Properties

Label 245.6.a.a
Level $245$
Weight $6$
Character orbit 245.a
Self dual yes
Analytic conductor $39.294$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 245.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(39.2940358542\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 8 q^{2} - q^{3} + 32 q^{4} - 25 q^{5} + 8 q^{6} - 242 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 q^{2} - q^{3} + 32 q^{4} - 25 q^{5} + 8 q^{6} - 242 q^{9} + 200 q^{10} - 453 q^{11} - 32 q^{12} + 969 q^{13} + 25 q^{15} - 1024 q^{16} - 1637 q^{17} + 1936 q^{18} + 1550 q^{19} - 800 q^{20} + 3624 q^{22} - 1654 q^{23} + 625 q^{25} - 7752 q^{26} + 485 q^{27} - 4985 q^{29} - 200 q^{30} - 1192 q^{31} + 8192 q^{32} + 453 q^{33} + 13096 q^{34} - 7744 q^{36} - 11018 q^{37} - 12400 q^{38} - 969 q^{39} + 1728 q^{41} - 10814 q^{43} - 14496 q^{44} + 6050 q^{45} + 13232 q^{46} - 26237 q^{47} + 1024 q^{48} - 5000 q^{50} + 1637 q^{51} + 31008 q^{52} + 25936 q^{53} - 3880 q^{54} + 11325 q^{55} - 1550 q^{57} + 39880 q^{58} + 4580 q^{59} + 800 q^{60} + 12488 q^{61} + 9536 q^{62} - 32768 q^{64} - 24225 q^{65} - 3624 q^{66} - 15848 q^{67} - 52384 q^{68} + 1654 q^{69} + 51792 q^{71} - 4846 q^{73} + 88144 q^{74} - 625 q^{75} + 49600 q^{76} + 7752 q^{78} + 62765 q^{79} + 25600 q^{80} + 58321 q^{81} - 13824 q^{82} + 23644 q^{83} + 40925 q^{85} + 86512 q^{86} + 4985 q^{87} + 147300 q^{89} - 48400 q^{90} - 52928 q^{92} + 1192 q^{93} + 209896 q^{94} - 38750 q^{95} - 8192 q^{96} + 8343 q^{97} + 109626 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
−8.00000 −1.00000 32.0000 −25.0000 8.00000 0 0 −242.000 200.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(7\) \(-1\)

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 245.6.a.a 1
7.b odd 2 1 35.6.a.a 1
21.c even 2 1 315.6.a.a 1
28.d even 2 1 560.6.a.c 1
35.c odd 2 1 175.6.a.a 1
35.f even 4 2 175.6.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.a.a 1 7.b odd 2 1
175.6.a.a 1 35.c odd 2 1
175.6.b.b 2 35.f even 4 2
245.6.a.a 1 1.a even 1 1 trivial
315.6.a.a 1 21.c even 2 1
560.6.a.c 1 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(245))\):

\( T_{2} + 8 \) Copy content Toggle raw display
\( T_{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 8 \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T + 25 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 453 \) Copy content Toggle raw display
$13$ \( T - 969 \) Copy content Toggle raw display
$17$ \( T + 1637 \) Copy content Toggle raw display
$19$ \( T - 1550 \) Copy content Toggle raw display
$23$ \( T + 1654 \) Copy content Toggle raw display
$29$ \( T + 4985 \) Copy content Toggle raw display
$31$ \( T + 1192 \) Copy content Toggle raw display
$37$ \( T + 11018 \) Copy content Toggle raw display
$41$ \( T - 1728 \) Copy content Toggle raw display
$43$ \( T + 10814 \) Copy content Toggle raw display
$47$ \( T + 26237 \) Copy content Toggle raw display
$53$ \( T - 25936 \) Copy content Toggle raw display
$59$ \( T - 4580 \) Copy content Toggle raw display
$61$ \( T - 12488 \) Copy content Toggle raw display
$67$ \( T + 15848 \) Copy content Toggle raw display
$71$ \( T - 51792 \) Copy content Toggle raw display
$73$ \( T + 4846 \) Copy content Toggle raw display
$79$ \( T - 62765 \) Copy content Toggle raw display
$83$ \( T - 23644 \) Copy content Toggle raw display
$89$ \( T - 147300 \) Copy content Toggle raw display
$97$ \( T - 8343 \) Copy content Toggle raw display
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