LMFDB - Newform orbit 350.6.c.d

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,-80,0,0,286,0,-672] 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} + 10 i q^{3} - 16 q^{4} - 40 q^{6} - 49 i q^{7} - 64 i q^{8} + 143 q^{9} - 336 q^{11} - 160 i q^{12} + 584 i q^{13} + 196 q^{14} + 256 q^{16} + 1458 i q^{17} + 572 i q^{18} - 470 q^{19} + 490 q^{21} + \cdots - 48048 q^{99} +O(q^{100}) $ q + 4i q^2 + 10i q^3 - 16 * q^4 - 40 * q^6 - 49i q^7 - 64i q^8 + 143 * q^9 - 336 * q^11 - 160i q^12 + 584i q^13 + 196 * q^14 + 256 * q^16 + 1458i q^17 + 572i q^18 - 470 * q^19 + 490 * q^21 - 1344i q^22 - 4200i q^23 + 640 * q^24 - 2336 * q^26 + 3860i q^27 + 784i q^28 - 4866 * q^29 - 7372 * q^31 + 1024i q^32 - 3360i q^33 - 5832 * q^34 - 2288 * q^36 - 14330i q^37 - 1880i q^38 - 5840 * q^39 + 6222 * q^41 + 1960i q^42 + 3704i q^43 + 5376 * q^44 + 16800 * q^46 + 1812i q^47 + 2560i q^48 - 2401 * q^49 - 14580 * q^51 - 9344i q^52 - 37242i q^53 - 15440 * q^54 - 3136 * q^56 - 4700i q^57 - 19464i q^58 - 34302 * q^59 + 24476 * q^61 - 29488i q^62 - 7007i q^63 - 4096 * q^64 + 13440 * q^66 + 17452i q^67 - 23328i q^68 + 42000 * q^69 + 28224 * q^71 - 9152i q^72 + 3602i q^73 + 57320 * q^74 + 7520 * q^76 + 16464i q^77 - 23360i q^78 - 42872 * q^79 - 3851 * q^81 + 24888i q^82 - 35202i q^83 - 7840 * q^84 - 14816 * q^86 - 48660i q^87 + 21504i q^88 - 26730 * q^89 + 28616 * q^91 + 67200i q^92 - 73720i q^93 - 7248 * q^94 - 10240 * q^96 + 16978i q^97 - 9604i q^98 - 48048 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4} - 80 q^{6} + 286 q^{9} - 672 q^{11} + 392 q^{14} + 512 q^{16} - 940 q^{19} + 980 q^{21} + 1280 q^{24} - 4672 q^{26} - 9732 q^{29} - 14744 q^{31} - 11664 q^{34} - 4576 q^{36} - 11680 q^{39}+ \cdots - 96096 q^{99}+O(q^{100}) $ 2 * q - 32 * q^4 - 80 * q^6 + 286 * q^9 - 672 * q^11 + 392 * q^14 + 512 * q^16 - 940 * q^19 + 980 * q^21 + 1280 * q^24 - 4672 * q^26 - 9732 * q^29 - 14744 * q^31 - 11664 * q^34 - 4576 * q^36 - 11680 * q^39 + 12444 * q^41 + 10752 * q^44 + 33600 * q^46 - 4802 * q^49 - 29160 * q^51 - 30880 * q^54 - 6272 * q^56 - 68604 * q^59 + 48952 * q^61 - 8192 * q^64 + 26880 * q^66 + 84000 * q^69 + 56448 * q^71 + 114640 * q^74 + 15040 * q^76 - 85744 * q^79 - 7702 * q^81 - 15680 * q^84 - 29632 * q^86 - 53460 * q^89 + 57232 * q^91 - 14496 * q^94 - 20480 * q^96 - 96096 * 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

−

10.0000i

−16.0000

−40.0000

49.0000i

64.0000i

143.000

99.2

4.00000i

10.0000i

−16.0000

−40.0000

−

49.0000i

−

64.0000i

143.000

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

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
14.6.a.a	✓	1	5.c	odd	4	1
98.6.a.a		1	35.f	even	4	1
98.6.c.c		2	35.l	odd	12	2
98.6.c.d		2	35.k	even	12	2
112.6.a.c		1	20.e	even	4	1
126.6.a.f		1	15.e	even	4	1
350.6.a.i		1	5.c	odd	4	1
350.6.c.d		2	1.a	even	1	1	trivial
350.6.c.d		2	5.b	even	2	1	inner
448.6.a.e		1	40.i	odd	4	1
448.6.a.l		1	40.k	even	4	1
784.6.a.i		1	140.j	odd	4	1
882.6.a.x		1	105.k	odd	4	1
1008.6.a.b		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} + 100 $

T3^2 + 100

$ T_{11} + 336 $

T11 + 336

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 16 $ T^2 + 16
$3$	$ T^{2} + 100 $ T^2 + 100
$5$	$ T^{2} $ T^2
$7$	$ T^{2} + 2401 $ T^2 + 2401
$11$	$ (T + 336)^{2} $ (T + 336)^2
$13$	$ T^{2} + 341056 $ T^2 + 341056
$17$	$ T^{2} + 2125764 $ T^2 + 2125764
$19$	$ (T + 470)^{2} $ (T + 470)^2
$23$	$ T^{2} + 17640000 $ T^2 + 17640000
$29$	$ (T + 4866)^{2} $ (T + 4866)^2
$31$	$ (T + 7372)^{2} $ (T + 7372)^2
$37$	$ T^{2} + 205348900 $ T^2 + 205348900
$41$	$ (T - 6222)^{2} $ (T - 6222)^2
$43$	$ T^{2} + 13719616 $ T^2 + 13719616
$47$	$ T^{2} + 3283344 $ T^2 + 3283344
$53$	$ T^{2} + 1386966564 $ T^2 + 1386966564
$59$	$ (T + 34302)^{2} $ (T + 34302)^2
$61$	$ (T - 24476)^{2} $ (T - 24476)^2
$67$	$ T^{2} + 304572304 $ T^2 + 304572304
$71$	$ (T - 28224)^{2} $ (T - 28224)^2
$73$	$ T^{2} + 12974404 $ T^2 + 12974404
$79$	$ (T + 42872)^{2} $ (T + 42872)^2
$83$	$ T^{2} + 1239180804 $ T^2 + 1239180804
$89$	$ (T + 26730)^{2} $ (T + 26730)^2
$97$	$ T^{2} + 288252484 $ T^2 + 288252484
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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)

\(f(q)\)	\(=\)	\( q + 4 i q^{2} + 10 i q^{3} - 16 q^{4} - 40 q^{6} - 49 i q^{7} - 64 i q^{8} + 143 q^{9} - 336 q^{11} - 160 i q^{12} + 584 i q^{13} + 196 q^{14} + 256 q^{16} + 1458 i q^{17} + 572 i q^{18} - 470 q^{19} + 490 q^{21} + \cdots - 48048 q^{99} +O(q^{100}) \) q + 4i q^2 + 10i q^3 - 16 * q^4 - 40 * q^6 - 49i q^7 - 64i q^8 + 143 * q^9 - 336 * q^11 - 160i q^12 + 584i q^13 + 196 * q^14 + 256 * q^16 + 1458i q^17 + 572i q^18 - 470 * q^19 + 490 * q^21 - 1344i q^22 - 4200i q^23 + 640 * q^24 - 2336 * q^26 + 3860i q^27 + 784i q^28 - 4866 * q^29 - 7372 * q^31 + 1024i q^32 - 3360i q^33 - 5832 * q^34 - 2288 * q^36 - 14330i q^37 - 1880i q^38 - 5840 * q^39 + 6222 * q^41 + 1960i q^42 + 3704i q^43 + 5376 * q^44 + 16800 * q^46 + 1812i q^47 + 2560i q^48 - 2401 * q^49 - 14580 * q^51 - 9344i q^52 - 37242i q^53 - 15440 * q^54 - 3136 * q^56 - 4700i q^57 - 19464i q^58 - 34302 * q^59 + 24476 * q^61 - 29488i q^62 - 7007i q^63 - 4096 * q^64 + 13440 * q^66 + 17452i q^67 - 23328i q^68 + 42000 * q^69 + 28224 * q^71 - 9152i q^72 + 3602i q^73 + 57320 * q^74 + 7520 * q^76 + 16464i q^77 - 23360i q^78 - 42872 * q^79 - 3851 * q^81 + 24888i q^82 - 35202i q^83 - 7840 * q^84 - 14816 * q^86 - 48660i q^87 + 21504i q^88 - 26730 * q^89 + 28616 * q^91 + 67200i q^92 - 73720i q^93 - 7248 * q^94 - 10240 * q^96 + 16978i q^97 - 9604i q^98 - 48048 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4} - 80 q^{6} + 286 q^{9} - 672 q^{11} + 392 q^{14} + 512 q^{16} - 940 q^{19} + 980 q^{21} + 1280 q^{24} - 4672 q^{26} - 9732 q^{29} - 14744 q^{31} - 11664 q^{34} - 4576 q^{36} - 11680 q^{39}+ \cdots - 96096 q^{99}+O(q^{100}) \) 2 * q - 32 * q^4 - 80 * q^6 + 286 * q^9 - 672 * q^11 + 392 * q^14 + 512 * q^16 - 940 * q^19 + 980 * q^21 + 1280 * q^24 - 4672 * q^26 - 9732 * q^29 - 14744 * q^31 - 11664 * q^34 - 4576 * q^36 - 11680 * q^39 + 12444 * q^41 + 10752 * q^44 + 33600 * q^46 - 4802 * q^49 - 29160 * q^51 - 30880 * q^54 - 6272 * q^56 - 68604 * q^59 + 48952 * q^61 - 8192 * q^64 + 26880 * q^66 + 84000 * q^69 + 56448 * q^71 + 114640 * q^74 + 15040 * q^76 - 85744 * q^79 - 7702 * q^81 - 15680 * q^84 - 29632 * q^86 - 53460 * q^89 + 57232 * q^91 - 14496 * q^94 - 20480 * q^96 - 96096 * q^99

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