Properties

Label 420.2.bv.a
Level $420$
Weight $2$
Character orbit 420.bv
Analytic conductor $3.354$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [420,2,Mod(53,420)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(420, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 6, 9, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("420.53");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 420.bv (of order \(12\), 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: \(3.35371688489\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: 8.0.303595776.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - 2 \beta_{4} - \beta_{3} - 2) q^{5} + (\beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{7} + (\beta_{6} - 2 \beta_{4} - \beta_{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - 2 \beta_{4} - \beta_{3} - 2) q^{5} + (\beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{7} + (\beta_{6} - 2 \beta_{4} - \beta_{2} - 2) q^{9} + ( - \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{11} + ( - 2 \beta_{5} - 2 \beta_{3} - 2) q^{13} + (2 \beta_{7} + \beta_{6} - 2 \beta_1 + 1) q^{15} + ( - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{2} + 2 \beta_1) q^{17} + ( - 2 \beta_{6} - \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - 1) q^{19} + ( - \beta_{7} + 2 \beta_{4} - 3 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{21} + (\beta_{7} - \beta_{6} + \beta_{2}) q^{23} + ( - 4 \beta_{5} + 3 \beta_{4}) q^{25} + (2 \beta_{7} - 3 \beta_{5} - 3 \beta_{3} - 2 \beta_1) q^{27} + (2 \beta_{7} + \beta_{5} + \beta_{3} - 2 \beta_1 - 4) q^{29} - 6 \beta_{4} q^{31} + ( - \beta_{7} - \beta_{6} - \beta_{4} - 6 \beta_{3} + \beta_{2} - 1) q^{33} + ( - 3 \beta_{7} + \beta_{6} - 2 \beta_{5} - 3 \beta_{4} - \beta_{3} + 3 \beta_1 - 1) q^{35} + (4 \beta_{4} - 4 \beta_{3} + 4) q^{37} + ( - 2 \beta_{4} + 2 \beta_{2} - 2 \beta_1) q^{39} + ( - 4 \beta_{6} + 3 \beta_{5} + 3 \beta_{3} - 2) q^{41} + (\beta_{7} + \beta_{6} + 4 \beta_{5} + 4 \beta_{3} - \beta_1 + 4) q^{43} + ( - 3 \beta_{5} + 4 \beta_{4} + 2 \beta_{2} + \beta_1) q^{45} + (5 \beta_{4} + 5 \beta_{3} + 5) q^{47} + ( - 2 \beta_{6} + \beta_{4} + 4 \beta_{3} - 2 \beta_{2} - 1) q^{49} + ( - 2 \beta_{7} - 6 \beta_{4} - 6 \beta_{3} - 6) q^{51} + (5 \beta_{5} + 5 \beta_{4} - 2 \beta_{2} - 2 \beta_1) q^{53} + ( - 2 \beta_{7} + 4 \beta_{6} + \beta_{5} + \beta_{3} + 2 \beta_1 + 1) q^{55} + (\beta_{7} + 3 \beta_{6} + 6 \beta_{5} + 6 \beta_{3} - \beta_1 + 3) q^{57} + ( - \beta_{5} - 9 \beta_{4} + 2 \beta_1) q^{59} + (2 \beta_{7} - 6 \beta_{4} + \beta_{3} - 6) q^{61} + ( - 2 \beta_{7} + 3 \beta_{6} - 2 \beta_{4} + 3 \beta_{3} - \beta_{2} + 3 \beta_1 + 3) q^{63} + (2 \beta_{4} + 6 \beta_{3} + 2) q^{65} + (6 \beta_{5} + 6 \beta_{4} - \beta_{2} - \beta_1) q^{67} + (\beta_{6} + 3 \beta_{5} + 3 \beta_{3} - 2) q^{69} + ( - 2 \beta_{6} - \beta_{5} - \beta_{3} - 1) q^{71} + ( - 2 \beta_{5} + 2 \beta_{4} - 4 \beta_{2} + 4 \beta_1) q^{73} + ( - 3 \beta_{7} - 4 \beta_{6} - 4 \beta_{4} + 4 \beta_{2} - 4) q^{75} + ( - 2 \beta_{5} + 7 \beta_{4} + 5 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 5) q^{77} + (2 \beta_{6} + \beta_{4} + 9 \beta_{3} - 2 \beta_{2} + 1) q^{79} + (\beta_{4} + 5 \beta_{2}) q^{81} + (\beta_{7} - \beta_{6} - 5 \beta_{5} - 5 \beta_{3} - \beta_1 + 5) q^{83} + (2 \beta_{7} + 6 \beta_{6} + 4 \beta_{5} + 4 \beta_{3} - 2 \beta_1 + 4) q^{85} + (5 \beta_{4} + \beta_{2} - 4 \beta_1) q^{87} + ( - 10 \beta_{7} - 5 \beta_{3}) q^{89} + ( - 2 \beta_{5} - 2 \beta_{4} + 4 \beta_1 - 4) q^{91} + 6 \beta_{7} q^{93} + ( - 5 \beta_{5} - \beta_{4} - 4 \beta_{2} - 2 \beta_1) q^{95} + (2 \beta_{7} - 2 \beta_{6} + 7 \beta_{5} + 7 \beta_{3} - 2 \beta_1 - 7) q^{97} + (\beta_{7} + 5 \beta_{6} + 3 \beta_{5} + 3 \beta_{3} - \beta_1 + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{5} + 6 q^{7} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{5} + 6 q^{7} - 10 q^{9} - 16 q^{13} + 4 q^{15} - 4 q^{17} + 14 q^{21} + 2 q^{23} - 12 q^{25} - 32 q^{29} + 24 q^{31} - 2 q^{33} + 16 q^{37} + 4 q^{39} + 28 q^{43} - 20 q^{45} + 20 q^{47} - 24 q^{51} - 16 q^{53} - 8 q^{55} + 12 q^{57} + 36 q^{59} - 24 q^{61} + 22 q^{63} + 8 q^{65} - 22 q^{67} - 20 q^{69} - 8 q^{75} + 16 q^{77} - 14 q^{81} + 44 q^{83} + 8 q^{85} - 22 q^{87} - 24 q^{91} + 12 q^{95} - 48 q^{97} + 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 32\nu^{4} + 16\nu^{2} + 45 ) / 144 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 32\nu^{5} + 16\nu^{3} + 45\nu ) / 432 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{6} - 16\nu^{4} - 80\nu^{2} - 225 ) / 144 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 13\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} - 13 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{7} + 16\nu^{5} + 80\nu^{3} + 225\nu ) / 144 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 2\beta_{4} - \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} - 3\beta_{5} - 3\beta_{3} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{7} + 15\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16\beta_{6} - 13 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 48\beta_{5} - 13\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/420\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(241\) \(281\) \(337\)
\(\chi(n)\) \(1\) \(-1 - \beta_{4}\) \(-1\) \(-\beta_{3} - \beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−0.396143 + 1.68614i
1.26217 1.18614i
−1.26217 + 1.18614i
0.396143 1.68614i
−1.26217 1.18614i
0.396143 + 1.68614i
−0.396143 1.68614i
1.26217 + 1.18614i
0 −0.396143 + 1.68614i 0 −0.133975 2.23205i 0 0.576028 2.58228i 0 −2.68614 1.33591i 0
53.2 0 1.26217 1.18614i 0 −0.133975 2.23205i 0 1.79000 + 1.94831i 0 0.186141 2.99422i 0
137.1 0 −1.26217 + 1.18614i 0 −1.86603 1.23205i 0 2.58228 + 0.576028i 0 0.186141 2.99422i 0
137.2 0 0.396143 1.68614i 0 −1.86603 1.23205i 0 −1.94831 + 1.79000i 0 −2.68614 1.33591i 0
233.1 0 −1.26217 1.18614i 0 −1.86603 + 1.23205i 0 2.58228 0.576028i 0 0.186141 + 2.99422i 0
233.2 0 0.396143 + 1.68614i 0 −1.86603 + 1.23205i 0 −1.94831 1.79000i 0 −2.68614 + 1.33591i 0
317.1 0 −0.396143 1.68614i 0 −0.133975 + 2.23205i 0 0.576028 + 2.58228i 0 −2.68614 + 1.33591i 0
317.2 0 1.26217 + 1.18614i 0 −0.133975 + 2.23205i 0 1.79000 1.94831i 0 0.186141 + 2.99422i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
15.e even 4 1 inner
105.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 420.2.bv.a 8
3.b odd 2 1 420.2.bv.b yes 8
5.c odd 4 1 420.2.bv.b yes 8
7.c even 3 1 inner 420.2.bv.a 8
15.e even 4 1 inner 420.2.bv.a 8
21.h odd 6 1 420.2.bv.b yes 8
35.l odd 12 1 420.2.bv.b yes 8
105.x even 12 1 inner 420.2.bv.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.bv.a 8 1.a even 1 1 trivial
420.2.bv.a 8 7.c even 3 1 inner
420.2.bv.a 8 15.e even 4 1 inner
420.2.bv.a 8 105.x even 12 1 inner
420.2.bv.b yes 8 3.b odd 2 1
420.2.bv.b yes 8 5.c odd 4 1
420.2.bv.b yes 8 21.h odd 6 1
420.2.bv.b yes 8 35.l odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(420, [\chi])\):

\( T_{11}^{8} - 24T_{11}^{6} + 476T_{11}^{4} - 2400T_{11}^{2} + 10000 \) Copy content Toggle raw display
\( T_{17}^{8} + 4T_{17}^{7} + 8T_{17}^{6} + 192T_{17}^{5} - 16T_{17}^{4} - 3840T_{17}^{3} + 3200T_{17}^{2} - 32000T_{17} + 160000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 5 T^{6} + 16 T^{4} + 45 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{4} + 4 T^{3} + 11 T^{2} + 20 T + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 6 T^{7} + 18 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} - 24 T^{6} + 476 T^{4} + \cdots + 10000 \) Copy content Toggle raw display
$13$ \( (T^{2} + 4 T + 8)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} + 4 T^{7} + 8 T^{6} + \cdots + 160000 \) Copy content Toggle raw display
$19$ \( T^{8} - 40 T^{6} + 1596 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{8} - 2 T^{7} + 2 T^{6} - 24 T^{5} + \cdots + 625 \) Copy content Toggle raw display
$29$ \( (T^{2} + 8 T + 5)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 6 T + 36)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 8 T^{3} + 32 T^{2} - 256 T + 1024)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 106 T^{2} + 1225)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 14 T^{3} + 98 T^{2} - 266 T + 361)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 10 T^{3} + 50 T^{2} - 500 T + 2500)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 10000 \) Copy content Toggle raw display
$59$ \( (T^{4} - 18 T^{3} + 254 T^{2} - 1260 T + 4900)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 12 T^{3} + 119 T^{2} + 300 T + 625)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 22 T^{7} + 242 T^{6} + \cdots + 9150625 \) Copy content Toggle raw display
$71$ \( (T^{4} + 24 T^{2} + 100)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 7744 T^{4} + \cdots + 59969536 \) Copy content Toggle raw display
$79$ \( T^{8} - 184 T^{6} + \cdots + 24010000 \) Copy content Toggle raw display
$83$ \( (T^{4} - 22 T^{3} + 242 T^{2} - 1210 T + 3025)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 275 T^{2} + 75625)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 24 T^{3} + 288 T^{2} + 1200 T + 2500)^{2} \) Copy content Toggle raw display
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