Properties

Label 10.14.a.a
Level 10
Weight 14
Character orbit 10.a
Self dual yes
Analytic conductor 10.723
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 14 \)
Character orbit: \([\chi]\) = 10.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.7230928952\)
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

\(f(q)\) \(=\) \( q - 64q^{2} - 26q^{3} + 4096q^{4} - 15625q^{5} + 1664q^{6} + 538538q^{7} - 262144q^{8} - 1593647q^{9} + O(q^{10}) \) \( q - 64q^{2} - 26q^{3} + 4096q^{4} - 15625q^{5} + 1664q^{6} + 538538q^{7} - 262144q^{8} - 1593647q^{9} + 1000000q^{10} - 3994848q^{11} - 106496q^{12} - 23834446q^{13} - 34466432q^{14} + 406250q^{15} + 16777216q^{16} - 192273222q^{17} + 101993408q^{18} + 166485740q^{19} - 64000000q^{20} - 14001988q^{21} + 255670272q^{22} - 366866946q^{23} + 6815744q^{24} + 244140625q^{25} + 1525404544q^{26} + 82887220q^{27} + 2205851648q^{28} + 989855670q^{29} - 26000000q^{30} - 3445048468q^{31} - 1073741824q^{32} + 103866048q^{33} + 12305486208q^{34} - 8414656250q^{35} - 6527578112q^{36} - 29429851822q^{37} - 10655087360q^{38} + 619695596q^{39} + 4096000000q^{40} + 7043712582q^{41} + 896127232q^{42} + 8228005214q^{43} - 16362897408q^{44} + 24900734375q^{45} + 23479484544q^{46} + 45741859938q^{47} - 436207616q^{48} + 193134167037q^{49} - 15625000000q^{50} + 4999103772q^{51} - 97625890816q^{52} - 90591954486q^{53} - 5304782080q^{54} + 62419500000q^{55} - 141174505472q^{56} - 4328629240q^{57} - 63350762880q^{58} + 126033098940q^{59} + 1664000000q^{60} - 292123673038q^{61} + 220483101952q^{62} - 858239468086q^{63} + 68719476736q^{64} + 372413218750q^{65} - 6647427072q^{66} + 572402067098q^{67} - 787551117312q^{68} + 9538540596q^{69} + 538538000000q^{70} - 1284329422908q^{71} + 417764999168q^{72} + 196494986594q^{73} + 1883510516608q^{74} - 6347656250q^{75} + 681925591040q^{76} - 2151377452224q^{77} - 39660518144q^{78} + 3776797097000q^{79} - 262144000000q^{80} + 2538632998261q^{81} - 450797605248q^{82} - 4556844205746q^{83} - 57352142848q^{84} + 3004269093750q^{85} - 526592333696q^{86} - 25736247420q^{87} + 1047225434112q^{88} + 3748393684890q^{89} - 1593647000000q^{90} - 12835754879948q^{91} - 1502687010816q^{92} + 89571260168q^{93} - 2927479036032q^{94} - 2601339687500q^{95} + 27917287424q^{96} - 2743981383742q^{97} - 12360586690368q^{98} + 6366377530656q^{99} + O(q^{100}) \)

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
−64.0000 −26.0000 4096.00 −15625.0 1664.00 538538. −262144. −1.59365e6 1.00000e6
\(n\): 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 10.14.a.a 1
3.b odd 2 1 90.14.a.i 1
4.b odd 2 1 80.14.a.b 1
5.b even 2 1 50.14.a.d 1
5.c odd 4 2 50.14.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.14.a.a 1 1.a even 1 1 trivial
50.14.a.d 1 5.b even 2 1
50.14.b.c 2 5.c odd 4 2
80.14.a.b 1 4.b odd 2 1
90.14.a.i 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 \( T_{3} + 26 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(10))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 64 T \)
$3$ \( 1 + 26 T + 1594323 T^{2} \)
$5$ \( 1 + 15625 T \)
$7$ \( 1 - 538538 T + 96889010407 T^{2} \)
$11$ \( 1 + 3994848 T + 34522712143931 T^{2} \)
$13$ \( 1 + 23834446 T + 302875106592253 T^{2} \)
$17$ \( 1 + 192273222 T + 9904578032905937 T^{2} \)
$19$ \( 1 - 166485740 T + 42052983462257059 T^{2} \)
$23$ \( 1 + 366866946 T + 504036361936467383 T^{2} \)
$29$ \( 1 - 989855670 T + 10260628712958602189 T^{2} \)
$31$ \( 1 + 3445048468 T + 24417546297445042591 T^{2} \)
$37$ \( 1 + 29429851822 T + \)\(24\!\cdots\!97\)\( T^{2} \)
$41$ \( 1 - 7043712582 T + \)\(92\!\cdots\!21\)\( T^{2} \)
$43$ \( 1 - 8228005214 T + \)\(17\!\cdots\!43\)\( T^{2} \)
$47$ \( 1 - 45741859938 T + \)\(54\!\cdots\!27\)\( T^{2} \)
$53$ \( 1 + 90591954486 T + \)\(26\!\cdots\!73\)\( T^{2} \)
$59$ \( 1 - 126033098940 T + \)\(10\!\cdots\!79\)\( T^{2} \)
$61$ \( 1 + 292123673038 T + \)\(16\!\cdots\!81\)\( T^{2} \)
$67$ \( 1 - 572402067098 T + \)\(54\!\cdots\!87\)\( T^{2} \)
$71$ \( 1 + 1284329422908 T + \)\(11\!\cdots\!11\)\( T^{2} \)
$73$ \( 1 - 196494986594 T + \)\(16\!\cdots\!33\)\( T^{2} \)
$79$ \( 1 - 3776797097000 T + \)\(46\!\cdots\!39\)\( T^{2} \)
$83$ \( 1 + 4556844205746 T + \)\(88\!\cdots\!63\)\( T^{2} \)
$89$ \( 1 - 3748393684890 T + \)\(21\!\cdots\!69\)\( T^{2} \)
$97$ \( 1 + 2743981383742 T + \)\(67\!\cdots\!77\)\( T^{2} \)
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