LMFDB - Embedded newform 180.2.k.b.163.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [180,2,Mod(127,180)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(180, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("180.127");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 180 = 2^{2} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	180.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:	$1.43730723638$
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			163.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	180.163
Dual form			180.2.k.b.127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(-1.00000 - 2.00000i) q^{5} +(-2.00000 - 2.00000i) q^{8} +O(q^{10})$	$q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(-1.00000 - 2.00000i) q^{5} +(-2.00000 - 2.00000i) q^{8} +(-3.00000 - 1.00000i) q^{10} +(5.00000 + 5.00000i) q^{13} -4.00000 q^{16} +(3.00000 - 3.00000i) q^{17} +(-4.00000 + 2.00000i) q^{20} +(-3.00000 + 4.00000i) q^{25} +10.0000 q^{26} +10.0000i q^{29} +(-4.00000 + 4.00000i) q^{32} -6.00000i q^{34} +(5.00000 - 5.00000i) q^{37} +(-2.00000 + 6.00000i) q^{40} -10.0000 q^{41} -7.00000i q^{49} +(1.00000 + 7.00000i) q^{50} +(10.0000 - 10.0000i) q^{52} +(9.00000 + 9.00000i) q^{53} +(10.0000 + 10.0000i) q^{58} -12.0000 q^{61} +8.00000i q^{64} +(5.00000 - 15.0000i) q^{65} +(-6.00000 - 6.00000i) q^{68} +(-5.00000 - 5.00000i) q^{73} -10.0000i q^{74} +(4.00000 + 8.00000i) q^{80} +(-10.0000 + 10.0000i) q^{82} +(-9.00000 - 3.00000i) q^{85} -10.0000i q^{89} +(-5.00000 + 5.00000i) q^{97} +(-7.00000 - 7.00000i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 6 q^{10} + 10 q^{13} - 8 q^{16} + 6 q^{17} - 8 q^{20} - 6 q^{25} + 20 q^{26} - 8 q^{32} + 10 q^{37} - 4 q^{40} - 20 q^{41} + 2 q^{50} + 20 q^{52} + 18 q^{53} + 20 q^{58} - 24 q^{61} + 10 q^{65} - 12 q^{68} - 10 q^{73} + 8 q^{80} - 20 q^{82} - 18 q^{85} - 10 q^{97} - 14 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^5 - 4 * q^8 - 6 * q^10 + 10 * q^13 - 8 * q^16 + 6 * q^17 - 8 * q^20 - 6 * q^25 + 20 * q^26 - 8 * q^32 + 10 * q^37 - 4 * q^40 - 20 * q^41 + 2 * q^50 + 20 * q^52 + 18 * q^53 + 20 * q^58 - 24 * q^61 + 10 * q^65 - 12 * q^68 - 10 * q^73 + 8 * q^80 - 20 * q^82 - 18 * q^85 - 10 * q^97 - 14 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/180\mathbb{Z}\right)^\times$.

$n$	$37$	$91$	$101$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$	$1$

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
$3$		0			0
$4$		−	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		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$	−2.00000	−	2.00000i	−0.707107	−	0.707107i
$9$		0			0
$10$	−3.00000	−	1.00000i	−0.948683	−	0.316228i
$11$		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	$0.187167\pi$
$13$	5.00000	+	5.00000i	1.38675	+	1.38675i	0.554700	+	0.832050i	$0.312833\pi$
$14$		0			0
$15$		0			0
$16$	−4.00000			−1.00000
$17$	3.00000	−	3.00000i	0.727607	−	0.727607i	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	3.00000	−	3.00000i	0.727607	−	0.727607i	0.970143	+	0.242536i	$0.0779791\pi$
$18$		0			0
$19$		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
$29$			10.0000i			1.85695i	0.371391	+	0.928477i	$0.378881\pi$
$29$			10.0000i			1.85695i	−0.371391	+	0.928477i	$0.621119\pi$
$30$		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
$34$		−	6.00000i		−	1.02899i
$35$		0			0
$36$		0			0
$37$	5.00000	−	5.00000i	0.821995	−	0.821995i	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	5.00000	−	5.00000i	0.821995	−	0.821995i	0.986394	+	0.164399i	$0.0525685\pi$
$38$		0			0
$39$		0			0
$40$	−2.00000	+	6.00000i	−0.316228	+	0.948683i
$41$	−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
$45$		0			0
$46$		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
$49$		−	7.00000i		−	1.00000i
$50$	1.00000	+	7.00000i	0.141421	+	0.989949i
$51$		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
$58$	10.0000	+	10.0000i	1.31306	+	1.31306i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	−12.0000			−1.53644			−0.768221	−	0.640184i	$-0.778858\pi$
$61$	−12.0000			−1.53644			−0.768221	+	0.640184i	$0.778858\pi$
$62$		0			0
$63$		0			0
$64$			8.00000i			1.00000i
$65$	5.00000	−	15.0000i	0.620174	−	1.86052i
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$	−6.00000	−	6.00000i	−0.727607	−	0.727607i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−5.00000	−	5.00000i	−0.585206	−	0.585206i	0.351123	−	0.936329i	$-0.385800\pi$
$73$	−5.00000	−	5.00000i	−0.585206	−	0.585206i	−0.936329	+	0.351123i	$0.885800\pi$
$74$		−	10.0000i		−	1.16248i
$75$		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
$81$		0			0
$82$	−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
$85$	−9.00000	−	3.00000i	−0.976187	−	0.325396i
$86$		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	$-0.177808\pi$
$90$		0			0
$91$		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	$0.366875\pi$
$98$	−7.00000	−	7.00000i	−0.707107	−	0.707107i
$99$		0			0
$100$	8.00000	+	6.00000i	0.800000	+	0.600000i
$101$	−20.0000			−1.99007			−0.995037	−	0.0995037i	$-0.968274\pi$
$101$	−20.0000			−1.99007			−0.995037	+	0.0995037i	$0.968274\pi$
$102$		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	0.752577	−	0.658505i	$-0.228811\pi$
$113$	1.00000	+	1.00000i	0.0940721	+	0.0940721i	−0.658505	+	0.752577i	$0.728811\pi$
$114$		0			0
$115$		0			0
$116$	20.0000			1.85695
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$	−12.0000	+	12.0000i	−1.08643	+	1.08643i
$123$		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
$136$	−12.0000			−1.02899
$137$	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
$144$		0			0
$145$	20.0000	−	10.0000i	1.66091	−	0.830455i
$146$	−10.0000			−0.827606
$147$		0			0
$148$	−10.0000	−	10.0000i	−0.821995	−	0.821995i
$149$			20.0000i			1.63846i	0.573462	+	0.819232i	$0.305600\pi$
$149$			20.0000i			1.63846i	−0.573462	+	0.819232i	$0.694400\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		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
$160$	12.0000	+	4.00000i	0.948683	+	0.316228i
$161$		0			0
$162$		0			0
$163$		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
$170$	−12.0000	+	6.00000i	−0.920358	+	0.460179i
$171$		0			0
$172$		0			0
$173$	11.0000	+	11.0000i	0.836315	+	0.836315i	0.988372	−	0.152057i	$-0.0485898\pi$
$173$	11.0000	+	11.0000i	0.836315	+	0.836315i	−0.152057	+	0.988372i	$0.548590\pi$
$174$		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	$0.5\pi$
$180$		0			0
$181$	18.0000			1.33793			0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000			1.33793			0.668965	+	0.743294i	$0.266738\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−15.0000	−	5.00000i	−1.10282	−	0.367607i
$186$		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
$193$	−5.00000	−	5.00000i	−0.359908	−	0.359908i	0.503871	−	0.863779i	$-0.331909\pi$
$193$	−5.00000	−	5.00000i	−0.359908	−	0.359908i	−0.863779	+	0.503871i	$0.831909\pi$
$194$			10.0000i			0.717958i
$195$		0			0
$196$	−14.0000			−1.00000
$197$	−13.0000	+	13.0000i	−0.926212	+	0.926212i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$	−13.0000	+	13.0000i	−0.926212	+	0.926212i	0.0712470	+	0.997459i	$0.477302\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	14.0000	−	2.00000i	0.989949	−	0.141421i
$201$		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
$221$	30.0000			2.01802
$222$		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
$225$		0			0
$226$	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
$229$		−	4.00000i		−	0.264327i	−0.991228	−	0.132164i	$-0.957808\pi$
$229$		−	4.00000i		−	0.264327i	0.991228	−	0.132164i	$-0.0421925\pi$
$230$		0			0
$231$		0			0
$232$	20.0000	−	20.0000i	1.31306	−	1.31306i
$233$	−21.0000	−	21.0000i	−1.37576	−	1.37576i	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−21.0000	−	21.0000i	−1.37576	−	1.37576i	−0.524097	−	0.851658i	$-0.675597\pi$
$234$		0			0
$235$		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
$241$	−8.00000			−0.515325			−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000			−0.515325			−0.257663	+	0.966235i	$0.582952\pi$
$242$	11.0000	−	11.0000i	0.707107	−	0.707107i
$243$		0			0
$244$			24.0000i			1.53644i
$245$	−14.0000	+	7.00000i	−0.894427	+	0.447214i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	13.0000	−	9.00000i	0.822192	−	0.569210i
$251$		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
$257$	17.0000	−	17.0000i	1.06043	−	1.06043i	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	17.0000	−	17.0000i	1.06043	−	1.06043i	0.998053	−	0.0623783i	$-0.0198685\pi$
$258$		0			0
$259$		0			0
$260$	−30.0000	−	10.0000i	−1.86052	−	0.620174i
$261$		0			0
$262$		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
$265$	9.00000	−	27.0000i	0.552866	−	1.65860i
$266$		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			$\pi$
$272$	−12.0000	+	12.0000i	−0.727607	+	0.727607i
$273$		0			0
$274$		−	14.0000i		−	0.845771i
$275$		0			0
$276$		0			0
$277$	5.00000	−	5.00000i	0.300421	−	0.300421i	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	5.00000	−	5.00000i	0.300421	−	0.300421i	0.841178	+	0.540758i	$0.181862\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	10.0000			0.596550			0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000			0.596550			0.298275	+	0.954480i	$0.403589\pi$
$282$		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
$288$		0			0
$289$		−	1.00000i		−	0.0588235i
$290$	10.0000	−	30.0000i	0.587220	−	1.76166i
$291$		0			0
$292$	−10.0000	+	10.0000i	−0.585206	+	0.585206i
$293$	−19.0000	−	19.0000i	−1.10999	−	1.10999i	−0.993151	−	0.116841i	$-0.962723\pi$
$293$	−19.0000	−	19.0000i	−1.10999	−	1.10999i	−0.116841	−	0.993151i	$-0.537277\pi$
$294$		0			0
$295$		0			0
$296$	−20.0000			−1.16248
$297$		0			0
$298$	20.0000	+	20.0000i	1.15857	+	1.15857i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	12.0000	+	24.0000i	0.687118	+	1.37424i
$306$		0			0
$307$		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
$324$		0			0
$325$	−35.0000	+	5.00000i	−1.94145	+	0.277350i
$326$		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
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−25.0000	+	25.0000i	−1.36184	+	1.36184i	−0.490261	+	0.871576i	$0.663099\pi$
$337$	−25.0000	+	25.0000i	−1.36184	+	1.36184i	−0.871576	+	0.490261i	$0.836901\pi$
$338$	37.0000	+	37.0000i	2.01253	+	2.01253i
$339$		0			0
$340$	−6.00000	+	18.0000i	−0.325396	+	0.976187i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	22.0000			1.18273
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$		−	36.0000i		−	1.92704i	−0.267644	−	0.963518i	$-0.586245\pi$
$349$		−	36.0000i		−	1.92704i	0.267644	−	0.963518i	$-0.413755\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	9.00000	+	9.00000i	0.479022	+	0.479022i	0.904819	−	0.425797i	$-0.140006\pi$
$353$	9.00000	+	9.00000i	0.479022	+	0.479022i	−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	$0.5\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$	18.0000	−	18.0000i	0.946059	−	0.946059i
$363$		0			0
$364$		0			0
$365$	−5.00000	+	15.0000i	−0.261712	+	0.785136i
$366$		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
$369$		0			0
$370$	−20.0000	+	10.0000i	−1.03975	+	0.519875i
$371$		0			0
$372$		0			0
$373$	25.0000	+	25.0000i	1.29445	+	1.29445i	0.932005	+	0.362446i	$0.118058\pi$
$373$	25.0000	+	25.0000i	1.29445	+	1.29445i	0.362446	+	0.932005i	$0.381942\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−50.0000	+	50.0000i	−2.57513	+	2.57513i
$378$		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
$386$	−10.0000			−0.508987
$387$		0			0
$388$	10.0000	+	10.0000i	0.507673	+	0.507673i
$389$		−	20.0000i		−	1.01404i	−0.861934	−	0.507020i	$-0.830747\pi$
$389$		−	20.0000i		−	1.01404i	0.861934	−	0.507020i	$-0.169253\pi$
$390$		0			0
$391$		0			0
$392$	−14.0000	+	14.0000i	−0.707107	+	0.707107i
$393$		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	$-0.402636\pi$
$397$	25.0000	−	25.0000i	1.25471	−	1.25471i	0.953583	−	0.301131i	$-0.0973643\pi$
$398$		0			0
$399$		0			0
$400$	12.0000	−	16.0000i	0.600000	−	0.800000i
$401$	40.0000			1.99750			0.998752	−	0.0499376i	$-0.0159023\pi$
$401$	40.0000			1.99750			0.998752	+	0.0499376i	$0.0159023\pi$
$402$		0			0
$403$		0			0
$404$			40.0000i			1.99007i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	6.00000i		−	0.296681i	−0.988936	−	0.148340i	$-0.952607\pi$
$409$		−	6.00000i		−	0.296681i	0.988936	−	0.148340i	$-0.0473931\pi$
$410$	30.0000	+	10.0000i	1.48159	+	0.493865i
$411$		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
$425$	3.00000	+	21.0000i	0.145521	+	1.01865i
$426$		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
$442$	30.0000	−	30.0000i	1.42695	−	1.42695i
$443$		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
$449$		−	40.0000i		−	1.88772i	−0.330350	−	0.943858i	$-0.607167\pi$
$449$		−	40.0000i		−	1.88772i	0.330350	−	0.943858i	$-0.392833\pi$
$450$		0			0
$451$		0			0
$452$	2.00000	−	2.00000i	0.0940721	−	0.0940721i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	25.0000	−	25.0000i	1.16945	−	1.16945i	0.187112	−	0.982339i	$-0.440087\pi$
$457$	25.0000	−	25.0000i	1.16945	−	1.16945i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$	−4.00000	−	4.00000i	−0.186908	−	0.186908i
$459$		0			0
$460$		0			0
$461$	20.0000			0.931493			0.465746	−	0.884918i	$-0.345786\pi$
$461$	20.0000			0.931493			0.465746	+	0.884918i	$0.345786\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		−	40.0000i		−	1.85695i
$465$		0			0
$466$	−42.0000			−1.94561
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	50.0000			2.27980
$482$	−8.00000	+	8.00000i	−0.364390	+	0.364390i
$483$		0			0
$484$		−	22.0000i		−	1.00000i
$485$	15.0000	+	5.00000i	0.681115	+	0.227038i
$486$		0			0
$487$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$487$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$488$	24.0000	+	24.0000i	1.08643	+	1.08643i
$489$		0			0
$490$	−7.00000	+	21.0000i	−0.316228	+	0.948683i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	30.0000	+	30.0000i	1.35113	+	1.35113i
$494$		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	$-0.928865\pi$
$509$		−	10.0000i		−	0.443242i	0.975133	−	0.221621i	$-0.0711348\pi$
$510$		0			0
$511$		0			0
$512$	16.0000	−	16.0000i	0.707107	−	0.707107i
$513$		0			0
$514$		−	34.0000i		−	1.49968i
$515$		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	$0.839940\pi$
$522$		0			0
$523$		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	$-0.858537\pi$
$541$	−42.0000			−1.80572			−0.902861	+	0.429934i	$0.858537\pi$
$542$		0			0
$543$		0			0
$544$			24.0000i			1.02899i
$545$	−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
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		−	10.0000i		−	0.424859i
$555$		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
$562$	10.0000	−	10.0000i	0.421825	−	0.421825i
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$		0			0
$565$	1.00000	−	3.00000i	0.0420703	−	0.126211i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			40.0000i			1.67689i	0.544988	+	0.838444i	$0.316534\pi$
$569$			40.0000i			1.67689i	−0.544988	+	0.838444i	$0.683466\pi$
$570$		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
$576$		0			0
$577$	−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	$0.986745\pi$
$578$	−1.00000	−	1.00000i	−0.0415945	−	0.0415945i
$579$		0			0
$580$	−20.0000	−	40.0000i	−0.830455	−	1.66091i
$581$		0			0
$582$		0			0
$583$		0			0
$584$			20.0000i			0.827606i
$585$		0			0
$586$	−38.0000			−1.56977
$587$		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
$592$	−20.0000	+	20.0000i	−0.821995	+	0.821995i
$593$	31.0000	+	31.0000i	1.27302	+	1.27302i	0.944497	+	0.328521i	$0.106550\pi$
$593$	31.0000	+	31.0000i	1.27302	+	1.27302i	0.328521	+	0.944497i	$0.393450\pi$
$594$		0			0
$595$		0			0
$596$	40.0000			1.63846
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	48.0000			1.95796			0.978980	−	0.203954i	$-0.0653794\pi$
$601$	48.0000			1.95796			0.978980	+	0.203954i	$0.0653794\pi$
$602$		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
$610$	36.0000	+	12.0000i	1.45760	+	0.485866i
$611$		0			0
$612$		0			0
$613$	35.0000	+	35.0000i	1.41364	+	1.41364i	0.727013	+	0.686624i	$0.240908\pi$
$613$	35.0000	+	35.0000i	1.41364	+	1.41364i	0.686624	+	0.727013i	$0.259092\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	3.00000	−	3.00000i	0.120775	−	0.120775i	−0.644136	−	0.764911i	$-0.722783\pi$
$617$	3.00000	−	3.00000i	0.120775	−	0.120775i	0.764911	+	0.644136i	$0.222783\pi$
$618$		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
$629$		−	30.0000i		−	1.19618i
$630$		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
$637$	35.0000	−	35.0000i	1.38675	−	1.38675i
$638$		0			0
$639$		0			0
$640$	8.00000	−	24.0000i	0.316228	−	0.948683i
$641$	50.0000			1.97488			0.987441	−	0.157991i	$-0.0505015\pi$
$641$	50.0000			1.97488			0.987441	+	0.157991i	$0.0505015\pi$
$642$		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	$0.750000\pi$
$648$		0			0
$649$		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	0.860927	−	0.508729i	$-0.169885\pi$
$653$	9.00000	+	9.00000i	0.352197	+	0.352197i	−0.508729	+	0.860927i	$0.669885\pi$
$654$		0			0
$655$		0			0
$656$	40.0000			1.56174
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−12.0000			−0.466746			−0.233373	−	0.972387i	$-0.574976\pi$
$661$	−12.0000			−0.466746			−0.233373	+	0.972387i	$0.574976\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−35.0000	−	35.0000i	−1.34915	−	1.34915i	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−35.0000	−	35.0000i	−1.34915	−	1.34915i	−0.462566	−	0.886585i	$-0.653071\pi$
$674$			50.0000i			1.92593i
$675$		0			0
$676$	74.0000			2.84615
$677$	−27.0000	+	27.0000i	−1.03769	+	1.03769i	−0.0384331	+	0.999261i	$0.512237\pi$
$677$	−27.0000	+	27.0000i	−1.03769	+	1.03769i	−0.999261	+	0.0384331i	$0.987763\pi$
$678$		0			0
$679$		0			0
$680$	12.0000	+	24.0000i	0.460179	+	0.920358i
$681$		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
$685$	−21.0000	−	7.00000i	−0.802369	−	0.267456i
$686$		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			$\pi$
$692$	22.0000	−	22.0000i	0.836315	−	0.836315i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−30.0000	+	30.0000i	−1.13633	+	1.13633i
$698$	−36.0000	−	36.0000i	−1.36262	−	1.36262i
$699$		0			0
$700$		0			0
$701$	−10.0000			−0.377695			−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000			−0.377695			−0.188847	+	0.982006i	$0.560475\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	18.0000			0.677439
$707$		0			0
$708$		0			0
$709$		−	44.0000i		−	1.65245i	−0.563337	−	0.826227i	$-0.690483\pi$
$709$		−	44.0000i		−	1.65245i	0.563337	−	0.826227i	$-0.309517\pi$
$710$		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	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	−19.0000	+	19.0000i	−0.707107	+	0.707107i
$723$		0			0
$724$		−	36.0000i		−	1.33793i
$725$	−40.0000	−	30.0000i	−1.48556	−	1.11417i
$726$		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
$729$		0			0
$730$	10.0000	+	20.0000i	0.370117	+	0.740233i
$731$		0			0
$732$		0			0
$733$	−25.0000	−	25.0000i	−0.923396	−	0.923396i	0.0738717	−	0.997268i	$-0.476464\pi$
$733$	−25.0000	−	25.0000i	−0.923396	−	0.923396i	−0.997268	+	0.0738717i	$0.976464\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	−10.0000	+	30.0000i	−0.367607	+	1.10282i
$741$		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
$745$	40.0000	−	20.0000i	1.46549	−	0.732743i
$746$	50.0000			1.83063
$747$		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
$754$			100.000i			3.64179i
$755$		0			0
$756$		0			0
$757$	−35.0000	+	35.0000i	−1.27210	+	1.27210i	−0.327111	+	0.944986i	$0.606075\pi$
$757$	−35.0000	+	35.0000i	−1.27210	+	1.27210i	−0.944986	+	0.327111i	$0.893925\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−40.0000			−1.45000			−0.724999	−	0.688749i	$-0.758160\pi$
$761$	−40.0000			−1.45000			−0.724999	+	0.688749i	$0.758160\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$			24.0000i			0.865462i	0.901523	+	0.432731i	$0.142450\pi$
$769$			24.0000i			0.865462i	−0.901523	+	0.432731i	$0.857550\pi$
$770$		0			0
$771$		0			0
$772$	−10.0000	+	10.0000i	−0.359908	+	0.359908i
$773$	39.0000	+	39.0000i	1.40273	+	1.40273i	0.791285	+	0.611448i	$0.209412\pi$
$773$	39.0000	+	39.0000i	1.40273	+	1.40273i	0.611448	+	0.791285i	$0.290588\pi$
$774$		0			0
$775$		0			0
$776$	20.0000			0.717958
$777$		0			0
$778$	−20.0000	−	20.0000i	−0.717035	−	0.717035i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$			28.0000i			1.00000i
$785$	15.0000	+	5.00000i	0.535373	+	0.178458i
$786$		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
$793$	−60.0000	−	60.0000i	−2.13066	−	2.13066i
$794$		−	50.0000i		−	1.77443i
$795$		0			0
$796$		0			0
$797$	37.0000	−	37.0000i	1.31061	−	1.31061i	0.389640	−	0.920967i	$-0.372599\pi$
$797$	37.0000	−	37.0000i	1.31061	−	1.31061i	0.920967	−	0.389640i	$-0.127401\pi$
$798$		0

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:	\( N \)	\(=\)	\( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	180.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(-1.00000 - 2.00000i) q^{5} +(-2.00000 - 2.00000i) q^{8} +O(q^{10})\)	\(q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(-1.00000 - 2.00000i) q^{5} +(-2.00000 - 2.00000i) q^{8} +(-3.00000 - 1.00000i) q^{10} +(5.00000 + 5.00000i) q^{13} -4.00000 q^{16} +(3.00000 - 3.00000i) q^{17} +(-4.00000 + 2.00000i) q^{20} +(-3.00000 + 4.00000i) q^{25} +10.0000 q^{26} +10.0000i q^{29} +(-4.00000 + 4.00000i) q^{32} -6.00000i q^{34} +(5.00000 - 5.00000i) q^{37} +(-2.00000 + 6.00000i) q^{40} -10.0000 q^{41} -7.00000i q^{49} +(1.00000 + 7.00000i) q^{50} +(10.0000 - 10.0000i) q^{52} +(9.00000 + 9.00000i) q^{53} +(10.0000 + 10.0000i) q^{58} -12.0000 q^{61} +8.00000i q^{64} +(5.00000 - 15.0000i) q^{65} +(-6.00000 - 6.00000i) q^{68} +(-5.00000 - 5.00000i) q^{73} -10.0000i q^{74} +(4.00000 + 8.00000i) q^{80} +(-10.0000 + 10.0000i) q^{82} +(-9.00000 - 3.00000i) q^{85} -10.0000i q^{89} +(-5.00000 + 5.00000i) q^{97} +(-7.00000 - 7.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{5} - 4 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^5 - 4 * q^8	\( 2 q + 2 q^{2} - 2 q^{5} - 4 q^{8} - 6 q^{10} + 10 q^{13} - 8 q^{16} + 6 q^{17} - 8 q^{20} - 6 q^{25} + 20 q^{26} - 8 q^{32} + 10 q^{37} - 4 q^{40} - 20 q^{41} + 2 q^{50} + 20 q^{52} + 18 q^{53} + 20 q^{58} - 24 q^{61} + 10 q^{65} - 12 q^{68} - 10 q^{73} + 8 q^{80} - 20 q^{82} - 18 q^{85} - 10 q^{97} - 14 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^5 - 4 * q^8 - 6 * q^10 + 10 * q^13 - 8 * q^16 + 6 * q^17 - 8 * q^20 - 6 * q^25 + 20 * q^26 - 8 * q^32 + 10 * q^37 - 4 * q^40 - 20 * q^41 + 2 * q^50 + 20 * q^52 + 18 * q^53 + 20 * q^58 - 24 * q^61 + 10 * q^65 - 12 * q^68 - 10 * q^73 + 8 * q^80 - 20 * q^82 - 18 * q^85 - 10 * q^97 - 14 * q^98

Introduction

L-functions

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Varieties

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Database

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