LMFDB - Newform orbit 174.2.g.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [174,2,Mod(7,174)] 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("174.7"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(174, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([0, 6])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 174 = 2 \cdot 3 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	174.g (of order $7$, degree $6$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	no
Analytic conductor:	$1.38939699517$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{14})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

$q$-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 $\zeta_{14}$. We also show the integral $q$-expansion of the trace form.

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/174\mathbb{Z}\right)^\times$.

$n$	$31$	$59$
$\chi(n)$	$-\zeta_{14}$	$1$

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} $

7.1

0.900969	−	0.433884i
0.900969	+	0.433884i
−0.623490	+	0.781831i
−0.623490	−	0.781831i
0.222521	+	0.974928i
0.222521	−	0.974928i

−0.222521

−

0.974928i

0.900969

0.433884i

−0.900969

0.433884i

0.455927

1.99755i

0.222521

−

0.974928i

1.84601

0.888992i

0.623490

0.781831i

0.623490

0.781831i

1.84601

−

0.888992i

25.1

−0.222521

0.974928i

0.900969

−

0.433884i

−0.900969

−

0.433884i

0.455927

−

1.99755i

0.222521

0.974928i

1.84601

−

0.888992i

0.623490

−

0.781831i

0.623490

−

0.781831i

1.84601

0.888992i

49.1

−0.900969

0.433884i

−0.623490

−

0.781831i

0.623490

−

0.781831i

−2.42543

1.16802i

0.900969

0.433884i

1.67845

2.10471i

−0.222521

0.974928i

−0.222521

0.974928i

1.67845

−

2.10471i

103.1

−0.900969

−

0.433884i

−0.623490

0.781831i

0.623490

0.781831i

−2.42543

−

1.16802i

0.900969

−

0.433884i

1.67845

−

2.10471i

−0.222521

−

0.974928i

−0.222521

−

0.974928i

1.67845

2.10471i

139.1

0.623490

−

0.781831i

0.222521

−

0.974928i

−0.222521

−

0.974928i

1.46950

−

1.84270i

−0.623490

−

0.781831i

−0.524459

2.29780i

−0.900969

−

0.433884i

−0.900969

−

0.433884i

−0.524459

−

2.29780i

169.1

0.623490

0.781831i

0.222521

0.974928i

−0.222521

0.974928i

1.46950

1.84270i

−0.623490

0.781831i

−0.524459

−

2.29780i

−0.900969

0.433884i

−0.900969

0.433884i

−0.524459

2.29780i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
29.d	even	7	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	174.2.g.a	✓	6
3.b	odd	2	1		522.2.k.d		6
29.d	even	7	1	inner	174.2.g.a	✓	6
29.d	even	7	1		5046.2.a.bb		3
29.e	even	14	1		5046.2.a.ba		3
87.j	odd	14	1		522.2.k.d		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
174.2.g.a	✓	6	1.a	even	1	1	trivial
174.2.g.a	✓	6	29.d	even	7	1	inner
522.2.k.d		6	3.b	odd	2	1
522.2.k.d		6	87.j	odd	14	1
5046.2.a.ba		3	29.e	even	14	1
5046.2.a.bb		3	29.d	even	7	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{6} + T_{5}^{5} + T_{5}^{4} + 15T_{5}^{3} + 29T_{5}^{2} - 13T_{5} + 169 $ T5^6 + T5^5 + T5^4 + 15*T5^3 + 29*T5^2 - 13*T5 + 169 acting on $S_{2}^{\mathrm{new}}(174, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} + T^{5} + T^{4} + \cdots + 1 $ T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$3$	$ T^{6} - T^{5} + T^{4} + \cdots + 1 $ T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$5$	$ T^{6} + T^{5} + \cdots + 169 $ T^6 + T^5 + T^4 + 15T^3 + 29T^2 - 13*T + 169
$7$	$ T^{6} - 6 T^{5} + \cdots + 169 $ T^6 - 6T^5 + 22T^4 - 55T^3 + 120T^2 - 195*T + 169
$11$	$ T^{6} - T^{5} + 15 T^{4} + \cdots + 1 $ T^6 - T^5 + 15T^4 + 13T^3 + T^2 - T + 1
$13$	$ T^{6} - 4 T^{5} + \cdots + 64 $ T^6 - 4T^5 + 16T^4 - 8T^3 + 32T^2 + 96*T + 64
$17$	$ (T^{3} - 28 T + 56)^{2} $ (T^3 - 28*T + 56)^2
$19$	$ T^{6} + 4 T^{5} + \cdots + 64 $ T^6 + 4T^5 + 16T^4 + 8T^3 + 32T^2 - 96*T + 64
$23$	$ T^{6} + 4 T^{5} + \cdots + 10816 $ T^6 + 4T^5 - 12T^4 + 176T^3 + 2272T^2 - 1664*T + 10816
$29$	$ T^{6} - T^{5} + \cdots + 24389 $ T^6 - T^5 - 27T^4 + 69T^3 - 783T^2 - 841T + 24389
$31$	$ T^{6} + 14 T^{5} + \cdots + 49 $ T^6 + 14T^5 + 126T^4 + 259T^3 + 784T^2 + 343*T + 49
$37$	$ T^{6} + 14 T^{5} + \cdots + 3136 $ T^6 + 14T^5 + 112T^4 + 1064T^3 + 7840T^2 + 6272*T + 3136
$41$	$ (T^{3} - 8 T^{2} - 44 T + 8)^{2} $ (T^3 - 8T^2 - 44T + 8)^2
$43$	$ T^{6} + 2 T^{5} + \cdots + 53824 $ T^6 + 2T^5 + 88T^4 + 8T^3 + 1920T^2 + 20416*T + 53824
$47$	$ T^{6} + 2 T^{5} + \cdots + 118336 $ T^6 + 2T^5 + 32T^4 - 440T^3 + 1248T^2 - 8256*T + 118336
$53$	$ T^{6} + 10 T^{5} + \cdots + 121801 $ T^6 + 10T^5 + 170T^4 + 692T^3 + 123T^2 + 10470*T + 121801
$59$	$ (T^{3} - 9 T^{2} + \cdots + 449)^{2} $ (T^3 - 9T^2 - 64T + 449)^2
$61$	$ T^{6} + 10 T^{5} + \cdots + 118336 $ T^6 + 10T^5 - 12T^4 + 104T^3 + 8656T^2 - 15136*T + 118336
$67$	$ T^{6} + 28 T^{5} + \cdots + 529984 $ T^6 + 28T^5 + 560T^4 + 6944T^3 + 54096T^2 + 244608*T + 529984
$71$	$ T^{6} + 24 T^{5} + \cdots + 4096 $ T^6 + 24T^5 + 240T^4 + 1280T^3 + 5632T^2 + 6144*T + 4096
$73$	$ T^{6} + 10 T^{5} + \cdots + 12769 $ T^6 + 10T^5 + 100T^4 - 484T^3 + 3693T^2 - 11300*T + 12769
$79$	$ T^{6} - 20 T^{5} + \cdots + 692224 $ T^6 - 20T^5 + 400T^4 - 4416T^3 + 59648T^2 - 332800*T + 692224
$83$	$ T^{6} - 26 T^{5} + \cdots + 253009 $ T^6 - 26T^5 + 298T^4 - 2680T^3 + 28863T^2 - 107642*T + 253009
$89$	$ T^{6} - 24 T^{5} + \cdots + 4096 $ T^6 - 24T^5 + 240T^4 - 1280T^3 + 5632T^2 - 6144*T + 4096
$97$	$ T^{6} - 22 T^{5} + \cdots + 1681 $ T^6 - 22T^5 + 232T^4 - 1520T^3 + 6693T^2 - 4264*T + 1681
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 174 = 2 \cdot 3 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	174.g (of order \(7\), degree \(6\), minimal)

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

Introduction

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Newspace parameters

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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	174.2.g.a

Level	$174$
Weight	$2$
Character orbit	174.g
Analytic conductor	$1.389$
Analytic rank	$0$
Dimension	$6$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(1.38939699517\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	\(\Q(\zeta_{14})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 7.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{6} + T^{5} + T^{4} + \cdots + 1 \) T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$3$	\( T^{6} - T^{5} + T^{4} + \cdots + 1 \) T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$5$	\( T^{6} + T^{5} + \cdots + 169 \) T^6 + T^5 + T^4 + 15T^3 + 29T^2 - 13*T + 169
$7$	\( T^{6} - 6 T^{5} + \cdots + 169 \) T^6 - 6T^5 + 22T^4 - 55T^3 + 120T^2 - 195*T + 169
$11$	\( T^{6} - T^{5} + 15 T^{4} + \cdots + 1 \) T^6 - T^5 + 15T^4 + 13T^3 + T^2 - T + 1
$13$	\( T^{6} - 4 T^{5} + \cdots + 64 \) T^6 - 4T^5 + 16T^4 - 8T^3 + 32T^2 + 96*T + 64
$17$	\( (T^{3} - 28 T + 56)^{2} \) (T^3 - 28*T + 56)^2
$19$	\( T^{6} + 4 T^{5} + \cdots + 64 \) T^6 + 4T^5 + 16T^4 + 8T^3 + 32T^2 - 96*T + 64
$23$	\( T^{6} + 4 T^{5} + \cdots + 10816 \) T^6 + 4T^5 - 12T^4 + 176T^3 + 2272T^2 - 1664*T + 10816
$29$	\( T^{6} - T^{5} + \cdots + 24389 \) T^6 - T^5 - 27T^4 + 69T^3 - 783T^2 - 841T + 24389
$31$	\( T^{6} + 14 T^{5} + \cdots + 49 \) T^6 + 14T^5 + 126T^4 + 259T^3 + 784T^2 + 343*T + 49
$37$	\( T^{6} + 14 T^{5} + \cdots + 3136 \) T^6 + 14T^5 + 112T^4 + 1064T^3 + 7840T^2 + 6272*T + 3136
$41$	\( (T^{3} - 8 T^{2} - 44 T + 8)^{2} \) (T^3 - 8T^2 - 44T + 8)^2
$43$	\( T^{6} + 2 T^{5} + \cdots + 53824 \) T^6 + 2T^5 + 88T^4 + 8T^3 + 1920T^2 + 20416*T + 53824
$47$	\( T^{6} + 2 T^{5} + \cdots + 118336 \) T^6 + 2T^5 + 32T^4 - 440T^3 + 1248T^2 - 8256*T + 118336
$53$	\( T^{6} + 10 T^{5} + \cdots + 121801 \) T^6 + 10T^5 + 170T^4 + 692T^3 + 123T^2 + 10470*T + 121801
$59$	\( (T^{3} - 9 T^{2} + \cdots + 449)^{2} \) (T^3 - 9T^2 - 64T + 449)^2
$61$	\( T^{6} + 10 T^{5} + \cdots + 118336 \) T^6 + 10T^5 - 12T^4 + 104T^3 + 8656T^2 - 15136*T + 118336
$67$	\( T^{6} + 28 T^{5} + \cdots + 529984 \) T^6 + 28T^5 + 560T^4 + 6944T^3 + 54096T^2 + 244608*T + 529984
$71$	\( T^{6} + 24 T^{5} + \cdots + 4096 \) T^6 + 24T^5 + 240T^4 + 1280T^3 + 5632T^2 + 6144*T + 4096
$73$	\( T^{6} + 10 T^{5} + \cdots + 12769 \) T^6 + 10T^5 + 100T^4 - 484T^3 + 3693T^2 - 11300*T + 12769
$79$	\( T^{6} - 20 T^{5} + \cdots + 692224 \) T^6 - 20T^5 + 400T^4 - 4416T^3 + 59648T^2 - 332800*T + 692224
$83$	\( T^{6} - 26 T^{5} + \cdots + 253009 \) T^6 - 26T^5 + 298T^4 - 2680T^3 + 28863T^2 - 107642*T + 253009
$89$	\( T^{6} - 24 T^{5} + \cdots + 4096 \) T^6 - 24T^5 + 240T^4 - 1280T^3 + 5632T^2 - 6144*T + 4096
$97$	\( T^{6} - 22 T^{5} + \cdots + 1681 \) T^6 - 22T^5 + 232T^4 - 1520T^3 + 6693T^2 - 4264*T + 1681
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\(n\)	\(31\)	\(59\)
\(\chi(n)\)	\(-\zeta_{14}\)	\(1\)