Properties

Label 7.7.b.a
Level 7
Weight 7
Character orbit 7.b
Self dual Yes
Analytic conductor 1.610
Analytic rank 0
Dimension 1
CM disc. -7
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: Yes
Analytic conductor: \(1.61037858534\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 9q^{2} \) \(\mathstrut +\mathstrut 17q^{4} \) \(\mathstrut -\mathstrut 343q^{7} \) \(\mathstrut -\mathstrut 423q^{8} \) \(\mathstrut +\mathstrut 729q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 9q^{2} \) \(\mathstrut +\mathstrut 17q^{4} \) \(\mathstrut -\mathstrut 343q^{7} \) \(\mathstrut -\mathstrut 423q^{8} \) \(\mathstrut +\mathstrut 729q^{9} \) \(\mathstrut +\mathstrut 1962q^{11} \) \(\mathstrut -\mathstrut 3087q^{14} \) \(\mathstrut -\mathstrut 4895q^{16} \) \(\mathstrut +\mathstrut 6561q^{18} \) \(\mathstrut +\mathstrut 17658q^{22} \) \(\mathstrut -\mathstrut 22734q^{23} \) \(\mathstrut +\mathstrut 15625q^{25} \) \(\mathstrut -\mathstrut 5831q^{28} \) \(\mathstrut -\mathstrut 21222q^{29} \) \(\mathstrut -\mathstrut 16983q^{32} \) \(\mathstrut +\mathstrut 12393q^{36} \) \(\mathstrut +\mathstrut 101194q^{37} \) \(\mathstrut -\mathstrut 126614q^{43} \) \(\mathstrut +\mathstrut 33354q^{44} \) \(\mathstrut -\mathstrut 204606q^{46} \) \(\mathstrut +\mathstrut 117649q^{49} \) \(\mathstrut +\mathstrut 140625q^{50} \) \(\mathstrut +\mathstrut 50346q^{53} \) \(\mathstrut +\mathstrut 145089q^{56} \) \(\mathstrut -\mathstrut 190998q^{58} \) \(\mathstrut -\mathstrut 250047q^{63} \) \(\mathstrut +\mathstrut 160433q^{64} \) \(\mathstrut -\mathstrut 53926q^{67} \) \(\mathstrut -\mathstrut 242478q^{71} \) \(\mathstrut -\mathstrut 308367q^{72} \) \(\mathstrut +\mathstrut 910746q^{74} \) \(\mathstrut -\mathstrut 672966q^{77} \) \(\mathstrut +\mathstrut 929378q^{79} \) \(\mathstrut +\mathstrut 531441q^{81} \) \(\mathstrut -\mathstrut 1139526q^{86} \) \(\mathstrut -\mathstrut 829926q^{88} \) \(\mathstrut -\mathstrut 386478q^{92} \) \(\mathstrut +\mathstrut 1058841q^{98} \) \(\mathstrut +\mathstrut 1430298q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
9.00000 0 17.0000 0 0 −343.000 −423.000 729.000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2} \) \(\mathstrut -\mathstrut 9 \) acting on \(S_{7}^{\mathrm{new}}(7, [\chi])\).