LMFDB - Embedded newform 116.2.e.a.75.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [116,2,Mod(75,116)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(116, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("116.75");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 116 = 2^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	116.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:	$0.926264663447$
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			75.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	116.75
Dual form			116.2.e.a.99.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.00000i) q^{2} -2.00000i q^{4} -4.00000i q^{5} +(2.00000 + 2.00000i) q^{8} -3.00000i q^{9} +O(q^{10})$	$q+(-1.00000 + 1.00000i) q^{2} -2.00000i q^{4} -4.00000i q^{5} +(2.00000 + 2.00000i) q^{8} -3.00000i q^{9} +(4.00000 + 4.00000i) q^{10} +6.00000i q^{13} -4.00000 q^{16} +(5.00000 - 5.00000i) q^{17} +(3.00000 + 3.00000i) q^{18} -8.00000 q^{20} -11.0000 q^{25} +(-6.00000 - 6.00000i) q^{26} +(-2.00000 + 5.00000i) q^{29} +(4.00000 - 4.00000i) q^{32} +10.0000i q^{34} -6.00000 q^{36} +(7.00000 + 7.00000i) q^{37} +(8.00000 - 8.00000i) q^{40} +(1.00000 + 1.00000i) q^{41} -12.0000 q^{45} +7.00000 q^{49} +(11.0000 - 11.0000i) q^{50} +12.0000 q^{52} -4.00000 q^{53} +(-3.00000 - 7.00000i) q^{58} +(-1.00000 + 1.00000i) q^{61} +8.00000i q^{64} +24.0000 q^{65} +(-10.0000 - 10.0000i) q^{68} +(6.00000 - 6.00000i) q^{72} +(11.0000 + 11.0000i) q^{73} -14.0000 q^{74} +16.0000i q^{80} -9.00000 q^{81} -2.00000 q^{82} +(-20.0000 - 20.0000i) q^{85} +(-3.00000 + 3.00000i) q^{89} +(12.0000 - 12.0000i) q^{90} +(-5.00000 - 5.00000i) q^{97} +(-7.00000 + 7.00000i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 4 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 4 * q^8	$ 2 q - 2 q^{2} + 4 q^{8} + 8 q^{10} - 8 q^{16} + 10 q^{17} + 6 q^{18} - 16 q^{20} - 22 q^{25} - 12 q^{26} - 4 q^{29} + 8 q^{32} - 12 q^{36} + 14 q^{37} + 16 q^{40} + 2 q^{41} - 24 q^{45} + 14 q^{49} + 22 q^{50} + 24 q^{52} - 8 q^{53} - 6 q^{58} - 2 q^{61} + 48 q^{65} - 20 q^{68} + 12 q^{72} + 22 q^{73} - 28 q^{74} - 18 q^{81} - 4 q^{82} - 40 q^{85} - 6 q^{89} + 24 q^{90} - 10 q^{97} - 14 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 4 * q^8 + 8 * q^10 - 8 * q^16 + 10 * q^17 + 6 * q^18 - 16 * q^20 - 22 * q^25 - 12 * q^26 - 4 * q^29 + 8 * q^32 - 12 * q^36 + 14 * q^37 + 16 * q^40 + 2 * q^41 - 24 * q^45 + 14 * q^49 + 22 * q^50 + 24 * q^52 - 8 * q^53 - 6 * q^58 - 2 * q^61 + 48 * q^65 - 20 * q^68 + 12 * q^72 + 22 * q^73 - 28 * q^74 - 18 * q^81 - 4 * q^82 - 40 * q^85 - 6 * q^89 + 24 * q^90 - 10 * q^97 - 14 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/116\mathbb{Z}\right)^\times$.

$n$	$59$	$89$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	−1.00000	+	1.00000i	−0.707107	+	0.707107i
$2$	−1.00000	+	1.00000i	−0.707107	+	0.707107i
$3$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$3$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$4$		−	2.00000i		−	1.00000i
$5$		−	4.00000i		−	1.78885i	−0.447214	−	0.894427i	$-0.647584\pi$
$5$		−	4.00000i		−	1.78885i	0.447214	−	0.894427i	$-0.352416\pi$
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$	2.00000	+	2.00000i	0.707107	+	0.707107i
$9$		−	3.00000i		−	1.00000i
$10$	4.00000	+	4.00000i	1.26491	+	1.26491i
$11$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$11$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$12$		0			0
$13$			6.00000i			1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$			6.00000i			1.66410i	−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		0.707107	−	0.707107i	$-0.250000\pi$
$19$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$20$	−8.00000			−1.78885
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	−11.0000			−2.20000
$26$	−6.00000	−	6.00000i	−1.17670	−	1.17670i
$27$		0			0
$28$		0			0
$29$	−2.00000	+	5.00000i	−0.371391	+	0.928477i
$29$	−2.00000	+	5.00000i	−0.371391	+	0.928477i
$30$		0			0
$31$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$31$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$32$	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	0.986394	+	0.164399i	$0.0525685\pi$
$37$	7.00000	+	7.00000i	1.15079	+	1.15079i	0.164399	+	0.986394i	$0.447432\pi$
$38$		0			0
$39$		0			0
$40$	8.00000	−	8.00000i	1.26491	−	1.26491i
$41$	1.00000	+	1.00000i	0.156174	+	0.156174i	0.780869	−	0.624695i	$-0.214777\pi$
$41$	1.00000	+	1.00000i	0.156174	+	0.156174i	−0.624695	+	0.780869i	$0.714777\pi$
$42$		0			0
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$	−12.0000			−1.78885
$46$		0			0
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$	11.0000	−	11.0000i	1.55563	−	1.55563i
$51$		0			0
$52$	12.0000			1.66410
$53$	−4.00000			−0.549442			−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000			−0.549442			−0.274721	+	0.961524i	$0.588586\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−3.00000	−	7.00000i	−0.393919	−	0.919145i
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$	−1.00000	+	1.00000i	−0.128037	+	0.128037i	−0.768221	−	0.640184i	$-0.778858\pi$
$61$	−1.00000	+	1.00000i	−0.128037	+	0.128037i	0.640184	+	0.768221i	$0.278858\pi$
$62$		0			0
$63$		0			0
$64$			8.00000i			1.00000i
$65$	24.0000			2.97683
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$	−10.0000	−	10.0000i	−1.21268	−	1.21268i
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	6.00000	−	6.00000i	0.707107	−	0.707107i
$73$	11.0000	+	11.0000i	1.28745	+	1.28745i	0.936329	+	0.351123i	$0.114200\pi$
$73$	11.0000	+	11.0000i	1.28745	+	1.28745i	0.351123	+	0.936329i	$0.385800\pi$
$74$	−14.0000			−1.62747
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$79$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$80$			16.0000i			1.78885i
$81$	−9.00000			−1.00000
$82$	−2.00000			−0.220863
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$	−20.0000	−	20.0000i	−2.16930	−	2.16930i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.00000	+	3.00000i	−0.317999	+	0.317999i	−0.847998	−	0.529999i	$-0.822192\pi$
$89$	−3.00000	+	3.00000i	−0.317999	+	0.317999i	0.529999	+	0.847998i	$0.322192\pi$
$90$	12.0000	−	12.0000i	1.26491	−	1.26491i
$91$		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	0.406138	−	0.913812i	$-0.366875\pi$
$97$	−5.00000	−	5.00000i	−0.507673	−	0.507673i	−0.913812	+	0.406138i	$0.866875\pi$
$98$	−7.00000	+	7.00000i	−0.707107	+	0.707107i
$99$		0			0
$100$			22.0000i			2.20000i
$101$	−11.0000	+	11.0000i	−1.09454	+	1.09454i	−0.0995037	+	0.995037i	$0.531726\pi$
$101$	−11.0000	+	11.0000i	−1.09454	+	1.09454i	−0.995037	+	0.0995037i	$0.968274\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$	−12.0000	+	12.0000i	−1.17670	+	1.17670i
$105$		0			0
$106$	4.00000	−	4.00000i	0.388514	−	0.388514i
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$		−	20.0000i		−	1.91565i	−0.287348	−	0.957826i	$-0.592774\pi$
$109$		−	20.0000i		−	1.91565i	0.287348	−	0.957826i	$-0.407226\pi$
$110$		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
$116$	10.0000	+	4.00000i	0.928477	+	0.371391i
$117$	18.0000			1.66410
$118$		0			0
$119$		0			0
$120$		0			0
$121$		−	11.0000i		−	1.00000i
$122$		−	2.00000i		−	0.181071i
$123$		0			0
$124$		0			0
$125$			24.0000i			2.14663i
$126$		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
$130$	−24.0000	+	24.0000i	−2.10494	+	2.10494i
$131$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$131$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	20.0000			1.71499
$137$	15.0000	−	15.0000i	1.28154	−	1.28154i	0.341743	−	0.939793i	$-0.388983\pi$
$137$	15.0000	−	15.0000i	1.28154	−	1.28154i	0.939793	−	0.341743i	$-0.111017\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$			12.0000i			1.00000i
$145$	20.0000	+	8.00000i	1.66091	+	0.664364i
$146$	−22.0000			−1.82073
$147$		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			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		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	$-0.158947\pi$
$157$	5.00000	+	5.00000i	0.399043	+	0.399043i	−0.478852	+	0.877896i	$0.658947\pi$
$158$		0			0
$159$		0			0
$160$	−16.0000	−	16.0000i	−1.26491	−	1.26491i
$161$		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	$0.250000\pi$
$164$	2.00000	−	2.00000i	0.156174	−	0.156174i
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−23.0000			−1.76923
$170$	40.0000			3.06786
$171$		0			0
$172$		0			0
$173$			26.0000i			1.97674i	0.152057	+	0.988372i	$0.451410\pi$
$173$			26.0000i			1.97674i	−0.152057	+	0.988372i	$0.548590\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		−	6.00000i		−	0.449719i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$			24.0000i			1.78885i
$181$	−18.0000			−1.33793			−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000			−1.33793			−0.668965	+	0.743294i	$0.733262\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	28.0000	−	28.0000i	2.05860	−	2.05860i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$191$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$192$		0			0
$193$	−19.0000	+	19.0000i	−1.36765	+	1.36765i	−0.503871	+	0.863779i	$0.668091\pi$
$193$	−19.0000	+	19.0000i	−1.36765	+	1.36765i	−0.863779	+	0.503871i	$0.831909\pi$
$194$	10.0000			0.717958
$195$		0			0
$196$		−	14.0000i		−	1.00000i
$197$	2.00000			0.142494			0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000			0.142494			0.0712470	+	0.997459i	$0.477302\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−22.0000	−	22.0000i	−1.55563	−	1.55563i
$201$		0			0
$202$		−	22.0000i		−	1.54791i
$203$		0			0
$204$		0			0
$205$	4.00000	−	4.00000i	0.279372	−	0.279372i
$206$		0			0
$207$		0			0
$208$		−	24.0000i		−	1.66410i
$209$		0			0
$210$		0			0
$211$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$211$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$212$			8.00000i			0.549442i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	20.0000	+	20.0000i	1.35457	+	1.35457i
$219$		0			0
$220$		0			0
$221$	30.0000	+	30.0000i	2.01802	+	2.01802i
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$			33.0000i			2.20000i
$226$	2.00000			0.133038
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$	17.0000	+	17.0000i	1.12339	+	1.12339i	0.991228	+	0.132164i	$0.0421925\pi$
$229$	17.0000	+	17.0000i	1.12339	+	1.12339i	0.132164	+	0.991228i	$0.457808\pi$
$230$		0			0
$231$		0			0
$232$	−14.0000	+	6.00000i	−0.919145	+	0.393919i
$233$	−26.0000			−1.70332			−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−26.0000			−1.70332			−0.851658	+	0.524097i	$0.824403\pi$
$234$	−18.0000	+	18.0000i	−1.17670	+	1.17670i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$			8.00000i			0.515325i	0.966235	+	0.257663i	$0.0829523\pi$
$241$			8.00000i			0.515325i	−0.966235	+	0.257663i	$0.917048\pi$
$242$	11.0000	+	11.0000i	0.707107	+	0.707107i
$243$		0			0
$244$	2.00000	+	2.00000i	0.128037	+	0.128037i
$245$		−	28.0000i		−	1.78885i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−24.0000	−	24.0000i	−1.51789	−	1.51789i
$251$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$251$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$	−32.0000			−1.99611			−0.998053	−	0.0623783i	$-0.980131\pi$
$257$	−32.0000			−1.99611			−0.998053	+	0.0623783i	$0.980131\pi$
$258$		0			0
$259$		0			0
$260$		−	48.0000i		−	2.97683i
$261$	15.0000	+	6.00000i	0.928477	+	0.371391i
$262$		0			0
$263$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$263$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$264$		0			0
$265$			16.0000i			0.982872i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−23.0000	−	23.0000i	−1.40233	−	1.40233i	−0.792624	−	0.609711i	$-0.791286\pi$
$269$	−23.0000	−	23.0000i	−1.40233	−	1.40233i	−0.609711	−	0.792624i	$-0.708714\pi$
$270$		0			0
$271$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$271$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$272$	−20.0000	+	20.0000i	−1.21268	+	1.21268i
$273$		0			0
$274$			30.0000i			1.81237i
$275$		0			0
$276$		0			0
$277$	−28.0000			−1.68236			−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000			−1.68236			−0.841178	+	0.540758i	$0.818138\pi$
$278$		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			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$		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
$290$	−28.0000	+	12.0000i	−1.64422	+	0.704664i
$291$		0			0
$292$	22.0000	−	22.0000i	1.28745	−	1.28745i
$293$	−15.0000	+	15.0000i	−0.876309	+	0.876309i	−0.993151	−	0.116841i	$-0.962723\pi$
$293$	−15.0000	+	15.0000i	−0.876309	+	0.876309i	0.116841	+	0.993151i	$0.462723\pi$
$294$		0			0
$295$		0			0
$296$			28.0000i			1.62747i
$297$		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
$305$	4.00000	+	4.00000i	0.229039	+	0.229039i
$306$	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		0.707107	−	0.707107i	$-0.250000\pi$
$311$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$312$		0			0
$313$	24.0000			1.35656			0.678280	−	0.734803i	$-0.262726\pi$
$313$	24.0000			1.35656			0.678280	+	0.734803i	$0.262726\pi$
$314$	−10.0000			−0.564333
$315$		0			0
$316$		0			0
$317$	3.00000	+	3.00000i	0.168497	+	0.168497i	0.786318	−	0.617822i	$-0.211985\pi$
$317$	3.00000	+	3.00000i	0.168497	+	0.168497i	−0.617822	+	0.786318i	$0.711985\pi$
$318$		0			0
$319$		0			0
$320$	32.0000			1.78885
$321$		0			0
$322$		0			0
$323$		0			0
$324$			18.0000i			1.00000i
$325$		−	66.0000i		−	3.66102i
$326$		0			0
$327$		0			0
$328$			4.00000i			0.220863i
$329$		0			0
$330$		0			0
$331$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$331$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$332$		0			0
$333$	21.0000	−	21.0000i	1.15079	−	1.15079i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	25.0000	+	25.0000i	1.36184	+	1.36184i	0.871576	+	0.490261i	$0.163099\pi$
$337$	25.0000	+	25.0000i	1.36184	+	1.36184i	0.490261	+	0.871576i	$0.336901\pi$
$338$	23.0000	−	23.0000i	1.25104	−	1.25104i
$339$		0			0
$340$	−40.0000	+	40.0000i	−2.16930	+	2.16930i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−26.0000	−	26.0000i	−1.39777	−	1.39777i
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$	36.0000			1.92704			0.963518	−	0.267644i	$-0.0862451\pi$
$349$	36.0000			1.92704			0.963518	+	0.267644i	$0.0862451\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			34.0000i			1.80964i	0.425797	+	0.904819i	$0.359994\pi$
$353$			34.0000i			1.80964i	−0.425797	+	0.904819i	$0.640006\pi$
$354$		0			0
$355$		0			0
$356$	6.00000	+	6.00000i	0.317999	+	0.317999i
$357$		0			0
$358$		0			0
$359$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$359$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$360$	−24.0000	−	24.0000i	−1.26491	−	1.26491i
$361$		−	19.0000i		−	1.00000i
$362$	18.0000	−	18.0000i	0.946059	−	0.946059i
$363$		0			0
$364$		0			0
$365$	44.0000	−	44.0000i	2.30307	−	2.30307i
$366$		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
$369$	3.00000	−	3.00000i	0.156174	−	0.156174i
$370$			56.0000i			2.91130i
$371$		0			0
$372$		0			0
$373$	14.0000			0.724893			0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000			0.724893			0.362446	+	0.932005i	$0.381942\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−30.0000	−	12.0000i	−1.54508	−	0.618031i
$378$		0			0
$379$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$379$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$		−	38.0000i		−	1.93415i
$387$		0			0
$388$	−10.0000	+	10.0000i	−0.507673	+	0.507673i
$389$	−27.0000	−	27.0000i	−1.36895	−	1.36895i	−0.861934	−	0.507020i	$-0.830747\pi$
$389$	−27.0000	−	27.0000i	−1.36895	−	1.36895i	−0.507020	−	0.861934i	$-0.669253\pi$
$390$		0			0
$391$		0			0
$392$	14.0000	+	14.0000i	0.707107	+	0.707107i
$393$		0			0
$394$	−2.00000	+	2.00000i	−0.100759	+	0.100759i
$395$		0			0
$396$		0			0
$397$	−38.0000			−1.90717			−0.953583	−	0.301131i	$-0.902636\pi$
$397$	−38.0000			−1.90717			−0.953583	+	0.301131i	$0.902636\pi$
$398$		0			0
$399$		0			0
$400$	44.0000			2.20000
$401$	−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
$404$	22.0000	+	22.0000i	1.09454	+	1.09454i
$405$			36.0000i			1.78885i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−17.0000	+	17.0000i	−0.840596	+	0.840596i	−0.988936	−	0.148340i	$-0.952607\pi$
$409$	−17.0000	+	17.0000i	−0.840596	+	0.840596i	0.148340	+	0.988936i	$0.452607\pi$
$410$			8.00000i			0.395092i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	24.0000	+	24.0000i	1.17670	+	1.17670i
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−1.00000	−	1.00000i	−0.0487370	−	0.0487370i	0.682318	−	0.731055i	$-0.260972\pi$
$421$	−1.00000	−	1.00000i	−0.0487370	−	0.0487370i	−0.731055	+	0.682318i	$0.760972\pi$
$422$		0			0
$423$		0			0
$424$	−8.00000	−	8.00000i	−0.388514	−	0.388514i
$425$	−55.0000	+	55.0000i	−2.66789	+	2.66789i
$426$		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
$436$	−40.0000			−1.91565
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$		−	21.0000i		−	1.00000i
$442$	−60.0000			−2.85391
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		0			0
$445$	12.0000	+	12.0000i	0.568855	+	0.568855i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	13.0000	−	13.0000i	0.613508	−	0.613508i	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	13.0000	−	13.0000i	0.613508	−	0.613508i	0.943858	+	0.330350i	$0.107167\pi$
$450$	−33.0000	−	33.0000i	−1.55563	−	1.55563i
$451$		0			0
$452$	−2.00000	+	2.00000i	−0.0940721	+	0.0940721i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	42.0000i		−	1.96468i	−0.187112	−	0.982339i	$-0.559913\pi$
$457$		−	42.0000i		−	1.96468i	0.187112	−	0.982339i	$-0.440087\pi$
$458$	−34.0000			−1.58872
$459$		0			0
$460$		0			0
$461$	9.00000	+	9.00000i	0.419172	+	0.419172i	0.884918	−	0.465746i	$-0.154214\pi$
$461$	9.00000	+	9.00000i	0.419172	+	0.419172i	−0.465746	+	0.884918i	$0.654214\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$	8.00000	−	20.0000i	0.371391	−	0.928477i
$465$		0			0
$466$	26.0000	−	26.0000i	1.20443	−	1.20443i
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$		−	36.0000i		−	1.66410i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$			12.0000i			0.549442i
$478$		0			0
$479$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$479$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$480$		0			0
$481$	−42.0000	+	42.0000i	−1.91504	+	1.91504i
$482$	−8.00000	−	8.00000i	−0.364390	−	0.364390i
$483$		0			0
$484$	−22.0000			−1.00000
$485$	−20.0000	+	20.0000i	−0.908153	+	0.908153i
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	−4.00000			−0.181071
$489$		0			0
$490$	28.0000	+	28.0000i	1.26491	+	1.26491i
$491$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$491$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$492$		0			0
$493$	15.0000	+	35.0000i	0.675566	+	1.57632i
$494$		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	$0.5\pi$
$500$	48.0000			2.14663
$501$		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
$505$	44.0000	+	44.0000i	1.95797	+	1.95797i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	44.0000			1.95027			0.975133	−	0.221621i	$-0.0711348\pi$
$509$	44.0000			1.95027			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
$514$	32.0000	−	32.0000i	1.41146	−	1.41146i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	48.0000	+	48.0000i	2.10494	+	2.10494i
$521$		−	22.0000i		−	0.963837i	−0.876216	−	0.481919i	$-0.839940\pi$
$521$		−	22.0000i		−	0.963837i	0.876216	−	0.481919i	$-0.160060\pi$
$522$	−21.0000	+	9.00000i	−0.919145	+	0.393919i
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	23.0000			1.00000
$530$	−16.0000	−	16.0000i	−0.694996	−	0.694996i
$531$		0			0
$532$		0			0
$533$	−6.00000	+	6.00000i	−0.259889	+	0.259889i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	46.0000			1.98320
$539$		0			0
$540$		0			0
$541$	31.0000	−	31.0000i	1.33279	−	1.33279i	0.429934	−	0.902861i	$-0.358537\pi$
$541$	31.0000	−	31.0000i	1.33279	−	1.33279i	0.902861	−	0.429934i	$-0.141463\pi$
$542$		0			0
$543$		0			0
$544$		−	40.0000i		−	1.71499i
$545$	−80.0000			−3.42682
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	−30.0000	−	30.0000i	−1.28154	−	1.28154i
$549$	3.00000	+	3.00000i	0.128037	+	0.128037i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	28.0000	−	28.0000i	1.18961	−	1.18961i
$555$		0			0
$556$		0			0
$557$			28.0000i			1.18640i	0.805056	+	0.593199i	$0.202135\pi$
$557$			28.0000i			1.18640i	−0.805056	+	0.593199i	$0.797865\pi$
$558$		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
$565$	−4.00000	+	4.00000i	−0.168281	+	0.168281i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−33.0000	−	33.0000i	−1.38343	−	1.38343i	−0.838444	−	0.544988i	$-0.816534\pi$
$569$	−33.0000	−	33.0000i	−1.38343	−	1.38343i	−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
$576$	24.0000			1.00000
$577$	23.0000	+	23.0000i	0.957503	+	0.957503i	0.999133	−	0.0416305i	$-0.0132552\pi$
$577$	23.0000	+	23.0000i	0.957503	+	0.957503i	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	33.0000	+	33.0000i	1.37262	+	1.37262i
$579$		0			0
$580$	16.0000	−	40.0000i	0.664364	−	1.66091i
$581$		0			0
$582$		0			0
$583$		0			0
$584$			44.0000i			1.82073i
$585$		−	72.0000i		−	2.97683i
$586$		−	30.0000i		−	1.23929i
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−28.0000	−	28.0000i	−1.15079	−	1.15079i
$593$		−	46.0000i		−	1.88899i	−0.328521	−	0.944497i	$-0.606550\pi$
$593$		−	46.0000i		−	1.88899i	0.328521	−	0.944497i	$-0.393450\pi$
$594$		0			0
$595$		0			0
$596$	−28.0000			−1.14692
$597$		0			0
$598$		0			0
$599$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$599$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$600$		0			0
$601$	−19.0000	+	19.0000i	−0.775026	+	0.775026i	−0.978980	−	0.203954i	$-0.934621\pi$
$601$	−19.0000	+	19.0000i	−0.775026	+	0.775026i	0.203954	+	0.978980i	$0.434621\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−44.0000			−1.78885
$606$		0			0
$607$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$607$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$608$		0			0
$609$		0			0
$610$	−8.00000			−0.323911
$611$		0			0
$612$	−30.0000	+	30.0000i	−1.21268	+	1.21268i
$613$		−	36.0000i		−	1.45403i	−0.686624	−	0.727013i	$-0.740908\pi$
$613$		−	36.0000i		−	1.45403i	0.686624	−	0.727013i	$-0.259092\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	35.0000	+	35.0000i	1.40905	+	1.40905i	0.764911	+	0.644136i	$0.222783\pi$
$617$	35.0000	+	35.0000i	1.40905	+	1.40905i	0.644136	+	0.764911i	$0.277217\pi$
$618$		0			0
$619$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$619$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	41.0000			1.64000
$626$	−24.0000	+	24.0000i	−0.959233	+	0.959233i
$627$		0			0
$628$	10.0000	−	10.0000i	0.399043	−	0.399043i
$629$	70.0000			2.79108
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$		0			0
$634$	−6.00000			−0.238290
$635$		0			0
$636$		0			0
$637$			42.0000i			1.66410i
$638$		0			0
$639$		0			0
$640$	−32.0000	+	32.0000i	−1.26491	+	1.26491i
$641$	−21.0000	+	21.0000i	−0.829450	+	0.829450i	−0.987441	−	0.157991i	$-0.949498\pi$
$641$	−21.0000	+	21.0000i	−0.829450	+	0.829450i	0.157991	+	0.987441i	$0.449498\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	−18.0000	−	18.0000i	−0.707107	−	0.707107i
$649$		0			0
$650$	66.0000	+	66.0000i	2.58873	+	2.58873i
$651$		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
$656$	−4.00000	−	4.00000i	−0.156174	−	0.156174i
$657$	33.0000	−	33.0000i	1.28745	−	1.28745i
$658$		0			0
$659$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$659$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$660$		0			0
$661$	12.0000			0.466746			0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000			0.466746			0.233373	+	0.972387i	$0.425024\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$			42.0000i			1.62747i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$			24.0000i			0.925132i	0.886585	+	0.462566i	$0.153071\pi$
$673$			24.0000i			0.925132i	−0.886585	+	0.462566i	$0.846929\pi$
$674$	−50.0000			−1.92593
$675$		0			0
$676$			46.0000i			1.76923i
$677$	−25.0000	−	25.0000i	−0.960828	−	0.960828i	0.0384331	−	0.999261i	$-0.487763\pi$
$677$	−25.0000	−	25.0000i	−0.960828	−	0.960828i	−0.999261	+	0.0384331i	$0.987763\pi$
$678$		0			0
$679$		0			0
$680$		−	80.0000i		−	3.06786i
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$	−60.0000	−	60.0000i	−2.29248	−	2.29248i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	24.0000i		−	0.914327i
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$	52.0000			1.97674
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	10.0000			0.378777
$698$	−36.0000	+	36.0000i	−1.36262	+	1.36262i
$699$		0			0
$700$		0			0
$701$			10.0000i			0.377695i	0.982006	+	0.188847i	$0.0604752\pi$
$701$			10.0000i			0.377695i	−0.982006	+	0.188847i	$0.939525\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−34.0000	−	34.0000i	−1.27961	−	1.27961i
$707$		0			0
$708$		0			0
$709$			30.0000i			1.12667i	0.826227	+	0.563337i	$0.190483\pi$
$709$			30.0000i			1.12667i	−0.826227	+	0.563337i	$0.809517\pi$
$710$		0			0
$711$		0			0
$712$	−12.0000			−0.449719
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$	48.0000			1.78885
$721$		0			0
$722$	19.0000	+	19.0000i	0.707107	+	0.707107i
$723$		0			0
$724$			36.0000i			1.33793i
$725$	22.0000	−	55.0000i	0.817059	−	2.04265i
$726$		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
$730$			88.0000i			3.25703i
$731$		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
$738$			6.00000i			0.220863i
$739$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$739$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$740$	−56.0000	−	56.0000i	−2.05860	−	2.05860i
$741$		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
$745$	−56.0000			−2.05168
$746$	−14.0000	+	14.0000i	−0.512576	+	0.512576i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$751$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$752$		0			0
$753$		0			0
$754$	42.0000	−	18.0000i	1.52955	−	0.655521i
$755$		0			0
$756$		0			0
$757$	−17.0000	+	17.0000i	−0.617876	+	0.617876i	−0.944986	−	0.327111i	$-0.893925\pi$
$757$	−17.0000	+	17.0000i	−0.617876	+	0.617876i	0.327111	+	0.944986i	$0.393925\pi$
$758$		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
$765$	−60.0000	+	60.0000i	−2.16930	+	2.16930i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	13.0000	+	13.0000i	0.468792	+	0.468792i	0.901523	−	0.432731i	$-0.142450\pi$
$769$	13.0000	+	13.0000i	0.468792	+	0.468792i	−0.432731	+	0.901523i	$0.642450\pi$
$770$		0			0
$771$		0			0
$772$	38.0000	+	38.0000i	1.36765	+	1.36765i
$773$	39.0000	−	39.0000i	1.40273	−	1.40273i	0.611448	−	0.791285i	$-0.290588\pi$
$773$	39.0000	−	39.0000i	1.40273	−	1.40273i	0.791285	−	0.611448i	$-0.209412\pi$
$774$		0			0
$775$		0			0
$776$		−	20.0000i		−	0.717958i
$777$		0			0
$778$	54.0000			1.93599
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−28.0000			−1.00000
$785$	20.0000	−	20.0000i	0.713831	−	0.713831i
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$		−	4.00000i		−	0.142494i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−6.00000	−	6.00000i	−0.213066	−	0.213066i
$794$	38.0000	−	38.0000i	1.34857	−	1.34857i
$795$		0			0
$796$		0			0
$797$	−37.0000	+	37.0000i	−1.31061	+	1.31061i	−0.389640	+	0.920967i	$0.627401\pi$
$797$	−37.0000	+	37.0000i	−1.31061	+	1.31061i	−0.920967	+	0.389640i	$0.872599\pi$
$798$		0			0
$799$		0			0
$800$	−44.0000	+	44.0000i	−1.55563	+	1.55563i
$801$	9.00000	+	9.00000i	0.317999	+	0.317999i
$802$	2.00000	−	2.00000i	0.0706225	−	0.0706225i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−44.0000			−1.54791
$809$	23.0000	+	23.0000i	0.808637	+	0.808637i	0.984428	−	0.175791i	$-0.0562482\pi$
$809$	23.0000	+	23.0000i	0.808637	+	0.808637i	−0.175791	+	0.984428i	$0.556248\pi$
$810$	−36.0000	−	36.0000i	−1.26491	−	1.26491i
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		−	34.0000i		−	1.18878i
$819$		0			0
$820$	−8.00000	−	8.00000i	−0.279372	−	0.279372i
$821$		−	50.0000i		−	1.74501i	−0.488603	−	0.872506i	$-0.662493\pi$
$821$		−	50.0000i		−	1.74501i	0.488603	−	0.872506i	$-0.337507\pi$
$822$		0			0
$823$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$823$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$827$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$828$		0			0
$829$	17.0000	−	17.0000i	0.590434	−	0.590434i	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	17.0000	−	17.0000i	0.590434	−	0.590434i	0.937749	+	0.347314i	$0.112906\pi$
$830$		0			0
$831$		0			0
$832$	−48.0000			−1.66410
$833$	35.0000	−	35.0000i	1.21268	−	1.21268i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$839$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$840$		0			0
$841$	−21.0000	−	20.0000i	−0.724138	−	0.689655i
$842$	2.00000			0.0689246
$843$		0			0
$844$		0			0
$845$			92.0000i			3.16490i
$846$		0			0
$847$		0			0
$848$	16.0000			0.549442
$849$		0			0
$850$		−	110.000i		−	3.77297i
$851$		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
$857$	−8.00000			−0.273275			−0.136637	−	0.990621i	$-0.543630\pi$
$857$	−8.00000			−0.273275			−0.136637	+	0.990621i	$0.543630\pi$
$858$		0			0
$859$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$859$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	104.000			3.53611
$866$	−10.0000			−0.339814
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	40.0000	−	40.0000i	1.35457	−	1.35457i
$873$	−15.0000	+	15.0000i	−0.507673	+	0.507673i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−12.0000			−0.405211			−0.202606	−	0.979260i	$-0.564941\pi$
$877$	−12.0000			−0.405211			−0.202606	+	0.979260i	$0.564941\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	9.00000	−	9.00000i	0.303218	−	0.303218i	−0.539054	−	0.842271i	$-0.681218\pi$
$881$	9.00000	−	9.00000i	0.303218	−	0.303218i	0.842271	+	0.539054i	$0.181218\pi$
$882$	21.0000	+	21.0000i	0.707107	+	0.707107i
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$	60.0000	−	60.0000i	2.01802	−	2.01802i
$885$		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
$890$	−24.0000			−0.804482
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$			26.0000i			0.867631i
$899$		0			0
$900$	66.0000			2.20000
$901$	−20.0000	+	20.0000i	−0.666297	+	0.666297i
$902$		0			0
$903$		0			0
$904$		−	4.00000i		−	0.133038i
$905$			72.0000i			2.39336i
$906$		0			0
$907$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$907$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$908$		0			0
$909$	33.0000	+	33.0000i	1.09454	+	1.09454i
$910$		0			0
$911$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$911$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$912$		0			0
$913$		0			0
$914$	42.0000	+	42.0000i	1.38924	+	1.38924i
$915$		0			0
$916$	34.0000	−	34.0000i	1.12339	−	1.12339i
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$	−18.0000			−0.592798
$923$		0			0
$924$		0			0
$925$	−77.0000	−	77.0000i	−2.53174	−	2.53174i
$926$		0			0
$927$		0			0
$928$	12.0000	+	28.0000i	0.393919	+	0.919145i
$929$	−40.0000			−1.31236			−0.656179	−	0.754606i	$-0.727828\pi$
$929$	−40.0000			−1.31236			−0.656179	+	0.754606i	$0.727828\pi$
$930$		0			0
$931$		0			0
$932$			52.0000i			1.70332i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	36.0000	+	36.0000i	1.17670	+	1.17670i
$937$		−	38.0000i		−	1.24141i	−0.784046	−	0.620703i	$-0.786847\pi$
$937$		−	38.0000i		−	1.24141i	0.784046	−	0.620703i	$-0.213153\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			20.0000i			0.651981i	0.945373	+	0.325991i	$0.105698\pi$
$941$			20.0000i			0.651981i	−0.945373	+	0.325991i	$0.894302\pi$
$942$		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
$949$	−66.0000	+	66.0000i	−2.14245	+	2.14245i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	26.0000			0.842223			0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000			0.842223			0.421111	+	0.907009i	$0.361640\pi$
$954$	−12.0000	−	12.0000i	−0.388514	−	0.388514i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$		−	31.0000i		−	1.00000i
$962$		−	84.0000i		−	2.70827i
$963$		0			0
$964$	16.0000			0.515325
$965$	76.0000	+	76.0000i	2.44653	+	2.44653i
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$	22.0000	−	22.0000i	0.707107	−	0.707107i
$969$		0			0
$970$		−	40.0000i		−	1.28432i
$971$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$971$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	4.00000	−	4.00000i	0.128037	−	0.128037i
$977$	−8.00000			−0.255943			−0.127971	−	0.991778i	$-0.540847\pi$
$977$	−8.00000			−0.255943			−0.127971	+	0.991778i	$0.540847\pi$
$978$		0			0
$979$		0			0
$980$	−56.0000			−1.78885
$981$	−60.0000			−1.91565
$982$		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
$985$		−	8.00000i		−	0.254901i
$986$	−50.0000	−	20.0000i	−1.59232	−	0.636930i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	37.0000	−	37.0000i	1.17180	−	1.17180i	0.190022	−	0.981780i	$-0.439144\pi$
$997$	37.0000	−	37.0000i	1.17180	−	1.17180i	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	116.2.e.a.75.1	✓	2
4.3	odd	2	CM	116.2.e.a.75.1	✓	2
29.12	odd	4	inner	116.2.e.a.99.1	yes	2
116.99	even	4	inner	116.2.e.a.99.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.e.a.75.1	✓	2	1.1	even	1	trivial
116.2.e.a.75.1	✓	2	4.3	odd	2	CM
116.2.e.a.99.1	yes	2	29.12	odd	4	inner
116.2.e.a.99.1	yes	2	116.99	even	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 116 = 2^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	116.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.00000i) q^{2} -2.00000i q^{4} -4.00000i q^{5} +(2.00000 + 2.00000i) q^{8} -3.00000i q^{9} +O(q^{10})\)	\(q+(-1.00000 + 1.00000i) q^{2} -2.00000i q^{4} -4.00000i q^{5} +(2.00000 + 2.00000i) q^{8} -3.00000i q^{9} +(4.00000 + 4.00000i) q^{10} +6.00000i q^{13} -4.00000 q^{16} +(5.00000 - 5.00000i) q^{17} +(3.00000 + 3.00000i) q^{18} -8.00000 q^{20} -11.0000 q^{25} +(-6.00000 - 6.00000i) q^{26} +(-2.00000 + 5.00000i) q^{29} +(4.00000 - 4.00000i) q^{32} +10.0000i q^{34} -6.00000 q^{36} +(7.00000 + 7.00000i) q^{37} +(8.00000 - 8.00000i) q^{40} +(1.00000 + 1.00000i) q^{41} -12.0000 q^{45} +7.00000 q^{49} +(11.0000 - 11.0000i) q^{50} +12.0000 q^{52} -4.00000 q^{53} +(-3.00000 - 7.00000i) q^{58} +(-1.00000 + 1.00000i) q^{61} +8.00000i q^{64} +24.0000 q^{65} +(-10.0000 - 10.0000i) q^{68} +(6.00000 - 6.00000i) q^{72} +(11.0000 + 11.0000i) q^{73} -14.0000 q^{74} +16.0000i q^{80} -9.00000 q^{81} -2.00000 q^{82} +(-20.0000 - 20.0000i) q^{85} +(-3.00000 + 3.00000i) q^{89} +(12.0000 - 12.0000i) q^{90} +(-5.00000 - 5.00000i) q^{97} +(-7.00000 + 7.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 4 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 4 * q^8	\( 2 q - 2 q^{2} + 4 q^{8} + 8 q^{10} - 8 q^{16} + 10 q^{17} + 6 q^{18} - 16 q^{20} - 22 q^{25} - 12 q^{26} - 4 q^{29} + 8 q^{32} - 12 q^{36} + 14 q^{37} + 16 q^{40} + 2 q^{41} - 24 q^{45} + 14 q^{49} + 22 q^{50} + 24 q^{52} - 8 q^{53} - 6 q^{58} - 2 q^{61} + 48 q^{65} - 20 q^{68} + 12 q^{72} + 22 q^{73} - 28 q^{74} - 18 q^{81} - 4 q^{82} - 40 q^{85} - 6 q^{89} + 24 q^{90} - 10 q^{97} - 14 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 4 * q^8 + 8 * q^10 - 8 * q^16 + 10 * q^17 + 6 * q^18 - 16 * q^20 - 22 * q^25 - 12 * q^26 - 4 * q^29 + 8 * q^32 - 12 * q^36 + 14 * q^37 + 16 * q^40 + 2 * q^41 - 24 * q^45 + 14 * q^49 + 22 * q^50 + 24 * q^52 - 8 * q^53 - 6 * q^58 - 2 * q^61 + 48 * q^65 - 20 * q^68 + 12 * q^72 + 22 * q^73 - 28 * q^74 - 18 * q^81 - 4 * q^82 - 40 * q^85 - 6 * q^89 + 24 * q^90 - 10 * q^97 - 14 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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