LMFDB - Embedded newform 3528.1.fb.a.1333.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3528,1,Mod(397,3528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3528, base_ring=CyclotomicField(42))

chi = DirichletCharacter(H, H._module([0, 21, 0, 29]))

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

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

chi := DirichletCharacter("3528.397");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3528.fb (of order $42$, degree $12$, 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:	$1.76070136457$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$2$ over $\Q(\zeta_{42})$
Coefficient field:	$\Q(\zeta_{84})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} + x^{22} - x^{18} - x^{16} + x^{12} - x^{8} - x^{6} + x^{2} + 1 $ x^24 + x^22 - x^18 - x^16 + x^12 - x^8 - x^6 + x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{42}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{42} - \cdots)$

Embedding invariants

Embedding label			1333.2
Root			$0.294755 + 0.955573i$ of defining polynomial
Character	$\chi$	$=$	3528.1333
Dual form			3528.1.fb.a.397.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.149042 - 0.988831i) q^{2} +(-0.955573 - 0.294755i) q^{4} +(0.139129 + 1.85654i) q^{5} +(0.0747301 - 0.997204i) q^{7} +(-0.433884 + 0.900969i) q^{8} +O(q^{10})$	\(q+(0.149042 - 0.988831i) q^{2} +(-0.955573 - 0.294755i) q^{4} +(0.139129 + 1.85654i) q^{5} +(0.0747301 - 0.997204i) q^{7} +(-0.433884 + 0.900969i) q^{8} +(1.85654 + 0.139129i) q^{10} +(-1.77904 + 0.698220i) q^{11} +(-0.974928 - 0.222521i) q^{14} +(0.826239 + 0.563320i) q^{16} +(0.414278 - 1.81507i) q^{20} +(0.425270 + 1.86323i) q^{22} +(-2.43856 + 0.367554i) q^{25} +(-0.365341 + 0.930874i) q^{28} +(-1.92808 - 0.440071i) q^{29} +(-0.975699 - 0.563320i) q^{31} +(0.680173 - 0.733052i) q^{32} +1.86175 q^{35} +(-1.73305 - 0.680173i) q^{40} +(1.90580 - 0.142820i) q^{44} +(-0.988831 - 0.149042i) q^{49} +2.46610i q^{50} +(-0.294755 + 0.955573i) q^{53} +(-1.54379 - 3.20571i) q^{55} +(0.866025 + 0.500000i) q^{56} +(-0.722521 + 1.84095i) q^{58} +(0.129436 - 1.72721i) q^{59} +(-0.702449 + 0.880843i) q^{62} +(-0.623490 - 0.781831i) q^{64} +(0.277479 - 1.84095i) q^{70} +(0.129334 + 0.858075i) q^{73} +(0.563320 + 1.82624i) q^{77} +(0.0747301 + 0.129436i) q^{79} +(-0.930874 + 1.61232i) q^{80} +(-0.185853 - 0.233052i) q^{83} +(0.142820 - 1.90580i) q^{88} +1.36035i q^{97} +(-0.294755 + 0.955573i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 2 q^{4} + 2 q^{7}+O(q^{10}) $ 24 * q - 2 * q^4 + 2 * q^7	$ 24 q - 2 q^{4} + 2 q^{7} + 6 q^{10} + 2 q^{16} + 10 q^{22} + 4 q^{25} - 2 q^{28} - 6 q^{31} - 22 q^{40} + 2 q^{49} - 14 q^{55} - 16 q^{58} + 4 q^{64} + 8 q^{70} + 2 q^{79} - 2 q^{88}+O(q^{100}) $ 24 * q - 2 * q^4 + 2 * q^7 + 6 * q^10 + 2 * q^16 + 10 * q^22 + 4 * q^25 - 2 * q^28 - 6 * q^31 - 22 * q^40 + 2 * q^49 - 14 * q^55 - 16 * q^58 + 4 * q^64 + 8 * q^70 + 2 * q^79 - 2 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$	$1765$	$2647$
$\chi(n)$	$1$	$e\left(\frac{13}{42}\right)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	0.149042	−	0.988831i	0.149042	−	0.988831i
$2$	0.149042	−	0.988831i	0.149042	−	0.988831i
$3$		0			0
$3$		0			0
$4$	−0.955573	−	0.294755i	−0.955573	−	0.294755i
$5$	0.139129	+	1.85654i	0.139129	+	1.85654i	0.433884	+	0.900969i	$0.357143\pi$
$5$	0.139129	+	1.85654i	0.139129	+	1.85654i	−0.294755	+	0.955573i	$0.595238\pi$
$6$		0			0
$7$	0.0747301	−	0.997204i	0.0747301	−	0.997204i
$7$	0.0747301	−	0.997204i	0.0747301	−	0.997204i
$8$	−0.433884	+	0.900969i	−0.433884	+	0.900969i
$9$		0			0
$10$	1.85654	+	0.139129i	1.85654	+	0.139129i
$11$	−1.77904	+	0.698220i	−1.77904	+	0.698220i	−0.781831	+	0.623490i	$0.785714\pi$
$11$	−1.77904	+	0.698220i	−1.77904	+	0.698220i	−0.997204	+	0.0747301i	$0.976190\pi$
$12$		0			0
$13$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$13$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$14$	−0.974928	−	0.222521i	−0.974928	−	0.222521i
$15$		0			0
$16$	0.826239	+	0.563320i	0.826239	+	0.563320i
$17$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$17$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$18$		0			0
$19$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$19$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$20$	0.414278	−	1.81507i	0.414278	−	1.81507i
$21$		0			0
$22$	0.425270	+	1.86323i	0.425270	+	1.86323i
$23$		0			0		−0.680173	−	0.733052i	$-0.738095\pi$
$23$		0			0		0.680173	+	0.733052i	$0.261905\pi$
$24$		0			0
$25$	−2.43856	+	0.367554i	−2.43856	+	0.367554i
$26$		0			0
$27$		0			0
$28$	−0.365341	+	0.930874i	−0.365341	+	0.930874i
$29$	−1.92808	−	0.440071i	−1.92808	−	0.440071i	−0.997204	−	0.0747301i	$-0.976190\pi$
$29$	−1.92808	−	0.440071i	−1.92808	−	0.440071i	−0.930874	−	0.365341i	$-0.880952\pi$
$30$		0			0
$31$	−0.975699	−	0.563320i	−0.975699	−	0.563320i	−0.0747301	−	0.997204i	$-0.523810\pi$
$31$	−0.975699	−	0.563320i	−0.975699	−	0.563320i	−0.900969	+	0.433884i	$0.857143\pi$
$32$	0.680173	−	0.733052i	0.680173	−	0.733052i
$33$		0			0
$34$		0			0
$35$	1.86175			1.86175
$36$		0			0
$37$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$37$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$38$		0			0
$39$		0			0
$40$	−1.73305	−	0.680173i	−1.73305	−	0.680173i
$41$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$41$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$42$		0			0
$43$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$43$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$44$	1.90580	−	0.142820i	1.90580	−	0.142820i
$45$		0			0
$46$		0			0
$47$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$47$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$48$		0			0
$49$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$50$			2.46610i			2.46610i
$51$		0			0
$52$		0			0
$53$	−0.294755	+	0.955573i	−0.294755	+	0.955573i	0.680173	+	0.733052i	$0.261905\pi$
$53$	−0.294755	+	0.955573i	−0.294755	+	0.955573i	−0.974928	+	0.222521i	$0.928571\pi$
$54$		0			0
$55$	−1.54379	−	3.20571i	−1.54379	−	3.20571i
$56$	0.866025	+	0.500000i	0.866025	+	0.500000i
$57$		0			0
$58$	−0.722521	+	1.84095i	−0.722521	+	1.84095i
$59$	0.129436	−	1.72721i	0.129436	−	1.72721i	−0.433884	−	0.900969i	$-0.642857\pi$
$59$	0.129436	−	1.72721i	0.129436	−	1.72721i	0.563320	−	0.826239i	$-0.309524\pi$
$60$		0			0
$61$		0			0		−0.294755	−	0.955573i	$-0.595238\pi$
$61$		0			0		0.294755	+	0.955573i	$0.404762\pi$
$62$	−0.702449	+	0.880843i	−0.702449	+	0.880843i
$63$		0			0
$64$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$65$		0			0
$66$		0			0
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$		0			0
$69$		0			0
$70$	0.277479	−	1.84095i	0.277479	−	1.84095i
$71$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$71$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$72$		0			0
$73$	0.129334	+	0.858075i	0.129334	+	0.858075i	0.955573	+	0.294755i	$0.0952381\pi$
$73$	0.129334	+	0.858075i	0.129334	+	0.858075i	−0.826239	+	0.563320i	$0.809524\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.563320	+	1.82624i	0.563320	+	1.82624i
$78$		0			0
$79$	0.0747301	+	0.129436i	0.0747301	+	0.129436i	0.900969	−	0.433884i	$-0.142857\pi$
$79$	0.0747301	+	0.129436i	0.0747301	+	0.129436i	−0.826239	+	0.563320i	$0.809524\pi$
$80$	−0.930874	+	1.61232i	−0.930874	+	1.61232i
$81$		0			0
$82$		0			0
$83$	−0.185853	−	0.233052i	−0.185853	−	0.233052i	0.680173	−	0.733052i	$-0.261905\pi$
$83$	−0.185853	−	0.233052i	−0.185853	−	0.233052i	−0.866025	+	0.500000i	$0.833333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	0.142820	−	1.90580i	0.142820	−	1.90580i
$89$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$89$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$			1.36035i			1.36035i	0.733052	+	0.680173i	$0.238095\pi$
$97$			1.36035i			1.36035i	−0.733052	+	0.680173i	$0.761905\pi$
$98$	−0.294755	+	0.955573i	−0.294755	+	0.955573i
$99$		0			0
$100$	2.43856	+	0.367554i	2.43856	+	0.367554i
$101$	−1.61105	+	1.09839i	−1.61105	+	1.09839i	−0.680173	+	0.733052i	$0.738095\pi$
$101$	−1.61105	+	1.09839i	−1.61105	+	1.09839i	−0.930874	+	0.365341i	$0.880952\pi$
$102$		0			0
$103$	−1.94440	+	0.145713i	−1.94440	+	0.145713i	−0.988831	−	0.149042i	$-0.952381\pi$
$103$	−1.94440	+	0.145713i	−1.94440	+	0.145713i	−0.955573	+	0.294755i	$0.904762\pi$
$104$		0			0
$105$		0			0
$106$	0.900969	+	0.433884i	0.900969	+	0.433884i
$107$	−0.139129	−	0.0546039i	−0.139129	−	0.0546039i	0.294755	−	0.955573i	$-0.404762\pi$
$107$	−0.139129	−	0.0546039i	−0.139129	−	0.0546039i	−0.433884	+	0.900969i	$0.642857\pi$
$108$		0			0
$109$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$109$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$110$	−3.40000	+	1.04876i	−3.40000	+	1.04876i
$111$		0			0
$112$	0.623490	−	0.781831i	0.623490	−	0.781831i
$113$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$113$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$114$		0			0
$115$		0			0
$116$	1.71271	+	0.988831i	1.71271	+	0.988831i
$117$		0			0
$118$	−1.68862	−	0.385418i	−1.68862	−	0.385418i
$119$		0			0
$120$		0			0
$121$	1.94440	−	1.80414i	1.94440	−	1.80414i
$122$		0			0
$123$		0			0
$124$	0.766310	+	0.825886i	0.766310	+	0.825886i
$125$	−0.607374	−	2.66108i	−0.607374	−	2.66108i
$126$		0			0
$127$	−0.367711	+	1.61105i	−0.367711	+	1.61105i	0.365341	+	0.930874i	$0.380952\pi$
$127$	−0.367711	+	1.61105i	−0.367711	+	1.61105i	−0.733052	+	0.680173i	$0.761905\pi$
$128$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$129$		0			0
$130$		0			0
$131$	1.64786	+	1.12349i	1.64786	+	1.12349i	0.866025	+	0.500000i	$0.166667\pi$
$131$	1.64786	+	1.12349i	1.64786	+	1.12349i	0.781831	+	0.623490i	$0.214286\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.997204	−	0.0747301i	$-0.976190\pi$
$137$		0			0		0.997204	+	0.0747301i	$0.0238095\pi$
$138$		0			0
$139$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$139$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$140$	−1.77904	−	0.548760i	−1.77904	−	0.548760i
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	0.548760	−	3.64078i	0.548760	−	3.64078i
$146$	0.867767			0.867767
$147$		0			0
$148$		0			0
$149$	−0.268565	+	1.78181i	−0.268565	+	1.78181i	0.294755	+	0.955573i	$0.404762\pi$
$149$	−0.268565	+	1.78181i	−0.268565	+	1.78181i	−0.563320	+	0.826239i	$0.690476\pi$
$150$		0			0
$151$	0.955573	+	0.294755i	0.955573	+	0.294755i	0.733052	−	0.680173i	$-0.238095\pi$
$151$	0.955573	+	0.294755i	0.955573	+	0.294755i	0.222521	+	0.974928i	$0.428571\pi$
$152$		0			0
$153$		0			0
$154$	1.88980	−	0.284841i	1.88980	−	0.284841i
$155$	0.910080	−	1.88980i	0.910080	−	1.88980i
$156$		0			0
$157$		0			0		−0.997204	−	0.0747301i	$-0.976190\pi$
$157$		0			0		0.997204	+	0.0747301i	$0.0238095\pi$
$158$	0.139129	−	0.0546039i	0.139129	−	0.0546039i
$159$		0			0
$160$	1.45557	+	1.16078i	1.45557	+	1.16078i
$161$		0			0
$162$		0			0
$163$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$163$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$164$		0			0
$165$		0			0
$166$	−0.258149	+	0.149042i	−0.258149	+	0.149042i
$167$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$167$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$168$		0			0
$169$	0.222521	+	0.974928i	0.222521	+	0.974928i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.14625	−	1.06356i	1.14625	−	1.06356i	0.149042	−	0.988831i	$-0.452381\pi$
$173$	1.14625	−	1.06356i	1.14625	−	1.06356i	0.997204	−	0.0747301i	$-0.0238095\pi$
$174$		0			0
$175$	0.184292	+	2.45921i	0.184292	+	2.45921i
$176$	−1.86323	−	0.425270i	−1.86323	−	0.425270i
$177$		0			0
$178$		0			0
$179$	0.848162	−	0.914101i	0.848162	−	0.914101i	−0.149042	−	0.988831i	$-0.547619\pi$
$179$	0.848162	−	0.914101i	0.848162	−	0.914101i	0.997204	+	0.0747301i	$0.0238095\pi$
$180$		0			0
$181$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$181$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.997204	−	0.0747301i	$-0.0238095\pi$
$191$		0			0		−0.997204	+	0.0747301i	$0.976190\pi$
$192$		0			0
$193$	0.123490	−	0.0841939i	0.123490	−	0.0841939i	−0.500000	−	0.866025i	$-0.666667\pi$
$193$	0.123490	−	0.0841939i	0.123490	−	0.0841939i	0.623490	+	0.781831i	$0.285714\pi$
$194$	1.34515	+	0.202749i	1.34515	+	0.202749i
$195$		0			0
$196$	0.900969	+	0.433884i	0.900969	+	0.433884i
$197$		−	0.445042i		−	0.445042i	−0.974928	−	0.222521i	$-0.928571\pi$
$197$		−	0.445042i		−	0.445042i	0.974928	−	0.222521i	$-0.0714286\pi$
$198$		0			0
$199$	−0.488831	−	0.716983i	−0.488831	−	0.716983i	0.500000	−	0.866025i	$-0.333333\pi$
$199$	−0.488831	−	0.716983i	−0.488831	−	0.716983i	−0.988831	+	0.149042i	$0.952381\pi$
$200$	0.726897	−	2.35654i	0.726897	−	2.35654i
$201$		0			0
$202$	0.846011	+	1.75676i	0.846011	+	1.75676i
$203$	−0.582926	+	1.88980i	−0.582926	+	1.88980i
$204$		0			0
$205$		0			0
$206$	−0.145713	+	1.94440i	−0.145713	+	1.94440i
$207$		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	$0.285714\pi$
$212$	0.563320	−	0.826239i	0.563320	−	0.826239i
$213$		0			0
$214$	−0.0747301	+	0.129436i	−0.0747301	+	0.129436i
$215$		0			0
$216$		0			0
$217$	−0.634659	+	0.930874i	−0.634659	+	0.930874i
$218$		0			0
$219$		0			0
$220$	0.530303	+	3.51833i	0.530303	+	3.51833i
$221$		0			0
$222$		0			0
$223$	−1.32624	+	0.302705i	−1.32624	+	0.302705i	−0.826239	−	0.563320i	$-0.809524\pi$
$223$	−1.32624	+	0.302705i	−1.32624	+	0.302705i	−0.500000	+	0.866025i	$0.666667\pi$
$224$	−0.680173	−	0.733052i	−0.680173	−	0.733052i
$225$		0			0
$226$		0			0
$227$	0.294755	−	0.510531i	0.294755	−	0.510531i	−0.680173	−	0.733052i	$-0.738095\pi$
$227$	0.294755	−	0.510531i	0.294755	−	0.510531i	0.974928	+	0.222521i	$0.0714286\pi$
$228$		0			0
$229$		0			0		0.563320	−	0.826239i	$-0.309524\pi$
$229$		0			0		−0.563320	+	0.826239i	$0.690476\pi$
$230$		0			0
$231$		0			0
$232$	1.23305	−	1.54620i	1.23305	−	1.54620i
$233$		0			0		−0.294755	−	0.955573i	$-0.595238\pi$
$233$		0			0		0.294755	+	0.955573i	$0.404762\pi$
$234$		0			0
$235$		0			0
$236$	−0.632789	+	1.61232i	−0.632789	+	1.61232i
$237$		0			0
$238$		0			0
$239$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$239$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$240$		0			0
$241$	0.0878620	−	0.284841i	0.0878620	−	0.284841i	−0.900969	−	0.433884i	$-0.857143\pi$
$241$	0.0878620	−	0.284841i	0.0878620	−	0.284841i	0.988831	+	0.149042i	$0.0476190\pi$
$242$	−1.49419	−	2.19158i	−1.49419	−	2.19158i
$243$		0			0
$244$		0			0
$245$	0.139129	−	1.85654i	0.139129	−	1.85654i
$246$		0			0
$247$		0			0
$248$	0.930874	−	0.634659i	0.930874	−	0.634659i
$249$		0			0
$250$	−2.72188	+	0.203977i	−2.72188	+	0.203977i
$251$	−1.01507	+	0.488831i	−1.01507	+	0.488831i	−0.866025	−	0.500000i	$-0.833333\pi$
$251$	−1.01507	+	0.488831i	−1.01507	+	0.488831i	−0.149042	+	0.988831i	$0.547619\pi$
$252$		0			0
$253$		0			0
$254$	1.53825	+	0.603718i	1.53825	+	0.603718i
$255$		0			0
$256$	0.365341	+	0.930874i	0.365341	+	0.930874i
$257$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$257$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	1.35654	−	1.46200i	1.35654	−	1.46200i
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$	−1.81507	−	0.414278i	−1.81507	−	0.414278i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.294755	−	0.0444272i	0.294755	−	0.0444272i		−	1.00000i	$-0.5\pi$
$269$	0.294755	−	0.0444272i	0.294755	−	0.0444272i	0.294755	+	0.955573i	$0.404762\pi$
$270$		0			0
$271$	0.925270	+	0.997204i	0.925270	+	0.997204i	1.00000			$0$
$271$	0.925270	+	0.997204i	0.925270	+	0.997204i	−0.0747301	+	0.997204i	$0.523810\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.08165	−	2.35654i	4.08165	−	2.35654i
$276$		0			0
$277$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$277$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$278$		0			0
$279$		0			0
$280$	−0.807782	+	1.67738i	−0.807782	+	1.67738i
$281$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$281$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$282$		0			0
$283$		0			0		0.930874	−	0.365341i	$-0.119048\pi$
$283$		0			0		−0.930874	+	0.365341i	$0.880952\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$290$	−3.51833	−	1.08526i	−3.51833	−	1.08526i
$291$		0			0
$292$	0.129334	−	0.858075i	0.129334	−	0.858075i
$293$	−1.36035			−1.36035			−0.680173	−	0.733052i	$-0.738095\pi$
$293$	−1.36035			−1.36035			−0.680173	+	0.733052i	$0.738095\pi$
$294$		0			0
$295$	3.22464			3.22464
$296$		0			0
$297$		0			0
$298$	1.72188	+	0.531130i	1.72188	+	0.531130i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	0.433884	−	0.900969i	0.433884	−	0.900969i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$307$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$308$		−	1.91115i		−	1.91115i
$309$		0			0
$310$	−1.73305	−	1.18157i	−1.73305	−	1.18157i
$311$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$311$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$312$		0			0
$313$	1.72721	−	0.997204i	1.72721	−	0.997204i	0.826239	−	0.563320i	$-0.190476\pi$
$313$	1.72721	−	0.997204i	1.72721	−	0.997204i	0.900969	−	0.433884i	$-0.142857\pi$
$314$		0			0
$315$		0			0
$316$	−0.0332580	−	0.145713i	−0.0332580	−	0.145713i
$317$	0.997204	+	1.07473i	0.997204	+	1.07473i	0.997204	+	0.0747301i	$0.0238095\pi$
$317$	0.997204	+	1.07473i	0.997204	+	1.07473i			1.00000i	$0.5\pi$
$318$		0			0
$319$	3.73738	−	0.563320i	3.73738	−	0.563320i
$320$	1.36476	−	1.26631i	1.36476	−	1.26631i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$331$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$332$	0.108903	+	0.277479i	0.108903	+	0.277479i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.900969	−	0.433884i	0.900969	−	0.433884i	0.0747301	−	0.997204i	$-0.476190\pi$
$337$	0.900969	−	0.433884i	0.900969	−	0.433884i	0.826239	+	0.563320i	$0.190476\pi$
$338$	0.997204	−	0.0747301i	0.997204	−	0.0747301i
$339$		0			0
$340$		0			0
$341$	2.12912	+	0.320914i	2.12912	+	0.320914i
$342$		0			0
$343$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$344$		0			0
$345$		0			0
$346$	−0.880843	−	1.29196i	−0.880843	−	1.29196i
$347$	0.367554	−	1.19158i	0.367554	−	1.19158i	−0.563320	−	0.826239i	$-0.690476\pi$
$347$	0.367554	−	1.19158i	0.367554	−	1.19158i	0.930874	−	0.365341i	$-0.119048\pi$
$348$		0			0
$349$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$349$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$350$	2.45921	+	0.184292i	2.45921	+	0.184292i
$351$		0			0
$352$	−0.698220	+	1.77904i	−0.698220	+	1.77904i
$353$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$353$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−0.777479	−	0.974928i	−0.777479	−	0.974928i
$359$		0			0		0.563320	−	0.826239i	$-0.309524\pi$
$359$		0			0		−0.563320	+	0.826239i	$0.690476\pi$
$360$		0			0
$361$	0.500000	−	0.866025i	0.500000	−	0.866025i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.57506	+	0.359497i	−1.57506	+	0.359497i
$366$		0			0
$367$	−0.258149	−	1.71271i	−0.258149	−	1.71271i	−0.623490	−	0.781831i	$-0.714286\pi$
$367$	−0.258149	−	1.71271i	−0.258149	−	1.71271i	0.365341	−	0.930874i	$-0.380952\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.930874	+	0.365341i	0.930874	+	0.365341i
$372$		0			0
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$379$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$383$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$384$		0			0
$385$	−3.31211	+	1.29991i	−3.31211	+	1.29991i
$386$	−0.0648483	−	0.134659i	−0.0648483	−	0.134659i
$387$		0			0
$388$	0.400969	−	1.29991i	0.400969	−	1.29991i
$389$	−0.702449	−	1.03030i	−0.702449	−	1.03030i	−0.997204	−	0.0747301i	$-0.976190\pi$
$389$	−0.702449	−	1.03030i	−0.702449	−	1.03030i	0.294755	−	0.955573i	$-0.404762\pi$
$390$		0			0
$391$		0			0
$392$	0.563320	−	0.826239i	0.563320	−	0.826239i
$393$		0			0
$394$	−0.440071	−	0.0663300i	−0.440071	−	0.0663300i
$395$	−0.229907	+	0.156748i	−0.229907	+	0.156748i
$396$		0			0
$397$		0			0		0.997204	−	0.0747301i	$-0.0238095\pi$
$397$		0			0		−0.997204	+	0.0747301i	$0.976190\pi$
$398$	−0.781831	+	0.376510i	−0.781831	+	0.376510i
$399$		0			0
$400$	−2.22188	−	1.07000i	−2.22188	−	1.07000i
$401$		0			0		−0.930874	−	0.365341i	$-0.880952\pi$
$401$		0			0		0.930874	+	0.365341i	$0.119048\pi$
$402$		0			0
$403$		0			0
$404$	1.86323	−	0.574730i	1.86323	−	0.574730i
$405$		0			0
$406$	1.78181	+	0.858075i	1.78181	+	0.858075i
$407$		0			0
$408$		0			0
$409$	−1.17809	+	1.26968i	−1.17809	+	1.26968i	−0.222521	+	0.974928i	$0.571429\pi$
$409$	−1.17809	+	1.26968i	−1.17809	+	1.26968i	−0.955573	+	0.294755i	$0.904762\pi$
$410$		0			0
$411$		0			0
$412$	1.90097	+	0.433884i	1.90097	+	0.433884i
$413$	−1.71271	−	0.258149i	−1.71271	−	0.258149i
$414$		0			0
$415$	0.406813	−	0.377467i	0.406813	−	0.377467i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−0.347948	−	1.52446i	−0.347948	−	1.52446i	−0.781831	−	0.623490i	$-0.785714\pi$
$419$	−0.347948	−	1.52446i	−0.347948	−	1.52446i	0.433884	−	0.900969i	$-0.357143\pi$
$420$		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
$423$		0			0
$424$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	0.116853	+	0.0931869i	0.116853	+	0.0931869i
$429$		0			0
$430$		0			0
$431$		0			0		−0.997204	−	0.0747301i	$-0.976190\pi$
$431$		0			0		0.997204	+	0.0747301i	$0.0238095\pi$
$432$		0			0
$433$	0.678448	−	1.40881i	0.678448	−	1.40881i	−0.222521	−	0.974928i	$-0.571429\pi$
$433$	0.678448	−	1.40881i	0.678448	−	1.40881i	0.900969	−	0.433884i	$-0.142857\pi$
$434$	0.825886	+	0.766310i	0.825886	+	0.766310i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−0.0878620	+	0.582926i	−0.0878620	+	0.582926i	0.900969	+	0.433884i	$0.142857\pi$
$439$	−0.0878620	+	0.582926i	−0.0878620	+	0.582926i	−0.988831	+	0.149042i	$0.952381\pi$
$440$	3.55807			3.55807
$441$		0			0
$442$		0			0
$443$	−0.218511	+	1.44973i	−0.218511	+	1.44973i	0.563320	+	0.826239i	$0.309524\pi$
$443$	−0.218511	+	1.44973i	−0.218511	+	1.44973i	−0.781831	+	0.623490i	$0.785714\pi$
$444$		0			0
$445$		0			0
$446$	0.101659	+	1.35654i	0.101659	+	1.35654i
$447$		0			0
$448$	−0.826239	+	0.563320i	−0.826239	+	0.563320i
$449$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$449$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$	−0.460898	−	0.367554i	−0.460898	−	0.367554i
$455$		0			0
$456$		0			0
$457$	−1.63402	−	1.11406i	−1.63402	−	1.11406i	−0.900969	−	0.433884i	$-0.857143\pi$
$457$	−1.63402	−	1.11406i	−1.63402	−	1.11406i	−0.733052	−	0.680173i	$-0.761905\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−0.193096	+	0.846011i	−0.193096	+	0.846011i	0.781831	+	0.623490i	$0.214286\pi$
$461$	−0.193096	+	0.846011i	−0.193096	+	0.846011i	−0.974928	+	0.222521i	$0.928571\pi$
$462$		0			0
$463$	0.277479	+	1.21572i	0.277479	+	1.21572i	0.900969	+	0.433884i	$0.142857\pi$
$463$	0.277479	+	1.21572i	0.277479	+	1.21572i	−0.623490	+	0.781831i	$0.714286\pi$
$464$	−1.34515	−	1.44973i	−1.34515	−	1.44973i
$465$		0			0
$466$		0			0
$467$	−1.42935	+	1.32624i	−1.42935	+	1.32624i	−0.563320	+	0.826239i	$0.690476\pi$
$467$	−1.42935	+	1.32624i	−1.42935	+	1.32624i	−0.866025	+	0.500000i	$0.833333\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	1.50000	+	0.866025i	1.50000	+	0.866025i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$479$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$480$		0			0
$481$		0			0
$482$	−0.268565	−	0.129334i	−0.268565	−	0.129334i
$483$		0			0
$484$	−2.38980	+	1.15087i	−2.38980	+	1.15087i
$485$	−2.52554	+	0.189263i	−2.52554	+	0.189263i
$486$		0			0
$487$	−1.36534	+	0.930874i	−1.36534	+	0.930874i	−0.365341	+	0.930874i	$0.619048\pi$
$487$	−1.36534	+	0.930874i	−1.36534	+	0.930874i	−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$	−1.81507	−	0.414278i	−1.81507	−	0.414278i
$491$			0.730682i			0.730682i	0.930874	+	0.365341i	$0.119048\pi$
$491$			0.730682i			0.730682i	−0.930874	+	0.365341i	$0.880952\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.488831	−	1.01507i	−0.488831	−	1.01507i
$497$		0			0
$498$		0			0
$499$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$499$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$500$	−0.203977	+	2.72188i	−0.203977	+	2.72188i
$501$		0			0
$502$	0.332083	+	1.07659i	0.332083	+	1.07659i
$503$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$503$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$504$		0			0
$505$	−2.26335	−	2.83816i	−2.26335	−	2.83816i
$506$		0			0
$507$		0			0
$508$	0.826239	−	1.43109i	0.826239	−	1.43109i
$509$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$509$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$510$		0			0
$511$	0.865341	−	0.0648483i	0.865341	−	0.0648483i
$512$	0.974928	−	0.222521i	0.974928	−	0.222521i
$513$		0			0
$514$		0			0
$515$	−0.541044	−	3.58959i	−0.541044	−	3.58959i
$516$		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			0		0.563320	−	0.826239i	$-0.309524\pi$
$523$		0			0		−0.563320	+	0.826239i	$0.690476\pi$
$524$	−1.24349	−	1.55929i	−1.24349	−	1.55929i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.0747301	+	0.997204i	−0.0747301	+	0.997204i
$530$	−0.680173	+	1.73305i	−0.680173	+	1.73305i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	0.0820177	−	0.265895i	0.0820177	−	0.265895i
$536$		0			0
$537$		0			0
$538$		−	0.298085i		−	0.298085i
$539$	1.86323	−	0.425270i	1.86323	−	0.425270i
$540$		0			0
$541$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$541$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$542$	1.12397	−	0.766310i	1.12397	−	0.766310i
$543$		0			0
$544$		0			0
$545$		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
$549$		0			0
$550$	−1.72188	−	4.38729i	−1.72188	−	4.38729i
$551$		0			0
$552$		0			0
$553$	0.134659	−	0.0648483i	0.134659	−	0.0648483i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	0.632789	+	0.365341i	0.632789	+	0.365341i	0.781831	−	0.623490i	$-0.214286\pi$
$557$	0.632789	+	0.365341i	0.632789	+	0.365341i	−0.149042	+	0.988831i	$0.547619\pi$
$558$		0			0
$559$		0			0
$560$	1.53825	+	1.04876i	1.53825	+	1.04876i
$561$		0			0
$562$		0			0
$563$	−1.97213	+	0.297251i	−1.97213	+	0.297251i	−0.974928	+	0.222521i	$0.928571\pi$
$563$	−1.97213	+	0.297251i	−1.97213	+	0.297251i	−0.997204	+	0.0747301i	$0.976190\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$569$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$570$		0			0
$571$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$571$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	1.04876	−	0.411608i	1.04876	−	0.411608i	0.222521	−	0.974928i	$-0.428571\pi$
$577$	1.04876	−	0.411608i	1.04876	−	0.411608i	0.826239	+	0.563320i	$0.190476\pi$
$578$	0.997204	+	0.0747301i	0.997204	+	0.0747301i
$579$		0			0
$580$	−1.59752	+	3.31728i	−1.59752	+	3.31728i
$581$	−0.246289	+	0.167917i	−0.246289	+	0.167917i
$582$		0			0
$583$	−0.142820	−	1.90580i	−0.142820	−	1.90580i
$584$	−0.829215	−	0.255779i	−0.829215	−	0.255779i
$585$		0			0
$586$	−0.202749	+	1.34515i	−0.202749	+	1.34515i
$587$	0.589510			0.589510			0.294755	−	0.955573i	$-0.404762\pi$
$587$	0.589510			0.589510			0.294755	+	0.955573i	$0.404762\pi$
$588$		0			0
$589$		0			0
$590$	0.480608	−	3.18862i	0.480608	−	3.18862i
$591$		0			0
$592$		0			0
$593$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$593$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$594$		0			0
$595$		0			0
$596$	0.781831	−	1.62349i	0.781831	−	1.62349i
$597$		0			0
$598$		0			0
$599$		0			0		0.930874	−	0.365341i	$-0.119048\pi$
$599$		0			0		−0.930874	+	0.365341i	$0.880952\pi$
$600$		0			0
$601$	0.880843	+	0.702449i	0.880843	+	0.702449i	0.955573	−	0.294755i	$-0.0952381\pi$
$601$	0.880843	+	0.702449i	0.880843	+	0.702449i	−0.0747301	+	0.997204i	$0.523810\pi$
$602$		0			0
$603$		0			0
$604$	−0.826239	−	0.563320i	−0.826239	−	0.563320i
$605$	3.61999	+	3.35886i	3.61999	+	3.35886i
$606$		0			0
$607$	0.510531	−	0.294755i	0.510531	−	0.294755i	−0.222521	−	0.974928i	$-0.571429\pi$
$607$	0.510531	−	0.294755i	0.510531	−	0.294755i	0.733052	+	0.680173i	$0.238095\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$613$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$614$		0			0
$615$		0			0
$616$	−1.88980	−	0.284841i	−1.88980	−	0.284841i
$617$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$617$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$618$		0			0
$619$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$619$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$620$	−1.42668	+	1.53759i	−1.42668	+	1.53759i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	2.49936	−	0.770951i	2.49936	−	0.770951i
$626$	−0.728639	−	1.85654i	−0.728639	−	1.85654i
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−1.32091	+	0.636119i	−1.32091	+	0.636119i	−0.955573	−	0.294755i	$-0.904762\pi$
$631$	−1.32091	+	0.636119i	−1.32091	+	0.636119i	−0.365341	+	0.930874i	$0.619048\pi$
$632$	−0.149042	+	0.0111692i	−0.149042	+	0.0111692i
$633$		0			0
$634$	1.21135	−	0.825886i	1.21135	−	0.825886i
$635$	−3.04213	−	0.458528i	−3.04213	−	0.458528i
$636$		0			0
$637$		0			0
$638$		−	3.77960i		−	3.77960i
$639$		0			0
$640$	−1.04876	−	1.53825i	−1.04876	−	1.53825i
$641$		0			0		0.294755	−	0.955573i	$-0.404762\pi$
$641$		0			0		−0.294755	+	0.955573i	$0.595238\pi$
$642$		0			0
$643$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$643$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$647$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$648$		0			0
$649$	0.975699	+	3.16314i	0.975699	+	3.16314i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	0.411608	−	0.603718i	0.411608	−	0.603718i	−0.563320	−	0.826239i	$-0.690476\pi$
$653$	0.411608	−	0.603718i	0.411608	−	0.603718i	0.974928	+	0.222521i	$0.0714286\pi$
$654$		0			0
$655$	−1.85654	+	3.21562i	−1.85654	+	3.21562i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−0.433884	+	0.0990311i	−0.433884	+	0.0990311i	−0.433884	−	0.900969i	$-0.642857\pi$
$659$	−0.433884	+	0.0990311i	−0.433884	+	0.0990311i			1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		−0.149042	−	0.988831i	$-0.547619\pi$
$661$		0			0		0.149042	+	0.988831i	$0.452381\pi$
$662$		0			0
$663$		0			0
$664$	0.290611	−	0.0663300i	0.290611	−	0.0663300i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.19158	+	1.49419i	−1.19158	+	1.49419i	−0.365341	+	0.930874i	$0.619048\pi$
$673$	−1.19158	+	1.49419i	−1.19158	+	1.49419i	−0.826239	+	0.563320i	$0.809524\pi$
$674$	−0.294755	−	0.955573i	−0.294755	−	0.955573i
$675$		0			0
$676$	0.0747301	−	0.997204i	0.0747301	−	0.997204i
$677$	0.215372	−	0.548760i	0.215372	−	0.548760i	−0.781831	−	0.623490i	$-0.785714\pi$
$677$	0.215372	−	0.548760i	0.215372	−	0.548760i	0.997204	+	0.0747301i	$0.0238095\pi$
$678$		0			0
$679$	1.35654	+	0.101659i	1.35654	+	0.101659i
$680$		0			0
$681$		0			0
$682$	0.634659	−	2.05751i	0.634659	−	2.05751i
$683$	0.411608	+	0.603718i	0.411608	+	0.603718i	0.974928	−	0.222521i	$-0.0714286\pi$
$683$	0.411608	+	0.603718i	0.411608	+	0.603718i	−0.563320	+	0.826239i	$0.690476\pi$
$684$		0			0
$685$		0			0
$686$	0.930874	+	0.365341i	0.930874	+	0.365341i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.997204	−	0.0747301i	$-0.0238095\pi$
$691$		0			0		−0.997204	+	0.0747301i	$0.976190\pi$
$692$	−1.40881	+	0.678448i	−1.40881	+	0.678448i
$693$		0			0
$694$	−1.12349	−	0.541044i	−1.12349	−	0.541044i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.548760	−	2.40427i	0.548760	−	2.40427i
$701$	−1.29196	+	1.03030i	−1.29196	+	1.03030i	−0.294755	+	0.955573i	$0.595238\pi$
$701$	−1.29196	+	1.03030i	−1.29196	+	1.03030i	−0.997204	+	0.0747301i	$0.976190\pi$
$702$		0			0
$703$		0			0
$704$	1.65510	+	0.955573i	1.65510	+	0.955573i
$705$		0			0
$706$		0			0
$707$	0.974928	+	1.68862i	0.974928	+	1.68862i
$708$		0			0
$709$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$709$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−1.07992	+	0.623490i	−1.07992	+	0.623490i
$717$		0			0
$718$		0			0
$719$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$719$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$720$		0			0
$721$			1.94986i			1.94986i
$722$	−0.781831	−	0.623490i	−0.781831	−	0.623490i
$723$		0			0
$724$		0			0
$725$	4.86348	+	0.364468i	4.86348	+	0.364468i
$726$		0			0
$727$	−0.129334	+	0.268565i	−0.129334	+	0.268565i	−0.955573	−	0.294755i	$-0.904762\pi$
$727$	−0.129334	+	0.268565i	−0.129334	+	0.268565i	0.826239	+	0.563320i	$0.190476\pi$
$728$		0			0
$729$		0			0
$730$	0.120731	+	1.61105i	0.120731	+	1.61105i
$731$		0			0
$732$		0			0
$733$		0			0		0.149042	−	0.988831i	$-0.452381\pi$
$733$		0			0		−0.149042	+	0.988831i	$0.547619\pi$
$734$	−1.73205			−1.73205
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$739$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$740$		0			0
$741$		0			0
$742$	0.500000	−	0.866025i	0.500000	−	0.866025i
$743$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$743$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$744$		0			0
$745$	−3.34537	−	0.250701i	−3.34537	−	0.250701i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−0.0648483	+	0.134659i	−0.0648483	+	0.134659i
$750$		0			0
$751$	−1.57906	−	1.07659i	−1.57906	−	1.07659i	−0.955573	−	0.294755i	$-0.904762\pi$
$751$	−1.57906	−	1.07659i	−1.57906	−	1.07659i	−0.623490	−	0.781831i	$-0.714286\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−0.414278	+	1.81507i	−0.414278	+	1.81507i
$756$		0			0
$757$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$757$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$761$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	1.55929	−	1.24349i	1.55929	−	1.24349i	0.733052	−	0.680173i	$-0.238095\pi$
$769$	1.55929	−	1.24349i	1.55929	−	1.24349i	0.826239	−	0.563320i	$-0.190476\pi$
$770$	0.791745	+	3.46886i	0.791745	+	3.46886i
$771$		0			0
$772$	−0.142820	+	0.0440542i	−0.142820	+	0.0440542i
$773$	−0.712362	−	1.81507i	−0.712362	−	1.81507i	−0.563320	−	0.826239i	$-0.690476\pi$
$773$	−0.712362	−	1.81507i	−0.712362	−	1.81507i	−0.149042	−	0.988831i	$-0.547619\pi$
$774$		0			0
$775$	2.58635	+	1.01507i	2.58635	+	1.01507i
$776$	−1.22563	−	0.590232i	−1.22563	−	0.590232i
$777$		0			0
$778$	−1.12349	+	0.541044i	−1.12349	+	0.541044i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$785$		0			0
$786$		0			0
$787$		0			0		−0.563320	−	0.826239i	$-0.690476\pi$
$787$		0			0		0.563320	+	0.826239i	$0.309524\pi$
$788$	−0.131178	+	0.425270i	−0.131178	+	0.425270i
$789$		0			0
$790$	0.120731	+	0.250701i	0.120731	+	0.250701i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	0.255779	+	0.829215i	0.255779	+	0.829215i
$797$	0.848162	−	1.06356i	0.848162	−	1.06356i	−0.149042	−	0.988831i	$-0.547619\pi$
$797$	0.848162	−	1.06356i	0.848162	−	1.06356i	0.997204	−	0.0747301i	$-0.0238095\pi$
$798$		0			0
$799$		0			0
$800$	−1.38921	+	2.03759i	−1.38921	+	2.03759i
$801$		0			0
$802$		0			0
$803$	−0.829215	−	1.43624i	−0.829215	−	1.43624i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−0.290611	−	1.92808i	−0.290611	−	1.92808i
$809$		0			0		−0.149042	−	0.988831i	$-0.547619\pi$
$809$		0			0		0.149042	+	0.988831i	$0.452381\pi$
$810$		0			0
$811$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$811$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$812$	1.11406	−	1.63402i	1.11406	−	1.63402i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	1.07992	+	1.35417i	1.07992	+	1.35417i
$819$		0			0
$820$		0			0
$821$	−0.0440542	−	0.142820i	−0.0440542	−	0.142820i	0.930874	−	0.365341i	$-0.119048\pi$
$821$	−0.0440542	−	0.142820i	−0.0440542	−	0.142820i	−0.974928	+	0.222521i	$0.928571\pi$
$822$		0			0
$823$	0.134659	−	1.79690i	0.134659	−	1.79690i	−0.365341	−	0.930874i	$-0.619048\pi$
$823$	0.134659	−	1.79690i	0.134659	−	1.79690i	0.500000	−	0.866025i	$-0.333333\pi$
$824$	0.712362	−	1.81507i	0.712362	−	1.81507i
$825$		0			0
$826$	−0.510531	+	1.65510i	−0.510531	+	1.65510i
$827$	0.317031	+	0.658322i	0.317031	+	0.658322i	0.997204	−	0.0747301i	$-0.0238095\pi$
$827$	0.317031	+	0.658322i	0.317031	+	0.658322i	−0.680173	+	0.733052i	$0.738095\pi$
$828$		0			0
$829$		0			0		0.294755	−	0.955573i	$-0.404762\pi$
$829$		0			0		−0.294755	+	0.955573i	$0.595238\pi$
$830$	−0.312619	−	0.458528i	−0.312619	−	0.458528i
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−1.55929	+	0.116853i	−1.55929	+	0.116853i
$839$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$839$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$840$		0			0
$841$	2.62285	+	1.26310i	2.62285	+	1.26310i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.77904	+	0.548760i	−1.77904	+	0.548760i
$846$		0			0
$847$	−1.65379	−	2.07379i	−1.65379	−	2.07379i
$848$	−0.781831	+	0.623490i	−0.781831	+	0.623490i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$853$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$854$		0			0
$855$		0			0
$856$	0.109562	−	0.101659i	0.109562	−	0.101659i
$857$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$857$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$858$		0			0
$859$		0			0		−0.680173	−	0.733052i	$-0.738095\pi$
$859$		0			0		0.680173	+	0.733052i	$0.261905\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$863$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$864$		0			0
$865$	2.13402	+	1.98008i	2.13402	+	1.98008i
$866$	−1.29196	−	0.880843i	−1.29196	−	0.880843i
$867$		0			0
$868$	0.880843	−	0.702449i	0.880843	−	0.702449i
$869$	−0.223322	−	0.178094i	−0.223322	−	0.178094i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.69903	+	0.406813i	−2.69903	+	0.406813i
$876$		0			0
$877$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$877$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$878$	0.563320	+	0.173761i	0.563320	+	0.173761i
$879$		0			0
$880$	0.530303	−	3.51833i	0.530303	−	3.51833i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$	1.40097	+	0.432142i	1.40097	+	0.432142i
$887$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$887$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$888$		0			0
$889$	1.57906	+	0.487076i	1.57906	+	0.487076i
$890$		0			0
$891$		0			0
$892$	1.35654	+	0.101659i	1.35654	+	0.101659i
$893$		0			0
$894$		0			0
$895$	1.81507	+	1.44747i	1.81507	+	1.44747i
$896$	0.433884	+	0.900969i	0.433884	+	0.900969i
$897$		0			0
$898$		0			0
$899$	1.63332	+	1.51550i	1.63332	+	1.51550i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$907$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$908$	−0.432142	+	0.400969i	−0.432142	+	0.400969i
$909$		0			0
$910$		0			0
$911$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$911$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$912$		0			0
$913$	0.493360	+	0.284841i	0.493360	+	0.284841i
$914$	−1.34515	+	1.44973i	−1.34515	+	1.44973i
$915$		0			0
$916$		0			0
$917$	1.24349	−	1.55929i	1.24349	−	1.55929i
$918$		0			0
$919$	−0.425270	+	0.131178i	−0.425270	+	0.131178i	−0.500000	−	0.866025i	$-0.666667\pi$
$919$	−0.425270	+	0.131178i	−0.425270	+	0.131178i	0.0747301	+	0.997204i	$0.476190\pi$
$920$		0			0
$921$		0			0
$922$	0.807782	+	0.317031i	0.807782	+	0.317031i
$923$		0			0
$924$		0			0
$925$		0			0
$926$	1.24349	−	0.0931869i	1.24349	−	0.0931869i
$927$		0			0
$928$	−1.63402	+	1.11406i	−1.63402	+	1.11406i
$929$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$929$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$	1.09839	+	1.61105i	1.09839	+	1.61105i
$935$		0			0
$936$		0			0
$937$	0.488831	+	1.01507i	0.488831	+	1.01507i	0.988831	+	0.149042i	$0.0476190\pi$
$937$	0.488831	+	1.01507i	0.488831	+	1.01507i	−0.500000	+	0.866025i	$0.666667\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.101659	+	1.35654i	−0.101659	+	1.35654i	0.680173	+	0.733052i	$0.261905\pi$
$941$	−0.101659	+	1.35654i	−0.101659	+	1.35654i	−0.781831	+	0.623490i	$0.785714\pi$
$942$		0			0
$943$		0			0
$944$	1.07992	−	1.35417i	1.07992	−	1.35417i
$945$		0			0
$946$		0			0
$947$	1.01507	−	1.48883i	1.01507	−	1.48883i	0.149042	−	0.988831i	$-0.452381\pi$
$947$	1.01507	−	1.48883i	1.01507	−	1.48883i	0.866025	−	0.500000i	$-0.166667\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$953$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.134659	+	0.233236i	0.134659	+	0.233236i
$962$		0			0
$963$		0			0
$964$	−0.167917	+	0.246289i	−0.167917	+	0.246289i
$965$	0.173490	+	0.217550i	0.173490	+	0.217550i
$966$		0			0
$967$	0.0931869	−	0.116853i	0.0931869	−	0.116853i	−0.733052	−	0.680173i	$-0.761905\pi$
$967$	0.0931869	−	0.116853i	0.0931869	−	0.116853i	0.826239	+	0.563320i	$0.190476\pi$
$968$	0.781831	+	2.53464i	0.781831	+	2.53464i
$969$		0			0
$970$	−0.189263	+	2.52554i	−0.189263	+	2.52554i
$971$	−0.411608	+	1.04876i	−0.411608	+	1.04876i	0.563320	+	0.826239i	$0.309524\pi$
$971$	−0.411608	+	1.04876i	−0.411608	+	1.04876i	−0.974928	+	0.222521i	$0.928571\pi$
$972$		0			0
$973$		0			0
$974$	0.716983	+	1.48883i	0.716983	+	1.48883i
$975$		0			0
$976$		0			0
$977$		0			0		−0.563320	−	0.826239i	$-0.690476\pi$
$977$		0			0		0.563320	+	0.826239i	$0.309524\pi$
$978$		0			0
$979$		0			0
$980$	−0.680173	+	1.73305i	−0.680173	+	1.73305i
$981$		0			0
$982$	0.722521	+	0.108903i	0.722521	+	0.108903i
$983$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$983$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$984$		0			0
$985$	0.826239	−	0.0619180i	0.826239	−	0.0619180i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	0.698220	+	1.77904i	0.698220	+	1.77904i	0.623490	+	0.781831i	$0.285714\pi$
$991$	0.698220	+	1.77904i	0.698220	+	1.77904i	0.0747301	+	0.997204i	$0.476190\pi$
$992$	−1.07659	+	0.332083i	−1.07659	+	0.332083i
$993$		0			0
$994$		0			0
$995$	1.26310	−	1.00729i	1.26310	−	1.00729i
$996$		0			0
$997$		0			0		0.680173	−	0.733052i	$-0.261905\pi$
$997$		0			0		−0.680173	+	0.733052i	$0.738095\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3528.1.fb.a.1333.2	yes	24
3.2	odd	2	inner	3528.1.fb.a.1333.1	yes	24
8.5	even	2	inner	3528.1.fb.a.1333.1	yes	24
24.5	odd	2	CM	3528.1.fb.a.1333.2	yes	24
49.5	odd	42	inner	3528.1.fb.a.397.1	✓	24
147.5	even	42	inner	3528.1.fb.a.397.2	yes	24
392.5	odd	42	inner	3528.1.fb.a.397.2	yes	24
1176.5	even	42	inner	3528.1.fb.a.397.1	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3528.1.fb.a.397.1	✓	24	49.5	odd	42	inner
3528.1.fb.a.397.1	✓	24	1176.5	even	42	inner
3528.1.fb.a.397.2	yes	24	147.5	even	42	inner
3528.1.fb.a.397.2	yes	24	392.5	odd	42	inner
3528.1.fb.a.1333.1	yes	24	3.2	odd	2	inner
3528.1.fb.a.1333.1	yes	24	8.5	even	2	inner
3528.1.fb.a.1333.2	yes	24	1.1	even	1	trivial
3528.1.fb.a.1333.2	yes	24	24.5	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:	\( N \)	\(=\)	\( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3528.fb (of order \(42\), degree \(12\), minimal)

\(f(q)\)	\(=\)	\(q+(0.149042 - 0.988831i) q^{2} +(-0.955573 - 0.294755i) q^{4} +(0.139129 + 1.85654i) q^{5} +(0.0747301 - 0.997204i) q^{7} +(-0.433884 + 0.900969i) q^{8} +O(q^{10})\)	\(q+(0.149042 - 0.988831i) q^{2} +(-0.955573 - 0.294755i) q^{4} +(0.139129 + 1.85654i) q^{5} +(0.0747301 - 0.997204i) q^{7} +(-0.433884 + 0.900969i) q^{8} +(1.85654 + 0.139129i) q^{10} +(-1.77904 + 0.698220i) q^{11} +(-0.974928 - 0.222521i) q^{14} +(0.826239 + 0.563320i) q^{16} +(0.414278 - 1.81507i) q^{20} +(0.425270 + 1.86323i) q^{22} +(-2.43856 + 0.367554i) q^{25} +(-0.365341 + 0.930874i) q^{28} +(-1.92808 - 0.440071i) q^{29} +(-0.975699 - 0.563320i) q^{31} +(0.680173 - 0.733052i) q^{32} +1.86175 q^{35} +(-1.73305 - 0.680173i) q^{40} +(1.90580 - 0.142820i) q^{44} +(-0.988831 - 0.149042i) q^{49} +2.46610i q^{50} +(-0.294755 + 0.955573i) q^{53} +(-1.54379 - 3.20571i) q^{55} +(0.866025 + 0.500000i) q^{56} +(-0.722521 + 1.84095i) q^{58} +(0.129436 - 1.72721i) q^{59} +(-0.702449 + 0.880843i) q^{62} +(-0.623490 - 0.781831i) q^{64} +(0.277479 - 1.84095i) q^{70} +(0.129334 + 0.858075i) q^{73} +(0.563320 + 1.82624i) q^{77} +(0.0747301 + 0.129436i) q^{79} +(-0.930874 + 1.61232i) q^{80} +(-0.185853 - 0.233052i) q^{83} +(0.142820 - 1.90580i) q^{88} +1.36035i q^{97} +(-0.294755 + 0.955573i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 2 q^{4} + 2 q^{7}+O(q^{10}) \) 24 * q - 2 * q^4 + 2 * q^7	\( 24 q - 2 q^{4} + 2 q^{7} + 6 q^{10} + 2 q^{16} + 10 q^{22} + 4 q^{25} - 2 q^{28} - 6 q^{31} - 22 q^{40} + 2 q^{49} - 14 q^{55} - 16 q^{58} + 4 q^{64} + 8 q^{70} + 2 q^{79} - 2 q^{88}+O(q^{100}) \) 24 * q - 2 * q^4 + 2 * q^7 + 6 * q^10 + 2 * q^16 + 10 * q^22 + 4 * q^25 - 2 * q^28 - 6 * q^31 - 22 * q^40 + 2 * q^49 - 14 * q^55 - 16 * q^58 + 4 * q^64 + 8 * q^70 + 2 * q^79 - 2 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more