LMFDB - Newform orbit 7225.2.a.bd

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7225,2,Mod(1,7225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7225, 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("7225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$7225 = 5^{2} \cdot 17^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	7225.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:	$57.6919154604$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.199789929.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - x^{5} - 9x^{4} + 5x^{3} + 21x^{2} - 3x - 3$ x^6 - x^5 - 9x^4 + 5x^3 + 21x^2 - 3x - 3
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 $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} + \beta_1 + 1) q^{4} + ( - \beta_{4} - \beta_{3} - \beta_{2} - 1) q^{6} + (\beta_{4} + 1) q^{7} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{8} + ( - \beta_{5} + \beta_{3} + \beta_{2} + 1) q^{9}+ \cdots + (3 \beta_{5} + 2 \beta_{4} - \beta_{3} + \cdots - 7) q^{99}+O(q^{100})$ q - b1 * q^2 + b3 * q^3 + (b2 + b1 + 1) * q^4 + (-b4 - b3 - b2 - 1) * q^6 + (b4 + 1) * q^7 + (-b3 - b2 - b1 - 1) * q^8 + (-b5 + b3 + b2 + 1) * q^9 + (b5 + b4 + b3 - 1) * q^11 + (b5 + b4 + 2b3 + b2 + 2b1 - 1) * q^12 + (b5 + b1) * q^13 + (-b5 - 2b3 - b1) q^14 + (b4 + 2b3 + b1) q^16 + (-b5 - 2b3 - b2 - 2b1) * q^18 + (-b2 - 1) * q^19 + (b5 + b3 + 3b1 - 2) q^21 + (-2b4 - 3b3 - b2 + b1) * q^22 + (-b5 - b4 + 2b2 + 2b1 + 2) * q^23 + (-b4 - 3b3 - 2b2 - 2b1 - 3) q^24 + (b5 - b4 - b2 - b1 - 2) * q^26 + (-b5 - b4 + b3 + b1 + 2) * q^27 + (-b5 + b4 + 2b3 + 3b2 + b1 + 2) * q^28 + (-b4 + b2 + b1 + 3) * q^29 + (b4 - 2b3 - 2b2 - b1 + 2) * q^31 + (-b5 - 2b4 - 2b3 - b2 + b1 - 3) * q^32 + (b5 + b4 - 2b3 + 2b2 + 4b1 + 2) q^33 + (b5 + 3b4 + b3 + 2b2 + 3b1 + 3) q^36 + (-2b2 - b1 + 1) q^37 + (b3 + 2b1 - 2) q^38 + (b5 + 2b4 - b3 + 2b2 + b1 + 1) * q^39 + (2b3 + 2b2 - 3b1 + 2) q^41 + (b5 - 2b4 - b3 - 4b2 - b1 - 9) * q^42 + (-b5 - b4 - b3 + b2 - b1) * q^43 + (b4 + 6b3 + 2b2) * q^44 + (b4 - 2b2 - 6b1 - 3) * q^46 + (b5 - b4 - b3 - 2b2 + 3) q^47 + (-b5 + b4 + 3b3 + 3b2 + 3b1 + 7) q^48 + (2b4 + b2 - 2b1 + 3) * q^49 + (-b4 + 3b3 + b2 + 2b1 + 2) * q^52 + (3b4 + 3b3 - b2 - b1) * q^53 + (b3 - 2b2 - 3b1 - 5) * q^54 + (-b4 - 3b3 - 3b2 - 4b1) q^56 + (-b5 - b3 - 2b1 + 2) q^57 + (b5 + b3 - b2 - 5b1 - 1) q^58 + (2b4 + 2b3 - 3b2 - 5b1 + 2) * q^59 + (2b3 - 3b1 + 3) * q^61 + (-b5 + 2b4 + 2b3 + 3b2 + b1 + 1) q^62 + (b4 + 5b2 + b1 + 4) q^63 + (b5 + b4 + 3b3 + b2 + b1 - 4) q^64 + (b4 - 2b3 - 2b2 - 8b1 - 5) q^66 + (4b3 + 2b2 - 3b1 + 3) q^67 + (b4 + 6b3 + b2) q^69 + (b5 - 2b4 - 2b3 + 2b1 + 2) q^71 + (-2b4 - 5b3 - 2b2 - 4b1 - 5) * q^72 + (-b4 + b3 + 2b2 + b1 + 5) q^73 + (2b3 + b2 + 2b1 - 1) * q^74 + (-b4 - b3 - b2 - 5) * q^76 + (2b3 - b2 + 3b1 + 2) * q^77 + (-b5 - 5b3 - 4b1 + 3) * q^78 + (-b5 - 2b4 - b3 + 2) q^79 + (3b3 - 2b2 - 4b1 + 4) q^81 + (-2b4 - 4b3 + b2 - b1 + 11) * q^82 + (b4 + 3b3 + 3b2 - 3b1) q^83 + (b5 + 7b3 + 2b2 + 8b1 + 1) q^84 + (2b4 + 2b3 + 2b2 + 5) q^86 + (b4 + 4b3 + b2 - b1 + 1) q^87 + (-b5 - 2b4 - 4b3 - 4b2 - 4b1 - 2) * q^88 + (2b4 - 2b3 + b2 + 3b1 + 4) q^89 + (3b3 - 2b2 + 3b1 - 4) q^91 + (b5 + 2b4 + 2b2 + 7b1 + 10) q^92 + (b5 - b4 - b3 - 3b2 - b1 - 7) q^93 + (2b5 + 5b3 + b2 - b1 - 2) * q^94 + (-2b5 - 2b3 - 2b2 - 9b1 - 1) * q^96 + (-3b5 - b4 - 3b3 + 3b2 + 2b1 - 2) * q^97 + (-2b5 - 5b3 + 2b2 - 2b1 + 8) * q^98 + (3b5 + 2b4 - b3 + 3b2 + 8b1 - 7) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$6 q - q^{2} + 2 q^{3} + 7 q^{4} - 8 q^{6} + 6 q^{7} - 9 q^{8} + 10 q^{9} - 6 q^{11} - 2 q^{12} - q^{13} - 3 q^{14} + 5 q^{16} - 4 q^{18} - 6 q^{19} - 9 q^{21} - 5 q^{22} + 16 q^{23} - 26 q^{24} - 15 q^{26}+ \cdots - 42 q^{99}+O(q^{100})$ 6 * q - q^2 + 2 * q^3 + 7 * q^4 - 8 * q^6 + 6 * q^7 - 9 * q^8 + 10 * q^9 - 6 * q^11 - 2 * q^12 - q^13 - 3 * q^14 + 5 * q^16 - 4 * q^18 - 6 * q^19 - 9 * q^21 - 5 * q^22 + 16 * q^23 - 26 * q^24 - 15 * q^26 + 17 * q^27 + 19 * q^28 + 19 * q^29 + 7 * q^31 - 19 * q^32 + 10 * q^33 + 21 * q^36 + 5 * q^37 - 8 * q^38 + 3 * q^39 + 13 * q^41 - 59 * q^42 - q^43 + 12 * q^44 - 24 * q^46 + 14 * q^47 + 53 * q^48 + 16 * q^49 + 20 * q^52 + 5 * q^53 - 31 * q^54 - 10 * q^56 + 10 * q^57 - 11 * q^58 + 11 * q^59 + 19 * q^61 + 13 * q^62 + 25 * q^63 - 19 * q^64 - 42 * q^66 + 23 * q^67 + 12 * q^69 + 8 * q^71 - 44 * q^72 + 33 * q^73 - 32 * q^76 + 19 * q^77 + 6 * q^78 + 12 * q^79 + 26 * q^81 + 57 * q^82 + 3 * q^83 + 26 * q^84 + 34 * q^86 + 13 * q^87 - 22 * q^88 + 23 * q^89 - 15 * q^91 + 65 * q^92 - 47 * q^93 - 7 * q^94 - 15 * q^96 - 10 * q^97 + 40 * q^98 - 42 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{6} - x^{5} - 9x^{4} + 5x^{3} + 21x^{2} - 3x - 3$ x^6 - x^5 - 9*x^4 + 5*x^3 + 21*x^2 - 3*x - 3 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$\nu^{2} - \nu - 3$ v^2 - v - 3
$\beta_{3}$	$=$	$\nu^{3} - \nu^{2} - 4\nu + 2$ v^3 - v^2 - 4*v + 2
$\beta_{4}$	$=$	$\nu^{4} - 2\nu^{3} - 4\nu^{2} + 7\nu$ v^4 - 2v^3 - 4v^2 + 7*v
$\beta_{5}$	$=$	$\nu^{5} - 2\nu^{4} - 6\nu^{3} + 9\nu^{2} + 8\nu - 4$ v^5 - 2v^4 - 6v^3 + 9v^2 + 8v - 4

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{2} + \beta _1 + 3$ b2 + b1 + 3
$\nu^{3}$	$=$	$\beta_{3} + \beta_{2} + 5\beta _1 + 1$ b3 + b2 + 5*b1 + 1
$\nu^{4}$	$=$	$\beta_{4} + 2\beta_{3} + 6\beta_{2} + 7\beta _1 + 14$ b4 + 2b3 + 6b2 + 7*b1 + 14
$\nu^{5}$	$=$	$\beta_{5} + 2\beta_{4} + 10\beta_{3} + 9\beta_{2} + 27\beta _1 + 11$ b5 + 2b4 + 10b3 + 9b2 + 27b1 + 11

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

2.63233
2.08568
0.451353
−0.330308
−1.78030
−2.05876

−2.63233

2.78135

4.92916

−7.32143

3.24325

−7.71051

4.73590

1.2

−2.08568

−1.61993

2.35007

3.37867

−1.02312

−0.730139

−0.375818

1.3

−0.451353

0.0828196

−1.79628

−0.0373808

3.20219

1.71346

−2.99314

1.4

0.330308

3.17609

−1.89090

1.04909

−1.66459

−1.28519

7.08755

1.5

1.78030

0.309133

1.16947

0.550349

−2.80925

−1.47860

−2.90444

1.6

2.05876

−2.72946

2.23848

−5.61929

5.05151

0.490977

4.44994

Atkin-Lehner signs

$p$	Sign
$5$	$-1$
$17$	$+1$

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	7225.2.a.bd	yes	6
5.b	even	2	1		7225.2.a.be	yes	6
17.b	even	2	1		7225.2.a.bc	✓	6
85.c	even	2	1		7225.2.a.bf	yes	6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
7225.2.a.bc	✓	6	17.b	even	2	1
7225.2.a.bd	yes	6	1.a	even	1	1	trivial
7225.2.a.be	yes	6	5.b	even	2	1
7225.2.a.bf	yes	6	85.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(7225))$ :

T_{2}^{6} + T_{2}^{5} - 9T_{2}^{4} - 5T_{2}^{3} + 21T_{2}^{2} + 3T_{2} - 3

T_{3}^{6} - 2T_{3}^{5} - 12T_{3}^{4} + 17T_{3}^{3} + 34T_{3}^{2} - 15T_{3} + 1

T_{7}^{6} - 6T_{7}^{5} - 11T_{7}^{4} + 82T_{7}^{3} + 54T_{7}^{2} - 280T_{7} - 251

T_{11}^{6} + 6T_{11}^{5} - 32T_{11}^{4} - 161T_{11}^{3} + 348T_{11}^{2} + 747T_{11} - 1317

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{6} + T^{5} - 9 T^{4} + \cdots - 3$ T^6 + T^5 - 9T^4 - 5T^3 + 21T^2 + 3T - 3
$3$	$T^{6} - 2 T^{5} + \cdots + 1$ T^6 - 2T^5 - 12T^4 + 17T^3 + 34T^2 - 15*T + 1
$5$	$T^{6}$ T^6
$7$	$T^{6} - 6 T^{5} + \cdots - 251$ T^6 - 6T^5 - 11T^4 + 82T^3 + 54T^2 - 280*T - 251
$11$	$T^{6} + 6 T^{5} + \cdots - 1317$ T^6 + 6T^5 - 32T^4 - 161T^3 + 348T^2 + 747*T - 1317
$13$	$T^{6} + T^{5} + \cdots + 529$ T^6 + T^5 - 44T^4 + 23T^3 + 490T^2 - 992T + 529
$17$	$T^{6}$ T^6
$19$	$T^{6} + 6 T^{5} + \cdots + 27$ T^6 + 6T^5 - 2T^4 - 50T^3 - 21T^2 + 100*T + 27
$23$	$T^{6} - 16 T^{5} + \cdots + 81$ T^6 - 16T^5 + 35T^4 + 582T^3 - 3456T^2 + 5292*T + 81
$29$	$T^{6} - 19 T^{5} + \cdots - 216$ T^6 - 19T^5 + 119T^4 - 213T^3 - 396T^2 + 1236*T - 216
$31$	$T^{6} - 7 T^{5} + \cdots - 8523$ T^6 - 7T^5 - 87T^4 + 389T^3 + 2173T^2 - 3279*T - 8523
$37$	$T^{6} - 5 T^{5} + \cdots - 293$ T^6 - 5T^5 - 49T^4 + 121T^3 + 809T^2 + 433*T - 293
$41$	$T^{6} - 13 T^{5} + \cdots + 53691$ T^6 - 13T^5 - 97T^4 + 1441T^3 + 717T^2 - 31341*T + 53691
$43$	$T^{6} + T^{5} + \cdots + 1657$ T^6 + T^5 - 75T^4 + 11T^3 + 1267T^2 - 2829T + 1657
$47$	$T^{6} - 14 T^{5} + \cdots + 35829$ T^6 - 14T^5 - 40T^4 + 1137T^3 - 1854T^2 - 17355*T + 35829
$53$	$T^{6} - 5 T^{5} + \cdots + 170109$ T^6 - 5T^5 - 242T^4 + 1477T^3 + 13044T^2 - 103896*T + 170109
$59$	$T^{6} - 11 T^{5} + \cdots - 115377$ T^6 - 11T^5 - 233T^4 + 2275T^3 + 9741T^2 - 50421*T - 115377
$61$	$T^{6} - 19 T^{5} + \cdots - 2699$ T^6 - 19T^5 + 57T^4 + 667T^3 - 4013T^2 + 6417*T - 2699
$67$	$T^{6} - 23 T^{5} + \cdots + 338167$ T^6 - 23T^5 - 17T^4 + 3347T^3 - 16319T^2 - 57305*T + 338167
$71$	$T^{6} - 8 T^{5} + \cdots + 11691$ T^6 - 8T^5 - 159T^4 + 474T^3 + 8418T^2 + 20574*T + 11691
$73$	$T^{6} - 33 T^{5} + \cdots - 34184$ T^6 - 33T^5 + 382T^4 - 1637T^3 - 456T^2 + 19844*T - 34184
$79$	$T^{6} - 12 T^{5} + \cdots - 2393$ T^6 - 12T^5 - 38T^4 + 580T^3 + 573T^2 - 4732*T - 2393
$83$	$T^{6} - 3 T^{5} + \cdots + 14961$ T^6 - 3T^5 - 278T^4 + 1411T^3 + 14940T^2 - 89184*T + 14961
$89$	$T^{6} - 23 T^{5} + \cdots + 843021$ T^6 - 23T^5 - 67T^4 + 4263T^3 - 14085T^2 - 171009*T + 843021
$97$	$T^{6} + 10 T^{5} + \cdots + 137939$ T^6 + 10T^5 - 387T^4 - 3492T^3 + 30728T^2 + 200880*T + 137939
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