Properties

Label 35.10.a.a
Level $35$
Weight $10$
Character orbit 35.a
Self dual yes
Analytic conductor $18.026$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 35.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.0262542657\)
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

\(f(q)\) \(=\) \( q + 28q^{2} - 116q^{3} + 272q^{4} + 625q^{5} - 3248q^{6} + 2401q^{7} - 6720q^{8} - 6227q^{9} + O(q^{10}) \) \( q + 28q^{2} - 116q^{3} + 272q^{4} + 625q^{5} - 3248q^{6} + 2401q^{7} - 6720q^{8} - 6227q^{9} + 17500q^{10} - 25548q^{11} - 31552q^{12} - 42306q^{13} + 67228q^{14} - 72500q^{15} - 327424q^{16} - 526342q^{17} - 174356q^{18} - 350060q^{19} + 170000q^{20} - 278516q^{21} - 715344q^{22} - 621976q^{23} + 779520q^{24} + 390625q^{25} - 1184568q^{26} + 3005560q^{27} + 653072q^{28} + 6720430q^{29} - 2030000q^{30} - 6412208q^{31} - 5727232q^{32} + 2963568q^{33} - 14737576q^{34} + 1500625q^{35} - 1693744q^{36} - 2317682q^{37} - 9801680q^{38} + 4907496q^{39} - 4200000q^{40} - 10224678q^{41} - 7798448q^{42} + 30114004q^{43} - 6949056q^{44} - 3891875q^{45} - 17415328q^{46} - 23644912q^{47} + 37981184q^{48} + 5764801q^{49} + 10937500q^{50} + 61055672q^{51} - 11507232q^{52} + 57292654q^{53} + 84155680q^{54} - 15967500q^{55} - 16134720q^{56} + 40606960q^{57} + 188172040q^{58} + 84934780q^{59} - 19720000q^{60} + 14677822q^{61} - 179541824q^{62} - 14951027q^{63} + 7278592q^{64} - 26441250q^{65} + 82979904q^{66} - 244557812q^{67} - 143165024q^{68} + 72149216q^{69} + 42017500q^{70} + 61901952q^{71} + 41845440q^{72} - 283763726q^{73} - 64895096q^{74} - 45312500q^{75} - 95216320q^{76} - 61340748q^{77} + 137409888q^{78} + 276107480q^{79} - 204640000q^{80} - 226078919q^{81} - 286290984q^{82} - 72995956q^{83} - 75756352q^{84} - 328963750q^{85} + 843192112q^{86} - 779569880q^{87} + 171682560q^{88} - 896368470q^{89} - 108972500q^{90} - 101576706q^{91} - 169177472q^{92} + 743816128q^{93} - 662057536q^{94} - 218787500q^{95} + 664358912q^{96} + 1205809578q^{97} + 161414428q^{98} + 159087396q^{99} + O(q^{100}) \)

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
28.0000 −116.000 272.000 625.000 −3248.00 2401.00 −6720.00 −6227.00 17500.0
\(n\): 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 35.10.a.a 1
3.b odd 2 1 315.10.a.a 1
5.b even 2 1 175.10.a.a 1
5.c odd 4 2 175.10.b.a 2
7.b odd 2 1 245.10.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.10.a.a 1 1.a even 1 1 trivial
175.10.a.a 1 5.b even 2 1
175.10.b.a 2 5.c odd 4 2
245.10.a.b 1 7.b odd 2 1
315.10.a.a 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 28 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(35))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -28 + T \)
$3$ \( 116 + T \)
$5$ \( -625 + T \)
$7$ \( -2401 + T \)
$11$ \( 25548 + T \)
$13$ \( 42306 + T \)
$17$ \( 526342 + T \)
$19$ \( 350060 + T \)
$23$ \( 621976 + T \)
$29$ \( -6720430 + T \)
$31$ \( 6412208 + T \)
$37$ \( 2317682 + T \)
$41$ \( 10224678 + T \)
$43$ \( -30114004 + T \)
$47$ \( 23644912 + T \)
$53$ \( -57292654 + T \)
$59$ \( -84934780 + T \)
$61$ \( -14677822 + T \)
$67$ \( 244557812 + T \)
$71$ \( -61901952 + T \)
$73$ \( 283763726 + T \)
$79$ \( -276107480 + T \)
$83$ \( 72995956 + T \)
$89$ \( 896368470 + T \)
$97$ \( -1205809578 + T \)
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