Properties

Label 1815.2.c.g
Level $1815$
Weight $2$
Character orbit 1815.c
Analytic conductor $14.493$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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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: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

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

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
0.258819 0.965926i
−0.965926 + 0.258819i
0.965926 + 0.258819i
−0.258819 0.965926i
0.965926 0.258819i
−0.258819 + 0.965926i
0.258819 + 0.965926i
−0.965926 0.258819i
1.41421i 1.00000i 0 −2.22474 0.224745i −1.41421 1.73205i 2.82843i −1.00000 −0.317837 + 3.14626i
364.2 1.41421i 1.00000i 0 0.224745 + 2.22474i −1.41421 1.73205i 2.82843i −1.00000 3.14626 0.317837i
364.3 1.41421i 1.00000i 0 −2.22474 + 0.224745i 1.41421 1.73205i 2.82843i −1.00000 0.317837 + 3.14626i
364.4 1.41421i 1.00000i 0 0.224745 2.22474i 1.41421 1.73205i 2.82843i −1.00000 −3.14626 0.317837i
364.5 1.41421i 1.00000i 0 −2.22474 0.224745i 1.41421 1.73205i 2.82843i −1.00000 0.317837 3.14626i
364.6 1.41421i 1.00000i 0 0.224745 + 2.22474i 1.41421 1.73205i 2.82843i −1.00000 −3.14626 + 0.317837i
364.7 1.41421i 1.00000i 0 −2.22474 + 0.224745i −1.41421 1.73205i 2.82843i −1.00000 −0.317837 3.14626i
364.8 1.41421i 1.00000i 0 0.224745 2.22474i −1.41421 1.73205i 2.82843i −1.00000 3.14626 + 0.317837i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 364.8
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.g 8
5.b even 2 1 inner 1815.2.c.g 8
5.c odd 4 1 9075.2.a.cr 4
5.c odd 4 1 9075.2.a.cy 4
11.b odd 2 1 inner 1815.2.c.g 8
55.d odd 2 1 inner 1815.2.c.g 8
55.e even 4 1 9075.2.a.cr 4
55.e even 4 1 9075.2.a.cy 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.c.g 8 1.a even 1 1 trivial
1815.2.c.g 8 5.b even 2 1 inner
1815.2.c.g 8 11.b odd 2 1 inner
1815.2.c.g 8 55.d odd 2 1 inner
9075.2.a.cr 4 5.c odd 4 1
9075.2.a.cr 4 55.e even 4 1
9075.2.a.cy 4 5.c odd 4 1
9075.2.a.cy 4 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 \(S_{2}^{\mathrm{new}}(1815, [\chi])\):

\( T_{2}^{2} + 2 \)
\( T_{19}^{2} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 + T^{2} )^{4} \)
$3$ \( ( 1 + T^{2} )^{4} \)
$5$ \( ( 25 + 20 T + 8 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$7$ \( ( 3 + T^{2} )^{4} \)
$11$ \( T^{8} \)
$13$ \( ( 16 + 40 T^{2} + T^{4} )^{2} \)
$17$ \( ( 100 + 28 T^{2} + T^{4} )^{2} \)
$19$ \( ( -3 + T^{2} )^{4} \)
$23$ \( ( 4 + T^{2} )^{4} \)
$29$ \( ( 2116 - 100 T^{2} + T^{4} )^{2} \)
$31$ \( ( -15 + 6 T + T^{2} )^{4} \)
$37$ \( ( 529 + 50 T^{2} + T^{4} )^{2} \)
$41$ \( ( 400 - 88 T^{2} + T^{4} )^{2} \)
$43$ \( ( 32 + T^{2} )^{4} \)
$47$ \( ( 54 + T^{2} )^{4} \)
$53$ \( ( 100 + T^{2} )^{4} \)
$59$ \( ( -150 + T^{2} )^{4} \)
$61$ \( ( 2025 - 198 T^{2} + T^{4} )^{2} \)
$67$ \( ( 3249 + 210 T^{2} + T^{4} )^{2} \)
$71$ \( ( 30 + 12 T + T^{2} )^{4} \)
$73$ \( ( 2025 + 198 T^{2} + T^{4} )^{2} \)
$79$ \( ( 4761 - 150 T^{2} + T^{4} )^{2} \)
$83$ \( ( 162 + T^{2} )^{4} \)
$89$ \( ( -18 - 12 T + T^{2} )^{4} \)
$97$ \( ( 25 + T^{2} )^{4} \)
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