Properties

Label 3234.2.a.be
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

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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: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−0.618034
1.61803
1.00000 1.00000 1.00000 −1.23607 1.00000 0 1.00000 1.00000 −1.23607
1.2 1.00000 1.00000 1.00000 3.23607 1.00000 0 1.00000 1.00000 3.23607
\(n\): 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.be yes 2
3.b odd 2 1 9702.2.a.ck 2
7.b odd 2 1 3234.2.a.bb 2
21.c even 2 1 9702.2.a.cw 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3234.2.a.bb 2 7.b odd 2 1
3234.2.a.be yes 2 1.a even 1 1 trivial
9702.2.a.ck 2 3.b odd 2 1
9702.2.a.cw 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3234))\):

\( T_{5}^{2} - 2 T_{5} - 4 \)
\( T_{13} \)
\( T_{17}^{2} - 6 T_{17} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{2} \)
$3$ \( ( 1 - T )^{2} \)
$5$ \( 1 - 2 T + 6 T^{2} - 10 T^{3} + 25 T^{4} \)
$7$ 1
$11$ \( ( 1 + T )^{2} \)
$13$ \( ( 1 + 13 T^{2} )^{2} \)
$17$ \( 1 - 6 T + 38 T^{2} - 102 T^{3} + 289 T^{4} \)
$19$ \( 1 - 2 T - 6 T^{2} - 38 T^{3} + 361 T^{4} \)
$23$ \( 1 - 4 T + 30 T^{2} - 92 T^{3} + 529 T^{4} \)
$29$ \( 1 + 38 T^{2} + 841 T^{4} \)
$31$ \( 1 - 10 T + 82 T^{2} - 310 T^{3} + 961 T^{4} \)
$37$ \( 1 + 4 T - 2 T^{2} + 148 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 6 T + 86 T^{2} - 246 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 4 T + 70 T^{2} - 172 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 6 T + 58 T^{2} - 282 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 + 6 T + 53 T^{2} )^{2} \)
$59$ \( 1 - 12 T + 134 T^{2} - 708 T^{3} + 3481 T^{4} \)
$61$ \( ( 1 + 61 T^{2} )^{2} \)
$67$ \( 1 - 4 T - 42 T^{2} - 268 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 4 T + 126 T^{2} - 284 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 18 T + 182 T^{2} - 1314 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 4 T - 18 T^{2} + 316 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 2 T + 162 T^{2} - 166 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 4 T + 162 T^{2} - 356 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 97 T^{2} )^{2} \)
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