Properties

Label 1815.4.a.j
Level $1815$
Weight $4$
Character orbit 1815.a
Self dual yes
Analytic conductor $107.088$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1815.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(107.088466660\)
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: \( 2^{3}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 12\sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} -8 q^{4} -5 q^{5} + \beta q^{7} + 9 q^{9} +O(q^{10})\) \( q + 3 q^{3} -8 q^{4} -5 q^{5} + \beta q^{7} + 9 q^{9} -24 q^{12} + \beta q^{13} -15 q^{15} + 64 q^{16} -2 \beta q^{17} -3 \beta q^{19} + 40 q^{20} + 3 \beta q^{21} -72 q^{23} + 25 q^{25} + 27 q^{27} -8 \beta q^{28} -11 \beta q^{29} -20 q^{31} -5 \beta q^{35} -72 q^{36} + 214 q^{37} + 3 \beta q^{39} -\beta q^{41} + 9 \beta q^{43} -45 q^{45} -264 q^{47} + 192 q^{48} + 377 q^{49} -6 \beta q^{51} -8 \beta q^{52} + 78 q^{53} -9 \beta q^{57} + 480 q^{59} + 120 q^{60} + 4 \beta q^{61} + 9 \beta q^{63} -512 q^{64} -5 \beta q^{65} -524 q^{67} + 16 \beta q^{68} -216 q^{69} + 492 q^{71} + 35 \beta q^{73} + 75 q^{75} + 24 \beta q^{76} -11 \beta q^{79} -320 q^{80} + 81 q^{81} -22 \beta q^{83} -24 \beta q^{84} + 10 \beta q^{85} -33 \beta q^{87} -1206 q^{89} + 720 q^{91} + 576 q^{92} -60 q^{93} + 15 \beta q^{95} -1186 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{3} - 16q^{4} - 10q^{5} + 18q^{9} + O(q^{10}) \) \( 2q + 6q^{3} - 16q^{4} - 10q^{5} + 18q^{9} - 48q^{12} - 30q^{15} + 128q^{16} + 80q^{20} - 144q^{23} + 50q^{25} + 54q^{27} - 40q^{31} - 144q^{36} + 428q^{37} - 90q^{45} - 528q^{47} + 384q^{48} + 754q^{49} + 156q^{53} + 960q^{59} + 240q^{60} - 1024q^{64} - 1048q^{67} - 432q^{69} + 984q^{71} + 150q^{75} - 640q^{80} + 162q^{81} - 2412q^{89} + 1440q^{91} + 1152q^{92} - 120q^{93} - 2372q^{97} + O(q^{100}) \)

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
0 3.00000 −8.00000 −5.00000 0 −26.8328 0 9.00000 0
1.2 0 3.00000 −8.00000 −5.00000 0 26.8328 0 9.00000 0
\(n\): 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 1815.4.a.j 2
11.b odd 2 1 inner 1815.4.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.4.a.j 2 1.a even 1 1 trivial
1815.4.a.j 2 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1815))\):

\( T_{2} \)
\( T_{7}^{2} - 720 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -3 + T )^{2} \)
$5$ \( ( 5 + T )^{2} \)
$7$ \( -720 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( -720 + T^{2} \)
$17$ \( -2880 + T^{2} \)
$19$ \( -6480 + T^{2} \)
$23$ \( ( 72 + T )^{2} \)
$29$ \( -87120 + T^{2} \)
$31$ \( ( 20 + T )^{2} \)
$37$ \( ( -214 + T )^{2} \)
$41$ \( -720 + T^{2} \)
$43$ \( -58320 + T^{2} \)
$47$ \( ( 264 + T )^{2} \)
$53$ \( ( -78 + T )^{2} \)
$59$ \( ( -480 + T )^{2} \)
$61$ \( -11520 + T^{2} \)
$67$ \( ( 524 + T )^{2} \)
$71$ \( ( -492 + T )^{2} \)
$73$ \( -882000 + T^{2} \)
$79$ \( -87120 + T^{2} \)
$83$ \( -348480 + T^{2} \)
$89$ \( ( 1206 + T )^{2} \)
$97$ \( ( 1186 + T )^{2} \)
show more
show less