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: \( x^{12} + 8x^{6} + 64 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + (\beta_{11} + \beta_{5}) q^{2} + \beta_{5} q^{3} + ( - \beta_{10} - \beta_{4}) q^{4} - \beta_{2} q^{5} - 2 \beta_{4} q^{6} + \beta_{10} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{11} + \beta_{5}) q^{2} + \beta_{5} q^{3} + ( - \beta_{10} - \beta_{4}) q^{4} - \beta_{2} q^{5} - 2 \beta_{4} q^{6} + \beta_{10} q^{9} + \beta_1 q^{10} + \beta_{3} q^{12} + (\beta_{7} + \beta_1) q^{13} - \beta_{7} q^{15} + (\beta_{8} + \beta_{2}) q^{16} - \beta_{9} q^{18} - q^{20} + \beta_{4} q^{25} + ( - 2 \beta_{6} - 2) q^{26} + 2 \beta_{6} q^{30} + ( - \beta_{7} - \beta_1) q^{32} + \beta_{8} q^{36} + \beta_{9} q^{37} - 2 q^{39} + (\beta_{6} + 1) q^{45} - \beta_1 q^{48} + \beta_{6} q^{49} - \beta_{3} q^{50} - \beta_{11} q^{52} + \beta_{7} q^{53} - \beta_{5} q^{60} + (\beta_{6} + 1) q^{64} + ( - \beta_{9} - \beta_{3}) q^{65} - \beta_1 q^{67} - 2 \beta_{8} q^{74} + \beta_{9} q^{75} + ( - 2 \beta_{11} - 2 \beta_{5}) q^{78} + ( - \beta_{10} - \beta_{4}) q^{80} - \beta_{2} q^{81} + \beta_{11} q^{90} + 2 q^{96} + ( - \beta_{11} - \beta_{5}) q^{97} - \beta_{5} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{20} - 12 q^{26} - 12 q^{30} - 24 q^{39} + 6 q^{45} - 6 q^{49} + 6 q^{64} + 24 q^{96}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{8} ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{9} ) / 16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{10} ) / 32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{11} ) / 32 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{7} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16\beta_{8} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 16\beta_{9} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 32\beta_{10} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 32\beta_{11} \) Copy content Toggle raw display

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 \( T_{2}^{12} + 8T_{2}^{6} + 64 \) acting on \(S_{1}^{\mathrm{new}}(1805, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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