LMFDB - Embedded newform 3864.2.u.e.1609.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3864,2,Mod(1609,3864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3864, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3864.1609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3864.u (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:	$30.8541953410$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.2992527616.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 13x^{6} + 52x^{4} + 61x^{2} + 1 $ x^8 + 13x^6 + 52x^4 + 61*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1609.1
Root			$0.128950i$ of defining polynomial
Character	$\chi$	$=$	3864.1609
Dual form			3864.2.u.e.1609.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -2.98337 q^{5} +(-2.64261 + 0.128950i) q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -2.98337 q^{5} +(-2.64261 + 0.128950i) q^{7} -1.00000 q^{9} -1.65924i q^{11} -2.64261i q^{13} +2.98337i q^{15} -4.34076 q^{17} -4.94445 q^{19} +(0.128950 + 2.64261i) q^{21} +(-2.98337 - 3.75493i) q^{23} +3.90051 q^{25} +1.00000i q^{27} -0.686549 q^{29} -5.62598i q^{31} -1.65924 q^{33} +(7.88388 - 0.384707i) q^{35} -6.59867i q^{37} -2.64261 q^{39} -1.91714i q^{41} +5.83994i q^{43} +2.98337 q^{45} +5.74210i q^{47} +(6.96674 - 0.681530i) q^{49} +4.34076i q^{51} +6.46757i q^{53} +4.95012i q^{55} +4.94445i q^{57} +3.66490i q^{59} -6.35145 q^{61} +(2.64261 - 0.128950i) q^{63} +7.88388i q^{65} +1.27453i q^{67} +(-3.75493 + 2.98337i) q^{69} -1.66992 q^{71} -4.39968i q^{73} -3.90051i q^{75} +(0.213959 + 4.38471i) q^{77} -6.34076i q^{79} +1.00000 q^{81} -1.02229 q^{83} +12.9501 q^{85} +0.686549i q^{87} -5.12515 q^{89} +(0.340765 + 6.98337i) q^{91} -5.62598 q^{93} +14.7511 q^{95} +14.1418 q^{97} +1.65924i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{5} - 2 q^{7} - 8 q^{9}+O(q^{10}) $ 8 * q + 2 * q^5 - 2 * q^7 - 8 * q^9	$ 8 q + 2 q^{5} - 2 q^{7} - 8 q^{9} - 28 q^{17} - 8 q^{19} - 2 q^{21} + 2 q^{23} + 6 q^{25} + 20 q^{29} - 20 q^{33} + 12 q^{35} - 2 q^{39} - 2 q^{45} + 4 q^{49} + 14 q^{61} + 2 q^{63} + 18 q^{69} + 38 q^{71} + 2 q^{77} + 8 q^{81} + 12 q^{83} + 26 q^{85} + 42 q^{89} - 4 q^{91} - 10 q^{95} + 56 q^{97}+O(q^{100}) $ 8 * q + 2 * q^5 - 2 * q^7 - 8 * q^9 - 28 * q^17 - 8 * q^19 - 2 * q^21 + 2 * q^23 + 6 * q^25 + 20 * q^29 - 20 * q^33 + 12 * q^35 - 2 * q^39 - 2 * q^45 + 4 * q^49 + 14 * q^61 + 2 * q^63 + 18 * q^69 + 38 * q^71 + 2 * q^77 + 8 * q^81 + 12 * q^83 + 26 * q^85 + 42 * q^89 - 4 * q^91 - 10 * q^95 + 56 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3864\mathbb{Z}\right)^\times$.

$n$	$967$	$1289$	$1933$	$2761$	$2857$
$\chi(n)$	$1$	$1$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	−2.98337			−1.33420			−0.667102	−	0.744966i	$-0.732466\pi$
$5$	−2.98337			−1.33420			−0.667102	+	0.744966i	$0.732466\pi$
$6$		0			0
$7$	−2.64261	+	0.128950i	−0.998812	+	0.0487386i
$7$	−2.64261	+	0.128950i	−0.998812	+	0.0487386i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$		−	1.65924i		−	0.500278i	−0.968210	−	0.250139i	$-0.919524\pi$
$11$		−	1.65924i		−	0.500278i	0.968210	−	0.250139i	$-0.0804764\pi$
$12$		0			0
$13$		−	2.64261i		−	0.732927i	−0.930432	−	0.366464i	$-0.880568\pi$
$13$		−	2.64261i		−	0.732927i	0.930432	−	0.366464i	$-0.119432\pi$
$14$		0			0
$15$			2.98337i			0.770303i
$16$		0			0
$17$	−4.34076			−1.05279			−0.526395	−	0.850240i	$-0.676457\pi$
$17$	−4.34076			−1.05279			−0.526395	+	0.850240i	$0.676457\pi$
$18$		0			0
$19$	−4.94445			−1.13433			−0.567167	−	0.823603i	$-0.691961\pi$
$19$	−4.94445			−1.13433			−0.567167	+	0.823603i	$0.691961\pi$
$20$		0			0
$21$	0.128950	+	2.64261i	0.0281393	+	0.576664i
$22$		0			0
$23$	−2.98337	−	3.75493i	−0.622076	−	0.782957i
$23$	−2.98337	−	3.75493i	−0.622076	−	0.782957i
$24$		0			0
$25$	3.90051			0.780102
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	−0.686549			−0.127489			−0.0637444	−	0.997966i	$-0.520304\pi$
$29$	−0.686549			−0.127489			−0.0637444	+	0.997966i	$0.520304\pi$
$30$		0			0
$31$		−	5.62598i		−	1.01046i	−0.862986	−	0.505228i	$-0.831408\pi$
$31$		−	5.62598i		−	1.01046i	0.862986	−	0.505228i	$-0.168592\pi$
$32$		0			0
$33$	−1.65924			−0.288836
$34$		0			0
$35$	7.88388	−	0.384707i	1.33262	−	0.0650273i
$36$		0			0
$37$		−	6.59867i		−	1.08481i	−0.840116	−	0.542407i	$-0.817513\pi$
$37$		−	6.59867i		−	1.08481i	0.840116	−	0.542407i	$-0.182487\pi$
$38$		0			0
$39$	−2.64261			−0.423156
$40$		0			0
$41$		−	1.91714i		−	0.299406i	−0.988731	−	0.149703i	$-0.952168\pi$
$41$		−	1.91714i		−	0.299406i	0.988731	−	0.149703i	$-0.0478318\pi$
$42$		0			0
$43$			5.83994i			0.890582i	0.895386	+	0.445291i	$0.146900\pi$
$43$			5.83994i			0.890582i	−0.895386	+	0.445291i	$0.853100\pi$
$44$		0			0
$45$	2.98337			0.444735
$46$		0			0
$47$			5.74210i			0.837571i	0.908085	+	0.418786i	$0.137544\pi$
$47$			5.74210i			0.837571i	−0.908085	+	0.418786i	$0.862456\pi$
$48$		0			0
$49$	6.96674	−	0.681530i	0.995249	−	0.0973614i
$50$		0			0
$51$			4.34076i			0.607829i
$52$		0			0
$53$			6.46757i			0.888389i	0.895930	+	0.444195i	$0.146510\pi$
$53$			6.46757i			0.888389i	−0.895930	+	0.444195i	$0.853490\pi$
$54$		0			0
$55$			4.95012i			0.667473i
$56$		0			0
$57$			4.94445i			0.654908i
$58$		0			0
$59$			3.66490i			0.477130i	0.971127	+	0.238565i	$0.0766769\pi$
$59$			3.66490i			0.477130i	−0.971127	+	0.238565i	$0.923323\pi$
$60$		0			0
$61$	−6.35145			−0.813220			−0.406610	−	0.913602i	$-0.633289\pi$
$61$	−6.35145			−0.813220			−0.406610	+	0.913602i	$0.633289\pi$
$62$		0			0
$63$	2.64261	−	0.128950i	0.332937	−	0.0162462i
$64$		0			0
$65$			7.88388i			0.977875i
$66$		0			0
$67$			1.27453i			0.155708i	0.996965	+	0.0778542i	$0.0248069\pi$
$67$			1.27453i			0.155708i	−0.996965	+	0.0778542i	$0.975193\pi$
$68$		0			0
$69$	−3.75493	+	2.98337i	−0.452040	+	0.359156i
$70$		0			0
$71$	−1.66992			−0.198183			−0.0990916	−	0.995078i	$-0.531594\pi$
$71$	−1.66992			−0.198183			−0.0990916	+	0.995078i	$0.531594\pi$
$72$		0			0
$73$		−	4.39968i		−	0.514944i	−0.966286	−	0.257472i	$-0.917111\pi$
$73$		−	4.39968i		−	0.514944i	0.966286	−	0.257472i	$-0.0828895\pi$
$74$		0			0
$75$		−	3.90051i		−	0.450392i
$76$		0			0
$77$	0.213959	+	4.38471i	0.0243829	+	0.499684i
$78$		0			0
$79$		−	6.34076i		−	0.713392i	−0.934221	−	0.356696i	$-0.883903\pi$
$79$		−	6.34076i		−	0.713392i	0.934221	−	0.356696i	$-0.116097\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−1.02229			−0.112211			−0.0561057	−	0.998425i	$-0.517868\pi$
$83$	−1.02229			−0.112211			−0.0561057	+	0.998425i	$0.517868\pi$
$84$		0			0
$85$	12.9501			1.40464
$86$		0			0
$87$			0.686549i			0.0736057i
$88$		0			0
$89$	−5.12515			−0.543265			−0.271632	−	0.962401i	$-0.587564\pi$
$89$	−5.12515			−0.543265			−0.271632	+	0.962401i	$0.587564\pi$
$90$		0			0
$91$	0.340765	+	6.98337i	0.0357219	+	0.732056i
$92$		0			0
$93$	−5.62598			−0.583387
$94$		0			0
$95$	14.7511			1.51343
$96$		0			0
$97$	14.1418			1.43588			0.717940	−	0.696105i	$-0.245085\pi$
$97$	14.1418			1.43588			0.717940	+	0.696105i	$0.245085\pi$
$98$		0			0
$99$			1.65924i			0.166759i
$100$		0			0
$101$		−	2.52649i		−	0.251395i	−0.992069	−	0.125697i	$-0.959883\pi$
$101$		−	2.52649i		−	0.251395i	0.992069	−	0.125697i	$-0.0401168\pi$
$102$		0			0
$103$	0.769413			0.0758125			0.0379063	−	0.999281i	$-0.487931\pi$
$103$	0.769413			0.0758125			0.0379063	+	0.999281i	$0.487931\pi$
$104$		0			0
$105$	−0.384707	−	7.88388i	−0.0375435	−	0.769388i
$106$		0			0
$107$		−	13.3241i		−	1.28809i	−0.764986	−	0.644046i	$-0.777254\pi$
$107$		−	13.3241i		−	1.28809i	0.764986	−	0.644046i	$-0.222746\pi$
$108$		0			0
$109$			1.53243i			0.146780i	0.997303	+	0.0733900i	$0.0233818\pi$
$109$			1.53243i			0.146780i	−0.997303	+	0.0733900i	$0.976618\pi$
$110$		0			0
$111$	−6.59867			−0.626318
$112$		0			0
$113$			2.81764i			0.265062i	0.991179	+	0.132531i	$0.0423103\pi$
$113$			2.81764i			0.265062i	−0.991179	+	0.132531i	$0.957690\pi$
$114$		0			0
$115$	8.90051	+	11.2023i	0.829977	+	1.04462i
$116$		0			0
$117$			2.64261i			0.244309i
$118$		0			0
$119$	11.4709	−	0.559743i	1.05154	−	0.0513115i
$120$		0			0
$121$	8.24694			0.749722
$122$		0			0
$123$	−1.91714			−0.172862
$124$		0			0
$125$	3.28019			0.293390
$126$		0			0
$127$	−2.93376			−0.260329			−0.130165	−	0.991492i	$-0.541551\pi$
$127$	−2.93376			−0.260329			−0.130165	+	0.991492i	$0.541551\pi$
$128$		0			0
$129$	5.83994			0.514178
$130$		0			0
$131$			11.2023i			0.978754i	0.872072	+	0.489377i	$0.162776\pi$
$131$			11.2023i			0.978754i	−0.872072	+	0.489377i	$0.837224\pi$
$132$		0			0
$133$	13.0662	−	0.637588i	1.13299	−	0.0552859i
$134$		0			0
$135$		−	2.98337i		−	0.256768i
$136$		0			0
$137$			13.0812i			1.11760i	0.829301	+	0.558802i	$0.188739\pi$
$137$			13.0812i			1.11760i	−0.829301	+	0.558802i	$0.811261\pi$
$138$		0			0
$139$			12.3464i			1.04721i	0.851961	+	0.523605i	$0.175413\pi$
$139$			12.3464i			1.04721i	−0.851961	+	0.523605i	$0.824587\pi$
$140$		0			0
$141$	5.74210			0.483572
$142$		0			0
$143$	−4.38471			−0.366668
$144$		0			0
$145$	2.04823			0.170096
$146$		0			0
$147$	−0.681530	−	6.96674i	−0.0562116	−	0.574607i
$148$		0			0
$149$			12.4493i			1.01989i	0.860208	+	0.509943i	$0.170333\pi$
$149$			12.4493i			1.01989i	−0.860208	+	0.509943i	$0.829667\pi$
$150$		0			0
$151$	−1.80102			−0.146565			−0.0732823	−	0.997311i	$-0.523347\pi$
$151$	−1.80102			−0.146565			−0.0732823	+	0.997311i	$0.523347\pi$
$152$		0			0
$153$	4.34076			0.350930
$154$		0			0
$155$			16.7844i			1.34815i
$156$		0			0
$157$	−1.51746			−0.121106			−0.0605531	−	0.998165i	$-0.519286\pi$
$157$	−1.51746			−0.121106			−0.0605531	+	0.998165i	$0.519286\pi$
$158$		0			0
$159$	6.46757			0.512912
$160$		0			0
$161$	8.36808	+	9.53810i	0.659497	+	0.751707i
$162$		0			0
$163$	−15.5871			−1.22087			−0.610436	−	0.792065i	$-0.709006\pi$
$163$	−15.5871			−1.22087			−0.610436	+	0.792065i	$0.709006\pi$
$164$		0			0
$165$	4.95012			0.385366
$166$		0			0
$167$			23.9642i			1.85440i	0.374563	+	0.927201i	$0.377793\pi$
$167$			23.9642i			1.85440i	−0.374563	+	0.927201i	$0.622207\pi$
$168$		0			0
$169$	6.01663			0.462818
$170$		0			0
$171$	4.94445			0.378111
$172$		0			0
$173$		−	18.7455i		−	1.42519i	−0.701575	−	0.712596i	$-0.747519\pi$
$173$		−	18.7455i		−	1.42519i	0.701575	−	0.712596i	$-0.252481\pi$
$174$		0			0
$175$	−10.3075	+	0.502971i	−0.779174	+	0.0380211i
$176$		0			0
$177$	3.66490			0.275471
$178$		0			0
$179$	−20.1907			−1.50913			−0.754563	−	0.656227i	$-0.772151\pi$
$179$	−20.1907			−1.50913			−0.754563	+	0.656227i	$0.772151\pi$
$180$		0			0
$181$	−2.85228			−0.212008			−0.106004	−	0.994366i	$-0.533806\pi$
$181$	−2.85228			−0.212008			−0.106004	+	0.994366i	$0.533806\pi$
$182$		0			0
$183$			6.35145i			0.469513i
$184$		0			0
$185$			19.6863i			1.44736i
$186$		0			0
$187$			7.20235i			0.526688i
$188$		0			0
$189$	−0.128950	−	2.64261i	−0.00937975	−	0.192221i
$190$		0			0
$191$		−	4.34578i		−	0.314450i	−0.987563	−	0.157225i	$-0.949745\pi$
$191$		−	4.34578i		−	0.314450i	0.987563	−	0.157225i	$-0.0502548\pi$
$192$		0			0
$193$	16.7362			1.20469			0.602347	−	0.798234i	$-0.294232\pi$
$193$	16.7362			1.20469			0.602347	+	0.798234i	$0.294232\pi$
$194$		0			0
$195$	7.88388			0.564576
$196$		0			0
$197$	19.8656			1.41537			0.707683	−	0.706531i	$-0.249741\pi$
$197$	19.8656			1.41537			0.707683	+	0.706531i	$0.249741\pi$
$198$		0			0
$199$	26.8290			1.90186			0.950928	−	0.309413i	$-0.100132\pi$
$199$	26.8290			1.90186			0.950928	+	0.309413i	$0.100132\pi$
$200$		0			0
$201$	1.27453			0.0898983
$202$		0			0
$203$	1.81428	−	0.0885306i	0.127337	−	0.00621363i
$204$		0			0
$205$			5.71953i			0.399469i
$206$		0			0
$207$	2.98337	+	3.75493i	0.207359	+	0.260986i
$208$		0			0
$209$			8.20400i			0.567483i
$210$		0			0
$211$	6.35509			0.437503			0.218751	−	0.975781i	$-0.429802\pi$
$211$	6.35509			0.437503			0.218751	+	0.975781i	$0.429802\pi$
$212$		0			0
$213$			1.66992i			0.114421i
$214$		0			0
$215$		−	17.4227i		−	1.18822i
$216$		0			0
$217$	0.725471	+	14.8673i	0.0492482	+	1.00925i
$218$		0			0
$219$	−4.39968			−0.297303
$220$		0			0
$221$			11.4709i			0.771619i
$222$		0			0
$223$		−	21.6127i		−	1.44729i	−0.690170	−	0.723647i	$-0.742464\pi$
$223$		−	21.6127i		−	1.44729i	0.690170	−	0.723647i	$-0.257536\pi$
$224$		0			0
$225$	−3.90051			−0.260034
$226$		0			0
$227$	0.209670			0.0139163			0.00695816	−	0.999976i	$-0.497785\pi$
$227$	0.209670			0.0139163			0.00695816	+	0.999976i	$0.497785\pi$
$228$		0			0
$229$	13.0166			0.860163			0.430082	−	0.902790i	$-0.358485\pi$
$229$	13.0166			0.860163			0.430082	+	0.902790i	$0.358485\pi$
$230$		0			0
$231$	4.38471	−	0.213959i	0.288493	−	0.0140775i
$232$		0			0
$233$	1.74275			0.114171			0.0570856	−	0.998369i	$-0.481819\pi$
$233$	1.74275			0.114171			0.0570856	+	0.998369i	$0.481819\pi$
$234$		0			0
$235$		−	17.1308i		−	1.11749i
$236$		0			0
$237$	−6.34076			−0.411877
$238$		0			0
$239$	−2.14745			−0.138907			−0.0694534	−	0.997585i	$-0.522126\pi$
$239$	−2.14745			−0.138907			−0.0694534	+	0.997585i	$0.522126\pi$
$240$		0			0
$241$	24.0420			1.54868			0.774341	−	0.632768i	$-0.218082\pi$
$241$	24.0420			1.54868			0.774341	+	0.632768i	$0.218082\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	−20.7844	+	2.03326i	−1.32787	+	0.129900i
$246$		0			0
$247$			13.0662i			0.831385i
$248$		0			0
$249$			1.02229i			0.0647853i
$250$		0			0
$251$	0.136761			0.00863226			0.00431613	−	0.999991i	$-0.498626\pi$
$251$	0.136761			0.00863226			0.00431613	+	0.999991i	$0.498626\pi$
$252$		0			0
$253$	−6.23031	+	4.95012i	−0.391696	+	0.311211i
$254$		0			0
$255$		−	12.9501i		−	0.810968i
$256$		0			0
$257$			2.45260i			0.152989i	0.997070	+	0.0764944i	$0.0243727\pi$
$257$			2.45260i			0.152989i	−0.997070	+	0.0764944i	$0.975627\pi$
$258$		0			0
$259$	0.850900	+	17.4377i	0.0528723	+	1.08352i
$260$		0			0
$261$	0.686549			0.0424963
$262$		0			0
$263$		−	18.1186i		−	1.11724i	−0.829424	−	0.558619i	$-0.811331\pi$
$263$		−	18.1186i		−	1.11724i	0.829424	−	0.558619i	$-0.188669\pi$
$264$		0			0
$265$		−	19.2952i		−	1.18529i
$266$		0			0
$267$			5.12515i			0.313654i
$268$		0			0
$269$			10.4104i			0.634731i	0.948303	+	0.317366i	$0.102798\pi$
$269$			10.4104i			0.634731i	−0.948303	+	0.317366i	$0.897202\pi$
$270$		0			0
$271$		−	29.9874i		−	1.82160i	−0.412844	−	0.910802i	$-0.635464\pi$
$271$		−	29.9874i		−	1.82160i	0.412844	−	0.910802i	$-0.364536\pi$
$272$		0			0
$273$	6.98337	−	0.340765i	0.422653	−	0.0206240i
$274$		0			0
$275$		−	6.47186i		−	0.390268i
$276$		0			0
$277$	−5.63666			−0.338674			−0.169337	−	0.985558i	$-0.554163\pi$
$277$	−5.63666			−0.338674			−0.169337	+	0.985558i	$0.554163\pi$
$278$		0			0
$279$			5.62598i			0.336819i
$280$		0			0
$281$			24.3614i			1.45328i	0.687019	+	0.726640i	$0.258919\pi$
$281$			24.3614i			1.45328i	−0.687019	+	0.726640i	$0.741081\pi$
$282$		0			0
$283$	−17.9228			−1.06540			−0.532700	−	0.846304i	$-0.678823\pi$
$283$	−17.9228			−1.06540			−0.532700	+	0.846304i	$0.678823\pi$
$284$		0			0
$285$		−	14.7511i		−	0.873782i
$286$		0			0
$287$	0.247215	+	5.06624i	0.0145926	+	0.299050i
$288$		0			0
$289$	1.84224			0.108367
$290$		0			0
$291$		−	14.1418i		−	0.829006i
$292$		0			0
$293$	−24.2836			−1.41866			−0.709330	−	0.704876i	$-0.751003\pi$
$293$	−24.2836			−1.41866			−0.709330	+	0.704876i	$0.751003\pi$
$294$		0			0
$295$		−	10.9338i		−	0.636588i
$296$		0			0
$297$	1.65924			0.0962786
$298$		0			0
$299$	−9.92280	+	7.88388i	−0.573850	+	0.455937i
$300$		0			0
$301$	−0.753061	−	15.4327i	−0.0434057	−	0.889524i
$302$		0			0
$303$	−2.52649			−0.145143
$304$		0			0
$305$	18.9487			1.08500
$306$		0			0
$307$		−	17.3449i		−	0.989923i	−0.868915	−	0.494962i	$-0.835182\pi$
$307$		−	17.3449i		−	0.989923i	0.868915	−	0.494962i	$-0.164818\pi$
$308$		0			0
$309$		−	0.769413i		−	0.0437704i
$310$		0			0
$311$		−	20.4650i		−	1.16046i	−0.814452	−	0.580232i	$-0.802962\pi$
$311$		−	20.4650i		−	1.16046i	0.814452	−	0.580232i	$-0.197038\pi$
$312$		0			0
$313$	−4.57472			−0.258578			−0.129289	−	0.991607i	$-0.541270\pi$
$313$	−4.57472			−0.258578			−0.129289	+	0.991607i	$0.541270\pi$
$314$		0			0
$315$	−7.88388	+	0.384707i	−0.444206	+	0.0216758i
$316$		0			0
$317$	−7.78604			−0.437308			−0.218654	−	0.975802i	$-0.570167\pi$
$317$	−7.78604			−0.437308			−0.218654	+	0.975802i	$0.570167\pi$
$318$		0			0
$319$			1.13915i			0.0637799i
$320$		0			0
$321$	−13.3241			−0.743681
$322$		0			0
$323$	21.4627			1.19422
$324$		0			0
$325$		−	10.3075i		−	0.571758i
$326$		0			0
$327$	1.53243			0.0847435
$328$		0			0
$329$	−0.740445	−	15.1741i	−0.0408221	−	0.836576i
$330$		0			0
$331$	25.8337			1.41995			0.709975	−	0.704227i	$-0.248706\pi$
$331$	25.8337			1.41995			0.709975	+	0.704227i	$0.248706\pi$
$332$		0			0
$333$			6.59867i			0.361605i
$334$		0			0
$335$		−	3.80239i		−	0.207747i
$336$		0			0
$337$			14.8673i			0.809871i	0.914345	+	0.404935i	$0.132706\pi$
$337$			14.8673i			0.809871i	−0.914345	+	0.404935i	$0.867294\pi$
$338$		0			0
$339$	2.81764			0.153033
$340$		0			0
$341$	−9.33482			−0.505509
$342$		0			0
$343$	−18.3225	+	2.69938i	−0.989321	+	0.145753i
$344$		0			0
$345$	11.2023	−	8.90051i	0.603114	−	0.479187i
$346$		0			0
$347$	−5.32349			−0.285780			−0.142890	−	0.989739i	$-0.545639\pi$
$347$	−5.32349			−0.285780			−0.142890	+	0.989739i	$0.545639\pi$
$348$		0			0
$349$		−	26.2064i		−	1.40279i	−0.712771	−	0.701397i	$-0.752560\pi$
$349$		−	26.2064i		−	1.40279i	0.712771	−	0.701397i	$-0.247440\pi$
$350$		0			0
$351$	2.64261			0.141052
$352$		0			0
$353$			23.7509i			1.26413i	0.774915	+	0.632065i	$0.217793\pi$
$353$			23.7509i			1.26413i	−0.774915	+	0.632065i	$0.782207\pi$
$354$		0			0
$355$	4.98199			0.264417
$356$		0			0
$357$	−0.559743	−	11.4709i	−0.0296247	−	0.607106i
$358$		0			0
$359$			9.79033i			0.516714i	0.966050	+	0.258357i	$0.0831811\pi$
$359$			9.79033i			0.516714i	−0.966050	+	0.258357i	$0.916819\pi$
$360$		0			0
$361$	5.44758			0.286715
$362$		0			0
$363$		−	8.24694i		−	0.432852i
$364$		0			0
$365$			13.1259i			0.687040i
$366$		0			0
$367$	−19.6153			−1.02391			−0.511955	−	0.859012i	$-0.671079\pi$
$367$	−19.6153			−1.02391			−0.511955	+	0.859012i	$0.671079\pi$
$368$		0			0
$369$			1.91714i			0.0998021i
$370$		0			0
$371$	−0.833995	−	17.0912i	−0.0432989	−	0.887333i
$372$		0			0
$373$		−	27.9718i		−	1.44832i	−0.689630	−	0.724161i	$-0.742227\pi$
$373$		−	27.9718i		−	1.44832i	0.689630	−	0.724161i	$-0.257773\pi$
$374$		0			0
$375$		−	3.28019i		−	0.169389i
$376$		0			0
$377$			1.81428i			0.0934401i
$378$		0			0
$379$			15.4320i			0.792689i	0.918102	+	0.396345i	$0.129721\pi$
$379$			15.4320i			0.792689i	−0.918102	+	0.396345i	$0.870279\pi$
$380$		0			0
$381$			2.93376i			0.150301i
$382$		0			0
$383$	−35.8174			−1.83018			−0.915091	−	0.403247i	$-0.867881\pi$
$383$	−35.8174			−1.83018			−0.915091	+	0.403247i	$0.867881\pi$
$384$		0			0
$385$	−0.638319	−	13.0812i	−0.0325317	−	0.666680i
$386$		0			0
$387$		−	5.83994i		−	0.296861i
$388$		0			0
$389$		−	8.49890i		−	0.430911i	−0.976514	−	0.215456i	$-0.930876\pi$
$389$		−	8.49890i		−	0.430911i	0.976514	−	0.215456i	$-0.0691236\pi$
$390$		0			0
$391$	12.9501	+	16.2993i	0.654915	+	0.824289i
$392$		0			0
$393$	11.2023			0.565084
$394$		0			0
$395$			18.9169i			0.951810i
$396$		0			0
$397$			5.76347i			0.289260i	0.989486	+	0.144630i	$0.0461992\pi$
$397$			5.76347i			0.289260i	−0.989486	+	0.144630i	$0.953801\pi$
$398$		0			0
$399$	−0.637588	−	13.0662i	−0.0319193	−	0.654130i
$400$		0			0
$401$			11.4859i			0.573579i	0.957994	+	0.286789i	$0.0925880\pi$
$401$			11.4859i			0.573579i	−0.957994	+	0.286789i	$0.907412\pi$
$402$		0			0
$403$	−14.8673			−0.740591
$404$		0			0
$405$	−2.98337			−0.148245
$406$		0			0
$407$	−10.9487			−0.542709
$408$		0			0
$409$			18.1042i			0.895197i	0.894235	+	0.447598i	$0.147721\pi$
$409$			18.1042i			0.895197i	−0.894235	+	0.447598i	$0.852279\pi$
$410$		0			0
$411$	13.0812			0.645249
$412$		0			0
$413$	−0.472590	−	9.68489i	−0.0232546	−	0.476562i
$414$		0			0
$415$	3.04988			0.149713
$416$		0			0
$417$	12.3464			0.604607
$418$		0			0
$419$	13.2626			0.647923			0.323961	−	0.946070i	$-0.394985\pi$
$419$	13.2626			0.647923			0.323961	+	0.946070i	$0.394985\pi$
$420$		0			0
$421$			4.98839i			0.243119i	0.992584	+	0.121560i	$0.0387896\pi$
$421$			4.98839i			0.243119i	−0.992584	+	0.121560i	$0.961210\pi$
$422$		0			0
$423$		−	5.74210i		−	0.279190i
$424$		0			0
$425$	−16.9312			−0.821283
$426$		0			0
$427$	16.7844	−	0.819021i	0.812253	−	0.0396352i
$428$		0			0
$429$			4.38471i			0.211696i
$430$		0			0
$431$		−	26.0057i		−	1.25265i	−0.779563	−	0.626324i	$-0.784559\pi$
$431$		−	26.0057i		−	1.25265i	0.779563	−	0.626324i	$-0.215441\pi$
$432$		0			0
$433$	31.4766			1.51267			0.756334	−	0.654185i	$-0.226988\pi$
$433$	31.4766			1.51267			0.756334	+	0.654185i	$0.226988\pi$
$434$		0			0
$435$		−	2.04823i		−	0.0982051i
$436$		0			0
$437$	14.7511	+	18.5661i	0.705642	+	0.888135i
$438$		0			0
$439$		−	5.18802i		−	0.247611i	−0.992307	−	0.123805i	$-0.960490\pi$
$439$		−	5.18802i		−	0.247611i	0.992307	−	0.123805i	$-0.0395099\pi$
$440$		0			0
$441$	−6.96674	+	0.681530i	−0.331750	+	0.0324538i
$442$		0			0
$443$	−34.8354			−1.65508			−0.827539	−	0.561408i	$-0.810260\pi$
$443$	−34.8354			−1.65508			−0.827539	+	0.561408i	$0.810260\pi$
$444$		0			0
$445$	15.2902			0.724827
$446$		0			0
$447$	12.4493			0.588831
$448$		0			0
$449$	32.4524			1.53152			0.765761	−	0.643125i	$-0.222362\pi$
$449$	32.4524			1.53152			0.765761	+	0.643125i	$0.222362\pi$
$450$		0			0
$451$	−3.18098			−0.149786
$452$		0			0
$453$			1.80102i			0.0846191i
$454$		0			0
$455$	−1.01663	−	20.8340i	−0.0476603	−	0.976713i
$456$		0			0
$457$			26.9169i			1.25912i	0.776953	+	0.629559i	$0.216764\pi$
$457$			26.9169i			1.25912i	−0.776953	+	0.629559i	$0.783236\pi$
$458$		0			0
$459$		−	4.34076i		−	0.202610i
$460$		0			0
$461$			32.1731i			1.49845i	0.662315	+	0.749225i	$0.269574\pi$
$461$			32.1731i			1.49845i	−0.662315	+	0.749225i	$0.730426\pi$
$462$		0			0
$463$	−39.1572			−1.81979			−0.909895	−	0.414839i	$-0.863838\pi$
$463$	−39.1572			−1.81979			−0.909895	+	0.414839i	$0.863838\pi$
$464$		0			0
$465$	16.7844			0.778357
$466$		0			0
$467$	−14.4833			−0.670206			−0.335103	−	0.942181i	$-0.608771\pi$
$467$	−14.4833			−0.670206			−0.335103	+	0.942181i	$0.608771\pi$
$468$		0			0
$469$	−0.164351	−	3.36808i	−0.00758901	−	0.155523i
$470$		0			0
$471$			1.51746i			0.0699207i
$472$		0			0
$473$	9.68983			0.445539
$474$		0			0
$475$	−19.2859			−0.884896
$476$		0			0
$477$		−	6.46757i		−	0.296130i
$478$		0			0
$479$	−0.447578			−0.0204504			−0.0102252	−	0.999948i	$-0.503255\pi$
$479$	−0.447578			−0.0204504			−0.0102252	+	0.999948i	$0.503255\pi$
$480$		0			0
$481$	−17.4377			−0.795090
$482$		0			0
$483$	9.53810	−	8.36808i	0.433998	−	0.380761i
$484$		0			0
$485$	−42.1902			−1.91576
$486$		0			0
$487$	−23.2137			−1.05191			−0.525956	−	0.850512i	$-0.676292\pi$
$487$	−23.2137			−1.05191			−0.525956	+	0.850512i	$0.676292\pi$
$488$		0			0
$489$			15.5871i			0.704871i
$490$		0			0
$491$	−17.6034			−0.794431			−0.397215	−	0.917725i	$-0.630023\pi$
$491$	−17.6034			−0.794431			−0.397215	+	0.917725i	$0.630023\pi$
$492$		0			0
$493$	2.98015			0.134219
$494$		0			0
$495$		−	4.95012i		−	0.222491i
$496$		0			0
$497$	4.41294	−	0.215337i	0.197948	−	0.00965917i
$498$		0			0
$499$	−9.95177			−0.445502			−0.222751	−	0.974875i	$-0.571504\pi$
$499$	−9.95177			−0.445502			−0.222751	+	0.974875i	$0.571504\pi$
$500$		0			0
$501$	23.9642			1.07064
$502$		0			0
$503$	1.02796			0.0458345			0.0229173	−	0.999737i	$-0.492705\pi$
$503$	1.02796			0.0458345			0.0229173	+	0.999737i	$0.492705\pi$
$504$		0			0
$505$			7.53745i			0.335412i
$506$		0			0
$507$		−	6.01663i		−	0.267208i
$508$		0			0
$509$		−	6.76512i		−	0.299859i	−0.988697	−	0.149929i	$-0.952095\pi$
$509$		−	6.76512i		−	0.299859i	0.988697	−	0.149929i	$-0.0479047\pi$
$510$		0			0
$511$	0.567340	+	11.6266i	0.0250976	+	0.514332i
$512$		0			0
$513$		−	4.94445i		−	0.218303i
$514$		0			0
$515$	−2.29545			−0.101149
$516$		0			0
$517$	9.52749			0.419019
$518$		0			0
$519$	−18.7455			−0.822835
$520$		0			0
$521$	15.7388			0.689529			0.344764	−	0.938689i	$-0.387959\pi$
$521$	15.7388			0.689529			0.344764	+	0.938689i	$0.387959\pi$
$522$		0			0
$523$	15.2809			0.668188			0.334094	−	0.942540i	$-0.391570\pi$
$523$	15.2809			0.668188			0.334094	+	0.942540i	$0.391570\pi$
$524$		0			0
$525$	0.502971	+	10.3075i	0.0219515	+	0.449857i
$526$		0			0
$527$			24.4211i			1.06380i
$528$		0			0
$529$	−5.19898	+	22.4047i	−0.226043	+	0.974117i
$530$		0			0
$531$		−	3.66490i		−	0.159043i
$532$		0			0
$533$	−5.06624			−0.219443
$534$		0			0
$535$			39.7509i			1.71858i
$536$		0			0
$537$			20.1907i			0.871295i
$538$		0			0
$539$	−1.13082	−	11.5595i	−0.0487078	−	0.497901i
$540$		0			0
$541$	−29.3598			−1.26227			−0.631137	−	0.775671i	$-0.717411\pi$
$541$	−29.3598			−1.26227			−0.631137	+	0.775671i	$0.717411\pi$
$542$		0			0
$543$			2.85228i			0.122403i
$544$		0			0
$545$		−	4.57181i		−	0.195835i
$546$		0			0
$547$	−9.98172			−0.426787			−0.213394	−	0.976966i	$-0.568452\pi$
$547$	−9.98172			−0.426787			−0.213394	+	0.976966i	$0.568452\pi$
$548$		0			0
$549$	6.35145			0.271073
$550$		0			0
$551$	3.39461			0.144615
$552$		0			0
$553$	0.817643	+	16.7561i	0.0347697	+	0.712544i
$554$		0			0
$555$	19.6863			0.835636
$556$		0			0
$557$		−	22.5818i		−	0.956820i	−0.878137	−	0.478410i	$-0.841213\pi$
$557$		−	22.5818i		−	0.956820i	0.878137	−	0.478410i	$-0.158787\pi$
$558$		0			0
$559$	15.4327			0.652732
$560$		0			0
$561$	7.20235			0.304083
$562$		0			0
$563$	−12.8715			−0.542471			−0.271235	−	0.962513i	$-0.587432\pi$
$563$	−12.8715			−0.542471			−0.271235	+	0.962513i	$0.587432\pi$
$564$		0			0
$565$		−	8.40608i		−	0.353646i
$566$		0			0
$567$	−2.64261	+	0.128950i	−0.110979	+	0.00541540i
$568$		0			0
$569$			1.90023i			0.0796618i	0.999206	+	0.0398309i	$0.0126819\pi$
$569$			1.90023i			0.0796618i	−0.999206	+	0.0398309i	$0.987318\pi$
$570$		0			0
$571$		−	28.6890i		−	1.20060i	−0.799776	−	0.600298i	$-0.795048\pi$
$571$		−	28.6890i		−	1.20060i	0.799776	−	0.600298i	$-0.204952\pi$
$572$		0			0
$573$	−4.34578			−0.181548
$574$		0			0
$575$	−11.6367	−	14.6461i	−0.485282	−	0.610786i
$576$		0			0
$577$			31.1854i			1.29827i	0.760674	+	0.649134i	$0.224868\pi$
$577$			31.1854i			1.29827i	−0.760674	+	0.649134i	$0.775132\pi$
$578$		0			0
$579$		−	16.7362i		−	0.695531i
$580$		0			0
$581$	2.70152	−	0.131825i	0.112078	−	0.00546903i
$582$		0			0
$583$	10.7312			0.444442
$584$		0			0
$585$		−	7.88388i		−	0.325958i
$586$		0			0
$587$			27.1722i			1.12152i	0.827980	+	0.560758i	$0.189490\pi$
$587$			27.1722i			1.12152i	−0.827980	+	0.560758i	$0.810510\pi$
$588$		0			0
$589$			27.8174i			1.14619i
$590$		0			0
$591$		−	19.8656i		−	0.817161i
$592$		0			0
$593$			9.94537i			0.408408i	0.978928	+	0.204204i	$0.0654605\pi$
$593$			9.94537i			0.408408i	−0.978928	+	0.204204i	$0.934539\pi$
$594$		0			0
$595$	−34.2221	+	1.66992i	−1.40297	+	0.0684601i
$596$		0			0
$597$		−	26.8290i		−	1.09804i
$598$		0			0
$599$	−20.8905			−0.853561			−0.426781	−	0.904355i	$-0.640352\pi$
$599$	−20.8905			−0.853561			−0.426781	+	0.904355i	$0.640352\pi$
$600$		0			0
$601$		−	37.2236i		−	1.51838i	−0.650867	−	0.759192i	$-0.725594\pi$
$601$		−	37.2236i		−	1.51838i	0.650867	−	0.759192i	$-0.274406\pi$
$602$		0			0
$603$		−	1.27453i		−	0.0519028i
$604$		0			0
$605$	−24.6037			−1.00028
$606$		0			0
$607$		−	28.1335i		−	1.14190i	−0.820983	−	0.570952i	$-0.806574\pi$
$607$		−	28.1335i		−	1.14190i	0.820983	−	0.570952i	$-0.193426\pi$
$608$		0			0
$609$	−0.0885306	−	1.81428i	−0.00358744	−	0.0735183i
$610$		0			0
$611$	15.1741			0.613879
$612$		0			0
$613$		−	15.5738i		−	0.629020i	−0.949254	−	0.314510i	$-0.898160\pi$
$613$		−	15.5738i		−	0.629020i	0.949254	−	0.314510i	$-0.101840\pi$
$614$		0			0
$615$	5.71953			0.230634
$616$		0			0
$617$		−	22.0414i		−	0.887352i	−0.896187	−	0.443676i	$-0.853674\pi$
$617$		−	22.0414i		−	0.887352i	0.896187	−	0.443676i	$-0.146326\pi$
$618$		0			0
$619$	14.0490			0.564675			0.282338	−	0.959315i	$-0.408890\pi$
$619$	14.0490			0.564675			0.282338	+	0.959315i	$0.408890\pi$
$620$		0			0
$621$	3.75493	−	2.98337i	0.150680	−	0.119719i
$622$		0			0
$623$	13.5438	−	0.660890i	0.542619	−	0.0264780i
$624$		0			0
$625$	−29.2886			−1.17154
$626$		0			0
$627$	8.20400			0.327636
$628$		0			0
$629$			28.6433i			1.14208i
$630$		0			0
$631$			14.9558i			0.595380i	0.954663	+	0.297690i	$0.0962162\pi$
$631$			14.9558i			0.595380i	−0.954663	+	0.297690i	$0.903784\pi$
$632$		0			0
$633$		−	6.35509i		−	0.252592i
$634$		0			0
$635$	8.75251			0.347333
$636$		0			0
$637$	−1.80102	−	18.4104i	−0.0713588	−	0.729445i
$638$		0			0
$639$	1.66992			0.0660610
$640$		0			0
$641$			17.6999i			0.699103i	0.936917	+	0.349551i	$0.113666\pi$
$641$			17.6999i			0.699103i	−0.936917	+	0.349551i	$0.886334\pi$
$642$		0			0
$643$	−0.344406			−0.0135820			−0.00679102	−	0.999977i	$-0.502162\pi$
$643$	−0.344406			−0.0135820			−0.00679102	+	0.999977i	$0.502162\pi$
$644$		0			0
$645$	−17.4227			−0.686018
$646$		0			0
$647$			43.1715i			1.69724i	0.528999	+	0.848622i	$0.322568\pi$
$647$			43.1715i			1.69724i	−0.528999	+	0.848622i	$0.677432\pi$
$648$		0			0
$649$	6.08093			0.238698
$650$		0			0
$651$	14.8673	−	0.725471i	0.582694	−	0.0284335i
$652$		0			0
$653$	−21.5105			−0.841771			−0.420886	−	0.907114i	$-0.638281\pi$
$653$	−21.5105			−0.841771			−0.420886	+	0.907114i	$0.638281\pi$
$654$		0			0
$655$		−	33.4208i		−	1.30586i
$656$		0			0
$657$			4.39968i			0.171648i
$658$		0			0
$659$		−	15.3159i		−	0.596622i	−0.954469	−	0.298311i	$-0.903577\pi$
$659$		−	15.3159i		−	0.596622i	0.954469	−	0.298311i	$-0.0964233\pi$
$660$		0			0
$661$	25.0361			0.973790			0.486895	−	0.873460i	$-0.338129\pi$
$661$	25.0361			0.973790			0.486895	+	0.873460i	$0.338129\pi$
$662$		0			0
$663$	11.4709			0.445494
$664$		0			0
$665$	−38.9814	+	1.90216i	−1.51164	+	0.0737627i
$666$		0			0
$667$	2.04823	+	2.57794i	0.0793078	+	0.0998183i
$668$		0			0
$669$	−21.6127			−0.835596
$670$		0			0
$671$			10.5385i			0.406836i
$672$		0			0
$673$	−0.345053			−0.0133008			−0.00665041	−	0.999978i	$-0.502117\pi$
$673$	−0.345053			−0.0133008			−0.00665041	+	0.999978i	$0.502117\pi$
$674$		0			0
$675$			3.90051i			0.150131i
$676$		0			0
$677$	−10.8786			−0.418098			−0.209049	−	0.977905i	$-0.567037\pi$
$677$	−10.8786			−0.418098			−0.209049	+	0.977905i	$0.567037\pi$
$678$		0			0
$679$	−37.3712	+	1.82359i	−1.43417	+	0.0699828i
$680$		0			0
$681$		−	0.209670i		−	0.00803459i
$682$		0			0
$683$	−0.643254			−0.0246134			−0.0123067	−	0.999924i	$-0.503917\pi$
$683$	−0.643254			−0.0246134			−0.0123067	+	0.999924i	$0.503917\pi$
$684$		0			0
$685$		−	39.0261i		−	1.49111i
$686$		0			0
$687$		−	13.0166i		−	0.496615i
$688$		0			0
$689$	17.0912			0.651125
$690$		0			0
$691$		−	13.0763i		−	0.497445i	−0.968575	−	0.248722i	$-0.919989\pi$
$691$		−	13.0763i		−	0.497445i	0.968575	−	0.248722i	$-0.0800107\pi$
$692$		0			0
$693$	−0.213959	−	4.38471i	−0.00812762	−	0.166561i
$694$		0			0
$695$		−	36.8340i		−	1.39719i
$696$		0			0
$697$			8.32184i			0.315212i
$698$		0			0
$699$		−	1.74275i		−	0.0659167i
$700$		0			0
$701$		−	39.7742i		−	1.50225i	−0.660161	−	0.751125i	$-0.729512\pi$
$701$		−	39.7742i		−	1.50225i	0.660161	−	0.751125i	$-0.270488\pi$
$702$		0			0
$703$			32.6268i			1.23054i
$704$		0			0
$705$	−17.1308			−0.645184
$706$		0			0
$707$	0.325791	+	6.67651i	0.0122526	+	0.251096i
$708$		0			0
$709$		−	22.4806i		−	0.844277i	−0.906531	−	0.422139i	$-0.861280\pi$
$709$		−	22.4806i		−	0.844277i	0.906531	−	0.422139i	$-0.138720\pi$
$710$		0			0
$711$			6.34076i			0.237797i
$712$		0			0
$713$	−21.1252	+	16.7844i	−0.791143	+	0.628580i
$714$		0			0
$715$	13.0812			0.489210
$716$		0			0
$717$			2.14745i			0.0801978i
$718$		0			0
$719$			45.7438i			1.70596i	0.521947	+	0.852978i	$0.325206\pi$
$719$			45.7438i			1.70596i	−0.521947	+	0.852978i	$0.674794\pi$
$720$		0			0
$721$	−2.03326	+	0.0992160i	−0.0757224	+	0.00369500i
$722$		0			0
$723$		−	24.0420i		−	0.894132i
$724$		0			0
$725$	−2.67789			−0.0994543
$726$		0			0
$727$	41.9785			1.55690			0.778448	−	0.627709i	$-0.216007\pi$
$727$	41.9785			1.55690			0.778448	+	0.627709i	$0.216007\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		−	25.3498i		−	0.937596i
$732$		0			0
$733$	−5.14943			−0.190199			−0.0950993	−	0.995468i	$-0.530317\pi$
$733$	−5.14943			−0.190199			−0.0950993	+	0.995468i	$0.530317\pi$
$734$		0			0
$735$	2.03326	+	20.7844i	0.0749978	+	0.766644i
$736$		0			0
$737$	2.11474			0.0778976
$738$		0			0
$739$	−28.9847			−1.06622			−0.533111	−	0.846046i	$-0.678977\pi$
$739$	−28.9847			−1.06622			−0.533111	+	0.846046i	$0.678977\pi$
$740$		0			0
$741$	13.0662			0.480000
$742$		0			0
$743$		−	35.7119i		−	1.31014i	−0.755567	−	0.655072i	$-0.772638\pi$
$743$		−	35.7119i		−	1.31014i	0.755567	−	0.655072i	$-0.227362\pi$
$744$		0			0
$745$		−	37.1409i		−	1.36074i
$746$		0			0
$747$	1.02229			0.0374038
$748$		0			0
$749$	1.71815	+	35.2105i	0.0627799	+	1.28656i
$750$		0			0
$751$			36.6922i			1.33892i	0.742849	+	0.669459i	$0.233474\pi$
$751$			36.6922i			1.33892i	−0.742849	+	0.669459i	$0.766526\pi$
$752$		0			0
$753$		−	0.136761i		−	0.00498384i
$754$		0			0
$755$	5.37310			0.195547
$756$		0			0
$757$			9.16909i			0.333256i	0.986020	+	0.166628i	$0.0532880\pi$
$757$			9.16909i			0.333256i	−0.986020	+	0.166628i	$0.946712\pi$
$758$		0			0
$759$	4.95012	+	6.23031i	0.179678	+	0.226146i
$760$		0			0
$761$			9.01299i			0.326721i	0.986566	+	0.163360i	$0.0522333\pi$
$761$			9.01299i			0.326721i	−0.986566	+	0.163360i	$0.947767\pi$
$762$		0			0
$763$	−0.197607	−	4.04961i	−0.00715386	−	0.146606i
$764$		0			0
$765$	−12.9501			−0.468212
$766$		0			0
$767$	9.68489			0.349701
$768$		0			0
$769$	3.36735			0.121430			0.0607148	−	0.998155i	$-0.480662\pi$
$769$	3.36735			0.121430			0.0607148	+	0.998155i	$0.480662\pi$
$770$		0			0
$771$	2.45260			0.0883281
$772$		0			0
$773$	44.3759			1.59609			0.798045	−	0.602598i	$-0.205868\pi$
$773$	44.3759			1.59609			0.798045	+	0.602598i	$0.205868\pi$
$774$		0			0
$775$		−	21.9442i		−	0.788258i
$776$		0			0
$777$	17.4377	−	0.850900i	0.625573	−	0.0305258i
$778$		0			0
$779$			9.47918i			0.339627i
$780$		0			0
$781$			2.77079i			0.0991467i
$782$		0			0
$783$		−	0.686549i		−	0.0245352i
$784$		0			0
$785$	4.52713			0.161580
$786$		0			0
$787$	−21.2513			−0.757527			−0.378764	−	0.925493i	$-0.623651\pi$
$787$	−21.2513			−0.757527			−0.378764	+	0.925493i	$0.623651\pi$
$788$		0			0
$789$	−18.1186			−0.645038
$790$		0			0
$791$	−0.363336	−	7.44592i	−0.0129187	−	0.264747i
$792$		0			0
$793$			16.7844i			0.596031i
$794$		0			0
$795$	−19.2952			−0.684329
$796$		0			0
$797$	−26.7096			−0.946104			−0.473052	−	0.881035i	$-0.656848\pi$
$797$	−26.7096			−0.946104			−0.473052	+	0.881035i	$0.656848\pi$
$798$		0			0
$799$		−	24.9251i		−	0.881787i
$800$		0			0
$801$	5.12515			0.181088
$802$		0			0
$803$	−7.30010			−0.257615
$804$		0			0
$805$	−24.9651	−	28.4557i	−0.879904	−	1.00293i
$806$		0			0
$807$	10.4104			0.366462
$808$		0			0
$809$	48.9100			1.71958			0.859792	−	0.510645i	$-0.170593\pi$
$809$	48.9100			1.71958			0.859792	+	0.510645i	$0.170593\pi$
$810$		0			0
$811$			27.6330i			0.970327i	0.874424	+	0.485163i	$0.161240\pi$
$811$			27.6330i			0.970327i	−0.874424	+	0.485163i	$0.838760\pi$
$812$		0			0
$813$	−29.9874			−1.05170
$814$		0			0
$815$	46.5020			1.62889
$816$		0			0
$817$		−	28.8753i		−	1.01022i
$818$		0			0
$819$	−0.340765	−	6.98337i	−0.0119073	−	0.244019i
$820$		0			0
$821$	−26.7594			−0.933909			−0.466954	−	0.884281i	$-0.654649\pi$
$821$	−26.7594			−0.933909			−0.466954	+	0.884281i	$0.654649\pi$
$822$		0			0
$823$	15.4641			0.539044			0.269522	−	0.962994i	$-0.413134\pi$
$823$	15.4641			0.539044			0.269522	+	0.962994i	$0.413134\pi$
$824$		0			0
$825$	−6.47186			−0.225321
$826$		0			0
$827$			20.0220i			0.696234i	0.937451	+	0.348117i	$0.113179\pi$
$827$			20.0220i			0.696234i	−0.937451	+	0.348117i	$0.886821\pi$
$828$		0			0
$829$			26.9512i			0.936053i	0.883714	+	0.468027i	$0.155035\pi$
$829$			26.9512i			0.936053i	−0.883714	+	0.468027i	$0.844965\pi$
$830$		0			0
$831$			5.63666i			0.195534i
$832$		0			0
$833$	−30.2410	+	2.95836i	−1.04779	+	0.102501i
$834$		0			0
$835$		−	71.4940i		−	2.47415i
$836$		0			0
$837$	5.62598			0.194462
$838$		0			0
$839$	12.6783			0.437704			0.218852	−	0.975758i	$-0.429769\pi$
$839$	12.6783			0.437704			0.218852	+	0.975758i	$0.429769\pi$
$840$		0			0
$841$	−28.5287			−0.983747
$842$		0			0
$843$	24.3614			0.839051
$844$		0			0
$845$	−17.9498			−0.617493
$846$		0			0
$847$	−21.7934	+	1.06344i	−0.748831	+	0.0365404i
$848$		0			0
$849$			17.9228i			0.615109i
$850$		0			0
$851$	−24.7775	+	19.6863i	−0.849362	+	0.674837i
$852$		0			0
$853$			14.3306i			0.490670i	0.969438	+	0.245335i	$0.0788980\pi$
$853$			14.3306i			0.490670i	−0.969438	+	0.245335i	$0.921102\pi$
$854$		0			0
$855$	−14.7511			−0.504478
$856$		0			0
$857$			17.6366i			0.602454i	0.953552	+	0.301227i	$0.0973962\pi$
$857$			17.6366i			0.602454i	−0.953552	+	0.301227i	$0.902604\pi$
$858$		0			0
$859$			44.8107i			1.52892i	0.644670	+	0.764461i	$0.276995\pi$
$859$			44.8107i			1.52892i	−0.644670	+	0.764461i	$0.723005\pi$
$860$		0			0
$861$	5.06624	−	0.247215i	0.172657	−	0.00842507i
$862$		0			0
$863$	41.5123			1.41310			0.706548	−	0.707666i	$-0.250252\pi$
$863$	41.5123			1.41310			0.706548	+	0.707666i	$0.250252\pi$
$864$		0			0
$865$			55.9247i			1.90150i
$866$		0			0
$867$		−	1.84224i		−	0.0625657i
$868$		0			0
$869$	−10.5208			−0.356894
$870$		0			0
$871$	3.36808			0.114123
$872$		0			0
$873$	−14.1418			−0.478627
$874$		0			0
$875$	−8.66827	+	0.422982i	−0.293041	+	0.0142994i
$876$		0			0
$877$	15.9103			0.537252			0.268626	−	0.963245i	$-0.413430\pi$
$877$	15.9103			0.537252			0.268626	+	0.963245i	$0.413430\pi$
$878$		0			0
$879$			24.2836i			0.819064i
$880$		0			0
$881$	−12.8497			−0.432917			−0.216459	−	0.976292i	$-0.569451\pi$
$881$	−12.8497			−0.432917			−0.216459	+	0.976292i	$0.569451\pi$
$882$		0			0
$883$	−19.2287			−0.647096			−0.323548	−	0.946212i	$-0.604876\pi$
$883$	−19.2287			−0.647096			−0.323548	+	0.946212i	$0.604876\pi$
$884$		0			0
$885$	−10.9338			−0.367534
$886$		0			0
$887$			9.65099i			0.324049i	0.986787	+	0.162024i	$0.0518023\pi$
$887$			9.65099i			0.324049i	−0.986787	+	0.162024i	$0.948198\pi$
$888$		0			0
$889$	7.75278	−	0.378310i	0.260020	−	0.0126881i
$890$		0			0
$891$		−	1.65924i		−	0.0555865i
$892$		0			0
$893$		−	28.3915i		−	0.950086i
$894$		0			0
$895$	60.2365			2.01348
$896$		0			0
$897$	7.88388	+	9.92280i	0.263235	+	0.331313i
$898$		0			0
$899$			3.86251i			0.128822i
$900$		0			0
$901$		−	28.0742i		−	0.935287i
$902$		0			0
$903$	−15.4327	+	0.753061i	−0.513567	+	0.0250603i
$904$		0			0
$905$	8.50940			0.282862
$906$		0			0
$907$		−	40.9293i		−	1.35904i	−0.733659	−	0.679518i	$-0.762189\pi$
$907$		−	40.9293i		−	1.35904i	0.733659	−	0.679518i	$-0.237811\pi$
$908$		0			0
$909$			2.52649i			0.0837983i
$910$		0			0
$911$		−	4.24786i		−	0.140738i	−0.997521	−	0.0703690i	$-0.977582\pi$
$911$		−	4.24786i		−	0.140738i	0.997521	−	0.0703690i	$-0.0224177\pi$
$912$		0			0
$913$			1.69623i			0.0561369i
$914$		0			0
$915$		−	18.9487i		−	0.626426i
$916$		0			0
$917$	−1.44455	−	29.6034i	−0.0477031	−	0.977591i
$918$		0			0
$919$		−	6.00203i		−	0.197989i	−0.995088	−	0.0989943i	$-0.968437\pi$
$919$		−	6.00203i		−	0.197989i	0.995088	−	0.0989943i	$-0.0315625\pi$
$920$		0			0
$921$	−17.3449			−0.571533
$922$		0			0
$923$			4.41294i			0.145254i
$924$		0			0
$925$		−	25.7381i		−	0.846265i
$926$		0			0
$927$	−0.769413			−0.0252708
$928$		0			0
$929$		−	47.2883i		−	1.55148i	−0.631053	−	0.775739i	$-0.717377\pi$
$929$		−	47.2883i		−	1.55148i	0.631053	−	0.775739i	$-0.282623\pi$
$930$		0			0
$931$	−34.4467	+	3.36979i	−1.12895	+	0.110440i
$932$		0			0
$933$	−20.4650			−0.669994
$934$		0			0
$935$		−	21.4873i		−	0.702709i
$936$		0			0
$937$	−1.12022			−0.0365959			−0.0182979	−	0.999833i	$-0.505825\pi$
$937$	−1.12022			−0.0365959			−0.0182979	+	0.999833i	$0.505825\pi$
$938$		0			0
$939$			4.57472i			0.149290i
$940$		0			0
$941$	28.6364			0.933520			0.466760	−	0.884384i	$-0.345421\pi$
$941$	28.6364			0.933520			0.466760	+	0.884384i	$0.345421\pi$
$942$		0			0
$943$	−7.19871	+	5.71953i	−0.234422	+	0.186253i
$944$		0			0
$945$	0.384707	+	7.88388i	0.0125145	+	0.256463i
$946$		0			0
$947$	0.835926			0.0271639			0.0135820	−	0.999908i	$-0.495677\pi$
$947$	0.835926			0.0271639			0.0135820	+	0.999908i	$0.495677\pi$
$948$		0			0
$949$	−11.6266			−0.377416
$950$		0			0
$951$			7.78604i			0.252480i
$952$		0			0
$953$		−	50.9500i		−	1.65043i	−0.564818	−	0.825216i	$-0.691053\pi$
$953$		−	50.9500i		−	1.65043i	0.564818	−	0.825216i	$-0.308947\pi$
$954$		0			0
$955$			12.9651i			0.419540i
$956$		0			0
$957$	1.13915			0.0368234
$958$		0			0
$959$	−1.68683	−	34.5685i	−0.0544704	−	1.11627i
$960$		0			0
$961$	−0.651638			−0.0210206
$962$		0			0
$963$			13.3241i			0.429364i
$964$		0			0
$965$	−49.9302			−1.60731
$966$		0			0
$967$	−44.0827			−1.41760			−0.708802	−	0.705407i	$-0.750764\pi$
$967$	−44.0827			−1.41760			−0.708802	+	0.705407i	$0.750764\pi$
$968$		0			0
$969$		−	21.4627i		−	0.689481i
$970$		0			0
$971$	−29.7288			−0.954044			−0.477022	−	0.878891i	$-0.658284\pi$
$971$	−29.7288			−0.954044			−0.477022	+	0.878891i	$0.658284\pi$
$972$		0			0
$973$	−1.59208	−	32.6268i	−0.0510396	−	1.04597i
$974$		0			0
$975$	−10.3075			−0.330104
$976$		0			0
$977$			11.9354i			0.381848i	0.981605	+	0.190924i	$0.0611484\pi$
$977$			11.9354i			0.381848i	−0.981605	+	0.190924i	$0.938852\pi$
$978$		0			0
$979$			8.50383i			0.271784i
$980$		0			0
$981$		−	1.53243i		−	0.0489267i
$982$		0			0
$983$	33.5405			1.06978			0.534888	−	0.844923i	$-0.320354\pi$
$983$	33.5405			1.06978			0.534888	+	0.844923i	$0.320354\pi$
$984$		0			0
$985$	−59.2665			−1.88839
$986$		0			0
$987$	−15.1741	+	0.740445i	−0.482997	+	0.0235686i
$988$		0			0
$989$	21.9286	−	17.4227i	0.697287	−	0.554010i
$990$		0			0
$991$	15.0229			0.477219			0.238610	−	0.971116i	$-0.423308\pi$
$991$	15.0229			0.477219			0.238610	+	0.971116i	$0.423308\pi$
$992$		0			0
$993$		−	25.8337i		−	0.819808i
$994$		0			0
$995$	−80.0408			−2.53746
$996$		0			0
$997$			51.4584i			1.62970i	0.579669	+	0.814852i	$0.303182\pi$
$997$			51.4584i			1.62970i	−0.579669	+	0.814852i	$0.696818\pi$
$998$		0			0
$999$	6.59867			0.208773

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3864.2.u.e.1609.1	yes	8
7.6	odd	2		3864.2.u.c.1609.8	yes	8
23.22	odd	2		3864.2.u.c.1609.4	✓	8
161.160	even	2	inner	3864.2.u.e.1609.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.u.c.1609.4	✓	8	23.22	odd	2
3864.2.u.c.1609.8	yes	8	7.6	odd	2
3864.2.u.e.1609.1	yes	8	1.1	even	1	trivial
3864.2.u.e.1609.5	yes	8	161.160	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3864.u (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -2.98337 q^{5} +(-2.64261 + 0.128950i) q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -2.98337 q^{5} +(-2.64261 + 0.128950i) q^{7} -1.00000 q^{9} -1.65924i q^{11} -2.64261i q^{13} +2.98337i q^{15} -4.34076 q^{17} -4.94445 q^{19} +(0.128950 + 2.64261i) q^{21} +(-2.98337 - 3.75493i) q^{23} +3.90051 q^{25} +1.00000i q^{27} -0.686549 q^{29} -5.62598i q^{31} -1.65924 q^{33} +(7.88388 - 0.384707i) q^{35} -6.59867i q^{37} -2.64261 q^{39} -1.91714i q^{41} +5.83994i q^{43} +2.98337 q^{45} +5.74210i q^{47} +(6.96674 - 0.681530i) q^{49} +4.34076i q^{51} +6.46757i q^{53} +4.95012i q^{55} +4.94445i q^{57} +3.66490i q^{59} -6.35145 q^{61} +(2.64261 - 0.128950i) q^{63} +7.88388i q^{65} +1.27453i q^{67} +(-3.75493 + 2.98337i) q^{69} -1.66992 q^{71} -4.39968i q^{73} -3.90051i q^{75} +(0.213959 + 4.38471i) q^{77} -6.34076i q^{79} +1.00000 q^{81} -1.02229 q^{83} +12.9501 q^{85} +0.686549i q^{87} -5.12515 q^{89} +(0.340765 + 6.98337i) q^{91} -5.62598 q^{93} +14.7511 q^{95} +14.1418 q^{97} +1.65924i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{5} - 2 q^{7} - 8 q^{9}+O(q^{10}) \) 8 * q + 2 * q^5 - 2 * q^7 - 8 * q^9	\( 8 q + 2 q^{5} - 2 q^{7} - 8 q^{9} - 28 q^{17} - 8 q^{19} - 2 q^{21} + 2 q^{23} + 6 q^{25} + 20 q^{29} - 20 q^{33} + 12 q^{35} - 2 q^{39} - 2 q^{45} + 4 q^{49} + 14 q^{61} + 2 q^{63} + 18 q^{69} + 38 q^{71} + 2 q^{77} + 8 q^{81} + 12 q^{83} + 26 q^{85} + 42 q^{89} - 4 q^{91} - 10 q^{95} + 56 q^{97}+O(q^{100}) \) 8 * q + 2 * q^5 - 2 * q^7 - 8 * q^9 - 28 * q^17 - 8 * q^19 - 2 * q^21 + 2 * q^23 + 6 * q^25 + 20 * q^29 - 20 * q^33 + 12 * q^35 - 2 * q^39 - 2 * q^45 + 4 * q^49 + 14 * q^61 + 2 * q^63 + 18 * q^69 + 38 * q^71 + 2 * q^77 + 8 * q^81 + 12 * q^83 + 26 * q^85 + 42 * q^89 - 4 * q^91 - 10 * q^95 + 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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