Properties

Label 2475.2.a.bj
Level $2475$
Weight $2$
Character orbit 2475.a
Self dual yes
Analytic conductor $19.763$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.7629745003\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.48704.2
Defining polynomial: \(x^{4} - 2 x^{3} - 6 x^{2} + 4 x + 6\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 495)
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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 2 + \beta_{2} ) q^{4} + ( -1 - \beta_{1} - \beta_{3} ) q^{7} + ( 2 - 2 \beta_{1} - \beta_{3} ) q^{8} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 2 + \beta_{2} ) q^{4} + ( -1 - \beta_{1} - \beta_{3} ) q^{7} + ( 2 - 2 \beta_{1} - \beta_{3} ) q^{8} + q^{11} + ( -1 - \beta_{1} + \beta_{3} ) q^{13} + ( 1 + \beta_{2} - 2 \beta_{3} ) q^{14} + ( 3 - 2 \beta_{1} - 2 \beta_{3} ) q^{16} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{17} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{19} + ( 1 - \beta_{1} ) q^{22} + ( -2 - 2 \beta_{2} ) q^{23} + ( 3 + 4 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{26} + ( 3 - 5 \beta_{1} - 3 \beta_{3} ) q^{28} + ( 2 + 2 \beta_{2} + 2 \beta_{3} ) q^{29} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{31} + ( 3 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{32} + ( 3 - 4 \beta_{1} - \beta_{2} ) q^{34} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{37} + ( 4 + 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{38} + ( 2 - 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{41} + ( -3 + \beta_{1} + \beta_{3} ) q^{43} + ( 2 + \beta_{2} ) q^{44} + ( -6 + 6 \beta_{1} + 2 \beta_{3} ) q^{46} + ( -2 + 2 \beta_{1} - 2 \beta_{3} ) q^{47} + ( 7 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{49} + ( -3 - 3 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{52} + ( 4 - 2 \beta_{2} ) q^{53} + ( 13 - 4 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{56} + ( 8 - 2 \beta_{1} + 2 \beta_{3} ) q^{58} + ( 8 - 4 \beta_{1} - 2 \beta_{2} ) q^{59} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{61} + ( -4 + 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{62} + ( 2 - 6 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{64} + ( -2 + 4 \beta_{1} - 2 \beta_{2} ) q^{67} + ( 11 + \beta_{1} - \beta_{3} ) q^{68} + ( 4 - 2 \beta_{2} ) q^{71} + ( -1 - \beta_{1} + \beta_{3} ) q^{73} + ( -2 - 2 \beta_{2} + 2 \beta_{3} ) q^{74} + ( -6 - 4 \beta_{1} + 2 \beta_{3} ) q^{76} + ( -1 - \beta_{1} - \beta_{3} ) q^{77} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{79} + ( 8 + 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{82} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{83} + ( -5 + 4 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{86} + ( 2 - 2 \beta_{1} - \beta_{3} ) q^{88} + ( 2 + 4 \beta_{1} ) q^{89} + ( -8 + 6 \beta_{1} - 2 \beta_{3} ) q^{91} + ( -18 + 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{92} + ( -10 - 4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{94} + ( -4 + 4 \beta_{1} + 2 \beta_{2} ) q^{97} + ( 19 - 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} + 8q^{4} - 4q^{7} + 6q^{8} + O(q^{10}) \) \( 4q + 2q^{2} + 8q^{4} - 4q^{7} + 6q^{8} + 4q^{11} - 8q^{13} + 8q^{14} + 12q^{16} + 4q^{17} + 4q^{19} + 2q^{22} - 8q^{23} + 16q^{26} + 8q^{28} + 4q^{29} + 14q^{32} + 4q^{34} - 8q^{37} + 20q^{38} + 4q^{41} - 12q^{43} + 8q^{44} - 16q^{46} + 20q^{49} - 20q^{52} + 16q^{53} + 48q^{56} + 24q^{58} + 24q^{59} + 8q^{61} - 20q^{62} + 48q^{68} + 16q^{71} - 8q^{73} - 12q^{74} - 36q^{76} - 4q^{77} + 12q^{79} + 40q^{82} + 8q^{83} - 16q^{86} + 6q^{88} + 16q^{89} - 16q^{91} - 72q^{92} - 40q^{94} - 8q^{97} + 62q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 6 x^{2} + 4 x + 6\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 3 \nu^{2} - 3 \nu + 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 3 \beta_{2} + 9 \beta_{1} + 4\)

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
3.28632
1.26270
−0.852061
−1.69696
−2.28632 0 3.22727 0 0 −2.51962 −2.80595 0 0
1.2 −0.262696 0 −1.93099 0 0 −0.704647 1.03266 0 0
1.3 1.85206 0 1.43013 0 0 −4.90749 −1.05543 0 0
1.4 2.69696 0 5.27358 0 0 4.13176 8.82872 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(11\) \(-1\)

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 2475.2.a.bj 4
3.b odd 2 1 2475.2.a.bf 4
5.b even 2 1 495.2.a.f 4
5.c odd 4 2 2475.2.c.t 8
15.d odd 2 1 495.2.a.g yes 4
15.e even 4 2 2475.2.c.s 8
20.d odd 2 1 7920.2.a.cm 4
55.d odd 2 1 5445.2.a.bs 4
60.h even 2 1 7920.2.a.cn 4
165.d even 2 1 5445.2.a.bh 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.2.a.f 4 5.b even 2 1
495.2.a.g yes 4 15.d odd 2 1
2475.2.a.bf 4 3.b odd 2 1
2475.2.a.bj 4 1.a even 1 1 trivial
2475.2.c.s 8 15.e even 4 2
2475.2.c.t 8 5.c odd 4 2
5445.2.a.bh 4 165.d even 2 1
5445.2.a.bs 4 55.d odd 2 1
7920.2.a.cm 4 20.d odd 2 1
7920.2.a.cn 4 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2475))\):

\( T_{2}^{4} - 2 T_{2}^{3} - 6 T_{2}^{2} + 10 T_{2} + 3 \)
\( T_{7}^{4} + 4 T_{7}^{3} - 16 T_{7}^{2} - 64 T_{7} - 36 \)
\( T_{29}^{4} - 4 T_{29}^{3} - 88 T_{29}^{2} + 240 T_{29} - 144 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 3 + 10 T - 6 T^{2} - 2 T^{3} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( -36 - 64 T - 16 T^{2} + 4 T^{3} + T^{4} \)
$11$ \( ( -1 + T )^{4} \)
$13$ \( -292 - 168 T - 8 T^{2} + 8 T^{3} + T^{4} \)
$17$ \( -324 + 240 T - 40 T^{2} - 4 T^{3} + T^{4} \)
$19$ \( 288 + 192 T - 56 T^{2} - 4 T^{3} + T^{4} \)
$23$ \( -192 - 256 T - 32 T^{2} + 8 T^{3} + T^{4} \)
$29$ \( -144 + 240 T - 88 T^{2} - 4 T^{3} + T^{4} \)
$31$ \( 144 + 192 T - 88 T^{2} + T^{4} \)
$37$ \( 976 - 224 T - 56 T^{2} + 8 T^{3} + T^{4} \)
$41$ \( 2160 + 432 T - 120 T^{2} - 4 T^{3} + T^{4} \)
$43$ \( -36 + 32 T^{2} + 12 T^{3} + T^{4} \)
$47$ \( -576 + 576 T - 128 T^{2} + T^{4} \)
$53$ \( -240 + 128 T + 40 T^{2} - 16 T^{3} + T^{4} \)
$59$ \( -8496 + 1376 T + 88 T^{2} - 24 T^{3} + T^{4} \)
$61$ \( -2224 + 1056 T - 104 T^{2} - 8 T^{3} + T^{4} \)
$67$ \( 6208 + 576 T - 224 T^{2} + T^{4} \)
$71$ \( -240 + 128 T + 40 T^{2} - 16 T^{3} + T^{4} \)
$73$ \( -292 - 168 T - 8 T^{2} + 8 T^{3} + T^{4} \)
$79$ \( 160 + 192 T - 8 T^{2} - 12 T^{3} + T^{4} \)
$83$ \( 1836 + 360 T - 72 T^{2} - 8 T^{3} + T^{4} \)
$89$ \( 720 + 512 T - 24 T^{2} - 16 T^{3} + T^{4} \)
$97$ \( -2864 - 1184 T - 104 T^{2} + 8 T^{3} + T^{4} \)
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