LMFDB - Embedded newform 720.3.bh.e.577.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,3,Mod(433,720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("720.433");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	720.bh (of order $4$, 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:	$19.6185790339$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 20)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			577.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	720.577
Dual form			720.3.bh.e.433.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.00000 - 4.00000i) q^{5} +(7.00000 + 7.00000i) q^{7} +O(q^{10})$	$q+(3.00000 - 4.00000i) q^{5} +(7.00000 + 7.00000i) q^{7} +10.0000 q^{11} +(9.00000 - 9.00000i) q^{13} +(-1.00000 - 1.00000i) q^{17} +8.00000i q^{19} +(-23.0000 + 23.0000i) q^{23} +(-7.00000 - 24.0000i) q^{25} +8.00000i q^{29} +14.0000 q^{31} +(49.0000 - 7.00000i) q^{35} +(33.0000 + 33.0000i) q^{37} +14.0000 q^{41} +(15.0000 - 15.0000i) q^{43} +(-39.0000 - 39.0000i) q^{47} +49.0000i q^{49} +(7.00000 - 7.00000i) q^{53} +(30.0000 - 40.0000i) q^{55} -56.0000i q^{59} +42.0000 q^{61} +(-9.00000 - 63.0000i) q^{65} +(7.00000 + 7.00000i) q^{67} +98.0000 q^{71} +(49.0000 - 49.0000i) q^{73} +(70.0000 + 70.0000i) q^{77} -96.0000i q^{79} +(-63.0000 + 63.0000i) q^{83} +(-7.00000 + 1.00000i) q^{85} +112.000i q^{89} +126.000 q^{91} +(32.0000 + 24.0000i) q^{95} +(33.0000 + 33.0000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{5} + 14 q^{7}+O(q^{10}) $ 2 * q + 6 * q^5 + 14 * q^7	$ 2 q + 6 q^{5} + 14 q^{7} + 20 q^{11} + 18 q^{13} - 2 q^{17} - 46 q^{23} - 14 q^{25} + 28 q^{31} + 98 q^{35} + 66 q^{37} + 28 q^{41} + 30 q^{43} - 78 q^{47} + 14 q^{53} + 60 q^{55} + 84 q^{61} - 18 q^{65} + 14 q^{67} + 196 q^{71} + 98 q^{73} + 140 q^{77} - 126 q^{83} - 14 q^{85} + 252 q^{91} + 64 q^{95} + 66 q^{97}+O(q^{100}) $ 2 * q + 6 * q^5 + 14 * q^7 + 20 * q^11 + 18 * q^13 - 2 * q^17 - 46 * q^23 - 14 * q^25 + 28 * q^31 + 98 * q^35 + 66 * q^37 + 28 * q^41 + 30 * q^43 - 78 * q^47 + 14 * q^53 + 60 * q^55 + 84 * q^61 - 18 * q^65 + 14 * q^67 + 196 * q^71 + 98 * q^73 + 140 * q^77 - 126 * q^83 - 14 * q^85 + 252 * q^91 + 64 * q^95 + 66 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$	$1$

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
$5$	3.00000	−	4.00000i	0.600000	−	0.800000i
$5$	3.00000	−	4.00000i	0.600000	−	0.800000i
$6$		0			0
$7$	7.00000	+	7.00000i	1.00000	+	1.00000i	1.00000			$0$
$7$	7.00000	+	7.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	10.0000			0.909091			0.454545	−	0.890724i	$-0.349802\pi$
$11$	10.0000			0.909091			0.454545	+	0.890724i	$0.349802\pi$
$12$		0			0
$13$	9.00000	−	9.00000i	0.692308	−	0.692308i	−0.270432	−	0.962739i	$-0.587166\pi$
$13$	9.00000	−	9.00000i	0.692308	−	0.692308i	0.962739	+	0.270432i	$0.0871664\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.00000	−	1.00000i	−0.0588235	−	0.0588235i	0.677083	−	0.735907i	$-0.263244\pi$
$17$	−1.00000	−	1.00000i	−0.0588235	−	0.0588235i	−0.735907	+	0.677083i	$0.763244\pi$
$18$		0			0
$19$			8.00000i			0.421053i	0.977588	+	0.210526i	$0.0675178\pi$
$19$			8.00000i			0.421053i	−0.977588	+	0.210526i	$0.932482\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−23.0000	+	23.0000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$23$	−23.0000	+	23.0000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$24$		0			0
$25$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			8.00000i			0.275862i	0.990442	+	0.137931i	$0.0440452\pi$
$29$			8.00000i			0.275862i	−0.990442	+	0.137931i	$0.955955\pi$
$30$		0			0
$31$	14.0000			0.451613			0.225806	−	0.974172i	$-0.427498\pi$
$31$	14.0000			0.451613			0.225806	+	0.974172i	$0.427498\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	49.0000	−	7.00000i	1.40000	−	0.200000i
$36$		0			0
$37$	33.0000	+	33.0000i	0.891892	+	0.891892i	0.994701	−	0.102809i	$-0.0327831\pi$
$37$	33.0000	+	33.0000i	0.891892	+	0.891892i	−0.102809	+	0.994701i	$0.532783\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	14.0000			0.341463			0.170732	−	0.985318i	$-0.445387\pi$
$41$	14.0000			0.341463			0.170732	+	0.985318i	$0.445387\pi$
$42$		0			0
$43$	15.0000	−	15.0000i	0.348837	−	0.348837i	−0.510839	−	0.859676i	$-0.670665\pi$
$43$	15.0000	−	15.0000i	0.348837	−	0.348837i	0.859676	+	0.510839i	$0.170665\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−39.0000	−	39.0000i	−0.829787	−	0.829787i	0.157700	−	0.987487i	$-0.449592\pi$
$47$	−39.0000	−	39.0000i	−0.829787	−	0.829787i	−0.987487	+	0.157700i	$0.949592\pi$
$48$		0			0
$49$			49.0000i			1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	7.00000	−	7.00000i	0.132075	−	0.132075i	−0.637979	−	0.770054i	$-0.720229\pi$
$53$	7.00000	−	7.00000i	0.132075	−	0.132075i	0.770054	+	0.637979i	$0.220229\pi$
$54$		0			0
$55$	30.0000	−	40.0000i	0.545455	−	0.727273i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	56.0000i		−	0.949153i	−0.880214	−	0.474576i	$-0.842601\pi$
$59$		−	56.0000i		−	0.949153i	0.880214	−	0.474576i	$-0.157399\pi$
$60$		0			0
$61$	42.0000			0.688525			0.344262	−	0.938874i	$-0.388129\pi$
$61$	42.0000			0.688525			0.344262	+	0.938874i	$0.388129\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−9.00000	−	63.0000i	−0.138462	−	0.969231i
$66$		0			0
$67$	7.00000	+	7.00000i	0.104478	+	0.104478i	0.757413	−	0.652936i	$-0.226463\pi$
$67$	7.00000	+	7.00000i	0.104478	+	0.104478i	−0.652936	+	0.757413i	$0.726463\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	98.0000			1.38028			0.690141	−	0.723675i	$-0.257549\pi$
$71$	98.0000			1.38028			0.690141	+	0.723675i	$0.257549\pi$
$72$		0			0
$73$	49.0000	−	49.0000i	0.671233	−	0.671233i	−0.286767	−	0.958000i	$-0.592581\pi$
$73$	49.0000	−	49.0000i	0.671233	−	0.671233i	0.958000	+	0.286767i	$0.0925807\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	70.0000	+	70.0000i	0.909091	+	0.909091i
$78$		0			0
$79$		−	96.0000i		−	1.21519i	−0.794247	−	0.607595i	$-0.792134\pi$
$79$		−	96.0000i		−	1.21519i	0.794247	−	0.607595i	$-0.207866\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−63.0000	+	63.0000i	−0.759036	+	0.759036i	−0.976147	−	0.217111i	$-0.930337\pi$
$83$	−63.0000	+	63.0000i	−0.759036	+	0.759036i	0.217111	+	0.976147i	$0.430337\pi$
$84$		0			0
$85$	−7.00000	+	1.00000i	−0.0823529	+	0.0117647i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			112.000i			1.25843i	0.777233	+	0.629213i	$0.216623\pi$
$89$			112.000i			1.25843i	−0.777233	+	0.629213i	$0.783377\pi$
$90$		0			0
$91$	126.000			1.38462
$92$		0			0
$93$		0			0
$94$		0			0
$95$	32.0000	+	24.0000i	0.336842	+	0.252632i
$96$		0			0
$97$	33.0000	+	33.0000i	0.340206	+	0.340206i	0.856445	−	0.516239i	$-0.172668\pi$
$97$	33.0000	+	33.0000i	0.340206	+	0.340206i	−0.516239	+	0.856445i	$0.672668\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−26.0000			−0.257426			−0.128713	−	0.991682i	$-0.541085\pi$
$101$	−26.0000			−0.257426			−0.128713	+	0.991682i	$0.541085\pi$
$102$		0			0
$103$	−73.0000	+	73.0000i	−0.708738	+	0.708738i	−0.966270	−	0.257532i	$-0.917091\pi$
$103$	−73.0000	+	73.0000i	−0.708738	+	0.708738i	0.257532	+	0.966270i	$0.417091\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	121.000	+	121.000i	1.13084	+	1.13084i	0.990038	+	0.140804i	$0.0449686\pi$
$107$	121.000	+	121.000i	1.13084	+	1.13084i	0.140804	+	0.990038i	$0.455031\pi$
$108$		0			0
$109$		−	136.000i		−	1.24771i	−0.781542	−	0.623853i	$-0.785566\pi$
$109$		−	136.000i		−	1.24771i	0.781542	−	0.623853i	$-0.214434\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	127.000	−	127.000i	1.12389	−	1.12389i	0.132743	−	0.991150i	$-0.457621\pi$
$113$	127.000	−	127.000i	1.12389	−	1.12389i	0.991150	−	0.132743i	$-0.0423786\pi$
$114$		0			0
$115$	23.0000	+	161.000i	0.200000	+	1.40000i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	14.0000i		−	0.117647i
$120$		0			0
$121$	−21.0000			−0.173554
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−117.000	−	44.0000i	−0.936000	−	0.352000i
$126$		0			0
$127$	7.00000	+	7.00000i	0.0551181	+	0.0551181i	0.734129	−	0.679010i	$-0.237591\pi$
$127$	7.00000	+	7.00000i	0.0551181	+	0.0551181i	−0.679010	+	0.734129i	$0.737591\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−230.000			−1.75573			−0.877863	−	0.478913i	$-0.841031\pi$
$131$	−230.000			−1.75573			−0.877863	+	0.478913i	$0.841031\pi$
$132$		0			0
$133$	−56.0000	+	56.0000i	−0.421053	+	0.421053i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	63.0000	+	63.0000i	0.459854	+	0.459854i	0.898607	−	0.438753i	$-0.144580\pi$
$137$	63.0000	+	63.0000i	0.459854	+	0.459854i	−0.438753	+	0.898607i	$0.644580\pi$
$138$		0			0
$139$			88.0000i			0.633094i	0.948577	+	0.316547i	$0.102523\pi$
$139$			88.0000i			0.633094i	−0.948577	+	0.316547i	$0.897477\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	90.0000	−	90.0000i	0.629371	−	0.629371i
$144$		0			0
$145$	32.0000	+	24.0000i	0.220690	+	0.165517i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	168.000i		−	1.12752i	−0.825940	−	0.563758i	$-0.809355\pi$
$149$		−	168.000i		−	1.12752i	0.825940	−	0.563758i	$-0.190645\pi$
$150$		0			0
$151$	−130.000			−0.860927			−0.430464	−	0.902608i	$-0.641650\pi$
$151$	−130.000			−0.860927			−0.430464	+	0.902608i	$0.641650\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	42.0000	−	56.0000i	0.270968	−	0.361290i
$156$		0			0
$157$	−63.0000	−	63.0000i	−0.401274	−	0.401274i	0.477408	−	0.878682i	$-0.341576\pi$
$157$	−63.0000	−	63.0000i	−0.401274	−	0.401274i	−0.878682	+	0.477408i	$0.841576\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−322.000			−2.00000
$162$		0			0
$163$	−65.0000	+	65.0000i	−0.398773	+	0.398773i	−0.877800	−	0.479027i	$-0.840990\pi$
$163$	−65.0000	+	65.0000i	−0.398773	+	0.398773i	0.479027	+	0.877800i	$0.340990\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−103.000	−	103.000i	−0.616766	−	0.616766i	0.327934	−	0.944701i	$-0.393648\pi$
$167$	−103.000	−	103.000i	−0.616766	−	0.616766i	−0.944701	+	0.327934i	$0.893648\pi$
$168$		0			0
$169$			7.00000i			0.0414201i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−73.0000	+	73.0000i	−0.421965	+	0.421965i	−0.885880	−	0.463915i	$-0.846444\pi$
$173$	−73.0000	+	73.0000i	−0.421965	+	0.421965i	0.463915	+	0.885880i	$0.346444\pi$
$174$		0			0
$175$	119.000	−	217.000i	0.680000	−	1.24000i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			56.0000i			0.312849i	0.987690	+	0.156425i	$0.0499968\pi$
$179$			56.0000i			0.312849i	−0.987690	+	0.156425i	$0.950003\pi$
$180$		0			0
$181$	−70.0000			−0.386740			−0.193370	−	0.981126i	$-0.561942\pi$
$181$	−70.0000			−0.386740			−0.193370	+	0.981126i	$0.561942\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	231.000	−	33.0000i	1.24865	−	0.178378i
$186$		0			0
$187$	−10.0000	−	10.0000i	−0.0534759	−	0.0534759i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−142.000			−0.743455			−0.371728	−	0.928342i	$-0.621235\pi$
$191$	−142.000			−0.743455			−0.371728	+	0.928342i	$0.621235\pi$
$192$		0			0
$193$	−63.0000	+	63.0000i	−0.326425	+	0.326425i	−0.851225	−	0.524800i	$-0.824140\pi$
$193$	−63.0000	+	63.0000i	−0.326425	+	0.326425i	0.524800	+	0.851225i	$0.324140\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	63.0000	+	63.0000i	0.319797	+	0.319797i	0.848689	−	0.528892i	$-0.177392\pi$
$197$	63.0000	+	63.0000i	0.319797	+	0.319797i	−0.528892	+	0.848689i	$0.677392\pi$
$198$		0			0
$199$		−	336.000i		−	1.68844i	−0.535995	−	0.844221i	$-0.680063\pi$
$199$		−	336.000i		−	1.68844i	0.535995	−	0.844221i	$-0.319937\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−56.0000	+	56.0000i	−0.275862	+	0.275862i
$204$		0			0
$205$	42.0000	−	56.0000i	0.204878	−	0.273171i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			80.0000i			0.382775i
$210$		0			0
$211$	−314.000			−1.48815			−0.744076	−	0.668095i	$-0.767110\pi$
$211$	−314.000			−1.48815			−0.744076	+	0.668095i	$0.767110\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−15.0000	−	105.000i	−0.0697674	−	0.488372i
$216$		0			0
$217$	98.0000	+	98.0000i	0.451613	+	0.451613i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−18.0000			−0.0814480
$222$		0			0
$223$	135.000	−	135.000i	0.605381	−	0.605381i	−0.336354	−	0.941736i	$-0.609194\pi$
$223$	135.000	−	135.000i	0.605381	−	0.605381i	0.941736	+	0.336354i	$0.109194\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	281.000	+	281.000i	1.23789	+	1.23789i	0.960864	+	0.277022i	$0.0893474\pi$
$227$	281.000	+	281.000i	1.23789	+	1.23789i	0.277022	+	0.960864i	$0.410653\pi$
$228$		0			0
$229$			168.000i			0.733624i	0.930295	+	0.366812i	$0.119551\pi$
$229$			168.000i			0.733624i	−0.930295	+	0.366812i	$0.880449\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−273.000	+	273.000i	−1.17167	+	1.17167i	−0.189863	+	0.981811i	$0.560804\pi$
$233$	−273.000	+	273.000i	−1.17167	+	1.17167i	−0.981811	+	0.189863i	$0.939196\pi$
$234$		0			0
$235$	−273.000	+	39.0000i	−1.16170	+	0.165957i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	288.000i		−	1.20502i	−0.798111	−	0.602510i	$-0.794167\pi$
$239$		−	288.000i		−	1.20502i	0.798111	−	0.602510i	$-0.205833\pi$
$240$		0			0
$241$	−446.000			−1.85062			−0.925311	−	0.379209i	$-0.876196\pi$
$241$	−446.000			−1.85062			−0.925311	+	0.379209i	$0.876196\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	196.000	+	147.000i	0.800000	+	0.600000i
$246$		0			0
$247$	72.0000	+	72.0000i	0.291498	+	0.291498i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−150.000			−0.597610			−0.298805	−	0.954314i	$-0.596588\pi$
$251$	−150.000			−0.597610			−0.298805	+	0.954314i	$0.596588\pi$
$252$		0			0
$253$	−230.000	+	230.000i	−0.909091	+	0.909091i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−161.000	−	161.000i	−0.626459	−	0.626459i	0.320716	−	0.947175i	$-0.396076\pi$
$257$	−161.000	−	161.000i	−0.626459	−	0.626459i	−0.947175	+	0.320716i	$0.896076\pi$
$258$		0			0
$259$			462.000i			1.78378i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−151.000	+	151.000i	−0.574144	+	0.574144i	−0.933284	−	0.359139i	$-0.883070\pi$
$263$	−151.000	+	151.000i	−0.574144	+	0.574144i	0.359139	+	0.933284i	$0.383070\pi$
$264$		0			0
$265$	−7.00000	−	49.0000i	−0.0264151	−	0.184906i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	376.000i		−	1.39777i	−0.715234	−	0.698885i	$-0.753680\pi$
$269$		−	376.000i		−	1.39777i	0.715234	−	0.698885i	$-0.246320\pi$
$270$		0			0
$271$	−210.000			−0.774908			−0.387454	−	0.921889i	$-0.626645\pi$
$271$	−210.000			−0.774908			−0.387454	+	0.921889i	$0.626645\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−70.0000	−	240.000i	−0.254545	−	0.872727i
$276$		0			0
$277$	129.000	+	129.000i	0.465704	+	0.465704i	0.900520	−	0.434816i	$-0.143186\pi$
$277$	129.000	+	129.000i	0.465704	+	0.465704i	−0.434816	+	0.900520i	$0.643186\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	174.000			0.619217			0.309609	−	0.950864i	$-0.399802\pi$
$281$	174.000			0.619217			0.309609	+	0.950864i	$0.399802\pi$
$282$		0			0
$283$	−113.000	+	113.000i	−0.399293	+	0.399293i	−0.877984	−	0.478690i	$-0.841112\pi$
$283$	−113.000	+	113.000i	−0.399293	+	0.399293i	0.478690	+	0.877984i	$0.341112\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	98.0000	+	98.0000i	0.341463	+	0.341463i
$288$		0			0
$289$		−	287.000i		−	0.993080i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−345.000	+	345.000i	−1.17747	+	1.17747i	−0.197089	+	0.980386i	$0.563149\pi$
$293$	−345.000	+	345.000i	−1.17747	+	1.17747i	−0.980386	+	0.197089i	$0.936851\pi$
$294$		0			0
$295$	−224.000	−	168.000i	−0.759322	−	0.569492i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			414.000i			1.38462i
$300$		0			0
$301$	210.000			0.697674
$302$		0			0
$303$		0			0
$304$		0			0
$305$	126.000	−	168.000i	0.413115	−	0.550820i
$306$		0			0
$307$	327.000	+	327.000i	1.06515	+	1.06515i	0.997725	+	0.0674220i	$0.0214774\pi$
$307$	327.000	+	327.000i	1.06515	+	1.06515i	0.0674220	+	0.997725i	$0.478523\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	2.00000			0.00643087			0.00321543	−	0.999995i	$-0.498976\pi$
$311$	2.00000			0.00643087			0.00321543	+	0.999995i	$0.498976\pi$
$312$		0			0
$313$	81.0000	−	81.0000i	0.258786	−	0.258786i	−0.565774	−	0.824560i	$-0.691423\pi$
$313$	81.0000	−	81.0000i	0.258786	−	0.258786i	0.824560	+	0.565774i	$0.191423\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	159.000	+	159.000i	0.501577	+	0.501577i	0.911928	−	0.410351i	$-0.134594\pi$
$317$	159.000	+	159.000i	0.501577	+	0.501577i	−0.410351	+	0.911928i	$0.634594\pi$
$318$		0			0
$319$			80.0000i			0.250784i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	8.00000	−	8.00000i	0.0247678	−	0.0247678i
$324$		0			0
$325$	−279.000	−	153.000i	−0.858462	−	0.470769i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	546.000i		−	1.65957i
$330$		0			0
$331$	182.000			0.549849			0.274924	−	0.961466i	$-0.411347\pi$
$331$	182.000			0.549849			0.274924	+	0.961466i	$0.411347\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	49.0000	−	7.00000i	0.146269	−	0.0208955i
$336$		0			0
$337$	−447.000	−	447.000i	−1.32641	−	1.32641i	−0.908479	−	0.417930i	$-0.862756\pi$
$337$	−447.000	−	447.000i	−1.32641	−	1.32641i	−0.417930	−	0.908479i	$-0.637244\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	140.000			0.410557
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	25.0000	+	25.0000i	0.0720461	+	0.0720461i	0.742212	−	0.670166i	$-0.233777\pi$
$347$	25.0000	+	25.0000i	0.0720461	+	0.0720461i	−0.670166	+	0.742212i	$0.733777\pi$
$348$		0			0
$349$		−	200.000i		−	0.573066i	−0.958070	−	0.286533i	$-0.907497\pi$
$349$		−	200.000i		−	0.573066i	0.958070	−	0.286533i	$-0.0925028\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−321.000	+	321.000i	−0.909348	+	0.909348i	−0.996220	−	0.0868711i	$-0.972313\pi$
$353$	−321.000	+	321.000i	−0.909348	+	0.909348i	0.0868711	+	0.996220i	$0.472313\pi$
$354$		0			0
$355$	294.000	−	392.000i	0.828169	−	1.10423i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	112.000i		−	0.311978i	−0.987759	−	0.155989i	$-0.950144\pi$
$359$		−	112.000i		−	0.311978i	0.987759	−	0.155989i	$-0.0498564\pi$
$360$		0			0
$361$	297.000			0.822715
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−49.0000	−	343.000i	−0.134247	−	0.939726i
$366$		0			0
$367$	−377.000	−	377.000i	−1.02725	−	1.02725i	−0.999618	−	0.0276297i	$-0.991204\pi$
$367$	−377.000	−	377.000i	−1.02725	−	1.02725i	−0.0276297	−	0.999618i	$-0.508796\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	98.0000			0.264151
$372$		0			0
$373$	217.000	−	217.000i	0.581769	−	0.581769i	−0.353620	−	0.935389i	$-0.615049\pi$
$373$	217.000	−	217.000i	0.581769	−	0.581769i	0.935389	+	0.353620i	$0.115049\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	72.0000	+	72.0000i	0.190981	+	0.190981i
$378$		0			0
$379$			56.0000i			0.147757i	0.997267	+	0.0738786i	$0.0235377\pi$
$379$			56.0000i			0.147757i	−0.997267	+	0.0738786i	$0.976462\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	57.0000	−	57.0000i	0.148825	−	0.148825i	−0.628768	−	0.777593i	$-0.716440\pi$
$383$	57.0000	−	57.0000i	0.148825	−	0.148825i	0.777593	+	0.628768i	$0.216440\pi$
$384$		0			0
$385$	490.000	−	70.0000i	1.27273	−	0.181818i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			312.000i			0.802057i	0.916066	+	0.401028i	$0.131347\pi$
$389$			312.000i			0.802057i	−0.916066	+	0.401028i	$0.868653\pi$
$390$		0			0
$391$	46.0000			0.117647
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−384.000	−	288.000i	−0.972152	−	0.729114i
$396$		0			0
$397$	193.000	+	193.000i	0.486146	+	0.486146i	0.907088	−	0.420942i	$-0.138300\pi$
$397$	193.000	+	193.000i	0.486146	+	0.486146i	−0.420942	+	0.907088i	$0.638300\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	30.0000			0.0748130			0.0374065	−	0.999300i	$-0.488090\pi$
$401$	30.0000			0.0748130			0.0374065	+	0.999300i	$0.488090\pi$
$402$		0			0
$403$	126.000	−	126.000i	0.312655	−	0.312655i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	330.000	+	330.000i	0.810811	+	0.810811i
$408$		0			0
$409$			432.000i			1.05623i	0.849171	+	0.528117i	$0.177102\pi$
$409$			432.000i			1.05623i	−0.849171	+	0.528117i	$0.822898\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	392.000	−	392.000i	0.949153	−	0.949153i
$414$		0			0
$415$	63.0000	+	441.000i	0.151807	+	1.06265i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	168.000i		−	0.400955i	−0.979698	−	0.200477i	$-0.935751\pi$
$419$		−	168.000i		−	0.400955i	0.979698	−	0.200477i	$-0.0642493\pi$
$420$		0			0
$421$	−454.000			−1.07838			−0.539192	−	0.842183i	$-0.681270\pi$
$421$	−454.000			−1.07838			−0.539192	+	0.842183i	$0.681270\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−17.0000	+	31.0000i	−0.0400000	+	0.0729412i
$426$		0			0
$427$	294.000	+	294.000i	0.688525	+	0.688525i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−494.000			−1.14617			−0.573086	−	0.819495i	$-0.694254\pi$
$431$	−494.000			−1.14617			−0.573086	+	0.819495i	$0.694254\pi$
$432$		0			0
$433$	−511.000	+	511.000i	−1.18014	+	1.18014i	−0.200431	+	0.979708i	$0.564234\pi$
$433$	−511.000	+	511.000i	−1.18014	+	1.18014i	−0.979708	+	0.200431i	$0.935766\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−184.000	−	184.000i	−0.421053	−	0.421053i
$438$		0			0
$439$		−	176.000i		−	0.400911i	−0.979703	−	0.200456i	$-0.935758\pi$
$439$		−	176.000i		−	0.400911i	0.979703	−	0.200456i	$-0.0642422\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	177.000	−	177.000i	0.399549	−	0.399549i	−0.478525	−	0.878074i	$-0.658828\pi$
$443$	177.000	−	177.000i	0.399549	−	0.399549i	0.878074	+	0.478525i	$0.158828\pi$
$444$		0			0
$445$	448.000	+	336.000i	1.00674	+	0.755056i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			608.000i			1.35412i	0.735928	+	0.677060i	$0.236746\pi$
$449$			608.000i			1.35412i	−0.735928	+	0.677060i	$0.763254\pi$
$450$		0			0
$451$	140.000			0.310421
$452$		0			0
$453$		0			0
$454$		0			0
$455$	378.000	−	504.000i	0.830769	−	1.10769i
$456$		0			0
$457$	609.000	+	609.000i	1.33260	+	1.33260i	0.903033	+	0.429571i	$0.141335\pi$
$457$	609.000	+	609.000i	1.33260	+	1.33260i	0.429571	+	0.903033i	$0.358665\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−490.000			−1.06291			−0.531453	−	0.847088i	$-0.678354\pi$
$461$	−490.000			−1.06291			−0.531453	+	0.847088i	$0.678354\pi$
$462$		0			0
$463$	7.00000	−	7.00000i	0.0151188	−	0.0151188i	−0.699507	−	0.714626i	$-0.746597\pi$
$463$	7.00000	−	7.00000i	0.0151188	−	0.0151188i	0.714626	+	0.699507i	$0.246597\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	25.0000	+	25.0000i	0.0535332	+	0.0535332i	0.733367	−	0.679833i	$-0.237948\pi$
$467$	25.0000	+	25.0000i	0.0535332	+	0.0535332i	−0.679833	+	0.733367i	$0.737948\pi$
$468$		0			0
$469$			98.0000i			0.208955i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	150.000	−	150.000i	0.317125	−	0.317125i
$474$		0			0
$475$	192.000	−	56.0000i	0.404211	−	0.117895i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	128.000i		−	0.267223i	−0.991034	−	0.133612i	$-0.957343\pi$
$479$		−	128.000i		−	0.267223i	0.991034	−	0.133612i	$-0.0426575\pi$
$480$		0			0
$481$	594.000			1.23493
$482$		0			0
$483$		0			0
$484$		0			0
$485$	231.000	−	33.0000i	0.476289	−	0.0680412i
$486$		0			0
$487$	−249.000	−	249.000i	−0.511294	−	0.511294i	0.403629	−	0.914923i	$-0.367749\pi$
$487$	−249.000	−	249.000i	−0.511294	−	0.511294i	−0.914923	+	0.403629i	$0.867749\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	650.000			1.32383			0.661914	−	0.749579i	$-0.269744\pi$
$491$	650.000			1.32383			0.661914	+	0.749579i	$0.269744\pi$
$492$		0			0
$493$	8.00000	−	8.00000i	0.0162272	−	0.0162272i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	686.000	+	686.000i	1.38028	+	1.38028i
$498$		0			0
$499$		−	632.000i		−	1.26653i	−0.773934	−	0.633267i	$-0.781714\pi$
$499$		−	632.000i		−	1.26653i	0.773934	−	0.633267i	$-0.218286\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−471.000	+	471.000i	−0.936382	+	0.936382i	−0.998094	−	0.0617123i	$-0.980344\pi$
$503$	−471.000	+	471.000i	−0.936382	+	0.936382i	0.0617123	+	0.998094i	$0.480344\pi$
$504$		0			0
$505$	−78.0000	+	104.000i	−0.154455	+	0.205941i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			8.00000i			0.0157171i	0.999969	+	0.00785855i	$0.00250148\pi$
$509$			8.00000i			0.0157171i	−0.999969	+	0.00785855i	$0.997499\pi$
$510$		0			0
$511$	686.000			1.34247
$512$		0			0
$513$		0			0
$514$		0			0
$515$	73.0000	+	511.000i	0.141748	+	0.992233i
$516$		0			0
$517$	−390.000	−	390.000i	−0.754352	−	0.754352i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	366.000			0.702495			0.351248	−	0.936283i	$-0.385758\pi$
$521$	366.000			0.702495			0.351248	+	0.936283i	$0.385758\pi$
$522$		0			0
$523$	−273.000	+	273.000i	−0.521989	+	0.521989i	−0.918172	−	0.396183i	$-0.870335\pi$
$523$	−273.000	+	273.000i	−0.521989	+	0.521989i	0.396183	+	0.918172i	$0.370335\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−14.0000	−	14.0000i	−0.0265655	−	0.0265655i
$528$		0			0
$529$		−	529.000i		−	1.00000i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	126.000	−	126.000i	0.236398	−	0.236398i
$534$		0			0
$535$	847.000	−	121.000i	1.58318	−	0.226168i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			490.000i			0.909091i
$540$		0			0
$541$	394.000			0.728281			0.364140	−	0.931344i	$-0.381363\pi$
$541$	394.000			0.728281			0.364140	+	0.931344i	$0.381363\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−544.000	−	408.000i	−0.998165	−	0.748624i
$546$		0			0
$547$	231.000	+	231.000i	0.422303	+	0.422303i	0.885996	−	0.463693i	$-0.153476\pi$
$547$	231.000	+	231.000i	0.422303	+	0.422303i	−0.463693	+	0.885996i	$0.653476\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−64.0000			−0.116152
$552$		0			0
$553$	672.000	−	672.000i	1.21519	−	1.21519i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	735.000	+	735.000i	1.31957	+	1.31957i	0.914116	+	0.405453i	$0.132886\pi$
$557$	735.000	+	735.000i	1.31957	+	1.31957i	0.405453	+	0.914116i	$0.367114\pi$
$558$		0			0
$559$		−	270.000i		−	0.483005i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	609.000	−	609.000i	1.08171	−	1.08171i	0.0853545	−	0.996351i	$-0.472798\pi$
$563$	609.000	−	609.000i	1.08171	−	1.08171i	0.996351	−	0.0853545i	$-0.0272023\pi$
$564$		0			0
$565$	−127.000	−	889.000i	−0.224779	−	1.57345i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	560.000i		−	0.984183i	−0.870544	−	0.492091i	$-0.836233\pi$
$569$		−	560.000i		−	0.984183i	0.870544	−	0.492091i	$-0.163767\pi$
$570$		0			0
$571$	−938.000			−1.64273			−0.821366	−	0.570401i	$-0.806788\pi$
$571$	−938.000			−1.64273			−0.821366	+	0.570401i	$0.806788\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	713.000	+	391.000i	1.24000	+	0.680000i
$576$		0			0
$577$	97.0000	+	97.0000i	0.168111	+	0.168111i	0.786149	−	0.618038i	$-0.212072\pi$
$577$	97.0000	+	97.0000i	0.168111	+	0.168111i	−0.618038	+	0.786149i	$0.712072\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−882.000			−1.51807
$582$		0			0
$583$	70.0000	−	70.0000i	0.120069	−	0.120069i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−39.0000	−	39.0000i	−0.0664395	−	0.0664395i	0.673106	−	0.739546i	$-0.264960\pi$
$587$	−39.0000	−	39.0000i	−0.0664395	−	0.0664395i	−0.739546	+	0.673106i	$0.764960\pi$
$588$		0			0
$589$			112.000i			0.190153i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	479.000	−	479.000i	0.807757	−	0.807757i	−0.176537	−	0.984294i	$-0.556489\pi$
$593$	479.000	−	479.000i	0.807757	−	0.807757i	0.984294	+	0.176537i	$0.0564895\pi$
$594$		0			0
$595$	−56.0000	−	42.0000i	−0.0941176	−	0.0705882i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	1040.00i		−	1.73623i	−0.496366	−	0.868114i	$-0.665332\pi$
$599$		−	1040.00i		−	1.73623i	0.496366	−	0.868114i	$-0.334668\pi$
$600$		0			0
$601$	−430.000			−0.715474			−0.357737	−	0.933822i	$-0.616452\pi$
$601$	−430.000			−0.715474			−0.357737	+	0.933822i	$0.616452\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−63.0000	+	84.0000i	−0.104132	+	0.138843i
$606$		0			0
$607$	423.000	+	423.000i	0.696870	+	0.696870i	0.963734	−	0.266864i	$-0.0859875\pi$
$607$	423.000	+	423.000i	0.696870	+	0.696870i	−0.266864	+	0.963734i	$0.585988\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−702.000			−1.14894
$612$		0			0
$613$	249.000	−	249.000i	0.406199	−	0.406199i	−0.474212	−	0.880411i	$-0.657267\pi$
$613$	249.000	−	249.000i	0.406199	−	0.406199i	0.880411	+	0.474212i	$0.157267\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−321.000	−	321.000i	−0.520259	−	0.520259i	0.397390	−	0.917650i	$-0.369916\pi$
$617$	−321.000	−	321.000i	−0.520259	−	0.520259i	−0.917650	+	0.397390i	$0.869916\pi$
$618$		0			0
$619$			600.000i			0.969305i	0.874707	+	0.484653i	$0.161054\pi$
$619$			600.000i			0.969305i	−0.874707	+	0.484653i	$0.838946\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−784.000	+	784.000i	−1.25843	+	1.25843i
$624$		0			0
$625$	−527.000	+	336.000i	−0.843200	+	0.537600i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	66.0000i		−	0.104928i
$630$		0			0
$631$	638.000			1.01109			0.505547	−	0.862799i	$-0.331291\pi$
$631$	638.000			1.01109			0.505547	+	0.862799i	$0.331291\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	49.0000	−	7.00000i	0.0771654	−	0.0110236i
$636$		0			0
$637$	441.000	+	441.000i	0.692308	+	0.692308i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−482.000			−0.751950			−0.375975	−	0.926630i	$-0.622692\pi$
$641$	−482.000			−0.751950			−0.375975	+	0.926630i	$0.622692\pi$
$642$		0			0
$643$	−33.0000	+	33.0000i	−0.0513219	+	0.0513219i	−0.732302	−	0.680980i	$-0.761554\pi$
$643$	−33.0000	+	33.0000i	−0.0513219	+	0.0513219i	0.680980	+	0.732302i	$0.261554\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−7.00000	−	7.00000i	−0.0108192	−	0.0108192i	0.701677	−	0.712496i	$-0.252435\pi$
$647$	−7.00000	−	7.00000i	−0.0108192	−	0.0108192i	−0.712496	+	0.701677i	$0.752435\pi$
$648$		0			0
$649$		−	560.000i		−	0.862866i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	471.000	−	471.000i	0.721286	−	0.721286i	−0.247581	−	0.968867i	$-0.579636\pi$
$653$	471.000	−	471.000i	0.721286	−	0.721286i	0.968867	+	0.247581i	$0.0796356\pi$
$654$		0			0
$655$	−690.000	+	920.000i	−1.05344	+	1.40458i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	328.000i		−	0.497724i	−0.968539	−	0.248862i	$-0.919943\pi$
$659$		−	328.000i		−	0.497724i	0.968539	−	0.248862i	$-0.0800565\pi$
$660$		0			0
$661$	−742.000			−1.12254			−0.561271	−	0.827632i	$-0.689687\pi$
$661$	−742.000			−1.12254			−0.561271	+	0.827632i	$0.689687\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	56.0000	+	392.000i	0.0842105	+	0.589474i
$666$		0			0
$667$	−184.000	−	184.000i	−0.275862	−	0.275862i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	420.000			0.625931
$672$		0			0
$673$	−287.000	+	287.000i	−0.426449	+	0.426449i	−0.887417	−	0.460968i	$-0.847502\pi$
$673$	−287.000	+	287.000i	−0.426449	+	0.426449i	0.460968	+	0.887417i	$0.347502\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−577.000	−	577.000i	−0.852290	−	0.852290i	0.138125	−	0.990415i	$-0.455892\pi$
$677$	−577.000	−	577.000i	−0.852290	−	0.852290i	−0.990415	+	0.138125i	$0.955892\pi$
$678$		0			0
$679$			462.000i			0.680412i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−399.000	+	399.000i	−0.584187	+	0.584187i	−0.936051	−	0.351864i	$-0.885548\pi$
$683$	−399.000	+	399.000i	−0.584187	+	0.584187i	0.351864	+	0.936051i	$0.385548\pi$
$684$		0			0
$685$	441.000	−	63.0000i	0.643796	−	0.0919708i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	126.000i		−	0.182874i
$690$		0			0
$691$	−378.000			−0.547033			−0.273517	−	0.961867i	$-0.588187\pi$
$691$	−378.000			−0.547033			−0.273517	+	0.961867i	$0.588187\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	352.000	+	264.000i	0.506475	+	0.379856i
$696$		0			0
$697$	−14.0000	−	14.0000i	−0.0200861	−	0.0200861i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	470.000			0.670471			0.335235	−	0.942134i	$-0.391184\pi$
$701$	470.000			0.670471			0.335235	+	0.942134i	$0.391184\pi$
$702$		0			0
$703$	−264.000	+	264.000i	−0.375533	+	0.375533i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−182.000	−	182.000i	−0.257426	−	0.257426i
$708$		0			0
$709$			1192.00i			1.68124i	0.541624	+	0.840621i	$0.317810\pi$
$709$			1192.00i			1.68124i	−0.541624	+	0.840621i	$0.682190\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−322.000	+	322.000i	−0.451613	+	0.451613i
$714$		0			0
$715$	−90.0000	−	630.000i	−0.125874	−	0.881119i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	608.000i		−	0.845619i	−0.906219	−	0.422809i	$-0.861044\pi$
$719$		−	608.000i		−	0.845619i	0.906219	−	0.422809i	$-0.138956\pi$
$720$		0			0
$721$	−1022.00			−1.41748
$722$		0			0
$723$		0			0
$724$		0			0
$725$	192.000	−	56.0000i	0.264828	−	0.0772414i
$726$		0			0
$727$	−441.000	−	441.000i	−0.606602	−	0.606602i	0.335454	−	0.942057i	$-0.391110\pi$
$727$	−441.000	−	441.000i	−0.606602	−	0.606602i	−0.942057	+	0.335454i	$0.891110\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−30.0000			−0.0410397
$732$		0			0
$733$	361.000	−	361.000i	0.492497	−	0.492497i	−0.416595	−	0.909092i	$-0.636777\pi$
$733$	361.000	−	361.000i	0.492497	−	0.492497i	0.909092	+	0.416595i	$0.136777\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	70.0000	+	70.0000i	0.0949796	+	0.0949796i
$738$		0			0
$739$		−	920.000i		−	1.24493i	−0.782649	−	0.622463i	$-0.786132\pi$
$739$		−	920.000i		−	1.24493i	0.782649	−	0.622463i	$-0.213868\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−343.000	+	343.000i	−0.461642	+	0.461642i	−0.899193	−	0.437551i	$-0.855846\pi$
$743$	−343.000	+	343.000i	−0.461642	+	0.461642i	0.437551	+	0.899193i	$0.355846\pi$
$744$		0			0
$745$	−672.000	−	504.000i	−0.902013	−	0.676510i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			1694.00i			2.26168i
$750$		0			0
$751$	−786.000			−1.04660			−0.523302	−	0.852147i	$-0.675300\pi$
$751$	−786.000			−1.04660			−0.523302	+	0.852147i	$0.675300\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−390.000	+	520.000i	−0.516556	+	0.688742i
$756$		0			0
$757$	−63.0000	−	63.0000i	−0.0832232	−	0.0832232i	0.664270	−	0.747493i	$-0.268743\pi$
$757$	−63.0000	−	63.0000i	−0.0832232	−	0.0832232i	−0.747493	+	0.664270i	$0.768743\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	398.000			0.522996			0.261498	−	0.965204i	$-0.415784\pi$
$761$	398.000			0.522996			0.261498	+	0.965204i	$0.415784\pi$
$762$		0			0
$763$	952.000	−	952.000i	1.24771	−	1.24771i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−504.000	−	504.000i	−0.657106	−	0.657106i
$768$		0			0
$769$			704.000i			0.915475i	0.889087	+	0.457737i	$0.151340\pi$
$769$			704.000i			0.915475i	−0.889087	+	0.457737i	$0.848660\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−825.000	+	825.000i	−1.06727	+	1.06727i	−0.0697026	+	0.997568i	$0.522205\pi$
$773$	−825.000	+	825.000i	−1.06727	+	1.06727i	−0.997568	+	0.0697026i	$0.977795\pi$
$774$		0			0
$775$	−98.0000	−	336.000i	−0.126452	−	0.433548i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			112.000i			0.143774i
$780$		0			0
$781$	980.000			1.25480
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−441.000	+	63.0000i	−0.561783	+	0.0802548i
$786$		0			0
$787$	71.0000	+	71.0000i	0.0902160	+	0.0902160i	0.750775	−	0.660559i	$-0.229680\pi$
$787$	71.0000	+	71.0000i	0.0902160	+	0.0902160i	−0.660559	+	0.750775i	$0.729680\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1778.00			2.24779
$792$		0			0
$793$	378.000	−	378.000i	0.476671	−	0.476671i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−33.0000	−	33.0000i	−0.0414053	−	0.0414053i	0.686101	−	0.727506i	$-0.259321\pi$
$797$	−33.0000	−	33.0000i	−0.0414053	−	0.0414053i	−0.727506	+	0.686101i	$0.759321\pi$
$798$		0			0
$799$			78.0000i			0.0976220i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	490.000	−	490.000i	0.610212	−	0.610212i
$804$		0			0
$805$	−966.000	+	1288.00i	−1.20000	+	1.60000i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	368.000i		−	0.454883i	−0.973792	−	0.227441i	$-0.926964\pi$
$809$		−	368.000i		−	0.454883i	0.973792	−	0.227441i	$-0.0730360\pi$
$810$		0			0
$811$	886.000			1.09248			0.546239	−	0.837629i	$-0.316059\pi$
$811$	886.000			1.09248			0.546239	+	0.837629i	$0.316059\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	65.0000	+	455.000i	0.0797546	+	0.558282i
$816$		0			0
$817$	120.000	+	120.000i	0.146879	+	0.146879i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	614.000			0.747868			0.373934	−	0.927455i	$-0.378009\pi$
$821$	614.000			0.747868			0.373934	+	0.927455i	$0.378009\pi$
$822$		0			0
$823$	−1001.00	+	1001.00i	−1.21628	+	1.21628i	−0.247358	+	0.968924i	$0.579562\pi$
$823$	−1001.00	+	1001.00i	−1.21628	+	1.21628i	−0.968924	+	0.247358i	$0.920438\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−615.000	−	615.000i	−0.743652	−	0.743652i	0.229627	−	0.973279i	$-0.426249\pi$
$827$	−615.000	−	615.000i	−0.743652	−	0.743652i	−0.973279	+	0.229627i	$0.926249\pi$
$828$		0			0
$829$		−	616.000i		−	0.743064i	−0.928420	−	0.371532i	$-0.878833\pi$
$829$		−	616.000i		−	0.743064i	0.928420	−	0.371532i	$-0.121167\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	49.0000	−	49.0000i	0.0588235	−	0.0588235i
$834$		0			0
$835$	−721.000	+	103.000i	−0.863473	+	0.123353i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			1424.00i			1.69726i	0.528988	+	0.848629i	$0.322572\pi$
$839$			1424.00i			1.69726i	−0.528988	+	0.848629i	$0.677428\pi$
$840$		0			0
$841$	777.000			0.923900
$842$		0			0
$843$		0			0
$844$		0			0
$845$	28.0000	+	21.0000i	0.0331361	+	0.0248521i
$846$		0			0
$847$	−147.000	−	147.000i	−0.173554	−	0.173554i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1518.00			−1.78378
$852$		0			0
$853$	−935.000	+	935.000i	−1.09613	+	1.09613i	−0.101273	+	0.994859i	$0.532291\pi$
$853$	−935.000	+	935.000i	−1.09613	+	1.09613i	−0.994859	+	0.101273i	$0.967709\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	63.0000	+	63.0000i	0.0735123	+	0.0735123i	0.742907	−	0.669395i	$-0.233446\pi$
$857$	63.0000	+	63.0000i	0.0735123	+	0.0735123i	−0.669395	+	0.742907i	$0.733446\pi$
$858$		0			0
$859$		−	392.000i		−	0.456345i	−0.973621	−	0.228172i	$-0.926725\pi$
$859$		−	392.000i		−	0.456345i	0.973621	−	0.228172i	$-0.0732749\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	217.000	−	217.000i	0.251448	−	0.251448i	−0.570116	−	0.821564i	$-0.693102\pi$
$863$	217.000	−	217.000i	0.251448	−	0.251448i	0.821564	+	0.570116i	$0.193102\pi$
$864$		0			0
$865$	73.0000	+	511.000i	0.0843931	+	0.590751i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	960.000i		−	1.10472i
$870$		0			0
$871$	126.000			0.144661
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−511.000	−	1127.00i	−0.584000	−	1.28800i
$876$		0			0
$877$	−831.000	−	831.000i	−0.947548	−	0.947548i	0.0511429	−	0.998691i	$-0.483714\pi$
$877$	−831.000	−	831.000i	−0.947548	−	0.947548i	−0.998691	+	0.0511429i	$0.983714\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−322.000			−0.365494			−0.182747	−	0.983160i	$-0.558499\pi$
$881$	−322.000			−0.365494			−0.182747	+	0.983160i	$0.558499\pi$
$882$		0			0
$883$	287.000	−	287.000i	0.325028	−	0.325028i	−0.525664	−	0.850692i	$-0.676183\pi$
$883$	287.000	−	287.000i	0.325028	−	0.325028i	0.850692	+	0.525664i	$0.176183\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−903.000	−	903.000i	−1.01804	−	1.01804i	−0.999834	−	0.0182040i	$-0.994205\pi$
$887$	−903.000	−	903.000i	−1.01804	−	1.01804i	−0.0182040	−	0.999834i	$-0.505795\pi$
$888$		0			0
$889$			98.0000i			0.110236i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	312.000	−	312.000i	0.349384	−	0.349384i
$894$		0			0
$895$	224.000	+	168.000i	0.250279	+	0.187709i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			112.000i			0.124583i
$900$		0			0
$901$	−14.0000			−0.0155383
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−210.000	+	280.000i	−0.232044	+	0.309392i
$906$		0			0
$907$	−1081.00	−	1081.00i	−1.19184	−	1.19184i	−0.976550	−	0.215291i	$-0.930930\pi$
$907$	−1081.00	−	1081.00i	−1.19184	−	1.19184i	−0.215291	−	0.976550i	$-0.569070\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−238.000			−0.261251			−0.130626	−	0.991432i	$-0.541699\pi$
$911$	−238.000			−0.261251			−0.130626	+	0.991432i	$0.541699\pi$
$912$		0			0
$913$	−630.000	+	630.000i	−0.690033	+	0.690033i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1610.00	−	1610.00i	−1.75573	−	1.75573i
$918$		0			0
$919$			1552.00i			1.68879i	0.535719	+	0.844396i	$0.320040\pi$
$919$			1552.00i			1.68879i	−0.535719	+	0.844396i	$0.679960\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	882.000	−	882.000i	0.955580	−	0.955580i
$924$		0			0
$925$	561.000	−	1023.00i	0.606486	−	1.10595i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			224.000i			0.241119i	0.992706	+	0.120560i	$0.0384689\pi$
$929$			224.000i			0.241119i	−0.992706	+	0.120560i	$0.961531\pi$
$930$		0			0
$931$	−392.000			−0.421053
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−70.0000	+	10.0000i	−0.0748663	+	0.0106952i
$936$		0			0
$937$	−383.000	−	383.000i	−0.408751	−	0.408751i	0.472552	−	0.881303i	$-0.343333\pi$
$937$	−383.000	−	383.000i	−0.408751	−	0.408751i	−0.881303	+	0.472552i	$0.843333\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−42.0000			−0.0446334			−0.0223167	−	0.999751i	$-0.507104\pi$
$941$	−42.0000			−0.0446334			−0.0223167	+	0.999751i	$0.507104\pi$
$942$		0			0
$943$	−322.000	+	322.000i	−0.341463	+	0.341463i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1337.00	+	1337.00i	1.41183	+	1.41183i	0.746977	+	0.664849i	$0.231504\pi$
$947$	1337.00	+	1337.00i	1.41183	+	1.41183i	0.664849	+	0.746977i	$0.268496\pi$
$948$		0			0
$949$		−	882.000i		−	0.929399i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−273.000	+	273.000i	−0.286464	+	0.286464i	−0.835680	−	0.549216i	$-0.814926\pi$
$953$	−273.000	+	273.000i	−0.286464	+	0.286464i	0.549216	+	0.835680i	$0.314926\pi$
$954$		0			0
$955$	−426.000	+	568.000i	−0.446073	+	0.594764i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			882.000i			0.919708i
$960$		0			0
$961$	−765.000			−0.796046
$962$		0			0
$963$		0			0
$964$		0			0
$965$	63.0000	+	441.000i	0.0652850	+	0.456995i
$966$		0			0
$967$	743.000	+	743.000i	0.768356	+	0.768356i	0.977817	−	0.209461i	$-0.0671710\pi$
$967$	743.000	+	743.000i	0.768356	+	0.768356i	−0.209461	+	0.977817i	$0.567171\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	266.000			0.273944			0.136972	−	0.990575i	$-0.456263\pi$
$971$	266.000			0.273944			0.136972	+	0.990575i	$0.456263\pi$
$972$		0			0
$973$	−616.000	+	616.000i	−0.633094	+	0.633094i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−417.000	−	417.000i	−0.426817	−	0.426817i	0.460726	−	0.887543i	$-0.347589\pi$
$977$	−417.000	−	417.000i	−0.426817	−	0.426817i	−0.887543	+	0.460726i	$0.847589\pi$
$978$		0			0
$979$			1120.00i			1.14402i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−631.000	+	631.000i	−0.641913	+	0.641913i	−0.951025	−	0.309113i	$-0.899968\pi$
$983$	−631.000	+	631.000i	−0.641913	+	0.641913i	0.309113	+	0.951025i	$0.399968\pi$
$984$		0			0
$985$	441.000	−	63.0000i	0.447716	−	0.0639594i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			690.000i			0.697674i
$990$		0			0
$991$	14.0000			0.0141271			0.00706357	−	0.999975i	$-0.497752\pi$
$991$	14.0000			0.0141271			0.00706357	+	0.999975i	$0.497752\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1344.00	−	1008.00i	−1.35075	−	1.01307i
$996$		0			0
$997$	−671.000	−	671.000i	−0.673019	−	0.673019i	0.285392	−	0.958411i	$-0.407876\pi$
$997$	−671.000	−	671.000i	−0.673019	−	0.673019i	−0.958411	+	0.285392i	$0.907876\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.3.bh.e.577.1		2
3.2	odd	2		80.3.p.a.17.1		2
4.3	odd	2		180.3.l.a.37.1		2
5.3	odd	4	inner	720.3.bh.e.433.1		2
12.11	even	2		20.3.f.a.17.1	yes	2
15.2	even	4		400.3.p.d.193.1		2
15.8	even	4		80.3.p.a.33.1		2
15.14	odd	2		400.3.p.d.257.1		2
20.3	even	4		180.3.l.a.73.1		2
20.7	even	4		900.3.l.a.793.1		2
20.19	odd	2		900.3.l.a.757.1		2
24.5	odd	2		320.3.p.g.257.1		2
24.11	even	2		320.3.p.c.257.1		2
60.23	odd	4		20.3.f.a.13.1	✓	2
60.47	odd	4		100.3.f.a.93.1		2
60.59	even	2		100.3.f.a.57.1		2
84.83	odd	2		980.3.l.a.197.1		2
120.53	even	4		320.3.p.g.193.1		2
120.83	odd	4		320.3.p.c.193.1		2
420.83	even	4		980.3.l.a.393.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.3.f.a.13.1	✓	2	60.23	odd	4
20.3.f.a.17.1	yes	2	12.11	even	2
80.3.p.a.17.1		2	3.2	odd	2
80.3.p.a.33.1		2	15.8	even	4
100.3.f.a.57.1		2	60.59	even	2
100.3.f.a.93.1		2	60.47	odd	4
180.3.l.a.37.1		2	4.3	odd	2
180.3.l.a.73.1		2	20.3	even	4
320.3.p.c.193.1		2	120.83	odd	4
320.3.p.c.257.1		2	24.11	even	2
320.3.p.g.193.1		2	120.53	even	4
320.3.p.g.257.1		2	24.5	odd	2
400.3.p.d.193.1		2	15.2	even	4
400.3.p.d.257.1		2	15.14	odd	2
720.3.bh.e.433.1		2	5.3	odd	4	inner
720.3.bh.e.577.1		2	1.1	even	1	trivial
900.3.l.a.757.1		2	20.19	odd	2
900.3.l.a.793.1		2	20.7	even	4
980.3.l.a.197.1		2	84.83	odd	2
980.3.l.a.393.1		2	420.83	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	720.bh (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(3.00000 - 4.00000i) q^{5} +(7.00000 + 7.00000i) q^{7} +O(q^{10})\)	\(q+(3.00000 - 4.00000i) q^{5} +(7.00000 + 7.00000i) q^{7} +10.0000 q^{11} +(9.00000 - 9.00000i) q^{13} +(-1.00000 - 1.00000i) q^{17} +8.00000i q^{19} +(-23.0000 + 23.0000i) q^{23} +(-7.00000 - 24.0000i) q^{25} +8.00000i q^{29} +14.0000 q^{31} +(49.0000 - 7.00000i) q^{35} +(33.0000 + 33.0000i) q^{37} +14.0000 q^{41} +(15.0000 - 15.0000i) q^{43} +(-39.0000 - 39.0000i) q^{47} +49.0000i q^{49} +(7.00000 - 7.00000i) q^{53} +(30.0000 - 40.0000i) q^{55} -56.0000i q^{59} +42.0000 q^{61} +(-9.00000 - 63.0000i) q^{65} +(7.00000 + 7.00000i) q^{67} +98.0000 q^{71} +(49.0000 - 49.0000i) q^{73} +(70.0000 + 70.0000i) q^{77} -96.0000i q^{79} +(-63.0000 + 63.0000i) q^{83} +(-7.00000 + 1.00000i) q^{85} +112.000i q^{89} +126.000 q^{91} +(32.0000 + 24.0000i) q^{95} +(33.0000 + 33.0000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{5} + 14 q^{7}+O(q^{10}) \) 2 * q + 6 * q^5 + 14 * q^7	\( 2 q + 6 q^{5} + 14 q^{7} + 20 q^{11} + 18 q^{13} - 2 q^{17} - 46 q^{23} - 14 q^{25} + 28 q^{31} + 98 q^{35} + 66 q^{37} + 28 q^{41} + 30 q^{43} - 78 q^{47} + 14 q^{53} + 60 q^{55} + 84 q^{61} - 18 q^{65} + 14 q^{67} + 196 q^{71} + 98 q^{73} + 140 q^{77} - 126 q^{83} - 14 q^{85} + 252 q^{91} + 64 q^{95} + 66 q^{97}+O(q^{100}) \) 2 * q + 6 * q^5 + 14 * q^7 + 20 * q^11 + 18 * q^13 - 2 * q^17 - 46 * q^23 - 14 * q^25 + 28 * q^31 + 98 * q^35 + 66 * q^37 + 28 * q^41 + 30 * q^43 - 78 * q^47 + 14 * q^53 + 60 * q^55 + 84 * q^61 - 18 * q^65 + 14 * q^67 + 196 * q^71 + 98 * q^73 + 140 * q^77 - 126 * q^83 - 14 * q^85 + 252 * q^91 + 64 * q^95 + 66 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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