# Properties

 Label 26.2.b.a Level $26$ Weight $2$ Character orbit 26.b 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(25,26)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(26, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

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

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

chi := DirichletCharacter("26.25");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$26 = 2 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 26.b (of order $$2$$, degree $$1$$, 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: $$\Q(\sqrt{-1})$$ 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: $\mathrm{SU}(2)[C_{2}]$

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

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{3} - q^{4} - 3 i q^{5} - i q^{6} + 3 i q^{7} - i q^{8} - 2 q^{9} +O(q^{10})$$ q + i * q^2 - q^3 - q^4 - 3*i * q^5 - i * q^6 + 3*i * q^7 - i * q^8 - 2 * q^9 $$q + i q^{2} - q^{3} - q^{4} - 3 i q^{5} - i q^{6} + 3 i q^{7} - i q^{8} - 2 q^{9} + 3 q^{10} + q^{12} + (3 i + 2) q^{13} - 3 q^{14} + 3 i q^{15} + q^{16} + 3 q^{17} - 2 i q^{18} - 6 i q^{19} + 3 i q^{20} - 3 i q^{21} - 6 q^{23} + i q^{24} - 4 q^{25} + (2 i - 3) q^{26} + 5 q^{27} - 3 i q^{28} - 3 q^{30} + i q^{32} + 3 i q^{34} + 9 q^{35} + 2 q^{36} + 3 i q^{37} + 6 q^{38} + ( - 3 i - 2) q^{39} - 3 q^{40} + 3 q^{42} - q^{43} + 6 i q^{45} - 6 i q^{46} + 3 i q^{47} - q^{48} - 2 q^{49} - 4 i q^{50} - 3 q^{51} + ( - 3 i - 2) q^{52} - 6 q^{53} + 5 i q^{54} + 3 q^{56} + 6 i q^{57} - 6 i q^{59} - 3 i q^{60} - 8 q^{61} - 6 i q^{63} - q^{64} + ( - 6 i + 9) q^{65} - 12 i q^{67} - 3 q^{68} + 6 q^{69} + 9 i q^{70} + 15 i q^{71} + 2 i q^{72} + 6 i q^{73} - 3 q^{74} + 4 q^{75} + 6 i q^{76} + ( - 2 i + 3) q^{78} + 10 q^{79} - 3 i q^{80} + q^{81} + 6 i q^{83} + 3 i q^{84} - 9 i q^{85} - i q^{86} - 6 i q^{89} - 6 q^{90} + (6 i - 9) q^{91} + 6 q^{92} - 3 q^{94} - 18 q^{95} - i q^{96} - 12 i q^{97} - 2 i q^{98} +O(q^{100})$$ q + i * q^2 - q^3 - q^4 - 3*i * q^5 - i * q^6 + 3*i * q^7 - i * q^8 - 2 * q^9 + 3 * q^10 + q^12 + (3*i + 2) * q^13 - 3 * q^14 + 3*i * q^15 + q^16 + 3 * q^17 - 2*i * q^18 - 6*i * q^19 + 3*i * q^20 - 3*i * q^21 - 6 * q^23 + i * q^24 - 4 * q^25 + (2*i - 3) * q^26 + 5 * q^27 - 3*i * q^28 - 3 * q^30 + i * q^32 + 3*i * q^34 + 9 * q^35 + 2 * q^36 + 3*i * q^37 + 6 * q^38 + (-3*i - 2) * q^39 - 3 * q^40 + 3 * q^42 - q^43 + 6*i * q^45 - 6*i * q^46 + 3*i * q^47 - q^48 - 2 * q^49 - 4*i * q^50 - 3 * q^51 + (-3*i - 2) * q^52 - 6 * q^53 + 5*i * q^54 + 3 * q^56 + 6*i * q^57 - 6*i * q^59 - 3*i * q^60 - 8 * q^61 - 6*i * q^63 - q^64 + (-6*i + 9) * q^65 - 12*i * q^67 - 3 * q^68 + 6 * q^69 + 9*i * q^70 + 15*i * q^71 + 2*i * q^72 + 6*i * q^73 - 3 * q^74 + 4 * q^75 + 6*i * q^76 + (-2*i + 3) * q^78 + 10 * q^79 - 3*i * q^80 + q^81 + 6*i * q^83 + 3*i * q^84 - 9*i * q^85 - i * q^86 - 6*i * q^89 - 6 * q^90 + (6*i - 9) * q^91 + 6 * q^92 - 3 * q^94 - 18 * q^95 - i * q^96 - 12*i * q^97 - 2*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 2 q^{4} - 4 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 2 * q^4 - 4 * q^9 $$2 q - 2 q^{3} - 2 q^{4} - 4 q^{9} + 6 q^{10} + 2 q^{12} + 4 q^{13} - 6 q^{14} + 2 q^{16} + 6 q^{17} - 12 q^{23} - 8 q^{25} - 6 q^{26} + 10 q^{27} - 6 q^{30} + 18 q^{35} + 4 q^{36} + 12 q^{38} - 4 q^{39} - 6 q^{40} + 6 q^{42} - 2 q^{43} - 2 q^{48} - 4 q^{49} - 6 q^{51} - 4 q^{52} - 12 q^{53} + 6 q^{56} - 16 q^{61} - 2 q^{64} + 18 q^{65} - 6 q^{68} + 12 q^{69} - 6 q^{74} + 8 q^{75} + 6 q^{78} + 20 q^{79} + 2 q^{81} - 12 q^{90} - 18 q^{91} + 12 q^{92} - 6 q^{94} - 36 q^{95}+O(q^{100})$$ 2 * q - 2 * q^3 - 2 * q^4 - 4 * q^9 + 6 * q^10 + 2 * q^12 + 4 * q^13 - 6 * q^14 + 2 * q^16 + 6 * q^17 - 12 * q^23 - 8 * q^25 - 6 * q^26 + 10 * q^27 - 6 * q^30 + 18 * q^35 + 4 * q^36 + 12 * q^38 - 4 * q^39 - 6 * q^40 + 6 * q^42 - 2 * q^43 - 2 * q^48 - 4 * q^49 - 6 * q^51 - 4 * q^52 - 12 * q^53 + 6 * q^56 - 16 * q^61 - 2 * q^64 + 18 * q^65 - 6 * q^68 + 12 * q^69 - 6 * q^74 + 8 * q^75 + 6 * q^78 + 20 * q^79 + 2 * q^81 - 12 * q^90 - 18 * q^91 + 12 * q^92 - 6 * q^94 - 36 * q^95

## Character values

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

 $$n$$ $$15$$ $$\chi(n)$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 − 1.00000i 1.00000i
1.00000i −1.00000 −1.00000 3.00000i 1.00000i 3.00000i 1.00000i −2.00000 3.00000
25.2 1.00000i −1.00000 −1.00000 3.00000i 1.00000i 3.00000i 1.00000i −2.00000 3.00000
 $$n$$: 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.b even 2 1 inner

## Twists

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

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} + 9$$
$7$ $$T^{2} + 9$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 4T + 13$$
$17$ $$(T - 3)^{2}$$
$19$ $$T^{2} + 36$$
$23$ $$(T + 6)^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 9$$
$41$ $$T^{2}$$
$43$ $$(T + 1)^{2}$$
$47$ $$T^{2} + 9$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} + 36$$
$61$ $$(T + 8)^{2}$$
$67$ $$T^{2} + 144$$
$71$ $$T^{2} + 225$$
$73$ $$T^{2} + 36$$
$79$ $$(T - 10)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$T^{2} + 36$$
$97$ $$T^{2} + 144$$