Newform orbit 700.2.e.c

• Label: 700.2.e.c
• Level: $700$
• Weight: $2$
• Character orbit: 700.e
• Analytic conductor: $5.590$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$700 = 2^{2} \cdot 5^{2} \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	700.e (of order $2$, degree $1$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$5.58952814149$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1})$
Defining polynomial:	$x^{2} + 1$
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $i = \sqrt{-1}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + i * q^3 - i * q^7 + 2 * q^9 + 3 * q^11 - i * q^13 + 3*i * q^17 - 2 * q^19 + q^21 - 6*i * q^23 + 5*i * q^27 + 9 * q^29 + 8 * q^31 + 3*i * q^33 + 10*i * q^37 + q^39 + 2*i * q^43 + 3*i * q^47 - q^49 - 3 * q^51 - 2*i * q^57 - 12 * q^59 + 8 * q^61 - 2*i * q^63 - 8*i * q^67 + 6 * q^69 + 14*i * q^73 - 3*i * q^77 - 5 * q^79 + q^81 - 12*i * q^83 + 9*i * q^87 - 12 * q^89 - q^91 + 8*i * q^93 - 17*i * q^97 + 6 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 4 q^{9} + 6 q^{11} - 4 q^{19} + 2 q^{21} + 18 q^{29} + 16 q^{31} + 2 q^{39} - 2 q^{49} - 6 q^{51} - 24 q^{59} + 16 q^{61} + 12 q^{69} - 10 q^{79} + 2 q^{81} - 24 q^{89} - 2 q^{91} + 12 q^{99}+O(q^{100})$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/700\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$477$
$\chi(n)$	$1$	$1$	$-1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

Label  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

449.1

	−	1.00000i
		1.00000i

0

−

1.00000i

0

0

0

1.00000i

0

2.00000

0

449.2

0

1.00000i

0

0

0

−

1.00000i

0

2.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	700.2.e.c		2
3.b	odd	2	1		6300.2.k.c		2
4.b	odd	2	1		2800.2.g.j		2
5.b	even	2	1	inner	700.2.e.c		2
5.c	odd	4	1		140.2.a.a	✓	1
5.c	odd	4	1		700.2.a.d		1
7.b	odd	2	1		4900.2.e.l		2
15.d	odd	2	1		6300.2.k.c		2
15.e	even	4	1		1260.2.a.c		1
15.e	even	4	1		6300.2.a.d		1
20.d	odd	2	1		2800.2.g.j		2
20.e	even	4	1		560.2.a.c		1
20.e	even	4	1		2800.2.a.y		1
35.c	odd	2	1		4900.2.e.l		2
35.f	even	4	1		980.2.a.c		1
35.f	even	4	1		4900.2.a.p		1
35.k	even	12	2		980.2.i.h		2
35.l	odd	12	2		980.2.i.d		2
40.i	odd	4	1		2240.2.a.g		1
40.k	even	4	1		2240.2.a.r		1
60.l	odd	4	1		5040.2.a.h		1
105.k	odd	4	1		8820.2.a.r		1
140.j	odd	4	1		3920.2.a.u		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
140.2.a.a	✓	1	5.c	odd	4	1
560.2.a.c		1	20.e	even	4	1
700.2.a.d		1	5.c	odd	4	1
700.2.e.c		2	1.a	even	1	1	trivial
700.2.e.c		2	5.b	even	2	1	inner
980.2.a.c		1	35.f	even	4	1
980.2.i.d		2	35.l	odd	12	2
980.2.i.h		2	35.k	even	12	2
1260.2.a.c		1	15.e	even	4	1
2240.2.a.g		1	40.i	odd	4	1
2240.2.a.r		1	40.k	even	4	1
2800.2.a.y		1	20.e	even	4	1
2800.2.g.j		2	4.b	odd	2	1
2800.2.g.j		2	20.d	odd	2	1
3920.2.a.u		1	140.j	odd	4	1
4900.2.a.p		1	35.f	even	4	1
4900.2.e.l		2	7.b	odd	2	1
4900.2.e.l		2	35.c	odd	2	1
5040.2.a.h		1	60.l	odd	4	1
6300.2.a.d		1	15.e	even	4	1
6300.2.k.c		2	3.b	odd	2	1
6300.2.k.c		2	15.d	odd	2	1
8820.2.a.r		1	105.k	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{2} + 1$ acting on $S_{2}^{\mathrm{new}}(700, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2} + 1$
$5$	$T^{2}$
$7$	$T^{2} + 1$
$11$	$(T - 3)^{2}$