LMFDB - Embedded newform 3060.2.w.b.2177.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3060,2,Mod(953,3060)] 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("3060.953"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3060, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 3, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3060.w (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$24.4342230185$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2177.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	3060.2177
Dual form			3060.2.w.b.953.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.12132 + 0.707107i) q^{5} +(2.00000 + 2.00000i) q^{7} +2.82843i q^{11} +(3.00000 - 3.00000i) q^{13} +(-0.707107 + 0.707107i) q^{17} -4.00000i q^{19} +(4.00000 - 3.00000i) q^{25} +7.07107 q^{29} +(-5.65685 - 2.82843i) q^{35} +(1.00000 + 1.00000i) q^{37} +7.07107i q^{41} +(4.00000 - 4.00000i) q^{43} +(-2.82843 + 2.82843i) q^{47} +1.00000i q^{49} +(7.07107 + 7.07107i) q^{53} +(-2.00000 - 6.00000i) q^{55} +2.82843 q^{59} -8.00000 q^{61} +(-4.24264 + 8.48528i) q^{65} +(4.00000 + 4.00000i) q^{67} -11.3137i q^{71} +(-5.00000 + 5.00000i) q^{73} +(-5.65685 + 5.65685i) q^{77} +16.0000i q^{79} +(-2.82843 - 2.82843i) q^{83} +(1.00000 - 2.00000i) q^{85} -9.89949 q^{89} +12.0000 q^{91} +(2.82843 + 8.48528i) q^{95} +(7.00000 + 7.00000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{7} + 12 q^{13} + 16 q^{25} + 4 q^{37} + 16 q^{43} - 8 q^{55} - 32 q^{61} + 16 q^{67} - 20 q^{73} + 4 q^{85} + 48 q^{91} + 28 q^{97}+O(q^{100}) $ 4 * q + 8 * q^7 + 12 * q^13 + 16 * q^25 + 4 * q^37 + 16 * q^43 - 8 * q^55 - 32 * q^61 + 16 * q^67 - 20 * q^73 + 4 * q^85 + 48 * q^91 + 28 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3060\mathbb{Z}\right)^\times$.

$n$	$1261$	$1361$	$1531$	$1837$
$\chi(n)$	$1$	$-1$	$1$	$e\left(\frac{1}{4}\right)$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.12132	+	0.707107i	−0.948683	+	0.316228i
$5$	−2.12132	+	0.707107i	−0.948683	+	0.316228i
$6$		0			0
$7$	2.00000	+	2.00000i	0.755929	+	0.755929i	0.975579	−	0.219650i	$-0.0704915\pi$
$7$	2.00000	+	2.00000i	0.755929	+	0.755929i	−0.219650	+	0.975579i	$0.570491\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			2.82843i			0.852803i	0.904534	+	0.426401i	$0.140219\pi$
$11$			2.82843i			0.852803i	−0.904534	+	0.426401i	$0.859781\pi$
$12$		0			0
$13$	3.00000	−	3.00000i	0.832050	−	0.832050i	−0.155747	−	0.987797i	$-0.549778\pi$
$13$	3.00000	−	3.00000i	0.832050	−	0.832050i	0.987797	+	0.155747i	$0.0497784\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.707107	+	0.707107i	−0.171499	+	0.171499i
$17$	−0.707107	+	0.707107i	−0.171499	+	0.171499i
$18$		0			0
$19$		−	4.00000i		−	0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$		−	4.00000i		−	0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$23$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$24$		0			0
$25$	4.00000	−	3.00000i	0.800000	−	0.600000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	7.07107			1.31306			0.656532	−	0.754298i	$-0.272023\pi$
$29$	7.07107			1.31306			0.656532	+	0.754298i	$0.272023\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−5.65685	−	2.82843i	−0.956183	−	0.478091i
$36$		0			0
$37$	1.00000	+	1.00000i	0.164399	+	0.164399i	0.784512	−	0.620113i	$-0.212913\pi$
$37$	1.00000	+	1.00000i	0.164399	+	0.164399i	−0.620113	+	0.784512i	$0.712913\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			7.07107i			1.10432i	0.833740	+	0.552158i	$0.186195\pi$
$41$			7.07107i			1.10432i	−0.833740	+	0.552158i	$0.813805\pi$
$42$		0			0
$43$	4.00000	−	4.00000i	0.609994	−	0.609994i	−0.332950	−	0.942944i	$-0.608044\pi$
$43$	4.00000	−	4.00000i	0.609994	−	0.609994i	0.942944	+	0.332950i	$0.108044\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.82843	+	2.82843i	−0.412568	+	0.412568i	−0.882632	−	0.470064i	$-0.844231\pi$
$47$	−2.82843	+	2.82843i	−0.412568	+	0.412568i	0.470064	+	0.882632i	$0.344231\pi$
$48$		0			0
$49$			1.00000i			0.142857i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	7.07107	+	7.07107i	0.971286	+	0.971286i	0.999599	−	0.0283132i	$-0.00901359\pi$
$53$	7.07107	+	7.07107i	0.971286	+	0.971286i	−0.0283132	+	0.999599i	$0.509014\pi$
$54$		0			0
$55$	−2.00000	−	6.00000i	−0.269680	−	0.809040i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.82843			0.368230			0.184115	−	0.982905i	$-0.441058\pi$
$59$	2.82843			0.368230			0.184115	+	0.982905i	$0.441058\pi$
$60$		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
$63$		0			0
$64$		0			0
$65$	−4.24264	+	8.48528i	−0.526235	+	1.05247i
$66$		0			0
$67$	4.00000	+	4.00000i	0.488678	+	0.488678i	0.907889	−	0.419211i	$-0.137693\pi$
$67$	4.00000	+	4.00000i	0.488678	+	0.488678i	−0.419211	+	0.907889i	$0.637693\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	11.3137i		−	1.34269i	−0.741145	−	0.671345i	$-0.765717\pi$
$71$		−	11.3137i		−	1.34269i	0.741145	−	0.671345i	$-0.234283\pi$
$72$		0			0
$73$	−5.00000	+	5.00000i	−0.585206	+	0.585206i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	−5.00000	+	5.00000i	−0.585206	+	0.585206i	0.351123	+	0.936329i	$0.385800\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−5.65685	+	5.65685i	−0.644658	+	0.644658i
$78$		0			0
$79$			16.0000i			1.80014i	0.435745	+	0.900070i	$0.356485\pi$
$79$			16.0000i			1.80014i	−0.435745	+	0.900070i	$0.643515\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−2.82843	−	2.82843i	−0.310460	−	0.310460i	0.534628	−	0.845088i	$-0.320452\pi$
$83$	−2.82843	−	2.82843i	−0.310460	−	0.310460i	−0.845088	+	0.534628i	$0.820452\pi$
$84$		0			0
$85$	1.00000	−	2.00000i	0.108465	−	0.216930i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−9.89949			−1.04934			−0.524672	−	0.851304i	$-0.675812\pi$
$89$	−9.89949			−1.04934			−0.524672	+	0.851304i	$0.675812\pi$
$90$		0			0
$91$	12.0000			1.25794
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.82843	+	8.48528i	0.290191	+	0.870572i
$96$		0			0
$97$	7.00000	+	7.00000i	0.710742	+	0.710742i	0.966691	−	0.255948i	$-0.0823876\pi$
$97$	7.00000	+	7.00000i	0.710742	+	0.710742i	−0.255948	+	0.966691i	$0.582388\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			9.89949i			0.985037i	0.870302	+	0.492518i	$0.163924\pi$
$101$			9.89949i			0.985037i	−0.870302	+	0.492518i	$0.836076\pi$
$102$		0			0
$103$	2.00000	−	2.00000i	0.197066	−	0.197066i	−0.601675	−	0.798741i	$-0.705500\pi$
$103$	2.00000	−	2.00000i	0.197066	−	0.197066i	0.798741	+	0.601675i	$0.205500\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.82843	−	2.82843i	0.273434	−	0.273434i	−0.557047	−	0.830481i	$-0.688066\pi$
$107$	2.82843	−	2.82843i	0.273434	−	0.273434i	0.830481	+	0.557047i	$0.188066\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−2.82843	−	2.82843i	−0.266076	−	0.266076i	0.561441	−	0.827517i	$-0.310247\pi$
$113$	−2.82843	−	2.82843i	−0.266076	−	0.266076i	−0.827517	+	0.561441i	$0.810247\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.82843			−0.259281
$120$		0			0
$121$	3.00000			0.272727
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−6.36396	+	9.19239i	−0.569210	+	0.822192i
$126$		0			0
$127$	−6.00000	−	6.00000i	−0.532414	−	0.532414i	0.388876	−	0.921290i	$-0.372863\pi$
$127$	−6.00000	−	6.00000i	−0.532414	−	0.532414i	−0.921290	+	0.388876i	$0.872863\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			14.1421i			1.23560i	0.786334	+	0.617802i	$0.211977\pi$
$131$			14.1421i			1.23560i	−0.786334	+	0.617802i	$0.788023\pi$
$132$		0			0
$133$	8.00000	−	8.00000i	0.693688	−	0.693688i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.48528	−	8.48528i	0.724947	−	0.724947i	−0.244662	−	0.969608i	$-0.578677\pi$
$137$	8.48528	−	8.48528i	0.724947	−	0.724947i	0.969608	+	0.244662i	$0.0786770\pi$
$138$		0			0
$139$			16.0000i			1.35710i	0.734553	+	0.678551i	$0.237392\pi$
$139$			16.0000i			1.35710i	−0.734553	+	0.678551i	$0.762608\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	8.48528	+	8.48528i	0.709575	+	0.709575i
$144$		0			0
$145$	−15.0000	+	5.00000i	−1.24568	+	0.415227i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	12.7279			1.04271			0.521356	−	0.853339i	$-0.325426\pi$
$149$	12.7279			1.04271			0.521356	+	0.853339i	$0.325426\pi$
$150$		0			0
$151$	20.0000			1.62758			0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000			1.62758			0.813788	+	0.581161i	$0.197401\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	15.0000	+	15.0000i	1.19713	+	1.19713i	0.975022	+	0.222108i	$0.0712939\pi$
$157$	15.0000	+	15.0000i	1.19713	+	1.19713i	0.222108	+	0.975022i	$0.428706\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−12.0000	+	12.0000i	−0.939913	+	0.939913i	−0.998294	−	0.0583818i	$-0.981406\pi$
$163$	−12.0000	+	12.0000i	−0.939913	+	0.939913i	0.0583818	+	0.998294i	$0.481406\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−11.3137	+	11.3137i	−0.875481	+	0.875481i	−0.993063	−	0.117582i	$-0.962486\pi$
$167$	−11.3137	+	11.3137i	−0.875481	+	0.875481i	0.117582	+	0.993063i	$0.462486\pi$
$168$		0			0
$169$		−	5.00000i		−	0.384615i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	14.1421	+	14.1421i	1.07521	+	1.07521i	0.996932	+	0.0782748i	$0.0249412\pi$
$173$	14.1421	+	14.1421i	1.07521	+	1.07521i	0.0782748	+	0.996932i	$0.475059\pi$
$174$		0			0
$175$	14.0000	+	2.00000i	1.05830	+	0.151186i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2.82843			0.211407			0.105703	−	0.994398i	$-0.466291\pi$
$179$	2.82843			0.211407			0.105703	+	0.994398i	$0.466291\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2.82843	−	1.41421i	−0.207950	−	0.103975i
$186$		0			0
$187$	−2.00000	−	2.00000i	−0.146254	−	0.146254i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	11.3137i		−	0.818631i	−0.912393	−	0.409316i	$-0.865768\pi$
$191$		−	11.3137i		−	0.818631i	0.912393	−	0.409316i	$-0.134232\pi$
$192$		0			0
$193$	13.0000	−	13.0000i	0.935760	−	0.935760i	−0.0622972	−	0.998058i	$-0.519843\pi$
$193$	13.0000	−	13.0000i	0.935760	−	0.935760i	0.998058	+	0.0622972i	$0.0198427\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−8.48528	+	8.48528i	−0.604551	+	0.604551i	−0.941517	−	0.336966i	$-0.890599\pi$
$197$	−8.48528	+	8.48528i	−0.604551	+	0.604551i	0.336966	+	0.941517i	$0.390599\pi$
$198$		0			0
$199$			4.00000i			0.283552i	0.989899	+	0.141776i	$0.0452813\pi$
$199$			4.00000i			0.283552i	−0.989899	+	0.141776i	$0.954719\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	14.1421	+	14.1421i	0.992583	+	0.992583i
$204$		0			0
$205$	−5.00000	−	15.0000i	−0.349215	−	1.04765i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	11.3137			0.782586
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−5.65685	+	11.3137i	−0.385794	+	0.771589i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$			4.24264i			0.285391i
$222$		0			0
$223$	−2.00000	+	2.00000i	−0.133930	+	0.133930i	−0.770894	−	0.636964i	$-0.780190\pi$
$223$	−2.00000	+	2.00000i	−0.133930	+	0.133930i	0.636964	+	0.770894i	$0.280190\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$		0			0
$229$		−	10.0000i		−	0.660819i	−0.943838	−	0.330409i	$-0.892813\pi$
$229$		−	10.0000i		−	0.660819i	0.943838	−	0.330409i	$-0.107187\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−4.24264	−	4.24264i	−0.277945	−	0.277945i	0.554343	−	0.832288i	$-0.312969\pi$
$233$	−4.24264	−	4.24264i	−0.277945	−	0.277945i	−0.832288	+	0.554343i	$0.812969\pi$
$234$		0			0
$235$	4.00000	−	8.00000i	0.260931	−	0.521862i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−11.3137			−0.731823			−0.365911	−	0.930650i	$-0.619243\pi$
$239$	−11.3137			−0.731823			−0.365911	+	0.930650i	$0.619243\pi$
$240$		0			0
$241$	26.0000			1.67481			0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000			1.67481			0.837404	+	0.546585i	$0.184072\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.707107	−	2.12132i	−0.0451754	−	0.135526i
$246$		0			0
$247$	−12.0000	−	12.0000i	−0.763542	−	0.763542i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	2.82843i		−	0.178529i	−0.996008	−	0.0892644i	$-0.971548\pi$
$251$		−	2.82843i		−	0.178529i	0.996008	−	0.0892644i	$-0.0284516\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−8.48528	+	8.48528i	−0.529297	+	0.529297i	−0.920363	−	0.391066i	$-0.872107\pi$
$257$	−8.48528	+	8.48528i	−0.529297	+	0.529297i	0.391066	+	0.920363i	$0.372107\pi$
$258$		0			0
$259$			4.00000i			0.248548i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−16.9706	−	16.9706i	−1.04645	−	1.04645i	−0.998867	−	0.0475824i	$-0.984848\pi$
$263$	−16.9706	−	16.9706i	−1.04645	−	1.04645i	−0.0475824	−	0.998867i	$-0.515152\pi$
$264$		0			0
$265$	−20.0000	−	10.0000i	−1.22859	−	0.614295i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	21.2132			1.29339			0.646696	−	0.762748i	$-0.276150\pi$
$269$	21.2132			1.29339			0.646696	+	0.762748i	$0.276150\pi$
$270$		0			0
$271$	20.0000			1.21491			0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000			1.21491			0.607457	+	0.794353i	$0.292190\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	8.48528	+	11.3137i	0.511682	+	0.682242i
$276$		0			0
$277$	−21.0000	−	21.0000i	−1.26177	−	1.26177i	−0.950236	−	0.311532i	$-0.899158\pi$
$277$	−21.0000	−	21.0000i	−1.26177	−	1.26177i	−0.311532	−	0.950236i	$-0.600842\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	1.41421i		−	0.0843649i	−0.999110	−	0.0421825i	$-0.986569\pi$
$281$		−	1.41421i		−	0.0843649i	0.999110	−	0.0421825i	$-0.0134311\pi$
$282$		0			0
$283$	−12.0000	+	12.0000i	−0.713326	+	0.713326i	−0.967230	−	0.253904i	$-0.918285\pi$
$283$	−12.0000	+	12.0000i	−0.713326	+	0.713326i	0.253904	+	0.967230i	$0.418285\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−14.1421	+	14.1421i	−0.834784	+	0.834784i
$288$		0			0
$289$		−	1.00000i		−	0.0588235i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2.82843	+	2.82843i	0.165238	+	0.165238i	0.784883	−	0.619644i	$-0.212723\pi$
$293$	2.82843	+	2.82843i	0.165238	+	0.165238i	−0.619644	+	0.784883i	$0.712723\pi$
$294$		0			0
$295$	−6.00000	+	2.00000i	−0.349334	+	0.116445i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	16.0000			0.922225
$302$		0			0
$303$		0			0
$304$		0			0
$305$	16.9706	−	5.65685i	0.971732	−	0.323911i
$306$		0			0
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	11.3137i		−	0.641542i	−0.947157	−	0.320771i	$-0.896058\pi$
$311$		−	11.3137i		−	0.641542i	0.947157	−	0.320771i	$-0.103942\pi$
$312$		0			0
$313$	−9.00000	+	9.00000i	−0.508710	+	0.508710i	−0.914130	−	0.405420i	$-0.867125\pi$
$313$	−9.00000	+	9.00000i	−0.508710	+	0.508710i	0.405420	+	0.914130i	$0.367125\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	4.24264	−	4.24264i	0.238290	−	0.238290i	−0.577851	−	0.816142i	$-0.696109\pi$
$317$	4.24264	−	4.24264i	0.238290	−	0.238290i	0.816142	+	0.577851i	$0.196109\pi$
$318$		0			0
$319$			20.0000i			1.11979i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2.82843	+	2.82843i	0.157378	+	0.157378i
$324$		0			0
$325$	3.00000	−	21.0000i	0.166410	−	1.16487i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−11.3137			−0.623745
$330$		0			0
$331$	28.0000			1.53902			0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000			1.53902			0.769510	+	0.638635i	$0.220501\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−11.3137	−	5.65685i	−0.618134	−	0.309067i
$336$		0			0
$337$	9.00000	+	9.00000i	0.490261	+	0.490261i	0.908388	−	0.418127i	$-0.137313\pi$
$337$	9.00000	+	9.00000i	0.490261	+	0.490261i	−0.418127	+	0.908388i	$0.637313\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	12.0000	−	12.0000i	0.647939	−	0.647939i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−22.6274	+	22.6274i	−1.21470	+	1.21470i	−0.245241	+	0.969462i	$0.578867\pi$
$347$	−22.6274	+	22.6274i	−1.21470	+	1.21470i	−0.969462	+	0.245241i	$0.921133\pi$
$348$		0			0
$349$		−	16.0000i		−	0.856460i	−0.903670	−	0.428230i	$-0.859137\pi$
$349$		−	16.0000i		−	0.856460i	0.903670	−	0.428230i	$-0.140863\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.89949	−	9.89949i	−0.526897	−	0.526897i	0.392749	−	0.919646i	$-0.371524\pi$
$353$	−9.89949	−	9.89949i	−0.526897	−	0.526897i	−0.919646	+	0.392749i	$0.871524\pi$
$354$		0			0
$355$	8.00000	+	24.0000i	0.424596	+	1.27379i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−22.6274			−1.19423			−0.597115	−	0.802156i	$-0.703686\pi$
$359$	−22.6274			−1.19423			−0.597115	+	0.802156i	$0.703686\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$		0			0
$363$		0			0
$364$		0			0
$365$	7.07107	−	14.1421i	0.370117	−	0.740233i
$366$		0			0
$367$	−18.0000	−	18.0000i	−0.939592	−	0.939592i	0.0586842	−	0.998277i	$-0.481309\pi$
$367$	−18.0000	−	18.0000i	−0.939592	−	0.939592i	−0.998277	+	0.0586842i	$0.981309\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			28.2843i			1.46845i
$372$		0			0
$373$	−21.0000	+	21.0000i	−1.08734	+	1.08734i	−0.0915371	+	0.995802i	$0.529178\pi$
$373$	−21.0000	+	21.0000i	−1.08734	+	1.08734i	−0.995802	+	0.0915371i	$0.970822\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	21.2132	−	21.2132i	1.09254	−	1.09254i
$378$		0			0
$379$			16.0000i			0.821865i	0.911666	+	0.410932i	$0.134797\pi$
$379$			16.0000i			0.821865i	−0.911666	+	0.410932i	$0.865203\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−8.48528	−	8.48528i	−0.433578	−	0.433578i	0.456266	−	0.889843i	$-0.349187\pi$
$383$	−8.48528	−	8.48528i	−0.433578	−	0.433578i	−0.889843	+	0.456266i	$0.849187\pi$
$384$		0			0
$385$	8.00000	−	16.0000i	0.407718	−	0.815436i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.41421			−0.0717035			−0.0358517	−	0.999357i	$-0.511414\pi$
$389$	−1.41421			−0.0717035			−0.0358517	+	0.999357i	$0.511414\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−11.3137	−	33.9411i	−0.569254	−	1.70776i
$396$		0			0
$397$	−5.00000	−	5.00000i	−0.250943	−	0.250943i	0.570414	−	0.821357i	$-0.306783\pi$
$397$	−5.00000	−	5.00000i	−0.250943	−	0.250943i	−0.821357	+	0.570414i	$0.806783\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	26.8701i		−	1.34183i	−0.741536	−	0.670913i	$-0.765902\pi$
$401$		−	26.8701i		−	1.34183i	0.741536	−	0.670913i	$-0.234098\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−2.82843	+	2.82843i	−0.140200	+	0.140200i
$408$		0			0
$409$			8.00000i			0.395575i	0.980245	+	0.197787i	$0.0633755\pi$
$409$			8.00000i			0.395575i	−0.980245	+	0.197787i	$0.936624\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	5.65685	+	5.65685i	0.278356	+	0.278356i
$414$		0			0
$415$	8.00000	+	4.00000i	0.392705	+	0.196352i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−8.48528			−0.414533			−0.207267	−	0.978285i	$-0.566457\pi$
$419$	−8.48528			−0.414533			−0.207267	+	0.978285i	$0.566457\pi$
$420$		0			0
$421$	−6.00000			−0.292422			−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000			−0.292422			−0.146211	+	0.989253i	$0.546708\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.707107	+	4.94975i	−0.0342997	+	0.240098i
$426$		0			0
$427$	−16.0000	−	16.0000i	−0.774294	−	0.774294i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	33.9411i		−	1.63489i	−0.576009	−	0.817443i	$-0.695391\pi$
$431$		−	33.9411i		−	1.63489i	0.576009	−	0.817443i	$-0.304609\pi$
$432$		0			0
$433$	−9.00000	+	9.00000i	−0.432512	+	0.432512i	−0.889482	−	0.456970i	$-0.848935\pi$
$433$	−9.00000	+	9.00000i	−0.432512	+	0.432512i	0.456970	+	0.889482i	$0.348935\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		−	4.00000i		−	0.190910i	−0.995434	−	0.0954548i	$-0.969569\pi$
$439$		−	4.00000i		−	0.190910i	0.995434	−	0.0954548i	$-0.0304305\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−5.65685	−	5.65685i	−0.268765	−	0.268765i	0.559837	−	0.828603i	$-0.310864\pi$
$443$	−5.65685	−	5.65685i	−0.268765	−	0.268765i	−0.828603	+	0.559837i	$0.810864\pi$
$444$		0			0
$445$	21.0000	−	7.00000i	0.995495	−	0.331832i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	26.8701			1.26808			0.634038	−	0.773302i	$-0.281396\pi$
$449$	26.8701			1.26808			0.634038	+	0.773302i	$0.281396\pi$
$450$		0			0
$451$	−20.0000			−0.941763
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−25.4558	+	8.48528i	−1.19339	+	0.397796i
$456$		0			0
$457$	−17.0000	−	17.0000i	−0.795226	−	0.795226i	0.187112	−	0.982339i	$-0.440087\pi$
$457$	−17.0000	−	17.0000i	−0.795226	−	0.795226i	−0.982339	+	0.187112i	$0.940087\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	4.24264i		−	0.197599i	−0.995107	−	0.0987997i	$-0.968500\pi$
$461$		−	4.24264i		−	0.197599i	0.995107	−	0.0987997i	$-0.0315003\pi$
$462$		0			0
$463$	−18.0000	+	18.0000i	−0.836531	+	0.836531i	−0.988401	−	0.151870i	$-0.951471\pi$
$463$	−18.0000	+	18.0000i	−0.836531	+	0.836531i	0.151870	+	0.988401i	$0.451471\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	8.48528	−	8.48528i	0.392652	−	0.392652i	−0.482980	−	0.875632i	$-0.660445\pi$
$467$	8.48528	−	8.48528i	0.392652	−	0.392652i	0.875632	+	0.482980i	$0.160445\pi$
$468$		0			0
$469$			16.0000i			0.738811i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	11.3137	+	11.3137i	0.520205	+	0.520205i
$474$		0			0
$475$	−12.0000	−	16.0000i	−0.550598	−	0.734130i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	11.3137			0.516937			0.258468	−	0.966020i	$-0.416782\pi$
$479$	11.3137			0.516937			0.258468	+	0.966020i	$0.416782\pi$
$480$		0			0
$481$	6.00000			0.273576
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−19.7990	−	9.89949i	−0.899026	−	0.449513i
$486$		0			0
$487$	−2.00000	−	2.00000i	−0.0906287	−	0.0906287i	0.660339	−	0.750968i	$-0.270413\pi$
$487$	−2.00000	−	2.00000i	−0.0906287	−	0.0906287i	−0.750968	+	0.660339i	$0.770413\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	42.4264i		−	1.91468i	−0.288969	−	0.957338i	$-0.593312\pi$
$491$		−	42.4264i		−	1.91468i	0.288969	−	0.957338i	$-0.406688\pi$
$492$		0			0
$493$	−5.00000	+	5.00000i	−0.225189	+	0.225189i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	22.6274	−	22.6274i	1.01498	−	1.01498i
$498$		0			0
$499$			44.0000i			1.96971i	0.173379	+	0.984855i	$0.444532\pi$
$499$			44.0000i			1.96971i	−0.173379	+	0.984855i	$0.555468\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	8.48528	+	8.48528i	0.378340	+	0.378340i	0.870503	−	0.492163i	$-0.163794\pi$
$503$	8.48528	+	8.48528i	0.378340	+	0.378340i	−0.492163	+	0.870503i	$0.663794\pi$
$504$		0			0
$505$	−7.00000	−	21.0000i	−0.311496	−	0.934488i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.41421			−0.0626839			−0.0313420	−	0.999509i	$-0.509978\pi$
$509$	−1.41421			−0.0626839			−0.0313420	+	0.999509i	$0.509978\pi$
$510$		0			0
$511$	−20.0000			−0.884748
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.82843	+	5.65685i	−0.124635	+	0.249271i
$516$		0			0
$517$	−8.00000	−	8.00000i	−0.351840	−	0.351840i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	24.0416i		−	1.05328i	−0.850087	−	0.526641i	$-0.823451\pi$
$521$		−	24.0416i		−	1.05328i	0.850087	−	0.526641i	$-0.176549\pi$
$522$		0			0
$523$	−20.0000	+	20.0000i	−0.874539	+	0.874539i	−0.992963	−	0.118424i	$-0.962216\pi$
$523$	−20.0000	+	20.0000i	−0.874539	+	0.874539i	0.118424	+	0.992963i	$0.462216\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$		−	23.0000i		−	1.00000i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	21.2132	+	21.2132i	0.918846	+	0.918846i
$534$		0			0
$535$	−4.00000	+	8.00000i	−0.172935	+	0.345870i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2.82843			−0.121829
$540$		0			0
$541$	32.0000			1.37579			0.687894	−	0.725811i	$-0.258536\pi$
$541$	32.0000			1.37579			0.687894	+	0.725811i	$0.258536\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	20.0000	+	20.0000i	0.855138	+	0.855138i	0.990761	−	0.135622i	$-0.0433034\pi$
$547$	20.0000	+	20.0000i	0.855138	+	0.855138i	−0.135622	+	0.990761i	$0.543303\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	28.2843i		−	1.20495i
$552$		0			0
$553$	−32.0000	+	32.0000i	−1.36078	+	1.36078i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	32.5269	−	32.5269i	1.37821	−	1.37821i	0.530566	−	0.847644i	$-0.321980\pi$
$557$	32.5269	−	32.5269i	1.37821	−	1.37821i	0.847644	−	0.530566i	$-0.178020\pi$
$558$		0			0
$559$		−	24.0000i		−	1.01509i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	16.9706	+	16.9706i	0.715224	+	0.715224i	0.967623	−	0.252399i	$-0.0812196\pi$
$563$	16.9706	+	16.9706i	0.715224	+	0.715224i	−0.252399	+	0.967623i	$0.581220\pi$
$564$		0			0
$565$	8.00000	+	4.00000i	0.336563	+	0.168281i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	18.3848			0.770730			0.385365	−	0.922764i	$-0.374076\pi$
$569$	18.3848			0.770730			0.385365	+	0.922764i	$0.374076\pi$
$570$		0			0
$571$	−28.0000			−1.17176			−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000			−1.17176			−0.585882	+	0.810397i	$0.699252\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	15.0000	+	15.0000i	0.624458	+	0.624458i	0.946668	−	0.322210i	$-0.104426\pi$
$577$	15.0000	+	15.0000i	0.624458	+	0.624458i	−0.322210	+	0.946668i	$0.604426\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	11.3137i		−	0.469372i
$582$		0			0
$583$	−20.0000	+	20.0000i	−0.828315	+	0.828315i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−25.4558	+	25.4558i	−1.05068	+	1.05068i	−0.0520296	+	0.998646i	$0.516569\pi$
$587$	−25.4558	+	25.4558i	−1.05068	+	1.05068i	−0.998646	+	0.0520296i	$0.983431\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	25.4558	+	25.4558i	1.04535	+	1.04535i	0.998922	+	0.0464244i	$0.0147827\pi$
$593$	25.4558	+	25.4558i	1.04535	+	1.04535i	0.0464244	+	0.998922i	$0.485217\pi$
$594$		0			0
$595$	6.00000	−	2.00000i	0.245976	−	0.0819920i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	5.65685			0.231133			0.115566	−	0.993300i	$-0.463132\pi$
$599$	5.65685			0.231133			0.115566	+	0.993300i	$0.463132\pi$
$600$		0			0
$601$		0			0			−	1.00000i	$-0.5\pi$
$601$		0			0				1.00000i	$0.5\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−6.36396	+	2.12132i	−0.258732	+	0.0862439i
$606$		0			0
$607$	−30.0000	−	30.0000i	−1.21766	−	1.21766i	−0.968448	−	0.249214i	$-0.919828\pi$
$607$	−30.0000	−	30.0000i	−1.21766	−	1.21766i	−0.249214	−	0.968448i	$-0.580172\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			16.9706i			0.686555i
$612$		0			0
$613$	−19.0000	+	19.0000i	−0.767403	+	0.767403i	−0.977649	−	0.210246i	$-0.932574\pi$
$613$	−19.0000	+	19.0000i	−0.767403	+	0.767403i	0.210246	+	0.977649i	$0.432574\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4.24264	+	4.24264i	−0.170802	+	0.170802i	−0.787332	−	0.616530i	$-0.788538\pi$
$617$	−4.24264	+	4.24264i	−0.170802	+	0.170802i	0.616530	+	0.787332i	$0.288538\pi$
$618$		0			0
$619$		−	32.0000i		−	1.28619i	−0.765787	−	0.643094i	$-0.777650\pi$
$619$		−	32.0000i		−	1.28619i	0.765787	−	0.643094i	$-0.222350\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−19.7990	−	19.7990i	−0.793230	−	0.793230i
$624$		0			0
$625$	7.00000	−	24.0000i	0.280000	−	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1.41421			−0.0563884
$630$		0			0
$631$	−8.00000			−0.318475			−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000			−0.318475			−0.159237	+	0.987240i	$0.550904\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	16.9706	+	8.48528i	0.673456	+	0.336728i
$636$		0			0
$637$	3.00000	+	3.00000i	0.118864	+	0.118864i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			12.7279i			0.502723i	0.967893	+	0.251361i	$0.0808782\pi$
$641$			12.7279i			0.502723i	−0.967893	+	0.251361i	$0.919122\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−2.82843	+	2.82843i	−0.111197	+	0.111197i	−0.760516	−	0.649319i	$-0.775054\pi$
$647$	−2.82843	+	2.82843i	−0.111197	+	0.111197i	0.649319	+	0.760516i	$0.275054\pi$
$648$		0			0
$649$			8.00000i			0.314027i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	4.24264	+	4.24264i	0.166027	+	0.166027i	0.785231	−	0.619203i	$-0.212544\pi$
$653$	4.24264	+	4.24264i	0.166027	+	0.166027i	−0.619203	+	0.785231i	$0.712544\pi$
$654$		0			0
$655$	−10.0000	−	30.0000i	−0.390732	−	1.17220i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−14.1421			−0.550899			−0.275450	−	0.961315i	$-0.588827\pi$
$659$	−14.1421			−0.550899			−0.275450	+	0.961315i	$0.588827\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−11.3137	+	22.6274i	−0.438727	+	0.877454i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	22.6274i		−	0.873522i
$672$		0			0
$673$	21.0000	−	21.0000i	0.809491	−	0.809491i	−0.175066	−	0.984557i	$-0.556014\pi$
$673$	21.0000	−	21.0000i	0.809491	−	0.809491i	0.984557	+	0.175066i	$0.0560139\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	12.7279	−	12.7279i	0.489174	−	0.489174i	−0.418872	−	0.908045i	$-0.637574\pi$
$677$	12.7279	−	12.7279i	0.489174	−	0.489174i	0.908045	+	0.418872i	$0.137574\pi$
$678$		0			0
$679$			28.0000i			1.07454i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−2.82843	−	2.82843i	−0.108227	−	0.108227i	0.650920	−	0.759147i	$-0.274383\pi$
$683$	−2.82843	−	2.82843i	−0.108227	−	0.108227i	−0.759147	+	0.650920i	$0.774383\pi$
$684$		0			0
$685$	−12.0000	+	24.0000i	−0.458496	+	0.916993i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	42.4264			1.61632
$690$		0			0
$691$	12.0000			0.456502			0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000			0.456502			0.228251	+	0.973602i	$0.426699\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−11.3137	−	33.9411i	−0.429153	−	1.28746i
$696$		0			0
$697$	−5.00000	−	5.00000i	−0.189389	−	0.189389i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	24.0416i		−	0.908040i	−0.890992	−	0.454020i	$-0.849989\pi$
$701$		−	24.0416i		−	0.908040i	0.890992	−	0.454020i	$-0.150011\pi$
$702$		0			0
$703$	4.00000	−	4.00000i	0.150863	−	0.150863i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−19.7990	+	19.7990i	−0.744618	+	0.744618i
$708$		0			0
$709$		−	38.0000i		−	1.42712i	−0.700594	−	0.713560i	$-0.747082\pi$
$709$		−	38.0000i		−	1.42712i	0.700594	−	0.713560i	$-0.252918\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−24.0000	−	12.0000i	−0.897549	−	0.448775i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	11.3137			0.421930			0.210965	−	0.977494i	$-0.432339\pi$
$719$	11.3137			0.421930			0.210965	+	0.977494i	$0.432339\pi$
$720$		0			0
$721$	8.00000			0.297936
$722$		0			0
$723$		0			0
$724$		0			0
$725$	28.2843	−	21.2132i	1.05045	−	0.787839i
$726$		0			0
$727$	−18.0000	−	18.0000i	−0.667583	−	0.667583i	0.289573	−	0.957156i	$-0.406487\pi$
$727$	−18.0000	−	18.0000i	−0.667583	−	0.667583i	−0.957156	+	0.289573i	$0.906487\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			5.65685i			0.209226i
$732$		0			0
$733$	3.00000	−	3.00000i	0.110808	−	0.110808i	−0.649529	−	0.760337i	$-0.725034\pi$
$733$	3.00000	−	3.00000i	0.110808	−	0.110808i	0.760337	+	0.649529i	$0.225034\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−11.3137	+	11.3137i	−0.416746	+	0.416746i
$738$		0			0
$739$			52.0000i			1.91285i	0.291977	+	0.956425i	$0.405687\pi$
$739$			52.0000i			1.91285i	−0.291977	+	0.956425i	$0.594313\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−19.7990	−	19.7990i	−0.726354	−	0.726354i	0.243537	−	0.969892i	$-0.421692\pi$
$743$	−19.7990	−	19.7990i	−0.726354	−	0.726354i	−0.969892	+	0.243537i	$0.921692\pi$
$744$		0			0
$745$	−27.0000	+	9.00000i	−0.989203	+	0.329734i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	11.3137			0.413394
$750$		0			0
$751$	−24.0000			−0.875772			−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000			−0.875772			−0.437886	+	0.899030i	$0.644273\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−42.4264	+	14.1421i	−1.54406	+	0.514685i
$756$		0			0
$757$	33.0000	+	33.0000i	1.19941	+	1.19941i	0.974345	+	0.225061i	$0.0722580\pi$
$757$	33.0000	+	33.0000i	1.19941	+	1.19941i	0.225061	+	0.974345i	$0.427742\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	7.07107i		−	0.256326i	−0.991753	−	0.128163i	$-0.959092\pi$
$761$		−	7.07107i		−	0.256326i	0.991753	−	0.128163i	$-0.0409081\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	8.48528	−	8.48528i	0.306386	−	0.306386i
$768$		0			0
$769$			24.0000i			0.865462i	0.901523	+	0.432731i	$0.142450\pi$
$769$			24.0000i			0.865462i	−0.901523	+	0.432731i	$0.857550\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	21.2132	+	21.2132i	0.762986	+	0.762986i	0.976861	−	0.213875i	$-0.0686086\pi$
$773$	21.2132	+	21.2132i	0.762986	+	0.762986i	−0.213875	+	0.976861i	$0.568609\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	28.2843			1.01339
$780$		0			0
$781$	32.0000			1.14505
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−42.4264	−	21.2132i	−1.51426	−	0.757132i
$786$		0			0
$787$	28.0000	+	28.0000i	0.998092	+	0.998092i	0.999998	−	0.00190598i	$-0.000606691\pi$
$787$	28.0000	+	28.0000i	0.998092	+	0.998092i	−0.00190598	+	0.999998i	$0.500607\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	11.3137i		−	0.402269i
$792$		0			0
$793$	−24.0000	+	24.0000i	−0.852265	+	0.852265i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−2.82843	+	2.82843i	−0.100188	+	0.100188i	−0.755424	−	0.655236i	$-0.772569\pi$
$797$	−2.82843	+	2.82843i	−0.100188	+	0.100188i	0.655236	+	0.755424i	$0.272569\pi$
$798$		0			0
$799$		−	4.00000i		−	0.141510i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−14.1421	−	14.1421i	−0.499065	−	0.499065i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−52.3259			−1.83968			−0.919840	−	0.392293i	$-0.871682\pi$
$809$	−52.3259			−1.83968			−0.919840	+	0.392293i	$0.871682\pi$
$810$		0			0
$811$	8.00000			0.280918			0.140459	−	0.990086i	$-0.455142\pi$
$811$	8.00000			0.280918			0.140459	+	0.990086i	$0.455142\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	16.9706	−	33.9411i	0.594453	−	1.18891i
$816$		0			0
$817$	−16.0000	−	16.0000i	−0.559769	−	0.559769i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	55.1543i		−	1.92490i	−0.271460	−	0.962450i	$-0.587507\pi$
$821$		−	55.1543i		−	1.92490i	0.271460	−	0.962450i	$-0.412493\pi$
$822$		0			0
$823$	−6.00000	+	6.00000i	−0.209147	+	0.209147i	−0.803905	−	0.594758i	$-0.797248\pi$
$823$	−6.00000	+	6.00000i	−0.209147	+	0.209147i	0.594758	+	0.803905i	$0.297248\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	33.9411	−	33.9411i	1.18025	−	1.18025i	0.200569	−	0.979680i	$-0.435721\pi$
$827$	33.9411	−	33.9411i	1.18025	−	1.18025i	0.979680	−	0.200569i	$-0.0642791\pi$
$828$		0			0
$829$		−	40.0000i		−	1.38926i	−0.719368	−	0.694629i	$-0.755569\pi$
$829$		−	40.0000i		−	1.38926i	0.719368	−	0.694629i	$-0.244431\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.707107	−	0.707107i	−0.0244998	−	0.0244998i
$834$		0			0
$835$	16.0000	−	32.0000i	0.553703	−	1.10741i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	28.2843			0.976481			0.488241	−	0.872709i	$-0.337639\pi$
$839$	28.2843			0.976481			0.488241	+	0.872709i	$0.337639\pi$
$840$		0			0
$841$	21.0000			0.724138
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.53553	+	10.6066i	0.121626	+	0.364878i
$846$		0			0
$847$	6.00000	+	6.00000i	0.206162	+	0.206162i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1.00000	+	1.00000i	−0.0342393	+	0.0342393i	−0.724019	−	0.689780i	$-0.757707\pi$
$853$	−1.00000	+	1.00000i	−0.0342393	+	0.0342393i	0.689780	+	0.724019i	$0.257707\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	15.5563	−	15.5563i	0.531395	−	0.531395i	−0.389593	−	0.920987i	$-0.627384\pi$
$857$	15.5563	−	15.5563i	0.531395	−	0.531395i	0.920987	+	0.389593i	$0.127384\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	16.9706	+	16.9706i	0.577685	+	0.577685i	0.934265	−	0.356580i	$-0.116057\pi$
$863$	16.9706	+	16.9706i	0.577685	+	0.577685i	−0.356580	+	0.934265i	$0.616057\pi$
$864$		0			0
$865$	−40.0000	−	20.0000i	−1.36004	−	0.680020i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−45.2548			−1.53517
$870$		0			0
$871$	24.0000			0.813209
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−31.1127	+	5.65685i	−1.05180	+	0.191237i
$876$		0			0
$877$	−27.0000	−	27.0000i	−0.911725	−	0.911725i	0.0846827	−	0.996408i	$-0.473012\pi$
$877$	−27.0000	−	27.0000i	−0.911725	−	0.911725i	−0.996408	+	0.0846827i	$0.973012\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	24.0416i		−	0.809983i	−0.914320	−	0.404992i	$-0.867274\pi$
$881$		−	24.0416i		−	0.809983i	0.914320	−	0.404992i	$-0.132726\pi$
$882$		0			0
$883$	24.0000	−	24.0000i	0.807664	−	0.807664i	−0.176616	−	0.984280i	$-0.556515\pi$
$883$	24.0000	−	24.0000i	0.807664	−	0.807664i	0.984280	+	0.176616i	$0.0565149\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	5.65685	−	5.65685i	0.189939	−	0.189939i	−0.605731	−	0.795670i	$-0.707119\pi$
$887$	5.65685	−	5.65685i	0.189939	−	0.189939i	0.795670	+	0.605731i	$0.207119\pi$
$888$		0			0
$889$		−	24.0000i		−	0.804934i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	11.3137	+	11.3137i	0.378599	+	0.378599i
$894$		0			0
$895$	−6.00000	+	2.00000i	−0.200558	+	0.0668526i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−10.0000			−0.333148
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−16.0000	−	16.0000i	−0.531271	−	0.531271i	0.389679	−	0.920951i	$-0.372586\pi$
$907$	−16.0000	−	16.0000i	−0.531271	−	0.531271i	−0.920951	+	0.389679i	$0.872586\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	22.6274i		−	0.749680i	−0.927090	−	0.374840i	$-0.877698\pi$
$911$		−	22.6274i		−	0.749680i	0.927090	−	0.374840i	$-0.122302\pi$
$912$		0			0
$913$	8.00000	−	8.00000i	0.264761	−	0.264761i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−28.2843	+	28.2843i	−0.934029	+	0.934029i
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−33.9411	−	33.9411i	−1.11719	−	1.11719i
$924$		0			0
$925$	7.00000	+	1.00000i	0.230159	+	0.0328798i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−18.3848			−0.603185			−0.301592	−	0.953437i	$-0.597518\pi$
$929$	−18.3848			−0.603185			−0.301592	+	0.953437i	$0.597518\pi$
$930$		0			0
$931$	4.00000			0.131095
$932$		0			0
$933$		0			0
$934$		0			0
$935$	5.65685	+	2.82843i	0.184999	+	0.0924995i
$936$		0			0
$937$	−35.0000	−	35.0000i	−1.14340	−	1.14340i	−0.987824	−	0.155576i	$-0.950277\pi$
$937$	−35.0000	−	35.0000i	−1.14340	−	1.14340i	−0.155576	−	0.987824i	$-0.549723\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	57.9828i		−	1.89018i	−0.326805	−	0.945092i	$-0.605972\pi$
$941$		−	57.9828i		−	1.89018i	0.326805	−	0.945092i	$-0.394028\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−28.2843	+	28.2843i	−0.919115	+	0.919115i	−0.996965	−	0.0778498i	$-0.975195\pi$
$947$	−28.2843	+	28.2843i	−0.919115	+	0.919115i	0.0778498	+	0.996965i	$0.475195\pi$
$948$		0			0
$949$			30.0000i			0.973841i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−25.4558	−	25.4558i	−0.824596	−	0.824596i	0.162168	−	0.986763i	$-0.448151\pi$
$953$	−25.4558	−	25.4558i	−0.824596	−	0.824596i	−0.986763	+	0.162168i	$0.948151\pi$
$954$		0			0
$955$	8.00000	+	24.0000i	0.258874	+	0.776622i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	33.9411			1.09602
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−18.3848	+	36.7696i	−0.591827	+	1.18365i
$966$		0			0
$967$	−30.0000	−	30.0000i	−0.964735	−	0.964735i	0.0346641	−	0.999399i	$-0.488964\pi$
$967$	−30.0000	−	30.0000i	−0.964735	−	0.964735i	−0.999399	+	0.0346641i	$0.988964\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			31.1127i			0.998454i	0.866471	+	0.499227i	$0.166383\pi$
$971$			31.1127i			0.998454i	−0.866471	+	0.499227i	$0.833617\pi$
$972$		0			0
$973$	−32.0000	+	32.0000i	−1.02587	+	1.02587i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	9.89949	−	9.89949i	0.316713	−	0.316713i	−0.530790	−	0.847503i	$-0.678105\pi$
$977$	9.89949	−	9.89949i	0.316713	−	0.316713i	0.847503	+	0.530790i	$0.178105\pi$
$978$		0			0
$979$		−	28.0000i		−	0.894884i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−19.7990	−	19.7990i	−0.631490	−	0.631490i	0.316952	−	0.948442i	$-0.397341\pi$
$983$	−19.7990	−	19.7990i	−0.631490	−	0.631490i	−0.948442	+	0.316952i	$0.897341\pi$
$984$		0			0
$985$	12.0000	−	24.0000i	0.382352	−	0.764704i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	36.0000			1.14358			0.571789	−	0.820401i	$-0.306250\pi$
$991$	36.0000			1.14358			0.571789	+	0.820401i	$0.306250\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−2.82843	−	8.48528i	−0.0896672	−	0.269002i
$996$		0			0
$997$	−15.0000	−	15.0000i	−0.475055	−	0.475055i	0.428491	−	0.903546i	$-0.359045\pi$
$997$	−15.0000	−	15.0000i	−0.475055	−	0.475055i	−0.903546	+	0.428491i	$0.859045\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3060.2.w.b.2177.1	yes	4
3.2	odd	2	inner	3060.2.w.b.2177.2	yes	4
5.3	odd	4	inner	3060.2.w.b.953.2	yes	4
15.8	even	4	inner	3060.2.w.b.953.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3060.2.w.b.953.1	✓	4	15.8	even	4	inner
3060.2.w.b.953.2	yes	4	5.3	odd	4	inner
3060.2.w.b.2177.1	yes	4	1.1	even	1	trivial
3060.2.w.b.2177.2	yes	4	3.2	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3060.w (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.12132 + 0.707107i) q^{5} +(2.00000 + 2.00000i) q^{7} +2.82843i q^{11} +(3.00000 - 3.00000i) q^{13} +(-0.707107 + 0.707107i) q^{17} -4.00000i q^{19} +(4.00000 - 3.00000i) q^{25} +7.07107 q^{29} +(-5.65685 - 2.82843i) q^{35} +(1.00000 + 1.00000i) q^{37} +7.07107i q^{41} +(4.00000 - 4.00000i) q^{43} +(-2.82843 + 2.82843i) q^{47} +1.00000i q^{49} +(7.07107 + 7.07107i) q^{53} +(-2.00000 - 6.00000i) q^{55} +2.82843 q^{59} -8.00000 q^{61} +(-4.24264 + 8.48528i) q^{65} +(4.00000 + 4.00000i) q^{67} -11.3137i q^{71} +(-5.00000 + 5.00000i) q^{73} +(-5.65685 + 5.65685i) q^{77} +16.0000i q^{79} +(-2.82843 - 2.82843i) q^{83} +(1.00000 - 2.00000i) q^{85} -9.89949 q^{89} +12.0000 q^{91} +(2.82843 + 8.48528i) q^{95} +(7.00000 + 7.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{7} + 12 q^{13} + 16 q^{25} + 4 q^{37} + 16 q^{43} - 8 q^{55} - 32 q^{61} + 16 q^{67} - 20 q^{73} + 4 q^{85} + 48 q^{91} + 28 q^{97}+O(q^{100}) \) 4 * q + 8 * q^7 + 12 * q^13 + 16 * q^25 + 4 * q^37 + 16 * q^43 - 8 * q^55 - 32 * q^61 + 16 * q^67 - 20 * q^73 + 4 * q^85 + 48 * q^91 + 28 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more