Properties

Label 588.7.d.b
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: \( x^{8} - x^{7} + 82x^{6} - 165x^{5} + 5606x^{4} - 7807x^{3} + 102447x^{2} + 132594x + 1162084 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{4}\cdot 7^{2} \)
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.

\(f(q)\) \(=\) \( q - 9 \beta_1 q^{3} + (\beta_{6} + 3 \beta_1) q^{5} - 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 9 \beta_1 q^{3} + (\beta_{6} + 3 \beta_1) q^{5} - 243 q^{9} + ( - 2 \beta_{7} + \beta_{5} + 2 \beta_{4} - 33) q^{11} + (7 \beta_{6} - 4 \beta_{3} - 3 \beta_{2} - 273 \beta_1) q^{13} + ( - 9 \beta_{5} + 9 \beta_{4} + 99) q^{15} + (14 \beta_{6} + 2 \beta_{3} - 20 \beta_{2} - 218 \beta_1) q^{17} + (29 \beta_{6} - 17 \beta_{3} + 3 \beta_{2} - 1681 \beta_1) q^{19} + ( - 46 \beta_{7} - 14 \beta_{5} + 14 \beta_{4} + 3896) q^{23} + (50 \beta_{7} - 31 \beta_{5} - 34 \beta_{4} - 10634) q^{25} + 2187 \beta_1 q^{27} + (40 \beta_{7} + 33 \beta_{5} + 57 \beta_{4} + 16187) q^{29} + ( - 36 \beta_{6} + 132 \beta_{3} + 21 \beta_{2} - 3490 \beta_1) q^{31} + ( - 9 \beta_{6} + 9 \beta_{3} + 45 \beta_{2} + 288 \beta_1) q^{33} + (65 \beta_{7} - 104 \beta_{5} - 116 \beta_{4} - 2382) q^{37} + ( - 63 \beta_{7} - 72 \beta_{5} + 90 \beta_{4} - 7272) q^{39} + (388 \beta_{6} - 272 \beta_{3} - 28 \beta_{2} - 6718 \beta_1) q^{41} + ( - 26 \beta_{7} - 251 \beta_{5} + 388 \beta_{4} - 8858) q^{43} + ( - 243 \beta_{6} - 729 \beta_1) q^{45} + (418 \beta_{6} - 126 \beta_{3} + 162 \beta_{2} - 7458 \beta_1) q^{47} + ( - 162 \beta_{7} + 72 \beta_{5} + 306 \beta_{4} - 5814) q^{51} + (114 \beta_{7} - 103 \beta_{5} + 325 \beta_{4} - 97819) q^{53} + (837 \beta_{6} + 300 \beta_{3} - 120 \beta_{2} - 9039 \beta_1) q^{55} + ( - 126 \beta_{7} - 441 \beta_{5} + 234 \beta_{4} - 44838) q^{57} + ( - 191 \beta_{6} + 319 \beta_{3} - 199 \beta_{2} - 46024 \beta_1) q^{59} + (960 \beta_{6} + 294 \beta_{3} - 128 \beta_{2} + 83612 \beta_1) q^{61} + (110 \beta_{7} - 24 \beta_{5} - 546 \beta_{4} - 147296) q^{65} + ( - 436 \beta_{7} + 339 \beta_{5} - 426 \beta_{4} - 57828) q^{67} + (36 \beta_{6} + 828 \beta_{3} + 414 \beta_{2} - 35226 \beta_1) q^{69} + (68 \beta_{7} - 458 \beta_{5} + 560 \beta_{4} + 26324) q^{71} + (311 \beta_{6} - 1835 \beta_{3} + 651 \beta_{2} - 104263 \beta_1) q^{73} + ( - 117 \beta_{6} - 315 \beta_{3} - 1035 \beta_{2} + 96129 \beta_1) q^{75} + (553 \beta_{7} - 97 \beta_{5} + 1853 \beta_{4} - 173127) q^{79} + 59049 q^{81} + ( - 1285 \beta_{6} - 909 \beta_{3} + 189 \beta_{2} - 117516 \beta_1) q^{83} + ( - 40 \beta_{7} + 890 \beta_{5} - 4000 \beta_{4} - 163290) q^{85} + ( - 1089 \beta_{6} - 1530 \beta_{3} + 450 \beta_{2} - 145107 \beta_1) q^{87} + ( - 1614 \beta_{6} + 1572 \beta_{3} - 540 \beta_{2} + 118698 \beta_1) q^{89} + (1377 \beta_{7} + 1323 \beta_{5} - 513 \beta_{4} - 94689) q^{93} + (1060 \beta_{7} - 1936 \beta_{5} + 1216 \beta_{4} - 760394) q^{95} + ( - 2295 \beta_{6} + 1185 \beta_{3} + 305 \beta_{2} - 274460 \beta_1) q^{97} + (486 \beta_{7} - 243 \beta_{5} - 486 \beta_{4} + 8019) 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 \( 8 q - 1944 q^{9} - 252 q^{11} + 756 q^{15} + 31296 q^{23} - 85396 q^{25} + 129468 q^{29} - 19732 q^{37} - 58212 q^{39} - 71764 q^{43} - 45576 q^{51} - 783420 q^{53} - 359964 q^{57} - 1178904 q^{65} - 459524 q^{67} + 208488 q^{71} - 1387616 q^{79} + 472392 q^{81} - 1302600 q^{85} - 757728 q^{93} - 6095136 q^{95} + 61236 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 82x^{6} - 165x^{5} + 5606x^{4} - 7807x^{3} + 102447x^{2} + 132594x + 1162084 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 9659521 \nu^{7} - 77530555 \nu^{6} + 720692020 \nu^{5} - 6152911039 \nu^{4} + 55995523440 \nu^{3} - 452112814595 \nu^{2} + \cdots - 2940259618219 ) / 3136225324001 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 117104250 \nu^{7} - 15366386696 \nu^{6} - 220137367995 \nu^{5} - 1407936162666 \nu^{4} - 9257243671427 \nu^{3} + \cdots - 720628335681947 ) / 3136225324001 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1165235138 \nu^{7} - 24628162010 \nu^{6} - 47869345266 \nu^{5} - 2232961192814 \nu^{4} + 1362037855790 \nu^{3} + \cdots - 974144033632308 ) / 3136225324001 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 16140060 \nu^{7} - 72852936 \nu^{6} + 1084173660 \nu^{5} - 438571260 \nu^{4} + 85988122680 \nu^{3} + 46834269660 \nu^{2} + \cdots + 14768088989220 ) / 40730199013 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 32917023 \nu^{7} + 79537827 \nu^{6} + 2211129903 \nu^{5} - 894448983 \nu^{4} + 62303568087 \nu^{3} + 95516666703 \nu^{2} + \cdots + 4200958317262 ) / 81460398026 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3051472253 \nu^{7} - 22494447805 \nu^{6} + 154260852199 \nu^{5} - 2196721109627 \nu^{4} + 14885837292119 \nu^{3} + \cdots - 655450188971886 ) / 6272450648002 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 248107131 \nu^{7} - 844858047 \nu^{6} - 16666060491 \nu^{5} + 6741775251 \nu^{4} - 789458296947 \nu^{3} + \cdots + 10317857972954 ) / 81460398026 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + 8\beta_{6} + 5\beta_{5} - 2\beta_{4} - 5\beta_{3} + 2\beta_{2} - 46\beta _1 + 40 ) / 336 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 20\beta_{6} + 5\beta_{5} + 12\beta_{4} + 19\beta_{3} - 16\beta_{2} - 6856\beta _1 - 6848 ) / 336 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -31\beta_{7} - 267\beta_{5} + 34\beta_{4} + 5368 ) / 168 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 173 \beta_{7} - 940 \beta_{6} + 753 \beta_{5} + 760 \beta_{4} - 1859 \beta_{3} + 1340 \beta_{2} + 363476 \beta _1 - 363296 ) / 336 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1383 \beta_{7} - 14480 \beta_{6} + 16099 \beta_{5} + 118 \beta_{4} + 18983 \beta_{3} - 14834 \beta_{2} - 942338 \beta _1 - 956936 ) / 336 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2195\beta_{7} - 10335\beta_{5} - 6332\beta_{4} + 3106880 ) / 24 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 84519 \beta_{7} + 761216 \beta_{6} + 1036467 \beta_{5} + 95366 \beta_{4} - 1300871 \beta_{3} + 1047314 \beta_{2} + 93805778 \beta _1 - 94662360 ) / 336 \) Copy content Toggle raw display

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
3.42459 5.93157i
2.91150 5.04287i
−4.24382 + 7.35052i
−1.59227 + 2.75789i
−1.59227 2.75789i
−4.24382 7.35052i
2.91150 + 5.04287i
3.42459 + 5.93157i
0 15.5885i 0 232.601i 0 0 0 −243.000 0
97.2 0 15.5885i 0 31.3783i 0 0 0 −243.000 0
97.3 0 15.5885i 0 78.6431i 0 0 0 −243.000 0
97.4 0 15.5885i 0 209.584i 0 0 0 −243.000 0
97.5 0 15.5885i 0 209.584i 0 0 0 −243.000 0
97.6 0 15.5885i 0 78.6431i 0 0 0 −243.000 0
97.7 0 15.5885i 0 31.3783i 0 0 0 −243.000 0
97.8 0 15.5885i 0 232.601i 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.b 8
7.b odd 2 1 inner 588.7.d.b 8
7.c even 3 1 84.7.m.a 8
7.c even 3 1 588.7.m.c 8
7.d odd 6 1 84.7.m.a 8
7.d odd 6 1 588.7.m.c 8
21.g even 6 1 252.7.z.d 8
21.h odd 6 1 252.7.z.d 8
28.f even 6 1 336.7.bh.e 8
28.g odd 6 1 336.7.bh.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.7.m.a 8 7.c even 3 1
84.7.m.a 8 7.d odd 6 1
252.7.z.d 8 21.g even 6 1
252.7.z.d 8 21.h odd 6 1
336.7.bh.e 8 28.f even 6 1
336.7.bh.e 8 28.g odd 6 1
588.7.d.b 8 1.a even 1 1 trivial
588.7.d.b 8 7.b odd 2 1 inner
588.7.m.c 8 7.c even 3 1
588.7.m.c 8 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 105198T_{5}^{6} + 3085400025T_{5}^{4} + 17634944835000T_{5}^{2} + 14471729102250000 \) 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
$5$ \( T^{8} + 105198 T^{6} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 126 T^{3} + \cdots + 816739643556)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 12743190 T^{6} + \cdots + 65\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{8} + 161776824 T^{6} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{8} + 186839790 T^{6} + \cdots + 51\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( (T^{4} - 15648 T^{3} + \cdots - 35\!\cdots\!08)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 64734 T^{3} + \cdots + 88\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 4493536716 T^{6} + \cdots + 49\!\cdots\!69 \) Copy content Toggle raw display
$37$ \( (T^{4} + 9866 T^{3} + \cdots + 42\!\cdots\!08)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 32134086384 T^{6} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( (T^{4} + 35882 T^{3} + \cdots - 86\!\cdots\!40)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 36305948184 T^{6} + \cdots + 15\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( (T^{4} + 391710 T^{3} + \cdots - 30\!\cdots\!16)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 63639768966 T^{6} + \cdots + 90\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{8} + 194783837760 T^{6} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{4} + 229762 T^{3} + \cdots - 11\!\cdots\!52)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 104244 T^{3} + \cdots - 95\!\cdots\!20)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 929679101142 T^{6} + \cdots + 76\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{4} + 693808 T^{3} + \cdots + 44\!\cdots\!05)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 503985708366 T^{6} + \cdots + 47\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{8} + 1061082102456 T^{6} + \cdots + 87\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{8} + 1777407002550 T^{6} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
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