LMFDB - Newform orbit 137.2.a.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [137,2,Mod(1,137)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(137, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("137.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$137$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	137.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$1.09395050769$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.725.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{3} - 3x^{2} + x + 1$ x^4 - x^3 - 3*x^2 + x + 1
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 $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + (\beta_1 - 1) q^{2} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{3} + (\beta_{2} - \beta_1) q^{4} + ( - 2 \beta_{3} + \beta_{2}) q^{5} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{6} + ( - \beta_{3} + \beta_1 - 3) q^{7}+ \cdots + ( - 3 \beta_{3} + \beta_{2} + 8 \beta_1 + 4) q^{99}+O(q^{100})$ q + (b1 - 1) * q^2 + (b3 - b2 - b1 - 1) * q^3 + (b2 - b1) * q^4 + (-2b3 + b2) q^5 + (-2b3 + b2 - b1 + 1) q^6 + (-b3 + b1 - 3) * q^7 + (b3 - 2b2 - b1) q^8 + (2b2 + b1) q^9 + (3b3 - 3b2 - b1 - 1) * q^10 + (4b3 + b2 - b1 - 2) q^11 + (b3 - 2b2 + 2b1 - 1) * q^12 + (b2 + 2b1 - 3) q^13 + (b3 - 4b1 + 4) q^14 + (4b1 - 3) q^15 + (-3b3 + b1 + 1) q^16 + (-b3 - 3b2 + 1) q^17 + (2b3 - b2 + 2b1 - 1) * q^18 + (-2b3 - 3b2) * q^19 + (-2b3 + 3b2 - b1 + 3) * q^20 + (-4b3 + 4b2 + 3b1 + 2) q^21 + (-3b3 + 2b2 + 3b1) q^22 + (b2 - 3b1) q^23 + (b3 + 3b2 + 3) q^24 + (3b3 + b2 - 4b1 + 1) * q^25 + (b3 + b2 - 2b1 + 4) q^26 + (-4b3 - b2 + b1 + 1) q^27 + (b3 - 3b2 + 3b1 - 2) * q^28 + (-b3 - 3b2 + 3b1 + 4) * q^29 + (4b2 - 3b1 + 7) * q^30 + (5b3 - b2 - b1 - 6) q^31 + (b3 + 2b2) q^32 + (-b3 - 4b2 - 4b1 + 5) * q^33 + (-2b3 + 2b2 - 3b1 + 2) q^34 + (9b3 - 5b2 - 2b1 + 1) q^35 + (-3b3 + b2 - 2b1 + 4) * q^36 + (2b3 + 3b2 + 2b1 - 4) q^37 + (-b3 + b2 - 5b1 + 3) q^38 + (-5b3 + b2 - b1 + 2) q^39 + (-b3 + 6b1 - 5) q^40 + (-2b3 + 2b2 + 5b1 - 3) q^41 + (8b3 - 5b2 + 2b1 - 3) q^42 + (2b3 - 2b2 - 5b1 - 2) q^43 + (-3b3 - 4b2 + b1 + 5) * q^44 + (-b3 - 5b1 + 3) q^45 + (b3 - 4b2 + b1 - 4) q^46 + (-3b1 - 2) q^47 + (2b2 + 3b1 - 4) * q^48 + (7b3 - b2 - 7b1 + 4) * q^49 + (-2b3 - 2b2 + 5b1 - 6) q^50 + (b3 + 6b2 + b1 + 1) q^51 + (-4b2 + 2b1 - 1) * q^52 + (-b3 - 2b2 + 1) q^53 + (3b3 - 2b2 - 4b1 + 1) q^54 + (-6b3 + b2 + 3b1 - 5) * q^55 + (-6b3 + 7b2 + 4b1) q^56 + (8b2 + 4b1 + 1) * q^57 + (-2b3 + 5b2 + 2) * q^58 + (-7b3 + 2b2 + 4b1 + 2) q^59 + (4b3 - 7b2 + 3b1 - 8) q^60 + (-3b3 - b2 - b1 + 4) q^61 + (-6b3 + 5b2 - 2b1 + 6) q^62 + (2b3 - 6b2 - 3b1 - 1) q^63 + (7b3 - b2 + b1 - 4) q^64 + (7b3 - 6b2 - 4b1) q^65 + (-3b3 - b2 - 5) q^66 + (-b3 + 6b2 + 4b1 - 5) * q^67 + (6b3 - b2 + 2b1 - 9) * q^68 + (3b3 - 2b2 + 6b1 - 1) q^69 + (-14b3 + 12b2 + 5b1 + 2) q^70 + 4b3 q^71 + (-4b2 - 2b1 - 5) * q^72 + (-2b3 + 4b2 - 3b1 - 7) q^73 + (b3 + b2 + b1 + 3) * q^74 + (5b3 - 6b2 + b1 + 1) * q^75 + (6b3 - b2 + 3b1 - 9) * q^76 + (-13b3 + b2 + 5b1) * q^77 + (6b3 - 7b2 - 2b1 - 4) q^78 + (-8b3 + b2 - 3b1 + 5) * q^79 + (5b3 - b2 - 4b1 + 5) * q^80 + (-b2 + 2b1 - 4) q^81 + (4b3 + b2 - 3b1 + 6) * q^82 + (3b3 + 8b2 + 3b1 - 7) q^83 + (-5b3 + 7b2 - 6b1 + 6) q^84 + (3b3 - 2b2 + 5b1 - 4) q^85 + (-4b3 - b2 - 2b1 - 1) * q^86 + (b3 + 3b2 - 8b1 - 2) * q^87 + (5b3 - 2b2 - 8b1) q^88 + (3b3 + 2) q^89 + (b3 - 6b2 + 2b1 - 8) * q^90 + (4b3 - 3b2 - 9b1 + 10) q^91 + (-5b3 + 4b2 - b1 + 9) * q^92 + (-5b3 + 3b2 - 2b1 + 12) q^93 + (-3b2 - 2b1 - 1) * q^94 + (7b3 - 3b2 + 4b1 - 2) q^95 + (-5b2 - 2b1 - 1) * q^96 + (b3 - 5b2 + 5b1 - 1) * q^97 + (-8b3 + b2 + 10b1 - 10) * q^98 + (-3b3 + b2 + 8b1 + 4) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 3 q^{2} - 5 q^{3} + q^{4} - 2 q^{5} + q^{6} - 13 q^{7} - 3 q^{8} + 5 q^{9} - 5 q^{10} + q^{11} - 4 q^{12} - 8 q^{13} + 14 q^{14} - 8 q^{15} - q^{16} - 4 q^{17} - 10 q^{19} + 13 q^{20} + 11 q^{21}+ \cdots + 20 q^{99}+O(q^{100})$ 4 * q - 3 * q^2 - 5 * q^3 + q^4 - 2 * q^5 + q^6 - 13 * q^7 - 3 * q^8 + 5 * q^9 - 5 * q^10 + q^11 - 4 * q^12 - 8 * q^13 + 14 * q^14 - 8 * q^15 - q^16 - 4 * q^17 - 10 * q^19 + 13 * q^20 + 11 * q^21 + q^22 - q^23 + 20 * q^24 + 8 * q^25 + 18 * q^26 - 5 * q^27 - 9 * q^28 + 11 * q^29 + 33 * q^30 - 17 * q^31 + 6 * q^32 + 6 * q^33 + 5 * q^34 + 10 * q^35 + 10 * q^36 - 4 * q^37 + 7 * q^38 - q^39 - 16 * q^40 - 7 * q^41 - 4 * q^42 - 13 * q^43 + 7 * q^44 + 5 * q^45 - 21 * q^46 - 11 * q^47 - 9 * q^48 + 21 * q^49 - 27 * q^50 + 19 * q^51 - 10 * q^52 - 2 * q^53 + 2 * q^54 - 27 * q^55 + 6 * q^56 + 24 * q^57 + 14 * q^58 + 2 * q^59 - 35 * q^60 + 7 * q^61 + 20 * q^62 - 15 * q^63 - 3 * q^64 - 2 * q^65 - 28 * q^66 - 6 * q^67 - 24 * q^68 + 4 * q^69 + 9 * q^70 + 8 * q^71 - 30 * q^72 - 27 * q^73 + 17 * q^74 + 3 * q^75 - 23 * q^76 - 19 * q^77 - 20 * q^78 + 3 * q^79 + 24 * q^80 - 16 * q^81 + 31 * q^82 - 3 * q^83 + 22 * q^84 - 9 * q^85 - 16 * q^86 - 8 * q^87 - 2 * q^88 + 14 * q^89 - 40 * q^90 + 33 * q^91 + 33 * q^92 + 42 * q^93 - 12 * q^94 + 4 * q^95 - 16 * q^96 - 7 * q^97 - 44 * q^98 + 20 * q^99

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

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

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

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

−1.35567
−0.477260
0.737640
2.09529

−2.35567

−2.45589

3.54920

3.42960

5.78527

−3.73764

−3.64941

3.03138

−8.07901

1.2

−1.47726

1.39026

0.182297

−3.53103

−2.05377

−5.09529

2.68522

−1.06719

5.21625

1.3

−0.262360

−1.16215

−1.93117

0.0425409

0.304901

−1.64433

1.03138

−1.64941

−0.0111610

1.4

1.09529

−2.77222

−0.800331

−1.94111

−3.03640

−2.52274

−3.06719

4.68522

−2.12608

Atkin-Lehner signs

$p$	Sign
$137$	$+1$

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	137.2.a.a	✓	4
3.b	odd	2	1		1233.2.a.d		4
4.b	odd	2	1		2192.2.a.j		4
5.b	even	2	1		3425.2.a.b		4
7.b	odd	2	1		6713.2.a.b		4
8.b	even	2	1		8768.2.a.w		4
8.d	odd	2	1		8768.2.a.r		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
137.2.a.a	✓	4	1.a	even	1	1	trivial
1233.2.a.d		4	3.b	odd	2	1
2192.2.a.j		4	4.b	odd	2	1
3425.2.a.b		4	5.b	even	2	1
6713.2.a.b		4	7.b	odd	2	1
8768.2.a.r		4	8.d	odd	2	1
8768.2.a.w		4	8.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{4} + 3T_{2}^{3} - 4T_{2} - 1$ T2^4 + 3*T2^3 - 4*T2 - 1 acting on $S_{2}^{\mathrm{new}}(\Gamma_0(137))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4} + 3 T^{3} + \cdots - 1$ T^4 + 3T^3 - 4T - 1
$3$	$T^{4} + 5 T^{3} + \cdots - 11$ T^4 + 5T^3 + 4T^2 - 10*T - 11
$5$	$T^{4} + 2 T^{3} + \cdots + 1$ T^4 + 2T^3 - 12T^2 - 23*T + 1
$7$	$T^{4} + 13 T^{3} + \cdots + 79$ T^4 + 13T^3 + 60T^2 + 116*T + 79
$11$	$T^{4} - T^{3} + \cdots + 101$ T^4 - T^3 - 38T^2 + 76T + 101
$13$	$T^{4} + 8 T^{3} + \cdots - 101$ T^4 + 8T^3 + 10T^2 - 49*T - 101
$17$	$T^{4} + 4 T^{3} + \cdots + 31$ T^4 + 4T^3 - 28T^2 - 109*T + 31
$19$	$T^{4} + 10 T^{3} + \cdots - 431$ T^4 + 10T^3 - 4T^2 - 235*T - 431
$23$	$T^{4} + T^{3} + \cdots + 121$ T^4 + T^3 - 38T^2 - 66T + 121
$29$	$T^{4} - 11 T^{3} + \cdots - 551$ T^4 - 11T^3 - 25T^2 + 377*T - 551
$31$	$T^{4} + 17 T^{3} + \cdots - 319$ T^4 + 17T^3 + 53T^2 - 203*T - 319
$37$	$T^{4} + 4 T^{3} + \cdots - 191$ T^4 + 4T^3 - 50T^2 - 213*T - 191
$41$	$T^{4} + 7 T^{3} + \cdots - 121$ T^4 + 7T^3 - 50T^2 - 286*T - 121
$43$	$T^{4} + 13 T^{3} + \cdots - 191$ T^4 + 13T^3 - 5T^2 - 239*T - 191
$47$	$T^{4} + 11 T^{3} + \cdots - 41$ T^4 + 11T^3 + 15T^2 - 67*T - 41
$53$	$T^{4} + 2 T^{3} + \cdots - 1$ T^4 + 2T^3 - 15T^2 - 36*T - 1
$59$	$T^{4} - 2 T^{3} + \cdots - 709$ T^4 - 2T^3 - 107T^2 + 608*T - 709
$61$	$T^{4} - 7 T^{3} + \cdots + 11$ T^4 - 7T^3 - 17T^2 + 133*T + 11
$67$	$T^{4} + 6 T^{3} + \cdots + 2831$ T^4 + 6T^3 - 123T^2 - 536*T + 2831
$71$	$(T^{2} - 4 T - 16)^{2}$ (T^2 - 4*T - 16)^2
$73$	$T^{4} + 27 T^{3} + \cdots - 10219$ T^4 + 27T^3 + 144T^2 - 1282*T - 10219
$79$	$T^{4} - 3 T^{3} + \cdots + 9329$ T^4 - 3T^3 - 255T^2 + 89*T + 9329
$83$	$T^{4} + 3 T^{3} + \cdots + 6449$ T^4 + 3T^3 - 260T^2 - 354*T + 6449
$89$	$(T^{2} - 7 T + 1)^{2}$ (T^2 - 7*T + 1)^2
$97$	$T^{4} + 7 T^{3} + \cdots + 211$ T^4 + 7T^3 - 206T^2 + 658*T + 211
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