Properties

Label 700.1.j.a
Level $700$
Weight $1$
Character orbit 700.j
Analytic conductor $0.349$
Analytic rank $0$
Dimension $4$
Projective image $D_{2}$
CM/RM discs -7, -20, 140
Inner twists $16$

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Newspace parameters

Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 700.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.349345508843\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-5}, \sqrt{-7})\)
Artin image: $OD_{16}:C_2$
Artin field: Galois closure of 16.0.960400000000000000.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{8}^{3} q^{2} -\zeta_{8}^{2} q^{4} -\zeta_{8}^{3} q^{7} -\zeta_{8} q^{8} + \zeta_{8}^{2} q^{9} +O(q^{10})\) \( q -\zeta_{8}^{3} q^{2} -\zeta_{8}^{2} q^{4} -\zeta_{8}^{3} q^{7} -\zeta_{8} q^{8} + \zeta_{8}^{2} q^{9} -\zeta_{8}^{2} q^{14} - q^{16} + \zeta_{8} q^{18} -2 \zeta_{8} q^{23} -\zeta_{8} q^{28} + 2 \zeta_{8}^{2} q^{29} + \zeta_{8}^{3} q^{32} + q^{36} + 2 \zeta_{8} q^{43} -2 q^{46} -\zeta_{8}^{2} q^{49} - q^{56} + 2 \zeta_{8} q^{58} + \zeta_{8} q^{63} + \zeta_{8}^{2} q^{64} + 2 \zeta_{8}^{3} q^{67} -\zeta_{8}^{3} q^{72} - q^{81} + 2 q^{86} + 2 \zeta_{8}^{3} q^{92} -\zeta_{8} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 4q^{16} + 4q^{36} - 8q^{46} - 4q^{56} - 4q^{81} + 8q^{86} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{8}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i 0 1.00000i 0 0 −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 0
307.2 0.707107 0.707107i 0 1.00000i 0 0 0.707107 0.707107i −0.707107 0.707107i 1.00000i 0
643.1 −0.707107 0.707107i 0 1.00000i 0 0 −0.707107 0.707107i 0.707107 0.707107i 1.00000i 0
643.2 0.707107 + 0.707107i 0 1.00000i 0 0 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
140.c even 2 1 RM by \(\Q(\sqrt{35}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
20.e even 4 2 inner
28.d even 2 1 inner
35.c odd 2 1 inner
35.f even 4 2 inner
140.j odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.1.j.a 4
4.b odd 2 1 inner 700.1.j.a 4
5.b even 2 1 inner 700.1.j.a 4
5.c odd 4 2 inner 700.1.j.a 4
7.b odd 2 1 CM 700.1.j.a 4
20.d odd 2 1 CM 700.1.j.a 4
20.e even 4 2 inner 700.1.j.a 4
28.d even 2 1 inner 700.1.j.a 4
35.c odd 2 1 inner 700.1.j.a 4
35.f even 4 2 inner 700.1.j.a 4
140.c even 2 1 RM 700.1.j.a 4
140.j odd 4 2 inner 700.1.j.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.1.j.a 4 1.a even 1 1 trivial
700.1.j.a 4 4.b odd 2 1 inner
700.1.j.a 4 5.b even 2 1 inner
700.1.j.a 4 5.c odd 4 2 inner
700.1.j.a 4 7.b odd 2 1 CM
700.1.j.a 4 20.d odd 2 1 CM
700.1.j.a 4 20.e even 4 2 inner
700.1.j.a 4 28.d even 2 1 inner
700.1.j.a 4 35.c odd 2 1 inner
700.1.j.a 4 35.f even 4 2 inner
700.1.j.a 4 140.c even 2 1 RM
700.1.j.a 4 140.j odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11} \) acting on \(S_{1}^{\mathrm{new}}(700, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( 1 + T^{4} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( 16 + T^{4} \)
$29$ \( ( 4 + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( 16 + T^{4} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( 16 + T^{4} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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