Properties

 Label 1274.4.a.f Level $1274$ Weight $4$ Character orbit 1274.a Self dual yes Analytic conductor $75.168$ 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] = [1274,4,Mod(1,1274)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1274, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

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

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

chi := DirichletCharacter("1274.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1274 = 2 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1274.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: $$75.1684333473$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + q^{3} + 4 q^{4} - 17 q^{5} + 2 q^{6} + 8 q^{8} - 26 q^{9}+O(q^{10})$$ q + 2 * q^2 + q^3 + 4 * q^4 - 17 * q^5 + 2 * q^6 + 8 * q^8 - 26 * q^9 $$q + 2 q^{2} + q^{3} + 4 q^{4} - 17 q^{5} + 2 q^{6} + 8 q^{8} - 26 q^{9} - 34 q^{10} + 2 q^{11} + 4 q^{12} - 13 q^{13} - 17 q^{15} + 16 q^{16} + 19 q^{17} - 52 q^{18} - 94 q^{19} - 68 q^{20} + 4 q^{22} - 72 q^{23} + 8 q^{24} + 164 q^{25} - 26 q^{26} - 53 q^{27} + 246 q^{29} - 34 q^{30} + 100 q^{31} + 32 q^{32} + 2 q^{33} + 38 q^{34} - 104 q^{36} - 11 q^{37} - 188 q^{38} - 13 q^{39} - 136 q^{40} + 280 q^{41} + 241 q^{43} + 8 q^{44} + 442 q^{45} - 144 q^{46} - 137 q^{47} + 16 q^{48} + 328 q^{50} + 19 q^{51} - 52 q^{52} - 232 q^{53} - 106 q^{54} - 34 q^{55} - 94 q^{57} + 492 q^{58} + 386 q^{59} - 68 q^{60} - 64 q^{61} + 200 q^{62} + 64 q^{64} + 221 q^{65} + 4 q^{66} - 670 q^{67} + 76 q^{68} - 72 q^{69} + 55 q^{71} - 208 q^{72} + 838 q^{73} - 22 q^{74} + 164 q^{75} - 376 q^{76} - 26 q^{78} + 1016 q^{79} - 272 q^{80} + 649 q^{81} + 560 q^{82} - 420 q^{83} - 323 q^{85} + 482 q^{86} + 246 q^{87} + 16 q^{88} + 934 q^{89} + 884 q^{90} - 288 q^{92} + 100 q^{93} - 274 q^{94} + 1598 q^{95} + 32 q^{96} + 1154 q^{97} - 52 q^{99}+O(q^{100})$$ q + 2 * q^2 + q^3 + 4 * q^4 - 17 * q^5 + 2 * q^6 + 8 * q^8 - 26 * q^9 - 34 * q^10 + 2 * q^11 + 4 * q^12 - 13 * q^13 - 17 * q^15 + 16 * q^16 + 19 * q^17 - 52 * q^18 - 94 * q^19 - 68 * q^20 + 4 * q^22 - 72 * q^23 + 8 * q^24 + 164 * q^25 - 26 * q^26 - 53 * q^27 + 246 * q^29 - 34 * q^30 + 100 * q^31 + 32 * q^32 + 2 * q^33 + 38 * q^34 - 104 * q^36 - 11 * q^37 - 188 * q^38 - 13 * q^39 - 136 * q^40 + 280 * q^41 + 241 * q^43 + 8 * q^44 + 442 * q^45 - 144 * q^46 - 137 * q^47 + 16 * q^48 + 328 * q^50 + 19 * q^51 - 52 * q^52 - 232 * q^53 - 106 * q^54 - 34 * q^55 - 94 * q^57 + 492 * q^58 + 386 * q^59 - 68 * q^60 - 64 * q^61 + 200 * q^62 + 64 * q^64 + 221 * q^65 + 4 * q^66 - 670 * q^67 + 76 * q^68 - 72 * q^69 + 55 * q^71 - 208 * q^72 + 838 * q^73 - 22 * q^74 + 164 * q^75 - 376 * q^76 - 26 * q^78 + 1016 * q^79 - 272 * q^80 + 649 * q^81 + 560 * q^82 - 420 * q^83 - 323 * q^85 + 482 * q^86 + 246 * q^87 + 16 * q^88 + 934 * q^89 + 884 * q^90 - 288 * q^92 + 100 * q^93 - 274 * q^94 + 1598 * q^95 + 32 * q^96 + 1154 * q^97 - 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 0 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
$$2$$ $$-1$$
$$7$$ $$-1$$
$$13$$ $$+1$$

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 1274.4.a.f 1
7.b odd 2 1 26.4.a.b 1
21.c even 2 1 234.4.a.a 1
28.d even 2 1 208.4.a.e 1
35.c odd 2 1 650.4.a.c 1
35.f even 4 2 650.4.b.d 2
56.e even 2 1 832.4.a.g 1
56.h odd 2 1 832.4.a.j 1
84.h odd 2 1 1872.4.a.b 1
91.b odd 2 1 338.4.a.b 1
91.i even 4 2 338.4.b.b 2
91.n odd 6 2 338.4.c.c 2
91.t odd 6 2 338.4.c.g 2
91.bc even 12 4 338.4.e.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.a.b 1 7.b odd 2 1
208.4.a.e 1 28.d even 2 1
234.4.a.a 1 21.c even 2 1
338.4.a.b 1 91.b odd 2 1
338.4.b.b 2 91.i even 4 2
338.4.c.c 2 91.n odd 6 2
338.4.c.g 2 91.t odd 6 2
338.4.e.c 4 91.bc even 12 4
650.4.a.c 1 35.c odd 2 1
650.4.b.d 2 35.f even 4 2
832.4.a.g 1 56.e even 2 1
832.4.a.j 1 56.h odd 2 1
1274.4.a.f 1 1.a even 1 1 trivial
1872.4.a.b 1 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1274))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{5} + 17$$ T5 + 17

Hecke characteristic polynomials

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