LMFDB - Newform orbit 350.6.c.h

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,-32,0,184,0,0,-572,0,1110] 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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
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} + 23 i q^{3} - 16 q^{4} + 92 q^{6} + 49 i q^{7} + 64 i q^{8} - 286 q^{9} + 555 q^{11} - 368 i q^{12} + 241 i q^{13} + 196 q^{14} + 256 q^{16} - 1491 i q^{17} + 1144 i q^{18} + 2038 q^{19} + \cdots - 158730 q^{99} +O(q^{100}) $ q - 4i q^2 + 23i q^3 - 16 * q^4 + 92 * q^6 + 49i q^7 + 64i q^8 - 286 * q^9 + 555 * q^11 - 368i q^12 + 241i q^13 + 196 * q^14 + 256 * q^16 - 1491i q^17 + 1144i q^18 + 2038 * q^19 - 1127 * q^21 - 2220i q^22 + 1230i q^23 - 1472 * q^24 + 964 * q^26 - 989i q^27 - 784i q^28 + 5001 * q^29 + 5696 * q^31 - 1024i q^32 + 12765i q^33 - 5964 * q^34 + 4576 * q^36 - 5602i q^37 - 8152i q^38 - 5543 * q^39 - 2424 * q^41 + 4508i q^42 - 602i q^43 - 8880 * q^44 + 4920 * q^46 - 23163i q^47 + 5888i q^48 - 2401 * q^49 + 34293 * q^51 - 3856i q^52 + 25296i q^53 - 3956 * q^54 - 3136 * q^56 + 46874i q^57 - 20004i q^58 - 5724 * q^59 - 36112 * q^61 - 22784i q^62 - 14014i q^63 - 4096 * q^64 + 51060 * q^66 + 66104i q^67 + 23856i q^68 - 28290 * q^69 + 16080 * q^71 - 18304i q^72 + 80482i q^73 - 22408 * q^74 - 32608 * q^76 + 27195i q^77 + 22172i q^78 + 64147 * q^79 - 46751 * q^81 + 9696i q^82 + 106284i q^83 + 18032 * q^84 - 2408 * q^86 + 115023i q^87 + 35520i q^88 + 71676 * q^89 - 11809 * q^91 - 19680i q^92 + 131008i q^93 - 92652 * q^94 + 23552 * q^96 + 151025i q^97 + 9604i q^98 - 158730 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4} + 184 q^{6} - 572 q^{9} + 1110 q^{11} + 392 q^{14} + 512 q^{16} + 4076 q^{19} - 2254 q^{21} - 2944 q^{24} + 1928 q^{26} + 10002 q^{29} + 11392 q^{31} - 11928 q^{34} + 9152 q^{36} - 11086 q^{39}+ \cdots - 317460 q^{99}+O(q^{100}) $ 2 * q - 32 * q^4 + 184 * q^6 - 572 * q^9 + 1110 * q^11 + 392 * q^14 + 512 * q^16 + 4076 * q^19 - 2254 * q^21 - 2944 * q^24 + 1928 * q^26 + 10002 * q^29 + 11392 * q^31 - 11928 * q^34 + 9152 * q^36 - 11086 * q^39 - 4848 * q^41 - 17760 * q^44 + 9840 * q^46 - 4802 * q^49 + 68586 * q^51 - 7912 * q^54 - 6272 * q^56 - 11448 * q^59 - 72224 * q^61 - 8192 * q^64 + 102120 * q^66 - 56580 * q^69 + 32160 * q^71 - 44816 * q^74 - 65216 * q^76 + 128294 * q^79 - 93502 * q^81 + 36064 * q^84 - 4816 * q^86 + 143352 * q^89 - 23618 * q^91 - 185304 * q^94 + 47104 * q^96 - 317460 * 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

23.0000i

−16.0000

92.0000

49.0000i

64.0000i

−286.000

99.2

4.00000i

−

23.0000i

−16.0000

92.0000

−

49.0000i

−

64.0000i

−286.000

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.h		2
5.b	even	2	1	inner	350.6.c.h		2
5.c	odd	4	1		70.6.a.a	✓	1
5.c	odd	4	1		350.6.a.n		1
15.e	even	4	1		630.6.a.j		1
20.e	even	4	1		560.6.a.i		1
35.f	even	4	1		490.6.a.i		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.6.a.a	✓	1	5.c	odd	4	1
350.6.a.n		1	5.c	odd	4	1
350.6.c.h		2	1.a	even	1	1	trivial
350.6.c.h		2	5.b	even	2	1	inner
490.6.a.i		1	35.f	even	4	1
560.6.a.i		1	20.e	even	4	1
630.6.a.j		1	15.e	even	4	1

Hecke kernels

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

$ T_{3}^{2} + 529 $

T3^2 + 529

$ T_{11} - 555 $

T11 - 555

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 16 $ T^2 + 16
$3$	$ T^{2} + 529 $ T^2 + 529
$5$	$ T^{2} $ T^2
$7$	$ T^{2} + 2401 $ T^2 + 2401
$11$	$ (T - 555)^{2} $ (T - 555)^2
$13$	$ T^{2} + 58081 $ T^2 + 58081
$17$	$ T^{2} + 2223081 $ T^2 + 2223081
$19$	$ (T - 2038)^{2} $ (T - 2038)^2
$23$	$ T^{2} + 1512900 $ T^2 + 1512900
$29$	$ (T - 5001)^{2} $ (T - 5001)^2
$31$	$ (T - 5696)^{2} $ (T - 5696)^2
$37$	$ T^{2} + 31382404 $ T^2 + 31382404
$41$	$ (T + 2424)^{2} $ (T + 2424)^2
$43$	$ T^{2} + 362404 $ T^2 + 362404
$47$	$ T^{2} + 536524569 $ T^2 + 536524569
$53$	$ T^{2} + 639887616 $ T^2 + 639887616
$59$	$ (T + 5724)^{2} $ (T + 5724)^2
$61$	$ (T + 36112)^{2} $ (T + 36112)^2
$67$	$ T^{2} + 4369738816 $ T^2 + 4369738816
$71$	$ (T - 16080)^{2} $ (T - 16080)^2
$73$	$ T^{2} + 6477352324 $ T^2 + 6477352324
$79$	$ (T - 64147)^{2} $ (T - 64147)^2
$83$	$ T^{2} + 11296288656 $ T^2 + 11296288656
$89$	$ (T - 71676)^{2} $ (T - 71676)^2
$97$	$ T^{2} + 22808550625 $ T^2 + 22808550625
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - 4 i q^{2} + 23 i q^{3} - 16 q^{4} + 92 q^{6} + 49 i q^{7} + 64 i q^{8} - 286 q^{9} + 555 q^{11} - 368 i q^{12} + 241 i q^{13} + 196 q^{14} + 256 q^{16} - 1491 i q^{17} + 1144 i q^{18} + 2038 q^{19} + \cdots - 158730 q^{99} +O(q^{100}) \) q - 4i q^2 + 23i q^3 - 16 * q^4 + 92 * q^6 + 49i q^7 + 64i q^8 - 286 * q^9 + 555 * q^11 - 368i q^12 + 241i q^13 + 196 * q^14 + 256 * q^16 - 1491i q^17 + 1144i q^18 + 2038 * q^19 - 1127 * q^21 - 2220i q^22 + 1230i q^23 - 1472 * q^24 + 964 * q^26 - 989i q^27 - 784i q^28 + 5001 * q^29 + 5696 * q^31 - 1024i q^32 + 12765i q^33 - 5964 * q^34 + 4576 * q^36 - 5602i q^37 - 8152i q^38 - 5543 * q^39 - 2424 * q^41 + 4508i q^42 - 602i q^43 - 8880 * q^44 + 4920 * q^46 - 23163i q^47 + 5888i q^48 - 2401 * q^49 + 34293 * q^51 - 3856i q^52 + 25296i q^53 - 3956 * q^54 - 3136 * q^56 + 46874i q^57 - 20004i q^58 - 5724 * q^59 - 36112 * q^61 - 22784i q^62 - 14014i q^63 - 4096 * q^64 + 51060 * q^66 + 66104i q^67 + 23856i q^68 - 28290 * q^69 + 16080 * q^71 - 18304i q^72 + 80482i q^73 - 22408 * q^74 - 32608 * q^76 + 27195i q^77 + 22172i q^78 + 64147 * q^79 - 46751 * q^81 + 9696i q^82 + 106284i q^83 + 18032 * q^84 - 2408 * q^86 + 115023i q^87 + 35520i q^88 + 71676 * q^89 - 11809 * q^91 - 19680i q^92 + 131008i q^93 - 92652 * q^94 + 23552 * q^96 + 151025i q^97 + 9604i q^98 - 158730 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4} + 184 q^{6} - 572 q^{9} + 1110 q^{11} + 392 q^{14} + 512 q^{16} + 4076 q^{19} - 2254 q^{21} - 2944 q^{24} + 1928 q^{26} + 10002 q^{29} + 11392 q^{31} - 11928 q^{34} + 9152 q^{36} - 11086 q^{39}+ \cdots - 317460 q^{99}+O(q^{100}) \) 2 * q - 32 * q^4 + 184 * q^6 - 572 * q^9 + 1110 * q^11 + 392 * q^14 + 512 * q^16 + 4076 * q^19 - 2254 * q^21 - 2944 * q^24 + 1928 * q^26 + 10002 * q^29 + 11392 * q^31 - 11928 * q^34 + 9152 * q^36 - 11086 * q^39 - 4848 * q^41 - 17760 * q^44 + 9840 * q^46 - 4802 * q^49 + 68586 * q^51 - 7912 * q^54 - 6272 * q^56 - 11448 * q^59 - 72224 * q^61 - 8192 * q^64 + 102120 * q^66 - 56580 * q^69 + 32160 * q^71 - 44816 * q^74 - 65216 * q^76 + 128294 * q^79 - 93502 * q^81 + 36064 * q^84 - 4816 * q^86 + 143352 * q^89 - 23618 * q^91 - 185304 * q^94 + 47104 * q^96 - 317460 * q^99

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