Properties

Label 700.1.d.a
Level $700$
Weight $1$
Character orbit 700.d
Analytic conductor $0.349$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -35
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 700.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.349345508843\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
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)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.140.1
Artin image: $C_4\times S_3$
Artin field: Galois closure of 12.0.1200500000000.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -i q^{3} -i q^{7} +O(q^{10})\) \( q -i q^{3} -i q^{7} - q^{11} -i q^{13} + i q^{17} - q^{21} -i q^{27} + q^{29} + i q^{33} - q^{39} + i q^{47} - q^{49} + q^{51} + 2 q^{71} + 2 i q^{73} + i q^{77} + q^{79} - q^{81} + 2 i q^{83} -i q^{87} - q^{91} + i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 2q^{11} - 2q^{21} + 2q^{29} - 2q^{39} - 2q^{49} + 2q^{51} + 4q^{71} + 2q^{79} - 2q^{81} - 2q^{91} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
601.1
1.00000i
1.00000i
0 1.00000i 0 0 0 1.00000i 0 0 0
601.2 0 1.00000i 0 0 0 1.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
5.b even 2 1 inner
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.1.d.a 2
4.b odd 2 1 2800.1.f.c 2
5.b even 2 1 inner 700.1.d.a 2
5.c odd 4 1 140.1.h.a 1
5.c odd 4 1 140.1.h.b yes 1
7.b odd 2 1 inner 700.1.d.a 2
15.e even 4 1 1260.1.p.a 1
15.e even 4 1 1260.1.p.b 1
20.d odd 2 1 2800.1.f.c 2
20.e even 4 1 560.1.p.a 1
20.e even 4 1 560.1.p.b 1
28.d even 2 1 2800.1.f.c 2
35.c odd 2 1 CM 700.1.d.a 2
35.f even 4 1 140.1.h.a 1
35.f even 4 1 140.1.h.b yes 1
35.k even 12 2 980.1.n.a 2
35.k even 12 2 980.1.n.b 2
35.l odd 12 2 980.1.n.a 2
35.l odd 12 2 980.1.n.b 2
40.i odd 4 1 2240.1.p.b 1
40.i odd 4 1 2240.1.p.c 1
40.k even 4 1 2240.1.p.a 1
40.k even 4 1 2240.1.p.d 1
105.k odd 4 1 1260.1.p.a 1
105.k odd 4 1 1260.1.p.b 1
140.c even 2 1 2800.1.f.c 2
140.j odd 4 1 560.1.p.a 1
140.j odd 4 1 560.1.p.b 1
140.w even 12 2 3920.1.br.a 2
140.w even 12 2 3920.1.br.b 2
140.x odd 12 2 3920.1.br.a 2
140.x odd 12 2 3920.1.br.b 2
280.s even 4 1 2240.1.p.b 1
280.s even 4 1 2240.1.p.c 1
280.y odd 4 1 2240.1.p.a 1
280.y odd 4 1 2240.1.p.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.1.h.a 1 5.c odd 4 1
140.1.h.a 1 35.f even 4 1
140.1.h.b yes 1 5.c odd 4 1
140.1.h.b yes 1 35.f even 4 1
560.1.p.a 1 20.e even 4 1
560.1.p.a 1 140.j odd 4 1
560.1.p.b 1 20.e even 4 1
560.1.p.b 1 140.j odd 4 1
700.1.d.a 2 1.a even 1 1 trivial
700.1.d.a 2 5.b even 2 1 inner
700.1.d.a 2 7.b odd 2 1 inner
700.1.d.a 2 35.c odd 2 1 CM
980.1.n.a 2 35.k even 12 2
980.1.n.a 2 35.l odd 12 2
980.1.n.b 2 35.k even 12 2
980.1.n.b 2 35.l odd 12 2
1260.1.p.a 1 15.e even 4 1
1260.1.p.a 1 105.k odd 4 1
1260.1.p.b 1 15.e even 4 1
1260.1.p.b 1 105.k odd 4 1
2240.1.p.a 1 40.k even 4 1
2240.1.p.a 1 280.y odd 4 1
2240.1.p.b 1 40.i odd 4 1
2240.1.p.b 1 280.s even 4 1
2240.1.p.c 1 40.i odd 4 1
2240.1.p.c 1 280.s even 4 1
2240.1.p.d 1 40.k even 4 1
2240.1.p.d 1 280.y odd 4 1
2800.1.f.c 2 4.b odd 2 1
2800.1.f.c 2 20.d odd 2 1
2800.1.f.c 2 28.d even 2 1
2800.1.f.c 2 140.c even 2 1
3920.1.br.a 2 140.w even 12 2
3920.1.br.a 2 140.x odd 12 2
3920.1.br.b 2 140.w even 12 2
3920.1.br.b 2 140.x odd 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(700, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( 1 + T^{2} \)
$17$ \( 1 + T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( -1 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( T^{2} \)
$43$ \( T^{2} \)
$47$ \( 1 + T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( T^{2} \)
$67$ \( T^{2} \)
$71$ \( ( -2 + T )^{2} \)
$73$ \( 4 + T^{2} \)
$79$ \( ( -1 + T )^{2} \)
$83$ \( 4 + T^{2} \)
$89$ \( T^{2} \)
$97$ \( 1 + T^{2} \)
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