# Properties

 Label 49.4.a.c Level $49$ Weight $4$ Character orbit 49.a Self dual yes Analytic conductor $2.891$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} - 7 q^{3} - 4 q^{4} - 7 q^{5} - 14 q^{6} - 24 q^{8} + 22 q^{9}+O(q^{10})$$ q + 2 * q^2 - 7 * q^3 - 4 * q^4 - 7 * q^5 - 14 * q^6 - 24 * q^8 + 22 * q^9 $$q + 2 q^{2} - 7 q^{3} - 4 q^{4} - 7 q^{5} - 14 q^{6} - 24 q^{8} + 22 q^{9} - 14 q^{10} - 5 q^{11} + 28 q^{12} + 14 q^{13} + 49 q^{15} - 16 q^{16} + 21 q^{17} + 44 q^{18} - 49 q^{19} + 28 q^{20} - 10 q^{22} - 159 q^{23} + 168 q^{24} - 76 q^{25} + 28 q^{26} + 35 q^{27} + 58 q^{29} + 98 q^{30} - 147 q^{31} + 160 q^{32} + 35 q^{33} + 42 q^{34} - 88 q^{36} + 219 q^{37} - 98 q^{38} - 98 q^{39} + 168 q^{40} - 350 q^{41} - 124 q^{43} + 20 q^{44} - 154 q^{45} - 318 q^{46} - 525 q^{47} + 112 q^{48} - 152 q^{50} - 147 q^{51} - 56 q^{52} + 303 q^{53} + 70 q^{54} + 35 q^{55} + 343 q^{57} + 116 q^{58} + 105 q^{59} - 196 q^{60} + 413 q^{61} - 294 q^{62} + 448 q^{64} - 98 q^{65} + 70 q^{66} + 415 q^{67} - 84 q^{68} + 1113 q^{69} - 432 q^{71} - 528 q^{72} + 1113 q^{73} + 438 q^{74} + 532 q^{75} + 196 q^{76} - 196 q^{78} - 103 q^{79} + 112 q^{80} - 839 q^{81} - 700 q^{82} - 1092 q^{83} - 147 q^{85} - 248 q^{86} - 406 q^{87} + 120 q^{88} + 329 q^{89} - 308 q^{90} + 636 q^{92} + 1029 q^{93} - 1050 q^{94} + 343 q^{95} - 1120 q^{96} + 882 q^{97} - 110 q^{99}+O(q^{100})$$ q + 2 * q^2 - 7 * q^3 - 4 * q^4 - 7 * q^5 - 14 * q^6 - 24 * q^8 + 22 * q^9 - 14 * q^10 - 5 * q^11 + 28 * q^12 + 14 * q^13 + 49 * q^15 - 16 * q^16 + 21 * q^17 + 44 * q^18 - 49 * q^19 + 28 * q^20 - 10 * q^22 - 159 * q^23 + 168 * q^24 - 76 * q^25 + 28 * q^26 + 35 * q^27 + 58 * q^29 + 98 * q^30 - 147 * q^31 + 160 * q^32 + 35 * q^33 + 42 * q^34 - 88 * q^36 + 219 * q^37 - 98 * q^38 - 98 * q^39 + 168 * q^40 - 350 * q^41 - 124 * q^43 + 20 * q^44 - 154 * q^45 - 318 * q^46 - 525 * q^47 + 112 * q^48 - 152 * q^50 - 147 * q^51 - 56 * q^52 + 303 * q^53 + 70 * q^54 + 35 * q^55 + 343 * q^57 + 116 * q^58 + 105 * q^59 - 196 * q^60 + 413 * q^61 - 294 * q^62 + 448 * q^64 - 98 * q^65 + 70 * q^66 + 415 * q^67 - 84 * q^68 + 1113 * q^69 - 432 * q^71 - 528 * q^72 + 1113 * q^73 + 438 * q^74 + 532 * q^75 + 196 * q^76 - 196 * q^78 - 103 * q^79 + 112 * q^80 - 839 * q^81 - 700 * q^82 - 1092 * q^83 - 147 * q^85 - 248 * q^86 - 406 * q^87 + 120 * q^88 + 329 * q^89 - 308 * q^90 + 636 * q^92 + 1029 * q^93 - 1050 * q^94 + 343 * q^95 - 1120 * q^96 + 882 * q^97 - 110 * 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 −7.00000 −4.00000 −7.00000 −14.0000 0 −24.0000 22.0000 −14.0000
 $$n$$: 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.c 1
3.b odd 2 1 441.4.a.e 1
4.b odd 2 1 784.4.a.r 1
5.b even 2 1 1225.4.a.d 1
7.b odd 2 1 49.4.a.d 1
7.c even 3 2 49.4.c.a 2
7.d odd 6 2 7.4.c.a 2
21.c even 2 1 441.4.a.d 1
21.g even 6 2 63.4.e.b 2
21.h odd 6 2 441.4.e.k 2
28.d even 2 1 784.4.a.b 1
28.f even 6 2 112.4.i.c 2
35.c odd 2 1 1225.4.a.c 1
35.i odd 6 2 175.4.e.a 2
35.k even 12 4 175.4.k.a 4
56.j odd 6 2 448.4.i.f 2
56.m even 6 2 448.4.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.c.a 2 7.d odd 6 2
49.4.a.c 1 1.a even 1 1 trivial
49.4.a.d 1 7.b odd 2 1
49.4.c.a 2 7.c even 3 2
63.4.e.b 2 21.g even 6 2
112.4.i.c 2 28.f even 6 2
175.4.e.a 2 35.i odd 6 2
175.4.k.a 4 35.k even 12 4
441.4.a.d 1 21.c even 2 1
441.4.a.e 1 3.b odd 2 1
441.4.e.k 2 21.h odd 6 2
448.4.i.a 2 56.m even 6 2
448.4.i.f 2 56.j odd 6 2
784.4.a.b 1 28.d even 2 1
784.4.a.r 1 4.b odd 2 1
1225.4.a.c 1 35.c odd 2 1
1225.4.a.d 1 5.b even 2 1

## Hecke kernels

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

 $$T_{2} - 2$$ T2 - 2 $$T_{3} + 7$$ T3 + 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T + 7$$
$5$ $$T + 7$$
$7$ $$T$$
$11$ $$T + 5$$
$13$ $$T - 14$$
$17$ $$T - 21$$
$19$ $$T + 49$$
$23$ $$T + 159$$
$29$ $$T - 58$$
$31$ $$T + 147$$
$37$ $$T - 219$$
$41$ $$T + 350$$
$43$ $$T + 124$$
$47$ $$T + 525$$
$53$ $$T - 303$$
$59$ $$T - 105$$
$61$ $$T - 413$$
$67$ $$T - 415$$
$71$ $$T + 432$$
$73$ $$T - 1113$$
$79$ $$T + 103$$
$83$ $$T + 1092$$
$89$ $$T - 329$$
$97$ $$T - 882$$
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