Properties

Label 3549.2.a.s
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: 3.3.169.1
Defining polynomial: \(x^{3} - x^{2} - 4 x - 1\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 4 x - 1\):

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

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
2.65109
−0.273891
−1.37720
−1.65109 −1.00000 0.726109 −2.92498 1.65109 1.00000 2.10331 1.00000 4.82942
1.2 1.27389 −1.00000 −0.377203 −1.10331 −1.27389 1.00000 −3.02830 1.00000 −1.40550
1.3 2.37720 −1.00000 3.65109 4.02830 −2.37720 1.00000 3.92498 1.00000 9.57608
\(n\): 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.s 3
13.b even 2 1 3549.2.a.h 3
13.e even 6 2 273.2.k.d 6
39.h odd 6 2 819.2.o.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.k.d 6 13.e even 6 2
819.2.o.d 6 39.h odd 6 2
3549.2.a.h 3 13.b even 2 1
3549.2.a.s 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3549))\):

\( T_{2}^{3} - 2 T_{2}^{2} - 3 T_{2} + 5 \)
\( T_{5}^{3} - 13 T_{5} - 13 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 5 - 3 T - 2 T^{2} + T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( -13 - 13 T + T^{3} \)
$7$ \( ( -1 + T )^{3} \)
$11$ \( -5 + 17 T - 8 T^{2} + T^{3} \)
$13$ \( T^{3} \)
$17$ \( 1 + T - 4 T^{2} + T^{3} \)
$19$ \( 47 - T - 7 T^{2} + T^{3} \)
$23$ \( -1 + 14 T - 9 T^{2} + T^{3} \)
$29$ \( 5 - 14 T + 7 T^{2} + T^{3} \)
$31$ \( 281 - 40 T - 7 T^{2} + T^{3} \)
$37$ \( 13 - 13 T + T^{3} \)
$41$ \( -200 - 68 T + 2 T^{2} + T^{3} \)
$43$ \( 229 + 116 T + 19 T^{2} + T^{3} \)
$47$ \( 547 + 14 T - 17 T^{2} + T^{3} \)
$53$ \( 13 + 39 T - 13 T^{2} + T^{3} \)
$59$ \( 25 - 10 T - 3 T^{2} + T^{3} \)
$61$ \( 13 + 39 T - 13 T^{2} + T^{3} \)
$67$ \( -395 - 74 T + 5 T^{2} + T^{3} \)
$71$ \( -515 - 113 T + 8 T^{2} + T^{3} \)
$73$ \( -73 - 81 T - 2 T^{2} + T^{3} \)
$79$ \( -337 - 290 T + T^{2} + T^{3} \)
$83$ \( 229 - 185 T + 2 T^{2} + T^{3} \)
$89$ \( 1019 - T - 19 T^{2} + T^{3} \)
$97$ \( 599 + 230 T + 27 T^{2} + T^{3} \)
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