LMFDB - Embedded newform 2432.2.c.j.1217.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2432,2,Mod(1217,2432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2432, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2432.1217");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2432 = 2^{7} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2432.c (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:	$19.4196177716$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 170x^{16} + 6593x^{12} + 64168x^{8} + 95760x^{4} + 4096 $ x^20 + 170x^16 + 6593x^12 + 64168x^8 + 95760x^4 + 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{19} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1217.14
Root			$-0.814454 + 0.814454i$ of defining polynomial
Character	$\chi$	$=$	2432.1217
Dual form			2432.2.c.j.1217.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.15181i q^{3} +1.07587i q^{5} +0.238565 q^{7} +1.67333 q^{9} +O(q^{10})$	$q+1.15181i q^{3} +1.07587i q^{5} +0.238565 q^{7} +1.67333 q^{9} -2.84249i q^{11} +1.31444i q^{13} -1.23921 q^{15} -4.47279 q^{17} -1.00000i q^{19} +0.274782i q^{21} +1.94338 q^{23} +3.84249 q^{25} +5.38280i q^{27} +8.83944i q^{29} +6.64877 q^{31} +3.27402 q^{33} +0.256666i q^{35} +4.06763i q^{37} -1.51399 q^{39} +2.78211 q^{41} -1.21363i q^{43} +1.80029i q^{45} +10.2695 q^{47} -6.94309 q^{49} -5.15181i q^{51} +8.36231i q^{53} +3.05817 q^{55} +1.15181 q^{57} +9.25916i q^{59} -4.66035i q^{61} +0.399197 q^{63} -1.41417 q^{65} -4.31849i q^{67} +2.23841i q^{69} +2.78069 q^{71} +0.134460 q^{73} +4.42583i q^{75} -0.678119i q^{77} -12.7841 q^{79} -1.17998 q^{81} -13.4028i q^{83} -4.81216i q^{85} -10.1814 q^{87} -9.20650 q^{89} +0.313579i q^{91} +7.65814i q^{93} +1.07587 q^{95} -0.247499 q^{97} -4.75643i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 28 q^{9}+O(q^{10}) $ 20 * q - 28 * q^9	$ 20 q - 28 q^{9} + 8 q^{17} - 20 q^{25} - 16 q^{33} + 24 q^{41} + 52 q^{49} - 8 q^{57} - 48 q^{65} - 24 q^{73} + 68 q^{81} + 40 q^{89} - 56 q^{97}+O(q^{100}) $ 20 * q - 28 * q^9 + 8 * q^17 - 20 * q^25 - 16 * q^33 + 24 * q^41 + 52 * q^49 - 8 * q^57 - 48 * q^65 - 24 * q^73 + 68 * q^81 + 40 * q^89 - 56 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2432\mathbb{Z}\right)^\times$.

$n$	$1407$	$1921$	$2053$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.15181i	0.664999i	0.943103	+	0.332500i	$0.107892\pi$
$3$	1.15181i	0.664999i	−0.943103	+	0.332500i	$0.892108\pi$
$4$	0	0
$5$	1.07587i	0.481146i	0.970631	+	0.240573i	$0.0773353\pi$
$5$	1.07587i	0.481146i	−0.970631	+	0.240573i	$0.922665\pi$
$6$	0	0
$7$	0.238565	0.0901690	0.0450845	−	0.998983i	$-0.485644\pi$
$7$	0.238565	0.0901690	0.0450845	+	0.998983i	$0.485644\pi$
$8$	0	0
$9$	1.67333	0.557776
$10$	0	0
$11$	− 2.84249i	− 0.857044i	−0.903531	−	0.428522i	$-0.859034\pi$
$11$	− 2.84249i	− 0.857044i	0.903531	−	0.428522i	$-0.140966\pi$
$12$	0	0
$13$	1.31444i	0.364560i	0.983247	+	0.182280i	$0.0583477\pi$
$13$	1.31444i	0.364560i	−0.983247	+	0.182280i	$0.941652\pi$
$14$	0	0
$15$	−1.23921	−0.319962
$16$	0	0
$17$	−4.47279	−1.08481	−0.542405	−	0.840117i	$-0.682486\pi$
$17$	−4.47279	−1.08481	−0.542405	+	0.840117i	$0.682486\pi$
$18$	0	0
$19$	− 1.00000i	− 0.229416i
$19$	− 1.00000i	− 0.229416i
$20$	0	0
$21$	0.274782i	0.0599623i
$22$	0	0
$23$	1.94338	0.405222	0.202611	−	0.979259i	$-0.435057\pi$
$23$	1.94338	0.405222	0.202611	+	0.979259i	$0.435057\pi$
$24$	0	0
$25$	3.84249	0.768499
$26$	0	0
$27$	5.38280i	1.03592i
$28$	0	0
$29$	8.83944i	1.64144i	0.571329	+	0.820721i	$0.306428\pi$
$29$	8.83944i	1.64144i	−0.571329	+	0.820721i	$0.693572\pi$
$30$	0	0
$31$	6.64877	1.19415	0.597077	−	0.802184i	$-0.296329\pi$
$31$	6.64877	1.19415	0.597077	+	0.802184i	$0.296329\pi$
$32$	0	0
$33$	3.27402	0.569933
$34$	0	0
$35$	0.256666i	0.0433845i
$36$	0	0
$37$	4.06763i	0.668715i	0.942446	+	0.334357i	$0.108519\pi$
$37$	4.06763i	0.668715i	−0.942446	+	0.334357i	$0.891481\pi$
$38$	0	0
$39$	−1.51399	−0.242432
$40$	0	0
$41$	2.78211	0.434492	0.217246	−	0.976117i	$-0.430293\pi$
$41$	2.78211	0.434492	0.217246	+	0.976117i	$0.430293\pi$
$42$	0	0
$43$	− 1.21363i	− 0.185077i	−0.995709	−	0.0925386i	$-0.970502\pi$
$43$	− 1.21363i	− 0.185077i	0.995709	−	0.0925386i	$-0.0294982\pi$
$44$	0	0
$45$	1.80029i	0.268372i
$46$	0	0
$47$	10.2695	1.49796	0.748978	−	0.662595i	$-0.230545\pi$
$47$	10.2695	1.49796	0.748978	+	0.662595i	$0.230545\pi$
$48$	0	0
$49$	−6.94309	−0.991870
$50$	0	0
$51$	− 5.15181i	− 0.721398i
$52$	0	0
$53$	8.36231i	1.14865i	0.818627	+	0.574326i	$0.194736\pi$
$53$	8.36231i	1.14865i	−0.818627	+	0.574326i	$0.805264\pi$
$54$	0	0
$55$	3.05817	0.412363
$56$	0	0
$57$	1.15181	0.152561
$58$	0	0
$59$	9.25916i	1.20544i	0.797953	+	0.602720i	$0.205916\pi$
$59$	9.25916i	1.20544i	−0.797953	+	0.602720i	$0.794084\pi$
$60$	0	0
$61$	− 4.66035i	− 0.596697i	−0.954457	−	0.298349i	$-0.903564\pi$
$61$	− 4.66035i	− 0.596697i	0.954457	−	0.298349i	$-0.0964358\pi$
$62$	0	0
$63$	0.399197	0.0502941
$64$	0	0
$65$	−1.41417	−0.175407
$66$	0	0
$67$	− 4.31849i	− 0.527587i	−0.964579	−	0.263794i	$-0.915026\pi$
$67$	− 4.31849i	− 0.527587i	0.964579	−	0.263794i	$-0.0849738\pi$
$68$	0	0
$69$	2.23841i	0.269472i
$70$	0	0
$71$	2.78069	0.330007	0.165003	−	0.986293i	$-0.447236\pi$
$71$	2.78069	0.330007	0.165003	+	0.986293i	$0.447236\pi$
$72$	0	0
$73$	0.134460	0.0157373	0.00786866	−	0.999969i	$-0.497495\pi$
$73$	0.134460	0.0157373	0.00786866	+	0.999969i	$0.497495\pi$
$74$	0	0
$75$	4.42583i	0.511051i
$76$	0	0
$77$	− 0.678119i	− 0.0772788i
$78$	0	0
$79$	−12.7841	−1.43832	−0.719162	−	0.694843i	$-0.755474\pi$
$79$	−12.7841	−1.43832	−0.719162	+	0.694843i	$0.755474\pi$
$80$	0	0
$81$	−1.17998	−0.131109
$82$	0	0
$83$	− 13.4028i	− 1.47115i	−0.677445	−	0.735573i	$-0.736913\pi$
$83$	− 13.4028i	− 1.47115i	0.677445	−	0.735573i	$-0.263087\pi$
$84$	0	0
$85$	− 4.81216i	− 0.521952i
$86$	0	0
$87$	−10.1814	−1.09156
$88$	0	0
$89$	−9.20650	−0.975887	−0.487944	−	0.872875i	$-0.662253\pi$
$89$	−9.20650	−0.975887	−0.487944	+	0.872875i	$0.662253\pi$
$90$	0	0
$91$	0.313579i	0.0328720i
$92$	0	0
$93$	7.65814i	0.794112i
$94$	0	0
$95$	1.07587	0.110382
$96$	0	0
$97$	−0.247499	−0.0251297	−0.0125648	−	0.999921i	$-0.504000\pi$
$97$	−0.247499	−0.0251297	−0.0125648	+	0.999921i	$0.504000\pi$
$98$	0	0
$99$	− 4.75643i	− 0.478039i
$100$	0	0
$101$	16.8026i	1.67193i	0.548786	+	0.835963i	$0.315090\pi$
$101$	16.8026i	1.67193i	−0.548786	+	0.835963i	$0.684910\pi$
$102$	0	0
$103$	5.40957	0.533020	0.266510	−	0.963832i	$-0.414129\pi$
$103$	5.40957	0.533020	0.266510	+	0.963832i	$0.414129\pi$
$104$	0	0
$105$	−0.295631	−0.0288506
$106$	0	0
$107$	6.39113i	0.617853i	0.951086	+	0.308927i	$0.0999698\pi$
$107$	6.39113i	0.617853i	−0.951086	+	0.308927i	$0.900030\pi$
$108$	0	0
$109$	14.8779i	1.42505i	0.701648	+	0.712524i	$0.252448\pi$
$109$	14.8779i	1.42505i	−0.701648	+	0.712524i	$0.747552\pi$
$110$	0	0
$111$	−4.68515	−0.444695
$112$	0	0
$113$	8.59073	0.808148	0.404074	−	0.914726i	$-0.367594\pi$
$113$	8.59073	0.808148	0.404074	+	0.914726i	$0.367594\pi$
$114$	0	0
$115$	2.09083i	0.194971i
$116$	0	0
$117$	2.19949i	0.203343i
$118$	0	0
$119$	−1.06705	−0.0978163
$120$	0	0
$121$	2.92023	0.265476
$122$	0	0
$123$	3.20446i	0.288937i
$124$	0	0
$125$	9.51342i	0.850906i
$126$	0	0
$127$	1.15983	0.102918	0.0514592	−	0.998675i	$-0.483613\pi$
$127$	1.15983	0.102918	0.0514592	+	0.998675i	$0.483613\pi$
$128$	0	0
$129$	1.39788	0.123076
$130$	0	0
$131$	3.85559i	0.336864i	0.985713	+	0.168432i	$0.0538704\pi$
$131$	3.85559i	0.336864i	−0.985713	+	0.168432i	$0.946130\pi$
$132$	0	0
$133$	− 0.238565i	− 0.0206862i
$134$	0	0
$135$	−5.79122	−0.498428
$136$	0	0
$137$	1.42976	0.122152	0.0610761	−	0.998133i	$-0.480547\pi$
$137$	1.42976	0.122152	0.0610761	+	0.998133i	$0.480547\pi$
$138$	0	0
$139$	3.58190i	0.303813i	0.988395	+	0.151907i	$0.0485413\pi$
$139$	3.58190i	0.303813i	−0.988395	+	0.151907i	$0.951459\pi$
$140$	0	0
$141$	11.8285i	0.996139i
$142$	0	0
$143$	3.73629	0.312444
$144$	0	0
$145$	−9.51013	−0.789773
$146$	0	0
$147$	− 7.99713i	− 0.659592i
$148$	0	0
$149$	16.2763i	1.33341i	0.745322	+	0.666705i	$0.232296\pi$
$149$	16.2763i	1.33341i	−0.745322	+	0.666705i	$0.767704\pi$
$150$	0	0
$151$	−14.3242	−1.16569	−0.582845	−	0.812583i	$-0.698061\pi$
$151$	−14.3242	−1.16569	−0.582845	+	0.812583i	$0.698061\pi$
$152$	0	0
$153$	−7.48445	−0.605082
$154$	0	0
$155$	7.15325i	0.574563i
$156$	0	0
$157$	− 3.73495i	− 0.298081i	−0.988831	−	0.149041i	$-0.952381\pi$
$157$	− 3.73495i	− 0.298081i	0.988831	−	0.149041i	$-0.0476185\pi$
$158$	0	0
$159$	−9.63181	−0.763852
$160$	0	0
$161$	0.463621	0.0365385
$162$	0	0
$163$	3.98861i	0.312412i	0.987724	+	0.156206i	$0.0499264\pi$
$163$	3.98861i	0.312412i	−0.987724	+	0.156206i	$0.950074\pi$
$164$	0	0
$165$	3.52243i	0.274221i
$166$	0	0
$167$	8.17158	0.632336	0.316168	−	0.948703i	$-0.397604\pi$
$167$	8.17158	0.632336	0.316168	+	0.948703i	$0.397604\pi$
$168$	0	0
$169$	11.2722	0.867096
$170$	0	0
$171$	− 1.67333i	− 0.127963i
$172$	0	0
$173$	− 0.351356i	− 0.0267131i	−0.999911	−	0.0133565i	$-0.995748\pi$
$173$	− 0.351356i	− 0.0267131i	0.999911	−	0.0133565i	$-0.00425164\pi$
$174$	0	0
$175$	0.916684	0.0692948
$176$	0	0
$177$	−10.6648	−0.801616
$178$	0	0
$179$	− 15.2740i	− 1.14163i	−0.821077	−	0.570817i	$-0.806627\pi$
$179$	− 15.2740i	− 1.14163i	0.821077	−	0.570817i	$-0.193373\pi$
$180$	0	0
$181$	8.94630i	0.664974i	0.943108	+	0.332487i	$0.107888\pi$
$181$	8.94630i	0.664974i	−0.943108	+	0.332487i	$0.892112\pi$
$182$	0	0
$183$	5.36785	0.396803
$184$	0	0
$185$	−4.37626	−0.321749
$186$	0	0
$187$	12.7139i	0.929730i
$188$	0	0
$189$	1.28415i	0.0934079i
$190$	0	0
$191$	−18.3946	−1.33098	−0.665492	−	0.746405i	$-0.731778\pi$
$191$	−18.3946	−1.33098	−0.665492	+	0.746405i	$0.731778\pi$
$192$	0	0
$193$	−5.09902	−0.367035	−0.183518	−	0.983016i	$-0.558748\pi$
$193$	−5.09902	−0.367035	−0.183518	+	0.983016i	$0.558748\pi$
$194$	0	0
$195$	− 1.62886i	− 0.116645i
$196$	0	0
$197$	− 20.2916i	− 1.44572i	−0.690995	−	0.722860i	$-0.742827\pi$
$197$	− 20.2916i	− 1.44572i	0.690995	−	0.722860i	$-0.257173\pi$
$198$	0	0
$199$	0.310999	0.0220461	0.0110231	−	0.999939i	$-0.496491\pi$
$199$	0.310999	0.0220461	0.0110231	+	0.999939i	$0.496491\pi$
$200$	0	0
$201$	4.97408	0.350845
$202$	0	0
$203$	2.10878i	0.148007i
$204$	0	0
$205$	2.99320i	0.209054i
$206$	0	0
$207$	3.25191	0.226023
$208$	0	0
$209$	−2.84249	−0.196619
$210$	0	0
$211$	− 14.3284i	− 0.986406i	−0.869914	−	0.493203i	$-0.835826\pi$
$211$	− 14.3284i	− 0.986406i	0.869914	−	0.493203i	$-0.164174\pi$
$212$	0	0
$213$	3.20283i	0.219454i
$214$	0	0
$215$	1.30572	0.0890491
$216$	0	0
$217$	1.58616	0.107676
$218$	0	0
$219$	0.154872i	0.0104653i
$220$	0	0
$221$	− 5.87921i	− 0.395479i
$222$	0	0
$223$	12.8621	0.861312	0.430656	−	0.902516i	$-0.358282\pi$
$223$	12.8621	0.861312	0.430656	+	0.902516i	$0.358282\pi$
$224$	0	0
$225$	6.42976	0.428650
$226$	0	0
$227$	− 18.4491i	− 1.22451i	−0.790659	−	0.612256i	$-0.790262\pi$
$227$	− 18.4491i	− 1.22451i	0.790659	−	0.612256i	$-0.209738\pi$
$228$	0	0
$229$	− 4.73279i	− 0.312751i	−0.987698	−	0.156376i	$-0.950019\pi$
$229$	− 4.73279i	− 0.312751i	0.987698	−	0.156376i	$-0.0499811\pi$
$230$	0	0
$231$	0.781066	0.0513903
$232$	0	0
$233$	4.27809	0.280267	0.140133	−	0.990133i	$-0.455247\pi$
$233$	4.27809	0.280267	0.140133	+	0.990133i	$0.455247\pi$
$234$	0	0
$235$	11.0487i	0.720735i
$236$	0	0
$237$	− 14.7249i	− 0.956483i
$238$	0	0
$239$	−20.3905	−1.31895	−0.659474	−	0.751727i	$-0.729221\pi$
$239$	−20.3905	−1.31895	−0.659474	+	0.751727i	$0.729221\pi$
$240$	0	0
$241$	25.5877	1.64825	0.824123	−	0.566410i	$-0.191668\pi$
$241$	25.5877	1.64825	0.824123	+	0.566410i	$0.191668\pi$
$242$	0	0
$243$	14.7893i	0.948732i
$244$	0	0
$245$	− 7.46989i	− 0.477234i
$246$	0	0
$247$	1.31444	0.0836358
$248$	0	0
$249$	15.4375	0.978311
$250$	0	0
$251$	28.9847i	1.82950i	0.404024	+	0.914749i	$0.367611\pi$
$251$	28.9847i	1.82950i	−0.404024	+	0.914749i	$0.632389\pi$
$252$	0	0
$253$	− 5.52404i	− 0.347293i
$254$	0	0
$255$	5.54270	0.347098
$256$	0	0
$257$	5.10203	0.318256	0.159128	−	0.987258i	$-0.449132\pi$
$257$	5.10203	0.318256	0.159128	+	0.987258i	$0.449132\pi$
$258$	0	0
$259$	0.970394i	0.0602974i
$260$	0	0
$261$	14.7913i	0.915558i
$262$	0	0
$263$	11.6953	0.721160	0.360580	−	0.932728i	$-0.382579\pi$
$263$	11.6953	0.721160	0.360580	+	0.932728i	$0.382579\pi$
$264$	0	0
$265$	−8.99680	−0.552669
$266$	0	0
$267$	− 10.6042i	− 0.648964i
$268$	0	0
$269$	− 0.163433i	− 0.00996466i	−0.999988	−	0.00498233i	$-0.998414\pi$
$269$	− 0.163433i	− 0.00996466i	0.999988	−	0.00498233i	$-0.00158593\pi$
$270$	0	0
$271$	−1.01216	−0.0614846	−0.0307423	−	0.999527i	$-0.509787\pi$
$271$	−1.01216	−0.0614846	−0.0307423	+	0.999527i	$0.509787\pi$
$272$	0	0
$273$	−0.361184	−0.0218599
$274$	0	0
$275$	− 10.9223i	− 0.658637i
$276$	0	0
$277$	− 3.14825i	− 0.189160i	−0.995517	−	0.0945800i	$-0.969849\pi$
$277$	− 3.14825i	− 0.189160i	0.995517	−	0.0945800i	$-0.0301508\pi$
$278$	0	0
$279$	11.1256	0.666071
$280$	0	0
$281$	−24.6734	−1.47189	−0.735946	−	0.677040i	$-0.763262\pi$
$281$	−24.6734	−1.47189	−0.735946	+	0.677040i	$0.763262\pi$
$282$	0	0
$283$	− 13.8095i	− 0.820889i	−0.911886	−	0.410445i	$-0.865374\pi$
$283$	− 13.8095i	− 0.820889i	0.911886	−	0.410445i	$-0.134626\pi$
$284$	0	0
$285$	1.23921i	0.0734042i
$286$	0	0
$287$	0.663713	0.0391777
$288$	0	0
$289$	3.00584	0.176814
$290$	0	0
$291$	− 0.285072i	− 0.0167112i
$292$	0	0
$293$	− 8.24682i	− 0.481784i	−0.970552	−	0.240892i	$-0.922560\pi$
$293$	− 8.24682i	− 0.481784i	0.970552	−	0.240892i	$-0.0774400\pi$
$294$	0	0
$295$	−9.96169	−0.579992
$296$	0	0
$297$	15.3006	0.887829
$298$	0	0
$299$	2.55445i	0.147728i
$300$	0	0
$301$	− 0.289530i	− 0.0166882i
$302$	0	0
$303$	−19.3535	−1.11183
$304$	0	0
$305$	5.01396	0.287098
$306$	0	0
$307$	− 17.4540i	− 0.996153i	−0.867133	−	0.498076i	$-0.834040\pi$
$307$	− 17.4540i	− 0.996153i	0.867133	−	0.498076i	$-0.165960\pi$
$308$	0	0
$309$	6.23080i	0.354458i
$310$	0	0
$311$	10.5256	0.596851	0.298425	−	0.954433i	$-0.403539\pi$
$311$	10.5256	0.596851	0.298425	+	0.954433i	$0.403539\pi$
$312$	0	0
$313$	−14.3498	−0.811098	−0.405549	−	0.914073i	$-0.632920\pi$
$313$	−14.3498	−0.811098	−0.405549	+	0.914073i	$0.632920\pi$
$314$	0	0
$315$	0.429486i	0.0241988i
$316$	0	0
$317$	16.3470i	0.918138i	0.888401	+	0.459069i	$0.151817\pi$
$317$	16.3470i	0.918138i	−0.888401	+	0.459069i	$0.848183\pi$
$318$	0	0
$319$	25.1260	1.40679
$320$	0	0
$321$	−7.36138	−0.410872
$322$	0	0
$323$	4.47279i	0.248873i
$324$	0	0
$325$	5.05073i	0.280164i
$326$	0	0
$327$	−17.1366	−0.947656
$328$	0	0
$329$	2.44993	0.135069
$330$	0	0
$331$	− 15.0118i	− 0.825123i	−0.910930	−	0.412562i	$-0.864634\pi$
$331$	− 15.0118i	− 0.825123i	0.910930	−	0.412562i	$-0.135366\pi$
$332$	0	0
$333$	6.80649i	0.372993i
$334$	0	0
$335$	4.64615	0.253846
$336$	0	0
$337$	0.462187	0.0251769	0.0125885	−	0.999921i	$-0.495993\pi$
$337$	0.462187	0.0251769	0.0125885	+	0.999921i	$0.495993\pi$
$338$	0	0
$339$	9.89491i	0.537418i
$340$	0	0
$341$	− 18.8991i	− 1.02344i
$342$	0	0
$343$	−3.32633	−0.179605
$344$	0	0
$345$	−2.40824	−0.129655
$346$	0	0
$347$	17.2669i	0.926935i	0.886114	+	0.463468i	$0.153395\pi$
$347$	17.2669i	0.926935i	−0.886114	+	0.463468i	$0.846605\pi$
$348$	0	0
$349$	− 26.1466i	− 1.39960i	−0.714341	−	0.699798i	$-0.753273\pi$
$349$	− 26.1466i	− 1.39960i	0.714341	−	0.699798i	$-0.246727\pi$
$350$	0	0
$351$	−7.07536	−0.377655
$352$	0	0
$353$	12.3930	0.659614	0.329807	−	0.944048i	$-0.393016\pi$
$353$	12.3930	0.659614	0.329807	+	0.944048i	$0.393016\pi$
$354$	0	0
$355$	2.99167i	0.158781i
$356$	0	0
$357$	− 1.22904i	− 0.0650478i
$358$	0	0
$359$	−14.1664	−0.747673	−0.373837	−	0.927495i	$-0.621958\pi$
$359$	−14.1664	−0.747673	−0.373837	+	0.927495i	$0.621958\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	3.36356i	0.176541i
$364$	0	0
$365$	0.144662i	0.00757195i
$366$	0	0
$367$	−3.00328	−0.156770	−0.0783850	−	0.996923i	$-0.524976\pi$
$367$	−3.00328	−0.156770	−0.0783850	+	0.996923i	$0.524976\pi$
$368$	0	0
$369$	4.65538	0.242349
$370$	0	0
$371$	1.99495i	0.103573i
$372$	0	0
$373$	− 38.0432i	− 1.96980i	−0.173119	−	0.984901i	$-0.555385\pi$
$373$	− 38.0432i	− 1.96980i	0.173119	−	0.984901i	$-0.444615\pi$
$374$	0	0
$375$	−10.9577	−0.565852
$376$	0	0
$377$	−11.6189	−0.598404
$378$	0	0
$379$	28.6291i	1.47058i	0.677753	+	0.735290i	$0.262954\pi$
$379$	28.6291i	1.47058i	−0.677753	+	0.735290i	$0.737046\pi$
$380$	0	0
$381$	1.33591i	0.0684407i
$382$	0	0
$383$	24.1407	1.23353	0.616767	−	0.787146i	$-0.288442\pi$
$383$	24.1407	1.23353	0.616767	+	0.787146i	$0.288442\pi$
$384$	0	0
$385$	0.729571	0.0371824
$386$	0	0
$387$	− 2.03081i	− 0.103232i
$388$	0	0
$389$	16.2953i	0.826206i	0.910684	+	0.413103i	$0.135555\pi$
$389$	16.2953i	0.826206i	−0.910684	+	0.413103i	$0.864445\pi$
$390$	0	0
$391$	−8.69231	−0.439589
$392$	0	0
$393$	−4.44091	−0.224014
$394$	0	0
$395$	− 13.7541i	− 0.692043i
$396$	0	0
$397$	− 36.8139i	− 1.84764i	−0.382829	−	0.923819i	$-0.625050\pi$
$397$	− 36.8139i	− 1.84764i	0.382829	−	0.923819i	$-0.374950\pi$
$398$	0	0
$399$	0.274782	0.0137563
$400$	0	0
$401$	−23.6899	−1.18302	−0.591508	−	0.806299i	$-0.701467\pi$
$401$	−23.6899	−1.18302	−0.591508	+	0.806299i	$0.701467\pi$
$402$	0	0
$403$	8.73941i	0.435341i
$404$	0	0
$405$	− 1.26951i	− 0.0630827i
$406$	0	0
$407$	11.5622	0.573118
$408$	0	0
$409$	0.102363	0.00506154	0.00253077	−	0.999997i	$-0.499194\pi$
$409$	0.102363	0.00506154	0.00253077	+	0.999997i	$0.499194\pi$
$410$	0	0
$411$	1.64681i	0.0812311i
$412$	0	0
$413$	2.20891i	0.108693i
$414$	0	0
$415$	14.4197	0.707836
$416$	0	0
$417$	−4.12568	−0.202035
$418$	0	0
$419$	− 25.7326i	− 1.25712i	−0.777761	−	0.628560i	$-0.783645\pi$
$419$	− 25.7326i	− 1.25712i	0.777761	−	0.628560i	$-0.216355\pi$
$420$	0	0
$421$	− 18.0014i	− 0.877334i	−0.898650	−	0.438667i	$-0.855451\pi$
$421$	− 18.0014i	− 0.877334i	0.898650	−	0.438667i	$-0.144549\pi$
$422$	0	0
$423$	17.1842	0.835524
$424$	0	0
$425$	−17.1867	−0.833675
$426$	0	0
$427$	− 1.11180i	− 0.0538036i
$428$	0	0
$429$	4.30350i	0.207775i
$430$	0	0
$431$	27.6392	1.33133	0.665667	−	0.746249i	$-0.268147\pi$
$431$	27.6392	1.33133	0.665667	+	0.746249i	$0.268147\pi$
$432$	0	0
$433$	38.5219	1.85124	0.925621	−	0.378451i	$-0.123543\pi$
$433$	38.5219	1.85124	0.925621	+	0.378451i	$0.123543\pi$
$434$	0	0
$435$	− 10.9539i	− 0.525198i
$436$	0	0
$437$	− 1.94338i	− 0.0929643i
$438$	0	0
$439$	23.1302	1.10394	0.551971	−	0.833863i	$-0.313876\pi$
$439$	23.1302	1.10394	0.551971	+	0.833863i	$0.313876\pi$
$440$	0	0
$441$	−11.6181	−0.553241
$442$	0	0
$443$	19.2436i	0.914290i	0.889392	+	0.457145i	$0.151128\pi$
$443$	19.2436i	0.914290i	−0.889392	+	0.457145i	$0.848872\pi$
$444$	0	0
$445$	− 9.90505i	− 0.469544i
$446$	0	0
$447$	−18.7473	−0.886716
$448$	0	0
$449$	11.0578	0.521851	0.260926	−	0.965359i	$-0.415972\pi$
$449$	11.0578	0.521851	0.260926	+	0.965359i	$0.415972\pi$
$450$	0	0
$451$	− 7.90812i	− 0.372379i
$452$	0	0
$453$	− 16.4988i	− 0.775183i
$454$	0	0
$455$	−0.337372	−0.0158162
$456$	0	0
$457$	4.08310	0.190999	0.0954996	−	0.995429i	$-0.469555\pi$
$457$	4.08310	0.190999	0.0954996	+	0.995429i	$0.469555\pi$
$458$	0	0
$459$	− 24.0761i	− 1.12378i
$460$	0	0
$461$	− 7.59017i	− 0.353509i	−0.984255	−	0.176755i	$-0.943440\pi$
$461$	− 7.59017i	− 0.353509i	0.984255	−	0.176755i	$-0.0565599\pi$
$462$	0	0
$463$	37.2637	1.73179	0.865894	−	0.500227i	$-0.166750\pi$
$463$	37.2637	1.73179	0.865894	+	0.500227i	$0.166750\pi$
$464$	0	0
$465$	−8.23920	−0.382084
$466$	0	0
$467$	8.92480i	0.412991i	0.978448	+	0.206495i	$0.0662058\pi$
$467$	8.92480i	0.412991i	−0.978448	+	0.206495i	$0.933794\pi$
$468$	0	0
$469$	− 1.03024i	− 0.0475720i
$470$	0	0
$471$	4.30196	0.198224
$472$	0	0
$473$	−3.44974	−0.158619
$474$	0	0
$475$	− 3.84249i	− 0.176306i
$476$	0	0
$477$	13.9929i	0.640690i
$478$	0	0
$479$	−19.6241	−0.896650	−0.448325	−	0.893871i	$-0.647979\pi$
$479$	−19.6241	−0.896650	−0.448325	+	0.893871i	$0.647979\pi$
$480$	0	0
$481$	−5.34666	−0.243787
$482$	0	0
$483$	0.534005i	0.0242981i
$484$	0	0
$485$	− 0.266277i	− 0.0120910i
$486$	0	0
$487$	13.5448	0.613775	0.306887	−	0.951746i	$-0.400712\pi$
$487$	13.5448	0.613775	0.306887	+	0.951746i	$0.400712\pi$
$488$	0	0
$489$	−4.59413	−0.207754
$490$	0	0
$491$	− 9.58255i	− 0.432454i	−0.976343	−	0.216227i	$-0.930625\pi$
$491$	− 9.58255i	− 0.432454i	0.976343	−	0.216227i	$-0.0693752\pi$
$492$	0	0
$493$	− 39.5369i	− 1.78065i
$494$	0	0
$495$	5.11732	0.230006
$496$	0	0
$497$	0.663374	0.0297564
$498$	0	0
$499$	− 7.24834i	− 0.324480i	−0.986751	−	0.162240i	$-0.948128\pi$
$499$	− 7.24834i	− 0.324480i	0.986751	−	0.162240i	$-0.0518719\pi$
$500$	0	0
$501$	9.41213i	0.420503i
$502$	0	0
$503$	−2.19200	−0.0977367	−0.0488683	−	0.998805i	$-0.515561\pi$
$503$	−2.19200	−0.0977367	−0.0488683	+	0.998805i	$0.515561\pi$
$504$	0	0
$505$	−18.0775	−0.804440
$506$	0	0
$507$	12.9835i	0.576618i
$508$	0	0
$509$	− 13.5765i	− 0.601766i	−0.953661	−	0.300883i	$-0.902719\pi$
$509$	− 13.5765i	− 0.601766i	0.953661	−	0.300883i	$-0.0972814\pi$
$510$	0	0
$511$	0.0320774	0.00141902
$512$	0	0
$513$	5.38280	0.237656
$514$	0	0
$515$	5.82002i	0.256461i
$516$	0	0
$517$	− 29.1909i	− 1.28381i
$518$	0	0
$519$	0.404696	0.0177642
$520$	0	0
$521$	6.57594	0.288097	0.144049	−	0.989571i	$-0.453988\pi$
$521$	6.57594	0.288097	0.144049	+	0.989571i	$0.453988\pi$
$522$	0	0
$523$	− 22.2161i	− 0.971443i	−0.874114	−	0.485722i	$-0.838557\pi$
$523$	− 22.2161i	− 0.971443i	0.874114	−	0.485722i	$-0.161443\pi$
$524$	0	0
$525$	1.05585i	0.0460810i
$526$	0	0
$527$	−29.7386	−1.29543
$528$	0	0
$529$	−19.2233	−0.835795
$530$	0	0
$531$	15.4936i	0.672366i
$532$	0	0
$533$	3.65691i	0.158398i
$534$	0	0
$535$	−6.87605	−0.297278
$536$	0	0
$537$	17.5928	0.759185
$538$	0	0
$539$	19.7357i	0.850076i
$540$	0	0
$541$	− 22.5004i	− 0.967368i	−0.875243	−	0.483684i	$-0.839298\pi$
$541$	− 22.5004i	− 0.967368i	0.875243	−	0.483684i	$-0.160702\pi$
$542$	0	0
$543$	−10.3045	−0.442207
$544$	0	0
$545$	−16.0068	−0.685656
$546$	0	0
$547$	24.1551i	1.03280i	0.856348	+	0.516399i	$0.172728\pi$
$547$	24.1551i	1.03280i	−0.856348	+	0.516399i	$0.827272\pi$
$548$	0	0
$549$	− 7.79830i	− 0.332823i
$550$	0	0
$551$	8.83944	0.376573
$552$	0	0
$553$	−3.04984	−0.129692
$554$	0	0
$555$	− 5.04063i	− 0.213963i
$556$	0	0
$557$	21.4469i	0.908734i	0.890815	+	0.454367i	$0.150135\pi$
$557$	21.4469i	0.908734i	−0.890815	+	0.454367i	$0.849865\pi$
$558$	0	0
$559$	1.59525	0.0674717
$560$	0	0
$561$	−14.6440	−0.618270
$562$	0	0
$563$	− 15.3594i	− 0.647322i	−0.946173	−	0.323661i	$-0.895086\pi$
$563$	− 15.3594i	− 0.647322i	0.946173	−	0.323661i	$-0.104914\pi$
$564$	0	0
$565$	9.24256i	0.388837i
$566$	0	0
$567$	−0.281503	−0.0118220
$568$	0	0
$569$	−2.05613	−0.0861973	−0.0430986	−	0.999071i	$-0.513723\pi$
$569$	−2.05613	−0.0861973	−0.0430986	+	0.999071i	$0.513723\pi$
$570$	0	0
$571$	35.4718i	1.48445i	0.670151	+	0.742225i	$0.266229\pi$
$571$	35.4718i	1.48445i	−0.670151	+	0.742225i	$0.733771\pi$
$572$	0	0
$573$	− 21.1871i	− 0.885103i
$574$	0	0
$575$	7.46741	0.311413
$576$	0	0
$577$	14.6910	0.611595	0.305798	−	0.952097i	$-0.401077\pi$
$577$	14.6910	0.611595	0.305798	+	0.952097i	$0.401077\pi$
$578$	0	0
$579$	− 5.87311i	− 0.244078i
$580$	0	0
$581$	− 3.19743i	− 0.132652i
$582$	0	0
$583$	23.7698	0.984445
$584$	0	0
$585$	−2.36638	−0.0978376
$586$	0	0
$587$	1.71339i	0.0707193i	0.999375	+	0.0353596i	$0.0112577\pi$
$587$	1.71339i	0.0707193i	−0.999375	+	0.0353596i	$0.988742\pi$
$588$	0	0
$589$	− 6.64877i	− 0.273958i
$590$	0	0
$591$	23.3722	0.961402
$592$	0	0
$593$	20.3947	0.837509	0.418754	−	0.908100i	$-0.362467\pi$
$593$	20.3947	0.837509	0.418754	+	0.908100i	$0.362467\pi$
$594$	0	0
$595$	− 1.14801i	− 0.0470639i
$596$	0	0
$597$	0.358212i	0.0146607i
$598$	0	0
$599$	−21.9447	−0.896637	−0.448318	−	0.893874i	$-0.647977\pi$
$599$	−21.9447	−0.896637	−0.448318	+	0.893874i	$0.647977\pi$
$600$	0	0
$601$	3.41894	0.139461	0.0697306	−	0.997566i	$-0.477786\pi$
$601$	3.41894	0.139461	0.0697306	+	0.997566i	$0.477786\pi$
$602$	0	0
$603$	− 7.22625i	− 0.294276i
$604$	0	0
$605$	3.14180i	0.127732i
$606$	0	0
$607$	−23.1235	−0.938555	−0.469278	−	0.883051i	$-0.655486\pi$
$607$	−23.1235	−0.938555	−0.469278	+	0.883051i	$0.655486\pi$
$608$	0	0
$609$	−2.42892	−0.0984247
$610$	0	0
$611$	13.4986i	0.546095i
$612$	0	0
$613$	− 46.3763i	− 1.87312i	−0.350504	−	0.936561i	$-0.613990\pi$
$613$	− 46.3763i	− 1.87312i	0.350504	−	0.936561i	$-0.386010\pi$
$614$	0	0
$615$	−3.44760	−0.139021
$616$	0	0
$617$	25.2123	1.01501	0.507504	−	0.861649i	$-0.330568\pi$
$617$	25.2123	1.01501	0.507504	+	0.861649i	$0.330568\pi$
$618$	0	0
$619$	− 37.2031i	− 1.49532i	−0.664082	−	0.747660i	$-0.731178\pi$
$619$	− 37.2031i	− 1.49532i	0.664082	−	0.747660i	$-0.268822\pi$
$620$	0	0
$621$	10.4608i	0.419778i
$622$	0	0
$623$	−2.19635	−0.0879948
$624$	0	0
$625$	8.97722	0.359089
$626$	0	0
$627$	− 3.27402i	− 0.130752i
$628$	0	0
$629$	− 18.1937i	− 0.725429i
$630$	0	0
$631$	−40.8412	−1.62586	−0.812931	−	0.582360i	$-0.802129\pi$
$631$	−40.8412	−1.62586	−0.812931	+	0.582360i	$0.802129\pi$
$632$	0	0
$633$	16.5036	0.655959
$634$	0	0
$635$	1.24783i	0.0495188i
$636$	0	0
$637$	− 9.12627i	− 0.361596i
$638$	0	0
$639$	4.65300	0.184070
$640$	0	0
$641$	5.87446	0.232027	0.116014	−	0.993248i	$-0.462988\pi$
$641$	5.87446	0.232027	0.116014	+	0.993248i	$0.462988\pi$
$642$	0	0
$643$	21.3344i	0.841347i	0.907212	+	0.420673i	$0.138206\pi$
$643$	21.3344i	0.841347i	−0.907212	+	0.420673i	$0.861794\pi$
$644$	0	0
$645$	1.50394i	0.0592176i
$646$	0	0
$647$	−48.1625	−1.89346	−0.946732	−	0.322024i	$-0.895637\pi$
$647$	−48.1625	−1.89346	−0.946732	+	0.322024i	$0.895637\pi$
$648$	0	0
$649$	26.3191	1.03311
$650$	0	0
$651$	1.82696i	0.0716043i
$652$	0	0
$653$	− 11.3441i	− 0.443929i	−0.975055	−	0.221965i	$-0.928753\pi$
$653$	− 11.3441i	− 0.443929i	0.975055	−	0.221965i	$-0.0712469\pi$
$654$	0	0
$655$	−4.14813	−0.162081
$656$	0	0
$657$	0.224995	0.00877791
$658$	0	0
$659$	− 24.6320i	− 0.959526i	−0.877398	−	0.479763i	$-0.840723\pi$
$659$	− 24.6320i	− 0.959526i	0.877398	−	0.479763i	$-0.159277\pi$
$660$	0	0
$661$	− 47.3409i	− 1.84135i	−0.390335	−	0.920673i	$-0.627641\pi$
$661$	− 47.3409i	− 1.84135i	0.390335	−	0.920673i	$-0.372359\pi$
$662$	0	0
$663$	6.77175	0.262993
$664$	0	0
$665$	0.256666	0.00995308
$666$	0	0
$667$	17.1784i	0.665149i
$668$	0	0
$669$	14.8148i	0.572772i
$670$	0	0
$671$	−13.2470	−0.511396
$672$	0	0
$673$	−35.1767	−1.35596	−0.677981	−	0.735079i	$-0.737145\pi$
$673$	−35.1767	−1.35596	−0.677981	+	0.735079i	$0.737145\pi$
$674$	0	0
$675$	20.6834i	0.796103i
$676$	0	0
$677$	2.71085i	0.104186i	0.998642	+	0.0520932i	$0.0165893\pi$
$677$	2.71085i	0.104186i	−0.998642	+	0.0520932i	$0.983411\pi$
$678$	0	0
$679$	−0.0590444	−0.00226592
$680$	0	0
$681$	21.2499	0.814300
$682$	0	0
$683$	− 48.4820i	− 1.85511i	−0.373682	−	0.927557i	$-0.621905\pi$
$683$	− 48.4820i	− 1.85511i	0.373682	−	0.927557i	$-0.378095\pi$
$684$	0	0
$685$	1.53824i	0.0587730i
$686$	0	0
$687$	5.45128	0.207979
$688$	0	0
$689$	−10.9917	−0.418752
$690$	0	0
$691$	− 41.3295i	− 1.57225i	−0.618068	−	0.786124i	$-0.712085\pi$
$691$	− 41.3295i	− 1.57225i	0.618068	−	0.786124i	$-0.287915\pi$
$692$	0	0
$693$	− 1.13472i	− 0.0431043i
$694$	0	0
$695$	−3.85368	−0.146178
$696$	0	0
$697$	−12.4438	−0.471342
$698$	0	0
$699$	4.92755i	0.186377i
$700$	0	0
$701$	− 15.9692i	− 0.603147i	−0.953443	−	0.301573i	$-0.902488\pi$
$701$	− 15.9692i	− 0.603147i	0.953443	−	0.301573i	$-0.0975118\pi$
$702$	0	0
$703$	4.06763	0.153414
$704$	0	0
$705$	−12.7260	−0.479288
$706$	0	0
$707$	4.00852i	0.150756i
$708$	0	0
$709$	− 1.27532i	− 0.0478957i	−0.999713	−	0.0239478i	$-0.992376\pi$
$709$	− 1.27532i	− 0.0478957i	0.999713	−	0.0239478i	$-0.00762356\pi$
$710$	0	0
$711$	−21.3920	−0.802263
$712$	0	0
$713$	12.9211	0.483898
$714$	0	0
$715$	4.01978i	0.150331i
$716$	0	0
$717$	− 23.4860i	− 0.877100i
$718$	0	0
$719$	38.6245	1.44045	0.720225	−	0.693740i	$-0.244038\pi$
$719$	38.6245	1.44045	0.720225	+	0.693740i	$0.244038\pi$
$720$	0	0
$721$	1.29053	0.0480619
$722$	0	0
$723$	29.4722i	1.09608i
$724$	0	0
$725$	33.9655i	1.26145i
$726$	0	0
$727$	−3.96945	−0.147219	−0.0736095	−	0.997287i	$-0.523452\pi$
$727$	−3.96945	−0.147219	−0.0736095	+	0.997287i	$0.523452\pi$
$728$	0	0
$729$	−20.5744	−0.762015
$730$	0	0
$731$	5.42832i	0.200774i
$732$	0	0
$733$	− 23.9821i	− 0.885800i	−0.896571	−	0.442900i	$-0.853950\pi$
$733$	− 23.9821i	− 0.885800i	0.896571	−	0.442900i	$-0.146050\pi$
$734$	0	0
$735$	8.60391	0.317360
$736$	0	0
$737$	−12.2753	−0.452165
$738$	0	0
$739$	34.2553i	1.26010i	0.776554	+	0.630051i	$0.216966\pi$
$739$	34.2553i	1.26010i	−0.776554	+	0.630051i	$0.783034\pi$
$740$	0	0
$741$	1.51399i	0.0556177i
$742$	0	0
$743$	31.0058	1.13749	0.568746	−	0.822513i	$-0.307429\pi$
$743$	31.0058	1.13749	0.568746	+	0.822513i	$0.307429\pi$
$744$	0	0
$745$	−17.5113	−0.641564
$746$	0	0
$747$	− 22.4273i	− 0.820571i
$748$	0	0
$749$	1.52470i	0.0557112i
$750$	0	0
$751$	−36.0417	−1.31518	−0.657590	−	0.753376i	$-0.728424\pi$
$751$	−36.0417	−1.31518	−0.657590	+	0.753376i	$0.728424\pi$
$752$	0	0
$753$	−33.3849	−1.21661
$754$	0	0
$755$	− 15.4111i	− 0.560867i
$756$	0	0
$757$	26.6253i	0.967712i	0.875148	+	0.483856i	$0.160764\pi$
$757$	26.6253i	0.967712i	−0.875148	+	0.483856i	$0.839236\pi$
$758$	0	0
$759$	6.36265	0.230950
$760$	0	0
$761$	40.0064	1.45023	0.725115	−	0.688628i	$-0.241787\pi$
$761$	40.0064	1.45023	0.725115	+	0.688628i	$0.241787\pi$
$762$	0	0
$763$	3.54935i	0.128495i
$764$	0	0
$765$	− 8.05233i	− 0.291133i
$766$	0	0
$767$	−12.1706	−0.439455
$768$	0	0
$769$	19.1147	0.689295	0.344648	−	0.938732i	$-0.387998\pi$
$769$	19.1147	0.689295	0.344648	+	0.938732i	$0.387998\pi$
$770$	0	0
$771$	5.87658i	0.211640i
$772$	0	0
$773$	42.2244i	1.51871i	0.650679	+	0.759353i	$0.274484\pi$
$773$	42.2244i	1.51871i	−0.650679	+	0.759353i	$0.725516\pi$
$774$	0	0
$775$	25.5479	0.917706
$776$	0	0
$777$	−1.11771	−0.0400977
$778$	0	0
$779$	− 2.78211i	− 0.0996793i
$780$	0	0
$781$	− 7.90408i	− 0.282830i
$782$	0	0
$783$	−47.5809	−1.70040
$784$	0	0
$785$	4.01833	0.143421
$786$	0	0
$787$	43.6009i	1.55420i	0.629375	+	0.777102i	$0.283311\pi$
$787$	43.6009i	1.55420i	−0.629375	+	0.777102i	$0.716689\pi$
$788$	0	0
$789$	13.4707i	0.479571i
$790$	0	0
$791$	2.04945	0.0728699
$792$	0	0
$793$	6.12575	0.217532
$794$	0	0
$795$	− 10.3626i	− 0.367524i
$796$	0	0
$797$	14.6901i	0.520351i	0.965561	+	0.260176i	$0.0837805\pi$
$797$	14.6901i	0.520351i	−0.965561	+	0.260176i	$0.916220\pi$
$798$	0	0
$799$	−45.9332	−1.62500
$800$	0	0
$801$	−15.4055	−0.544327
$802$	0	0
$803$	− 0.382201i	− 0.0134876i
$804$	0	0
$805$	0.498799i	0.0175803i
$806$	0	0
$807$	0.188244	0.00662649
$808$	0	0
$809$	4.76004	0.167354	0.0836771	−	0.996493i	$-0.473334\pi$
$809$	4.76004	0.167354	0.0836771	+	0.996493i	$0.473334\pi$
$810$	0	0
$811$	− 14.6172i	− 0.513279i	−0.966507	−	0.256640i	$-0.917385\pi$
$811$	− 14.6172i	− 0.513279i	0.966507	−	0.256640i	$-0.0826154\pi$
$812$	0	0
$813$	− 1.16582i	− 0.0408872i
$814$	0	0
$815$	−4.29125	−0.150316
$816$	0	0
$817$	−1.21363	−0.0424596
$818$	0	0
$819$	0.524721i	0.0183352i
$820$	0	0
$821$	− 32.3079i	− 1.12755i	−0.825928	−	0.563776i	$-0.809348\pi$
$821$	− 32.3079i	− 1.12755i	0.825928	−	0.563776i	$-0.190652\pi$
$822$	0	0
$823$	−9.79982	−0.341600	−0.170800	−	0.985306i	$-0.554635\pi$
$823$	−9.79982	−0.341600	−0.170800	+	0.985306i	$0.554635\pi$
$824$	0	0
$825$	12.5804	0.437993
$826$	0	0
$827$	− 11.2066i	− 0.389691i	−0.980834	−	0.194845i	$-0.937579\pi$
$827$	− 11.2066i	− 0.389691i	0.980834	−	0.194845i	$-0.0624205\pi$
$828$	0	0
$829$	45.4323i	1.57793i	0.614438	+	0.788965i	$0.289383\pi$
$829$	45.4323i	1.57793i	−0.614438	+	0.788965i	$0.710617\pi$
$830$	0	0
$831$	3.62619	0.125791
$832$	0	0
$833$	31.0550	1.07599
$834$	0	0
$835$	8.79160i	0.304246i
$836$	0	0
$837$	35.7890i	1.23705i
$838$	0	0
$839$	−16.8997	−0.583441	−0.291721	−	0.956504i	$-0.594228\pi$
$839$	−16.8997	−0.583441	−0.291721	+	0.956504i	$0.594228\pi$
$840$	0	0
$841$	−49.1356	−1.69433
$842$	0	0
$843$	− 28.4191i	− 0.978807i
$844$	0	0
$845$	12.1275i	0.417200i
$846$	0	0
$847$	0.696664	0.0239377
$848$	0	0
$849$	15.9059	0.545890
$850$	0	0
$851$	7.90494i	0.270978i
$852$	0	0
$853$	− 37.7943i	− 1.29405i	−0.762467	−	0.647027i	$-0.776012\pi$
$853$	− 37.7943i	− 1.29405i	0.762467	−	0.647027i	$-0.223988\pi$
$854$	0	0
$855$	1.80029	0.0615687
$856$	0	0
$857$	38.8619	1.32750	0.663749	−	0.747956i	$-0.268964\pi$
$857$	38.8619	1.32750	0.663749	+	0.747956i	$0.268964\pi$
$858$	0	0
$859$	− 41.1795i	− 1.40503i	−0.711671	−	0.702513i	$-0.752061\pi$
$859$	− 41.1795i	− 1.40503i	0.711671	−	0.702513i	$-0.247939\pi$
$860$	0	0
$861$	0.764473i	0.0260532i
$862$	0	0
$863$	25.5909	0.871125	0.435562	−	0.900159i	$-0.356550\pi$
$863$	25.5909	0.871125	0.435562	+	0.900159i	$0.356550\pi$
$864$	0	0
$865$	0.378015	0.0128529
$866$	0	0
$867$	3.46216i	0.117581i
$868$	0	0
$869$	36.3387i	1.23271i
$870$	0	0
$871$	5.67639	0.192337
$872$	0	0
$873$	−0.414146	−0.0140167
$874$	0	0
$875$	2.26957i	0.0767253i
$876$	0	0
$877$	− 14.4790i	− 0.488920i	−0.969659	−	0.244460i	$-0.921389\pi$
$877$	− 14.4790i	− 0.488920i	0.969659	−	0.244460i	$-0.0786107\pi$
$878$	0	0
$879$	9.49879	0.320386
$880$	0	0
$881$	48.3407	1.62864	0.814320	−	0.580416i	$-0.197110\pi$
$881$	48.3407	1.62864	0.814320	+	0.580416i	$0.197110\pi$
$882$	0	0
$883$	17.2669i	0.581077i	0.956863	+	0.290539i	$0.0938344\pi$
$883$	17.2669i	0.581077i	−0.956863	+	0.290539i	$0.906166\pi$
$884$	0	0
$885$	− 11.4740i	− 0.385694i
$886$	0	0
$887$	51.0650	1.71460	0.857298	−	0.514820i	$-0.172141\pi$
$887$	51.0650	1.71460	0.857298	+	0.514820i	$0.172141\pi$
$888$	0	0
$889$	0.276695	0.00928006
$890$	0	0
$891$	3.35410i	0.112366i
$892$	0	0
$893$	− 10.2695i	− 0.343655i
$894$	0	0
$895$	16.4329	0.549292
$896$	0	0
$897$	−2.94225	−0.0982388
$898$	0	0
$899$	58.7714i	1.96014i
$900$	0	0
$901$	− 37.4028i	− 1.24607i
$902$	0	0
$903$	0.333484	0.0110977
$904$	0	0
$905$	−9.62510	−0.319949
$906$	0	0
$907$	− 30.0974i	− 0.999367i	−0.866208	−	0.499684i	$-0.833450\pi$
$907$	− 30.0974i	− 0.999367i	0.866208	−	0.499684i	$-0.166550\pi$
$908$	0	0
$909$	28.1164i	0.932561i
$910$	0	0
$911$	9.15413	0.303290	0.151645	−	0.988435i	$-0.451543\pi$
$911$	9.15413	0.303290	0.151645	+	0.988435i	$0.451543\pi$
$912$	0	0
$913$	−38.0973	−1.26084
$914$	0	0
$915$	5.77514i	0.190920i
$916$	0	0
$917$	0.919807i	0.0303747i
$918$	0	0
$919$	−7.49270	−0.247161	−0.123581	−	0.992335i	$-0.539438\pi$
$919$	−7.49270	−0.247161	−0.123581	+	0.992335i	$0.539438\pi$
$920$	0	0
$921$	20.1037	0.662440
$922$	0	0
$923$	3.65505i	0.120307i
$924$	0	0
$925$	15.6299i	0.513906i
$926$	0	0
$927$	9.05198	0.297306
$928$	0	0
$929$	−12.0792	−0.396306	−0.198153	−	0.980171i	$-0.563494\pi$
$929$	−12.0792	−0.396306	−0.198153	+	0.980171i	$0.563494\pi$
$930$	0	0
$931$	6.94309i	0.227550i
$932$	0	0
$933$	12.1235i	0.396905i
$934$	0	0
$935$	−13.6785	−0.447336
$936$	0	0
$937$	−16.6732	−0.544690	−0.272345	−	0.962200i	$-0.587799\pi$
$937$	−16.6732	−0.544690	−0.272345	+	0.962200i	$0.587799\pi$
$938$	0	0
$939$	− 16.5283i	− 0.539380i
$940$	0	0
$941$	29.5263i	0.962529i	0.876575	+	0.481265i	$0.159822\pi$
$941$	29.5263i	0.962529i	−0.876575	+	0.481265i	$0.840178\pi$
$942$	0	0
$943$	5.40668	0.176066
$944$	0	0
$945$	−1.38158	−0.0449428
$946$	0	0
$947$	− 24.2099i	− 0.786717i	−0.919385	−	0.393358i	$-0.871313\pi$
$947$	− 24.2099i	− 0.786717i	0.919385	−	0.393358i	$-0.128687\pi$
$948$	0	0
$949$	0.176739i	0.00573720i
$950$	0	0
$951$	−18.8287	−0.610561
$952$	0	0
$953$	−18.5304	−0.600258	−0.300129	−	0.953899i	$-0.597030\pi$
$953$	−18.5304	−0.600258	−0.300129	+	0.953899i	$0.597030\pi$
$954$	0	0
$955$	− 19.7903i	− 0.640398i
$956$	0	0
$957$	28.9405i	0.935513i
$958$	0	0
$959$	0.341089	0.0110143
$960$	0	0
$961$	13.2062	0.426005
$962$	0	0
$963$	10.6945i	0.344624i
$964$	0	0
$965$	− 5.48590i	− 0.176597i
$966$	0	0
$967$	54.2821	1.74559	0.872797	−	0.488084i	$-0.162304\pi$
$967$	54.2821	1.74559	0.872797	+	0.488084i	$0.162304\pi$
$968$	0	0
$969$	−5.15181	−0.165500
$970$	0	0
$971$	7.69284i	0.246875i	0.992352	+	0.123437i	$0.0393918\pi$
$971$	7.69284i	0.246875i	−0.992352	+	0.123437i	$0.960608\pi$
$972$	0	0
$973$	0.854516i	0.0273945i
$974$	0	0
$975$	−5.81749	−0.186309
$976$	0	0
$977$	−32.8587	−1.05124	−0.525621	−	0.850719i	$-0.676167\pi$
$977$	−32.8587	−1.05124	−0.525621	+	0.850719i	$0.676167\pi$
$978$	0	0
$979$	26.1694i	0.836378i
$980$	0	0
$981$	24.8957i	0.794858i
$982$	0	0
$983$	−10.4144	−0.332169	−0.166084	−	0.986112i	$-0.553112\pi$
$983$	−10.4144	−0.332169	−0.166084	+	0.986112i	$0.553112\pi$
$984$	0	0
$985$	21.8313	0.695602
$986$	0	0
$987$	2.82186i	0.0898209i
$988$	0	0
$989$	− 2.35855i	− 0.0749974i
$990$	0	0
$991$	−62.1164	−1.97319	−0.986596	−	0.163183i	$-0.947824\pi$
$991$	−62.1164	−1.97319	−0.986596	+	0.163183i	$0.947824\pi$
$992$	0	0
$993$	17.2908	0.548706
$994$	0	0
$995$	0.334596i	0.0106074i
$996$	0	0
$997$	5.60362i	0.177468i	0.996055	+	0.0887341i	$0.0282822\pi$
$997$	5.60362i	0.177468i	−0.996055	+	0.0887341i	$0.971718\pi$
$998$	0	0
$999$	−21.8952	−0.692735

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2432.2.c.j.1217.14	yes	20
4.3	odd	2	inner	2432.2.c.j.1217.8	yes	20
8.3	odd	2	inner	2432.2.c.j.1217.13	yes	20
8.5	even	2	inner	2432.2.c.j.1217.7	✓	20
16.3	odd	4		4864.2.a.bs.1.8		10
16.5	even	4		4864.2.a.bs.1.7		10
16.11	odd	4		4864.2.a.bt.1.3		10
16.13	even	4		4864.2.a.bt.1.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.j.1217.7	✓	20	8.5	even	2	inner
2432.2.c.j.1217.8	yes	20	4.3	odd	2	inner
2432.2.c.j.1217.13	yes	20	8.3	odd	2	inner
2432.2.c.j.1217.14	yes	20	1.1	even	1	trivial
4864.2.a.bs.1.7		10	16.5	even	4
4864.2.a.bs.1.8		10	16.3	odd	4
4864.2.a.bt.1.3		10	16.11	odd	4
4864.2.a.bt.1.4		10	16.13	even	4

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:	\( N \)	\(=\)	\( 2432 = 2^{7} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2432.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.15181i q^{3} +1.07587i q^{5} +0.238565 q^{7} +1.67333 q^{9} +O(q^{10})\)	\(q+1.15181i q^{3} +1.07587i q^{5} +0.238565 q^{7} +1.67333 q^{9} -2.84249i q^{11} +1.31444i q^{13} -1.23921 q^{15} -4.47279 q^{17} -1.00000i q^{19} +0.274782i q^{21} +1.94338 q^{23} +3.84249 q^{25} +5.38280i q^{27} +8.83944i q^{29} +6.64877 q^{31} +3.27402 q^{33} +0.256666i q^{35} +4.06763i q^{37} -1.51399 q^{39} +2.78211 q^{41} -1.21363i q^{43} +1.80029i q^{45} +10.2695 q^{47} -6.94309 q^{49} -5.15181i q^{51} +8.36231i q^{53} +3.05817 q^{55} +1.15181 q^{57} +9.25916i q^{59} -4.66035i q^{61} +0.399197 q^{63} -1.41417 q^{65} -4.31849i q^{67} +2.23841i q^{69} +2.78069 q^{71} +0.134460 q^{73} +4.42583i q^{75} -0.678119i q^{77} -12.7841 q^{79} -1.17998 q^{81} -13.4028i q^{83} -4.81216i q^{85} -10.1814 q^{87} -9.20650 q^{89} +0.313579i q^{91} +7.65814i q^{93} +1.07587 q^{95} -0.247499 q^{97} -4.75643i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 28 q^{9}+O(q^{10}) \) 20 * q - 28 * q^9	\( 20 q - 28 q^{9} + 8 q^{17} - 20 q^{25} - 16 q^{33} + 24 q^{41} + 52 q^{49} - 8 q^{57} - 48 q^{65} - 24 q^{73} + 68 q^{81} + 40 q^{89} - 56 q^{97}+O(q^{100}) \) 20 * q - 28 * q^9 + 8 * q^17 - 20 * q^25 - 16 * q^33 + 24 * q^41 + 52 * q^49 - 8 * q^57 - 48 * q^65 - 24 * q^73 + 68 * q^81 + 40 * q^89 - 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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