Properties

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

Related objects

Downloads

Learn more

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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 275)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( 1 + \beta ) q^{4} + ( -2 - \beta ) q^{7} + 3 q^{8} +O(q^{10})\) \( q + \beta q^{2} + ( 1 + \beta ) q^{4} + ( -2 - \beta ) q^{7} + 3 q^{8} + q^{11} -5 q^{13} + ( -3 - 3 \beta ) q^{14} + ( -2 + \beta ) q^{16} + ( 3 - 3 \beta ) q^{17} - q^{19} + \beta q^{22} + ( -6 + \beta ) q^{23} -5 \beta q^{26} + ( -5 - 4 \beta ) q^{28} + ( 3 + 3 \beta ) q^{29} + ( 5 - 4 \beta ) q^{31} + ( -3 - \beta ) q^{32} -9 q^{34} + ( -5 - 2 \beta ) q^{37} -\beta q^{38} + ( 3 - 2 \beta ) q^{41} + ( -2 + 4 \beta ) q^{43} + ( 1 + \beta ) q^{44} + ( 3 - 5 \beta ) q^{46} -3 q^{47} + 5 \beta q^{49} + ( -5 - 5 \beta ) q^{52} + \beta q^{53} + ( -6 - 3 \beta ) q^{56} + ( 9 + 6 \beta ) q^{58} + ( 9 - 4 \beta ) q^{59} + ( -4 + 3 \beta ) q^{61} + ( -12 + \beta ) q^{62} + ( 1 - 6 \beta ) q^{64} + 4 q^{67} + ( -6 - 3 \beta ) q^{68} -2 \beta q^{71} + ( 4 - 3 \beta ) q^{73} + ( -6 - 7 \beta ) q^{74} + ( -1 - \beta ) q^{76} + ( -2 - \beta ) q^{77} + ( -7 + 3 \beta ) q^{79} + ( -6 + \beta ) q^{82} + ( 3 + 5 \beta ) q^{83} + ( 12 + 2 \beta ) q^{86} + 3 q^{88} + ( -3 - \beta ) q^{89} + ( 10 + 5 \beta ) q^{91} + ( -3 - 4 \beta ) q^{92} -3 \beta q^{94} + ( -14 + \beta ) q^{97} + ( 15 + 5 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + 3q^{4} - 5q^{7} + 6q^{8} + O(q^{10}) \) \( 2q + q^{2} + 3q^{4} - 5q^{7} + 6q^{8} + 2q^{11} - 10q^{13} - 9q^{14} - 3q^{16} + 3q^{17} - 2q^{19} + q^{22} - 11q^{23} - 5q^{26} - 14q^{28} + 9q^{29} + 6q^{31} - 7q^{32} - 18q^{34} - 12q^{37} - q^{38} + 4q^{41} + 3q^{44} + q^{46} - 6q^{47} + 5q^{49} - 15q^{52} + q^{53} - 15q^{56} + 24q^{58} + 14q^{59} - 5q^{61} - 23q^{62} - 4q^{64} + 8q^{67} - 15q^{68} - 2q^{71} + 5q^{73} - 19q^{74} - 3q^{76} - 5q^{77} - 11q^{79} - 11q^{82} + 11q^{83} + 26q^{86} + 6q^{88} - 7q^{89} + 25q^{91} - 10q^{92} - 3q^{94} - 27q^{97} + 35q^{98} + O(q^{100}) \)

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
−1.30278
2.30278
−1.30278 0 −0.302776 0 0 −0.697224 3.00000 0 0
1.2 2.30278 0 3.30278 0 0 −4.30278 3.00000 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.t 2
3.b odd 2 1 275.2.a.e 2
5.b even 2 1 2475.2.a.o 2
5.c odd 4 2 2475.2.c.k 4
12.b even 2 1 4400.2.a.bs 2
15.d odd 2 1 275.2.a.f yes 2
15.e even 4 2 275.2.b.c 4
33.d even 2 1 3025.2.a.n 2
60.h even 2 1 4400.2.a.bh 2
60.l odd 4 2 4400.2.b.y 4
165.d even 2 1 3025.2.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.a.e 2 3.b odd 2 1
275.2.a.f yes 2 15.d odd 2 1
275.2.b.c 4 15.e even 4 2
2475.2.a.o 2 5.b even 2 1
2475.2.a.t 2 1.a even 1 1 trivial
2475.2.c.k 4 5.c odd 4 2
3025.2.a.h 2 165.d even 2 1
3025.2.a.n 2 33.d even 2 1
4400.2.a.bh 2 60.h even 2 1
4400.2.a.bs 2 12.b even 2 1
4400.2.b.y 4 60.l odd 4 2

Hecke kernels

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

\( T_{2}^{2} - T_{2} - 3 \)
\( T_{7}^{2} + 5 T_{7} + 3 \)
\( T_{29}^{2} - 9 T_{29} - 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -3 - T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 3 + 5 T + T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( ( 5 + T )^{2} \)
$17$ \( -27 - 3 T + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( 27 + 11 T + T^{2} \)
$29$ \( -9 - 9 T + T^{2} \)
$31$ \( -43 - 6 T + T^{2} \)
$37$ \( 23 + 12 T + T^{2} \)
$41$ \( -9 - 4 T + T^{2} \)
$43$ \( -52 + T^{2} \)
$47$ \( ( 3 + T )^{2} \)
$53$ \( -3 - T + T^{2} \)
$59$ \( -3 - 14 T + T^{2} \)
$61$ \( -23 + 5 T + T^{2} \)
$67$ \( ( -4 + T )^{2} \)
$71$ \( -12 + 2 T + T^{2} \)
$73$ \( -23 - 5 T + T^{2} \)
$79$ \( 1 + 11 T + T^{2} \)
$83$ \( -51 - 11 T + T^{2} \)
$89$ \( 9 + 7 T + T^{2} \)
$97$ \( 179 + 27 T + T^{2} \)
show more
show less