Properties

Label 3630.2.a.bs
Level $3630$
Weight $2$
Character orbit 3630.a
Self dual yes
Analytic conductor $28.986$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

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: \(4\)
Coefficient field: 4.4.52625.1
Defining polynomial: \(x^{4} - x^{3} - 24 x^{2} + 19 x + 121\)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + ( -\beta_{2} + \beta_{3} ) q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + ( -\beta_{2} + \beta_{3} ) q^{7} + q^{8} + q^{9} - q^{10} + q^{12} + ( 1 - \beta_{1} + \beta_{3} ) q^{13} + ( -\beta_{2} + \beta_{3} ) q^{14} - q^{15} + q^{16} + ( 2 + \beta_{1} - \beta_{2} ) q^{17} + q^{18} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{19} - q^{20} + ( -\beta_{2} + \beta_{3} ) q^{21} + ( -\beta_{1} - \beta_{3} ) q^{23} + q^{24} + q^{25} + ( 1 - \beta_{1} + \beta_{3} ) q^{26} + q^{27} + ( -\beta_{2} + \beta_{3} ) q^{28} + ( -4 \beta_{2} - \beta_{3} ) q^{29} - q^{30} + ( 4 - \beta_{3} ) q^{31} + q^{32} + ( 2 + \beta_{1} - \beta_{2} ) q^{34} + ( \beta_{2} - \beta_{3} ) q^{35} + q^{36} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{37} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{38} + ( 1 - \beta_{1} + \beta_{3} ) q^{39} - q^{40} + ( -2 \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{41} + ( -\beta_{2} + \beta_{3} ) q^{42} + ( 2 - \beta_{1} + 3 \beta_{2} ) q^{43} - q^{45} + ( -\beta_{1} - \beta_{3} ) q^{46} + ( 3 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{47} + q^{48} + ( 12 - \beta_{1} + 8 \beta_{2} ) q^{49} + q^{50} + ( 2 + \beta_{1} - \beta_{2} ) q^{51} + ( 1 - \beta_{1} + \beta_{3} ) q^{52} + ( 4 + 3 \beta_{2} - \beta_{3} ) q^{53} + q^{54} + ( -\beta_{2} + \beta_{3} ) q^{56} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{57} + ( -4 \beta_{2} - \beta_{3} ) q^{58} + ( -1 - 7 \beta_{2} ) q^{59} - q^{60} + ( -4 - 2 \beta_{2} - 2 \beta_{3} ) q^{61} + ( 4 - \beta_{3} ) q^{62} + ( -\beta_{2} + \beta_{3} ) q^{63} + q^{64} + ( -1 + \beta_{1} - \beta_{3} ) q^{65} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{67} + ( 2 + \beta_{1} - \beta_{2} ) q^{68} + ( -\beta_{1} - \beta_{3} ) q^{69} + ( \beta_{2} - \beta_{3} ) q^{70} + ( 2 + 2 \beta_{1} ) q^{71} + q^{72} + ( -6 - 8 \beta_{2} ) q^{73} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{74} + q^{75} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{76} + ( 1 - \beta_{1} + \beta_{3} ) q^{78} + ( -4 + 3 \beta_{1} - 3 \beta_{2} ) q^{79} - q^{80} + q^{81} + ( -2 \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{82} + 8 q^{83} + ( -\beta_{2} + \beta_{3} ) q^{84} + ( -2 - \beta_{1} + \beta_{2} ) q^{85} + ( 2 - \beta_{1} + 3 \beta_{2} ) q^{86} + ( -4 \beta_{2} - \beta_{3} ) q^{87} + ( -8 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{89} - q^{90} + ( 11 - \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{91} + ( -\beta_{1} - \beta_{3} ) q^{92} + ( 4 - \beta_{3} ) q^{93} + ( 3 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{94} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{95} + q^{96} + ( -2 - 4 \beta_{2} + 2 \beta_{3} ) q^{97} + ( 12 - \beta_{1} + 8 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} + 4q^{3} + 4q^{4} - 4q^{5} + 4q^{6} + q^{7} + 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{2} + 4q^{3} + 4q^{4} - 4q^{5} + 4q^{6} + q^{7} + 4q^{8} + 4q^{9} - 4q^{10} + 4q^{12} + 2q^{13} + q^{14} - 4q^{15} + 4q^{16} + 11q^{17} + 4q^{18} + q^{19} - 4q^{20} + q^{21} + 4q^{24} + 4q^{25} + 2q^{26} + 4q^{27} + q^{28} + 9q^{29} - 4q^{30} + 17q^{31} + 4q^{32} + 11q^{34} - q^{35} + 4q^{36} + 8q^{37} + q^{38} + 2q^{39} - 4q^{40} - 11q^{41} + q^{42} + q^{43} - 4q^{45} + 10q^{47} + 4q^{48} + 31q^{49} + 4q^{50} + 11q^{51} + 2q^{52} + 11q^{53} + 4q^{54} + q^{56} + q^{57} + 9q^{58} + 10q^{59} - 4q^{60} - 10q^{61} + 17q^{62} + q^{63} + 4q^{64} - 2q^{65} + 9q^{67} + 11q^{68} - q^{70} + 10q^{71} + 4q^{72} - 8q^{73} + 8q^{74} + 4q^{75} + q^{76} + 2q^{78} - 7q^{79} - 4q^{80} + 4q^{81} - 11q^{82} + 32q^{83} + q^{84} - 11q^{85} + q^{86} + 9q^{87} - 23q^{89} - 4q^{90} + 48q^{91} + 17q^{93} + 10q^{94} - q^{95} + 4q^{96} - 2q^{97} + 31q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 24 x^{2} + 19 x + 121\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 14 \nu - 27 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{3} + 10 \nu^{2} - 42 \nu - 125 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 3 \beta_{2} + 11\)
\(\nu^{3}\)\(=\)\(-2 \beta_{3} + 10 \beta_{2} + 14 \beta_{1} + 5\)

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
−2.15753
3.55745
−4.17549
3.77556
1.00000 1.00000 1.00000 −1.00000 1.00000 −5.10899 1.00000 1.00000 −1.00000
1.2 1.00000 1.00000 1.00000 −1.00000 1.00000 −1.58059 1.00000 1.00000 −1.00000
1.3 1.00000 1.00000 1.00000 −1.00000 1.00000 3.19863 1.00000 1.00000 −1.00000
1.4 1.00000 1.00000 1.00000 −1.00000 1.00000 4.49096 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.bs 4
11.b odd 2 1 3630.2.a.bq 4
11.d odd 10 2 330.2.m.f 8
33.f even 10 2 990.2.n.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.m.f 8 11.d odd 10 2
990.2.n.i 8 33.f even 10 2
3630.2.a.bq 4 11.b odd 2 1
3630.2.a.bs 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3630))\):

\( T_{7}^{4} - T_{7}^{3} - 29 T_{7}^{2} + 34 T_{7} + 116 \)
\( T_{13}^{4} - 2 T_{13}^{3} - 41 T_{13}^{2} + 42 T_{13} + 121 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( ( 1 + T )^{4} \)
$7$ \( 116 + 34 T - 29 T^{2} - T^{3} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( 121 + 42 T - 41 T^{2} - 2 T^{3} + T^{4} \)
$17$ \( 16 + 44 T + 21 T^{2} - 11 T^{3} + T^{4} \)
$19$ \( -4 + 54 T - 39 T^{2} - T^{3} + T^{4} \)
$23$ \( 55 + 130 T - 65 T^{2} + T^{4} \)
$29$ \( -724 + 486 T - 49 T^{2} - 9 T^{3} + T^{4} \)
$31$ \( -164 - 38 T + 79 T^{2} - 17 T^{3} + T^{4} \)
$37$ \( 1741 + 348 T - 71 T^{2} - 8 T^{3} + T^{4} \)
$41$ \( -1804 - 1554 T - 129 T^{2} + 11 T^{3} + T^{4} \)
$43$ \( -4 + 54 T - 39 T^{2} - T^{3} + T^{4} \)
$47$ \( -745 + 360 T - 15 T^{2} - 10 T^{3} + T^{4} \)
$53$ \( -44 + 64 T + T^{2} - 11 T^{3} + T^{4} \)
$59$ \( ( -55 - 5 T + T^{2} )^{2} \)
$61$ \( 1280 - 160 T - 100 T^{2} + 10 T^{3} + T^{4} \)
$67$ \( 76 - 74 T - 89 T^{2} - 9 T^{3} + T^{4} \)
$71$ \( 1280 + 480 T - 60 T^{2} - 10 T^{3} + T^{4} \)
$73$ \( ( -76 + 4 T + T^{2} )^{2} \)
$79$ \( 9076 - 932 T - 201 T^{2} + 7 T^{3} + T^{4} \)
$83$ \( ( -8 + T )^{4} \)
$89$ \( -6724 - 1178 T + 79 T^{2} + 23 T^{3} + T^{4} \)
$97$ \( 1136 + 328 T - 136 T^{2} + 2 T^{3} + T^{4} \)
show more
show less