# Properties

 Label 49.2.e.a Level $49$ Weight $2$ Character orbit 49.e Analytic conductor $0.391$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [49,2,Mod(8,49)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(49, base_ring=CyclotomicField(14))

chi = DirichletCharacter(H, H._module([12]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("49.8");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 49.e (of order $$7$$, degree $$6$$, 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: $$0.391266969904$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{14})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$$ x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

## $q$-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 $$\zeta_{14}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{14}^{5} - \zeta_{14}^{4} + \cdots - 1) q^{2}+ \cdots + (\zeta_{14}^{5} - \zeta_{14}^{4} + \cdots - 2) q^{9}+O(q^{10})$$ q + (z^5 - z^4 + z - 1) * q^2 + (z^5 + z^3 + z - 1) * q^3 + (z^3 + z^2 - z - 1) * q^4 + (-z^4 - z + 1) * q^5 + (-z^5 - z) * q^6 + (-2*z^5 + 2*z^4 - z^3 + 2) * q^7 + (-2*z^5 + 2*z^4 - 2*z^3 - z + 1) * q^8 + (z^5 - z^4 - 2*z^2 + 2*z - 2) * q^9 $$q + (\zeta_{14}^{5} - \zeta_{14}^{4} + \cdots - 1) q^{2}+ \cdots + (5 \zeta_{14}^{5} - 6 \zeta_{14}^{4} + \cdots - 4) q^{99}+O(q^{100})$$ q + (z^5 - z^4 + z - 1) * q^2 + (z^5 + z^3 + z - 1) * q^3 + (z^3 + z^2 - z - 1) * q^4 + (-z^4 - z + 1) * q^5 + (-z^5 - z) * q^6 + (-2*z^5 + 2*z^4 - z^3 + 2) * q^7 + (-2*z^5 + 2*z^4 - 2*z^3 - z + 1) * q^8 + (z^5 - z^4 - 2*z^2 + 2*z - 2) * q^9 + (z^4 - z^3 + z^2) * q^10 + (-2*z^5 + z^4 - 3*z^3 + 3*z^2 - z + 2) * q^11 + (-z^3 - 2*z^2 - z) * q^12 + (2*z^5 + 5*z^3 - 5*z^2 - 2) * q^13 + (4*z^5 - 3*z^4 + z^3 - 2*z^2 + 3*z - 1) * q^14 + (-z^5 + z^4 + z^2 + z + 1) * q^15 + (-z^5 - z^3 - z) * q^16 + (2*z^4 - 3*z^3 - 3*z + 2) * q^17 + (-z^5 - 2*z^4 + 2*z^3 + z^2 - 1) * q^18 + (-2*z^5 - z^4 + z^3 + 2*z^2 - 1) * q^19 + (z^4 - z^3 + 3*z^2 - z + 1) * q^20 + (3*z^5 + 2*z^3 + z^2 + 2*z - 1) * q^21 + (2*z^3 - z^2 + z - 2) * q^22 + (-3*z^5 + z^3 + z^2 - z - 1) * q^23 + (2*z^2 + z + 2) * q^24 + (2*z^5 - 2*z^4 + z^2 + 2*z + 1) * q^25 + (3*z^5 - 5*z^3 + 5) * q^26 + (z^3 + 2*z^2 - 2*z - 1) * q^27 + (2*z^5 - 4*z^4 + 6*z^3 - z^2 + 3*z - 5) * q^28 + (-2*z^4 - 3*z^3 - 3*z - 2) * q^29 + (z^5 - z^4 + z^3 - z^2 - 1) * q^30 + (z^5 - z^2 - 2) * q^31 + (3*z^4 + 2*z^3 - 2*z^2 + 2*z + 3) * q^32 + (2*z^5 + z^4 + 4*z^3 + z^2 + 2*z) * q^33 + (4*z^5 - 4*z^4 - 2*z^2 + 7*z - 2) * q^34 + (-2*z^5 - z^4 + z^3 - 4*z^2 + 2*z - 1) * q^35 + (-4*z^5 + 3*z^4 - 3*z + 4) * q^36 + (4*z^4 - 6*z^3 + 4*z^2 - 6*z + 4) * q^37 + (-4*z^5 + 7*z^4 - z^3 + z^2 - 7*z + 4) * q^38 + (-2*z^5 - 2*z^4 - 7*z^3 - 2*z^2 - 2*z) * q^39 + (-z^5 + z^4 - 3*z^2 + 2*z - 3) * q^40 + (-2*z^5 - 4*z^4 - 2*z^3) * q^41 + (-2*z^5 - z^4 - z - 2) * q^42 + (3*z^5 - 2*z^4 + 2*z^3 - 2*z^2 + 3*z) * q^43 + (z^5 + 4*z^4 + z^3 + z - 1) * q^44 + (z^5 + 3*z^3 - 4*z^2 + 4*z - 3) * q^45 + (-2*z^5 + 7*z^4 - 2*z^3 - 6*z + 6) * q^46 + (5*z^5 - 4*z^4 + z^3 - z^2 + 4*z - 5) * q^47 + (-2*z^5 + z^4 - z^3 + 2*z^2 + 3) * q^48 + (-7*z^5 + 7*z^4 - 7*z^3 + 7*z^2 - 7*z + 7) * q^49 + (-2*z^5 - 2*z^4 + 2*z^3 + 2*z^2 - 5) * q^50 + (-2*z^5 - 2*z^4 - z^3 + z^2 + 2*z + 2) * q^51 + (z^5 - 13*z^4 + z^3 + 3*z - 3) * q^52 + (6*z^5 + 8*z^3 - 4*z^2 + 4*z - 8) * q^53 + (-3*z^5 + 6*z^4 - 3*z^3 - 4*z + 4) * q^54 + (-3*z^5 + 3*z^4 - 7*z^3 + 3*z^2 - 3*z) * q^55 + (-4*z^5 + 3*z^4 - 5*z^3 + 4*z^2 - 8*z + 4) * q^56 + (z^5 + 3*z^4 + z^3 - z + 1) * q^57 + (-4*z^5 + 4*z^4 + 2*z^2 - z + 2) * q^58 + (z^5 - z^4 + 7*z^3 - z^2 + z) * q^59 + (3*z^5 - z^4 + 3*z^3 - 3*z^2 + z - 3) * q^60 + (-3*z^4 + 2*z^3 - 3*z^2 + 2*z - 3) * q^61 + (-z^5 + 1) * q^62 + (4*z^5 - 4*z^3 - z - 1) * q^63 + (6*z^5 - 6*z^4 - z^2 + 6*z - 1) * q^64 + (5*z^5 - 6*z^4 + 13*z^3 - 6*z^2 + 5*z) * q^65 + (-2*z^3 - z^2 - 2*z) * q^66 + (-2*z^5 + z^4 - z^3 + 2*z^2 - 1) * q^67 + (-6*z^5 - z^4 + z^3 + 6*z^2 - 3) * q^68 + (3*z^4 - z^3 + z^2 - z + 3) * q^69 + (2*z^5 - z^4 + z^3 + z^2 - z + 2) * q^70 + (6*z^5 - z^3 + 3*z^2 - 3*z + 1) * q^71 + (5*z^5 - z^3 + z^2 - z + 1) * q^72 + (5*z^5 - 5*z^4 - 3*z^2 + 9*z - 3) * q^73 + (4*z^5 - 4*z^4 - 4*z^2 + 10*z - 4) * q^74 + (2*z^5 + 4*z^3 - z^2 + z - 4) * q^75 + (8*z^5 - 2*z^3 - 5*z^2 + 5*z + 2) * q^76 + (-2*z^5 - 4*z^4 - z^3 + z^2 - z + 6) * q^77 + (-3*z^4 + 5*z^3 + 2*z^2 + 5*z - 3) * q^78 + (-2*z^5 + z^4 - z^3 + 2*z^2 + 11) * q^79 + (z^5 - z^2 - 2) * q^80 + (-3*z^4 - 4*z^3 - 4*z - 3) * q^81 + (-4*z^5 + 4*z^4 + 2*z^3 + 4*z^2 - 4*z) * q^82 + (-9*z^5 + 9*z^4 + z^2 - z + 1) * q^83 + (-z^5 + 2*z^4 - 3*z^3 - 3*z^2 - 5*z - 1) * q^84 + (z^5 + 3*z^4 - 3*z^3 + 3*z^2 - 3*z - 1) * q^85 + (-4*z^4 + z^3 + 2*z^2 + z - 4) * q^86 + (-10*z^5 + 2*z^4 - 5*z^3 + 5*z^2 - 2*z + 10) * q^87 + (z^5 - 6*z^4 + 2*z^3 - 6*z^2 + z) * q^88 + (-9*z^5 + 9*z^4 + 8*z^2 - 9*z + 8) * q^89 + (z^5 - 3*z^4 + z^3 - z + 1) * q^90 + (-2*z^5 + 11*z^4 + z^3 + z^2 - 3*z - 9) * q^91 + (9*z^5 - 5*z^4 - 5*z^2 + 9*z) * q^92 + (-3*z^5 - z^4 - 3*z^3 - 2*z + 2) * q^93 + (-8*z^5 + 3*z^3 + 4*z^2 - 4*z - 3) * q^94 + (-z^5 - z^4 - z^3) * q^95 + (8*z^5 - 3*z^4 + 3*z^3 - 3*z^2 + 3*z - 8) * q^96 + (z^5 - 2*z^4 + 2*z^3 - z^2 - 1) * q^97 + (7*z^5 - 7*z^2 + 7*z) * q^98 + (5*z^5 - 6*z^4 + 6*z^3 - 5*z^2 - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{2} - 3 q^{3} - 7 q^{4} + 6 q^{5} - 2 q^{6} + 7 q^{7} - q^{8} - 6 q^{9}+O(q^{10})$$ 6 * q - 3 * q^2 - 3 * q^3 - 7 * q^4 + 6 * q^5 - 2 * q^6 + 7 * q^7 - q^8 - 6 * q^9 $$6 q - 3 q^{2} - 3 q^{3} - 7 q^{4} + 6 q^{5} - 2 q^{6} + 7 q^{7} - q^{8} - 6 q^{9} - 3 q^{10} + 2 q^{11} + 7 q^{14} + 4 q^{15} - 3 q^{16} + 4 q^{17} - 4 q^{18} - 8 q^{19} - 8 q^{22} - 10 q^{23} + 11 q^{24} + 11 q^{25} + 28 q^{26} - 9 q^{27} - 14 q^{28} - 16 q^{29} - 2 q^{30} - 10 q^{31} + 21 q^{32} + 6 q^{33} + 5 q^{34} + 14 q^{36} + 4 q^{37} + 4 q^{38} - 7 q^{39} - 15 q^{40} - 14 q^{42} + 12 q^{43} - 7 q^{44} - 6 q^{45} + 19 q^{46} - 15 q^{47} + 12 q^{48} + 7 q^{49} - 30 q^{50} + 12 q^{51} - 26 q^{53} + 8 q^{54} - 19 q^{55} + 4 q^{57} + q^{58} + 11 q^{59} - 7 q^{60} - 8 q^{61} + 5 q^{62} - 7 q^{63} + 13 q^{64} + 35 q^{65} - 3 q^{66} - 12 q^{67} - 28 q^{68} + 12 q^{69} + 14 q^{70} + 5 q^{71} + 8 q^{72} + 4 q^{73} - 2 q^{74} - 16 q^{75} + 28 q^{76} + 35 q^{77} - 7 q^{78} + 60 q^{79} - 10 q^{80} - 23 q^{81} - 14 q^{82} - 14 q^{83} - 14 q^{84} - 17 q^{85} - 20 q^{86} + 36 q^{87} + 16 q^{88} + 13 q^{89} + 10 q^{90} - 70 q^{91} + 28 q^{92} + 5 q^{93} - 31 q^{94} - q^{95} - 28 q^{96} + 21 q^{98} - 2 q^{99}+O(q^{100})$$ 6 * q - 3 * q^2 - 3 * q^3 - 7 * q^4 + 6 * q^5 - 2 * q^6 + 7 * q^7 - q^8 - 6 * q^9 - 3 * q^10 + 2 * q^11 + 7 * q^14 + 4 * q^15 - 3 * q^16 + 4 * q^17 - 4 * q^18 - 8 * q^19 - 8 * q^22 - 10 * q^23 + 11 * q^24 + 11 * q^25 + 28 * q^26 - 9 * q^27 - 14 * q^28 - 16 * q^29 - 2 * q^30 - 10 * q^31 + 21 * q^32 + 6 * q^33 + 5 * q^34 + 14 * q^36 + 4 * q^37 + 4 * q^38 - 7 * q^39 - 15 * q^40 - 14 * q^42 + 12 * q^43 - 7 * q^44 - 6 * q^45 + 19 * q^46 - 15 * q^47 + 12 * q^48 + 7 * q^49 - 30 * q^50 + 12 * q^51 - 26 * q^53 + 8 * q^54 - 19 * q^55 + 4 * q^57 + q^58 + 11 * q^59 - 7 * q^60 - 8 * q^61 + 5 * q^62 - 7 * q^63 + 13 * q^64 + 35 * q^65 - 3 * q^66 - 12 * q^67 - 28 * q^68 + 12 * q^69 + 14 * q^70 + 5 * q^71 + 8 * q^72 + 4 * q^73 - 2 * q^74 - 16 * q^75 + 28 * q^76 + 35 * q^77 - 7 * q^78 + 60 * q^79 - 10 * q^80 - 23 * q^81 - 14 * q^82 - 14 * q^83 - 14 * q^84 - 17 * q^85 - 20 * q^86 + 36 * q^87 + 16 * q^88 + 13 * q^89 + 10 * q^90 - 70 * q^91 + 28 * q^92 + 5 * q^93 - 31 * q^94 - q^95 - 28 * q^96 + 21 * q^98 - 2 * q^99

## Character values

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

 $$n$$ $$3$$ $$\chi(n)$$ $$-\zeta_{14}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
8.1
 −0.623490 + 0.781831i 0.222521 + 0.974928i 0.900969 + 0.433884i 0.900969 − 0.433884i 0.222521 − 0.974928i −0.623490 − 0.781831i
−0.500000 0.626980i −0.500000 + 0.240787i 0.301938 1.32288i 2.52446 1.21572i 0.400969 + 0.193096i −1.14795 + 2.38374i −2.42543 + 1.16802i −1.67845 + 2.10471i −2.02446 0.974928i
15.1 −0.500000 + 2.19064i −0.500000 + 0.626980i −2.74698 1.32288i 0.153989 0.193096i −1.12349 1.40881i 2.06853 1.64960i 1.46950 1.84270i 0.524459 + 2.29780i 0.346011 + 0.433884i
22.1 −0.500000 + 0.240787i −0.500000 + 2.19064i −1.05496 + 1.32288i 0.321552 1.40881i −0.277479 1.21572i 2.57942 0.588735i 0.455927 1.99755i −1.84601 0.888992i 0.178448 + 0.781831i
29.1 −0.500000 0.240787i −0.500000 2.19064i −1.05496 1.32288i 0.321552 + 1.40881i −0.277479 + 1.21572i 2.57942 + 0.588735i 0.455927 + 1.99755i −1.84601 + 0.888992i 0.178448 0.781831i
36.1 −0.500000 2.19064i −0.500000 0.626980i −2.74698 + 1.32288i 0.153989 + 0.193096i −1.12349 + 1.40881i 2.06853 + 1.64960i 1.46950 + 1.84270i 0.524459 2.29780i 0.346011 0.433884i
43.1 −0.500000 + 0.626980i −0.500000 0.240787i 0.301938 + 1.32288i 2.52446 + 1.21572i 0.400969 0.193096i −1.14795 2.38374i −2.42543 1.16802i −1.67845 2.10471i −2.02446 + 0.974928i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 8.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.2.e.a 6
3.b odd 2 1 441.2.u.a 6
4.b odd 2 1 784.2.u.a 6
7.b odd 2 1 343.2.e.a 6
7.c even 3 2 343.2.g.f 12
7.d odd 6 2 343.2.g.e 12
49.e even 7 1 inner 49.2.e.a 6
49.e even 7 1 2401.2.a.b 3
49.f odd 14 1 343.2.e.a 6
49.f odd 14 1 2401.2.a.a 3
49.g even 21 2 343.2.g.f 12
49.h odd 42 2 343.2.g.e 12
147.l odd 14 1 441.2.u.a 6
196.k odd 14 1 784.2.u.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.2.e.a 6 1.a even 1 1 trivial
49.2.e.a 6 49.e even 7 1 inner
343.2.e.a 6 7.b odd 2 1
343.2.e.a 6 49.f odd 14 1
343.2.g.e 12 7.d odd 6 2
343.2.g.e 12 49.h odd 42 2
343.2.g.f 12 7.c even 3 2
343.2.g.f 12 49.g even 21 2
441.2.u.a 6 3.b odd 2 1
441.2.u.a 6 147.l odd 14 1
784.2.u.a 6 4.b odd 2 1
784.2.u.a 6 196.k odd 14 1
2401.2.a.a 3 49.f odd 14 1
2401.2.a.b 3 49.e even 7 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 3T_{2}^{5} + 9T_{2}^{4} + 13T_{2}^{3} + 11T_{2}^{2} + 5T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(49, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 3 T^{5} + \cdots + 1$$
$3$ $$T^{6} + 3 T^{5} + \cdots + 1$$
$5$ $$T^{6} - 6 T^{5} + \cdots + 1$$
$7$ $$T^{6} - 7 T^{5} + \cdots + 343$$
$11$ $$T^{6} - 2 T^{5} + \cdots + 169$$
$13$ $$T^{6} + 14 T^{4} + \cdots + 41209$$
$17$ $$T^{6} - 4 T^{5} + \cdots + 1681$$
$19$ $$(T^{3} + 4 T^{2} - 11 T - 1)^{2}$$
$23$ $$T^{6} + 10 T^{5} + \cdots + 169$$
$29$ $$T^{6} + 16 T^{5} + \cdots + 6889$$
$31$ $$(T^{3} + 5 T^{2} + 6 T + 1)^{2}$$
$37$ $$T^{6} - 4 T^{5} + \cdots + 10816$$
$41$ $$T^{6} + 56 T^{4} + \cdots + 3136$$
$43$ $$T^{6} - 12 T^{5} + \cdots + 841$$
$47$ $$T^{6} + 15 T^{5} + \cdots + 9409$$
$53$ $$T^{6} + 26 T^{5} + \cdots + 53824$$
$59$ $$T^{6} - 11 T^{5} + \cdots + 57121$$
$61$ $$T^{6} + 8 T^{5} + \cdots + 1849$$
$67$ $$(T^{3} + 6 T^{2} + 5 T - 13)^{2}$$
$71$ $$T^{6} - 5 T^{5} + \cdots + 85849$$
$73$ $$T^{6} - 4 T^{5} + \cdots + 5041$$
$79$ $$(T^{3} - 30 T^{2} + \cdots - 937)^{2}$$
$83$ $$T^{6} + 14 T^{5} + \cdots + 625681$$
$89$ $$T^{6} - 13 T^{5} + \cdots + 187489$$
$97$ $$(T^{3} - 7 T - 7)^{2}$$