LMFDB - Newform orbit 792.2.ce.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [792,2,Mod(139,792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(792, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([15, 15, 10, 21]))

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

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

chi := DirichletCharacter("792.139");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 792 = 2^{3} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	792.ce (of order $30$, degree $8$, 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:	$6.32415184009$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$2$ over $\Q(\zeta_{30})$
Coefficient field:	16.0.26873856000000000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 2x^{14} - 8x^{10} - 16x^{8} - 32x^{6} + 128x^{2} + 256 $ x^16 + 2x^14 - 8x^10 - 16x^8 - 32x^6 + 128*x^2 + 256
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{30}]$

$q$-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 + \beta_{8} q^{2} + ( - \beta_{13} - \beta_{12} + \cdots - \beta_1) q^{3}+ \cdots + ( - \beta_{14} - 2 \beta_{12} + \cdots - 2 \beta_1) q^{9}+O(q^{10}) $ q + b8 * q^2 + (-b13 - b12 - b10 + b8 + b6 + b5 - b2 - b1) * q^3 + (2b9 - 2b2) * q^4 + (-2b15 - 2b14 - 2b11 - b10 + 2b9 + 2b7 - 2b4 - 2b2) q^6 + (-2b13 - 2b10) * q^8 + (-b14 - 2b12 - b11 - 2b10 + 2b5 + 2b3 - 2b1) q^9	$ q + \beta_{8} q^{2} + ( - \beta_{13} - \beta_{12} + \cdots - \beta_1) q^{3}+ \cdots + (\beta_{9} + 7 \beta_{5} - \beta_{2}) q^{99}+O(q^{100}) $ q + b8 * q^2 + (-b13 - b12 - b10 + b8 + b6 + b5 - b2 - b1) * q^3 + (2b9 - 2b2) * q^4 + (-2b15 - 2b14 - 2b11 - b10 + 2b9 + 2b7 - 2b4 - 2b2) q^6 + (-2b13 - 2b10) * q^8 + (-b14 - 2b12 - b11 - 2b10 + 2b5 + 2b3 - 2b1) q^9 + (-3b14 - b13 - b10 + 3b9 - 3b4 + 3) q^11 + (2b14 - 2b6) * q^12 - 4b11 q^16 + (-3b15 + 2b13 + 2b12 + 2b10 - 2b8 - 2b6 - 2b5 - 3b4 - 3b2) q^17 + (-b12 + 4) * q^18 + (-b15 + 3b12 - 3b8 + b7 - 3b3) q^19 + (3b12 - 2b11 + 3b10 - 3b6 - 3b5 - 3b3 + 3b1) q^22 + (-4b7 + 2b3 + 4) * q^24 + (5b15 + 5b11 - 5b7 + 5b2 + 5) * q^25 + (b8 + 5b4 - b1) q^27 + (-4b12 + 4b3) * q^32 + (-5b7 - 2b3 + 5) * q^33 + (4b15 + 4b14 + 3b13 + 3b12 + 4b11 - 8b9 - 3b8 - 4b7 - 3b6 + 4b4 + 4b2 + 3b1) * q^34 + (-2b15 + 4b8) * q^36 + (-6b9 + b8 + b5 - 6b4 + 6b2 - b1) q^38 + (-4b13 - 8b12 - 3b11 - 8b10 - 3b9 + 4b8 + 8b6 + 4b5 + 4b3 - 8b1) * q^41 + (5b15 + 5b14 - 3b6 + 5b4 - 3b1) q^43 + (-2b12 - 6b7 + 2b3) q^44 + (4b15 + 4b4 + 4b1) q^48 + (-7b14 - 7b11) * q^49 + (5b12 + 5b10 - 5b5 - 5b3 + 5b1) q^50 + (8b14 + 5b13 - b12 + 7b11 + 4b10 - b9 + b5 + b4 + b3 - b1 - 1) * q^51 + (-5b13 - 5b12 - 5b10 + 5b8 + 5b6 + 5b5 - 2b2 - 5b1) * q^54 + (5b15 + 5b14 + 2b13 + 2b12 + 5b11 + 6b10 - 12b9 - 2b8 - 5b7 - 2b6 + 5b4 + 5b2 + 2b1) q^57 + (3b14 - 3b7 + 5b6 - 5b3 + 3) * q^59 + 8b4 q^64 + (-4b15 - 4b4 + 5b1) q^66 + (7b15 + 3b12 + 7b11 + 3b10 + 7b9 - 3b8 - 7b7 - 3b6 + 7b2 + 3b1 + 7) * q^67 + (6b15 + 6b14 + 4b13 + 6b11 - 6b7 + 4b6 + 6b2 + 6) q^68 + (8b9 + 2b5 - 8b2) q^72 + (-b14 - b11 + 6b10 + b9 - 6b8 + b7 - b4 - b2) * q^73 + (5b12 - 5b7 - 5b3) q^75 + (-2b14 + 12b13 + 6b12 + 12b10 + 2b9 - 6b8 - 6b6 - 6b5 - 2b4 - 2b2 + 6b1 + 2) q^76 + (-7b15 - 4b13 - 7b11 + 7b7 - 7b2 - 7) q^81 + (-8b15 + 3b13 - 3b12 - 8b11 + 16b7 + 3b3 - 8b2 - 8) q^82 + (-9b15 - b13 - 9b11 + 18b7 - b3 - 9b2 - 18) * q^83 + (-5b13 - 5b12 - 5b10 - 6b9 + 5b8 - 6b7 + 5b6 + 5b3 - 5b1 + 6) q^86 + (-6b8 + 4b4 + 6b1) q^88 + (-9b14 + 2b8 - 2b6 - 9b4 - 2b1) q^89 + (-4b13 - 4b12 - 4b10 + 8b9 + 4b8 + 4b6 - 4b1) q^96 + (6b12 - 5b9 - 6b5 + 5b2 + 5) * q^97 - 7b12 q^98 + (b9 + 7b5 - b2) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{3} - 4 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10}) $ 16 * q + 2 * q^3 - 4 * q^4 + 4 * q^6 - 2 * q^9	$ 16 q + 2 q^{3} - 4 q^{4} + 4 q^{6} - 2 q^{9} + 12 q^{11} + 8 q^{12} + 8 q^{16} + 64 q^{18} + 10 q^{19} + 4 q^{22} + 32 q^{24} + 10 q^{25} + 20 q^{27} + 40 q^{33} + 8 q^{34} + 4 q^{36} - 12 q^{38} + 18 q^{41} + 30 q^{43} - 48 q^{44} + 8 q^{48} - 14 q^{49} + 10 q^{51} + 4 q^{54} + 18 q^{57} + 36 q^{59} + 32 q^{64} - 8 q^{66} - 14 q^{67} + 36 q^{68} - 16 q^{72} - 40 q^{75} + 12 q^{76} - 14 q^{81} + 48 q^{82} - 90 q^{83} + 72 q^{86} + 16 q^{88} - 72 q^{89} - 32 q^{96} + 90 q^{97} - 2 q^{99}+O(q^{100}) $ 16 * q + 2 * q^3 - 4 * q^4 + 4 * q^6 - 2 * q^9 + 12 * q^11 + 8 * q^12 + 8 * q^16 + 64 * q^18 + 10 * q^19 + 4 * q^22 + 32 * q^24 + 10 * q^25 + 20 * q^27 + 40 * q^33 + 8 * q^34 + 4 * q^36 - 12 * q^38 + 18 * q^41 + 30 * q^43 - 48 * q^44 + 8 * q^48 - 14 * q^49 + 10 * q^51 + 4 * q^54 + 18 * q^57 + 36 * q^59 + 32 * q^64 - 8 * q^66 - 14 * q^67 + 36 * q^68 - 16 * q^72 - 40 * q^75 + 12 * q^76 - 14 * q^81 + 48 * q^82 - 90 * q^83 + 72 * q^86 + 16 * q^88 - 72 * q^89 - 32 * q^96 + 90 * q^97 - 2 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{16} + 2x^{14} - 8x^{10} - 16x^{8} - 32x^{6} + 128x^{2} + 256 $ x^16 + 2*x^14 - 8*x^10 - 16*x^8 - 32*x^6 + 128*x^2 + 256 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( \nu^{2} ) / 2 $ (v^2) / 2
$\beta_{3}$	$=$	$ ( \nu^{5} ) / 4 $ (v^5) / 4
$\beta_{4}$	$=$	$ ( \nu^{6} ) / 8 $ (v^6) / 8
$\beta_{5}$	$=$	$ ( \nu^{7} ) / 8 $ (v^7) / 8
$\beta_{6}$	$=$	$ ( \nu^{9} ) / 16 $ (v^9) / 16
$\beta_{7}$	$=$	$ ( \nu^{10} ) / 32 $ (v^10) / 32
$\beta_{8}$	$=$	$ ( \nu^{11} ) / 32 $ (v^11) / 32
$\beta_{9}$	$=$	$ ( \nu^{12} ) / 64 $ (v^12) / 64
$\beta_{10}$	$=$	$ ( \nu^{13} ) / 64 $ (v^13) / 64
$\beta_{11}$	$=$	$ ( \nu^{14} ) / 128 $ (v^14) / 128
$\beta_{12}$	$=$	$ ( \nu^{15} ) / 128 $ (v^15) / 128
$\beta_{13}$	$=$	$ ( -\nu^{13} + 32\nu^{3} ) / 64 $ (-v^13 + 32*v^3) / 64
$\beta_{14}$	$=$	$ ( -\nu^{14} + 32\nu^{4} ) / 128 $ (-v^14 + 32*v^4) / 128
$\beta_{15}$	$=$	$ ( -\nu^{14} + 4\nu^{10} + 8\nu^{8} - 64\nu^{2} - 128 ) / 128 $ (-v^14 + 4v^10 + 8v^8 - 64*v^2 - 128) / 128

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ 2\beta_{2} $ 2*b2
$\nu^{3}$	$=$	$ 2\beta_{13} + 2\beta_{10} $ 2b13 + 2b10
$\nu^{4}$	$=$	$ 4\beta_{14} + 4\beta_{11} $ 4b14 + 4b11
$\nu^{5}$	$=$	$ 4\beta_{3} $ 4*b3
$\nu^{6}$	$=$	$ 8\beta_{4} $ 8*b4
$\nu^{7}$	$=$	$ 8\beta_{5} $ 8*b5
$\nu^{8}$	$=$	$ 16\beta_{15} + 16\beta_{11} - 16\beta_{7} + 16\beta_{2} + 16 $ 16b15 + 16b11 - 16b7 + 16b2 + 16
$\nu^{9}$	$=$	$ 16\beta_{6} $ 16*b6
$\nu^{10}$	$=$	$ 32\beta_{7} $ 32*b7
$\nu^{11}$	$=$	$ 32\beta_{8} $ 32*b8
$\nu^{12}$	$=$	$ 64\beta_{9} $ 64*b9
$\nu^{13}$	$=$	$ 64\beta_{10} $ 64*b10
$\nu^{14}$	$=$	$ 128\beta_{11} $ 128*b11
$\nu^{15}$	$=$	$ 128\beta_{12} $ 128*b12

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/792\mathbb{Z}\right)^\times$.

$n$	$145$	$199$	$353$	$397$
$\chi(n)$	$-\beta_{9}$	$-1$	$-\beta_{7}$	$-1$

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} $

139.1

−0.575212	−	1.29195i
0.575212	+	1.29195i
1.05097	−	0.946294i
−1.05097	+	0.946294i
1.05097	+	0.946294i
−1.05097	−	0.946294i
−1.40647	+	0.147826i
1.40647	−	0.147826i
−1.40647	−	0.147826i
1.40647	+	0.147826i
0.294032	+	1.38331i
−0.294032	−	1.38331i
−0.575212	+	1.29195i
0.575212	−	1.29195i
0.294032	−	1.38331i
−0.294032	+	1.38331i

−1.40647

−

0.147826i

−0.381835

−

1.68944i

1.95630

0.415823i

0.287296

2.43258i

−2.68999

−

0.874032i

−2.70840

1.29017i

139.2

1.40647

0.147826i

1.72010

0.203149i

1.95630

0.415823i

2.38923

0.539996i

2.68999

0.874032i

2.91746

0.698871i

211.1

−0.294032

−

1.38331i

1.30194

1.14235i

−1.82709

0.813473i

1.19741

−

2.13687i

1.66251

2.28825i

0.390085

2.97453i

211.2

0.294032

1.38331i

−1.51099

0.846696i

−1.82709

0.813473i

−1.61552

−

1.84122i

−1.66251

−

2.28825i

1.56621

−

2.55871i

259.1

−0.294032

1.38331i

1.30194

−

1.14235i

−1.82709

−

0.813473i

1.19741

2.13687i

1.66251

−

2.28825i

0.390085

−

2.97453i

259.2

0.294032

−

1.38331i

−1.51099

−

0.846696i

−1.82709

−

0.813473i

−1.61552

1.84122i

−1.66251

2.28825i

1.56621

2.55871i

283.1

−0.575212

1.29195i

−0.684116

1.59122i

−1.33826

−

1.48629i

−1.66226

−

1.79913i

2.68999

−

0.874032i

−2.06397

−

2.17716i

283.2

0.575212

−

1.29195i

−1.27218

−

1.17540i

−1.33826

−

1.48629i

−2.25033

0.967486i

−2.68999

0.874032i

0.236879

2.99063i

403.1

−0.575212

−

1.29195i

−0.684116

−

1.59122i

−1.33826

1.48629i

−1.66226

1.79913i

2.68999

0.874032i

−2.06397

2.17716i

403.2

0.575212

1.29195i

−1.27218

1.17540i

−1.33826

1.48629i

−2.25033

−

0.967486i

−2.68999

−

0.874032i

0.236879

−

2.99063i

475.1

−1.05097

0.946294i

0.338333

−

1.69869i

0.209057

−

1.98904i

1.25188

2.10542i

1.66251

2.28825i

−2.77106

−

1.14944i

475.2

1.05097

−

0.946294i

1.48876

0.885212i

0.209057

−

1.98904i

2.40230

−

0.478475i

−1.66251

−

2.28825i

1.43280

2.63573i

547.1

−1.40647

0.147826i

−0.381835

1.68944i

1.95630

−

0.415823i

0.287296

−

2.43258i

−2.68999

0.874032i

−2.70840

−

1.29017i

547.2

1.40647

−

0.147826i

1.72010

−

0.203149i

1.95630

−

0.415823i

2.38923

−

0.539996i

2.68999

−

0.874032i

2.91746

−

0.698871i

787.1

−1.05097

−

0.946294i

0.338333

1.69869i

0.209057

1.98904i

1.25188

−

2.10542i

1.66251

−

2.28825i

−2.77106

1.14944i

787.2

1.05097

0.946294i

1.48876

−

0.885212i

0.209057

1.98904i

2.40230

0.478475i

−1.66251

2.28825i

1.43280

−

2.63573i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2}) $
99.o	odd	30	1	inner
792.ce	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	792.2.ce.a	✓	16
8.d	odd	2	1	CM	792.2.ce.a	✓	16
9.c	even	3	1		792.2.ce.b	yes	16
11.d	odd	10	1		792.2.ce.b	yes	16
72.p	odd	6	1		792.2.ce.b	yes	16
88.k	even	10	1		792.2.ce.b	yes	16
99.o	odd	30	1	inner	792.2.ce.a	✓	16
792.ce	even	30	1	inner	792.2.ce.a	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
792.2.ce.a	✓	16	1.a	even	1	1	trivial
792.2.ce.a	✓	16	8.d	odd	2	1	CM
792.2.ce.a	✓	16	99.o	odd	30	1	inner
792.2.ce.a	✓	16	792.ce	even	30	1	inner
792.2.ce.b	yes	16	9.c	even	3	1
792.2.ce.b	yes	16	11.d	odd	10	1
792.2.ce.b	yes	16	72.p	odd	6	1
792.2.ce.b	yes	16	88.k	even	10	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(792, [\chi])$:

$ T_{5} $

$ T_{7} $

$ T_{41}^{16} - 18 T_{41}^{15} + 221 T_{41}^{14} - 3264 T_{41}^{13} + 43774 T_{41}^{12} + \cdots + 243842434857025 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} + 2 T^{14} + \cdots + 256 $ T^16 + 2T^14 - 8T^10 - 16T^8 - 32T^6 + 128*T^2 + 256
$3$	$ T^{16} - 2 T^{15} + \cdots + 6561 $ T^16 - 2T^15 + 3T^14 - 10T^13 + 20T^12 - 28T^11 + 69T^10 - 140T^9 + 199T^8 - 420T^7 + 621T^6 - 756T^5 + 1620T^4 - 2430T^3 + 2187T^2 - 4374*T + 6561
$5$	$ T^{16} $ T^16
$7$	$ T^{16} $ T^16
$11$	$ (T^{8} - 6 T^{7} + \cdots + 14641)^{2} $ (T^8 - 6T^7 + 25T^6 - 84T^5 + 229T^4 - 924T^3 + 3025T^2 - 7986*T + 14641)^2
$13$	$ T^{16} $ T^16
$17$	$ T^{16} + \cdots + 13943122561 $ T^16 - 70T^14 - 510T^13 + 4369T^12 + 35700T^11 - 143285T^10 - 1491240T^9 + 5270766T^8 + 26322120T^7 + 22625T^6 - 1901487060T^5 + 11066011399T^4 - 23144493600T^3 + 10892087875T^2 + 9695630910T + 13943122561
$19$	$ T^{16} + \cdots + 151052709025 $ T^16 - 10T^15 - 77T^14 + 750T^13 + 6864T^12 - 45100T^11 - 461823T^10 + 3410160T^9 + 21620491T^8 - 243809740T^7 + 61688125T^6 + 4859828440T^5 - 7901699340T^4 - 20933681850T^3 + 159680320475T^2 - 195808275550*T + 151052709025
$23$	$ T^{16} $ T^16
$29$	$ T^{16} $ T^16
$31$	$ T^{16} $ T^16
$37$	$ T^{16} $ T^16
$41$	$ T^{16} + \cdots + 243842434857025 $ T^16 - 18T^15 + 221T^14 - 3264T^13 + 43774T^12 - 389622T^11 + 1995064T^10 - 24034026T^9 + 119780241T^8 - 1527803550T^7 + 8096357840T^6 + 44946942570T^5 + 1278792853870T^4 + 9555242666400T^3 + 45411391528225T^2 + 164376867830250*T + 243842434857025
$43$	$ T^{16} + \cdots + 40220976684001 $ T^16 - 30T^15 + 113T^14 + 5610T^13 - 40405T^12 - 954030T^11 + 13208498T^10 + 9744180T^9 - 851931341T^8 + 716107110T^7 + 41205765442T^6 - 66679449240T^5 - 1067019730150T^4 + 372805950720T^3 + 19738019090032T^2 + 48417752954460*T + 40220976684001
$47$	$ T^{16} $ T^16
$53$	$ T^{16} $ T^16
$59$	$ T^{16} + \cdots + 2136973261921 $ T^16 - 36T^15 + 284T^14 + 4344T^13 - 79854T^12 + 122352T^11 + 10759646T^10 - 193520778T^9 + 1218545727T^8 + 5340488208T^7 - 109477969894T^6 + 409761951798T^5 + 2621402327231T^4 - 12911909544774T^3 + 28579600874409T^2 + 689131370346*T + 2136973261921
$61$	$ T^{16} $ T^16
$67$	$ T^{16} + \cdots + 85\!\cdots\!25 $ T^16 + 14T^15 + 665T^14 + 6426T^13 + 247079T^12 + 2187766T^11 + 55690150T^10 + 389986884T^9 + 8212560331T^8 + 55255322850T^7 + 785865316810T^6 + 4362931376720T^5 + 47866571936030T^4 + 244493300950500T^3 + 1603497906780200T^2 + 3893294729434600*T + 8596378114389025
$71$	$ T^{16} $ T^16
$73$	$ T^{16} + \cdots + 45864304637761 $ T^16 - 150T^14 + 730T^13 + 69569T^12 - 109500T^11 - 18913725T^10 + 31334520T^9 + 4302533166T^8 + 6369433960T^7 - 2757120135T^6 + 3510443685180T^5 + 94937575732599T^4 + 582960336744800T^3 + 1218093937974555T^2 + 430708178627270T + 45864304637761
$79$	$ T^{16} $ T^16
$83$	$ T^{16} + \cdots + 94\!\cdots\!41 $ T^16 + 90T^15 + 4453T^14 + 153270T^13 + 4056055T^12 + 86697450T^11 + 1550034518T^10 + 23803879680T^9 + 319318569579T^8 + 3767193600210T^7 + 38882388247762T^6 + 346087388216520T^5 + 2613670803934750T^4 + 16970909231260620T^3 + 97596256315453712T^2 + 431423648736705180*T + 946249801719345841
$89$	$ (T^{4} + 18 T^{3} + \cdots + 401)^{4} $ (T^4 + 18T^3 - 121T^2 - 2178*T + 401)^4
$97$	$ T^{16} + \cdots + 93\!\cdots\!61 $ T^16 - 90T^15 + 4279T^14 - 140230T^13 + 3503335T^12 - 69516270T^11 + 1110972406T^10 - 14353843140T^9 + 148846313959T^8 - 1194150894410T^7 + 7041959295466T^6 - 33718242047680T^5 + 203712218940650T^4 - 1355872820350120T^3 + 5660501605176844T^2 - 11524995127576360*T + 9341392798076161
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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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 792 = 2^{3} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	792.ce (of order \(30\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{8} q^{2} + ( - \beta_{13} - \beta_{12} + \cdots - \beta_1) q^{3}+ \cdots + ( - \beta_{14} - 2 \beta_{12} + \cdots - 2 \beta_1) q^{9}+O(q^{10}) \) q + b8 * q^2 + (-b13 - b12 - b10 + b8 + b6 + b5 - b2 - b1) * q^3 + (2b9 - 2b2) * q^4 + (-2b15 - 2b14 - 2b11 - b10 + 2b9 + 2b7 - 2b4 - 2b2) q^6 + (-2b13 - 2b10) * q^8 + (-b14 - 2b12 - b11 - 2b10 + 2b5 + 2b3 - 2b1) q^9	\( q + \beta_{8} q^{2} + ( - \beta_{13} - \beta_{12} + \cdots - \beta_1) q^{3}+ \cdots + (\beta_{9} + 7 \beta_{5} - \beta_{2}) q^{99}+O(q^{100}) \) q + b8 * q^2 + (-b13 - b12 - b10 + b8 + b6 + b5 - b2 - b1) * q^3 + (2b9 - 2b2) * q^4 + (-2b15 - 2b14 - 2b11 - b10 + 2b9 + 2b7 - 2b4 - 2b2) q^6 + (-2b13 - 2b10) * q^8 + (-b14 - 2b12 - b11 - 2b10 + 2b5 + 2b3 - 2b1) q^9 + (-3b14 - b13 - b10 + 3b9 - 3b4 + 3) q^11 + (2b14 - 2b6) * q^12 - 4b11 q^16 + (-3b15 + 2b13 + 2b12 + 2b10 - 2b8 - 2b6 - 2b5 - 3b4 - 3b2) q^17 + (-b12 + 4) * q^18 + (-b15 + 3b12 - 3b8 + b7 - 3b3) q^19 + (3b12 - 2b11 + 3b10 - 3b6 - 3b5 - 3b3 + 3b1) q^22 + (-4b7 + 2b3 + 4) * q^24 + (5b15 + 5b11 - 5b7 + 5b2 + 5) * q^25 + (b8 + 5b4 - b1) q^27 + (-4b12 + 4b3) * q^32 + (-5b7 - 2b3 + 5) * q^33 + (4b15 + 4b14 + 3b13 + 3b12 + 4b11 - 8b9 - 3b8 - 4b7 - 3b6 + 4b4 + 4b2 + 3b1) * q^34 + (-2b15 + 4b8) * q^36 + (-6b9 + b8 + b5 - 6b4 + 6b2 - b1) q^38 + (-4b13 - 8b12 - 3b11 - 8b10 - 3b9 + 4b8 + 8b6 + 4b5 + 4b3 - 8b1) * q^41 + (5b15 + 5b14 - 3b6 + 5b4 - 3b1) q^43 + (-2b12 - 6b7 + 2b3) q^44 + (4b15 + 4b4 + 4b1) q^48 + (-7b14 - 7b11) * q^49 + (5b12 + 5b10 - 5b5 - 5b3 + 5b1) q^50 + (8b14 + 5b13 - b12 + 7b11 + 4b10 - b9 + b5 + b4 + b3 - b1 - 1) * q^51 + (-5b13 - 5b12 - 5b10 + 5b8 + 5b6 + 5b5 - 2b2 - 5b1) * q^54 + (5b15 + 5b14 + 2b13 + 2b12 + 5b11 + 6b10 - 12b9 - 2b8 - 5b7 - 2b6 + 5b4 + 5b2 + 2b1) q^57 + (3b14 - 3b7 + 5b6 - 5b3 + 3) * q^59 + 8b4 q^64 + (-4b15 - 4b4 + 5b1) q^66 + (7b15 + 3b12 + 7b11 + 3b10 + 7b9 - 3b8 - 7b7 - 3b6 + 7b2 + 3b1 + 7) * q^67 + (6b15 + 6b14 + 4b13 + 6b11 - 6b7 + 4b6 + 6b2 + 6) q^68 + (8b9 + 2b5 - 8b2) q^72 + (-b14 - b11 + 6b10 + b9 - 6b8 + b7 - b4 - b2) * q^73 + (5b12 - 5b7 - 5b3) q^75 + (-2b14 + 12b13 + 6b12 + 12b10 + 2b9 - 6b8 - 6b6 - 6b5 - 2b4 - 2b2 + 6b1 + 2) q^76 + (-7b15 - 4b13 - 7b11 + 7b7 - 7b2 - 7) q^81 + (-8b15 + 3b13 - 3b12 - 8b11 + 16b7 + 3b3 - 8b2 - 8) q^82 + (-9b15 - b13 - 9b11 + 18b7 - b3 - 9b2 - 18) * q^83 + (-5b13 - 5b12 - 5b10 - 6b9 + 5b8 - 6b7 + 5b6 + 5b3 - 5b1 + 6) q^86 + (-6b8 + 4b4 + 6b1) q^88 + (-9b14 + 2b8 - 2b6 - 9b4 - 2b1) q^89 + (-4b13 - 4b12 - 4b10 + 8b9 + 4b8 + 4b6 - 4b1) q^96 + (6b12 - 5b9 - 6b5 + 5b2 + 5) * q^97 - 7b12 q^98 + (b9 + 7b5 - b2) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{3} - 4 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10}) \) 16 * q + 2 * q^3 - 4 * q^4 + 4 * q^6 - 2 * q^9	\( 16 q + 2 q^{3} - 4 q^{4} + 4 q^{6} - 2 q^{9} + 12 q^{11} + 8 q^{12} + 8 q^{16} + 64 q^{18} + 10 q^{19} + 4 q^{22} + 32 q^{24} + 10 q^{25} + 20 q^{27} + 40 q^{33} + 8 q^{34} + 4 q^{36} - 12 q^{38} + 18 q^{41} + 30 q^{43} - 48 q^{44} + 8 q^{48} - 14 q^{49} + 10 q^{51} + 4 q^{54} + 18 q^{57} + 36 q^{59} + 32 q^{64} - 8 q^{66} - 14 q^{67} + 36 q^{68} - 16 q^{72} - 40 q^{75} + 12 q^{76} - 14 q^{81} + 48 q^{82} - 90 q^{83} + 72 q^{86} + 16 q^{88} - 72 q^{89} - 32 q^{96} + 90 q^{97} - 2 q^{99}+O(q^{100}) \) 16 * q + 2 * q^3 - 4 * q^4 + 4 * q^6 - 2 * q^9 + 12 * q^11 + 8 * q^12 + 8 * q^16 + 64 * q^18 + 10 * q^19 + 4 * q^22 + 32 * q^24 + 10 * q^25 + 20 * q^27 + 40 * q^33 + 8 * q^34 + 4 * q^36 - 12 * q^38 + 18 * q^41 + 30 * q^43 - 48 * q^44 + 8 * q^48 - 14 * q^49 + 10 * q^51 + 4 * q^54 + 18 * q^57 + 36 * q^59 + 32 * q^64 - 8 * q^66 - 14 * q^67 + 36 * q^68 - 16 * q^72 - 40 * q^75 + 12 * q^76 - 14 * q^81 + 48 * q^82 - 90 * q^83 + 72 * q^86 + 16 * q^88 - 72 * q^89 - 32 * q^96 + 90 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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

Label	792.2.ce.a

Level	$792$
Weight	$2$
Character orbit	792.ce
Analytic conductor	$6.324$
Analytic rank	$0$
Dimension	$16$
CM discriminant	-8
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(6.32415184009\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(2\) over \(\Q(\zeta_{30})\)
Coefficient field:	16.0.26873856000000000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} + 2x^{14} - 8x^{10} - 16x^{8} - 32x^{6} + 128x^{2} + 256 \) x^16 + 2x^14 - 8x^10 - 16x^8 - 32x^6 + 128*x^2 + 256
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{30}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( \nu^{2} ) / 2 \) (v^2) / 2
\(\beta_{3}\)	\(=\)	\( ( \nu^{5} ) / 4 \) (v^5) / 4
\(\beta_{4}\)	\(=\)	\( ( \nu^{6} ) / 8 \) (v^6) / 8
\(\beta_{5}\)	\(=\)	\( ( \nu^{7} ) / 8 \) (v^7) / 8
\(\beta_{6}\)	\(=\)	\( ( \nu^{9} ) / 16 \) (v^9) / 16
\(\beta_{7}\)	\(=\)	\( ( \nu^{10} ) / 32 \) (v^10) / 32
\(\beta_{8}\)	\(=\)	\( ( \nu^{11} ) / 32 \) (v^11) / 32
\(\beta_{9}\)	\(=\)	\( ( \nu^{12} ) / 64 \) (v^12) / 64
\(\beta_{10}\)	\(=\)	\( ( \nu^{13} ) / 64 \) (v^13) / 64
\(\beta_{11}\)	\(=\)	\( ( \nu^{14} ) / 128 \) (v^14) / 128
\(\beta_{12}\)	\(=\)	\( ( \nu^{15} ) / 128 \) (v^15) / 128
\(\beta_{13}\)	\(=\)	\( ( -\nu^{13} + 32\nu^{3} ) / 64 \) (-v^13 + 32*v^3) / 64
\(\beta_{14}\)	\(=\)	\( ( -\nu^{14} + 32\nu^{4} ) / 128 \) (-v^14 + 32*v^4) / 128
\(\beta_{15}\)	\(=\)	\( ( -\nu^{14} + 4\nu^{10} + 8\nu^{8} - 64\nu^{2} - 128 ) / 128 \) (-v^14 + 4v^10 + 8v^8 - 64*v^2 - 128) / 128

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( 2\beta_{2} \) 2*b2
\(\nu^{3}\)	\(=\)	\( 2\beta_{13} + 2\beta_{10} \) 2b13 + 2b10
\(\nu^{4}\)	\(=\)	\( 4\beta_{14} + 4\beta_{11} \) 4b14 + 4b11
\(\nu^{5}\)	\(=\)	\( 4\beta_{3} \) 4*b3
\(\nu^{6}\)	\(=\)	\( 8\beta_{4} \) 8*b4
\(\nu^{7}\)	\(=\)	\( 8\beta_{5} \) 8*b5
\(\nu^{8}\)	\(=\)	\( 16\beta_{15} + 16\beta_{11} - 16\beta_{7} + 16\beta_{2} + 16 \) 16b15 + 16b11 - 16b7 + 16b2 + 16
\(\nu^{9}\)	\(=\)	\( 16\beta_{6} \) 16*b6
\(\nu^{10}\)	\(=\)	\( 32\beta_{7} \) 32*b7
\(\nu^{11}\)	\(=\)	\( 32\beta_{8} \) 32*b8
\(\nu^{12}\)	\(=\)	\( 64\beta_{9} \) 64*b9
\(\nu^{13}\)	\(=\)	\( 64\beta_{10} \) 64*b10
\(\nu^{14}\)	\(=\)	\( 128\beta_{11} \) 128*b11
\(\nu^{15}\)	\(=\)	\( 128\beta_{12} \) 128*b12

\(n\)	\(145\)	\(199\)	\(353\)	\(397\)
\(\chi(n)\)	\(-\beta_{9}\)	\(-1\)	\(-\beta_{7}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 139.2
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
99.o	odd	30	1	inner
792.ce	even	30	1	inner

$p$	$F_p(T)$
$2$	\( T^{16} + 2 T^{14} + \cdots + 256 \) T^16 + 2T^14 - 8T^10 - 16T^8 - 32T^6 + 128*T^2 + 256
$3$	\( T^{16} - 2 T^{15} + \cdots + 6561 \) T^16 - 2T^15 + 3T^14 - 10T^13 + 20T^12 - 28T^11 + 69T^10 - 140T^9 + 199T^8 - 420T^7 + 621T^6 - 756T^5 + 1620T^4 - 2430T^3 + 2187T^2 - 4374*T + 6561
$5$	\( T^{16} \) T^16
$7$	\( T^{16} \) T^16
$11$	\( (T^{8} - 6 T^{7} + \cdots + 14641)^{2} \) (T^8 - 6T^7 + 25T^6 - 84T^5 + 229T^4 - 924T^3 + 3025T^2 - 7986*T + 14641)^2
$13$	\( T^{16} \) T^16
$17$	\( T^{16} + \cdots + 13943122561 \) T^16 - 70T^14 - 510T^13 + 4369T^12 + 35700T^11 - 143285T^10 - 1491240T^9 + 5270766T^8 + 26322120T^7 + 22625T^6 - 1901487060T^5 + 11066011399T^4 - 23144493600T^3 + 10892087875T^2 + 9695630910T + 13943122561
$19$	\( T^{16} + \cdots + 151052709025 \) T^16 - 10T^15 - 77T^14 + 750T^13 + 6864T^12 - 45100T^11 - 461823T^10 + 3410160T^9 + 21620491T^8 - 243809740T^7 + 61688125T^6 + 4859828440T^5 - 7901699340T^4 - 20933681850T^3 + 159680320475T^2 - 195808275550*T + 151052709025
$23$	\( T^{16} \) T^16
$29$	\( T^{16} \) T^16
$31$	\( T^{16} \) T^16
$37$	\( T^{16} \) T^16
$41$	\( T^{16} + \cdots + 243842434857025 \) T^16 - 18T^15 + 221T^14 - 3264T^13 + 43774T^12 - 389622T^11 + 1995064T^10 - 24034026T^9 + 119780241T^8 - 1527803550T^7 + 8096357840T^6 + 44946942570T^5 + 1278792853870T^4 + 9555242666400T^3 + 45411391528225T^2 + 164376867830250*T + 243842434857025
$43$	\( T^{16} + \cdots + 40220976684001 \) T^16 - 30T^15 + 113T^14 + 5610T^13 - 40405T^12 - 954030T^11 + 13208498T^10 + 9744180T^9 - 851931341T^8 + 716107110T^7 + 41205765442T^6 - 66679449240T^5 - 1067019730150T^4 + 372805950720T^3 + 19738019090032T^2 + 48417752954460*T + 40220976684001
$47$	\( T^{16} \) T^16
$53$	\( T^{16} \) T^16
$59$	\( T^{16} + \cdots + 2136973261921 \) T^16 - 36T^15 + 284T^14 + 4344T^13 - 79854T^12 + 122352T^11 + 10759646T^10 - 193520778T^9 + 1218545727T^8 + 5340488208T^7 - 109477969894T^6 + 409761951798T^5 + 2621402327231T^4 - 12911909544774T^3 + 28579600874409T^2 + 689131370346*T + 2136973261921
$61$	\( T^{16} \) T^16
$67$	\( T^{16} + \cdots + 85\!\cdots\!25 \) T^16 + 14T^15 + 665T^14 + 6426T^13 + 247079T^12 + 2187766T^11 + 55690150T^10 + 389986884T^9 + 8212560331T^8 + 55255322850T^7 + 785865316810T^6 + 4362931376720T^5 + 47866571936030T^4 + 244493300950500T^3 + 1603497906780200T^2 + 3893294729434600*T + 8596378114389025
$71$	\( T^{16} \) T^16
$73$	\( T^{16} + \cdots + 45864304637761 \) T^16 - 150T^14 + 730T^13 + 69569T^12 - 109500T^11 - 18913725T^10 + 31334520T^9 + 4302533166T^8 + 6369433960T^7 - 2757120135T^6 + 3510443685180T^5 + 94937575732599T^4 + 582960336744800T^3 + 1218093937974555T^2 + 430708178627270T + 45864304637761
$79$	\( T^{16} \) T^16
$83$	\( T^{16} + \cdots + 94\!\cdots\!41 \) T^16 + 90T^15 + 4453T^14 + 153270T^13 + 4056055T^12 + 86697450T^11 + 1550034518T^10 + 23803879680T^9 + 319318569579T^8 + 3767193600210T^7 + 38882388247762T^6 + 346087388216520T^5 + 2613670803934750T^4 + 16970909231260620T^3 + 97596256315453712T^2 + 431423648736705180*T + 946249801719345841
$89$	\( (T^{4} + 18 T^{3} + \cdots + 401)^{4} \) (T^4 + 18T^3 - 121T^2 - 2178*T + 401)^4
$97$	\( T^{16} + \cdots + 93\!\cdots\!61 \) T^16 - 90T^15 + 4279T^14 - 140230T^13 + 3503335T^12 - 69516270T^11 + 1110972406T^10 - 14353843140T^9 + 148846313959T^8 - 1194150894410T^7 + 7041959295466T^6 - 33718242047680T^5 + 203712218940650T^4 - 1355872820350120T^3 + 5660501605176844T^2 - 11524995127576360*T + 9341392798076161
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