# Properties

 Label 49.2.e.b Level $49$ Weight $2$ Character orbit 49.e Analytic conductor $0.391$ Analytic rank $0$ Dimension $12$ 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: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{7})$$ Coefficient field: $$\Q(\zeta_{21})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1$$ x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$7$$ 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 basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{11} - \beta_{9} + \cdots + \beta_{2}) q^{2}+ \cdots + (2 \beta_{10} - 2 \beta_{8} + \beta_{7} + \cdots - 2) q^{9}+O(q^{10})$$ q + (b11 - b9 - b5 + b4 + b2) * q^2 + (-b7 + b6 - b2 + b1) * q^3 + (-b11 - b4 + b3) * q^4 + (-b11 + b10 - b8 + b7 - b6 + 2*b5 - 2*b4 - b2 - b1 - 2) * q^5 + (b11 - b10 + b8 - b6 + b2 - b1) * q^6 + (-2*b10 + b9 + b8 - b7 + b6 + 2*b4 + 1) * q^7 + (-b10 - b5 + b4 + b3 + b2 + 1) * q^8 + (2*b10 - 2*b8 + b7 + 2*b5 - 2*b3 - 2) * q^9 $$q + (\beta_{11} - \beta_{9} + \cdots + \beta_{2}) q^{2}+ \cdots + (\beta_{11} - 10 \beta_{10} + 6 \beta_{9} + \cdots + 4) q^{99}+O(q^{100})$$ q + (b11 - b9 - b5 + b4 + b2) * q^2 + (-b7 + b6 - b2 + b1) * q^3 + (-b11 - b4 + b3) * q^4 + (-b11 + b10 - b8 + b7 - b6 + 2*b5 - 2*b4 - b2 - b1 - 2) * q^5 + (b11 - b10 + b8 - b6 + b2 - b1) * q^6 + (-2*b10 + b9 + b8 - b7 + b6 + 2*b4 + 1) * q^7 + (-b10 - b5 + b4 + b3 + b2 + 1) * q^8 + (2*b10 - 2*b8 + b7 + 2*b5 - 2*b3 - 2) * q^9 + (b10 + 2*b9 - b6 + b5 - b4 - 2*b3 - b2 + 1) * q^10 + (-b11 + b10 + b9 + b7 - b5 + b4 - b3 + b2) * q^11 + (2*b6 - 2*b5 - b2 + 2*b1 - 1) * q^12 + (b8 + b6 + b1 + 1) * q^13 + (b10 - b9 + 2*b7 - 3*b6 - b4 + b3 - 2*b2 - b1 - 4) * q^14 + (2*b10 - b9 + b8 + b7 - 2*b6 - 2*b5 - 3*b4 + b3 - 2*b2 + b1 - 2) * q^15 + (-2*b10 - b9 + 2*b8 + 2*b6 - 5*b5 + 2*b4 + b3 + 2*b2 + 4) * q^16 + (b11 - b10 - b7 + b4 + b2 - 3*b1 + 1) * q^17 + (-b10 + 2*b9 - 2*b7 + b6 + 3*b5 - b4 + 2) * q^18 + (2*b11 + b10 + b5 - b4 - b3 + b1 - 2) * q^19 + (2*b10 - b9 - b8 - 2*b6 + 2*b5 - 1) * q^20 + (-2*b11 - b10 - 3*b8 - b7 + 4*b5 + b4 + b3 + b2 - 2*b1 + 1) * q^21 + (-b11 - 2*b10 - 3*b8 - 3*b7 + 4*b5 + b4 + b3 - b2) * q^22 + (-b10 - b9 + b4) * q^23 + (-b9 + b6 - b5 - b4 + b2 + b1) * q^24 + (-3*b10 - 2*b9 + b8 - b7 + 5*b6 - 3*b5 + 2*b4 + b3 + 5*b2 + 2*b1 + 3) * q^25 + (-b10 - b9 + b8 + b7 - b5 + b4) * q^26 + (4*b10 + 3*b8 + 3*b7 - 4*b6 - 4*b4 - 3*b2) * q^27 + (-b10 + 2*b9 + 3*b6 + b5 - b3 - b2 + 2) * q^28 + (3*b11 - 4*b10 - b9 - b8 - 3*b7 + 3*b6 - 3*b5 + 3*b4 + 2*b2 + 1) * q^29 + (-b11 - 2*b10 + 2*b9 - 2*b7 + 2*b6 + b5 + b4 + 5) * q^30 + (2*b11 + 2*b10 - 2*b9 + 2*b7 - 5*b6 - 2*b5 + 3*b3 - 3*b1 - 3) * q^31 + (3*b10 - 2*b9 - 2*b8 - 3*b6 + 3*b5 - 3*b2 - b1 - 5) * q^32 + (-b11 - 3*b10 + 3*b8 - 6*b5 - b4 + 3*b2 + b1 + 6) * q^33 + (7*b10 - b9 - 2*b8 + b7 - 2*b6 + b5 - 3*b4 - 2*b3 - 2*b2 + b1 - 7) * q^34 + (2*b11 + b10 + 2*b9 + b8 - 2*b7 + b6 - b5 - b4 + b3 - b2 + b1 + 2) * q^35 + (2*b10 - b8 + 2*b7 - b6 + b5 + 3*b4 - 2*b3 + 3*b2 - b1 - 1) * q^36 + (-2*b11 + 2*b7 - 2*b4 - 4*b2 + 3*b1 - 4) * q^37 + (-2*b11 + 2*b9 + 3*b8 + 4*b6 - b5 + 3*b1 + 4) * q^38 + (b10 + b6 + b4 - b2) * q^39 + (2*b10 + b9 - 2*b8 - 2*b6 + 3*b5 - 2*b4 - 2*b3 - 2*b2 - b1 - 2) * q^40 + (-b11 - 5*b10 - b8 - 4*b5 + 3*b4 + 3*b3 + 3) * q^41 + (3*b11 - b9 + b8 + 2*b7 - 2*b6 - 4*b5 - b3 + 4*b2 - 2*b1 - 2) * q^42 + (-2*b11 + 6*b10 + 4*b9 + b8 - 3*b6 + 5*b5 - 5*b4 - 4*b3 - 6*b2 + 2*b1 - 1) * q^43 + (2*b11 + 3*b10 + 2*b8 + 3*b7 - 3*b6 + b5 - b4 - b3 + b2 - 3*b1 - 1) * q^44 + (b11 - 5*b10 + b8 + b7 + 5*b6 - 4*b5 + 6*b4 - b3 + 3*b2) * q^45 + (b10 + 2*b7 - 2*b6 + b5 - 3*b4 + b3 - 2*b2 - 2*b1 - 3) * q^46 + (-2*b11 - 2*b10 + 2*b9 - b8 - 2*b7 + 4*b5 + 5*b4 + 2*b3 + 5*b2 - b1) * q^47 + (-b11 - 6*b10 + 4*b9 - 4*b7 + 7*b6 + b5 + 3*b4 - b3 + b1 + 4) * q^48 + (-b11 - b10 - 2*b9 - 3*b8 - 3*b6 + 2*b5 - 3*b4 - 2*b3 + 3*b2 - b1 - 1) * q^49 + (-3*b11 - b10 - 3*b9 + 3*b7 + 2*b6 - b5 - 2*b4 - b3 + b1 - 4) * q^50 + (-3*b11 - 4*b10 + 3*b9 - b8 - 4*b7 + 6*b5 - 3*b4 + 4*b3 - 3*b2 - b1) * q^51 + (-b11 - b10 - b8 - b4 + b3 - 1) * q^52 + (2*b11 + 2*b9 - 2*b8 - 2*b7 - 2*b6 + 2*b5 + 2*b4 - 2*b3) * q^53 + (-b7 + b6 + 4*b4 - 2*b2 + b1 + 4) * q^54 + (3*b11 - 3*b10 + b5 + 3*b4 + 3*b2 - 3*b1 - 1) * q^55 + (-b11 + b10 + 2*b8 + 2*b7 + 2*b5 - 3*b4 - b3 - 3*b2 + b1 - 2) * q^56 + (b11 + 4*b10 + b8 + 4*b7 - 4*b6 + 3*b5 - b4 + b2 - 4*b1 - 1) * q^57 + (4*b10 - b9 + b7 + b6 - b5 - 4*b4 + b2 + b1 - 4) * q^58 + (-3*b11 + b10 - b6 - 4*b4 - b2 + 3*b1) * q^59 + (3*b11 - b10 - 3*b9 + 2*b8 - b7 - 2*b6 - 6*b5 + b3 + 2*b1 - 2) * q^60 + (2*b11 + 5*b10 - b9 - b8 - 2*b7 - 2*b6 + 2*b5 + 2*b4 - b2 - 3*b1 - 2) * q^61 + (-3*b11 - 2*b10 + 3*b9 - 2*b7 + 9*b6 - 5*b5 + 2*b4 + 2*b3 + 2*b2 + 9) * q^62 + (b11 + 3*b10 - 3*b9 + 2*b8 - 2*b7 - b6 - 5*b5 - b4 + 4*b3 - 3*b2 + 3*b1) * q^63 + (-3*b10 + b9 + 4*b8 - b6 - 3*b5 - b4 + 4*b3 - b2 - b1 + 3) * q^64 + (b11 + 3*b10 + b9 - 2*b8 - 4*b6 + 3*b5 - 3*b4 - b3 - 3*b2 - b1 - 2) * q^65 + (5*b10 - 6*b9 - 6*b8 - 4*b6 + 4*b5 - 3*b2 + 3*b1 - 9) * q^66 + (b11 + 2*b10 - 4*b9 + 4*b7 - b6 - 2*b5 - 2*b4 - b3 + b1 + 2) * q^67 + (-b11 - 2*b10 + b9 - b7 + 2*b6 + 3*b5 - 2*b4 + 5) * q^68 + (b11 - b10 - b9 - b8 - b7 - b6 + b5 + b4 - b1 - 1) * q^69 + (2*b11 - 3*b10 - 2*b9 + 3*b8 - 3*b7 + 6*b6 - 8*b5 + 9*b4 - b3 + 8*b2 + 4*b1 + 8) * q^70 + (3*b11 + 2*b10 - b9 + 3*b8 + 3*b7 + 6*b6 - 7*b5 + b4 - 3*b3 + 4*b2) * q^71 + (b11 + b9 - 2*b8 - 2*b7 + b6 + b4 - b3 + 2*b2) * q^72 + (-5*b10 + b7 + b6 + 3*b4 + b2 + 5) * q^73 + (-6*b10 + 4*b9 + b8 - 3*b6 + 3*b5 - b4 + b3 - 3*b2 - 4*b1 + 6) * q^74 + (-b11 + b8 + b7 + b6 + 2*b5 - b4 + b3 - 3*b2) * q^75 + (b11 - 5*b10 - 2*b8 - 2*b7 + 2*b6 - b5 + 6*b4 - b3 + 3*b2) * q^76 + (2*b10 - 3*b9 + b8 + b7 + 3*b6 - 7*b5 - 3*b3 - b2 + 7*b1 + 2) * q^77 + (b11 - 2*b10 - b7 + b4 + b2 - 2*b1 + 1) * q^78 + (-b11 + 2*b10 - b9 + b7 - 2*b6 - 4*b5 + 3*b4 - 1) * q^79 + (2*b11 + 5*b10 + 2*b9 - 2*b7 - 6*b6 + 7*b5 - 5*b4 + b3 - b1) * q^80 + (-2*b11 + b10 + b9 + b8 + 2*b7 + 2*b6 - 2*b5 - 2*b4 - 3*b2 + b1 - 2) * q^81 + (4*b11 - 5*b10 - 3*b9 - 2*b8 - 7*b5 + 4*b4 + 3*b3 + 5*b2 - 4*b1 + 4) * q^82 + (2*b10 - 5*b9 - 3*b8 - b7 + 4*b6 - 2*b5 - 3*b4 - 3*b3 + 4*b2 + 5*b1 - 2) * q^83 + (-5*b11 + 8*b10 - 2*b8 + 2*b7 - 2*b6 + 7*b5 - 4*b4 - 4*b2 - 1) * q^84 + (-2*b11 - 2*b10 + 2*b9 + 2*b8 - 2*b7 + 5*b6 + 2*b5 + 2*b4 + 2*b3 + 2*b2 + 2*b1 + 5) * q^85 + (4*b11 + b9 + b8 - 4*b7 - 7*b6 + 7*b5 + 4*b4 - b2 + b1) * q^86 + (-2*b11 + 2*b10 + 2*b9 - 5*b8 + 2*b7 - 3*b6 + 15*b5 + b4 - 2*b3 + b2 - 5*b1 - 3) * q^87 + (2*b11 + 3*b10 + b9 + b8 - 2*b6 + 3*b5 - b3 - 3*b2 - 2*b1 - 2) * q^88 + (-3*b10 + 2*b9 + 4*b8 - 3*b7 + 4*b6 - 2*b5 - b4 + 4*b3 + 4*b2 - 2*b1 + 3) * q^89 + (-4*b11 + b10 - 4*b8 + 2*b7 - 2*b6 + 5*b5 - 6*b4 + b3 - 8*b2 - 2*b1 - 6) * q^90 + (-2*b11 - b10 + b9 - b8 - b7 + b6 + b5 - 2*b1 - 1) * q^91 + (-b11 + b9 + b6 - b5 - b3 + b1 + 2) * q^92 + (4*b11 - 3*b10 + 4*b8 - 7*b5 - 3*b4 + b3 + 5*b2 - 3) * q^93 + (-4*b11 - 6*b10 + 3*b8 + 3*b7 + 3*b6 - 6*b5 + 2*b4 + 4*b3 + 3*b2) * q^94 + (-b11 - 4*b10 - b8 - 3*b5 - 2*b3 - 4*b2) * q^95 + (4*b11 + b10 - 4*b9 + b7 - 7*b6 - 5*b5 + 2*b4 - b3 + 2*b2 - 7) * q^96 + (-4*b11 - 2*b6 + 3*b5 - 3*b4 + 2*b3 - 2*b1 - 3) * q^97 + (-b11 + 9*b10 + b9 - 2*b8 + 2*b7 - 4*b6 + 7*b5 - 9*b4 - b3 - 4*b2 + 3*b1 - 9) * q^98 + (b11 - 10*b10 + 6*b9 - 6*b7 + 8*b6 + b5 + 5*b4 + 2*b3 - 2*b1 + 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 2 q^{2} + 2 q^{4} - 7 q^{5} - 7 q^{6} - 7 q^{7} + 6 q^{8} - 8 q^{9}+O(q^{10})$$ 12 * q - 2 * q^2 + 2 * q^4 - 7 * q^5 - 7 * q^6 - 7 * q^7 + 6 * q^8 - 8 * q^9 $$12 q - 2 q^{2} + 2 q^{4} - 7 q^{5} - 7 q^{6} - 7 q^{7} + 6 q^{8} - 8 q^{9} + 14 q^{10} - 8 q^{11} - 14 q^{12} + 7 q^{13} - 28 q^{14} - 7 q^{15} + 16 q^{16} + 20 q^{18} - 14 q^{19} + 7 q^{20} + 21 q^{21} + 13 q^{22} - 2 q^{23} + 3 q^{25} - 7 q^{26} + 21 q^{27} + 14 q^{28} - 11 q^{29} + 42 q^{30} - 14 q^{31} - 24 q^{32} + 35 q^{33} - 42 q^{34} + 21 q^{35} - 13 q^{36} - 30 q^{37} + 21 q^{38} + 21 q^{41} - 35 q^{42} + 17 q^{43} - 6 q^{44} - 49 q^{45} - 16 q^{46} - 21 q^{47} + 7 q^{49} - 46 q^{50} + 7 q^{51} - 7 q^{52} + 6 q^{53} + 42 q^{54} - 28 q^{55} - 14 q^{56} + 7 q^{57} - 32 q^{58} + 14 q^{59} - 28 q^{60} - 7 q^{61} + 56 q^{62} + 14 q^{63} + 14 q^{64} + 14 q^{65} - 28 q^{66} + 48 q^{67} + 56 q^{68} - 7 q^{69} + 21 q^{70} - 39 q^{71} - 4 q^{72} + 42 q^{73} + 61 q^{74} + 7 q^{75} - 28 q^{76} + 21 q^{77} - 16 q^{79} + 42 q^{80} - 25 q^{81} + 28 q^{82} - 7 q^{83} + 42 q^{84} + 28 q^{85} + 17 q^{86} + 7 q^{87} - 11 q^{88} - 14 q^{89} - 14 q^{90} - 21 q^{91} + 16 q^{92} - 70 q^{93} - 49 q^{94} - 7 q^{95} - 70 q^{96} - 28 q^{97} - 28 q^{98} - 18 q^{99}+O(q^{100})$$ 12 * q - 2 * q^2 + 2 * q^4 - 7 * q^5 - 7 * q^6 - 7 * q^7 + 6 * q^8 - 8 * q^9 + 14 * q^10 - 8 * q^11 - 14 * q^12 + 7 * q^13 - 28 * q^14 - 7 * q^15 + 16 * q^16 + 20 * q^18 - 14 * q^19 + 7 * q^20 + 21 * q^21 + 13 * q^22 - 2 * q^23 + 3 * q^25 - 7 * q^26 + 21 * q^27 + 14 * q^28 - 11 * q^29 + 42 * q^30 - 14 * q^31 - 24 * q^32 + 35 * q^33 - 42 * q^34 + 21 * q^35 - 13 * q^36 - 30 * q^37 + 21 * q^38 + 21 * q^41 - 35 * q^42 + 17 * q^43 - 6 * q^44 - 49 * q^45 - 16 * q^46 - 21 * q^47 + 7 * q^49 - 46 * q^50 + 7 * q^51 - 7 * q^52 + 6 * q^53 + 42 * q^54 - 28 * q^55 - 14 * q^56 + 7 * q^57 - 32 * q^58 + 14 * q^59 - 28 * q^60 - 7 * q^61 + 56 * q^62 + 14 * q^63 + 14 * q^64 + 14 * q^65 - 28 * q^66 + 48 * q^67 + 56 * q^68 - 7 * q^69 + 21 * q^70 - 39 * q^71 - 4 * q^72 + 42 * q^73 + 61 * q^74 + 7 * q^75 - 28 * q^76 + 21 * q^77 - 16 * q^79 + 42 * q^80 - 25 * q^81 + 28 * q^82 - 7 * q^83 + 42 * q^84 + 28 * q^85 + 17 * q^86 + 7 * q^87 - 11 * q^88 - 14 * q^89 - 14 * q^90 - 21 * q^91 + 16 * q^92 - 70 * q^93 - 49 * q^94 - 7 * q^95 - 70 * q^96 - 28 * q^97 - 28 * q^98 - 18 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{21}^{2} + \zeta_{21}$$ v^2 + v $$\beta_{2}$$ $$=$$ $$\zeta_{21}^{3}$$ v^3 $$\beta_{3}$$ $$=$$ $$\zeta_{21}^{5} + \zeta_{21}$$ v^5 + v $$\beta_{4}$$ $$=$$ $$\zeta_{21}^{6}$$ v^6 $$\beta_{5}$$ $$=$$ $$\zeta_{21}^{8} + \zeta_{21}$$ v^8 + v $$\beta_{6}$$ $$=$$ $$\zeta_{21}^{9}$$ v^9 $$\beta_{7}$$ $$=$$ $$\zeta_{21}^{11} + \zeta_{21}$$ v^11 + v $$\beta_{8}$$ $$=$$ $$\zeta_{21}^{8} + \zeta_{21}^{7}$$ v^8 + v^7 $$\beta_{9}$$ $$=$$ $$-\zeta_{21}^{10} - \zeta_{21}^{8}$$ -v^10 - v^8 $$\beta_{10}$$ $$=$$ $$-\zeta_{21}^{11} + \zeta_{21}^{9} - \zeta_{21}^{8} + \zeta_{21}^{6} - \zeta_{21}^{4} + \zeta_{21}^{3} - \zeta_{21} + 1$$ -v^11 + v^9 - v^8 + v^6 - v^4 + v^3 - v + 1 $$\beta_{11}$$ $$=$$ $$-\zeta_{21}^{11} + \zeta_{21}^{10} - \zeta_{21}^{8} + \zeta_{21}^{7} - \zeta_{21}^{5} + \zeta_{21}^{3} - \zeta_{21}^{2} + \zeta_{21} + 1$$ -v^11 + v^10 - v^8 + v^7 - v^5 + v^3 - v^2 + v + 1
 $$\zeta_{21}$$ $$=$$ $$( \beta_{11} + \beta_{9} - \beta_{8} + \beta_{7} + 3\beta_{5} + \beta_{3} - \beta_{2} + \beta _1 - 1 ) / 7$$ (b11 + b9 - b8 + b7 + 3*b5 + b3 - b2 + b1 - 1) / 7 $$\zeta_{21}^{2}$$ $$=$$ $$( -\beta_{11} - \beta_{9} + \beta_{8} - \beta_{7} - 3\beta_{5} - \beta_{3} + \beta_{2} + 6\beta _1 + 1 ) / 7$$ (-b11 - b9 + b8 - b7 - 3*b5 - b3 + b2 + 6*b1 + 1) / 7 $$\zeta_{21}^{3}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{21}^{4}$$ $$=$$ $$( \beta_{11} - 7 \beta_{10} + \beta_{9} - \beta_{8} - 6 \beta_{7} + 7 \beta_{6} - 4 \beta_{5} + 7 \beta_{4} + \cdots + 6 ) / 7$$ (b11 - 7*b10 + b9 - b8 - 6*b7 + 7*b6 - 4*b5 + 7*b4 + b3 + 6*b2 + b1 + 6) / 7 $$\zeta_{21}^{5}$$ $$=$$ $$( -\beta_{11} - \beta_{9} + \beta_{8} - \beta_{7} - 3\beta_{5} + 6\beta_{3} + \beta_{2} - \beta _1 + 1 ) / 7$$ (-b11 - b9 + b8 - b7 - 3*b5 + 6*b3 + b2 - b1 + 1) / 7 $$\zeta_{21}^{6}$$ $$=$$ $$\beta_{4}$$ b4 $$\zeta_{21}^{7}$$ $$=$$ $$( \beta_{11} + \beta_{9} + 6\beta_{8} + \beta_{7} - 4\beta_{5} + \beta_{3} - \beta_{2} + \beta _1 - 1 ) / 7$$ (b11 + b9 + 6*b8 + b7 - 4*b5 + b3 - b2 + b1 - 1) / 7 $$\zeta_{21}^{8}$$ $$=$$ $$( -\beta_{11} - \beta_{9} + \beta_{8} - \beta_{7} + 4\beta_{5} - \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 7$$ (-b11 - b9 + b8 - b7 + 4*b5 - b3 + b2 - b1 + 1) / 7 $$\zeta_{21}^{9}$$ $$=$$ $$\beta_{6}$$ b6 $$\zeta_{21}^{10}$$ $$=$$ $$( \beta_{11} - 6\beta_{9} - \beta_{8} + \beta_{7} - 4\beta_{5} + \beta_{3} - \beta_{2} + \beta _1 - 1 ) / 7$$ (b11 - 6*b9 - b8 + b7 - 4*b5 + b3 - b2 + b1 - 1) / 7 $$\zeta_{21}^{11}$$ $$=$$ $$( -\beta_{11} - \beta_{9} + \beta_{8} + 6\beta_{7} - 3\beta_{5} - \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 7$$ (-b11 - b9 + b8 + 6*b7 - 3*b5 - b3 + b2 - b1 + 1) / 7

## Character values

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

 $$n$$ $$3$$ $$\chi(n)$$ $$\beta_{4}$$

## 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
 −0.988831 + 0.149042i 0.365341 − 0.930874i −0.733052 − 0.680173i 0.955573 − 0.294755i 0.0747301 − 0.997204i 0.826239 + 0.563320i 0.0747301 + 0.997204i 0.826239 − 0.563320i −0.733052 + 0.680173i 0.955573 + 0.294755i −0.988831 − 0.149042i 0.365341 + 0.930874i
−1.19158 1.49419i 1.55929 0.750915i −0.367711 + 1.61105i −1.91647 + 0.922924i −2.98003 1.43511i 1.91879 1.82161i −0.598393 + 0.288171i −0.00295512 + 0.00370560i 3.66265 + 1.76384i
8.2 0.914101 + 1.14625i −0.880843 + 0.424191i −0.0332580 + 0.145713i −0.830509 + 0.399952i −1.29141 0.621909i −0.938402 2.47374i 2.44440 1.17716i −1.27452 + 1.59820i −1.21761 0.586371i
15.1 0.0332580 0.145713i −1.81507 + 2.27603i 1.78181 + 0.858075i 1.36967 1.71752i 0.271281 + 0.340175i −2.64558 + 0.0302261i 0.370666 0.464800i −1.21825 5.33750i −0.204712 0.256700i
15.2 0.367711 1.61105i 0.290611 0.364415i −0.658322 0.317031i −2.42463 + 3.04039i −0.480228 0.602187i −1.62586 2.08724i 1.30778 1.63991i 0.619220 + 2.71298i 4.00665 + 5.02418i
22.1 −1.78181 + 0.858075i 0.590232 2.58597i 1.19158 1.49419i 0.359497 1.57506i 1.16728 + 5.11418i 1.95991 + 1.77729i 0.0391023 0.171318i −3.63598 1.75100i 0.710963 + 3.11493i
22.2 0.658322 0.317031i 0.255779 1.12064i −0.914101 + 1.14625i −0.0575591 + 0.252183i −0.186893 0.818832i −2.16885 1.51528i −0.563561 + 2.46912i 1.51249 + 0.728379i 0.0420574 + 0.184265i
29.1 −1.78181 0.858075i 0.590232 + 2.58597i 1.19158 + 1.49419i 0.359497 + 1.57506i 1.16728 5.11418i 1.95991 1.77729i 0.0391023 + 0.171318i −3.63598 + 1.75100i 0.710963 3.11493i
29.2 0.658322 + 0.317031i 0.255779 + 1.12064i −0.914101 1.14625i −0.0575591 0.252183i −0.186893 + 0.818832i −2.16885 + 1.51528i −0.563561 2.46912i 1.51249 0.728379i 0.0420574 0.184265i
36.1 0.0332580 + 0.145713i −1.81507 2.27603i 1.78181 0.858075i 1.36967 + 1.71752i 0.271281 0.340175i −2.64558 0.0302261i 0.370666 + 0.464800i −1.21825 + 5.33750i −0.204712 + 0.256700i
36.2 0.367711 + 1.61105i 0.290611 + 0.364415i −0.658322 + 0.317031i −2.42463 3.04039i −0.480228 + 0.602187i −1.62586 + 2.08724i 1.30778 + 1.63991i 0.619220 2.71298i 4.00665 5.02418i
43.1 −1.19158 + 1.49419i 1.55929 + 0.750915i −0.367711 1.61105i −1.91647 0.922924i −2.98003 + 1.43511i 1.91879 + 1.82161i −0.598393 0.288171i −0.00295512 0.00370560i 3.66265 1.76384i
43.2 0.914101 1.14625i −0.880843 0.424191i −0.0332580 0.145713i −0.830509 0.399952i −1.29141 + 0.621909i −0.938402 + 2.47374i 2.44440 + 1.17716i −1.27452 1.59820i −1.21761 + 0.586371i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 8.2 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.b 12
3.b odd 2 1 441.2.u.b 12
4.b odd 2 1 784.2.u.b 12
7.b odd 2 1 343.2.e.b 12
7.c even 3 1 343.2.g.b 12
7.c even 3 1 343.2.g.d 12
7.d odd 6 1 343.2.g.a 12
7.d odd 6 1 343.2.g.c 12
49.e even 7 1 inner 49.2.e.b 12
49.e even 7 1 2401.2.a.c 6
49.f odd 14 1 343.2.e.b 12
49.f odd 14 1 2401.2.a.d 6
49.g even 21 1 343.2.g.b 12
49.g even 21 1 343.2.g.d 12
49.h odd 42 1 343.2.g.a 12
49.h odd 42 1 343.2.g.c 12
147.l odd 14 1 441.2.u.b 12
196.k odd 14 1 784.2.u.b 12

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + 2 T_{2}^{11} + 3 T_{2}^{10} + 4 T_{2}^{9} + 12 T_{2}^{8} + 6 T_{2}^{7} + 7 T_{2}^{6} + \cdots + 1$$ acting on $$S_{2}^{\mathrm{new}}(49, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + 2 T^{11} + \cdots + 1$$
$3$ $$T^{12} + 7 T^{10} + \cdots + 49$$
$5$ $$T^{12} + 7 T^{11} + \cdots + 49$$
$7$ $$T^{12} + 7 T^{11} + \cdots + 117649$$
$11$ $$T^{12} + 8 T^{11} + \cdots + 1849$$
$13$ $$T^{12} - 7 T^{11} + \cdots + 49$$
$17$ $$T^{12} + 35 T^{10} + \cdots + 3087049$$
$19$ $$(T^{6} + 7 T^{5} + \cdots + 2107)^{2}$$
$23$ $$T^{12} + 2 T^{11} + \cdots + 1$$
$29$ $$T^{12} + 11 T^{11} + \cdots + 1681$$
$31$ $$(T^{6} + 7 T^{5} + \cdots - 8183)^{2}$$
$37$ $$T^{12} + \cdots + 295118041$$
$41$ $$T^{12} - 21 T^{11} + \cdots + 10413529$$
$43$ $$T^{12} + \cdots + 200307409$$
$47$ $$T^{12} + \cdots + 3021810841$$
$53$ $$T^{12} - 6 T^{11} + \cdots + 2985984$$
$59$ $$T^{12} - 14 T^{11} + \cdots + 54125449$$
$61$ $$T^{12} + \cdots + 261242569$$
$67$ $$(T^{6} - 24 T^{5} + \cdots - 293)^{2}$$
$71$ $$T^{12} + \cdots + 312925003609$$
$73$ $$T^{12} + \cdots + 460917961$$
$79$ $$(T^{6} + 8 T^{5} + \cdots - 21629)^{2}$$
$83$ $$T^{12} + \cdots + 118178641$$
$89$ $$T^{12} + \cdots + 89755965649$$
$97$ $$(T^{6} + 14 T^{5} + \cdots + 18571)^{2}$$