Properties

Label 8400.2.a.dg
Level 8400
Weight 2
Character orbit 8400.a
Self dual yes
Analytic conductor 67.074
Analytic rank 1
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.0743376979\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} - q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} - q^{7} + q^{9} -2 q^{11} + ( 2 - \beta_{2} ) q^{13} + \beta_{2} q^{17} + ( -2 - \beta_{2} ) q^{19} + q^{21} + ( 1 - \beta_{1} ) q^{23} - q^{27} -2 \beta_{1} q^{29} + ( 2 \beta_{1} + \beta_{2} ) q^{31} + 2 q^{33} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{37} + ( -2 + \beta_{2} ) q^{39} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{41} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{43} + ( 2 - 2 \beta_{1} ) q^{47} + q^{49} -\beta_{2} q^{51} + ( -4 + 2 \beta_{1} + \beta_{2} ) q^{53} + ( 2 + \beta_{2} ) q^{57} + ( -6 - 2 \beta_{1} + 2 \beta_{2} ) q^{59} + ( -2 - 2 \beta_{2} ) q^{61} - q^{63} + ( 2 - 2 \beta_{1} ) q^{67} + ( -1 + \beta_{1} ) q^{69} -2 q^{71} + ( 6 + \beta_{2} ) q^{73} + 2 q^{77} + ( -4 - 2 \beta_{2} ) q^{79} + q^{81} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{83} + 2 \beta_{1} q^{87} + ( 5 + \beta_{1} ) q^{89} + ( -2 + \beta_{2} ) q^{91} + ( -2 \beta_{1} - \beta_{2} ) q^{93} + ( 6 - 4 \beta_{1} - \beta_{2} ) q^{97} -2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} - 3q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} - 3q^{7} + 3q^{9} - 6q^{11} + 6q^{13} - 6q^{19} + 3q^{21} + 4q^{23} - 3q^{27} + 2q^{29} - 2q^{31} + 6q^{33} + 4q^{37} - 6q^{39} + 2q^{41} + 4q^{43} + 8q^{47} + 3q^{49} - 14q^{53} + 6q^{57} - 16q^{59} - 6q^{61} - 3q^{63} + 8q^{67} - 4q^{69} - 6q^{71} + 18q^{73} + 6q^{77} - 12q^{79} + 3q^{81} + 8q^{83} - 2q^{87} + 14q^{89} - 6q^{91} + 2q^{93} + 22q^{97} - 6q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu - 1 \)
\(\beta_{2}\)\(=\)\( 2 \nu^{2} - 2 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 5\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.48119
2.17009
0.311108
0 −1.00000 0 0 0 −1.00000 0 1.00000 0
1.2 0 −1.00000 0 0 0 −1.00000 0 1.00000 0
1.3 0 −1.00000 0 0 0 −1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8400.2.a.dg 3
4.b odd 2 1 525.2.a.j 3
5.b even 2 1 8400.2.a.dj 3
5.c odd 4 2 1680.2.t.k 6
12.b even 2 1 1575.2.a.x 3
15.e even 4 2 5040.2.t.v 6
20.d odd 2 1 525.2.a.k 3
20.e even 4 2 105.2.d.b 6
28.d even 2 1 3675.2.a.bi 3
60.h even 2 1 1575.2.a.w 3
60.l odd 4 2 315.2.d.e 6
140.c even 2 1 3675.2.a.bj 3
140.j odd 4 2 735.2.d.b 6
140.w even 12 4 735.2.q.e 12
140.x odd 12 4 735.2.q.f 12
420.w even 4 2 2205.2.d.l 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.b 6 20.e even 4 2
315.2.d.e 6 60.l odd 4 2
525.2.a.j 3 4.b odd 2 1
525.2.a.k 3 20.d odd 2 1
735.2.d.b 6 140.j odd 4 2
735.2.q.e 12 140.w even 12 4
735.2.q.f 12 140.x odd 12 4
1575.2.a.w 3 60.h even 2 1
1575.2.a.x 3 12.b even 2 1
1680.2.t.k 6 5.c odd 4 2
2205.2.d.l 6 420.w even 4 2
3675.2.a.bi 3 28.d even 2 1
3675.2.a.bj 3 140.c even 2 1
5040.2.t.v 6 15.e even 4 2
8400.2.a.dg 3 1.a even 1 1 trivial
8400.2.a.dj 3 5.b even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)
\(7\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8400))\):

\( T_{11} + 2 \)
\( T_{13}^{3} - 6 T_{13}^{2} - 4 T_{13} + 8 \)
\( T_{17}^{3} - 16 T_{17} + 16 \)
\( T_{19}^{3} + 6 T_{19}^{2} - 4 T_{19} - 40 \)
\( T_{23}^{3} - 4 T_{23}^{2} - 8 T_{23} + 16 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + T )^{3} \)
$5$ 1
$7$ \( ( 1 + T )^{3} \)
$11$ \( ( 1 + 2 T + 11 T^{2} )^{3} \)
$13$ \( 1 - 6 T + 35 T^{2} - 148 T^{3} + 455 T^{4} - 1014 T^{5} + 2197 T^{6} \)
$17$ \( 1 + 35 T^{2} + 16 T^{3} + 595 T^{4} + 4913 T^{6} \)
$19$ \( 1 + 6 T + 53 T^{2} + 188 T^{3} + 1007 T^{4} + 2166 T^{5} + 6859 T^{6} \)
$23$ \( 1 - 4 T + 61 T^{2} - 168 T^{3} + 1403 T^{4} - 2116 T^{5} + 12167 T^{6} \)
$29$ \( 1 - 2 T + 35 T^{2} - 76 T^{3} + 1015 T^{4} - 1682 T^{5} + 24389 T^{6} \)
$31$ \( 1 + 2 T + 41 T^{2} - 60 T^{3} + 1271 T^{4} + 1922 T^{5} + 29791 T^{6} \)
$37$ \( 1 - 4 T + 31 T^{2} - 232 T^{3} + 1147 T^{4} - 5476 T^{5} + 50653 T^{6} \)
$41$ \( 1 - 2 T + 63 T^{2} + 36 T^{3} + 2583 T^{4} - 3362 T^{5} + 68921 T^{6} \)
$43$ \( 1 - 4 T - 15 T^{2} + 488 T^{3} - 645 T^{4} - 7396 T^{5} + 79507 T^{6} \)
$47$ \( 1 - 8 T + 109 T^{2} - 624 T^{3} + 5123 T^{4} - 17672 T^{5} + 103823 T^{6} \)
$53$ \( 1 + 14 T + 171 T^{2} + 1188 T^{3} + 9063 T^{4} + 39326 T^{5} + 148877 T^{6} \)
$59$ \( 1 + 16 T + 113 T^{2} + 608 T^{3} + 6667 T^{4} + 55696 T^{5} + 205379 T^{6} \)
$61$ \( 1 + 6 T + 131 T^{2} + 484 T^{3} + 7991 T^{4} + 22326 T^{5} + 226981 T^{6} \)
$67$ \( 1 - 8 T + 169 T^{2} - 944 T^{3} + 11323 T^{4} - 35912 T^{5} + 300763 T^{6} \)
$71$ \( ( 1 + 2 T + 71 T^{2} )^{3} \)
$73$ \( 1 - 18 T + 311 T^{2} - 2732 T^{3} + 22703 T^{4} - 95922 T^{5} + 389017 T^{6} \)
$79$ \( 1 + 12 T + 221 T^{2} + 1576 T^{3} + 17459 T^{4} + 74892 T^{5} + 493039 T^{6} \)
$83$ \( 1 - 8 T + 185 T^{2} - 1072 T^{3} + 15355 T^{4} - 55112 T^{5} + 571787 T^{6} \)
$89$ \( 1 - 14 T + 319 T^{2} - 2532 T^{3} + 28391 T^{4} - 110894 T^{5} + 704969 T^{6} \)
$97$ \( 1 - 22 T + 255 T^{2} - 2404 T^{3} + 24735 T^{4} - 206998 T^{5} + 912673 T^{6} \)
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