Properties

Label 3549.2.a.w
Level $3549$
Weight $2$
Character orbit 3549.a
Self dual yes
Analytic conductor $28.339$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3549.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.3389076774\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.17428.1
Defining polynomial: \(x^{4} - x^{3} - 6 x^{2} + 4 x + 6\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 273)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + q^{3} + ( 2 + \beta_{2} ) q^{4} + ( 1 + \beta_{2} ) q^{5} -\beta_{1} q^{6} - q^{7} + ( -2 \beta_{1} - \beta_{3} ) q^{8} + q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + q^{3} + ( 2 + \beta_{2} ) q^{4} + ( 1 + \beta_{2} ) q^{5} -\beta_{1} q^{6} - q^{7} + ( -2 \beta_{1} - \beta_{3} ) q^{8} + q^{9} + ( -3 \beta_{1} - \beta_{3} ) q^{10} + ( \beta_{1} + \beta_{3} ) q^{11} + ( 2 + \beta_{2} ) q^{12} + \beta_{1} q^{14} + ( 1 + \beta_{2} ) q^{15} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{16} -2 \beta_{1} q^{17} -\beta_{1} q^{18} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{19} + ( 8 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{20} - q^{21} + ( -2 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{22} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{23} + ( -2 \beta_{1} - \beta_{3} ) q^{24} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{25} + q^{27} + ( -2 - \beta_{2} ) q^{28} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{29} + ( -3 \beta_{1} - \beta_{3} ) q^{30} + ( -1 + 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{31} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{32} + ( \beta_{1} + \beta_{3} ) q^{33} + ( 8 + 2 \beta_{2} ) q^{34} + ( -1 - \beta_{2} ) q^{35} + ( 2 + \beta_{2} ) q^{36} + ( -4 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{37} + ( 2 + 2 \beta_{2} ) q^{38} + ( -2 - 7 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{40} + ( 4 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{41} + \beta_{1} q^{42} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{43} + ( 2 + 5 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{44} + ( 1 + \beta_{2} ) q^{45} + ( 6 - 4 \beta_{1} - 2 \beta_{3} ) q^{46} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{47} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{48} + q^{49} + ( -2 - 5 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{50} -2 \beta_{1} q^{51} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{53} -\beta_{1} q^{54} + ( 2 + 4 \beta_{1} + 2 \beta_{2} ) q^{55} + ( 2 \beta_{1} + \beta_{3} ) q^{56} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{57} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{58} + ( 6 - 3 \beta_{1} - \beta_{3} ) q^{59} + ( 8 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{60} + ( 2 + 4 \beta_{1} ) q^{61} + ( -14 - 2 \beta_{2} ) q^{62} - q^{63} + ( 4 \beta_{1} - \beta_{2} ) q^{64} + ( -2 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{66} + ( 6 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{67} + ( -8 \beta_{1} - 2 \beta_{3} ) q^{68} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{69} + ( 3 \beta_{1} + \beta_{3} ) q^{70} + ( -2 + 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{71} + ( -2 \beta_{1} - \beta_{3} ) q^{72} + ( 1 + 5 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{73} + ( -4 + 6 \beta_{1} - 4 \beta_{2} ) q^{74} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{75} + ( 2 - 4 \beta_{1} - 2 \beta_{2} ) q^{76} + ( -\beta_{1} - \beta_{3} ) q^{77} + ( 1 + 5 \beta_{1} - \beta_{2} + \beta_{3} ) q^{79} + ( 10 + 5 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{80} + q^{81} + ( 2 + \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{82} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{83} + ( -2 - \beta_{2} ) q^{84} + ( -6 \beta_{1} - 2 \beta_{3} ) q^{85} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{86} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{87} + ( -14 - 5 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{88} + ( 3 - 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{89} + ( -3 \beta_{1} - \beta_{3} ) q^{90} + ( 10 - 2 \beta_{1} + 4 \beta_{2} ) q^{92} + ( -1 + 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{93} + ( 8 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{94} + ( 3 - 3 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{95} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{96} + ( 5 - 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{97} -\beta_{1} q^{98} + ( \beta_{1} + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - q^{2} + 4q^{3} + 7q^{4} + 3q^{5} - q^{6} - 4q^{7} - 3q^{8} + 4q^{9} + O(q^{10}) \) \( 4q - q^{2} + 4q^{3} + 7q^{4} + 3q^{5} - q^{6} - 4q^{7} - 3q^{8} + 4q^{9} - 4q^{10} + 2q^{11} + 7q^{12} + q^{14} + 3q^{15} + 9q^{16} - 2q^{17} - q^{18} - 7q^{19} + 32q^{20} - 4q^{21} - 8q^{22} + 3q^{23} - 3q^{24} + 9q^{25} + 4q^{27} - 7q^{28} + q^{29} - 4q^{30} - 3q^{31} - 7q^{32} + 2q^{33} + 30q^{34} - 3q^{35} + 7q^{36} - 10q^{37} + 6q^{38} - 14q^{40} + 16q^{41} + q^{42} + 3q^{43} + 12q^{44} + 3q^{45} + 18q^{46} - 5q^{47} + 9q^{48} + 4q^{49} - 13q^{50} - 2q^{51} + 5q^{53} - q^{54} + 10q^{55} + 3q^{56} - 7q^{57} + 4q^{58} + 20q^{59} + 32q^{60} + 12q^{61} - 54q^{62} - 4q^{63} + 5q^{64} - 8q^{66} + 22q^{67} - 10q^{68} + 3q^{69} + 4q^{70} - 3q^{72} + 13q^{73} - 6q^{74} + 9q^{75} + 6q^{76} - 2q^{77} + 11q^{79} + 42q^{80} + 4q^{81} + 10q^{82} - q^{83} - 7q^{84} - 8q^{85} + 10q^{86} + q^{87} - 60q^{88} + 5q^{89} - 4q^{90} + 34q^{92} - 3q^{93} + 34q^{94} + 13q^{95} - 7q^{96} + 17q^{97} - q^{98} + 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 6 x^{2} + 4 x + 6\):

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

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
2.36865
−2.10710
1.52616
−0.787711
−2.61050 1.00000 4.81471 3.81471 −2.61050 −1.00000 −7.34780 1.00000 −9.95830
1.2 −1.43986 1.00000 0.0731828 −0.926817 −1.43986 −1.00000 2.77434 1.00000 1.33448
1.3 0.670843 1.00000 −1.54997 −2.54997 0.670843 −1.00000 −2.38147 1.00000 −1.71063
1.4 2.37951 1.00000 3.66208 2.66208 2.37951 −1.00000 3.95493 1.00000 6.33445
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(13\) \(1\)

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 3549.2.a.w 4
13.b even 2 1 273.2.a.e 4
39.d odd 2 1 819.2.a.k 4
52.b odd 2 1 4368.2.a.br 4
65.d even 2 1 6825.2.a.bg 4
91.b odd 2 1 1911.2.a.s 4
273.g even 2 1 5733.2.a.bf 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.a.e 4 13.b even 2 1
819.2.a.k 4 39.d odd 2 1
1911.2.a.s 4 91.b odd 2 1
3549.2.a.w 4 1.a even 1 1 trivial
4368.2.a.br 4 52.b odd 2 1
5733.2.a.bf 4 273.g even 2 1
6825.2.a.bg 4 65.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3549))\):

\( T_{2}^{4} + T_{2}^{3} - 7 T_{2}^{2} - 5 T_{2} + 6 \)
\( T_{5}^{4} - 3 T_{5}^{3} - 10 T_{5}^{2} + 20 T_{5} + 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 6 - 5 T - 7 T^{2} + T^{3} + T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( 24 + 20 T - 10 T^{2} - 3 T^{3} + T^{4} \)
$7$ \( ( 1 + T )^{4} \)
$11$ \( 96 + 32 T - 24 T^{2} - 2 T^{3} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( 96 - 40 T - 28 T^{2} + 2 T^{3} + T^{4} \)
$19$ \( 64 - 48 T - 12 T^{2} + 7 T^{3} + T^{4} \)
$23$ \( -288 + 256 T - 52 T^{2} - 3 T^{3} + T^{4} \)
$29$ \( 72 + 52 T - 30 T^{2} - T^{3} + T^{4} \)
$31$ \( 3968 - 160 T - 128 T^{2} + 3 T^{3} + T^{4} \)
$37$ \( -128 - 840 T - 84 T^{2} + 10 T^{3} + T^{4} \)
$41$ \( -1392 + 688 T - 16 T^{3} + T^{4} \)
$43$ \( -64 + 112 T - 44 T^{2} - 3 T^{3} + T^{4} \)
$47$ \( 144 - 16 T - 40 T^{2} + 5 T^{3} + T^{4} \)
$53$ \( -24 + 68 T - 38 T^{2} - 5 T^{3} + T^{4} \)
$59$ \( -1536 + 304 T + 80 T^{2} - 20 T^{3} + T^{4} \)
$61$ \( 496 + 688 T - 64 T^{2} - 12 T^{3} + T^{4} \)
$67$ \( -15488 + 3168 T - 40 T^{2} - 22 T^{3} + T^{4} \)
$71$ \( 10176 - 304 T - 232 T^{2} + T^{4} \)
$73$ \( -11672 + 3108 T - 166 T^{2} - 13 T^{3} + T^{4} \)
$79$ \( -3456 + 1440 T - 120 T^{2} - 11 T^{3} + T^{4} \)
$83$ \( -48 + 80 T - 36 T^{2} + T^{3} + T^{4} \)
$89$ \( -1704 + 1196 T - 162 T^{2} - 5 T^{3} + T^{4} \)
$97$ \( -1528 + 820 T - 14 T^{2} - 17 T^{3} + T^{4} \)
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