LMFDB - Embedded newform 980.2.k.c.883.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [980,2,Mod(687,980)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(980, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("980.687");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 980 = 2^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	980.k (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:	$7.82533939809$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			883.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	980.883
Dual form			980.2.k.c.687.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(1.00000 - 2.00000i) q^{5} +(-2.00000 - 2.00000i) q^{8} +3.00000i q^{9} +O(q^{10})$	$q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(1.00000 - 2.00000i) q^{5} +(-2.00000 - 2.00000i) q^{8} +3.00000i q^{9} +(-1.00000 - 3.00000i) q^{10} +(-5.00000 - 5.00000i) q^{13} -4.00000 q^{16} +(5.00000 - 5.00000i) q^{17} +(3.00000 + 3.00000i) q^{18} +(-4.00000 - 2.00000i) q^{20} +(-3.00000 - 4.00000i) q^{25} -10.0000 q^{26} +4.00000i q^{29} +(-4.00000 + 4.00000i) q^{32} -10.0000i q^{34} +6.00000 q^{36} +(7.00000 - 7.00000i) q^{37} +(-6.00000 + 2.00000i) q^{40} -10.0000 q^{41} +(6.00000 + 3.00000i) q^{45} +(-7.00000 - 1.00000i) q^{50} +(-10.0000 + 10.0000i) q^{52} +(9.00000 + 9.00000i) q^{53} +(4.00000 + 4.00000i) q^{58} +10.0000 q^{61} +8.00000i q^{64} +(-15.0000 + 5.00000i) q^{65} +(-10.0000 - 10.0000i) q^{68} +(6.00000 - 6.00000i) q^{72} +(5.00000 + 5.00000i) q^{73} -14.0000i q^{74} +(-4.00000 + 8.00000i) q^{80} -9.00000 q^{81} +(-10.0000 + 10.0000i) q^{82} +(-5.00000 - 15.0000i) q^{85} -10.0000i q^{89} +(9.00000 - 3.00000i) q^{90} +(-5.00000 + 5.00000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{5} - 4 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^5 - 4 * q^8	$ 2 q + 2 q^{2} + 2 q^{5} - 4 q^{8} - 2 q^{10} - 10 q^{13} - 8 q^{16} + 10 q^{17} + 6 q^{18} - 8 q^{20} - 6 q^{25} - 20 q^{26} - 8 q^{32} + 12 q^{36} + 14 q^{37} - 12 q^{40} - 20 q^{41} + 12 q^{45} - 14 q^{50} - 20 q^{52} + 18 q^{53} + 8 q^{58} + 20 q^{61} - 30 q^{65} - 20 q^{68} + 12 q^{72} + 10 q^{73} - 8 q^{80} - 18 q^{81} - 20 q^{82} - 10 q^{85} + 18 q^{90} - 10 q^{97}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^5 - 4 * q^8 - 2 * q^10 - 10 * q^13 - 8 * q^16 + 10 * q^17 + 6 * q^18 - 8 * q^20 - 6 * q^25 - 20 * q^26 - 8 * q^32 + 12 * q^36 + 14 * q^37 - 12 * q^40 - 20 * q^41 + 12 * q^45 - 14 * q^50 - 20 * q^52 + 18 * q^53 + 8 * q^58 + 20 * q^61 - 30 * q^65 - 20 * q^68 + 12 * q^72 + 10 * q^73 - 8 * q^80 - 18 * q^81 - 20 * q^82 - 10 * q^85 + 18 * q^90 - 10 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/980\mathbb{Z}\right)^\times$.

$n$	$101$	$197$	$491$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$-1$

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 $

$2$	1.00000	−	1.00000i	0.707107	−	0.707107i
$2$	1.00000	−	1.00000i	0.707107	−	0.707107i
$3$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$3$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$4$		−	2.00000i		−	1.00000i
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$6$		0			0
$7$		0			0
$7$		0			0
$8$	−2.00000	−	2.00000i	−0.707107	−	0.707107i
$9$			3.00000i			1.00000i
$10$	−1.00000	−	3.00000i	−0.316228	−	0.948683i
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	−5.00000	−	5.00000i	−1.38675	−	1.38675i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−5.00000	−	5.00000i	−1.38675	−	1.38675i	−0.554700	−	0.832050i	$-0.687167\pi$
$14$		0			0
$15$		0			0
$16$	−4.00000			−1.00000
$17$	5.00000	−	5.00000i	1.21268	−	1.21268i	0.242536	−	0.970143i	$-0.422021\pi$
$17$	5.00000	−	5.00000i	1.21268	−	1.21268i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	3.00000	+	3.00000i	0.707107	+	0.707107i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	−4.00000	−	2.00000i	−0.894427	−	0.447214i
$21$		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
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$	−10.0000			−1.96116
$27$		0			0
$28$		0			0
$29$			4.00000i			0.742781i	0.928477	+	0.371391i	$0.121119\pi$
$29$			4.00000i			0.742781i	−0.928477	+	0.371391i	$0.878881\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−4.00000	+	4.00000i	−0.707107	+	0.707107i
$33$		0			0
$34$		−	10.0000i		−	1.71499i
$35$		0			0
$36$	6.00000			1.00000
$37$	7.00000	−	7.00000i	1.15079	−	1.15079i	0.164399	−	0.986394i	$-0.447432\pi$
$37$	7.00000	−	7.00000i	1.15079	−	1.15079i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$		0			0
$39$		0			0
$40$	−6.00000	+	2.00000i	−0.948683	+	0.316228i
$41$	−10.0000			−1.56174			−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000			−1.56174			−0.780869	+	0.624695i	$0.785223\pi$
$42$		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
$45$	6.00000	+	3.00000i	0.894427	+	0.447214i
$46$		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
$49$		0			0
$50$	−7.00000	−	1.00000i	−0.989949	−	0.141421i
$51$		0			0
$52$	−10.0000	+	10.0000i	−1.38675	+	1.38675i
$53$	9.00000	+	9.00000i	1.23625	+	1.23625i	0.961524	+	0.274721i	$0.0885855\pi$
$53$	9.00000	+	9.00000i	1.23625	+	1.23625i	0.274721	+	0.961524i	$0.411414\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	4.00000	+	4.00000i	0.525226	+	0.525226i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	10.0000			1.28037			0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000			1.28037			0.640184	+	0.768221i	$0.278858\pi$
$62$		0			0
$63$		0			0
$64$			8.00000i			1.00000i
$65$	−15.0000	+	5.00000i	−1.86052	+	0.620174i
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$	−10.0000	−	10.0000i	−1.21268	−	1.21268i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	6.00000	−	6.00000i	0.707107	−	0.707107i
$73$	5.00000	+	5.00000i	0.585206	+	0.585206i	0.936329	−	0.351123i	$-0.114200\pi$
$73$	5.00000	+	5.00000i	0.585206	+	0.585206i	−0.351123	+	0.936329i	$0.614200\pi$
$74$		−	14.0000i		−	1.62747i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	−4.00000	+	8.00000i	−0.447214	+	0.894427i
$81$	−9.00000			−1.00000
$82$	−10.0000	+	10.0000i	−1.10432	+	1.10432i
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$	−5.00000	−	15.0000i	−0.542326	−	1.62698i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	10.0000i		−	1.06000i	−0.847998	−	0.529999i	$-0.822192\pi$
$89$		−	10.0000i		−	1.06000i	0.847998	−	0.529999i	$-0.177808\pi$
$90$	9.00000	−	3.00000i	0.948683	−	0.316228i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−5.00000	+	5.00000i	−0.507673	+	0.507673i	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	−5.00000	+	5.00000i	−0.507673	+	0.507673i	0.406138	+	0.913812i	$0.366875\pi$
$98$		0			0
$99$		0			0
$100$	−8.00000	+	6.00000i	−0.800000	+	0.600000i
$101$	20.0000			1.99007			0.995037	−	0.0995037i	$-0.0317255\pi$
$101$	20.0000			1.99007			0.995037	+	0.0995037i	$0.0317255\pi$
$102$		0			0
$103$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$103$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$104$			20.0000i			1.96116i
$105$		0			0
$106$	18.0000			1.74831
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$		0			0
$109$		−	6.00000i		−	0.574696i	−0.957826	−	0.287348i	$-0.907226\pi$
$109$		−	6.00000i		−	0.574696i	0.957826	−	0.287348i	$-0.0927736\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.00000	−	1.00000i	−0.0940721	−	0.0940721i	0.658505	−	0.752577i	$-0.271189\pi$
$113$	−1.00000	−	1.00000i	−0.0940721	−	0.0940721i	−0.752577	+	0.658505i	$0.771189\pi$
$114$		0			0
$115$		0			0
$116$	8.00000			0.742781
$117$	15.0000	−	15.0000i	1.38675	−	1.38675i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$	10.0000	−	10.0000i	0.905357	−	0.905357i
$123$		0			0
$124$		0			0
$125$	−11.0000	+	2.00000i	−0.983870	+	0.178885i
$126$		0			0
$127$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$	8.00000	+	8.00000i	0.707107	+	0.707107i
$129$		0			0
$130$	−10.0000	+	20.0000i	−0.877058	+	1.75412i
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−20.0000			−1.71499
$137$	7.00000	−	7.00000i	0.598050	−	0.598050i	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	7.00000	−	7.00000i	0.598050	−	0.598050i	0.939793	+	0.341743i	$0.111017\pi$
$138$		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
$144$		−	12.0000i		−	1.00000i
$145$	8.00000	+	4.00000i	0.664364	+	0.332182i
$146$	10.0000			0.827606
$147$		0			0
$148$	−14.0000	−	14.0000i	−1.15079	−	1.15079i
$149$		−	14.0000i		−	1.14692i	−0.819232	−	0.573462i	$-0.805600\pi$
$149$		−	14.0000i		−	1.14692i	0.819232	−	0.573462i	$-0.194400\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	15.0000	+	15.0000i	1.21268	+	1.21268i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−5.00000	+	5.00000i	−0.399043	+	0.399043i	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−5.00000	+	5.00000i	−0.399043	+	0.399043i	0.478852	+	0.877896i	$0.341053\pi$
$158$		0			0
$159$		0			0
$160$	4.00000	+	12.0000i	0.316228	+	0.948683i
$161$		0			0
$162$	−9.00000	+	9.00000i	−0.707107	+	0.707107i
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$			20.0000i			1.56174i
$165$		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
$169$			37.0000i			2.84615i
$170$	−20.0000	−	10.0000i	−1.53393	−	0.766965i
$171$		0			0
$172$		0			0
$173$	15.0000	+	15.0000i	1.14043	+	1.14043i	0.988372	+	0.152057i	$0.0485898\pi$
$173$	15.0000	+	15.0000i	1.14043	+	1.14043i	0.152057	+	0.988372i	$0.451410\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−10.0000	−	10.0000i	−0.749532	−	0.749532i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	6.00000	−	12.0000i	0.447214	−	0.894427i
$181$	20.0000			1.48659			0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000			1.48659			0.743294	+	0.668965i	$0.233262\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−7.00000	−	21.0000i	−0.514650	−	1.54395i
$186$		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
$193$	19.0000	+	19.0000i	1.36765	+	1.36765i	0.863779	+	0.503871i	$0.168091\pi$
$193$	19.0000	+	19.0000i	1.36765	+	1.36765i	0.503871	+	0.863779i	$0.331909\pi$
$194$			10.0000i			0.717958i
$195$		0			0
$196$		0			0
$197$	13.0000	−	13.0000i	0.926212	−	0.926212i	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	13.0000	−	13.0000i	0.926212	−	0.926212i	0.997459	+	0.0712470i	$0.0226979\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	−2.00000	+	14.0000i	−0.141421	+	0.989949i
$201$		0			0
$202$	20.0000	−	20.0000i	1.40720	−	1.40720i
$203$		0			0
$204$		0			0
$205$	−10.0000	+	20.0000i	−0.698430	+	1.39686i
$206$		0			0
$207$		0			0
$208$	20.0000	+	20.0000i	1.38675	+	1.38675i
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	18.0000	−	18.0000i	1.23625	−	1.23625i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−6.00000	−	6.00000i	−0.406371	−	0.406371i
$219$		0			0
$220$		0			0
$221$	−50.0000			−3.36336
$222$		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
$225$	12.0000	−	9.00000i	0.800000	−	0.600000i
$226$	−2.00000			−0.133038
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$			30.0000i			1.98246i	0.132164	+	0.991228i	$0.457808\pi$
$229$			30.0000i			1.98246i	−0.132164	+	0.991228i	$0.542192\pi$
$230$		0			0
$231$		0			0
$232$	8.00000	−	8.00000i	0.525226	−	0.525226i
$233$	21.0000	+	21.0000i	1.37576	+	1.37576i	0.851658	+	0.524097i	$0.175597\pi$
$233$	21.0000	+	21.0000i	1.37576	+	1.37576i	0.524097	+	0.851658i	$0.324403\pi$
$234$		−	30.0000i		−	1.96116i
$235$		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
$241$	−30.0000			−1.93247			−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−30.0000			−1.93247			−0.966235	+	0.257663i	$0.917048\pi$
$242$	11.0000	−	11.0000i	0.707107	−	0.707107i
$243$		0			0
$244$		−	20.0000i		−	1.28037i
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−9.00000	+	13.0000i	−0.569210	+	0.822192i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$	−15.0000	+	15.0000i	−0.935674	+	0.935674i	−0.998053	−	0.0623783i	$-0.980131\pi$
$257$	−15.0000	+	15.0000i	−0.935674	+	0.935674i	0.0623783	+	0.998053i	$0.480131\pi$
$258$		0			0
$259$		0			0
$260$	10.0000	+	30.0000i	0.620174	+	1.86052i
$261$	−12.0000			−0.742781
$262$		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
$265$	27.0000	−	9.00000i	1.65860	−	0.552866i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	20.0000i		−	1.21942i	−0.792624	−	0.609711i	$-0.791286\pi$
$269$		−	20.0000i		−	1.21942i	0.792624	−	0.609711i	$-0.208714\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−20.0000	+	20.0000i	−1.21268	+	1.21268i
$273$		0			0
$274$		−	14.0000i		−	0.845771i
$275$		0			0
$276$		0			0
$277$	23.0000	−	23.0000i	1.38194	−	1.38194i	0.540758	−	0.841178i	$-0.318138\pi$
$277$	23.0000	−	23.0000i	1.38194	−	1.38194i	0.841178	−	0.540758i	$-0.181862\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−32.0000			−1.90896			−0.954480	−	0.298275i	$-0.903589\pi$
$281$	−32.0000			−1.90896			−0.954480	+	0.298275i	$0.903589\pi$
$282$		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
$288$	−12.0000	−	12.0000i	−0.707107	−	0.707107i
$289$		−	33.0000i		−	1.94118i
$290$	12.0000	−	4.00000i	0.704664	−	0.234888i
$291$		0			0
$292$	10.0000	−	10.0000i	0.585206	−	0.585206i
$293$	−15.0000	−	15.0000i	−0.876309	−	0.876309i	0.116841	−	0.993151i	$-0.462723\pi$
$293$	−15.0000	−	15.0000i	−0.876309	−	0.876309i	−0.993151	+	0.116841i	$0.962723\pi$
$294$		0			0
$295$		0			0
$296$	−28.0000			−1.62747
$297$		0			0
$298$	−14.0000	−	14.0000i	−0.810998	−	0.810998i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	10.0000	−	20.0000i	0.572598	−	1.14520i
$306$	30.0000			1.71499
$307$		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
$313$	−25.0000	−	25.0000i	−1.41308	−	1.41308i	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−25.0000	−	25.0000i	−1.41308	−	1.41308i	−0.678280	−	0.734803i	$-0.737274\pi$
$314$			10.0000i			0.564333i
$315$		0			0
$316$		0			0
$317$	3.00000	−	3.00000i	0.168497	−	0.168497i	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	3.00000	−	3.00000i	0.168497	−	0.168497i	0.786318	+	0.617822i	$0.211985\pi$
$318$		0			0
$319$		0			0
$320$	16.0000	+	8.00000i	0.894427	+	0.447214i
$321$		0			0
$322$		0			0
$323$		0			0
$324$			18.0000i			1.00000i
$325$	−5.00000	+	35.0000i	−0.277350	+	1.94145i
$326$		0			0
$327$		0			0
$328$	20.0000	+	20.0000i	1.10432	+	1.10432i
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	21.0000	+	21.0000i	1.15079	+	1.15079i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	7.00000	−	7.00000i	0.381314	−	0.381314i	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	7.00000	−	7.00000i	0.381314	−	0.381314i	0.871576	+	0.490261i	$0.163099\pi$
$338$	37.0000	+	37.0000i	2.01253	+	2.01253i
$339$		0			0
$340$	−30.0000	+	10.0000i	−1.62698	+	0.542326i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	30.0000			1.61281
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$			10.0000i			0.535288i	0.963518	+	0.267644i	$0.0862451\pi$
$349$			10.0000i			0.535288i	−0.963518	+	0.267644i	$0.913755\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−25.0000	−	25.0000i	−1.33062	−	1.33062i	−0.904819	−	0.425797i	$-0.859994\pi$
$353$	−25.0000	−	25.0000i	−1.33062	−	1.33062i	−0.425797	−	0.904819i	$-0.640006\pi$
$354$		0			0
$355$		0			0
$356$	−20.0000			−1.06000
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	−6.00000	−	18.0000i	−0.316228	−	0.948683i
$361$	−19.0000			−1.00000
$362$	20.0000	−	20.0000i	1.05118	−	1.05118i
$363$		0			0
$364$		0			0
$365$	15.0000	−	5.00000i	0.785136	−	0.261712i
$366$		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
$369$		−	30.0000i		−	1.56174i
$370$	−28.0000	−	14.0000i	−1.45565	−	0.727825i
$371$		0			0
$372$		0			0
$373$	−11.0000	−	11.0000i	−0.569558	−	0.569558i	0.362446	−	0.932005i	$-0.381942\pi$
$373$	−11.0000	−	11.0000i	−0.569558	−	0.569558i	−0.932005	+	0.362446i	$0.881942\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	20.0000	−	20.0000i	1.03005	−	1.03005i
$378$		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
$386$	38.0000			1.93415
$387$		0			0
$388$	10.0000	+	10.0000i	0.507673	+	0.507673i
$389$			34.0000i			1.72387i	0.507020	+	0.861934i	$0.330747\pi$
$389$			34.0000i			1.72387i	−0.507020	+	0.861934i	$0.669253\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		−	26.0000i		−	1.30986i
$395$		0			0
$396$		0			0
$397$	−25.0000	+	25.0000i	−1.25471	+	1.25471i	−0.301131	+	0.953583i	$0.597364\pi$
$397$	−25.0000	+	25.0000i	−1.25471	+	1.25471i	−0.953583	+	0.301131i	$0.902636\pi$
$398$		0			0
$399$		0			0
$400$	12.0000	+	16.0000i	0.600000	+	0.800000i
$401$	−2.00000			−0.0998752			−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000			−0.0998752			−0.0499376	+	0.998752i	$0.515902\pi$
$402$		0			0
$403$		0			0
$404$		−	40.0000i		−	1.99007i
$405$	−9.00000	+	18.0000i	−0.447214	+	0.894427i
$406$		0			0
$407$		0			0
$408$		0			0
$409$			40.0000i			1.97787i	0.148340	+	0.988936i	$0.452607\pi$
$409$			40.0000i			1.97787i	−0.148340	+	0.988936i	$0.547393\pi$
$410$	10.0000	+	30.0000i	0.493865	+	1.48159i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	40.0000			1.96116
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−28.0000			−1.36464			−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000			−1.36464			−0.682318	+	0.731055i	$0.739028\pi$
$422$		0			0
$423$		0			0
$424$		−	36.0000i		−	1.74831i
$425$	−35.0000	−	5.00000i	−1.69775	−	0.242536i
$426$		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
$433$	5.00000	+	5.00000i	0.240285	+	0.240285i	0.816968	−	0.576683i	$-0.195653\pi$
$433$	5.00000	+	5.00000i	0.240285	+	0.240285i	−0.576683	+	0.816968i	$0.695653\pi$
$434$		0			0
$435$		0			0
$436$	−12.0000			−0.574696
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$		0			0
$442$	−50.0000	+	50.0000i	−2.37826	+	2.37826i
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		0			0
$445$	−20.0000	−	10.0000i	−0.948091	−	0.474045i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	14.0000i		−	0.660701i	−0.943858	−	0.330350i	$-0.892833\pi$
$449$		−	14.0000i		−	0.660701i	0.943858	−	0.330350i	$-0.107167\pi$
$450$	3.00000	−	21.0000i	0.141421	−	0.989949i
$451$		0			0
$452$	−2.00000	+	2.00000i	−0.0940721	+	0.0940721i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−17.0000	+	17.0000i	−0.795226	+	0.795226i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$	−17.0000	+	17.0000i	−0.795226	+	0.795226i	0.187112	+	0.982339i	$0.440087\pi$
$458$	30.0000	+	30.0000i	1.40181	+	1.40181i
$459$		0			0
$460$		0			0
$461$	−20.0000			−0.931493			−0.465746	−	0.884918i	$-0.654214\pi$
$461$	−20.0000			−0.931493			−0.465746	+	0.884918i	$0.654214\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		−	16.0000i		−	0.742781i
$465$		0			0
$466$	42.0000			1.94561
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$	−30.0000	−	30.0000i	−1.38675	−	1.38675i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−27.0000	+	27.0000i	−1.23625	+	1.23625i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	−70.0000			−3.19173
$482$	−30.0000	+	30.0000i	−1.36646	+	1.36646i
$483$		0			0
$484$		−	22.0000i		−	1.00000i
$485$	5.00000	+	15.0000i	0.227038	+	0.681115i
$486$		0			0
$487$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$487$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$488$	−20.0000	−	20.0000i	−0.905357	−	0.905357i
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	20.0000	+	20.0000i	0.900755	+	0.900755i
$494$		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	$0.5\pi$
$500$	4.00000	+	22.0000i	0.178885	+	0.983870i
$501$		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
$505$	20.0000	−	40.0000i	0.889988	−	1.77998i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			10.0000i			0.443242i	0.975133	+	0.221621i	$0.0711348\pi$
$509$			10.0000i			0.443242i	−0.975133	+	0.221621i	$0.928865\pi$
$510$		0			0
$511$		0			0
$512$	16.0000	−	16.0000i	0.707107	−	0.707107i
$513$		0			0
$514$			30.0000i			1.32324i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	40.0000	+	20.0000i	1.75412	+	0.877058i
$521$	−40.0000			−1.75243			−0.876216	−	0.481919i	$-0.839940\pi$
$521$	−40.0000			−1.75243			−0.876216	+	0.481919i	$0.839940\pi$
$522$	−12.0000	+	12.0000i	−0.525226	+	0.525226i
$523$		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
$529$			23.0000i			1.00000i
$530$	18.0000	−	36.0000i	0.781870	−	1.56374i
$531$		0			0
$532$		0			0
$533$	50.0000	+	50.0000i	2.16574	+	2.16574i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−20.0000	−	20.0000i	−0.862261	−	0.862261i
$539$		0			0
$540$		0			0
$541$	42.0000			1.80572			0.902861	−	0.429934i	$-0.141463\pi$
$541$	42.0000			1.80572			0.902861	+	0.429934i	$0.141463\pi$
$542$		0			0
$543$		0			0
$544$			40.0000i			1.71499i
$545$	−12.0000	−	6.00000i	−0.514024	−	0.257012i
$546$		0			0
$547$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$547$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$548$	−14.0000	−	14.0000i	−0.598050	−	0.598050i
$549$			30.0000i			1.28037i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		−	46.0000i		−	1.95435i
$555$		0			0
$556$		0			0
$557$	33.0000	−	33.0000i	1.39825	−	1.39825i	0.593199	−	0.805056i	$-0.297865\pi$
$557$	33.0000	−	33.0000i	1.39825	−	1.39825i	0.805056	−	0.593199i	$-0.202135\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−32.0000	+	32.0000i	−1.34984	+	1.34984i
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$		0			0
$565$	−3.00000	+	1.00000i	−0.126211	+	0.0420703i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	26.0000i		−	1.08998i	−0.838444	−	0.544988i	$-0.816534\pi$
$569$		−	26.0000i		−	1.08998i	0.838444	−	0.544988i	$-0.183466\pi$
$570$		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
$576$	−24.0000			−1.00000
$577$	−25.0000	+	25.0000i	−1.04076	+	1.04076i	−0.0416305	+	0.999133i	$0.513255\pi$
$577$	−25.0000	+	25.0000i	−1.04076	+	1.04076i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$	−33.0000	−	33.0000i	−1.37262	−	1.37262i
$579$		0			0
$580$	8.00000	−	16.0000i	0.332182	−	0.664364i
$581$		0			0
$582$		0			0
$583$		0			0
$584$		−	20.0000i		−	0.827606i
$585$	−15.0000	−	45.0000i	−0.620174	−	1.86052i
$586$	−30.0000			−1.23929
$587$		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
$592$	−28.0000	+	28.0000i	−1.15079	+	1.15079i
$593$	−15.0000	−	15.0000i	−0.615976	−	0.615976i	0.328521	−	0.944497i	$-0.393450\pi$
$593$	−15.0000	−	15.0000i	−0.615976	−	0.615976i	−0.944497	+	0.328521i	$0.893450\pi$
$594$		0			0
$595$		0			0
$596$	−28.0000			−1.14692
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	10.0000			0.407909			0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000			0.407909			0.203954	+	0.978980i	$0.434621\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	11.0000	−	22.0000i	0.447214	−	0.894427i
$606$		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
$610$	−10.0000	−	30.0000i	−0.404888	−	1.21466i
$611$		0			0
$612$	30.0000	−	30.0000i	1.21268	−	1.21268i
$613$	1.00000	+	1.00000i	0.0403896	+	0.0403896i	0.727013	−	0.686624i	$-0.240908\pi$
$613$	1.00000	+	1.00000i	0.0403896	+	0.0403896i	−0.686624	+	0.727013i	$0.740908\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−3.00000	+	3.00000i	−0.120775	+	0.120775i	−0.764911	−	0.644136i	$-0.777217\pi$
$617$	−3.00000	+	3.00000i	−0.120775	+	0.120775i	0.644136	+	0.764911i	$0.277217\pi$
$618$		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
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$	−50.0000			−1.99840
$627$		0			0
$628$	10.0000	+	10.0000i	0.399043	+	0.399043i
$629$		−	70.0000i		−	2.79108i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		−	6.00000i		−	0.238290i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	24.0000	−	8.00000i	0.948683	−	0.316228i
$641$	8.00000			0.315981			0.157991	−	0.987441i	$-0.449498\pi$
$641$	8.00000			0.315981			0.157991	+	0.987441i	$0.449498\pi$
$642$		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	$0.750000\pi$
$648$	18.0000	+	18.0000i	0.707107	+	0.707107i
$649$		0			0
$650$	30.0000	+	40.0000i	1.17670	+	1.56893i
$651$		0			0
$652$		0			0
$653$	−9.00000	−	9.00000i	−0.352197	−	0.352197i	0.508729	−	0.860927i	$-0.330115\pi$
$653$	−9.00000	−	9.00000i	−0.352197	−	0.352197i	−0.860927	+	0.508729i	$0.830115\pi$
$654$		0			0
$655$		0			0
$656$	40.0000			1.56174
$657$	−15.0000	+	15.0000i	−0.585206	+	0.585206i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	50.0000			1.94477			0.972387	−	0.233373i	$-0.0749763\pi$
$661$	50.0000			1.94477			0.972387	+	0.233373i	$0.0749763\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	42.0000			1.62747
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	11.0000	+	11.0000i	0.424019	+	0.424019i	0.886585	−	0.462566i	$-0.153071\pi$
$673$	11.0000	+	11.0000i	0.424019	+	0.424019i	−0.462566	+	0.886585i	$0.653071\pi$
$674$		−	14.0000i		−	0.539260i
$675$		0			0
$676$	74.0000			2.84615
$677$	25.0000	−	25.0000i	0.960828	−	0.960828i	−0.0384331	−	0.999261i	$-0.512237\pi$
$677$	25.0000	−	25.0000i	0.960828	−	0.960828i	0.999261	+	0.0384331i	$0.0122367\pi$
$678$		0			0
$679$		0			0
$680$	−20.0000	+	40.0000i	−0.766965	+	1.53393i
$681$		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
$685$	−7.00000	−	21.0000i	−0.267456	−	0.802369i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	90.0000i		−	3.42873i
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$	30.0000	−	30.0000i	1.14043	−	1.14043i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−50.0000	+	50.0000i	−1.89389	+	1.89389i
$698$	10.0000	+	10.0000i	0.378506	+	0.378506i
$699$		0			0
$700$		0			0
$701$	−52.0000			−1.96401			−0.982006	−	0.188847i	$-0.939525\pi$
$701$	−52.0000			−1.96401			−0.982006	+	0.188847i	$0.939525\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−50.0000			−1.88177
$707$		0			0
$708$		0			0
$709$			44.0000i			1.65245i	0.563337	+	0.826227i	$0.309517\pi$
$709$			44.0000i			1.65245i	−0.563337	+	0.826227i	$0.690483\pi$
$710$		0			0
$711$		0			0
$712$	−20.0000	+	20.0000i	−0.749532	+	0.749532i
$713$		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	$0.5\pi$
$720$	−24.0000	−	12.0000i	−0.894427	−	0.447214i
$721$		0			0
$722$	−19.0000	+	19.0000i	−0.707107	+	0.707107i
$723$		0			0
$724$		−	40.0000i		−	1.48659i
$725$	16.0000	−	12.0000i	0.594225	−	0.445669i
$726$		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
$729$		−	27.0000i		−	1.00000i
$730$	10.0000	−	20.0000i	0.370117	−	0.740233i
$731$		0			0
$732$		0			0
$733$	25.0000	+	25.0000i	0.923396	+	0.923396i	0.997268	−	0.0738717i	$-0.0235355\pi$
$733$	25.0000	+	25.0000i	0.923396	+	0.923396i	−0.0738717	+	0.997268i	$0.523536\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−30.0000	−	30.0000i	−1.10432	−	1.10432i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	−42.0000	+	14.0000i	−1.54395	+	0.514650i
$741$		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
$745$	−28.0000	−	14.0000i	−1.02584	−	0.512920i
$746$	−22.0000			−0.805477
$747$		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
$754$		−	40.0000i		−	1.45671i
$755$		0			0
$756$		0			0
$757$	17.0000	−	17.0000i	0.617876	−	0.617876i	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	17.0000	−	17.0000i	0.617876	−	0.617876i	0.944986	+	0.327111i	$0.106075\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	40.0000			1.45000			0.724999	−	0.688749i	$-0.241840\pi$
$761$	40.0000			1.45000			0.724999	+	0.688749i	$0.241840\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	45.0000	−	15.0000i	1.62698	−	0.542326i
$766$		0			0
$767$		0			0
$768$		0			0
$769$			50.0000i			1.80305i	0.432731	+	0.901523i	$0.357550\pi$
$769$			50.0000i			1.80305i	−0.432731	+	0.901523i	$0.642450\pi$
$770$		0			0
$771$		0			0
$772$	38.0000	−	38.0000i	1.36765	−	1.36765i
$773$	−5.00000	−	5.00000i	−0.179838	−	0.179838i	0.611448	−	0.791285i	$-0.290588\pi$
$773$	−5.00000	−	5.00000i	−0.179838	−	0.179838i	−0.791285	+	0.611448i	$0.790588\pi$
$774$		0			0
$775$		0			0
$776$	20.0000			0.717958
$777$		0			0
$778$	34.0000	+	34.0000i	1.21896	+	1.21896i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	5.00000	+	15.0000i	0.178458	+	0.535373i
$786$		0			0
$787$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$787$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$788$	−26.0000	−	26.0000i	−0.926212	−	0.926212i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−50.0000	−	50.0000i	−1.77555	−	1.77555i
$794$			50.0000i			1.77443i
$795$		0			0
$796$		0			0
$797$	−15.0000	+	15.0000i	−0.531327	+	0.531327i	−0.920967	−	0.389640i	$-0.872599\pi$
$797$	−15.0000	+	15.0000i	−0.531327	+	0.531327i	0.389640	+	0.920967i	$0.372599\pi$
$798$		0			0
$799$		0			0
$800$	28.0000	+	4.00000i	0.989949	+	0.141421i
$801$	30.0000			1.06000
$802$	−2.00000	+	2.00000i	−0.0706225	+	0.0706225i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−40.0000	−	40.0000i	−1.40720	−	1.40720i
$809$			56.0000i			1.96886i	0.175791	+	0.984428i	$0.443752\pi$
$809$			56.0000i			1.96886i	−0.175791	+	0.984428i	$0.556248\pi$
$810$	9.00000	+	27.0000i	0.316228	+	0.948683i
$811$		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
$818$	40.0000	+	40.0000i	1.39857	+	1.39857i
$819$		0			0
$820$	40.0000	+	20.0000i	1.39686	+	0.698430i
$821$	−28.0000			−0.977207			−0.488603	−	0.872506i	$-0.662493\pi$
$821$	−28.0000			−0.977207			−0.488603	+	0.872506i	$0.662493\pi$
$822$		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
$829$			20.0000i			0.694629i	0.937749	+	0.347314i	$0.112906\pi$
$829$			20.0000i			0.694629i	−0.937749	+	0.347314i	$0.887094\pi$
$830$		0			0
$831$		0			0
$832$	40.0000	−	40.0000i	1.38675	−	1.38675i
$833$		0			0
$834$		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
$841$	13.0000			0.448276
$842$	−28.0000	+	28.0000i	−0.964944	+	0.964944i
$843$		0			0
$844$		0			0
$845$	74.0000	+	37.0000i	2.54568	+	1.27284i
$846$		0			0
$847$		0			0
$848$	−36.0000	−	36.0000i	−1.23625	−	1.23625i
$849$		0			0
$850$	−40.0000	+	30.0000i	−1.37199	+	1.02899i
$851$		0			0
$852$		0			0
$853$	5.00000	+	5.00000i	0.171197	+	0.171197i	0.787505	−	0.616308i	$-0.211372\pi$
$853$	5.00000	+	5.00000i	0.171197	+	0.171197i	−0.616308	+	0.787505i	$0.711372\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	25.0000	−	25.0000i	0.853984	−	0.853984i	−0.136637	−	0.990621i	$-0.543630\pi$
$857$	25.0000	−	25.0000i	0.853984	−	0.853984i	0.990621	+	0.136637i	$0.0436295\pi$
$858$		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
$865$	45.0000	−	15.0000i	1.53005	−	0.510015i
$866$	10.0000			0.339814
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−12.0000	+	12.0000i	−0.406371	+	0.406371i
$873$	−15.0000	−	15.0000i	−0.507673	−	0.507673i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−23.0000	+	23.0000i	−0.776655	+	0.776655i	−0.979260	−	0.202606i	$-0.935059\pi$
$877$	−23.0000	+	23.0000i	−0.776655	+	0.776655i	0.202606	+	0.979260i	$0.435059\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	50.0000			1.68454			0.842271	−	0.539054i	$-0.181218\pi$
$881$	50.0000			1.68454			0.842271	+	0.539054i	$0.181218\pi$
$882$		0			0
$883$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$883$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$884$			100.000i			3.36336i
$885$		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
$890$	−30.0000	+	10.0000i	−1.00560	+	0.335201i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−14.0000	−	14.0000i	−0.467186	−	0.467186i
$899$		0			0
$900$	−18.0000	−	24.0000i	−0.600000	−	0.800000i
$901$	90.0000			2.99833
$902$		0			0
$903$		0			0
$904$			4.00000i			0.133038i
$905$	20.0000	−	40.0000i	0.664822	−	1.32964i
$906$		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
$909$			60.0000i			1.99007i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$			34.0000i			1.12462i
$915$		0			0
$916$	60.0000			1.98246
$917$		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
$922$	−20.0000	+	20.0000i	−0.658665	+	0.658665i
$923$		0			0
$924$		0			0
$925$	−49.0000	−	7.00000i	−1.61111	−	0.230159i
$926$		0			0
$927$		0			0
$928$	−16.0000	−	16.0000i	−0.525226	−	0.525226i
$929$			40.0000i			1.31236i	0.754606	+	0.656179i	$0.227828\pi$
$929$			40.0000i			1.31236i	−0.754606	+	0.656179i	$0.772172\pi$
$930$		0			0
$931$		0			0
$932$	42.0000	−	42.0000i	1.37576	−	1.37576i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−60.0000			−1.96116
$937$	5.00000	−	5.00000i	0.163343	−	0.163343i	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	5.00000	−	5.00000i	0.163343	−	0.163343i	0.784046	+	0.620703i	$0.213153\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−20.0000			−0.651981			−0.325991	−	0.945373i	$-0.605698\pi$
$941$	−20.0000			−0.651981			−0.325991	+	0.945373i	$0.605698\pi$
$942$		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
$949$		−	50.0000i		−	1.62307i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−41.0000	−	41.0000i	−1.32812	−	1.32812i	−0.907009	−	0.421111i	$-0.861640\pi$
$953$	−41.0000	−	41.0000i	−1.32812	−	1.32812i	−0.421111	−	0.907009i	$-0.638360\pi$
$954$			54.0000i			1.74831i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$	−70.0000	+	70.0000i	−2.25689	+	2.25689i
$963$		0			0
$964$			60.0000i			1.93247i
$965$	57.0000	−	19.0000i	1.83489	−	0.611632i
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$	−22.0000	−	22.0000i	−0.707107	−	0.707107i
$969$		0			0
$970$	20.0000	+	10.0000i	0.642161	+	0.321081i
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−40.0000			−1.28037
$977$	27.0000	−	27.0000i	0.863807	−	0.863807i	−0.127971	−	0.991778i	$-0.540847\pi$
$977$	27.0000	−	27.0000i	0.863807	−	0.863807i	0.991778	+	0.127971i	$0.0408466\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	18.0000			0.574696
$982$		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
$985$	−13.0000	−	39.0000i	−0.414214	−	1.24264i
$986$	40.0000			1.27386
$987$		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
$997$	25.0000	−	25.0000i	0.791758	−	0.791758i	−0.190022	−	0.981780i	$-0.560856\pi$
$997$	25.0000	−	25.0000i	0.791758	−	0.791758i	0.981780	+	0.190022i	$0.0608559\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	980.2.k.c.883.1	yes	2
4.3	odd	2	CM	980.2.k.c.883.1	yes	2
5.2	odd	4	inner	980.2.k.c.687.1	yes	2
7.2	even	3		980.2.x.a.263.1		4
7.3	odd	6		980.2.x.b.863.1		4
7.4	even	3		980.2.x.a.863.1		4
7.5	odd	6		980.2.x.b.263.1		4
7.6	odd	2		980.2.k.b.883.1	yes	2
20.7	even	4	inner	980.2.k.c.687.1	yes	2
28.3	even	6		980.2.x.b.863.1		4
28.11	odd	6		980.2.x.a.863.1		4
28.19	even	6		980.2.x.b.263.1		4
28.23	odd	6		980.2.x.a.263.1		4
28.27	even	2		980.2.k.b.883.1	yes	2
35.2	odd	12		980.2.x.a.67.1		4
35.12	even	12		980.2.x.b.67.1		4
35.17	even	12		980.2.x.b.667.1		4
35.27	even	4		980.2.k.b.687.1	✓	2
35.32	odd	12		980.2.x.a.667.1		4
140.27	odd	4		980.2.k.b.687.1	✓	2
140.47	odd	12		980.2.x.b.67.1		4
140.67	even	12		980.2.x.a.667.1		4
140.87	odd	12		980.2.x.b.667.1		4
140.107	even	12		980.2.x.a.67.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
980.2.k.b.687.1	✓	2	35.27	even	4
980.2.k.b.687.1	✓	2	140.27	odd	4
980.2.k.b.883.1	yes	2	7.6	odd	2
980.2.k.b.883.1	yes	2	28.27	even	2
980.2.k.c.687.1	yes	2	5.2	odd	4	inner
980.2.k.c.687.1	yes	2	20.7	even	4	inner
980.2.k.c.883.1	yes	2	1.1	even	1	trivial
980.2.k.c.883.1	yes	2	4.3	odd	2	CM
980.2.x.a.67.1		4	35.2	odd	12
980.2.x.a.67.1		4	140.107	even	12
980.2.x.a.263.1		4	7.2	even	3
980.2.x.a.263.1		4	28.23	odd	6
980.2.x.a.667.1		4	35.32	odd	12
980.2.x.a.667.1		4	140.67	even	12
980.2.x.a.863.1		4	7.4	even	3
980.2.x.a.863.1		4	28.11	odd	6
980.2.x.b.67.1		4	35.12	even	12
980.2.x.b.67.1		4	140.47	odd	12
980.2.x.b.263.1		4	7.5	odd	6
980.2.x.b.263.1		4	28.19	even	6
980.2.x.b.667.1		4	35.17	even	12
980.2.x.b.667.1		4	140.87	odd	12
980.2.x.b.863.1		4	7.3	odd	6
980.2.x.b.863.1		4	28.3	even	6

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:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	980.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(1.00000 - 2.00000i) q^{5} +(-2.00000 - 2.00000i) q^{8} +3.00000i q^{9} +O(q^{10})\)	\(q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(1.00000 - 2.00000i) q^{5} +(-2.00000 - 2.00000i) q^{8} +3.00000i q^{9} +(-1.00000 - 3.00000i) q^{10} +(-5.00000 - 5.00000i) q^{13} -4.00000 q^{16} +(5.00000 - 5.00000i) q^{17} +(3.00000 + 3.00000i) q^{18} +(-4.00000 - 2.00000i) q^{20} +(-3.00000 - 4.00000i) q^{25} -10.0000 q^{26} +4.00000i q^{29} +(-4.00000 + 4.00000i) q^{32} -10.0000i q^{34} +6.00000 q^{36} +(7.00000 - 7.00000i) q^{37} +(-6.00000 + 2.00000i) q^{40} -10.0000 q^{41} +(6.00000 + 3.00000i) q^{45} +(-7.00000 - 1.00000i) q^{50} +(-10.0000 + 10.0000i) q^{52} +(9.00000 + 9.00000i) q^{53} +(4.00000 + 4.00000i) q^{58} +10.0000 q^{61} +8.00000i q^{64} +(-15.0000 + 5.00000i) q^{65} +(-10.0000 - 10.0000i) q^{68} +(6.00000 - 6.00000i) q^{72} +(5.00000 + 5.00000i) q^{73} -14.0000i q^{74} +(-4.00000 + 8.00000i) q^{80} -9.00000 q^{81} +(-10.0000 + 10.0000i) q^{82} +(-5.00000 - 15.0000i) q^{85} -10.0000i q^{89} +(9.00000 - 3.00000i) q^{90} +(-5.00000 + 5.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{5} - 4 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^5 - 4 * q^8	\( 2 q + 2 q^{2} + 2 q^{5} - 4 q^{8} - 2 q^{10} - 10 q^{13} - 8 q^{16} + 10 q^{17} + 6 q^{18} - 8 q^{20} - 6 q^{25} - 20 q^{26} - 8 q^{32} + 12 q^{36} + 14 q^{37} - 12 q^{40} - 20 q^{41} + 12 q^{45} - 14 q^{50} - 20 q^{52} + 18 q^{53} + 8 q^{58} + 20 q^{61} - 30 q^{65} - 20 q^{68} + 12 q^{72} + 10 q^{73} - 8 q^{80} - 18 q^{81} - 20 q^{82} - 10 q^{85} + 18 q^{90} - 10 q^{97}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^5 - 4 * q^8 - 2 * q^10 - 10 * q^13 - 8 * q^16 + 10 * q^17 + 6 * q^18 - 8 * q^20 - 6 * q^25 - 20 * q^26 - 8 * q^32 + 12 * q^36 + 14 * q^37 - 12 * q^40 - 20 * q^41 + 12 * q^45 - 14 * q^50 - 20 * q^52 + 18 * q^53 + 8 * q^58 + 20 * q^61 - 30 * q^65 - 20 * q^68 + 12 * q^72 + 10 * q^73 - 8 * q^80 - 18 * q^81 - 20 * q^82 - 10 * q^85 + 18 * q^90 - 10 * q^97

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