Properties

Label 2475.2.c.r
Level $2475$
Weight $2$
Character orbit 2475.c
Analytic conductor $19.763$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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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: \(6\)
Coefficient field: 6.0.350464.1
Defining polynomial: \(x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + ( -2 + \beta_{3} ) q^{4} + ( -\beta_{2} - \beta_{5} ) q^{7} + ( 2 \beta_{2} - \beta_{4} - \beta_{5} ) q^{8} +O(q^{10})\) \( q -\beta_{2} q^{2} + ( -2 + \beta_{3} ) q^{4} + ( -\beta_{2} - \beta_{5} ) q^{7} + ( 2 \beta_{2} - \beta_{4} - \beta_{5} ) q^{8} - q^{11} + ( -\beta_{2} + \beta_{5} ) q^{13} + ( -4 - \beta_{1} ) q^{14} + ( 5 - 2 \beta_{1} - 2 \beta_{3} ) q^{16} + ( 3 \beta_{2} + \beta_{4} - \beta_{5} ) q^{17} + ( -3 - \beta_{1} + \beta_{3} ) q^{19} + \beta_{2} q^{22} + ( 2 \beta_{2} + 2 \beta_{5} ) q^{23} + ( -4 + \beta_{1} + 2 \beta_{3} ) q^{26} + ( 3 \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{28} + ( -3 - \beta_{1} - \beta_{3} ) q^{29} + ( 2 + 2 \beta_{3} ) q^{31} + ( -3 \beta_{2} + 4 \beta_{4} + 2 \beta_{5} ) q^{32} + ( 11 - 3 \beta_{3} ) q^{34} -\beta_{4} q^{37} + ( 6 \beta_{2} + \beta_{4} ) q^{38} + ( 5 - \beta_{1} - \beta_{3} ) q^{41} + ( \beta_{2} - 3 \beta_{5} ) q^{43} + ( 2 - \beta_{3} ) q^{44} + ( 8 + 2 \beta_{1} ) q^{46} + ( 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{47} + ( -3 - 2 \beta_{1} - 2 \beta_{3} ) q^{49} + ( 5 \beta_{2} - 4 \beta_{4} - \beta_{5} ) q^{52} + ( -2 \beta_{2} - \beta_{4} - 2 \beta_{5} ) q^{53} + ( 2 - \beta_{1} - 2 \beta_{3} ) q^{56} + ( 2 \beta_{2} + 3 \beta_{4} + 2 \beta_{5} ) q^{58} + ( 4 - 2 \beta_{1} ) q^{59} + ( -2 + 2 \beta_{1} ) q^{61} + ( 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{62} + ( -6 + 2 \beta_{1} + 5 \beta_{3} ) q^{64} + ( 2 \beta_{2} - 2 \beta_{5} ) q^{67} + ( -11 \beta_{2} + 5 \beta_{4} + \beta_{5} ) q^{68} + ( -2 - 2 \beta_{3} ) q^{71} + ( -3 \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{73} + ( 1 - \beta_{1} - \beta_{3} ) q^{74} + ( 17 - \beta_{1} - 3 \beta_{3} ) q^{76} + ( \beta_{2} + \beta_{5} ) q^{77} + ( -3 - \beta_{1} - 3 \beta_{3} ) q^{79} + ( -6 \beta_{2} + 3 \beta_{4} + 2 \beta_{5} ) q^{82} + ( 3 \beta_{2} + \beta_{4} - 3 \beta_{5} ) q^{83} + ( 4 - 3 \beta_{1} - 4 \beta_{3} ) q^{86} + ( -2 \beta_{2} + \beta_{4} + \beta_{5} ) q^{88} + ( -4 + 2 \beta_{1} + 2 \beta_{3} ) q^{89} + ( 2 + 2 \beta_{3} ) q^{91} + ( -6 \beta_{2} - 4 \beta_{4} + 2 \beta_{5} ) q^{92} + ( 6 - 2 \beta_{3} ) q^{94} + ( -2 \beta_{2} + 3 \beta_{4} + 2 \beta_{5} ) q^{97} + ( \beta_{2} + 6 \beta_{4} + 4 \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{4} + O(q^{10}) \) \( 6 q - 10 q^{4} - 6 q^{11} - 24 q^{14} + 26 q^{16} - 16 q^{19} - 20 q^{26} - 20 q^{29} + 16 q^{31} + 60 q^{34} + 28 q^{41} + 10 q^{44} + 48 q^{46} - 22 q^{49} + 8 q^{56} + 24 q^{59} - 12 q^{61} - 26 q^{64} - 16 q^{71} + 4 q^{74} + 96 q^{76} - 24 q^{79} + 16 q^{86} - 20 q^{89} + 16 q^{91} + 32 q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{5} - 16 \nu^{4} + 8 \nu^{3} + 2 \nu^{2} - 4 \nu - 76 \)\()/23\)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{5} + \nu^{4} + 11 \nu^{3} - 26 \nu^{2} + 6 \nu - 1 \)\()/23\)
\(\beta_{3}\)\(=\)\((\)\( -10 \nu^{5} + 34 \nu^{4} - 40 \nu^{3} - 10 \nu^{2} + 20 \nu + 81 \)\()/23\)
\(\beta_{4}\)\(=\)\((\)\( 14 \nu^{5} - 20 \nu^{4} + 10 \nu^{3} + 60 \nu^{2} + 64 \nu - 26 \)\()/23\)
\(\beta_{5}\)\(=\)\((\)\( -25 \nu^{5} + 39 \nu^{4} - 31 \nu^{3} - 48 \nu^{2} - 134 \nu + 53 \)\()/23\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_{1} + 1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} + 2 \beta_{4} + \beta_{2}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{5} + 2 \beta_{4} - \beta_{3} + 3 \beta_{2} - 2 \beta_{1} - 3\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{3} - 5 \beta_{1} - 13\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-5 \beta_{5} - 9 \beta_{4} - 3 \beta_{3} - 11 \beta_{2} - 8 \beta_{1} - 15\)\()/2\)

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
−0.854638 + 0.854638i
1.45161 + 1.45161i
0.403032 0.403032i
0.403032 + 0.403032i
1.45161 1.45161i
−0.854638 0.854638i
2.70928i 0 −5.34017 0 0 1.07838i 9.04945i 0 0
199.2 1.90321i 0 −1.62222 0 0 4.42864i 0.719004i 0 0
199.3 0.193937i 0 1.96239 0 0 3.35026i 0.768452i 0 0
199.4 0.193937i 0 1.96239 0 0 3.35026i 0.768452i 0 0
199.5 1.90321i 0 −1.62222 0 0 4.42864i 0.719004i 0 0
199.6 2.70928i 0 −5.34017 0 0 1.07838i 9.04945i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.6
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.r 6
3.b odd 2 1 825.2.c.g 6
5.b even 2 1 inner 2475.2.c.r 6
5.c odd 4 1 495.2.a.e 3
5.c odd 4 1 2475.2.a.bb 3
15.d odd 2 1 825.2.c.g 6
15.e even 4 1 165.2.a.c 3
15.e even 4 1 825.2.a.k 3
20.e even 4 1 7920.2.a.cj 3
55.e even 4 1 5445.2.a.z 3
60.l odd 4 1 2640.2.a.be 3
105.k odd 4 1 8085.2.a.bk 3
165.l odd 4 1 1815.2.a.m 3
165.l odd 4 1 9075.2.a.cf 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.c 3 15.e even 4 1
495.2.a.e 3 5.c odd 4 1
825.2.a.k 3 15.e even 4 1
825.2.c.g 6 3.b odd 2 1
825.2.c.g 6 15.d odd 2 1
1815.2.a.m 3 165.l odd 4 1
2475.2.a.bb 3 5.c odd 4 1
2475.2.c.r 6 1.a even 1 1 trivial
2475.2.c.r 6 5.b even 2 1 inner
2640.2.a.be 3 60.l odd 4 1
5445.2.a.z 3 55.e even 4 1
7920.2.a.cj 3 20.e even 4 1
8085.2.a.bk 3 105.k odd 4 1
9075.2.a.cf 3 165.l odd 4 1

Hecke kernels

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

\( T_{2}^{6} + 11 T_{2}^{4} + 27 T_{2}^{2} + 1 \)
\( T_{7}^{6} + 32 T_{7}^{4} + 256 T_{7}^{2} + 256 \)
\( T_{29}^{3} + 10 T_{29}^{2} + 12 T_{29} - 40 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 27 T^{2} + 11 T^{4} + T^{6} \)
$3$ \( T^{6} \)
$5$ \( T^{6} \)
$7$ \( 256 + 256 T^{2} + 32 T^{4} + T^{6} \)
$11$ \( ( 1 + T )^{6} \)
$13$ \( 64 + 176 T^{2} + 28 T^{4} + T^{6} \)
$17$ \( 33856 + 3440 T^{2} + 108 T^{4} + T^{6} \)
$19$ \( ( -160 - 16 T + 8 T^{2} + T^{3} )^{2} \)
$23$ \( 16384 + 4096 T^{2} + 128 T^{4} + T^{6} \)
$29$ \( ( -40 + 12 T + 10 T^{2} + T^{3} )^{2} \)
$31$ \( ( 128 - 32 T - 8 T^{2} + T^{3} )^{2} \)
$37$ \( ( 4 + T^{2} )^{3} \)
$41$ \( ( -8 + 44 T - 14 T^{2} + T^{3} )^{2} \)
$43$ \( 160000 + 9600 T^{2} + 176 T^{4} + T^{6} \)
$47$ \( 16384 + 3072 T^{2} + 128 T^{4} + T^{6} \)
$53$ \( 64 + 2608 T^{2} + 140 T^{4} + T^{6} \)
$59$ \( ( 320 - 16 T - 12 T^{2} + T^{3} )^{2} \)
$61$ \( ( -248 - 52 T + 6 T^{2} + T^{3} )^{2} \)
$67$ \( 4096 + 2816 T^{2} + 112 T^{4} + T^{6} \)
$71$ \( ( -128 - 32 T + 8 T^{2} + T^{3} )^{2} \)
$73$ \( 118336 + 9648 T^{2} + 188 T^{4} + T^{6} \)
$79$ \( ( -800 - 64 T + 12 T^{2} + T^{3} )^{2} \)
$83$ \( 256 + 14400 T^{2} + 240 T^{4} + T^{6} \)
$89$ \( ( -200 - 52 T + 10 T^{2} + T^{3} )^{2} \)
$97$ \( 64 + 11312 T^{2} + 268 T^{4} + T^{6} \)
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