LMFDB - Newform orbit 2116.4.a.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2116,4,Mod(1,2116)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2116, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2116.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$2116 = 2^{2} \cdot 23^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	2116.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:	$124.848041572$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - x^{2} - 7x + 6$ x^3 - x^2 - 7*x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 92)
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 $1,\beta_1,\beta_2$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - \beta_{2} - 1) q^{3} + ( - 3 \beta_{2} + \beta_1 + 4) q^{5} + ( - 3 \beta_{2} - 5 \beta_1 + 18) q^{7} + (2 \beta_{2} - 3 \beta_1 - 10) q^{9} + (11 \beta_{2} + 5 \beta_1 + 16) q^{11} + ( - 6 \beta_{2} - 11 \beta_1 - 9) q^{13}+ \cdots + ( - 62 \beta_{2} - 232 \beta_1 + 104) q^{99}+O(q^{100})$ q + (-b2 - 1) * q^3 + (-3b2 + b1 + 4) q^5 + (-3b2 - 5b1 + 18) * q^7 + (2b2 - 3b1 - 10) * q^9 + (11b2 + 5b1 + 16) * q^11 + (-6b2 - 11b1 - 9) * q^13 + (b2 - 11b1 + 48) q^15 + (-5b2 - 33b1 + 42) * q^17 + (16b2 + 24b1 + 18) * q^19 + (-25b2 + b1 + 10) q^21 + (-10b2 - 22b1 + 71) * q^25 + (29b2 + 12b1 - 7) * q^27 + (-6b2 - b1 + 105) q^29 + (59b2 + 8b1 - 69) * q^31 + (-17b2 + 23b1 - 172) * q^33 + (-100b2 - 32b1 + 108) * q^35 + (3b2 - 65b1 + 12) * q^37 + (-7b2 + 4b1 + 61) * q^39 + (-52b2 + 27b1 + 203) * q^41 + (-12b2 - 62b1 + 184) * q^43 + (10b2 - 2b1 - 216) * q^45 + (47b2 - 72b1 - 1) * q^47 + (-118b2 - 102b1 + 305) * q^49 + (-103b2 + 51b1 - 94) * q^51 + (-42b2 + 64b1 - 126) * q^53 + (14b2 + 146b1 - 388) * q^55 + (14b2 - 178) q^57 + (38b2 + 94b1 + 4) * q^59 + (-96b2 + 14b1 + 378) * q^61 + (98b2 + 58b1 - 92) * q^63 + (-73b2 - 113b1 + 12) * q^65 + (61b2 + 51b1 + 632) * q^67 + (-47b2 - 58b1 + 155) * q^71 + (-68b2 + 89b1 - 91) * q^73 + (-105b2 + 14b1 + 1) * q^75 + (240b2 - 36b1 - 260) * q^77 + (-176b2 + 134b1 + 254) * q^79 + (-52b2 + 144b1 - 139) * q^81 + (185b2 + 97b1 + 374) * q^83 + (-400b2 - 140b1 - 364) * q^85 + (-101b2 - 16b1 - 13) * q^87 + (-238b2 + 16b1 - 104) * q^89 + (-97b2 + 21b1 + 534) * q^91 + (26b2 + 161b1 - 843) * q^93 + (170b2 + 274b1 - 184) * q^95 + (-177b2 + 23b1 - 194) * q^97 + (-62b2 - 232b1 + 104) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$3 q - 4 q^{3} + 10 q^{5} + 46 q^{7} - 31 q^{9} + 64 q^{11} - 44 q^{13} + 134 q^{15} + 88 q^{17} + 94 q^{19} + 6 q^{21} + 181 q^{25} + 20 q^{27} + 308 q^{29} - 140 q^{31} - 510 q^{33} + 192 q^{35} - 26 q^{37}+ \cdots + 18 q^{99}+O(q^{100})$ 3 * q - 4 * q^3 + 10 * q^5 + 46 * q^7 - 31 * q^9 + 64 * q^11 - 44 * q^13 + 134 * q^15 + 88 * q^17 + 94 * q^19 + 6 * q^21 + 181 * q^25 + 20 * q^27 + 308 * q^29 - 140 * q^31 - 510 * q^33 + 192 * q^35 - 26 * q^37 + 180 * q^39 + 584 * q^41 + 478 * q^43 - 640 * q^45 - 28 * q^47 + 695 * q^49 - 334 * q^51 - 356 * q^53 - 1004 * q^55 - 520 * q^57 + 144 * q^59 + 1052 * q^61 - 120 * q^63 - 150 * q^65 + 2008 * q^67 + 360 * q^71 - 252 * q^73 - 88 * q^75 - 576 * q^77 + 720 * q^79 - 325 * q^81 + 1404 * q^83 - 1632 * q^85 - 156 * q^87 - 534 * q^89 + 1526 * q^91 - 2342 * q^93 - 108 * q^95 - 736 * q^97 + 18 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{3} - x^{2} - 7x + 6$ x^3 - x^2 - 7*x + 6 :

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

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

$\nu$	$=$	$( \beta_{2} + \beta_1 ) / 2$ (b2 + b1) / 2
$\nu^{2}$	$=$	$( -\beta_{2} + \beta _1 + 10 ) / 2$ (-b2 + b1 + 10) / 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

0.841083
2.75153
−2.59261

−6.13366

−14.8525

19.8565

10.6218

1.2

−1.18060

8.78065

−9.15411

−25.6062

1.3

3.31427

16.0718

35.2976

−16.0156

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$23$	$-1$

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	2116.4.a.a		3
23.b	odd	2	1		92.4.a.a	✓	3
69.c	even	2	1		828.4.a.f		3
92.b	even	2	1		368.4.a.k		3
115.c	odd	2	1		2300.4.a.b		3
115.e	even	4	2		2300.4.c.b		6
184.e	odd	2	1		1472.4.a.w		3
184.h	even	2	1		1472.4.a.p		3

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
92.4.a.a	✓	3	23.b	odd	2	1
368.4.a.k		3	92.b	even	2	1
828.4.a.f		3	69.c	even	2	1
1472.4.a.p		3	184.h	even	2	1
1472.4.a.w		3	184.e	odd	2	1
2116.4.a.a		3	1.a	even	1	1	trivial
2300.4.a.b		3	115.c	odd	2	1
2300.4.c.b		6	115.e	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(2116))$ :

T_{3}^{3} + 4T_{3}^{2} - 17T_{3} - 24

T_{5}^{3} - 10T_{5}^{2} - 228T_{5} + 2096

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{3}$ T^3
$3$	$T^{3} + 4 T^{2} + \cdots - 24$ T^3 + 4T^2 - 17T - 24
$5$	$T^{3} - 10 T^{2} + \cdots + 2096$ T^3 - 10T^2 - 228T + 2096
$7$	$T^{3} - 46 T^{2} + \cdots + 6416$ T^3 - 46T^2 + 196T + 6416
$11$	$T^{3} - 64 T^{2} + \cdots + 88184$ T^3 - 64T^2 - 1112T + 88184
$13$	$T^{3} + 44 T^{2} + \cdots - 3334$ T^3 + 44T^2 - 1739T - 3334
$17$	$T^{3} - 88 T^{2} + \cdots + 1617496$ T^3 - 88T^2 - 17920T + 1617496
$19$	$T^{3} - 94 T^{2} + \cdots + 184984$ T^3 - 94T^2 - 9364T + 184984
$23$	$T^{3}$ T^3
$29$	$T^{3} - 308 T^{2} + \cdots - 1008698$ T^3 - 308T^2 + 30877T - 1008698
$31$	$T^{3} + 140 T^{2} + \cdots - 1074768$ T^3 + 140T^2 - 66217T - 1074768
$37$	$T^{3} + 26 T^{2} + \cdots + 4672784$ T^3 + 26T^2 - 88484T + 4672784
$41$	$T^{3} - 584 T^{2} + \cdots + 21405186$ T^3 - 584T^2 + 19753T + 21405186
$43$	$T^{3} - 478 T^{2} + \cdots + 14441984$ T^3 - 478T^2 + 4704T + 14441984
$47$	$T^{3} + 28 T^{2} + \cdots - 25906224$ T^3 + 28T^2 - 199601T - 25906224
$53$	$T^{3} + 356 T^{2} + \cdots - 63184$ T^3 + 356T^2 - 116276T - 63184
$59$	$T^{3} - 144 T^{2} + \cdots - 15495744$ T^3 - 144T^2 - 157376T - 15495744
$61$	$T^{3} - 1052 T^{2} + \cdots + 55378432$ T^3 - 1052T^2 + 141172T + 55378432
$67$	$T^{3} - 2008 T^{2} + \cdots - 228170232$ T^3 - 2008T^2 + 1249512T - 228170232
$71$	$T^{3} - 360 T^{2} + \cdots + 7542816$ T^3 - 360T^2 - 38189T + 7542816
$73$	$T^{3} + 252 T^{2} + \cdots + 34560066$ T^3 + 252T^2 - 323855T + 34560066
$79$	$T^{3} - 720 T^{2} + \cdots + 932658192$ T^3 - 720T^2 - 1198556T + 932658192
$83$	$T^{3} - 1404 T^{2} + \cdots + 464610136$ T^3 - 1404T^2 - 59336T + 464610136
$89$	$T^{3} + 534 T^{2} + \cdots - 77599776$ T^3 + 534T^2 - 1225976T - 77599776
$97$	$T^{3} + 736 T^{2} + \cdots - 67270584$ T^3 + 736T^2 - 584152T - 67270584
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