Properties

Label 14.6.a.b
Level $14$
Weight $6$
Character orbit 14.a
Self dual yes
Analytic conductor $2.245$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 14.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.24537347738\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 4q^{2} + 8q^{3} + 16q^{4} + 10q^{5} + 32q^{6} - 49q^{7} + 64q^{8} - 179q^{9} + O(q^{10}) \) \( q + 4q^{2} + 8q^{3} + 16q^{4} + 10q^{5} + 32q^{6} - 49q^{7} + 64q^{8} - 179q^{9} + 40q^{10} - 340q^{11} + 128q^{12} - 294q^{13} - 196q^{14} + 80q^{15} + 256q^{16} + 1226q^{17} - 716q^{18} + 2432q^{19} + 160q^{20} - 392q^{21} - 1360q^{22} + 2000q^{23} + 512q^{24} - 3025q^{25} - 1176q^{26} - 3376q^{27} - 784q^{28} - 6746q^{29} + 320q^{30} + 8856q^{31} + 1024q^{32} - 2720q^{33} + 4904q^{34} - 490q^{35} - 2864q^{36} + 9182q^{37} + 9728q^{38} - 2352q^{39} + 640q^{40} - 14574q^{41} - 1568q^{42} + 8108q^{43} - 5440q^{44} - 1790q^{45} + 8000q^{46} - 312q^{47} + 2048q^{48} + 2401q^{49} - 12100q^{50} + 9808q^{51} - 4704q^{52} - 14634q^{53} - 13504q^{54} - 3400q^{55} - 3136q^{56} + 19456q^{57} - 26984q^{58} - 27656q^{59} + 1280q^{60} + 34338q^{61} + 35424q^{62} + 8771q^{63} + 4096q^{64} - 2940q^{65} - 10880q^{66} + 12316q^{67} + 19616q^{68} + 16000q^{69} - 1960q^{70} + 36920q^{71} - 11456q^{72} - 61718q^{73} + 36728q^{74} - 24200q^{75} + 38912q^{76} + 16660q^{77} - 9408q^{78} - 64752q^{79} + 2560q^{80} + 16489q^{81} - 58296q^{82} - 77056q^{83} - 6272q^{84} + 12260q^{85} + 32432q^{86} - 53968q^{87} - 21760q^{88} - 8166q^{89} - 7160q^{90} + 14406q^{91} + 32000q^{92} + 70848q^{93} - 1248q^{94} + 24320q^{95} + 8192q^{96} + 20650q^{97} + 9604q^{98} + 60860q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
4.00000 8.00000 16.0000 10.0000 32.0000 −49.0000 64.0000 −179.000 40.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 14.6.a.b 1
3.b odd 2 1 126.6.a.c 1
4.b odd 2 1 112.6.a.d 1
5.b even 2 1 350.6.a.b 1
5.c odd 4 2 350.6.c.f 2
7.b odd 2 1 98.6.a.b 1
7.c even 3 2 98.6.c.a 2
7.d odd 6 2 98.6.c.b 2
8.b even 2 1 448.6.a.f 1
8.d odd 2 1 448.6.a.k 1
12.b even 2 1 1008.6.a.n 1
21.c even 2 1 882.6.a.g 1
28.d even 2 1 784.6.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.6.a.b 1 1.a even 1 1 trivial
98.6.a.b 1 7.b odd 2 1
98.6.c.a 2 7.c even 3 2
98.6.c.b 2 7.d odd 6 2
112.6.a.d 1 4.b odd 2 1
126.6.a.c 1 3.b odd 2 1
350.6.a.b 1 5.b even 2 1
350.6.c.f 2 5.c odd 4 2
448.6.a.f 1 8.b even 2 1
448.6.a.k 1 8.d odd 2 1
784.6.a.h 1 28.d even 2 1
882.6.a.g 1 21.c even 2 1
1008.6.a.n 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 8 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(14))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 + T \)
$3$ \( -8 + T \)
$5$ \( -10 + T \)
$7$ \( 49 + T \)
$11$ \( 340 + T \)
$13$ \( 294 + T \)
$17$ \( -1226 + T \)
$19$ \( -2432 + T \)
$23$ \( -2000 + T \)
$29$ \( 6746 + T \)
$31$ \( -8856 + T \)
$37$ \( -9182 + T \)
$41$ \( 14574 + T \)
$43$ \( -8108 + T \)
$47$ \( 312 + T \)
$53$ \( 14634 + T \)
$59$ \( 27656 + T \)
$61$ \( -34338 + T \)
$67$ \( -12316 + T \)
$71$ \( -36920 + T \)
$73$ \( 61718 + T \)
$79$ \( 64752 + T \)
$83$ \( 77056 + T \)
$89$ \( 8166 + T \)
$97$ \( -20650 + T \)
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