Properties

Label 2475.2.c.q
Level $2475$
Weight $2$
Character orbit 2475.c
Analytic conductor $19.763$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.5161984.1
Defining polynomial: \( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 825)
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.

\(f(q)\) \(=\) \( q + (\beta_{4} + \beta_{3}) q^{2} + ( - \beta_1 - 3) q^{4} + ( - \beta_{5} - \beta_{4} + \beta_{3}) q^{7} + (2 \beta_{5} - \beta_{4} - 3 \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + \beta_{3}) q^{2} + ( - \beta_1 - 3) q^{4} + ( - \beta_{5} - \beta_{4} + \beta_{3}) q^{7} + (2 \beta_{5} - \beta_{4} - 3 \beta_{3}) q^{8} - q^{11} + (\beta_{5} - 2 \beta_{3}) q^{13} + (2 \beta_{2} - \beta_1 + 1) q^{14} + ( - 2 \beta_{2} + 3 \beta_1 + 5) q^{16} + (2 \beta_{5} + \beta_{4} + \beta_{3}) q^{17} + ( - \beta_{2} + \beta_1 + 1) q^{19} + ( - \beta_{4} - \beta_{3}) q^{22} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3}) q^{23} + ( - 2 \beta_{2} + 2 \beta_1 + 4) q^{26} + (2 \beta_{5} + \beta_{4} - 7 \beta_{3}) q^{28} + 2 \beta_{2} q^{29} + (\beta_1 + 6) q^{31} + ( - 4 \beta_{5} + \beta_{4} + 13 \beta_{3}) q^{32} + (3 \beta_1 - 1) q^{34} + (3 \beta_{5} - \beta_{3}) q^{37} + ( - 3 \beta_{5} + 7 \beta_{3}) q^{38} + ( - \beta_{2} - 1) q^{41} + ( - 3 \beta_{5} + 2 \beta_{4} - 4 \beta_{3}) q^{43} + (\beta_1 + 3) q^{44} + ( - \beta_{2} - 4 \beta_1 - 11) q^{46} + (\beta_{5} + 3 \beta_{3}) q^{47} - 2 \beta_{2} q^{49} + ( - 4 \beta_{5} + 2 \beta_{4} + 12 \beta_{3}) q^{52} + (4 \beta_{5} + 4 \beta_{4} - 2 \beta_{3}) q^{53} + ( - 4 \beta_{2} + \beta_1 + 9) q^{56} + (2 \beta_{5} + 2 \beta_{4} - 8 \beta_{3}) q^{58} + ( - 2 \beta_{2} + \beta_1 - 1) q^{59} + ( - 4 \beta_{2} - \beta_1) q^{61} + ( - 2 \beta_{5} + 6 \beta_{4} + 8 \beta_{3}) q^{62} + (8 \beta_{2} - 3 \beta_1 - 15) q^{64} + (\beta_{5} + 6 \beta_{4} + 4 \beta_{3}) q^{67} + ( - 2 \beta_{5} + \beta_{4} + 7 \beta_{3}) q^{68} + ( - \beta_1 - 9) q^{71} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3}) q^{73} + ( - \beta_{2} + 6 \beta_1 + 7) q^{74} + (5 \beta_{2} - 4 \beta_1 - 11) q^{76} + (\beta_{5} + \beta_{4} - \beta_{3}) q^{77} + ( - \beta_{2} + 2 \beta_1 - 1) q^{79} + ( - \beta_{5} - 2 \beta_{4} + 3 \beta_{3}) q^{82} + (6 \beta_{4} + 4 \beta_{3}) q^{83} + ( - 6 \beta_{2} - 8 \beta_1 - 10) q^{86} + ( - 2 \beta_{5} + \beta_{4} + 3 \beta_{3}) q^{88} + ( - 4 \beta_{2} + 2) q^{89} + (\beta_1 + 6) q^{91} + (5 \beta_{5} - 8 \beta_{4} - 13 \beta_{3}) q^{92} + (3 \beta_{2} + 2 \beta_1 - 1) q^{94} + (2 \beta_{5} + 6 \beta_{4} + 11 \beta_{3}) q^{97} + ( - 2 \beta_{5} - 2 \beta_{4} + 8 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 16 q^{4} - 6 q^{11} + 12 q^{14} + 20 q^{16} + 2 q^{19} + 16 q^{26} + 4 q^{29} + 34 q^{31} - 12 q^{34} - 8 q^{41} + 16 q^{44} - 60 q^{46} - 4 q^{49} + 44 q^{56} - 12 q^{59} - 6 q^{61} - 68 q^{64} - 52 q^{71} + 28 q^{74} - 48 q^{76} - 12 q^{79} - 56 q^{86} + 4 q^{89} + 34 q^{91} - 4 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{5} + 25\nu^{4} + 10\nu^{3} - 4\nu^{2} + 323 ) / 121 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -7\nu^{5} - 27\nu^{4} - 35\nu^{3} + 14\nu^{2} - 223 ) / 121 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -25\nu^{5} - 10\nu^{4} - 4\nu^{3} + 50\nu^{2} - 605\nu + 258 ) / 242 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -65\nu^{5} - 26\nu^{4} + 38\nu^{3} + 372\nu^{2} - 1331\nu + 574 ) / 242 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 75\nu^{5} + 30\nu^{4} + 12\nu^{3} - 392\nu^{2} + 1815\nu - 774 ) / 242 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} - 3\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{5} + 5\beta_{4} + 4\beta_{3} - 5\beta_{2} - 5\beta _1 + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{2} + 7\beta _1 - 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -29\beta_{5} - 25\beta_{4} - 32\beta_{3} - 25\beta_{2} - 29\beta _1 + 32 ) / 2 \) 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
0.432320 + 0.432320i
1.32001 1.32001i
−1.75233 1.75233i
−1.75233 + 1.75233i
1.32001 + 1.32001i
0.432320 0.432320i
2.76156i 0 −5.62620 0 0 1.86464i 10.0140i 0 0
199.2 2.12489i 0 −2.51514 0 0 3.64002i 1.09461i 0 0
199.3 1.36333i 0 0.141336 0 0 2.50466i 2.91934i 0 0
199.4 1.36333i 0 0.141336 0 0 2.50466i 2.91934i 0 0
199.5 2.12489i 0 −2.51514 0 0 3.64002i 1.09461i 0 0
199.6 2.76156i 0 −5.62620 0 0 1.86464i 10.0140i 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.q 6
3.b odd 2 1 825.2.c.f 6
5.b even 2 1 inner 2475.2.c.q 6
5.c odd 4 1 2475.2.a.z 3
5.c odd 4 1 2475.2.a.bd 3
15.d odd 2 1 825.2.c.f 6
15.e even 4 1 825.2.a.i 3
15.e even 4 1 825.2.a.m yes 3
165.l odd 4 1 9075.2.a.cd 3
165.l odd 4 1 9075.2.a.cj 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.a.i 3 15.e even 4 1
825.2.a.m yes 3 15.e even 4 1
825.2.c.f 6 3.b odd 2 1
825.2.c.f 6 15.d odd 2 1
2475.2.a.z 3 5.c odd 4 1
2475.2.a.bd 3 5.c odd 4 1
2475.2.c.q 6 1.a even 1 1 trivial
2475.2.c.q 6 5.b even 2 1 inner
9075.2.a.cd 3 165.l odd 4 1
9075.2.a.cj 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 \(S_{2}^{\mathrm{new}}(2475, [\chi])\):

\( T_{2}^{6} + 14T_{2}^{4} + 57T_{2}^{2} + 64 \) Copy content Toggle raw display
\( T_{7}^{6} + 23T_{7}^{4} + 151T_{7}^{2} + 289 \) Copy content Toggle raw display
\( T_{29}^{3} - 2T_{29}^{2} - 24T_{29} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 14 T^{4} + 57 T^{2} + 64 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 23 T^{4} + 151 T^{2} + \cdots + 289 \) Copy content Toggle raw display
$11$ \( (T + 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 25 T^{4} + 80 T^{2} + 64 \) Copy content Toggle raw display
$17$ \( T^{6} + 66 T^{4} + 449 T^{2} + \cdots + 484 \) Copy content Toggle raw display
$19$ \( (T^{3} - T^{2} - 19 T - 25)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 86 T^{4} + 1849 T^{2} + \cdots + 3364 \) Copy content Toggle raw display
$29$ \( (T^{3} - 2 T^{2} - 24 T - 16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 17 T^{2} + 88 T - 136)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 150 T^{4} + 5625 T^{2} + \cdots + 1156 \) Copy content Toggle raw display
$41$ \( (T^{3} + 4 T^{2} - T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 353 T^{4} + 38424 T^{2} + \cdots + 1210000 \) Copy content Toggle raw display
$47$ \( T^{6} + 50 T^{4} + 465 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$53$ \( T^{6} + 332 T^{4} + 31792 T^{2} + \cdots + 678976 \) Copy content Toggle raw display
$59$ \( (T^{3} + 6 T^{2} - 31 T - 136)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 3 T^{2} - 88 T + 244)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 433 T^{4} + 59096 T^{2} + \cdots + 2521744 \) Copy content Toggle raw display
$71$ \( (T^{3} + 26 T^{2} + 217 T + 580)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 188 T^{4} + 11376 T^{2} + \cdots + 222784 \) Copy content Toggle raw display
$79$ \( (T^{3} + 6 T^{2} - 37 T - 212)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 468 T^{4} + 62592 T^{2} + \cdots + 1763584 \) Copy content Toggle raw display
$89$ \( (T^{3} - 2 T^{2} - 100 T + 328)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 691 T^{4} + 130499 T^{2} + \cdots + 4635409 \) Copy content Toggle raw display
show more
show less