Properties

Label 4.10.a.a
Level $4$
Weight $10$
Character orbit 4.a
Self dual yes
Analytic conductor $2.060$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 4.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.06014334466\)
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

\(f(q)\) \(=\) \( q + 228q^{3} - 666q^{5} - 6328q^{7} + 32301q^{9} + O(q^{10}) \) \( q + 228q^{3} - 666q^{5} - 6328q^{7} + 32301q^{9} - 30420q^{11} - 32338q^{13} - 151848q^{15} + 590994q^{17} + 34676q^{19} - 1442784q^{21} + 1048536q^{23} - 1509569q^{25} + 2876904q^{27} + 4409406q^{29} - 7401184q^{31} - 6935760q^{33} + 4214448q^{35} + 10234502q^{37} - 7373064q^{39} + 18352746q^{41} - 252340q^{43} - 21512466q^{45} - 49517136q^{47} - 310023q^{49} + 134746632q^{51} - 66396906q^{53} + 20259720q^{55} + 7906128q^{57} - 61523748q^{59} + 35638622q^{61} - 204400728q^{63} + 21537108q^{65} + 181742372q^{67} + 239066208q^{69} + 90904968q^{71} - 262978678q^{73} - 344181732q^{75} + 192497760q^{77} - 116502832q^{79} + 20153529q^{81} - 9563724q^{83} - 393602004q^{85} + 1005344568q^{87} + 611826714q^{89} + 204634864q^{91} - 1687469952q^{93} - 23094216q^{95} - 259312798q^{97} - 982596420q^{99} + O(q^{100}) \)

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
0 228.000 0 −666.000 0 −6328.00 0 32301.0 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 4.10.a.a 1
3.b odd 2 1 36.10.a.b 1
4.b odd 2 1 16.10.a.a 1
5.b even 2 1 100.10.a.a 1
5.c odd 4 2 100.10.c.a 2
7.b odd 2 1 196.10.a.a 1
7.c even 3 2 196.10.e.a 2
7.d odd 6 2 196.10.e.b 2
8.b even 2 1 64.10.a.a 1
8.d odd 2 1 64.10.a.i 1
9.c even 3 2 324.10.e.e 2
9.d odd 6 2 324.10.e.b 2
12.b even 2 1 144.10.a.j 1
16.e even 4 2 256.10.b.j 2
16.f odd 4 2 256.10.b.b 2
20.d odd 2 1 400.10.a.k 1
20.e even 4 2 400.10.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.10.a.a 1 1.a even 1 1 trivial
16.10.a.a 1 4.b odd 2 1
36.10.a.b 1 3.b odd 2 1
64.10.a.a 1 8.b even 2 1
64.10.a.i 1 8.d odd 2 1
100.10.a.a 1 5.b even 2 1
100.10.c.a 2 5.c odd 4 2
144.10.a.j 1 12.b even 2 1
196.10.a.a 1 7.b odd 2 1
196.10.e.a 2 7.c even 3 2
196.10.e.b 2 7.d odd 6 2
256.10.b.b 2 16.f odd 4 2
256.10.b.j 2 16.e even 4 2
324.10.e.b 2 9.d odd 6 2
324.10.e.e 2 9.c even 3 2
400.10.a.k 1 20.d odd 2 1
400.10.c.a 2 20.e even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(\Gamma_0(4))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -228 + T \)
$5$ \( 666 + T \)
$7$ \( 6328 + T \)
$11$ \( 30420 + T \)
$13$ \( 32338 + T \)
$17$ \( -590994 + T \)
$19$ \( -34676 + T \)
$23$ \( -1048536 + T \)
$29$ \( -4409406 + T \)
$31$ \( 7401184 + T \)
$37$ \( -10234502 + T \)
$41$ \( -18352746 + T \)
$43$ \( 252340 + T \)
$47$ \( 49517136 + T \)
$53$ \( 66396906 + T \)
$59$ \( 61523748 + T \)
$61$ \( -35638622 + T \)
$67$ \( -181742372 + T \)
$71$ \( -90904968 + T \)
$73$ \( 262978678 + T \)
$79$ \( 116502832 + T \)
$83$ \( 9563724 + T \)
$89$ \( -611826714 + T \)
$97$ \( 259312798 + T \)
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