LMFDB - Embedded newform 2303.1.j.b.1268.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2303,1,Mod(281,2303)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2303, base_ring=CyclotomicField(14))

chi = DirichletCharacter(H, H._module([4, 7]))

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

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

chi := DirichletCharacter("2303.281");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2303 = 7^{2} \cdot 47 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2303.j (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:	$1.14934672409$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$4$ over $\Q(\zeta_{14})$
Coefficient field:	$\Q(\zeta_{35})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} - x^{23} + x^{19} - x^{18} + x^{17} - x^{16} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} + \cdots + 1 $ x^24 - x^23 + x^19 - x^18 + x^17 - x^16 + x^14 - x^13 + x^12 - x^11 + x^10 - x^8 + x^7 - x^6 + x^5 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{35}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{35} - \cdots)$

Embedding invariants

Embedding label			1268.3
Root			$-0.0448648 - 0.998993i$ of defining polynomial
Character	$\chi$	$=$	2303.1268
Dual form			2303.1.j.b.939.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.385338 - 0.483198i) q^{2} +(-1.35699 - 0.653491i) q^{3} +(0.137526 + 0.602539i) q^{4} +(-0.838665 + 0.403880i) q^{6} +(0.936235 - 0.351375i) q^{7} +(0.900969 + 0.433884i) q^{8} +(0.790876 + 0.991727i) q^{9} +(0.207133 - 0.907509i) q^{12} +(0.190983 - 0.587785i) q^{14} +(-0.210891 + 0.923976i) q^{17} +0.783955 q^{18} +(-1.50008 - 0.135010i) q^{21} +(-0.939065 - 1.17755i) q^{24} +(0.623490 + 0.781831i) q^{25} +(-0.0899761 - 0.394211i) q^{27} +(0.340473 + 0.515795i) q^{28} +(-0.222521 + 0.974928i) q^{32} +(0.365199 + 0.457945i) q^{34} +(-0.488788 + 0.612921i) q^{36} +(0.0199667 - 0.0874800i) q^{37} +(-0.643274 + 0.672812i) q^{42} +(0.623490 - 0.781831i) q^{47} +(0.753071 - 0.657939i) q^{49} +0.618034 q^{50} +(0.889987 - 1.11601i) q^{51} +(0.443250 + 1.94201i) q^{53} +(-0.225153 - 0.108428i) q^{54} +(0.995974 + 0.0896393i) q^{56} +(-0.241880 + 0.116483i) q^{59} +(0.245172 - 1.07417i) q^{61} +(1.08891 + 0.650596i) q^{63} +(0.385338 + 0.483198i) q^{64} -0.585734 q^{68} +(-0.335148 - 1.46838i) q^{71} +(0.282260 + 1.23666i) q^{72} +(-0.0345762 - 0.0433572i) q^{74} +(-0.335148 - 1.46838i) q^{75} +1.96786 q^{79} +(0.146743 - 0.642925i) q^{81} +(-0.277479 - 0.347948i) q^{83} +(-0.124951 - 0.922423i) q^{84} +(0.385338 + 0.483198i) q^{89} +(-0.137526 - 0.602539i) q^{94} +(0.939065 - 1.17755i) q^{96} +0.618034 q^{97} +(-0.0277280 - 0.617412i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 2 q^{2} + 2 q^{3} - 2 q^{4} + 4 q^{6} + q^{7} + 4 q^{8} - 2 q^{9} + 10 q^{12} + 18 q^{14} + 2 q^{17} - 18 q^{18} - 5 q^{21} - 2 q^{24} - 4 q^{25} - 3 q^{27} + 3 q^{28} - 4 q^{32} + 4 q^{34} + 4 q^{36}+ \cdots + 2 q^{98}+O(q^{100}) $ 24 * q + 2 * q^2 + 2 * q^3 - 2 * q^4 + 4 * q^6 + q^7 + 4 * q^8 - 2 * q^9 + 10 * q^12 + 18 * q^14 + 2 * q^17 - 18 * q^18 - 5 * q^21 - 2 * q^24 - 4 * q^25 - 3 * q^27 + 3 * q^28 - 4 * q^32 + 4 * q^34 + 4 * q^36 + 2 * q^37 - q^42 - 4 * q^47 + q^49 - 12 * q^50 - 13 * q^51 - 5 * q^53 - 2 * q^54 - q^56 - 5 * q^59 + 2 * q^61 - 2 * q^63 + 2 * q^64 - 4 * q^68 - 5 * q^71 + 2 * q^72 - 10 * q^74 - 5 * q^75 + 2 * q^79 - 8 * q^83 + q^84 + 2 * q^89 + 2 * q^94 + 2 * q^96 - 12 * q^97 + 2 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2303\mathbb{Z}\right)^\times$.

$n$	$99$	$2257$
$\chi(n)$	$-1$	$e\left(\frac{1}{7}\right)$

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 $

$2$	0.385338	−	0.483198i	0.385338	−	0.483198i	−0.550897	−	0.834573i	$-0.685714\pi$
$2$	0.385338	−	0.483198i	0.385338	−	0.483198i	0.936235	+	0.351375i	$0.114286\pi$
$3$	−1.35699	−	0.653491i	−1.35699	−	0.653491i	−0.393025	−	0.919528i	$-0.628571\pi$
$3$	−1.35699	−	0.653491i	−1.35699	−	0.653491i	−0.963963	+	0.266037i	$0.914286\pi$
$4$	0.137526	+	0.602539i	0.137526	+	0.602539i
$5$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$5$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$6$	−0.838665	+	0.403880i	−0.838665	+	0.403880i
$7$	0.936235	−	0.351375i	0.936235	−	0.351375i
$7$	0.936235	−	0.351375i	0.936235	−	0.351375i
$8$	0.900969	+	0.433884i	0.900969	+	0.433884i
$9$	0.790876	+	0.991727i	0.790876	+	0.991727i
$10$		0			0
$11$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$11$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$12$	0.207133	−	0.907509i	0.207133	−	0.907509i
$13$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$13$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$14$	0.190983	−	0.587785i	0.190983	−	0.587785i
$15$		0			0
$16$		0			0
$17$	−0.210891	+	0.923976i	−0.210891	+	0.923976i	0.753071	+	0.657939i	$0.228571\pi$
$17$	−0.210891	+	0.923976i	−0.210891	+	0.923976i	−0.963963	+	0.266037i	$0.914286\pi$
$18$	0.783955			0.783955
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$	−1.50008	−	0.135010i	−1.50008	−	0.135010i
$22$		0			0
$23$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$23$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$24$	−0.939065	−	1.17755i	−0.939065	−	1.17755i
$25$	0.623490	+	0.781831i	0.623490	+	0.781831i
$26$		0			0
$27$	−0.0899761	−	0.394211i	−0.0899761	−	0.394211i
$28$	0.340473	+	0.515795i	0.340473	+	0.515795i
$29$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$29$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$33$		0			0
$34$	0.365199	+	0.457945i	0.365199	+	0.457945i
$35$		0			0
$36$	−0.488788	+	0.612921i	−0.488788	+	0.612921i
$37$	0.0199667	−	0.0874800i	0.0199667	−	0.0874800i	−0.963963	−	0.266037i	$-0.914286\pi$
$37$	0.0199667	−	0.0874800i	0.0199667	−	0.0874800i	0.983930	+	0.178557i	$0.0571429\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$41$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$42$	−0.643274	+	0.672812i	−0.643274	+	0.672812i
$43$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$43$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.623490	−	0.781831i	0.623490	−	0.781831i
$47$	0.623490	−	0.781831i	0.623490	−	0.781831i
$48$		0			0
$49$	0.753071	−	0.657939i	0.753071	−	0.657939i
$50$	0.618034			0.618034
$51$	0.889987	−	1.11601i	0.889987	−	1.11601i
$52$		0			0
$53$	0.443250	+	1.94201i	0.443250	+	1.94201i	0.309017	+	0.951057i	$0.400000\pi$
$53$	0.443250	+	1.94201i	0.443250	+	1.94201i	0.134233	+	0.990950i	$0.457143\pi$
$54$	−0.225153	−	0.108428i	−0.225153	−	0.108428i
$55$		0			0
$56$	0.995974	+	0.0896393i	0.995974	+	0.0896393i
$57$		0			0
$58$		0			0
$59$	−0.241880	+	0.116483i	−0.241880	+	0.116483i	−0.550897	−	0.834573i	$-0.685714\pi$
$59$	−0.241880	+	0.116483i	−0.241880	+	0.116483i	0.309017	+	0.951057i	$0.400000\pi$
$60$		0			0
$61$	0.245172	−	1.07417i	0.245172	−	1.07417i	−0.691063	−	0.722795i	$-0.742857\pi$
$61$	0.245172	−	1.07417i	0.245172	−	1.07417i	0.936235	−	0.351375i	$-0.114286\pi$
$62$		0			0
$63$	1.08891	+	0.650596i	1.08891	+	0.650596i
$64$	0.385338	+	0.483198i	0.385338	+	0.483198i
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$	−0.585734			−0.585734
$69$		0			0
$70$		0			0
$71$	−0.335148	−	1.46838i	−0.335148	−	1.46838i	−0.809017	−	0.587785i	$-0.800000\pi$
$71$	−0.335148	−	1.46838i	−0.335148	−	1.46838i	0.473869	−	0.880596i	$-0.342857\pi$
$72$	0.282260	+	1.23666i	0.282260	+	1.23666i
$73$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$73$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$74$	−0.0345762	−	0.0433572i	−0.0345762	−	0.0433572i
$75$	−0.335148	−	1.46838i	−0.335148	−	1.46838i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	1.96786			1.96786			0.983930	−	0.178557i	$-0.0571429\pi$
$79$	1.96786			1.96786			0.983930	+	0.178557i	$0.0571429\pi$
$80$		0			0
$81$	0.146743	−	0.642925i	0.146743	−	0.642925i
$82$		0			0
$83$	−0.277479	−	0.347948i	−0.277479	−	0.347948i	0.623490	−	0.781831i	$-0.285714\pi$
$83$	−0.277479	−	0.347948i	−0.277479	−	0.347948i	−0.900969	+	0.433884i	$0.857143\pi$
$84$	−0.124951	−	0.922423i	−0.124951	−	0.922423i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.385338	+	0.483198i	0.385338	+	0.483198i	0.936235	−	0.351375i	$-0.114286\pi$
$89$	0.385338	+	0.483198i	0.385338	+	0.483198i	−0.550897	+	0.834573i	$0.685714\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$	−0.137526	−	0.602539i	−0.137526	−	0.602539i
$95$		0			0
$96$	0.939065	−	1.17755i	0.939065	−	1.17755i
$97$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$97$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$98$	−0.0277280	−	0.617412i	−0.0277280	−	0.617412i
$99$		0			0
$100$	−0.385338	+	0.483198i	−0.385338	+	0.483198i
$101$	−0.853882	−	0.411208i	−0.853882	−	0.411208i	−0.0448648	−	0.998993i	$-0.514286\pi$
$101$	−0.853882	−	0.411208i	−0.853882	−	0.411208i	−0.809017	+	0.587785i	$0.800000\pi$
$102$	−0.196308	−	0.860080i	−0.196308	−	0.860080i
$103$	0.0808436	+	0.0389322i	0.0808436	+	0.0389322i	0.473869	−	0.880596i	$-0.342857\pi$
$103$	0.0808436	+	0.0389322i	0.0808436	+	0.0389322i	−0.393025	+	0.919528i	$0.628571\pi$
$104$		0			0
$105$		0			0
$106$	1.10918	+	0.534151i	1.10918	+	0.534151i
$107$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$107$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$108$	0.225153	−	0.108428i	0.225153	−	0.108428i
$109$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$109$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$110$		0			0
$111$	−0.0842620	+	0.105661i	−0.0842620	+	0.105661i
$112$		0			0
$113$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$113$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−0.0369210	+	0.161761i	−0.0369210	+	0.161761i
$119$	0.127218	+	0.939160i	0.127218	+	0.939160i
$120$		0			0
$121$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$122$	−0.424563	−	0.532385i	−0.424563	−	0.532385i
$123$		0			0
$124$		0			0
$125$		0			0
$126$	0.733966	−	0.275462i	0.733966	−	0.275462i
$127$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$127$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$128$	−0.618034			−0.618034
$129$		0			0
$130$		0			0
$131$	0.992682	−	0.478050i	0.992682	−	0.478050i	0.134233	−	0.990950i	$-0.457143\pi$
$131$	0.992682	−	0.478050i	0.992682	−	0.478050i	0.858449	+	0.512899i	$0.171429\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−0.590905	+	0.740971i	−0.590905	+	0.740971i
$137$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$137$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$138$		0			0
$139$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$139$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$140$		0			0
$141$	−1.35699	+	0.653491i	−1.35699	+	0.653491i
$142$	−0.838665	−	0.403880i	−0.838665	−	0.403880i
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−1.45187	+	0.400690i	−1.45187	+	0.400690i
$148$	0.0554560			0.0554560
$149$	−0.490094	+	0.614559i	−0.490094	+	0.614559i	−0.963963	−	0.266037i	$-0.914286\pi$
$149$	−0.490094	+	0.614559i	−0.490094	+	0.614559i	0.473869	+	0.880596i	$0.342857\pi$
$150$	−0.838665	−	0.403880i	−0.838665	−	0.403880i
$151$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$151$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$152$		0			0
$153$	−1.08312	+	0.521603i	−1.08312	+	0.521603i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.68704	+	0.812434i	−1.68704	+	0.812434i	−0.691063	+	0.722795i	$0.742857\pi$
$157$	−1.68704	+	0.812434i	−1.68704	+	0.812434i	−0.995974	+	0.0896393i	$0.971429\pi$
$158$	0.758291	−	0.950866i	0.758291	−	0.950866i
$159$	0.667598	−	2.92494i	0.667598	−	2.92494i
$160$		0			0
$161$		0			0
$162$	−0.254115	−	0.318650i	−0.254115	−	0.318650i
$163$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$163$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$164$		0			0
$165$		0			0
$166$	−0.275051			−0.275051
$167$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$167$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$168$	−1.29295	−	0.772500i	−1.29295	−	0.772500i
$169$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.443250	+	1.94201i	0.443250	+	1.94201i	0.309017	+	0.951057i	$0.400000\pi$
$173$	0.443250	+	1.94201i	0.443250	+	1.94201i	0.134233	+	0.990950i	$0.457143\pi$
$174$		0			0
$175$	0.858449	+	0.512899i	0.858449	+	0.512899i
$176$		0			0
$177$	0.404349			0.404349
$178$	0.381966			0.381966
$179$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$179$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$180$		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
$183$	−1.03466	+	1.29742i	−1.03466	+	1.29742i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	0.556829	+	0.268155i	0.556829	+	0.268155i
$189$	−0.222755	−	0.337459i	−0.222755	−	0.337459i
$190$		0			0
$191$	−1.12349	−	0.541044i	−1.12349	−	0.541044i	−0.222521	−	0.974928i	$-0.571429\pi$
$191$	−1.12349	−	0.541044i	−1.12349	−	0.541044i	−0.900969	+	0.433884i	$0.857143\pi$
$192$	−0.207133	−	0.907509i	−0.207133	−	0.907509i
$193$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$193$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$194$	0.238152	−	0.298633i	0.238152	−	0.298633i
$195$		0			0
$196$	0.500000	+	0.363271i	0.500000	+	0.363271i
$197$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$197$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$198$		0			0
$199$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$199$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$200$	0.222521	+	0.974928i	0.222521	+	0.974928i
$201$		0			0
$202$	−0.527728	+	0.254140i	−0.527728	+	0.254140i
$203$		0			0
$204$	0.794834	+	0.382772i	0.794834	+	0.382772i
$205$		0			0
$206$	0.0499641	−	0.0240614i	0.0499641	−	0.0240614i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$211$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$212$	−1.10918	+	0.534151i	−1.10918	+	0.534151i
$213$	−0.504781	+	2.21159i	−0.504781	+	2.21159i
$214$		0			0
$215$		0			0
$216$	0.0899761	−	0.394211i	0.0899761	−	0.394211i
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$	0.0185860	+	0.0814305i	0.0185860	+	0.0814305i
$223$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$223$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$224$	0.134233	+	0.990950i	0.134233	+	0.990950i
$225$	−0.282260	+	1.23666i	−0.282260	+	1.23666i
$226$		0			0
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$		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
$233$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$233$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$234$		0			0
$235$		0			0
$236$	−0.103450	−	0.129723i	−0.103450	−	0.129723i
$237$	−2.67036	−	1.28598i	−2.67036	−	1.28598i
$238$	0.502823	+	0.300422i	0.502823	+	0.300422i
$239$	0.0808436	−	0.0389322i	0.0808436	−	0.0389322i	−0.393025	−	0.919528i	$-0.628571\pi$
$239$	0.0808436	−	0.0389322i	0.0808436	−	0.0389322i	0.473869	+	0.880596i	$0.342857\pi$
$240$		0			0
$241$	0.429004	+	1.87959i	0.429004	+	1.87959i	0.473869	+	0.880596i	$0.342857\pi$
$241$	0.429004	+	1.87959i	0.429004	+	1.87959i	−0.0448648	+	0.998993i	$0.514286\pi$
$242$	−0.556829	−	0.268155i	−0.556829	−	0.268155i
$243$	−0.871382	+	1.09268i	−0.871382	+	1.09268i
$244$	0.680946			0.680946
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	0.149155	+	0.653491i	0.149155	+	0.653491i
$250$		0			0
$251$	−1.77298	+	0.853822i	−1.77298	+	0.853822i	−0.809017	+	0.587785i	$0.800000\pi$
$251$	−1.77298	+	0.853822i	−1.77298	+	0.853822i	−0.963963	+	0.266037i	$0.914286\pi$
$252$	−0.242256	+	0.745586i	−0.242256	+	0.745586i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$257$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$257$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$258$		0			0
$259$	−0.0120447	−	0.0889176i	−0.0120447	−	0.0889176i
$260$		0			0
$261$		0			0
$262$	0.151525	−	0.663873i	0.151525	−	0.663873i
$263$	−1.10179			−1.10179			−0.550897	−	0.834573i	$-0.685714\pi$
$263$	−1.10179			−1.10179			−0.550897	+	0.834573i	$0.685714\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−0.207133	−	0.907509i	−0.207133	−	0.907509i
$268$		0			0
$269$	−1.12349	−	1.40881i	−1.12349	−	1.40881i	−0.900969	−	0.433884i	$-0.857143\pi$
$269$	−1.12349	−	1.40881i	−1.12349	−	1.40881i	−0.222521	−	0.974928i	$-0.571429\pi$
$270$		0			0
$271$	0.0199667	+	0.0874800i	0.0199667	+	0.0874800i	0.983930	−	0.178557i	$-0.0571429\pi$
$271$	0.0199667	+	0.0874800i	0.0199667	+	0.0874800i	−0.963963	+	0.266037i	$0.914286\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.0199667	−	0.0874800i	0.0199667	−	0.0874800i	−0.963963	−	0.266037i	$-0.914286\pi$
$277$	0.0199667	−	0.0874800i	0.0199667	−	0.0874800i	0.983930	+	0.178557i	$0.0571429\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$281$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$282$	−0.207133	+	0.907509i	−0.207133	+	0.907509i
$283$	1.16747	−	1.46396i	1.16747	−	1.46396i	0.309017	−	0.951057i	$-0.400000\pi$
$283$	1.16747	−	1.46396i	1.16747	−	1.46396i	0.858449	−	0.512899i	$-0.171429\pi$
$284$	0.838665	−	0.403880i	0.838665	−	0.403880i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−1.14285	+	0.550367i	−1.14285	+	0.550367i
$289$	0.0917132	+	0.0441667i	0.0917132	+	0.0441667i
$290$		0			0
$291$	−0.838665	−	0.403880i	−0.838665	−	0.403880i
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$	−0.365846	+	0.855940i	−0.365846	+	0.855940i
$295$		0			0
$296$	0.0559455	−	0.0701535i	0.0559455	−	0.0701535i
$297$		0			0
$298$	0.108102	+	0.473625i	0.108102	+	0.473625i
$299$		0			0
$300$	0.838665	−	0.403880i	0.838665	−	0.403880i
$301$		0			0
$302$		0			0
$303$	0.889987	+	1.11601i	0.889987	+	1.11601i
$304$		0			0
$305$		0			0
$306$	−0.165329	+	0.724356i	−0.165329	+	0.724356i
$307$	0.939065	−	1.17755i	0.939065	−	1.17755i	−0.0448648	−	0.998993i	$-0.514286\pi$
$307$	0.939065	−	1.17755i	0.939065	−	1.17755i	0.983930	−	0.178557i	$-0.0571429\pi$
$308$		0			0
$309$	−0.0842620	−	0.105661i	−0.0842620	−	0.105661i
$310$		0			0
$311$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$311$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$	−0.257512	+	1.12824i	−0.257512	+	1.12824i
$315$		0			0
$316$	0.270631	+	1.18571i	0.270631	+	1.18571i
$317$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$317$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$318$	−1.15608	−	1.44967i	−1.15608	−	1.44967i
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.407568			0.407568
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.309017	−	0.951057i	0.309017	−	0.951057i
$330$		0			0
$331$	0.443250	−	1.94201i	0.443250	−	1.94201i	0.134233	−	0.990950i	$-0.457143\pi$
$331$	0.443250	−	1.94201i	0.443250	−	1.94201i	0.309017	−	0.951057i	$-0.400000\pi$
$332$	0.171491	−	0.215044i	0.171491	−	0.215044i
$333$	0.102547	−	0.0493842i	0.102547	−	0.0493842i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.54687	+	0.744934i	−1.54687	+	0.744934i	−0.995974	−	0.0896393i	$-0.971429\pi$
$337$	−1.54687	+	0.744934i	−1.54687	+	0.744934i	−0.550897	+	0.834573i	$0.685714\pi$
$338$	−0.556829	−	0.268155i	−0.556829	−	0.268155i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.473869	−	0.880596i	0.473869	−	0.880596i
$344$		0			0
$345$		0			0
$346$	1.10918	+	0.534151i	1.10918	+	0.534151i
$347$	−0.437890	−	1.91852i	−0.437890	−	1.91852i	−0.393025	−	0.919528i	$-0.628571\pi$
$347$	−0.437890	−	1.91852i	−0.437890	−	1.91852i	−0.0448648	−	0.998993i	$-0.514286\pi$
$348$		0			0
$349$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$349$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$350$	0.578625	−	0.217162i	0.578625	−	0.217162i
$351$		0			0
$352$		0			0
$353$	−0.556829	+	0.268155i	−0.556829	+	0.268155i	−0.691063	−	0.722795i	$-0.742857\pi$
$353$	−0.556829	+	0.268155i	−0.556829	+	0.268155i	0.134233	+	0.990950i	$0.457143\pi$
$354$	0.155811	−	0.195381i	0.155811	−	0.195381i
$355$		0			0
$356$	−0.238152	+	0.298633i	−0.238152	+	0.298633i
$357$	0.441099	−	1.35756i	0.441099	−	1.35756i
$358$		0			0
$359$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$359$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$360$		0			0
$361$	1.00000			1.00000
$362$		0			0
$363$	−0.335148	+	1.46838i	−0.335148	+	1.46838i
$364$		0			0
$365$		0			0
$366$	0.228218	+	0.999888i	0.228218	+	0.999888i
$367$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$367$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.09736	+	1.66243i	1.09736	+	1.66243i
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	0.900969	−	0.433884i	0.900969	−	0.433884i
$377$		0			0
$378$	−0.248895	−	0.0224010i	−0.248895	−	0.0224010i
$379$	0.167386	−	0.209896i	0.167386	−	0.209896i	−0.691063	−	0.722795i	$-0.742857\pi$
$379$	0.167386	−	0.209896i	0.167386	−	0.209896i	0.858449	+	0.512899i	$0.171429\pi$
$380$		0			0
$381$		0			0
$382$	−0.694355	+	0.334384i	−0.694355	+	0.334384i
$383$	0.939065	+	1.17755i	0.939065	+	1.17755i	0.983930	+	0.178557i	$0.0571429\pi$
$383$	0.939065	+	1.17755i	0.939065	+	1.17755i	−0.0448648	+	0.998993i	$0.514286\pi$
$384$	0.838665	+	0.403880i	0.838665	+	0.403880i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	0.0849954	+	0.372389i	0.0849954	+	0.372389i
$389$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$389$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$390$		0			0
$391$		0			0
$392$	0.963963	−	0.266037i	0.963963	−	0.266037i
$393$	−1.65946			−1.65946
$394$	−0.694355	+	0.870693i	−0.694355	+	0.870693i
$395$		0			0
$396$		0			0
$397$	−1.35699	−	0.653491i	−1.35699	−	0.653491i	−0.393025	−	0.919528i	$-0.628571\pi$
$397$	−1.35699	−	0.653491i	−1.35699	−	0.653491i	−0.963963	+	0.266037i	$0.914286\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−0.490094	−	0.614559i	−0.490094	−	0.614559i	0.473869	−	0.880596i	$-0.342857\pi$
$401$	−0.490094	−	0.614559i	−0.490094	−	0.614559i	−0.963963	+	0.266037i	$0.914286\pi$
$402$		0			0
$403$		0			0
$404$	0.130338	−	0.571048i	0.130338	−	0.571048i
$405$		0			0
$406$		0			0
$407$		0			0
$408$	1.28607	−	0.619338i	1.28607	−	0.619338i
$409$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$409$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$410$		0			0
$411$		0			0
$412$	−0.0123401	+	0.0540656i	−0.0123401	+	0.0540656i
$413$	−0.185527	+	0.194046i	−0.185527	+	0.194046i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$419$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$420$		0			0
$421$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$421$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$422$		0			0
$423$	1.26847			1.26847
$424$	−0.443250	+	1.94201i	−0.443250	+	1.94201i
$425$	−0.853882	+	0.411208i	−0.853882	+	0.411208i
$426$	0.874126	+	1.09612i	0.874126	+	1.09612i
$427$	−0.147897	−	1.09182i	−0.147897	−	1.09182i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−1.54687	+	0.744934i	−1.54687	+	0.744934i	−0.995974	−	0.0896393i	$-0.971429\pi$
$431$	−1.54687	+	0.744934i	−1.54687	+	0.744934i	−0.550897	+	0.834573i	$0.685714\pi$
$432$		0			0
$433$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$433$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−1.12349	+	1.40881i	−1.12349	+	1.40881i	−0.222521	+	0.974928i	$0.571429\pi$
$439$	−1.12349	+	1.40881i	−1.12349	+	1.40881i	−0.900969	+	0.433884i	$0.857143\pi$
$440$		0			0
$441$	1.24808	+	0.226493i	1.24808	+	0.226493i
$442$		0			0
$443$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$443$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$444$	−0.0752531	−	0.0362400i	−0.0752531	−	0.0362400i
$445$		0			0
$446$		0			0
$447$	1.06666	−	0.513677i	1.06666	−	0.513677i
$448$	0.530551	+	0.316989i	0.530551	+	0.316989i
$449$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$449$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$450$	0.488788	+	0.612921i	0.488788	+	0.612921i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.853882	+	0.411208i	−0.853882	+	0.411208i	−0.809017	−	0.587785i	$-0.800000\pi$
$457$	−0.853882	+	0.411208i	−0.853882	+	0.411208i	−0.0448648	+	0.998993i	$0.514286\pi$
$458$		0			0
$459$	0.383217			0.383217
$460$		0			0
$461$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$461$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$462$		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
$466$		0			0
$467$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$467$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	2.82021			2.82021
$472$	−0.268467			−0.268467
$473$		0			0
$474$	−1.65037	+	0.794778i	−1.65037	+	0.794778i
$475$		0			0
$476$	−0.548384	+	0.205812i	−0.548384	+	0.205812i
$477$	−1.57538	+	1.97547i	−1.57538	+	1.97547i
$478$	0.0123401	−	0.0540656i	0.0123401	−	0.0540656i
$479$	1.07047	−	1.34232i	1.07047	−	1.34232i	0.134233	−	0.990950i	$-0.457143\pi$
$479$	1.07047	−	1.34232i	1.07047	−	1.34232i	0.936235	−	0.351375i	$-0.114286\pi$
$480$		0			0
$481$		0			0
$482$	1.07353	+	0.516983i	1.07353	+	0.516983i
$483$		0			0
$484$	0.556829	−	0.268155i	0.556829	−	0.268155i
$485$		0			0
$486$	0.192204	+	0.842101i	0.192204	+	0.842101i
$487$	−1.12349	−	0.541044i	−1.12349	−	0.541044i	−0.222521	−	0.974928i	$-0.571429\pi$
$487$	−1.12349	−	0.541044i	−1.12349	−	0.541044i	−0.900969	+	0.433884i	$0.857143\pi$
$488$	0.686957	−	0.861417i	0.686957	−	0.861417i
$489$		0			0
$490$		0			0
$491$	−1.38213			−1.38213			−0.691063	−	0.722795i	$-0.742857\pi$
$491$	−1.38213			−1.38213			−0.691063	+	0.722795i	$0.742857\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.829730	−	1.25699i	−0.829730	−	1.25699i
$498$	0.373241	+	0.179743i	0.373241	+	0.179743i
$499$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$499$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$500$		0			0
$501$		0			0
$502$	−0.270631	+	1.18571i	−0.270631	+	1.18571i
$503$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$503$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$504$	0.698794	+	1.05863i	0.698794	+	1.05863i
$505$		0			0
$506$		0			0
$507$	−0.335148	+	1.46838i	−0.335148	+	1.46838i
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−0.0476061	−	0.0284433i	−0.0476061	−	0.0284433i
$519$	0.667598	−	2.92494i	0.667598	−	2.92494i
$520$		0			0
$521$	1.87247			1.87247			0.936235	−	0.351375i	$-0.114286\pi$
$521$	1.87247			1.87247			0.936235	+	0.351375i	$0.114286\pi$
$522$		0			0
$523$	−1.12349	+	0.541044i	−1.12349	+	0.541044i	−0.900969	−	0.433884i	$-0.857143\pi$
$523$	−1.12349	+	0.541044i	−1.12349	+	0.541044i	−0.222521	+	0.974928i	$0.571429\pi$
$524$	0.424563	+	0.532385i	0.424563	+	0.532385i
$525$	−0.829730	−	1.25699i	−0.829730	−	1.25699i
$526$	−0.424563	+	0.532385i	−0.424563	+	0.532385i
$527$		0			0
$528$		0			0
$529$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$530$		0			0
$531$	−0.306817	−	0.147755i	−0.306817	−	0.147755i
$532$		0			0
$533$		0			0
$534$	−0.518323	−	0.249611i	−0.518323	−	0.249611i
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−1.11366			−1.11366
$539$		0			0
$540$		0			0
$541$	1.16747	−	1.46396i	1.16747	−	1.46396i	0.309017	−	0.951057i	$-0.400000\pi$
$541$	1.16747	−	1.46396i	1.16747	−	1.46396i	0.858449	−	0.512899i	$-0.171429\pi$
$542$	0.0499641	+	0.0240614i	0.0499641	+	0.0240614i
$543$		0			0
$544$	−0.853882	−	0.411208i	−0.853882	−	0.411208i
$545$		0			0
$546$		0			0
$547$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$547$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$548$		0			0
$549$	1.25918	−	0.606391i	1.25918	−	0.606391i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	1.84238	−	0.691456i	1.84238	−	0.691456i
$554$	−0.0345762	−	0.0433572i	−0.0345762	−	0.0433572i
$555$		0			0
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$563$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$564$	−0.580374	−	0.727766i	−0.580374	−	0.727766i
$565$		0			0
$566$	−0.257512	−	1.12824i	−0.257512	−	1.12824i
$567$	−0.0885214	−	0.653491i	−0.0885214	−	0.653491i
$568$	0.335148	−	1.46838i	0.335148	−	1.46838i
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	0.429004	−	1.87959i	0.429004	−	1.87959i	−0.0448648	−	0.998993i	$-0.514286\pi$
$571$	0.429004	−	1.87959i	0.429004	−	1.87959i	0.473869	−	0.880596i	$-0.342857\pi$
$572$		0			0
$573$	1.17099	+	1.46838i	1.17099	+	1.46838i
$574$		0			0
$575$		0			0
$576$	−0.174446	+	0.764300i	−0.174446	+	0.764300i
$577$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$577$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$578$	0.0566819	−	0.0272965i	0.0566819	−	0.0272965i
$579$		0			0
$580$		0			0
$581$	−0.382046	−	0.228262i	−0.382046	−	0.228262i
$582$	−0.518323	+	0.249611i	−0.518323	+	0.249611i
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$	−0.441099	−	0.819700i	−0.441099	−	0.819700i
$589$		0			0
$590$		0			0
$591$	2.44521	+	1.17755i	2.44521	+	1.17755i
$592$		0			0
$593$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$593$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$594$		0			0
$595$		0			0
$596$	−0.437696	−	0.210783i	−0.437696	−	0.210783i
$597$		0			0
$598$		0			0
$599$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$599$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$600$	0.335148	−	1.46838i	0.335148	−	1.46838i
$601$	0.167386	−	0.209896i	0.167386	−	0.209896i	−0.691063	−	0.722795i	$-0.742857\pi$
$601$	0.167386	−	0.209896i	0.167386	−	0.209896i	0.858449	+	0.512899i	$0.171429\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$	0.882199			0.882199
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−0.463243	−	0.580888i	−0.463243	−	0.580888i
$613$	1.07047	+	1.34232i	1.07047	+	1.34232i	0.936235	+	0.351375i	$0.114286\pi$
$613$	1.07047	+	1.34232i	1.07047	+	1.34232i	0.134233	+	0.990950i	$0.457143\pi$
$614$	−0.207133	−	0.907509i	−0.207133	−	0.907509i
$615$		0			0
$616$		0			0
$617$	0.429004	−	1.87959i	0.429004	−	1.87959i	−0.0448648	−	0.998993i	$-0.514286\pi$
$617$	0.429004	−	1.87959i	0.429004	−	1.87959i	0.473869	−	0.880596i	$-0.342857\pi$
$618$	−0.0835246			−0.0835246
$619$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$619$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	0.530551	+	0.316989i	0.530551	+	0.316989i
$624$		0			0
$625$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$626$		0			0
$627$		0			0
$628$	−0.721534	−	0.904774i	−0.721534	−	0.904774i
$629$	0.0766185	+	0.0368975i	0.0766185	+	0.0368975i
$630$		0			0
$631$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$631$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$632$	1.77298	+	0.853822i	1.77298	+	0.853822i
$633$		0			0
$634$		0			0
$635$		0			0
$636$	1.85420			1.85420
$637$		0			0
$638$		0			0
$639$	1.19117	−	1.49368i	1.19117	−	1.49368i
$640$		0			0
$641$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$641$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$642$		0			0
$643$	−1.54687	+	0.744934i	−1.54687	+	0.744934i	−0.995974	−	0.0896393i	$-0.971429\pi$
$643$	−1.54687	+	0.744934i	−1.54687	+	0.744934i	−0.550897	+	0.834573i	$0.685714\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−0.241880	+	0.116483i	−0.241880	+	0.116483i	−0.550897	−	0.834573i	$-0.685714\pi$
$647$	−0.241880	+	0.116483i	−0.241880	+	0.116483i	0.309017	+	0.951057i	$0.400000\pi$
$648$	0.411166	−	0.515586i	0.411166	−	0.515586i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	0.708207	−	0.341054i	0.708207	−	0.341054i	−0.0448648	−	0.998993i	$-0.514286\pi$
$653$	0.708207	−	0.341054i	0.708207	−	0.341054i	0.753071	+	0.657939i	$0.228571\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	−0.340473	−	0.515795i	−0.340473	−	0.515795i
$659$	0.0199667	+	0.0874800i	0.0199667	+	0.0874800i	0.983930	−	0.178557i	$-0.0571429\pi$
$659$	0.0199667	+	0.0874800i	0.0199667	+	0.0874800i	−0.963963	+	0.266037i	$0.914286\pi$
$660$		0			0
$661$	−1.20204	−	1.50731i	−1.20204	−	1.50731i	−0.809017	−	0.587785i	$-0.800000\pi$
$661$	−1.20204	−	1.50731i	−1.20204	−	1.50731i	−0.393025	−	0.919528i	$-0.628571\pi$
$662$	−0.767573	−	0.962506i	−0.767573	−	0.962506i
$663$		0			0
$664$	−0.0990311	−	0.433884i	−0.0990311	−	0.433884i
$665$		0			0
$666$	0.0156530	−	0.0685804i	0.0156530	−	0.0685804i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	0.465424	−	1.43243i	0.465424	−	1.43243i
$673$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$673$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$674$	−0.236117	+	1.03450i	−0.236117	+	1.03450i
$675$	0.252107	−	0.316133i	0.252107	−	0.316133i
$676$	0.556829	−	0.268155i	0.556829	−	0.268155i
$677$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$677$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$678$		0			0
$679$	0.578625	−	0.217162i	0.578625	−	0.217162i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−0.556829	−	0.268155i	−0.556829	−	0.268155i	0.134233	−	0.990950i	$-0.457143\pi$
$683$	−0.556829	−	0.268155i	−0.556829	−	0.268155i	−0.691063	+	0.722795i	$0.742857\pi$
$684$		0			0
$685$		0			0
$686$	−0.242903	−	0.568299i	−0.242903	−	0.568299i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$691$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$692$	−1.10918	+	0.534151i	−1.10918	+	0.534151i
$693$		0			0
$694$	−1.09576	−	0.527691i	−1.09576	−	0.527691i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$701$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$701$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−0.0849954	+	0.372389i	−0.0849954	+	0.372389i
$707$	−0.943922	−	0.0849545i	−0.943922	−	0.0849545i
$708$	0.0556083	+	0.243636i	0.0556083	+	0.243636i
$709$	−0.0597394	−	0.261736i	−0.0597394	−	0.261736i	0.936235	−	0.351375i	$-0.114286\pi$
$709$	−0.0597394	−	0.261736i	−0.0597394	−	0.261736i	−0.995974	+	0.0896393i	$0.971429\pi$
$710$		0			0
$711$	1.55633	+	1.95158i	1.55633	+	1.95158i
$712$	0.137526	+	0.602539i	0.137526	+	0.602539i
$713$		0			0
$714$	−0.486001	−	0.736260i	−0.486001	−	0.736260i
$715$		0			0
$716$		0			0
$717$	−0.135146			−0.135146
$718$		0			0
$719$	−1.68704	+	0.812434i	−1.68704	+	0.812434i	−0.691063	+	0.722795i	$0.742857\pi$
$719$	−1.68704	+	0.812434i	−1.68704	+	0.812434i	−0.995974	+	0.0896393i	$0.971429\pi$
$720$		0			0
$721$	0.0893684	+	0.00804330i	0.0893684	+	0.00804330i
$722$	0.385338	−	0.483198i	0.385338	−	0.483198i
$723$	0.646141	−	2.83093i	0.646141	−	2.83093i
$724$		0			0
$725$		0			0
$726$	0.580374	+	0.727766i	0.580374	+	0.727766i
$727$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$727$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$728$		0			0
$729$	1.30236	−	0.627183i	1.30236	−	0.627183i
$730$		0			0
$731$		0			0
$732$	−0.924036	−	0.444992i	−0.924036	−	0.444992i
$733$	−0.490094	+	0.614559i	−0.490094	+	0.614559i	−0.963963	−	0.266037i	$-0.914286\pi$
$733$	−0.490094	+	0.614559i	−0.490094	+	0.614559i	0.473869	+	0.880596i	$0.342857\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.245172	+	1.07417i	0.245172	+	1.07417i	0.936235	+	0.351375i	$0.114286\pi$
$739$	0.245172	+	1.07417i	0.245172	+	1.07417i	−0.691063	+	0.722795i	$0.742857\pi$
$740$		0			0
$741$		0			0
$742$	1.22614	+	0.110354i	1.22614	+	0.110354i
$743$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$743$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	0.125618	−	0.550367i	0.125618	−	0.550367i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$751$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$752$		0			0
$753$	2.96388			2.96388
$754$		0			0
$755$		0			0
$756$	0.172698	−	0.180627i	0.172698	−	0.180627i
$757$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$757$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$758$	−0.0369210	−	0.161761i	−0.0369210	−	0.161761i
$759$		0			0
$760$		0			0
$761$	−0.277479	−	1.21572i	−0.277479	−	1.21572i	−0.900969	−	0.433884i	$-0.857143\pi$
$761$	−0.277479	−	1.21572i	−0.277479	−	1.21572i	0.623490	−	0.781831i	$-0.285714\pi$
$762$		0			0
$763$		0			0
$764$	0.171491	−	0.751353i	0.171491	−	0.751353i
$765$		0			0
$766$	0.930848			0.930848
$767$		0			0
$768$	1.35699	−	0.653491i	1.35699	−	0.653491i
$769$	0.590905	+	0.740971i	0.590905	+	0.740971i	0.983930	−	0.178557i	$-0.0571429\pi$
$769$	0.590905	+	0.740971i	0.590905	+	0.740971i	−0.393025	+	0.919528i	$0.628571\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−0.0559455	+	0.0701535i	−0.0559455	+	0.0701535i	−0.809017	−	0.587785i	$-0.800000\pi$
$773$	−0.0559455	+	0.0701535i	−0.0559455	+	0.0701535i	0.753071	+	0.657939i	$0.228571\pi$
$774$		0			0
$775$		0			0
$776$	0.556829	+	0.268155i	0.556829	+	0.268155i
$777$	−0.0417623	+	0.128531i	−0.0417623	+	0.128531i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$	−0.639453	+	0.801848i	−0.639453	+	0.801848i
$787$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$787$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$788$	−0.247812	−	1.08574i	−0.247812	−	1.08574i
$789$	1.49512	+	0.720012i	1.49512	+	0.720012i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	−0.838665	+	0.403880i	−0.838665	+	0.403880i
$795$		0			0
$796$		0			0
$797$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$797$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$798$		0			0
$799$	0.590905	+	0.740971i	0.590905	+	0.740971i
$800$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$801$	−0.174446	+	0.764300i	−0.174446	+	0.764300i
$802$	−0.485806			−0.485806
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	0.603916	+	2.64593i	0.603916	+	2.64593i
$808$	−0.590905	−	0.740971i	−0.590905	−	0.740971i
$809$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$809$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$810$		0			0
$811$	0.360046	+	1.57747i	0.360046	+	1.57747i	0.753071	+	0.657939i	$0.228571\pi$
$811$	0.360046	+	1.57747i	0.360046	+	1.57747i	−0.393025	+	0.919528i	$0.628571\pi$
$812$		0			0
$813$	0.0300727	−	0.131757i	0.0300727	−	0.131757i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$821$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$822$		0			0
$823$	1.62349	−	0.781831i	1.62349	−	0.781831i	0.623490	−	0.781831i	$-0.285714\pi$
$823$	1.62349	−	0.781831i	1.62349	−	0.781831i	1.00000			$0$
$824$	0.0559455	+	0.0701535i	0.0559455	+	0.0701535i
$825$		0			0
$826$	0.0222722	+	0.164420i	0.0222722	+	0.164420i
$827$	1.24525	−	0.599682i	1.24525	−	0.599682i	0.309017	−	0.951057i	$-0.400000\pi$
$827$	1.24525	−	0.599682i	1.24525	−	0.599682i	0.936235	+	0.351375i	$0.114286\pi$
$828$		0			0
$829$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$829$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$830$		0			0
$831$	−0.0842620	+	0.105661i	−0.0842620	+	0.105661i
$832$		0			0
$833$	0.449103	+	0.834573i	0.449103	+	0.834573i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$839$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$840$		0			0
$841$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$	0.488788	−	0.612921i	0.488788	−	0.612921i
$847$	−0.550897	−	0.834573i	−0.550897	−	0.834573i
$848$		0			0
$849$	−2.54092	+	1.22364i	−2.54092	+	1.22364i
$850$	−0.130338	+	0.571048i	−0.130338	+	0.571048i
$851$		0			0
$852$	−1.40199			−1.40199
$853$	0.360046	−	1.57747i	0.360046	−	1.57747i	−0.393025	−	0.919528i	$-0.628571\pi$
$853$	0.360046	−	1.57747i	0.360046	−	1.57747i	0.753071	−	0.657939i	$-0.228571\pi$
$854$	−0.584557	−	0.349257i	−0.584557	−	0.349257i
$855$		0			0
$856$		0			0
$857$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$857$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$858$		0			0
$859$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$859$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$860$		0			0
$861$		0			0
$862$	−0.236117	+	1.03450i	−0.236117	+	1.03450i
$863$	1.50614			1.50614			0.753071	−	0.657939i	$-0.228571\pi$
$863$	1.50614			1.50614			0.753071	+	0.657939i	$0.228571\pi$
$864$	0.404349			0.404349
$865$		0			0
$866$		0			0
$867$	−0.0955911	−	0.119867i	−0.0955911	−	0.119867i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.488788	+	0.612921i	0.488788	+	0.612921i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$877$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$878$	0.247812	+	1.08574i	0.247812	+	1.08574i
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	0.590374	−	0.515795i	0.590374	−	0.515795i
$883$	−1.38213			−1.38213			−0.691063	−	0.722795i	$-0.742857\pi$
$883$	−1.38213			−1.38213			−0.691063	+	0.722795i	$0.742857\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$887$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$888$	−0.121762	+	0.0586375i	−0.121762	+	0.0586375i
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$	0.162817	−	0.713348i	0.162817	−	0.713348i
$895$		0			0
$896$	−0.578625	+	0.217162i	−0.578625	+	0.217162i
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.783955			−0.783955
$901$	−1.88784			−1.88784
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1.24698	+	1.56366i	1.24698	+	1.56366i	0.623490	+	0.781831i	$0.285714\pi$
$907$	1.24698	+	1.56366i	1.24698	+	1.56366i	0.623490	+	0.781831i	$0.285714\pi$
$908$		0			0
$909$	−0.267509	−	1.17203i	−0.267509	−	1.17203i
$910$		0			0
$911$	0.245172	−	1.07417i	0.245172	−	1.07417i	−0.691063	−	0.722795i	$-0.742857\pi$
$911$	0.245172	−	1.07417i	0.245172	−	1.07417i	0.936235	−	0.351375i	$-0.114286\pi$
$912$		0			0
$913$		0			0
$914$	−0.130338	+	0.571048i	−0.130338	+	0.571048i
$915$		0			0
$916$		0			0
$917$	0.761409	−	0.796371i	0.761409	−	0.796371i
$918$	0.147668	−	0.185170i	0.147668	−	0.185170i
$919$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$919$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$920$		0			0
$921$	−2.04382	+	0.984251i	−2.04382	+	0.984251i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.0808436	−	0.0389322i	0.0808436	−	0.0389322i
$926$		0			0
$927$	0.0253271	+	0.110965i	0.0253271	+	0.110965i
$928$		0			0
$929$	−0.490094	+	0.614559i	−0.490094	+	0.614559i	−0.963963	−	0.266037i	$-0.914286\pi$
$929$	−0.490094	+	0.614559i	−0.490094	+	0.614559i	0.473869	+	0.880596i	$0.342857\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$937$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.24525	−	0.599682i	1.24525	−	0.599682i	0.309017	−	0.951057i	$-0.400000\pi$
$941$	1.24525	−	0.599682i	1.24525	−	0.599682i	0.936235	+	0.351375i	$0.114286\pi$
$942$	1.08673	−	1.36272i	1.08673	−	1.36272i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.12349	+	0.541044i	−1.12349	+	0.541044i	−0.900969	−	0.433884i	$-0.857143\pi$
$947$	−1.12349	+	0.541044i	−1.12349	+	0.541044i	−0.222521	+	0.974928i	$0.571429\pi$
$948$	0.407609	−	1.78585i	0.407609	−	1.78585i
$949$		0			0
$950$		0			0
$951$		0			0
$952$	−0.292867	+	0.901352i	−0.292867	+	0.901352i
$953$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$953$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$954$	0.347488	+	1.52245i	0.347488	+	1.52245i
$955$		0			0
$956$	0.0345762	+	0.0433572i	0.0345762	+	0.0433572i
$957$		0			0
$958$	−0.236117	−	1.03450i	−0.236117	−	1.03450i
$959$		0			0
$960$		0			0
$961$	1.00000			1.00000
$962$		0			0
$963$		0			0
$964$	−1.07353	+	0.516983i	−1.07353	+	0.516983i
$965$		0			0
$966$		0			0
$967$	−0.861741	+	1.08059i	−0.861741	+	1.08059i	0.134233	+	0.990950i	$0.457143\pi$
$967$	−0.861741	+	1.08059i	−0.861741	+	1.08059i	−0.995974	+	0.0896393i	$0.971429\pi$
$968$	0.222521	−	0.974928i	0.222521	−	0.974928i
$969$		0			0
$970$		0			0
$971$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$971$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$972$	−0.778218	−	0.374770i	−0.778218	−	0.374770i
$973$		0			0
$974$	−0.694355	+	0.334384i	−0.694355	+	0.334384i
$975$		0			0
$976$		0			0
$977$	0.400969	+	0.193096i	0.400969	+	0.193096i	0.623490	−	0.781831i	$-0.285714\pi$
$977$	0.400969	+	0.193096i	0.400969	+	0.193096i	−0.222521	+	0.974928i	$0.571429\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−0.532585	+	0.667841i	−0.532585	+	0.667841i
$983$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$983$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−1.04084	+	1.08863i	−1.04084	+	1.08863i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−1.24196	+	1.55737i	−1.24196	+	1.55737i	−0.550897	+	0.834573i	$0.685714\pi$
$991$	−1.24196	+	1.55737i	−1.24196	+	1.55737i	−0.691063	+	0.722795i	$0.742857\pi$
$992$		0			0
$993$	−1.87057	+	2.34562i	−1.87057	+	2.34562i
$994$	−0.927100	−	0.0834405i	−0.927100	−	0.0834405i
$995$		0			0
$996$	−0.373241	+	0.179743i	−0.373241	+	0.179743i
$997$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$997$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$998$		0			0
$999$	−0.0362821			−0.0362821

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2303.1.j.b.1268.3	yes	24
47.46	odd	2	CM	2303.1.j.b.1268.3	yes	24
49.8	even	7	inner	2303.1.j.b.939.3	✓	24
2303.939	odd	14	inner	2303.1.j.b.939.3	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2303.1.j.b.939.3	✓	24	49.8	even	7	inner
2303.1.j.b.939.3	✓	24	2303.939	odd	14	inner
2303.1.j.b.1268.3	yes	24	1.1	even	1	trivial
2303.1.j.b.1268.3	yes	24	47.46	odd	2	CM

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:	\( N \)	\(=\)	\( 2303 = 7^{2} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2303.j (of order \(14\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(0.385338 - 0.483198i) q^{2} +(-1.35699 - 0.653491i) q^{3} +(0.137526 + 0.602539i) q^{4} +(-0.838665 + 0.403880i) q^{6} +(0.936235 - 0.351375i) q^{7} +(0.900969 + 0.433884i) q^{8} +(0.790876 + 0.991727i) q^{9} +(0.207133 - 0.907509i) q^{12} +(0.190983 - 0.587785i) q^{14} +(-0.210891 + 0.923976i) q^{17} +0.783955 q^{18} +(-1.50008 - 0.135010i) q^{21} +(-0.939065 - 1.17755i) q^{24} +(0.623490 + 0.781831i) q^{25} +(-0.0899761 - 0.394211i) q^{27} +(0.340473 + 0.515795i) q^{28} +(-0.222521 + 0.974928i) q^{32} +(0.365199 + 0.457945i) q^{34} +(-0.488788 + 0.612921i) q^{36} +(0.0199667 - 0.0874800i) q^{37} +(-0.643274 + 0.672812i) q^{42} +(0.623490 - 0.781831i) q^{47} +(0.753071 - 0.657939i) q^{49} +0.618034 q^{50} +(0.889987 - 1.11601i) q^{51} +(0.443250 + 1.94201i) q^{53} +(-0.225153 - 0.108428i) q^{54} +(0.995974 + 0.0896393i) q^{56} +(-0.241880 + 0.116483i) q^{59} +(0.245172 - 1.07417i) q^{61} +(1.08891 + 0.650596i) q^{63} +(0.385338 + 0.483198i) q^{64} -0.585734 q^{68} +(-0.335148 - 1.46838i) q^{71} +(0.282260 + 1.23666i) q^{72} +(-0.0345762 - 0.0433572i) q^{74} +(-0.335148 - 1.46838i) q^{75} +1.96786 q^{79} +(0.146743 - 0.642925i) q^{81} +(-0.277479 - 0.347948i) q^{83} +(-0.124951 - 0.922423i) q^{84} +(0.385338 + 0.483198i) q^{89} +(-0.137526 - 0.602539i) q^{94} +(0.939065 - 1.17755i) q^{96} +0.618034 q^{97} +(-0.0277280 - 0.617412i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 2 q^{2} + 2 q^{3} - 2 q^{4} + 4 q^{6} + q^{7} + 4 q^{8} - 2 q^{9} + 10 q^{12} + 18 q^{14} + 2 q^{17} - 18 q^{18} - 5 q^{21} - 2 q^{24} - 4 q^{25} - 3 q^{27} + 3 q^{28} - 4 q^{32} + 4 q^{34} + 4 q^{36}+ \cdots + 2 q^{98}+O(q^{100}) \) 24 * q + 2 * q^2 + 2 * q^3 - 2 * q^4 + 4 * q^6 + q^7 + 4 * q^8 - 2 * q^9 + 10 * q^12 + 18 * q^14 + 2 * q^17 - 18 * q^18 - 5 * q^21 - 2 * q^24 - 4 * q^25 - 3 * q^27 + 3 * q^28 - 4 * q^32 + 4 * q^34 + 4 * q^36 + 2 * q^37 - q^42 - 4 * q^47 + q^49 - 12 * q^50 - 13 * q^51 - 5 * q^53 - 2 * q^54 - q^56 - 5 * q^59 + 2 * q^61 - 2 * q^63 + 2 * q^64 - 4 * q^68 - 5 * q^71 + 2 * q^72 - 10 * q^74 - 5 * q^75 + 2 * q^79 - 8 * q^83 + q^84 + 2 * q^89 + 2 * q^94 + 2 * q^96 - 12 * q^97 + 2 * q^98

Introduction

L-functions

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