Properties

Label 1815.2.c.f
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: 8.0.49787136.1
Defining polynomial: \(x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} -\beta_{3} q^{3} + \beta_{4} q^{4} + ( 2 - \beta_{3} ) q^{5} -\beta_{7} q^{6} -2 \beta_{6} q^{7} + ( -\beta_{2} - \beta_{6} ) q^{8} - q^{9} +O(q^{10})\) \( q + \beta_{6} q^{2} -\beta_{3} q^{3} + \beta_{4} q^{4} + ( 2 - \beta_{3} ) q^{5} -\beta_{7} q^{6} -2 \beta_{6} q^{7} + ( -\beta_{2} - \beta_{6} ) q^{8} - q^{9} + ( 2 \beta_{6} - \beta_{7} ) q^{10} + ( -\beta_{1} + \beta_{3} ) q^{12} -2 \beta_{6} q^{13} + ( 4 - 2 \beta_{4} ) q^{14} + ( -1 - 2 \beta_{3} ) q^{15} + ( 1 + \beta_{4} ) q^{16} + ( 3 \beta_{2} - \beta_{6} ) q^{17} -\beta_{6} q^{18} + ( 2 \beta_{5} + 2 \beta_{7} ) q^{19} + ( -\beta_{1} + \beta_{3} + 2 \beta_{4} ) q^{20} + 2 \beta_{7} q^{21} + ( -2 \beta_{1} + 5 \beta_{3} ) q^{23} + ( -\beta_{5} + \beta_{7} ) q^{24} + ( 3 - 4 \beta_{3} ) q^{25} + ( 4 - 2 \beta_{4} ) q^{26} + \beta_{3} q^{27} + ( 2 \beta_{2} + 6 \beta_{6} ) q^{28} + 2 \beta_{5} q^{29} + ( -\beta_{6} - 2 \beta_{7} ) q^{30} + ( -1 + 2 \beta_{4} ) q^{31} + ( -3 \beta_{2} - 4 \beta_{6} ) q^{32} + ( 5 - \beta_{4} ) q^{34} + ( -4 \beta_{6} + 2 \beta_{7} ) q^{35} -\beta_{4} q^{36} + ( -2 \beta_{1} - 4 \beta_{3} ) q^{37} + ( 2 \beta_{1} - 8 \beta_{3} ) q^{38} + 2 \beta_{7} q^{39} + ( -2 \beta_{2} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{40} + ( -4 \beta_{5} + 4 \beta_{7} ) q^{41} + ( 2 \beta_{1} - 6 \beta_{3} ) q^{42} -2 \beta_{2} q^{43} + ( -2 + \beta_{3} ) q^{45} + ( -2 \beta_{5} + 9 \beta_{7} ) q^{46} + ( -2 \beta_{1} - \beta_{3} ) q^{47} -\beta_{1} q^{48} + ( -1 + 4 \beta_{4} ) q^{49} + ( 3 \beta_{6} - 4 \beta_{7} ) q^{50} + ( 3 \beta_{5} + \beta_{7} ) q^{51} + ( 2 \beta_{2} + 6 \beta_{6} ) q^{52} + 5 \beta_{3} q^{53} + \beta_{7} q^{54} + ( -2 + 2 \beta_{4} ) q^{56} + ( -2 \beta_{2} + 2 \beta_{6} ) q^{57} -2 \beta_{3} q^{58} + 2 \beta_{4} q^{59} + ( -2 \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{60} + ( -5 \beta_{5} + 5 \beta_{7} ) q^{61} + ( -2 \beta_{2} - 7 \beta_{6} ) q^{62} + 2 \beta_{6} q^{63} + ( 7 - 2 \beta_{4} ) q^{64} + ( -4 \beta_{6} + 2 \beta_{7} ) q^{65} + ( 6 \beta_{1} - 2 \beta_{3} ) q^{67} + ( 7 \beta_{2} + 6 \beta_{6} ) q^{68} + ( 3 - 2 \beta_{4} ) q^{69} + ( 8 + 2 \beta_{1} - 6 \beta_{3} - 4 \beta_{4} ) q^{70} + ( -2 - 4 \beta_{4} ) q^{71} + ( \beta_{2} + \beta_{6} ) q^{72} + ( -8 \beta_{2} - 4 \beta_{6} ) q^{73} -2 \beta_{5} q^{74} + ( -4 - 3 \beta_{3} ) q^{75} + ( 6 \beta_{5} - 8 \beta_{7} ) q^{76} + ( 2 \beta_{1} - 6 \beta_{3} ) q^{78} + ( -\beta_{5} - \beta_{7} ) q^{79} + ( 2 - \beta_{1} + 2 \beta_{4} ) q^{80} + q^{81} + ( 4 \beta_{1} - 8 \beta_{3} ) q^{82} + ( 6 \beta_{2} + 2 \beta_{6} ) q^{83} + ( 2 \beta_{5} - 6 \beta_{7} ) q^{84} + ( 6 \beta_{2} + 3 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{85} -2 q^{86} -2 \beta_{2} q^{87} + ( -6 - 6 \beta_{4} ) q^{89} + ( -2 \beta_{6} + \beta_{7} ) q^{90} + ( -8 + 4 \beta_{4} ) q^{91} + ( 5 \beta_{1} - 15 \beta_{3} ) q^{92} + ( -2 \beta_{1} + 3 \beta_{3} ) q^{93} + ( -2 \beta_{5} + 3 \beta_{7} ) q^{94} + ( -2 \beta_{2} + 4 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{95} + ( -3 \beta_{5} + 4 \beta_{7} ) q^{96} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{97} + ( -4 \beta_{2} - 13 \beta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{4} + 16q^{5} - 8q^{9} + O(q^{10}) \) \( 8q - 4q^{4} + 16q^{5} - 8q^{9} + 40q^{14} - 8q^{15} + 4q^{16} - 8q^{20} + 24q^{25} + 40q^{26} - 16q^{31} + 44q^{34} + 4q^{36} - 16q^{45} - 24q^{49} - 24q^{56} - 8q^{59} + 4q^{60} + 64q^{64} + 32q^{69} + 80q^{70} - 32q^{75} + 8q^{80} + 8q^{81} - 16q^{86} - 24q^{89} - 80q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{7} + 5 \nu^{5} + 15 \nu^{3} + 42 \nu \)\()/20\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{6} + 5 \nu^{4} + 15 \nu^{2} + 8 \)\()/20\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{7} - 5 \nu^{5} + 5 \nu^{3} - 16 \nu \)\()/40\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{6} - 3 \nu^{4} - \nu^{2} - 8 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} - 3 \nu^{5} - 5 \nu^{3} - 4 \nu \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( 7 \nu^{6} + 5 \nu^{4} + 15 \nu^{2} + 44 \)\()/20\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} - \nu^{5} - 3 \nu^{3} - 6 \nu \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} - \beta_{3} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} + \beta_{4} + 2 \beta_{2} - 1\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{7} - \beta_{5} + 5 \beta_{3}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-2 \beta_{6} - 3 \beta_{4} + \beta_{2} - 2\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(5 \beta_{7} - 6 \beta_{5} - 6 \beta_{3} + \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(5 \beta_{6} - 5 \beta_{2} - 9\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-10 \beta_{7} + 3 \beta_{5} - 3 \beta_{3} - 7 \beta_{1}\)\()/2\)

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
0.228425 1.39564i
−0.228425 + 1.39564i
1.09445 + 0.895644i
−1.09445 0.895644i
−1.09445 + 0.895644i
1.09445 0.895644i
−0.228425 1.39564i
0.228425 + 1.39564i
2.18890i 1.00000i −2.79129 2.00000 1.00000i −2.18890 4.37780i 1.73205i −1.00000 −2.18890 4.37780i
364.2 2.18890i 1.00000i −2.79129 2.00000 + 1.00000i 2.18890 4.37780i 1.73205i −1.00000 2.18890 4.37780i
364.3 0.456850i 1.00000i 1.79129 2.00000 1.00000i −0.456850 0.913701i 1.73205i −1.00000 −0.456850 0.913701i
364.4 0.456850i 1.00000i 1.79129 2.00000 + 1.00000i 0.456850 0.913701i 1.73205i −1.00000 0.456850 0.913701i
364.5 0.456850i 1.00000i 1.79129 2.00000 1.00000i 0.456850 0.913701i 1.73205i −1.00000 0.456850 + 0.913701i
364.6 0.456850i 1.00000i 1.79129 2.00000 + 1.00000i −0.456850 0.913701i 1.73205i −1.00000 −0.456850 + 0.913701i
364.7 2.18890i 1.00000i −2.79129 2.00000 1.00000i 2.18890 4.37780i 1.73205i −1.00000 2.18890 + 4.37780i
364.8 2.18890i 1.00000i −2.79129 2.00000 + 1.00000i −2.18890 4.37780i 1.73205i −1.00000 −2.18890 + 4.37780i
\(n\): 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.f 8
5.b even 2 1 inner 1815.2.c.f 8
5.c odd 4 1 9075.2.a.cs 4
5.c odd 4 1 9075.2.a.cz 4
11.b odd 2 1 inner 1815.2.c.f 8
55.d odd 2 1 inner 1815.2.c.f 8
55.e even 4 1 9075.2.a.cs 4
55.e even 4 1 9075.2.a.cz 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.c.f 8 1.a even 1 1 trivial
1815.2.c.f 8 5.b even 2 1 inner
1815.2.c.f 8 11.b odd 2 1 inner
1815.2.c.f 8 55.d odd 2 1 inner
9075.2.a.cs 4 5.c odd 4 1
9075.2.a.cs 4 55.e even 4 1
9075.2.a.cz 4 5.c odd 4 1
9075.2.a.cz 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 \(S_{2}^{\mathrm{new}}(1815, [\chi])\):

\( T_{2}^{4} + 5 T_{2}^{2} + 1 \)
\( T_{19}^{2} - 28 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 5 T^{2} + T^{4} )^{2} \)
$3$ \( ( 1 + T^{2} )^{4} \)
$5$ \( ( 5 - 4 T + T^{2} )^{4} \)
$7$ \( ( 16 + 20 T^{2} + T^{4} )^{2} \)
$11$ \( T^{8} \)
$13$ \( ( 16 + 20 T^{2} + T^{4} )^{2} \)
$17$ \( ( 625 + 62 T^{2} + T^{4} )^{2} \)
$19$ \( ( -28 + T^{2} )^{4} \)
$23$ \( ( 25 + 74 T^{2} + T^{4} )^{2} \)
$29$ \( ( 16 - 20 T^{2} + T^{4} )^{2} \)
$31$ \( ( -17 + 4 T + T^{2} )^{4} \)
$37$ \( ( 16 + 92 T^{2} + T^{4} )^{2} \)
$41$ \( ( -48 + T^{2} )^{4} \)
$43$ \( ( 16 + 20 T^{2} + T^{4} )^{2} \)
$47$ \( ( 289 + 50 T^{2} + T^{4} )^{2} \)
$53$ \( ( 25 + T^{2} )^{4} \)
$59$ \( ( -20 + 2 T + T^{2} )^{4} \)
$61$ \( ( -75 + T^{2} )^{4} \)
$67$ \( ( 35344 + 380 T^{2} + T^{4} )^{2} \)
$71$ \( ( -84 + T^{2} )^{4} \)
$73$ \( ( 6400 + 272 T^{2} + T^{4} )^{2} \)
$79$ \( ( -7 + T^{2} )^{4} \)
$83$ \( ( 400 + 152 T^{2} + T^{4} )^{2} \)
$89$ \( ( -180 + 6 T + T^{2} )^{4} \)
$97$ \( ( 400 + 44 T^{2} + T^{4} )^{2} \)
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