Properties

Label 2535.2.a.j
Level 2535
Weight 2
Character orbit 2535.a
Self dual yes
Analytic conductor 20.242
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2535 = 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 2535.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(20.2420769124\)
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

\(f(q)\) \(=\) \( q + q^{2} - q^{3} - q^{4} - q^{5} - q^{6} - 3q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} - q^{4} - q^{5} - q^{6} - 3q^{8} + q^{9} - q^{10} + 4q^{11} + q^{12} + q^{15} - q^{16} + 2q^{17} + q^{18} - 4q^{19} + q^{20} + 4q^{22} + 3q^{24} + q^{25} - q^{27} - 2q^{29} + q^{30} + 5q^{32} - 4q^{33} + 2q^{34} - q^{36} + 10q^{37} - 4q^{38} + 3q^{40} - 10q^{41} + 4q^{43} - 4q^{44} - q^{45} - 8q^{47} + q^{48} - 7q^{49} + q^{50} - 2q^{51} - 10q^{53} - q^{54} - 4q^{55} + 4q^{57} - 2q^{58} + 4q^{59} - q^{60} - 2q^{61} + 7q^{64} - 4q^{66} - 12q^{67} - 2q^{68} + 8q^{71} - 3q^{72} - 10q^{73} + 10q^{74} - q^{75} + 4q^{76} + q^{80} + q^{81} - 10q^{82} - 12q^{83} - 2q^{85} + 4q^{86} + 2q^{87} - 12q^{88} + 6q^{89} - q^{90} - 8q^{94} + 4q^{95} - 5q^{96} - 2q^{97} - 7q^{98} + 4q^{99} + O(q^{100}) \)

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

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 2535.2.a.j 1
3.b odd 2 1 7605.2.a.g 1
13.b even 2 1 15.2.a.a 1
39.d odd 2 1 45.2.a.a 1
52.b odd 2 1 240.2.a.d 1
65.d even 2 1 75.2.a.b 1
65.h odd 4 2 75.2.b.b 2
91.b odd 2 1 735.2.a.c 1
91.r even 6 2 735.2.i.e 2
91.s odd 6 2 735.2.i.d 2
104.e even 2 1 960.2.a.l 1
104.h odd 2 1 960.2.a.a 1
117.n odd 6 2 405.2.e.c 2
117.t even 6 2 405.2.e.f 2
143.d odd 2 1 1815.2.a.d 1
156.h even 2 1 720.2.a.c 1
195.e odd 2 1 225.2.a.b 1
195.s even 4 2 225.2.b.b 2
208.o odd 4 2 3840.2.k.r 2
208.p even 4 2 3840.2.k.m 2
221.b even 2 1 4335.2.a.c 1
247.d odd 2 1 5415.2.a.j 1
260.g odd 2 1 1200.2.a.e 1
260.p even 4 2 1200.2.f.h 2
273.g even 2 1 2205.2.a.i 1
299.c odd 2 1 7935.2.a.d 1
312.b odd 2 1 2880.2.a.y 1
312.h even 2 1 2880.2.a.bc 1
429.e even 2 1 5445.2.a.c 1
455.h odd 2 1 3675.2.a.j 1
520.b odd 2 1 4800.2.a.bz 1
520.p even 2 1 4800.2.a.t 1
520.bc even 4 2 4800.2.f.c 2
520.bg odd 4 2 4800.2.f.bf 2
715.c odd 2 1 9075.2.a.g 1
780.d even 2 1 3600.2.a.u 1
780.w odd 4 2 3600.2.f.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.2.a.a 1 13.b even 2 1
45.2.a.a 1 39.d odd 2 1
75.2.a.b 1 65.d even 2 1
75.2.b.b 2 65.h odd 4 2
225.2.a.b 1 195.e odd 2 1
225.2.b.b 2 195.s even 4 2
240.2.a.d 1 52.b odd 2 1
405.2.e.c 2 117.n odd 6 2
405.2.e.f 2 117.t even 6 2
720.2.a.c 1 156.h even 2 1
735.2.a.c 1 91.b odd 2 1
735.2.i.d 2 91.s odd 6 2
735.2.i.e 2 91.r even 6 2
960.2.a.a 1 104.h odd 2 1
960.2.a.l 1 104.e even 2 1
1200.2.a.e 1 260.g odd 2 1
1200.2.f.h 2 260.p even 4 2
1815.2.a.d 1 143.d odd 2 1
2205.2.a.i 1 273.g even 2 1
2535.2.a.j 1 1.a even 1 1 trivial
2880.2.a.y 1 312.b odd 2 1
2880.2.a.bc 1 312.h even 2 1
3600.2.a.u 1 780.d even 2 1
3600.2.f.e 2 780.w odd 4 2
3675.2.a.j 1 455.h odd 2 1
3840.2.k.m 2 208.p even 4 2
3840.2.k.r 2 208.o odd 4 2
4335.2.a.c 1 221.b even 2 1
4800.2.a.t 1 520.p even 2 1
4800.2.a.bz 1 520.b odd 2 1
4800.2.f.c 2 520.bc even 4 2
4800.2.f.bf 2 520.bg odd 4 2
5415.2.a.j 1 247.d odd 2 1
5445.2.a.c 1 429.e even 2 1
7605.2.a.g 1 3.b odd 2 1
7935.2.a.d 1 299.c odd 2 1
9075.2.a.g 1 715.c odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(13\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2535))\):

\( T_{2} - 1 \)
\( T_{7} \)
\( T_{11} - 4 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + 2 T^{2} \)
$3$ \( 1 + T \)
$5$ \( 1 + T \)
$7$ \( 1 + 7 T^{2} \)
$11$ \( 1 - 4 T + 11 T^{2} \)
$13$ \( \)
$17$ \( 1 - 2 T + 17 T^{2} \)
$19$ \( 1 + 4 T + 19 T^{2} \)
$23$ \( 1 + 23 T^{2} \)
$29$ \( 1 + 2 T + 29 T^{2} \)
$31$ \( 1 + 31 T^{2} \)
$37$ \( 1 - 10 T + 37 T^{2} \)
$41$ \( 1 + 10 T + 41 T^{2} \)
$43$ \( 1 - 4 T + 43 T^{2} \)
$47$ \( 1 + 8 T + 47 T^{2} \)
$53$ \( 1 + 10 T + 53 T^{2} \)
$59$ \( 1 - 4 T + 59 T^{2} \)
$61$ \( 1 + 2 T + 61 T^{2} \)
$67$ \( 1 + 12 T + 67 T^{2} \)
$71$ \( 1 - 8 T + 71 T^{2} \)
$73$ \( 1 + 10 T + 73 T^{2} \)
$79$ \( 1 + 79 T^{2} \)
$83$ \( 1 + 12 T + 83 T^{2} \)
$89$ \( 1 - 6 T + 89 T^{2} \)
$97$ \( 1 + 2 T + 97 T^{2} \)
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