Properties

Label 4.14.a.a
Level $4$
Weight $14$
Character orbit 4.a
Self dual yes
Analytic conductor $4.289$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 4.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.28923715808\)
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

\(f(q)\) \(=\) \( q + 468q^{3} + 56214q^{5} + 333032q^{7} - 1375299q^{9} + O(q^{10}) \) \( q + 468q^{3} + 56214q^{5} + 333032q^{7} - 1375299q^{9} - 6397380q^{11} + 15199742q^{13} + 26308152q^{15} + 43114194q^{17} - 365115484q^{19} + 155858976q^{21} - 57226824q^{23} + 1939310671q^{25} - 1389783096q^{27} - 46418994q^{29} - 5682185824q^{31} - 2993973840q^{33} + 18721060848q^{35} - 1887185098q^{37} + 7113479256q^{39} - 7336802934q^{41} - 26886674980q^{43} - 77311057986q^{45} + 101839834224q^{47} + 14021302617q^{49} + 20177442792q^{51} + 278731884294q^{53} - 359622319320q^{55} - 170874046512q^{57} + 59573945772q^{59} - 27484470418q^{61} - 458018576568q^{63} + 854438296788q^{65} + 784410054932q^{67} - 26782153632q^{69} - 360365227992q^{71} - 1592635413718q^{73} + 907597394028q^{75} - 2130532256160q^{77} - 23161184752q^{79} + 1542252338649q^{81} + 2050158110436q^{83} + 2423621301516q^{85} - 21724089192q^{87} - 3485391237126q^{89} + 5062000477744q^{91} - 2659262965632q^{93} - 20524601817576q^{95} + 6706667416802q^{97} + 8798310316620q^{99} + O(q^{100}) \)

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
0 468.000 0 56214.0 0 333032. 0 −1.37530e6 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 4.14.a.a 1
3.b odd 2 1 36.14.a.a 1
4.b odd 2 1 16.14.a.b 1
5.b even 2 1 100.14.a.a 1
5.c odd 4 2 100.14.c.a 2
7.b odd 2 1 196.14.a.a 1
7.c even 3 2 196.14.e.a 2
7.d odd 6 2 196.14.e.b 2
8.b even 2 1 64.14.a.c 1
8.d odd 2 1 64.14.a.g 1
12.b even 2 1 144.14.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.14.a.a 1 1.a even 1 1 trivial
16.14.a.b 1 4.b odd 2 1
36.14.a.a 1 3.b odd 2 1
64.14.a.c 1 8.b even 2 1
64.14.a.g 1 8.d odd 2 1
100.14.a.a 1 5.b even 2 1
100.14.c.a 2 5.c odd 4 2
144.14.a.a 1 12.b even 2 1
196.14.a.a 1 7.b odd 2 1
196.14.e.a 2 7.c even 3 2
196.14.e.b 2 7.d odd 6 2

Hecke kernels

This newform subspace is the entire newspace \(S_{14}^{\mathrm{new}}(\Gamma_0(4))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -468 + T \)
$5$ \( -56214 + T \)
$7$ \( -333032 + T \)
$11$ \( 6397380 + T \)
$13$ \( -15199742 + T \)
$17$ \( -43114194 + T \)
$19$ \( 365115484 + T \)
$23$ \( 57226824 + T \)
$29$ \( 46418994 + T \)
$31$ \( 5682185824 + T \)
$37$ \( 1887185098 + T \)
$41$ \( 7336802934 + T \)
$43$ \( 26886674980 + T \)
$47$ \( -101839834224 + T \)
$53$ \( -278731884294 + T \)
$59$ \( -59573945772 + T \)
$61$ \( 27484470418 + T \)
$67$ \( -784410054932 + T \)
$71$ \( 360365227992 + T \)
$73$ \( 1592635413718 + T \)
$79$ \( 23161184752 + T \)
$83$ \( -2050158110436 + T \)
$89$ \( 3485391237126 + T \)
$97$ \( -6706667416802 + T \)
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