LMFDB - Newform orbit 7200.2.h.m

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7200,2,Mod(1151,7200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7200, 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("7200.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7200.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:	$57.4922894553$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.426337261060096.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64 $ x^12 - 4x^9 - 3x^8 + 4x^7 + 8x^6 + 8x^5 - 12x^4 - 32*x^3 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{41}]$
Coefficient ring index:	$ 2^{15} $
Twist minimal:	no (minimal twist has level 1440)
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_{6} q^{7}+O(q^{10}) $ q + b6 * q^7	$ q + \beta_{6} q^{7} - \beta_{4} q^{11} + ( - \beta_{9} + 2) q^{13} - \beta_{5} q^{17} - 2 \beta_{6} q^{19} + ( - \beta_{4} - \beta_{3}) q^{23} + (2 \beta_{10} + \beta_{5}) q^{29} + ( - \beta_{11} - \beta_{6} + \beta_1) q^{31} + ( - \beta_{8} + 1) q^{37} + ( - 2 \beta_{10} + \beta_{2}) q^{41} + ( - 2 \beta_{11} + 2 \beta_1) q^{43} + ( - \beta_{7} - \beta_{4} + 2 \beta_{3}) q^{47} + (\beta_{9} - \beta_{8}) q^{49} + (2 \beta_{10} + 3 \beta_{5} + 2 \beta_{2}) q^{53} + ( - \beta_{7} + 2 \beta_{3}) q^{59} + (2 \beta_{9} + 2 \beta_{8}) q^{61} + ( - 2 \beta_{6} - 2 \beta_1) q^{67} + (\beta_{7} - \beta_{4} + 2 \beta_{3}) q^{71} + (\beta_{9} + \beta_{8} + 9) q^{73} + (\beta_{10} - \beta_{5} + 2 \beta_{2}) q^{77} + (\beta_{11} - \beta_{6} - 3 \beta_1) q^{79} + ( - \beta_{7} + \beta_{4} - 2 \beta_{3}) q^{83} + (2 \beta_{10} + \beta_{2}) q^{89} + (2 \beta_{11} + 3 \beta_{6} - \beta_1) q^{91} + (3 \beta_{9} - \beta_{8} + 3) q^{97}+O(q^{100}) $ q + b6 * q^7 - b4 * q^11 + (-b9 + 2) * q^13 - b5 * q^17 - 2b6 q^19 + (-b4 - b3) * q^23 + (2b10 + b5) q^29 + (-b11 - b6 + b1) * q^31 + (-b8 + 1) * q^37 + (-2b10 + b2) q^41 + (-2b11 + 2b1) * q^43 + (-b7 - b4 + 2b3) q^47 + (b9 - b8) * q^49 + (2b10 + 3b5 + 2b2) q^53 + (-b7 + 2b3) q^59 + (2b9 + 2b8) * q^61 + (-2b6 - 2b1) * q^67 + (b7 - b4 + 2b3) q^71 + (b9 + b8 + 9) * q^73 + (b10 - b5 + 2b2) q^77 + (b11 - b6 - 3b1) q^79 + (-b7 + b4 - 2b3) q^83 + (2b10 + b2) q^89 + (2b11 + 3b6 - b1) * q^91 + (3b9 - b8 + 3) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q + 24 q^{13} + 16 q^{37} + 4 q^{49} - 8 q^{61} + 104 q^{73} + 40 q^{97}+O(q^{100}) $ 12 * q + 24 * q^13 + 16 * q^37 + 4 * q^49 - 8 * q^61 + 104 * q^73 + 40 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64 $ x^12 - 4*x^9 - 3*x^8 + 4*x^7 + 8*x^6 + 8*x^5 - 12*x^4 - 32*x^3 + 64 :

$\beta_{1}$	$=$	$ ( \nu^{11} - 8 \nu^{10} + 20 \nu^{9} + 12 \nu^{8} + 45 \nu^{7} + 12 \nu^{6} - 116 \nu^{5} - 24 \nu^{4} + \cdots - 448 ) / 160 $ (v^11 - 8v^10 + 20v^9 + 12v^8 + 45v^7 + 12v^6 - 116v^5 - 24v^4 + 4v^3 - 32v^2 + 208v - 448) / 160
$\beta_{2}$	$=$	$ ( \nu^{10} + \nu^{9} - 2\nu^{7} - 3\nu^{6} + 5\nu^{5} + 4\nu^{4} + 2\nu^{3} - 8\nu^{2} - 24\nu ) / 16 $ (v^10 + v^9 - 2v^7 - 3v^6 + 5v^5 + 4v^4 + 2v^3 - 8v^2 - 24*v) / 16
$\beta_{3}$	$=$	$ ( \nu^{11} - 4\nu^{9} - 4\nu^{8} - 3\nu^{7} + 20\nu^{6} + 20\nu^{5} - 8\nu^{4} - 12\nu^{3} - 64\nu^{2} + 16\nu + 64 ) / 32 $ (v^11 - 4v^9 - 4v^8 - 3v^7 + 20v^6 + 20v^5 - 8v^4 - 12v^3 - 64v^2 + 16*v + 64) / 32
$\beta_{4}$	$=$	$ ( 3 \nu^{11} - 12 \nu^{10} + 12 \nu^{9} + 4 \nu^{8} - 9 \nu^{7} - 28 \nu^{5} + 56 \nu^{4} + 76 \nu^{3} + \cdots - 320 ) / 160 $ (3v^11 - 12v^10 + 12v^9 + 4v^8 - 9v^7 - 28v^5 + 56v^4 + 76v^3 - 112v^2 - 144v - 320) / 160
$\beta_{5}$	$=$	$ ( 7 \nu^{11} - 6 \nu^{10} - 30 \nu^{9} + 4 \nu^{8} + 15 \nu^{7} + 94 \nu^{6} + 18 \nu^{5} - 128 \nu^{4} + \cdots + 384 ) / 160 $ (7v^11 - 6v^10 - 30v^9 + 4v^8 + 15v^7 + 94v^6 + 18v^5 - 128v^4 - 232v^3 - 64v^2 + 256*v + 384) / 160
$\beta_{6}$	$=$	$ ( \nu^{11} + 12 \nu^{10} + 20 \nu^{9} + 12 \nu^{8} - 35 \nu^{7} - 48 \nu^{6} - 36 \nu^{5} + 136 \nu^{4} + \cdots - 448 ) / 160 $ (v^11 + 12v^10 + 20v^9 + 12v^8 - 35v^7 - 48v^6 - 36v^5 + 136v^4 + 164v^3 + 48v^2 - 432v - 448) / 160
$\beta_{7}$	$=$	$ ( \nu^{11} - 4 \nu^{10} - 4 \nu^{9} - 4 \nu^{8} + 13 \nu^{7} + 32 \nu^{6} + 4 \nu^{5} - 40 \nu^{4} + \cdots + 64 ) / 32 $ (v^11 - 4v^10 - 4v^9 - 4v^8 + 13v^7 + 32v^6 + 4v^5 - 40v^4 - 44v^3 + 48v^2 + 144v + 64) / 32
$\beta_{8}$	$=$	$ ( - 11 \nu^{11} + 4 \nu^{10} - 4 \nu^{9} + 12 \nu^{8} + 33 \nu^{7} - 40 \nu^{6} + 36 \nu^{5} + \cdots + 160 ) / 160 $ (-11v^11 + 4v^10 - 4v^9 + 12v^8 + 33v^7 - 40v^6 + 36v^5 + 88v^4 - 12v^3 + 144v^2 - 272*v + 160) / 160
$\beta_{9}$	$=$	$ ( - 7 \nu^{11} - 12 \nu^{10} + 12 \nu^{9} + 44 \nu^{8} + 21 \nu^{7} - 40 \nu^{6} - 108 \nu^{5} + \cdots - 320 ) / 160 $ (-7v^11 - 12v^10 + 12v^9 + 44v^8 + 21v^7 - 40v^6 - 108v^5 - 24v^4 + 196v^3 + 208v^2 + 176*v - 320) / 160
$\beta_{10}$	$=$	$ ( 11 \nu^{11} + 22 \nu^{10} + 30 \nu^{9} - 28 \nu^{8} - 85 \nu^{7} - 38 \nu^{6} + 94 \nu^{5} + \cdots - 448 ) / 160 $ (11v^11 + 22v^10 + 30v^9 - 28v^8 - 85v^7 - 38v^6 + 94v^5 + 256v^4 + 64v^3 - 352v^2 - 352*v - 448) / 160
$\beta_{11}$	$=$	$ ( -3\nu^{11} - \nu^{10} + 4\nu^{8} + 5\nu^{7} - \nu^{6} - 12\nu^{5} - 8\nu^{4} - 12\nu^{3} + 36\nu^{2} + 16\nu + 64 ) / 20 $ (-3v^11 - v^10 + 4v^8 + 5v^7 - v^6 - 12v^5 - 8v^4 - 12v^3 + 36v^2 + 16v + 64) / 20

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

$\nu$	$=$	$ ( \beta_{10} + \beta_{9} - \beta_{6} - \beta_{4} - \beta_{2} ) / 4 $ (b10 + b9 - b6 - b4 - b2) / 4
$\nu^{2}$	$=$	$ ( \beta_{7} + \beta_{6} - \beta_{3} - \beta_1 ) / 4 $ (b7 + b6 - b3 - b1) / 4
$\nu^{3}$	$=$	$ ( -\beta_{11} - \beta_{10} + \beta_{9} + \beta_{6} - 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 4 ) / 4 $ (-b11 - b10 + b9 + b6 - 2b5 - b4 + 2b3 - b2 + 4) / 4
$\nu^{4}$	$=$	$ ( 3\beta_{10} + \beta_{9} + 3\beta_{8} + \beta_{5} - 6\beta_{2} + 5 ) / 4 $ (3b10 + b9 + 3b8 + b5 - 6*b2 + 5) / 4
$\nu^{5}$	$=$	$ ( -\beta_{11} + \beta_{10} + 3\beta_{9} + 2\beta_{8} + 2\beta_{7} - 3\beta_{6} + \beta_{4} + 7\beta_{2} - 2\beta _1 - 6 ) / 4 $ (-b11 + b10 + 3b9 + 2b8 + 2b7 - 3b6 + b4 + 7b2 - 2b1 - 6) / 4
$\nu^{6}$	$=$	$ ( 2\beta_{11} + 3\beta_{7} + 7\beta_{6} + 5\beta_{3} + \beta_1 ) / 4 $ (2b11 + 3b7 + 7b6 + 5b3 + b1) / 4
$\nu^{7}$	$=$	$ ( - 5 \beta_{11} - 3 \beta_{10} - \beta_{9} + 8 \beta_{8} - \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + \cdots + 12 ) / 4 $ (-5b11 - 3b10 - b9 + 8b8 - b6 - 2b5 - 3b4 + 2b3 + b2 + 8*b1 + 12) / 4
$\nu^{8}$	$=$	$ ( 5\beta_{10} + 15\beta_{9} + 5\beta_{8} + 15\beta_{5} + 14\beta_{2} + 3 ) / 4 $ (5b10 + 15b9 + 5b8 + 15b5 + 14*b2 + 3) / 4
$\nu^{9}$	$=$	$ ( 5 \beta_{11} + 3 \beta_{10} + \beta_{9} - 6 \beta_{8} + 10 \beta_{7} - \beta_{6} - 4 \beta_{5} + \cdots + 26 ) / 4 $ (5b11 + 3b10 + b9 - 6b8 + 10b7 - b6 - 4b5 + 7b4 - 4b3 + 33b2 + 6*b1 + 26) / 4
$\nu^{10}$	$=$	$ ( -2\beta_{11} - 3\beta_{7} + 17\beta_{6} - 40\beta_{4} + 11\beta_{3} + 15\beta_1 ) / 4 $ (-2b11 - 3b7 + 17b6 - 40b4 + 11b3 + 15b1) / 4
$\nu^{11}$	$=$	$ ( - 27 \beta_{11} - \beta_{10} + 5 \beta_{9} + 4 \beta_{8} + 4 \beta_{7} + 5 \beta_{6} + 22 \beta_{5} + \cdots + 104 ) / 4 $ (-27b11 - b10 + 5b9 + 4b8 + 4b7 + 5b6 + 22b5 + 3b4 - 22b3 + 7b2 + 4b1 + 104) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$6401$	$6751$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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} $

1151.1

−1.35818	+	0.394157i
−0.394157	+	1.35818i
−0.0912546	+	1.41127i
1.41127	−	0.0912546i
−0.760198	+	1.19252i
1.19252	−	0.760198i
−0.760198	−	1.19252i
1.19252	+	0.760198i
−0.0912546	−	1.41127i
1.41127	+	0.0912546i
−1.35818	−	0.394157i
−0.394157	−	1.35818i

−

3.50466i

1151.2

−

3.50466i

1151.3

−

2.64002i

1151.4

−

2.64002i

1151.5

−

0.864641i

1151.6

−

0.864641i

0.864641i

0.864641i

2.64002i

2.64002i

3.50466i

3.50466i

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	7200.2.h.m		12
3.b	odd	2	1	inner	7200.2.h.m		12
4.b	odd	2	1	inner	7200.2.h.m		12
5.b	even	2	1		7200.2.h.l		12
5.c	odd	4	1		1440.2.o.a	✓	12
5.c	odd	4	1		1440.2.o.b	yes	12
12.b	even	2	1	inner	7200.2.h.m		12
15.d	odd	2	1		7200.2.h.l		12
15.e	even	4	1		1440.2.o.a	✓	12
15.e	even	4	1		1440.2.o.b	yes	12
20.d	odd	2	1		7200.2.h.l		12
20.e	even	4	1		1440.2.o.a	✓	12
20.e	even	4	1		1440.2.o.b	yes	12
40.i	odd	4	1		2880.2.o.e		12
40.i	odd	4	1		2880.2.o.f		12
40.k	even	4	1		2880.2.o.e		12
40.k	even	4	1		2880.2.o.f		12
60.h	even	2	1		7200.2.h.l		12
60.l	odd	4	1		1440.2.o.a	✓	12
60.l	odd	4	1		1440.2.o.b	yes	12
120.q	odd	4	1		2880.2.o.e		12
120.q	odd	4	1		2880.2.o.f		12
120.w	even	4	1		2880.2.o.e		12
120.w	even	4	1		2880.2.o.f		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1440.2.o.a	✓	12	5.c	odd	4	1
1440.2.o.a	✓	12	15.e	even	4	1
1440.2.o.a	✓	12	20.e	even	4	1
1440.2.o.a	✓	12	60.l	odd	4	1
1440.2.o.b	yes	12	5.c	odd	4	1
1440.2.o.b	yes	12	15.e	even	4	1
1440.2.o.b	yes	12	20.e	even	4	1
1440.2.o.b	yes	12	60.l	odd	4	1
2880.2.o.e		12	40.i	odd	4	1
2880.2.o.e		12	40.k	even	4	1
2880.2.o.e		12	120.q	odd	4	1
2880.2.o.e		12	120.w	even	4	1
2880.2.o.f		12	40.i	odd	4	1
2880.2.o.f		12	40.k	even	4	1
2880.2.o.f		12	120.q	odd	4	1
2880.2.o.f		12	120.w	even	4	1
7200.2.h.l		12	5.b	even	2	1
7200.2.h.l		12	15.d	odd	2	1
7200.2.h.l		12	20.d	odd	2	1
7200.2.h.l		12	60.h	even	2	1
7200.2.h.m		12	1.a	even	1	1	trivial
7200.2.h.m		12	3.b	odd	2	1	inner
7200.2.h.m		12	4.b	odd	2	1	inner
7200.2.h.m		12	12.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(7200, [\chi])$:

$ T_{7}^{6} + 20T_{7}^{4} + 100T_{7}^{2} + 64 $

$ T_{11}^{6} - 28T_{11}^{4} + 228T_{11}^{2} - 512 $

$ T_{13}^{3} - 6T_{13}^{2} + 2T_{13} + 4 $

$ T_{23}^{6} - 88T_{23}^{4} + 528T_{23}^{2} - 512 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ T^{12} $ T^12
$5$	$ T^{12} $ T^12
$7$	$ (T^{6} + 20 T^{4} + \cdots + 64)^{2} $ (T^6 + 20T^4 + 100T^2 + 64)^2
$11$	$ (T^{6} - 28 T^{4} + \cdots - 512)^{2} $ (T^6 - 28T^4 + 228T^2 - 512)^2
$13$	$ (T^{3} - 6 T^{2} + 2 T + 4)^{4} $ (T^3 - 6T^2 + 2T + 4)^4
$17$	$ (T^{6} + 34 T^{4} + \cdots + 128)^{2} $ (T^6 + 34T^4 + 288T^2 + 128)^2
$19$	$ (T^{6} + 80 T^{4} + \cdots + 4096)^{2} $ (T^6 + 80T^4 + 1600T^2 + 4096)^2
$23$	$ (T^{6} - 88 T^{4} + \cdots - 512)^{2} $ (T^6 - 88T^4 + 528T^2 - 512)^2
$29$	$ (T^{6} + 98 T^{4} + \cdots + 3200)^{2} $ (T^6 + 98T^4 + 2144T^2 + 3200)^2
$31$	$ (T^{6} + 88 T^{4} + \cdots + 6400)^{2} $ (T^6 + 88T^4 + 1424T^2 + 6400)^2
$37$	$ (T^{3} - 4 T^{2} - 14 T - 8)^{4} $ (T^3 - 4T^2 - 14T - 8)^4
$41$	$ (T^{6} + 102 T^{4} + \cdots + 13448)^{2} $ (T^6 + 102T^4 + 2828T^2 + 13448)^2
$43$	$ (T^{6} + 240 T^{4} + \cdots + 262144)^{2} $ (T^6 + 240T^4 + 16448T^2 + 262144)^2
$47$	$ (T^{6} - 176 T^{4} + \cdots - 51200)^{2} $ (T^6 - 176T^4 + 5696T^2 - 51200)^2
$53$	$ (T^{6} + 306 T^{4} + \cdots + 991232)^{2} $ (T^6 + 306T^4 + 30464T^2 + 991232)^2
$59$	$ (T^{6} - 220 T^{4} + \cdots - 359552)^{2} $ (T^6 - 220T^4 + 15652T^2 - 359552)^2
$61$	$ (T^{3} + 2 T^{2} + \cdots - 328)^{4} $ (T^3 + 2T^2 - 100T - 328)^4
$67$	$ (T^{6} + 224 T^{4} + \cdots + 262144)^{2} $ (T^6 + 224T^4 + 14592T^2 + 262144)^2
$71$	$ (T^{6} - 256 T^{4} + \cdots - 204800)^{2} $ (T^6 - 256T^4 + 15616T^2 - 204800)^2
$73$	$ (T^{3} - 26 T^{2} + \cdots - 464)^{4} $ (T^3 - 26T^2 + 200T - 464)^4
$79$	$ (T^{6} + 344 T^{4} + \cdots + 215296)^{2} $ (T^6 + 344T^4 + 29584T^2 + 215296)^2
$83$	$ (T^{6} - 256 T^{4} + \cdots - 204800)^{2} $ (T^6 - 256T^4 + 15616T^2 - 204800)^2
$89$	$ (T^{6} + 118 T^{4} + \cdots + 200)^{2} $ (T^6 + 118T^4 + 1356T^2 + 200)^2
$97$	$ (T^{3} - 10 T^{2} + \cdots + 848)^{4} $ (T^3 - 10T^2 - 88T + 848)^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:	\( N \)	\(=\)	\( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7200.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{6} q^{7}+O(q^{10}) \) q + b6 * q^7	\( q + \beta_{6} q^{7} - \beta_{4} q^{11} + ( - \beta_{9} + 2) q^{13} - \beta_{5} q^{17} - 2 \beta_{6} q^{19} + ( - \beta_{4} - \beta_{3}) q^{23} + (2 \beta_{10} + \beta_{5}) q^{29} + ( - \beta_{11} - \beta_{6} + \beta_1) q^{31} + ( - \beta_{8} + 1) q^{37} + ( - 2 \beta_{10} + \beta_{2}) q^{41} + ( - 2 \beta_{11} + 2 \beta_1) q^{43} + ( - \beta_{7} - \beta_{4} + 2 \beta_{3}) q^{47} + (\beta_{9} - \beta_{8}) q^{49} + (2 \beta_{10} + 3 \beta_{5} + 2 \beta_{2}) q^{53} + ( - \beta_{7} + 2 \beta_{3}) q^{59} + (2 \beta_{9} + 2 \beta_{8}) q^{61} + ( - 2 \beta_{6} - 2 \beta_1) q^{67} + (\beta_{7} - \beta_{4} + 2 \beta_{3}) q^{71} + (\beta_{9} + \beta_{8} + 9) q^{73} + (\beta_{10} - \beta_{5} + 2 \beta_{2}) q^{77} + (\beta_{11} - \beta_{6} - 3 \beta_1) q^{79} + ( - \beta_{7} + \beta_{4} - 2 \beta_{3}) q^{83} + (2 \beta_{10} + \beta_{2}) q^{89} + (2 \beta_{11} + 3 \beta_{6} - \beta_1) q^{91} + (3 \beta_{9} - \beta_{8} + 3) q^{97}+O(q^{100}) \) q + b6 * q^7 - b4 * q^11 + (-b9 + 2) * q^13 - b5 * q^17 - 2b6 q^19 + (-b4 - b3) * q^23 + (2b10 + b5) q^29 + (-b11 - b6 + b1) * q^31 + (-b8 + 1) * q^37 + (-2b10 + b2) q^41 + (-2b11 + 2b1) * q^43 + (-b7 - b4 + 2b3) q^47 + (b9 - b8) * q^49 + (2b10 + 3b5 + 2b2) q^53 + (-b7 + 2b3) q^59 + (2b9 + 2b8) * q^61 + (-2b6 - 2b1) * q^67 + (b7 - b4 + 2b3) q^71 + (b9 + b8 + 9) * q^73 + (b10 - b5 + 2b2) q^77 + (b11 - b6 - 3b1) q^79 + (-b7 + b4 - 2b3) q^83 + (2b10 + b2) q^89 + (2b11 + 3b6 - b1) * q^91 + (3b9 - b8 + 3) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q + 24 q^{13} + 16 q^{37} + 4 q^{49} - 8 q^{61} + 104 q^{73} + 40 q^{97}+O(q^{100}) \) 12 * q + 24 * q^13 + 16 * q^37 + 4 * q^49 - 8 * q^61 + 104 * q^73 + 40 * q^97

Introduction

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Newspace parameters

Newform invariants

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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	7200.2.h.m

Level	$7200$
Weight	$2$
Character orbit	7200.h
Analytic conductor	$57.492$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(57.4922894553\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	12.0.426337261060096.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64 \) x^12 - 4x^9 - 3x^8 + 4x^7 + 8x^6 + 8x^5 - 12x^4 - 32*x^3 + 64
Coefficient ring:	\(\Z[a_1, \ldots, a_{41}]\)
Coefficient ring index:	\( 2^{15} \)
Twist minimal:	no (minimal twist has level 1440)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{11} - 8 \nu^{10} + 20 \nu^{9} + 12 \nu^{8} + 45 \nu^{7} + 12 \nu^{6} - 116 \nu^{5} - 24 \nu^{4} + \cdots - 448 ) / 160 \) (v^11 - 8v^10 + 20v^9 + 12v^8 + 45v^7 + 12v^6 - 116v^5 - 24v^4 + 4v^3 - 32v^2 + 208v - 448) / 160
\(\beta_{2}\)	\(=\)	\( ( \nu^{10} + \nu^{9} - 2\nu^{7} - 3\nu^{6} + 5\nu^{5} + 4\nu^{4} + 2\nu^{3} - 8\nu^{2} - 24\nu ) / 16 \) (v^10 + v^9 - 2v^7 - 3v^6 + 5v^5 + 4v^4 + 2v^3 - 8v^2 - 24*v) / 16
\(\beta_{3}\)	\(=\)	\( ( \nu^{11} - 4\nu^{9} - 4\nu^{8} - 3\nu^{7} + 20\nu^{6} + 20\nu^{5} - 8\nu^{4} - 12\nu^{3} - 64\nu^{2} + 16\nu + 64 ) / 32 \) (v^11 - 4v^9 - 4v^8 - 3v^7 + 20v^6 + 20v^5 - 8v^4 - 12v^3 - 64v^2 + 16*v + 64) / 32
\(\beta_{4}\)	\(=\)	\( ( 3 \nu^{11} - 12 \nu^{10} + 12 \nu^{9} + 4 \nu^{8} - 9 \nu^{7} - 28 \nu^{5} + 56 \nu^{4} + 76 \nu^{3} + \cdots - 320 ) / 160 \) (3v^11 - 12v^10 + 12v^9 + 4v^8 - 9v^7 - 28v^5 + 56v^4 + 76v^3 - 112v^2 - 144v - 320) / 160
\(\beta_{5}\)	\(=\)	\( ( 7 \nu^{11} - 6 \nu^{10} - 30 \nu^{9} + 4 \nu^{8} + 15 \nu^{7} + 94 \nu^{6} + 18 \nu^{5} - 128 \nu^{4} + \cdots + 384 ) / 160 \) (7v^11 - 6v^10 - 30v^9 + 4v^8 + 15v^7 + 94v^6 + 18v^5 - 128v^4 - 232v^3 - 64v^2 + 256*v + 384) / 160
\(\beta_{6}\)	\(=\)	\( ( \nu^{11} + 12 \nu^{10} + 20 \nu^{9} + 12 \nu^{8} - 35 \nu^{7} - 48 \nu^{6} - 36 \nu^{5} + 136 \nu^{4} + \cdots - 448 ) / 160 \) (v^11 + 12v^10 + 20v^9 + 12v^8 - 35v^7 - 48v^6 - 36v^5 + 136v^4 + 164v^3 + 48v^2 - 432v - 448) / 160
\(\beta_{7}\)	\(=\)	\( ( \nu^{11} - 4 \nu^{10} - 4 \nu^{9} - 4 \nu^{8} + 13 \nu^{7} + 32 \nu^{6} + 4 \nu^{5} - 40 \nu^{4} + \cdots + 64 ) / 32 \) (v^11 - 4v^10 - 4v^9 - 4v^8 + 13v^7 + 32v^6 + 4v^5 - 40v^4 - 44v^3 + 48v^2 + 144v + 64) / 32
\(\beta_{8}\)	\(=\)	\( ( - 11 \nu^{11} + 4 \nu^{10} - 4 \nu^{9} + 12 \nu^{8} + 33 \nu^{7} - 40 \nu^{6} + 36 \nu^{5} + \cdots + 160 ) / 160 \) (-11v^11 + 4v^10 - 4v^9 + 12v^8 + 33v^7 - 40v^6 + 36v^5 + 88v^4 - 12v^3 + 144v^2 - 272*v + 160) / 160
\(\beta_{9}\)	\(=\)	\( ( - 7 \nu^{11} - 12 \nu^{10} + 12 \nu^{9} + 44 \nu^{8} + 21 \nu^{7} - 40 \nu^{6} - 108 \nu^{5} + \cdots - 320 ) / 160 \) (-7v^11 - 12v^10 + 12v^9 + 44v^8 + 21v^7 - 40v^6 - 108v^5 - 24v^4 + 196v^3 + 208v^2 + 176*v - 320) / 160
\(\beta_{10}\)	\(=\)	\( ( 11 \nu^{11} + 22 \nu^{10} + 30 \nu^{9} - 28 \nu^{8} - 85 \nu^{7} - 38 \nu^{6} + 94 \nu^{5} + \cdots - 448 ) / 160 \) (11v^11 + 22v^10 + 30v^9 - 28v^8 - 85v^7 - 38v^6 + 94v^5 + 256v^4 + 64v^3 - 352v^2 - 352*v - 448) / 160
\(\beta_{11}\)	\(=\)	\( ( -3\nu^{11} - \nu^{10} + 4\nu^{8} + 5\nu^{7} - \nu^{6} - 12\nu^{5} - 8\nu^{4} - 12\nu^{3} + 36\nu^{2} + 16\nu + 64 ) / 20 \) (-3v^11 - v^10 + 4v^8 + 5v^7 - v^6 - 12v^5 - 8v^4 - 12v^3 + 36v^2 + 16v + 64) / 20

\(\nu\)	\(=\)	\( ( \beta_{10} + \beta_{9} - \beta_{6} - \beta_{4} - \beta_{2} ) / 4 \) (b10 + b9 - b6 - b4 - b2) / 4
\(\nu^{2}\)	\(=\)	\( ( \beta_{7} + \beta_{6} - \beta_{3} - \beta_1 ) / 4 \) (b7 + b6 - b3 - b1) / 4
\(\nu^{3}\)	\(=\)	\( ( -\beta_{11} - \beta_{10} + \beta_{9} + \beta_{6} - 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 4 ) / 4 \) (-b11 - b10 + b9 + b6 - 2b5 - b4 + 2b3 - b2 + 4) / 4
\(\nu^{4}\)	\(=\)	\( ( 3\beta_{10} + \beta_{9} + 3\beta_{8} + \beta_{5} - 6\beta_{2} + 5 ) / 4 \) (3b10 + b9 + 3b8 + b5 - 6*b2 + 5) / 4
\(\nu^{5}\)	\(=\)	\( ( -\beta_{11} + \beta_{10} + 3\beta_{9} + 2\beta_{8} + 2\beta_{7} - 3\beta_{6} + \beta_{4} + 7\beta_{2} - 2\beta _1 - 6 ) / 4 \) (-b11 + b10 + 3b9 + 2b8 + 2b7 - 3b6 + b4 + 7b2 - 2b1 - 6) / 4
\(\nu^{6}\)	\(=\)	\( ( 2\beta_{11} + 3\beta_{7} + 7\beta_{6} + 5\beta_{3} + \beta_1 ) / 4 \) (2b11 + 3b7 + 7b6 + 5b3 + b1) / 4
\(\nu^{7}\)	\(=\)	\( ( - 5 \beta_{11} - 3 \beta_{10} - \beta_{9} + 8 \beta_{8} - \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + \cdots + 12 ) / 4 \) (-5b11 - 3b10 - b9 + 8b8 - b6 - 2b5 - 3b4 + 2b3 + b2 + 8*b1 + 12) / 4
\(\nu^{8}\)	\(=\)	\( ( 5\beta_{10} + 15\beta_{9} + 5\beta_{8} + 15\beta_{5} + 14\beta_{2} + 3 ) / 4 \) (5b10 + 15b9 + 5b8 + 15b5 + 14*b2 + 3) / 4
\(\nu^{9}\)	\(=\)	\( ( 5 \beta_{11} + 3 \beta_{10} + \beta_{9} - 6 \beta_{8} + 10 \beta_{7} - \beta_{6} - 4 \beta_{5} + \cdots + 26 ) / 4 \) (5b11 + 3b10 + b9 - 6b8 + 10b7 - b6 - 4b5 + 7b4 - 4b3 + 33b2 + 6*b1 + 26) / 4
\(\nu^{10}\)	\(=\)	\( ( -2\beta_{11} - 3\beta_{7} + 17\beta_{6} - 40\beta_{4} + 11\beta_{3} + 15\beta_1 ) / 4 \) (-2b11 - 3b7 + 17b6 - 40b4 + 11b3 + 15b1) / 4
\(\nu^{11}\)	\(=\)	\( ( - 27 \beta_{11} - \beta_{10} + 5 \beta_{9} + 4 \beta_{8} + 4 \beta_{7} + 5 \beta_{6} + 22 \beta_{5} + \cdots + 104 ) / 4 \) (-27b11 - b10 + 5b9 + 4b8 + 4b7 + 5b6 + 22b5 + 3b4 - 22b3 + 7b2 + 4b1 + 104) / 4

\(n\)	\(577\)	\(901\)	\(6401\)	\(6751\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1151.12
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( T^{12} \) T^12
$5$	\( T^{12} \) T^12
$7$	\( (T^{6} + 20 T^{4} + \cdots + 64)^{2} \) (T^6 + 20T^4 + 100T^2 + 64)^2
$11$	\( (T^{6} - 28 T^{4} + \cdots - 512)^{2} \) (T^6 - 28T^4 + 228T^2 - 512)^2
$13$	\( (T^{3} - 6 T^{2} + 2 T + 4)^{4} \) (T^3 - 6T^2 + 2T + 4)^4
$17$	\( (T^{6} + 34 T^{4} + \cdots + 128)^{2} \) (T^6 + 34T^4 + 288T^2 + 128)^2
$19$	\( (T^{6} + 80 T^{4} + \cdots + 4096)^{2} \) (T^6 + 80T^4 + 1600T^2 + 4096)^2
$23$	\( (T^{6} - 88 T^{4} + \cdots - 512)^{2} \) (T^6 - 88T^4 + 528T^2 - 512)^2
$29$	\( (T^{6} + 98 T^{4} + \cdots + 3200)^{2} \) (T^6 + 98T^4 + 2144T^2 + 3200)^2
$31$	\( (T^{6} + 88 T^{4} + \cdots + 6400)^{2} \) (T^6 + 88T^4 + 1424T^2 + 6400)^2
$37$	\( (T^{3} - 4 T^{2} - 14 T - 8)^{4} \) (T^3 - 4T^2 - 14T - 8)^4
$41$	\( (T^{6} + 102 T^{4} + \cdots + 13448)^{2} \) (T^6 + 102T^4 + 2828T^2 + 13448)^2
$43$	\( (T^{6} + 240 T^{4} + \cdots + 262144)^{2} \) (T^6 + 240T^4 + 16448T^2 + 262144)^2
$47$	\( (T^{6} - 176 T^{4} + \cdots - 51200)^{2} \) (T^6 - 176T^4 + 5696T^2 - 51200)^2
$53$	\( (T^{6} + 306 T^{4} + \cdots + 991232)^{2} \) (T^6 + 306T^4 + 30464T^2 + 991232)^2
$59$	\( (T^{6} - 220 T^{4} + \cdots - 359552)^{2} \) (T^6 - 220T^4 + 15652T^2 - 359552)^2
$61$	\( (T^{3} + 2 T^{2} + \cdots - 328)^{4} \) (T^3 + 2T^2 - 100T - 328)^4
$67$	\( (T^{6} + 224 T^{4} + \cdots + 262144)^{2} \) (T^6 + 224T^4 + 14592T^2 + 262144)^2
$71$	\( (T^{6} - 256 T^{4} + \cdots - 204800)^{2} \) (T^6 - 256T^4 + 15616T^2 - 204800)^2
$73$	\( (T^{3} - 26 T^{2} + \cdots - 464)^{4} \) (T^3 - 26T^2 + 200T - 464)^4
$79$	\( (T^{6} + 344 T^{4} + \cdots + 215296)^{2} \) (T^6 + 344T^4 + 29584T^2 + 215296)^2
$83$	\( (T^{6} - 256 T^{4} + \cdots - 204800)^{2} \) (T^6 - 256T^4 + 15616T^2 - 204800)^2
$89$	\( (T^{6} + 118 T^{4} + \cdots + 200)^{2} \) (T^6 + 118T^4 + 1356T^2 + 200)^2
$97$	\( (T^{3} - 10 T^{2} + \cdots + 848)^{4} \) (T^3 - 10T^2 - 88T + 848)^4
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