LMFDB - Newform orbit 350.4.c.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [350,4,Mod(99,350)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(350, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("350.99"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	350.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-8,0,-16,0,0,22,0,120] 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:	$20.6506685020$
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 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 + 2 i q^{2} + 4 i q^{3} - 4 q^{4} - 8 q^{6} - 7 i q^{7} - 8 i q^{8} + 11 q^{9} + 60 q^{11} - 16 i q^{12} + 38 i q^{13} + 14 q^{14} + 16 q^{16} - 42 i q^{17} + 22 i q^{18} + 52 q^{19} + 28 q^{21} + 120 i q^{22} + \cdots + 660 q^{99} +O(q^{100}) $ q + 2i q^2 + 4i q^3 - 4 * q^4 - 8 * q^6 - 7i q^7 - 8i q^8 + 11 * q^9 + 60 * q^11 - 16i q^12 + 38i q^13 + 14 * q^14 + 16 * q^16 - 42i q^17 + 22i q^18 + 52 * q^19 + 28 * q^21 + 120i q^22 + 120i q^23 + 32 * q^24 - 76 * q^26 + 152i q^27 + 28i q^28 + 234 * q^29 - 304 * q^31 + 32i q^32 + 240i q^33 + 84 * q^34 - 44 * q^36 + 106i q^37 + 104i q^38 - 152 * q^39 - 54 * q^41 + 56i q^42 - 196i q^43 - 240 * q^44 - 240 * q^46 - 336i q^47 + 64i q^48 - 49 * q^49 + 168 * q^51 - 152i q^52 + 438i q^53 - 304 * q^54 - 56 * q^56 + 208i q^57 + 468i q^58 + 444 * q^59 + 38 * q^61 - 608i q^62 - 77i q^63 - 64 * q^64 - 480 * q^66 + 988i q^67 + 168i q^68 - 480 * q^69 - 720 * q^71 - 88i q^72 + 146i q^73 - 212 * q^74 - 208 * q^76 - 420i q^77 - 304i q^78 + 808 * q^79 - 311 * q^81 - 108i q^82 + 612i q^83 - 112 * q^84 + 392 * q^86 + 936i q^87 - 480i q^88 - 1146 * q^89 + 266 * q^91 - 480i q^92 - 1216i q^93 + 672 * q^94 - 128 * q^96 + 70i q^97 - 98i q^98 + 660 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} - 16 q^{6} + 22 q^{9} + 120 q^{11} + 28 q^{14} + 32 q^{16} + 104 q^{19} + 56 q^{21} + 64 q^{24} - 152 q^{26} + 468 q^{29} - 608 q^{31} + 168 q^{34} - 88 q^{36} - 304 q^{39} - 108 q^{41} - 480 q^{44}+ \cdots + 1320 q^{99}+O(q^{100}) $ 2 * q - 8 * q^4 - 16 * q^6 + 22 * q^9 + 120 * q^11 + 28 * q^14 + 32 * q^16 + 104 * q^19 + 56 * q^21 + 64 * q^24 - 152 * q^26 + 468 * q^29 - 608 * q^31 + 168 * q^34 - 88 * q^36 - 304 * q^39 - 108 * q^41 - 480 * q^44 - 480 * q^46 - 98 * q^49 + 336 * q^51 - 608 * q^54 - 112 * q^56 + 888 * q^59 + 76 * q^61 - 128 * q^64 - 960 * q^66 - 960 * q^69 - 1440 * q^71 - 424 * q^74 - 416 * q^76 + 1616 * q^79 - 622 * q^81 - 224 * q^84 + 784 * q^86 - 2292 * q^89 + 532 * q^91 + 1344 * q^94 - 256 * q^96 + 1320 * 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

−

2.00000i

−

4.00000i

−4.00000

−8.00000

7.00000i

8.00000i

11.0000

99.2

2.00000i

4.00000i

−4.00000

−8.00000

−

7.00000i

−

8.00000i

11.0000

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	350.4.c.d		2
5.b	even	2	1	inner	350.4.c.d		2
5.c	odd	4	1		70.4.a.d	✓	1
5.c	odd	4	1		350.4.a.o		1
15.e	even	4	1		630.4.a.o		1
20.e	even	4	1		560.4.a.g		1
35.f	even	4	1		490.4.a.b		1
35.f	even	4	1		2450.4.a.bm		1
35.k	even	12	2		490.4.e.q		2
35.l	odd	12	2		490.4.e.k		2
40.i	odd	4	1		2240.4.a.m		1
40.k	even	4	1		2240.4.a.y		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.4.a.d	✓	1	5.c	odd	4	1
350.4.a.o		1	5.c	odd	4	1
350.4.c.d		2	1.a	even	1	1	trivial
350.4.c.d		2	5.b	even	2	1	inner
490.4.a.b		1	35.f	even	4	1
490.4.e.k		2	35.l	odd	12	2
490.4.e.q		2	35.k	even	12	2
560.4.a.g		1	20.e	even	4	1
630.4.a.o		1	15.e	even	4	1
2240.4.a.m		1	40.i	odd	4	1
2240.4.a.y		1	40.k	even	4	1
2450.4.a.bm		1	35.f	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(350, [\chi])$:

$ T_{3}^{2} + 16 $

T3^2 + 16

$ T_{11} - 60 $

T11 - 60

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 4 $ T^2 + 4
$3$	$ T^{2} + 16 $ T^2 + 16
$5$	$ T^{2} $ T^2
$7$	$ T^{2} + 49 $ T^2 + 49
$11$	$ (T - 60)^{2} $ (T - 60)^2
$13$	$ T^{2} + 1444 $ T^2 + 1444
$17$	$ T^{2} + 1764 $ T^2 + 1764
$19$	$ (T - 52)^{2} $ (T - 52)^2
$23$	$ T^{2} + 14400 $ T^2 + 14400
$29$	$ (T - 234)^{2} $ (T - 234)^2
$31$	$ (T + 304)^{2} $ (T + 304)^2
$37$	$ T^{2} + 11236 $ T^2 + 11236
$41$	$ (T + 54)^{2} $ (T + 54)^2
$43$	$ T^{2} + 38416 $ T^2 + 38416
$47$	$ T^{2} + 112896 $ T^2 + 112896
$53$	$ T^{2} + 191844 $ T^2 + 191844
$59$	$ (T - 444)^{2} $ (T - 444)^2
$61$	$ (T - 38)^{2} $ (T - 38)^2
$67$	$ T^{2} + 976144 $ T^2 + 976144
$71$	$ (T + 720)^{2} $ (T + 720)^2
$73$	$ T^{2} + 21316 $ T^2 + 21316
$79$	$ (T - 808)^{2} $ (T - 808)^2
$83$	$ T^{2} + 374544 $ T^2 + 374544
$89$	$ (T + 1146)^{2} $ (T + 1146)^2
$97$	$ T^{2} + 4900 $ T^2 + 4900
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Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + 2 i q^{2} + 4 i q^{3} - 4 q^{4} - 8 q^{6} - 7 i q^{7} - 8 i q^{8} + 11 q^{9} + 60 q^{11} - 16 i q^{12} + 38 i q^{13} + 14 q^{14} + 16 q^{16} - 42 i q^{17} + 22 i q^{18} + 52 q^{19} + 28 q^{21} + 120 i q^{22} + \cdots + 660 q^{99} +O(q^{100}) \) q + 2i q^2 + 4i q^3 - 4 * q^4 - 8 * q^6 - 7i q^7 - 8i q^8 + 11 * q^9 + 60 * q^11 - 16i q^12 + 38i q^13 + 14 * q^14 + 16 * q^16 - 42i q^17 + 22i q^18 + 52 * q^19 + 28 * q^21 + 120i q^22 + 120i q^23 + 32 * q^24 - 76 * q^26 + 152i q^27 + 28i q^28 + 234 * q^29 - 304 * q^31 + 32i q^32 + 240i q^33 + 84 * q^34 - 44 * q^36 + 106i q^37 + 104i q^38 - 152 * q^39 - 54 * q^41 + 56i q^42 - 196i q^43 - 240 * q^44 - 240 * q^46 - 336i q^47 + 64i q^48 - 49 * q^49 + 168 * q^51 - 152i q^52 + 438i q^53 - 304 * q^54 - 56 * q^56 + 208i q^57 + 468i q^58 + 444 * q^59 + 38 * q^61 - 608i q^62 - 77i q^63 - 64 * q^64 - 480 * q^66 + 988i q^67 + 168i q^68 - 480 * q^69 - 720 * q^71 - 88i q^72 + 146i q^73 - 212 * q^74 - 208 * q^76 - 420i q^77 - 304i q^78 + 808 * q^79 - 311 * q^81 - 108i q^82 + 612i q^83 - 112 * q^84 + 392 * q^86 + 936i q^87 - 480i q^88 - 1146 * q^89 + 266 * q^91 - 480i q^92 - 1216i q^93 + 672 * q^94 - 128 * q^96 + 70i q^97 - 98i q^98 + 660 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} - 16 q^{6} + 22 q^{9} + 120 q^{11} + 28 q^{14} + 32 q^{16} + 104 q^{19} + 56 q^{21} + 64 q^{24} - 152 q^{26} + 468 q^{29} - 608 q^{31} + 168 q^{34} - 88 q^{36} - 304 q^{39} - 108 q^{41} - 480 q^{44}+ \cdots + 1320 q^{99}+O(q^{100}) \) 2 * q - 8 * q^4 - 16 * q^6 + 22 * q^9 + 120 * q^11 + 28 * q^14 + 32 * q^16 + 104 * q^19 + 56 * q^21 + 64 * q^24 - 152 * q^26 + 468 * q^29 - 608 * q^31 + 168 * q^34 - 88 * q^36 - 304 * q^39 - 108 * q^41 - 480 * q^44 - 480 * q^46 - 98 * q^49 + 336 * q^51 - 608 * q^54 - 112 * q^56 + 888 * q^59 + 76 * q^61 - 128 * q^64 - 960 * q^66 - 960 * q^69 - 1440 * q^71 - 424 * q^74 - 416 * q^76 + 1616 * q^79 - 622 * q^81 - 224 * q^84 + 784 * q^86 - 2292 * q^89 + 532 * q^91 + 1344 * q^94 - 256 * q^96 + 1320 * q^99

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