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

Related objects

Downloads

Learn more about

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: \(x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16\)
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} + 15 \nu^{4} + 5 \nu^{2} + 12 \)\()/20\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} - 5 \nu^{5} + 5 \nu^{3} + 12 \nu \)\()/40\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{6} - 5 \nu^{4} - 15 \nu^{2} - 36 \)\()/20\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} - 7 \nu \)\()/10\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} + 13 \nu \)\()/10\)
\(\beta_{6}\)\(=\)\((\)\( -9 \nu^{6} - 15 \nu^{4} - 5 \nu^{2} - 48 \)\()/20\)
\(\beta_{7}\)\(=\)\((\)\( 11 \nu^{7} + 25 \nu^{5} + 55 \nu^{3} + 132 \nu \)\()/40\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} - \beta_{4}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} - 3 \beta_{3} - 3\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} - \beta_{5} + 5 \beta_{4} + 5 \beta_{2}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{3} + 3 \beta_{1}\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(\beta_{7} - 11 \beta_{2}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-5 \beta_{6} - 5 \beta_{1} - 9\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-7 \beta_{5} - 13 \beta_{4}\)\()/2\)

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 \(S_{2}^{\mathrm{new}}(2100, [\chi])\):

\( T_{11}^{4} + 2 T_{11}^{3} + 10 T_{11}^{2} - 12 T_{11} + 36 \)
\( T_{13}^{2} + 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$5$ 1
$7$ \( ( 1 - 7 T^{2} + 49 T^{4} )^{2} \)
$11$ \( ( 1 + 2 T - 12 T^{2} - 12 T^{3} + 91 T^{4} - 132 T^{5} - 1452 T^{6} + 2662 T^{7} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 - 19 T^{2} + 169 T^{4} )^{4} \)
$17$ \( 1 + 52 T^{2} + 1478 T^{4} + 33696 T^{6} + 638099 T^{8} + 9738144 T^{10} + 123444038 T^{12} + 1255153588 T^{14} + 6975757441 T^{16} \)
$19$ \( ( 1 - 6 T + 17 T^{2} + 114 T^{3} - 684 T^{4} + 2166 T^{5} + 6137 T^{6} - 41154 T^{7} + 130321 T^{8} )^{2} \)
$23$ \( 1 + 76 T^{2} + 3302 T^{4} + 107616 T^{6} + 2785331 T^{8} + 56928864 T^{10} + 924034982 T^{12} + 11250727564 T^{14} + 78310985281 T^{16} \)
$29$ \( ( 1 + 10 T + 76 T^{2} + 290 T^{3} + 841 T^{4} )^{4} \)
$31$ \( ( 1 + 2 T - 31 T^{2} - 54 T^{3} + 140 T^{4} - 1674 T^{5} - 29791 T^{6} + 59582 T^{7} + 923521 T^{8} )^{2} \)
$37$ \( 1 + 126 T^{2} + 9281 T^{4} + 485982 T^{6} + 19885620 T^{8} + 665309358 T^{10} + 17394088241 T^{12} + 323281527534 T^{14} + 3512479453921 T^{16} \)
$41$ \( ( 1 + 6 T + 28 T^{2} + 246 T^{3} + 1681 T^{4} )^{4} \)
$43$ \( ( 1 - 38 T^{2} + 3051 T^{4} - 70262 T^{6} + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 + 58 T^{2} + 1155 T^{4} + 128122 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( 1 + 148 T^{2} + 11258 T^{4} + 744144 T^{6} + 43918499 T^{8} + 2090300496 T^{10} + 88831035098 T^{12} + 3280325447092 T^{14} + 62259690411361 T^{16} \)
$59$ \( ( 1 - 6 T - 28 T^{2} + 324 T^{3} - 1509 T^{4} + 19116 T^{5} - 97468 T^{6} - 1232274 T^{7} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} )^{4} \)
$67$ \( 1 + 246 T^{2} + 36521 T^{4} + 3694182 T^{6} + 283952580 T^{8} + 16583182998 T^{10} + 735939090041 T^{12} + 22252762013574 T^{14} + 406067677556641 T^{16} \)
$71$ \( ( 1 - 22 T + 256 T^{2} - 1562 T^{3} + 5041 T^{4} )^{4} \)
$73$ \( ( 1 - 29 T^{2} - 4488 T^{4} - 154541 T^{6} + 28398241 T^{8} )^{2} \)
$79$ \( ( 1 + 6 T - 103 T^{2} - 114 T^{3} + 10236 T^{4} - 9006 T^{5} - 642823 T^{6} + 2958234 T^{7} + 38950081 T^{8} )^{2} \)
$83$ \( ( 1 - 188 T^{2} + 20346 T^{4} - 1295132 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 + 2 T - 348 T^{3} - 8261 T^{4} - 30972 T^{5} + 1409938 T^{7} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 - 18 T + 97 T^{2} )^{4}( 1 + 18 T + 97 T^{2} )^{4} \)
show more
show less