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 discriminant -7
Inner twists 2

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Newspace parameters

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

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: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

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

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7.7.b.a 1
3.b odd 2 1 63.7.d.a 1
4.b odd 2 1 112.7.c.a 1
5.b even 2 1 175.7.d.a 1
5.c odd 4 2 175.7.c.a 2
7.b odd 2 1 CM 7.7.b.a 1
7.c even 3 2 49.7.d.a 2
7.d odd 6 2 49.7.d.a 2
8.b even 2 1 448.7.c.a 1
8.d odd 2 1 448.7.c.b 1
21.c even 2 1 63.7.d.a 1
28.d even 2 1 112.7.c.a 1
35.c odd 2 1 175.7.d.a 1
35.f even 4 2 175.7.c.a 2
56.e even 2 1 448.7.c.b 1
56.h odd 2 1 448.7.c.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.7.b.a 1 1.a even 1 1 trivial
7.7.b.a 1 7.b odd 2 1 CM
49.7.d.a 2 7.c even 3 2
49.7.d.a 2 7.d odd 6 2
63.7.d.a 1 3.b odd 2 1
63.7.d.a 1 21.c even 2 1
112.7.c.a 1 4.b odd 2 1
112.7.c.a 1 28.d even 2 1
175.7.c.a 2 5.c odd 4 2
175.7.c.a 2 35.f even 4 2
175.7.d.a 1 5.b even 2 1
175.7.d.a 1 35.c odd 2 1
448.7.c.a 1 8.b even 2 1
448.7.c.a 1 56.h odd 2 1
448.7.c.b 1 8.d odd 2 1
448.7.c.b 1 56.e even 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 9 T + 64 T^{2} \)
$3$ \( ( 1 - 27 T )( 1 + 27 T ) \)
$5$ \( ( 1 - 125 T )( 1 + 125 T ) \)
$7$ \( 1 + 343 T \)
$11$ \( 1 - 1962 T + 1771561 T^{2} \)
$13$ \( ( 1 - 2197 T )( 1 + 2197 T ) \)
$17$ \( ( 1 - 4913 T )( 1 + 4913 T ) \)
$19$ \( ( 1 - 6859 T )( 1 + 6859 T ) \)
$23$ \( 1 + 22734 T + 148035889 T^{2} \)
$29$ \( 1 + 21222 T + 594823321 T^{2} \)
$31$ \( ( 1 - 29791 T )( 1 + 29791 T ) \)
$37$ \( 1 - 101194 T + 2565726409 T^{2} \)
$41$ \( ( 1 - 68921 T )( 1 + 68921 T ) \)
$43$ \( 1 + 126614 T + 6321363049 T^{2} \)
$47$ \( ( 1 - 103823 T )( 1 + 103823 T ) \)
$53$ \( 1 - 50346 T + 22164361129 T^{2} \)
$59$ \( ( 1 - 205379 T )( 1 + 205379 T ) \)
$61$ \( ( 1 - 226981 T )( 1 + 226981 T ) \)
$67$ \( 1 + 53926 T + 90458382169 T^{2} \)
$71$ \( 1 + 242478 T + 128100283921 T^{2} \)
$73$ \( ( 1 - 389017 T )( 1 + 389017 T ) \)
$79$ \( 1 - 929378 T + 243087455521 T^{2} \)
$83$ \( ( 1 - 571787 T )( 1 + 571787 T ) \)
$89$ \( ( 1 - 704969 T )( 1 + 704969 T ) \)
$97$ \( ( 1 - 912673 T )( 1 + 912673 T ) \)
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