LMFDB - Embedded newform 20.4.e.a.3.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [20,4,Mod(3,20)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(20, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("20.3");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 20 = 2^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	20.e (of order $4$, degree $2$, 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:	$1.18003820011$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			3.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	20.3
Dual form			20.4.e.a.7.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.00000 - 2.00000i) q^{2} -8.00000i q^{4} +(-2.00000 + 11.0000i) q^{5} +(-16.0000 - 16.0000i) q^{8} +27.0000i q^{9} +O(q^{10})$	\(q+(2.00000 - 2.00000i) q^{2} -8.00000i q^{4} +(-2.00000 + 11.0000i) q^{5} +(-16.0000 - 16.0000i) q^{8} +27.0000i q^{9} +(18.0000 + 26.0000i) q^{10} +(-37.0000 - 37.0000i) q^{13} -64.0000 q^{16} +(99.0000 - 99.0000i) q^{17} +(54.0000 + 54.0000i) q^{18} +(88.0000 + 16.0000i) q^{20} +(-117.000 - 44.0000i) q^{25} -148.000 q^{26} +284.000i q^{29} +(-128.000 + 128.000i) q^{32} -396.000i q^{34} +216.000 q^{36} +(-91.0000 + 91.0000i) q^{37} +(208.000 - 144.000i) q^{40} +472.000 q^{41} +(-297.000 - 54.0000i) q^{45} -343.000i q^{49} +(-322.000 + 146.000i) q^{50} +(-296.000 + 296.000i) q^{52} +(-27.0000 - 27.0000i) q^{53} +(568.000 + 568.000i) q^{58} -468.000 q^{61} +512.000i q^{64} +(481.000 - 333.000i) q^{65} +(-792.000 - 792.000i) q^{68} +(432.000 - 432.000i) q^{72} +(253.000 + 253.000i) q^{73} +364.000i q^{74} +(128.000 - 704.000i) q^{80} -729.000 q^{81} +(944.000 - 944.000i) q^{82} +(891.000 + 1287.00i) q^{85} -176.000i q^{89} +(-702.000 + 486.000i) q^{90} +(-611.000 + 611.000i) q^{97} +(-686.000 - 686.000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} - 4 q^{5} - 32 q^{8}+O(q^{10}) $ 2 * q + 4 * q^2 - 4 * q^5 - 32 * q^8	$ 2 q + 4 q^{2} - 4 q^{5} - 32 q^{8} + 36 q^{10} - 74 q^{13} - 128 q^{16} + 198 q^{17} + 108 q^{18} + 176 q^{20} - 234 q^{25} - 296 q^{26} - 256 q^{32} + 432 q^{36} - 182 q^{37} + 416 q^{40} + 944 q^{41} - 594 q^{45} - 644 q^{50} - 592 q^{52} - 54 q^{53} + 1136 q^{58} - 936 q^{61} + 962 q^{65} - 1584 q^{68} + 864 q^{72} + 506 q^{73} + 256 q^{80} - 1458 q^{81} + 1888 q^{82} + 1782 q^{85} - 1404 q^{90} - 1222 q^{97} - 1372 q^{98}+O(q^{100}) $ 2 * q + 4 * q^2 - 4 * q^5 - 32 * q^8 + 36 * q^10 - 74 * q^13 - 128 * q^16 + 198 * q^17 + 108 * q^18 + 176 * q^20 - 234 * q^25 - 296 * q^26 - 256 * q^32 + 432 * q^36 - 182 * q^37 + 416 * q^40 + 944 * q^41 - 594 * q^45 - 644 * q^50 - 592 * q^52 - 54 * q^53 + 1136 * q^58 - 936 * q^61 + 962 * q^65 - 1584 * q^68 + 864 * q^72 + 506 * q^73 + 256 * q^80 - 1458 * q^81 + 1888 * q^82 + 1782 * q^85 - 1404 * q^90 - 1222 * q^97 - 1372 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/20\mathbb{Z}\right)^\times$.

$n$	$11$	$17$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	2.00000	−	2.00000i	0.707107	−	0.707107i
$2$	2.00000	−	2.00000i	0.707107	−	0.707107i
$3$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$3$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$4$		−	8.00000i		−	1.00000i
$5$	−2.00000	+	11.0000i	−0.178885	+	0.983870i
$5$	−2.00000	+	11.0000i	−0.178885	+	0.983870i
$6$		0			0
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$	−16.0000	−	16.0000i	−0.707107	−	0.707107i
$9$			27.0000i			1.00000i
$10$	18.0000	+	26.0000i	0.569210	+	0.822192i
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	−37.0000	−	37.0000i	−0.789381	−	0.789381i	0.192012	−	0.981393i	$-0.438499\pi$
$13$	−37.0000	−	37.0000i	−0.789381	−	0.789381i	−0.981393	+	0.192012i	$0.938499\pi$
$14$		0			0
$15$		0			0
$16$	−64.0000			−1.00000
$17$	99.0000	−	99.0000i	1.41241	−	1.41241i	0.670540	−	0.741874i	$-0.266063\pi$
$17$	99.0000	−	99.0000i	1.41241	−	1.41241i	0.741874	−	0.670540i	$-0.233937\pi$
$18$	54.0000	+	54.0000i	0.707107	+	0.707107i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	88.0000	+	16.0000i	0.983870	+	0.178885i
$21$		0			0
$22$		0			0
$23$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$23$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$24$		0			0
$25$	−117.000	−	44.0000i	−0.936000	−	0.352000i
$26$	−148.000			−1.11635
$27$		0			0
$28$		0			0
$29$			284.000i			1.81853i	0.416214	+	0.909267i	$0.363357\pi$
$29$			284.000i			1.81853i	−0.416214	+	0.909267i	$0.636643\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−128.000	+	128.000i	−0.707107	+	0.707107i
$33$		0			0
$34$		−	396.000i		−	1.99745i
$35$		0			0
$36$	216.000			1.00000
$37$	−91.0000	+	91.0000i	−0.404333	+	0.404333i	−0.879757	−	0.475424i	$-0.842295\pi$
$37$	−91.0000	+	91.0000i	−0.404333	+	0.404333i	0.475424	+	0.879757i	$0.342295\pi$
$38$		0			0
$39$		0			0
$40$	208.000	−	144.000i	0.822192	−	0.569210i
$41$	472.000			1.79790			0.898951	−	0.438048i	$-0.144330\pi$
$41$	472.000			1.79790			0.898951	+	0.438048i	$0.144330\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$	−297.000	−	54.0000i	−0.983870	−	0.178885i
$46$		0			0
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$		0			0
$49$		−	343.000i		−	1.00000i
$50$	−322.000	+	146.000i	−0.910754	+	0.412950i
$51$		0			0
$52$	−296.000	+	296.000i	−0.789381	+	0.789381i
$53$	−27.0000	−	27.0000i	−0.0699761	−	0.0699761i	0.671253	−	0.741229i	$-0.265757\pi$
$53$	−27.0000	−	27.0000i	−0.0699761	−	0.0699761i	−0.741229	+	0.671253i	$0.765757\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	568.000	+	568.000i	1.28590	+	1.28590i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	−468.000			−0.982316			−0.491158	−	0.871071i	$-0.663426\pi$
$61$	−468.000			−0.982316			−0.491158	+	0.871071i	$0.663426\pi$
$62$		0			0
$63$		0			0
$64$			512.000i			1.00000i
$65$	481.000	−	333.000i	0.917857	−	0.635439i
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$	−792.000	−	792.000i	−1.41241	−	1.41241i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	432.000	−	432.000i	0.707107	−	0.707107i
$73$	253.000	+	253.000i	0.405636	+	0.405636i	0.880214	−	0.474578i	$-0.157399\pi$
$73$	253.000	+	253.000i	0.405636	+	0.405636i	−0.474578	+	0.880214i	$0.657399\pi$
$74$			364.000i			0.571813i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	128.000	−	704.000i	0.178885	−	0.983870i
$81$	−729.000			−1.00000
$82$	944.000	−	944.000i	1.27131	−	1.27131i
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$	891.000	+	1287.00i	1.13697	+	1.64229i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	176.000i		−	0.209618i	−0.994492	−	0.104809i	$-0.966577\pi$
$89$		−	176.000i		−	0.209618i	0.994492	−	0.104809i	$-0.0334231\pi$
$90$	−702.000	+	486.000i	−0.822192	+	0.569210i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−611.000	+	611.000i	−0.639563	+	0.639563i	−0.950448	−	0.310884i	$-0.899375\pi$
$97$	−611.000	+	611.000i	−0.639563	+	0.639563i	0.310884	+	0.950448i	$0.399375\pi$
$98$	−686.000	−	686.000i	−0.707107	−	0.707107i
$99$		0			0
$100$	−352.000	+	936.000i	−0.352000	+	0.936000i
$101$	−598.000			−0.589141			−0.294570	−	0.955630i	$-0.595177\pi$
$101$	−598.000			−0.589141			−0.294570	+	0.955630i	$0.595177\pi$
$102$		0			0
$103$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$103$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$104$			1184.00i			1.11635i
$105$		0			0
$106$	−108.000			−0.0989612
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$		0			0
$109$		−	1746.00i		−	1.53428i	−0.641480	−	0.767140i	$-0.721679\pi$
$109$		−	1746.00i		−	1.53428i	0.641480	−	0.767140i	$-0.278321\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−337.000	−	337.000i	−0.280551	−	0.280551i	0.552778	−	0.833329i	$-0.313568\pi$
$113$	−337.000	−	337.000i	−0.280551	−	0.280551i	−0.833329	+	0.552778i	$0.813568\pi$
$114$		0			0
$115$		0			0
$116$	2272.00			1.81853
$117$	999.000	−	999.000i	0.789381	−	0.789381i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1331.00			1.00000
$122$	−936.000	+	936.000i	−0.694602	+	0.694602i
$123$		0			0
$124$		0			0
$125$	718.000	−	1199.00i	0.513759	−	0.857935i
$126$		0			0
$127$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$	1024.00	+	1024.00i	0.707107	+	0.707107i
$129$		0			0
$130$	296.000	−	1628.00i	0.199699	−	1.09835i
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−3168.00			−1.99745
$137$	−2191.00	+	2191.00i	−1.36635	+	1.36635i	−0.500766	+	0.865583i	$0.666948\pi$
$137$	−2191.00	+	2191.00i	−1.36635	+	1.36635i	−0.865583	+	0.500766i	$0.833052\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		−	1728.00i		−	1.00000i
$145$	−3124.00	−	568.000i	−1.78920	−	0.325309i
$146$	1012.00			0.573656
$147$		0			0
$148$	728.000	+	728.000i	0.404333	+	0.404333i
$149$			3514.00i			1.93207i	0.258415	+	0.966034i	$0.416800\pi$
$149$			3514.00i			1.93207i	−0.258415	+	0.966034i	$0.583200\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	2673.00	+	2673.00i	1.41241	+	1.41241i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	1819.00	−	1819.00i	0.924662	−	0.924662i	−0.0726920	−	0.997354i	$-0.523159\pi$
$157$	1819.00	−	1819.00i	0.924662	−	0.924662i	0.997354	+	0.0726920i	$0.0231590\pi$
$158$		0			0
$159$		0			0
$160$	−1152.00	−	1664.00i	−0.569210	−	0.822192i
$161$		0			0
$162$	−1458.00	+	1458.00i	−0.707107	+	0.707107i
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$		−	3776.00i		−	1.79790i
$165$		0			0
$166$		0			0
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$			541.000i			0.246245i
$170$	4356.00	+	792.000i	1.96523	+	0.357315i
$171$		0			0
$172$		0			0
$173$	−3047.00	−	3047.00i	−1.33907	−	1.33907i	−0.896962	−	0.442108i	$-0.854231\pi$
$173$	−3047.00	−	3047.00i	−1.33907	−	1.33907i	−0.442108	−	0.896962i	$-0.645769\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−352.000	−	352.000i	−0.148222	−	0.148222i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	−432.000	+	2376.00i	−0.178885	+	0.983870i
$181$	3942.00			1.61882			0.809410	−	0.587243i	$-0.199787\pi$
$181$	3942.00			1.61882			0.809410	+	0.587243i	$0.199787\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−819.000	−	1183.00i	−0.325482	−	0.470140i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	−2717.00	−	2717.00i	−1.01334	−	1.01334i	−0.999910	−	0.0134266i	$-0.995726\pi$
$193$	−2717.00	−	2717.00i	−1.01334	−	1.01334i	−0.0134266	−	0.999910i	$-0.504274\pi$
$194$			2444.00i			0.904479i
$195$		0			0
$196$	−2744.00			−1.00000
$197$	3289.00	−	3289.00i	1.18950	−	1.18950i	0.212295	−	0.977206i	$-0.431906\pi$
$197$	3289.00	−	3289.00i	1.18950	−	1.18950i	0.977206	−	0.212295i	$-0.0680936\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	1168.00	+	2576.00i	0.412950	+	0.910754i
$201$		0			0
$202$	−1196.00	+	1196.00i	−0.416585	+	0.416585i
$203$		0			0
$204$		0			0
$205$	−944.000	+	5192.00i	−0.321619	+	1.76890i
$206$		0			0
$207$		0			0
$208$	2368.00	+	2368.00i	0.789381	+	0.789381i
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	−216.000	+	216.000i	−0.0699761	+	0.0699761i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−3492.00	−	3492.00i	−1.08490	−	1.08490i
$219$		0			0
$220$		0			0
$221$	−7326.00			−2.22986
$222$		0			0
$223$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$223$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$224$		0			0
$225$	1188.00	−	3159.00i	0.352000	−	0.936000i
$226$	−1348.00			−0.396759
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$			2684.00i			0.774514i	0.921972	+	0.387257i	$0.126577\pi$
$229$			2684.00i			0.774514i	−0.921972	+	0.387257i	$0.873423\pi$
$230$		0			0
$231$		0			0
$232$	4544.00	−	4544.00i	1.28590	−	1.28590i
$233$	3843.00	+	3843.00i	1.08053	+	1.08053i	0.996460	+	0.0840693i	$0.0267917\pi$
$233$	3843.00	+	3843.00i	1.08053	+	1.08053i	0.0840693	+	0.996460i	$0.473208\pi$
$234$		−	3996.00i		−	1.11635i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	5272.00			1.40913			0.704563	−	0.709641i	$-0.251143\pi$
$241$	5272.00			1.40913			0.704563	+	0.709641i	$0.251143\pi$
$242$	2662.00	−	2662.00i	0.707107	−	0.707107i
$243$		0			0
$244$			3744.00i			0.982316i
$245$	3773.00	+	686.000i	0.983870	+	0.178885i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−962.000	−	3834.00i	−0.243369	−	0.969934i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	4096.00			1.00000
$257$	−3281.00	+	3281.00i	−0.796355	+	0.796355i	−0.982519	−	0.186164i	$-0.940394\pi$
$257$	−3281.00	+	3281.00i	−0.796355	+	0.796355i	0.186164	+	0.982519i	$0.440394\pi$
$258$		0			0
$259$		0			0
$260$	−2664.00	−	3848.00i	−0.635439	−	0.917857i
$261$	−7668.00			−1.81853
$262$		0			0
$263$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$263$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$264$		0			0
$265$	351.000	−	243.000i	0.0813651	−	0.0563297i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	3406.00i		−	0.771998i	−0.922499	−	0.385999i	$-0.873857\pi$
$269$		−	3406.00i		−	0.771998i	0.922499	−	0.385999i	$-0.126143\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−6336.00	+	6336.00i	−1.41241	+	1.41241i
$273$		0			0
$274$			8764.00i			1.93231i
$275$		0			0
$276$		0			0
$277$	−5221.00	+	5221.00i	−1.13249	+	1.13249i	−0.142727	+	0.989762i	$0.545587\pi$
$277$	−5221.00	+	5221.00i	−1.13249	+	1.13249i	−0.989762	+	0.142727i	$0.954413\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	5792.00			1.22961			0.614807	−	0.788677i	$-0.289234\pi$
$281$	5792.00			1.22961			0.614807	+	0.788677i	$0.289234\pi$
$282$		0			0
$283$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$283$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−3456.00	−	3456.00i	−0.707107	−	0.707107i
$289$		−	14689.0i		−	2.98982i
$290$	−7384.00	+	5112.00i	−1.49518	+	1.03513i
$291$		0			0
$292$	2024.00	−	2024.00i	0.405636	−	0.405636i
$293$	2983.00	+	2983.00i	0.594774	+	0.594774i	0.938917	−	0.344143i	$-0.111831\pi$
$293$	2983.00	+	2983.00i	0.594774	+	0.594774i	−0.344143	+	0.938917i	$0.611831\pi$
$294$		0			0
$295$		0			0
$296$	2912.00			0.571813
$297$		0			0
$298$	7028.00	+	7028.00i	1.36618	+	1.36618i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	936.000	−	5148.00i	0.175722	−	0.966471i
$306$	10692.0			1.99745
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	−937.000	−	937.000i	−0.169209	−	0.169209i	0.617423	−	0.786632i	$-0.288177\pi$
$313$	−937.000	−	937.000i	−0.169209	−	0.169209i	−0.786632	+	0.617423i	$0.788177\pi$
$314$		−	7276.00i		−	1.30767i
$315$		0			0
$316$		0			0
$317$	2799.00	−	2799.00i	0.495923	−	0.495923i	−0.414243	−	0.910166i	$-0.635954\pi$
$317$	2799.00	−	2799.00i	0.495923	−	0.495923i	0.910166	+	0.414243i	$0.135954\pi$
$318$		0			0
$319$		0			0
$320$	−5632.00	−	1024.00i	−0.983870	−	0.178885i
$321$		0			0
$322$		0			0
$323$		0			0
$324$			5832.00i			1.00000i
$325$	2701.00	+	5957.00i	0.460999	+	1.01672i
$326$		0			0
$327$		0			0
$328$	−7552.00	−	7552.00i	−1.27131	−	1.27131i
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	−2457.00	−	2457.00i	−0.404333	−	0.404333i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−6391.00	+	6391.00i	−1.03306	+	1.03306i	−0.0336216	+	0.999435i	$0.510704\pi$
$337$	−6391.00	+	6391.00i	−1.03306	+	1.03306i	−0.999435	+	0.0336216i	$0.989296\pi$
$338$	1082.00	+	1082.00i	0.174121	+	0.174121i
$339$		0			0
$340$	10296.0	−	7128.00i	1.64229	−	1.13697i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−12188.0			−1.89373
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$			8964.00i			1.37488i	0.726243	+	0.687438i	$0.241265\pi$
$349$			8964.00i			1.37488i	−0.726243	+	0.687438i	$0.758735\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	8073.00	+	8073.00i	1.21723	+	1.21723i	0.968598	+	0.248633i	$0.0799813\pi$
$353$	8073.00	+	8073.00i	1.21723	+	1.21723i	0.248633	+	0.968598i	$0.420019\pi$
$354$		0			0
$355$		0			0
$356$	−1408.00			−0.209618
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	3888.00	+	5616.00i	0.569210	+	0.822192i
$361$	−6859.00			−1.00000
$362$	7884.00	−	7884.00i	1.14468	−	1.14468i
$363$		0			0
$364$		0			0
$365$	−3289.00	+	2277.00i	−0.471655	+	0.326530i
$366$		0			0
$367$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$367$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$368$		0			0
$369$			12744.0i			1.79790i
$370$	−4004.00	−	728.000i	−0.562589	−	0.102289i
$371$		0			0
$372$		0			0
$373$	−9647.00	−	9647.00i	−1.33915	−	1.33915i	−0.896884	−	0.442265i	$-0.854175\pi$
$373$	−9647.00	−	9647.00i	−1.33915	−	1.33915i	−0.442265	−	0.896884i	$-0.645825\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	10508.0	−	10508.0i	1.43552	−	1.43552i
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$		0			0
$385$		0			0
$386$	−10868.0			−1.43307
$387$		0			0
$388$	4888.00	+	4888.00i	0.639563	+	0.639563i
$389$			374.000i			0.0487469i	0.999703	+	0.0243735i	$0.00775908\pi$
$389$			374.000i			0.0487469i	−0.999703	+	0.0243735i	$0.992241\pi$
$390$		0			0
$391$		0			0
$392$	−5488.00	+	5488.00i	−0.707107	+	0.707107i
$393$		0			0
$394$		−	13156.0i		−	1.68221i
$395$		0			0
$396$		0			0
$397$	11089.0	−	11089.0i	1.40187	−	1.40187i	0.607699	−	0.794168i	$-0.292093\pi$
$397$	11089.0	−	11089.0i	1.40187	−	1.40187i	0.794168	−	0.607699i	$-0.207907\pi$
$398$		0			0
$399$		0			0
$400$	7488.00	+	2816.00i	0.936000	+	0.352000i
$401$	−2398.00			−0.298629			−0.149315	−	0.988790i	$-0.547707\pi$
$401$	−2398.00			−0.298629			−0.149315	+	0.988790i	$0.547707\pi$
$402$		0			0
$403$		0			0
$404$			4784.00i			0.589141i
$405$	1458.00	−	8019.00i	0.178885	−	0.983870i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	7146.00i		−	0.863929i	−0.901891	−	0.431964i	$-0.857821\pi$
$409$		−	7146.00i		−	0.863929i	0.901891	−	0.431964i	$-0.142179\pi$
$410$	8496.00	+	12272.0i	1.02338	+	1.47822i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	9472.00			1.11635
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	13412.0			1.55264			0.776319	−	0.630340i	$-0.217084\pi$
$421$	13412.0			1.55264			0.776319	+	0.630340i	$0.217084\pi$
$422$		0			0
$423$		0			0
$424$			864.000i			0.0989612i
$425$	−15939.0	+	7227.00i	−1.81919	+	0.824849i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	−11107.0	−	11107.0i	−1.23272	−	1.23272i	−0.962914	−	0.269807i	$-0.913040\pi$
$433$	−11107.0	−	11107.0i	−1.23272	−	1.23272i	−0.269807	−	0.962914i	$-0.586960\pi$
$434$		0			0
$435$		0			0
$436$	−13968.0			−1.53428
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	9261.00			1.00000
$442$	−14652.0	+	14652.0i	−1.57675	+	1.57675i
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		0			0
$445$	1936.00	+	352.000i	0.206236	+	0.0374975i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			16114.0i			1.69369i	0.531840	+	0.846845i	$0.321501\pi$
$449$			16114.0i			1.69369i	−0.531840	+	0.846845i	$0.678499\pi$
$450$	−3942.00	−	8694.00i	−0.412950	−	0.910754i
$451$		0			0
$452$	−2696.00	+	2696.00i	−0.280551	+	0.280551i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−13481.0	+	13481.0i	−1.37990	+	1.37990i	−0.535132	+	0.844768i	$0.679738\pi$
$457$	−13481.0	+	13481.0i	−1.37990	+	1.37990i	−0.844768	+	0.535132i	$0.820262\pi$
$458$	5368.00	+	5368.00i	0.547664	+	0.547664i
$459$		0			0
$460$		0			0
$461$	−2318.00			−0.234187			−0.117093	−	0.993121i	$-0.537358\pi$
$461$	−2318.00			−0.234187			−0.117093	+	0.993121i	$0.537358\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		−	18176.0i		−	1.81853i
$465$		0			0
$466$	15372.0			1.52810
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$	−7992.00	−	7992.00i	−0.789381	−	0.789381i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	729.000	−	729.000i	0.0699761	−	0.0699761i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	6734.00			0.638345
$482$	10544.0	−	10544.0i	0.996403	−	0.996403i
$483$		0			0
$484$		−	10648.0i		−	1.00000i
$485$	−5499.00	−	7943.00i	−0.514839	−	0.743656i
$486$		0			0
$487$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$487$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$488$	7488.00	+	7488.00i	0.694602	+	0.694602i
$489$		0			0
$490$	8918.00	−	6174.00i	0.822192	−	0.569210i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	28116.0	+	28116.0i	2.56852	+	2.56852i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	−9592.00	−	5744.00i	−0.857935	−	0.513759i
$501$		0			0
$502$		0			0
$503$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$503$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$504$		0			0
$505$	1196.00	−	6578.00i	0.105389	−	0.579638i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	17996.0i		−	1.56711i	−0.621323	−	0.783555i	$-0.713404\pi$
$509$		−	17996.0i		−	1.56711i	0.621323	−	0.783555i	$-0.286596\pi$
$510$		0			0
$511$		0			0
$512$	8192.00	−	8192.00i	0.707107	−	0.707107i
$513$		0			0
$514$			13124.0i			1.12622i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−13024.0	−	2368.00i	−1.09835	−	0.199699i
$521$	−23738.0			−1.99612			−0.998062	−	0.0622265i	$-0.980180\pi$
$521$	−23738.0			−1.99612			−0.998062	+	0.0622265i	$0.980180\pi$
$522$	−15336.0	+	15336.0i	−1.28590	+	1.28590i
$523$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$523$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$			12167.0i			1.00000i
$530$	216.000	−	1188.00i	0.0177027	−	0.0973649i
$531$		0			0
$532$		0			0
$533$	−17464.0	−	17464.0i	−1.41923	−	1.41923i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−6812.00	−	6812.00i	−0.545885	−	0.545885i
$539$		0			0
$540$		0			0
$541$	5922.00			0.470622			0.235311	−	0.971920i	$-0.424389\pi$
$541$	5922.00			0.470622			0.235311	+	0.971920i	$0.424389\pi$
$542$		0			0
$543$		0			0
$544$			25344.0i			1.99745i
$545$	19206.0	+	3492.00i	1.50953	+	0.274460i
$546$		0			0
$547$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$547$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$548$	17528.0	+	17528.0i	1.36635	+	1.36635i
$549$		−	12636.0i		−	0.982316i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$			20884.0i			1.60158i
$555$		0			0
$556$		0			0
$557$	−16731.0	+	16731.0i	−1.27274	+	1.27274i	−0.328093	+	0.944646i	$0.606406\pi$
$557$	−16731.0	+	16731.0i	−1.27274	+	1.27274i	−0.944646	+	0.328093i	$0.893594\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	11584.0	−	11584.0i	0.869469	−	0.869469i
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$		0			0
$565$	4381.00	−	3033.00i	0.326212	−	0.225839i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	26806.0i		−	1.97498i	−0.157669	−	0.987492i	$-0.550398\pi$
$569$		−	26806.0i		−	1.97498i	0.157669	−	0.987492i	$-0.449602\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−13824.0			−1.00000
$577$	15479.0	−	15479.0i	1.11681	−	1.11681i	0.124603	−	0.992207i	$-0.460234\pi$
$577$	15479.0	−	15479.0i	1.11681	−	1.11681i	0.992207	−	0.124603i	$-0.0397657\pi$
$578$	−29378.0	−	29378.0i	−2.11412	−	2.11412i
$579$		0			0
$580$	−4544.00	+	24992.0i	−0.325309	+	1.78920i
$581$		0			0
$582$		0			0
$583$		0			0
$584$		−	8096.00i		−	0.573656i
$585$	8991.00	+	12987.0i	0.635439	+	0.917857i
$586$	11932.0			0.841137
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	5824.00	−	5824.00i	0.404333	−	0.404333i
$593$	4433.00	+	4433.00i	0.306984	+	0.306984i	0.843738	−	0.536755i	$-0.180350\pi$
$593$	4433.00	+	4433.00i	0.306984	+	0.306984i	−0.536755	+	0.843738i	$0.680350\pi$
$594$		0			0
$595$		0			0
$596$	28112.0			1.93207
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−24048.0			−1.63218			−0.816089	−	0.577927i	$-0.803862\pi$
$601$	−24048.0			−1.63218			−0.816089	+	0.577927i	$0.803862\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2662.00	+	14641.0i	−0.178885	+	0.983870i
$606$		0			0
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$		0			0
$609$		0			0
$610$	−8424.00	−	12168.0i	−0.559144	−	0.807652i
$611$		0			0
$612$	21384.0	−	21384.0i	1.41241	−	1.41241i
$613$	−1837.00	−	1837.00i	−0.121037	−	0.121037i	0.643994	−	0.765031i	$-0.277276\pi$
$613$	−1837.00	−	1837.00i	−0.121037	−	0.121037i	−0.765031	+	0.643994i	$0.777276\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	5499.00	−	5499.00i	0.358803	−	0.358803i	−0.504569	−	0.863372i	$-0.668348\pi$
$617$	5499.00	−	5499.00i	0.358803	−	0.358803i	0.863372	+	0.504569i	$0.168348\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	11753.0	+	10296.0i	0.752192	+	0.658944i
$626$	−3748.00			−0.239297
$627$		0			0
$628$	−14552.0	−	14552.0i	−0.924662	−	0.924662i
$629$			18018.0i			1.14217i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		−	11196.0i		−	0.701341i
$635$		0			0
$636$		0			0
$637$	−12691.0	+	12691.0i	−0.789381	+	0.789381i
$638$		0			0
$639$		0			0
$640$	−13312.0	+	9216.00i	−0.822192	+	0.569210i
$641$	14872.0			0.916394			0.458197	−	0.888851i	$-0.348495\pi$
$641$	14872.0			0.916394			0.458197	+	0.888851i	$0.348495\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$647$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$648$	11664.0	+	11664.0i	0.707107	+	0.707107i
$649$		0			0
$650$	17316.0	+	6512.00i	1.04491	+	0.392956i
$651$		0			0
$652$		0			0
$653$	16173.0	+	16173.0i	0.969217	+	0.969217i	0.999540	−	0.0303236i	$-0.00965378\pi$
$653$	16173.0	+	16173.0i	0.969217	+	0.969217i	−0.0303236	+	0.999540i	$0.509654\pi$
$654$		0			0
$655$		0			0
$656$	−30208.0			−1.79790
$657$	−6831.00	+	6831.00i	−0.405636	+	0.405636i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−22068.0			−1.29856			−0.649278	−	0.760551i	$-0.724929\pi$
$661$	−22068.0			−1.29856			−0.649278	+	0.760551i	$0.724929\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−9828.00			−0.571813
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−19547.0	−	19547.0i	−1.11959	−	1.11959i	−0.991802	−	0.127784i	$-0.959214\pi$
$673$	−19547.0	−	19547.0i	−1.11959	−	1.11959i	−0.127784	−	0.991802i	$-0.540786\pi$
$674$			25564.0i			1.46096i
$675$		0			0
$676$	4328.00			0.246245
$677$	−15471.0	+	15471.0i	−0.878285	+	0.878285i	−0.993357	−	0.115072i	$-0.963290\pi$
$677$	−15471.0	+	15471.0i	−0.878285	+	0.878285i	0.115072	+	0.993357i	$0.463290\pi$
$678$		0			0
$679$		0			0
$680$	6336.00	−	34848.0i	0.357315	−	1.96523i
$681$		0			0
$682$		0			0
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$		0			0
$685$	−19719.0	−	28483.0i	−1.09989	−	1.58873i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			1998.00i			0.110476i
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$	−24376.0	+	24376.0i	−1.33907	+	1.33907i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	46728.0	−	46728.0i	2.53938	−	2.53938i
$698$	17928.0	+	17928.0i	0.972185	+	0.972185i
$699$		0			0
$700$		0			0
$701$	31252.0			1.68384			0.841920	−	0.539602i	$-0.181425\pi$
$701$	31252.0			1.68384			0.841920	+	0.539602i	$0.181425\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	32292.0			1.72142
$707$		0			0
$708$		0			0
$709$			8404.00i			0.445161i	0.974914	+	0.222580i	$0.0714479\pi$
$709$			8404.00i			0.445161i	−0.974914	+	0.222580i	$0.928552\pi$
$710$		0			0
$711$		0			0
$712$	−2816.00	+	2816.00i	−0.148222	+	0.148222i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	19008.0	+	3456.00i	0.983870	+	0.178885i
$721$		0			0
$722$	−13718.0	+	13718.0i	−0.707107	+	0.707107i
$723$		0			0
$724$		−	31536.0i		−	1.61882i
$725$	12496.0	−	33228.0i	0.640124	−	1.70215i
$726$		0			0
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$		−	19683.0i		−	1.00000i
$730$	−2024.00	+	11132.0i	−0.102619	+	0.564402i
$731$		0			0
$732$		0			0
$733$	14993.0	+	14993.0i	0.755497	+	0.755497i	0.975499	−	0.220003i	$-0.0706066\pi$
$733$	14993.0	+	14993.0i	0.755497	+	0.755497i	−0.220003	+	0.975499i	$0.570607\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	25488.0	+	25488.0i	1.27131	+	1.27131i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	−9464.00	+	6552.00i	−0.470140	+	0.325482i
$741$		0			0
$742$		0			0
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$		0			0
$745$	−38654.0	−	7028.00i	−1.90090	−	0.345619i
$746$	−38588.0			−1.89384
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$		−	42032.0i		−	2.03013i
$755$		0			0
$756$		0			0
$757$	−28781.0	+	28781.0i	−1.38185	+	1.38185i	−0.540527	+	0.841327i	$0.681775\pi$
$757$	−28781.0	+	28781.0i	−1.38185	+	1.38185i	−0.841327	+	0.540527i	$0.818225\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	31882.0			1.51869			0.759344	−	0.650689i	$-0.225520\pi$
$761$	31882.0			1.51869			0.759344	+	0.650689i	$0.225520\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−34749.0	+	24057.0i	−1.64229	+	1.13697i
$766$		0			0
$767$		0			0
$768$		0			0
$769$			41544.0i			1.94813i	0.226260	+	0.974067i	$0.427350\pi$
$769$			41544.0i			1.94813i	−0.226260	+	0.974067i	$0.572650\pi$
$770$		0			0
$771$		0			0
$772$	−21736.0	+	21736.0i	−1.01334	+	1.01334i
$773$	−28197.0	−	28197.0i	−1.31200	−	1.31200i	−0.919940	−	0.392060i	$-0.871763\pi$
$773$	−28197.0	−	28197.0i	−1.31200	−	1.31200i	−0.392060	−	0.919940i	$-0.628237\pi$
$774$		0			0
$775$		0			0
$776$	19552.0			0.904479
$777$		0			0
$778$	748.000	+	748.000i	0.0344693	+	0.0344693i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$			21952.0i			1.00000i
$785$	16371.0	+	23647.0i	0.744339	+	1.07516i
$786$		0			0
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$	−26312.0	−	26312.0i	−1.18950	−	1.18950i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	17316.0	+	17316.0i	0.775421	+	0.775421i
$794$		−	44356.0i		−	1.98254i
$795$		0			0
$796$		0			0
$797$	12839.0	−	12839.0i	0.570616	−	0.570616i	−0.361685	−	0.932300i	$-0.617798\pi$
$797$	12839.0	−	12839.0i	0.570616	−	0.570616i	0.932300	+	0.361685i	$0.117798\pi$
$798$		0			0
$799$		0			0
$800$	20608.0	−	9344.00i	0.910754	−	0.412950i
$801$	4752.00			0.209618
$802$	−4796.00	+	4796.00i	−0.211163	+	0.211163i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	9568.00	+	9568.00i	0.416585	+	0.416585i
$809$			39704.0i			1.72549i	0.505643	+	0.862743i	$0.331255\pi$
$809$			39704.0i			1.72549i	−0.505643	+	0.862743i	$0.668745\pi$
$810$	−13122.0	−	18954.0i	−0.569210	−	0.822192i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−14292.0	−	14292.0i	−0.610890	−	0.610890i
$819$		0			0
$820$	41536.0	+	7552.00i	1.76890	+	0.321619i
$821$	47012.0			1.99845			0.999227	−	0.0393212i	$-0.0125196\pi$
$821$	47012.0			1.99845			0.999227	+	0.0393212i	$0.0125196\pi$
$822$		0			0
$823$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$823$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$		−	23166.0i		−	0.970553i	−0.874361	−	0.485276i	$-0.838719\pi$
$829$		−	23166.0i		−	0.970553i	0.874361	−	0.485276i	$-0.161281\pi$
$830$		0			0
$831$		0			0
$832$	18944.0	−	18944.0i	0.789381	−	0.789381i
$833$	−33957.0	−	33957.0i	−1.41241	−	1.41241i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−56267.0			−2.30706
$842$	26824.0	−	26824.0i	1.09788	−	1.09788i
$843$		0			0
$844$		0			0
$845$	−5951.00	−	1082.00i	−0.242273	−	0.0440496i
$846$		0			0
$847$		0			0
$848$	1728.00	+	1728.00i	0.0699761	+	0.0699761i
$849$		0			0
$850$	−17424.0	+	46332.0i	−0.703104	+	1.86962i
$851$		0			0
$852$		0			0
$853$	32923.0	+	32923.0i	1.32153	+	1.32153i	0.912541	+	0.408986i	$0.134117\pi$
$853$	32923.0	+	32923.0i	1.32153	+	1.32153i	0.408986	+	0.912541i	$0.365883\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	12969.0	−	12969.0i	0.516934	−	0.516934i	−0.399708	−	0.916642i	$-0.630889\pi$
$857$	12969.0	−	12969.0i	0.516934	−	0.516934i	0.916642	+	0.399708i	$0.130889\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$	39611.0	−	27423.0i	1.55701	−	1.07793i
$866$	−44428.0			−1.74333
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−27936.0	+	27936.0i	−1.08490	+	1.08490i
$873$	−16497.0	−	16497.0i	−0.639563	−	0.639563i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	36179.0	−	36179.0i	1.39302	−	1.39302i	0.574550	−	0.818470i	$-0.305177\pi$
$877$	36179.0	−	36179.0i	1.39302	−	1.39302i	0.818470	−	0.574550i	$-0.194823\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−51808.0			−1.98122			−0.990611	−	0.136714i	$-0.956346\pi$
$881$	−51808.0			−1.98122			−0.990611	+	0.136714i	$0.956346\pi$
$882$	18522.0	−	18522.0i	0.707107	−	0.707107i
$883$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$883$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$884$			58608.0i			2.22986i
$885$		0			0
$886$		0			0
$887$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$887$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$888$		0			0
$889$		0			0
$890$	4576.00	−	3168.00i	0.172346	−	0.119316i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	32228.0	+	32228.0i	1.19762	+	1.19762i
$899$		0			0
$900$	−25272.0	−	9504.00i	−0.936000	−	0.352000i
$901$	−5346.00			−0.197670
$902$		0			0
$903$		0			0
$904$			10784.0i			0.396759i
$905$	−7884.00	+	43362.0i	−0.289583	+	1.59271i
$906$		0			0
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$		−	16146.0i		−	0.589141i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$			53924.0i			1.95147i
$915$		0			0
$916$	21472.0			0.774514
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$	−4636.00	+	4636.00i	−0.165595	+	0.165595i
$923$		0			0
$924$		0			0
$925$	14651.0	−	6643.00i	0.520780	−	0.236130i
$926$		0			0
$927$		0			0
$928$	−36352.0	−	36352.0i	−1.28590	−	1.28590i
$929$		−	30866.0i		−	1.09008i	−0.838411	−	0.545038i	$-0.816515\pi$
$929$		−	30866.0i		−	1.09008i	0.838411	−	0.545038i	$-0.183485\pi$
$930$		0			0
$931$		0			0
$932$	30744.0	−	30744.0i	1.08053	−	1.08053i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−31968.0			−1.11635
$937$	−38141.0	+	38141.0i	−1.32979	+	1.32979i	−0.424238	+	0.905551i	$0.639458\pi$
$937$	−38141.0	+	38141.0i	−1.32979	+	1.32979i	−0.905551	+	0.424238i	$0.860542\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−31378.0			−1.08703			−0.543514	−	0.839400i	$-0.682907\pi$
$941$	−31378.0			−1.08703			−0.543514	+	0.839400i	$0.682907\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$		0			0
$949$		−	18722.0i		−	0.640402i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	20623.0	+	20623.0i	0.700991	+	0.700991i	0.964623	−	0.263632i	$-0.0849205\pi$
$953$	20623.0	+	20623.0i	0.700991	+	0.700991i	−0.263632	+	0.964623i	$0.584921\pi$
$954$		−	2916.00i		−	0.0989612i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	29791.0			1.00000
$962$	13468.0	−	13468.0i	0.451378	−	0.451378i
$963$		0			0
$964$		−	42176.0i		−	1.40913i
$965$	35321.0	−	24453.0i	1.17826	−	0.815720i
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$	−21296.0	−	21296.0i	−0.707107	−	0.707107i
$969$		0			0
$970$	−26884.0	−	4888.00i	−0.889890	−	0.161798i
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	29952.0			0.982316
$977$	−39771.0	+	39771.0i	−1.30234	+	1.30234i	−0.375531	+	0.926810i	$0.622540\pi$
$977$	−39771.0	+	39771.0i	−1.30234	+	1.30234i	−0.926810	+	0.375531i	$0.877460\pi$
$978$		0			0
$979$		0			0
$980$	5488.00	−	30184.0i	0.178885	−	0.983870i
$981$	47142.0			1.53428
$982$		0			0
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$		0			0
$985$	29601.0	+	42757.0i	0.957529	+	1.38310i
$986$	112464.			3.63244
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−9361.00	+	9361.00i	−0.297358	+	0.297358i	−0.839978	−	0.542620i	$-0.817432\pi$
$997$	−9361.00	+	9361.00i	−0.297358	+	0.297358i	0.542620	+	0.839978i	$0.317432\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.4.e.a.3.1	✓	2
3.2	odd	2		180.4.k.a.163.1		2
4.3	odd	2	CM	20.4.e.a.3.1	✓	2
5.2	odd	4	inner	20.4.e.a.7.1	yes	2
5.3	odd	4		100.4.e.a.7.1		2
5.4	even	2		100.4.e.a.43.1		2
8.3	odd	2		320.4.n.c.63.1		2
8.5	even	2		320.4.n.c.63.1		2
12.11	even	2		180.4.k.a.163.1		2
15.2	even	4		180.4.k.a.127.1		2
20.3	even	4		100.4.e.a.7.1		2
20.7	even	4	inner	20.4.e.a.7.1	yes	2
20.19	odd	2		100.4.e.a.43.1		2
40.27	even	4		320.4.n.c.127.1		2
40.37	odd	4		320.4.n.c.127.1		2
60.47	odd	4		180.4.k.a.127.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.4.e.a.3.1	✓	2	1.1	even	1	trivial
20.4.e.a.3.1	✓	2	4.3	odd	2	CM
20.4.e.a.7.1	yes	2	5.2	odd	4	inner
20.4.e.a.7.1	yes	2	20.7	even	4	inner
100.4.e.a.7.1		2	5.3	odd	4
100.4.e.a.7.1		2	20.3	even	4
100.4.e.a.43.1		2	5.4	even	2
100.4.e.a.43.1		2	20.19	odd	2
180.4.k.a.127.1		2	15.2	even	4
180.4.k.a.127.1		2	60.47	odd	4
180.4.k.a.163.1		2	3.2	odd	2
180.4.k.a.163.1		2	12.11	even	2
320.4.n.c.63.1		2	8.3	odd	2
320.4.n.c.63.1		2	8.5	even	2
320.4.n.c.127.1		2	40.27	even	4
320.4.n.c.127.1		2	40.37	odd	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:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	20.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.00000 - 2.00000i) q^{2} -8.00000i q^{4} +(-2.00000 + 11.0000i) q^{5} +(-16.0000 - 16.0000i) q^{8} +27.0000i q^{9} +O(q^{10})\)	\(q+(2.00000 - 2.00000i) q^{2} -8.00000i q^{4} +(-2.00000 + 11.0000i) q^{5} +(-16.0000 - 16.0000i) q^{8} +27.0000i q^{9} +(18.0000 + 26.0000i) q^{10} +(-37.0000 - 37.0000i) q^{13} -64.0000 q^{16} +(99.0000 - 99.0000i) q^{17} +(54.0000 + 54.0000i) q^{18} +(88.0000 + 16.0000i) q^{20} +(-117.000 - 44.0000i) q^{25} -148.000 q^{26} +284.000i q^{29} +(-128.000 + 128.000i) q^{32} -396.000i q^{34} +216.000 q^{36} +(-91.0000 + 91.0000i) q^{37} +(208.000 - 144.000i) q^{40} +472.000 q^{41} +(-297.000 - 54.0000i) q^{45} -343.000i q^{49} +(-322.000 + 146.000i) q^{50} +(-296.000 + 296.000i) q^{52} +(-27.0000 - 27.0000i) q^{53} +(568.000 + 568.000i) q^{58} -468.000 q^{61} +512.000i q^{64} +(481.000 - 333.000i) q^{65} +(-792.000 - 792.000i) q^{68} +(432.000 - 432.000i) q^{72} +(253.000 + 253.000i) q^{73} +364.000i q^{74} +(128.000 - 704.000i) q^{80} -729.000 q^{81} +(944.000 - 944.000i) q^{82} +(891.000 + 1287.00i) q^{85} -176.000i q^{89} +(-702.000 + 486.000i) q^{90} +(-611.000 + 611.000i) q^{97} +(-686.000 - 686.000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} - 4 q^{5} - 32 q^{8}+O(q^{10}) \) 2 * q + 4 * q^2 - 4 * q^5 - 32 * q^8	\( 2 q + 4 q^{2} - 4 q^{5} - 32 q^{8} + 36 q^{10} - 74 q^{13} - 128 q^{16} + 198 q^{17} + 108 q^{18} + 176 q^{20} - 234 q^{25} - 296 q^{26} - 256 q^{32} + 432 q^{36} - 182 q^{37} + 416 q^{40} + 944 q^{41} - 594 q^{45} - 644 q^{50} - 592 q^{52} - 54 q^{53} + 1136 q^{58} - 936 q^{61} + 962 q^{65} - 1584 q^{68} + 864 q^{72} + 506 q^{73} + 256 q^{80} - 1458 q^{81} + 1888 q^{82} + 1782 q^{85} - 1404 q^{90} - 1222 q^{97} - 1372 q^{98}+O(q^{100}) \) 2 * q + 4 * q^2 - 4 * q^5 - 32 * q^8 + 36 * q^10 - 74 * q^13 - 128 * q^16 + 198 * q^17 + 108 * q^18 + 176 * q^20 - 234 * q^25 - 296 * q^26 - 256 * q^32 + 432 * q^36 - 182 * q^37 + 416 * q^40 + 944 * q^41 - 594 * q^45 - 644 * q^50 - 592 * q^52 - 54 * q^53 + 1136 * q^58 - 936 * q^61 + 962 * q^65 - 1584 * q^68 + 864 * q^72 + 506 * q^73 + 256 * q^80 - 1458 * q^81 + 1888 * q^82 + 1782 * q^85 - 1404 * q^90 - 1222 * q^97 - 1372 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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