LMFDB - Newform orbit 576.7.g.o

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,7,Mod(127,576)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0]))

N = Newforms(chi, 7, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("576.127");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	576.g (of order $2$, degree $1$, 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:	$132.511152165$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 80x^{3} + 1369x^{2} + 222x + 18 $ x^6 - 2x^5 + 2x^4 + 80x^3 + 1369x^2 + 222*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{25}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 288)
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 $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \beta_1 - 52) q^{5} + ( - \beta_{5} + \beta_{3}) q^{7}+O(q^{10}) $ q + (-b1 - 52) * q^5 + (-b5 + b3) * q^7	$ q + ( - \beta_1 - 52) q^{5} + ( - \beta_{5} + \beta_{3}) q^{7} + ( - 3 \beta_{5} + \beta_{4} + \beta_{3}) q^{11} + ( - \beta_{2} - 11 \beta_1 - 558) q^{13} + ( - 4 \beta_{2} + 10 \beta_1 + 760) q^{17} + (10 \beta_{5} + 16 \beta_{4} - 37 \beta_{3}) q^{19} + (22 \beta_{5} + 14 \beta_{4} - 30 \beta_{3}) q^{23} + ( - 10 \beta_{2} + 146 \beta_1 + 7687) q^{25} + ( - 8 \beta_{2} + 13 \beta_1 - 2940) q^{29} + ( - 29 \beta_{5} - 32 \beta_{4} + 99 \beta_{3}) q^{31} + (141 \beta_{5} + 81 \beta_{4} - 439 \beta_{3}) q^{35} + ( - 35 \beta_{2} + 127 \beta_1 + 6974) q^{37} + (36 \beta_{2} - 130 \beta_1 + 15656) q^{41} + ( - 30 \beta_{5} + \cdots + 315 \beta_{3}) q^{43}+ \cdots + ( - 140 \beta_{2} + 1020 \beta_1 + 64734) q^{97}+O(q^{100}) $ q + (-b1 - 52) * q^5 + (-b5 + b3) * q^7 + (-3b5 + b4 + b3) q^11 + (-b2 - 11b1 - 558) q^13 + (-4b2 + 10b1 + 760) * q^17 + (10b5 + 16b4 - 37b3) q^19 + (22b5 + 14b4 - 30b3) q^23 + (-10b2 + 146b1 + 7687) * q^25 + (-8b2 + 13b1 - 2940) * q^29 + (-29b5 - 32b4 + 99b3) q^31 + (141b5 + 81b4 - 439b3) q^35 + (-35b2 + 127b1 + 6974) * q^37 + (36b2 - 130b1 + 15656) * q^41 + (-30b5 - 112b4 + 315b3) q^43 + (118b5 + 238b4 + 178b3) q^47 + (-30b2 - 586b1 - 29823) * q^49 + (72b2 + b1 - 82092) q^53 + (304b5 + 224b4 - 346b3) q^55 + (-58b5 + 318b4 - 2602b3) q^59 + (109b2 + 687b1 + 34990) * q^61 + (-100b2 + 738b1 + 249176) * q^65 + (-308b5 + 160b4 + 1030b3) q^67 + (680b5 - 56b4 + 4200b3) q^71 + (278b2 + 242b1 + 7198) * q^73 + (-184b2 - 728b1 - 465088) * q^77 + (-1013b5 - 256b4 + 3813b3) q^79 + (467b5 - 753b4 - 8225b3) q^83 + (140b2 - 5116b1 - 271712) * q^85 + (-24b2 - 2444b1 + 619824) * q^89 + (1998b5 + 560b4 + 1857b3) q^91 + (-3234b5 - 1034b4 + 19706b3) q^95 + (-140b2 + 1020b1 + 64734) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 312 q^{5}+O(q^{10}) $ 6 * q - 312 * q^5	$ 6 q - 312 q^{5} - 3348 q^{13} + 4560 q^{17} + 46122 q^{25} - 17640 q^{29} + 41844 q^{37} + 93936 q^{41} - 178938 q^{49} - 492552 q^{53} + 209940 q^{61} + 1495056 q^{65} + 43188 q^{73} - 2790528 q^{77} - 1630272 q^{85} + 3718944 q^{89} + 388404 q^{97}+O(q^{100}) $ 6 * q - 312 * q^5 - 3348 * q^13 + 4560 * q^17 + 46122 * q^25 - 17640 * q^29 + 41844 * q^37 + 93936 * q^41 - 178938 * q^49 - 492552 * q^53 + 209940 * q^61 + 1495056 * q^65 + 43188 * q^73 - 2790528 * q^77 - 1630272 * q^85 + 3718944 * q^89 + 388404 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - 2x^{5} + 2x^{4} + 80x^{3} + 1369x^{2} + 222x + 18 $ x^6 - 2*x^5 + 2*x^4 + 80*x^3 + 1369*x^2 + 222*x + 18 :

$\beta_{1}$	$=$	$ ( -944\nu^{5} + 13432\nu^{4} - 36816\nu^{3} - 37760\nu^{2} - 5664\nu + 9758640 ) / 51557 $ (-944v^5 + 13432v^4 - 36816v^3 - 37760v^2 - 5664*v + 9758640) / 51557
$\beta_{2}$	$=$	$ ( -4656\nu^{5} - 17640\nu^{4} - 181584\nu^{3} - 186240\nu^{2} - 27936\nu - 37491472 ) / 51557 $ (-4656v^5 - 17640v^4 - 181584v^3 - 186240v^2 - 27936*v - 37491472) / 51557
$\beta_{3}$	$=$	$ ( 89792\nu^{5} - 187072\nu^{4} + 202240\nu^{3} + 6891328\nu^{2} + 122625728\nu + 9989952 ) / 154671 $ (89792v^5 - 187072v^4 + 202240v^3 + 6891328v^2 + 122625728*v + 9989952) / 154671
$\beta_{4}$	$=$	$ ( -523780\nu^{5} + 1082420\nu^{4} - 835760\nu^{3} - 43017596\nu^{2} - 703286740\nu - 57295020 ) / 154671 $ (-523780v^5 + 1082420v^4 - 835760v^3 - 43017596v^2 - 703286740*v - 57295020) / 154671
$\beta_{5}$	$=$	$ ( -197044\nu^{5} + 411108\nu^{4} - 466736\nu^{3} - 15924652\nu^{2} - 269897348\nu - 21987708 ) / 51557 $ (-197044v^5 + 411108v^4 - 466736v^3 - 15924652v^2 - 269897348*v - 21987708) / 51557

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( -8\beta_{5} + 8\beta_{4} - 5\beta_{3} + 4\beta_{2} + 12\beta _1 + 512 ) / 1536 $ (-8b5 + 8b4 - 5b3 + 4b2 + 12*b1 + 512) / 1536
$\nu^{2}$	$=$	$ ( -20\beta_{5} - 4\beta_{4} - 155\beta_{3} ) / 384 $ (-20b5 - 4b4 - 155*b3) / 384
$\nu^{3}$	$=$	$ ( -376\beta_{5} + 280\beta_{4} - 877\beta_{3} - 164\beta_{2} - 300\beta _1 - 62464 ) / 1536 $ (-376b5 + 280b4 - 877b3 - 164b2 - 300*b1 - 62464) / 1536
$\nu^{4}$	$=$	$ ( -59\beta_{2} + 291\beta _1 - 97984 ) / 96 $ (-59b2 + 291b1 - 97984) / 96
$\nu^{5}$	$=$	$ ( 17912\beta_{5} - 10328\beta_{4} + 59033\beta_{3} - 7060\beta_{2} - 6012\beta _1 - 3995648 ) / 1536 $ (17912b5 - 10328b4 + 59033b3 - 7060b2 - 6012*b1 - 3995648) / 1536

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$1$	$-1$	$1$

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} $

127.1

−0.0814691	−	0.0814691i
−0.0814691	+	0.0814691i
4.86558	−	4.86558i
4.86558	+	4.86558i
−3.78411	−	3.78411i
−3.78411	+	3.78411i

−241.287

−

486.395i

127.2

−241.287

486.395i

127.3

−20.9167

−

429.262i

127.4

−20.9167

429.262i

127.5

106.203

−

146.868i

127.6

106.203

146.868i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	576.7.g.o		6
3.b	odd	2	1		576.7.g.q		6
4.b	odd	2	1	inner	576.7.g.o		6
8.b	even	2	1		288.7.g.e	yes	6
8.d	odd	2	1		288.7.g.e	yes	6
12.b	even	2	1		576.7.g.q		6
24.f	even	2	1		288.7.g.d	✓	6
24.h	odd	2	1		288.7.g.d	✓	6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
288.7.g.d	✓	6	24.f	even	2	1
288.7.g.d	✓	6	24.h	odd	2	1
288.7.g.e	yes	6	8.b	even	2	1
288.7.g.e	yes	6	8.d	odd	2	1
576.7.g.o		6	1.a	even	1	1	trivial
576.7.g.o		6	4.b	odd	2	1	inner
576.7.g.q		6	3.b	odd	2	1
576.7.g.q		6	12.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{3} + 156T_{5}^{2} - 22800T_{5} - 536000 $ T5^3 + 156*T5^2 - 22800*T5 - 536000 acting on $S_{7}^{\mathrm{new}}(576, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} $ T^6
$5$	$ (T^{3} + 156 T^{2} + \cdots - 536000)^{2} $ (T^3 + 156T^2 - 22800T - 536000)^2
$7$	$ T^{6} + \cdots + 940320146329600 $ T^6 + 442416T^4 + 52671333120T^2 + 940320146329600
$11$	$ T^{6} + \cdots + 30\!\cdots\!76 $ T^6 + 5223168T^4 + 7269054087168T^2 + 3047078247767474176
$13$	$ (T^{3} + 1674 T^{2} + \cdots - 8974388168)^{2} $ (T^3 + 1674T^2 - 5148468T - 8974388168)^2
$17$	$ (T^{3} - 2280 T^{2} + \cdots + 138003319296)^{2} $ (T^3 - 2280T^2 - 43063872T + 138003319296)^2
$19$	$ T^{6} + \cdots + 13\!\cdots\!36 $ T^6 + 230017728T^4 + 12496189739839488T^2 + 137956733678665116942336
$23$	$ T^{6} + \cdots + 99\!\cdots\!64 $ T^6 + 312216576T^4 + 12395587272966144T^2 + 99987844431891360907264
$29$	$ (T^{3} + 8820 T^{2} + \cdots + 370691313344)^{2} $ (T^3 + 8820T^2 - 145018512T + 370691313344)^2
$31$	$ T^{6} + \cdots + 37\!\cdots\!24 $ T^6 + 1115649456T^4 + 184514610328253184T^2 + 3750387926743721278541824
$37$	$ (T^{3} + \cdots + 88423165557000)^{2} $ (T^3 - 20922T^2 - 3572819700T + 88423165557000)^2
$41$	$ (T^{3} + \cdots - 10872293646848)^{2} $ (T^3 - 46968T^2 - 3193425984T - 10872293646848)^2
$43$	$ T^{6} + \cdots + 40\!\cdots\!36 $ T^6 + 10897146048T^4 + 37034484560548343808T^2 + 40136310047252038773261991936
$47$	$ T^{6} + \cdots + 28\!\cdots\!00 $ T^6 + 43481140224T^4 + 455489086073733120000T^2 + 284718501877537728102400000000
$53$	$ (T^{3} + \cdots - 10\!\cdots\!40)^{2} $ (T^3 + 246276T^2 + 6959904240T - 1084720719896640)^2
$59$	$ T^{6} + \cdots + 21\!\cdots\!04 $ T^6 + 174096239616T^4 + 7494372265922437054464T^2 + 21597477189984174043852548603904
$61$	$ (T^{3} + \cdots + 35\!\cdots\!28)^{2} $ (T^3 - 104970T^2 - 39837322548T + 3558137472720328)^2
$67$	$ T^{6} + \cdots + 17\!\cdots\!00 $ T^6 + 81741159168T^4 + 2095368108826200047616T^2 + 17151957173597424551185717657600
$71$	$ T^{6} + \cdots + 53\!\cdots\!44 $ T^6 + 423053279232T^4 + 11914141411746615656448T^2 + 53024599909193350458055327744
$73$	$ (T^{3} + \cdots - 25\!\cdots\!40)^{2} $ (T^3 - 21594T^2 - 198001395060T - 25732432166795640)^2
$79$	$ T^{6} + \cdots + 32\!\cdots\!00 $ T^6 + 633265034928T^4 + 27997268823716031910656T^2 + 321198362864212682338934857830400
$83$	$ T^{6} + \cdots + 80\!\cdots\!76 $ T^6 + 1366493516544T^4 + 233580731246158226325504T^2 + 8000465239282408971586162917376
$89$	$ (T^{3} + \cdots - 11\!\cdots\!84)^{2} $ (T^3 - 1859472T^2 + 967579315968T - 117599737106755584)^2
$97$	$ (T^{3} + \cdots + 46\!\cdots\!96)^{2} $ (T^3 - 194202T^2 - 72515554932T + 4687432589589896)^2
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	576.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_1 - 52) q^{5} + ( - \beta_{5} + \beta_{3}) q^{7}+O(q^{10}) \) q + (-b1 - 52) * q^5 + (-b5 + b3) * q^7	\( q + ( - \beta_1 - 52) q^{5} + ( - \beta_{5} + \beta_{3}) q^{7} + ( - 3 \beta_{5} + \beta_{4} + \beta_{3}) q^{11} + ( - \beta_{2} - 11 \beta_1 - 558) q^{13} + ( - 4 \beta_{2} + 10 \beta_1 + 760) q^{17} + (10 \beta_{5} + 16 \beta_{4} - 37 \beta_{3}) q^{19} + (22 \beta_{5} + 14 \beta_{4} - 30 \beta_{3}) q^{23} + ( - 10 \beta_{2} + 146 \beta_1 + 7687) q^{25} + ( - 8 \beta_{2} + 13 \beta_1 - 2940) q^{29} + ( - 29 \beta_{5} - 32 \beta_{4} + 99 \beta_{3}) q^{31} + (141 \beta_{5} + 81 \beta_{4} - 439 \beta_{3}) q^{35} + ( - 35 \beta_{2} + 127 \beta_1 + 6974) q^{37} + (36 \beta_{2} - 130 \beta_1 + 15656) q^{41} + ( - 30 \beta_{5} + \cdots + 315 \beta_{3}) q^{43}+ \cdots + ( - 140 \beta_{2} + 1020 \beta_1 + 64734) q^{97}+O(q^{100}) \) q + (-b1 - 52) * q^5 + (-b5 + b3) * q^7 + (-3b5 + b4 + b3) q^11 + (-b2 - 11b1 - 558) q^13 + (-4b2 + 10b1 + 760) * q^17 + (10b5 + 16b4 - 37b3) q^19 + (22b5 + 14b4 - 30b3) q^23 + (-10b2 + 146b1 + 7687) * q^25 + (-8b2 + 13b1 - 2940) * q^29 + (-29b5 - 32b4 + 99b3) q^31 + (141b5 + 81b4 - 439b3) q^35 + (-35b2 + 127b1 + 6974) * q^37 + (36b2 - 130b1 + 15656) * q^41 + (-30b5 - 112b4 + 315b3) q^43 + (118b5 + 238b4 + 178b3) q^47 + (-30b2 - 586b1 - 29823) * q^49 + (72b2 + b1 - 82092) q^53 + (304b5 + 224b4 - 346b3) q^55 + (-58b5 + 318b4 - 2602b3) q^59 + (109b2 + 687b1 + 34990) * q^61 + (-100b2 + 738b1 + 249176) * q^65 + (-308b5 + 160b4 + 1030b3) q^67 + (680b5 - 56b4 + 4200b3) q^71 + (278b2 + 242b1 + 7198) * q^73 + (-184b2 - 728b1 - 465088) * q^77 + (-1013b5 - 256b4 + 3813b3) q^79 + (467b5 - 753b4 - 8225b3) q^83 + (140b2 - 5116b1 - 271712) * q^85 + (-24b2 - 2444b1 + 619824) * q^89 + (1998b5 + 560b4 + 1857b3) q^91 + (-3234b5 - 1034b4 + 19706b3) q^95 + (-140b2 + 1020b1 + 64734) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 312 q^{5}+O(q^{10}) \) 6 * q - 312 * q^5	\( 6 q - 312 q^{5} - 3348 q^{13} + 4560 q^{17} + 46122 q^{25} - 17640 q^{29} + 41844 q^{37} + 93936 q^{41} - 178938 q^{49} - 492552 q^{53} + 209940 q^{61} + 1495056 q^{65} + 43188 q^{73} - 2790528 q^{77} - 1630272 q^{85} + 3718944 q^{89} + 388404 q^{97}+O(q^{100}) \) 6 * q - 312 * q^5 - 3348 * q^13 + 4560 * q^17 + 46122 * q^25 - 17640 * q^29 + 41844 * q^37 + 93936 * q^41 - 178938 * q^49 - 492552 * q^53 + 209940 * q^61 + 1495056 * q^65 + 43188 * q^73 - 2790528 * q^77 - 1630272 * q^85 + 3718944 * q^89 + 388404 * q^97

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