# Properties

 Label 58.2.d.b Level $58$ Weight $2$ Character orbit 58.d Analytic conductor $0.463$ 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] = [58,2,Mod(7,58)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(58, base_ring=CyclotomicField(14))

chi = DirichletCharacter(H, H._module([6]))

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

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

chi := DirichletCharacter("58.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$58 = 2 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 58.d (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: $$0.463132331723$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{7})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 3 x^{11} + 13 x^{10} - 9 x^{9} - 5 x^{8} + 35 x^{7} + 197 x^{6} - 140 x^{5} - 80 x^{4} + \cdots + 4096$$ x^12 - 3*x^11 + 13*x^10 - 9*x^9 - 5*x^8 + 35*x^7 + 197*x^6 - 140*x^5 - 80*x^4 + 576*x^3 + 3328*x^2 + 3072*x + 4096 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + \beta_{11} q^{3} + \beta_{6} q^{4} + ( - \beta_{11} + \beta_{10} + \cdots + \beta_1) q^{5}+ \cdots + ( - \beta_{10} + \beta_{8} + \cdots - \beta_1) q^{9}+O(q^{10})$$ q + b2 * q^2 + b11 * q^3 + b6 * q^4 + (-b11 + b10 - b9 - b7 + b5 - b4 + b3 + b1) * q^5 - b5 * q^6 + (b9 + b7) * q^7 + b8 * q^8 + (-b10 + b8 + b7 - b1) * q^9 $$q + \beta_{2} q^{2} + \beta_{11} q^{3} + \beta_{6} q^{4} + ( - \beta_{11} + \beta_{10} + \cdots + \beta_1) q^{5}+ \cdots + (\beta_{11} + \beta_{10} + 2 \beta_{9} + \cdots - 4) q^{99}+O(q^{100})$$ q + b2 * q^2 + b11 * q^3 + b6 * q^4 + (-b11 + b10 - b9 - b7 + b5 - b4 + b3 + b1) * q^5 - b5 * q^6 + (b9 + b7) * q^7 + b8 * q^8 + (-b10 + b8 + b7 - b1) * q^9 + (-b8 - b6 - b3 - b2 - b1 - 1) * q^10 + (-b10 - b5 - b1) * q^11 + b9 * q^12 + (-b11 - b8 - b7 + b5 + b4 - b3 - b2 + b1 - 1) * q^13 + (-b11 - b9 - b7 + b5 + b1) * q^14 + (b11 + b9 - 4*b8 + b7 - 4*b6 - b5 + 4*b4 - 4*b3 - 4*b2 - b1 - 4) * q^15 + b3 * q^16 + (-b11 + b10 - b9 + 3*b8 - b7 - b6 + b4 + 3*b3 + b1 - 1) * q^17 + (b11 - b10 + b7 + b3) * q^18 + (b11 - b10 + b9 + 4*b6 - b5) * q^19 + (-b8 + b7 - b6 + b4 - b3 - b2) * q^20 + (-b7 - 4*b4 + 4*b2) * q^21 + (b11 + b9 + b7) * q^22 + (-b11 - b9 + b5 - 4*b4 + 4) * q^23 + (-b11 + b10 - b9 - b7 + b5 + b1) * q^24 + (b11 + b10 - b9 + b8 + b6 - b5 + b2) * q^25 + (b10 - b9 + b8 - b7 + b5 + 1) * q^26 + (-b11 + b10 + b9 + 4*b6 + 4*b2 - b1 + 4) * q^27 + (b11 - b1) * q^28 + (2*b11 + b9 + 4*b8 + b7 - b5 + b3 - b1) * q^29 + (-b11 + b1 + 4) * q^30 + (-b9 + b7 - b5 + 4*b2 + b1) * q^31 - b4 * q^32 + (-b11 + b10 - b9 + 4*b6 + b5 + 4*b3 + b1 + 4) * q^33 + (b9 + b6 - 4*b4 + 4*b3 - b1 + 1) * q^34 + (-b7 + 2*b5 + 4*b3 + 4*b2 + b1) * q^35 + (b11 - b10 - b5 - b4) * q^36 + (-b7 - 4*b6 + 3*b4 - 4*b3 - 3*b2) * q^37 + (b10 + 4*b8 - b7 + b1) * q^38 + (b11 - b10 + b9 - 4*b8 - b5 - 4*b3 - 4*b2 - 4) * q^39 + (-b10 + b2 + 1) * q^40 + (-b10 - b9 - 4*b8 + b7 - 3*b6 + 3*b4 - 4*b3 - 4) * q^41 + (b10 - 4*b8 + 4*b4 - 4*b3 - 4*b2 - 4) * q^42 + (b11 + b9 + b7 - b1) * q^43 + (-b11 - b9 - b7 + b1) * q^44 + (-2*b10 + 4*b8 - 4*b4 + 4*b3 + b2 + 1) * q^45 + (b11 - b10 - 4*b8 + b7 - 4*b6 + 4*b4 - 4*b3 - b1 - 4) * q^46 + (-b11 + b10 - b7 + 2*b5 - 4*b2 + 2*b1 - 4) * q^47 - b1 * q^48 + (-b10 + b9 - 3*b8 + b7 - 4*b6 - b5 + 4*b4 - 4*b3 - 4*b2 - b1) * q^49 + (-b10 + 2*b9 + b8 + b7 + b6 - 2*b5 + b3 - b1) * q^50 + (-2*b11 + 2*b10 - 3*b9 - 4*b8 - 3*b7 - 4*b6 + 2*b5 + 4*b4 - 4*b2 - 4) * q^51 + (b7 - b5 + b3 + b2 - b1) * q^52 + (2*b9 + b7 + b6 - b5 - 2*b1 + 1) * q^53 + (-2*b11 + b10 - b9 + 4*b8 + 4*b6 + 2*b5 + 4*b2 + b1) * q^54 + (-b11 - b10 + 4*b8 - b5 - 4*b4 + 4) * q^55 + (b7 - b5) * q^56 + (b11 - 2*b10 + 3*b9 + 4*b8 + 2*b7 + 4*b6 - 4*b4 + 4*b3 - b1) * q^57 + (-b11 - b5 - b4 + 4*b3 + b1) * q^58 + (b10 + 4*b8 - b7 + 4*b3 + 4) * q^59 + (-b7 + b5 + 4*b2) * q^60 + (-3*b11 + b10 - b9 - b8 - b7 - b6 + b5 - 3*b4 - b2 - 1) * q^61 + (b11 - 2*b10 + 2*b9 + 4*b6 - b5 - b1) * q^62 + (b11 - b10 + 2*b7 - 2*b5 - 4*b2) * q^63 + (-b8 - b6 + b4 - b3 - b2 - 1) * q^64 + (b10 - b9 - b7 + 3*b6 + b5 - 3*b4 + 3*b2) * q^65 + (-b10 + 4*b8 - 4*b4 + 4*b2 - b1) * q^66 + (b10 - b9 + 4*b8 - b7 + b5 + b1) * q^67 + (-b11 + b10 - b9 - 3*b8 - 4*b6 + b5 - 4*b3 - 3*b2 + b1 - 4) * q^68 + (2*b11 + 3*b10 - 4*b8 + 2*b7 + b5 + 4*b4 - 4*b3 + b1) * q^69 + (b10 - 2*b9 - b7 + 4*b6 - 4*b4) * q^70 + (3*b11 - 2*b10 - 4*b8 + 3*b7 - 3*b5 + 4*b4 - 4*b3 - 3*b1) * q^71 + (b11 + b9 - b8 - b6 - b5 + b4 - b3 - b2 - 1) * q^72 + (-2*b11 - 2*b9 - 8*b8 - 3*b7 - 8*b6 + 4*b5 + 7*b4 - 4*b3 - 4*b2 + 3*b1 - 7) * q^73 + (b10 - b8 + b4 + 3*b3 + 3*b2 + 3) * q^74 + (-b10 + b9 + 4*b8 + b7 - 4*b6 + 4*b4 + 4*b3) * q^75 + (-b11 + b10 - b7 + 4*b3) * q^76 + (b10 - b9 - 4*b8 + 4*b6 - 4*b2 + b1) * q^77 + (b10 - b7 - 4*b6 + 4*b4 - 4*b3 - 4*b2 + b1) * q^78 + (2*b10 - 2*b9 + 4*b8 + 4*b6 + 2*b5 - 8*b4 + 4*b3 + 8*b2 + 2*b1) * q^79 + (b11 + b6 + b2) * q^80 + (2*b11 + 2*b9 - b8 - 2*b7 - b6 - 3*b5 - 7*b4 + 3*b3 + 3*b2 + 2*b1 + 7) * q^81 + (2*b11 - 2*b10 + b9 + b7 + 3*b6 - b5 + b4 - b3 - b2 - b1 + 3) * q^82 + (b11 - 3*b10 + 3*b9 - b5 - 4*b3 - b1 - 4) * q^83 + (-b11 + 4*b8 + 4) * q^84 + (3*b11 - 3*b10 + 7*b6 + 7*b2 + 7) * q^85 + (-b11 - b9 + b1) * q^86 + (-4*b11 + 2*b10 - 4*b9 + 8*b8 - 3*b7 + 4*b5 - 4*b4 + 4*b3 + 4*b2 + 2*b1 + 4) * q^87 + (b11 + b9 - b1) * q^88 + (3*b9 + 4*b6 - b4 + b3 - 3*b1 + 4) * q^89 + (2*b11 - 4*b8 - 3*b6 - 3*b2 - 4) * q^90 + (4*b8 + 8*b3 + 4*b2 - b1 + 8) * q^91 + (b11 - b10 + b7 + 4*b6 - b5 + 4) * q^92 + (b7 - 2*b5 + 4*b4 + 4*b3 + 4*b2 - b1 - 4) * q^93 + (-b11 + b10 - 2*b9 - 2*b7 - 4*b6 + b5 - 4*b2) * q^94 + (-8*b8 + 3*b7 - 8*b6 + 4*b4 - 8*b3 - 4*b2) * q^95 + b7 * q^96 + (-b11 + 2*b10 - 2*b9 + 4*b8 + 8*b6 + b5 + b3 + 4*b2 + 2*b1 + 1) * q^97 + (b5 + b3 + 4*b2 + b1 + 4) * q^98 + (b11 + b10 + 2*b9 - 4*b8 - b7 - 4*b6 + 4*b4 - 4*b3 - b1 - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 2 q^{2} - 3 q^{3} - 2 q^{4} + 4 q^{6} + q^{7} - 2 q^{8} - 11 q^{9}+O(q^{10})$$ 12 * q - 2 * q^2 - 3 * q^3 - 2 * q^4 + 4 * q^6 + q^7 - 2 * q^8 - 11 * q^9 $$12 q - 2 q^{2} - 3 q^{3} - 2 q^{4} + 4 q^{6} + q^{7} - 2 q^{8} - 11 q^{9} - 7 q^{10} - 2 q^{11} + 4 q^{12} + q^{13} + q^{14} - 9 q^{15} - 2 q^{16} - 12 q^{17} - 11 q^{18} - 6 q^{19} + 7 q^{20} - 13 q^{21} - 2 q^{22} + 35 q^{23} + 4 q^{24} - 6 q^{25} + 8 q^{26} + 39 q^{27} - 6 q^{28} - 14 q^{29} + 54 q^{30} - 8 q^{31} - 2 q^{32} + 33 q^{33} - 5 q^{34} - 18 q^{35} - 4 q^{36} + 31 q^{37} + q^{38} - 22 q^{39} + 7 q^{40} - 30 q^{41} - 13 q^{42} - 5 q^{43} + 5 q^{44} - 20 q^{45} - 28 q^{46} - 33 q^{47} - 3 q^{48} + 37 q^{49} + q^{50} - 15 q^{51} - 6 q^{52} + 13 q^{53} - 24 q^{54} + 36 q^{55} + q^{56} - 38 q^{57} + 38 q^{59} - 9 q^{60} - 5 q^{61} - 8 q^{62} + 4 q^{63} - 2 q^{64} - 20 q^{65} - 30 q^{66} - 7 q^{67} - 19 q^{68} + 20 q^{69} - 18 q^{70} + 3 q^{71} + 3 q^{72} - 22 q^{73} + 31 q^{74} - 2 q^{75} + q^{76} + 10 q^{77} + 41 q^{78} - 60 q^{79} - 7 q^{80} + 88 q^{81} + 26 q^{82} - 39 q^{83} + 43 q^{84} + 38 q^{85} + 2 q^{86} + 9 q^{87} - 2 q^{88} + 39 q^{89} - 34 q^{90} + 61 q^{91} + 35 q^{92} - 54 q^{93} + 16 q^{94} + 55 q^{95} - 3 q^{96} - 19 q^{97} + 37 q^{98} - 8 q^{99}+O(q^{100})$$ 12 * q - 2 * q^2 - 3 * q^3 - 2 * q^4 + 4 * q^6 + q^7 - 2 * q^8 - 11 * q^9 - 7 * q^10 - 2 * q^11 + 4 * q^12 + q^13 + q^14 - 9 * q^15 - 2 * q^16 - 12 * q^17 - 11 * q^18 - 6 * q^19 + 7 * q^20 - 13 * q^21 - 2 * q^22 + 35 * q^23 + 4 * q^24 - 6 * q^25 + 8 * q^26 + 39 * q^27 - 6 * q^28 - 14 * q^29 + 54 * q^30 - 8 * q^31 - 2 * q^32 + 33 * q^33 - 5 * q^34 - 18 * q^35 - 4 * q^36 + 31 * q^37 + q^38 - 22 * q^39 + 7 * q^40 - 30 * q^41 - 13 * q^42 - 5 * q^43 + 5 * q^44 - 20 * q^45 - 28 * q^46 - 33 * q^47 - 3 * q^48 + 37 * q^49 + q^50 - 15 * q^51 - 6 * q^52 + 13 * q^53 - 24 * q^54 + 36 * q^55 + q^56 - 38 * q^57 + 38 * q^59 - 9 * q^60 - 5 * q^61 - 8 * q^62 + 4 * q^63 - 2 * q^64 - 20 * q^65 - 30 * q^66 - 7 * q^67 - 19 * q^68 + 20 * q^69 - 18 * q^70 + 3 * q^71 + 3 * q^72 - 22 * q^73 + 31 * q^74 - 2 * q^75 + q^76 + 10 * q^77 + 41 * q^78 - 60 * q^79 - 7 * q^80 + 88 * q^81 + 26 * q^82 - 39 * q^83 + 43 * q^84 + 38 * q^85 + 2 * q^86 + 9 * q^87 - 2 * q^88 + 39 * q^89 - 34 * q^90 + 61 * q^91 + 35 * q^92 - 54 * q^93 + 16 * q^94 + 55 * q^95 - 3 * q^96 - 19 * q^97 + 37 * q^98 - 8 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3 x^{11} + 13 x^{10} - 9 x^{9} - 5 x^{8} + 35 x^{7} + 197 x^{6} - 140 x^{5} - 80 x^{4} + \cdots + 4096$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{11} + 97 \nu^{10} - 291 \nu^{9} + 1067 \nu^{8} - 233 \nu^{7} - 1945 \nu^{6} + 1185 \nu^{5} + \cdots + 293952 ) / 443456$$ (v^11 + 97*v^10 - 291*v^9 + 1067*v^8 - 233*v^7 - 1945*v^6 + 1185*v^5 + 50792*v^4 - 53096*v^3 - 16252*v^2 + 163104*v + 293952) / 443456 $$\beta_{3}$$ $$=$$ $$( 53 \nu^{11} - 435 \nu^{10} + 1469 \nu^{9} - 3473 \nu^{8} + 443 \nu^{7} + 7123 \nu^{6} + \cdots - 838912 ) / 1773824$$ (53*v^11 - 435*v^10 + 1469*v^9 - 3473*v^8 + 443*v^7 + 7123*v^6 + 4093*v^5 - 70768*v^4 + 6384*v^3 + 592832*v^2 - 244288*v - 838912) / 1773824 $$\beta_{4}$$ $$=$$ $$( 30 \nu^{11} + 40 \nu^{10} - 79 \nu^{9} + 1137 \nu^{8} - 307 \nu^{7} - 8945 \nu^{6} + 25423 \nu^{5} + \cdots + 324672 ) / 443456$$ (30*v^11 + 40*v^10 - 79*v^9 + 1137*v^8 - 307*v^7 - 8945*v^6 + 25423*v^5 + 5981*v^4 - 66491*v^3 + 39372*v^2 + 274224*v + 324672) / 443456 $$\beta_{5}$$ $$=$$ $$( - 69 \nu^{11} + 195 \nu^{10} - 749 \nu^{9} + 177 \nu^{8} + 1317 \nu^{7} - 1587 \nu^{6} + \cdots - 54272 ) / 443456$$ (-69*v^11 + 195*v^10 - 749*v^9 + 177*v^8 + 1317*v^7 - 1587*v^6 - 15837*v^5 + 2656*v^4 + 140576*v^3 - 105168*v^2 - 250432*v - 54272) / 443456 $$\beta_{6}$$ $$=$$ $$( 325 \nu^{11} - 250 \nu^{10} + 750 \nu^{9} + 12256 \nu^{8} - 28698 \nu^{7} + 30394 \nu^{6} + \cdots + 1298688 ) / 1773824$$ (325*v^11 - 250*v^10 + 750*v^9 + 12256*v^8 - 28698*v^7 + 30394*v^6 + 118748*v^5 - 81487*v^4 - 241200*v^3 + 490080*v^2 + 1540352*v + 1298688) / 1773824 $$\beta_{7}$$ $$=$$ $$( - 25 \nu^{11} + 76 \nu^{10} - 269 \nu^{9} + 57 \nu^{8} + 495 \nu^{7} - 247 \nu^{6} - 12733 \nu^{5} + \cdots + 1024 ) / 110864$$ (-25*v^11 + 76*v^10 - 269*v^9 + 57*v^8 + 495*v^7 - 247*v^6 - 12733*v^5 + 13254*v^4 + 4207*v^3 - 39944*v^2 - 72720*v + 1024) / 110864 $$\beta_{8}$$ $$=$$ $$( 1961 \nu^{11} - 9535 \nu^{10} + 37297 \nu^{9} - 72085 \nu^{8} + 46567 \nu^{7} + 31327 \nu^{6} + \cdots - 3739648 ) / 7095296$$ (1961*v^11 - 9535*v^10 + 37297*v^9 - 72085*v^8 + 46567*v^7 + 31327*v^6 + 265585*v^5 - 880016*v^4 + 419888*v^3 + 289408*v^2 + 4524800*v - 3739648) / 7095296 $$\beta_{9}$$ $$=$$ $$( 130 \nu^{11} - 469 \nu^{10} + 1407 \nu^{9} - 157 \nu^{8} - 9995 \nu^{7} + 19513 \nu^{6} + \cdots - 122880 ) / 443456$$ (130*v^11 - 469*v^10 + 1407*v^9 - 157*v^8 - 9995*v^7 + 19513*v^6 + 10181*v^5 - 64091*v^4 + 22092*v^3 + 174384*v^2 + 232512*v - 122880) / 443456 $$\beta_{10}$$ $$=$$ $$( 725 \nu^{11} - 3475 \nu^{10} + 15181 \nu^{9} - 27073 \nu^{8} + 19019 \nu^{7} + 54723 \nu^{6} + \cdots - 1331200 ) / 1773824$$ (725*v^11 - 3475*v^10 + 15181*v^9 - 27073*v^8 + 19019*v^7 + 54723*v^6 - 35987*v^5 - 215200*v^4 + 302880*v^3 + 458752*v^2 + 300288*v - 1331200) / 1773824 $$\beta_{11}$$ $$=$$ $$( 913 \nu^{11} - 2951 \nu^{10} + 13609 \nu^{9} - 14093 \nu^{8} + 9327 \nu^{7} + 30183 \nu^{6} + \cdots + 2008064 ) / 1773824$$ (913*v^11 - 2951*v^10 + 13609*v^9 - 14093*v^8 + 9327*v^7 + 30183*v^6 + 151369*v^5 - 144192*v^4 + 210032*v^3 + 500352*v^2 + 2440960*v + 2008064) / 1773824
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{11} - \beta_{10} + \beta_{7} + 4\beta_{3}$$ b11 - b10 + b7 + 4*b3 $$\nu^{3}$$ $$=$$ $$-\beta_{11} - \beta_{9} + 4\beta_{8} + \beta_{7} + 4\beta_{6} + 6\beta_{5} + 4\beta_{3} + 4\beta_{2} - \beta_1$$ -b11 - b9 + 4*b8 + b7 + 4*b6 + 6*b5 + 4*b3 + 4*b2 - b1 $$\nu^{4}$$ $$=$$ $$- 14 \beta_{11} + 14 \beta_{10} - \beta_{9} - 12 \beta_{7} + 4 \beta_{6} + 12 \beta_{5} + 4 \beta_{4} + \cdots + 4$$ -14*b11 + 14*b10 - b9 - 12*b7 + 4*b6 + 12*b5 + 4*b4 - 4*b3 + 24*b2 + b1 + 4 $$\nu^{5}$$ $$=$$ $$18 \beta_{10} + 17 \beta_{9} - 56 \beta_{8} - 49 \beta_{7} - 52 \beta_{6} - 17 \beta_{5} + \cdots + 18 \beta_1$$ 18*b10 + 17*b9 - 56*b8 - 49*b7 - 52*b6 - 17*b5 + 8*b4 - 52*b3 - 8*b2 + 18*b1 $$\nu^{6}$$ $$=$$ $$175 \beta_{11} - 153 \beta_{10} + 43 \beta_{9} - 72 \beta_{8} + 43 \beta_{7} - 140 \beta_{6} - 153 \beta_{5} + \cdots - 72$$ 175*b11 - 153*b10 + 43*b9 - 72*b8 + 43*b7 - 140*b6 - 153*b5 - 124*b4 - 140*b2 - 72 $$\nu^{7}$$ $$=$$ $$247 \beta_{11} - 468 \beta_{10} - 234 \beta_{9} + 612 \beta_{8} + 468 \beta_{7} + 440 \beta_{6} + \cdots - 88$$ 247*b11 - 468*b10 - 234*b9 + 612*b8 + 468*b7 + 440*b6 - 440*b4 + 612*b3 - 247*b1 - 88 $$\nu^{8}$$ $$=$$ $$- 1782 \beta_{11} + 1142 \beta_{10} - 1142 \beta_{9} + 1872 \beta_{8} + 2808 \beta_{6} + 1782 \beta_{5} + \cdots + 884$$ -1782*b11 + 1142*b10 - 1142*b9 + 1872*b8 + 2808*b6 + 1782*b5 + 884*b3 + 1872*b2 - 335*b1 + 884 $$\nu^{9}$$ $$=$$ $$- 5771 \beta_{11} + 7849 \beta_{10} - 4568 \beta_{8} - 5771 \beta_{7} + 2666 \beta_{5} + 4568 \beta_{4} + \cdots + 2560$$ -5771*b11 + 7849*b10 - 4568*b8 - 5771*b7 + 2666*b5 + 4568*b4 - 5908*b3 + 2560*b2 + 2666*b1 + 2560 $$\nu^{10}$$ $$=$$ $$12417 \beta_{11} + 12417 \beta_{9} - 31396 \beta_{8} - 8331 \beta_{7} - 31396 \beta_{6} - 16862 \beta_{5} + \cdots - 8312$$ 12417*b11 + 12417*b9 - 31396*b8 - 8331*b7 - 31396*b6 - 16862*b5 + 8312*b4 - 20732*b3 - 20732*b2 + 8331*b1 - 8312 $$\nu^{11}$$ $$=$$ $$89754 \beta_{11} - 89754 \beta_{10} + 20729 \beta_{9} + 58342 \beta_{7} - 49668 \beta_{6} - 58342 \beta_{5} + \cdots - 49668$$ 89754*b11 - 89754*b10 + 20729*b9 + 58342*b7 - 49668*b6 - 58342*b5 - 33324*b4 + 33324*b3 - 67448*b2 - 20729*b1 - 49668

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/58\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$\chi(n)$$ $$\beta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 2.06920 − 0.996473i −1.56920 + 0.755686i 0.760453 + 3.33176i −0.260453 − 1.14112i 2.06920 + 0.996473i −1.56920 − 0.755686i 1.52179 + 1.90827i −1.02179 − 1.28129i 1.52179 − 1.90827i −1.02179 + 1.28129i 0.760453 − 3.33176i −0.260453 + 1.14112i
−0.222521 0.974928i −2.06920 0.996473i −0.900969 + 0.433884i −0.788529 3.45477i −0.511050 + 2.23905i 3.72857 + 1.79558i 0.623490 + 0.781831i 1.41815 + 1.77830i −3.19269 + 1.53752i
7.2 −0.222521 0.974928i 1.56920 + 0.755686i −0.900969 + 0.433884i 0.110081 + 0.482295i 0.387560 1.69801i −2.82760 1.36170i 0.623490 + 0.781831i 0.0208506 + 0.0261458i 0.445708 0.214642i
23.1 0.623490 0.781831i −0.760453 + 3.33176i −0.222521 0.974928i 1.00725 1.26305i 2.13074 + 2.67187i 0.338433 1.48277i −0.900969 0.433884i −7.81944 3.76565i −0.359484 1.57500i
23.2 0.623490 0.781831i 0.260453 1.14112i −0.222521 0.974928i −1.85326 + 2.32392i −0.729773 0.915107i −0.115912 + 0.507846i −0.900969 0.433884i 1.46859 + 0.707235i 0.661422 + 2.89788i
25.1 −0.222521 + 0.974928i −2.06920 + 0.996473i −0.900969 0.433884i −0.788529 + 3.45477i −0.511050 2.23905i 3.72857 1.79558i 0.623490 0.781831i 1.41815 1.77830i −3.19269 1.53752i
25.2 −0.222521 + 0.974928i 1.56920 0.755686i −0.900969 0.433884i 0.110081 0.482295i 0.387560 + 1.69801i −2.82760 + 1.36170i 0.623490 0.781831i 0.0208506 0.0261458i 0.445708 + 0.214642i
45.1 −0.900969 0.433884i −1.52179 + 1.90827i 0.623490 + 0.781831i 2.60002 + 1.25211i 2.19905 1.05901i −1.89765 + 2.37957i −0.222521 0.974928i −0.658071 2.88320i −1.79927 2.25622i
45.2 −0.900969 0.433884i 1.02179 1.28129i 0.623490 + 0.781831i −1.07557 0.517965i −1.47653 + 0.711061i 1.27416 1.59774i −0.222521 0.974928i 0.0699247 + 0.306360i 0.744314 + 0.933340i
49.1 −0.900969 + 0.433884i −1.52179 1.90827i 0.623490 0.781831i 2.60002 1.25211i 2.19905 + 1.05901i −1.89765 2.37957i −0.222521 + 0.974928i −0.658071 + 2.88320i −1.79927 + 2.25622i
49.2 −0.900969 + 0.433884i 1.02179 + 1.28129i 0.623490 0.781831i −1.07557 + 0.517965i −1.47653 0.711061i 1.27416 + 1.59774i −0.222521 + 0.974928i 0.0699247 0.306360i 0.744314 0.933340i
53.1 0.623490 + 0.781831i −0.760453 3.33176i −0.222521 + 0.974928i 1.00725 + 1.26305i 2.13074 2.67187i 0.338433 + 1.48277i −0.900969 + 0.433884i −7.81944 + 3.76565i −0.359484 + 1.57500i
53.2 0.623490 + 0.781831i 0.260453 + 1.14112i −0.222521 + 0.974928i −1.85326 2.32392i −0.729773 + 0.915107i −0.115912 0.507846i −0.900969 + 0.433884i 1.46859 0.707235i 0.661422 2.89788i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 7.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.d even 7 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 58.2.d.b 12
3.b odd 2 1 522.2.k.h 12
4.b odd 2 1 464.2.u.h 12
29.d even 7 1 inner 58.2.d.b 12
29.d even 7 1 1682.2.a.t 6
29.e even 14 1 1682.2.a.q 6
29.f odd 28 2 1682.2.b.i 12
87.j odd 14 1 522.2.k.h 12
116.j odd 14 1 464.2.u.h 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.2.d.b 12 1.a even 1 1 trivial
58.2.d.b 12 29.d even 7 1 inner
464.2.u.h 12 4.b odd 2 1
464.2.u.h 12 116.j odd 14 1
522.2.k.h 12 3.b odd 2 1
522.2.k.h 12 87.j odd 14 1
1682.2.a.q 6 29.e even 14 1
1682.2.a.t 6 29.d even 7 1
1682.2.b.i 12 29.f odd 28 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} + 3 T_{3}^{11} + 13 T_{3}^{10} + 9 T_{3}^{9} - 5 T_{3}^{8} - 35 T_{3}^{7} + 197 T_{3}^{6} + \cdots + 4096$$ acting on $$S_{2}^{\mathrm{new}}(58, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{6} + T^{5} + T^{4} + \cdots + 1)^{2}$$
$3$ $$T^{12} + 3 T^{11} + \cdots + 4096$$
$5$ $$T^{12} + 8 T^{10} + \cdots + 841$$
$7$ $$T^{12} - T^{11} + \cdots + 4096$$
$11$ $$T^{12} + 2 T^{11} + \cdots + 4096$$
$13$ $$T^{12} - T^{11} + \cdots + 49$$
$17$ $$(T^{6} + 6 T^{5} + \cdots - 1259)^{2}$$
$19$ $$T^{12} + 6 T^{11} + \cdots + 12845056$$
$23$ $$T^{12} - 35 T^{11} + \cdots + 7573504$$
$29$ $$T^{12} + \cdots + 594823321$$
$31$ $$T^{12} + 8 T^{11} + \cdots + 3444736$$
$37$ $$T^{12} - 31 T^{11} + \cdots + 68442529$$
$41$ $$(T^{6} + 15 T^{5} + \cdots - 9653)^{2}$$
$43$ $$T^{12} + 5 T^{11} + \cdots + 4096$$
$47$ $$T^{12} + 33 T^{11} + \cdots + 7573504$$
$53$ $$T^{12} - 13 T^{11} + \cdots + 35724529$$
$59$ $$(T^{6} - 19 T^{5} + \cdots - 3584)^{2}$$
$61$ $$T^{12} + 5 T^{11} + \cdots + 97160449$$
$67$ $$T^{12} + 7 T^{11} + \cdots + 4096$$
$71$ $$T^{12} + \cdots + 3720024064$$
$73$ $$T^{12} + \cdots + 490724067289$$
$79$ $$T^{12} + \cdots + 113844158464$$
$83$ $$T^{12} + 39 T^{11} + \cdots + 200704$$
$89$ $$T^{12} + \cdots + 126405049$$
$97$ $$T^{12} + \cdots + 36316162624$$