LMFDB - Newform orbit 350.8.c.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [350,8,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, 8); 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, 8, names="a")

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	350.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-128,0,-1056] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$109.334758919$
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 + 8 i q^{2} + 66 i q^{3} - 64 q^{4} - 528 q^{6} - 343 i q^{7} - 512 i q^{8} - 2169 q^{9} + 40 q^{11} - 4224 i q^{12} + 4452 i q^{13} + 2744 q^{14} + 4096 q^{16} + 36502 i q^{17} - 17352 i q^{18} + 46222 q^{19} + \cdots - 86760 q^{99} +O(q^{100}) $ q + 8i q^2 + 66i q^3 - 64 * q^4 - 528 * q^6 - 343i q^7 - 512i q^8 - 2169 * q^9 + 40 * q^11 - 4224i q^12 + 4452i q^13 + 2744 * q^14 + 4096 * q^16 + 36502i q^17 - 17352i q^18 + 46222 * q^19 + 22638 * q^21 + 320i q^22 + 105200i q^23 + 33792 * q^24 - 35616 * q^26 + 1188i q^27 + 21952i q^28 + 126334 * q^29 - 170964 * q^31 + 32768i q^32 + 2640i q^33 - 292016 * q^34 + 138816 * q^36 + 20954i q^37 + 369776i q^38 - 293832 * q^39 + 318486 * q^41 + 181104i q^42 - 77744i q^43 - 2560 * q^44 - 841600 * q^46 + 703716i q^47 + 270336i q^48 - 117649 * q^49 - 2409132 * q^51 - 284928i q^52 - 1603278i q^53 - 9504 * q^54 - 175616 * q^56 + 3050652i q^57 + 1010672i q^58 + 1171894 * q^59 - 2068872 * q^61 - 1367712i q^62 + 743967i q^63 - 262144 * q^64 - 21120 * q^66 - 994268i q^67 - 2336128i q^68 - 6943200 * q^69 + 33280 * q^71 + 1110528i q^72 + 2971454i q^73 - 167632 * q^74 - 2958208 * q^76 - 13720i q^77 - 2350656i q^78 + 2376168 * q^79 - 4822011 * q^81 + 2547888i q^82 + 2122358i q^83 - 1448832 * q^84 + 621952 * q^86 + 8338044i q^87 - 20480i q^88 - 6920346 * q^89 + 1527036 * q^91 - 6732800i q^92 - 11283624i q^93 - 5629728 * q^94 - 2162688 * q^96 + 4952710i q^97 - 941192i q^98 - 86760 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 128 q^{4} - 1056 q^{6} - 4338 q^{9} + 80 q^{11} + 5488 q^{14} + 8192 q^{16} + 92444 q^{19} + 45276 q^{21} + 67584 q^{24} - 71232 q^{26} + 252668 q^{29} - 341928 q^{31} - 584032 q^{34} + 277632 q^{36}+ \cdots - 173520 q^{99}+O(q^{100}) $ 2 * q - 128 * q^4 - 1056 * q^6 - 4338 * q^9 + 80 * q^11 + 5488 * q^14 + 8192 * q^16 + 92444 * q^19 + 45276 * q^21 + 67584 * q^24 - 71232 * q^26 + 252668 * q^29 - 341928 * q^31 - 584032 * q^34 + 277632 * q^36 - 587664 * q^39 + 636972 * q^41 - 5120 * q^44 - 1683200 * q^46 - 235298 * q^49 - 4818264 * q^51 - 19008 * q^54 - 351232 * q^56 + 2343788 * q^59 - 4137744 * q^61 - 524288 * q^64 - 42240 * q^66 - 13886400 * q^69 + 66560 * q^71 - 335264 * q^74 - 5916416 * q^76 + 4752336 * q^79 - 9644022 * q^81 - 2897664 * q^84 + 1243904 * q^86 - 13840692 * q^89 + 3054072 * q^91 - 11259456 * q^94 - 4325376 * q^96 - 173520 * 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

−

8.00000i

−

66.0000i

−64.0000

−528.000

343.000i

512.000i

−2169.00

99.2

8.00000i

66.0000i

−64.0000

−528.000

−

343.000i

−

512.000i

−2169.00

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	350.8.c.b		2
5.b	even	2	1	inner	350.8.c.b		2
5.c	odd	4	1		14.8.a.b	✓	1
5.c	odd	4	1		350.8.a.d		1
15.e	even	4	1		126.8.a.c		1
20.e	even	4	1		112.8.a.d		1
35.f	even	4	1		98.8.a.c		1
35.k	even	12	2		98.8.c.a		2
35.l	odd	12	2		98.8.c.b		2
40.i	odd	4	1		448.8.a.i		1
40.k	even	4	1		448.8.a.b		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
14.8.a.b	✓	1	5.c	odd	4	1
98.8.a.c		1	35.f	even	4	1
98.8.c.a		2	35.k	even	12	2
98.8.c.b		2	35.l	odd	12	2
112.8.a.d		1	20.e	even	4	1
126.8.a.c		1	15.e	even	4	1
350.8.a.d		1	5.c	odd	4	1
350.8.c.b		2	1.a	even	1	1	trivial
350.8.c.b		2	5.b	even	2	1	inner
448.8.a.b		1	40.k	even	4	1
448.8.a.i		1	40.i	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{2} + 4356 $ T3^2 + 4356 acting on $S_{8}^{\mathrm{new}}(350, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 64 $ T^2 + 64
$3$	$ T^{2} + 4356 $ T^2 + 4356
$5$	$ T^{2} $ T^2
$7$	$ T^{2} + 117649 $ T^2 + 117649
$11$	$ (T - 40)^{2} $ (T - 40)^2
$13$	$ T^{2} + 19820304 $ T^2 + 19820304
$17$	$ T^{2} + 1332396004 $ T^2 + 1332396004
$19$	$ (T - 46222)^{2} $ (T - 46222)^2
$23$	$ T^{2} + 11067040000 $ T^2 + 11067040000
$29$	$ (T - 126334)^{2} $ (T - 126334)^2
$31$	$ (T + 170964)^{2} $ (T + 170964)^2
$37$	$ T^{2} + 439070116 $ T^2 + 439070116
$41$	$ (T - 318486)^{2} $ (T - 318486)^2
$43$	$ T^{2} + 6044129536 $ T^2 + 6044129536
$47$	$ T^{2} + 495216208656 $ T^2 + 495216208656
$53$	$ T^{2} + 2570500345284 $ T^2 + 2570500345284
$59$	$ (T - 1171894)^{2} $ (T - 1171894)^2
$61$	$ (T + 2068872)^{2} $ (T + 2068872)^2
$67$	$ T^{2} + 988568855824 $ T^2 + 988568855824
$71$	$ (T - 33280)^{2} $ (T - 33280)^2
$73$	$ T^{2} + 8829538874116 $ T^2 + 8829538874116
$79$	$ (T - 2376168)^{2} $ (T - 2376168)^2
$83$	$ T^{2} + 4504403480164 $ T^2 + 4504403480164
$89$	$ (T + 6920346)^{2} $ (T + 6920346)^2
$97$	$ T^{2} + 24529336344100 $ T^2 + 24529336344100
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + 8 i q^{2} + 66 i q^{3} - 64 q^{4} - 528 q^{6} - 343 i q^{7} - 512 i q^{8} - 2169 q^{9} + 40 q^{11} - 4224 i q^{12} + 4452 i q^{13} + 2744 q^{14} + 4096 q^{16} + 36502 i q^{17} - 17352 i q^{18} + 46222 q^{19} + \cdots - 86760 q^{99} +O(q^{100}) \) q + 8i q^2 + 66i q^3 - 64 * q^4 - 528 * q^6 - 343i q^7 - 512i q^8 - 2169 * q^9 + 40 * q^11 - 4224i q^12 + 4452i q^13 + 2744 * q^14 + 4096 * q^16 + 36502i q^17 - 17352i q^18 + 46222 * q^19 + 22638 * q^21 + 320i q^22 + 105200i q^23 + 33792 * q^24 - 35616 * q^26 + 1188i q^27 + 21952i q^28 + 126334 * q^29 - 170964 * q^31 + 32768i q^32 + 2640i q^33 - 292016 * q^34 + 138816 * q^36 + 20954i q^37 + 369776i q^38 - 293832 * q^39 + 318486 * q^41 + 181104i q^42 - 77744i q^43 - 2560 * q^44 - 841600 * q^46 + 703716i q^47 + 270336i q^48 - 117649 * q^49 - 2409132 * q^51 - 284928i q^52 - 1603278i q^53 - 9504 * q^54 - 175616 * q^56 + 3050652i q^57 + 1010672i q^58 + 1171894 * q^59 - 2068872 * q^61 - 1367712i q^62 + 743967i q^63 - 262144 * q^64 - 21120 * q^66 - 994268i q^67 - 2336128i q^68 - 6943200 * q^69 + 33280 * q^71 + 1110528i q^72 + 2971454i q^73 - 167632 * q^74 - 2958208 * q^76 - 13720i q^77 - 2350656i q^78 + 2376168 * q^79 - 4822011 * q^81 + 2547888i q^82 + 2122358i q^83 - 1448832 * q^84 + 621952 * q^86 + 8338044i q^87 - 20480i q^88 - 6920346 * q^89 + 1527036 * q^91 - 6732800i q^92 - 11283624i q^93 - 5629728 * q^94 - 2162688 * q^96 + 4952710i q^97 - 941192i q^98 - 86760 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 128 q^{4} - 1056 q^{6} - 4338 q^{9} + 80 q^{11} + 5488 q^{14} + 8192 q^{16} + 92444 q^{19} + 45276 q^{21} + 67584 q^{24} - 71232 q^{26} + 252668 q^{29} - 341928 q^{31} - 584032 q^{34} + 277632 q^{36}+ \cdots - 173520 q^{99}+O(q^{100}) \) 2 * q - 128 * q^4 - 1056 * q^6 - 4338 * q^9 + 80 * q^11 + 5488 * q^14 + 8192 * q^16 + 92444 * q^19 + 45276 * q^21 + 67584 * q^24 - 71232 * q^26 + 252668 * q^29 - 341928 * q^31 - 584032 * q^34 + 277632 * q^36 - 587664 * q^39 + 636972 * q^41 - 5120 * q^44 - 1683200 * q^46 - 235298 * q^49 - 4818264 * q^51 - 19008 * q^54 - 351232 * q^56 + 2343788 * q^59 - 4137744 * q^61 - 524288 * q^64 - 42240 * q^66 - 13886400 * q^69 + 66560 * q^71 - 335264 * q^74 - 5916416 * q^76 + 4752336 * q^79 - 9644022 * q^81 - 2897664 * q^84 + 1243904 * q^86 - 13840692 * q^89 + 3054072 * q^91 - 11259456 * q^94 - 4325376 * q^96 - 173520 * q^99

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