Properties

Label 3630.2.a.bm
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: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 330)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} -2 \beta q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} -2 \beta q^{7} + q^{8} + q^{9} + q^{10} - q^{12} + ( -1 + \beta ) q^{13} -2 \beta q^{14} - q^{15} + q^{16} + ( -2 + 3 \beta ) q^{17} + q^{18} + ( 2 + 2 \beta ) q^{19} + q^{20} + 2 \beta q^{21} + ( 2 + \beta ) q^{23} - q^{24} + q^{25} + ( -1 + \beta ) q^{26} - q^{27} -2 \beta q^{28} + ( 3 - 5 \beta ) q^{29} - q^{30} + ( 1 - 3 \beta ) q^{31} + q^{32} + ( -2 + 3 \beta ) q^{34} -2 \beta q^{35} + q^{36} + ( 4 - 3 \beta ) q^{37} + ( 2 + 2 \beta ) q^{38} + ( 1 - \beta ) q^{39} + q^{40} + ( -2 - 6 \beta ) q^{41} + 2 \beta q^{42} + ( 4 - 3 \beta ) q^{43} + q^{45} + ( 2 + \beta ) q^{46} + ( 13 - \beta ) q^{47} - q^{48} + ( -3 + 4 \beta ) q^{49} + q^{50} + ( 2 - 3 \beta ) q^{51} + ( -1 + \beta ) q^{52} + ( 6 - 4 \beta ) q^{53} - q^{54} -2 \beta q^{56} + ( -2 - 2 \beta ) q^{57} + ( 3 - 5 \beta ) q^{58} + ( 2 + 3 \beta ) q^{59} - q^{60} + ( 4 + 4 \beta ) q^{61} + ( 1 - 3 \beta ) q^{62} -2 \beta q^{63} + q^{64} + ( -1 + \beta ) q^{65} + ( -5 + 7 \beta ) q^{67} + ( -2 + 3 \beta ) q^{68} + ( -2 - \beta ) q^{69} -2 \beta q^{70} + ( -8 + 12 \beta ) q^{71} + q^{72} + ( 2 - 4 \beta ) q^{73} + ( 4 - 3 \beta ) q^{74} - q^{75} + ( 2 + 2 \beta ) q^{76} + ( 1 - \beta ) q^{78} + ( -4 + 9 \beta ) q^{79} + q^{80} + q^{81} + ( -2 - 6 \beta ) q^{82} + ( 4 + 4 \beta ) q^{83} + 2 \beta q^{84} + ( -2 + 3 \beta ) q^{85} + ( 4 - 3 \beta ) q^{86} + ( -3 + 5 \beta ) q^{87} + ( 2 - 6 \beta ) q^{89} + q^{90} -2 q^{91} + ( 2 + \beta ) q^{92} + ( -1 + 3 \beta ) q^{93} + ( 13 - \beta ) q^{94} + ( 2 + 2 \beta ) q^{95} - q^{96} + ( 8 + 6 \beta ) q^{97} + ( -3 + 4 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} + 2q^{10} - 2q^{12} - q^{13} - 2q^{14} - 2q^{15} + 2q^{16} - q^{17} + 2q^{18} + 6q^{19} + 2q^{20} + 2q^{21} + 5q^{23} - 2q^{24} + 2q^{25} - q^{26} - 2q^{27} - 2q^{28} + q^{29} - 2q^{30} - q^{31} + 2q^{32} - q^{34} - 2q^{35} + 2q^{36} + 5q^{37} + 6q^{38} + q^{39} + 2q^{40} - 10q^{41} + 2q^{42} + 5q^{43} + 2q^{45} + 5q^{46} + 25q^{47} - 2q^{48} - 2q^{49} + 2q^{50} + q^{51} - q^{52} + 8q^{53} - 2q^{54} - 2q^{56} - 6q^{57} + q^{58} + 7q^{59} - 2q^{60} + 12q^{61} - q^{62} - 2q^{63} + 2q^{64} - q^{65} - 3q^{67} - q^{68} - 5q^{69} - 2q^{70} - 4q^{71} + 2q^{72} + 5q^{74} - 2q^{75} + 6q^{76} + q^{78} + q^{79} + 2q^{80} + 2q^{81} - 10q^{82} + 12q^{83} + 2q^{84} - q^{85} + 5q^{86} - q^{87} - 2q^{89} + 2q^{90} - 4q^{91} + 5q^{92} + q^{93} + 25q^{94} + 6q^{95} - 2q^{96} + 22q^{97} - 2q^{98} + O(q^{100}) \)

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
1.61803
−0.618034
1.00000 −1.00000 1.00000 1.00000 −1.00000 −3.23607 1.00000 1.00000 1.00000
1.2 1.00000 −1.00000 1.00000 1.00000 −1.00000 1.23607 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.bm 2
11.b odd 2 1 3630.2.a.be 2
11.d odd 10 2 330.2.m.c 4
33.f even 10 2 990.2.n.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.m.c 4 11.d odd 10 2
990.2.n.d 4 33.f even 10 2
3630.2.a.be 2 11.b odd 2 1
3630.2.a.bm 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3630))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( -4 + 2 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( -1 + T + T^{2} \)
$17$ \( -11 + T + T^{2} \)
$19$ \( 4 - 6 T + T^{2} \)
$23$ \( 5 - 5 T + T^{2} \)
$29$ \( -31 - T + T^{2} \)
$31$ \( -11 + T + T^{2} \)
$37$ \( -5 - 5 T + T^{2} \)
$41$ \( -20 + 10 T + T^{2} \)
$43$ \( -5 - 5 T + T^{2} \)
$47$ \( 155 - 25 T + T^{2} \)
$53$ \( -4 - 8 T + T^{2} \)
$59$ \( 1 - 7 T + T^{2} \)
$61$ \( 16 - 12 T + T^{2} \)
$67$ \( -59 + 3 T + T^{2} \)
$71$ \( -176 + 4 T + T^{2} \)
$73$ \( -20 + T^{2} \)
$79$ \( -101 - T + T^{2} \)
$83$ \( 16 - 12 T + T^{2} \)
$89$ \( -44 + 2 T + T^{2} \)
$97$ \( 76 - 22 T + T^{2} \)
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