LMFDB - Embedded newform 432.7.q.b.17.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [432,7,Mod(17,432)] 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("432.17"); S:= CuspForms(chi, 7); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(432, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 5])) N = Newforms(chi, 7, names="a")

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	432.q (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,-432] 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:	$99.3833641238$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 370x^{10} + 51793x^{8} + 3491832x^{6} + 117603792x^{4} + 1832032512x^{2} + 10453017600 $ x^12 + 370x^10 + 51793x^8 + 3491832x^6 + 117603792x^4 + 1832032512*x^2 + 10453017600
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{20} $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.5
Root			$7.20150i$ of defining polynomial
Character	$\chi$	$=$	432.17
Dual form			432.7.q.b.305.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(39.5602 + 22.8401i) q^{5} +(245.097 + 424.521i) q^{7} +(-873.336 + 504.221i) q^{11} +(466.801 - 808.523i) q^{13} -8090.59i q^{17} +7727.36 q^{19} +(-11848.0 - 6840.45i) q^{23} +(-6769.16 - 11724.5i) q^{25} +(1964.70 - 1134.32i) q^{29} +(17062.6 - 29553.2i) q^{31} +22392.2i q^{35} +92058.0 q^{37} +(31021.6 + 17910.3i) q^{41} +(-34570.9 - 59878.5i) q^{43} +(13211.1 - 7627.46i) q^{47} +(-61321.0 + 106211. i) q^{49} +236591. i q^{53} -46065.9 q^{55} +(-221890. - 128108. i) q^{59} +(-19919.6 - 34501.8i) q^{61} +(36933.5 - 21323.6i) q^{65} +(160204. - 277482. i) q^{67} +404593. i q^{71} +393719. q^{73} +(-428105. - 247167. i) q^{77} +(449184. + 778010. i) q^{79} +(154916. - 89441.0i) q^{83} +(184790. - 320066. i) q^{85} -826458. i q^{89} +457647. q^{91} +(305696. + 176494. i) q^{95} +(-317981. - 550760. i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 432 q^{5} - 240 q^{7} + 378 q^{11} + 1680 q^{13} + 2820 q^{19} - 76248 q^{23} + 8094 q^{25} - 97092 q^{29} - 21480 q^{31} - 25536 q^{37} + 410562 q^{41} - 71430 q^{43} + 347652 q^{47} - 135954 q^{49}+ \cdots - 38874 q^{97}+O(q^{100}) $ 12 * q - 432 * q^5 - 240 * q^7 + 378 * q^11 + 1680 * q^13 + 2820 * q^19 - 76248 * q^23 + 8094 * q^25 - 97092 * q^29 - 21480 * q^31 - 25536 * q^37 + 410562 * q^41 - 71430 * q^43 + 347652 * q^47 - 135954 * q^49 - 580392 * q^55 + 369738 * q^59 + 135744 * q^61 + 753840 * q^65 + 289938 * q^67 - 977700 * q^73 + 159192 * q^77 + 764796 * q^79 + 396900 * q^83 + 1619568 * q^85 - 355584 * q^91 - 2089260 * q^95 - 38874 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

$n$	$271$	$325$	$353$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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
$5$	39.5602	+	22.8401i	0.316482	+	0.182721i	0.649823	−	0.760085i	$-0.274843\pi$
$5$	39.5602	+	22.8401i	0.316482	+	0.182721i	−0.333341	+	0.942806i	$0.608176\pi$
$6$		0			0
$7$	245.097	+	424.521i	0.714570	+	1.23767i	0.963125	+	0.269053i	$0.0867108\pi$
$7$	245.097	+	424.521i	0.714570	+	1.23767i	−0.248556	+	0.968618i	$0.579956\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−873.336	+	504.221i	−0.656151	+	0.378829i	−0.790809	−	0.612063i	$-0.790340\pi$
$11$	−873.336	+	504.221i	−0.656151	+	0.378829i	0.134658	+	0.990892i	$0.457006\pi$
$12$		0			0
$13$	466.801	−	808.523i	0.212472	−	0.368012i	−0.740016	−	0.672590i	$-0.765182\pi$
$13$	466.801	−	808.523i	0.212472	−	0.368012i	0.952488	+	0.304577i	$0.0985152\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	8090.59i		−	1.64677i	−0.567482	−	0.823386i	$-0.692082\pi$
$17$		−	8090.59i		−	1.64677i	0.567482	−	0.823386i	$-0.307918\pi$
$18$		0			0
$19$	7727.36			1.12660			0.563301	−	0.826252i	$-0.309531\pi$
$19$	7727.36			1.12660			0.563301	+	0.826252i	$0.309531\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−11848.0	−	6840.45i	−0.973782	−	0.562213i	−0.0733950	−	0.997303i	$-0.523383\pi$
$23$	−11848.0	−	6840.45i	−0.973782	−	0.562213i	−0.900387	+	0.435090i	$0.856717\pi$
$24$		0			0
$25$	−6769.16	−	11724.5i	−0.433226	−	0.750370i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1964.70	−	1134.32i	0.0805570	−	0.0465096i	−0.459180	−	0.888343i	$-0.651857\pi$
$29$	1964.70	−	1134.32i	0.0805570	−	0.0465096i	0.539737	+	0.841833i	$0.318524\pi$
$30$		0			0
$31$	17062.6	−	29553.2i	0.572742	−	0.992019i	−0.423541	−	0.905877i	$-0.639213\pi$
$31$	17062.6	−	29553.2i	0.572742	−	0.992019i	0.996283	−	0.0861417i	$-0.0274538\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			22392.2i			0.522267i
$36$		0			0
$37$	92058.0			1.81743			0.908713	−	0.417422i	$-0.137066\pi$
$37$	92058.0			1.81743			0.908713	+	0.417422i	$0.137066\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	31021.6	+	17910.3i	0.450103	+	0.259867i	0.707874	−	0.706339i	$-0.249655\pi$
$41$	31021.6	+	17910.3i	0.450103	+	0.259867i	−0.257771	+	0.966206i	$0.582988\pi$
$42$		0			0
$43$	−34570.9	−	59878.5i	−0.434816	−	0.753123i	0.562465	−	0.826821i	$-0.309853\pi$
$43$	−34570.9	−	59878.5i	−0.434816	−	0.753123i	−0.997281	+	0.0736985i	$0.976520\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	13211.1	−	7627.46i	0.127247	−	0.0734659i	−0.435025	−	0.900418i	$-0.643261\pi$
$47$	13211.1	−	7627.46i	0.127247	−	0.0734659i	0.562272	+	0.826952i	$0.309927\pi$
$48$		0			0
$49$	−61321.0	+	106211.i	−0.521220	+	0.902779i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			236591.i			1.58917i	0.607153	+	0.794585i	$0.292312\pi$
$53$			236591.i			1.58917i	−0.607153	+	0.794585i	$0.707688\pi$
$54$		0			0
$55$	−46065.9			−0.276880
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−221890.	−	128108.i	−1.08039	−	0.623766i	−0.149392	−	0.988778i	$-0.547732\pi$
$59$	−221890.	−	128108.i	−1.08039	−	0.623766i	−0.931003	+	0.365012i	$0.881065\pi$
$60$		0			0
$61$	−19919.6	−	34501.8i	−0.0877589	−	0.152003i	0.818805	−	0.574072i	$-0.194637\pi$
$61$	−19919.6	−	34501.8i	−0.0877589	−	0.152003i	−0.906563	+	0.422069i	$0.861304\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	36933.5	−	21323.6i	0.134487	−	0.0776462i
$66$		0			0
$67$	160204.	−	277482.i	0.532660	−	0.922593i	−0.466613	−	0.884461i	$-0.654526\pi$
$67$	160204.	−	277482.i	0.532660	−	0.922593i	0.999273	−	0.0381319i	$-0.0121407\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			404593.i			1.13043i	0.824944	+	0.565215i	$0.191207\pi$
$71$			404593.i			1.13043i	−0.824944	+	0.565215i	$0.808793\pi$
$72$		0			0
$73$	393719.			1.01209			0.506044	−	0.862508i	$-0.331107\pi$
$73$	393719.			1.01209			0.506044	+	0.862508i	$0.331107\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−428105.	−	247167.i	−0.937731	−	0.541399i
$78$		0			0
$79$	449184.	+	778010.i	0.911052	+	1.57799i	0.812581	+	0.582848i	$0.198062\pi$
$79$	449184.	+	778010.i	0.911052	+	1.57799i	0.0984709	+	0.995140i	$0.468605\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	154916.	−	89441.0i	0.270934	−	0.156424i	−0.358378	−	0.933576i	$-0.616670\pi$
$83$	154916.	−	89441.0i	0.270934	−	0.156424i	0.629312	+	0.777153i	$0.283337\pi$
$84$		0			0
$85$	184790.	−	320066.i	0.300900	−	0.521173i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	826458.i		−	1.17233i	−0.810191	−	0.586166i	$-0.800636\pi$
$89$		−	826458.i		−	1.17233i	0.810191	−	0.586166i	$-0.199364\pi$
$90$		0			0
$91$	457647.			0.607304
$92$		0			0
$93$		0			0
$94$		0			0
$95$	305696.	+	176494.i	0.356549	+	0.205854i
$96$		0			0
$97$	−317981.	−	550760.i	−0.348406	−	0.603458i	0.637560	−	0.770401i	$-0.279944\pi$
$97$	−317981.	−	550760.i	−0.348406	−	0.603458i	−0.985967	+	0.166943i	$0.946610\pi$
$98$		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	432.7.q.b.17.5		12
3.2	odd	2		144.7.q.c.113.3		12
4.3	odd	2		54.7.d.a.17.6		12
9.2	odd	6	inner	432.7.q.b.305.5		12
9.7	even	3		144.7.q.c.65.3		12
12.11	even	2		18.7.d.a.5.2	✓	12
36.7	odd	6		18.7.d.a.11.2	yes	12
36.11	even	6		54.7.d.a.35.6		12
36.23	even	6		162.7.b.c.161.5		12
36.31	odd	6		162.7.b.c.161.8		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.7.d.a.5.2	✓	12	12.11	even	2
18.7.d.a.11.2	yes	12	36.7	odd	6
54.7.d.a.17.6		12	4.3	odd	2
54.7.d.a.35.6		12	36.11	even	6
144.7.q.c.65.3		12	9.7	even	3
144.7.q.c.113.3		12	3.2	odd	2
162.7.b.c.161.5		12	36.23	even	6
162.7.b.c.161.8		12	36.31	odd	6
432.7.q.b.17.5		12	1.1	even	1	trivial
432.7.q.b.305.5		12	9.2	odd	6	inner

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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	432.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(39.5602 + 22.8401i) q^{5} +(245.097 + 424.521i) q^{7} +(-873.336 + 504.221i) q^{11} +(466.801 - 808.523i) q^{13} -8090.59i q^{17} +7727.36 q^{19} +(-11848.0 - 6840.45i) q^{23} +(-6769.16 - 11724.5i) q^{25} +(1964.70 - 1134.32i) q^{29} +(17062.6 - 29553.2i) q^{31} +22392.2i q^{35} +92058.0 q^{37} +(31021.6 + 17910.3i) q^{41} +(-34570.9 - 59878.5i) q^{43} +(13211.1 - 7627.46i) q^{47} +(-61321.0 + 106211. i) q^{49} +236591. i q^{53} -46065.9 q^{55} +(-221890. - 128108. i) q^{59} +(-19919.6 - 34501.8i) q^{61} +(36933.5 - 21323.6i) q^{65} +(160204. - 277482. i) q^{67} +404593. i q^{71} +393719. q^{73} +(-428105. - 247167. i) q^{77} +(449184. + 778010. i) q^{79} +(154916. - 89441.0i) q^{83} +(184790. - 320066. i) q^{85} -826458. i q^{89} +457647. q^{91} +(305696. + 176494. i) q^{95} +(-317981. - 550760. i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 432 q^{5} - 240 q^{7} + 378 q^{11} + 1680 q^{13} + 2820 q^{19} - 76248 q^{23} + 8094 q^{25} - 97092 q^{29} - 21480 q^{31} - 25536 q^{37} + 410562 q^{41} - 71430 q^{43} + 347652 q^{47} - 135954 q^{49}+ \cdots - 38874 q^{97}+O(q^{100}) \) 12 * q - 432 * q^5 - 240 * q^7 + 378 * q^11 + 1680 * q^13 + 2820 * q^19 - 76248 * q^23 + 8094 * q^25 - 97092 * q^29 - 21480 * q^31 - 25536 * q^37 + 410562 * q^41 - 71430 * q^43 + 347652 * q^47 - 135954 * q^49 - 580392 * q^55 + 369738 * q^59 + 135744 * q^61 + 753840 * q^65 + 289938 * q^67 - 977700 * q^73 + 159192 * q^77 + 764796 * q^79 + 396900 * q^83 + 1619568 * q^85 - 355584 * q^91 - 2089260 * q^95 - 38874 * q^97

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