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Properties

Label 588.6.i.o
Level $588$
Weight $6$
Character orbit 588.i
Analytic conductor $94.306$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [588,6,Mod(361,588)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(588, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.361");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 588.i (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: \(94.3056860500\)
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: \( x^{8} - 2x^{7} + 703x^{6} + 2770x^{5} + 427565x^{4} + 718170x^{3} + 42175732x^{2} - 40929504x + 3559792896 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 84)
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.

\(f(q)\) \(=\) \( q + (9 \beta_1 - 9) q^{3} + (\beta_{4} - \beta_{3}) q^{5} - 81 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (9 \beta_1 - 9) q^{3} + (\beta_{4} - \beta_{3}) q^{5} - 81 \beta_1 q^{9} + (\beta_{6} + \beta_{2} + 116 \beta_1 - 116) q^{11} + (\beta_{7} + \beta_{6} - 3 \beta_{5} + \cdots + 149) q^{13}+ \cdots + ( - 81 \beta_{7} - 81 \beta_{6} + \cdots + 9396) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 36 q^{3} - 324 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 36 q^{3} - 324 q^{9} - 462 q^{11} + 1204 q^{13} - 228 q^{17} - 358 q^{19} - 2148 q^{23} - 5454 q^{25} + 5832 q^{27} - 11064 q^{29} - 830 q^{31} - 4158 q^{33} - 3914 q^{37} - 5418 q^{39} + 16632 q^{41} - 29036 q^{43} - 41700 q^{47} - 2052 q^{51} + 22164 q^{53} - 7784 q^{55} + 6444 q^{57} - 32886 q^{59} - 83732 q^{61} - 93192 q^{65} - 80034 q^{67} + 38664 q^{69} + 89544 q^{71} + 22470 q^{73} - 49086 q^{75} - 75286 q^{79} - 26244 q^{81} + 34836 q^{83} + 278504 q^{85} + 49788 q^{87} - 28944 q^{89} - 7470 q^{93} - 144120 q^{95} + 433356 q^{97} + 74844 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 703x^{6} + 2770x^{5} + 427565x^{4} + 718170x^{3} + 42175732x^{2} - 40929504x + 3559792896 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 202509581 \nu^{7} - 15579673694 \nu^{6} - 97000119635 \nu^{5} - 10465172425490 \nu^{4} + \cdots - 71\!\cdots\!48 ) / 59\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11365172267 \nu^{7} + 2784758446418 \nu^{6} + 17338097559845 \nu^{5} + \cdots + 11\!\cdots\!16 ) / 16\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4074390615 \nu^{7} + 75239346984 \nu^{6} + 2524519237925 \nu^{5} + 9170584770500 \nu^{4} + \cdots - 91\!\cdots\!92 ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1106077550275 \nu^{7} - 64933012777282 \nu^{6} - 404277402169405 \nu^{5} + \cdots - 27\!\cdots\!84 ) / 36\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3324229413173 \nu^{7} + 34725520373122 \nu^{6} + \cdots + 27\!\cdots\!04 ) / 12\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 65758365195 \nu^{7} - 47855487758 \nu^{6} - 297951557195 \nu^{5} + 436249749491390 \nu^{4} + \cdots - 20\!\cdots\!96 ) / 19\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 307834459245 \nu^{7} - 1338826811442 \nu^{6} + 231778524130995 \nu^{5} + \cdots - 16\!\cdots\!04 ) / 19\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} - 13\beta_{4} - 16\beta_{2} - 134\beta _1 + 134 ) / 252 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} - 31\beta_{5} + 215\beta_{4} - 215\beta_{3} - 22066\beta_1 ) / 63 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -187\beta_{7} - 187\beta_{6} - 1220\beta_{5} + 305\beta_{3} + 1220\beta_{2} + 1220\beta _1 - 56950 ) / 36 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -209\beta_{6} - 16332\beta_{4} + 3187\beta_{2} + 1323253\beta _1 - 1323253 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 995191\beta_{7} + 5329364\beta_{5} - 1049131\beta_{4} + 1049131\beta_{3} + 421240394\beta_1 ) / 252 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 273940 \beta_{7} + 273940 \beta_{6} + 3189875 \beta_{5} + 12867895 \beta_{3} - 3189875 \beta_{2} + \cdots + 1041120367 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 638040437\beta_{6} + 1984662401\beta_{4} - 3473034028\beta_{2} - 365369003882\beta _1 + 365369003882 ) / 252 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{1}\)

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
−11.2416 + 19.4709i
13.1471 − 22.7714i
−5.49618 + 9.51967i
4.59067 − 7.95128i
−11.2416 − 19.4709i
13.1471 + 22.7714i
−5.49618 − 9.51967i
4.59067 + 7.95128i
0 −4.50000 + 7.79423i 0 −46.4128 − 80.3893i 0 0 0 −40.5000 − 70.1481i 0
361.2 0 −4.50000 + 7.79423i 0 −15.9808 − 27.6796i 0 0 0 −40.5000 − 70.1481i 0
361.3 0 −4.50000 + 7.79423i 0 23.0577 + 39.9371i 0 0 0 −40.5000 − 70.1481i 0
361.4 0 −4.50000 + 7.79423i 0 39.3359 + 68.1317i 0 0 0 −40.5000 − 70.1481i 0
373.1 0 −4.50000 − 7.79423i 0 −46.4128 + 80.3893i 0 0 0 −40.5000 + 70.1481i 0
373.2 0 −4.50000 − 7.79423i 0 −15.9808 + 27.6796i 0 0 0 −40.5000 + 70.1481i 0
373.3 0 −4.50000 − 7.79423i 0 23.0577 − 39.9371i 0 0 0 −40.5000 + 70.1481i 0
373.4 0 −4.50000 − 7.79423i 0 39.3359 − 68.1317i 0 0 0 −40.5000 + 70.1481i 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 588.6.i.o 8
7.b odd 2 1 84.6.i.c ✓ 8
7.c even 3 1 588.6.a.p 4
7.c even 3 1 inner 588.6.i.o 8
7.d odd 6 1 84.6.i.c ✓ 8
7.d odd 6 1 588.6.a.n 4
21.c even 2 1 252.6.k.f 8
21.g even 6 1 252.6.k.f 8
28.d even 2 1 336.6.q.i 8
28.f even 6 1 336.6.q.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.6.i.c ✓ 8 7.b odd 2 1
84.6.i.c ✓ 8 7.d odd 6 1
252.6.k.f 8 21.c even 2 1
252.6.k.f 8 21.g even 6 1
336.6.q.i 8 28.d even 2 1
336.6.q.i 8 28.f even 6 1
588.6.a.n 4 7.d odd 6 1
588.6.a.p 4 7.c even 3 1
588.6.i.o 8 1.a even 1 1 trivial
588.6.i.o 8 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 8977 T_{5}^{6} - 165000 T_{5}^{5} + 69822853 T_{5}^{4} - 740602500 T_{5}^{3} + \cdots + 115856721032976 \) acting on \(S_{6}^{\mathrm{new}}(588, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9 T + 81)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 115856721032976 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 25\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( (T^{4} - 602 T^{3} + \cdots + 755795447424)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 23\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 74\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 18\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 333378202056336)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 10\!\cdots\!49 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 15\!\cdots\!88)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 359515701753932)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 58\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 88\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 38\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 32\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 41\!\cdots\!92)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 34\!\cdots\!61 \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 53\!\cdots\!08)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 37\!\cdots\!72)^{2} \) Copy content Toggle raw display
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