LMFDB - Newform orbit 375.2.g.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [375,2,Mod(76,375)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(375, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("375.76");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 375 = 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	375.g (of order $5$, degree $4$, not 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:	$2.99439007580$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + x^{2} - x + 1 $ x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/375\mathbb{Z}\right)^\times$.

$n$	$127$	$251$
$\chi(n)$	$-\zeta_{10}^{3}$	$1$

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} $

76.1

0.809017	−	0.587785i
−0.309017	+	0.951057i
−0.309017	−	0.951057i
0.809017	+	0.587785i

0.809017

0.587785i

−0.309017

−

0.951057i

−0.309017

−

0.951057i

0.309017

−

0.951057i

−4.47214

0.927051

−

2.85317i

−0.809017

0.587785i

151.1

−0.309017

−

0.951057i

0.809017

−

0.587785i

0.809017

−

0.587785i

−0.809017

−

0.587785i

4.47214

−2.42705

−

1.76336i

0.309017

−

0.951057i

226.1

−0.309017

0.951057i

0.809017

0.587785i

0.809017

0.587785i

−0.809017

0.587785i

4.47214

−2.42705

1.76336i

0.309017

0.951057i

301.1

0.809017

−

0.587785i

−0.309017

0.951057i

−0.309017

0.951057i

0.309017

0.951057i

−4.47214

0.927051

2.85317i

−0.809017

−

0.587785i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
25.d	even	5	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	375.2.g.a		4
5.b	even	2	1		75.2.g.a	✓	4
5.c	odd	4	2		375.2.i.a		8
15.d	odd	2	1		225.2.h.a		4
25.d	even	5	1	inner	375.2.g.a		4
25.d	even	5	1		1875.2.a.a		2
25.e	even	10	1		75.2.g.a	✓	4
25.e	even	10	1		1875.2.a.d		2
25.f	odd	20	2		375.2.i.a		8
25.f	odd	20	2		1875.2.b.b		4
75.h	odd	10	1		225.2.h.a		4
75.h	odd	10	1		5625.2.a.a		2
75.j	odd	10	1		5625.2.a.h		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
75.2.g.a	✓	4	5.b	even	2	1
75.2.g.a	✓	4	25.e	even	10	1
225.2.h.a		4	15.d	odd	2	1
225.2.h.a		4	75.h	odd	10	1
375.2.g.a		4	1.a	even	1	1	trivial
375.2.g.a		4	25.d	even	5	1	inner
375.2.i.a		8	5.c	odd	4	2
375.2.i.a		8	25.f	odd	20	2
1875.2.a.a		2	25.d	even	5	1
1875.2.a.d		2	25.e	even	10	1
1875.2.b.b		4	25.f	odd	20	2
5625.2.a.a		2	75.h	odd	10	1
5625.2.a.h		2	75.j	odd	10	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{4} - T_{2}^{3} + T_{2}^{2} - T_{2} + 1 $ T2^4 - T2^3 + T2^2 - T2 + 1 acting on $S_{2}^{\mathrm{new}}(375, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} - T^{3} + T^{2} - T + 1 $ T^4 - T^3 + T^2 - T + 1
$3$	$ T^{4} - T^{3} + T^{2} - T + 1 $ T^4 - T^3 + T^2 - T + 1
$5$	$ T^{4} $ T^4
$7$	$ (T^{2} - 20)^{2} $ (T^2 - 20)^2
$11$	$ T^{4} + 6 T^{3} + 16 T^{2} + 16 T + 16 $ T^4 + 6T^3 + 16T^2 + 16*T + 16
$13$	$ T^{4} - 2 T^{3} + 24 T^{2} - 133 T + 361 $ T^4 - 2T^3 + 24T^2 - 133*T + 361
$17$	$ T^{4} - 8 T^{3} + 34 T^{2} - 77 T + 121 $ T^4 - 8T^3 + 34T^2 - 77*T + 121
$19$	$ T^{4} + 4 T^{3} + 16 T^{2} + 24 T + 16 $ T^4 + 4T^3 + 16T^2 + 24*T + 16
$23$	$ T^{4} + 10 T^{3} + 60 T^{2} + \cdots + 400 $ T^4 + 10T^3 + 60T^2 + 200*T + 400
$29$	$ T^{4} + 8 T^{3} + 34 T^{2} + 87 T + 841 $ T^4 + 8T^3 + 34T^2 + 87*T + 841
$31$	$ T^{4} + 40 T^{2} + 200 T + 400 $ T^4 + 40T^2 + 200T + 400
$37$	$ T^{4} - 15 T^{3} + 100 T^{2} + \cdots + 625 $ T^4 - 15T^3 + 100T^2 - 250*T + 625
$41$	$ T^{4} + 10 T^{2} - 25 T + 25 $ T^4 + 10T^2 - 25T + 25
$43$	$ (T^{2} + 2 T - 44)^{2} $ (T^2 + 2*T - 44)^2
$47$	$ T^{4} - 2 T^{3} + 24 T^{2} + 32 T + 16 $ T^4 - 2T^3 + 24T^2 + 32*T + 16
$53$	$ T^{4} - 5 T^{3} + 10 T^{2} + 25 $ T^4 - 5T^3 + 10T^2 + 25
$59$	$ T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256 $ T^4 - 4T^3 + 16T^2 - 64*T + 256
$61$	$ T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1 $ T^4 - 2T^3 + 4T^2 - 3*T + 1
$67$	$ T^{4} + 2 T^{3} + 24 T^{2} - 32 T + 16 $ T^4 + 2T^3 + 24T^2 - 32*T + 16
$71$	$ T^{4} - 8 T^{3} + 24 T^{2} + 8 T + 16 $ T^4 - 8T^3 + 24T^2 + 8*T + 16
$73$	$ T^{4} - 10 T^{3} + 100 T^{2} + \cdots + 625 $ T^4 - 10T^3 + 100T^2 - 375*T + 625
$79$	$ T^{4} $ T^4
$83$	$ T^{4} - 18 T^{3} + 124 T^{2} + \cdots + 1936 $ T^4 - 18T^3 + 124T^2 + 88*T + 1936
$89$	$ T^{4} + 9 T^{3} + 46 T^{2} + \cdots + 1681 $ T^4 + 9T^3 + 46T^2 + 164*T + 1681
$97$	$ T^{4} + 2 T^{3} + 64 T^{2} - 247 T + 361 $ T^4 + 2T^3 + 64T^2 - 247*T + 361
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Overview	Random
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Classical	Maass
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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:	\( N \)	\(=\)	\( 375 = 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	375.g (of order \(5\), degree \(4\), not minimal)

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

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Newspace parameters

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Inner twists

Twists

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	375.2.g.a

Level	$375$
Weight	$2$
Character orbit	375.g
Analytic conductor	$2.994$
Analytic rank	$0$
Dimension	$4$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(2.99439007580\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\zeta_{10})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{4} - x^{3} + x^{2} - x + 1 \) x^4 - x^3 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 75)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$p$	$F_p(T)$
$2$	\( T^{4} - T^{3} + T^{2} - T + 1 \) T^4 - T^3 + T^2 - T + 1
$3$	\( T^{4} - T^{3} + T^{2} - T + 1 \) T^4 - T^3 + T^2 - T + 1
$5$	\( T^{4} \) T^4
$7$	\( (T^{2} - 20)^{2} \) (T^2 - 20)^2
$11$	\( T^{4} + 6 T^{3} + 16 T^{2} + 16 T + 16 \) T^4 + 6T^3 + 16T^2 + 16*T + 16
$13$	\( T^{4} - 2 T^{3} + 24 T^{2} - 133 T + 361 \) T^4 - 2T^3 + 24T^2 - 133*T + 361
$17$	\( T^{4} - 8 T^{3} + 34 T^{2} - 77 T + 121 \) T^4 - 8T^3 + 34T^2 - 77*T + 121
$19$	\( T^{4} + 4 T^{3} + 16 T^{2} + 24 T + 16 \) T^4 + 4T^3 + 16T^2 + 24*T + 16
$23$	\( T^{4} + 10 T^{3} + 60 T^{2} + \cdots + 400 \) T^4 + 10T^3 + 60T^2 + 200*T + 400
$29$	\( T^{4} + 8 T^{3} + 34 T^{2} + 87 T + 841 \) T^4 + 8T^3 + 34T^2 + 87*T + 841
$31$	\( T^{4} + 40 T^{2} + 200 T + 400 \) T^4 + 40T^2 + 200T + 400
$37$	\( T^{4} - 15 T^{3} + 100 T^{2} + \cdots + 625 \) T^4 - 15T^3 + 100T^2 - 250*T + 625
$41$	\( T^{4} + 10 T^{2} - 25 T + 25 \) T^4 + 10T^2 - 25T + 25
$43$	\( (T^{2} + 2 T - 44)^{2} \) (T^2 + 2*T - 44)^2
$47$	\( T^{4} - 2 T^{3} + 24 T^{2} + 32 T + 16 \) T^4 - 2T^3 + 24T^2 + 32*T + 16
$53$	\( T^{4} - 5 T^{3} + 10 T^{2} + 25 \) T^4 - 5T^3 + 10T^2 + 25
$59$	\( T^{4} - 4 T^{3} + 16 T^{2} - 64 T + 256 \) T^4 - 4T^3 + 16T^2 - 64*T + 256
$61$	\( T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1 \) T^4 - 2T^3 + 4T^2 - 3*T + 1
$67$	\( T^{4} + 2 T^{3} + 24 T^{2} - 32 T + 16 \) T^4 + 2T^3 + 24T^2 - 32*T + 16
$71$	\( T^{4} - 8 T^{3} + 24 T^{2} + 8 T + 16 \) T^4 - 8T^3 + 24T^2 + 8*T + 16
$73$	\( T^{4} - 10 T^{3} + 100 T^{2} + \cdots + 625 \) T^4 - 10T^3 + 100T^2 - 375*T + 625
$79$	\( T^{4} \) T^4
$83$	\( T^{4} - 18 T^{3} + 124 T^{2} + \cdots + 1936 \) T^4 - 18T^3 + 124T^2 + 88*T + 1936
$89$	\( T^{4} + 9 T^{3} + 46 T^{2} + \cdots + 1681 \) T^4 + 9T^3 + 46T^2 + 164*T + 1681
$97$	\( T^{4} + 2 T^{3} + 64 T^{2} - 247 T + 361 \) T^4 + 2T^3 + 64T^2 - 247*T + 361
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\(n\)	\(127\)	\(251\)
\(\chi(n)\)	\(-\zeta_{10}^{3}\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Significant digits:
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