LMFDB - Newform orbit 350.4.c.a

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)

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);

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")

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,-40,0,0,-146,0,18] 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:	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 + 2 i q^{2} + 10 i q^{3} - 4 q^{4} - 20 q^{6} - 7 i q^{7} - 8 i q^{8} - 73 q^{9} + 9 q^{11} - 40 i q^{12} - 52 i q^{13} + 14 q^{14} + 16 q^{16} - 96 i q^{17} - 146 i q^{18} + 10 q^{19} + 70 q^{21} + 18 i q^{22} + \cdots - 657 q^{99} +O(q^{100}) $ q + 2i q^2 + 10i q^3 - 4 * q^4 - 20 * q^6 - 7i q^7 - 8i q^8 - 73 * q^9 + 9 * q^11 - 40i q^12 - 52i q^13 + 14 * q^14 + 16 * q^16 - 96i q^17 - 146i q^18 + 10 * q^19 + 70 * q^21 + 18i q^22 + 75i q^23 + 80 * q^24 + 104 * q^26 - 460i q^27 + 28i q^28 - 189 * q^29 - 232 * q^31 + 32i q^32 + 90i q^33 + 192 * q^34 + 292 * q^36 - 305i q^37 + 20i q^38 + 520 * q^39 - 438 * q^41 + 140i q^42 + 353i q^43 - 36 * q^44 - 150 * q^46 + 486i q^47 + 160i q^48 - 49 * q^49 + 960 * q^51 + 208i q^52 - 354i q^53 + 920 * q^54 - 56 * q^56 + 100i q^57 - 378i q^58 + 672 * q^59 + 206 * q^61 - 464i q^62 + 511i q^63 - 64 * q^64 - 180 * q^66 - 599i q^67 + 384i q^68 - 750 * q^69 - 471 * q^71 + 584i q^72 + 614i q^73 + 610 * q^74 - 40 * q^76 - 63i q^77 + 1040i q^78 - 743 * q^79 + 2629 * q^81 - 876i q^82 + 996i q^83 - 280 * q^84 - 706 * q^86 - 1890i q^87 - 72i q^88 - 180 * q^89 - 364 * q^91 - 300i q^92 - 2320i q^93 - 972 * q^94 - 320 * q^96 + 184i q^97 - 98i q^98 - 657 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} - 40 q^{6} - 146 q^{9} + 18 q^{11} + 28 q^{14} + 32 q^{16} + 20 q^{19} + 140 q^{21} + 160 q^{24} + 208 q^{26} - 378 q^{29} - 464 q^{31} + 384 q^{34} + 584 q^{36} + 1040 q^{39} - 876 q^{41}+ \cdots - 1314 q^{99}+O(q^{100}) $ 2 * q - 8 * q^4 - 40 * q^6 - 146 * q^9 + 18 * q^11 + 28 * q^14 + 32 * q^16 + 20 * q^19 + 140 * q^21 + 160 * q^24 + 208 * q^26 - 378 * q^29 - 464 * q^31 + 384 * q^34 + 584 * q^36 + 1040 * q^39 - 876 * q^41 - 72 * q^44 - 300 * q^46 - 98 * q^49 + 1920 * q^51 + 1840 * q^54 - 112 * q^56 + 1344 * q^59 + 412 * q^61 - 128 * q^64 - 360 * q^66 - 1500 * q^69 - 942 * q^71 + 1220 * q^74 - 80 * q^76 - 1486 * q^79 + 5258 * q^81 - 560 * q^84 - 1412 * q^86 - 360 * q^89 - 728 * q^91 - 1944 * q^94 - 640 * q^96 - 1314 * 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

−

10.0000i

−4.00000

−20.0000

7.00000i

8.00000i

−73.0000

99.2

2.00000i

10.0000i

−4.00000

−20.0000

−

7.00000i

−

8.00000i

−73.0000

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.a		2
5.b	even	2	1	inner	350.4.c.a		2
5.c	odd	4	1		350.4.a.j	✓	1
5.c	odd	4	1		350.4.a.k	yes	1
35.f	even	4	1		2450.4.a.a		1
35.f	even	4	1		2450.4.a.bp		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
350.4.a.j	✓	1	5.c	odd	4	1
350.4.a.k	yes	1	5.c	odd	4	1
350.4.c.a		2	1.a	even	1	1	trivial
350.4.c.a		2	5.b	even	2	1	inner
2450.4.a.a		1	35.f	even	4	1
2450.4.a.bp		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} + 100 $

T3^2 + 100

$ T_{11} - 9 $

T11 - 9

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 4 $ T^2 + 4
$3$	$ T^{2} + 100 $ T^2 + 100
$5$	$ T^{2} $ T^2
$7$	$ T^{2} + 49 $ T^2 + 49
$11$	$ (T - 9)^{2} $ (T - 9)^2
$13$	$ T^{2} + 2704 $ T^2 + 2704
$17$	$ T^{2} + 9216 $ T^2 + 9216
$19$	$ (T - 10)^{2} $ (T - 10)^2
$23$	$ T^{2} + 5625 $ T^2 + 5625
$29$	$ (T + 189)^{2} $ (T + 189)^2
$31$	$ (T + 232)^{2} $ (T + 232)^2
$37$	$ T^{2} + 93025 $ T^2 + 93025
$41$	$ (T + 438)^{2} $ (T + 438)^2
$43$	$ T^{2} + 124609 $ T^2 + 124609
$47$	$ T^{2} + 236196 $ T^2 + 236196
$53$	$ T^{2} + 125316 $ T^2 + 125316
$59$	$ (T - 672)^{2} $ (T - 672)^2
$61$	$ (T - 206)^{2} $ (T - 206)^2
$67$	$ T^{2} + 358801 $ T^2 + 358801
$71$	$ (T + 471)^{2} $ (T + 471)^2
$73$	$ T^{2} + 376996 $ T^2 + 376996
$79$	$ (T + 743)^{2} $ (T + 743)^2
$83$	$ T^{2} + 992016 $ T^2 + 992016
$89$	$ (T + 180)^{2} $ (T + 180)^2
$97$	$ T^{2} + 33856 $ T^2 + 33856
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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)

\(f(q)\)	\(=\)	\( q + 2 i q^{2} + 10 i q^{3} - 4 q^{4} - 20 q^{6} - 7 i q^{7} - 8 i q^{8} - 73 q^{9} + 9 q^{11} - 40 i q^{12} - 52 i q^{13} + 14 q^{14} + 16 q^{16} - 96 i q^{17} - 146 i q^{18} + 10 q^{19} + 70 q^{21} + 18 i q^{22} + \cdots - 657 q^{99} +O(q^{100}) \) q + 2i q^2 + 10i q^3 - 4 * q^4 - 20 * q^6 - 7i q^7 - 8i q^8 - 73 * q^9 + 9 * q^11 - 40i q^12 - 52i q^13 + 14 * q^14 + 16 * q^16 - 96i q^17 - 146i q^18 + 10 * q^19 + 70 * q^21 + 18i q^22 + 75i q^23 + 80 * q^24 + 104 * q^26 - 460i q^27 + 28i q^28 - 189 * q^29 - 232 * q^31 + 32i q^32 + 90i q^33 + 192 * q^34 + 292 * q^36 - 305i q^37 + 20i q^38 + 520 * q^39 - 438 * q^41 + 140i q^42 + 353i q^43 - 36 * q^44 - 150 * q^46 + 486i q^47 + 160i q^48 - 49 * q^49 + 960 * q^51 + 208i q^52 - 354i q^53 + 920 * q^54 - 56 * q^56 + 100i q^57 - 378i q^58 + 672 * q^59 + 206 * q^61 - 464i q^62 + 511i q^63 - 64 * q^64 - 180 * q^66 - 599i q^67 + 384i q^68 - 750 * q^69 - 471 * q^71 + 584i q^72 + 614i q^73 + 610 * q^74 - 40 * q^76 - 63i q^77 + 1040i q^78 - 743 * q^79 + 2629 * q^81 - 876i q^82 + 996i q^83 - 280 * q^84 - 706 * q^86 - 1890i q^87 - 72i q^88 - 180 * q^89 - 364 * q^91 - 300i q^92 - 2320i q^93 - 972 * q^94 - 320 * q^96 + 184i q^97 - 98i q^98 - 657 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} - 40 q^{6} - 146 q^{9} + 18 q^{11} + 28 q^{14} + 32 q^{16} + 20 q^{19} + 140 q^{21} + 160 q^{24} + 208 q^{26} - 378 q^{29} - 464 q^{31} + 384 q^{34} + 584 q^{36} + 1040 q^{39} - 876 q^{41}+ \cdots - 1314 q^{99}+O(q^{100}) \) 2 * q - 8 * q^4 - 40 * q^6 - 146 * q^9 + 18 * q^11 + 28 * q^14 + 32 * q^16 + 20 * q^19 + 140 * q^21 + 160 * q^24 + 208 * q^26 - 378 * q^29 - 464 * q^31 + 384 * q^34 + 584 * q^36 + 1040 * q^39 - 876 * q^41 - 72 * q^44 - 300 * q^46 - 98 * q^49 + 1920 * q^51 + 1840 * q^54 - 112 * q^56 + 1344 * q^59 + 412 * q^61 - 128 * q^64 - 360 * q^66 - 1500 * q^69 - 942 * q^71 + 1220 * q^74 - 80 * q^76 - 1486 * q^79 + 5258 * q^81 - 560 * q^84 - 1412 * q^86 - 360 * q^89 - 728 * q^91 - 1944 * q^94 - 640 * q^96 - 1314 * q^99

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