LMFDB - Newform orbit 3871.1.bx.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3871,1,Mod(116,3871)] 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("3871.116"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3871, base_ring=CyclotomicField(78)) chi = DirichletCharacter(H, H._module([52, 19])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 3871 = 7^{2} \cdot 79 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3871.bx (of order $78$, degree $24$, not minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$1.93188066390$
Analytic rank:	$0$
Dimension:	$24$
Coefficient field:	$\Q(\zeta_{39})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} - x^{23} + x^{21} - x^{20} + x^{18} - x^{17} + x^{15} - x^{14} + x^{12} - x^{10} + x^{9} + \cdots + 1 $ x^24 - x^23 + x^21 - x^20 + x^18 - x^17 + x^15 - x^14 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{78}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{78} - \cdots)$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

The $q$-expansion and trace form are shown below.

$f(q)$	$=$	$ q + ( - \zeta_{78}^{11} + \zeta_{78}^{4}) q^{2} + (\zeta_{78}^{22} + \cdots + \zeta_{78}^{8}) q^{4} + ( - \zeta_{78}^{33} + \cdots + \zeta_{78}^{12}) q^{8} - \zeta_{78}^{7} q^{9} + ( - \zeta_{78}^{34} - \zeta_{78}^{26}) q^{11} + \cdots + (\zeta_{78}^{33} - \zeta_{78}^{2}) q^{99} +O(q^{100}) $ q + (-z^11 + z^4) * q^2 + (z^22 - z^15 + z^8) * q^4 + (-z^33 + z^26 - z^19 + z^12) * q^8 - z^7 * q^9 + (-z^34 - z^26) * q^11 + (-z^37 + z^30 - z^23 + z^16 - z^5) * q^16 + (z^18 - z^11) * q^18 + (-z^38 + z^37 - z^30 - z^6) * q^22 + (-z^14 - z^12) * q^23 + z^5 * q^25 + (-z^14 - z^11) * q^29 + (z^34 - z^27 + z^20 + z^16 - z^9 + z^2) * q^32 + (-z^29 + z^22 - z^15) * q^36 + (-z^36 + z^6) * q^37 + (-z^28 + z^16) * q^43 + (-z^34 + z^17 - z^10 + z^9 + z^3 - z^2) * q^44 + (z^25 + z^23 - z^18 - z^16) * q^46 + (-z^16 + z^9) * q^50 + (z^6 - 1) * q^53 + (z^25 + z^22 - z^18 - z^15) * q^58 + (z^38 - z^31 - z^27 + z^24 + z^20 - z^13 + z^6) * q^64 + (z^35 + z^15) * q^67 + (z^30 + z^15) * q^71 + (-z^33 + z^26 - z^19 - z) * q^72 + (-z^17 + z^10 - z^8 + z) * q^74 + z^32 * q^79 + z^14 * q^81 + (-z^32 - z^27 + z^20 - 1) * q^86 + (-z^38 - z^28 + z^21 - z^20 - z^14 + z^13 + z^7 - z^6) * q^88 + (-z^36 - z^34 + z^29 + z^27 - z^22 - z^20) * q^92 + (z^33 - z^2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 2 q^{2} - 15 q^{8} + q^{9} + 11 q^{11} + 2 q^{16} - q^{18} + 2 q^{22} + q^{23} - q^{25} - q^{46} + q^{50} - 26 q^{53} - 15 q^{64} + q^{67} - 12 q^{72} + q^{79} + q^{81} - 26 q^{86} + 11 q^{88}+ \cdots + q^{99}+O(q^{100}) $ 24 * q + 2 * q^2 - 15 * q^8 + q^9 + 11 * q^11 + 2 * q^16 - q^18 + 2 * q^22 + q^23 - q^25 - q^46 + q^50 - 26 * q^53 - 15 * q^64 + q^67 - 12 * q^72 + q^79 + q^81 - 26 * q^86 + 11 * q^88 + q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3871\mathbb{Z}\right)^\times$.

$n$	$1030$	$2845$
$\chi(n)$	$-\zeta_{78}^{20}$	$\zeta_{78}^{26}$

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} $

116.1

−0.996757	+	0.0804666i
−0.0402659	−	0.999189i
−0.632445	−	0.774605i
0.948536	+	0.316668i
−0.919979	−	0.391967i
0.799443	+	0.600742i
−0.200026	+	0.979791i
0.692724	−	0.721202i
−0.919979	+	0.391967i
−0.632445	+	0.774605i
−0.845190	+	0.534466i
0.278217	+	0.960518i
0.987050	+	0.160411i
0.987050	−	0.160411i
−0.200026	−	0.979791i
0.799443	−	0.600742i
0.948536	−	0.316668i
−0.996757	−	0.0804666i
0.428693	+	0.903450i
0.692724	+	0.721202i

0.316091

0.457937i

0.244812

−

0.645517i

0.913255

−

0.225097i

−0.845190

0.534466i

226.1

1.41574

0.743039i

0.884156

1.28092i

0.107239

0.883189i

0.278217

0.960518i

508.1

0.0285570

−

0.0752986i

0.743656

0.658822i

0.142152

−

0.0746073i

−0.996757

0.0804666i

1255.1

−0.641762

0.568552i

−0.0319291

0.262959i

−0.616065

−

0.892525i

−0.632445

0.774605i

1292.1

0.237952

1.95971i

−2.81289

0.693315i

−1.32800

−

3.50165i

0.948536

−

0.316668i

1341.1

−0.152466

1.25567i

−0.582515

−

0.143577i

−0.179438

0.473138i

−0.200026

−

0.979791i

1451.1

1.49217

1.32194i

0.358488

2.95242i

−2.23556

3.23877i

0.987050

−

0.160411i

1488.1

−1.84195

−

0.453999i

2.30120

1.20776i

−2.27038

−

2.01139i

0.799443

0.600742i

1549.1

0.237952

−

1.95971i

−2.81289

−

0.693315i

−1.32800

3.50165i

0.948536

0.316668i

1745.1

0.0285570

0.0752986i

0.743656

−

0.658822i

0.142152

0.0746073i

−0.996757

−

0.0804666i

1880.1

−1.62920

−

0.855072i

1.35509

1.96319i

−0.307268

−

2.53058i

0.692724

−

0.721202i

1892.1

0.388427

0.0957386i

−0.743747

−

0.390348i

−0.550962

−

0.488110i

−0.919979

0.391967i

1978.1

0.599417

1.58053i

−1.39027

1.23167i

−1.28330

−

0.673526i

0.428693

0.903450i

2186.1

0.599417

−

1.58053i

−1.39027

−

1.23167i

−1.28330

0.673526i

0.428693

−

0.903450i

2321.1

1.49217

−

1.32194i

0.358488

−

2.95242i

−2.23556

−

3.23877i

0.987050

0.160411i

2725.1

−0.152466

−

1.25567i

−0.582515

0.143577i

−0.179438

−

0.473138i

−0.200026

0.979791i

2958.1

−0.641762

−

0.568552i

−0.0319291

−

0.262959i

−0.616065

0.892525i

−0.632445

−

0.774605i

2970.1

0.316091

−

0.457937i

0.244812

0.645517i

0.913255

0.225097i

−0.845190

−

0.534466i

3056.1

0.787025

−

1.14020i

−0.326048

−

0.859718i

0.108331

0.0267011i

−0.0402659

0.999189i

3166.1

−1.84195

0.453999i

2.30120

−

1.20776i

−2.27038

2.01139i

0.799443

−

0.600742i

See all 24 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by $\Q(\sqrt{-7}) $
553.bd	odd	78	1	inner
553.be	even	78	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3871.1.bx.a		24
7.b	odd	2	1	CM	3871.1.bx.a		24
7.c	even	3	1		3871.1.cc.a	✓	24
7.c	even	3	1		3871.1.cg.a		24
7.d	odd	6	1		3871.1.cc.a	✓	24
7.d	odd	6	1		3871.1.cg.a		24
79.h	odd	78	1		3871.1.cg.a		24
553.bd	odd	78	1	inner	3871.1.bx.a		24
553.be	even	78	1	inner	3871.1.bx.a		24
553.bf	even	78	1		3871.1.cc.a	✓	24
553.bl	even	78	1		3871.1.cg.a		24
553.bm	odd	78	1		3871.1.cc.a	✓	24

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3871.1.bx.a		24	1.a	even	1	1	trivial
3871.1.bx.a		24	7.b	odd	2	1	CM
3871.1.bx.a		24	553.bd	odd	78	1	inner
3871.1.bx.a		24	553.be	even	78	1	inner
3871.1.cc.a	✓	24	7.c	even	3	1
3871.1.cc.a	✓	24	7.d	odd	6	1
3871.1.cc.a	✓	24	553.bf	even	78	1
3871.1.cc.a	✓	24	553.bm	odd	78	1
3871.1.cg.a		24	7.c	even	3	1
3871.1.cg.a		24	7.d	odd	6	1
3871.1.cg.a		24	79.h	odd	78	1
3871.1.cg.a		24	553.bl	even	78	1

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(3871, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{24} - 2 T^{23} + \cdots + 1 $ T^24 - 2T^23 + 3T^22 + 9T^21 - 21T^20 + 33T^19 + 72T^18 - 177T^17 + 256T^16 + 211T^15 - 678T^14 + 1002T^13 + 520T^12 - 2040T^11 + 4912T^10 - 4742T^9 + 4754T^8 - 1724T^7 + 2243T^6 - 3152T^5 + 2540T^4 - 1109T^3 + 224T^2 - 15*T + 1
$3$	$ T^{24} $ T^24
$5$	$ T^{24} $ T^24
$7$	$ T^{24} $ T^24
$11$	$ T^{24} - 11 T^{23} + \cdots + 1 $ T^24 - 11T^23 + 68T^22 - 295T^21 + 993T^20 - 2724T^19 + 6286T^18 - 12446T^17 + 21446T^16 - 32464T^15 + 43470T^14 - 51702T^13 + 54769T^12 - 51702T^11 + 43457T^10 - 32334T^9 + 21433T^8 - 13317T^7 + 7131T^6 - 1879T^5 + 122T^4 - 308T^3 + 198T^2 - 24*T + 1
$13$	$ T^{24} $ T^24
$17$	$ T^{24} $ T^24
$19$	$ T^{24} $ T^24
$23$	$ T^{24} - T^{23} + \cdots + 1 $ T^24 - T^23 + 13T^22 - 12T^21 + 103T^20 - 91T^19 + 521T^18 - 430T^17 + 1924T^16 - 1494T^15 + 5069T^14 - 3575T^13 + 9855T^12 - 6279T^11 + 13155T^10 - 6889T^9 + 11817T^8 - 5058T^7 + 5747T^6 - 845T^5 + 1013T^4 + 14T^3 + 156T^2 + 12T + 1
$29$	$ T^{24} - 26 T^{21} + \cdots + 169 $ T^24 - 26T^21 + 273T^18 - 13T^16 - 1482T^15 + 143T^13 + 4420T^12 - 338T^10 - 7098T^9 + 169T^8 - 507T^7 + 5577T^6 + 676T^5 + 1521T^4 - 1690T^3 + 676T^2 - 338T + 169
$31$	$ T^{24} $ T^24
$37$	$ (T^{12} - 13 T^{9} + \cdots + 13)^{2} $ (T^12 - 13T^9 + 52T^6 + 13T^4 - 65T^3 + 26*T + 13)^2
$41$	$ T^{24} $ T^24
$43$	$ T^{24} + 26 T^{19} + \cdots + 169 $ T^24 + 26T^19 + 26T^18 + 26T^15 + 611T^14 + 286T^13 + 195T^12 + 338T^10 + 2028T^9 - 3549T^8 - 338T^7 + 507T^6 - 845T^5 + 3549T^4 - 3042T^3 + 1859T^2 - 676T + 169
$47$	$ T^{24} $ T^24
$53$	$ (T^{12} + 13 T^{11} + \cdots + 13)^{2} $ (T^12 + 13T^11 + 78T^10 + 286T^9 + 715T^8 + 1287T^7 + 1716T^6 + 1716T^5 + 1287T^4 + 715T^3 + 286T^2 + 78*T + 13)^2
$59$	$ T^{24} $ T^24
$61$	$ T^{24} $ T^24
$67$	$ T^{24} - T^{23} + \cdots + 1 $ T^24 - T^23 - 12T^21 + 12T^20 + 105T^18 - 105T^17 + 13T^16 - 454T^15 + 441T^14 - 351T^13 + 1756T^12 - 1404T^11 - 14T^10 - 1624T^9 + 1820T^8 + 3353T^7 - 1624T^6 - 1495T^5 + 1091T^4 - 454T^3 + 104T^2 + 12T + 1
$71$	$ (T^{12} + 13 T^{5} + \cdots + 13)^{2} $ (T^12 + 13T^5 + 91T^4 + 182T^3 + 156T^2 + 65*T + 13)^2
$73$	$ T^{24} $ T^24
$79$	$ T^{24} - T^{23} + \cdots + 1 $ T^24 - T^23 + T^21 - T^20 + T^18 - T^17 + T^15 - T^14 + T^12 - T^10 + T^9 - T^7 + T^6 - T^4 + T^3 - T + 1
$83$	$ T^{24} $ T^24
$89$	$ T^{24} $ T^24
$97$	$ T^{24} $ T^24
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 3871 = 7^{2} \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3871.bx (of order \(78\), degree \(24\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{78}^{11} + \zeta_{78}^{4}) q^{2} + (\zeta_{78}^{22} + \cdots + \zeta_{78}^{8}) q^{4} + ( - \zeta_{78}^{33} + \cdots + \zeta_{78}^{12}) q^{8} - \zeta_{78}^{7} q^{9} + ( - \zeta_{78}^{34} - \zeta_{78}^{26}) q^{11} + \cdots + (\zeta_{78}^{33} - \zeta_{78}^{2}) q^{99} +O(q^{100}) \) q + (-z^11 + z^4) * q^2 + (z^22 - z^15 + z^8) * q^4 + (-z^33 + z^26 - z^19 + z^12) * q^8 - z^7 * q^9 + (-z^34 - z^26) * q^11 + (-z^37 + z^30 - z^23 + z^16 - z^5) * q^16 + (z^18 - z^11) * q^18 + (-z^38 + z^37 - z^30 - z^6) * q^22 + (-z^14 - z^12) * q^23 + z^5 * q^25 + (-z^14 - z^11) * q^29 + (z^34 - z^27 + z^20 + z^16 - z^9 + z^2) * q^32 + (-z^29 + z^22 - z^15) * q^36 + (-z^36 + z^6) * q^37 + (-z^28 + z^16) * q^43 + (-z^34 + z^17 - z^10 + z^9 + z^3 - z^2) * q^44 + (z^25 + z^23 - z^18 - z^16) * q^46 + (-z^16 + z^9) * q^50 + (z^6 - 1) * q^53 + (z^25 + z^22 - z^18 - z^15) * q^58 + (z^38 - z^31 - z^27 + z^24 + z^20 - z^13 + z^6) * q^64 + (z^35 + z^15) * q^67 + (z^30 + z^15) * q^71 + (-z^33 + z^26 - z^19 - z) * q^72 + (-z^17 + z^10 - z^8 + z) * q^74 + z^32 * q^79 + z^14 * q^81 + (-z^32 - z^27 + z^20 - 1) * q^86 + (-z^38 - z^28 + z^21 - z^20 - z^14 + z^13 + z^7 - z^6) * q^88 + (-z^36 - z^34 + z^29 + z^27 - z^22 - z^20) * q^92 + (z^33 - z^2) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 2 q^{2} - 15 q^{8} + q^{9} + 11 q^{11} + 2 q^{16} - q^{18} + 2 q^{22} + q^{23} - q^{25} - q^{46} + q^{50} - 26 q^{53} - 15 q^{64} + q^{67} - 12 q^{72} + q^{79} + q^{81} - 26 q^{86} + 11 q^{88}+ \cdots + q^{99}+O(q^{100}) \) 24 * q + 2 * q^2 - 15 * q^8 + q^9 + 11 * q^11 + 2 * q^16 - q^18 + 2 * q^22 + q^23 - q^25 - q^46 + q^50 - 26 * q^53 - 15 * q^64 + q^67 - 12 * q^72 + q^79 + q^81 - 26 * q^86 + 11 * q^88 + q^99

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