LMFDB - Newform orbit 322.2.e.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [322,2,Mod(93,322)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(322, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 0]))

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

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

chi := DirichletCharacter("322.93");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 322 = 2 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	322.e (of order $3$, degree $2$, 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:	$2.57118294509$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.1767277521.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + x^{6} - 10x^{5} + 38x^{4} - 40x^{3} + 64x^{2} - 38x + 7 $ x^8 - 2x^7 + x^6 - 10x^5 + 38x^4 - 40x^3 + 64x^2 - 38x + 7
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
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 basis $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{5} q^{2} + (\beta_{6} - \beta_{5} + \beta_{2} + 1) q^{3} + (\beta_{5} - 1) q^{4} + ( - \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3}) q^{5} + ( - \beta_{2} - 1) q^{6} + ( - \beta_{6} + \beta_{3}) q^{7} + q^{8} + (\beta_{7} + 2 \beta_{6} - \beta_{5}) q^{9}+O(q^{10}) $ q - b5 * q^2 + (b6 - b5 + b2 + 1) * q^3 + (b5 - 1) * q^4 + (-b7 + b5 - b4 + b3) * q^5 + (-b2 - 1) * q^6 + (-b6 + b3) * q^7 + q^8 + (b7 + 2b6 - b5) q^9	\( q - \beta_{5} q^{2} + (\beta_{6} - \beta_{5} + \beta_{2} + 1) q^{3} + (\beta_{5} - 1) q^{4} + ( - \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3}) q^{5} + ( - \beta_{2} - 1) q^{6} + ( - \beta_{6} + \beta_{3}) q^{7} + q^{8} + (\beta_{7} + 2 \beta_{6} - \beta_{5}) q^{9} + (\beta_{7} - \beta_{5} + \beta_{4} + \beta_1 + 1) q^{10} + (\beta_{7} + \beta_{6} + \beta_{2} + \beta_1) q^{11} + ( - \beta_{6} + \beta_{5}) q^{12} + (\beta_{3} + \beta_{2} - 2) q^{13} + (\beta_{6} + \beta_{4} - \beta_{3} + \beta_{2}) q^{14} + (3 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 1) q^{15} - \beta_{5} q^{16} + ( - 3 \beta_{7} - \beta_{4} - 3 \beta_1) q^{17} + ( - \beta_{7} - 2 \beta_{6} + \beta_{5} - 2 \beta_{2} - \beta_1 - 1) q^{18} + (\beta_{7} + 2 \beta_{6} + 3 \beta_{5} + \beta_{4} - \beta_{3}) q^{19} + ( - \beta_{3} - \beta_1 - 1) q^{20} + (\beta_{7} + \beta_{5} + 2 \beta_{4} + \beta_{2} + 2 \beta_1 + 2) q^{21} + ( - \beta_{2} - \beta_1) q^{22} - \beta_{5} q^{23} + (\beta_{6} - \beta_{5} + \beta_{2} + 1) q^{24} + ( - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{2} - 2 \beta_1 - 2) q^{25} + (\beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3}) q^{26} + ( - \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 3) q^{27} + ( - \beta_{4} - \beta_{2}) q^{28} + ( - \beta_{3} - 2 \beta_{2}) q^{29} + (2 \beta_{7} + 3 \beta_{6} + \beta_{5} + 3 \beta_{4} - 3 \beta_{3}) q^{30} + ( - \beta_{7} - \beta_{6} - 4 \beta_{5} - 2 \beta_{4} - \beta_{2} - \beta_1 + 4) q^{31} + (\beta_{5} - 1) q^{32} + (2 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{3}) q^{33} + (\beta_{3} + 3 \beta_1) q^{34} + ( - 4 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_1 + 2) q^{35} + (2 \beta_{2} + \beta_1 + 1) q^{36} + ( - \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3}) q^{37} + ( - \beta_{7} - 2 \beta_{6} - 3 \beta_{5} - \beta_{4} - 2 \beta_{2} - \beta_1 + 3) q^{38} + (2 \beta_{7} - \beta_{6} + 2 \beta_{4} - \beta_{2} + 2 \beta_1) q^{39} + ( - \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3}) q^{40} + ( - \beta_{3} + \beta_{2} - 4) q^{41} + (\beta_{7} + \beta_{6} - 3 \beta_{5} - 2 \beta_{3} - \beta_1 + 1) q^{42} + (2 \beta_{3} + \beta_{2} + \beta_1 - 5) q^{43} + ( - \beta_{7} - \beta_{6}) q^{44} + (5 \beta_{7} + 6 \beta_{6} + 5 \beta_{4} + 6 \beta_{2} + 5 \beta_1) q^{45} + (\beta_{5} - 1) q^{46} + ( - 2 \beta_{7} + \beta_{6} + 5 \beta_{5} - \beta_{4} + \beta_{3}) q^{47} + ( - \beta_{2} - 1) q^{48} + ( - 3 \beta_{7} + \beta_{5} - 2 \beta_{4} + \beta_{3} - 3 \beta_1 + 2) q^{49} + (\beta_{2} + 2 \beta_1 + 2) q^{50} + ( - 4 \beta_{7} - 6 \beta_{6} - 4 \beta_{5} - 5 \beta_{4} + 5 \beta_{3}) q^{51} + ( - \beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_{2} + 2) q^{52} + (2 \beta_{7} - 3 \beta_{6} - \beta_{5} + 2 \beta_{4} - 3 \beta_{2} + 2 \beta_1 + 1) q^{53} + ( - 3 \beta_{7} - 2 \beta_{6} + 3 \beta_{5} - \beta_{4} + \beta_{3}) q^{54} + (2 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 1) q^{55} + ( - \beta_{6} + \beta_{3}) q^{56} + ( - 3 \beta_{3} - \beta_{2} - 4 \beta_1 - 1) q^{57} + ( - 2 \beta_{6} - \beta_{4} + \beta_{3}) q^{58} + (\beta_{7} - \beta_{6} - 4 \beta_{5} - 4 \beta_{4} - \beta_{2} + \beta_1 + 4) q^{59} + ( - 2 \beta_{7} - 3 \beta_{6} - \beta_{5} - 3 \beta_{4} - 3 \beta_{2} - 2 \beta_1 + 1) q^{60} + (3 \beta_{7} + 4 \beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{3}) q^{61} + (2 \beta_{3} + \beta_{2} + \beta_1 - 4) q^{62} + (5 \beta_{7} + 4 \beta_{6} - \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 5) q^{63} + q^{64} + (2 \beta_{7} - 4 \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3}) q^{65} + ( - 2 \beta_{7} - 3 \beta_{6} + 2 \beta_{5} - \beta_{4} - 3 \beta_{2} - 2 \beta_1 - 2) q^{66} + (\beta_{6} - 4 \beta_{5} - 2 \beta_{4} + \beta_{2} + 4) q^{67} + (3 \beta_{7} + \beta_{4} - \beta_{3}) q^{68} + ( - \beta_{2} - 1) q^{69} + ( - \beta_{7} + \beta_{6} - 3 \beta_{5} - \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + 1) q^{70} + ( - 5 \beta_{3} - \beta_{2}) q^{71} + (\beta_{7} + 2 \beta_{6} - \beta_{5}) q^{72} + (\beta_{7} + 3 \beta_{6} + 4 \beta_{4} + 3 \beta_{2} + \beta_1) q^{73} + (\beta_{7} - \beta_{6} - \beta_{4} - \beta_{2} + \beta_1) q^{74} + ( - 3 \beta_{7} - 7 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} + 2 \beta_{3}) q^{75} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 3) q^{76} + (2 \beta_{7} + \beta_{6} + 3 \beta_{5} + \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 1) q^{77} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{78} + ( - \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{3}) q^{79} + (\beta_{7} - \beta_{5} + \beta_{4} + \beta_1 + 1) q^{80} + ( - 3 \beta_{7} - 5 \beta_{6} + 2 \beta_{5} - 5 \beta_{4} - 5 \beta_{2} - 3 \beta_1 - 2) q^{81} + (\beta_{6} + 4 \beta_{5} - \beta_{4} + \beta_{3}) q^{82} + ( - 5 \beta_{3} - 4 \beta_{2}) q^{83} + ( - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 - 3) q^{84} + (\beta_{3} - \beta_{2} - 5 \beta_1 - 12) q^{85} + (\beta_{7} + \beta_{6} + 5 \beta_{5} + 2 \beta_{4} - 2 \beta_{3}) q^{86} + ( - 3 \beta_{7} - 2 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} - 2 \beta_{2} - 3 \beta_1 - 5) q^{87} + (\beta_{7} + \beta_{6} + \beta_{2} + \beta_1) q^{88} + ( - \beta_{7} - 4 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3}) q^{89} + ( - 5 \beta_{3} - 6 \beta_{2} - 5 \beta_1) q^{90} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{3} - \beta_1 + 4) q^{91} + q^{92} + ( - 4 \beta_{7} + \beta_{6} - 4 \beta_{5} - 5 \beta_{4} + 5 \beta_{3}) q^{93} + (2 \beta_{7} - \beta_{6} - 5 \beta_{5} + \beta_{4} - \beta_{2} + 2 \beta_1 + 5) q^{94} + (7 \beta_{6} + \beta_{5} + 7 \beta_{2} - 1) q^{95} + ( - \beta_{6} + \beta_{5}) q^{96} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 9) q^{97} + ( - 3 \beta_{5} + \beta_{4} + \beta_{3} + 3 \beta_1 + 1) q^{98} + ( - 4 \beta_{3} - 6 \beta_{2} - 3 \beta_1 - 8) q^{99}+O(q^{100}) \) q - b5 * q^2 + (b6 - b5 + b2 + 1) * q^3 + (b5 - 1) * q^4 + (-b7 + b5 - b4 + b3) * q^5 + (-b2 - 1) * q^6 + (-b6 + b3) * q^7 + q^8 + (b7 + 2b6 - b5) q^9 + (b7 - b5 + b4 + b1 + 1) * q^10 + (b7 + b6 + b2 + b1) * q^11 + (-b6 + b5) * q^12 + (b3 + b2 - 2) * q^13 + (b6 + b4 - b3 + b2) * q^14 + (3b3 + 3b2 + 2b1 - 1) q^15 - b5 * q^16 + (-3b7 - b4 - 3b1) * q^17 + (-b7 - 2b6 + b5 - 2b2 - b1 - 1) * q^18 + (b7 + 2b6 + 3b5 + b4 - b3) * q^19 + (-b3 - b1 - 1) * q^20 + (b7 + b5 + 2b4 + b2 + 2b1 + 2) * q^21 + (-b2 - b1) * q^22 - b5 * q^23 + (b6 - b5 + b2 + 1) * q^24 + (-2b7 - b6 + 2b5 - b2 - 2b1 - 2) q^25 + (b6 + 2b5 + b4 - b3) q^26 + (-b3 - 2b2 - 3b1 - 3) * q^27 + (-b4 - b2) * q^28 + (-b3 - 2b2) q^29 + (2b7 + 3b6 + b5 + 3b4 - 3b3) * q^30 + (-b7 - b6 - 4b5 - 2b4 - b2 - b1 + 4) * q^31 + (b5 - 1) * q^32 + (2b7 + 3b6 - 2b5 + b4 - b3) q^33 + (b3 + 3b1) q^34 + (-4b6 + b5 - b4 - b3 - 3b2 - b1 + 2) * q^35 + (2b2 + b1 + 1) q^36 + (-b7 + b6 + b4 - b3) * q^37 + (-b7 - 2b6 - 3b5 - b4 - 2b2 - b1 + 3) q^38 + (2b7 - b6 + 2b4 - b2 + 2b1) q^39 + (-b7 + b5 - b4 + b3) * q^40 + (-b3 + b2 - 4) * q^41 + (b7 + b6 - 3b5 - 2b3 - b1 + 1) * q^42 + (2b3 + b2 + b1 - 5) q^43 + (-b7 - b6) * q^44 + (5b7 + 6b6 + 5b4 + 6b2 + 5b1) q^45 + (b5 - 1) * q^46 + (-2b7 + b6 + 5b5 - b4 + b3) * q^47 + (-b2 - 1) * q^48 + (-3b7 + b5 - 2b4 + b3 - 3b1 + 2) q^49 + (b2 + 2b1 + 2) q^50 + (-4b7 - 6b6 - 4b5 - 5b4 + 5b3) q^51 + (-b6 - 2b5 - b4 - b2 + 2) q^52 + (2b7 - 3b6 - b5 + 2b4 - 3b2 + 2b1 + 1) q^53 + (-3b7 - 2b6 + 3b5 - b4 + b3) q^54 + (2b3 + 3b2 + 3b1 + 1) q^55 + (-b6 + b3) * q^56 + (-3b3 - b2 - 4b1 - 1) * q^57 + (-2b6 - b4 + b3) q^58 + (b7 - b6 - 4b5 - 4b4 - b2 + b1 + 4) * q^59 + (-2b7 - 3b6 - b5 - 3b4 - 3b2 - 2b1 + 1) q^60 + (3b7 + 4b6 + 2b5 - b4 + b3) q^61 + (2b3 + b2 + b1 - 4) q^62 + (5b7 + 4b6 - b5 + 3b4 - 2b3 + 3b2 + 2b1 + 5) * q^63 + q^64 + (2b7 - 4b6 - b5 + b4 - b3) * q^65 + (-2b7 - 3b6 + 2b5 - b4 - 3b2 - 2b1 - 2) q^66 + (b6 - 4b5 - 2b4 + b2 + 4) * q^67 + (3b7 + b4 - b3) q^68 + (-b2 - 1) * q^69 + (-b7 + b6 - 3b5 - b4 + 2b3 + 4b2 + 1) q^70 + (-5b3 - b2) q^71 + (b7 + 2b6 - b5) q^72 + (b7 + 3b6 + 4b4 + 3b2 + b1) q^73 + (b7 - b6 - b4 - b2 + b1) * q^74 + (-3b7 - 7b6 + 3b5 - 2b4 + 2b3) q^75 + (b3 + 2b2 + b1 - 3) q^76 + (2b7 + b6 + 3b5 + b3 + 3b2 + 3b1 - 1) * q^77 + (-2b3 + b2 - 2b1) * q^78 + (-b7 - 2b6 - 2b5 + b4 - b3) * q^79 + (b7 - b5 + b4 + b1 + 1) * q^80 + (-3b7 - 5b6 + 2b5 - 5b4 - 5b2 - 3b1 - 2) * q^81 + (b6 + 4b5 - b4 + b3) q^82 + (-5b3 - 4b2) * q^83 + (-2b7 - b6 + 2b5 - 2b4 + 2b3 - b2 - b1 - 3) * q^84 + (b3 - b2 - 5b1 - 12) q^85 + (b7 + b6 + 5b5 + 2b4 - 2b3) q^86 + (-3b7 - 2b6 + 5b5 - 2b4 - 2b2 - 3b1 - 5) * q^87 + (b7 + b6 + b2 + b1) * q^88 + (-b7 - 4b6 + b5 + b4 - b3) q^89 + (-5b3 - 6b2 - 5b1) q^90 + (-2b7 + 2b6 + 2b5 - b4 - b3 - b1 + 4) q^91 + q^92 + (-4b7 + b6 - 4b5 - 5b4 + 5b3) * q^93 + (2b7 - b6 - 5b5 + b4 - b2 + 2b1 + 5) q^94 + (7b6 + b5 + 7b2 - 1) * q^95 + (-b6 + b5) * q^96 + (-b3 + 2b2 + 2b1 + 9) * q^97 + (-3b5 + b4 + b3 + 3b1 + 1) * q^98 + (-4b3 - 6b2 - 3b1 - 8) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{2} + 5 q^{3} - 4 q^{4} + 5 q^{5} - 10 q^{6} + q^{7} + 8 q^{8} - 7 q^{9}+O(q^{10}) $ 8 * q - 4 * q^2 + 5 * q^3 - 4 * q^4 + 5 * q^5 - 10 * q^6 + q^7 + 8 * q^8 - 7 * q^9	\( 8 q - 4 q^{2} + 5 q^{3} - 4 q^{4} + 5 q^{5} - 10 q^{6} + q^{7} + 8 q^{8} - 7 q^{9} + 5 q^{10} + 2 q^{11} + 5 q^{12} - 14 q^{13} + q^{14} + 2 q^{15} - 4 q^{16} - 3 q^{17} - 7 q^{18} + 9 q^{19} - 10 q^{20} + 25 q^{21} - 4 q^{22} - 4 q^{23} + 5 q^{24} - 11 q^{25} + 7 q^{26} - 34 q^{27} - 2 q^{28} - 4 q^{29} - q^{30} + 14 q^{31} - 4 q^{32} - 13 q^{33} + 6 q^{34} + 16 q^{35} + 14 q^{36} + 9 q^{38} + q^{39} + 5 q^{40} - 30 q^{41} - 8 q^{42} - 36 q^{43} + 2 q^{44} + 11 q^{45} - 4 q^{46} + 21 q^{47} - 10 q^{48} + 17 q^{49} + 22 q^{50} - 6 q^{51} + 7 q^{52} + 3 q^{53} + 17 q^{54} + 20 q^{55} + q^{56} - 18 q^{57} + 2 q^{58} + 16 q^{59} - q^{60} + q^{61} - 28 q^{62} + 37 q^{63} + 8 q^{64} - 2 q^{65} - 13 q^{66} + 17 q^{67} - 3 q^{68} - 10 q^{69} + 4 q^{70} - 2 q^{71} - 7 q^{72} + 4 q^{73} + 22 q^{75} - 18 q^{76} + 13 q^{77} - 2 q^{78} - 5 q^{79} + 5 q^{80} - 16 q^{81} + 15 q^{82} - 8 q^{83} - 17 q^{84} - 108 q^{85} + 18 q^{86} - 25 q^{87} + 2 q^{88} + 9 q^{89} - 22 q^{90} + 38 q^{91} + 8 q^{92} - 13 q^{93} + 21 q^{94} + 3 q^{95} + 5 q^{96} + 80 q^{97} + 2 q^{98} - 82 q^{99}+O(q^{100}) \) 8 * q - 4 * q^2 + 5 * q^3 - 4 * q^4 + 5 * q^5 - 10 * q^6 + q^7 + 8 * q^8 - 7 * q^9 + 5 * q^10 + 2 * q^11 + 5 * q^12 - 14 * q^13 + q^14 + 2 * q^15 - 4 * q^16 - 3 * q^17 - 7 * q^18 + 9 * q^19 - 10 * q^20 + 25 * q^21 - 4 * q^22 - 4 * q^23 + 5 * q^24 - 11 * q^25 + 7 * q^26 - 34 * q^27 - 2 * q^28 - 4 * q^29 - q^30 + 14 * q^31 - 4 * q^32 - 13 * q^33 + 6 * q^34 + 16 * q^35 + 14 * q^36 + 9 * q^38 + q^39 + 5 * q^40 - 30 * q^41 - 8 * q^42 - 36 * q^43 + 2 * q^44 + 11 * q^45 - 4 * q^46 + 21 * q^47 - 10 * q^48 + 17 * q^49 + 22 * q^50 - 6 * q^51 + 7 * q^52 + 3 * q^53 + 17 * q^54 + 20 * q^55 + q^56 - 18 * q^57 + 2 * q^58 + 16 * q^59 - q^60 + q^61 - 28 * q^62 + 37 * q^63 + 8 * q^64 - 2 * q^65 - 13 * q^66 + 17 * q^67 - 3 * q^68 - 10 * q^69 + 4 * q^70 - 2 * q^71 - 7 * q^72 + 4 * q^73 + 22 * q^75 - 18 * q^76 + 13 * q^77 - 2 * q^78 - 5 * q^79 + 5 * q^80 - 16 * q^81 + 15 * q^82 - 8 * q^83 - 17 * q^84 - 108 * q^85 + 18 * q^86 - 25 * q^87 + 2 * q^88 + 9 * q^89 - 22 * q^90 + 38 * q^91 + 8 * q^92 - 13 * q^93 + 21 * q^94 + 3 * q^95 + 5 * q^96 + 80 * q^97 + 2 * q^98 - 82 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 2x^{7} + x^{6} - 10x^{5} + 38x^{4} - 40x^{3} + 64x^{2} - 38x + 7 $ x^8 - 2*x^7 + x^6 - 10*x^5 + 38*x^4 - 40*x^3 + 64*x^2 - 38*x + 7 :

$\beta_{1}$	$=$	$ ( -14\nu^{7} + 688\nu^{6} - 619\nu^{5} - 193\nu^{4} - 6480\nu^{3} + 17846\nu^{2} - 10595\nu + 23150 ) / 8102 $ (-14v^7 + 688v^6 - 619v^5 - 193v^4 - 6480v^3 + 17846v^2 - 10595*v + 23150) / 8102
$\beta_{2}$	$=$	$ ( 74\nu^{7} - 743\nu^{6} + 957\nu^{5} - 716\nu^{4} + 8788\nu^{3} - 23147\nu^{2} + 13756\nu - 21668 ) / 8102 $ (74v^7 - 743v^6 + 957v^5 - 716v^4 + 8788v^3 - 23147v^2 + 13756*v - 21668) / 8102
$\beta_{3}$	$=$	$ ( -139\nu^{7} + 465\nu^{6} - 648\nu^{5} + 688\nu^{4} - 5887\nu^{3} + 15724\nu^{2} - 9416\nu - 2218 ) / 8102 $ (-139v^7 + 465v^6 - 648v^5 + 688v^4 - 5887v^3 + 15724v^2 - 9416*v - 2218) / 8102
$\beta_{4}$	$=$	$ ( -1269\nu^{7} + 2176\nu^{6} - 262\nu^{5} + 11731\nu^{4} - 43953\nu^{3} + 36565\nu^{2} - 52069\nu + 14432 ) / 8102 $ (-1269v^7 + 2176v^6 - 262v^5 + 11731v^4 - 43953v^3 + 36565v^2 - 52069*v + 14432) / 8102
$\beta_{5}$	$=$	$ ( -962\nu^{7} + 1557\nu^{6} - 288\nu^{5} + 9308\nu^{4} - 33224\nu^{3} + 25443\nu^{2} - 49196\nu + 18369 ) / 4051 $ (-962v^7 + 1557v^6 - 288v^5 + 9308v^4 - 33224v^3 + 25443v^2 - 49196*v + 18369) / 4051
$\beta_{6}$	$=$	$ ( - 4270 \nu^{7} + 7290 \nu^{6} - 2449 \nu^{5} + 42410 \nu^{4} - 149399 \nu^{3} + 128118 \nu^{2} - 241837 \nu + 88979 ) / 8102 $ (-4270v^7 + 7290v^6 - 2449v^5 + 42410v^4 - 149399v^3 + 128118v^2 - 241837*v + 88979) / 8102
$\beta_{7}$	$=$	$ ( 4805\nu^{7} - 8118\nu^{6} + 2087\nu^{5} - 47477\nu^{4} + 167278\nu^{3} - 139939\nu^{2} + 265634\nu - 98045 ) / 8102 $ (4805v^7 - 8118v^6 + 2087v^5 - 47477v^4 + 167278v^3 - 139939v^2 + 265634*v - 98045) / 8102

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( \beta_{7} + \beta_{6} - \beta_{5} + 2\beta_{4} - \beta_{3} - \beta_{2} - \beta _1 + 2 ) / 3 $ (b7 + b6 - b5 + 2*b4 - b3 - b2 - b1 + 2) / 3
$\nu^{2}$	$=$	$ ( -5\beta_{7} - 5\beta_{6} - \beta_{5} - \beta_{4} + 2\beta_{3} - 4\beta_{2} - 4\beta _1 + 2 ) / 3 $ (-5b7 - 5b6 - b5 - b4 + 2b3 - 4b2 - 4*b1 + 2) / 3
$\nu^{3}$	$=$	$ -\beta_{7} + \beta_{6} - 6\beta_{5} + 2\beta_{4} + \beta_{3} + 3\beta_{2} + \beta _1 + 6 $ -b7 + b6 - 6b5 + 2b4 + b3 + 3*b2 + b1 + 6
$\nu^{4}$	$=$	$ ( -7\beta_{7} + 5\beta_{6} - 38\beta_{5} + 16\beta_{4} - 23\beta_{3} - 17\beta_{2} - 17\beta _1 + 1 ) / 3 $ (-7b7 + 5b6 - 38b5 + 16b4 - 23b3 - 17b2 - 17*b1 + 1) / 3
$\nu^{5}$	$=$	$ ( -70\beta_{7} - 70\beta_{6} - 11\beta_{5} - 14\beta_{4} + 4\beta_{3} - 14\beta_{2} - 8\beta _1 - 17 ) / 3 $ (-70b7 - 70b6 - 11b5 - 14b4 + 4b3 - 14b2 - 8*b1 - 17) / 3
$\nu^{6}$	$=$	$ 17\beta_{7} + 37\beta_{6} - 60\beta_{5} + 35\beta_{4} - 16\beta_{3} + 50\beta_{2} + 46\beta _1 + 7 $ 17b7 + 37b6 - 60b5 + 35b4 - 16b3 + 50b2 + 46*b1 + 7
$\nu^{7}$	$=$	$ ( -44\beta_{7} - 38\beta_{6} - 22\beta_{5} - 7\beta_{4} - 301\beta_{3} - 283\beta_{2} - 97\beta _1 - 565 ) / 3 $ (-44b7 - 38b6 - 22b5 - 7b4 - 301b3 - 283b2 - 97*b1 - 565) / 3

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/322\mathbb{Z}\right)^\times$.

$n$	$185$	$281$
$\chi(n)$	$-\beta_{5}$	$1$

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} $

93.1

0.373419	+	0.0835272i
2.11692	+	0.978886i
0.0512865	−	1.21608i
−1.54162	+	1.88572i
0.373419	−	0.0835272i
2.11692	−	0.978886i
0.0512865	+	1.21608i
−1.54162	−	1.88572i

−0.500000

0.866025i

−0.685837

−

1.18790i

−0.500000

−

0.866025i

1.57146

−

2.72186i

1.37167

−1.66774

2.05393i

1.00000

0.559256

−

0.968659i

1.57146

2.72186i

93.2

−0.500000

0.866025i

0.313577

0.543132i

−0.500000

−

0.866025i

0.475705

−

0.823946i

−0.627155

2.62597

0.322894i

1.00000

1.30334

−

2.25745i

0.475705

0.823946i

93.3

−0.500000

0.866025i

1.26826

2.19669i

−0.500000

−

0.866025i

−1.34706

2.33318i

−2.53652

−2.28677

−

1.33067i

1.00000

−1.71698

2.97389i

−1.34706

−

2.33318i

93.4

−0.500000

0.866025i

1.60400

2.77821i

−0.500000

−

0.866025i

1.79990

−

3.11751i

−3.20800

1.82854

−

1.91218i

1.00000

−3.64562

6.31440i

1.79990

3.11751i

277.1

−0.500000

−

0.866025i

−0.685837

1.18790i

−0.500000

0.866025i

1.57146

2.72186i

1.37167

−1.66774

−

2.05393i

1.00000

0.559256

0.968659i

1.57146

−

2.72186i

277.2

−0.500000

−

0.866025i

0.313577

−

0.543132i

−0.500000

0.866025i

0.475705

0.823946i

−0.627155

2.62597

−

0.322894i

1.00000

1.30334

2.25745i

0.475705

−

0.823946i

277.3

−0.500000

−

0.866025i

1.26826

−

2.19669i

−0.500000

0.866025i

−1.34706

−

2.33318i

−2.53652

−2.28677

1.33067i

1.00000

−1.71698

−

2.97389i

−1.34706

2.33318i

277.4

−0.500000

−

0.866025i

1.60400

−

2.77821i

−0.500000

0.866025i

1.79990

3.11751i

−3.20800

1.82854

1.91218i

1.00000

−3.64562

−

6.31440i

1.79990

−

3.11751i

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	322.2.e.b	✓	8
7.c	even	3	1	inner	322.2.e.b	✓	8
7.c	even	3	1		2254.2.a.w		4
7.d	odd	6	1		2254.2.a.ba		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
322.2.e.b	✓	8	1.a	even	1	1	trivial
322.2.e.b	✓	8	7.c	even	3	1	inner
2254.2.a.w		4	7.c	even	3	1
2254.2.a.ba		4	7.d	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} - 5T_{3}^{7} + 22T_{3}^{6} - 37T_{3}^{5} + 71T_{3}^{4} - 37T_{3}^{3} + 142T_{3}^{2} - 77T_{3} + 49 $ T3^8 - 5*T3^7 + 22*T3^6 - 37*T3^5 + 71*T3^4 - 37*T3^3 + 142*T3^2 - 77*T3 + 49 acting on $S_{2}^{\mathrm{new}}(322, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} + T + 1)^{4} $ (T^2 + T + 1)^4
$3$	$ T^{8} - 5 T^{7} + 22 T^{6} - 37 T^{5} + \cdots + 49 $ T^8 - 5T^7 + 22T^6 - 37T^5 + 71T^4 - 37T^3 + 142T^2 - 77*T + 49
$5$	$ T^{8} - 5 T^{7} + 28 T^{6} - 59 T^{5} + \cdots + 841 $ T^8 - 5T^7 + 28T^6 - 59T^5 + 223T^4 - 401T^3 + 1282T^2 - 1073*T + 841
$7$	$ T^{8} - T^{7} - 8 T^{6} - 5 T^{5} + \cdots + 2401 $ T^8 - T^7 - 8T^6 - 5T^5 + 83T^4 - 35T^3 - 392T^2 - 343T + 2401
$11$	$ T^{8} - 2 T^{7} + 16 T^{6} - 6 T^{5} + \cdots + 9 $ T^8 - 2T^7 + 16T^6 - 6T^5 + 177T^4 - 192T^3 + 189T^2 - 45*T + 9
$13$	$ (T^{4} + 7 T^{3} + 6 T^{2} - 20 T + 7)^{2} $ (T^4 + 7T^3 + 6T^2 - 20*T + 7)^2
$17$	$ T^{8} + 3 T^{7} + 81 T^{6} + \cdots + 1814409 $ T^8 + 3T^7 + 81T^6 + 4T^5 + 4167T^4 - 162T^3 + 109084T^2 - 148170*T + 1814409
$19$	$ T^{8} - 9 T^{7} + 78 T^{6} + \cdots + 95481 $ T^8 - 9T^7 + 78T^6 - 317T^5 + 1623T^4 - 5127T^3 + 21952T^2 - 44805*T + 95481
$23$	$ (T^{2} + T + 1)^{4} $ (T^2 + T + 1)^4
$29$	$ (T^{4} + 2 T^{3} - 27 T^{2} - 56 T - 1)^{2} $ (T^4 + 2T^3 - 27T^2 - 56*T - 1)^2
$31$	$ T^{8} - 14 T^{7} + 154 T^{6} + \cdots + 11449 $ T^8 - 14T^7 + 154T^6 - 650T^5 + 2305T^4 - 1694T^3 + 5455T^2 - 3317*T + 11449
$37$	$ T^{8} + 30 T^{6} + 10 T^{5} + \cdots + 729 $ T^8 + 30T^6 + 10T^5 + 873T^4 + 150T^3 + 835T^2 - 135T + 729
$41$	$ (T^{4} + 15 T^{3} + 66 T^{2} + 62 T - 63)^{2} $ (T^4 + 15T^3 + 66T^2 + 62*T - 63)^2
$43$	$ (T^{4} + 18 T^{3} + 90 T^{2} + 99 T - 81)^{2} $ (T^4 + 18T^3 + 90T^2 + 99*T - 81)^2
$47$	$ T^{8} - 21 T^{7} + 324 T^{6} + \cdots + 35721 $ T^8 - 21T^7 + 324T^6 - 2259T^5 + 11799T^4 - 19521T^3 + 31914T^2 + 18711*T + 35721
$53$	$ T^{8} - 3 T^{7} + 144 T^{6} + \cdots + 13689 $ T^8 - 3T^7 + 144T^6 - 33T^5 + 18765T^4 - 28863T^3 + 63756T^2 + 25623*T + 13689
$59$	$ T^{8} - 16 T^{7} + 334 T^{6} + \cdots + 1320201 $ T^8 - 16T^7 + 334T^6 - 1878T^5 + 32241T^4 - 158682T^3 + 2353347T^2 - 1795887*T + 1320201
$61$	$ T^{8} - T^{7} + 199 T^{6} + \cdots + 8346321 $ T^8 - T^7 + 199T^6 + 846T^5 + 35991T^4 + 69930T^3 + 676998T^2 - 936036T + 8346321
$67$	$ T^{8} - 17 T^{7} + 229 T^{6} + \cdots + 82369 $ T^8 - 17T^7 + 229T^6 - 1124T^5 + 4771T^4 - 6638T^3 + 19924T^2 - 14924*T + 82369
$71$	$ (T^{4} + T^{3} - 216 T^{2} - 196 T + 5243)^{2} $ (T^4 + T^3 - 216T^2 - 196T + 5243)^2
$73$	$ T^{8} - 4 T^{7} + 154 T^{6} + \cdots + 19245769 $ T^8 - 4T^7 + 154T^6 + 98T^5 + 15565T^4 + 3770T^3 + 656935T^2 + 995849*T + 19245769
$79$	$ T^{8} + 5 T^{7} + 67 T^{6} + \cdots + 145161 $ T^8 + 5T^7 + 67T^6 - 78T^5 + 1713T^4 - 1038T^3 + 20358T^2 - 25146*T + 145161
$83$	$ (T^{4} + 4 T^{3} - 261 T^{2} - 574 T + 17003)^{2} $ (T^4 + 4T^3 - 261T^2 - 574*T + 17003)^2
$89$	$ T^{8} - 9 T^{7} + 180 T^{6} + \cdots + 12131289 $ T^8 - 9T^7 + 180T^6 - 171T^5 + 11097T^4 + 10125T^3 + 626778T^2 + 1849473*T + 12131289
$97$	$ (T^{4} - 40 T^{3} + 519 T^{2} - 2126 T - 637)^{2} $ (T^4 - 40T^3 + 519T^2 - 2126*T - 637)^2
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 322 = 2 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	322.e (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{5} q^{2} + (\beta_{6} - \beta_{5} + \beta_{2} + 1) q^{3} + (\beta_{5} - 1) q^{4} + ( - \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3}) q^{5} + ( - \beta_{2} - 1) q^{6} + ( - \beta_{6} + \beta_{3}) q^{7} + q^{8} + (\beta_{7} + 2 \beta_{6} - \beta_{5}) q^{9}+O(q^{10}) \) q - b5 * q^2 + (b6 - b5 + b2 + 1) * q^3 + (b5 - 1) * q^4 + (-b7 + b5 - b4 + b3) * q^5 + (-b2 - 1) * q^6 + (-b6 + b3) * q^7 + q^8 + (b7 + 2b6 - b5) q^9	\( q - \beta_{5} q^{2} + (\beta_{6} - \beta_{5} + \beta_{2} + 1) q^{3} + (\beta_{5} - 1) q^{4} + ( - \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3}) q^{5} + ( - \beta_{2} - 1) q^{6} + ( - \beta_{6} + \beta_{3}) q^{7} + q^{8} + (\beta_{7} + 2 \beta_{6} - \beta_{5}) q^{9} + (\beta_{7} - \beta_{5} + \beta_{4} + \beta_1 + 1) q^{10} + (\beta_{7} + \beta_{6} + \beta_{2} + \beta_1) q^{11} + ( - \beta_{6} + \beta_{5}) q^{12} + (\beta_{3} + \beta_{2} - 2) q^{13} + (\beta_{6} + \beta_{4} - \beta_{3} + \beta_{2}) q^{14} + (3 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 1) q^{15} - \beta_{5} q^{16} + ( - 3 \beta_{7} - \beta_{4} - 3 \beta_1) q^{17} + ( - \beta_{7} - 2 \beta_{6} + \beta_{5} - 2 \beta_{2} - \beta_1 - 1) q^{18} + (\beta_{7} + 2 \beta_{6} + 3 \beta_{5} + \beta_{4} - \beta_{3}) q^{19} + ( - \beta_{3} - \beta_1 - 1) q^{20} + (\beta_{7} + \beta_{5} + 2 \beta_{4} + \beta_{2} + 2 \beta_1 + 2) q^{21} + ( - \beta_{2} - \beta_1) q^{22} - \beta_{5} q^{23} + (\beta_{6} - \beta_{5} + \beta_{2} + 1) q^{24} + ( - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{2} - 2 \beta_1 - 2) q^{25} + (\beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3}) q^{26} + ( - \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 3) q^{27} + ( - \beta_{4} - \beta_{2}) q^{28} + ( - \beta_{3} - 2 \beta_{2}) q^{29} + (2 \beta_{7} + 3 \beta_{6} + \beta_{5} + 3 \beta_{4} - 3 \beta_{3}) q^{30} + ( - \beta_{7} - \beta_{6} - 4 \beta_{5} - 2 \beta_{4} - \beta_{2} - \beta_1 + 4) q^{31} + (\beta_{5} - 1) q^{32} + (2 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{3}) q^{33} + (\beta_{3} + 3 \beta_1) q^{34} + ( - 4 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_1 + 2) q^{35} + (2 \beta_{2} + \beta_1 + 1) q^{36} + ( - \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3}) q^{37} + ( - \beta_{7} - 2 \beta_{6} - 3 \beta_{5} - \beta_{4} - 2 \beta_{2} - \beta_1 + 3) q^{38} + (2 \beta_{7} - \beta_{6} + 2 \beta_{4} - \beta_{2} + 2 \beta_1) q^{39} + ( - \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3}) q^{40} + ( - \beta_{3} + \beta_{2} - 4) q^{41} + (\beta_{7} + \beta_{6} - 3 \beta_{5} - 2 \beta_{3} - \beta_1 + 1) q^{42} + (2 \beta_{3} + \beta_{2} + \beta_1 - 5) q^{43} + ( - \beta_{7} - \beta_{6}) q^{44} + (5 \beta_{7} + 6 \beta_{6} + 5 \beta_{4} + 6 \beta_{2} + 5 \beta_1) q^{45} + (\beta_{5} - 1) q^{46} + ( - 2 \beta_{7} + \beta_{6} + 5 \beta_{5} - \beta_{4} + \beta_{3}) q^{47} + ( - \beta_{2} - 1) q^{48} + ( - 3 \beta_{7} + \beta_{5} - 2 \beta_{4} + \beta_{3} - 3 \beta_1 + 2) q^{49} + (\beta_{2} + 2 \beta_1 + 2) q^{50} + ( - 4 \beta_{7} - 6 \beta_{6} - 4 \beta_{5} - 5 \beta_{4} + 5 \beta_{3}) q^{51} + ( - \beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_{2} + 2) q^{52} + (2 \beta_{7} - 3 \beta_{6} - \beta_{5} + 2 \beta_{4} - 3 \beta_{2} + 2 \beta_1 + 1) q^{53} + ( - 3 \beta_{7} - 2 \beta_{6} + 3 \beta_{5} - \beta_{4} + \beta_{3}) q^{54} + (2 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 1) q^{55} + ( - \beta_{6} + \beta_{3}) q^{56} + ( - 3 \beta_{3} - \beta_{2} - 4 \beta_1 - 1) q^{57} + ( - 2 \beta_{6} - \beta_{4} + \beta_{3}) q^{58} + (\beta_{7} - \beta_{6} - 4 \beta_{5} - 4 \beta_{4} - \beta_{2} + \beta_1 + 4) q^{59} + ( - 2 \beta_{7} - 3 \beta_{6} - \beta_{5} - 3 \beta_{4} - 3 \beta_{2} - 2 \beta_1 + 1) q^{60} + (3 \beta_{7} + 4 \beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{3}) q^{61} + (2 \beta_{3} + \beta_{2} + \beta_1 - 4) q^{62} + (5 \beta_{7} + 4 \beta_{6} - \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 5) q^{63} + q^{64} + (2 \beta_{7} - 4 \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3}) q^{65} + ( - 2 \beta_{7} - 3 \beta_{6} + 2 \beta_{5} - \beta_{4} - 3 \beta_{2} - 2 \beta_1 - 2) q^{66} + (\beta_{6} - 4 \beta_{5} - 2 \beta_{4} + \beta_{2} + 4) q^{67} + (3 \beta_{7} + \beta_{4} - \beta_{3}) q^{68} + ( - \beta_{2} - 1) q^{69} + ( - \beta_{7} + \beta_{6} - 3 \beta_{5} - \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + 1) q^{70} + ( - 5 \beta_{3} - \beta_{2}) q^{71} + (\beta_{7} + 2 \beta_{6} - \beta_{5}) q^{72} + (\beta_{7} + 3 \beta_{6} + 4 \beta_{4} + 3 \beta_{2} + \beta_1) q^{73} + (\beta_{7} - \beta_{6} - \beta_{4} - \beta_{2} + \beta_1) q^{74} + ( - 3 \beta_{7} - 7 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} + 2 \beta_{3}) q^{75} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 3) q^{76} + (2 \beta_{7} + \beta_{6} + 3 \beta_{5} + \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 1) q^{77} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{78} + ( - \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{3}) q^{79} + (\beta_{7} - \beta_{5} + \beta_{4} + \beta_1 + 1) q^{80} + ( - 3 \beta_{7} - 5 \beta_{6} + 2 \beta_{5} - 5 \beta_{4} - 5 \beta_{2} - 3 \beta_1 - 2) q^{81} + (\beta_{6} + 4 \beta_{5} - \beta_{4} + \beta_{3}) q^{82} + ( - 5 \beta_{3} - 4 \beta_{2}) q^{83} + ( - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 - 3) q^{84} + (\beta_{3} - \beta_{2} - 5 \beta_1 - 12) q^{85} + (\beta_{7} + \beta_{6} + 5 \beta_{5} + 2 \beta_{4} - 2 \beta_{3}) q^{86} + ( - 3 \beta_{7} - 2 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} - 2 \beta_{2} - 3 \beta_1 - 5) q^{87} + (\beta_{7} + \beta_{6} + \beta_{2} + \beta_1) q^{88} + ( - \beta_{7} - 4 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3}) q^{89} + ( - 5 \beta_{3} - 6 \beta_{2} - 5 \beta_1) q^{90} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{3} - \beta_1 + 4) q^{91} + q^{92} + ( - 4 \beta_{7} + \beta_{6} - 4 \beta_{5} - 5 \beta_{4} + 5 \beta_{3}) q^{93} + (2 \beta_{7} - \beta_{6} - 5 \beta_{5} + \beta_{4} - \beta_{2} + 2 \beta_1 + 5) q^{94} + (7 \beta_{6} + \beta_{5} + 7 \beta_{2} - 1) q^{95} + ( - \beta_{6} + \beta_{5}) q^{96} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 9) q^{97} + ( - 3 \beta_{5} + \beta_{4} + \beta_{3} + 3 \beta_1 + 1) q^{98} + ( - 4 \beta_{3} - 6 \beta_{2} - 3 \beta_1 - 8) q^{99}+O(q^{100}) \) q - b5 * q^2 + (b6 - b5 + b2 + 1) * q^3 + (b5 - 1) * q^4 + (-b7 + b5 - b4 + b3) * q^5 + (-b2 - 1) * q^6 + (-b6 + b3) * q^7 + q^8 + (b7 + 2b6 - b5) q^9 + (b7 - b5 + b4 + b1 + 1) * q^10 + (b7 + b6 + b2 + b1) * q^11 + (-b6 + b5) * q^12 + (b3 + b2 - 2) * q^13 + (b6 + b4 - b3 + b2) * q^14 + (3b3 + 3b2 + 2b1 - 1) q^15 - b5 * q^16 + (-3b7 - b4 - 3b1) * q^17 + (-b7 - 2b6 + b5 - 2b2 - b1 - 1) * q^18 + (b7 + 2b6 + 3b5 + b4 - b3) * q^19 + (-b3 - b1 - 1) * q^20 + (b7 + b5 + 2b4 + b2 + 2b1 + 2) * q^21 + (-b2 - b1) * q^22 - b5 * q^23 + (b6 - b5 + b2 + 1) * q^24 + (-2b7 - b6 + 2b5 - b2 - 2b1 - 2) q^25 + (b6 + 2b5 + b4 - b3) q^26 + (-b3 - 2b2 - 3b1 - 3) * q^27 + (-b4 - b2) * q^28 + (-b3 - 2b2) q^29 + (2b7 + 3b6 + b5 + 3b4 - 3b3) * q^30 + (-b7 - b6 - 4b5 - 2b4 - b2 - b1 + 4) * q^31 + (b5 - 1) * q^32 + (2b7 + 3b6 - 2b5 + b4 - b3) q^33 + (b3 + 3b1) q^34 + (-4b6 + b5 - b4 - b3 - 3b2 - b1 + 2) * q^35 + (2b2 + b1 + 1) q^36 + (-b7 + b6 + b4 - b3) * q^37 + (-b7 - 2b6 - 3b5 - b4 - 2b2 - b1 + 3) q^38 + (2b7 - b6 + 2b4 - b2 + 2b1) q^39 + (-b7 + b5 - b4 + b3) * q^40 + (-b3 + b2 - 4) * q^41 + (b7 + b6 - 3b5 - 2b3 - b1 + 1) * q^42 + (2b3 + b2 + b1 - 5) q^43 + (-b7 - b6) * q^44 + (5b7 + 6b6 + 5b4 + 6b2 + 5b1) q^45 + (b5 - 1) * q^46 + (-2b7 + b6 + 5b5 - b4 + b3) * q^47 + (-b2 - 1) * q^48 + (-3b7 + b5 - 2b4 + b3 - 3b1 + 2) q^49 + (b2 + 2b1 + 2) q^50 + (-4b7 - 6b6 - 4b5 - 5b4 + 5b3) q^51 + (-b6 - 2b5 - b4 - b2 + 2) q^52 + (2b7 - 3b6 - b5 + 2b4 - 3b2 + 2b1 + 1) q^53 + (-3b7 - 2b6 + 3b5 - b4 + b3) q^54 + (2b3 + 3b2 + 3b1 + 1) q^55 + (-b6 + b3) * q^56 + (-3b3 - b2 - 4b1 - 1) * q^57 + (-2b6 - b4 + b3) q^58 + (b7 - b6 - 4b5 - 4b4 - b2 + b1 + 4) * q^59 + (-2b7 - 3b6 - b5 - 3b4 - 3b2 - 2b1 + 1) q^60 + (3b7 + 4b6 + 2b5 - b4 + b3) q^61 + (2b3 + b2 + b1 - 4) q^62 + (5b7 + 4b6 - b5 + 3b4 - 2b3 + 3b2 + 2b1 + 5) * q^63 + q^64 + (2b7 - 4b6 - b5 + b4 - b3) * q^65 + (-2b7 - 3b6 + 2b5 - b4 - 3b2 - 2b1 - 2) q^66 + (b6 - 4b5 - 2b4 + b2 + 4) * q^67 + (3b7 + b4 - b3) q^68 + (-b2 - 1) * q^69 + (-b7 + b6 - 3b5 - b4 + 2b3 + 4b2 + 1) q^70 + (-5b3 - b2) q^71 + (b7 + 2b6 - b5) q^72 + (b7 + 3b6 + 4b4 + 3b2 + b1) q^73 + (b7 - b6 - b4 - b2 + b1) * q^74 + (-3b7 - 7b6 + 3b5 - 2b4 + 2b3) q^75 + (b3 + 2b2 + b1 - 3) q^76 + (2b7 + b6 + 3b5 + b3 + 3b2 + 3b1 - 1) * q^77 + (-2b3 + b2 - 2b1) * q^78 + (-b7 - 2b6 - 2b5 + b4 - b3) * q^79 + (b7 - b5 + b4 + b1 + 1) * q^80 + (-3b7 - 5b6 + 2b5 - 5b4 - 5b2 - 3b1 - 2) * q^81 + (b6 + 4b5 - b4 + b3) q^82 + (-5b3 - 4b2) * q^83 + (-2b7 - b6 + 2b5 - 2b4 + 2b3 - b2 - b1 - 3) * q^84 + (b3 - b2 - 5b1 - 12) q^85 + (b7 + b6 + 5b5 + 2b4 - 2b3) q^86 + (-3b7 - 2b6 + 5b5 - 2b4 - 2b2 - 3b1 - 5) * q^87 + (b7 + b6 + b2 + b1) * q^88 + (-b7 - 4b6 + b5 + b4 - b3) q^89 + (-5b3 - 6b2 - 5b1) q^90 + (-2b7 + 2b6 + 2b5 - b4 - b3 - b1 + 4) q^91 + q^92 + (-4b7 + b6 - 4b5 - 5b4 + 5b3) * q^93 + (2b7 - b6 - 5b5 + b4 - b2 + 2b1 + 5) q^94 + (7b6 + b5 + 7b2 - 1) * q^95 + (-b6 + b5) * q^96 + (-b3 + 2b2 + 2b1 + 9) * q^97 + (-3b5 + b4 + b3 + 3b1 + 1) * q^98 + (-4b3 - 6b2 - 3b1 - 8) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} + 5 q^{3} - 4 q^{4} + 5 q^{5} - 10 q^{6} + q^{7} + 8 q^{8} - 7 q^{9}+O(q^{10}) \) 8 * q - 4 * q^2 + 5 * q^3 - 4 * q^4 + 5 * q^5 - 10 * q^6 + q^7 + 8 * q^8 - 7 * q^9	\( 8 q - 4 q^{2} + 5 q^{3} - 4 q^{4} + 5 q^{5} - 10 q^{6} + q^{7} + 8 q^{8} - 7 q^{9} + 5 q^{10} + 2 q^{11} + 5 q^{12} - 14 q^{13} + q^{14} + 2 q^{15} - 4 q^{16} - 3 q^{17} - 7 q^{18} + 9 q^{19} - 10 q^{20} + 25 q^{21} - 4 q^{22} - 4 q^{23} + 5 q^{24} - 11 q^{25} + 7 q^{26} - 34 q^{27} - 2 q^{28} - 4 q^{29} - q^{30} + 14 q^{31} - 4 q^{32} - 13 q^{33} + 6 q^{34} + 16 q^{35} + 14 q^{36} + 9 q^{38} + q^{39} + 5 q^{40} - 30 q^{41} - 8 q^{42} - 36 q^{43} + 2 q^{44} + 11 q^{45} - 4 q^{46} + 21 q^{47} - 10 q^{48} + 17 q^{49} + 22 q^{50} - 6 q^{51} + 7 q^{52} + 3 q^{53} + 17 q^{54} + 20 q^{55} + q^{56} - 18 q^{57} + 2 q^{58} + 16 q^{59} - q^{60} + q^{61} - 28 q^{62} + 37 q^{63} + 8 q^{64} - 2 q^{65} - 13 q^{66} + 17 q^{67} - 3 q^{68} - 10 q^{69} + 4 q^{70} - 2 q^{71} - 7 q^{72} + 4 q^{73} + 22 q^{75} - 18 q^{76} + 13 q^{77} - 2 q^{78} - 5 q^{79} + 5 q^{80} - 16 q^{81} + 15 q^{82} - 8 q^{83} - 17 q^{84} - 108 q^{85} + 18 q^{86} - 25 q^{87} + 2 q^{88} + 9 q^{89} - 22 q^{90} + 38 q^{91} + 8 q^{92} - 13 q^{93} + 21 q^{94} + 3 q^{95} + 5 q^{96} + 80 q^{97} + 2 q^{98} - 82 q^{99}+O(q^{100}) \) 8 * q - 4 * q^2 + 5 * q^3 - 4 * q^4 + 5 * q^5 - 10 * q^6 + q^7 + 8 * q^8 - 7 * q^9 + 5 * q^10 + 2 * q^11 + 5 * q^12 - 14 * q^13 + q^14 + 2 * q^15 - 4 * q^16 - 3 * q^17 - 7 * q^18 + 9 * q^19 - 10 * q^20 + 25 * q^21 - 4 * q^22 - 4 * q^23 + 5 * q^24 - 11 * q^25 + 7 * q^26 - 34 * q^27 - 2 * q^28 - 4 * q^29 - q^30 + 14 * q^31 - 4 * q^32 - 13 * q^33 + 6 * q^34 + 16 * q^35 + 14 * q^36 + 9 * q^38 + q^39 + 5 * q^40 - 30 * q^41 - 8 * q^42 - 36 * q^43 + 2 * q^44 + 11 * q^45 - 4 * q^46 + 21 * q^47 - 10 * q^48 + 17 * q^49 + 22 * q^50 - 6 * q^51 + 7 * q^52 + 3 * q^53 + 17 * q^54 + 20 * q^55 + q^56 - 18 * q^57 + 2 * q^58 + 16 * q^59 - q^60 + q^61 - 28 * q^62 + 37 * q^63 + 8 * q^64 - 2 * q^65 - 13 * q^66 + 17 * q^67 - 3 * q^68 - 10 * q^69 + 4 * q^70 - 2 * q^71 - 7 * q^72 + 4 * q^73 + 22 * q^75 - 18 * q^76 + 13 * q^77 - 2 * q^78 - 5 * q^79 + 5 * q^80 - 16 * q^81 + 15 * q^82 - 8 * q^83 - 17 * q^84 - 108 * q^85 + 18 * q^86 - 25 * q^87 + 2 * q^88 + 9 * q^89 - 22 * q^90 + 38 * q^91 + 8 * q^92 - 13 * q^93 + 21 * q^94 + 3 * q^95 + 5 * q^96 + 80 * q^97 + 2 * q^98 - 82 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	322.2.e.b

Level	$322$
Weight	$2$
Character orbit	322.e
Analytic conductor	$2.571$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(2.57118294509\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{3})\)
Coefficient field:	8.0.1767277521.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - 2x^{7} + x^{6} - 10x^{5} + 38x^{4} - 40x^{3} + 64x^{2} - 38x + 7 \) x^8 - 2x^7 + x^6 - 10x^5 + 38x^4 - 40x^3 + 64x^2 - 38x + 7
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( ( -14\nu^{7} + 688\nu^{6} - 619\nu^{5} - 193\nu^{4} - 6480\nu^{3} + 17846\nu^{2} - 10595\nu + 23150 ) / 8102 \) (-14v^7 + 688v^6 - 619v^5 - 193v^4 - 6480v^3 + 17846v^2 - 10595*v + 23150) / 8102
\(\beta_{2}\)	\(=\)	\( ( 74\nu^{7} - 743\nu^{6} + 957\nu^{5} - 716\nu^{4} + 8788\nu^{3} - 23147\nu^{2} + 13756\nu - 21668 ) / 8102 \) (74v^7 - 743v^6 + 957v^5 - 716v^4 + 8788v^3 - 23147v^2 + 13756*v - 21668) / 8102
\(\beta_{3}\)	\(=\)	\( ( -139\nu^{7} + 465\nu^{6} - 648\nu^{5} + 688\nu^{4} - 5887\nu^{3} + 15724\nu^{2} - 9416\nu - 2218 ) / 8102 \) (-139v^7 + 465v^6 - 648v^5 + 688v^4 - 5887v^3 + 15724v^2 - 9416*v - 2218) / 8102
\(\beta_{4}\)	\(=\)	\( ( -1269\nu^{7} + 2176\nu^{6} - 262\nu^{5} + 11731\nu^{4} - 43953\nu^{3} + 36565\nu^{2} - 52069\nu + 14432 ) / 8102 \) (-1269v^7 + 2176v^6 - 262v^5 + 11731v^4 - 43953v^3 + 36565v^2 - 52069*v + 14432) / 8102
\(\beta_{5}\)	\(=\)	\( ( -962\nu^{7} + 1557\nu^{6} - 288\nu^{5} + 9308\nu^{4} - 33224\nu^{3} + 25443\nu^{2} - 49196\nu + 18369 ) / 4051 \) (-962v^7 + 1557v^6 - 288v^5 + 9308v^4 - 33224v^3 + 25443v^2 - 49196*v + 18369) / 4051
\(\beta_{6}\)	\(=\)	\( ( - 4270 \nu^{7} + 7290 \nu^{6} - 2449 \nu^{5} + 42410 \nu^{4} - 149399 \nu^{3} + 128118 \nu^{2} - 241837 \nu + 88979 ) / 8102 \) (-4270v^7 + 7290v^6 - 2449v^5 + 42410v^4 - 149399v^3 + 128118v^2 - 241837*v + 88979) / 8102
\(\beta_{7}\)	\(=\)	\( ( 4805\nu^{7} - 8118\nu^{6} + 2087\nu^{5} - 47477\nu^{4} + 167278\nu^{3} - 139939\nu^{2} + 265634\nu - 98045 ) / 8102 \) (4805v^7 - 8118v^6 + 2087v^5 - 47477v^4 + 167278v^3 - 139939v^2 + 265634*v - 98045) / 8102

\(\nu\)	\(=\)	\( ( \beta_{7} + \beta_{6} - \beta_{5} + 2\beta_{4} - \beta_{3} - \beta_{2} - \beta _1 + 2 ) / 3 \) (b7 + b6 - b5 + 2*b4 - b3 - b2 - b1 + 2) / 3
\(\nu^{2}\)	\(=\)	\( ( -5\beta_{7} - 5\beta_{6} - \beta_{5} - \beta_{4} + 2\beta_{3} - 4\beta_{2} - 4\beta _1 + 2 ) / 3 \) (-5b7 - 5b6 - b5 - b4 + 2b3 - 4b2 - 4*b1 + 2) / 3
\(\nu^{3}\)	\(=\)	\( -\beta_{7} + \beta_{6} - 6\beta_{5} + 2\beta_{4} + \beta_{3} + 3\beta_{2} + \beta _1 + 6 \) -b7 + b6 - 6b5 + 2b4 + b3 + 3*b2 + b1 + 6
\(\nu^{4}\)	\(=\)	\( ( -7\beta_{7} + 5\beta_{6} - 38\beta_{5} + 16\beta_{4} - 23\beta_{3} - 17\beta_{2} - 17\beta _1 + 1 ) / 3 \) (-7b7 + 5b6 - 38b5 + 16b4 - 23b3 - 17b2 - 17*b1 + 1) / 3
\(\nu^{5}\)	\(=\)	\( ( -70\beta_{7} - 70\beta_{6} - 11\beta_{5} - 14\beta_{4} + 4\beta_{3} - 14\beta_{2} - 8\beta _1 - 17 ) / 3 \) (-70b7 - 70b6 - 11b5 - 14b4 + 4b3 - 14b2 - 8*b1 - 17) / 3
\(\nu^{6}\)	\(=\)	\( 17\beta_{7} + 37\beta_{6} - 60\beta_{5} + 35\beta_{4} - 16\beta_{3} + 50\beta_{2} + 46\beta _1 + 7 \) 17b7 + 37b6 - 60b5 + 35b4 - 16b3 + 50b2 + 46*b1 + 7
\(\nu^{7}\)	\(=\)	\( ( -44\beta_{7} - 38\beta_{6} - 22\beta_{5} - 7\beta_{4} - 301\beta_{3} - 283\beta_{2} - 97\beta _1 - 565 ) / 3 \) (-44b7 - 38b6 - 22b5 - 7b4 - 301b3 - 283b2 - 97*b1 - 565) / 3

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 277.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{2} + T + 1)^{4} \) (T^2 + T + 1)^4
$3$	\( T^{8} - 5 T^{7} + 22 T^{6} - 37 T^{5} + \cdots + 49 \) T^8 - 5T^7 + 22T^6 - 37T^5 + 71T^4 - 37T^3 + 142T^2 - 77*T + 49
$5$	\( T^{8} - 5 T^{7} + 28 T^{6} - 59 T^{5} + \cdots + 841 \) T^8 - 5T^7 + 28T^6 - 59T^5 + 223T^4 - 401T^3 + 1282T^2 - 1073*T + 841
$7$	\( T^{8} - T^{7} - 8 T^{6} - 5 T^{5} + \cdots + 2401 \) T^8 - T^7 - 8T^6 - 5T^5 + 83T^4 - 35T^3 - 392T^2 - 343T + 2401
$11$	\( T^{8} - 2 T^{7} + 16 T^{6} - 6 T^{5} + \cdots + 9 \) T^8 - 2T^7 + 16T^6 - 6T^5 + 177T^4 - 192T^3 + 189T^2 - 45*T + 9
$13$	\( (T^{4} + 7 T^{3} + 6 T^{2} - 20 T + 7)^{2} \) (T^4 + 7T^3 + 6T^2 - 20*T + 7)^2
$17$	\( T^{8} + 3 T^{7} + 81 T^{6} + \cdots + 1814409 \) T^8 + 3T^7 + 81T^6 + 4T^5 + 4167T^4 - 162T^3 + 109084T^2 - 148170*T + 1814409
$19$	\( T^{8} - 9 T^{7} + 78 T^{6} + \cdots + 95481 \) T^8 - 9T^7 + 78T^6 - 317T^5 + 1623T^4 - 5127T^3 + 21952T^2 - 44805*T + 95481
$23$	\( (T^{2} + T + 1)^{4} \) (T^2 + T + 1)^4
$29$	\( (T^{4} + 2 T^{3} - 27 T^{2} - 56 T - 1)^{2} \) (T^4 + 2T^3 - 27T^2 - 56*T - 1)^2
$31$	\( T^{8} - 14 T^{7} + 154 T^{6} + \cdots + 11449 \) T^8 - 14T^7 + 154T^6 - 650T^5 + 2305T^4 - 1694T^3 + 5455T^2 - 3317*T + 11449
$37$	\( T^{8} + 30 T^{6} + 10 T^{5} + \cdots + 729 \) T^8 + 30T^6 + 10T^5 + 873T^4 + 150T^3 + 835T^2 - 135T + 729
$41$	\( (T^{4} + 15 T^{3} + 66 T^{2} + 62 T - 63)^{2} \) (T^4 + 15T^3 + 66T^2 + 62*T - 63)^2
$43$	\( (T^{4} + 18 T^{3} + 90 T^{2} + 99 T - 81)^{2} \) (T^4 + 18T^3 + 90T^2 + 99*T - 81)^2
$47$	\( T^{8} - 21 T^{7} + 324 T^{6} + \cdots + 35721 \) T^8 - 21T^7 + 324T^6 - 2259T^5 + 11799T^4 - 19521T^3 + 31914T^2 + 18711*T + 35721
$53$	\( T^{8} - 3 T^{7} + 144 T^{6} + \cdots + 13689 \) T^8 - 3T^7 + 144T^6 - 33T^5 + 18765T^4 - 28863T^3 + 63756T^2 + 25623*T + 13689
$59$	\( T^{8} - 16 T^{7} + 334 T^{6} + \cdots + 1320201 \) T^8 - 16T^7 + 334T^6 - 1878T^5 + 32241T^4 - 158682T^3 + 2353347T^2 - 1795887*T + 1320201
$61$	\( T^{8} - T^{7} + 199 T^{6} + \cdots + 8346321 \) T^8 - T^7 + 199T^6 + 846T^5 + 35991T^4 + 69930T^3 + 676998T^2 - 936036T + 8346321
$67$	\( T^{8} - 17 T^{7} + 229 T^{6} + \cdots + 82369 \) T^8 - 17T^7 + 229T^6 - 1124T^5 + 4771T^4 - 6638T^3 + 19924T^2 - 14924*T + 82369
$71$	\( (T^{4} + T^{3} - 216 T^{2} - 196 T + 5243)^{2} \) (T^4 + T^3 - 216T^2 - 196T + 5243)^2
$73$	\( T^{8} - 4 T^{7} + 154 T^{6} + \cdots + 19245769 \) T^8 - 4T^7 + 154T^6 + 98T^5 + 15565T^4 + 3770T^3 + 656935T^2 + 995849*T + 19245769
$79$	\( T^{8} + 5 T^{7} + 67 T^{6} + \cdots + 145161 \) T^8 + 5T^7 + 67T^6 - 78T^5 + 1713T^4 - 1038T^3 + 20358T^2 - 25146*T + 145161
$83$	\( (T^{4} + 4 T^{3} - 261 T^{2} - 574 T + 17003)^{2} \) (T^4 + 4T^3 - 261T^2 - 574*T + 17003)^2
$89$	\( T^{8} - 9 T^{7} + 180 T^{6} + \cdots + 12131289 \) T^8 - 9T^7 + 180T^6 - 171T^5 + 11097T^4 + 10125T^3 + 626778T^2 + 1849473*T + 12131289
$97$	\( (T^{4} - 40 T^{3} + 519 T^{2} - 2126 T - 637)^{2} \) (T^4 - 40T^3 + 519T^2 - 2126*T - 637)^2
show more
show less

\(n\)	\(185\)	\(281\)
\(\chi(n)\)	\(-\beta_{5}\)	\(1\)