LMFDB - Embedded newform 3724.1.dj.a.2165.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3724,1,Mod(37,3724)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3724.37"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3724, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([0, 32, 21])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 3724 = 2^{2} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3724.dj (of order $42$, degree $12$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.85851810705$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{21}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{21} - \cdots)$

Embedding invariants

Embedding label			2165.1
Root			$0.826239 + 0.563320i$ of defining polynomial
Character	$\chi$	$=$	3724.2165
Dual form			3724.1.dj.a.1481.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.603718 + 1.53825i) q^{5} +(0.623490 - 0.781831i) q^{7} +(0.955573 + 0.294755i) q^{9} +(-0.425270 + 0.131178i) q^{11} +(0.123490 + 0.0841939i) q^{17} +(-0.500000 + 0.866025i) q^{19} +(1.57906 - 1.07659i) q^{23} +(-1.26868 + 1.17716i) q^{25} +(1.57906 + 0.487076i) q^{35} +(0.0931869 + 0.116853i) q^{43} +(0.123490 + 1.64786i) q^{45} +(-0.914101 - 0.848162i) q^{47} +(-0.222521 - 0.974928i) q^{49} +(-0.458528 - 0.574976i) q^{55} +(-0.0332580 + 0.443797i) q^{61} +(0.826239 - 0.563320i) q^{63} +(-1.46610 + 1.36035i) q^{73} +(-0.162592 + 0.414278i) q^{77} +(0.826239 + 0.563320i) q^{81} +(-0.367711 - 1.61105i) q^{83} +(-0.0549581 + 0.240787i) q^{85} +(-1.63402 - 0.246289i) q^{95} -0.445042 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - q^{5} - 2 q^{7} + q^{9} - 5 q^{11} - 8 q^{17} - 6 q^{19} - q^{23} - q^{35} + 2 q^{43} - 8 q^{45} + 2 q^{47} - 2 q^{49} - 3 q^{55} + 2 q^{61} + q^{63} + 2 q^{73} + 2 q^{77} + q^{81} + 2 q^{83}+ \cdots - 4 q^{99}+O(q^{100}) $ 12 * q - q^5 - 2 * q^7 + q^9 - 5 * q^11 - 8 * q^17 - 6 * q^19 - q^23 - q^35 + 2 * q^43 - 8 * q^45 + 2 * q^47 - 2 * q^49 - 3 * q^55 + 2 * q^61 + q^63 + 2 * q^73 + 2 * q^77 + q^81 + 2 * q^83 - 2 * q^85 - q^95 - 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3724\mathbb{Z}\right)^\times$.

$n$	$1863$	$3041$	$3137$
$\chi(n)$	$1$	$e\left(\frac{1}{21}\right)$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$3$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$4$		0			0
$5$	0.603718	+	1.53825i	0.603718	+	1.53825i	0.826239	+	0.563320i	$0.190476\pi$
$5$	0.603718	+	1.53825i	0.603718	+	1.53825i	−0.222521	+	0.974928i	$0.571429\pi$
$6$		0			0
$7$	0.623490	−	0.781831i	0.623490	−	0.781831i
$7$	0.623490	−	0.781831i	0.623490	−	0.781831i
$8$		0			0
$9$	0.955573	+	0.294755i	0.955573	+	0.294755i
$10$		0			0
$11$	−0.425270	+	0.131178i	−0.425270	+	0.131178i	−0.500000	−	0.866025i	$-0.666667\pi$
$11$	−0.425270	+	0.131178i	−0.425270	+	0.131178i	0.0747301	+	0.997204i	$0.476190\pi$
$12$		0			0
$13$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$13$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.123490	+	0.0841939i	0.123490	+	0.0841939i	0.623490	−	0.781831i	$-0.285714\pi$
$17$	0.123490	+	0.0841939i	0.123490	+	0.0841939i	−0.500000	+	0.866025i	$0.666667\pi$
$18$		0			0
$19$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$19$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.57906	−	1.07659i	1.57906	−	1.07659i	0.623490	−	0.781831i	$-0.285714\pi$
$23$	1.57906	−	1.07659i	1.57906	−	1.07659i	0.955573	−	0.294755i	$-0.0952381\pi$
$24$		0			0
$25$	−1.26868	+	1.17716i	−1.26868	+	1.17716i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$29$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$30$		0			0
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.57906	+	0.487076i	1.57906	+	0.487076i
$36$		0			0
$37$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$37$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$38$		0			0
$39$		0			0
$40$		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
$43$	0.0931869	+	0.116853i	0.0931869	+	0.116853i	0.826239	−	0.563320i	$-0.190476\pi$
$43$	0.0931869	+	0.116853i	0.0931869	+	0.116853i	−0.733052	+	0.680173i	$0.761905\pi$
$44$		0			0
$45$	0.123490	+	1.64786i	0.123490	+	1.64786i
$46$		0			0
$47$	−0.914101	−	0.848162i	−0.914101	−	0.848162i	0.0747301	−	0.997204i	$-0.476190\pi$
$47$	−0.914101	−	0.848162i	−0.914101	−	0.848162i	−0.988831	+	0.149042i	$0.952381\pi$
$48$		0			0
$49$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$53$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$54$		0			0
$55$	−0.458528	−	0.574976i	−0.458528	−	0.574976i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$59$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$60$		0			0
$61$	−0.0332580	+	0.443797i	−0.0332580	+	0.443797i	0.955573	+	0.294755i	$0.0952381\pi$
$61$	−0.0332580	+	0.443797i	−0.0332580	+	0.443797i	−0.988831	+	0.149042i	$0.952381\pi$
$62$		0			0
$63$	0.826239	−	0.563320i	0.826239	−	0.563320i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$		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
$73$	−1.46610	+	1.36035i	−1.46610	+	1.36035i	−0.733052	+	0.680173i	$0.761905\pi$
$73$	−1.46610	+	1.36035i	−1.46610	+	1.36035i	−0.733052	+	0.680173i	$0.761905\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.162592	+	0.414278i	−0.162592	+	0.414278i
$78$		0			0
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$		0			0
$81$	0.826239	+	0.563320i	0.826239	+	0.563320i
$82$		0			0
$83$	−0.367711	−	1.61105i	−0.367711	−	1.61105i	−0.733052	−	0.680173i	$-0.761905\pi$
$83$	−0.367711	−	1.61105i	−0.367711	−	1.61105i	0.365341	−	0.930874i	$-0.380952\pi$
$84$		0			0
$85$	−0.0549581	+	0.240787i	−0.0549581	+	0.240787i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$89$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.63402	−	0.246289i	−1.63402	−	0.246289i
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$	−0.445042			−0.445042
$100$		0			0
$101$	−0.722521	−	0.108903i	−0.722521	−	0.108903i	−0.222521	−	0.974928i	$-0.571429\pi$
$101$	−0.722521	−	0.108903i	−0.722521	−	0.108903i	−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$103$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$107$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$108$		0			0
$109$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$109$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$110$		0			0
$111$		0			0
$112$		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
$115$	2.60937	+	1.77904i	2.60937	+	1.77904i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.142820	−	0.0440542i	0.142820	−	0.0440542i
$120$		0			0
$121$	−0.662592	+	0.451748i	−0.662592	+	0.451748i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.08786	−	0.523887i	−1.08786	−	0.523887i
$126$		0			0
$127$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$127$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1.44973	−	0.218511i	1.44973	−	0.218511i	0.623490	−	0.781831i	$-0.285714\pi$
$131$	1.44973	−	0.218511i	1.44973	−	0.218511i	0.826239	+	0.563320i	$0.190476\pi$
$132$		0			0
$133$	0.365341	+	0.930874i	0.365341	+	0.930874i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.730682	−	1.86175i	0.730682	−	1.86175i	0.365341	−	0.930874i	$-0.380952\pi$
$137$	0.730682	−	1.86175i	0.730682	−	1.86175i	0.365341	−	0.930874i	$-0.380952\pi$
$138$		0			0
$139$	−1.12349	+	1.40881i	−1.12349	+	1.40881i	−0.222521	+	0.974928i	$0.571429\pi$
$139$	−1.12349	+	1.40881i	−1.12349	+	1.40881i	−0.900969	+	0.433884i	$0.857143\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.44973	+	1.34515i	1.44973	+	1.34515i	0.826239	+	0.563320i	$0.190476\pi$
$149$	1.44973	+	1.34515i	1.44973	+	1.34515i	0.623490	+	0.781831i	$0.285714\pi$
$150$		0			0
$151$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$151$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$152$		0			0
$153$	0.0931869	+	0.116853i	0.0931869	+	0.116853i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.365341	+	0.930874i	−0.365341	+	0.930874i	0.623490	+	0.781831i	$0.285714\pi$
$157$	−0.365341	+	0.930874i	−0.365341	+	0.930874i	−0.988831	+	0.149042i	$0.952381\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	0.142820	−	1.90580i	0.142820	−	1.90580i
$162$		0			0
$163$	−0.722521	+	0.108903i	−0.722521	+	0.108903i	−0.500000	−	0.866025i	$-0.666667\pi$
$163$	−0.722521	+	0.108903i	−0.722521	+	0.108903i	−0.222521	+	0.974928i	$0.571429\pi$
$164$		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
$169$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$170$		0			0
$171$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$172$		0			0
$173$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$173$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$174$		0			0
$175$	0.129334	+	1.72584i	0.129334	+	1.72584i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$179$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$180$		0			0
$181$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$181$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−0.0635609	−	0.0196059i	−0.0635609	−	0.0196059i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−0.365341	−	0.930874i	−0.365341	−	0.930874i	−0.988831	−	0.149042i	$-0.952381\pi$
$191$	−0.365341	−	0.930874i	−0.365341	−	0.930874i	0.623490	−	0.781831i	$-0.285714\pi$
$192$		0			0
$193$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$193$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	0.149460			0.149460			0.0747301	−	0.997204i	$-0.476190\pi$
$197$	0.149460			0.149460			0.0747301	+	0.997204i	$0.476190\pi$
$198$		0			0
$199$	−0.147791	−	0.0222759i	−0.147791	−	0.0222759i	0.0747301	−	0.997204i	$-0.476190\pi$
$199$	−0.147791	−	0.0222759i	−0.147791	−	0.0222759i	−0.222521	+	0.974928i	$0.571429\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	1.82624	−	0.563320i	1.82624	−	0.563320i
$208$		0			0
$209$	0.0990311	−	0.433884i	0.0990311	−	0.433884i
$210$		0			0
$211$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$211$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.123490	+	0.213891i	−0.123490	+	0.213891i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$223$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$224$		0			0
$225$	−1.55929	+	0.750915i	−1.55929	+	0.750915i
$226$		0			0
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$		0			0
$229$	1.95557	−	0.294755i	1.95557	−	0.294755i	0.955573	−	0.294755i	$-0.0952381\pi$
$229$	1.95557	−	0.294755i	1.95557	−	0.294755i	1.00000			$0$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	0.0546039	−	0.728639i	0.0546039	−	0.728639i	−0.900969	−	0.433884i	$-0.857143\pi$
$233$	0.0546039	−	0.728639i	0.0546039	−	0.728639i	0.955573	−	0.294755i	$-0.0952381\pi$
$234$		0			0
$235$	0.752824	−	1.91816i	0.752824	−	1.91816i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.03030	+	1.29196i	1.03030	+	1.29196i	0.955573	+	0.294755i	$0.0952381\pi$
$239$	1.03030	+	1.29196i	1.03030	+	1.29196i	0.0747301	+	0.997204i	$0.476190\pi$
$240$		0			0
$241$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$241$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.36534	−	0.930874i	1.36534	−	0.930874i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−0.914101	−	1.14625i	−0.914101	−	1.14625i	−0.988831	−	0.149042i	$-0.952381\pi$
$251$	−0.914101	−	1.14625i	−0.914101	−	1.14625i	0.0747301	−	0.997204i	$-0.476190\pi$
$252$		0			0
$253$	−0.530303	+	0.664979i	−0.530303	+	0.664979i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$257$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	0.733052	+	1.26968i	0.733052	+	1.26968i	0.955573	+	0.294755i	$0.0952381\pi$
$263$	0.733052	+	1.26968i	0.733052	+	1.26968i	−0.222521	+	0.974928i	$0.571429\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$269$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$270$		0			0
$271$	−1.63402	+	1.11406i	−1.63402	+	1.11406i	−0.733052	+	0.680173i	$0.761905\pi$
$271$	−1.63402	+	1.11406i	−1.63402	+	1.11406i	−0.900969	+	0.433884i	$0.857143\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.385113	−	0.667035i	0.385113	−	0.667035i
$276$		0			0
$277$	−1.48883	−	1.01507i	−1.48883	−	1.01507i	−0.988831	−	0.149042i	$-0.952381\pi$
$277$	−1.48883	−	1.01507i	−1.48883	−	1.01507i	−0.500000	−	0.866025i	$-0.666667\pi$
$278$		0			0
$279$		0			0
$280$		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
$283$	1.82624	−	0.563320i	1.82624	−	0.563320i	0.826239	−	0.563320i	$-0.190476\pi$
$283$	1.82624	−	0.563320i	1.82624	−	0.563320i	1.00000			$0$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.357180	−	0.910080i	−0.357180	−	0.910080i
$290$		0			0
$291$		0			0
$292$		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
$301$	0.149460			0.149460
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.702749	+	0.216769i	−0.702749	+	0.216769i
$306$		0			0
$307$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$307$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1.63402	−	1.11406i	−1.63402	−	1.11406i	−0.900969	−	0.433884i	$-0.857143\pi$
$311$	−1.63402	−	1.11406i	−1.63402	−	1.11406i	−0.733052	−	0.680173i	$-0.761905\pi$
$312$		0			0
$313$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.365341	−	0.930874i	$-0.380952\pi$
$313$	0.988831	−	1.71271i	0.988831	−	1.71271i	0.623490	−	0.781831i	$-0.285714\pi$
$314$		0			0
$315$	1.36534	+	0.930874i	1.36534	+	0.930874i
$316$		0			0
$317$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$317$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−0.134659	+	0.0648483i	−0.134659	+	0.0648483i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.23305	+	0.185853i	−1.23305	+	0.185853i
$330$		0			0
$331$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$331$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$337$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−0.147791	−	1.97213i	−0.147791	−	1.97213i	−0.222521	−	0.974928i	$-0.571429\pi$
$347$	−0.147791	−	1.97213i	−0.147791	−	1.97213i	0.0747301	−	0.997204i	$-0.476190\pi$
$348$		0			0
$349$	−0.914101	−	1.14625i	−0.914101	−	1.14625i	−0.988831	−	0.149042i	$-0.952381\pi$
$349$	−0.914101	−	1.14625i	−0.914101	−	1.14625i	0.0747301	−	0.997204i	$-0.476190\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.455573	−	1.16078i	0.455573	−	1.16078i	−0.500000	−	0.866025i	$-0.666667\pi$
$353$	0.455573	−	1.16078i	0.455573	−	1.16078i	0.955573	−	0.294755i	$-0.0952381\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.78181	−	0.268565i	1.78181	−	0.268565i	0.826239	−	0.563320i	$-0.190476\pi$
$359$	1.78181	−	0.268565i	1.78181	−	0.268565i	0.955573	+	0.294755i	$0.0952381\pi$
$360$		0			0
$361$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.97766	−	1.43397i	−2.97766	−	1.43397i
$366$		0			0
$367$	0.326239	−	0.302705i	0.326239	−	0.302705i	−0.500000	−	0.866025i	$-0.666667\pi$
$367$	0.326239	−	0.302705i	0.326239	−	0.302705i	0.826239	+	0.563320i	$0.190476\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$379$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$383$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$384$		0			0
$385$	−0.735422			−0.735422
$386$		0			0
$387$	0.0546039	+	0.139129i	0.0546039	+	0.139129i
$388$		0			0
$389$	−1.88980	−	0.284841i	−1.88980	−	0.284841i	−0.900969	−	0.433884i	$-0.857143\pi$
$389$	−1.88980	−	0.284841i	−1.88980	−	0.284841i	−0.988831	+	0.149042i	$0.952381\pi$
$390$		0			0
$391$	0.285640			0.285640
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.698220	+	1.77904i	0.698220	+	1.77904i	0.623490	+	0.781831i	$0.285714\pi$
$397$	0.698220	+	1.77904i	0.698220	+	1.77904i	0.0747301	+	0.997204i	$0.476190\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$401$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−0.367711	+	1.61105i	−0.367711	+	1.61105i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$409$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	2.25620	−	1.53825i	2.25620	−	1.53825i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	0.900969	+	0.433884i	0.900969	+	0.433884i	0.826239	−	0.563320i	$-0.190476\pi$
$419$	0.900969	+	0.433884i	0.900969	+	0.433884i	0.0747301	+	0.997204i	$0.476190\pi$
$420$		0			0
$421$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$421$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$422$		0			0
$423$	−0.623490	−	1.07992i	−0.623490	−	1.07992i
$424$		0			0
$425$	−0.255779	+	0.0385525i	−0.255779	+	0.0385525i
$426$		0			0
$427$	0.326239	+	0.302705i	0.326239	+	0.302705i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$431$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$432$		0			0
$433$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$433$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.142820	+	1.90580i	0.142820	+	1.90580i
$438$		0			0
$439$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$439$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$440$		0			0
$441$	0.0747301	−	0.997204i	0.0747301	−	0.997204i
$442$		0			0
$443$	−0.535628	−	0.496990i	−0.535628	−	0.496990i	0.365341	−	0.930874i	$-0.380952\pi$
$443$	−0.535628	−	0.496990i	−0.535628	−	0.496990i	−0.900969	+	0.433884i	$0.857143\pi$
$444$		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			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.23305	+	0.185853i	−1.23305	+	0.185853i	−0.733052	−	0.680173i	$-0.761905\pi$
$457$	−1.23305	+	0.185853i	−1.23305	+	0.185853i	−0.500000	+	0.866025i	$0.666667\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.12349	+	0.541044i	−1.12349	+	0.541044i	−0.900969	−	0.433884i	$-0.857143\pi$
$461$	−1.12349	+	0.541044i	−1.12349	+	0.541044i	−0.222521	+	0.974928i	$0.571429\pi$
$462$		0			0
$463$	1.62349	+	0.781831i	1.62349	+	0.781831i	1.00000			$0$
$463$	1.62349	+	0.781831i	1.62349	+	0.781831i	0.623490	+	0.781831i	$0.285714\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−0.367711	+	0.250701i	−0.367711	+	0.250701i	−0.733052	−	0.680173i	$-0.761905\pi$
$467$	−0.367711	+	0.250701i	−0.367711	+	0.250701i	0.365341	+	0.930874i	$0.380952\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.0549581	−	0.0374698i	−0.0549581	−	0.0374698i
$474$		0			0
$475$	−0.385113	−	1.68729i	−0.385113	−	1.68729i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−0.955573	+	0.294755i	−0.955573	+	0.294755i	−0.733052	−	0.680173i	$-0.761905\pi$
$479$	−0.955573	+	0.294755i	−0.955573	+	0.294755i	−0.222521	+	0.974928i	$0.571429\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$487$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.97766			−1.97766			−0.988831	−	0.149042i	$-0.952381\pi$
$491$	−1.97766			−1.97766			−0.988831	+	0.149042i	$0.952381\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−0.268680	−	0.684585i	−0.268680	−	0.684585i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−1.40097	−	0.432142i	−1.40097	−	0.432142i	−0.500000	−	0.866025i	$-0.666667\pi$
$499$	−1.40097	−	0.432142i	−1.40097	−	0.432142i	−0.900969	+	0.433884i	$0.857143\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−0.162592	+	0.712362i	−0.162592	+	0.712362i	0.826239	+	0.563320i	$0.190476\pi$
$503$	−0.162592	+	0.712362i	−0.162592	+	0.712362i	−0.988831	+	0.149042i	$0.952381\pi$
$504$		0			0
$505$	−0.268680	−	1.17716i	−0.268680	−	1.17716i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$	0.149460	+	1.99441i	0.149460	+	1.99441i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	0.500000	+	0.240787i	0.500000	+	0.240787i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$		0			0
$523$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$523$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.969059	−	2.46912i	0.969059	−	2.46912i
$530$		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
$539$	0.222521	+	0.385418i	0.222521	+	0.385418i
$540$		0			0
$541$	−1.21135	−	1.12397i	−1.21135	−	1.12397i	−0.988831	−	0.149042i	$-0.952381\pi$
$541$	−1.21135	−	1.12397i	−1.21135	−	1.12397i	−0.222521	−	0.974928i	$-0.571429\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$547$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$548$		0			0
$549$	−0.162592	+	0.414278i	−0.162592	+	0.414278i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	0.733052	+	1.26968i	0.733052	+	1.26968i	0.955573	+	0.294755i	$0.0952381\pi$
$557$	0.733052	+	1.26968i	0.733052	+	1.26968i	−0.222521	+	0.974928i	$0.571429\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$563$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.955573	−	0.294755i	0.955573	−	0.294755i
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$	−0.826239	−	0.563320i	−0.826239	−	0.563320i	0.0747301	−	0.997204i	$-0.476190\pi$
$571$	−0.826239	−	0.563320i	−0.826239	−	0.563320i	−0.900969	+	0.433884i	$0.857143\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.736007	+	3.22466i	−0.736007	+	3.22466i
$576$		0			0
$577$	1.19158	−	0.367554i	1.19158	−	0.367554i	0.365341	−	0.930874i	$-0.380952\pi$
$577$	1.19158	−	0.367554i	1.19158	−	0.367554i	0.826239	+	0.563320i	$0.190476\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1.48883	−	0.716983i	−1.48883	−	0.716983i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.97766			−1.97766			−0.988831	−	0.149042i	$-0.952381\pi$
$587$	−1.97766			−1.97766			−0.988831	+	0.149042i	$0.952381\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.603718	+	1.53825i	0.603718	+	1.53825i	0.826239	+	0.563320i	$0.190476\pi$
$593$	0.603718	+	1.53825i	0.603718	+	1.53825i	−0.222521	+	0.974928i	$0.571429\pi$
$594$		0			0
$595$	0.153989	+	0.193096i	0.153989	+	0.193096i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$599$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$600$		0			0
$601$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$601$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.09492	−	0.746503i	−1.09492	−	0.746503i
$606$		0			0
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1.07473	−	0.997204i	1.07473	−	0.997204i	0.0747301	−	0.997204i	$-0.476190\pi$
$613$	1.07473	−	0.997204i	1.07473	−	0.997204i	1.00000			$0$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.48883	+	0.716983i	−1.48883	+	0.716983i	−0.988831	−	0.149042i	$-0.952381\pi$
$617$	−1.48883	+	0.716983i	−1.48883	+	0.716983i	−0.500000	+	0.866025i	$0.666667\pi$
$618$		0			0
$619$	−0.365341	−	0.632789i	−0.365341	−	0.632789i	0.623490	−	0.781831i	$-0.285714\pi$
$619$	−0.365341	−	0.632789i	−0.365341	−	0.632789i	−0.988831	+	0.149042i	$0.952381\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.0197720	−	0.263838i	0.0197720	−	0.263838i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−1.23305	−	1.54620i	−1.23305	−	1.54620i	−0.733052	−	0.680173i	$-0.761905\pi$
$631$	−1.23305	−	1.54620i	−1.23305	−	1.54620i	−0.500000	−	0.866025i	$-0.666667\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$641$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$642$		0			0
$643$	−0.623490	−	0.781831i	−0.623490	−	0.781831i	0.365341	−	0.930874i	$-0.380952\pi$
$643$	−0.623490	−	0.781831i	−0.623490	−	0.781831i	−0.988831	+	0.149042i	$0.952381\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.698220	−	1.77904i	0.698220	−	1.77904i	0.0747301	−	0.997204i	$-0.476190\pi$
$647$	0.698220	−	1.77904i	0.698220	−	1.77904i	0.623490	−	0.781831i	$-0.285714\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.95557	−	0.294755i	1.95557	−	0.294755i	0.955573	−	0.294755i	$-0.0952381\pi$
$653$	1.95557	−	0.294755i	1.95557	−	0.294755i	1.00000			$0$
$654$		0			0
$655$	1.21135	+	2.09812i	1.21135	+	2.09812i
$656$		0			0
$657$	−1.80194	+	0.867767i	−1.80194	+	0.867767i
$658$		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			0		0.733052	−	0.680173i	$-0.238095\pi$
$661$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.21135	+	1.12397i	−1.21135	+	1.12397i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−0.0440730	−	0.193096i	−0.0440730	−	0.193096i
$672$		0			0
$673$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$673$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$677$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$683$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$684$		0			0
$685$	3.30496			3.30496
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.603718	+	1.53825i	0.603718	+	1.53825i	0.826239	+	0.563320i	$0.190476\pi$
$691$	0.603718	+	1.53825i	0.603718	+	1.53825i	−0.222521	+	0.974928i	$0.571429\pi$
$692$		0			0
$693$	−0.277479	+	0.347948i	−0.277479	+	0.347948i
$694$		0			0
$695$	−2.84537	−	0.877681i	−2.84537	−	0.877681i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−0.162592	−	0.712362i	−0.162592	−	0.712362i	−0.988831	−	0.149042i	$-0.952381\pi$
$701$	−0.162592	−	0.712362i	−0.162592	−	0.712362i	0.826239	−	0.563320i	$-0.190476\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−0.535628	+	0.496990i	−0.535628	+	0.496990i
$708$		0			0
$709$	−1.63402	+	1.11406i	−1.63402	+	1.11406i	−0.733052	+	0.680173i	$0.761905\pi$
$709$	−1.63402	+	1.11406i	−1.63402	+	1.11406i	−0.900969	+	0.433884i	$0.857143\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	0.988831	−	0.149042i	0.988831	−	0.149042i	0.365341	−	0.930874i	$-0.380952\pi$
$719$	0.988831	−	0.149042i	0.988831	−	0.149042i	0.623490	+	0.781831i	$0.285714\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−0.914101	+	1.14625i	−0.914101	+	1.14625i	0.0747301	+	0.997204i	$0.476190\pi$
$727$	−0.914101	+	1.14625i	−0.914101	+	1.14625i	−0.988831	+	0.149042i	$0.952381\pi$
$728$		0			0
$729$	0.623490	+	0.781831i	0.623490	+	0.781831i
$730$		0			0
$731$	0.00166935	+	0.0222759i	0.00166935	+	0.0222759i
$732$		0			0
$733$	1.32091	+	1.22563i	1.32091	+	1.22563i	0.955573	+	0.294755i	$0.0952381\pi$
$733$	1.32091	+	1.22563i	1.32091	+	1.22563i	0.365341	+	0.930874i	$0.380952\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−0.109562	−	1.46200i	−0.109562	−	1.46200i	−0.733052	−	0.680173i	$-0.761905\pi$
$739$	−0.109562	−	1.46200i	−0.109562	−	1.46200i	0.623490	−	0.781831i	$-0.285714\pi$
$740$		0			0
$741$		0			0
$742$		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
$745$	−1.19395	+	3.04213i	−1.19395	+	3.04213i
$746$		0			0
$747$	0.123490	−	1.64786i	0.123490	−	1.64786i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$751$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.400969	+	0.193096i	0.400969	+	0.193096i	0.623490	−	0.781831i	$-0.285714\pi$
$757$	0.400969	+	0.193096i	0.400969	+	0.193096i	−0.222521	+	0.974928i	$0.571429\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.57906	−	1.07659i	1.57906	−	1.07659i	0.623490	−	0.781831i	$-0.285714\pi$
$761$	1.57906	−	1.07659i	1.57906	−	1.07659i	0.955573	−	0.294755i	$-0.0952381\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−0.123490	+	0.213891i	−0.123490	+	0.213891i
$766$		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		0.955573	−	0.294755i	$-0.0952381\pi$
$773$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.65248			−1.65248
$786$		0			0
$787$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$787$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		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
$799$	−0.0414721	−	0.181701i	−0.0414721	−	0.181701i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	0.445042	−	0.770835i	0.445042	−	0.770835i
$804$		0			0
$805$	3.01782	−	0.930874i	3.01782	−	0.930874i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.733052	−	0.680173i	0.733052	−	0.680173i	−0.222521	−	0.974928i	$-0.571429\pi$
$809$	0.733052	−	0.680173i	0.733052	−	0.680173i	0.955573	+	0.294755i	$0.0952381\pi$
$810$		0			0
$811$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$811$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−0.603718	−	1.04567i	−0.603718	−	1.04567i
$816$		0			0
$817$	−0.147791	+	0.0222759i	−0.147791	+	0.0222759i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	0.0931869	−	1.24349i	0.0931869	−	1.24349i	−0.733052	−	0.680173i	$-0.761905\pi$
$821$	0.0931869	−	1.24349i	0.0931869	−	1.24349i	0.826239	−	0.563320i	$-0.190476\pi$
$822$		0			0
$823$	0.455573	−	1.16078i	0.455573	−	1.16078i	−0.500000	−	0.866025i	$-0.666667\pi$
$823$	0.455573	−	1.16078i	0.455573	−	1.16078i	0.955573	−	0.294755i	$-0.0952381\pi$
$824$		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
$829$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$829$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.0546039	−	0.139129i	0.0546039	−	0.139129i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		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
$845$	0.123490	−	1.64786i	0.123490	−	1.64786i
$846$		0			0
$847$	−0.0599289	+	0.799695i	−0.0599289	+	0.799695i
$848$		0			0
$849$		0			0
$850$		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
$855$	−1.48883	−	0.716983i	−1.48883	−	0.716983i
$856$		0			0
$857$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$857$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$858$		0			0
$859$	−0.826239	+	0.563320i	−0.826239	+	0.563320i	−0.900969	−	0.433884i	$-0.857143\pi$
$859$	−0.826239	+	0.563320i	−0.826239	+	0.563320i	0.0747301	+	0.997204i	$0.476190\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.08786	+	0.523887i	−1.08786	+	0.523887i
$876$		0			0
$877$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$877$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.91115			1.91115			0.955573	−	0.294755i	$-0.0952381\pi$
$881$	1.91115			1.91115			0.955573	+	0.294755i	$0.0952381\pi$
$882$		0			0
$883$	1.65248			1.65248			0.826239	−	0.563320i	$-0.190476\pi$
$883$	1.65248			1.65248			0.826239	+	0.563320i	$0.190476\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$887$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−0.425270	−	0.131178i	−0.425270	−	0.131178i
$892$		0			0
$893$	1.19158	−	0.367554i	1.19158	−	0.367554i
$894$		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			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$907$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$908$		0			0
$909$	−0.658322	−	0.317031i	−0.658322	−	0.317031i
$910$		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
$913$	0.367711	+	0.636894i	0.367711	+	0.636894i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	0.733052	−	1.26968i	0.733052	−	1.26968i
$918$		0			0
$919$	0.0111692	−	0.149042i	0.0111692	−	0.149042i	−0.988831	−	0.149042i	$-0.952381\pi$
$919$	0.0111692	−	0.149042i	0.0111692	−	0.149042i	1.00000			$0$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.07473	+	0.997204i	1.07473	+	0.997204i	1.00000			$0$
$929$	1.07473	+	0.997204i	1.07473	+	0.997204i	0.0747301	+	0.997204i	$0.476190\pi$
$930$		0			0
$931$	0.955573	+	0.294755i	0.955573	+	0.294755i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−0.00821405	−	0.109609i	−0.00821405	−	0.109609i
$936$		0			0
$937$	−1.23305	−	1.54620i	−1.23305	−	1.54620i	−0.733052	−	0.680173i	$-0.761905\pi$
$937$	−1.23305	−	1.54620i	−1.23305	−	1.54620i	−0.500000	−	0.866025i	$-0.666667\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$941$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.440071	−	0.0663300i	0.440071	−	0.0663300i	0.0747301	−	0.997204i	$-0.476190\pi$
$947$	0.440071	−	0.0663300i	0.440071	−	0.0663300i	0.365341	+	0.930874i	$0.380952\pi$
$948$		0			0
$949$		0			0
$950$		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
$955$	1.21135	−	1.12397i	1.21135	−	1.12397i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$960$		0			0
$961$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−0.277479	+	1.21572i	−0.277479	+	1.21572i	0.623490	+	0.781831i	$0.285714\pi$
$967$	−0.277479	+	1.21572i	−0.277479	+	1.21572i	−0.900969	+	0.433884i	$0.857143\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$971$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$972$		0			0
$973$	0.400969	+	1.75676i	0.400969	+	1.75676i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$977$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$983$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$984$		0			0
$985$	0.0902318	+	0.229907i	0.0902318	+	0.229907i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	0.272950	+	0.0841939i	0.272950	+	0.0841939i
$990$		0			0
$991$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$991$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−0.0549581	−	0.240787i	−0.0549581	−	0.240787i
$996$		0			0
$997$	0.603718	+	0.411608i	0.603718	+	0.411608i	0.826239	−	0.563320i	$-0.190476\pi$
$997$	0.603718	+	0.411608i	0.603718	+	0.411608i	−0.222521	+	0.974928i	$0.571429\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3724.1.dj.a.2165.1	yes	12
19.18	odd	2	CM	3724.1.dj.a.2165.1	yes	12
49.11	even	21	inner	3724.1.dj.a.1481.1	✓	12
931.550	odd	42	inner	3724.1.dj.a.1481.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3724.1.dj.a.1481.1	✓	12	49.11	even	21	inner
3724.1.dj.a.1481.1	✓	12	931.550	odd	42	inner
3724.1.dj.a.2165.1	yes	12	1.1	even	1	trivial
3724.1.dj.a.2165.1	yes	12	19.18	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3724 = 2^{2} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3724.dj (of order \(42\), degree \(12\), minimal)

\(f(q)\)	\(=\)	\(q+(0.603718 + 1.53825i) q^{5} +(0.623490 - 0.781831i) q^{7} +(0.955573 + 0.294755i) q^{9} +(-0.425270 + 0.131178i) q^{11} +(0.123490 + 0.0841939i) q^{17} +(-0.500000 + 0.866025i) q^{19} +(1.57906 - 1.07659i) q^{23} +(-1.26868 + 1.17716i) q^{25} +(1.57906 + 0.487076i) q^{35} +(0.0931869 + 0.116853i) q^{43} +(0.123490 + 1.64786i) q^{45} +(-0.914101 - 0.848162i) q^{47} +(-0.222521 - 0.974928i) q^{49} +(-0.458528 - 0.574976i) q^{55} +(-0.0332580 + 0.443797i) q^{61} +(0.826239 - 0.563320i) q^{63} +(-1.46610 + 1.36035i) q^{73} +(-0.162592 + 0.414278i) q^{77} +(0.826239 + 0.563320i) q^{81} +(-0.367711 - 1.61105i) q^{83} +(-0.0549581 + 0.240787i) q^{85} +(-1.63402 - 0.246289i) q^{95} -0.445042 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{5} - 2 q^{7} + q^{9} - 5 q^{11} - 8 q^{17} - 6 q^{19} - q^{23} - q^{35} + 2 q^{43} - 8 q^{45} + 2 q^{47} - 2 q^{49} - 3 q^{55} + 2 q^{61} + q^{63} + 2 q^{73} + 2 q^{77} + q^{81} + 2 q^{83}+ \cdots - 4 q^{99}+O(q^{100}) \) 12 * q - q^5 - 2 * q^7 + q^9 - 5 * q^11 - 8 * q^17 - 6 * q^19 - q^23 - q^35 + 2 * q^43 - 8 * q^45 + 2 * q^47 - 2 * q^49 - 3 * q^55 + 2 * q^61 + q^63 + 2 * q^73 + 2 * q^77 + q^81 + 2 * q^83 - 2 * q^85 - q^95 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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