# Properties

 Label 49.4.c.c 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + ( - 2 \zeta_{6} + 2) q^{3} + ( - 7 \zeta_{6} + 7) q^{4} - 16 \zeta_{6} q^{5} + 2 q^{6} + 15 q^{8} + 23 \zeta_{6} q^{9} +O(q^{10})$$ q + z * q^2 + (-2*z + 2) * q^3 + (-7*z + 7) * q^4 - 16*z * q^5 + 2 * q^6 + 15 * q^8 + 23*z * q^9 $$q + \zeta_{6} q^{2} + ( - 2 \zeta_{6} + 2) q^{3} + ( - 7 \zeta_{6} + 7) q^{4} - 16 \zeta_{6} q^{5} + 2 q^{6} + 15 q^{8} + 23 \zeta_{6} q^{9} + ( - 16 \zeta_{6} + 16) q^{10} + ( - 8 \zeta_{6} + 8) q^{11} - 14 \zeta_{6} q^{12} + 28 q^{13} - 32 q^{15} - 41 \zeta_{6} q^{16} + (54 \zeta_{6} - 54) q^{17} + (23 \zeta_{6} - 23) q^{18} + 110 \zeta_{6} q^{19} - 112 q^{20} + 8 q^{22} - 48 \zeta_{6} q^{23} + ( - 30 \zeta_{6} + 30) q^{24} + (131 \zeta_{6} - 131) q^{25} + 28 \zeta_{6} q^{26} + 100 q^{27} - 110 q^{29} - 32 \zeta_{6} q^{30} + (12 \zeta_{6} - 12) q^{31} + ( - 161 \zeta_{6} + 161) q^{32} - 16 \zeta_{6} q^{33} - 54 q^{34} + 161 q^{36} + 246 \zeta_{6} q^{37} + (110 \zeta_{6} - 110) q^{38} + ( - 56 \zeta_{6} + 56) q^{39} - 240 \zeta_{6} q^{40} + 182 q^{41} + 128 q^{43} - 56 \zeta_{6} q^{44} + ( - 368 \zeta_{6} + 368) q^{45} + ( - 48 \zeta_{6} + 48) q^{46} - 324 \zeta_{6} q^{47} - 82 q^{48} - 131 q^{50} + 108 \zeta_{6} q^{51} + ( - 196 \zeta_{6} + 196) q^{52} + ( - 162 \zeta_{6} + 162) q^{53} + 100 \zeta_{6} q^{54} - 128 q^{55} + 220 q^{57} - 110 \zeta_{6} q^{58} + (810 \zeta_{6} - 810) q^{59} + (224 \zeta_{6} - 224) q^{60} + 488 \zeta_{6} q^{61} - 12 q^{62} - 167 q^{64} - 448 \zeta_{6} q^{65} + ( - 16 \zeta_{6} + 16) q^{66} + (244 \zeta_{6} - 244) q^{67} + 378 \zeta_{6} q^{68} - 96 q^{69} - 768 q^{71} + 345 \zeta_{6} q^{72} + ( - 702 \zeta_{6} + 702) q^{73} + (246 \zeta_{6} - 246) q^{74} + 262 \zeta_{6} q^{75} + 770 q^{76} + 56 q^{78} - 440 \zeta_{6} q^{79} + (656 \zeta_{6} - 656) q^{80} + (421 \zeta_{6} - 421) q^{81} + 182 \zeta_{6} q^{82} - 1302 q^{83} + 864 q^{85} + 128 \zeta_{6} q^{86} + (220 \zeta_{6} - 220) q^{87} + ( - 120 \zeta_{6} + 120) q^{88} - 730 \zeta_{6} q^{89} + 368 q^{90} - 336 q^{92} + 24 \zeta_{6} q^{93} + ( - 324 \zeta_{6} + 324) q^{94} + ( - 1760 \zeta_{6} + 1760) q^{95} - 322 \zeta_{6} q^{96} + 294 q^{97} + 184 q^{99} +O(q^{100})$$ q + z * q^2 + (-2*z + 2) * q^3 + (-7*z + 7) * q^4 - 16*z * q^5 + 2 * q^6 + 15 * q^8 + 23*z * q^9 + (-16*z + 16) * q^10 + (-8*z + 8) * q^11 - 14*z * q^12 + 28 * q^13 - 32 * q^15 - 41*z * q^16 + (54*z - 54) * q^17 + (23*z - 23) * q^18 + 110*z * q^19 - 112 * q^20 + 8 * q^22 - 48*z * q^23 + (-30*z + 30) * q^24 + (131*z - 131) * q^25 + 28*z * q^26 + 100 * q^27 - 110 * q^29 - 32*z * q^30 + (12*z - 12) * q^31 + (-161*z + 161) * q^32 - 16*z * q^33 - 54 * q^34 + 161 * q^36 + 246*z * q^37 + (110*z - 110) * q^38 + (-56*z + 56) * q^39 - 240*z * q^40 + 182 * q^41 + 128 * q^43 - 56*z * q^44 + (-368*z + 368) * q^45 + (-48*z + 48) * q^46 - 324*z * q^47 - 82 * q^48 - 131 * q^50 + 108*z * q^51 + (-196*z + 196) * q^52 + (-162*z + 162) * q^53 + 100*z * q^54 - 128 * q^55 + 220 * q^57 - 110*z * q^58 + (810*z - 810) * q^59 + (224*z - 224) * q^60 + 488*z * q^61 - 12 * q^62 - 167 * q^64 - 448*z * q^65 + (-16*z + 16) * q^66 + (244*z - 244) * q^67 + 378*z * q^68 - 96 * q^69 - 768 * q^71 + 345*z * q^72 + (-702*z + 702) * q^73 + (246*z - 246) * q^74 + 262*z * q^75 + 770 * q^76 + 56 * q^78 - 440*z * q^79 + (656*z - 656) * q^80 + (421*z - 421) * q^81 + 182*z * q^82 - 1302 * q^83 + 864 * q^85 + 128*z * q^86 + (220*z - 220) * q^87 + (-120*z + 120) * q^88 - 730*z * q^89 + 368 * q^90 - 336 * q^92 + 24*z * q^93 + (-324*z + 324) * q^94 + (-1760*z + 1760) * q^95 - 322*z * q^96 + 294 * q^97 + 184 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 2 q^{3} + 7 q^{4} - 16 q^{5} + 4 q^{6} + 30 q^{8} + 23 q^{9}+O(q^{10})$$ 2 * q + q^2 + 2 * q^3 + 7 * q^4 - 16 * q^5 + 4 * q^6 + 30 * q^8 + 23 * q^9 $$2 q + q^{2} + 2 q^{3} + 7 q^{4} - 16 q^{5} + 4 q^{6} + 30 q^{8} + 23 q^{9} + 16 q^{10} + 8 q^{11} - 14 q^{12} + 56 q^{13} - 64 q^{15} - 41 q^{16} - 54 q^{17} - 23 q^{18} + 110 q^{19} - 224 q^{20} + 16 q^{22} - 48 q^{23} + 30 q^{24} - 131 q^{25} + 28 q^{26} + 200 q^{27} - 220 q^{29} - 32 q^{30} - 12 q^{31} + 161 q^{32} - 16 q^{33} - 108 q^{34} + 322 q^{36} + 246 q^{37} - 110 q^{38} + 56 q^{39} - 240 q^{40} + 364 q^{41} + 256 q^{43} - 56 q^{44} + 368 q^{45} + 48 q^{46} - 324 q^{47} - 164 q^{48} - 262 q^{50} + 108 q^{51} + 196 q^{52} + 162 q^{53} + 100 q^{54} - 256 q^{55} + 440 q^{57} - 110 q^{58} - 810 q^{59} - 224 q^{60} + 488 q^{61} - 24 q^{62} - 334 q^{64} - 448 q^{65} + 16 q^{66} - 244 q^{67} + 378 q^{68} - 192 q^{69} - 1536 q^{71} + 345 q^{72} + 702 q^{73} - 246 q^{74} + 262 q^{75} + 1540 q^{76} + 112 q^{78} - 440 q^{79} - 656 q^{80} - 421 q^{81} + 182 q^{82} - 2604 q^{83} + 1728 q^{85} + 128 q^{86} - 220 q^{87} + 120 q^{88} - 730 q^{89} + 736 q^{90} - 672 q^{92} + 24 q^{93} + 324 q^{94} + 1760 q^{95} - 322 q^{96} + 588 q^{97} + 368 q^{99}+O(q^{100})$$ 2 * q + q^2 + 2 * q^3 + 7 * q^4 - 16 * q^5 + 4 * q^6 + 30 * q^8 + 23 * q^9 + 16 * q^10 + 8 * q^11 - 14 * q^12 + 56 * q^13 - 64 * q^15 - 41 * q^16 - 54 * q^17 - 23 * q^18 + 110 * q^19 - 224 * q^20 + 16 * q^22 - 48 * q^23 + 30 * q^24 - 131 * q^25 + 28 * q^26 + 200 * q^27 - 220 * q^29 - 32 * q^30 - 12 * q^31 + 161 * q^32 - 16 * q^33 - 108 * q^34 + 322 * q^36 + 246 * q^37 - 110 * q^38 + 56 * q^39 - 240 * q^40 + 364 * q^41 + 256 * q^43 - 56 * q^44 + 368 * q^45 + 48 * q^46 - 324 * q^47 - 164 * q^48 - 262 * q^50 + 108 * q^51 + 196 * q^52 + 162 * q^53 + 100 * q^54 - 256 * q^55 + 440 * q^57 - 110 * q^58 - 810 * q^59 - 224 * q^60 + 488 * q^61 - 24 * q^62 - 334 * q^64 - 448 * q^65 + 16 * q^66 - 244 * q^67 + 378 * q^68 - 192 * q^69 - 1536 * q^71 + 345 * q^72 + 702 * q^73 - 246 * q^74 + 262 * q^75 + 1540 * q^76 + 112 * q^78 - 440 * q^79 - 656 * q^80 - 421 * q^81 + 182 * q^82 - 2604 * q^83 + 1728 * q^85 + 128 * q^86 - 220 * q^87 + 120 * q^88 - 730 * q^89 + 736 * q^90 - 672 * q^92 + 24 * q^93 + 324 * q^94 + 1760 * q^95 - 322 * q^96 + 588 * q^97 + 368 * 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
0.500000 + 0.866025i 1.00000 1.73205i 3.50000 6.06218i −8.00000 13.8564i 2.00000 0 15.0000 11.5000 + 19.9186i 8.00000 13.8564i
30.1 0.500000 0.866025i 1.00000 + 1.73205i 3.50000 + 6.06218i −8.00000 + 13.8564i 2.00000 0 15.0000 11.5000 19.9186i 8.00000 + 13.8564i
 $$n$$: 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.c 2
3.b odd 2 1 441.4.e.h 2
7.b odd 2 1 49.4.c.b 2
7.c even 3 1 7.4.a.a 1
7.c even 3 1 inner 49.4.c.c 2
7.d odd 6 1 49.4.a.b 1
7.d odd 6 1 49.4.c.b 2
21.c even 2 1 441.4.e.e 2
21.g even 6 1 441.4.a.i 1
21.g even 6 1 441.4.e.e 2
21.h odd 6 1 63.4.a.b 1
21.h odd 6 1 441.4.e.h 2
28.f even 6 1 784.4.a.g 1
28.g odd 6 1 112.4.a.f 1
35.i odd 6 1 1225.4.a.j 1
35.j even 6 1 175.4.a.b 1
35.l odd 12 2 175.4.b.b 2
56.k odd 6 1 448.4.a.e 1
56.p even 6 1 448.4.a.i 1
77.h odd 6 1 847.4.a.b 1
84.n even 6 1 1008.4.a.c 1
91.r even 6 1 1183.4.a.b 1
105.o odd 6 1 1575.4.a.e 1
119.j even 6 1 2023.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.a.a 1 7.c even 3 1
49.4.a.b 1 7.d odd 6 1
49.4.c.b 2 7.b odd 2 1
49.4.c.b 2 7.d odd 6 1
49.4.c.c 2 1.a even 1 1 trivial
49.4.c.c 2 7.c even 3 1 inner
63.4.a.b 1 21.h odd 6 1
112.4.a.f 1 28.g odd 6 1
175.4.a.b 1 35.j even 6 1
175.4.b.b 2 35.l odd 12 2
441.4.a.i 1 21.g even 6 1
441.4.e.e 2 21.c even 2 1
441.4.e.e 2 21.g even 6 1
441.4.e.h 2 3.b odd 2 1
441.4.e.h 2 21.h odd 6 1
448.4.a.e 1 56.k odd 6 1
448.4.a.i 1 56.p even 6 1
784.4.a.g 1 28.f even 6 1
847.4.a.b 1 77.h odd 6 1
1008.4.a.c 1 84.n even 6 1
1183.4.a.b 1 91.r even 6 1
1225.4.a.j 1 35.i odd 6 1
1575.4.a.e 1 105.o odd 6 1
2023.4.a.a 1 119.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 $$S_{4}^{\mathrm{new}}(49, [\chi])$$:

 $$T_{2}^{2} - T_{2} + 1$$ T2^2 - T2 + 1 $$T_{3}^{2} - 2T_{3} + 4$$ T3^2 - 2*T3 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} - 2T + 4$$
$5$ $$T^{2} + 16T + 256$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 8T + 64$$
$13$ $$(T - 28)^{2}$$
$17$ $$T^{2} + 54T + 2916$$
$19$ $$T^{2} - 110T + 12100$$
$23$ $$T^{2} + 48T + 2304$$
$29$ $$(T + 110)^{2}$$
$31$ $$T^{2} + 12T + 144$$
$37$ $$T^{2} - 246T + 60516$$
$41$ $$(T - 182)^{2}$$
$43$ $$(T - 128)^{2}$$
$47$ $$T^{2} + 324T + 104976$$
$53$ $$T^{2} - 162T + 26244$$
$59$ $$T^{2} + 810T + 656100$$
$61$ $$T^{2} - 488T + 238144$$
$67$ $$T^{2} + 244T + 59536$$
$71$ $$(T + 768)^{2}$$
$73$ $$T^{2} - 702T + 492804$$
$79$ $$T^{2} + 440T + 193600$$
$83$ $$(T + 1302)^{2}$$
$89$ $$T^{2} + 730T + 532900$$
$97$ $$(T - 294)^{2}$$