LMFDB - Newform orbit 845.2.k.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [845,2,Mod(268,845)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(845, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([3, 1]))

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

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

chi := DirichletCharacter("845.268");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 845 = 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	845.k (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:	$6.74735897080$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.619810816.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1 $ x^8 - 2x^5 + 14x^4 - 8x^3 + 2x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 65)
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_{5} q^{2} + (\beta_{3} - \beta_{2} - 1) q^{3} + ( - \beta_{7} - \beta_{5} + \beta_{4} + 1) q^{4} + ( - \beta_{7} + \beta_{6} + \cdots - \beta_{3}) q^{5}+ \cdots + ( - \beta_{5} + \beta_{4} + \cdots - \beta_1) q^{9}+O(q^{10}) $ q - b5 * q^2 + (b3 - b2 - 1) * q^3 + (-b7 - b5 + b4 + 1) * q^4 + (-b7 + b6 - b5 + b4 - b3) * q^5 + (-b6 + b5 + b3 + b2 + 1) * q^6 + (2b6 - 2b5 + 2b4 - 2b3 + b1) * q^7 + (-b7 - b5 - b3) * q^8 + (-b5 + b4 - b3 + b2 - b1) * q^9	$ q - \beta_{5} q^{2} + (\beta_{3} - \beta_{2} - 1) q^{3} + ( - \beta_{7} - \beta_{5} + \beta_{4} + 1) q^{4} + ( - \beta_{7} + \beta_{6} + \cdots - \beta_{3}) q^{5}+ \cdots + (\beta_{7} + \beta_{5} - \beta_{4} + \cdots - \beta_1) q^{99}+O(q^{100}) $ q - b5 * q^2 + (b3 - b2 - 1) * q^3 + (-b7 - b5 + b4 + 1) * q^4 + (-b7 + b6 - b5 + b4 - b3) * q^5 + (-b6 + b5 + b3 + b2 + 1) * q^6 + (2b6 - 2b5 + 2b4 - 2b3 + b1) * q^7 + (-b7 - b5 - b3) * q^8 + (-b5 + b4 - b3 + b2 - b1) * q^9 + (-b7 - 2b5 - 2b3 + 2b2 + b1 - 1) q^10 + (-b6 - b4 + b2 - 1) * q^11 + (b7 + b6 - b4 - b3 - b1) * q^12 + (2b6 - 3b5 + 3b4 - 3b3 + 2b2 + 3b1) * q^14 + (b7 - 2b4 - 2b3) * q^15 + (-2b4 - 2b3 - 1) * q^16 + (-b7 + b6 - 2b5 + b4 - 2b3 - 2b2 + b1 - 2) q^17 + (-b6 - 2b2 - b1) q^18 + (-b5 + b4 + 2b2 - 2) q^19 + (-b7 + 2b6 - b5 - 2b3 - b2 + b1 + 3) * q^20 + (b7 - b6 - b4 - b2 + b1 + 1) * q^21 + (-b7 + b6 + b5 - 2b2 - b1 + 2) q^22 + (-b7 + b6 - b5 + 2b4 - 2b2 - b1 + 2) * q^23 - b3 * q^24 + (b7 + 2b6 + b3 - 2b2 + b1 + 1) * q^25 + (2b7 - b6 + b5 - 2b4 - b2 + 2b1 + 1) q^27 + (4b6 - 2b5 + 2b4 - 2b3 + 3b1) q^28 + (-2b6 + b5 - b4 + b3 + 2b2 - b1) * q^29 + (-b7 + b5 + b4 - b3 - b2 + 3) * q^30 + (b6 - b5 + b3 - b2 - 1) * q^31 + (3b5 + 2) q^32 + (-b7 + b5 + b4 + 2) * q^33 + (-2b7 - 2b5 + 2b4 - 4b3 + b2 + 2b1 + 1) q^34 + (b7 + b6 + 3b5 - 2b4 + 3b3 + b2 - 3) q^35 + (-4b6 + 3b5 - 3b4 + 3b3 - 5b2) q^36 + (b5 - b4 + b3 - 6b2 - b1) q^37 + (2b6 + b5 + b4 - b2 + 1) q^38 + (b6 - 2b5 + b4 + b2 + b1 + 2) q^40 + (2b7 - b6 + 3b5 - 2b4 + 2b3 - 2b2 - 2b1 - 2) * q^41 + (b6 - b5 + 2b4 - 3b2 + 3) * q^42 + (-b7 - 3b6 - b5 - 2b4 - b1) * q^43 + (-2b6 - b5 - b4 + 2b2 - 2) * q^44 + (-2b7 - 3b6 + b5 - b4 + 4b3 - 3b2 - 1) * q^45 + (-4b6 - 3b5 - b4 + 2b2 - 2) q^46 + (2b2 + 3b1) * q^47 + (-2b7 - 2b6 + 2b4 + b3 - b2 + 2b1 - 1) * q^48 + (4b5 - 2b4 - 2b3 - 1) q^49 + (b7 - b5 + 2b4 + b3 + 6b2 + 3b1 + 2) q^50 + (-3b5 + 3b4 - 3b3 + 4b2) * q^51 + (-b7 + 2b6 - 3b5 + b4 - 4b3 - 3b2 + b1 - 3) * q^53 + (b7 + 3b6 + 3b4 - 3b2 + b1 + 3) q^54 + (-2b6 - b5 + b4 + 3b2 + b1 + 1) * q^55 + (2b6 + b5 - b4 + b3 + 4b2 + b1) * q^56 + (b7 + 3b5 - 3b4 - 2b3 + 6) q^57 + (2b5 - 2b4 + 2b3 - 4b2 - 3b1) q^58 + (-4b6 + 4b5 + 5b3 - 2b2 - 2) * q^59 + (-b7 - b6 - 3b5 + 2b4 + 2b3 - b2 - 7) q^60 + (3b7 - 2b5 - 2b4 + b3) q^61 + (-b7 - b6 + b4 + b3 + 4b2 + b1 + 4) q^62 + (-5b7 - 5b5 + 3b4 - 2b3 - 2) * q^63 + (3b7 + b5 + b4 + 4b3 - 7) * q^64 + (b7 - 2b5 - 2b4 - b3 - 6) * q^66 + (b7 - 2b5 + b3) q^67 + (3b6 - 3b5 - 4b3) q^68 + (3b7 - 4b4 - b3 - 2) * q^69 + (2b7 + b6 + 7b5 - 2b4 + 4b3 + 7b2 + b1 - 1) q^70 + (-b7 + 2b6 - 3b5 + b4 - 7b3 + 2b2 + b1 + 2) * q^71 + (-3b6 + 3b5 - 3b4 + 3b3 - 2b2 - 2b1) * q^72 + (3b7 + 2b5 - 2b4 + b3) q^73 + (-8b6 + 2b5 - 2b4 + 2b3 + 2b2 - b1) q^74 + (-b7 - 2b6 - 3b5 + 3b4 + 3b3 + 5b2 - b1 - 5) q^75 + (2b7 - b6 + 2b5 - 3b4 + b2 + 2b1 - 1) * q^76 + (-b6 + b5 + 2b3 + 3b2 + 3) * q^77 + (-6b6 + 5b5 - 5b4 + 5b3 + 2b2 - 2b1) * q^79 + (b7 - b6 - 3b5 + 3b4 + 3b3 + 4b2 - 2) * q^80 + (-b7 + 2b5 + 4b4 + 3b3 - 1) q^81 + (3b7 - 6b6 + 9b5 - 3b4 + 6b3 - b2 - 3b1 - 1) * q^82 + (-4b6 - 4b2 + 3b1) q^83 + (-b7 - b6 - 4b5 + 3b4 + 3b2 - b1 - 3) q^84 + (3b7 + 2b5 - 4b4 + 3b3 + 3b2 - b1 + 1) q^85 + (-4b7 - 2b6 - b5 - b4 - 6b2 - 4b1 + 6) * q^86 + (b6 + 3b5 - 2b4 - 3b2 + 3) q^87 + (-b5 + b4 - b2 + 1) * q^88 + (b7 - 2b6 + 3b5 - b4 - 2b3 + 5b2 - b1 + 5) * q^89 + (-2b7 - 3b6 - 3b4 + 2b3 - 2b2 - 3b1 + 1) * q^90 + (-2b7 + b5 - b4 - 7b2 - 2b1 + 7) q^92 + (-2b6 + 6b2 - b1) * q^93 + (8b6 - 3b5 + 3b4 - 3b3 + 3b1) q^94 + (b7 - 3b6 + 3b5 - 2b4 + 2b3 + 2b1) q^95 + (3b6 - 3b5 - b3 - 5b2 - 5) q^96 + (-2b7 - 3b5 - 3b4 - 5b3 - 2) * q^97 + (2b7 + 5b5 - 4b4 - 2b3 - 10) * q^98 + (b7 + b5 - b4 + b3 - b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{2} - 6 q^{3} + 8 q^{4} - 2 q^{5} + 6 q^{6}+O(q^{10}) $ 8 * q + 4 * q^2 - 6 * q^3 + 8 * q^4 - 2 * q^5 + 6 * q^6	$ 8 q + 4 q^{2} - 6 q^{3} + 8 q^{4} - 2 q^{5} + 6 q^{6} - 6 q^{10} - 6 q^{11} + 2 q^{12} + 2 q^{15} - 8 q^{16} - 16 q^{17} - 14 q^{19} + 22 q^{20} + 12 q^{21} + 10 q^{22} + 14 q^{23} - 2 q^{24} + 12 q^{25} + 12 q^{27} + 14 q^{30} - 2 q^{31} + 4 q^{32} + 8 q^{33} - 24 q^{35} + 2 q^{38} + 22 q^{40} - 16 q^{41} + 24 q^{42} + 6 q^{43} - 10 q^{44} - 6 q^{45} - 2 q^{46} - 14 q^{48} - 24 q^{49} + 20 q^{50} - 24 q^{53} + 20 q^{54} + 10 q^{55} + 40 q^{57} - 22 q^{59} - 46 q^{60} + 20 q^{61} + 30 q^{62} - 16 q^{63} - 48 q^{64} - 36 q^{66} + 12 q^{67} + 4 q^{68} - 4 q^{69} - 20 q^{70} + 10 q^{71} + 4 q^{73} - 30 q^{75} - 6 q^{76} + 24 q^{77} - 2 q^{80} - 20 q^{81} - 20 q^{82} - 16 q^{84} + 20 q^{85} + 46 q^{86} + 16 q^{87} + 10 q^{88} + 28 q^{89} + 14 q^{90} + 50 q^{92} - 2 q^{95} - 30 q^{96} - 12 q^{97} - 92 q^{98} + 2 q^{99}+O(q^{100}) $ 8 * q + 4 * q^2 - 6 * q^3 + 8 * q^4 - 2 * q^5 + 6 * q^6 - 6 * q^10 - 6 * q^11 + 2 * q^12 + 2 * q^15 - 8 * q^16 - 16 * q^17 - 14 * q^19 + 22 * q^20 + 12 * q^21 + 10 * q^22 + 14 * q^23 - 2 * q^24 + 12 * q^25 + 12 * q^27 + 14 * q^30 - 2 * q^31 + 4 * q^32 + 8 * q^33 - 24 * q^35 + 2 * q^38 + 22 * q^40 - 16 * q^41 + 24 * q^42 + 6 * q^43 - 10 * q^44 - 6 * q^45 - 2 * q^46 - 14 * q^48 - 24 * q^49 + 20 * q^50 - 24 * q^53 + 20 * q^54 + 10 * q^55 + 40 * q^57 - 22 * q^59 - 46 * q^60 + 20 * q^61 + 30 * q^62 - 16 * q^63 - 48 * q^64 - 36 * q^66 + 12 * q^67 + 4 * q^68 - 4 * q^69 - 20 * q^70 + 10 * q^71 + 4 * q^73 - 30 * q^75 - 6 * q^76 + 24 * q^77 - 2 * q^80 - 20 * q^81 - 20 * q^82 - 16 * q^84 + 20 * q^85 + 46 * q^86 + 16 * q^87 + 10 * q^88 + 28 * q^89 + 14 * q^90 + 50 * q^92 - 2 * q^95 - 30 * q^96 - 12 * q^97 - 92 * q^98 + 2 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1 $ x^8 - 2*x^5 + 14*x^4 - 8*x^3 + 2*x^2 + 2*x + 1 :

$\beta_{1}$	$=$	$ ( 64\nu^{7} + 16\nu^{6} + 4\nu^{5} - 127\nu^{4} + 944\nu^{3} - 276\nu^{2} + 378\nu + 63 ) / 319 $ (64v^7 + 16v^6 + 4v^5 - 127v^4 + 944v^3 - 276v^2 + 378*v + 63) / 319
$\beta_{2}$	$=$	$ ( -63\nu^{7} + 64\nu^{6} + 16\nu^{5} + 130\nu^{4} - 1009\nu^{3} + 1448\nu^{2} - 402\nu - 67 ) / 319 $ (-63v^7 + 64v^6 + 16v^5 + 130v^4 - 1009v^3 + 1448v^2 - 402*v - 67) / 319
$\beta_{3}$	$=$	$ ( -67\nu^{7} + 63\nu^{6} - 64\nu^{5} + 118\nu^{4} - 1068\nu^{3} + 1545\nu^{2} - 1263\nu + 268 ) / 319 $ (-67v^7 + 63v^6 - 64v^5 + 118v^4 - 1068v^3 + 1545v^2 - 1263*v + 268) / 319
$\beta_{4}$	$=$	$ ( 83\nu^{7} - 59\nu^{6} + 65\nu^{5} - 70\nu^{4} + 1304\nu^{3} - 1614\nu^{2} + 1198\nu + 306 ) / 319 $ (83v^7 - 59v^6 + 65v^5 - 70v^4 + 1304v^3 - 1614v^2 + 1198*v + 306) / 319
$\beta_{5}$	$=$	$ ( -172\nu^{7} - 43\nu^{6} + 69\nu^{5} + 441\nu^{4} - 2218\nu^{3} + 662\nu^{2} + 619\nu - 269 ) / 319 $ (-172v^7 - 43v^6 + 69v^5 + 441v^4 - 2218v^3 + 662v^2 + 619*v - 269) / 319
$\beta_{6}$	$=$	$ ( -196\nu^{7} - 49\nu^{6} - 92\nu^{5} + 369\nu^{4} - 2572\nu^{3} + 1244\nu^{2} - 1038\nu - 173 ) / 319 $ (-196v^7 - 49v^6 - 92v^5 + 369v^4 - 2572v^3 + 1244v^2 - 1038*v - 173) / 319
$\beta_{7}$	$=$	$ \nu^{7} - 2\nu^{4} + 14\nu^{3} - 8\nu^{2} + \nu + 2 $ v^7 - 2v^4 + 14v^3 - 8*v^2 + v + 2

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

$\nu$	$=$	$ ( -\beta_{7} - \beta_{5} + \beta_{4} + \beta_1 ) / 2 $ (-b7 - b5 + b4 + b1) / 2
$\nu^{2}$	$=$	$ \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 2\beta_{2} $ b6 - b5 + b4 - b3 + 2*b2
$\nu^{3}$	$=$	$ ( 3\beta_{7} + 5\beta_{5} - 5\beta_{4} - 2\beta_{2} + 3\beta _1 + 2 ) / 2 $ (3b7 + 5b5 - 5b4 - 2b2 + 3*b1 + 2) / 2
$\nu^{4}$	$=$	$ -\beta_{7} - \beta_{5} + 5\beta_{4} + 4\beta_{3} - 7 $ -b7 - b5 + 5b4 + 4b3 - 7
$\nu^{5}$	$=$	$ ( 11\beta_{7} + 2\beta_{6} + 9\beta_{5} - 11\beta_{4} - 12\beta_{3} + 12\beta_{2} - 11\beta _1 + 12 ) / 2 $ (11b7 + 2b6 + 9b5 - 11b4 - 12b3 + 12b2 - 11*b1 + 12) / 2
$\nu^{6}$	$=$	$ -15\beta_{6} + 16\beta_{5} - 16\beta_{4} + 16\beta_{3} - 28\beta_{2} + 7\beta_1 $ -15b6 + 16b5 - 16b4 + 16b3 - 28b2 + 7b1
$\nu^{7}$	$=$	$ ( -43\beta_{7} + 16\beta_{6} - 89\beta_{5} + 105\beta_{4} + 60\beta_{2} - 43\beta _1 - 60 ) / 2 $ (-43b7 + 16b6 - 89b5 + 105b4 + 60b2 - 43b1 - 60) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/845\mathbb{Z}\right)^\times$.

$n$	$171$	$677$
$\chi(n)$	$\beta_{2}$	$-\beta_{2}$

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} $

268.1

0.561103	+	0.561103i
−1.49094	−	1.49094i
−0.252709	−	0.252709i
1.18254	+	1.18254i
0.561103	−	0.561103i
−1.49094	+	1.49094i
−0.252709	+	0.252709i
1.18254	−	1.18254i

−2.03032

−1.33000

−

1.33000i

2.12221

1.45220

−

1.70032i

2.70032

2.70032i

−

1.61845i

−0.248119

0.537789i

−2.94844

3.45220i

268.2

0.134632

−2.15558

−

2.15558i

−1.98187

−1.82630

1.29021i

−0.290209

−

0.290209i

1.90970i

−0.536087

6.29303i

−0.245878

0.173703i

268.3

1.57942

0.725850

0.725850i

0.494582

−2.23127

−

0.146426i

1.14643

1.14643i

−

4.24997i

−2.37769

−

1.94628i

−3.52412

−

0.231269i

268.4

2.31627

−0.240275

−

0.240275i

3.36509

1.60536

1.55654i

−0.556540

−

0.556540i

3.95872i

3.16190

−

2.88454i

3.71844

3.60536i

577.1