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: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (\beta_{3} - 2) q^{4} + ( - \beta_{5} - \beta_{2}) q^{7} + ( - \beta_{5} - \beta_{4} + 2 \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + (\beta_{3} - 2) q^{4} + ( - \beta_{5} - \beta_{2}) q^{7} + ( - \beta_{5} - \beta_{4} + 2 \beta_{2}) q^{8} - q^{11} + (\beta_{5} - \beta_{2}) q^{13} + ( - \beta_1 - 4) q^{14} + ( - 2 \beta_{3} - 2 \beta_1 + 5) q^{16} + ( - \beta_{5} + \beta_{4} + 3 \beta_{2}) q^{17} + (\beta_{3} - \beta_1 - 3) q^{19} + \beta_{2} q^{22} + (2 \beta_{5} + 2 \beta_{2}) q^{23} + (2 \beta_{3} + \beta_1 - 4) q^{26} + ( - \beta_{5} + 2 \beta_{4} + 3 \beta_{2}) q^{28} + ( - \beta_{3} - \beta_1 - 3) q^{29} + (2 \beta_{3} + 2) q^{31} + (2 \beta_{5} + 4 \beta_{4} - 3 \beta_{2}) q^{32} + ( - 3 \beta_{3} + 11) q^{34} - \beta_{4} q^{37} + (\beta_{4} + 6 \beta_{2}) q^{38} + ( - \beta_{3} - \beta_1 + 5) q^{41} + ( - 3 \beta_{5} + \beta_{2}) q^{43} + ( - \beta_{3} + 2) q^{44} + (2 \beta_1 + 8) q^{46} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{2}) q^{47} + ( - 2 \beta_{3} - 2 \beta_1 - 3) q^{49} + ( - \beta_{5} - 4 \beta_{4} + 5 \beta_{2}) q^{52} + ( - 2 \beta_{5} - \beta_{4} - 2 \beta_{2}) q^{53} + ( - 2 \beta_{3} - \beta_1 + 2) q^{56} + (2 \beta_{5} + 3 \beta_{4} + 2 \beta_{2}) q^{58} + ( - 2 \beta_1 + 4) q^{59} + (2 \beta_1 - 2) q^{61} + ( - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{2}) q^{62} + (5 \beta_{3} + 2 \beta_1 - 6) q^{64} + ( - 2 \beta_{5} + 2 \beta_{2}) q^{67} + (\beta_{5} + 5 \beta_{4} - 11 \beta_{2}) q^{68} + ( - 2 \beta_{3} - 2) q^{71} + ( - \beta_{5} + 2 \beta_{4} - 3 \beta_{2}) q^{73} + ( - \beta_{3} - \beta_1 + 1) q^{74} + ( - 3 \beta_{3} - \beta_1 + 17) q^{76} + (\beta_{5} + \beta_{2}) q^{77} + ( - 3 \beta_{3} - \beta_1 - 3) q^{79} + (2 \beta_{5} + 3 \beta_{4} - 6 \beta_{2}) q^{82} + ( - 3 \beta_{5} + \beta_{4} + 3 \beta_{2}) q^{83} + ( - 4 \beta_{3} - 3 \beta_1 + 4) q^{86} + (\beta_{5} + \beta_{4} - 2 \beta_{2}) q^{88} + (2 \beta_{3} + 2 \beta_1 - 4) q^{89} + (2 \beta_{3} + 2) q^{91} + (2 \beta_{5} - 4 \beta_{4} - 6 \beta_{2}) q^{92} + ( - 2 \beta_{3} + 6) q^{94} + (2 \beta_{5} + 3 \beta_{4} - 2 \beta_{2}) q^{97} + (4 \beta_{5} + 6 \beta_{4} + \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 10 q^{4} - 6 q^{11} - 24 q^{14} + 26 q^{16} - 16 q^{19} - 20 q^{26} - 20 q^{29} + 16 q^{31} + 60 q^{34} + 28 q^{41} + 10 q^{44} + 48 q^{46} - 22 q^{49} + 8 q^{56} + 24 q^{59} - 12 q^{61} - 26 q^{64} - 16 q^{71} + 4 q^{74} + 96 q^{76} - 24 q^{79} + 16 q^{86} - 20 q^{89} + 16 q^{91} + 32 q^{94}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( 2\nu^{5} - 16\nu^{4} + 8\nu^{3} + 2\nu^{2} - 4\nu - 76 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{5} + \nu^{4} + 11\nu^{3} - 26\nu^{2} + 6\nu - 1 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -10\nu^{5} + 34\nu^{4} - 40\nu^{3} - 10\nu^{2} + 20\nu + 81 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 14\nu^{5} - 20\nu^{4} + 10\nu^{3} + 60\nu^{2} + 64\nu - 26 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -25\nu^{5} + 39\nu^{4} - 31\nu^{3} - 48\nu^{2} - 134\nu + 53 ) / 23 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} + 2\beta_{2} + \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 2\beta_{4} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} + 2\beta_{4} - \beta_{3} + 3\beta_{2} - 2\beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{3} - 5\beta _1 - 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{5} - 9\beta_{4} - 3\beta_{3} - 11\beta_{2} - 8\beta _1 - 15 ) / 2 \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(2475, [\chi])\):

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

Hecke characteristic polynomials

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