LMFDB - Newform orbit 351.2.e.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [351,2,Mod(118,351)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(351, 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("351.118"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 351 = 3^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	351.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$2.80274911095$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 3 x^{10} - x^{9} - 2 x^{8} + 9 x^{7} + 24 x^{6} + 27 x^{5} - 18 x^{4} - 27 x^{3} + \cdots + 729 $ x^12 - x^11 - 3x^10 - x^9 - 2x^8 + 9x^7 + 24x^6 + 27x^5 - 18x^4 - 27x^3 - 243x^2 - 243*x + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 117)
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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - x^{11} - 3 x^{10} - x^{9} - 2 x^{8} + 9 x^{7} + 24 x^{6} + 27 x^{5} - 18 x^{4} - 27 x^{3} + \cdots + 729 $ x^12 - x^11 - 3*x^10 - x^9 - 2*x^8 + 9*x^7 + 24*x^6 + 27*x^5 - 18*x^4 - 27*x^3 - 243*x^2 - 243*x + 729 :

$\beta_{1}$	$=$	$ ( \nu^{11} - 12 \nu^{10} - 31 \nu^{9} - 28 \nu^{8} + 133 \nu^{6} + 345 \nu^{5} + 456 \nu^{4} + \cdots - 3321 ) / 162 $ (v^11 - 12v^10 - 31v^9 - 28v^8 + 133v^6 + 345v^5 + 456v^4 + 144v^3 - 639v^2 - 2430*v - 3321) / 162
$\beta_{2}$	$=$	$ ( - 28 \nu^{11} - 53 \nu^{10} - 42 \nu^{9} + 46 \nu^{8} + 299 \nu^{7} + 603 \nu^{6} + 660 \nu^{5} + \cdots + 1944 ) / 729 $ (-28v^11 - 53v^10 - 42v^9 + 46v^8 + 299v^7 + 603v^6 + 660v^5 - 135v^4 - 1710v^3 - 4752v^2 - 4536*v + 1944) / 729
$\beta_{3}$	$=$	$ ( 61 \nu^{11} + 152 \nu^{10} + 117 \nu^{9} - 52 \nu^{8} - 686 \nu^{7} - 1605 \nu^{6} - 1965 \nu^{5} + \cdots - 2187 ) / 1458 $ (61v^11 + 152v^10 + 117v^9 - 52v^8 - 686v^7 - 1605v^6 - 1965v^5 - 342v^4 + 3762v^3 + 12177v^2 + 14904*v - 2187) / 1458
$\beta_{4}$	$=$	$ ( 47 \nu^{11} + 13 \nu^{10} - 129 \nu^{9} - 245 \nu^{8} - 424 \nu^{7} - 174 \nu^{6} + 984 \nu^{5} + \cdots - 22356 ) / 729 $ (47v^11 + 13v^10 - 129v^9 - 245v^8 - 424v^7 - 174v^6 + 984v^5 + 2763v^4 + 3231v^3 + 3564v^2 - 7290*v - 22356) / 729
$\beta_{5}$	$=$	$ ( - 103 \nu^{11} - 116 \nu^{10} + 87 \nu^{9} + 364 \nu^{8} + 1046 \nu^{7} + 1329 \nu^{6} + \cdots + 30861 ) / 1458 $ (-103v^11 - 116v^10 + 87v^9 + 364v^8 + 1046v^7 + 1329v^6 - 51v^5 - 3690v^4 - 6948v^3 - 12393v^2 + 486*v + 30861) / 1458
$\beta_{6}$	$=$	$ ( - 133 \nu^{11} - 176 \nu^{10} + 51 \nu^{9} + 394 \nu^{8} + 1358 \nu^{7} + 1941 \nu^{6} + \cdots + 33777 ) / 1458 $ (-133v^11 - 176v^10 + 51v^9 + 394v^8 + 1358v^7 + 1941v^6 + 795v^5 - 3744v^4 - 8676v^3 - 17253v^2 - 6318*v + 33777) / 1458
$\beta_{7}$	$=$	$ ( - 68 \nu^{11} - 37 \nu^{10} + 156 \nu^{9} + 320 \nu^{8} + 673 \nu^{7} + 399 \nu^{6} - 1083 \nu^{5} + \cdots + 31104 ) / 729 $ (-68v^11 - 37v^10 + 156v^9 + 320v^8 + 673v^7 + 399v^6 - 1083v^5 - 3843v^4 - 5040v^3 - 5832v^2 + 8505*v + 31104) / 729
$\beta_{8}$	$=$	$ ( - 107 \nu^{11} - 91 \nu^{10} + 159 \nu^{9} + 440 \nu^{8} + 1060 \nu^{7} + 1008 \nu^{6} - 822 \nu^{5} + \cdots + 39609 ) / 729 $ (-107v^11 - 91v^10 + 159v^9 + 440v^8 + 1060v^7 + 1008v^6 - 822v^5 - 4698v^4 - 7524v^3 - 10881v^2 + 6237*v + 39609) / 729
$\beta_{9}$	$=$	$ ( - 25 \nu^{11} - 24 \nu^{10} + 31 \nu^{9} + 94 \nu^{8} + 252 \nu^{7} + 263 \nu^{6} - 117 \nu^{5} + \cdots + 8505 ) / 162 $ (-25v^11 - 24v^10 + 31v^9 + 94v^8 + 252v^7 + 263v^6 - 117v^5 - 996v^4 - 1674v^3 - 2709v^2 + 810*v + 8505) / 162
$\beta_{10}$	$=$	$ ( 99 \nu^{11} + 86 \nu^{10} - 155 \nu^{9} - 414 \nu^{8} - 986 \nu^{7} - 913 \nu^{6} + 795 \nu^{5} + \cdots - 37503 ) / 486 $ (99v^11 + 86v^10 - 155v^9 - 414v^8 - 986v^7 - 913v^6 + 795v^5 + 4494v^4 + 6822v^3 + 10197v^2 - 5940*v - 37503) / 486
$\beta_{11}$	$=$	$ ( 307 \nu^{11} + 254 \nu^{10} - 501 \nu^{9} - 1324 \nu^{8} - 3092 \nu^{7} - 2823 \nu^{6} + \cdots - 118341 ) / 1458 $ (307v^11 + 254v^10 - 501v^9 - 1324v^8 - 3092v^7 - 2823v^6 + 2733v^5 + 14166v^4 + 21744v^3 + 31833v^2 - 20412*v - 118341) / 1458

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

$\nu$	$=$	$ ( \beta_{11} + \beta_{9} + 2\beta_{7} - 2\beta_{6} - \beta_{4} + 2\beta_1 ) / 3 $ (b11 + b9 + 2b7 - 2b6 - b4 + 2*b1) / 3
$\nu^{2}$	$=$	$ ( \beta_{11} + \beta_{9} + 3\beta_{8} - \beta_{7} + \beta_{6} + 2\beta_{4} + 3\beta_{3} - 3\beta_{2} + 2\beta_1 ) / 3 $ (b11 + b9 + 3b8 - b7 + b6 + 2b4 + 3b3 - 3b2 + 2*b1) / 3
$\nu^{3}$	$=$	$ ( \beta_{11} - 3 \beta_{10} + 4 \beta_{9} - 6 \beta_{8} - \beta_{7} - 2 \beta_{6} + 3 \beta_{5} - \beta_{4} + \cdots + 6 ) / 3 $ (b11 - 3b10 + 4b9 - 6b8 - b7 - 2b6 + 3b5 - b4 + 3b3 + 2*b1 + 6) / 3
$\nu^{4}$	$=$	$ ( - 2 \beta_{11} + 6 \beta_{10} + 7 \beta_{9} + 6 \beta_{8} - \beta_{7} - 5 \beta_{6} - 12 \beta_{5} + \cdots + 3 ) / 3 $ (-2b11 + 6b10 + 7b9 + 6b8 - b7 - 5b6 - 12b5 - 4b4 - 3b3 - 3b2 + 5b1 + 3) / 3
$\nu^{5}$	$=$	$ ( \beta_{11} + \beta_{9} - 6 \beta_{8} + 17 \beta_{7} + 10 \beta_{6} - 18 \beta_{5} - 7 \beta_{4} + \cdots + 20 \beta_1 ) / 3 $ (b11 + b9 - 6b8 + 17b7 + 10b6 - 18b5 - 7b4 + 12b3 - 3b2 + 20b1) / 3
$\nu^{6}$	$=$	$ ( \beta_{11} + 15 \beta_{10} + 4 \beta_{9} + 3 \beta_{8} - \beta_{7} - 2 \beta_{6} + 57 \beta_{5} + \cdots + 33 ) / 3 $ (b11 + 15b10 + 4b9 + 3b8 - b7 - 2b6 + 57b5 - b4 + 12b3 - 27b2 + 11b1 + 33) / 3
$\nu^{7}$	$=$	$ ( - 38 \beta_{11} + 24 \beta_{10} + 61 \beta_{9} - 39 \beta_{8} + 8 \beta_{7} - 5 \beta_{6} - 48 \beta_{5} + \cdots - 15 ) / 3 $ (-38b11 + 24b10 + 61b9 - 39b8 + 8b7 - 5b6 - 48b5 + 32b4 + 33b3 - 3b2 + 23*b1 - 15) / 3
$\nu^{8}$	$=$	$ ( - 53 \beta_{11} - 27 \beta_{10} - 35 \beta_{9} + 12 \beta_{8} - 73 \beta_{7} + 28 \beta_{6} - 81 \beta_{5} + \cdots - 18 ) / 3 $ (-53b11 - 27b10 - 35b9 + 12b8 - 73b7 + 28b6 - 81b5 + 11b4 + 57b3 + 6b2 + 29*b1 - 18) / 3
$\nu^{9}$	$=$	$ ( - 17 \beta_{11} + 78 \beta_{10} + 67 \beta_{9} - 24 \beta_{8} + 80 \beta_{7} + 61 \beta_{6} + \cdots - 201 ) / 3 $ (-17b11 + 78b10 + 67b9 - 24b8 + 80b7 + 61b6 - 87b5 - 82b4 - 15b3 - 171b2 + 119*b1 - 201) / 3
$\nu^{10}$	$=$	$ ( - 155 \beta_{11} + 231 \beta_{10} + 34 \beta_{9} - 138 \beta_{8} + 44 \beta_{7} + 184 \beta_{6} + \cdots + 291 ) / 3 $ (-155b11 + 231b10 + 34b9 - 138b8 + 44b7 + 184b6 + 105b5 - 67b4 + 150b3 + 78b2 - 13*b1 + 291) / 3
$\nu^{11}$	$=$	$ ( - 512 \beta_{11} + 135 \beta_{10} - 71 \beta_{9} - 348 \beta_{8} + 53 \beta_{7} + 46 \beta_{6} + \cdots + 99 ) / 3 $ (-512b11 + 135b10 - 71b9 - 348b8 + 53b7 + 46b6 - 36b5 + 326b4 + 48b3 - 120b2 + 38*b1 + 99) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/351\mathbb{Z}\right)^\times$.

$n$	$28$	$326$
$\chi(n)$	$1$	$-1 - \beta_{5}$

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} $

118.1

−1.70946	+	0.278853i
1.70358	+	0.312736i
0.471837	−	1.66654i
1.70010	−	0.331167i
−0.372202	+	1.69159i
−1.29386	−	1.15149i
−1.70946	−	0.278853i
1.70358	−	0.312736i
0.471837	+	1.66654i
1.70010	+	0.331167i
−0.372202	−	1.69159i
−1.29386	+	1.15149i

−1.25245

−

2.16930i

−2.13724

3.70181i

0.495441

−

0.858128i

−1.58167

−

2.73953i

5.69734

−2.48205

118.2

−1.10985

−

1.92231i

−1.46353

2.53491i

−1.13044

1.95798i

−0.183027

−

0.317013i

2.05779

5.01848

118.3

−0.248047

−

0.429630i

0.876945

−

1.51891i

−0.663441

1.14911i

−1.56464

−

2.71004i

−1.86228

0.658258

118.4

−0.129090

−

0.223591i

0.966671

−

1.67432i

1.73641

−

3.00755i

2.21165

3.83070i

−1.01551

−0.896615

118.5

0.636681

1.10276i

0.189274

−

0.327833i

−0.134264

0.232553i

0.577621

1.00047i

3.02875

−0.341935

118.6

1.10275

1.91002i

−1.43212

2.48050i

−1.80370

3.12410i

0.540062

0.935414i

−1.90608

−7.95614

235.1