Properties

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

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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: 4.4.8112.1
Defining polynomial: \(x^{4} - 5 x^{2} + 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} - q^{3} + ( 2 - \beta_{3} ) q^{4} + q^{5} + \beta_{2} q^{6} + \beta_{1} q^{7} + ( \beta_{1} - 3 \beta_{2} ) q^{8} + q^{9} +O(q^{10})\) \( q -\beta_{2} q^{2} - q^{3} + ( 2 - \beta_{3} ) q^{4} + q^{5} + \beta_{2} q^{6} + \beta_{1} q^{7} + ( \beta_{1} - 3 \beta_{2} ) q^{8} + q^{9} -\beta_{2} q^{10} + ( -2 + \beta_{3} ) q^{12} + ( \beta_{1} + 2 \beta_{2} ) q^{13} + ( 1 - \beta_{3} ) q^{14} - q^{15} + ( 9 - 2 \beta_{3} ) q^{16} + \beta_{1} q^{17} -\beta_{2} q^{18} -2 \beta_{2} q^{19} + ( 2 - \beta_{3} ) q^{20} -\beta_{1} q^{21} + ( -2 + 2 \beta_{3} ) q^{23} + ( -\beta_{1} + 3 \beta_{2} ) q^{24} + q^{25} + ( -7 + \beta_{3} ) q^{26} - q^{27} + ( -\beta_{1} - 4 \beta_{2} ) q^{28} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{29} + \beta_{2} q^{30} + 4 q^{31} -9 \beta_{2} q^{32} + ( 1 - \beta_{3} ) q^{34} + \beta_{1} q^{35} + ( 2 - \beta_{3} ) q^{36} -2 \beta_{3} q^{37} + ( 8 - 2 \beta_{3} ) q^{38} + ( -\beta_{1} - 2 \beta_{2} ) q^{39} + ( \beta_{1} - 3 \beta_{2} ) q^{40} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{41} + ( -1 + \beta_{3} ) q^{42} + ( \beta_{1} + 4 \beta_{2} ) q^{43} + q^{45} + ( -2 \beta_{1} + 8 \beta_{2} ) q^{46} + ( -9 + 2 \beta_{3} ) q^{48} + ( 3 + 2 \beta_{3} ) q^{49} -\beta_{2} q^{50} -\beta_{1} q^{51} + ( -3 \beta_{1} + 6 \beta_{2} ) q^{52} + ( 4 + 2 \beta_{3} ) q^{53} + \beta_{2} q^{54} + ( 13 - \beta_{3} ) q^{56} + 2 \beta_{2} q^{57} + ( -10 + 4 \beta_{3} ) q^{58} + ( 2 - 2 \beta_{3} ) q^{59} + ( -2 + \beta_{3} ) q^{60} + 2 \beta_{1} q^{61} -4 \beta_{2} q^{62} + \beta_{1} q^{63} + ( 18 - 5 \beta_{3} ) q^{64} + ( \beta_{1} + 2 \beta_{2} ) q^{65} + ( -6 - 2 \beta_{3} ) q^{67} + ( -\beta_{1} - 4 \beta_{2} ) q^{68} + ( 2 - 2 \beta_{3} ) q^{69} + ( 1 - \beta_{3} ) q^{70} + ( -2 + 2 \beta_{3} ) q^{71} + ( \beta_{1} - 3 \beta_{2} ) q^{72} + ( -\beta_{1} + 2 \beta_{2} ) q^{73} + ( 2 \beta_{1} - 6 \beta_{2} ) q^{74} - q^{75} + ( 2 \beta_{1} - 10 \beta_{2} ) q^{76} + ( 7 - \beta_{3} ) q^{78} -2 \beta_{2} q^{79} + ( 9 - 2 \beta_{3} ) q^{80} + q^{81} + 6 q^{82} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{83} + ( \beta_{1} + 4 \beta_{2} ) q^{84} + \beta_{1} q^{85} + ( -15 + 3 \beta_{3} ) q^{86} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{87} + 6 q^{89} -\beta_{2} q^{90} + ( 8 + 4 \beta_{3} ) q^{91} + ( -30 + 6 \beta_{3} ) q^{92} -4 q^{93} -2 \beta_{2} q^{95} + 9 \beta_{2} q^{96} + ( 12 + 2 \beta_{3} ) q^{97} + ( -2 \beta_{1} + 3 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} + 8q^{4} + 4q^{5} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} + 8q^{4} + 4q^{5} + 4q^{9} - 8q^{12} + 4q^{14} - 4q^{15} + 36q^{16} + 8q^{20} - 8q^{23} + 4q^{25} - 28q^{26} - 4q^{27} + 16q^{31} + 4q^{34} + 8q^{36} + 32q^{38} - 4q^{42} + 4q^{45} - 36q^{48} + 12q^{49} + 16q^{53} + 52q^{56} - 40q^{58} + 8q^{59} - 8q^{60} + 72q^{64} - 24q^{67} + 8q^{69} + 4q^{70} - 8q^{71} - 4q^{75} + 28q^{78} + 36q^{80} + 4q^{81} + 24q^{82} - 60q^{86} + 24q^{89} + 32q^{91} - 120q^{92} - 16q^{93} + 48q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 5 x^{2} + 3\):

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

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
−0.835000
2.07431
−2.07431
0.835000
−2.75782 −1.00000 5.60555 1.00000 2.75782 −1.67000 −9.94345 1.00000 −2.75782
1.2 −0.628052 −1.00000 −1.60555 1.00000 0.628052 4.14863 2.26447 1.00000 −0.628052
1.3 0.628052 −1.00000 −1.60555 1.00000 −0.628052 −4.14863 −2.26447 1.00000 0.628052
1.4 2.75782 −1.00000 5.60555 1.00000 −2.75782 1.67000 9.94345 1.00000 2.75782
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1815.2.a.s 4
3.b odd 2 1 5445.2.a.bo 4
5.b even 2 1 9075.2.a.dc 4
11.b odd 2 1 inner 1815.2.a.s 4
33.d even 2 1 5445.2.a.bo 4
55.d odd 2 1 9075.2.a.dc 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.a.s 4 1.a even 1 1 trivial
1815.2.a.s 4 11.b odd 2 1 inner
5445.2.a.bo 4 3.b odd 2 1
5445.2.a.bo 4 33.d even 2 1
9075.2.a.dc 4 5.b even 2 1
9075.2.a.dc 4 55.d odd 2 1

Hecke kernels

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

\( T_{2}^{4} - 8 T_{2}^{2} + 3 \)
\( T_{7}^{4} - 20 T_{7}^{2} + 48 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 3 - 8 T^{2} + T^{4} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( ( -1 + T )^{4} \)
$7$ \( 48 - 20 T^{2} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( 432 - 44 T^{2} + T^{4} \)
$17$ \( 48 - 20 T^{2} + T^{4} \)
$19$ \( 48 - 32 T^{2} + T^{4} \)
$23$ \( ( -48 + 4 T + T^{2} )^{2} \)
$29$ \( 3888 - 128 T^{2} + T^{4} \)
$31$ \( ( -4 + T )^{4} \)
$37$ \( ( -52 + T^{2} )^{2} \)
$41$ \( 432 - 96 T^{2} + T^{4} \)
$43$ \( 3888 - 132 T^{2} + T^{4} \)
$47$ \( T^{4} \)
$53$ \( ( -36 - 8 T + T^{2} )^{2} \)
$59$ \( ( -48 - 4 T + T^{2} )^{2} \)
$61$ \( 768 - 80 T^{2} + T^{4} \)
$67$ \( ( -16 + 12 T + T^{2} )^{2} \)
$71$ \( ( -48 + 4 T + T^{2} )^{2} \)
$73$ \( 432 - 60 T^{2} + T^{4} \)
$79$ \( 48 - 32 T^{2} + T^{4} \)
$83$ \( 48 - 188 T^{2} + T^{4} \)
$89$ \( ( -6 + T )^{4} \)
$97$ \( ( 92 - 24 T + T^{2} )^{2} \)
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