LMFDB - Newform orbit 225.2.h.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,2,Mod(46,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 6]))

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

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

chi := DirichletCharacter("225.46");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	225.h (of order $5$, degree $4$, 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:	$1.79663404548$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{5})$
Coefficient field:	8.0.26265625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} + 2x^{6} + x^{4} + 8x^{2} - 24x + 16 $ x^8 - 3x^7 + 2x^6 + x^4 + 8x^2 - 24x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 5 $
Twist minimal:	no (minimal twist has level 75)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$q$-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_{7} + \beta_{3}) q^{2} + (\beta_{6} - \beta_{4} + 2 \beta_{3} + \cdots - 1) q^{4}+ \cdots + ( - \beta_{5} + \beta_{3} + \beta_{2} + \cdots - 1) q^{8}+O(q^{10}) $ q + (b7 + b3) * q^2 + (b6 - b4 + 2b3 + 2b1 - 1) * q^4 + (b6 + b4 + b1) * q^5 + (b7 - 2b6 - b3 - b2 + 1) q^7 + (-b5 + b3 + b2 + 5b1 - 1) q^8	$ q + (\beta_{7} + \beta_{3}) q^{2} + (\beta_{6} - \beta_{4} + 2 \beta_{3} + \cdots - 1) q^{4}+ \cdots + (\beta_{7} - 13 \beta_{6} + \cdots - 13 \beta_1) q^{98}+O(q^{100}) $ q + (b7 + b3) * q^2 + (b6 - b4 + 2b3 + 2b1 - 1) * q^4 + (b6 + b4 + b1) * q^5 + (b7 - 2b6 - b3 - b2 + 1) q^7 + (-b5 + b3 + b2 + 5b1 - 1) q^8 + (b7 + b4 - 4b1 + 1) q^10 + (-3b6 - b4 - 3b3 - b2 - 3b1) q^11 + (b7 + b6 + b4 + b3 + 2b1 - 2) q^13 + (b6 + 2b4 + 4b3 + 2b2 + b1) q^14 + (-2b7 + b6 - 3b5 + b4 - 2b3 + 3b2 + 5b1 - 2) q^16 + b5 * q^17 + (2b5 + b2 - 2b1) * q^19 + (b7 + 3b6 + 3b5 - 2b4 + 2b3 - 2b2 - 3b1) * q^20 + (-2b7 - 2b6 + 2b5 + b4 - b3 - b1 + 2) q^22 + (-b7 - 2b6 - b4 - b3 - b2 - 2b1) * q^23 + (-b6 - b5 + b4 + b3 - b2 + 3b1 - 3) q^25 + (-b7 + 3b6 - b5 + b4 + 2b3 + b2 + b1 - 2) * q^26 + (-b7 + 2b6 + b5 - 3b4 - b3 - b1 - 2) * q^28 + (4b6 - b4 + 5b3 + 5b1 - 4) q^29 + (2b3 - b2 - 5b1 - 2) * q^31 + (-2b7 - 7b6 - 2b5 + 2b4 - 9b3 + 2b2 + 2b1 + 1) q^32 + (3b6 - b1 + 1) q^34 + (2b7 - 2b6 - 2b5 + b4 - 4b3 - b2 - b1 + 3) * q^35 + (-2b7 - 2b6 + b5 - 3b4 - 2b3 - b2 - 2b1 + 1) q^37 + (-b7 + 8b6 + 2b5 - 3b4 - b3 - 2b2 - b1 - 1) * q^38 + (5b7 + 6b6 + b5 - b4 + 9b3 - 4b2 + 2b1 - 2) q^40 + (-5b6 - 4b1 + 4) * q^41 + (-b6 + 2b5 - 2b4 - b3 - 2b1 + 5) q^43 + (2b5 - 3b3 - b2 - 10b1 + 3) q^44 + (-b7 - 5b6 + b5 - 4b3 - 4b1 + 5) q^46 + (2b7 - 2b5 + b4 + b3 + b1) * q^47 + (-b7 + b6 - 4b5 + 4b4 + b2 + 4b1 + 1) q^49 + (-3b7 - 5b6 - 2b5 + 3b4 - 4b3 + 4b2 - 4b1 + 3) q^50 - 3b3 q^52 + (2b7 + b6 - 2b5 + b4 + 3b3 + 3b1 - 1) * q^53 + (-b7 - 3b6 - 2b5 - 2b4 - 7b3 + 2b2 - 2b1 + 6) * q^55 + (-2b5 - 3b3 + b2 + b1 + 3) * q^56 + (-6b5 + b3 + b2 + 10b1 - 1) * q^58 + (-3b7 - 3b6 - 3b4 - 3b3 + 3b1 - 3) q^59 + (-4b6 - 2b4 - b3 - 2b2 - 4b1) * q^61 + (-b7 - 2b6 + 5b5 - 6b4 - b3 - 5b2 - 8b1 + 3) q^62 + (-2b7 - 4b6 + 5b4 - 10b3 + 5b2 - 4b1) * q^64 + (2b7 - 2b6 - b5 + 2b3 - 2b2 - 2b1 - 3) q^65 + (-2b5 - b3 + 1) q^67 + (2b7 + b5 - b4 + 2b3 - 2b2 - b1 + 2) q^68 + (2b7 - 2b6 + 2b5 + 4b4 + 9b3 + b2 + b1 - 4) q^70 + (-4b7 + 2b6 + 4b5 - b4 - 4b3 - 4b1 - 2) q^71 + (4b7 + b6 - 2b4 - b3 - 2b2 + b1) q^73 + (-5b6 - b5 + b4 - 5b3 + b1 + 7) * q^74 + (5b7 - b6 + 4b3 - 5b2 + 8) q^76 + (-5b7 + 6b6 + b4 + 5b3 + b2 + 6b1) * q^77 + (-3b7 + 10b6 + 3b5 + 6b3 + 6b1 - 10) q^79 + (2b7 + 10b6 - b5 - 3b4 + 17b3 - 2b2 + 10b1 - 3) * q^80 + (-b7 + 4b5 - 4b4 - b3 + b2 - 4b1) q^82 + (4b2 + 5b1) * q^83 + (b7 + b4 + b1 - 4) * q^85 + (4b7 + 6b6 - b4 + 6b3 - b2 + 6b1) * q^86 + (4b5 - 4b4 - 4b2 - 15b1 + 11) * q^88 + (4b6 + b4 - 7b3 + b2 + 4b1) q^89 + (3b6 - 3b5 + 3b4 + 3b2 + 3) * q^91 + (2b5 - b2 - 7b1) * q^92 + (7b3 + b2 + 6b1 - 7) * q^94 + (b7 + b6 + 3b4 + 2b3 + 3b2 - 8) q^95 + (4b7 + 5b6 - 4b5 + 4b4 + 7b3 + 7b1 - 5) * q^97 + (b7 - 13b6 + 3b4 - 7b3 + 3b2 - 13b1) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} + q^{4} + 5 q^{5} + 4 q^{7}+O(q^{10}) $ 8 * q + q^2 + q^4 + 5 * q^5 + 4 * q^7	$ 8 q + q^{2} + q^{4} + 5 q^{5} + 4 q^{7} - 16 q^{11} - 8 q^{13} + 8 q^{14} - 17 q^{16} + q^{17} - 5 q^{19} + 10 q^{20} + 13 q^{22} - 7 q^{23} - 15 q^{25} - 6 q^{26} - 17 q^{28} - 5 q^{29} - 19 q^{31} - 24 q^{32} + 12 q^{34} + 10 q^{35} - q^{37} + 10 q^{38} + 25 q^{40} + 14 q^{41} + 32 q^{43} + 3 q^{44} + 16 q^{46} + q^{47} + 16 q^{49} - 10 q^{50} - 6 q^{52} + 3 q^{53} + 15 q^{55} + 15 q^{56} + 5 q^{58} - 30 q^{59} - 14 q^{61} + 17 q^{62} - 44 q^{64} - 25 q^{65} + 4 q^{67} + 22 q^{68} - 15 q^{70} - 21 q^{71} + 2 q^{73} + 38 q^{74} + 80 q^{76} + 37 q^{77} - 30 q^{79} + 50 q^{80} - 12 q^{82} - 2 q^{83} - 30 q^{85} + 34 q^{86} + 70 q^{88} + 21 q^{91} - 9 q^{92} - 33 q^{94} - 65 q^{95} - 6 q^{97} - 73 q^{98}+O(q^{100}) $ 8 * q + q^2 + q^4 + 5 * q^5 + 4 * q^7 - 16 * q^11 - 8 * q^13 + 8 * q^14 - 17 * q^16 + q^17 - 5 * q^19 + 10 * q^20 + 13 * q^22 - 7 * q^23 - 15 * q^25 - 6 * q^26 - 17 * q^28 - 5 * q^29 - 19 * q^31 - 24 * q^32 + 12 * q^34 + 10 * q^35 - q^37 + 10 * q^38 + 25 * q^40 + 14 * q^41 + 32 * q^43 + 3 * q^44 + 16 * q^46 + q^47 + 16 * q^49 - 10 * q^50 - 6 * q^52 + 3 * q^53 + 15 * q^55 + 15 * q^56 + 5 * q^58 - 30 * q^59 - 14 * q^61 + 17 * q^62 - 44 * q^64 - 25 * q^65 + 4 * q^67 + 22 * q^68 - 15 * q^70 - 21 * q^71 + 2 * q^73 + 38 * q^74 + 80 * q^76 + 37 * q^77 - 30 * q^79 + 50 * q^80 - 12 * q^82 - 2 * q^83 - 30 * q^85 + 34 * q^86 + 70 * q^88 + 21 * q^91 - 9 * q^92 - 33 * q^94 - 65 * q^95 - 6 * q^97 - 73 * q^98

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 3x^{7} + 2x^{6} + x^{4} + 8x^{2} - 24x + 16 $ x^8 - 3*x^7 + 2*x^6 + x^4 + 8*x^2 - 24*x + 16 :

$\beta_{1}$	$=$	$ ( \nu^{7} - \nu^{6} + \nu^{3} + 2\nu^{2} + 4\nu - 8 ) / 8 $ (v^7 - v^6 + v^3 + 2v^2 + 4v - 8) / 8
$\beta_{2}$	$=$	$ ( \nu^{7} - \nu^{6} + \nu^{3} - 2\nu^{2} + 12\nu - 12 ) / 4 $ (v^7 - v^6 + v^3 - 2v^2 + 12v - 12) / 4
$\beta_{3}$	$=$	$ ( -7\nu^{7} + 9\nu^{6} + 2\nu^{5} + 4\nu^{4} + \nu^{3} - 4\nu^{2} - 60\nu + 64 ) / 8 $ (-7v^7 + 9v^6 + 2v^5 + 4v^4 + v^3 - 4v^2 - 60v + 64) / 8
$\beta_{4}$	$=$	$ ( 11\nu^{7} - 15\nu^{6} - 4\nu^{5} - 8\nu^{4} + 3\nu^{3} + 2\nu^{2} + 92\nu - 96 ) / 8 $ (11v^7 - 15v^6 - 4v^5 - 8v^4 + 3v^3 + 2v^2 + 92*v - 96) / 8
$\beta_{5}$	$=$	$ ( 11\nu^{7} - 13\nu^{6} - 6\nu^{5} - 8\nu^{4} + 3\nu^{3} + 4\nu^{2} + 96\nu - 88 ) / 8 $ (11v^7 - 13v^6 - 6v^5 - 8v^4 + 3v^3 + 4v^2 + 96*v - 88) / 8
$\beta_{6}$	$=$	$ ( 13\nu^{7} - 21\nu^{6} - 4\nu^{5} - 4\nu^{4} + 5\nu^{3} + 10\nu^{2} + 120\nu - 144 ) / 8 $ (13v^7 - 21v^6 - 4v^5 - 4v^4 + 5v^3 + 10v^2 + 120*v - 144) / 8
$\beta_{7}$	$=$	$ ( 19\nu^{7} - 31\nu^{6} - 4\nu^{5} - 8\nu^{4} + 11\nu^{3} + 18\nu^{2} + 180\nu - 224 ) / 8 $ (19v^7 - 31v^6 - 4v^5 - 8v^4 + 11v^3 + 18v^2 + 180*v - 224) / 8

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

$\nu$	$=$	$ ( \beta_{7} - \beta_{6} + \beta_{5} - 2\beta_{4} - \beta_{3} + \beta_{2} - 4\beta _1 + 4 ) / 5 $ (b7 - b6 + b5 - 2b4 - b3 + b2 - 4b1 + 4) / 5
$\nu^{2}$	$=$	$ ( 2\beta_{7} - 2\beta_{6} + 2\beta_{5} - 4\beta_{4} - 2\beta_{3} - 3\beta_{2} + 2\beta _1 + 3 ) / 5 $ (2b7 - 2b6 + 2b5 - 4b4 - 2b3 - 3b2 + 2*b1 + 3) / 5
$\nu^{3}$	$=$	$ ( 2\beta_{7} - 2\beta_{6} + 2\beta_{5} + \beta_{4} + 8\beta_{3} + 2\beta_{2} + 7\beta _1 + 3 ) / 5 $ (2b7 - 2b6 + 2b5 + b4 + 8b3 + 2b2 + 7b1 + 3) / 5
$\nu^{4}$	$=$	$ -\beta_{7} + 2\beta_{6} - \beta_{4} + \beta_{2} + 2\beta _1 + 1 $ -b7 + 2b6 - b4 + b2 + 2b1 + 1
$\nu^{5}$	$=$	$ ( 6\beta_{7} - 11\beta_{6} - 9\beta_{5} + 3\beta_{4} - 11\beta_{3} + 6\beta_{2} + 6\beta _1 + 19 ) / 5 $ (6b7 - 11b6 - 9b5 + 3b4 - 11b3 + 6b2 + 6*b1 + 19) / 5
$\nu^{6}$	$=$	$ ( 2\beta_{7} - 7\beta_{6} + 7\beta_{5} - 9\beta_{4} - 7\beta_{3} + 7\beta_{2} + 12\beta _1 - 12 ) / 5 $ (2b7 - 7b6 + 7b5 - 9b4 - 7b3 + 7b2 + 12*b1 - 12) / 5
$\nu^{7}$	$=$	$ ( -8\beta_{7} + 3\beta_{6} - 3\beta_{5} + 6\beta_{4} - 7\beta_{3} + 7\beta_{2} + 57\beta _1 + 3 ) / 5 $ (-8b7 + 3b6 - 3b5 + 6b4 - 7b3 + 7b2 + 57*b1 + 3) / 5

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1 + \beta_{1} + \beta_{3} + \beta_{6}$

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} $

46.1

−1.21700	+	0.720348i
1.40799	−	0.132563i
−0.0272949	−	1.41395i
1.33631	+	0.462894i
−0.0272949	+	1.41395i
1.33631	−	0.462894i
−1.21700	−	0.720348i
1.40799	+	0.132563i

−0.655837

2.01846i

−2.02602

−

1.47199i

2.21700

−

0.291365i

4.35840

0.865884

−

0.629102i

−0.865884

4.66602i

46.2

0.346820

−

1.06740i

0.598970

0.435177i

−0.407987

−

2.19853i

1.11373

2.48822

−

1.80780i

−2.48822

−

0.327009i

91.1

−1.38048

−

1.00297i

0.281722

0.867051i

1.02729

1.98612i

−3.94243

−0.573870

1.76619i

0.573870

−

3.77214i

91.2

2.18949

1.59076i

1.64533

5.06380i

−0.336312

−

2.21063i

0.470294

−2.78023

8.55667i

2.78023

−

5.37515i

136.1