Properties

Label 3630.2.a.bf
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: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + ( 3 - \beta ) q^{7} - q^{8} + q^{9} +O(q^{10})\) \( q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + ( 3 - \beta ) q^{7} - q^{8} + q^{9} - q^{10} - q^{12} + 3 \beta q^{13} + ( -3 + \beta ) q^{14} - q^{15} + q^{16} -4 \beta q^{17} - q^{18} + 3 \beta q^{19} + q^{20} + ( -3 + \beta ) q^{21} + ( -7 - \beta ) q^{23} + q^{24} + q^{25} -3 \beta q^{26} - q^{27} + ( 3 - \beta ) q^{28} + ( -4 - 2 \beta ) q^{29} + q^{30} + ( -4 - 2 \beta ) q^{31} - q^{32} + 4 \beta q^{34} + ( 3 - \beta ) q^{35} + q^{36} + ( -8 + 3 \beta ) q^{37} -3 \beta q^{38} -3 \beta q^{39} - q^{40} + ( -7 + 5 \beta ) q^{41} + ( 3 - \beta ) q^{42} + ( 2 - 2 \beta ) q^{43} + q^{45} + ( 7 + \beta ) q^{46} + ( -6 + 11 \beta ) q^{47} - q^{48} + ( 3 - 5 \beta ) q^{49} - q^{50} + 4 \beta q^{51} + 3 \beta q^{52} + ( -5 - 5 \beta ) q^{53} + q^{54} + ( -3 + \beta ) q^{56} -3 \beta q^{57} + ( 4 + 2 \beta ) q^{58} + ( 2 - 7 \beta ) q^{59} - q^{60} + ( -6 + 6 \beta ) q^{61} + ( 4 + 2 \beta ) q^{62} + ( 3 - \beta ) q^{63} + q^{64} + 3 \beta q^{65} + ( -6 - 6 \beta ) q^{67} -4 \beta q^{68} + ( 7 + \beta ) q^{69} + ( -3 + \beta ) q^{70} + ( -8 + 10 \beta ) q^{71} - q^{72} -2 q^{73} + ( 8 - 3 \beta ) q^{74} - q^{75} + 3 \beta q^{76} + 3 \beta q^{78} + ( 10 - 4 \beta ) q^{79} + q^{80} + q^{81} + ( 7 - 5 \beta ) q^{82} + 4 \beta q^{83} + ( -3 + \beta ) q^{84} -4 \beta q^{85} + ( -2 + 2 \beta ) q^{86} + ( 4 + 2 \beta ) q^{87} + ( -3 + 9 \beta ) q^{89} - q^{90} + ( -3 + 6 \beta ) q^{91} + ( -7 - \beta ) q^{92} + ( 4 + 2 \beta ) q^{93} + ( 6 - 11 \beta ) q^{94} + 3 \beta q^{95} + q^{96} -2 q^{97} + ( -3 + 5 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 5q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 5q^{7} - 2q^{8} + 2q^{9} - 2q^{10} - 2q^{12} + 3q^{13} - 5q^{14} - 2q^{15} + 2q^{16} - 4q^{17} - 2q^{18} + 3q^{19} + 2q^{20} - 5q^{21} - 15q^{23} + 2q^{24} + 2q^{25} - 3q^{26} - 2q^{27} + 5q^{28} - 10q^{29} + 2q^{30} - 10q^{31} - 2q^{32} + 4q^{34} + 5q^{35} + 2q^{36} - 13q^{37} - 3q^{38} - 3q^{39} - 2q^{40} - 9q^{41} + 5q^{42} + 2q^{43} + 2q^{45} + 15q^{46} - q^{47} - 2q^{48} + q^{49} - 2q^{50} + 4q^{51} + 3q^{52} - 15q^{53} + 2q^{54} - 5q^{56} - 3q^{57} + 10q^{58} - 3q^{59} - 2q^{60} - 6q^{61} + 10q^{62} + 5q^{63} + 2q^{64} + 3q^{65} - 18q^{67} - 4q^{68} + 15q^{69} - 5q^{70} - 6q^{71} - 2q^{72} - 4q^{73} + 13q^{74} - 2q^{75} + 3q^{76} + 3q^{78} + 16q^{79} + 2q^{80} + 2q^{81} + 9q^{82} + 4q^{83} - 5q^{84} - 4q^{85} - 2q^{86} + 10q^{87} + 3q^{89} - 2q^{90} - 15q^{92} + 10q^{93} + q^{94} + 3q^{95} + 2q^{96} - 4q^{97} - q^{98} + O(q^{100}) \)

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 1.38197 −1.00000 1.00000 −1.00000
1.2 −1.00000 −1.00000 1.00000 1.00000 1.00000 3.61803 −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.bf 2
11.b odd 2 1 3630.2.a.bl 2
11.d odd 10 2 330.2.m.a 4
33.f even 10 2 990.2.n.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.m.a 4 11.d odd 10 2
990.2.n.h 4 33.f even 10 2
3630.2.a.bf 2 1.a even 1 1 trivial
3630.2.a.bl 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3630))\):

\( T_{7}^{2} - 5 T_{7} + 5 \)
\( T_{13}^{2} - 3 T_{13} - 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( 5 - 5 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( -9 - 3 T + T^{2} \)
$17$ \( -16 + 4 T + T^{2} \)
$19$ \( -9 - 3 T + T^{2} \)
$23$ \( 55 + 15 T + T^{2} \)
$29$ \( 20 + 10 T + T^{2} \)
$31$ \( 20 + 10 T + T^{2} \)
$37$ \( 31 + 13 T + T^{2} \)
$41$ \( -11 + 9 T + T^{2} \)
$43$ \( -4 - 2 T + T^{2} \)
$47$ \( -151 + T + T^{2} \)
$53$ \( 25 + 15 T + T^{2} \)
$59$ \( -59 + 3 T + T^{2} \)
$61$ \( -36 + 6 T + T^{2} \)
$67$ \( 36 + 18 T + T^{2} \)
$71$ \( -116 + 6 T + T^{2} \)
$73$ \( ( 2 + T )^{2} \)
$79$ \( 44 - 16 T + T^{2} \)
$83$ \( -16 - 4 T + T^{2} \)
$89$ \( -99 - 3 T + T^{2} \)
$97$ \( ( 2 + T )^{2} \)
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