LMFDB - Newform orbit 8325.2.a.ca

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8325,2,Mod(1,8325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8325, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8325.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$8325 = 3^{2} \cdot 5^{2} \cdot 37$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	8325.a (trivial)

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:	yes
Analytic conductor:	$66.4754596827$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.528933.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{5} - 2x^{4} - 6x^{3} + 4x^{2} + 7x - 1$ x^5 - 2x^4 - 6x^3 + 4x^2 + 7x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 2775)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$ -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 $1,\beta_1,\beta_2,\beta_3,\beta_4$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta_{3} q^{2} + (\beta_{4} - \beta_{2} + 2) q^{4} + (\beta_{4} - \beta_{3} - \beta_1 + 1) q^{7} + ( - \beta_{4} - \beta_{3} - \beta_1) q^{8} + (\beta_{3} - \beta_{2}) q^{11} + (\beta_{4} + 3) q^{13}+ \cdots + (2 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + \cdots - 6) q^{98}+O(q^{100})$ q - b3 * q^2 + (b4 - b2 + 2) * q^4 + (b4 - b3 - b1 + 1) * q^7 + (-b4 - b3 - b1) * q^8 + (b3 - b2) * q^11 + (b4 + 3) * q^13 + (-2b2 + b1 + 2) q^14 + (2b4 + b3 - 2b2 + b1 + 2) * q^16 + (b4 - 2) * q^17 + (-b4 + 2b3 + 1) q^19 + (-b3 - b1 - 2) * q^22 + (2b4 - b1) q^23 + (-2b4 - 3b3 + b2 - 2) * q^26 + (-b4 - 3b3 - b1 + 2) q^28 + (-2b3 - b2) q^29 + (2b4 - 2b3 - 2b2 + 1) q^31 + (-2b4 - 3b3 + 3b2 - b1 - 4) q^32 + (-2b4 + 2b3 + b2 - 2) * q^34 + q^37 + (-b3 + b2 - 6) * q^38 + (2b4 + b3 - 2b2 + 2) * q^41 + (-b4 - 2b3 - b2 + 1) q^43 + (2b4 + b3 - b2 + b1 + 4) q^44 + (-3b4 + b3 + b1 - 4) q^46 + (-b4 - b3 + 2b2 - b1) q^47 + (-b4 + 2b3 - 2b1 + 4) * q^49 + (4b4 + 3b3 - 4b2 + b1 + 8) q^52 + (b4 + b3 - 2b2 + b1) q^53 + (6b4 - b3 - 2b2 - b1 + 10) * q^56 + (3b4 - b3 - 3b2 - b1 + 10) * q^58 + (b4 + b3 + 4) * q^59 + (4b4 - b3 - b2 - b1 + 7) q^61 + (-3b3 - 2b2 - 2b1 + 8) q^62 + (b4 + 6b3 + 2b1 + 6) * q^64 + (-b4 + b3 + b2 + b1 + 3) * q^67 + (-b4 + 3b3 + b2 + b1 - 2) q^68 + (2b4 - 2b3 + b2 + b1 + 4) * q^71 + (-b3 - b2 - b1 + 6) * q^73 - b3 * q^74 + (2b4 + 3b3 + b1) * q^76 + (b4 - 3b3 + b2 - 2) q^77 + (-b4 + b3 - b2 - b1 + 2) * q^79 + (-3b4 - 4b3 + b2 - 2b1 - 4) q^82 + (3b4 + b3 - 2b2 - 2b1 + 2) q^83 + (5b4 - 2b3 - 4b2 - b1 + 12) q^86 + (-5b4 - 4b3 + 4b2 - 2) q^88 + (2b4 + b2 - b1 - 6) q^89 + (-b4 - b3 + b2 - 3b1 + 3) q^91 + (3b3 + b1 + 2) q^92 + (2b4 + 3b3 - 2b2 + 3b1 + 2) * q^94 + (-2b4 - 2b3 + 3b2 + 3) q^97 + (2b4 - 2b3 - 3b2 + 2b1 - 6) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$5 q - 2 q^{2} + 8 q^{4} - 3 q^{8} + q^{11} + 14 q^{13} + 10 q^{14} + 10 q^{16} - 11 q^{17} + 10 q^{19} - 14 q^{22} - 4 q^{23} - 13 q^{26} + 3 q^{28} - 5 q^{29} - 3 q^{31} - 23 q^{32} - 3 q^{34} + 5 q^{37}+ \cdots - 35 q^{98}+O(q^{100})$ 5 * q - 2 * q^2 + 8 * q^4 - 3 * q^8 + q^11 + 14 * q^13 + 10 * q^14 + 10 * q^16 - 11 * q^17 + 10 * q^19 - 14 * q^22 - 4 * q^23 - 13 * q^26 + 3 * q^28 - 5 * q^29 - 3 * q^31 - 23 * q^32 - 3 * q^34 + 5 * q^37 - 31 * q^38 + 8 * q^41 + q^43 + 21 * q^44 - 13 * q^46 - q^47 + 21 * q^49 + 40 * q^52 + q^53 + 38 * q^56 + 40 * q^58 + 21 * q^59 + 26 * q^61 + 28 * q^62 + 45 * q^64 + 21 * q^67 + 17 * q^71 + 25 * q^73 - 2 * q^74 + 6 * q^76 - 16 * q^77 + 10 * q^79 - 28 * q^82 + 3 * q^83 + 45 * q^86 - 9 * q^88 - 33 * q^89 + 9 * q^91 + 18 * q^92 + 18 * q^94 + 16 * q^97 - 35 * q^98

Basis of coefficient ring in terms of a root $\nu$ of $x^{5} - 2x^{4} - 6x^{3} + 4x^{2} + 7x - 1$ x^5 - 2*x^4 - 6*x^3 + 4*x^2 + 7*x - 1 :

$\beta_{1}$	$=$	$\nu^{3} - 3\nu^{2} - \nu + 4$ v^3 - 3*v^2 - v + 4
$\beta_{2}$	$=$	$\nu^{4} - 3\nu^{3} - 2\nu^{2} + 5\nu - 1$ v^4 - 3v^3 - 2v^2 + 5*v - 1
$\beta_{3}$	$=$	$\nu^{4} - 3\nu^{3} - 3\nu^{2} + 6\nu + 2$ v^4 - 3v^3 - 3v^2 + 6*v + 2
$\beta_{4}$	$=$	$\nu^{4} - 2\nu^{3} - 5\nu^{2} + 2\nu + 3$ v^4 - 2v^3 - 5v^2 + 2*v + 3

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

$\nu$	$=$	$( -\beta_{4} + \beta_{2} + \beta_1 ) / 2$ (-b4 + b2 + b1) / 2
$\nu^{2}$	$=$	$( -\beta_{4} - 2\beta_{3} + 3\beta_{2} + \beta _1 + 6 ) / 2$ (-b4 - 2b3 + 3b2 + b1 + 6) / 2
$\nu^{3}$	$=$	$-2\beta_{4} - 3\beta_{3} + 5\beta_{2} + 3\beta _1 + 5$ -2b4 - 3b3 + 5b2 + 3b1 + 5
$\nu^{4}$	$=$	$( -9\beta_{4} - 22\beta_{3} + 33\beta_{2} + 15\beta _1 + 44 ) / 2$ (-9b4 - 22b3 + 33b2 + 15b1 + 44) / 2

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

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}

1.1

0.134674
−1.50450
1.27987
3.26942
−1.17947

−2.74664

5.54401

−2.38589

−9.73410

1.2

−1.52233

0.317493

4.77701

2.56133

1.3

−1.15875

−0.657292

−4.20154

3.07914

1.4

1.03520

−0.928359

−1.11914

−3.03144

1.5

2.39252

3.72415

2.92955

4.12506

Atkin-Lehner signs

$p$	Sign
$3$	$-1$
$5$	$-1$
$37$	$-1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	8325.2.a.ca		5
3.b	odd	2	1		2775.2.a.bc	yes	5
5.b	even	2	1		8325.2.a.cg		5
15.d	odd	2	1		2775.2.a.z	✓	5

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2775.2.a.z	✓	5	15.d	odd	2	1
2775.2.a.bc	yes	5	3.b	odd	2	1
8325.2.a.ca		5	1.a	even	1	1	trivial
8325.2.a.cg		5	5.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(8325))$ :

T_{2}^{5} + 2T_{2}^{4} - 7T_{2}^{3} - 13T_{2}^{2} + 6T_{2} + 12

T_{7}^{5} - 28T_{7}^{3} - 15T_{7}^{2} + 157T_{7} + 157

T_{11}^{5} - T_{11}^{4} - 18T_{11}^{3} + 31T_{11}^{2} + 18T_{11} - 12

T_{13}^{5} - 14T_{13}^{4} + 64T_{13}^{3} - 91T_{13}^{2} - 63T_{13} + 167

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{5} + 2 T^{4} + \cdots + 12$ T^5 + 2T^4 - 7T^3 - 13T^2 + 6T + 12
$3$	$T^{5}$ T^5
$5$	$T^{5}$ T^5
$7$	$T^{5} - 28 T^{3} + \cdots + 157$ T^5 - 28T^3 - 15T^2 + 157*T + 157
$11$	$T^{5} - T^{4} + \cdots - 12$ T^5 - T^4 - 18T^3 + 31T^2 + 18*T - 12
$13$	$T^{5} - 14 T^{4} + \cdots + 167$ T^5 - 14T^4 + 64T^3 - 91T^2 - 63T + 167
$17$	$T^{5} + 11 T^{4} + \cdots - 48$ T^5 + 11T^4 + 34T^3 + 19T^2 - 48T - 48
$19$	$T^{5} - 10 T^{4} + \cdots - 349$ T^5 - 10T^4 + 6T^3 + 117T^2 - 105T - 349
$23$	$T^{5} + 4 T^{4} + \cdots + 516$ T^5 + 4T^4 - 51T^3 - 109T^2 + 294T + 516
$29$	$T^{5} + 5 T^{4} + \cdots - 228$ T^5 + 5T^4 - 45T^3 - 317T^2 - 504T - 228
$31$	$T^{5} + 3 T^{4} + \cdots + 2395$ T^5 + 3T^4 - 82T^3 - 366T^2 + 481T + 2395
$37$	$(T - 1)^{5}$ (T - 1)^5
$41$	$T^{5} - 8 T^{4} + \cdots - 60$ T^5 - 8T^4 - 57T^3 + 107T^2 + 366T - 60
$43$	$T^{5} - T^{4} + \cdots + 1161$ T^5 - T^4 - 95T^3 - 66T^2 + 1512*T + 1161
$47$	$T^{5} + T^{4} + \cdots - 48$ T^5 + T^4 - 88T^3 + 157T^2 + 408*T - 48
$53$	$T^{5} - T^{4} + \cdots + 48$ T^5 - T^4 - 88T^3 - 157T^2 + 408*T + 48
$59$	$T^{5} - 21 T^{4} + \cdots - 108$ T^5 - 21T^4 + 146T^3 - 395T^2 + 378T - 108
$61$	$T^{5} - 26 T^{4} + \cdots + 30839$ T^5 - 26T^4 + 112T^3 + 1805T^2 - 16311T + 30839
$67$	$T^{5} - 21 T^{4} + \cdots + 911$ T^5 - 21T^4 + 141T^3 - 252T^2 - 476T + 911
$71$	$T^{5} - 17 T^{4} + \cdots - 684$ T^5 - 17T^4 - 26T^3 + 1151T^2 - 2706T - 684
$73$	$T^{5} - 25 T^{4} + \cdots + 4$ T^5 - 25T^4 + 198T^3 - 493T^2 - 80T + 4
$79$	$T^{5} - 10 T^{4} + \cdots - 2384$ T^5 - 10T^4 - 35T^3 + 469T^2 - 352T - 2384
$83$	$T^{5} - 3 T^{4} + \cdots - 17196$ T^5 - 3T^4 - 166T^3 + 617T^2 + 5058T - 17196
$89$	$T^{5} + 33 T^{4} + \cdots - 7284$ T^5 + 33T^4 + 344T^3 + 1093T^2 - 1122T - 7284
$97$	$T^{5} - 16 T^{4} + \cdots + 5464$ T^5 - 16T^4 - 51T^3 + 1442T^2 - 5624T + 5464
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