LMFDB - Newform orbit 136.4.b.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [136,4,Mod(33,136)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(136, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("136.33");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 136 = 2^{3} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	136.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$8.02425976078$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 81x^{4} + 222x^{2} + 64 $ x^6 + 81x^4 + 222x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8} $
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 a basis $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_1 q^{3} - \beta_{4} q^{5} + (\beta_{4} + \beta_{2} - \beta_1) q^{7} + (\beta_{3} - 7) q^{9}+O(q^{10}) $ q - b1 * q^3 - b4 * q^5 + (b4 + b2 - b1) * q^7 + (b3 - 7) * q^9	$ q - \beta_1 q^{3} - \beta_{4} q^{5} + (\beta_{4} + \beta_{2} - \beta_1) q^{7} + (\beta_{3} - 7) q^{9} + (\beta_{2} - \beta_1) q^{11} + (\beta_{5} - 12) q^{13} + (\beta_{5} - \beta_{3} - 4) q^{15} + (\beta_{5} + \beta_{4} + 2 \beta_{3} + \cdots - 21) q^{17}+ \cdots + ( - 22 \beta_{4} - 2 \beta_{2} + 73 \beta_1) q^{99}+O(q^{100}) $ q - b1 * q^3 - b4 * q^5 + (b4 + b2 - b1) * q^7 + (b3 - 7) * q^9 + (b2 - b1) * q^11 + (b5 - 12) * q^13 + (b5 - b3 - 4) * q^15 + (b5 + b4 + 2b3 - b2 + 2b1 - 21) * q^17 + (-b5 + 3b3 - 4) q^19 + (5b3 - 34) q^21 + (-7b4 - 2b2 + 3b1) q^23 + (3b5 - b3 - 19) q^25 + (-10b4 - 7b2 - 2b1) q^27 + (-9b4 + 6b2 + 4b1) q^29 + (15b4 + 6b2 + 3b1) q^31 + (b5 + 4b3 - 38) q^33 + (-2b5 - 4b3 + 68) * q^35 + (9b4 + 12b2 - 28b1) q^37 + (18b4 - b2 + 2b1) * q^39 + (-2b4 - 2b2 - 24b1) q^41 + (b5 - 13b3 + 32) q^43 + (b4 + 6b2 - 24b1) * q^45 + (-5b5 + b3 - 12) q^47 + (-3b5 + 6b3 - 3) * q^49 + (-2b5 - 2b4 - 4b3 - 15b2 + 47b1 + 76) q^51 + (-b5 - 7b3 + 154) q^53 + (b5 - 5b3 - 76) q^55 + (-48b4 - 20b2 + 68b1) q^57 + (-9b5 - 11b3 + 280) * q^59 + (-5b4 - 14b2 - 52b1) q^61 + (-23b4 - 8b2 + 97b1) q^63 + (66b4 + 10b2 + 72b1) q^65 + (-7b5 + 5b3 - 52) * q^67 + (5b5 - 16b3 + 82) * q^69 + (7b4 - 19b2 - 117b1) q^71 + (-8b4 - 22b2 + 44b1) q^73 + (64b4 + 4b2 - 29b1) q^75 + (-5b5 + 2b3 - 278) * q^77 + (-27b4 + 34b2 - 25b1) q^79 + (3b5 - 2b3 - 269) * q^81 + (-b5 - 23b3 + 216) q^83 + (-5b5 + 63b4 + 7b3 + 22b2 + 24b1 + 224) q^85 + (15b5 + 5b3 + 76) * q^87 + (-3b5 - 14b3 + 4) * q^89 + (-20b4 + 21b2 - 22b1) q^91 + (-9b5 + 30b3 + 138) * q^93 + (-68b4 + 8b2 - 144b1) q^95 + (-70b4 + 8b2 + 116b1) q^97 + (-22b4 - 2b2 + 73b1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 42 q^{9}+O(q^{10}) $ 6 * q - 42 * q^9	$ 6 q - 42 q^{9} - 72 q^{13} - 24 q^{15} - 126 q^{17} - 24 q^{19} - 204 q^{21} - 114 q^{25} - 228 q^{33} + 408 q^{35} + 192 q^{43} - 72 q^{47} - 18 q^{49} + 456 q^{51} + 924 q^{53} - 456 q^{55} + 1680 q^{59} - 312 q^{67} + 492 q^{69} - 1668 q^{77} - 1614 q^{81} + 1296 q^{83} + 1344 q^{85} + 456 q^{87} + 24 q^{89} + 828 q^{93}+O(q^{100}) $ 6 * q - 42 * q^9 - 72 * q^13 - 24 * q^15 - 126 * q^17 - 24 * q^19 - 204 * q^21 - 114 * q^25 - 228 * q^33 + 408 * q^35 + 192 * q^43 - 72 * q^47 - 18 * q^49 + 456 * q^51 + 924 * q^53 - 456 * q^55 + 1680 * q^59 - 312 * q^67 + 492 * q^69 - 1668 * q^77 - 1614 * q^81 + 1296 * q^83 + 1344 * q^85 + 456 * q^87 + 24 * q^89 + 828 * q^93

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} + 81x^{4} + 222x^{2} + 64 $ x^6 + 81*x^4 + 222*x^2 + 64 :

$\beta_{1}$	$=$	$ ( \nu^{5} + 85\nu^{3} + 562\nu ) / 68 $ (v^5 + 85v^3 + 562v) / 68
$\beta_{2}$	$=$	$ ( -\nu^{5} - 85\nu^{3} - 426\nu ) / 34 $ (-v^5 - 85v^3 - 426v) / 34
$\beta_{3}$	$=$	$ ( 5\nu^{4} + 391\nu^{2} + 362 ) / 17 $ (5v^4 + 391v^2 + 362) / 17
$\beta_{4}$	$=$	$ ( 15\nu^{5} + 1207\nu^{3} + 2650\nu ) / 68 $ (15v^5 + 1207v^3 + 2650*v) / 68
$\beta_{5}$	$=$	$ ( -9\nu^{4} - 731\nu^{2} - 1386 ) / 17 $ (-9v^4 - 731v^2 - 1386) / 17

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( \beta_{2} + 2\beta_1 ) / 4 $ (b2 + 2*b1) / 4
$\nu^{2}$	$=$	$ ( -5\beta_{5} - 9\beta_{3} - 216 ) / 8 $ (-5b5 - 9b3 - 216) / 8
$\nu^{3}$	$=$	$ ( -4\beta_{4} - 85\beta_{2} - 110\beta_1 ) / 4 $ (-4b4 - 85b2 - 110*b1) / 4
$\nu^{4}$	$=$	$ ( 391\beta_{5} + 731\beta_{3} + 16312 ) / 8 $ (391b5 + 731b3 + 16312) / 8
$\nu^{5}$	$=$	$ ( 340\beta_{4} + 6663\beta_{2} + 8498\beta_1 ) / 4 $ (340b4 + 6663b2 + 8498*b1) / 4

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/136\mathbb{Z}\right)^\times$.

$n$	$69$	$103$	$105$
$\chi(n)$	$1$	$1$	$-1$

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} $

33.1

		1.58186i
		0.572006i
		8.84141i
	−	8.84141i
	−	0.572006i
	−	1.58186i

−

8.27144i

6.42821i

−

24.9151i

−41.4167

33.2

−

4.49442i

−

18.9829i

7.78768i

6.80022

33.3

−

3.65834i

5.50702i

18.8836i

13.6165

33.4

3.65834i

−

5.50702i

−

18.8836i

13.6165

33.5

4.49442i

18.9829i

−

7.78768i

6.80022

33.6

8.27144i

−

6.42821i

24.9151i

−41.4167

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
17.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	136.4.b.a	✓	6
3.b	odd	2	1		1224.4.c.c		6
4.b	odd	2	1		272.4.b.e		6
17.b	even	2	1	inner	136.4.b.a	✓	6
17.c	even	4	2		2312.4.a.h		6
51.c	odd	2	1		1224.4.c.c		6
68.d	odd	2	1		272.4.b.e		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
136.4.b.a	✓	6	1.a	even	1	1	trivial
136.4.b.a	✓	6	17.b	even	2	1	inner
272.4.b.e		6	4.b	odd	2	1
272.4.b.e		6	68.d	odd	2	1
1224.4.c.c		6	3.b	odd	2	1
1224.4.c.c		6	51.c	odd	2	1
2312.4.a.h		6	17.c	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{6} + 102T_{3}^{4} + 2568T_{3}^{2} + 18496 $ T3^6 + 102*T3^4 + 2568*T3^2 + 18496 acting on $S_{4}^{\mathrm{new}}(136, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} + 102 T^{4} + \cdots + 18496 $ T^6 + 102T^4 + 2568T^2 + 18496
$5$	$ T^{6} + 432 T^{4} + \cdots + 451584 $ T^6 + 432T^4 + 27072T^2 + 451584
$7$	$ T^{6} + 1038 T^{4} + \cdots + 13424896 $ T^6 + 1038T^4 + 280632T^2 + 13424896
$11$	$ T^{6} + 1062 T^{4} + \cdots + 25482304 $ T^6 + 1062T^4 + 320712T^2 + 25482304
$13$	$ (T^{3} + 36 T^{2} + \cdots + 28016)^{2} $ (T^3 + 36T^2 - 3108T + 28016)^2
$17$	$ T^{6} + \cdots + 118587876497 $ T^6 + 126T^5 - 2193T^4 - 886652T^3 - 10774209T^2 + 3041333694*T + 118587876497
$19$	$ (T^{3} + 12 T^{2} + \cdots + 178496)^{2} $ (T^3 + 12T^2 - 13968T + 178496)^2
$23$	$ T^{6} + \cdots + 3953894400 $ T^6 + 19422T^4 + 92332728T^2 + 3953894400
$29$	$ T^{6} + \cdots + 1770986054656 $ T^6 + 92208T^4 + 2095177152T^2 + 1770986054656
$31$	$ T^{6} + \cdots + 10987474749696 $ T^6 + 93582T^4 + 1963014264T^2 + 10987474749696
$37$	$ T^{6} + \cdots + 7252378264576 $ T^6 + 205104T^4 + 6040482240T^2 + 7252378264576
$41$	$ T^{6} + \cdots + 8454416891904 $ T^6 + 62496T^4 + 1270536192T^2 + 8454416891904
$43$	$ (T^{3} - 96 T^{2} + \cdots - 20641152)^{2} $ (T^3 - 96T^2 - 162864T - 20641152)^2
$47$	$ (T^{3} + 36 T^{2} + \cdots - 11699200)^{2} $ (T^3 + 36T^2 - 92928T - 11699200)^2
$53$	$ (T^{3} - 462 T^{2} + \cdots + 1630776)^{2} $ (T^3 - 462T^2 + 29052T + 1630776)^2
$59$	$ (T^{3} - 840 T^{2} + \cdots + 116758016)^{2} $ (T^3 - 840T^2 - 82032T + 116758016)^2
$61$	$ T^{6} + \cdots + 11\!\cdots\!76 $ T^6 + 428592T^4 + 40641622464T^2 + 1105521004389376
$67$	$ (T^{3} + 156 T^{2} + \cdots - 49226176)^{2} $ (T^3 + 156T^2 - 215568T - 49226176)^2
$71$	$ T^{6} + \cdots + 47\!\cdots\!76 $ T^6 + 1739790T^4 + 787343311224T^2 + 47250141207192576
$73$	$ T^{6} + \cdots + 25\!\cdots\!64 $ T^6 + 616896T^4 + 91224932352T^2 + 2599674984792064
$79$	$ T^{6} + \cdots + 10\!\cdots\!44 $ T^6 + 1893870T^4 + 996368004024T^2 + 107610430551122944
$83$	$ (T^{3} - 648 T^{2} + \cdots - 10110464)^{2} $ (T^3 - 648T^2 - 321456T - 10110464)^2
$89$	$ (T^{3} - 12 T^{2} + \cdots + 2334896)^{2} $ (T^3 - 12T^2 - 174948T + 2334896)^2
$97$	$ T^{6} + \cdots + 13\!\cdots\!16 $ T^6 + 3573984T^4 + 3985838211072T^2 + 1398109715946274816
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 136 = 2^{3} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	136.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_1 q^{3} - \beta_{4} q^{5} + (\beta_{4} + \beta_{2} - \beta_1) q^{7} + (\beta_{3} - 7) q^{9}+O(q^{10}) \) q - b1 * q^3 - b4 * q^5 + (b4 + b2 - b1) * q^7 + (b3 - 7) * q^9	\( q - \beta_1 q^{3} - \beta_{4} q^{5} + (\beta_{4} + \beta_{2} - \beta_1) q^{7} + (\beta_{3} - 7) q^{9} + (\beta_{2} - \beta_1) q^{11} + (\beta_{5} - 12) q^{13} + (\beta_{5} - \beta_{3} - 4) q^{15} + (\beta_{5} + \beta_{4} + 2 \beta_{3} + \cdots - 21) q^{17}+ \cdots + ( - 22 \beta_{4} - 2 \beta_{2} + 73 \beta_1) q^{99}+O(q^{100}) \) q - b1 * q^3 - b4 * q^5 + (b4 + b2 - b1) * q^7 + (b3 - 7) * q^9 + (b2 - b1) * q^11 + (b5 - 12) * q^13 + (b5 - b3 - 4) * q^15 + (b5 + b4 + 2b3 - b2 + 2b1 - 21) * q^17 + (-b5 + 3b3 - 4) q^19 + (5b3 - 34) q^21 + (-7b4 - 2b2 + 3b1) q^23 + (3b5 - b3 - 19) q^25 + (-10b4 - 7b2 - 2b1) q^27 + (-9b4 + 6b2 + 4b1) q^29 + (15b4 + 6b2 + 3b1) q^31 + (b5 + 4b3 - 38) q^33 + (-2b5 - 4b3 + 68) * q^35 + (9b4 + 12b2 - 28b1) q^37 + (18b4 - b2 + 2b1) * q^39 + (-2b4 - 2b2 - 24b1) q^41 + (b5 - 13b3 + 32) q^43 + (b4 + 6b2 - 24b1) * q^45 + (-5b5 + b3 - 12) q^47 + (-3b5 + 6b3 - 3) * q^49 + (-2b5 - 2b4 - 4b3 - 15b2 + 47b1 + 76) q^51 + (-b5 - 7b3 + 154) q^53 + (b5 - 5b3 - 76) q^55 + (-48b4 - 20b2 + 68b1) q^57 + (-9b5 - 11b3 + 280) * q^59 + (-5b4 - 14b2 - 52b1) q^61 + (-23b4 - 8b2 + 97b1) q^63 + (66b4 + 10b2 + 72b1) q^65 + (-7b5 + 5b3 - 52) * q^67 + (5b5 - 16b3 + 82) * q^69 + (7b4 - 19b2 - 117b1) q^71 + (-8b4 - 22b2 + 44b1) q^73 + (64b4 + 4b2 - 29b1) q^75 + (-5b5 + 2b3 - 278) * q^77 + (-27b4 + 34b2 - 25b1) q^79 + (3b5 - 2b3 - 269) * q^81 + (-b5 - 23b3 + 216) q^83 + (-5b5 + 63b4 + 7b3 + 22b2 + 24b1 + 224) q^85 + (15b5 + 5b3 + 76) * q^87 + (-3b5 - 14b3 + 4) * q^89 + (-20b4 + 21b2 - 22b1) q^91 + (-9b5 + 30b3 + 138) * q^93 + (-68b4 + 8b2 - 144b1) q^95 + (-70b4 + 8b2 + 116b1) q^97 + (-22b4 - 2b2 + 73b1) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 42 q^{9}+O(q^{10}) \) 6 * q - 42 * q^9	\( 6 q - 42 q^{9} - 72 q^{13} - 24 q^{15} - 126 q^{17} - 24 q^{19} - 204 q^{21} - 114 q^{25} - 228 q^{33} + 408 q^{35} + 192 q^{43} - 72 q^{47} - 18 q^{49} + 456 q^{51} + 924 q^{53} - 456 q^{55} + 1680 q^{59} - 312 q^{67} + 492 q^{69} - 1668 q^{77} - 1614 q^{81} + 1296 q^{83} + 1344 q^{85} + 456 q^{87} + 24 q^{89} + 828 q^{93}+O(q^{100}) \) 6 * q - 42 * q^9 - 72 * q^13 - 24 * q^15 - 126 * q^17 - 24 * q^19 - 204 * q^21 - 114 * q^25 - 228 * q^33 + 408 * q^35 + 192 * q^43 - 72 * q^47 - 18 * q^49 + 456 * q^51 + 924 * q^53 - 456 * q^55 + 1680 * q^59 - 312 * q^67 + 492 * q^69 - 1668 * q^77 - 1614 * q^81 + 1296 * q^83 + 1344 * q^85 + 456 * q^87 + 24 * q^89 + 828 * q^93

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