LMFDB - Newform orbit 320.8.c.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [320,8,Mod(129,320)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("320.129"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	320.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-550,0,0,0,2062,0,2648] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$99.9632081549$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 40)
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 $\beta = 2i$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + 17 \beta q^{3} + (25 \beta - 275) q^{5} + 53 \beta q^{7} + 1031 q^{9} + 1324 q^{11} + 4414 \beta q^{13} + ( - 4675 \beta - 1700) q^{15} - 12000 \beta q^{17} - 4876 q^{19} - 3604 q^{21} - 23323 \beta q^{23} + \cdots + 1365044 q^{99} +O(q^{100}) $ q + 17b q^3 + (25b - 275) q^5 + 53b q^7 + 1031 * q^9 + 1324 * q^11 + 4414b q^13 + (-4675b - 1700) q^15 - 12000b q^17 - 4876 * q^19 - 3604 * q^21 - 23323b q^23 + (-13750b + 73125) q^25 + 54706b q^27 - 110902 * q^29 - 247680 * q^31 + 22508b q^33 + (-14575b - 5300) q^35 - 180046b q^37 - 300152 * q^39 + 104402 * q^41 - 356811b q^43 + (25775b - 283525) q^45 + 78441b q^47 + 812307 * q^49 + 816000 * q^51 - 533134b q^53 + (33100b - 364100) q^55 - 82892b q^57 + 832572 * q^59 - 529070 * q^61 + 54643b q^63 + (-1213850b - 441400) q^65 + 2087209b q^67 + 1585964 * q^69 + 5176568 * q^71 - 118988b q^73 + (1243125b + 935000) q^75 + 70172b q^77 + 3742736 * q^79 - 1465211 * q^81 - 3930943b q^83 + (3300000b + 1200000) q^85 - 1885334b q^87 - 4300854 * q^89 - 935768 * q^91 - 4210560b q^93 + (-121900b + 1340900) q^95 - 573896b q^97 + 1365044 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 550 q^{5} + 2062 q^{9} + 2648 q^{11} - 3400 q^{15} - 9752 q^{19} - 7208 q^{21} + 146250 q^{25} - 221804 q^{29} - 495360 q^{31} - 10600 q^{35} - 600304 q^{39} + 208804 q^{41} - 567050 q^{45} + 1624614 q^{49}+ \cdots + 2730088 q^{99}+O(q^{100}) $ 2 * q - 550 * q^5 + 2062 * q^9 + 2648 * q^11 - 3400 * q^15 - 9752 * q^19 - 7208 * q^21 + 146250 * q^25 - 221804 * q^29 - 495360 * q^31 - 10600 * q^35 - 600304 * q^39 + 208804 * q^41 - 567050 * q^45 + 1624614 * q^49 + 1632000 * q^51 - 728200 * q^55 + 1665144 * q^59 - 1058140 * q^61 - 882800 * q^65 + 3171928 * q^69 + 10353136 * q^71 + 1870000 * q^75 + 7485472 * q^79 - 2930422 * q^81 + 2400000 * q^85 - 8601708 * q^89 - 1871536 * q^91 + 2681800 * q^95 + 2730088 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$1$	$-1$	$1$

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} $

129.1

	−	1.00000i
		1.00000i

−

34.0000i

−275.000

−

50.0000i

−

106.000i

1031.00

129.2

34.0000i

−275.000

50.0000i

106.000i

1031.00

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	320.8.c.b		2
4.b	odd	2	1		320.8.c.a		2
5.b	even	2	1	inner	320.8.c.b		2
8.b	even	2	1		40.8.c.a	✓	2
8.d	odd	2	1		80.8.c.b		2
20.d	odd	2	1		320.8.c.a		2
24.h	odd	2	1		360.8.f.a		2
40.e	odd	2	1		80.8.c.b		2
40.f	even	2	1		40.8.c.a	✓	2
40.i	odd	4	1		200.8.a.c		1
40.i	odd	4	1		200.8.a.f		1
40.k	even	4	1		400.8.a.h		1
40.k	even	4	1		400.8.a.n		1
120.i	odd	2	1		360.8.f.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
40.8.c.a	✓	2	8.b	even	2	1
40.8.c.a	✓	2	40.f	even	2	1
80.8.c.b		2	8.d	odd	2	1
80.8.c.b		2	40.e	odd	2	1
200.8.a.c		1	40.i	odd	4	1
200.8.a.f		1	40.i	odd	4	1
320.8.c.a		2	4.b	odd	2	1
320.8.c.a		2	20.d	odd	2	1
320.8.c.b		2	1.a	even	1	1	trivial
320.8.c.b		2	5.b	even	2	1	inner
360.8.f.a		2	24.h	odd	2	1
360.8.f.a		2	120.i	odd	2	1
400.8.a.h		1	40.k	even	4	1
400.8.a.n		1	40.k	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{8}^{\mathrm{new}}(320, [\chi])$:

$ T_{3}^{2} + 1156 $

T3^2 + 1156

$ T_{11} - 1324 $

T11 - 1324

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} + 1156 $ T^2 + 1156
$5$	$ T^{2} + 550T + 78125 $ T^2 + 550*T + 78125
$7$	$ T^{2} + 11236 $ T^2 + 11236
$11$	$ (T - 1324)^{2} $ (T - 1324)^2
$13$	$ T^{2} + 77933584 $ T^2 + 77933584
$17$	$ T^{2} + 576000000 $ T^2 + 576000000
$19$	$ (T + 4876)^{2} $ (T + 4876)^2
$23$	$ T^{2} + 2175849316 $ T^2 + 2175849316
$29$	$ (T + 110902)^{2} $ (T + 110902)^2
$31$	$ (T + 247680)^{2} $ (T + 247680)^2
$37$	$ T^{2} + 129666248464 $ T^2 + 129666248464
$41$	$ (T - 104402)^{2} $ (T - 104402)^2
$43$	$ T^{2} + 509256358884 $ T^2 + 509256358884
$47$	$ T^{2} + 24611961924 $ T^2 + 24611961924
$53$	$ T^{2} + 1136927447824 $ T^2 + 1136927447824
$59$	$ (T - 832572)^{2} $ (T - 832572)^2
$61$	$ (T + 529070)^{2} $ (T + 529070)^2
$67$	$ T^{2} + 17425765638724 $ T^2 + 17425765638724
$71$	$ (T - 5176568)^{2} $ (T - 5176568)^2
$73$	$ T^{2} + 56632576576 $ T^2 + 56632576576
$79$	$ (T - 3742736)^{2} $ (T - 3742736)^2
$83$	$ T^{2} + 61809251476996 $ T^2 + 61809251476996
$89$	$ (T + 4300854)^{2} $ (T + 4300854)^2
$97$	$ T^{2} + 1317426475264 $ T^2 + 1317426475264
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Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	320.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + 17 \beta q^{3} + (25 \beta - 275) q^{5} + 53 \beta q^{7} + 1031 q^{9} + 1324 q^{11} + 4414 \beta q^{13} + ( - 4675 \beta - 1700) q^{15} - 12000 \beta q^{17} - 4876 q^{19} - 3604 q^{21} - 23323 \beta q^{23} + \cdots + 1365044 q^{99} +O(q^{100}) \) q + 17b q^3 + (25b - 275) q^5 + 53b q^7 + 1031 * q^9 + 1324 * q^11 + 4414b q^13 + (-4675b - 1700) q^15 - 12000b q^17 - 4876 * q^19 - 3604 * q^21 - 23323b q^23 + (-13750b + 73125) q^25 + 54706b q^27 - 110902 * q^29 - 247680 * q^31 + 22508b q^33 + (-14575b - 5300) q^35 - 180046b q^37 - 300152 * q^39 + 104402 * q^41 - 356811b q^43 + (25775b - 283525) q^45 + 78441b q^47 + 812307 * q^49 + 816000 * q^51 - 533134b q^53 + (33100b - 364100) q^55 - 82892b q^57 + 832572 * q^59 - 529070 * q^61 + 54643b q^63 + (-1213850b - 441400) q^65 + 2087209b q^67 + 1585964 * q^69 + 5176568 * q^71 - 118988b q^73 + (1243125b + 935000) q^75 + 70172b q^77 + 3742736 * q^79 - 1465211 * q^81 - 3930943b q^83 + (3300000b + 1200000) q^85 - 1885334b q^87 - 4300854 * q^89 - 935768 * q^91 - 4210560b q^93 + (-121900b + 1340900) q^95 - 573896b q^97 + 1365044 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 550 q^{5} + 2062 q^{9} + 2648 q^{11} - 3400 q^{15} - 9752 q^{19} - 7208 q^{21} + 146250 q^{25} - 221804 q^{29} - 495360 q^{31} - 10600 q^{35} - 600304 q^{39} + 208804 q^{41} - 567050 q^{45} + 1624614 q^{49}+ \cdots + 2730088 q^{99}+O(q^{100}) \) 2 * q - 550 * q^5 + 2062 * q^9 + 2648 * q^11 - 3400 * q^15 - 9752 * q^19 - 7208 * q^21 + 146250 * q^25 - 221804 * q^29 - 495360 * q^31 - 10600 * q^35 - 600304 * q^39 + 208804 * q^41 - 567050 * q^45 + 1624614 * q^49 + 1632000 * q^51 - 728200 * q^55 + 1665144 * q^59 - 1058140 * q^61 - 882800 * q^65 + 3171928 * q^69 + 10353136 * q^71 + 1870000 * q^75 + 7485472 * q^79 - 2930422 * q^81 + 2400000 * q^85 - 8601708 * q^89 - 1871536 * q^91 + 2681800 * q^95 + 2730088 * q^99

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