Properties

Label 2475.2.c.j
Level $2475$
Weight $2$
Character orbit 2475.c
Analytic conductor $19.763$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.7629745003\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \(x^{4} + 9 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -3 + \beta_{3} ) q^{4} + \beta_{2} q^{7} + ( -\beta_{1} + 4 \beta_{2} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -3 + \beta_{3} ) q^{4} + \beta_{2} q^{7} + ( -\beta_{1} + 4 \beta_{2} ) q^{8} + q^{11} + ( \beta_{1} + 3 \beta_{2} ) q^{13} + ( 1 - \beta_{3} ) q^{14} + ( 3 - 3 \beta_{3} ) q^{16} + ( -\beta_{1} - 4 \beta_{2} ) q^{17} -3 q^{19} + \beta_{1} q^{22} + ( -\beta_{1} + 4 \beta_{2} ) q^{23} + ( -2 - 2 \beta_{3} ) q^{26} + ( \beta_{1} - 2 \beta_{2} ) q^{28} + ( 2 - 2 \beta_{3} ) q^{29} -3 \beta_{3} q^{31} + ( \beta_{1} - 4 \beta_{2} ) q^{32} + ( 1 + 3 \beta_{3} ) q^{34} + ( 5 \beta_{1} + 2 \beta_{2} ) q^{37} -3 \beta_{1} q^{38} + ( 3 - 3 \beta_{3} ) q^{41} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{43} + ( -3 + \beta_{3} ) q^{44} + ( 9 - 5 \beta_{3} ) q^{46} + ( -\beta_{1} - 8 \beta_{2} ) q^{47} + 6 q^{49} -2 \beta_{2} q^{52} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{53} + ( -5 + \beta_{3} ) q^{56} + ( 2 \beta_{1} - 8 \beta_{2} ) q^{58} + ( 9 - \beta_{3} ) q^{59} + ( -6 + \beta_{3} ) q^{61} -12 \beta_{2} q^{62} + ( -3 - \beta_{3} ) q^{64} + ( 3 \beta_{1} + \beta_{2} ) q^{67} + ( -\beta_{1} + 4 \beta_{2} ) q^{68} + ( -1 + 5 \beta_{3} ) q^{71} + ( 4 \beta_{1} - 6 \beta_{2} ) q^{73} + ( -23 + 3 \beta_{3} ) q^{74} + ( 9 - 3 \beta_{3} ) q^{76} + \beta_{2} q^{77} + ( 11 - 3 \beta_{3} ) q^{79} + ( 3 \beta_{1} - 12 \beta_{2} ) q^{82} -12 q^{86} + ( -\beta_{1} + 4 \beta_{2} ) q^{88} + ( -14 + 2 \beta_{3} ) q^{89} + ( -2 - \beta_{3} ) q^{91} + ( 7 \beta_{1} - 12 \beta_{2} ) q^{92} + ( -3 + 7 \beta_{3} ) q^{94} + ( -2 \beta_{1} - 3 \beta_{2} ) q^{97} + 6 \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 10q^{4} + O(q^{10}) \) \( 4q - 10q^{4} + 4q^{11} + 2q^{14} + 6q^{16} - 12q^{19} - 12q^{26} + 4q^{29} - 6q^{31} + 10q^{34} + 6q^{41} - 10q^{44} + 26q^{46} + 24q^{49} - 18q^{56} + 34q^{59} - 22q^{61} - 14q^{64} + 6q^{71} - 86q^{74} + 30q^{76} + 38q^{79} - 48q^{86} - 52q^{89} - 10q^{91} + 2q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9 x^{2} + 16\):

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(2026\) \(2377\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
2.56155i
1.56155i
1.56155i
2.56155i
2.56155i 0 −4.56155 0 0 1.00000i 6.56155i 0 0
199.2 1.56155i 0 −0.438447 0 0 1.00000i 2.43845i 0 0
199.3 1.56155i 0 −0.438447 0 0 1.00000i 2.43845i 0 0
199.4 2.56155i 0 −4.56155 0 0 1.00000i 6.56155i 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2475.2.c.j 4
3.b odd 2 1 2475.2.c.i 4
5.b even 2 1 inner 2475.2.c.j 4
5.c odd 4 1 2475.2.a.q yes 2
5.c odd 4 1 2475.2.a.u yes 2
15.d odd 2 1 2475.2.c.i 4
15.e even 4 1 2475.2.a.p 2
15.e even 4 1 2475.2.a.v yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2475.2.a.p 2 15.e even 4 1
2475.2.a.q yes 2 5.c odd 4 1
2475.2.a.u yes 2 5.c odd 4 1
2475.2.a.v yes 2 15.e even 4 1
2475.2.c.i 4 3.b odd 2 1
2475.2.c.i 4 15.d odd 2 1
2475.2.c.j 4 1.a even 1 1 trivial
2475.2.c.j 4 5.b even 2 1 inner

Hecke kernels

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

\( T_{2}^{4} + 9 T_{2}^{2} + 16 \)
\( T_{7}^{2} + 1 \)
\( T_{29}^{2} - 2 T_{29} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + 9 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( -1 + T )^{4} \)
$13$ \( 4 + 21 T^{2} + T^{4} \)
$17$ \( 64 + 33 T^{2} + T^{4} \)
$19$ \( ( 3 + T )^{4} \)
$23$ \( 256 + 49 T^{2} + T^{4} \)
$29$ \( ( -16 - 2 T + T^{2} )^{2} \)
$31$ \( ( -36 + 3 T + T^{2} )^{2} \)
$37$ \( 11236 + 213 T^{2} + T^{4} \)
$41$ \( ( -36 - 3 T + T^{2} )^{2} \)
$43$ \( 1296 + 81 T^{2} + T^{4} \)
$47$ \( 2704 + 121 T^{2} + T^{4} \)
$53$ \( 64 + 52 T^{2} + T^{4} \)
$59$ \( ( 68 - 17 T + T^{2} )^{2} \)
$61$ \( ( 26 + 11 T + T^{2} )^{2} \)
$67$ \( 1444 + 77 T^{2} + T^{4} \)
$71$ \( ( -104 - 3 T + T^{2} )^{2} \)
$73$ \( 16 + 264 T^{2} + T^{4} \)
$79$ \( ( 52 - 19 T + T^{2} )^{2} \)
$83$ \( T^{4} \)
$89$ \( ( 152 + 26 T + T^{2} )^{2} \)
$97$ \( 169 + 42 T^{2} + T^{4} \)
show more
show less