# Properties

 Label 49.4.a.b Level $49$ Weight $4$ Character orbit 49.a Self dual yes Analytic conductor $2.891$ 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] = [49,4,Mod(1,49)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(49, 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("49.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 49.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: $$2.89109359028$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + 2 q^{3} - 7 q^{4} - 16 q^{5} - 2 q^{6} + 15 q^{8} - 23 q^{9}+O(q^{10})$$ q - q^2 + 2 * q^3 - 7 * q^4 - 16 * q^5 - 2 * q^6 + 15 * q^8 - 23 * q^9 $$q - q^{2} + 2 q^{3} - 7 q^{4} - 16 q^{5} - 2 q^{6} + 15 q^{8} - 23 q^{9} + 16 q^{10} - 8 q^{11} - 14 q^{12} - 28 q^{13} - 32 q^{15} + 41 q^{16} - 54 q^{17} + 23 q^{18} + 110 q^{19} + 112 q^{20} + 8 q^{22} + 48 q^{23} + 30 q^{24} + 131 q^{25} + 28 q^{26} - 100 q^{27} - 110 q^{29} + 32 q^{30} - 12 q^{31} - 161 q^{32} - 16 q^{33} + 54 q^{34} + 161 q^{36} - 246 q^{37} - 110 q^{38} - 56 q^{39} - 240 q^{40} - 182 q^{41} + 128 q^{43} + 56 q^{44} + 368 q^{45} - 48 q^{46} - 324 q^{47} + 82 q^{48} - 131 q^{50} - 108 q^{51} + 196 q^{52} - 162 q^{53} + 100 q^{54} + 128 q^{55} + 220 q^{57} + 110 q^{58} - 810 q^{59} + 224 q^{60} + 488 q^{61} + 12 q^{62} - 167 q^{64} + 448 q^{65} + 16 q^{66} + 244 q^{67} + 378 q^{68} + 96 q^{69} - 768 q^{71} - 345 q^{72} + 702 q^{73} + 246 q^{74} + 262 q^{75} - 770 q^{76} + 56 q^{78} + 440 q^{79} - 656 q^{80} + 421 q^{81} + 182 q^{82} + 1302 q^{83} + 864 q^{85} - 128 q^{86} - 220 q^{87} - 120 q^{88} - 730 q^{89} - 368 q^{90} - 336 q^{92} - 24 q^{93} + 324 q^{94} - 1760 q^{95} - 322 q^{96} - 294 q^{97} + 184 q^{99}+O(q^{100})$$ q - q^2 + 2 * q^3 - 7 * q^4 - 16 * q^5 - 2 * q^6 + 15 * q^8 - 23 * q^9 + 16 * q^10 - 8 * q^11 - 14 * q^12 - 28 * q^13 - 32 * q^15 + 41 * q^16 - 54 * q^17 + 23 * q^18 + 110 * q^19 + 112 * q^20 + 8 * q^22 + 48 * q^23 + 30 * q^24 + 131 * q^25 + 28 * q^26 - 100 * q^27 - 110 * q^29 + 32 * q^30 - 12 * q^31 - 161 * q^32 - 16 * q^33 + 54 * q^34 + 161 * q^36 - 246 * q^37 - 110 * q^38 - 56 * q^39 - 240 * q^40 - 182 * q^41 + 128 * q^43 + 56 * q^44 + 368 * q^45 - 48 * q^46 - 324 * q^47 + 82 * q^48 - 131 * q^50 - 108 * q^51 + 196 * q^52 - 162 * q^53 + 100 * q^54 + 128 * q^55 + 220 * q^57 + 110 * q^58 - 810 * q^59 + 224 * q^60 + 488 * q^61 + 12 * q^62 - 167 * q^64 + 448 * q^65 + 16 * q^66 + 244 * q^67 + 378 * q^68 + 96 * q^69 - 768 * q^71 - 345 * q^72 + 702 * q^73 + 246 * q^74 + 262 * q^75 - 770 * q^76 + 56 * q^78 + 440 * q^79 - 656 * q^80 + 421 * q^81 + 182 * q^82 + 1302 * q^83 + 864 * q^85 - 128 * q^86 - 220 * q^87 - 120 * q^88 - 730 * q^89 - 368 * q^90 - 336 * q^92 - 24 * q^93 + 324 * q^94 - 1760 * q^95 - 322 * q^96 - 294 * q^97 + 184 * 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
−1.00000 2.00000 −7.00000 −16.0000 −2.00000 0 15.0000 −23.0000 16.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$

## 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 49.4.a.b 1
3.b odd 2 1 441.4.a.i 1
4.b odd 2 1 784.4.a.g 1
5.b even 2 1 1225.4.a.j 1
7.b odd 2 1 7.4.a.a 1
7.c even 3 2 49.4.c.b 2
7.d odd 6 2 49.4.c.c 2
21.c even 2 1 63.4.a.b 1
21.g even 6 2 441.4.e.h 2
21.h odd 6 2 441.4.e.e 2
28.d even 2 1 112.4.a.f 1
35.c odd 2 1 175.4.a.b 1
35.f even 4 2 175.4.b.b 2
56.e even 2 1 448.4.a.e 1
56.h odd 2 1 448.4.a.i 1
77.b even 2 1 847.4.a.b 1
84.h odd 2 1 1008.4.a.c 1
91.b odd 2 1 1183.4.a.b 1
105.g even 2 1 1575.4.a.e 1
119.d odd 2 1 2023.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.a.a 1 7.b odd 2 1
49.4.a.b 1 1.a even 1 1 trivial
49.4.c.b 2 7.c even 3 2
49.4.c.c 2 7.d odd 6 2
63.4.a.b 1 21.c even 2 1
112.4.a.f 1 28.d even 2 1
175.4.a.b 1 35.c odd 2 1
175.4.b.b 2 35.f even 4 2
441.4.a.i 1 3.b odd 2 1
441.4.e.e 2 21.h odd 6 2
441.4.e.h 2 21.g even 6 2
448.4.a.e 1 56.e even 2 1
448.4.a.i 1 56.h odd 2 1
784.4.a.g 1 4.b odd 2 1
847.4.a.b 1 77.b even 2 1
1008.4.a.c 1 84.h odd 2 1
1183.4.a.b 1 91.b odd 2 1
1225.4.a.j 1 5.b even 2 1
1575.4.a.e 1 105.g even 2 1
2023.4.a.a 1 119.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(49))$$:

 $$T_{2} + 1$$ T2 + 1 $$T_{3} - 2$$ T3 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 2$$
$5$ $$T + 16$$
$7$ $$T$$
$11$ $$T + 8$$
$13$ $$T + 28$$
$17$ $$T + 54$$
$19$ $$T - 110$$
$23$ $$T - 48$$
$29$ $$T + 110$$
$31$ $$T + 12$$
$37$ $$T + 246$$
$41$ $$T + 182$$
$43$ $$T - 128$$
$47$ $$T + 324$$
$53$ $$T + 162$$
$59$ $$T + 810$$
$61$ $$T - 488$$
$67$ $$T - 244$$
$71$ $$T + 768$$
$73$ $$T - 702$$
$79$ $$T - 440$$
$83$ $$T - 1302$$
$89$ $$T + 730$$
$97$ $$T + 294$$