# Properties

 Label 26.4.a.b Level $26$ Weight $4$ Character orbit 26.a Self dual yes Analytic conductor $1.534$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [26,4,Mod(1,26)]

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([0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("26.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$26 = 2 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 26.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: $$1.53404966015$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} - q^{3} + 4 q^{4} + 17 q^{5} - 2 q^{6} - 35 q^{7} + 8 q^{8} - 26 q^{9}+O(q^{10})$$ q + 2 * q^2 - q^3 + 4 * q^4 + 17 * q^5 - 2 * q^6 - 35 * q^7 + 8 * q^8 - 26 * q^9 $$q + 2 q^{2} - q^{3} + 4 q^{4} + 17 q^{5} - 2 q^{6} - 35 q^{7} + 8 q^{8} - 26 q^{9} + 34 q^{10} + 2 q^{11} - 4 q^{12} + 13 q^{13} - 70 q^{14} - 17 q^{15} + 16 q^{16} - 19 q^{17} - 52 q^{18} + 94 q^{19} + 68 q^{20} + 35 q^{21} + 4 q^{22} - 72 q^{23} - 8 q^{24} + 164 q^{25} + 26 q^{26} + 53 q^{27} - 140 q^{28} + 246 q^{29} - 34 q^{30} - 100 q^{31} + 32 q^{32} - 2 q^{33} - 38 q^{34} - 595 q^{35} - 104 q^{36} - 11 q^{37} + 188 q^{38} - 13 q^{39} + 136 q^{40} - 280 q^{41} + 70 q^{42} + 241 q^{43} + 8 q^{44} - 442 q^{45} - 144 q^{46} + 137 q^{47} - 16 q^{48} + 882 q^{49} + 328 q^{50} + 19 q^{51} + 52 q^{52} - 232 q^{53} + 106 q^{54} + 34 q^{55} - 280 q^{56} - 94 q^{57} + 492 q^{58} - 386 q^{59} - 68 q^{60} + 64 q^{61} - 200 q^{62} + 910 q^{63} + 64 q^{64} + 221 q^{65} - 4 q^{66} - 670 q^{67} - 76 q^{68} + 72 q^{69} - 1190 q^{70} + 55 q^{71} - 208 q^{72} - 838 q^{73} - 22 q^{74} - 164 q^{75} + 376 q^{76} - 70 q^{77} - 26 q^{78} + 1016 q^{79} + 272 q^{80} + 649 q^{81} - 560 q^{82} + 420 q^{83} + 140 q^{84} - 323 q^{85} + 482 q^{86} - 246 q^{87} + 16 q^{88} - 934 q^{89} - 884 q^{90} - 455 q^{91} - 288 q^{92} + 100 q^{93} + 274 q^{94} + 1598 q^{95} - 32 q^{96} - 1154 q^{97} + 1764 q^{98} - 52 q^{99}+O(q^{100})$$ q + 2 * q^2 - q^3 + 4 * q^4 + 17 * q^5 - 2 * q^6 - 35 * q^7 + 8 * q^8 - 26 * q^9 + 34 * q^10 + 2 * q^11 - 4 * q^12 + 13 * q^13 - 70 * q^14 - 17 * q^15 + 16 * q^16 - 19 * q^17 - 52 * q^18 + 94 * q^19 + 68 * q^20 + 35 * q^21 + 4 * q^22 - 72 * q^23 - 8 * q^24 + 164 * q^25 + 26 * q^26 + 53 * q^27 - 140 * q^28 + 246 * q^29 - 34 * q^30 - 100 * q^31 + 32 * q^32 - 2 * q^33 - 38 * q^34 - 595 * q^35 - 104 * q^36 - 11 * q^37 + 188 * q^38 - 13 * q^39 + 136 * q^40 - 280 * q^41 + 70 * q^42 + 241 * q^43 + 8 * q^44 - 442 * q^45 - 144 * q^46 + 137 * q^47 - 16 * q^48 + 882 * q^49 + 328 * q^50 + 19 * q^51 + 52 * q^52 - 232 * q^53 + 106 * q^54 + 34 * q^55 - 280 * q^56 - 94 * q^57 + 492 * q^58 - 386 * q^59 - 68 * q^60 + 64 * q^61 - 200 * q^62 + 910 * q^63 + 64 * q^64 + 221 * q^65 - 4 * q^66 - 670 * q^67 - 76 * q^68 + 72 * q^69 - 1190 * q^70 + 55 * q^71 - 208 * q^72 - 838 * q^73 - 22 * q^74 - 164 * q^75 + 376 * q^76 - 70 * q^77 - 26 * q^78 + 1016 * q^79 + 272 * q^80 + 649 * q^81 - 560 * q^82 + 420 * q^83 + 140 * q^84 - 323 * q^85 + 482 * q^86 - 246 * q^87 + 16 * q^88 - 934 * q^89 - 884 * q^90 - 455 * q^91 - 288 * q^92 + 100 * q^93 + 274 * q^94 + 1598 * q^95 - 32 * q^96 - 1154 * q^97 + 1764 * q^98 - 52 * 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
 0
2.00000 −1.00000 4.00000 17.0000 −2.00000 −35.0000 8.00000 −26.0000 34.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$13$$ $$-1$$

## 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 26.4.a.b 1
3.b odd 2 1 234.4.a.a 1
4.b odd 2 1 208.4.a.e 1
5.b even 2 1 650.4.a.c 1
5.c odd 4 2 650.4.b.d 2
7.b odd 2 1 1274.4.a.f 1
8.b even 2 1 832.4.a.j 1
8.d odd 2 1 832.4.a.g 1
12.b even 2 1 1872.4.a.b 1
13.b even 2 1 338.4.a.b 1
13.c even 3 2 338.4.c.c 2
13.d odd 4 2 338.4.b.b 2
13.e even 6 2 338.4.c.g 2
13.f odd 12 4 338.4.e.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.a.b 1 1.a even 1 1 trivial
208.4.a.e 1 4.b odd 2 1
234.4.a.a 1 3.b odd 2 1
338.4.a.b 1 13.b even 2 1
338.4.b.b 2 13.d odd 4 2
338.4.c.c 2 13.c even 3 2
338.4.c.g 2 13.e even 6 2
338.4.e.c 4 13.f odd 12 4
650.4.a.c 1 5.b even 2 1
650.4.b.d 2 5.c odd 4 2
832.4.a.g 1 8.d odd 2 1
832.4.a.j 1 8.b even 2 1
1274.4.a.f 1 7.b odd 2 1
1872.4.a.b 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 1$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(26))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T + 1$$
$5$ $$T - 17$$
$7$ $$T + 35$$
$11$ $$T - 2$$
$13$ $$T - 13$$
$17$ $$T + 19$$
$19$ $$T - 94$$
$23$ $$T + 72$$
$29$ $$T - 246$$
$31$ $$T + 100$$
$37$ $$T + 11$$
$41$ $$T + 280$$
$43$ $$T - 241$$
$47$ $$T - 137$$
$53$ $$T + 232$$
$59$ $$T + 386$$
$61$ $$T - 64$$
$67$ $$T + 670$$
$71$ $$T - 55$$
$73$ $$T + 838$$
$79$ $$T - 1016$$
$83$ $$T - 420$$
$89$ $$T + 934$$
$97$ $$T + 1154$$