LMFDB - Embedded newform 2432.2.c.i.1217.6

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} + 8 x^{14} - 8 x^{13} - 3 x^{12} + 20 x^{11} - 24 x^{10} + 28 x^{8} - 96 x^{6} + \cdots + 256 $ x^16 - 4x^15 + 8x^14 - 8x^13 - 3x^12 + 20x^11 - 24x^10 + 28x^8 - 96x^6 + 160x^5 - 48x^4 - 256x^3 + 512x^2 - 512*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1217.6
Root			$-1.39580 + 0.227497i$ of defining polynomial
Character	$\chi$	$=$	2432.1217
Dual form			2432.2.c.i.1217.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.65222i q^{3} +4.30533i q^{5} -2.73423 q^{7} +0.270160 q^{9} +O(q^{10})$	$q-1.65222i q^{3} +4.30533i q^{5} -2.73423 q^{7} +0.270160 q^{9} +1.72984i q^{11} +0.546394i q^{13} +7.11337 q^{15} -3.82843 q^{17} +1.00000i q^{19} +4.51756i q^{21} +0.546394 q^{23} -13.5359 q^{25} -5.40303i q^{27} +0.0738068i q^{29} +1.49730 q^{31} +2.85808 q^{33} -11.7718i q^{35} +8.56253i q^{37} +0.902764 q^{39} -1.90604 q^{41} -9.07622i q^{43} +1.16313i q^{45} +5.33004 q^{47} +0.476019 q^{49} +6.32541i q^{51} -10.1336i q^{53} -7.44754 q^{55} +1.65222 q^{57} +6.03429i q^{59} +1.31074i q^{61} -0.738679 q^{63} -2.35241 q^{65} -9.02097i q^{67} -0.902764i q^{69} -14.6512 q^{71} -8.50162 q^{73} +22.3643i q^{75} -4.72978i q^{77} -7.68543 q^{79} -8.11654 q^{81} -16.8061i q^{83} -16.4827i q^{85} +0.121945 q^{87} -14.0166 q^{89} -1.49397i q^{91} -2.47387i q^{93} -4.30533 q^{95} +7.30445 q^{97} +0.467333i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{9}+O(q^{10}) $ 16 * q - 8 * q^9	$ 16 q - 8 q^{9} - 16 q^{17} - 40 q^{25} + 48 q^{33} - 16 q^{41} + 16 q^{49} + 8 q^{57} + 16 q^{65} + 16 q^{73} - 64 q^{81} + 16 q^{89} + 80 q^{97}+O(q^{100}) $ 16 * q - 8 * q^9 - 16 * q^17 - 40 * q^25 + 48 * q^33 - 16 * q^41 + 16 * q^49 + 8 * q^57 + 16 * q^65 + 16 * q^73 - 64 * q^81 + 16 * q^89 + 80 * 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
$3$	− 1.65222i	− 0.953911i	−0.878927	−	0.476956i	$-0.841740\pi$
$3$	− 1.65222i	− 0.953911i	0.878927	−	0.476956i	$-0.158260\pi$
$4$	0	0
$5$	4.30533i	1.92540i	0.270564	+	0.962702i	$0.412790\pi$
$5$	4.30533i	1.92540i	−0.270564	+	0.962702i	$0.587210\pi$
$6$	0	0
$7$	−2.73423	−1.03344	−0.516721	−	0.856154i	$-0.672848\pi$
$7$	−2.73423	−1.03344	−0.516721	+	0.856154i	$0.672848\pi$
$8$	0	0
$9$	0.270160	0.0900532
$10$	0	0
$11$	1.72984i	0.521567i	0.965397	+	0.260783i	$0.0839808\pi$
$11$	1.72984i	0.521567i	−0.965397	+	0.260783i	$0.916019\pi$
$12$	0	0
$13$	0.546394i	0.151542i	0.997125	+	0.0757712i	$0.0241419\pi$
$13$	0.546394i	0.151542i	−0.997125	+	0.0757712i	$0.975858\pi$
$14$	0	0
$15$	7.11337	1.83666
$16$	0	0
$17$	−3.82843	−0.928530	−0.464265	−	0.885696i	$-0.653681\pi$
$17$	−3.82843	−0.928530	−0.464265	+	0.885696i	$0.653681\pi$
$18$	0	0
$19$	1.00000i	0.229416i
$19$	1.00000i	0.229416i
$20$	0	0
$21$	4.51756i	0.985812i
$22$	0	0
$23$	0.546394	0.113931	0.0569655	−	0.998376i	$-0.481857\pi$
$23$	0.546394	0.113931	0.0569655	+	0.998376i	$0.481857\pi$
$24$	0	0
$25$	−13.5359	−2.70718
$26$	0	0
$27$	− 5.40303i	− 1.03981i
$28$	0	0
$29$	0.0738068i	0.0137056i	0.999977	+	0.00685279i	$0.00218133\pi$
$29$	0.0738068i	0.0137056i	−0.999977	+	0.00685279i	$0.997819\pi$
$30$	0	0
$31$	1.49730	0.268922	0.134461	−	0.990919i	$-0.457070\pi$
$31$	1.49730	0.268922	0.134461	+	0.990919i	$0.457070\pi$
$32$	0	0
$33$	2.85808	0.497528
$34$	0	0
$35$	− 11.7718i	− 1.98979i
$36$	0	0
$37$	8.56253i	1.40767i	0.710363	+	0.703836i	$0.248531\pi$
$37$	8.56253i	1.40767i	−0.710363	+	0.703836i	$0.751469\pi$
$38$	0	0
$39$	0.902764	0.144558
$40$	0	0
$41$	−1.90604	−0.297674	−0.148837	−	0.988862i	$-0.547553\pi$
$41$	−1.90604	−0.297674	−0.148837	+	0.988862i	$0.547553\pi$
$42$	0	0
$43$	− 9.07622i	− 1.38411i	−0.721844	−	0.692056i	$-0.756705\pi$
$43$	− 9.07622i	− 1.38411i	0.721844	−	0.692056i	$-0.243295\pi$
$44$	0	0
$45$	1.16313i	0.173389i
$46$	0	0
$47$	5.33004	0.777467	0.388733	−	0.921350i	$-0.372913\pi$
$47$	5.33004	0.777467	0.388733	+	0.921350i	$0.372913\pi$
$48$	0	0
$49$	0.476019	0.0680027
$50$	0	0
$51$	6.32541i	0.885735i
$52$	0	0
$53$	− 10.1336i	− 1.39196i	−0.718060	−	0.695981i	$-0.754970\pi$
$53$	− 10.1336i	− 1.39196i	0.718060	−	0.695981i	$-0.245030\pi$
$54$	0	0
$55$	−7.44754	−1.00423
$56$	0	0
$57$	1.65222	0.218842
$58$	0	0
$59$	6.03429i	0.785597i	0.919625	+	0.392799i	$0.128493\pi$
$59$	6.03429i	0.785597i	−0.919625	+	0.392799i	$0.871507\pi$
$60$	0	0
$61$	1.31074i	0.167823i	0.996473	+	0.0839116i	$0.0267413\pi$
$61$	1.31074i	0.167823i	−0.996473	+	0.0839116i	$0.973259\pi$
$62$	0	0
$63$	−0.738679	−0.0930648
$64$	0	0
$65$	−2.35241	−0.291780
$66$	0	0
$67$	− 9.02097i	− 1.10209i	−0.834477	−	0.551043i	$-0.814230\pi$
$67$	− 9.02097i	− 1.10209i	0.834477	−	0.551043i	$-0.185770\pi$
$68$	0	0
$69$	− 0.902764i	− 0.108680i
$70$	0	0
$71$	−14.6512	−1.73878	−0.869388	−	0.494129i	$-0.835487\pi$
$71$	−14.6512	−1.73878	−0.869388	+	0.494129i	$0.835487\pi$
$72$	0	0
$73$	−8.50162	−0.995039	−0.497520	−	0.867453i	$-0.665756\pi$
$73$	−8.50162	−0.995039	−0.497520	+	0.867453i	$0.665756\pi$
$74$	0	0
$75$	22.3643i	2.58241i
$76$	0	0
$77$	− 4.72978i	− 0.539009i
$78$	0	0
$79$	−7.68543	−0.864679	−0.432339	−	0.901711i	$-0.642312\pi$
$79$	−7.68543	−0.864679	−0.432339	+	0.901711i	$0.642312\pi$
$80$	0	0
$81$	−8.11654	−0.901837
$82$	0	0
$83$	− 16.8061i	− 1.84471i	−0.386349	−	0.922353i	$-0.626264\pi$
$83$	− 16.8061i	− 1.84471i	0.386349	−	0.922353i	$-0.373736\pi$
$84$	0	0
$85$	− 16.4827i	− 1.78780i
$86$	0	0
$87$	0.121945	0.0130739
$88$	0	0
$89$	−14.0166	−1.48575	−0.742876	−	0.669429i	$-0.766539\pi$
$89$	−14.0166	−1.48575	−0.742876	+	0.669429i	$0.766539\pi$
$90$	0	0
$91$	− 1.49397i	− 0.156610i
$92$	0	0
$93$	− 2.47387i	− 0.256528i
$94$	0	0
$95$	−4.30533	−0.441718
$96$	0	0
$97$	7.30445	0.741654	0.370827	−	0.928702i	$-0.379074\pi$
$97$	7.30445	0.741654	0.370827	+	0.928702i	$0.379074\pi$
$98$	0	0
$99$	0.467333i	0.0469687i
$100$	0	0
$101$	− 9.85107i	− 0.980218i	−0.871661	−	0.490109i	$-0.836957\pi$
$101$	− 9.85107i	− 0.980218i	0.871661	−	0.490109i	$-0.163043\pi$
$102$	0	0
$103$	−12.8457	−1.26572	−0.632860	−	0.774266i	$-0.718119\pi$
$103$	−12.8457	−1.26572	−0.632860	+	0.774266i	$0.718119\pi$
$104$	0	0
$105$	−19.4496	−1.89809
$106$	0	0
$107$	7.81048i	0.755067i	0.925996	+	0.377534i	$0.123228\pi$
$107$	7.81048i	0.755067i	−0.925996	+	0.377534i	$0.876772\pi$
$108$	0	0
$109$	16.4792i	1.57842i	0.614123	+	0.789210i	$0.289510\pi$
$109$	16.4792i	1.57842i	−0.614123	+	0.789210i	$0.710490\pi$
$110$	0	0
$111$	14.1472	1.34279
$112$	0	0
$113$	−14.8673	−1.39860	−0.699301	−	0.714827i	$-0.746505\pi$
$113$	−14.8673	−1.39860	−0.699301	+	0.714827i	$0.746505\pi$
$114$	0	0
$115$	2.35241i	0.219363i
$116$	0	0
$117$	0.147614i	0.0136469i
$118$	0	0
$119$	10.4678	0.959582
$120$	0	0
$121$	8.00765	0.727968
$122$	0	0
$123$	3.14921i	0.283955i
$124$	0	0
$125$	− 36.7499i	− 3.28701i
$126$	0	0
$127$	−10.1080	−0.896937	−0.448468	−	0.893799i	$-0.648030\pi$
$127$	−10.1080	−0.896937	−0.448468	+	0.893799i	$0.648030\pi$
$128$	0	0
$129$	−14.9959	−1.32032
$130$	0	0
$131$	− 0.356822i	− 0.0311757i	−0.999879	−	0.0155879i	$-0.995038\pi$
$131$	− 0.356822i	− 0.0311757i	0.999879	−	0.0155879i	$-0.00496197\pi$
$132$	0	0
$133$	− 2.73423i	− 0.237088i
$134$	0	0
$135$	23.2619	2.00206
$136$	0	0
$137$	8.68953	0.742397	0.371198	−	0.928554i	$-0.378947\pi$
$137$	8.68953	0.742397	0.371198	+	0.928554i	$0.378947\pi$
$138$	0	0
$139$	14.4673i	1.22710i	0.789655	+	0.613552i	$0.210260\pi$
$139$	14.4673i	1.22710i	−0.789655	+	0.613552i	$0.789740\pi$
$140$	0	0
$141$	− 8.80642i	− 0.741634i
$142$	0	0
$143$	−0.945174	−0.0790394
$144$	0	0
$145$	−0.317763	−0.0263888
$146$	0	0
$147$	− 0.786489i	− 0.0648685i
$148$	0	0
$149$	− 9.77380i	− 0.800701i	−0.916362	−	0.400350i	$-0.868888\pi$
$149$	− 9.77380i	− 0.800701i	0.916362	−	0.400350i	$-0.131112\pi$
$150$	0	0
$151$	8.61067	0.700726	0.350363	−	0.936614i	$-0.386058\pi$
$151$	8.61067	0.700726	0.350363	+	0.936614i	$0.386058\pi$
$152$	0	0
$153$	−1.03429	−0.0836171
$154$	0	0
$155$	6.44636i	0.517784i
$156$	0	0
$157$	18.8862i	1.50728i	0.657286	+	0.753641i	$0.271704\pi$
$157$	18.8862i	1.50728i	−0.657286	+	0.753641i	$0.728296\pi$
$158$	0	0
$159$	−16.7430	−1.32781
$160$	0	0
$161$	−1.49397	−0.117741
$162$	0	0
$163$	20.6867i	1.62031i	0.586216	+	0.810155i	$0.300617\pi$
$163$	20.6867i	1.62031i	−0.586216	+	0.810155i	$0.699383\pi$
$164$	0	0
$165$	12.3050i	0.957943i
$166$	0	0
$167$	20.3835	1.57732	0.788661	−	0.614829i	$-0.210775\pi$
$167$	20.3835	1.57732	0.788661	+	0.614829i	$0.210775\pi$
$168$	0	0
$169$	12.7015	0.977035
$170$	0	0
$171$	0.270160i	0.0206596i
$172$	0	0
$173$	8.96704i	0.681751i	0.940108	+	0.340876i	$0.110724\pi$
$173$	8.96704i	0.681751i	−0.940108	+	0.340876i	$0.889276\pi$
$174$	0	0
$175$	37.0103	2.79771
$176$	0	0
$177$	9.96999	0.749390
$178$	0	0
$179$	− 10.6702i	− 0.797526i	−0.917054	−	0.398763i	$-0.869440\pi$
$179$	− 10.6702i	− 0.797526i	0.917054	−	0.398763i	$-0.130560\pi$
$180$	0	0
$181$	14.4424i	1.07350i	0.843742	+	0.536749i	$0.180348\pi$
$181$	14.4424i	1.07350i	−0.843742	+	0.536749i	$0.819652\pi$
$182$	0	0
$183$	2.16564	0.160088
$184$	0	0
$185$	−36.8646	−2.71034
$186$	0	0
$187$	− 6.62257i	− 0.484290i
$188$	0	0
$189$	14.7731i	1.07459i
$190$	0	0
$191$	−21.8574	−1.58154	−0.790772	−	0.612111i	$-0.790321\pi$
$191$	−21.8574	−1.58154	−0.790772	+	0.612111i	$0.790321\pi$
$192$	0	0
$193$	0.601599	0.0433040	0.0216520	−	0.999766i	$-0.493107\pi$
$193$	0.601599	0.0433040	0.0216520	+	0.999766i	$0.493107\pi$
$194$	0	0
$195$	3.88670i	0.278333i
$196$	0	0
$197$	− 6.28441i	− 0.447746i	−0.974618	−	0.223873i	$-0.928130\pi$
$197$	− 6.28441i	− 0.447746i	0.974618	−	0.223873i	$-0.0718701\pi$
$198$	0	0
$199$	−24.0262	−1.70317	−0.851586	−	0.524214i	$-0.824359\pi$
$199$	−24.0262	−1.70317	−0.851586	+	0.524214i	$0.824359\pi$
$200$	0	0
$201$	−14.9047	−1.05129
$202$	0	0
$203$	− 0.201805i	− 0.0141639i
$204$	0	0
$205$	− 8.20616i	− 0.573143i
$206$	0	0
$207$	0.147614	0.0102598
$208$	0	0
$209$	−1.72984	−0.119656
$210$	0	0
$211$	− 6.25382i	− 0.430531i	−0.976556	−	0.215265i	$-0.930938\pi$
$211$	− 6.25382i	− 0.430531i	0.976556	−	0.215265i	$-0.0690617\pi$
$212$	0	0
$213$	24.2070i	1.65864i
$214$	0	0
$215$	39.0762	2.66497
$216$	0	0
$217$	−4.09396	−0.277916
$218$	0	0
$219$	14.0466i	0.949179i
$220$	0	0
$221$	− 2.09183i	− 0.140712i
$222$	0	0
$223$	−12.3449	−0.826674	−0.413337	−	0.910578i	$-0.635637\pi$
$223$	−12.3449	−0.826674	−0.413337	+	0.910578i	$0.635637\pi$
$224$	0	0
$225$	−3.65685	−0.243790
$226$	0	0
$227$	− 2.30886i	− 0.153244i	−0.997060	−	0.0766222i	$-0.975586\pi$
$227$	− 2.30886i	− 0.153244i	0.997060	−	0.0766222i	$-0.0244135\pi$
$228$	0	0
$229$	− 2.11976i	− 0.140078i	−0.997544	−	0.0700388i	$-0.977688\pi$
$229$	− 2.11976i	− 0.140078i	0.997544	−	0.0700388i	$-0.0223123\pi$
$230$	0	0
$231$	−7.81466	−0.514167
$232$	0	0
$233$	25.2134	1.65178	0.825891	−	0.563829i	$-0.190672\pi$
$233$	25.2134	1.65178	0.825891	+	0.563829i	$0.190672\pi$
$234$	0	0
$235$	22.9476i	1.49694i
$236$	0	0
$237$	12.6981i	0.824827i
$238$	0	0
$239$	−22.7927	−1.47434	−0.737170	−	0.675707i	$-0.763838\pi$
$239$	−22.7927	−1.47434	−0.737170	+	0.675707i	$0.763838\pi$
$240$	0	0
$241$	2.98064	0.192000	0.0960000	−	0.995381i	$-0.469395\pi$
$241$	2.98064	0.192000	0.0960000	+	0.995381i	$0.469395\pi$
$242$	0	0
$243$	− 2.79877i	− 0.179541i
$244$	0	0
$245$	2.04942i	0.130933i
$246$	0	0
$247$	−0.546394	−0.0347662
$248$	0	0
$249$	−27.7674	−1.75969
$250$	0	0
$251$	8.84638i	0.558378i	0.960236	+	0.279189i	$0.0900656\pi$
$251$	8.84638i	0.558378i	−0.960236	+	0.279189i	$0.909934\pi$
$252$	0	0
$253$	0.945174i	0.0594226i
$254$	0	0
$255$	−27.2330	−1.70540
$256$	0	0
$257$	−6.04194	−0.376886	−0.188443	−	0.982084i	$-0.560344\pi$
$257$	−6.04194	−0.376886	−0.188443	+	0.982084i	$0.560344\pi$
$258$	0	0
$259$	− 23.4119i	− 1.45475i
$260$	0	0
$261$	0.0199396i	0.00123423i
$262$	0	0
$263$	13.1317	0.809735	0.404868	−	0.914375i	$-0.367318\pi$
$263$	13.1317	0.809735	0.404868	+	0.914375i	$0.367318\pi$
$264$	0	0
$265$	43.6287	2.68009
$266$	0	0
$267$	23.1585i	1.41728i
$268$	0	0
$269$	13.4659i	0.821028i	0.911854	+	0.410514i	$0.134651\pi$
$269$	13.4659i	0.821028i	−0.911854	+	0.410514i	$0.865349\pi$
$270$	0	0
$271$	3.02026	0.183468	0.0917339	−	0.995784i	$-0.470759\pi$
$271$	3.02026	0.183468	0.0917339	+	0.995784i	$0.470759\pi$
$272$	0	0
$273$	−2.46837	−0.149392
$274$	0	0
$275$	− 23.4150i	− 1.41197i
$276$	0	0
$277$	− 8.16722i	− 0.490721i	−0.969432	−	0.245360i	$-0.921094\pi$
$277$	− 8.16722i	− 0.490721i	0.969432	−	0.245360i	$-0.0789063\pi$
$278$	0	0
$279$	0.404509	0.0242173
$280$	0	0
$281$	20.4956	1.22266	0.611332	−	0.791374i	$-0.290634\pi$
$281$	20.4956	1.22266	0.611332	+	0.791374i	$0.290634\pi$
$282$	0	0
$283$	− 19.9690i	− 1.18703i	−0.804823	−	0.593515i	$-0.797740\pi$
$283$	− 19.9690i	− 1.18703i	0.804823	−	0.593515i	$-0.202260\pi$
$284$	0	0
$285$	7.11337i	0.421360i
$286$	0	0
$287$	5.21157	0.307629
$288$	0	0
$289$	−2.34315	−0.137832
$290$	0	0
$291$	− 12.0686i	− 0.707472i
$292$	0	0
$293$	10.8432i	0.633465i	0.948515	+	0.316733i	$0.102586\pi$
$293$	10.8432i	0.633465i	−0.948515	+	0.316733i	$0.897414\pi$
$294$	0	0
$295$	−25.9796	−1.51259
$296$	0	0
$297$	9.34638	0.542332
$298$	0	0
$299$	0.298546i	0.0172654i
$300$	0	0
$301$	24.8165i	1.43040i
$302$	0	0
$303$	−16.2762	−0.935041
$304$	0	0
$305$	−5.64318	−0.323127
$306$	0	0
$307$	5.36573i	0.306238i	0.988208	+	0.153119i	$0.0489318\pi$
$307$	5.36573i	0.306238i	−0.988208	+	0.153119i	$0.951068\pi$
$308$	0	0
$309$	21.2239i	1.20739i
$310$	0	0
$311$	−10.3271	−0.585598	−0.292799	−	0.956174i	$-0.594587\pi$
$311$	−10.3271	−0.585598	−0.292799	+	0.956174i	$0.594587\pi$
$312$	0	0
$313$	29.2020	1.65060	0.825298	−	0.564698i	$-0.191007\pi$
$313$	29.2020	1.65060	0.825298	+	0.564698i	$0.191007\pi$
$314$	0	0
$315$	− 3.18026i	− 0.179187i
$316$	0	0
$317$	− 11.5115i	− 0.646551i	−0.946305	−	0.323276i	$-0.895216\pi$
$317$	− 11.5115i	− 0.646551i	0.946305	−	0.323276i	$-0.104784\pi$
$318$	0	0
$319$	−0.127674	−0.00714837
$320$	0	0
$321$	12.9047	0.720267
$322$	0	0
$323$	− 3.82843i	− 0.213019i
$324$	0	0
$325$	− 7.39594i	− 0.410253i
$326$	0	0
$327$	27.2273	1.50567
$328$	0	0
$329$	−14.5736	−0.803467
$330$	0	0
$331$	27.0629i	1.48751i	0.668451	+	0.743756i	$0.266958\pi$
$331$	27.0629i	1.48751i	−0.668451	+	0.743756i	$0.733042\pi$
$332$	0	0
$333$	2.31325i	0.126765i
$334$	0	0
$335$	38.8383	2.12196
$336$	0	0
$337$	−26.1851	−1.42639	−0.713197	−	0.700963i	$-0.752754\pi$
$337$	−26.1851	−1.42639	−0.713197	+	0.700963i	$0.752754\pi$
$338$	0	0
$339$	24.5642i	1.33414i
$340$	0	0
$341$	2.59008i	0.140261i
$342$	0	0
$343$	17.8381	0.963165
$344$	0	0
$345$	3.88670	0.209253
$346$	0	0
$347$	27.2734i	1.46411i	0.681244	+	0.732056i	$0.261439\pi$
$347$	27.2734i	1.46411i	−0.681244	+	0.732056i	$0.738561\pi$
$348$	0	0
$349$	− 6.11780i	− 0.327478i	−0.986504	−	0.163739i	$-0.947644\pi$
$349$	− 6.11780i	− 0.327478i	0.986504	−	0.163739i	$-0.0523555\pi$
$350$	0	0
$351$	2.95218	0.157576
$352$	0	0
$353$	−8.51030	−0.452958	−0.226479	−	0.974016i	$-0.572721\pi$
$353$	−8.51030	−0.452958	−0.226479	+	0.974016i	$0.572721\pi$
$354$	0	0
$355$	− 63.0783i	− 3.34785i
$356$	0	0
$357$	− 17.2951i	− 0.915356i
$358$	0	0
$359$	−13.0097	−0.686628	−0.343314	−	0.939221i	$-0.611550\pi$
$359$	−13.0097	−0.686628	−0.343314	+	0.939221i	$0.611550\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	− 13.2304i	− 0.694417i
$364$	0	0
$365$	− 36.6023i	− 1.91585i
$366$	0	0
$367$	26.1953	1.36738	0.683692	−	0.729771i	$-0.260373\pi$
$367$	26.1953	1.36738	0.683692	+	0.729771i	$0.260373\pi$
$368$	0	0
$369$	−0.514936	−0.0268065
$370$	0	0
$371$	27.7077i	1.43851i
$372$	0	0
$373$	33.4436i	1.73165i	0.500351	+	0.865823i	$0.333204\pi$
$373$	33.4436i	1.73165i	−0.500351	+	0.865823i	$0.666796\pi$
$374$	0	0
$375$	−60.7191	−3.13552
$376$	0	0
$377$	−0.0403276	−0.00207698
$378$	0	0
$379$	− 28.6269i	− 1.47046i	−0.677816	−	0.735231i	$-0.737073\pi$
$379$	− 28.6269i	− 1.47046i	0.677816	−	0.735231i	$-0.262927\pi$
$380$	0	0
$381$	16.7006i	0.855598i
$382$	0	0
$383$	−20.1066	−1.02740	−0.513701	−	0.857969i	$-0.671726\pi$
$383$	−20.1066	−1.02740	−0.513701	+	0.857969i	$0.671726\pi$
$384$	0	0
$385$	20.3633	1.03781
$386$	0	0
$387$	− 2.45203i	− 0.124644i
$388$	0	0
$389$	− 25.6141i	− 1.29868i	−0.760496	−	0.649342i	$-0.775044\pi$
$389$	− 25.6141i	− 1.29868i	0.760496	−	0.649342i	$-0.224956\pi$
$390$	0	0
$391$	−2.09183	−0.105788
$392$	0	0
$393$	−0.589550	−0.0297389
$394$	0	0
$395$	− 33.0884i	− 1.66486i
$396$	0	0
$397$	− 4.40161i	− 0.220911i	−0.993881	−	0.110455i	$-0.964769\pi$
$397$	− 4.40161i	− 0.220911i	0.993881	−	0.110455i	$-0.0352309\pi$
$398$	0	0
$399$	−4.51756	−0.226161
$400$	0	0
$401$	−9.38031	−0.468430	−0.234215	−	0.972185i	$-0.575252\pi$
$401$	−9.38031	−0.468430	−0.234215	+	0.972185i	$0.575252\pi$
$402$	0	0
$403$	0.818114i	0.0407532i
$404$	0	0
$405$	− 34.9444i	− 1.73640i
$406$	0	0
$407$	−14.8118	−0.734194
$408$	0	0
$409$	0.216515	0.0107060	0.00535299	−	0.999986i	$-0.498296\pi$
$409$	0.216515	0.0107060	0.00535299	+	0.999986i	$0.498296\pi$
$410$	0	0
$411$	− 14.3570i	− 0.708181i
$412$	0	0
$413$	− 16.4991i	− 0.811869i
$414$	0	0
$415$	72.3557	3.55180
$416$	0	0
$417$	23.9033	1.17055
$418$	0	0
$419$	18.7827i	0.917593i	0.888541	+	0.458796i	$0.151719\pi$
$419$	18.7827i	0.917593i	−0.888541	+	0.458796i	$0.848281\pi$
$420$	0	0
$421$	− 2.19130i	− 0.106798i	−0.998573	−	0.0533988i	$-0.982995\pi$
$421$	− 2.19130i	− 0.106798i	0.998573	−	0.0533988i	$-0.0170055\pi$
$422$	0	0
$423$	1.43996	0.0700134
$424$	0	0
$425$	51.8212	2.51370
$426$	0	0
$427$	− 3.58387i	− 0.173436i
$428$	0	0
$429$	1.56164i	0.0753966i
$430$	0	0
$431$	−20.2673	−0.976240	−0.488120	−	0.872777i	$-0.662317\pi$
$431$	−20.2673	−0.976240	−0.488120	+	0.872777i	$0.662317\pi$
$432$	0	0
$433$	10.8766	0.522696	0.261348	−	0.965245i	$-0.415833\pi$
$433$	10.8766	0.522696	0.261348	+	0.965245i	$0.415833\pi$
$434$	0	0
$435$	0.525015i	0.0251725i
$436$	0	0
$437$	0.546394i	0.0261376i
$438$	0	0
$439$	10.3917	0.495970	0.247985	−	0.968764i	$-0.420232\pi$
$439$	10.3917	0.495970	0.247985	+	0.968764i	$0.420232\pi$
$440$	0	0
$441$	0.128601	0.00612386
$442$	0	0
$443$	− 26.2254i	− 1.24601i	−0.782219	−	0.623004i	$-0.785912\pi$
$443$	− 26.2254i	− 1.24601i	0.782219	−	0.623004i	$-0.214088\pi$
$444$	0	0
$445$	− 60.3460i	− 2.86067i
$446$	0	0
$447$	−16.1485	−0.763797
$448$	0	0
$449$	26.6928	1.25971	0.629855	−	0.776713i	$-0.283114\pi$
$449$	26.6928	1.25971	0.629855	+	0.776713i	$0.283114\pi$
$450$	0	0
$451$	− 3.29715i	− 0.155257i
$452$	0	0
$453$	− 14.2267i	− 0.668431i
$454$	0	0
$455$	6.43203	0.301538
$456$	0	0
$457$	−14.1819	−0.663401	−0.331700	−	0.943385i	$-0.607622\pi$
$457$	−14.1819	−0.663401	−0.331700	+	0.943385i	$0.607622\pi$
$458$	0	0
$459$	20.6851i	0.965499i
$460$	0	0
$461$	− 34.9375i	− 1.62720i	−0.581425	−	0.813600i	$-0.697505\pi$
$461$	− 34.9375i	− 1.62720i	0.581425	−	0.813600i	$-0.302495\pi$
$462$	0	0
$463$	−36.9587	−1.71762	−0.858808	−	0.512298i	$-0.828794\pi$
$463$	−36.9587	−1.71762	−0.858808	+	0.512298i	$0.828794\pi$
$464$	0	0
$465$	10.6508	0.493920
$466$	0	0
$467$	− 5.58065i	− 0.258242i	−0.991629	−	0.129121i	$-0.958785\pi$
$467$	− 5.58065i	− 0.258242i	0.991629	−	0.129121i	$-0.0412155\pi$
$468$	0	0
$469$	24.6654i	1.13894i
$470$	0	0
$471$	31.2042	1.43781
$472$	0	0
$473$	15.7004	0.721906
$474$	0	0
$475$	− 13.5359i	− 0.621070i
$476$	0	0
$477$	− 2.73770i	− 0.125351i
$478$	0	0
$479$	35.0828	1.60297	0.801487	−	0.598012i	$-0.204042\pi$
$479$	35.0828	1.60297	0.801487	+	0.598012i	$0.204042\pi$
$480$	0	0
$481$	−4.67851	−0.213322
$482$	0	0
$483$	2.46837i	0.112315i
$484$	0	0
$485$	31.4481i	1.42798i
$486$	0	0
$487$	−30.5808	−1.38575	−0.692874	−	0.721059i	$-0.743656\pi$
$487$	−30.5808	−1.38575	−0.692874	+	0.721059i	$0.743656\pi$
$488$	0	0
$489$	34.1791	1.54563
$490$	0	0
$491$	12.7282i	0.574417i	0.957868	+	0.287208i	$0.0927273\pi$
$491$	12.7282i	0.574417i	−0.957868	+	0.287208i	$0.907273\pi$
$492$	0	0
$493$	− 0.282564i	− 0.0127260i
$494$	0	0
$495$	−2.01202	−0.0904338
$496$	0	0
$497$	40.0597	1.79693
$498$	0	0
$499$	− 13.4286i	− 0.601148i	−0.953758	−	0.300574i	$-0.902822\pi$
$499$	− 13.4286i	− 0.601148i	0.953758	−	0.300574i	$-0.0971783\pi$
$500$	0	0
$501$	− 33.6781i	− 1.50462i
$502$	0	0
$503$	−30.3977	−1.35537	−0.677683	−	0.735354i	$-0.737016\pi$
$503$	−30.3977	−1.35537	−0.677683	+	0.735354i	$0.737016\pi$
$504$	0	0
$505$	42.4122	1.88732
$506$	0	0
$507$	− 20.9856i	− 0.932005i
$508$	0	0
$509$	21.5175i	0.953745i	0.878972	+	0.476873i	$0.158230\pi$
$509$	21.5175i	0.953745i	−0.878972	+	0.476873i	$0.841770\pi$
$510$	0	0
$511$	23.2454	1.02832
$512$	0	0
$513$	5.40303	0.238550
$514$	0	0
$515$	− 55.3049i	− 2.43702i
$516$	0	0
$517$	9.22013i	0.405501i
$518$	0	0
$519$	14.8155	0.650330
$520$	0	0
$521$	14.0851	0.617081	0.308540	−	0.951211i	$-0.400160\pi$
$521$	14.0851	0.617081	0.308540	+	0.951211i	$0.400160\pi$
$522$	0	0
$523$	− 22.8507i	− 0.999190i	−0.866259	−	0.499595i	$-0.833482\pi$
$523$	− 22.8507i	− 0.999190i	0.866259	−	0.499595i	$-0.166518\pi$
$524$	0	0
$525$	− 61.1492i	− 2.66877i
$526$	0	0
$527$	−5.73229	−0.249703
$528$	0	0
$529$	−22.7015	−0.987020
$530$	0	0
$531$	1.63022i	0.0707455i
$532$	0	0
$533$	− 1.04145i	− 0.0451103i
$534$	0	0
$535$	−33.6267	−1.45381
$536$	0	0
$537$	−17.6295	−0.760769
$538$	0	0
$539$	0.823436i	0.0354679i
$540$	0	0
$541$	36.2741i	1.55955i	0.626062	+	0.779774i	$0.284666\pi$
$541$	36.2741i	1.55955i	−0.626062	+	0.779774i	$0.715334\pi$
$542$	0	0
$543$	23.8621	1.02402
$544$	0	0
$545$	−70.9484	−3.03910
$546$	0	0
$547$	22.1698i	0.947913i	0.880548	+	0.473957i	$0.157175\pi$
$547$	22.1698i	0.947913i	−0.880548	+	0.473957i	$0.842825\pi$
$548$	0	0
$549$	0.354109i	0.0151130i
$550$	0	0
$551$	−0.0738068	−0.00314427
$552$	0	0
$553$	21.0138	0.893596
$554$	0	0
$555$	60.9085i	2.58542i
$556$	0	0
$557$	15.3385i	0.649915i	0.945729	+	0.324957i	$0.105350\pi$
$557$	15.3385i	0.649915i	−0.945729	+	0.324957i	$0.894650\pi$
$558$	0	0
$559$	4.95919	0.209752
$560$	0	0
$561$	−10.9420	−0.461970
$562$	0	0
$563$	− 8.47021i	− 0.356977i	−0.983942	−	0.178488i	$-0.942879\pi$
$563$	− 8.47021i	− 0.356977i	0.983942	−	0.178488i	$-0.0571207\pi$
$564$	0	0
$565$	− 64.0089i	− 2.69287i
$566$	0	0
$567$	22.1925	0.931997
$568$	0	0
$569$	−25.7940	−1.08134	−0.540670	−	0.841235i	$-0.681829\pi$
$569$	−25.7940	−1.08134	−0.540670	+	0.841235i	$0.681829\pi$
$570$	0	0
$571$	12.3855i	0.518318i	0.965835	+	0.259159i	$0.0834453\pi$
$571$	12.3855i	0.518318i	−0.965835	+	0.259159i	$0.916555\pi$
$572$	0	0
$573$	36.1133i	1.50865i
$574$	0	0
$575$	−7.39594	−0.308432
$576$	0	0
$577$	26.8555	1.11801	0.559005	−	0.829164i	$-0.311183\pi$
$577$	26.8555	1.11801	0.559005	+	0.829164i	$0.311183\pi$
$578$	0	0
$579$	− 0.993976i	− 0.0413082i
$580$	0	0
$581$	45.9517i	1.90640i
$582$	0	0
$583$	17.5296	0.726001
$584$	0	0
$585$	−0.635526	−0.0262758
$586$	0	0
$587$	7.02574i	0.289983i	0.989433	+	0.144992i	$0.0463156\pi$
$587$	7.02574i	0.289983i	−0.989433	+	0.144992i	$0.953684\pi$
$588$	0	0
$589$	1.49730i	0.0616950i
$590$	0	0
$591$	−10.3833	−0.427110
$592$	0	0
$593$	12.2750	0.504074	0.252037	−	0.967718i	$-0.418900\pi$
$593$	12.2750	0.504074	0.252037	+	0.967718i	$0.418900\pi$
$594$	0	0
$595$	45.0674i	1.84758i
$596$	0	0
$597$	39.6967i	1.62468i
$598$	0	0
$599$	−1.13112	−0.0462163	−0.0231081	−	0.999733i	$-0.507356\pi$
$599$	−1.13112	−0.0462163	−0.0231081	+	0.999733i	$0.507356\pi$
$600$	0	0
$601$	15.4182	0.628921	0.314460	−	0.949271i	$-0.398176\pi$
$601$	15.4182	0.628921	0.314460	+	0.949271i	$0.398176\pi$
$602$	0	0
$603$	− 2.43710i	− 0.0992464i
$604$	0	0
$605$	34.4756i	1.40163i
$606$	0	0
$607$	41.2922	1.67600	0.838000	−	0.545671i	$-0.183725\pi$
$607$	41.2922	1.67600	0.838000	+	0.545671i	$0.183725\pi$
$608$	0	0
$609$	−0.333426	−0.0135111
$610$	0	0
$611$	2.91230i	0.117819i
$612$	0	0
$613$	− 19.0972i	− 0.771329i	−0.922639	−	0.385664i	$-0.873972\pi$
$613$	− 19.0972i	− 0.771329i	0.922639	−	0.385664i	$-0.126028\pi$
$614$	0	0
$615$	−13.5584	−0.546728
$616$	0	0
$617$	−40.7423	−1.64022	−0.820112	−	0.572202i	$-0.806089\pi$
$617$	−40.7423	−1.64022	−0.820112	+	0.572202i	$0.806089\pi$
$618$	0	0
$619$	35.8186i	1.43967i	0.694145	+	0.719835i	$0.255782\pi$
$619$	35.8186i	1.43967i	−0.694145	+	0.719835i	$0.744218\pi$
$620$	0	0
$621$	− 2.95218i	− 0.118467i
$622$	0	0
$623$	38.3245	1.53544
$624$	0	0
$625$	90.5412	3.62165
$626$	0	0
$627$	2.85808i	0.114141i
$628$	0	0
$629$	− 32.7810i	− 1.30707i
$630$	0	0
$631$	7.14251	0.284339	0.142169	−	0.989842i	$-0.454592\pi$
$631$	7.14251	0.284339	0.142169	+	0.989842i	$0.454592\pi$
$632$	0	0
$633$	−10.3327	−0.410688
$634$	0	0
$635$	− 43.5182i	− 1.72697i
$636$	0	0
$637$	0.260094i	0.0103053i
$638$	0	0
$639$	−3.95816	−0.156582
$640$	0	0
$641$	−9.03591	−0.356897	−0.178449	−	0.983949i	$-0.557108\pi$
$641$	−9.03591	−0.356897	−0.178449	+	0.983949i	$0.557108\pi$
$642$	0	0
$643$	− 33.5484i	− 1.32302i	−0.749936	−	0.661510i	$-0.769916\pi$
$643$	− 33.5484i	− 1.32302i	0.749936	−	0.661510i	$-0.230084\pi$
$644$	0	0
$645$	− 64.5626i	− 2.54215i
$646$	0	0
$647$	29.9641	1.17801	0.589004	−	0.808130i	$-0.299520\pi$
$647$	29.9641	1.17801	0.589004	+	0.808130i	$0.299520\pi$
$648$	0	0
$649$	−10.4384	−0.409741
$650$	0	0
$651$	6.76413i	0.265107i
$652$	0	0
$653$	5.44946i	0.213254i	0.994299	+	0.106627i	$0.0340050\pi$
$653$	5.44946i	0.213254i	−0.994299	+	0.106627i	$0.965995\pi$
$654$	0	0
$655$	1.53624	0.0600258
$656$	0	0
$657$	−2.29679	−0.0896065
$658$	0	0
$659$	− 12.1018i	− 0.471420i	−0.971823	−	0.235710i	$-0.924259\pi$
$659$	− 12.1018i	− 0.471420i	0.971823	−	0.235710i	$-0.0757415\pi$
$660$	0	0
$661$	33.3474i	1.29706i	0.761188	+	0.648531i	$0.224616\pi$
$661$	33.3474i	1.29706i	−0.761188	+	0.648531i	$0.775384\pi$
$662$	0	0
$663$	−3.45617	−0.134226
$664$	0	0
$665$	11.7718	0.456490
$666$	0	0
$667$	0.0403276i	0.00156149i
$668$	0	0
$669$	20.3965i	0.788574i
$670$	0	0
$671$	−2.26737	−0.0875309
$672$	0	0
$673$	37.9052	1.46114	0.730570	−	0.682838i	$-0.239255\pi$
$673$	37.9052	1.46114	0.730570	+	0.682838i	$0.239255\pi$
$674$	0	0
$675$	73.1349i	2.81496i
$676$	0	0
$677$	47.5427i	1.82721i	0.406598	+	0.913607i	$0.366715\pi$
$677$	47.5427i	1.82721i	−0.406598	+	0.913607i	$0.633285\pi$
$678$	0	0
$679$	−19.9720	−0.766457
$680$	0	0
$681$	−3.81475	−0.146182
$682$	0	0
$683$	29.7347i	1.13777i	0.822418	+	0.568883i	$0.192624\pi$
$683$	29.7347i	1.13777i	−0.822418	+	0.568883i	$0.807376\pi$
$684$	0	0
$685$	37.4113i	1.42941i
$686$	0	0
$687$	−3.50231	−0.133622
$688$	0	0
$689$	5.53696	0.210941
$690$	0	0
$691$	2.82857i	0.107604i	0.998552	+	0.0538019i	$0.0171340\pi$
$691$	2.82857i	0.107604i	−0.998552	+	0.0538019i	$0.982866\pi$
$692$	0	0
$693$	− 1.27780i	− 0.0485395i
$694$	0	0
$695$	−62.2867	−2.36267
$696$	0	0
$697$	7.29715	0.276399
$698$	0	0
$699$	− 41.6581i	− 1.57565i
$700$	0	0
$701$	38.0093i	1.43559i	0.696253	+	0.717796i	$0.254849\pi$
$701$	38.0093i	1.43559i	−0.696253	+	0.717796i	$0.745151\pi$
$702$	0	0
$703$	−8.56253	−0.322942
$704$	0	0
$705$	37.9146	1.42795
$706$	0	0
$707$	26.9351i	1.01300i
$708$	0	0
$709$	− 0.225499i	− 0.00846877i	−0.999991	−	0.00423439i	$-0.998652\pi$
$709$	− 0.225499i	− 0.00846877i	0.999991	−	0.00423439i	$-0.00134785\pi$
$710$	0	0
$711$	−2.07629	−0.0778671
$712$	0	0
$713$	0.818114	0.0306386
$714$	0	0
$715$	− 4.06929i	− 0.152183i
$716$	0	0
$717$	37.6587i	1.40639i
$718$	0	0
$719$	−26.7607	−0.998006	−0.499003	−	0.866600i	$-0.666300\pi$
$719$	−26.7607	−0.998006	−0.499003	+	0.866600i	$0.666300\pi$
$720$	0	0
$721$	35.1230	1.30805
$722$	0	0
$723$	− 4.92468i	− 0.183151i
$724$	0	0
$725$	− 0.999041i	− 0.0371035i
$726$	0	0
$727$	10.0434	0.372487	0.186244	−	0.982504i	$-0.440369\pi$
$727$	10.0434	0.372487	0.186244	+	0.982504i	$0.440369\pi$
$728$	0	0
$729$	−28.9738	−1.07310
$730$	0	0
$731$	34.7477i	1.28519i
$732$	0	0
$733$	− 1.32974i	− 0.0491152i	−0.999698	−	0.0245576i	$-0.992182\pi$
$733$	− 1.32974i	− 0.0491152i	0.999698	−	0.0245576i	$-0.00781771\pi$
$734$	0	0
$735$	3.38610	0.124898
$736$	0	0
$737$	15.6048	0.574812
$738$	0	0
$739$	13.0730i	0.480898i	0.970662	+	0.240449i	$0.0772947\pi$
$739$	13.0730i	0.480898i	−0.970662	+	0.240449i	$0.922705\pi$
$740$	0	0
$741$	0.902764i	0.0331639i
$742$	0	0
$743$	11.0577	0.405666	0.202833	−	0.979213i	$-0.434985\pi$
$743$	11.0577	0.405666	0.202833	+	0.979213i	$0.434985\pi$
$744$	0	0
$745$	42.0795	1.54167
$746$	0	0
$747$	− 4.54032i	− 0.166122i
$748$	0	0
$749$	− 21.3557i	− 0.780319i
$750$	0	0
$751$	−43.0977	−1.57266	−0.786330	−	0.617807i	$-0.788021\pi$
$751$	−43.0977	−1.57266	−0.786330	+	0.617807i	$0.788021\pi$
$752$	0	0
$753$	14.6162	0.532643
$754$	0	0
$755$	37.0718i	1.34918i
$756$	0	0
$757$	− 7.53688i	− 0.273933i	−0.990576	−	0.136966i	$-0.956265\pi$
$757$	− 7.53688i	− 0.273933i	0.990576	−	0.136966i	$-0.0437352\pi$
$758$	0	0
$759$	1.56164	0.0566839
$760$	0	0
$761$	32.3702	1.17342	0.586710	−	0.809797i	$-0.300423\pi$
$761$	32.3702	1.17342	0.586710	+	0.809797i	$0.300423\pi$
$762$	0	0
$763$	− 45.0579i	− 1.63121i
$764$	0	0
$765$	− 4.45295i	− 0.160997i
$766$	0	0
$767$	−3.29710	−0.119051
$768$	0	0
$769$	11.1972	0.403780	0.201890	−	0.979408i	$-0.435292\pi$
$769$	11.1972	0.403780	0.201890	+	0.979408i	$0.435292\pi$
$770$	0	0
$771$	9.98263i	0.359516i
$772$	0	0
$773$	47.2345i	1.69891i	0.527664	+	0.849453i	$0.323068\pi$
$773$	47.2345i	1.69891i	−0.527664	+	0.849453i	$0.676932\pi$
$774$	0	0
$775$	−20.2673	−0.728022
$776$	0	0
$777$	−38.6817	−1.38770
$778$	0	0
$779$	− 1.90604i	− 0.0682911i
$780$	0	0
$781$	− 25.3442i	− 0.906888i
$782$	0	0
$783$	0.398780	0.0142512
$784$	0	0
$785$	−81.3114	−2.90213
$786$	0	0
$787$	35.6791i	1.27182i	0.771762	+	0.635911i	$0.219376\pi$
$787$	35.6791i	1.27182i	−0.771762	+	0.635911i	$0.780624\pi$
$788$	0	0
$789$	− 21.6965i	− 0.772415i
$790$	0	0
$791$	40.6508	1.44537
$792$	0	0
$793$	−0.716181	−0.0254323
$794$	0	0
$795$	− 72.0843i	− 2.55657i
$796$	0	0
$797$	12.4881i	0.442351i	0.975234	+	0.221175i	$0.0709893\pi$
$797$	12.4881i	0.442351i	−0.975234	+	0.221175i	$0.929011\pi$
$798$	0	0
$799$	−20.4057	−0.721901
$800$	0	0
$801$	−3.78671	−0.133797
$802$	0	0
$803$	− 14.7064i	− 0.518979i
$804$	0	0
$805$	− 6.43203i	− 0.226699i
$806$	0	0
$807$	22.2486	0.783188
$808$	0	0
$809$	−25.1660	−0.884789	−0.442394	−	0.896821i	$-0.645871\pi$
$809$	−25.1660	−0.884789	−0.442394	+	0.896821i	$0.645871\pi$
$810$	0	0
$811$	− 2.99454i	− 0.105152i	−0.998617	−	0.0525762i	$-0.983257\pi$
$811$	− 2.99454i	− 0.105152i	0.998617	−	0.0525762i	$-0.0167432\pi$
$812$	0	0
$813$	− 4.99015i	− 0.175012i
$814$	0	0
$815$	−89.0633	−3.11975
$816$	0	0
$817$	9.07622	0.317537
$818$	0	0
$819$	− 0.403609i	− 0.0141033i
$820$	0	0
$821$	− 3.63700i	− 0.126932i	−0.997984	−	0.0634660i	$-0.979785\pi$
$821$	− 3.63700i	− 0.126932i	0.997984	−	0.0634660i	$-0.0202154\pi$
$822$	0	0
$823$	16.7620	0.584287	0.292144	−	0.956374i	$-0.405631\pi$
$823$	16.7620	0.584287	0.292144	+	0.956374i	$0.405631\pi$
$824$	0	0
$825$	−38.6867	−1.34690
$826$	0	0
$827$	1.31834i	0.0458432i	0.999737	+	0.0229216i	$0.00729681\pi$
$827$	1.31834i	0.0458432i	−0.999737	+	0.0229216i	$0.992703\pi$
$828$	0	0
$829$	18.2603i	0.634205i	0.948391	+	0.317102i	$0.102710\pi$
$829$	18.2603i	0.634205i	−0.948391	+	0.317102i	$0.897290\pi$
$830$	0	0
$831$	−13.4941	−0.468104
$832$	0	0
$833$	−1.82240	−0.0631425
$834$	0	0
$835$	87.7577i	3.03698i
$836$	0	0
$837$	− 8.08994i	− 0.279629i
$838$	0	0
$839$	−24.0265	−0.829486	−0.414743	−	0.909939i	$-0.636129\pi$
$839$	−24.0265	−0.829486	−0.414743	+	0.909939i	$0.636129\pi$
$840$	0	0
$841$	28.9946	0.999812
$842$	0	0
$843$	− 33.8633i	− 1.16631i
$844$	0	0
$845$	54.6840i	1.88119i
$846$	0	0
$847$	−21.8948	−0.752313
$848$	0	0
$849$	−32.9932	−1.13232
$850$	0	0
$851$	4.67851i	0.160377i
$852$	0	0
$853$	11.1739i	0.382586i	0.981533	+	0.191293i	$0.0612681\pi$
$853$	11.1739i	0.382586i	−0.981533	+	0.191293i	$0.938732\pi$
$854$	0	0
$855$	−1.16313	−0.0397781
$856$	0	0
$857$	−28.6182	−0.977578	−0.488789	−	0.872402i	$-0.662561\pi$
$857$	−28.6182	−0.977578	−0.488789	+	0.872402i	$0.662561\pi$
$858$	0	0
$859$	16.1420i	0.550758i	0.961336	+	0.275379i	$0.0888033\pi$
$859$	16.1420i	0.550758i	−0.961336	+	0.275379i	$0.911197\pi$
$860$	0	0
$861$	− 8.61067i	− 0.293451i
$862$	0	0
$863$	−3.17360	−0.108031	−0.0540154	−	0.998540i	$-0.517202\pi$
$863$	−3.17360	−0.108031	−0.0540154	+	0.998540i	$0.517202\pi$
$864$	0	0
$865$	−38.6061	−1.31265
$866$	0	0
$867$	3.87140i	0.131480i
$868$	0	0
$869$	− 13.2946i	− 0.450988i
$870$	0	0
$871$	4.92900	0.167013
$872$	0	0
$873$	1.97337	0.0667883
$874$	0	0
$875$	100.483i	3.39694i
$876$	0	0
$877$	− 30.9086i	− 1.04371i	−0.853034	−	0.521855i	$-0.825240\pi$
$877$	− 30.9086i	− 1.04371i	0.853034	−	0.521855i	$-0.174760\pi$
$878$	0	0
$879$	17.9153	0.604270
$880$	0	0
$881$	−13.2076	−0.444976	−0.222488	−	0.974935i	$-0.571418\pi$
$881$	−13.2076	−0.444976	−0.222488	+	0.974935i	$0.571418\pi$
$882$	0	0
$883$	2.47616i	0.0833294i	0.999132	+	0.0416647i	$0.0132661\pi$
$883$	2.47616i	0.0833294i	−0.999132	+	0.0416647i	$0.986734\pi$
$884$	0	0
$885$	42.9241i	1.44288i
$886$	0	0
$887$	−27.5413	−0.924745	−0.462372	−	0.886686i	$-0.653002\pi$
$887$	−27.5413	−0.924745	−0.462372	+	0.886686i	$0.653002\pi$
$888$	0	0
$889$	27.6375	0.926932
$890$	0	0
$891$	− 14.0403i	− 0.470368i
$892$	0	0
$893$	5.33004i	0.178363i
$894$	0	0
$895$	45.9387	1.53556
$896$	0	0
$897$	0.493265	0.0164696
$898$	0	0
$899$	0.110511i	0.00368574i
$900$	0	0
$901$	38.7959i	1.29248i
$902$	0	0
$903$	41.0024	1.36447
$904$	0	0
$905$	−62.1795	−2.06692
$906$	0	0
$907$	10.7841i	0.358079i	0.983842	+	0.179039i	$0.0572990\pi$
$907$	10.7841i	0.358079i	−0.983842	+	0.179039i	$0.942701\pi$
$908$	0	0
$909$	− 2.66136i	− 0.0882718i
$910$	0	0
$911$	−14.3560	−0.475634	−0.237817	−	0.971310i	$-0.576432\pi$
$911$	−14.3560	−0.475634	−0.237817	+	0.971310i	$0.576432\pi$
$912$	0	0
$913$	29.0718	0.962137
$914$	0	0
$915$	9.32379i	0.308235i
$916$	0	0
$917$	0.975635i	0.0322183i
$918$	0	0
$919$	32.1519	1.06059	0.530297	−	0.847812i	$-0.322081\pi$
$919$	32.1519	1.06059	0.530297	+	0.847812i	$0.322081\pi$
$920$	0	0
$921$	8.86537	0.292124
$922$	0	0
$923$	− 8.00532i	− 0.263498i
$924$	0	0
$925$	− 115.902i	− 3.81082i
$926$	0	0
$927$	−3.47038	−0.113982
$928$	0	0
$929$	14.8371	0.486790	0.243395	−	0.969927i	$-0.421739\pi$
$929$	14.8371	0.486790	0.243395	+	0.969927i	$0.421739\pi$
$930$	0	0
$931$	0.476019i	0.0156009i
$932$	0	0
$933$	17.0627i	0.558608i
$934$	0	0
$935$	28.5124	0.932454
$936$	0	0
$937$	18.1301	0.592284	0.296142	−	0.955144i	$-0.404300\pi$
$937$	18.1301	0.592284	0.296142	+	0.955144i	$0.404300\pi$
$938$	0	0
$939$	− 48.2482i	− 1.57452i
$940$	0	0
$941$	30.2219i	0.985206i	0.870254	+	0.492603i	$0.163954\pi$
$941$	30.2219i	0.985206i	−0.870254	+	0.492603i	$0.836046\pi$
$942$	0	0
$943$	−1.04145	−0.0339143
$944$	0	0
$945$	−63.6033	−2.06902
$946$	0	0
$947$	− 37.5706i	− 1.22088i	−0.792062	−	0.610441i	$-0.790992\pi$
$947$	− 37.5706i	− 1.22088i	0.792062	−	0.610441i	$-0.209008\pi$
$948$	0	0
$949$	− 4.64523i	− 0.150791i
$950$	0	0
$951$	−19.0196	−0.616752
$952$	0	0
$953$	−46.0319	−1.49112	−0.745559	−	0.666440i	$-0.767817\pi$
$953$	−46.0319	−1.49112	−0.745559	+	0.666440i	$0.767817\pi$
$954$	0	0
$955$	− 94.1033i	− 3.04511i
$956$	0	0
$957$	0.210946i	0.00681891i
$958$	0	0
$959$	−23.7592	−0.767224
$960$	0	0
$961$	−28.7581	−0.927681
$962$	0	0
$963$	2.11008i	0.0679962i
$964$	0	0
$965$	2.59008i	0.0833778i
$966$	0	0
$967$	55.6384	1.78921	0.894605	−	0.446858i	$-0.147457\pi$
$967$	55.6384	1.78921	0.894605	+	0.446858i	$0.147457\pi$
$968$	0	0
$969$	−6.32541	−0.203202
$970$	0	0
$971$	− 3.24836i	− 0.104245i	−0.998641	−	0.0521224i	$-0.983401\pi$
$971$	− 3.24836i	− 0.104245i	0.998641	−	0.0521224i	$-0.0165986\pi$
$972$	0	0
$973$	− 39.5570i	− 1.26814i
$974$	0	0
$975$	−12.2197	−0.391345
$976$	0	0
$977$	37.9387	1.21377	0.606884	−	0.794791i	$-0.292419\pi$
$977$	37.9387	1.21377	0.606884	+	0.794791i	$0.292419\pi$
$978$	0	0
$979$	− 24.2464i	− 0.774918i
$980$	0	0
$981$	4.45201i	0.142142i
$982$	0	0
$983$	40.6822	1.29756	0.648780	−	0.760976i	$-0.275280\pi$
$983$	40.6822	1.29756	0.648780	+	0.760976i	$0.275280\pi$
$984$	0	0
$985$	27.0565	0.862092
$986$	0	0
$987$	24.0788i	0.766436i
$988$	0	0
$989$	− 4.95919i	− 0.157693i
$990$	0	0
$991$	38.5615	1.22494	0.612472	−	0.790492i	$-0.290175\pi$
$991$	38.5615	1.22494	0.612472	+	0.790492i	$0.290175\pi$
$992$	0	0
$993$	44.7140	1.41895
$994$	0	0
$995$	− 103.441i	− 3.27930i
$996$	0	0
$997$	− 53.2516i	− 1.68649i	−0.537526	−	0.843247i	$-0.680641\pi$
$997$	− 53.2516i	− 1.68649i	0.537526	−	0.843247i	$-0.319359\pi$
$998$	0	0
$999$	46.2636	1.46372

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2432.2.c.i.1217.6	yes	16
4.3	odd	2	inner	2432.2.c.i.1217.12	yes	16
8.3	odd	2	inner	2432.2.c.i.1217.5	✓	16
8.5	even	2	inner	2432.2.c.i.1217.11	yes	16
16.3	odd	4		4864.2.a.bm.1.4		8
16.5	even	4		4864.2.a.bm.1.3		8
16.11	odd	4		4864.2.a.br.1.5		8
16.13	even	4		4864.2.a.br.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.i.1217.5	✓	16	8.3	odd	2	inner
2432.2.c.i.1217.6	yes	16	1.1	even	1	trivial
2432.2.c.i.1217.11	yes	16	8.5	even	2	inner
2432.2.c.i.1217.12	yes	16	4.3	odd	2	inner
4864.2.a.bm.1.3		8	16.5	even	4
4864.2.a.bm.1.4		8	16.3	odd	4
4864.2.a.br.1.5		8	16.11	odd	4
4864.2.a.br.1.6		8	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)

\(f(q)\)	\(=\)	\(q-1.65222i q^{3} +4.30533i q^{5} -2.73423 q^{7} +0.270160 q^{9} +O(q^{10})\)	\(q-1.65222i q^{3} +4.30533i q^{5} -2.73423 q^{7} +0.270160 q^{9} +1.72984i q^{11} +0.546394i q^{13} +7.11337 q^{15} -3.82843 q^{17} +1.00000i q^{19} +4.51756i q^{21} +0.546394 q^{23} -13.5359 q^{25} -5.40303i q^{27} +0.0738068i q^{29} +1.49730 q^{31} +2.85808 q^{33} -11.7718i q^{35} +8.56253i q^{37} +0.902764 q^{39} -1.90604 q^{41} -9.07622i q^{43} +1.16313i q^{45} +5.33004 q^{47} +0.476019 q^{49} +6.32541i q^{51} -10.1336i q^{53} -7.44754 q^{55} +1.65222 q^{57} +6.03429i q^{59} +1.31074i q^{61} -0.738679 q^{63} -2.35241 q^{65} -9.02097i q^{67} -0.902764i q^{69} -14.6512 q^{71} -8.50162 q^{73} +22.3643i q^{75} -4.72978i q^{77} -7.68543 q^{79} -8.11654 q^{81} -16.8061i q^{83} -16.4827i q^{85} +0.121945 q^{87} -14.0166 q^{89} -1.49397i q^{91} -2.47387i q^{93} -4.30533 q^{95} +7.30445 q^{97} +0.467333i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{9}+O(q^{10}) \) 16 * q - 8 * q^9	\( 16 q - 8 q^{9} - 16 q^{17} - 40 q^{25} + 48 q^{33} - 16 q^{41} + 16 q^{49} + 8 q^{57} + 16 q^{65} + 16 q^{73} - 64 q^{81} + 16 q^{89} + 80 q^{97}+O(q^{100}) \) 16 * q - 8 * q^9 - 16 * q^17 - 40 * q^25 + 48 * q^33 - 16 * q^41 + 16 * q^49 + 8 * q^57 + 16 * q^65 + 16 * q^73 - 64 * q^81 + 16 * q^89 + 80 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more