LMFDB - Newform orbit 1110.4.a.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1110,4,Mod(1,1110)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1110, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1110.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$1110 = 2 \cdot 3 \cdot 5 \cdot 37$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1110.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:	$65.4921201064$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{61})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 15$ x^2 - x - 15
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
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 $\beta = \frac{1}{2}(1 + \sqrt{61})$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + 2 q^{2} + 3 q^{3} + 4 q^{4} + 5 q^{5} + 6 q^{6} - 15 q^{7} + 8 q^{8} + 9 q^{9} + 10 q^{10} + (3 \beta - 40) q^{11} + 12 q^{12} + ( - 10 \beta - 13) q^{13} - 30 q^{14} + 15 q^{15} + 16 q^{16} + (5 \beta - 35) q^{17}+ \cdots + (27 \beta - 360) q^{99}+O(q^{100})$ q + 2 * q^2 + 3 * q^3 + 4 * q^4 + 5 * q^5 + 6 * q^6 - 15 * q^7 + 8 * q^8 + 9 * q^9 + 10 * q^10 + (3b - 40) q^11 + 12 * q^12 + (-10b - 13) q^13 - 30 * q^14 + 15 * q^15 + 16 * q^16 + (5b - 35) q^17 + 18 * q^18 + (19b - 37) q^19 + 20 * q^20 - 45 * q^21 + (6b - 80) q^22 + (-26b - 51) q^23 + 24 * q^24 + 25 * q^25 + (-20b - 26) q^26 + 27 * q^27 - 60 * q^28 + (49b - 51) q^29 + 30 * q^30 + (-26b - 181) q^31 + 32 * q^32 + (9b - 120) q^33 + (10b - 70) q^34 - 75 * q^35 + 36 * q^36 - 37 * q^37 + (38b - 74) q^38 + (-30b - 39) q^39 + 40 * q^40 + (-92b + 105) q^41 - 90 * q^42 + (53b - 203) q^43 + (12b - 160) q^44 + 45 * q^45 + (-52b - 102) q^46 + (-16b - 65) q^47 + 48 * q^48 - 118 * q^49 + 50 * q^50 + (15b - 105) q^51 + (-40b - 52) q^52 + (120b - 284) q^53 + 54 * q^54 + (15b - 200) q^55 - 120 * q^56 + (57b - 111) q^57 + (98b - 102) q^58 + (-157b + 70) q^59 + 60 * q^60 + (-68b + 25) q^61 + (-52b - 362) q^62 - 135 * q^63 + 64 * q^64 + (-50b - 65) q^65 + (18b - 240) q^66 + (73b + 37) q^67 + (20b - 140) q^68 + (-78b - 153) q^69 - 150 * q^70 + (93b - 69) q^71 + 72 * q^72 + (47b + 58) q^73 - 74 * q^74 + 75 * q^75 + (76b - 148) q^76 + (-45b + 600) q^77 + (-60b - 78) q^78 + (27b - 40) q^79 + 80 * q^80 + 81 * q^81 + (-184b + 210) q^82 + (-129b - 630) q^83 - 180 * q^84 + (25b - 175) q^85 + (106b - 406) q^86 + (147b - 153) q^87 + (24b - 320) q^88 + (148b - 583) q^89 + 90 * q^90 + (150b + 195) q^91 + (-104b - 204) q^92 + (-78b - 543) q^93 + (-32b - 130) q^94 + (95b - 185) q^95 + 96 * q^96 + (-76b + 1197) q^97 - 236 * q^98 + (27b - 360) q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} + 10 q^{5} + 12 q^{6} - 30 q^{7} + 16 q^{8} + 18 q^{9} + 20 q^{10} - 77 q^{11} + 24 q^{12} - 36 q^{13} - 60 q^{14} + 30 q^{15} + 32 q^{16} - 65 q^{17} + 36 q^{18} - 55 q^{19}+ \cdots - 693 q^{99}+O(q^{100})$ 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 + 10 * q^5 + 12 * q^6 - 30 * q^7 + 16 * q^8 + 18 * q^9 + 20 * q^10 - 77 * q^11 + 24 * q^12 - 36 * q^13 - 60 * q^14 + 30 * q^15 + 32 * q^16 - 65 * q^17 + 36 * q^18 - 55 * q^19 + 40 * q^20 - 90 * q^21 - 154 * q^22 - 128 * q^23 + 48 * q^24 + 50 * q^25 - 72 * q^26 + 54 * q^27 - 120 * q^28 - 53 * q^29 + 60 * q^30 - 388 * q^31 + 64 * q^32 - 231 * q^33 - 130 * q^34 - 150 * q^35 + 72 * q^36 - 74 * q^37 - 110 * q^38 - 108 * q^39 + 80 * q^40 + 118 * q^41 - 180 * q^42 - 353 * q^43 - 308 * q^44 + 90 * q^45 - 256 * q^46 - 146 * q^47 + 96 * q^48 - 236 * q^49 + 100 * q^50 - 195 * q^51 - 144 * q^52 - 448 * q^53 + 108 * q^54 - 385 * q^55 - 240 * q^56 - 165 * q^57 - 106 * q^58 - 17 * q^59 + 120 * q^60 - 18 * q^61 - 776 * q^62 - 270 * q^63 + 128 * q^64 - 180 * q^65 - 462 * q^66 + 147 * q^67 - 260 * q^68 - 384 * q^69 - 300 * q^70 - 45 * q^71 + 144 * q^72 + 163 * q^73 - 148 * q^74 + 150 * q^75 - 220 * q^76 + 1155 * q^77 - 216 * q^78 - 53 * q^79 + 160 * q^80 + 162 * q^81 + 236 * q^82 - 1389 * q^83 - 360 * q^84 - 325 * q^85 - 706 * q^86 - 159 * q^87 - 616 * q^88 - 1018 * q^89 + 180 * q^90 + 540 * q^91 - 512 * q^92 - 1164 * q^93 - 292 * q^94 - 275 * q^95 + 192 * q^96 + 2318 * q^97 - 472 * q^98 - 693 * q^99

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

−3.40512
4.40512

2.00000

3.00000

4.00000

5.00000

6.00000

−15.0000

8.00000

9.00000

10.0000

1.2

2.00000

3.00000

4.00000

5.00000

6.00000

−15.0000

8.00000

9.00000

10.0000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$-1$
$5$	$-1$
$37$	$+1$

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	1110.4.a.d	✓	2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1110.4.a.d	✓	2	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{7} + 15$ T7 + 15 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(1110))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T - 2)^{2}$ (T - 2)^2
$3$	$(T - 3)^{2}$ (T - 3)^2
$5$	$(T - 5)^{2}$ (T - 5)^2
$7$	$(T + 15)^{2}$ (T + 15)^2
$11$	$T^{2} + 77T + 1345$ T^2 + 77*T + 1345
$13$	$T^{2} + 36T - 1201$ T^2 + 36*T - 1201
$17$	$T^{2} + 65T + 675$ T^2 + 65*T + 675
$19$	$T^{2} + 55T - 4749$ T^2 + 55*T - 4749
$23$	$T^{2} + 128T - 6213$ T^2 + 128*T - 6213
$29$	$T^{2} + 53T - 35913$ T^2 + 53*T - 35913
$31$	$T^{2} + 388T + 27327$ T^2 + 388*T + 27327
$37$	$(T + 37)^{2}$ (T + 37)^2
$41$	$T^{2} - 118T - 125595$ T^2 - 118*T - 125595
$43$	$T^{2} + 353T - 11685$ T^2 + 353*T - 11685
$47$	$T^{2} + 146T + 1425$ T^2 + 146*T + 1425
$53$	$T^{2} + 448T - 169424$ T^2 + 448*T - 169424
$59$	$T^{2} + 17T - 375825$ T^2 + 17*T - 375825
$61$	$T^{2} + 18T - 70435$ T^2 + 18*T - 70435
$67$	$T^{2} - 147T - 75865$ T^2 - 147*T - 75865
$71$	$T^{2} + 45T - 131391$ T^2 + 45*T - 131391
$73$	$T^{2} - 163T - 27045$ T^2 - 163*T - 27045
$79$	$T^{2} + 53T - 10415$ T^2 + 53*T - 10415
$83$	$T^{2} + 1389 T + 228555$ T^2 + 1389*T + 228555
$89$	$T^{2} + 1018T - 74955$ T^2 + 1018*T - 74955
$97$	$T^{2} - 2318 T + 1255197$ T^2 - 2318*T + 1255197
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