Properties

Label 1815.2.c.c
Level $1815$
Weight $2$
Character orbit 1815.c
Analytic conductor $14.493$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1815.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.4928479669\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} -\beta_{1} q^{3} -3 q^{4} + ( -1 - 2 \beta_{1} ) q^{5} -\beta_{3} q^{6} + \beta_{2} q^{8} - q^{9} +O(q^{10})\) \( q -\beta_{2} q^{2} -\beta_{1} q^{3} -3 q^{4} + ( -1 - 2 \beta_{1} ) q^{5} -\beta_{3} q^{6} + \beta_{2} q^{8} - q^{9} + ( \beta_{2} - 2 \beta_{3} ) q^{10} + 3 \beta_{1} q^{12} -2 \beta_{2} q^{13} + ( -2 + \beta_{1} ) q^{15} - q^{16} + 2 \beta_{2} q^{17} + \beta_{2} q^{18} + ( 3 + 6 \beta_{1} ) q^{20} + 4 \beta_{1} q^{23} + \beta_{3} q^{24} + ( -3 + 4 \beta_{1} ) q^{25} -10 q^{26} + \beta_{1} q^{27} -4 \beta_{3} q^{29} + ( 2 \beta_{2} + \beta_{3} ) q^{30} + 3 \beta_{2} q^{32} + 10 q^{34} + 3 q^{36} -8 \beta_{1} q^{37} -2 \beta_{3} q^{39} + ( -\beta_{2} + 2 \beta_{3} ) q^{40} -4 \beta_{3} q^{41} + 4 \beta_{2} q^{43} + ( 1 + 2 \beta_{1} ) q^{45} + 4 \beta_{3} q^{46} -12 \beta_{1} q^{47} + \beta_{1} q^{48} + 7 q^{49} + ( 3 \beta_{2} + 4 \beta_{3} ) q^{50} + 2 \beta_{3} q^{51} + 6 \beta_{2} q^{52} + 4 \beta_{1} q^{53} + \beta_{3} q^{54} + 20 \beta_{1} q^{58} + ( 6 - 3 \beta_{1} ) q^{60} + 13 q^{64} + ( 2 \beta_{2} - 4 \beta_{3} ) q^{65} -12 \beta_{1} q^{67} -6 \beta_{2} q^{68} + 4 q^{69} + 12 q^{71} -\beta_{2} q^{72} + 6 \beta_{2} q^{73} -8 \beta_{3} q^{74} + ( 4 + 3 \beta_{1} ) q^{75} + 10 \beta_{1} q^{78} + ( 1 + 2 \beta_{1} ) q^{80} + q^{81} + 20 \beta_{1} q^{82} + ( -2 \beta_{2} + 4 \beta_{3} ) q^{85} + 20 q^{86} + 4 \beta_{2} q^{87} -6 q^{89} + ( -\beta_{2} + 2 \beta_{3} ) q^{90} -12 \beta_{1} q^{92} -12 \beta_{3} q^{94} + 3 \beta_{3} q^{96} -8 \beta_{1} q^{97} -7 \beta_{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{4} - 4q^{5} - 4q^{9} + O(q^{10}) \) \( 4q - 12q^{4} - 4q^{5} - 4q^{9} - 8q^{15} - 4q^{16} + 12q^{20} - 12q^{25} - 40q^{26} + 40q^{34} + 12q^{36} + 4q^{45} + 28q^{49} + 24q^{60} + 52q^{64} + 16q^{69} + 48q^{71} + 16q^{75} + 4q^{80} + 4q^{81} + 80q^{86} - 24q^{89} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1815\mathbb{Z}\right)^\times\).

\(n\) \(727\) \(1211\) \(1696\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
364.1
0.618034i
1.61803i
1.61803i
0.618034i
2.23607i 1.00000i −3.00000 −1.00000 2.00000i −2.23607 0 2.23607i −1.00000 −4.47214 + 2.23607i
364.2 2.23607i 1.00000i −3.00000 −1.00000 + 2.00000i 2.23607 0 2.23607i −1.00000 4.47214 + 2.23607i
364.3 2.23607i 1.00000i −3.00000 −1.00000 2.00000i 2.23607 0 2.23607i −1.00000 4.47214 2.23607i
364.4 2.23607i 1.00000i −3.00000 −1.00000 + 2.00000i −2.23607 0 2.23607i −1.00000 −4.47214 2.23607i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.b odd 2 1 inner
55.d 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.c.c 4
5.b even 2 1 inner 1815.2.c.c 4
5.c odd 4 1 9075.2.a.bj 2
5.c odd 4 1 9075.2.a.bq 2
11.b odd 2 1 inner 1815.2.c.c 4
55.d odd 2 1 inner 1815.2.c.c 4
55.e even 4 1 9075.2.a.bj 2
55.e even 4 1 9075.2.a.bq 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.c.c 4 1.a even 1 1 trivial
1815.2.c.c 4 5.b even 2 1 inner
1815.2.c.c 4 11.b odd 2 1 inner
1815.2.c.c 4 55.d odd 2 1 inner
9075.2.a.bj 2 5.c odd 4 1
9075.2.a.bj 2 55.e even 4 1
9075.2.a.bq 2 5.c odd 4 1
9075.2.a.bq 2 55.e even 4 1

Hecke kernels

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

\( T_{2}^{2} + 5 \)
\( T_{19} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 5 + T^{2} )^{2} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( ( 5 + 2 T + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( ( 20 + T^{2} )^{2} \)
$17$ \( ( 20 + T^{2} )^{2} \)
$19$ \( T^{4} \)
$23$ \( ( 16 + T^{2} )^{2} \)
$29$ \( ( -80 + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( ( 64 + T^{2} )^{2} \)
$41$ \( ( -80 + T^{2} )^{2} \)
$43$ \( ( 80 + T^{2} )^{2} \)
$47$ \( ( 144 + T^{2} )^{2} \)
$53$ \( ( 16 + T^{2} )^{2} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( ( 144 + T^{2} )^{2} \)
$71$ \( ( -12 + T )^{4} \)
$73$ \( ( 180 + T^{2} )^{2} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( ( 6 + T )^{4} \)
$97$ \( ( 64 + T^{2} )^{2} \)
show more
show less