LMFDB - Embedded newform 960.2.h.c.191.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [960,2,Mod(191,960)] 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("960.191"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(960, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	960.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$7.66563859404$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 240)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.4
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	960.191
Dual form			960.2.h.c.191.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 + 1.22474i) q^{3} +1.00000i q^{5} -2.44949i q^{7} +3.00000i q^{9} +4.89898 q^{11} +2.00000 q^{13} +(-1.22474 + 1.22474i) q^{15} -6.00000i q^{17} +4.89898i q^{19} +(3.00000 - 3.00000i) q^{21} +2.44949 q^{23} -1.00000 q^{25} +(-3.67423 + 3.67423i) q^{27} +9.79796i q^{31} +(6.00000 + 6.00000i) q^{33} +2.44949 q^{35} -2.00000 q^{37} +(2.44949 + 2.44949i) q^{39} -6.00000i q^{41} -2.44949i q^{43} -3.00000 q^{45} +12.2474 q^{47} +1.00000 q^{49} +(7.34847 - 7.34847i) q^{51} +6.00000i q^{53} +4.89898i q^{55} +(-6.00000 + 6.00000i) q^{57} -9.79796 q^{59} -8.00000 q^{61} +7.34847 q^{63} +2.00000i q^{65} +7.34847i q^{67} +(3.00000 + 3.00000i) q^{69} -4.89898 q^{71} -14.0000 q^{73} +(-1.22474 - 1.22474i) q^{75} -12.0000i q^{77} +4.89898i q^{79} -9.00000 q^{81} +7.34847 q^{83} +6.00000 q^{85} -12.0000i q^{89} -4.89898i q^{91} +(-12.0000 + 12.0000i) q^{93} -4.89898 q^{95} +10.0000 q^{97} +14.6969i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{13} + 12 q^{21} - 4 q^{25} + 24 q^{33} - 8 q^{37} - 12 q^{45} + 4 q^{49} - 24 q^{57} - 32 q^{61} + 12 q^{69} - 56 q^{73} - 36 q^{81} + 24 q^{85} - 48 q^{93} + 40 q^{97}+O(q^{100}) $ 4 * q + 8 * q^13 + 12 * q^21 - 4 * q^25 + 24 * q^33 - 8 * q^37 - 12 * q^45 + 4 * q^49 - 24 * q^57 - 32 * q^61 + 12 * q^69 - 56 * q^73 - 36 * q^81 + 24 * q^85 - 48 * q^93 + 40 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	1.22474	+	1.22474i	0.707107	+	0.707107i
$3$	1.22474	+	1.22474i	0.707107	+	0.707107i
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$		−	2.44949i		−	0.925820i	−0.886405	−	0.462910i	$-0.846805\pi$
$7$		−	2.44949i		−	0.925820i	0.886405	−	0.462910i	$-0.153195\pi$
$8$		0			0
$9$			3.00000i			1.00000i
$10$		0			0
$11$	4.89898			1.47710			0.738549	−	0.674200i	$-0.235511\pi$
$11$	4.89898			1.47710			0.738549	+	0.674200i	$0.235511\pi$
$12$		0			0
$13$	2.00000			0.554700			0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000			0.554700			0.277350	+	0.960769i	$0.410544\pi$
$14$		0			0
$15$	−1.22474	+	1.22474i	−0.316228	+	0.316228i
$16$		0			0
$17$		−	6.00000i		−	1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$		−	6.00000i		−	1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$		0			0
$19$			4.89898i			1.12390i	0.827170	+	0.561951i	$0.189949\pi$
$19$			4.89898i			1.12390i	−0.827170	+	0.561951i	$0.810051\pi$
$20$		0			0
$21$	3.00000	−	3.00000i	0.654654	−	0.654654i
$22$		0			0
$23$	2.44949			0.510754			0.255377	−	0.966842i	$-0.417800\pi$
$23$	2.44949			0.510754			0.255377	+	0.966842i	$0.417800\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−3.67423	+	3.67423i	−0.707107	+	0.707107i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$			9.79796i			1.75977i	0.475191	+	0.879883i	$0.342379\pi$
$31$			9.79796i			1.75977i	−0.475191	+	0.879883i	$0.657621\pi$
$32$		0			0
$33$	6.00000	+	6.00000i	1.04447	+	1.04447i
$34$		0			0
$35$	2.44949			0.414039
$36$		0			0
$37$	−2.00000			−0.328798			−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000			−0.328798			−0.164399	+	0.986394i	$0.552568\pi$
$38$		0			0
$39$	2.44949	+	2.44949i	0.392232	+	0.392232i
$40$		0			0
$41$		−	6.00000i		−	0.937043i	−0.883452	−	0.468521i	$-0.844787\pi$
$41$		−	6.00000i		−	0.937043i	0.883452	−	0.468521i	$-0.155213\pi$
$42$		0			0
$43$		−	2.44949i		−	0.373544i	−0.982403	−	0.186772i	$-0.940197\pi$
$43$		−	2.44949i		−	0.373544i	0.982403	−	0.186772i	$-0.0598025\pi$
$44$		0			0
$45$	−3.00000			−0.447214
$46$		0			0
$47$	12.2474			1.78647			0.893237	−	0.449586i	$-0.148429\pi$
$47$	12.2474			1.78647			0.893237	+	0.449586i	$0.148429\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$	7.34847	−	7.34847i	1.02899	−	1.02899i
$52$		0			0
$53$			6.00000i			0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$		0			0
$55$			4.89898i			0.660578i
$56$		0			0
$57$	−6.00000	+	6.00000i	−0.794719	+	0.794719i
$58$		0			0
$59$	−9.79796			−1.27559			−0.637793	−	0.770208i	$-0.720152\pi$
$59$	−9.79796			−1.27559			−0.637793	+	0.770208i	$0.720152\pi$
$60$		0			0
$61$	−8.00000			−1.02430			−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000			−1.02430			−0.512148	+	0.858898i	$0.671150\pi$
$62$		0			0
$63$	7.34847			0.925820
$64$		0			0
$65$			2.00000i			0.248069i
$66$		0			0
$67$			7.34847i			0.897758i	0.893592	+	0.448879i	$0.148177\pi$
$67$			7.34847i			0.897758i	−0.893592	+	0.448879i	$0.851823\pi$
$68$		0			0
$69$	3.00000	+	3.00000i	0.361158	+	0.361158i
$70$		0			0
$71$	−4.89898			−0.581402			−0.290701	−	0.956814i	$-0.593888\pi$
$71$	−4.89898			−0.581402			−0.290701	+	0.956814i	$0.593888\pi$
$72$		0			0
$73$	−14.0000			−1.63858			−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000			−1.63858			−0.819288	+	0.573382i	$0.805631\pi$
$74$		0			0
$75$	−1.22474	−	1.22474i	−0.141421	−	0.141421i
$76$		0			0
$77$		−	12.0000i		−	1.36753i
$78$		0			0
$79$			4.89898i			0.551178i	0.961276	+	0.275589i	$0.0888729\pi$
$79$			4.89898i			0.551178i	−0.961276	+	0.275589i	$0.911127\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$	7.34847			0.806599			0.403300	−	0.915068i	$-0.367863\pi$
$83$	7.34847			0.806599			0.403300	+	0.915068i	$0.367863\pi$
$84$		0			0
$85$	6.00000			0.650791
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	12.0000i		−	1.27200i	−0.771690	−	0.635999i	$-0.780588\pi$
$89$		−	12.0000i		−	1.27200i	0.771690	−	0.635999i	$-0.219412\pi$
$90$		0			0
$91$		−	4.89898i		−	0.513553i
$92$		0			0
$93$	−12.0000	+	12.0000i	−1.24434	+	1.24434i
$94$		0			0
$95$	−4.89898			−0.502625
$96$		0			0
$97$	10.0000			1.01535			0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000			1.01535			0.507673	+	0.861550i	$0.330506\pi$
$98$		0			0
$99$			14.6969i			1.47710i
$100$		0			0
$101$		−	12.0000i		−	1.19404i	−0.802225	−	0.597022i	$-0.796350\pi$
$101$		−	12.0000i		−	1.19404i	0.802225	−	0.597022i	$-0.203650\pi$
$102$		0			0
$103$		−	12.2474i		−	1.20678i	−0.797447	−	0.603388i	$-0.793817\pi$
$103$		−	12.2474i		−	1.20678i	0.797447	−	0.603388i	$-0.206183\pi$
$104$		0			0
$105$	3.00000	+	3.00000i	0.292770	+	0.292770i
$106$		0			0
$107$	2.44949			0.236801			0.118401	−	0.992966i	$-0.462223\pi$
$107$	2.44949			0.236801			0.118401	+	0.992966i	$0.462223\pi$
$108$		0			0
$109$	−4.00000			−0.383131			−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000			−0.383131			−0.191565	+	0.981480i	$0.561356\pi$
$110$		0			0
$111$	−2.44949	−	2.44949i	−0.232495	−	0.232495i
$112$		0			0
$113$			6.00000i			0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$			6.00000i			0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$		0			0
$115$			2.44949i			0.228416i
$116$		0			0
$117$			6.00000i			0.554700i
$118$		0			0
$119$	−14.6969			−1.34727
$120$		0			0
$121$	13.0000			1.18182
$122$		0			0
$123$	7.34847	−	7.34847i	0.662589	−	0.662589i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$		−	7.34847i		−	0.652071i	−0.945357	−	0.326036i	$-0.894287\pi$
$127$		−	7.34847i		−	0.652071i	0.945357	−	0.326036i	$-0.105713\pi$
$128$		0			0
$129$	3.00000	−	3.00000i	0.264135	−	0.264135i
$130$		0			0
$131$	4.89898			0.428026			0.214013	−	0.976831i	$-0.431347\pi$
$131$	4.89898			0.428026			0.214013	+	0.976831i	$0.431347\pi$
$132$		0			0
$133$	12.0000			1.04053
$134$		0			0
$135$	−3.67423	−	3.67423i	−0.316228	−	0.316228i
$136$		0			0
$137$			6.00000i			0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$			6.00000i			0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$		0			0
$139$			4.89898i			0.415526i	0.978179	+	0.207763i	$0.0666183\pi$
$139$			4.89898i			0.415526i	−0.978179	+	0.207763i	$0.933382\pi$
$140$		0			0
$141$	15.0000	+	15.0000i	1.26323	+	1.26323i
$142$		0			0
$143$	9.79796			0.819346
$144$		0			0
$145$		0			0
$146$		0			0
$147$	1.22474	+	1.22474i	0.101015	+	0.101015i
$148$		0			0
$149$		−	18.0000i		−	1.47462i	−0.675556	−	0.737309i	$-0.736096\pi$
$149$		−	18.0000i		−	1.47462i	0.675556	−	0.737309i	$-0.263904\pi$
$150$		0			0
$151$		−	19.5959i		−	1.59469i	−0.603522	−	0.797347i	$-0.706236\pi$
$151$		−	19.5959i		−	1.59469i	0.603522	−	0.797347i	$-0.293764\pi$
$152$		0			0
$153$	18.0000			1.45521
$154$		0			0
$155$	−9.79796			−0.786991
$156$		0			0
$157$	10.0000			0.798087			0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000			0.798087			0.399043	+	0.916932i	$0.369342\pi$
$158$		0			0
$159$	−7.34847	+	7.34847i	−0.582772	+	0.582772i
$160$		0			0
$161$		−	6.00000i		−	0.472866i
$162$		0			0
$163$		−	7.34847i		−	0.575577i	−0.957694	−	0.287788i	$-0.907080\pi$
$163$		−	7.34847i		−	0.575577i	0.957694	−	0.287788i	$-0.0929199\pi$
$164$		0			0
$165$	−6.00000	+	6.00000i	−0.467099	+	0.467099i
$166$		0			0
$167$	−17.1464			−1.32683			−0.663415	−	0.748251i	$-0.730894\pi$
$167$	−17.1464			−1.32683			−0.663415	+	0.748251i	$0.730894\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$	−14.6969			−1.12390
$172$		0			0
$173$		−	18.0000i		−	1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$		−	18.0000i		−	1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$		0			0
$175$			2.44949i			0.185164i
$176$		0			0
$177$	−12.0000	−	12.0000i	−0.901975	−	0.901975i
$178$		0			0
$179$	−9.79796			−0.732334			−0.366167	−	0.930549i	$-0.619330\pi$
$179$	−9.79796			−0.732334			−0.366167	+	0.930549i	$0.619330\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$		0			0
$183$	−9.79796	−	9.79796i	−0.724286	−	0.724286i
$184$		0			0
$185$		−	2.00000i		−	0.147043i
$186$		0			0
$187$		−	29.3939i		−	2.14949i
$188$		0			0
$189$	9.00000	+	9.00000i	0.654654	+	0.654654i
$190$		0			0
$191$	14.6969			1.06343			0.531717	−	0.846922i	$-0.321547\pi$
$191$	14.6969			1.06343			0.531717	+	0.846922i	$0.321547\pi$
$192$		0			0
$193$	−10.0000			−0.719816			−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000			−0.719816			−0.359908	+	0.932988i	$0.617192\pi$
$194$		0			0
$195$	−2.44949	+	2.44949i	−0.175412	+	0.175412i
$196$		0			0
$197$			18.0000i			1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$			18.0000i			1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$		0			0
$199$			14.6969i			1.04184i	0.853606	+	0.520919i	$0.174411\pi$
$199$			14.6969i			1.04184i	−0.853606	+	0.520919i	$0.825589\pi$
$200$		0			0
$201$	−9.00000	+	9.00000i	−0.634811	+	0.634811i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	6.00000			0.419058
$206$		0			0
$207$			7.34847i			0.510754i
$208$		0			0
$209$			24.0000i			1.66011i
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$	−6.00000	−	6.00000i	−0.411113	−	0.411113i
$214$		0			0
$215$	2.44949			0.167054
$216$		0			0
$217$	24.0000			1.62923
$218$		0			0
$219$	−17.1464	−	17.1464i	−1.15865	−	1.15865i
$220$		0			0
$221$		−	12.0000i		−	0.807207i
$222$		0			0
$223$			2.44949i			0.164030i	0.996631	+	0.0820150i	$0.0261355\pi$
$223$			2.44949i			0.164030i	−0.996631	+	0.0820150i	$0.973864\pi$
$224$		0			0
$225$		−	3.00000i		−	0.200000i
$226$		0			0
$227$	−2.44949			−0.162578			−0.0812892	−	0.996691i	$-0.525904\pi$
$227$	−2.44949			−0.162578			−0.0812892	+	0.996691i	$0.525904\pi$
$228$		0			0
$229$	−22.0000			−1.45380			−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000			−1.45380			−0.726900	+	0.686743i	$0.759040\pi$
$230$		0			0
$231$	14.6969	−	14.6969i	0.966988	−	0.966988i
$232$		0			0
$233$		−	18.0000i		−	1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$		−	18.0000i		−	1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$		0			0
$235$			12.2474i			0.798935i
$236$		0			0
$237$	−6.00000	+	6.00000i	−0.389742	+	0.389742i
$238$		0			0
$239$	−9.79796			−0.633777			−0.316889	−	0.948463i	$-0.602638\pi$
$239$	−9.79796			−0.633777			−0.316889	+	0.948463i	$0.602638\pi$
$240$		0			0
$241$	4.00000			0.257663			0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000			0.257663			0.128831	+	0.991667i	$0.458877\pi$
$242$		0			0
$243$	−11.0227	−	11.0227i	−0.707107	−	0.707107i
$244$		0			0
$245$			1.00000i			0.0638877i
$246$		0			0
$247$			9.79796i			0.623429i
$248$		0			0
$249$	9.00000	+	9.00000i	0.570352	+	0.570352i
$250$		0			0
$251$	−24.4949			−1.54610			−0.773052	−	0.634343i	$-0.781271\pi$
$251$	−24.4949			−1.54610			−0.773052	+	0.634343i	$0.781271\pi$
$252$		0			0
$253$	12.0000			0.754434
$254$		0			0
$255$	7.34847	+	7.34847i	0.460179	+	0.460179i
$256$		0			0
$257$		−	6.00000i		−	0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$		−	6.00000i		−	0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$		0			0
$259$			4.89898i			0.304408i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2.44949			0.151042			0.0755210	−	0.997144i	$-0.475938\pi$
$263$	2.44949			0.151042			0.0755210	+	0.997144i	$0.475938\pi$
$264$		0			0
$265$	−6.00000			−0.368577
$266$		0			0
$267$	14.6969	−	14.6969i	0.899438	−	0.899438i
$268$		0			0
$269$			18.0000i			1.09748i	0.835993	+	0.548740i	$0.184892\pi$
$269$			18.0000i			1.09748i	−0.835993	+	0.548740i	$0.815108\pi$
$270$		0			0
$271$		−	9.79796i		−	0.595184i	−0.954693	−	0.297592i	$-0.903817\pi$
$271$		−	9.79796i		−	0.595184i	0.954693	−	0.297592i	$-0.0961834\pi$
$272$		0			0
$273$	6.00000	−	6.00000i	0.363137	−	0.363137i
$274$		0			0
$275$	−4.89898			−0.295420
$276$		0			0
$277$	2.00000			0.120168			0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000			0.120168			0.0600842	+	0.998193i	$0.480863\pi$
$278$		0			0
$279$	−29.3939			−1.75977
$280$		0			0
$281$			6.00000i			0.357930i	0.983855	+	0.178965i	$0.0572749\pi$
$281$			6.00000i			0.357930i	−0.983855	+	0.178965i	$0.942725\pi$
$282$		0			0
$283$		−	26.9444i		−	1.60168i	−0.598880	−	0.800839i	$-0.704387\pi$
$283$		−	26.9444i		−	1.60168i	0.598880	−	0.800839i	$-0.295613\pi$
$284$		0			0
$285$	−6.00000	−	6.00000i	−0.355409	−	0.355409i
$286$		0			0
$287$	−14.6969			−0.867533
$288$		0			0
$289$	−19.0000			−1.11765
$290$		0			0
$291$	12.2474	+	12.2474i	0.717958	+	0.717958i
$292$		0			0
$293$		−	6.00000i		−	0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$		−	6.00000i		−	0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$		0			0
$295$		−	9.79796i		−	0.570459i
$296$		0			0
$297$	−18.0000	+	18.0000i	−1.04447	+	1.04447i
$298$		0			0
$299$	4.89898			0.283315
$300$		0			0
$301$	−6.00000			−0.345834
$302$		0			0
$303$	14.6969	−	14.6969i	0.844317	−	0.844317i
$304$		0			0
$305$		−	8.00000i		−	0.458079i
$306$		0			0
$307$			2.44949i			0.139800i	0.997554	+	0.0698999i	$0.0222680\pi$
$307$			2.44949i			0.139800i	−0.997554	+	0.0698999i	$0.977732\pi$
$308$		0			0
$309$	15.0000	−	15.0000i	0.853320	−	0.853320i
$310$		0			0
$311$	24.4949			1.38898			0.694489	−	0.719503i	$-0.255630\pi$
$311$	24.4949			1.38898			0.694489	+	0.719503i	$0.255630\pi$
$312$		0			0
$313$	−10.0000			−0.565233			−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000			−0.565233			−0.282617	+	0.959233i	$0.591202\pi$
$314$		0			0
$315$			7.34847i			0.414039i
$316$		0			0
$317$		−	18.0000i		−	1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$		−	18.0000i		−	1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	3.00000	+	3.00000i	0.167444	+	0.167444i
$322$		0			0
$323$	29.3939			1.63552
$324$		0			0
$325$	−2.00000			−0.110940
$326$		0			0
$327$	−4.89898	−	4.89898i	−0.270914	−	0.270914i
$328$		0			0
$329$		−	30.0000i		−	1.65395i
$330$		0			0
$331$			19.5959i			1.07709i	0.842597	+	0.538545i	$0.181026\pi$
$331$			19.5959i			1.07709i	−0.842597	+	0.538545i	$0.818974\pi$
$332$		0			0
$333$		−	6.00000i		−	0.328798i
$334$		0			0
$335$	−7.34847			−0.401490
$336$		0			0
$337$	−14.0000			−0.762629			−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000			−0.762629			−0.381314	+	0.924445i	$0.624528\pi$
$338$		0			0
$339$	−7.34847	+	7.34847i	−0.399114	+	0.399114i
$340$		0			0
$341$			48.0000i			2.59935i
$342$		0			0
$343$		−	19.5959i		−	1.05808i
$344$		0			0
$345$	−3.00000	+	3.00000i	−0.161515	+	0.161515i
$346$		0			0
$347$	7.34847			0.394486			0.197243	−	0.980355i	$-0.436801\pi$
$347$	7.34847			0.394486			0.197243	+	0.980355i	$0.436801\pi$
$348$		0			0
$349$	−2.00000			−0.107058			−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000			−0.107058			−0.0535288	+	0.998566i	$0.517047\pi$
$350$		0			0
$351$	−7.34847	+	7.34847i	−0.392232	+	0.392232i
$352$		0			0
$353$		−	6.00000i		−	0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$		−	6.00000i		−	0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$		0			0
$355$		−	4.89898i		−	0.260011i
$356$		0			0
$357$	−18.0000	−	18.0000i	−0.952661	−	0.952661i
$358$		0			0
$359$	−9.79796			−0.517116			−0.258558	−	0.965996i	$-0.583247\pi$
$359$	−9.79796			−0.517116			−0.258558	+	0.965996i	$0.583247\pi$
$360$		0			0
$361$	−5.00000			−0.263158
$362$		0			0
$363$	15.9217	+	15.9217i	0.835672	+	0.835672i
$364$		0			0
$365$		−	14.0000i		−	0.732793i
$366$		0			0
$367$			7.34847i			0.383587i	0.981435	+	0.191793i	$0.0614304\pi$
$367$			7.34847i			0.383587i	−0.981435	+	0.191793i	$0.938570\pi$
$368$		0			0
$369$	18.0000			0.937043
$370$		0			0
$371$	14.6969			0.763027
$372$		0			0
$373$	−26.0000			−1.34623			−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000			−1.34623			−0.673114	+	0.739538i	$0.735044\pi$
$374$		0			0
$375$	1.22474	−	1.22474i	0.0632456	−	0.0632456i
$376$		0			0
$377$		0			0
$378$		0			0
$379$			14.6969i			0.754931i	0.926024	+	0.377466i	$0.123204\pi$
$379$			14.6969i			0.754931i	−0.926024	+	0.377466i	$0.876796\pi$
$380$		0			0
$381$	9.00000	−	9.00000i	0.461084	−	0.461084i
$382$		0			0
$383$	−2.44949			−0.125163			−0.0625815	−	0.998040i	$-0.519933\pi$
$383$	−2.44949			−0.125163			−0.0625815	+	0.998040i	$0.519933\pi$
$384$		0			0
$385$	12.0000			0.611577
$386$		0			0
$387$	7.34847			0.373544
$388$		0			0
$389$			18.0000i			0.912636i	0.889817	+	0.456318i	$0.150832\pi$
$389$			18.0000i			0.912636i	−0.889817	+	0.456318i	$0.849168\pi$
$390$		0			0
$391$		−	14.6969i		−	0.743256i
$392$		0			0
$393$	6.00000	+	6.00000i	0.302660	+	0.302660i
$394$		0			0
$395$	−4.89898			−0.246494
$396$		0			0
$397$	−2.00000			−0.100377			−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000			−0.100377			−0.0501886	+	0.998740i	$0.515982\pi$
$398$		0			0
$399$	14.6969	+	14.6969i	0.735767	+	0.735767i
$400$		0			0
$401$			24.0000i			1.19850i	0.800561	+	0.599251i	$0.204535\pi$
$401$			24.0000i			1.19850i	−0.800561	+	0.599251i	$0.795465\pi$
$402$		0			0
$403$			19.5959i			0.976142i
$404$		0			0
$405$		−	9.00000i		−	0.447214i
$406$		0			0
$407$	−9.79796			−0.485667
$408$		0			0
$409$	16.0000			0.791149			0.395575	−	0.918434i	$-0.370545\pi$
$409$	16.0000			0.791149			0.395575	+	0.918434i	$0.370545\pi$
$410$		0			0
$411$	−7.34847	+	7.34847i	−0.362473	+	0.362473i
$412$		0			0
$413$			24.0000i			1.18096i
$414$		0			0
$415$			7.34847i			0.360722i
$416$		0			0
$417$	−6.00000	+	6.00000i	−0.293821	+	0.293821i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	8.00000			0.389896			0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000			0.389896			0.194948	+	0.980814i	$0.437546\pi$
$422$		0			0
$423$			36.7423i			1.78647i
$424$		0			0
$425$			6.00000i			0.291043i
$426$		0			0
$427$			19.5959i			0.948313i
$428$		0			0
$429$	12.0000	+	12.0000i	0.579365	+	0.579365i
$430$		0			0
$431$	−24.4949			−1.17988			−0.589939	−	0.807448i	$-0.700848\pi$
$431$	−24.4949			−1.17988			−0.589939	+	0.807448i	$0.700848\pi$
$432$		0			0
$433$	2.00000			0.0961139			0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000			0.0961139			0.0480569	+	0.998845i	$0.484697\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			12.0000i			0.574038i
$438$		0			0
$439$			24.4949i			1.16908i	0.811366	+	0.584539i	$0.198725\pi$
$439$			24.4949i			1.16908i	−0.811366	+	0.584539i	$0.801275\pi$
$440$		0			0
$441$			3.00000i			0.142857i
$442$		0			0
$443$	−17.1464			−0.814651			−0.407326	−	0.913283i	$-0.633539\pi$
$443$	−17.1464			−0.814651			−0.407326	+	0.913283i	$0.633539\pi$
$444$		0			0
$445$	12.0000			0.568855
$446$		0			0
$447$	22.0454	−	22.0454i	1.04271	−	1.04271i
$448$		0			0
$449$			6.00000i			0.283158i	0.989927	+	0.141579i	$0.0452178\pi$
$449$			6.00000i			0.283158i	−0.989927	+	0.141579i	$0.954782\pi$
$450$		0			0
$451$		−	29.3939i		−	1.38410i
$452$		0			0
$453$	24.0000	−	24.0000i	1.12762	−	1.12762i
$454$		0			0
$455$	4.89898			0.229668
$456$		0			0
$457$	22.0000			1.02912			0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000			1.02912			0.514558	+	0.857455i	$0.327956\pi$
$458$		0			0
$459$	22.0454	+	22.0454i	1.02899	+	1.02899i
$460$		0			0
$461$			12.0000i			0.558896i	0.960161	+	0.279448i	$0.0901514\pi$
$461$			12.0000i			0.558896i	−0.960161	+	0.279448i	$0.909849\pi$
$462$		0			0
$463$			2.44949i			0.113837i	0.998379	+	0.0569187i	$0.0181276\pi$
$463$			2.44949i			0.113837i	−0.998379	+	0.0569187i	$0.981872\pi$
$464$		0			0
$465$	−12.0000	−	12.0000i	−0.556487	−	0.556487i
$466$		0			0
$467$	−17.1464			−0.793442			−0.396721	−	0.917939i	$-0.629852\pi$
$467$	−17.1464			−0.793442			−0.396721	+	0.917939i	$0.629852\pi$
$468$		0			0
$469$	18.0000			0.831163
$470$		0			0
$471$	12.2474	+	12.2474i	0.564333	+	0.564333i
$472$		0			0
$473$		−	12.0000i		−	0.551761i
$474$		0			0
$475$		−	4.89898i		−	0.224781i
$476$		0			0
$477$	−18.0000			−0.824163
$478$		0			0
$479$	29.3939			1.34304			0.671520	−	0.740986i	$-0.265642\pi$
$479$	29.3939			1.34304			0.671520	+	0.740986i	$0.265642\pi$
$480$		0			0
$481$	−4.00000			−0.182384
$482$		0			0
$483$	7.34847	−	7.34847i	0.334367	−	0.334367i
$484$		0			0
$485$			10.0000i			0.454077i
$486$		0			0
$487$			22.0454i			0.998973i	0.866322	+	0.499486i	$0.166478\pi$
$487$			22.0454i			0.998973i	−0.866322	+	0.499486i	$0.833522\pi$
$488$		0			0
$489$	9.00000	−	9.00000i	0.406994	−	0.406994i
$490$		0			0
$491$	14.6969			0.663264			0.331632	−	0.943409i	$-0.392401\pi$
$491$	14.6969			0.663264			0.331632	+	0.943409i	$0.392401\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−14.6969			−0.660578
$496$		0			0
$497$			12.0000i			0.538274i
$498$		0			0
$499$		−	4.89898i		−	0.219308i	−0.993970	−	0.109654i	$-0.965026\pi$
$499$		−	4.89898i		−	0.219308i	0.993970	−	0.109654i	$-0.0349744\pi$
$500$		0			0
$501$	−21.0000	−	21.0000i	−0.938211	−	0.938211i
$502$		0			0
$503$	36.7423			1.63826			0.819130	−	0.573608i	$-0.194457\pi$
$503$	36.7423			1.63826			0.819130	+	0.573608i	$0.194457\pi$
$504$		0			0
$505$	12.0000			0.533993
$506$		0			0
$507$	−11.0227	−	11.0227i	−0.489535	−	0.489535i
$508$		0			0
$509$		−	24.0000i		−	1.06378i	−0.846813	−	0.531891i	$-0.821482\pi$
$509$		−	24.0000i		−	1.06378i	0.846813	−	0.531891i	$-0.178518\pi$
$510$		0			0
$511$			34.2929i			1.51703i
$512$		0			0
$513$	−18.0000	−	18.0000i	−0.794719	−	0.794719i
$514$		0			0
$515$	12.2474			0.539687
$516$		0			0
$517$	60.0000			2.63880
$518$		0			0
$519$	22.0454	−	22.0454i	0.967686	−	0.967686i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		−	2.44949i		−	0.107109i	−0.998565	−	0.0535544i	$-0.982945\pi$
$523$		−	2.44949i		−	0.107109i	0.998565	−	0.0535544i	$-0.0170550\pi$
$524$		0			0
$525$	−3.00000	+	3.00000i	−0.130931	+	0.130931i
$526$		0			0
$527$	58.7878			2.56083
$528$		0			0
$529$	−17.0000			−0.739130
$530$		0			0
$531$		−	29.3939i		−	1.27559i
$532$		0			0
$533$		−	12.0000i		−	0.519778i
$534$		0			0
$535$			2.44949i			0.105901i
$536$		0			0
$537$	−12.0000	−	12.0000i	−0.517838	−	0.517838i
$538$		0			0
$539$	4.89898			0.211014
$540$		0			0
$541$	2.00000			0.0859867			0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000			0.0859867			0.0429934	+	0.999075i	$0.486311\pi$
$542$		0			0
$543$	−12.2474	−	12.2474i	−0.525588	−	0.525588i
$544$		0			0
$545$		−	4.00000i		−	0.171341i
$546$		0			0
$547$			41.6413i			1.78045i	0.455517	+	0.890227i	$0.349455\pi$
$547$			41.6413i			1.78045i	−0.455517	+	0.890227i	$0.650545\pi$
$548$		0			0
$549$		−	24.0000i		−	1.02430i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	12.0000			0.510292
$554$		0			0
$555$	2.44949	−	2.44949i	0.103975	−	0.103975i
$556$		0			0
$557$			18.0000i			0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$			18.0000i			0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$		0			0
$559$		−	4.89898i		−	0.207205i
$560$		0			0
$561$	36.0000	−	36.0000i	1.51992	−	1.51992i
$562$		0			0
$563$	−12.2474			−0.516168			−0.258084	−	0.966122i	$-0.583091\pi$
$563$	−12.2474			−0.516168			−0.258084	+	0.966122i	$0.583091\pi$
$564$		0			0
$565$	−6.00000			−0.252422
$566$		0			0
$567$			22.0454i			0.925820i
$568$		0			0
$569$			30.0000i			1.25767i	0.777541	+	0.628833i	$0.216467\pi$
$569$			30.0000i			1.25767i	−0.777541	+	0.628833i	$0.783533\pi$
$570$		0			0
$571$		−	9.79796i		−	0.410032i	−0.978759	−	0.205016i	$-0.934275\pi$
$571$		−	9.79796i		−	0.410032i	0.978759	−	0.205016i	$-0.0657246\pi$
$572$		0			0
$573$	18.0000	+	18.0000i	0.751961	+	0.751961i
$574$		0			0
$575$	−2.44949			−0.102151
$576$		0			0
$577$	38.0000			1.58196			0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000			1.58196			0.790980	+	0.611842i	$0.209571\pi$
$578$		0			0
$579$	−12.2474	−	12.2474i	−0.508987	−	0.508987i
$580$		0			0
$581$		−	18.0000i		−	0.746766i
$582$		0			0
$583$			29.3939i			1.21737i
$584$		0			0
$585$	−6.00000			−0.248069
$586$		0			0
$587$	36.7423			1.51652			0.758259	−	0.651953i	$-0.226050\pi$
$587$	36.7423			1.51652			0.758259	+	0.651953i	$0.226050\pi$
$588$		0			0
$589$	−48.0000			−1.97781
$590$		0			0
$591$	−22.0454	+	22.0454i	−0.906827	+	0.906827i
$592$		0			0
$593$			42.0000i			1.72473i	0.506284	+	0.862367i	$0.331019\pi$
$593$			42.0000i			1.72473i	−0.506284	+	0.862367i	$0.668981\pi$
$594$		0			0
$595$		−	14.6969i		−	0.602516i
$596$		0			0
$597$	−18.0000	+	18.0000i	−0.736691	+	0.736691i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−28.0000			−1.14214			−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000			−1.14214			−0.571072	+	0.820900i	$0.693472\pi$
$602$		0			0
$603$	−22.0454			−0.897758
$604$		0			0
$605$			13.0000i			0.528525i
$606$		0			0
$607$		−	26.9444i		−	1.09364i	−0.837251	−	0.546819i	$-0.815838\pi$
$607$		−	26.9444i		−	1.09364i	0.837251	−	0.546819i	$-0.184162\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	24.4949			0.990957
$612$		0			0
$613$	−38.0000			−1.53481			−0.767403	−	0.641165i	$-0.778451\pi$
$613$	−38.0000			−1.53481			−0.767403	+	0.641165i	$0.778451\pi$
$614$		0			0
$615$	7.34847	+	7.34847i	0.296319	+	0.296319i
$616$		0			0
$617$		−	6.00000i		−	0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$		−	6.00000i		−	0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$		0			0
$619$			4.89898i			0.196907i	0.995142	+	0.0984533i	$0.0313895\pi$
$619$			4.89898i			0.196907i	−0.995142	+	0.0984533i	$0.968610\pi$
$620$		0			0
$621$	−9.00000	+	9.00000i	−0.361158	+	0.361158i
$622$		0			0
$623$	−29.3939			−1.17764
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−29.3939	+	29.3939i	−1.17388	+	1.17388i
$628$		0			0
$629$			12.0000i			0.478471i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	7.34847			0.291615
$636$		0			0
$637$	2.00000			0.0792429
$638$		0			0
$639$		−	14.6969i		−	0.581402i
$640$		0			0
$641$			18.0000i			0.710957i	0.934684	+	0.355479i	$0.115682\pi$
$641$			18.0000i			0.710957i	−0.934684	+	0.355479i	$0.884318\pi$
$642$		0			0
$643$			36.7423i			1.44898i	0.689287	+	0.724488i	$0.257924\pi$
$643$			36.7423i			1.44898i	−0.689287	+	0.724488i	$0.742076\pi$
$644$		0			0
$645$	3.00000	+	3.00000i	0.118125	+	0.118125i
$646$		0			0
$647$	26.9444			1.05929			0.529647	−	0.848218i	$-0.322325\pi$
$647$	26.9444			1.05929			0.529647	+	0.848218i	$0.322325\pi$
$648$		0			0
$649$	−48.0000			−1.88416
$650$		0			0
$651$	29.3939	+	29.3939i	1.15204	+	1.15204i
$652$		0			0
$653$		−	18.0000i		−	0.704394i	−0.935926	−	0.352197i	$-0.885435\pi$
$653$		−	18.0000i		−	0.704394i	0.935926	−	0.352197i	$-0.114565\pi$
$654$		0			0
$655$			4.89898i			0.191419i
$656$		0			0
$657$		−	42.0000i		−	1.63858i
$658$		0			0
$659$	−29.3939			−1.14502			−0.572511	−	0.819897i	$-0.694031\pi$
$659$	−29.3939			−1.14502			−0.572511	+	0.819897i	$0.694031\pi$
$660$		0			0
$661$	40.0000			1.55582			0.777910	−	0.628376i	$-0.216280\pi$
$661$	40.0000			1.55582			0.777910	+	0.628376i	$0.216280\pi$
$662$		0			0
$663$	14.6969	−	14.6969i	0.570782	−	0.570782i
$664$		0			0
$665$			12.0000i			0.465340i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−3.00000	+	3.00000i	−0.115987	+	0.115987i
$670$		0			0
$671$	−39.1918			−1.51298
$672$		0			0
$673$	26.0000			1.00223			0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000			1.00223			0.501113	+	0.865382i	$0.332924\pi$
$674$		0			0
$675$	3.67423	−	3.67423i	0.141421	−	0.141421i
$676$		0			0
$677$			18.0000i			0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$			18.0000i			0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$		0			0
$679$		−	24.4949i		−	0.940028i
$680$		0			0
$681$	−3.00000	−	3.00000i	−0.114960	−	0.114960i
$682$		0			0
$683$	31.8434			1.21845			0.609226	−	0.792996i	$-0.291480\pi$
$683$	31.8434			1.21845			0.609226	+	0.792996i	$0.291480\pi$
$684$		0			0
$685$	−6.00000			−0.229248
$686$		0			0
$687$	−26.9444	−	26.9444i	−1.02799	−	1.02799i
$688$		0			0
$689$			12.0000i			0.457164i
$690$		0			0
$691$		−	29.3939i		−	1.11820i	−0.829102	−	0.559098i	$-0.811148\pi$
$691$		−	29.3939i		−	1.11820i	0.829102	−	0.559098i	$-0.188852\pi$
$692$		0			0
$693$	36.0000			1.36753
$694$		0			0
$695$	−4.89898			−0.185829
$696$		0			0
$697$	−36.0000			−1.36360
$698$		0			0
$699$	22.0454	−	22.0454i	0.833834	−	0.833834i
$700$		0			0
$701$		−	30.0000i		−	1.13308i	−0.824033	−	0.566542i	$-0.808281\pi$
$701$		−	30.0000i		−	1.13308i	0.824033	−	0.566542i	$-0.191719\pi$
$702$		0			0
$703$		−	9.79796i		−	0.369537i
$704$		0			0
$705$	−15.0000	+	15.0000i	−0.564933	+	0.564933i
$706$		0			0
$707$	−29.3939			−1.10547
$708$		0			0
$709$	−10.0000			−0.375558			−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000			−0.375558			−0.187779	+	0.982211i	$0.560129\pi$
$710$		0			0
$711$	−14.6969			−0.551178
$712$		0			0
$713$			24.0000i			0.898807i
$714$		0			0
$715$			9.79796i			0.366423i
$716$		0			0
$717$	−12.0000	−	12.0000i	−0.448148	−	0.448148i
$718$		0			0
$719$	−9.79796			−0.365402			−0.182701	−	0.983169i	$-0.558484\pi$
$719$	−9.79796			−0.365402			−0.182701	+	0.983169i	$0.558484\pi$
$720$		0			0
$721$	−30.0000			−1.11726
$722$		0			0
$723$	4.89898	+	4.89898i	0.182195	+	0.182195i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	26.9444i		−	0.999312i	−0.866224	−	0.499656i	$-0.833460\pi$
$727$		−	26.9444i		−	0.999312i	0.866224	−	0.499656i	$-0.166540\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$		0			0
$731$	−14.6969			−0.543586
$732$		0			0
$733$	−34.0000			−1.25582			−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000			−1.25582			−0.627909	+	0.778287i	$0.716089\pi$
$734$		0			0
$735$	−1.22474	+	1.22474i	−0.0451754	+	0.0451754i
$736$		0			0
$737$			36.0000i			1.32608i
$738$		0			0
$739$			14.6969i			0.540636i	0.962771	+	0.270318i	$0.0871288\pi$
$739$			14.6969i			0.540636i	−0.962771	+	0.270318i	$0.912871\pi$
$740$		0			0
$741$	−12.0000	+	12.0000i	−0.440831	+	0.440831i
$742$		0			0
$743$	−41.6413			−1.52767			−0.763836	−	0.645410i	$-0.776686\pi$
$743$	−41.6413			−1.52767			−0.763836	+	0.645410i	$0.776686\pi$
$744$		0			0
$745$	18.0000			0.659469
$746$		0			0
$747$			22.0454i			0.806599i
$748$		0			0
$749$		−	6.00000i		−	0.219235i
$750$		0			0
$751$			19.5959i			0.715065i	0.933901	+	0.357533i	$0.116382\pi$
$751$			19.5959i			0.715065i	−0.933901	+	0.357533i	$0.883618\pi$
$752$		0			0
$753$	−30.0000	−	30.0000i	−1.09326	−	1.09326i
$754$		0			0
$755$	19.5959			0.713168
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$		0			0
$759$	14.6969	+	14.6969i	0.533465	+	0.533465i
$760$		0			0
$761$		−	48.0000i		−	1.74000i	−0.493053	−	0.869999i	$-0.664119\pi$
$761$		−	48.0000i		−	1.74000i	0.493053	−	0.869999i	$-0.335881\pi$
$762$		0			0
$763$			9.79796i			0.354710i
$764$		0			0
$765$			18.0000i			0.650791i
$766$		0			0
$767$	−19.5959			−0.707568
$768$		0			0
$769$	14.0000			0.504853			0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000			0.504853			0.252426	+	0.967616i	$0.418771\pi$
$770$		0			0
$771$	7.34847	−	7.34847i	0.264649	−	0.264649i
$772$		0			0
$773$			42.0000i			1.51064i	0.655359	+	0.755318i	$0.272517\pi$
$773$			42.0000i			1.51064i	−0.655359	+	0.755318i	$0.727483\pi$
$774$		0			0
$775$		−	9.79796i		−	0.351953i
$776$		0			0
$777$	−6.00000	+	6.00000i	−0.215249	+	0.215249i
$778$		0			0
$779$	29.3939			1.05314
$780$		0			0
$781$	−24.0000			−0.858788
$782$		0			0
$783$		0			0
$784$		0			0
$785$			10.0000i			0.356915i
$786$		0			0
$787$		−	46.5403i		−	1.65898i	−0.558520	−	0.829491i	$-0.688630\pi$
$787$		−	46.5403i		−	1.65898i	0.558520	−	0.829491i	$-0.311370\pi$
$788$		0			0
$789$	3.00000	+	3.00000i	0.106803	+	0.106803i
$790$		0			0
$791$	14.6969			0.522563
$792$		0			0
$793$	−16.0000			−0.568177
$794$		0			0
$795$	−7.34847	−	7.34847i	−0.260623	−	0.260623i
$796$		0			0
$797$		−	42.0000i		−	1.48772i	−0.668338	−	0.743858i	$-0.732994\pi$
$797$		−	42.0000i		−	1.48772i	0.668338	−	0.743858i	$-0.267006\pi$
$798$		0			0
$799$		−	73.4847i		−	2.59970i
$800$		0			0
$801$	36.0000			1.27200
$802$		0			0
$803$	−68.5857			−2.42034
$804$		0			0
$805$	6.00000			0.211472
$806$		0			0
$807$	−22.0454	+	22.0454i	−0.776035	+	0.776035i
$808$		0			0
$809$			12.0000i			0.421898i	0.977497	+	0.210949i	$0.0676553\pi$
$809$			12.0000i			0.421898i	−0.977497	+	0.210949i	$0.932345\pi$
$810$		0			0
$811$			39.1918i			1.37621i	0.725610	+	0.688106i	$0.241557\pi$
$811$			39.1918i			1.37621i	−0.725610	+	0.688106i	$0.758443\pi$
$812$		0			0
$813$	12.0000	−	12.0000i	0.420858	−	0.420858i
$814$		0			0
$815$	7.34847			0.257406
$816$		0			0
$817$	12.0000			0.419827
$818$		0			0
$819$	14.6969			0.513553
$820$		0			0
$821$		−	6.00000i		−	0.209401i	−0.994504	−	0.104701i	$-0.966612\pi$
$821$		−	6.00000i		−	0.209401i	0.994504	−	0.104701i	$-0.0333885\pi$
$822$		0			0
$823$			36.7423i			1.28076i	0.768059	+	0.640379i	$0.221223\pi$
$823$			36.7423i			1.28076i	−0.768059	+	0.640379i	$0.778777\pi$
$824$		0			0
$825$	−6.00000	−	6.00000i	−0.208893	−	0.208893i
$826$		0			0
$827$	−17.1464			−0.596240			−0.298120	−	0.954528i	$-0.596360\pi$
$827$	−17.1464			−0.596240			−0.298120	+	0.954528i	$0.596360\pi$
$828$		0			0
$829$	20.0000			0.694629			0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000			0.694629			0.347314	+	0.937749i	$0.387094\pi$
$830$		0			0
$831$	2.44949	+	2.44949i	0.0849719	+	0.0849719i
$832$		0			0
$833$		−	6.00000i		−	0.207888i
$834$		0			0
$835$		−	17.1464i		−	0.593377i
$836$		0			0
$837$	−36.0000	−	36.0000i	−1.24434	−	1.24434i
$838$		0			0
$839$	19.5959			0.676526			0.338263	−	0.941052i	$-0.390161\pi$
$839$	19.5959			0.676526			0.338263	+	0.941052i	$0.390161\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$	−7.34847	+	7.34847i	−0.253095	+	0.253095i
$844$		0			0
$845$		−	9.00000i		−	0.309609i
$846$		0			0
$847$		−	31.8434i		−	1.09415i
$848$		0			0
$849$	33.0000	−	33.0000i	1.13256	−	1.13256i
$850$		0			0
$851$	−4.89898			−0.167935
$852$		0			0
$853$	−14.0000			−0.479351			−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000			−0.479351			−0.239675	+	0.970853i	$0.577041\pi$
$854$		0			0
$855$		−	14.6969i		−	0.502625i
$856$		0			0
$857$			6.00000i			0.204956i	0.994735	+	0.102478i	$0.0326771\pi$
$857$			6.00000i			0.204956i	−0.994735	+	0.102478i	$0.967323\pi$
$858$		0			0
$859$		−	53.8888i		−	1.83866i	−0.393486	−	0.919331i	$-0.628731\pi$
$859$		−	53.8888i		−	1.83866i	0.393486	−	0.919331i	$-0.371269\pi$
$860$		0			0
$861$	−18.0000	−	18.0000i	−0.613438	−	0.613438i
$862$		0			0
$863$	7.34847			0.250145			0.125072	−	0.992148i	$-0.460084\pi$
$863$	7.34847			0.250145			0.125072	+	0.992148i	$0.460084\pi$
$864$		0			0
$865$	18.0000			0.612018
$866$		0			0
$867$	−23.2702	−	23.2702i	−0.790296	−	0.790296i
$868$		0			0
$869$			24.0000i			0.814144i
$870$		0			0
$871$			14.6969i			0.497987i
$872$		0			0
$873$			30.0000i			1.01535i
$874$		0			0
$875$	−2.44949			−0.0828079
$876$		0			0
$877$	58.0000			1.95852			0.979260	−	0.202606i	$-0.0649409\pi$
$877$	58.0000			1.95852			0.979260	+	0.202606i	$0.0649409\pi$
$878$		0			0
$879$	7.34847	−	7.34847i	0.247858	−	0.247858i
$880$		0			0
$881$			18.0000i			0.606435i	0.952921	+	0.303218i	$0.0980609\pi$
$881$			18.0000i			0.606435i	−0.952921	+	0.303218i	$0.901939\pi$
$882$		0			0
$883$		−	56.3383i		−	1.89593i	−0.318369	−	0.947967i	$-0.603135\pi$
$883$		−	56.3383i		−	1.89593i	0.318369	−	0.947967i	$-0.396865\pi$
$884$		0			0
$885$	12.0000	−	12.0000i	0.403376	−	0.403376i
$886$		0			0
$887$	−41.6413			−1.39818			−0.699089	−	0.715034i	$-0.746411\pi$
$887$	−41.6413			−1.39818			−0.699089	+	0.715034i	$0.746411\pi$
$888$		0			0
$889$	−18.0000			−0.603701
$890$		0			0
$891$	−44.0908			−1.47710
$892$		0			0
$893$			60.0000i			2.00782i
$894$		0			0
$895$		−	9.79796i		−	0.327510i
$896$		0			0
$897$	6.00000	+	6.00000i	0.200334	+	0.200334i
$898$		0			0
$899$		0			0
$900$		0			0
$901$	36.0000			1.19933
$902$		0			0
$903$	−7.34847	−	7.34847i	−0.244542	−	0.244542i
$904$		0			0
$905$		−	10.0000i		−	0.332411i
$906$		0			0
$907$		−	12.2474i		−	0.406670i	−0.979109	−	0.203335i	$-0.934822\pi$
$907$		−	12.2474i		−	0.406670i	0.979109	−	0.203335i	$-0.0651780\pi$
$908$		0			0
$909$	36.0000			1.19404
$910$		0			0
$911$	−24.4949			−0.811552			−0.405776	−	0.913973i	$-0.632999\pi$
$911$	−24.4949			−0.811552			−0.405776	+	0.913973i	$0.632999\pi$
$912$		0			0
$913$	36.0000			1.19143
$914$		0			0
$915$	9.79796	−	9.79796i	0.323911	−	0.323911i
$916$		0			0
$917$		−	12.0000i		−	0.396275i
$918$		0			0
$919$			34.2929i			1.13122i	0.824674	+	0.565608i	$0.191359\pi$
$919$			34.2929i			1.13122i	−0.824674	+	0.565608i	$0.808641\pi$
$920$		0			0
$921$	−3.00000	+	3.00000i	−0.0988534	+	0.0988534i
$922$		0			0
$923$	−9.79796			−0.322504
$924$		0			0
$925$	2.00000			0.0657596
$926$		0			0
$927$	36.7423			1.20678
$928$		0			0
$929$		−	54.0000i		−	1.77168i	−0.463988	−	0.885841i	$-0.653582\pi$
$929$		−	54.0000i		−	1.77168i	0.463988	−	0.885841i	$-0.346418\pi$
$930$		0			0
$931$			4.89898i			0.160558i
$932$		0			0
$933$	30.0000	+	30.0000i	0.982156	+	0.982156i
$934$		0			0
$935$	29.3939			0.961283
$936$		0			0
$937$	38.0000			1.24141			0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000			1.24141			0.620703	+	0.784046i	$0.286847\pi$
$938$		0			0
$939$	−12.2474	−	12.2474i	−0.399680	−	0.399680i
$940$		0			0
$941$			12.0000i			0.391189i	0.980685	+	0.195594i	$0.0626636\pi$
$941$			12.0000i			0.391189i	−0.980685	+	0.195594i	$0.937336\pi$
$942$		0			0
$943$		−	14.6969i		−	0.478598i
$944$		0			0
$945$	−9.00000	+	9.00000i	−0.292770	+	0.292770i
$946$		0			0
$947$	−22.0454			−0.716379			−0.358190	−	0.933649i	$-0.616606\pi$
$947$	−22.0454			−0.716379			−0.358190	+	0.933649i	$0.616606\pi$
$948$		0			0
$949$	−28.0000			−0.908918
$950$		0			0
$951$	22.0454	−	22.0454i	0.714871	−	0.714871i
$952$		0			0
$953$			30.0000i			0.971795i	0.874016	+	0.485898i	$0.161507\pi$
$953$			30.0000i			0.971795i	−0.874016	+	0.485898i	$0.838493\pi$
$954$		0			0
$955$			14.6969i			0.475582i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	14.6969			0.474589
$960$		0			0
$961$	−65.0000			−2.09677
$962$		0			0
$963$			7.34847i			0.236801i
$964$		0			0
$965$		−	10.0000i		−	0.321911i
$966$		0			0
$967$			56.3383i			1.81172i	0.423581	+	0.905858i	$0.360773\pi$
$967$			56.3383i			1.81172i	−0.423581	+	0.905858i	$0.639227\pi$
$968$		0			0
$969$	36.0000	+	36.0000i	1.15649	+	1.15649i
$970$		0			0
$971$	−53.8888			−1.72937			−0.864687	−	0.502312i	$-0.832483\pi$
$971$	−53.8888			−1.72937			−0.864687	+	0.502312i	$0.832483\pi$
$972$		0			0
$973$	12.0000			0.384702
$974$		0			0
$975$	−2.44949	−	2.44949i	−0.0784465	−	0.0784465i
$976$		0			0
$977$		−	18.0000i		−	0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$		−	18.0000i		−	0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$		0			0
$979$		−	58.7878i		−	1.87886i
$980$		0			0
$981$		−	12.0000i		−	0.383131i
$982$		0			0
$983$	−31.8434			−1.01565			−0.507823	−	0.861462i	$-0.669550\pi$
$983$	−31.8434			−1.01565			−0.507823	+	0.861462i	$0.669550\pi$
$984$		0			0
$985$	−18.0000			−0.573528
$986$		0			0
$987$	36.7423	−	36.7423i	1.16952	−	1.16952i
$988$		0			0
$989$		−	6.00000i		−	0.190789i
$990$		0			0
$991$		−	9.79796i		−	0.311242i	−0.987817	−	0.155621i	$-0.950262\pi$
$991$		−	9.79796i		−	0.311242i	0.987817	−	0.155621i	$-0.0497379\pi$
$992$		0			0
$993$	−24.0000	+	24.0000i	−0.761617	+	0.761617i
$994$		0			0
$995$	−14.6969			−0.465924
$996$		0			0
$997$	−14.0000			−0.443384			−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000			−0.443384			−0.221692	+	0.975117i	$0.571158\pi$
$998$		0			0
$999$	7.34847	−	7.34847i	0.232495	−	0.232495i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.2.h.c.191.4		4
3.2	odd	2	inner	960.2.h.c.191.2		4
4.3	odd	2	inner	960.2.h.c.191.1		4
8.3	odd	2		240.2.h.a.191.4	yes	4
8.5	even	2		240.2.h.a.191.1	✓	4
12.11	even	2	inner	960.2.h.c.191.3		4
24.5	odd	2		240.2.h.a.191.3	yes	4
24.11	even	2		240.2.h.a.191.2	yes	4
40.3	even	4		1200.2.o.e.1199.2		4
40.13	odd	4		1200.2.o.e.1199.3		4
40.19	odd	2		1200.2.h.k.1151.1		4
40.27	even	4		1200.2.o.f.1199.3		4
40.29	even	2		1200.2.h.k.1151.4		4
40.37	odd	4		1200.2.o.f.1199.2		4
120.29	odd	2		1200.2.h.k.1151.2		4
120.53	even	4		1200.2.o.f.1199.4		4
120.59	even	2		1200.2.h.k.1151.3		4
120.77	even	4		1200.2.o.e.1199.1		4
120.83	odd	4		1200.2.o.f.1199.1		4
120.107	odd	4		1200.2.o.e.1199.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.2.h.a.191.1	✓	4	8.5	even	2
240.2.h.a.191.2	yes	4	24.11	even	2
240.2.h.a.191.3	yes	4	24.5	odd	2
240.2.h.a.191.4	yes	4	8.3	odd	2
960.2.h.c.191.1		4	4.3	odd	2	inner
960.2.h.c.191.2		4	3.2	odd	2	inner
960.2.h.c.191.3		4	12.11	even	2	inner
960.2.h.c.191.4		4	1.1	even	1	trivial
1200.2.h.k.1151.1		4	40.19	odd	2
1200.2.h.k.1151.2		4	120.29	odd	2
1200.2.h.k.1151.3		4	120.59	even	2
1200.2.h.k.1151.4		4	40.29	even	2
1200.2.o.e.1199.1		4	120.77	even	4
1200.2.o.e.1199.2		4	40.3	even	4
1200.2.o.e.1199.3		4	40.13	odd	4
1200.2.o.e.1199.4		4	120.107	odd	4
1200.2.o.f.1199.1		4	120.83	odd	4
1200.2.o.f.1199.2		4	40.37	odd	4
1200.2.o.f.1199.3		4	40.27	even	4
1200.2.o.f.1199.4		4	120.53	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	960.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 + 1.22474i) q^{3} +1.00000i q^{5} -2.44949i q^{7} +3.00000i q^{9} +4.89898 q^{11} +2.00000 q^{13} +(-1.22474 + 1.22474i) q^{15} -6.00000i q^{17} +4.89898i q^{19} +(3.00000 - 3.00000i) q^{21} +2.44949 q^{23} -1.00000 q^{25} +(-3.67423 + 3.67423i) q^{27} +9.79796i q^{31} +(6.00000 + 6.00000i) q^{33} +2.44949 q^{35} -2.00000 q^{37} +(2.44949 + 2.44949i) q^{39} -6.00000i q^{41} -2.44949i q^{43} -3.00000 q^{45} +12.2474 q^{47} +1.00000 q^{49} +(7.34847 - 7.34847i) q^{51} +6.00000i q^{53} +4.89898i q^{55} +(-6.00000 + 6.00000i) q^{57} -9.79796 q^{59} -8.00000 q^{61} +7.34847 q^{63} +2.00000i q^{65} +7.34847i q^{67} +(3.00000 + 3.00000i) q^{69} -4.89898 q^{71} -14.0000 q^{73} +(-1.22474 - 1.22474i) q^{75} -12.0000i q^{77} +4.89898i q^{79} -9.00000 q^{81} +7.34847 q^{83} +6.00000 q^{85} -12.0000i q^{89} -4.89898i q^{91} +(-12.0000 + 12.0000i) q^{93} -4.89898 q^{95} +10.0000 q^{97} +14.6969i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{13} + 12 q^{21} - 4 q^{25} + 24 q^{33} - 8 q^{37} - 12 q^{45} + 4 q^{49} - 24 q^{57} - 32 q^{61} + 12 q^{69} - 56 q^{73} - 36 q^{81} + 24 q^{85} - 48 q^{93} + 40 q^{97}+O(q^{100}) \) 4 * q + 8 * q^13 + 12 * q^21 - 4 * q^25 + 24 * q^33 - 8 * q^37 - 12 * q^45 + 4 * q^49 - 24 * q^57 - 32 * q^61 + 12 * q^69 - 56 * q^73 - 36 * q^81 + 24 * q^85 - 48 * q^93 + 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more