LMFDB - Newform orbit 352.2.e.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [352,2,Mod(351,352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(352, 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("352.351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 352 = 2^{5} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	352.e (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:	$2.81073415115$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.653473922154496.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{10} + 13x^{8} - 28x^{6} + 52x^{4} - 64x^{2} + 64 $ x^12 - 4x^10 + 13x^8 - 28x^6 + 52x^4 - 64*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{15} $
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_{2} q^{3} - \beta_{5} q^{5} + \beta_{8} q^{7} + \beta_{3} q^{9}+O(q^{10}) $ q - b2 * q^3 - b5 * q^5 + b8 * q^7 + b3 * q^9	$ q - \beta_{2} q^{3} - \beta_{5} q^{5} + \beta_{8} q^{7} + \beta_{3} q^{9} + \beta_{7} q^{11} - \beta_{10} q^{13} + (\beta_{4} - \beta_{2}) q^{15} - \beta_{11} q^{17} + ( - \beta_{9} - \beta_{7}) q^{19} + (\beta_{11} - \beta_{10} - \beta_1) q^{21} + ( - \beta_{7} - \beta_{6} + \cdots - \beta_{2}) q^{23}+ \cdots + ( - \beta_{9} - 2 \beta_{8} + \cdots + 3 \beta_{2}) q^{99}+O(q^{100}) $ q - b2 * q^3 - b5 * q^5 + b8 * q^7 + b3 * q^9 + b7 * q^11 - b10 * q^13 + (b4 - b2) * q^15 - b11 * q^17 + (-b9 - b7) * q^19 + (b11 - b10 - b1) * q^21 + (-b7 - b6 + b4 - b2) * q^23 + b3 * q^25 + (-b7 - b6 + 2b4 - b2) q^27 + (b11 + b1) * q^29 + (b7 + b6 - b4 - b2) * q^31 + (b3 - b1 - 1) * q^33 + (b9 + b6) * q^35 - b5 * q^37 - b8 * q^39 + (-b11 + 2b10) q^41 + (b7 - b6) * q^43 + (-2b5 - 2) q^45 + (b7 + b6 + 4b2) q^47 + (2b5 - 2b3 + 3) * q^49 + (b9 + 2b8 + b6) q^51 + (2b5 - 2b3) * q^53 + (-b8 - 2b6 - b4 - b2) q^55 + (-b11 + 2b1) q^57 + (-4b4 + 3b2) * q^59 + (-b11 + b1) * q^61 + (-2b9 - 2b8 - 2b7) q^63 + (b11 + 2b10) q^65 + (2b7 + 2b6 - 2b4 + 3b2) * q^67 + (b5 - 2b3) q^69 + (b7 + b6 - 5b4 + 3b2) * q^71 + (b11 - 2b10) q^73 + (-b7 - b6 + 2b4 + 2b2) * q^75 + (b11 + b10 + 2b5 - 2b3 - b1 + 2) * q^77 + (2b9 + 2b8 + 2b6) q^79 + (2b5 + 1) q^81 + (-2b8 - b7 + b6) q^83 + (-b11 + 3b10 - b1) q^85 + (2b7 - 2b6) * q^87 + (4b5 - 3b3 - 1) * q^89 + (-2b7 - 2b6 - 2b4 - 4b2) * q^91 + (-b5 + 4b3 - 6) q^93 + (2b9 - b8 + 2b6) * q^95 + (-4b5 + b3 - 5) q^97 + (-b9 - 2b8 + 2b4 + 3b2) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{9}+O(q^{10}) $ 12 * q - 4 * q^9	$ 12 q - 4 q^{9} - 4 q^{25} - 16 q^{33} - 24 q^{45} + 44 q^{49} + 8 q^{53} + 8 q^{69} + 32 q^{77} + 12 q^{81} - 88 q^{93} - 64 q^{97}+O(q^{100}) $ 12 * q - 4 * q^9 - 4 * q^25 - 16 * q^33 - 24 * q^45 + 44 * q^49 + 8 * q^53 + 8 * q^69 + 32 * q^77 + 12 * q^81 - 88 * q^93 - 64 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 4x^{10} + 13x^{8} - 28x^{6} + 52x^{4} - 64x^{2} + 64 $ x^12 - 4*x^10 + 13*x^8 - 28*x^6 + 52*x^4 - 64*x^2 + 64 :

$\beta_{1}$	$=$	$ ( \nu^{7} - 2\nu^{5} + 5\nu^{3} - 2\nu ) / 2 $ (v^7 - 2v^5 + 5v^3 - 2*v) / 2
$\beta_{2}$	$=$	$ ( \nu^{10} - 6\nu^{8} + 9\nu^{6} - 14\nu^{4} ) / 32 $ (v^10 - 6v^8 + 9v^6 - 14*v^4) / 32
$\beta_{3}$	$=$	$ ( \nu^{8} - 4\nu^{6} + 9\nu^{4} - 12\nu^{2} + 16 ) / 4 $ (v^8 - 4v^6 + 9v^4 - 12*v^2 + 16) / 4
$\beta_{4}$	$=$	$ ( 3\nu^{10} - 10\nu^{8} + 27\nu^{6} - 34\nu^{4} + 64\nu^{2} - 32 ) / 32 $ (3v^10 - 10v^8 + 27v^6 - 34v^4 + 64*v^2 - 32) / 32
$\beta_{5}$	$=$	$ ( \nu^{10} - 2\nu^{8} + 5\nu^{6} - 10\nu^{4} + 12\nu^{2} - 8 ) / 8 $ (v^10 - 2v^8 + 5v^6 - 10v^4 + 12v^2 - 8) / 8
$\beta_{6}$	$=$	$ ( - 3 \nu^{11} + 3 \nu^{10} + 2 \nu^{9} - 18 \nu^{8} - 11 \nu^{7} + 43 \nu^{6} + 26 \nu^{5} - 106 \nu^{4} + \cdots - 224 ) / 64 $ (-3v^11 + 3v^10 + 2v^9 - 18v^8 - 11v^7 + 43v^6 + 26v^5 - 106v^4 - 48v^3 + 208v^2 + 96*v - 224) / 64
$\beta_{7}$	$=$	$ ( 3 \nu^{11} + 3 \nu^{10} - 2 \nu^{9} - 18 \nu^{8} + 11 \nu^{7} + 43 \nu^{6} - 26 \nu^{5} - 106 \nu^{4} + \cdots - 224 ) / 64 $ (3v^11 + 3v^10 - 2v^9 - 18v^8 + 11v^7 + 43v^6 - 26v^5 - 106v^4 + 48v^3 + 208v^2 - 96*v - 224) / 64
$\beta_{8}$	$=$	$ ( -\nu^{11} + 6\nu^{9} - 17\nu^{7} + 30\nu^{5} - 40\nu^{3} + 48\nu ) / 16 $ (-v^11 + 6v^9 - 17v^7 + 30v^5 - 40v^3 + 48*v) / 16
$\beta_{9}$	$=$	$ ( - 5 \nu^{11} - 3 \nu^{10} - 2 \nu^{9} + 18 \nu^{8} + 3 \nu^{7} - 43 \nu^{6} - 26 \nu^{5} + 106 \nu^{4} + \cdots + 224 ) / 64 $ (-5v^11 - 3v^10 - 2v^9 + 18v^8 + 3v^7 - 43v^6 - 26v^5 + 106v^4 + 112v^3 - 208v^2 - 288*v + 224) / 64
$\beta_{10}$	$=$	$ ( \nu^{11} - 4\nu^{9} + 9\nu^{7} - 20\nu^{5} + 32\nu^{3} - 24\nu ) / 8 $ (v^11 - 4v^9 + 9v^7 - 20v^5 + 32v^3 - 24*v) / 8
$\beta_{11}$	$=$	$ ( \nu^{11} - 2\nu^{9} + 5\nu^{7} - 2\nu^{5} + 12\nu^{3} + 8\nu ) / 8 $ (v^11 - 2v^9 + 5v^7 - 2v^5 + 12v^3 + 8*v) / 8

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

$\nu$	$=$	$ ( \beta_{10} - \beta_{9} + \beta_{8} - 2\beta_{7} + \beta_{6} + \beta_1 ) / 8 $ (b10 - b9 + b8 - 2*b7 + b6 + b1) / 8
$\nu^{2}$	$=$	$ ( \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - 2\beta_{2} + 3 ) / 4 $ (b7 + b6 - b5 + b4 + b3 - 2*b2 + 3) / 4
$\nu^{3}$	$=$	$ ( \beta_{11} + \beta_{10} + \beta_{9} + 2\beta_{8} + \beta_{6} + \beta_1 ) / 4 $ (b11 + b10 + b9 + 2*b8 + b6 + b1) / 4
$\nu^{4}$	$=$	$ ( -2\beta_{5} + 3\beta_{4} + \beta_{3} - \beta_{2} - 3 ) / 2 $ (-2b5 + 3b4 + b3 - b2 - 3) / 2
$\nu^{5}$	$=$	$ ( 8\beta_{11} - 5\beta_{10} + 5\beta_{9} - \beta_{8} + 2\beta_{7} + 3\beta_{6} - \beta_1 ) / 8 $ (8b11 - 5b10 + 5b9 - b8 + 2b7 + 3*b6 - b1) / 8
$\nu^{6}$	$=$	$ ( -5\beta_{7} - 5\beta_{6} - 3\beta_{5} + 11\beta_{4} - 5\beta_{3} - 6\beta_{2} - 7 ) / 4 $ (-5b7 - 5b6 - 3b5 + 11b4 - 5b3 - 6b2 - 7) / 4
$\nu^{7}$	$=$	$ ( 3\beta_{11} - 9\beta_{10} - \beta_{9} - 10\beta_{8} - \beta_{6} + 3\beta_1 ) / 4 $ (3b11 - 9b10 - b9 - 10b8 - b6 + 3b1) / 4
$\nu^{8}$	$=$	$ ( -4\beta_{7} - 4\beta_{6} + 6\beta_{5} + \beta_{4} - 5\beta_{3} - 15\beta_{2} - 1 ) / 2 $ (-4b7 - 4b6 + 6b5 + b4 - 5b3 - 15*b2 - 1) / 2
$\nu^{9}$	$=$	$ ( -8\beta_{11} - 19\beta_{10} - 13\beta_{9} - 7\beta_{8} + 14\beta_{7} - 27\beta_{6} + 25\beta_1 ) / 8 $ (-8b11 - 19b10 - 13b9 - 7b8 + 14b7 - 27b6 + 25*b1) / 8
$\nu^{10}$	$=$	$ ( -3\beta_{7} - 3\beta_{6} + 43\beta_{5} - 3\beta_{4} + 13\beta_{3} - 26\beta_{2} - 33 ) / 4 $ (-3b7 - 3b6 + 43b5 - 3b4 + 13b3 - 26b2 - 33) / 4
$\nu^{11}$	$=$	$ ( 5\beta_{11} + 5\beta_{10} - 11\beta_{9} + 14\beta_{8} + 24\beta_{7} - 35\beta_{6} - 7\beta_1 ) / 4 $ (5b11 + 5b10 - 11b9 + 14b8 + 24b7 - 35b6 - 7*b1) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/352\mathbb{Z}\right)^\times$.

$n$	$133$	$287$	$321$
$\chi(n)$	$1$	$-1$	$-1$

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} $

351.1

0.892524	−	1.09700i
−0.892524	+	1.09700i
−1.35489	+	0.405301i
1.35489	−	0.405301i
−1.16947	−	0.795191i
1.16947	+	0.795191i
−1.16947	+	0.795191i
1.16947	−	0.795191i
−1.35489	−	0.405301i
1.35489	+	0.405301i
0.892524	+	1.09700i
−0.892524	−	1.09700i

−

2.81361i

0.289169

−4.38799

−4.91638

351.2

−

2.81361i

0.289169

4.38799

−4.91638

351.3

−

1.34292i

2.48929

−1.62121

1.19656

351.4

−

1.34292i

2.48929

1.62121

1.19656

351.5

−

0.529317i

−2.77846

−3.18077

2.71982

351.6

−

0.529317i

−2.77846

3.18077

2.71982

351.7

0.529317i

−2.77846

−3.18077

2.71982

351.8

0.529317i

−2.77846

3.18077

2.71982

351.9

1.34292i

2.48929

−1.62121

1.19656

351.10

1.34292i

2.48929

1.62121

1.19656

351.11

2.81361i

0.289169

−4.38799

−4.91638

351.12

2.81361i

0.289169

4.38799

−4.91638

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	352.2.e.a	✓	12
3.b	odd	2	1		3168.2.o.e		12
4.b	odd	2	1	inner	352.2.e.a	✓	12
8.b	even	2	1		704.2.e.d		12
8.d	odd	2	1		704.2.e.d		12
11.b	odd	2	1	inner	352.2.e.a	✓	12
12.b	even	2	1		3168.2.o.e		12
16.e	even	4	1		2816.2.g.d		12
16.e	even	4	1		2816.2.g.i		12
16.f	odd	4	1		2816.2.g.d		12
16.f	odd	4	1		2816.2.g.i		12
33.d	even	2	1		3168.2.o.e		12
44.c	even	2	1	inner	352.2.e.a	✓	12
88.b	odd	2	1		704.2.e.d		12
88.g	even	2	1		704.2.e.d		12
132.d	odd	2	1		3168.2.o.e		12
176.i	even	4	1		2816.2.g.d		12
176.i	even	4	1		2816.2.g.i		12
176.l	odd	4	1		2816.2.g.d		12
176.l	odd	4	1		2816.2.g.i		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
352.2.e.a	✓	12	1.a	even	1	1	trivial
352.2.e.a	✓	12	4.b	odd	2	1	inner
352.2.e.a	✓	12	11.b	odd	2	1	inner
352.2.e.a	✓	12	44.c	even	2	1	inner
704.2.e.d		12	8.b	even	2	1
704.2.e.d		12	8.d	odd	2	1
704.2.e.d		12	88.b	odd	2	1
704.2.e.d		12	88.g	even	2	1
2816.2.g.d		12	16.e	even	4	1
2816.2.g.d		12	16.f	odd	4	1
2816.2.g.d		12	176.i	even	4	1
2816.2.g.d		12	176.l	odd	4	1
2816.2.g.i		12	16.e	even	4	1
2816.2.g.i		12	16.f	odd	4	1
2816.2.g.i		12	176.i	even	4	1
2816.2.g.i		12	176.l	odd	4	1
3168.2.o.e		12	3.b	odd	2	1
3168.2.o.e		12	12.b	even	2	1
3168.2.o.e		12	33.d	even	2	1
3168.2.o.e		12	132.d	odd	2	1

Hecke kernels

This newform subspace is the entire newspace $S_{2}^{\mathrm{new}}(352, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ (T^{6} + 10 T^{4} + 17 T^{2} + 4)^{2} $ (T^6 + 10T^4 + 17T^2 + 4)^2
$5$	$ (T^{3} - 7 T + 2)^{4} $ (T^3 - 7*T + 2)^4
$7$	$ (T^{6} - 32 T^{4} + \cdots - 512)^{2} $ (T^6 - 32T^4 + 272T^2 - 512)^2
$11$	$ T^{12} - 6 T^{10} + \cdots + 1771561 $ T^12 - 6T^10 - 73T^8 + 460T^6 - 8833T^4 - 87846*T^2 + 1771561
$13$	$ (T^{6} + 40 T^{4} + \cdots + 128)^{2} $ (T^6 + 40T^4 + 144T^2 + 128)^2
$17$	$ (T^{6} + 64 T^{4} + \cdots + 8192)^{2} $ (T^6 + 64T^4 + 1296T^2 + 8192)^2
$19$	$ (T^{6} - 104 T^{4} + \cdots - 15488)^{2} $ (T^6 - 104T^4 + 2960T^2 - 15488)^2
$23$	$ (T^{6} + 50 T^{4} + \cdots + 3844)^{2} $ (T^6 + 50T^4 + 777T^2 + 3844)^2
$29$	$ (T^{6} + 136 T^{4} + \cdots + 32768)^{2} $ (T^6 + 136T^4 + 4096T^2 + 32768)^2
$31$	$ (T^{6} + 98 T^{4} + \cdots + 1156)^{2} $ (T^6 + 98T^4 + 1369T^2 + 1156)^2
$37$	$ (T^{3} - 7 T + 2)^{4} $ (T^3 - 7*T + 2)^4
$41$	$ (T^{6} + 160 T^{4} + \cdots + 147968)^{2} $ (T^6 + 160T^4 + 8464T^2 + 147968)^2
$43$	$ (T^{6} - 72 T^{4} + \cdots - 2048)^{2} $ (T^6 - 72T^4 + 1280T^2 - 2048)^2
$47$	$ (T^{6} + 156 T^{4} + \cdots + 118336)^{2} $ (T^6 + 156T^4 + 7664T^2 + 118336)^2
$53$	$ (T^{3} - 2 T^{2} - 68 T + 8)^{4} $ (T^3 - 2T^2 - 68T + 8)^4
$59$	$ (T^{6} + 186 T^{4} + \cdots + 1156)^{2} $ (T^6 + 186T^4 + 5985T^2 + 1156)^2
$61$	$ (T^{6} + 136 T^{4} + \cdots + 2048)^{2} $ (T^6 + 136T^4 + 2304T^2 + 2048)^2
$67$	$ (T^{6} + 202 T^{4} + \cdots + 131044)^{2} $ (T^6 + 202T^4 + 11249T^2 + 131044)^2
$71$	$ (T^{6} + 242 T^{4} + \cdots + 24964)^{2} $ (T^6 + 242T^4 + 7689T^2 + 24964)^2
$73$	$ (T^{6} + 160 T^{4} + \cdots + 147968)^{2} $ (T^6 + 160T^4 + 8464T^2 + 147968)^2
$79$	$ (T^{2} - 128)^{6} $ (T^2 - 128)^6
$83$	$ (T^{6} - 264 T^{4} + \cdots - 131072)^{2} $ (T^6 - 264T^4 + 14336T^2 - 131072)^2
$89$	$ (T^{3} - 187 T + 358)^{4} $ (T^3 - 187*T + 358)^4
$97$	$ (T^{3} + 16 T^{2} + \cdots - 722)^{4} $ (T^3 + 16T^2 - 19T - 722)^4
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 352 = 2^{5} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	352.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{2} q^{3} - \beta_{5} q^{5} + \beta_{8} q^{7} + \beta_{3} q^{9}+O(q^{10}) \) q - b2 * q^3 - b5 * q^5 + b8 * q^7 + b3 * q^9	\( q - \beta_{2} q^{3} - \beta_{5} q^{5} + \beta_{8} q^{7} + \beta_{3} q^{9} + \beta_{7} q^{11} - \beta_{10} q^{13} + (\beta_{4} - \beta_{2}) q^{15} - \beta_{11} q^{17} + ( - \beta_{9} - \beta_{7}) q^{19} + (\beta_{11} - \beta_{10} - \beta_1) q^{21} + ( - \beta_{7} - \beta_{6} + \cdots - \beta_{2}) q^{23}+ \cdots + ( - \beta_{9} - 2 \beta_{8} + \cdots + 3 \beta_{2}) q^{99}+O(q^{100}) \) q - b2 * q^3 - b5 * q^5 + b8 * q^7 + b3 * q^9 + b7 * q^11 - b10 * q^13 + (b4 - b2) * q^15 - b11 * q^17 + (-b9 - b7) * q^19 + (b11 - b10 - b1) * q^21 + (-b7 - b6 + b4 - b2) * q^23 + b3 * q^25 + (-b7 - b6 + 2b4 - b2) q^27 + (b11 + b1) * q^29 + (b7 + b6 - b4 - b2) * q^31 + (b3 - b1 - 1) * q^33 + (b9 + b6) * q^35 - b5 * q^37 - b8 * q^39 + (-b11 + 2b10) q^41 + (b7 - b6) * q^43 + (-2b5 - 2) q^45 + (b7 + b6 + 4b2) q^47 + (2b5 - 2b3 + 3) * q^49 + (b9 + 2b8 + b6) q^51 + (2b5 - 2b3) * q^53 + (-b8 - 2b6 - b4 - b2) q^55 + (-b11 + 2b1) q^57 + (-4b4 + 3b2) * q^59 + (-b11 + b1) * q^61 + (-2b9 - 2b8 - 2b7) q^63 + (b11 + 2b10) q^65 + (2b7 + 2b6 - 2b4 + 3b2) * q^67 + (b5 - 2b3) q^69 + (b7 + b6 - 5b4 + 3b2) * q^71 + (b11 - 2b10) q^73 + (-b7 - b6 + 2b4 + 2b2) * q^75 + (b11 + b10 + 2b5 - 2b3 - b1 + 2) * q^77 + (2b9 + 2b8 + 2b6) q^79 + (2b5 + 1) q^81 + (-2b8 - b7 + b6) q^83 + (-b11 + 3b10 - b1) q^85 + (2b7 - 2b6) * q^87 + (4b5 - 3b3 - 1) * q^89 + (-2b7 - 2b6 - 2b4 - 4b2) * q^91 + (-b5 + 4b3 - 6) q^93 + (2b9 - b8 + 2b6) * q^95 + (-4b5 + b3 - 5) q^97 + (-b9 - 2b8 + 2b4 + 3b2) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{9}+O(q^{10}) \) 12 * q - 4 * q^9	\( 12 q - 4 q^{9} - 4 q^{25} - 16 q^{33} - 24 q^{45} + 44 q^{49} + 8 q^{53} + 8 q^{69} + 32 q^{77} + 12 q^{81} - 88 q^{93} - 64 q^{97}+O(q^{100}) \) 12 * q - 4 * q^9 - 4 * q^25 - 16 * q^33 - 24 * q^45 + 44 * q^49 + 8 * q^53 + 8 * q^69 + 32 * q^77 + 12 * q^81 - 88 * q^93 - 64 * q^97

Introduction

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Newspace parameters

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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	352.2.e.a

Level	$352$
Weight	$2$
Character orbit	352.e
Analytic conductor	$2.811$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(2.81073415115\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	12.0.653473922154496.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 4x^{10} + 13x^{8} - 28x^{6} + 52x^{4} - 64x^{2} + 64 \) x^12 - 4x^10 + 13x^8 - 28x^6 + 52x^4 - 64*x^2 + 64
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{15} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{7} - 2\nu^{5} + 5\nu^{3} - 2\nu ) / 2 \) (v^7 - 2v^5 + 5v^3 - 2*v) / 2
\(\beta_{2}\)	\(=\)	\( ( \nu^{10} - 6\nu^{8} + 9\nu^{6} - 14\nu^{4} ) / 32 \) (v^10 - 6v^8 + 9v^6 - 14*v^4) / 32
\(\beta_{3}\)	\(=\)	\( ( \nu^{8} - 4\nu^{6} + 9\nu^{4} - 12\nu^{2} + 16 ) / 4 \) (v^8 - 4v^6 + 9v^4 - 12*v^2 + 16) / 4
\(\beta_{4}\)	\(=\)	\( ( 3\nu^{10} - 10\nu^{8} + 27\nu^{6} - 34\nu^{4} + 64\nu^{2} - 32 ) / 32 \) (3v^10 - 10v^8 + 27v^6 - 34v^4 + 64*v^2 - 32) / 32
\(\beta_{5}\)	\(=\)	\( ( \nu^{10} - 2\nu^{8} + 5\nu^{6} - 10\nu^{4} + 12\nu^{2} - 8 ) / 8 \) (v^10 - 2v^8 + 5v^6 - 10v^4 + 12v^2 - 8) / 8
\(\beta_{6}\)	\(=\)	\( ( - 3 \nu^{11} + 3 \nu^{10} + 2 \nu^{9} - 18 \nu^{8} - 11 \nu^{7} + 43 \nu^{6} + 26 \nu^{5} - 106 \nu^{4} + \cdots - 224 ) / 64 \) (-3v^11 + 3v^10 + 2v^9 - 18v^8 - 11v^7 + 43v^6 + 26v^5 - 106v^4 - 48v^3 + 208v^2 + 96*v - 224) / 64
\(\beta_{7}\)	\(=\)	\( ( 3 \nu^{11} + 3 \nu^{10} - 2 \nu^{9} - 18 \nu^{8} + 11 \nu^{7} + 43 \nu^{6} - 26 \nu^{5} - 106 \nu^{4} + \cdots - 224 ) / 64 \) (3v^11 + 3v^10 - 2v^9 - 18v^8 + 11v^7 + 43v^6 - 26v^5 - 106v^4 + 48v^3 + 208v^2 - 96*v - 224) / 64
\(\beta_{8}\)	\(=\)	\( ( -\nu^{11} + 6\nu^{9} - 17\nu^{7} + 30\nu^{5} - 40\nu^{3} + 48\nu ) / 16 \) (-v^11 + 6v^9 - 17v^7 + 30v^5 - 40v^3 + 48*v) / 16
\(\beta_{9}\)	\(=\)	\( ( - 5 \nu^{11} - 3 \nu^{10} - 2 \nu^{9} + 18 \nu^{8} + 3 \nu^{7} - 43 \nu^{6} - 26 \nu^{5} + 106 \nu^{4} + \cdots + 224 ) / 64 \) (-5v^11 - 3v^10 - 2v^9 + 18v^8 + 3v^7 - 43v^6 - 26v^5 + 106v^4 + 112v^3 - 208v^2 - 288*v + 224) / 64
\(\beta_{10}\)	\(=\)	\( ( \nu^{11} - 4\nu^{9} + 9\nu^{7} - 20\nu^{5} + 32\nu^{3} - 24\nu ) / 8 \) (v^11 - 4v^9 + 9v^7 - 20v^5 + 32v^3 - 24*v) / 8
\(\beta_{11}\)	\(=\)	\( ( \nu^{11} - 2\nu^{9} + 5\nu^{7} - 2\nu^{5} + 12\nu^{3} + 8\nu ) / 8 \) (v^11 - 2v^9 + 5v^7 - 2v^5 + 12v^3 + 8*v) / 8

\(\nu\)	\(=\)	\( ( \beta_{10} - \beta_{9} + \beta_{8} - 2\beta_{7} + \beta_{6} + \beta_1 ) / 8 \) (b10 - b9 + b8 - 2*b7 + b6 + b1) / 8
\(\nu^{2}\)	\(=\)	\( ( \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - 2\beta_{2} + 3 ) / 4 \) (b7 + b6 - b5 + b4 + b3 - 2*b2 + 3) / 4
\(\nu^{3}\)	\(=\)	\( ( \beta_{11} + \beta_{10} + \beta_{9} + 2\beta_{8} + \beta_{6} + \beta_1 ) / 4 \) (b11 + b10 + b9 + 2*b8 + b6 + b1) / 4
\(\nu^{4}\)	\(=\)	\( ( -2\beta_{5} + 3\beta_{4} + \beta_{3} - \beta_{2} - 3 ) / 2 \) (-2b5 + 3b4 + b3 - b2 - 3) / 2
\(\nu^{5}\)	\(=\)	\( ( 8\beta_{11} - 5\beta_{10} + 5\beta_{9} - \beta_{8} + 2\beta_{7} + 3\beta_{6} - \beta_1 ) / 8 \) (8b11 - 5b10 + 5b9 - b8 + 2b7 + 3*b6 - b1) / 8
\(\nu^{6}\)	\(=\)	\( ( -5\beta_{7} - 5\beta_{6} - 3\beta_{5} + 11\beta_{4} - 5\beta_{3} - 6\beta_{2} - 7 ) / 4 \) (-5b7 - 5b6 - 3b5 + 11b4 - 5b3 - 6b2 - 7) / 4
\(\nu^{7}\)	\(=\)	\( ( 3\beta_{11} - 9\beta_{10} - \beta_{9} - 10\beta_{8} - \beta_{6} + 3\beta_1 ) / 4 \) (3b11 - 9b10 - b9 - 10b8 - b6 + 3b1) / 4
\(\nu^{8}\)	\(=\)	\( ( -4\beta_{7} - 4\beta_{6} + 6\beta_{5} + \beta_{4} - 5\beta_{3} - 15\beta_{2} - 1 ) / 2 \) (-4b7 - 4b6 + 6b5 + b4 - 5b3 - 15*b2 - 1) / 2
\(\nu^{9}\)	\(=\)	\( ( -8\beta_{11} - 19\beta_{10} - 13\beta_{9} - 7\beta_{8} + 14\beta_{7} - 27\beta_{6} + 25\beta_1 ) / 8 \) (-8b11 - 19b10 - 13b9 - 7b8 + 14b7 - 27b6 + 25*b1) / 8
\(\nu^{10}\)	\(=\)	\( ( -3\beta_{7} - 3\beta_{6} + 43\beta_{5} - 3\beta_{4} + 13\beta_{3} - 26\beta_{2} - 33 ) / 4 \) (-3b7 - 3b6 + 43b5 - 3b4 + 13b3 - 26b2 - 33) / 4
\(\nu^{11}\)	\(=\)	\( ( 5\beta_{11} + 5\beta_{10} - 11\beta_{9} + 14\beta_{8} + 24\beta_{7} - 35\beta_{6} - 7\beta_1 ) / 4 \) (5b11 + 5b10 - 11b9 + 14b8 + 24b7 - 35b6 - 7*b1) / 4

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 351.12
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( (T^{6} + 10 T^{4} + 17 T^{2} + 4)^{2} \) (T^6 + 10T^4 + 17T^2 + 4)^2
$5$	\( (T^{3} - 7 T + 2)^{4} \) (T^3 - 7*T + 2)^4
$7$	\( (T^{6} - 32 T^{4} + \cdots - 512)^{2} \) (T^6 - 32T^4 + 272T^2 - 512)^2
$11$	\( T^{12} - 6 T^{10} + \cdots + 1771561 \) T^12 - 6T^10 - 73T^8 + 460T^6 - 8833T^4 - 87846*T^2 + 1771561
$13$	\( (T^{6} + 40 T^{4} + \cdots + 128)^{2} \) (T^6 + 40T^4 + 144T^2 + 128)^2
$17$	\( (T^{6} + 64 T^{4} + \cdots + 8192)^{2} \) (T^6 + 64T^4 + 1296T^2 + 8192)^2
$19$	\( (T^{6} - 104 T^{4} + \cdots - 15488)^{2} \) (T^6 - 104T^4 + 2960T^2 - 15488)^2
$23$	\( (T^{6} + 50 T^{4} + \cdots + 3844)^{2} \) (T^6 + 50T^4 + 777T^2 + 3844)^2
$29$	\( (T^{6} + 136 T^{4} + \cdots + 32768)^{2} \) (T^6 + 136T^4 + 4096T^2 + 32768)^2
$31$	\( (T^{6} + 98 T^{4} + \cdots + 1156)^{2} \) (T^6 + 98T^4 + 1369T^2 + 1156)^2
$37$	\( (T^{3} - 7 T + 2)^{4} \) (T^3 - 7*T + 2)^4
$41$	\( (T^{6} + 160 T^{4} + \cdots + 147968)^{2} \) (T^6 + 160T^4 + 8464T^2 + 147968)^2
$43$	\( (T^{6} - 72 T^{4} + \cdots - 2048)^{2} \) (T^6 - 72T^4 + 1280T^2 - 2048)^2
$47$	\( (T^{6} + 156 T^{4} + \cdots + 118336)^{2} \) (T^6 + 156T^4 + 7664T^2 + 118336)^2
$53$	\( (T^{3} - 2 T^{2} - 68 T + 8)^{4} \) (T^3 - 2T^2 - 68T + 8)^4
$59$	\( (T^{6} + 186 T^{4} + \cdots + 1156)^{2} \) (T^6 + 186T^4 + 5985T^2 + 1156)^2
$61$	\( (T^{6} + 136 T^{4} + \cdots + 2048)^{2} \) (T^6 + 136T^4 + 2304T^2 + 2048)^2
$67$	\( (T^{6} + 202 T^{4} + \cdots + 131044)^{2} \) (T^6 + 202T^4 + 11249T^2 + 131044)^2
$71$	\( (T^{6} + 242 T^{4} + \cdots + 24964)^{2} \) (T^6 + 242T^4 + 7689T^2 + 24964)^2
$73$	\( (T^{6} + 160 T^{4} + \cdots + 147968)^{2} \) (T^6 + 160T^4 + 8464T^2 + 147968)^2
$79$	\( (T^{2} - 128)^{6} \) (T^2 - 128)^6
$83$	\( (T^{6} - 264 T^{4} + \cdots - 131072)^{2} \) (T^6 - 264T^4 + 14336T^2 - 131072)^2
$89$	\( (T^{3} - 187 T + 358)^{4} \) (T^3 - 187*T + 358)^4
$97$	\( (T^{3} + 16 T^{2} + \cdots - 722)^{4} \) (T^3 + 16T^2 - 19T - 722)^4
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\(n\)	\(133\)	\(287\)	\(321\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)