LMFDB - Newform orbit 50.8.b.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [50,8,Mod(49,50)] 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("50.49"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(50, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 50 = 2 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	50.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-128,0,-912] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	no
Analytic conductor:	$15.6192512742$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
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 $i = \sqrt{-1}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + 8 i q^{2} + 57 i q^{3} - 64 q^{4} - 456 q^{6} + 1174 i q^{7} - 512 i q^{8} - 1062 q^{9} - 7563 q^{11} - 3648 i q^{12} + 5372 i q^{13} - 9392 q^{14} + 4096 q^{16} - 24021 i q^{17} - 8496 i q^{18} + 51235 q^{19} + \cdots + 8031906 q^{99} +O(q^{100}) $ q + 8i q^2 + 57i q^3 - 64 * q^4 - 456 * q^6 + 1174i q^7 - 512i q^8 - 1062 * q^9 - 7563 * q^11 - 3648i q^12 + 5372i q^13 - 9392 * q^14 + 4096 * q^16 - 24021i q^17 - 8496i q^18 + 51235 * q^19 - 66918 * q^21 - 60504i q^22 - 57618i q^23 + 29184 * q^24 - 42976 * q^26 + 64125i q^27 - 75136i q^28 - 47040 * q^29 - 192358 * q^31 + 32768i q^32 - 431091i q^33 + 192168 * q^34 + 67968 * q^36 - 197066i q^37 + 409880i q^38 - 306204 * q^39 - 237723 * q^41 - 535344i q^42 + 653012i q^43 + 484032 * q^44 + 460944 * q^46 + 826884i q^47 + 233472i q^48 - 554733 * q^49 + 1369197 * q^51 - 343808i q^52 + 569022i q^53 - 513000 * q^54 + 601088 * q^56 + 2920395i q^57 - 376320i q^58 - 1501080 * q^59 - 2068918 * q^61 - 1538864i q^62 - 1246788i q^63 - 262144 * q^64 + 3448728 * q^66 + 3444349i q^67 + 1537344i q^68 + 3284226 * q^69 + 4121052 * q^71 + 543744i q^72 - 83653i q^73 + 1576528 * q^74 - 3279040 * q^76 - 8878962i q^77 - 2449632i q^78 - 1454030 * q^79 - 5977719 * q^81 - 1901784i q^82 + 1626567i q^83 + 4282752 * q^84 - 5224096 * q^86 - 2681280i q^87 + 3872256i q^88 - 6004335 * q^89 - 6306728 * q^91 + 3687552i q^92 - 10964406i q^93 - 6615072 * q^94 - 1867776 * q^96 - 3411746i q^97 - 4437864i q^98 + 8031906 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 128 q^{4} - 912 q^{6} - 2124 q^{9} - 15126 q^{11} - 18784 q^{14} + 8192 q^{16} + 102470 q^{19} - 133836 q^{21} + 58368 q^{24} - 85952 q^{26} - 94080 q^{29} - 384716 q^{31} + 384336 q^{34} + 135936 q^{36}+ \cdots + 16063812 q^{99}+O(q^{100}) $ 2 * q - 128 * q^4 - 912 * q^6 - 2124 * q^9 - 15126 * q^11 - 18784 * q^14 + 8192 * q^16 + 102470 * q^19 - 133836 * q^21 + 58368 * q^24 - 85952 * q^26 - 94080 * q^29 - 384716 * q^31 + 384336 * q^34 + 135936 * q^36 - 612408 * q^39 - 475446 * q^41 + 968064 * q^44 + 921888 * q^46 - 1109466 * q^49 + 2738394 * q^51 - 1026000 * q^54 + 1202176 * q^56 - 3002160 * q^59 - 4137836 * q^61 - 524288 * q^64 + 6897456 * q^66 + 6568452 * q^69 + 8242104 * q^71 + 3153056 * q^74 - 6558080 * q^76 - 2908060 * q^79 - 11955438 * q^81 + 8565504 * q^84 - 10448192 * q^86 - 12008670 * q^89 - 12613456 * q^91 - 13230144 * q^94 - 3735552 * q^96 + 16063812 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/50\mathbb{Z}\right)^\times$.

$n$	$27$
$\chi(n)$	$-1$

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} $

49.1

	−	1.00000i
		1.00000i

−

8.00000i

−

57.0000i

−64.0000

−456.000

−

1174.00i

512.000i

−1062.00

49.2

8.00000i

57.0000i

−64.0000

−456.000

1174.00i

−

512.000i

−1062.00

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	50.8.b.b		2
3.b	odd	2	1		450.8.c.q		2
4.b	odd	2	1		400.8.c.d		2
5.b	even	2	1	inner	50.8.b.b		2
5.c	odd	4	1		50.8.a.c	✓	1
5.c	odd	4	1		50.8.a.f	yes	1
15.d	odd	2	1		450.8.c.q		2
15.e	even	4	1		450.8.a.l		1
15.e	even	4	1		450.8.a.p		1
20.d	odd	2	1		400.8.c.d		2
20.e	even	4	1		400.8.a.d		1
20.e	even	4	1		400.8.a.q		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
50.8.a.c	✓	1	5.c	odd	4	1
50.8.a.f	yes	1	5.c	odd	4	1
50.8.b.b		2	1.a	even	1	1	trivial
50.8.b.b		2	5.b	even	2	1	inner
400.8.a.d		1	20.e	even	4	1
400.8.a.q		1	20.e	even	4	1
400.8.c.d		2	4.b	odd	2	1
400.8.c.d		2	20.d	odd	2	1
450.8.a.l		1	15.e	even	4	1
450.8.a.p		1	15.e	even	4	1
450.8.c.q		2	3.b	odd	2	1
450.8.c.q		2	15.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{2} + 3249 $ T3^2 + 3249 acting on $S_{8}^{\mathrm{new}}(50, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 64 $ T^2 + 64
$3$	$ T^{2} + 3249 $ T^2 + 3249
$5$	$ T^{2} $ T^2
$7$	$ T^{2} + 1378276 $ T^2 + 1378276
$11$	$ (T + 7563)^{2} $ (T + 7563)^2
$13$	$ T^{2} + 28858384 $ T^2 + 28858384
$17$	$ T^{2} + 577008441 $ T^2 + 577008441
$19$	$ (T - 51235)^{2} $ (T - 51235)^2
$23$	$ T^{2} + 3319833924 $ T^2 + 3319833924
$29$	$ (T + 47040)^{2} $ (T + 47040)^2
$31$	$ (T + 192358)^{2} $ (T + 192358)^2
$37$	$ T^{2} + 38835008356 $ T^2 + 38835008356
$41$	$ (T + 237723)^{2} $ (T + 237723)^2
$43$	$ T^{2} + 426424672144 $ T^2 + 426424672144
$47$	$ T^{2} + 683737149456 $ T^2 + 683737149456
$53$	$ T^{2} + 323786036484 $ T^2 + 323786036484
$59$	$ (T + 1501080)^{2} $ (T + 1501080)^2
$61$	$ (T + 2068918)^{2} $ (T + 2068918)^2
$67$	$ T^{2} + 11863540033801 $ T^2 + 11863540033801
$71$	$ (T - 4121052)^{2} $ (T - 4121052)^2
$73$	$ T^{2} + 6997824409 $ T^2 + 6997824409
$79$	$ (T + 1454030)^{2} $ (T + 1454030)^2
$83$	$ T^{2} + 2645720205489 $ T^2 + 2645720205489
$89$	$ (T + 6004335)^{2} $ (T + 6004335)^2
$97$	$ T^{2} + 11640010768516 $ T^2 + 11640010768516
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Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	50.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + 8 i q^{2} + 57 i q^{3} - 64 q^{4} - 456 q^{6} + 1174 i q^{7} - 512 i q^{8} - 1062 q^{9} - 7563 q^{11} - 3648 i q^{12} + 5372 i q^{13} - 9392 q^{14} + 4096 q^{16} - 24021 i q^{17} - 8496 i q^{18} + 51235 q^{19} + \cdots + 8031906 q^{99} +O(q^{100}) \) q + 8i q^2 + 57i q^3 - 64 * q^4 - 456 * q^6 + 1174i q^7 - 512i q^8 - 1062 * q^9 - 7563 * q^11 - 3648i q^12 + 5372i q^13 - 9392 * q^14 + 4096 * q^16 - 24021i q^17 - 8496i q^18 + 51235 * q^19 - 66918 * q^21 - 60504i q^22 - 57618i q^23 + 29184 * q^24 - 42976 * q^26 + 64125i q^27 - 75136i q^28 - 47040 * q^29 - 192358 * q^31 + 32768i q^32 - 431091i q^33 + 192168 * q^34 + 67968 * q^36 - 197066i q^37 + 409880i q^38 - 306204 * q^39 - 237723 * q^41 - 535344i q^42 + 653012i q^43 + 484032 * q^44 + 460944 * q^46 + 826884i q^47 + 233472i q^48 - 554733 * q^49 + 1369197 * q^51 - 343808i q^52 + 569022i q^53 - 513000 * q^54 + 601088 * q^56 + 2920395i q^57 - 376320i q^58 - 1501080 * q^59 - 2068918 * q^61 - 1538864i q^62 - 1246788i q^63 - 262144 * q^64 + 3448728 * q^66 + 3444349i q^67 + 1537344i q^68 + 3284226 * q^69 + 4121052 * q^71 + 543744i q^72 - 83653i q^73 + 1576528 * q^74 - 3279040 * q^76 - 8878962i q^77 - 2449632i q^78 - 1454030 * q^79 - 5977719 * q^81 - 1901784i q^82 + 1626567i q^83 + 4282752 * q^84 - 5224096 * q^86 - 2681280i q^87 + 3872256i q^88 - 6004335 * q^89 - 6306728 * q^91 + 3687552i q^92 - 10964406i q^93 - 6615072 * q^94 - 1867776 * q^96 - 3411746i q^97 - 4437864i q^98 + 8031906 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 128 q^{4} - 912 q^{6} - 2124 q^{9} - 15126 q^{11} - 18784 q^{14} + 8192 q^{16} + 102470 q^{19} - 133836 q^{21} + 58368 q^{24} - 85952 q^{26} - 94080 q^{29} - 384716 q^{31} + 384336 q^{34} + 135936 q^{36}+ \cdots + 16063812 q^{99}+O(q^{100}) \) 2 * q - 128 * q^4 - 912 * q^6 - 2124 * q^9 - 15126 * q^11 - 18784 * q^14 + 8192 * q^16 + 102470 * q^19 - 133836 * q^21 + 58368 * q^24 - 85952 * q^26 - 94080 * q^29 - 384716 * q^31 + 384336 * q^34 + 135936 * q^36 - 612408 * q^39 - 475446 * q^41 + 968064 * q^44 + 921888 * q^46 - 1109466 * q^49 + 2738394 * q^51 - 1026000 * q^54 + 1202176 * q^56 - 3002160 * q^59 - 4137836 * q^61 - 524288 * q^64 + 6897456 * q^66 + 6568452 * q^69 + 8242104 * q^71 + 3153056 * q^74 - 6558080 * q^76 - 2908060 * q^79 - 11955438 * q^81 + 8565504 * q^84 - 10448192 * q^86 - 12008670 * q^89 - 12613456 * q^91 - 13230144 * q^94 - 3735552 * q^96 + 16063812 * q^99

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