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Properties

Label 2100.1.bq.b
Level $2100$
Weight $1$
Character orbit 2100.bq
Analytic conductor $1.048$
Analytic rank $0$
Dimension $8$
Projective image $D_{10}$
CM discriminant -84
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2100,1,Mod(419,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 5, 9, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.419");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2100.bq (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: \(1.04803652653\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{10} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{20}^{7} q^{2} - \zeta_{20}^{9} q^{3} - \zeta_{20}^{4} q^{4} - \zeta_{20}^{4} q^{5} + \zeta_{20}^{6} q^{6} - \zeta_{20}^{5} q^{7} + \zeta_{20} q^{8} - \zeta_{20}^{8} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{20}^{7} q^{2} - \zeta_{20}^{9} q^{3} - \zeta_{20}^{4} q^{4} - \zeta_{20}^{4} q^{5} + \zeta_{20}^{6} q^{6} - \zeta_{20}^{5} q^{7} + \zeta_{20} q^{8} - \zeta_{20}^{8} q^{9} + \zeta_{20} q^{10} + ( - \zeta_{20}^{3} - \zeta_{20}) q^{11} - \zeta_{20}^{3} q^{12} + \zeta_{20}^{2} q^{14} - \zeta_{20}^{3} q^{15} + \zeta_{20}^{8} q^{16} + ( - \zeta_{20}^{8} + \zeta_{20}^{4}) q^{17} + \zeta_{20}^{5} q^{18} + (\zeta_{20}^{9} + \zeta_{20}^{3}) q^{19} + \zeta_{20}^{8} q^{20} - \zeta_{20}^{4} q^{21} + ( - \zeta_{20}^{8} + 1) q^{22} + (\zeta_{20}^{3} - \zeta_{20}) q^{23} + q^{24} + \zeta_{20}^{8} q^{25} - \zeta_{20}^{7} q^{27} + \zeta_{20}^{9} q^{28} + q^{30} - \zeta_{20}^{5} q^{32} + ( - \zeta_{20}^{2} - 1) q^{33} + (\zeta_{20}^{5} - \zeta_{20}) q^{34} + \zeta_{20}^{9} q^{35} - \zeta_{20}^{2} q^{36} + ( - \zeta_{20}^{4} - \zeta_{20}^{2}) q^{37} + ( - \zeta_{20}^{6} - 1) q^{38} - \zeta_{20}^{5} q^{40} + ( - \zeta_{20}^{6} + 1) q^{41} + \zeta_{20} q^{42} + (\zeta_{20}^{7} + \zeta_{20}^{5}) q^{44} - \zeta_{20}^{2} q^{45} + ( - \zeta_{20}^{8} - 1) q^{46} + \zeta_{20}^{7} q^{48} - q^{49} - \zeta_{20}^{5} q^{50} + ( - \zeta_{20}^{7} + \zeta_{20}^{3}) q^{51} + \zeta_{20}^{4} q^{54} + (\zeta_{20}^{7} + \zeta_{20}^{5}) q^{55} - \zeta_{20}^{6} q^{56} + (\zeta_{20}^{8} + \zeta_{20}^{2}) q^{57} + \zeta_{20}^{7} q^{60} - \zeta_{20}^{3} q^{63} + \zeta_{20}^{2} q^{64} + ( - \zeta_{20}^{9} - \zeta_{20}^{7}) q^{66} + ( - \zeta_{20}^{8} - \zeta_{20}^{2}) q^{68} + (\zeta_{20}^{2} - 1) q^{69} - \zeta_{20}^{6} q^{70} - \zeta_{20}^{9} q^{72} + ( - \zeta_{20}^{9} + \zeta_{20}) q^{74} + \zeta_{20}^{7} q^{75} + ( - \zeta_{20}^{7} + \zeta_{20}^{3}) q^{76} + (\zeta_{20}^{8} + \zeta_{20}^{6}) q^{77} + \zeta_{20}^{2} q^{80} - \zeta_{20}^{6} q^{81} + (\zeta_{20}^{7} + \zeta_{20}^{3}) q^{82} + \zeta_{20}^{8} q^{84} + ( - \zeta_{20}^{8} - \zeta_{20}^{2}) q^{85} + ( - \zeta_{20}^{4} - \zeta_{20}^{2}) q^{88} + (\zeta_{20}^{4} + 1) q^{89} - \zeta_{20}^{9} q^{90} + ( - \zeta_{20}^{7} + \zeta_{20}^{5}) q^{92} + ( - \zeta_{20}^{7} + \zeta_{20}^{3}) q^{95} - \zeta_{20}^{4} q^{96} - \zeta_{20}^{7} q^{98} + (\zeta_{20}^{9} - \zeta_{20}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{4} + 2 q^{5} + 2 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{4} + 2 q^{5} + 2 q^{6} + 2 q^{9} + 2 q^{14} - 2 q^{16} - 2 q^{20} + 2 q^{21} + 10 q^{22} + 8 q^{24} - 2 q^{25} + 8 q^{30} - 10 q^{33} - 2 q^{36} - 10 q^{38} + 6 q^{41} - 2 q^{45} - 6 q^{46} - 8 q^{49} - 2 q^{54} - 2 q^{56} + 2 q^{64} - 6 q^{69} - 2 q^{70} + 2 q^{80} - 2 q^{81} - 2 q^{84} + 6 q^{89} + 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{20}^{4}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
419.1
−0.587785 + 0.809017i
0.587785 − 0.809017i
0.951057 − 0.309017i
−0.951057 + 0.309017i
0.951057 + 0.309017i
−0.951057 − 0.309017i
−0.587785 − 0.809017i
0.587785 + 0.809017i
−0.951057 + 0.309017i −0.587785 − 0.809017i 0.809017 − 0.587785i 0.809017 − 0.587785i 0.809017 + 0.587785i 1.00000i −0.587785 + 0.809017i −0.309017 + 0.951057i −0.587785 + 0.809017i
419.2 0.951057 − 0.309017i 0.587785 + 0.809017i 0.809017 − 0.587785i 0.809017 − 0.587785i 0.809017 + 0.587785i − 1.00000i 0.587785 − 0.809017i −0.309017 + 0.951057i 0.587785 − 0.809017i
839.1 −0.587785 − 0.809017i 0.951057 + 0.309017i −0.309017 + 0.951057i −0.309017 + 0.951057i −0.309017 − 0.951057i 1.00000i 0.951057 − 0.309017i 0.809017 + 0.587785i 0.951057 − 0.309017i
839.2 0.587785 + 0.809017i −0.951057 − 0.309017i −0.309017 + 0.951057i −0.309017 + 0.951057i −0.309017 − 0.951057i − 1.00000i −0.951057 + 0.309017i 0.809017 + 0.587785i −0.951057 + 0.309017i
1259.1 −0.587785 + 0.809017i 0.951057 − 0.309017i −0.309017 − 0.951057i −0.309017 − 0.951057i −0.309017 + 0.951057i − 1.00000i 0.951057 + 0.309017i 0.809017 − 0.587785i 0.951057 + 0.309017i
1259.2 0.587785 − 0.809017i −0.951057 + 0.309017i −0.309017 − 0.951057i −0.309017 − 0.951057i −0.309017 + 0.951057i 1.00000i −0.951057 − 0.309017i 0.809017 − 0.587785i −0.951057 − 0.309017i
1679.1 −0.951057 − 0.309017i −0.587785 + 0.809017i 0.809017 + 0.587785i 0.809017 + 0.587785i 0.809017 − 0.587785i − 1.00000i −0.587785 − 0.809017i −0.309017 − 0.951057i −0.587785 − 0.809017i
1679.2 0.951057 + 0.309017i 0.587785 − 0.809017i 0.809017 + 0.587785i 0.809017 + 0.587785i 0.809017 − 0.587785i 1.00000i 0.587785 + 0.809017i −0.309017 − 0.951057i 0.587785 + 0.809017i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 419.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
84.h odd 2 1 CM by \(\Q(\sqrt{-21}) \)
4.b odd 2 1 inner
21.c even 2 1 inner
25.e even 10 1 inner
100.h odd 10 1 inner
525.w even 10 1 inner
2100.bq odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.1.bq.b yes 8
3.b odd 2 1 2100.1.bq.a ✓ 8
4.b odd 2 1 inner 2100.1.bq.b yes 8
7.b odd 2 1 2100.1.bq.a ✓ 8
12.b even 2 1 2100.1.bq.a ✓ 8
21.c even 2 1 inner 2100.1.bq.b yes 8
25.e even 10 1 inner 2100.1.bq.b yes 8
28.d even 2 1 2100.1.bq.a ✓ 8
75.h odd 10 1 2100.1.bq.a ✓ 8
84.h odd 2 1 CM 2100.1.bq.b yes 8
100.h odd 10 1 inner 2100.1.bq.b yes 8
175.m odd 10 1 2100.1.bq.a ✓ 8
300.r even 10 1 2100.1.bq.a ✓ 8
525.w even 10 1 inner 2100.1.bq.b yes 8
700.ba even 10 1 2100.1.bq.a ✓ 8
2100.bq odd 10 1 inner 2100.1.bq.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.1.bq.a ✓ 8 3.b odd 2 1
2100.1.bq.a ✓ 8 7.b odd 2 1
2100.1.bq.a ✓ 8 12.b even 2 1
2100.1.bq.a ✓ 8 28.d even 2 1
2100.1.bq.a ✓ 8 75.h odd 10 1
2100.1.bq.a ✓ 8 175.m odd 10 1
2100.1.bq.a ✓ 8 300.r even 10 1
2100.1.bq.a ✓ 8 700.ba even 10 1
2100.1.bq.b yes 8 1.a even 1 1 trivial
2100.1.bq.b yes 8 4.b odd 2 1 inner
2100.1.bq.b yes 8 21.c even 2 1 inner
2100.1.bq.b yes 8 25.e even 10 1 inner
2100.1.bq.b yes 8 84.h odd 2 1 CM
2100.1.bq.b yes 8 100.h odd 10 1 inner
2100.1.bq.b yes 8 525.w even 10 1 inner
2100.1.bq.b yes 8 2100.bq odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{4} - 5T_{17} + 5 \) acting on \(S_{1}^{\mathrm{new}}(2100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{6} + T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} - T^{6} + T^{4} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} + 10 T^{4} + \cdots + 25 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} - 5 T + 5)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 10 T^{4} + \cdots + 25 \) Copy content Toggle raw display
$23$ \( T^{8} - 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} - 5 T + 5)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 3 T^{3} + 4 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T^{4} - 3 T^{3} + 4 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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