Properties

Label 3549.2.a.m
Level $3549$
Weight $2$
Character orbit 3549.a
Self dual yes
Analytic conductor $28.339$
Analytic rank $0$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.473.1
Defining polynomial: \(x^{3} - 5 x - 1\)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} - q^{3} + ( 1 + \beta_{2} ) q^{4} + ( 1 + \beta_{1} - \beta_{2} ) q^{5} + \beta_{1} q^{6} + q^{7} + ( -1 - \beta_{1} ) q^{8} + q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} - q^{3} + ( 1 + \beta_{2} ) q^{4} + ( 1 + \beta_{1} - \beta_{2} ) q^{5} + \beta_{1} q^{6} + q^{7} + ( -1 - \beta_{1} ) q^{8} + q^{9} + ( -2 + \beta_{1} - \beta_{2} ) q^{10} + ( 3 + \beta_{1} - \beta_{2} ) q^{11} + ( -1 - \beta_{2} ) q^{12} -\beta_{1} q^{14} + ( -1 - \beta_{1} + \beta_{2} ) q^{15} + ( 1 + \beta_{1} - \beta_{2} ) q^{16} + ( 4 - \beta_{1} ) q^{17} -\beta_{1} q^{18} + ( 1 - 2 \beta_{1} ) q^{19} + ( -4 + 2 \beta_{1} + \beta_{2} ) q^{20} - q^{21} + ( -2 - \beta_{1} - \beta_{2} ) q^{22} + ( -2 - \beta_{2} ) q^{23} + ( 1 + \beta_{1} ) q^{24} + ( 3 - \beta_{1} - 2 \beta_{2} ) q^{25} - q^{27} + ( 1 + \beta_{2} ) q^{28} + ( 3 + \beta_{1} ) q^{29} + ( 2 - \beta_{1} + \beta_{2} ) q^{30} + ( 2 + 3 \beta_{1} - \beta_{2} ) q^{31} + ( 3 \beta_{1} - \beta_{2} ) q^{32} + ( -3 - \beta_{1} + \beta_{2} ) q^{33} + ( 3 - 4 \beta_{1} + \beta_{2} ) q^{34} + ( 1 + \beta_{1} - \beta_{2} ) q^{35} + ( 1 + \beta_{2} ) q^{36} + ( -1 + \beta_{1} + 3 \beta_{2} ) q^{37} + ( 6 - \beta_{1} + 2 \beta_{2} ) q^{38} -3 q^{40} + ( 2 - 4 \beta_{1} ) q^{41} + \beta_{1} q^{42} + ( -2 - \beta_{2} ) q^{43} + ( -2 + 2 \beta_{1} + 3 \beta_{2} ) q^{44} + ( 1 + \beta_{1} - \beta_{2} ) q^{45} + ( 1 + 4 \beta_{1} ) q^{46} + ( 2 - \beta_{1} + 3 \beta_{2} ) q^{47} + ( -1 - \beta_{1} + \beta_{2} ) q^{48} + q^{49} + ( 5 + \beta_{1} + \beta_{2} ) q^{50} + ( -4 + \beta_{1} ) q^{51} + ( -5 + 2 \beta_{2} ) q^{53} + \beta_{1} q^{54} + ( 10 + \beta_{1} - 4 \beta_{2} ) q^{55} + ( -1 - \beta_{1} ) q^{56} + ( -1 + 2 \beta_{1} ) q^{57} + ( -3 - 3 \beta_{1} - \beta_{2} ) q^{58} + ( 2 - 2 \beta_{1} + 5 \beta_{2} ) q^{59} + ( 4 - 2 \beta_{1} - \beta_{2} ) q^{60} + ( 7 - 2 \beta_{1} - 2 \beta_{2} ) q^{61} + ( -8 - 3 \beta_{2} ) q^{62} + q^{63} + ( -10 - \beta_{2} ) q^{64} + ( 2 + \beta_{1} + \beta_{2} ) q^{66} + ( 7 - 3 \beta_{1} ) q^{67} + ( 3 - 3 \beta_{1} + 4 \beta_{2} ) q^{68} + ( 2 + \beta_{2} ) q^{69} + ( -2 + \beta_{1} - \beta_{2} ) q^{70} + ( 7 + 2 \beta_{1} + \beta_{2} ) q^{71} + ( -1 - \beta_{1} ) q^{72} + ( -5 + \beta_{2} ) q^{73} + ( -6 - 5 \beta_{1} - \beta_{2} ) q^{74} + ( -3 + \beta_{1} + 2 \beta_{2} ) q^{75} + ( -1 - 6 \beta_{1} + \beta_{2} ) q^{76} + ( 3 + \beta_{1} - \beta_{2} ) q^{77} + ( -3 \beta_{1} - \beta_{2} ) q^{79} + ( 8 - \beta_{1} - 2 \beta_{2} ) q^{80} + q^{81} + ( 12 - 2 \beta_{1} + 4 \beta_{2} ) q^{82} + ( 11 - 3 \beta_{1} - \beta_{2} ) q^{83} + ( -1 - \beta_{2} ) q^{84} + ( 2 + 5 \beta_{1} - 5 \beta_{2} ) q^{85} + ( 1 + 4 \beta_{1} ) q^{86} + ( -3 - \beta_{1} ) q^{87} + ( -5 - 2 \beta_{1} ) q^{88} + ( -1 + 4 \beta_{1} ) q^{89} + ( -2 + \beta_{1} - \beta_{2} ) q^{90} + ( -8 - \beta_{1} - 2 \beta_{2} ) q^{92} + ( -2 - 3 \beta_{1} + \beta_{2} ) q^{93} + ( -8 \beta_{1} + \beta_{2} ) q^{94} + ( -3 + 3 \beta_{1} - 3 \beta_{2} ) q^{95} + ( -3 \beta_{1} + \beta_{2} ) q^{96} + ( -8 - 3 \beta_{1} + 3 \beta_{2} ) q^{97} -\beta_{1} q^{98} + ( 3 + \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} + 4q^{4} + 2q^{5} + 3q^{7} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} + 4q^{4} + 2q^{5} + 3q^{7} - 3q^{8} + 3q^{9} - 7q^{10} + 8q^{11} - 4q^{12} - 2q^{15} + 2q^{16} + 12q^{17} + 3q^{19} - 11q^{20} - 3q^{21} - 7q^{22} - 7q^{23} + 3q^{24} + 7q^{25} - 3q^{27} + 4q^{28} + 9q^{29} + 7q^{30} + 5q^{31} - q^{32} - 8q^{33} + 10q^{34} + 2q^{35} + 4q^{36} + 20q^{38} - 9q^{40} + 6q^{41} - 7q^{43} - 3q^{44} + 2q^{45} + 3q^{46} + 9q^{47} - 2q^{48} + 3q^{49} + 16q^{50} - 12q^{51} - 13q^{53} + 26q^{55} - 3q^{56} - 3q^{57} - 10q^{58} + 11q^{59} + 11q^{60} + 19q^{61} - 27q^{62} + 3q^{63} - 31q^{64} + 7q^{66} + 21q^{67} + 13q^{68} + 7q^{69} - 7q^{70} + 22q^{71} - 3q^{72} - 14q^{73} - 19q^{74} - 7q^{75} - 2q^{76} + 8q^{77} - q^{79} + 22q^{80} + 3q^{81} + 40q^{82} + 32q^{83} - 4q^{84} + q^{85} + 3q^{86} - 9q^{87} - 15q^{88} - 3q^{89} - 7q^{90} - 26q^{92} - 5q^{93} + q^{94} - 12q^{95} + q^{96} - 21q^{97} + 8q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 5 x - 1\):

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

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
2.33006
−0.201640
−2.12842
−2.33006 −1.00000 3.42917 0.900885 2.33006 1.00000 −3.33006 1.00000 −2.09911
1.2 0.201640 −1.00000 −1.95934 3.75770 −0.201640 1.00000 −0.798360 1.00000 0.757702
1.3 2.12842 −1.00000 2.53017 −2.65859 −2.12842 1.00000 1.12842 1.00000 −5.65859
\(n\): 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.m 3
13.b even 2 1 3549.2.a.l 3
13.c even 3 2 273.2.k.b 6
39.i odd 6 2 819.2.o.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.k.b 6 13.c even 3 2
819.2.o.f 6 39.i odd 6 2
3549.2.a.l 3 13.b even 2 1
3549.2.a.m 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3549))\):

\( T_{2}^{3} - 5 T_{2} + 1 \)
\( T_{5}^{3} - 2 T_{5}^{2} - 9 T_{5} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 5 T + T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( 9 - 9 T - 2 T^{2} + T^{3} \)
$7$ \( ( -1 + T )^{3} \)
$11$ \( 11 + 11 T - 8 T^{2} + T^{3} \)
$13$ \( T^{3} \)
$17$ \( -43 + 43 T - 12 T^{2} + T^{3} \)
$19$ \( 27 - 17 T - 3 T^{2} + T^{3} \)
$23$ \( -15 + 8 T + 7 T^{2} + T^{3} \)
$29$ \( -13 + 22 T - 9 T^{2} + T^{3} \)
$31$ \( 169 - 36 T - 5 T^{2} + T^{3} \)
$37$ \( 127 - 89 T + T^{3} \)
$41$ \( 216 - 68 T - 6 T^{2} + T^{3} \)
$43$ \( -15 + 8 T + 7 T^{2} + T^{3} \)
$47$ \( 405 - 44 T - 9 T^{2} + T^{3} \)
$53$ \( 3 + 23 T + 13 T^{2} + T^{3} \)
$59$ \( 1635 - 158 T - 11 T^{2} + T^{3} \)
$61$ \( 275 + 55 T - 19 T^{2} + T^{3} \)
$67$ \( -1 + 102 T - 21 T^{2} + T^{3} \)
$71$ \( -219 + 127 T - 22 T^{2} + T^{3} \)
$73$ \( 71 + 57 T + 14 T^{2} + T^{3} \)
$79$ \( 163 - 62 T + T^{2} + T^{3} \)
$83$ \( -365 + 279 T - 32 T^{2} + T^{3} \)
$89$ \( -143 - 77 T + 3 T^{2} + T^{3} \)
$97$ \( -373 + 54 T + 21 T^{2} + T^{3} \)
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