# Properties

 Label 26.4.a.c 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 4 q^{3} + 4 q^{4} - 18 q^{5} + 8 q^{6} + 20 q^{7} + 8 q^{8} - 11 q^{9}+O(q^{10})$$ q + 2 * q^2 + 4 * q^3 + 4 * q^4 - 18 * q^5 + 8 * q^6 + 20 * q^7 + 8 * q^8 - 11 * q^9 $$q + 2 q^{2} + 4 q^{3} + 4 q^{4} - 18 q^{5} + 8 q^{6} + 20 q^{7} + 8 q^{8} - 11 q^{9} - 36 q^{10} - 48 q^{11} + 16 q^{12} + 13 q^{13} + 40 q^{14} - 72 q^{15} + 16 q^{16} + 66 q^{17} - 22 q^{18} - 16 q^{19} - 72 q^{20} + 80 q^{21} - 96 q^{22} + 168 q^{23} + 32 q^{24} + 199 q^{25} + 26 q^{26} - 152 q^{27} + 80 q^{28} + 6 q^{29} - 144 q^{30} + 20 q^{31} + 32 q^{32} - 192 q^{33} + 132 q^{34} - 360 q^{35} - 44 q^{36} + 254 q^{37} - 32 q^{38} + 52 q^{39} - 144 q^{40} - 390 q^{41} + 160 q^{42} - 124 q^{43} - 192 q^{44} + 198 q^{45} + 336 q^{46} - 468 q^{47} + 64 q^{48} + 57 q^{49} + 398 q^{50} + 264 q^{51} + 52 q^{52} + 558 q^{53} - 304 q^{54} + 864 q^{55} + 160 q^{56} - 64 q^{57} + 12 q^{58} - 96 q^{59} - 288 q^{60} - 826 q^{61} + 40 q^{62} - 220 q^{63} + 64 q^{64} - 234 q^{65} - 384 q^{66} - 160 q^{67} + 264 q^{68} + 672 q^{69} - 720 q^{70} - 420 q^{71} - 88 q^{72} + 362 q^{73} + 508 q^{74} + 796 q^{75} - 64 q^{76} - 960 q^{77} + 104 q^{78} + 776 q^{79} - 288 q^{80} - 311 q^{81} - 780 q^{82} + 320 q^{84} - 1188 q^{85} - 248 q^{86} + 24 q^{87} - 384 q^{88} + 1626 q^{89} + 396 q^{90} + 260 q^{91} + 672 q^{92} + 80 q^{93} - 936 q^{94} + 288 q^{95} + 128 q^{96} - 1294 q^{97} + 114 q^{98} + 528 q^{99}+O(q^{100})$$ q + 2 * q^2 + 4 * q^3 + 4 * q^4 - 18 * q^5 + 8 * q^6 + 20 * q^7 + 8 * q^8 - 11 * q^9 - 36 * q^10 - 48 * q^11 + 16 * q^12 + 13 * q^13 + 40 * q^14 - 72 * q^15 + 16 * q^16 + 66 * q^17 - 22 * q^18 - 16 * q^19 - 72 * q^20 + 80 * q^21 - 96 * q^22 + 168 * q^23 + 32 * q^24 + 199 * q^25 + 26 * q^26 - 152 * q^27 + 80 * q^28 + 6 * q^29 - 144 * q^30 + 20 * q^31 + 32 * q^32 - 192 * q^33 + 132 * q^34 - 360 * q^35 - 44 * q^36 + 254 * q^37 - 32 * q^38 + 52 * q^39 - 144 * q^40 - 390 * q^41 + 160 * q^42 - 124 * q^43 - 192 * q^44 + 198 * q^45 + 336 * q^46 - 468 * q^47 + 64 * q^48 + 57 * q^49 + 398 * q^50 + 264 * q^51 + 52 * q^52 + 558 * q^53 - 304 * q^54 + 864 * q^55 + 160 * q^56 - 64 * q^57 + 12 * q^58 - 96 * q^59 - 288 * q^60 - 826 * q^61 + 40 * q^62 - 220 * q^63 + 64 * q^64 - 234 * q^65 - 384 * q^66 - 160 * q^67 + 264 * q^68 + 672 * q^69 - 720 * q^70 - 420 * q^71 - 88 * q^72 + 362 * q^73 + 508 * q^74 + 796 * q^75 - 64 * q^76 - 960 * q^77 + 104 * q^78 + 776 * q^79 - 288 * q^80 - 311 * q^81 - 780 * q^82 + 320 * q^84 - 1188 * q^85 - 248 * q^86 + 24 * q^87 - 384 * q^88 + 1626 * q^89 + 396 * q^90 + 260 * q^91 + 672 * q^92 + 80 * q^93 - 936 * q^94 + 288 * q^95 + 128 * q^96 - 1294 * q^97 + 114 * q^98 + 528 * 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 4.00000 4.00000 −18.0000 8.00000 20.0000 8.00000 −11.0000 −36.0000
 $$n$$: 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.c 1
3.b odd 2 1 234.4.a.e 1
4.b odd 2 1 208.4.a.b 1
5.b even 2 1 650.4.a.b 1
5.c odd 4 2 650.4.b.f 2
7.b odd 2 1 1274.4.a.d 1
8.b even 2 1 832.4.a.d 1
8.d odd 2 1 832.4.a.o 1
12.b even 2 1 1872.4.a.q 1
13.b even 2 1 338.4.a.c 1
13.c even 3 2 338.4.c.a 2
13.d odd 4 2 338.4.b.d 2
13.e even 6 2 338.4.c.e 2
13.f odd 12 4 338.4.e.a 4

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T - 4$$
$5$ $$T + 18$$
$7$ $$T - 20$$
$11$ $$T + 48$$
$13$ $$T - 13$$
$17$ $$T - 66$$
$19$ $$T + 16$$
$23$ $$T - 168$$
$29$ $$T - 6$$
$31$ $$T - 20$$
$37$ $$T - 254$$
$41$ $$T + 390$$
$43$ $$T + 124$$
$47$ $$T + 468$$
$53$ $$T - 558$$
$59$ $$T + 96$$
$61$ $$T + 826$$
$67$ $$T + 160$$
$71$ $$T + 420$$
$73$ $$T - 362$$
$79$ $$T - 776$$
$83$ $$T$$
$89$ $$T - 1626$$
$97$ $$T + 1294$$