Properties

Label 3234.2.a.ba
Level 3234
Weight 2
Character orbit 3234.a
Self dual yes
Analytic conductor 25.824
Analytic rank 0
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
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 \(\beta = \sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} + \beta q^{5} - q^{6} - q^{8} + q^{9} +O(q^{10})\) \( q - q^{2} + q^{3} + q^{4} + \beta q^{5} - q^{6} - q^{8} + q^{9} -\beta q^{10} + q^{11} + q^{12} + 4 q^{13} + \beta q^{15} + q^{16} + 3 q^{17} - q^{18} + 2 \beta q^{19} + \beta q^{20} - q^{22} + \beta q^{23} - q^{24} + 2 q^{25} -4 q^{26} + q^{27} + 2 q^{29} -\beta q^{30} + 4 q^{31} - q^{32} + q^{33} -3 q^{34} + q^{36} + ( -4 - 2 \beta ) q^{37} -2 \beta q^{38} + 4 q^{39} -\beta q^{40} -9 q^{41} + ( 4 - 2 \beta ) q^{43} + q^{44} + \beta q^{45} -\beta q^{46} + ( -4 - 3 \beta ) q^{47} + q^{48} -2 q^{50} + 3 q^{51} + 4 q^{52} + 4 q^{53} - q^{54} + \beta q^{55} + 2 \beta q^{57} -2 q^{58} + ( 4 + 4 \beta ) q^{59} + \beta q^{60} + ( -4 + 3 \beta ) q^{61} -4 q^{62} + q^{64} + 4 \beta q^{65} - q^{66} + ( -3 - 4 \beta ) q^{67} + 3 q^{68} + \beta q^{69} + ( -8 - 2 \beta ) q^{71} - q^{72} + ( 10 - 2 \beta ) q^{73} + ( 4 + 2 \beta ) q^{74} + 2 q^{75} + 2 \beta q^{76} -4 q^{78} + ( 8 - \beta ) q^{79} + \beta q^{80} + q^{81} + 9 q^{82} + ( -5 + 4 \beta ) q^{83} + 3 \beta q^{85} + ( -4 + 2 \beta ) q^{86} + 2 q^{87} - q^{88} + ( -8 - 2 \beta ) q^{89} -\beta q^{90} + \beta q^{92} + 4 q^{93} + ( 4 + 3 \beta ) q^{94} + 14 q^{95} - q^{96} + ( 1 - 4 \beta ) q^{97} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} - 2q^{8} + 2q^{9} + 2q^{11} + 2q^{12} + 8q^{13} + 2q^{16} + 6q^{17} - 2q^{18} - 2q^{22} - 2q^{24} + 4q^{25} - 8q^{26} + 2q^{27} + 4q^{29} + 8q^{31} - 2q^{32} + 2q^{33} - 6q^{34} + 2q^{36} - 8q^{37} + 8q^{39} - 18q^{41} + 8q^{43} + 2q^{44} - 8q^{47} + 2q^{48} - 4q^{50} + 6q^{51} + 8q^{52} + 8q^{53} - 2q^{54} - 4q^{58} + 8q^{59} - 8q^{61} - 8q^{62} + 2q^{64} - 2q^{66} - 6q^{67} + 6q^{68} - 16q^{71} - 2q^{72} + 20q^{73} + 8q^{74} + 4q^{75} - 8q^{78} + 16q^{79} + 2q^{81} + 18q^{82} - 10q^{83} - 8q^{86} + 4q^{87} - 2q^{88} - 16q^{89} + 8q^{93} + 8q^{94} + 28q^{95} - 2q^{96} + 2q^{97} + 2q^{99} + O(q^{100}) \)

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
−2.64575
2.64575
−1.00000 1.00000 1.00000 −2.64575 −1.00000 0 −1.00000 1.00000 2.64575
1.2 −1.00000 1.00000 1.00000 2.64575 −1.00000 0 −1.00000 1.00000 −2.64575
\(n\): 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.ba 2
3.b odd 2 1 9702.2.a.dm 2
7.b odd 2 1 3234.2.a.w 2
7.d odd 6 2 462.2.i.e 4
21.c even 2 1 9702.2.a.db 2
21.g even 6 2 1386.2.k.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.e 4 7.d odd 6 2
1386.2.k.r 4 21.g even 6 2
3234.2.a.w 2 7.b odd 2 1
3234.2.a.ba 2 1.a even 1 1 trivial
9702.2.a.db 2 21.c even 2 1
9702.2.a.dm 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3234))\):

\( T_{5}^{2} - 7 \)
\( T_{13} - 4 \)
\( T_{17} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( 1 - T )^{2} \)
$5$ \( 1 + 3 T^{2} + 25 T^{4} \)
$7$ 1
$11$ \( ( 1 - T )^{2} \)
$13$ \( ( 1 - 4 T + 13 T^{2} )^{2} \)
$17$ \( ( 1 - 3 T + 17 T^{2} )^{2} \)
$19$ \( 1 + 10 T^{2} + 361 T^{4} \)
$23$ \( 1 + 39 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - 2 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 4 T + 31 T^{2} )^{2} \)
$37$ \( 1 + 8 T + 62 T^{2} + 296 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 9 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 8 T + 74 T^{2} - 344 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 8 T + 47 T^{2} + 376 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 - 4 T + 53 T^{2} )^{2} \)
$59$ \( 1 - 8 T + 22 T^{2} - 472 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 8 T + 75 T^{2} + 488 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 6 T + 31 T^{2} + 402 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 16 T + 178 T^{2} + 1136 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 20 T + 218 T^{2} - 1460 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 16 T + 215 T^{2} - 1264 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 10 T + 79 T^{2} + 830 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 16 T + 214 T^{2} + 1424 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 2 T + 83 T^{2} - 194 T^{3} + 9409 T^{4} \)
show more
show less