Properties

Label 3234.2.a.bi
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: 3.3.621.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + \beta_{2} q^{5} + q^{6} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} + q^{3} + q^{4} + \beta_{2} q^{5} + q^{6} + q^{8} + q^{9} + \beta_{2} q^{10} + q^{11} + q^{12} + 2 \beta_{2} q^{13} + \beta_{2} q^{15} + q^{16} + q^{17} + q^{18} + ( -1 + \beta_{1} - \beta_{2} ) q^{19} + \beta_{2} q^{20} + q^{22} + ( 3 + \beta_{1} ) q^{23} + q^{24} + ( 1 - \beta_{1} - \beta_{2} ) q^{25} + 2 \beta_{2} q^{26} + q^{27} + ( 3 - \beta_{1} - \beta_{2} ) q^{29} + \beta_{2} q^{30} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{31} + q^{32} + q^{33} + q^{34} + q^{36} + ( 3 + \beta_{1} - \beta_{2} ) q^{37} + ( -1 + \beta_{1} - \beta_{2} ) q^{38} + 2 \beta_{2} q^{39} + \beta_{2} q^{40} + ( 4 - \beta_{1} - \beta_{2} ) q^{41} + ( 3 + \beta_{1} - \beta_{2} ) q^{43} + q^{44} + \beta_{2} q^{45} + ( 3 + \beta_{1} ) q^{46} + ( -5 + \beta_{1} ) q^{47} + q^{48} + ( 1 - \beta_{1} - \beta_{2} ) q^{50} + q^{51} + 2 \beta_{2} q^{52} + 2 \beta_{2} q^{53} + q^{54} + \beta_{2} q^{55} + ( -1 + \beta_{1} - \beta_{2} ) q^{57} + ( 3 - \beta_{1} - \beta_{2} ) q^{58} + ( 3 + \beta_{1} - 3 \beta_{2} ) q^{59} + \beta_{2} q^{60} + ( -2 - 2 \beta_{1} - 3 \beta_{2} ) q^{61} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{62} + q^{64} + ( 12 - 2 \beta_{1} - 2 \beta_{2} ) q^{65} + q^{66} + ( 2 - \beta_{1} - \beta_{2} ) q^{67} + q^{68} + ( 3 + \beta_{1} ) q^{69} + ( 1 - \beta_{1} - \beta_{2} ) q^{71} + q^{72} + 2 \beta_{1} q^{73} + ( 3 + \beta_{1} - \beta_{2} ) q^{74} + ( 1 - \beta_{1} - \beta_{2} ) q^{75} + ( -1 + \beta_{1} - \beta_{2} ) q^{76} + 2 \beta_{2} q^{78} + ( 4 + 3 \beta_{2} ) q^{79} + \beta_{2} q^{80} + q^{81} + ( 4 - \beta_{1} - \beta_{2} ) q^{82} + ( -2 + \beta_{1} - 3 \beta_{2} ) q^{83} + \beta_{2} q^{85} + ( 3 + \beta_{1} - \beta_{2} ) q^{86} + ( 3 - \beta_{1} - \beta_{2} ) q^{87} + q^{88} + ( -12 + 2 \beta_{2} ) q^{89} + \beta_{2} q^{90} + ( 3 + \beta_{1} ) q^{92} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{93} + ( -5 + \beta_{1} ) q^{94} + ( -8 + 2 \beta_{1} - 2 \beta_{2} ) q^{95} + q^{96} + ( 3 + 2 \beta_{1} + 2 \beta_{2} ) q^{97} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{3} + 3q^{4} + 3q^{6} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{3} + 3q^{4} + 3q^{6} + 3q^{8} + 3q^{9} + 3q^{11} + 3q^{12} + 3q^{16} + 3q^{17} + 3q^{18} - 3q^{19} + 3q^{22} + 9q^{23} + 3q^{24} + 3q^{25} + 3q^{27} + 9q^{29} + 6q^{31} + 3q^{32} + 3q^{33} + 3q^{34} + 3q^{36} + 9q^{37} - 3q^{38} + 12q^{41} + 9q^{43} + 3q^{44} + 9q^{46} - 15q^{47} + 3q^{48} + 3q^{50} + 3q^{51} + 3q^{54} - 3q^{57} + 9q^{58} + 9q^{59} - 6q^{61} + 6q^{62} + 3q^{64} + 36q^{65} + 3q^{66} + 6q^{67} + 3q^{68} + 9q^{69} + 3q^{71} + 3q^{72} + 9q^{74} + 3q^{75} - 3q^{76} + 12q^{79} + 3q^{81} + 12q^{82} - 6q^{83} + 9q^{86} + 9q^{87} + 3q^{88} - 36q^{89} + 9q^{92} + 6q^{93} - 15q^{94} - 24q^{95} + 3q^{96} + 9q^{97} + 3q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 6 x - 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{2} + \beta_{1} + 8\)\()/2\)

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
−0.523976
2.66908
−2.14510
1.00000 1.00000 1.00000 −3.20147 1.00000 0 1.00000 1.00000 −3.20147
1.2 1.00000 1.00000 1.00000 0.454904 1.00000 0 1.00000 1.00000 0.454904
1.3 1.00000 1.00000 1.00000 2.74657 1.00000 0 1.00000 1.00000 2.74657
\(n\): 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.bi 3
3.b odd 2 1 9702.2.a.dt 3
7.b odd 2 1 3234.2.a.bg 3
7.c even 3 2 462.2.i.f 6
21.c even 2 1 9702.2.a.du 3
21.h odd 6 2 1386.2.k.w 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.f 6 7.c even 3 2
1386.2.k.w 6 21.h odd 6 2
3234.2.a.bg 3 7.b odd 2 1
3234.2.a.bi 3 1.a even 1 1 trivial
9702.2.a.dt 3 3.b odd 2 1
9702.2.a.du 3 21.c even 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3234))\):

\( T_{5}^{3} - 9 T_{5} + 4 \)
\( T_{13}^{3} - 36 T_{13} + 32 \)
\( T_{17} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{3} \)
$3$ \( ( 1 - T )^{3} \)
$5$ \( 1 + 6 T^{2} + 4 T^{3} + 30 T^{4} + 125 T^{6} \)
$7$ 1
$11$ \( ( 1 - T )^{3} \)
$13$ \( 1 + 3 T^{2} + 32 T^{3} + 39 T^{4} + 2197 T^{6} \)
$17$ \( ( 1 - T + 17 T^{2} )^{3} \)
$19$ \( 1 + 3 T + 21 T^{2} + 150 T^{3} + 399 T^{4} + 1083 T^{5} + 6859 T^{6} \)
$23$ \( 1 - 9 T + 72 T^{2} - 393 T^{3} + 1656 T^{4} - 4761 T^{5} + 12167 T^{6} \)
$29$ \( 1 - 9 T + 87 T^{2} - 430 T^{3} + 2523 T^{4} - 7569 T^{5} + 24389 T^{6} \)
$31$ \( 1 - 6 T - 3 T^{2} + 140 T^{3} - 93 T^{4} - 5766 T^{5} + 29791 T^{6} \)
$37$ \( 1 - 9 T + 99 T^{2} - 502 T^{3} + 3663 T^{4} - 12321 T^{5} + 50653 T^{6} \)
$41$ \( 1 - 12 T + 144 T^{2} - 902 T^{3} + 5904 T^{4} - 20172 T^{5} + 68921 T^{6} \)
$43$ \( 1 - 9 T + 117 T^{2} - 610 T^{3} + 5031 T^{4} - 16641 T^{5} + 79507 T^{6} \)
$47$ \( 1 + 15 T + 192 T^{2} + 1391 T^{3} + 9024 T^{4} + 33135 T^{5} + 103823 T^{6} \)
$53$ \( 1 + 123 T^{2} + 32 T^{3} + 6519 T^{4} + 148877 T^{6} \)
$59$ \( 1 - 9 T + 81 T^{2} - 294 T^{3} + 4779 T^{4} - 31329 T^{5} + 205379 T^{6} \)
$61$ \( 1 + 6 T + 54 T^{2} + 506 T^{3} + 3294 T^{4} + 22326 T^{5} + 226981 T^{6} \)
$67$ \( 1 - 6 T + 186 T^{2} - 720 T^{3} + 12462 T^{4} - 26934 T^{5} + 300763 T^{6} \)
$71$ \( 1 - 3 T + 189 T^{2} - 362 T^{3} + 13419 T^{4} - 15123 T^{5} + 357911 T^{6} \)
$73$ \( 1 + 123 T^{2} - 192 T^{3} + 8979 T^{4} + 389017 T^{6} \)
$79$ \( 1 - 12 T + 204 T^{2} - 1528 T^{3} + 16116 T^{4} - 74892 T^{5} + 493039 T^{6} \)
$83$ \( 1 + 6 T + 138 T^{2} + 1184 T^{3} + 11454 T^{4} + 41334 T^{5} + 571787 T^{6} \)
$89$ \( 1 + 36 T + 663 T^{2} + 7736 T^{3} + 59007 T^{4} + 285156 T^{5} + 704969 T^{6} \)
$97$ \( 1 - 9 T + 210 T^{2} - 1753 T^{3} + 20370 T^{4} - 84681 T^{5} + 912673 T^{6} \)
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