Properties

Label 4.12.a.a
Level 4
Weight 12
Character orbit 4.a
Self dual yes
Analytic conductor 3.073
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 4 = 2^{2} \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 4.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.07337272224\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q - 516q^{3} - 10530q^{5} + 49304q^{7} + 89109q^{9} + O(q^{10}) \) \( q - 516q^{3} - 10530q^{5} + 49304q^{7} + 89109q^{9} - 309420q^{11} - 1723594q^{13} + 5433480q^{15} - 2279502q^{17} + 4550444q^{19} - 25440864q^{21} - 7282872q^{23} + 62052775q^{25} + 45427608q^{27} - 69040026q^{29} - 141740704q^{31} + 159660720q^{33} - 519171120q^{35} + 711366974q^{37} + 889374504q^{39} - 1225262214q^{41} - 33606220q^{43} - 938317770q^{45} + 123214608q^{47} + 453557673q^{49} + 1176223032q^{51} + 1106121582q^{53} + 3258192600q^{55} - 2348029104q^{57} - 9062779932q^{59} - 3854150458q^{61} + 4393430136q^{63} + 18149444820q^{65} - 15313764676q^{67} + 3757961952q^{69} + 20619626328q^{71} - 2063718694q^{73} - 32019231900q^{75} - 15255643680q^{77} + 13689871472q^{79} - 39226037751q^{81} + 65570428908q^{83} + 24003156060q^{85} + 35624653416q^{87} - 29715508854q^{89} - 84980078576q^{91} + 73138203264q^{93} - 47916175320q^{95} - 23439626206q^{97} - 27572106780q^{99} + O(q^{100}) \)

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 −516.000 0 −10530.0 0 49304.0 0 89109.0 0
\(n\): 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 4.12.a.a 1
3.b odd 2 1 36.12.a.d 1
4.b odd 2 1 16.12.a.c 1
5.b even 2 1 100.12.a.b 1
5.c odd 4 2 100.12.c.a 2
7.b odd 2 1 196.12.a.a 1
7.c even 3 2 196.12.e.b 2
7.d odd 6 2 196.12.e.a 2
8.b even 2 1 64.12.a.g 1
8.d odd 2 1 64.12.a.a 1
12.b even 2 1 144.12.a.n 1
16.e even 4 2 256.12.b.b 2
16.f odd 4 2 256.12.b.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.12.a.a 1 1.a even 1 1 trivial
16.12.a.c 1 4.b odd 2 1
36.12.a.d 1 3.b odd 2 1
64.12.a.a 1 8.d odd 2 1
64.12.a.g 1 8.b even 2 1
100.12.a.b 1 5.b even 2 1
100.12.c.a 2 5.c odd 4 2
144.12.a.n 1 12.b even 2 1
196.12.a.a 1 7.b odd 2 1
196.12.e.a 2 7.d odd 6 2
196.12.e.b 2 7.c even 3 2
256.12.b.b 2 16.e even 4 2
256.12.b.f 2 16.f odd 4 2

Atkin-Lehner signs

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

Hecke kernels

This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(\Gamma_0(4))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 + 516 T + 177147 T^{2} \)
$5$ \( 1 + 10530 T + 48828125 T^{2} \)
$7$ \( 1 - 49304 T + 1977326743 T^{2} \)
$11$ \( 1 + 309420 T + 285311670611 T^{2} \)
$13$ \( 1 + 1723594 T + 1792160394037 T^{2} \)
$17$ \( 1 + 2279502 T + 34271896307633 T^{2} \)
$19$ \( 1 - 4550444 T + 116490258898219 T^{2} \)
$23$ \( 1 + 7282872 T + 952809757913927 T^{2} \)
$29$ \( 1 + 69040026 T + 12200509765705829 T^{2} \)
$31$ \( 1 + 141740704 T + 25408476896404831 T^{2} \)
$37$ \( 1 - 711366974 T + 177917621779460413 T^{2} \)
$41$ \( 1 + 1225262214 T + 550329031716248441 T^{2} \)
$43$ \( 1 + 33606220 T + 929293739471222707 T^{2} \)
$47$ \( 1 - 123214608 T + 2472159215084012303 T^{2} \)
$53$ \( 1 - 1106121582 T + 9269035929372191597 T^{2} \)
$59$ \( 1 + 9062779932 T + 30155888444737842659 T^{2} \)
$61$ \( 1 + 3854150458 T + 43513917611435838661 T^{2} \)
$67$ \( 1 + 15313764676 T + \)\(12\!\cdots\!83\)\( T^{2} \)
$71$ \( 1 - 20619626328 T + \)\(23\!\cdots\!71\)\( T^{2} \)
$73$ \( 1 + 2063718694 T + \)\(31\!\cdots\!77\)\( T^{2} \)
$79$ \( 1 - 13689871472 T + \)\(74\!\cdots\!79\)\( T^{2} \)
$83$ \( 1 - 65570428908 T + \)\(12\!\cdots\!67\)\( T^{2} \)
$89$ \( 1 + 29715508854 T + \)\(27\!\cdots\!89\)\( T^{2} \)
$97$ \( 1 + 23439626206 T + \)\(71\!\cdots\!53\)\( T^{2} \)
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