Properties

 Label 26.8.a.c Level $26$ Weight $8$ Character orbit 26.a Self dual yes Analytic conductor $8.122$ Analytic rank $1$ 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] = [26,8,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, 8, names="a")

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

chi := DirichletCharacter("26.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$26 = 2 \cdot 13$$ Weight: $$k$$ $$=$$ $$8$$ 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: $$8.12201066259$$ Analytic rank: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + 8 q^{2} - 27 q^{3} + 64 q^{4} - 245 q^{5} - 216 q^{6} - 587 q^{7} + 512 q^{8} - 1458 q^{9}+O(q^{10})$$ q + 8 * q^2 - 27 * q^3 + 64 * q^4 - 245 * q^5 - 216 * q^6 - 587 * q^7 + 512 * q^8 - 1458 * q^9 $$q + 8 q^{2} - 27 q^{3} + 64 q^{4} - 245 q^{5} - 216 q^{6} - 587 q^{7} + 512 q^{8} - 1458 q^{9} - 1960 q^{10} - 3874 q^{11} - 1728 q^{12} - 2197 q^{13} - 4696 q^{14} + 6615 q^{15} + 4096 q^{16} + 5229 q^{17} - 11664 q^{18} - 6522 q^{19} - 15680 q^{20} + 15849 q^{21} - 30992 q^{22} - 500 q^{23} - 13824 q^{24} - 18100 q^{25} - 17576 q^{26} + 98415 q^{27} - 37568 q^{28} + 226954 q^{29} + 52920 q^{30} + 130156 q^{31} + 32768 q^{32} + 104598 q^{33} + 41832 q^{34} + 143815 q^{35} - 93312 q^{36} - 377769 q^{37} - 52176 q^{38} + 59319 q^{39} - 125440 q^{40} - 539760 q^{41} + 126792 q^{42} + 13987 q^{43} - 247936 q^{44} + 357210 q^{45} - 4000 q^{46} - 526879 q^{47} - 110592 q^{48} - 478974 q^{49} - 144800 q^{50} - 141183 q^{51} - 140608 q^{52} - 1649940 q^{53} + 787320 q^{54} + 949130 q^{55} - 300544 q^{56} + 176094 q^{57} + 1815632 q^{58} - 81194 q^{59} + 423360 q^{60} - 1126952 q^{61} + 1041248 q^{62} + 855846 q^{63} + 262144 q^{64} + 538265 q^{65} + 836784 q^{66} + 478798 q^{67} + 334656 q^{68} + 13500 q^{69} + 1150520 q^{70} + 940007 q^{71} - 746496 q^{72} + 1671926 q^{73} - 3022152 q^{74} + 488700 q^{75} - 417408 q^{76} + 2274038 q^{77} + 474552 q^{78} - 5801188 q^{79} - 1003520 q^{80} + 531441 q^{81} - 4318080 q^{82} + 7398816 q^{83} + 1014336 q^{84} - 1281105 q^{85} + 111896 q^{86} - 6127758 q^{87} - 1983488 q^{88} - 953754 q^{89} + 2857680 q^{90} + 1289639 q^{91} - 32000 q^{92} - 3514212 q^{93} - 4215032 q^{94} + 1597890 q^{95} - 884736 q^{96} - 10318690 q^{97} - 3831792 q^{98} + 5648292 q^{99}+O(q^{100})$$ q + 8 * q^2 - 27 * q^3 + 64 * q^4 - 245 * q^5 - 216 * q^6 - 587 * q^7 + 512 * q^8 - 1458 * q^9 - 1960 * q^10 - 3874 * q^11 - 1728 * q^12 - 2197 * q^13 - 4696 * q^14 + 6615 * q^15 + 4096 * q^16 + 5229 * q^17 - 11664 * q^18 - 6522 * q^19 - 15680 * q^20 + 15849 * q^21 - 30992 * q^22 - 500 * q^23 - 13824 * q^24 - 18100 * q^25 - 17576 * q^26 + 98415 * q^27 - 37568 * q^28 + 226954 * q^29 + 52920 * q^30 + 130156 * q^31 + 32768 * q^32 + 104598 * q^33 + 41832 * q^34 + 143815 * q^35 - 93312 * q^36 - 377769 * q^37 - 52176 * q^38 + 59319 * q^39 - 125440 * q^40 - 539760 * q^41 + 126792 * q^42 + 13987 * q^43 - 247936 * q^44 + 357210 * q^45 - 4000 * q^46 - 526879 * q^47 - 110592 * q^48 - 478974 * q^49 - 144800 * q^50 - 141183 * q^51 - 140608 * q^52 - 1649940 * q^53 + 787320 * q^54 + 949130 * q^55 - 300544 * q^56 + 176094 * q^57 + 1815632 * q^58 - 81194 * q^59 + 423360 * q^60 - 1126952 * q^61 + 1041248 * q^62 + 855846 * q^63 + 262144 * q^64 + 538265 * q^65 + 836784 * q^66 + 478798 * q^67 + 334656 * q^68 + 13500 * q^69 + 1150520 * q^70 + 940007 * q^71 - 746496 * q^72 + 1671926 * q^73 - 3022152 * q^74 + 488700 * q^75 - 417408 * q^76 + 2274038 * q^77 + 474552 * q^78 - 5801188 * q^79 - 1003520 * q^80 + 531441 * q^81 - 4318080 * q^82 + 7398816 * q^83 + 1014336 * q^84 - 1281105 * q^85 + 111896 * q^86 - 6127758 * q^87 - 1983488 * q^88 - 953754 * q^89 + 2857680 * q^90 + 1289639 * q^91 - 32000 * q^92 - 3514212 * q^93 - 4215032 * q^94 + 1597890 * q^95 - 884736 * q^96 - 10318690 * q^97 - 3831792 * q^98 + 5648292 * 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
8.00000 −27.0000 64.0000 −245.000 −216.000 −587.000 512.000 −1458.00 −1960.00
 $$n$$: 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 26.8.a.c 1
3.b odd 2 1 234.8.a.b 1
4.b odd 2 1 208.8.a.a 1
13.b even 2 1 338.8.a.b 1
13.d odd 4 2 338.8.b.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.8.a.c 1 1.a even 1 1 trivial
208.8.a.a 1 4.b odd 2 1
234.8.a.b 1 3.b odd 2 1
338.8.a.b 1 13.b even 2 1
338.8.b.c 2 13.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 27$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(26))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 8$$
$3$ $$T + 27$$
$5$ $$T + 245$$
$7$ $$T + 587$$
$11$ $$T + 3874$$
$13$ $$T + 2197$$
$17$ $$T - 5229$$
$19$ $$T + 6522$$
$23$ $$T + 500$$
$29$ $$T - 226954$$
$31$ $$T - 130156$$
$37$ $$T + 377769$$
$41$ $$T + 539760$$
$43$ $$T - 13987$$
$47$ $$T + 526879$$
$53$ $$T + 1649940$$
$59$ $$T + 81194$$
$61$ $$T + 1126952$$
$67$ $$T - 478798$$
$71$ $$T - 940007$$
$73$ $$T - 1671926$$
$79$ $$T + 5801188$$
$83$ $$T - 7398816$$
$89$ $$T + 953754$$
$97$ $$T + 10318690$$