LMFDB - Embedded newform 3060.2.z.a.2809.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3060,2,Mod(829,3060)] 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("3060.829"); 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, 0, 2, 1])) 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.z (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-2,0,0] 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:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 340)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2809.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3060.2809
Dual form			3060.2.z.a.829.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 2.00000i) q^{5} +(2.00000 + 2.00000i) q^{11} -2.00000i q^{13} +(1.00000 + 4.00000i) q^{17} +(-4.00000 + 4.00000i) q^{23} +(-3.00000 + 4.00000i) q^{25} +(5.00000 - 5.00000i) q^{29} +(4.00000 - 4.00000i) q^{31} +(5.00000 + 5.00000i) q^{37} +(5.00000 + 5.00000i) q^{41} +4.00000 q^{43} -8.00000i q^{47} +7.00000i q^{49} -6.00000 q^{53} +(2.00000 - 6.00000i) q^{55} +12.0000i q^{59} +(3.00000 + 3.00000i) q^{61} +(-4.00000 + 2.00000i) q^{65} -4.00000i q^{67} +(8.00000 - 8.00000i) q^{71} +(-1.00000 - 1.00000i) q^{73} +(-12.0000 - 12.0000i) q^{79} -4.00000 q^{83} +(7.00000 - 6.00000i) q^{85} +16.0000 q^{89} +(-5.00000 - 5.00000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 4 q^{11} + 2 q^{17} - 8 q^{23} - 6 q^{25} + 10 q^{29} + 8 q^{31} + 10 q^{37} + 10 q^{41} + 8 q^{43} - 12 q^{53} + 4 q^{55} + 6 q^{61} - 8 q^{65} + 16 q^{71} - 2 q^{73} - 24 q^{79} - 8 q^{83}+ \cdots - 10 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 4 * q^11 + 2 * q^17 - 8 * q^23 - 6 * q^25 + 10 * q^29 + 8 * q^31 + 10 * q^37 + 10 * q^41 + 8 * q^43 - 12 * q^53 + 4 * q^55 + 6 * q^61 - 8 * q^65 + 16 * q^71 - 2 * q^73 - 24 * q^79 - 8 * q^83 + 14 * q^85 + 32 * q^89 - 10 * q^97

Character values

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

$n$	$1261$	$1361$	$1531$	$1837$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$6$		0			0
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.00000	+	2.00000i	0.603023	+	0.603023i	0.941113	−	0.338091i	$-0.109781\pi$
$11$	2.00000	+	2.00000i	0.603023	+	0.603023i	−0.338091	+	0.941113i	$0.609781\pi$
$12$		0			0
$13$		−	2.00000i		−	0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$		−	2.00000i		−	0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.00000	+	4.00000i	0.242536	+	0.970143i
$17$	1.00000	+	4.00000i	0.242536	+	0.970143i
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.00000	+	4.00000i	−0.834058	+	0.834058i	−0.988069	−	0.154011i	$-0.950781\pi$
$23$	−4.00000	+	4.00000i	−0.834058	+	0.834058i	0.154011	+	0.988069i	$0.450781\pi$
$24$		0			0
$25$	−3.00000	+	4.00000i	−0.600000	+	0.800000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	5.00000	−	5.00000i	0.928477	−	0.928477i	−0.0691309	−	0.997608i	$-0.522023\pi$
$29$	5.00000	−	5.00000i	0.928477	−	0.928477i	0.997608	+	0.0691309i	$0.0220226\pi$
$30$		0			0
$31$	4.00000	−	4.00000i	0.718421	−	0.718421i	−0.249861	−	0.968282i	$-0.580385\pi$
$31$	4.00000	−	4.00000i	0.718421	−	0.718421i	0.968282	+	0.249861i	$0.0803848\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	5.00000	+	5.00000i	0.821995	+	0.821995i	0.986394	−	0.164399i	$-0.0525685\pi$
$37$	5.00000	+	5.00000i	0.821995	+	0.821995i	−0.164399	+	0.986394i	$0.552568\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5.00000	+	5.00000i	0.780869	+	0.780869i	0.979977	−	0.199109i	$-0.0638047\pi$
$41$	5.00000	+	5.00000i	0.780869	+	0.780869i	−0.199109	+	0.979977i	$0.563805\pi$
$42$		0			0
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	8.00000i		−	1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$		−	8.00000i		−	1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$		0			0
$49$			7.00000i			1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$	2.00000	−	6.00000i	0.269680	−	0.809040i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			12.0000i			1.56227i	0.624364	+	0.781133i	$0.285358\pi$
$59$			12.0000i			1.56227i	−0.624364	+	0.781133i	$0.714642\pi$
$60$		0			0
$61$	3.00000	+	3.00000i	0.384111	+	0.384111i	0.872581	−	0.488470i	$-0.162445\pi$
$61$	3.00000	+	3.00000i	0.384111	+	0.384111i	−0.488470	+	0.872581i	$0.662445\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.00000	+	2.00000i	−0.496139	+	0.248069i
$66$		0			0
$67$		−	4.00000i		−	0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$		−	4.00000i		−	0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	8.00000	−	8.00000i	0.949425	−	0.949425i	−0.0493559	−	0.998781i	$-0.515717\pi$
$71$	8.00000	−	8.00000i	0.949425	−	0.949425i	0.998781	+	0.0493559i	$0.0157169\pi$
$72$		0			0
$73$	−1.00000	−	1.00000i	−0.117041	−	0.117041i	0.646160	−	0.763202i	$-0.276374\pi$
$73$	−1.00000	−	1.00000i	−0.117041	−	0.117041i	−0.763202	+	0.646160i	$0.776374\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−12.0000	−	12.0000i	−1.35011	−	1.35011i	−0.885537	−	0.464568i	$-0.846210\pi$
$79$	−12.0000	−	12.0000i	−1.35011	−	1.35011i	−0.464568	−	0.885537i	$-0.653790\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.00000			−0.439057			−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000			−0.439057			−0.219529	+	0.975606i	$0.570452\pi$
$84$		0			0
$85$	7.00000	−	6.00000i	0.759257	−	0.650791i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	16.0000			1.69600			0.847998	−	0.529999i	$-0.177808\pi$
$89$	16.0000			1.69600			0.847998	+	0.529999i	$0.177808\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−5.00000	−	5.00000i	−0.507673	−	0.507673i	0.406138	−	0.913812i	$-0.366875\pi$
$97$	−5.00000	−	5.00000i	−0.507673	−	0.507673i	−0.913812	+	0.406138i	$0.866875\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−8.00000			−0.796030			−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000			−0.796030			−0.398015	+	0.917379i	$0.630301\pi$
$102$		0			0
$103$		−	12.0000i		−	1.18240i	−0.806527	−	0.591198i	$-0.798655\pi$
$103$		−	12.0000i		−	1.18240i	0.806527	−	0.591198i	$-0.201345\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	14.0000	+	14.0000i	1.35343	+	1.35343i	0.881791	+	0.471640i	$0.156338\pi$
$107$	14.0000	+	14.0000i	1.35343	+	1.35343i	0.471640	+	0.881791i	$0.343662\pi$
$108$		0			0
$109$	3.00000	+	3.00000i	0.287348	+	0.287348i	0.836031	−	0.548683i	$-0.184871\pi$
$109$	3.00000	+	3.00000i	0.287348	+	0.287348i	−0.548683	+	0.836031i	$0.684871\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	11.0000	−	11.0000i	1.03479	−	1.03479i	0.0354205	−	0.999372i	$-0.488723\pi$
$113$	11.0000	−	11.0000i	1.03479	−	1.03479i	0.999372	−	0.0354205i	$-0.0112770\pi$
$114$		0			0
$115$	12.0000	+	4.00000i	1.11901	+	0.373002i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$		−	3.00000i		−	0.272727i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.0000	+	2.00000i	0.983870	+	0.178885i
$126$		0			0
$127$	20.0000			1.77471			0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000			1.77471			0.887357	+	0.461084i	$0.152539\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	6.00000	−	6.00000i	0.524222	−	0.524222i	−0.394621	−	0.918844i	$-0.629124\pi$
$131$	6.00000	−	6.00000i	0.524222	−	0.524222i	0.918844	+	0.394621i	$0.129124\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	6.00000i		−	0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$		−	6.00000i		−	0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$		0			0
$139$	2.00000	−	2.00000i	0.169638	−	0.169638i	−0.617182	−	0.786820i	$-0.711726\pi$
$139$	2.00000	−	2.00000i	0.169638	−	0.169638i	0.786820	+	0.617182i	$0.211726\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	4.00000	−	4.00000i	0.334497	−	0.334497i
$144$		0			0
$145$	−15.0000	−	5.00000i	−1.24568	−	0.415227i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	22.0000			1.80231			0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000			1.80231			0.901155	+	0.433497i	$0.142720\pi$
$150$		0			0
$151$			8.00000i			0.651031i	0.945537	+	0.325515i	$0.105538\pi$
$151$			8.00000i			0.651031i	−0.945537	+	0.325515i	$0.894462\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−12.0000	−	4.00000i	−0.963863	−	0.321288i
$156$		0			0
$157$			16.0000i			1.27694i	0.769647	+	0.638470i	$0.220432\pi$
$157$			16.0000i			1.27694i	−0.769647	+	0.638470i	$0.779568\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	6.00000	−	6.00000i	0.469956	−	0.469956i	−0.431944	−	0.901900i	$-0.642172\pi$
$163$	6.00000	−	6.00000i	0.469956	−	0.469956i	0.901900	+	0.431944i	$0.142172\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−8.00000	−	8.00000i	−0.619059	−	0.619059i	0.326231	−	0.945290i	$-0.394221\pi$
$167$	−8.00000	−	8.00000i	−0.619059	−	0.619059i	−0.945290	+	0.326231i	$0.894221\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−7.00000	−	7.00000i	−0.532200	−	0.532200i	0.389026	−	0.921227i	$-0.372811\pi$
$173$	−7.00000	−	7.00000i	−0.532200	−	0.532200i	−0.921227	+	0.389026i	$0.872811\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	17.0000	+	17.0000i	1.26360	+	1.26360i	0.949335	+	0.314265i	$0.101758\pi$
$181$	17.0000	+	17.0000i	1.26360	+	1.26360i	0.314265	+	0.949335i	$0.398242\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	5.00000	−	15.0000i	0.367607	−	1.10282i
$186$		0			0
$187$	−6.00000	+	10.0000i	−0.438763	+	0.731272i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4.00000			−0.289430			−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000			−0.289430			−0.144715	+	0.989473i	$0.546227\pi$
$192$		0			0
$193$	−5.00000	+	5.00000i	−0.359908	+	0.359908i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	−5.00000	+	5.00000i	−0.359908	+	0.359908i	0.503871	+	0.863779i	$0.331909\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	5.00000	−	5.00000i	0.356235	−	0.356235i	−0.506188	−	0.862423i	$-0.668946\pi$
$197$	5.00000	−	5.00000i	0.356235	−	0.356235i	0.862423	+	0.506188i	$0.168946\pi$
$198$		0			0
$199$	−12.0000	+	12.0000i	−0.850657	+	0.850657i	−0.990214	−	0.139557i	$-0.955432\pi$
$199$	−12.0000	+	12.0000i	−0.850657	+	0.850657i	0.139557	+	0.990214i	$0.455432\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	5.00000	−	15.0000i	0.349215	−	1.04765i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−10.0000	−	10.0000i	−0.688428	−	0.688428i	0.273456	−	0.961884i	$-0.411833\pi$
$211$	−10.0000	−	10.0000i	−0.688428	−	0.688428i	−0.961884	+	0.273456i	$0.911833\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−4.00000	−	8.00000i	−0.272798	−	0.545595i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	8.00000	−	2.00000i	0.538138	−	0.134535i
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2.00000	+	2.00000i	−0.132745	+	0.132745i	−0.770357	−	0.637613i	$-0.779922\pi$
$227$	−2.00000	+	2.00000i	−0.132745	+	0.132745i	0.637613	+	0.770357i	$0.279922\pi$
$228$		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
$233$	1.00000	+	1.00000i	0.0655122	+	0.0655122i	0.739104	−	0.673592i	$-0.235249\pi$
$233$	1.00000	+	1.00000i	0.0655122	+	0.0655122i	−0.673592	+	0.739104i	$0.735249\pi$
$234$		0			0
$235$	−16.0000	+	8.00000i	−1.04372	+	0.521862i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	12.0000			0.776215			0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000			0.776215			0.388108	+	0.921614i	$0.373129\pi$
$240$		0			0
$241$	7.00000	−	7.00000i	0.450910	−	0.450910i	−0.444747	−	0.895656i	$-0.646706\pi$
$241$	7.00000	−	7.00000i	0.450910	−	0.450910i	0.895656	+	0.444747i	$0.146706\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	14.0000	−	7.00000i	0.894427	−	0.447214i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$	−16.0000			−1.00591
$254$		0			0
$255$		0			0
$256$		0			0
$257$	24.0000			1.49708			0.748539	−	0.663090i	$-0.230755\pi$
$257$	24.0000			1.49708			0.748539	+	0.663090i	$0.230755\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	12.0000			0.739952			0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000			0.739952			0.369976	+	0.929041i	$0.379366\pi$
$264$		0			0
$265$	6.00000	+	12.0000i	0.368577	+	0.737154i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	5.00000	−	5.00000i	0.304855	−	0.304855i	−0.538055	−	0.842910i	$-0.680841\pi$
$269$	5.00000	−	5.00000i	0.304855	−	0.304855i	0.842910	+	0.538055i	$0.180841\pi$
$270$		0			0
$271$	4.00000			0.242983			0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000			0.242983			0.121491	+	0.992592i	$0.461232\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−14.0000	+	2.00000i	−0.844232	+	0.120605i
$276$		0			0
$277$	−3.00000	−	3.00000i	−0.180253	−	0.180253i	0.611213	−	0.791466i	$-0.290682\pi$
$277$	−3.00000	−	3.00000i	−0.180253	−	0.180253i	−0.791466	+	0.611213i	$0.790682\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			24.0000i			1.43172i	0.698244	+	0.715860i	$0.253965\pi$
$281$			24.0000i			1.43172i	−0.698244	+	0.715860i	$0.746035\pi$
$282$		0			0
$283$	−10.0000	+	10.0000i	−0.594438	+	0.594438i	−0.938827	−	0.344389i	$-0.888086\pi$
$283$	−10.0000	+	10.0000i	−0.594438	+	0.594438i	0.344389	+	0.938827i	$0.388086\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−15.0000	+	8.00000i	−0.882353	+	0.470588i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			16.0000i			0.934730i	0.884064	+	0.467365i	$0.154797\pi$
$293$			16.0000i			0.934730i	−0.884064	+	0.467365i	$0.845203\pi$
$294$		0			0
$295$	24.0000	−	12.0000i	1.39733	−	0.698667i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	8.00000	+	8.00000i	0.462652	+	0.462652i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	3.00000	−	9.00000i	0.171780	−	0.515339i
$306$		0			0
$307$			32.0000i			1.82634i	0.407583	+	0.913168i	$0.366372\pi$
$307$			32.0000i			1.82634i	−0.407583	+	0.913168i	$0.633628\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−12.0000	+	12.0000i	−0.680458	+	0.680458i	−0.960103	−	0.279646i	$-0.909783\pi$
$311$	−12.0000	+	12.0000i	−0.680458	+	0.680458i	0.279646	+	0.960103i	$0.409783\pi$
$312$		0			0
$313$	−17.0000	+	17.0000i	−0.960897	+	0.960897i	−0.999264	−	0.0383669i	$-0.987784\pi$
$313$	−17.0000	+	17.0000i	−0.960897	+	0.960897i	0.0383669	+	0.999264i	$0.487784\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−17.0000	+	17.0000i	−0.954815	+	0.954815i	−0.999022	−	0.0442073i	$-0.985924\pi$
$317$	−17.0000	+	17.0000i	−0.954815	+	0.954815i	0.0442073	+	0.999022i	$0.485924\pi$
$318$		0			0
$319$	20.0000			1.11979
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	8.00000	+	6.00000i	0.443760	+	0.332820i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	4.00000i		−	0.219860i	−0.993939	−	0.109930i	$-0.964937\pi$
$331$		−	4.00000i		−	0.219860i	0.993939	−	0.109930i	$-0.0350627\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−8.00000	+	4.00000i	−0.437087	+	0.218543i
$336$		0			0
$337$	13.0000	+	13.0000i	0.708155	+	0.708155i	0.966147	−	0.257992i	$-0.0830608\pi$
$337$	13.0000	+	13.0000i	0.708155	+	0.708155i	−0.257992	+	0.966147i	$0.583061\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	16.0000			0.866449
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−14.0000	+	14.0000i	−0.751559	+	0.751559i	−0.974770	−	0.223211i	$-0.928346\pi$
$347$	−14.0000	+	14.0000i	−0.751559	+	0.751559i	0.223211	+	0.974770i	$0.428346\pi$
$348$		0			0
$349$		−	24.0000i		−	1.28469i	−0.766415	−	0.642345i	$-0.777962\pi$
$349$		−	24.0000i		−	1.28469i	0.766415	−	0.642345i	$-0.222038\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	24.0000i		−	1.27739i	−0.769460	−	0.638696i	$-0.779474\pi$
$353$		−	24.0000i		−	1.27739i	0.769460	−	0.638696i	$-0.220526\pi$
$354$		0			0
$355$	−24.0000	−	8.00000i	−1.27379	−	0.424596i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	20.0000i		−	1.05556i	−0.849381	−	0.527780i	$-0.823025\pi$
$359$		−	20.0000i		−	1.05556i	0.849381	−	0.527780i	$-0.176975\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.00000	+	3.00000i	−0.0523424	+	0.157027i
$366$		0			0
$367$	−8.00000	+	8.00000i	−0.417597	+	0.417597i	−0.884375	−	0.466778i	$-0.845415\pi$
$367$	−8.00000	+	8.00000i	−0.417597	+	0.417597i	0.466778	+	0.884375i	$0.345415\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	26.0000i		−	1.34623i	−0.739538	−	0.673114i	$-0.764956\pi$
$373$		−	26.0000i		−	1.34623i	0.739538	−	0.673114i	$-0.235044\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−10.0000	−	10.0000i	−0.515026	−	0.515026i
$378$		0			0
$379$	2.00000	−	2.00000i	0.102733	−	0.102733i	−0.653872	−	0.756605i	$-0.726857\pi$
$379$	2.00000	−	2.00000i	0.102733	−	0.102733i	0.756605	+	0.653872i	$0.226857\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4.00000			0.204390			0.102195	−	0.994764i	$-0.467413\pi$
$383$	4.00000			0.204390			0.102195	+	0.994764i	$0.467413\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$			6.00000i			0.304212i	0.988364	+	0.152106i	$0.0486055\pi$
$389$			6.00000i			0.304212i	−0.988364	+	0.152106i	$0.951394\pi$
$390$		0			0
$391$	−20.0000	−	12.0000i	−1.01144	−	0.606866i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−12.0000	+	36.0000i	−0.603786	+	1.81136i
$396$		0			0
$397$	7.00000	−	7.00000i	0.351320	−	0.351320i	−0.509281	−	0.860601i	$-0.670088\pi$
$397$	7.00000	−	7.00000i	0.351320	−	0.351320i	0.860601	+	0.509281i	$0.170088\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.00000	−	9.00000i	−0.449439	−	0.449439i	0.445729	−	0.895168i	$-0.352944\pi$
$401$	−9.00000	−	9.00000i	−0.449439	−	0.449439i	−0.895168	+	0.445729i	$0.852944\pi$
$402$		0			0
$403$	−8.00000	−	8.00000i	−0.398508	−	0.398508i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			20.0000i			0.991363i
$408$		0			0
$409$	−14.0000			−0.692255			−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000			−0.692255			−0.346128	+	0.938187i	$0.612504\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	4.00000	+	8.00000i	0.196352	+	0.392705i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−26.0000	−	26.0000i	−1.27018	−	1.27018i	−0.945991	−	0.324192i	$-0.894908\pi$
$419$	−26.0000	−	26.0000i	−1.27018	−	1.27018i	−0.324192	−	0.945991i	$-0.605092\pi$
$420$		0			0
$421$	−8.00000			−0.389896			−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000			−0.389896			−0.194948	+	0.980814i	$0.562454\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−19.0000	−	8.00000i	−0.921635	−	0.388057i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	4.00000	+	4.00000i	0.192673	+	0.192673i	0.796850	−	0.604177i	$-0.206498\pi$
$431$	4.00000	+	4.00000i	0.192673	+	0.192673i	−0.604177	+	0.796850i	$0.706498\pi$
$432$		0			0
$433$	16.0000			0.768911			0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000			0.768911			0.384455	+	0.923144i	$0.374389\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−28.0000	+	28.0000i	−1.33637	+	1.33637i	−0.436816	+	0.899551i	$0.643894\pi$
$439$	−28.0000	+	28.0000i	−1.33637	+	1.33637i	−0.899551	+	0.436816i	$0.856106\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	12.0000i		−	0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$		−	12.0000i		−	0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$		0			0
$445$	−16.0000	−	32.0000i	−0.758473	−	1.51695i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	23.0000	+	23.0000i	1.08544	+	1.08544i	0.995992	+	0.0894454i	$0.0285095\pi$
$449$	23.0000	+	23.0000i	1.08544	+	1.08544i	0.0894454	+	0.995992i	$0.471491\pi$
$450$		0			0
$451$			20.0000i			0.941763i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	32.0000			1.49690			0.748448	−	0.663193i	$-0.230799\pi$
$457$	32.0000			1.49690			0.748448	+	0.663193i	$0.230799\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	24.0000i		−	1.11779i	−0.829238	−	0.558896i	$-0.811225\pi$
$461$		−	24.0000i		−	1.11779i	0.829238	−	0.558896i	$-0.188775\pi$
$462$		0			0
$463$			12.0000i			0.557687i	0.960337	+	0.278844i	$0.0899511\pi$
$463$			12.0000i			0.557687i	−0.960337	+	0.278844i	$0.910049\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	8.00000	+	8.00000i	0.367840	+	0.367840i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$479$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$480$		0			0
$481$	10.0000	−	10.0000i	0.455961	−	0.455961i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−5.00000	+	15.0000i	−0.227038	+	0.681115i
$486$		0			0
$487$	8.00000	−	8.00000i	0.362515	−	0.362515i	−0.502223	−	0.864738i	$-0.667484\pi$
$487$	8.00000	−	8.00000i	0.362515	−	0.362515i	0.864738	+	0.502223i	$0.167484\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	25.0000	+	15.0000i	1.12594	+	0.675566i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−6.00000	−	6.00000i	−0.268597	−	0.268597i	0.559938	−	0.828535i	$-0.310825\pi$
$499$	−6.00000	−	6.00000i	−0.268597	−	0.268597i	−0.828535	+	0.559938i	$0.810825\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−8.00000	+	8.00000i	−0.356702	+	0.356702i	−0.862596	−	0.505894i	$-0.831163\pi$
$503$	−8.00000	+	8.00000i	−0.356702	+	0.356702i	0.505894	+	0.862596i	$0.331163\pi$
$504$		0			0
$505$	8.00000	+	16.0000i	0.355995	+	0.711991i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−18.0000			−0.797836			−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000			−0.797836			−0.398918	+	0.916987i	$0.630614\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−24.0000	+	12.0000i	−1.05757	+	0.528783i
$516$		0			0
$517$	16.0000	−	16.0000i	0.703679	−	0.703679i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−29.0000	−	29.0000i	−1.27051	−	1.27051i	−0.945819	−	0.324694i	$-0.894739\pi$
$521$	−29.0000	−	29.0000i	−1.27051	−	1.27051i	−0.324694	−	0.945819i	$-0.605261\pi$
$522$		0			0
$523$		−	36.0000i		−	1.57417i	−0.616844	−	0.787085i	$-0.711589\pi$
$523$		−	36.0000i		−	1.57417i	0.616844	−	0.787085i	$-0.288411\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	20.0000	+	12.0000i	0.871214	+	0.522728i
$528$		0			0
$529$		−	9.00000i		−	0.391304i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	10.0000	−	10.0000i	0.433148	−	0.433148i
$534$		0			0
$535$	14.0000	−	42.0000i	0.605273	−	1.81582i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−14.0000	+	14.0000i	−0.603023	+	0.603023i
$540$		0			0
$541$	−27.0000	+	27.0000i	−1.16082	+	1.16082i	−0.176524	+	0.984296i	$0.556485\pi$
$541$	−27.0000	+	27.0000i	−1.16082	+	1.16082i	−0.984296	+	0.176524i	$0.943515\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	3.00000	−	9.00000i	0.128506	−	0.385518i
$546$		0			0
$547$	22.0000	+	22.0000i	0.940652	+	0.940652i	0.998335	−	0.0576829i	$-0.0183712\pi$
$547$	22.0000	+	22.0000i	0.940652	+	0.940652i	−0.0576829	+	0.998335i	$0.518371\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		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
$559$		−	8.00000i		−	0.338364i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−44.0000			−1.85438			−0.927189	−	0.374593i	$-0.877783\pi$
$563$	−44.0000			−1.85438			−0.927189	+	0.374593i	$0.877783\pi$
$564$		0			0
$565$	−33.0000	−	11.0000i	−1.38832	−	0.462773i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	−2.00000	−	2.00000i	−0.0836974	−	0.0836974i	0.664019	−	0.747716i	$-0.268850\pi$
$571$	−2.00000	−	2.00000i	−0.0836974	−	0.0836974i	−0.747716	+	0.664019i	$0.768850\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.00000	−	28.0000i	−0.166812	−	1.16768i
$576$		0			0
$577$			2.00000i			0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$			2.00000i			0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−12.0000	−	12.0000i	−0.496989	−	0.496989i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	24.0000			0.990586			0.495293	−	0.868726i	$-0.335061\pi$
$587$	24.0000			0.990586			0.495293	+	0.868726i	$0.335061\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−22.0000			−0.903432			−0.451716	−	0.892162i	$-0.649188\pi$
$593$	−22.0000			−0.903432			−0.451716	+	0.892162i	$0.649188\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−24.0000			−0.980613			−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000			−0.980613			−0.490307	+	0.871550i	$0.663115\pi$
$600$		0			0
$601$	3.00000	+	3.00000i	0.122373	+	0.122373i	0.765641	−	0.643268i	$-0.222422\pi$
$601$	3.00000	+	3.00000i	0.122373	+	0.122373i	−0.643268	+	0.765641i	$0.722422\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−6.00000	+	3.00000i	−0.243935	+	0.121967i
$606$		0			0
$607$	16.0000	+	16.0000i	0.649420	+	0.649420i	0.952853	−	0.303433i	$-0.0981328\pi$
$607$	16.0000	+	16.0000i	0.649420	+	0.649420i	−0.303433	+	0.952853i	$0.598133\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−16.0000			−0.647291
$612$		0			0
$613$		−	16.0000i		−	0.646234i	−0.946359	−	0.323117i	$-0.895269\pi$
$613$		−	16.0000i		−	0.646234i	0.946359	−	0.323117i	$-0.104731\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	23.0000	+	23.0000i	0.925945	+	0.925945i	0.997441	−	0.0714958i	$-0.0227772\pi$
$617$	23.0000	+	23.0000i	0.925945	+	0.925945i	−0.0714958	+	0.997441i	$0.522777\pi$
$618$		0			0
$619$	−14.0000	−	14.0000i	−0.562708	−	0.562708i	0.367368	−	0.930076i	$-0.380259\pi$
$619$	−14.0000	−	14.0000i	−0.562708	−	0.562708i	−0.930076	+	0.367368i	$0.880259\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−15.0000	+	25.0000i	−0.598089	+	0.996815i
$630$		0			0
$631$		−	4.00000i		−	0.159237i	−0.996825	−	0.0796187i	$-0.974630\pi$
$631$		−	4.00000i		−	0.159237i	0.996825	−	0.0796187i	$-0.0253703\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−20.0000	−	40.0000i	−0.793676	−	1.58735i
$636$		0			0
$637$	14.0000			0.554700
$638$		0			0
$639$		0			0
$640$		0			0
$641$	15.0000	−	15.0000i	0.592464	−	0.592464i	−0.345832	−	0.938296i	$-0.612403\pi$
$641$	15.0000	−	15.0000i	0.592464	−	0.592464i	0.938296	+	0.345832i	$0.112403\pi$
$642$		0			0
$643$	−22.0000	−	22.0000i	−0.867595	−	0.867595i	0.124610	−	0.992206i	$-0.460232\pi$
$643$	−22.0000	−	22.0000i	−0.867595	−	0.867595i	−0.992206	+	0.124610i	$0.960232\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			16.0000i			0.629025i	0.949253	+	0.314512i	$0.101841\pi$
$647$			16.0000i			0.629025i	−0.949253	+	0.314512i	$0.898159\pi$
$648$		0			0
$649$	−24.0000	+	24.0000i	−0.942082	+	0.942082i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−7.00000	+	7.00000i	−0.273931	+	0.273931i	−0.830681	−	0.556749i	$-0.812048\pi$
$653$	−7.00000	+	7.00000i	−0.273931	+	0.273931i	0.556749	+	0.830681i	$0.312048\pi$
$654$		0			0
$655$	−18.0000	−	6.00000i	−0.703318	−	0.234439i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−40.0000			−1.55818			−0.779089	−	0.626913i	$-0.784318\pi$
$659$	−40.0000			−1.55818			−0.779089	+	0.626913i	$0.784318\pi$
$660$		0			0
$661$		−	24.0000i		−	0.933492i	−0.884391	−	0.466746i	$-0.845426\pi$
$661$		−	24.0000i		−	0.933492i	0.884391	−	0.466746i	$-0.154574\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			40.0000i			1.54881i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			12.0000i			0.463255i
$672$		0			0
$673$	−19.0000	+	19.0000i	−0.732396	+	0.732396i	−0.971094	−	0.238698i	$-0.923279\pi$
$673$	−19.0000	+	19.0000i	−0.732396	+	0.732396i	0.238698	+	0.971094i	$0.423279\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−5.00000	−	5.00000i	−0.192166	−	0.192166i	0.604466	−	0.796631i	$-0.293387\pi$
$677$	−5.00000	−	5.00000i	−0.192166	−	0.192166i	−0.796631	+	0.604466i	$0.793387\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	22.0000	+	22.0000i	0.841807	+	0.841807i	0.989094	−	0.147287i	$-0.0470541\pi$
$683$	22.0000	+	22.0000i	0.841807	+	0.841807i	−0.147287	+	0.989094i	$0.547054\pi$
$684$		0			0
$685$	−12.0000	+	6.00000i	−0.458496	+	0.229248i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			12.0000i			0.457164i
$690$		0			0
$691$	−26.0000	−	26.0000i	−0.989087	−	0.989087i	0.0108545	−	0.999941i	$-0.496545\pi$
$691$	−26.0000	−	26.0000i	−0.989087	−	0.989087i	−0.999941	+	0.0108545i	$0.996545\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−6.00000	−	2.00000i	−0.227593	−	0.0758643i
$696$		0			0
$697$	−15.0000	+	25.0000i	−0.568166	+	0.946943i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−8.00000			−0.302156			−0.151078	−	0.988522i	$-0.548274\pi$
$701$	−8.00000			−0.302156			−0.151078	+	0.988522i	$0.548274\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	15.0000	−	15.0000i	0.563337	−	0.563337i	−0.366917	−	0.930254i	$-0.619587\pi$
$709$	15.0000	−	15.0000i	0.563337	−	0.563337i	0.930254	+	0.366917i	$0.119587\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			32.0000i			1.19841i
$714$		0			0
$715$	−12.0000	−	4.00000i	−0.448775	−	0.149592i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−4.00000	+	4.00000i	−0.149175	+	0.149175i	−0.777749	−	0.628575i	$-0.783639\pi$
$719$	−4.00000	+	4.00000i	−0.149175	+	0.149175i	0.628575	+	0.777749i	$0.283639\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	5.00000	+	35.0000i	0.185695	+	1.29987i
$726$		0			0
$727$			16.0000i			0.593407i	0.954970	+	0.296704i	$0.0958873\pi$
$727$			16.0000i			0.593407i	−0.954970	+	0.296704i	$0.904113\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4.00000	+	16.0000i	0.147945	+	0.591781i
$732$		0			0
$733$	34.0000			1.25582			0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000			1.25582			0.627909	+	0.778287i	$0.283911\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	8.00000	−	8.00000i	0.294684	−	0.294684i
$738$		0			0
$739$			40.0000i			1.47142i	0.677295	+	0.735712i	$0.263152\pi$
$739$			40.0000i			1.47142i	−0.677295	+	0.735712i	$0.736848\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−12.0000	−	12.0000i	−0.440237	−	0.440237i	0.451854	−	0.892092i	$-0.350763\pi$
$743$	−12.0000	−	12.0000i	−0.440237	−	0.440237i	−0.892092	+	0.451854i	$0.850763\pi$
$744$		0			0
$745$	−22.0000	−	44.0000i	−0.806018	−	1.61204i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	4.00000	−	4.00000i	0.145962	−	0.145962i	−0.630349	−	0.776312i	$-0.717088\pi$
$751$	4.00000	−	4.00000i	0.145962	−	0.145962i	0.776312	+	0.630349i	$0.217088\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	16.0000	−	8.00000i	0.582300	−	0.291150i
$756$		0			0
$757$	16.0000			0.581530			0.290765	−	0.956795i	$-0.406090\pi$
$757$	16.0000			0.581530			0.290765	+	0.956795i	$0.406090\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	8.00000			0.290000			0.145000	−	0.989432i	$-0.453682\pi$
$761$	8.00000			0.290000			0.145000	+	0.989432i	$0.453682\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	24.0000			0.866590
$768$		0			0
$769$	24.0000			0.865462			0.432731	−	0.901523i	$-0.357550\pi$
$769$	24.0000			0.865462			0.432731	+	0.901523i	$0.357550\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	32.0000			1.15096			0.575480	−	0.817816i	$-0.304815\pi$
$773$	32.0000			1.15096			0.575480	+	0.817816i	$0.304815\pi$
$774$		0			0
$775$	4.00000	+	28.0000i	0.143684	+	1.00579i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	32.0000			1.14505
$782$		0			0
$783$		0			0
$784$		0			0
$785$	32.0000	−	16.0000i	1.14213	−	0.571064i
$786$		0			0
$787$	14.0000	+	14.0000i	0.499046	+	0.499046i	0.911141	−	0.412095i	$-0.135203\pi$
$787$	14.0000	+	14.0000i	0.499046	+	0.499046i	−0.412095	+	0.911141i	$0.635203\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	6.00000	−	6.00000i	0.213066	−	0.213066i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−30.0000			−1.06265			−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000			−1.06265			−0.531327	+	0.847167i	$0.678307\pi$
$798$		0			0
$799$	32.0000	−	8.00000i	1.13208	−	0.283020i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	4.00000i		−	0.141157i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−27.0000	−	27.0000i	−0.949269	−	0.949269i	0.0495045	−	0.998774i	$-0.484236\pi$
$809$	−27.0000	−	27.0000i	−0.949269	−	0.949269i	−0.998774	+	0.0495045i	$0.984236\pi$
$810$		0			0
$811$	−38.0000	+	38.0000i	−1.33436	+	1.33436i	−0.432936	+	0.901425i	$0.642522\pi$
$811$	−38.0000	+	38.0000i	−1.33436	+	1.33436i	−0.901425	+	0.432936i	$0.857478\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−18.0000	−	6.00000i	−0.630512	−	0.210171i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−25.0000	+	25.0000i	−0.872506	+	0.872506i	−0.992745	−	0.120239i	$-0.961634\pi$
$821$	−25.0000	+	25.0000i	−0.872506	+	0.872506i	0.120239	+	0.992745i	$0.461634\pi$
$822$		0			0
$823$	−20.0000	+	20.0000i	−0.697156	+	0.697156i	−0.963796	−	0.266640i	$-0.914087\pi$
$823$	−20.0000	+	20.0000i	−0.697156	+	0.697156i	0.266640	+	0.963796i	$0.414087\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−6.00000	+	6.00000i	−0.208640	+	0.208640i	−0.803689	−	0.595049i	$-0.797133\pi$
$827$	−6.00000	+	6.00000i	−0.208640	+	0.208640i	0.595049	+	0.803689i	$0.297133\pi$
$828$		0			0
$829$	−14.0000			−0.486240			−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000			−0.486240			−0.243120	+	0.969996i	$0.578171\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−28.0000	+	7.00000i	−0.970143	+	0.242536i
$834$		0			0
$835$	−8.00000	+	24.0000i	−0.276851	+	0.830554i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$839$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$840$		0			0
$841$		−	21.0000i		−	0.724138i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−9.00000	−	18.0000i	−0.309609	−	0.619219i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−40.0000			−1.37118
$852$		0			0
$853$	21.0000	+	21.0000i	0.719026	+	0.719026i	0.968406	−	0.249380i	$-0.0802267\pi$
$853$	21.0000	+	21.0000i	0.719026	+	0.719026i	−0.249380	+	0.968406i	$0.580227\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	9.00000	−	9.00000i	0.307434	−	0.307434i	−0.536479	−	0.843913i	$-0.680246\pi$
$857$	9.00000	−	9.00000i	0.307434	−	0.307434i	0.843913	+	0.536479i	$0.180246\pi$
$858$		0			0
$859$		−	36.0000i		−	1.22830i	−0.789188	−	0.614152i	$-0.789498\pi$
$859$		−	36.0000i		−	1.22830i	0.789188	−	0.614152i	$-0.210502\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	12.0000i		−	0.408485i	−0.978920	−	0.204242i	$-0.934527\pi$
$863$		−	12.0000i		−	0.408485i	0.978920	−	0.204242i	$-0.0654731\pi$
$864$		0			0
$865$	−7.00000	+	21.0000i	−0.238007	+	0.714021i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	48.0000i		−	1.62829i
$870$		0			0
$871$	−8.00000			−0.271070
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−33.0000	+	33.0000i	−1.11433	+	1.11433i	−0.121773	+	0.992558i	$0.538858\pi$
$877$	−33.0000	+	33.0000i	−1.11433	+	1.11433i	−0.992558	+	0.121773i	$0.961142\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−23.0000	+	23.0000i	−0.774890	+	0.774890i	−0.978957	−	0.204067i	$-0.934584\pi$
$881$	−23.0000	+	23.0000i	−0.774890	+	0.774890i	0.204067	+	0.978957i	$0.434584\pi$
$882$		0			0
$883$		−	4.00000i		−	0.134611i	−0.997732	−	0.0673054i	$-0.978560\pi$
$883$		−	4.00000i		−	0.134611i	0.997732	−	0.0673054i	$-0.0214402\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	4.00000	+	4.00000i	0.134307	+	0.134307i	0.771064	−	0.636757i	$-0.219725\pi$
$887$	4.00000	+	4.00000i	0.134307	+	0.134307i	−0.636757	+	0.771064i	$0.719725\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	40.0000i		−	1.33407i
$900$		0			0
$901$	−6.00000	−	24.0000i	−0.199889	−	0.799556i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	17.0000	−	51.0000i	0.565099	−	1.69530i
$906$		0			0
$907$	22.0000	−	22.0000i	0.730498	−	0.730498i	−0.240220	−	0.970718i	$-0.577220\pi$
$907$	22.0000	−	22.0000i	0.730498	−	0.730498i	0.970718	+	0.240220i	$0.0772197\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−16.0000	−	16.0000i	−0.530104	−	0.530104i	0.390499	−	0.920603i	$-0.372302\pi$
$911$	−16.0000	−	16.0000i	−0.530104	−	0.530104i	−0.920603	+	0.390499i	$0.872302\pi$
$912$		0			0
$913$	−8.00000	−	8.00000i	−0.264761	−	0.264761i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	12.0000			0.395843			0.197922	−	0.980218i	$-0.436581\pi$
$919$	12.0000			0.395843			0.197922	+	0.980218i	$0.436581\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−16.0000	−	16.0000i	−0.526646	−	0.526646i
$924$		0			0
$925$	−35.0000	+	5.00000i	−1.15079	+	0.164399i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	9.00000	+	9.00000i	0.295280	+	0.295280i	0.839162	−	0.543882i	$-0.183046\pi$
$929$	9.00000	+	9.00000i	0.295280	+	0.295280i	−0.543882	+	0.839162i	$0.683046\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	26.0000	+	2.00000i	0.850291	+	0.0654070i
$936$		0			0
$937$	10.0000			0.326686			0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000			0.326686			0.163343	+	0.986569i	$0.447772\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	21.0000	+	21.0000i	0.684580	+	0.684580i	0.961029	−	0.276448i	$-0.0891575\pi$
$941$	21.0000	+	21.0000i	0.684580	+	0.684580i	−0.276448	+	0.961029i	$0.589157\pi$
$942$		0			0
$943$	−40.0000			−1.30258
$944$		0			0
$945$		0			0
$946$		0			0
$947$	30.0000	+	30.0000i	0.974869	+	0.974869i	0.999692	−	0.0248229i	$-0.00790217\pi$
$947$	30.0000	+	30.0000i	0.974869	+	0.974869i	−0.0248229	+	0.999692i	$0.507902\pi$
$948$		0			0
$949$	−2.00000	+	2.00000i	−0.0649227	+	0.0649227i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	30.0000i		−	0.971795i	−0.874016	−	0.485898i	$-0.838493\pi$
$953$		−	30.0000i		−	0.971795i	0.874016	−	0.485898i	$-0.161507\pi$
$954$		0			0
$955$	4.00000	+	8.00000i	0.129437	+	0.258874i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$		−	1.00000i		−	0.0322581i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	15.0000	+	5.00000i	0.482867	+	0.160956i
$966$		0			0
$967$	−48.0000			−1.54358			−0.771788	−	0.635880i	$-0.780637\pi$
$967$	−48.0000			−1.54358			−0.771788	+	0.635880i	$0.780637\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			4.00000i			0.128366i	0.997938	+	0.0641831i	$0.0204442\pi$
$971$			4.00000i			0.128366i	−0.997938	+	0.0641831i	$0.979556\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	42.0000			1.34370			0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000			1.34370			0.671850	+	0.740688i	$0.265500\pi$
$978$		0			0
$979$	32.0000	+	32.0000i	1.02272	+	1.02272i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$	−15.0000	−	5.00000i	−0.477940	−	0.159313i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−16.0000	+	16.0000i	−0.508770	+	0.508770i
$990$		0			0
$991$	−36.0000	+	36.0000i	−1.14358	+	1.14358i	−0.155787	+	0.987791i	$0.549791\pi$
$991$	−36.0000	+	36.0000i	−1.14358	+	1.14358i	−0.987791	+	0.155787i	$0.950209\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	36.0000	+	12.0000i	1.14128	+	0.380426i
$996$		0			0
$997$	3.00000	−	3.00000i	0.0950110	−	0.0950110i	−0.658004	−	0.753015i	$-0.728599\pi$
$997$	3.00000	−	3.00000i	0.0950110	−	0.0950110i	0.753015	+	0.658004i	$0.228599\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3060.2.z.a.2809.1		2
3.2	odd	2		340.2.m.b.89.1	yes	2
5.4	even	2		3060.2.z.b.2809.1		2
15.2	even	4		1700.2.o.b.701.1		2
15.8	even	4		1700.2.o.a.701.1		2
15.14	odd	2		340.2.m.a.89.1	✓	2
17.13	even	4		3060.2.z.b.829.1		2
51.47	odd	4		340.2.m.a.149.1	yes	2
85.64	even	4	inner	3060.2.z.a.829.1		2
255.47	even	4		1700.2.o.b.1101.1		2
255.98	even	4		1700.2.o.a.1101.1		2
255.149	odd	4		340.2.m.b.149.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
340.2.m.a.89.1	✓	2	15.14	odd	2
340.2.m.a.149.1	yes	2	51.47	odd	4
340.2.m.b.89.1	yes	2	3.2	odd	2
340.2.m.b.149.1	yes	2	255.149	odd	4
1700.2.o.a.701.1		2	15.8	even	4
1700.2.o.a.1101.1		2	255.98	even	4
1700.2.o.b.701.1		2	15.2	even	4
1700.2.o.b.1101.1		2	255.47	even	4
3060.2.z.a.829.1		2	85.64	even	4	inner
3060.2.z.a.2809.1		2	1.1	even	1	trivial
3060.2.z.b.829.1		2	17.13	even	4
3060.2.z.b.2809.1		2	5.4	even	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:	\( N \)	\(=\)	\( 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3060.z (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 2.00000i) q^{5} +(2.00000 + 2.00000i) q^{11} -2.00000i q^{13} +(1.00000 + 4.00000i) q^{17} +(-4.00000 + 4.00000i) q^{23} +(-3.00000 + 4.00000i) q^{25} +(5.00000 - 5.00000i) q^{29} +(4.00000 - 4.00000i) q^{31} +(5.00000 + 5.00000i) q^{37} +(5.00000 + 5.00000i) q^{41} +4.00000 q^{43} -8.00000i q^{47} +7.00000i q^{49} -6.00000 q^{53} +(2.00000 - 6.00000i) q^{55} +12.0000i q^{59} +(3.00000 + 3.00000i) q^{61} +(-4.00000 + 2.00000i) q^{65} -4.00000i q^{67} +(8.00000 - 8.00000i) q^{71} +(-1.00000 - 1.00000i) q^{73} +(-12.0000 - 12.0000i) q^{79} -4.00000 q^{83} +(7.00000 - 6.00000i) q^{85} +16.0000 q^{89} +(-5.00000 - 5.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 4 q^{11} + 2 q^{17} - 8 q^{23} - 6 q^{25} + 10 q^{29} + 8 q^{31} + 10 q^{37} + 10 q^{41} + 8 q^{43} - 12 q^{53} + 4 q^{55} + 6 q^{61} - 8 q^{65} + 16 q^{71} - 2 q^{73} - 24 q^{79} - 8 q^{83}+ \cdots - 10 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 4 * q^11 + 2 * q^17 - 8 * q^23 - 6 * q^25 + 10 * q^29 + 8 * q^31 + 10 * q^37 + 10 * q^41 + 8 * q^43 - 12 * q^53 + 4 * q^55 + 6 * q^61 - 8 * q^65 + 16 * q^71 - 2 * q^73 - 24 * q^79 - 8 * q^83 + 14 * q^85 + 32 * q^89 - 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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