Properties

Label 10.20.a.a
Level 10
Weight 20
Character orbit 10.a
Self dual yes
Analytic conductor 22.882
Analytic rank 0
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 20 \)
Character orbit: \([\chi]\) = 10.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.8816696556\)
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

\(f(q)\) \(=\) \( q - 512q^{2} - 26622q^{3} + 262144q^{4} - 1953125q^{5} + 13630464q^{6} - 39884026q^{7} - 134217728q^{8} - 453530583q^{9} + O(q^{10}) \) \( q - 512q^{2} - 26622q^{3} + 262144q^{4} - 1953125q^{5} + 13630464q^{6} - 39884026q^{7} - 134217728q^{8} - 453530583q^{9} + 1000000000q^{10} - 10161579168q^{11} - 6978797568q^{12} - 26970649702q^{13} + 20420621312q^{14} + 51996093750q^{15} + 68719476736q^{16} - 80154753126q^{17} + 232207658496q^{18} - 1169772071260q^{19} - 512000000000q^{20} + 1061792540172q^{21} + 5202728534016q^{22} + 13795883851698q^{23} + 3573144354816q^{24} + 3814697265625q^{25} + 13808972647424q^{26} + 43015615955100q^{27} - 10455358111744q^{28} + 65324757765390q^{29} - 26622000000000q^{30} - 8926539984748q^{31} - 35184372088832q^{32} + 270521560610496q^{33} + 41039233600512q^{34} + 77898488281250q^{35} - 118890321149952q^{36} + 525454617064394q^{37} + 598923300485120q^{38} + 718012636366644q^{39} + 262144000000000q^{40} - 2635226882131818q^{41} - 543637780568064q^{42} - 1501708702325062q^{43} - 2663797009416192q^{44} + 885801919921875q^{45} - 7063492532069376q^{46} - 3651608570665986q^{47} - 1829449909665792q^{48} - 9808159655404467q^{49} - 1953125000000000q^{50} + 2133879837720372q^{51} - 7070193995481088q^{52} + 43306800238889538q^{53} - 22023995369011200q^{54} + 19846834312500000q^{55} + 5353143353212928q^{56} + 31141672081083720q^{57} - 33446275975879680q^{58} + 51652090463616180q^{59} + 13630464000000000q^{60} + 45200043953043002q^{61} + 4570388472190976q^{62} + 18088625564167158q^{63} + 18014398509481984q^{64} + 52677050199218750q^{65} - 138507039032573952q^{66} + 322077213275888894q^{67} - 21012087603462144q^{68} - 367274019899904156q^{69} - 39884026000000000q^{70} + 393293311705873692q^{71} + 60871844428775424q^{72} - 672469661893471342q^{73} - 269032763936969728q^{74} - 101554870605468750q^{75} - 306648729848381440q^{76} + 405284687737570368q^{77} - 367622469819721728q^{78} - 482639101471927720q^{79} - 134217728000000000q^{80} - 618040607229726939q^{81} + 1349236163651490816q^{82} - 313265345629507302q^{83} + 278342543650848768q^{84} + 156552252199218750q^{85} + 768874855590431744q^{86} - 1739075701230212580q^{87} + 1363864068821090304q^{88} - 4230101056729722390q^{89} - 453530583000000000q^{90} + 1075698093951460252q^{91} + 3616508176419520512q^{92} + 237642347473961256q^{93} + 1869623588180984832q^{94} + 2284711076679687500q^{95} + 936678353748885504q^{96} + 354705113301714434q^{97} + 5021777743567087104q^{98} + 4608586924263694944q^{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
−512.000 −26622.0 262144. −1.95313e6 1.36305e7 −3.98840e7 −1.34218e8 −4.53531e8 1.00000e9
\(n\): 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 10.20.a.a 1
3.b odd 2 1 90.20.a.e 1
4.b odd 2 1 80.20.a.c 1
5.b even 2 1 50.20.a.e 1
5.c odd 4 2 50.20.b.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.20.a.a 1 1.a even 1 1 trivial
50.20.a.e 1 5.b even 2 1
50.20.b.e 2 5.c odd 4 2
80.20.a.c 1 4.b odd 2 1
90.20.a.e 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 \( T_{3} + 26622 \) acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(10))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 512 T \)
$3$ \( 1 + 26622 T + 1162261467 T^{2} \)
$5$ \( 1 + 1953125 T \)
$7$ \( 1 + 39884026 T + 11398895185373143 T^{2} \)
$11$ \( 1 + 10161579168 T + 61159090448414546291 T^{2} \)
$13$ \( 1 + 26970649702 T + \)\(14\!\cdots\!77\)\( T^{2} \)
$17$ \( 1 + 80154753126 T + \)\(23\!\cdots\!53\)\( T^{2} \)
$19$ \( 1 + 1169772071260 T + \)\(19\!\cdots\!79\)\( T^{2} \)
$23$ \( 1 - 13795883851698 T + \)\(74\!\cdots\!87\)\( T^{2} \)
$29$ \( 1 - 65324757765390 T + \)\(61\!\cdots\!69\)\( T^{2} \)
$31$ \( 1 + 8926539984748 T + \)\(21\!\cdots\!71\)\( T^{2} \)
$37$ \( 1 - 525454617064394 T + \)\(62\!\cdots\!73\)\( T^{2} \)
$41$ \( 1 + 2635226882131818 T + \)\(43\!\cdots\!61\)\( T^{2} \)
$43$ \( 1 + 1501708702325062 T + \)\(10\!\cdots\!07\)\( T^{2} \)
$47$ \( 1 + 3651608570665986 T + \)\(58\!\cdots\!83\)\( T^{2} \)
$53$ \( 1 - 43306800238889538 T + \)\(57\!\cdots\!17\)\( T^{2} \)
$59$ \( 1 - 51652090463616180 T + \)\(44\!\cdots\!39\)\( T^{2} \)
$61$ \( 1 - 45200043953043002 T + \)\(83\!\cdots\!41\)\( T^{2} \)
$67$ \( 1 - 322077213275888894 T + \)\(49\!\cdots\!03\)\( T^{2} \)
$71$ \( 1 - 393293311705873692 T + \)\(14\!\cdots\!31\)\( T^{2} \)
$73$ \( 1 + 672469661893471342 T + \)\(25\!\cdots\!37\)\( T^{2} \)
$79$ \( 1 + 482639101471927720 T + \)\(11\!\cdots\!19\)\( T^{2} \)
$83$ \( 1 + 313265345629507302 T + \)\(29\!\cdots\!47\)\( T^{2} \)
$89$ \( 1 + 4230101056729722390 T + \)\(10\!\cdots\!09\)\( T^{2} \)
$97$ \( 1 - 354705113301714434 T + \)\(56\!\cdots\!33\)\( T^{2} \)
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