Properties

Label 2100.2.bc.e
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: 8.0.49787136.1
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + \beta_1 q^{7} - \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + \beta_1 q^{7} - \beta_{3} q^{9} + ( - \beta_{7} - \beta_{3} - 1) q^{11} + ( - \beta_{6} - \beta_1) q^{13} + ( - \beta_{6} + \beta_{2}) q^{17} + (2 \beta_{5} - 3 \beta_{3}) q^{19} + ( - \beta_{7} + \beta_{5}) q^{21} + (\beta_{4} - \beta_1) q^{23} + ( - \beta_{4} - \beta_{2}) q^{27} + (\beta_{7} - \beta_{5} - 5) q^{29} + ( - 2 \beta_{7} - \beta_{3} - 1) q^{31} + ( - \beta_{4} + \beta_1) q^{33} + ( - 2 \beta_{4} + \beta_1) q^{37} + \beta_{7} q^{39} + ( - 3 \beta_{7} + 3 \beta_{5} - 3) q^{41} + ( - 3 \beta_{6} + 2 \beta_{4} + 2 \beta_{2} - 3 \beta_1) q^{43} + 6 \beta_{4} q^{47} + (7 \beta_{3} + 7) q^{49} + (\beta_{5} + \beta_{3}) q^{51} + ( - 2 \beta_{6} + 2 \beta_{2}) q^{53} + ( - 2 \beta_{6} - 3 \beta_{4} - 3 \beta_{2} - 2 \beta_1) q^{57} + ( - 3 \beta_{7} + 3 \beta_{3} + 3) q^{59} + 8 \beta_{3} q^{61} - \beta_{6} q^{63} + (\beta_{6} - 2 \beta_{2}) q^{67} + (\beta_{7} - \beta_{5} + 1) q^{69} + ( - \beta_{7} + \beta_{5} + 11) q^{71} - 5 \beta_{6} q^{73} + ( - \beta_{6} + 7 \beta_{4} + 7 \beta_{2} - \beta_1) q^{77} + (2 \beta_{5} + 3 \beta_{3}) q^{79} + ( - \beta_{3} - 1) q^{81} + ( - 3 \beta_{6} - 3 \beta_{4} - 3 \beta_{2} - 3 \beta_1) q^{83} + (\beta_{6} + 5 \beta_{2}) q^{87} + ( - 5 \beta_{5} + \beta_{3}) q^{89} - 7 \beta_{3} q^{91} + ( - \beta_{4} + 2 \beta_1) q^{93} + ( - 8 \beta_{4} - 8 \beta_{2}) q^{97} + ( - \beta_{7} + \beta_{5} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{9} - 4 q^{11} + 12 q^{19} - 40 q^{29} - 4 q^{31} - 24 q^{41} + 28 q^{49} - 4 q^{51} + 12 q^{59} - 32 q^{61} + 8 q^{69} + 88 q^{71} - 12 q^{79} - 4 q^{81} - 4 q^{89} + 28 q^{91} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 15\nu^{4} + 5\nu^{2} + 12 ) / 20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 5\nu^{5} + 5\nu^{3} + 12\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{6} - 5\nu^{4} - 15\nu^{2} - 36 ) / 20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 7\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 13\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -9\nu^{6} - 15\nu^{4} - 5\nu^{2} - 48 ) / 20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11\nu^{7} + 25\nu^{5} + 55\nu^{3} + 132\nu ) / 40 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - 3\beta_{3} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - \beta_{5} + 5\beta_{4} + 5\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{3} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{7} - 11\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -5\beta_{6} - 5\beta _1 - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -7\beta_{5} - 13\beta_{4} ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(\beta_{3}\)

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
1.09445 + 0.895644i
−0.228425 1.39564i
−1.09445 0.895644i
0.228425 + 1.39564i
1.09445 0.895644i
−0.228425 + 1.39564i
−1.09445 + 0.895644i
0.228425 1.39564i
0 −0.866025 0.500000i 0 0 0 −2.29129 + 1.32288i 0 0.500000 + 0.866025i 0
949.2 0 −0.866025 0.500000i 0 0 0 2.29129 1.32288i 0 0.500000 + 0.866025i 0
949.3 0 0.866025 + 0.500000i 0 0 0 −2.29129 + 1.32288i 0 0.500000 + 0.866025i 0
949.4 0 0.866025 + 0.500000i 0 0 0 2.29129 1.32288i 0 0.500000 + 0.866025i 0
1549.1 0 −0.866025 + 0.500000i 0 0 0 −2.29129 1.32288i 0 0.500000 0.866025i 0
1549.2 0 −0.866025 + 0.500000i 0 0 0 2.29129 + 1.32288i 0 0.500000 0.866025i 0
1549.3 0 0.866025 0.500000i 0 0 0 −2.29129 1.32288i 0 0.500000 0.866025i 0
1549.4 0 0.866025 0.500000i 0 0 0 2.29129 + 1.32288i 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.e 8
5.b even 2 1 inner 2100.2.bc.e 8
5.c odd 4 1 420.2.q.c 4
5.c odd 4 1 2100.2.q.h 4
7.c even 3 1 inner 2100.2.bc.e 8
15.e even 4 1 1260.2.s.f 4
20.e even 4 1 1680.2.bg.q 4
35.f even 4 1 2940.2.q.t 4
35.j even 6 1 inner 2100.2.bc.e 8
35.k even 12 1 2940.2.a.m 2
35.k even 12 1 2940.2.q.t 4
35.l odd 12 1 420.2.q.c 4
35.l odd 12 1 2100.2.q.h 4
35.l odd 12 1 2940.2.a.s 2
105.w odd 12 1 8820.2.a.bj 2
105.x even 12 1 1260.2.s.f 4
105.x even 12 1 8820.2.a.be 2
140.w even 12 1 1680.2.bg.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.c 4 5.c odd 4 1
420.2.q.c 4 35.l odd 12 1
1260.2.s.f 4 15.e even 4 1
1260.2.s.f 4 105.x even 12 1
1680.2.bg.q 4 20.e even 4 1
1680.2.bg.q 4 140.w even 12 1
2100.2.q.h 4 5.c odd 4 1
2100.2.q.h 4 35.l odd 12 1
2100.2.bc.e 8 1.a even 1 1 trivial
2100.2.bc.e 8 5.b even 2 1 inner
2100.2.bc.e 8 7.c even 3 1 inner
2100.2.bc.e 8 35.j even 6 1 inner
2940.2.a.m 2 35.k even 12 1
2940.2.a.s 2 35.l odd 12 1
2940.2.q.t 4 35.f even 4 1
2940.2.q.t 4 35.k even 12 1
8820.2.a.be 2 105.x even 12 1
8820.2.a.bj 2 105.w odd 12 1

Hecke kernels

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

\( T_{11}^{4} + 2T_{11}^{3} + 10T_{11}^{2} - 12T_{11} + 36 \) Copy content Toggle raw display
\( T_{13}^{2} + 7 \) 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
$7$ \( (T^{4} - 7 T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 2 T^{3} + 10 T^{2} - 12 T + 36)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 7)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} - 16 T^{6} + 220 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$19$ \( (T^{4} - 6 T^{3} + 55 T^{2} + 114 T + 361)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 16 T^{6} + 220 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$29$ \( (T^{2} + 10 T + 18)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2 T^{3} + 31 T^{2} - 54 T + 729)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 22 T^{6} + 475 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$41$ \( (T^{2} + 6 T - 54)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 134 T^{2} + 3481)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 36 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} - 64 T^{6} + 3520 T^{4} + \cdots + 331776 \) Copy content Toggle raw display
$59$ \( (T^{4} - 6 T^{3} + 90 T^{2} + 324 T + 2916)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 8 T + 64)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} - 22 T^{6} + 475 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$71$ \( (T^{2} - 22 T + 114)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} - 175 T^{2} + 30625)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 6 T^{3} + 55 T^{2} - 114 T + 361)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 144 T^{2} + 2916)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 2 T^{3} + 178 T^{2} - 348 T + 30276)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 64)^{4} \) Copy content Toggle raw display
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