Properties

Label 18.26.a.d
Level $18$
Weight $26$
Character orbit 18.a
Self dual yes
Analytic conductor $71.279$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 18.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(71.2794203914\)
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

\(f(q)\) \(=\) \( q + 4096 q^{2} + 16777216 q^{4} + 292754850 q^{5} + 3580644032 q^{7} + 68719476736 q^{8} + O(q^{10}) \) \( q + 4096 q^{2} + 16777216 q^{4} + 292754850 q^{5} + 3580644032 q^{7} + 68719476736 q^{8} + 1199123865600 q^{10} - 15111573238212 q^{11} + 1221071681246 q^{13} + 14666317955072 q^{14} + 281474976710656 q^{16} - 2518250853863682 q^{17} - 7992693407413060 q^{19} + 4911611353497600 q^{20} - 61897003983716352 q^{22} + 99645642629247624 q^{23} - 212317821678430625 q^{25} + 5001509606383616 q^{26} + 60073238343974912 q^{28} + 2080672742244316890 q^{29} - 4937672075835729208 q^{31} + 1152921504606846976 q^{32} - 10314755497425641472 q^{34} + 1048250906491555200 q^{35} + 19829154107621718182 q^{37} - 32738072196763893760 q^{38} + 20117960103926169600 q^{40} - 224696060863159376442 q^{41} - 72221008334482349884 q^{43} - 253530128317302177792 q^{44} + 408148552209398267904 q^{46} - 189872435947262116992 q^{47} - 1328247607980067683783 q^{49} - 869653797594851840000 q^{50} + 20486183347747291136 q^{52} + 2645676034335389555874 q^{53} - 4423986356616768328200 q^{55} + 246059984256921239552 q^{56} + 8522435552232721981440 q^{58} + 16454608826354674865340 q^{59} - 35546954389065591688738 q^{61} - 20224704822623146835968 q^{62} + 4722366482869645213696 q^{64} + 357474656882420543100 q^{65} + 106703750402023286661692 q^{67} - 42249238517455427469312 q^{68} + 4293635712989410099200 q^{70} - 73672004836753334994312 q^{71} - 262402855870448192600374 q^{73} + 81220215224818557673472 q^{74} - 134095143717944908840960 q^{76} - 54109164529534712150784 q^{77} - 1002642123108883497568840 q^{79} + 82403164585681590681600 q^{80} - 920355065295500805906432 q^{82} - 1558588706101147601425596 q^{83} - 737230150985234144357700 q^{85} - 295817250138039705124864 q^{86} - 1038459405587669720236032 q^{88} - 2181670205644666928498490 q^{89} + 4372223028097696223872 q^{91} + 1671776469849695305334784 q^{92} - 777717497639985631199232 q^{94} - 2339899759583199268341000 q^{95} - 440165308375605500117758 q^{97} - 5440502202286357232775168 q^{98} + O(q^{100}) \)

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
4096.00 0 1.67772e7 2.92755e8 0 3.58064e9 6.87195e10 0 1.19912e12
\(n\): 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 18.26.a.d 1
3.b odd 2 1 6.26.a.a 1
12.b even 2 1 48.26.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.26.a.a 1 3.b odd 2 1
18.26.a.d 1 1.a even 1 1 trivial
48.26.a.c 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 292754850 \) acting on \(S_{26}^{\mathrm{new}}(\Gamma_0(18))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4096 + T \)
$3$ \( T \)
$5$ \( -292754850 + T \)
$7$ \( -3580644032 + T \)
$11$ \( 15111573238212 + T \)
$13$ \( -1221071681246 + T \)
$17$ \( 2518250853863682 + T \)
$19$ \( 7992693407413060 + T \)
$23$ \( -99645642629247624 + T \)
$29$ \( -2080672742244316890 + T \)
$31$ \( 4937672075835729208 + T \)
$37$ \( -19829154107621718182 + T \)
$41$ \( \)\(22\!\cdots\!42\)\( + T \)
$43$ \( 72221008334482349884 + T \)
$47$ \( \)\(18\!\cdots\!92\)\( + T \)
$53$ \( -\)\(26\!\cdots\!74\)\( + T \)
$59$ \( -\)\(16\!\cdots\!40\)\( + T \)
$61$ \( \)\(35\!\cdots\!38\)\( + T \)
$67$ \( -\)\(10\!\cdots\!92\)\( + T \)
$71$ \( \)\(73\!\cdots\!12\)\( + T \)
$73$ \( \)\(26\!\cdots\!74\)\( + T \)
$79$ \( \)\(10\!\cdots\!40\)\( + T \)
$83$ \( \)\(15\!\cdots\!96\)\( + T \)
$89$ \( \)\(21\!\cdots\!90\)\( + T \)
$97$ \( \)\(44\!\cdots\!58\)\( + T \)
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