LMFDB - Newform orbit 1156.4.a.h

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1156,4,Mod(1,1156)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1156, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1156.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$1156 = 2^{2} \cdot 17^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1156.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$68.2062079666$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.5999648.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 21x^{2} + 2$ x^4 - 21*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2^{4}$
Twist minimal:	no (minimal twist has level 68)
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.

$f(q)$	$=$	$q + \beta_1 q^{3} + \beta_{2} q^{5} + ( - \beta_{2} - \beta_1) q^{7} + (\beta_{3} + 15) q^{9} + ( - 2 \beta_{2} - \beta_1) q^{11} + ( - \beta_{3} - 8) q^{13} - 8 q^{15} + ( - 2 \beta_{3} - 40) q^{19} + ( - \beta_{3} - 34) q^{21}+ \cdots + (50 \beta_{2} - 41 \beta_1) q^{99}+O(q^{100})$ q + b1 * q^3 + b2 * q^5 + (-b2 - b1) * q^7 + (b3 + 15) * q^9 + (-2b2 - b1) q^11 + (-b3 - 8) * q^13 - 8 * q^15 + (-2b3 - 40) q^19 + (-b3 - 34) * q^21 + (13b2 - 13b1) * q^23 + (-2b3 - 41) q^25 + (4b2 + 30b1) * q^27 + (5b2 - 20b1) * q^29 + (-19b2 - 19b1) * q^31 + (-b3 - 26) * q^33 + (2b3 - 76) q^35 + (-17b2 + 28b1) * q^37 + (-4b2 - 50b1) * q^39 + (-14b2 - 40b1) * q^41 + (-4b3 + 36) q^43 + (-27b2 - 8b1) * q^45 + (4b3 + 280) q^47 + (-b3 - 233) * q^49 + (12b3 - 6) q^53 + (4b3 - 160) q^55 + (-8b2 - 124b1) * q^57 + (12b3 + 84) q^59 + (-25b2 + 92b1) * q^61 + (23b2 - 49b1) * q^63 + (34b2 + 8b1) * q^65 + (6b3 + 80) q^67 + (-13b3 - 650) q^69 + (19b2 + 21b1) * q^71 + (-20b2 - 92b1) * q^73 + (-8b2 - 125b1) * q^75 + (-3b3 + 186) q^77 + (81b2 - 9b1) * q^79 + (3b3 + 823) q^81 + (8b3 - 556) q^83 + (-20b3 - 880) q^87 + (5b3 - 784) q^89 + (-30b2 + 42b1) * q^91 + (-19b3 - 646) q^93 + (44b2 + 16b1) * q^95 + (-138b2 - 84b1) * q^97 + (50b2 - 41b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 60 q^{9} - 32 q^{13} - 32 q^{15} - 160 q^{19} - 136 q^{21} - 164 q^{25} - 104 q^{33} - 304 q^{35} + 144 q^{43} + 1120 q^{47} - 932 q^{49} - 24 q^{53} - 640 q^{55} + 336 q^{59} + 320 q^{67} - 2600 q^{69}+ \cdots - 2584 q^{93}+O(q^{100})$ 4 * q + 60 * q^9 - 32 * q^13 - 32 * q^15 - 160 * q^19 - 136 * q^21 - 164 * q^25 - 104 * q^33 - 304 * q^35 + 144 * q^43 + 1120 * q^47 - 932 * q^49 - 24 * q^53 - 640 * q^55 + 336 * q^59 + 320 * q^67 - 2600 * q^69 + 744 * q^77 + 3292 * q^81 - 2224 * q^83 - 3520 * q^87 - 3136 * q^89 - 2584 * q^93

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

$\beta_{1}$	$=$	$2\nu$ 2*v
$\beta_{2}$	$=$	$2\nu^{3} - 42\nu$ 2v^3 - 42v
$\beta_{3}$	$=$	$4\nu^{2} - 42$ 4*v^2 - 42

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

$\nu$	$=$	$( \beta_1 ) / 2$ (b1) / 2
$\nu^{2}$	$=$	$( \beta_{3} + 42 ) / 4$ (b3 + 42) / 4
$\nu^{3}$	$=$	$( \beta_{2} + 21\beta_1 ) / 2$ (b2 + 21*b1) / 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

−4.57212
−0.309312
0.309312
4.57212

−9.14425

0.874867

8.26938

56.6173

1.2

−0.618624

12.9319

−12.3133

−26.6173

1.3

0.618624

−12.9319

12.3133

−26.6173

1.4

9.14425

−0.874867

−8.26938

56.6173

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$17$	$+1$

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1156.4.a.h		4
17.b	even	2	1	inner	1156.4.a.h		4
17.c	even	4	2		68.4.b.a	✓	4
51.f	odd	4	2		612.4.b.b		4
68.f	odd	4	2		272.4.b.c		4
85.f	odd	4	2		1700.4.g.a		8
85.i	odd	4	2		1700.4.g.a		8
85.j	even	4	2		1700.4.c.a		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
68.4.b.a	✓	4	17.c	even	4	2
272.4.b.c		4	68.f	odd	4	2
612.4.b.b		4	51.f	odd	4	2
1156.4.a.h		4	1.a	even	1	1	trivial
1156.4.a.h		4	17.b	even	2	1	inner
1700.4.c.a		4	85.j	even	4	2
1700.4.g.a		8	85.f	odd	4	2
1700.4.g.a		8	85.i	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{4} - 84T_{3}^{2} + 32$ T3^4 - 84*T3^2 + 32 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(1156))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4}$ T^4
$3$	$T^{4} - 84T^{2} + 32$ T^4 - 84*T^2 + 32
$5$	$T^{4} - 168T^{2} + 128$ T^4 - 168*T^2 + 128
$7$	$T^{4} - 220 T^{2} + 10368$ T^4 - 220*T^2 + 10368
$11$	$T^{4} - 692 T^{2} + 34848$ T^4 - 692*T^2 + 34848
$13$	$(T^{2} + 16 T - 1668)^{2}$ (T^2 + 16*T - 1668)^2
$17$	$T^{4}$ T^4
$19$	$(T^{2} + 80 T - 5328)^{2}$ (T^2 + 80*T - 5328)^2
$23$	$T^{4} - 47996 T^{2} + 526436352$ T^4 - 47996*T^2 + 526436352
$29$	$T^{4} - 41000 T^{2} + 208080000$ T^4 - 41000*T^2 + 208080000
$31$	$T^{4} + \cdots + 1351168128$ T^4 - 79420*T^2 + 1351168128
$37$	$T^{4} + \cdots + 4128133248$ T^4 - 129640*T^2 + 4128133248
$41$	$T^{4} + \cdots + 3053242368$ T^4 - 149408*T^2 + 3053242368
$43$	$(T^{2} - 72 T - 26416)^{2}$ (T^2 - 72*T - 26416)^2
$47$	$(T^{2} - 560 T + 50688)^{2}$ (T^2 - 560*T + 50688)^2
$53$	$(T^{2} + 12 T - 249372)^{2}$ (T^2 + 12*T - 249372)^2
$59$	$(T^{2} - 168 T - 242352)^{2}$ (T^2 - 168*T - 242352)^2
$61$	$T^{4} + \cdots + 107699974272$ T^4 - 889576*T^2 + 107699974272
$67$	$(T^{2} - 160 T - 55952)^{2}$ (T^2 - 160*T - 55952)^2
$71$	$T^{4} + \cdots + 1666260992$ T^4 - 84924*T^2 + 1666260992
$73$	$T^{4} + \cdots + 27614380032$ T^4 - 719296*T^2 + 27614380032
$79$	$T^{4} + \cdots + 26013892608$ T^4 - 1132380*T^2 + 26013892608
$83$	$(T^{2} + 1112 T + 198288)^{2}$ (T^2 + 1112*T + 198288)^2
$89$	$(T^{2} + 1568 T + 571356)^{2}$ (T^2 + 1568*T + 571356)^2
$97$	$T^{4} + \cdots + 1258180190208$ T^4 - 3421152*T^2 + 1258180190208
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