# Properties

 Label 26.2.c.a Level $26$ Weight $2$ Character orbit 26.c Analytic conductor $0.208$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [26,2,Mod(3,26)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(26, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("26.3");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$26 = 2 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 26.c (of order $$3$$, 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: $$0.207611045255$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} - q^{5} - 4 \zeta_{6} q^{7} + q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})$$ q + (z - 1) * q^2 - z * q^4 - q^5 - 4*z * q^7 + q^8 + 3*z * q^9 $$q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} - q^{5} - 4 \zeta_{6} q^{7} + q^{8} + 3 \zeta_{6} q^{9} + ( - \zeta_{6} + 1) q^{10} + (4 \zeta_{6} - 4) q^{11} + (\zeta_{6} + 3) q^{13} + 4 q^{14} + (\zeta_{6} - 1) q^{16} - 3 \zeta_{6} q^{17} - 3 q^{18} + \zeta_{6} q^{20} - 4 \zeta_{6} q^{22} + ( - 4 \zeta_{6} + 4) q^{23} - 4 q^{25} + (3 \zeta_{6} - 4) q^{26} + (4 \zeta_{6} - 4) q^{28} + ( - \zeta_{6} + 1) q^{29} + 4 q^{31} - \zeta_{6} q^{32} + 3 q^{34} + 4 \zeta_{6} q^{35} + ( - 3 \zeta_{6} + 3) q^{36} + (3 \zeta_{6} - 3) q^{37} - q^{40} + ( - 9 \zeta_{6} + 9) q^{41} + 8 \zeta_{6} q^{43} + 4 q^{44} - 3 \zeta_{6} q^{45} + 4 \zeta_{6} q^{46} - 8 q^{47} + (9 \zeta_{6} - 9) q^{49} + ( - 4 \zeta_{6} + 4) q^{50} + ( - 4 \zeta_{6} + 1) q^{52} - 9 q^{53} + ( - 4 \zeta_{6} + 4) q^{55} - 4 \zeta_{6} q^{56} + \zeta_{6} q^{58} + 4 \zeta_{6} q^{59} - 7 \zeta_{6} q^{61} + (4 \zeta_{6} - 4) q^{62} + ( - 12 \zeta_{6} + 12) q^{63} + q^{64} + ( - \zeta_{6} - 3) q^{65} + (4 \zeta_{6} - 4) q^{67} + (3 \zeta_{6} - 3) q^{68} - 4 q^{70} + 8 \zeta_{6} q^{71} + 3 \zeta_{6} q^{72} + 11 q^{73} - 3 \zeta_{6} q^{74} + 16 q^{77} - 4 q^{79} + ( - \zeta_{6} + 1) q^{80} + (9 \zeta_{6} - 9) q^{81} + 9 \zeta_{6} q^{82} + 3 \zeta_{6} q^{85} - 8 q^{86} + (4 \zeta_{6} - 4) q^{88} + ( - 6 \zeta_{6} + 6) q^{89} + 3 q^{90} + ( - 16 \zeta_{6} + 4) q^{91} - 4 q^{92} + ( - 8 \zeta_{6} + 8) q^{94} - 2 \zeta_{6} q^{97} - 9 \zeta_{6} q^{98} - 12 q^{99} +O(q^{100})$$ q + (z - 1) * q^2 - z * q^4 - q^5 - 4*z * q^7 + q^8 + 3*z * q^9 + (-z + 1) * q^10 + (4*z - 4) * q^11 + (z + 3) * q^13 + 4 * q^14 + (z - 1) * q^16 - 3*z * q^17 - 3 * q^18 + z * q^20 - 4*z * q^22 + (-4*z + 4) * q^23 - 4 * q^25 + (3*z - 4) * q^26 + (4*z - 4) * q^28 + (-z + 1) * q^29 + 4 * q^31 - z * q^32 + 3 * q^34 + 4*z * q^35 + (-3*z + 3) * q^36 + (3*z - 3) * q^37 - q^40 + (-9*z + 9) * q^41 + 8*z * q^43 + 4 * q^44 - 3*z * q^45 + 4*z * q^46 - 8 * q^47 + (9*z - 9) * q^49 + (-4*z + 4) * q^50 + (-4*z + 1) * q^52 - 9 * q^53 + (-4*z + 4) * q^55 - 4*z * q^56 + z * q^58 + 4*z * q^59 - 7*z * q^61 + (4*z - 4) * q^62 + (-12*z + 12) * q^63 + q^64 + (-z - 3) * q^65 + (4*z - 4) * q^67 + (3*z - 3) * q^68 - 4 * q^70 + 8*z * q^71 + 3*z * q^72 + 11 * q^73 - 3*z * q^74 + 16 * q^77 - 4 * q^79 + (-z + 1) * q^80 + (9*z - 9) * q^81 + 9*z * q^82 + 3*z * q^85 - 8 * q^86 + (4*z - 4) * q^88 + (-6*z + 6) * q^89 + 3 * q^90 + (-16*z + 4) * q^91 - 4 * q^92 + (-8*z + 8) * q^94 - 2*z * q^97 - 9*z * q^98 - 12 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} - 2 q^{5} - 4 q^{7} + 2 q^{8} + 3 q^{9}+O(q^{10})$$ 2 * q - q^2 - q^4 - 2 * q^5 - 4 * q^7 + 2 * q^8 + 3 * q^9 $$2 q - q^{2} - q^{4} - 2 q^{5} - 4 q^{7} + 2 q^{8} + 3 q^{9} + q^{10} - 4 q^{11} + 7 q^{13} + 8 q^{14} - q^{16} - 3 q^{17} - 6 q^{18} + q^{20} - 4 q^{22} + 4 q^{23} - 8 q^{25} - 5 q^{26} - 4 q^{28} + q^{29} + 8 q^{31} - q^{32} + 6 q^{34} + 4 q^{35} + 3 q^{36} - 3 q^{37} - 2 q^{40} + 9 q^{41} + 8 q^{43} + 8 q^{44} - 3 q^{45} + 4 q^{46} - 16 q^{47} - 9 q^{49} + 4 q^{50} - 2 q^{52} - 18 q^{53} + 4 q^{55} - 4 q^{56} + q^{58} + 4 q^{59} - 7 q^{61} - 4 q^{62} + 12 q^{63} + 2 q^{64} - 7 q^{65} - 4 q^{67} - 3 q^{68} - 8 q^{70} + 8 q^{71} + 3 q^{72} + 22 q^{73} - 3 q^{74} + 32 q^{77} - 8 q^{79} + q^{80} - 9 q^{81} + 9 q^{82} + 3 q^{85} - 16 q^{86} - 4 q^{88} + 6 q^{89} + 6 q^{90} - 8 q^{91} - 8 q^{92} + 8 q^{94} - 2 q^{97} - 9 q^{98} - 24 q^{99}+O(q^{100})$$ 2 * q - q^2 - q^4 - 2 * q^5 - 4 * q^7 + 2 * q^8 + 3 * q^9 + q^10 - 4 * q^11 + 7 * q^13 + 8 * q^14 - q^16 - 3 * q^17 - 6 * q^18 + q^20 - 4 * q^22 + 4 * q^23 - 8 * q^25 - 5 * q^26 - 4 * q^28 + q^29 + 8 * q^31 - q^32 + 6 * q^34 + 4 * q^35 + 3 * q^36 - 3 * q^37 - 2 * q^40 + 9 * q^41 + 8 * q^43 + 8 * q^44 - 3 * q^45 + 4 * q^46 - 16 * q^47 - 9 * q^49 + 4 * q^50 - 2 * q^52 - 18 * q^53 + 4 * q^55 - 4 * q^56 + q^58 + 4 * q^59 - 7 * q^61 - 4 * q^62 + 12 * q^63 + 2 * q^64 - 7 * q^65 - 4 * q^67 - 3 * q^68 - 8 * q^70 + 8 * q^71 + 3 * q^72 + 22 * q^73 - 3 * q^74 + 32 * q^77 - 8 * q^79 + q^80 - 9 * q^81 + 9 * q^82 + 3 * q^85 - 16 * q^86 - 4 * q^88 + 6 * q^89 + 6 * q^90 - 8 * q^91 - 8 * q^92 + 8 * q^94 - 2 * q^97 - 9 * q^98 - 24 * q^99

## Character values

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

 $$n$$ $$15$$ $$\chi(n)$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −1.00000 0 −2.00000 + 3.46410i 1.00000 1.50000 2.59808i 0.500000 + 0.866025i
9.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −1.00000 0 −2.00000 3.46410i 1.00000 1.50000 + 2.59808i 0.500000 0.866025i
 $$n$$: 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 26.2.c.a 2
3.b odd 2 1 234.2.h.c 2
4.b odd 2 1 208.2.i.b 2
5.b even 2 1 650.2.e.c 2
5.c odd 4 2 650.2.o.c 4
7.b odd 2 1 1274.2.g.a 2
7.c even 3 1 1274.2.e.n 2
7.c even 3 1 1274.2.h.b 2
7.d odd 6 1 1274.2.e.m 2
7.d odd 6 1 1274.2.h.a 2
8.b even 2 1 832.2.i.e 2
8.d odd 2 1 832.2.i.f 2
12.b even 2 1 1872.2.t.k 2
13.b even 2 1 338.2.c.e 2
13.c even 3 1 inner 26.2.c.a 2
13.c even 3 1 338.2.a.e 1
13.d odd 4 2 338.2.e.b 4
13.e even 6 1 338.2.a.c 1
13.e even 6 1 338.2.c.e 2
13.f odd 12 2 338.2.b.b 2
13.f odd 12 2 338.2.e.b 4
39.h odd 6 1 3042.2.a.k 1
39.i odd 6 1 234.2.h.c 2
39.i odd 6 1 3042.2.a.e 1
39.k even 12 2 3042.2.b.e 2
52.i odd 6 1 2704.2.a.i 1
52.j odd 6 1 208.2.i.b 2
52.j odd 6 1 2704.2.a.h 1
52.l even 12 2 2704.2.f.g 2
65.l even 6 1 8450.2.a.s 1
65.n even 6 1 650.2.e.c 2
65.n even 6 1 8450.2.a.f 1
65.q odd 12 2 650.2.o.c 4
91.g even 3 1 1274.2.e.n 2
91.h even 3 1 1274.2.h.b 2
91.m odd 6 1 1274.2.e.m 2
91.n odd 6 1 1274.2.g.a 2
91.v odd 6 1 1274.2.h.a 2
104.n odd 6 1 832.2.i.f 2
104.r even 6 1 832.2.i.e 2
156.p even 6 1 1872.2.t.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.c.a 2 1.a even 1 1 trivial
26.2.c.a 2 13.c even 3 1 inner
208.2.i.b 2 4.b odd 2 1
208.2.i.b 2 52.j odd 6 1
234.2.h.c 2 3.b odd 2 1
234.2.h.c 2 39.i odd 6 1
338.2.a.c 1 13.e even 6 1
338.2.a.e 1 13.c even 3 1
338.2.b.b 2 13.f odd 12 2
338.2.c.e 2 13.b even 2 1
338.2.c.e 2 13.e even 6 1
338.2.e.b 4 13.d odd 4 2
338.2.e.b 4 13.f odd 12 2
650.2.e.c 2 5.b even 2 1
650.2.e.c 2 65.n even 6 1
650.2.o.c 4 5.c odd 4 2
650.2.o.c 4 65.q odd 12 2
832.2.i.e 2 8.b even 2 1
832.2.i.e 2 104.r even 6 1
832.2.i.f 2 8.d odd 2 1
832.2.i.f 2 104.n odd 6 1
1274.2.e.m 2 7.d odd 6 1
1274.2.e.m 2 91.m odd 6 1
1274.2.e.n 2 7.c even 3 1
1274.2.e.n 2 91.g even 3 1
1274.2.g.a 2 7.b odd 2 1
1274.2.g.a 2 91.n odd 6 1
1274.2.h.a 2 7.d odd 6 1
1274.2.h.a 2 91.v odd 6 1
1274.2.h.b 2 7.c even 3 1
1274.2.h.b 2 91.h even 3 1
1872.2.t.k 2 12.b even 2 1
1872.2.t.k 2 156.p even 6 1
2704.2.a.h 1 52.j odd 6 1
2704.2.a.i 1 52.i odd 6 1
2704.2.f.g 2 52.l even 12 2
3042.2.a.e 1 39.i odd 6 1
3042.2.a.k 1 39.h odd 6 1
3042.2.b.e 2 39.k even 12 2
8450.2.a.f 1 65.n even 6 1
8450.2.a.s 1 65.l even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(26, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + 4T + 16$$
$11$ $$T^{2} + 4T + 16$$
$13$ $$T^{2} - 7T + 13$$
$17$ $$T^{2} + 3T + 9$$
$19$ $$T^{2}$$
$23$ $$T^{2} - 4T + 16$$
$29$ $$T^{2} - T + 1$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 3T + 9$$
$41$ $$T^{2} - 9T + 81$$
$43$ $$T^{2} - 8T + 64$$
$47$ $$(T + 8)^{2}$$
$53$ $$(T + 9)^{2}$$
$59$ $$T^{2} - 4T + 16$$
$61$ $$T^{2} + 7T + 49$$
$67$ $$T^{2} + 4T + 16$$
$71$ $$T^{2} - 8T + 64$$
$73$ $$(T - 11)^{2}$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2} - 6T + 36$$
$97$ $$T^{2} + 2T + 4$$