Properties

Label 3549.2.a.q
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: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + q^{3} + ( \beta_{1} + \beta_{2} ) q^{4} + ( 1 - \beta_{1} + \beta_{2} ) q^{5} + \beta_{1} q^{6} - q^{7} + ( 1 + \beta_{2} ) q^{8} + q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + q^{3} + ( \beta_{1} + \beta_{2} ) q^{4} + ( 1 - \beta_{1} + \beta_{2} ) q^{5} + \beta_{1} q^{6} - q^{7} + ( 1 + \beta_{2} ) q^{8} + q^{9} + ( -3 + \beta_{1} - \beta_{2} ) q^{10} + ( -3 - \beta_{1} - \beta_{2} ) q^{11} + ( \beta_{1} + \beta_{2} ) q^{12} -\beta_{1} q^{14} + ( 1 - \beta_{1} + \beta_{2} ) q^{15} + ( -1 - 2 \beta_{2} ) q^{16} -2 \beta_{2} q^{17} + \beta_{1} q^{18} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{19} + ( 1 - \beta_{1} - \beta_{2} ) q^{20} - q^{21} + ( -1 - 5 \beta_{1} - \beta_{2} ) q^{22} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{23} + ( 1 + \beta_{2} ) q^{24} + ( 3 - 4 \beta_{1} + 2 \beta_{2} ) q^{25} + q^{27} + ( -\beta_{1} - \beta_{2} ) q^{28} + ( -2 + 4 \beta_{1} ) q^{29} + ( -3 + \beta_{1} - \beta_{2} ) q^{30} + ( -4 - 4 \beta_{2} ) q^{31} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{32} + ( -3 - \beta_{1} - \beta_{2} ) q^{33} + ( 2 - 2 \beta_{1} ) q^{34} + ( -1 + \beta_{1} - \beta_{2} ) q^{35} + ( \beta_{1} + \beta_{2} ) q^{36} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{37} + ( -2 - 6 \beta_{1} - 2 \beta_{2} ) q^{38} + ( 5 - 3 \beta_{1} + \beta_{2} ) q^{40} + ( -5 + \beta_{1} + 3 \beta_{2} ) q^{41} -\beta_{1} q^{42} + ( 4 + 4 \beta_{1} ) q^{43} + ( -3 - 5 \beta_{1} - 3 \beta_{2} ) q^{44} + ( 1 - \beta_{1} + \beta_{2} ) q^{45} + ( -2 - 4 \beta_{1} - 2 \beta_{2} ) q^{46} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{47} + ( -1 - 2 \beta_{2} ) q^{48} + q^{49} + ( -10 + \beta_{1} - 4 \beta_{2} ) q^{50} -2 \beta_{2} q^{51} -6 q^{53} + \beta_{1} q^{54} + ( -4 + 4 \beta_{1} - 2 \beta_{2} ) q^{55} + ( -1 - \beta_{2} ) q^{56} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{57} + ( 8 + 2 \beta_{1} + 4 \beta_{2} ) q^{58} + ( -3 + \beta_{1} - \beta_{2} ) q^{59} + ( 1 - \beta_{1} - \beta_{2} ) q^{60} + ( 4 + 2 \beta_{1} + 8 \beta_{2} ) q^{61} + ( 4 - 8 \beta_{1} ) q^{62} - q^{63} + ( -2 - 5 \beta_{1} + \beta_{2} ) q^{64} + ( -1 - 5 \beta_{1} - \beta_{2} ) q^{66} + ( -6 + 2 \beta_{1} + 2 \beta_{2} ) q^{67} + ( -4 + 2 \beta_{2} ) q^{68} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{69} + ( 3 - \beta_{1} + \beta_{2} ) q^{70} + ( 1 + 3 \beta_{1} - 5 \beta_{2} ) q^{71} + ( 1 + \beta_{2} ) q^{72} + ( -6 + 2 \beta_{1} - 4 \beta_{2} ) q^{73} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{74} + ( 3 - 4 \beta_{1} + 2 \beta_{2} ) q^{75} + ( -6 - 6 \beta_{1} - 2 \beta_{2} ) q^{76} + ( 3 + \beta_{1} + \beta_{2} ) q^{77} + ( -6 + 2 \beta_{1} ) q^{79} + ( -9 + 5 \beta_{1} - \beta_{2} ) q^{80} + q^{81} + ( -1 - \beta_{1} + \beta_{2} ) q^{82} + ( -5 + 7 \beta_{1} + 5 \beta_{2} ) q^{83} + ( -\beta_{1} - \beta_{2} ) q^{84} + ( -8 + 4 \beta_{1} ) q^{85} + ( 8 + 8 \beta_{1} + 4 \beta_{2} ) q^{86} + ( -2 + 4 \beta_{1} ) q^{87} + ( -5 - \beta_{1} - 3 \beta_{2} ) q^{88} + ( -3 + 3 \beta_{1} + 5 \beta_{2} ) q^{89} + ( -3 + \beta_{1} - \beta_{2} ) q^{90} + ( -6 - 4 \beta_{1} ) q^{92} + ( -4 - 4 \beta_{2} ) q^{93} + ( 5 + 3 \beta_{1} + 3 \beta_{2} ) q^{94} + ( -4 + 4 \beta_{1} ) q^{95} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{96} + 2 \beta_{2} q^{97} + \beta_{1} q^{98} + ( -3 - \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{2} + 3q^{3} + q^{4} + 2q^{5} + q^{6} - 3q^{7} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + q^{2} + 3q^{3} + q^{4} + 2q^{5} + q^{6} - 3q^{7} + 3q^{8} + 3q^{9} - 8q^{10} - 10q^{11} + q^{12} - q^{14} + 2q^{15} - 3q^{16} + q^{18} - 8q^{19} + 2q^{20} - 3q^{21} - 8q^{22} - 2q^{23} + 3q^{24} + 5q^{25} + 3q^{27} - q^{28} - 2q^{29} - 8q^{30} - 12q^{31} - 3q^{32} - 10q^{33} + 4q^{34} - 2q^{35} + q^{36} - 4q^{37} - 12q^{38} + 12q^{40} - 14q^{41} - q^{42} + 16q^{43} - 14q^{44} + 2q^{45} - 10q^{46} - 3q^{48} + 3q^{49} - 29q^{50} - 18q^{53} + q^{54} - 8q^{55} - 3q^{56} - 8q^{57} + 26q^{58} - 8q^{59} + 2q^{60} + 14q^{61} + 4q^{62} - 3q^{63} - 11q^{64} - 8q^{66} - 16q^{67} - 12q^{68} - 2q^{69} + 8q^{70} + 6q^{71} + 3q^{72} - 16q^{73} + 8q^{74} + 5q^{75} - 24q^{76} + 10q^{77} - 16q^{79} - 22q^{80} + 3q^{81} - 4q^{82} - 8q^{83} - q^{84} - 20q^{85} + 32q^{86} - 2q^{87} - 16q^{88} - 6q^{89} - 8q^{90} - 22q^{92} - 12q^{93} + 18q^{94} - 8q^{95} - 3q^{96} + q^{98} - 10q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

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

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
−1.48119
0.311108
2.17009
−1.48119 1.00000 0.193937 4.15633 −1.48119 −1.00000 2.67513 1.00000 −6.15633
1.2 0.311108 1.00000 −1.90321 −1.52543 0.311108 −1.00000 −1.21432 1.00000 −0.474572
1.3 2.17009 1.00000 2.70928 −0.630898 2.17009 −1.00000 1.53919 1.00000 −1.36910
\(n\): 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.q 3
13.b even 2 1 3549.2.a.k 3
13.d odd 4 2 273.2.c.b 6
39.f even 4 2 819.2.c.c 6
52.f even 4 2 4368.2.h.o 6
91.i even 4 2 1911.2.c.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.c.b 6 13.d odd 4 2
819.2.c.c 6 39.f even 4 2
1911.2.c.h 6 91.i even 4 2
3549.2.a.k 3 13.b even 2 1
3549.2.a.q 3 1.a even 1 1 trivial
4368.2.h.o 6 52.f even 4 2

Hecke kernels

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

\( T_{2}^{3} - T_{2}^{2} - 3 T_{2} + 1 \)
\( T_{5}^{3} - 2 T_{5}^{2} - 8 T_{5} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T - T^{2} + T^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( -4 - 8 T - 2 T^{2} + T^{3} \)
$7$ \( ( 1 + T )^{3} \)
$11$ \( 20 + 28 T + 10 T^{2} + T^{3} \)
$13$ \( T^{3} \)
$17$ \( -16 - 16 T + T^{3} \)
$19$ \( -32 + 8 T^{2} + T^{3} \)
$23$ \( -8 - 20 T + 2 T^{2} + T^{3} \)
$29$ \( -40 - 52 T + 2 T^{2} + T^{3} \)
$31$ \( -320 - 16 T + 12 T^{2} + T^{3} \)
$37$ \( -32 - 16 T + 4 T^{2} + T^{3} \)
$41$ \( 20 + 32 T + 14 T^{2} + T^{3} \)
$43$ \( 128 + 32 T - 16 T^{2} + T^{3} \)
$47$ \( -52 - 28 T + T^{3} \)
$53$ \( ( 6 + T )^{3} \)
$59$ \( 4 + 12 T + 8 T^{2} + T^{3} \)
$61$ \( 2392 - 172 T - 14 T^{2} + T^{3} \)
$67$ \( 32 + 64 T + 16 T^{2} + T^{3} \)
$71$ \( 740 - 148 T - 6 T^{2} + T^{3} \)
$73$ \( -208 - 8 T + 16 T^{2} + T^{3} \)
$79$ \( 80 + 72 T + 16 T^{2} + T^{3} \)
$83$ \( -1252 - 172 T + 8 T^{2} + T^{3} \)
$89$ \( 76 - 88 T + 6 T^{2} + T^{3} \)
$97$ \( 16 - 16 T + T^{3} \)
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