LMFDB - Embedded newform 3360.2.z.a.1231.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3360,2,Mod(1231,3360)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3360.1231"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3360.z (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$26.8297350792$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 840)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1231.1
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	3360.1231
Dual form			3360.2.z.a.1231.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -1.00000 q^{5} +(-1.00000 - 2.44949i) q^{7} -1.00000 q^{9} -2.00000 q^{11} +2.89898 q^{13} +1.00000i q^{15} -4.89898i q^{17} -2.89898i q^{19} +(-2.44949 + 1.00000i) q^{21} -6.00000i q^{23} +1.00000 q^{25} +1.00000i q^{27} +0.898979i q^{29} +2.00000 q^{31} +2.00000i q^{33} +(1.00000 + 2.44949i) q^{35} +11.7980i q^{37} -2.89898i q^{39} -6.89898i q^{41} -0.898979 q^{43} +1.00000 q^{45} +1.10102 q^{47} +(-5.00000 + 4.89898i) q^{49} -4.89898 q^{51} +9.79796i q^{53} +2.00000 q^{55} -2.89898 q^{57} -12.8990i q^{59} -8.89898 q^{61} +(1.00000 + 2.44949i) q^{63} -2.89898 q^{65} -4.89898 q^{67} -6.00000 q^{69} +3.10102i q^{71} +0.898979i q^{73} -1.00000i q^{75} +(2.00000 + 4.89898i) q^{77} -4.00000i q^{79} +1.00000 q^{81} -4.00000i q^{83} +4.89898i q^{85} +0.898979 q^{87} -10.8990i q^{89} +(-2.89898 - 7.10102i) q^{91} -2.00000i q^{93} +2.89898i q^{95} +8.89898i q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} - 4 q^{7} - 4 q^{9} - 8 q^{11} - 8 q^{13} + 4 q^{25} + 8 q^{31} + 4 q^{35} + 16 q^{43} + 4 q^{45} + 24 q^{47} - 20 q^{49} + 8 q^{55} + 8 q^{57} - 16 q^{61} + 4 q^{63} + 8 q^{65} - 24 q^{69}+ \cdots + 8 q^{99}+O(q^{100}) $ 4 * q - 4 * q^5 - 4 * q^7 - 4 * q^9 - 8 * q^11 - 8 * q^13 + 4 * q^25 + 8 * q^31 + 4 * q^35 + 16 * q^43 + 4 * q^45 + 24 * q^47 - 20 * q^49 + 8 * q^55 + 8 * q^57 - 16 * q^61 + 4 * q^63 + 8 * q^65 - 24 * q^69 + 8 * q^77 + 4 * q^81 - 16 * q^87 + 8 * q^91 + 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times$.

$n$	$421$	$1121$	$1471$	$1921$	$2017$
$\chi(n)$	$-1$	$1$	$-1$	$-1$	$1$

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
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$	−1.00000	−	2.44949i	−0.377964	−	0.925820i
$7$	−1.00000	−	2.44949i	−0.377964	−	0.925820i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−2.00000			−0.603023			−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000			−0.603023			−0.301511	+	0.953463i	$0.597491\pi$
$12$		0			0
$13$	2.89898			0.804032			0.402016	−	0.915633i	$-0.368310\pi$
$13$	2.89898			0.804032			0.402016	+	0.915633i	$0.368310\pi$
$14$		0			0
$15$			1.00000i			0.258199i
$16$		0			0
$17$		−	4.89898i		−	1.18818i	−0.804400	−	0.594089i	$-0.797513\pi$
$17$		−	4.89898i		−	1.18818i	0.804400	−	0.594089i	$-0.202487\pi$
$18$		0			0
$19$		−	2.89898i		−	0.665072i	−0.943091	−	0.332536i	$-0.892096\pi$
$19$		−	2.89898i		−	0.665072i	0.943091	−	0.332536i	$-0.107904\pi$
$20$		0			0
$21$	−2.44949	+	1.00000i	−0.534522	+	0.218218i
$22$		0			0
$23$		−	6.00000i		−	1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$		−	6.00000i		−	1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$			0.898979i			0.166936i	0.996510	+	0.0834681i	$0.0265997\pi$
$29$			0.898979i			0.166936i	−0.996510	+	0.0834681i	$0.973400\pi$
$30$		0			0
$31$	2.00000			0.359211			0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000			0.359211			0.179605	+	0.983739i	$0.442518\pi$
$32$		0			0
$33$			2.00000i			0.348155i
$34$		0			0
$35$	1.00000	+	2.44949i	0.169031	+	0.414039i
$36$		0			0
$37$			11.7980i			1.93957i	0.243956	+	0.969786i	$0.421555\pi$
$37$			11.7980i			1.93957i	−0.243956	+	0.969786i	$0.578445\pi$
$38$		0			0
$39$		−	2.89898i		−	0.464208i
$40$		0			0
$41$		−	6.89898i		−	1.07744i	−0.842485	−	0.538720i	$-0.818908\pi$
$41$		−	6.89898i		−	1.07744i	0.842485	−	0.538720i	$-0.181092\pi$
$42$		0			0
$43$	−0.898979			−0.137093			−0.0685465	−	0.997648i	$-0.521836\pi$
$43$	−0.898979			−0.137093			−0.0685465	+	0.997648i	$0.521836\pi$
$44$		0			0
$45$	1.00000			0.149071
$46$		0			0
$47$	1.10102			0.160600			0.0803002	−	0.996771i	$-0.474412\pi$
$47$	1.10102			0.160600			0.0803002	+	0.996771i	$0.474412\pi$
$48$		0			0
$49$	−5.00000	+	4.89898i	−0.714286	+	0.699854i
$50$		0			0
$51$	−4.89898			−0.685994
$52$		0			0
$53$			9.79796i			1.34585i	0.739709	+	0.672927i	$0.234963\pi$
$53$			9.79796i			1.34585i	−0.739709	+	0.672927i	$0.765037\pi$
$54$		0			0
$55$	2.00000			0.269680
$56$		0			0
$57$	−2.89898			−0.383979
$58$		0			0
$59$		−	12.8990i		−	1.67930i	−0.543125	−	0.839652i	$-0.682759\pi$
$59$		−	12.8990i		−	1.67930i	0.543125	−	0.839652i	$-0.317241\pi$
$60$		0			0
$61$	−8.89898			−1.13940			−0.569699	−	0.821854i	$-0.692940\pi$
$61$	−8.89898			−1.13940			−0.569699	+	0.821854i	$0.692940\pi$
$62$		0			0
$63$	1.00000	+	2.44949i	0.125988	+	0.308607i
$64$		0			0
$65$	−2.89898			−0.359574
$66$		0			0
$67$	−4.89898			−0.598506			−0.299253	−	0.954174i	$-0.596737\pi$
$67$	−4.89898			−0.598506			−0.299253	+	0.954174i	$0.596737\pi$
$68$		0			0
$69$	−6.00000			−0.722315
$70$		0			0
$71$			3.10102i			0.368023i	0.982924	+	0.184012i	$0.0589084\pi$
$71$			3.10102i			0.368023i	−0.982924	+	0.184012i	$0.941092\pi$
$72$		0			0
$73$			0.898979i			0.105218i	0.998615	+	0.0526088i	$0.0167536\pi$
$73$			0.898979i			0.105218i	−0.998615	+	0.0526088i	$0.983246\pi$
$74$		0			0
$75$		−	1.00000i		−	0.115470i
$76$		0			0
$77$	2.00000	+	4.89898i	0.227921	+	0.558291i
$78$		0			0
$79$		−	4.00000i		−	0.450035i	−0.974355	−	0.225018i	$-0.927756\pi$
$79$		−	4.00000i		−	0.450035i	0.974355	−	0.225018i	$-0.0722440\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	4.00000i		−	0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$		−	4.00000i		−	0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$		0			0
$85$			4.89898i			0.531369i
$86$		0			0
$87$	0.898979			0.0963807
$88$		0			0
$89$		−	10.8990i		−	1.15529i	−0.816288	−	0.577645i	$-0.803972\pi$
$89$		−	10.8990i		−	1.15529i	0.816288	−	0.577645i	$-0.196028\pi$
$90$		0			0
$91$	−2.89898	−	7.10102i	−0.303896	−	0.744389i
$92$		0			0
$93$		−	2.00000i		−	0.207390i
$94$		0			0
$95$			2.89898i			0.297429i
$96$		0			0
$97$			8.89898i			0.903554i	0.892131	+	0.451777i	$0.149210\pi$
$97$			8.89898i			0.903554i	−0.892131	+	0.451777i	$0.850790\pi$
$98$		0			0
$99$	2.00000			0.201008
$100$		0			0
$101$	−15.7980			−1.57196			−0.785978	−	0.618255i	$-0.787840\pi$
$101$	−15.7980			−1.57196			−0.785978	+	0.618255i	$0.787840\pi$
$102$		0			0
$103$	−19.7980			−1.95075			−0.975375	−	0.220551i	$-0.929214\pi$
$103$	−19.7980			−1.95075			−0.975375	+	0.220551i	$0.929214\pi$
$104$		0			0
$105$	2.44949	−	1.00000i	0.239046	−	0.0975900i
$106$		0			0
$107$	−11.7980			−1.14055			−0.570276	−	0.821453i	$-0.693164\pi$
$107$	−11.7980			−1.14055			−0.570276	+	0.821453i	$0.693164\pi$
$108$		0			0
$109$			4.00000i			0.383131i	0.981480	+	0.191565i	$0.0613564\pi$
$109$			4.00000i			0.383131i	−0.981480	+	0.191565i	$0.938644\pi$
$110$		0			0
$111$	11.7980			1.11981
$112$		0			0
$113$	17.7980			1.67429			0.837146	−	0.546980i	$-0.184223\pi$
$113$	17.7980			1.67429			0.837146	+	0.546980i	$0.184223\pi$
$114$		0			0
$115$			6.00000i			0.559503i
$116$		0			0
$117$	−2.89898			−0.268011
$118$		0			0
$119$	−12.0000	+	4.89898i	−1.10004	+	0.449089i
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$	−6.89898			−0.622060
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$			8.89898i			0.789657i	0.918755	+	0.394828i	$0.129196\pi$
$127$			8.89898i			0.789657i	−0.918755	+	0.394828i	$0.870804\pi$
$128$		0			0
$129$			0.898979i			0.0791507i
$130$		0			0
$131$			16.8990i			1.47647i	0.674543	+	0.738235i	$0.264341\pi$
$131$			16.8990i			1.47647i	−0.674543	+	0.738235i	$0.735659\pi$
$132$		0			0
$133$	−7.10102	+	2.89898i	−0.615737	+	0.251373i
$134$		0			0
$135$		−	1.00000i		−	0.0860663i
$136$		0			0
$137$	−5.79796			−0.495353			−0.247677	−	0.968843i	$-0.579667\pi$
$137$	−5.79796			−0.495353			−0.247677	+	0.968843i	$0.579667\pi$
$138$		0			0
$139$		−	2.89898i		−	0.245888i	−0.992414	−	0.122944i	$-0.960766\pi$
$139$		−	2.89898i		−	0.245888i	0.992414	−	0.122944i	$-0.0392336\pi$
$140$		0			0
$141$		−	1.10102i		−	0.0927227i
$142$		0			0
$143$	−5.79796			−0.484850
$144$		0			0
$145$		−	0.898979i		−	0.0746562i
$146$		0			0
$147$	4.89898	+	5.00000i	0.404061	+	0.412393i
$148$		0			0
$149$			7.10102i			0.581738i	0.956763	+	0.290869i	$0.0939444\pi$
$149$			7.10102i			0.581738i	−0.956763	+	0.290869i	$0.906056\pi$
$150$		0			0
$151$			13.7980i			1.12286i	0.827524	+	0.561431i	$0.189749\pi$
$151$			13.7980i			1.12286i	−0.827524	+	0.561431i	$0.810251\pi$
$152$		0			0
$153$			4.89898i			0.396059i
$154$		0			0
$155$	−2.00000			−0.160644
$156$		0			0
$157$	12.6969			1.01333			0.506663	−	0.862144i	$-0.330879\pi$
$157$	12.6969			1.01333			0.506663	+	0.862144i	$0.330879\pi$
$158$		0			0
$159$	9.79796			0.777029
$160$		0			0
$161$	−14.6969	+	6.00000i	−1.15828	+	0.472866i
$162$		0			0
$163$	6.69694			0.524545			0.262272	−	0.964994i	$-0.415528\pi$
$163$	6.69694			0.524545			0.262272	+	0.964994i	$0.415528\pi$
$164$		0			0
$165$		−	2.00000i		−	0.155700i
$166$		0			0
$167$	1.10102			0.0851995			0.0425998	−	0.999092i	$-0.486436\pi$
$167$	1.10102			0.0851995			0.0425998	+	0.999092i	$0.486436\pi$
$168$		0			0
$169$	−4.59592			−0.353532
$170$		0			0
$171$			2.89898i			0.221691i
$172$		0			0
$173$	−7.79796			−0.592868			−0.296434	−	0.955053i	$-0.595797\pi$
$173$	−7.79796			−0.592868			−0.296434	+	0.955053i	$0.595797\pi$
$174$		0			0
$175$	−1.00000	−	2.44949i	−0.0755929	−	0.185164i
$176$		0			0
$177$	−12.8990			−0.969547
$178$		0			0
$179$	10.0000			0.747435			0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000			0.747435			0.373718	+	0.927543i	$0.378083\pi$
$180$		0			0
$181$	11.1010			0.825132			0.412566	−	0.910928i	$-0.364633\pi$
$181$	11.1010			0.825132			0.412566	+	0.910928i	$0.364633\pi$
$182$		0			0
$183$			8.89898i			0.657831i
$184$		0			0
$185$		−	11.7980i		−	0.867403i
$186$		0			0
$187$			9.79796i			0.716498i
$188$		0			0
$189$	2.44949	−	1.00000i	0.178174	−	0.0727393i
$190$		0			0
$191$		−	16.8990i		−	1.22277i	−0.791334	−	0.611384i	$-0.790613\pi$
$191$		−	16.8990i		−	1.22277i	0.791334	−	0.611384i	$-0.209387\pi$
$192$		0			0
$193$	−19.7980			−1.42509			−0.712544	−	0.701627i	$-0.752457\pi$
$193$	−19.7980			−1.42509			−0.712544	+	0.701627i	$0.752457\pi$
$194$		0			0
$195$			2.89898i			0.207600i
$196$		0			0
$197$		−	5.79796i		−	0.413087i	−0.978437	−	0.206544i	$-0.933778\pi$
$197$		−	5.79796i		−	0.413087i	0.978437	−	0.206544i	$-0.0662216\pi$
$198$		0			0
$199$	10.0000			0.708881			0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000			0.708881			0.354441	+	0.935079i	$0.384671\pi$
$200$		0			0
$201$			4.89898i			0.345547i
$202$		0			0
$203$	2.20204	−	0.898979i	0.154553	−	0.0630960i
$204$		0			0
$205$			6.89898i			0.481846i
$206$		0			0
$207$			6.00000i			0.417029i
$208$		0			0
$209$			5.79796i			0.401053i
$210$		0			0
$211$	−22.0000			−1.51454			−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000			−1.51454			−0.757271	+	0.653101i	$0.773468\pi$
$212$		0			0
$213$	3.10102			0.212478
$214$		0			0
$215$	0.898979			0.0613099
$216$		0			0
$217$	−2.00000	−	4.89898i	−0.135769	−	0.332564i
$218$		0			0
$219$	0.898979			0.0607474
$220$		0			0
$221$		−	14.2020i		−	0.955333i
$222$		0			0
$223$	−6.00000			−0.401790			−0.200895	−	0.979613i	$-0.564385\pi$
$223$	−6.00000			−0.401790			−0.200895	+	0.979613i	$0.564385\pi$
$224$		0			0
$225$	−1.00000			−0.0666667
$226$		0			0
$227$		−	12.0000i		−	0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$		−	12.0000i		−	0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$		0			0
$229$	24.4949			1.61867			0.809334	−	0.587348i	$-0.199828\pi$
$229$	24.4949			1.61867			0.809334	+	0.587348i	$0.199828\pi$
$230$		0			0
$231$	4.89898	−	2.00000i	0.322329	−	0.131590i
$232$		0			0
$233$	−16.0000			−1.04819			−0.524097	−	0.851658i	$-0.675597\pi$
$233$	−16.0000			−1.04819			−0.524097	+	0.851658i	$0.675597\pi$
$234$		0			0
$235$	−1.10102			−0.0718227
$236$		0			0
$237$	−4.00000			−0.259828
$238$		0			0
$239$			26.6969i			1.72688i	0.504450	+	0.863441i	$0.331695\pi$
$239$			26.6969i			1.72688i	−0.504450	+	0.863441i	$0.668305\pi$
$240$		0			0
$241$		−	20.0000i		−	1.28831i	−0.764894	−	0.644157i	$-0.777208\pi$
$241$		−	20.0000i		−	1.28831i	0.764894	−	0.644157i	$-0.222792\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	5.00000	−	4.89898i	0.319438	−	0.312984i
$246$		0			0
$247$		−	8.40408i		−	0.534739i
$248$		0			0
$249$	−4.00000			−0.253490
$250$		0			0
$251$		−	3.10102i		−	0.195735i	−0.995199	−	0.0978673i	$-0.968798\pi$
$251$		−	3.10102i		−	0.195735i	0.995199	−	0.0978673i	$-0.0312021\pi$
$252$		0			0
$253$			12.0000i			0.754434i
$254$		0			0
$255$	4.89898			0.306786
$256$		0			0
$257$		−	24.8990i		−	1.55316i	−0.630021	−	0.776578i	$-0.716954\pi$
$257$		−	24.8990i		−	1.55316i	0.630021	−	0.776578i	$-0.283046\pi$
$258$		0			0
$259$	28.8990	−	11.7980i	1.79570	−	0.733090i
$260$		0			0
$261$		−	0.898979i		−	0.0556454i
$262$		0			0
$263$			7.79796i			0.480843i	0.970669	+	0.240421i	$0.0772856\pi$
$263$			7.79796i			0.480843i	−0.970669	+	0.240421i	$0.922714\pi$
$264$		0			0
$265$		−	9.79796i		−	0.601884i
$266$		0			0
$267$	−10.8990			−0.667007
$268$		0			0
$269$	3.79796			0.231566			0.115783	−	0.993275i	$-0.463062\pi$
$269$	3.79796			0.231566			0.115783	+	0.993275i	$0.463062\pi$
$270$		0			0
$271$	−4.20204			−0.255256			−0.127628	−	0.991822i	$-0.540736\pi$
$271$	−4.20204			−0.255256			−0.127628	+	0.991822i	$0.540736\pi$
$272$		0			0
$273$	−7.10102	+	2.89898i	−0.429773	+	0.175454i
$274$		0			0
$275$	−2.00000			−0.120605
$276$		0			0
$277$		−	15.7980i		−	0.949207i	−0.880200	−	0.474604i	$-0.842591\pi$
$277$		−	15.7980i		−	0.949207i	0.880200	−	0.474604i	$-0.157409\pi$
$278$		0			0
$279$	−2.00000			−0.119737
$280$		0			0
$281$	8.20204			0.489293			0.244646	−	0.969612i	$-0.421328\pi$
$281$	8.20204			0.489293			0.244646	+	0.969612i	$0.421328\pi$
$282$		0			0
$283$			9.79796i			0.582428i	0.956658	+	0.291214i	$0.0940592\pi$
$283$			9.79796i			0.582428i	−0.956658	+	0.291214i	$0.905941\pi$
$284$		0			0
$285$	2.89898			0.171721
$286$		0			0
$287$	−16.8990	+	6.89898i	−0.997515	+	0.407234i
$288$		0			0
$289$	−7.00000			−0.411765
$290$		0			0
$291$	8.89898			0.521667
$292$		0			0
$293$	−21.5959			−1.26165			−0.630823	−	0.775926i	$-0.717283\pi$
$293$	−21.5959			−1.26165			−0.630823	+	0.775926i	$0.717283\pi$
$294$		0			0
$295$			12.8990i			0.751008i
$296$		0			0
$297$		−	2.00000i		−	0.116052i
$298$		0			0
$299$		−	17.3939i		−	1.00591i
$300$		0			0
$301$	0.898979	+	2.20204i	0.0518163	+	0.126924i
$302$		0			0
$303$			15.7980i			0.907569i
$304$		0			0
$305$	8.89898			0.509554
$306$		0			0
$307$			15.5959i			0.890106i	0.895504	+	0.445053i	$0.146815\pi$
$307$			15.5959i			0.890106i	−0.895504	+	0.445053i	$0.853185\pi$
$308$		0			0
$309$			19.7980i			1.12627i
$310$		0			0
$311$	−8.00000			−0.453638			−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000			−0.453638			−0.226819	+	0.973937i	$0.572833\pi$
$312$		0			0
$313$			28.4949i			1.61063i	0.592849	+	0.805313i	$0.298003\pi$
$313$			28.4949i			1.61063i	−0.592849	+	0.805313i	$0.701997\pi$
$314$		0			0
$315$	−1.00000	−	2.44949i	−0.0563436	−	0.138013i
$316$		0			0
$317$		−	12.0000i		−	0.673987i	−0.941507	−	0.336994i	$-0.890590\pi$
$317$		−	12.0000i		−	0.673987i	0.941507	−	0.336994i	$-0.109410\pi$
$318$		0			0
$319$		−	1.79796i		−	0.100666i
$320$		0			0
$321$			11.7980i			0.658498i
$322$		0			0
$323$	−14.2020			−0.790223
$324$		0			0
$325$	2.89898			0.160806
$326$		0			0
$327$	4.00000			0.221201
$328$		0			0
$329$	−1.10102	−	2.69694i	−0.0607012	−	0.148687i
$330$		0			0
$331$	−23.3939			−1.28584			−0.642922	−	0.765932i	$-0.722278\pi$
$331$	−23.3939			−1.28584			−0.642922	+	0.765932i	$0.722278\pi$
$332$		0			0
$333$		−	11.7980i		−	0.646524i
$334$		0			0
$335$	4.89898			0.267660
$336$		0			0
$337$	31.7980			1.73215			0.866073	−	0.499918i	$-0.166637\pi$
$337$	31.7980			1.73215			0.866073	+	0.499918i	$0.166637\pi$
$338$		0			0
$339$		−	17.7980i		−	0.966652i
$340$		0			0
$341$	−4.00000			−0.216612
$342$		0			0
$343$	17.0000	+	7.34847i	0.917914	+	0.396780i
$344$		0			0
$345$	6.00000			0.323029
$346$		0			0
$347$	−25.5959			−1.37406			−0.687030	−	0.726629i	$-0.741086\pi$
$347$	−25.5959			−1.37406			−0.687030	+	0.726629i	$0.741086\pi$
$348$		0			0
$349$	30.6969			1.64317			0.821585	−	0.570086i	$-0.193090\pi$
$349$	30.6969			1.64317			0.821585	+	0.570086i	$0.193090\pi$
$350$		0			0
$351$			2.89898i			0.154736i
$352$		0			0
$353$			34.6969i			1.84673i	0.383921	+	0.923366i	$0.374573\pi$
$353$			34.6969i			1.84673i	−0.383921	+	0.923366i	$0.625427\pi$
$354$		0			0
$355$		−	3.10102i		−	0.164585i
$356$		0			0
$357$	4.89898	+	12.0000i	0.259281	+	0.635107i
$358$		0			0
$359$		−	7.10102i		−	0.374778i	−0.982286	−	0.187389i	$-0.939998\pi$
$359$		−	7.10102i		−	0.374778i	0.982286	−	0.187389i	$-0.0600024\pi$
$360$		0			0
$361$	10.5959			0.557680
$362$		0			0
$363$			7.00000i			0.367405i
$364$		0			0
$365$		−	0.898979i		−	0.0470547i
$366$		0			0
$367$	−15.7980			−0.824647			−0.412323	−	0.911038i	$-0.635283\pi$
$367$	−15.7980			−0.824647			−0.412323	+	0.911038i	$0.635283\pi$
$368$		0			0
$369$			6.89898i			0.359147i
$370$		0			0
$371$	24.0000	−	9.79796i	1.24602	−	0.508685i
$372$		0			0
$373$		−	14.0000i		−	0.724893i	−0.932005	−	0.362446i	$-0.881942\pi$
$373$		−	14.0000i		−	0.724893i	0.932005	−	0.362446i	$-0.118058\pi$
$374$		0			0
$375$			1.00000i			0.0516398i
$376$		0			0
$377$			2.60612i			0.134222i
$378$		0			0
$379$	−31.3939			−1.61260			−0.806298	−	0.591510i	$-0.798532\pi$
$379$	−31.3939			−1.61260			−0.806298	+	0.591510i	$0.798532\pi$
$380$		0			0
$381$	8.89898			0.455909
$382$		0			0
$383$	−9.10102			−0.465040			−0.232520	−	0.972592i	$-0.574697\pi$
$383$	−9.10102			−0.465040			−0.232520	+	0.972592i	$0.574697\pi$
$384$		0			0
$385$	−2.00000	−	4.89898i	−0.101929	−	0.249675i
$386$		0			0
$387$	0.898979			0.0456977
$388$		0			0
$389$		−	26.6969i		−	1.35359i	−0.736172	−	0.676794i	$-0.763369\pi$
$389$		−	26.6969i		−	1.35359i	0.736172	−	0.676794i	$-0.236631\pi$
$390$		0			0
$391$	−29.3939			−1.48651
$392$		0			0
$393$	16.8990			0.852441
$394$		0			0
$395$			4.00000i			0.201262i
$396$		0			0
$397$	12.6969			0.637241			0.318621	−	0.947882i	$-0.396780\pi$
$397$	12.6969			0.637241			0.318621	+	0.947882i	$0.396780\pi$
$398$		0			0
$399$	2.89898	+	7.10102i	0.145131	+	0.355496i
$400$		0			0
$401$	−11.7980			−0.589162			−0.294581	−	0.955627i	$-0.595180\pi$
$401$	−11.7980			−0.589162			−0.294581	+	0.955627i	$0.595180\pi$
$402$		0			0
$403$	5.79796			0.288817
$404$		0			0
$405$	−1.00000			−0.0496904
$406$		0			0
$407$		−	23.5959i		−	1.16961i
$408$		0			0
$409$		−	4.00000i		−	0.197787i	−0.995098	−	0.0988936i	$-0.968470\pi$
$409$		−	4.00000i		−	0.197787i	0.995098	−	0.0988936i	$-0.0315304\pi$
$410$		0			0
$411$			5.79796i			0.285992i
$412$		0			0
$413$	−31.5959	+	12.8990i	−1.55473	+	0.634717i
$414$		0			0
$415$			4.00000i			0.196352i
$416$		0			0
$417$	−2.89898			−0.141964
$418$		0			0
$419$		−	34.2929i		−	1.67532i	−0.546195	−	0.837658i	$-0.683924\pi$
$419$		−	34.2929i		−	1.67532i	0.546195	−	0.837658i	$-0.316076\pi$
$420$		0			0
$421$			6.20204i			0.302269i	0.988513	+	0.151134i	$0.0482927\pi$
$421$			6.20204i			0.302269i	−0.988513	+	0.151134i	$0.951707\pi$
$422$		0			0
$423$	−1.10102			−0.0535334
$424$		0			0
$425$		−	4.89898i		−	0.237635i
$426$		0			0
$427$	8.89898	+	21.7980i	0.430652	+	1.05488i
$428$		0			0
$429$			5.79796i			0.279928i
$430$		0			0
$431$		−	3.10102i		−	0.149371i	−0.997207	−	0.0746855i	$-0.976205\pi$
$431$		−	3.10102i		−	0.149371i	0.997207	−	0.0746855i	$-0.0237953\pi$
$432$		0			0
$433$		−	5.30306i		−	0.254849i	−0.991848	−	0.127424i	$-0.959329\pi$
$433$		−	5.30306i		−	0.254849i	0.991848	−	0.127424i	$-0.0406710\pi$
$434$		0			0
$435$	−0.898979			−0.0431028
$436$		0			0
$437$	−17.3939			−0.832062
$438$		0			0
$439$	23.7980			1.13581			0.567907	−	0.823093i	$-0.307753\pi$
$439$	23.7980			1.13581			0.567907	+	0.823093i	$0.307753\pi$
$440$		0			0
$441$	5.00000	−	4.89898i	0.238095	−	0.233285i
$442$		0			0
$443$	6.00000			0.285069			0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000			0.285069			0.142534	+	0.989790i	$0.454475\pi$
$444$		0			0
$445$			10.8990i			0.516661i
$446$		0			0
$447$	7.10102			0.335867
$448$		0			0
$449$	23.7980			1.12310			0.561548	−	0.827444i	$-0.310206\pi$
$449$	23.7980			1.12310			0.561548	+	0.827444i	$0.310206\pi$
$450$		0			0
$451$			13.7980i			0.649721i
$452$		0			0
$453$	13.7980			0.648285
$454$		0			0
$455$	2.89898	+	7.10102i	0.135906	+	0.332901i
$456$		0			0
$457$	11.7980			0.551885			0.275943	−	0.961174i	$-0.411010\pi$
$457$	11.7980			0.551885			0.275943	+	0.961174i	$0.411010\pi$
$458$		0			0
$459$	4.89898			0.228665
$460$		0			0
$461$	4.20204			0.195709			0.0978543	−	0.995201i	$-0.468802\pi$
$461$	4.20204			0.195709			0.0978543	+	0.995201i	$0.468802\pi$
$462$		0			0
$463$		−	26.6969i		−	1.24071i	−0.784320	−	0.620356i	$-0.786988\pi$
$463$		−	26.6969i		−	1.24071i	0.784320	−	0.620356i	$-0.213012\pi$
$464$		0			0
$465$			2.00000i			0.0927478i
$466$		0			0
$467$			21.7980i			1.00869i	0.863502	+	0.504345i	$0.168266\pi$
$467$			21.7980i			1.00869i	−0.863502	+	0.504345i	$0.831734\pi$
$468$		0			0
$469$	4.89898	+	12.0000i	0.226214	+	0.554109i
$470$		0			0
$471$		−	12.6969i		−	0.585044i
$472$		0			0
$473$	1.79796			0.0826702
$474$		0			0
$475$		−	2.89898i		−	0.133014i
$476$		0			0
$477$		−	9.79796i		−	0.448618i
$478$		0			0
$479$	27.5959			1.26089			0.630445	−	0.776234i	$-0.282873\pi$
$479$	27.5959			1.26089			0.630445	+	0.776234i	$0.282873\pi$
$480$		0			0
$481$			34.2020i			1.55948i
$482$		0			0
$483$	6.00000	+	14.6969i	0.273009	+	0.668734i
$484$		0			0
$485$		−	8.89898i		−	0.404082i
$486$		0			0
$487$			15.1010i			0.684293i	0.939647	+	0.342146i	$0.111154\pi$
$487$			15.1010i			0.684293i	−0.939647	+	0.342146i	$0.888846\pi$
$488$		0			0
$489$		−	6.69694i		−	0.302846i
$490$		0			0
$491$	25.5959			1.15513			0.577564	−	0.816346i	$-0.304003\pi$
$491$	25.5959			1.15513			0.577564	+	0.816346i	$0.304003\pi$
$492$		0			0
$493$	4.40408			0.198350
$494$		0			0
$495$	−2.00000			−0.0898933
$496$		0			0
$497$	7.59592	−	3.10102i	0.340723	−	0.139100i
$498$		0			0
$499$	−10.0000			−0.447661			−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000			−0.447661			−0.223831	+	0.974628i	$0.571856\pi$
$500$		0			0
$501$		−	1.10102i		−	0.0491900i
$502$		0			0
$503$	−10.4949			−0.467944			−0.233972	−	0.972243i	$-0.575172\pi$
$503$	−10.4949			−0.467944			−0.233972	+	0.972243i	$0.575172\pi$
$504$		0			0
$505$	15.7980			0.703000
$506$		0			0
$507$			4.59592i			0.204112i
$508$		0			0
$509$	−37.5959			−1.66641			−0.833205	−	0.552964i	$-0.813497\pi$
$509$	−37.5959			−1.66641			−0.833205	+	0.552964i	$0.813497\pi$
$510$		0			0
$511$	2.20204	−	0.898979i	0.0974126	−	0.0397685i
$512$		0			0
$513$	2.89898			0.127993
$514$		0			0
$515$	19.7980			0.872402
$516$		0			0
$517$	−2.20204			−0.0968457
$518$		0			0
$519$			7.79796i			0.342292i
$520$		0			0
$521$		−	33.1010i		−	1.45018i	−0.688653	−	0.725091i	$-0.741798\pi$
$521$		−	33.1010i		−	1.45018i	0.688653	−	0.725091i	$-0.258202\pi$
$522$		0			0
$523$			3.59592i			0.157239i	0.996905	+	0.0786193i	$0.0250511\pi$
$523$			3.59592i			0.157239i	−0.996905	+	0.0786193i	$0.974949\pi$
$524$		0			0
$525$	−2.44949	+	1.00000i	−0.106904	+	0.0436436i
$526$		0			0
$527$		−	9.79796i		−	0.426806i
$528$		0			0
$529$	−13.0000			−0.565217
$530$		0			0
$531$			12.8990i			0.559768i
$532$		0			0
$533$		−	20.0000i		−	0.866296i
$534$		0			0
$535$	11.7980			0.510070
$536$		0			0
$537$		−	10.0000i		−	0.431532i
$538$		0			0
$539$	10.0000	−	9.79796i	0.430730	−	0.422028i
$540$		0			0
$541$		−	41.3939i		−	1.77966i	−0.456290	−	0.889831i	$-0.650822\pi$
$541$		−	41.3939i		−	1.77966i	0.456290	−	0.889831i	$-0.349178\pi$
$542$		0			0
$543$		−	11.1010i		−	0.476390i
$544$		0			0
$545$		−	4.00000i		−	0.171341i
$546$		0			0
$547$	30.2929			1.29523			0.647615	−	0.761968i	$-0.275767\pi$
$547$	30.2929			1.29523			0.647615	+	0.761968i	$0.275767\pi$
$548$		0			0
$549$	8.89898			0.379799
$550$		0			0
$551$	2.60612			0.111025
$552$		0			0
$553$	−9.79796	+	4.00000i	−0.416652	+	0.170097i
$554$		0			0
$555$	−11.7980			−0.500795
$556$		0			0
$557$		−	5.79796i		−	0.245667i	−0.992427	−	0.122834i	$-0.960802\pi$
$557$		−	5.79796i		−	0.245667i	0.992427	−	0.122834i	$-0.0391982\pi$
$558$		0			0
$559$	−2.60612			−0.110227
$560$		0			0
$561$	9.79796			0.413670
$562$		0			0
$563$			2.20204i			0.0928050i	0.998923	+	0.0464025i	$0.0147757\pi$
$563$			2.20204i			0.0928050i	−0.998923	+	0.0464025i	$0.985224\pi$
$564$		0			0
$565$	−17.7980			−0.748766
$566$		0			0
$567$	−1.00000	−	2.44949i	−0.0419961	−	0.102869i
$568$		0			0
$569$	17.5959			0.737659			0.368830	−	0.929497i	$-0.379759\pi$
$569$	17.5959			0.737659			0.368830	+	0.929497i	$0.379759\pi$
$570$		0			0
$571$	−15.7980			−0.661124			−0.330562	−	0.943784i	$-0.607238\pi$
$571$	−15.7980			−0.661124			−0.330562	+	0.943784i	$0.607238\pi$
$572$		0			0
$573$	−16.8990			−0.705965
$574$		0			0
$575$		−	6.00000i		−	0.250217i
$576$		0			0
$577$		−	24.8990i		−	1.03656i	−0.855212	−	0.518279i	$-0.826573\pi$
$577$		−	24.8990i		−	1.03656i	0.855212	−	0.518279i	$-0.173427\pi$
$578$		0			0
$579$			19.7980i			0.822775i
$580$		0			0
$581$	−9.79796	+	4.00000i	−0.406488	+	0.165948i
$582$		0			0
$583$		−	19.5959i		−	0.811580i
$584$		0			0
$585$	2.89898			0.119858
$586$		0			0
$587$		−	45.7980i		−	1.89028i	−0.326660	−	0.945142i	$-0.605923\pi$
$587$		−	45.7980i		−	1.89028i	0.326660	−	0.945142i	$-0.394077\pi$
$588$		0			0
$589$		−	5.79796i		−	0.238901i
$590$		0			0
$591$	−5.79796			−0.238496
$592$		0			0
$593$		−	12.8990i		−	0.529698i	−0.964290	−	0.264849i	$-0.914678\pi$
$593$		−	12.8990i		−	0.529698i	0.964290	−	0.264849i	$-0.0853220\pi$
$594$		0			0
$595$	12.0000	−	4.89898i	0.491952	−	0.200839i
$596$		0			0
$597$		−	10.0000i		−	0.409273i
$598$		0			0
$599$		−	0.898979i		−	0.0367313i	−0.999831	−	0.0183657i	$-0.994154\pi$
$599$		−	0.898979i		−	0.0367313i	0.999831	−	0.0183657i	$-0.00584630\pi$
$600$		0			0
$601$			13.7980i			0.562830i	0.959586	+	0.281415i	$0.0908038\pi$
$601$			13.7980i			0.562830i	−0.959586	+	0.281415i	$0.909196\pi$
$602$		0			0
$603$	4.89898			0.199502
$604$		0			0
$605$	7.00000			0.284590
$606$		0			0
$607$	−22.0000			−0.892952			−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000			−0.892952			−0.446476	+	0.894795i	$0.647321\pi$
$608$		0			0
$609$	−0.898979	−	2.20204i	−0.0364285	−	0.0892312i
$610$		0			0
$611$	3.19184			0.129128
$612$		0			0
$613$		−	0.202041i		−	0.00816036i	−0.999992	−	0.00408018i	$-0.998701\pi$
$613$		−	0.202041i		−	0.00816036i	0.999992	−	0.00408018i	$-0.00129877\pi$
$614$		0			0
$615$	6.89898			0.278194
$616$		0			0
$617$	41.7980			1.68272			0.841361	−	0.540473i	$-0.181755\pi$
$617$	41.7980			1.68272			0.841361	+	0.540473i	$0.181755\pi$
$618$		0			0
$619$		−	44.2929i		−	1.78028i	−0.455687	−	0.890140i	$-0.650606\pi$
$619$		−	44.2929i		−	1.78028i	0.455687	−	0.890140i	$-0.349394\pi$
$620$		0			0
$621$	6.00000			0.240772
$622$		0			0
$623$	−26.6969	+	10.8990i	−1.06959	+	0.436658i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	5.79796			0.231548
$628$		0			0
$629$	57.7980			2.30456
$630$		0			0
$631$			13.7980i			0.549288i	0.961546	+	0.274644i	$0.0885600\pi$
$631$			13.7980i			0.549288i	−0.961546	+	0.274644i	$0.911440\pi$
$632$		0			0
$633$			22.0000i			0.874421i
$634$		0			0
$635$		−	8.89898i		−	0.353145i
$636$		0			0
$637$	−14.4949	+	14.2020i	−0.574309	+	0.562705i
$638$		0			0
$639$		−	3.10102i		−	0.122674i
$640$		0			0
$641$	−45.5959			−1.80093			−0.900465	−	0.434928i	$-0.856774\pi$
$641$	−45.5959			−1.80093			−0.900465	+	0.434928i	$0.856774\pi$
$642$		0			0
$643$			36.0000i			1.41970i	0.704352	+	0.709851i	$0.251238\pi$
$643$			36.0000i			1.41970i	−0.704352	+	0.709851i	$0.748762\pi$
$644$		0			0
$645$		−	0.898979i		−	0.0353973i
$646$		0			0
$647$	−32.6969			−1.28545			−0.642725	−	0.766097i	$-0.722196\pi$
$647$	−32.6969			−1.28545			−0.642725	+	0.766097i	$0.722196\pi$
$648$		0			0
$649$			25.7980i			1.01266i
$650$		0			0
$651$	−4.89898	+	2.00000i	−0.192006	+	0.0783862i
$652$		0			0
$653$		−	44.0000i		−	1.72185i	−0.508729	−	0.860927i	$-0.669885\pi$
$653$		−	44.0000i		−	1.72185i	0.508729	−	0.860927i	$-0.330115\pi$
$654$		0			0
$655$		−	16.8990i		−	0.660298i
$656$		0			0
$657$		−	0.898979i		−	0.0350725i
$658$		0			0
$659$	30.0000			1.16863			0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000			1.16863			0.584317	+	0.811525i	$0.301362\pi$
$660$		0			0
$661$	−1.30306			−0.0506832			−0.0253416	−	0.999679i	$-0.508067\pi$
$661$	−1.30306			−0.0506832			−0.0253416	+	0.999679i	$0.508067\pi$
$662$		0			0
$663$	−14.2020			−0.551562
$664$		0			0
$665$	7.10102	−	2.89898i	0.275366	−	0.112418i
$666$		0			0
$667$	5.39388			0.208852
$668$		0			0
$669$			6.00000i			0.231973i
$670$		0			0
$671$	17.7980			0.687083
$672$		0			0
$673$	−13.5959			−0.524084			−0.262042	−	0.965056i	$-0.584396\pi$
$673$	−13.5959			−0.524084			−0.262042	+	0.965056i	$0.584396\pi$
$674$		0			0
$675$			1.00000i			0.0384900i
$676$		0			0
$677$	−31.7980			−1.22209			−0.611047	−	0.791594i	$-0.709252\pi$
$677$	−31.7980			−1.22209			−0.611047	+	0.791594i	$0.709252\pi$
$678$		0			0
$679$	21.7980	−	8.89898i	0.836529	−	0.341511i
$680$		0			0
$681$	−12.0000			−0.459841
$682$		0			0
$683$	12.2020			0.466898			0.233449	−	0.972369i	$-0.424999\pi$
$683$	12.2020			0.466898			0.233449	+	0.972369i	$0.424999\pi$
$684$		0			0
$685$	5.79796			0.221529
$686$		0			0
$687$		−	24.4949i		−	0.934539i
$688$		0			0
$689$			28.4041i			1.08211i
$690$		0			0
$691$		−	34.4949i		−	1.31225i	−0.754653	−	0.656124i	$-0.772195\pi$
$691$		−	34.4949i		−	1.31225i	0.754653	−	0.656124i	$-0.227805\pi$
$692$		0			0
$693$	−2.00000	−	4.89898i	−0.0759737	−	0.186097i
$694$		0			0
$695$			2.89898i			0.109965i
$696$		0			0
$697$	−33.7980			−1.28019
$698$		0			0
$699$			16.0000i			0.605176i
$700$		0			0
$701$			23.1010i			0.872514i	0.899822	+	0.436257i	$0.143696\pi$
$701$			23.1010i			0.872514i	−0.899822	+	0.436257i	$0.856304\pi$
$702$		0			0
$703$	34.2020			1.28995
$704$		0			0
$705$			1.10102i			0.0414668i
$706$		0			0
$707$	15.7980	+	38.6969i	0.594143	+	1.45535i
$708$		0			0
$709$		−	29.7980i		−	1.11909i	−0.828801	−	0.559543i	$-0.810977\pi$
$709$		−	29.7980i		−	1.11909i	0.828801	−	0.559543i	$-0.189023\pi$
$710$		0			0
$711$			4.00000i			0.150012i
$712$		0			0
$713$		−	12.0000i		−	0.449404i
$714$		0			0
$715$	5.79796			0.216831
$716$		0			0
$717$	26.6969			0.997015
$718$		0			0
$719$	26.2020			0.977171			0.488586	−	0.872516i	$-0.337513\pi$
$719$	26.2020			0.977171			0.488586	+	0.872516i	$0.337513\pi$
$720$		0			0
$721$	19.7980	+	48.4949i	0.737315	+	1.80604i
$722$		0			0
$723$	−20.0000			−0.743808
$724$		0			0
$725$			0.898979i			0.0333873i
$726$		0			0
$727$	39.3939			1.46104			0.730519	−	0.682892i	$-0.239278\pi$
$727$	39.3939			1.46104			0.730519	+	0.682892i	$0.239278\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$			4.40408i			0.162891i
$732$		0			0
$733$	1.50510			0.0555922			0.0277961	−	0.999614i	$-0.491151\pi$
$733$	1.50510			0.0555922			0.0277961	+	0.999614i	$0.491151\pi$
$734$		0			0
$735$	−4.89898	−	5.00000i	−0.180702	−	0.184428i
$736$		0			0
$737$	9.79796			0.360912
$738$		0			0
$739$	10.0000			0.367856			0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000			0.367856			0.183928	+	0.982940i	$0.441119\pi$
$740$		0			0
$741$	−8.40408			−0.308732
$742$		0			0
$743$		−	6.00000i		−	0.220119i	−0.993925	−	0.110059i	$-0.964896\pi$
$743$		−	6.00000i		−	0.220119i	0.993925	−	0.110059i	$-0.0351041\pi$
$744$		0			0
$745$		−	7.10102i		−	0.260161i
$746$		0			0
$747$			4.00000i			0.146352i
$748$		0			0
$749$	11.7980	+	28.8990i	0.431088	+	1.05595i
$750$		0			0
$751$		−	13.7980i		−	0.503495i	−0.967793	−	0.251747i	$-0.918995\pi$
$751$		−	13.7980i		−	0.503495i	0.967793	−	0.251747i	$-0.0810052\pi$
$752$		0			0
$753$	−3.10102			−0.113007
$754$		0			0
$755$		−	13.7980i		−	0.502159i
$756$		0			0
$757$			5.59592i			0.203387i	0.994816	+	0.101694i	$0.0324261\pi$
$757$			5.59592i			0.203387i	−0.994816	+	0.101694i	$0.967574\pi$
$758$		0			0
$759$	12.0000			0.435572
$760$		0			0
$761$			0.696938i			0.0252640i	0.999920	+	0.0126320i	$0.00402100\pi$
$761$			0.696938i			0.0252640i	−0.999920	+	0.0126320i	$0.995979\pi$
$762$		0			0
$763$	9.79796	−	4.00000i	0.354710	−	0.144810i
$764$		0			0
$765$		−	4.89898i		−	0.177123i
$766$		0			0
$767$		−	37.3939i		−	1.35021i
$768$		0			0
$769$		−	51.5959i		−	1.86060i	−0.366804	−	0.930298i	$-0.619548\pi$
$769$		−	51.5959i		−	1.86060i	0.366804	−	0.930298i	$-0.380452\pi$
$770$		0			0
$771$	−24.8990			−0.896715
$772$		0			0
$773$	19.7980			0.712083			0.356042	−	0.934470i	$-0.384126\pi$
$773$	19.7980			0.712083			0.356042	+	0.934470i	$0.384126\pi$
$774$		0			0
$775$	2.00000			0.0718421
$776$		0			0
$777$	−11.7980	−	28.8990i	−0.423249	−	1.03675i
$778$		0			0
$779$	−20.0000			−0.716574
$780$		0			0
$781$		−	6.20204i		−	0.221926i
$782$		0			0
$783$	−0.898979			−0.0321269
$784$		0			0
$785$	−12.6969			−0.453173
$786$		0			0
$787$			35.5959i			1.26886i	0.772981	+	0.634429i	$0.218765\pi$
$787$			35.5959i			1.26886i	−0.772981	+	0.634429i	$0.781235\pi$
$788$		0			0
$789$	7.79796			0.277615
$790$		0			0
$791$	−17.7980	−	43.5959i	−0.632823	−	1.55009i
$792$		0			0
$793$	−25.7980			−0.916112
$794$		0			0
$795$	−9.79796			−0.347498
$796$		0			0
$797$	−25.5959			−0.906654			−0.453327	−	0.891344i	$-0.649763\pi$
$797$	−25.5959			−0.906654			−0.453327	+	0.891344i	$0.649763\pi$
$798$		0			0
$799$		−	5.39388i		−	0.190822i
$800$		0			0
$801$			10.8990i			0.385097i
$802$		0			0
$803$		−	1.79796i		−	0.0634486i
$804$		0			0
$805$	14.6969	−	6.00000i	0.517999	−	0.211472i
$806$		0			0
$807$		−	3.79796i		−	0.133694i
$808$		0			0
$809$	23.7980			0.836692			0.418346	−	0.908288i	$-0.362610\pi$
$809$	23.7980			0.836692			0.418346	+	0.908288i	$0.362610\pi$
$810$		0			0
$811$			13.1010i			0.460039i	0.973186	+	0.230020i	$0.0738790\pi$
$811$			13.1010i			0.460039i	−0.973186	+	0.230020i	$0.926121\pi$
$812$		0			0
$813$			4.20204i			0.147372i
$814$		0			0
$815$	−6.69694			−0.234584
$816$		0			0
$817$			2.60612i			0.0911767i
$818$		0			0
$819$	2.89898	+	7.10102i	0.101299	+	0.248130i
$820$		0			0
$821$			10.6969i			0.373326i	0.982424	+	0.186663i	$0.0597672\pi$
$821$			10.6969i			0.373326i	−0.982424	+	0.186663i	$0.940233\pi$
$822$		0			0
$823$		−	40.4949i		−	1.41156i	−0.708429	−	0.705782i	$-0.750596\pi$
$823$		−	40.4949i		−	1.41156i	0.708429	−	0.705782i	$-0.249404\pi$
$824$		0			0
$825$			2.00000i			0.0696311i
$826$		0			0
$827$	−33.1918			−1.15419			−0.577097	−	0.816676i	$-0.695814\pi$
$827$	−33.1918			−1.15419			−0.577097	+	0.816676i	$0.695814\pi$
$828$		0			0
$829$	−9.30306			−0.323109			−0.161554	−	0.986864i	$-0.551651\pi$
$829$	−9.30306			−0.323109			−0.161554	+	0.986864i	$0.551651\pi$
$830$		0			0
$831$	−15.7980			−0.548025
$832$		0			0
$833$	24.0000	+	24.4949i	0.831551	+	0.848698i
$834$		0			0
$835$	−1.10102			−0.0381024
$836$		0			0
$837$			2.00000i			0.0691301i
$838$		0			0
$839$	−26.2020			−0.904595			−0.452297	−	0.891867i	$-0.649395\pi$
$839$	−26.2020			−0.904595			−0.452297	+	0.891867i	$0.649395\pi$
$840$		0			0
$841$	28.1918			0.972132
$842$		0			0
$843$		−	8.20204i		−	0.282493i
$844$		0			0
$845$	4.59592			0.158104
$846$		0			0
$847$	7.00000	+	17.1464i	0.240523	+	0.589158i
$848$		0			0
$849$	9.79796			0.336265
$850$		0			0
$851$	70.7878			2.42657
$852$		0			0
$853$	−44.6969			−1.53039			−0.765197	−	0.643796i	$-0.777358\pi$
$853$	−44.6969			−1.53039			−0.765197	+	0.643796i	$0.777358\pi$
$854$		0			0
$855$		−	2.89898i		−	0.0991430i
$856$		0			0
$857$		−	12.4949i		−	0.426818i	−0.976963	−	0.213409i	$-0.931543\pi$
$857$		−	12.4949i		−	0.426818i	0.976963	−	0.213409i	$-0.0684566\pi$
$858$		0			0
$859$		−	9.10102i		−	0.310523i	−0.987873	−	0.155261i	$-0.950378\pi$
$859$		−	9.10102i		−	0.310523i	0.987873	−	0.155261i	$-0.0496220\pi$
$860$		0			0
$861$	6.89898	+	16.8990i	0.235117	+	0.575916i
$862$		0			0
$863$		−	6.00000i		−	0.204242i	−0.994772	−	0.102121i	$-0.967437\pi$
$863$		−	6.00000i		−	0.204242i	0.994772	−	0.102121i	$-0.0325630\pi$
$864$		0			0
$865$	7.79796			0.265139
$866$		0			0
$867$			7.00000i			0.237732i
$868$		0			0
$869$			8.00000i			0.271381i
$870$		0			0
$871$	−14.2020			−0.481218
$872$		0			0
$873$		−	8.89898i		−	0.301185i
$874$		0			0
$875$	1.00000	+	2.44949i	0.0338062	+	0.0828079i
$876$		0			0
$877$		−	22.0000i		−	0.742887i	−0.928456	−	0.371444i	$-0.878863\pi$
$877$		−	22.0000i		−	0.742887i	0.928456	−	0.371444i	$-0.121137\pi$
$878$		0			0
$879$			21.5959i			0.728412i
$880$		0			0
$881$			0.696938i			0.0234805i	0.999931	+	0.0117402i	$0.00373712\pi$
$881$			0.696938i			0.0234805i	−0.999931	+	0.0117402i	$0.996263\pi$
$882$		0			0
$883$	40.4949			1.36276			0.681381	−	0.731929i	$-0.261380\pi$
$883$	40.4949			1.36276			0.681381	+	0.731929i	$0.261380\pi$
$884$		0			0
$885$	12.8990			0.433594
$886$		0			0
$887$	42.4949			1.42684			0.713420	−	0.700737i	$-0.247145\pi$
$887$	42.4949			1.42684			0.713420	+	0.700737i	$0.247145\pi$
$888$		0			0
$889$	21.7980	−	8.89898i	0.731080	−	0.298462i
$890$		0			0
$891$	−2.00000			−0.0670025
$892$		0			0
$893$		−	3.19184i		−	0.106811i
$894$		0			0
$895$	−10.0000			−0.334263
$896$		0			0
$897$	−17.3939			−0.580765
$898$		0			0
$899$			1.79796i			0.0599653i
$900$		0			0
$901$	48.0000			1.59911
$902$		0			0
$903$	2.20204	−	0.898979i	0.0732793	−	0.0299162i
$904$		0			0
$905$	−11.1010			−0.369010
$906$		0			0
$907$	36.4949			1.21179			0.605897	−	0.795543i	$-0.292815\pi$
$907$	36.4949			1.21179			0.605897	+	0.795543i	$0.292815\pi$
$908$		0			0
$909$	15.7980			0.523985
$910$		0			0
$911$			23.1010i			0.765371i	0.923879	+	0.382685i	$0.125001\pi$
$911$			23.1010i			0.765371i	−0.923879	+	0.382685i	$0.874999\pi$
$912$		0			0
$913$			8.00000i			0.264761i
$914$		0			0
$915$		−	8.89898i		−	0.294191i
$916$		0			0
$917$	41.3939	−	16.8990i	1.36695	−	0.558053i
$918$		0			0
$919$		−	25.3939i		−	0.837667i	−0.908063	−	0.418833i	$-0.862439\pi$
$919$		−	25.3939i		−	0.837667i	0.908063	−	0.418833i	$-0.137561\pi$
$920$		0			0
$921$	15.5959			0.513903
$922$		0			0
$923$			8.98979i			0.295903i
$924$		0			0
$925$			11.7980i			0.387915i
$926$		0			0
$927$	19.7980			0.650250
$928$		0			0
$929$		−	38.4949i		−	1.26298i	−0.775385	−	0.631488i	$-0.782444\pi$
$929$		−	38.4949i		−	1.26298i	0.775385	−	0.631488i	$-0.217556\pi$
$930$		0			0
$931$	14.2020	+	14.4949i	0.465453	+	0.475051i
$932$		0			0
$933$			8.00000i			0.261908i
$934$		0			0
$935$		−	9.79796i		−	0.320428i
$936$		0			0
$937$			36.4949i			1.19224i	0.802897	+	0.596118i	$0.203291\pi$
$937$			36.4949i			1.19224i	−0.802897	+	0.596118i	$0.796709\pi$
$938$		0			0
$939$	28.4949			0.929896
$940$		0			0
$941$	−15.7980			−0.514999			−0.257499	−	0.966278i	$-0.582899\pi$
$941$	−15.7980			−0.514999			−0.257499	+	0.966278i	$0.582899\pi$
$942$		0			0
$943$	−41.3939			−1.34797
$944$		0			0
$945$	−2.44949	+	1.00000i	−0.0796819	+	0.0325300i
$946$		0			0
$947$	−31.7980			−1.03329			−0.516647	−	0.856198i	$-0.672820\pi$
$947$	−31.7980			−1.03329			−0.516647	+	0.856198i	$0.672820\pi$
$948$		0			0
$949$			2.60612i			0.0845983i
$950$		0			0
$951$	−12.0000			−0.389127
$952$		0			0
$953$	24.0000			0.777436			0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000			0.777436			0.388718	+	0.921357i	$0.372918\pi$
$954$		0			0
$955$			16.8990i			0.546838i
$956$		0			0
$957$	−1.79796			−0.0581198
$958$		0			0
$959$	5.79796	+	14.2020i	0.187226	+	0.458608i
$960$		0			0
$961$	−27.0000			−0.870968
$962$		0			0
$963$	11.7980			0.380184
$964$		0			0
$965$	19.7980			0.637319
$966$		0			0
$967$			56.4949i			1.81675i	0.418153	+	0.908377i	$0.362678\pi$
$967$			56.4949i			1.81675i	−0.418153	+	0.908377i	$0.637322\pi$
$968$		0			0
$969$			14.2020i			0.456235i
$970$		0			0
$971$		−	3.10102i		−	0.0995165i	−0.998761	−	0.0497582i	$-0.984155\pi$
$971$		−	3.10102i		−	0.0995165i	0.998761	−	0.0497582i	$-0.0158451\pi$
$972$		0			0
$973$	−7.10102	+	2.89898i	−0.227648	+	0.0929370i
$974$		0			0
$975$		−	2.89898i		−	0.0928416i
$976$		0			0
$977$	23.1918			0.741973			0.370986	−	0.928638i	$-0.379020\pi$
$977$	23.1918			0.741973			0.370986	+	0.928638i	$0.379020\pi$
$978$		0			0
$979$			21.7980i			0.696666i
$980$		0			0
$981$		−	4.00000i		−	0.127710i
$982$		0			0
$983$	−2.89898			−0.0924631			−0.0462315	−	0.998931i	$-0.514721\pi$
$983$	−2.89898			−0.0924631			−0.0462315	+	0.998931i	$0.514721\pi$
$984$		0			0
$985$			5.79796i			0.184738i
$986$		0			0
$987$	−2.69694	+	1.10102i	−0.0858445	+	0.0350459i
$988$		0			0
$989$			5.39388i			0.171515i
$990$		0			0
$991$		−	20.0000i		−	0.635321i	−0.948205	−	0.317660i	$-0.897103\pi$
$991$		−	20.0000i		−	0.635321i	0.948205	−	0.317660i	$-0.102897\pi$
$992$		0			0
$993$			23.3939i			0.742382i
$994$		0			0
$995$	−10.0000			−0.317021
$996$		0			0
$997$	−16.2929			−0.516000			−0.258000	−	0.966145i	$-0.583063\pi$
$997$	−16.2929			−0.516000			−0.258000	+	0.966145i	$0.583063\pi$
$998$		0			0
$999$	−11.7980			−0.373271

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3360.2.z.a.1231.1		4
4.3	odd	2		840.2.z.a.811.2	✓	4
7.6	odd	2		3360.2.z.b.1231.3		4
8.3	odd	2		3360.2.z.b.1231.2		4
8.5	even	2		840.2.z.b.811.3	yes	4
28.27	even	2		840.2.z.b.811.2	yes	4
56.13	odd	2		840.2.z.a.811.3	yes	4
56.27	even	2	inner	3360.2.z.a.1231.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.z.a.811.2	✓	4	4.3	odd	2
840.2.z.a.811.3	yes	4	56.13	odd	2
840.2.z.b.811.2	yes	4	28.27	even	2
840.2.z.b.811.3	yes	4	8.5	even	2
3360.2.z.a.1231.1		4	1.1	even	1	trivial
3360.2.z.a.1231.4		4	56.27	even	2	inner
3360.2.z.b.1231.2		4	8.3	odd	2
3360.2.z.b.1231.3		4	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3360.z (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -1.00000 q^{5} +(-1.00000 - 2.44949i) q^{7} -1.00000 q^{9} -2.00000 q^{11} +2.89898 q^{13} +1.00000i q^{15} -4.89898i q^{17} -2.89898i q^{19} +(-2.44949 + 1.00000i) q^{21} -6.00000i q^{23} +1.00000 q^{25} +1.00000i q^{27} +0.898979i q^{29} +2.00000 q^{31} +2.00000i q^{33} +(1.00000 + 2.44949i) q^{35} +11.7980i q^{37} -2.89898i q^{39} -6.89898i q^{41} -0.898979 q^{43} +1.00000 q^{45} +1.10102 q^{47} +(-5.00000 + 4.89898i) q^{49} -4.89898 q^{51} +9.79796i q^{53} +2.00000 q^{55} -2.89898 q^{57} -12.8990i q^{59} -8.89898 q^{61} +(1.00000 + 2.44949i) q^{63} -2.89898 q^{65} -4.89898 q^{67} -6.00000 q^{69} +3.10102i q^{71} +0.898979i q^{73} -1.00000i q^{75} +(2.00000 + 4.89898i) q^{77} -4.00000i q^{79} +1.00000 q^{81} -4.00000i q^{83} +4.89898i q^{85} +0.898979 q^{87} -10.8990i q^{89} +(-2.89898 - 7.10102i) q^{91} -2.00000i q^{93} +2.89898i q^{95} +8.89898i q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} - 4 q^{7} - 4 q^{9} - 8 q^{11} - 8 q^{13} + 4 q^{25} + 8 q^{31} + 4 q^{35} + 16 q^{43} + 4 q^{45} + 24 q^{47} - 20 q^{49} + 8 q^{55} + 8 q^{57} - 16 q^{61} + 4 q^{63} + 8 q^{65} - 24 q^{69}+ \cdots + 8 q^{99}+O(q^{100}) \) 4 * q - 4 * q^5 - 4 * q^7 - 4 * q^9 - 8 * q^11 - 8 * q^13 + 4 * q^25 + 8 * q^31 + 4 * q^35 + 16 * q^43 + 4 * q^45 + 24 * q^47 - 20 * q^49 + 8 * q^55 + 8 * q^57 - 16 * q^61 + 4 * q^63 + 8 * q^65 - 24 * q^69 + 8 * q^77 + 4 * q^81 - 16 * q^87 + 8 * q^91 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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