LMFDB - Newform orbit 3479.1.ea.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3479,1,Mod(99,3479)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3479, base_ring=CyclotomicField(70))

chi = DirichletCharacter(H, H._module([0, 13]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3479.99");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3479 = 7^{2} \cdot 71 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3479.ea (of order $70$, degree $24$, 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:	$1.73624717895$
Analytic rank:	$0$
Dimension:	$24$
Coefficient field:	$\Q(\zeta_{35})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} - x^{23} + x^{19} - x^{18} + x^{17} - x^{16} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} + \cdots + 1 $ x^24 - x^23 + x^19 - x^18 + x^17 - x^16 + x^14 - x^13 + x^12 - x^11 + x^10 - x^8 + x^7 - x^6 + x^5 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{70}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{70} - \cdots)$

$q$-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_{70}^{27} + \zeta_{70}^{10}) q^{2} + (\zeta_{70}^{20} + \cdots + \zeta_{70}^{2}) q^{4}+ \cdots - \zeta_{70}^{32} q^{9} +O(q^{10}) $ q + (-z^27 + z^10) * q^2 + (z^20 - z^19 + z^2) * q^4 + (z^30 - z^29 + z^12 - z^11) * q^8 - z^32 * q^9	$ q + ( - \zeta_{70}^{27} + \zeta_{70}^{10}) q^{2} + (\zeta_{70}^{20} + \cdots + \zeta_{70}^{2}) q^{4}+ \cdots + ( - \zeta_{70}^{31} - \zeta_{70}^{20}) q^{99}+O(q^{100}) $ q + (-z^27 + z^10) * q^2 + (z^20 - z^19 + z^2) * q^4 + (z^30 - z^29 + z^12 - z^11) * q^8 - z^32 * q^9 + (-z^34 - z^23) * q^11 + (z^22 - z^21 - z^5 + z^4 - z^3) * q^16 + (-z^24 + z^7) * q^18 + (-z^33 - z^26 - z^15 + z^9) * q^22 + (-z^4 - z) * q^23 + z^21 * q^25 + (-z^28 - z^18) * q^29 + (z^32 - z^31 + z^30 - z^15 + z^14 - z^13) * q^32 + (-z^34 + z^17 - z^16) * q^36 + (z^17 + z^13) * q^37 + (z^14 + z^2) * q^43 + (-z^25 + z^19 - z^18 + z^8 - z^7 + z) * q^44 + (z^31 + z^28 - z^14 - z^11) * q^46 + (z^31 + z^13) * q^50 + (-z^22 - z^9) * q^53 + (-z^28 - z^20 - z^10 + z^3) * q^58 + (-z^25 + z^24 - z^23 + z^22 - z^7 + z^6 - z^5) * q^64 + (z^24 + z^15) * q^67 + z^8 * q^71 + (z^27 - z^26 + z^9 - z^8) * q^72 + (z^27 + z^23 + z^9 + z^5) * q^74 + (z^25 - z^16) * q^79 - z^29 * q^81 + (-z^29 + z^24 + z^12 + z^6) * q^86 + (z^34 + z^29 - z^28 + z^18 - z^17 + z^11 - z^10 + 1) * q^88 + (-z^24 + z^23 - z^21 + z^20 - z^6 - z^3) * q^92 + (-z^31 - z^20) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 3 q^{2} - 2 q^{4} - q^{8} - q^{9}+O(q^{10}) $ 24 * q - 3 * q^2 - 2 * q^4 - q^8 - q^9	$ 24 q - 3 q^{2} - 2 q^{4} - q^{8} - q^{9} - 7 q^{16} + 5 q^{18} - 5 q^{22} + 6 q^{25} + 5 q^{29} - 11 q^{32} - 3 q^{36} - 2 q^{37} - 5 q^{43} - 12 q^{44} - 2 q^{50} + 13 q^{58} - 10 q^{64} + 5 q^{67} + q^{71} - 4 q^{72} + q^{74} + 3 q^{79} + q^{81} + 4 q^{86} + 35 q^{88} - 12 q^{92} + 5 q^{99}+O(q^{100}) $ 24 * q - 3 * q^2 - 2 * q^4 - q^8 - q^9 - 7 * q^16 + 5 * q^18 - 5 * q^22 + 6 * q^25 + 5 * q^29 - 11 * q^32 - 3 * q^36 - 2 * q^37 - 5 * q^43 - 12 * q^44 - 2 * q^50 + 13 * q^58 - 10 * q^64 + 5 * q^67 + q^71 - 4 * q^72 + q^74 + 3 * q^79 + q^81 + 4 * q^86 + 35 * q^88 - 12 * q^92 + 5 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3479\mathbb{Z}\right)^\times$.

$n$	$640$	$1569$
$\chi(n)$	$1$	$-\zeta_{70}^{12}$

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} $

99.1

0.936235	+	0.351375i
0.983930	+	0.178557i
0.936235	−	0.351375i
0.473869	−	0.880596i
−0.691063	+	0.722795i
−0.550897	−	0.834573i
−0.995974	+	0.0896393i
0.983930	−	0.178557i
0.134233	−	0.990950i
0.753071	+	0.657939i
−0.963963	−	0.266037i
−0.393025	+	0.919528i
−0.963963	+	0.266037i
−0.550897	+	0.834573i
0.858449	+	0.512899i
−0.691063	−	0.722795i
0.858449	−	0.512899i
−0.0448648	+	0.998993i
0.753071	−	0.657939i
−0.393025	−	0.919528i

−1.86493

−

0.699921i

2.23501

1.95267i

−1.85751

−

3.45182i

−0.473869

0.880596i

197.1

−0.0882877

−

0.0160218i

−0.928697

−

0.348546i

0.153436

0.0916740i

−0.858449

0.512899i

246.1

−1.86493

0.699921i

2.23501

−

1.95267i

−1.85751

3.45182i

−0.473869

−

0.880596i

295.1

−0.913584

1.69772i

−1.49673

−

2.26745i

3.29673

−

0.296711i

0.995974

0.0896393i

491.1

0.761409

−

0.796371i

−0.00959883

−

0.213735i

0.652209

0.569818i

−0.753071

0.657939i

589.1

−0.945834

−

1.43288i

−0.765510

1.79100i

1.60102

−

0.290542i

−0.983930

−

0.178557i

834.1

1.37656

−

0.123893i

0.895642

−

0.162535i

−0.119548

0.0329933i

0.963963

0.266037i

883.1

−0.0882877

0.0160218i

−0.928697

0.348546i

0.153436

−

0.0916740i

−0.858449

−

0.512899i

1128.1

0.251348

−

1.85552i

−2.41583

−

0.666726i

−1.10841

2.59326i

0.393025

0.919528i

1275.1

1.48194

1.29473i

0.385581

2.84647i

−2.02992

3.07520i

0.550897

0.834573i

1569.1

−1.45187

−

0.400690i

1.08891

0.650596i

−0.279430

−

0.292261i

0.691063

−

0.722795i

1618.1

−0.372484

0.871471i

0.0703460

0.0735762i

−0.977627

0.366910i

−0.936235

−

0.351375i

1765.1

−1.45187

0.400690i

1.08891

−

0.650596i

−0.279430

0.292261i

0.691063

0.722795i

1961.1

−0.945834

1.43288i

−0.765510

−

1.79100i

1.60102

0.290542i

−0.983930

0.178557i

2010.1

0.230465

0.137696i

−0.439715

−

0.817127i

0.0232213

−

0.517062i

0.0448648

0.998993i

2402.1

0.761409

0.796371i

−0.00959883

0.213735i

0.652209

−

0.569818i

−0.753071

−

0.657939i

2598.1

0.230465

−

0.137696i

−0.439715

0.817127i

0.0232213

0.517062i

0.0448648

−

0.998993i

2696.1

0.0352660

−

0.785259i

0.380587

0.0342534i

0.145834

−

1.07659i

−0.134233

−

0.990950i

2745.1

1.48194

−

1.29473i

0.385581

−

2.84647i

−2.02992

−

3.07520i

0.550897

−

0.834573i

2892.1

−0.372484

−

0.871471i

0.0703460

−

0.0735762i

−0.977627

−

0.366910i

−0.936235

0.351375i

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}) $
71.h	odd	70	1	inner
497.z	even	70	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3479.1.ea.a	✓	24
7.b	odd	2	1	CM	3479.1.ea.a	✓	24
7.c	even	3	2		3479.1.ex.a		48
7.d	odd	6	2		3479.1.ex.a		48
71.h	odd	70	1	inner	3479.1.ea.a	✓	24
497.z	even	70	1	inner	3479.1.ea.a	✓	24
497.bd	odd	210	2		3479.1.ex.a		48
497.bf	even	210	2		3479.1.ex.a		48

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3479.1.ea.a	✓	24	1.a	even	1	1	trivial
3479.1.ea.a	✓	24	7.b	odd	2	1	CM
3479.1.ea.a	✓	24	71.h	odd	70	1	inner
3479.1.ea.a	✓	24	497.z	even	70	1	inner
3479.1.ex.a		48	7.c	even	3	2
3479.1.ex.a		48	7.d	odd	6	2
3479.1.ex.a		48	497.bd	odd	210	2
3479.1.ex.a		48	497.bf	even	210	2

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(3479, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{24} + 3 T^{23} + \cdots + 1 $ T^24 + 3T^23 + 5T^22 + 5T^21 + 7T^20 + 22T^19 + 50T^18 + 68T^17 + 90T^16 - 218T^15 - 634T^14 - 764T^13 + 96T^12 + 915T^11 - 244T^10 - 202T^9 + 1271T^8 + 523T^7 + 3136T^6 - 153T^5 + 1500T^4 - 460T^3 + 7T^2 + 16*T + 1
$3$	$ T^{24} $ T^24
$5$	$ T^{24} $ T^24
$7$	$ T^{24} $ T^24
$11$	$ T^{24} + 2 T^{21} + \cdots + 1 $ T^24 + 2T^21 - 5T^20 - 7T^19 + 32T^18 + 15T^17 - 38T^16 + 120T^15 - 186T^14 - 816T^13 + 1011T^12 + 1569T^11 - 2536T^10 - 93T^9 + 1947T^8 - 1367T^7 + 699T^6 - 422T^5 + 334T^4 - 85T^3 + 36T^2 - T + 1
$13$	$ T^{24} $ T^24
$17$	$ T^{24} $ T^24
$19$	$ T^{24} $ T^24
$23$	$ T^{24} - 5 T^{22} + \cdots + 1 $ T^24 - 5T^22 + 13T^20 - 14T^19 - 47T^18 + 70T^17 + 219T^16 - 84T^15 - 566T^14 - 259T^13 + 1441T^12 - 343T^11 - 940T^10 - 175T^9 + 2101T^8 - 2065T^7 + 1020T^6 + 126T^5 - 170T^4 + 91T^3 + 13T^2 - 7*T + 1
$29$	$ T^{24} - 5 T^{23} + \cdots + 1 $ T^24 - 5T^23 + 14T^22 - 28T^21 + 42T^20 - 67T^19 + 125T^18 - 219T^17 + 283T^16 - 126T^15 - 201T^14 + 375T^13 - 41T^12 - 803T^11 + 1495T^10 - 1442T^9 + 649T^8 + 157T^7 - 183T^6 + 47T^5 - 14T^4 + 14T^3 - 3T + 1
$31$	$ T^{24} $ T^24
$37$	$ T^{24} + 2 T^{23} + \cdots + 1 $ T^24 + 2T^23 + 5T^22 + 10T^21 + 27T^20 + 38T^19 + 85T^18 + 142T^17 + 315T^16 + 368T^15 + 976T^14 + 1849T^13 + 4071T^12 + 5585T^11 + 7136T^10 + 6387T^9 + 6041T^8 + 4737T^7 + 3326T^6 + 1418T^5 + 1980T^4 + 165T^3 + 117T^2 + 19*T + 1
$41$	$ T^{24} $ T^24
$43$	$ T^{24} + 5 T^{23} + \cdots + 1 $ T^24 + 5T^23 + 14T^22 + 33T^21 + 72T^20 + 132T^19 + 210T^18 + 324T^17 + 473T^16 + 636T^15 + 1009T^14 + 1080T^13 - 226T^12 - 2022T^11 - 2580T^10 - 188T^9 + 2344T^8 + 2438T^7 + 3277T^6 + 1623T^5 - 434T^4 - 544T^3 + 180T^2 - 17*T + 1
$47$	$ T^{24} $ T^24
$53$	$ T^{24} - 5 T^{21} + \cdots + 1 $ T^24 - 5T^21 - 12T^20 - 14T^19 + 18T^18 + 85T^17 + 179T^16 + 176T^15 - 221T^14 - 1124T^13 - 2069T^12 - 1833T^11 + 1580T^10 + 9105T^9 + 16766T^8 + 18415T^7 + 13495T^6 + 6886T^5 + 2490T^4 + 636T^3 + 113T^2 + 13*T + 1
$59$	$ T^{24} $ T^24
$61$	$ T^{24} $ T^24
$67$	$ T^{24} - 5 T^{23} + \cdots + 1 $ T^24 - 5T^23 + 15T^22 - 42T^21 + 105T^20 - 223T^19 + 452T^18 - 865T^17 + 1412T^16 - 2060T^15 + 2584T^14 - 2654T^13 + 2426T^12 - 1949T^11 + 1859T^10 - 2167T^9 + 1467T^8 + 177T^7 - 191T^6 + 187T^5 + 179T^4 - 15T^3 + 16T^2 - 9*T + 1
$71$	$ T^{24} - T^{23} + \cdots + 1 $ T^24 - T^23 + T^19 - T^18 + T^17 - T^16 + T^14 - T^13 + T^12 - T^11 + T^10 - T^8 + T^7 - T^6 + T^5 - T + 1
$73$	$ T^{24} $ T^24
$79$	$ T^{24} - 3 T^{23} + \cdots + 1 $ T^24 - 3T^23 + 5T^22 - 5T^21 + 7T^20 - 22T^19 + 15T^18 + 37T^17 - 85T^16 - 97T^15 + 591T^14 - 951T^13 + 1321T^12 + 65T^11 - 734T^10 + 202T^9 + 1271T^8 + 212T^7 + 1596T^6 + 2253T^5 - 145T^4 - 660T^3 + 182T^2 - 16*T + 1
$83$	$ T^{24} $ T^24
$89$	$ T^{24} $ T^24
$97$	$ T^{24} $ T^24
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Level:	\( N \)	\(=\)	\( 3479 = 7^{2} \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3479.ea (of order \(70\), degree \(24\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{70}^{27} + \zeta_{70}^{10}) q^{2} + (\zeta_{70}^{20} + \cdots + \zeta_{70}^{2}) q^{4}+ \cdots - \zeta_{70}^{32} q^{9} +O(q^{10}) \) q + (-z^27 + z^10) * q^2 + (z^20 - z^19 + z^2) * q^4 + (z^30 - z^29 + z^12 - z^11) * q^8 - z^32 * q^9	\( q + ( - \zeta_{70}^{27} + \zeta_{70}^{10}) q^{2} + (\zeta_{70}^{20} + \cdots + \zeta_{70}^{2}) q^{4}+ \cdots + ( - \zeta_{70}^{31} - \zeta_{70}^{20}) q^{99}+O(q^{100}) \) q + (-z^27 + z^10) * q^2 + (z^20 - z^19 + z^2) * q^4 + (z^30 - z^29 + z^12 - z^11) * q^8 - z^32 * q^9 + (-z^34 - z^23) * q^11 + (z^22 - z^21 - z^5 + z^4 - z^3) * q^16 + (-z^24 + z^7) * q^18 + (-z^33 - z^26 - z^15 + z^9) * q^22 + (-z^4 - z) * q^23 + z^21 * q^25 + (-z^28 - z^18) * q^29 + (z^32 - z^31 + z^30 - z^15 + z^14 - z^13) * q^32 + (-z^34 + z^17 - z^16) * q^36 + (z^17 + z^13) * q^37 + (z^14 + z^2) * q^43 + (-z^25 + z^19 - z^18 + z^8 - z^7 + z) * q^44 + (z^31 + z^28 - z^14 - z^11) * q^46 + (z^31 + z^13) * q^50 + (-z^22 - z^9) * q^53 + (-z^28 - z^20 - z^10 + z^3) * q^58 + (-z^25 + z^24 - z^23 + z^22 - z^7 + z^6 - z^5) * q^64 + (z^24 + z^15) * q^67 + z^8 * q^71 + (z^27 - z^26 + z^9 - z^8) * q^72 + (z^27 + z^23 + z^9 + z^5) * q^74 + (z^25 - z^16) * q^79 - z^29 * q^81 + (-z^29 + z^24 + z^12 + z^6) * q^86 + (z^34 + z^29 - z^28 + z^18 - z^17 + z^11 - z^10 + 1) * q^88 + (-z^24 + z^23 - z^21 + z^20 - z^6 - z^3) * q^92 + (-z^31 - z^20) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 3 q^{2} - 2 q^{4} - q^{8} - q^{9}+O(q^{10}) \) 24 * q - 3 * q^2 - 2 * q^4 - q^8 - q^9	\( 24 q - 3 q^{2} - 2 q^{4} - q^{8} - q^{9} - 7 q^{16} + 5 q^{18} - 5 q^{22} + 6 q^{25} + 5 q^{29} - 11 q^{32} - 3 q^{36} - 2 q^{37} - 5 q^{43} - 12 q^{44} - 2 q^{50} + 13 q^{58} - 10 q^{64} + 5 q^{67} + q^{71} - 4 q^{72} + q^{74} + 3 q^{79} + q^{81} + 4 q^{86} + 35 q^{88} - 12 q^{92} + 5 q^{99}+O(q^{100}) \) 24 * q - 3 * q^2 - 2 * q^4 - q^8 - q^9 - 7 * q^16 + 5 * q^18 - 5 * q^22 + 6 * q^25 + 5 * q^29 - 11 * q^32 - 3 * q^36 - 2 * q^37 - 5 * q^43 - 12 * q^44 - 2 * q^50 + 13 * q^58 - 10 * q^64 + 5 * q^67 + q^71 - 4 * q^72 + q^74 + 3 * q^79 + q^81 + 4 * q^86 + 35 * q^88 - 12 * q^92 + 5 * q^99

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