LMFDB - Newform orbit 3380.1.cs.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,1,Mod(7,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, base_ring=CyclotomicField(156))

chi = DirichletCharacter(H, H._module([78, 39, 107]))

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

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

chi := DirichletCharacter("3380.7");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3380.cs (of order $156$, degree $48$, 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:	$1.68683974270$
Analytic rank:	$0$
Dimension:	$48$
Coefficient field:	$\Q(\zeta_{156})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{48} + x^{46} - x^{42} - x^{40} + x^{36} + x^{34} - x^{30} - x^{28} + x^{24} - x^{20} - x^{18} + \cdots + 1 $ x^48 + x^46 - x^42 - x^40 + x^36 + x^34 - x^30 - x^28 + x^24 - x^20 - x^18 + x^14 + x^12 - x^8 - x^6 + x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{156}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{156} - \cdots)$

$q$-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_{156}^{41} q^{2} - \zeta_{156}^{4} q^{4} - \zeta_{156}^{46} q^{5} + \zeta_{156}^{45} q^{8} + \zeta_{156}^{67} q^{9} +O(q^{10}) $ q - z^41 * q^2 - z^4 * q^4 - z^46 * q^5 + z^45 * q^8 + z^67 * q^9	\( q - \zeta_{156}^{41} q^{2} - \zeta_{156}^{4} q^{4} - \zeta_{156}^{46} q^{5} + \zeta_{156}^{45} q^{8} + \zeta_{156}^{67} q^{9} - \zeta_{156}^{9} q^{10} - \zeta_{156}^{20} q^{13} + \zeta_{156}^{8} q^{16} + (\zeta_{156}^{54} - \zeta_{156}^{23}) q^{17} + \zeta_{156}^{30} q^{18} + \zeta_{156}^{50} q^{20} - \zeta_{156}^{14} q^{25} + \zeta_{156}^{61} q^{26} + ( - \zeta_{156}^{3} + \zeta_{156}) q^{29} - \zeta_{156}^{49} q^{32} + (\zeta_{156}^{64} + \zeta_{156}^{17}) q^{34} - \zeta_{156}^{71} q^{36} + (\zeta_{156}^{73} + \zeta_{156}^{63}) q^{37} + \zeta_{156}^{13} q^{40} + (\zeta_{156}^{51} - \zeta_{156}^{16}) q^{41} + \zeta_{156}^{35} q^{45} - \zeta_{156}^{38} q^{49} + \zeta_{156}^{55} q^{50} + \zeta_{156}^{24} q^{52} + (\zeta_{156}^{65} - \zeta_{156}^{64}) q^{53} + (\zeta_{156}^{44} - \zeta_{156}^{42}) q^{58} + (\zeta_{156}^{71} + \zeta_{156}^{21}) q^{61} - \zeta_{156}^{12} q^{64} + \zeta_{156}^{66} q^{65} + ( - \zeta_{156}^{58} + \zeta_{156}^{27}) q^{68} - \zeta_{156}^{34} q^{72} - \zeta_{156}^{63} q^{73} + (\zeta_{156}^{36} + \zeta_{156}^{26}) q^{74} - \zeta_{156}^{54} q^{80} - \zeta_{156}^{56} q^{81} + (\zeta_{156}^{57} + \zeta_{156}^{14}) q^{82} + (\zeta_{156}^{69} + \zeta_{156}^{22}) q^{85} + (\zeta_{156}^{47} - \zeta_{156}^{44}) q^{89} - \zeta_{156}^{76} q^{90} + (\zeta_{156}^{76} + \zeta_{156}^{70}) q^{97} - \zeta_{156} q^{98} +O(q^{100}) \) q - z^41 * q^2 - z^4 * q^4 - z^46 * q^5 + z^45 * q^8 + z^67 * q^9 - z^9 * q^10 - z^20 * q^13 + z^8 * q^16 + (z^54 - z^23) * q^17 + z^30 * q^18 + z^50 * q^20 - z^14 * q^25 + z^61 * q^26 + (-z^3 + z) * q^29 - z^49 * q^32 + (z^64 + z^17) * q^34 - z^71 * q^36 + (z^73 + z^63) * q^37 + z^13 * q^40 + (z^51 - z^16) * q^41 + z^35 * q^45 - z^38 * q^49 + z^55 * q^50 + z^24 * q^52 + (z^65 - z^64) * q^53 + (z^44 - z^42) * q^58 + (z^71 + z^21) * q^61 - z^12 * q^64 + z^66 * q^65 + (-z^58 + z^27) * q^68 - z^34 * q^72 - z^63 * q^73 + (z^36 + z^26) * q^74 - z^54 * q^80 - z^56 * q^81 + (z^57 + z^14) * q^82 + (z^69 + z^22) * q^85 + (z^47 - z^44) * q^89 - z^76 * q^90 + (z^76 + z^70) * q^97 - z * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 2 q^{4} + 2 q^{5}+O(q^{10}) $ 48 * q - 2 * q^4 + 2 * q^5	$ 48 q - 2 q^{4} + 2 q^{5} - 2 q^{13} + 2 q^{16} + 4 q^{17} + 4 q^{18} - 2 q^{20} + 2 q^{25} + 2 q^{34} - 2 q^{41} + 2 q^{49} - 4 q^{52} - 2 q^{53} - 2 q^{58} + 4 q^{64} + 4 q^{65} + 2 q^{68} + 2 q^{72} + 20 q^{74} - 4 q^{80} - 2 q^{81} - 2 q^{82} - 2 q^{85} - 2 q^{89} - 2 q^{90}+O(q^{100}) $ 48 * q - 2 * q^4 + 2 * q^5 - 2 * q^13 + 2 * q^16 + 4 * q^17 + 4 * q^18 - 2 * q^20 + 2 * q^25 + 2 * q^34 - 2 * q^41 + 2 * q^49 - 4 * q^52 - 2 * q^53 - 2 * q^58 + 4 * q^64 + 4 * q^65 + 2 * q^68 + 2 * q^72 + 20 * q^74 - 4 * q^80 - 2 * q^81 - 2 * q^82 - 2 * q^85 - 2 * q^89 - 2 * q^90

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times$.

$n$	$677$	$1691$	$1861$
$\chi(n)$	$-\zeta_{156}^{39}$	$-1$	$-\zeta_{156}^{43}$

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

−0.774605	−	0.632445i
−0.600742	−	0.799443i
0.391967	−	0.919979i
0.534466	+	0.845190i
−0.979791	−	0.200026i
−0.160411	−	0.987050i
−0.774605	+	0.632445i
−0.960518	+	0.278217i
0.316668	−	0.948536i
−0.600742	+	0.799443i
−0.721202	−	0.692724i
−0.721202	+	0.692724i
0.721202	−	0.692724i
0.721202	+	0.692724i
0.600742	−	0.799443i
−0.316668	+	0.948536i
0.960518	−	0.278217i
0.774605	−	0.632445i
0.160411	+	0.987050i
0.979791	+	0.200026i

−0.979791

0.200026i

0.919979

−

0.391967i

−0.996757

−

0.0804666i

−0.822984

0.568065i

0.316668

−

0.948536i

0.992709

−

0.120537i

123.1

0.960518

0.278217i

0.845190

0.534466i

−0.200026

0.979791i

0.663123

0.748511i

−0.721202

0.692724i

−0.464723

0.885456i

167.1

0.721202

−

0.692724i

0.0402659

−

0.999189i

0.948536

−

0.316668i

−0.663123

−

0.748511i

−0.960518

−

0.278217i

0.464723

−

0.885456i

223.1

0.903450

0.428693i

0.632445

0.774605i

0.692724

−

0.721202i

0.239316

0.970942i

−0.0804666

−

0.996757i

0.935016

−

0.354605i

267.1

−0.391967

0.919979i

−0.692724

−

0.721202i

0.987050

−

0.160411i

0.935016

−

0.354605i

−0.600742

−

0.799443i

−0.239316

0.970942i

383.1

0.316668

0.948536i

−0.799443

0.600742i

0.428693

−

0.903450i

−0.822984

−

0.568065i

−0.979791

−

0.200026i

0.992709

0.120537i

483.1

−0.979791

−

0.200026i

0.919979

0.391967i

−0.996757

0.0804666i

−0.822984

−

0.568065i

0.316668

0.948536i

0.992709

0.120537i

527.1

0.534466

0.845190i

−0.428693

0.903450i

−0.919979

0.391967i

−0.992709

0.120537i

−0.999189

0.0402659i

−0.822984

−

0.568065i

643.1

−0.600742

0.799443i

−0.278217

−

0.960518i

−0.632445

0.774605i

0.935016

0.354605i

−0.391967

−

0.919979i

−0.239316

−

0.970942i

687.1

0.960518

−

0.278217i

0.845190

−

0.534466i

−0.200026

−

0.979791i

0.663123

−

0.748511i

−0.721202

−

0.692724i

−0.464723

−

0.885456i

743.1

0.999189

−

0.0402659i

0.996757

−

0.0804666i

0.799443

0.600742i

0.992709

−

0.120537i

−0.534466

−

0.845190i

0.822984

0.568065i

787.1

0.999189

0.0402659i

0.996757

0.0804666i

0.799443

−

0.600742i

0.992709

0.120537i

−0.534466

0.845190i

0.822984

−

0.568065i

903.1

−0.999189

−

0.0402659i

0.996757

0.0804666i

0.799443

−

0.600742i

−0.992709

−

0.120537i

0.534466

−

0.845190i

−0.822984

0.568065i

947.1

−0.999189

0.0402659i

0.996757

−

0.0804666i

0.799443

0.600742i

−0.992709

0.120537i

0.534466

0.845190i

−0.822984

−

0.568065i

1003.1

−0.960518

0.278217i

0.845190

−

0.534466i

−0.200026

−

0.979791i

−0.663123

0.748511i

0.721202

0.692724i

0.464723

0.885456i

1047.1

0.600742

−

0.799443i

−0.278217

−

0.960518i

−0.632445

0.774605i

−0.935016

−

0.354605i

0.391967

0.919979i

0.239316

0.970942i

1163.1

−0.534466

−

0.845190i

−0.428693

0.903450i

−0.919979

0.391967i

0.992709

−

0.120537i

0.999189

−

0.0402659i

0.822984

0.568065i

1207.1

0.979791

0.200026i

0.919979

0.391967i

−0.996757

0.0804666i

0.822984

0.568065i

−0.316668

−

0.948536i

−0.992709

−

0.120537i

1307.1

−0.316668

−

0.948536i

−0.799443

0.600742i

0.428693

−

0.903450i

0.822984

0.568065i

0.979791

0.200026i

−0.992709

−

0.120537i

1423.1

0.391967

−

0.919979i

−0.692724

−

0.721202i

0.987050

−

0.160411i

−0.935016

0.354605i

0.600742

0.799443i

0.239316

−

0.970942i

See all 48 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
845.bi	even	156	1	inner
3380.cs	odd	156	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3380.1.cs.a	✓	48
4.b	odd	2	1	CM	3380.1.cs.a	✓	48
5.c	odd	4	1		3380.1.cz.a	yes	48
20.e	even	4	1		3380.1.cz.a	yes	48
169.l	odd	156	1		3380.1.cz.a	yes	48
676.w	even	156	1		3380.1.cz.a	yes	48
845.bi	even	156	1	inner	3380.1.cs.a	✓	48
3380.cs	odd	156	1	inner	3380.1.cs.a	✓	48

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3380.1.cs.a	✓	48	1.a	even	1	1	trivial
3380.1.cs.a	✓	48	4.b	odd	2	1	CM
3380.1.cs.a	✓	48	845.bi	even	156	1	inner
3380.1.cs.a	✓	48	3380.cs	odd	156	1	inner
3380.1.cz.a	yes	48	5.c	odd	4	1
3380.1.cz.a	yes	48	20.e	even	4	1
3380.1.cz.a	yes	48	169.l	odd	156	1
3380.1.cz.a	yes	48	676.w	even	156	1

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(3380, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{48} + T^{46} + \cdots + 1 $ T^48 + T^46 - T^42 - T^40 + T^36 + T^34 - T^30 - T^28 + T^24 - T^20 - T^18 + T^14 + T^12 - T^8 - T^6 + T^2 + 1
$3$	$ T^{48} $ T^48
$5$	$ (T^{24} - T^{23} + T^{21} + \cdots + 1)^{2} $ (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)^2
$7$	$ T^{48} $ T^48
$11$	$ T^{48} $ T^48
$13$	$ (T^{24} + T^{23} - T^{21} + \cdots + 1)^{2} $ (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)^2
$17$	$ T^{48} - 4 T^{47} + \cdots + 1 $ T^48 - 4T^47 + 11T^46 - 26T^45 + 56T^44 - 60T^43 + T^42 + 210T^41 - 781T^40 + 2132T^39 - 3146T^38 + 3276T^37 - 1000T^36 - 8216T^35 + 34458T^34 - 63552T^33 + 99350T^32 - 128848T^31 + 111275T^30 + 63940T^29 - 200362T^28 + 388046T^27 - 454597T^26 + 201032T^25 + 912159T^24 - 1434992T^23 + 1231711T^22 - 988288T^21 - 349992T^20 + 5037170T^19 - 6155940T^18 + 1129614T^17 + 207505T^16 + 1435440T^15 + 1222437T^14 - 3173846T^13 + 3401880T^12 - 3582852T^11 + 3603327T^10 - 1921712T^9 + 1571218T^8 - 423822T^7 + 126453T^6 + 25844T^5 - 4684T^4 - 96T^3 + 636T^2 + 50T + 1
$19$	$ T^{48} $ T^48
$23$	$ T^{48} $ T^48
$29$	$ T^{48} + T^{46} + \cdots + 1 $ T^48 + T^46 - T^42 - T^40 - 117T^38 + 1600T^36 - 3860T^34 + 1638T^32 + 1052T^30 + 14039T^28 - 389610T^26 + 3845908T^24 - 16565679T^22 + 36082916T^20 - 42057991T^18 + 27392157T^16 - 12308165T^14 + 3665299T^12 + 1101321T^10 + 1117349T^8 + 218828T^6 + 26442T^4 + 313*T^2 + 1
$31$	$ T^{48} $ T^48
$37$	$ T^{48} - 3 T^{46} + \cdots + 1 $ T^48 - 3T^46 + 27T^42 - 81T^40 - 117T^38 - 636T^36 - 2616T^34 + 6474T^32 - 272T^30 - 45789T^28 + 541866T^26 + 3036008T^24 + 6816381T^22 + 8727372T^20 + 7138509T^18 + 4169841T^16 + 1924719T^14 + 751531T^12 + 30225T^10 + 110205T^8 - 26012T^6 + 2574T^4 - 87T^2 + 1
$41$	$ T^{48} + 2 T^{47} + \cdots + 1 $ T^48 + 2T^47 - T^46 - 8T^45 - 4T^44 + 26T^43 + 41T^42 + 22T^41 - 144T^40 - 260T^39 + 247T^38 + 1196T^37 + 261T^36 - 4212T^35 - 291T^34 + 2074T^33 - 4408T^32 - 16164T^31 - 1701T^30 + 57712T^29 + 64019T^28 - 61254T^27 + 2379T^26 + 230490T^25 + 34919T^24 - 1014260T^23 - 1283529T^22 + 2425258T^21 + 8498458T^20 + 10596372T^19 + 8684944T^18 + 5728520T^17 + 3505246T^16 + 2574918T^15 + 1748040T^14 - 167024T^13 - 155440T^12 - 79326T^11 - 42302T^10 + 68198T^9 + 32252T^8 - 548888T^7 + 585170T^6 - 273798T^5 + 70945T^4 - 10452T^3 + 830T^2 - 22T + 1
$43$	$ T^{48} $ T^48
$47$	$ T^{48} $ T^48
$53$	$ T^{48} + 2 T^{47} + \cdots + 1 $ T^48 + 2T^47 - 11T^46 - 28T^45 + 52T^44 + 180T^43 - 119T^42 - 674T^41 + 129T^40 + 1664T^39 - 468T^38 - 3848T^37 + 2510T^36 + 11986T^35 - 626T^34 - 26188T^33 - 24982T^32 - 3590T^31 + 7984T^30 + 59234T^29 + 264506T^28 + 242296T^27 - 341978T^26 - 596778T^25 - 1057055T^24 - 515814T^23 + 2405078T^22 + 1388190T^21 + 256849T^20 + 863220T^19 - 4118190T^18 - 1805230T^17 + 2981541T^16 - 203288T^15 + 2774835T^14 + 1891188T^13 - 181245T^12 + 91702T^11 + 1967901T^10 - 863928T^9 + 1547298T^8 - 487918T^7 + 312284T^6 - 41962T^5 + 14846T^4 - 440T^3 + 600T^2 + 24*T + 1
$59$	$ T^{48} $ T^48
$61$	$ T^{48} - 3 T^{46} + \cdots + 1 $ T^48 - 3T^46 + 52T^44 - 129T^42 + 1089T^40 - 2080T^38 + 11363T^36 - 15876T^34 + 64168T^32 - 69900T^30 + 245385T^28 - 267943T^26 + 588316T^24 + 435630T^22 - 604938T^20 - 348178T^18 + 2489253T^16 + 581273T^14 - 897779T^12 - 135876T^10 + 178754T^8 - 21826T^6 + 56T^2 + 1
$67$	$ T^{48} $ T^48
$71$	$ T^{48} $ T^48
$73$	$ (T^{24} - T^{22} + T^{20} + \cdots + 1)^{2} $ (T^24 - T^22 + T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1)^2
$79$	$ T^{48} $ T^48
$83$	$ T^{48} $ T^48
$89$	$ T^{48} + 2 T^{47} + \cdots + 1 $ T^48 + 2T^47 + 2T^46 + 4T^45 - 31T^44 - 70T^43 - 70T^42 - 140T^41 + 692T^40 + 1664T^39 + 1664T^38 + 3328T^37 - 7539T^36 - 21736T^35 - 21738T^34 - 43524T^33 + 59683T^32 + 206692T^31 + 206874T^30 + 413894T^29 - 108085T^28 - 1038498T^27 - 1032746T^26 - 2038036T^25 - 196975T^24 + 3546348T^23 + 3465488T^22 + 6886494T^21 + 6223247T^20 - 1428206T^19 - 1618024T^18 - 4000656T^17 - 4924985T^16 + 429338T^15 + 1204534T^14 + 2501590T^13 + 2261130T^12 - 562406T^11 - 325052T^10 - 434408T^9 - 69697T^8 + 75080T^7 + 25186T^6 + 13972T^5 + 6923T^4 + 1008T^3 + 98T^2 + 14*T + 1
$97$	$ (T^{24} - 13 T^{20} + \cdots + 169)^{2} $ (T^24 - 13T^20 + 39T^17 + 130T^16 - 13T^14 - 1053T^13 - 481T^12 + 1859T^10 + 1014T^9 + 1352T^8 - 1014T^7 + 1014T^6 + 3549T^5 - 338T^4 - 507T^3 + 1352T^2 - 507T + 169)^2
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3380.cs (of order \(156\), degree \(48\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{156}^{41} q^{2} - \zeta_{156}^{4} q^{4} - \zeta_{156}^{46} q^{5} + \zeta_{156}^{45} q^{8} + \zeta_{156}^{67} q^{9} +O(q^{10}) \) q - z^41 * q^2 - z^4 * q^4 - z^46 * q^5 + z^45 * q^8 + z^67 * q^9	\( q - \zeta_{156}^{41} q^{2} - \zeta_{156}^{4} q^{4} - \zeta_{156}^{46} q^{5} + \zeta_{156}^{45} q^{8} + \zeta_{156}^{67} q^{9} - \zeta_{156}^{9} q^{10} - \zeta_{156}^{20} q^{13} + \zeta_{156}^{8} q^{16} + (\zeta_{156}^{54} - \zeta_{156}^{23}) q^{17} + \zeta_{156}^{30} q^{18} + \zeta_{156}^{50} q^{20} - \zeta_{156}^{14} q^{25} + \zeta_{156}^{61} q^{26} + ( - \zeta_{156}^{3} + \zeta_{156}) q^{29} - \zeta_{156}^{49} q^{32} + (\zeta_{156}^{64} + \zeta_{156}^{17}) q^{34} - \zeta_{156}^{71} q^{36} + (\zeta_{156}^{73} + \zeta_{156}^{63}) q^{37} + \zeta_{156}^{13} q^{40} + (\zeta_{156}^{51} - \zeta_{156}^{16}) q^{41} + \zeta_{156}^{35} q^{45} - \zeta_{156}^{38} q^{49} + \zeta_{156}^{55} q^{50} + \zeta_{156}^{24} q^{52} + (\zeta_{156}^{65} - \zeta_{156}^{64}) q^{53} + (\zeta_{156}^{44} - \zeta_{156}^{42}) q^{58} + (\zeta_{156}^{71} + \zeta_{156}^{21}) q^{61} - \zeta_{156}^{12} q^{64} + \zeta_{156}^{66} q^{65} + ( - \zeta_{156}^{58} + \zeta_{156}^{27}) q^{68} - \zeta_{156}^{34} q^{72} - \zeta_{156}^{63} q^{73} + (\zeta_{156}^{36} + \zeta_{156}^{26}) q^{74} - \zeta_{156}^{54} q^{80} - \zeta_{156}^{56} q^{81} + (\zeta_{156}^{57} + \zeta_{156}^{14}) q^{82} + (\zeta_{156}^{69} + \zeta_{156}^{22}) q^{85} + (\zeta_{156}^{47} - \zeta_{156}^{44}) q^{89} - \zeta_{156}^{76} q^{90} + (\zeta_{156}^{76} + \zeta_{156}^{70}) q^{97} - \zeta_{156} q^{98} +O(q^{100}) \) q - z^41 * q^2 - z^4 * q^4 - z^46 * q^5 + z^45 * q^8 + z^67 * q^9 - z^9 * q^10 - z^20 * q^13 + z^8 * q^16 + (z^54 - z^23) * q^17 + z^30 * q^18 + z^50 * q^20 - z^14 * q^25 + z^61 * q^26 + (-z^3 + z) * q^29 - z^49 * q^32 + (z^64 + z^17) * q^34 - z^71 * q^36 + (z^73 + z^63) * q^37 + z^13 * q^40 + (z^51 - z^16) * q^41 + z^35 * q^45 - z^38 * q^49 + z^55 * q^50 + z^24 * q^52 + (z^65 - z^64) * q^53 + (z^44 - z^42) * q^58 + (z^71 + z^21) * q^61 - z^12 * q^64 + z^66 * q^65 + (-z^58 + z^27) * q^68 - z^34 * q^72 - z^63 * q^73 + (z^36 + z^26) * q^74 - z^54 * q^80 - z^56 * q^81 + (z^57 + z^14) * q^82 + (z^69 + z^22) * q^85 + (z^47 - z^44) * q^89 - z^76 * q^90 + (z^76 + z^70) * q^97 - z * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 2 q^{4} + 2 q^{5}+O(q^{10}) \) 48 * q - 2 * q^4 + 2 * q^5	\( 48 q - 2 q^{4} + 2 q^{5} - 2 q^{13} + 2 q^{16} + 4 q^{17} + 4 q^{18} - 2 q^{20} + 2 q^{25} + 2 q^{34} - 2 q^{41} + 2 q^{49} - 4 q^{52} - 2 q^{53} - 2 q^{58} + 4 q^{64} + 4 q^{65} + 2 q^{68} + 2 q^{72} + 20 q^{74} - 4 q^{80} - 2 q^{81} - 2 q^{82} - 2 q^{85} - 2 q^{89} - 2 q^{90}+O(q^{100}) \) 48 * q - 2 * q^4 + 2 * q^5 - 2 * q^13 + 2 * q^16 + 4 * q^17 + 4 * q^18 - 2 * q^20 + 2 * q^25 + 2 * q^34 - 2 * q^41 + 2 * q^49 - 4 * q^52 - 2 * q^53 - 2 * q^58 + 4 * q^64 + 4 * q^65 + 2 * q^68 + 2 * q^72 + 20 * q^74 - 4 * q^80 - 2 * q^81 - 2 * q^82 - 2 * q^85 - 2 * q^89 - 2 * q^90

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

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	3380.1.cs.a

Level	$3380$
Weight	$1$
Character orbit	3380.cs
Analytic conductor	$1.687$
Analytic rank	$0$
Dimension	$48$
Projective image	$D_{156}$
CM discriminant	-4
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.68683974270\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Coefficient field:	\(\Q(\zeta_{156})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{48} + x^{46} - x^{42} - x^{40} + x^{36} + x^{34} - x^{30} - x^{28} + x^{24} - x^{20} - x^{18} + \cdots + 1 \) x^48 + x^46 - x^42 - x^40 + x^36 + x^34 - x^30 - x^28 + x^24 - x^20 - x^18 + x^14 + x^12 - x^8 - x^6 + x^2 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{156}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{156} - \cdots)\)

\(n\)	\(677\)	\(1691\)	\(1861\)
\(\chi(n)\)	\(-\zeta_{156}^{39}\)	\(-1\)	\(-\zeta_{156}^{43}\)

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

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
845.bi	even	156	1	inner
3380.cs	odd	156	1	inner

$p$	$F_p(T)$
$2$	\( T^{48} + T^{46} + \cdots + 1 \) T^48 + T^46 - T^42 - T^40 + T^36 + T^34 - T^30 - T^28 + T^24 - T^20 - T^18 + T^14 + T^12 - T^8 - T^6 + T^2 + 1
$3$	\( T^{48} \) T^48
$5$	\( (T^{24} - T^{23} + T^{21} + \cdots + 1)^{2} \) (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)^2
$7$	\( T^{48} \) T^48
$11$	\( T^{48} \) T^48
$13$	\( (T^{24} + T^{23} - T^{21} + \cdots + 1)^{2} \) (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)^2
$17$	\( T^{48} - 4 T^{47} + \cdots + 1 \) T^48 - 4T^47 + 11T^46 - 26T^45 + 56T^44 - 60T^43 + T^42 + 210T^41 - 781T^40 + 2132T^39 - 3146T^38 + 3276T^37 - 1000T^36 - 8216T^35 + 34458T^34 - 63552T^33 + 99350T^32 - 128848T^31 + 111275T^30 + 63940T^29 - 200362T^28 + 388046T^27 - 454597T^26 + 201032T^25 + 912159T^24 - 1434992T^23 + 1231711T^22 - 988288T^21 - 349992T^20 + 5037170T^19 - 6155940T^18 + 1129614T^17 + 207505T^16 + 1435440T^15 + 1222437T^14 - 3173846T^13 + 3401880T^12 - 3582852T^11 + 3603327T^10 - 1921712T^9 + 1571218T^8 - 423822T^7 + 126453T^6 + 25844T^5 - 4684T^4 - 96T^3 + 636T^2 + 50T + 1
$19$	\( T^{48} \) T^48
$23$	\( T^{48} \) T^48
$29$	\( T^{48} + T^{46} + \cdots + 1 \) T^48 + T^46 - T^42 - T^40 - 117T^38 + 1600T^36 - 3860T^34 + 1638T^32 + 1052T^30 + 14039T^28 - 389610T^26 + 3845908T^24 - 16565679T^22 + 36082916T^20 - 42057991T^18 + 27392157T^16 - 12308165T^14 + 3665299T^12 + 1101321T^10 + 1117349T^8 + 218828T^6 + 26442T^4 + 313*T^2 + 1
$31$	\( T^{48} \) T^48
$37$	\( T^{48} - 3 T^{46} + \cdots + 1 \) T^48 - 3T^46 + 27T^42 - 81T^40 - 117T^38 - 636T^36 - 2616T^34 + 6474T^32 - 272T^30 - 45789T^28 + 541866T^26 + 3036008T^24 + 6816381T^22 + 8727372T^20 + 7138509T^18 + 4169841T^16 + 1924719T^14 + 751531T^12 + 30225T^10 + 110205T^8 - 26012T^6 + 2574T^4 - 87T^2 + 1
$41$	\( T^{48} + 2 T^{47} + \cdots + 1 \) T^48 + 2T^47 - T^46 - 8T^45 - 4T^44 + 26T^43 + 41T^42 + 22T^41 - 144T^40 - 260T^39 + 247T^38 + 1196T^37 + 261T^36 - 4212T^35 - 291T^34 + 2074T^33 - 4408T^32 - 16164T^31 - 1701T^30 + 57712T^29 + 64019T^28 - 61254T^27 + 2379T^26 + 230490T^25 + 34919T^24 - 1014260T^23 - 1283529T^22 + 2425258T^21 + 8498458T^20 + 10596372T^19 + 8684944T^18 + 5728520T^17 + 3505246T^16 + 2574918T^15 + 1748040T^14 - 167024T^13 - 155440T^12 - 79326T^11 - 42302T^10 + 68198T^9 + 32252T^8 - 548888T^7 + 585170T^6 - 273798T^5 + 70945T^4 - 10452T^3 + 830T^2 - 22T + 1
$43$	\( T^{48} \) T^48
$47$	\( T^{48} \) T^48
$53$	\( T^{48} + 2 T^{47} + \cdots + 1 \) T^48 + 2T^47 - 11T^46 - 28T^45 + 52T^44 + 180T^43 - 119T^42 - 674T^41 + 129T^40 + 1664T^39 - 468T^38 - 3848T^37 + 2510T^36 + 11986T^35 - 626T^34 - 26188T^33 - 24982T^32 - 3590T^31 + 7984T^30 + 59234T^29 + 264506T^28 + 242296T^27 - 341978T^26 - 596778T^25 - 1057055T^24 - 515814T^23 + 2405078T^22 + 1388190T^21 + 256849T^20 + 863220T^19 - 4118190T^18 - 1805230T^17 + 2981541T^16 - 203288T^15 + 2774835T^14 + 1891188T^13 - 181245T^12 + 91702T^11 + 1967901T^10 - 863928T^9 + 1547298T^8 - 487918T^7 + 312284T^6 - 41962T^5 + 14846T^4 - 440T^3 + 600T^2 + 24*T + 1
$59$	\( T^{48} \) T^48
$61$	\( T^{48} - 3 T^{46} + \cdots + 1 \) T^48 - 3T^46 + 52T^44 - 129T^42 + 1089T^40 - 2080T^38 + 11363T^36 - 15876T^34 + 64168T^32 - 69900T^30 + 245385T^28 - 267943T^26 + 588316T^24 + 435630T^22 - 604938T^20 - 348178T^18 + 2489253T^16 + 581273T^14 - 897779T^12 - 135876T^10 + 178754T^8 - 21826T^6 + 56T^2 + 1
$67$	\( T^{48} \) T^48
$71$	\( T^{48} \) T^48
$73$	\( (T^{24} - T^{22} + T^{20} + \cdots + 1)^{2} \) (T^24 - T^22 + T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1)^2
$79$	\( T^{48} \) T^48
$83$	\( T^{48} \) T^48
$89$	\( T^{48} + 2 T^{47} + \cdots + 1 \) T^48 + 2T^47 + 2T^46 + 4T^45 - 31T^44 - 70T^43 - 70T^42 - 140T^41 + 692T^40 + 1664T^39 + 1664T^38 + 3328T^37 - 7539T^36 - 21736T^35 - 21738T^34 - 43524T^33 + 59683T^32 + 206692T^31 + 206874T^30 + 413894T^29 - 108085T^28 - 1038498T^27 - 1032746T^26 - 2038036T^25 - 196975T^24 + 3546348T^23 + 3465488T^22 + 6886494T^21 + 6223247T^20 - 1428206T^19 - 1618024T^18 - 4000656T^17 - 4924985T^16 + 429338T^15 + 1204534T^14 + 2501590T^13 + 2261130T^12 - 562406T^11 - 325052T^10 - 434408T^9 - 69697T^8 + 75080T^7 + 25186T^6 + 13972T^5 + 6923T^4 + 1008T^3 + 98T^2 + 14*T + 1
$97$	\( (T^{24} - 13 T^{20} + \cdots + 169)^{2} \) (T^24 - 13T^20 + 39T^17 + 130T^16 - 13T^14 - 1053T^13 - 481T^12 + 1859T^10 + 1014T^9 + 1352T^8 - 1014T^7 + 1014T^6 + 3549T^5 - 338T^4 - 507T^3 + 1352T^2 - 507T + 169)^2
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