LMFDB - Newform orbit 370.2.q.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [370,2,Mod(97,370)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("370.97"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(370, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([3, 5])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 370 = 2 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	370.q (of order $12$, degree $4$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,4,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$2.95446487479$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
Coefficient field:	$\Q(\zeta_{24})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-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 primitive root of unity $\zeta_{24}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \zeta_{24}^{4} q^{2} + (\zeta_{24}^{6} + \cdots + \zeta_{24}^{4}) q^{3} + (\zeta_{24}^{4} - 1) q^{4} + ( - \zeta_{24}^{6} + 2 \zeta_{24}^{5} + \cdots - \zeta_{24}) q^{5} + (\zeta_{24}^{6} + \zeta_{24}^{5} + \cdots - 1) q^{6}+ \cdots + ( - 4 \zeta_{24}^{7} + \cdots - 4 \zeta_{24}) q^{99}+O(q^{100}) $ q + z^4 * q^2 + (z^6 + z^5 + z^4) * q^3 + (z^4 - 1) * q^4 + (-z^6 + 2z^5 + z^4 + z^2 - z) q^5 + (z^6 + z^5 + z^4 - z^2 - z - 1) * q^6 + (z^6 - z^5 - z^4 - 2z^2 + 2) q^7 - q^8 + (2z^7 + 2z^5 + z^4 - 2z^3 - 2z - 2) * q^9 + (z^5 + z^4 + z^2 - 2z - 1) q^10 + (z^6 - 2z^5 + 2z^4 - 2z^3 + 2z - 1) * q^11 + (-z^2 - z - 1) * q^12 + (-z^7 - 2z^6 - z^5 + z^4 - z^3 + z^2 + 2z - 1) * q^13 + (-z^6 - z^5 + z^4 - z^2 + z + 1) * q^14 + (z^7 + 2z^6 + 2z^5 + 2z^4 - z^3 - 2z^2 - 3z - 1) q^15 - z^4 * q^16 + (z^7 - 4z^6 - z^4 + 4z^2 - z + 2) * q^17 + (-z^4 - 2z^3 - 2z - 1) * q^18 + (-2z^5 - z^4 + z^2 + 4z + 1) * q^19 + (z^6 - z^5 - z - 1) * q^20 + (-2z^7 - z^6 - z^4 + z^2 + 2z + 2) * q^21 + (-2z^7 + z^6 + z^4 - z^2 + 2z - 2) * q^22 + (-3z^7 + 2z^5 + z^3 + z - 1) * q^23 + (-z^6 - z^5 - z^4) * q^24 + (2z^7 + 2z^5 + 2z^3 - z^2 - 4z) * q^25 + (-2z^7 - z^6 + z^5 + z^3 + 2z^2 + z - 1) * q^26 + (-z^6 - 3z^5 - z^3 - 3z - 1) * q^27 + (-2z^6 + 2z^4 + z^2 + z - 1) * q^28 + (-2z^7 + 4z^6 + 2z^4 + z^3 - 2z^2 - 4) * q^29 + (-z^5 + z^4 - z^3 - 2z^2 - 2z - 2) * q^30 + (z^6 - 4z^4 + 2z^3 - 4z^2 + 1) q^31 + (-z^4 + 1) * q^32 + (-z^7 + 2z^6 - z^5 - z^4 + z^3 - z^2 + 2z - 1) * q^33 + (z^7 - z^5 + z^4 - z^3 + 4z^2 + 1) q^34 + (-3z^7 - 4z^6 + 2z^5 + 2z^4 - z^3 + 2z^2 + z - 1) q^35 + (-2z^7 - 2z^5 - 2z^4 + 1) q^36 + (3z^7 - 7z^3) * q^37 + (z^6 + 2z^5 + 2z + 1) * q^38 + (-2z^7 + 4z^3 + 2z^2 + 2z + 2) * q^39 + (z^6 - 2z^5 - z^4 - z^2 + z) q^40 + (-z^7 + z^5 + z^4 + 5z^3 - 2z^2 + 4z + 1) q^41 + (-2z^7 + 2z^5 + z^4 + 2z^3 + z^2 + 1) q^42 + (3z^7 + 3z^6 - z^5 - 2z^3 - 6z^2 - 2z + 2) q^43 + (-2z^7 + 2z^5 - z^4 + 2z^3 - z^2 - 1) q^44 + (2z^7 - z^5 - 5z^4 - 2z^3 - 3z^2 - 4z + 1) q^45 + (-2z^7 + 3z^5 - z^4 + 3z^3 - 2z) * q^46 + (10z^7 + 2z^6 + z^5 + 2z^4 - 5z^3 - 2z^2 - z - 2) q^47 + (-z^6 - z^5 - z^4 + z^2 + z + 1) * q^48 + (2z^7 - 2z^5 - 3z^4 + 2z^3 - 2z + 6) q^49 + (4z^7 - z^6 - 2z^5 - 2z^3 - 2z) * q^50 + (5z^4 + 3z^3 + 5z^2) q^51 + (-z^7 + z^6 + 2z^5 - z^4 + 2z^3 + z^2 - z) * q^52 + (6z^7 - 4z^5 - 2z^2 + 2z + 2) * q^53 + (-z^7 - z^6 - 6z^5 - z^4 + z^2 + 3z) * q^54 + (-3z^7 + 4z^6 - 2z^3 + 2z^2 - 3z + 2) q^55 + (-z^6 + z^5 + z^4 + 2z^2 - 2) q^56 + (3z^7 + 3z^6 + 2z^5 + z^4 + 2z^3 + 3z^2 + 3z) * q^57 + (-z^7 + 2z^6 - 2z^4 + 2z^3 - 4z^2 - 2) * q^58 + (-2z^7 - 6z^6 - 6z^4 + 4z^2 + 2) * q^59 + (-z^7 - 2z^6 - 3z^5 - z^4 + z - 1) * q^60 + (-5z^7 + 2z^6 - 2z^5 - 4z^4 + 5z^3 + 2z^2 + 4z + 2) q^61 + (2z^7 - 3z^6 - 3z^4 - z^2 + 4) q^62 + (-3z^6 + 3z^5 + 5z^4 + 5z^2 + 3z - 3) q^63 + q^64 + (-4z^7 + 3z^6 - z^5 - 2z^4 + 5z^3 + 2z^2 - z + 3) q^65 + (z^6 + z^5 - 2z^4 + z^3 - 2z^2 + z + 1) * q^66 + (5z^7 + 2z^6 - z^5 - 5z^3 - 2z^2 + 2z - 2) q^67 + (4z^6 - z^5 + 2z^4 - z^3 + z - 1) * q^68 + (z^7 + 2z^6 + 3z^5 + z^3 - 2z^2 + 2) q^69 + (-4z^7 - 2z^6 + 3z^5 + z^4 + 3z^3 + 4z^2 - 2z - 2) * q^70 + (2z^7 - 4z^6 + 2z^5 + z^4 + z^3 + 2z^2 - 3z - 1) q^71 + (-2z^7 - 2z^5 - z^4 + 2z^3 + 2z + 2) * q^72 + (2z^7 - 3z^6 - 3z^5 - 2z^4 - z^3 + 2z^2 + 3z + 3) * q^73 + (-4z^7 - 3z^3) * q^74 + (z^7 - 3z^6 + z^4 - 4z^3 - 2z^2 - 6z - 3) * q^75 + (z^6 + 4z^5 + z^4 - z^2 - 2z) * q^76 + (5z^7 - 2z^6 - 3z^5 + 3z^4 - 5z^3 - z^2 + 6z - 1) * q^77 + (2z^7 + 2z^6 + 2z^5 + 2z^4 + 2z^3) q^78 + (-3z^7 + 4z^6 - z^5 + 3z^3 - 4z^2 + 2z - 4) q^79 + (-z^5 - z^4 - z^2 + 2z + 1) q^80 + (-2z^7 - 2z^6 - 2z^5 - 2z^4 + 4z^3 + z^2 - 2z + 2) * q^81 + (4z^7 - 2z^6 + 5z^5 + 2z^4 + z^3 - z - 1) * q^82 + (-8z^7 - 6z^6 + 6z^5 - 6z^4 + 3z^2 - 3z + 3) * q^83 + (z^6 + 2z^5 + 2z^4 + 2z^3 - 2z - 1) * q^84 + (6z^7 - 4z^6 + 3z^5 - 4z^4 + 2z^3 + 6z^2 + 4) * q^85 + (z^7 - 3z^6 - 3z^5 + 2z^4 - 3z^3 - 3z^2 + z) q^86 + (z^7 - z^5 - 3z^4 - 2z^3 - 6z^2 - z - 3) q^87 + (-z^6 + 2z^5 - 2z^4 + 2z^3 - 2z + 1) * q^88 + (-4z^7 + 8z^3 + z^2 - 4z + 1) q^89 + (-3z^6 - 5z^5 - 4z^4 - 2z^3 + z + 5) * q^90 + (4z^7 - 6z^6 + 6z^4 - 8z^3 + 4z - 6) q^91 + (z^7 + z^5 - z^4 + 2z^3 - 3z + 1) * q^92 + (-z^7 - 6z^6 - z^5 - 5z^4 - z^3 + 3z^2 + 2z + 5) * q^93 + (5z^7 - 10z^3 - 2z^2 - z - 2) q^94 + (-2z^7 + 6z^6 + 3z^5 + z^3 + 3z + 2) * q^95 + (z^2 + z + 1) * q^96 + (8z^7 + 6z^6 + 9z^5 - 6z^4 + z^3 - z + 3) * q^97 + (4z^7 - 4z^5 + 3z^4 - 2z^3 + 2z + 3) q^98 + (-4z^7 + 3z^6 + z^4 + 3z^2 - 4z) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{2} + 4 q^{3} - 4 q^{4} + 4 q^{5} - 4 q^{6} + 12 q^{7} - 8 q^{8} - 12 q^{9} - 4 q^{10} - 8 q^{12} - 4 q^{13} + 12 q^{14} - 4 q^{16} + 12 q^{17} - 12 q^{18} + 4 q^{19} - 8 q^{20} + 12 q^{21} - 12 q^{22}+ \cdots + 4 q^{99}+O(q^{100}) $ 8 * q + 4 * q^2 + 4 * q^3 - 4 * q^4 + 4 * q^5 - 4 * q^6 + 12 * q^7 - 8 * q^8 - 12 * q^9 - 4 * q^10 - 8 * q^12 - 4 * q^13 + 12 * q^14 - 4 * q^16 + 12 * q^17 - 12 * q^18 + 4 * q^19 - 8 * q^20 + 12 * q^21 - 12 * q^22 - 8 * q^23 - 4 * q^24 - 8 * q^26 - 8 * q^27 - 24 * q^29 - 12 * q^30 - 8 * q^31 + 4 * q^32 - 12 * q^33 + 12 * q^34 + 8 * q^38 + 16 * q^39 - 4 * q^40 + 12 * q^41 + 12 * q^42 + 16 * q^43 - 12 * q^44 - 12 * q^45 - 4 * q^46 - 8 * q^47 + 4 * q^48 + 36 * q^49 + 20 * q^51 - 4 * q^52 + 16 * q^53 - 4 * q^54 + 16 * q^55 - 12 * q^56 + 4 * q^57 - 24 * q^58 - 8 * q^59 - 12 * q^60 + 20 * q^62 - 4 * q^63 + 8 * q^64 + 16 * q^65 - 16 * q^67 + 16 * q^69 - 12 * q^70 - 4 * q^71 + 12 * q^72 + 16 * q^73 - 20 * q^75 + 4 * q^76 + 4 * q^77 + 8 * q^78 - 32 * q^79 + 4 * q^80 + 8 * q^81 + 16 * q^85 + 8 * q^86 - 36 * q^87 + 8 * q^89 + 24 * q^90 - 24 * q^91 + 4 * q^92 + 20 * q^93 - 16 * q^94 + 16 * q^95 + 8 * q^96 + 36 * q^98 + 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/370\mathbb{Z}\right)^\times$.

$n$	$261$	$297$
$\chi(n)$	$-\zeta_{24}^{2}$	$-\zeta_{24}^{6}$

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} $

97.1

−0.965926	+	0.258819i
0.965926	−	0.258819i
−0.965926	−	0.258819i
0.965926	+	0.258819i
−0.258819	+	0.965926i
0.258819	−	0.965926i
−0.258819	−	0.965926i
0.258819	+	0.965926i

0.500000

−

0.866025i

0.241181

−

0.900100i

−0.500000

−

0.866025i

1.81431

1.30701i

−0.658919

−

0.658919i

0.0267682

−

0.0999004i

−1.00000

1.84607

1.06583i

2.03906

−

0.917738i

97.2

0.500000

−

0.866025i

0.758819

−

2.83195i

−0.500000

−

0.866025i

0.917738

−

2.03906i

−2.07313

−

2.07313i

−0.490870

1.83195i

−1.00000

−4.84607

−

2.79788i

−1.30701

−

1.81431i

103.1

0.500000

0.866025i

0.241181

0.900100i

−0.500000

0.866025i

1.81431

−

1.30701i

−0.658919

0.658919i

0.0267682

0.0999004i

−1.00000

1.84607

−

1.06583i

2.03906

0.917738i

103.2

0.500000

0.866025i

0.758819

2.83195i

−0.500000

0.866025i

0.917738

2.03906i

−2.07313

2.07313i

−0.490870

−

1.83195i

−1.00000

−4.84607

2.79788i

−1.30701

1.81431i

267.1

0.500000

0.866025i

−0.465926

0.124844i

−0.500000

0.866025i

−2.03906

0.917738i

−0.341081

−

0.341081i

4.19798

−

1.12484i

−1.00000

−2.39658

1.38366i

−1.81431

−

1.30701i

267.2

0.500000

0.866025i

1.46593

−

0.392794i

−0.500000

0.866025i

1.30701

1.81431i

1.07313

1.07313i

2.26612

−

0.607206i

−1.00000

−0.603425

0.348387i

−0.917738

2.03906i

273.1

0.500000

−

0.866025i

−0.465926

−

0.124844i

−0.500000

−

0.866025i

−2.03906

−

0.917738i

−0.341081

0.341081i

4.19798

1.12484i

−1.00000

−2.39658

−

1.38366i

−1.81431

1.30701i

273.2

0.500000

−

0.866025i

1.46593

0.392794i

−0.500000

−

0.866025i

1.30701

−

1.81431i

1.07313

−

1.07313i

2.26612

0.607206i

−1.00000

−0.603425

−

0.348387i

−0.917738

−

2.03906i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
185.p	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	370.2.q.c	✓	8
5.c	odd	4	1		370.2.r.c	yes	8
37.g	odd	12	1		370.2.r.c	yes	8
185.p	even	12	1	inner	370.2.q.c	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
370.2.q.c	✓	8	1.a	even	1	1	trivial
370.2.q.c	✓	8	185.p	even	12	1	inner
370.2.r.c	yes	8	5.c	odd	4	1
370.2.r.c	yes	8	37.g	odd	12	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} - 4T_{3}^{7} + 14T_{3}^{6} - 24T_{3}^{5} + 14T_{3}^{4} - 4T_{3}^{2} + 8T_{3} + 4 $ T3^8 - 4*T3^7 + 14*T3^6 - 24*T3^5 + 14*T3^4 - 4*T3^2 + 8*T3 + 4 acting on $S_{2}^{\mathrm{new}}(370, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} - T + 1)^{4} $ (T^2 - T + 1)^4
$3$	$ T^{8} - 4 T^{7} + \cdots + 4 $ T^8 - 4T^7 + 14T^6 - 24T^5 + 14T^4 - 4T^2 + 8T + 4
$5$	$ T^{8} - 4 T^{7} + \cdots + 625 $ T^8 - 4T^7 + 8T^6 + 8T^5 - 41T^4 + 40T^3 + 200T^2 - 500*T + 625
$7$	$ T^{8} - 12 T^{7} + \cdots + 4 $ T^8 - 12T^7 + 54T^6 - 120T^5 + 206T^4 - 384T^3 + 396T^2 - 24*T + 4
$11$	$ T^{8} + 48 T^{6} + \cdots + 16 $ T^8 + 48T^6 + 584T^4 + 1728*T^2 + 16
$13$	$ T^{8} + 4 T^{7} + \cdots + 64 $ T^8 + 4T^7 + 28T^6 + 16T^5 + 280T^4 + 448T^3 + 928T^2 + 256*T + 64
$17$	$ T^{8} - 12 T^{7} + \cdots + 9409 $ T^8 - 12T^7 + 30T^6 + 216T^5 - 493T^4 - 3240T^3 + 9054T^2 + 17460*T + 9409
$19$	$ T^{8} - 4 T^{7} + \cdots + 10000 $ T^8 - 4T^7 + 8T^6 - 112T^5 + 124T^4 + 1120T^3 + 800T^2 + 4000*T + 10000
$23$	$ (T^{4} + 4 T^{3} + \cdots + 142)^{2} $ (T^4 + 4T^3 - 22T^2 - 52*T + 142)^2
$29$	$ T^{8} + 24 T^{7} + \cdots + 81 $ T^8 + 24T^7 + 288T^6 + 1944T^5 + 8082T^4 + 19224T^3 + 23328T^2 + 1944*T + 81
$31$	$ T^{8} + 8 T^{7} + \cdots + 85264 $ T^8 + 8T^7 + 32T^6 - 144T^5 + 1352T^4 + 6816T^3 + 21632T^2 - 60736*T + 85264
$37$	$ T^{8} - 529 T^{4} + 1874161 $ T^8 - 529*T^4 + 1874161
$41$	$ T^{8} - 12 T^{7} + \cdots + 1270129 $ T^8 - 12T^7 - 26T^6 + 888T^5 + 2235T^4 - 80808T^3 + 480886T^2 - 1230684*T + 1270129
$43$	$ (T^{4} - 8 T^{3} + \cdots - 386)^{2} $ (T^4 - 8T^3 - 58T^2 + 404*T - 386)^2
$47$	$ T^{8} + 8 T^{7} + \cdots + 20394256 $ T^8 + 8T^7 + 32T^6 - 816T^5 + 14072T^4 + 24672T^3 + 80000T^2 - 1806400*T + 20394256
$53$	$ T^{8} - 16 T^{7} + \cdots + 160000 $ T^8 - 16T^7 - 40T^6 + 992T^5 + 3376T^4 + 8320T^3 + 80000T^2 + 64000*T + 160000
$59$	$ T^{8} + 8 T^{7} + \cdots + 2262016 $ T^8 + 8T^7 + 152T^6 + 2112T^5 + 14048T^4 + 98304T^3 + 666368T^2 + 2069504*T + 2262016
$61$	$ T^{8} - 48 T^{6} + \cdots + 8427409 $ T^8 - 48T^6 - 696T^5 - 3433T^4 + 39000T^3 + 443856T^2 + 2229504T + 8427409
$67$	$ T^{8} + 16 T^{7} + \cdots + 35344 $ T^8 + 16T^7 + 164T^6 + 1296T^5 + 6764T^4 + 31200T^3 + 129776T^2 + 123328*T + 35344
$71$	$ T^{8} + 4 T^{7} + \cdots + 21316 $ T^8 + 4T^7 + 62T^6 + 160T^5 + 2950T^4 + 9080T^3 + 22868T^2 + 25112*T + 21316
$73$	$ T^{8} - 16 T^{7} + \cdots + 341056 $ T^8 - 16T^7 + 128T^6 - 496T^5 + 1312T^4 - 5696T^3 + 46208T^2 - 177536*T + 341056
$79$	$ T^{8} + 32 T^{7} + \cdots + 10000 $ T^8 + 32T^7 + 380T^6 + 2144T^5 + 7756T^4 + 18880T^3 + 26000T^2 + 16000*T + 10000
$83$	$ T^{8} - 126 T^{6} + \cdots + 240436036 $ T^8 - 126T^6 + 2520T^5 - 18514T^4 - 368640T^3 + 6174756T^2 - 41866200T + 240436036
$89$	$ T^{8} - 8 T^{7} + \cdots + 332929 $ T^8 - 8T^7 + 218T^6 - 1452T^5 + 15083T^4 - 98076T^3 + 100046T^2 + 514684*T + 332929
$97$	$ T^{8} + \cdots + 1120843441 $ T^8 + 836T^6 + 246006T^4 + 29339684*T^2 + 1120843441
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 370 = 2 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	370.q (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{24}^{4} q^{2} + (\zeta_{24}^{6} + \cdots + \zeta_{24}^{4}) q^{3} + (\zeta_{24}^{4} - 1) q^{4} + ( - \zeta_{24}^{6} + 2 \zeta_{24}^{5} + \cdots - \zeta_{24}) q^{5} + (\zeta_{24}^{6} + \zeta_{24}^{5} + \cdots - 1) q^{6}+ \cdots + ( - 4 \zeta_{24}^{7} + \cdots - 4 \zeta_{24}) q^{99}+O(q^{100}) \) q + z^4 * q^2 + (z^6 + z^5 + z^4) * q^3 + (z^4 - 1) * q^4 + (-z^6 + 2z^5 + z^4 + z^2 - z) q^5 + (z^6 + z^5 + z^4 - z^2 - z - 1) * q^6 + (z^6 - z^5 - z^4 - 2z^2 + 2) q^7 - q^8 + (2z^7 + 2z^5 + z^4 - 2z^3 - 2z - 2) * q^9 + (z^5 + z^4 + z^2 - 2z - 1) q^10 + (z^6 - 2z^5 + 2z^4 - 2z^3 + 2z - 1) * q^11 + (-z^2 - z - 1) * q^12 + (-z^7 - 2z^6 - z^5 + z^4 - z^3 + z^2 + 2z - 1) * q^13 + (-z^6 - z^5 + z^4 - z^2 + z + 1) * q^14 + (z^7 + 2z^6 + 2z^5 + 2z^4 - z^3 - 2z^2 - 3z - 1) q^15 - z^4 * q^16 + (z^7 - 4z^6 - z^4 + 4z^2 - z + 2) * q^17 + (-z^4 - 2z^3 - 2z - 1) * q^18 + (-2z^5 - z^4 + z^2 + 4z + 1) * q^19 + (z^6 - z^5 - z - 1) * q^20 + (-2z^7 - z^6 - z^4 + z^2 + 2z + 2) * q^21 + (-2z^7 + z^6 + z^4 - z^2 + 2z - 2) * q^22 + (-3z^7 + 2z^5 + z^3 + z - 1) * q^23 + (-z^6 - z^5 - z^4) * q^24 + (2z^7 + 2z^5 + 2z^3 - z^2 - 4z) * q^25 + (-2z^7 - z^6 + z^5 + z^3 + 2z^2 + z - 1) * q^26 + (-z^6 - 3z^5 - z^3 - 3z - 1) * q^27 + (-2z^6 + 2z^4 + z^2 + z - 1) * q^28 + (-2z^7 + 4z^6 + 2z^4 + z^3 - 2z^2 - 4) * q^29 + (-z^5 + z^4 - z^3 - 2z^2 - 2z - 2) * q^30 + (z^6 - 4z^4 + 2z^3 - 4z^2 + 1) q^31 + (-z^4 + 1) * q^32 + (-z^7 + 2z^6 - z^5 - z^4 + z^3 - z^2 + 2z - 1) * q^33 + (z^7 - z^5 + z^4 - z^3 + 4z^2 + 1) q^34 + (-3z^7 - 4z^6 + 2z^5 + 2z^4 - z^3 + 2z^2 + z - 1) q^35 + (-2z^7 - 2z^5 - 2z^4 + 1) q^36 + (3z^7 - 7z^3) * q^37 + (z^6 + 2z^5 + 2z + 1) * q^38 + (-2z^7 + 4z^3 + 2z^2 + 2z + 2) * q^39 + (z^6 - 2z^5 - z^4 - z^2 + z) q^40 + (-z^7 + z^5 + z^4 + 5z^3 - 2z^2 + 4z + 1) q^41 + (-2z^7 + 2z^5 + z^4 + 2z^3 + z^2 + 1) q^42 + (3z^7 + 3z^6 - z^5 - 2z^3 - 6z^2 - 2z + 2) q^43 + (-2z^7 + 2z^5 - z^4 + 2z^3 - z^2 - 1) q^44 + (2z^7 - z^5 - 5z^4 - 2z^3 - 3z^2 - 4z + 1) q^45 + (-2z^7 + 3z^5 - z^4 + 3z^3 - 2z) * q^46 + (10z^7 + 2z^6 + z^5 + 2z^4 - 5z^3 - 2z^2 - z - 2) q^47 + (-z^6 - z^5 - z^4 + z^2 + z + 1) * q^48 + (2z^7 - 2z^5 - 3z^4 + 2z^3 - 2z + 6) q^49 + (4z^7 - z^6 - 2z^5 - 2z^3 - 2z) * q^50 + (5z^4 + 3z^3 + 5z^2) q^51 + (-z^7 + z^6 + 2z^5 - z^4 + 2z^3 + z^2 - z) * q^52 + (6z^7 - 4z^5 - 2z^2 + 2z + 2) * q^53 + (-z^7 - z^6 - 6z^5 - z^4 + z^2 + 3z) * q^54 + (-3z^7 + 4z^6 - 2z^3 + 2z^2 - 3z + 2) q^55 + (-z^6 + z^5 + z^4 + 2z^2 - 2) q^56 + (3z^7 + 3z^6 + 2z^5 + z^4 + 2z^3 + 3z^2 + 3z) * q^57 + (-z^7 + 2z^6 - 2z^4 + 2z^3 - 4z^2 - 2) * q^58 + (-2z^7 - 6z^6 - 6z^4 + 4z^2 + 2) * q^59 + (-z^7 - 2z^6 - 3z^5 - z^4 + z - 1) * q^60 + (-5z^7 + 2z^6 - 2z^5 - 4z^4 + 5z^3 + 2z^2 + 4z + 2) q^61 + (2z^7 - 3z^6 - 3z^4 - z^2 + 4) q^62 + (-3z^6 + 3z^5 + 5z^4 + 5z^2 + 3z - 3) q^63 + q^64 + (-4z^7 + 3z^6 - z^5 - 2z^4 + 5z^3 + 2z^2 - z + 3) q^65 + (z^6 + z^5 - 2z^4 + z^3 - 2z^2 + z + 1) * q^66 + (5z^7 + 2z^6 - z^5 - 5z^3 - 2z^2 + 2z - 2) q^67 + (4z^6 - z^5 + 2z^4 - z^3 + z - 1) * q^68 + (z^7 + 2z^6 + 3z^5 + z^3 - 2z^2 + 2) q^69 + (-4z^7 - 2z^6 + 3z^5 + z^4 + 3z^3 + 4z^2 - 2z - 2) * q^70 + (2z^7 - 4z^6 + 2z^5 + z^4 + z^3 + 2z^2 - 3z - 1) q^71 + (-2z^7 - 2z^5 - z^4 + 2z^3 + 2z + 2) * q^72 + (2z^7 - 3z^6 - 3z^5 - 2z^4 - z^3 + 2z^2 + 3z + 3) * q^73 + (-4z^7 - 3z^3) * q^74 + (z^7 - 3z^6 + z^4 - 4z^3 - 2z^2 - 6z - 3) * q^75 + (z^6 + 4z^5 + z^4 - z^2 - 2z) * q^76 + (5z^7 - 2z^6 - 3z^5 + 3z^4 - 5z^3 - z^2 + 6z - 1) * q^77 + (2z^7 + 2z^6 + 2z^5 + 2z^4 + 2z^3) q^78 + (-3z^7 + 4z^6 - z^5 + 3z^3 - 4z^2 + 2z - 4) q^79 + (-z^5 - z^4 - z^2 + 2z + 1) q^80 + (-2z^7 - 2z^6 - 2z^5 - 2z^4 + 4z^3 + z^2 - 2z + 2) * q^81 + (4z^7 - 2z^6 + 5z^5 + 2z^4 + z^3 - z - 1) * q^82 + (-8z^7 - 6z^6 + 6z^5 - 6z^4 + 3z^2 - 3z + 3) * q^83 + (z^6 + 2z^5 + 2z^4 + 2z^3 - 2z - 1) * q^84 + (6z^7 - 4z^6 + 3z^5 - 4z^4 + 2z^3 + 6z^2 + 4) * q^85 + (z^7 - 3z^6 - 3z^5 + 2z^4 - 3z^3 - 3z^2 + z) q^86 + (z^7 - z^5 - 3z^4 - 2z^3 - 6z^2 - z - 3) q^87 + (-z^6 + 2z^5 - 2z^4 + 2z^3 - 2z + 1) * q^88 + (-4z^7 + 8z^3 + z^2 - 4z + 1) q^89 + (-3z^6 - 5z^5 - 4z^4 - 2z^3 + z + 5) * q^90 + (4z^7 - 6z^6 + 6z^4 - 8z^3 + 4z - 6) q^91 + (z^7 + z^5 - z^4 + 2z^3 - 3z + 1) * q^92 + (-z^7 - 6z^6 - z^5 - 5z^4 - z^3 + 3z^2 + 2z + 5) * q^93 + (5z^7 - 10z^3 - 2z^2 - z - 2) q^94 + (-2z^7 + 6z^6 + 3z^5 + z^3 + 3z + 2) * q^95 + (z^2 + z + 1) * q^96 + (8z^7 + 6z^6 + 9z^5 - 6z^4 + z^3 - z + 3) * q^97 + (4z^7 - 4z^5 + 3z^4 - 2z^3 + 2z + 3) q^98 + (-4z^7 + 3z^6 + z^4 + 3z^2 - 4z) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} + 4 q^{3} - 4 q^{4} + 4 q^{5} - 4 q^{6} + 12 q^{7} - 8 q^{8} - 12 q^{9} - 4 q^{10} - 8 q^{12} - 4 q^{13} + 12 q^{14} - 4 q^{16} + 12 q^{17} - 12 q^{18} + 4 q^{19} - 8 q^{20} + 12 q^{21} - 12 q^{22}+ \cdots + 4 q^{99}+O(q^{100}) \) 8 * q + 4 * q^2 + 4 * q^3 - 4 * q^4 + 4 * q^5 - 4 * q^6 + 12 * q^7 - 8 * q^8 - 12 * q^9 - 4 * q^10 - 8 * q^12 - 4 * q^13 + 12 * q^14 - 4 * q^16 + 12 * q^17 - 12 * q^18 + 4 * q^19 - 8 * q^20 + 12 * q^21 - 12 * q^22 - 8 * q^23 - 4 * q^24 - 8 * q^26 - 8 * q^27 - 24 * q^29 - 12 * q^30 - 8 * q^31 + 4 * q^32 - 12 * q^33 + 12 * q^34 + 8 * q^38 + 16 * q^39 - 4 * q^40 + 12 * q^41 + 12 * q^42 + 16 * q^43 - 12 * q^44 - 12 * q^45 - 4 * q^46 - 8 * q^47 + 4 * q^48 + 36 * q^49 + 20 * q^51 - 4 * q^52 + 16 * q^53 - 4 * q^54 + 16 * q^55 - 12 * q^56 + 4 * q^57 - 24 * q^58 - 8 * q^59 - 12 * q^60 + 20 * q^62 - 4 * q^63 + 8 * q^64 + 16 * q^65 - 16 * q^67 + 16 * q^69 - 12 * q^70 - 4 * q^71 + 12 * q^72 + 16 * q^73 - 20 * q^75 + 4 * q^76 + 4 * q^77 + 8 * q^78 - 32 * q^79 + 4 * q^80 + 8 * q^81 + 16 * q^85 + 8 * q^86 - 36 * q^87 + 8 * q^89 + 24 * q^90 - 24 * q^91 + 4 * q^92 + 20 * q^93 - 16 * q^94 + 16 * q^95 + 8 * q^96 + 36 * q^98 + 4 * q^99

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	370.2.q.c

Level	$370$
Weight	$2$
Character orbit	370.q
Analytic conductor	$2.954$
Analytic rank	$0$
Dimension	$8$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(2.95446487479\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{12})\)
Coefficient field:	\(\Q(\zeta_{24})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{4} + 1 \) x^8 - x^4 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

\(n\)	\(261\)	\(297\)
\(\chi(n)\)	\(-\zeta_{24}^{2}\)	\(-\zeta_{24}^{6}\)

\(n\):		e.g. 2-40 or 80-90
Embeddings:		e.g. 1-3 or 97.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{2} - T + 1)^{4} \) (T^2 - T + 1)^4
$3$	\( T^{8} - 4 T^{7} + \cdots + 4 \) T^8 - 4T^7 + 14T^6 - 24T^5 + 14T^4 - 4T^2 + 8T + 4
$5$	\( T^{8} - 4 T^{7} + \cdots + 625 \) T^8 - 4T^7 + 8T^6 + 8T^5 - 41T^4 + 40T^3 + 200T^2 - 500*T + 625
$7$	\( T^{8} - 12 T^{7} + \cdots + 4 \) T^8 - 12T^7 + 54T^6 - 120T^5 + 206T^4 - 384T^3 + 396T^2 - 24*T + 4
$11$	\( T^{8} + 48 T^{6} + \cdots + 16 \) T^8 + 48T^6 + 584T^4 + 1728*T^2 + 16
$13$	\( T^{8} + 4 T^{7} + \cdots + 64 \) T^8 + 4T^7 + 28T^6 + 16T^5 + 280T^4 + 448T^3 + 928T^2 + 256*T + 64
$17$	\( T^{8} - 12 T^{7} + \cdots + 9409 \) T^8 - 12T^7 + 30T^6 + 216T^5 - 493T^4 - 3240T^3 + 9054T^2 + 17460*T + 9409
$19$	\( T^{8} - 4 T^{7} + \cdots + 10000 \) T^8 - 4T^7 + 8T^6 - 112T^5 + 124T^4 + 1120T^3 + 800T^2 + 4000*T + 10000
$23$	\( (T^{4} + 4 T^{3} + \cdots + 142)^{2} \) (T^4 + 4T^3 - 22T^2 - 52*T + 142)^2
$29$	\( T^{8} + 24 T^{7} + \cdots + 81 \) T^8 + 24T^7 + 288T^6 + 1944T^5 + 8082T^4 + 19224T^3 + 23328T^2 + 1944*T + 81
$31$	\( T^{8} + 8 T^{7} + \cdots + 85264 \) T^8 + 8T^7 + 32T^6 - 144T^5 + 1352T^4 + 6816T^3 + 21632T^2 - 60736*T + 85264
$37$	\( T^{8} - 529 T^{4} + 1874161 \) T^8 - 529*T^4 + 1874161
$41$	\( T^{8} - 12 T^{7} + \cdots + 1270129 \) T^8 - 12T^7 - 26T^6 + 888T^5 + 2235T^4 - 80808T^3 + 480886T^2 - 1230684*T + 1270129
$43$	\( (T^{4} - 8 T^{3} + \cdots - 386)^{2} \) (T^4 - 8T^3 - 58T^2 + 404*T - 386)^2
$47$	\( T^{8} + 8 T^{7} + \cdots + 20394256 \) T^8 + 8T^7 + 32T^6 - 816T^5 + 14072T^4 + 24672T^3 + 80000T^2 - 1806400*T + 20394256
$53$	\( T^{8} - 16 T^{7} + \cdots + 160000 \) T^8 - 16T^7 - 40T^6 + 992T^5 + 3376T^4 + 8320T^3 + 80000T^2 + 64000*T + 160000
$59$	\( T^{8} + 8 T^{7} + \cdots + 2262016 \) T^8 + 8T^7 + 152T^6 + 2112T^5 + 14048T^4 + 98304T^3 + 666368T^2 + 2069504*T + 2262016
$61$	\( T^{8} - 48 T^{6} + \cdots + 8427409 \) T^8 - 48T^6 - 696T^5 - 3433T^4 + 39000T^3 + 443856T^2 + 2229504T + 8427409
$67$	\( T^{8} + 16 T^{7} + \cdots + 35344 \) T^8 + 16T^7 + 164T^6 + 1296T^5 + 6764T^4 + 31200T^3 + 129776T^2 + 123328*T + 35344
$71$	\( T^{8} + 4 T^{7} + \cdots + 21316 \) T^8 + 4T^7 + 62T^6 + 160T^5 + 2950T^4 + 9080T^3 + 22868T^2 + 25112*T + 21316
$73$	\( T^{8} - 16 T^{7} + \cdots + 341056 \) T^8 - 16T^7 + 128T^6 - 496T^5 + 1312T^4 - 5696T^3 + 46208T^2 - 177536*T + 341056
$79$	\( T^{8} + 32 T^{7} + \cdots + 10000 \) T^8 + 32T^7 + 380T^6 + 2144T^5 + 7756T^4 + 18880T^3 + 26000T^2 + 16000*T + 10000
$83$	\( T^{8} - 126 T^{6} + \cdots + 240436036 \) T^8 - 126T^6 + 2520T^5 - 18514T^4 - 368640T^3 + 6174756T^2 - 41866200T + 240436036
$89$	\( T^{8} - 8 T^{7} + \cdots + 332929 \) T^8 - 8T^7 + 218T^6 - 1452T^5 + 15083T^4 - 98076T^3 + 100046T^2 + 514684*T + 332929
$97$	\( T^{8} + \cdots + 1120843441 \) T^8 + 836T^6 + 246006T^4 + 29339684*T^2 + 1120843441
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