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Properties

Label 210.3.o.a
Level $210$
Weight $3$
Character orbit 210.o
Analytic conductor $5.722$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [210,3,Mod(31,210)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(210, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("210.31");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 210.o (of order \(6\), degree \(2\), 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: \(5.72208555157\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.3317760000.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{4} + \beta_{2}) q^{2} + (\beta_{3} + 1) q^{3} + (2 \beta_{3} - 2) q^{4} + \beta_1 q^{5} + (2 \beta_{4} + \beta_{2}) q^{6} + (2 \beta_{6} + 3 \beta_{4} + \cdots + \beta_1) q^{7}+ \cdots + 3 \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + \beta_{2}) q^{2} + (\beta_{3} + 1) q^{3} + (2 \beta_{3} - 2) q^{4} + \beta_1 q^{5} + (2 \beta_{4} + \beta_{2}) q^{6} + (2 \beta_{6} + 3 \beta_{4} + \cdots + \beta_1) q^{7}+ \cdots + ( - 6 \beta_{7} - 3 \beta_{6} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{3} - 8 q^{4} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{3} - 8 q^{4} + 12 q^{9} - 4 q^{11} - 24 q^{12} - 40 q^{14} - 16 q^{16} + 84 q^{17} + 108 q^{19} - 48 q^{22} + 12 q^{23} + 20 q^{25} - 96 q^{26} + 72 q^{29} - 132 q^{31} - 12 q^{33} + 100 q^{35} - 48 q^{36} - 96 q^{37} - 168 q^{38} + 24 q^{39} - 72 q^{42} - 112 q^{43} - 8 q^{44} + 8 q^{46} - 24 q^{47} + 156 q^{49} + 84 q^{51} + 48 q^{52} + 32 q^{53} + 16 q^{56} + 216 q^{57} + 104 q^{58} + 132 q^{59} + 96 q^{61} + 64 q^{64} + 20 q^{65} - 72 q^{66} - 120 q^{67} - 168 q^{68} + 8 q^{71} + 24 q^{73} - 16 q^{74} + 60 q^{75} - 216 q^{77} - 192 q^{78} + 12 q^{79} - 36 q^{81} + 24 q^{82} + 120 q^{85} - 40 q^{86} + 108 q^{87} + 48 q^{88} + 492 q^{89} - 308 q^{91} - 48 q^{92} - 132 q^{93} + 480 q^{94} - 40 q^{95} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 14\nu^{4} + 7\nu^{2} - 36 ) / 63 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{7} + 7\nu^{5} + 35\nu^{3} + 81\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{6} + 7\nu^{4} - 28\nu^{2} + 144 ) / 63 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{7} - 7\nu^{5} - 35\nu^{3} + 180\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 13\nu ) / 21 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -8\nu^{6} + 14\nu^{4} + 7\nu^{2} + 162 ) / 63 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19\nu^{7} - 49\nu^{5} + 133\nu^{3} - 684\nu ) / 189 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 2\beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + \beta_{5} + 7\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{3} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{7} - 19\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{6} + 7\beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 29\beta_{5} - 13\beta_{4} - 13\beta_{2} ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(31\) \(71\) \(127\)
\(\chi(n)\) \(\beta_{3}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
1.01575 − 1.40294i
−1.72286 + 0.178197i
−1.01575 + 1.40294i
1.72286 − 0.178197i
1.01575 + 1.40294i
−1.72286 − 0.178197i
−1.01575 − 1.40294i
1.72286 + 0.178197i
−0.707107 − 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i −1.93649 + 1.11803i − 2.44949i −5.10237 − 4.79227i 2.82843 1.50000 + 2.59808i 2.73861 + 1.58114i
31.2 −0.707107 − 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 1.93649 − 1.11803i − 2.44949i 6.51658 − 2.55620i 2.82843 1.50000 + 2.59808i −2.73861 − 1.58114i
31.3 0.707107 + 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i −1.93649 + 1.11803i 2.44949i −6.51658 + 2.55620i −2.82843 1.50000 + 2.59808i −2.73861 − 1.58114i
31.4 0.707107 + 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 1.93649 − 1.11803i 2.44949i 5.10237 + 4.79227i −2.82843 1.50000 + 2.59808i 2.73861 + 1.58114i
61.1 −0.707107 + 1.22474i 1.50000 − 0.866025i −1.00000 − 1.73205i −1.93649 − 1.11803i 2.44949i −5.10237 + 4.79227i 2.82843 1.50000 − 2.59808i 2.73861 − 1.58114i
61.2 −0.707107 + 1.22474i 1.50000 − 0.866025i −1.00000 − 1.73205i 1.93649 + 1.11803i 2.44949i 6.51658 + 2.55620i 2.82843 1.50000 − 2.59808i −2.73861 + 1.58114i
61.3 0.707107 − 1.22474i 1.50000 − 0.866025i −1.00000 − 1.73205i −1.93649 − 1.11803i − 2.44949i −6.51658 − 2.55620i −2.82843 1.50000 − 2.59808i −2.73861 + 1.58114i
61.4 0.707107 − 1.22474i 1.50000 − 0.866025i −1.00000 − 1.73205i 1.93649 + 1.11803i − 2.44949i 5.10237 − 4.79227i −2.82843 1.50000 − 2.59808i 2.73861 − 1.58114i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.3.o.a ✓ 8
3.b odd 2 1 630.3.v.b 8
5.b even 2 1 1050.3.p.b 8
5.c odd 4 2 1050.3.q.c 16
7.c even 3 1 1470.3.f.a 8
7.d odd 6 1 inner 210.3.o.a ✓ 8
7.d odd 6 1 1470.3.f.a 8
21.g even 6 1 630.3.v.b 8
35.i odd 6 1 1050.3.p.b 8
35.k even 12 2 1050.3.q.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.o.a ✓ 8 1.a even 1 1 trivial
210.3.o.a ✓ 8 7.d odd 6 1 inner
630.3.v.b 8 3.b odd 2 1
630.3.v.b 8 21.g even 6 1
1050.3.p.b 8 5.b even 2 1
1050.3.p.b 8 35.i odd 6 1
1050.3.q.c 16 5.c odd 4 2
1050.3.q.c 16 35.k even 12 2
1470.3.f.a 8 7.c even 3 1
1470.3.f.a 8 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{8} + 4 T_{11}^{7} + 316 T_{11}^{6} + 2896 T_{11}^{5} + 93448 T_{11}^{4} + 576448 T_{11}^{3} + \cdots + 22505536 \) acting on \(S_{3}^{\mathrm{new}}(210, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 3)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} - 5 T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 78 T^{6} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{8} + 4 T^{7} + \cdots + 22505536 \) Copy content Toggle raw display
$13$ \( T^{8} + 492 T^{6} + \cdots + 33189121 \) Copy content Toggle raw display
$17$ \( T^{8} - 84 T^{7} + \cdots + 15808576 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 3571138081 \) Copy content Toggle raw display
$23$ \( T^{8} - 12 T^{7} + \cdots + 1032256 \) Copy content Toggle raw display
$29$ \( (T^{4} - 36 T^{3} + \cdots + 20104)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 132 T^{7} + \cdots + 56085121 \) Copy content Toggle raw display
$37$ \( T^{8} + 96 T^{7} + \cdots + 686911681 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 6360766467136 \) Copy content Toggle raw display
$43$ \( (T^{4} + 56 T^{3} + \cdots - 877199)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 17622397993216 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 1469648896 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 19263180552256 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 981652934656 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 848699720001 \) Copy content Toggle raw display
$71$ \( (T^{4} - 4 T^{3} + \cdots - 2872184)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 52818460352161 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 11\!\cdots\!21 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 89\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 91315148362816 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 34\!\cdots\!56 \) Copy content Toggle raw display
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