LMFDB - Newform orbit 234.6.a.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [234,6,Mod(1,234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(234, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("234.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$234 = 2 \cdot 3^{2} \cdot 13$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	234.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:	$37.5298138362$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{849})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 212$ x^2 - x - 212
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$3$
Twist minimal:	no (minimal twist has level 26)
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 $\beta = \frac{1}{2}(-1 + 3\sqrt{849})$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - 4 q^{2} + 16 q^{4} + ( - \beta - 37) q^{5} + (3 \beta + 79) q^{7} - 64 q^{8} + (4 \beta + 148) q^{10} + (8 \beta + 114) q^{11} - 169 q^{13} + ( - 12 \beta - 316) q^{14} + 256 q^{16} + ( - 43 \beta + 73) q^{17}+ \cdots + ( - 1860 \beta - 26496) q^{98}+O(q^{100})$ q - 4 * q^2 + 16 * q^4 + (-b - 37) * q^5 + (3b + 79) q^7 - 64 * q^8 + (4b + 148) q^10 + (8b + 114) q^11 - 169 * q^13 + (-12b - 316) q^14 + 256 * q^16 + (-43b + 73) q^17 + (-20b - 1258) q^19 + (-16b - 592) q^20 + (-32b - 456) q^22 + (36b + 1540) q^23 + (73b + 154) q^25 + 676 * q^26 + (48b + 1264) q^28 + (88b - 906) q^29 + (-126b + 1336) q^31 - 1024 * q^32 + (172b - 292) q^34 + (-187b - 8653) q^35 + (-59b + 8873) q^37 + (80b + 5032) q^38 + (64b + 2368) q^40 + (154b - 5740) q^41 + (-73b - 2071) q^43 + (128b + 1824) q^44 + (-144b - 6160) q^46 + (-135b + 12677) q^47 + (465b + 6624) q^49 + (-292b - 616) q^50 - 2704 * q^52 + (266b + 2440) q^53 + (-402b - 19498) q^55 + (-192b - 5056) q^56 + (-352b + 3624) q^58 + (152b + 11786) q^59 + (-134b + 48348) q^61 + (504b - 5344) q^62 + 4096 * q^64 + (169b + 6253) q^65 + (-92b + 36174) q^67 + (-688b + 1168) q^68 + (748b + 34612) q^70 + (-1073b + 17803) q^71 + (-1028b + 23062) q^73 + (236b - 35492) q^74 + (-320b - 20128) q^76 + (950b + 54846) q^77 + (352b + 26188) q^79 + (-256b - 9472) q^80 + (-616b + 22960) q^82 + (378b + 19068) q^83 + (1475b + 79429) q^85 + (292b + 8284) q^86 + (-512b - 7296) q^88 + (-940b + 52066) q^89 + (-507b - 13351) q^91 + (576b + 24640) q^92 + (540b - 50708) q^94 + (1978b + 84746) q^95 + (1908b + 53150) q^97 + (-1860b - 26496) q^98
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 8 q^{2} + 32 q^{4} - 73 q^{5} + 155 q^{7} - 128 q^{8} + 292 q^{10} + 220 q^{11} - 338 q^{13} - 620 q^{14} + 512 q^{16} + 189 q^{17} - 2496 q^{19} - 1168 q^{20} - 880 q^{22} + 3044 q^{23} + 235 q^{25}+ \cdots - 51132 q^{98}+O(q^{100})$ 2 * q - 8 * q^2 + 32 * q^4 - 73 * q^5 + 155 * q^7 - 128 * q^8 + 292 * q^10 + 220 * q^11 - 338 * q^13 - 620 * q^14 + 512 * q^16 + 189 * q^17 - 2496 * q^19 - 1168 * q^20 - 880 * q^22 + 3044 * q^23 + 235 * q^25 + 1352 * q^26 + 2480 * q^28 - 1900 * q^29 + 2798 * q^31 - 2048 * q^32 - 756 * q^34 - 17119 * q^35 + 17805 * q^37 + 9984 * q^38 + 4672 * q^40 - 11634 * q^41 - 4069 * q^43 + 3520 * q^44 - 12176 * q^46 + 25489 * q^47 + 12783 * q^49 - 940 * q^50 - 5408 * q^52 + 4614 * q^53 - 38594 * q^55 - 9920 * q^56 + 7600 * q^58 + 23420 * q^59 + 96830 * q^61 - 11192 * q^62 + 8192 * q^64 + 12337 * q^65 + 72440 * q^67 + 3024 * q^68 + 68476 * q^70 + 36679 * q^71 + 47152 * q^73 - 71220 * q^74 - 39936 * q^76 + 108742 * q^77 + 52024 * q^79 - 18688 * q^80 + 46536 * q^82 + 37758 * q^83 + 157383 * q^85 + 16276 * q^86 - 14080 * q^88 + 105072 * q^89 - 26195 * q^91 + 48704 * q^92 - 101956 * q^94 + 167514 * q^95 + 104392 * q^97 - 51132 * q^98

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

15.0688
−14.0688

−4.00000

16.0000

−80.2064

208.619

−64.0000

320.826

1.2

−4.00000

16.0000

7.20641

−53.6192

−64.0000

−28.8256

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$3$	$-1$
$13$	$+1$

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	234.6.a.h		2
3.b	odd	2	1		26.6.a.c	✓	2
12.b	even	2	1		208.6.a.g		2
15.d	odd	2	1		650.6.a.b		2
15.e	even	4	2		650.6.b.h		4
24.f	even	2	1		832.6.a.m		2
24.h	odd	2	1		832.6.a.k		2
39.d	odd	2	1		338.6.a.f		2
39.f	even	4	2		338.6.b.b		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
26.6.a.c	✓	2	3.b	odd	2	1
208.6.a.g		2	12.b	even	2	1
234.6.a.h		2	1.a	even	1	1	trivial
338.6.a.f		2	39.d	odd	2	1
338.6.b.b		4	39.f	even	4	2
650.6.a.b		2	15.d	odd	2	1
650.6.b.h		4	15.e	even	4	2
832.6.a.k		2	24.h	odd	2	1
832.6.a.m		2	24.f	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{6}^{\mathrm{new}}(\Gamma_0(234))$ :

T_{5}^{2} + 73T_{5} - 578

T_{7}^{2} - 155T_{7} - 11186

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T + 4)^{2}$ (T + 4)^2
$3$	$T^{2}$ T^2
$5$	$T^{2} + 73T - 578$ T^2 + 73*T - 578
$7$	$T^{2} - 155T - 11186$ T^2 - 155*T - 11186
$11$	$T^{2} - 220T - 110156$ T^2 - 220*T - 110156
$13$	$(T + 169)^{2}$ (T + 169)^2
$17$	$T^{2} - 189 T - 3523122$ T^2 - 189*T - 3523122
$19$	$T^{2} + 2496 T + 793404$ T^2 + 2496*T + 793404
$23$	$T^{2} - 3044 T - 159200$ T^2 - 3044*T - 159200
$29$	$T^{2} + 1900 T - 13890476$ T^2 + 1900*T - 13890476
$31$	$T^{2} - 2798 T - 28369928$ T^2 - 2798*T - 28369928
$37$	$T^{2} - 17805 T + 72604926$ T^2 - 17805*T + 72604926
$41$	$T^{2} + 11634 T - 11466000$ T^2 + 11634*T - 11466000
$43$	$T^{2} + 4069 T - 6040532$ T^2 + 4069*T - 6040532
$47$	$T^{2} - 25489 T + 127607974$ T^2 - 25489*T + 127607974
$53$	$T^{2} - 4614 T - 129839400$ T^2 - 4614*T - 129839400
$59$	$T^{2} - 23420 T + 92989684$ T^2 - 23420*T + 92989684
$61$	$T^{2} + \cdots + 2309711776$ T^2 - 96830*T + 2309711776
$67$	$T^{2} + \cdots + 1295720044$ T^2 - 72440*T + 1295720044
$71$	$T^{2} + \cdots - 1862988962$ T^2 - 36679*T - 1862988962
$73$	$T^{2} + \cdots - 1462893860$ T^2 - 47152*T - 1462893860
$79$	$T^{2} - 52024 T + 439936528$ T^2 - 52024*T + 439936528
$83$	$T^{2} - 37758 T + 83472480$ T^2 - 37758*T + 83472480
$89$	$T^{2} + \cdots + 1072134396$ T^2 - 105072*T + 1072134396
$97$	$T^{2} + \cdots - 4229773940$ T^2 - 104392*T - 4229773940
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