LMFDB - Newform orbit 396.2.h.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [396,2,Mod(307,396)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(396, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("396.307");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 396 = 2^{2} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	396.h (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:	$3.16207592004$
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:	no (minimal twist has level 132)
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_1 q^{2} + \beta_{2} q^{4} + ( - \beta_{10} + \beta_{7} + \cdots + \beta_1) q^{5} + (\beta_{11} + \beta_{3}) q^{7} + (\beta_{10} + \beta_{9} + \beta_{3}) q^{8} + ( - \beta_{11} + \beta_{10} + \cdots - \beta_{4}) q^{10}+ \cdots + ( - 2 \beta_{11} + 2 \beta_{10} + \cdots - \beta_1) q^{98}+O(q^{100}) $ q + b1 * q^2 + b2 * q^4 + (-b10 + b7 - b5 - b4 + b1) * q^5 + (b11 + b3) * q^7 + (b10 + b9 + b3) * q^8 + (-b11 + b10 + b9 - b4) * q^10 + (-b11 + b8 + b2) * q^11 + (-b4 + b3) * q^13 + (2b10 - b8 - 2b7 + b6 + b5 + 2b4 + b2 - 2b1 - 1) * q^14 + (b10 + b6 + 2b5 + b4 + b2 - b1) q^16 + (b10 - b9 - b3 - b1) * q^17 + (-b11 + 2b10 + b4 + 2b3) * q^19 + (-b10 + b8 + 2b7 - b6 + b5 - b4 - b2 + b1 + 3) q^20 + (-b10 + b9 + 2b7 - b2) q^22 + (-b7 - b5) * q^23 + (-3b10 + 2b8 + 2b7 - b6 - 2b5 - 3b4 - 2b2 + 3b1 + 3) q^25 + (b10 - b8 - b6 + b5 + b4 - b2 - b1 + 1) * q^26 + (b11 + b10 - b9 + b4) * q^28 + (b9 - 3b4 + 3b3) * q^29 + (-b10 + 2b8 - 2b7 + b6 - 2b5 - b4 + 2b2 + b1) * q^31 + (2b11 - b10 + b9 - b3) q^32 + (-b10 + b6 - 2b5 - b4 - 2b2 + b1 + 4) * q^34 + (2b11 - b10 + b4 + 2b3 - 3b1) q^35 + (3b10 - 2b8 - 2b7 + b6 + 2b5 + 3b4 + 2b2 - 3b1 - 4) q^37 + (b10 - 2b8 + 2b7 + 2b6 + 2b5 + b4 - b1) * q^38 + (b11 - 3b10 + b9 - b4 + 2b1) * q^40 + (-2b10 - 3b9 + b4 + b3 + 2b1) q^41 + (-b11 - 2b10 + b4 - 2b3 - 4b1) q^43 + (-b10 + b9 + b8 + b6 + b5 - 2b4 - b3 + b2 - 3) q^44 + (-b11 + b10 - b9 + b4) * q^46 + (-b10 + 2b8 - 3b7 - 3b5 - b4 + 2b2 + b1) * q^47 + (-2b10 + 2b7 - 2b5 - 2b4 + 2b1 - 1) q^49 + (-2b11 - 2b10 - 2b3 + b1) q^50 + (b11 - b10 - b9 - b4 + 2b3 + 2b1) * q^52 + (-5b10 + 4b8 + 3b7 - 2b6 - 3b5 - 5b4 - 4b2 + 5b1 + 4) * q^53 + (-b10 + 2b7 - b6 + 2b5 - 2b4 - 3b3 + 3b1) q^55 + (b10 - b8 - 2b7 + 3b6 - b5 + b4 + b2 - b1 + 1) * q^56 + (3b10 - 2b8 - 3b6 + 4b5 + 3b4 - 2b2 - 3b1 + 2) q^58 + (b10 - 2b8 - 2b6 + b4 - 2b2 - b1) q^59 + (-2b10 - 2b9 - b4 + 3b3 + 2b1) * q^61 + (-2b11 + 2b10 + 4b4 - 2b3 - 2b1) q^62 + (b10 + 2b8 - 4b7 + b6 + b4 + 3b2 - b1 - 2) q^64 + (-b10 + b4 + b1) * q^65 + (b10 - 2b8 - b6 + b4 - 2b2 - b1) * q^67 + (-2b11 - 2b9 - 4b3 + 4b1) * q^68 + (4b10 - 2b8 - 4b7 + 2b6 + 2b5 + 4b4 - 4b1 - 6) q^70 + (b10 - 2b8 + b7 + 2b6 + b5 + b4 - 2b2 - b1) q^71 + (4b10 + 2b9 - 4b3 - 4b1) * q^73 + (2b11 + 2b10 + 2b3 - 2b1) * q^74 + (2b11 + 2b9 - 4b4 - 2b3 + 2b1) q^76 + (2b10 - b9 - b7 + b5 + 3b4 - 3b3 - 2b1 - 4) * q^77 + (b11 - 2b10 - b3 - 2b1) * q^79 + (b10 + b8 - 2b7 - 3b6 + b5 + b4 + 3b2 - b1 - 5) q^80 + (b10 - 4b8 - b6 - 2b5 + b4 - b1 - 4) * q^82 + (4b10 + 2b4 + 6b3) q^83 + (-6b9 + 4b4 - 4b3) q^85 + (-3b10 + 2b8 + 2b7 - 2b6 - 2b5 - 3b4 - 4b2 + 3b1 - 4) * q^86 + (b11 - 2b10 + b9 + 2b8 - 3b6 - 2b3 - b2 - 3b1 + 2) q^88 + (4b10 - 4b8 - 2b7 + 2b6 + 2b5 + 4b4 + 4b2 - 4b1 - 2) * q^89 + (b10 - 2b8 - b6 + b4 - 2b2 - b1) * q^91 + (-b10 - b8 + 2b7 + b6 - b5 - b4 - b2 + b1 + 1) q^92 + (-3b11 + 3b10 - b9 + 5b4 - 2b1) * q^94 + (2b11 - 4b10 - 4b4 - 6b3 + 4b1) q^95 + (2b10 - 2b7 + 2b5 + 2b4 - 2b1 - 4) q^97 + (-2b11 + 2b10 + 2b9 - 2b4 - b1) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{4} - 4 q^{14} + 4 q^{16} + 28 q^{20} - 4 q^{22} + 20 q^{25} + 12 q^{26} + 40 q^{34} - 32 q^{37} + 8 q^{38} - 36 q^{44} - 12 q^{49} + 16 q^{53} + 20 q^{56} + 24 q^{58} - 20 q^{64} - 64 q^{70}+ \cdots - 48 q^{97}+O(q^{100}) $ 12 * q + 4 * q^4 - 4 * q^14 + 4 * q^16 + 28 * q^20 - 4 * q^22 + 20 * q^25 + 12 * q^26 + 40 * q^34 - 32 * q^37 + 8 * q^38 - 36 * q^44 - 12 * q^49 + 16 * q^53 + 20 * q^56 + 24 * q^58 - 20 * q^64 - 64 * q^70 - 48 * q^77 - 52 * q^80 - 32 * q^82 - 72 * q^86 + 12 * q^88 + 8 * q^89 + 12 * q^92 - 48 * q^97

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 $ v
$\beta_{2}$	$=$	$ \nu^{2} $ v^2
$\beta_{3}$	$=$	$ ( -\nu^{9} - \nu^{5} + 2\nu^{3} - 8\nu ) / 16 $ (-v^9 - v^5 + 2v^3 - 8v) / 16
$\beta_{4}$	$=$	$ ( \nu^{11} - 2\nu^{9} + 9\nu^{7} + 4\nu^{5} + 12\nu^{3} + 16\nu ) / 64 $ (v^11 - 2v^9 + 9v^7 + 4v^5 + 12v^3 + 16*v) / 64
$\beta_{5}$	$=$	$ ( \nu^{11} - 2\nu^{9} - 4\nu^{8} - 7\nu^{7} + 4\nu^{5} + 28\nu^{4} - 4\nu^{3} - 24\nu^{2} - 16\nu - 32 ) / 64 $ (v^11 - 2v^9 - 4v^8 - 7v^7 + 4v^5 + 28v^4 - 4v^3 - 24v^2 - 16v - 32) / 64
$\beta_{6}$	$=$	$ ( \nu^{8} + \nu^{4} - 2\nu^{2} + 8 ) / 8 $ (v^8 + v^4 - 2*v^2 + 8) / 8
$\beta_{7}$	$=$	$ ( -\nu^{11} + 2\nu^{10} + 2\nu^{9} + 7\nu^{7} - 14\nu^{6} - 4\nu^{5} + 12\nu^{4} + 4\nu^{3} + 48\nu^{2} + 16\nu - 64 ) / 64 $ (-v^11 + 2v^10 + 2v^9 + 7v^7 - 14v^6 - 4v^5 + 12v^4 + 4v^3 + 48v^2 + 16*v - 64) / 64
$\beta_{8}$	$=$	$ ( - \nu^{11} + 4 \nu^{10} + 2 \nu^{9} - 4 \nu^{8} + 7 \nu^{7} + 4 \nu^{6} - 4 \nu^{5} + 20 \nu^{4} + \cdots - 96 ) / 64 $ (-v^11 + 4v^10 + 2v^9 - 4v^8 + 7v^7 + 4v^6 - 4v^5 + 20v^4 + 4v^3 + 8v^2 + 16v - 96) / 64
$\beta_{9}$	$=$	$ ( 3\nu^{11} - 2\nu^{9} - 5\nu^{7} + 16\nu^{5} + 60\nu^{3} - 48\nu ) / 64 $ (3v^11 - 2v^9 - 5v^7 + 16v^5 + 60v^3 - 48v) / 64
$\beta_{10}$	$=$	$ ( -3\nu^{11} + 6\nu^{9} + 5\nu^{7} - 12\nu^{5} - 4\nu^{3} + 80\nu ) / 64 $ (-3v^11 + 6v^9 + 5v^7 - 12v^5 - 4v^3 + 80v) / 64
$\beta_{11}$	$=$	$ ( -3\nu^{11} + 2\nu^{9} + 5\nu^{7} + 16\nu^{5} - 28\nu^{3} + 48\nu ) / 64 $ (-3v^11 + 2v^9 + 5v^7 + 16v^5 - 28v^3 + 48v) / 64

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} $ b2
$\nu^{3}$	$=$	$ \beta_{10} + \beta_{9} + \beta_{3} $ b10 + b9 + b3
$\nu^{4}$	$=$	$ \beta_{10} + \beta_{6} + 2\beta_{5} + \beta_{4} + \beta_{2} - \beta_1 $ b10 + b6 + 2*b5 + b4 + b2 - b1
$\nu^{5}$	$=$	$ 2\beta_{11} - \beta_{10} + \beta_{9} - \beta_{3} $ 2*b11 - b10 + b9 - b3
$\nu^{6}$	$=$	$ \beta_{10} + 2\beta_{8} - 4\beta_{7} + \beta_{6} + \beta_{4} + 3\beta_{2} - \beta _1 - 2 $ b10 + 2b8 - 4b7 + b6 + b4 + 3*b2 - b1 - 2
$\nu^{7}$	$=$	$ \beta_{10} - \beta_{9} + 6\beta_{4} - \beta_{3} - 4\beta_1 $ b10 - b9 + 6b4 - b3 - 4b1
$\nu^{8}$	$=$	$ -\beta_{10} + 7\beta_{6} - 2\beta_{5} - \beta_{4} + \beta_{2} + \beta _1 - 8 $ -b10 + 7b6 - 2b5 - b4 + b2 + b1 - 8
$\nu^{9}$	$=$	$ -2\beta_{11} + 3\beta_{10} + \beta_{9} - 13\beta_{3} - 8\beta_1 $ -2b11 + 3b10 + b9 - 13b3 - 8b1
$\nu^{10}$	$=$	$ -15\beta_{10} + 14\beta_{8} + 4\beta_{7} + \beta_{6} - 12\beta_{5} - 15\beta_{4} - 9\beta_{2} + 15\beta _1 + 18 $ -15b10 + 14b8 + 4b7 + b6 - 12b5 - 15b4 - 9b2 + 15*b1 + 18
$\nu^{11}$	$=$	$ -12\beta_{11} - 11\beta_{10} - 5\beta_{9} + 10\beta_{4} - 25\beta_{3} + 4\beta_1 $ -12b11 - 11b10 - 5b9 + 10b4 - 25b3 + 4b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/396\mathbb{Z}\right)^\times$.

$n$	$145$	$199$	$353$
$\chi(n)$	$-1$	$-1$	$1$

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} $

307.1

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

−1.37027

−

0.349801i

1.75528

0.958643i

−0.406728

−2.27740

−2.06987

−

1.92760i

0.557328

0.142274i

307.2

−1.37027

0.349801i

1.75528

−

0.958643i

−0.406728

−2.27740

−2.06987

1.92760i

0.557328

−

0.142274i

307.3

−1.19877

−

0.750295i

0.874114

1.79887i

3.34596

3.56257

0.301817

−

2.81228i

−4.01105

−

2.51046i

307.4

−1.19877

0.750295i

0.874114

−

1.79887i

3.34596

3.56257

0.301817

2.81228i

−4.01105

2.51046i

307.5

−0.430469

−

1.34711i

−1.62939

1.15978i

−2.93923

−0.348612

2.26374

1.69572i

1.26525

3.95946i

307.6

−0.430469

1.34711i

−1.62939

−

1.15978i

−2.93923

−0.348612

2.26374

−

1.69572i

1.26525

−

3.95946i

307.7

0.430469

−

1.34711i

−1.62939

−

1.15978i

−2.93923

0.348612

−2.26374

1.69572i

−1.26525

3.95946i

307.8

0.430469

1.34711i

−1.62939

1.15978i

−2.93923

0.348612

−2.26374

−

1.69572i

−1.26525

−

3.95946i

307.9

1.19877

−

0.750295i

0.874114

−

1.79887i

3.34596

−3.56257

−0.301817

−

2.81228i

4.01105

−

2.51046i

307.10

1.19877

0.750295i

0.874114

1.79887i

3.34596

−3.56257

−0.301817

2.81228i

4.01105

2.51046i

307.11

1.37027

−

0.349801i

1.75528

−

0.958643i

−0.406728

2.27740

2.06987

−

1.92760i

−0.557328

0.142274i

307.12

1.37027

0.349801i

1.75528

0.958643i

−0.406728

2.27740

2.06987

1.92760i

−0.557328

−

0.142274i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
11.b	odd	2	1	inner
44.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	396.2.h.d		12
3.b	odd	2	1		132.2.h.a	✓	12
4.b	odd	2	1	inner	396.2.h.d		12
11.b	odd	2	1	inner	396.2.h.d		12
12.b	even	2	1		132.2.h.a	✓	12
24.f	even	2	1		2112.2.o.g		12
24.h	odd	2	1		2112.2.o.g		12
33.d	even	2	1		132.2.h.a	✓	12
44.c	even	2	1	inner	396.2.h.d		12
132.d	odd	2	1		132.2.h.a	✓	12
264.m	even	2	1		2112.2.o.g		12
264.p	odd	2	1		2112.2.o.g		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
132.2.h.a	✓	12	3.b	odd	2	1
132.2.h.a	✓	12	12.b	even	2	1
132.2.h.a	✓	12	33.d	even	2	1
132.2.h.a	✓	12	132.d	odd	2	1
396.2.h.d		12	1.a	even	1	1	trivial
396.2.h.d		12	4.b	odd	2	1	inner
396.2.h.d		12	11.b	odd	2	1	inner
396.2.h.d		12	44.c	even	2	1	inner
2112.2.o.g		12	24.f	even	2	1
2112.2.o.g		12	24.h	odd	2	1
2112.2.o.g		12	264.m	even	2	1
2112.2.o.g		12	264.p	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{3} - 10T_{5} - 4 $ T5^3 - 10*T5 - 4 acting on $S_{2}^{\mathrm{new}}(396, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} - 2 T^{10} + \cdots + 64 $ T^12 - 2T^10 + T^8 + 4T^6 + 4T^4 - 32T^2 + 64
$3$	$ T^{12} $ T^12
$5$	$ (T^{3} - 10 T - 4)^{4} $ (T^3 - 10*T - 4)^4
$7$	$ (T^{6} - 18 T^{4} + 68 T^{2} - 8)^{2} $ (T^6 - 18T^4 + 68T^2 - 8)^2
$11$	$ T^{12} + 2 T^{10} + \cdots + 1771561 $ T^12 + 2T^10 + 167T^8 - 420T^6 + 20207T^4 + 29282*T^2 + 1771561
$13$	$ (T^{6} + 16 T^{4} + \cdots + 32)^{2} $ (T^6 + 16T^4 + 52T^2 + 32)^2
$17$	$ (T^{6} + 66 T^{4} + \cdots + 32)^{2} $ (T^6 + 66T^4 + 576T^2 + 32)^2
$19$	$ (T^{6} - 66 T^{4} + \cdots - 800)^{2} $ (T^6 - 66T^4 + 512T^2 - 800)^2
$23$	$ (T^{6} + 20 T^{4} + \cdots + 16)^{2} $ (T^6 + 20T^4 + 100T^2 + 16)^2
$29$	$ (T^{6} + 138 T^{4} + \cdots + 86528)^{2} $ (T^6 + 138T^4 + 6080T^2 + 86528)^2
$31$	$ (T^{6} + 104 T^{4} + \cdots + 25600)^{2} $ (T^6 + 104T^4 + 3216T^2 + 25600)^2
$37$	$ (T^{3} + 8 T^{2} + \cdots - 128)^{4} $ (T^3 + 8T^2 - 12T - 128)^4
$41$	$ (T^{6} + 170 T^{4} + \cdots + 80000)^{2} $ (T^6 + 170T^4 + 7808T^2 + 80000)^2
$43$	$ (T^{6} - 130 T^{4} + \cdots - 32)^{2} $ (T^6 - 130T^4 + 576T^2 - 32)^2
$47$	$ (T^{6} + 176 T^{4} + \cdots + 65536)^{2} $ (T^6 + 176T^4 + 6372T^2 + 65536)^2
$53$	$ (T^{3} - 4 T^{2} + \cdots + 652)^{4} $ (T^3 - 4T^2 - 114T + 652)^4
$59$	$ (T^{6} + 100 T^{4} + \cdots + 256)^{2} $ (T^6 + 100T^4 + 320T^2 + 256)^2
$61$	$ (T^{6} + 184 T^{4} + \cdots + 12800)^{2} $ (T^6 + 184T^4 + 5652T^2 + 12800)^2
$67$	$ (T^{6} + 72 T^{4} + \cdots + 256)^{2} $ (T^6 + 72T^4 + 656T^2 + 256)^2
$71$	$ (T^{6} + 128 T^{4} + \cdots + 6400)^{2} $ (T^6 + 128T^4 + 3396T^2 + 6400)^2
$73$	$ (T^{6} + 456 T^{4} + \cdots + 2580992)^{2} $ (T^6 + 456T^4 + 63552T^2 + 2580992)^2
$79$	$ (T^{6} - 66 T^{4} + \cdots - 9800)^{2} $ (T^6 - 66T^4 + 1412T^2 - 9800)^2
$83$	$ (T^{6} - 320 T^{4} + \cdots - 346112)^{2} $ (T^6 - 320T^4 + 24896T^2 - 346112)^2
$89$	$ (T^{3} - 2 T^{2} + \cdots - 200)^{4} $ (T^3 - 2T^2 - 124T - 200)^4
$97$	$ (T^{3} + 12 T^{2} + \cdots - 64)^{4} $ (T^3 + 12T^2 + 8T - 64)^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:	\( N \)	\(=\)	\( 396 = 2^{2} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	396.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + \beta_{2} q^{4} + ( - \beta_{10} + \beta_{7} + \cdots + \beta_1) q^{5} + (\beta_{11} + \beta_{3}) q^{7} + (\beta_{10} + \beta_{9} + \beta_{3}) q^{8} + ( - \beta_{11} + \beta_{10} + \cdots - \beta_{4}) q^{10}+ \cdots + ( - 2 \beta_{11} + 2 \beta_{10} + \cdots - \beta_1) q^{98}+O(q^{100}) \) q + b1 * q^2 + b2 * q^4 + (-b10 + b7 - b5 - b4 + b1) * q^5 + (b11 + b3) * q^7 + (b10 + b9 + b3) * q^8 + (-b11 + b10 + b9 - b4) * q^10 + (-b11 + b8 + b2) * q^11 + (-b4 + b3) * q^13 + (2b10 - b8 - 2b7 + b6 + b5 + 2b4 + b2 - 2b1 - 1) * q^14 + (b10 + b6 + 2b5 + b4 + b2 - b1) q^16 + (b10 - b9 - b3 - b1) * q^17 + (-b11 + 2b10 + b4 + 2b3) * q^19 + (-b10 + b8 + 2b7 - b6 + b5 - b4 - b2 + b1 + 3) q^20 + (-b10 + b9 + 2b7 - b2) q^22 + (-b7 - b5) * q^23 + (-3b10 + 2b8 + 2b7 - b6 - 2b5 - 3b4 - 2b2 + 3b1 + 3) q^25 + (b10 - b8 - b6 + b5 + b4 - b2 - b1 + 1) * q^26 + (b11 + b10 - b9 + b4) * q^28 + (b9 - 3b4 + 3b3) * q^29 + (-b10 + 2b8 - 2b7 + b6 - 2b5 - b4 + 2b2 + b1) * q^31 + (2b11 - b10 + b9 - b3) q^32 + (-b10 + b6 - 2b5 - b4 - 2b2 + b1 + 4) * q^34 + (2b11 - b10 + b4 + 2b3 - 3b1) q^35 + (3b10 - 2b8 - 2b7 + b6 + 2b5 + 3b4 + 2b2 - 3b1 - 4) q^37 + (b10 - 2b8 + 2b7 + 2b6 + 2b5 + b4 - b1) * q^38 + (b11 - 3b10 + b9 - b4 + 2b1) * q^40 + (-2b10 - 3b9 + b4 + b3 + 2b1) q^41 + (-b11 - 2b10 + b4 - 2b3 - 4b1) q^43 + (-b10 + b9 + b8 + b6 + b5 - 2b4 - b3 + b2 - 3) q^44 + (-b11 + b10 - b9 + b4) * q^46 + (-b10 + 2b8 - 3b7 - 3b5 - b4 + 2b2 + b1) * q^47 + (-2b10 + 2b7 - 2b5 - 2b4 + 2b1 - 1) q^49 + (-2b11 - 2b10 - 2b3 + b1) q^50 + (b11 - b10 - b9 - b4 + 2b3 + 2b1) * q^52 + (-5b10 + 4b8 + 3b7 - 2b6 - 3b5 - 5b4 - 4b2 + 5b1 + 4) * q^53 + (-b10 + 2b7 - b6 + 2b5 - 2b4 - 3b3 + 3b1) q^55 + (b10 - b8 - 2b7 + 3b6 - b5 + b4 + b2 - b1 + 1) * q^56 + (3b10 - 2b8 - 3b6 + 4b5 + 3b4 - 2b2 - 3b1 + 2) q^58 + (b10 - 2b8 - 2b6 + b4 - 2b2 - b1) q^59 + (-2b10 - 2b9 - b4 + 3b3 + 2b1) * q^61 + (-2b11 + 2b10 + 4b4 - 2b3 - 2b1) q^62 + (b10 + 2b8 - 4b7 + b6 + b4 + 3b2 - b1 - 2) q^64 + (-b10 + b4 + b1) * q^65 + (b10 - 2b8 - b6 + b4 - 2b2 - b1) * q^67 + (-2b11 - 2b9 - 4b3 + 4b1) * q^68 + (4b10 - 2b8 - 4b7 + 2b6 + 2b5 + 4b4 - 4b1 - 6) q^70 + (b10 - 2b8 + b7 + 2b6 + b5 + b4 - 2b2 - b1) q^71 + (4b10 + 2b9 - 4b3 - 4b1) * q^73 + (2b11 + 2b10 + 2b3 - 2b1) * q^74 + (2b11 + 2b9 - 4b4 - 2b3 + 2b1) q^76 + (2b10 - b9 - b7 + b5 + 3b4 - 3b3 - 2b1 - 4) * q^77 + (b11 - 2b10 - b3 - 2b1) * q^79 + (b10 + b8 - 2b7 - 3b6 + b5 + b4 + 3b2 - b1 - 5) q^80 + (b10 - 4b8 - b6 - 2b5 + b4 - b1 - 4) * q^82 + (4b10 + 2b4 + 6b3) q^83 + (-6b9 + 4b4 - 4b3) q^85 + (-3b10 + 2b8 + 2b7 - 2b6 - 2b5 - 3b4 - 4b2 + 3b1 - 4) * q^86 + (b11 - 2b10 + b9 + 2b8 - 3b6 - 2b3 - b2 - 3b1 + 2) q^88 + (4b10 - 4b8 - 2b7 + 2b6 + 2b5 + 4b4 + 4b2 - 4b1 - 2) * q^89 + (b10 - 2b8 - b6 + b4 - 2b2 - b1) * q^91 + (-b10 - b8 + 2b7 + b6 - b5 - b4 - b2 + b1 + 1) q^92 + (-3b11 + 3b10 - b9 + 5b4 - 2b1) * q^94 + (2b11 - 4b10 - 4b4 - 6b3 + 4b1) q^95 + (2b10 - 2b7 + 2b5 + 2b4 - 2b1 - 4) q^97 + (-2b11 + 2b10 + 2b9 - 2b4 - b1) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{4} - 4 q^{14} + 4 q^{16} + 28 q^{20} - 4 q^{22} + 20 q^{25} + 12 q^{26} + 40 q^{34} - 32 q^{37} + 8 q^{38} - 36 q^{44} - 12 q^{49} + 16 q^{53} + 20 q^{56} + 24 q^{58} - 20 q^{64} - 64 q^{70}+ \cdots - 48 q^{97}+O(q^{100}) \) 12 * q + 4 * q^4 - 4 * q^14 + 4 * q^16 + 28 * q^20 - 4 * q^22 + 20 * q^25 + 12 * q^26 + 40 * q^34 - 32 * q^37 + 8 * q^38 - 36 * q^44 - 12 * q^49 + 16 * q^53 + 20 * q^56 + 24 * q^58 - 20 * q^64 - 64 * q^70 - 48 * q^77 - 52 * q^80 - 32 * q^82 - 72 * q^86 + 12 * q^88 + 8 * q^89 + 12 * q^92 - 48 * q^97

Introduction

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Varieties

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Database

Properties

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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	396.2.h.d

Level	$396$
Weight	$2$
Character orbit	396.h
Analytic conductor	$3.162$
Analytic rank	$0$
Dimension	$12$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(3.16207592004\)
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:	no (minimal twist has level 132)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( \nu^{2} \) v^2
\(\beta_{3}\)	\(=\)	\( ( -\nu^{9} - \nu^{5} + 2\nu^{3} - 8\nu ) / 16 \) (-v^9 - v^5 + 2v^3 - 8v) / 16
\(\beta_{4}\)	\(=\)	\( ( \nu^{11} - 2\nu^{9} + 9\nu^{7} + 4\nu^{5} + 12\nu^{3} + 16\nu ) / 64 \) (v^11 - 2v^9 + 9v^7 + 4v^5 + 12v^3 + 16*v) / 64
\(\beta_{5}\)	\(=\)	\( ( \nu^{11} - 2\nu^{9} - 4\nu^{8} - 7\nu^{7} + 4\nu^{5} + 28\nu^{4} - 4\nu^{3} - 24\nu^{2} - 16\nu - 32 ) / 64 \) (v^11 - 2v^9 - 4v^8 - 7v^7 + 4v^5 + 28v^4 - 4v^3 - 24v^2 - 16v - 32) / 64
\(\beta_{6}\)	\(=\)	\( ( \nu^{8} + \nu^{4} - 2\nu^{2} + 8 ) / 8 \) (v^8 + v^4 - 2*v^2 + 8) / 8
\(\beta_{7}\)	\(=\)	\( ( -\nu^{11} + 2\nu^{10} + 2\nu^{9} + 7\nu^{7} - 14\nu^{6} - 4\nu^{5} + 12\nu^{4} + 4\nu^{3} + 48\nu^{2} + 16\nu - 64 ) / 64 \) (-v^11 + 2v^10 + 2v^9 + 7v^7 - 14v^6 - 4v^5 + 12v^4 + 4v^3 + 48v^2 + 16*v - 64) / 64
\(\beta_{8}\)	\(=\)	\( ( - \nu^{11} + 4 \nu^{10} + 2 \nu^{9} - 4 \nu^{8} + 7 \nu^{7} + 4 \nu^{6} - 4 \nu^{5} + 20 \nu^{4} + \cdots - 96 ) / 64 \) (-v^11 + 4v^10 + 2v^9 - 4v^8 + 7v^7 + 4v^6 - 4v^5 + 20v^4 + 4v^3 + 8v^2 + 16v - 96) / 64
\(\beta_{9}\)	\(=\)	\( ( 3\nu^{11} - 2\nu^{9} - 5\nu^{7} + 16\nu^{5} + 60\nu^{3} - 48\nu ) / 64 \) (3v^11 - 2v^9 - 5v^7 + 16v^5 + 60v^3 - 48v) / 64
\(\beta_{10}\)	\(=\)	\( ( -3\nu^{11} + 6\nu^{9} + 5\nu^{7} - 12\nu^{5} - 4\nu^{3} + 80\nu ) / 64 \) (-3v^11 + 6v^9 + 5v^7 - 12v^5 - 4v^3 + 80v) / 64
\(\beta_{11}\)	\(=\)	\( ( -3\nu^{11} + 2\nu^{9} + 5\nu^{7} + 16\nu^{5} - 28\nu^{3} + 48\nu ) / 64 \) (-3v^11 + 2v^9 + 5v^7 + 16v^5 - 28v^3 + 48v) / 64

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{2} \) b2
\(\nu^{3}\)	\(=\)	\( \beta_{10} + \beta_{9} + \beta_{3} \) b10 + b9 + b3
\(\nu^{4}\)	\(=\)	\( \beta_{10} + \beta_{6} + 2\beta_{5} + \beta_{4} + \beta_{2} - \beta_1 \) b10 + b6 + 2*b5 + b4 + b2 - b1
\(\nu^{5}\)	\(=\)	\( 2\beta_{11} - \beta_{10} + \beta_{9} - \beta_{3} \) 2*b11 - b10 + b9 - b3
\(\nu^{6}\)	\(=\)	\( \beta_{10} + 2\beta_{8} - 4\beta_{7} + \beta_{6} + \beta_{4} + 3\beta_{2} - \beta _1 - 2 \) b10 + 2b8 - 4b7 + b6 + b4 + 3*b2 - b1 - 2
\(\nu^{7}\)	\(=\)	\( \beta_{10} - \beta_{9} + 6\beta_{4} - \beta_{3} - 4\beta_1 \) b10 - b9 + 6b4 - b3 - 4b1
\(\nu^{8}\)	\(=\)	\( -\beta_{10} + 7\beta_{6} - 2\beta_{5} - \beta_{4} + \beta_{2} + \beta _1 - 8 \) -b10 + 7b6 - 2b5 - b4 + b2 + b1 - 8
\(\nu^{9}\)	\(=\)	\( -2\beta_{11} + 3\beta_{10} + \beta_{9} - 13\beta_{3} - 8\beta_1 \) -2b11 + 3b10 + b9 - 13b3 - 8b1
\(\nu^{10}\)	\(=\)	\( -15\beta_{10} + 14\beta_{8} + 4\beta_{7} + \beta_{6} - 12\beta_{5} - 15\beta_{4} - 9\beta_{2} + 15\beta _1 + 18 \) -15b10 + 14b8 + 4b7 + b6 - 12b5 - 15b4 - 9b2 + 15*b1 + 18
\(\nu^{11}\)	\(=\)	\( -12\beta_{11} - 11\beta_{10} - 5\beta_{9} + 10\beta_{4} - 25\beta_{3} + 4\beta_1 \) -12b11 - 11b10 - 5b9 + 10b4 - 25b3 + 4b1

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 307.12
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} - 2 T^{10} + \cdots + 64 \) T^12 - 2T^10 + T^8 + 4T^6 + 4T^4 - 32T^2 + 64
$3$	\( T^{12} \) T^12
$5$	\( (T^{3} - 10 T - 4)^{4} \) (T^3 - 10*T - 4)^4
$7$	\( (T^{6} - 18 T^{4} + 68 T^{2} - 8)^{2} \) (T^6 - 18T^4 + 68T^2 - 8)^2
$11$	\( T^{12} + 2 T^{10} + \cdots + 1771561 \) T^12 + 2T^10 + 167T^8 - 420T^6 + 20207T^4 + 29282*T^2 + 1771561
$13$	\( (T^{6} + 16 T^{4} + \cdots + 32)^{2} \) (T^6 + 16T^4 + 52T^2 + 32)^2
$17$	\( (T^{6} + 66 T^{4} + \cdots + 32)^{2} \) (T^6 + 66T^4 + 576T^2 + 32)^2
$19$	\( (T^{6} - 66 T^{4} + \cdots - 800)^{2} \) (T^6 - 66T^4 + 512T^2 - 800)^2
$23$	\( (T^{6} + 20 T^{4} + \cdots + 16)^{2} \) (T^6 + 20T^4 + 100T^2 + 16)^2
$29$	\( (T^{6} + 138 T^{4} + \cdots + 86528)^{2} \) (T^6 + 138T^4 + 6080T^2 + 86528)^2
$31$	\( (T^{6} + 104 T^{4} + \cdots + 25600)^{2} \) (T^6 + 104T^4 + 3216T^2 + 25600)^2
$37$	\( (T^{3} + 8 T^{2} + \cdots - 128)^{4} \) (T^3 + 8T^2 - 12T - 128)^4
$41$	\( (T^{6} + 170 T^{4} + \cdots + 80000)^{2} \) (T^6 + 170T^4 + 7808T^2 + 80000)^2
$43$	\( (T^{6} - 130 T^{4} + \cdots - 32)^{2} \) (T^6 - 130T^4 + 576T^2 - 32)^2
$47$	\( (T^{6} + 176 T^{4} + \cdots + 65536)^{2} \) (T^6 + 176T^4 + 6372T^2 + 65536)^2
$53$	\( (T^{3} - 4 T^{2} + \cdots + 652)^{4} \) (T^3 - 4T^2 - 114T + 652)^4
$59$	\( (T^{6} + 100 T^{4} + \cdots + 256)^{2} \) (T^6 + 100T^4 + 320T^2 + 256)^2
$61$	\( (T^{6} + 184 T^{4} + \cdots + 12800)^{2} \) (T^6 + 184T^4 + 5652T^2 + 12800)^2
$67$	\( (T^{6} + 72 T^{4} + \cdots + 256)^{2} \) (T^6 + 72T^4 + 656T^2 + 256)^2
$71$	\( (T^{6} + 128 T^{4} + \cdots + 6400)^{2} \) (T^6 + 128T^4 + 3396T^2 + 6400)^2
$73$	\( (T^{6} + 456 T^{4} + \cdots + 2580992)^{2} \) (T^6 + 456T^4 + 63552T^2 + 2580992)^2
$79$	\( (T^{6} - 66 T^{4} + \cdots - 9800)^{2} \) (T^6 - 66T^4 + 1412T^2 - 9800)^2
$83$	\( (T^{6} - 320 T^{4} + \cdots - 346112)^{2} \) (T^6 - 320T^4 + 24896T^2 - 346112)^2
$89$	\( (T^{3} - 2 T^{2} + \cdots - 200)^{4} \) (T^3 - 2T^2 - 124T - 200)^4
$97$	\( (T^{3} + 12 T^{2} + \cdots - 64)^{4} \) (T^3 + 12T^2 + 8T - 64)^4
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\(n\)	\(145\)	\(199\)	\(353\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)