Properties

Label 240.10.a.g
Level 240
Weight 10
Character orbit 240.a
Self dual yes
Analytic conductor 123.609
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 240.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(123.608600679\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 81q^{3} - 625q^{5} + 5988q^{7} + 6561q^{9} + O(q^{10}) \) \( q + 81q^{3} - 625q^{5} + 5988q^{7} + 6561q^{9} + 14648q^{11} + 37906q^{13} - 50625q^{15} - 441098q^{17} - 441820q^{19} + 485028q^{21} - 2264136q^{23} + 390625q^{25} + 531441q^{27} - 1049350q^{29} + 7910568q^{31} + 1186488q^{33} - 3742500q^{35} - 20992558q^{37} + 3070386q^{39} + 13285562q^{41} + 23130764q^{43} - 4100625q^{45} + 13873688q^{47} - 4497463q^{49} - 35728938q^{51} - 57635174q^{53} - 9155000q^{55} - 35787420q^{57} + 32042120q^{59} + 110664022q^{61} + 39287268q^{63} - 23691250q^{65} + 118568268q^{67} - 183395016q^{69} - 276679712q^{71} - 264023294q^{73} + 31640625q^{75} + 87712224q^{77} - 448202760q^{79} + 43046721q^{81} - 851015796q^{83} + 275686250q^{85} - 84997350q^{87} + 189894930q^{89} + 226981128q^{91} + 640756008q^{93} + 276137500q^{95} - 1014149278q^{97} + 96105528q^{99} + O(q^{100}) \)

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
0 81.0000 0 −625.000 0 5988.00 0 6561.00 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)

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 240.10.a.g 1
4.b odd 2 1 15.10.a.b 1
12.b even 2 1 45.10.a.a 1
20.d odd 2 1 75.10.a.a 1
20.e even 4 2 75.10.b.b 2
60.h even 2 1 225.10.a.f 1
60.l odd 4 2 225.10.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.10.a.b 1 4.b odd 2 1
45.10.a.a 1 12.b even 2 1
75.10.a.a 1 20.d odd 2 1
75.10.b.b 2 20.e even 4 2
225.10.a.f 1 60.h even 2 1
225.10.b.b 2 60.l odd 4 2
240.10.a.g 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 5988 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(240))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 81 T \)
$5$ \( 1 + 625 T \)
$7$ \( 1 - 5988 T + 40353607 T^{2} \)
$11$ \( 1 - 14648 T + 2357947691 T^{2} \)
$13$ \( 1 - 37906 T + 10604499373 T^{2} \)
$17$ \( 1 + 441098 T + 118587876497 T^{2} \)
$19$ \( 1 + 441820 T + 322687697779 T^{2} \)
$23$ \( 1 + 2264136 T + 1801152661463 T^{2} \)
$29$ \( 1 + 1049350 T + 14507145975869 T^{2} \)
$31$ \( 1 - 7910568 T + 26439622160671 T^{2} \)
$37$ \( 1 + 20992558 T + 129961739795077 T^{2} \)
$41$ \( 1 - 13285562 T + 327381934393961 T^{2} \)
$43$ \( 1 - 23130764 T + 502592611936843 T^{2} \)
$47$ \( 1 - 13873688 T + 1119130473102767 T^{2} \)
$53$ \( 1 + 57635174 T + 3299763591802133 T^{2} \)
$59$ \( 1 - 32042120 T + 8662995818654939 T^{2} \)
$61$ \( 1 - 110664022 T + 11694146092834141 T^{2} \)
$67$ \( 1 - 118568268 T + 27206534396294947 T^{2} \)
$71$ \( 1 + 276679712 T + 45848500718449031 T^{2} \)
$73$ \( 1 + 264023294 T + 58871586708267913 T^{2} \)
$79$ \( 1 + 448202760 T + 119851595982618319 T^{2} \)
$83$ \( 1 + 851015796 T + 186940255267540403 T^{2} \)
$89$ \( 1 - 189894930 T + 350356403707485209 T^{2} \)
$97$ \( 1 + 1014149278 T + 760231058654565217 T^{2} \)
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