LMFDB - Embedded newform 20.6.e.a.3.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [20,6,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, 6, names="a")

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

chi := DirichletCharacter("20.3");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 20 = 2^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
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:	$3.20767639626$
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			20.6.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)

$f(q)$	$=$	$q+(4.00000 - 4.00000i) q^{2} -32.0000i q^{4} +(38.0000 - 41.0000i) q^{5} +(-128.000 - 128.000i) q^{8} +243.000i q^{9} +O(q^{10})$	\(q+(4.00000 - 4.00000i) q^{2} -32.0000i q^{4} +(38.0000 - 41.0000i) q^{5} +(-128.000 - 128.000i) q^{8} +243.000i q^{9} +(-12.0000 - 316.000i) q^{10} +(719.000 + 719.000i) q^{13} -1024.00 q^{16} +(-717.000 + 717.000i) q^{17} +(972.000 + 972.000i) q^{18} +(-1312.00 - 1216.00i) q^{20} +(-237.000 - 3116.00i) q^{25} +5752.00 q^{26} -8564.00i q^{29} +(-4096.00 + 4096.00i) q^{32} +5736.00i q^{34} +7776.00 q^{36} +(-11767.0 + 11767.0i) q^{37} +(-10112.0 + 384.000i) q^{40} +4952.00 q^{41} +(9963.00 + 9234.00i) q^{45} -16807.0i q^{49} +(-13412.0 - 11516.0i) q^{50} +(23008.0 - 23008.0i) q^{52} +(23769.0 + 23769.0i) q^{53} +(-34256.0 - 34256.0i) q^{58} -54948.0 q^{61} +32768.0i q^{64} +(56801.0 - 2157.00i) q^{65} +(22944.0 + 22944.0i) q^{68} +(31104.0 - 31104.0i) q^{72} +(-34331.0 - 34331.0i) q^{73} +94136.0i q^{74} +(-38912.0 + 41984.0i) q^{80} -59049.0 q^{81} +(19808.0 - 19808.0i) q^{82} +(2151.00 + 56643.0i) q^{85} -140464. i q^{89} +(76788.0 - 2916.00i) q^{90} +(34333.0 - 34333.0i) q^{97} +(-67228.0 - 67228.0i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{2} + 76 q^{5} - 256 q^{8}+O(q^{10}) $ 2 * q + 8 * q^2 + 76 * q^5 - 256 * q^8	$ 2 q + 8 q^{2} + 76 q^{5} - 256 q^{8} - 24 q^{10} + 1438 q^{13} - 2048 q^{16} - 1434 q^{17} + 1944 q^{18} - 2624 q^{20} - 474 q^{25} + 11504 q^{26} - 8192 q^{32} + 15552 q^{36} - 23534 q^{37} - 20224 q^{40} + 9904 q^{41} + 19926 q^{45} - 26824 q^{50} + 46016 q^{52} + 47538 q^{53} - 68512 q^{58} - 109896 q^{61} + 113602 q^{65} + 45888 q^{68} + 62208 q^{72} - 68662 q^{73} - 77824 q^{80} - 118098 q^{81} + 39616 q^{82} + 4302 q^{85} + 153576 q^{90} + 68666 q^{97} - 134456 q^{98}+O(q^{100}) $ 2 * q + 8 * q^2 + 76 * q^5 - 256 * q^8 - 24 * q^10 + 1438 * q^13 - 2048 * q^16 - 1434 * q^17 + 1944 * q^18 - 2624 * q^20 - 474 * q^25 + 11504 * q^26 - 8192 * q^32 + 15552 * q^36 - 23534 * q^37 - 20224 * q^40 + 9904 * q^41 + 19926 * q^45 - 26824 * q^50 + 46016 * q^52 + 47538 * q^53 - 68512 * q^58 - 109896 * q^61 + 113602 * q^65 + 45888 * q^68 + 62208 * q^72 - 68662 * q^73 - 77824 * q^80 - 118098 * q^81 + 39616 * q^82 + 4302 * q^85 + 153576 * q^90 + 68666 * q^97 - 134456 * 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 $

$2$	4.00000	−	4.00000i	0.707107	−	0.707107i
$2$	4.00000	−	4.00000i	0.707107	−	0.707107i
$3$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$3$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$4$		−	32.0000i		−	1.00000i
$5$	38.0000	−	41.0000i	0.679765	−	0.733430i
$5$	38.0000	−	41.0000i	0.679765	−	0.733430i
$6$		0			0
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$	−128.000	−	128.000i	−0.707107	−	0.707107i
$9$			243.000i			1.00000i
$10$	−12.0000	−	316.000i	−0.0379473	−	0.999280i
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	719.000	+	719.000i	1.17997	+	1.17997i	0.979752	+	0.200217i	$0.0641648\pi$
$13$	719.000	+	719.000i	1.17997	+	1.17997i	0.200217	+	0.979752i	$0.435835\pi$
$14$		0			0
$15$		0			0
$16$	−1024.00			−1.00000
$17$	−717.000	+	717.000i	−0.601723	+	0.601723i	−0.940770	−	0.339046i	$-0.889896\pi$
$17$	−717.000	+	717.000i	−0.601723	+	0.601723i	0.339046	+	0.940770i	$0.389896\pi$
$18$	972.000	+	972.000i	0.707107	+	0.707107i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	−1312.00	−	1216.00i	−0.733430	−	0.679765i
$21$		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
$25$	−237.000	−	3116.00i	−0.0758400	−	0.997120i
$26$	5752.00			1.66873
$27$		0			0
$28$		0			0
$29$		−	8564.00i		−	1.89096i	−0.325684	−	0.945479i	$-0.605595\pi$
$29$		−	8564.00i		−	1.89096i	0.325684	−	0.945479i	$-0.394405\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−4096.00	+	4096.00i	−0.707107	+	0.707107i
$33$		0			0
$34$			5736.00i			0.850965i
$35$		0			0
$36$	7776.00			1.00000
$37$	−11767.0	+	11767.0i	−1.41306	+	1.41306i	−0.678011	+	0.735052i	$0.737158\pi$
$37$	−11767.0	+	11767.0i	−1.41306	+	1.41306i	−0.735052	+	0.678011i	$0.762842\pi$
$38$		0			0
$39$		0			0
$40$	−10112.0	+	384.000i	−0.999280	+	0.0379473i
$41$	4952.00			0.460067			0.230033	−	0.973183i	$-0.426116\pi$
$41$	4952.00			0.460067			0.230033	+	0.973183i	$0.426116\pi$
$42$		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
$45$	9963.00	+	9234.00i	0.733430	+	0.679765i
$46$		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
$49$		−	16807.0i		−	1.00000i
$50$	−13412.0	−	11516.0i	−0.758697	−	0.651443i
$51$		0			0
$52$	23008.0	−	23008.0i	1.17997	−	1.17997i
$53$	23769.0	+	23769.0i	1.16231	+	1.16231i	0.983969	+	0.178339i	$0.0570723\pi$
$53$	23769.0	+	23769.0i	1.16231	+	1.16231i	0.178339	+	0.983969i	$0.442928\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−34256.0	−	34256.0i	−1.33711	−	1.33711i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	−54948.0			−1.89072			−0.945360	−	0.326028i	$-0.894290\pi$
$61$	−54948.0			−1.89072			−0.945360	+	0.326028i	$0.894290\pi$
$62$		0			0
$63$		0			0
$64$			32768.0i			1.00000i
$65$	56801.0	−	2157.00i	1.66753	−	0.0633238i
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$	22944.0	+	22944.0i	0.601723	+	0.601723i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	31104.0	−	31104.0i	0.707107	−	0.707107i
$73$	−34331.0	−	34331.0i	−0.754014	−	0.754014i	0.221212	−	0.975226i	$-0.428999\pi$
$73$	−34331.0	−	34331.0i	−0.754014	−	0.754014i	−0.975226	+	0.221212i	$0.928999\pi$
$74$			94136.0i			1.99837i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	−38912.0	+	41984.0i	−0.679765	+	0.733430i
$81$	−59049.0			−1.00000
$82$	19808.0	−	19808.0i	0.325316	−	0.325316i
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$	2151.00	+	56643.0i	0.0322919	+	0.850352i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	140464.i		−	1.87971i	−0.341579	−	0.939853i	$-0.610962\pi$
$89$		−	140464.i		−	1.87971i	0.341579	−	0.939853i	$-0.389038\pi$
$90$	76788.0	−	2916.00i	0.999280	−	0.0379473i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	34333.0	−	34333.0i	0.370495	−	0.370495i	−0.497162	−	0.867657i	$-0.665625\pi$
$97$	34333.0	−	34333.0i	0.370495	−	0.370495i	0.867657	+	0.497162i	$0.165625\pi$
$98$	−67228.0	−	67228.0i	−0.707107	−	0.707107i
$99$		0			0
$100$	−99712.0	+	7584.00i	−0.997120	+	0.0758400i
$101$	98002.0			0.955942			0.477971	−	0.878376i	$-0.341372\pi$
$101$	98002.0			0.955942			0.477971	+	0.878376i	$0.341372\pi$
$102$		0			0
$103$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$103$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$104$		−	184064.i		−	1.66873i
$105$		0			0
$106$	190152.			1.64375
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$		0			0
$109$			246486.i			1.98713i	0.113269	+	0.993564i	$0.463868\pi$
$109$			246486.i			1.98713i	−0.113269	+	0.993564i	$0.536132\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	62719.0	+	62719.0i	0.462065	+	0.462065i	0.899332	−	0.437267i	$-0.144053\pi$
$113$	62719.0	+	62719.0i	0.462065	+	0.462065i	−0.437267	+	0.899332i	$0.644053\pi$
$114$		0			0
$115$		0			0
$116$	−274048.			−1.89096
$117$	−174717.	+	174717.i	−1.17997	+	1.17997i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	161051.			1.00000
$122$	−219792.	+	219792.i	−1.33694	+	1.33694i
$123$		0			0
$124$		0			0
$125$	−136762.	−	108691.i	−0.782871	−	0.622184i
$126$		0			0
$127$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$	131072.	+	131072.i	0.707107	+	0.707107i
$129$		0			0
$130$	218576.	−	235832.i	1.13434	−	1.22390i
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	183552.			0.850965
$137$	254233.	−	254233.i	1.15726	−	1.15726i	0.172196	−	0.985063i	$-0.444914\pi$
$137$	254233.	−	254233.i	1.15726	−	1.15726i	0.985063	−	0.172196i	$-0.0550863\pi$
$138$		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
$144$		−	248832.i		−	1.00000i
$145$	−351124.	−	325432.i	−1.38689	−	1.28541i
$146$	−274648.			−1.06634
$147$		0			0
$148$	376544.	+	376544.i	1.41306	+	1.41306i
$149$		−	47614.0i		−	0.175699i	−0.996134	−	0.0878494i	$-0.972001\pi$
$149$		−	47614.0i		−	0.175699i	0.996134	−	0.0878494i	$-0.0279994\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	−174231.	−	174231.i	−0.601723	−	0.601723i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	61183.0	−	61183.0i	0.198099	−	0.198099i	−0.601086	−	0.799185i	$-0.705265\pi$
$157$	61183.0	−	61183.0i	0.198099	−	0.198099i	0.799185	+	0.601086i	$0.205265\pi$
$158$		0			0
$159$		0			0
$160$	12288.0	+	323584.i	0.0379473	+	0.999280i
$161$		0			0
$162$	−236196.	+	236196.i	−0.707107	+	0.707107i
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$		−	158464.i		−	0.460067i
$165$		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
$169$			662629.i			1.78465i
$170$	235176.	+	217968.i	0.624124	+	0.578456i
$171$		0			0
$172$		0			0
$173$	−12331.0	−	12331.0i	−0.0313244	−	0.0313244i	0.691271	−	0.722596i	$-0.257051\pi$
$173$	−12331.0	−	12331.0i	−0.0313244	−	0.0313244i	−0.722596	+	0.691271i	$0.757051\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−561856.	−	561856.i	−1.32915	−	1.32915i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	295488.	−	318816.i	0.679765	−	0.733430i
$181$	439902.			0.998067			0.499033	−	0.866583i	$-0.333689\pi$
$181$	439902.			0.998067			0.499033	+	0.866583i	$0.333689\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	35301.0	+	929593.i	0.0758329	+	1.99693i
$186$		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
$193$	−205181.	−	205181.i	−0.396501	−	0.396501i	0.480496	−	0.876997i	$-0.340457\pi$
$193$	−205181.	−	205181.i	−0.396501	−	0.396501i	−0.876997	+	0.480496i	$0.840457\pi$
$194$		−	274664.i		−	0.523959i
$195$		0			0
$196$	−537824.			−1.00000
$197$	320333.	−	320333.i	0.588080	−	0.588080i	−0.349031	−	0.937111i	$-0.613489\pi$
$197$	320333.	−	320333.i	0.588080	−	0.588080i	0.937111	+	0.349031i	$0.113489\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	−368512.	+	429184.i	−0.651443	+	0.758697i
$201$		0			0
$202$	392008.	−	392008.i	0.675953	−	0.675953i
$203$		0			0
$204$		0			0
$205$	188176.	−	203032.i	0.312737	−	0.337427i
$206$		0			0
$207$		0			0
$208$	−736256.	−	736256.i	−1.17997	−	1.17997i
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	760608.	−	760608.i	1.16231	−	1.16231i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	985944.	+	985944.i	1.40511	+	1.40511i
$219$		0			0
$220$		0			0
$221$	−1.03105e6			−1.42003
$222$		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
$225$	757188.	−	57591.0i	0.997120	−	0.0758400i
$226$	501752.			0.653459
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$		−	976564.i		−	1.23059i	−0.788298	−	0.615293i	$-0.789038\pi$
$229$		−	976564.i		−	1.23059i	0.788298	−	0.615293i	$-0.210962\pi$
$230$		0			0
$231$		0			0
$232$	−1.09619e6	+	1.09619e6i	−1.33711	+	1.33711i
$233$	458619.	+	458619.i	0.553429	+	0.553429i	0.927429	−	0.373999i	$-0.122014\pi$
$233$	458619.	+	458619.i	0.553429	+	0.553429i	−0.373999	+	0.927429i	$0.622014\pi$
$234$			1.39774e6i			1.66873i
$235$		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
$241$	−1.73905e6			−1.92872			−0.964360	−	0.264595i	$-0.914762\pi$
$241$	−1.73905e6			−1.92872			−0.964360	+	0.264595i	$0.914762\pi$
$242$	644204.	−	644204.i	0.707107	−	0.707107i
$243$		0			0
$244$			1.75834e6i			1.89072i
$245$	−689087.	−	638666.i	−0.733430	−	0.679765i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−981812.	+	112284.i	−0.993524	+	0.113623i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	1.04858e6			1.00000
$257$	−1.33282e6	+	1.33282e6i	−1.25874	+	1.25874i	−0.307052	+	0.951693i	$0.599343\pi$
$257$	−1.33282e6	+	1.33282e6i	−1.25874	+	1.25874i	−0.951693	+	0.307052i	$0.900657\pi$
$258$		0			0
$259$		0			0
$260$	−69024.0	−	1.81763e6i	−0.0633238	−	1.66753i
$261$	2.08105e6			1.89096
$262$		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
$265$	1.87775e6	−	71307.0i	1.64257	−	0.0623760i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	2.35141e6i		−	1.98129i	−0.136458	−	0.990646i	$-0.543572\pi$
$269$		−	2.35141e6i		−	1.98129i	0.136458	−	0.990646i	$-0.456428\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	734208.	−	734208.i	0.601723	−	0.601723i
$273$		0			0
$274$		−	2.03386e6i		−	1.63661i
$275$		0			0
$276$		0			0
$277$	−866617.	+	866617.i	−0.678622	+	0.678622i	−0.959688	−	0.281067i	$-0.909312\pi$
$277$	−866617.	+	866617.i	−0.678622	+	0.678622i	0.281067	+	0.959688i	$0.409312\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	149152.			0.112684			0.0563421	−	0.998412i	$-0.482056\pi$
$281$	149152.			0.112684			0.0563421	+	0.998412i	$0.482056\pi$
$282$		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
$288$	−995328.	−	995328.i	−0.707107	−	0.707107i
$289$			391679.i			0.275858i
$290$	−2.70622e6	+	102768.i	−1.88960	+	0.0717568i
$291$		0			0
$292$	−1.09859e6	+	1.09859e6i	−0.754014	+	0.754014i
$293$	2.03682e6	+	2.03682e6i	1.38606	+	1.38606i	0.833412	+	0.552653i	$0.186384\pi$
$293$	2.03682e6	+	2.03682e6i	1.38606	+	1.38606i	0.552653	+	0.833412i	$0.313616\pi$
$294$		0			0
$295$		0			0
$296$	3.01235e6			1.99837
$297$		0			0
$298$	−190456.	−	190456.i	−0.124238	−	0.124238i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.08802e6	+	2.25287e6i	−1.28524	+	1.38671i
$306$	−1.39385e6			−0.850965
$307$		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
$313$	486719.	+	486719.i	0.280813	+	0.280813i	0.833433	−	0.552620i	$-0.186372\pi$
$313$	486719.	+	486719.i	0.280813	+	0.280813i	−0.552620	+	0.833433i	$0.686372\pi$
$314$		−	489464.i		−	0.280154i
$315$		0			0
$316$		0			0
$317$	−1.42272e6	+	1.42272e6i	−0.795189	+	0.795189i	−0.982333	−	0.187144i	$-0.940077\pi$
$317$	−1.42272e6	+	1.42272e6i	−0.795189	+	0.795189i	0.187144	+	0.982333i	$0.440077\pi$
$318$		0			0
$319$		0			0
$320$	1.34349e6	+	1.24518e6i	0.733430	+	0.679765i
$321$		0			0
$322$		0			0
$323$		0			0
$324$			1.88957e6i			1.00000i
$325$	2.07000e6	−	2.41081e6i	1.08708	−	1.26606i
$326$		0			0
$327$		0			0
$328$	−633856.	−	633856.i	−0.325316	−	0.325316i
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	−2.85938e6	−	2.85938e6i	−1.41306	−	1.41306i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	2.88623e6	−	2.88623e6i	1.38438	−	1.38438i	0.547727	−	0.836657i	$-0.315493\pi$
$337$	2.88623e6	−	2.88623e6i	1.38438	−	1.38438i	0.836657	−	0.547727i	$-0.184507\pi$
$338$	2.65052e6	+	2.65052e6i	1.26194	+	1.26194i
$339$		0			0
$340$	1.81258e6	−	68832.0i	0.850352	−	0.0322919i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−98648.0			−0.0442994
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$			975636.i			0.428770i	0.976749	+	0.214385i	$0.0687747\pi$
$349$			975636.i			0.428770i	−0.976749	+	0.214385i	$0.931225\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3.27023e6	−	3.27023e6i	−1.39682	−	1.39682i	−0.808959	−	0.587865i	$-0.799969\pi$
$353$	−3.27023e6	−	3.27023e6i	−1.39682	−	1.39682i	−0.587865	−	0.808959i	$-0.700031\pi$
$354$		0			0
$355$		0			0
$356$	−4.49485e6			−1.87971
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	−93312.0	−	2.45722e6i	−0.0379473	−	0.999280i
$361$	−2.47610e6			−1.00000
$362$	1.75961e6	−	1.75961e6i	0.705740	−	0.705740i
$363$		0			0
$364$		0			0
$365$	−2.71215e6	+	102993.i	−1.06557	+	0.0404646i
$366$		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
$369$			1.20334e6i			0.460067i
$370$	3.85958e6	+	3.57717e6i	1.46567	+	1.35842i
$371$		0			0
$372$		0			0
$373$	3.33167e6	+	3.33167e6i	1.23991	+	1.23991i	0.960038	+	0.279871i	$0.0902918\pi$
$373$	3.33167e6	+	3.33167e6i	1.23991	+	1.23991i	0.279871	+	0.960038i	$0.409708\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	6.15752e6	−	6.15752e6i	2.23127	−	2.23127i
$378$		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
$386$	−1.64145e6			−0.560737
$387$		0			0
$388$	−1.09866e6	−	1.09866e6i	−0.370495	−	0.370495i
$389$			5.28629e6i			1.77124i	0.464414	+	0.885618i	$0.346265\pi$
$389$			5.28629e6i			1.77124i	−0.464414	+	0.885618i	$0.653735\pi$
$390$		0			0
$391$		0			0
$392$	−2.15130e6	+	2.15130e6i	−0.707107	+	0.707107i
$393$		0			0
$394$		−	2.56266e6i		−	0.831670i
$395$		0			0
$396$		0			0
$397$	−3.00767e6	+	3.00767e6i	−0.957753	+	0.957753i	−0.999143	−	0.0413901i	$-0.986821\pi$
$397$	−3.00767e6	+	3.00767e6i	−0.957753	+	0.957753i	0.0413901	+	0.999143i	$0.486821\pi$
$398$		0			0
$399$		0			0
$400$	242688.	+	3.19078e6i	0.0758400	+	0.997120i
$401$	1.59200e6			0.494405			0.247202	−	0.968964i	$-0.420489\pi$
$401$	1.59200e6			0.494405			0.247202	+	0.968964i	$0.420489\pi$
$402$		0			0
$403$		0			0
$404$		−	3.13606e6i		−	0.955942i
$405$	−2.24386e6	+	2.42101e6i	−0.679765	+	0.733430i
$406$		0			0
$407$		0			0
$408$		0			0
$409$			4.58449e6i			1.35513i	0.735461	+	0.677567i	$0.236966\pi$
$409$			4.58449e6i			1.35513i	−0.735461	+	0.677567i	$0.763034\pi$
$410$	−59424.0	−	1.56483e6i	−0.0174583	−	0.459736i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−5.89005e6			−1.66873
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	4.18485e6			1.15073			0.575367	−	0.817895i	$-0.304859\pi$
$421$	4.18485e6			1.15073			0.575367	+	0.817895i	$0.304859\pi$
$422$		0			0
$423$		0			0
$424$		−	6.08486e6i		−	1.64375i
$425$	2.40410e6	+	2.06424e6i	0.645625	+	0.554356i
$426$		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
$433$	−3.62613e6	−	3.62613e6i	−0.929445	−	0.929445i	0.0682249	−	0.997670i	$-0.478266\pi$
$433$	−3.62613e6	−	3.62613e6i	−0.929445	−	0.929445i	−0.997670	+	0.0682249i	$0.978266\pi$
$434$		0			0
$435$		0			0
$436$	7.88755e6			1.98713
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	4.08410e6			1.00000
$442$	−4.12418e6	+	4.12418e6i	−1.00411	+	1.00411i
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		0			0
$445$	−5.75902e6	−	5.33763e6i	−1.37863	−	1.27776i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	8.48961e6i		−	1.98734i	−0.112340	−	0.993670i	$-0.535835\pi$
$449$		−	8.48961e6i		−	1.98734i	0.112340	−	0.993670i	$-0.464165\pi$
$450$	2.79839e6	−	3.25912e6i	0.651443	−	0.758697i
$451$		0			0
$452$	2.00701e6	−	2.00701e6i	0.462065	−	0.462065i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	979183.	−	979183.i	0.219318	−	0.219318i	−0.588893	−	0.808211i	$-0.700436\pi$
$457$	979183.	−	979183.i	0.219318	−	0.219318i	0.808211	+	0.588893i	$0.200436\pi$
$458$	−3.90626e6	−	3.90626e6i	−0.870156	−	0.870156i
$459$		0			0
$460$		0			0
$461$	6.86580e6			1.50466			0.752331	−	0.658785i	$-0.228929\pi$
$461$	6.86580e6			1.50466			0.752331	+	0.658785i	$0.228929\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$			8.76954e6i			1.89096i
$465$		0			0
$466$	3.66895e6			0.782667
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$	5.59094e6	+	5.59094e6i	1.17997	+	1.17997i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−5.77587e6	+	5.77587e6i	−1.16231	+	1.16231i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	−1.69209e7			−3.33474
$482$	−6.95619e6	+	6.95619e6i	−1.36381	+	1.36381i
$483$		0			0
$484$		−	5.15363e6i		−	1.00000i
$485$	−102999.	−	2.71231e6i	−0.0198829	−	0.523582i
$486$		0			0
$487$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$487$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$488$	7.03334e6	+	7.03334e6i	1.33694	+	1.33694i
$489$		0			0
$490$	−5.31101e6	+	201684.i	−0.999280	+	0.0379473i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	6.14039e6	+	6.14039e6i	1.13783	+	1.13783i
$494$		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	$0.5\pi$
$500$	−3.47811e6	+	4.37638e6i	−0.622184	+	0.782871i
$501$		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
$505$	3.72408e6	−	4.01808e6i	0.649816	−	0.701117i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	5.12076e6i		−	0.876073i	−0.898957	−	0.438037i	$-0.855674\pi$
$509$		−	5.12076e6i		−	0.876073i	0.898957	−	0.438037i	$-0.144326\pi$
$510$		0			0
$511$		0			0
$512$	4.19430e6	−	4.19430e6i	0.707107	−	0.707107i
$513$		0			0
$514$			1.06625e7i			1.78013i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−7.54662e6	−	6.99443e6i	−1.22390	−	1.13434i
$521$	7.27410e6			1.17405			0.587023	−	0.809570i	$-0.300300\pi$
$521$	7.27410e6			1.17405			0.587023	+	0.809570i	$0.300300\pi$
$522$	8.32421e6	−	8.32421e6i	1.33711	−	1.33711i
$523$		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
$529$			6.43634e6i			1.00000i
$530$	7.22578e6	−	7.79623e6i	1.11736	−	1.20558i
$531$		0			0
$532$		0			0
$533$	3.56049e6	+	3.56049e6i	0.542865	+	0.542865i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−9.40566e6	−	9.40566e6i	−1.40098	−	1.40098i
$539$		0			0
$540$		0			0
$541$	−8.25380e6			−1.21244			−0.606221	−	0.795297i	$-0.707315\pi$
$541$	−8.25380e6			−1.21244			−0.606221	+	0.795297i	$0.707315\pi$
$542$		0			0
$543$		0			0
$544$		−	5.87366e6i		−	0.850965i
$545$	1.01059e7	+	9.36647e6i	1.45742	+	1.35078i
$546$		0			0
$547$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$547$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$548$	−8.13546e6	−	8.13546e6i	−1.15726	−	1.15726i
$549$		−	1.33524e7i		−	1.89072i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$			6.93294e6i			0.959716i
$555$		0			0
$556$		0			0
$557$	−7.56357e6	+	7.56357e6i	−1.03297	+	1.03297i	−0.0335347	+	0.999438i	$0.510676\pi$
$557$	−7.56357e6	+	7.56357e6i	−1.03297	+	1.03297i	−0.999438	+	0.0335347i	$0.989324\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	596608.	−	596608.i	0.0796798	−	0.0796798i
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$		0			0
$565$	4.95480e6	−	188157.i	0.652988	−	0.0247970i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			3.96659e6i			0.513613i	0.966463	+	0.256807i	$0.0826703\pi$
$569$			3.96659e6i			0.513613i	−0.966463	+	0.256807i	$0.917330\pi$
$570$		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
$576$	−7.96262e6			−1.00000
$577$	6.17138e6	−	6.17138e6i	0.771690	−	0.771690i	−0.206712	−	0.978402i	$-0.566276\pi$
$577$	6.17138e6	−	6.17138e6i	0.771690	−	0.771690i	0.978402	+	0.206712i	$0.0662762\pi$
$578$	1.56672e6	+	1.56672e6i	0.195061	+	0.195061i
$579$		0			0
$580$	−1.04138e7	+	1.12360e7i	−1.28541	+	1.38689i
$581$		0			0
$582$		0			0
$583$		0			0
$584$			8.78874e6i			1.06634i
$585$	524151.	+	1.38026e7i	0.0633238	+	1.66753i
$586$	1.62946e7			1.96019
$587$		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
$592$	1.20494e7	−	1.20494e7i	1.41306	−	1.41306i
$593$	−7.63843e6	−	7.63843e6i	−0.892005	−	0.892005i	0.102706	−	0.994712i	$-0.467250\pi$
$593$	−7.63843e6	−	7.63843e6i	−0.892005	−	0.892005i	−0.994712	+	0.102706i	$0.967250\pi$
$594$		0			0
$595$		0			0
$596$	−1.52365e6			−0.175699
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−9.16325e6			−1.03482			−0.517408	−	0.855739i	$-0.673103\pi$
$601$	−9.16325e6			−1.03482			−0.517408	+	0.855739i	$0.673103\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	6.11994e6	−	6.60309e6i	0.679765	−	0.733430i
$606$		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
$610$	659376.	+	1.73636e7i	0.0717478	+	1.88936i
$611$		0			0
$612$	−5.57539e6	+	5.57539e6i	−0.601723	+	0.601723i
$613$	1.87272e6	+	1.87272e6i	0.201290	+	0.201290i	0.800552	−	0.599263i	$-0.204540\pi$
$613$	1.87272e6	+	1.87272e6i	0.201290	+	0.201290i	−0.599263	+	0.800552i	$0.704540\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−5.54472e6	+	5.54472e6i	−0.586363	+	0.586363i	−0.936644	−	0.350282i	$-0.886086\pi$
$617$	−5.54472e6	+	5.54472e6i	−0.586363	+	0.586363i	0.350282	+	0.936644i	$0.386086\pi$
$618$		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
$625$	−9.65329e6	+	1.47698e6i	−0.988497	+	0.151243i
$626$	3.89375e6			0.397130
$627$		0			0
$628$	−1.95786e6	−	1.95786e6i	−0.198099	−	0.198099i
$629$		−	1.68739e7i		−	1.70055i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$			1.13817e7i			1.12457i
$635$		0			0
$636$		0			0
$637$	1.20842e7	−	1.20842e7i	1.17997	−	1.17997i
$638$		0			0
$639$		0			0
$640$	1.03547e7	−	393216.i	0.999280	−	0.0379473i
$641$	−1.48270e7			−1.42531			−0.712655	−	0.701514i	$-0.752508\pi$
$641$	−1.48270e7			−1.42531			−0.712655	+	0.701514i	$0.752508\pi$
$642$		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	$0.750000\pi$
$648$	7.55827e6	+	7.55827e6i	0.707107	+	0.707107i
$649$		0			0
$650$	−1.36322e6	−	1.79232e7i	−0.126556	−	1.66392i
$651$		0			0
$652$		0			0
$653$	−1.46642e7	−	1.46642e7i	−1.34579	−	1.34579i	−0.890182	−	0.455605i	$-0.849423\pi$
$653$	−1.46642e7	−	1.46642e7i	−1.34579	−	1.34579i	−0.455605	−	0.890182i	$-0.650577\pi$
$654$		0			0
$655$		0			0
$656$	−5.07085e6			−0.460067
$657$	8.34243e6	−	8.34243e6i	0.754014	−	0.754014i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	2.07531e7			1.84747			0.923737	−	0.383027i	$-0.125119\pi$
$661$	2.07531e7			1.84747			0.923737	+	0.383027i	$0.125119\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−2.28750e7			−1.99837
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.65977e7	+	1.65977e7i	1.41257	+	1.41257i	0.740383	+	0.672185i	$0.234644\pi$
$673$	1.65977e7	+	1.65977e7i	1.41257	+	1.41257i	0.672185	+	0.740383i	$0.265356\pi$
$674$		−	2.30899e7i		−	1.95781i
$675$		0			0
$676$	2.12041e7			1.78465
$677$	−1.39839e7	+	1.39839e7i	−1.17262	+	1.17262i	−0.191032	+	0.981584i	$0.561183\pi$
$677$	−1.39839e7	+	1.39839e7i	−1.17262	+	1.17262i	−0.981584	+	0.191032i	$0.938817\pi$
$678$		0			0
$679$		0			0
$680$	6.97498e6	−	7.52563e6i	0.578456	−	0.624124i
$681$		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
$685$	−762699.	−	2.00844e7i	−0.0621050	−	1.63543i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			3.41798e7i			2.74297i
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$	−394592.	+	394592.i	−0.0313244	+	0.0313244i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−3.55058e6	+	3.55058e6i	−0.276833	+	0.276833i
$698$	3.90254e6	+	3.90254e6i	0.303186	+	0.303186i
$699$		0			0
$700$		0			0
$701$	1.51373e7			1.16346			0.581731	−	0.813381i	$-0.302376\pi$
$701$	1.51373e7			1.16346			0.581731	+	0.813381i	$0.302376\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−2.61618e7			−1.97541
$707$		0			0
$708$		0			0
$709$			2.64712e7i			1.97769i	0.148942	+	0.988846i	$0.452413\pi$
$709$			2.64712e7i			1.97769i	−0.148942	+	0.988846i	$0.547587\pi$
$710$		0			0
$711$		0			0
$712$	−1.79794e7	+	1.79794e7i	−1.32915	+	1.32915i
$713$		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	$0.5\pi$
$720$	−1.02021e7	−	9.45562e6i	−0.733430	−	0.679765i
$721$		0			0
$722$	−9.90440e6	+	9.90440e6i	−0.707107	+	0.707107i
$723$		0			0
$724$		−	1.40769e7i		−	0.998067i
$725$	−2.66854e7	+	2.02967e6i	−1.88551	+	0.143410i
$726$		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
$729$		−	1.43489e7i		−	1.00000i
$730$	−1.04366e7	+	1.12606e7i	−0.724858	+	0.782084i
$731$		0			0
$732$		0			0
$733$	1.88199e7	+	1.88199e7i	1.29377	+	1.29377i	0.932438	+	0.361331i	$0.117678\pi$
$733$	1.88199e7	+	1.88199e7i	1.29377	+	1.29377i	0.361331	+	0.932438i	$0.382322\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	4.81334e6	+	4.81334e6i	0.325316	+	0.325316i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	2.97470e7	−	1.12963e6i	1.99693	−	0.0758329i
$741$		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
$745$	−1.95217e6	−	1.80933e6i	−0.128863	−	0.119434i
$746$	2.66534e7			1.75350
$747$		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
$754$		−	4.92601e7i		−	3.15549i
$755$		0			0
$756$		0			0
$757$	1.71972e7	−	1.71972e7i	1.09073	−	1.09073i	0.0952804	−	0.995450i	$-0.469625\pi$
$757$	1.71972e7	−	1.71972e7i	1.09073	−	1.09073i	0.995450	−	0.0952804i	$-0.0303748\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.95198e7			1.22184			0.610919	−	0.791693i	$-0.290800\pi$
$761$	1.95198e7			1.22184			0.610919	+	0.791693i	$0.290800\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−1.37642e7	+	522693.i	−0.850352	+	0.0322919i
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	2.57727e7i		−	1.57161i	−0.618477	−	0.785803i	$-0.712250\pi$
$769$		−	2.57727e7i		−	1.57161i	0.618477	−	0.785803i	$-0.287750\pi$
$770$		0			0
$771$		0			0
$772$	−6.56579e6	+	6.56579e6i	−0.396501	+	0.396501i
$773$	−1.88791e7	−	1.88791e7i	−1.13640	−	1.13640i	−0.989090	−	0.147312i	$-0.952938\pi$
$773$	−1.88791e7	−	1.88791e7i	−1.13640	−	1.13640i	−0.147312	−	0.989090i	$-0.547062\pi$
$774$		0			0
$775$		0			0
$776$	−8.78925e6			−0.523959
$777$		0			0
$778$	2.11451e7	+	2.11451e7i	1.25245	+	1.25245i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$			1.72104e7i			1.00000i
$785$	−183549.	−	4.83346e6i	−0.0106311	−	0.279952i
$786$		0			0
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$	−1.02507e7	−	1.02507e7i	−0.588080	−	0.588080i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−3.95076e7	−	3.95076e7i	−2.23099	−	2.23099i
$794$			2.40613e7i			1.35447i
$795$		0			0
$796$		0			0
$797$	−8.81492e6	+	8.81492e6i	−0.491555	+	0.491555i	−0.908796	−	0.417241i	$-0.862997\pi$
$797$	−8.81492e6	+	8.81492e6i	−0.491555	+	0.491555i	0.417241	+	0.908796i	$0.362997\pi$
$798$		0			0
$799$		0			0
$800$	1.37339e7	+	1.17924e7i	0.758697	+	0.651443i
$801$	3.41328e7			1.87971
$802$	6.36801e6	−	6.36801e6i	0.349597	−	0.349597i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−1.25443e7	−	1.25443e7i	−0.675953	−	0.675953i
$809$			2.36197e7i			1.26883i	0.772992	+	0.634415i	$0.218759\pi$
$809$			2.36197e7i			1.26883i	−0.772992	+	0.634415i	$0.781241\pi$
$810$	708588.	+	1.86595e7i	0.0379473	+	0.999280i
$811$		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
$818$	1.83379e7	+	1.83379e7i	0.958225	+	0.958225i
$819$		0			0
$820$	−6.49702e6	−	6.02163e6i	−0.337427	−	0.312737i
$821$	−2.14631e7			−1.11131			−0.555655	−	0.831413i	$-0.687533\pi$
$821$	−2.14631e7			−1.11131			−0.555655	+	0.831413i	$0.687533\pi$
$822$		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
$829$			7.96819e6i			0.402692i	0.979520	+	0.201346i	$0.0645316\pi$
$829$			7.96819e6i			0.402692i	−0.979520	+	0.201346i	$0.935468\pi$
$830$		0			0
$831$		0			0
$832$	−2.35602e7	+	2.35602e7i	−1.17997	+	1.17997i
$833$	1.20506e7	+	1.20506e7i	0.601723	+	0.601723i
$834$		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
$841$	−5.28309e7			−2.57572
$842$	1.67394e7	−	1.67394e7i	0.813692	−	0.813692i
$843$		0			0
$844$		0			0
$845$	2.71678e7	+	2.51799e7i	1.30892	+	1.21314i
$846$		0			0
$847$		0			0
$848$	−2.43395e7	−	2.43395e7i	−1.16231	−	1.16231i
$849$		0			0
$850$	1.78734e7	−	1.35943e6i	0.848515	−	0.0645372i
$851$		0			0
$852$		0			0
$853$	2.46885e7	+	2.46885e7i	1.16178	+	1.16178i	0.984087	+	0.177690i	$0.0568623\pi$
$853$	2.46885e7	+	2.46885e7i	1.16178	+	1.16178i	0.177690	+	0.984087i	$0.443138\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	3.02544e7	−	3.02544e7i	1.40714	−	1.40714i	0.632929	−	0.774210i	$-0.281852\pi$
$857$	3.02544e7	−	3.02544e7i	1.40714	−	1.40714i	0.774210	−	0.632929i	$-0.218148\pi$
$858$		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
$865$	−974149.	+	36993.0i	−0.0442675	+	0.00168104i
$866$	−2.90090e7			−1.31443
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	3.15502e7	−	3.15502e7i	1.40511	−	1.40511i
$873$	8.34292e6	+	8.34292e6i	0.370495	+	0.370495i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−7.49062e6	+	7.49062e6i	−0.328866	+	0.328866i	−0.852155	−	0.523289i	$-0.824705\pi$
$877$	−7.49062e6	+	7.49062e6i	−0.328866	+	0.328866i	0.523289	+	0.852155i	$0.324705\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.33972e7			0.581531			0.290765	−	0.956794i	$-0.406090\pi$
$881$	1.33972e7			0.581531			0.290765	+	0.956794i	$0.406090\pi$
$882$	1.63364e7	−	1.63364e7i	0.707107	−	0.707107i
$883$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$883$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$884$			3.29935e7i			1.42003i
$885$		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
$890$	−4.43866e7	+	1.68557e6i	−1.87835	+	0.0713298i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−3.39585e7	−	3.39585e7i	−1.40526	−	1.40526i
$899$		0			0
$900$	−1.84291e6	−	2.42300e7i	−0.0758400	−	0.997120i
$901$	−3.40847e7			−1.39878
$902$		0			0
$903$		0			0
$904$		−	1.60561e7i		−	0.653459i
$905$	1.67163e7	−	1.80360e7i	0.678450	−	0.732012i
$906$		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
$909$			2.38145e7i			0.955942i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		−	7.83346e6i		−	0.310162i
$915$		0			0
$916$	−3.12500e7			−1.23059
$917$		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
$922$	2.74632e7	−	2.74632e7i	1.06396	−	1.06396i
$923$		0			0
$924$		0			0
$925$	3.94548e7	+	3.38772e7i	1.51616	+	1.30183i
$926$		0			0
$927$		0			0
$928$	3.50781e7	+	3.50781e7i	1.33711	+	1.33711i
$929$		−	4.76633e7i		−	1.81194i	−0.423337	−	0.905972i	$-0.639141\pi$
$929$		−	4.76633e7i		−	1.81194i	0.423337	−	0.905972i	$-0.360859\pi$
$930$		0			0
$931$		0			0
$932$	1.46758e7	−	1.46758e7i	0.553429	−	0.553429i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	4.47276e7			1.66873
$937$	−3.18165e7	+	3.18165e7i	−1.18387	+	1.18387i	−0.205135	+	0.978734i	$0.565763\pi$
$937$	−3.18165e7	+	3.18165e7i	−1.18387	+	1.18387i	−0.978734	+	0.205135i	$0.934237\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	4.85570e6			0.178763			0.0893816	−	0.995997i	$-0.471511\pi$
$941$	4.85570e6			0.178763			0.0893816	+	0.995997i	$0.471511\pi$
$942$		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
$949$		−	4.93680e7i		−	1.77943i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−7.20948e6	−	7.20948e6i	−0.257141	−	0.257141i	0.566749	−	0.823890i	$-0.308201\pi$
$953$	−7.20948e6	−	7.20948e6i	−0.257141	−	0.257141i	−0.823890	+	0.566749i	$0.808201\pi$
$954$			4.62069e7i			1.64375i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	2.86292e7			1.00000
$962$	−6.76838e7	+	6.76838e7i	−2.35802	+	2.35802i
$963$		0			0
$964$			5.56495e7i			1.92872i
$965$	−1.62093e7	+	615543.i	−0.560333	+	0.0212785i
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$	−2.06145e7	−	2.06145e7i	−0.707107	−	0.707107i
$969$		0			0
$970$	−1.12612e7	−	1.04372e7i	−0.384288	−	0.356169i
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	5.62668e7			1.89072
$977$	−6.04587e6	+	6.04587e6i	−0.202639	+	0.202639i	−0.801130	−	0.598491i	$-0.795767\pi$
$977$	−6.04587e6	+	6.04587e6i	−0.202639	+	0.202639i	0.598491	+	0.801130i	$0.295767\pi$
$978$		0			0
$979$		0			0
$980$	−2.04373e7	+	2.20508e7i	−0.679765	+	0.733430i
$981$	−5.98961e7			−1.98713
$982$		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
$985$	−960999.	−	2.53063e7i	−0.0315597	−	0.831071i
$986$	4.91231e7			1.60914
$987$		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
$997$	−4.37429e7	+	4.37429e7i	−1.39370	+	1.39370i	−0.576856	+	0.816846i	$0.695720\pi$
$997$	−4.37429e7	+	4.37429e7i	−1.39370	+	1.39370i	−0.816846	+	0.576856i	$0.804280\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.6.e.a.3.1	✓	2
4.3	odd	2	CM	20.6.e.a.3.1	✓	2
5.2	odd	4	inner	20.6.e.a.7.1	yes	2
5.3	odd	4		100.6.e.b.7.1		2
5.4	even	2		100.6.e.b.43.1		2
20.3	even	4		100.6.e.b.7.1		2
20.7	even	4	inner	20.6.e.a.7.1	yes	2
20.19	odd	2		100.6.e.b.43.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.6.e.a.3.1	✓	2	1.1	even	1	trivial
20.6.e.a.3.1	✓	2	4.3	odd	2	CM
20.6.e.a.7.1	yes	2	5.2	odd	4	inner
20.6.e.a.7.1	yes	2	20.7	even	4	inner
100.6.e.b.7.1		2	5.3	odd	4
100.6.e.b.7.1		2	20.3	even	4
100.6.e.b.43.1		2	5.4	even	2
100.6.e.b.43.1		2	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	20.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(4.00000 - 4.00000i) q^{2} -32.0000i q^{4} +(38.0000 - 41.0000i) q^{5} +(-128.000 - 128.000i) q^{8} +243.000i q^{9} +O(q^{10})\)	\(q+(4.00000 - 4.00000i) q^{2} -32.0000i q^{4} +(38.0000 - 41.0000i) q^{5} +(-128.000 - 128.000i) q^{8} +243.000i q^{9} +(-12.0000 - 316.000i) q^{10} +(719.000 + 719.000i) q^{13} -1024.00 q^{16} +(-717.000 + 717.000i) q^{17} +(972.000 + 972.000i) q^{18} +(-1312.00 - 1216.00i) q^{20} +(-237.000 - 3116.00i) q^{25} +5752.00 q^{26} -8564.00i q^{29} +(-4096.00 + 4096.00i) q^{32} +5736.00i q^{34} +7776.00 q^{36} +(-11767.0 + 11767.0i) q^{37} +(-10112.0 + 384.000i) q^{40} +4952.00 q^{41} +(9963.00 + 9234.00i) q^{45} -16807.0i q^{49} +(-13412.0 - 11516.0i) q^{50} +(23008.0 - 23008.0i) q^{52} +(23769.0 + 23769.0i) q^{53} +(-34256.0 - 34256.0i) q^{58} -54948.0 q^{61} +32768.0i q^{64} +(56801.0 - 2157.00i) q^{65} +(22944.0 + 22944.0i) q^{68} +(31104.0 - 31104.0i) q^{72} +(-34331.0 - 34331.0i) q^{73} +94136.0i q^{74} +(-38912.0 + 41984.0i) q^{80} -59049.0 q^{81} +(19808.0 - 19808.0i) q^{82} +(2151.00 + 56643.0i) q^{85} -140464. i q^{89} +(76788.0 - 2916.00i) q^{90} +(34333.0 - 34333.0i) q^{97} +(-67228.0 - 67228.0i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{2} + 76 q^{5} - 256 q^{8}+O(q^{10}) \) 2 * q + 8 * q^2 + 76 * q^5 - 256 * q^8	\( 2 q + 8 q^{2} + 76 q^{5} - 256 q^{8} - 24 q^{10} + 1438 q^{13} - 2048 q^{16} - 1434 q^{17} + 1944 q^{18} - 2624 q^{20} - 474 q^{25} + 11504 q^{26} - 8192 q^{32} + 15552 q^{36} - 23534 q^{37} - 20224 q^{40} + 9904 q^{41} + 19926 q^{45} - 26824 q^{50} + 46016 q^{52} + 47538 q^{53} - 68512 q^{58} - 109896 q^{61} + 113602 q^{65} + 45888 q^{68} + 62208 q^{72} - 68662 q^{73} - 77824 q^{80} - 118098 q^{81} + 39616 q^{82} + 4302 q^{85} + 153576 q^{90} + 68666 q^{97} - 134456 q^{98}+O(q^{100}) \) 2 * q + 8 * q^2 + 76 * q^5 - 256 * q^8 - 24 * q^10 + 1438 * q^13 - 2048 * q^16 - 1434 * q^17 + 1944 * q^18 - 2624 * q^20 - 474 * q^25 + 11504 * q^26 - 8192 * q^32 + 15552 * q^36 - 23534 * q^37 - 20224 * q^40 + 9904 * q^41 + 19926 * q^45 - 26824 * q^50 + 46016 * q^52 + 47538 * q^53 - 68512 * q^58 - 109896 * q^61 + 113602 * q^65 + 45888 * q^68 + 62208 * q^72 - 68662 * q^73 - 77824 * q^80 - 118098 * q^81 + 39616 * q^82 + 4302 * q^85 + 153576 * q^90 + 68666 * q^97 - 134456 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more