LMFDB - Newform orbit 912.2.d.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,2,Mod(191,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("912.191");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	912.d (of order $2$, degree $1$, 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:	$7.28235666434$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.2593100598870016.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2x^{10} + x^{8} + 4x^{6} + 4x^{4} - 32x^{2} + 64 $ x^12 - 2x^10 + x^8 + 4x^6 + 4x^4 - 32x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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_{9} q^{3} + (\beta_{10} + \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{4} - \beta_1) q^{7} + ( - \beta_{10} + \beta_{5}) q^{9}+O(q^{10}) $ q - b9 * q^3 + (b10 + b3 - b2) * q^5 + (-b4 - b1) * q^7 + (-b10 + b5) * q^9	$ q - \beta_{9} q^{3} + (\beta_{10} + \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{4} - \beta_1) q^{7} + ( - \beta_{10} + \beta_{5}) q^{9} + (\beta_{11} + \beta_{9} + \cdots + 2 \beta_{6}) q^{11}+ \cdots + ( - 3 \beta_{11} - \beta_{9} + \cdots + 3 \beta_1) q^{99}+O(q^{100}) $ q - b9 * q^3 + (b10 + b3 - b2) * q^5 + (-b4 - b1) * q^7 + (-b10 + b5) * q^9 + (b11 + b9 - b8 + 2b6) q^11 + (b3 + b2) * q^13 + (-b11 + b8 - b6 - b4 + 2b1) q^15 + (b10 + 2b7 + b3 - b2) q^17 - b1 * q^19 + (-b7 + 2b3 + b2 - 1) q^21 + (-b11 + b9 - b8 + 3b6) q^23 + (-b5 + b3 + b2) * q^25 + (2b11 + b8 + 2b1) * q^27 + (-3b10 - b7 + b3 - b2) q^29 + (b9 + b8 - 2b1) q^31 + (3b10 + 2b7 + 2b3 - b2 - 1) q^33 + (-2b11 - 3b6) * q^35 + (-2b5 + 3b3 + 3b2 - 2) q^37 + (b6 + b4 + 3b1) q^39 + (-b10 - 3b7) q^41 + (b4 - 3b1) q^43 + (-2b10 - 2b7 - b5 + b3 - b2 + 1) * q^45 + (b11 - b9 + b8 - 2b6) q^47 + (b5 + b3 + b2) * q^49 + (-b11 - 2b9 + b8 - 5b6 + b4 + 2b1) q^51 + (-b10 - b7 + 2b3 - 2b2) * q^53 + (-2b9 - 2b8 - b4 - 5b1) q^55 + b2 * q^57 + (-2b11 + b9 - b8 - 2b6) * q^59 + (3b5 + b3 + b2 - 1) q^61 + (2b9 + 3b6 + 6b1) q^63 + (-2b10 - 2b7 - b3 + b2) * q^65 + (-2b9 - 2b8 + 4b4 - 4b1) * q^67 + (-b10 + 3b7 - 2b5 + 3b3 + b2) q^69 + (4b11 - b9 + b8) q^71 + (-b5 - 3b3 - 3b2 + 3) * q^73 + (-b11 - 2b8 + b6 + b4 + 2b1) * q^75 + (7b10 + 4b7 + 3b3 - 3b2) * q^77 + (2b4 - 8b1) * q^79 + (4b10 + 2b5 - 4b2 + 3) q^81 + (5b11 + b9 - b8 + 3b6) * q^83 + (b5 + 3b3 + 3b2 - 3) * q^85 + (3b11 + b9 - 3b8 + b6 - 2b4 + 6b1) * q^87 + (3b10 + 3b7) * q^89 + (3b9 + 3b8 + 2b1) q^91 + (b10 - b5 + 2b2 + 3) q^93 + (-b9 + b8 - b6) * q^95 + (4b5 - b3 - b2) q^97 + (-3b11 - b9 + 3b8 - 5b6 + b4 + 3b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{9}+O(q^{10}) $ 12 * q - 4 * q^9	$ 12 q - 4 q^{9} - 12 q^{21} + 4 q^{25} - 12 q^{33} - 16 q^{37} + 16 q^{45} - 4 q^{49} - 24 q^{61} + 8 q^{69} + 40 q^{73} + 28 q^{81} - 40 q^{85} + 40 q^{93} - 16 q^{97}+O(q^{100}) $ 12 * q - 4 * q^9 - 12 * q^21 + 4 * q^25 - 12 * q^33 - 16 * q^37 + 16 * q^45 - 4 * q^49 - 24 * q^61 + 8 * q^69 + 40 * q^73 + 28 * q^81 - 40 * q^85 + 40 * q^93 - 16 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 2x^{10} + x^{8} + 4x^{6} + 4x^{4} - 32x^{2} + 64 $ x^12 - 2*x^10 + x^8 + 4*x^6 + 4*x^4 - 32*x^2 + 64 :

$\beta_{1}$	$=$	$ ( -\nu^{8} - \nu^{4} + 2\nu^{2} - 8 ) / 16 $ (-v^8 - v^4 + 2*v^2 - 8) / 16
$\beta_{2}$	$=$	$ ( \nu^{11} - 2 \nu^{10} + 2 \nu^{9} - 4 \nu^{8} + 9 \nu^{7} + 14 \nu^{6} + 8 \nu^{5} + 16 \nu^{4} + \cdots + 32 ) / 128 $ (v^11 - 2v^10 + 2v^9 - 4v^8 + 9v^7 + 14v^6 + 8v^5 + 16v^4 + 4v^3 - 72v^2 + 48v + 32) / 128
$\beta_{3}$	$=$	$ ( - \nu^{11} - 2 \nu^{10} - 2 \nu^{9} - 4 \nu^{8} - 9 \nu^{7} + 14 \nu^{6} - 8 \nu^{5} + 16 \nu^{4} + \cdots + 32 ) / 128 $ (-v^11 - 2v^10 - 2v^9 - 4v^8 - 9v^7 + 14v^6 - 8v^5 + 16v^4 - 4v^3 - 72v^2 - 48v + 32) / 128
$\beta_{4}$	$=$	$ ( \nu^{10} - \nu^{8} + \nu^{6} + 5\nu^{4} + 18\nu^{2} - 24 ) / 16 $ (v^10 - v^8 + v^6 + 5v^4 + 18v^2 - 24) / 16
$\beta_{5}$	$=$	$ ( -3\nu^{10} + 2\nu^{8} + 5\nu^{6} - 12\nu^{2} + 48 ) / 32 $ (-3v^10 + 2v^8 + 5v^6 - 12v^2 + 48) / 32
$\beta_{6}$	$=$	$ ( -\nu^{11} - 2\nu^{9} + 7\nu^{7} - 8\nu^{5} + 12\nu^{3} - 16\nu ) / 64 $ (-v^11 - 2v^9 + 7v^7 - 8v^5 + 12v^3 - 16*v) / 64
$\beta_{7}$	$=$	$ ( \nu^{11} + 2\nu^{9} - 7\nu^{7} + 8\nu^{5} + 52\nu^{3} - 48\nu ) / 64 $ (v^11 + 2v^9 - 7v^7 + 8v^5 + 52v^3 - 48*v) / 64
$\beta_{8}$	$=$	$ ( - 3 \nu^{11} + 2 \nu^{10} + 2 \nu^{9} - 4 \nu^{8} + 5 \nu^{7} - 14 \nu^{6} - 16 \nu^{5} + 40 \nu^{4} + \cdots - 96 ) / 128 $ (-3v^11 + 2v^10 + 2v^9 - 4v^8 + 5v^7 - 14v^6 - 16v^5 + 40v^4 + 4v^3 + 24v^2 + 112*v - 96) / 128
$\beta_{9}$	$=$	$ ( 3 \nu^{11} + 2 \nu^{10} - 2 \nu^{9} - 4 \nu^{8} - 5 \nu^{7} - 14 \nu^{6} + 16 \nu^{5} + 40 \nu^{4} + \cdots - 96 ) / 128 $ (3v^11 + 2v^10 - 2v^9 - 4v^8 - 5v^7 - 14v^6 + 16v^5 + 40v^4 - 4v^3 + 24v^2 - 112*v - 96) / 128
$\beta_{10}$	$=$	$ ( \nu^{11} - 2\nu^{9} + \nu^{7} + 4\nu^{5} + 4\nu^{3} ) / 32 $ (v^11 - 2v^9 + v^7 + 4v^5 + 4*v^3) / 32
$\beta_{11}$	$=$	$ ( -\nu^{11} - \nu^{7} + 10\nu^{5} - 16\nu^{3} + 16\nu ) / 32 $ (-v^11 - v^7 + 10v^5 - 16v^3 + 16*v) / 32

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

$\nu$	$=$	$ ( \beta_{10} - \beta_{9} + \beta_{8} - \beta_{6} ) / 2 $ (b10 - b9 + b8 - b6) / 2
$\nu^{2}$	$=$	$ ( \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 2 $ (b5 + b4 - b3 - b2 + b1 + 1) / 2
$\nu^{3}$	$=$	$ ( \beta_{10} - \beta_{9} + \beta_{8} + 2\beta_{7} + \beta_{6} ) / 2 $ (b10 - b9 + b8 + 2*b7 + b6) / 2
$\nu^{4}$	$=$	$ ( 2\beta_{9} + 2\beta_{8} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3\beta _1 + 1 ) / 2 $ (2b9 + 2b8 + b5 + b4 + b3 + b2 - 3*b1 + 1) / 2
$\nu^{5}$	$=$	$ ( 4\beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + 2\beta_{7} + \beta_{6} - 2\beta_{3} + 2\beta_{2} ) / 2 $ (4b11 + b10 + b9 - b8 + 2b7 + b6 - 2b3 + 2b2) / 2
$\nu^{6}$	$=$	$ ( -4\beta_{9} - 4\beta_{8} + \beta_{5} + 5\beta_{4} + 3\beta_{3} + 3\beta_{2} - 3\beta _1 - 3 ) / 2 $ (-4b9 - 4b8 + b5 + 5b4 + 3b3 + 3b2 - 3b1 - 3) / 2
$\nu^{7}$	$=$	$ ( -3\beta_{10} + 3\beta_{9} - 3\beta_{8} - 2\beta_{7} + 9\beta_{6} - 8\beta_{3} + 8\beta_{2} ) / 2 $ (-3b10 + 3b9 - 3b8 - 2b7 + 9b6 - 8b3 + 8*b2) / 2
$\nu^{8}$	$=$	$ ( -2\beta_{9} - 2\beta_{8} + \beta_{5} + \beta_{4} - 3\beta_{3} - 3\beta_{2} - 27\beta _1 - 15 ) / 2 $ (-2b9 - 2b8 + b5 + b4 - 3b3 - 3b2 - 27*b1 - 15) / 2
$\nu^{9}$	$=$	$ ( -4\beta_{11} - 23\beta_{10} + 5\beta_{9} - 5\beta_{8} + 2\beta_{7} - 7\beta_{6} - 14\beta_{3} + 14\beta_{2} ) / 2 $ (-4b11 - 23b10 + 5b9 - 5b8 + 2b7 - 7b6 - 14b3 + 14b2) / 2
$\nu^{10}$	$=$	$ ( -8\beta_{9} - 8\beta_{8} - 23\beta_{5} + 5\beta_{4} + 7\beta_{3} + 7\beta_{2} - 27\beta _1 + 13 ) / 2 $ (-8b9 - 8b8 - 23b5 + 5b4 + 7b3 + 7b2 - 27*b1 + 13) / 2
$\nu^{11}$	$=$	$ ( -24\beta_{11} + 13\beta_{10} + 7\beta_{9} - 7\beta_{8} - 10\beta_{7} - 31\beta_{6} - 12\beta_{3} + 12\beta_{2} ) / 2 $ (-24b11 + 13b10 + 7b9 - 7b8 - 10b7 - 31b6 - 12b3 + 12b2) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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} $

191.1

−1.37027	−	0.349801i
−1.37027	+	0.349801i
0.430469	−	1.34711i
0.430469	+	1.34711i
−1.19877	−	0.750295i
−1.19877	+	0.750295i
1.19877	+	0.750295i
1.19877	−	0.750295i
−0.430469	+	1.34711i
−0.430469	−	1.34711i
1.37027	+	0.349801i
1.37027	−	0.349801i

−1.72007

−

0.203364i

2.74054i

−

1.91729i

2.91729

0.699602i

191.2

−1.72007

0.203364i

−

2.74054i

1.91729i

2.91729

−

0.699602i

191.3

−0.916638

−

1.46962i

−

0.860938i

2.31955i

−1.31955

2.69421i

191.4

−0.916638

1.46962i

0.860938i

−

2.31955i

−1.31955

−

2.69421i

191.5

−0.448478

−

1.67298i

−

2.39755i

−

3.59774i

−2.59774

1.50059i

191.6

−0.448478

1.67298i

2.39755i

3.59774i

−2.59774

−

1.50059i

191.7

0.448478

−

1.67298i

2.39755i

−

3.59774i

−2.59774

−

1.50059i

191.8

0.448478

1.67298i

−

2.39755i

3.59774i

−2.59774

1.50059i

191.9

0.916638

−

1.46962i

0.860938i

2.31955i

−1.31955

−

2.69421i

191.10

0.916638

1.46962i

−

0.860938i

−

2.31955i

−1.31955

2.69421i

191.11

1.72007

−

0.203364i

−

2.74054i

−

1.91729i

2.91729

−

0.699602i

191.12

1.72007

0.203364i

2.74054i

1.91729i

2.91729

0.699602i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
12.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	912.2.d.a	✓	12
3.b	odd	2	1	inner	912.2.d.a	✓	12
4.b	odd	2	1	inner	912.2.d.a	✓	12
12.b	even	2	1	inner	912.2.d.a	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
912.2.d.a	✓	12	1.a	even	1	1	trivial
912.2.d.a	✓	12	3.b	odd	2	1	inner
912.2.d.a	✓	12	4.b	odd	2	1	inner
912.2.d.a	✓	12	12.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{6} + 14T_{5}^{4} + 53T_{5}^{2} + 32 $ T5^6 + 14*T5^4 + 53*T5^2 + 32 acting on $S_{2}^{\mathrm{new}}(912, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ T^{12} + 2 T^{10} + \cdots + 729 $ T^12 + 2T^10 - 5T^8 - 44T^6 - 45T^4 + 162*T^2 + 729
$5$	$ (T^{6} + 14 T^{4} + \cdots + 32)^{2} $ (T^6 + 14T^4 + 53T^2 + 32)^2
$7$	$ (T^{6} + 22 T^{4} + \cdots + 256)^{2} $ (T^6 + 22T^4 + 137T^2 + 256)^2
$11$	$ (T^{6} - 36 T^{4} + 17 T^{2} - 2)^{2} $ (T^6 - 36T^4 + 17T^2 - 2)^2
$13$	$ (T^{3} - 10 T + 4)^{4} $ (T^3 - 10*T + 4)^4
$17$	$ (T^{6} + 70 T^{4} + \cdots + 200)^{2} $ (T^6 + 70T^4 + 1173T^2 + 200)^2
$19$	$ (T^{2} + 1)^{6} $ (T^2 + 1)^6
$23$	$ (T^{6} - 106 T^{4} + \cdots - 8192)^{2} $ (T^6 - 106T^4 + 1792T^2 - 8192)^2
$29$	$ (T^{6} + 130 T^{4} + \cdots + 76832)^{2} $ (T^6 + 130T^4 + 5536T^2 + 76832)^2
$31$	$ (T^{6} + 32 T^{4} + \cdots + 256)^{2} $ (T^6 + 32T^4 + 196T^2 + 256)^2
$37$	$ (T^{3} + 4 T^{2} + \cdots - 428)^{4} $ (T^3 + 4T^2 - 82T - 428)^4
$41$	$ (T^{6} + 130 T^{4} + \cdots + 72200)^{2} $ (T^6 + 130T^4 + 5412T^2 + 72200)^2
$43$	$ (T^{6} + 38 T^{4} + \cdots + 16)^{2} $ (T^6 + 38T^4 + 105T^2 + 16)^2
$47$	$ (T^{6} - 60 T^{4} + \cdots - 1922)^{2} $ (T^6 - 60T^4 + 761T^2 - 1922)^2
$53$	$ (T^{6} + 98 T^{4} + \cdots + 2888)^{2} $ (T^6 + 98T^4 + 1220T^2 + 2888)^2
$59$	$ (T^{6} - 104 T^{4} + \cdots - 3200)^{2} $ (T^6 - 104T^4 + 2612T^2 - 3200)^2
$61$	$ (T^{3} + 6 T^{2} + \cdots - 200)^{4} $ (T^3 + 6T^2 - 91T - 200)^4
$67$	$ (T^{6} + 272 T^{4} + \cdots + 1024)^{2} $ (T^6 + 272T^4 + 11328T^2 + 1024)^2
$71$	$ (T^{6} - 256 T^{4} + \cdots - 549152)^{2} $ (T^6 - 256T^4 + 21076T^2 - 549152)^2
$73$	$ (T^{3} - 10 T^{2} + \cdots - 80)^{4} $ (T^3 - 10T^2 - 83T - 80)^4
$79$	$ (T^{6} + 228 T^{4} + \cdots + 43264)^{2} $ (T^6 + 228T^4 + 7232T^2 + 43264)^2
$83$	$ (T^{6} - 370 T^{4} + \cdots - 850208)^{2} $ (T^6 - 370T^4 + 33856T^2 - 850208)^2
$89$	$ (T^{6} + 162 T^{4} + \cdots + 5832)^{2} $ (T^6 + 162T^4 + 5508T^2 + 5832)^2
$97$	$ (T^{3} + 4 T^{2} + \cdots - 652)^{4} $ (T^3 + 4T^2 - 114T - 652)^4
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	912.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{9} q^{3} + (\beta_{10} + \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{4} - \beta_1) q^{7} + ( - \beta_{10} + \beta_{5}) q^{9}+O(q^{10}) \) q - b9 * q^3 + (b10 + b3 - b2) * q^5 + (-b4 - b1) * q^7 + (-b10 + b5) * q^9	\( q - \beta_{9} q^{3} + (\beta_{10} + \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{4} - \beta_1) q^{7} + ( - \beta_{10} + \beta_{5}) q^{9} + (\beta_{11} + \beta_{9} + \cdots + 2 \beta_{6}) q^{11}+ \cdots + ( - 3 \beta_{11} - \beta_{9} + \cdots + 3 \beta_1) q^{99}+O(q^{100}) \) q - b9 * q^3 + (b10 + b3 - b2) * q^5 + (-b4 - b1) * q^7 + (-b10 + b5) * q^9 + (b11 + b9 - b8 + 2b6) q^11 + (b3 + b2) * q^13 + (-b11 + b8 - b6 - b4 + 2b1) q^15 + (b10 + 2b7 + b3 - b2) q^17 - b1 * q^19 + (-b7 + 2b3 + b2 - 1) q^21 + (-b11 + b9 - b8 + 3b6) q^23 + (-b5 + b3 + b2) * q^25 + (2b11 + b8 + 2b1) * q^27 + (-3b10 - b7 + b3 - b2) q^29 + (b9 + b8 - 2b1) q^31 + (3b10 + 2b7 + 2b3 - b2 - 1) q^33 + (-2b11 - 3b6) * q^35 + (-2b5 + 3b3 + 3b2 - 2) q^37 + (b6 + b4 + 3b1) q^39 + (-b10 - 3b7) q^41 + (b4 - 3b1) q^43 + (-2b10 - 2b7 - b5 + b3 - b2 + 1) * q^45 + (b11 - b9 + b8 - 2b6) q^47 + (b5 + b3 + b2) * q^49 + (-b11 - 2b9 + b8 - 5b6 + b4 + 2b1) q^51 + (-b10 - b7 + 2b3 - 2b2) * q^53 + (-2b9 - 2b8 - b4 - 5b1) q^55 + b2 * q^57 + (-2b11 + b9 - b8 - 2b6) * q^59 + (3b5 + b3 + b2 - 1) q^61 + (2b9 + 3b6 + 6b1) q^63 + (-2b10 - 2b7 - b3 + b2) * q^65 + (-2b9 - 2b8 + 4b4 - 4b1) * q^67 + (-b10 + 3b7 - 2b5 + 3b3 + b2) q^69 + (4b11 - b9 + b8) q^71 + (-b5 - 3b3 - 3b2 + 3) * q^73 + (-b11 - 2b8 + b6 + b4 + 2b1) * q^75 + (7b10 + 4b7 + 3b3 - 3b2) * q^77 + (2b4 - 8b1) * q^79 + (4b10 + 2b5 - 4b2 + 3) q^81 + (5b11 + b9 - b8 + 3b6) * q^83 + (b5 + 3b3 + 3b2 - 3) * q^85 + (3b11 + b9 - 3b8 + b6 - 2b4 + 6b1) * q^87 + (3b10 + 3b7) * q^89 + (3b9 + 3b8 + 2b1) q^91 + (b10 - b5 + 2b2 + 3) q^93 + (-b9 + b8 - b6) * q^95 + (4b5 - b3 - b2) q^97 + (-3b11 - b9 + 3b8 - 5b6 + b4 + 3b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{9}+O(q^{10}) \) 12 * q - 4 * q^9	\( 12 q - 4 q^{9} - 12 q^{21} + 4 q^{25} - 12 q^{33} - 16 q^{37} + 16 q^{45} - 4 q^{49} - 24 q^{61} + 8 q^{69} + 40 q^{73} + 28 q^{81} - 40 q^{85} + 40 q^{93} - 16 q^{97}+O(q^{100}) \) 12 * q - 4 * q^9 - 12 * q^21 + 4 * q^25 - 12 * q^33 - 16 * q^37 + 16 * q^45 - 4 * q^49 - 24 * q^61 + 8 * q^69 + 40 * q^73 + 28 * q^81 - 40 * q^85 + 40 * q^93 - 16 * q^97

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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	912.2.d.a

Level	$912$
Weight	$2$
Character orbit	912.d
Analytic conductor	$7.282$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(7.28235666434\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	12.0.2593100598870016.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 2x^{10} + x^{8} + 4x^{6} + 4x^{4} - 32x^{2} + 64 \) x^12 - 2x^10 + x^8 + 4x^6 + 4x^4 - 32x^2 + 64
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{5} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( -\nu^{8} - \nu^{4} + 2\nu^{2} - 8 ) / 16 \) (-v^8 - v^4 + 2*v^2 - 8) / 16
\(\beta_{2}\)	\(=\)	\( ( \nu^{11} - 2 \nu^{10} + 2 \nu^{9} - 4 \nu^{8} + 9 \nu^{7} + 14 \nu^{6} + 8 \nu^{5} + 16 \nu^{4} + \cdots + 32 ) / 128 \) (v^11 - 2v^10 + 2v^9 - 4v^8 + 9v^7 + 14v^6 + 8v^5 + 16v^4 + 4v^3 - 72v^2 + 48v + 32) / 128
\(\beta_{3}\)	\(=\)	\( ( - \nu^{11} - 2 \nu^{10} - 2 \nu^{9} - 4 \nu^{8} - 9 \nu^{7} + 14 \nu^{6} - 8 \nu^{5} + 16 \nu^{4} + \cdots + 32 ) / 128 \) (-v^11 - 2v^10 - 2v^9 - 4v^8 - 9v^7 + 14v^6 - 8v^5 + 16v^4 - 4v^3 - 72v^2 - 48v + 32) / 128
\(\beta_{4}\)	\(=\)	\( ( \nu^{10} - \nu^{8} + \nu^{6} + 5\nu^{4} + 18\nu^{2} - 24 ) / 16 \) (v^10 - v^8 + v^6 + 5v^4 + 18v^2 - 24) / 16
\(\beta_{5}\)	\(=\)	\( ( -3\nu^{10} + 2\nu^{8} + 5\nu^{6} - 12\nu^{2} + 48 ) / 32 \) (-3v^10 + 2v^8 + 5v^6 - 12v^2 + 48) / 32
\(\beta_{6}\)	\(=\)	\( ( -\nu^{11} - 2\nu^{9} + 7\nu^{7} - 8\nu^{5} + 12\nu^{3} - 16\nu ) / 64 \) (-v^11 - 2v^9 + 7v^7 - 8v^5 + 12v^3 - 16*v) / 64
\(\beta_{7}\)	\(=\)	\( ( \nu^{11} + 2\nu^{9} - 7\nu^{7} + 8\nu^{5} + 52\nu^{3} - 48\nu ) / 64 \) (v^11 + 2v^9 - 7v^7 + 8v^5 + 52v^3 - 48*v) / 64
\(\beta_{8}\)	\(=\)	\( ( - 3 \nu^{11} + 2 \nu^{10} + 2 \nu^{9} - 4 \nu^{8} + 5 \nu^{7} - 14 \nu^{6} - 16 \nu^{5} + 40 \nu^{4} + \cdots - 96 ) / 128 \) (-3v^11 + 2v^10 + 2v^9 - 4v^8 + 5v^7 - 14v^6 - 16v^5 + 40v^4 + 4v^3 + 24v^2 + 112*v - 96) / 128
\(\beta_{9}\)	\(=\)	\( ( 3 \nu^{11} + 2 \nu^{10} - 2 \nu^{9} - 4 \nu^{8} - 5 \nu^{7} - 14 \nu^{6} + 16 \nu^{5} + 40 \nu^{4} + \cdots - 96 ) / 128 \) (3v^11 + 2v^10 - 2v^9 - 4v^8 - 5v^7 - 14v^6 + 16v^5 + 40v^4 - 4v^3 + 24v^2 - 112*v - 96) / 128
\(\beta_{10}\)	\(=\)	\( ( \nu^{11} - 2\nu^{9} + \nu^{7} + 4\nu^{5} + 4\nu^{3} ) / 32 \) (v^11 - 2v^9 + v^7 + 4v^5 + 4*v^3) / 32
\(\beta_{11}\)	\(=\)	\( ( -\nu^{11} - \nu^{7} + 10\nu^{5} - 16\nu^{3} + 16\nu ) / 32 \) (-v^11 - v^7 + 10v^5 - 16v^3 + 16*v) / 32

\(\nu\)	\(=\)	\( ( \beta_{10} - \beta_{9} + \beta_{8} - \beta_{6} ) / 2 \) (b10 - b9 + b8 - b6) / 2
\(\nu^{2}\)	\(=\)	\( ( \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 2 \) (b5 + b4 - b3 - b2 + b1 + 1) / 2
\(\nu^{3}\)	\(=\)	\( ( \beta_{10} - \beta_{9} + \beta_{8} + 2\beta_{7} + \beta_{6} ) / 2 \) (b10 - b9 + b8 + 2*b7 + b6) / 2
\(\nu^{4}\)	\(=\)	\( ( 2\beta_{9} + 2\beta_{8} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3\beta _1 + 1 ) / 2 \) (2b9 + 2b8 + b5 + b4 + b3 + b2 - 3*b1 + 1) / 2
\(\nu^{5}\)	\(=\)	\( ( 4\beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + 2\beta_{7} + \beta_{6} - 2\beta_{3} + 2\beta_{2} ) / 2 \) (4b11 + b10 + b9 - b8 + 2b7 + b6 - 2b3 + 2b2) / 2
\(\nu^{6}\)	\(=\)	\( ( -4\beta_{9} - 4\beta_{8} + \beta_{5} + 5\beta_{4} + 3\beta_{3} + 3\beta_{2} - 3\beta _1 - 3 ) / 2 \) (-4b9 - 4b8 + b5 + 5b4 + 3b3 + 3b2 - 3b1 - 3) / 2
\(\nu^{7}\)	\(=\)	\( ( -3\beta_{10} + 3\beta_{9} - 3\beta_{8} - 2\beta_{7} + 9\beta_{6} - 8\beta_{3} + 8\beta_{2} ) / 2 \) (-3b10 + 3b9 - 3b8 - 2b7 + 9b6 - 8b3 + 8*b2) / 2
\(\nu^{8}\)	\(=\)	\( ( -2\beta_{9} - 2\beta_{8} + \beta_{5} + \beta_{4} - 3\beta_{3} - 3\beta_{2} - 27\beta _1 - 15 ) / 2 \) (-2b9 - 2b8 + b5 + b4 - 3b3 - 3b2 - 27*b1 - 15) / 2
\(\nu^{9}\)	\(=\)	\( ( -4\beta_{11} - 23\beta_{10} + 5\beta_{9} - 5\beta_{8} + 2\beta_{7} - 7\beta_{6} - 14\beta_{3} + 14\beta_{2} ) / 2 \) (-4b11 - 23b10 + 5b9 - 5b8 + 2b7 - 7b6 - 14b3 + 14b2) / 2
\(\nu^{10}\)	\(=\)	\( ( -8\beta_{9} - 8\beta_{8} - 23\beta_{5} + 5\beta_{4} + 7\beta_{3} + 7\beta_{2} - 27\beta _1 + 13 ) / 2 \) (-8b9 - 8b8 - 23b5 + 5b4 + 7b3 + 7b2 - 27*b1 + 13) / 2
\(\nu^{11}\)	\(=\)	\( ( -24\beta_{11} + 13\beta_{10} + 7\beta_{9} - 7\beta_{8} - 10\beta_{7} - 31\beta_{6} - 12\beta_{3} + 12\beta_{2} ) / 2 \) (-24b11 + 13b10 + 7b9 - 7b8 - 10b7 - 31b6 - 12b3 + 12b2) / 2

\(n\)	\(97\)	\(229\)	\(305\)	\(799\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 191.12
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( T^{12} + 2 T^{10} + \cdots + 729 \) T^12 + 2T^10 - 5T^8 - 44T^6 - 45T^4 + 162*T^2 + 729
$5$	\( (T^{6} + 14 T^{4} + \cdots + 32)^{2} \) (T^6 + 14T^4 + 53T^2 + 32)^2
$7$	\( (T^{6} + 22 T^{4} + \cdots + 256)^{2} \) (T^6 + 22T^4 + 137T^2 + 256)^2
$11$	\( (T^{6} - 36 T^{4} + 17 T^{2} - 2)^{2} \) (T^6 - 36T^4 + 17T^2 - 2)^2
$13$	\( (T^{3} - 10 T + 4)^{4} \) (T^3 - 10*T + 4)^4
$17$	\( (T^{6} + 70 T^{4} + \cdots + 200)^{2} \) (T^6 + 70T^4 + 1173T^2 + 200)^2
$19$	\( (T^{2} + 1)^{6} \) (T^2 + 1)^6
$23$	\( (T^{6} - 106 T^{4} + \cdots - 8192)^{2} \) (T^6 - 106T^4 + 1792T^2 - 8192)^2
$29$	\( (T^{6} + 130 T^{4} + \cdots + 76832)^{2} \) (T^6 + 130T^4 + 5536T^2 + 76832)^2
$31$	\( (T^{6} + 32 T^{4} + \cdots + 256)^{2} \) (T^6 + 32T^4 + 196T^2 + 256)^2
$37$	\( (T^{3} + 4 T^{2} + \cdots - 428)^{4} \) (T^3 + 4T^2 - 82T - 428)^4
$41$	\( (T^{6} + 130 T^{4} + \cdots + 72200)^{2} \) (T^6 + 130T^4 + 5412T^2 + 72200)^2
$43$	\( (T^{6} + 38 T^{4} + \cdots + 16)^{2} \) (T^6 + 38T^4 + 105T^2 + 16)^2
$47$	\( (T^{6} - 60 T^{4} + \cdots - 1922)^{2} \) (T^6 - 60T^4 + 761T^2 - 1922)^2
$53$	\( (T^{6} + 98 T^{4} + \cdots + 2888)^{2} \) (T^6 + 98T^4 + 1220T^2 + 2888)^2
$59$	\( (T^{6} - 104 T^{4} + \cdots - 3200)^{2} \) (T^6 - 104T^4 + 2612T^2 - 3200)^2
$61$	\( (T^{3} + 6 T^{2} + \cdots - 200)^{4} \) (T^3 + 6T^2 - 91T - 200)^4
$67$	\( (T^{6} + 272 T^{4} + \cdots + 1024)^{2} \) (T^6 + 272T^4 + 11328T^2 + 1024)^2
$71$	\( (T^{6} - 256 T^{4} + \cdots - 549152)^{2} \) (T^6 - 256T^4 + 21076T^2 - 549152)^2
$73$	\( (T^{3} - 10 T^{2} + \cdots - 80)^{4} \) (T^3 - 10T^2 - 83T - 80)^4
$79$	\( (T^{6} + 228 T^{4} + \cdots + 43264)^{2} \) (T^6 + 228T^4 + 7232T^2 + 43264)^2
$83$	\( (T^{6} - 370 T^{4} + \cdots - 850208)^{2} \) (T^6 - 370T^4 + 33856T^2 - 850208)^2
$89$	\( (T^{6} + 162 T^{4} + \cdots + 5832)^{2} \) (T^6 + 162T^4 + 5508T^2 + 5832)^2
$97$	\( (T^{3} + 4 T^{2} + \cdots - 652)^{4} \) (T^3 + 4T^2 - 114T - 652)^4
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