Properties

Label 3630.2.a.bk
Level $3630$
Weight $2$
Character orbit 3630.a
Self dual yes
Analytic conductor $28.986$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3630 = 2 \cdot 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3630.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.9856959337\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + \beta q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + \beta q^{7} + q^{8} + q^{9} - q^{10} - q^{12} -3 q^{13} + \beta q^{14} + q^{15} + q^{16} + \beta q^{17} + q^{18} + 3 q^{19} - q^{20} -\beta q^{21} + ( -3 + 3 \beta ) q^{23} - q^{24} + q^{25} -3 q^{26} - q^{27} + \beta q^{28} + ( 3 - 2 \beta ) q^{29} + q^{30} + ( 1 - 3 \beta ) q^{31} + q^{32} + \beta q^{34} -\beta q^{35} + q^{36} + ( 4 + 3 \beta ) q^{37} + 3 q^{38} + 3 q^{39} - q^{40} + ( 9 - \beta ) q^{41} -\beta q^{42} + ( -3 - 3 \beta ) q^{43} - q^{45} + ( -3 + 3 \beta ) q^{46} + 6 q^{47} - q^{48} -4 q^{49} + q^{50} -\beta q^{51} -3 q^{52} + ( -3 - 3 \beta ) q^{53} - q^{54} + \beta q^{56} -3 q^{57} + ( 3 - 2 \beta ) q^{58} + ( 9 - 3 \beta ) q^{59} + q^{60} + ( 3 + 5 \beta ) q^{61} + ( 1 - 3 \beta ) q^{62} + \beta q^{63} + q^{64} + 3 q^{65} + ( 4 + 6 \beta ) q^{67} + \beta q^{68} + ( 3 - 3 \beta ) q^{69} -\beta q^{70} + 3 q^{71} + q^{72} + ( -3 - \beta ) q^{73} + ( 4 + 3 \beta ) q^{74} - q^{75} + 3 q^{76} + 3 q^{78} -6 \beta q^{79} - q^{80} + q^{81} + ( 9 - \beta ) q^{82} + ( 3 + 2 \beta ) q^{83} -\beta q^{84} -\beta q^{85} + ( -3 - 3 \beta ) q^{86} + ( -3 + 2 \beta ) q^{87} + ( -6 + 6 \beta ) q^{89} - q^{90} -3 \beta q^{91} + ( -3 + 3 \beta ) q^{92} + ( -1 + 3 \beta ) q^{93} + 6 q^{94} -3 q^{95} - q^{96} + ( 5 - 3 \beta ) q^{97} -4 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} + 2q^{8} + 2q^{9} - 2q^{10} - 2q^{12} - 6q^{13} + 2q^{15} + 2q^{16} + 2q^{18} + 6q^{19} - 2q^{20} - 6q^{23} - 2q^{24} + 2q^{25} - 6q^{26} - 2q^{27} + 6q^{29} + 2q^{30} + 2q^{31} + 2q^{32} + 2q^{36} + 8q^{37} + 6q^{38} + 6q^{39} - 2q^{40} + 18q^{41} - 6q^{43} - 2q^{45} - 6q^{46} + 12q^{47} - 2q^{48} - 8q^{49} + 2q^{50} - 6q^{52} - 6q^{53} - 2q^{54} - 6q^{57} + 6q^{58} + 18q^{59} + 2q^{60} + 6q^{61} + 2q^{62} + 2q^{64} + 6q^{65} + 8q^{67} + 6q^{69} + 6q^{71} + 2q^{72} - 6q^{73} + 8q^{74} - 2q^{75} + 6q^{76} + 6q^{78} - 2q^{80} + 2q^{81} + 18q^{82} + 6q^{83} - 6q^{86} - 6q^{87} - 12q^{89} - 2q^{90} - 6q^{92} - 2q^{93} + 12q^{94} - 6q^{95} - 2q^{96} + 10q^{97} - 8q^{98} + O(q^{100}) \)

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
−1.73205
1.73205
1.00000 −1.00000 1.00000 −1.00000 −1.00000 −1.73205 1.00000 1.00000 −1.00000
1.2 1.00000 −1.00000 1.00000 −1.00000 −1.00000 1.73205 1.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)
\(11\) \(1\)

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 3630.2.a.bk yes 2
11.b odd 2 1 3630.2.a.ba 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3630.2.a.ba 2 11.b odd 2 1
3630.2.a.bk yes 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3630))\):

\( T_{7}^{2} - 3 \)
\( T_{13} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( -3 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( 3 + T )^{2} \)
$17$ \( -3 + T^{2} \)
$19$ \( ( -3 + T )^{2} \)
$23$ \( -18 + 6 T + T^{2} \)
$29$ \( -3 - 6 T + T^{2} \)
$31$ \( -26 - 2 T + T^{2} \)
$37$ \( -11 - 8 T + T^{2} \)
$41$ \( 78 - 18 T + T^{2} \)
$43$ \( -18 + 6 T + T^{2} \)
$47$ \( ( -6 + T )^{2} \)
$53$ \( -18 + 6 T + T^{2} \)
$59$ \( 54 - 18 T + T^{2} \)
$61$ \( -66 - 6 T + T^{2} \)
$67$ \( -92 - 8 T + T^{2} \)
$71$ \( ( -3 + T )^{2} \)
$73$ \( 6 + 6 T + T^{2} \)
$79$ \( -108 + T^{2} \)
$83$ \( -3 - 6 T + T^{2} \)
$89$ \( -72 + 12 T + T^{2} \)
$97$ \( -2 - 10 T + T^{2} \)
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