LMFDB - Newform orbit 725.2.a.k

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [725,2,Mod(1,725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(725, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("725.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$725 = 5^{2} \cdot 29$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	725.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:	$5.78915414654$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.240881.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{5} - 2x^{4} - 5x^{3} + 9x^{2} + 5x - 7$ x^5 - 2x^4 - 5x^3 + 9x^2 + 5x - 7
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
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.

$f(q)$	$=$	$q + \beta_1 q^{2} + ( - \beta_{4} + 1) q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{4} - \beta_{3} + 2 \beta_1 - 1) q^{6} + ( - \beta_{3} + \beta_1 + 1) q^{7} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{8}+ \cdots + (4 \beta_{4} + 6 \beta_{3} - \beta_{2} + \cdots - 1) q^{99}+O(q^{100})$ q + b1 * q^2 + (-b4 + 1) * q^3 + (b2 + 1) * q^4 + (-b4 - b3 + 2b1 - 1) q^6 + (-b3 + b1 + 1) * q^7 + (b3 + b2 - b1 + 1) * q^8 + (-2b4 - 2b2 + b1 + 1) * q^9 + (b4 + 2b3 + b2 - b1) q^11 + (-b3 + b1 + 1) * q^12 + (b4 + b2 - 2b1 + 2) q^13 + (-b4 - b2 + 2b1 + 1) q^14 + (b4 + b3 - 2) * q^16 + (b4 + 2b3 - b1 + 2) q^17 + (-2b4 - 4b3 - b2 + 3b1 - 1) q^18 + (-2b3 + b2 + 1) q^19 + (-2b4 - 3b3 + b2 + 3b1 - 1) q^21 + (3b4 + 2b3 + 4b2 - 3b1 + 3) * q^22 + (2b4 + b3 - b2 - 2b1 + 1) * q^23 + (b4 + 2b3 - b2 - 2b1 + 3) * q^24 + (b4 + 2b3 - b2 + b1 - 4) q^26 + (-3b4 + b3 - 4b2 + 2b1 + 3) q^27 + (-b4 + b2 + 2) * q^28 + q^29 + (-b4 - b3 - 2b2 + b1 - 1) q^31 + (2b4 - b3 - 2b1 + 1) * q^32 + (3b4 + 4b3 - 4b1) q^33 + (3b4 + b3 + 3b2 - b1 + 2) * q^34 + (-2b4 - 3b3 - 2b2 + 3b1 - 4) * q^36 + (-2b2 + b1 + 2) q^37 + (-2b4 + b3 - 3b2 + 3b1 - 3) q^38 + (2b4 + b3 + 2b2 - 4b1 + 1) q^39 + (b2 - 2b1 + 2) q^41 + (-5b4 - b3 - 2b2 + 4b1 + 2) q^42 + (-b4 - b3 - 3b2 + 5) q^43 + (3b4 + 3b3 + 3b2 + 2) q^44 + (3b4 + b3 - b2 - 2b1 - 3) * q^46 + (-b4 + 2b2 - 2b1 + 4) * q^47 + (3b4 + 2b3 + b2 - 2b1 - 4) q^48 + (-2b4 - b3 - 2b2 + 2b1 - 2) q^49 + (5b3 - 5b1 + 2) * q^51 + (b4 + 2b2 - 3b1 + 3) * q^52 + (-b4 - 3b3 + 2b1 - 2) * q^53 + (-2b4 - 7b3 + 5b1 + 1) q^54 + (b4 + 3b2 - b1 - 2) q^56 + (-5b3 + 2b2 + 3b1 - 1) q^57 + b1 * q^58 + (-b3 - 2b2 + 4b1 - 4) * q^59 + (b4 - 4b3 - 3b2 + b1 - 1) * q^61 + (-2b4 - 3b3 - 3b2 + b1 - 2) q^62 + (-3b4 - 7b3 - b2 + 9b1 - 4) q^63 + (-b4 - 4b2 - 2) q^64 + (7b4 + 3b3 + 4b2 - 7b1 - 1) * q^66 + (-2b4 + 3b1 + 3) * q^67 + (2b4 + 2b3 + 4b2 + 1) q^68 + (2b4 + 5b3 + 3b2 - 8b1 - 2) * q^69 + (-3b4 - 2b3 + b2 - 1) * q^71 + (-b4 + 4b3 - 3b2 - 5b1 + 1) q^72 + (b4 + 3b2 + 4) q^73 + (-2b3 - b2 + 2b1 + 1) * q^74 + (-b4 - b3 - 2b1 + 4) q^76 + (4b4 + 3b3 + 3b2 - 3b1 - 4) * q^77 + (3b4 + 4b3 - 2b1 - 6) q^78 + (2b4 - b3 - b2 - 4b1) * q^79 + (-6b4 + 4b3 - b2 - b1 + 8) * q^81 + (b3 - b2 + 2b1 - 5) q^82 + (5b4 + 2b3 + 2b2 - 3b1 + 4) * q^83 + (-2b4 - b3 - 2b2 + 2b1 + 5) q^84 + (-2b4 - 4b3 - 5b2 + 7b1 - 6) * q^86 + (-b4 + 1) * q^87 + (2b3 + b2 + 2b1 + 6) * q^88 + (-2b4 - b3 - b2 - 4) q^89 + (3b4 + b3 + 2b2 - 5b1 + 3) q^91 + (b2 - 3b1 - 4) q^92 + (-3b4 - b3 - b2 + 2b1) * q^93 + (-b4 + b3 + 5b1 - 5) q^94 + (3b4 + 5b2 - 5b1 - 4) q^96 + (-b4 - 2b3 + 6b2 + 6) * q^97 + (-3b4 - 4b3 - 2b2 + b1) q^98 + (4b4 + 6b3 - b2 - 12b1 - 1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$5 q + 2 q^{2} + 6 q^{3} + 4 q^{4} - q^{6} + 6 q^{7} + 3 q^{8} + 11 q^{9} - 2 q^{11} + 6 q^{12} + 4 q^{13} + 11 q^{14} - 10 q^{16} + 9 q^{17} + 2 q^{19} - q^{21} + 4 q^{22} + q^{23} + 13 q^{24} - 16 q^{26}+ \cdots - 26 q^{99}+O(q^{100})$ 5 * q + 2 * q^2 + 6 * q^3 + 4 * q^4 - q^6 + 6 * q^7 + 3 * q^8 + 11 * q^9 - 2 * q^11 + 6 * q^12 + 4 * q^13 + 11 * q^14 - 10 * q^16 + 9 * q^17 + 2 * q^19 - q^21 + 4 * q^22 + q^23 + 13 * q^24 - 16 * q^26 + 27 * q^27 + 10 * q^28 + 5 * q^29 - q^31 - 2 * q^32 - 7 * q^33 + 3 * q^34 - 13 * q^36 + 14 * q^37 - 3 * q^38 - 6 * q^39 + 5 * q^41 + 24 * q^42 + 28 * q^43 + 7 * q^44 - 20 * q^46 + 15 * q^47 - 26 * q^48 - 3 * q^49 + 5 * q^51 + 6 * q^52 - 8 * q^53 + 10 * q^54 - 16 * q^56 - 6 * q^57 + 2 * q^58 - 11 * q^59 - 5 * q^61 - 6 * q^62 - 5 * q^63 - 5 * q^64 - 27 * q^66 + 23 * q^67 + q^68 - 26 * q^69 - 5 * q^71 + 3 * q^72 + 16 * q^73 + 8 * q^74 + 16 * q^76 - 30 * q^77 - 33 * q^78 - 10 * q^79 + 49 * q^81 - 19 * q^82 + 9 * q^83 + 32 * q^84 - 13 * q^86 + 6 * q^87 + 35 * q^88 - 18 * q^89 + q^91 - 27 * q^92 + 7 * q^93 - 13 * q^94 - 38 * q^96 + 23 * q^97 + 3 * q^98 - 26 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{5} - 2x^{4} - 5x^{3} + 9x^{2} + 5x - 7$ x^5 - 2*x^4 - 5*x^3 + 9*x^2 + 5*x - 7 :

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

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{2} + 3$ b2 + 3
$\nu^{3}$	$=$	$\beta_{3} + \beta_{2} + 3\beta _1 + 1$ b3 + b2 + 3*b1 + 1
$\nu^{4}$	$=$	$\beta_{4} + \beta_{3} + 6\beta_{2} + 12$ b4 + b3 + 6*b2 + 12

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

−1.88481
−1.06634
0.838718
1.78154
2.33090

−1.88481

1.10085

1.55251

−2.07489

1.70908

0.843434

−1.78814

1.2

−1.06634

3.37897

−0.862915

−3.60314

−2.91576

3.05285

8.41742

1.3

0.838718

−1.90376

−1.29655

−1.59672

2.46832

−2.76488

0.624302

1.4

1.78154

3.10566

1.17387

5.53284

3.64565

−1.47177

6.64510

1.5

2.33090

0.318289

3.43308

0.741899

1.09271

3.34037

−2.89869

Atkin-Lehner signs

$p$	Sign
$5$	$+1$
$29$	$-1$

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	725.2.a.k	yes	5
3.b	odd	2	1		6525.2.a.bm		5
5.b	even	2	1		725.2.a.h	✓	5
5.c	odd	4	2		725.2.b.f		10
15.d	odd	2	1		6525.2.a.bq		5

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
725.2.a.h	✓	5	5.b	even	2	1
725.2.a.k	yes	5	1.a	even	1	1	trivial
725.2.b.f		10	5.c	odd	4	2
6525.2.a.bm		5	3.b	odd	2	1
6525.2.a.bq		5	15.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(725))$ :

T_{2}^{5} - 2T_{2}^{4} - 5T_{2}^{3} + 9T_{2}^{2} + 5T_{2} - 7

T_{3}^{5} - 6T_{3}^{4} + 5T_{3}^{3} + 21T_{3}^{2} - 29T_{3} + 7

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{5} - 2 T^{4} + \cdots - 7$ T^5 - 2T^4 - 5T^3 + 9T^2 + 5T - 7
$3$	$T^{5} - 6 T^{4} + \cdots + 7$ T^5 - 6T^4 + 5T^3 + 21T^2 - 29T + 7
$5$	$T^{5}$ T^5
$7$	$T^{5} - 6 T^{4} + \cdots + 49$ T^5 - 6T^4 + 2T^3 + 45T^2 - 90T + 49
$11$	$T^{5} + 2 T^{4} + \cdots + 307$ T^5 + 2T^4 - 35T^3 - 64T^2 + 190T + 307
$13$	$T^{5} - 4 T^{4} + \cdots - 1$ T^5 - 4T^4 - 18T^3 + 27T^2 - 6T - 1
$17$	$T^{5} - 9 T^{4} + \cdots - 439$ T^5 - 9T^4 - 3T^3 + 155T^2 - 118T - 439
$19$	$T^{5} - 2 T^{4} + \cdots + 5$ T^5 - 2T^4 - 47T^3 + 81T^2 + 137T + 5
$23$	$T^{5} - T^{4} + \cdots - 581$ T^5 - T^4 - 62T^3 + 111T^2 + 375*T - 581
$29$	$(T - 1)^{5}$ (T - 1)^5
$31$	$T^{5} + T^{4} + \cdots - 73$ T^5 + T^4 - 30T^3 + 43T^2 + 49*T - 73
$37$	$T^{5} - 14 T^{4} + \cdots - 63$ T^5 - 14T^4 + 53T^3 - 11T^2 - 141T - 63
$41$	$T^{5} - 5 T^{4} + \cdots + 9$ T^5 - 5T^4 - 13T^3 + 22T^2 + 45T + 9
$43$	$T^{5} - 28 T^{4} + \cdots + 22833$ T^5 - 28T^4 + 235T^3 - 68T^2 - 7104T + 22833
$47$	$T^{5} - 15 T^{4} + \cdots + 2263$ T^5 - 15T^4 + 37T^3 + 316T^2 - 1713T + 2263
$53$	$T^{5} + 8 T^{4} + \cdots + 863$ T^5 + 8T^4 - 55T^3 - 295T^2 + 365T + 863
$59$	$T^{5} + 11 T^{4} + \cdots + 2205$ T^5 + 11T^4 - 36T^3 - 413T^2 + 669T + 2205
$61$	$T^{5} + 5 T^{4} + \cdots - 16381$ T^5 + 5T^4 - 233T^3 - 547T^2 + 11628T - 16381
$67$	$T^{5} - 23 T^{4} + \cdots - 353$ T^5 - 23T^4 + 135T^3 + 50T^2 - 1209T - 353
$71$	$T^{5} + 5 T^{4} + \cdots + 2837$ T^5 + 5T^4 - 98T^3 - 241T^2 + 1857T + 2837
$73$	$T^{5} - 16 T^{4} + \cdots + 9$ T^5 - 16T^4 + 30T^3 + 271T^2 - 642T + 9
$79$	$T^{5} + 10 T^{4} + \cdots + 10035$ T^5 + 10T^4 - 146T^3 - 1045T^2 + 5586T + 10035
$83$	$T^{5} - 9 T^{4} + \cdots - 18113$ T^5 - 9T^4 - 163T^3 + 1791T^2 - 804T - 18113
$89$	$T^{5} + 18 T^{4} + \cdots - 25$ T^5 + 18T^4 + 86T^3 + 51T^2 - 102T - 25
$97$	$T^{5} - 23 T^{4} + \cdots - 196403$ T^5 - 23T^4 - 101T^3 + 4168T^2 + 271T - 196403
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