LMFDB - Newform orbit 504.4.s.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [504,4,Mod(289,504)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(504, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

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

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

chi := DirichletCharacter("504.289");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 504 = 2^{3} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	504.s (of order $3$, degree $2$, 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:	$29.7369626429$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.11163123.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 14x^{4} + 49x^{2} + 27 $ x^6 + 14x^4 + 49x^2 + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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_{5} + \beta_{4} + \beta_{3} + \cdots - 1) q^{5}+ \cdots + ( - 3 \beta_{5} - \beta_{4} + \beta_{3} + \cdots - 4) q^{7}+O(q^{10}) $ q + (-b5 + b4 + b3 + b2 + b1 - 1) * q^5 + (-3b5 - b4 + b3 + 6b1 - 4) * q^7	$ q + ( - \beta_{5} + \beta_{4} + \beta_{3} + \cdots - 1) q^{5}+ \cdots + (263 \beta_{3} - 19 \beta_{2} - 20) q^{97}+O(q^{100}) $ q + (-b5 + b4 + b3 + b2 + b1 - 1) * q^5 + (-3b5 - b4 + b3 + 6b1 - 4) * q^7 + 3b4 q^11 + (-b3 - 3b2 - 8) q^13 + (-13b5 - b4 - 15b1) * q^17 + (2b5 + 3b4 - 2b3 + 3b2 - 28b1 + 28) q^19 + (-7b5 - 2b4 + 7b3 - 2b2 + 64b1 - 64) q^23 + (14b5 + 10b4 - 92b1) q^25 + (7b3 - 3b2 - 60) * q^29 + (26b5 + b4 + 122b1) * q^31 + (16b5 - 3b4 + 25b3 + 10b2 - 40b1 + 106) q^35 + (-14b5 - 12b4 + 14b3 - 12b2 - 187b1 + 187) q^37 + (23b3 - 11b2 - 8) * q^41 + (56b3 + 108) q^43 + (50b5 + 3b4 - 50b3 + 3b2 + 86b1 - 86) q^47 + (30b5 - 2b4 - 3b3 - b2 + 204b1 - 353) * q^49 + (28b5 + 34b4 + 273b1) q^53 + (3b3 - 30b2 - 540) * q^55 + (29b5 - 590b1) * q^59 + (-66b5 - 4b4 + 66b3 - 4b2 + 185b1 - 185) q^61 + (-7b5 - 39b4 + 7b3 - 39b2 + 568b1 - 568) q^65 + (-7b5 + 34b4 + 174b1) q^67 + (-14b3 + 6b2 - 220) * q^71 + (98b5 + 38b4 + 193b1) q^73 + (-69b5 + 21b4 + 93b3 + 27b2 - 459b1 + 432) q^77 + (7b5 + 46b4 - 7b3 + 46b2 + 44b1 - 44) q^79 + (-2b3 - 30b2 + 800) * q^83 + (140b3 - 18b2 - 273) * q^85 + (-72b5 - 52b4 + 72b3 - 52b2 - 315b1 + 315) q^89 + (-76b5 + 40b4 + 23b3 + 7b2 - 408b1 - 22) q^91 + (-49b5 + 56b4 - 440b1) q^95 + (263b3 - 19b2 - 20) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{5} - 4 q^{7}+O(q^{10}) $ 6 * q - 3 * q^5 - 4 * q^7	$ 6 q - 3 q^{5} - 4 q^{7} - 3 q^{11} - 52 q^{13} - 31 q^{17} + 89 q^{19} - 201 q^{23} - 300 q^{25} - 380 q^{29} + 339 q^{31} + 473 q^{35} + 535 q^{37} - 116 q^{41} + 536 q^{43} - 205 q^{47} - 1530 q^{49} + 757 q^{53} - 3306 q^{55} - 1799 q^{59} - 625 q^{61} - 1750 q^{65} + 495 q^{67} - 1280 q^{71} + 443 q^{73} + 1131 q^{77} - 79 q^{79} + 4744 q^{83} - 1954 q^{85} + 821 q^{89} - 1352 q^{91} - 1327 q^{95} - 684 q^{97}+O(q^{100}) $ 6 * q - 3 * q^5 - 4 * q^7 - 3 * q^11 - 52 * q^13 - 31 * q^17 + 89 * q^19 - 201 * q^23 - 300 * q^25 - 380 * q^29 + 339 * q^31 + 473 * q^35 + 535 * q^37 - 116 * q^41 + 536 * q^43 - 205 * q^47 - 1530 * q^49 + 757 * q^53 - 3306 * q^55 - 1799 * q^59 - 625 * q^61 - 1750 * q^65 + 495 * q^67 - 1280 * q^71 + 443 * q^73 + 1131 * q^77 - 79 * q^79 + 4744 * q^83 - 1954 * q^85 + 821 * q^89 - 1352 * q^91 - 1327 * q^95 - 684 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} + 14x^{4} + 49x^{2} + 27 $ x^6 + 14*x^4 + 49*x^2 + 27 :

$\beta_{1}$	$=$	$ ( \nu^{3} + 7\nu + 3 ) / 6 $ (v^3 + 7*v + 3) / 6
$\beta_{2}$	$=$	$ ( 2\nu^{4} + 8\nu^{2} - 27 ) / 3 $ (2v^4 + 8v^2 - 27) / 3
$\beta_{3}$	$=$	$ ( 2\nu^{4} + 20\nu^{2} + 27 ) / 3 $ (2v^4 + 20v^2 + 27) / 3
$\beta_{4}$	$=$	$ ( 2\nu^{5} - 2\nu^{4} + 23\nu^{3} - 8\nu^{2} + 81\nu + 27 ) / 6 $ (2v^5 - 2v^4 + 23v^3 - 8v^2 + 81*v + 27) / 6
$\beta_{5}$	$=$	$ ( 2\nu^{5} + 2\nu^{4} + 23\nu^{3} + 20\nu^{2} + 45\nu + 27 ) / 6 $ (2v^5 + 2v^4 + 23v^3 + 20v^2 + 45*v + 27) / 6

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

$\nu$	$=$	$ ( -2\beta_{5} + 2\beta_{4} + \beta_{3} + \beta_{2} ) / 12 $ (-2b5 + 2b4 + b3 + b2) / 12
$\nu^{2}$	$=$	$ ( \beta_{3} - \beta_{2} - 18 ) / 4 $ (b3 - b2 - 18) / 4
$\nu^{3}$	$=$	$ ( 14\beta_{5} - 14\beta_{4} - 7\beta_{3} - 7\beta_{2} + 72\beta _1 - 36 ) / 12 $ (14b5 - 14b4 - 7b3 - 7b2 + 72*b1 - 36) / 12
$\nu^{4}$	$=$	$ ( -2\beta_{3} + 5\beta_{2} + 63 ) / 2 $ (-2b3 + 5b2 + 63) / 2
$\nu^{5}$	$=$	$ ( -40\beta_{5} + 58\beta_{4} + 20\beta_{3} + 29\beta_{2} - 414\beta _1 + 207 ) / 6 $ (-40b5 + 58b4 + 20b3 + 29b2 - 414*b1 + 207) / 6

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/504\mathbb{Z}\right)^\times$.

$n$	$73$	$127$	$253$	$281$
$\chi(n)$	$-1 + \beta_{1}$	$1$	$1$	$1$

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} $

289.1

		2.13755i
		0.821510i
	−	2.95906i
	−	2.13755i
	−	0.821510i
		2.95906i

−7.90468

13.6913i

−0.861792

18.5002i

289.2

−3.34580

5.79509i

−8.65024

−

16.3760i

289.3

9.75047

−

16.8883i

7.51203

16.9284i

361.1

−7.90468

−

13.6913i

−0.861792

−

18.5002i

361.2

−3.34580

−

5.79509i

−8.65024

16.3760i

361.3

9.75047

16.8883i

7.51203

−

16.9284i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	504.4.s.h		6
3.b	odd	2	1		56.4.i.b	✓	6
7.c	even	3	1	inner	504.4.s.h		6
12.b	even	2	1		112.4.i.e		6
21.c	even	2	1		392.4.i.m		6
21.g	even	6	1		392.4.a.l		3
21.g	even	6	1		392.4.i.m		6
21.h	odd	6	1		56.4.i.b	✓	6
21.h	odd	6	1		392.4.a.i		3
24.f	even	2	1		448.4.i.m		6
24.h	odd	2	1		448.4.i.j		6
84.j	odd	6	1		784.4.a.bb		3
84.n	even	6	1		112.4.i.e		6
84.n	even	6	1		784.4.a.be		3
168.s	odd	6	1		448.4.i.j		6
168.v	even	6	1		448.4.i.m		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
56.4.i.b	✓	6	3.b	odd	2	1
56.4.i.b	✓	6	21.h	odd	6	1
112.4.i.e		6	12.b	even	2	1
112.4.i.e		6	84.n	even	6	1
392.4.a.i		3	21.h	odd	6	1
392.4.a.l		3	21.g	even	6	1
392.4.i.m		6	21.c	even	2	1
392.4.i.m		6	21.g	even	6	1
448.4.i.j		6	24.h	odd	2	1
448.4.i.j		6	168.s	odd	6	1
448.4.i.m		6	24.f	even	2	1
448.4.i.m		6	168.v	even	6	1
504.4.s.h		6	1.a	even	1	1	trivial
504.4.s.h		6	7.c	even	3	1	inner
784.4.a.bb		3	84.j	odd	6	1
784.4.a.be		3	84.n	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{6} + 3T_{5}^{5} + 342T_{5}^{4} + 3127T_{5}^{3} + 117078T_{5}^{2} + 686979T_{5} + 4255969 $ T5^6 + 3*T5^5 + 342*T5^4 + 3127*T5^3 + 117078*T5^2 + 686979*T5 + 4255969 acting on $S_{4}^{\mathrm{new}}(504, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} $ T^6
$5$	$ T^{6} + 3 T^{5} + \cdots + 4255969 $ T^6 + 3T^5 + 342T^4 + 3127T^3 + 117078T^2 + 686979*T + 4255969
$7$	$ T^{6} + 4 T^{5} + \cdots + 40353607 $ T^6 + 4T^5 + 773T^4 + 2296T^3 + 265139T^2 + 470596*T + 40353607
$11$	$ T^{6} + \cdots + 1492817769 $ T^6 + 3T^5 + 2322T^4 - 84213T^3 + 5234058T^2 - 89367381*T + 1492817769
$13$	$ (T^{3} + 26 T^{2} + \cdots + 26328)^{2} $ (T^3 + 26T^2 - 2244T + 26328)^2
$17$	$ T^{6} + \cdots + 15768076041 $ T^6 + 31T^5 + 7398T^4 + 51595T^3 + 45327670T^2 + 808300527*T + 15768076041
$19$	$ T^{6} - 89 T^{5} + \cdots + 507555841 $ T^6 - 89T^5 + 7538T^4 - 79145T^3 + 2151770T^2 + 8628607*T + 507555841
$23$	$ T^{6} + \cdots + 7342261969 $ T^6 + 201T^5 + 29466T^4 + 2026561T^3 + 102351138T^2 + 936987345*T + 7342261969
$29$	$ (T^{3} + 190 T^{2} + \cdots - 48504)^{2} $ (T^3 + 190T^2 + 8476T - 48504)^2
$31$	$ T^{6} + \cdots + 6524157303049 $ T^6 - 339T^5 + 103842T^4 - 8864267T^3 + 988632618T^2 + 28298458197*T + 6524157303049
$37$	$ T^{6} + \cdots + 783910139769 $ T^6 - 535T^5 + 229702T^4 - 32010579T^3 + 3668531574T^2 + 50044729401*T + 783910139769
$41$	$ (T^{3} + 58 T^{2} + \cdots - 3860392)^{2} $ (T^3 + 58T^2 - 42436T - 3860392)^2
$43$	$ (T^{3} - 268 T^{2} + \cdots + 24343488)^{2} $ (T^3 - 268T^2 - 105680T + 24343488)^2
$47$	$ T^{6} + \cdots + 126279990328969 $ T^6 + 205T^5 + 128066T^4 + 4836469T^3 + 9706728266T^2 + 966880316917*T + 126279990328969
$53$	$ T^{6} + \cdots + 55\!\cdots\!81 $ T^6 - 757T^5 + 676374T^4 - 71108593T^3 + 67195802038T^2 - 7714534739925*T + 5574535047770481
$59$	$ T^{6} + \cdots + 38\!\cdots\!89 $ T^6 + 1799T^5 + 2192362T^4 + 1484892927T^3 + 736214189538T^2 + 205327618146063*T + 38677758242224689
$61$	$ T^{6} + \cdots + 273487351727209 $ T^6 + 625T^5 + 434726T^4 + 5511781T^3 + 12280806326T^2 + 729318214753*T + 273487351727209
$67$	$ T^{6} + \cdots + 12\!\cdots\!41 $ T^6 - 495T^5 + 471738T^4 - 112131207T^3 + 106926434514T^2 - 25432000297623*T + 12583695258139041
$71$	$ (T^{3} + 640 T^{2} + \cdots + 7291392)^{2} $ (T^3 + 640T^2 + 122304T + 7291392)^2
$73$	$ T^{6} + \cdots + 10\!\cdots\!09 $ T^6 - 443T^5 + 760358T^4 + 450580081T^3 + 273768389510T^2 + 56602638956773*T + 10068094929970609
$79$	$ T^{6} + \cdots + 21\!\cdots\!41 $ T^6 + 79T^5 + 538682T^4 + 252300103T^3 + 295120754690T^2 + 78365449600711*T + 21662385405723841
$83$	$ (T^{3} - 2372 T^{2} + \cdots - 266787264)^{2} $ (T^3 - 2372T^2 + 1641456T - 266787264)^2
$89$	$ T^{6} + \cdots + 93\!\cdots\!01 $ T^6 - 821T^5 + 1219686T^4 + 254421247T^3 + 377182094854T^2 - 52805694643605*T + 9365719791669201
$97$	$ (T^{3} + 342 T^{2} + \cdots + 217321448)^{2} $ (T^3 + 342T^2 - 2726340T + 217321448)^2
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Elliptic curves over $\Q$
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Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	504.s (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{5} + \beta_{4} + \beta_{3} + \cdots - 1) q^{5}+ \cdots + ( - 3 \beta_{5} - \beta_{4} + \beta_{3} + \cdots - 4) q^{7}+O(q^{10}) \) q + (-b5 + b4 + b3 + b2 + b1 - 1) * q^5 + (-3b5 - b4 + b3 + 6b1 - 4) * q^7	\( q + ( - \beta_{5} + \beta_{4} + \beta_{3} + \cdots - 1) q^{5}+ \cdots + (263 \beta_{3} - 19 \beta_{2} - 20) q^{97}+O(q^{100}) \) q + (-b5 + b4 + b3 + b2 + b1 - 1) * q^5 + (-3b5 - b4 + b3 + 6b1 - 4) * q^7 + 3b4 q^11 + (-b3 - 3b2 - 8) q^13 + (-13b5 - b4 - 15b1) * q^17 + (2b5 + 3b4 - 2b3 + 3b2 - 28b1 + 28) q^19 + (-7b5 - 2b4 + 7b3 - 2b2 + 64b1 - 64) q^23 + (14b5 + 10b4 - 92b1) q^25 + (7b3 - 3b2 - 60) * q^29 + (26b5 + b4 + 122b1) * q^31 + (16b5 - 3b4 + 25b3 + 10b2 - 40b1 + 106) q^35 + (-14b5 - 12b4 + 14b3 - 12b2 - 187b1 + 187) q^37 + (23b3 - 11b2 - 8) * q^41 + (56b3 + 108) q^43 + (50b5 + 3b4 - 50b3 + 3b2 + 86b1 - 86) q^47 + (30b5 - 2b4 - 3b3 - b2 + 204b1 - 353) * q^49 + (28b5 + 34b4 + 273b1) q^53 + (3b3 - 30b2 - 540) * q^55 + (29b5 - 590b1) * q^59 + (-66b5 - 4b4 + 66b3 - 4b2 + 185b1 - 185) q^61 + (-7b5 - 39b4 + 7b3 - 39b2 + 568b1 - 568) q^65 + (-7b5 + 34b4 + 174b1) q^67 + (-14b3 + 6b2 - 220) * q^71 + (98b5 + 38b4 + 193b1) q^73 + (-69b5 + 21b4 + 93b3 + 27b2 - 459b1 + 432) q^77 + (7b5 + 46b4 - 7b3 + 46b2 + 44b1 - 44) q^79 + (-2b3 - 30b2 + 800) * q^83 + (140b3 - 18b2 - 273) * q^85 + (-72b5 - 52b4 + 72b3 - 52b2 - 315b1 + 315) q^89 + (-76b5 + 40b4 + 23b3 + 7b2 - 408b1 - 22) q^91 + (-49b5 + 56b4 - 440b1) q^95 + (263b3 - 19b2 - 20) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{5} - 4 q^{7}+O(q^{10}) \) 6 * q - 3 * q^5 - 4 * q^7	\( 6 q - 3 q^{5} - 4 q^{7} - 3 q^{11} - 52 q^{13} - 31 q^{17} + 89 q^{19} - 201 q^{23} - 300 q^{25} - 380 q^{29} + 339 q^{31} + 473 q^{35} + 535 q^{37} - 116 q^{41} + 536 q^{43} - 205 q^{47} - 1530 q^{49} + 757 q^{53} - 3306 q^{55} - 1799 q^{59} - 625 q^{61} - 1750 q^{65} + 495 q^{67} - 1280 q^{71} + 443 q^{73} + 1131 q^{77} - 79 q^{79} + 4744 q^{83} - 1954 q^{85} + 821 q^{89} - 1352 q^{91} - 1327 q^{95} - 684 q^{97}+O(q^{100}) \) 6 * q - 3 * q^5 - 4 * q^7 - 3 * q^11 - 52 * q^13 - 31 * q^17 + 89 * q^19 - 201 * q^23 - 300 * q^25 - 380 * q^29 + 339 * q^31 + 473 * q^35 + 535 * q^37 - 116 * q^41 + 536 * q^43 - 205 * q^47 - 1530 * q^49 + 757 * q^53 - 3306 * q^55 - 1799 * q^59 - 625 * q^61 - 1750 * q^65 + 495 * q^67 - 1280 * q^71 + 443 * q^73 + 1131 * q^77 - 79 * q^79 + 4744 * q^83 - 1954 * q^85 + 821 * q^89 - 1352 * q^91 - 1327 * q^95 - 684 * q^97

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