LMFDB - Newform orbit 357.2.k.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [357,2,Mod(64,357)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(357, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("357.64");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$357 = 3 \cdot 7 \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	357.k (of order $4$ , 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:	$2.85065935216$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{20} + 32 x^{18} + 426 x^{16} + 3072 x^{14} + 13121 x^{12} + 34148 x^{10} + 53608 x^{8} + 48276 x^{6} + \cdots + 4$ x^20 + 32x^18 + 426x^16 + 3072x^14 + 13121x^12 + 34148x^10 + 53608x^8 + 48276x^6 + 22224x^4 + 3920*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2^{4}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$ -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 $1,\beta_1,\ldots,\beta_{19}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta_1 q^{2} + \beta_{9} q^{3} + (\beta_{8} - \beta_{7} - 1) q^{4} + ( - \beta_{12} + \beta_{9}) q^{5} - \beta_{3} q^{6} - \beta_{10} q^{7} + (\beta_{18} + \beta_{17} + \cdots - 2 \beta_1) q^{8}+ \cdots + ( - \beta_{18} - \beta_{16} + \cdots + \beta_{2}) q^{99}+O(q^{100})$ q + b1 * q^2 + b9 * q^3 + (b8 - b7 - 1) * q^4 + (-b12 + b9) * q^5 - b3 * q^6 - b10 * q^7 + (b18 + b17 + b16 + b13 - b11 - b10 - b9 - b5 + b4 - 2b1) q^8 - b11 * q^9 + (b14 + b7 - b4 - b3 + b1) * q^10 + (b17 + b16 + b8 - b3) * q^11 + (b18 - b9 - b2) * q^12 + (b15 + b10 - b9 - b5 - b4 - b3 + b2 - 1) * q^13 - b2 * q^14 + (b13 - b11) * q^15 + (-b19 + b18 - b17 - 2b15 - b10 + b9 - 2b8 + 3b7 - b5 - b4 + b3 - b2 + 2) q^16 + (-b18 + b17 + b15 - b10 - b6 + b5 - b3 + b2 + b1) * q^17 + b7 * q^18 + (b16 + b14 - b13 + b12 - b6 + b3 + b2) * q^19 + (b18 - b16 + 2b11 - b9 + b8 - 2b7 + b5 - 2b2 + 2b1 - 2) * q^20 + q^21 + (-b19 + 3b18 + b16 - b15 + b13 + b12 + b11 - 3b9 - b8 + 2b7 + b6 - 2b5 - 3b2 - 2b1 - 1) * q^22 + (-b16 - 2b14 - b10 - b8 + b7 - b4 + b3 + b1) q^23 + (-b19 - b17 - b16 - b14 + b11 + b10 - b8 - b6 + 2b3 + 1) q^24 + (-b18 - b17 - b16 - b14 - b12 + b11 + b10 + b9 + b3 + b2 - b1) * q^25 + (b18 + b17 + b14 + b13 + b12 - 2b11 - 3b10 - 3b9 - b6 - b1) q^26 + b10 * q^27 + (-b17 + b10 + b3) * q^28 + (-b18 - b16 + b15 - b13 - 2b12 + b11 + 3b9 + b8 - 2b7 + b5 + 2b1 - 1) * q^29 + (b19 - b15 + b7 - b3 + b2) * q^30 + (-b19 + b16 + b12 - b11 - b8 + b7 + b6 + b5 - b2 - b1 + 1) * q^31 + (-b18 - b17 - 3b16 - b13 + 3b11 - b3 - b2 + 4b1) q^32 + (b18 - b17 - b8 + b7) * q^33 + (b18 + b17 + 2b16 - b15 + b13 + b12 - b11 - 2b9 + b8 + b6 - b3 - 3b2 - 2) q^34 + (-b15 + 1) * q^35 + (-b16 + b11 + b1) * q^36 + (-b18 - b15 + b13 + b12 - 2b11 + b9 + b2 + 2) q^37 + (-b19 - b15 - b14 + b12 - b10 + b9 + b7 - 2) * q^38 + (b19 + b12 + b11 - b9 + b7 - b6 - b1 - 1) * q^39 + (-b19 + b16 + b15 + b13 - 2b11 + 2b10 + b8 - 4b7 - b6 + 4b3 - 4b1 - 2) q^40 + (b19 + b17 + b16 - b15 + b14 - b13 + b8 - 3b7 + b6 + 3b4 - 3b1) q^41 + b1 * q^42 + (b14 + b12 - b11 + b6 - b5 + b4 - b1) * q^43 + (-2b19 - 3b17 - 3b16 - b15 - 2b14 - b13 + 5b11 + 3b10 - 3b8 + b7 - 2b6 - b4 + 6b3 + b1 + 5) q^44 + (-b14 + b10) * q^45 + (-3b18 - b16 + b15 - b13 - b12 + 5b9 + b8 - b7 - b2 + b1) * q^46 + (-b19 + b18 - b17 - b15 + b14 - b12 + 3b10 - 3b9 - 3b7) q^47 + (b19 - 2b18 - b16 - 2b12 - b11 + 2b9 + b8 - b7 - b6 + b5 + 3b2 + b1 + 1) * q^48 + b11 * q^49 + (2b19 - 2b18 + 2b17 + b15 + 2b14 - 2b12 - 2b10 + 2b9 + b8 - 4b3 + 4b2 + 3) q^50 + (b16 + b12 - b8 + b7 + b6 + b4 - b3 - b1 + 1) * q^51 + (-b19 - 2b14 + 2b12 + b10 - b9 - 3b8 + 2b7 - b5 - b4 + 2b3 - 2b2 + 3) * q^52 + (b16 + b13 - 3b11 + 2b6 + b1) * q^53 + b2 * q^54 + (b19 + b15 + 2b14 - 2b12 - 2b10 + 2b9 + b8 - b3 + b2 - 4) * q^55 + (b19 - b18 - b16 - b12 + b11 + b9 + b8 - b6 + 2b2 - 1) q^56 + (-b17 - b15 + b14 - b13 - b7 + b4 - b1) * q^57 + (2b17 + 3b16 + 2b15 + 2b14 + 2b13 - 4b11 + 3b8 - 3b7 - 2b3 - 3b1 - 4) * q^58 + (-b14 - b13 - b12 - 2b11 + b6 + b5 - b4 + b1) q^59 + (b18 + b17 - b16 + b11 - 2b10 - 2b9 + b6 - 2b3 - 2b2 + 2b1) q^60 + (3b19 + 2b17 + b16 + b15 + b14 + b13 + b8 - b7 + 3b6 + b4 - 3b3 - b1) * q^61 + (-3b17 - b16 - 2b14 + 2b11 + 3b10 - b8 + 2b7 - b4 + 2b3 + 2b1 + 2) q^62 + b9 * q^63 + (-b19 + b14 - b12 + 2b8 - 7b7 + b5 + b4 + 2b3 - 2b2 - 4) * q^64 + (2b19 - 3b18 - b16 + 2b15 - 2b13 + 2b11 - 2b9 + b8 + b7 - 2b6 + 2b5 + 3b2 - b1 - 2) q^65 + (-b18 - b17 - 3b16 - b14 - b13 - b12 + 3b11 - b10 - b9 - 2b6 + b5 - b4 + 2b3 + 2b2 + 3b1) * q^66 + (b19 + b18 - b17 + b10 - b9 - b8 - 2b7 + b5 + b4 + b3 - b2 + 3) q^67 + (-b19 + 2b18 - 2b17 - 2b14 + b12 + 2b11 + 5b10 - 3b9 - 2b8 + 3b7 - b6 - b5 + 3b3 - b2 - 3b1 - 2) * q^68 + (b19 - b18 + b17 + 2b15 - b7 - b3 + b2 + 1) q^69 + (-b13 + b6 - b3 - b2 + b1) * q^70 + (b18 + 2b15 - 2b13 - b9 - b5) * q^71 + (-b18 + b17 + b15 - b10 + b9 + b8 - 2b7 + b5 + b4 - 1) q^72 + (-2b18 + 2b15 - 2b13 + 2b12 + 4b11 + 2b7 + 2b5 + 2b2 - 2b1 - 4) q^73 + (2b19 + b17 + b16 - b15 - b13 - b11 - b10 + b8 + b7 + 2b6 + b4 - 6b3 + b1 - 1) q^74 + (b17 + b16 + b15 + b13 - b11 - b10 + b8 - b7 + b3 - b1 - 1) * q^75 + (b16 + b14 - 2b13 + b12 + 3b11 - b10 - b9 - b6 - b5 + b4 - 4b1) q^76 + (-b18 - b17 - b16 + b1) * q^77 + (-b16 - b15 - b14 - b13 + 3b11 + 2b10 - b8 + b4 + b3 + 3) * q^78 + (-b19 - b17 - b15 - 3b14 - b13 - b10 + 3b7 - b6 - 2b4 + b3 + 3b1) * q^79 + (b19 + 2b16 - 3b15 + 3b13 - 6b11 + 6b9 - 2b8 + 5b7 - b6 + 4b2 - 5b1 + 6) q^80 - q^81 + (-2b19 + 2b18 + 4b16 + b15 - b13 - b12 - 5b11 + 2b9 - 4b8 + 3b7 + 2b6 - b5 - b2 - 3b1 + 5) q^82 + (b18 + b17 + b14 + b12 + 2b11 + 2b6 - b5 + b4) * q^83 + (b8 - b7 - 1) * q^84 + (2b19 - 2b18 + b17 + b16 + b15 + 2b14 - b13 - b12 - b10 + b8 + b7 + b6 - b5 + b4 - b3 + 2b2 - 1) * q^85 + (b15 - b10 + b9 - b8 - b7 + b5 + b4 - b3 + b2 + 1) * q^86 + (b18 + b17 + b16 + b14 + 2b13 + b12 - 3b11 - b10 - b9 + b6 - 2b3 - 2b2) * q^87 + (-6b18 - 2b16 - 2b12 + b11 + 8b9 + 2b8 - 4b7 + 6b2 + 4b1 - 1) * q^88 + (2b19 - b18 + b17 - b14 + b12 - b10 + b9 - 2b8 + 2b7 - 4) q^89 + (-b12 + b7 - b5 + b2 - b1) * q^90 + (b19 - b14 - b11 + b10 + b7 + b6 + b1 - 1) * q^91 + (2b19 + b17 + 2b16 + 2b15 + 2b13 - 4b11 + 5b10 + 2b8 + 2b6 - b4 - 6b3 - 4) q^92 + (-b18 - b17 - b13 + b10 + b9 + b6 + b5 - b4 + b3 + b2 + b1) * q^93 + (b16 - 7b11 - 3b10 - 3b9 - b6 + b5 - b4 + 4b3 + 4b2 - 2b1) * q^94 + (2b16 + b15 + 4b14 + b13 + b11 - 5b10 + 2b8 - b7 + b4 + b3 - b1 + 1) * q^95 + (3b17 + b16 + b14 - 3b10 + b8 + b7 - 4b3 + b1) q^96 + (-b19 + 2b18 + b16 - b15 + b13 - 4b11 - 4b9 - b8 + b6 - 2b2 + 4) * q^97 - b7 * q^98 + (-b18 - b16 + b8 + b2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$20 q - 24 q^{4} + 8 q^{5} - 4 q^{6} + 16 q^{10} + 4 q^{11} - 12 q^{13} + 4 q^{14} + 40 q^{16} + 4 q^{17} + 8 q^{18} - 52 q^{20} + 20 q^{21} - 24 q^{22} - 4 q^{23} + 4 q^{24} - 8 q^{29} + 8 q^{31} - 4 q^{33}+ \cdots + 4 q^{99}+O(q^{100})$ 20 * q - 24 * q^4 + 8 * q^5 - 4 * q^6 + 16 * q^10 + 4 * q^11 - 12 * q^13 + 4 * q^14 + 40 * q^16 + 4 * q^17 + 8 * q^18 - 52 * q^20 + 20 * q^21 - 24 * q^22 - 4 * q^23 + 4 * q^24 - 8 * q^29 + 8 * q^31 - 4 * q^33 - 44 * q^34 + 12 * q^35 + 24 * q^37 - 64 * q^38 - 12 * q^39 - 52 * q^40 - 20 * q^41 + 72 * q^44 - 8 * q^45 + 28 * q^46 - 32 * q^47 + 32 * q^48 + 104 * q^50 + 8 * q^51 + 48 * q^52 - 4 * q^54 - 36 * q^55 - 4 * q^56 - 16 * q^57 - 60 * q^58 + 28 * q^61 + 36 * q^62 - 112 * q^64 - 4 * q^65 + 40 * q^67 - 52 * q^68 + 36 * q^69 + 16 * q^71 - 24 * q^72 - 72 * q^73 - 24 * q^74 - 8 * q^75 + 40 * q^78 - 8 * q^79 + 120 * q^80 - 20 * q^81 + 108 * q^82 - 24 * q^84 + 40 * q^85 - 28 * q^88 - 64 * q^89 + 16 * q^90 - 12 * q^91 - 56 * q^92 + 60 * q^95 + 16 * q^96 + 60 * q^97 - 8 * q^98 + 4 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{20} + 32 x^{18} + 426 x^{16} + 3072 x^{14} + 13121 x^{12} + 34148 x^{10} + 53608 x^{8} + 48276 x^{6} + \cdots + 4$ x^20 + 32*x^18 + 426*x^16 + 3072*x^14 + 13121*x^12 + 34148*x^10 + 53608*x^8 + 48276*x^6 + 22224*x^4 + 3920*x^2 + 4 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( 17 \nu^{19} - 406 \nu^{18} - 85 \nu^{17} - 13920 \nu^{16} - 11305 \nu^{15} - 197026 \nu^{14} + \cdots - 7424 ) / 173536$ (17v^19 - 406v^18 - 85v^17 - 13920v^16 - 11305v^15 - 197026v^14 - 169405v^13 - 1483756v^12 - 1155388v^11 - 6392876v^10 - 4186454v^9 - 15808248v^8 - 8127156v^7 - 21418240v^6 - 7693996v^5 - 14177288v^4 - 2688720v^3 - 3417592v^2 - 125460*v - 7424) / 173536
$\beta_{3}$	$=$	$( 17 \nu^{19} + 406 \nu^{18} - 85 \nu^{17} + 13920 \nu^{16} - 11305 \nu^{15} + 197026 \nu^{14} + \cdots + 7424 ) / 173536$ (17v^19 + 406v^18 - 85v^17 + 13920v^16 - 11305v^15 + 197026v^14 - 169405v^13 + 1483756v^12 - 1155388v^11 + 6392876v^10 - 4186454v^9 + 15808248v^8 - 8127156v^7 + 21418240v^6 - 7693996v^5 + 14177288v^4 - 2688720v^3 + 3417592v^2 - 125460*v + 7424) / 173536
$\beta_{4}$	$=$	$( - 657 \nu^{19} + 93 \nu^{18} - 22873 \nu^{17} + 811 \nu^{16} - 335075 \nu^{15} - 21013 \nu^{14} + \cdots + 183892 ) / 173536$ (-657v^19 + 93v^18 - 22873v^17 + 811v^16 - 335075v^15 - 21013v^14 - 2688045v^13 - 387635v^12 - 12881216v^11 - 2547520v^10 - 37768438v^9 - 7984012v^8 - 66745560v^7 - 11785792v^6 - 67236060v^5 - 6339716v^4 - 33904536v^3 + 213940v^2 - 5767660*v + 183892) / 173536
$\beta_{5}$	$=$	$( 657 \nu^{19} + 93 \nu^{18} + 22873 \nu^{17} + 811 \nu^{16} + 335075 \nu^{15} - 21013 \nu^{14} + \cdots + 183892 ) / 173536$ (657v^19 + 93v^18 + 22873v^17 + 811v^16 + 335075v^15 - 21013v^14 + 2688045v^13 - 387635v^12 + 12881216v^11 - 2547520v^10 + 37768438v^9 - 7984012v^8 + 66745560v^7 - 11785792v^6 + 67236060v^5 - 6339716v^4 + 33904536v^3 + 213940v^2 + 5767660*v + 183892) / 173536
$\beta_{6}$	$=$	$( 195 \nu^{19} + 5724 \nu^{17} + 68424 \nu^{15} + 434332 \nu^{13} + 1631879 \nu^{11} + \cdots + 2034172 \nu ) / 43384$ (195v^19 + 5724v^17 + 68424v^15 + 434332v^13 + 1631879v^11 + 3979738v^9 + 7011006v^7 + 9112434v^5 + 7096832v^3 + 2034172v) / 43384
$\beta_{7}$	$=$	$( - 69 \nu^{18} - 2033 \nu^{16} - 24295 \nu^{14} - 151965 \nu^{12} - 537850 \nu^{10} - 1090814 \nu^{8} + \cdots - 368 ) / 7888$ (-69v^18 - 2033v^16 - 24295v^14 - 151965v^12 - 537850v^10 - 1090814v^8 - 1213762v^6 - 645676v^4 - 112992*v^2 - 368) / 7888
$\beta_{8}$	$=$	$( - 69 \nu^{18} - 2033 \nu^{16} - 24295 \nu^{14} - 151965 \nu^{12} - 537850 \nu^{10} - 1090814 \nu^{8} + \cdots + 23296 ) / 7888$ (-69v^18 - 2033v^16 - 24295v^14 - 151965v^12 - 537850v^10 - 1090814v^8 - 1213762v^6 - 645676v^4 - 105104*v^2 + 23296) / 7888
$\beta_{9}$	$=$	$( 1856 \nu^{19} - 17 \nu^{18} + 58986 \nu^{17} + 85 \nu^{16} + 776736 \nu^{15} + 11305 \nu^{14} + \cdots + 125460 ) / 173536$ (1856v^19 - 17v^18 + 58986v^17 + 85v^16 + 776736v^15 + 11305v^14 + 5504606v^13 + 169405v^12 + 22868820v^11 + 1155388v^10 + 56985812v^9 + 4186454v^8 + 83688200v^7 + 8127156v^6 + 68182016v^5 + 7693996v^4 + 27070456v^3 + 2688720v^2 + 3857928*v + 125460) / 173536
$\beta_{10}$	$=$	$( 1856 \nu^{19} + 17 \nu^{18} + 58986 \nu^{17} - 85 \nu^{16} + 776736 \nu^{15} - 11305 \nu^{14} + \cdots - 125460 ) / 173536$ (1856v^19 + 17v^18 + 58986v^17 - 85v^16 + 776736v^15 - 11305v^14 + 5504606v^13 - 169405v^12 + 22868820v^11 - 1155388v^10 + 56985812v^9 - 4186454v^8 + 83688200v^7 - 8127156v^6 + 68182016v^5 - 7693996v^4 + 27070456v^3 - 2688720v^2 + 3857928*v - 125460) / 173536
$\beta_{11}$	$=$	$( - 92 \nu^{19} - 2875 \nu^{17} - 37159 \nu^{15} - 258329 \nu^{13} - 1055167 \nu^{11} + \cdots - 247648 \nu ) / 7888$ (-92v^19 - 2875v^17 - 37159v^15 - 258329v^13 - 1055167v^11 - 2603766v^9 - 3841122v^7 - 3227630v^5 - 1398932v^3 - 247648v) / 7888
$\beta_{12}$	$=$	$( - 2100 \nu^{19} + 1443 \nu^{18} - 64146 \nu^{17} + 41273 \nu^{16} - 807790 \nu^{15} + \cdots - 183588 ) / 173536$ (-2100v^19 + 1443v^18 - 64146v^17 + 41273v^16 - 807790v^15 + 472715v^14 - 5465008v^13 + 2776963v^12 - 21821542v^11 + 8940326v^10 - 53485294v^9 + 15716856v^8 - 80846048v^7 + 14100488v^6 - 72362140v^5 + 5126080v^4 - 33679164v^3 - 225372v^2 - 5757608*v - 183588) / 173536
$\beta_{13}$	$=$	$( - 1401 \nu^{19} - 40207 \nu^{17} - 459813 \nu^{15} - 2656383 \nu^{13} - 8010836 \nu^{11} + \cdots + 4062736 \nu ) / 86768$ (-1401v^19 - 40207v^17 - 459813v^15 - 2656383v^13 - 8010836v^11 - 10924586v^9 + 67858v^7 + 16648736v^5 + 15490296v^3 + 4062736v) / 86768
$\beta_{14}$	$=$	$( - 2100 \nu^{19} - 1443 \nu^{18} - 64146 \nu^{17} - 41273 \nu^{16} - 807790 \nu^{15} + \cdots + 183588 ) / 173536$ (-2100v^19 - 1443v^18 - 64146v^17 - 41273v^16 - 807790v^15 - 472715v^14 - 5465008v^13 - 2776963v^12 - 21821542v^11 - 8940326v^10 - 53485294v^9 - 15716856v^8 - 80846048v^7 - 14100488v^6 - 72362140v^5 - 5126080v^4 - 33679164v^3 + 225372v^2 - 5757608*v + 183588) / 173536
$\beta_{15}$	$=$	$( - 887 \nu^{18} - 25870 \nu^{16} - 303983 \nu^{14} - 1847226 \nu^{12} - 6214991 \nu^{10} + \cdots - 65534 ) / 43384$ (-887v^18 - 25870v^16 - 303983v^14 - 1847226v^12 - 6214991v^10 - 11535258v^8 - 11040655v^6 - 4635064v^4 - 696044*v^2 - 65534) / 43384
$\beta_{16}$	$=$	$( - 207 \nu^{19} - 6592 \nu^{17} - 87182 \nu^{15} - 623022 \nu^{13} - 2627651 \nu^{11} + \cdots - 734688 \nu ) / 7888$ (-207v^19 - 6592v^17 - 87182v^15 - 623022v^13 - 2627651v^11 - 6720484v^9 - 10309604v^7 - 9037214v^5 - 4083804v^3 - 734688v) / 7888
$\beta_{17}$	$=$	$( 5179 \nu^{19} - 172 \nu^{18} + 162953 \nu^{17} - 4882 \nu^{16} + 2121877 \nu^{15} - 58518 \nu^{14} + \cdots - 369024 ) / 173536$ (5179v^19 - 172v^18 + 162953v^17 - 4882v^16 + 2121877v^15 - 58518v^14 + 14860657v^13 - 402904v^12 + 61058196v^11 - 1840258v^10 + 150962734v^9 - 5789606v^8 + 221519204v^7 - 11477916v^6 + 182674764v^5 - 11971228v^4 + 75105056v^3 - 4840668v^2 + 11440900*v - 369024) / 173536
$\beta_{18}$	$=$	$( 5179 \nu^{19} + 172 \nu^{18} + 162953 \nu^{17} + 4882 \nu^{16} + 2121877 \nu^{15} + 58518 \nu^{14} + \cdots + 369024 ) / 173536$ (5179v^19 + 172v^18 + 162953v^17 + 4882v^16 + 2121877v^15 + 58518v^14 + 14860657v^13 + 402904v^12 + 61058196v^11 + 1840258v^10 + 150962734v^9 + 5789606v^8 + 221519204v^7 + 11477916v^6 + 182674764v^5 + 11971228v^4 + 75105056v^3 + 4840668v^2 + 11440900*v + 369024) / 173536
$\beta_{19}$	$=$	$( 3257 \nu^{18} + 99193 \nu^{16} + 1236549 \nu^{14} + 8160989 \nu^{12} + 30879656 \nu^{10} + \cdots - 118040 ) / 86768$ (3257v^18 + 99193v^16 + 1236549v^14 + 8160989v^12 + 30879656v^10 + 67910398v^8 + 83620342v^6 + 51533280v^4 + 11580160*v^2 - 118040) / 86768

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{8} - \beta_{7} - 3$ b8 - b7 - 3
$\nu^{3}$	$=$	$\beta_{18} + \beta_{17} + \beta_{16} + \beta_{13} - \beta_{11} - \beta_{10} - \beta_{9} - \beta_{5} + \beta_{4} - 6\beta_1$ b18 + b17 + b16 + b13 - b11 - b10 - b9 - b5 + b4 - 6*b1
$\nu^{4}$	$=$	$- \beta_{19} + \beta_{18} - \beta_{17} - 2 \beta_{15} - \beta_{10} + \beta_{9} - 8 \beta_{8} + \cdots + 16$ -b19 + b18 - b17 - 2b15 - b10 + b9 - 8b8 + 9*b7 - b5 - b4 + b3 - b2 + 16
$\nu^{5}$	$=$	$- 9 \beta_{18} - 9 \beta_{17} - 11 \beta_{16} - 9 \beta_{13} + 11 \beta_{11} + 8 \beta_{10} + \cdots + 40 \beta_1$ -9b18 - 9b17 - 11b16 - 9b13 + 11b11 + 8b10 + 8b9 + 8b5 - 8b4 - b3 - b2 + 40b1
$\nu^{6}$	$=$	$9 \beta_{19} - 10 \beta_{18} + 10 \beta_{17} + 20 \beta_{15} + \beta_{14} - \beta_{12} + 10 \beta_{10} + \cdots - 100$ 9b19 - 10b18 + 10b17 + 20b15 + b14 - b12 + 10b10 - 10b9 + 58b8 - 73b7 + 11b5 + 11b4 - 8b3 + 8b2 - 100
$\nu^{7}$	$=$	$66 \beta_{18} + 66 \beta_{17} + 93 \beta_{16} + 69 \beta_{13} - 101 \beta_{11} - 53 \beta_{10} + \cdots - 278 \beta_1$ 66b18 + 66b17 + 93b16 + 69b13 - 101b11 - 53b10 - 53b9 - 57b5 + 57b4 + 13b3 + 13b2 - 278b1
$\nu^{8}$	$=$	$- 63 \beta_{19} + 80 \beta_{18} - 80 \beta_{17} - 162 \beta_{15} - 9 \beta_{14} + 9 \beta_{12} + \cdots + 666$ -63b19 + 80b18 - 80b17 - 162b15 - 9b14 + 9b12 - 78b10 + 78b9 - 410b8 + 565b7 - 93b5 - 93b4 + 46b3 - 46b2 + 666
$\nu^{9}$	$=$	$- 456 \beta_{18} - 456 \beta_{17} - 725 \beta_{16} + 4 \beta_{14} - 509 \beta_{13} + 4 \beta_{12} + \cdots + 1970 \beta_1$ -456b18 - 456b17 - 725b16 + 4b14 - 509b13 + 4b12 + 849b11 + 325b10 + 325b9 + 2b6 + 401b5 - 401b4 - 127b3 - 127b2 + 1970*b1
$\nu^{10}$	$=$	$403 \beta_{19} - 598 \beta_{18} + 598 \beta_{17} + 1236 \beta_{15} + 51 \beta_{14} - 51 \beta_{12} + \cdots - 4570$ 403b19 - 598b18 + 598b17 + 1236b15 + 51b14 - 51b12 + 566b10 - 566b9 + 2882b8 - 4279b7 + 729b5 + 729b4 - 200b3 + 200b2 - 4570
$\nu^{11}$	$=$	$3082 \beta_{18} + 3082 \beta_{17} + 5475 \beta_{16} - 80 \beta_{14} + 3715 \beta_{13} + \cdots - 14108 \beta_1$ 3082b18 + 3082b17 + 5475b16 - 80b14 + 3715b13 - 80b12 - 6811b11 - 1883b10 - 1883b9 - 40b6 - 2831b5 + 2831b4 + 1135b3 + 1135b2 - 14108*b1
$\nu^{12}$	$=$	$- 2449 \beta_{19} + 4340 \beta_{18} - 4340 \beta_{17} - 9230 \beta_{15} - 171 \beta_{14} + 171 \beta_{12} + \cdots + 31876$ -2449b19 + 4340b18 - 4340b17 - 9230b15 - 171b14 + 171b12 - 4054b10 + 4054b9 - 20272b8 + 32069b7 - 5555b5 - 5555b4 + 366b3 - 366b2 + 31876
$\nu^{13}$	$=$	$- 20638 \beta_{18} - 20638 \beta_{17} - 40749 \beta_{16} + 1044 \beta_{14} - 27053 \beta_{13} + \cdots + 101698 \beta_1$ -20638b18 - 20638b17 - 40749b16 + 1044b14 - 27053b13 + 1044b12 + 53233b11 + 10241b10 + 10241b9 + 550b6 + 20101b5 - 20101b4 - 9793b3 - 9793b2 + 101698*b1
$\nu^{14}$	$=$	$14223 \beta_{19} - 30956 \beta_{18} + 30956 \beta_{17} + 68352 \beta_{15} - 507 \beta_{14} + \cdots - 224670$ 14223b19 - 30956b18 + 30956b17 + 68352b15 - 507b14 + 507b12 + 29356b10 - 29356b9 + 142974b8 - 239067b7 + 41793b5 + 41793b4 + 5430b3 - 5430b2 - 224670
$\nu^{15}$	$=$	$137544 \beta_{18} + 137544 \beta_{17} + 300979 \beta_{16} - 11344 \beta_{14} + 197103 \beta_{13} + \cdots - 736476 \beta_1$ 137544b18 + 137544b17 + 300979b16 - 11344b14 + 197103b13 - 11344b12 - 409579b11 - 50549b10 - 50549b9 - 6440b6 - 143481b5 + 143481b4 + 83019b3 + 83019b2 - 736476*b1
$\nu^{16}$	$=$	$- 77985 \beta_{19} + 217960 \beta_{18} - 217960 \beta_{17} - 504522 \beta_{15} + 17281 \beta_{14} + \cdots + 1595596$ -77985b19 + 217960b18 - 217960b17 - 504522b15 + 17281b14 - 17281b12 - 216726b10 + 216726b9 - 1011564b8 + 1777141b7 - 312323b5 - 312323b4 - 97966b3 + 97966b2 + 1595596
$\nu^{17}$	$=$	$- 913598 \beta_{18} - 913598 \beta_{17} - 2213061 \beta_{16} + 111644 \beta_{14} - 1438101 \beta_{13} + \cdots + 5352442 \beta_1$ -913598b18 - 913598b17 - 2213061b16 + 111644b14 - 1438101b13 + 111644b12 + 3119569b11 + 203761b10 + 203761b9 + 68602b6 + 1028845b5 - 1028845b4 - 695617b3 - 695617b2 + 5352442*b1
$\nu^{18}$	$=$	$389095 \beta_{19} - 1517444 \beta_{18} + 1517444 \beta_{17} + 3719764 \beta_{15} - 226891 \beta_{14} + \cdots - 11400710$ 389095b19 - 1517444b18 + 1517444b17 + 3719764b15 - 226891b14 + 226891b12 + 1632384b10 - 1632384b9 + 7179638b8 - 13190023b7 + 2324705b5 + 2324705b4 + 1131606b3 - 1131606b2 - 11400710
$\nu^{19}$	$=$	$6048032 \beta_{18} + 6048032 \beta_{17} + 16224911 \beta_{16} - 1034152 \beta_{14} + \cdots - 39014696 \beta_1$ 6048032b18 + 6048032b17 + 16224911b16 - 1034152b14 + 10510307b13 - 1034152b12 - 23596271b11 - 360837b10 - 360837b9 - 684876b6 - 7406529b5 + 7406529b4 + 5773571b3 + 5773571b2 - 39014696*b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/357\mathbb{Z}\right)^\times$ .

$n$	$52$	$190$	$239$
$\chi(n)$	$1$	$-\beta_{11}$	$1$

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}

64.1

	−	2.75017i
	−	2.65813i
	−	1.71520i
	−	1.51109i
	−	1.23743i
		0.0320370i
		0.661473i
		0.901740i
		1.80873i
		2.46803i
	−	2.46803i
	−	1.80873i
	−	0.901740i
	−	0.661473i
	−	0.0320370i
		1.23743i
		1.51109i
		1.71520i
		2.65813i
		2.75017i

−

2.75017i

0.707107

0.707107i

−5.56342

2.93829

2.93829i

1.94466

−

1.94466i

0.707107

−

0.707107i

9.79999i

1.00000i

8.08080

−

8.08080i

64.2

−

2.65813i

−0.707107

−

0.707107i

−5.06565

0.521373

0.521373i

−1.87958

1.87958i

−0.707107

0.707107i

8.14889i

1.00000i

1.38588

−

1.38588i

64.3

−

1.71520i

−0.707107

−

0.707107i

−0.941909

2.50881

2.50881i

−1.21283

1.21283i

−0.707107

0.707107i

−

1.81484i

1.00000i

4.30311

−

4.30311i

64.4

−

1.51109i

−0.707107

−

0.707107i

−0.283389

−1.65084

−

1.65084i

−1.06850

1.06850i

−0.707107

0.707107i

−

2.59395i

1.00000i

−2.49457

2.49457i

64.5

−

1.23743i

0.707107

0.707107i

0.468767

−1.28416

−

1.28416i

0.874995

−

0.874995i

0.707107

−

0.707107i

−

3.05493i

1.00000i

−1.58906

1.58906i

64.6

0.0320370i

0.707107

0.707107i

1.99897

1.76367

1.76367i

−0.0226536

0.0226536i

0.707107

−

0.707107i

0.128115i

1.00000i

−0.0565026

0.0565026i

64.7

0.661473i

−0.707107

−

0.707107i

1.56245

−1.66372

−

1.66372i

0.467732

−

0.467732i

−0.707107

0.707107i

2.35647i

1.00000i

1.10050

−

1.10050i

64.8

0.901740i

0.707107

0.707107i

1.18686

−0.446227

−

0.446227i

−0.637627

0.637627i

0.707107

−

0.707107i

2.87372i

1.00000i

0.402381

−

0.402381i

64.9

1.80873i

−0.707107

−

0.707107i

−1.27151

0.163056

0.163056i

1.27897

−

1.27897i

−0.707107

0.707107i

1.31765i

1.00000i

−0.294925

0.294925i

64.10

2.46803i

0.707107

0.707107i

−4.09119

1.14975

1.14975i

−1.74516

1.74516i

0.707107

−

0.707107i

−

5.16112i

1.00000i

−2.83762

2.83762i

106.1

−

2.46803i

0.707107

−

0.707107i

−4.09119

1.14975

−

1.14975i

−1.74516

−

1.74516i

0.707107

0.707107i

5.16112i

−

1.00000i

−2.83762

−

2.83762i

106.2

−

1.80873i

−0.707107

0.707107i

−1.27151

0.163056

−

0.163056i

1.27897

1.27897i

−0.707107

−

0.707107i

−

1.31765i

−

1.00000i

−0.294925

−

0.294925i

106.3

−

0.901740i

0.707107

−

0.707107i

1.18686

−0.446227

0.446227i

−0.637627

−

0.637627i

0.707107

0.707107i

−

2.87372i

−

1.00000i

0.402381

0.402381i

106.4

−

0.661473i

−0.707107

0.707107i

1.56245

−1.66372

1.66372i

0.467732

0.467732i

−0.707107

−

0.707107i

−

2.35647i

−

1.00000i

1.10050

1.10050i

106.5

−

0.0320370i

0.707107

−

0.707107i

1.99897

1.76367

−

1.76367i

−0.0226536

−

0.0226536i

0.707107

0.707107i

−

0.128115i

−

1.00000i

−0.0565026

−

0.0565026i

106.6

1.23743i

0.707107

−

0.707107i

0.468767

−1.28416

1.28416i

0.874995

0.874995i

0.707107

0.707107i

3.05493i

−

1.00000i

−1.58906

−

1.58906i

106.7

1.51109i

−0.707107

0.707107i

−0.283389

−1.65084

1.65084i

−1.06850

−

1.06850i

−0.707107

−

0.707107i

2.59395i

−

1.00000i

−2.49457

−

2.49457i

106.8

1.71520i

−0.707107

0.707107i

−0.941909

2.50881

−

2.50881i

−1.21283

−

1.21283i

−0.707107

−

0.707107i

1.81484i

−

1.00000i

4.30311

4.30311i

106.9

2.65813i

−0.707107

0.707107i

−5.06565

0.521373

−

0.521373i

−1.87958

−

1.87958i

−0.707107

−

0.707107i

−

8.14889i

−

1.00000i

1.38588

1.38588i

106.10

2.75017i

0.707107

−

0.707107i

−5.56342

2.93829

−

2.93829i

1.94466

1.94466i

0.707107

0.707107i

−

9.79999i

−

1.00000i

8.08080

8.08080i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
17.c	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	357.2.k.b	✓	20
3.b	odd	2	1		1071.2.n.b		20
17.c	even	4	1	inner	357.2.k.b	✓	20
17.d	even	8	1		6069.2.a.bd		10
17.d	even	8	1		6069.2.a.be		10
51.f	odd	4	1		1071.2.n.b		20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
357.2.k.b	✓	20	1.a	even	1	1	trivial
357.2.k.b	✓	20	17.c	even	4	1	inner
1071.2.n.b		20	3.b	odd	2	1
1071.2.n.b		20	51.f	odd	4	1
6069.2.a.bd		10	17.d	even	8	1
6069.2.a.be		10	17.d	even	8	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{20} + 32 T_{2}^{18} + 426 T_{2}^{16} + 3072 T_{2}^{14} + 13121 T_{2}^{12} + 34148 T_{2}^{10} + \cdots + 4$ T2^20 + 32*T2^18 + 426*T2^16 + 3072*T2^14 + 13121*T2^12 + 34148*T2^10 + 53608*T2^8 + 48276*T2^6 + 22224*T2^4 + 3920*T2^2 + 4 acting on $S_{2}^{\mathrm{new}}(357, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{20} + 32 T^{18} + \cdots + 4$ T^20 + 32T^18 + 426T^16 + 3072T^14 + 13121T^12 + 34148T^10 + 53608T^8 + 48276T^6 + 22224T^4 + 3920*T^2 + 4
$3$	$(T^{4} + 1)^{5}$ (T^4 + 1)^5
$5$	$T^{20} - 8 T^{19} + \cdots + 4096$ T^20 - 8T^19 + 32T^18 - 36T^17 + 63T^16 - 496T^15 + 2600T^14 - 1952T^13 + 1283T^12 - 9504T^11 + 64456T^10 - 28244T^9 + 12177T^8 - 68912T^7 + 401024T^6 - 180608T^5 + 38272T^4 + 7168T^3 + 73728T^2 - 24576*T + 4096
$7$	$(T^{4} + 1)^{5}$ (T^4 + 1)^5
$11$	$T^{20} + \cdots + 101687056$ T^20 - 4T^19 + 8T^18 - 68T^17 + 1763T^16 - 8224T^15 + 21104T^14 - 76728T^13 + 782675T^12 - 3815204T^11 + 10827656T^10 - 22931396T^9 + 80767305T^8 - 325866280T^7 + 929761568T^6 - 1638157968T^5 + 1744536568T^4 - 887250400T^3 + 64161792T^2 + 114231552*T + 101687056
$13$	$(T^{10} + 6 T^{9} + \cdots - 91664)^{2}$ (T^10 + 6T^9 - 71T^8 - 344T^7 + 2005T^6 + 6506T^5 - 25673T^4 - 44584T^3 + 121544T^2 + 92032*T - 91664)^2
$17$	$T^{20} + \cdots + 2015993900449$ T^20 - 4T^19 - 4T^18 + 244T^17 - 875T^16 + 712T^15 + 28256T^14 - 119112T^13 + 238426T^12 + 1914448T^11 - 9832248T^10 + 32545616T^9 + 68905114T^8 - 585197256T^7 + 2359969376T^6 + 1010938184T^5 - 21120372875T^4 + 100122636212T^3 - 27903029764T^2 - 474351505988*T + 2015993900449
$19$	$T^{20} + \cdots + 527712784$ T^20 + 206T^18 + 16871T^16 + 711840T^14 + 16798779T^12 + 225951210T^10 + 1698679657T^8 + 6706719912T^6 + 12298224904T^4 + 7569606528*T^2 + 527712784
$23$	$T^{20} + 4 T^{19} + \cdots + 234256$ T^20 + 4T^19 + 8T^18 - 132T^17 + 6315T^16 + 21536T^15 + 44336T^14 - 719384T^13 + 10408075T^12 + 21829844T^11 + 50931272T^10 - 774226596T^9 + 3729418825T^8 - 4000306472T^7 + 2162128416T^6 - 152151184T^5 + 702660984T^4 - 725711712T^3 + 376915968T^2 - 13288704*T + 234256
$29$	$T^{20} + 8 T^{19} + \cdots + 32444416$ T^20 + 8T^19 + 32T^18 + 72T^17 + 9840T^16 + 85584T^15 + 372384T^14 + 88864T^13 + 13544288T^12 + 119713440T^11 + 528740480T^10 - 16624064T^9 + 5092789776T^8 + 45153656320T^7 + 200084033536T^6 - 25740777472T^5 + 6976557568T^4 + 29911187456T^3 + 56147345408T^2 + 1908752384*T + 32444416
$31$	$T^{20} - 8 T^{19} + \cdots + 5456896$ T^20 - 8T^19 + 32T^18 - 116T^17 + 7900T^16 - 73352T^15 + 340744T^14 - 88040T^13 - 967660T^12 + 1249704T^11 + 29560480T^10 + 50533520T^9 + 43331396T^8 + 87395008T^7 + 456126976T^6 + 976512256T^5 + 1173515392T^4 + 793183232T^3 + 314703872T^2 + 58605568*T + 5456896
$37$	$T^{20} + \cdots + 180162895936$ T^20 - 24T^19 + 288T^18 - 1876T^17 + 15856T^16 - 220168T^15 + 2477192T^14 - 16080272T^13 + 74194832T^12 - 407067096T^11 + 3526706720T^10 - 22145354736T^9 + 85703335940T^8 - 173398196704T^7 + 220097619968T^6 - 224272440512T^5 + 386692938784T^4 - 686752015872T^3 + 823136878592T^2 - 544607608832*T + 180162895936
$41$	$T^{20} + \cdots + 69\!\cdots\!64$ T^20 + 20T^19 + 200T^18 + 1032T^17 + 30491T^16 + 561192T^15 + 5658152T^14 + 28520904T^13 + 244321351T^12 + 3481631036T^11 + 35147441248T^10 + 176340136024T^9 + 826646358625T^8 + 7760577341584T^7 + 78361790430848T^6 + 392510313875008T^5 + 1255118384322848T^4 + 5494868623494400T^3 + 55485256030963712T^2 + 277730711359507456*T + 695088691572523264
$43$	$T^{20} + 182 T^{18} + \cdots + 200704$ T^20 + 182T^18 + 10879T^16 + 303700T^14 + 4408303T^12 + 34384310T^10 + 147382833T^8 + 341474656T^6 + 388022656T^4 + 157513728*T^2 + 200704
$47$	$(T^{10} + 16 T^{9} + \cdots - 9831296)^{2}$ (T^10 + 16T^9 - 124T^8 - 3316T^7 - 8046T^6 + 141884T^5 + 761074T^4 - 904960T^3 - 12319648T^2 - 22000384*T - 9831296)^2
$53$	$T^{20} + \cdots + 533764670464$ T^20 + 312T^18 + 38784T^16 + 2480964T^14 + 89044208T^12 + 1858893728T^10 + 22594524932T^8 + 154578383552T^6 + 553067028864T^4 + 925655010304*T^2 + 533764670464
$59$	$T^{20} + \cdots + 1849600000000$ T^20 + 344T^18 + 46924T^16 + 3325820T^14 + 134070740T^12 + 3155325336T^10 + 42924925764T^8 + 325717846400T^6 + 1295926720000T^4 + 2503552000000*T^2 + 1849600000000
$61$	$T^{20} + \cdots + 67\!\cdots\!44$ T^20 - 28T^19 + 392T^18 - 2076T^17 + 23600T^16 - 546704T^15 + 8211400T^14 - 40738592T^13 + 185123148T^12 - 2955215912T^11 + 50404680320T^10 - 223406027776T^9 + 644842385732T^8 - 6139509904832T^7 + 118693101777408T^6 - 457186098704640T^5 + 1040583488725120T^4 - 5223734686139392T^3 + 84577303811268608T^2 - 337979990551908352*T + 675302172485714944
$67$	$(T^{10} - 20 T^{9} + \cdots + 4614400)^{2}$ (T^10 - 20T^9 - 40T^8 + 2812T^7 - 10596T^6 - 93240T^5 + 630308T^4 - 194368T^3 - 4986816T^2 + 6507520*T + 4614400)^2
$71$	$T^{20} + \cdots + 10\!\cdots\!16$ T^20 - 16T^19 + 128T^18 + 200T^17 + 17384T^16 - 267648T^15 + 2077216T^14 + 2254368T^13 + 78907632T^12 - 1207873184T^11 + 9384070656T^10 + 7510235776T^9 + 99289351952T^8 - 1652531567744T^7 + 14018564683776T^6 + 11368629288192T^5 - 6396880452992T^4 - 169925542690816T^3 + 4098800848504832T^2 + 9225173857221632*T + 10381552537129216
$73$	$T^{20} + \cdots + 18\!\cdots\!36$ T^20 + 72T^19 + 2592T^18 + 58528T^17 + 939344T^16 + 12256768T^15 + 160471040T^14 + 2289432576T^13 + 30505413120T^12 + 322408050688T^11 + 2515930955776T^10 + 13985111261184T^9 + 56373637324800T^8 + 205433168855040T^7 + 1062734027816960T^6 + 5702446736736256T^5 + 19657367891542016T^4 + 26824319657574400T^3 + 1590703803072512T^2 - 23964751870558208*T + 180520512715227136
$79$	$T^{20} + \cdots + 114770254495744$ T^20 + 8T^19 + 32T^18 - 2580T^17 + 53576T^16 + 57480T^15 + 2073608T^14 - 90703168T^13 + 1229274544T^12 - 5304298488T^11 + 42799353120T^10 - 835040612848T^9 + 9763956827716T^8 - 65614693459072T^7 + 283630209124352T^6 - 806684758151168T^5 + 1511992393424896T^4 - 1763292538601472T^3 + 1297854243012608T^2 - 545811435880448*T + 114770254495744
$83$	$T^{20} + \cdots + 39382954541056$ T^20 + 628T^18 + 163388T^16 + 22933056T^14 + 1892953392T^12 + 93684323264T^10 + 2699697032256T^8 + 41268609333248T^6 + 264200320606208T^4 + 253086345068544*T^2 + 39382954541056
$89$	$(T^{10} + 32 T^{9} + \cdots - 4681216)^{2}$ (T^10 + 32T^9 + 4T^8 - 8256T^7 - 53532T^6 + 524992T^5 + 4350368T^4 - 11243008T^3 - 78750848T^2 + 178915840*T - 4681216)^2
$97$	$T^{20} + \cdots + 96348160000$ T^20 - 60T^19 + 1800T^18 - 33096T^17 + 408780T^16 - 3481696T^15 + 20770368T^14 - 89110528T^13 + 336993776T^12 - 1583609024T^11 + 8222266496T^10 - 30401699200T^9 + 77048085056T^8 - 159528972288T^7 + 465287139328T^6 - 1472760885248T^5 + 3387629615104T^4 - 4614891192320T^3 + 3486489804800T^2 - 819654656000*T + 96348160000
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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 $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 64.10
Significant digits:
Format:

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	357.2.k.b

Level	$357$
Weight	$2$
Character orbit	357.k
Analytic conductor	$2.851$
Analytic rank	$0$
Dimension	$20$
Inner twists	$2$