -5.54412i q^{80} +1.00000 q^{81} -1.40426 q^{82} -3.55883i q^{83} -6.17756i q^{84} -10.5836i q^{85} -4.44461 q^{86} -8.19556i q^{87} -3.82498i q^{88} -17.2933 q^{89} -0.815098 q^{90} +4.98085i q^{91} -15.5474i q^{92} +4.52666 q^{93} -0.720689i q^{94} +11.4580i q^{95} +4.53866i q^{96} +16.7957i q^{97} +1.97084i q^{98} +2.33159 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{3} - 6 q^{4} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^3 - 6 * q^4 + 12 * q^9	$ 12 q - 12 q^{3} - 6 q^{4} + 12 q^{9} + 18 q^{10} - 2 q^{11} + 6 q^{12} + 4 q^{13} + 4 q^{14} + 10 q^{16} + 2 q^{17} + 4 q^{19} + 12 q^{25} - 12 q^{27} - 18 q^{30} + 2 q^{31} + 2 q^{33} - 4 q^{35} - 6 q^{36} - 2 q^{37} - 4 q^{39} - 40 q^{40} - 4 q^{42} - 36 q^{44} + 34 q^{47} - 10 q^{48} + 10 q^{49} - 2 q^{51} - 6 q^{52} + 2 q^{56} - 4 q^{57} + 24 q^{58} + 28 q^{64} - 38 q^{67} + 32 q^{68} - 26 q^{71} - 12 q^{75} - 28 q^{76} + 12 q^{81} - 50 q^{82} + 6 q^{86} - 4 q^{89} + 18 q^{90} - 2 q^{93} - 2 q^{99}+O(q^{100}) $ 12 * q - 12 * q^3 - 6 * q^4 + 12 * q^9 + 18 * q^10 - 2 * q^11 + 6 * q^12 + 4 * q^13 + 4 * q^14 + 10 * q^16 + 2 * q^17 + 4 * q^19 + 12 * q^25 - 12 * q^27 - 18 * q^30 + 2 * q^31 + 2 * q^33 - 4 * q^35 - 6 * q^36 - 2 * q^37 - 4 * q^39 - 40 * q^40 - 4 * q^42 - 36 * q^44 + 34 * q^47 - 10 * q^48 + 10 * q^49 - 2 * q^51 - 6 * q^52 + 2 * q^56 - 4 * q^57 + 24 * q^58 + 28 * q^64 - 38 * q^67 + 32 * q^68 - 26 * q^71 - 12 * q^75 - 28 * q^76 + 12 * q^81 - 50 * q^82 + 6 * q^86 - 4 * q^89 + 18 * q^90 - 2 * q^93 - 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/471\mathbb{Z}\right)^\times$.

$n$	$158$	$319$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$		−	0.430002i		−	0.304057i	−0.988376	−	0.152029i	$-0.951419\pi$
$2$		−	0.430002i		−	0.304057i	0.988376	−	0.152029i	$-0.0485806\pi$
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$	1.81510			0.907549
$5$		−	1.89557i		−	0.847725i	−0.905727	−	0.423862i	$-0.860674\pi$
$5$		−	1.89557i		−	0.847725i	0.905727	−	0.423862i	$-0.139326\pi$
$6$			0.430002i			0.175548i
$7$			3.40343i			1.28637i	0.765709	+	0.643187i	$0.222388\pi$
$7$			3.40343i			1.28637i	−0.765709	+	0.643187i	$0.777612\pi$
$8$		−	1.64050i		−	0.580004i
$9$	1.00000			0.333333
$10$	−0.815098			−0.257757
$11$	2.33159			0.703002			0.351501	−	0.936187i	$-0.385671\pi$
$11$	2.33159			0.703002			0.351501	+	0.936187i	$0.385671\pi$
$12$	−1.81510			−0.523974
$13$	1.46348			0.405896			0.202948	−	0.979189i	$-0.434948\pi$
$13$	1.46348			0.405896			0.202948	+	0.979189i	$0.434948\pi$
$14$	1.46348			0.391132
$15$			1.89557i			0.489434i
$16$	2.92478			0.731195
$17$	5.58332			1.35415			0.677077	−	0.735912i	$-0.263246\pi$
$17$	5.58332			1.35415			0.677077	+	0.735912i	$0.263246\pi$
$18$		−	0.430002i		−	0.101352i
$19$	−6.04462			−1.38673			−0.693365	−	0.720586i	$-0.743873\pi$
$19$	−6.04462			−1.38673			−0.693365	+	0.720586i	$0.743873\pi$
$20$		−	3.44065i		−	0.769352i
$21$		−	3.40343i		−	0.742689i
$22$		−	1.00259i		−	0.213753i
$23$		−	8.56557i		−	1.78605i	−0.450011	−	0.893023i	$-0.648580\pi$
$23$		−	8.56557i		−	1.78605i	0.450011	−	0.893023i	$-0.351420\pi$
$24$			1.64050i			0.334866i
$25$	1.40682			0.281363
$26$		−	0.629299i		−	0.123416i
$27$	−1.00000			−0.192450
$28$			6.17756i			1.16745i
$29$			8.19556i			1.52188i	0.648824	+	0.760938i	$0.275261\pi$
$29$			8.19556i			1.52188i	−0.648824	+	0.760938i	$0.724739\pi$
$30$	0.815098			0.148816
$31$	−4.52666			−0.813011			−0.406506	−	0.913648i	$-0.633253\pi$
$31$	−4.52666			−0.813011			−0.406506	+	0.913648i	$0.633253\pi$
$32$		−	4.53866i		−	0.802329i
$33$	−2.33159			−0.405878
$34$		−	2.40084i		−	0.411740i
$35$	6.45143			1.09049
$36$	1.81510			0.302516
$37$	9.83969			1.61764			0.808818	−	0.588059i	$-0.200108\pi$
$37$	9.83969			1.61764			0.808818	+	0.588059i	$0.200108\pi$
$38$			2.59920i			0.421646i
$39$	−1.46348			−0.234344
$40$	−3.10968			−0.491684
$41$		−	3.26571i		−	0.510019i	−0.966939	−	0.255009i	$-0.917921\pi$
$41$		−	3.26571i		−	0.510019i	0.966939	−	0.255009i	$-0.0820786\pi$
$42$	−1.46348			−0.225820
$43$		−	10.3363i		−	1.57626i	−0.615507	−	0.788132i	$-0.711049\pi$
$43$		−	10.3363i		−	1.57626i	0.615507	−	0.788132i	$-0.288951\pi$
$44$	4.23207			0.638009
$45$		−	1.89557i		−	0.282575i
$46$	−3.68321			−0.543060
$47$	1.67601			0.244472			0.122236	−	0.992501i	$-0.460994\pi$
$47$	1.67601			0.244472			0.122236	+	0.992501i	$0.460994\pi$
$48$	−2.92478			−0.422156
$49$	−4.58332			−0.654760
$50$		−	0.604933i		−	0.0855505i
$51$	−5.58332			−0.781821
$52$	2.65636			0.368371
$53$			0.511004i			0.0701918i	0.999384	+	0.0350959i	$0.0111737\pi$
$53$			0.511004i			0.0701918i	−0.999384	+	0.0350959i	$0.988826\pi$
$54$			0.430002i			0.0585158i
$55$		−	4.41970i		−	0.595952i
$56$	5.58332			0.746103
$57$	6.04462			0.800629
$58$	3.52410			0.462737
$59$			9.95014i			1.29540i	0.761896	+	0.647699i	$0.224269\pi$
$59$			9.95014i			1.29540i	−0.761896	+	0.647699i	$0.775731\pi$
$60$			3.44065i			0.444185i
$61$			1.38457i			0.177275i	0.996064	+	0.0886377i	$0.0282513\pi$
$61$			1.38457i			0.177275i	−0.996064	+	0.0886377i	$0.971749\pi$
$62$			1.94647i			0.247202i
$63$			3.40343i			0.428792i
$64$	3.89793			0.487241
$65$		−	2.77413i		−	0.344088i
$66$			1.00259i			0.123410i
$67$	−11.0544			−1.35051			−0.675255	−	0.737584i	$-0.735967\pi$
$67$	−11.0544			−1.35051			−0.675255	+	0.737584i	$0.735967\pi$
$68$	10.1343			1.22896
$69$			8.56557i			1.03117i
$70$		−	2.77413i		−	0.331572i
$71$	−12.7906			−1.51797			−0.758985	−	0.651108i	$-0.774304\pi$
$71$	−12.7906			−1.51797			−0.758985	+	0.651108i	$0.774304\pi$
$72$		−	1.64050i		−	0.193335i
$73$			16.3878i			1.91805i	0.283319	+	0.959026i	$0.408565\pi$
$73$			16.3878i			1.91805i	−0.283319	+	0.959026i	$0.591435\pi$
$74$		−	4.23109i		−	0.491854i
$75$	−1.40682			−0.162445
$76$	−10.9716			−1.25853
$77$			7.93541i			0.904324i
$78$			0.629299i			0.0712541i
$79$			1.85932i			0.209190i	0.994515	+	0.104595i	$0.0333546\pi$
$79$			1.85932i			0.209190i	−0.994515	+	0.104595i	$0.966645\pi$
$80$		−	5.54412i		−	0.619852i
$81$	1.00000			0.111111
$82$	−1.40426			−0.155075
$83$		−	3.55883i		−	0.390633i	−0.980740	−	0.195316i	$-0.937427\pi$
$83$		−	3.55883i		−	0.390633i	0.980740	−	0.195316i	$-0.0625734\pi$
$84$		−	6.17756i		−	0.674027i
$85$		−	10.5836i		−	1.14795i
$86$	−4.44461			−0.479274
$87$		−	8.19556i		−	0.878656i
$88$		−	3.82498i		−	0.407744i
$89$	−17.2933			−1.83309			−0.916544	−	0.399935i	$-0.869033\pi$
$89$	−17.2933			−1.83309			−0.916544	+	0.399935i	$0.869033\pi$
$90$	−0.815098			−0.0859189
$91$			4.98085i			0.522135i
$92$		−	15.5474i		−	1.62092i
$93$	4.52666			0.469392
$94$		−	0.720689i		−	0.0743334i
$95$			11.4580i			1.17557i
$96$			4.53866i			0.463225i
$97$			16.7957i			1.70535i	0.522445	+	0.852673i	$0.325020\pi$
$97$			16.7957i			1.70535i	−0.522445	+	0.852673i	$0.674980\pi$
$98$			1.97084i			0.199085i
$99$	2.33159			0.234334
$100$	2.55351			0.255351
$101$	−4.98795			−0.496320			−0.248160	−	0.968719i	$-0.579826\pi$
$101$	−4.98795			−0.496320			−0.248160	+	0.968719i	$0.579826\pi$
$102$			2.40084i			0.237718i
$103$			6.76460i			0.666536i	0.942832	+	0.333268i	$0.108151\pi$
$103$			6.76460i			0.666536i	−0.942832	+	0.333268i	$0.891849\pi$
$104$		−	2.40084i		−	0.235422i
$105$	−6.45143			−0.629596
$106$	0.219733			0.0213423
$107$		−	6.66267i		−	0.644104i	−0.946722	−	0.322052i	$-0.895627\pi$
$107$		−	6.66267i		−	0.644104i	0.946722	−	0.322052i	$-0.104373\pi$
$108$	−1.81510			−0.174658
$109$	−0.573532			−0.0549344			−0.0274672	−	0.999623i	$-0.508744\pi$
$109$	−0.573532			−0.0549344			−0.0274672	+	0.999623i	$0.508744\pi$
$110$	−1.90048			−0.181204
$111$	−9.83969			−0.933942
$112$			9.95427i			0.940591i
$113$	−6.06645			−0.570684			−0.285342	−	0.958426i	$-0.592107\pi$
$113$	−6.06645			−0.570684			−0.285342	+	0.958426i	$0.592107\pi$
$114$		−	2.59920i		−	0.243437i
$115$	−16.2366			−1.51407
$116$			14.8757i			1.38118i
$117$	1.46348			0.135299
$118$	4.27858			0.393875
$119$			19.0024i			1.74195i
$120$	3.10968			0.283874
$121$	−5.56367			−0.505788
$122$	0.595366			0.0539019
$123$			3.26571i			0.294460i
$124$	−8.21633			−0.737848
$125$		−	12.1446i		−	1.08624i
$126$	1.46348			0.130377
$127$	8.53426			0.757293			0.378647	−	0.925541i	$-0.376390\pi$
$127$	8.53426			0.757293			0.378647	+	0.925541i	$0.376390\pi$
$128$		−	10.7534i		−	0.950478i
$129$			10.3363i			0.910056i
$130$	−1.19288			−0.104623
$131$		−	12.2174i		−	1.06744i	−0.845662	−	0.533720i	$-0.820794\pi$
$131$		−	12.2174i		−	1.06744i	0.845662	−	0.533720i	$-0.179206\pi$
$132$	−4.23207			−0.368355
$133$		−	20.5724i		−	1.78386i
$134$			4.75342i			0.410633i
$135$			1.89557i			0.163145i
$136$		−	9.15943i		−	0.785415i
$137$			5.16215i			0.441032i	0.975383	+	0.220516i	$0.0707741\pi$
$137$			5.16215i			0.441032i	−0.975383	+	0.220516i	$0.929226\pi$
$138$	3.68321			0.313536
$139$			14.3404i			1.21634i	0.793808	+	0.608169i	$0.208096\pi$
$139$			14.3404i			1.21634i	−0.793808	+	0.608169i	$0.791904\pi$
$140$	11.7100			0.989675
$141$	−1.67601			−0.141146
$142$			5.50000i			0.461550i
$143$	3.41224			0.285346
$144$	2.92478			0.243732
$145$	15.5352			1.29013
$146$	7.04680			0.583197
$147$	4.58332			0.378026
$148$	17.8600			1.46808
$149$		−	4.65528i		−	0.381375i	−0.981651	−	0.190688i	$-0.938928\pi$
$149$		−	4.65528i		−	0.381375i	0.981651	−	0.190688i	$-0.0610717\pi$
$150$			0.604933i			0.0493926i
$151$			16.0404i			1.30535i	0.757637	+	0.652676i	$0.226354\pi$
$151$			16.0404i			1.30535i	−0.757637	+	0.652676i	$0.773646\pi$
$152$			9.91619i			0.804310i
$153$	5.58332			0.451385
$154$	3.41224			0.274966
$155$			8.58059i			0.689210i
$156$	−2.65636			−0.212679
$157$	−10.3784	−	7.02061i	−0.828286	−	0.560305i
$157$	−10.3784	−	7.02061i	−0.828286	−	0.560305i
$158$	0.799512			0.0636057
$159$		−	0.511004i		−	0.0405252i
$160$	−8.60334			−0.680154
$161$	29.1523			2.29752
$162$		−	0.430002i		−	0.0337841i
$163$			0.401866i			0.0314766i	0.999876	+	0.0157383i	$0.00500986\pi$
$163$			0.401866i			0.0314766i	−0.999876	+	0.0157383i	$0.994990\pi$
$164$		−	5.92759i		−	0.462867i
$165$			4.41970i			0.344073i
$166$	−1.53030			−0.118775
$167$	−11.7129			−0.906373			−0.453187	−	0.891416i	$-0.649713\pi$
$167$	−11.7129			−0.906373			−0.453187	+	0.891416i	$0.649713\pi$
$168$	−5.58332			−0.430763
$169$	−10.8582			−0.835248
$170$	−4.55096			−0.349042
$171$	−6.04462			−0.462244
$172$		−	18.7613i		−	1.43054i
$173$	10.2952			0.782731			0.391366	−	0.920235i	$-0.372003\pi$
$173$	10.2952			0.782731			0.391366	+	0.920235i	$0.372003\pi$
$174$	−3.52410			−0.267162
$175$			4.78799i			0.361938i
$176$	6.81940			0.514032
$177$		−	9.95014i		−	0.747898i
$178$			7.43615i			0.557363i
$179$		−	5.43511i		−	0.406239i	−0.979154	−	0.203120i	$-0.934892\pi$
$179$		−	5.43511i		−	0.406239i	0.979154	−	0.203120i	$-0.0651080\pi$
$180$		−	3.44065i		−	0.256451i
$181$		−	14.0203i		−	1.04212i	−0.853519	−	0.521061i	$-0.825536\pi$
$181$		−	14.0203i		−	1.04212i	0.853519	−	0.521061i	$-0.174464\pi$
$182$	2.14177			0.158759
$183$		−	1.38457i		−	0.102350i
$184$	−14.0518			−1.03591
$185$		−	18.6518i		−	1.37131i
$186$		−	1.94647i		−	0.142722i
$187$	13.0180			0.951973
$188$	3.04213			0.221870
$189$		−	3.40343i		−	0.247563i
$190$	4.92696			0.357439
$191$			14.4241i			1.04369i	0.853040	+	0.521846i	$0.174757\pi$
$191$			14.4241i			1.04369i	−0.853040	+	0.521846i	$0.825243\pi$
$192$	−3.89793			−0.281309
$193$	17.8252			1.28308			0.641541	−	0.767088i	$-0.278295\pi$
$193$	17.8252			1.28308			0.641541	+	0.767088i	$0.278295\pi$
$194$	7.22219			0.518523
$195$			2.77413i			0.198659i
$196$	−8.31918			−0.594227
$197$	6.96932			0.496544			0.248272	−	0.968690i	$-0.420137\pi$
$197$	6.96932			0.496544			0.248272	+	0.968690i	$0.420137\pi$
$198$		−	1.00259i		−	0.0712510i
$199$	8.18766			0.580408			0.290204	−	0.956965i	$-0.406277\pi$
$199$	8.18766			0.580408			0.290204	+	0.956965i	$0.406277\pi$
$200$		−	2.30788i		−	0.163192i
$201$	11.0544			0.779718
$202$			2.14483i			0.150910i
$203$	−27.8930			−1.95770
$204$	−10.1343			−0.709541
$205$	−6.19039			−0.432356
$206$	2.90879			0.202665
$207$		−	8.56557i		−	0.595349i
$208$	4.28036			0.296789
$209$	−14.0936			−0.974875
$210$			2.77413i			0.191433i
$211$		−	9.61430i		−	0.661876i	−0.943653	−	0.330938i	$-0.892635\pi$
$211$		−	9.61430i		−	0.661876i	0.943653	−	0.330938i	$-0.107365\pi$
$212$			0.927522i			0.0637025i
$213$	12.7906			0.876400
$214$	−2.86496			−0.195845
$215$	−19.5931			−1.33624
$216$			1.64050i			0.111622i
$217$		−	15.4061i		−	1.04584i
$218$			0.246620i			0.0167032i
$219$		−	16.3878i		−	1.10739i
$220$		−	8.02219i		−	0.540856i
$221$	8.17108			0.549646
$222$			4.23109i			0.283972i
$223$		−	8.59828i		−	0.575784i	−0.957663	−	0.287892i	$-0.907046\pi$
$223$		−	8.59828i		−	0.575784i	0.957663	−	0.287892i	$-0.0929544\pi$
$224$	15.4470			1.03210
$225$	1.40682			0.0937877
$226$			2.60859i			0.173521i
$227$		−	9.31730i		−	0.618411i	−0.950995	−	0.309206i	$-0.899937\pi$
$227$		−	9.31730i		−	0.618411i	0.950995	−	0.309206i	$-0.100063\pi$
$228$	10.9716			0.726611
$229$		−	26.7104i		−	1.76507i	−0.470243	−	0.882537i	$-0.655834\pi$
$229$		−	26.7104i		−	1.76507i	0.470243	−	0.882537i	$-0.344166\pi$
$230$			6.98179i			0.460365i
$231$		−	7.93541i		−	0.522112i
$232$	13.4448			0.882695
$233$	−29.4687			−1.93056			−0.965279	−	0.261220i	$-0.915875\pi$
$233$	−29.4687			−1.93056			−0.965279	+	0.261220i	$0.915875\pi$
$234$		−	0.629299i		−	0.0411386i
$235$		−	3.17700i		−	0.207245i
$236$			18.0605i			1.17564i
$237$		−	1.85932i		−	0.120776i
$238$	8.17108			0.529652
$239$	6.45437			0.417499			0.208749	−	0.977969i	$-0.433061\pi$
$239$	6.45437			0.417499			0.208749	+	0.977969i	$0.433061\pi$
$240$			5.54412i			0.357872i
$241$			11.7121i			0.754445i	0.926123	+	0.377222i	$0.123121\pi$
$241$			11.7121i			0.754445i	−0.926123	+	0.377222i	$0.876879\pi$
$242$			2.39239i			0.153788i
$243$	−1.00000			−0.0641500
$244$			2.51312i			0.160886i
$245$			8.68800i			0.555056i
$246$	1.40426			0.0895325
$247$	−8.84618			−0.562869
$248$			7.42598i			0.471550i
$249$			3.55883i			0.225532i
$250$	−5.22219			−0.330280
$251$			7.43809i			0.469488i	0.972057	+	0.234744i	$0.0754252\pi$
$251$			7.43809i			0.469488i	−0.972057	+	0.234744i	$0.924575\pi$
$252$			6.17756i			0.389149i
$253$		−	19.9714i		−	1.25559i
$254$		−	3.66975i		−	0.230261i
$255$			10.5836i			0.662769i
$256$	3.17186			0.198241
$257$	−16.6835			−1.04069			−0.520344	−	0.853957i	$-0.674196\pi$
$257$	−16.6835			−1.04069			−0.520344	+	0.853957i	$0.674196\pi$
$258$	4.44461			0.276709
$259$			33.4887i			2.08089i
$260$		−	5.03532i		−	0.312277i
$261$			8.19556i			0.507292i
$262$	−5.25350			−0.324563
$263$	0.658517			0.0406059			0.0203030	−	0.999794i	$-0.493537\pi$
$263$	0.658517			0.0406059			0.0203030	+	0.999794i	$0.493537\pi$
$264$			3.82498i			0.235411i
$265$	0.968644			0.0595033
$266$	−8.84618			−0.542394
$267$	17.2933			1.05833
$268$	−20.0648			−1.22566
$269$		−	5.01356i		−	0.305682i	−0.988251	−	0.152841i	$-0.951158\pi$
$269$		−	5.01356i		−	0.305682i	0.988251	−	0.152841i	$-0.0488422\pi$
$270$	0.815098			0.0496053
$271$			11.5998i			0.704640i	0.935880	+	0.352320i	$0.114607\pi$
$271$			11.5998i			0.704640i	−0.935880	+	0.352320i	$0.885393\pi$
$272$	16.3300			0.990151
$273$		−	4.98085i		−	0.301455i
$274$	2.21973			0.134099
$275$	3.28012			0.197799
$276$			15.5474i			0.935841i
$277$	29.8124			1.79125			0.895625	−	0.444809i	$-0.146728\pi$
$277$	29.8124			1.79125			0.895625	+	0.444809i	$0.146728\pi$
$278$	6.16640			0.369836
$279$	−4.52666			−0.271004
$280$		−	10.5836i		−	0.632490i
$281$	20.7979			1.24070			0.620349	−	0.784326i	$-0.286991\pi$
$281$	20.7979			1.24070			0.620349	+	0.784326i	$0.286991\pi$
$282$			0.720689i			0.0429164i
$283$	−2.62618			−0.156110			−0.0780550	−	0.996949i	$-0.524871\pi$
$283$	−2.62618			−0.156110			−0.0780550	+	0.996949i	$0.524871\pi$
$284$	−23.2163			−1.37763
$285$		−	11.4580i		−	0.678713i
$286$		−	1.46727i		−	0.0867615i
$287$	11.1146			0.656075
$288$		−	4.53866i		−	0.267443i
$289$	14.1735			0.833734
$290$		−	6.68018i		−	0.392274i
$291$		−	16.7957i		−	0.984582i
$292$			29.7455i			1.74073i
$293$			6.59824i			0.385474i	0.981250	+	0.192737i	$0.0617364\pi$
$293$			6.59824i			0.385474i	−0.981250	+	0.192737i	$0.938264\pi$
$294$		−	1.97084i		−	0.114941i
$295$	18.8612			1.09814
$296$		−	16.1420i		−	0.938235i
$297$	−2.33159			−0.135293
$298$	−2.00178			−0.115960
$299$		−	12.5355i		−	0.724949i
$300$	−2.55351			−0.147427
$301$	35.1787			2.02767
$302$	6.89742			0.396902
$303$	4.98795			0.286551
$304$	−17.6792			−1.01397
$305$	2.62454			0.150281
$306$		−	2.40084i		−	0.137247i
$307$		−	13.1559i		−	0.750847i	−0.926853	−	0.375424i	$-0.877497\pi$
$307$		−	13.1559i		−	0.750847i	0.926853	−	0.375424i	$-0.122503\pi$
$308$			14.4036i			0.820719i
$309$		−	6.76460i		−	0.384825i
$310$	3.68967			0.209559
$311$	6.93457			0.393223			0.196612	−	0.980481i	$-0.437006\pi$
$311$	6.93457			0.393223			0.196612	+	0.980481i	$0.437006\pi$
$312$			2.40084i			0.135921i
$313$	−5.77187			−0.326245			−0.163123	−	0.986606i	$-0.552157\pi$
$313$	−5.77187			−0.326245			−0.163123	+	0.986606i	$0.552157\pi$
$314$	−3.01887	+	4.46273i	−0.170365	+	0.251846i
$315$	6.45143			0.363497
$316$			3.37485i			0.189850i
$317$	−27.8675			−1.56520			−0.782598	−	0.622527i	$-0.786106\pi$
$317$	−27.8675			−1.56520			−0.782598	+	0.622527i	$0.786106\pi$
$318$	−0.219733			−0.0123220
$319$			19.1087i			1.06988i
$320$		−	7.38879i		−	0.413046i
$321$			6.66267i			0.371874i
$322$		−	12.5355i		−	0.698579i
$323$	−33.7491			−1.87785
$324$	1.81510			0.100839
$325$	2.05885			0.114204
$326$	0.172803			0.00957068
$327$	0.573532			0.0317164
$328$	−5.35740			−0.295813
$329$			5.70419i			0.314482i
$330$	1.90048			0.104618
$331$	16.7263			0.919363			0.459681	−	0.888084i	$-0.347964\pi$
$331$	16.7263			0.919363			0.459681	+	0.888084i	$0.347964\pi$
$332$		−	6.45963i		−	0.354518i
$333$	9.83969			0.539212
$334$			5.03658i			0.275589i
$335$			20.9544i			1.14486i
$336$		−	9.95427i		−	0.543050i
$337$		−	35.6386i		−	1.94136i	−0.240379	−	0.970679i	$-0.577272\pi$
$337$		−	35.6386i		−	1.94136i	0.240379	−	0.970679i	$-0.422728\pi$
$338$			4.66906i			0.253963i
$339$	6.06645			0.329485
$340$		−	19.2102i		−	1.04182i
$341$	−10.5543			−0.571549
$342$			2.59920i			0.140549i
$343$			8.22499i			0.444108i
$344$	−16.9566			−0.914239
$345$	16.2366			0.874152
$346$		−	4.42697i		−	0.237995i
$347$	8.96081			0.481041			0.240521	−	0.970644i	$-0.422682\pi$
$347$	8.96081			0.481041			0.240521	+	0.970644i	$0.422682\pi$
$348$		−	14.8757i		−	0.797423i
$349$	−14.3533			−0.768314			−0.384157	−	0.923268i	$-0.625508\pi$
$349$	−14.3533			−0.768314			−0.384157	+	0.923268i	$0.625508\pi$
$350$	2.05885			0.110050
$351$	−1.46348			−0.0781148
$352$		−	10.5823i		−	0.564039i
$353$	−9.54221			−0.507881			−0.253940	−	0.967220i	$-0.581727\pi$
$353$	−9.54221			−0.507881			−0.253940	+	0.967220i	$0.581727\pi$
$354$	−4.27858			−0.227404
$355$			24.2455i			1.28682i
$356$	−31.3891			−1.66362
$357$		−	19.0024i		−	1.00572i
$358$	−2.33711			−0.123520
$359$		−	6.10398i		−	0.322156i	−0.986942	−	0.161078i	$-0.948503\pi$
$359$		−	6.10398i		−	0.322156i	0.986942	−	0.161078i	$-0.0514971\pi$
$360$	−3.10968			−0.163895
$361$	17.5374			0.923023
$362$	−6.02876			−0.316865
$363$	5.56367			0.292017
$364$			9.04073i			0.473863i
$365$	31.0643			1.62598
$366$	−0.595366			−0.0311203
$367$		−	28.4733i		−	1.48629i	−0.669129	−	0.743146i	$-0.733333\pi$
$367$		−	28.4733i		−	1.48629i	0.669129	−	0.743146i	$-0.266667\pi$
$368$		−	25.0524i		−	1.30595i
$369$		−	3.26571i		−	0.170006i
$370$	−8.02032			−0.416957
$371$	−1.73916			−0.0902929
$372$	8.21633			0.425997
$373$			11.8904i			0.615663i	0.951441	+	0.307832i	$0.0996033\pi$
$373$			11.8904i			0.615663i	−0.951441	+	0.307832i	$0.900397\pi$
$374$		−	5.59778i		−	0.289454i
$375$			12.1446i			0.627143i
$376$		−	2.74950i		−	0.141795i
$377$			11.9940i			0.617724i
$378$	−1.46348			−0.0752733
$379$		−	7.43615i		−	0.381970i	−0.981593	−	0.190985i	$-0.938832\pi$
$379$		−	7.43615i		−	0.381970i	0.981593	−	0.190985i	$-0.0611681\pi$
$380$			20.7974i			1.06688i
$381$	−8.53426			−0.437224
$382$	6.20240			0.317342
$383$			24.6404i			1.25907i	0.776974	+	0.629533i	$0.216754\pi$
$383$			24.6404i			1.25907i	−0.776974	+	0.629533i	$0.783246\pi$
$384$			10.7534i			0.548759i
$385$	15.0421			0.766618
$386$		−	7.66485i		−	0.390131i
$387$		−	10.3363i		−	0.525421i
$388$			30.4859i			1.54769i
$389$	−31.2079			−1.58230			−0.791151	−	0.611621i	$-0.790518\pi$
$389$	−31.2079			−1.58230			−0.791151	+	0.611621i	$0.790518\pi$
$390$	1.19288			0.0604038
$391$		−	47.8244i		−	2.41858i
$392$			7.51893i			0.379764i
$393$			12.2174i			0.616286i
$394$		−	2.99682i		−	0.150978i
$395$	3.52447			0.177336
$396$	4.23207			0.212670
$397$		−	11.0593i		−	0.555052i	−0.960718	−	0.277526i	$-0.910486\pi$
$397$		−	11.0593i		−	0.555052i	0.960718	−	0.277526i	$-0.0895145\pi$
$398$		−	3.52071i		−	0.176477i
$399$			20.5724i			1.02991i
$400$	4.11462			0.205731
$401$			27.3231i			1.36445i	0.731141	+	0.682226i	$0.238988\pi$
$401$			27.3231i			1.36445i	−0.731141	+	0.682226i	$0.761012\pi$
$402$		−	4.75342i		−	0.237079i
$403$	−6.62467			−0.329998
$404$	−9.05363			−0.450435
$405$		−	1.89557i		−	0.0941916i
$406$			11.9940i			0.595254i
$407$	22.9422			1.13720
$408$			9.15943i			0.453460i
$409$			15.5309i			0.767951i	0.923343	+	0.383976i	$0.125445\pi$
$409$			15.5309i			0.767951i	−0.923343	+	0.383976i	$0.874555\pi$
$410$			2.66188i			0.131461i
$411$		−	5.16215i		−	0.254630i
$412$			12.2784i			0.604914i
$413$	−33.8646			−1.66637
$414$	−3.68321			−0.181020
$415$	−6.74602			−0.331149
$416$		−	6.64224i		−	0.325662i
$417$		−	14.3404i		−	0.702253i
$418$			6.06027i			0.296418i
$419$	13.5614			0.662517			0.331259	−	0.943540i	$-0.392527\pi$
$419$	13.5614			0.662517			0.331259	+	0.943540i	$0.392527\pi$
$420$	−11.7100			−0.571389
$421$			20.5983i			1.00390i	0.864896	+	0.501951i	$0.167384\pi$
$421$			20.5983i			1.00390i	−0.864896	+	0.501951i	$0.832616\pi$
$422$	−4.13417			−0.201248
$423$	1.67601			0.0814906
$424$	0.838302			0.0407115
$425$	7.85470			0.381009
$426$		−	5.50000i		−	0.266476i
$427$	−4.71227			−0.228043
$428$		−	12.0934i		−	0.584556i
$429$	−3.41224			−0.164745
$430$			8.42506i			0.406293i
$431$	20.4731			0.986156			0.493078	−	0.869985i	$-0.335872\pi$
$431$	20.4731			0.986156			0.493078	+	0.869985i	$0.335872\pi$
$432$	−2.92478			−0.140719
$433$		−	4.41970i		−	0.212397i	−0.994345	−	0.106199i	$-0.966132\pi$
$433$		−	4.41970i		−	0.212397i	0.994345	−	0.106199i	$-0.0338679\pi$
$434$	−6.62467			−0.317994
$435$	−15.5352			−0.744858
$436$	−1.04102			−0.0498557
$437$			51.7756i			2.47676i
$438$	−7.04680			−0.336709
$439$			11.3844i			0.543346i	0.962390	+	0.271673i	$0.0875769\pi$
$439$			11.3844i			0.543346i	−0.962390	+	0.271673i	$0.912423\pi$
$440$	−7.25051			−0.345655
$441$	−4.58332			−0.218253
$442$		−	3.51358i		−	0.167124i
$443$			15.8749i			0.754239i	0.926165	+	0.377119i	$0.123085\pi$
$443$			15.8749i			0.754239i	−0.926165	+	0.377119i	$0.876915\pi$
$444$	−17.8600			−0.847599
$445$			32.7807i			1.55395i
$446$	−3.69728			−0.175071
$447$			4.65528i			0.220187i
$448$			13.2663i			0.626774i
$449$		−	20.7533i		−	0.979407i	−0.871889	−	0.489704i	$-0.837105\pi$
$449$		−	20.7533i		−	0.979407i	0.871889	−	0.489704i	$-0.162895\pi$
$450$		−	0.604933i		−	0.0285168i
$451$		−	7.61432i		−	0.358544i
$452$	−11.0112			−0.517924
$453$		−	16.0404i		−	0.753646i
$454$	−4.00646			−0.188032
$455$	9.44155			0.442627
$456$		−	9.91619i		−	0.464368i
$457$	−34.9190			−1.63344			−0.816721	−	0.577033i	$-0.804210\pi$
$457$	−34.9190			−1.63344			−0.816721	+	0.577033i	$0.804210\pi$
$458$	−11.4855			−0.536683
$459$	−5.58332			−0.260607
$460$	−29.4711			−1.37410
$461$	−20.7979			−0.968655			−0.484327	−	0.874887i	$-0.660936\pi$
$461$	−20.7979			−0.968655			−0.484327	+	0.874887i	$0.660936\pi$
$462$	−3.41224			−0.158752
$463$			19.3484i			0.899197i	0.893231	+	0.449598i	$0.148433\pi$
$463$			19.3484i			0.899197i	−0.893231	+	0.449598i	$0.851567\pi$
$464$			23.9702i			1.11279i
$465$		−	8.58059i		−	0.397915i
$466$			12.6716i			0.587000i
$467$	−1.54096			−0.0713070			−0.0356535	−	0.999364i	$-0.511351\pi$
$467$	−1.54096			−0.0713070			−0.0356535	+	0.999364i	$0.511351\pi$
$468$	2.65636			0.122790
$469$		−	37.6229i		−	1.73726i
$470$	−1.36612			−0.0630142
$471$	10.3784	+	7.02061i	0.478211	+	0.323492i
$472$	16.3232			0.751336
$473$		−	24.0999i		−	1.10812i
$474$	−0.799512			−0.0367228
$475$	−8.50366			−0.390175
$476$			34.4913i			1.58091i
$477$			0.511004i			0.0233973i
$478$		−	2.77539i		−	0.126943i
$479$			20.7399i			0.947631i	0.880624	+	0.473816i	$0.157124\pi$
$479$			20.7399i			0.947631i	−0.880624	+	0.473816i	$0.842876\pi$
$480$	8.60334			0.392687
$481$	14.4002			0.656592
$482$	5.03624			0.229394
$483$	−29.1523			−1.32648
$484$	−10.0986			−0.459027
$485$	31.8374			1.44566
$486$			0.430002i			0.0195053i
$487$	4.82336			0.218567			0.109284	−	0.994011i	$-0.465144\pi$
$487$	4.82336			0.218567			0.109284	+	0.994011i	$0.465144\pi$
$488$	2.27138			0.102821
$489$		−	0.401866i		−	0.0181730i
$490$	3.73586			0.168769
$491$			4.31637i			0.194795i	0.995246	+	0.0973974i	$0.0310518\pi$
$491$			4.31637i			0.194795i	−0.995246	+	0.0973974i	$0.968948\pi$
$492$			5.92759i			0.267237i
$493$			45.7584i			2.06086i
$494$			3.80387i			0.171144i
$495$		−	4.41970i		−	0.198651i
$496$	−13.2395			−0.594470
$497$		−	43.5320i		−	1.95268i
$498$	1.53030			0.0685746
$499$		−	1.06041i		−	0.0474704i	−0.999718	−	0.0237352i	$-0.992444\pi$
$499$		−	1.06041i		−	0.0474704i	0.999718	−	0.0237352i	$-0.00755585\pi$
$500$		−	22.0436i		−	0.985819i
$501$	11.7129			0.523295
$502$	3.19839			0.142751
$503$		−	26.4704i		−	1.18025i	−0.807310	−	0.590127i	$-0.799077\pi$
$503$		−	26.4704i		−	1.18025i	0.807310	−	0.590127i	$-0.200923\pi$
$504$	5.58332			0.248701
$505$			9.45502i			0.420743i
$506$	−8.58776			−0.381772
$507$	10.8582			0.482231
$508$	15.4905			0.687281
$509$		−	31.4921i		−	1.39586i	−0.716166	−	0.697930i	$-0.754105\pi$
$509$		−	31.4921i		−	1.39586i	0.716166	−	0.697930i	$-0.245895\pi$
$510$	4.55096			0.201520
$511$	−55.7748			−2.46733
$512$		−	22.8708i		−	1.01075i
$513$	6.04462			0.266876
$514$			7.17394i			0.316429i
$515$	12.8228			0.565039
$516$			18.7613i			0.825921i
$517$	3.90778			0.171864
$518$	14.4002			0.632708
$519$	−10.2952			−0.451910
$520$	−4.55096			−0.199573
$521$			32.7748i			1.43589i	0.696099	+	0.717945i	$0.254917\pi$
$521$			32.7748i			1.43589i	−0.696099	+	0.717945i	$0.745083\pi$
$522$	3.52410			0.154246
$523$	14.9835			0.655185			0.327592	−	0.944819i	$-0.393763\pi$
$523$	14.9835			0.655185			0.327592	+	0.944819i	$0.393763\pi$
$524$		−	22.1758i		−	0.968754i
$525$		−	4.78799i		−	0.208965i
$526$		−	0.283164i		−	0.0123465i
$527$	−25.2738			−1.10094
$528$	−6.81940			−0.296776
$529$	−50.3691			−2.18996
$530$		−	0.416519i		−	0.0180924i
$531$			9.95014i			0.431799i
$532$		−	37.3410i		−	1.61894i
$533$		−	4.77931i		−	0.207015i
$534$		−	7.43615i		−	0.321794i
$535$	−12.6296			−0.546023
$536$			18.1347i			0.783302i
$537$			5.43511i			0.234542i
$538$	−2.15584			−0.0929448
$539$	−10.6864			−0.460298
$540$			3.44065i			0.148062i
$541$		−	30.8678i		−	1.32711i	−0.748127	−	0.663556i	$-0.769046\pi$
$541$		−	30.8678i		−	1.32711i	0.748127	−	0.663556i	$-0.230954\pi$
$542$	4.98795			0.214251
$543$			14.0203i			0.601669i
$544$		−	25.3408i		−	1.08648i
$545$			1.08717i			0.0465692i
$546$	−2.14177			−0.0916595
$547$	−3.79917			−0.162441			−0.0812204	−	0.996696i	$-0.525882\pi$
$547$	−3.79917			−0.162441			−0.0812204	+	0.996696i	$0.525882\pi$
$548$			9.36981i			0.400258i
$549$			1.38457i			0.0590918i
$550$		−	1.41046i		−	0.0601422i
$551$		−	49.5390i		−	2.11043i
$552$	14.0518			0.598085
$553$	−6.32807			−0.269097
$554$		−	12.8194i		−	0.544643i
$555$			18.6518i			0.791726i
$556$			26.0292i			1.10389i
$557$	12.8088			0.542727			0.271363	−	0.962477i	$-0.412526\pi$
$557$	12.8088			0.542727			0.271363	+	0.962477i	$0.412526\pi$
$558$			1.94647i			0.0824007i
$559$		−	15.1269i		−	0.639800i
$560$	18.8690			0.797362
$561$	−13.0180			−0.549622
$562$		−	8.94313i		−	0.377243i
$563$			19.1750i			0.808129i	0.914731	+	0.404064i	$0.132403\pi$
$563$			19.1750i			0.808129i	−0.914731	+	0.404064i	$0.867597\pi$
$564$	−3.04213			−0.128097
$565$			11.4994i			0.483783i
$566$			1.12926i			0.0474664i
$567$			3.40343i			0.142931i
$568$			20.9830i			0.880428i
$569$			36.7249i			1.53959i	0.638292	+	0.769794i	$0.279641\pi$
$569$			36.7249i			1.53959i	−0.638292	+	0.769794i	$0.720359\pi$
$570$	−4.92696			−0.206368
$571$	−12.0534			−0.504419			−0.252209	−	0.967673i	$-0.581157\pi$
$571$	−12.0534			−0.504419			−0.252209	+	0.967673i	$0.581157\pi$
$572$	6.19356			0.258966
$573$		−	14.4241i		−	0.602576i
$574$		−	4.77931i		−	0.199484i
$575$		−	12.0502i		−	0.502527i
$576$	3.89793			0.162414
$577$	36.5742			1.52261			0.761303	−	0.648397i	$-0.224560\pi$
$577$	36.5742			1.52261			0.761303	+	0.648397i	$0.224560\pi$
$578$		−	6.09462i		−	0.253503i
$579$	−17.8252			−0.740788
$580$	28.1980			1.17086
$581$	12.1122			0.502500
$582$	−7.22219			−0.299369
$583$			1.19145i			0.0493450i
$584$	26.8842			1.11248
$585$		−	2.77413i		−	0.114696i
$586$	2.83726			0.117206
$587$		−	6.24832i		−	0.257896i	−0.991651	−	0.128948i	$-0.958840\pi$
$587$		−	6.24832i		−	0.257896i	0.991651	−	0.128948i	$-0.0411600\pi$
$588$	8.31918			0.343077
$589$	27.3619			1.12743
$590$		−	8.11034i		−	0.333898i
$591$	−6.96932			−0.286680
$592$	28.7789			1.18281
$593$	−3.11961			−0.128107			−0.0640534	−	0.997946i	$-0.520403\pi$
$593$	−3.11961			−0.128107			−0.0640534	+	0.997946i	$0.520403\pi$
$594$			1.00259i			0.0411368i
$595$	36.0204			1.47669
$596$		−	8.44979i		−	0.346117i
$597$	−8.18766			−0.335099
$598$	−5.39031			−0.220426
$599$		−	30.7303i		−	1.25561i	−0.778372	−	0.627803i	$-0.783954\pi$
$599$		−	30.7303i		−	1.25561i	0.778372	−	0.627803i	$-0.216046\pi$
$600$			2.30788i			0.0942188i
$601$	−2.53244			−0.103300			−0.0516502	−	0.998665i	$-0.516448\pi$
$601$	−2.53244			−0.103300			−0.0516502	+	0.998665i	$0.516448\pi$
$602$		−	15.1269i		−	0.616526i
$603$	−11.0544			−0.450170
$604$			29.1150i			1.18467i
$605$			10.5463i			0.428769i
$606$		−	2.14483i		−	0.0871277i
$607$		−	27.0321i		−	1.09720i	−0.836085	−	0.548599i	$-0.815161\pi$
$607$		−	27.0321i		−	1.09720i	0.836085	−	0.548599i	$-0.184839\pi$
$608$			27.4345i			1.11261i
$609$	27.8930			1.13028
$610$		−	1.12856i		−	0.0456940i
$611$	2.45281			0.0992302
$612$	10.1343			0.409654
$613$		−	46.4153i		−	1.87470i	−0.348395	−	0.937348i	$-0.613273\pi$
$613$		−	46.4153i		−	1.87470i	0.348395	−	0.937348i	$-0.386727\pi$
$614$	−5.65706			−0.228300
$615$	6.19039			0.249621
$616$	13.0180			0.524512
$617$	26.5938			1.07062			0.535312	−	0.844654i	$-0.320194\pi$
$617$	26.5938			1.07062			0.535312	+	0.844654i	$0.320194\pi$
$618$	−2.90879			−0.117009
$619$	−17.7745			−0.714417			−0.357209	−	0.934025i	$-0.616271\pi$
$619$	−17.7745			−0.714417			−0.357209	+	0.934025i	$0.616271\pi$
$620$			15.5746i			0.625492i
$621$			8.56557i			0.343725i
$622$		−	2.98188i		−	0.119562i
$623$		−	58.8565i		−	2.35804i
$624$	−4.28036			−0.171351
$625$	−15.9868			−0.639472
$626$			2.48192i			0.0991973i
$627$	14.0936			0.562844
$628$	−18.8378	−	12.7431i	−0.751710	−	0.508505i
$629$	54.9382			2.19053
$630$		−	2.77413i		−	0.110524i
$631$	23.4843			0.934894			0.467447	−	0.884021i	$-0.345174\pi$
$631$	23.4843			0.934894			0.467447	+	0.884021i	$0.345174\pi$
$632$	3.05022			0.121331
$633$			9.61430i			0.382134i
$634$			11.9831i			0.475909i
$635$		−	16.1773i		−	0.641976i
$636$		−	0.927522i		−	0.0367787i
$637$	−6.70760			−0.265765
$638$	8.21678			0.325305
$639$	−12.7906			−0.505990
$640$	−20.3839			−0.805744
$641$	26.9271			1.06356			0.531778	−	0.846884i	$-0.321524\pi$
$641$	26.9271			1.06356			0.531778	+	0.846884i	$0.321524\pi$
$642$	2.86496			0.113071
$643$		−	13.4443i		−	0.530190i	−0.964222	−	0.265095i	$-0.914597\pi$
$643$		−	13.4443i		−	0.530190i	0.964222	−	0.265095i	$-0.0854034\pi$
$644$	52.9143			2.08512
$645$	19.5931			0.771477
$646$			14.5122i			0.570973i
$647$	−34.7069			−1.36447			−0.682234	−	0.731134i	$-0.738991\pi$
$647$	−34.7069			−1.36447			−0.682234	+	0.731134i	$0.738991\pi$
$648$		−	1.64050i		−	0.0644449i
$649$			23.1997i			0.910668i
$650$		−	0.885307i		−	0.0347246i
$651$			15.4061i			0.603814i
$652$			0.729426i			0.0285665i
$653$	2.82575			0.110580			0.0552901	−	0.998470i	$-0.482392\pi$
$653$	2.82575			0.110580			0.0552901	+	0.998470i	$0.482392\pi$
$654$		−	0.246620i		−	0.00964360i
$655$	−23.1589			−0.904894
$656$		−	9.55149i		−	0.372923i
$657$			16.3878i			0.639351i
$658$	2.45281			0.0956206
$659$	−0.589317			−0.0229565			−0.0114783	−	0.999934i	$-0.503654\pi$
$659$	−0.589317			−0.0229565			−0.0114783	+	0.999934i	$0.503654\pi$
$660$			8.02219i			0.312263i
$661$	11.0556			0.430012			0.215006	−	0.976613i	$-0.431023\pi$
$661$	11.0556			0.430012			0.215006	+	0.976613i	$0.431023\pi$
$662$		−	7.19236i		−	0.279539i
$663$	−8.17108			−0.317338
$664$	−5.83826			−0.226569
$665$	−38.9965			−1.51222
$666$		−	4.23109i		−	0.163951i
$667$	70.1996			2.71814
$668$	−21.2601			−0.822578
$669$			8.59828i			0.332429i
$670$	9.01043			0.348103
$671$			3.22825i			0.124625i
$672$	−15.4470			−0.595881
$673$			0.328899i			0.0126781i	0.999980	+	0.00633906i	$0.00201780\pi$
$673$			0.328899i			0.0126781i	−0.999980	+	0.00633906i	$0.997982\pi$
$674$	−15.3247			−0.590284
$675$	−1.40682			−0.0541483
$676$	−19.7087			−0.758029
$677$	21.1742			0.813793			0.406896	−	0.913474i	$-0.366611\pi$
$677$	21.1742			0.813793			0.406896	+	0.913474i	$0.366611\pi$
$678$		−	2.60859i		−	0.100182i
$679$	−57.1630			−2.19371
$680$	−17.3623			−0.665816
$681$			9.31730i			0.357040i
$682$			4.53838i			0.173784i
$683$			46.2689i			1.77043i	0.465182	+	0.885215i	$0.345989\pi$
$683$			46.2689i			1.77043i	−0.465182	+	0.885215i	$0.654011\pi$
$684$	−10.9716			−0.419509
$685$	9.78521			0.373874
$686$	3.53676			0.135034
$687$			26.7104i			1.01907i
$688$		−	30.2313i		−	1.15256i
$689$			0.747844i			0.0284906i
$690$		−	6.98179i		−	0.265792i
$691$		−	18.5354i		−	0.705118i	−0.935790	−	0.352559i	$-0.885312\pi$
$691$		−	18.5354i		−	0.705118i	0.935790	−	0.352559i	$-0.114688\pi$
$692$	18.6868			0.710367
$693$			7.93541i			0.301441i
$694$		−	3.85316i		−	0.146264i
$695$	27.1832			1.03112
$696$	−13.4448			−0.509624
$697$		−	18.2335i		−	0.690644i
$698$			6.17194i			0.233611i
$699$	29.4687			1.11461
$700$			8.69068i			0.328477i
$701$		−	12.4010i		−	0.468381i	−0.972191	−	0.234190i	$-0.924756\pi$
$701$		−	12.4010i		−	0.468381i	0.972191	−	0.234190i	$-0.0752439\pi$
$702$			0.629299i			0.0237514i
$703$	−59.4772			−2.24323
$704$	9.08838			0.342531
$705$			3.17700i			0.119653i
$706$			4.10317i			0.154425i
$707$		−	16.9761i		−	0.638454i
$708$		−	18.0605i		−	0.678755i
$709$	30.0005			1.12669			0.563347	−	0.826220i	$-0.309513\pi$
$709$	30.0005			1.12669			0.563347	+	0.826220i	$0.309513\pi$
$710$	10.4256			0.391267
$711$			1.85932i			0.0697300i
$712$			28.3697i			1.06320i
$713$			38.7734i			1.45208i
$714$	−8.17108			−0.305795
$715$		−	6.46814i		−	0.241895i
$716$		−	9.86526i		−	0.368682i
$717$	−6.45437			−0.241043
$718$	−2.62472			−0.0979538
$719$			43.6502i			1.62788i	0.580949	+	0.813940i	$0.302681\pi$
$719$			43.6502i			1.62788i	−0.580949	+	0.813940i	$0.697319\pi$
$720$		−	5.54412i		−	0.206617i
$721$	−23.0228			−0.857415
$722$		−	7.54113i		−	0.280652i
$723$		−	11.7121i		−	0.435579i
$724$		−	25.4483i		−	0.945777i
$725$			11.5296i			0.428200i
$726$		−	2.39239i		−	0.0887898i
$727$	3.78327			0.140314			0.0701569	−	0.997536i	$-0.477650\pi$
$727$	3.78327			0.140314			0.0701569	+	0.997536i	$0.477650\pi$
$728$	8.17108			0.302840
$729$	1.00000			0.0370370
$730$		−	13.3577i		−	0.494391i
$731$		−	57.7106i		−	2.13450i
$732$		−	2.51312i		−	0.0928877i
$733$	1.20321			0.0444417			0.0222209	−	0.999753i	$-0.492926\pi$
$733$	1.20321			0.0444417			0.0222209	+	0.999753i	$0.492926\pi$
$734$	−12.2436			−0.451918
$735$		−	8.68800i		−	0.320462i
$736$	−38.8762			−1.43300
$737$	−25.7744			−0.949412
$738$	−1.40426			−0.0516916
$739$	26.4164			0.971744			0.485872	−	0.874030i	$-0.338502\pi$
$739$	26.4164			0.971744			0.485872	+	0.874030i	$0.338502\pi$
$740$		−	33.8549i		−	1.24453i
$741$	8.84618			0.324973
$742$			0.747844i			0.0274542i
$743$	12.4404			0.456394			0.228197	−	0.973615i	$-0.426717\pi$
$743$	12.4404			0.456394			0.228197	+	0.973615i	$0.426717\pi$
$744$		−	7.42598i		−	0.272249i
$745$	−8.82440			−0.323301
$746$	5.11291			0.187197
$747$		−	3.55883i		−	0.130211i
$748$	23.6290			0.863963
$749$	22.6759			0.828560
$750$	5.22219			0.190687
$751$			21.0227i			0.767129i	0.923514	+	0.383564i	$0.125304\pi$
$751$			21.0227i			0.767129i	−0.923514	+	0.383564i	$0.874696\pi$
$752$	4.90197			0.178756
$753$		−	7.43809i		−	0.271059i
$754$	5.15746			0.187823
$755$	30.4058			1.10658
$756$		−	6.17756i		−	0.224676i
$757$			0.649241i			0.0235971i	0.999930	+	0.0117985i	$0.00375568\pi$
$757$			0.649241i			0.0235971i	−0.999930	+	0.0117985i	$0.996244\pi$
$758$	−3.19756			−0.116141
$759$			19.9714i			0.724918i
$760$	18.7968			0.681833
$761$		−	13.4001i		−	0.485753i	−0.970057	−	0.242877i	$-0.921909\pi$
$761$		−	13.4001i		−	0.485753i	0.970057	−	0.242877i	$-0.0780910\pi$
$762$			3.66975i			0.132941i
$763$		−	1.95197i		−	0.0706662i
$764$			26.1812i			0.947202i
$765$		−	10.5836i		−	0.382650i
$766$	10.5954			0.382828
$767$			14.5618i			0.525797i
$768$	−3.17186			−0.114455
$769$	14.0166			0.505453			0.252726	−	0.967538i	$-0.418673\pi$
$769$	14.0166			0.505453			0.252726	+	0.967538i	$0.418673\pi$
$770$		−	6.46814i		−	0.233096i
$771$	16.6835			0.600842
$772$	32.3544			1.16446
$773$	−13.8201			−0.497073			−0.248537	−	0.968623i	$-0.579950\pi$
$773$	−13.8201			−0.497073			−0.248537	+	0.968623i	$0.579950\pi$
$774$	−4.44461			−0.159758
$775$	−6.36817			−0.228751
$776$	27.5533			0.989108
$777$		−	33.4887i		−	1.20140i
$778$			13.4194i			0.481110i
$779$			19.7400i			0.707259i
$780$			5.03532i			0.180293i
$781$	−29.8226			−1.06714
$782$	−20.5646			−0.735387
$783$		−	8.19556i		−	0.292885i
$784$	−13.4052			−0.478757
$785$	−13.3080	+	19.6730i	−0.474985	+	0.702158i
$786$	5.25350			0.187386
$787$			41.1510i			1.46688i	0.679757	+	0.733438i	$0.262085\pi$
$787$			41.1510i			1.46688i	−0.679757	+	0.733438i	$0.737915\pi$
$788$	12.6500			0.450638
$789$	−0.658517			−0.0234438
$790$		−	1.51553i		−	0.0539202i
$791$		−	20.6467i		−	0.734113i
$792$		−	3.82498i		−	0.135915i
$793$			2.02628i			0.0719555i
$794$	−4.75553			−0.168768
$795$	−0.968644			−0.0343542
$796$	14.8614			0.526749
$797$	20.1706			0.714479			0.357239	−	0.934013i	$-0.383718\pi$
$797$	20.1706			0.714479			0.357239	+	0.934013i	$0.383718\pi$
$798$	8.84618			0.313151
$799$	9.35772			0.331052
$800$		−	6.38505i		−	0.225746i
$801$	−17.2933			−0.611029
$802$	11.7490			0.414872
$803$			38.2098i			1.34839i
$804$	20.0648			0.707632
$805$		−	55.2602i		−	1.94767i
$806$			2.84862i			0.100338i
$807$			5.01356i			0.176486i
$808$			8.18274i			0.287868i
$809$			21.3875i			0.751945i	0.926631	+	0.375973i	$0.122691\pi$
$809$			21.3875i			0.751945i	−0.926631	+	0.375973i	$0.877309\pi$
$810$	−0.815098			−0.0286396
$811$			16.4792i			0.578664i	0.957229	+	0.289332i	$0.0934332\pi$
$811$			16.4792i			0.578664i	−0.957229	+	0.289332i	$0.906567\pi$
$812$	−50.6285			−1.77671
$813$		−	11.5998i		−	0.406824i
$814$		−	9.86518i		−	0.345774i
$815$	0.761765			0.0266835
$816$	−16.3300			−0.571664
$817$			62.4787i			2.18585i
$818$	6.67829			0.233501
$819$			4.98085i			0.174045i
$820$	−11.2362			−0.392384
$821$	−1.28228			−0.0447518			−0.0223759	−	0.999750i	$-0.507123\pi$
$821$	−1.28228			−0.0447518			−0.0223759	+	0.999750i	$0.507123\pi$
$822$	−2.21973			−0.0774221
$823$		−	3.83740i		−	0.133763i	−0.997761	−	0.0668817i	$-0.978695\pi$
$823$		−	3.83740i		−	0.133763i	0.997761	−	0.0668817i	$-0.0213050\pi$
$824$	11.0973			0.386593
$825$	−3.28012			−0.114199
$826$			14.5618i			0.506671i
$827$	41.7457			1.45164			0.725819	−	0.687885i	$-0.241461\pi$
$827$	41.7457			1.45164			0.725819	+	0.687885i	$0.241461\pi$
$828$		−	15.5474i		−	0.540308i
$829$	6.24546			0.216914			0.108457	−	0.994101i	$-0.465409\pi$
$829$	6.24546			0.216914			0.108457	+	0.994101i	$0.465409\pi$
$830$			2.90080i			0.100688i
$831$	−29.8124			−1.03418
$832$	5.70454			0.197769
$833$	−25.5901			−0.886646
$834$	−6.16640			−0.213525
$835$			22.2027i			0.768355i
$836$	−25.5813			−0.884747
$837$	4.52666			0.156464
$838$		−	5.83142i		−	0.201443i
$839$		−	8.81092i		−	0.304187i	−0.988366	−	0.152093i	$-0.951399\pi$
$839$		−	8.81092i		−	0.304187i	0.988366	−	0.152093i	$-0.0486014\pi$
$840$			10.5836i			0.365168i
$841$	−38.1671			−1.31611
$842$	8.85733			0.305244
$843$	−20.7979			−0.716318
$844$		−	17.4509i		−	0.600685i
$845$			20.5825i			0.708060i
$846$		−	0.720689i		−	0.0247778i
$847$		−	18.9355i		−	0.650633i
$848$			1.49457i			0.0513239i
$849$	2.62618			0.0901302
$850$		−	3.37754i		−	0.115849i
$851$		−	84.2826i		−	2.88917i
$852$	23.2163			0.795376
$853$	0.736874			0.0252301			0.0126150	−	0.999920i	$-0.495984\pi$
$853$	0.736874			0.0252301			0.0126150	+	0.999920i	$0.495984\pi$
$854$			2.02628i			0.0693380i
$855$			11.4580i			0.391855i
$856$	−10.9301			−0.373583
$857$		−	15.8246i		−	0.540559i	−0.962782	−	0.270279i	$-0.912884\pi$
$857$		−	15.8246i		−	0.540559i	0.962782	−	0.270279i	$-0.0871161\pi$
$858$			1.46727i			0.0500918i
$859$		−	15.4988i		−	0.528812i	−0.964411	−	0.264406i	$-0.914824\pi$
$859$		−	15.4988i		−	0.528812i	0.964411	−	0.264406i	$-0.0851759\pi$
$860$	−35.5634			−1.21270
$861$	−11.1146			−0.378785
$862$		−	8.80348i		−	0.299848i
$863$		−	21.5139i		−	0.732341i	−0.930548	−	0.366170i	$-0.880669\pi$
$863$		−	21.5139i		−	0.732341i	0.930548	−	0.366170i	$-0.119331\pi$
$864$			4.53866i			0.154408i
$865$		−	19.5153i		−	0.663541i
$866$	−1.90048			−0.0645809
$867$	−14.1735			−0.481356
$868$		−	27.9637i		−	0.949149i
$869$			4.33518i			0.147061i
$870$			6.68018i			0.226479i
$871$	−16.1779			−0.548167
$872$			0.940879i			0.0318622i
$873$			16.7957i			0.568449i
$874$	22.2636			0.753078
$875$	41.3332			1.39732
$876$		−	29.7455i		−	1.00501i
$877$			29.0792i			0.981935i	0.871178	+	0.490967i	$0.163357\pi$
$877$			29.0792i			0.981935i	−0.871178	+	0.490967i	$0.836643\pi$
$878$	4.89530			0.165208
$879$		−	6.59824i		−	0.222553i
$880$		−	12.9266i		−	0.435757i
$881$		−	36.5364i		−	1.23094i	−0.788159	−	0.615471i	$-0.788966\pi$
$881$		−	36.5364i		−	1.23094i	0.788159	−	0.615471i	$-0.211034\pi$
$882$			1.97084i			0.0663615i
$883$			57.4310i			1.93271i	0.257219	+	0.966353i	$0.417194\pi$
$883$			57.4310i			1.93271i	−0.257219	+	0.966353i	$0.582806\pi$
$884$	14.8313			0.498831
$885$	−18.8612			−0.634012
$886$	6.82623			0.229332
$887$			14.3057i			0.480338i	0.970731	+	0.240169i	$0.0772028\pi$
$887$			14.3057i			0.480338i	−0.970731	+	0.240169i	$0.922797\pi$
$888$			16.1420i			0.541690i
$889$			29.0457i			0.974163i
$890$	14.0958			0.472491
$891$	2.33159			0.0781114
$892$		−	15.6067i		−	0.522552i
$893$	−10.1309			−0.339016
$894$	2.00178			0.0669495
$895$	−10.3026			−0.344379
$896$	36.5985			1.22267
$897$			12.5355i			0.418550i
$898$	−8.92394			−0.297796
$899$		−	37.0985i		−	1.23730i
$900$	2.55351			0.0851169
$901$			2.85310i			0.0950505i
$902$	−3.27417			−0.109018
$903$	−35.1787			−1.17067
$904$			9.95201i			0.330999i
$905$	−26.5765			−0.883432
$906$	−6.89742			−0.229151
$907$	−22.3760			−0.742983			−0.371492	−	0.928436i	$-0.621154\pi$
$907$	−22.3760			−0.742983			−0.371492	+	0.928436i	$0.621154\pi$
$908$		−	16.9118i		−	0.561239i
$909$	−4.98795			−0.165440
$910$		−	4.05988i		−	0.134584i
$911$	33.8820			1.12256			0.561280	−	0.827626i	$-0.310309\pi$
$911$	33.8820			1.12256			0.561280	+	0.827626i	$0.310309\pi$
$912$	17.6792			0.585416
$913$		−	8.29776i		−	0.274616i
$914$			15.0152i			0.496660i
$915$	−2.62454			−0.0867647
$916$		−	48.4820i		−	1.60189i
$917$	41.5810			1.37313
$918$			2.40084i			0.0792395i
$919$			0.595727i			0.0196512i	0.999952	+	0.00982561i	$0.00312764\pi$
$919$			0.595727i			0.0196512i	−0.999952	+	0.00982561i	$0.996872\pi$
$920$			26.6362i			0.878170i
$921$			13.1559i			0.433502i
$922$			8.94313i			0.294526i
$923$	−18.7188			−0.616138
$924$		−	14.4036i		−	0.473842i
$925$	13.8426			0.455143
$926$	8.31985			0.273407
$927$			6.76460i			0.222179i
$928$	37.1968			1.22105
$929$	−34.9634			−1.14711			−0.573556	−	0.819166i	$-0.694437\pi$
$929$	−34.9634			−1.14711			−0.573556	+	0.819166i	$0.694437\pi$
$930$	−3.68967			−0.120989
$931$	27.7044			0.907976
$932$	−53.4886			−1.75208
$933$	−6.93457			−0.227028
$934$			0.662614i			0.0216814i
$935$		−	24.6766i		−	0.807011i
$936$		−	2.40084i		−	0.0784738i
$937$			23.7710i			0.776566i	0.921540	+	0.388283i	$0.126932\pi$
$937$			23.7710i			0.776566i	−0.921540	+	0.388283i	$0.873068\pi$
$938$	−16.1779			−0.528227
$939$	5.77187			0.188358
$940$		−	5.76657i		−	0.188085i
$941$	35.4342			1.15512			0.577562	−	0.816347i	$-0.304004\pi$
$941$	35.4342			1.15512			0.577562	+	0.816347i	$0.304004\pi$
$942$	3.01887	−	4.46273i	0.0983602	−	0.145404i
$943$	−27.9727			−0.910917
$944$			29.1020i			0.947188i
$945$	−6.45143			−0.209865
$946$	−10.3630			−0.336931
$947$		−	36.2703i		−	1.17863i	−0.807904	−	0.589314i	$-0.799398\pi$
$947$		−	36.2703i		−	1.17863i	0.807904	−	0.589314i	$-0.200602\pi$
$948$		−	3.37485i		−	0.109610i
$949$			23.9833i			0.778530i
$950$			3.65659i			0.118635i
$951$	27.8675			0.903666
$952$	31.1735			1.01034
$953$	2.11516			0.0685168			0.0342584	−	0.999413i	$-0.489093\pi$
$953$	2.11516			0.0685168			0.0342584	+	0.999413i	$0.489093\pi$
$954$	0.219733			0.00711411
$955$	27.3419			0.884764
$956$	11.7153			0.378901
$957$		−	19.1087i		−	0.617697i
$958$	8.91820			0.288134
$959$	−17.5690			−0.567332
$960$			7.38879i			0.238472i
$961$	−10.5094			−0.339013
$962$		−	6.19211i		−	0.199642i
$963$		−	6.66267i		−	0.214701i
$964$			21.2587i			0.684696i
$965$		−	33.7888i		−	1.08770i
$966$			12.5355i			0.403325i
$967$	29.9993			0.964713			0.482357	−	0.875975i	$-0.339781\pi$
$967$	29.9993			0.964713			0.482357	+	0.875975i	$0.339781\pi$
$968$			9.12719i			0.293359i
$969$	33.7491			1.08418
$970$		−	13.6902i		−	0.439564i
$971$		−	17.4791i		−	0.560930i	−0.959864	−	0.280465i	$-0.909511\pi$
$971$		−	17.4791i		−	0.560930i	0.959864	−	0.280465i	$-0.0904887\pi$
$972$	−1.81510			−0.0582193
$973$	−48.8065			−1.56467
$974$		−	2.07405i		−	0.0664569i
$975$	−2.05885			−0.0659358
$976$			4.04955i			0.129623i
$977$	−50.8097			−1.62555			−0.812773	−	0.582580i	$-0.802043\pi$
$977$	−50.8097			−1.62555			−0.812773	+	0.582580i	$0.802043\pi$
$978$	−0.172803			−0.00552563
$979$	−40.3210			−1.28866
$980$			15.7696i			0.503741i
$981$	−0.573532			−0.0183115
$982$	1.85605			0.0592288
$983$			12.5747i			0.401072i	0.979686	+	0.200536i	$0.0642684\pi$
$983$			12.5747i			0.401072i	−0.979686	+	0.200536i	$0.935732\pi$
$984$	5.35740			0.170788
$985$		−	13.2108i		−	0.420932i
$986$	19.6762			0.626618
$987$		−	5.70419i		−	0.181566i
$988$	−16.0567			−0.510831
$989$	−88.5359			−2.81528
$990$	−1.90048			−0.0604012
$991$	1.37020			0.0435258			0.0217629	−	0.999763i	$-0.493072\pi$
$991$	1.37020			0.0435258			0.0217629	+	0.999763i	$0.493072\pi$
$992$			20.5449i			0.652303i
$993$	−16.7263			−0.530794
$994$	−18.7188			−0.593726
$995$		−	15.5203i		−	0.492026i
$996$			6.45963i			0.204681i
$997$		−	3.75209i		−	0.118830i	−0.998233	−	0.0594149i	$-0.981076\pi$
$997$		−	3.75209i		−	0.118830i	0.998233	−	0.0594149i	$-0.0189235\pi$
$998$	−0.455977			−0.0144337
$999$	−9.83969			−0.311314

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.2.b.a.313.5	✓	12
3.2	odd	2		1413.2.b.c.784.8		12
157.156	even	2	inner	471.2.b.a.313.8	yes	12
471.470	odd	2		1413.2.b.c.784.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.2.b.a.313.5	✓	12	1.1	even	1	trivial
471.2.b.a.313.8	yes	12	157.156	even	2	inner
1413.2.b.c.784.5		12	471.470	odd	2
1413.2.b.c.784.8		12	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 471 = 3 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	471.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.430002i q^{2} -1.00000 q^{3} +1.81510 q^{4} -1.89557i q^{5} +0.430002i q^{6} +3.40343i q^{7} -1.64050i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.430002i q^{2} -1.00000 q^{3} +1.81510 q^{4} -1.89557i q^{5} +0.430002i q^{6} +3.40343i q^{7} -1.64050i q^{8} +1.00000 q^{9} -0.815098 q^{10} +2.33159 q^{11} -1.81510 q^{12} +1.46348 q^{13} +1.46348 q^{14} +1.89557i q^{15} +2.92478 q^{16} +5.58332 q^{17} -0.430002i q^{18} -6.04462 q^{19} -3.44065i q^{20} -3.40343i q^{21} -1.00259i q^{22} -8.56557i q^{23} +1.64050i q^{24} +1.40682 q^{25} -0.629299i q^{26} -1.00000 q^{27} +6.17756i q^{28} +8.19556i q^{29} +0.815098 q^{30} -4.52666 q^{31} -4.53866i q^{32} -2.33159 q^{33} -2.40084i q^{34} +6.45143 q^{35} +1.81510 q^{36} +9.83969 q^{37} +2.59920i q^{38} -1.46348 q^{39} -3.10968 q^{40} -3.26571i q^{41} -1.46348 q^{42} -10.3363i q^{43} +4.23207 q^{44} -1.89557i q^{45} -3.68321 q^{46} +1.67601 q^{47} -2.92478 q^{48} -4.58332 q^{49} -0.604933i q^{50} -5.58332 q^{51} +2.65636 q^{52} +0.511004i q^{53} +0.430002i q^{54} -4.41970i q^{55} +5.58332 q^{56} +6.04462 q^{57} +3.52410 q^{58} +9.95014i q^{59} +3.44065i q^{60} +1.38457i q^{61} +1.94647i q^{62} +3.40343i q^{63} +3.89793 q^{64} -2.77413i q^{65} +1.00259i q^{66} -11.0544 q^{67} +10.1343 q^{68} +8.56557i q^{69} -2.77413i q^{70} -12.7906 q^{71} -1.64050i q^{72} +16.3878i q^{73} -4.23109i q^{74} -1.40682 q^{75} -10.9716 q^{76} +7.93541i q^{77} +0.629299i q^{78} +1.85932i q^{79} -5.54412i q^{80} +1.00000 q^{81} -1.40426 q^{82} -3.55883i q^{83} -6.17756i q^{84} -10.5836i q^{85} -4.44461 q^{86} -8.19556i q^{87} -3.82498i q^{88} -17.2933 q^{89} -0.815098 q^{90} +4.98085i q^{91} -15.5474i q^{92} +4.52666 q^{93} -0.720689i q^{94} +11.4580i q^{95} +4.53866i q^{96} +16.7957i q^{97} +1.97084i q^{98} +2.33159 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} - 6 q^{4} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^3 - 6 * q^4 + 12 * q^9	\( 12 q - 12 q^{3} - 6 q^{4} + 12 q^{9} + 18 q^{10} - 2 q^{11} + 6 q^{12} + 4 q^{13} + 4 q^{14} + 10 q^{16} + 2 q^{17} + 4 q^{19} + 12 q^{25} - 12 q^{27} - 18 q^{30} + 2 q^{31} + 2 q^{33} - 4 q^{35} - 6 q^{36} - 2 q^{37} - 4 q^{39} - 40 q^{40} - 4 q^{42} - 36 q^{44} + 34 q^{47} - 10 q^{48} + 10 q^{49} - 2 q^{51} - 6 q^{52} + 2 q^{56} - 4 q^{57} + 24 q^{58} + 28 q^{64} - 38 q^{67} + 32 q^{68} - 26 q^{71} - 12 q^{75} - 28 q^{76} + 12 q^{81} - 50 q^{82} + 6 q^{86} - 4 q^{89} + 18 q^{90} - 2 q^{93} - 2 q^{99}+O(q^{100}) \) 12 * q - 12 * q^3 - 6 * q^4 + 12 * q^9 + 18 * q^10 - 2 * q^11 + 6 * q^12 + 4 * q^13 + 4 * q^14 + 10 * q^16 + 2 * q^17 + 4 * q^19 + 12 * q^25 - 12 * q^27 - 18 * q^30 + 2 * q^31 + 2 * q^33 - 4 * q^35 - 6 * q^36 - 2 * q^37 - 4 * q^39 - 40 * q^40 - 4 * q^42 - 36 * q^44 + 34 * q^47 - 10 * q^48 + 10 * q^49 - 2 * q^51 - 6 * q^52 + 2 * q^56 - 4 * q^57 + 24 * q^58 + 28 * q^64 - 38 * q^67 + 32 * q^68 - 26 * q^71 - 12 * q^75 - 28 * q^76 + 12 * q^81 - 50 * q^82 + 6 * q^86 - 4 * q^89 + 18 * q^90 - 2 * q^93 - 2 * q^99

Introduction

L-functions

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