LMFDB - Embedded newform 2160.2.f.d.1729.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2160,2,Mod(1729,2160)] 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("2160.1729"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2160.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-2,0,0,0,0,0,10,0,0,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$17.2476868366$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 270)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1729.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2160.1729
Dual form			2160.2.f.d.1729.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 2.00000i) q^{5} +4.00000i q^{7} +5.00000 q^{11} -3.00000i q^{13} +1.00000i q^{17} -6.00000 q^{19} +1.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} +9.00000 q^{29} +5.00000 q^{31} +(8.00000 - 4.00000i) q^{35} +2.00000i q^{37} -2.00000 q^{41} -1.00000i q^{43} +13.0000i q^{47} -9.00000 q^{49} +(-5.00000 - 10.0000i) q^{55} -4.00000 q^{59} +8.00000 q^{61} +(-6.00000 + 3.00000i) q^{65} -4.00000i q^{67} +6.00000 q^{71} -2.00000i q^{73} +20.0000i q^{77} +9.00000 q^{79} +4.00000i q^{83} +(2.00000 - 1.00000i) q^{85} +14.0000 q^{89} +12.0000 q^{91} +(6.00000 + 12.0000i) q^{95} +10.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 10 q^{11} - 12 q^{19} - 6 q^{25} + 18 q^{29} + 10 q^{31} + 16 q^{35} - 4 q^{41} - 18 q^{49} - 10 q^{55} - 8 q^{59} + 16 q^{61} - 12 q^{65} + 12 q^{71} + 18 q^{79} + 4 q^{85} + 28 q^{89}+ \cdots + 12 q^{95}+O(q^{100}) $ 2 * q - 2 * q^5 + 10 * q^11 - 12 * q^19 - 6 * q^25 + 18 * q^29 + 10 * q^31 + 16 * q^35 - 4 * q^41 - 18 * q^49 - 10 * q^55 - 8 * q^59 + 16 * q^61 - 12 * q^65 + 12 * q^71 + 18 * q^79 + 4 * q^85 + 28 * q^89 + 24 * q^91 + 12 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		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
$7$			4.00000i			1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$			4.00000i			1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	5.00000			1.50756			0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000			1.50756			0.753778	+	0.657129i	$0.228229\pi$
$12$		0			0
$13$		−	3.00000i		−	0.832050i	−0.909353	−	0.416025i	$-0.863423\pi$
$13$		−	3.00000i		−	0.832050i	0.909353	−	0.416025i	$-0.136577\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			1.00000i			0.242536i	0.992620	+	0.121268i	$0.0386960\pi$
$17$			1.00000i			0.242536i	−0.992620	+	0.121268i	$0.961304\pi$
$18$		0			0
$19$	−6.00000			−1.37649			−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000			−1.37649			−0.688247	+	0.725476i	$0.741620\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1.00000i			0.208514i	0.994550	+	0.104257i	$0.0332465\pi$
$23$			1.00000i			0.208514i	−0.994550	+	0.104257i	$0.966753\pi$
$24$		0			0
$25$	−3.00000	+	4.00000i	−0.600000	+	0.800000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	9.00000			1.67126			0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000			1.67126			0.835629	+	0.549294i	$0.185103\pi$
$30$		0			0
$31$	5.00000			0.898027			0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000			0.898027			0.449013	+	0.893525i	$0.351776\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	8.00000	−	4.00000i	1.35225	−	0.676123i
$36$		0			0
$37$			2.00000i			0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$			2.00000i			0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.00000			−0.312348			−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000			−0.312348			−0.156174	+	0.987730i	$0.549916\pi$
$42$		0			0
$43$		−	1.00000i		−	0.152499i	−0.997089	−	0.0762493i	$-0.975706\pi$
$43$		−	1.00000i		−	0.152499i	0.997089	−	0.0762493i	$-0.0242945\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			13.0000i			1.89624i	0.317905	+	0.948122i	$0.397021\pi$
$47$			13.0000i			1.89624i	−0.317905	+	0.948122i	$0.602979\pi$
$48$		0			0
$49$	−9.00000			−1.28571
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$	−5.00000	−	10.0000i	−0.674200	−	1.34840i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.00000			−0.520756			−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000			−0.520756			−0.260378	+	0.965507i	$0.583847\pi$
$60$		0			0
$61$	8.00000			1.02430			0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000			1.02430			0.512148	+	0.858898i	$0.328850\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−6.00000	+	3.00000i	−0.744208	+	0.372104i
$66$		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
$71$	6.00000			0.712069			0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000			0.712069			0.356034	+	0.934473i	$0.384129\pi$
$72$		0			0
$73$		−	2.00000i		−	0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$		−	2.00000i		−	0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			20.0000i			2.27921i
$78$		0			0
$79$	9.00000			1.01258			0.506290	−	0.862364i	$-0.331017\pi$
$79$	9.00000			1.01258			0.506290	+	0.862364i	$0.331017\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			4.00000i			0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$			4.00000i			0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$		0			0
$85$	2.00000	−	1.00000i	0.216930	−	0.108465i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	14.0000			1.48400			0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000			1.48400			0.741999	+	0.670402i	$0.233878\pi$
$90$		0			0
$91$	12.0000			1.25794
$92$		0			0
$93$		0			0
$94$		0			0
$95$	6.00000	+	12.0000i	0.615587	+	1.23117i
$96$		0			0
$97$			10.0000i			1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$			10.0000i			1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	17.0000			1.69156			0.845782	−	0.533529i	$-0.179135\pi$
$101$	17.0000			1.69156			0.845782	+	0.533529i	$0.179135\pi$
$102$		0			0
$103$			10.0000i			0.985329i	0.870219	+	0.492665i	$0.163977\pi$
$103$			10.0000i			0.985329i	−0.870219	+	0.492665i	$0.836023\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	6.00000i		−	0.580042i	−0.957020	−	0.290021i	$-0.906338\pi$
$107$		−	6.00000i		−	0.580042i	0.957020	−	0.290021i	$-0.0936623\pi$
$108$		0			0
$109$	8.00000			0.766261			0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000			0.766261			0.383131	+	0.923694i	$0.374846\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			3.00000i			0.282216i	0.989994	+	0.141108i	$0.0450665\pi$
$113$			3.00000i			0.282216i	−0.989994	+	0.141108i	$0.954933\pi$
$114$		0			0
$115$	2.00000	−	1.00000i	0.186501	−	0.0932505i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−4.00000			−0.366679
$120$		0			0
$121$	14.0000			1.27273
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.0000	+	2.00000i	0.983870	+	0.178885i
$126$		0			0
$127$			4.00000i			0.354943i	0.984126	+	0.177471i	$0.0567917\pi$
$127$			4.00000i			0.354943i	−0.984126	+	0.177471i	$0.943208\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.00000			0.262111			0.131056	−	0.991375i	$-0.458163\pi$
$131$	3.00000			0.262111			0.131056	+	0.991375i	$0.458163\pi$
$132$		0			0
$133$		−	24.0000i		−	2.08106i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	18.0000i		−	1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$		−	18.0000i		−	1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$		0			0
$139$	−8.00000			−0.678551			−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000			−0.678551			−0.339276	+	0.940687i	$0.610182\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	15.0000i		−	1.25436i
$144$		0			0
$145$	−9.00000	−	18.0000i	−0.747409	−	1.49482i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	13.0000			1.06500			0.532501	−	0.846430i	$-0.321252\pi$
$149$	13.0000			1.06500			0.532501	+	0.846430i	$0.321252\pi$
$150$		0			0
$151$	−13.0000			−1.05792			−0.528962	−	0.848645i	$-0.677419\pi$
$151$	−13.0000			−1.05792			−0.528962	+	0.848645i	$0.677419\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−5.00000	−	10.0000i	−0.401610	−	0.803219i
$156$		0			0
$157$			13.0000i			1.03751i	0.854922	+	0.518756i	$0.173605\pi$
$157$			13.0000i			1.03751i	−0.854922	+	0.518756i	$0.826395\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−4.00000			−0.315244
$162$		0			0
$163$			19.0000i			1.48819i	0.668071	+	0.744097i	$0.267120\pi$
$163$			19.0000i			1.48819i	−0.668071	+	0.744097i	$0.732880\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	12.0000i		−	0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$		−	12.0000i		−	0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$		0			0
$169$	4.00000			0.307692
$170$		0			0
$171$		0			0
$172$		0			0
$173$			16.0000i			1.21646i	0.793762	+	0.608229i	$0.208120\pi$
$173$			16.0000i			1.21646i	−0.793762	+	0.608229i	$0.791880\pi$
$174$		0			0
$175$	−16.0000	−	12.0000i	−1.20949	−	0.907115i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	20.0000			1.49487			0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000			1.49487			0.747435	+	0.664335i	$0.231285\pi$
$180$		0			0
$181$	−22.0000			−1.63525			−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000			−1.63525			−0.817624	+	0.575753i	$0.804709\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	4.00000	−	2.00000i	0.294086	−	0.147043i
$186$		0			0
$187$			5.00000i			0.365636i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.00000			0.434145			0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000			0.434145			0.217072	+	0.976156i	$0.430349\pi$
$192$		0			0
$193$			4.00000i			0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$			4.00000i			0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	22.0000i		−	1.56744i	−0.621117	−	0.783718i	$-0.713321\pi$
$197$		−	22.0000i		−	1.56744i	0.621117	−	0.783718i	$-0.286679\pi$
$198$		0			0
$199$	3.00000			0.212664			0.106332	−	0.994331i	$-0.466089\pi$
$199$	3.00000			0.212664			0.106332	+	0.994331i	$0.466089\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			36.0000i			2.52670i
$204$		0			0
$205$	2.00000	+	4.00000i	0.139686	+	0.279372i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−30.0000			−2.07514
$210$		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
$215$	−2.00000	+	1.00000i	−0.136399	+	0.0681994i
$216$		0			0
$217$			20.0000i			1.35769i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3.00000			0.201802
$222$		0			0
$223$		−	22.0000i		−	1.47323i	−0.676313	−	0.736614i	$-0.736423\pi$
$223$		−	22.0000i		−	1.47323i	0.676313	−	0.736614i	$-0.263577\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			2.00000i			0.132745i	0.997795	+	0.0663723i	$0.0211425\pi$
$227$			2.00000i			0.132745i	−0.997795	+	0.0663723i	$0.978857\pi$
$228$		0			0
$229$	−2.00000			−0.132164			−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000			−0.132164			−0.0660819	+	0.997814i	$0.521050\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	14.0000i		−	0.917170i	−0.888650	−	0.458585i	$-0.848356\pi$
$233$		−	14.0000i		−	0.917170i	0.888650	−	0.458585i	$-0.151644\pi$
$234$		0			0
$235$	26.0000	−	13.0000i	1.69605	−	0.848026i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	16.0000			1.03495			0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000			1.03495			0.517477	+	0.855697i	$0.326871\pi$
$240$		0			0
$241$	−23.0000			−1.48156			−0.740780	−	0.671748i	$-0.765544\pi$
$241$	−23.0000			−1.48156			−0.740780	+	0.671748i	$0.765544\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	9.00000	+	18.0000i	0.574989	+	1.14998i
$246$		0			0
$247$			18.0000i			1.14531i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−15.0000			−0.946792			−0.473396	−	0.880850i	$-0.656972\pi$
$251$	−15.0000			−0.946792			−0.473396	+	0.880850i	$0.656972\pi$
$252$		0			0
$253$			5.00000i			0.314347i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			21.0000i			1.30994i	0.755653	+	0.654972i	$0.227320\pi$
$257$			21.0000i			1.30994i	−0.755653	+	0.654972i	$0.772680\pi$
$258$		0			0
$259$	−8.00000			−0.497096
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	8.00000i		−	0.493301i	−0.969104	−	0.246651i	$-0.920670\pi$
$263$		−	8.00000i		−	0.493301i	0.969104	−	0.246651i	$-0.0793300\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−9.00000			−0.548740			−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000			−0.548740			−0.274370	+	0.961624i	$0.588469\pi$
$270$		0			0
$271$	−16.0000			−0.971931			−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000			−0.971931			−0.485965	+	0.873978i	$0.661532\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−15.0000	+	20.0000i	−0.904534	+	1.20605i
$276$		0			0
$277$			26.0000i			1.56219i	0.624413	+	0.781094i	$0.285338\pi$
$277$			26.0000i			1.56219i	−0.624413	+	0.781094i	$0.714662\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$		−	20.0000i		−	1.18888i	−0.804141	−	0.594438i	$-0.797374\pi$
$283$		−	20.0000i		−	1.18888i	0.804141	−	0.594438i	$-0.202626\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	8.00000i		−	0.472225i
$288$		0			0
$289$	16.0000			0.941176
$290$		0			0
$291$		0			0
$292$		0			0
$293$			6.00000i			0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$			6.00000i			0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$		0			0
$295$	4.00000	+	8.00000i	0.232889	+	0.465778i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.00000			0.173494
$300$		0			0
$301$	4.00000			0.230556
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−8.00000	−	16.0000i	−0.458079	−	0.916157i
$306$		0			0
$307$			21.0000i			1.19853i	0.800549	+	0.599267i	$0.204541\pi$
$307$			21.0000i			1.19853i	−0.800549	+	0.599267i	$0.795459\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−18.0000			−1.02069			−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000			−1.02069			−0.510343	+	0.859971i	$0.670482\pi$
$312$		0			0
$313$		−	14.0000i		−	0.791327i	−0.918396	−	0.395663i	$-0.870515\pi$
$313$		−	14.0000i		−	0.791327i	0.918396	−	0.395663i	$-0.129485\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			4.00000i			0.224662i	0.993671	+	0.112331i	$0.0358318\pi$
$317$			4.00000i			0.224662i	−0.993671	+	0.112331i	$0.964168\pi$
$318$		0			0
$319$	45.0000			2.51952
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	6.00000i		−	0.333849i
$324$		0			0
$325$	12.0000	+	9.00000i	0.665640	+	0.499230i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−52.0000			−2.86685
$330$		0			0
$331$	−10.0000			−0.549650			−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000			−0.549650			−0.274825	+	0.961494i	$0.588620\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−8.00000	+	4.00000i	−0.437087	+	0.218543i
$336$		0			0
$337$		−	16.0000i		−	0.871576i	−0.900049	−	0.435788i	$-0.856470\pi$
$337$		−	16.0000i		−	0.871576i	0.900049	−	0.435788i	$-0.143530\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	25.0000			1.35383
$342$		0			0
$343$		−	8.00000i		−	0.431959i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			6.00000i			0.322097i	0.986947	+	0.161048i	$0.0514875\pi$
$347$			6.00000i			0.322097i	−0.986947	+	0.161048i	$0.948512\pi$
$348$		0			0
$349$	−16.0000			−0.856460			−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000			−0.856460			−0.428230	+	0.903670i	$0.640863\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	11.0000i		−	0.585471i	−0.956193	−	0.292735i	$-0.905434\pi$
$353$		−	11.0000i		−	0.585471i	0.956193	−	0.292735i	$-0.0945655\pi$
$354$		0			0
$355$	−6.00000	−	12.0000i	−0.318447	−	0.636894i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	6.00000			0.316668			0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000			0.316668			0.158334	+	0.987386i	$0.449388\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−4.00000	+	2.00000i	−0.209370	+	0.104685i
$366$		0			0
$367$			22.0000i			1.14839i	0.818718	+	0.574195i	$0.194685\pi$
$367$			22.0000i			1.14839i	−0.818718	+	0.574195i	$0.805315\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	21.0000i		−	1.08734i	−0.839299	−	0.543669i	$-0.817035\pi$
$373$		−	21.0000i		−	1.08734i	0.839299	−	0.543669i	$-0.182965\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	27.0000i		−	1.39057i
$378$		0			0
$379$	−28.0000			−1.43826			−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000			−1.43826			−0.719132	+	0.694874i	$0.755460\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	19.0000i		−	0.970855i	−0.874277	−	0.485427i	$-0.838664\pi$
$383$		−	19.0000i		−	0.970855i	0.874277	−	0.485427i	$-0.161336\pi$
$384$		0			0
$385$	40.0000	−	20.0000i	2.03859	−	1.01929i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5.00000			0.253510			0.126755	−	0.991934i	$-0.459544\pi$
$389$	5.00000			0.253510			0.126755	+	0.991934i	$0.459544\pi$
$390$		0			0
$391$	−1.00000			−0.0505722
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−9.00000	−	18.0000i	−0.452839	−	0.905678i
$396$		0			0
$397$		−	13.0000i		−	0.652451i	−0.945292	−	0.326226i	$-0.894223\pi$
$397$		−	13.0000i		−	0.652451i	0.945292	−	0.326226i	$-0.105777\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−16.0000			−0.799002			−0.399501	−	0.916733i	$-0.630817\pi$
$401$	−16.0000			−0.799002			−0.399501	+	0.916733i	$0.630817\pi$
$402$		0			0
$403$		−	15.0000i		−	0.747203i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			10.0000i			0.495682i
$408$		0			0
$409$	−11.0000			−0.543915			−0.271957	−	0.962309i	$-0.587671\pi$
$409$	−11.0000			−0.543915			−0.271957	+	0.962309i	$0.587671\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	16.0000i		−	0.787309i
$414$		0			0
$415$	8.00000	−	4.00000i	0.392705	−	0.196352i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−23.0000			−1.12362			−0.561812	−	0.827265i	$-0.689895\pi$
$419$	−23.0000			−1.12362			−0.561812	+	0.827265i	$0.689895\pi$
$420$		0			0
$421$	−26.0000			−1.26716			−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000			−1.26716			−0.633581	+	0.773676i	$0.718416\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−4.00000	−	3.00000i	−0.194029	−	0.145521i
$426$		0			0
$427$			32.0000i			1.54859i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	8.00000			0.385346			0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000			0.385346			0.192673	+	0.981263i	$0.438284\pi$
$432$		0			0
$433$			22.0000i			1.05725i	0.848855	+	0.528626i	$0.177293\pi$
$433$			22.0000i			1.05725i	−0.848855	+	0.528626i	$0.822707\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	6.00000i		−	0.287019i
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	30.0000i		−	1.42534i	−0.701498	−	0.712672i	$-0.747485\pi$
$443$		−	30.0000i		−	1.42534i	0.701498	−	0.712672i	$-0.252515\pi$
$444$		0			0
$445$	−14.0000	−	28.0000i	−0.663664	−	1.32733i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$	−10.0000			−0.470882
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−12.0000	−	24.0000i	−0.562569	−	1.12514i
$456$		0			0
$457$		−	16.0000i		−	0.748448i	−0.927338	−	0.374224i	$-0.877909\pi$
$457$		−	16.0000i		−	0.748448i	0.927338	−	0.374224i	$-0.122091\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−6.00000			−0.279448			−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000			−0.279448			−0.139724	+	0.990190i	$0.544622\pi$
$462$		0			0
$463$			6.00000i			0.278844i	0.990233	+	0.139422i	$0.0445244\pi$
$463$			6.00000i			0.278844i	−0.990233	+	0.139422i	$0.955476\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	6.00000i		−	0.277647i	−0.990317	−	0.138823i	$-0.955668\pi$
$467$		−	6.00000i		−	0.277647i	0.990317	−	0.138823i	$-0.0443321\pi$
$468$		0			0
$469$	16.0000			0.738811
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	5.00000i		−	0.229900i
$474$		0			0
$475$	18.0000	−	24.0000i	0.825897	−	1.10120i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−2.00000			−0.0913823			−0.0456912	−	0.998956i	$-0.514549\pi$
$479$	−2.00000			−0.0913823			−0.0456912	+	0.998956i	$0.514549\pi$
$480$		0			0
$481$	6.00000			0.273576
$482$		0			0
$483$		0			0
$484$		0			0
$485$	20.0000	−	10.0000i	0.908153	−	0.454077i
$486$		0			0
$487$		−	12.0000i		−	0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$		−	12.0000i		−	0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−24.0000			−1.08310			−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000			−1.08310			−0.541552	+	0.840667i	$0.682163\pi$
$492$		0			0
$493$			9.00000i			0.405340i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			24.0000i			1.07655i
$498$		0			0
$499$	−12.0000			−0.537194			−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000			−0.537194			−0.268597	+	0.963253i	$0.586560\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			9.00000i			0.401290i	0.979664	+	0.200645i	$0.0643038\pi$
$503$			9.00000i			0.401290i	−0.979664	+	0.200645i	$0.935696\pi$
$504$		0			0
$505$	−17.0000	−	34.0000i	−0.756490	−	1.51298i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	21.0000			0.930809			0.465404	−	0.885098i	$-0.345909\pi$
$509$	21.0000			0.930809			0.465404	+	0.885098i	$0.345909\pi$
$510$		0			0
$511$	8.00000			0.353899
$512$		0			0
$513$		0			0
$514$		0			0
$515$	20.0000	−	10.0000i	0.881305	−	0.440653i
$516$		0			0
$517$			65.0000i			2.85870i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−22.0000			−0.963837			−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000			−0.963837			−0.481919	+	0.876216i	$0.660060\pi$
$522$		0			0
$523$		−	23.0000i		−	1.00572i	−0.864368	−	0.502860i	$-0.832281\pi$
$523$		−	23.0000i		−	1.00572i	0.864368	−	0.502860i	$-0.167719\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			5.00000i			0.217803i
$528$		0			0
$529$	22.0000			0.956522
$530$		0			0
$531$		0			0
$532$		0			0
$533$			6.00000i			0.259889i
$534$		0			0
$535$	−12.0000	+	6.00000i	−0.518805	+	0.259403i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−45.0000			−1.93829
$540$		0			0
$541$	10.0000			0.429934			0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000			0.429934			0.214967	+	0.976621i	$0.431036\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−8.00000	−	16.0000i	−0.342682	−	0.685365i
$546$		0			0
$547$			25.0000i			1.06892i	0.845193	+	0.534461i	$0.179486\pi$
$547$			25.0000i			1.06892i	−0.845193	+	0.534461i	$0.820514\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−54.0000			−2.30048
$552$		0			0
$553$			36.0000i			1.53088i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			42.0000i			1.77960i	0.456354	+	0.889799i	$0.349155\pi$
$557$			42.0000i			1.77960i	−0.456354	+	0.889799i	$0.650845\pi$
$558$		0			0
$559$	−3.00000			−0.126886
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	4.00000i		−	0.168580i	−0.996441	−	0.0842900i	$-0.973138\pi$
$563$		−	4.00000i		−	0.168580i	0.996441	−	0.0842900i	$-0.0268622\pi$
$564$		0			0
$565$	6.00000	−	3.00000i	0.252422	−	0.126211i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	20.0000			0.838444			0.419222	−	0.907884i	$-0.362303\pi$
$569$	20.0000			0.838444			0.419222	+	0.907884i	$0.362303\pi$
$570$		0			0
$571$	32.0000			1.33916			0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000			1.33916			0.669579	+	0.742741i	$0.266474\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.00000	−	3.00000i	−0.166812	−	0.125109i
$576$		0			0
$577$			32.0000i			1.33218i	0.745873	+	0.666089i	$0.232033\pi$
$577$			32.0000i			1.33218i	−0.745873	+	0.666089i	$0.767967\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−16.0000			−0.663792
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	36.0000i		−	1.48588i	−0.669359	−	0.742940i	$-0.733431\pi$
$587$		−	36.0000i		−	1.48588i	0.669359	−	0.742940i	$-0.266569\pi$
$588$		0			0
$589$	−30.0000			−1.23613
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	21.0000i		−	0.862367i	−0.902264	−	0.431183i	$-0.858096\pi$
$593$		−	21.0000i		−	0.862367i	0.902264	−	0.431183i	$-0.141904\pi$
$594$		0			0
$595$	4.00000	+	8.00000i	0.163984	+	0.327968i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−42.0000			−1.71607			−0.858037	−	0.513588i	$-0.828316\pi$
$599$	−42.0000			−1.71607			−0.858037	+	0.513588i	$0.828316\pi$
$600$		0			0
$601$	−19.0000			−0.775026			−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000			−0.775026			−0.387513	+	0.921864i	$0.626666\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−14.0000	−	28.0000i	−0.569181	−	1.13836i
$606$		0			0
$607$		−	10.0000i		−	0.405887i	−0.979190	−	0.202944i	$-0.934949\pi$
$607$		−	10.0000i		−	0.405887i	0.979190	−	0.202944i	$-0.0650509\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	39.0000			1.57777
$612$		0			0
$613$		−	23.0000i		−	0.928961i	−0.885583	−	0.464481i	$-0.846241\pi$
$613$		−	23.0000i		−	0.928961i	0.885583	−	0.464481i	$-0.153759\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	29.0000i		−	1.16750i	−0.811935	−	0.583748i	$-0.801586\pi$
$617$		−	29.0000i		−	1.16750i	0.811935	−	0.583748i	$-0.198414\pi$
$618$		0			0
$619$	26.0000			1.04503			0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000			1.04503			0.522514	+	0.852631i	$0.324994\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			56.0000i			2.24359i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−2.00000			−0.0797452
$630$		0			0
$631$	4.00000			0.159237			0.0796187	−	0.996825i	$-0.474630\pi$
$631$	4.00000			0.159237			0.0796187	+	0.996825i	$0.474630\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	8.00000	−	4.00000i	0.317470	−	0.158735i
$636$		0			0
$637$			27.0000i			1.06978i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−20.0000			−0.789953			−0.394976	−	0.918691i	$-0.629247\pi$
$641$	−20.0000			−0.789953			−0.394976	+	0.918691i	$0.629247\pi$
$642$		0			0
$643$		−	21.0000i		−	0.828159i	−0.910241	−	0.414080i	$-0.864104\pi$
$643$		−	21.0000i		−	0.828159i	0.910241	−	0.414080i	$-0.135896\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$	−20.0000			−0.785069
$650$		0			0
$651$		0			0
$652$		0			0
$653$			26.0000i			1.01746i	0.860927	+	0.508729i	$0.169885\pi$
$653$			26.0000i			1.01746i	−0.860927	+	0.508729i	$0.830115\pi$
$654$		0			0
$655$	−3.00000	−	6.00000i	−0.117220	−	0.234439i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	20.0000			0.779089			0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000			0.779089			0.389545	+	0.921008i	$0.372632\pi$
$660$		0			0
$661$	44.0000			1.71140			0.855701	−	0.517471i	$-0.173126\pi$
$661$	44.0000			1.71140			0.855701	+	0.517471i	$0.173126\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−48.0000	+	24.0000i	−1.86136	+	0.930680i
$666$		0			0
$667$			9.00000i			0.348481i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	40.0000			1.54418
$672$		0			0
$673$			48.0000i			1.85026i	0.379646	+	0.925132i	$0.376046\pi$
$673$			48.0000i			1.85026i	−0.379646	+	0.925132i	$0.623954\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			4.00000i			0.153732i	0.997041	+	0.0768662i	$0.0244914\pi$
$677$			4.00000i			0.153732i	−0.997041	+	0.0768662i	$0.975509\pi$
$678$		0			0
$679$	−40.0000			−1.53506
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	44.0000i		−	1.68361i	−0.539779	−	0.841807i	$-0.681492\pi$
$683$		−	44.0000i		−	1.68361i	0.539779	−	0.841807i	$-0.318508\pi$
$684$		0			0
$685$	−36.0000	+	18.0000i	−1.37549	+	0.687745i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−22.0000			−0.836919			−0.418460	−	0.908235i	$-0.637430\pi$
$691$	−22.0000			−0.836919			−0.418460	+	0.908235i	$0.637430\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.00000	+	16.0000i	0.303457	+	0.606915i
$696$		0			0
$697$		−	2.00000i		−	0.0757554i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	31.0000			1.17085			0.585427	−	0.810725i	$-0.300927\pi$
$701$	31.0000			1.17085			0.585427	+	0.810725i	$0.300927\pi$
$702$		0			0
$703$		−	12.0000i		−	0.452589i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			68.0000i			2.55740i
$708$		0			0
$709$	24.0000			0.901339			0.450669	−	0.892691i	$-0.351185\pi$
$709$	24.0000			0.901339			0.450669	+	0.892691i	$0.351185\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			5.00000i			0.187251i
$714$		0			0
$715$	−30.0000	+	15.0000i	−1.12194	+	0.560968i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−20.0000			−0.745874			−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000			−0.745874			−0.372937	+	0.927857i	$0.621649\pi$
$720$		0			0
$721$	−40.0000			−1.48968
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−27.0000	+	36.0000i	−1.00275	+	1.33701i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.00000			0.0369863
$732$		0			0
$733$		−	46.0000i		−	1.69905i	−0.527549	−	0.849524i	$-0.676889\pi$
$733$		−	46.0000i		−	1.69905i	0.527549	−	0.849524i	$-0.323111\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	20.0000i		−	0.736709i
$738$		0			0
$739$	10.0000			0.367856			0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000			0.367856			0.183928	+	0.982940i	$0.441119\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	9.00000i		−	0.330178i	−0.986279	−	0.165089i	$-0.947209\pi$
$743$		−	9.00000i		−	0.330178i	0.986279	−	0.165089i	$-0.0527911\pi$
$744$		0			0
$745$	−13.0000	−	26.0000i	−0.476283	−	0.952566i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	24.0000			0.876941
$750$		0			0
$751$	7.00000			0.255434			0.127717	−	0.991811i	$-0.459235\pi$
$751$	7.00000			0.255434			0.127717	+	0.991811i	$0.459235\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	13.0000	+	26.0000i	0.473118	+	0.946237i
$756$		0			0
$757$		−	43.0000i		−	1.56286i	−0.623992	−	0.781431i	$-0.714490\pi$
$757$		−	43.0000i		−	1.56286i	0.623992	−	0.781431i	$-0.285510\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	12.0000			0.435000			0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000			0.435000			0.217500	+	0.976060i	$0.430210\pi$
$762$		0			0
$763$			32.0000i			1.15848i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			12.0000i			0.433295i
$768$		0			0
$769$	−3.00000			−0.108183			−0.0540914	−	0.998536i	$-0.517226\pi$
$769$	−3.00000			−0.108183			−0.0540914	+	0.998536i	$0.517226\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			18.0000i			0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$			18.0000i			0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$		0			0
$775$	−15.0000	+	20.0000i	−0.538816	+	0.718421i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	12.0000			0.429945
$780$		0			0
$781$	30.0000			1.07348
$782$		0			0
$783$		0			0
$784$		0			0
$785$	26.0000	−	13.0000i	0.927980	−	0.463990i
$786$		0			0
$787$			13.0000i			0.463400i	0.972787	+	0.231700i	$0.0744288\pi$
$787$			13.0000i			0.463400i	−0.972787	+	0.231700i	$0.925571\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−12.0000			−0.426671
$792$		0			0
$793$		−	24.0000i		−	0.852265i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	14.0000i		−	0.495905i	−0.968772	−	0.247953i	$-0.920242\pi$
$797$		−	14.0000i		−	0.495905i	0.968772	−	0.247953i	$-0.0797578\pi$
$798$		0			0
$799$	−13.0000			−0.459907
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	10.0000i		−	0.352892i
$804$		0			0
$805$	4.00000	+	8.00000i	0.140981	+	0.281963i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−12.0000			−0.421898			−0.210949	−	0.977497i	$-0.567655\pi$
$809$	−12.0000			−0.421898			−0.210949	+	0.977497i	$0.567655\pi$
$810$		0			0
$811$	26.0000			0.912983			0.456492	−	0.889728i	$-0.349106\pi$
$811$	26.0000			0.912983			0.456492	+	0.889728i	$0.349106\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	38.0000	−	19.0000i	1.33108	−	0.665541i
$816$		0			0
$817$			6.00000i			0.209913i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−46.0000			−1.60541			−0.802706	−	0.596376i	$-0.796607\pi$
$821$	−46.0000			−1.60541			−0.802706	+	0.596376i	$0.796607\pi$
$822$		0			0
$823$		−	24.0000i		−	0.836587i	−0.908312	−	0.418294i	$-0.862628\pi$
$823$		−	24.0000i		−	0.836587i	0.908312	−	0.418294i	$-0.137372\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	22.0000i		−	0.765015i	−0.923952	−	0.382507i	$-0.875061\pi$
$827$		−	22.0000i		−	0.765015i	0.923952	−	0.382507i	$-0.124939\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	9.00000i		−	0.311832i
$834$		0			0
$835$	−24.0000	+	12.0000i	−0.830554	+	0.415277i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−34.0000			−1.17381			−0.586905	−	0.809656i	$-0.699654\pi$
$839$	−34.0000			−1.17381			−0.586905	+	0.809656i	$0.699654\pi$
$840$		0			0
$841$	52.0000			1.79310
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−4.00000	−	8.00000i	−0.137604	−	0.275208i
$846$		0			0
$847$			56.0000i			1.92418i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2.00000			−0.0685591
$852$		0			0
$853$		−	19.0000i		−	0.650548i	−0.945620	−	0.325274i	$-0.894544\pi$
$853$		−	19.0000i		−	0.650548i	0.945620	−	0.325274i	$-0.105456\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	6.00000i		−	0.204956i	−0.994735	−	0.102478i	$-0.967323\pi$
$857$		−	6.00000i		−	0.204956i	0.994735	−	0.102478i	$-0.0326771\pi$
$858$		0			0
$859$	2.00000			0.0682391			0.0341196	−	0.999418i	$-0.489137\pi$
$859$	2.00000			0.0682391			0.0341196	+	0.999418i	$0.489137\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			39.0000i			1.32758i	0.747921	+	0.663788i	$0.231052\pi$
$863$			39.0000i			1.32758i	−0.747921	+	0.663788i	$0.768948\pi$
$864$		0			0
$865$	32.0000	−	16.0000i	1.08803	−	0.544016i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	45.0000			1.52652
$870$		0			0
$871$	−12.0000			−0.406604
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−8.00000	+	44.0000i	−0.270449	+	1.48747i
$876$		0			0
$877$			1.00000i			0.0337676i	0.999857	+	0.0168838i	$0.00537454\pi$
$877$			1.00000i			0.0337676i	−0.999857	+	0.0168838i	$0.994625\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$		0			0
$883$		−	36.0000i		−	1.21150i	−0.795656	−	0.605748i	$-0.792874\pi$
$883$		−	36.0000i		−	1.21150i	0.795656	−	0.605748i	$-0.207126\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	51.0000i		−	1.71241i	−0.516634	−	0.856206i	$-0.672815\pi$
$887$		−	51.0000i		−	1.71241i	0.516634	−	0.856206i	$-0.327185\pi$
$888$		0			0
$889$	−16.0000			−0.536623
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	78.0000i		−	2.61017i
$894$		0			0
$895$	−20.0000	−	40.0000i	−0.668526	−	1.33705i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	45.0000			1.50083
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	22.0000	+	44.0000i	0.731305	+	1.46261i
$906$		0			0
$907$		−	3.00000i		−	0.0996134i	−0.998759	−	0.0498067i	$-0.984139\pi$
$907$		−	3.00000i		−	0.0996134i	0.998759	−	0.0498067i	$-0.0158605\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	36.0000			1.19273			0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000			1.19273			0.596367	+	0.802712i	$0.296610\pi$
$912$		0			0
$913$			20.0000i			0.661903i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			12.0000i			0.396275i
$918$		0			0
$919$	33.0000			1.08857			0.544285	−	0.838901i	$-0.316801\pi$
$919$	33.0000			1.08857			0.544285	+	0.838901i	$0.316801\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	18.0000i		−	0.592477i
$924$		0			0
$925$	−8.00000	−	6.00000i	−0.263038	−	0.197279i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	6.00000			0.196854			0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000			0.196854			0.0984268	+	0.995144i	$0.468619\pi$
$930$		0			0
$931$	54.0000			1.76978
$932$		0			0
$933$		0			0
$934$		0			0
$935$	10.0000	−	5.00000i	0.327035	−	0.163517i
$936$		0			0
$937$			40.0000i			1.30674i	0.757037	+	0.653372i	$0.226646\pi$
$937$			40.0000i			1.30674i	−0.757037	+	0.653372i	$0.773354\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	31.0000			1.01057			0.505286	−	0.862952i	$-0.331387\pi$
$941$	31.0000			1.01057			0.505286	+	0.862952i	$0.331387\pi$
$942$		0			0
$943$		−	2.00000i		−	0.0651290i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	48.0000i		−	1.55979i	−0.625910	−	0.779895i	$-0.715272\pi$
$947$		−	48.0000i		−	1.55979i	0.625910	−	0.779895i	$-0.284728\pi$
$948$		0			0
$949$	−6.00000			−0.194768
$950$		0			0
$951$		0			0
$952$		0			0
$953$			61.0000i			1.97598i	0.154506	+	0.987992i	$0.450622\pi$
$953$			61.0000i			1.97598i	−0.154506	+	0.987992i	$0.549378\pi$
$954$		0			0
$955$	−6.00000	−	12.0000i	−0.194155	−	0.388311i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	72.0000			2.32500
$960$		0			0
$961$	−6.00000			−0.193548
$962$		0			0
$963$		0			0
$964$		0			0
$965$	8.00000	−	4.00000i	0.257529	−	0.128765i
$966$		0			0
$967$			2.00000i			0.0643157i	0.999483	+	0.0321578i	$0.0102379\pi$
$967$			2.00000i			0.0643157i	−0.999483	+	0.0321578i	$0.989762\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	9.00000			0.288824			0.144412	−	0.989518i	$-0.453871\pi$
$971$	9.00000			0.288824			0.144412	+	0.989518i	$0.453871\pi$
$972$		0			0
$973$		−	32.0000i		−	1.02587i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	39.0000i		−	1.24772i	−0.781536	−	0.623860i	$-0.785563\pi$
$977$		−	39.0000i		−	1.24772i	0.781536	−	0.623860i	$-0.214437\pi$
$978$		0			0
$979$	70.0000			2.23721
$980$		0			0
$981$		0			0
$982$		0			0
$983$			29.0000i			0.924956i	0.886631	+	0.462478i	$0.153040\pi$
$983$			29.0000i			0.924956i	−0.886631	+	0.462478i	$0.846960\pi$
$984$		0			0
$985$	−44.0000	+	22.0000i	−1.40196	+	0.700978i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.00000			0.0317982
$990$		0			0
$991$	17.0000			0.540023			0.270011	−	0.962857i	$-0.412973\pi$
$991$	17.0000			0.540023			0.270011	+	0.962857i	$0.412973\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−3.00000	−	6.00000i	−0.0951064	−	0.190213i
$996$		0			0
$997$		−	15.0000i		−	0.475055i	−0.971381	−	0.237527i	$-0.923663\pi$
$997$		−	15.0000i		−	0.475055i	0.971381	−	0.237527i	$-0.0763369\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.2.f.d.1729.1		2
3.2	odd	2		2160.2.f.e.1729.2		2
4.3	odd	2		270.2.c.a.109.2	yes	2
5.4	even	2	inner	2160.2.f.d.1729.2		2
12.11	even	2		270.2.c.b.109.1	yes	2
15.14	odd	2		2160.2.f.e.1729.1		2
20.3	even	4		1350.2.a.l.1.1		1
20.7	even	4		1350.2.a.j.1.1		1
20.19	odd	2		270.2.c.a.109.1	✓	2
36.7	odd	6		810.2.i.d.109.2		4
36.11	even	6		810.2.i.c.109.1		4
36.23	even	6		810.2.i.c.379.2		4
36.31	odd	6		810.2.i.d.379.1		4
60.23	odd	4		1350.2.a.b.1.1		1
60.47	odd	4		1350.2.a.v.1.1		1
60.59	even	2		270.2.c.b.109.2	yes	2
180.59	even	6		810.2.i.c.379.1		4
180.79	odd	6		810.2.i.d.109.1		4
180.119	even	6		810.2.i.c.109.2		4
180.139	odd	6		810.2.i.d.379.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.2.c.a.109.1	✓	2	20.19	odd	2
270.2.c.a.109.2	yes	2	4.3	odd	2
270.2.c.b.109.1	yes	2	12.11	even	2
270.2.c.b.109.2	yes	2	60.59	even	2
810.2.i.c.109.1		4	36.11	even	6
810.2.i.c.109.2		4	180.119	even	6
810.2.i.c.379.1		4	180.59	even	6
810.2.i.c.379.2		4	36.23	even	6
810.2.i.d.109.1		4	180.79	odd	6
810.2.i.d.109.2		4	36.7	odd	6
810.2.i.d.379.1		4	36.31	odd	6
810.2.i.d.379.2		4	180.139	odd	6
1350.2.a.b.1.1		1	60.23	odd	4
1350.2.a.j.1.1		1	20.7	even	4
1350.2.a.l.1.1		1	20.3	even	4
1350.2.a.v.1.1		1	60.47	odd	4
2160.2.f.d.1729.1		2	1.1	even	1	trivial
2160.2.f.d.1729.2		2	5.4	even	2	inner
2160.2.f.e.1729.1		2	15.14	odd	2
2160.2.f.e.1729.2		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2160.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 2.00000i) q^{5} +4.00000i q^{7} +5.00000 q^{11} -3.00000i q^{13} +1.00000i q^{17} -6.00000 q^{19} +1.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} +9.00000 q^{29} +5.00000 q^{31} +(8.00000 - 4.00000i) q^{35} +2.00000i q^{37} -2.00000 q^{41} -1.00000i q^{43} +13.0000i q^{47} -9.00000 q^{49} +(-5.00000 - 10.0000i) q^{55} -4.00000 q^{59} +8.00000 q^{61} +(-6.00000 + 3.00000i) q^{65} -4.00000i q^{67} +6.00000 q^{71} -2.00000i q^{73} +20.0000i q^{77} +9.00000 q^{79} +4.00000i q^{83} +(2.00000 - 1.00000i) q^{85} +14.0000 q^{89} +12.0000 q^{91} +(6.00000 + 12.0000i) q^{95} +10.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 10 q^{11} - 12 q^{19} - 6 q^{25} + 18 q^{29} + 10 q^{31} + 16 q^{35} - 4 q^{41} - 18 q^{49} - 10 q^{55} - 8 q^{59} + 16 q^{61} - 12 q^{65} + 12 q^{71} + 18 q^{79} + 4 q^{85} + 28 q^{89}+ \cdots + 12 q^{95}+O(q^{100}) \) 2 * q - 2 * q^5 + 10 * q^11 - 12 * q^19 - 6 * q^25 + 18 * q^29 + 10 * q^31 + 16 * q^35 - 4 * q^41 - 18 * q^49 - 10 * q^55 - 8 * q^59 + 16 * q^61 - 12 * q^65 + 12 * q^71 + 18 * q^79 + 4 * q^85 + 28 * q^89 + 24 * q^91 + 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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