LMFDB - Embedded newform 2700.2.s.c.2449.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2700,2,Mod(1549,2700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2700, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 3]))

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

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

chi := DirichletCharacter("2700.1549");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2700.s (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$21.5596085457$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 $ x^12 - 16x^8 - 24x^7 + 96x^5 + 304x^4 + 384x^3 + 288x^2 + 144*x + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2449.3
Root			$0.583700 - 2.17840i$ of defining polynomial
Character	$\chi$	$=$	2700.2449
Dual form			2700.2.s.c.1549.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.36788 - 1.36710i) q^{7} +(2.76210 - 4.78410i) q^{11} +(3.05205 - 1.76210i) q^{13} +5.52420i q^{17} -7.52420 q^{19} +(0.635828 - 0.367095i) q^{23} +(2.23419 - 3.86973i) q^{29} +(-3.76210 - 6.51615i) q^{31} +6.05582i q^{37} +(-0.527909 - 0.914365i) q^{41} +(-3.05205 - 1.76210i) q^{43} +(1.04788 + 0.604996i) q^{47} +(0.237900 + 0.412055i) q^{49} +1.46838i q^{53} +(0.734191 + 1.27166i) q^{59} +(-4.52791 + 7.84257i) q^{61} +(-3.68787 + 2.12920i) q^{67} -10.0558 q^{71} -8.00000i q^{73} +(-13.0806 + 7.55211i) q^{77} +(1.00000 - 1.73205i) q^{79} +(4.56032 + 2.63290i) q^{83} -3.00000 q^{89} -9.63583 q^{91} +(-8.19986 - 4.73419i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 24 q^{19} + 6 q^{29} - 12 q^{31} + 6 q^{41} + 36 q^{49} - 12 q^{59} - 42 q^{61} - 96 q^{71} + 12 q^{79} - 36 q^{89}+O(q^{100}) $ 12 * q - 24 * q^19 + 6 * q^29 - 12 * q^31 + 6 * q^41 + 36 * q^49 - 12 * q^59 - 42 * q^61 - 96 * q^71 + 12 * q^79 - 36 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times$.

$n$	$1001$	$1351$	$2377$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$-1$

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
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.36788	−	1.36710i	−0.894974	−	0.516714i	−0.0194079	−	0.999812i	$-0.506178\pi$
$7$	−2.36788	−	1.36710i	−0.894974	−	0.516714i	−0.875566	+	0.483098i	$0.839511\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.76210	−	4.78410i	0.832804	−	1.44246i	−0.0630012	−	0.998013i	$-0.520067\pi$
$11$	2.76210	−	4.78410i	0.832804	−	1.44246i	0.895806	−	0.444446i	$-0.146599\pi$
$12$		0			0
$13$	3.05205	−	1.76210i	0.846485	−	0.488719i	−0.0129781	−	0.999916i	$-0.504131\pi$
$13$	3.05205	−	1.76210i	0.846485	−	0.488719i	0.859463	+	0.511197i	$0.170798\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			5.52420i			1.33982i	0.742444	+	0.669908i	$0.233666\pi$
$17$			5.52420i			1.33982i	−0.742444	+	0.669908i	$0.766334\pi$
$18$		0			0
$19$	−7.52420			−1.72617			−0.863085	−	0.505059i	$-0.831471\pi$
$19$	−7.52420			−1.72617			−0.863085	+	0.505059i	$0.831471\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.635828	−	0.367095i	0.132579	−	0.0765447i	−0.432243	−	0.901757i	$-0.642278\pi$
$23$	0.635828	−	0.367095i	0.132579	−	0.0765447i	0.564823	+	0.825212i	$0.308945\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.23419	−	3.86973i	0.414879	−	0.718591i	−0.580537	−	0.814234i	$-0.697157\pi$
$29$	2.23419	−	3.86973i	0.414879	−	0.718591i	0.995416	+	0.0956427i	$0.0304906\pi$
$30$		0			0
$31$	−3.76210	−	6.51615i	−0.675693	−	1.17033i	−0.976266	−	0.216576i	$-0.930511\pi$
$31$	−3.76210	−	6.51615i	−0.675693	−	1.17033i	0.300573	−	0.953759i	$-0.402822\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$			6.05582i			0.995570i	0.867300	+	0.497785i	$0.165853\pi$
$37$			6.05582i			0.995570i	−0.867300	+	0.497785i	$0.834147\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.527909	−	0.914365i	−0.0824455	−	0.142800i	0.821854	−	0.569698i	$-0.192940\pi$
$41$	−0.527909	−	0.914365i	−0.0824455	−	0.142800i	−0.904300	+	0.426898i	$0.859606\pi$
$42$		0			0
$43$	−3.05205	−	1.76210i	−0.465433	−	0.268718i	0.248893	−	0.968531i	$-0.419933\pi$
$43$	−3.05205	−	1.76210i	−0.465433	−	0.268718i	−0.714326	+	0.699813i	$0.753267\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.04788	+	0.604996i	0.152849	+	0.0882477i	0.574474	−	0.818523i	$-0.305207\pi$
$47$	1.04788	+	0.604996i	0.152849	+	0.0882477i	−0.421625	+	0.906771i	$0.638540\pi$
$48$		0			0
$49$	0.237900	+	0.412055i	0.0339857	+	0.0588650i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			1.46838i			0.201698i	0.994902	+	0.100849i	$0.0321559\pi$
$53$			1.46838i			0.201698i	−0.994902	+	0.100849i	$0.967844\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.734191	+	1.27166i	0.0955835	+	0.165556i	0.909852	−	0.414933i	$-0.136195\pi$
$59$	0.734191	+	1.27166i	0.0955835	+	0.165556i	−0.814268	+	0.580488i	$0.802862\pi$
$60$		0			0
$61$	−4.52791	+	7.84257i	−0.579739	+	1.00414i	0.415770	+	0.909470i	$0.363512\pi$
$61$	−4.52791	+	7.84257i	−0.579739	+	1.00414i	−0.995509	+	0.0946680i	$0.969821\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−3.68787	+	2.12920i	−0.450546	+	0.260123i	−0.708061	−	0.706152i	$-0.750430\pi$
$67$	−3.68787	+	2.12920i	−0.450546	+	0.260123i	0.257515	+	0.966274i	$0.417096\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−10.0558			−1.19341			−0.596703	−	0.802462i	$-0.703523\pi$
$71$	−10.0558			−1.19341			−0.596703	+	0.802462i	$0.703523\pi$
$72$		0			0
$73$		−	8.00000i		−	0.936329i	−0.883641	−	0.468165i	$-0.844915\pi$
$73$		−	8.00000i		−	0.936329i	0.883641	−	0.468165i	$-0.155085\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−13.0806	+	7.55211i	−1.49068	+	0.860643i
$78$		0			0
$79$	1.00000	−	1.73205i	0.112509	−	0.194871i	−0.804272	−	0.594261i	$-0.797445\pi$
$79$	1.00000	−	1.73205i	0.112509	−	0.194871i	0.916781	+	0.399390i	$0.130778\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4.56032	+	2.63290i	0.500561	+	0.288999i	0.728945	−	0.684572i	$-0.240011\pi$
$83$	4.56032	+	2.63290i	0.500561	+	0.288999i	−0.228384	+	0.973571i	$0.573344\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.00000			−0.317999			−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000			−0.317999			−0.159000	+	0.987279i	$0.550827\pi$
$90$		0			0
$91$	−9.63583			−1.01011
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−8.19986	−	4.73419i	−0.832570	−	0.480684i	0.0221621	−	0.999754i	$-0.492945\pi$
$97$	−8.19986	−	4.73419i	−0.832570	−	0.480684i	−0.854732	+	0.519070i	$0.826278\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.73419	−	11.6640i	0.670077	−	1.16061i	−0.307805	−	0.951450i	$-0.599594\pi$
$101$	6.73419	−	11.6640i	0.670077	−	1.16061i	0.977882	−	0.209158i	$-0.0670723\pi$
$102$		0			0
$103$	−16.0843	+	9.28630i	−1.58484	+	0.915006i	−0.590699	+	0.806892i	$0.701148\pi$
$103$	−16.0843	+	9.28630i	−1.58484	+	0.915006i	−0.994138	+	0.108114i	$0.965519\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	19.2510i		−	1.86106i	−0.366213	−	0.930531i	$-0.619346\pi$
$107$		−	19.2510i		−	1.86106i	0.366213	−	0.930531i	$-0.380654\pi$
$108$		0			0
$109$	0.524200			0.0502092			0.0251046	−	0.999685i	$-0.492008\pi$
$109$	0.524200			0.0502092			0.0251046	+	0.999685i	$0.492008\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6.46781	+	3.73419i	−0.608440	+	0.351283i	−0.772355	−	0.635191i	$-0.780921\pi$
$113$	−6.46781	+	3.73419i	−0.608440	+	0.351283i	0.163915	+	0.986475i	$0.447588\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	7.55211	−	13.0806i	0.692301	−	1.19910i
$120$		0			0
$121$	−9.75839	−	16.9020i	−0.887126	−	1.53655i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$			8.25839i			0.732814i	0.930455	+	0.366407i	$0.119412\pi$
$127$			8.25839i			0.732814i	−0.930455	+	0.366407i	$0.880588\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−8.04840	−	13.9402i	−0.703192	−	1.21796i	−0.967340	−	0.253482i	$-0.918424\pi$
$131$	−8.04840	−	13.9402i	−0.703192	−	1.21796i	0.264148	−	0.964482i	$-0.414909\pi$
$132$		0			0
$133$	17.8164	+	10.2863i	1.54488	+	0.891935i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−16.0360	−	9.25839i	−1.37005	−	0.790998i	−0.379115	−	0.925350i	$-0.623771\pi$
$137$	−16.0360	−	9.25839i	−1.37005	−	0.790998i	−0.990934	+	0.134352i	$0.957105\pi$
$138$		0			0
$139$	−2.00000	−	3.46410i	−0.169638	−	0.293821i	0.768655	−	0.639664i	$-0.220926\pi$
$139$	−2.00000	−	3.46410i	−0.169638	−	0.293821i	−0.938293	+	0.345843i	$0.887593\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	19.4684i		−	1.62803i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.78630	−	11.7542i	−0.555955	−	0.962943i	−0.997829	−	0.0658653i	$-0.979019\pi$
$149$	−6.78630	−	11.7542i	−0.555955	−	0.962943i	0.441873	−	0.897078i	$-0.354314\pi$
$150$		0			0
$151$	6.23048	−	10.7915i	0.507029	−	0.878201i	−0.492937	−	0.870065i	$-0.664077\pi$
$151$	6.23048	−	10.7915i	0.507029	−	0.878201i	0.999967	−	0.00813598i	$-0.00258979\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.36834	+	0.790009i	−0.109205	+	0.0630496i	−0.553608	−	0.832778i	$-0.686749\pi$
$157$	−1.36834	+	0.790009i	−0.109205	+	0.0630496i	0.444403	+	0.895827i	$0.353416\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.00742			−0.158207
$162$		0			0
$163$			4.47580i			0.350572i	0.984518	+	0.175286i	$0.0560850\pi$
$163$			4.47580i			0.350572i	−0.984518	+	0.175286i	$0.943915\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−14.5406	+	8.39500i	−1.12518	+	0.649625i	−0.942719	−	0.333588i	$-0.891741\pi$
$167$	−14.5406	+	8.39500i	−1.12518	+	0.649625i	−0.182464	+	0.983212i	$0.558407\pi$
$168$		0			0
$169$	−0.290009	+	0.502310i	−0.0223084	+	0.0386392i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	19.9605	+	11.5242i	1.51757	+	0.876169i	0.999787	+	0.0206561i	$0.00657550\pi$
$173$	19.9605	+	11.5242i	1.51757	+	0.876169i	0.517782	+	0.855513i	$0.326758\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	12.9926			0.971111			0.485556	−	0.874206i	$-0.338617\pi$
$179$	12.9926			0.971111			0.485556	+	0.874206i	$0.338617\pi$
$180$		0			0
$181$	−24.5652			−1.82592			−0.912958	−	0.408054i	$-0.866207\pi$
$181$	−24.5652			−1.82592			−0.912958	+	0.408054i	$0.866207\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	26.4283	+	15.2584i	1.93263	+	1.11580i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	7.55211	−	13.0806i	0.546451	−	0.946482i	−0.452063	−	0.891986i	$-0.649312\pi$
$191$	7.55211	−	13.0806i	0.546451	−	0.946482i	0.998514	−	0.0544954i	$-0.0173550\pi$
$192$		0			0
$193$	12.6202	−	7.28630i	0.908425	−	0.524479i	0.0285008	−	0.999594i	$-0.490927\pi$
$193$	12.6202	−	7.28630i	0.908425	−	0.524479i	0.879924	+	0.475114i	$0.157593\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			4.53162i			0.322864i	0.986884	+	0.161432i	$0.0516113\pi$
$197$			4.53162i			0.322864i	−0.986884	+	0.161432i	$0.948389\pi$
$198$		0			0
$199$	7.41256			0.525463			0.262731	−	0.964869i	$-0.415377\pi$
$199$	7.41256			0.525463			0.262731	+	0.964869i	$0.415377\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−10.5806	+	6.10870i	−0.742612	+	0.428747i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−20.7826	+	35.9965i	−1.43756	+	2.48993i
$210$		0			0
$211$	6.70628	+	11.6156i	0.461680	+	0.799652i	0.999045	−	0.0436972i	$-0.0139137\pi$
$211$	6.70628	+	11.6156i	0.461680	+	0.799652i	−0.537365	+	0.843350i	$0.680580\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$			20.5726i			1.39656i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	9.73419	+	16.8601i	0.654793	+	1.13413i
$222$		0			0
$223$	−7.61237	−	4.39500i	−0.509762	−	0.294311i	0.222974	−	0.974824i	$-0.428424\pi$
$223$	−7.61237	−	4.39500i	−0.509762	−	0.294311i	−0.732736	+	0.680513i	$0.761757\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.51244	+	2.02791i	0.233129	+	0.134597i	0.612015	−	0.790846i	$-0.290359\pi$
$227$	3.51244	+	2.02791i	0.233129	+	0.134597i	−0.378886	+	0.925443i	$0.623693\pi$
$228$		0			0
$229$	1.29001	+	2.23436i	0.0852462	+	0.147651i	0.905496	−	0.424355i	$-0.139499\pi$
$229$	1.29001	+	2.23436i	0.0852462	+	0.147651i	−0.820250	+	0.572005i	$0.806166\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	1.94418i		−	0.127368i	−0.997970	−	0.0636838i	$-0.979715\pi$
$233$		−	1.94418i		−	0.127368i	0.997970	−	0.0636838i	$-0.0202849\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3.73419	−	6.46781i	−0.241545	−	0.418368i	0.719610	−	0.694379i	$-0.244321\pi$
$239$	−3.73419	−	6.46781i	−0.241545	−	0.418368i	−0.961154	+	0.276011i	$0.910987\pi$
$240$		0			0
$241$	−1.20628	+	2.08934i	−0.0777035	+	0.134586i	−0.902259	−	0.431195i	$-0.858092\pi$
$241$	−1.20628	+	2.08934i	−0.0777035	+	0.134586i	0.824555	+	0.565781i	$0.191425\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−22.9642	+	13.2584i	−1.46118	+	0.843611i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	6.99258			0.441368			0.220684	−	0.975345i	$-0.429171\pi$
$251$	6.99258			0.441368			0.220684	+	0.975345i	$0.429171\pi$
$252$		0			0
$253$		−	4.05582i		−	0.254987i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	24.3325	−	14.0484i	1.51782	−	0.876315i	0.518042	−	0.855355i	$-0.326661\pi$
$257$	24.3325	−	14.0484i	1.51782	−	0.876315i	0.999780	−	0.0209598i	$-0.00667221\pi$
$258$		0			0
$259$	8.27888	−	14.3394i	0.514425	−	0.891010i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.61283	−	3.81792i	−0.407764	−	0.235423i	0.282064	−	0.959395i	$-0.408981\pi$
$263$	−6.61283	−	3.81792i	−0.407764	−	0.235423i	−0.689829	+	0.723973i	$0.742314\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−18.6210			−1.13534			−0.567671	−	0.823255i	$-0.692155\pi$
$269$	−18.6210			−1.13534			−0.567671	+	0.823255i	$0.692155\pi$
$270$		0			0
$271$	6.57260			0.399257			0.199628	−	0.979872i	$-0.436026\pi$
$271$	6.57260			0.399257			0.199628	+	0.979872i	$0.436026\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.32000	−	0.762100i	−0.0793108	−	0.0457901i	0.459820	−	0.888012i	$-0.347914\pi$
$277$	−1.32000	−	0.762100i	−0.0793108	−	0.0457901i	−0.539131	+	0.842222i	$0.681247\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	7.26210	−	12.5783i	0.433221	−	0.750360i	−0.563928	−	0.825824i	$-0.690710\pi$
$281$	7.26210	−	12.5783i	0.433221	−	0.750360i	0.997149	+	0.0754640i	$0.0240438\pi$
$282$		0			0
$283$	14.4922	−	8.36710i	0.861474	−	0.497372i	−0.00303167	−	0.999995i	$-0.500965\pi$
$283$	14.4922	−	8.36710i	0.861474	−	0.497372i	0.864506	+	0.502623i	$0.167632\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			2.88681i			0.170403i
$288$		0			0
$289$	−13.5168			−0.795105
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−17.7197	+	10.2305i	−1.03520	+	0.597671i	−0.918469	−	0.395493i	$-0.870574\pi$
$293$	−17.7197	+	10.2305i	−1.03520	+	0.597671i	−0.116728	+	0.993164i	$0.537240\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.29372	−	2.24078i	0.0748176	−	0.129588i
$300$		0			0
$301$	4.81792	+	8.34488i	0.277700	+	0.480991i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	18.2026i		−	1.03888i	−0.854508	−	0.519438i	$-0.826141\pi$
$307$		−	18.2026i		−	1.03888i	0.854508	−	0.519438i	$-0.173859\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.55211	+	2.68833i	0.0880120	+	0.152441i	0.906671	−	0.421839i	$-0.138615\pi$
$311$	1.55211	+	2.68833i	0.0880120	+	0.152441i	−0.818659	+	0.574280i	$0.805282\pi$
$312$		0			0
$313$	−16.5447	−	9.55211i	−0.935164	−	0.539917i	−0.0467228	−	0.998908i	$-0.514878\pi$
$313$	−16.5447	−	9.55211i	−0.935164	−	0.539917i	−0.888441	+	0.458991i	$0.848211\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	4.33655	+	2.50371i	0.243565	+	0.140622i	0.616814	−	0.787109i	$-0.288423\pi$
$317$	4.33655	+	2.50371i	0.243565	+	0.140622i	−0.373249	+	0.927731i	$0.621756\pi$
$318$		0			0
$319$	−12.3421	−	21.3772i	−0.691026	−	1.19689i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	41.5652i		−	2.31275i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.65417	−	2.86511i	−0.0911976	−	0.157959i
$330$		0			0
$331$	1.74161	−	3.01656i	0.0957275	−	0.165805i	−0.814184	−	0.580606i	$-0.802816\pi$
$331$	1.74161	−	3.01656i	0.0957275	−	0.165805i	0.909912	+	0.414801i	$0.136149\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	10.0286	−	5.79001i	0.546292	−	0.315402i	−0.201333	−	0.979523i	$-0.564527\pi$
$337$	10.0286	−	5.79001i	0.546292	−	0.315402i	0.747625	+	0.664121i	$0.231194\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−41.5652			−2.25088
$342$		0			0
$343$			17.8384i			0.963183i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−22.9159	+	13.2305i	−1.23019	+	0.710249i	−0.967069	−	0.254514i	$-0.918085\pi$
$347$	−22.9159	+	13.2305i	−1.23019	+	0.710249i	−0.263119	+	0.964763i	$0.584751\pi$
$348$		0			0
$349$	−13.0168	+	22.5457i	−0.696772	+	1.20685i	0.272807	+	0.962069i	$0.412048\pi$
$349$	−13.0168	+	22.5457i	−0.696772	+	1.20685i	−0.969580	+	0.244776i	$0.921285\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	20.7846	+	12.0000i	1.10625	+	0.638696i	0.937856	−	0.347024i	$-0.112808\pi$
$353$	20.7846	+	12.0000i	1.10625	+	0.638696i	0.168397	+	0.985719i	$0.446141\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	6.04098			0.318831			0.159415	−	0.987212i	$-0.449039\pi$
$359$	6.04098			0.318831			0.159415	+	0.987212i	$0.449039\pi$
$360$		0			0
$361$	37.6136			1.97966
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−19.4518	−	11.2305i	−1.01537	−	0.586226i	−0.102613	−	0.994721i	$-0.532720\pi$
$367$	−19.4518	−	11.2305i	−1.01537	−	0.586226i	−0.912761	+	0.408495i	$0.866054\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2.00742	−	3.47695i	0.104220	−	0.180514i
$372$		0			0
$373$	2.19245	−	1.26581i	0.113521	−	0.0655411i	−0.442165	−	0.896934i	$-0.645789\pi$
$373$	2.19245	−	1.26581i	0.113521	−	0.0655411i	0.555685	+	0.831393i	$0.312456\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	15.7475i		−	0.811036i
$378$		0			0
$379$	21.0484			1.08118			0.540592	−	0.841285i	$-0.318200\pi$
$379$	21.0484			1.08118			0.540592	+	0.841285i	$0.318200\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−6.61283	+	3.81792i	−0.337900	+	0.195086i	−0.659343	−	0.751842i	$-0.729165\pi$
$383$	−6.61283	+	3.81792i	−0.337900	+	0.195086i	0.321443	+	0.946929i	$0.395832\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	9.46467	−	16.3933i	0.479878	−	0.831173i	−0.519856	−	0.854254i	$-0.674014\pi$
$389$	9.46467	−	16.3933i	0.479878	−	0.831173i	0.999734	+	0.0230811i	$0.00734760\pi$
$390$		0			0
$391$	2.02791	+	3.51244i	0.102556	+	0.177632i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	29.1600i		−	1.46350i	−0.681573	−	0.731750i	$-0.738704\pi$
$397$		−	29.1600i		−	1.46350i	0.681573	−	0.731750i	$-0.261296\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	16.3142	+	28.2570i	0.814693	+	1.41109i	0.909548	+	0.415598i	$0.136428\pi$
$401$	16.3142	+	28.2570i	0.814693	+	1.41109i	−0.0948557	+	0.995491i	$0.530239\pi$
$402$		0			0
$403$	−22.9642	−	13.2584i	−1.14393	−	0.660447i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	28.9716	+	16.7268i	1.43607	+	0.829115i
$408$		0			0
$409$	16.1042	+	27.8933i	0.796302	+	1.37924i	0.922009	+	0.387169i	$0.126547\pi$
$409$	16.1042	+	27.8933i	0.796302	+	1.37924i	−0.125707	+	0.992067i	$0.540120\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	4.01484i		−	0.197557i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	10.7063	+	18.5438i	0.523036	+	0.905925i	0.999641	+	0.0268073i	$0.00853405\pi$
$419$	10.7063	+	18.5438i	0.523036	+	0.905925i	−0.476605	+	0.879118i	$0.658133\pi$
$420$		0			0
$421$	0.944182	−	1.63537i	0.0460166	−	0.0797031i	−0.842100	−	0.539322i	$-0.818681\pi$
$421$	0.944182	−	1.63537i	0.0460166	−	0.0797031i	0.888116	+	0.459619i	$0.152014\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	21.4431	−	12.3802i	1.03770	−	0.599118i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	2.11164			0.101714			0.0508569	−	0.998706i	$-0.483805\pi$
$431$	2.11164			0.101714			0.0508569	+	0.998706i	$0.483805\pi$
$432$		0			0
$433$			9.52420i			0.457704i	0.973461	+	0.228852i	$0.0734972\pi$
$433$			9.52420i			0.457704i	−0.973461	+	0.228852i	$0.926503\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−4.78410	+	2.76210i	−0.228854	+	0.132129i
$438$		0			0
$439$	−7.20257	+	12.4752i	−0.343760	+	0.595410i	−0.985128	−	0.171824i	$-0.945034\pi$
$439$	−7.20257	+	12.4752i	−0.343760	+	0.595410i	0.641368	+	0.767234i	$0.278367\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	16.6363	+	9.60500i	0.790416	+	0.456347i	0.840109	−	0.542417i	$-0.182491\pi$
$443$	16.6363	+	9.60500i	0.790416	+	0.456347i	−0.0496927	+	0.998765i	$0.515824\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	33.5800			1.58474			0.792369	−	0.610041i	$-0.208847\pi$
$449$	33.5800			1.58474			0.792369	+	0.610041i	$0.208847\pi$
$450$		0			0
$451$	−5.83255			−0.274644
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.87707	−	1.08373i	−0.0878056	−	0.0506946i	0.455454	−	0.890259i	$-0.349477\pi$
$457$	−1.87707	−	1.08373i	−0.0878056	−	0.0506946i	−0.543260	+	0.839565i	$0.682810\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−12.8700	+	22.2915i	−0.599417	+	1.03822i	0.393490	+	0.919329i	$0.371267\pi$
$461$	−12.8700	+	22.2915i	−0.599417	+	1.03822i	−0.992907	+	0.118892i	$0.962066\pi$
$462$		0			0
$463$	−10.4406	+	6.02791i	−0.485218	+	0.280141i	−0.722588	−	0.691279i	$-0.757048\pi$
$463$	−10.4406	+	6.02791i	−0.485218	+	0.280141i	0.237371	+	0.971419i	$0.423714\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			14.4610i			0.669174i	0.942365	+	0.334587i	$0.108597\pi$
$467$			14.4610i			0.669174i	−0.942365	+	0.334587i	$0.891403\pi$
$468$		0			0
$469$	11.6433			0.537635
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−16.8601	+	9.73419i	−0.775229	+	0.447579i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	9.99258	−	17.3077i	0.456573	−	0.790807i	−0.542204	−	0.840247i	$-0.682410\pi$
$479$	9.99258	−	17.3077i	0.456573	−	0.790807i	0.998777	+	0.0494395i	$0.0157435\pi$
$480$		0			0
$481$	10.6710	+	18.4826i	0.486554	+	0.842736i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	18.1526i		−	0.822574i	−0.911506	−	0.411287i	$-0.865079\pi$
$487$		−	18.1526i		−	0.822574i	0.911506	−	0.411287i	$-0.134921\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	19.7900	+	34.2773i	0.893111	+	1.54691i	0.836126	+	0.548538i	$0.184815\pi$
$491$	19.7900	+	34.2773i	0.893111	+	1.54691i	0.0569849	+	0.998375i	$0.481851\pi$
$492$		0			0
$493$	21.3772	+	12.3421i	0.962779	+	0.555861i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	23.8110	+	13.7473i	1.06807	+	0.616649i
$498$		0			0
$499$	−21.3142	−	36.9173i	−0.954155	−	1.65264i	−0.736290	−	0.676666i	$-0.763424\pi$
$499$	−21.3142	−	36.9173i	−0.954155	−	1.65264i	−0.217865	−	0.975979i	$-0.569909\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			21.1952i			0.945045i	0.881319	+	0.472523i	$0.156657\pi$
$503$			21.1952i			0.945045i	−0.881319	+	0.472523i	$0.843343\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	11.0168	+	19.0816i	0.488310	+	0.845778i	0.999910	−	0.0134460i	$-0.00428012\pi$
$509$	11.0168	+	19.0816i	0.488310	+	0.845778i	−0.511599	+	0.859224i	$0.670947\pi$
$510$		0			0
$511$	−10.9368	+	18.9430i	−0.483814	+	0.837990i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	5.78872	−	3.34212i	0.254587	−	0.146986i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.40515			−0.0615606			−0.0307803	−	0.999526i	$-0.509799\pi$
$521$	−1.40515			−0.0615606			−0.0307803	+	0.999526i	$0.509799\pi$
$522$		0			0
$523$			11.8532i			0.518306i	0.965836	+	0.259153i	$0.0834434\pi$
$523$			11.8532i			0.518306i	−0.965836	+	0.259153i	$0.916557\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	35.9965	−	20.7826i	1.56803	−	0.905304i
$528$		0			0
$529$	−11.2305	+	19.4518i	−0.488282	+	0.845729i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3.22240	−	1.86046i	−0.139578	−	0.0805853i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.62842			0.113214
$540$		0			0
$541$	−8.98516			−0.386302			−0.193151	−	0.981169i	$-0.561871\pi$
$541$	−8.98516			−0.386302			−0.193151	+	0.981169i	$0.561871\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	5.37159	+	3.10129i	0.229672	+	0.132601i	0.610421	−	0.792077i	$-0.291000\pi$
$547$	5.37159	+	3.10129i	0.229672	+	0.132601i	−0.380749	+	0.924679i	$0.624334\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−16.8105	+	29.1166i	−0.716151	+	1.24041i
$552$		0			0
$553$	−4.73576	+	2.73419i	−0.201385	+	0.116270i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	6.00000i		−	0.254228i	−0.991888	−	0.127114i	$-0.959429\pi$
$557$		−	6.00000i		−	0.254228i	0.991888	−	0.127114i	$-0.0405714\pi$
$558$		0			0
$559$	−12.4200			−0.525309
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3.70072	+	2.13661i	−0.155967	+	0.0900475i	−0.575952	−	0.817483i	$-0.695369\pi$
$563$	−3.70072	+	2.13661i	−0.155967	+	0.0900475i	0.419985	+	0.907531i	$0.362035\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−8.04840	+	13.9402i	−0.337406	+	0.584405i	−0.983944	−	0.178477i	$-0.942883\pi$
$569$	−8.04840	+	13.9402i	−0.337406	+	0.584405i	0.646538	+	0.762882i	$0.276216\pi$
$570$		0			0
$571$	−5.46838	−	9.47152i	−0.228845	−	0.396371i	0.728621	−	0.684917i	$-0.240161\pi$
$571$	−5.46838	−	9.47152i	−0.228845	−	0.396371i	−0.957466	+	0.288546i	$0.906828\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	37.6620i		−	1.56789i	−0.620831	−	0.783944i	$-0.713205\pi$
$577$		−	37.6620i		−	1.56789i	0.620831	−	0.783944i	$-0.286795\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−7.19886	−	12.4688i	−0.298659	−	0.517293i
$582$		0			0
$583$	7.02488	+	4.05582i	0.290941	+	0.167975i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	18.8771	+	10.8987i	0.779142	+	0.449838i	0.836126	−	0.548537i	$-0.184815\pi$
$587$	18.8771	+	10.8987i	0.779142	+	0.449838i	−0.0569839	+	0.998375i	$0.518148\pi$
$588$		0			0
$589$	28.3068	+	49.0288i	1.16636	+	2.02020i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			28.2643i			1.16067i	0.814377	+	0.580337i	$0.197079\pi$
$593$			28.2643i			1.16067i	−0.814377	+	0.580337i	$0.802921\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−7.05582	−	12.2210i	−0.288293	−	0.499338i	0.685109	−	0.728440i	$-0.259754\pi$
$599$	−7.05582	−	12.2210i	−0.288293	−	0.499338i	−0.973402	+	0.229102i	$0.926421\pi$
$600$		0			0
$601$	14.1042	−	24.4292i	0.575323	−	0.996489i	−0.420683	−	0.907207i	$-0.638210\pi$
$601$	14.1042	−	24.4292i	0.575323	−	0.996489i	0.996006	−	0.0892812i	$-0.0284570\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	24.5692	−	14.1850i	0.997232	−	0.575752i	0.0898036	−	0.995959i	$-0.471376\pi$
$607$	24.5692	−	14.1850i	0.997232	−	0.575752i	0.907428	+	0.420208i	$0.138043\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	4.26425			0.172513
$612$		0			0
$613$		−	24.5726i		−	0.992478i	−0.868186	−	0.496239i	$-0.834714\pi$
$613$		−	24.5726i		−	0.992478i	0.868186	−	0.496239i	$-0.165286\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	23.0609	−	13.3142i	0.928396	−	0.536010i	0.0420923	−	0.999114i	$-0.486598\pi$
$617$	23.0609	−	13.3142i	0.928396	−	0.536010i	0.886304	+	0.463104i	$0.153264\pi$
$618$		0			0
$619$	15.2863	−	26.4766i	0.614408	−	1.06419i	−0.376080	−	0.926587i	$-0.622728\pi$
$619$	15.2863	−	26.4766i	0.614408	−	1.06419i	0.990488	−	0.137599i	$-0.0439385\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	7.10364	+	4.10129i	0.284601	+	0.164315i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−33.4535			−1.33388
$630$		0			0
$631$	27.4684			1.09350			0.546750	−	0.837296i	$-0.315865\pi$
$631$	27.4684			1.09350			0.546750	+	0.837296i	$0.315865\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.45216	+	0.838408i	0.0575369	+	0.0332189i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	6.26952	−	10.8591i	0.247631	−	0.428910i	−0.715237	−	0.698882i	$-0.753681\pi$
$641$	6.26952	−	10.8591i	0.247631	−	0.428910i	0.962868	+	0.269972i	$0.0870146\pi$
$642$		0			0
$643$	8.88403	−	5.12920i	0.350352	−	0.202276i	−0.314488	−	0.949261i	$-0.601833\pi$
$643$	8.88403	−	5.12920i	0.350352	−	0.202276i	0.664840	+	0.746986i	$0.268500\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	16.7900i		−	0.660083i	−0.943966	−	0.330042i	$-0.892937\pi$
$647$		−	16.7900i		−	0.660083i	0.943966	−	0.330042i	$-0.107063\pi$
$648$		0			0
$649$	8.11164			0.318410
$650$		0			0
$651$		0			0
$652$		0			0
$653$	43.3239	−	25.0131i	1.69540	−	0.978837i	0.745377	−	0.666643i	$-0.232269\pi$
$653$	43.3239	−	25.0131i	1.69540	−	0.978837i	0.950018	−	0.312194i	$-0.101064\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−2.78259	+	4.81959i	−0.108394	+	0.187744i	−0.915120	−	0.403182i	$-0.867904\pi$
$659$	−2.78259	+	4.81959i	−0.108394	+	0.187744i	0.806726	+	0.590926i	$0.201238\pi$
$660$		0			0
$661$	19.6284	+	33.9974i	0.763457	+	1.32235i	0.941059	+	0.338244i	$0.109833\pi$
$661$	19.6284	+	33.9974i	0.763457	+	1.32235i	−0.177602	+	0.984102i	$0.556834\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	3.28065i		−	0.127027i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	25.0131	+	43.3239i	0.965619	+	1.67250i
$672$		0			0
$673$	13.8081	+	7.97209i	0.532262	+	0.307302i	0.741937	−	0.670470i	$-0.233907\pi$
$673$	13.8081	+	7.97209i	0.532262	+	0.307302i	−0.209675	+	0.977771i	$0.567241\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−18.0963	−	10.4479i	−0.695497	−	0.401545i	0.110171	−	0.993913i	$-0.464860\pi$
$677$	−18.0963	−	10.4479i	−0.695497	−	0.401545i	−0.805668	+	0.592368i	$0.798193\pi$
$678$		0			0
$679$	12.9442	+	22.4200i	0.496752	+	0.860400i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	19.6358i		−	0.751344i	−0.926753	−	0.375672i	$-0.877412\pi$
$683$		−	19.6358i		−	0.751344i	0.926753	−	0.375672i	$-0.122588\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.58744	+	4.48157i	0.0985734	+	0.170734i
$690$		0			0
$691$	21.5726	−	37.3648i	0.820660	−	1.42143i	−0.0845309	−	0.996421i	$-0.526939\pi$
$691$	21.5726	−	37.3648i	0.820660	−	1.42143i	0.905191	−	0.425005i	$-0.139728\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	5.05113	−	2.91627i	0.191325	−	0.110462i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−27.0410			−1.02132			−0.510662	−	0.859782i	$-0.670600\pi$
$701$	−27.0410			−1.02132			−0.510662	+	0.859782i	$0.670600\pi$
$702$		0			0
$703$		−	45.5652i		−	1.71852i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−31.8915	+	18.4126i	−1.19940	+	0.692476i
$708$		0			0
$709$	21.7510	−	37.6738i	0.816875	−	1.41487i	−0.0910989	−	0.995842i	$-0.529038\pi$
$709$	21.7510	−	37.6738i	0.816875	−	1.41487i	0.907974	−	0.419027i	$-0.137629\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−4.78410	−	2.76210i	−0.179166	−	0.103441i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−4.40515			−0.164284			−0.0821421	−	0.996621i	$-0.526176\pi$
$719$	−4.40515			−0.164284			−0.0821421	+	0.996621i	$0.526176\pi$
$720$		0			0
$721$	50.7810			1.89118
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−10.8094	−	6.24083i	−0.400900	−	0.231460i	0.285972	−	0.958238i	$-0.407683\pi$
$727$	−10.8094	−	6.24083i	−0.400900	−	0.231460i	−0.686872	+	0.726778i	$0.741017\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	9.73419	−	16.8601i	0.360032	−	0.623594i
$732$		0			0
$733$	−33.8041	+	19.5168i	−1.24858	+	0.720869i	−0.970826	−	0.239783i	$-0.922924\pi$
$733$	−33.8041	+	19.5168i	−1.24858	+	0.720869i	−0.277755	+	0.960652i	$0.589590\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			23.5242i			0.866525i
$738$		0			0
$739$	12.2935			0.452224			0.226112	−	0.974101i	$-0.427398\pi$
$739$	12.2935			0.452224			0.226112	+	0.974101i	$0.427398\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−35.3607	+	20.4155i	−1.29726	+	0.748972i	−0.979930	−	0.199344i	$-0.936119\pi$
$743$	−35.3607	+	20.4155i	−1.29726	+	0.748972i	−0.317328	+	0.948316i	$0.602786\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−26.3179	+	45.5840i	−0.961636	+	1.66560i
$750$		0			0
$751$	7.54469	+	13.0678i	0.275310	+	0.476850i	0.970213	−	0.242253i	$-0.0778863\pi$
$751$	7.54469	+	13.0678i	0.275310	+	0.476850i	−0.694904	+	0.719103i	$0.744553\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$			34.4610i			1.25251i	0.779620	+	0.626253i	$0.215412\pi$
$757$			34.4610i			1.25251i	−0.779620	+	0.626253i	$0.784588\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−4.17837	−	7.23716i	−0.151466	−	0.262347i	0.780301	−	0.625405i	$-0.215066\pi$
$761$	−4.17837	−	7.23716i	−0.151466	−	0.262347i	−0.931767	+	0.363058i	$0.881733\pi$
$762$		0			0
$763$	−1.24124	−	0.716631i	−0.0449359	−	0.0259438i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	4.48157	+	2.58744i	0.161820	+	0.0934269i
$768$		0			0
$769$	−9.17837	−	15.8974i	−0.330981	−	0.573275i	0.651724	−	0.758456i	$-0.274046\pi$
$769$	−9.17837	−	15.8974i	−0.330981	−	0.573275i	−0.982704	+	0.185181i	$0.940713\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	10.0968i		−	0.363157i	−0.983376	−	0.181578i	$-0.941879\pi$
$773$		−	10.0968i		−	0.363157i	0.983376	−	0.181578i	$-0.0581206\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.97209	+	6.87986i	0.142315	+	0.246497i
$780$		0			0
$781$	−27.7752	+	48.1080i	−0.993874	+	1.72144i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−12.6202	+	7.28630i	−0.449863	+	0.259729i	−0.707772	−	0.706441i	$-0.750300\pi$
$787$	−12.6202	+	7.28630i	−0.449863	+	0.259729i	0.257909	+	0.966169i	$0.416966\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	20.4200			0.726051
$792$		0			0
$793$			31.9145i			1.13332i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.79324	−	1.03533i	0.0635198	−	0.0366732i	−0.467904	−	0.883779i	$-0.654991\pi$
$797$	1.79324	−	1.03533i	0.0635198	−	0.0366732i	0.531424	+	0.847106i	$0.321657\pi$
$798$		0			0
$799$	−3.34212	+	5.78872i	−0.118236	+	0.204790i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−38.2728	−	22.0968i	−1.35062	−	0.779779i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	3.37158			0.118539			0.0592693	−	0.998242i	$-0.481123\pi$
$809$	3.37158			0.118539			0.0592693	+	0.998242i	$0.481123\pi$
$810$		0			0
$811$	−12.9368			−0.454271			−0.227136	−	0.973863i	$-0.572936\pi$
$811$	−12.9368			−0.454271			−0.227136	+	0.973863i	$0.572936\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	22.9642	+	13.2584i	0.803416	+	0.463852i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−10.9963	+	19.0461i	−0.383773	+	0.664715i	−0.991598	−	0.129356i	$-0.958709\pi$
$821$	−10.9963	+	19.0461i	−0.383773	+	0.664715i	0.607825	+	0.794071i	$0.292042\pi$
$822$		0			0
$823$	−12.9052	+	7.45082i	−0.449847	+	0.259719i	−0.707766	−	0.706447i	$-0.750297\pi$
$823$	−12.9052	+	7.45082i	−0.449847	+	0.259719i	0.257919	+	0.966167i	$0.416963\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	56.5785i		−	1.96743i	−0.179747	−	0.983713i	$-0.557528\pi$
$827$		−	56.5785i		−	1.96743i	0.179747	−	0.983713i	$-0.442472\pi$
$828$		0			0
$829$	−27.0558			−0.939687			−0.469844	−	0.882750i	$-0.655690\pi$
$829$	−27.0558			−0.939687			−0.469844	+	0.882750i	$0.655690\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2.27628	+	1.31421i	−0.0788683	+	0.0455346i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	17.2863	−	29.9407i	0.596789	−	1.03367i	−0.396502	−	0.918034i	$-0.629776\pi$
$839$	17.2863	−	29.9407i	0.596789	−	1.03367i	0.993292	−	0.115636i	$-0.0368905\pi$
$840$		0			0
$841$	4.51678	+	7.82329i	0.155751	+	0.269769i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$			53.3626i			1.83356i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.22306	+	3.85046i	0.0762056	+	0.131992i
$852$		0			0
$853$	14.7032	+	8.48887i	0.503427	+	0.290653i	0.730127	−	0.683311i	$-0.239461\pi$
$853$	14.7032	+	8.48887i	0.503427	+	0.290653i	−0.226701	+	0.973964i	$0.572794\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	2.95537	+	1.70628i	0.100953	+	0.0582855i	0.549627	−	0.835410i	$-0.314770\pi$
$857$	2.95537	+	1.70628i	0.100953	+	0.0582855i	−0.448673	+	0.893696i	$0.648103\pi$
$858$		0			0
$859$	−4.91627	−	8.51524i	−0.167741	−	0.290536i	0.769884	−	0.638184i	$-0.220314\pi$
$859$	−4.91627	−	8.51524i	−0.167741	−	0.290536i	−0.937625	+	0.347647i	$0.886981\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	48.9836i		−	1.66742i	−0.552202	−	0.833711i	$-0.686212\pi$
$863$		−	48.9836i		−	1.66742i	0.552202	−	0.833711i	$-0.313788\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−5.52420	−	9.56819i	−0.187396	−	0.324579i
$870$		0			0
$871$	−7.50371	+	12.9968i	−0.254253	+	0.440380i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−10.9720	+	6.33470i	−0.370499	+	0.213908i	−0.673676	−	0.739027i	$-0.735286\pi$
$877$	−10.9720	+	6.33470i	−0.370499	+	0.213908i	0.303178	+	0.952934i	$0.401952\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	26.0894			0.878974			0.439487	−	0.898249i	$-0.355160\pi$
$881$	26.0894			0.878974			0.439487	+	0.898249i	$0.355160\pi$
$882$		0			0
$883$		−	2.69321i		−	0.0906337i	−0.998973	−	0.0453169i	$-0.985570\pi$
$883$		−	2.69321i		−	0.0906337i	0.998973	−	0.0453169i	$-0.0144297\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	21.2676	−	12.2789i	0.714098	−	0.412284i	−0.0984788	−	0.995139i	$-0.531398\pi$
$887$	21.2676	−	12.2789i	0.714098	−	0.412284i	0.812576	+	0.582855i	$0.198064\pi$
$888$		0			0
$889$	11.2900	−	19.5549i	0.378655	−	0.655849i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−7.88448	−	4.55211i	−0.263844	−	0.152330i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−33.6210			−1.12132
$900$		0			0
$901$	−8.11164			−0.270238
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−29.7653	−	17.1850i	−0.988341	−	0.570619i	−0.0835631	−	0.996502i	$-0.526630\pi$
$907$	−29.7653	−	17.1850i	−0.988341	−	0.570619i	−0.904778	+	0.425884i	$0.859963\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	18.7752	−	32.5196i	0.622049	−	1.07742i	−0.367054	−	0.930199i	$-0.619634\pi$
$911$	18.7752	−	32.5196i	0.622049	−	1.07742i	0.989104	−	0.147221i	$-0.0470330\pi$
$912$		0			0
$913$	25.1921	−	14.5447i	0.833738	−	0.481359i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			44.0117i			1.45340i
$918$		0			0
$919$	−10.1116			−0.333552			−0.166776	−	0.985995i	$-0.553336\pi$
$919$	−10.1116			−0.333552			−0.166776	+	0.985995i	$0.553336\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−30.6908	+	17.7194i	−1.01020	+	0.583240i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−12.4126	+	21.4992i	−0.407243	+	0.705366i	−0.994580	−	0.103977i	$-0.966843\pi$
$929$	−12.4126	+	21.4992i	−0.407243	+	0.705366i	0.587337	+	0.809343i	$0.300176\pi$
$930$		0			0
$931$	−1.79001	−	3.10039i	−0.0586652	−	0.101611i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			1.56518i			0.0511322i	0.999673	+	0.0255661i	$0.00813883\pi$
$937$			1.56518i			0.0511322i	−0.999673	+	0.0255661i	$0.991861\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−5.23419	−	9.06588i	−0.170630	−	0.295539i	0.768010	−	0.640437i	$-0.221247\pi$
$941$	−5.23419	−	9.06588i	−0.170630	−	0.295539i	−0.938640	+	0.344898i	$0.887914\pi$
$942$		0			0
$943$	−0.671318	−	0.387586i	−0.0218611	−	0.0126215i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−2.76709	−	1.59758i	−0.0899182	−	0.0519143i	0.454367	−	0.890815i	$-0.349866\pi$
$947$	−2.76709	−	1.59758i	−0.0899182	−	0.0519143i	−0.544285	+	0.838900i	$0.683199\pi$
$948$		0			0
$949$	−14.0968	−	24.4164i	−0.457601	−	0.792589i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	33.9293i		−	1.09908i	−0.835468	−	0.549540i	$-0.814803\pi$
$953$		−	33.9293i		−	1.09908i	0.835468	−	0.549540i	$-0.185197\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	25.3142	+	43.8455i	0.817438	+	1.41584i
$960$		0			0
$961$	−12.8068	+	22.1820i	−0.413122	+	0.715549i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	33.2778	−	19.2129i	1.07014	−	0.617846i	0.141921	−	0.989878i	$-0.454672\pi$
$967$	33.2778	−	19.2129i	1.07014	−	0.617846i	0.928220	+	0.372032i	$0.121339\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−29.7326			−0.954166			−0.477083	−	0.878858i	$-0.658306\pi$
$971$	−29.7326			−0.954166			−0.477083	+	0.878858i	$0.658306\pi$
$972$		0			0
$973$			10.9368i			0.350617i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	28.5596	−	16.4889i	0.913701	−	0.527526i	0.0320812	−	0.999485i	$-0.489786\pi$
$977$	28.5596	−	16.4889i	0.913701	−	0.527526i	0.881620	+	0.471960i	$0.156453\pi$
$978$		0			0
$979$	−8.28630	+	14.3523i	−0.264831	+	0.458701i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	6.38905	+	3.68872i	0.203779	+	0.117652i	0.598417	−	0.801185i	$-0.295797\pi$
$983$	6.38905	+	3.68872i	0.203779	+	0.117652i	−0.394638	+	0.918837i	$0.629130\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2.58744			−0.0822757
$990$		0			0
$991$	−4.51678			−0.143480			−0.0717401	−	0.997423i	$-0.522855\pi$
$991$	−4.51678			−0.143480			−0.0717401	+	0.997423i	$0.522855\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	42.1134	+	24.3142i	1.33375	+	0.770039i	0.985872	−	0.167502i	$-0.0535702\pi$
$997$	42.1134	+	24.3142i	1.33375	+	0.770039i	0.347874	+	0.937541i	$0.386904\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.2.s.c.2449.3		12
3.2	odd	2		900.2.s.c.349.4		12
5.2	odd	4		2700.2.i.c.1801.2		6
5.3	odd	4		540.2.i.b.181.2		6
5.4	even	2	inner	2700.2.s.c.2449.4		12
9.2	odd	6		8100.2.d.p.649.4		6
9.4	even	3	inner	2700.2.s.c.1549.4		12
9.5	odd	6		900.2.s.c.49.3		12
9.7	even	3		8100.2.d.o.649.4		6
15.2	even	4		900.2.i.c.601.3		6
15.8	even	4		180.2.i.b.61.1	✓	6
15.14	odd	2		900.2.s.c.349.3		12
20.3	even	4		2160.2.q.i.721.2		6
45.2	even	12		8100.2.a.v.1.2		3
45.4	even	6	inner	2700.2.s.c.1549.3		12
45.7	odd	12		8100.2.a.u.1.2		3
45.13	odd	12		540.2.i.b.361.2		6
45.14	odd	6		900.2.s.c.49.4		12
45.22	odd	12		2700.2.i.c.901.2		6
45.23	even	12		180.2.i.b.121.1	yes	6
45.29	odd	6		8100.2.d.p.649.3		6
45.32	even	12		900.2.i.c.301.3		6
45.34	even	6		8100.2.d.o.649.3		6
45.38	even	12		1620.2.a.i.1.2		3
45.43	odd	12		1620.2.a.j.1.2		3
60.23	odd	4		720.2.q.k.241.3		6
180.23	odd	12		720.2.q.k.481.3		6
180.43	even	12		6480.2.a.bw.1.2		3
180.83	odd	12		6480.2.a.bt.1.2		3
180.103	even	12		2160.2.q.i.1441.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.i.b.61.1	✓	6	15.8	even	4
180.2.i.b.121.1	yes	6	45.23	even	12
540.2.i.b.181.2		6	5.3	odd	4
540.2.i.b.361.2		6	45.13	odd	12
720.2.q.k.241.3		6	60.23	odd	4
720.2.q.k.481.3		6	180.23	odd	12
900.2.i.c.301.3		6	45.32	even	12
900.2.i.c.601.3		6	15.2	even	4
900.2.s.c.49.3		12	9.5	odd	6
900.2.s.c.49.4		12	45.14	odd	6
900.2.s.c.349.3		12	15.14	odd	2
900.2.s.c.349.4		12	3.2	odd	2
1620.2.a.i.1.2		3	45.38	even	12
1620.2.a.j.1.2		3	45.43	odd	12
2160.2.q.i.721.2		6	20.3	even	4
2160.2.q.i.1441.2		6	180.103	even	12
2700.2.i.c.901.2		6	45.22	odd	12
2700.2.i.c.1801.2		6	5.2	odd	4
2700.2.s.c.1549.3		12	45.4	even	6	inner
2700.2.s.c.1549.4		12	9.4	even	3	inner
2700.2.s.c.2449.3		12	1.1	even	1	trivial
2700.2.s.c.2449.4		12	5.4	even	2	inner
6480.2.a.bt.1.2		3	180.83	odd	12
6480.2.a.bw.1.2		3	180.43	even	12
8100.2.a.u.1.2		3	45.7	odd	12
8100.2.a.v.1.2		3	45.2	even	12
8100.2.d.o.649.3		6	45.34	even	6
8100.2.d.o.649.4		6	9.7	even	3
8100.2.d.p.649.3		6	45.29	odd	6
8100.2.d.p.649.4		6	9.2	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2700.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.36788 - 1.36710i) q^{7} +(2.76210 - 4.78410i) q^{11} +(3.05205 - 1.76210i) q^{13} +5.52420i q^{17} -7.52420 q^{19} +(0.635828 - 0.367095i) q^{23} +(2.23419 - 3.86973i) q^{29} +(-3.76210 - 6.51615i) q^{31} +6.05582i q^{37} +(-0.527909 - 0.914365i) q^{41} +(-3.05205 - 1.76210i) q^{43} +(1.04788 + 0.604996i) q^{47} +(0.237900 + 0.412055i) q^{49} +1.46838i q^{53} +(0.734191 + 1.27166i) q^{59} +(-4.52791 + 7.84257i) q^{61} +(-3.68787 + 2.12920i) q^{67} -10.0558 q^{71} -8.00000i q^{73} +(-13.0806 + 7.55211i) q^{77} +(1.00000 - 1.73205i) q^{79} +(4.56032 + 2.63290i) q^{83} -3.00000 q^{89} -9.63583 q^{91} +(-8.19986 - 4.73419i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 24 q^{19} + 6 q^{29} - 12 q^{31} + 6 q^{41} + 36 q^{49} - 12 q^{59} - 42 q^{61} - 96 q^{71} + 12 q^{79} - 36 q^{89}+O(q^{100}) \) 12 * q - 24 * q^19 + 6 * q^29 - 12 * q^31 + 6 * q^41 + 36 * q^49 - 12 * q^59 - 42 * q^61 - 96 * q^71 + 12 * q^79 - 36 * q^89

Introduction

L-functions

Modular forms

Varieties

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