Properties

Label 3630.2.a.bh
Level $3630$
Weight $2$
Character orbit 3630.a
Self dual yes
Analytic conductor $28.986$
Analytic rank $1$
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: \(1\)
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.

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

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 −3.73205 −1.00000 1.00000 −1.00000
1.2 −1.00000 1.00000 1.00000 1.00000 −1.00000 −0.267949 −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.bh 2
11.b odd 2 1 3630.2.a.bp yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3630.2.a.bh 2 1.a even 1 1 trivial
3630.2.a.bp yes 2 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3630))\):

\( T_{7}^{2} + 4 T_{7} + 1 \)
\( T_{13} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( 1 + 4 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( 3 + T )^{2} \)
$17$ \( 1 + 4 T + T^{2} \)
$19$ \( -11 - 2 T + T^{2} \)
$23$ \( 6 + 6 T + T^{2} \)
$29$ \( ( -3 + T )^{2} \)
$31$ \( -18 - 6 T + T^{2} \)
$37$ \( -3 + T^{2} \)
$41$ \( 6 + 6 T + T^{2} \)
$43$ \( 22 + 10 T + T^{2} \)
$47$ \( -44 - 4 T + T^{2} \)
$53$ \( 46 + 14 T + T^{2} \)
$59$ \( -26 - 2 T + T^{2} \)
$61$ \( -18 + 6 T + T^{2} \)
$67$ \( 52 + 16 T + T^{2} \)
$71$ \( -39 + 6 T + T^{2} \)
$73$ \( -26 + 2 T + T^{2} \)
$79$ \( 4 + 8 T + T^{2} \)
$83$ \( -107 - 2 T + T^{2} \)
$89$ \( -72 + 12 T + T^{2} \)
$97$ \( 286 + 34 T + T^{2} \)
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