LMFDB - Embedded newform 1400.2.q.e.401.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1400,2,Mod(401,1400)] 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("1400.401"); 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, 0, 2])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1400 = 2^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1400.q (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,1,0,0,0,5] 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:	$11.1790562830$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			401.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1400.401
Dual form			1400.2.q.e.1201.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{3} +(2.50000 + 0.866025i) q^{7} +(1.00000 - 1.73205i) q^{9} +(-1.00000 - 1.73205i) q^{11} +(2.00000 + 3.46410i) q^{17} +(1.00000 - 1.73205i) q^{19} +(0.500000 + 2.59808i) q^{21} +(0.500000 - 0.866025i) q^{23} +5.00000 q^{27} +9.00000 q^{29} +(-2.00000 - 3.46410i) q^{31} +(1.00000 - 1.73205i) q^{33} +(2.00000 - 3.46410i) q^{37} +1.00000 q^{41} -9.00000 q^{43} +(5.50000 + 4.33013i) q^{49} +(-2.00000 + 3.46410i) q^{51} +(-5.00000 - 8.66025i) q^{53} +2.00000 q^{57} +(5.00000 + 8.66025i) q^{59} +(-4.50000 + 7.79423i) q^{61} +(4.00000 - 3.46410i) q^{63} +(2.50000 + 4.33013i) q^{67} +1.00000 q^{69} +14.0000 q^{71} +(6.00000 + 10.3923i) q^{73} +(-1.00000 - 5.19615i) q^{77} +(-7.00000 + 12.1244i) q^{79} +(-0.500000 - 0.866025i) q^{81} -11.0000 q^{83} +(4.50000 + 7.79423i) q^{87} +(7.50000 - 12.9904i) q^{89} +(2.00000 - 3.46410i) q^{93} +18.0000 q^{97} -4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + 5 q^{7} + 2 q^{9} - 2 q^{11} + 4 q^{17} + 2 q^{19} + q^{21} + q^{23} + 10 q^{27} + 18 q^{29} - 4 q^{31} + 2 q^{33} + 4 q^{37} + 2 q^{41} - 18 q^{43} + 11 q^{49} - 4 q^{51} - 10 q^{53} + 4 q^{57}+ \cdots - 8 q^{99}+O(q^{100}) $ 2 * q + q^3 + 5 * q^7 + 2 * q^9 - 2 * q^11 + 4 * q^17 + 2 * q^19 + q^21 + q^23 + 10 * q^27 + 18 * q^29 - 4 * q^31 + 2 * q^33 + 4 * q^37 + 2 * q^41 - 18 * q^43 + 11 * q^49 - 4 * q^51 - 10 * q^53 + 4 * q^57 + 10 * q^59 - 9 * q^61 + 8 * q^63 + 5 * q^67 + 2 * q^69 + 28 * q^71 + 12 * q^73 - 2 * q^77 - 14 * q^79 - q^81 - 22 * q^83 + 9 * q^87 + 15 * q^89 + 4 * q^93 + 36 * q^97 - 8 * q^99

Character values

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

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

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
$3$	0.500000	+	0.866025i	0.288675	+	0.500000i	0.973494	−	0.228714i	$-0.0734519\pi$
$3$	0.500000	+	0.866025i	0.288675	+	0.500000i	−0.684819	+	0.728714i	$0.740119\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	2.50000	+	0.866025i	0.944911	+	0.327327i
$7$	2.50000	+	0.866025i	0.944911	+	0.327327i
$8$		0			0
$9$	1.00000	−	1.73205i	0.333333	−	0.577350i
$10$		0			0
$11$	−1.00000	−	1.73205i	−0.301511	−	0.522233i	0.674967	−	0.737848i	$-0.264158\pi$
$11$	−1.00000	−	1.73205i	−0.301511	−	0.522233i	−0.976478	+	0.215615i	$0.930824\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.00000	+	3.46410i	0.485071	+	0.840168i	0.999853	−	0.0171533i	$-0.00546033\pi$
$17$	2.00000	+	3.46410i	0.485071	+	0.840168i	−0.514782	+	0.857321i	$0.672127\pi$
$18$		0			0
$19$	1.00000	−	1.73205i	0.229416	−	0.397360i	−0.728219	−	0.685344i	$-0.759652\pi$
$19$	1.00000	−	1.73205i	0.229416	−	0.397360i	0.957635	+	0.287984i	$0.0929851\pi$
$20$		0			0
$21$	0.500000	+	2.59808i	0.109109	+	0.566947i
$22$		0			0
$23$	0.500000	−	0.866025i	0.104257	−	0.180579i	−0.809177	−	0.587565i	$-0.800087\pi$
$23$	0.500000	−	0.866025i	0.104257	−	0.180579i	0.913434	+	0.406986i	$0.133420\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	5.00000			0.962250
$28$		0			0
$29$	9.00000			1.67126			0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000			1.67126			0.835629	+	0.549294i	$0.185103\pi$
$30$		0			0
$31$	−2.00000	−	3.46410i	−0.359211	−	0.622171i	0.628619	−	0.777714i	$-0.283621\pi$
$31$	−2.00000	−	3.46410i	−0.359211	−	0.622171i	−0.987829	+	0.155543i	$0.950287\pi$
$32$		0			0
$33$	1.00000	−	1.73205i	0.174078	−	0.301511i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2.00000	−	3.46410i	0.328798	−	0.569495i	−0.653476	−	0.756948i	$-0.726690\pi$
$37$	2.00000	−	3.46410i	0.328798	−	0.569495i	0.982274	+	0.187453i	$0.0600231\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.00000			0.156174			0.0780869	−	0.996947i	$-0.475119\pi$
$41$	1.00000			0.156174			0.0780869	+	0.996947i	$0.475119\pi$
$42$		0			0
$43$	−9.00000			−1.37249			−0.686244	−	0.727372i	$-0.740742\pi$
$43$	−9.00000			−1.37249			−0.686244	+	0.727372i	$0.740742\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$	5.50000	+	4.33013i	0.785714	+	0.618590i
$50$		0			0
$51$	−2.00000	+	3.46410i	−0.280056	+	0.485071i
$52$		0			0
$53$	−5.00000	−	8.66025i	−0.686803	−	1.18958i	−0.972867	−	0.231367i	$-0.925680\pi$
$53$	−5.00000	−	8.66025i	−0.686803	−	1.18958i	0.286064	−	0.958211i	$-0.407653\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	2.00000			0.264906
$58$		0			0
$59$	5.00000	+	8.66025i	0.650945	+	1.12747i	0.982894	+	0.184172i	$0.0589603\pi$
$59$	5.00000	+	8.66025i	0.650945	+	1.12747i	−0.331949	+	0.943297i	$0.607706\pi$
$60$		0			0
$61$	−4.50000	+	7.79423i	−0.576166	+	0.997949i	0.419748	+	0.907641i	$0.362118\pi$
$61$	−4.50000	+	7.79423i	−0.576166	+	0.997949i	−0.995914	+	0.0903080i	$0.971215\pi$
$62$		0			0
$63$	4.00000	−	3.46410i	0.503953	−	0.436436i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	2.50000	+	4.33013i	0.305424	+	0.529009i	0.977356	−	0.211604i	$-0.0678686\pi$
$67$	2.50000	+	4.33013i	0.305424	+	0.529009i	−0.671932	+	0.740613i	$0.734535\pi$
$68$		0			0
$69$	1.00000			0.120386
$70$		0			0
$71$	14.0000			1.66149			0.830747	−	0.556650i	$-0.187914\pi$
$71$	14.0000			1.66149			0.830747	+	0.556650i	$0.187914\pi$
$72$		0			0
$73$	6.00000	+	10.3923i	0.702247	+	1.21633i	0.967676	+	0.252197i	$0.0811531\pi$
$73$	6.00000	+	10.3923i	0.702247	+	1.21633i	−0.265429	+	0.964130i	$0.585514\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.00000	−	5.19615i	−0.113961	−	0.592157i
$78$		0			0
$79$	−7.00000	+	12.1244i	−0.787562	+	1.36410i	0.139895	+	0.990166i	$0.455323\pi$
$79$	−7.00000	+	12.1244i	−0.787562	+	1.36410i	−0.927457	+	0.373930i	$0.878010\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	−11.0000			−1.20741			−0.603703	−	0.797209i	$-0.706309\pi$
$83$	−11.0000			−1.20741			−0.603703	+	0.797209i	$0.706309\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	4.50000	+	7.79423i	0.482451	+	0.835629i
$88$		0			0
$89$	7.50000	−	12.9904i	0.794998	−	1.37698i	−0.127842	−	0.991795i	$-0.540805\pi$
$89$	7.50000	−	12.9904i	0.794998	−	1.37698i	0.922840	−	0.385183i	$-0.125862\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	2.00000	−	3.46410i	0.207390	−	0.359211i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	18.0000			1.82762			0.913812	−	0.406138i	$-0.133125\pi$
$97$	18.0000			1.82762			0.913812	+	0.406138i	$0.133125\pi$
$98$		0			0
$99$	−4.00000			−0.402015

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.q.e.401.1		2
5.2	odd	4		1400.2.bh.b.849.2		4
5.3	odd	4		1400.2.bh.b.849.1		4
5.4	even	2		280.2.q.a.121.1	yes	2
7.2	even	3		9800.2.a.r.1.1		1
7.4	even	3	inner	1400.2.q.e.1201.1		2
7.5	odd	6		9800.2.a.bc.1.1		1
15.14	odd	2		2520.2.bi.a.1801.1		2
20.19	odd	2		560.2.q.h.401.1		2
35.4	even	6		280.2.q.a.81.1	✓	2
35.9	even	6		1960.2.a.i.1.1		1
35.18	odd	12		1400.2.bh.b.249.2		4
35.19	odd	6		1960.2.a.e.1.1		1
35.24	odd	6		1960.2.q.k.361.1		2
35.32	odd	12		1400.2.bh.b.249.1		4
35.34	odd	2		1960.2.q.k.961.1		2
105.74	odd	6		2520.2.bi.a.361.1		2
140.19	even	6		3920.2.a.y.1.1		1
140.39	odd	6		560.2.q.h.81.1		2
140.79	odd	6		3920.2.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.q.a.81.1	✓	2	35.4	even	6
280.2.q.a.121.1	yes	2	5.4	even	2
560.2.q.h.81.1		2	140.39	odd	6
560.2.q.h.401.1		2	20.19	odd	2
1400.2.q.e.401.1		2	1.1	even	1	trivial
1400.2.q.e.1201.1		2	7.4	even	3	inner
1400.2.bh.b.249.1		4	35.32	odd	12
1400.2.bh.b.249.2		4	35.18	odd	12
1400.2.bh.b.849.1		4	5.3	odd	4
1400.2.bh.b.849.2		4	5.2	odd	4
1960.2.a.e.1.1		1	35.19	odd	6
1960.2.a.i.1.1		1	35.9	even	6
1960.2.q.k.361.1		2	35.24	odd	6
1960.2.q.k.961.1		2	35.34	odd	2
2520.2.bi.a.361.1		2	105.74	odd	6
2520.2.bi.a.1801.1		2	15.14	odd	2
3920.2.a.m.1.1		1	140.79	odd	6
3920.2.a.y.1.1		1	140.19	even	6
9800.2.a.r.1.1		1	7.2	even	3
9800.2.a.bc.1.1		1	7.5	odd	6

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

Level:	\( N \)	\(=\)	\( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1400.q (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{3} +(2.50000 + 0.866025i) q^{7} +(1.00000 - 1.73205i) q^{9} +(-1.00000 - 1.73205i) q^{11} +(2.00000 + 3.46410i) q^{17} +(1.00000 - 1.73205i) q^{19} +(0.500000 + 2.59808i) q^{21} +(0.500000 - 0.866025i) q^{23} +5.00000 q^{27} +9.00000 q^{29} +(-2.00000 - 3.46410i) q^{31} +(1.00000 - 1.73205i) q^{33} +(2.00000 - 3.46410i) q^{37} +1.00000 q^{41} -9.00000 q^{43} +(5.50000 + 4.33013i) q^{49} +(-2.00000 + 3.46410i) q^{51} +(-5.00000 - 8.66025i) q^{53} +2.00000 q^{57} +(5.00000 + 8.66025i) q^{59} +(-4.50000 + 7.79423i) q^{61} +(4.00000 - 3.46410i) q^{63} +(2.50000 + 4.33013i) q^{67} +1.00000 q^{69} +14.0000 q^{71} +(6.00000 + 10.3923i) q^{73} +(-1.00000 - 5.19615i) q^{77} +(-7.00000 + 12.1244i) q^{79} +(-0.500000 - 0.866025i) q^{81} -11.0000 q^{83} +(4.50000 + 7.79423i) q^{87} +(7.50000 - 12.9904i) q^{89} +(2.00000 - 3.46410i) q^{93} +18.0000 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 5 q^{7} + 2 q^{9} - 2 q^{11} + 4 q^{17} + 2 q^{19} + q^{21} + q^{23} + 10 q^{27} + 18 q^{29} - 4 q^{31} + 2 q^{33} + 4 q^{37} + 2 q^{41} - 18 q^{43} + 11 q^{49} - 4 q^{51} - 10 q^{53} + 4 q^{57}+ \cdots - 8 q^{99}+O(q^{100}) \) 2 * q + q^3 + 5 * q^7 + 2 * q^9 - 2 * q^11 + 4 * q^17 + 2 * q^19 + q^21 + q^23 + 10 * q^27 + 18 * q^29 - 4 * q^31 + 2 * q^33 + 4 * q^37 + 2 * q^41 - 18 * q^43 + 11 * q^49 - 4 * q^51 - 10 * q^53 + 4 * q^57 + 10 * q^59 - 9 * q^61 + 8 * q^63 + 5 * q^67 + 2 * q^69 + 28 * q^71 + 12 * q^73 - 2 * q^77 - 14 * q^79 - q^81 - 22 * q^83 + 9 * q^87 + 15 * q^89 + 4 * q^93 + 36 * q^97 - 8 * q^99

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