LMFDB - Newform orbit 405.3.g.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,3,Mod(82,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("405.82");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	405.g (of order $4$, 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:	$11.0354507066$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 2x^{6} + 16x^{5} + 145x^{4} - 130x^{3} + 98x^{2} + 560x + 1600 $ x^8 - 2x^7 + 2x^6 + 16x^5 + 145x^4 - 130x^3 + 98x^2 + 560*x + 1600
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-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 $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{6} q^{2} + ( - \beta_{6} - \beta_{5} + \cdots - \beta_1) q^{4}+ \cdots + ( - \beta_{5} + \beta_{4} + \beta_{3} + \cdots - 4) q^{8}+O(q^{10}) $ q - b6 * q^2 + (-b6 - b5 - 4b2 - b1) q^4 + (b6 + b3 - b2) * q^5 + (-b6 - 3b2 + 3) q^7 + (-b5 + b4 + b3 - 4b2 - 4b1 - 4) * q^8	$ q - \beta_{6} q^{2} + ( - \beta_{6} - \beta_{5} + \cdots - \beta_1) q^{4}+ \cdots + ( - 7 \beta_{5} + \beta_{4} + \cdots - 52) q^{98}+O(q^{100}) $ q - b6 * q^2 + (-b6 - b5 - 4b2 - b1) q^4 + (b6 + b3 - b2) * q^5 + (-b6 - 3b2 + 3) q^7 + (-b5 + b4 + b3 - 4b2 - 4b1 - 4) * q^8 + (b7 + 3b6 + b5 + 4b2 + 4) * q^10 + (-b7 - b4 + b3 - 3) * q^11 + (b4 - 4b2 + 2b1 - 4) * q^13 + (-4b6 - b5 - 8b2 - 4b1) q^14 + (b7 + 8b6 + b4 + 5b3 - 8b1 - 8) q^16 + (b7 + 3b6 + b5 + b3 - 5b2 + 5) * q^17 + (-b7 + b6 + b4 - 5b2 + b1) q^19 + (b6 + 4b5 - b4 + 20b2 + 7b1) q^20 + (b7 + b6 - 5b5 - 5b3 - 4b2 + 4) q^22 + (b5 + b4 - b3 + 9b2 + 9) q^23 + (-b7 - 3b6 + b5 + b4 + b3 - b1 + 15) q^25 + (4b6 + 3b3 - 4b1 + 16) q^26 + (-4b5 + b4 + 4b3 - 16b2 - 14b1 - 16) * q^28 + (-3b6 + 5b5 + 7b2 - 3b1) * q^29 + (2b6 - 5b3 - 2b1 - 11) q^31 + (b7 + 22b6 + 9b5 + 9b3 + 28b2 - 28) * q^32 + (b7 + 2b6 + 8b5 - b4 + 16b2 + 2b1) * q^34 + (b7 + 6b6 + 4b5 + 3b3 + b2 + 3b1 + 1) * q^35 + (b7 - 11b6 - b5 - b3 - 4b2 + 4) * q^37 + (-4b5 + 4b3 + 8b2 - 7b1 + 8) * q^38 + (-8b6 + b5 - 8b3 + 8b2 + 26b1 + 24) * q^40 + (-b7 - b6 - b4 - 4b3 + b1 + 41) q^41 + (-4b5 - b4 + 4b3 - 4b2 - 9b1 - 4) * q^43 + (-b7 - 11b6 + 2b5 + b4 + 36b2 - 11b1) * q^44 + (-b7 - 9b6 - b4 + 5b3 + 9b1 - 8) q^46 + (7b6 - b5 - b3 - 40b2 + 40) * q^47 + (-7b6 - b5 + 23b2 - 7b1) q^49 + (b7 - 15b6 - 8b5 - b4 + 6b3 - 32b2 - 6b1 - 8) q^50 + (-b7 + 6b6 + 4b5 + 4b3 + 4b2 - 4) * q^52 + (2b5 - 4b4 - 2b3 - 17b2 + 13b1 - 17) q^53 + (-4b7 + 11b6 + 6b5 - b4 - 3b3 + 7b2 - 12b1 + 28) * q^55 + (4b7 + 24b6 + 4b4 + 15b3 - 24b1 - 48) q^56 + (-3b5 - 5b4 + 3b3 - 44b2 + 11b1 - 44) q^58 + (-b7 - 15b6 + 2b5 + b4 - 5b2 - 15b1) * q^59 + (-3b7 - 6b6 - 3b4 + 6b3 + 6b1 + 8) q^61 + (-5b7 + 5b6 + 2b5 + 2b3 + 36b2 - 36) q^62 + (5b7 + 38b6 + 7b5 - 5b4 + 72b2 + 38b1) * q^64 + (b7 - 6b6 + b5 + 5b4 - 4b3 + 8b2 + 10b1 - 16) q^65 + (-6b7 - 6b6 - 7b5 - 7b3 - 3b2 + 3) q^67 + (3b5 - 4b4 - 3b3 + 4b2 + 28b1 + 4) q^68 + (3b7 + 10b6 + 11b5 - 4b4 - 3b3 + 20b2 + 20b1 + 20) q^70 + (-b7 + 7b6 - b4 + 3b3 - 7b1 - 29) q^71 + (12b5 - 3b4 - 12b3 + 13b2 + 9b1 + 13) q^73 + (-b7 - 15b6 - 6b5 + b4 - 80b2 - 15b1) * q^74 + (11b6 + 7b3 - 11b1 - 4) q^76 + (-5b7 + b6 - 2b5 - 2b3 + 5b2 - 5) * q^77 + (-4b7 - 21b6 - 12b5 + 4b4 + 48b2 - 21b1) * q^79 + (-4b7 - 46b6 - 8b5 - b4 - 10b3 - 36b2 + 24b1 + 92) * q^80 + (-4b7 - 55b6 - 6b5 - 6b3 + 8b2 - 8) q^82 + (-4b5 + b4 + 4b3 + 36b2 - b1 + 36) q^83 + (b7 - 14b6 + 2b5 + 8b3 - 29b2 + b1 + 3) * q^85 + (4b7 + 19b6 + 4b4 + 4b3 - 19b1 - 40) q^86 + (4b5 + 2b4 - 4b3 - 80b2 + 10b1 - 80) q^88 + (5b7 + 2b6 + 11b5 - 5b4 + 33b2 + 2b1) * q^89 + (3b7 - 2b6 + 3b4 + 3b3 + 2b1 - 8) q^91 + (b7 - 4b6 - 10b5 - 10b3 - 56b2 + 56) * q^92 + (-b7 - 35b6 + 7b5 + b4 + 64b2 - 35b1) * q^94 + (-3b7 + 7b6 + 3b5 + 5b4 - 4b3 - 8b2 + 16b1 - 13) q^95 + (3b7 - 17b6 - 13b5 - 13b3 + 20b2 - 20) q^97 + (-7b5 + b4 + 7b3 - 52b2 + 7b1 - 52) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{2} + 2 q^{5} + 26 q^{7} - 36 q^{8}+O(q^{10}) $ 8 * q + 2 * q^2 + 2 * q^5 + 26 * q^7 - 36 * q^8	$ 8 q + 2 q^{2} + 2 q^{5} + 26 q^{7} - 36 q^{8} + 26 q^{10} - 20 q^{11} - 28 q^{13} - 76 q^{16} + 38 q^{17} + 12 q^{20} + 10 q^{22} + 68 q^{23} + 128 q^{25} + 124 q^{26} - 140 q^{28} - 116 q^{31} - 232 q^{32} + 14 q^{35} + 50 q^{37} + 66 q^{38} + 228 q^{40} + 316 q^{41} - 34 q^{43} - 8 q^{46} + 302 q^{47} - 22 q^{50} - 28 q^{52} - 118 q^{53} + 166 q^{55} - 420 q^{56} - 318 q^{58} + 112 q^{61} - 290 q^{62} - 112 q^{65} + 8 q^{67} + 76 q^{68} + 168 q^{70} - 248 q^{71} + 74 q^{73} - 48 q^{76} - 50 q^{77} + 836 q^{80} + 22 q^{82} + 302 q^{83} + 86 q^{85} - 380 q^{86} - 636 q^{88} - 44 q^{91} + 416 q^{92} - 102 q^{95} - 178 q^{97} - 374 q^{98}+O(q^{100}) $ 8 * q + 2 * q^2 + 2 * q^5 + 26 * q^7 - 36 * q^8 + 26 * q^10 - 20 * q^11 - 28 * q^13 - 76 * q^16 + 38 * q^17 + 12 * q^20 + 10 * q^22 + 68 * q^23 + 128 * q^25 + 124 * q^26 - 140 * q^28 - 116 * q^31 - 232 * q^32 + 14 * q^35 + 50 * q^37 + 66 * q^38 + 228 * q^40 + 316 * q^41 - 34 * q^43 - 8 * q^46 + 302 * q^47 - 22 * q^50 - 28 * q^52 - 118 * q^53 + 166 * q^55 - 420 * q^56 - 318 * q^58 + 112 * q^61 - 290 * q^62 - 112 * q^65 + 8 * q^67 + 76 * q^68 + 168 * q^70 - 248 * q^71 + 74 * q^73 - 48 * q^76 - 50 * q^77 + 836 * q^80 + 22 * q^82 + 302 * q^83 + 86 * q^85 - 380 * q^86 - 636 * q^88 - 44 * q^91 + 416 * q^92 - 102 * q^95 - 178 * q^97 - 374 * q^98

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 2x^{7} + 2x^{6} + 16x^{5} + 145x^{4} - 130x^{3} + 98x^{2} + 560x + 1600 $ x^8 - 2*x^7 + 2*x^6 + 16*x^5 + 145*x^4 - 130*x^3 + 98*x^2 + 560*x + 1600 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( - 37873 \nu^{7} + 228706 \nu^{6} - 314106 \nu^{5} - 377848 \nu^{4} - 3410625 \nu^{3} + \cdots - 10194840 ) / 105082840 $ (-37873v^7 + 228706v^6 - 314106v^5 - 377848v^4 - 3410625v^3 + 35276490v^2 - 16757594*v - 10194840) / 105082840
$\beta_{3}$	$=$	$ ( - 2135 \nu^{7} + 4014 \nu^{6} - 14863 \nu^{5} + 152321 \nu^{4} - 587856 \nu^{3} + 226750 \nu^{2} + \cdots + 13383248 ) / 2627071 $ (-2135v^7 + 4014v^6 - 14863v^5 + 152321v^4 - 587856v^3 + 226750v^2 + 1725200*v + 13383248) / 2627071
$\beta_{4}$	$=$	$ ( 35313 \nu^{7} - 334636 \nu^{6} + 2178916 \nu^{5} - 2404962 \nu^{4} + 2902625 \nu^{3} + \cdots - 60728000 ) / 26270710 $ (35313v^7 - 334636v^6 + 2178916v^5 - 2404962v^4 + 2902625v^3 - 15932190v^2 + 215087494*v - 60728000) / 26270710
$\beta_{5}$	$=$	$ ( 18753 \nu^{7} - 198911 \nu^{6} + 285591 \nu^{5} + 117728 \nu^{4} - 383500 \nu^{3} + \cdots + 2620240 ) / 13135355 $ (18753v^7 - 198911v^6 + 285591v^5 + 117728v^4 - 383500v^3 - 20510380v^2 + 2245484*v + 2620240) / 13135355
$\beta_{6}$	$=$	$ ( 3824 \nu^{7} - 5959 \nu^{6} + 5703 \nu^{5} + 52024 \nu^{4} + 758825 \nu^{3} - 326151 \nu^{2} + \cdots + 1514920 ) / 2627071 $ (3824v^7 - 5959v^6 + 5703v^5 + 52024v^4 + 758825v^3 - 326151v^2 + 275351*v + 1514920) / 2627071
$\beta_{7}$	$=$	$ ( - 437163 \nu^{7} + 844056 \nu^{6} - 792806 \nu^{5} - 7623698 \nu^{4} - 54732105 \nu^{3} + \cdots - 205585680 ) / 26270710 $ (-437163v^7 + 844056v^6 - 792806v^5 - 7623698v^4 - 54732105v^3 + 42614890v^2 - 38027494*v - 205585680) / 26270710

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{6} + \beta_{5} + 8\beta_{2} + \beta_1 $ b6 + b5 + 8*b2 + b1
$\nu^{3}$	$=$	$ \beta_{7} + 12\beta_{6} + \beta_{5} + \beta_{3} + 4\beta_{2} - 4 $ b7 + 12b6 + b5 + b3 + 4b2 - 4
$\nu^{4}$	$=$	$ \beta_{7} + 20\beta_{6} + \beta_{4} + 17\beta_{3} - 20\beta _1 - 88 $ b7 + 20b6 + b4 + 17b3 - 20*b1 - 88
$\nu^{5}$	$=$	$ -25\beta_{5} + 17\beta_{4} + 25\beta_{3} - 92\beta_{2} - 166\beta _1 - 92 $ -25b5 + 17b4 + 25b3 - 92b2 - 166*b1 - 92
$\nu^{6}$	$=$	$ -25\beta_{7} - 342\beta_{6} - 251\beta_{5} + 25\beta_{4} - 1128\beta_{2} - 342\beta_1 $ -25b7 - 342b6 - 251b5 + 25b4 - 1128b2 - 342b1
$\nu^{7}$	$=$	$ -251\beta_{7} - 2414\beta_{6} - 467\beta_{5} - 467\beta_{3} - 1732\beta_{2} + 1732 $ -251b7 - 2414b6 - 467b5 - 467b3 - 1732*b2 + 1732

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$-\beta_{2}$	$1$

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} $

82.1

−2.19641	+	2.19641i
−1.08828	+	1.08828i
1.50511	−	1.50511i
2.77958	−	2.77958i
−2.19641	−	2.19641i
−1.08828	−	1.08828i
1.50511	+	1.50511i
2.77958	+	2.77958i

−2.19641

−

2.19641i

5.64845i

−3.84486

3.19641i

0.803588

0.803588i

3.62067

−

3.62067i

15.4655

1.42426i

82.2

−1.08828

−

1.08828i

−

1.63130i

4.54303

2.08828i

1.91172

1.91172i

−6.12842

6.12842i

−2.67145

−

7.21670i

82.3

1.50511

1.50511i

0.530684i

4.97442

−

0.505105i

4.50511

4.50511i

5.22169

−

5.22169i

8.24726

6.72679i

82.4

2.77958

2.77958i

11.4522i

−4.67259

−

1.77958i

5.77958

5.77958i

−20.7139

20.7139i

−8.04135

−

17.9344i

163.1