Properties

Label 10.12.a.b
Level 10
Weight 12
Character orbit 10.a
Self dual yes
Analytic conductor 7.683
Analytic rank 0
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: \(0\)
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

\(f(q)\) \(=\) \( q - 32q^{2} + 738q^{3} + 1024q^{4} - 3125q^{5} - 23616q^{6} + 25574q^{7} - 32768q^{8} + 367497q^{9} + O(q^{10}) \) \( q - 32q^{2} + 738q^{3} + 1024q^{4} - 3125q^{5} - 23616q^{6} + 25574q^{7} - 32768q^{8} + 367497q^{9} + 100000q^{10} + 769152q^{11} + 755712q^{12} - 918982q^{13} - 818368q^{14} - 2306250q^{15} + 1048576q^{16} + 10312794q^{17} - 11759904q^{18} - 5521660q^{19} - 3200000q^{20} + 18873612q^{21} - 24612864q^{22} - 39973422q^{23} - 24182784q^{24} + 9765625q^{25} + 29407424q^{26} + 140478300q^{27} + 26187776q^{28} - 15269010q^{29} + 73800000q^{30} - 241583788q^{31} - 33554432q^{32} + 567634176q^{33} - 330009408q^{34} - 79918750q^{35} + 376316928q^{36} - 25751446q^{37} + 176693120q^{38} - 678208716q^{39} + 102400000q^{40} - 1217700138q^{41} - 603955584q^{42} - 683436262q^{43} + 787611648q^{44} - 1148428125q^{45} + 1279149504q^{46} + 1537395294q^{47} + 773849088q^{48} - 1323297267q^{49} - 312500000q^{50} + 7610841972q^{51} - 941037568q^{52} + 3572891298q^{53} - 4495305600q^{54} - 2403600000q^{55} - 838008832q^{56} - 4074985080q^{57} + 488608320q^{58} - 1069039020q^{59} - 2361600000q^{60} - 2091535078q^{61} + 7730681216q^{62} + 9398368278q^{63} + 1073741824q^{64} + 2871818750q^{65} - 18164293632q^{66} - 1462369186q^{67} + 10560301056q^{68} - 29500385436q^{69} + 2557400000q^{70} + 9660178332q^{71} - 12042141696q^{72} - 5603447662q^{73} + 824046272q^{74} + 7207031250q^{75} - 5654179840q^{76} + 19670293248q^{77} + 21702678912q^{78} + 5026936280q^{79} - 3276800000q^{80} + 38571994341q^{81} + 38966404416q^{82} - 38405955462q^{83} + 19326578688q^{84} - 32227481250q^{85} + 21869960384q^{86} - 11268529380q^{87} - 25203572736q^{88} + 35558583210q^{89} + 36749700000q^{90} - 23502045668q^{91} - 40932784128q^{92} - 178288835544q^{93} - 49196649408q^{94} + 17255187500q^{95} - 24763170816q^{96} + 10572232514q^{97} + 42345512544q^{98} + 282661052544q^{99} + O(q^{100}) \)

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 738.000 1024.00 −3125.00 −23616.0 25574.0 −32768.0 367497. 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.b 1
3.b odd 2 1 90.12.a.k 1
4.b odd 2 1 80.12.a.a 1
5.b even 2 1 50.12.a.c 1
5.c odd 4 2 50.12.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.12.a.b 1 1.a even 1 1 trivial
50.12.a.c 1 5.b even 2 1
50.12.b.a 2 5.c odd 4 2
80.12.a.a 1 4.b odd 2 1
90.12.a.k 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 \( T_{3} - 738 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(10))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 32 T \)
$3$ \( 1 - 738 T + 177147 T^{2} \)
$5$ \( 1 + 3125 T \)
$7$ \( 1 - 25574 T + 1977326743 T^{2} \)
$11$ \( 1 - 769152 T + 285311670611 T^{2} \)
$13$ \( 1 + 918982 T + 1792160394037 T^{2} \)
$17$ \( 1 - 10312794 T + 34271896307633 T^{2} \)
$19$ \( 1 + 5521660 T + 116490258898219 T^{2} \)
$23$ \( 1 + 39973422 T + 952809757913927 T^{2} \)
$29$ \( 1 + 15269010 T + 12200509765705829 T^{2} \)
$31$ \( 1 + 241583788 T + 25408476896404831 T^{2} \)
$37$ \( 1 + 25751446 T + 177917621779460413 T^{2} \)
$41$ \( 1 + 1217700138 T + 550329031716248441 T^{2} \)
$43$ \( 1 + 683436262 T + 929293739471222707 T^{2} \)
$47$ \( 1 - 1537395294 T + 2472159215084012303 T^{2} \)
$53$ \( 1 - 3572891298 T + 9269035929372191597 T^{2} \)
$59$ \( 1 + 1069039020 T + 30155888444737842659 T^{2} \)
$61$ \( 1 + 2091535078 T + 43513917611435838661 T^{2} \)
$67$ \( 1 + 1462369186 T + \)\(12\!\cdots\!83\)\( T^{2} \)
$71$ \( 1 - 9660178332 T + \)\(23\!\cdots\!71\)\( T^{2} \)
$73$ \( 1 + 5603447662 T + \)\(31\!\cdots\!77\)\( T^{2} \)
$79$ \( 1 - 5026936280 T + \)\(74\!\cdots\!79\)\( T^{2} \)
$83$ \( 1 + 38405955462 T + \)\(12\!\cdots\!67\)\( T^{2} \)
$89$ \( 1 - 35558583210 T + \)\(27\!\cdots\!89\)\( T^{2} \)
$97$ \( 1 - 10572232514 T + \)\(71\!\cdots\!53\)\( T^{2} \)
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