LMFDB - Newform orbit 738.2.e.j

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [738,2,Mod(247,738)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(738, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("738.247"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 738 = 2 \cdot 3^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	738.e (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [14,-7,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.89295966917$
Analytic rank:	$0$
Dimension:	$14$
Relative dimension:	$7$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 4 x^{13} + 12 x^{12} - 31 x^{11} + 73 x^{10} - 150 x^{9} + 297 x^{8} - 549 x^{7} + 891 x^{6} + \cdots + 2187 $ x^14 - 4x^13 + 12x^12 - 31x^11 + 73x^10 - 150x^9 + 297x^8 - 549x^7 + 891x^6 - 1350x^5 + 1971x^4 - 2511x^3 + 2916x^2 - 2916*x + 2187
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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 $1,\beta_1,\ldots,\beta_{13}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $\nu$ of $ x^{14} - 4 x^{13} + 12 x^{12} - 31 x^{11} + 73 x^{10} - 150 x^{9} + 297 x^{8} - 549 x^{7} + 891 x^{6} + \cdots + 2187 $ x^14 - 4*x^13 + 12*x^12 - 31*x^11 + 73*x^10 - 150*x^9 + 297*x^8 - 549*x^7 + 891*x^6 - 1350*x^5 + 1971*x^4 - 2511*x^3 + 2916*x^2 - 2916*x + 2187 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( - 8 \nu^{13} + 419 \nu^{12} - 573 \nu^{11} + 1769 \nu^{10} - 5102 \nu^{9} + 10668 \nu^{8} + \cdots + 250047 ) / 159651 $ (-8v^13 + 419v^12 - 573v^11 + 1769v^10 - 5102v^9 + 10668v^8 - 19449v^7 + 38331v^6 - 57186v^5 + 82323v^4 - 124389v^3 + 172449v^2 - 114696*v + 250047) / 159651
$\beta_{3}$	$=$	$ ( - 130 \nu^{13} + 841 \nu^{12} - 1044 \nu^{11} + 331 \nu^{10} - 892 \nu^{9} + 5601 \nu^{8} + \cdots + 457812 ) / 478953 $ (-130v^13 + 841v^12 - 1044v^11 + 331v^10 - 892v^9 + 5601v^8 - 13005v^7 + 11376v^6 - 66960v^5 + 103410v^4 - 204552v^3 + 423792v^2 - 710775*v + 457812) / 478953
$\beta_{4}$	$=$	$ ( 203 \nu^{13} - 914 \nu^{12} + 1701 \nu^{11} - 623 \nu^{10} + 5345 \nu^{9} - 3192 \nu^{8} + \cdots + 925830 ) / 478953 $ (203v^13 - 914v^12 + 1701v^11 - 623v^10 + 5345v^9 - 3192v^8 - 11304v^7 + 35271v^6 - 47358v^5 + 146907v^4 - 177822v^3 + 480897v^2 - 672867*v + 925830) / 478953
$\beta_{5}$	$=$	$ ( 203 \nu^{13} - 914 \nu^{12} + 1701 \nu^{11} - 623 \nu^{10} + 5345 \nu^{9} - 3192 \nu^{8} + \cdots + 446877 ) / 478953 $ (203v^13 - 914v^12 + 1701v^11 - 623v^10 + 5345v^9 - 3192v^8 - 11304v^7 + 35271v^6 - 47358v^5 + 146907v^4 - 177822v^3 + 1944v^2 - 672867*v + 446877) / 478953
$\beta_{6}$	$=$	$ ( - 253 \nu^{13} - 1340 \nu^{12} + 3204 \nu^{11} - 10166 \nu^{10} + 30548 \nu^{9} - 48174 \nu^{8} + \cdots - 1744497 ) / 478953 $ (-253v^13 - 1340v^12 + 3204v^11 - 10166v^10 + 30548v^9 - 48174v^8 + 121833v^7 - 221931v^6 + 388665v^5 - 500121v^4 + 803250v^3 - 829602v^2 + 878445*v - 1744497) / 478953
$\beta_{7}$	$=$	$ ( - 124 \nu^{13} + 472 \nu^{12} - 231 \nu^{11} + 2125 \nu^{10} - 3745 \nu^{9} + 3294 \nu^{8} + \cdots + 17496 ) / 159651 $ (-124v^13 + 472v^12 - 231v^11 + 2125v^10 - 3745v^9 + 3294v^8 - 4824v^7 + 9729v^6 + 4509v^5 - 4158v^4 + 2565v^3 - 61803v^2 + 155763*v + 17496) / 159651
$\beta_{8}$	$=$	$ ( \nu^{13} - 4 \nu^{12} + 12 \nu^{11} - 31 \nu^{10} + 73 \nu^{9} - 150 \nu^{8} + 297 \nu^{7} + \cdots - 2916 ) / 729 $ (v^13 - 4v^12 + 12v^11 - 31v^10 + 73v^9 - 150v^8 + 297v^7 - 549v^6 + 891v^5 - 1350v^4 + 1971v^3 - 2511v^2 + 2916v - 2916) / 729
$\beta_{9}$	$=$	$ ( - 1100 \nu^{13} - 313 \nu^{12} - 495 \nu^{11} - 838 \nu^{10} + 7159 \nu^{9} - 16437 \nu^{8} + \cdots - 289413 ) / 478953 $ (-1100v^13 - 313v^12 - 495v^11 - 838v^10 + 7159v^9 - 16437v^8 + 25047v^7 - 62028v^6 + 147069v^5 - 54243v^4 + 242298v^3 - 274347v^2 + 52488*v - 289413) / 478953
$\beta_{10}$	$=$	$ ( - 43 \nu^{13} + 53 \nu^{12} - 169 \nu^{11} + 502 \nu^{10} - 1052 \nu^{9} + 1897 \nu^{8} + \cdots - 1944 ) / 17739 $ (-43v^13 + 53v^12 - 169v^11 + 502v^10 - 1052v^9 + 1897v^8 - 3771v^7 + 5562v^6 - 7947v^5 + 12069v^4 - 16929v^3 + 10152v^2 - 25191*v - 1944) / 17739
$\beta_{11}$	$=$	$ ( - 1370 \nu^{13} + 1838 \nu^{12} - 6858 \nu^{11} + 15011 \nu^{10} - 31334 \nu^{9} + 65040 \nu^{8} + \cdots + 207036 ) / 478953 $ (-1370v^13 + 1838v^12 - 6858v^11 + 15011v^10 - 31334v^9 + 65040v^8 - 136143v^7 + 195390v^6 - 260361v^5 + 511218v^4 - 740637v^3 + 805059v^2 - 998001*v + 207036) / 478953
$\beta_{12}$	$=$	$ ( - 1381 \nu^{13} + 4522 \nu^{12} - 12546 \nu^{11} + 32308 \nu^{10} - 67969 \nu^{9} + 131502 \nu^{8} + \cdots + 1422279 ) / 478953 $ (-1381v^13 + 4522v^12 - 12546v^11 + 32308v^10 - 67969v^9 + 131502v^8 - 267102v^7 + 428688v^6 - 703134v^5 + 1049409v^4 - 1249452v^3 + 1453869v^2 - 2042658*v + 1422279) / 478953
$\beta_{13}$	$=$	$ ( 26 \nu^{13} - 80 \nu^{12} + 207 \nu^{11} - 509 \nu^{10} + 1073 \nu^{9} - 2112 \nu^{8} + 4302 \nu^{7} + \cdots - 25515 ) / 6561 $ (26v^13 - 80v^12 + 207v^11 - 509v^10 + 1073v^9 - 2112v^8 + 4302v^7 - 7443v^6 + 10800v^5 - 15822v^4 + 22005v^3 - 26973v^2 + 29889*v - 25515) / 6561

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ -\beta_{5} + \beta_{4} - 1 $ -b5 + b4 - 1
$\nu^{3}$	$=$	$ \beta_{12} - \beta_{11} + \beta_{9} + \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} + 2\beta_{2} - \beta _1 - 1 $ b12 - b11 + b9 + b8 - b7 - b5 + b4 - b3 + 2*b2 - b1 - 1
$\nu^{4}$	$=$	$ 2 \beta_{13} + 2 \beta_{12} + \beta_{11} - 3 \beta_{10} + 2 \beta_{9} - \beta_{8} + \beta_{7} + \beta_{5} + \cdots - 3 $ 2b13 + 2b12 + b11 - 3b10 + 2b9 - b8 + b7 + b5 - b4 - b3 + 2*b2 - 3
$\nu^{5}$	$=$	$ \beta_{13} + 6\beta_{11} - 6\beta_{10} + 3\beta_{7} - 4\beta_{3} + 2\beta_{2} - 8\beta _1 + 1 $ b13 + 6b11 - 6b10 + 3b7 - 4b3 + 2b2 - 8b1 + 1
$\nu^{6}$	$=$	$ - 6 \beta_{13} - 5 \beta_{12} - 4 \beta_{11} - 3 \beta_{10} + 4 \beta_{9} - 2 \beta_{8} - \beta_{7} + \cdots - 7 $ -6b13 - 5b12 - 4b11 - 3b10 + 4b9 - 2b8 - b7 + 2b5 - 2b4 - 7b3 + 2b2 - 7*b1 - 7
$\nu^{7}$	$=$	$ 2 \beta_{13} - 4 \beta_{12} - 11 \beta_{11} - 3 \beta_{10} + 5 \beta_{9} - 25 \beta_{8} + 7 \beta_{7} + \cdots - 3 $ 2b13 - 4b12 - 11b11 - 3b10 + 5b9 - 25b8 + 7b7 + 9b6 + b5 - 10b4 - 4b3 - 10b2 - 3b1 - 3
$\nu^{8}$	$=$	$ 7 \beta_{13} + 3 \beta_{12} + 3 \beta_{11} + 12 \beta_{10} - 15 \beta_{9} - 6 \beta_{8} - 9 \beta_{7} + \cdots + 49 $ 7b13 + 3b12 + 3b11 + 12b10 - 15b9 - 6b8 - 9b7 + 27b6 + 6b5 - 6b4 - 4b3 + 29b2 + 4*b1 + 49
$\nu^{9}$	$=$	$ - 27 \beta_{13} - 2 \beta_{12} - 7 \beta_{11} + 6 \beta_{10} - 29 \beta_{9} + 19 \beta_{8} - 13 \beta_{7} + \cdots + 50 $ -27b13 - 2b12 - 7b11 + 6b10 - 29b9 + 19b8 - 13b7 + 36b6 + 44b5 + b4 + 29b3 - 16b2 + 77b1 + 50
$\nu^{10}$	$=$	$ - \beta_{13} + 17 \beta_{12} - 5 \beta_{11} - 3 \beta_{10} - 64 \beta_{9} - 40 \beta_{8} + 85 \beta_{7} + \cdots + 36 $ -b13 + 17b12 - 5b11 - 3b10 - 64b9 - 40b8 + 85b7 + 45b6 + 7b5 + 56b4 + 23b3 - 172b2 + 63b1 + 36
$\nu^{11}$	$=$	$ 49 \beta_{13} + 15 \beta_{12} - 63 \beta_{11} + 192 \beta_{10} - 12 \beta_{9} + 78 \beta_{8} + \cdots - 146 $ 49b13 + 15b12 - 63b11 + 192b10 - 12b9 + 78b8 + 24b7 + 9b6 + 3b5 + 123b4 + 68b3 + 236b2 + 79*b1 - 146
$\nu^{12}$	$=$	$ - 48 \beta_{13} + 28 \beta_{12} - 37 \beta_{11} - 66 \beta_{10} + 46 \beta_{9} + 175 \beta_{8} + \cdots - 244 $ -48b13 + 28b12 - 37b11 - 66b10 + 46b9 + 175b8 + 83b7 - 9b6 + 140b5 - 104b4 + 254b3 + 317b2 + 53*b1 - 244
$\nu^{13}$	$=$	$ 131 \beta_{13} + 227 \beta_{12} + 460 \beta_{11} - 759 \beta_{10} - 349 \beta_{9} - 28 \beta_{8} + \cdots + 75 $ 131b13 + 227b12 + 460b11 - 759b10 - 349b9 - 28b8 + 298b7 + 40b5 - 67b4 - 274b3 - 334b2 - 573b1 + 75

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/738\mathbb{Z}\right)^\times$.

$n$	$83$	$703$
$\chi(n)$	$-1 + \beta_{2}$	$1$

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} $

247.1

−1.24171	+	1.20754i
−0.932393	−	1.45967i
−0.170865	+	1.72360i
0.132454	−	1.72698i
1.19349	−	1.25522i
1.33574	+	1.10264i
1.68329	+	0.408093i
−1.24171	−	1.20754i
−0.932393	+	1.45967i
−0.170865	−	1.72360i
0.132454	+	1.72698i
1.19349	+	1.25522i
1.33574	−	1.10264i
1.68329	−	0.408093i

−0.500000

−

0.866025i

−1.24171

−

1.20754i

−0.500000

0.866025i

1.44490

−

2.50264i

−0.424909

1.67912i

−2.37697

−

4.11703i

1.00000

0.0836779

2.99883i

−2.88980

247.2

−0.500000

−

0.866025i

−0.932393

1.45967i

−0.500000

0.866025i

0.285913

−

0.495215i

1.73031

0.0776401i

0.871584

1.50963i

1.00000

−1.26129

−

2.72198i

−0.571825

247.3

−0.500000

−

0.866025i

−0.170865

−

1.72360i

−0.500000

0.866025i

−0.205691

0.356267i

−1.40725

1.00977i

0.217780

0.377206i

1.00000

−2.94161

0.589007i

0.411382

247.4

−0.500000

−

0.866025i

0.132454

1.72698i

−0.500000

0.866025i

1.61620

−

2.79935i

1.42938

−

0.978198i

1.41515

2.45112i

1.00000

−2.96491

0.457489i

−3.23241

247.5

−0.500000

−

0.866025i

1.19349

1.25522i

−0.500000

0.866025i

−1.81416

3.14221i

0.490312

−

1.66120i

−0.330242

−

0.571996i

1.00000

−0.151175

2.99619i

3.62831

247.6

−0.500000

−

0.866025i

1.33574

−

1.10264i

−0.500000

0.866025i

−0.726439

1.25823i

−1.62278

−

0.605464i

0.415712

0.720035i

1.00000

0.568385

−

2.94566i

1.45288

247.7

−0.500000

−

0.866025i

1.68329

−

0.408093i

−0.500000

0.866025i

1.89927

−

3.28963i

−1.19506

−

1.25372i

−0.713019

−

1.23498i

1.00000

2.66692

−

1.37388i

−3.79853

493.1

−0.500000

0.866025i

−1.24171

1.20754i

−0.500000

−

0.866025i

1.44490

2.50264i

−0.424909

−

1.67912i

−2.37697

4.11703i

1.00000

0.0836779

−

2.99883i

−2.88980

493.2

−0.500000

0.866025i

−0.932393

−

1.45967i

−0.500000

−

0.866025i

0.285913

0.495215i

1.73031

−

0.0776401i

0.871584

−

1.50963i

1.00000

−1.26129

2.72198i

−0.571825

493.3

−0.500000

0.866025i

−0.170865

1.72360i

−0.500000

−

0.866025i

−0.205691

−

0.356267i

−1.40725

−

1.00977i

0.217780

−

0.377206i

1.00000

−2.94161

−

0.589007i

0.411382

493.4

−0.500000

0.866025i

0.132454

−

1.72698i

−0.500000

−

0.866025i

1.61620

2.79935i

1.42938

0.978198i

1.41515

−

2.45112i

1.00000

−2.96491

−

0.457489i

−3.23241

493.5

−0.500000

0.866025i

1.19349

−

1.25522i

−0.500000

−

0.866025i

−1.81416

−

3.14221i

0.490312

1.66120i

−0.330242

0.571996i

1.00000

−0.151175

−

2.99619i

3.62831

493.6

−0.500000

0.866025i

1.33574

1.10264i

−0.500000

−

0.866025i

−0.726439

−

1.25823i

−1.62278

0.605464i

0.415712

−

0.720035i

1.00000

0.568385

2.94566i

1.45288

493.7

−0.500000

0.866025i

1.68329

0.408093i

−0.500000

−

0.866025i

1.89927

3.28963i

−1.19506

1.25372i

−0.713019

1.23498i

1.00000

2.66692

1.37388i

−3.79853

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	738.2.e.j	✓	14
3.b	odd	2	1		2214.2.e.j		14
9.c	even	3	1	inner	738.2.e.j	✓	14
9.c	even	3	1		6642.2.a.bv		7
9.d	odd	6	1		2214.2.e.j		14
9.d	odd	6	1		6642.2.a.bu		7

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
738.2.e.j	✓	14	1.a	even	1	1	trivial
738.2.e.j	✓	14	9.c	even	3	1	inner
2214.2.e.j		14	3.b	odd	2	1
2214.2.e.j		14	9.d	odd	6	1
6642.2.a.bu		7	9.d	odd	6	1
6642.2.a.bv		7	9.c	even	3	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(738, [\chi])$:

$ T_{5}^{14} - 5 T_{5}^{13} + 37 T_{5}^{12} - 102 T_{5}^{11} + 567 T_{5}^{10} - 1356 T_{5}^{9} + \cdots + 1936 $

T5^14 - 5*T5^13 + 37*T5^12 - 102*T5^11 + 567*T5^10 - 1356*T5^9 + 5389*T5^8 - 6470*T5^7 + 16792*T5^6 - 2040*T5^5 + 38628*T5^4 - 4944*T5^3 + 10000*T5^2 + 1408*T5 + 1936

$ T_{7}^{14} + T_{7}^{13} + 19 T_{7}^{12} - 36 T_{7}^{11} + 273 T_{7}^{10} - 270 T_{7}^{9} + 829 T_{7}^{8} + \cdots + 64 $

T7^14 + T7^13 + 19*T7^12 - 36*T7^11 + 273*T7^10 - 270*T7^9 + 829*T7^8 - 494*T7^7 + 1708*T7^6 - 864*T7^5 + 1176*T7^4 - 288*T7^3 + 448*T7^2 - 128*T7 + 64

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} + T + 1)^{7} $ (T^2 + T + 1)^7
$3$	$ T^{14} - 4 T^{13} + \cdots + 2187 $ T^14 - 4T^13 + 12T^12 - 31T^11 + 73T^10 - 150T^9 + 297T^8 - 549T^7 + 891T^6 - 1350T^5 + 1971T^4 - 2511T^3 + 2916T^2 - 2916*T + 2187
$5$	$ T^{14} - 5 T^{13} + \cdots + 1936 $ T^14 - 5T^13 + 37T^12 - 102T^11 + 567T^10 - 1356T^9 + 5389T^8 - 6470T^7 + 16792T^6 - 2040T^5 + 38628T^4 - 4944T^3 + 10000T^2 + 1408*T + 1936
$7$	$ T^{14} + T^{13} + \cdots + 64 $ T^14 + T^13 + 19T^12 - 36T^11 + 273T^10 - 270T^9 + 829T^8 - 494T^7 + 1708T^6 - 864T^5 + 1176T^4 - 288T^3 + 448T^2 - 128T + 64
$11$	$ T^{14} - 10 T^{13} + \cdots + 1227664 $ T^14 - 10T^13 + 85T^12 - 428T^11 + 2153T^10 - 8339T^9 + 32631T^8 - 95370T^7 + 250800T^6 - 458708T^5 + 772628T^4 - 876368T^3 + 1255888T^2 - 1041520*T + 1227664
$13$	$ T^{14} - 4 T^{13} + \cdots + 808201 $ T^14 - 4T^13 + 49T^12 - 282T^11 + 2136T^10 - 9243T^9 + 36136T^8 - 96100T^7 + 243814T^6 - 478671T^5 + 967056T^4 - 1434582T^3 + 1893589T^2 - 1417723*T + 808201
$17$	$ (T^{7} + 13 T^{6} + \cdots - 7688)^{2} $ (T^7 + 13T^6 - 6T^5 - 559T^4 - 1114T^3 + 3828T^2 + 3760T - 7688)^2
$19$	$ (T^{7} + T^{6} - 60 T^{5} + \cdots - 4335)^{2} $ (T^7 + T^6 - 60T^5 - 65T^4 + 1033T^3 + 1197T^2 - 3978*T - 4335)^2
$23$	$ T^{14} + \cdots + 20597129289 $ T^14 - 15T^13 + 231T^12 - 1996T^11 + 19662T^10 - 138687T^9 + 1064326T^8 - 5916798T^7 + 34297872T^6 - 151137843T^5 + 707502426T^4 - 2412346680T^3 + 7674632793T^2 - 14017879458*T + 20597129289
$29$	$ T^{14} - 30 T^{13} + \cdots + 4100625 $ T^14 - 30T^13 + 561T^12 - 6648T^11 + 58164T^10 - 362925T^9 + 1713654T^8 - 5713164T^7 + 14107266T^6 - 21724281T^5 + 24149340T^4 - 8674128T^3 + 24487839T^2 - 9349425*T + 4100625
$31$	$ T^{14} + \cdots + 1782106225 $ T^14 + 13T^13 + 253T^12 + 2282T^11 + 33704T^10 + 275039T^9 + 2372208T^8 + 10110372T^7 + 41764650T^6 + 56748335T^5 + 187906520T^4 - 69975742T^3 + 1394463079T^2 - 1296507080*T + 1782106225
$37$	$ (T^{7} - 7 T^{6} + \cdots + 147880)^{2} $ (T^7 - 7T^6 - 102T^5 + 609T^4 + 3162T^3 - 16920T^2 - 29656T + 147880)^2
$41$	$ (T^{2} + T + 1)^{7} $ (T^2 + T + 1)^7
$43$	$ T^{14} - 5 T^{13} + \cdots + 1089936 $ T^14 - 5T^13 + 67T^12 + 76T^11 + 1567T^10 + 2782T^9 + 25981T^8 + 63364T^7 + 228100T^6 + 340224T^5 + 739836T^4 + 802800T^3 + 1533600T^2 + 1165104*T + 1089936
$47$	$ T^{14} + \cdots + 793210896 $ T^14 - 3T^13 + 165T^12 + 2T^11 + 20547T^10 - 11736T^9 + 678997T^8 + 397656T^7 + 15509940T^6 + 9953352T^5 + 165577236T^4 + 332090640T^3 + 1106875872T^2 + 964222704*T + 793210896
$53$	$ (T^{7} + 12 T^{6} + \cdots - 311796)^{2} $ (T^7 + 12T^6 - 111T^5 - 1285T^4 + 4578T^3 + 39780T^2 - 72036T - 311796)^2
$59$	$ T^{14} + \cdots + 154876879936 $ T^14 - 13T^13 + 373T^12 - 918T^11 + 48999T^10 + 8832T^9 + 5455177T^8 + 15955598T^7 + 261774868T^6 + 132423984T^5 + 4407473256T^4 - 3630215136T^3 + 67905423424T^2 - 90423817792*T + 154876879936
$61$	$ T^{14} + \cdots + 9981608464 $ T^14 - 55T^13 + 1867T^12 - 41268T^11 + 681771T^10 - 8271570T^9 + 77049265T^8 - 517206004T^7 + 2554456360T^6 - 7471115232T^5 + 15641573604T^4 - 1340960640T^3 + 145185139120T^2 + 37724861168*T + 9981608464
$67$	$ T^{14} - 15 T^{13} + \cdots + 85766121 $ T^14 - 15T^13 + 219T^12 - 932T^11 + 6606T^10 - 9093T^9 + 119236T^8 - 50466T^7 + 1458144T^6 + 1067661T^5 + 11854080T^4 + 4723110T^3 + 36756909T^2 + 85766121
$71$	$ (T^{7} + 12 T^{6} + \cdots + 325368)^{2} $ (T^7 + 12T^6 - 231T^5 - 2743T^4 + 6810T^3 + 123420T^2 + 363456T + 325368)^2
$73$	$ (T^{7} - 270 T^{5} + \cdots + 93192)^{2} $ (T^7 - 270T^5 - 412T^4 + 12849T^3 - 11316T^2 - 74652*T + 93192)^2
$79$	$ T^{14} + \cdots + 1821923856 $ T^14 + 12T^13 + 255T^12 + 1054T^11 + 26433T^10 + 144123T^9 + 1269541T^8 + 3535056T^7 + 20081628T^6 + 35323908T^5 + 222179604T^4 + 206609472T^3 + 838343664T^2 - 486597600*T + 1821923856
$83$	$ T^{14} + \cdots + 23364956736 $ T^14 - 36T^13 + 855T^12 - 12438T^11 + 138975T^10 - 1145403T^9 + 7966827T^8 - 45291114T^7 + 239017752T^6 - 1059712416T^5 + 4190876712T^4 - 12031149888T^3 + 26830235232T^2 - 29726212032*T + 23364956736
$89$	$ (T^{7} + 16 T^{6} + \cdots - 9980)^{2} $ (T^7 + 16T^6 - 105T^5 - 1741T^4 - 3682T^3 + 10380T^2 + 25132T - 9980)^2
$97$	$ T^{14} + \cdots + 229263326405136 $ T^14 + T^13 + 433T^12 - 1586T^11 + 122941T^10 - 586976T^9 + 21683005T^8 - 134244002T^7 + 2812070968T^6 - 16354927176T^5 + 229201471260T^4 - 1263501838128T^3 + 12575858951664T^2 - 46357166081952T + 229263326405136
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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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 738 = 2 \cdot 3^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	738.e (of order \(3\), degree \(2\), minimal)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

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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	738.2.e.j

Level	$738$
Weight	$2$
Character orbit	738.e
Analytic conductor	$5.893$
Analytic rank	$0$
Dimension	$14$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(5.89295966917\)
Analytic rank:	\(0\)
Dimension:	\(14\)
Relative dimension:	\(7\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{14} - 4 x^{13} + 12 x^{12} - 31 x^{11} + 73 x^{10} - 150 x^{9} + 297 x^{8} - 549 x^{7} + 891 x^{6} + \cdots + 2187 \) x^14 - 4x^13 + 12x^12 - 31x^11 + 73x^10 - 150x^9 + 297x^8 - 549x^7 + 891x^6 - 1350x^5 + 1971x^4 - 2511x^3 + 2916x^2 - 2916*x + 2187
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( - 8 \nu^{13} + 419 \nu^{12} - 573 \nu^{11} + 1769 \nu^{10} - 5102 \nu^{9} + 10668 \nu^{8} + \cdots + 250047 ) / 159651 \) (-8v^13 + 419v^12 - 573v^11 + 1769v^10 - 5102v^9 + 10668v^8 - 19449v^7 + 38331v^6 - 57186v^5 + 82323v^4 - 124389v^3 + 172449v^2 - 114696*v + 250047) / 159651
\(\beta_{3}\)	\(=\)	\( ( - 130 \nu^{13} + 841 \nu^{12} - 1044 \nu^{11} + 331 \nu^{10} - 892 \nu^{9} + 5601 \nu^{8} + \cdots + 457812 ) / 478953 \) (-130v^13 + 841v^12 - 1044v^11 + 331v^10 - 892v^9 + 5601v^8 - 13005v^7 + 11376v^6 - 66960v^5 + 103410v^4 - 204552v^3 + 423792v^2 - 710775*v + 457812) / 478953
\(\beta_{4}\)	\(=\)	\( ( 203 \nu^{13} - 914 \nu^{12} + 1701 \nu^{11} - 623 \nu^{10} + 5345 \nu^{9} - 3192 \nu^{8} + \cdots + 925830 ) / 478953 \) (203v^13 - 914v^12 + 1701v^11 - 623v^10 + 5345v^9 - 3192v^8 - 11304v^7 + 35271v^6 - 47358v^5 + 146907v^4 - 177822v^3 + 480897v^2 - 672867*v + 925830) / 478953
\(\beta_{5}\)	\(=\)	\( ( 203 \nu^{13} - 914 \nu^{12} + 1701 \nu^{11} - 623 \nu^{10} + 5345 \nu^{9} - 3192 \nu^{8} + \cdots + 446877 ) / 478953 \) (203v^13 - 914v^12 + 1701v^11 - 623v^10 + 5345v^9 - 3192v^8 - 11304v^7 + 35271v^6 - 47358v^5 + 146907v^4 - 177822v^3 + 1944v^2 - 672867*v + 446877) / 478953
\(\beta_{6}\)	\(=\)	\( ( - 253 \nu^{13} - 1340 \nu^{12} + 3204 \nu^{11} - 10166 \nu^{10} + 30548 \nu^{9} - 48174 \nu^{8} + \cdots - 1744497 ) / 478953 \) (-253v^13 - 1340v^12 + 3204v^11 - 10166v^10 + 30548v^9 - 48174v^8 + 121833v^7 - 221931v^6 + 388665v^5 - 500121v^4 + 803250v^3 - 829602v^2 + 878445*v - 1744497) / 478953
\(\beta_{7}\)	\(=\)	\( ( - 124 \nu^{13} + 472 \nu^{12} - 231 \nu^{11} + 2125 \nu^{10} - 3745 \nu^{9} + 3294 \nu^{8} + \cdots + 17496 ) / 159651 \) (-124v^13 + 472v^12 - 231v^11 + 2125v^10 - 3745v^9 + 3294v^8 - 4824v^7 + 9729v^6 + 4509v^5 - 4158v^4 + 2565v^3 - 61803v^2 + 155763*v + 17496) / 159651
\(\beta_{8}\)	\(=\)	\( ( \nu^{13} - 4 \nu^{12} + 12 \nu^{11} - 31 \nu^{10} + 73 \nu^{9} - 150 \nu^{8} + 297 \nu^{7} + \cdots - 2916 ) / 729 \) (v^13 - 4v^12 + 12v^11 - 31v^10 + 73v^9 - 150v^8 + 297v^7 - 549v^6 + 891v^5 - 1350v^4 + 1971v^3 - 2511v^2 + 2916v - 2916) / 729
\(\beta_{9}\)	\(=\)	\( ( - 1100 \nu^{13} - 313 \nu^{12} - 495 \nu^{11} - 838 \nu^{10} + 7159 \nu^{9} - 16437 \nu^{8} + \cdots - 289413 ) / 478953 \) (-1100v^13 - 313v^12 - 495v^11 - 838v^10 + 7159v^9 - 16437v^8 + 25047v^7 - 62028v^6 + 147069v^5 - 54243v^4 + 242298v^3 - 274347v^2 + 52488*v - 289413) / 478953
\(\beta_{10}\)	\(=\)	\( ( - 43 \nu^{13} + 53 \nu^{12} - 169 \nu^{11} + 502 \nu^{10} - 1052 \nu^{9} + 1897 \nu^{8} + \cdots - 1944 ) / 17739 \) (-43v^13 + 53v^12 - 169v^11 + 502v^10 - 1052v^9 + 1897v^8 - 3771v^7 + 5562v^6 - 7947v^5 + 12069v^4 - 16929v^3 + 10152v^2 - 25191*v - 1944) / 17739
\(\beta_{11}\)	\(=\)	\( ( - 1370 \nu^{13} + 1838 \nu^{12} - 6858 \nu^{11} + 15011 \nu^{10} - 31334 \nu^{9} + 65040 \nu^{8} + \cdots + 207036 ) / 478953 \) (-1370v^13 + 1838v^12 - 6858v^11 + 15011v^10 - 31334v^9 + 65040v^8 - 136143v^7 + 195390v^6 - 260361v^5 + 511218v^4 - 740637v^3 + 805059v^2 - 998001*v + 207036) / 478953
\(\beta_{12}\)	\(=\)	\( ( - 1381 \nu^{13} + 4522 \nu^{12} - 12546 \nu^{11} + 32308 \nu^{10} - 67969 \nu^{9} + 131502 \nu^{8} + \cdots + 1422279 ) / 478953 \) (-1381v^13 + 4522v^12 - 12546v^11 + 32308v^10 - 67969v^9 + 131502v^8 - 267102v^7 + 428688v^6 - 703134v^5 + 1049409v^4 - 1249452v^3 + 1453869v^2 - 2042658*v + 1422279) / 478953
\(\beta_{13}\)	\(=\)	\( ( 26 \nu^{13} - 80 \nu^{12} + 207 \nu^{11} - 509 \nu^{10} + 1073 \nu^{9} - 2112 \nu^{8} + 4302 \nu^{7} + \cdots - 25515 ) / 6561 \) (26v^13 - 80v^12 + 207v^11 - 509v^10 + 1073v^9 - 2112v^8 + 4302v^7 - 7443v^6 + 10800v^5 - 15822v^4 + 22005v^3 - 26973v^2 + 29889*v - 25515) / 6561

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( -\beta_{5} + \beta_{4} - 1 \) -b5 + b4 - 1
\(\nu^{3}\)	\(=\)	\( \beta_{12} - \beta_{11} + \beta_{9} + \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} + 2\beta_{2} - \beta _1 - 1 \) b12 - b11 + b9 + b8 - b7 - b5 + b4 - b3 + 2*b2 - b1 - 1
\(\nu^{4}\)	\(=\)	\( 2 \beta_{13} + 2 \beta_{12} + \beta_{11} - 3 \beta_{10} + 2 \beta_{9} - \beta_{8} + \beta_{7} + \beta_{5} + \cdots - 3 \) 2b13 + 2b12 + b11 - 3b10 + 2b9 - b8 + b7 + b5 - b4 - b3 + 2*b2 - 3
\(\nu^{5}\)	\(=\)	\( \beta_{13} + 6\beta_{11} - 6\beta_{10} + 3\beta_{7} - 4\beta_{3} + 2\beta_{2} - 8\beta _1 + 1 \) b13 + 6b11 - 6b10 + 3b7 - 4b3 + 2b2 - 8b1 + 1
\(\nu^{6}\)	\(=\)	\( - 6 \beta_{13} - 5 \beta_{12} - 4 \beta_{11} - 3 \beta_{10} + 4 \beta_{9} - 2 \beta_{8} - \beta_{7} + \cdots - 7 \) -6b13 - 5b12 - 4b11 - 3b10 + 4b9 - 2b8 - b7 + 2b5 - 2b4 - 7b3 + 2b2 - 7*b1 - 7
\(\nu^{7}\)	\(=\)	\( 2 \beta_{13} - 4 \beta_{12} - 11 \beta_{11} - 3 \beta_{10} + 5 \beta_{9} - 25 \beta_{8} + 7 \beta_{7} + \cdots - 3 \) 2b13 - 4b12 - 11b11 - 3b10 + 5b9 - 25b8 + 7b7 + 9b6 + b5 - 10b4 - 4b3 - 10b2 - 3b1 - 3
\(\nu^{8}\)	\(=\)	\( 7 \beta_{13} + 3 \beta_{12} + 3 \beta_{11} + 12 \beta_{10} - 15 \beta_{9} - 6 \beta_{8} - 9 \beta_{7} + \cdots + 49 \) 7b13 + 3b12 + 3b11 + 12b10 - 15b9 - 6b8 - 9b7 + 27b6 + 6b5 - 6b4 - 4b3 + 29b2 + 4*b1 + 49
\(\nu^{9}\)	\(=\)	\( - 27 \beta_{13} - 2 \beta_{12} - 7 \beta_{11} + 6 \beta_{10} - 29 \beta_{9} + 19 \beta_{8} - 13 \beta_{7} + \cdots + 50 \) -27b13 - 2b12 - 7b11 + 6b10 - 29b9 + 19b8 - 13b7 + 36b6 + 44b5 + b4 + 29b3 - 16b2 + 77b1 + 50
\(\nu^{10}\)	\(=\)	\( - \beta_{13} + 17 \beta_{12} - 5 \beta_{11} - 3 \beta_{10} - 64 \beta_{9} - 40 \beta_{8} + 85 \beta_{7} + \cdots + 36 \) -b13 + 17b12 - 5b11 - 3b10 - 64b9 - 40b8 + 85b7 + 45b6 + 7b5 + 56b4 + 23b3 - 172b2 + 63b1 + 36
\(\nu^{11}\)	\(=\)	\( 49 \beta_{13} + 15 \beta_{12} - 63 \beta_{11} + 192 \beta_{10} - 12 \beta_{9} + 78 \beta_{8} + \cdots - 146 \) 49b13 + 15b12 - 63b11 + 192b10 - 12b9 + 78b8 + 24b7 + 9b6 + 3b5 + 123b4 + 68b3 + 236b2 + 79*b1 - 146
\(\nu^{12}\)	\(=\)	\( - 48 \beta_{13} + 28 \beta_{12} - 37 \beta_{11} - 66 \beta_{10} + 46 \beta_{9} + 175 \beta_{8} + \cdots - 244 \) -48b13 + 28b12 - 37b11 - 66b10 + 46b9 + 175b8 + 83b7 - 9b6 + 140b5 - 104b4 + 254b3 + 317b2 + 53*b1 - 244
\(\nu^{13}\)	\(=\)	\( 131 \beta_{13} + 227 \beta_{12} + 460 \beta_{11} - 759 \beta_{10} - 349 \beta_{9} - 28 \beta_{8} + \cdots + 75 \) 131b13 + 227b12 + 460b11 - 759b10 - 349b9 - 28b8 + 298b7 + 40b5 - 67b4 - 274b3 - 334b2 - 573b1 + 75

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 247.7
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{2} + T + 1)^{7} \) (T^2 + T + 1)^7
$3$	\( T^{14} - 4 T^{13} + \cdots + 2187 \) T^14 - 4T^13 + 12T^12 - 31T^11 + 73T^10 - 150T^9 + 297T^8 - 549T^7 + 891T^6 - 1350T^5 + 1971T^4 - 2511T^3 + 2916T^2 - 2916*T + 2187
$5$	\( T^{14} - 5 T^{13} + \cdots + 1936 \) T^14 - 5T^13 + 37T^12 - 102T^11 + 567T^10 - 1356T^9 + 5389T^8 - 6470T^7 + 16792T^6 - 2040T^5 + 38628T^4 - 4944T^3 + 10000T^2 + 1408*T + 1936
$7$	\( T^{14} + T^{13} + \cdots + 64 \) T^14 + T^13 + 19T^12 - 36T^11 + 273T^10 - 270T^9 + 829T^8 - 494T^7 + 1708T^6 - 864T^5 + 1176T^4 - 288T^3 + 448T^2 - 128T + 64
$11$	\( T^{14} - 10 T^{13} + \cdots + 1227664 \) T^14 - 10T^13 + 85T^12 - 428T^11 + 2153T^10 - 8339T^9 + 32631T^8 - 95370T^7 + 250800T^6 - 458708T^5 + 772628T^4 - 876368T^3 + 1255888T^2 - 1041520*T + 1227664
$13$	\( T^{14} - 4 T^{13} + \cdots + 808201 \) T^14 - 4T^13 + 49T^12 - 282T^11 + 2136T^10 - 9243T^9 + 36136T^8 - 96100T^7 + 243814T^6 - 478671T^5 + 967056T^4 - 1434582T^3 + 1893589T^2 - 1417723*T + 808201
$17$	\( (T^{7} + 13 T^{6} + \cdots - 7688)^{2} \) (T^7 + 13T^6 - 6T^5 - 559T^4 - 1114T^3 + 3828T^2 + 3760T - 7688)^2
$19$	\( (T^{7} + T^{6} - 60 T^{5} + \cdots - 4335)^{2} \) (T^7 + T^6 - 60T^5 - 65T^4 + 1033T^3 + 1197T^2 - 3978*T - 4335)^2
$23$	\( T^{14} + \cdots + 20597129289 \) T^14 - 15T^13 + 231T^12 - 1996T^11 + 19662T^10 - 138687T^9 + 1064326T^8 - 5916798T^7 + 34297872T^6 - 151137843T^5 + 707502426T^4 - 2412346680T^3 + 7674632793T^2 - 14017879458*T + 20597129289
$29$	\( T^{14} - 30 T^{13} + \cdots + 4100625 \) T^14 - 30T^13 + 561T^12 - 6648T^11 + 58164T^10 - 362925T^9 + 1713654T^8 - 5713164T^7 + 14107266T^6 - 21724281T^5 + 24149340T^4 - 8674128T^3 + 24487839T^2 - 9349425*T + 4100625
$31$	\( T^{14} + \cdots + 1782106225 \) T^14 + 13T^13 + 253T^12 + 2282T^11 + 33704T^10 + 275039T^9 + 2372208T^8 + 10110372T^7 + 41764650T^6 + 56748335T^5 + 187906520T^4 - 69975742T^3 + 1394463079T^2 - 1296507080*T + 1782106225
$37$	\( (T^{7} - 7 T^{6} + \cdots + 147880)^{2} \) (T^7 - 7T^6 - 102T^5 + 609T^4 + 3162T^3 - 16920T^2 - 29656T + 147880)^2
$41$	\( (T^{2} + T + 1)^{7} \) (T^2 + T + 1)^7
$43$	\( T^{14} - 5 T^{13} + \cdots + 1089936 \) T^14 - 5T^13 + 67T^12 + 76T^11 + 1567T^10 + 2782T^9 + 25981T^8 + 63364T^7 + 228100T^6 + 340224T^5 + 739836T^4 + 802800T^3 + 1533600T^2 + 1165104*T + 1089936
$47$	\( T^{14} + \cdots + 793210896 \) T^14 - 3T^13 + 165T^12 + 2T^11 + 20547T^10 - 11736T^9 + 678997T^8 + 397656T^7 + 15509940T^6 + 9953352T^5 + 165577236T^4 + 332090640T^3 + 1106875872T^2 + 964222704*T + 793210896
$53$	\( (T^{7} + 12 T^{6} + \cdots - 311796)^{2} \) (T^7 + 12T^6 - 111T^5 - 1285T^4 + 4578T^3 + 39780T^2 - 72036T - 311796)^2
$59$	\( T^{14} + \cdots + 154876879936 \) T^14 - 13T^13 + 373T^12 - 918T^11 + 48999T^10 + 8832T^9 + 5455177T^8 + 15955598T^7 + 261774868T^6 + 132423984T^5 + 4407473256T^4 - 3630215136T^3 + 67905423424T^2 - 90423817792*T + 154876879936
$61$	\( T^{14} + \cdots + 9981608464 \) T^14 - 55T^13 + 1867T^12 - 41268T^11 + 681771T^10 - 8271570T^9 + 77049265T^8 - 517206004T^7 + 2554456360T^6 - 7471115232T^5 + 15641573604T^4 - 1340960640T^3 + 145185139120T^2 + 37724861168*T + 9981608464
$67$	\( T^{14} - 15 T^{13} + \cdots + 85766121 \) T^14 - 15T^13 + 219T^12 - 932T^11 + 6606T^10 - 9093T^9 + 119236T^8 - 50466T^7 + 1458144T^6 + 1067661T^5 + 11854080T^4 + 4723110T^3 + 36756909T^2 + 85766121
$71$	\( (T^{7} + 12 T^{6} + \cdots + 325368)^{2} \) (T^7 + 12T^6 - 231T^5 - 2743T^4 + 6810T^3 + 123420T^2 + 363456T + 325368)^2
$73$	\( (T^{7} - 270 T^{5} + \cdots + 93192)^{2} \) (T^7 - 270T^5 - 412T^4 + 12849T^3 - 11316T^2 - 74652*T + 93192)^2
$79$	\( T^{14} + \cdots + 1821923856 \) T^14 + 12T^13 + 255T^12 + 1054T^11 + 26433T^10 + 144123T^9 + 1269541T^8 + 3535056T^7 + 20081628T^6 + 35323908T^5 + 222179604T^4 + 206609472T^3 + 838343664T^2 - 486597600*T + 1821923856
$83$	\( T^{14} + \cdots + 23364956736 \) T^14 - 36T^13 + 855T^12 - 12438T^11 + 138975T^10 - 1145403T^9 + 7966827T^8 - 45291114T^7 + 239017752T^6 - 1059712416T^5 + 4190876712T^4 - 12031149888T^3 + 26830235232T^2 - 29726212032*T + 23364956736
$89$	\( (T^{7} + 16 T^{6} + \cdots - 9980)^{2} \) (T^7 + 16T^6 - 105T^5 - 1741T^4 - 3682T^3 + 10380T^2 + 25132T - 9980)^2
$97$	\( T^{14} + \cdots + 229263326405136 \) T^14 + T^13 + 433T^12 - 1586T^11 + 122941T^10 - 586976T^9 + 21683005T^8 - 134244002T^7 + 2812070968T^6 - 16354927176T^5 + 229201471260T^4 - 1263501838128T^3 + 12575858951664T^2 - 46357166081952T + 229263326405136
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\(n\)	\(83\)	\(703\)
\(\chi(n)\)	\(-1 + \beta_{2}\)	\(1\)