Properties

Label 14.2.a.a
Level $14$
Weight $2$
Character orbit 14.a
Self dual yes
Analytic conductor $0.112$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(0.111790562830\)
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

\(f(q)\) \(=\) \( q - q^{2} - 2 q^{3} + q^{4} + 2 q^{6} + q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - 2 q^{3} + q^{4} + 2 q^{6} + q^{7} - q^{8} + q^{9} - 2 q^{12} - 4 q^{13} - q^{14} + q^{16} + 6 q^{17} - q^{18} + 2 q^{19} - 2 q^{21} + 2 q^{24} - 5 q^{25} + 4 q^{26} + 4 q^{27} + q^{28} - 6 q^{29} - 4 q^{31} - q^{32} - 6 q^{34} + q^{36} + 2 q^{37} - 2 q^{38} + 8 q^{39} + 6 q^{41} + 2 q^{42} + 8 q^{43} - 12 q^{47} - 2 q^{48} + q^{49} + 5 q^{50} - 12 q^{51} - 4 q^{52} + 6 q^{53} - 4 q^{54} - q^{56} - 4 q^{57} + 6 q^{58} - 6 q^{59} + 8 q^{61} + 4 q^{62} + q^{63} + q^{64} - 4 q^{67} + 6 q^{68} - q^{72} + 2 q^{73} - 2 q^{74} + 10 q^{75} + 2 q^{76} - 8 q^{78} + 8 q^{79} - 11 q^{81} - 6 q^{82} - 6 q^{83} - 2 q^{84} - 8 q^{86} + 12 q^{87} - 6 q^{89} - 4 q^{91} + 8 q^{93} + 12 q^{94} + 2 q^{96} - 10 q^{97} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Expression as an eta quotient

\(f(z) = \eta(z)\eta(2z)\eta(7z)\eta(14z)=q\prod_{n=1}^\infty(1 - q^{n})^{}(1 - q^{2n})^{}(1 - q^{7n})^{}(1 - q^{14n})^{}\)

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
−1.00000 −2.00000 1.00000 0 2.00000 1.00000 −1.00000 1.00000 0
\(n\): 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 14.2.a.a 1
3.b odd 2 1 126.2.a.b 1
4.b odd 2 1 112.2.a.c 1
5.b even 2 1 350.2.a.f 1
5.c odd 4 2 350.2.c.d 2
7.b odd 2 1 98.2.a.a 1
7.c even 3 2 98.2.c.b 2
7.d odd 6 2 98.2.c.a 2
8.b even 2 1 448.2.a.g 1
8.d odd 2 1 448.2.a.a 1
9.c even 3 2 1134.2.f.l 2
9.d odd 6 2 1134.2.f.f 2
11.b odd 2 1 1694.2.a.e 1
12.b even 2 1 1008.2.a.h 1
13.b even 2 1 2366.2.a.j 1
13.d odd 4 2 2366.2.d.b 2
15.d odd 2 1 3150.2.a.i 1
15.e even 4 2 3150.2.g.j 2
16.e even 4 2 1792.2.b.c 2
16.f odd 4 2 1792.2.b.g 2
17.b even 2 1 4046.2.a.f 1
19.b odd 2 1 5054.2.a.c 1
20.d odd 2 1 2800.2.a.g 1
20.e even 4 2 2800.2.g.h 2
21.c even 2 1 882.2.a.i 1
21.g even 6 2 882.2.g.d 2
21.h odd 6 2 882.2.g.c 2
23.b odd 2 1 7406.2.a.a 1
24.f even 2 1 4032.2.a.r 1
24.h odd 2 1 4032.2.a.w 1
28.d even 2 1 784.2.a.b 1
28.f even 6 2 784.2.i.i 2
28.g odd 6 2 784.2.i.c 2
35.c odd 2 1 2450.2.a.t 1
35.f even 4 2 2450.2.c.c 2
56.e even 2 1 3136.2.a.z 1
56.h odd 2 1 3136.2.a.e 1
84.h odd 2 1 7056.2.a.bd 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.2.a.a 1 1.a even 1 1 trivial
98.2.a.a 1 7.b odd 2 1
98.2.c.a 2 7.d odd 6 2
98.2.c.b 2 7.c even 3 2
112.2.a.c 1 4.b odd 2 1
126.2.a.b 1 3.b odd 2 1
350.2.a.f 1 5.b even 2 1
350.2.c.d 2 5.c odd 4 2
448.2.a.a 1 8.d odd 2 1
448.2.a.g 1 8.b even 2 1
784.2.a.b 1 28.d even 2 1
784.2.i.c 2 28.g odd 6 2
784.2.i.i 2 28.f even 6 2
882.2.a.i 1 21.c even 2 1
882.2.g.c 2 21.h odd 6 2
882.2.g.d 2 21.g even 6 2
1008.2.a.h 1 12.b even 2 1
1134.2.f.f 2 9.d odd 6 2
1134.2.f.l 2 9.c even 3 2
1694.2.a.e 1 11.b odd 2 1
1792.2.b.c 2 16.e even 4 2
1792.2.b.g 2 16.f odd 4 2
2366.2.a.j 1 13.b even 2 1
2366.2.d.b 2 13.d odd 4 2
2450.2.a.t 1 35.c odd 2 1
2450.2.c.c 2 35.f even 4 2
2800.2.a.g 1 20.d odd 2 1
2800.2.g.h 2 20.e even 4 2
3136.2.a.e 1 56.h odd 2 1
3136.2.a.z 1 56.e even 2 1
3150.2.a.i 1 15.d odd 2 1
3150.2.g.j 2 15.e even 4 2
4032.2.a.r 1 24.f even 2 1
4032.2.a.w 1 24.h odd 2 1
4046.2.a.f 1 17.b even 2 1
5054.2.a.c 1 19.b odd 2 1
7056.2.a.bd 1 84.h odd 2 1
7406.2.a.a 1 23.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T + 2 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 4 \) Copy content Toggle raw display
$17$ \( T - 6 \) Copy content Toggle raw display
$19$ \( T - 2 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T - 2 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T - 8 \) Copy content Toggle raw display
$47$ \( T + 12 \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T + 6 \) Copy content Toggle raw display
$61$ \( T - 8 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 2 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T + 6 \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T + 10 \) Copy content Toggle raw display
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