Properties

Label 735.4.a.i
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

\(f(q)\) \(=\) \( q + 3 q^{2} + 3 q^{3} + q^{4} + 5 q^{5} + 9 q^{6} - 21 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{2} + 3 q^{3} + q^{4} + 5 q^{5} + 9 q^{6} - 21 q^{8} + 9 q^{9} + 15 q^{10} - 24 q^{11} + 3 q^{12} - 74 q^{13} + 15 q^{15} - 71 q^{16} - 54 q^{17} + 27 q^{18} + 124 q^{19} + 5 q^{20} - 72 q^{22} - 120 q^{23} - 63 q^{24} + 25 q^{25} - 222 q^{26} + 27 q^{27} - 78 q^{29} + 45 q^{30} - 200 q^{31} - 45 q^{32} - 72 q^{33} - 162 q^{34} + 9 q^{36} - 70 q^{37} + 372 q^{38} - 222 q^{39} - 105 q^{40} - 330 q^{41} + 92 q^{43} - 24 q^{44} + 45 q^{45} - 360 q^{46} + 24 q^{47} - 213 q^{48} + 75 q^{50} - 162 q^{51} - 74 q^{52} + 450 q^{53} + 81 q^{54} - 120 q^{55} + 372 q^{57} - 234 q^{58} - 24 q^{59} + 15 q^{60} + 322 q^{61} - 600 q^{62} + 433 q^{64} - 370 q^{65} - 216 q^{66} - 196 q^{67} - 54 q^{68} - 360 q^{69} - 288 q^{71} - 189 q^{72} + 430 q^{73} - 210 q^{74} + 75 q^{75} + 124 q^{76} - 666 q^{78} - 520 q^{79} - 355 q^{80} + 81 q^{81} - 990 q^{82} - 156 q^{83} - 270 q^{85} + 276 q^{86} - 234 q^{87} + 504 q^{88} - 1026 q^{89} + 135 q^{90} - 120 q^{92} - 600 q^{93} + 72 q^{94} + 620 q^{95} - 135 q^{96} + 286 q^{97} - 216 q^{99}+O(q^{100}) \) 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
3.00000 3.00000 1.00000 5.00000 9.00000 0 −21.0000 9.00000 15.0000
\(n\): 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.i 1
3.b odd 2 1 2205.4.a.c 1
7.b odd 2 1 15.4.a.b 1
21.c even 2 1 45.4.a.b 1
28.d even 2 1 240.4.a.f 1
35.c odd 2 1 75.4.a.a 1
35.f even 4 2 75.4.b.a 2
56.e even 2 1 960.4.a.l 1
56.h odd 2 1 960.4.a.bi 1
63.l odd 6 2 405.4.e.d 2
63.o even 6 2 405.4.e.k 2
77.b even 2 1 1815.4.a.a 1
84.h odd 2 1 720.4.a.r 1
105.g even 2 1 225.4.a.g 1
105.k odd 4 2 225.4.b.d 2
140.c even 2 1 1200.4.a.o 1
140.j odd 4 2 1200.4.f.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.b 1 7.b odd 2 1
45.4.a.b 1 21.c even 2 1
75.4.a.a 1 35.c odd 2 1
75.4.b.a 2 35.f even 4 2
225.4.a.g 1 105.g even 2 1
225.4.b.d 2 105.k odd 4 2
240.4.a.f 1 28.d even 2 1
405.4.e.d 2 63.l odd 6 2
405.4.e.k 2 63.o even 6 2
720.4.a.r 1 84.h odd 2 1
735.4.a.i 1 1.a even 1 1 trivial
960.4.a.l 1 56.e even 2 1
960.4.a.bi 1 56.h odd 2 1
1200.4.a.o 1 140.c even 2 1
1200.4.f.m 2 140.j odd 4 2
1815.4.a.a 1 77.b even 2 1
2205.4.a.c 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(735))\):

\( T_{2} - 3 \) Copy content Toggle raw display
\( T_{11} + 24 \) Copy content Toggle raw display
\( T_{13} + 74 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 3 \) Copy content Toggle raw display
$3$ \( T - 3 \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 24 \) Copy content Toggle raw display
$13$ \( T + 74 \) Copy content Toggle raw display
$17$ \( T + 54 \) Copy content Toggle raw display
$19$ \( T - 124 \) Copy content Toggle raw display
$23$ \( T + 120 \) Copy content Toggle raw display
$29$ \( T + 78 \) Copy content Toggle raw display
$31$ \( T + 200 \) Copy content Toggle raw display
$37$ \( T + 70 \) Copy content Toggle raw display
$41$ \( T + 330 \) Copy content Toggle raw display
$43$ \( T - 92 \) Copy content Toggle raw display
$47$ \( T - 24 \) Copy content Toggle raw display
$53$ \( T - 450 \) Copy content Toggle raw display
$59$ \( T + 24 \) Copy content Toggle raw display
$61$ \( T - 322 \) Copy content Toggle raw display
$67$ \( T + 196 \) Copy content Toggle raw display
$71$ \( T + 288 \) Copy content Toggle raw display
$73$ \( T - 430 \) Copy content Toggle raw display
$79$ \( T + 520 \) Copy content Toggle raw display
$83$ \( T + 156 \) Copy content Toggle raw display
$89$ \( T + 1026 \) Copy content Toggle raw display
$97$ \( T - 286 \) Copy content Toggle raw display
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