Properties

Label 3234.2.a.bh
Level 3234
Weight 2
Character orbit 3234.a
Self dual yes
Analytic conductor 25.824
Analytic rank 0
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3234.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.8236200137\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.2700.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 462)
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} -\beta_{1} q^{5} + q^{6} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} + q^{3} + q^{4} -\beta_{1} q^{5} + q^{6} + q^{8} + q^{9} -\beta_{1} q^{10} - q^{11} + q^{12} -\beta_{1} q^{15} + q^{16} + ( -1 - 2 \beta_{2} ) q^{17} + q^{18} + ( 3 + \beta_{2} ) q^{19} -\beta_{1} q^{20} - q^{22} + ( 1 + \beta_{1} - \beta_{2} ) q^{23} + q^{24} + ( 5 + 2 \beta_{1} + \beta_{2} ) q^{25} + q^{27} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{29} -\beta_{1} q^{30} + ( 2 + 2 \beta_{2} ) q^{31} + q^{32} - q^{33} + ( -1 - 2 \beta_{2} ) q^{34} + q^{36} + ( 7 + \beta_{2} ) q^{37} + ( 3 + \beta_{2} ) q^{38} -\beta_{1} q^{40} + ( -4 - 2 \beta_{1} + \beta_{2} ) q^{41} + ( 1 + 3 \beta_{2} ) q^{43} - q^{44} -\beta_{1} q^{45} + ( 1 + \beta_{1} - \beta_{2} ) q^{46} + ( -1 + \beta_{1} + \beta_{2} ) q^{47} + q^{48} + ( 5 + 2 \beta_{1} + \beta_{2} ) q^{50} + ( -1 - 2 \beta_{2} ) q^{51} + 8 q^{53} + q^{54} + \beta_{1} q^{55} + ( 3 + \beta_{2} ) q^{57} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{58} + ( 3 + 2 \beta_{1} + \beta_{2} ) q^{59} -\beta_{1} q^{60} + ( 2 - \beta_{1} - 2 \beta_{2} ) q^{61} + ( 2 + 2 \beta_{2} ) q^{62} + q^{64} - q^{66} + ( 2 + 2 \beta_{1} - \beta_{2} ) q^{67} + ( -1 - 2 \beta_{2} ) q^{68} + ( 1 + \beta_{1} - \beta_{2} ) q^{69} + ( 3 - 3 \beta_{2} ) q^{71} + q^{72} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{73} + ( 7 + \beta_{2} ) q^{74} + ( 5 + 2 \beta_{1} + \beta_{2} ) q^{75} + ( 3 + \beta_{2} ) q^{76} + ( 4 - \beta_{1} ) q^{79} -\beta_{1} q^{80} + q^{81} + ( -4 - 2 \beta_{1} + \beta_{2} ) q^{82} + ( -6 + 2 \beta_{1} - \beta_{2} ) q^{83} + ( 3 \beta_{1} - 4 \beta_{2} ) q^{85} + ( 1 + 3 \beta_{2} ) q^{86} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{87} - q^{88} + ( 4 + 2 \beta_{1} ) q^{89} -\beta_{1} q^{90} + ( 1 + \beta_{1} - \beta_{2} ) q^{92} + ( 2 + 2 \beta_{2} ) q^{93} + ( -1 + \beta_{1} + \beta_{2} ) q^{94} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{95} + q^{96} -7 q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{3} + 3q^{4} + 3q^{6} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{3} + 3q^{4} + 3q^{6} + 3q^{8} + 3q^{9} - 3q^{11} + 3q^{12} + 3q^{16} - 3q^{17} + 3q^{18} + 9q^{19} - 3q^{22} + 3q^{23} + 3q^{24} + 15q^{25} + 3q^{27} + 9q^{29} + 6q^{31} + 3q^{32} - 3q^{33} - 3q^{34} + 3q^{36} + 21q^{37} + 9q^{38} - 12q^{41} + 3q^{43} - 3q^{44} + 3q^{46} - 3q^{47} + 3q^{48} + 15q^{50} - 3q^{51} + 24q^{53} + 3q^{54} + 9q^{57} + 9q^{58} + 9q^{59} + 6q^{61} + 6q^{62} + 3q^{64} - 3q^{66} + 6q^{67} - 3q^{68} + 3q^{69} + 9q^{71} + 3q^{72} + 21q^{74} + 15q^{75} + 9q^{76} + 12q^{79} + 3q^{81} - 12q^{82} - 18q^{83} + 3q^{86} + 9q^{87} - 3q^{88} + 12q^{89} + 3q^{92} + 6q^{93} - 3q^{94} + 3q^{96} - 21q^{97} - 3q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 15 x - 20\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 10 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 10\)

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
4.41883
−1.61323
−2.80560
1.00000 1.00000 1.00000 −4.41883 1.00000 0 1.00000 1.00000 −4.41883
1.2 1.00000 1.00000 1.00000 1.61323 1.00000 0 1.00000 1.00000 1.61323
1.3 1.00000 1.00000 1.00000 2.80560 1.00000 0 1.00000 1.00000 2.80560
\(n\): 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 3234.2.a.bh 3
3.b odd 2 1 9702.2.a.dw 3
7.b odd 2 1 3234.2.a.bf 3
7.d odd 6 2 462.2.i.g 6
21.c even 2 1 9702.2.a.dv 3
21.g even 6 2 1386.2.k.v 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.g 6 7.d odd 6 2
1386.2.k.v 6 21.g even 6 2
3234.2.a.bf 3 7.b odd 2 1
3234.2.a.bh 3 1.a even 1 1 trivial
9702.2.a.dv 3 21.c even 2 1
9702.2.a.dw 3 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(-1\)
\(11\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3234))\):

\( T_{5}^{3} - 15 T_{5} + 20 \)
\( T_{13} \)
\( T_{17}^{3} + 3 T_{17}^{2} - 57 T_{17} - 139 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{3} \)
$3$ \( ( 1 - T )^{3} \)
$5$ \( 1 + 20 T^{3} + 125 T^{6} \)
$7$ 1
$11$ \( ( 1 + T )^{3} \)
$13$ \( ( 1 + 13 T^{2} )^{3} \)
$17$ \( 1 + 3 T - 6 T^{2} - 37 T^{3} - 102 T^{4} + 867 T^{5} + 4913 T^{6} \)
$19$ \( 1 - 9 T + 69 T^{2} - 314 T^{3} + 1311 T^{4} - 3249 T^{5} + 6859 T^{6} \)
$23$ \( 1 - 3 T + 42 T^{2} - 49 T^{3} + 966 T^{4} - 1587 T^{5} + 12167 T^{6} \)
$29$ \( 1 - 9 T + 39 T^{2} - 174 T^{3} + 1131 T^{4} - 7569 T^{5} + 24389 T^{6} \)
$31$ \( 1 - 6 T + 45 T^{2} - 180 T^{3} + 1395 T^{4} - 5766 T^{5} + 29791 T^{6} \)
$37$ \( 1 - 21 T + 243 T^{2} - 1782 T^{3} + 8991 T^{4} - 28749 T^{5} + 50653 T^{6} \)
$41$ \( 1 + 12 T + 96 T^{2} + 678 T^{3} + 3936 T^{4} + 20172 T^{5} + 68921 T^{6} \)
$43$ \( 1 - 3 T - 3 T^{2} + 146 T^{3} - 129 T^{4} - 5547 T^{5} + 79507 T^{6} \)
$47$ \( 1 + 3 T + 114 T^{2} + 273 T^{3} + 5358 T^{4} + 6627 T^{5} + 103823 T^{6} \)
$53$ \( ( 1 - 8 T + 53 T^{2} )^{3} \)
$59$ \( 1 - 9 T + 129 T^{2} - 1014 T^{3} + 7611 T^{4} - 31329 T^{5} + 205379 T^{6} \)
$61$ \( 1 - 6 T + 120 T^{2} - 830 T^{3} + 7320 T^{4} - 22326 T^{5} + 226981 T^{6} \)
$67$ \( 1 - 6 T + 138 T^{2} - 592 T^{3} + 9246 T^{4} - 26934 T^{5} + 300763 T^{6} \)
$71$ \( 1 - 9 T + 105 T^{2} - 1170 T^{3} + 7455 T^{4} - 45369 T^{5} + 357911 T^{6} \)
$73$ \( 1 + 99 T^{2} + 480 T^{3} + 7227 T^{4} + 389017 T^{6} \)
$79$ \( 1 - 12 T + 270 T^{2} - 1880 T^{3} + 21330 T^{4} - 74892 T^{5} + 493039 T^{6} \)
$83$ \( 1 + 18 T + 282 T^{2} + 2824 T^{3} + 23406 T^{4} + 124002 T^{5} + 571787 T^{6} \)
$89$ \( 1 - 12 T + 255 T^{2} - 2120 T^{3} + 22695 T^{4} - 95052 T^{5} + 704969 T^{6} \)
$97$ \( ( 1 + 7 T + 97 T^{2} )^{3} \)
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