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Properties

Label 1470.2.m.a
Level $1470$
Weight $2$
Character orbit 1470.m
Analytic conductor $11.738$
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] = [1470,2,Mod(97,1470)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1470, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1470.97");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1470.m (of order \(4\), 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: \(11.7380090971\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - \beta_{3} q^{3} - \beta_{2} q^{4} + (2 \beta_{2} - 1) q^{5} + \beta_{2} q^{6} + \beta_1 q^{8} - \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - \beta_{3} q^{3} - \beta_{2} q^{4} + (2 \beta_{2} - 1) q^{5} + \beta_{2} q^{6} + \beta_1 q^{8} - \beta_{2} q^{9} + ( - \beta_{3} - 2 \beta_1) q^{10} + ( - \beta_{7} - \beta_{3} + \beta_1) q^{11} - \beta_1 q^{12} + (\beta_{7} - \beta_{6} + \beta_{2} - 1) q^{13} + (\beta_{3} + 2 \beta_1) q^{15} - q^{16} + ( - \beta_{5} - \beta_{2} - 1) q^{17} + \beta_1 q^{18} + ( - \beta_{3} + \beta_1 + 2) q^{19} + (\beta_{2} + 2) q^{20} + ( - \beta_{4} + \beta_{2} - 1) q^{22} + ( - 3 \beta_{2} - 3) q^{23} + q^{24} + ( - 4 \beta_{2} - 3) q^{25} + (\beta_{5} + \beta_{4} - \beta_{3} - \beta_1) q^{26} - \beta_1 q^{27} + ( - \beta_{6} + \beta_{5} + \cdots + 2 \beta_1) q^{29}+ \cdots + (\beta_{6} - \beta_{3} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{5} - 8 q^{13} - 8 q^{16} - 8 q^{17} + 16 q^{19} + 16 q^{20} - 8 q^{22} - 24 q^{23} + 8 q^{24} - 24 q^{25} - 16 q^{30} + 8 q^{33} - 8 q^{36} + 8 q^{37} - 8 q^{38} + 32 q^{43} + 16 q^{45} - 16 q^{47} + 8 q^{52} - 32 q^{53} + 8 q^{54} + 8 q^{57} - 16 q^{58} - 16 q^{59} - 8 q^{62} - 8 q^{65} - 16 q^{67} - 8 q^{68} + 32 q^{71} - 16 q^{73} - 8 q^{78} + 8 q^{80} - 8 q^{81} + 16 q^{82} + 24 q^{85} - 32 q^{86} + 16 q^{87} + 8 q^{88} - 64 q^{89} - 24 q^{92} + 8 q^{93} - 32 q^{94} - 16 q^{95} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{16}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{16}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{16}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{16}^{7} + \zeta_{16}^{5} + 2\zeta_{16}^{3} - 2\zeta_{16} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -2\zeta_{16}^{7} + 2\zeta_{16}^{5} + \zeta_{16}^{3} + \zeta_{16} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 2\zeta_{16}^{7} + \zeta_{16}^{5} + \zeta_{16}^{3} + 2\zeta_{16} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{16}^{7} - 2\zeta_{16}^{5} + 2\zeta_{16}^{3} + \zeta_{16} \) Copy content Toggle raw display

Display \(\zeta_{16}^j\) in terms of \(\beta_i\)

\(\zeta_{16}\)\(=\) \( ( \beta_{7} + 3\beta_{6} + \beta_{5} - 3\beta_{4} ) / 14 \) Copy content Toggle raw display
\(\zeta_{16}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{16}^{3}\)\(=\) \( ( 3\beta_{7} + \beta_{6} + \beta_{5} + 3\beta_{4} ) / 14 \) Copy content Toggle raw display
\(\zeta_{16}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{16}^{5}\)\(=\) \( ( -3\beta_{7} + \beta_{6} + 3\beta_{5} + \beta_{4} ) / 14 \) Copy content Toggle raw display
\(\zeta_{16}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{16}^{7}\)\(=\) \( ( -\beta_{7} + 3\beta_{6} - 3\beta_{5} + \beta_{4} ) / 14 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{16}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times\).

\(n\) \(491\) \(1081\) \(1177\)
\(\chi(n)\) \(1\) \(-1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
0.923880 − 0.382683i
−0.923880 + 0.382683i
−0.382683 − 0.923880i
0.382683 + 0.923880i
0.923880 + 0.382683i
−0.923880 − 0.382683i
−0.382683 + 0.923880i
0.382683 − 0.923880i
−0.707107 − 0.707107i 0.707107 + 0.707107i 1.00000i −1.00000 − 2.00000i − 1.00000i 0 0.707107 − 0.707107i 1.00000i −0.707107 + 2.12132i
97.2 −0.707107 − 0.707107i 0.707107 + 0.707107i 1.00000i −1.00000 − 2.00000i − 1.00000i 0 0.707107 − 0.707107i 1.00000i −0.707107 + 2.12132i
97.3 0.707107 + 0.707107i −0.707107 − 0.707107i 1.00000i −1.00000 − 2.00000i − 1.00000i 0 −0.707107 + 0.707107i 1.00000i 0.707107 − 2.12132i
97.4 0.707107 + 0.707107i −0.707107 − 0.707107i 1.00000i −1.00000 − 2.00000i − 1.00000i 0 −0.707107 + 0.707107i 1.00000i 0.707107 − 2.12132i
1273.1 −0.707107 + 0.707107i 0.707107 − 0.707107i − 1.00000i −1.00000 + 2.00000i 1.00000i 0 0.707107 + 0.707107i − 1.00000i −0.707107 − 2.12132i
1273.2 −0.707107 + 0.707107i 0.707107 − 0.707107i − 1.00000i −1.00000 + 2.00000i 1.00000i 0 0.707107 + 0.707107i − 1.00000i −0.707107 − 2.12132i
1273.3 0.707107 − 0.707107i −0.707107 + 0.707107i − 1.00000i −1.00000 + 2.00000i 1.00000i 0 −0.707107 − 0.707107i − 1.00000i 0.707107 + 2.12132i
1273.4 0.707107 − 0.707107i −0.707107 + 0.707107i − 1.00000i −1.00000 + 2.00000i 1.00000i 0 −0.707107 − 0.707107i − 1.00000i 0.707107 + 2.12132i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.2.m.a ✓ 8
5.c odd 4 1 1470.2.m.b yes 8
7.b odd 2 1 1470.2.m.b yes 8
35.f even 4 1 inner 1470.2.m.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.m.a ✓ 8 1.a even 1 1 trivial
1470.2.m.a ✓ 8 35.f even 4 1 inner
1470.2.m.b yes 8 5.c odd 4 1
1470.2.m.b yes 8 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1470, [\chi])\):

\( T_{11}^{4} - 24T_{11}^{2} - 8T_{11} + 62 \) Copy content Toggle raw display
\( T_{13}^{8} + 8T_{13}^{7} + 32T_{13}^{6} - 80T_{13}^{5} + 152T_{13}^{4} + 1504T_{13}^{3} + 10368T_{13}^{2} - 45504T_{13} + 99856 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 2 T + 5)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 24 T^{2} + \cdots + 62)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 8 T^{7} + \cdots + 99856 \) Copy content Toggle raw display
$17$ \( T^{8} + 8 T^{7} + \cdots + 3844 \) Copy content Toggle raw display
$19$ \( (T^{2} - 4 T + 2)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 6 T + 18)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} + 136 T^{6} + \cdots + 64516 \) Copy content Toggle raw display
$31$ \( T^{8} + 128 T^{6} + \cdots + 3844 \) Copy content Toggle raw display
$37$ \( T^{8} - 8 T^{7} + \cdots + 454276 \) Copy content Toggle raw display
$41$ \( T^{8} + 208 T^{6} + \cdots + 565504 \) Copy content Toggle raw display
$43$ \( T^{8} - 32 T^{7} + \cdots + 18645124 \) Copy content Toggle raw display
$47$ \( T^{8} + 16 T^{7} + \cdots + 8655364 \) Copy content Toggle raw display
$53$ \( T^{8} + 32 T^{7} + \cdots + 312299584 \) Copy content Toggle raw display
$59$ \( (T^{4} + 8 T^{3} + \cdots + 124)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 192 T^{6} + \cdots + 984064 \) Copy content Toggle raw display
$67$ \( T^{8} + 16 T^{7} + \cdots + 12630916 \) Copy content Toggle raw display
$71$ \( (T^{4} - 16 T^{3} + \cdots + 3844)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 16 T^{7} + \cdots + 3717184 \) Copy content Toggle raw display
$79$ \( T^{8} + 208 T^{6} + \cdots + 565504 \) Copy content Toggle raw display
$83$ \( T^{8} + 384 T^{5} + \cdots + 3268864 \) Copy content Toggle raw display
$89$ \( (T^{4} + 32 T^{3} + \cdots - 2168)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 32 T^{7} + \cdots + 764854336 \) Copy content Toggle raw display
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