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Properties

Label 1805.1.r.a
Level $1805$
Weight $1$
Character orbit 1805.r
Analytic conductor $0.901$
Analytic rank $0$
Dimension $12$
Projective image $D_{4}$
CM discriminant -19
Inner twists $24$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1805,1,Mod(28,1805)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1805, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([27, 16]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1805.28");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1805 = 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1805.r (of order \(36\), degree \(12\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.900812347803\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{36})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{6} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.2375.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{36}^{7} q^{4} + \zeta_{36}^{2} q^{5} + (\zeta_{36}^{12} + \zeta_{36}^{3}) q^{7} - \zeta_{36} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{36}^{7} q^{4} + \zeta_{36}^{2} q^{5} + (\zeta_{36}^{12} + \zeta_{36}^{3}) q^{7} - \zeta_{36} q^{9} + \zeta_{36}^{14} q^{16} + ( - \zeta_{36}^{17} + \zeta_{36}^{8}) q^{17} - \zeta_{36}^{9} q^{20} + ( - \zeta_{36}^{16} + \zeta_{36}^{7}) q^{23} + \zeta_{36}^{4} q^{25} + ( - \zeta_{36}^{10} + \zeta_{36}) q^{28} + (\zeta_{36}^{14} + \zeta_{36}^{5}) q^{35} + \zeta_{36}^{8} q^{36} + ( - \zeta_{36}^{11} - \zeta_{36}^{2}) q^{43} - \zeta_{36}^{3} q^{45} + ( - \zeta_{36}^{10} - \zeta_{36}) q^{47} + \zeta_{36}^{15} q^{49} + ( - \zeta_{36}^{13} - \zeta_{36}^{4}) q^{63} + \zeta_{36}^{3} q^{64} + ( - \zeta_{36}^{15} - \zeta_{36}^{6}) q^{68} + ( - \zeta_{36}^{14} + \zeta_{36}^{5}) q^{73} + \zeta_{36}^{16} q^{80} + \zeta_{36}^{2} q^{81} + (\zeta_{36}^{12} - \zeta_{36}^{3}) q^{83} + (\zeta_{36}^{10} + \zeta_{36}) q^{85} + ( - \zeta_{36}^{14} - \zeta_{36}^{5}) q^{92} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{7} - 6 q^{68} - 6 q^{83}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(362\) \(1446\)
\(\chi(n)\) \(-\zeta_{36}^{9}\) \(\zeta_{36}^{16}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
0.984808 + 0.173648i
−0.342020 + 0.939693i
0.342020 − 0.939693i
0.984808 − 0.173648i
−0.642788 − 0.766044i
0.342020 + 0.939693i
0.642788 − 0.766044i
−0.984808 + 0.173648i
−0.984808 − 0.173648i
0.642788 + 0.766044i
−0.342020 − 0.939693i
−0.642788 + 0.766044i
0 0 −0.342020 − 0.939693i 0.939693 + 0.342020i 0 0.366025 + 1.36603i 0 −0.984808 − 0.173648i 0
62.1 0 0 −0.642788 − 0.766044i −0.766044 − 0.642788i 0 0.366025 − 1.36603i 0 0.342020 − 0.939693i 0
423.1 0 0 0.642788 + 0.766044i −0.766044 − 0.642788i 0 −1.36603 − 0.366025i 0 −0.342020 + 0.939693i 0
967.1 0 0 −0.342020 + 0.939693i 0.939693 − 0.342020i 0 0.366025 − 1.36603i 0 −0.984808 + 0.173648i 0
1137.1 0 0 0.984808 − 0.173648i −0.173648 + 0.984808i 0 0.366025 − 1.36603i 0 0.642788 + 0.766044i 0
1182.1 0 0 0.642788 − 0.766044i −0.766044 + 0.642788i 0 −1.36603 + 0.366025i 0 −0.342020 − 0.939693i 0
1317.1 0 0 −0.984808 − 0.173648i −0.173648 − 0.984808i 0 −1.36603 + 0.366025i 0 −0.642788 + 0.766044i 0
1328.1 0 0 0.342020 − 0.939693i 0.939693 − 0.342020i 0 −1.36603 − 0.366025i 0 0.984808 − 0.173648i 0
1472.1 0 0 0.342020 + 0.939693i 0.939693 + 0.342020i 0 −1.36603 + 0.366025i 0 0.984808 + 0.173648i 0
1498.1 0 0 −0.984808 + 0.173648i −0.173648 + 0.984808i 0 −1.36603 − 0.366025i 0 −0.642788 − 0.766044i 0
1543.1 0 0 −0.642788 + 0.766044i −0.766044 + 0.642788i 0 0.366025 + 1.36603i 0 0.342020 + 0.939693i 0
1678.1 0 0 0.984808 + 0.173648i −0.173648 − 0.984808i 0 0.366025 + 1.36603i 0 0.642788 − 0.766044i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
5.c odd 4 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.g even 4 1 inner
95.l even 12 2 inner
95.m odd 12 2 inner
95.q odd 36 3 inner
95.r even 36 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1805.1.r.a 12
5.c odd 4 1 inner 1805.1.r.a 12
19.b odd 2 1 CM 1805.1.r.a 12
19.c even 3 2 inner 1805.1.r.a 12
19.d odd 6 2 inner 1805.1.r.a 12
19.e even 9 1 1805.1.f.a ✓ 2
19.e even 9 2 1805.1.m.a 4
19.e even 9 3 inner 1805.1.r.a 12
19.f odd 18 1 1805.1.f.a ✓ 2
19.f odd 18 2 1805.1.m.a 4
19.f odd 18 3 inner 1805.1.r.a 12
95.g even 4 1 inner 1805.1.r.a 12
95.l even 12 2 inner 1805.1.r.a 12
95.m odd 12 2 inner 1805.1.r.a 12
95.q odd 36 1 1805.1.f.a ✓ 2
95.q odd 36 2 1805.1.m.a 4
95.q odd 36 3 inner 1805.1.r.a 12
95.r even 36 1 1805.1.f.a ✓ 2
95.r even 36 2 1805.1.m.a 4
95.r even 36 3 inner 1805.1.r.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1805.1.f.a ✓ 2 19.e even 9 1
1805.1.f.a ✓ 2 19.f odd 18 1
1805.1.f.a ✓ 2 95.q odd 36 1
1805.1.f.a ✓ 2 95.r even 36 1
1805.1.m.a 4 19.e even 9 2
1805.1.m.a 4 19.f odd 18 2
1805.1.m.a 4 95.q odd 36 2
1805.1.m.a 4 95.r even 36 2
1805.1.r.a 12 1.a even 1 1 trivial
1805.1.r.a 12 5.c odd 4 1 inner
1805.1.r.a 12 19.b odd 2 1 CM
1805.1.r.a 12 19.c even 3 2 inner
1805.1.r.a 12 19.d odd 6 2 inner
1805.1.r.a 12 19.e even 9 3 inner
1805.1.r.a 12 19.f odd 18 3 inner
1805.1.r.a 12 95.g even 4 1 inner
1805.1.r.a 12 95.l even 12 2 inner
1805.1.r.a 12 95.m odd 12 2 inner
1805.1.r.a 12 95.q odd 36 3 inner
1805.1.r.a 12 95.r even 36 3 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1805, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} - T^{3} + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{3} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} - 4 T^{9} + \cdots + 64 \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( T^{12} + 4 T^{9} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} \) Copy content Toggle raw display
$37$ \( T^{12} \) Copy content Toggle raw display
$41$ \( T^{12} \) Copy content Toggle raw display
$43$ \( T^{12} - 4 T^{9} + \cdots + 64 \) Copy content Toggle raw display
$47$ \( T^{12} - 4 T^{9} + \cdots + 64 \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( T^{12} \) Copy content Toggle raw display
$61$ \( T^{12} \) Copy content Toggle raw display
$67$ \( T^{12} \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( T^{12} - 4 T^{9} + \cdots + 64 \) Copy content Toggle raw display
$79$ \( T^{12} \) Copy content Toggle raw display
$83$ \( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{3} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( T^{12} \) Copy content Toggle raw display
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