Properties

Label 7.5.b.a
Level 7
Weight 5
Character orbit 7.b
Self dual yes
Analytic conductor 0.724
Analytic rank 0
Dimension 1
CM discriminant -7
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 7.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.723589741587\)
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

\(f(q)\) \(=\) \( q + q^{2} - 15q^{4} + 49q^{7} - 31q^{8} + 81q^{9} + O(q^{10}) \) \( q + q^{2} - 15q^{4} + 49q^{7} - 31q^{8} + 81q^{9} - 206q^{11} + 49q^{14} + 209q^{16} + 81q^{18} - 206q^{22} - 734q^{23} + 625q^{25} - 735q^{28} + 1234q^{29} + 705q^{32} - 1215q^{36} - 1294q^{37} - 334q^{43} + 3090q^{44} - 734q^{46} + 2401q^{49} + 625q^{50} - 5582q^{53} - 1519q^{56} + 1234q^{58} + 3969q^{63} - 2639q^{64} + 4946q^{67} + 2914q^{71} - 2511q^{72} - 1294q^{74} - 10094q^{77} - 3646q^{79} + 6561q^{81} - 334q^{86} + 6386q^{88} + 11010q^{92} + 2401q^{98} - 16686q^{99} + O(q^{100}) \)

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
1.00000 0 −15.0000 0 0 49.0000 −31.0000 81.0000 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 7.5.b.a 1
3.b odd 2 1 63.5.d.a 1
4.b odd 2 1 112.5.c.a 1
5.b even 2 1 175.5.d.a 1
5.c odd 4 2 175.5.c.a 2
7.b odd 2 1 CM 7.5.b.a 1
7.c even 3 2 49.5.d.a 2
7.d odd 6 2 49.5.d.a 2
8.b even 2 1 448.5.c.b 1
8.d odd 2 1 448.5.c.a 1
12.b even 2 1 1008.5.f.a 1
21.c even 2 1 63.5.d.a 1
28.d even 2 1 112.5.c.a 1
35.c odd 2 1 175.5.d.a 1
35.f even 4 2 175.5.c.a 2
56.e even 2 1 448.5.c.a 1
56.h odd 2 1 448.5.c.b 1
84.h odd 2 1 1008.5.f.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.5.b.a 1 1.a even 1 1 trivial
7.5.b.a 1 7.b odd 2 1 CM
49.5.d.a 2 7.c even 3 2
49.5.d.a 2 7.d odd 6 2
63.5.d.a 1 3.b odd 2 1
63.5.d.a 1 21.c even 2 1
112.5.c.a 1 4.b odd 2 1
112.5.c.a 1 28.d even 2 1
175.5.c.a 2 5.c odd 4 2
175.5.c.a 2 35.f even 4 2
175.5.d.a 1 5.b even 2 1
175.5.d.a 1 35.c odd 2 1
448.5.c.a 1 8.d odd 2 1
448.5.c.a 1 56.e even 2 1
448.5.c.b 1 8.b even 2 1
448.5.c.b 1 56.h odd 2 1
1008.5.f.a 1 12.b even 2 1
1008.5.f.a 1 84.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(7, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + 16 T^{2} \)
$3$ \( ( 1 - 9 T )( 1 + 9 T ) \)
$5$ \( ( 1 - 25 T )( 1 + 25 T ) \)
$7$ \( 1 - 49 T \)
$11$ \( 1 + 206 T + 14641 T^{2} \)
$13$ \( ( 1 - 169 T )( 1 + 169 T ) \)
$17$ \( ( 1 - 289 T )( 1 + 289 T ) \)
$19$ \( ( 1 - 361 T )( 1 + 361 T ) \)
$23$ \( 1 + 734 T + 279841 T^{2} \)
$29$ \( 1 - 1234 T + 707281 T^{2} \)
$31$ \( ( 1 - 961 T )( 1 + 961 T ) \)
$37$ \( 1 + 1294 T + 1874161 T^{2} \)
$41$ \( ( 1 - 1681 T )( 1 + 1681 T ) \)
$43$ \( 1 + 334 T + 3418801 T^{2} \)
$47$ \( ( 1 - 2209 T )( 1 + 2209 T ) \)
$53$ \( 1 + 5582 T + 7890481 T^{2} \)
$59$ \( ( 1 - 3481 T )( 1 + 3481 T ) \)
$61$ \( ( 1 - 3721 T )( 1 + 3721 T ) \)
$67$ \( 1 - 4946 T + 20151121 T^{2} \)
$71$ \( 1 - 2914 T + 25411681 T^{2} \)
$73$ \( ( 1 - 5329 T )( 1 + 5329 T ) \)
$79$ \( 1 + 3646 T + 38950081 T^{2} \)
$83$ \( ( 1 - 6889 T )( 1 + 6889 T ) \)
$89$ \( ( 1 - 7921 T )( 1 + 7921 T ) \)
$97$ \( ( 1 - 9409 T )( 1 + 9409 T ) \)
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