Properties

Label 7.3.b.a
Level 7
Weight 3
Character orbit 7.b
Self dual Yes
Analytic conductor 0.191
Analytic rank 0
Dimension 1
CM disc. -7
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 7 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 7.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.190736185052\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\)  \(=\)  \(q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut 5q^{4} \) \(\mathstrut -\mathstrut 7q^{7} \) \(\mathstrut -\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut 9q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut 5q^{4} \) \(\mathstrut -\mathstrut 7q^{7} \) \(\mathstrut -\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut 9q^{9} \) \(\mathstrut -\mathstrut 6q^{11} \) \(\mathstrut +\mathstrut 21q^{14} \) \(\mathstrut -\mathstrut 11q^{16} \) \(\mathstrut -\mathstrut 27q^{18} \) \(\mathstrut +\mathstrut 18q^{22} \) \(\mathstrut +\mathstrut 18q^{23} \) \(\mathstrut +\mathstrut 25q^{25} \) \(\mathstrut -\mathstrut 35q^{28} \) \(\mathstrut -\mathstrut 54q^{29} \) \(\mathstrut +\mathstrut 45q^{32} \) \(\mathstrut +\mathstrut 45q^{36} \) \(\mathstrut -\mathstrut 38q^{37} \) \(\mathstrut +\mathstrut 58q^{43} \) \(\mathstrut -\mathstrut 30q^{44} \) \(\mathstrut -\mathstrut 54q^{46} \) \(\mathstrut +\mathstrut 49q^{49} \) \(\mathstrut -\mathstrut 75q^{50} \) \(\mathstrut -\mathstrut 6q^{53} \) \(\mathstrut +\mathstrut 21q^{56} \) \(\mathstrut +\mathstrut 162q^{58} \) \(\mathstrut -\mathstrut 63q^{63} \) \(\mathstrut -\mathstrut 91q^{64} \) \(\mathstrut -\mathstrut 118q^{67} \) \(\mathstrut +\mathstrut 114q^{71} \) \(\mathstrut -\mathstrut 27q^{72} \) \(\mathstrut +\mathstrut 114q^{74} \) \(\mathstrut +\mathstrut 42q^{77} \) \(\mathstrut -\mathstrut 94q^{79} \) \(\mathstrut +\mathstrut 81q^{81} \) \(\mathstrut -\mathstrut 174q^{86} \) \(\mathstrut +\mathstrut 18q^{88} \) \(\mathstrut +\mathstrut 90q^{92} \) \(\mathstrut -\mathstrut 147q^{98} \) \(\mathstrut -\mathstrut 54q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1
0
−3.00000 0 5.00000 0 0 −7.00000 −3.00000 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Type Proved
1.a Even 1 trivial yes
7.b Odd 1 CM by \(\Q(\sqrt{-7}) \) yes

Hecke kernels

There are no other newforms in \(S_{3}^{\mathrm{new}}(7, \chi)\).