Properties

Label 1.26.a.a
Level 1
Weight 26
Character orbit 1.a
Self dual yes
Analytic conductor 3.960
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.95996779952\)
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

\(f(q)\) \(=\) \( q - 48q^{2} - 195804q^{3} - 33552128q^{4} - 741989850q^{5} + 9398592q^{6} + 39080597192q^{7} + 3221114880q^{8} - 808949403027q^{9} + O(q^{10}) \) \( q - 48q^{2} - 195804q^{3} - 33552128q^{4} - 741989850q^{5} + 9398592q^{6} + 39080597192q^{7} + 3221114880q^{8} - 808949403027q^{9} + 35615512800q^{10} + 8419515299052q^{11} + 6569640870912q^{12} - 81651045335314q^{13} - 1875868665216q^{14} + 145284580589400q^{15} + 1125667983917056q^{16} - 2519900028948078q^{17} + 38829571345296q^{18} - 6082056370308940q^{19} + 24895338421900800q^{20} - 7652137252582368q^{21} - 404136734354496q^{22} - 94995280296320424q^{23} - 630707177963520q^{24} + 252525713626069375q^{25} + 3919250176095072q^{26} + 324298027793675880q^{27} - 1311237199302424576q^{28} - 271246959476737410q^{29} - 6973659868291200q^{30} + 4291666067521509152q^{31} - 162114743433166848q^{32} - 1648574773615577808q^{33} + 120955201389507744q^{34} - 28997406448402501200q^{35} + 27141973915885491456q^{36} + 20301484446109126982q^{37} + 291938705774829120q^{38} + 15987601280835822456q^{39} - 2390034546643968000q^{40} - 183744249574071224598q^{41} + 367302588123953664q^{42} + 300901824185586335756q^{43} - 282492655011750982656q^{44} + 600232246209593275950q^{45} + 4559773454223380352q^{46} - 924361048064704868688q^{47} - 220410293922895233024q^{48} + 186224457219393384057q^{49} - 12121234254051330000q^{50} + 493406505268149464712q^{51} + 2739566324424258248192q^{52} - 990292205554990470954q^{53} - 15566305334096442240q^{54} - 6247194893816298622200q^{55} + 125883093134437416960q^{56} + 1190890965531971687760q^{57} + 13019854054883395680q^{58} + 13052569416454201837980q^{59} - 4874606844361864243200q^{60} + 9015451224701414617502q^{61} - 205999971241032439296q^{62} - 31614225768407052500184q^{63} - 37763368313237157183488q^{64} + 60584246880692834562900q^{65} + 79131589133547734784q^{66} - 26689067808908579702428q^{67} + 84548008318469618409984q^{68} + 18600455863140724300896q^{69} + 1391875509523320057600q^{70} - 192390516186217637440248q^{71} - 2605718959257386741760q^{72} + 42404584838092453858826q^{73} - 974471253413238095136q^{74} - 49445544830838887902500q^{75} + 204065933839820954424320q^{76} + 329039685954132631461984q^{77} - 767404861480119477888q^{78} - 271681055025772277197360q^{79} - 835234218536418793881600q^{80} + 621914763766378892976441q^{81} + 8819723979555418780704q^{82} - 931454457307013524361484q^{83} + 256745488572211941679104q^{84} + 1869740244494180053008300q^{85} - 14443287560908144116288q^{86} + 53111239653383091827640q^{87} + 27120226012164047093760q^{88} - 1763635518049807316502630q^{89} - 28811147818060477245600q^{90} - 3190971613055137006838288q^{91} + 3187293803898020795062272q^{92} - 840325382684981577998208q^{93} + 44369330307105833697024q^{94} + 4512824093897074844259000q^{95} + 31742715223187801505792q^{96} + 2829240869926872086187362q^{97} - 8938773946530882434736q^{98} - 6810961874944808779030404q^{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
−48.0000 −195804. −3.35521e7 −7.41990e8 9.39859e6 3.90806e10 3.22111e9 −8.08949e11 3.56155e10
\(n\): 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 1.26.a.a 1
3.b odd 2 1 9.26.a.a 1
4.b odd 2 1 16.26.a.b 1
5.b even 2 1 25.26.a.a 1
5.c odd 4 2 25.26.b.a 2
7.b odd 2 1 49.26.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.26.a.a 1 1.a even 1 1 trivial
9.26.a.a 1 3.b odd 2 1
16.26.a.b 1 4.b odd 2 1
25.26.a.a 1 5.b even 2 1
25.26.b.a 2 5.c odd 4 2
49.26.a.a 1 7.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{26}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 48 T + 33554432 T^{2} \)
$3$ \( 1 + 195804 T + 847288609443 T^{2} \)
$5$ \( 1 + 741989850 T + 298023223876953125 T^{2} \)
$7$ \( 1 - 39080597192 T + \)\(13\!\cdots\!07\)\( T^{2} \)
$11$ \( 1 - 8419515299052 T + \)\(10\!\cdots\!51\)\( T^{2} \)
$13$ \( 1 + 81651045335314 T + \)\(70\!\cdots\!93\)\( T^{2} \)
$17$ \( 1 + 2519900028948078 T + \)\(57\!\cdots\!57\)\( T^{2} \)
$19$ \( 1 + 6082056370308940 T + \)\(93\!\cdots\!99\)\( T^{2} \)
$23$ \( 1 + 94995280296320424 T + \)\(11\!\cdots\!43\)\( T^{2} \)
$29$ \( 1 + 271246959476737410 T + \)\(36\!\cdots\!49\)\( T^{2} \)
$31$ \( 1 - 4291666067521509152 T + \)\(19\!\cdots\!51\)\( T^{2} \)
$37$ \( 1 - 20301484446109126982 T + \)\(16\!\cdots\!57\)\( T^{2} \)
$41$ \( 1 + \)\(18\!\cdots\!98\)\( T + \)\(20\!\cdots\!01\)\( T^{2} \)
$43$ \( 1 - \)\(30\!\cdots\!56\)\( T + \)\(68\!\cdots\!43\)\( T^{2} \)
$47$ \( 1 + \)\(92\!\cdots\!88\)\( T + \)\(63\!\cdots\!07\)\( T^{2} \)
$53$ \( 1 + \)\(99\!\cdots\!54\)\( T + \)\(12\!\cdots\!93\)\( T^{2} \)
$59$ \( 1 - \)\(13\!\cdots\!80\)\( T + \)\(18\!\cdots\!99\)\( T^{2} \)
$61$ \( 1 - \)\(90\!\cdots\!02\)\( T + \)\(42\!\cdots\!01\)\( T^{2} \)
$67$ \( 1 + \)\(26\!\cdots\!28\)\( T + \)\(44\!\cdots\!07\)\( T^{2} \)
$71$ \( 1 + \)\(19\!\cdots\!48\)\( T + \)\(19\!\cdots\!51\)\( T^{2} \)
$73$ \( 1 - \)\(42\!\cdots\!26\)\( T + \)\(38\!\cdots\!93\)\( T^{2} \)
$79$ \( 1 + \)\(27\!\cdots\!60\)\( T + \)\(27\!\cdots\!99\)\( T^{2} \)
$83$ \( 1 + \)\(93\!\cdots\!84\)\( T + \)\(94\!\cdots\!43\)\( T^{2} \)
$89$ \( 1 + \)\(17\!\cdots\!30\)\( T + \)\(54\!\cdots\!49\)\( T^{2} \)
$97$ \( 1 - \)\(28\!\cdots\!62\)\( T + \)\(46\!\cdots\!57\)\( T^{2} \)
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