Properties

Label 28.4.a.a
Level $28$
Weight $4$
Character orbit 28.a
Self dual yes
Analytic conductor $1.652$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 28.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.65205348016\)
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

\(f(q)\) \(=\) \( q - 10 q^{3} - 8 q^{5} - 7 q^{7} + 73 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 10 q^{3} - 8 q^{5} - 7 q^{7} + 73 q^{9} - 40 q^{11} - 12 q^{13} + 80 q^{15} - 58 q^{17} + 26 q^{19} + 70 q^{21} - 64 q^{23} - 61 q^{25} - 460 q^{27} - 62 q^{29} + 252 q^{31} + 400 q^{33} + 56 q^{35} + 26 q^{37} + 120 q^{39} + 6 q^{41} + 416 q^{43} - 584 q^{45} - 396 q^{47} + 49 q^{49} + 580 q^{51} - 450 q^{53} + 320 q^{55} - 260 q^{57} + 274 q^{59} - 576 q^{61} - 511 q^{63} + 96 q^{65} - 476 q^{67} + 640 q^{69} - 448 q^{71} - 158 q^{73} + 610 q^{75} + 280 q^{77} - 936 q^{79} + 2629 q^{81} + 530 q^{83} + 464 q^{85} + 620 q^{87} - 390 q^{89} + 84 q^{91} - 2520 q^{93} - 208 q^{95} + 214 q^{97} - 2920 q^{99}+O(q^{100}) \) 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
0 −10.0000 0 −8.00000 0 −7.00000 0 73.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)

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 28.4.a.a 1
3.b odd 2 1 252.4.a.d 1
4.b odd 2 1 112.4.a.g 1
5.b even 2 1 700.4.a.n 1
5.c odd 4 2 700.4.e.a 2
7.b odd 2 1 196.4.a.d 1
7.c even 3 2 196.4.e.f 2
7.d odd 6 2 196.4.e.a 2
8.b even 2 1 448.4.a.p 1
8.d odd 2 1 448.4.a.a 1
12.b even 2 1 1008.4.a.o 1
21.c even 2 1 1764.4.a.c 1
21.g even 6 2 1764.4.k.m 2
21.h odd 6 2 1764.4.k.d 2
28.d even 2 1 784.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.a.a 1 1.a even 1 1 trivial
112.4.a.g 1 4.b odd 2 1
196.4.a.d 1 7.b odd 2 1
196.4.e.a 2 7.d odd 6 2
196.4.e.f 2 7.c even 3 2
252.4.a.d 1 3.b odd 2 1
448.4.a.a 1 8.d odd 2 1
448.4.a.p 1 8.b even 2 1
700.4.a.n 1 5.b even 2 1
700.4.e.a 2 5.c odd 4 2
784.4.a.a 1 28.d even 2 1
1008.4.a.o 1 12.b even 2 1
1764.4.a.c 1 21.c even 2 1
1764.4.k.d 2 21.h odd 6 2
1764.4.k.m 2 21.g even 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 10 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(28))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 10 \) Copy content Toggle raw display
$5$ \( T + 8 \) Copy content Toggle raw display
$7$ \( T + 7 \) Copy content Toggle raw display
$11$ \( T + 40 \) Copy content Toggle raw display
$13$ \( T + 12 \) Copy content Toggle raw display
$17$ \( T + 58 \) Copy content Toggle raw display
$19$ \( T - 26 \) Copy content Toggle raw display
$23$ \( T + 64 \) Copy content Toggle raw display
$29$ \( T + 62 \) Copy content Toggle raw display
$31$ \( T - 252 \) Copy content Toggle raw display
$37$ \( T - 26 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T - 416 \) Copy content Toggle raw display
$47$ \( T + 396 \) Copy content Toggle raw display
$53$ \( T + 450 \) Copy content Toggle raw display
$59$ \( T - 274 \) Copy content Toggle raw display
$61$ \( T + 576 \) Copy content Toggle raw display
$67$ \( T + 476 \) Copy content Toggle raw display
$71$ \( T + 448 \) Copy content Toggle raw display
$73$ \( T + 158 \) Copy content Toggle raw display
$79$ \( T + 936 \) Copy content Toggle raw display
$83$ \( T - 530 \) Copy content Toggle raw display
$89$ \( T + 390 \) Copy content Toggle raw display
$97$ \( T - 214 \) Copy content Toggle raw display
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