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Properties

Label 1428.2.q.g
Level $1428$
Weight $2$
Character orbit 1428.q
Analytic conductor $11.403$
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] = [1428,2,Mod(205,1428)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1428, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1428.205");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1428 = 2^{2} \cdot 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1428.q (of order \(3\), 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: \(11.4026374086\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.1405125225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 4x^{6} - 2x^{5} + 43x^{4} - 28x^{3} - 7x^{2} - 49x + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + ( - \beta_{7} + \beta_{6} - \beta_{4} + 1) q^{5} + (\beta_{6} - \beta_{2} - \beta_1) q^{7} + (\beta_{4} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + ( - \beta_{7} + \beta_{6} - \beta_{4} + 1) q^{5} + (\beta_{6} - \beta_{2} - \beta_1) q^{7} + (\beta_{4} - 1) q^{9} + (\beta_{6} - \beta_{2}) q^{11} + ( - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{13} + (\beta_{2} + \beta_1 + 1) q^{15} - \beta_{4} q^{17} + ( - \beta_{7} - \beta_{5}) q^{19} + ( - \beta_{7} + \beta_{6} - \beta_1) q^{21} + ( - \beta_{7} - \beta_{4} + 1) q^{23} + ( - \beta_{7} + \beta_{6} + \cdots - \beta_1) q^{25}+ \cdots + \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} + 2 q^{5} + 3 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} + 2 q^{5} + 3 q^{7} - 4 q^{9} + q^{11} + 4 q^{13} + 4 q^{15} - 4 q^{17} + q^{19} + 3 q^{23} - 2 q^{25} - 8 q^{27} - 6 q^{29} + 10 q^{31} - q^{33} - 30 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{41} - 4 q^{43} + 2 q^{45} + 12 q^{47} + 5 q^{49} + 4 q^{51} - 31 q^{53} - 20 q^{55} + 2 q^{57} + 9 q^{59} - 3 q^{61} - 3 q^{63} + q^{65} + 18 q^{67} + 6 q^{69} + 22 q^{71} + 3 q^{73} + 2 q^{75} - 23 q^{77} + q^{79} - 4 q^{81} - 32 q^{83} - 4 q^{85} - 3 q^{87} - 4 q^{89} - 23 q^{91} - 10 q^{93} - 11 q^{95} - 38 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 4x^{6} - 2x^{5} + 43x^{4} - 28x^{3} - 7x^{2} - 49x + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -4\nu^{7} + 32\nu^{6} + 31\nu^{5} - 109\nu^{4} - 507\nu^{3} + 486\nu^{2} + 861\nu + 665 ) / 483 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 12\nu^{7} - 27\nu^{6} - 70\nu^{5} + 51\nu^{4} + 578\nu^{3} - 423\nu^{2} - 812\nu + 98 ) / 483 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -14\nu^{7} + 20\nu^{6} + 51\nu^{5} + 67\nu^{4} - 544\nu^{3} + 321\nu^{2} + 518\nu + 637 ) / 483 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -90\nu^{7} + 76\nu^{6} + 456\nu^{5} + 687\nu^{4} - 3116\nu^{3} - 1140\nu^{2} - 189\nu + 4256 ) / 483 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 104\nu^{7} - 96\nu^{6} - 507\nu^{5} - 754\nu^{4} + 3660\nu^{3} + 819\nu^{2} + 1120\nu - 4893 ) / 483 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{7} + 5\nu^{6} + 23\nu^{5} + 37\nu^{4} - 170\nu^{3} - 33\nu^{2} - 63\nu + 231 ) / 21 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -281\nu^{7} + 247\nu^{6} + 1390\nu^{5} + 2158\nu^{4} - 9598\nu^{3} - 2877\nu^{2} - 1862\nu + 12222 ) / 483 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{7} - 3\beta_{6} - \beta_{5} - 7\beta_{4} + 2\beta_{3} + 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{7} + 3\beta_{6} + 8\beta_{5} - 4\beta_{4} - 4\beta_{3} - 6\beta_{2} - 3\beta _1 + 18 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{7} - 4\beta_{6} + 3\beta_{5} - 10\beta_{4} + 3\beta_{3} + 6\beta_{2} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 51\beta_{7} - 39\beta_{6} + 14\beta_{5} - 91\beta_{4} - 28\beta_{3} - 12\beta_{2} + 12\beta _1 + 105 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 51\beta_{7} + 51\beta_{6} + 130\beta_{5} - 65\beta_{4} - 65\beta_{3} + 15\beta_{2} + 66\beta _1 + 30 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 297\beta_{7} - 312\beta_{6} - 17\beta_{5} - 536\beta_{4} - 17\beta_{3} + 297\beta_{2} + 312\beta_1 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1428\mathbb{Z}\right)^\times\).

\(n\) \(409\) \(715\) \(953\) \(1261\)
\(\chi(n)\) \(-1 + \beta_{4}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
205.1
−1.78862 + 1.61001i
2.28862 − 0.743984i
1.09796 − 0.0565565i
−0.597959 + 0.922582i
−1.78862 − 1.61001i
2.28862 + 0.743984i
1.09796 + 0.0565565i
−0.597959 − 0.922582i
0 0.500000 + 0.866025i 0 −0.895644 + 1.55130i 0 1.55575 + 2.14001i 0 −0.500000 + 0.866025i 0
205.2 0 0.500000 + 0.866025i 0 −0.895644 + 1.55130i 0 2.63118 + 0.277320i 0 −0.500000 + 0.866025i 0
205.3 0 0.500000 + 0.866025i 0 1.39564 − 2.41733i 0 −2.41508 + 1.08045i 0 −0.500000 + 0.866025i 0
205.4 0 0.500000 + 0.866025i 0 1.39564 − 2.41733i 0 −0.271847 − 2.63175i 0 −0.500000 + 0.866025i 0
613.1 0 0.500000 − 0.866025i 0 −0.895644 − 1.55130i 0 1.55575 − 2.14001i 0 −0.500000 − 0.866025i 0
613.2 0 0.500000 − 0.866025i 0 −0.895644 − 1.55130i 0 2.63118 − 0.277320i 0 −0.500000 − 0.866025i 0
613.3 0 0.500000 − 0.866025i 0 1.39564 + 2.41733i 0 −2.41508 − 1.08045i 0 −0.500000 − 0.866025i 0
613.4 0 0.500000 − 0.866025i 0 1.39564 + 2.41733i 0 −0.271847 + 2.63175i 0 −0.500000 − 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 205.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1428.2.q.g ✓ 8
7.c even 3 1 inner 1428.2.q.g ✓ 8
7.c even 3 1 9996.2.a.y 4
7.d odd 6 1 9996.2.a.bc 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1428.2.q.g ✓ 8 1.a even 1 1 trivial
1428.2.q.g ✓ 8 7.c even 3 1 inner
9996.2.a.y 4 7.c even 3 1
9996.2.a.bc 4 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - T_{5}^{3} + 6T_{5}^{2} + 5T_{5} + 25 \) acting on \(S_{2}^{\mathrm{new}}(1428, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{3} + 6 T^{2} + \cdots + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 3 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} - T^{7} + 9 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( (T^{4} - 2 T^{3} + \cdots + 249)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$19$ \( T^{8} - T^{7} + \cdots + 2025 \) Copy content Toggle raw display
$23$ \( T^{8} - 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( (T^{4} + 3 T^{3} + \cdots - 189)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 10 T^{7} + \cdots + 5625 \) Copy content Toggle raw display
$37$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 4 T^{3} + \cdots - 147)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 2 T^{3} + \cdots + 249)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 12 T^{7} + \cdots + 164025 \) Copy content Toggle raw display
$53$ \( T^{8} + 31 T^{7} + \cdots + 986049 \) Copy content Toggle raw display
$59$ \( T^{8} - 9 T^{7} + \cdots + 82369 \) Copy content Toggle raw display
$61$ \( T^{8} + 3 T^{7} + \cdots + 59049 \) Copy content Toggle raw display
$67$ \( T^{8} - 18 T^{7} + \cdots + 18225 \) Copy content Toggle raw display
$71$ \( (T^{4} - 11 T^{3} + \cdots - 1701)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 3 T^{7} + \cdots + 105452361 \) Copy content Toggle raw display
$79$ \( T^{8} - T^{7} + \cdots + 81 \) Copy content Toggle raw display
$83$ \( (T^{4} + 16 T^{3} + \cdots + 1401)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 4 T^{7} + \cdots + 61261929 \) Copy content Toggle raw display
$97$ \( (T^{4} + 19 T^{3} + \cdots + 697)^{2} \) Copy content Toggle raw display
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