LMFDB - Newform orbit 8281.2.a.ci

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8281,2,Mod(1,8281)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8281, 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("8281.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$8281 = 7^{2} \cdot 13^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	8281.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:	$66.1241179138$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.8446345216.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - 8x^{6} + 19x^{4} - 14x^{2} + 1$ x^8 - 8x^6 + 19x^4 - 14*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 637)
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 $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - \beta_{5} - 1) q^{2} + ( - \beta_{4} - \beta_1) q^{3} + ( - \beta_{6} - \beta_{5} - \beta_{2} + 1) q^{4} + ( - \beta_{7} + \beta_{3}) q^{5} + (\beta_{4} - \beta_{3} + 2 \beta_1) q^{6} + (\beta_{6} - \beta_{5} + 1) q^{8}+ \cdots + ( - \beta_{6} - \beta_{5} - \beta_{2} + 3) q^{99}+O(q^{100})$ q + (-b5 - 1) * q^2 + (-b4 - b1) * q^3 + (-b6 - b5 - b2 + 1) * q^4 + (-b7 + b3) * q^5 + (b4 - b3 + 2b1) q^6 + (b6 - b5 + 1) * q^8 + (b5 + 1) * q^9 + (2b7 - 2b3 + b1) * q^10 + (b5 + 1) * q^11 + (-2b7 - b4 + 2b3) * q^12 + (-2b6 - b2 + 1) q^15 + (-b6 - b5 + b2) * q^16 + (2b4 - b3 + b1) q^17 + (b6 + b5 + b2 - 3) * q^18 + (2b7 + 2b4 - b3) * q^19 + (-3b7 - b4 + 2b3 - 2b1) q^20 + (b6 + b5 + b2 - 3) * q^22 + (2b6 + b5 + b2 - 1) q^23 + (3b7 - 2b4 - 3b3) q^24 + (-2b6 + b5 - 1) q^25 + (2b4 + b3 + b1) q^27 + (2b6 - 2b5 - 3b2 - 2) q^29 + (4b6 - b5 + b2 - 4) q^30 + (3b7 - b4) q^31 + (-b6 - 2b2) q^32 + (-b4 + b3 - 2b1) q^33 + (-2b4 + 3b3 - 4b1) q^34 + (-b6 + 3b5 + 1) q^36 + (b6 + 4b5 + 3b2 + 3) * q^37 + (-b7 + b4 + 5b3 - 6b1) * q^38 + (2b7 + b4 - 2b3 + 3b1) q^40 + (-2b7 - 2b4 - 2b3 + 2b1) * q^41 + (2b5 + 5b2 - 3) * q^43 + (-b6 + 3b5 + 1) q^44 + (-2b7 + 2b3 - b1) * q^45 + (-3b6 + 3b5 + 2) * q^46 + (2b7 + b4 - 2b3 - 4b1) q^47 + (-4b7 + 2b4 + b1) * q^48 + (5b6 + 3b5 + b2 - 3) * q^50 + (b6 - b5 + 3b2 - 6) q^51 + (b6 - b5 - 4b2 - 1) q^53 + (2b7 + b3 - 4b1) * q^54 + (-2b7 + 2b3 - b1) * q^55 + (3b6 + 2b2 - 6) * q^57 + (-6b6 - 2b5 + b2 + 5) * q^58 + (2b7 - 4b4 + b3 - 3b1) q^59 + (-5b6 + 2b5 + 9) * q^60 + (-2b7 + b4 + 3b3 + b1) * q^61 + (-4b7 + 3b4 + 2b3 - b1) q^62 + (4b6 + 2b5 - 3) * q^64 + (2b7 + 3b4 - 2b3 + 2b1) * q^66 + (-2b6 + b5 + b2 - 2) q^67 + (5b7 + 3b4 - 3b3 + 2b1) * q^68 + (5b7 - 4b3) * q^69 + (b6 - 4b5 - 5b2 - 3) * q^71 + (3b6 + 3b5 + b2 - 2) * q^72 + (4b7 - 7b4 - 3b3 - b1) q^73 + (2b6 + 5b5 + b2 - 7) * q^74 + (-6b7 + 3b4 + 5b3) q^75 + (9b7 + 6b4 - 3b3 - b1) q^76 + (-b5 + 7b2 - 1) q^79 + (-b4 + b3) * q^80 + (-b6 - 4b5 - b2 - 9) q^81 + (-4b7 - 6b4 - 2b3 + 6b1) * q^82 + (-3b7 + 2b4 - 3b3 - b1) q^83 + (4b6 - b5 - 5) q^85 + (2b6 + 7b5 - 3b2 + 4) q^86 + (9b7 - 3b4 - 3b3 + 4b1) * q^87 + (3b6 + 3b5 + b2 - 2) * q^88 + (-10b7 + 2b4 + 3b3 + 4b1) * q^89 + (5b7 + b4 - 4b3 + 2b1) q^90 + (5b6 + 2b5 + b2 - 9) * q^92 + (3b6 - 4b2 - 1) * q^93 + (b7 + 4b4 + 5b3 - 4b1) q^94 + (5b6 - b5 + b2 - 7) q^95 + (-b7 - b4 + 4b3) q^96 + (-b4 + b3 + 6b1) q^97 + (-b6 - b5 - b2 + 3) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 4 q^{2} + 12 q^{4} + 12 q^{8} + 4 q^{9} + 4 q^{11} + 8 q^{15} + 4 q^{16} - 28 q^{18} - 28 q^{22} - 12 q^{23} - 12 q^{25} - 8 q^{29} - 28 q^{30} - 4 q^{36} + 8 q^{37} - 32 q^{43} - 4 q^{44} + 4 q^{46}+ \cdots + 28 q^{99}+O(q^{100})$ 8 * q - 4 * q^2 + 12 * q^4 + 12 * q^8 + 4 * q^9 + 4 * q^11 + 8 * q^15 + 4 * q^16 - 28 * q^18 - 28 * q^22 - 12 * q^23 - 12 * q^25 - 8 * q^29 - 28 * q^30 - 4 * q^36 + 8 * q^37 - 32 * q^43 - 4 * q^44 + 4 * q^46 - 36 * q^50 - 44 * q^51 - 4 * q^53 - 48 * q^57 + 48 * q^58 + 64 * q^60 - 32 * q^64 - 20 * q^67 - 8 * q^71 - 28 * q^72 - 76 * q^74 - 4 * q^79 - 56 * q^81 - 36 * q^85 + 4 * q^86 - 28 * q^88 - 80 * q^92 - 8 * q^93 - 52 * q^95 + 28 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{8} - 8x^{6} + 19x^{4} - 14x^{2} + 1$ x^8 - 8*x^6 + 19*x^4 - 14*x^2 + 1 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$\nu^{2} - 2$ v^2 - 2
$\beta_{3}$	$=$	$\nu^{3} - 3\nu$ v^3 - 3*v
$\beta_{4}$	$=$	$\nu^{5} - 6\nu^{3} + 7\nu$ v^5 - 6v^3 + 7v
$\beta_{5}$	$=$	$\nu^{6} - 6\nu^{4} + 7\nu^{2}$ v^6 - 6v^4 + 7v^2
$\beta_{6}$	$=$	$-\nu^{6} + 7\nu^{4} - 12\nu^{2} + 3$ -v^6 + 7v^4 - 12v^2 + 3
$\beta_{7}$	$=$	$\nu^{7} - 7\nu^{5} + 13\nu^{3} - 6\nu$ v^7 - 7v^5 + 13v^3 - 6*v

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{2} + 2$ b2 + 2
$\nu^{3}$	$=$	$\beta_{3} + 3\beta_1$ b3 + 3*b1
$\nu^{4}$	$=$	$\beta_{6} + \beta_{5} + 5\beta_{2} + 7$ b6 + b5 + 5*b2 + 7
$\nu^{5}$	$=$	$\beta_{4} + 6\beta_{3} + 11\beta_1$ b4 + 6b3 + 11b1
$\nu^{6}$	$=$	$6\beta_{6} + 7\beta_{5} + 23\beta_{2} + 28$ 6b6 + 7b5 + 23*b2 + 28
$\nu^{7}$	$=$	$\beta_{7} + 7\beta_{4} + 29\beta_{3} + 44\beta_1$ b7 + 7b4 + 29b3 + 44*b1

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

1.11758
−1.11758
0.282452
−0.282452
2.09282
−2.09282
−1.51373
1.51373

−2.33152

−2.30901

3.43596

−3.37112

5.38349

−3.34797

2.33152

7.85981

1.2

−2.33152

2.30901

3.43596

3.37112

−5.38349

−3.34797

2.33152

−7.85981

1.3

−1.52077

−2.12621

0.312752

0.589391

3.23349

2.56592

1.52077

−0.896331

1.4

−1.52077

2.12621

0.312752

−0.589391

−3.23349

2.56592

1.52077

0.896331

1.5

−0.579810

−1.89204

−1.66382

1.47362

1.09702

2.12432

0.579810

−0.854419

1.6

−0.579810

1.89204

−1.66382

−1.47362

−1.09702

2.12432

0.579810

0.854419

1.7

2.43210

−0.753592

3.91511

−0.341537

−1.83281

4.65773

−2.43210

−0.830652

1.8

2.43210

0.753592

3.91511

0.341537

1.83281

4.65773

−2.43210

0.830652

Atkin-Lehner signs

$p$	Sign
$7$	$+1$
$13$	$+1$

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	8281.2.a.ci		8
7.b	odd	2	1	inner	8281.2.a.ci		8
13.b	even	2	1		8281.2.a.cl		8
13.c	even	3	2		637.2.f.l	✓	16
91.b	odd	2	1		8281.2.a.cl		8
91.g	even	3	2		637.2.g.m		16
91.h	even	3	2		637.2.h.m		16
91.m	odd	6	2		637.2.g.m		16
91.n	odd	6	2		637.2.f.l	✓	16
91.v	odd	6	2		637.2.h.m		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
637.2.f.l	✓	16	13.c	even	3	2
637.2.f.l	✓	16	91.n	odd	6	2
637.2.g.m		16	91.g	even	3	2
637.2.g.m		16	91.m	odd	6	2
637.2.h.m		16	91.h	even	3	2
637.2.h.m		16	91.v	odd	6	2
8281.2.a.ci		8	1.a	even	1	1	trivial
8281.2.a.ci		8	7.b	odd	2	1	inner
8281.2.a.cl		8	13.b	even	2	1
8281.2.a.cl		8	91.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(8281))$ :

T_{2}^{4} + 2T_{2}^{3} - 5T_{2}^{2} - 12T_{2} - 5

T_{3}^{8} - 14T_{3}^{6} + 67T_{3}^{4} - 120T_{3}^{2} + 49

T_{5}^{8} - 14T_{5}^{6} + 31T_{5}^{4} - 12T_{5}^{2} + 1

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

T_{17}^{8} - 58T_{17}^{6} + 966T_{17}^{4} - 3866T_{17}^{2} + 3721

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{4} + 2 T^{3} - 5 T^{2} + \cdots - 5)^{2}$ (T^4 + 2T^3 - 5T^2 - 12*T - 5)^2
$3$	$T^{8} - 14 T^{6} + \cdots + 49$ T^8 - 14T^6 + 67T^4 - 120*T^2 + 49
$5$	$T^{8} - 14 T^{6} + \cdots + 1$ T^8 - 14T^6 + 31T^4 - 12*T^2 + 1
$7$	$T^{8}$ T^8
$11$	$(T^{4} - 2 T^{3} - 5 T^{2} + \cdots - 5)^{2}$ (T^4 - 2T^3 - 5T^2 + 12*T - 5)^2
$13$	$T^{8}$ T^8
$17$	$T^{8} - 58 T^{6} + \cdots + 3721$ T^8 - 58T^6 + 966T^4 - 3866*T^2 + 3721
$19$	$T^{8} - 94 T^{6} + \cdots + 1225$ T^8 - 94T^6 + 2311T^4 - 6252*T^2 + 1225
$23$	$(T^{4} + 6 T^{3} + \cdots + 100)^{2}$ (T^4 + 6T^3 - 14T^2 - 56*T + 100)^2
$29$	$(T^{4} + 4 T^{3} + \cdots + 1781)^{2}$ (T^4 + 4T^3 - 85T^2 - 190*T + 1781)^2
$31$	$T^{8} - 70 T^{6} + \cdots + 26569$ T^8 - 70T^6 + 1518T^4 - 11726*T^2 + 26569
$37$	$(T^{4} - 4 T^{3} + \cdots - 380)^{2}$ (T^4 - 4T^3 - 86T^2 + 396*T - 380)^2
$41$	$T^{8} - 176 T^{6} + \cdots + 4096$ T^8 - 176T^6 + 3008T^4 - 7168*T^2 + 4096
$43$	$(T^{4} + 16 T^{3} + \cdots - 3205)^{2}$ (T^4 + 16T^3 - 15T^2 - 1044*T - 3205)^2
$47$	$T^{8} - 226 T^{6} + \cdots + 27889$ T^8 - 226T^6 + 13374T^4 - 131954*T^2 + 27889
$53$	$(T^{4} + 2 T^{3} + \cdots + 271)^{2}$ (T^4 + 2T^3 - 80T^2 - 70*T + 271)^2
$59$	$T^{8} - 194 T^{6} + \cdots + 169$ T^8 - 194T^6 + 7494T^4 - 3346*T^2 + 169
$61$	$T^{8} - 108 T^{6} + \cdots + 19600$ T^8 - 108T^6 + 3196T^4 - 18992*T^2 + 19600
$67$	$(T^{4} + 10 T^{3} + \cdots - 283)^{2}$ (T^4 + 10T^3 - 6T^2 - 222*T - 283)^2
$71$	$(T^{4} + 4 T^{3} + \cdots + 5956)^{2}$ (T^4 + 4T^3 - 166T^2 - 268*T + 5956)^2
$73$	$T^{8} - 428 T^{6} + \cdots + 2226064$ T^8 - 428T^6 + 45484T^4 - 706640*T^2 + 2226064
$79$	$(T^{4} + 2 T^{3} + \cdots + 8164)^{2}$ (T^4 + 2T^3 - 278T^2 - 712*T + 8164)^2
$83$	$T^{8} - 350 T^{6} + \cdots + 405769$ T^8 - 350T^6 + 27838T^4 - 257574*T^2 + 405769
$89$	$T^{8} - 606 T^{6} + \cdots + 45225625$ T^8 - 606T^6 + 119687T^4 - 8007708*T^2 + 45225625
$97$	$T^{8} - 364 T^{6} + \cdots + 310249$ T^8 - 364T^6 + 30631T^4 - 355846*T^2 + 310249
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