LMFDB - Newform orbit 464.2.u.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [464,2,Mod(49,464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(464, base_ring=CyclotomicField(14))

chi = DirichletCharacter(H, H._module([0, 0, 12]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("464.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$464 = 2^{4} \cdot 29$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	464.u (of order $7$ , degree $6$ , 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:	$3.70505865379$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{7})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} - 3 x^{11} + 13 x^{10} - 9 x^{9} - 5 x^{8} + 35 x^{7} + 197 x^{6} - 140 x^{5} - 80 x^{4} + \cdots + 4096$ x^12 - 3x^11 + 13x^10 - 9x^9 - 5x^8 + 35x^7 + 197x^6 - 140x^5 - 80x^4 + 576x^3 + 3328x^2 + 3072*x + 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 58)
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

$q$ -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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta_{11} q^{3} + ( - \beta_{11} + \beta_{10} + \cdots + \beta_1) q^{5} + ( - \beta_{9} - \beta_{7}) q^{7} + ( - \beta_{10} + \beta_{8} + \cdots - \beta_1) q^{9} + (\beta_{10} + \beta_{5} + \beta_1) q^{11}+ \cdots + ( - \beta_{11} - \beta_{10} - 2 \beta_{9} + \cdots + 4) q^{99}+O(q^{100})$ q - b11 * q^3 + (-b11 + b10 - b9 - b7 + b5 - b4 + b3 + b1) * q^5 + (-b9 - b7) * q^7 + (-b10 + b8 + b7 - b1) * q^9 + (b10 + b5 + b1) * q^11 + (-b11 - b8 - b7 + b5 + b4 - b3 - b2 + b1 - 1) * q^13 + (-b11 - b9 + 4b8 - b7 + 4b6 + b5 - 4b4 + 4b3 + 4b2 + b1 + 4) q^15 + (-b11 + b10 - b9 + 3b8 - b7 - b6 + b4 + 3b3 + b1 - 1) * q^17 + (-b11 + b10 - b9 - 4b6 + b5) q^19 + (-b7 - 4b4 + 4b2) * q^21 + (b11 + b9 - b5 + 4b4 - 4) q^23 + (b11 + b10 - b9 + b8 + b6 - b5 + b2) * q^25 + (b11 - b10 - b9 - 4b6 - 4b2 + b1 - 4) * q^27 + (2b11 + b9 + 4b8 + b7 - b5 + b3 - b1) * q^29 + (b9 - b7 + b5 - 4b2 - b1) q^31 + (-b11 + b10 - b9 + 4b6 + b5 + 4b3 + b1 + 4) * q^33 + (b7 - 2b5 - 4b3 - 4b2 - b1) q^35 + (-b7 - 4b6 + 3b4 - 4b3 - 3b2) * q^37 + (-b11 + b10 - b9 + 4b8 + b5 + 4b3 + 4b2 + 4) q^39 + (-b10 - b9 - 4b8 + b7 - 3b6 + 3b4 - 4b3 - 4) * q^41 + (-b11 - b9 - b7 + b1) * q^43 + (-2b10 + 4b8 - 4b4 + 4b3 + b2 + 1) * q^45 + (b11 - b10 + b7 - 2b5 + 4b2 - 2b1 + 4) q^47 + (-b10 + b9 - 3b8 + b7 - 4b6 - b5 + 4b4 - 4b3 - 4b2 - b1) q^49 + (2b11 - 2b10 + 3b9 + 4b8 + 3b7 + 4b6 - 2b5 - 4b4 + 4b2 + 4) q^51 + (2b9 + b7 + b6 - b5 - 2b1 + 1) * q^53 + (b11 + b10 - 4b8 + b5 + 4b4 - 4) * q^55 + (b11 - 2b10 + 3b9 + 4b8 + 2b7 + 4b6 - 4b4 + 4b3 - b1) q^57 + (-b10 - 4b8 + b7 - 4b3 - 4) * q^59 + (-3b11 + b10 - b9 - b8 - b7 - b6 + b5 - 3b4 - b2 - 1) * q^61 + (-b11 + b10 - 2b7 + 2b5 + 4b2) q^63 + (b10 - b9 - b7 + 3b6 + b5 - 3b4 + 3b2) q^65 + (-b10 + b9 - 4b8 + b7 - b5 - b1) q^67 + (2b11 + 3b10 - 4b8 + 2b7 + b5 + 4b4 - 4b3 + b1) * q^69 + (-3b11 + 2b10 + 4b8 - 3b7 + 3b5 - 4b4 + 4b3 + 3b1) * q^71 + (-2b11 - 2b9 - 8b8 - 3b7 - 8b6 + 4b5 + 7b4 - 4b3 - 4b2 + 3b1 - 7) * q^73 + (b10 - b9 - 4b8 - b7 + 4b6 - 4b4 - 4b3) * q^75 + (b10 - b9 - 4b8 + 4b6 - 4b2 + b1) q^77 + (-2b10 + 2b9 - 4b8 - 4b6 - 2b5 + 8b4 - 4b3 - 8b2 - 2b1) q^79 + (2b11 + 2b9 - b8 - 2b7 - b6 - 3b5 - 7b4 + 3b3 + 3b2 + 2b1 + 7) * q^81 + (-b11 + 3b10 - 3b9 + b5 + 4b3 + b1 + 4) q^83 + (3b11 - 3b10 + 7b6 + 7b2 + 7) * q^85 + (4b11 - 2b10 + 4b9 - 8b8 + 3b7 - 4b5 + 4b4 - 4b3 - 4b2 - 2b1 - 4) * q^87 + (3b9 + 4b6 - b4 + b3 - 3b1 + 4) q^89 + (-4b8 - 8b3 - 4b2 + b1 - 8) q^91 + (b7 - 2b5 + 4b4 + 4b3 + 4b2 - b1 - 4) * q^93 + (8b8 - 3b7 + 8b6 - 4b4 + 8b3 + 4b2) * q^95 + (-b11 + 2b10 - 2b9 + 4b8 + 8b6 + b5 + b3 + 4b2 + 2b1 + 1) * q^97 + (-b11 - b10 - 2b9 + 4b8 + b7 + 4b6 - 4b4 + 4b3 + b1 + 4) q^99
$\operatorname{Tr}(f)(q)$	$=$	$12 q + 3 q^{3} - q^{7} - 11 q^{9} + 2 q^{11} + q^{13} + 9 q^{15} - 12 q^{17} + 6 q^{19} - 13 q^{21} - 35 q^{23} - 6 q^{25} - 39 q^{27} - 14 q^{29} + 8 q^{31} + 33 q^{33} + 18 q^{35} + 31 q^{37} + 22 q^{39}+ \cdots + 8 q^{99}+O(q^{100})$ 12 * q + 3 * q^3 - q^7 - 11 * q^9 + 2 * q^11 + q^13 + 9 * q^15 - 12 * q^17 + 6 * q^19 - 13 * q^21 - 35 * q^23 - 6 * q^25 - 39 * q^27 - 14 * q^29 + 8 * q^31 + 33 * q^33 + 18 * q^35 + 31 * q^37 + 22 * q^39 - 30 * q^41 + 5 * q^43 - 20 * q^45 + 33 * q^47 + 37 * q^49 + 15 * q^51 + 13 * q^53 - 36 * q^55 - 38 * q^57 - 38 * q^59 - 5 * q^61 - 4 * q^63 - 20 * q^65 + 7 * q^67 + 20 * q^69 - 3 * q^71 - 22 * q^73 + 2 * q^75 + 10 * q^77 + 60 * q^79 + 88 * q^81 + 39 * q^83 + 38 * q^85 - 9 * q^87 + 39 * q^89 - 61 * q^91 - 54 * q^93 - 55 * q^95 - 19 * q^97 + 8 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{12} - 3 x^{11} + 13 x^{10} - 9 x^{9} - 5 x^{8} + 35 x^{7} + 197 x^{6} - 140 x^{5} - 80 x^{4} + \cdots + 4096$ x^12 - 3*x^11 + 13*x^10 - 9*x^9 - 5*x^8 + 35*x^7 + 197*x^6 - 140*x^5 - 80*x^4 + 576*x^3 + 3328*x^2 + 3072*x + 4096 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( \nu^{11} + 97 \nu^{10} - 291 \nu^{9} + 1067 \nu^{8} - 233 \nu^{7} - 1945 \nu^{6} + 1185 \nu^{5} + \cdots + 293952 ) / 443456$ (v^11 + 97v^10 - 291v^9 + 1067v^8 - 233v^7 - 1945v^6 + 1185v^5 + 50792v^4 - 53096v^3 - 16252v^2 + 163104v + 293952) / 443456
$\beta_{3}$	$=$	$( 53 \nu^{11} - 435 \nu^{10} + 1469 \nu^{9} - 3473 \nu^{8} + 443 \nu^{7} + 7123 \nu^{6} + \cdots - 838912 ) / 1773824$ (53v^11 - 435v^10 + 1469v^9 - 3473v^8 + 443v^7 + 7123v^6 + 4093v^5 - 70768v^4 + 6384v^3 + 592832v^2 - 244288*v - 838912) / 1773824
$\beta_{4}$	$=$	$( 30 \nu^{11} + 40 \nu^{10} - 79 \nu^{9} + 1137 \nu^{8} - 307 \nu^{7} - 8945 \nu^{6} + 25423 \nu^{5} + \cdots + 324672 ) / 443456$ (30v^11 + 40v^10 - 79v^9 + 1137v^8 - 307v^7 - 8945v^6 + 25423v^5 + 5981v^4 - 66491v^3 + 39372v^2 + 274224*v + 324672) / 443456
$\beta_{5}$	$=$	$( - 69 \nu^{11} + 195 \nu^{10} - 749 \nu^{9} + 177 \nu^{8} + 1317 \nu^{7} - 1587 \nu^{6} + \cdots - 54272 ) / 443456$ (-69v^11 + 195v^10 - 749v^9 + 177v^8 + 1317v^7 - 1587v^6 - 15837v^5 + 2656v^4 + 140576v^3 - 105168v^2 - 250432*v - 54272) / 443456
$\beta_{6}$	$=$	$( 325 \nu^{11} - 250 \nu^{10} + 750 \nu^{9} + 12256 \nu^{8} - 28698 \nu^{7} + 30394 \nu^{6} + \cdots + 1298688 ) / 1773824$ (325v^11 - 250v^10 + 750v^9 + 12256v^8 - 28698v^7 + 30394v^6 + 118748v^5 - 81487v^4 - 241200v^3 + 490080v^2 + 1540352*v + 1298688) / 1773824
$\beta_{7}$	$=$	$( - 25 \nu^{11} + 76 \nu^{10} - 269 \nu^{9} + 57 \nu^{8} + 495 \nu^{7} - 247 \nu^{6} - 12733 \nu^{5} + \cdots + 1024 ) / 110864$ (-25v^11 + 76v^10 - 269v^9 + 57v^8 + 495v^7 - 247v^6 - 12733v^5 + 13254v^4 + 4207v^3 - 39944v^2 - 72720*v + 1024) / 110864
$\beta_{8}$	$=$	$( 1961 \nu^{11} - 9535 \nu^{10} + 37297 \nu^{9} - 72085 \nu^{8} + 46567 \nu^{7} + 31327 \nu^{6} + \cdots - 3739648 ) / 7095296$ (1961v^11 - 9535v^10 + 37297v^9 - 72085v^8 + 46567v^7 + 31327v^6 + 265585v^5 - 880016v^4 + 419888v^3 + 289408v^2 + 4524800*v - 3739648) / 7095296
$\beta_{9}$	$=$	$( 130 \nu^{11} - 469 \nu^{10} + 1407 \nu^{9} - 157 \nu^{8} - 9995 \nu^{7} + 19513 \nu^{6} + \cdots - 122880 ) / 443456$ (130v^11 - 469v^10 + 1407v^9 - 157v^8 - 9995v^7 + 19513v^6 + 10181v^5 - 64091v^4 + 22092v^3 + 174384v^2 + 232512*v - 122880) / 443456
$\beta_{10}$	$=$	$( 725 \nu^{11} - 3475 \nu^{10} + 15181 \nu^{9} - 27073 \nu^{8} + 19019 \nu^{7} + 54723 \nu^{6} + \cdots - 1331200 ) / 1773824$ (725v^11 - 3475v^10 + 15181v^9 - 27073v^8 + 19019v^7 + 54723v^6 - 35987v^5 - 215200v^4 + 302880v^3 + 458752v^2 + 300288*v - 1331200) / 1773824
$\beta_{11}$	$=$	$( 913 \nu^{11} - 2951 \nu^{10} + 13609 \nu^{9} - 14093 \nu^{8} + 9327 \nu^{7} + 30183 \nu^{6} + \cdots + 2008064 ) / 1773824$ (913v^11 - 2951v^10 + 13609v^9 - 14093v^8 + 9327v^7 + 30183v^6 + 151369v^5 - 144192v^4 + 210032v^3 + 500352v^2 + 2440960*v + 2008064) / 1773824

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{11} - \beta_{10} + \beta_{7} + 4\beta_{3}$ b11 - b10 + b7 + 4*b3
$\nu^{3}$	$=$	$-\beta_{11} - \beta_{9} + 4\beta_{8} + \beta_{7} + 4\beta_{6} + 6\beta_{5} + 4\beta_{3} + 4\beta_{2} - \beta_1$ -b11 - b9 + 4b8 + b7 + 4b6 + 6b5 + 4b3 + 4*b2 - b1
$\nu^{4}$	$=$	$- 14 \beta_{11} + 14 \beta_{10} - \beta_{9} - 12 \beta_{7} + 4 \beta_{6} + 12 \beta_{5} + 4 \beta_{4} + \cdots + 4$ -14b11 + 14b10 - b9 - 12b7 + 4b6 + 12b5 + 4b4 - 4b3 + 24b2 + b1 + 4
$\nu^{5}$	$=$	$18 \beta_{10} + 17 \beta_{9} - 56 \beta_{8} - 49 \beta_{7} - 52 \beta_{6} - 17 \beta_{5} + \cdots + 18 \beta_1$ 18b10 + 17b9 - 56b8 - 49b7 - 52b6 - 17b5 + 8b4 - 52b3 - 8b2 + 18b1
$\nu^{6}$	$=$	$175 \beta_{11} - 153 \beta_{10} + 43 \beta_{9} - 72 \beta_{8} + 43 \beta_{7} - 140 \beta_{6} - 153 \beta_{5} + \cdots - 72$ 175b11 - 153b10 + 43b9 - 72b8 + 43b7 - 140b6 - 153b5 - 124b4 - 140*b2 - 72
$\nu^{7}$	$=$	$247 \beta_{11} - 468 \beta_{10} - 234 \beta_{9} + 612 \beta_{8} + 468 \beta_{7} + 440 \beta_{6} + \cdots - 88$ 247b11 - 468b10 - 234b9 + 612b8 + 468b7 + 440b6 - 440b4 + 612b3 - 247*b1 - 88
$\nu^{8}$	$=$	$- 1782 \beta_{11} + 1142 \beta_{10} - 1142 \beta_{9} + 1872 \beta_{8} + 2808 \beta_{6} + 1782 \beta_{5} + \cdots + 884$ -1782b11 + 1142b10 - 1142b9 + 1872b8 + 2808b6 + 1782b5 + 884b3 + 1872b2 - 335*b1 + 884
$\nu^{9}$	$=$	$- 5771 \beta_{11} + 7849 \beta_{10} - 4568 \beta_{8} - 5771 \beta_{7} + 2666 \beta_{5} + 4568 \beta_{4} + \cdots + 2560$ -5771b11 + 7849b10 - 4568b8 - 5771b7 + 2666b5 + 4568b4 - 5908b3 + 2560b2 + 2666*b1 + 2560
$\nu^{10}$	$=$	$12417 \beta_{11} + 12417 \beta_{9} - 31396 \beta_{8} - 8331 \beta_{7} - 31396 \beta_{6} - 16862 \beta_{5} + \cdots - 8312$ 12417b11 + 12417b9 - 31396b8 - 8331b7 - 31396b6 - 16862b5 + 8312b4 - 20732b3 - 20732b2 + 8331b1 - 8312
$\nu^{11}$	$=$	$89754 \beta_{11} - 89754 \beta_{10} + 20729 \beta_{9} + 58342 \beta_{7} - 49668 \beta_{6} - 58342 \beta_{5} + \cdots - 49668$ 89754b11 - 89754b10 + 20729b9 + 58342b7 - 49668b6 - 58342b5 - 33324b4 + 33324b3 - 67448b2 - 20729b1 - 49668

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

Character values

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

$n$	$117$	$175$	$321$
$\chi(n)$	$1$	$1$	$\beta_{6}$

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}

49.1

−1.02179	+	1.28129i
1.52179	−	1.90827i
−1.56920	+	0.755686i
2.06920	−	0.996473i
−0.260453	−	1.14112i
0.760453	+	3.33176i
−1.02179	−	1.28129i
1.52179	+	1.90827i
−1.56920	−	0.755686i
2.06920	+	0.996473i
−0.260453	+	1.14112i
0.760453	−	3.33176i

−1.02179

−

1.28129i

−1.07557

0.517965i

−1.27416

−

1.59774i

0.0699247

−

0.306360i

49.2

1.52179

1.90827i

2.60002

−

1.25211i

1.89765

2.37957i

−0.658071

2.88320i

65.1

−1.56920

−

0.755686i

0.110081

0.482295i

2.82760

1.36170i

0.0208506

0.0261458i

65.2

2.06920

0.996473i

−0.788529

−

3.45477i

−3.72857

−

1.79558i

1.41815

1.77830i

81.1

−0.260453

1.14112i

−1.85326

2.32392i

0.115912

−

0.507846i

1.46859

0.707235i

81.2

0.760453

−

3.33176i

1.00725

−

1.26305i

−0.338433

1.48277i

−7.81944

−

3.76565i

161.1