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Properties

Label 588.4.f.c
Level $588$
Weight $4$
Character orbit 588.f
Analytic conductor $34.693$
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] = [588,4,Mod(293,588)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(588, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.293");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 588.f (of order \(2\), degree \(1\), 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: \(34.6931230834\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 14 x^{10} - 32 x^{9} + 70 x^{8} + 224 x^{7} - 50 x^{6} + 2016 x^{5} + 5670 x^{4} + \cdots + 531441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{7} \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + \beta_{7} q^{5} + (\beta_{10} + \beta_{9} + 14) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + \beta_{7} q^{5} + (\beta_{10} + \beta_{9} + 14) q^{9} + (\beta_{11} - 2 \beta_{10} + \cdots + \beta_{4}) q^{11}+ \cdots + (18 \beta_{11} - 12 \beta_{10} + \cdots + 768) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 168 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 168 q^{9} + 132 q^{15} + 888 q^{25} - 480 q^{37} + 864 q^{39} + 600 q^{51} + 180 q^{57} - 3960 q^{67} + 876 q^{79} - 2016 q^{81} - 6144 q^{85} - 1764 q^{93} + 9216 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 14 x^{10} - 32 x^{9} + 70 x^{8} + 224 x^{7} - 50 x^{6} + 2016 x^{5} + 5670 x^{4} + \cdots + 531441 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 533 \nu^{11} + 1872 \nu^{10} + 901 \nu^{9} - 14984 \nu^{8} - 64409 \nu^{7} + 67052 \nu^{6} + \cdots - 151874028 ) / 23147208 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 36 \nu^{11} + 109 \nu^{10} - 180 \nu^{9} - 1949 \nu^{8} - 2588 \nu^{7} + 1681 \nu^{6} + \cdots - 7722297 ) / 1285956 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 109 \nu^{11} - 324 \nu^{10} + 797 \nu^{9} + 5108 \nu^{8} + 6383 \nu^{7} - 9188 \nu^{6} + \cdots + 19131876 ) / 2571912 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 14 \nu^{11} + 115 \nu^{9} + 448 \nu^{8} + 154 \nu^{7} - 544 \nu^{6} - 4970 \nu^{5} + \cdots + 944784 ) / 275562 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1390 \nu^{11} - 9567 \nu^{10} - 3422 \nu^{9} + 107683 \nu^{8} + 323254 \nu^{7} - 595255 \nu^{6} + \cdots + 546734691 ) / 23147208 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3281 \nu^{11} + 9792 \nu^{10} - 6325 \nu^{9} - 131272 \nu^{8} - 342955 \nu^{7} - 217436 \nu^{6} + \cdots - 616235364 ) / 23147208 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 125 \nu^{11} + 477 \nu^{10} - 454 \nu^{9} - 9949 \nu^{8} - 18097 \nu^{7} + 9712 \nu^{6} + \cdots - 40743810 ) / 826686 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 44 \nu^{11} - 81 \nu^{10} + 454 \nu^{9} + 1813 \nu^{8} + 1780 \nu^{7} - 136 \nu^{6} + \cdots + 5471874 ) / 275562 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1373 \nu^{11} - 2916 \nu^{10} + 10393 \nu^{9} + 58516 \nu^{8} + 61759 \nu^{7} + \cdots + 198640836 ) / 7715736 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1543 \nu^{11} - 3645 \nu^{10} + 4187 \nu^{9} + 67601 \nu^{8} + 75293 \nu^{7} + 120523 \nu^{6} + \cdots + 214682481 ) / 7715736 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 4909 \nu^{11} - 18306 \nu^{10} + 23285 \nu^{9} + 248618 \nu^{8} + 346895 \nu^{7} + \cdots + 755236710 ) / 7715736 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} + \beta_{4} + 3\beta_{3} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} + \beta_{9} + \beta_{5} + 13\beta_{2} + 2\beta _1 + 14 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} - 2\beta_{10} - 3\beta_{9} - \beta_{8} + 9\beta_{4} + 24 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{10} - 21\beta_{9} + 9\beta_{8} + 5\beta_{5} - 2\beta_{4} + 21\beta_{3} - 70\beta_{2} + 37\beta _1 + 56 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 14 \beta_{11} - 12 \beta_{10} + 19 \beta_{9} - 14 \beta_{8} - 42 \beta_{7} + 17 \beta_{6} - 12 \beta_{5} + \cdots + 560 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 16\beta_{11} + 24\beta_{10} - 216\beta_{9} + 110\beta_{8} + 116\beta_{4} + 745 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 70 \beta_{11} - 164 \beta_{10} - 251 \beta_{9} + 214 \beta_{8} - 210 \beta_{7} - 644 \beta_{6} + \cdots - 3920 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 448 \beta_{11} + 45 \beta_{10} - 803 \beta_{9} + 182 \beta_{8} - 1344 \beta_{7} + 1840 \beta_{6} + \cdots + 2128 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 311\beta_{11} - 1966\beta_{10} - 2389\beta_{9} + 3721\beta_{8} + 517\beta_{4} + 10104 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 5376 \beta_{11} + 3415 \beta_{10} - 4681 \beta_{9} + 8175 \beta_{8} - 16128 \beta_{7} + 1344 \beta_{6} + \cdots - 39466 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 10136 \beta_{11} + 2608 \beta_{10} - 28519 \beta_{9} + 38248 \beta_{8} - 30408 \beta_{7} + 45761 \beta_{6} + \cdots + 261184 ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
293.1
−2.14770 − 2.09461i
−2.14770 + 2.09461i
−1.67777 − 2.48698i
−1.67777 + 2.48698i
−2.92030 − 0.686929i
−2.92030 + 0.686929i
0.865250 − 2.87252i
0.865250 + 2.87252i
2.99268 + 0.209499i
2.99268 − 0.209499i
2.88784 + 0.812653i
2.88784 − 0.812653i
0 −5.03553 − 1.28196i 0 −19.3778 0 0 0 23.7132 + 12.9107i 0
293.2 0 −5.03553 + 1.28196i 0 −19.3778 0 0 0 23.7132 − 12.9107i 0
293.3 0 −4.67045 − 2.27749i 0 14.8312 0 0 0 16.6261 + 21.2737i 0
293.4 0 −4.67045 + 2.27749i 0 14.8312 0 0 0 16.6261 − 21.2737i 0
293.5 0 −3.78555 − 3.55944i 0 −1.23911 0 0 0 1.66071 + 26.9489i 0
293.6 0 −3.78555 + 3.55944i 0 −1.23911 0 0 0 1.66071 − 26.9489i 0
293.7 0 3.78555 − 3.55944i 0 1.23911 0 0 0 1.66071 − 26.9489i 0
293.8 0 3.78555 + 3.55944i 0 1.23911 0 0 0 1.66071 + 26.9489i 0
293.9 0 4.67045 − 2.27749i 0 −14.8312 0 0 0 16.6261 − 21.2737i 0
293.10 0 4.67045 + 2.27749i 0 −14.8312 0 0 0 16.6261 + 21.2737i 0
293.11 0 5.03553 − 1.28196i 0 19.3778 0 0 0 23.7132 − 12.9107i 0
293.12 0 5.03553 + 1.28196i 0 19.3778 0 0 0 23.7132 + 12.9107i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 293.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.4.f.c 12
3.b odd 2 1 inner 588.4.f.c 12
7.b odd 2 1 inner 588.4.f.c 12
7.c even 3 1 84.4.k.c ✓ 12
7.c even 3 1 588.4.k.c 12
7.d odd 6 1 84.4.k.c ✓ 12
7.d odd 6 1 588.4.k.c 12
21.c even 2 1 inner 588.4.f.c 12
21.g even 6 1 84.4.k.c ✓ 12
21.g even 6 1 588.4.k.c 12
21.h odd 6 1 84.4.k.c ✓ 12
21.h odd 6 1 588.4.k.c 12
28.f even 6 1 336.4.bc.c 12
28.g odd 6 1 336.4.bc.c 12
84.j odd 6 1 336.4.bc.c 12
84.n even 6 1 336.4.bc.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.k.c ✓ 12 7.c even 3 1
84.4.k.c ✓ 12 7.d odd 6 1
84.4.k.c ✓ 12 21.g even 6 1
84.4.k.c ✓ 12 21.h odd 6 1
336.4.bc.c 12 28.f even 6 1
336.4.bc.c 12 28.g odd 6 1
336.4.bc.c 12 84.j odd 6 1
336.4.bc.c 12 84.n even 6 1
588.4.f.c 12 1.a even 1 1 trivial
588.4.f.c 12 3.b odd 2 1 inner
588.4.f.c 12 7.b odd 2 1 inner
588.4.f.c 12 21.c even 2 1 inner
588.4.k.c 12 7.c even 3 1
588.4.k.c 12 7.d odd 6 1
588.4.k.c 12 21.g even 6 1
588.4.k.c 12 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(588, [\chi])\):

\( T_{5}^{6} - 597T_{5}^{4} + 83511T_{5}^{2} - 126819 \) Copy content Toggle raw display
\( T_{13}^{6} + 11724T_{13}^{4} + 42650496T_{13}^{2} + 49422707712 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 387420489 \) Copy content Toggle raw display
$5$ \( (T^{6} - 597 T^{4} + \cdots - 126819)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} + 5967 T^{4} + \cdots + 3424113)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 11724 T^{4} + \cdots + 49422707712)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 19902 T^{4} + \cdots - 3494624364)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 15402 T^{4} + \cdots + 33716904588)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 31338 T^{4} + \cdots + 671126148)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 55377 T^{4} + \cdots + 42952073472)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 2248345829547)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 120 T^{2} + \cdots - 486598)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 18213269969664)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 100848 T + 12209024)^{4} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 13\!\cdots\!44)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 13410336541833)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 19\!\cdots\!99)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 687780167594688)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + 990 T^{2} + \cdots - 106815638)^{4} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 15\!\cdots\!72)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 13\!\cdots\!72)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 219 T^{2} + \cdots + 48980551)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 18\!\cdots\!04)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 11\!\cdots\!76)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 59\!\cdots\!52)^{2} \) Copy content Toggle raw display
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