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Properties

Label 1764.4.k.bb
Level $1764$
Weight $4$
Character orbit 1764.k
Analytic conductor $104.079$
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] = [1764,4,Mod(361,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.361");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1764.k (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: \(104.079369250\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 27x^{6} + 10x^{5} + 446x^{4} + 62x^{3} + 3061x^{2} + 2142x + 14161 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{8}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 588)
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{5} + ( - \beta_{4} - 3 \beta_{2}) q^{11} + (3 \beta_{7} + 3 \beta_{4}) q^{13} + ( - 3 \beta_{4} - 3 \beta_{3} + \cdots - 12 \beta_1) q^{17}+ \cdots + (48 \beta_{7} - 18 \beta_{6} + \cdots - 504) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 48 q^{17} + 192 q^{19} + 192 q^{23} - 324 q^{25} - 192 q^{29} + 48 q^{31} - 256 q^{37} + 2016 q^{41} - 224 q^{43} - 864 q^{47} - 648 q^{53} - 4704 q^{55} - 336 q^{59} + 960 q^{61} - 360 q^{65} - 720 q^{67} + 2688 q^{71} + 672 q^{73} + 1984 q^{79} + 6240 q^{83} + 1360 q^{85} - 2160 q^{89} - 3744 q^{95} - 4032 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 27x^{6} + 10x^{5} + 446x^{4} + 62x^{3} + 3061x^{2} + 2142x + 14161 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 21796 \nu^{7} + 134695 \nu^{6} - 629156 \nu^{5} + 1627866 \nu^{4} - 5942222 \nu^{3} + \cdots + 283035907 ) / 256662294 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1205557 \nu^{7} - 11454035 \nu^{6} + 53501428 \nu^{5} - 460363998 \nu^{4} + \cdots - 24068474591 ) / 2309960646 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 319577 \nu^{7} - 1805905 \nu^{6} + 8435324 \nu^{5} - 86212482 \nu^{4} + 79669538 \nu^{3} + \cdots - 3794765653 ) / 329994378 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1120543 \nu^{7} + 13834525 \nu^{6} - 64620620 \nu^{5} + 391076178 \nu^{4} + \cdots + 29070621265 ) / 769986882 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 553591 \nu^{7} - 2047537 \nu^{6} + 18671351 \nu^{5} - 13516500 \nu^{4} + 207897296 \nu^{3} + \cdots + 431668097 ) / 164997189 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4603973 \nu^{7} + 12181892 \nu^{6} - 132244027 \nu^{5} + 14215002 \nu^{4} + \cdots - 7476734776 ) / 1154980323 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1576285 \nu^{7} - 11501593 \nu^{6} + 52895627 \nu^{5} - 153395616 \nu^{4} + 356720156 \nu^{3} + \cdots - 3319150975 ) / 384993441 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -5\beta_{3} + 7\beta_{2} + 14\beta_1 ) / 28 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 7\beta_{7} - 7\beta_{6} - 8\beta_{5} + 350\beta _1 - 350 ) / 28 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 21\beta_{7} - 98\beta_{6} - 115\beta_{5} + 21\beta_{4} + 115\beta_{3} - 98\beta_{2} - 644 ) / 28 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 203\beta_{4} + 324\beta_{3} - 231\beta_{2} - 5754\beta_1 ) / 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -763\beta_{7} + 1757\beta_{6} + 2554\beta_{5} - 18354\beta _1 + 18354 ) / 28 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2492\beta_{7} + 3115\beta_{6} + 4769\beta_{5} - 2492\beta_{4} - 4769\beta_{3} + 3115\beta_{2} + 57904 ) / 14 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -21364\beta_{4} - 58301\beta_{3} + 37051\beta_{2} + 473550\beta_1 ) / 28 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−1.25866 + 2.18006i
−1.65506 + 2.86665i
2.46576 − 4.27083i
1.44795 − 2.50793i
−1.25866 − 2.18006i
−1.65506 − 2.86665i
2.46576 + 4.27083i
1.44795 + 2.50793i
0 0 0 −9.57016 − 16.5760i 0 0 0 0 0
361.2 0 0 0 −4.08470 − 7.07491i 0 0 0 0 0
361.3 0 0 0 5.32752 + 9.22754i 0 0 0 0 0
361.4 0 0 0 8.32734 + 14.4234i 0 0 0 0 0
1549.1 0 0 0 −9.57016 + 16.5760i 0 0 0 0 0
1549.2 0 0 0 −4.08470 + 7.07491i 0 0 0 0 0
1549.3 0 0 0 5.32752 − 9.22754i 0 0 0 0 0
1549.4 0 0 0 8.32734 − 14.4234i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.4
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 1764.4.k.bb 8
3.b odd 2 1 588.4.i.l 8
7.b odd 2 1 1764.4.k.bd 8
7.c even 3 1 1764.4.a.bc 4
7.c even 3 1 inner 1764.4.k.bb 8
7.d odd 6 1 1764.4.a.ba 4
7.d odd 6 1 1764.4.k.bd 8
21.c even 2 1 588.4.i.k 8
21.g even 6 1 588.4.a.k yes 4
21.g even 6 1 588.4.i.k 8
21.h odd 6 1 588.4.a.j ✓ 4
21.h odd 6 1 588.4.i.l 8
84.j odd 6 1 2352.4.a.cl 4
84.n even 6 1 2352.4.a.cq 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.4.a.j ✓ 4 21.h odd 6 1
588.4.a.k yes 4 21.g even 6 1
588.4.i.k 8 21.c even 2 1
588.4.i.k 8 21.g even 6 1
588.4.i.l 8 3.b odd 2 1
588.4.i.l 8 21.h odd 6 1
1764.4.a.ba 4 7.d odd 6 1
1764.4.a.bc 4 7.c even 3 1
1764.4.k.bb 8 1.a even 1 1 trivial
1764.4.k.bb 8 7.c even 3 1 inner
1764.4.k.bd 8 7.b odd 2 1
1764.4.k.bd 8 7.d odd 6 1
2352.4.a.cl 4 84.j odd 6 1
2352.4.a.cq 4 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{8} + 412T_{5}^{6} - 1152T_{5}^{5} + 141996T_{5}^{4} - 237312T_{5}^{3} + 11763952T_{5}^{2} + 15982848T_{5} + 769951504 \) Copy content Toggle raw display
\( T_{11}^{8} + 4136 T_{11}^{6} - 165888 T_{11}^{5} + 16373424 T_{11}^{4} - 343056384 T_{11}^{3} + \cdots + 537394557184 \) Copy content Toggle raw display
\( T_{13}^{4} - 7092T_{13}^{2} + 31104T_{13} + 10008036 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 412 T^{6} + \cdots + 769951504 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 537394557184 \) Copy content Toggle raw display
$13$ \( (T^{4} - 7092 T^{2} + \cdots + 10008036)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 23669042327056 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 17\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( (T^{4} + 96 T^{3} + \cdots + 38719552)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 36\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 21\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( (T^{4} - 1008 T^{3} + \cdots + 1611829828)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 112 T^{3} + \cdots - 789373952)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 42\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 36\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{4} - 1344 T^{3} + \cdots - 17989567344)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 82\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( (T^{4} - 3120 T^{3} + \cdots + 74256064768)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 97\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( (T^{4} + 2016 T^{3} + \cdots - 580580611196)^{2} \) Copy content Toggle raw display
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