LMFDB - Embedded newform 344.1.s.a.299.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [344,1,Mod(11,344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(344, base_ring=CyclotomicField(14))

chi = DirichletCharacter(H, H._module([7, 7, 10]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("344.11");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 344 = 2^{3} \cdot 43 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	344.s (of order $14$, degree $6$, 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:	$0.171678364346$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{14})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{7}$
Projective field:	Galois closure of 7.1.3236537881088.1

Embedding invariants

Embedding label			299.1
Root			$0.222521 + 0.974928i$ of defining polynomial
Character	$\chi$	$=$	344.299
Dual form			344.1.s.a.107.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.900969 + 0.433884i) q^{2} +(-1.12349 - 0.541044i) q^{3} +(0.623490 - 0.781831i) q^{4} +1.24698 q^{6} +(-0.222521 + 0.974928i) q^{8} +(0.346011 + 0.433884i) q^{9} +O(q^{10})$	\(q+(-0.900969 + 0.433884i) q^{2} +(-1.12349 - 0.541044i) q^{3} +(0.623490 - 0.781831i) q^{4} +1.24698 q^{6} +(-0.222521 + 0.974928i) q^{8} +(0.346011 + 0.433884i) q^{9} +(-1.12349 - 1.40881i) q^{11} +(-1.12349 + 0.541044i) q^{12} +(-0.222521 - 0.974928i) q^{16} +(-0.277479 - 1.21572i) q^{17} +(-0.500000 - 0.240787i) q^{18} +(0.777479 - 0.974928i) q^{19} +(1.62349 + 0.781831i) q^{22} +(0.777479 - 0.974928i) q^{24} +(-0.900969 - 0.433884i) q^{25} +(0.123490 + 0.541044i) q^{27} +(0.623490 + 0.781831i) q^{32} +(0.500000 + 2.19064i) q^{33} +(0.777479 + 0.974928i) q^{34} +0.554958 q^{36} +(-0.277479 + 1.21572i) q^{38} +(0.400969 - 0.193096i) q^{41} +(-0.900969 + 0.433884i) q^{43} -1.80194 q^{44} +(-0.277479 + 1.21572i) q^{48} +1.00000 q^{49} +1.00000 q^{50} +(-0.346011 + 1.51597i) q^{51} +(-0.346011 - 0.433884i) q^{54} +(-1.40097 + 0.674671i) q^{57} +(0.400969 + 1.75676i) q^{59} +(-0.900969 - 0.433884i) q^{64} +(-1.40097 - 1.75676i) q^{66} +(1.24698 - 1.56366i) q^{67} +(-1.12349 - 0.541044i) q^{68} +(-0.500000 + 0.240787i) q^{72} +(-0.445042 + 1.94986i) q^{73} +(0.777479 + 0.974928i) q^{75} +(-0.277479 - 1.21572i) q^{76} +(0.277479 - 1.21572i) q^{81} +(-0.277479 + 0.347948i) q^{82} +(1.62349 + 0.781831i) q^{83} +(0.623490 - 0.781831i) q^{86} +(1.62349 - 0.781831i) q^{88} +(0.400969 + 0.193096i) q^{89} +(-0.277479 - 1.21572i) q^{96} +(-1.12349 - 1.40881i) q^{97} +(-0.900969 + 0.433884i) q^{98} +(0.222521 - 0.974928i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{2} - 2 q^{3} - q^{4} - 2 q^{6} - q^{8} - 3 q^{9}+O(q^{10}) $ 6 * q - q^2 - 2 * q^3 - q^4 - 2 * q^6 - q^8 - 3 * q^9	$ 6 q - q^{2} - 2 q^{3} - q^{4} - 2 q^{6} - q^{8} - 3 q^{9} - 2 q^{11} - 2 q^{12} - q^{16} - 2 q^{17} - 3 q^{18} + 5 q^{19} + 5 q^{22} + 5 q^{24} - q^{25} - 4 q^{27} - q^{32} + 3 q^{33} + 5 q^{34} + 4 q^{36} - 2 q^{38} - 2 q^{41} - q^{43} - 2 q^{44} - 2 q^{48} + 6 q^{49} + 6 q^{50} + 3 q^{51} + 3 q^{54} - 4 q^{57} - 2 q^{59} - q^{64} - 4 q^{66} - 2 q^{67} - 2 q^{68} - 3 q^{72} - 2 q^{73} + 5 q^{75} - 2 q^{76} + 2 q^{81} - 2 q^{82} + 5 q^{83} - q^{86} + 5 q^{88} - 2 q^{89} - 2 q^{96} - 2 q^{97} - q^{98} + q^{99}+O(q^{100}) $ 6 * q - q^2 - 2 * q^3 - q^4 - 2 * q^6 - q^8 - 3 * q^9 - 2 * q^11 - 2 * q^12 - q^16 - 2 * q^17 - 3 * q^18 + 5 * q^19 + 5 * q^22 + 5 * q^24 - q^25 - 4 * q^27 - q^32 + 3 * q^33 + 5 * q^34 + 4 * q^36 - 2 * q^38 - 2 * q^41 - q^43 - 2 * q^44 - 2 * q^48 + 6 * q^49 + 6 * q^50 + 3 * q^51 + 3 * q^54 - 4 * q^57 - 2 * q^59 - q^64 - 4 * q^66 - 2 * q^67 - 2 * q^68 - 3 * q^72 - 2 * q^73 + 5 * q^75 - 2 * q^76 + 2 * q^81 - 2 * q^82 + 5 * q^83 - q^86 + 5 * q^88 - 2 * q^89 - 2 * q^96 - 2 * q^97 - q^98 + q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/344\mathbb{Z}\right)^\times$.

$n$	$87$	$89$	$173$
$\chi(n)$	$-1$	$e\left(\frac{1}{7}\right)$	$-1$

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 $

$2$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$2$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$3$	−1.12349	−	0.541044i	−1.12349	−	0.541044i	−0.222521	−	0.974928i	$-0.571429\pi$
$3$	−1.12349	−	0.541044i	−1.12349	−	0.541044i	−0.900969	+	0.433884i	$0.857143\pi$
$4$	0.623490	−	0.781831i	0.623490	−	0.781831i
$5$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$5$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$6$	1.24698			1.24698
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$9$	0.346011	+	0.433884i	0.346011	+	0.433884i
$10$		0			0
$11$	−1.12349	−	1.40881i	−1.12349	−	1.40881i	−0.900969	−	0.433884i	$-0.857143\pi$
$11$	−1.12349	−	1.40881i	−1.12349	−	1.40881i	−0.222521	−	0.974928i	$-0.571429\pi$
$12$	−1.12349	+	0.541044i	−1.12349	+	0.541044i
$13$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$13$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$14$		0			0
$15$		0			0
$16$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$17$	−0.277479	−	1.21572i	−0.277479	−	1.21572i	−0.900969	−	0.433884i	$-0.857143\pi$
$17$	−0.277479	−	1.21572i	−0.277479	−	1.21572i	0.623490	−	0.781831i	$-0.285714\pi$
$18$	−0.500000	−	0.240787i	−0.500000	−	0.240787i
$19$	0.777479	−	0.974928i	0.777479	−	0.974928i	−0.222521	−	0.974928i	$-0.571429\pi$
$19$	0.777479	−	0.974928i	0.777479	−	0.974928i	1.00000			$0$
$20$		0			0
$21$		0			0
$22$	1.62349	+	0.781831i	1.62349	+	0.781831i
$23$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$23$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$24$	0.777479	−	0.974928i	0.777479	−	0.974928i
$25$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$26$		0			0
$27$	0.123490	+	0.541044i	0.123490	+	0.541044i
$28$		0			0
$29$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$29$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$30$		0			0
$31$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$31$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$32$	0.623490	+	0.781831i	0.623490	+	0.781831i
$33$	0.500000	+	2.19064i	0.500000	+	2.19064i
$34$	0.777479	+	0.974928i	0.777479	+	0.974928i
$35$		0			0
$36$	0.554958			0.554958
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	−0.277479	+	1.21572i	−0.277479	+	1.21572i
$39$		0			0
$40$		0			0
$41$	0.400969	−	0.193096i	0.400969	−	0.193096i	−0.222521	−	0.974928i	$-0.571429\pi$
$41$	0.400969	−	0.193096i	0.400969	−	0.193096i	0.623490	+	0.781831i	$0.285714\pi$
$42$		0			0
$43$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$43$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$44$	−1.80194			−1.80194
$45$		0			0
$46$		0			0
$47$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$47$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$48$	−0.277479	+	1.21572i	−0.277479	+	1.21572i
$49$	1.00000			1.00000
$50$	1.00000			1.00000
$51$	−0.346011	+	1.51597i	−0.346011	+	1.51597i
$52$		0			0
$53$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$53$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$54$	−0.346011	−	0.433884i	−0.346011	−	0.433884i
$55$		0			0
$56$		0			0
$57$	−1.40097	+	0.674671i	−1.40097	+	0.674671i
$58$		0			0
$59$	0.400969	+	1.75676i	0.400969	+	1.75676i	0.623490	+	0.781831i	$0.285714\pi$
$59$	0.400969	+	1.75676i	0.400969	+	1.75676i	−0.222521	+	0.974928i	$0.571429\pi$
$60$		0			0
$61$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$61$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$62$		0			0
$63$		0			0
$64$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$65$		0			0
$66$	−1.40097	−	1.75676i	−1.40097	−	1.75676i
$67$	1.24698	−	1.56366i	1.24698	−	1.56366i	0.623490	−	0.781831i	$-0.285714\pi$
$67$	1.24698	−	1.56366i	1.24698	−	1.56366i	0.623490	−	0.781831i	$-0.285714\pi$
$68$	−1.12349	−	0.541044i	−1.12349	−	0.541044i
$69$		0			0
$70$		0			0
$71$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$71$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$72$	−0.500000	+	0.240787i	−0.500000	+	0.240787i
$73$	−0.445042	+	1.94986i	−0.445042	+	1.94986i	−0.222521	+	0.974928i	$0.571429\pi$
$73$	−0.445042	+	1.94986i	−0.445042	+	1.94986i	−0.222521	+	0.974928i	$0.571429\pi$
$74$		0			0
$75$	0.777479	+	0.974928i	0.777479	+	0.974928i
$76$	−0.277479	−	1.21572i	−0.277479	−	1.21572i
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	0.277479	−	1.21572i	0.277479	−	1.21572i
$82$	−0.277479	+	0.347948i	−0.277479	+	0.347948i
$83$	1.62349	+	0.781831i	1.62349	+	0.781831i	1.00000			$0$
$83$	1.62349	+	0.781831i	1.62349	+	0.781831i	0.623490	+	0.781831i	$0.285714\pi$
$84$		0			0
$85$		0			0
$86$	0.623490	−	0.781831i	0.623490	−	0.781831i
$87$		0			0
$88$	1.62349	−	0.781831i	1.62349	−	0.781831i
$89$	0.400969	+	0.193096i	0.400969	+	0.193096i	0.623490	−	0.781831i	$-0.285714\pi$
$89$	0.400969	+	0.193096i	0.400969	+	0.193096i	−0.222521	+	0.974928i	$0.571429\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−0.277479	−	1.21572i	−0.277479	−	1.21572i
$97$	−1.12349	−	1.40881i	−1.12349	−	1.40881i	−0.900969	−	0.433884i	$-0.857143\pi$
$97$	−1.12349	−	1.40881i	−1.12349	−	1.40881i	−0.222521	−	0.974928i	$-0.571429\pi$
$98$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$99$	0.222521	−	0.974928i	0.222521	−	0.974928i
$100$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$101$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$101$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$102$	−0.346011	−	1.51597i	−0.346011	−	1.51597i
$103$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$103$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.400969	+	0.193096i	0.400969	+	0.193096i	0.623490	−	0.781831i	$-0.285714\pi$
$107$	0.400969	+	0.193096i	0.400969	+	0.193096i	−0.222521	+	0.974928i	$0.571429\pi$
$108$	0.500000	+	0.240787i	0.500000	+	0.240787i
$109$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$109$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.277479	−	1.21572i	−0.277479	−	1.21572i	−0.900969	−	0.433884i	$-0.857143\pi$
$113$	−0.277479	−	1.21572i	−0.277479	−	1.21572i	0.623490	−	0.781831i	$-0.285714\pi$
$114$	0.969501	−	1.21572i	0.969501	−	1.21572i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−1.12349	−	1.40881i	−1.12349	−	1.40881i
$119$		0			0
$120$		0			0
$121$	−0.500000	+	2.19064i	−0.500000	+	2.19064i
$122$		0			0
$123$	−0.554958			−0.554958
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$127$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$128$	1.00000			1.00000
$129$	1.24698			1.24698
$130$		0			0
$131$	0.400969	−	0.193096i	0.400969	−	0.193096i	−0.222521	−	0.974928i	$-0.571429\pi$
$131$	0.400969	−	0.193096i	0.400969	−	0.193096i	0.623490	+	0.781831i	$0.285714\pi$
$132$	2.02446	+	0.974928i	2.02446	+	0.974928i
$133$		0			0
$134$	−0.445042	+	1.94986i	−0.445042	+	1.94986i
$135$		0			0
$136$	1.24698			1.24698
$137$	0.400969	−	1.75676i	0.400969	−	1.75676i	−0.222521	−	0.974928i	$-0.571429\pi$
$137$	0.400969	−	1.75676i	0.400969	−	1.75676i	0.623490	−	0.781831i	$-0.285714\pi$
$138$		0			0
$139$	0.0990311	+	0.433884i	0.0990311	+	0.433884i	1.00000			$0$
$139$	0.0990311	+	0.433884i	0.0990311	+	0.433884i	−0.900969	+	0.433884i	$0.857143\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.346011	−	0.433884i	0.346011	−	0.433884i
$145$		0			0
$146$	−0.445042	−	1.94986i	−0.445042	−	1.94986i
$147$	−1.12349	−	0.541044i	−1.12349	−	0.541044i
$148$		0			0
$149$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$149$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$150$	−1.12349	−	0.541044i	−1.12349	−	0.541044i
$151$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$151$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$152$	0.777479	+	0.974928i	0.777479	+	0.974928i
$153$	0.431468	−	0.541044i	0.431468	−	0.541044i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$157$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	0.277479	+	1.21572i	0.277479	+	1.21572i
$163$	−0.277479	−	0.347948i	−0.277479	−	0.347948i	0.623490	−	0.781831i	$-0.285714\pi$
$163$	−0.277479	−	0.347948i	−0.277479	−	0.347948i	−0.900969	+	0.433884i	$0.857143\pi$
$164$	0.0990311	−	0.433884i	0.0990311	−	0.433884i
$165$		0			0
$166$	−1.80194			−1.80194
$167$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$167$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$168$		0			0
$169$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$170$		0			0
$171$	0.692021			0.692021
$172$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$		0			0
$176$	−1.12349	+	1.40881i	−1.12349	+	1.40881i
$177$	0.500000	−	2.19064i	0.500000	−	2.19064i
$178$	−0.445042			−0.445042
$179$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$179$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$180$		0			0
$181$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$181$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−1.40097	+	1.75676i	−1.40097	+	1.75676i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$191$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$192$	0.777479	+	0.974928i	0.777479	+	0.974928i
$193$	1.62349	+	0.781831i	1.62349	+	0.781831i	1.00000			$0$
$193$	1.62349	+	0.781831i	1.62349	+	0.781831i	0.623490	+	0.781831i	$0.285714\pi$
$194$	1.62349	+	0.781831i	1.62349	+	0.781831i
$195$		0			0
$196$	0.623490	−	0.781831i	0.623490	−	0.781831i
$197$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$197$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$198$	0.222521	+	0.974928i	0.222521	+	0.974928i
$199$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$199$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$200$	0.623490	−	0.781831i	0.623490	−	0.781831i
$201$	−2.24698	+	1.08209i	−2.24698	+	1.08209i
$202$		0			0
$203$		0			0
$204$	0.969501	+	1.21572i	0.969501	+	1.21572i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2.24698			−2.24698
$210$		0			0
$211$	−0.277479	+	0.347948i	−0.277479	+	0.347948i	−0.900969	−	0.433884i	$-0.857143\pi$
$211$	−0.277479	+	0.347948i	−0.277479	+	0.347948i	0.623490	+	0.781831i	$0.285714\pi$
$212$		0			0
$213$		0			0
$214$	−0.445042			−0.445042
$215$		0			0
$216$	−0.554958			−0.554958
$217$		0			0
$218$		0			0
$219$	1.55496	−	1.94986i	1.55496	−	1.94986i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$223$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$224$		0			0
$225$	−0.123490	−	0.541044i	−0.123490	−	0.541044i
$226$	0.777479	+	0.974928i	0.777479	+	0.974928i
$227$	1.62349	−	0.781831i	1.62349	−	0.781831i	0.623490	−	0.781831i	$-0.285714\pi$
$227$	1.62349	−	0.781831i	1.62349	−	0.781831i	1.00000			$0$
$228$	−0.346011	+	1.51597i	−0.346011	+	1.51597i
$229$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$229$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.12349	−	0.541044i	−1.12349	−	0.541044i	−0.222521	−	0.974928i	$-0.571429\pi$
$233$	−1.12349	−	0.541044i	−1.12349	−	0.541044i	−0.900969	+	0.433884i	$0.857143\pi$
$234$		0			0
$235$		0			0
$236$	1.62349	+	0.781831i	1.62349	+	0.781831i
$237$		0			0
$238$		0			0
$239$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$239$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$240$		0			0
$241$	0.400969	+	1.75676i	0.400969	+	1.75676i	0.623490	+	0.781831i	$0.285714\pi$
$241$	0.400969	+	1.75676i	0.400969	+	1.75676i	−0.222521	+	0.974928i	$0.571429\pi$
$242$	−0.500000	−	2.19064i	−0.500000	−	2.19064i
$243$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$244$		0			0
$245$		0			0
$246$	0.500000	−	0.240787i	0.500000	−	0.240787i
$247$		0			0
$248$		0			0
$249$	−1.40097	−	1.75676i	−1.40097	−	1.75676i
$250$		0			0
$251$	2.00000			2.00000			1.00000			$0$
$251$	2.00000			2.00000			1.00000			$0$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$257$	2.00000			2.00000			1.00000			$0$
$257$	2.00000			2.00000			1.00000			$0$
$258$	−1.12349	+	0.541044i	−1.12349	+	0.541044i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−0.277479	+	0.347948i	−0.277479	+	0.347948i
$263$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$263$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$264$	−2.24698			−2.24698
$265$		0			0
$266$		0			0
$267$	−0.346011	−	0.433884i	−0.346011	−	0.433884i
$268$	−0.445042	−	1.94986i	−0.445042	−	1.94986i
$269$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$269$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$270$		0			0
$271$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$271$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$272$	−1.12349	+	0.541044i	−1.12349	+	0.541044i
$273$		0			0
$274$	0.400969	+	1.75676i	0.400969	+	1.75676i
$275$	0.400969	+	1.75676i	0.400969	+	1.75676i
$276$		0			0
$277$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$277$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$278$	−0.277479	−	0.347948i	−0.277479	−	0.347948i
$279$		0			0
$280$		0			0
$281$	1.24698	+	1.56366i	1.24698	+	1.56366i	0.623490	+	0.781831i	$0.285714\pi$
$281$	1.24698	+	1.56366i	1.24698	+	1.56366i	0.623490	+	0.781831i	$0.285714\pi$
$282$		0			0
$283$	−1.12349	−	0.541044i	−1.12349	−	0.541044i	−0.222521	−	0.974928i	$-0.571429\pi$
$283$	−1.12349	−	0.541044i	−1.12349	−	0.541044i	−0.900969	+	0.433884i	$0.857143\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.123490	+	0.541044i	−0.123490	+	0.541044i
$289$	−0.500000	+	0.240787i	−0.500000	+	0.240787i
$290$		0			0
$291$	0.500000	+	2.19064i	0.500000	+	2.19064i
$292$	1.24698	+	1.56366i	1.24698	+	1.56366i
$293$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$293$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$294$	1.24698			1.24698
$295$		0			0
$296$		0			0
$297$	0.623490	−	0.781831i	0.623490	−	0.781831i
$298$		0			0
$299$		0			0
$300$	1.24698			1.24698
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−1.12349	−	0.541044i	−1.12349	−	0.541044i
$305$		0			0
$306$	−0.153989	+	0.674671i	−0.153989	+	0.674671i
$307$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$307$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$311$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$312$		0			0
$313$	0.400969	−	0.193096i	0.400969	−	0.193096i	−0.222521	−	0.974928i	$-0.571429\pi$
$313$	0.400969	−	0.193096i	0.400969	−	0.193096i	0.623490	+	0.781831i	$0.285714\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$317$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−0.346011	−	0.433884i	−0.346011	−	0.433884i
$322$		0			0
$323$	−1.40097	−	0.674671i	−1.40097	−	0.674671i
$324$	−0.777479	−	0.974928i	−0.777479	−	0.974928i
$325$		0			0
$326$	0.400969	+	0.193096i	0.400969	+	0.193096i
$327$		0			0
$328$	0.0990311	+	0.433884i	0.0990311	+	0.433884i
$329$		0			0
$330$		0			0
$331$	0.400969	−	1.75676i	0.400969	−	1.75676i	−0.222521	−	0.974928i	$-0.571429\pi$
$331$	0.400969	−	1.75676i	0.400969	−	1.75676i	0.623490	−	0.781831i	$-0.285714\pi$
$332$	1.62349	−	0.781831i	1.62349	−	0.781831i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$337$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$338$	1.00000			1.00000
$339$	−0.346011	+	1.51597i	−0.346011	+	1.51597i
$340$		0			0
$341$		0			0
$342$	−0.623490	+	0.300257i	−0.623490	+	0.300257i
$343$		0			0
$344$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$345$		0			0
$346$		0			0
$347$	−1.12349	−	0.541044i	−1.12349	−	0.541044i	−0.222521	−	0.974928i	$-0.571429\pi$
$347$	−1.12349	−	0.541044i	−1.12349	−	0.541044i	−0.900969	+	0.433884i	$0.857143\pi$
$348$		0			0
$349$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$349$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$350$		0			0
$351$		0			0
$352$	0.400969	−	1.75676i	0.400969	−	1.75676i
$353$	−0.277479	−	0.347948i	−0.277479	−	0.347948i	0.623490	−	0.781831i	$-0.285714\pi$
$353$	−0.277479	−	0.347948i	−0.277479	−	0.347948i	−0.900969	+	0.433884i	$0.857143\pi$
$354$	0.500000	+	2.19064i	0.500000	+	2.19064i
$355$		0			0
$356$	0.400969	−	0.193096i	0.400969	−	0.193096i
$357$		0			0
$358$	1.62349	−	0.781831i	1.62349	−	0.781831i
$359$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$359$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$360$		0			0
$361$	−0.123490	−	0.541044i	−0.123490	−	0.541044i
$362$		0			0
$363$	1.74698	−	2.19064i	1.74698	−	2.19064i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$367$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$368$		0			0
$369$	0.222521	+	0.107160i	0.222521	+	0.107160i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$373$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$374$	0.500000	−	2.19064i	0.500000	−	2.19064i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.400969	−	1.75676i	0.400969	−	1.75676i	−0.222521	−	0.974928i	$-0.571429\pi$
$379$	0.400969	−	1.75676i	0.400969	−	1.75676i	0.623490	−	0.781831i	$-0.285714\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$383$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$384$	−1.12349	−	0.541044i	−1.12349	−	0.541044i
$385$		0			0
$386$	−1.80194			−1.80194
$387$	−0.500000	−	0.240787i	−0.500000	−	0.240787i
$388$	−1.80194			−1.80194
$389$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$389$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$390$		0			0
$391$		0			0
$392$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$393$	−0.554958			−0.554958
$394$		0			0
$395$		0			0
$396$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$397$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$397$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$398$		0			0
$399$		0			0
$400$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$401$	1.62349	−	0.781831i	1.62349	−	0.781831i	0.623490	−	0.781831i	$-0.285714\pi$
$401$	1.62349	−	0.781831i	1.62349	−	0.781831i	1.00000			$0$
$402$	1.55496	−	1.94986i	1.55496	−	1.94986i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−1.40097	−	0.674671i	−1.40097	−	0.674671i
$409$	0.400969	+	0.193096i	0.400969	+	0.193096i	0.623490	−	0.781831i	$-0.285714\pi$
$409$	0.400969	+	0.193096i	0.400969	+	0.193096i	−0.222521	+	0.974928i	$0.571429\pi$
$410$		0			0
$411$	−1.40097	+	1.75676i	−1.40097	+	1.75676i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.123490	−	0.541044i	0.123490	−	0.541044i
$418$	2.02446	−	0.974928i	2.02446	−	0.974928i
$419$	−0.277479	−	0.347948i	−0.277479	−	0.347948i	0.623490	−	0.781831i	$-0.285714\pi$
$419$	−0.277479	−	0.347948i	−0.277479	−	0.347948i	−0.900969	+	0.433884i	$0.857143\pi$
$420$		0			0
$421$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$421$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$422$	0.0990311	−	0.433884i	0.0990311	−	0.433884i
$423$		0			0
$424$		0			0
$425$	−0.277479	+	1.21572i	−0.277479	+	1.21572i
$426$		0			0
$427$		0			0
$428$	0.400969	−	0.193096i	0.400969	−	0.193096i
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	0.500000	−	0.240787i	0.500000	−	0.240787i
$433$	0.400969	+	0.193096i	0.400969	+	0.193096i	0.623490	−	0.781831i	$-0.285714\pi$
$433$	0.400969	+	0.193096i	0.400969	+	0.193096i	−0.222521	+	0.974928i	$0.571429\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−0.554958	+	2.43143i	−0.554958	+	2.43143i
$439$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$439$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$440$		0			0
$441$	0.346011	+	0.433884i	0.346011	+	0.433884i
$442$		0			0
$443$	0.0990311	−	0.433884i	0.0990311	−	0.433884i	−0.900969	−	0.433884i	$-0.857143\pi$
$443$	0.0990311	−	0.433884i	0.0990311	−	0.433884i	1.00000			$0$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.12349	+	1.40881i	−1.12349	+	1.40881i	−0.222521	+	0.974928i	$0.571429\pi$
$449$	−1.12349	+	1.40881i	−1.12349	+	1.40881i	−0.900969	+	0.433884i	$0.857143\pi$
$450$	0.346011	+	0.433884i	0.346011	+	0.433884i
$451$	−0.722521	−	0.347948i	−0.722521	−	0.347948i
$452$	−1.12349	−	0.541044i	−1.12349	−	0.541044i
$453$		0			0
$454$	−1.12349	+	1.40881i	−1.12349	+	1.40881i
$455$		0			0
$456$	−0.346011	−	1.51597i	−0.346011	−	1.51597i
$457$	0.0990311	+	0.433884i	0.0990311	+	0.433884i	1.00000			$0$
$457$	0.0990311	+	0.433884i	0.0990311	+	0.433884i	−0.900969	+	0.433884i	$0.857143\pi$
$458$		0			0
$459$	0.623490	−	0.300257i	0.623490	−	0.300257i
$460$		0			0
$461$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$461$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$462$		0			0
$463$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$463$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$464$		0			0
$465$		0			0
$466$	1.24698			1.24698
$467$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$467$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−1.80194			−1.80194
$473$	1.62349	+	0.781831i	1.62349	+	0.781831i
$474$		0			0
$475$	−1.12349	+	0.541044i	−1.12349	+	0.541044i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$		0			0
$482$	−1.12349	−	1.40881i	−1.12349	−	1.40881i
$483$		0			0
$484$	1.40097	+	1.75676i	1.40097	+	1.75676i
$485$		0			0
$486$	0.222521	−	0.974928i	0.222521	−	0.974928i
$487$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$487$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$488$		0			0
$489$	0.123490	+	0.541044i	0.123490	+	0.541044i
$490$		0			0
$491$	−1.12349	−	0.541044i	−1.12349	−	0.541044i	−0.222521	−	0.974928i	$-0.571429\pi$
$491$	−1.12349	−	0.541044i	−1.12349	−	0.541044i	−0.900969	+	0.433884i	$0.857143\pi$
$492$	−0.346011	+	0.433884i	−0.346011	+	0.433884i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	2.02446	+	0.974928i	2.02446	+	0.974928i
$499$	0.400969	+	1.75676i	0.400969	+	1.75676i	0.623490	+	0.781831i	$0.285714\pi$
$499$	0.400969	+	1.75676i	0.400969	+	1.75676i	−0.222521	+	0.974928i	$0.571429\pi$
$500$		0			0
$501$		0			0
$502$	−1.80194	+	0.867767i	−1.80194	+	0.867767i
$503$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$503$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.777479	+	0.974928i	0.777479	+	0.974928i
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$	0.623490	−	0.781831i	0.623490	−	0.781831i
$513$	0.623490	+	0.300257i	0.623490	+	0.300257i
$514$	−1.80194	+	0.867767i	−1.80194	+	0.867767i
$515$		0			0
$516$	0.777479	−	0.974928i	0.777479	−	0.974928i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	0.0990311	−	0.433884i	0.0990311	−	0.433884i	−0.900969	−	0.433884i	$-0.857143\pi$
$521$	0.0990311	−	0.433884i	0.0990311	−	0.433884i	1.00000			$0$
$522$		0			0
$523$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$523$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$524$	0.0990311	−	0.433884i	0.0990311	−	0.433884i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	2.02446	−	0.974928i	2.02446	−	0.974928i
$529$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$530$		0			0
$531$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$532$		0			0
$533$		0			0
$534$	0.500000	+	0.240787i	0.500000	+	0.240787i
$535$		0			0
$536$	1.24698	+	1.56366i	1.24698	+	1.56366i
$537$	2.02446	+	0.974928i	2.02446	+	0.974928i
$538$		0			0
$539$	−1.12349	−	1.40881i	−1.12349	−	1.40881i
$540$		0			0
$541$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$541$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$542$		0			0
$543$		0			0
$544$	0.777479	−	0.974928i	0.777479	−	0.974928i
$545$		0			0
$546$		0			0
$547$	0.400969	−	0.193096i	0.400969	−	0.193096i	−0.222521	−	0.974928i	$-0.571429\pi$
$547$	0.400969	−	0.193096i	0.400969	−	0.193096i	0.623490	+	0.781831i	$0.285714\pi$
$548$	−1.12349	−	1.40881i	−1.12349	−	1.40881i
$549$		0			0
$550$	−1.12349	−	1.40881i	−1.12349	−	1.40881i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	0.400969	+	0.193096i	0.400969	+	0.193096i
$557$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$557$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	2.52446	−	1.21572i	2.52446	−	1.21572i
$562$	−1.80194	−	0.867767i	−1.80194	−	0.867767i
$563$	−0.277479	+	0.347948i	−0.277479	+	0.347948i	−0.900969	−	0.433884i	$-0.857143\pi$
$563$	−0.277479	+	0.347948i	−0.277479	+	0.347948i	0.623490	+	0.781831i	$0.285714\pi$
$564$		0			0
$565$		0			0
$566$	1.24698			1.24698
$567$		0			0
$568$		0			0
$569$	0.400969	+	1.75676i	0.400969	+	1.75676i	0.623490	+	0.781831i	$0.285714\pi$
$569$	0.400969	+	1.75676i	0.400969	+	1.75676i	−0.222521	+	0.974928i	$0.571429\pi$
$570$		0			0
$571$	1.62349	−	0.781831i	1.62349	−	0.781831i	0.623490	−	0.781831i	$-0.285714\pi$
$571$	1.62349	−	0.781831i	1.62349	−	0.781831i	1.00000			$0$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.123490	−	0.541044i	−0.123490	−	0.541044i
$577$	1.62349	+	0.781831i	1.62349	+	0.781831i	1.00000			$0$
$577$	1.62349	+	0.781831i	1.62349	+	0.781831i	0.623490	+	0.781831i	$0.285714\pi$
$578$	0.346011	−	0.433884i	0.346011	−	0.433884i
$579$	−1.40097	−	1.75676i	−1.40097	−	1.75676i
$580$		0			0
$581$		0			0
$582$	−1.40097	−	1.75676i	−1.40097	−	1.75676i
$583$		0			0
$584$	−1.80194	−	0.867767i	−1.80194	−	0.867767i
$585$		0			0
$586$		0			0
$587$	−1.12349	+	1.40881i	−1.12349	+	1.40881i	−0.222521	+	0.974928i	$0.571429\pi$
$587$	−1.12349	+	1.40881i	−1.12349	+	1.40881i	−0.900969	+	0.433884i	$0.857143\pi$
$588$	−1.12349	+	0.541044i	−1.12349	+	0.541044i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.777479	+	0.974928i	0.777479	+	0.974928i	1.00000			$0$
$593$	0.777479	+	0.974928i	0.777479	+	0.974928i	−0.222521	+	0.974928i	$0.571429\pi$
$594$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$599$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$600$	−1.12349	+	0.541044i	−1.12349	+	0.541044i
$601$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$601$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$602$		0			0
$603$	1.10992			1.10992
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$607$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$608$	1.24698			1.24698
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−0.153989	−	0.674671i	−0.153989	−	0.674671i
$613$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$613$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$614$	−1.12349	+	0.541044i	−1.12349	+	0.541044i
$615$		0			0
$616$		0			0
$617$	0.777479	−	0.974928i	0.777479	−	0.974928i	−0.222521	−	0.974928i	$-0.571429\pi$
$617$	0.777479	−	0.974928i	0.777479	−	0.974928i	1.00000			$0$
$618$		0			0
$619$	−0.277479	−	1.21572i	−0.277479	−	1.21572i	−0.900969	−	0.433884i	$-0.857143\pi$
$619$	−0.277479	−	1.21572i	−0.277479	−	1.21572i	0.623490	−	0.781831i	$-0.285714\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.623490	+	0.781831i	0.623490	+	0.781831i
$626$	−0.277479	+	0.347948i	−0.277479	+	0.347948i
$627$	2.52446	+	1.21572i	2.52446	+	1.21572i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$631$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$632$		0			0
$633$	0.500000	−	0.240787i	0.500000	−	0.240787i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.12349	+	1.40881i	−1.12349	+	1.40881i	−0.222521	+	0.974928i	$0.571429\pi$
$641$	−1.12349	+	1.40881i	−1.12349	+	1.40881i	−0.900969	+	0.433884i	$0.857143\pi$
$642$	0.500000	+	0.240787i	0.500000	+	0.240787i
$643$	−1.12349	+	0.541044i	−1.12349	+	0.541044i	−0.900969	−	0.433884i	$-0.857143\pi$
$643$	−1.12349	+	0.541044i	−1.12349	+	0.541044i	−0.222521	+	0.974928i	$0.571429\pi$
$644$		0			0
$645$		0			0
$646$	1.55496			1.55496
$647$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$647$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$648$	1.12349	+	0.541044i	1.12349	+	0.541044i
$649$	2.02446	−	2.53859i	2.02446	−	2.53859i
$650$		0			0
$651$		0			0
$652$	−0.445042			−0.445042
$653$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$653$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$654$		0			0
$655$		0			0
$656$	−0.277479	−	0.347948i	−0.277479	−	0.347948i
$657$	−1.00000	+	0.481575i	−1.00000	+	0.481575i
$658$		0			0
$659$	−1.12349	+	0.541044i	−1.12349	+	0.541044i	−0.900969	−	0.433884i	$-0.857143\pi$
$659$	−1.12349	+	0.541044i	−1.12349	+	0.541044i	−0.222521	+	0.974928i	$0.571429\pi$
$660$		0			0
$661$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$661$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$662$	0.400969	+	1.75676i	0.400969	+	1.75676i
$663$		0			0
$664$	−1.12349	+	1.40881i	−1.12349	+	1.40881i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.277479	+	0.347948i	−0.277479	+	0.347948i	−0.900969	−	0.433884i	$-0.857143\pi$
$673$	−0.277479	+	0.347948i	−0.277479	+	0.347948i	0.623490	+	0.781831i	$0.285714\pi$
$674$	1.62349	−	0.781831i	1.62349	−	0.781831i
$675$	0.123490	−	0.541044i	0.123490	−	0.541044i
$676$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$677$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$677$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$678$	−0.346011	−	1.51597i	−0.346011	−	1.51597i
$679$		0			0
$680$		0			0
$681$	−2.24698			−2.24698
$682$		0			0
$683$	−0.277479	+	1.21572i	−0.277479	+	1.21572i	0.623490	+	0.781831i	$0.285714\pi$
$683$	−0.277479	+	1.21572i	−0.277479	+	1.21572i	−0.900969	+	0.433884i	$0.857143\pi$
$684$	0.431468	−	0.541044i	0.431468	−	0.541044i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	0.623490	+	0.781831i	0.623490	+	0.781831i
$689$		0			0
$690$		0			0
$691$	−1.80194	−	0.867767i	−1.80194	−	0.867767i	−0.900969	−	0.433884i	$-0.857143\pi$
$691$	−1.80194	−	0.867767i	−1.80194	−	0.867767i	−0.900969	−	0.433884i	$-0.857143\pi$
$692$		0			0
$693$		0			0
$694$	1.24698			1.24698
$695$		0			0
$696$		0			0
$697$	−0.346011	−	0.433884i	−0.346011	−	0.433884i
$698$		0			0
$699$	0.969501	+	1.21572i	0.969501	+	1.21572i
$700$		0			0
$701$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$701$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$702$		0			0
$703$		0			0
$704$	0.400969	+	1.75676i	0.400969	+	1.75676i
$705$		0			0
$706$	0.400969	+	0.193096i	0.400969	+	0.193096i
$707$		0			0
$708$	−1.40097	−	1.75676i	−1.40097	−	1.75676i
$709$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$709$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$710$		0			0
$711$		0			0
$712$	−0.277479	+	0.347948i	−0.277479	+	0.347948i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−1.12349	+	1.40881i	−1.12349	+	1.40881i
$717$		0			0
$718$		0			0
$719$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$719$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$720$		0			0
$721$		0			0
$722$	0.346011	+	0.433884i	0.346011	+	0.433884i
$723$	0.500000	−	2.19064i	0.500000	−	2.19064i
$724$		0			0
$725$		0			0
$726$	−0.623490	+	2.73169i	−0.623490	+	2.73169i
$727$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$727$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	0.777479	+	0.974928i	0.777479	+	0.974928i
$732$		0			0
$733$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$733$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3.60388			−3.60388
$738$	−0.246980			−0.246980
$739$	0.0990311	−	0.433884i	0.0990311	−	0.433884i	−0.900969	−	0.433884i	$-0.857143\pi$
$739$	0.0990311	−	0.433884i	0.0990311	−	0.433884i	1.00000			$0$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$743$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	0.222521	+	0.974928i	0.222521	+	0.974928i
$748$	0.500000	+	2.19064i	0.500000	+	2.19064i
$749$		0			0
$750$		0			0
$751$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$751$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$752$		0			0
$753$	−2.24698	−	1.08209i	−2.24698	−	1.08209i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$757$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$758$	0.400969	+	1.75676i	0.400969	+	1.75676i
$759$		0			0
$760$		0			0
$761$	0.0990311	−	0.433884i	0.0990311	−	0.433884i	−0.900969	−	0.433884i	$-0.857143\pi$
$761$	0.0990311	−	0.433884i	0.0990311	−	0.433884i	1.00000			$0$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	1.24698			1.24698
$769$	0.400969	−	1.75676i	0.400969	−	1.75676i	−0.222521	−	0.974928i	$-0.571429\pi$
$769$	0.400969	−	1.75676i	0.400969	−	1.75676i	0.623490	−	0.781831i	$-0.285714\pi$
$770$		0			0
$771$	−2.24698	−	1.08209i	−2.24698	−	1.08209i
$772$	1.62349	−	0.781831i	1.62349	−	0.781831i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$	0.554958			0.554958
$775$		0			0
$776$	1.62349	−	0.781831i	1.62349	−	0.781831i
$777$		0			0
$778$		0			0
$779$	0.123490	−	0.541044i	0.123490	−	0.541044i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$785$		0			0
$786$	0.500000	−	0.240787i	0.500000	−	0.240787i
$787$	−0.277479	+	1.21572i	−0.277479	+	1.21572i	0.623490	+	0.781831i	$0.285714\pi$
$787$	−0.277479	+	1.21572i	−0.277479	+	1.21572i	−0.900969	+	0.433884i	$0.857143\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	0.900969	+	0.433884i	0.900969	+	0.433884i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$797$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$798$		0			0
$799$		0			0
$800$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$801$	0.0549581	+	0.240787i	0.0549581	+	0.240787i
$802$	−1.12349	+	1.40881i	−1.12349	+	1.40881i
$803$	3.24698	−	1.56366i	3.24698	−	1.56366i
$804$	−0.554958	+	2.43143i	−0.554958	+	2.43143i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.277479	+	1.21572i	−0.277479	+	1.21572i	0.623490	+	0.781831i	$0.285714\pi$
$809$	−0.277479	+	1.21572i	−0.277479	+	1.21572i	−0.900969	+	0.433884i	$0.857143\pi$
$810$		0			0
$811$	2.00000			2.00000			1.00000			$0$
$811$	2.00000			2.00000			1.00000			$0$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	1.55496			1.55496
$817$	−0.277479	+	1.21572i	−0.277479	+	1.21572i
$818$	−0.445042			−0.445042
$819$		0			0
$820$		0			0
$821$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$821$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$822$	0.500000	−	2.19064i	0.500000	−	2.19064i
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$	0.500000	−	2.19064i	0.500000	−	2.19064i
$826$		0			0
$827$	0.0990311	+	0.433884i	0.0990311	+	0.433884i	1.00000			$0$
$827$	0.0990311	+	0.433884i	0.0990311	+	0.433884i	−0.900969	+	0.433884i	$0.857143\pi$
$828$		0			0
$829$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$829$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.277479	−	1.21572i	−0.277479	−	1.21572i
$834$	0.123490	+	0.541044i	0.123490	+	0.541044i
$835$		0			0
$836$	−1.40097	+	1.75676i	−1.40097	+	1.75676i
$837$		0			0
$838$	0.400969	+	0.193096i	0.400969	+	0.193096i
$839$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$839$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$840$		0			0
$841$	0.623490	−	0.781831i	0.623490	−	0.781831i
$842$		0			0
$843$	−0.554958	−	2.43143i	−0.554958	−	2.43143i
$844$	0.0990311	+	0.433884i	0.0990311	+	0.433884i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	0.969501	+	1.21572i	0.969501	+	1.21572i
$850$	−0.277479	−	1.21572i	−0.277479	−	1.21572i
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$	−0.277479	+	0.347948i	−0.277479	+	0.347948i
$857$	−1.12349	−	0.541044i	−1.12349	−	0.541044i	−0.222521	−	0.974928i	$-0.571429\pi$
$857$	−1.12349	−	0.541044i	−1.12349	−	0.541044i	−0.900969	+	0.433884i	$0.857143\pi$
$858$		0			0
$859$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$859$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$863$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$864$	−0.346011	+	0.433884i	−0.346011	+	0.433884i
$865$		0			0
$866$	−0.445042			−0.445042
$867$	0.692021			0.692021
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.222521	−	0.974928i	0.222521	−	0.974928i
$874$		0			0
$875$		0			0
$876$	−0.554958	−	2.43143i	−0.554958	−	2.43143i
$877$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$877$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.12349	−	0.541044i	−1.12349	−	0.541044i	−0.222521	−	0.974928i	$-0.571429\pi$
$881$	−1.12349	−	0.541044i	−1.12349	−	0.541044i	−0.900969	+	0.433884i	$0.857143\pi$
$882$	−0.500000	−	0.240787i	−0.500000	−	0.240787i
$883$	0.777479	+	0.974928i	0.777479	+	0.974928i	1.00000			$0$
$883$	0.777479	+	0.974928i	0.777479	+	0.974928i	−0.222521	+	0.974928i	$0.571429\pi$
$884$		0			0
$885$		0			0
$886$	0.0990311	+	0.433884i	0.0990311	+	0.433884i
$887$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$887$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−2.02446	+	0.974928i	−2.02446	+	0.974928i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	0.400969	−	1.75676i	0.400969	−	1.75676i
$899$		0			0
$900$	−0.500000	−	0.240787i	−0.500000	−	0.240787i
$901$		0			0
$902$	0.801938			0.801938
$903$		0			0
$904$	1.24698			1.24698
$905$		0			0
$906$		0			0
$907$	−0.277479	+	0.347948i	−0.277479	+	0.347948i	−0.900969	−	0.433884i	$-0.857143\pi$
$907$	−0.277479	+	0.347948i	−0.277479	+	0.347948i	0.623490	+	0.781831i	$0.285714\pi$
$908$	0.400969	−	1.75676i	0.400969	−	1.75676i
$909$		0			0
$910$		0			0
$911$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$911$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$912$	0.969501	+	1.21572i	0.969501	+	1.21572i
$913$	−0.722521	−	3.16557i	−0.722521	−	3.16557i
$914$	−0.277479	−	0.347948i	−0.277479	−	0.347948i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−0.431468	+	0.541044i	−0.431468	+	0.541044i
$919$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$919$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$920$		0			0
$921$	−1.40097	−	0.674671i	−1.40097	−	0.674671i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−0.445042	−	1.94986i	−0.445042	−	1.94986i	−0.222521	−	0.974928i	$-0.571429\pi$
$929$	−0.445042	−	1.94986i	−0.445042	−	1.94986i	−0.222521	−	0.974928i	$-0.571429\pi$
$930$		0			0
$931$	0.777479	−	0.974928i	0.777479	−	0.974928i
$932$	−1.12349	+	0.541044i	−1.12349	+	0.541044i
$933$		0			0
$934$	0.400969	−	0.193096i	0.400969	−	0.193096i
$935$		0			0
$936$		0			0
$937$	−1.12349	−	1.40881i	−1.12349	−	1.40881i	−0.900969	−	0.433884i	$-0.857143\pi$
$937$	−1.12349	−	1.40881i	−1.12349	−	1.40881i	−0.222521	−	0.974928i	$-0.571429\pi$
$938$		0			0
$939$	−0.554958			−0.554958
$940$		0			0
$941$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$941$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$942$		0			0
$943$		0			0
$944$	1.62349	−	0.781831i	1.62349	−	0.781831i
$945$		0			0
$946$	−1.80194			−1.80194
$947$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$947$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$948$		0			0
$949$		0			0
$950$	0.777479	−	0.974928i	0.777479	−	0.974928i
$951$		0			0
$952$		0			0
$953$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$953$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.623490	−	0.781831i	0.623490	−	0.781831i
$962$		0			0
$963$	0.0549581	+	0.240787i	0.0549581	+	0.240787i
$964$	1.62349	+	0.781831i	1.62349	+	0.781831i
$965$		0			0
$966$		0			0
$967$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$967$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$968$	−2.02446	−	0.974928i	−2.02446	−	0.974928i
$969$	1.20895	+	1.51597i	1.20895	+	1.51597i
$970$		0			0
$971$	0.400969	+	0.193096i	0.400969	+	0.193096i	0.623490	−	0.781831i	$-0.285714\pi$
$971$	0.400969	+	0.193096i	0.400969	+	0.193096i	−0.222521	+	0.974928i	$0.571429\pi$
$972$	0.222521	+	0.974928i	0.222521	+	0.974928i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.12349	+	0.541044i	−1.12349	+	0.541044i	−0.900969	−	0.433884i	$-0.857143\pi$
$977$	−1.12349	+	0.541044i	−1.12349	+	0.541044i	−0.222521	+	0.974928i	$0.571429\pi$
$978$	−0.346011	−	0.433884i	−0.346011	−	0.433884i
$979$	−0.178448	−	0.781831i	−0.178448	−	0.781831i
$980$		0			0
$981$		0			0
$982$	1.24698			1.24698
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$	0.123490	−	0.541044i	0.123490	−	0.541044i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$991$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$992$		0			0
$993$	−1.40097	+	1.75676i	−1.40097	+	1.75676i
$994$		0			0
$995$		0			0
$996$	−2.24698			−2.24698
$997$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$997$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$998$	−1.12349	−	1.40881i	−1.12349	−	1.40881i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	344.1.s.a.299.1	yes	6
3.2	odd	2		3096.1.dm.a.1675.1		6
4.3	odd	2		1376.1.be.a.815.1		6
8.3	odd	2	CM	344.1.s.a.299.1	yes	6
8.5	even	2		1376.1.be.a.815.1		6
24.11	even	2		3096.1.dm.a.1675.1		6
43.21	even	7	inner	344.1.s.a.107.1	✓	6
129.107	odd	14		3096.1.dm.a.451.1		6
172.107	odd	14		1376.1.be.a.623.1		6
344.21	even	14		1376.1.be.a.623.1		6
344.107	odd	14	inner	344.1.s.a.107.1	✓	6
1032.107	even	14		3096.1.dm.a.451.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
344.1.s.a.107.1	✓	6	43.21	even	7	inner
344.1.s.a.107.1	✓	6	344.107	odd	14	inner
344.1.s.a.299.1	yes	6	1.1	even	1	trivial
344.1.s.a.299.1	yes	6	8.3	odd	2	CM
1376.1.be.a.623.1		6	172.107	odd	14
1376.1.be.a.623.1		6	344.21	even	14
1376.1.be.a.815.1		6	4.3	odd	2
1376.1.be.a.815.1		6	8.5	even	2
3096.1.dm.a.451.1		6	129.107	odd	14
3096.1.dm.a.451.1		6	1032.107	even	14
3096.1.dm.a.1675.1		6	3.2	odd	2
3096.1.dm.a.1675.1		6	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 344 = 2^{3} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	344.s (of order \(14\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.900969 + 0.433884i) q^{2} +(-1.12349 - 0.541044i) q^{3} +(0.623490 - 0.781831i) q^{4} +1.24698 q^{6} +(-0.222521 + 0.974928i) q^{8} +(0.346011 + 0.433884i) q^{9} +O(q^{10})\)	\(q+(-0.900969 + 0.433884i) q^{2} +(-1.12349 - 0.541044i) q^{3} +(0.623490 - 0.781831i) q^{4} +1.24698 q^{6} +(-0.222521 + 0.974928i) q^{8} +(0.346011 + 0.433884i) q^{9} +(-1.12349 - 1.40881i) q^{11} +(-1.12349 + 0.541044i) q^{12} +(-0.222521 - 0.974928i) q^{16} +(-0.277479 - 1.21572i) q^{17} +(-0.500000 - 0.240787i) q^{18} +(0.777479 - 0.974928i) q^{19} +(1.62349 + 0.781831i) q^{22} +(0.777479 - 0.974928i) q^{24} +(-0.900969 - 0.433884i) q^{25} +(0.123490 + 0.541044i) q^{27} +(0.623490 + 0.781831i) q^{32} +(0.500000 + 2.19064i) q^{33} +(0.777479 + 0.974928i) q^{34} +0.554958 q^{36} +(-0.277479 + 1.21572i) q^{38} +(0.400969 - 0.193096i) q^{41} +(-0.900969 + 0.433884i) q^{43} -1.80194 q^{44} +(-0.277479 + 1.21572i) q^{48} +1.00000 q^{49} +1.00000 q^{50} +(-0.346011 + 1.51597i) q^{51} +(-0.346011 - 0.433884i) q^{54} +(-1.40097 + 0.674671i) q^{57} +(0.400969 + 1.75676i) q^{59} +(-0.900969 - 0.433884i) q^{64} +(-1.40097 - 1.75676i) q^{66} +(1.24698 - 1.56366i) q^{67} +(-1.12349 - 0.541044i) q^{68} +(-0.500000 + 0.240787i) q^{72} +(-0.445042 + 1.94986i) q^{73} +(0.777479 + 0.974928i) q^{75} +(-0.277479 - 1.21572i) q^{76} +(0.277479 - 1.21572i) q^{81} +(-0.277479 + 0.347948i) q^{82} +(1.62349 + 0.781831i) q^{83} +(0.623490 - 0.781831i) q^{86} +(1.62349 - 0.781831i) q^{88} +(0.400969 + 0.193096i) q^{89} +(-0.277479 - 1.21572i) q^{96} +(-1.12349 - 1.40881i) q^{97} +(-0.900969 + 0.433884i) q^{98} +(0.222521 - 0.974928i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} - 2 q^{3} - q^{4} - 2 q^{6} - q^{8} - 3 q^{9}+O(q^{10}) \) 6 * q - q^2 - 2 * q^3 - q^4 - 2 * q^6 - q^8 - 3 * q^9	\( 6 q - q^{2} - 2 q^{3} - q^{4} - 2 q^{6} - q^{8} - 3 q^{9} - 2 q^{11} - 2 q^{12} - q^{16} - 2 q^{17} - 3 q^{18} + 5 q^{19} + 5 q^{22} + 5 q^{24} - q^{25} - 4 q^{27} - q^{32} + 3 q^{33} + 5 q^{34} + 4 q^{36} - 2 q^{38} - 2 q^{41} - q^{43} - 2 q^{44} - 2 q^{48} + 6 q^{49} + 6 q^{50} + 3 q^{51} + 3 q^{54} - 4 q^{57} - 2 q^{59} - q^{64} - 4 q^{66} - 2 q^{67} - 2 q^{68} - 3 q^{72} - 2 q^{73} + 5 q^{75} - 2 q^{76} + 2 q^{81} - 2 q^{82} + 5 q^{83} - q^{86} + 5 q^{88} - 2 q^{89} - 2 q^{96} - 2 q^{97} - q^{98} + q^{99}+O(q^{100}) \) 6 * q - q^2 - 2 * q^3 - q^4 - 2 * q^6 - q^8 - 3 * q^9 - 2 * q^11 - 2 * q^12 - q^16 - 2 * q^17 - 3 * q^18 + 5 * q^19 + 5 * q^22 + 5 * q^24 - q^25 - 4 * q^27 - q^32 + 3 * q^33 + 5 * q^34 + 4 * q^36 - 2 * q^38 - 2 * q^41 - q^43 - 2 * q^44 - 2 * q^48 + 6 * q^49 + 6 * q^50 + 3 * q^51 + 3 * q^54 - 4 * q^57 - 2 * q^59 - q^64 - 4 * q^66 - 2 * q^67 - 2 * q^68 - 3 * q^72 - 2 * q^73 + 5 * q^75 - 2 * q^76 + 2 * q^81 - 2 * q^82 + 5 * q^83 - q^86 + 5 * q^88 - 2 * q^89 - 2 * q^96 - 2 * q^97 - q^98 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more