Properties

Label 2100.2.d.j
Level 2100
Weight 2
Character orbit 2100.d
Analytic conductor 16.769
Analytic rank 0
Dimension 8
CM discriminant -35
Inner twists 8

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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.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.121550625.1
Defining polynomial: \(x^{8} - x^{7} - 4 x^{6} - 9 x^{5} + 23 x^{4} + 18 x^{3} - 16 x^{2} + 8 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + ( -\beta_{2} + \beta_{3} ) q^{7} -\beta_{1} q^{9} +O(q^{10})\) \( q + \beta_{3} q^{3} + ( -\beta_{2} + \beta_{3} ) q^{7} -\beta_{1} q^{9} + \beta_{4} q^{11} + ( \beta_{2} + \beta_{3} - \beta_{6} ) q^{13} + ( -\beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} ) q^{17} + ( -2 - \beta_{1} - \beta_{4} ) q^{21} + ( \beta_{2} - \beta_{5} ) q^{27} + ( \beta_{1} + \beta_{4} + \beta_{7} ) q^{29} + ( 2 \beta_{2} - \beta_{3} - \beta_{6} ) q^{33} + ( -5 - \beta_{1} + 2 \beta_{4} ) q^{39} + ( \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{47} -7 q^{49} + ( -7 + 2 \beta_{1} + \beta_{7} ) q^{51} + ( -\beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} ) q^{63} + ( -\beta_{1} + 2 \beta_{4} - \beta_{7} ) q^{71} + ( -4 \beta_{2} + 4 \beta_{3} ) q^{73} + ( \beta_{2} + 4 \beta_{3} - \beta_{5} + \beta_{6} ) q^{77} + ( -2 - 5 \beta_{1} - \beta_{4} + \beta_{7} ) q^{79} + ( 4 + \beta_{4} + \beta_{7} ) q^{81} + ( 2 \beta_{3} - 2 \beta_{5} ) q^{83} + ( 4 \beta_{2} - 2 \beta_{3} + 3 \beta_{5} + \beta_{6} ) q^{87} + ( 2 - 5 \beta_{1} - \beta_{4} + \beta_{7} ) q^{91} + ( -3 \beta_{2} + 5 \beta_{3} - \beta_{6} ) q^{97} + ( -3 + \beta_{1} + 3 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{9} + O(q^{10}) \) \( 8q + 2q^{9} - 14q^{21} - 38q^{39} - 56q^{49} - 58q^{51} - 4q^{79} + 34q^{81} + 28q^{91} - 26q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} - 4 x^{6} - 9 x^{5} + 23 x^{4} + 18 x^{3} - 16 x^{2} + 8 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -83 \nu^{7} + 165 \nu^{6} + 110 \nu^{5} + 599 \nu^{4} - 2255 \nu^{3} + 1540 \nu^{2} - 1560 \nu - 2200 \)\()/816\)
\(\beta_{2}\)\(=\)\((\)\( -263 \nu^{7} + 309 \nu^{6} + 818 \nu^{5} + 2891 \nu^{4} - 5447 \nu^{3} - 2624 \nu^{2} - 3552 \nu - 4120 \)\()/2448\)
\(\beta_{3}\)\(=\)\((\)\( 377 \nu^{7} - 747 \nu^{6} - 974 \nu^{5} - 2357 \nu^{4} + 11705 \nu^{3} - 1600 \nu^{2} - 6624 \nu + 4792 \)\()/2448\)
\(\beta_{4}\)\(=\)\((\)\( 11 \nu^{7} - 2 \nu^{6} - 41 \nu^{5} - 131 \nu^{4} + 152 \nu^{3} + 293 \nu^{2} + 90 \nu + 4 \)\()/68\)
\(\beta_{5}\)\(=\)\((\)\( 159 \nu^{7} - 253 \nu^{6} - 418 \nu^{5} - 1059 \nu^{4} + 3775 \nu^{3} - 480 \nu^{2} - 2368 \nu + 5096 \)\()/816\)
\(\beta_{6}\)\(=\)\((\)\( -121 \nu^{7} + 294 \nu^{6} + 43 \nu^{5} + 625 \nu^{4} - 3712 \nu^{3} + 4121 \nu^{2} + 1050 \nu - 3920 \)\()/612\)
\(\beta_{7}\)\(=\)\((\)\( -673 \nu^{7} + 1215 \nu^{6} + 1762 \nu^{5} + 4621 \nu^{4} - 19597 \nu^{3} + 3860 \nu^{2} + 8616 \nu - 10760 \)\()/816\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} + \beta_{2} - 3 \beta_{1}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{7} + \beta_{4} + 9 \beta_{3} - \beta_{1} + 6\)\()/6\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{7} - 5 \beta_{5} + 2 \beta_{4} + \beta_{3} + 4 \beta_{2} - \beta_{1} + 20\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{7} + 9 \beta_{6} + 9 \beta_{5} + 4 \beta_{4} + 9 \beta_{3} + 27 \beta_{2} - 25 \beta_{1} - 12\)\()/6\)
\(\nu^{5}\)\(=\)\((\)\(39 \beta_{7} - 44 \beta_{6} - 33 \beta_{5} + 48 \beta_{4} + 165 \beta_{3} + 22 \beta_{2} + 9 \beta_{1} + 174\)\()/12\)
\(\nu^{6}\)\(=\)\((\)\(-10 \beta_{7} - 18 \beta_{5} + 20 \beta_{4} - 27 \beta_{3} + 45 \beta_{2} - 10 \beta_{1} + 81\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(117 \beta_{7} + 34 \beta_{6} + 279 \beta_{5} + 234 \beta_{4} + 279 \beta_{3} + 592 \beta_{2} - 429 \beta_{1} - 702\)\()/12\)

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
0.553538 + 0.676408i
0.553538 0.676408i
−1.44918 + 1.77086i
−1.44918 1.77086i
−0.862555 0.141174i
−0.862555 + 0.141174i
2.25820 0.369600i
2.25820 + 0.369600i
0 −1.70466 0.306808i 0 0 0 2.64575i 0 2.81174 + 1.04601i 0
1301.2 0 −1.70466 + 0.306808i 0 0 0 2.64575i 0 2.81174 1.04601i 0
1301.3 0 −0.586627 1.62968i 0 0 0 2.64575i 0 −2.31174 + 1.91203i 0
1301.4 0 −0.586627 + 1.62968i 0 0 0 2.64575i 0 −2.31174 1.91203i 0
1301.5 0 0.586627 1.62968i 0 0 0 2.64575i 0 −2.31174 1.91203i 0
1301.6 0 0.586627 + 1.62968i 0 0 0 2.64575i 0 −2.31174 + 1.91203i 0
1301.7 0 1.70466 0.306808i 0 0 0 2.64575i 0 2.81174 1.04601i 0
1301.8 0 1.70466 + 0.306808i 0 0 0 2.64575i 0 2.81174 + 1.04601i 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
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
15.d odd 2 1 inner
21.c even 2 1 inner
105.g 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.j 8
3.b odd 2 1 inner 2100.2.d.j 8
5.b even 2 1 inner 2100.2.d.j 8
5.c odd 4 2 420.2.f.a 8
7.b odd 2 1 inner 2100.2.d.j 8
15.d odd 2 1 inner 2100.2.d.j 8
15.e even 4 2 420.2.f.a 8
20.e even 4 2 1680.2.k.d 8
21.c even 2 1 inner 2100.2.d.j 8
35.c odd 2 1 CM 2100.2.d.j 8
35.f even 4 2 420.2.f.a 8
60.l odd 4 2 1680.2.k.d 8
105.g even 2 1 inner 2100.2.d.j 8
105.k odd 4 2 420.2.f.a 8
140.j odd 4 2 1680.2.k.d 8
420.w even 4 2 1680.2.k.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.f.a 8 5.c odd 4 2
420.2.f.a 8 15.e even 4 2
420.2.f.a 8 35.f even 4 2
420.2.f.a 8 105.k odd 4 2
1680.2.k.d 8 20.e even 4 2
1680.2.k.d 8 60.l odd 4 2
1680.2.k.d 8 140.j odd 4 2
1680.2.k.d 8 420.w even 4 2
2100.2.d.j 8 1.a even 1 1 trivial
2100.2.d.j 8 3.b odd 2 1 inner
2100.2.d.j 8 5.b even 2 1 inner
2100.2.d.j 8 7.b odd 2 1 inner
2100.2.d.j 8 15.d odd 2 1 inner
2100.2.d.j 8 21.c even 2 1 inner
2100.2.d.j 8 35.c odd 2 1 CM
2100.2.d.j 8 105.g even 2 1 inner

Hecke kernels

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

\( T_{11}^{4} + 31 T_{11}^{2} + 4 \)
\( T_{13}^{4} + 71 T_{13}^{2} + 1024 \)
\( T_{17}^{4} - 97 T_{17}^{2} + 2116 \)
\( T_{37} \)
\( T_{41} \)
\( T_{43} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - T^{2} - 8 T^{4} - 9 T^{6} + 81 T^{8} \)
$5$ 1
$7$ \( ( 1 + 7 T^{2} )^{4} \)
$11$ \( ( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} )^{2}( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} )^{2} \)
$13$ \( ( 1 + 19 T^{2} + 192 T^{4} + 3211 T^{6} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 - 29 T^{2} + 552 T^{4} - 8381 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 19 T^{2} )^{8} \)
$23$ \( ( 1 - 23 T^{2} )^{8} \)
$29$ \( ( 1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4} )^{2}( 1 + 9 T + 52 T^{2} + 261 T^{3} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 31 T^{2} )^{8} \)
$37$ \( ( 1 + 37 T^{2} )^{8} \)
$41$ \( ( 1 + 41 T^{2} )^{8} \)
$43$ \( ( 1 + 43 T^{2} )^{8} \)
$47$ \( ( 1 + 31 T^{2} - 1248 T^{4} + 68479 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 - 53 T^{2} )^{8} \)
$59$ \( ( 1 + 59 T^{2} )^{8} \)
$61$ \( ( 1 - 61 T^{2} )^{8} \)
$67$ \( ( 1 + 67 T^{2} )^{8} \)
$71$ \( ( 1 - 12 T + 71 T^{2} )^{4}( 1 + 12 T + 71 T^{2} )^{4} \)
$73$ \( ( 1 - 34 T^{2} + 5329 T^{4} )^{4} \)
$79$ \( ( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 + 86 T^{2} + 6889 T^{4} )^{4} \)
$89$ \( ( 1 + 89 T^{2} )^{8} \)
$97$ \( ( 1 - 149 T^{2} + 12792 T^{4} - 1401941 T^{6} + 88529281 T^{8} )^{2} \)
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