Properties

Label 1805.1.h.b
Level $1805$
Weight $1$
Character orbit 1805.h
Analytic conductor $0.901$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -95
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.900812347803\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
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: $C_3\times D_8$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-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.

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

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

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

Character values

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

\(n\) \(362\) \(1446\)
\(\chi(n)\) \(-1\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
69.1
0.707107 + 1.22474i
−0.707107 1.22474i
0.707107 1.22474i
−0.707107 + 1.22474i
−0.707107 1.22474i 0.707107 + 1.22474i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 1.73205i 0 0 −0.500000 + 0.866025i 0.707107 1.22474i
69.2 0.707107 + 1.22474i −0.707107 1.22474i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 1.73205i 0 0 −0.500000 + 0.866025i −0.707107 + 1.22474i
654.1 −0.707107 + 1.22474i 0.707107 1.22474i −0.500000 0.866025i 0.500000 0.866025i 1.00000 + 1.73205i 0 0 −0.500000 0.866025i 0.707107 + 1.22474i
654.2 0.707107 1.22474i −0.707107 + 1.22474i −0.500000 0.866025i 0.500000 0.866025i 1.00000 + 1.73205i 0 0 −0.500000 0.866025i −0.707107 1.22474i
\(n\): e.g. 2-40 or 990-1000
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 1 inner
19.d odd 6 1 inner
95.h odd 6 1 inner
95.i even 6 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 2 T_{2}^{2} + 4 \) acting on \(S_{1}^{\mathrm{new}}(1805, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + 2 T^{2} + T^{4} \)
$3$ \( 4 + 2 T^{2} + T^{4} \)
$5$ \( ( 1 - T + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( 4 + 2 T^{2} + T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( ( -2 + T^{2} )^{2} \)
$41$ \( T^{4} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( 4 + 2 T^{2} + T^{4} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( 4 + 2 T^{2} + T^{4} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( 4 + 2 T^{2} + T^{4} \)
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