LMFDB - Newform orbit 275.6.b.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [275,6,Mod(199,275)] 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("275.199"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 275 = 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	275.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$44.1055504486$
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:	no (minimal twist has level 11)
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 + 4 i q^{2} - 15 i q^{3} + 16 q^{4} + 60 q^{6} - 10 i q^{7} + 192 i q^{8} + 18 q^{9} - 121 q^{11} - 240 i q^{12} - 1148 i q^{13} + 40 q^{14} - 256 q^{16} - 686 i q^{17} + 72 i q^{18} + 384 q^{19} - 150 q^{21} + \cdots - 2178 q^{99} +O(q^{100}) $ q + 4i q^2 - 15i q^3 + 16 * q^4 + 60 * q^6 - 10i q^7 + 192i q^8 + 18 * q^9 - 121 * q^11 - 240i q^12 - 1148i q^13 + 40 * q^14 - 256 * q^16 - 686i q^17 + 72i q^18 + 384 * q^19 - 150 * q^21 - 484i q^22 + 3709i q^23 + 2880 * q^24 + 4592 * q^26 - 3915i q^27 - 160i q^28 + 5424 * q^29 - 6443 * q^31 + 5120i q^32 + 1815i q^33 + 2744 * q^34 + 288 * q^36 - 12063i q^37 + 1536i q^38 - 17220 * q^39 - 1528 * q^41 - 600i q^42 - 4026i q^43 - 1936 * q^44 - 14836 * q^46 - 7168i q^47 + 3840i q^48 + 16707 * q^49 - 10290 * q^51 - 18368i q^52 - 29862i q^53 + 15660 * q^54 + 1920 * q^56 - 5760i q^57 + 21696i q^58 + 6461 * q^59 - 16980 * q^61 - 25772i q^62 - 180i q^63 - 28672 * q^64 - 7260 * q^66 - 29999i q^67 - 10976i q^68 + 55635 * q^69 + 31023 * q^71 + 3456i q^72 + 1924i q^73 + 48252 * q^74 + 6144 * q^76 + 1210i q^77 - 68880i q^78 - 65138 * q^79 - 54351 * q^81 - 6112i q^82 - 102714i q^83 - 2400 * q^84 + 16104 * q^86 - 81360i q^87 - 23232i q^88 - 17415 * q^89 - 11480 * q^91 + 59344i q^92 + 96645i q^93 + 28672 * q^94 + 76800 * q^96 - 66905i q^97 + 66828i q^98 - 2178 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 32 q^{4} + 120 q^{6} + 36 q^{9} - 242 q^{11} + 80 q^{14} - 512 q^{16} + 768 q^{19} - 300 q^{21} + 5760 q^{24} + 9184 q^{26} + 10848 q^{29} - 12886 q^{31} + 5488 q^{34} + 576 q^{36} - 34440 q^{39}+ \cdots - 4356 q^{99}+O(q^{100}) $ 2 * q + 32 * q^4 + 120 * q^6 + 36 * q^9 - 242 * q^11 + 80 * q^14 - 512 * q^16 + 768 * q^19 - 300 * q^21 + 5760 * q^24 + 9184 * q^26 + 10848 * q^29 - 12886 * q^31 + 5488 * q^34 + 576 * q^36 - 34440 * q^39 - 3056 * q^41 - 3872 * q^44 - 29672 * q^46 + 33414 * q^49 - 20580 * q^51 + 31320 * q^54 + 3840 * q^56 + 12922 * q^59 - 33960 * q^61 - 57344 * q^64 - 14520 * q^66 + 111270 * q^69 + 62046 * q^71 + 96504 * q^74 + 12288 * q^76 - 130276 * q^79 - 108702 * q^81 - 4800 * q^84 + 32208 * q^86 - 34830 * q^89 - 22960 * q^91 + 57344 * q^94 + 153600 * q^96 - 4356 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/275\mathbb{Z}\right)^\times$.

$n$	$101$	$177$
$\chi(n)$	$1$	$-1$

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} $

199.1

	−	1.00000i
		1.00000i

−

4.00000i

15.0000i

16.0000

60.0000

10.0000i

−

192.000i

18.0000

199.2

4.00000i

−

15.0000i

16.0000

60.0000

−

10.0000i

192.000i

18.0000

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	275.6.b.a		2
5.b	even	2	1	inner	275.6.b.a		2
5.c	odd	4	1		11.6.a.a	✓	1
5.c	odd	4	1		275.6.a.a		1
15.e	even	4	1		99.6.a.c		1
20.e	even	4	1		176.6.a.c		1
35.f	even	4	1		539.6.a.c		1
40.i	odd	4	1		704.6.a.h		1
40.k	even	4	1		704.6.a.c		1
55.e	even	4	1		121.6.a.b		1
165.l	odd	4	1		1089.6.a.c		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
11.6.a.a	✓	1	5.c	odd	4	1
99.6.a.c		1	15.e	even	4	1
121.6.a.b		1	55.e	even	4	1
176.6.a.c		1	20.e	even	4	1
275.6.a.a		1	5.c	odd	4	1
275.6.b.a		2	1.a	even	1	1	trivial
275.6.b.a		2	5.b	even	2	1	inner
539.6.a.c		1	35.f	even	4	1
704.6.a.c		1	40.k	even	4	1
704.6.a.h		1	40.i	odd	4	1
1089.6.a.c		1	165.l	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{2} + 16 $ T2^2 + 16 acting on $S_{6}^{\mathrm{new}}(275, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 16 $ T^2 + 16
$3$	$ T^{2} + 225 $ T^2 + 225
$5$	$ T^{2} $ T^2
$7$	$ T^{2} + 100 $ T^2 + 100
$11$	$ (T + 121)^{2} $ (T + 121)^2
$13$	$ T^{2} + 1317904 $ T^2 + 1317904
$17$	$ T^{2} + 470596 $ T^2 + 470596
$19$	$ (T - 384)^{2} $ (T - 384)^2
$23$	$ T^{2} + 13756681 $ T^2 + 13756681
$29$	$ (T - 5424)^{2} $ (T - 5424)^2
$31$	$ (T + 6443)^{2} $ (T + 6443)^2
$37$	$ T^{2} + 145515969 $ T^2 + 145515969
$41$	$ (T + 1528)^{2} $ (T + 1528)^2
$43$	$ T^{2} + 16208676 $ T^2 + 16208676
$47$	$ T^{2} + 51380224 $ T^2 + 51380224
$53$	$ T^{2} + 891739044 $ T^2 + 891739044
$59$	$ (T - 6461)^{2} $ (T - 6461)^2
$61$	$ (T + 16980)^{2} $ (T + 16980)^2
$67$	$ T^{2} + 899940001 $ T^2 + 899940001
$71$	$ (T - 31023)^{2} $ (T - 31023)^2
$73$	$ T^{2} + 3701776 $ T^2 + 3701776
$79$	$ (T + 65138)^{2} $ (T + 65138)^2
$83$	$ T^{2} + 10550165796 $ T^2 + 10550165796
$89$	$ (T + 17415)^{2} $ (T + 17415)^2
$97$	$ T^{2} + 4476279025 $ T^2 + 4476279025
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Elliptic curves over $\Q$ over number fields
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Level:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	275.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + 4 i q^{2} - 15 i q^{3} + 16 q^{4} + 60 q^{6} - 10 i q^{7} + 192 i q^{8} + 18 q^{9} - 121 q^{11} - 240 i q^{12} - 1148 i q^{13} + 40 q^{14} - 256 q^{16} - 686 i q^{17} + 72 i q^{18} + 384 q^{19} - 150 q^{21} + \cdots - 2178 q^{99} +O(q^{100}) \) q + 4i q^2 - 15i q^3 + 16 * q^4 + 60 * q^6 - 10i q^7 + 192i q^8 + 18 * q^9 - 121 * q^11 - 240i q^12 - 1148i q^13 + 40 * q^14 - 256 * q^16 - 686i q^17 + 72i q^18 + 384 * q^19 - 150 * q^21 - 484i q^22 + 3709i q^23 + 2880 * q^24 + 4592 * q^26 - 3915i q^27 - 160i q^28 + 5424 * q^29 - 6443 * q^31 + 5120i q^32 + 1815i q^33 + 2744 * q^34 + 288 * q^36 - 12063i q^37 + 1536i q^38 - 17220 * q^39 - 1528 * q^41 - 600i q^42 - 4026i q^43 - 1936 * q^44 - 14836 * q^46 - 7168i q^47 + 3840i q^48 + 16707 * q^49 - 10290 * q^51 - 18368i q^52 - 29862i q^53 + 15660 * q^54 + 1920 * q^56 - 5760i q^57 + 21696i q^58 + 6461 * q^59 - 16980 * q^61 - 25772i q^62 - 180i q^63 - 28672 * q^64 - 7260 * q^66 - 29999i q^67 - 10976i q^68 + 55635 * q^69 + 31023 * q^71 + 3456i q^72 + 1924i q^73 + 48252 * q^74 + 6144 * q^76 + 1210i q^77 - 68880i q^78 - 65138 * q^79 - 54351 * q^81 - 6112i q^82 - 102714i q^83 - 2400 * q^84 + 16104 * q^86 - 81360i q^87 - 23232i q^88 - 17415 * q^89 - 11480 * q^91 + 59344i q^92 + 96645i q^93 + 28672 * q^94 + 76800 * q^96 - 66905i q^97 + 66828i q^98 - 2178 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 32 q^{4} + 120 q^{6} + 36 q^{9} - 242 q^{11} + 80 q^{14} - 512 q^{16} + 768 q^{19} - 300 q^{21} + 5760 q^{24} + 9184 q^{26} + 10848 q^{29} - 12886 q^{31} + 5488 q^{34} + 576 q^{36} - 34440 q^{39}+ \cdots - 4356 q^{99}+O(q^{100}) \) 2 * q + 32 * q^4 + 120 * q^6 + 36 * q^9 - 242 * q^11 + 80 * q^14 - 512 * q^16 + 768 * q^19 - 300 * q^21 + 5760 * q^24 + 9184 * q^26 + 10848 * q^29 - 12886 * q^31 + 5488 * q^34 + 576 * q^36 - 34440 * q^39 - 3056 * q^41 - 3872 * q^44 - 29672 * q^46 + 33414 * q^49 - 20580 * q^51 + 31320 * q^54 + 3840 * q^56 + 12922 * q^59 - 33960 * q^61 - 57344 * q^64 - 14520 * q^66 + 111270 * q^69 + 62046 * q^71 + 96504 * q^74 + 12288 * q^76 - 130276 * q^79 - 108702 * q^81 - 4800 * q^84 + 32208 * q^86 - 34830 * q^89 - 22960 * q^91 + 57344 * q^94 + 153600 * q^96 - 4356 * q^99

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