Properties

Label 825.2.c.g
Level $825$
Weight $2$
Character orbit 825.c
Analytic conductor $6.588$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.58765816676\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{4} + \beta_{2} - 1) q^{4} + \beta_{2} q^{6} + (\beta_{5} + \beta_1) q^{7} + ( - \beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{4} + \beta_{2} - 1) q^{4} + \beta_{2} q^{6} + (\beta_{5} + \beta_1) q^{7} + ( - \beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{8} - q^{9} + q^{11} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{12} + ( - \beta_{5} + \beta_1) q^{13} + ( - \beta_{4} + \beta_{2} + 4) q^{14} + ( - 4 \beta_{2} + 3) q^{16} + ( - \beta_{5} + 2 \beta_{3} + 3 \beta_1) q^{17} + \beta_1 q^{18} + (2 \beta_{4} - 2) q^{19} + (\beta_{4} - \beta_{2}) q^{21} - \beta_1 q^{22} + (2 \beta_{5} + 2 \beta_1) q^{23} + ( - \beta_{4} - 2 \beta_{2} + 2) q^{24} + ( - \beta_{4} - 3 \beta_{2} + 2) q^{26} - \beta_{3} q^{27} + (\beta_{5} - 4 \beta_{3} - 3 \beta_1) q^{28} + (2 \beta_{2} + 4) q^{29} + (2 \beta_{4} + 2 \beta_{2} + 4) q^{31} + (2 \beta_{5} + 8 \beta_{3} - 3 \beta_1) q^{32} + \beta_{3} q^{33} + ( - 3 \beta_{4} - 3 \beta_{2} + 8) q^{34} + ( - \beta_{4} - \beta_{2} + 1) q^{36} + 2 \beta_{3} q^{37} + (2 \beta_{3} + 6 \beta_1) q^{38} + ( - \beta_{4} - \beta_{2}) q^{39} + (2 \beta_{2} - 4) q^{41} + (\beta_{5} + 4 \beta_{3} + \beta_1) q^{42} + (3 \beta_{5} - \beta_1) q^{43} + (\beta_{4} + \beta_{2} - 1) q^{44} + ( - 2 \beta_{4} + 2 \beta_{2} + 8) q^{46} + ( - 2 \beta_{5} + 4 \beta_{3} + 2 \beta_1) q^{47} + (3 \beta_{3} - 4 \beta_1) q^{48} + ( - 4 \beta_{2} - 5) q^{49} + ( - \beta_{4} - 3 \beta_{2} - 2) q^{51} + (\beta_{5} + 8 \beta_{3} - 5 \beta_1) q^{52} + ( - 2 \beta_{5} - 2 \beta_{3} - 2 \beta_1) q^{53} - \beta_{2} q^{54} + (\beta_{4} + 3 \beta_{2}) q^{56} + ( - 2 \beta_{5} - 2 \beta_{3}) q^{57} + ( - 2 \beta_{5} - 6 \beta_{3} - 2 \beta_1) q^{58} + ( - 2 \beta_{4} + 2 \beta_{2} - 4) q^{59} + ( - 2 \beta_{4} + 2 \beta_{2} - 2) q^{61} + ( - 2 \beta_{5} - 4 \beta_{3} + 2 \beta_1) q^{62} + ( - \beta_{5} - \beta_1) q^{63} + (3 \beta_{4} + 7 \beta_{2} - 1) q^{64} + \beta_{2} q^{66} + (2 \beta_{5} - 2 \beta_1) q^{67} + (\beta_{5} + 10 \beta_{3} - 11 \beta_1) q^{68} + (2 \beta_{4} - 2 \beta_{2}) q^{69} + (2 \beta_{4} + 2 \beta_{2} + 4) q^{71} + (\beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{72} + (\beta_{5} - 4 \beta_{3} + 3 \beta_1) q^{73} + 2 \beta_{2} q^{74} + ( - 2 \beta_{4} - 4 \beta_{2} + 14) q^{76} + (\beta_{5} + \beta_1) q^{77} + (\beta_{5} + 2 \beta_{3} - 3 \beta_1) q^{78} + ( - 2 \beta_{4} - 4 \beta_{2} - 6) q^{79} + q^{81} + ( - 2 \beta_{5} - 6 \beta_{3} + 6 \beta_1) q^{82} + ( - 3 \beta_{5} + 2 \beta_{3} + 3 \beta_1) q^{83} + (\beta_{4} + 3 \beta_{2} + 4) q^{84} + (\beta_{4} + 7 \beta_{2}) q^{86} + (4 \beta_{3} + 2 \beta_1) q^{87} + ( - \beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{88} + ( - 4 \beta_{2} + 2) q^{89} + (2 \beta_{4} + 2 \beta_{2} + 4) q^{91} + (2 \beta_{5} - 8 \beta_{3} - 6 \beta_1) q^{92} + ( - 2 \beta_{5} + 4 \beta_{3} + 2 \beta_1) q^{93} + ( - 2 \beta_{4} - 2 \beta_{2} + 4) q^{94} + (2 \beta_{4} + 3 \beta_{2} - 8) q^{96} + ( - 2 \beta_{5} - 6 \beta_{3} + 2 \beta_1) q^{97} + (4 \beta_{5} + 12 \beta_{3} + \beta_1) q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{4} - 2 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 10 q^{4} - 2 q^{6} - 6 q^{9} + 6 q^{11} + 24 q^{14} + 26 q^{16} - 16 q^{19} + 18 q^{24} + 20 q^{26} + 20 q^{29} + 16 q^{31} + 60 q^{34} + 10 q^{36} + 4 q^{39} - 28 q^{41} - 10 q^{44} + 48 q^{46} - 22 q^{49} - 4 q^{51} + 2 q^{54} - 8 q^{56} - 24 q^{59} - 12 q^{61} - 26 q^{64} - 2 q^{66} + 16 q^{71} - 4 q^{74} + 96 q^{76} - 24 q^{79} + 6 q^{81} + 16 q^{84} - 16 q^{86} + 20 q^{89} + 16 q^{91} + 32 q^{94} - 58 q^{96} - 6 q^{99}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( -3\nu^{5} + \nu^{4} + 11\nu^{3} - 26\nu^{2} + 6\nu - 1 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} - 4\nu^{2} + 8\nu - 9 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -6\nu^{5} + 25\nu^{4} - 24\nu^{3} - 6\nu^{2} + 12\nu + 67 ) / 23 \) 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
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 4\beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} + \beta_{4} + 4\beta_{3} - 3\beta_{2} + 3\beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{4} - 3\beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{5} + 5\beta_{4} - 18\beta_{3} - 11\beta_{2} - 11\beta _1 - 18 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 1.00000i −5.34017 0 −2.70928 1.07838i 9.04945i −1.00000 0
199.2 1.90321i 1.00000i −1.62222 0 1.90321 4.42864i 0.719004i −1.00000 0
199.3 0.193937i 1.00000i 1.96239 0 −0.193937 3.35026i 0.768452i −1.00000 0
199.4 0.193937i 1.00000i 1.96239 0 −0.193937 3.35026i 0.768452i −1.00000 0
199.5 1.90321i 1.00000i −1.62222 0 1.90321 4.42864i 0.719004i −1.00000 0
199.6 2.70928i 1.00000i −5.34017 0 −2.70928 1.07838i 9.04945i −1.00000 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 825.2.c.g 6
3.b odd 2 1 2475.2.c.r 6
5.b even 2 1 inner 825.2.c.g 6
5.c odd 4 1 165.2.a.c 3
5.c odd 4 1 825.2.a.k 3
15.d odd 2 1 2475.2.c.r 6
15.e even 4 1 495.2.a.e 3
15.e even 4 1 2475.2.a.bb 3
20.e even 4 1 2640.2.a.be 3
35.f even 4 1 8085.2.a.bk 3
55.e even 4 1 1815.2.a.m 3
55.e even 4 1 9075.2.a.cf 3
60.l odd 4 1 7920.2.a.cj 3
165.l odd 4 1 5445.2.a.z 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.c 3 5.c odd 4 1
495.2.a.e 3 15.e even 4 1
825.2.a.k 3 5.c odd 4 1
825.2.c.g 6 1.a even 1 1 trivial
825.2.c.g 6 5.b even 2 1 inner
1815.2.a.m 3 55.e even 4 1
2475.2.a.bb 3 15.e even 4 1
2475.2.c.r 6 3.b odd 2 1
2475.2.c.r 6 15.d odd 2 1
2640.2.a.be 3 20.e even 4 1
5445.2.a.z 3 165.l odd 4 1
7920.2.a.cj 3 60.l odd 4 1
8085.2.a.bk 3 35.f even 4 1
9075.2.a.cf 3 55.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{6} + 11T_{2}^{4} + 27T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{6} + 32T_{7}^{4} + 256T_{7}^{2} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 11 T^{4} + 27 T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{3} \) 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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