Properties

Label 48.26.a.c
Level $48$
Weight $26$
Character orbit 48.a
Self dual yes
Analytic conductor $190.078$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 48.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(190.078454377\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 531441 q^{3} - 292754850 q^{5} - 3580644032 q^{7} + 282429536481 q^{9} + O(q^{10}) \) \( q + 531441 q^{3} - 292754850 q^{5} - 3580644032 q^{7} + 282429536481 q^{9} - 15111573238212 q^{11} + 1221071681246 q^{13} - 155581930238850 q^{15} + 2518250853863682 q^{17} + 7992693407413060 q^{19} - 1902901045010112 q^{21} + 99645642629247624 q^{23} - 212317821678430625 q^{25} + 150094635296999121 q^{27} - 2080672742244316890 q^{29} + 4937672075835729208 q^{31} - 8030909593288623492 q^{33} + 1048250906491555200 q^{35} + 19829154107621718182 q^{37} + 648927555353055486 q^{39} + 224696060863159376442 q^{41} + 72221008334482349884 q^{43} - 82682616588064682850 q^{45} - 189872435947262116992 q^{47} - 1328247607980067683783 q^{49} + 1338301752028169025762 q^{51} - 2645676034335389555874 q^{53} + 4423986356616768328200 q^{55} + 4247644977129004019460 q^{57} + 16454608826354674865340 q^{59} - 35546954389065591688738 q^{61} - 1011279634261218931392 q^{63} - 357474656882420543100 q^{65} - 106703750402023286661692 q^{67} + 52955779964529986546184 q^{69} - 73672004836753334994312 q^{71} - 262402855870448192600374 q^{73} - 112834395470606849780625 q^{75} + 54109164529534712150784 q^{77} + 1002642123108883497568840 q^{79} + 79766443076872509863361 q^{81} - 1558588706101147601425596 q^{83} - 737230150985234144357700 q^{85} - 1105754802811062012338490 q^{87} + 2181670205644666928498490 q^{89} - 4372223028097696223872 q^{91} + 2624081385654215766028728 q^{93} - 2339899759583199268341000 q^{95} - 440165308375605500117758 q^{97} - 4267954625166899357211972 q^{99} + O(q^{100}) \)

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 531441. 0 −2.92755e8 0 −3.58064e9 0 2.82430e11 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)

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 48.26.a.c 1
4.b odd 2 1 6.26.a.a 1
12.b even 2 1 18.26.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.26.a.a 1 4.b odd 2 1
18.26.a.d 1 12.b even 2 1
48.26.a.c 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 292754850 \) acting on \(S_{26}^{\mathrm{new}}(\Gamma_0(48))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -531441 + T \)
$5$ \( 292754850 + T \)
$7$ \( 3580644032 + T \)
$11$ \( 15111573238212 + T \)
$13$ \( -1221071681246 + T \)
$17$ \( -2518250853863682 + T \)
$19$ \( -7992693407413060 + T \)
$23$ \( -99645642629247624 + T \)
$29$ \( 2080672742244316890 + T \)
$31$ \( -4937672075835729208 + T \)
$37$ \( -19829154107621718182 + T \)
$41$ \( -\)\(22\!\cdots\!42\)\( + T \)
$43$ \( -72221008334482349884 + T \)
$47$ \( \)\(18\!\cdots\!92\)\( + T \)
$53$ \( \)\(26\!\cdots\!74\)\( + T \)
$59$ \( -\)\(16\!\cdots\!40\)\( + T \)
$61$ \( \)\(35\!\cdots\!38\)\( + T \)
$67$ \( \)\(10\!\cdots\!92\)\( + T \)
$71$ \( \)\(73\!\cdots\!12\)\( + T \)
$73$ \( \)\(26\!\cdots\!74\)\( + T \)
$79$ \( -\)\(10\!\cdots\!40\)\( + T \)
$83$ \( \)\(15\!\cdots\!96\)\( + T \)
$89$ \( -\)\(21\!\cdots\!90\)\( + T \)
$97$ \( \)\(44\!\cdots\!58\)\( + T \)
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