Properties

Label 3630.2.a.bb
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.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + 2 \beta q^{7} - q^{8} + q^{9} +O(q^{10})\) \( q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + 2 \beta q^{7} - q^{8} + q^{9} + q^{10} - q^{12} + ( -5 + \beta ) q^{13} -2 \beta q^{14} + q^{15} + q^{16} + ( -2 + 5 \beta ) q^{17} - q^{18} + ( 2 - 2 \beta ) q^{19} - q^{20} -2 \beta q^{21} + ( 2 + 3 \beta ) q^{23} + q^{24} + q^{25} + ( 5 - \beta ) q^{26} - q^{27} + 2 \beta q^{28} + ( 1 + \beta ) q^{29} - q^{30} + ( -3 + \beta ) q^{31} - q^{32} + ( 2 - 5 \beta ) q^{34} -2 \beta q^{35} + q^{36} + ( -8 + \beta ) q^{37} + ( -2 + 2 \beta ) q^{38} + ( 5 - \beta ) q^{39} + q^{40} + ( 2 - 2 \beta ) q^{41} + 2 \beta q^{42} + ( -8 + 5 \beta ) q^{43} - q^{45} + ( -2 - 3 \beta ) q^{46} + ( 7 - 7 \beta ) q^{47} - q^{48} + ( -3 + 4 \beta ) q^{49} - q^{50} + ( 2 - 5 \beta ) q^{51} + ( -5 + \beta ) q^{52} + ( 2 + 4 \beta ) q^{53} + q^{54} -2 \beta q^{56} + ( -2 + 2 \beta ) q^{57} + ( -1 - \beta ) q^{58} + ( 6 + 5 \beta ) q^{59} + q^{60} + ( 3 - \beta ) q^{62} + 2 \beta q^{63} + q^{64} + ( 5 - \beta ) q^{65} + ( 3 - 5 \beta ) q^{67} + ( -2 + 5 \beta ) q^{68} + ( -2 - 3 \beta ) q^{69} + 2 \beta q^{70} + ( 4 + 4 \beta ) q^{71} - q^{72} + ( 6 - 12 \beta ) q^{73} + ( 8 - \beta ) q^{74} - q^{75} + ( 2 - 2 \beta ) q^{76} + ( -5 + \beta ) q^{78} + ( -4 + 5 \beta ) q^{79} - q^{80} + q^{81} + ( -2 + 2 \beta ) q^{82} -4 q^{83} -2 \beta q^{84} + ( 2 - 5 \beta ) q^{85} + ( 8 - 5 \beta ) q^{86} + ( -1 - \beta ) q^{87} + ( 14 + 2 \beta ) q^{89} + q^{90} + ( 2 - 8 \beta ) q^{91} + ( 2 + 3 \beta ) q^{92} + ( 3 - \beta ) q^{93} + ( -7 + 7 \beta ) q^{94} + ( -2 + 2 \beta ) q^{95} + q^{96} + ( -4 + 14 \beta ) q^{97} + ( 3 - 4 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} + 2q^{10} - 2q^{12} - 9q^{13} - 2q^{14} + 2q^{15} + 2q^{16} + q^{17} - 2q^{18} + 2q^{19} - 2q^{20} - 2q^{21} + 7q^{23} + 2q^{24} + 2q^{25} + 9q^{26} - 2q^{27} + 2q^{28} + 3q^{29} - 2q^{30} - 5q^{31} - 2q^{32} - q^{34} - 2q^{35} + 2q^{36} - 15q^{37} - 2q^{38} + 9q^{39} + 2q^{40} + 2q^{41} + 2q^{42} - 11q^{43} - 2q^{45} - 7q^{46} + 7q^{47} - 2q^{48} - 2q^{49} - 2q^{50} - q^{51} - 9q^{52} + 8q^{53} + 2q^{54} - 2q^{56} - 2q^{57} - 3q^{58} + 17q^{59} + 2q^{60} + 5q^{62} + 2q^{63} + 2q^{64} + 9q^{65} + q^{67} + q^{68} - 7q^{69} + 2q^{70} + 12q^{71} - 2q^{72} + 15q^{74} - 2q^{75} + 2q^{76} - 9q^{78} - 3q^{79} - 2q^{80} + 2q^{81} - 2q^{82} - 8q^{83} - 2q^{84} - q^{85} + 11q^{86} - 3q^{87} + 30q^{89} + 2q^{90} - 4q^{91} + 7q^{92} + 5q^{93} - 7q^{94} - 2q^{95} + 2q^{96} + 6q^{97} + 2q^{98} + O(q^{100}) \)

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

\( T_{7}^{2} - 2 T_{7} - 4 \)
\( T_{13}^{2} + 9 T_{13} + 19 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( -4 - 2 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 19 + 9 T + T^{2} \)
$17$ \( -31 - T + T^{2} \)
$19$ \( -4 - 2 T + T^{2} \)
$23$ \( 1 - 7 T + T^{2} \)
$29$ \( 1 - 3 T + T^{2} \)
$31$ \( 5 + 5 T + T^{2} \)
$37$ \( 55 + 15 T + T^{2} \)
$41$ \( -4 - 2 T + T^{2} \)
$43$ \( -1 + 11 T + T^{2} \)
$47$ \( -49 - 7 T + T^{2} \)
$53$ \( -4 - 8 T + T^{2} \)
$59$ \( 41 - 17 T + T^{2} \)
$61$ \( T^{2} \)
$67$ \( -31 - T + T^{2} \)
$71$ \( 16 - 12 T + T^{2} \)
$73$ \( -180 + T^{2} \)
$79$ \( -29 + 3 T + T^{2} \)
$83$ \( ( 4 + T )^{2} \)
$89$ \( 220 - 30 T + T^{2} \)
$97$ \( -236 - 6 T + T^{2} \)
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