Properties

Label 735.4.a.e
Level $735$
Weight $4$
Character orbit 735.a
Self dual yes
Analytic conductor $43.366$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 735.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.3664038542\)
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

\(f(q)\) \(=\) \( q + q^{2} - 3 q^{3} - 7 q^{4} - 5 q^{5} - 3 q^{6} - 15 q^{8} + 9 q^{9} + O(q^{10}) \) \( q + q^{2} - 3 q^{3} - 7 q^{4} - 5 q^{5} - 3 q^{6} - 15 q^{8} + 9 q^{9} - 5 q^{10} + 52 q^{11} + 21 q^{12} - 22 q^{13} + 15 q^{15} + 41 q^{16} + 14 q^{17} + 9 q^{18} + 20 q^{19} + 35 q^{20} + 52 q^{22} - 168 q^{23} + 45 q^{24} + 25 q^{25} - 22 q^{26} - 27 q^{27} + 230 q^{29} + 15 q^{30} + 288 q^{31} + 161 q^{32} - 156 q^{33} + 14 q^{34} - 63 q^{36} - 34 q^{37} + 20 q^{38} + 66 q^{39} + 75 q^{40} - 122 q^{41} - 188 q^{43} - 364 q^{44} - 45 q^{45} - 168 q^{46} - 256 q^{47} - 123 q^{48} + 25 q^{50} - 42 q^{51} + 154 q^{52} - 338 q^{53} - 27 q^{54} - 260 q^{55} - 60 q^{57} + 230 q^{58} - 100 q^{59} - 105 q^{60} - 742 q^{61} + 288 q^{62} - 167 q^{64} + 110 q^{65} - 156 q^{66} - 84 q^{67} - 98 q^{68} + 504 q^{69} - 328 q^{71} - 135 q^{72} + 38 q^{73} - 34 q^{74} - 75 q^{75} - 140 q^{76} + 66 q^{78} - 240 q^{79} - 205 q^{80} + 81 q^{81} - 122 q^{82} - 1212 q^{83} - 70 q^{85} - 188 q^{86} - 690 q^{87} - 780 q^{88} - 330 q^{89} - 45 q^{90} + 1176 q^{92} - 864 q^{93} - 256 q^{94} - 100 q^{95} - 483 q^{96} - 866 q^{97} + 468 q^{99} + O(q^{100}) \)

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
1.00000 −3.00000 −7.00000 −5.00000 −3.00000 0 −15.0000 9.00000 −5.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(7\) \(-1\)

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 735.4.a.e 1
3.b odd 2 1 2205.4.a.l 1
7.b odd 2 1 15.4.a.a 1
21.c even 2 1 45.4.a.c 1
28.d even 2 1 240.4.a.e 1
35.c odd 2 1 75.4.a.b 1
35.f even 4 2 75.4.b.b 2
56.e even 2 1 960.4.a.ba 1
56.h odd 2 1 960.4.a.b 1
63.l odd 6 2 405.4.e.g 2
63.o even 6 2 405.4.e.i 2
77.b even 2 1 1815.4.a.e 1
84.h odd 2 1 720.4.a.n 1
105.g even 2 1 225.4.a.f 1
105.k odd 4 2 225.4.b.e 2
140.c even 2 1 1200.4.a.t 1
140.j odd 4 2 1200.4.f.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.a 1 7.b odd 2 1
45.4.a.c 1 21.c even 2 1
75.4.a.b 1 35.c odd 2 1
75.4.b.b 2 35.f even 4 2
225.4.a.f 1 105.g even 2 1
225.4.b.e 2 105.k odd 4 2
240.4.a.e 1 28.d even 2 1
405.4.e.g 2 63.l odd 6 2
405.4.e.i 2 63.o even 6 2
720.4.a.n 1 84.h odd 2 1
735.4.a.e 1 1.a even 1 1 trivial
960.4.a.b 1 56.h odd 2 1
960.4.a.ba 1 56.e even 2 1
1200.4.a.t 1 140.c even 2 1
1200.4.f.b 2 140.j odd 4 2
1815.4.a.e 1 77.b even 2 1
2205.4.a.l 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(735))\):

\( T_{2} - 1 \)
\( T_{11} - 52 \)
\( T_{13} + 22 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( 3 + T \)
$5$ \( 5 + T \)
$7$ \( T \)
$11$ \( -52 + T \)
$13$ \( 22 + T \)
$17$ \( -14 + T \)
$19$ \( -20 + T \)
$23$ \( 168 + T \)
$29$ \( -230 + T \)
$31$ \( -288 + T \)
$37$ \( 34 + T \)
$41$ \( 122 + T \)
$43$ \( 188 + T \)
$47$ \( 256 + T \)
$53$ \( 338 + T \)
$59$ \( 100 + T \)
$61$ \( 742 + T \)
$67$ \( 84 + T \)
$71$ \( 328 + T \)
$73$ \( -38 + T \)
$79$ \( 240 + T \)
$83$ \( 1212 + T \)
$89$ \( 330 + T \)
$97$ \( 866 + T \)
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