Properties

Label 1900.1.bb.a
Level $1900$
Weight $1$
Character orbit 1900.bb
Analytic conductor $0.948$
Analytic rank $0$
Dimension $8$
Projective image $D_{15}$
CM discriminant -19
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1900,1,Mod(341,1900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1900, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 2, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1900.341");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1900.bb (of order \(10\), degree \(4\), 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: \(0.948223524003\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{15})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{15}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{15} + \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{30}^{7} q^{5} + ( - \zeta_{30}^{13} + \zeta_{30}^{2}) q^{7} + \zeta_{30}^{12} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{30}^{7} q^{5} + ( - \zeta_{30}^{13} + \zeta_{30}^{2}) q^{7} + \zeta_{30}^{12} q^{9} + ( - \zeta_{30}^{5} - \zeta_{30}) q^{11} + ( - \zeta_{30}^{13} - \zeta_{30}^{5}) q^{17} - \zeta_{30}^{9} q^{19} + (\zeta_{30}^{6} + 1) q^{23} + \zeta_{30}^{14} q^{25} + ( - \zeta_{30}^{9} - \zeta_{30}^{5}) q^{35} + (\zeta_{30}^{14} - \zeta_{30}) q^{43} + \zeta_{30}^{4} q^{45} - \zeta_{30}^{6} q^{47} + ( - \zeta_{30}^{11} + \zeta_{30}^{4} + 1) q^{49} + (\zeta_{30}^{12} + \zeta_{30}^{8}) q^{55} + (\zeta_{30}^{4} + \zeta_{30}^{2}) q^{61} + (\zeta_{30}^{14} + \zeta_{30}^{10}) q^{63} + \zeta_{30}^{3} q^{73} + (\zeta_{30}^{14} - \zeta_{30}^{7} - \zeta_{30}^{3}) q^{77} - \zeta_{30}^{9} q^{81} + ( - \zeta_{30}^{3} + 1) q^{83} + (\zeta_{30}^{12} - \zeta_{30}^{5}) q^{85} - \zeta_{30} q^{95} + ( - \zeta_{30}^{13} + \zeta_{30}^{2}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{5} + 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{5} + 2 q^{7} - 2 q^{9} - 3 q^{11} - 3 q^{17} - 2 q^{19} + 6 q^{23} + q^{25} - 6 q^{35} + 2 q^{43} + q^{45} + 2 q^{47} + 10 q^{49} - q^{55} + 2 q^{61} - 3 q^{63} + 2 q^{73} - 2 q^{77} - 2 q^{81} + 6 q^{83} - 6 q^{85} + q^{95} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(\zeta_{30}^{6}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
341.1
−0.978148 0.207912i
0.669131 0.743145i
0.913545 0.406737i
−0.104528 + 0.994522i
0.913545 + 0.406737i
−0.104528 0.994522i
−0.978148 + 0.207912i
0.669131 + 0.743145i
0 0 0 −0.104528 0.994522i 0 1.82709 0 −0.809017 + 0.587785i 0
341.2 0 0 0 0.913545 + 0.406737i 0 −0.209057 0 −0.809017 + 0.587785i 0
721.1 0 0 0 −0.978148 0.207912i 0 1.33826 0 0.309017 + 0.951057i 0
721.2 0 0 0 0.669131 0.743145i 0 −1.95630 0 0.309017 + 0.951057i 0
1481.1 0 0 0 −0.978148 + 0.207912i 0 1.33826 0 0.309017 0.951057i 0
1481.2 0 0 0 0.669131 + 0.743145i 0 −1.95630 0 0.309017 0.951057i 0
1861.1 0 0 0 −0.104528 + 0.994522i 0 1.82709 0 −0.809017 0.587785i 0
1861.2 0 0 0 0.913545 0.406737i 0 −0.209057 0 −0.809017 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 341.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
25.d even 5 1 inner
475.o odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.1.bb.a 8
19.b odd 2 1 CM 1900.1.bb.a 8
25.d even 5 1 inner 1900.1.bb.a 8
475.o odd 10 1 inner 1900.1.bb.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1900.1.bb.a 8 1.a even 1 1 trivial
1900.1.bb.a 8 19.b odd 2 1 CM
1900.1.bb.a 8 25.d even 5 1 inner
1900.1.bb.a 8 475.o odd 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1900, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - T^{7} + T^{5} - T^{4} + T^{3} - T + 1 \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{3} - 4 T^{2} + 4 T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 3 T^{7} + 8 T^{6} + 11 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} + 3 T^{7} + 8 T^{6} + 11 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} - T^{3} - 4 T^{2} + 4 T + 1)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} - 2 T^{7} + 3 T^{6} + T^{5} + T^{3} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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