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Properties

Label 35.6.a.a
Level $35$
Weight $6$
Character orbit 35.a
Self dual yes
Analytic conductor $5.613$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 35.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.61343369345\)
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

\(f(q)\) \(=\) \( q - 8q^{2} + q^{3} + 32q^{4} + 25q^{5} - 8q^{6} + 49q^{7} - 242q^{9} + O(q^{10}) \) \( q - 8q^{2} + q^{3} + 32q^{4} + 25q^{5} - 8q^{6} + 49q^{7} - 242q^{9} - 200q^{10} - 453q^{11} + 32q^{12} - 969q^{13} - 392q^{14} + 25q^{15} - 1024q^{16} + 1637q^{17} + 1936q^{18} - 1550q^{19} + 800q^{20} + 49q^{21} + 3624q^{22} - 1654q^{23} + 625q^{25} + 7752q^{26} - 485q^{27} + 1568q^{28} - 4985q^{29} - 200q^{30} + 1192q^{31} + 8192q^{32} - 453q^{33} - 13096q^{34} + 1225q^{35} - 7744q^{36} - 11018q^{37} + 12400q^{38} - 969q^{39} - 1728q^{41} - 392q^{42} - 10814q^{43} - 14496q^{44} - 6050q^{45} + 13232q^{46} + 26237q^{47} - 1024q^{48} + 2401q^{49} - 5000q^{50} + 1637q^{51} - 31008q^{52} + 25936q^{53} + 3880q^{54} - 11325q^{55} - 1550q^{57} + 39880q^{58} - 4580q^{59} + 800q^{60} - 12488q^{61} - 9536q^{62} - 11858q^{63} - 32768q^{64} - 24225q^{65} + 3624q^{66} - 15848q^{67} + 52384q^{68} - 1654q^{69} - 9800q^{70} + 51792q^{71} + 4846q^{73} + 88144q^{74} + 625q^{75} - 49600q^{76} - 22197q^{77} + 7752q^{78} + 62765q^{79} - 25600q^{80} + 58321q^{81} + 13824q^{82} - 23644q^{83} + 1568q^{84} + 40925q^{85} + 86512q^{86} - 4985q^{87} - 147300q^{89} + 48400q^{90} - 47481q^{91} - 52928q^{92} + 1192q^{93} - 209896q^{94} - 38750q^{95} + 8192q^{96} - 8343q^{97} - 19208q^{98} + 109626q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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 49.0000 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 35.6.a.a ✓ 1
3.b odd 2 1 315.6.a.a 1
4.b odd 2 1 560.6.a.c 1
5.b even 2 1 175.6.a.a 1
5.c odd 4 2 175.6.b.b 2
7.b odd 2 1 245.6.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.a.a ✓ 1 1.a even 1 1 trivial
175.6.a.a 1 5.b even 2 1
175.6.b.b 2 5.c odd 4 2
245.6.a.a 1 7.b odd 2 1
315.6.a.a 1 3.b odd 2 1
560.6.a.c 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 8 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(35))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 8 + T \)
$3$ \( -1 + T \)
$5$ \( -25 + T \)
$7$ \( -49 + T \)
$11$ \( 453 + T \)
$13$ \( 969 + T \)
$17$ \( -1637 + T \)
$19$ \( 1550 + T \)
$23$ \( 1654 + T \)
$29$ \( 4985 + T \)
$31$ \( -1192 + T \)
$37$ \( 11018 + T \)
$41$ \( 1728 + T \)
$43$ \( 10814 + T \)
$47$ \( -26237 + T \)
$53$ \( -25936 + T \)
$59$ \( 4580 + T \)
$61$ \( 12488 + T \)
$67$ \( 15848 + T \)
$71$ \( -51792 + T \)
$73$ \( -4846 + T \)
$79$ \( -62765 + T \)
$83$ \( 23644 + T \)
$89$ \( 147300 + T \)
$97$ \( 8343 + T \)
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