Properties

Label 3549.2.a.p
Level $3549$
Weight $2$
Character orbit 3549.a
Self dual yes
Analytic conductor $28.339$
Analytic rank $1$
Dimension $3$
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: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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
−1.24698
0.445042
1.80194
−1.24698 1.00000 −0.445042 −1.00000 −1.24698 1.00000 3.04892 1.00000 1.24698
1.2 0.445042 1.00000 −1.80194 −1.00000 0.445042 1.00000 −1.69202 1.00000 −0.445042
1.3 1.80194 1.00000 1.24698 −1.00000 1.80194 1.00000 −1.35690 1.00000 −1.80194
\(n\): 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.p yes 3
13.b even 2 1 3549.2.a.j 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3549.2.a.j 3 13.b even 2 1
3549.2.a.p yes 3 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2}^{3} - T_{2}^{2} - 2 T_{2} + 1 \)
\( T_{5} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T - T^{2} + T^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( ( -1 + T )^{3} \)
$11$ \( -1 - 2 T + T^{2} + T^{3} \)
$13$ \( T^{3} \)
$17$ \( 7 + 14 T + 7 T^{2} + T^{3} \)
$19$ \( -13 - 18 T + 3 T^{2} + T^{3} \)
$23$ \( 43 + 41 T + 12 T^{2} + T^{3} \)
$29$ \( -29 - 16 T + T^{2} + T^{3} \)
$31$ \( 1 - 2 T - T^{2} + T^{3} \)
$37$ \( -91 - 21 T + 7 T^{2} + T^{3} \)
$41$ \( -13 - 23 T - 8 T^{2} + T^{3} \)
$43$ \( ( 8 + T )^{3} \)
$47$ \( 377 - 15 T - 12 T^{2} + T^{3} \)
$53$ \( 449 - 121 T - T^{2} + T^{3} \)
$59$ \( -91 + 119 T - 21 T^{2} + T^{3} \)
$61$ \( 169 + 117 T + 20 T^{2} + T^{3} \)
$67$ \( 27 + 27 T - 12 T^{2} + T^{3} \)
$71$ \( -27 - 9 T + 6 T^{2} + T^{3} \)
$73$ \( -13 - 18 T + 3 T^{2} + T^{3} \)
$79$ \( 41 - 64 T + 9 T^{2} + T^{3} \)
$83$ \( 13 - 99 T + 2 T^{2} + T^{3} \)
$89$ \( 29 + 55 T + 19 T^{2} + T^{3} \)
$97$ \( -617 - T + 16 T^{2} + T^{3} \)
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