LMFDB - Embedded newform 1260.2.s.h.361.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1260,2,Mod(361,1260)] 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("1260.361"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1260, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1260.s (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,3,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$10.0611506547$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.4406832.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 6x^{4} + 7x^{3} + 24x^{2} + 5x + 1 $ x^6 - x^5 + 6x^4 + 7x^3 + 24x^2 + 5x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.3
Root			$-0.827721 - 1.43366i$ of defining polynomial
Character	$\chi$	$=$	1260.361
Dual form			1260.2.s.h.541.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{5} +(2.35341 - 1.20891i) q^{7} +(2.85341 - 4.94225i) q^{11} -6.70682 q^{13} +(1.72365 - 2.98545i) q^{17} +(-0.629755 - 1.09077i) q^{19} +(-3.98316 - 6.89904i) q^{23} +(-0.500000 + 0.866025i) q^{25} +7.18780 q^{29} +(0.500000 - 0.866025i) q^{31} +(2.22365 + 1.43366i) q^{35} +(4.48316 + 7.76507i) q^{37} -10.2258 q^{41} -4.44731 q^{43} +(0.740489 + 1.28257i) q^{47} +(4.07706 - 5.69013i) q^{49} +(3.44731 - 5.97091i) q^{53} +5.70682 q^{55} +(-0.146592 + 0.253904i) q^{59} +(0.129755 + 0.224743i) q^{61} +(-3.35341 - 5.80827i) q^{65} +(3.35341 - 5.80827i) q^{67} +11.7068 q^{71} +(-0.483164 + 0.836864i) q^{73} +(0.740489 - 15.0806i) q^{77} +(5.81755 + 10.0763i) q^{79} +14.8609 q^{83} +3.44731 q^{85} +(-6.30071 - 10.9132i) q^{89} +(-15.7839 + 8.10795i) q^{91} +(0.629755 - 1.09077i) q^{95} -13.1541 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{5} - q^{7} + 2 q^{11} - 10 q^{13} - 2 q^{17} - q^{19} - 6 q^{23} - 3 q^{25} + 24 q^{29} + 3 q^{31} + q^{35} + 9 q^{37} - 20 q^{41} - 2 q^{43} + 10 q^{47} - 3 q^{49} - 4 q^{53} + 4 q^{55} - 16 q^{59}+ \cdots - 24 q^{97}+O(q^{100}) $ 6 * q + 3 * q^5 - q^7 + 2 * q^11 - 10 * q^13 - 2 * q^17 - q^19 - 6 * q^23 - 3 * q^25 + 24 * q^29 + 3 * q^31 + q^35 + 9 * q^37 - 20 * q^41 - 2 * q^43 + 10 * q^47 - 3 * q^49 - 4 * q^53 + 4 * q^55 - 16 * q^59 - 2 * q^61 - 5 * q^65 + 5 * q^67 + 40 * q^71 + 15 * q^73 + 10 * q^77 + 13 * q^79 + 4 * q^83 - 4 * q^85 + 2 * q^89 - 37 * q^91 + q^95 - 24 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times$.

$n$	$281$	$631$	$757$	$1081$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display 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
$3$		0			0
$4$		0			0
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$6$		0			0
$7$	2.35341	−	1.20891i	0.889505	−	0.456926i
$7$	2.35341	−	1.20891i	0.889505	−	0.456926i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.85341	−	4.94225i	0.860335	−	1.49014i	−0.0112708	−	0.999936i	$-0.503588\pi$
$11$	2.85341	−	4.94225i	0.860335	−	1.49014i	0.871606	−	0.490207i	$-0.163079\pi$
$12$		0			0
$13$	−6.70682			−1.86014			−0.930068	−	0.367387i	$-0.880252\pi$
$13$	−6.70682			−1.86014			−0.930068	+	0.367387i	$0.880252\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.72365	−	2.98545i	0.418047	−	0.724079i	−0.577696	−	0.816252i	$-0.696048\pi$
$17$	1.72365	−	2.98545i	0.418047	−	0.724079i	0.995743	+	0.0921731i	$0.0293813\pi$
$18$		0			0
$19$	−0.629755	−	1.09077i	−0.144476	−	0.250239i	0.784701	−	0.619874i	$-0.212816\pi$
$19$	−0.629755	−	1.09077i	−0.144476	−	0.250239i	−0.929177	+	0.369634i	$0.879483\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.98316	−	6.89904i	−0.830547	−	1.43855i	−0.897605	−	0.440801i	$-0.854695\pi$
$23$	−3.98316	−	6.89904i	−0.830547	−	1.43855i	0.0670581	−	0.997749i	$-0.478639\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	7.18780			1.33474			0.667370	−	0.744726i	$-0.267420\pi$
$29$	7.18780			1.33474			0.667370	+	0.744726i	$0.267420\pi$
$30$		0			0
$31$	0.500000	−	0.866025i	0.0898027	−	0.155543i	−0.817625	−	0.575751i	$-0.804710\pi$
$31$	0.500000	−	0.866025i	0.0898027	−	0.155543i	0.907428	+	0.420208i	$0.138043\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.22365	+	1.43366i	0.375866	+	0.242332i
$36$		0			0
$37$	4.48316	+	7.76507i	0.737028	+	1.27657i	0.953828	+	0.300353i	$0.0971046\pi$
$37$	4.48316	+	7.76507i	0.737028	+	1.27657i	−0.216800	+	0.976216i	$0.569562\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−10.2258			−1.59701			−0.798504	−	0.601990i	$-0.794375\pi$
$41$	−10.2258			−1.59701			−0.798504	+	0.601990i	$0.794375\pi$
$42$		0			0
$43$	−4.44731			−0.678208			−0.339104	−	0.940749i	$-0.610124\pi$
$43$	−4.44731			−0.678208			−0.339104	+	0.940749i	$0.610124\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.740489	+	1.28257i	0.108011	+	0.187081i	0.914965	−	0.403534i	$-0.132218\pi$
$47$	0.740489	+	1.28257i	0.108011	+	0.187081i	−0.806953	+	0.590616i	$0.798885\pi$
$48$		0			0
$49$	4.07706	−	5.69013i	0.582437	−	0.812876i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.44731	−	5.97091i	0.473524	−	0.820167i	−0.526017	−	0.850474i	$-0.676315\pi$
$53$	3.44731	−	5.97091i	0.473524	−	0.820167i	0.999541	+	0.0303067i	$0.00964841\pi$
$54$		0			0
$55$	5.70682			0.769507
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.146592	+	0.253904i	−0.0190846	+	0.0330555i	−0.875410	−	0.483381i	$-0.839409\pi$
$59$	−0.146592	+	0.253904i	−0.0190846	+	0.0330555i	0.856325	+	0.516437i	$0.172742\pi$
$60$		0			0
$61$	0.129755	+	0.224743i	0.0166135	+	0.0287754i	0.874213	−	0.485543i	$-0.161378\pi$
$61$	0.129755	+	0.224743i	0.0166135	+	0.0287754i	−0.857599	+	0.514319i	$0.828045\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.35341	−	5.80827i	−0.415939	−	0.720428i
$66$		0			0
$67$	3.35341	−	5.80827i	0.409684	−	0.709594i	−0.585170	−	0.810911i	$-0.698972\pi$
$67$	3.35341	−	5.80827i	0.409684	−	0.709594i	0.994854	+	0.101317i	$0.0323056\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	11.7068			1.38934			0.694672	−	0.719327i	$-0.255550\pi$
$71$	11.7068			1.38934			0.694672	+	0.719327i	$0.255550\pi$
$72$		0			0
$73$	−0.483164	+	0.836864i	−0.0565500	+	0.0979475i	−0.892915	−	0.450226i	$-0.851343\pi$
$73$	−0.483164	+	0.836864i	−0.0565500	+	0.0979475i	0.836365	+	0.548174i	$0.184677\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.740489	−	15.0806i	0.0843866	−	1.71860i
$78$		0			0
$79$	5.81755	+	10.0763i	0.654526	+	1.13367i	0.982013	+	0.188816i	$0.0604649\pi$
$79$	5.81755	+	10.0763i	0.654526	+	1.13367i	−0.327487	+	0.944856i	$0.606202\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	14.8609			1.63120			0.815600	−	0.578616i	$-0.196407\pi$
$83$	14.8609			1.63120			0.815600	+	0.578616i	$0.196407\pi$
$84$		0			0
$85$	3.44731			0.373913
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.30071	−	10.9132i	−0.667874	−	1.15679i	−0.978497	−	0.206259i	$-0.933871\pi$
$89$	−6.30071	−	10.9132i	−0.667874	−	1.15679i	0.310623	−	0.950533i	$-0.399462\pi$
$90$		0			0
$91$	−15.7839	+	8.10795i	−1.65460	+	0.849945i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.629755	−	1.09077i	0.0646115	−	0.111910i
$96$		0			0
$97$	−13.1541			−1.33560			−0.667799	−	0.744341i	$-0.732764\pi$
$97$	−13.1541			−1.33560			−0.667799	+	0.744341i	$0.732764\pi$
$98$		0			0
$99$		0			0

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1260.2.s.h.361.3	yes	6
3.2	odd	2		1260.2.s.g.361.3	✓	6
7.2	even	3	inner	1260.2.s.h.541.3	yes	6
7.3	odd	6		8820.2.a.bp.1.1		3
7.4	even	3		8820.2.a.bn.1.1		3
21.2	odd	6		1260.2.s.g.541.3	yes	6
21.11	odd	6		8820.2.a.bq.1.3		3
21.17	even	6		8820.2.a.bo.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1260.2.s.g.361.3	✓	6	3.2	odd	2
1260.2.s.g.541.3	yes	6	21.2	odd	6
1260.2.s.h.361.3	yes	6	1.1	even	1	trivial
1260.2.s.h.541.3	yes	6	7.2	even	3	inner
8820.2.a.bn.1.1		3	7.4	even	3
8820.2.a.bo.1.3		3	21.17	even	6
8820.2.a.bp.1.1		3	7.3	odd	6
8820.2.a.bq.1.3		3	21.11	odd	6

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:	\( N \)	\(=\)	\( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1260.s (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{5} +(2.35341 - 1.20891i) q^{7} +(2.85341 - 4.94225i) q^{11} -6.70682 q^{13} +(1.72365 - 2.98545i) q^{17} +(-0.629755 - 1.09077i) q^{19} +(-3.98316 - 6.89904i) q^{23} +(-0.500000 + 0.866025i) q^{25} +7.18780 q^{29} +(0.500000 - 0.866025i) q^{31} +(2.22365 + 1.43366i) q^{35} +(4.48316 + 7.76507i) q^{37} -10.2258 q^{41} -4.44731 q^{43} +(0.740489 + 1.28257i) q^{47} +(4.07706 - 5.69013i) q^{49} +(3.44731 - 5.97091i) q^{53} +5.70682 q^{55} +(-0.146592 + 0.253904i) q^{59} +(0.129755 + 0.224743i) q^{61} +(-3.35341 - 5.80827i) q^{65} +(3.35341 - 5.80827i) q^{67} +11.7068 q^{71} +(-0.483164 + 0.836864i) q^{73} +(0.740489 - 15.0806i) q^{77} +(5.81755 + 10.0763i) q^{79} +14.8609 q^{83} +3.44731 q^{85} +(-6.30071 - 10.9132i) q^{89} +(-15.7839 + 8.10795i) q^{91} +(0.629755 - 1.09077i) q^{95} -13.1541 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{5} - q^{7} + 2 q^{11} - 10 q^{13} - 2 q^{17} - q^{19} - 6 q^{23} - 3 q^{25} + 24 q^{29} + 3 q^{31} + q^{35} + 9 q^{37} - 20 q^{41} - 2 q^{43} + 10 q^{47} - 3 q^{49} - 4 q^{53} + 4 q^{55} - 16 q^{59}+ \cdots - 24 q^{97}+O(q^{100}) \) 6 * q + 3 * q^5 - q^7 + 2 * q^11 - 10 * q^13 - 2 * q^17 - q^19 - 6 * q^23 - 3 * q^25 + 24 * q^29 + 3 * q^31 + q^35 + 9 * q^37 - 20 * q^41 - 2 * q^43 + 10 * q^47 - 3 * q^49 - 4 * q^53 + 4 * q^55 - 16 * q^59 - 2 * q^61 - 5 * q^65 + 5 * q^67 + 40 * q^71 + 15 * q^73 + 10 * q^77 + 13 * q^79 + 4 * q^83 - 4 * q^85 + 2 * q^89 - 37 * q^91 + q^95 - 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more