LMFDB - Newform orbit 350.6.c.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [350,6,Mod(99,350)] 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("350.99"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(350, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	350.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-32,0,64,0,0,358,0,-680] 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:	$56.1343369345$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 14)
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} + 8 i q^{3} - 16 q^{4} + 32 q^{6} + 49 i q^{7} + 64 i q^{8} + 179 q^{9} - 340 q^{11} - 128 i q^{12} - 294 i q^{13} + 196 q^{14} + 256 q^{16} - 1226 i q^{17} - 716 i q^{18} - 2432 q^{19} - 392 q^{21} + \cdots - 60860 q^{99} +O(q^{100}) $ q - 4i q^2 + 8i q^3 - 16 * q^4 + 32 * q^6 + 49i q^7 + 64i q^8 + 179 * q^9 - 340 * q^11 - 128i q^12 - 294i q^13 + 196 * q^14 + 256 * q^16 - 1226i q^17 - 716i q^18 - 2432 * q^19 - 392 * q^21 + 1360i q^22 + 2000i q^23 - 512 * q^24 - 1176 * q^26 + 3376i q^27 - 784i q^28 + 6746 * q^29 + 8856 * q^31 - 1024i q^32 - 2720i q^33 - 4904 * q^34 - 2864 * q^36 - 9182i q^37 + 9728i q^38 + 2352 * q^39 - 14574 * q^41 + 1568i q^42 + 8108i q^43 + 5440 * q^44 + 8000 * q^46 + 312i q^47 + 2048i q^48 - 2401 * q^49 + 9808 * q^51 + 4704i q^52 - 14634i q^53 + 13504 * q^54 - 3136 * q^56 - 19456i q^57 - 26984i q^58 + 27656 * q^59 + 34338 * q^61 - 35424i q^62 + 8771i q^63 - 4096 * q^64 - 10880 * q^66 - 12316i q^67 + 19616i q^68 - 16000 * q^69 + 36920 * q^71 + 11456i q^72 - 61718i q^73 - 36728 * q^74 + 38912 * q^76 - 16660i q^77 - 9408i q^78 + 64752 * q^79 + 16489 * q^81 + 58296i q^82 - 77056i q^83 + 6272 * q^84 + 32432 * q^86 + 53968i q^87 - 21760i q^88 + 8166 * q^89 + 14406 * q^91 - 32000i q^92 + 70848i q^93 + 1248 * q^94 + 8192 * q^96 - 20650i q^97 + 9604i q^98 - 60860 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4} + 64 q^{6} + 358 q^{9} - 680 q^{11} + 392 q^{14} + 512 q^{16} - 4864 q^{19} - 784 q^{21} - 1024 q^{24} - 2352 q^{26} + 13492 q^{29} + 17712 q^{31} - 9808 q^{34} - 5728 q^{36} + 4704 q^{39}+ \cdots - 121720 q^{99}+O(q^{100}) $ 2 * q - 32 * q^4 + 64 * q^6 + 358 * q^9 - 680 * q^11 + 392 * q^14 + 512 * q^16 - 4864 * q^19 - 784 * q^21 - 1024 * q^24 - 2352 * q^26 + 13492 * q^29 + 17712 * q^31 - 9808 * q^34 - 5728 * q^36 + 4704 * q^39 - 29148 * q^41 + 10880 * q^44 + 16000 * q^46 - 4802 * q^49 + 19616 * q^51 + 27008 * q^54 - 6272 * q^56 + 55312 * q^59 + 68676 * q^61 - 8192 * q^64 - 21760 * q^66 - 32000 * q^69 + 73840 * q^71 - 73456 * q^74 + 77824 * q^76 + 129504 * q^79 + 32978 * q^81 + 12544 * q^84 + 64864 * q^86 + 16332 * q^89 + 28812 * q^91 + 2496 * q^94 + 16384 * q^96 - 121720 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/350\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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} $

99.1

		1.00000i
	−	1.00000i

−

4.00000i

8.00000i

−16.0000

32.0000

49.0000i

64.0000i

179.000

99.2

4.00000i

−

8.00000i

−16.0000

32.0000

−

49.0000i

−

64.0000i

179.000

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	350.6.c.f		2
5.b	even	2	1	inner	350.6.c.f		2
5.c	odd	4	1		14.6.a.b	✓	1
5.c	odd	4	1		350.6.a.b		1
15.e	even	4	1		126.6.a.c		1
20.e	even	4	1		112.6.a.d		1
35.f	even	4	1		98.6.a.b		1
35.k	even	12	2		98.6.c.b		2
35.l	odd	12	2		98.6.c.a		2
40.i	odd	4	1		448.6.a.f		1
40.k	even	4	1		448.6.a.k		1
60.l	odd	4	1		1008.6.a.n		1
105.k	odd	4	1		882.6.a.g		1
140.j	odd	4	1		784.6.a.h		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
14.6.a.b	✓	1	5.c	odd	4	1
98.6.a.b		1	35.f	even	4	1
98.6.c.a		2	35.l	odd	12	2
98.6.c.b		2	35.k	even	12	2
112.6.a.d		1	20.e	even	4	1
126.6.a.c		1	15.e	even	4	1
350.6.a.b		1	5.c	odd	4	1
350.6.c.f		2	1.a	even	1	1	trivial
350.6.c.f		2	5.b	even	2	1	inner
448.6.a.f		1	40.i	odd	4	1
448.6.a.k		1	40.k	even	4	1
784.6.a.h		1	140.j	odd	4	1
882.6.a.g		1	105.k	odd	4	1
1008.6.a.n		1	60.l	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{6}^{\mathrm{new}}(350, [\chi])$:

$ T_{3}^{2} + 64 $

T3^2 + 64

$ T_{11} + 340 $

T11 + 340

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 16 $ T^2 + 16
$3$	$ T^{2} + 64 $ T^2 + 64
$5$	$ T^{2} $ T^2
$7$	$ T^{2} + 2401 $ T^2 + 2401
$11$	$ (T + 340)^{2} $ (T + 340)^2
$13$	$ T^{2} + 86436 $ T^2 + 86436
$17$	$ T^{2} + 1503076 $ T^2 + 1503076
$19$	$ (T + 2432)^{2} $ (T + 2432)^2
$23$	$ T^{2} + 4000000 $ T^2 + 4000000
$29$	$ (T - 6746)^{2} $ (T - 6746)^2
$31$	$ (T - 8856)^{2} $ (T - 8856)^2
$37$	$ T^{2} + 84309124 $ T^2 + 84309124
$41$	$ (T + 14574)^{2} $ (T + 14574)^2
$43$	$ T^{2} + 65739664 $ T^2 + 65739664
$47$	$ T^{2} + 97344 $ T^2 + 97344
$53$	$ T^{2} + 214153956 $ T^2 + 214153956
$59$	$ (T - 27656)^{2} $ (T - 27656)^2
$61$	$ (T - 34338)^{2} $ (T - 34338)^2
$67$	$ T^{2} + 151683856 $ T^2 + 151683856
$71$	$ (T - 36920)^{2} $ (T - 36920)^2
$73$	$ T^{2} + 3809111524 $ T^2 + 3809111524
$79$	$ (T - 64752)^{2} $ (T - 64752)^2
$83$	$ T^{2} + 5937627136 $ T^2 + 5937627136
$89$	$ (T - 8166)^{2} $ (T - 8166)^2
$97$	$ T^{2} + 426422500 $ T^2 + 426422500
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - 4 i q^{2} + 8 i q^{3} - 16 q^{4} + 32 q^{6} + 49 i q^{7} + 64 i q^{8} + 179 q^{9} - 340 q^{11} - 128 i q^{12} - 294 i q^{13} + 196 q^{14} + 256 q^{16} - 1226 i q^{17} - 716 i q^{18} - 2432 q^{19} - 392 q^{21} + \cdots - 60860 q^{99} +O(q^{100}) \) q - 4i q^2 + 8i q^3 - 16 * q^4 + 32 * q^6 + 49i q^7 + 64i q^8 + 179 * q^9 - 340 * q^11 - 128i q^12 - 294i q^13 + 196 * q^14 + 256 * q^16 - 1226i q^17 - 716i q^18 - 2432 * q^19 - 392 * q^21 + 1360i q^22 + 2000i q^23 - 512 * q^24 - 1176 * q^26 + 3376i q^27 - 784i q^28 + 6746 * q^29 + 8856 * q^31 - 1024i q^32 - 2720i q^33 - 4904 * q^34 - 2864 * q^36 - 9182i q^37 + 9728i q^38 + 2352 * q^39 - 14574 * q^41 + 1568i q^42 + 8108i q^43 + 5440 * q^44 + 8000 * q^46 + 312i q^47 + 2048i q^48 - 2401 * q^49 + 9808 * q^51 + 4704i q^52 - 14634i q^53 + 13504 * q^54 - 3136 * q^56 - 19456i q^57 - 26984i q^58 + 27656 * q^59 + 34338 * q^61 - 35424i q^62 + 8771i q^63 - 4096 * q^64 - 10880 * q^66 - 12316i q^67 + 19616i q^68 - 16000 * q^69 + 36920 * q^71 + 11456i q^72 - 61718i q^73 - 36728 * q^74 + 38912 * q^76 - 16660i q^77 - 9408i q^78 + 64752 * q^79 + 16489 * q^81 + 58296i q^82 - 77056i q^83 + 6272 * q^84 + 32432 * q^86 + 53968i q^87 - 21760i q^88 + 8166 * q^89 + 14406 * q^91 - 32000i q^92 + 70848i q^93 + 1248 * q^94 + 8192 * q^96 - 20650i q^97 + 9604i q^98 - 60860 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4} + 64 q^{6} + 358 q^{9} - 680 q^{11} + 392 q^{14} + 512 q^{16} - 4864 q^{19} - 784 q^{21} - 1024 q^{24} - 2352 q^{26} + 13492 q^{29} + 17712 q^{31} - 9808 q^{34} - 5728 q^{36} + 4704 q^{39}+ \cdots - 121720 q^{99}+O(q^{100}) \) 2 * q - 32 * q^4 + 64 * q^6 + 358 * q^9 - 680 * q^11 + 392 * q^14 + 512 * q^16 - 4864 * q^19 - 784 * q^21 - 1024 * q^24 - 2352 * q^26 + 13492 * q^29 + 17712 * q^31 - 9808 * q^34 - 5728 * q^36 + 4704 * q^39 - 29148 * q^41 + 10880 * q^44 + 16000 * q^46 - 4802 * q^49 + 19616 * q^51 + 27008 * q^54 - 6272 * q^56 + 55312 * q^59 + 68676 * q^61 - 8192 * q^64 - 21760 * q^66 - 32000 * q^69 + 73840 * q^71 - 73456 * q^74 + 77824 * q^76 + 129504 * q^79 + 32978 * q^81 + 12544 * q^84 + 64864 * q^86 + 16332 * q^89 + 28812 * q^91 + 2496 * q^94 + 16384 * q^96 - 121720 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more