Properties

Label 3234.2.a.bk
Level 3234
Weight 2
Character orbit 3234.a
Self dual yes
Analytic conductor 25.824
Analytic rank 0
Dimension 4
CM no
Inner twists 1

Related objects

Downloads

Learn more about

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: \(4\)
Coefficient field: 4.4.14336.1
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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} + ( -\beta_{1} + \beta_{3} ) q^{13} + \beta_{2} q^{15} + q^{16} + ( \beta_{2} - 2 \beta_{3} ) q^{17} - q^{18} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{19} + \beta_{2} q^{20} + q^{22} + ( -2 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{23} - q^{24} + ( 3 + 2 \beta_{3} ) q^{25} + ( \beta_{1} - \beta_{3} ) q^{26} + q^{27} + 2 \beta_{3} q^{29} -\beta_{2} q^{30} + ( 4 - \beta_{2} - \beta_{3} ) q^{31} - q^{32} - q^{33} + ( -\beta_{2} + 2 \beta_{3} ) q^{34} + q^{36} + ( 2 + \beta_{1} - 4 \beta_{3} ) q^{37} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{38} + ( -\beta_{1} + \beta_{3} ) q^{39} -\beta_{2} q^{40} + ( -\beta_{2} + 2 \beta_{3} ) q^{41} + ( 2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{43} - q^{44} + \beta_{2} q^{45} + ( 2 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{46} + ( 4 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{47} + q^{48} + ( -3 - 2 \beta_{3} ) q^{50} + ( \beta_{2} - 2 \beta_{3} ) q^{51} + ( -\beta_{1} + \beta_{3} ) q^{52} + ( 2 + \beta_{1} + 4 \beta_{3} ) q^{53} - q^{54} -\beta_{2} q^{55} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{57} -2 \beta_{3} q^{58} + ( 8 + 2 \beta_{3} ) q^{59} + \beta_{2} q^{60} + ( 4 - \beta_{1} - \beta_{3} ) q^{61} + ( -4 + \beta_{2} + \beta_{3} ) q^{62} + q^{64} + ( -4 + \beta_{1} - 8 \beta_{3} ) q^{65} + q^{66} + ( 4 - \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{67} + ( \beta_{2} - 2 \beta_{3} ) q^{68} + ( -2 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{69} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{71} - q^{72} + ( 2 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{73} + ( -2 - \beta_{1} + 4 \beta_{3} ) q^{74} + ( 3 + 2 \beta_{3} ) q^{75} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{76} + ( \beta_{1} - \beta_{3} ) q^{78} + ( 4 + \beta_{1} - 6 \beta_{3} ) q^{79} + \beta_{2} q^{80} + q^{81} + ( \beta_{2} - 2 \beta_{3} ) q^{82} + ( \beta_{2} + 3 \beta_{3} ) q^{83} + ( 8 - 2 \beta_{1} + 2 \beta_{3} ) q^{85} + ( -2 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{86} + 2 \beta_{3} q^{87} + q^{88} + ( 4 - 3 \beta_{1} - \beta_{3} ) q^{89} -\beta_{2} q^{90} + ( -2 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{92} + ( 4 - \beta_{2} - \beta_{3} ) q^{93} + ( -4 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{94} + ( -4 + \beta_{1} + 6 \beta_{3} ) q^{95} - q^{96} + ( -4 + 2 \beta_{1} - \beta_{3} ) q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} + 4q^{3} + 4q^{4} - 4q^{6} - 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{2} + 4q^{3} + 4q^{4} - 4q^{6} - 4q^{8} + 4q^{9} - 4q^{11} + 4q^{12} + 4q^{16} - 4q^{18} + 4q^{22} - 8q^{23} - 4q^{24} + 12q^{25} + 4q^{27} + 16q^{31} - 4q^{32} - 4q^{33} + 4q^{36} + 8q^{37} + 8q^{43} - 4q^{44} + 8q^{46} + 16q^{47} + 4q^{48} - 12q^{50} + 8q^{53} - 4q^{54} + 32q^{59} + 16q^{61} - 16q^{62} + 4q^{64} - 16q^{65} + 4q^{66} + 16q^{67} - 8q^{69} - 4q^{72} - 8q^{74} + 12q^{75} + 16q^{79} + 4q^{81} + 32q^{85} - 8q^{86} + 4q^{88} + 16q^{89} - 8q^{92} + 16q^{93} - 16q^{94} - 16q^{95} - 4q^{96} - 16q^{97} - 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 8 x^{2} + 14\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - 4 \nu \)
\(\beta_{3}\)\(=\)\( \nu^{2} - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{2} + 2 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.32685
1.60804
−1.60804
2.32685
−1.00000 1.00000 1.00000 −3.29066 −1.00000 0 −1.00000 1.00000 3.29066
1.2 −1.00000 1.00000 1.00000 −2.27411 −1.00000 0 −1.00000 1.00000 2.27411
1.3 −1.00000 1.00000 1.00000 2.27411 −1.00000 0 −1.00000 1.00000 −2.27411
1.4 −1.00000 1.00000 1.00000 3.29066 −1.00000 0 −1.00000 1.00000 −3.29066
\(n\): 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.bk yes 4
3.b odd 2 1 9702.2.a.ec 4
7.b odd 2 1 3234.2.a.bj 4
21.c even 2 1 9702.2.a.eb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3234.2.a.bj 4 7.b odd 2 1
3234.2.a.bk yes 4 1.a even 1 1 trivial
9702.2.a.eb 4 21.c even 2 1
9702.2.a.ec 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3234))\):

\( T_{5}^{4} - 16 T_{5}^{2} + 56 \)
\( T_{13}^{4} - 36 T_{13}^{2} - 32 T_{13} + 164 \)
\( T_{17}^{4} - 32 T_{17}^{2} + 32 T_{17} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( ( 1 - T )^{4} \)
$5$ \( 1 + 4 T^{2} + 46 T^{4} + 100 T^{6} + 625 T^{8} \)
$7$ 1
$11$ \( ( 1 + T )^{4} \)
$13$ \( 1 + 16 T^{2} - 32 T^{3} + 242 T^{4} - 416 T^{5} + 2704 T^{6} + 28561 T^{8} \)
$17$ \( 1 + 36 T^{2} + 32 T^{3} + 638 T^{4} + 544 T^{5} + 10404 T^{6} + 83521 T^{8} \)
$19$ \( 1 + 40 T^{2} + 80 T^{3} + 794 T^{4} + 1520 T^{5} + 14440 T^{6} + 130321 T^{8} \)
$23$ \( 1 + 8 T + 36 T^{2} - 56 T^{3} - 570 T^{4} - 1288 T^{5} + 19044 T^{6} + 97336 T^{7} + 279841 T^{8} \)
$29$ \( ( 1 + 50 T^{2} + 841 T^{4} )^{2} \)
$31$ \( 1 - 16 T + 200 T^{2} - 1568 T^{3} + 10378 T^{4} - 48608 T^{5} + 192200 T^{6} - 476656 T^{7} + 923521 T^{8} \)
$37$ \( 1 - 8 T + 76 T^{2} - 408 T^{3} + 2486 T^{4} - 15096 T^{5} + 104044 T^{6} - 405224 T^{7} + 1874161 T^{8} \)
$41$ \( 1 + 132 T^{2} - 32 T^{3} + 7454 T^{4} - 1312 T^{5} + 221892 T^{6} + 2825761 T^{8} \)
$43$ \( 1 - 8 T + 116 T^{2} - 424 T^{3} + 5110 T^{4} - 18232 T^{5} + 214484 T^{6} - 636056 T^{7} + 3418801 T^{8} \)
$47$ \( 1 - 16 T + 248 T^{2} - 2304 T^{3} + 18890 T^{4} - 108288 T^{5} + 547832 T^{6} - 1661168 T^{7} + 4879681 T^{8} \)
$53$ \( 1 - 8 T + 140 T^{2} - 1048 T^{3} + 9334 T^{4} - 55544 T^{5} + 393260 T^{6} - 1191016 T^{7} + 7890481 T^{8} \)
$59$ \( ( 1 - 16 T + 174 T^{2} - 944 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( 1 - 16 T + 304 T^{2} - 2864 T^{3} + 29362 T^{4} - 174704 T^{5} + 1131184 T^{6} - 3631696 T^{7} + 13845841 T^{8} \)
$67$ \( 1 - 16 T + 124 T^{2} - 1104 T^{3} + 11510 T^{4} - 73968 T^{5} + 556636 T^{6} - 4812208 T^{7} + 20151121 T^{8} \)
$71$ \( 1 + 140 T^{2} - 256 T^{3} + 12422 T^{4} - 18176 T^{5} + 705740 T^{6} + 25411681 T^{8} \)
$73$ \( 1 + 116 T^{2} - 448 T^{3} + 6462 T^{4} - 32704 T^{5} + 618164 T^{6} + 28398241 T^{8} \)
$79$ \( 1 - 16 T + 236 T^{2} - 2448 T^{3} + 24582 T^{4} - 193392 T^{5} + 1472876 T^{6} - 7888624 T^{7} + 38950081 T^{8} \)
$83$ \( 1 + 280 T^{2} - 48 T^{3} + 32794 T^{4} - 3984 T^{5} + 1928920 T^{6} + 47458321 T^{8} \)
$89$ \( 1 - 16 T + 160 T^{2} - 1904 T^{3} + 24642 T^{4} - 169456 T^{5} + 1267360 T^{6} - 11279504 T^{7} + 62742241 T^{8} \)
$97$ \( 1 + 16 T + 352 T^{2} + 3984 T^{3} + 51458 T^{4} + 386448 T^{5} + 3311968 T^{6} + 14602768 T^{7} + 88529281 T^{8} \)
show more
show less