# Properties

 Label 49.6.c.b Level $49$ Weight $6$ Character orbit 49.c Analytic conductor $7.859$ 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] = [49,6,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, 6, names="a")

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

chi := DirichletCharacter("49.18");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ 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: $$7.85880717084$$ 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: $$\Z[a_1, \ldots, a_{9}]$$ 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.

 $$f(q)$$ $$=$$ $$q + 10 \zeta_{6} q^{2} + (14 \zeta_{6} - 14) q^{3} + (68 \zeta_{6} - 68) q^{4} - 56 \zeta_{6} q^{5} - 140 q^{6} - 360 q^{8} + 47 \zeta_{6} q^{9} +O(q^{10})$$ q + 10*z * q^2 + (14*z - 14) * q^3 + (68*z - 68) * q^4 - 56*z * q^5 - 140 * q^6 - 360 * q^8 + 47*z * q^9 $$q + 10 \zeta_{6} q^{2} + (14 \zeta_{6} - 14) q^{3} + (68 \zeta_{6} - 68) q^{4} - 56 \zeta_{6} q^{5} - 140 q^{6} - 360 q^{8} + 47 \zeta_{6} q^{9} + ( - 560 \zeta_{6} + 560) q^{10} + (232 \zeta_{6} - 232) q^{11} - 952 \zeta_{6} q^{12} + 140 q^{13} + 784 q^{15} - 1424 \zeta_{6} q^{16} + (1722 \zeta_{6} - 1722) q^{17} + (470 \zeta_{6} - 470) q^{18} - 98 \zeta_{6} q^{19} + 3808 q^{20} - 2320 q^{22} - 1824 \zeta_{6} q^{23} + ( - 5040 \zeta_{6} + 5040) q^{24} + (11 \zeta_{6} - 11) q^{25} + 1400 \zeta_{6} q^{26} - 4060 q^{27} + 3418 q^{29} + 7840 \zeta_{6} q^{30} + (7644 \zeta_{6} - 7644) q^{31} + ( - 2720 \zeta_{6} + 2720) q^{32} - 3248 \zeta_{6} q^{33} - 17220 q^{34} - 3196 q^{36} + 10398 \zeta_{6} q^{37} + ( - 980 \zeta_{6} + 980) q^{38} + (1960 \zeta_{6} - 1960) q^{39} + 20160 \zeta_{6} q^{40} + 17962 q^{41} + 10880 q^{43} - 15776 \zeta_{6} q^{44} + ( - 2632 \zeta_{6} + 2632) q^{45} + ( - 18240 \zeta_{6} + 18240) q^{46} + 9324 \zeta_{6} q^{47} + 19936 q^{48} - 110 q^{50} - 24108 \zeta_{6} q^{51} + (9520 \zeta_{6} - 9520) q^{52} + (2262 \zeta_{6} - 2262) q^{53} - 40600 \zeta_{6} q^{54} + 12992 q^{55} + 1372 q^{57} + 34180 \zeta_{6} q^{58} + (2730 \zeta_{6} - 2730) q^{59} + (53312 \zeta_{6} - 53312) q^{60} + 25648 \zeta_{6} q^{61} - 76440 q^{62} - 18368 q^{64} - 7840 \zeta_{6} q^{65} + ( - 32480 \zeta_{6} + 32480) q^{66} + ( - 48404 \zeta_{6} + 48404) q^{67} - 117096 \zeta_{6} q^{68} + 25536 q^{69} - 58560 q^{71} - 16920 \zeta_{6} q^{72} + ( - 68082 \zeta_{6} + 68082) q^{73} + (103980 \zeta_{6} - 103980) q^{74} - 154 \zeta_{6} q^{75} + 6664 q^{76} - 19600 q^{78} - 31784 \zeta_{6} q^{79} + (79744 \zeta_{6} - 79744) q^{80} + ( - 45419 \zeta_{6} + 45419) q^{81} + 179620 \zeta_{6} q^{82} + 20538 q^{83} + 96432 q^{85} + 108800 \zeta_{6} q^{86} + (47852 \zeta_{6} - 47852) q^{87} + ( - 83520 \zeta_{6} + 83520) q^{88} - 50582 \zeta_{6} q^{89} + 26320 q^{90} + 124032 q^{92} - 107016 \zeta_{6} q^{93} + (93240 \zeta_{6} - 93240) q^{94} + (5488 \zeta_{6} - 5488) q^{95} + 38080 \zeta_{6} q^{96} + 58506 q^{97} - 10904 q^{99} +O(q^{100})$$ q + 10*z * q^2 + (14*z - 14) * q^3 + (68*z - 68) * q^4 - 56*z * q^5 - 140 * q^6 - 360 * q^8 + 47*z * q^9 + (-560*z + 560) * q^10 + (232*z - 232) * q^11 - 952*z * q^12 + 140 * q^13 + 784 * q^15 - 1424*z * q^16 + (1722*z - 1722) * q^17 + (470*z - 470) * q^18 - 98*z * q^19 + 3808 * q^20 - 2320 * q^22 - 1824*z * q^23 + (-5040*z + 5040) * q^24 + (11*z - 11) * q^25 + 1400*z * q^26 - 4060 * q^27 + 3418 * q^29 + 7840*z * q^30 + (7644*z - 7644) * q^31 + (-2720*z + 2720) * q^32 - 3248*z * q^33 - 17220 * q^34 - 3196 * q^36 + 10398*z * q^37 + (-980*z + 980) * q^38 + (1960*z - 1960) * q^39 + 20160*z * q^40 + 17962 * q^41 + 10880 * q^43 - 15776*z * q^44 + (-2632*z + 2632) * q^45 + (-18240*z + 18240) * q^46 + 9324*z * q^47 + 19936 * q^48 - 110 * q^50 - 24108*z * q^51 + (9520*z - 9520) * q^52 + (2262*z - 2262) * q^53 - 40600*z * q^54 + 12992 * q^55 + 1372 * q^57 + 34180*z * q^58 + (2730*z - 2730) * q^59 + (53312*z - 53312) * q^60 + 25648*z * q^61 - 76440 * q^62 - 18368 * q^64 - 7840*z * q^65 + (-32480*z + 32480) * q^66 + (-48404*z + 48404) * q^67 - 117096*z * q^68 + 25536 * q^69 - 58560 * q^71 - 16920*z * q^72 + (-68082*z + 68082) * q^73 + (103980*z - 103980) * q^74 - 154*z * q^75 + 6664 * q^76 - 19600 * q^78 - 31784*z * q^79 + (79744*z - 79744) * q^80 + (-45419*z + 45419) * q^81 + 179620*z * q^82 + 20538 * q^83 + 96432 * q^85 + 108800*z * q^86 + (47852*z - 47852) * q^87 + (-83520*z + 83520) * q^88 - 50582*z * q^89 + 26320 * q^90 + 124032 * q^92 - 107016*z * q^93 + (93240*z - 93240) * q^94 + (5488*z - 5488) * q^95 + 38080*z * q^96 + 58506 * q^97 - 10904 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 10 q^{2} - 14 q^{3} - 68 q^{4} - 56 q^{5} - 280 q^{6} - 720 q^{8} + 47 q^{9}+O(q^{10})$$ 2 * q + 10 * q^2 - 14 * q^3 - 68 * q^4 - 56 * q^5 - 280 * q^6 - 720 * q^8 + 47 * q^9 $$2 q + 10 q^{2} - 14 q^{3} - 68 q^{4} - 56 q^{5} - 280 q^{6} - 720 q^{8} + 47 q^{9} + 560 q^{10} - 232 q^{11} - 952 q^{12} + 280 q^{13} + 1568 q^{15} - 1424 q^{16} - 1722 q^{17} - 470 q^{18} - 98 q^{19} + 7616 q^{20} - 4640 q^{22} - 1824 q^{23} + 5040 q^{24} - 11 q^{25} + 1400 q^{26} - 8120 q^{27} + 6836 q^{29} + 7840 q^{30} - 7644 q^{31} + 2720 q^{32} - 3248 q^{33} - 34440 q^{34} - 6392 q^{36} + 10398 q^{37} + 980 q^{38} - 1960 q^{39} + 20160 q^{40} + 35924 q^{41} + 21760 q^{43} - 15776 q^{44} + 2632 q^{45} + 18240 q^{46} + 9324 q^{47} + 39872 q^{48} - 220 q^{50} - 24108 q^{51} - 9520 q^{52} - 2262 q^{53} - 40600 q^{54} + 25984 q^{55} + 2744 q^{57} + 34180 q^{58} - 2730 q^{59} - 53312 q^{60} + 25648 q^{61} - 152880 q^{62} - 36736 q^{64} - 7840 q^{65} + 32480 q^{66} + 48404 q^{67} - 117096 q^{68} + 51072 q^{69} - 117120 q^{71} - 16920 q^{72} + 68082 q^{73} - 103980 q^{74} - 154 q^{75} + 13328 q^{76} - 39200 q^{78} - 31784 q^{79} - 79744 q^{80} + 45419 q^{81} + 179620 q^{82} + 41076 q^{83} + 192864 q^{85} + 108800 q^{86} - 47852 q^{87} + 83520 q^{88} - 50582 q^{89} + 52640 q^{90} + 248064 q^{92} - 107016 q^{93} - 93240 q^{94} - 5488 q^{95} + 38080 q^{96} + 117012 q^{97} - 21808 q^{99}+O(q^{100})$$ 2 * q + 10 * q^2 - 14 * q^3 - 68 * q^4 - 56 * q^5 - 280 * q^6 - 720 * q^8 + 47 * q^9 + 560 * q^10 - 232 * q^11 - 952 * q^12 + 280 * q^13 + 1568 * q^15 - 1424 * q^16 - 1722 * q^17 - 470 * q^18 - 98 * q^19 + 7616 * q^20 - 4640 * q^22 - 1824 * q^23 + 5040 * q^24 - 11 * q^25 + 1400 * q^26 - 8120 * q^27 + 6836 * q^29 + 7840 * q^30 - 7644 * q^31 + 2720 * q^32 - 3248 * q^33 - 34440 * q^34 - 6392 * q^36 + 10398 * q^37 + 980 * q^38 - 1960 * q^39 + 20160 * q^40 + 35924 * q^41 + 21760 * q^43 - 15776 * q^44 + 2632 * q^45 + 18240 * q^46 + 9324 * q^47 + 39872 * q^48 - 220 * q^50 - 24108 * q^51 - 9520 * q^52 - 2262 * q^53 - 40600 * q^54 + 25984 * q^55 + 2744 * q^57 + 34180 * q^58 - 2730 * q^59 - 53312 * q^60 + 25648 * q^61 - 152880 * q^62 - 36736 * q^64 - 7840 * q^65 + 32480 * q^66 + 48404 * q^67 - 117096 * q^68 + 51072 * q^69 - 117120 * q^71 - 16920 * q^72 + 68082 * q^73 - 103980 * q^74 - 154 * q^75 + 13328 * q^76 - 39200 * q^78 - 31784 * q^79 - 79744 * q^80 + 45419 * q^81 + 179620 * q^82 + 41076 * q^83 + 192864 * q^85 + 108800 * q^86 - 47852 * q^87 + 83520 * q^88 - 50582 * q^89 + 52640 * q^90 + 248064 * q^92 - 107016 * q^93 - 93240 * q^94 - 5488 * q^95 + 38080 * q^96 + 117012 * q^97 - 21808 * 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
5.00000 + 8.66025i −7.00000 + 12.1244i −34.0000 + 58.8897i −28.0000 48.4974i −140.000 0 −360.000 23.5000 + 40.7032i 280.000 484.974i
30.1 5.00000 8.66025i −7.00000 12.1244i −34.0000 58.8897i −28.0000 + 48.4974i −140.000 0 −360.000 23.5000 40.7032i 280.000 + 484.974i
 $$n$$: 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 49.6.c.b 2
7.b odd 2 1 49.6.c.c 2
7.c even 3 1 49.6.a.a 1
7.c even 3 1 inner 49.6.c.b 2
7.d odd 6 1 7.6.a.a 1
7.d odd 6 1 49.6.c.c 2
21.g even 6 1 63.6.a.e 1
21.h odd 6 1 441.6.a.k 1
28.f even 6 1 112.6.a.g 1
28.g odd 6 1 784.6.a.c 1
35.i odd 6 1 175.6.a.b 1
35.k even 12 2 175.6.b.a 2
56.j odd 6 1 448.6.a.m 1
56.m even 6 1 448.6.a.c 1
77.i even 6 1 847.6.a.b 1
84.j odd 6 1 1008.6.a.y 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.a 1 7.d odd 6 1
49.6.a.a 1 7.c even 3 1
49.6.c.b 2 1.a even 1 1 trivial
49.6.c.b 2 7.c even 3 1 inner
49.6.c.c 2 7.b odd 2 1
49.6.c.c 2 7.d odd 6 1
63.6.a.e 1 21.g even 6 1
112.6.a.g 1 28.f even 6 1
175.6.a.b 1 35.i odd 6 1
175.6.b.a 2 35.k even 12 2
441.6.a.k 1 21.h odd 6 1
448.6.a.c 1 56.m even 6 1
448.6.a.m 1 56.j odd 6 1
784.6.a.c 1 28.g odd 6 1
847.6.a.b 1 77.i even 6 1
1008.6.a.y 1 84.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(49, [\chi])$$:

 $$T_{2}^{2} - 10T_{2} + 100$$ T2^2 - 10*T2 + 100 $$T_{3}^{2} + 14T_{3} + 196$$ T3^2 + 14*T3 + 196

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 10T + 100$$
$3$ $$T^{2} + 14T + 196$$
$5$ $$T^{2} + 56T + 3136$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 232T + 53824$$
$13$ $$(T - 140)^{2}$$
$17$ $$T^{2} + 1722 T + 2965284$$
$19$ $$T^{2} + 98T + 9604$$
$23$ $$T^{2} + 1824 T + 3326976$$
$29$ $$(T - 3418)^{2}$$
$31$ $$T^{2} + 7644 T + 58430736$$
$37$ $$T^{2} - 10398 T + 108118404$$
$41$ $$(T - 17962)^{2}$$
$43$ $$(T - 10880)^{2}$$
$47$ $$T^{2} - 9324 T + 86936976$$
$53$ $$T^{2} + 2262 T + 5116644$$
$59$ $$T^{2} + 2730 T + 7452900$$
$61$ $$T^{2} - 25648 T + 657819904$$
$67$ $$T^{2} - 48404 T + 2342947216$$
$71$ $$(T + 58560)^{2}$$
$73$ $$T^{2} - 68082 T + 4635158724$$
$79$ $$T^{2} + 31784 T + 1010222656$$
$83$ $$(T - 20538)^{2}$$
$89$ $$T^{2} + 50582 T + 2558538724$$
$97$ $$(T - 58506)^{2}$$