LMFDB - Newform orbit 375.2.i.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [375,2,Mod(49,375)]

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, 7]))

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

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

chi := DirichletCharacter("375.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 375 = 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	375.i (of order $10$, 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:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Character values

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

$n$	$127$	$251$
$\chi(n)$	$\zeta_{20}^{2}$	$1$

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} $

49.1

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

−0.587785

0.809017i

−0.951057

0.309017i

0.309017

0.951057i

0.309017

−

0.951057i

−

4.47214i

−2.85317

−

0.927051i

0.809017

−

0.587785i

49.2

0.587785

−

0.809017i

0.951057

−

0.309017i

0.309017

0.951057i

0.309017

−

0.951057i

4.47214i

2.85317

0.927051i

0.809017

−

0.587785i

199.1

−0.587785

−

0.809017i

−0.951057

−

0.309017i

0.309017

−

0.951057i

0.309017

0.951057i

4.47214i

−2.85317

0.927051i

0.809017

0.587785i

199.2

0.587785

0.809017i

0.951057

0.309017i

0.309017

−

0.951057i

0.309017

0.951057i

−

4.47214i

2.85317

−

0.927051i

0.809017

0.587785i

274.1

−0.951057

−

0.309017i

0.587785

−

0.809017i

−0.809017

−

0.587785i

−0.809017

0.587785i

4.47214i

1.76336

2.42705i

−0.309017

−

0.951057i

274.2

0.951057

0.309017i

−0.587785

0.809017i

−0.809017

−

0.587785i

−0.809017

0.587785i

−

4.47214i

−1.76336

−

2.42705i

−0.309017

−

0.951057i

349.1

−0.951057

0.309017i

0.587785

0.809017i

−0.809017

0.587785i

−0.809017

−

0.587785i

−

4.47214i

1.76336

−

2.42705i

−0.309017

0.951057i

349.2

0.951057

−

0.309017i

−0.587785

−

0.809017i

−0.809017

0.587785i

−0.809017

−

0.587785i

4.47214i

−1.76336

2.42705i

−0.309017

0.951057i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
25.d	even	5	1	inner
25.e	even	10	1	inner

Twists

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

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

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{8} - T_{2}^{6} + T_{2}^{4} - T_{2}^{2} + 1 $ T2^8 - T2^6 + T2^4 - T2^2 + 1 acting on $S_{2}^{\mathrm{new}}(375, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} - T^{6} + T^{4} - T^{2} + 1 $ T^8 - T^6 + T^4 - T^2 + 1
$3$	$ T^{8} - T^{6} + T^{4} - T^{2} + 1 $ T^8 - T^6 + T^4 - T^2 + 1
$5$	$ T^{8} $ T^8
$7$	$ (T^{2} + 20)^{4} $ (T^2 + 20)^4
$11$	$ (T^{4} + 6 T^{3} + 16 T^{2} + 16 T + 16)^{2} $ (T^4 + 6T^3 + 16T^2 + 16*T + 16)^2
$13$	$ T^{8} - 44 T^{6} + 766 T^{4} + \cdots + 130321 $ T^8 - 44T^6 + 766T^4 + 361*T^2 + 130321
$17$	$ T^{8} - 4 T^{6} + 166 T^{4} + \cdots + 14641 $ T^8 - 4T^6 + 166T^4 - 2299*T^2 + 14641
$19$	$ (T^{4} - 4 T^{3} + 16 T^{2} - 24 T + 16)^{2} $ (T^4 - 4T^3 + 16T^2 - 24*T + 16)^2
$23$	$ T^{8} - 20 T^{6} + 400 T^{4} + \cdots + 160000 $ T^8 - 20T^6 + 400T^4 - 8000*T^2 + 160000
$29$	$ (T^{4} - 8 T^{3} + 34 T^{2} - 87 T + 841)^{2} $ (T^4 - 8T^3 + 34T^2 - 87*T + 841)^2
$31$	$ (T^{4} + 40 T^{2} + 200 T + 400)^{2} $ (T^4 + 40T^2 + 200T + 400)^2
$37$	$ T^{8} + 25 T^{6} + 3750 T^{4} + \cdots + 390625 $ T^8 + 25T^6 + 3750T^4 - 62500*T^2 + 390625
$41$	$ (T^{4} + 10 T^{2} - 25 T + 25)^{2} $ (T^4 + 10T^2 - 25T + 25)^2
$43$	$ (T^{4} + 92 T^{2} + 1936)^{2} $ (T^4 + 92*T^2 + 1936)^2
$47$	$ T^{8} - 44 T^{6} + 736 T^{4} + \cdots + 256 $ T^8 - 44T^6 + 736T^4 + 256*T^2 + 256
$53$	$ T^{8} + 5 T^{6} + 150 T^{4} + \cdots + 625 $ T^8 + 5T^6 + 150T^4 - 500*T^2 + 625
$59$	$ (T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256)^{2} $ (T^4 + 4T^3 + 16T^2 + 64*T + 256)^2
$61$	$ (T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1)^{2} $ (T^4 - 2T^3 + 4T^2 - 3*T + 1)^2
$67$	$ T^{8} - 44 T^{6} + 736 T^{4} + \cdots + 256 $ T^8 - 44T^6 + 736T^4 + 256*T^2 + 256
$71$	$ (T^{4} - 8 T^{3} + 24 T^{2} + 8 T + 16)^{2} $ (T^4 - 8T^3 + 24T^2 + 8*T + 16)^2
$73$	$ T^{8} - 100 T^{6} + 3750 T^{4} + \cdots + 390625 $ T^8 - 100T^6 + 3750T^4 + 15625*T^2 + 390625
$79$	$ T^{8} $ T^8
$83$	$ T^{8} + 76 T^{6} + 22416 T^{4} + \cdots + 3748096 $ T^8 + 76T^6 + 22416T^4 - 472384*T^2 + 3748096
$89$	$ (T^{4} - 9 T^{3} + 46 T^{2} - 164 T + 1681)^{2} $ (T^4 - 9T^3 + 46T^2 - 164*T + 1681)^2
$97$	$ T^{8} - 124 T^{6} + 5806 T^{4} + \cdots + 130321 $ T^8 - 124T^6 + 5806T^4 + 14801*T^2 + 130321
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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.i (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{20} q^{2} + \zeta_{20}^{7} q^{3} - \zeta_{20}^{2} q^{4} + (\zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{2} - 1) q^{6} + ( - 4 \zeta_{20}^{7} + 2 \zeta_{20}^{5} - 4 \zeta_{20}^{3}) q^{7} - 3 \zeta_{20}^{3} q^{8} - \zeta_{20}^{4} q^{9} +O(q^{10}) \) q + z * q^2 + z^7 * q^3 - z^2 * q^4 + (z^6 - z^4 + z^2 - 1) * q^6 + (-4z^7 + 2z^5 - 4z^3) q^7 - 3z^3 q^8 - z^4 * q^9	\( q + \zeta_{20} q^{2} + \zeta_{20}^{7} q^{3} - \zeta_{20}^{2} q^{4} + (\zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{2} - 1) q^{6} + ( - 4 \zeta_{20}^{7} + 2 \zeta_{20}^{5} - 4 \zeta_{20}^{3}) q^{7} - 3 \zeta_{20}^{3} q^{8} - \zeta_{20}^{4} q^{9} + (2 \zeta_{20}^{2} - 2) q^{11} + ( - \zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} + \zeta_{20}) q^{12} + ( - 5 \zeta_{20}^{7} + 4 \zeta_{20}^{5} - 4 \zeta_{20}^{3} + 5 \zeta_{20}) q^{13} + ( - 2 \zeta_{20}^{6} - 4 \zeta_{20}^{2} + 4) q^{14} - \zeta_{20}^{4} q^{16} + ( - 3 \zeta_{20}^{5} + \zeta_{20}^{3} - 3 \zeta_{20}) q^{17} - \zeta_{20}^{5} q^{18} + ( - 2 \zeta_{20}^{4} + 2 \zeta_{20}^{2}) q^{19} + (4 \zeta_{20}^{4} - 2 \zeta_{20}^{2} + 4) q^{21} + (2 \zeta_{20}^{3} - 2 \zeta_{20}) q^{22} + (4 \zeta_{20}^{7} - 4 \zeta_{20}^{5} - 2 \zeta_{20}) q^{23} + 3 q^{24} + ( - \zeta_{20}^{6} + \zeta_{20}^{4} + 5) q^{26} + \zeta_{20} q^{27} + (2 \zeta_{20}^{7} + 4 \zeta_{20}^{3} - 4 \zeta_{20}) q^{28} + (\zeta_{20}^{4} + 5 \zeta_{20}^{2} + 1) q^{29} + (4 \zeta_{20}^{6} - 6 \zeta_{20}^{4} + 6 \zeta_{20}^{2} - 4) q^{31} + 5 \zeta_{20}^{5} q^{32} + ( - 2 \zeta_{20}^{5} + 2 \zeta_{20}^{3} - 2 \zeta_{20}) q^{33} + ( - 3 \zeta_{20}^{6} + \zeta_{20}^{4} - 3 \zeta_{20}^{2}) q^{34} + \zeta_{20}^{6} q^{36} + ( - 5 \zeta_{20}^{5} + 5 \zeta_{20}^{3}) q^{37} + ( - 2 \zeta_{20}^{5} + 2 \zeta_{20}^{3}) q^{38} + (5 \zeta_{20}^{6} + \zeta_{20}^{2} - 1) q^{39} + ( - \zeta_{20}^{6} - 2 \zeta_{20}^{4} - \zeta_{20}^{2}) q^{41} + (4 \zeta_{20}^{5} - 2 \zeta_{20}^{3} + 4 \zeta_{20}) q^{42} + ( - 6 \zeta_{20}^{7} + 2 \zeta_{20}^{5} - 6 \zeta_{20}^{3}) q^{43} + ( - 2 \zeta_{20}^{4} + 2 \zeta_{20}^{2}) q^{44} + ( - 4 \zeta_{20}^{4} + 2 \zeta_{20}^{2} - 4) q^{46} + (2 \zeta_{20}^{7} - 2 \zeta_{20}^{3} + 2 \zeta_{20}) q^{47} + \zeta_{20} q^{48} - 13 q^{49} + ( - 3 \zeta_{20}^{6} + 3 \zeta_{20}^{4} + 2) q^{51} + (\zeta_{20}^{7} - \zeta_{20}^{5} - 5 \zeta_{20}) q^{52} + (3 \zeta_{20}^{7} + \zeta_{20}^{3} - \zeta_{20}) q^{53} + \zeta_{20}^{2} q^{54} + (6 \zeta_{20}^{6} + 6 \zeta_{20}^{4} - 6 \zeta_{20}^{2} - 6) q^{56} + (2 \zeta_{20}^{7} - 2 \zeta_{20}^{5} + 2 \zeta_{20}^{3}) q^{57} + (\zeta_{20}^{5} + 5 \zeta_{20}^{3} + \zeta_{20}) q^{58} + 4 \zeta_{20}^{4} q^{59} + ( - \zeta_{20}^{6} - \zeta_{20}^{2} + 1) q^{61} + (4 \zeta_{20}^{7} - 6 \zeta_{20}^{5} + 6 \zeta_{20}^{3} - 4 \zeta_{20}) q^{62} + (2 \zeta_{20}^{7} + 2 \zeta_{20}^{5} - 2 \zeta_{20}^{3} - 2 \zeta_{20}) q^{63} + 7 \zeta_{20}^{6} q^{64} + ( - 2 \zeta_{20}^{6} + 2 \zeta_{20}^{4} - 2 \zeta_{20}^{2}) q^{66} + (2 \zeta_{20}^{5} - 4 \zeta_{20}^{3} + 2 \zeta_{20}) q^{67} + (3 \zeta_{20}^{7} - \zeta_{20}^{5} + 3 \zeta_{20}^{3}) q^{68} + ( - 2 \zeta_{20}^{6} - 2 \zeta_{20}^{4} + 2 \zeta_{20}^{2} + 2) q^{69} + (2 \zeta_{20}^{4} + 2 \zeta_{20}^{2} + 2) q^{71} + 3 \zeta_{20}^{7} q^{72} + ( - 5 \zeta_{20}^{7} + 5 \zeta_{20}^{5} + 5 \zeta_{20}) q^{73} + ( - 5 \zeta_{20}^{6} + 5 \zeta_{20}^{4}) q^{74} + (2 \zeta_{20}^{6} - 2 \zeta_{20}^{4}) q^{76} + (4 \zeta_{20}^{7} - 4 \zeta_{20}^{5} + 8 \zeta_{20}) q^{77} + (5 \zeta_{20}^{7} + \zeta_{20}^{3} - \zeta_{20}) q^{78} + (\zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{2} - 1) q^{81} + ( - \zeta_{20}^{7} - 2 \zeta_{20}^{5} - \zeta_{20}^{3}) q^{82} + (4 \zeta_{20}^{5} + 6 \zeta_{20}^{3} + 4 \zeta_{20}) q^{83} + ( - 4 \zeta_{20}^{6} + 2 \zeta_{20}^{4} - 4 \zeta_{20}^{2}) q^{84} + ( - 4 \zeta_{20}^{6} - 6 \zeta_{20}^{2} + 6) q^{86} + (6 \zeta_{20}^{7} - 5 \zeta_{20}^{5} + 5 \zeta_{20}^{3} - 6 \zeta_{20}) q^{87} + ( - 6 \zeta_{20}^{5} + 6 \zeta_{20}^{3}) q^{88} + (6 \zeta_{20}^{6} - \zeta_{20}^{2} + 1) q^{89} + ( - 18 \zeta_{20}^{6} + 4 \zeta_{20}^{4} - 18 \zeta_{20}^{2}) q^{91} + (4 \zeta_{20}^{5} - 2 \zeta_{20}^{3} + 4 \zeta_{20}) q^{92} + (2 \zeta_{20}^{7} - 6 \zeta_{20}^{5} + 2 \zeta_{20}^{3}) q^{93} + (2 \zeta_{20}^{6} - 4 \zeta_{20}^{4} + 4 \zeta_{20}^{2} - 2) q^{94} - 5 \zeta_{20}^{2} q^{96} + ( - 4 \zeta_{20}^{7} + 3 \zeta_{20}^{3} - 3 \zeta_{20}) q^{97} - 13 \zeta_{20} q^{98} + ( - 2 \zeta_{20}^{6} + 2 \zeta_{20}^{4}) q^{99} +O(q^{100}) \) q + z * q^2 + z^7 * q^3 - z^2 * q^4 + (z^6 - z^4 + z^2 - 1) * q^6 + (-4z^7 + 2z^5 - 4z^3) q^7 - 3z^3 q^8 - z^4 * q^9 + (2z^2 - 2) q^11 + (-z^7 + z^5 - z^3 + z) * q^12 + (-5z^7 + 4z^5 - 4z^3 + 5z) * q^13 + (-2z^6 - 4z^2 + 4) * q^14 - z^4 * q^16 + (-3z^5 + z^3 - 3z) * q^17 - z^5 * q^18 + (-2z^4 + 2z^2) * q^19 + (4z^4 - 2z^2 + 4) * q^21 + (2z^3 - 2z) * q^22 + (4z^7 - 4z^5 - 2z) q^23 + 3 * q^24 + (-z^6 + z^4 + 5) * q^26 + z * q^27 + (2z^7 + 4z^3 - 4z) q^28 + (z^4 + 5z^2 + 1) q^29 + (4z^6 - 6z^4 + 6z^2 - 4) q^31 + 5z^5 q^32 + (-2z^5 + 2z^3 - 2z) q^33 + (-3z^6 + z^4 - 3z^2) * q^34 + z^6 * q^36 + (-5z^5 + 5z^3) * q^37 + (-2z^5 + 2z^3) * q^38 + (5z^6 + z^2 - 1) q^39 + (-z^6 - 2z^4 - z^2) q^41 + (4z^5 - 2z^3 + 4z) q^42 + (-6z^7 + 2z^5 - 6z^3) q^43 + (-2z^4 + 2z^2) * q^44 + (-4z^4 + 2z^2 - 4) * q^46 + (2z^7 - 2z^3 + 2z) q^47 + z * q^48 - 13 * q^49 + (-3z^6 + 3z^4 + 2) * q^51 + (z^7 - z^5 - 5z) q^52 + (3z^7 + z^3 - z) q^53 + z^2 * q^54 + (6z^6 + 6z^4 - 6z^2 - 6) q^56 + (2z^7 - 2z^5 + 2z^3) q^57 + (z^5 + 5z^3 + z) q^58 + 4z^4 q^59 + (-z^6 - z^2 + 1) * q^61 + (4z^7 - 6z^5 + 6z^3 - 4z) * q^62 + (2z^7 + 2z^5 - 2z^3 - 2z) * q^63 + 7z^6 q^64 + (-2z^6 + 2z^4 - 2z^2) q^66 + (2z^5 - 4z^3 + 2z) q^67 + (3z^7 - z^5 + 3z^3) * q^68 + (-2z^6 - 2z^4 + 2z^2 + 2) q^69 + (2z^4 + 2z^2 + 2) * q^71 + 3z^7 q^72 + (-5z^7 + 5z^5 + 5z) q^73 + (-5z^6 + 5z^4) * q^74 + (2z^6 - 2z^4) * q^76 + (4z^7 - 4z^5 + 8z) q^77 + (5z^7 + z^3 - z) q^78 + (z^6 - z^4 + z^2 - 1) * q^81 + (-z^7 - 2z^5 - z^3) q^82 + (4z^5 + 6z^3 + 4z) q^83 + (-4z^6 + 2z^4 - 4z^2) q^84 + (-4z^6 - 6z^2 + 6) * q^86 + (6z^7 - 5z^5 + 5z^3 - 6z) * q^87 + (-6z^5 + 6z^3) * q^88 + (6z^6 - z^2 + 1) q^89 + (-18z^6 + 4z^4 - 18z^2) q^91 + (4z^5 - 2z^3 + 4z) q^92 + (2z^7 - 6z^5 + 2z^3) q^93 + (2z^6 - 4z^4 + 4z^2 - 2) q^94 - 5z^2 q^96 + (-4z^7 + 3z^3 - 3z) q^97 - 13z q^98 + (-2z^6 + 2z^4) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{4} - 2 q^{6} + 2 q^{9}+O(q^{10}) \) 8 * q - 2 * q^4 - 2 * q^6 + 2 * q^9	\( 8 q - 2 q^{4} - 2 q^{6} + 2 q^{9} - 12 q^{11} + 20 q^{14} + 2 q^{16} + 8 q^{19} + 20 q^{21} + 24 q^{24} + 36 q^{26} + 16 q^{29} - 14 q^{34} + 2 q^{36} + 4 q^{39} + 8 q^{44} - 20 q^{46} - 104 q^{49} + 4 q^{51} + 2 q^{54} - 60 q^{56} - 8 q^{59} + 4 q^{61} + 14 q^{64} - 12 q^{66} + 20 q^{69} + 16 q^{71} - 20 q^{74} + 8 q^{76} - 2 q^{81} - 20 q^{84} + 28 q^{86} + 18 q^{89} - 80 q^{91} + 4 q^{94} - 10 q^{96} - 8 q^{99}+O(q^{100}) \) 8 * q - 2 * q^4 - 2 * q^6 + 2 * q^9 - 12 * q^11 + 20 * q^14 + 2 * q^16 + 8 * q^19 + 20 * q^21 + 24 * q^24 + 36 * q^26 + 16 * q^29 - 14 * q^34 + 2 * q^36 + 4 * q^39 + 8 * q^44 - 20 * q^46 - 104 * q^49 + 4 * q^51 + 2 * q^54 - 60 * q^56 - 8 * q^59 + 4 * q^61 + 14 * q^64 - 12 * q^66 + 20 * q^69 + 16 * q^71 - 20 * q^74 + 8 * q^76 - 2 * q^81 - 20 * q^84 + 28 * q^86 + 18 * q^89 - 80 * q^91 + 4 * q^94 - 10 * q^96 - 8 * q^99

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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	375.2.i.a

Level	$375$
Weight	$2$
Character orbit	375.i
Analytic conductor	$2.994$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(2.99439007580\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{10})\)
Coefficient field:	\(\Q(\zeta_{20})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 75)
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 349.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} - T^{6} + T^{4} - T^{2} + 1 \) T^8 - T^6 + T^4 - T^2 + 1
$3$	\( T^{8} - T^{6} + T^{4} - T^{2} + 1 \) T^8 - T^6 + T^4 - T^2 + 1
$5$	\( T^{8} \) T^8
$7$	\( (T^{2} + 20)^{4} \) (T^2 + 20)^4
$11$	\( (T^{4} + 6 T^{3} + 16 T^{2} + 16 T + 16)^{2} \) (T^4 + 6T^3 + 16T^2 + 16*T + 16)^2
$13$	\( T^{8} - 44 T^{6} + 766 T^{4} + \cdots + 130321 \) T^8 - 44T^6 + 766T^4 + 361*T^2 + 130321
$17$	\( T^{8} - 4 T^{6} + 166 T^{4} + \cdots + 14641 \) T^8 - 4T^6 + 166T^4 - 2299*T^2 + 14641
$19$	\( (T^{4} - 4 T^{3} + 16 T^{2} - 24 T + 16)^{2} \) (T^4 - 4T^3 + 16T^2 - 24*T + 16)^2
$23$	\( T^{8} - 20 T^{6} + 400 T^{4} + \cdots + 160000 \) T^8 - 20T^6 + 400T^4 - 8000*T^2 + 160000
$29$	\( (T^{4} - 8 T^{3} + 34 T^{2} - 87 T + 841)^{2} \) (T^4 - 8T^3 + 34T^2 - 87*T + 841)^2
$31$	\( (T^{4} + 40 T^{2} + 200 T + 400)^{2} \) (T^4 + 40T^2 + 200T + 400)^2
$37$	\( T^{8} + 25 T^{6} + 3750 T^{4} + \cdots + 390625 \) T^8 + 25T^6 + 3750T^4 - 62500*T^2 + 390625
$41$	\( (T^{4} + 10 T^{2} - 25 T + 25)^{2} \) (T^4 + 10T^2 - 25T + 25)^2
$43$	\( (T^{4} + 92 T^{2} + 1936)^{2} \) (T^4 + 92*T^2 + 1936)^2
$47$	\( T^{8} - 44 T^{6} + 736 T^{4} + \cdots + 256 \) T^8 - 44T^6 + 736T^4 + 256*T^2 + 256
$53$	\( T^{8} + 5 T^{6} + 150 T^{4} + \cdots + 625 \) T^8 + 5T^6 + 150T^4 - 500*T^2 + 625
$59$	\( (T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256)^{2} \) (T^4 + 4T^3 + 16T^2 + 64*T + 256)^2
$61$	\( (T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1)^{2} \) (T^4 - 2T^3 + 4T^2 - 3*T + 1)^2
$67$	\( T^{8} - 44 T^{6} + 736 T^{4} + \cdots + 256 \) T^8 - 44T^6 + 736T^4 + 256*T^2 + 256
$71$	\( (T^{4} - 8 T^{3} + 24 T^{2} + 8 T + 16)^{2} \) (T^4 - 8T^3 + 24T^2 + 8*T + 16)^2
$73$	\( T^{8} - 100 T^{6} + 3750 T^{4} + \cdots + 390625 \) T^8 - 100T^6 + 3750T^4 + 15625*T^2 + 390625
$79$	\( T^{8} \) T^8
$83$	\( T^{8} + 76 T^{6} + 22416 T^{4} + \cdots + 3748096 \) T^8 + 76T^6 + 22416T^4 - 472384*T^2 + 3748096
$89$	\( (T^{4} - 9 T^{3} + 46 T^{2} - 164 T + 1681)^{2} \) (T^4 - 9T^3 + 46T^2 - 164*T + 1681)^2
$97$	\( T^{8} - 124 T^{6} + 5806 T^{4} + \cdots + 130321 \) T^8 - 124T^6 + 5806T^4 + 14801*T^2 + 130321
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\(n\)	\(127\)	\(251\)
\(\chi(n)\)	\(\zeta_{20}^{2}\)	\(1\)