Properties

Label 3549.2.a.g
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.

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

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
1.80194
−1.24698
0.445042
−2.24698 −1.00000 3.04892 0 2.24698 1.00000 −2.35690 1.00000 0
1.2 −0.554958 −1.00000 −1.69202 0 0.554958 1.00000 2.04892 1.00000 0
1.3 0.801938 −1.00000 −1.35690 0 −0.801938 1.00000 −2.69202 1.00000 0
\(n\): 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.g 3
13.b even 2 1 3549.2.a.r yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3549.2.a.g 3 1.a even 1 1 trivial
3549.2.a.r yes 3 13.b even 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - T + 2 T^{2} + T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( T^{3} \)
$7$ \( ( -1 + T )^{3} \)
$11$ \( -1 - 2 T + T^{2} + T^{3} \)
$13$ \( T^{3} \)
$17$ \( 8 - 8 T - 2 T^{2} + T^{3} \)
$19$ \( 104 - 16 T - 6 T^{2} + T^{3} \)
$23$ \( -13 - 58 T - T^{2} + T^{3} \)
$29$ \( -43 + 6 T + 9 T^{2} + T^{3} \)
$31$ \( -8 + 24 T - 10 T^{2} + T^{3} \)
$37$ \( -91 - 14 T + 7 T^{2} + T^{3} \)
$41$ \( 232 + 124 T + 20 T^{2} + T^{3} \)
$43$ \( -29 + 40 T - 13 T^{2} + T^{3} \)
$47$ \( -56 - 84 T + T^{3} \)
$53$ \( -377 - 92 T + 5 T^{2} + T^{3} \)
$59$ \( -64 - 32 T + 4 T^{2} + T^{3} \)
$61$ \( -104 - 116 T - 4 T^{2} + T^{3} \)
$67$ \( -127 + 104 T - 19 T^{2} + T^{3} \)
$71$ \( 7 + 14 T + 7 T^{2} + T^{3} \)
$73$ \( -728 - 56 T + 14 T^{2} + T^{3} \)
$79$ \( -43 - 64 T + 9 T^{2} + T^{3} \)
$83$ \( -232 - 64 T + 2 T^{2} + T^{3} \)
$89$ \( 56 - 84 T + T^{3} \)
$97$ \( 344 - 44 T - 8 T^{2} + T^{3} \)
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