Properties

Label 10.12.a.a
Level 10
Weight 12
Character orbit 10.a
Self dual yes
Analytic conductor 7.683
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 10.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.68343180560\)
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

\(f(q)\) \(=\) \( q - 32q^{2} - 12q^{3} + 1024q^{4} + 3125q^{5} + 384q^{6} - 14176q^{7} - 32768q^{8} - 177003q^{9} + O(q^{10}) \) \( q - 32q^{2} - 12q^{3} + 1024q^{4} + 3125q^{5} + 384q^{6} - 14176q^{7} - 32768q^{8} - 177003q^{9} - 100000q^{10} - 756348q^{11} - 12288q^{12} - 905482q^{13} + 453632q^{14} - 37500q^{15} + 1048576q^{16} + 2803794q^{17} + 5664096q^{18} - 5428660q^{19} + 3200000q^{20} + 170112q^{21} + 24203136q^{22} - 10236672q^{23} + 393216q^{24} + 9765625q^{25} + 28975424q^{26} + 4249800q^{27} - 14516224q^{28} - 197498010q^{29} + 1200000q^{30} - 44362288q^{31} - 33554432q^{32} + 9076176q^{33} - 89721408q^{34} - 44300000q^{35} - 181251072q^{36} + 576737054q^{37} + 173717120q^{38} + 10865784q^{39} - 102400000q^{40} + 930058362q^{41} - 5443584q^{42} + 1605598988q^{43} - 774500352q^{44} - 553134375q^{45} + 327573504q^{46} - 1803684456q^{47} - 12582912q^{48} - 1776367767q^{49} - 312500000q^{50} - 33645528q^{51} - 927213568q^{52} + 1558674798q^{53} - 135993600q^{54} - 2363587500q^{55} + 464519168q^{56} + 65143920q^{57} + 6319936320q^{58} - 9501997020q^{59} - 38400000q^{60} + 6736320422q^{61} + 1419593216q^{62} + 2509194528q^{63} + 1073741824q^{64} - 2829631250q^{65} - 290437632q^{66} + 8402906564q^{67} + 2871085056q^{68} + 122840064q^{69} + 1417600000q^{70} - 4806306168q^{71} + 5800034304q^{72} + 7462713338q^{73} - 18455585728q^{74} - 117187500q^{75} - 5558947840q^{76} + 10721989248q^{77} - 347705088q^{78} - 20644540720q^{79} + 3276800000q^{80} + 31304552841q^{81} - 29761867584q^{82} - 68013349212q^{83} + 174194688q^{84} + 8761856250q^{85} - 51379167616q^{86} + 2369976120q^{87} + 24784011264q^{88} + 69871323210q^{89} + 17700300000q^{90} + 12836112832q^{91} - 10482352128q^{92} + 532347456q^{93} + 57717902592q^{94} - 16964562500q^{95} + 402653184q^{96} + 39960952514q^{97} + 56843768544q^{98} + 133875865044q^{99} + O(q^{100}) \)

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
−32.0000 −12.0000 1024.00 3125.00 384.000 −14176.0 −32768.0 −177003. −100000.
\(n\): 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 10.12.a.a 1
3.b odd 2 1 90.12.a.g 1
4.b odd 2 1 80.12.a.d 1
5.b even 2 1 50.12.a.d 1
5.c odd 4 2 50.12.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.12.a.a 1 1.a even 1 1 trivial
50.12.a.d 1 5.b even 2 1
50.12.b.c 2 5.c odd 4 2
80.12.a.d 1 4.b odd 2 1
90.12.a.g 1 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 12 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(10))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 32 T \)
$3$ \( 1 + 12 T + 177147 T^{2} \)
$5$ \( 1 - 3125 T \)
$7$ \( 1 + 14176 T + 1977326743 T^{2} \)
$11$ \( 1 + 756348 T + 285311670611 T^{2} \)
$13$ \( 1 + 905482 T + 1792160394037 T^{2} \)
$17$ \( 1 - 2803794 T + 34271896307633 T^{2} \)
$19$ \( 1 + 5428660 T + 116490258898219 T^{2} \)
$23$ \( 1 + 10236672 T + 952809757913927 T^{2} \)
$29$ \( 1 + 197498010 T + 12200509765705829 T^{2} \)
$31$ \( 1 + 44362288 T + 25408476896404831 T^{2} \)
$37$ \( 1 - 576737054 T + 177917621779460413 T^{2} \)
$41$ \( 1 - 930058362 T + 550329031716248441 T^{2} \)
$43$ \( 1 - 1605598988 T + 929293739471222707 T^{2} \)
$47$ \( 1 + 1803684456 T + 2472159215084012303 T^{2} \)
$53$ \( 1 - 1558674798 T + 9269035929372191597 T^{2} \)
$59$ \( 1 + 9501997020 T + 30155888444737842659 T^{2} \)
$61$ \( 1 - 6736320422 T + 43513917611435838661 T^{2} \)
$67$ \( 1 - 8402906564 T + \)\(12\!\cdots\!83\)\( T^{2} \)
$71$ \( 1 + 4806306168 T + \)\(23\!\cdots\!71\)\( T^{2} \)
$73$ \( 1 - 7462713338 T + \)\(31\!\cdots\!77\)\( T^{2} \)
$79$ \( 1 + 20644540720 T + \)\(74\!\cdots\!79\)\( T^{2} \)
$83$ \( 1 + 68013349212 T + \)\(12\!\cdots\!67\)\( T^{2} \)
$89$ \( 1 - 69871323210 T + \)\(27\!\cdots\!89\)\( T^{2} \)
$97$ \( 1 - 39960952514 T + \)\(71\!\cdots\!53\)\( T^{2} \)
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