Properties

 Label 26.3.d.a Level $26$ Weight $3$ Character orbit 26.d Analytic conductor $0.708$ Analytic rank $0$ Dimension $2$ Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [26,3,Mod(5,26)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(26, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([3]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("26.5");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$26 = 2 \cdot 13$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 26.d (of order $$4$$, degree $$2$$, minimal)

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: no Analytic conductor: $$0.708448687337$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (i + 1) q^{2} + 2 i q^{4} + ( - 3 i - 3) q^{5} + ( - 2 i + 2) q^{7} + (2 i - 2) q^{8} - 9 q^{9}+O(q^{10})$$ q + (i + 1) * q^2 + 2*i * q^4 + (-3*i - 3) * q^5 + (-2*i + 2) * q^7 + (2*i - 2) * q^8 - 9 * q^9 $$q + (i + 1) q^{2} + 2 i q^{4} + ( - 3 i - 3) q^{5} + ( - 2 i + 2) q^{7} + (2 i - 2) q^{8} - 9 q^{9} - 6 i q^{10} + ( - 6 i + 6) q^{11} + 13 i q^{13} + 4 q^{14} - 4 q^{16} + 6 i q^{17} + ( - 9 i - 9) q^{18} + (26 i + 26) q^{19} + ( - 6 i + 6) q^{20} + 12 q^{22} - 24 i q^{23} - 7 i q^{25} + (13 i - 13) q^{26} + (4 i + 4) q^{28} - 48 q^{29} + ( - 14 i - 14) q^{31} + ( - 4 i - 4) q^{32} + (6 i - 6) q^{34} - 12 q^{35} - 18 i q^{36} + ( - 37 i + 37) q^{37} + 52 i q^{38} + 12 q^{40} + ( - 9 i - 9) q^{41} + 36 i q^{43} + (12 i + 12) q^{44} + (27 i + 27) q^{45} + ( - 24 i + 24) q^{46} + ( - 42 i + 42) q^{47} + 41 i q^{49} + ( - 7 i + 7) q^{50} - 26 q^{52} + 30 q^{53} - 36 q^{55} + 8 i q^{56} + ( - 48 i - 48) q^{58} + (54 i - 54) q^{59} - 18 q^{61} - 28 i q^{62} + (18 i - 18) q^{63} - 8 i q^{64} + ( - 39 i + 39) q^{65} + ( - 22 i - 22) q^{67} - 12 q^{68} + ( - 12 i - 12) q^{70} + (6 i + 6) q^{71} + ( - 18 i + 18) q^{72} + ( - 17 i + 17) q^{73} + 74 q^{74} + (52 i - 52) q^{76} - 24 i q^{77} - 108 q^{79} + (12 i + 12) q^{80} + 81 q^{81} - 18 i q^{82} + (78 i + 78) q^{83} + ( - 18 i + 18) q^{85} + (36 i - 36) q^{86} + 24 i q^{88} + (9 i - 9) q^{89} + 54 i q^{90} + (26 i + 26) q^{91} + 48 q^{92} + 84 q^{94} - 156 i q^{95} + ( - 47 i - 47) q^{97} + (41 i - 41) q^{98} + (54 i - 54) q^{99} +O(q^{100})$$ q + (i + 1) * q^2 + 2*i * q^4 + (-3*i - 3) * q^5 + (-2*i + 2) * q^7 + (2*i - 2) * q^8 - 9 * q^9 - 6*i * q^10 + (-6*i + 6) * q^11 + 13*i * q^13 + 4 * q^14 - 4 * q^16 + 6*i * q^17 + (-9*i - 9) * q^18 + (26*i + 26) * q^19 + (-6*i + 6) * q^20 + 12 * q^22 - 24*i * q^23 - 7*i * q^25 + (13*i - 13) * q^26 + (4*i + 4) * q^28 - 48 * q^29 + (-14*i - 14) * q^31 + (-4*i - 4) * q^32 + (6*i - 6) * q^34 - 12 * q^35 - 18*i * q^36 + (-37*i + 37) * q^37 + 52*i * q^38 + 12 * q^40 + (-9*i - 9) * q^41 + 36*i * q^43 + (12*i + 12) * q^44 + (27*i + 27) * q^45 + (-24*i + 24) * q^46 + (-42*i + 42) * q^47 + 41*i * q^49 + (-7*i + 7) * q^50 - 26 * q^52 + 30 * q^53 - 36 * q^55 + 8*i * q^56 + (-48*i - 48) * q^58 + (54*i - 54) * q^59 - 18 * q^61 - 28*i * q^62 + (18*i - 18) * q^63 - 8*i * q^64 + (-39*i + 39) * q^65 + (-22*i - 22) * q^67 - 12 * q^68 + (-12*i - 12) * q^70 + (6*i + 6) * q^71 + (-18*i + 18) * q^72 + (-17*i + 17) * q^73 + 74 * q^74 + (52*i - 52) * q^76 - 24*i * q^77 - 108 * q^79 + (12*i + 12) * q^80 + 81 * q^81 - 18*i * q^82 + (78*i + 78) * q^83 + (-18*i + 18) * q^85 + (36*i - 36) * q^86 + 24*i * q^88 + (9*i - 9) * q^89 + 54*i * q^90 + (26*i + 26) * q^91 + 48 * q^92 + 84 * q^94 - 156*i * q^95 + (-47*i - 47) * q^97 + (41*i - 41) * q^98 + (54*i - 54) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 6 q^{5} + 4 q^{7} - 4 q^{8} - 18 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 6 * q^5 + 4 * q^7 - 4 * q^8 - 18 * q^9 $$2 q + 2 q^{2} - 6 q^{5} + 4 q^{7} - 4 q^{8} - 18 q^{9} + 12 q^{11} + 8 q^{14} - 8 q^{16} - 18 q^{18} + 52 q^{19} + 12 q^{20} + 24 q^{22} - 26 q^{26} + 8 q^{28} - 96 q^{29} - 28 q^{31} - 8 q^{32} - 12 q^{34} - 24 q^{35} + 74 q^{37} + 24 q^{40} - 18 q^{41} + 24 q^{44} + 54 q^{45} + 48 q^{46} + 84 q^{47} + 14 q^{50} - 52 q^{52} + 60 q^{53} - 72 q^{55} - 96 q^{58} - 108 q^{59} - 36 q^{61} - 36 q^{63} + 78 q^{65} - 44 q^{67} - 24 q^{68} - 24 q^{70} + 12 q^{71} + 36 q^{72} + 34 q^{73} + 148 q^{74} - 104 q^{76} - 216 q^{79} + 24 q^{80} + 162 q^{81} + 156 q^{83} + 36 q^{85} - 72 q^{86} - 18 q^{89} + 52 q^{91} + 96 q^{92} + 168 q^{94} - 94 q^{97} - 82 q^{98} - 108 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 6 * q^5 + 4 * q^7 - 4 * q^8 - 18 * q^9 + 12 * q^11 + 8 * q^14 - 8 * q^16 - 18 * q^18 + 52 * q^19 + 12 * q^20 + 24 * q^22 - 26 * q^26 + 8 * q^28 - 96 * q^29 - 28 * q^31 - 8 * q^32 - 12 * q^34 - 24 * q^35 + 74 * q^37 + 24 * q^40 - 18 * q^41 + 24 * q^44 + 54 * q^45 + 48 * q^46 + 84 * q^47 + 14 * q^50 - 52 * q^52 + 60 * q^53 - 72 * q^55 - 96 * q^58 - 108 * q^59 - 36 * q^61 - 36 * q^63 + 78 * q^65 - 44 * q^67 - 24 * q^68 - 24 * q^70 + 12 * q^71 + 36 * q^72 + 34 * q^73 + 148 * q^74 - 104 * q^76 - 216 * q^79 + 24 * q^80 + 162 * q^81 + 156 * q^83 + 36 * q^85 - 72 * q^86 - 18 * q^89 + 52 * q^91 + 96 * q^92 + 168 * q^94 - 94 * q^97 - 82 * q^98 - 108 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/26\mathbb{Z}\right)^\times$$.

 $$n$$ $$15$$ $$\chi(n)$$ $$i$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 − 1.00000i 1.00000i
1.00000 1.00000i 0 2.00000i −3.00000 + 3.00000i 0 2.00000 + 2.00000i −2.00000 2.00000i −9.00000 6.00000i
21.1 1.00000 + 1.00000i 0 2.00000i −3.00000 3.00000i 0 2.00000 2.00000i −2.00000 + 2.00000i −9.00000 6.00000i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 26.3.d.a 2
3.b odd 2 1 234.3.i.a 2
4.b odd 2 1 208.3.t.b 2
5.b even 2 1 650.3.k.b 2
5.c odd 4 1 650.3.f.b 2
5.c odd 4 1 650.3.f.e 2
13.b even 2 1 338.3.d.a 2
13.c even 3 2 338.3.f.b 4
13.d odd 4 1 inner 26.3.d.a 2
13.d odd 4 1 338.3.d.a 2
13.e even 6 2 338.3.f.g 4
13.f odd 12 2 338.3.f.b 4
13.f odd 12 2 338.3.f.g 4
39.f even 4 1 234.3.i.a 2
52.f even 4 1 208.3.t.b 2
65.f even 4 1 650.3.f.b 2
65.g odd 4 1 650.3.k.b 2
65.k even 4 1 650.3.f.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.3.d.a 2 1.a even 1 1 trivial
26.3.d.a 2 13.d odd 4 1 inner
208.3.t.b 2 4.b odd 2 1
208.3.t.b 2 52.f even 4 1
234.3.i.a 2 3.b odd 2 1
234.3.i.a 2 39.f even 4 1
338.3.d.a 2 13.b even 2 1
338.3.d.a 2 13.d odd 4 1
338.3.f.b 4 13.c even 3 2
338.3.f.b 4 13.f odd 12 2
338.3.f.g 4 13.e even 6 2
338.3.f.g 4 13.f odd 12 2
650.3.f.b 2 5.c odd 4 1
650.3.f.b 2 65.f even 4 1
650.3.f.e 2 5.c odd 4 1
650.3.f.e 2 65.k even 4 1
650.3.k.b 2 5.b even 2 1
650.3.k.b 2 65.g odd 4 1

Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(26, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 2$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 6T + 18$$
$7$ $$T^{2} - 4T + 8$$
$11$ $$T^{2} - 12T + 72$$
$13$ $$T^{2} + 169$$
$17$ $$T^{2} + 36$$
$19$ $$T^{2} - 52T + 1352$$
$23$ $$T^{2} + 576$$
$29$ $$(T + 48)^{2}$$
$31$ $$T^{2} + 28T + 392$$
$37$ $$T^{2} - 74T + 2738$$
$41$ $$T^{2} + 18T + 162$$
$43$ $$T^{2} + 1296$$
$47$ $$T^{2} - 84T + 3528$$
$53$ $$(T - 30)^{2}$$
$59$ $$T^{2} + 108T + 5832$$
$61$ $$(T + 18)^{2}$$
$67$ $$T^{2} + 44T + 968$$
$71$ $$T^{2} - 12T + 72$$
$73$ $$T^{2} - 34T + 578$$
$79$ $$(T + 108)^{2}$$
$83$ $$T^{2} - 156T + 12168$$
$89$ $$T^{2} + 18T + 162$$
$97$ $$T^{2} + 94T + 4418$$