# Properties

 Label 49.4.c.a Level $49$ Weight $4$ Character orbit 49.c Analytic conductor $2.891$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [49,4,Mod(18,49)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(49, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4]))

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

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

chi := DirichletCharacter("49.18");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 49.c (of order $$3$$, degree $$2$$, not minimal)

## 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: no Analytic conductor: $$2.89109359028$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 7) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 \zeta_{6} q^{2} + ( - 7 \zeta_{6} + 7) q^{3} + ( - 4 \zeta_{6} + 4) q^{4} + 7 \zeta_{6} q^{5} - 14 q^{6} - 24 q^{8} - 22 \zeta_{6} q^{9} +O(q^{10})$$ q - 2*z * q^2 + (-7*z + 7) * q^3 + (-4*z + 4) * q^4 + 7*z * q^5 - 14 * q^6 - 24 * q^8 - 22*z * q^9 $$q - 2 \zeta_{6} q^{2} + ( - 7 \zeta_{6} + 7) q^{3} + ( - 4 \zeta_{6} + 4) q^{4} + 7 \zeta_{6} q^{5} - 14 q^{6} - 24 q^{8} - 22 \zeta_{6} q^{9} + ( - 14 \zeta_{6} + 14) q^{10} + ( - 5 \zeta_{6} + 5) q^{11} - 28 \zeta_{6} q^{12} + 14 q^{13} + 49 q^{15} + 16 \zeta_{6} q^{16} + (21 \zeta_{6} - 21) q^{17} + (44 \zeta_{6} - 44) q^{18} + 49 \zeta_{6} q^{19} + 28 q^{20} - 10 q^{22} + 159 \zeta_{6} q^{23} + (168 \zeta_{6} - 168) q^{24} + ( - 76 \zeta_{6} + 76) q^{25} - 28 \zeta_{6} q^{26} + 35 q^{27} + 58 q^{29} - 98 \zeta_{6} q^{30} + ( - 147 \zeta_{6} + 147) q^{31} + (160 \zeta_{6} - 160) q^{32} - 35 \zeta_{6} q^{33} + 42 q^{34} - 88 q^{36} - 219 \zeta_{6} q^{37} + ( - 98 \zeta_{6} + 98) q^{38} + ( - 98 \zeta_{6} + 98) q^{39} - 168 \zeta_{6} q^{40} - 350 q^{41} - 124 q^{43} - 20 \zeta_{6} q^{44} + ( - 154 \zeta_{6} + 154) q^{45} + ( - 318 \zeta_{6} + 318) q^{46} + 525 \zeta_{6} q^{47} + 112 q^{48} - 152 q^{50} + 147 \zeta_{6} q^{51} + ( - 56 \zeta_{6} + 56) q^{52} + (303 \zeta_{6} - 303) q^{53} - 70 \zeta_{6} q^{54} + 35 q^{55} + 343 q^{57} - 116 \zeta_{6} q^{58} + (105 \zeta_{6} - 105) q^{59} + ( - 196 \zeta_{6} + 196) q^{60} - 413 \zeta_{6} q^{61} - 294 q^{62} + 448 q^{64} + 98 \zeta_{6} q^{65} + (70 \zeta_{6} - 70) q^{66} + (415 \zeta_{6} - 415) q^{67} + 84 \zeta_{6} q^{68} + 1113 q^{69} - 432 q^{71} + 528 \zeta_{6} q^{72} + (1113 \zeta_{6} - 1113) q^{73} + (438 \zeta_{6} - 438) q^{74} - 532 \zeta_{6} q^{75} + 196 q^{76} - 196 q^{78} + 103 \zeta_{6} q^{79} + (112 \zeta_{6} - 112) q^{80} + ( - 839 \zeta_{6} + 839) q^{81} + 700 \zeta_{6} q^{82} - 1092 q^{83} - 147 q^{85} + 248 \zeta_{6} q^{86} + ( - 406 \zeta_{6} + 406) q^{87} + (120 \zeta_{6} - 120) q^{88} - 329 \zeta_{6} q^{89} - 308 q^{90} + 636 q^{92} - 1029 \zeta_{6} q^{93} + ( - 1050 \zeta_{6} + 1050) q^{94} + (343 \zeta_{6} - 343) q^{95} + 1120 \zeta_{6} q^{96} + 882 q^{97} - 110 q^{99} +O(q^{100})$$ q - 2*z * q^2 + (-7*z + 7) * q^3 + (-4*z + 4) * q^4 + 7*z * q^5 - 14 * q^6 - 24 * q^8 - 22*z * q^9 + (-14*z + 14) * q^10 + (-5*z + 5) * q^11 - 28*z * q^12 + 14 * q^13 + 49 * q^15 + 16*z * q^16 + (21*z - 21) * q^17 + (44*z - 44) * q^18 + 49*z * q^19 + 28 * q^20 - 10 * q^22 + 159*z * q^23 + (168*z - 168) * q^24 + (-76*z + 76) * q^25 - 28*z * q^26 + 35 * q^27 + 58 * q^29 - 98*z * q^30 + (-147*z + 147) * q^31 + (160*z - 160) * q^32 - 35*z * q^33 + 42 * q^34 - 88 * q^36 - 219*z * q^37 + (-98*z + 98) * q^38 + (-98*z + 98) * q^39 - 168*z * q^40 - 350 * q^41 - 124 * q^43 - 20*z * q^44 + (-154*z + 154) * q^45 + (-318*z + 318) * q^46 + 525*z * q^47 + 112 * q^48 - 152 * q^50 + 147*z * q^51 + (-56*z + 56) * q^52 + (303*z - 303) * q^53 - 70*z * q^54 + 35 * q^55 + 343 * q^57 - 116*z * q^58 + (105*z - 105) * q^59 + (-196*z + 196) * q^60 - 413*z * q^61 - 294 * q^62 + 448 * q^64 + 98*z * q^65 + (70*z - 70) * q^66 + (415*z - 415) * q^67 + 84*z * q^68 + 1113 * q^69 - 432 * q^71 + 528*z * q^72 + (1113*z - 1113) * q^73 + (438*z - 438) * q^74 - 532*z * q^75 + 196 * q^76 - 196 * q^78 + 103*z * q^79 + (112*z - 112) * q^80 + (-839*z + 839) * q^81 + 700*z * q^82 - 1092 * q^83 - 147 * q^85 + 248*z * q^86 + (-406*z + 406) * q^87 + (120*z - 120) * q^88 - 329*z * q^89 - 308 * q^90 + 636 * q^92 - 1029*z * q^93 + (-1050*z + 1050) * q^94 + (343*z - 343) * q^95 + 1120*z * q^96 + 882 * q^97 - 110 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 7 q^{3} + 4 q^{4} + 7 q^{5} - 28 q^{6} - 48 q^{8} - 22 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 7 * q^3 + 4 * q^4 + 7 * q^5 - 28 * q^6 - 48 * q^8 - 22 * q^9 $$2 q - 2 q^{2} + 7 q^{3} + 4 q^{4} + 7 q^{5} - 28 q^{6} - 48 q^{8} - 22 q^{9} + 14 q^{10} + 5 q^{11} - 28 q^{12} + 28 q^{13} + 98 q^{15} + 16 q^{16} - 21 q^{17} - 44 q^{18} + 49 q^{19} + 56 q^{20} - 20 q^{22} + 159 q^{23} - 168 q^{24} + 76 q^{25} - 28 q^{26} + 70 q^{27} + 116 q^{29} - 98 q^{30} + 147 q^{31} - 160 q^{32} - 35 q^{33} + 84 q^{34} - 176 q^{36} - 219 q^{37} + 98 q^{38} + 98 q^{39} - 168 q^{40} - 700 q^{41} - 248 q^{43} - 20 q^{44} + 154 q^{45} + 318 q^{46} + 525 q^{47} + 224 q^{48} - 304 q^{50} + 147 q^{51} + 56 q^{52} - 303 q^{53} - 70 q^{54} + 70 q^{55} + 686 q^{57} - 116 q^{58} - 105 q^{59} + 196 q^{60} - 413 q^{61} - 588 q^{62} + 896 q^{64} + 98 q^{65} - 70 q^{66} - 415 q^{67} + 84 q^{68} + 2226 q^{69} - 864 q^{71} + 528 q^{72} - 1113 q^{73} - 438 q^{74} - 532 q^{75} + 392 q^{76} - 392 q^{78} + 103 q^{79} - 112 q^{80} + 839 q^{81} + 700 q^{82} - 2184 q^{83} - 294 q^{85} + 248 q^{86} + 406 q^{87} - 120 q^{88} - 329 q^{89} - 616 q^{90} + 1272 q^{92} - 1029 q^{93} + 1050 q^{94} - 343 q^{95} + 1120 q^{96} + 1764 q^{97} - 220 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 7 * q^3 + 4 * q^4 + 7 * q^5 - 28 * q^6 - 48 * q^8 - 22 * q^9 + 14 * q^10 + 5 * q^11 - 28 * q^12 + 28 * q^13 + 98 * q^15 + 16 * q^16 - 21 * q^17 - 44 * q^18 + 49 * q^19 + 56 * q^20 - 20 * q^22 + 159 * q^23 - 168 * q^24 + 76 * q^25 - 28 * q^26 + 70 * q^27 + 116 * q^29 - 98 * q^30 + 147 * q^31 - 160 * q^32 - 35 * q^33 + 84 * q^34 - 176 * q^36 - 219 * q^37 + 98 * q^38 + 98 * q^39 - 168 * q^40 - 700 * q^41 - 248 * q^43 - 20 * q^44 + 154 * q^45 + 318 * q^46 + 525 * q^47 + 224 * q^48 - 304 * q^50 + 147 * q^51 + 56 * q^52 - 303 * q^53 - 70 * q^54 + 70 * q^55 + 686 * q^57 - 116 * q^58 - 105 * q^59 + 196 * q^60 - 413 * q^61 - 588 * q^62 + 896 * q^64 + 98 * q^65 - 70 * q^66 - 415 * q^67 + 84 * q^68 + 2226 * q^69 - 864 * q^71 + 528 * q^72 - 1113 * q^73 - 438 * q^74 - 532 * q^75 + 392 * q^76 - 392 * q^78 + 103 * q^79 - 112 * q^80 + 839 * q^81 + 700 * q^82 - 2184 * q^83 - 294 * q^85 + 248 * q^86 + 406 * q^87 - 120 * q^88 - 329 * q^89 - 616 * q^90 + 1272 * q^92 - 1029 * q^93 + 1050 * q^94 - 343 * q^95 + 1120 * q^96 + 1764 * q^97 - 220 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/49\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
18.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.00000 1.73205i 3.50000 6.06218i 2.00000 3.46410i 3.50000 + 6.06218i −14.0000 0 −24.0000 −11.0000 19.0526i 7.00000 12.1244i
30.1 −1.00000 + 1.73205i 3.50000 + 6.06218i 2.00000 + 3.46410i 3.50000 6.06218i −14.0000 0 −24.0000 −11.0000 + 19.0526i 7.00000 + 12.1244i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{2}^{2} + 2T_{2} + 4$$ T2^2 + 2*T2 + 4 $$T_{3}^{2} - 7T_{3} + 49$$ T3^2 - 7*T3 + 49

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2} - 7T + 49$$
$5$ $$T^{2} - 7T + 49$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 5T + 25$$
$13$ $$(T - 14)^{2}$$
$17$ $$T^{2} + 21T + 441$$
$19$ $$T^{2} - 49T + 2401$$
$23$ $$T^{2} - 159T + 25281$$
$29$ $$(T - 58)^{2}$$
$31$ $$T^{2} - 147T + 21609$$
$37$ $$T^{2} + 219T + 47961$$
$41$ $$(T + 350)^{2}$$
$43$ $$(T + 124)^{2}$$
$47$ $$T^{2} - 525T + 275625$$
$53$ $$T^{2} + 303T + 91809$$
$59$ $$T^{2} + 105T + 11025$$
$61$ $$T^{2} + 413T + 170569$$
$67$ $$T^{2} + 415T + 172225$$
$71$ $$(T + 432)^{2}$$
$73$ $$T^{2} + 1113 T + 1238769$$
$79$ $$T^{2} - 103T + 10609$$
$83$ $$(T + 1092)^{2}$$
$89$ $$T^{2} + 329T + 108241$$
$97$ $$(T - 882)^{2}$$