Properties

Label 2100.2.d.l
Level 2100
Weight 2
Character orbit 2100.d
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.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.14786560000.1
Defining polynomial: \(x^{8} - 6 x^{6} + 22 x^{4} - 54 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( \beta_{1} + \beta_{3} - \beta_{6} ) q^{3} + ( \beta_{3} - \beta_{5} ) q^{7} + ( 2 - \beta_{2} + \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( \beta_{1} + \beta_{3} - \beta_{6} ) q^{3} + ( \beta_{3} - \beta_{5} ) q^{7} + ( 2 - \beta_{2} + \beta_{5} ) q^{9} + ( 1 - \beta_{2} - \beta_{4} + \beta_{5} ) q^{11} + ( \beta_{3} + 2 \beta_{6} ) q^{13} + \beta_{7} q^{17} + ( 2 \beta_{3} + \beta_{6} ) q^{19} + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{7} ) q^{21} + ( -2 + \beta_{2} + 3 \beta_{4} - 2 \beta_{5} ) q^{23} + ( \beta_{1} + 2 \beta_{3} - 3 \beta_{6} + \beta_{7} ) q^{27} + ( -1 - \beta_{2} + 3 \beta_{4} - \beta_{5} ) q^{29} + ( -3 \beta_{3} + \beta_{6} ) q^{31} + ( -\beta_{1} - \beta_{3} - 2 \beta_{6} ) q^{33} -\beta_{5} q^{37} + ( -1 + 2 \beta_{2} + 3 \beta_{4} - 2 \beta_{5} ) q^{39} + ( -2 \beta_{1} - \beta_{3} + \beta_{6} - 2 \beta_{7} ) q^{41} + ( 5 + 2 \beta_{5} ) q^{43} + ( 4 \beta_{1} + 2 \beta_{3} - 2 \beta_{6} + 3 \beta_{7} ) q^{47} + ( 3 + 2 \beta_{3} - 4 \beta_{6} ) q^{49} + ( -1 + \beta_{2} - \beta_{4} + 2 \beta_{5} ) q^{51} + ( 3 - 4 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} ) q^{53} + ( -2 + \beta_{2} + 3 \beta_{4} - \beta_{5} ) q^{57} + ( 6 \beta_{1} + 3 \beta_{3} - 3 \beta_{6} + \beta_{7} ) q^{59} + ( -5 \beta_{3} + 5 \beta_{6} ) q^{61} + ( -3 - \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{63} -3 \beta_{5} q^{67} + ( \beta_{1} + 5 \beta_{3} + 3 \beta_{6} + \beta_{7} ) q^{69} + ( -1 + 3 \beta_{2} - \beta_{4} - \beta_{5} ) q^{71} -\beta_{3} q^{73} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{77} + ( -3 - 2 \beta_{5} ) q^{79} + ( -2 \beta_{2} - 2 \beta_{4} + 5 \beta_{5} ) q^{81} + ( 2 \beta_{1} + \beta_{3} - \beta_{6} + 2 \beta_{7} ) q^{83} + ( -\beta_{1} + 7 \beta_{3} + 2 \beta_{7} ) q^{87} + ( -2 \beta_{1} - \beta_{3} + \beta_{6} + 3 \beta_{7} ) q^{89} + ( -4 - 3 \beta_{3} - 2 \beta_{5} - 4 \beta_{6} ) q^{91} + ( 3 + \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{93} + ( 5 \beta_{3} - 3 \beta_{6} ) q^{97} + ( -2 - 2 \beta_{2} - 3 \beta_{4} + 2 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 12q^{9} + O(q^{10}) \) \( 8q + 12q^{9} - 4q^{21} + 12q^{39} + 40q^{43} + 24q^{49} - 8q^{51} - 20q^{63} - 24q^{79} - 16q^{81} - 32q^{91} + 20q^{93} - 36q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 6 x^{6} + 22 x^{4} - 54 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 1 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + 3 \nu^{5} - 5 \nu^{3} + 9 \nu \)\()/54\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} + 3 \nu^{4} - 5 \nu^{2} + 27 \)\()/18\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{6} + 6 \nu^{4} - 13 \nu^{2} + 27 \)\()/9\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - 3 \nu^{5} + 13 \nu^{3} - 15 \nu \)\()/18\)
\(\beta_{7}\)\(=\)\((\)\( -5 \nu^{7} + 21 \nu^{5} - 29 \nu^{3} + 45 \nu \)\()/54\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 1\)
\(\nu^{3}\)\(=\)\(\beta_{7} + 2 \beta_{6} - \beta_{3} + \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{5} + 2 \beta_{4} + 2 \beta_{2} - 4\)
\(\nu^{5}\)\(=\)\(3 \beta_{7} + 3 \beta_{6} + 6 \beta_{3} - \beta_{1}\)
\(\nu^{6}\)\(=\)\(-3 \beta_{5} + 12 \beta_{4} - \beta_{2} - 10\)
\(\nu^{7}\)\(=\)\(-4 \beta_{7} + \beta_{6} + 31 \beta_{3} - \beta_{1}\)

Character values

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

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1301.1
−1.67601 + 0.437016i
−1.67601 0.437016i
−1.30038 + 1.14412i
−1.30038 1.14412i
1.30038 + 1.14412i
1.30038 1.14412i
1.67601 + 0.437016i
1.67601 0.437016i
0 −1.67601 0.437016i 0 0 0 −2.23607 + 1.41421i 0 2.61803 + 1.46489i 0
1301.2 0 −1.67601 + 0.437016i 0 0 0 −2.23607 1.41421i 0 2.61803 1.46489i 0
1301.3 0 −1.30038 1.14412i 0 0 0 2.23607 1.41421i 0 0.381966 + 2.97558i 0
1301.4 0 −1.30038 + 1.14412i 0 0 0 2.23607 + 1.41421i 0 0.381966 2.97558i 0
1301.5 0 1.30038 1.14412i 0 0 0 2.23607 1.41421i 0 0.381966 2.97558i 0
1301.6 0 1.30038 + 1.14412i 0 0 0 2.23607 + 1.41421i 0 0.381966 + 2.97558i 0
1301.7 0 1.67601 0.437016i 0 0 0 −2.23607 + 1.41421i 0 2.61803 1.46489i 0
1301.8 0 1.67601 + 0.437016i 0 0 0 −2.23607 1.41421i 0 2.61803 + 1.46489i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1301.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.d.l yes 8
3.b odd 2 1 inner 2100.2.d.l yes 8
5.b even 2 1 2100.2.d.k 8
5.c odd 4 2 2100.2.f.i 16
7.b odd 2 1 inner 2100.2.d.l yes 8
15.d odd 2 1 2100.2.d.k 8
15.e even 4 2 2100.2.f.i 16
21.c even 2 1 inner 2100.2.d.l yes 8
35.c odd 2 1 2100.2.d.k 8
35.f even 4 2 2100.2.f.i 16
105.g even 2 1 2100.2.d.k 8
105.k odd 4 2 2100.2.f.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.d.k 8 5.b even 2 1
2100.2.d.k 8 15.d odd 2 1
2100.2.d.k 8 35.c odd 2 1
2100.2.d.k 8 105.g even 2 1
2100.2.d.l yes 8 1.a even 1 1 trivial
2100.2.d.l yes 8 3.b odd 2 1 inner
2100.2.d.l yes 8 7.b odd 2 1 inner
2100.2.d.l yes 8 21.c even 2 1 inner
2100.2.f.i 16 5.c odd 4 2
2100.2.f.i 16 15.e even 4 2
2100.2.f.i 16 35.f even 4 2
2100.2.f.i 16 105.k odd 4 2

Hecke kernels

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

\( T_{11}^{4} + 16 T_{11}^{2} + 19 \)
\( T_{13}^{4} + 36 T_{13}^{2} + 4 \)
\( T_{17}^{4} - 22 T_{17}^{2} + 76 \)
\( T_{37}^{2} - 5 \)
\( T_{41}^{4} - 90 T_{41}^{2} + 1900 \)
\( T_{43}^{2} - 10 T_{43} + 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 6 T^{2} + 22 T^{4} - 54 T^{6} + 81 T^{8} \)
$5$ 1
$7$ \( ( 1 - 6 T^{2} + 49 T^{4} )^{2} \)
$11$ \( ( 1 - 28 T^{2} + 393 T^{4} - 3388 T^{6} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 - 16 T^{2} + 82 T^{4} - 2704 T^{6} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 + 46 T^{2} + 1062 T^{4} + 13294 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 46 T^{2} + 1126 T^{4} - 16606 T^{6} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 - 12 T^{2} - 31 T^{4} - 6348 T^{6} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 - 12 T^{2} + 1313 T^{4} - 10092 T^{6} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 - 94 T^{2} + 4006 T^{4} - 90334 T^{6} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 + 69 T^{2} + 1369 T^{4} )^{4} \)
$41$ \( ( 1 + 74 T^{2} + 4606 T^{4} + 124394 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 - 10 T + 91 T^{2} - 430 T^{3} + 1849 T^{4} )^{4} \)
$47$ \( ( 1 - 34 T^{2} + 1582 T^{4} - 75106 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 - 32 T^{2} + 5374 T^{4} - 89888 T^{6} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 + 76 T^{2} + 3906 T^{4} + 264556 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 94 T^{2} + 6526 T^{4} - 349774 T^{6} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 + 89 T^{2} + 4489 T^{4} )^{4} \)
$71$ \( ( 1 - 164 T^{2} + 13681 T^{4} - 826724 T^{6} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 - 144 T^{2} + 5329 T^{4} )^{4} \)
$79$ \( ( 1 + 6 T + 147 T^{2} + 474 T^{3} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 + 242 T^{2} + 28294 T^{4} + 1667138 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 + 116 T^{2} + 6706 T^{4} + 918836 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 - 294 T^{2} + 38222 T^{4} - 2766246 T^{6} + 88529281 T^{8} )^{2} \)
show more
show less