Properties

Label 5175.2.a.bv
Level $5175$
Weight $2$
Character orbit 5175.a
Self dual yes
Analytic conductor $41.323$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5175 = 3^{2} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5175.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(41.3225830460\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.5744.1
Defining polynomial: \( x^{4} - 5x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 115)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + \beta_{2} + \beta_1) q^{4} + ( - \beta_{3} - \beta_{2} - 1) q^{7} + (\beta_{2} + \beta_1 + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} + \beta_{2} + \beta_1) q^{4} + ( - \beta_{3} - \beta_{2} - 1) q^{7} + (\beta_{2} + \beta_1 + 1) q^{8} + ( - \beta_{3} + \beta_{2} - 1) q^{11} + ( - \beta_{2} - \beta_1 - 3) q^{13} + (\beta_{3} - 3 \beta_1 + 1) q^{14} + ( - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{16} + ( - \beta_{2} + \beta_1 + 4) q^{17} + ( - \beta_{3} - \beta_1 - 1) q^{19} + (\beta_{3} - \beta_1 - 1) q^{22} - q^{23} + ( - \beta_{3} - \beta_{2} - 5 \beta_1 - 1) q^{26} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 - 4) q^{28} + (2 \beta_{3} - 1) q^{29} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 1) q^{31} + (2 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{32} + (\beta_{3} + \beta_{2} + 4 \beta_1 + 3) q^{34} + (\beta_{3} + \beta_{2} + 4 \beta_1 - 1) q^{37} + ( - \beta_{2} - 3 \beta_1 - 2) q^{38} + (2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 3) q^{41} + (\beta_{3} - 4 \beta_{2} - 3 \beta_1 + 1) q^{43} + ( - 3 \beta_{2} - \beta_1) q^{44} - \beta_1 q^{46} + (3 \beta_{2} - 3 \beta_1 - 1) q^{47} + (4 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 1) q^{49} + ( - 4 \beta_{3} - 3 \beta_{2} - 6 \beta_1 - 3) q^{52} + ( - 4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{53} + ( - \beta_{3} - \beta_{2} - 2 \beta_1 - 3) q^{56} + ( - 2 \beta_{3} + \beta_1) q^{58} + 2 \beta_1 q^{59} + ( - 3 \beta_{3} + 2 \beta_{2} - \beta_1 - 3) q^{61} + ( - 4 \beta_{3} - 2 \beta_{2} + \beta_1 - 6) q^{62} + ( - 2 \beta_{3} - 2 \beta_1 - 5) q^{64} + ( - \beta_{3} - \beta_{2} + 1) q^{67} + (3 \beta_{3} + 6 \beta_{2} + 7 \beta_1 - 1) q^{68} + ( - 2 \beta_{3} - 2 \beta_{2} + 4 \beta_1 + 7) q^{71} + ( - 4 \beta_{3} + \beta_{2} - \beta_1 - 5) q^{73} + (3 \beta_{3} + 4 \beta_{2} + 5 \beta_1 + 7) q^{74} + ( - \beta_{3} - 3 \beta_{2} - 4 \beta_1 - 3) q^{76} + (2 \beta_{3} + 2 \beta_{2}) q^{77} + (\beta_{3} + 2 \beta_{2} - \beta_1 + 5) q^{79} + ( - 6 \beta_{3} - 4 \beta_{2} - \beta_1 - 6) q^{82} + (4 \beta_{3} - 3 \beta_{2} - 5 \beta_1) q^{83} + ( - 4 \beta_{3} - 3 \beta_{2} - 5 \beta_1 - 2) q^{86} + ( - 3 \beta_{3} - \beta_{2} - 2 \beta_1 + 3) q^{88} + ( - \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 1) q^{89} + (3 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 5) q^{91} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{92} + ( - 3 \beta_{3} - 3 \beta_{2} - \beta_1 - 9) q^{94} + (3 \beta_{3} + \beta_{2} - 2 \beta_1 - 9) q^{97} + ( - 6 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 6) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 6 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 6 q^{7} + 6 q^{8} - 2 q^{11} - 14 q^{13} + 4 q^{14} + 2 q^{16} + 14 q^{17} - 4 q^{19} - 4 q^{22} - 4 q^{23} - 6 q^{26} - 18 q^{28} - 4 q^{29} + 2 q^{32} + 14 q^{34} - 2 q^{37} - 10 q^{38} + 8 q^{41} - 4 q^{43} - 6 q^{44} + 2 q^{47} - 18 q^{52} + 4 q^{53} - 14 q^{56} - 8 q^{61} - 28 q^{62} - 20 q^{64} + 2 q^{67} + 8 q^{68} + 24 q^{71} - 18 q^{73} + 36 q^{74} - 18 q^{76} + 4 q^{77} + 24 q^{79} - 32 q^{82} - 6 q^{83} - 14 q^{86} + 10 q^{88} + 8 q^{89} + 26 q^{91} - 2 q^{92} - 42 q^{94} - 34 q^{97} - 28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} - 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 5\nu - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + \nu^{2} + 4\nu - 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} + 5\beta _1 + 1 \) 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
−1.92022
−0.751024
0.291367
2.37988
−1.92022 0 1.68725 0 0 −4.60747 0.600553 0 0
1.2 −0.751024 0 −1.43596 0 0 −0.315061 2.58049 0 0
1.3 0.291367 0 −1.91511 0 0 1.20647 −1.14073 0 0
1.4 2.37988 0 3.66382 0 0 −2.28394 3.95969 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(23\) \(1\)

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 5175.2.a.bv 4
3.b odd 2 1 575.2.a.i 4
5.b even 2 1 5175.2.a.bw 4
5.c odd 4 2 1035.2.b.e 8
12.b even 2 1 9200.2.a.cq 4
15.d odd 2 1 575.2.a.j 4
15.e even 4 2 115.2.b.b 8
60.h even 2 1 9200.2.a.ck 4
60.l odd 4 2 1840.2.e.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.b.b 8 15.e even 4 2
575.2.a.i 4 3.b odd 2 1
575.2.a.j 4 15.d odd 2 1
1035.2.b.e 8 5.c odd 4 2
1840.2.e.d 8 60.l odd 4 2
5175.2.a.bv 4 1.a even 1 1 trivial
5175.2.a.bw 4 5.b even 2 1
9200.2.a.ck 4 60.h even 2 1
9200.2.a.cq 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5175))\):

\( T_{2}^{4} - 5T_{2}^{2} - 2T_{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + 6T_{7}^{3} + 4T_{7}^{2} - 12T_{7} - 4 \) Copy content Toggle raw display
\( T_{11}^{4} + 2T_{11}^{3} - 16T_{11}^{2} - 44T_{11} - 28 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 5 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 6 T^{3} + 4 T^{2} - 12 T - 4 \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{3} - 16 T^{2} - 44 T - 28 \) Copy content Toggle raw display
$13$ \( T^{4} + 14 T^{3} + 66 T^{2} + 118 T + 61 \) Copy content Toggle raw display
$17$ \( T^{4} - 14 T^{3} + 58 T^{2} - 64 T + 20 \) Copy content Toggle raw display
$19$ \( T^{4} + 4 T^{3} - 6 T^{2} - 28 T - 20 \) Copy content Toggle raw display
$23$ \( (T + 1)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 4 T^{3} - 22 T^{2} - 4 T + 5 \) Copy content Toggle raw display
$31$ \( T^{4} - 74 T^{2} - 256 T - 167 \) Copy content Toggle raw display
$37$ \( T^{4} + 2 T^{3} - 72 T^{2} - 380 T - 476 \) Copy content Toggle raw display
$41$ \( T^{4} - 8 T^{3} - 94 T^{2} + \cdots + 2485 \) Copy content Toggle raw display
$43$ \( T^{4} + 4 T^{3} - 118 T^{2} + \cdots + 1964 \) Copy content Toggle raw display
$47$ \( T^{4} - 2 T^{3} - 138 T^{2} + \cdots + 4513 \) Copy content Toggle raw display
$53$ \( T^{4} - 4 T^{3} - 104 T^{2} + \cdots + 2192 \) Copy content Toggle raw display
$59$ \( T^{4} - 20 T^{2} - 16 T + 16 \) Copy content Toggle raw display
$61$ \( T^{4} + 8 T^{3} - 102 T^{2} + \cdots - 2756 \) Copy content Toggle raw display
$67$ \( T^{4} - 2 T^{3} - 8 T^{2} + 12 T + 4 \) Copy content Toggle raw display
$71$ \( T^{4} - 24 T^{3} + 66 T^{2} + \cdots - 7435 \) Copy content Toggle raw display
$73$ \( T^{4} + 18 T^{3} - 22 T^{2} + \cdots - 8339 \) Copy content Toggle raw display
$79$ \( T^{4} - 24 T^{3} + 178 T^{2} + \cdots + 28 \) Copy content Toggle raw display
$83$ \( T^{4} + 6 T^{3} - 270 T^{2} + \cdots + 14948 \) Copy content Toggle raw display
$89$ \( T^{4} - 8 T^{3} - 86 T^{2} + \cdots + 2380 \) Copy content Toggle raw display
$97$ \( T^{4} + 34 T^{3} + 348 T^{2} + \cdots - 4676 \) Copy content Toggle raw display
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