Properties

Label 90.4.a.e
Level $90$
Weight $4$
Character orbit 90.a
Self dual yes
Analytic conductor $5.310$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 90.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.31017190052\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + 5 q^{5} + 14 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 4 q^{4} + 5 q^{5} + 14 q^{7} + 8 q^{8} + 10 q^{10} - 6 q^{11} + 68 q^{13} + 28 q^{14} + 16 q^{16} - 78 q^{17} + 44 q^{19} + 20 q^{20} - 12 q^{22} - 120 q^{23} + 25 q^{25} + 136 q^{26} + 56 q^{28} - 126 q^{29} - 244 q^{31} + 32 q^{32} - 156 q^{34} + 70 q^{35} - 304 q^{37} + 88 q^{38} + 40 q^{40} + 480 q^{41} + 104 q^{43} - 24 q^{44} - 240 q^{46} - 600 q^{47} - 147 q^{49} + 50 q^{50} + 272 q^{52} + 258 q^{53} - 30 q^{55} + 112 q^{56} - 252 q^{58} - 534 q^{59} + 362 q^{61} - 488 q^{62} + 64 q^{64} + 340 q^{65} - 268 q^{67} - 312 q^{68} + 140 q^{70} + 972 q^{71} + 470 q^{73} - 608 q^{74} + 176 q^{76} - 84 q^{77} + 1244 q^{79} + 80 q^{80} + 960 q^{82} - 396 q^{83} - 390 q^{85} + 208 q^{86} - 48 q^{88} + 972 q^{89} + 952 q^{91} - 480 q^{92} - 1200 q^{94} + 220 q^{95} - 46 q^{97} - 294 q^{98}+O(q^{100}) \) 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
2.00000 0 4.00000 5.00000 0 14.0000 8.00000 0 10.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)

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 90.4.a.e yes 1
3.b odd 2 1 90.4.a.b 1
4.b odd 2 1 720.4.a.t 1
5.b even 2 1 450.4.a.c 1
5.c odd 4 2 450.4.c.f 2
9.c even 3 2 810.4.e.a 2
9.d odd 6 2 810.4.e.u 2
12.b even 2 1 720.4.a.e 1
15.d odd 2 1 450.4.a.m 1
15.e even 4 2 450.4.c.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.4.a.b 1 3.b odd 2 1
90.4.a.e yes 1 1.a even 1 1 trivial
450.4.a.c 1 5.b even 2 1
450.4.a.m 1 15.d odd 2 1
450.4.c.f 2 5.c odd 4 2
450.4.c.g 2 15.e even 4 2
720.4.a.e 1 12.b even 2 1
720.4.a.t 1 4.b odd 2 1
810.4.e.a 2 9.c even 3 2
810.4.e.u 2 9.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(90))\):

\( T_{7} - 14 \) Copy content Toggle raw display
\( T_{11} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T - 14 \) Copy content Toggle raw display
$11$ \( T + 6 \) Copy content Toggle raw display
$13$ \( T - 68 \) Copy content Toggle raw display
$17$ \( T + 78 \) Copy content Toggle raw display
$19$ \( T - 44 \) Copy content Toggle raw display
$23$ \( T + 120 \) Copy content Toggle raw display
$29$ \( T + 126 \) Copy content Toggle raw display
$31$ \( T + 244 \) Copy content Toggle raw display
$37$ \( T + 304 \) Copy content Toggle raw display
$41$ \( T - 480 \) Copy content Toggle raw display
$43$ \( T - 104 \) Copy content Toggle raw display
$47$ \( T + 600 \) Copy content Toggle raw display
$53$ \( T - 258 \) Copy content Toggle raw display
$59$ \( T + 534 \) Copy content Toggle raw display
$61$ \( T - 362 \) Copy content Toggle raw display
$67$ \( T + 268 \) Copy content Toggle raw display
$71$ \( T - 972 \) Copy content Toggle raw display
$73$ \( T - 470 \) Copy content Toggle raw display
$79$ \( T - 1244 \) Copy content Toggle raw display
$83$ \( T + 396 \) Copy content Toggle raw display
$89$ \( T - 972 \) Copy content Toggle raw display
$97$ \( T + 46 \) Copy content Toggle raw display
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