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Properties

Label 588.7.d.a
Level $588$
Weight $7$
Character orbit 588.d
Analytic conductor $135.272$
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] = [588,7,Mod(97,588)]
 
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, 0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.97");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 588.d (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: \(135.271801168\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2 x^{7} + 1061 x^{6} + 35442 x^{5} + 1155979 x^{4} + 17325616 x^{3} + 201523590 x^{2} + \cdots + 5192355364 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{4}\cdot 7^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 9 \beta_1 q^{3} + (\beta_{3} + 24 \beta_1) q^{5} - 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 \beta_1 q^{3} + (\beta_{3} + 24 \beta_1) q^{5} - 243 q^{9} + ( - 3 \beta_{6} + \beta_{5} + \cdots - 93) q^{11}+ \cdots + (729 \beta_{6} - 243 \beta_{5} + \cdots + 22599) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 1944 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 1944 q^{9} - 756 q^{11} - 5292 q^{15} - 31200 q^{23} - 6772 q^{25} - 68604 q^{29} - 31828 q^{37} - 50652 q^{39} - 170044 q^{43} - 15336 q^{51} - 392820 q^{53} + 66420 q^{57} - 29784 q^{65} - 1549676 q^{67} - 721896 q^{71} + 987736 q^{79} + 472392 q^{81} - 1329816 q^{85} - 414504 q^{93} - 897888 q^{95} + 183708 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2 x^{7} + 1061 x^{6} + 35442 x^{5} + 1155979 x^{4} + 17325616 x^{3} + 201523590 x^{2} + \cdots + 5192355364 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 896877863432 \nu^{7} + 6041228263913 \nu^{6} + \cdots - 68\!\cdots\!51 ) / 36\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 195287483112311 \nu^{7} + \cdots + 18\!\cdots\!85 ) / 36\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 16\!\cdots\!65 \nu^{7} + \cdots - 97\!\cdots\!00 ) / 14\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 29605556337 \nu^{7} + 560019920991 \nu^{6} - 34363256068383 \nu^{5} + \cdots + 12\!\cdots\!68 ) / 21\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12257972979 \nu^{7} + 206989766085 \nu^{6} - 14227865187261 \nu^{5} + \cdots + 75\!\cdots\!48 ) / 83\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 107724840993 \nu^{7} + 2057535839451 \nu^{6} - 125036538879087 \nu^{5} + \cdots + 32\!\cdots\!04 ) / 21\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 74\!\cdots\!99 \nu^{7} + \cdots - 44\!\cdots\!72 ) / 14\!\cdots\!68 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 5\beta_{7} + 2\beta_{6} + \beta_{5} - 8\beta_{4} - 23\beta_{3} + \beta_{2} + 54\beta _1 + 46 ) / 168 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 32\beta_{7} - 17\beta_{6} + 2\beta_{5} + 61\beta_{4} - 164\beta_{3} + 19\beta_{2} + 11211\beta _1 - 11150 ) / 42 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -1324\beta_{6} - 403\beta_{5} + 5254\beta_{4} - 594218 ) / 42 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 57705 \beta_{7} - 27625 \beta_{6} - 2455 \beta_{5} + 103059 \beta_{4} + 284007 \beta_{3} + \cdots - 14084622 ) / 42 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2351184 \beta_{7} + 1074301 \beta_{6} + 202582 \beta_{5} - 4143477 \beta_{4} + 11558712 \beta_{3} + \cdots + 528845724 ) / 42 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 43604781\beta_{6} + 6213779\beta_{5} - 165392183\beta_{4} + 21699607978 ) / 21 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3743568769 \beta_{7} + 1730791115 \beta_{6} + 281986539 \beta_{5} - 6617111097 \beta_{4} + \cdots + 858504367554 ) / 42 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
−4.36471 + 7.55990i
−8.68580 + 15.0442i
−5.94197 + 10.2918i
19.9925 − 34.6280i
19.9925 + 34.6280i
−5.94197 − 10.2918i
−8.68580 − 15.0442i
−4.36471 − 7.55990i
0 − 15.5885i 0 − 192.844i 0 0 0 −243.000 0
97.2 0 − 15.5885i 0 − 107.627i 0 0 0 −243.000 0
97.3 0 − 15.5885i 0 − 0.0904912i 0 0 0 −243.000 0
97.4 0 − 15.5885i 0 130.820i 0 0 0 −243.000 0
97.5 0 15.5885i 0 − 130.820i 0 0 0 −243.000 0
97.6 0 15.5885i 0 0.0904912i 0 0 0 −243.000 0
97.7 0 15.5885i 0 107.627i 0 0 0 −243.000 0
97.8 0 15.5885i 0 192.844i 0 0 0 −243.000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.7.d.a 8
7.b odd 2 1 inner 588.7.d.a 8
7.c even 3 1 84.7.m.b ✓ 8
7.c even 3 1 588.7.m.b 8
7.d odd 6 1 84.7.m.b ✓ 8
7.d odd 6 1 588.7.m.b 8
21.g even 6 1 252.7.z.e 8
21.h odd 6 1 252.7.z.e 8
28.f even 6 1 336.7.bh.a 8
28.g odd 6 1 336.7.bh.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.7.m.b ✓ 8 7.c even 3 1
84.7.m.b ✓ 8 7.d odd 6 1
252.7.z.e 8 21.g even 6 1
252.7.z.e 8 21.h odd 6 1
336.7.bh.a 8 28.f even 6 1
336.7.bh.a 8 28.g odd 6 1
588.7.d.a 8 1.a even 1 1 trivial
588.7.d.a 8 7.b odd 2 1 inner
588.7.m.b 8 7.c even 3 1
588.7.m.b 8 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 65886T_{5}^{6} + 1265454009T_{5}^{4} + 7372220567400T_{5}^{2} + 60368490000 \) acting on \(S_{7}^{\mathrm{new}}(588, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 243)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 60368490000 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 1445218428420)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 79\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 90\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 35\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots - 37367406041472)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 86\!\cdots\!52)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 33\!\cdots\!49 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 17\!\cdots\!92)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 59\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 27\!\cdots\!20)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 20\!\cdots\!68)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 26\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 19\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots - 26\!\cdots\!48)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 59\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 36\!\cdots\!33)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 15\!\cdots\!84 \) Copy content Toggle raw display
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