LMFDB - Embedded newform 2352.1.cv.a.1985.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,1,Mod(65,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(42))

chi = DirichletCharacter(H, H._module([0, 0, 21, 20]))

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

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

chi := DirichletCharacter("2352.65");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2352.cv (of order $42$, degree $12$, not 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:	$1.17380090971$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{21})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 $ x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 588)
Projective image:	$D_{21}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{21} - \cdots)$

Embedding invariants

Embedding label			1985.1
Root			$0.0747301 + 0.997204i$ of defining polynomial
Character	$\chi$	$=$	2352.1985
Dual form			2352.1.cv.a.737.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.365341 - 0.930874i) q^{3} +(-0.955573 + 0.294755i) q^{7} +(-0.733052 + 0.680173i) q^{9} +O(q^{10})$	$q+(-0.365341 - 0.930874i) q^{3} +(-0.955573 + 0.294755i) q^{7} +(-0.733052 + 0.680173i) q^{9} +(-0.162592 + 0.712362i) q^{13} +(0.955573 + 1.65510i) q^{19} +(0.623490 + 0.781831i) q^{21} +(0.955573 + 0.294755i) q^{25} +(0.900969 + 0.433884i) q^{27} +(0.826239 - 1.43109i) q^{31} +(0.123490 + 0.0841939i) q^{37} +(0.722521 - 0.108903i) q^{39} +(0.914101 + 1.14625i) q^{43} +(0.826239 - 0.563320i) q^{49} +(1.19158 - 1.49419i) q^{57} +(-1.48883 - 1.01507i) q^{61} +(0.500000 - 0.866025i) q^{63} +(-0.988831 + 1.71271i) q^{67} +(0.142820 + 0.0440542i) q^{73} +(-0.0747301 - 0.997204i) q^{75} +(0.826239 + 1.43109i) q^{79} +(0.0747301 - 0.997204i) q^{81} +(-0.0546039 - 0.728639i) q^{91} +(-1.63402 - 0.246289i) q^{93} -0.445042 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - q^{3} - q^{7} + q^{9}+O(q^{10}) $ 12 * q - q^3 - q^7 + q^9	$ 12 q - q^{3} - q^{7} + q^{9} + 2 q^{13} + q^{19} - 2 q^{21} + q^{25} + 2 q^{27} + q^{31} - 8 q^{37} + 8 q^{39} - 2 q^{43} + q^{49} + 2 q^{57} - 5 q^{61} + 6 q^{63} + q^{67} - q^{73} - q^{75} + q^{79} + q^{81} + q^{91} - q^{93} - 4 q^{97}+O(q^{100}) $ 12 * q - q^3 - q^7 + q^9 + 2 * q^13 + q^19 - 2 * q^21 + q^25 + 2 * q^27 + q^31 - 8 * q^37 + 8 * q^39 - 2 * q^43 + q^49 + 2 * q^57 - 5 * q^61 + 6 * q^63 + q^67 - q^73 - q^75 + q^79 + q^81 + q^91 - q^93 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$-1$	$1$	$1$	$e\left(\frac{8}{21}\right)$

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			0
$2$		0			0
$3$	−0.365341	−	0.930874i	−0.365341	−	0.930874i
$3$	−0.365341	−	0.930874i	−0.365341	−	0.930874i
$4$		0			0
$5$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$5$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$6$		0			0
$7$	−0.955573	+	0.294755i	−0.955573	+	0.294755i
$7$	−0.955573	+	0.294755i	−0.955573	+	0.294755i
$8$		0			0
$9$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$10$		0			0
$11$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$11$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$12$		0			0
$13$	−0.162592	+	0.712362i	−0.162592	+	0.712362i	0.826239	+	0.563320i	$0.190476\pi$
$13$	−0.162592	+	0.712362i	−0.162592	+	0.712362i	−0.988831	+	0.149042i	$0.952381\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$17$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$18$		0			0
$19$	0.955573	+	1.65510i	0.955573	+	1.65510i	0.733052	+	0.680173i	$0.238095\pi$
$19$	0.955573	+	1.65510i	0.955573	+	1.65510i	0.222521	+	0.974928i	$0.428571\pi$
$20$		0			0
$21$	0.623490	+	0.781831i	0.623490	+	0.781831i
$22$		0			0
$23$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$23$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$24$		0			0
$25$	0.955573	+	0.294755i	0.955573	+	0.294755i
$26$		0			0
$27$	0.900969	+	0.433884i	0.900969	+	0.433884i
$28$		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
$31$	0.826239	−	1.43109i	0.826239	−	1.43109i	−0.0747301	−	0.997204i	$-0.523810\pi$
$31$	0.826239	−	1.43109i	0.826239	−	1.43109i	0.900969	−	0.433884i	$-0.142857\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	0.123490	+	0.0841939i	0.123490	+	0.0841939i	0.623490	−	0.781831i	$-0.285714\pi$
$37$	0.123490	+	0.0841939i	0.123490	+	0.0841939i	−0.500000	+	0.866025i	$0.666667\pi$
$38$		0			0
$39$	0.722521	−	0.108903i	0.722521	−	0.108903i
$40$		0			0
$41$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$41$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$42$		0			0
$43$	0.914101	+	1.14625i	0.914101	+	1.14625i	0.988831	+	0.149042i	$0.0476190\pi$
$43$	0.914101	+	1.14625i	0.914101	+	1.14625i	−0.0747301	+	0.997204i	$0.523810\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$47$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$48$		0			0
$49$	0.826239	−	0.563320i	0.826239	−	0.563320i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$53$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	1.19158	−	1.49419i	1.19158	−	1.49419i
$58$		0			0
$59$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$59$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$60$		0			0
$61$	−1.48883	−	1.01507i	−1.48883	−	1.01507i	−0.988831	−	0.149042i	$-0.952381\pi$
$61$	−1.48883	−	1.01507i	−1.48883	−	1.01507i	−0.500000	−	0.866025i	$-0.666667\pi$
$62$		0			0
$63$	0.500000	−	0.866025i	0.500000	−	0.866025i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−0.988831	+	1.71271i	−0.988831	+	1.71271i	−0.365341	+	0.930874i	$0.619048\pi$
$67$	−0.988831	+	1.71271i	−0.988831	+	1.71271i	−0.623490	+	0.781831i	$0.714286\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$71$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$72$		0			0
$73$	0.142820	+	0.0440542i	0.142820	+	0.0440542i	0.365341	−	0.930874i	$-0.380952\pi$
$73$	0.142820	+	0.0440542i	0.142820	+	0.0440542i	−0.222521	+	0.974928i	$0.571429\pi$
$74$		0			0
$75$	−0.0747301	−	0.997204i	−0.0747301	−	0.997204i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	0.826239	+	1.43109i	0.826239	+	1.43109i	0.900969	+	0.433884i	$0.142857\pi$
$79$	0.826239	+	1.43109i	0.826239	+	1.43109i	−0.0747301	+	0.997204i	$0.523810\pi$
$80$		0			0
$81$	0.0747301	−	0.997204i	0.0747301	−	0.997204i
$82$		0			0
$83$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$83$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$89$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$90$		0			0
$91$	−0.0546039	−	0.728639i	−0.0546039	−	0.728639i
$92$		0			0
$93$	−1.63402	−	0.246289i	−1.63402	−	0.246289i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$97$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$101$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$102$		0			0
$103$	−1.44973	−	0.218511i	−1.44973	−	0.218511i	−0.623490	−	0.781831i	$-0.714286\pi$
$103$	−1.44973	−	0.218511i	−1.44973	−	0.218511i	−0.826239	+	0.563320i	$0.809524\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$107$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$108$		0			0
$109$	1.44973	+	1.34515i	1.44973	+	1.34515i	0.826239	+	0.563320i	$0.190476\pi$
$109$	1.44973	+	1.34515i	1.44973	+	1.34515i	0.623490	+	0.781831i	$0.285714\pi$
$110$		0			0
$111$	0.0332580	−	0.145713i	0.0332580	−	0.145713i
$112$		0			0
$113$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$113$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.365341	−	0.632789i	−0.365341	−	0.632789i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	1.72188	−	0.829215i	1.72188	−	0.829215i	0.733052	−	0.680173i	$-0.238095\pi$
$127$	1.72188	−	0.829215i	1.72188	−	0.829215i	0.988831	−	0.149042i	$-0.0476190\pi$
$128$		0			0
$129$	0.733052	−	1.26968i	0.733052	−	1.26968i
$130$		0			0
$131$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$131$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$132$		0			0
$133$	−1.40097	−	1.29991i	−1.40097	−	1.29991i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$137$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$138$		0			0
$139$	1.23305	−	1.54620i	1.23305	−	1.54620i	0.500000	−	0.866025i	$-0.333333\pi$
$139$	1.23305	−	1.54620i	1.23305	−	1.54620i	0.733052	−	0.680173i	$-0.238095\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−0.826239	−	0.563320i	−0.826239	−	0.563320i
$148$		0			0
$149$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$149$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$150$		0			0
$151$	0.367711	−	0.250701i	0.367711	−	0.250701i	−0.365341	−	0.930874i	$-0.619048\pi$
$151$	0.367711	−	0.250701i	0.367711	−	0.250701i	0.733052	+	0.680173i	$0.238095\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.23305	+	0.185853i	−1.23305	+	0.185853i	−0.733052	−	0.680173i	$-0.761905\pi$
$157$	−1.23305	+	0.185853i	−1.23305	+	0.185853i	−0.500000	+	0.866025i	$0.666667\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.455573	+	1.16078i	−0.455573	+	1.16078i	0.500000	+	0.866025i	$0.333333\pi$
$163$	−0.455573	+	1.16078i	−0.455573	+	1.16078i	−0.955573	+	0.294755i	$0.904762\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$167$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$168$		0			0
$169$	0.419945	+	0.202235i	0.419945	+	0.202235i
$170$		0			0
$171$	−1.82624	−	0.563320i	−1.82624	−	0.563320i
$172$		0			0
$173$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$173$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$174$		0			0
$175$	−1.00000			−1.00000
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$179$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$180$		0			0
$181$	0.440071	+	1.92808i	0.440071	+	1.92808i	0.365341	+	0.930874i	$0.380952\pi$
$181$	0.440071	+	1.92808i	0.440071	+	1.92808i	0.0747301	+	0.997204i	$0.476190\pi$
$182$		0			0
$183$	−0.400969	+	1.75676i	−0.400969	+	1.75676i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$190$		0			0
$191$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$191$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$192$		0			0
$193$	0.266948	+	0.680173i	0.266948	+	0.680173i	1.00000			$0$
$193$	0.266948	+	0.680173i	0.266948	+	0.680173i	−0.733052	+	0.680173i	$0.761905\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	0.162592	+	0.414278i	0.162592	+	0.414278i	0.988831	−	0.149042i	$-0.0476190\pi$
$199$	0.162592	+	0.414278i	0.162592	+	0.414278i	−0.826239	+	0.563320i	$0.809524\pi$
$200$		0			0
$201$	1.95557	+	0.294755i	1.95557	+	0.294755i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−0.400969	−	1.75676i	−0.400969	−	1.75676i	−0.623490	−	0.781831i	$-0.714286\pi$
$211$	−0.400969	−	1.75676i	−0.400969	−	1.75676i	0.222521	−	0.974928i	$-0.428571\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−0.367711	+	1.61105i	−0.367711	+	1.61105i
$218$		0			0
$219$	−0.0111692	−	0.149042i	−0.0111692	−	0.149042i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.62349	−	0.781831i	−1.62349	−	0.781831i	−0.623490	−	0.781831i	$-0.714286\pi$
$223$	−1.62349	−	0.781831i	−1.62349	−	0.781831i	−1.00000			$\pi$
$224$		0			0
$225$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$226$		0			0
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$		0			0
$229$	−0.535628	+	1.36476i	−0.535628	+	1.36476i	0.365341	+	0.930874i	$0.380952\pi$
$229$	−0.535628	+	1.36476i	−0.535628	+	1.36476i	−0.900969	+	0.433884i	$0.857143\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$233$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	1.03030	−	1.29196i	1.03030	−	1.29196i
$238$		0			0
$239$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$239$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$240$		0			0
$241$	1.03030	−	0.702449i	1.03030	−	0.702449i	0.0747301	−	0.997204i	$-0.476190\pi$
$241$	1.03030	−	0.702449i	1.03030	−	0.702449i	0.955573	+	0.294755i	$0.0952381\pi$
$242$		0			0
$243$	−0.955573	+	0.294755i	−0.955573	+	0.294755i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−1.33440	+	0.411608i	−1.33440	+	0.411608i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$251$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$257$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$258$		0			0
$259$	−0.142820	−	0.0440542i	−0.142820	−	0.0440542i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$269$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$270$		0			0
$271$	−0.0931869	−	1.24349i	−0.0931869	−	1.24349i	−0.826239	−	0.563320i	$-0.809524\pi$
$271$	−0.0931869	−	1.24349i	−0.0931869	−	1.24349i	0.733052	−	0.680173i	$-0.238095\pi$
$272$		0			0
$273$	−0.658322	+	0.317031i	−0.658322	+	0.317031i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.0546039	−	0.728639i	0.0546039	−	0.728639i	−0.900969	−	0.433884i	$-0.857143\pi$
$277$	0.0546039	−	0.728639i	0.0546039	−	0.728639i	0.955573	−	0.294755i	$-0.0952381\pi$
$278$		0			0
$279$	0.367711	+	1.61105i	0.367711	+	1.61105i
$280$		0			0
$281$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$281$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$282$		0			0
$283$	0.109562	+	0.101659i	0.109562	+	0.101659i	0.733052	−	0.680173i	$-0.238095\pi$
$283$	0.109562	+	0.101659i	0.109562	+	0.101659i	−0.623490	+	0.781831i	$0.714286\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$290$		0			0
$291$	0.162592	+	0.414278i	0.162592	+	0.414278i
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−1.21135	−	0.825886i	−1.21135	−	0.825886i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−0.326239	+	1.42935i	−0.326239	+	1.42935i	0.500000	+	0.866025i	$0.333333\pi$
$307$	−0.326239	+	1.42935i	−0.326239	+	1.42935i	−0.826239	+	0.563320i	$0.809524\pi$
$308$		0			0
$309$	0.326239	+	1.42935i	0.326239	+	1.42935i
$310$		0			0
$311$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$311$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$312$		0			0
$313$	−0.826239	−	1.43109i	−0.826239	−	1.43109i	−0.900969	−	0.433884i	$-0.857143\pi$
$313$	−0.826239	−	1.43109i	−0.826239	−	1.43109i	0.0747301	−	0.997204i	$-0.476190\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$317$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−0.365341	+	0.632789i	−0.365341	+	0.632789i
$326$		0			0
$327$	0.722521	−	1.84095i	0.722521	−	1.84095i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.57906	−	1.07659i	−1.57906	−	1.07659i	−0.955573	−	0.294755i	$-0.904762\pi$
$331$	−1.57906	−	1.07659i	−1.57906	−	1.07659i	−0.623490	−	0.781831i	$-0.714286\pi$
$332$		0			0
$333$	−0.147791	+	0.0222759i	−0.147791	+	0.0222759i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.23305	−	1.54620i	−1.23305	−	1.54620i	−0.733052	−	0.680173i	$-0.761905\pi$
$337$	−1.23305	−	1.54620i	−1.23305	−	1.54620i	−0.500000	−	0.866025i	$-0.666667\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$347$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$348$		0			0
$349$	−0.277479	−	0.347948i	−0.277479	−	0.347948i	0.623490	−	0.781831i	$-0.285714\pi$
$349$	−0.277479	−	0.347948i	−0.277479	−	0.347948i	−0.900969	+	0.433884i	$0.857143\pi$
$350$		0			0
$351$	−0.455573	+	0.571270i	−0.455573	+	0.571270i
$352$		0			0
$353$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$353$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$359$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$360$		0			0
$361$	−1.32624	+	2.29711i	−1.32624	+	2.29711i
$362$		0			0
$363$	0.900969	−	0.433884i	0.900969	−	0.433884i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	1.88980	+	0.582926i	1.88980	+	0.582926i	0.988831	+	0.149042i	$0.0476190\pi$
$367$	1.88980	+	0.582926i	1.88980	+	0.582926i	0.900969	+	0.433884i	$0.142857\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	0.733052	+	1.26968i	0.733052	+	1.26968i	0.955573	+	0.294755i	$0.0952381\pi$
$373$	0.733052	+	1.26968i	0.733052	+	1.26968i	−0.222521	+	0.974928i	$0.571429\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.425270	−	1.86323i	0.425270	−	1.86323i	−0.0747301	−	0.997204i	$-0.523810\pi$
$379$	0.425270	−	1.86323i	0.425270	−	1.86323i	0.500000	−	0.866025i	$-0.333333\pi$
$380$		0			0
$381$	−1.40097	−	1.29991i	−1.40097	−	1.29991i
$382$		0			0
$383$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$383$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−1.44973	−	0.218511i	−1.44973	−	0.218511i
$388$		0			0
$389$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$389$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−0.147791	−	0.0222759i	−0.147791	−	0.0222759i	0.0747301	−	0.997204i	$-0.476190\pi$
$397$	−0.147791	−	0.0222759i	−0.147791	−	0.0222759i	−0.222521	+	0.974928i	$0.571429\pi$
$398$		0			0
$399$	−0.698220	+	1.77904i	−0.698220	+	1.77904i
$400$		0			0
$401$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$401$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$402$		0			0
$403$	0.885113	+	0.821265i	0.885113	+	0.821265i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.123490	−	1.64786i	0.123490	−	1.64786i	−0.500000	−	0.866025i	$-0.666667\pi$
$409$	0.123490	−	1.64786i	0.123490	−	1.64786i	0.623490	−	0.781831i	$-0.285714\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−1.88980	−	0.582926i	−1.88980	−	0.582926i
$418$		0			0
$419$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$419$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$420$		0			0
$421$	−0.658322	+	0.317031i	−0.658322	+	0.317031i	−0.733052	−	0.680173i	$-0.761905\pi$
$421$	−0.658322	+	0.317031i	−0.658322	+	0.317031i	0.0747301	+	0.997204i	$0.476190\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.72188	+	0.531130i	1.72188	+	0.531130i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$431$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$432$		0			0
$433$	1.19158	−	1.49419i	1.19158	−	1.49419i	0.365341	−	0.930874i	$-0.380952\pi$
$433$	1.19158	−	1.49419i	1.19158	−	1.49419i	0.826239	−	0.563320i	$-0.190476\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−1.19158	+	0.367554i	−1.19158	+	0.367554i	−0.826239	−	0.563320i	$-0.809524\pi$
$439$	−1.19158	+	0.367554i	−1.19158	+	0.367554i	−0.365341	+	0.930874i	$0.619048\pi$
$440$		0			0
$441$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$442$		0			0
$443$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$443$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$449$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−0.367711	−	0.250701i	−0.367711	−	0.250701i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.603718	−	1.53825i	0.603718	−	1.53825i	−0.222521	−	0.974928i	$-0.571429\pi$
$457$	0.603718	−	1.53825i	0.603718	−	1.53825i	0.826239	−	0.563320i	$-0.190476\pi$
$458$		0			0
$459$		0			0
$460$		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
$463$	−1.78181	−	0.858075i	−1.78181	−	0.858075i	−0.955573	−	0.294755i	$-0.904762\pi$
$463$	−1.78181	−	0.858075i	−1.78181	−	0.858075i	−0.826239	−	0.563320i	$-0.809524\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$467$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$468$		0			0
$469$	0.440071	−	1.92808i	0.440071	−	1.92808i
$470$		0			0
$471$	0.623490	+	1.07992i	0.623490	+	1.07992i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.425270	+	1.86323i	0.425270	+	1.86323i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$479$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$480$		0			0
$481$	−0.0800550	+	0.0742802i	−0.0800550	+	0.0742802i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−0.0546039	−	0.139129i	−0.0546039	−	0.139129i	0.900969	−	0.433884i	$-0.142857\pi$
$487$	−0.0546039	−	0.139129i	−0.0546039	−	0.139129i	−0.955573	+	0.294755i	$0.904762\pi$
$488$		0			0
$489$	1.24698			1.24698
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	1.21135	−	1.12397i	1.21135	−	1.12397i	0.222521	−	0.974928i	$-0.428571\pi$
$499$	1.21135	−	1.12397i	1.21135	−	1.12397i	0.988831	−	0.149042i	$-0.0476190\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		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
$507$	0.0348320	−	0.464800i	0.0348320	−	0.464800i
$508$		0			0
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$	−0.149460			−0.149460
$512$		0			0
$513$	0.142820	+	1.90580i	0.142820	+	1.90580i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$521$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$	−0.0546039	+	0.139129i	−0.0546039	+	0.139129i	−0.955573	−	0.294755i	$-0.904762\pi$
$523$	−0.0546039	+	0.139129i	−0.0546039	+	0.139129i	0.900969	+	0.433884i	$0.142857\pi$
$524$		0			0
$525$	0.365341	+	0.930874i	0.365341	+	0.930874i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.988831	+	0.149042i	−0.988831	+	0.149042i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.82624	−	0.563320i	1.82624	−	0.563320i	0.826239	−	0.563320i	$-0.190476\pi$
$541$	1.82624	−	0.563320i	1.82624	−	0.563320i	1.00000			$0$
$542$		0			0
$543$	1.63402	−	1.11406i	1.63402	−	1.11406i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.12349	−	1.40881i	1.12349	−	1.40881i	0.222521	−	0.974928i	$-0.428571\pi$
$547$	1.12349	−	1.40881i	1.12349	−	1.40881i	0.900969	−	0.433884i	$-0.142857\pi$
$548$		0			0
$549$	1.78181	−	0.268565i	1.78181	−	0.268565i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−1.21135	−	1.12397i	−1.21135	−	1.12397i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$	−0.965168	+	0.464800i	−0.965168	+	0.464800i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$563$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.222521	+	0.974928i	0.222521	+	0.974928i
$568$		0			0
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$	−0.0111692	+	0.149042i	−0.0111692	+	0.149042i	0.988831	+	0.149042i	$0.0476190\pi$
$571$	−0.0111692	+	0.149042i	−0.0111692	+	0.149042i	−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	1.44973	+	1.34515i	1.44973	+	1.34515i	0.826239	+	0.563320i	$0.190476\pi$
$577$	1.44973	+	1.34515i	1.44973	+	1.34515i	0.623490	+	0.781831i	$0.285714\pi$
$578$		0			0
$579$	0.535628	−	0.496990i	0.535628	−	0.496990i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		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			0
$589$	3.15813			3.15813
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$593$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	0.326239	−	0.302705i	0.326239	−	0.302705i
$598$		0			0
$599$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$599$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$600$		0			0
$601$	0.222521	−	0.974928i	0.222521	−	0.974928i	−0.733052	−	0.680173i	$-0.761905\pi$
$601$	0.222521	−	0.974928i	0.222521	−	0.974928i	0.955573	−	0.294755i	$-0.0952381\pi$
$602$		0			0
$603$	−0.440071	−	1.92808i	−0.440071	−	1.92808i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$607$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.72188	−	0.531130i	−1.72188	−	0.531130i	−0.733052	−	0.680173i	$-0.761905\pi$
$613$	−1.72188	−	0.531130i	−1.72188	−	0.531130i	−0.988831	+	0.149042i	$0.952381\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$617$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$618$		0			0
$619$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$619$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.826239	+	0.563320i	0.826239	+	0.563320i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	1.12349	+	1.40881i	1.12349	+	1.40881i	0.900969	+	0.433884i	$0.142857\pi$
$631$	1.12349	+	1.40881i	1.12349	+	1.40881i	0.222521	+	0.974928i	$0.428571\pi$
$632$		0			0
$633$	−1.48883	+	1.01507i	−1.48883	+	1.01507i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	0.266948	+	0.680173i	0.266948	+	0.680173i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$641$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$642$		0			0
$643$	−0.455573	−	0.571270i	−0.455573	−	0.571270i	0.500000	−	0.866025i	$-0.333333\pi$
$643$	−0.455573	−	0.571270i	−0.455573	−	0.571270i	−0.955573	+	0.294755i	$0.904762\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$647$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	1.63402	−	0.246289i	1.63402	−	0.246289i
$652$		0			0
$653$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$653$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−0.134659	+	0.0648483i	−0.134659	+	0.0648483i
$658$		0			0
$659$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$659$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$660$		0			0
$661$	−0.955573	−	0.294755i	−0.955573	−	0.294755i	−0.222521	−	0.974928i	$-0.571429\pi$
$661$	−0.955573	−	0.294755i	−0.955573	−	0.294755i	−0.733052	+	0.680173i	$0.761905\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−0.134659	+	1.79690i	−0.134659	+	1.79690i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.162592	+	0.712362i	−0.162592	+	0.712362i	0.826239	+	0.563320i	$0.190476\pi$
$673$	−0.162592	+	0.712362i	−0.162592	+	0.712362i	−0.988831	+	0.149042i	$0.952381\pi$
$674$		0			0
$675$	0.733052	+	0.680173i	0.733052	+	0.680173i
$676$		0			0
$677$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$677$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$678$		0			0
$679$	0.425270	−	0.131178i	0.425270	−	0.131178i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$683$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	1.46610			1.46610
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−0.988831	−	0.149042i	−0.988831	−	0.149042i	−0.365341	−	0.930874i	$-0.619048\pi$
$691$	−0.988831	−	0.149042i	−0.988831	−	0.149042i	−0.623490	+	0.781831i	$0.714286\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$701$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$702$		0			0
$703$	−0.0213459	+	0.284841i	−0.0213459	+	0.284841i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.0332580	−	0.443797i	−0.0332580	−	0.443797i	−0.988831	−	0.149042i	$-0.952381\pi$
$709$	−0.0332580	−	0.443797i	−0.0332580	−	0.443797i	0.955573	−	0.294755i	$-0.0952381\pi$
$710$		0			0
$711$	−1.57906	−	0.487076i	−1.57906	−	0.487076i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$719$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$720$		0			0
$721$	1.44973	−	0.218511i	1.44973	−	0.218511i
$722$		0			0
$723$	−1.03030	−	0.702449i	−1.03030	−	0.702449i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	0.623490	−	0.781831i	0.623490	−	0.781831i	−0.365341	−	0.930874i	$-0.619048\pi$
$727$	0.623490	−	0.781831i	0.623490	−	0.781831i	0.988831	+	0.149042i	$0.0476190\pi$
$728$		0			0
$729$	0.623490	+	0.781831i	0.623490	+	0.781831i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	1.82624	−	0.563320i	1.82624	−	0.563320i	0.826239	−	0.563320i	$-0.190476\pi$
$733$	1.82624	−	0.563320i	1.82624	−	0.563320i	1.00000			$0$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−0.603718	+	0.411608i	−0.603718	+	0.411608i	−0.826239	−	0.563320i	$-0.809524\pi$
$739$	−0.603718	+	0.411608i	−0.603718	+	0.411608i	0.222521	+	0.974928i	$0.428571\pi$
$740$		0			0
$741$	0.870666	+	1.09178i	0.870666	+	1.09178i
$742$		0			0
$743$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$743$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.535628	−	1.36476i	0.535628	−	1.36476i	−0.365341	−	0.930874i	$-0.619048\pi$
$751$	0.535628	−	1.36476i	0.535628	−	1.36476i	0.900969	−	0.433884i	$-0.142857\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.12349	−	0.541044i	−1.12349	−	0.541044i	−0.222521	−	0.974928i	$-0.571429\pi$
$757$	−1.12349	−	0.541044i	−1.12349	−	0.541044i	−0.900969	+	0.433884i	$0.857143\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$761$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$762$		0			0
$763$	−1.78181	−	0.858075i	−1.78181	−	0.858075i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	0.222521	+	0.974928i	0.222521	+	0.974928i	0.955573	+	0.294755i	$0.0952381\pi$
$769$	0.222521	+	0.974928i	0.222521	+	0.974928i	−0.733052	+	0.680173i	$0.761905\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$773$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$774$		0			0
$775$	1.21135	−	1.12397i	1.21135	−	1.12397i
$776$		0			0
$777$	0.0111692	+	0.149042i	0.0111692	+	0.149042i
$778$		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			0
$787$	0.658322	+	1.67738i	0.658322	+	1.67738i	0.733052	+	0.680173i	$0.238095\pi$
$787$	0.658322	+	1.67738i	0.658322	+	1.67738i	−0.0747301	+	0.997204i	$0.523810\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	0.965168	−	0.895545i	0.965168	−	0.895545i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$797$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$809$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$810$		0			0
$811$	1.12349	+	0.541044i	1.12349	+	0.541044i	0.900969	−	0.433884i	$-0.142857\pi$
$811$	1.12349	+	0.541044i	1.12349	+	0.541044i	0.222521	+	0.974928i	$0.428571\pi$
$812$		0			0
$813$	−1.12349	+	0.541044i	−1.12349	+	0.541044i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1.02366	+	2.60825i	−1.02366	+	2.60825i
$818$		0			0
$819$	0.535628	+	0.496990i	0.535628	+	0.496990i
$820$		0			0
$821$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$821$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$822$		0			0
$823$	−0.440071	+	0.0663300i	−0.440071	+	0.0663300i	−0.365341	−	0.930874i	$-0.619048\pi$
$823$	−0.440071	+	0.0663300i	−0.440071	+	0.0663300i	−0.0747301	+	0.997204i	$0.523810\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$827$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$828$		0			0
$829$	1.36534	−	0.930874i	1.36534	−	0.930874i	0.365341	−	0.930874i	$-0.380952\pi$
$829$	1.36534	−	0.930874i	1.36534	−	0.930874i	1.00000			$0$
$830$		0			0
$831$	−0.698220	+	0.215372i	−0.698220	+	0.215372i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1.36534	−	0.930874i	1.36534	−	0.930874i
$838$		0			0
$839$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$839$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$840$		0			0
$841$	0.623490	−	0.781831i	0.623490	−	0.781831i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−0.365341	−	0.930874i	−0.365341	−	0.930874i
$848$		0			0
$849$	0.0546039	−	0.139129i	0.0546039	−	0.139129i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	1.32091	−	0.636119i	1.32091	−	0.636119i	0.365341	−	0.930874i	$-0.380952\pi$
$853$	1.32091	−	0.636119i	1.32091	−	0.636119i	0.955573	+	0.294755i	$0.0952381\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$857$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$858$		0			0
$859$	0.0332580	+	0.443797i	0.0332580	+	0.443797i	0.988831	+	0.149042i	$0.0476190\pi$
$859$	0.0332580	+	0.443797i	0.0332580	+	0.443797i	−0.955573	+	0.294755i	$0.904762\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$863$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	0.222521	+	0.974928i	0.222521	+	0.974928i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−1.05929	−	0.982878i	−1.05929	−	0.982878i
$872$		0			0
$873$	0.326239	−	0.302705i	0.326239	−	0.302705i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.440071	+	0.0663300i	0.440071	+	0.0663300i	0.365341	−	0.930874i	$-0.380952\pi$
$877$	0.440071	+	0.0663300i	0.440071	+	0.0663300i	0.0747301	+	0.997204i	$0.476190\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	−0.730682			−0.730682			−0.365341	−	0.930874i	$-0.619048\pi$
$883$	−0.730682			−0.730682			−0.365341	+	0.930874i	$0.619048\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$887$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$888$		0			0
$889$	−1.40097	+	1.29991i	−1.40097	+	1.29991i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−0.326239	+	1.42935i	−0.326239	+	1.42935i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−0.698220	−	0.215372i	−0.698220	−	0.215372i	−0.0747301	−	0.997204i	$-0.523810\pi$
$907$	−0.698220	−	0.215372i	−0.698220	−	0.215372i	−0.623490	+	0.781831i	$0.714286\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$911$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	1.63402	+	1.11406i	1.63402	+	1.11406i	0.900969	+	0.433884i	$0.142857\pi$
$919$	1.63402	+	1.11406i	1.63402	+	1.11406i	0.733052	+	0.680173i	$0.238095\pi$
$920$		0			0
$921$	1.44973	−	0.218511i	1.44973	−	0.218511i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.0931869	+	0.116853i	0.0931869	+	0.116853i
$926$		0			0
$927$	1.21135	−	0.825886i	1.21135	−	0.825886i
$928$		0			0
$929$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$929$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$930$		0			0
$931$	1.72188	+	0.829215i	1.72188	+	0.829215i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.914101	−	1.14625i	−0.914101	−	1.14625i	−0.988831	−	0.149042i	$-0.952381\pi$
$937$	−0.914101	−	1.14625i	−0.914101	−	1.14625i	0.0747301	−	0.997204i	$-0.476190\pi$
$938$		0			0
$939$	−1.03030	+	1.29196i	−1.03030	+	1.29196i
$940$		0			0
$941$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$941$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$947$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$948$		0			0
$949$	−0.0546039	+	0.0945768i	−0.0546039	+	0.0945768i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$953$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.865341	−	1.49881i	−0.865341	−	1.49881i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−0.326239	+	1.42935i	−0.326239	+	1.42935i	0.500000	+	0.866025i	$0.333333\pi$
$967$	−0.326239	+	1.42935i	−0.326239	+	1.42935i	−0.826239	+	0.563320i	$0.809524\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$971$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$972$		0			0
$973$	−0.722521	+	1.84095i	−0.722521	+	1.84095i
$974$		0			0
$975$	0.722521	+	0.108903i	0.722521	+	0.108903i
$976$		0			0
$977$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$977$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−1.97766			−1.97766
$982$		0			0
$983$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$983$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	1.40097	+	1.29991i	1.40097	+	1.29991i	0.900969	+	0.433884i	$0.142857\pi$
$991$	1.40097	+	1.29991i	1.40097	+	1.29991i	0.500000	+	0.866025i	$0.333333\pi$
$992$		0			0
$993$	−0.425270	+	1.86323i	−0.425270	+	1.86323i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.0747301	+	0.997204i	−0.0747301	+	0.997204i	0.826239	+	0.563320i	$0.190476\pi$
$997$	−0.0747301	+	0.997204i	−0.0747301	+	0.997204i	−0.900969	+	0.433884i	$0.857143\pi$
$998$		0			0
$999$	0.0747301	+	0.129436i	0.0747301	+	0.129436i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.1.cv.a.1985.1		12
3.2	odd	2	CM	2352.1.cv.a.1985.1		12
4.3	odd	2		588.1.z.a.221.1	yes	12
12.11	even	2		588.1.z.a.221.1	yes	12
49.2	even	21	inner	2352.1.cv.a.737.1		12
147.2	odd	42	inner	2352.1.cv.a.737.1		12
196.51	odd	42		588.1.z.a.149.1	✓	12
588.443	even	42		588.1.z.a.149.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.1.z.a.149.1	✓	12	196.51	odd	42
588.1.z.a.149.1	✓	12	588.443	even	42
588.1.z.a.221.1	yes	12	4.3	odd	2
588.1.z.a.221.1	yes	12	12.11	even	2
2352.1.cv.a.737.1		12	49.2	even	21	inner
2352.1.cv.a.737.1		12	147.2	odd	42	inner
2352.1.cv.a.1985.1		12	1.1	even	1	trivial
2352.1.cv.a.1985.1		12	3.2	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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2352.cv (of order \(42\), degree \(12\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.365341 - 0.930874i) q^{3} +(-0.955573 + 0.294755i) q^{7} +(-0.733052 + 0.680173i) q^{9} +O(q^{10})\)	\(q+(-0.365341 - 0.930874i) q^{3} +(-0.955573 + 0.294755i) q^{7} +(-0.733052 + 0.680173i) q^{9} +(-0.162592 + 0.712362i) q^{13} +(0.955573 + 1.65510i) q^{19} +(0.623490 + 0.781831i) q^{21} +(0.955573 + 0.294755i) q^{25} +(0.900969 + 0.433884i) q^{27} +(0.826239 - 1.43109i) q^{31} +(0.123490 + 0.0841939i) q^{37} +(0.722521 - 0.108903i) q^{39} +(0.914101 + 1.14625i) q^{43} +(0.826239 - 0.563320i) q^{49} +(1.19158 - 1.49419i) q^{57} +(-1.48883 - 1.01507i) q^{61} +(0.500000 - 0.866025i) q^{63} +(-0.988831 + 1.71271i) q^{67} +(0.142820 + 0.0440542i) q^{73} +(-0.0747301 - 0.997204i) q^{75} +(0.826239 + 1.43109i) q^{79} +(0.0747301 - 0.997204i) q^{81} +(-0.0546039 - 0.728639i) q^{91} +(-1.63402 - 0.246289i) q^{93} -0.445042 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{3} - q^{7} + q^{9}+O(q^{10}) \) 12 * q - q^3 - q^7 + q^9	\( 12 q - q^{3} - q^{7} + q^{9} + 2 q^{13} + q^{19} - 2 q^{21} + q^{25} + 2 q^{27} + q^{31} - 8 q^{37} + 8 q^{39} - 2 q^{43} + q^{49} + 2 q^{57} - 5 q^{61} + 6 q^{63} + q^{67} - q^{73} - q^{75} + q^{79} + q^{81} + q^{91} - q^{93} - 4 q^{97}+O(q^{100}) \) 12 * q - q^3 - q^7 + q^9 + 2 * q^13 + q^19 - 2 * q^21 + q^25 + 2 * q^27 + q^31 - 8 * q^37 + 8 * q^39 - 2 * q^43 + q^49 + 2 * q^57 - 5 * q^61 + 6 * q^63 + q^67 - q^73 - q^75 + q^79 + q^81 + q^91 - q^93 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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