Properties

Label 14.6.a.a
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

\(f(q)\) \(=\) \( q - 4q^{2} + 10q^{3} + 16q^{4} + 84q^{5} - 40q^{6} + 49q^{7} - 64q^{8} - 143q^{9} + O(q^{10}) \) \( q - 4q^{2} + 10q^{3} + 16q^{4} + 84q^{5} - 40q^{6} + 49q^{7} - 64q^{8} - 143q^{9} - 336q^{10} - 336q^{11} + 160q^{12} + 584q^{13} - 196q^{14} + 840q^{15} + 256q^{16} - 1458q^{17} + 572q^{18} + 470q^{19} + 1344q^{20} + 490q^{21} + 1344q^{22} - 4200q^{23} - 640q^{24} + 3931q^{25} - 2336q^{26} - 3860q^{27} + 784q^{28} + 4866q^{29} - 3360q^{30} - 7372q^{31} - 1024q^{32} - 3360q^{33} + 5832q^{34} + 4116q^{35} - 2288q^{36} + 14330q^{37} - 1880q^{38} + 5840q^{39} - 5376q^{40} + 6222q^{41} - 1960q^{42} + 3704q^{43} - 5376q^{44} - 12012q^{45} + 16800q^{46} - 1812q^{47} + 2560q^{48} + 2401q^{49} - 15724q^{50} - 14580q^{51} + 9344q^{52} - 37242q^{53} + 15440q^{54} - 28224q^{55} - 3136q^{56} + 4700q^{57} - 19464q^{58} + 34302q^{59} + 13440q^{60} + 24476q^{61} + 29488q^{62} - 7007q^{63} + 4096q^{64} + 49056q^{65} + 13440q^{66} - 17452q^{67} - 23328q^{68} - 42000q^{69} - 16464q^{70} + 28224q^{71} + 9152q^{72} + 3602q^{73} - 57320q^{74} + 39310q^{75} + 7520q^{76} - 16464q^{77} - 23360q^{78} + 42872q^{79} + 21504q^{80} - 3851q^{81} - 24888q^{82} - 35202q^{83} + 7840q^{84} - 122472q^{85} - 14816q^{86} + 48660q^{87} + 21504q^{88} + 26730q^{89} + 48048q^{90} + 28616q^{91} - 67200q^{92} - 73720q^{93} + 7248q^{94} + 39480q^{95} - 10240q^{96} - 16978q^{97} - 9604q^{98} + 48048q^{99} + O(q^{100}) \)

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 10.0000 16.0000 84.0000 −40.0000 49.0000 −64.0000 −143.000 −336.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 10 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(14))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T \)
$3$ \( 1 - 10 T + 243 T^{2} \)
$5$ \( 1 - 84 T + 3125 T^{2} \)
$7$ \( 1 - 49 T \)
$11$ \( 1 + 336 T + 161051 T^{2} \)
$13$ \( 1 - 584 T + 371293 T^{2} \)
$17$ \( 1 + 1458 T + 1419857 T^{2} \)
$19$ \( 1 - 470 T + 2476099 T^{2} \)
$23$ \( 1 + 4200 T + 6436343 T^{2} \)
$29$ \( 1 - 4866 T + 20511149 T^{2} \)
$31$ \( 1 + 7372 T + 28629151 T^{2} \)
$37$ \( 1 - 14330 T + 69343957 T^{2} \)
$41$ \( 1 - 6222 T + 115856201 T^{2} \)
$43$ \( 1 - 3704 T + 147008443 T^{2} \)
$47$ \( 1 + 1812 T + 229345007 T^{2} \)
$53$ \( 1 + 37242 T + 418195493 T^{2} \)
$59$ \( 1 - 34302 T + 714924299 T^{2} \)
$61$ \( 1 - 24476 T + 844596301 T^{2} \)
$67$ \( 1 + 17452 T + 1350125107 T^{2} \)
$71$ \( 1 - 28224 T + 1804229351 T^{2} \)
$73$ \( 1 - 3602 T + 2073071593 T^{2} \)
$79$ \( 1 - 42872 T + 3077056399 T^{2} \)
$83$ \( 1 + 35202 T + 3939040643 T^{2} \)
$89$ \( 1 - 26730 T + 5584059449 T^{2} \)
$97$ \( 1 + 16978 T + 8587340257 T^{2} \)
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