LMFDB - Newform orbit 630.2.i.i

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [630,2,Mod(121,630)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(630, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("630.121");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$630 = 2 \cdot 3^{2} \cdot 5 \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	630.i (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:	$5.03057532734$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} - x^{15} + 2 x^{14} - 4 x^{13} + 5 x^{12} + 2 x^{11} - 35 x^{10} + 81 x^{9} - 66 x^{8} + \cdots + 6561$ x^16 - x^15 + 2x^14 - 4x^13 + 5x^12 + 2x^11 - 35x^10 + 81x^9 - 66x^8 + 243x^7 - 315x^6 + 54x^5 + 405x^4 - 972x^3 + 1458x^2 - 2187x + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

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

Basis of coefficient ring in terms of a root $\nu$ of $x^{16} - x^{15} + 2 x^{14} - 4 x^{13} + 5 x^{12} + 2 x^{11} - 35 x^{10} + 81 x^{9} - 66 x^{8} + \cdots + 6561$ x^16 - x^15 + 2*x^14 - 4*x^13 + 5*x^12 + 2*x^11 - 35*x^10 + 81*x^9 - 66*x^8 + 243*x^7 - 315*x^6 + 54*x^5 + 405*x^4 - 972*x^3 + 1458*x^2 - 2187*x + 6561 :

$\beta_{1}$	$=$	$( 6 \nu^{15} + 47 \nu^{14} - 161 \nu^{13} + 13 \nu^{12} - 287 \nu^{11} + 1486 \nu^{10} - 1109 \nu^{9} + \cdots - 188811 ) / 43740$ (6v^15 + 47v^14 - 161v^13 + 13v^12 - 287v^11 + 1486v^10 - 1109v^9 - 961v^8 - 408v^7 + 741v^6 + 3951v^5 - 36378v^4 + 38637v^3 + 7209v^2 + 96957*v - 188811) / 43740
$\beta_{2}$	$=$	$( - 193 \nu^{15} + 520 \nu^{14} - 2882 \nu^{13} + 5890 \nu^{12} - 14819 \nu^{11} + 17755 \nu^{10} + \cdots - 448335 ) / 787320$ (-193v^15 + 520v^14 - 2882v^13 + 5890v^12 - 14819v^11 + 17755v^10 - 22498v^9 + 10029v^8 + 46911v^7 - 149940v^6 + 260487v^5 - 621135v^4 + 664524v^3 - 767880v^2 + 412614*v - 448335) / 787320
$\beta_{3}$	$=$	$( 119 \nu^{15} + 1900 \nu^{14} + 586 \nu^{13} + 1870 \nu^{12} - 22943 \nu^{11} + 10315 \nu^{10} + \cdots + 3881925 ) / 787320$ (119v^15 + 1900v^14 + 586v^13 + 1870v^12 - 22943v^11 + 10315v^10 - 1846v^9 + 40233v^8 - 5433v^7 - 121320v^6 + 522819v^5 - 514215v^4 - 164592v^3 - 1598940v^2 + 2188458*v + 3881925) / 787320
$\beta_{4}$	$=$	$( 53 \nu^{15} - 173 \nu^{14} + 37 \nu^{13} - 317 \nu^{12} + 1474 \nu^{11} - 899 \nu^{10} + \cdots - 39366 ) / 131220$ (53v^15 - 173v^14 + 37v^13 - 317v^12 + 1474v^11 - 899v^10 - 1447v^9 - 12v^8 - 717v^7 + 5841v^6 - 36702v^5 + 36207v^4 + 13041v^3 + 88209v^2 - 175689*v - 39366) / 131220
$\beta_{5}$	$=$	$( - 92 \nu^{15} + 341 \nu^{14} - 208 \nu^{13} + 209 \nu^{12} - 736 \nu^{11} - 2242 \nu^{10} + \cdots + 428652 ) / 196830$ (-92v^15 + 341v^14 - 208v^13 + 209v^12 - 736v^11 - 2242v^10 + 4303v^9 - 8238v^8 + 16206v^7 - 19377v^6 + 23688v^5 + 32346v^4 - 67959v^3 - 50058v^2 - 226719*v + 428652) / 196830
$\beta_{6}$	$=$	$( 349 \nu^{15} - 532 \nu^{14} + 1106 \nu^{13} - 718 \nu^{12} + 767 \nu^{11} - 12511 \nu^{10} + \cdots + 3833811 ) / 787320$ (349v^15 - 532v^14 + 1106v^13 - 718v^12 + 767v^11 - 12511v^10 + 14734v^9 + 12471v^8 + 48093v^7 - 38736v^6 - 39411v^5 + 445203v^4 - 459432v^3 + 260496v^2 - 1296162*v + 3833811) / 787320
$\beta_{7}$	$=$	$( - \nu^{15} + \nu^{14} - 2 \nu^{13} + 4 \nu^{12} - 5 \nu^{11} - 2 \nu^{10} + 35 \nu^{9} - 81 \nu^{8} + \cdots + 2187 ) / 2187$ (-v^15 + v^14 - 2v^13 + 4v^12 - 5v^11 - 2v^10 + 35v^9 - 81v^8 + 66v^7 - 243v^6 + 315v^5 - 54v^4 - 405v^3 + 972v^2 - 1458*v + 2187) / 2187
$\beta_{8}$	$=$	$( - 247 \nu^{15} + 1105 \nu^{14} - 3143 \nu^{13} + 3145 \nu^{12} - 7466 \nu^{11} + 20005 \nu^{10} + \cdots - 2777490 ) / 393660$ (-247v^15 + 1105v^14 - 3143v^13 + 3145v^12 - 7466v^11 + 20005v^10 - 19357v^9 - 24v^8 + 2739v^7 - 27765v^6 + 160938v^5 - 558765v^4 + 827091v^3 - 633015v^2 + 1508301*v - 2777490) / 393660
$\beta_{9}$	$=$	$( 229 \nu^{15} - 385 \nu^{14} - 349 \nu^{13} - 1345 \nu^{12} + 32 \nu^{11} + 4205 \nu^{10} + \cdots - 174960 ) / 393660$ (229v^15 - 385v^14 - 349v^13 - 1345v^12 + 32v^11 + 4205v^10 - 3071v^9 + 14658v^8 - 17913v^7 + 6705v^6 - 66816v^5 - 80325v^4 + 132273v^3 + 239355v^2 + 748683*v - 174960) / 393660
$\beta_{10}$	$=$	$( - 727 \nu^{15} + 982 \nu^{14} - 4148 \nu^{13} + 9448 \nu^{12} - 3971 \nu^{11} - 899 \nu^{10} + \cdots + 234009 ) / 787320$ (-727v^15 + 982v^14 - 4148v^13 + 9448v^12 - 3971v^11 - 899v^10 + 3608v^9 - 57327v^8 + 121413v^7 - 249354v^6 + 257463v^5 - 23193v^4 + 580446v^3 - 92826v^2 - 1892484*v + 234009) / 787320
$\beta_{11}$	$=$	$( - 619 \nu^{15} - 1520 \nu^{14} + 514 \nu^{13} + 6550 \nu^{12} + 4903 \nu^{11} - 935 \nu^{10} + \cdots - 2219805 ) / 787320$ (-619v^15 - 1520v^14 + 514v^13 + 6550v^12 + 4903v^11 - 935v^10 - 33094v^9 + 11127v^8 - 26547v^7 - 103500v^6 - 1899v^5 + 397035v^4 + 1028052v^3 - 680400v^2 - 1853118*v - 2219805) / 787320
$\beta_{12}$	$=$	$( - 503 \nu^{15} + 905 \nu^{14} + 1193 \nu^{13} - 55 \nu^{12} + 4346 \nu^{11} - 24115 \nu^{10} + \cdots + 3171150 ) / 393660$ (-503v^15 + 905v^14 + 1193v^13 - 55v^12 + 4346v^11 - 24115v^10 + 41647v^9 - 55416v^8 + 63771v^7 - 44865v^6 - 19458v^5 + 556335v^4 - 1062801v^3 + 786105v^2 - 2186271*v + 3171150) / 393660
$\beta_{13}$	$=$	$( 499 \nu^{15} - 367 \nu^{14} + 1091 \nu^{13} - 3523 \nu^{12} + 5522 \nu^{11} - 7801 \nu^{10} + \cdots + 1570266 ) / 393660$ (499v^15 - 367v^14 + 1091v^13 - 3523v^12 + 5522v^11 - 7801v^10 + 15379v^9 + 636v^8 + 15513v^7 + 59139v^6 - 182646v^5 + 180333v^4 - 487377v^3 + 840051v^2 - 629127*v + 1570266) / 393660
$\beta_{14}$	$=$	$( 1291 \nu^{15} - 88 \nu^{14} - 286 \nu^{13} - 2902 \nu^{12} - 11947 \nu^{11} + 33671 \nu^{10} + \cdots - 640791 ) / 787320$ (1291v^15 - 88v^14 - 286v^13 - 2902v^12 - 11947v^11 + 33671v^10 - 37874v^9 + 106629v^8 - 84333v^7 + 119916v^6 + 163791v^5 - 967923v^4 + 577692v^3 - 1334556v^2 + 4547502*v - 640791) / 787320
$\beta_{15}$	$=$	$( 376 \nu^{15} - 883 \nu^{14} + 269 \nu^{13} - 1717 \nu^{12} + 3413 \nu^{11} + 3176 \nu^{10} + \cdots - 968841 ) / 196830$ (376v^15 - 883v^14 + 269v^13 - 1717v^12 + 3413v^11 + 3176v^10 - 11159v^9 + 22839v^8 - 32178v^7 + 53361v^6 - 140499v^5 - 13338v^4 + 190107v^3 + 17739v^2 + 271917*v - 968841) / 196830

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

$\nu$	$=$	$( - \beta_{15} + \beta_{14} - \beta_{13} + \beta_{10} + 2 \beta_{9} - 2 \beta_{8} - 3 \beta_{7} + \cdots + 1 ) / 3$ (-b15 + b14 - b13 + b10 + 2b9 - 2b8 - 3*b7 - b6 + b5 - b4 - b3 + b1 + 1) / 3
$\nu^{2}$	$=$	$( - 2 \beta_{15} - \beta_{14} + \beta_{13} - \beta_{10} + \beta_{9} + 2 \beta_{8} + \beta_{6} - 4 \beta_{5} + \cdots - 4 ) / 3$ (-2b15 - b14 + b13 - b10 + b9 + 2b8 + b6 - 4b5 + 4b4 + b3 - b1 - 4) / 3
$\nu^{3}$	$=$	$( - \beta_{15} - 5 \beta_{14} + 5 \beta_{13} - 6 \beta_{12} + \beta_{10} - \beta_{9} + \beta_{8} + \cdots + 7 ) / 3$ (-b15 - 5b14 + 5b13 - 6b12 + b10 - b9 + b8 - b6 + b5 - 4b4 + 2b3 - 3b2 - 2*b1 + 7) / 3
$\nu^{4}$	$=$	$( \beta_{15} - \beta_{14} - 5 \beta_{13} + 3 \beta_{12} - \beta_{10} + \beta_{9} + 2 \beta_{8} - 3 \beta_{7} + \cdots - 7 ) / 3$ (b15 - b14 - 5b13 + 3b12 - b10 + b9 + 2b8 - 3b7 + 10b6 - 4b5 + b4 + b3 - 3b2 + 5b1 - 7) / 3
$\nu^{5}$	$=$	$( - 7 \beta_{15} - 2 \beta_{14} - \beta_{13} - 9 \beta_{12} - 6 \beta_{11} - 2 \beta_{10} - 10 \beta_{9} + \cdots - 23 ) / 3$ (-7b15 - 2b14 - b13 - 9b12 - 6b11 - 2b10 - 10b9 + 7b8 + 18b7 + 11b6 - 11b5 + 17b4 + 2b3 - 9b2 - 8b1 - 23) / 3
$\nu^{6}$	$=$	$( - 2 \beta_{15} - 22 \beta_{14} + \beta_{13} - 12 \beta_{12} - 18 \beta_{11} - 4 \beta_{10} - 23 \beta_{9} + \cdots + 32 ) / 3$ (-2b15 - 22b14 + b13 - 12b12 - 18b11 - 4b10 - 23b9 + 17b8 - 6b7 + 22b6 - 34b5 + b4 + 4b3 - 6b2 - 10*b1 + 32) / 3
$\nu^{7}$	$=$	$( - 40 \beta_{15} + 37 \beta_{14} + 2 \beta_{13} - 21 \beta_{12} + 15 \beta_{11} - 5 \beta_{10} + \cdots - 26 ) / 3$ (-40b15 + 37b14 + 2b13 - 21b12 + 15b11 - 5b10 + 17b9 - 41b8 - 27b7 - 4b6 + 67b5 - 25b4 - 52b3 + 21b2 + 13*b1 - 26) / 3
$\nu^{8}$	$=$	$( - 23 \beta_{15} + 47 \beta_{14} + 25 \beta_{13} + 27 \beta_{12} + 36 \beta_{11} - 52 \beta_{10} + \cdots - 163 ) / 3$ (-23b15 + 47b14 + 25b13 + 27b12 + 36b11 - 52b10 + 16b9 + 68b8 + 81b7 + 25b6 - b5 + 64b4 - 20b3 + 18b2 - 70b1 - 163) / 3
$\nu^{9}$	$=$	$( 80 \beta_{15} - 59 \beta_{14} + 95 \beta_{13} - 60 \beta_{12} + 37 \beta_{10} - 55 \beta_{9} + 28 \beta_{8} + \cdots + 97 ) / 3$ (80b15 - 59b14 + 95b13 - 60b12 + 37b10 - 55b9 + 28b8 + 117b7 - 10b6 + 37b5 - 256b4 + 11b3 - 57b2 - 47b1 + 97) / 3
$\nu^{10}$	$=$	$( 82 \beta_{15} + 197 \beta_{14} - 32 \beta_{13} + 147 \beta_{12} + 171 \beta_{11} - 91 \beta_{10} + \cdots + 101 ) / 3$ (82b15 + 197b14 - 32b13 + 147b12 + 171b11 - 91b10 - 26b9 - 34b8 + 195b7 + 127b6 + 176b5 - 224b4 - 134b3 + 42b2 + 212*b1 + 101) / 3
$\nu^{11}$	$=$	$( - 106 \beta_{15} + 520 \beta_{14} + 8 \beta_{13} + 117 \beta_{12} + 228 \beta_{11} + 52 \beta_{10} + \cdots - 59 ) / 3$ (-106b15 + 520b14 + 8b13 + 117b12 + 228b11 + 52b10 + 80b9 - 128b8 + 279b7 - 259b6 + 394b5 + 116b4 - 259b3 - 54b2 - 35*b1 - 59) / 3
$\nu^{12}$	$=$	$( 232 \beta_{15} - 4 \beta_{14} - 8 \beta_{13} + 483 \beta_{12} + 18 \beta_{11} + 482 \beta_{10} + \cdots + 743 ) / 3$ (232b15 - 4b14 - 8b13 + 483b12 + 18b11 + 482b10 + 76b9 - 37b8 - 969b7 - 248b6 - 844b5 - 449b4 + 184b3 + 66b2 + 179*b1 + 743) / 3
$\nu^{13}$	$=$	$( - 598 \beta_{15} + 478 \beta_{14} + 929 \beta_{13} - 435 \beta_{12} + 1032 \beta_{11} + 157 \beta_{10} + \cdots + 532 ) / 3$ (-598b15 + 478b14 + 929b13 - 435b12 + 1032b11 + 157b10 + 431b9 - 1283b8 - 1008b7 - 1354b6 + 1444b5 - 1573b4 - 394b3 + 273b2 + 742*b1 + 532) / 3
$\nu^{14}$	$=$	$( - 32 \beta_{15} + 713 \beta_{14} + 934 \beta_{13} + 1575 \beta_{12} + 1368 \beta_{11} - 403 \beta_{10} + \cdots - 2152 ) / 3$ (-32b15 + 713b14 + 934b13 + 1575b12 + 1368b11 - 403b10 + 655b9 + 1337b8 - 585b7 - 704b6 - b5 + 1747b4 + 970b3 - 234b2 + 119b1 - 2152) / 3
$\nu^{15}$	$=$	$( 2627 \beta_{15} - 2318 \beta_{14} + 617 \beta_{13} + 174 \beta_{12} - 1548 \beta_{11} + 3871 \beta_{10} + \cdots - 1532 ) / 3$ (2627b15 - 2318b14 + 617b13 + 174b12 - 1548b11 + 3871b10 - 1504b9 - 350b8 - 3375b7 - 172b6 - 3527b5 - 805b4 + 3854b3 - 2397b2 + 2005*b1 - 1532) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/630\mathbb{Z}\right)^\times$ .

$n$	$127$	$281$	$451$
$\chi(n)$	$1$	$-\beta_{4}$	$-\beta_{4}$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

\iota_m(\nu)

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

a_{8}

a_{9}

a_{10}

121.1

−0.579249	−	1.63232i
0.702194	−	1.58333i
−1.73198	−	0.0153002i
1.52536	−	0.820531i
−1.03843	+	1.38624i
−0.803168	+	1.53458i
1.67659	+	0.434811i
0.748691	+	1.56188i
−0.579249	+	1.63232i
0.702194	+	1.58333i
−1.73198	+	0.0153002i
1.52536	+	0.820531i
−1.03843	−	1.38624i
−0.803168	−	1.53458i
1.67659	−	0.434811i
0.748691	−	1.56188i

−1.00000

−1.70326

0.314515i

1.00000

0.500000

0.866025i

1.70326

−

0.314515i

2.48140

0.917950i

−1.00000

2.80216

−

1.07140i

−0.500000

−

0.866025i

121.2

−1.00000

−1.02010

1.39978i

1.00000

0.500000

0.866025i

1.02010

−

1.39978i

−2.52336

0.795395i

−1.00000

−0.918776

−

2.85585i

−0.500000

−

0.866025i

121.3

−1.00000

−0.879242

−

1.49229i

1.00000

0.500000

0.866025i

0.879242

1.49229i

2.58337

−

0.571125i

−1.00000

−1.45387

2.62417i

−0.500000

−

0.866025i

121.4

−1.00000

0.0520808

1.73127i

1.00000

0.500000

0.866025i

−0.0520808

−

1.73127i

0.226513

2.63604i

−1.00000

−2.99458

0.180332i

−0.500000

−

0.866025i

121.5

−1.00000

0.681302

−

1.59243i

1.00000

0.500000

0.866025i

−0.681302

1.59243i

−1.52280

2.16358i

−1.00000

−2.07165

−

2.16985i

−0.500000

−

0.866025i

121.6

−1.00000

0.927397

−

1.46285i

1.00000

0.500000

0.866025i

−0.927397

1.46285i

0.832221

−

2.51146i

−1.00000

−1.27987

−

2.71329i

−0.500000

−

0.866025i

121.7

−1.00000

1.21485

1.23456i

1.00000

0.500000

0.866025i

−1.21485

−

1.23456i

−1.09594

−

2.40809i

−1.00000

−0.0482768

2.99961i

−0.500000

−

0.866025i

121.8

−1.00000

1.72697

−

0.132553i

1.00000

0.500000

0.866025i

−1.72697

0.132553i

1.01860

2.44181i

−1.00000

2.96486

−

0.457832i

−0.500000

−

0.866025i

151.1

−1.00000

−1.70326

−

0.314515i

1.00000

0.500000

−

0.866025i

1.70326

0.314515i

2.48140

−

0.917950i

−1.00000

2.80216

1.07140i

−0.500000

0.866025i

151.2

−1.00000

−1.02010

−

1.39978i

1.00000

0.500000

−

0.866025i

1.02010

1.39978i

−2.52336

−

0.795395i

−1.00000

−0.918776

2.85585i

−0.500000

0.866025i

151.3

−1.00000

−0.879242

1.49229i

1.00000

0.500000

−

0.866025i

0.879242

−

1.49229i

2.58337

0.571125i

−1.00000

−1.45387

−

2.62417i

−0.500000

0.866025i

151.4

−1.00000

0.0520808

−

1.73127i

1.00000

0.500000

−

0.866025i

−0.0520808

1.73127i

0.226513

−

2.63604i

−1.00000

−2.99458

−

0.180332i

−0.500000

0.866025i

151.5

−1.00000

0.681302

1.59243i

1.00000

0.500000

−

0.866025i

−0.681302

−

1.59243i

−1.52280

−

2.16358i

−1.00000

−2.07165

2.16985i

−0.500000

0.866025i

151.6

−1.00000

0.927397

1.46285i

1.00000

0.500000

−

0.866025i

−0.927397

−

1.46285i

0.832221

2.51146i

−1.00000

−1.27987

2.71329i

−0.500000

0.866025i

151.7

−1.00000

1.21485

−

1.23456i

1.00000

0.500000

−

0.866025i

−1.21485

1.23456i

−1.09594

2.40809i

−1.00000

−0.0482768

−

2.99961i

−0.500000

0.866025i

151.8

−1.00000

1.72697

0.132553i

1.00000

0.500000

−

0.866025i

−1.72697

−

0.132553i

1.01860

−

2.44181i

−1.00000

2.96486

0.457832i

−0.500000

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
63.h	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	630.2.i.i	✓	16
3.b	odd	2	1		1890.2.i.i		16
7.c	even	3	1		630.2.l.i	yes	16
9.c	even	3	1		630.2.l.i	yes	16
9.d	odd	6	1		1890.2.l.i		16
21.h	odd	6	1		1890.2.l.i		16
63.h	even	3	1	inner	630.2.i.i	✓	16
63.j	odd	6	1		1890.2.i.i		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
630.2.i.i	✓	16	1.a	even	1	1	trivial
630.2.i.i	✓	16	63.h	even	3	1	inner
630.2.l.i	yes	16	7.c	even	3	1
630.2.l.i	yes	16	9.c	even	3	1
1890.2.i.i		16	3.b	odd	2	1
1890.2.i.i		16	63.j	odd	6	1
1890.2.l.i		16	9.d	odd	6	1
1890.2.l.i		16	21.h	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(630, [\chi])$ :

T_{11}^{16} - T_{11}^{15} + 53 T_{11}^{14} + 116 T_{11}^{13} + 2018 T_{11}^{12} + 4688 T_{11}^{11} + \cdots + 2916

T_{13}^{16} - 2 T_{13}^{15} + 54 T_{13}^{14} - 8 T_{13}^{13} + 1834 T_{13}^{12} + 477 T_{13}^{11} + \cdots + 2298256

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T + 1)^{16}$ (T + 1)^16
$3$	$T^{16} - 2 T^{15} + \cdots + 6561$ T^16 - 2T^15 + 5T^14 - 5T^13 + 8T^12 - 8T^11 - 20T^10 + 51T^9 - 183T^8 + 153T^7 - 180T^6 - 216T^5 + 648T^4 - 1215T^3 + 3645T^2 - 4374*T + 6561
$5$	$(T^{2} - T + 1)^{8}$ (T^2 - T + 1)^8
$7$	$T^{16} - 4 T^{15} + \cdots + 5764801$ T^16 - 4T^15 + 15T^14 - 46T^13 + 85T^12 - 234T^11 + 295T^10 + 350T^9 - 39T^8 + 2450T^7 + 14455T^6 - 80262T^5 + 204085T^4 - 773122T^3 + 1764735T^2 - 3294172*T + 5764801
$11$	$T^{16} - T^{15} + \cdots + 2916$ T^16 - T^15 + 53T^14 + 116T^13 + 2018T^12 + 4688T^11 + 39862T^10 + 156501T^9 + 562494T^8 + 1194714T^7 + 1937466T^6 + 1987308T^5 + 1493181T^4 + 627426T^3 + 188406T^2 + 28188T + 2916
$13$	$T^{16} - 2 T^{15} + \cdots + 2298256$ T^16 - 2T^15 + 54T^14 - 8T^13 + 1834T^12 + 477T^11 + 34540T^10 + 52582T^9 + 433476T^8 + 684178T^7 + 3205075T^6 + 6671493T^5 + 14639257T^4 + 16719280T^3 + 15234300T^2 + 6900832*T + 2298256
$17$	$T^{16} + \cdots + 8707129344$ T^16 - 11T^15 + 172T^14 - 999T^13 + 10767T^12 - 48267T^11 + 453195T^10 - 1438209T^9 + 11605383T^8 - 26237682T^7 + 215458947T^6 - 330564807T^5 + 2370323817T^4 - 2178566928T^3 + 17265612672T^2 - 11972302848*T + 8707129344
$19$	$T^{16} + \cdots + 1174158756$ T^16 + 2T^15 + 98T^14 + 280T^13 + 6918T^12 + 19245T^11 + 235788T^10 + 637458T^9 + 5715978T^8 + 14051506T^7 + 75895571T^6 + 153664667T^5 + 652539913T^4 + 1231454664T^3 + 2158008918T^2 + 1769359176*T + 1174158756
$23$	$T^{16} + \cdots + 927567936$ T^16 - 11T^15 + 176T^14 - 1007T^13 + 11912T^12 - 56534T^11 + 516169T^10 - 1244817T^9 + 8326524T^8 - 101754T^7 + 97089705T^6 + 44990154T^5 + 547995969T^4 + 642142404T^3 + 2201299416T^2 + 1413280224*T + 927567936
$29$	$T^{16} + \cdots + 17639292969$ T^16 - 17T^15 + 278T^14 - 2735T^13 + 28451T^12 - 225260T^11 + 1821232T^10 - 11447856T^9 + 70777128T^8 - 351438732T^7 + 1743315804T^6 - 6789615372T^5 + 24575037471T^4 - 58028658471T^3 + 110947789962T^2 - 48483252837*T + 17639292969
$31$	$(T^{8} - 15 T^{7} + \cdots + 27294)^{2}$ (T^8 - 15T^7 + 46T^6 + 292T^5 - 1946T^4 + 1420T^3 + 13629T^2 - 36302*T + 27294)^2
$37$	$T^{16} + \cdots + 147622500$ T^16 + 2T^15 + 187T^14 - 50T^13 + 23858T^12 - 11501T^11 + 1541301T^10 - 1916286T^9 + 71688606T^8 - 89262833T^7 + 1552868072T^6 - 3683050217T^5 + 23529518011T^4 - 29017050460T^3 + 34261002250T^2 - 2308986000*T + 147622500
$41$	$T^{16} + \cdots + 86812992387801$ T^16 - 7T^15 + 270T^14 - 901T^13 + 39076T^12 - 84663T^11 + 3702942T^10 - 2777112T^9 + 245168532T^8 - 9845874T^7 + 11620565451T^6 + 9476498448T^5 + 381671616141T^4 + 402751302138T^3 + 8136771273864T^2 + 12647350897902*T + 86812992387801
$43$	$T^{16} + \cdots + 285914281$ T^16 + 13T^15 + 234T^14 + 1787T^13 + 23078T^12 + 158697T^11 + 1454068T^10 + 7267054T^9 + 48039750T^8 + 205695460T^7 + 1045232041T^6 + 2987533494T^5 + 7532421827T^4 + 6805283222T^3 + 4526436618T^2 + 1330433938*T + 285914281
$47$	$(T^{8} - 5 T^{7} + \cdots - 423063)^{2}$ (T^8 - 5T^7 - 209T^6 + 966T^5 + 9951T^4 - 50121T^3 - 37818T^2 + 406404*T - 423063)^2
$53$	$T^{16} + \cdots + 1238515248996$ T^16 - 18T^15 + 399T^14 - 2630T^13 + 38448T^12 - 106899T^11 + 2838763T^10 - 1726038T^9 + 108882912T^8 + 66347235T^7 + 3097006380T^6 + 3808368603T^5 + 45774547983T^4 + 92330519760T^3 + 506947936338T^2 + 716382524376*T + 1238515248996
$59$	$(T^{8} + T^{7} - 155 T^{6} + \cdots + 1458)^{2}$ (T^8 + T^7 - 155T^6 + 246T^5 + 5748T^4 - 22311T^3 + 25713T^2 - 11124T + 1458)^2
$61$	$(T^{8} - 27 T^{7} + \cdots - 502368)^{2}$ (T^8 - 27T^7 + 156T^6 + 2006T^5 - 33888T^4 + 203910T^3 - 611453T^2 + 898440*T - 502368)^2
$67$	$(T^{8} - 10 T^{7} + \cdots + 40406508)^{2}$ (T^8 - 10T^7 - 353T^6 + 2779T^5 + 44516T^4 - 227405T^3 - 2320437T^2 + 5028960*T + 40406508)^2
$71$	$(T^{8} + 19 T^{7} + \cdots - 7422678)^{2}$ (T^8 + 19T^7 - 168T^6 - 5670T^5 - 28107T^4 + 131949T^3 + 1116099T^2 + 14580*T - 7422678)^2
$73$	$T^{16} + 8 T^{15} + \cdots + 2143296$ T^16 + 8T^15 + 194T^14 + 2596T^13 + 38910T^12 + 345057T^11 + 2413146T^10 + 10754460T^9 + 35783472T^8 + 66554746T^7 + 92478587T^6 + 76189157T^5 + 52769077T^4 + 21750732T^3 + 12384024T^2 + 3882528*T + 2143296
$79$	$(T^{8} - 25 T^{7} + \cdots - 406534)^{2}$ (T^8 - 25T^7 + 175T^6 + 726T^5 - 18231T^4 + 117846T^3 - 377959T^2 + 615718*T - 406534)^2
$83$	$T^{16} + \cdots + 2857962683601$ T^16 - 2T^15 + 285T^14 - 950T^13 + 59479T^12 - 181731T^11 + 5425806T^10 - 12293469T^9 + 344131362T^8 - 646842591T^7 + 11108005821T^6 - 8778869019T^5 + 229179273264T^4 - 211647489975T^3 + 1490295654990T^2 + 1411385241717*T + 2857962683601
$89$	$T^{16} + \cdots + 99588211776$ T^16 + 6T^15 + 351T^14 + 110T^13 + 80562T^12 + 66627T^11 + 7690357T^10 + 2106420T^9 + 520688268T^8 + 643546485T^7 + 11015185194T^6 + 21704323815T^5 + 188866782837T^4 + 298622599884T^3 + 812199516120T^2 - 261580946400*T + 99588211776
$97$	$T^{16} + \cdots + 1545178274704$ T^16 - 26T^15 + 720T^14 - 12640T^13 + 253103T^12 - 3815676T^11 + 50715262T^10 - 482307254T^9 + 3675013929T^8 - 19206109952T^7 + 76868712919T^6 - 161539486848T^5 + 267155797613T^4 + 299843403512T^3 + 2147276158380T^2 + 1714109041504*T + 1545178274704
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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 121.8
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	630.2.i.i

Level	$630$
Weight	$2$
Character orbit	630.i
Analytic conductor	$5.031$
Analytic rank	$0$
Dimension	$16$
Inner twists	$2$