Properties

Label 10.10.a.a
Level 10
Weight 10
Character orbit 10.a
Self dual yes
Analytic conductor 5.150
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: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 10.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.15035836164\)
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

\(f(q)\) \(=\) \( q - 16q^{2} - 204q^{3} + 256q^{4} + 625q^{5} + 3264q^{6} + 5432q^{7} - 4096q^{8} + 21933q^{9} + O(q^{10}) \) \( q - 16q^{2} - 204q^{3} + 256q^{4} + 625q^{5} + 3264q^{6} + 5432q^{7} - 4096q^{8} + 21933q^{9} - 10000q^{10} + 73932q^{11} - 52224q^{12} - 114514q^{13} - 86912q^{14} - 127500q^{15} + 65536q^{16} + 41682q^{17} - 350928q^{18} + 1057460q^{19} + 160000q^{20} - 1108128q^{21} - 1182912q^{22} + 1599336q^{23} + 835584q^{24} + 390625q^{25} + 1832224q^{26} - 459000q^{27} + 1390592q^{28} + 2184510q^{29} + 2040000q^{30} - 9619648q^{31} - 1048576q^{32} - 15082128q^{33} - 666912q^{34} + 3395000q^{35} + 5614848q^{36} + 4799942q^{37} - 16919360q^{38} + 23360856q^{39} - 2560000q^{40} + 9531882q^{41} + 17730048q^{42} - 13464484q^{43} + 18926592q^{44} + 13708125q^{45} - 25589376q^{46} + 11441952q^{47} - 13369344q^{48} - 10846983q^{49} - 6250000q^{50} - 8503128q^{51} - 29315584q^{52} + 53615766q^{53} + 7344000q^{54} + 46207500q^{55} - 22249472q^{56} - 215721840q^{57} - 34952160q^{58} + 81862620q^{59} - 32640000q^{60} - 104691298q^{61} + 153914368q^{62} + 119140056q^{63} + 16777216q^{64} - 71571250q^{65} + 241314048q^{66} + 140571092q^{67} + 10670592q^{68} - 326264544q^{69} - 54320000q^{70} + 97098792q^{71} - 89837568q^{72} + 171848906q^{73} - 76799072q^{74} - 79687500q^{75} + 270709760q^{76} + 401598624q^{77} - 373773696q^{78} - 117380080q^{79} + 40960000q^{80} - 338071239q^{81} - 152510112q^{82} + 323637636q^{83} - 283680768q^{84} + 26051250q^{85} + 215431744q^{86} - 445640040q^{87} - 302825472q^{88} - 894379110q^{89} - 219330000q^{90} - 622040048q^{91} + 409430016q^{92} + 1962408192q^{93} - 183071232q^{94} + 660912500q^{95} + 213909504q^{96} + 232678562q^{97} + 173551728q^{98} + 1621550556q^{99} + O(q^{100}) \)

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
−16.0000 −204.000 256.000 625.000 3264.00 5432.00 −4096.00 21933.0 −10000.0
\(n\): 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 10.10.a.a 1
3.b odd 2 1 90.10.a.g 1
4.b odd 2 1 80.10.a.e 1
5.b even 2 1 50.10.a.f 1
5.c odd 4 2 50.10.b.e 2
8.b even 2 1 320.10.a.j 1
8.d odd 2 1 320.10.a.a 1
20.d odd 2 1 400.10.a.a 1
20.e even 4 2 400.10.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.10.a.a 1 1.a even 1 1 trivial
50.10.a.f 1 5.b even 2 1
50.10.b.e 2 5.c odd 4 2
80.10.a.e 1 4.b odd 2 1
90.10.a.g 1 3.b odd 2 1
320.10.a.a 1 8.d odd 2 1
320.10.a.j 1 8.b even 2 1
400.10.a.a 1 20.d odd 2 1
400.10.c.b 2 20.e even 4 2

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 204 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(10))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 16 T \)
$3$ \( 1 + 204 T + 19683 T^{2} \)
$5$ \( 1 - 625 T \)
$7$ \( 1 - 5432 T + 40353607 T^{2} \)
$11$ \( 1 - 73932 T + 2357947691 T^{2} \)
$13$ \( 1 + 114514 T + 10604499373 T^{2} \)
$17$ \( 1 - 41682 T + 118587876497 T^{2} \)
$19$ \( 1 - 1057460 T + 322687697779 T^{2} \)
$23$ \( 1 - 1599336 T + 1801152661463 T^{2} \)
$29$ \( 1 - 2184510 T + 14507145975869 T^{2} \)
$31$ \( 1 + 9619648 T + 26439622160671 T^{2} \)
$37$ \( 1 - 4799942 T + 129961739795077 T^{2} \)
$41$ \( 1 - 9531882 T + 327381934393961 T^{2} \)
$43$ \( 1 + 13464484 T + 502592611936843 T^{2} \)
$47$ \( 1 - 11441952 T + 1119130473102767 T^{2} \)
$53$ \( 1 - 53615766 T + 3299763591802133 T^{2} \)
$59$ \( 1 - 81862620 T + 8662995818654939 T^{2} \)
$61$ \( 1 + 104691298 T + 11694146092834141 T^{2} \)
$67$ \( 1 - 140571092 T + 27206534396294947 T^{2} \)
$71$ \( 1 - 97098792 T + 45848500718449031 T^{2} \)
$73$ \( 1 - 171848906 T + 58871586708267913 T^{2} \)
$79$ \( 1 + 117380080 T + 119851595982618319 T^{2} \)
$83$ \( 1 - 323637636 T + 186940255267540403 T^{2} \)
$89$ \( 1 + 894379110 T + 350356403707485209 T^{2} \)
$97$ \( 1 - 232678562 T + 760231058654565217 T^{2} \)
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