Properties

Label 2100.2.bc.f
Level $2100$
Weight $2$
Character orbit 2100.bc
Analytic conductor $16.769$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.bc (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{5} + \beta_{3} - \beta_1) q^{7} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{5} + \beta_{3} - \beta_1) q^{7} + \beta_{2} q^{9} + ( - 2 \beta_{7} + \beta_{4}) q^{11} + ( - \beta_{6} + 2 \beta_{5} - \beta_{3}) q^{13} + (\beta_{6} + \beta_{5}) q^{17} - 7 \beta_{2} q^{19} + (\beta_{7} + 1) q^{21} + ( - 2 \beta_{6} + \beta_{5}) q^{23} - \beta_{3} q^{27} + (\beta_{7} - 2 \beta_{4} + 6) q^{29} + ( - 4 \beta_{7} + 2 \beta_{4} - \beta_{2} + 1) q^{31} + ( - 2 \beta_{6} + \beta_{5}) q^{33} + ( - 2 \beta_{6} + \beta_{5} - \beta_{3} + \beta_1) q^{37} + ( - 2 \beta_{7} + \beta_{4} + \beta_{2} - 1) q^{39} + (\beta_{7} - 2 \beta_{4}) q^{41} + ( - \beta_{6} + 2 \beta_{5} - \beta_{3}) q^{43} + ( - 6 \beta_{3} + 6 \beta_1) q^{47} + (2 \beta_{4} + 5 \beta_{2} - 5) q^{49} + ( - \beta_{7} - \beta_{4}) q^{51} + ( - 2 \beta_{6} - 2 \beta_{5}) q^{53} + 7 \beta_{3} q^{57} + (2 \beta_{7} - \beta_{4} + 6 \beta_{2} - 6) q^{59} + ( - 2 \beta_{7} - 2 \beta_{4} + 4 \beta_{2}) q^{61} + (\beta_{6} - \beta_{5} - \beta_1) q^{63} + ( - \beta_{6} - \beta_{5} - \beta_1) q^{67} + ( - \beta_{7} + 2 \beta_{4}) q^{69} + (3 \beta_{7} - 6 \beta_{4}) q^{71} + ( - \beta_{6} - \beta_{5} - 5 \beta_1) q^{73} + ( - \beta_{6} + 2 \beta_{5} + 6 \beta_{3} - 12 \beta_1) q^{77} + 11 \beta_{2} q^{79} + (\beta_{2} - 1) q^{81} + (\beta_{6} - 2 \beta_{5} - 6 \beta_{3}) q^{83} + (\beta_{6} + \beta_{5} - 6 \beta_1) q^{87} + ( - \beta_{7} - \beta_{4} - 6 \beta_{2}) q^{89} + ( - 2 \beta_{4} - 5 \beta_{2} + 12) q^{91} + ( - 4 \beta_{6} + 2 \beta_{5} + \beta_{3} - \beta_1) q^{93} + ( - 2 \beta_{6} + 4 \beta_{5} - 8 \beta_{3}) q^{97} + ( - \beta_{7} + 2 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{9} - 28 q^{19} + 8 q^{21} + 48 q^{29} + 4 q^{31} - 4 q^{39} - 20 q^{49} - 24 q^{59} + 16 q^{61} + 44 q^{79} - 4 q^{81} - 24 q^{89} + 76 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + 2\zeta_{24}^{5} + 2\zeta_{24}^{3} - \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 2\zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{7} + 2\beta_{6} - \beta_{5} + \beta_{4} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -\beta_{7} - \beta_{6} + 2\beta_{5} + 2\beta_{4} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( 2\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} - 2\beta_{6} + \beta_{5} + \beta_{4} ) / 6 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
949.1
0.965926 + 0.258819i
−0.965926 0.258819i
0.258819 0.965926i
−0.258819 + 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
0.258819 + 0.965926i
−0.258819 0.965926i
0 −0.866025 0.500000i 0 0 0 −2.09077 1.62132i 0 0.500000 + 0.866025i 0
949.2 0 −0.866025 0.500000i 0 0 0 0.358719 + 2.62132i 0 0.500000 + 0.866025i 0
949.3 0 0.866025 + 0.500000i 0 0 0 −0.358719 2.62132i 0 0.500000 + 0.866025i 0
949.4 0 0.866025 + 0.500000i 0 0 0 2.09077 + 1.62132i 0 0.500000 + 0.866025i 0
1549.1 0 −0.866025 + 0.500000i 0 0 0 −2.09077 + 1.62132i 0 0.500000 0.866025i 0
1549.2 0 −0.866025 + 0.500000i 0 0 0 0.358719 2.62132i 0 0.500000 0.866025i 0
1549.3 0 0.866025 0.500000i 0 0 0 −0.358719 + 2.62132i 0 0.500000 0.866025i 0
1549.4 0 0.866025 0.500000i 0 0 0 2.09077 1.62132i 0 0.500000 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1549.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.bc.f 8
5.b even 2 1 inner 2100.2.bc.f 8
5.c odd 4 1 420.2.q.d 4
5.c odd 4 1 2100.2.q.k 4
7.c even 3 1 inner 2100.2.bc.f 8
15.e even 4 1 1260.2.s.e 4
20.e even 4 1 1680.2.bg.t 4
35.f even 4 1 2940.2.q.q 4
35.j even 6 1 inner 2100.2.bc.f 8
35.k even 12 1 2940.2.a.p 2
35.k even 12 1 2940.2.q.q 4
35.l odd 12 1 420.2.q.d 4
35.l odd 12 1 2100.2.q.k 4
35.l odd 12 1 2940.2.a.r 2
105.w odd 12 1 8820.2.a.bf 2
105.x even 12 1 1260.2.s.e 4
105.x even 12 1 8820.2.a.bk 2
140.w even 12 1 1680.2.bg.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.d 4 5.c odd 4 1
420.2.q.d 4 35.l odd 12 1
1260.2.s.e 4 15.e even 4 1
1260.2.s.e 4 105.x even 12 1
1680.2.bg.t 4 20.e even 4 1
1680.2.bg.t 4 140.w even 12 1
2100.2.q.k 4 5.c odd 4 1
2100.2.q.k 4 35.l odd 12 1
2100.2.bc.f 8 1.a even 1 1 trivial
2100.2.bc.f 8 5.b even 2 1 inner
2100.2.bc.f 8 7.c even 3 1 inner
2100.2.bc.f 8 35.j even 6 1 inner
2940.2.a.p 2 35.k even 12 1
2940.2.a.r 2 35.l odd 12 1
2940.2.q.q 4 35.f even 4 1
2940.2.q.q 4 35.k even 12 1
8820.2.a.bf 2 105.w odd 12 1
8820.2.a.bk 2 105.x even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2100, [\chi])\):

\( T_{11}^{4} + 18T_{11}^{2} + 324 \) Copy content Toggle raw display
\( T_{13}^{4} + 38T_{13}^{2} + 289 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 10 T^{6} + 51 T^{4} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{4} + 18 T^{2} + 324)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 38 T^{2} + 289)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 18 T^{2} + 324)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 7 T + 49)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 18 T^{2} + 324)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 12 T + 18)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 2 T^{3} + 75 T^{2} + 142 T + 5041)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 38 T^{6} + 1155 T^{4} + \cdots + 83521 \) Copy content Toggle raw display
$41$ \( (T^{2} - 18)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 38 T^{2} + 289)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 36 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 72 T^{2} + 5184)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 12 T^{3} + 126 T^{2} + 216 T + 324)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 8 T^{3} + 120 T^{2} + 448 T + 3136)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 38 T^{6} + 1155 T^{4} + \cdots + 83521 \) Copy content Toggle raw display
$71$ \( (T^{2} - 162)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} - 86 T^{6} + 7347 T^{4} + \cdots + 2401 \) Copy content Toggle raw display
$79$ \( (T^{2} - 11 T + 121)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 108 T^{2} + 324)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 12 T^{3} + 126 T^{2} + 216 T + 324)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 272 T^{2} + 64)^{2} \) Copy content Toggle raw display
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