# Properties

 Label 17.2.d.a Level $17$ Weight $2$ Character orbit 17.d Analytic conductor $0.136$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [17,2,Mod(2,17)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(17, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([7]))

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

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

chi := DirichletCharacter("17.2");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 17.d (of order $$8$$, degree $$4$$, 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: $$0.135745683436$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-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 $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/17\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$\zeta_{8}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i
−0.292893 0.292893i −2.41421 + 1.00000i 1.82843i 0.707107 + 1.70711i 1.00000 + 0.414214i 0.414214 1.00000i −1.12132 + 1.12132i 2.70711 2.70711i 0.292893 0.707107i
8.1 −1.70711 + 1.70711i 0.414214 1.00000i 3.82843i −0.707107 0.292893i 1.00000 + 2.41421i −2.41421 + 1.00000i 3.12132 + 3.12132i 1.29289 + 1.29289i 1.70711 0.707107i
9.1 −0.292893 + 0.292893i −2.41421 1.00000i 1.82843i 0.707107 1.70711i 1.00000 0.414214i 0.414214 + 1.00000i −1.12132 1.12132i 2.70711 + 2.70711i 0.292893 + 0.707107i
15.1 −1.70711 1.70711i 0.414214 + 1.00000i 3.82843i −0.707107 + 0.292893i 1.00000 2.41421i −2.41421 1.00000i 3.12132 3.12132i 1.29289 1.29289i 1.70711 + 0.707107i
 $$n$$: 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
17.d even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 17.2.d.a 4
3.b odd 2 1 153.2.l.c 4
4.b odd 2 1 272.2.v.d 4
5.b even 2 1 425.2.m.a 4
5.c odd 4 1 425.2.n.a 4
5.c odd 4 1 425.2.n.b 4
7.b odd 2 1 833.2.l.a 4
7.c even 3 2 833.2.v.b 8
7.d odd 6 2 833.2.v.a 8
17.b even 2 1 289.2.d.a 4
17.c even 4 1 289.2.d.b 4
17.c even 4 1 289.2.d.c 4
17.d even 8 1 inner 17.2.d.a 4
17.d even 8 1 289.2.d.a 4
17.d even 8 1 289.2.d.b 4
17.d even 8 1 289.2.d.c 4
17.e odd 16 2 289.2.a.f 4
17.e odd 16 2 289.2.b.b 4
17.e odd 16 4 289.2.c.c 8
51.g odd 8 1 153.2.l.c 4
51.i even 16 2 2601.2.a.bb 4
68.g odd 8 1 272.2.v.d 4
68.i even 16 2 4624.2.a.bp 4
85.k odd 8 1 425.2.n.b 4
85.m even 8 1 425.2.m.a 4
85.n odd 8 1 425.2.n.a 4
85.p odd 16 2 7225.2.a.u 4
119.l odd 8 1 833.2.l.a 4
119.q even 24 2 833.2.v.b 8
119.r odd 24 2 833.2.v.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.2.d.a 4 1.a even 1 1 trivial
17.2.d.a 4 17.d even 8 1 inner
153.2.l.c 4 3.b odd 2 1
153.2.l.c 4 51.g odd 8 1
272.2.v.d 4 4.b odd 2 1
272.2.v.d 4 68.g odd 8 1
289.2.a.f 4 17.e odd 16 2
289.2.b.b 4 17.e odd 16 2
289.2.c.c 8 17.e odd 16 4
289.2.d.a 4 17.b even 2 1
289.2.d.a 4 17.d even 8 1
289.2.d.b 4 17.c even 4 1
289.2.d.b 4 17.d even 8 1
289.2.d.c 4 17.c even 4 1
289.2.d.c 4 17.d even 8 1
425.2.m.a 4 5.b even 2 1
425.2.m.a 4 85.m even 8 1
425.2.n.a 4 5.c odd 4 1
425.2.n.a 4 85.n odd 8 1
425.2.n.b 4 5.c odd 4 1
425.2.n.b 4 85.k odd 8 1
833.2.l.a 4 7.b odd 2 1
833.2.l.a 4 119.l odd 8 1
833.2.v.a 8 7.d odd 6 2
833.2.v.a 8 119.r odd 24 2
833.2.v.b 8 7.c even 3 2
833.2.v.b 8 119.q even 24 2
2601.2.a.bb 4 51.i even 16 2
4624.2.a.bp 4 68.i even 16 2
7225.2.a.u 4 85.p odd 16 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(17, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 4 T^{3} + 8 T^{2} + 4 T + 1$$
$3$ $$T^{4} + 4 T^{3} + 4 T^{2} + 8$$
$5$ $$T^{4} + 2 T^{2} + 4 T + 2$$
$7$ $$T^{4} + 4 T^{3} + 4 T^{2} + 8$$
$11$ $$T^{4} + 4 T^{3} + 12 T^{2} + 16 T + 8$$
$13$ $$(T^{2} + 2)^{2}$$
$17$ $$T^{4} + 2T^{2} + 289$$
$19$ $$T^{4} - 8 T^{3} + 32 T^{2} - 32 T + 16$$
$23$ $$T^{4} - 4 T^{3} + 12 T^{2} - 112 T + 392$$
$29$ $$T^{4} + 4 T^{3} + 22 T^{2} + 12 T + 2$$
$31$ $$T^{4} + 12 T^{3} + 108 T^{2} + \cdots + 648$$
$37$ $$T^{4} + 50 T^{2} + 500 T + 1250$$
$41$ $$T^{4} + 4 T^{3} + 54 T^{2} - 140 T + 98$$
$43$ $$T^{4} + 8 T^{3} + 32 T^{2} + 32 T + 16$$
$47$ $$T^{4} + 144T^{2} + 3136$$
$53$ $$(T^{2} + 2 T + 2)^{2}$$
$59$ $$T^{4} + 1296$$
$61$ $$T^{4} + 50 T^{2} + 500 T + 1250$$
$67$ $$(T^{2} - 8 T + 8)^{2}$$
$71$ $$T^{4} - 20 T^{3} + 100 T^{2} + \cdots + 5000$$
$73$ $$T^{4} + 28 T^{3} + 294 T^{2} + \cdots + 4802$$
$79$ $$T^{4} + 4 T^{3} + 12 T^{2} + 112 T + 392$$
$83$ $$T^{4} - 16 T^{3} + 128 T^{2} + \cdots + 16$$
$89$ $$T^{4} + 132T^{2} + 3844$$
$97$ $$T^{4} - 24 T^{3} + 242 T^{2} + \cdots + 4418$$