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Properties

Label 3800.1.cq.b
Level $3800$
Weight $1$
Character orbit 3800.cq
Analytic conductor $1.896$
Analytic rank $0$
Dimension $12$
Projective image $D_{9}$
CM discriminant -8
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3800,1,Mod(99,3800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3800, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 9, 9, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3800.99");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3800.cq (of order \(18\), degree \(6\), 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: \(1.89644704801\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.69564674215936.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{36} q^{2} + ( - \zeta_{36}^{5} + \zeta_{36}^{3}) q^{3} + \zeta_{36}^{2} q^{4} + ( - \zeta_{36}^{6} + \zeta_{36}^{4}) q^{6} + \zeta_{36}^{3} q^{8} + (\zeta_{36}^{10} + \cdots + \zeta_{36}^{6}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{36} q^{2} + ( - \zeta_{36}^{5} + \zeta_{36}^{3}) q^{3} + \zeta_{36}^{2} q^{4} + ( - \zeta_{36}^{6} + \zeta_{36}^{4}) q^{6} + \zeta_{36}^{3} q^{8} + (\zeta_{36}^{10} + \cdots + \zeta_{36}^{6}) q^{9}+ \cdots + ( - \zeta_{36}^{16} + \zeta_{36}^{14} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{6} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{6} + 6 q^{9} + 6 q^{24} - 6 q^{36} - 6 q^{41} - 12 q^{44} + 6 q^{49} + 6 q^{51} + 6 q^{54} + 6 q^{59} + 6 q^{64} - 6 q^{66} + 6 q^{81} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(-\zeta_{36}^{2}\) \(-1\) \(-1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
−0.342020 + 0.939693i
0.342020 − 0.939693i
−0.342020 − 0.939693i
0.342020 + 0.939693i
−0.642788 − 0.766044i
0.642788 + 0.766044i
−0.642788 + 0.766044i
0.642788 − 0.766044i
−0.984808 − 0.173648i
0.984808 + 0.173648i
−0.984808 + 0.173648i
0.984808 − 0.173648i
−0.342020 + 0.939693i 1.85083 − 0.326352i −0.766044 − 0.642788i 0 −0.326352 + 1.85083i 0 0.866025 − 0.500000i 2.37939 − 0.866025i 0
99.2 0.342020 − 0.939693i −1.85083 + 0.326352i −0.766044 − 0.642788i 0 −0.326352 + 1.85083i 0 −0.866025 + 0.500000i 2.37939 − 0.866025i 0
499.1 −0.342020 − 0.939693i 1.85083 + 0.326352i −0.766044 + 0.642788i 0 −0.326352 − 1.85083i 0 0.866025 + 0.500000i 2.37939 + 0.866025i 0
499.2 0.342020 + 0.939693i −1.85083 − 0.326352i −0.766044 + 0.642788i 0 −0.326352 − 1.85083i 0 −0.866025 − 0.500000i 2.37939 + 0.866025i 0
899.1 −0.642788 − 0.766044i 0.524005 − 1.43969i −0.173648 + 0.984808i 0 −1.43969 + 0.524005i 0 0.866025 − 0.500000i −1.03209 − 0.866025i 0
899.2 0.642788 + 0.766044i −0.524005 + 1.43969i −0.173648 + 0.984808i 0 −1.43969 + 0.524005i 0 −0.866025 + 0.500000i −1.03209 − 0.866025i 0
1099.1 −0.642788 + 0.766044i 0.524005 + 1.43969i −0.173648 − 0.984808i 0 −1.43969 − 0.524005i 0 0.866025 + 0.500000i −1.03209 + 0.866025i 0
1099.2 0.642788 − 0.766044i −0.524005 − 1.43969i −0.173648 − 0.984808i 0 −1.43969 − 0.524005i 0 −0.866025 − 0.500000i −1.03209 + 0.866025i 0
1499.1 −0.984808 − 0.173648i −0.223238 + 0.266044i 0.939693 + 0.342020i 0 0.266044 − 0.223238i 0 −0.866025 − 0.500000i 0.152704 + 0.866025i 0
1499.2 0.984808 + 0.173648i 0.223238 − 0.266044i 0.939693 + 0.342020i 0 0.266044 − 0.223238i 0 0.866025 + 0.500000i 0.152704 + 0.866025i 0
2099.1 −0.984808 + 0.173648i −0.223238 − 0.266044i 0.939693 − 0.342020i 0 0.266044 + 0.223238i 0 −0.866025 + 0.500000i 0.152704 − 0.866025i 0
2099.2 0.984808 − 0.173648i 0.223238 + 0.266044i 0.939693 − 0.342020i 0 0.266044 + 0.223238i 0 0.866025 − 0.500000i 0.152704 − 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
5.b even 2 1 inner
19.e even 9 1 inner
40.e odd 2 1 inner
95.p even 18 1 inner
152.u odd 18 1 inner
760.bz odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3800.1.cq.b 12
5.b even 2 1 inner 3800.1.cq.b 12
5.c odd 4 1 152.1.u.a ✓ 6
5.c odd 4 1 3800.1.cv.c 6
8.d odd 2 1 CM 3800.1.cq.b 12
15.e even 4 1 1368.1.eh.a 6
19.e even 9 1 inner 3800.1.cq.b 12
20.e even 4 1 608.1.bg.a 6
40.e odd 2 1 inner 3800.1.cq.b 12
40.i odd 4 1 608.1.bg.a 6
40.k even 4 1 152.1.u.a ✓ 6
40.k even 4 1 3800.1.cv.c 6
95.g even 4 1 2888.1.u.e 6
95.l even 12 1 2888.1.u.a 6
95.l even 12 1 2888.1.u.f 6
95.m odd 12 1 2888.1.u.b 6
95.m odd 12 1 2888.1.u.g 6
95.p even 18 1 inner 3800.1.cq.b 12
95.q odd 36 1 152.1.u.a ✓ 6
95.q odd 36 1 2888.1.f.d 3
95.q odd 36 2 2888.1.k.b 6
95.q odd 36 1 2888.1.u.b 6
95.q odd 36 1 2888.1.u.g 6
95.q odd 36 1 3800.1.cv.c 6
95.r even 36 1 2888.1.f.c 3
95.r even 36 2 2888.1.k.c 6
95.r even 36 1 2888.1.u.a 6
95.r even 36 1 2888.1.u.e 6
95.r even 36 1 2888.1.u.f 6
120.q odd 4 1 1368.1.eh.a 6
152.u odd 18 1 inner 3800.1.cq.b 12
285.bi even 36 1 1368.1.eh.a 6
380.bj even 36 1 608.1.bg.a 6
760.y odd 4 1 2888.1.u.e 6
760.bu odd 12 1 2888.1.u.a 6
760.bu odd 12 1 2888.1.u.f 6
760.bw even 12 1 2888.1.u.b 6
760.bw even 12 1 2888.1.u.g 6
760.bz odd 18 1 inner 3800.1.cq.b 12
760.cn odd 36 1 2888.1.f.c 3
760.cn odd 36 2 2888.1.k.c 6
760.cn odd 36 1 2888.1.u.a 6
760.cn odd 36 1 2888.1.u.e 6
760.cn odd 36 1 2888.1.u.f 6
760.cp even 36 1 152.1.u.a ✓ 6
760.cp even 36 1 2888.1.f.d 3
760.cp even 36 2 2888.1.k.b 6
760.cp even 36 1 2888.1.u.b 6
760.cp even 36 1 2888.1.u.g 6
760.cp even 36 1 3800.1.cv.c 6
760.cq odd 36 1 608.1.bg.a 6
2280.fl odd 36 1 1368.1.eh.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.u.a ✓ 6 5.c odd 4 1
152.1.u.a ✓ 6 40.k even 4 1
152.1.u.a ✓ 6 95.q odd 36 1
152.1.u.a ✓ 6 760.cp even 36 1
608.1.bg.a 6 20.e even 4 1
608.1.bg.a 6 40.i odd 4 1
608.1.bg.a 6 380.bj even 36 1
608.1.bg.a 6 760.cq odd 36 1
1368.1.eh.a 6 15.e even 4 1
1368.1.eh.a 6 120.q odd 4 1
1368.1.eh.a 6 285.bi even 36 1
1368.1.eh.a 6 2280.fl odd 36 1
2888.1.f.c 3 95.r even 36 1
2888.1.f.c 3 760.cn odd 36 1
2888.1.f.d 3 95.q odd 36 1
2888.1.f.d 3 760.cp even 36 1
2888.1.k.b 6 95.q odd 36 2
2888.1.k.b 6 760.cp even 36 2
2888.1.k.c 6 95.r even 36 2
2888.1.k.c 6 760.cn odd 36 2
2888.1.u.a 6 95.l even 12 1
2888.1.u.a 6 95.r even 36 1
2888.1.u.a 6 760.bu odd 12 1
2888.1.u.a 6 760.cn odd 36 1
2888.1.u.b 6 95.m odd 12 1
2888.1.u.b 6 95.q odd 36 1
2888.1.u.b 6 760.bw even 12 1
2888.1.u.b 6 760.cp even 36 1
2888.1.u.e 6 95.g even 4 1
2888.1.u.e 6 95.r even 36 1
2888.1.u.e 6 760.y odd 4 1
2888.1.u.e 6 760.cn odd 36 1
2888.1.u.f 6 95.l even 12 1
2888.1.u.f 6 95.r even 36 1
2888.1.u.f 6 760.bu odd 12 1
2888.1.u.f 6 760.cn odd 36 1
2888.1.u.g 6 95.m odd 12 1
2888.1.u.g 6 95.q odd 36 1
2888.1.u.g 6 760.bw even 12 1
2888.1.u.g 6 760.cp even 36 1
3800.1.cq.b 12 1.a even 1 1 trivial
3800.1.cq.b 12 5.b even 2 1 inner
3800.1.cq.b 12 8.d odd 2 1 CM
3800.1.cq.b 12 19.e even 9 1 inner
3800.1.cq.b 12 40.e odd 2 1 inner
3800.1.cq.b 12 95.p even 18 1 inner
3800.1.cq.b 12 152.u odd 18 1 inner
3800.1.cq.b 12 760.bz odd 18 1 inner
3800.1.cv.c 6 5.c odd 4 1
3800.1.cv.c 6 40.k even 4 1
3800.1.cv.c 6 95.q odd 36 1
3800.1.cv.c 6 760.cp even 36 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 3T_{3}^{10} - 6T_{3}^{8} + 8T_{3}^{6} + 69T_{3}^{4} + 3T_{3}^{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(3800, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$3$ \( T^{12} - 3 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} + 3 T^{4} + 2 T^{3} + \cdots + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$19$ \( (T^{6} - T^{3} + 1)^{2} \) 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
$37$ \( T^{12} \) Copy content Toggle raw display
$41$ \( (T^{6} + 3 T^{5} + 6 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$47$ \( T^{12} \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( (T^{6} - 3 T^{5} + 6 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} \) Copy content Toggle raw display
$67$ \( T^{12} - 3 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( T^{12} + 6 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{12} \) Copy content Toggle raw display
$83$ \( T^{12} - 6 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} - 3 T^{10} + \cdots + 1 \) Copy content Toggle raw display
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