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Properties

Label 210.2.d.a
Level $210$
Weight $2$
Character orbit 210.d
Analytic conductor $1.677$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [210,2,Mod(209,210)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(210, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("210.209");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 210.d (of order \(2\), degree \(1\), 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: \(1.67685844245\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.10070523904.11
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 10x^{4} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - \beta_{7} q^{3} + q^{4} + ( - \beta_{7} + \beta_{6} + \cdots + \beta_1) q^{5}+ \cdots + (\beta_{4} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta_{7} q^{3} + q^{4} + ( - \beta_{7} + \beta_{6} + \cdots + \beta_1) q^{5}+ \cdots + (2 \beta_{6} + 2 \beta_{5} + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} + 8 q^{16} - 8 q^{21} + 16 q^{23} - 8 q^{32} + 16 q^{35} - 24 q^{39} + 8 q^{42} - 16 q^{46} - 8 q^{49} - 16 q^{51} + 16 q^{53} - 40 q^{57} + 8 q^{63} + 8 q^{64} - 8 q^{65} - 16 q^{70} - 16 q^{77} + 24 q^{78} - 16 q^{79} + 40 q^{81} - 8 q^{84} + 32 q^{85} + 16 q^{91} + 16 q^{92} - 32 q^{93} + 24 q^{95} + 8 q^{98} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 10x^{4} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - \nu^{2} ) / 18 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 7\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} + 19\nu^{2} ) / 18 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + \nu^{6} + 3\nu^{5} + 9\nu^{4} + \nu^{3} - \nu^{2} - 3\nu - 45 ) / 36 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - \nu^{6} - 3\nu^{5} + 9\nu^{4} - \nu^{3} + \nu^{2} + 3\nu - 45 ) / 36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - 10\nu^{3} ) / 27 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{7} + 2\beta_{6} - 2\beta_{5} + 2\beta_{3} + 2\beta_{2} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{6} + 2\beta_{5} + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6\beta_{3} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{4} + 19\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -3\beta_{7} + 20\beta_{6} - 20\beta_{5} + 20\beta_{3} + 20\beta_{2} + 20\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(71\) \(127\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
−1.68014 + 0.420861i
−1.68014 − 0.420861i
−0.420861 + 1.68014i
−0.420861 − 1.68014i
0.420861 + 1.68014i
0.420861 − 1.68014i
1.68014 + 0.420861i
1.68014 − 0.420861i
−1.00000 −1.68014 − 0.420861i 1.00000 −1.08495 + 1.95522i 1.68014 + 0.420861i 0.595188 − 2.57794i −1.00000 2.64575 + 1.41421i 1.08495 − 1.95522i
209.2 −1.00000 −1.68014 + 0.420861i 1.00000 −1.08495 − 1.95522i 1.68014 − 0.420861i 0.595188 + 2.57794i −1.00000 2.64575 − 1.41421i 1.08495 + 1.95522i
209.3 −1.00000 −0.420861 − 1.68014i 1.00000 1.95522 − 1.08495i 0.420861 + 1.68014i 2.37608 + 1.16372i −1.00000 −2.64575 + 1.41421i −1.95522 + 1.08495i
209.4 −1.00000 −0.420861 + 1.68014i 1.00000 1.95522 + 1.08495i 0.420861 − 1.68014i 2.37608 − 1.16372i −1.00000 −2.64575 − 1.41421i −1.95522 − 1.08495i
209.5 −1.00000 0.420861 − 1.68014i 1.00000 −1.95522 − 1.08495i −0.420861 + 1.68014i −2.37608 − 1.16372i −1.00000 −2.64575 − 1.41421i 1.95522 + 1.08495i
209.6 −1.00000 0.420861 + 1.68014i 1.00000 −1.95522 + 1.08495i −0.420861 − 1.68014i −2.37608 + 1.16372i −1.00000 −2.64575 + 1.41421i 1.95522 − 1.08495i
209.7 −1.00000 1.68014 − 0.420861i 1.00000 1.08495 + 1.95522i −1.68014 + 0.420861i −0.595188 + 2.57794i −1.00000 2.64575 − 1.41421i −1.08495 − 1.95522i
209.8 −1.00000 1.68014 + 0.420861i 1.00000 1.08495 − 1.95522i −1.68014 − 0.420861i −0.595188 − 2.57794i −1.00000 2.64575 + 1.41421i −1.08495 + 1.95522i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
15.d odd 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.2.d.a ✓ 8
3.b odd 2 1 210.2.d.b yes 8
4.b odd 2 1 1680.2.k.e 8
5.b even 2 1 210.2.d.b yes 8
5.c odd 4 2 1050.2.b.f 16
7.b odd 2 1 inner 210.2.d.a ✓ 8
12.b even 2 1 1680.2.k.f 8
15.d odd 2 1 inner 210.2.d.a ✓ 8
15.e even 4 2 1050.2.b.f 16
20.d odd 2 1 1680.2.k.f 8
21.c even 2 1 210.2.d.b yes 8
28.d even 2 1 1680.2.k.e 8
35.c odd 2 1 210.2.d.b yes 8
35.f even 4 2 1050.2.b.f 16
60.h even 2 1 1680.2.k.e 8
84.h odd 2 1 1680.2.k.f 8
105.g even 2 1 inner 210.2.d.a ✓ 8
105.k odd 4 2 1050.2.b.f 16
140.c even 2 1 1680.2.k.f 8
420.o odd 2 1 1680.2.k.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.d.a ✓ 8 1.a even 1 1 trivial
210.2.d.a ✓ 8 7.b odd 2 1 inner
210.2.d.a ✓ 8 15.d odd 2 1 inner
210.2.d.a ✓ 8 105.g even 2 1 inner
210.2.d.b yes 8 3.b odd 2 1
210.2.d.b yes 8 5.b even 2 1
210.2.d.b yes 8 21.c even 2 1
210.2.d.b yes 8 35.c odd 2 1
1050.2.b.f 16 5.c odd 4 2
1050.2.b.f 16 15.e even 4 2
1050.2.b.f 16 35.f even 4 2
1050.2.b.f 16 105.k odd 4 2
1680.2.k.e 8 4.b odd 2 1
1680.2.k.e 8 28.d even 2 1
1680.2.k.e 8 60.h even 2 1
1680.2.k.e 8 420.o odd 2 1
1680.2.k.f 8 12.b even 2 1
1680.2.k.f 8 20.d odd 2 1
1680.2.k.f 8 84.h odd 2 1
1680.2.k.f 8 140.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23}^{2} - 4T_{23} - 24 \) acting on \(S_{2}^{\mathrm{new}}(210, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 10T^{4} + 81 \) Copy content Toggle raw display
$5$ \( T^{8} + 22T^{4} + 625 \) Copy content Toggle raw display
$7$ \( T^{8} + 4 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 12 T^{2} + 8)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 24 T^{2} + 32)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 52 T^{2} + 648)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 4 T - 24)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 64 T^{2} + 16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 40 T^{2} + 288)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 128 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 80 T^{2} + 1152)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 128 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 80 T^{2} + 1152)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 4 T - 108)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 20 T^{2} + 72)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 108 T^{2} + 648)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 128 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 256 T^{2} + 15376)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 40 T^{2} + 288)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 4 T - 24)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 108 T^{2} + 2888)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 416 T^{2} + 41472)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 280 T^{2} + 14112)^{2} \) Copy content Toggle raw display
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