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Properties

Label 1008.2.r.m
Level $1008$
Weight $2$
Character orbit 1008.r
Analytic conductor $8.049$
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] = [1008,2,Mod(337,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, 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("1008.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.r (of order \(3\), degree \(2\), not 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: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.508277025.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 15x^{5} + 21x^{4} + 3x^{3} - 22x^{2} + 3x + 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 504)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{3} + ( - \beta_{7} + 2 \beta_{5} + \cdots + \beta_{2}) q^{5}+ \cdots + ( - \beta_{7} - \beta_{6} - \beta_{4} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 1) q^{3} + ( - \beta_{7} + 2 \beta_{5} + \cdots + \beta_{2}) q^{5}+ \cdots + ( - 8 \beta_{7} - 2 \beta_{6} + \cdots + 5 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} + 4 q^{5} + 4 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} + 4 q^{5} + 4 q^{7} + 10 q^{9} + 6 q^{11} - 3 q^{13} - 4 q^{15} - 16 q^{17} + 4 q^{19} - q^{21} + 5 q^{23} - 14 q^{25} - 5 q^{27} + q^{29} - 11 q^{31} + 8 q^{35} + 54 q^{37} + 12 q^{39} + 2 q^{41} + 11 q^{43} + 26 q^{45} - 7 q^{47} - 4 q^{49} - 17 q^{51} - 8 q^{53} - 12 q^{55} - 13 q^{57} - 9 q^{59} - 7 q^{61} + 5 q^{63} - 9 q^{65} + 12 q^{67} + 4 q^{69} + 24 q^{71} + 26 q^{73} + 23 q^{75} - 6 q^{77} + 22 q^{79} + 34 q^{81} + 6 q^{83} - 11 q^{85} - 37 q^{87} - 28 q^{89} - 6 q^{91} - 13 q^{93} + 23 q^{95} - q^{97} + 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 5x^{6} - 15x^{5} + 21x^{4} + 3x^{3} - 22x^{2} + 3x + 19 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{7} + 79\nu^{6} - 177\nu^{5} + 459\nu^{4} - 1008\nu^{3} + 1011\nu^{2} - 752\nu - 478 ) / 933 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 35\nu^{7} + 164\nu^{6} - 395\nu^{5} + 260\nu^{4} - 2687\nu^{3} + 2894\nu^{2} + 1604\nu - 2193 ) / 1866 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 217\nu^{7} + 146\nu^{6} + 39\nu^{5} - 876\nu^{4} - 4095\nu^{3} + 6498\nu^{2} + 1610\nu - 2525 ) / 5598 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 241\nu^{7} - 328\nu^{6} + 1101\nu^{5} - 3630\nu^{4} + 1953\nu^{3} - 5166\nu^{2} + 6122\nu + 343 ) / 5598 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -145\nu^{7} + 298\nu^{6} - 585\nu^{5} + 1944\nu^{4} - 2019\nu^{3} - 438\nu^{2} + 730\nu + 1799 ) / 1866 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 701\nu^{7} - 1016\nu^{6} + 1863\nu^{5} - 6966\nu^{4} + 2181\nu^{3} + 10122\nu^{2} - 6296\nu - 6265 ) / 5598 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{7} - 5\beta_{6} - \beta_{5} + \beta_{4} + 2\beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{6} + \beta_{5} - \beta_{4} - 3\beta_{3} + 6\beta_{2} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5\beta_{7} + 12\beta_{6} - \beta_{5} + 12\beta_{4} - 8\beta_{3} - 8\beta_{2} - \beta _1 - 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -29\beta_{7} - 35\beta_{6} - 7\beta_{5} + 32\beta_{4} - \beta_{3} + 2\beta_{2} - 18\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -38\beta_{7} - 77\beta_{6} + 20\beta_{5} - 13\beta_{4} - 10\beta_{3} + 92\beta_{2} + 52\beta _1 + 60 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
−0.577806 − 2.22188i
−0.734668 + 0.348716i
1.86526 + 0.199842i
0.947217 + 0.807294i
−0.577806 + 2.22188i
−0.734668 − 0.348716i
1.86526 − 0.199842i
0.947217 − 0.807294i
0 −1.71311 + 0.255482i 0 1.81197 − 3.13842i 0 0.500000 + 0.866025i 0 2.86946 − 0.875335i 0
337.2 0 0.434663 + 1.67662i 0 −1.21814 + 2.10988i 0 0.500000 + 0.866025i 0 −2.62214 + 1.45753i 0
337.3 0 1.60570 − 0.649414i 0 −0.468293 + 0.811107i 0 0.500000 + 0.866025i 0 2.15652 − 2.08552i 0
337.4 0 1.67275 + 0.449358i 0 1.87447 − 3.24667i 0 0.500000 + 0.866025i 0 2.59615 + 1.50332i 0
673.1 0 −1.71311 − 0.255482i 0 1.81197 + 3.13842i 0 0.500000 − 0.866025i 0 2.86946 + 0.875335i 0
673.2 0 0.434663 − 1.67662i 0 −1.21814 − 2.10988i 0 0.500000 − 0.866025i 0 −2.62214 − 1.45753i 0
673.3 0 1.60570 + 0.649414i 0 −0.468293 − 0.811107i 0 0.500000 − 0.866025i 0 2.15652 + 2.08552i 0
673.4 0 1.67275 − 0.449358i 0 1.87447 + 3.24667i 0 0.500000 − 0.866025i 0 2.59615 − 1.50332i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.r.m 8
3.b odd 2 1 3024.2.r.l 8
4.b odd 2 1 504.2.r.d ✓ 8
9.c even 3 1 inner 1008.2.r.m 8
9.c even 3 1 9072.2.a.ce 4
9.d odd 6 1 3024.2.r.l 8
9.d odd 6 1 9072.2.a.cl 4
12.b even 2 1 1512.2.r.d 8
36.f odd 6 1 504.2.r.d ✓ 8
36.f odd 6 1 4536.2.a.x 4
36.h even 6 1 1512.2.r.d 8
36.h even 6 1 4536.2.a.ba 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.r.d ✓ 8 4.b odd 2 1
504.2.r.d ✓ 8 36.f odd 6 1
1008.2.r.m 8 1.a even 1 1 trivial
1008.2.r.m 8 9.c even 3 1 inner
1512.2.r.d 8 12.b even 2 1
1512.2.r.d 8 36.h even 6 1
3024.2.r.l 8 3.b odd 2 1
3024.2.r.l 8 9.d odd 6 1
4536.2.a.x 4 36.f odd 6 1
4536.2.a.ba 4 36.h even 6 1
9072.2.a.ce 4 9.c even 3 1
9072.2.a.cl 4 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{8} - 4T_{5}^{7} + 25T_{5}^{6} - 22T_{5}^{5} + 166T_{5}^{4} - 13T_{5}^{3} + 1120T_{5}^{2} + 899T_{5} + 961 \) Copy content Toggle raw display
\( T_{11}^{8} - 6T_{11}^{7} + 45T_{11}^{6} - 108T_{11}^{5} + 621T_{11}^{4} - 1377T_{11}^{3} + 6075T_{11}^{2} - 4374T_{11} + 2916 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} - 4 T^{7} + \cdots + 961 \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} - 6 T^{7} + \cdots + 2916 \) Copy content Toggle raw display
$13$ \( T^{8} + 3 T^{7} + \cdots + 6084 \) Copy content Toggle raw display
$17$ \( (T^{4} + 8 T^{3} + \cdots + 274)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 2 T^{3} - 21 T^{2} + \cdots - 59)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 5 T^{7} + \cdots + 121801 \) Copy content Toggle raw display
$29$ \( T^{8} - T^{7} + \cdots + 676 \) Copy content Toggle raw display
$31$ \( T^{8} + 11 T^{7} + \cdots + 100 \) Copy content Toggle raw display
$37$ \( (T^{4} - 27 T^{3} + \cdots + 1686)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 2 T^{7} + \cdots + 4032064 \) Copy content Toggle raw display
$43$ \( T^{8} - 11 T^{7} + \cdots + 252004 \) Copy content Toggle raw display
$47$ \( T^{8} + 7 T^{7} + \cdots + 1032256 \) Copy content Toggle raw display
$53$ \( (T^{4} + 4 T^{3} + \cdots - 458)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 9 T^{7} + \cdots + 9216 \) Copy content Toggle raw display
$61$ \( T^{8} + 7 T^{7} + \cdots + 961 \) Copy content Toggle raw display
$67$ \( T^{8} - 12 T^{7} + \cdots + 144 \) Copy content Toggle raw display
$71$ \( (T^{4} - 12 T^{3} + \cdots - 6045)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 13 T^{3} + \cdots - 4394)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} - 22 T^{7} + \cdots + 108472225 \) Copy content Toggle raw display
$83$ \( T^{8} - 6 T^{7} + \cdots + 90000 \) Copy content Toggle raw display
$89$ \( (T^{4} + 14 T^{3} + \cdots - 500)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + T^{7} + \cdots + 190053796 \) Copy content Toggle raw display
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