LMFDB - Newform orbit 464.4.e.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [464,4,Mod(289,464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(464, 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("464.289");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 464 = 2^{4} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	464.e (of order $2$, degree $1$, not 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:	$27.3768862427$
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} + 38x^{4} + 301x^{2} + 560 $ x^6 + 38x^4 + 301x^2 + 560
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 29)
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_{2} q^{3} + (\beta_{4} + 4) q^{5} + (\beta_{4} + 5) q^{7} + ( - \beta_{5} + 7) q^{9}+O(q^{10}) $ q + b2 * q^3 + (b4 + 4) * q^5 + (b4 + 5) * q^7 + (-b5 + 7) * q^9	$ q + \beta_{2} q^{3} + (\beta_{4} + 4) q^{5} + (\beta_{4} + 5) q^{7} + ( - \beta_{5} + 7) q^{9} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{11} + ( - 3 \beta_{5} - 3 \beta_{4} + 5) q^{13} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{15} + ( - \beta_{3} + 11 \beta_{2} - 2 \beta_1) q^{17} + (\beta_{3} - 7 \beta_{2} + 5 \beta_1) q^{19} + ( - \beta_{3} + 3 \beta_{2} + 2 \beta_1) q^{21} + ( - 2 \beta_{5} - 7 \beta_{4} + 11) q^{23} + ( - 4 \beta_{5} + 2 \beta_{4} + 6) q^{25} + (\beta_{3} + 18 \beta_{2} - 7 \beta_1) q^{27} + (7 \beta_{5} + \beta_{4} + \beta_{3} + \cdots - 11) q^{29}+ \cdots + ( - 15 \beta_{3} - 41 \beta_{2} - 61 \beta_1) q^{99}+O(q^{100}) $ q + b2 * q^3 + (b4 + 4) * q^5 + (b4 + 5) * q^7 + (-b5 + 7) * q^9 + (-b3 - 2b2 - b1) q^11 + (-3b5 - 3b4 + 5) * q^13 + (-b3 + 2b2 + 2b1) * q^15 + (-b3 + 11b2 - 2b1) * q^17 + (b3 - 7b2 + 5b1) * q^19 + (-b3 + 3b2 + 2b1) * q^21 + (-2b5 - 7b4 + 11) * q^23 + (-4b5 + 2b4 + 6) * q^25 + (b3 + 18b2 - 7b1) * q^27 + (7b5 + b4 + b3 + 13b2 - 10b1 - 11) q^29 + (35b2 + 23b1) * q^31 + (-2b5 - 10b4 + 25) * q^33 + (-4b5 + 3b4 + 135) * q^35 + (6b3 + 46b2 + 22b1) q^37 + (6b3 - 37b2 - 27b1) q^39 + (b3 - 43b2 - 32b1) * q^41 + (-7b3 + 8b2 - b1) * q^43 + (-9b5 + 14b4 + 65) * q^45 + (7b3 + 36b2 - 26b1) q^47 + (-4b5 + 4b4 - 203) * q^49 + (-14b5 - 9b4 - 239) * q^51 + (-7b5 - 37b4 + 155) * q^53 + (37b2 - 59b1) * q^55 + (7b5 + 6b4 + 171) * q^57 + (-16b5 - b4 - 205) q^59 + (-13b3 - 13b2 + 32b1) q^61 + (-10b5 + 14b4 + 72) * q^63 + (-15b5 + 32b4 - 214) * q^65 + (-22b5 + 18b4 + 124) * q^67 + (9b3 - 7b2 - 28b1) q^69 + (-16b5 - 38b4 + 142) * q^71 + (16b3 - 28b2 - 70b1) q^73 + (2b3 - 62b2 - 24b1) q^75 + (-b3 + 35b2 - 60b1) * q^77 + (9b3 - 42b2 - 42b1) q^79 + (-33b5 + 18b4 - 188) * q^81 + (30b5 + 33b4 + 375) * q^83 + (-15b3 + 57b2 - 42b1) q^85 + (2b5 + 21b4 - 8b3 + 99b2 + 51b1 - 289) q^87 + (7b3 + 71b2 - 56b1) q^89 + (-18b5 + 29b4 - 209) * q^91 + (-58b5 - 23b4 - 608) * q^93 + (17b3 - 31b2 + 77b1) q^95 + (9b3 - 215b2 + 58b1) q^97 + (-15b3 - 41b2 - 61b1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 22 q^{5} + 28 q^{7} + 40 q^{9}+O(q^{10}) $ 6 * q + 22 * q^5 + 28 * q^7 + 40 * q^9	$ 6 q + 22 q^{5} + 28 q^{7} + 40 q^{9} + 30 q^{13} + 76 q^{23} + 24 q^{25} - 54 q^{29} + 166 q^{33} + 796 q^{35} + 344 q^{45} - 1234 q^{49} - 1444 q^{51} + 990 q^{53} + 1028 q^{57} - 1260 q^{59} + 384 q^{63} - 1378 q^{65} + 664 q^{67} + 896 q^{71} - 1230 q^{81} + 2244 q^{83} - 1772 q^{87} - 1348 q^{91} - 3718 q^{93}+O(q^{100}) $ 6 * q + 22 * q^5 + 28 * q^7 + 40 * q^9 + 30 * q^13 + 76 * q^23 + 24 * q^25 - 54 * q^29 + 166 * q^33 + 796 * q^35 + 344 * q^45 - 1234 * q^49 - 1444 * q^51 + 990 * q^53 + 1028 * q^57 - 1260 * q^59 + 384 * q^63 - 1378 * q^65 + 664 * q^67 + 896 * q^71 - 1230 * q^81 + 2244 * q^83 - 1772 * q^87 - 1348 * q^91 - 3718 * q^93

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} + 38x^{4} + 301x^{2} + 560 $ x^6 + 38*x^4 + 301*x^2 + 560 :

$\beta_{1}$	$=$	$ 2\nu $ 2*v
$\beta_{2}$	$=$	$ ( \nu^{5} + 34\nu^{3} + 165\nu ) / 20 $ (v^5 + 34v^3 + 165v) / 20
$\beta_{3}$	$=$	$ ( \nu^{5} + 14\nu^{3} - 255\nu ) / 20 $ (v^5 + 14v^3 - 255v) / 20
$\beta_{4}$	$=$	$ ( \nu^{4} + 29\nu^{2} + 85 ) / 5 $ (v^4 + 29*v^2 + 85) / 5
$\beta_{5}$	$=$	$ ( \nu^{4} + 39\nu^{2} + 215 ) / 5 $ (v^4 + 39*v^2 + 215) / 5

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

$\nu$	$=$	$ ( \beta_1 ) / 2 $ (b1) / 2
$\nu^{2}$	$=$	$ ( \beta_{5} - \beta_{4} - 26 ) / 2 $ (b5 - b4 - 26) / 2
$\nu^{3}$	$=$	$ ( -2\beta_{3} + 2\beta_{2} - 21\beta_1 ) / 2 $ (-2b3 + 2b2 - 21*b1) / 2
$\nu^{4}$	$=$	$ ( -29\beta_{5} + 39\beta_{4} + 584 ) / 2 $ (-29b5 + 39b4 + 584) / 2
$\nu^{5}$	$=$	$ ( 68\beta_{3} - 28\beta_{2} + 549\beta_1 ) / 2 $ (68b3 - 28b2 + 549*b1) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/464\mathbb{Z}\right)^\times$.

$n$	$117$	$175$	$321$
$\chi(n)$	$1$	$1$	$-1$

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} $

289.1

	−	1.65524i
		2.70440i
		5.28644i
	−	5.28644i
	−	2.70440i
		1.65524i

−

6.56740i

6.61042

7.61042

−16.1308

289.2

−

4.08054i

−10.7216

−9.72164

10.3492

289.3

−

1.10381i

15.1112

16.1112

25.7816

289.4

1.10381i

15.1112

16.1112

25.7816

289.5

4.08054i

−10.7216

−9.72164

10.3492

289.6

6.56740i

6.61042

7.61042

−16.1308

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	464.4.e.a		6
4.b	odd	2	1		29.4.b.a	✓	6
12.b	even	2	1		261.4.c.c		6
29.b	even	2	1	inner	464.4.e.a		6
116.d	odd	2	1		29.4.b.a	✓	6
116.e	even	4	2		841.4.a.c		6
348.b	even	2	1		261.4.c.c		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
29.4.b.a	✓	6	4.b	odd	2	1
29.4.b.a	✓	6	116.d	odd	2	1
261.4.c.c		6	12.b	even	2	1
261.4.c.c		6	348.b	even	2	1
464.4.e.a		6	1.a	even	1	1	trivial
464.4.e.a		6	29.b	even	2	1	inner
841.4.a.c		6	116.e	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{6} + 61T_{3}^{4} + 791T_{3}^{2} + 875 $ T3^6 + 61*T3^4 + 791*T3^2 + 875 acting on $S_{4}^{\mathrm{new}}(464, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} + 61 T^{4} + \cdots + 875 $ T^6 + 61T^4 + 791T^2 + 875
$5$	$ (T^{3} - 11 T^{2} + \cdots + 1071)^{2} $ (T^3 - 11T^2 - 133T + 1071)^2
$7$	$ (T^{3} - 14 T^{2} + \cdots + 1192)^{2} $ (T^3 - 14T^2 - 108T + 1192)^2
$11$	$ T^{6} + 3909 T^{4} + \cdots + 193452035 $ T^6 + 3909T^4 + 3919671T^2 + 193452035
$13$	$ (T^{3} - 15 T^{2} + \cdots + 119791)^{2} $ (T^3 - 15T^2 - 4641T + 119791)^2
$17$	$ T^{6} + \cdots + 8077890240 $ T^6 + 12932T^4 + 40558896T^2 + 8077890240
$19$	$ T^{6} + \cdots + 5493277440 $ T^6 + 12284T^4 + 28619584T^2 + 5493277440
$23$	$ (T^{3} - 38 T^{2} + \cdots - 188856)^{2} $ (T^3 - 38T^2 - 8428T - 188856)^2
$29$	$ T^{6} + \cdots + 14507145975869 $ T^6 + 54T^5 - 11861T^4 - 5298300T^3 - 289277929T^2 + 32120459334*T + 14507145975869
$31$	$ T^{6} + \cdots + 1328705534835 $ T^6 + 135813T^4 + 3963474151T^2 + 1328705534835
$37$	$ T^{6} + \cdots + 844118984540160 $ T^6 + 310128T^4 + 29438424064T^2 + 844118984540160
$41$	$ T^{6} + \cdots + 16572819704000 $ T^6 + 236388T^4 + 11410815024T^2 + 16572819704000
$43$	$ T^{6} + \cdots + 207644764642875 $ T^6 + 180669T^4 + 10707373159T^2 + 207644764642875
$47$	$ T^{6} + \cdots + 328073602115 $ T^6 + 335333T^4 + 1488882199T^2 + 328073602115
$53$	$ (T^{3} - 495 T^{2} + \cdots - 9329121)^{2} $ (T^3 - 495T^2 - 152081T - 9329121)^2
$59$	$ (T^{3} + 630 T^{2} + \cdots - 6664392)^{2} $ (T^3 + 630T^2 + 18676T - 6664392)^2
$61$	$ T^{6} + \cdots + 757905028251840 $ T^6 + 697956T^4 + 70588615216T^2 + 757905028251840
$67$	$ (T^{3} - 332 T^{2} + \cdots - 16765696)^{2} $ (T^3 - 332T^2 - 275968T - 16765696)^2
$71$	$ (T^{3} - 448 T^{2} + \cdots + 5011200)^{2} $ (T^3 - 448T^2 - 238432T + 5011200)^2
$73$	$ T^{6} + \cdots + 10\!\cdots\!40 $ T^6 + 1499888T^4 + 564679327744T^2 + 10400839666237440
$79$	$ T^{6} + \cdots + 424851885738315 $ T^6 + 581229T^4 + 74821656231T^2 + 424851885738315
$83$	$ (T^{3} - 1122 T^{2} + \cdots + 43495704)^{2} $ (T^3 - 1122T^2 - 75852T + 43495704)^2
$89$	$ T^{6} + \cdots + 18\!\cdots\!40 $ T^6 + 965508T^4 + 150078203184T^2 + 1826762940175040
$97$	$ T^{6} + \cdots + 18\!\cdots\!60 $ T^6 + 4099044T^4 + 5125383625264T^2 + 1898924295636768960
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Classical	Maass
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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:	\( N \)	\(=\)	\( 464 = 2^{4} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	464.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{2} q^{3} + (\beta_{4} + 4) q^{5} + (\beta_{4} + 5) q^{7} + ( - \beta_{5} + 7) q^{9}+O(q^{10}) \) q + b2 * q^3 + (b4 + 4) * q^5 + (b4 + 5) * q^7 + (-b5 + 7) * q^9	\( q + \beta_{2} q^{3} + (\beta_{4} + 4) q^{5} + (\beta_{4} + 5) q^{7} + ( - \beta_{5} + 7) q^{9} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{11} + ( - 3 \beta_{5} - 3 \beta_{4} + 5) q^{13} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{15} + ( - \beta_{3} + 11 \beta_{2} - 2 \beta_1) q^{17} + (\beta_{3} - 7 \beta_{2} + 5 \beta_1) q^{19} + ( - \beta_{3} + 3 \beta_{2} + 2 \beta_1) q^{21} + ( - 2 \beta_{5} - 7 \beta_{4} + 11) q^{23} + ( - 4 \beta_{5} + 2 \beta_{4} + 6) q^{25} + (\beta_{3} + 18 \beta_{2} - 7 \beta_1) q^{27} + (7 \beta_{5} + \beta_{4} + \beta_{3} + \cdots - 11) q^{29}+ \cdots + ( - 15 \beta_{3} - 41 \beta_{2} - 61 \beta_1) q^{99}+O(q^{100}) \) q + b2 * q^3 + (b4 + 4) * q^5 + (b4 + 5) * q^7 + (-b5 + 7) * q^9 + (-b3 - 2b2 - b1) q^11 + (-3b5 - 3b4 + 5) * q^13 + (-b3 + 2b2 + 2b1) * q^15 + (-b3 + 11b2 - 2b1) * q^17 + (b3 - 7b2 + 5b1) * q^19 + (-b3 + 3b2 + 2b1) * q^21 + (-2b5 - 7b4 + 11) * q^23 + (-4b5 + 2b4 + 6) * q^25 + (b3 + 18b2 - 7b1) * q^27 + (7b5 + b4 + b3 + 13b2 - 10b1 - 11) q^29 + (35b2 + 23b1) * q^31 + (-2b5 - 10b4 + 25) * q^33 + (-4b5 + 3b4 + 135) * q^35 + (6b3 + 46b2 + 22b1) q^37 + (6b3 - 37b2 - 27b1) q^39 + (b3 - 43b2 - 32b1) * q^41 + (-7b3 + 8b2 - b1) * q^43 + (-9b5 + 14b4 + 65) * q^45 + (7b3 + 36b2 - 26b1) q^47 + (-4b5 + 4b4 - 203) * q^49 + (-14b5 - 9b4 - 239) * q^51 + (-7b5 - 37b4 + 155) * q^53 + (37b2 - 59b1) * q^55 + (7b5 + 6b4 + 171) * q^57 + (-16b5 - b4 - 205) q^59 + (-13b3 - 13b2 + 32b1) q^61 + (-10b5 + 14b4 + 72) * q^63 + (-15b5 + 32b4 - 214) * q^65 + (-22b5 + 18b4 + 124) * q^67 + (9b3 - 7b2 - 28b1) q^69 + (-16b5 - 38b4 + 142) * q^71 + (16b3 - 28b2 - 70b1) q^73 + (2b3 - 62b2 - 24b1) q^75 + (-b3 + 35b2 - 60b1) * q^77 + (9b3 - 42b2 - 42b1) q^79 + (-33b5 + 18b4 - 188) * q^81 + (30b5 + 33b4 + 375) * q^83 + (-15b3 + 57b2 - 42b1) q^85 + (2b5 + 21b4 - 8b3 + 99b2 + 51b1 - 289) q^87 + (7b3 + 71b2 - 56b1) q^89 + (-18b5 + 29b4 - 209) * q^91 + (-58b5 - 23b4 - 608) * q^93 + (17b3 - 31b2 + 77b1) q^95 + (9b3 - 215b2 + 58b1) q^97 + (-15b3 - 41b2 - 61b1) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 22 q^{5} + 28 q^{7} + 40 q^{9}+O(q^{10}) \) 6 * q + 22 * q^5 + 28 * q^7 + 40 * q^9	\( 6 q + 22 q^{5} + 28 q^{7} + 40 q^{9} + 30 q^{13} + 76 q^{23} + 24 q^{25} - 54 q^{29} + 166 q^{33} + 796 q^{35} + 344 q^{45} - 1234 q^{49} - 1444 q^{51} + 990 q^{53} + 1028 q^{57} - 1260 q^{59} + 384 q^{63} - 1378 q^{65} + 664 q^{67} + 896 q^{71} - 1230 q^{81} + 2244 q^{83} - 1772 q^{87} - 1348 q^{91} - 3718 q^{93}+O(q^{100}) \) 6 * q + 22 * q^5 + 28 * q^7 + 40 * q^9 + 30 * q^13 + 76 * q^23 + 24 * q^25 - 54 * q^29 + 166 * q^33 + 796 * q^35 + 344 * q^45 - 1234 * q^49 - 1444 * q^51 + 990 * q^53 + 1028 * q^57 - 1260 * q^59 + 384 * q^63 - 1378 * q^65 + 664 * q^67 + 896 * q^71 - 1230 * q^81 + 2244 * q^83 - 1772 * q^87 - 1348 * q^91 - 3718 * q^93

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