Properties

Label 1805.1.o.b
Level $1805$
Weight $1$
Character orbit 1805.o
Analytic conductor $0.901$
Analytic rank $0$
Dimension $12$
Projective image $D_{4}$
CM discriminant -95
Inner twists $24$

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Newspace parameters

Level: \( N \) \(=\) \( 1805 = 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1805.o (of order \(18\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.900812347803\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: 12.0.101559956668416.2
Defining polynomial: \(x^{12} + 8 x^{6} + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.475.1
Artin image: $C_9\times D_8$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{72} - \cdots)\)

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{5} + \beta_{11} ) q^{2} + \beta_{5} q^{3} + ( -\beta_{4} - \beta_{10} ) q^{4} -\beta_{2} q^{5} -2 \beta_{4} q^{6} + \beta_{10} q^{9} +O(q^{10})\) \( q + ( \beta_{5} + \beta_{11} ) q^{2} + \beta_{5} q^{3} + ( -\beta_{4} - \beta_{10} ) q^{4} -\beta_{2} q^{5} -2 \beta_{4} q^{6} + \beta_{10} q^{9} + \beta_{1} q^{10} + \beta_{3} q^{12} + ( \beta_{1} + \beta_{7} ) q^{13} -\beta_{7} q^{15} + ( \beta_{2} + \beta_{8} ) q^{16} -\beta_{9} q^{18} - q^{20} + \beta_{4} q^{25} + ( -2 - 2 \beta_{6} ) q^{26} + 2 \beta_{6} q^{30} + ( -\beta_{1} - \beta_{7} ) q^{32} + \beta_{8} q^{36} + \beta_{9} q^{37} -2 q^{39} + ( 1 + \beta_{6} ) q^{45} -\beta_{1} q^{48} + \beta_{6} q^{49} -\beta_{3} q^{50} -\beta_{11} q^{52} + \beta_{7} q^{53} -\beta_{5} q^{60} + ( 1 + \beta_{6} ) q^{64} + ( -\beta_{3} - \beta_{9} ) q^{65} -\beta_{1} q^{67} -2 \beta_{8} q^{74} + \beta_{9} q^{75} + ( -2 \beta_{5} - 2 \beta_{11} ) q^{78} + ( -\beta_{4} - \beta_{10} ) q^{80} -\beta_{2} q^{81} + \beta_{11} q^{90} + 2 q^{96} + ( -\beta_{5} - \beta_{11} ) q^{97} -\beta_{5} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + O(q^{10}) \) \( 12 q - 12 q^{20} - 12 q^{26} - 12 q^{30} - 24 q^{39} + 6 q^{45} - 6 q^{49} + 6 q^{64} + 24 q^{96} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 8 x^{6} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(\beta_{4}\)\(=\)\( \nu^{4} \)\(/4\)
\(\beta_{5}\)\(=\)\( \nu^{5} \)\(/4\)
\(\beta_{6}\)\(=\)\( \nu^{6} \)\(/8\)
\(\beta_{7}\)\(=\)\( \nu^{7} \)\(/8\)
\(\beta_{8}\)\(=\)\( \nu^{8} \)\(/16\)
\(\beta_{9}\)\(=\)\( \nu^{9} \)\(/16\)
\(\beta_{10}\)\(=\)\( \nu^{10} \)\(/32\)
\(\beta_{11}\)\(=\)\( \nu^{11} \)\(/32\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)
\(\nu^{4}\)\(=\)\(4 \beta_{4}\)
\(\nu^{5}\)\(=\)\(4 \beta_{5}\)
\(\nu^{6}\)\(=\)\(8 \beta_{6}\)
\(\nu^{7}\)\(=\)\(8 \beta_{7}\)
\(\nu^{8}\)\(=\)\(16 \beta_{8}\)
\(\nu^{9}\)\(=\)\(16 \beta_{9}\)
\(\nu^{10}\)\(=\)\(32 \beta_{10}\)
\(\nu^{11}\)\(=\)\(32 \beta_{11}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1805\mathbb{Z}\right)^\times\).

\(n\) \(362\) \(1446\)
\(\chi(n)\) \(-1\) \(\beta_{4} + \beta_{10}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
299.1
1.32893 + 0.483690i
−1.32893 0.483690i
0.245576 1.39273i
−0.245576 + 1.39273i
1.08335 + 0.909039i
−1.08335 0.909039i
1.32893 0.483690i
−1.32893 + 0.483690i
1.08335 0.909039i
−1.08335 + 0.909039i
0.245576 + 1.39273i
−0.245576 1.39273i
−1.32893 + 0.483690i −0.245576 + 1.39273i 0.766044 0.642788i −0.766044 0.642788i −0.347296 1.96962i 0 0 −0.939693 0.342020i 1.32893 + 0.483690i
299.2 1.32893 0.483690i 0.245576 1.39273i 0.766044 0.642788i −0.766044 0.642788i −0.347296 1.96962i 0 0 −0.939693 0.342020i −1.32893 0.483690i
694.1 −0.245576 1.39273i 1.08335 0.909039i −0.939693 + 0.342020i 0.939693 + 0.342020i −1.53209 1.28558i 0 0 0.173648 0.984808i 0.245576 1.39273i
694.2 0.245576 + 1.39273i −1.08335 + 0.909039i −0.939693 + 0.342020i 0.939693 + 0.342020i −1.53209 1.28558i 0 0 0.173648 0.984808i −0.245576 + 1.39273i
849.1 −1.08335 + 0.909039i −1.32893 0.483690i 0.173648 0.984808i −0.173648 0.984808i 1.87939 0.684040i 0 0 0.766044 + 0.642788i 1.08335 + 0.909039i
849.2 1.08335 0.909039i 1.32893 + 0.483690i 0.173648 0.984808i −0.173648 0.984808i 1.87939 0.684040i 0 0 0.766044 + 0.642788i −1.08335 0.909039i
984.1 −1.32893 0.483690i −0.245576 1.39273i 0.766044 + 0.642788i −0.766044 + 0.642788i −0.347296 + 1.96962i 0 0 −0.939693 + 0.342020i 1.32893 0.483690i
984.2 1.32893 + 0.483690i 0.245576 + 1.39273i 0.766044 + 0.642788i −0.766044 + 0.642788i −0.347296 + 1.96962i 0 0 −0.939693 + 0.342020i −1.32893 + 0.483690i
1029.1 −1.08335 0.909039i −1.32893 + 0.483690i 0.173648 + 0.984808i −0.173648 + 0.984808i 1.87939 + 0.684040i 0 0 0.766044 0.642788i 1.08335 0.909039i
1029.2 1.08335 + 0.909039i 1.32893 0.483690i 0.173648 + 0.984808i −0.173648 + 0.984808i 1.87939 + 0.684040i 0 0 0.766044 0.642788i −1.08335 + 0.909039i
1199.1 −0.245576 + 1.39273i 1.08335 + 0.909039i −0.939693 0.342020i 0.939693 0.342020i −1.53209 + 1.28558i 0 0 0.173648 + 0.984808i 0.245576 + 1.39273i
1199.2 0.245576 1.39273i −1.08335 0.909039i −0.939693 0.342020i 0.939693 0.342020i −1.53209 + 1.28558i 0 0 0.173648 + 0.984808i −0.245576 1.39273i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1199.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
95.d odd 2 1 CM by \(\Q(\sqrt{-95}) \)
5.b even 2 1 inner
19.b odd 2 1 inner
19.c even 3 2 inner
19.d odd 6 2 inner
19.e even 9 3 inner
19.f odd 18 3 inner
95.h odd 6 2 inner
95.i even 6 2 inner
95.o odd 18 3 inner
95.p even 18 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1805.1.o.b 12
5.b even 2 1 inner 1805.1.o.b 12
19.b odd 2 1 inner 1805.1.o.b 12
19.c even 3 2 inner 1805.1.o.b 12
19.d odd 6 2 inner 1805.1.o.b 12
19.e even 9 1 95.1.d.b 2
19.e even 9 2 1805.1.h.b 4
19.e even 9 3 inner 1805.1.o.b 12
19.f odd 18 1 95.1.d.b 2
19.f odd 18 2 1805.1.h.b 4
19.f odd 18 3 inner 1805.1.o.b 12
57.j even 18 1 855.1.g.c 2
57.l odd 18 1 855.1.g.c 2
76.k even 18 1 1520.1.m.b 2
76.l odd 18 1 1520.1.m.b 2
95.d odd 2 1 CM 1805.1.o.b 12
95.h odd 6 2 inner 1805.1.o.b 12
95.i even 6 2 inner 1805.1.o.b 12
95.o odd 18 1 95.1.d.b 2
95.o odd 18 2 1805.1.h.b 4
95.o odd 18 3 inner 1805.1.o.b 12
95.p even 18 1 95.1.d.b 2
95.p even 18 2 1805.1.h.b 4
95.p even 18 3 inner 1805.1.o.b 12
95.q odd 36 2 475.1.c.b 2
95.r even 36 2 475.1.c.b 2
285.bd odd 18 1 855.1.g.c 2
285.bf even 18 1 855.1.g.c 2
380.ba odd 18 1 1520.1.m.b 2
380.bb even 18 1 1520.1.m.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.1.d.b 2 19.e even 9 1
95.1.d.b 2 19.f odd 18 1
95.1.d.b 2 95.o odd 18 1
95.1.d.b 2 95.p even 18 1
475.1.c.b 2 95.q odd 36 2
475.1.c.b 2 95.r even 36 2
855.1.g.c 2 57.j even 18 1
855.1.g.c 2 57.l odd 18 1
855.1.g.c 2 285.bd odd 18 1
855.1.g.c 2 285.bf even 18 1
1520.1.m.b 2 76.k even 18 1
1520.1.m.b 2 76.l odd 18 1
1520.1.m.b 2 380.ba odd 18 1
1520.1.m.b 2 380.bb even 18 1
1805.1.h.b 4 19.e even 9 2
1805.1.h.b 4 19.f odd 18 2
1805.1.h.b 4 95.o odd 18 2
1805.1.h.b 4 95.p even 18 2
1805.1.o.b 12 1.a even 1 1 trivial
1805.1.o.b 12 5.b even 2 1 inner
1805.1.o.b 12 19.b odd 2 1 inner
1805.1.o.b 12 19.c even 3 2 inner
1805.1.o.b 12 19.d odd 6 2 inner
1805.1.o.b 12 19.e even 9 3 inner
1805.1.o.b 12 19.f odd 18 3 inner
1805.1.o.b 12 95.d odd 2 1 CM
1805.1.o.b 12 95.h odd 6 2 inner
1805.1.o.b 12 95.i even 6 2 inner
1805.1.o.b 12 95.o odd 18 3 inner
1805.1.o.b 12 95.p even 18 3 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 8 T_{2}^{6} + 64 \) acting on \(S_{1}^{\mathrm{new}}(1805, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 64 + 8 T^{6} + T^{12} \)
$3$ \( 64 + 8 T^{6} + T^{12} \)
$5$ \( ( 1 - T^{3} + T^{6} )^{2} \)
$7$ \( T^{12} \)
$11$ \( T^{12} \)
$13$ \( 64 + 8 T^{6} + T^{12} \)
$17$ \( T^{12} \)
$19$ \( T^{12} \)
$23$ \( T^{12} \)
$29$ \( T^{12} \)
$31$ \( T^{12} \)
$37$ \( ( -2 + T^{2} )^{6} \)
$41$ \( T^{12} \)
$43$ \( T^{12} \)
$47$ \( T^{12} \)
$53$ \( 64 + 8 T^{6} + T^{12} \)
$59$ \( T^{12} \)
$61$ \( T^{12} \)
$67$ \( 64 + 8 T^{6} + T^{12} \)
$71$ \( T^{12} \)
$73$ \( T^{12} \)
$79$ \( T^{12} \)
$83$ \( T^{12} \)
$89$ \( T^{12} \)
$97$ \( 64 + 8 T^{6} + T^{12} \)
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