Properties

Label 3549.2.a.a
Level $3549$
Weight $2$
Character orbit 3549.a
Self dual yes
Analytic conductor $28.339$
Analytic rank $1$
Dimension $1$
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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 273)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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
0
−2.00000 1.00000 2.00000 −3.00000 −2.00000 −1.00000 0 1.00000 6.00000
\(n\): 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.a 1
13.b even 2 1 3549.2.a.e 1
13.d odd 4 2 273.2.c.a 2
39.f even 4 2 819.2.c.a 2
52.f even 4 2 4368.2.h.e 2
91.i even 4 2 1911.2.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.c.a 2 13.d odd 4 2
819.2.c.a 2 39.f even 4 2
1911.2.c.a 2 91.i even 4 2
3549.2.a.a 1 1.a even 1 1 trivial
3549.2.a.e 1 13.b even 2 1
4368.2.h.e 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3549))\):

\( T_{2} + 2 \)
\( T_{5} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( -1 + T \)
$5$ \( 3 + T \)
$7$ \( 1 + T \)
$11$ \( T \)
$13$ \( T \)
$17$ \( -2 + T \)
$19$ \( 1 + T \)
$23$ \( -1 + T \)
$29$ \( -5 + T \)
$31$ \( 5 + T \)
$37$ \( 8 + T \)
$41$ \( -10 + T \)
$43$ \( 9 + T \)
$47$ \( -7 + T \)
$53$ \( -9 + T \)
$59$ \( 4 + T \)
$61$ \( 8 + T \)
$67$ \( 2 + T \)
$71$ \( T \)
$73$ \( -9 + T \)
$79$ \( -15 + T \)
$83$ \( 9 + T \)
$89$ \( 9 + T \)
$97$ \( -13 + T \)
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