Properties

Label 1815.2.a.r
Level $1815$
Weight $2$
Character orbit 1815.a
Self dual yes
Analytic conductor $14.493$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1815.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4928479669\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.5725.1
Defining polynomial: \(x^{4} - x^{3} - 8 x^{2} + 6 x + 11\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 8 x^{2} + 6 x + 11\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 5 \nu + 1 \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{2} - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 4\)
\(\nu^{3}\)\(=\)\(3 \beta_{2} + 5 \beta_{1} - 1\)

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
−0.933531
−2.48008
2.55157
1.86205
−2.55157 −1.00000 4.51049 −1.00000 2.55157 4.19499 −6.40567 1.00000 2.55157
1.2 −1.86205 −1.00000 1.46722 −1.00000 1.86205 −2.63089 0.992053 1.00000 1.86205
1.3 0.933531 −1.00000 −1.12852 −1.00000 −0.933531 2.04108 −2.92057 1.00000 −0.933531
1.4 2.48008 −1.00000 4.15081 −1.00000 −2.48008 4.39482 5.33418 1.00000 −2.48008
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(11\) \(-1\)

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 1815.2.a.r 4
3.b odd 2 1 5445.2.a.br 4
5.b even 2 1 9075.2.a.dg 4
11.b odd 2 1 1815.2.a.v 4
11.d odd 10 2 165.2.m.b 8
33.d even 2 1 5445.2.a.bk 4
33.f even 10 2 495.2.n.b 8
55.d odd 2 1 9075.2.a.cq 4
55.h odd 10 2 825.2.n.i 8
55.l even 20 4 825.2.bx.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.m.b 8 11.d odd 10 2
495.2.n.b 8 33.f even 10 2
825.2.n.i 8 55.h odd 10 2
825.2.bx.g 16 55.l even 20 4
1815.2.a.r 4 1.a even 1 1 trivial
1815.2.a.v 4 11.b odd 2 1
5445.2.a.bk 4 33.d even 2 1
5445.2.a.br 4 3.b odd 2 1
9075.2.a.cq 4 55.d odd 2 1
9075.2.a.dg 4 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1815))\):

\( T_{2}^{4} + T_{2}^{3} - 8 T_{2}^{2} - 6 T_{2} + 11 \)
\( T_{7}^{4} - 8 T_{7}^{3} + 8 T_{7}^{2} + 57 T_{7} - 99 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 11 - 6 T - 8 T^{2} + T^{3} + T^{4} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( ( 1 + T )^{4} \)
$7$ \( -99 + 57 T + 8 T^{2} - 8 T^{3} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( 99 - 150 T + 76 T^{2} - 15 T^{3} + T^{4} \)
$17$ \( 99 + 102 T - 20 T^{2} - 6 T^{3} + T^{4} \)
$19$ \( 9 + 12 T - 10 T^{2} - T^{3} + T^{4} \)
$23$ \( 341 - 29 T - 39 T^{2} + T^{3} + T^{4} \)
$29$ \( -619 - 268 T + 48 T^{2} + 17 T^{3} + T^{4} \)
$31$ \( -25 + 125 T + 35 T^{2} - 15 T^{3} + T^{4} \)
$37$ \( 1151 - 21 T - 83 T^{2} + T^{3} + T^{4} \)
$41$ \( 11 - 37 T + 38 T^{2} - 12 T^{3} + T^{4} \)
$43$ \( 9 - 57 T + 50 T^{2} - 14 T^{3} + T^{4} \)
$47$ \( -3509 + 1081 T - 34 T^{2} - 14 T^{3} + T^{4} \)
$53$ \( 1711 + 83 T - 82 T^{2} - 2 T^{3} + T^{4} \)
$59$ \( -99 - 189 T + T^{2} + 11 T^{3} + T^{4} \)
$61$ \( 1111 - 106 T - 88 T^{2} + T^{3} + T^{4} \)
$67$ \( 1049 + 180 T - 74 T^{2} - 5 T^{3} + T^{4} \)
$71$ \( -821 - 699 T - 145 T^{2} + 3 T^{3} + T^{4} \)
$73$ \( -151 - 3350 T + 674 T^{2} - 45 T^{3} + T^{4} \)
$79$ \( -841 - 970 T - 229 T^{2} + T^{4} \)
$83$ \( -1975 - 625 T + 5 T^{2} + 15 T^{3} + T^{4} \)
$89$ \( 99 + 156 T - 55 T^{2} - 2 T^{3} + T^{4} \)
$97$ \( 891 + 819 T + 232 T^{2} + 26 T^{3} + T^{4} \)
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