LMFDB - Embedded newform 1400.2.bh.h.249.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1400,2,Mod(249,1400)] 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("1400.249"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1400, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1400 = 2^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1400.bh (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$11.1790562830$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{36})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{6} + 1 $ x^12 - x^6 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			249.4
Root			$0.342020 + 0.939693i$ of defining polynomial
Character	$\chi$	$=$	1400.249
Dual form			1400.2.bh.h.849.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.761570 + 0.439693i) q^{3} +(-2.27038 + 1.35844i) q^{7} +(-1.11334 - 1.92836i) q^{9} +(1.59240 - 2.75811i) q^{11} +0.490200i q^{13} +(-0.829748 - 0.479055i) q^{17} +(-1.05303 - 1.82391i) q^{19} +(-2.32635 + 0.0362770i) q^{21} +(5.44804 - 3.14543i) q^{23} -4.59627i q^{27} -0.588526 q^{29} +(3.73783 - 6.47410i) q^{31} +(2.42544 - 1.40033i) q^{33} +(1.30753 - 0.754900i) q^{37} +(-0.215537 + 0.373321i) q^{39} -4.29086 q^{41} -5.92127i q^{43} +(-4.47086 + 2.58125i) q^{47} +(3.30928 - 6.16836i) q^{49} +(-0.421274 - 0.729669i) q^{51} +(11.4125 + 6.58899i) q^{53} -1.85204i q^{57} +(2.31908 - 4.01676i) q^{59} +(-6.96451 - 12.0629i) q^{61} +(5.14728 + 2.86571i) q^{63} +(-6.30147 - 3.63816i) q^{67} +5.53209 q^{69} +12.1925 q^{71} +(8.77141 + 5.06418i) q^{73} +(0.131381 + 8.42514i) q^{77} +(-1.26604 - 2.19285i) q^{79} +(-1.31908 + 2.28471i) q^{81} -4.80066i q^{83} +(-0.448204 - 0.258770i) q^{87} +(-5.40420 - 9.36035i) q^{89} +(-0.665907 - 1.11294i) q^{91} +(5.69323 - 3.28699i) q^{93} +10.5544i q^{97} -7.09152 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{11} + 12 q^{19} - 30 q^{21} - 48 q^{29} + 6 q^{31} + 12 q^{39} + 12 q^{41} + 30 q^{51} - 6 q^{59} - 18 q^{61} + 48 q^{69} + 36 q^{71} - 6 q^{79} + 18 q^{81} + 12 q^{89} - 102 q^{91} + 36 q^{99}+O(q^{100}) $ 12 * q + 12 * q^11 + 12 * q^19 - 30 * q^21 - 48 * q^29 + 6 * q^31 + 12 * q^39 + 12 * q^41 + 30 * q^51 - 6 * q^59 - 18 * q^61 + 48 * q^69 + 36 * q^71 - 6 * q^79 + 18 * q^81 + 12 * q^89 - 102 * q^91 + 36 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times$.

$n$	$351$	$701$	$801$	$1177$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$. 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
$3$	0.761570	+	0.439693i	0.439693	+	0.253857i	0.703467	−	0.710728i	$-0.251634\pi$
$3$	0.761570	+	0.439693i	0.439693	+	0.253857i	−0.263775	+	0.964584i	$0.584968\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.27038	+	1.35844i	−0.858124	+	0.513442i
$7$	−2.27038	+	1.35844i	−0.858124	+	0.513442i
$8$		0			0
$9$	−1.11334	−	1.92836i	−0.371114	−	0.642788i
$10$		0			0
$11$	1.59240	−	2.75811i	0.480126	−	0.831602i	−0.519615	−	0.854401i	$-0.673924\pi$
$11$	1.59240	−	2.75811i	0.480126	−	0.831602i	0.999740	+	0.0227990i	$0.00725777\pi$
$12$		0			0
$13$			0.490200i			0.135957i	0.997687	+	0.0679785i	$0.0216549\pi$
$13$			0.490200i			0.135957i	−0.997687	+	0.0679785i	$0.978345\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.829748	−	0.479055i	−0.201244	−	0.116188i	0.395992	−	0.918254i	$-0.370401\pi$
$17$	−0.829748	−	0.479055i	−0.201244	−	0.116188i	−0.597235	+	0.802066i	$0.703734\pi$
$18$		0			0
$19$	−1.05303	−	1.82391i	−0.241582	−	0.418433i	0.719583	−	0.694407i	$-0.244333\pi$
$19$	−1.05303	−	1.82391i	−0.241582	−	0.418433i	−0.961165	+	0.275974i	$0.911000\pi$
$20$		0			0
$21$	−2.32635	+	0.0362770i	−0.507652	+	0.00791629i
$22$		0			0
$23$	5.44804	−	3.14543i	1.13600	−	0.655867i	0.190560	−	0.981676i	$-0.438970\pi$
$23$	5.44804	−	3.14543i	1.13600	−	0.655867i	0.945436	+	0.325808i	$0.105636\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	4.59627i		−	0.884552i
$28$		0			0
$29$	−0.588526			−0.109287			−0.0546433	−	0.998506i	$-0.517402\pi$
$29$	−0.588526			−0.109287			−0.0546433	+	0.998506i	$0.517402\pi$
$30$		0			0
$31$	3.73783	−	6.47410i	0.671333	−	1.16278i	−0.306193	−	0.951970i	$-0.599055\pi$
$31$	3.73783	−	6.47410i	0.671333	−	1.16278i	0.977526	−	0.210814i	$-0.0676114\pi$
$32$		0			0
$33$	2.42544	−	1.40033i	0.422215	−	0.243766i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.30753	−	0.754900i	0.214956	−	0.124105i	−0.388657	−	0.921383i	$-0.627061\pi$
$37$	1.30753	−	0.754900i	0.214956	−	0.124105i	0.603612	+	0.797278i	$0.293727\pi$
$38$		0			0
$39$	−0.215537	+	0.373321i	−0.0345136	+	0.0597793i
$40$		0			0
$41$	−4.29086			−0.670120			−0.335060	−	0.942197i	$-0.608757\pi$
$41$	−4.29086			−0.670120			−0.335060	+	0.942197i	$0.608757\pi$
$42$		0			0
$43$		−	5.92127i		−	0.902986i	−0.892275	−	0.451493i	$-0.850892\pi$
$43$		−	5.92127i		−	0.902986i	0.892275	−	0.451493i	$-0.149108\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.47086	+	2.58125i	−0.652142	+	0.376514i	−0.789276	−	0.614038i	$-0.789544\pi$
$47$	−4.47086	+	2.58125i	−0.652142	+	0.376514i	0.137134	+	0.990552i	$0.456211\pi$
$48$		0			0
$49$	3.30928	−	6.16836i	0.472754	−	0.881194i
$50$		0			0
$51$	−0.421274	−	0.729669i	−0.0589902	−	0.102174i
$52$		0			0
$53$	11.4125	+	6.58899i	1.56762	+	0.905068i	0.996445	+	0.0842415i	$0.0268467\pi$
$53$	11.4125	+	6.58899i	1.56762	+	0.905068i	0.571178	+	0.820826i	$0.306487\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	1.85204i		−	0.245309i
$58$		0			0
$59$	2.31908	−	4.01676i	0.301918	−	0.522938i	−0.674652	−	0.738136i	$-0.735706\pi$
$59$	2.31908	−	4.01676i	0.301918	−	0.522938i	0.976570	+	0.215198i	$0.0690397\pi$
$60$		0			0
$61$	−6.96451	−	12.0629i	−0.891714	−	1.54449i	−0.837820	−	0.545947i	$-0.816170\pi$
$61$	−6.96451	−	12.0629i	−0.891714	−	1.54449i	−0.0538941	−	0.998547i	$-0.517163\pi$
$62$		0			0
$63$	5.14728	+	2.86571i	0.648496	+	0.361046i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−6.30147	−	3.63816i	−0.769847	−	0.444471i	0.0629729	−	0.998015i	$-0.479942\pi$
$67$	−6.30147	−	3.63816i	−0.769847	−	0.444471i	−0.832820	+	0.553544i	$0.813275\pi$
$68$		0			0
$69$	5.53209			0.665985
$70$		0			0
$71$	12.1925			1.44699			0.723494	−	0.690331i	$-0.242535\pi$
$71$	12.1925			1.44699			0.723494	+	0.690331i	$0.242535\pi$
$72$		0			0
$73$	8.77141	+	5.06418i	1.02662	+	0.592717i	0.916013	−	0.401148i	$-0.131389\pi$
$73$	8.77141	+	5.06418i	1.02662	+	0.592717i	0.110603	+	0.993865i	$0.464722\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.131381	+	8.42514i	0.0149723	+	0.960134i
$78$		0			0
$79$	−1.26604	−	2.19285i	−0.142441	−	0.246715i	0.785974	−	0.618259i	$-0.212162\pi$
$79$	−1.26604	−	2.19285i	−0.142441	−	0.246715i	−0.928415	+	0.371544i	$0.878828\pi$
$80$		0			0
$81$	−1.31908	+	2.28471i	−0.146564	+	0.253857i
$82$		0			0
$83$		−	4.80066i		−	0.526941i	−0.964667	−	0.263470i	$-0.915133\pi$
$83$		−	4.80066i		−	0.526941i	0.964667	−	0.263470i	$-0.0848671\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−0.448204	−	0.258770i	−0.0480525	−	0.0277431i
$88$		0			0
$89$	−5.40420	−	9.36035i	−0.572844	−	0.992195i	−0.996272	−	0.0862653i	$-0.972507\pi$
$89$	−5.40420	−	9.36035i	−0.572844	−	0.992195i	0.423428	−	0.905930i	$-0.360827\pi$
$90$		0			0
$91$	−0.665907	−	1.11294i	−0.0698061	−	0.116668i
$92$		0			0
$93$	5.69323	−	3.28699i	0.590361	−	0.340845i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			10.5544i			1.07163i	0.844334	+	0.535817i	$0.179996\pi$
$97$			10.5544i			1.07163i	−0.844334	+	0.535817i	$0.820004\pi$
$98$		0			0
$99$	−7.09152			−0.712724

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	1400.2.bh.h.249.4		12
5.2	odd	4		1400.2.q.i.1201.3	yes	6
5.3	odd	4		1400.2.q.k.1201.1	yes	6
5.4	even	2	inner	1400.2.bh.h.249.3		12
7.2	even	3	inner	1400.2.bh.h.849.3		12
35.2	odd	12		1400.2.q.i.401.3	✓	6
35.3	even	12		9800.2.a.ch.1.1		3
35.9	even	6	inner	1400.2.bh.h.849.4		12
35.17	even	12		9800.2.a.cb.1.3		3
35.18	odd	12		9800.2.a.cc.1.3		3
35.23	odd	12		1400.2.q.k.401.1	yes	6
35.32	odd	12		9800.2.a.ci.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1400.2.q.i.401.3	✓	6	35.2	odd	12
1400.2.q.i.1201.3	yes	6	5.2	odd	4
1400.2.q.k.401.1	yes	6	35.23	odd	12
1400.2.q.k.1201.1	yes	6	5.3	odd	4
1400.2.bh.h.249.3		12	5.4	even	2	inner
1400.2.bh.h.249.4		12	1.1	even	1	trivial
1400.2.bh.h.849.3		12	7.2	even	3	inner
1400.2.bh.h.849.4		12	35.9	even	6	inner
9800.2.a.cb.1.3		3	35.17	even	12
9800.2.a.cc.1.3		3	35.18	odd	12
9800.2.a.ch.1.1		3	35.3	even	12
9800.2.a.ci.1.1		3	35.32	odd	12

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Higher genus families
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Belyi maps

Level:	\( N \)	\(=\)	\( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1400.bh (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.761570 + 0.439693i) q^{3} +(-2.27038 + 1.35844i) q^{7} +(-1.11334 - 1.92836i) q^{9} +(1.59240 - 2.75811i) q^{11} +0.490200i q^{13} +(-0.829748 - 0.479055i) q^{17} +(-1.05303 - 1.82391i) q^{19} +(-2.32635 + 0.0362770i) q^{21} +(5.44804 - 3.14543i) q^{23} -4.59627i q^{27} -0.588526 q^{29} +(3.73783 - 6.47410i) q^{31} +(2.42544 - 1.40033i) q^{33} +(1.30753 - 0.754900i) q^{37} +(-0.215537 + 0.373321i) q^{39} -4.29086 q^{41} -5.92127i q^{43} +(-4.47086 + 2.58125i) q^{47} +(3.30928 - 6.16836i) q^{49} +(-0.421274 - 0.729669i) q^{51} +(11.4125 + 6.58899i) q^{53} -1.85204i q^{57} +(2.31908 - 4.01676i) q^{59} +(-6.96451 - 12.0629i) q^{61} +(5.14728 + 2.86571i) q^{63} +(-6.30147 - 3.63816i) q^{67} +5.53209 q^{69} +12.1925 q^{71} +(8.77141 + 5.06418i) q^{73} +(0.131381 + 8.42514i) q^{77} +(-1.26604 - 2.19285i) q^{79} +(-1.31908 + 2.28471i) q^{81} -4.80066i q^{83} +(-0.448204 - 0.258770i) q^{87} +(-5.40420 - 9.36035i) q^{89} +(-0.665907 - 1.11294i) q^{91} +(5.69323 - 3.28699i) q^{93} +10.5544i q^{97} -7.09152 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{11} + 12 q^{19} - 30 q^{21} - 48 q^{29} + 6 q^{31} + 12 q^{39} + 12 q^{41} + 30 q^{51} - 6 q^{59} - 18 q^{61} + 48 q^{69} + 36 q^{71} - 6 q^{79} + 18 q^{81} + 12 q^{89} - 102 q^{91} + 36 q^{99}+O(q^{100}) \) 12 * q + 12 * q^11 + 12 * q^19 - 30 * q^21 - 48 * q^29 + 6 * q^31 + 12 * q^39 + 12 * q^41 + 30 * q^51 - 6 * q^59 - 18 * q^61 + 48 * q^69 + 36 * q^71 - 6 * q^79 + 18 * q^81 + 12 * q^89 - 102 * q^91 + 36 * q^99

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