LMFDB - Embedded newform 720.3.bh.c.433.1

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 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			433.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	720.433
Dual form			720.3.bh.c.577.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.00000i q^{5} +(-2.00000 + 2.00000i) q^{7} +O(q^{10})$	$q-5.00000i q^{5} +(-2.00000 + 2.00000i) q^{7} -8.00000 q^{11} +(3.00000 + 3.00000i) q^{13} +(-7.00000 + 7.00000i) q^{17} -20.0000i q^{19} +(-2.00000 - 2.00000i) q^{23} -25.0000 q^{25} +40.0000i q^{29} -52.0000 q^{31} +(10.0000 + 10.0000i) q^{35} +(-3.00000 + 3.00000i) q^{37} +8.00000 q^{41} +(42.0000 + 42.0000i) q^{43} +(-18.0000 + 18.0000i) q^{47} +41.0000i q^{49} +(-53.0000 - 53.0000i) q^{53} +40.0000i q^{55} +20.0000i q^{59} -48.0000 q^{61} +(15.0000 - 15.0000i) q^{65} +(-62.0000 + 62.0000i) q^{67} -28.0000 q^{71} +(-47.0000 - 47.0000i) q^{73} +(16.0000 - 16.0000i) q^{77} +(18.0000 + 18.0000i) q^{83} +(35.0000 + 35.0000i) q^{85} +80.0000i q^{89} -12.0000 q^{91} -100.000 q^{95} +(-63.0000 + 63.0000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{7}+O(q^{10}) $ 2 * q - 4 * q^7	$ 2 q - 4 q^{7} - 16 q^{11} + 6 q^{13} - 14 q^{17} - 4 q^{23} - 50 q^{25} - 104 q^{31} + 20 q^{35} - 6 q^{37} + 16 q^{41} + 84 q^{43} - 36 q^{47} - 106 q^{53} - 96 q^{61} + 30 q^{65} - 124 q^{67} - 56 q^{71} - 94 q^{73} + 32 q^{77} + 36 q^{83} + 70 q^{85} - 24 q^{91} - 200 q^{95} - 126 q^{97}+O(q^{100}) $ 2 * q - 4 * q^7 - 16 * q^11 + 6 * q^13 - 14 * q^17 - 4 * q^23 - 50 * q^25 - 104 * q^31 + 20 * q^35 - 6 * q^37 + 16 * q^41 + 84 * q^43 - 36 * q^47 - 106 * q^53 - 96 * q^61 + 30 * q^65 - 124 * q^67 - 56 * q^71 - 94 * q^73 + 32 * q^77 + 36 * q^83 + 70 * q^85 - 24 * q^91 - 200 * q^95 - 126 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$	$1$

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
$5$		−	5.00000i		−	1.00000i
$5$		−	5.00000i		−	1.00000i
$6$		0			0
$7$	−2.00000	+	2.00000i	−0.285714	+	0.285714i	−0.835383	−	0.549669i	$-0.814754\pi$
$7$	−2.00000	+	2.00000i	−0.285714	+	0.285714i	0.549669	+	0.835383i	$0.314754\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−8.00000			−0.727273			−0.363636	−	0.931541i	$-0.618465\pi$
$11$	−8.00000			−0.727273			−0.363636	+	0.931541i	$0.618465\pi$
$12$		0			0
$13$	3.00000	+	3.00000i	0.230769	+	0.230769i	0.813014	−	0.582245i	$-0.197825\pi$
$13$	3.00000	+	3.00000i	0.230769	+	0.230769i	−0.582245	+	0.813014i	$0.697825\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−7.00000	+	7.00000i	−0.411765	+	0.411765i	−0.882353	−	0.470588i	$-0.844042\pi$
$17$	−7.00000	+	7.00000i	−0.411765	+	0.411765i	0.470588	+	0.882353i	$0.344042\pi$
$18$		0			0
$19$		−	20.0000i		−	1.05263i	−0.850289	−	0.526316i	$-0.823573\pi$
$19$		−	20.0000i		−	1.05263i	0.850289	−	0.526316i	$-0.176427\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.00000	−	2.00000i	−0.0869565	−	0.0869565i	0.662291	−	0.749247i	$-0.269584\pi$
$23$	−2.00000	−	2.00000i	−0.0869565	−	0.0869565i	−0.749247	+	0.662291i	$0.769584\pi$
$24$		0			0
$25$	−25.0000			−1.00000
$26$		0			0
$27$		0			0
$28$		0			0
$29$			40.0000i			1.37931i	0.724138	+	0.689655i	$0.242238\pi$
$29$			40.0000i			1.37931i	−0.724138	+	0.689655i	$0.757762\pi$
$30$		0			0
$31$	−52.0000			−1.67742			−0.838710	−	0.544579i	$-0.816690\pi$
$31$	−52.0000			−1.67742			−0.838710	+	0.544579i	$0.816690\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	10.0000	+	10.0000i	0.285714	+	0.285714i
$36$		0			0
$37$	−3.00000	+	3.00000i	−0.0810811	+	0.0810811i	−0.746484	−	0.665403i	$-0.768260\pi$
$37$	−3.00000	+	3.00000i	−0.0810811	+	0.0810811i	0.665403	+	0.746484i	$0.268260\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	8.00000			0.195122			0.0975610	−	0.995230i	$-0.468896\pi$
$41$	8.00000			0.195122			0.0975610	+	0.995230i	$0.468896\pi$
$42$		0			0
$43$	42.0000	+	42.0000i	0.976744	+	0.976744i	0.999736	−	0.0229915i	$-0.00731906\pi$
$43$	42.0000	+	42.0000i	0.976744	+	0.976744i	−0.0229915	+	0.999736i	$0.507319\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−18.0000	+	18.0000i	−0.382979	+	0.382979i	−0.872174	−	0.489195i	$-0.837290\pi$
$47$	−18.0000	+	18.0000i	−0.382979	+	0.382979i	0.489195	+	0.872174i	$0.337290\pi$
$48$		0			0
$49$			41.0000i			0.836735i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−53.0000	−	53.0000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$53$	−53.0000	−	53.0000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$54$		0			0
$55$			40.0000i			0.727273i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			20.0000i			0.338983i	0.985532	+	0.169492i	$0.0542125\pi$
$59$			20.0000i			0.338983i	−0.985532	+	0.169492i	$0.945787\pi$
$60$		0			0
$61$	−48.0000			−0.786885			−0.393443	−	0.919349i	$-0.628716\pi$
$61$	−48.0000			−0.786885			−0.393443	+	0.919349i	$0.628716\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	15.0000	−	15.0000i	0.230769	−	0.230769i
$66$		0			0
$67$	−62.0000	+	62.0000i	−0.925373	+	0.925373i	−0.997403	−	0.0720294i	$-0.977052\pi$
$67$	−62.0000	+	62.0000i	−0.925373	+	0.925373i	0.0720294	+	0.997403i	$0.477052\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−28.0000			−0.394366			−0.197183	−	0.980367i	$-0.563179\pi$
$71$	−28.0000			−0.394366			−0.197183	+	0.980367i	$0.563179\pi$
$72$		0			0
$73$	−47.0000	−	47.0000i	−0.643836	−	0.643836i	0.307661	−	0.951496i	$-0.400454\pi$
$73$	−47.0000	−	47.0000i	−0.643836	−	0.643836i	−0.951496	+	0.307661i	$0.900454\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	16.0000	−	16.0000i	0.207792	−	0.207792i
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	18.0000	+	18.0000i	0.216867	+	0.216867i	0.807177	−	0.590310i	$-0.200994\pi$
$83$	18.0000	+	18.0000i	0.216867	+	0.216867i	−0.590310	+	0.807177i	$0.700994\pi$
$84$		0			0
$85$	35.0000	+	35.0000i	0.411765	+	0.411765i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			80.0000i			0.898876i	0.893311	+	0.449438i	$0.148376\pi$
$89$			80.0000i			0.898876i	−0.893311	+	0.449438i	$0.851624\pi$
$90$		0			0
$91$	−12.0000			−0.131868
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−100.000			−1.05263
$96$		0			0
$97$	−63.0000	+	63.0000i	−0.649485	+	0.649485i	−0.952868	−	0.303384i	$-0.901884\pi$
$97$	−63.0000	+	63.0000i	−0.649485	+	0.649485i	0.303384	+	0.952868i	$0.401884\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−62.0000			−0.613861			−0.306931	−	0.951732i	$-0.599302\pi$
$101$	−62.0000			−0.613861			−0.306931	+	0.951732i	$0.599302\pi$
$102$		0			0
$103$	−118.000	−	118.000i	−1.14563	−	1.14563i	−0.987403	−	0.158229i	$-0.949422\pi$
$103$	−118.000	−	118.000i	−1.14563	−	1.14563i	−0.158229	−	0.987403i	$-0.550578\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	142.000	−	142.000i	1.32710	−	1.32710i	0.419217	−	0.907886i	$-0.362305\pi$
$107$	142.000	−	142.000i	1.32710	−	1.32710i	0.907886	−	0.419217i	$-0.137695\pi$
$108$		0			0
$109$			10.0000i			0.0917431i	0.998947	+	0.0458716i	$0.0146065\pi$
$109$			10.0000i			0.0917431i	−0.998947	+	0.0458716i	$0.985394\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−23.0000	−	23.0000i	−0.203540	−	0.203540i	0.597975	−	0.801515i	$-0.295972\pi$
$113$	−23.0000	−	23.0000i	−0.203540	−	0.203540i	−0.801515	+	0.597975i	$0.795972\pi$
$114$		0			0
$115$	−10.0000	+	10.0000i	−0.0869565	+	0.0869565i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	28.0000i		−	0.235294i
$120$		0			0
$121$	−57.0000			−0.471074
$122$		0			0
$123$		0			0
$124$		0			0
$125$			125.000i			1.00000i
$126$		0			0
$127$	118.000	−	118.000i	0.929134	−	0.929134i	−0.0685161	−	0.997650i	$-0.521826\pi$
$127$	118.000	−	118.000i	0.929134	−	0.929134i	0.997650	+	0.0685161i	$0.0218265\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−128.000			−0.977099			−0.488550	−	0.872536i	$-0.662474\pi$
$131$	−128.000			−0.977099			−0.488550	+	0.872536i	$0.662474\pi$
$132$		0			0
$133$	40.0000	+	40.0000i	0.300752	+	0.300752i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	63.0000	−	63.0000i	0.459854	−	0.459854i	−0.438753	−	0.898607i	$-0.644580\pi$
$137$	63.0000	−	63.0000i	0.459854	−	0.459854i	0.898607	+	0.438753i	$0.144580\pi$
$138$		0			0
$139$			140.000i			1.00719i	0.863939	+	0.503597i	$0.167990\pi$
$139$			140.000i			1.00719i	−0.863939	+	0.503597i	$0.832010\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−24.0000	−	24.0000i	−0.167832	−	0.167832i
$144$		0			0
$145$	200.000			1.37931
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	150.000i		−	1.00671i	−0.864079	−	0.503356i	$-0.832099\pi$
$149$		−	150.000i		−	1.00671i	0.864079	−	0.503356i	$-0.167901\pi$
$150$		0			0
$151$	−52.0000			−0.344371			−0.172185	−	0.985065i	$-0.555083\pi$
$151$	−52.0000			−0.344371			−0.172185	+	0.985065i	$0.555083\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			260.000i			1.67742i
$156$		0			0
$157$	27.0000	−	27.0000i	0.171975	−	0.171975i	−0.615872	−	0.787846i	$-0.711196\pi$
$157$	27.0000	−	27.0000i	0.171975	−	0.171975i	0.787846	+	0.615872i	$0.211196\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	8.00000			0.0496894
$162$		0			0
$163$	82.0000	+	82.0000i	0.503067	+	0.503067i	0.912390	−	0.409322i	$-0.134235\pi$
$163$	82.0000	+	82.0000i	0.503067	+	0.503067i	−0.409322	+	0.912390i	$0.634235\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	62.0000	−	62.0000i	0.371257	−	0.371257i	−0.496678	−	0.867935i	$-0.665447\pi$
$167$	62.0000	−	62.0000i	0.371257	−	0.371257i	0.867935	+	0.496678i	$0.165447\pi$
$168$		0			0
$169$		−	151.000i		−	0.893491i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	107.000	+	107.000i	0.618497	+	0.618497i	0.945146	−	0.326649i	$-0.105919\pi$
$173$	107.000	+	107.000i	0.618497	+	0.618497i	−0.326649	+	0.945146i	$0.605919\pi$
$174$		0			0
$175$	50.0000	−	50.0000i	0.285714	−	0.285714i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			220.000i			1.22905i	0.788897	+	0.614525i	$0.210652\pi$
$179$			220.000i			1.22905i	−0.788897	+	0.614525i	$0.789348\pi$
$180$		0			0
$181$	2.00000			0.0110497			0.00552486	−	0.999985i	$-0.498241\pi$
$181$	2.00000			0.0110497			0.00552486	+	0.999985i	$0.498241\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	15.0000	+	15.0000i	0.0810811	+	0.0810811i
$186$		0			0
$187$	56.0000	−	56.0000i	0.299465	−	0.299465i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	212.000			1.10995			0.554974	−	0.831868i	$-0.312728\pi$
$191$	212.000			1.10995			0.554974	+	0.831868i	$0.312728\pi$
$192$		0			0
$193$	−57.0000	−	57.0000i	−0.295337	−	0.295337i	0.543847	−	0.839184i	$-0.316967\pi$
$193$	−57.0000	−	57.0000i	−0.295337	−	0.295337i	−0.839184	+	0.543847i	$0.816967\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	3.00000	−	3.00000i	0.0152284	−	0.0152284i	−0.699452	−	0.714680i	$-0.746572\pi$
$197$	3.00000	−	3.00000i	0.0152284	−	0.0152284i	0.714680	+	0.699452i	$0.246572\pi$
$198$		0			0
$199$		−	120.000i		−	0.603015i	−0.953464	−	0.301508i	$-0.902510\pi$
$199$		−	120.000i		−	0.603015i	0.953464	−	0.301508i	$-0.0974898\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−80.0000	−	80.0000i	−0.394089	−	0.394089i
$204$		0			0
$205$		−	40.0000i		−	0.195122i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			160.000i			0.765550i
$210$		0			0
$211$	328.000			1.55450			0.777251	−	0.629190i	$-0.216613\pi$
$211$	328.000			1.55450			0.777251	+	0.629190i	$0.216613\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	210.000	−	210.000i	0.976744	−	0.976744i
$216$		0			0
$217$	104.000	−	104.000i	0.479263	−	0.479263i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−42.0000			−0.190045
$222$		0			0
$223$	−138.000	−	138.000i	−0.618834	−	0.618834i	0.326398	−	0.945232i	$-0.394165\pi$
$223$	−138.000	−	138.000i	−0.618834	−	0.618834i	−0.945232	+	0.326398i	$0.894165\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2.00000	−	2.00000i	0.00881057	−	0.00881057i	−0.702688	−	0.711498i	$-0.748017\pi$
$227$	2.00000	−	2.00000i	0.00881057	−	0.00881057i	0.711498	+	0.702688i	$0.248017\pi$
$228$		0			0
$229$		−	120.000i		−	0.524017i	−0.965066	−	0.262009i	$-0.915615\pi$
$229$		−	120.000i		−	0.524017i	0.965066	−	0.262009i	$-0.0843849\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−183.000	−	183.000i	−0.785408	−	0.785408i	0.195330	−	0.980738i	$-0.437422\pi$
$233$	−183.000	−	183.000i	−0.785408	−	0.785408i	−0.980738	+	0.195330i	$0.937422\pi$
$234$		0			0
$235$	90.0000	+	90.0000i	0.382979	+	0.382979i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	120.000i		−	0.502092i	−0.967975	−	0.251046i	$-0.919225\pi$
$239$		−	120.000i		−	0.502092i	0.967975	−	0.251046i	$-0.0807746\pi$
$240$		0			0
$241$	232.000			0.962656			0.481328	−	0.876541i	$-0.340155\pi$
$241$	232.000			0.962656			0.481328	+	0.876541i	$0.340155\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	205.000			0.836735
$246$		0			0
$247$	60.0000	−	60.0000i	0.242915	−	0.242915i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−48.0000			−0.191235			−0.0956175	−	0.995418i	$-0.530483\pi$
$251$	−48.0000			−0.191235			−0.0956175	+	0.995418i	$0.530483\pi$
$252$		0			0
$253$	16.0000	+	16.0000i	0.0632411	+	0.0632411i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	313.000	−	313.000i	1.21790	−	1.21790i	0.249532	−	0.968366i	$-0.419723\pi$
$257$	313.000	−	313.000i	1.21790	−	1.21790i	0.968366	−	0.249532i	$-0.0802769\pi$
$258$		0			0
$259$		−	12.0000i		−	0.0463320i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−262.000	−	262.000i	−0.996198	−	0.996198i	0.00379508	−	0.999993i	$-0.498792\pi$
$263$	−262.000	−	262.000i	−0.996198	−	0.996198i	−0.999993	+	0.00379508i	$0.998792\pi$
$264$		0			0
$265$	−265.000	+	265.000i	−1.00000	+	1.00000i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			10.0000i			0.0371747i	0.999827	+	0.0185874i	$0.00591688\pi$
$269$			10.0000i			0.0371747i	−0.999827	+	0.0185874i	$0.994083\pi$
$270$		0			0
$271$	−252.000			−0.929889			−0.464945	−	0.885340i	$-0.653926\pi$
$271$	−252.000			−0.929889			−0.464945	+	0.885340i	$0.653926\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	200.000			0.727273
$276$		0			0
$277$	267.000	−	267.000i	0.963899	−	0.963899i	−0.0354718	−	0.999371i	$-0.511293\pi$
$277$	267.000	−	267.000i	0.963899	−	0.963899i	0.999371	+	0.0354718i	$0.0112934\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−312.000			−1.11032			−0.555160	−	0.831743i	$-0.687343\pi$
$281$	−312.000			−1.11032			−0.555160	+	0.831743i	$0.687343\pi$
$282$		0			0
$283$	262.000	+	262.000i	0.925795	+	0.925795i	0.997431	−	0.0716358i	$-0.0228219\pi$
$283$	262.000	+	262.000i	0.925795	+	0.925795i	−0.0716358	+	0.997431i	$0.522822\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−16.0000	+	16.0000i	−0.0557491	+	0.0557491i
$288$		0			0
$289$			191.000i			0.660900i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−243.000	−	243.000i	−0.829352	−	0.829352i	0.158075	−	0.987427i	$-0.449471\pi$
$293$	−243.000	−	243.000i	−0.829352	−	0.829352i	−0.987427	+	0.158075i	$0.949471\pi$
$294$		0			0
$295$	100.000			0.338983
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	12.0000i		−	0.0401338i
$300$		0			0
$301$	−168.000			−0.558140
$302$		0			0
$303$		0			0
$304$		0			0
$305$			240.000i			0.786885i
$306$		0			0
$307$	18.0000	−	18.0000i	0.0586319	−	0.0586319i	−0.677183	−	0.735815i	$-0.736799\pi$
$307$	18.0000	−	18.0000i	0.0586319	−	0.0586319i	0.735815	+	0.677183i	$0.236799\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−388.000			−1.24759			−0.623794	−	0.781589i	$-0.714410\pi$
$311$	−388.000			−1.24759			−0.623794	+	0.781589i	$0.714410\pi$
$312$		0			0
$313$	183.000	+	183.000i	0.584665	+	0.584665i	0.936182	−	0.351517i	$-0.114334\pi$
$313$	183.000	+	183.000i	0.584665	+	0.584665i	−0.351517	+	0.936182i	$0.614334\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	213.000	−	213.000i	0.671924	−	0.671924i	−0.286235	−	0.958159i	$-0.592404\pi$
$317$	213.000	−	213.000i	0.671924	−	0.671924i	0.958159	+	0.286235i	$0.0924038\pi$
$318$		0			0
$319$		−	320.000i		−	1.00313i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	140.000	+	140.000i	0.433437	+	0.433437i
$324$		0			0
$325$	−75.0000	−	75.0000i	−0.230769	−	0.230769i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	72.0000i		−	0.218845i
$330$		0			0
$331$	−232.000			−0.700906			−0.350453	−	0.936580i	$-0.613972\pi$
$331$	−232.000			−0.700906			−0.350453	+	0.936580i	$0.613972\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	310.000	+	310.000i	0.925373	+	0.925373i
$336$		0			0
$337$	417.000	−	417.000i	1.23739	−	1.23739i	0.276324	−	0.961064i	$-0.410884\pi$
$337$	417.000	−	417.000i	1.23739	−	1.23739i	0.961064	−	0.276324i	$-0.0891164\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	416.000			1.21994
$342$		0			0
$343$	−180.000	−	180.000i	−0.524781	−	0.524781i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	202.000	−	202.000i	0.582133	−	0.582133i	−0.353356	−	0.935489i	$-0.614960\pi$
$347$	202.000	−	202.000i	0.582133	−	0.582133i	0.935489	+	0.353356i	$0.114960\pi$
$348$		0			0
$349$			440.000i			1.26074i	0.776293	+	0.630372i	$0.217098\pi$
$349$			440.000i			1.26074i	−0.776293	+	0.630372i	$0.782902\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	447.000	+	447.000i	1.26629	+	1.26629i	0.947991	+	0.318298i	$0.103111\pi$
$353$	447.000	+	447.000i	1.26629	+	1.26629i	0.318298	+	0.947991i	$0.396889\pi$
$354$		0			0
$355$			140.000i			0.394366i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			400.000i			1.11421i	0.830443	+	0.557103i	$0.188087\pi$
$359$			400.000i			1.11421i	−0.830443	+	0.557103i	$0.811913\pi$
$360$		0			0
$361$	−39.0000			−0.108033
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−235.000	+	235.000i	−0.643836	+	0.643836i
$366$		0			0
$367$	118.000	−	118.000i	0.321526	−	0.321526i	−0.527826	−	0.849352i	$-0.676993\pi$
$367$	118.000	−	118.000i	0.321526	−	0.321526i	0.849352	+	0.527826i	$0.176993\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	212.000			0.571429
$372$		0			0
$373$	−107.000	−	107.000i	−0.286863	−	0.286863i	0.548975	−	0.835839i	$-0.315018\pi$
$373$	−107.000	−	107.000i	−0.286863	−	0.286863i	−0.835839	+	0.548975i	$0.815018\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−120.000	+	120.000i	−0.318302	+	0.318302i
$378$		0			0
$379$			340.000i			0.897098i	0.893758	+	0.448549i	$0.148059\pi$
$379$			340.000i			0.897098i	−0.893758	+	0.448549i	$0.851941\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−342.000	−	342.000i	−0.892950	−	0.892950i	0.101849	−	0.994800i	$-0.467524\pi$
$383$	−342.000	−	342.000i	−0.892950	−	0.892950i	−0.994800	+	0.101849i	$0.967524\pi$
$384$		0			0
$385$	−80.0000	−	80.0000i	−0.207792	−	0.207792i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			390.000i			1.00257i	0.865282	+	0.501285i	$0.167139\pi$
$389$			390.000i			1.00257i	−0.865282	+	0.501285i	$0.832861\pi$
$390$		0			0
$391$	28.0000			0.0716113
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−323.000	+	323.000i	−0.813602	+	0.813602i	−0.985172	−	0.171570i	$-0.945116\pi$
$397$	−323.000	+	323.000i	−0.813602	+	0.813602i	0.171570	+	0.985172i	$0.445116\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−642.000			−1.60100			−0.800499	−	0.599334i	$-0.795432\pi$
$401$	−642.000			−1.60100			−0.800499	+	0.599334i	$0.795432\pi$
$402$		0			0
$403$	−156.000	−	156.000i	−0.387097	−	0.387097i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	24.0000	−	24.0000i	0.0589681	−	0.0589681i
$408$		0			0
$409$			150.000i			0.366748i	0.983043	+	0.183374i	$0.0587020\pi$
$409$			150.000i			0.366748i	−0.983043	+	0.183374i	$0.941298\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−40.0000	−	40.0000i	−0.0968523	−	0.0968523i
$414$		0			0
$415$	90.0000	−	90.0000i	0.216867	−	0.216867i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	300.000i		−	0.715990i	−0.933723	−	0.357995i	$-0.883460\pi$
$419$		−	300.000i		−	0.715990i	0.933723	−	0.357995i	$-0.116540\pi$
$420$		0			0
$421$	−208.000			−0.494062			−0.247031	−	0.969008i	$-0.579455\pi$
$421$	−208.000			−0.494062			−0.247031	+	0.969008i	$0.579455\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	175.000	−	175.000i	0.411765	−	0.411765i
$426$		0			0
$427$	96.0000	−	96.0000i	0.224824	−	0.224824i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−788.000			−1.82831			−0.914153	−	0.405369i	$-0.867143\pi$
$431$	−788.000			−1.82831			−0.914153	+	0.405369i	$0.867143\pi$
$432$		0			0
$433$	−367.000	−	367.000i	−0.847575	−	0.847575i	0.142255	−	0.989830i	$-0.454565\pi$
$433$	−367.000	−	367.000i	−0.847575	−	0.847575i	−0.989830	+	0.142255i	$0.954565\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−40.0000	+	40.0000i	−0.0915332	+	0.0915332i
$438$		0			0
$439$			560.000i			1.27563i	0.770191	+	0.637813i	$0.220161\pi$
$439$			560.000i			1.27563i	−0.770191	+	0.637813i	$0.779839\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	378.000	+	378.000i	0.853273	+	0.853273i	0.990535	−	0.137262i	$-0.0438301\pi$
$443$	378.000	+	378.000i	0.853273	+	0.853273i	−0.137262	+	0.990535i	$0.543830\pi$
$444$		0			0
$445$	400.000			0.898876
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	410.000i		−	0.913140i	−0.889687	−	0.456570i	$-0.849078\pi$
$449$		−	410.000i		−	0.913140i	0.889687	−	0.456570i	$-0.150922\pi$
$450$		0			0
$451$	−64.0000			−0.141907
$452$		0			0
$453$		0			0
$454$		0			0
$455$			60.0000i			0.131868i
$456$		0			0
$457$	−393.000	+	393.000i	−0.859956	+	0.859956i	−0.991333	−	0.131376i	$-0.958060\pi$
$457$	−393.000	+	393.000i	−0.859956	+	0.859956i	0.131376	+	0.991333i	$0.458060\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−622.000			−1.34924			−0.674620	−	0.738165i	$-0.735693\pi$
$461$	−622.000			−1.34924			−0.674620	+	0.738165i	$0.735693\pi$
$462$		0			0
$463$	−278.000	−	278.000i	−0.600432	−	0.600432i	0.339995	−	0.940427i	$-0.389575\pi$
$463$	−278.000	−	278.000i	−0.600432	−	0.600432i	−0.940427	+	0.339995i	$0.889575\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−38.0000	+	38.0000i	−0.0813704	+	0.0813704i	−0.746621	−	0.665250i	$-0.768325\pi$
$467$	−38.0000	+	38.0000i	−0.0813704	+	0.0813704i	0.665250	+	0.746621i	$0.268325\pi$
$468$		0			0
$469$		−	248.000i		−	0.528785i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−336.000	−	336.000i	−0.710359	−	0.710359i
$474$		0			0
$475$			500.000i			1.05263i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			440.000i			0.918580i	0.888286	+	0.459290i	$0.151896\pi$
$479$			440.000i			0.918580i	−0.888286	+	0.459290i	$0.848104\pi$
$480$		0			0
$481$	−18.0000			−0.0374220
$482$		0			0
$483$		0			0
$484$		0			0
$485$	315.000	+	315.000i	0.649485	+	0.649485i
$486$		0			0
$487$	−522.000	+	522.000i	−1.07187	+	1.07187i	−0.0746595	+	0.997209i	$0.523787\pi$
$487$	−522.000	+	522.000i	−1.07187	+	1.07187i	−0.997209	+	0.0746595i	$0.976213\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−328.000			−0.668024			−0.334012	−	0.942569i	$-0.608403\pi$
$491$	−328.000			−0.668024			−0.334012	+	0.942569i	$0.608403\pi$
$492$		0			0
$493$	−280.000	−	280.000i	−0.567951	−	0.567951i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	56.0000	−	56.0000i	0.112676	−	0.112676i
$498$		0			0
$499$			380.000i			0.761523i	0.924673	+	0.380762i	$0.124338\pi$
$499$			380.000i			0.761523i	−0.924673	+	0.380762i	$0.875662\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−42.0000	−	42.0000i	−0.0834990	−	0.0834990i	0.664124	−	0.747623i	$-0.268805\pi$
$503$	−42.0000	−	42.0000i	−0.0834990	−	0.0834990i	−0.747623	+	0.664124i	$0.768805\pi$
$504$		0			0
$505$			310.000i			0.613861i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	440.000i		−	0.864440i	−0.901768	−	0.432220i	$-0.857730\pi$
$509$		−	440.000i		−	0.864440i	0.901768	−	0.432220i	$-0.142270\pi$
$510$		0			0
$511$	188.000			0.367906
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−590.000	+	590.000i	−1.14563	+	1.14563i
$516$		0			0
$517$	144.000	−	144.000i	0.278530	−	0.278530i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	258.000			0.495202			0.247601	−	0.968862i	$-0.420358\pi$
$521$	258.000			0.495202			0.247601	+	0.968862i	$0.420358\pi$
$522$		0			0
$523$	−258.000	−	258.000i	−0.493308	−	0.493308i	0.416039	−	0.909347i	$-0.363418\pi$
$523$	−258.000	−	258.000i	−0.493308	−	0.493308i	−0.909347	+	0.416039i	$0.863418\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	364.000	−	364.000i	0.690702	−	0.690702i
$528$		0			0
$529$		−	521.000i		−	0.984877i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	24.0000	+	24.0000i	0.0450281	+	0.0450281i
$534$		0			0
$535$	−710.000	−	710.000i	−1.32710	−	1.32710i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	328.000i		−	0.608534i
$540$		0			0
$541$	−338.000			−0.624769			−0.312384	−	0.949956i	$-0.601128\pi$
$541$	−338.000			−0.624769			−0.312384	+	0.949956i	$0.601128\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	50.0000			0.0917431
$546$		0			0
$547$	558.000	−	558.000i	1.02011	−	1.02011i	0.0203161	−	0.999794i	$-0.493533\pi$
$547$	558.000	−	558.000i	1.02011	−	1.02011i	0.999794	−	0.0203161i	$-0.00646725\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	800.000			1.45191
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	3.00000	−	3.00000i	0.00538600	−	0.00538600i	−0.704409	−	0.709795i	$-0.748788\pi$
$557$	3.00000	−	3.00000i	0.00538600	−	0.00538600i	0.709795	+	0.704409i	$0.248788\pi$
$558$		0			0
$559$			252.000i			0.450805i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−42.0000	−	42.0000i	−0.0746004	−	0.0746004i	0.668822	−	0.743422i	$-0.266799\pi$
$563$	−42.0000	−	42.0000i	−0.0746004	−	0.0746004i	−0.743422	+	0.668822i	$0.766799\pi$
$564$		0			0
$565$	−115.000	+	115.000i	−0.203540	+	0.203540i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			950.000i			1.66960i	0.550557	+	0.834798i	$0.314416\pi$
$569$			950.000i			1.66960i	−0.550557	+	0.834798i	$0.685584\pi$
$570$		0			0
$571$	−392.000			−0.686515			−0.343257	−	0.939241i	$-0.611530\pi$
$571$	−392.000			−0.686515			−0.343257	+	0.939241i	$0.611530\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	50.0000	+	50.0000i	0.0869565	+	0.0869565i
$576$		0			0
$577$	−473.000	+	473.000i	−0.819757	+	0.819757i	−0.986073	−	0.166315i	$-0.946813\pi$
$577$	−473.000	+	473.000i	−0.819757	+	0.819757i	0.166315	+	0.986073i	$0.446813\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−72.0000			−0.123924
$582$		0			0
$583$	424.000	+	424.000i	0.727273	+	0.727273i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−198.000	+	198.000i	−0.337308	+	0.337308i	−0.855353	−	0.518045i	$-0.826660\pi$
$587$	−198.000	+	198.000i	−0.337308	+	0.337308i	0.518045	+	0.855353i	$0.326660\pi$
$588$		0			0
$589$			1040.00i			1.76570i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	47.0000	+	47.0000i	0.0792580	+	0.0792580i	0.745624	−	0.666366i	$-0.232151\pi$
$593$	47.0000	+	47.0000i	0.0792580	+	0.0792580i	−0.666366	+	0.745624i	$0.732151\pi$
$594$		0			0
$595$	−140.000			−0.235294
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	520.000i		−	0.868114i	−0.900886	−	0.434057i	$-0.857082\pi$
$599$		−	520.000i		−	0.868114i	0.900886	−	0.434057i	$-0.142918\pi$
$600$		0			0
$601$	−328.000			−0.545757			−0.272879	−	0.962048i	$-0.587976\pi$
$601$	−328.000			−0.545757			−0.272879	+	0.962048i	$0.587976\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			285.000i			0.471074i
$606$		0			0
$607$	−462.000	+	462.000i	−0.761120	+	0.761120i	−0.976525	−	0.215405i	$-0.930893\pi$
$607$	−462.000	+	462.000i	−0.761120	+	0.761120i	0.215405	+	0.976525i	$0.430893\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−108.000			−0.176759
$612$		0			0
$613$	723.000	+	723.000i	1.17945	+	1.17945i	0.979886	+	0.199560i	$0.0639512\pi$
$613$	723.000	+	723.000i	1.17945	+	1.17945i	0.199560	+	0.979886i	$0.436049\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−327.000	+	327.000i	−0.529984	+	0.529984i	−0.920567	−	0.390584i	$-0.872273\pi$
$617$	−327.000	+	327.000i	−0.529984	+	0.529984i	0.390584	+	0.920567i	$0.372273\pi$
$618$		0			0
$619$		−	660.000i		−	1.06624i	−0.846041	−	0.533118i	$-0.821020\pi$
$619$		−	660.000i		−	1.06624i	0.846041	−	0.533118i	$-0.178980\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−160.000	−	160.000i	−0.256822	−	0.256822i
$624$		0			0
$625$	625.000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	42.0000i		−	0.0667727i
$630$		0			0
$631$	548.000			0.868463			0.434231	−	0.900801i	$-0.357020\pi$
$631$	548.000			0.868463			0.434231	+	0.900801i	$0.357020\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−590.000	−	590.000i	−0.929134	−	0.929134i
$636$		0			0
$637$	−123.000	+	123.000i	−0.193093	+	0.193093i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	568.000			0.886115			0.443058	−	0.896493i	$-0.353894\pi$
$641$	568.000			0.886115			0.443058	+	0.896493i	$0.353894\pi$
$642$		0			0
$643$	342.000	+	342.000i	0.531882	+	0.531882i	0.921132	−	0.389250i	$-0.127266\pi$
$643$	342.000	+	342.000i	0.531882	+	0.531882i	−0.389250	+	0.921132i	$0.627266\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−118.000	+	118.000i	−0.182380	+	0.182380i	−0.792392	−	0.610012i	$-0.791165\pi$
$647$	−118.000	+	118.000i	−0.182380	+	0.182380i	0.610012	+	0.792392i	$0.291165\pi$
$648$		0			0
$649$		−	160.000i		−	0.246533i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−453.000	−	453.000i	−0.693721	−	0.693721i	0.269327	−	0.963049i	$-0.413199\pi$
$653$	−453.000	−	453.000i	−0.693721	−	0.693721i	−0.963049	+	0.269327i	$0.913199\pi$
$654$		0			0
$655$			640.000i			0.977099i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	140.000i		−	0.212443i	−0.994342	−	0.106222i	$-0.966125\pi$
$659$		−	140.000i		−	0.212443i	0.994342	−	0.106222i	$-0.0338753\pi$
$660$		0			0
$661$	512.000			0.774584			0.387292	−	0.921957i	$-0.373411\pi$
$661$	512.000			0.774584			0.387292	+	0.921957i	$0.373411\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	200.000	−	200.000i	0.300752	−	0.300752i
$666$		0			0
$667$	80.0000	−	80.0000i	0.119940	−	0.119940i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	384.000			0.572280
$672$		0			0
$673$	193.000	+	193.000i	0.286776	+	0.286776i	0.835804	−	0.549028i	$-0.185002\pi$
$673$	193.000	+	193.000i	0.286776	+	0.286776i	−0.549028	+	0.835804i	$0.685002\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−157.000	+	157.000i	−0.231905	+	0.231905i	−0.813488	−	0.581582i	$-0.802434\pi$
$677$	−157.000	+	157.000i	−0.231905	+	0.231905i	0.581582	+	0.813488i	$0.302434\pi$
$678$		0			0
$679$		−	252.000i		−	0.371134i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	438.000	+	438.000i	0.641288	+	0.641288i	0.950872	−	0.309584i	$-0.100190\pi$
$683$	438.000	+	438.000i	0.641288	+	0.641288i	−0.309584	+	0.950872i	$0.600190\pi$
$684$		0			0
$685$	−315.000	−	315.000i	−0.459854	−	0.459854i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	318.000i		−	0.461538i
$690$		0			0
$691$	−1032.00			−1.49349			−0.746744	−	0.665112i	$-0.768384\pi$
$691$	−1032.00			−1.49349			−0.746744	+	0.665112i	$0.768384\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	700.000			1.00719
$696$		0			0
$697$	−56.0000	+	56.0000i	−0.0803443	+	0.0803443i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	128.000			0.182596			0.0912981	−	0.995824i	$-0.470898\pi$
$701$	128.000			0.182596			0.0912981	+	0.995824i	$0.470898\pi$
$702$		0			0
$703$	60.0000	+	60.0000i	0.0853485	+	0.0853485i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	124.000	−	124.000i	0.175389	−	0.175389i
$708$		0			0
$709$		−	760.000i		−	1.07193i	−0.844239	−	0.535966i	$-0.819947\pi$
$709$		−	760.000i		−	1.07193i	0.844239	−	0.535966i	$-0.180053\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	104.000	+	104.000i	0.145863	+	0.145863i
$714$		0			0
$715$	−120.000	+	120.000i	−0.167832	+	0.167832i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			1160.00i			1.61335i	0.590994	+	0.806676i	$0.298736\pi$
$719$			1160.00i			1.61335i	−0.590994	+	0.806676i	$0.701264\pi$
$720$		0			0
$721$	472.000			0.654646
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	1000.00i		−	1.37931i
$726$		0			0
$727$	558.000	−	558.000i	0.767538	−	0.767538i	−0.210135	−	0.977672i	$-0.567390\pi$
$727$	558.000	−	558.000i	0.767538	−	0.767538i	0.977672	+	0.210135i	$0.0673902\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−588.000			−0.804378
$732$		0			0
$733$	−827.000	−	827.000i	−1.12824	−	1.12824i	−0.990463	−	0.137777i	$-0.956004\pi$
$733$	−827.000	−	827.000i	−1.12824	−	1.12824i	−0.137777	−	0.990463i	$-0.543996\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	496.000	−	496.000i	0.672999	−	0.672999i
$738$		0			0
$739$		−	700.000i		−	0.947226i	−0.880733	−	0.473613i	$-0.842950\pi$
$739$		−	700.000i		−	0.947226i	0.880733	−	0.473613i	$-0.157050\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−382.000	−	382.000i	−0.514132	−	0.514132i	0.401658	−	0.915790i	$-0.368434\pi$
$743$	−382.000	−	382.000i	−0.514132	−	0.514132i	−0.915790	+	0.401658i	$0.868434\pi$
$744$		0			0
$745$	−750.000			−1.00671
$746$		0			0
$747$		0			0
$748$		0			0
$749$			568.000i			0.758344i
$750$		0			0
$751$	588.000			0.782956			0.391478	−	0.920187i	$-0.371964\pi$
$751$	588.000			0.782956			0.391478	+	0.920187i	$0.371964\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			260.000i			0.344371i
$756$		0			0
$757$	987.000	−	987.000i	1.30383	−	1.30383i	0.378043	−	0.925788i	$-0.376597\pi$
$757$	987.000	−	987.000i	1.30383	−	1.30383i	0.925788	−	0.378043i	$-0.123403\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	158.000			0.207622			0.103811	−	0.994597i	$-0.466896\pi$
$761$	158.000			0.207622			0.103811	+	0.994597i	$0.466896\pi$
$762$		0			0
$763$	−20.0000	−	20.0000i	−0.0262123	−	0.0262123i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−60.0000	+	60.0000i	−0.0782269	+	0.0782269i
$768$		0			0
$769$		−	80.0000i		−	0.104031i	−0.998646	−	0.0520156i	$-0.983435\pi$
$769$		−	80.0000i		−	0.104031i	0.998646	−	0.0520156i	$-0.0165646\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−243.000	−	243.000i	−0.314360	−	0.314360i	0.532236	−	0.846596i	$-0.321352\pi$
$773$	−243.000	−	243.000i	−0.314360	−	0.314360i	−0.846596	+	0.532236i	$0.821352\pi$
$774$		0			0
$775$	1300.00			1.67742
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	160.000i		−	0.205392i
$780$		0			0
$781$	224.000			0.286812
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−135.000	−	135.000i	−0.171975	−	0.171975i
$786$		0			0
$787$	−262.000	+	262.000i	−0.332910	+	0.332910i	−0.853690	−	0.520781i	$-0.825641\pi$
$787$	−262.000	+	262.000i	−0.332910	+	0.332910i	0.520781	+	0.853690i	$0.325641\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	92.0000			0.116308
$792$		0			0
$793$	−144.000	−	144.000i	−0.181589	−	0.181589i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−267.000	+	267.000i	−0.335006	+	0.335006i	−0.854484	−	0.519478i	$-0.826127\pi$
$797$	−267.000	+	267.000i	−0.335006	+	0.335006i	0.519478	+	0.854484i	$0.326127\pi$
$798$		0			0
$799$		−	252.000i		−	0.315394i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	376.000	+	376.000i	0.468244	+	0.468244i
$804$		0			0
$805$		−	40.0000i		−	0.0496894i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			560.000i			0.692213i	0.938195	+	0.346106i	$0.112496\pi$
$809$			560.000i			0.692213i	−0.938195	+	0.346106i	$0.887504\pi$
$810$		0			0
$811$	208.000			0.256473			0.128237	−	0.991744i	$-0.459068\pi$
$811$	208.000			0.256473			0.128237	+	0.991744i	$0.459068\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	410.000	−	410.000i	0.503067	−	0.503067i
$816$		0			0
$817$	840.000	−	840.000i	1.02815	−	1.02815i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1568.00			1.90987			0.954933	−	0.296821i	$-0.0959266\pi$
$821$	1568.00			1.90987			0.954933	+	0.296821i	$0.0959266\pi$
$822$		0			0
$823$	562.000	+	562.000i	0.682868	+	0.682868i	0.960645	−	0.277778i	$-0.0895979\pi$
$823$	562.000	+	562.000i	0.682868	+	0.682868i	−0.277778	+	0.960645i	$0.589598\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	762.000	−	762.000i	0.921403	−	0.921403i	−0.0757260	−	0.997129i	$-0.524127\pi$
$827$	762.000	−	762.000i	0.921403	−	0.921403i	0.997129	+	0.0757260i	$0.0241274\pi$
$828$		0			0
$829$		−	170.000i		−	0.205066i	−0.994730	−	0.102533i	$-0.967305\pi$
$829$		−	170.000i		−	0.205066i	0.994730	−	0.102533i	$-0.0326948\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−287.000	−	287.000i	−0.344538	−	0.344538i
$834$		0			0
$835$	−310.000	−	310.000i	−0.371257	−	0.371257i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			280.000i			0.333731i	0.985980	+	0.166865i	$0.0533645\pi$
$839$			280.000i			0.333731i	−0.985980	+	0.166865i	$0.946635\pi$
$840$		0			0
$841$	−759.000			−0.902497
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−755.000			−0.893491
$846$		0			0
$847$	114.000	−	114.000i	0.134593	−	0.134593i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	12.0000			0.0141011
$852$		0			0
$853$	1123.00	+	1123.00i	1.31653	+	1.31653i	0.916504	+	0.400026i	$0.130999\pi$
$853$	1123.00	+	1123.00i	1.31653	+	1.31653i	0.400026	+	0.916504i	$0.369001\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−417.000	+	417.000i	−0.486581	+	0.486581i	−0.907226	−	0.420644i	$-0.861804\pi$
$857$	−417.000	+	417.000i	−0.486581	+	0.486581i	0.420644	+	0.907226i	$0.361804\pi$
$858$		0			0
$859$		−	1300.00i		−	1.51339i	−0.653769	−	0.756694i	$-0.726813\pi$
$859$		−	1300.00i		−	1.51339i	0.653769	−	0.756694i	$-0.273187\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−242.000	−	242.000i	−0.280417	−	0.280417i	0.552858	−	0.833275i	$-0.313537\pi$
$863$	−242.000	−	242.000i	−0.280417	−	0.280417i	−0.833275	+	0.552858i	$0.813537\pi$
$864$		0			0
$865$	535.000	−	535.000i	0.618497	−	0.618497i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−372.000			−0.427095
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−250.000	−	250.000i	−0.285714	−	0.285714i
$876$		0			0
$877$	−453.000	+	453.000i	−0.516534	+	0.516534i	−0.916521	−	0.399987i	$-0.869015\pi$
$877$	−453.000	+	453.000i	−0.516534	+	0.516534i	0.399987	+	0.916521i	$0.369015\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−712.000			−0.808173			−0.404086	−	0.914721i	$-0.632410\pi$
$881$	−712.000			−0.808173			−0.404086	+	0.914721i	$0.632410\pi$
$882$		0			0
$883$	−118.000	−	118.000i	−0.133635	−	0.133635i	0.637125	−	0.770760i	$-0.280123\pi$
$883$	−118.000	−	118.000i	−0.133635	−	0.133635i	−0.770760	+	0.637125i	$0.780123\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1158.00	+	1158.00i	−1.30552	+	1.30552i	−0.380914	+	0.924611i	$0.624390\pi$
$887$	−1158.00	+	1158.00i	−1.30552	+	1.30552i	−0.924611	+	0.380914i	$0.875610\pi$
$888$		0			0
$889$			472.000i			0.530934i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	360.000	+	360.000i	0.403135	+	0.403135i
$894$		0			0
$895$	1100.00			1.22905
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	2080.00i		−	2.31368i
$900$		0			0
$901$	742.000			0.823529
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	10.0000i		−	0.0110497i
$906$		0			0
$907$	−142.000	+	142.000i	−0.156560	+	0.156560i	−0.781040	−	0.624480i	$-0.785311\pi$
$907$	−142.000	+	142.000i	−0.156560	+	0.156560i	0.624480	+	0.781040i	$0.285311\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1172.00			1.28650			0.643249	−	0.765657i	$-0.277586\pi$
$911$	1172.00			1.28650			0.643249	+	0.765657i	$0.277586\pi$
$912$		0			0
$913$	−144.000	−	144.000i	−0.157722	−	0.157722i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	256.000	−	256.000i	0.279171	−	0.279171i
$918$		0			0
$919$			920.000i			1.00109i	0.865711	+	0.500544i	$0.166867\pi$
$919$			920.000i			1.00109i	−0.865711	+	0.500544i	$0.833133\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−84.0000	−	84.0000i	−0.0910076	−	0.0910076i
$924$		0			0
$925$	75.0000	−	75.0000i	0.0810811	−	0.0810811i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	1190.00i		−	1.28095i	−0.767980	−	0.640474i	$-0.778738\pi$
$929$		−	1190.00i		−	1.28095i	0.767980	−	0.640474i	$-0.221262\pi$
$930$		0			0
$931$	820.000			0.880773
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−280.000	−	280.000i	−0.299465	−	0.299465i
$936$		0			0
$937$	−233.000	+	233.000i	−0.248666	+	0.248666i	−0.820423	−	0.571757i	$-0.806262\pi$
$937$	−233.000	+	233.000i	−0.248666	+	0.248666i	0.571757	+	0.820423i	$0.306262\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	78.0000			0.0828905			0.0414453	−	0.999141i	$-0.486804\pi$
$941$	78.0000			0.0828905			0.0414453	+	0.999141i	$0.486804\pi$
$942$		0			0
$943$	−16.0000	−	16.0000i	−0.0169671	−	0.0169671i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	62.0000	−	62.0000i	0.0654699	−	0.0654699i	−0.673614	−	0.739084i	$-0.735259\pi$
$947$	62.0000	−	62.0000i	0.0654699	−	0.0654699i	0.739084	+	0.673614i	$0.235259\pi$
$948$		0			0
$949$		−	282.000i		−	0.297155i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1017.00	+	1017.00i	1.06716	+	1.06716i	0.997576	+	0.0695800i	$0.0221659\pi$
$953$	1017.00	+	1017.00i	1.06716	+	1.06716i	0.0695800	+	0.997576i	$0.477834\pi$
$954$		0			0
$955$		−	1060.00i		−	1.10995i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			252.000i			0.262774i
$960$		0			0
$961$	1743.00			1.81374
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−285.000	+	285.000i	−0.295337	+	0.295337i
$966$		0			0
$967$	−502.000	+	502.000i	−0.519131	+	0.519131i	−0.917309	−	0.398177i	$-0.869643\pi$
$967$	−502.000	+	502.000i	−0.519131	+	0.519131i	0.398177	+	0.917309i	$0.369643\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	992.000			1.02163			0.510814	−	0.859691i	$-0.329344\pi$
$971$	992.000			1.02163			0.510814	+	0.859691i	$0.329344\pi$
$972$		0			0
$973$	−280.000	−	280.000i	−0.287770	−	0.287770i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	783.000	−	783.000i	0.801433	−	0.801433i	−0.181887	−	0.983320i	$-0.558220\pi$
$977$	783.000	−	783.000i	0.801433	−	0.801433i	0.983320	+	0.181887i	$0.0582204\pi$
$978$		0			0
$979$		−	640.000i		−	0.653728i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1058.00	+	1058.00i	1.07630	+	1.07630i	0.996838	+	0.0794589i	$0.0253192\pi$
$983$	1058.00	+	1058.00i	1.07630	+	1.07630i	0.0794589	+	0.996838i	$0.474681\pi$
$984$		0			0
$985$	−15.0000	−	15.0000i	−0.0152284	−	0.0152284i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	168.000i		−	0.169869i
$990$		0			0
$991$	68.0000			0.0686176			0.0343088	−	0.999411i	$-0.489077\pi$
$991$	68.0000			0.0686176			0.0343088	+	0.999411i	$0.489077\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−600.000			−0.603015
$996$		0			0
$997$	−773.000	+	773.000i	−0.775326	+	0.775326i	−0.979032	−	0.203706i	$-0.934701\pi$
$997$	−773.000	+	773.000i	−0.775326	+	0.775326i	0.203706	+	0.979032i	$0.434701\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.3.bh.c.433.1		2
3.2	odd	2		80.3.p.c.33.1		2
4.3	odd	2		90.3.g.b.73.1		2
5.2	odd	4	inner	720.3.bh.c.577.1		2
12.11	even	2		10.3.c.a.3.1	✓	2
15.2	even	4		80.3.p.c.17.1		2
15.8	even	4		400.3.p.b.257.1		2
15.14	odd	2		400.3.p.b.193.1		2
20.3	even	4		450.3.g.b.307.1		2
20.7	even	4		90.3.g.b.37.1		2
20.19	odd	2		450.3.g.b.343.1		2
24.5	odd	2		320.3.p.a.193.1		2
24.11	even	2		320.3.p.h.193.1		2
60.23	odd	4		50.3.c.c.7.1		2
60.47	odd	4		10.3.c.a.7.1	yes	2
60.59	even	2		50.3.c.c.43.1		2
84.83	odd	2		490.3.f.b.393.1		2
120.77	even	4		320.3.p.a.257.1		2
120.107	odd	4		320.3.p.h.257.1		2
420.167	even	4		490.3.f.b.197.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.3.c.a.3.1	✓	2	12.11	even	2
10.3.c.a.7.1	yes	2	60.47	odd	4
50.3.c.c.7.1		2	60.23	odd	4
50.3.c.c.43.1		2	60.59	even	2
80.3.p.c.17.1		2	15.2	even	4
80.3.p.c.33.1		2	3.2	odd	2
90.3.g.b.37.1		2	20.7	even	4
90.3.g.b.73.1		2	4.3	odd	2
320.3.p.a.193.1		2	24.5	odd	2
320.3.p.a.257.1		2	120.77	even	4
320.3.p.h.193.1		2	24.11	even	2
320.3.p.h.257.1		2	120.107	odd	4
400.3.p.b.193.1		2	15.14	odd	2
400.3.p.b.257.1		2	15.8	even	4
450.3.g.b.307.1		2	20.3	even	4
450.3.g.b.343.1		2	20.19	odd	2
490.3.f.b.197.1		2	420.167	even	4
490.3.f.b.393.1		2	84.83	odd	2
720.3.bh.c.433.1		2	1.1	even	1	trivial
720.3.bh.c.577.1		2	5.2	odd	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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)

\(f(q)\)	\(=\)	\(q-5.00000i q^{5} +(-2.00000 + 2.00000i) q^{7} +O(q^{10})\)	\(q-5.00000i q^{5} +(-2.00000 + 2.00000i) q^{7} -8.00000 q^{11} +(3.00000 + 3.00000i) q^{13} +(-7.00000 + 7.00000i) q^{17} -20.0000i q^{19} +(-2.00000 - 2.00000i) q^{23} -25.0000 q^{25} +40.0000i q^{29} -52.0000 q^{31} +(10.0000 + 10.0000i) q^{35} +(-3.00000 + 3.00000i) q^{37} +8.00000 q^{41} +(42.0000 + 42.0000i) q^{43} +(-18.0000 + 18.0000i) q^{47} +41.0000i q^{49} +(-53.0000 - 53.0000i) q^{53} +40.0000i q^{55} +20.0000i q^{59} -48.0000 q^{61} +(15.0000 - 15.0000i) q^{65} +(-62.0000 + 62.0000i) q^{67} -28.0000 q^{71} +(-47.0000 - 47.0000i) q^{73} +(16.0000 - 16.0000i) q^{77} +(18.0000 + 18.0000i) q^{83} +(35.0000 + 35.0000i) q^{85} +80.0000i q^{89} -12.0000 q^{91} -100.000 q^{95} +(-63.0000 + 63.0000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{7}+O(q^{10}) \) 2 * q - 4 * q^7	\( 2 q - 4 q^{7} - 16 q^{11} + 6 q^{13} - 14 q^{17} - 4 q^{23} - 50 q^{25} - 104 q^{31} + 20 q^{35} - 6 q^{37} + 16 q^{41} + 84 q^{43} - 36 q^{47} - 106 q^{53} - 96 q^{61} + 30 q^{65} - 124 q^{67} - 56 q^{71} - 94 q^{73} + 32 q^{77} + 36 q^{83} + 70 q^{85} - 24 q^{91} - 200 q^{95} - 126 q^{97}+O(q^{100}) \) 2 * q - 4 * q^7 - 16 * q^11 + 6 * q^13 - 14 * q^17 - 4 * q^23 - 50 * q^25 - 104 * q^31 + 20 * q^35 - 6 * q^37 + 16 * q^41 + 84 * q^43 - 36 * q^47 - 106 * q^53 - 96 * q^61 + 30 * q^65 - 124 * q^67 - 56 * q^71 - 94 * q^73 + 32 * q^77 + 36 * q^83 + 70 * q^85 - 24 * q^91 - 200 * q^95 - 126 * q^97

Introduction

L-functions

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