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Properties

Label 1008.3.dc.e
Level $1008$
Weight $3$
Character orbit 1008.dc
Analytic conductor $27.466$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,3,Mod(305,1008)]
 
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, 3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.305");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1008.dc (of order \(6\), 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: \(27.4660106475\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 20x^{10} + 307x^{8} - 1824x^{6} + 8289x^{4} - 1674x^{2} + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 63)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{11} q^{5} + (4 \beta_{8} - \beta_{3} - \beta_{2} + 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{11} q^{5} + (4 \beta_{8} - \beta_{3} - \beta_{2} + 2) q^{7} + ( - 2 \beta_{4} - \beta_1) q^{11} + ( - \beta_{5} - \beta_{3} + \beta_{2} - 4) q^{13} + (\beta_{6} - 7 \beta_1) q^{17} + (\beta_{9} + 3 \beta_{8} + \beta_{5} + \cdots + 1) q^{19}+ \cdots + ( - 2 \beta_{5} + 19 \beta_{3} + \cdots + 41) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{7} - 52 q^{13} - 26 q^{19} + 106 q^{25} - 22 q^{31} - 146 q^{37} - 108 q^{43} + 114 q^{49} - 16 q^{55} - 136 q^{61} - 2 q^{67} - 482 q^{73} - 42 q^{79} - 288 q^{85} - 222 q^{91} + 568 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 20x^{10} + 307x^{8} - 1824x^{6} + 8289x^{4} - 1674x^{2} + 324 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 680\nu^{10} - 10438\nu^{8} - 90901\nu^{6} + 2229417\nu^{4} - 27566757\nu^{2} - 5393898 ) / 15067458 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 400\nu^{10} - 6140\nu^{8} + 94249\nu^{6} - 165780\nu^{4} + 33480\nu^{2} + 17064783 ) / 2511243 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4940\nu^{11} + 75829\nu^{9} - 1038413\nu^{7} + 2047383\nu^{5} - 413478\nu^{3} - 142067574\nu ) / 15067458 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6940\nu^{10} + 106529\nu^{8} - 1509658\nu^{6} + 5387526\nu^{4} - 28204551\nu^{2} - 3890862 ) / 15067458 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -7340\nu^{11} + 112669\nu^{9} - 1603907\nu^{7} + 3042063\nu^{5} - 614358\nu^{3} - 289658646\nu ) / 15067458 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -9517\nu^{11} + 187940\nu^{9} - 2884879\nu^{7} + 16793514\nu^{5} - 77891733\nu^{3} + 15730578\nu ) / 15067458 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -9517\nu^{10} + 187940\nu^{8} - 2884879\nu^{6} + 16793514\nu^{4} - 77891733\nu^{2} + 663120 ) / 15067458 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -9517\nu^{10} + 187940\nu^{8} - 2884879\nu^{6} + 16793514\nu^{4} - 75739239\nu^{2} + 663120 ) / 2152494 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 88370 \nu^{11} + 1775020 \nu^{9} - 27246557 \nu^{7} + 162605637 \nu^{5} - 735657039 \nu^{3} + 148569174 \nu ) / 15067458 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 60380\nu^{11} - 1205860\nu^{9} + 18509951\nu^{7} - 109848699\nu^{5} + 499768677\nu^{3} - 100930482\nu ) / 5022486 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - 7\beta_{8} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} - \beta_{10} - 10\beta_{7} + \beta_{6} - \beta_{4} + 10\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 11\beta_{9} - 77\beta_{8} + 4\beta_{5} + 11\beta_{3} + 2\beta_{2} - 73 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -17\beta_{11} - 23\beta_{10} - 110\beta_{7} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 40\beta_{5} + 127\beta_{3} - 40\beta_{2} - 867 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -247\beta_{6} + 367\beta_{4} - 1288\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -1535\beta_{9} + 11105\beta_{8} - 614\beta_{5} - 1228\beta_{2} - 614 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3377\beta_{11} + 5219\beta_{10} + 15710\beta_{7} - 3377\beta_{6} + 5219\beta_{4} - 15710\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -19087\beta_{9} + 139135\beta_{8} - 17192\beta_{5} - 19087\beta_{3} - 8596\beta_{2} + 121943 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 44875\beta_{11} + 70663\beta_{10} + 196396\beta_{7} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(\beta_{8}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
305.1
−0.389477 − 0.224865i
3.11017 + 1.79566i
−2.27490 − 1.31341i
2.27490 + 1.31341i
−3.11017 − 1.79566i
0.389477 + 0.224865i
−0.389477 + 0.224865i
3.11017 − 1.79566i
−2.27490 + 1.31341i
2.27490 − 1.31341i
−3.11017 + 1.79566i
0.389477 − 0.224865i
0 0 0 −7.49023 + 4.32449i 0 −6.25176 − 3.14888i 0 0 0
305.2 0 0 0 −5.33753 + 3.08163i 0 −0.220803 − 6.99652i 0 0 0
305.3 0 0 0 −3.37744 + 1.94997i 0 6.97257 + 0.619118i 0 0 0
305.4 0 0 0 3.37744 − 1.94997i 0 6.97257 + 0.619118i 0 0 0
305.5 0 0 0 5.33753 − 3.08163i 0 −0.220803 − 6.99652i 0 0 0
305.6 0 0 0 7.49023 − 4.32449i 0 −6.25176 − 3.14888i 0 0 0
737.1 0 0 0 −7.49023 − 4.32449i 0 −6.25176 + 3.14888i 0 0 0
737.2 0 0 0 −5.33753 − 3.08163i 0 −0.220803 + 6.99652i 0 0 0
737.3 0 0 0 −3.37744 − 1.94997i 0 6.97257 − 0.619118i 0 0 0
737.4 0 0 0 3.37744 + 1.94997i 0 6.97257 − 0.619118i 0 0 0
737.5 0 0 0 5.33753 + 3.08163i 0 −0.220803 + 6.99652i 0 0 0
737.6 0 0 0 7.49023 + 4.32449i 0 −6.25176 + 3.14888i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 305.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.3.dc.e 12
3.b odd 2 1 inner 1008.3.dc.e 12
4.b odd 2 1 63.3.q.a ✓ 12
7.c even 3 1 inner 1008.3.dc.e 12
12.b even 2 1 63.3.q.a ✓ 12
21.h odd 6 1 inner 1008.3.dc.e 12
28.d even 2 1 441.3.q.d 12
28.f even 6 1 441.3.b.c 6
28.f even 6 1 441.3.q.d 12
28.g odd 6 1 63.3.q.a ✓ 12
28.g odd 6 1 441.3.b.d 6
84.h odd 2 1 441.3.q.d 12
84.j odd 6 1 441.3.b.c 6
84.j odd 6 1 441.3.q.d 12
84.n even 6 1 63.3.q.a ✓ 12
84.n even 6 1 441.3.b.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.q.a ✓ 12 4.b odd 2 1
63.3.q.a ✓ 12 12.b even 2 1
63.3.q.a ✓ 12 28.g odd 6 1
63.3.q.a ✓ 12 84.n even 6 1
441.3.b.c 6 28.f even 6 1
441.3.b.c 6 84.j odd 6 1
441.3.b.d 6 28.g odd 6 1
441.3.b.d 6 84.n even 6 1
441.3.q.d 12 28.d even 2 1
441.3.q.d 12 28.f even 6 1
441.3.q.d 12 84.h odd 2 1
441.3.q.d 12 84.j odd 6 1
1008.3.dc.e 12 1.a even 1 1 trivial
1008.3.dc.e 12 3.b odd 2 1 inner
1008.3.dc.e 12 7.c even 3 1 inner
1008.3.dc.e 12 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{12} - 128T_{5}^{10} + 11827T_{5}^{8} - 496860T_{5}^{6} + 15234345T_{5}^{4} - 196944426T_{5}^{2} + 1867795524 \) Copy content Toggle raw display
\( T_{11}^{12} - 212T_{11}^{10} + 32467T_{11}^{8} - 2263200T_{11}^{6} + 115191585T_{11}^{4} - 2382632874T_{11}^{2} + 36466485444 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 1867795524 \) Copy content Toggle raw display
$7$ \( (T^{6} - T^{5} + \cdots + 117649)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 36466485444 \) Copy content Toggle raw display
$13$ \( (T^{3} + 13 T^{2} + \cdots - 2842)^{4} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 619689727770624 \) Copy content Toggle raw display
$19$ \( (T^{6} + 13 T^{5} + \cdots + 8076964)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 367505463066624 \) Copy content Toggle raw display
$29$ \( (T^{6} + 2126 T^{4} + \cdots + 64706688)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 11 T^{5} + \cdots + 115154361)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 73 T^{5} + \cdots + 166255236)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 3194 T^{4} + \cdots + 6223392)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 27 T^{2} + \cdots - 26278)^{4} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 82\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 31\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{6} + 68 T^{5} + \cdots + 13645977856)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + T^{5} + \cdots + 19642624)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 26262 T^{4} + \cdots + 95234699592)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 241 T^{5} + \cdots + 283799121984)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 21 T^{5} + \cdots + 345392465401)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 19476 T^{4} + \cdots + 31505922)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( (T^{3} - 142 T^{2} + \cdots + 125832)^{4} \) Copy content Toggle raw display
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