Properties

Label 7.9.b.a
Level 7
Weight 9
Character orbit 7.b
Self dual Yes
Analytic conductor 2.852
Analytic rank 0
Dimension 1
CM disc. -7
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 7 \)
Weight: \( k \) = \( 9 \)
Character orbit: \([\chi]\) = 7.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(2.85165027043\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 31q^{2} \) \(\mathstrut +\mathstrut 705q^{4} \) \(\mathstrut +\mathstrut 2401q^{7} \) \(\mathstrut -\mathstrut 13919q^{8} \) \(\mathstrut +\mathstrut 6561q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 31q^{2} \) \(\mathstrut +\mathstrut 705q^{4} \) \(\mathstrut +\mathstrut 2401q^{7} \) \(\mathstrut -\mathstrut 13919q^{8} \) \(\mathstrut +\mathstrut 6561q^{9} \) \(\mathstrut +\mathstrut 13154q^{11} \) \(\mathstrut -\mathstrut 74431q^{14} \) \(\mathstrut +\mathstrut 251009q^{16} \) \(\mathstrut -\mathstrut 203391q^{18} \) \(\mathstrut -\mathstrut 407774q^{22} \) \(\mathstrut -\mathstrut 20926q^{23} \) \(\mathstrut +\mathstrut 390625q^{25} \) \(\mathstrut +\mathstrut 1692705q^{28} \) \(\mathstrut +\mathstrut 108194q^{29} \) \(\mathstrut -\mathstrut 4218015q^{32} \) \(\mathstrut +\mathstrut 4625505q^{36} \) \(\mathstrut -\mathstrut 2073886q^{37} \) \(\mathstrut -\mathstrut 6726046q^{43} \) \(\mathstrut +\mathstrut 9273570q^{44} \) \(\mathstrut +\mathstrut 648706q^{46} \) \(\mathstrut +\mathstrut 5764801q^{49} \) \(\mathstrut -\mathstrut 12109375q^{50} \) \(\mathstrut +\mathstrut 15377762q^{53} \) \(\mathstrut -\mathstrut 33419519q^{56} \) \(\mathstrut -\mathstrut 3354014q^{58} \) \(\mathstrut +\mathstrut 15752961q^{63} \) \(\mathstrut +\mathstrut 66500161q^{64} \) \(\mathstrut -\mathstrut 15839326q^{67} \) \(\mathstrut -\mathstrut 42331966q^{71} \) \(\mathstrut -\mathstrut 91322559q^{72} \) \(\mathstrut +\mathstrut 64290466q^{74} \) \(\mathstrut +\mathstrut 31582754q^{77} \) \(\mathstrut -\mathstrut 64606846q^{79} \) \(\mathstrut +\mathstrut 43046721q^{81} \) \(\mathstrut +\mathstrut 208507426q^{86} \) \(\mathstrut -\mathstrut 183090526q^{88} \) \(\mathstrut -\mathstrut 14752830q^{92} \) \(\mathstrut -\mathstrut 178708831q^{98} \) \(\mathstrut +\mathstrut 86303394q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−31.0000 0 705.000 0 0 2401.00 −13919.0 6561.00 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 \(T_{2} \) \(\mathstrut +\mathstrut 31 \) acting on \(S_{9}^{\mathrm{new}}(7, [\chi])\).