LMFDB - Newform orbit 690.2.r.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [690,2,Mod(49,690)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(690, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 11, 16]))

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

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

chi := DirichletCharacter("690.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 690 = 2 \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	690.r (of order $22$, degree $10$, 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:	$5.50967773947$
Analytic rank:	$0$
Dimension:	$120$
Relative dimension:	$12$ over $\Q(\zeta_{22})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $q$-expansion, but we have computed the trace expansion.

$\operatorname{Tr}(f)(q) = $ $ 120 q + 12 q^{4} + 12 q^{6} + 12 q^{9}+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 120 q + 12 q^{4} + 12 q^{6} + 12 q^{9} + 18 q^{10} - 8 q^{14} - 4 q^{15} - 12 q^{16} + 22 q^{20} + 14 q^{21} + 120 q^{24} + 52 q^{25} + 16 q^{29} + 8 q^{31} - 36 q^{34} - 90 q^{35} - 12 q^{36} + 22 q^{39} + 4 q^{40} + 16 q^{41} - 4 q^{49} - 4 q^{50} + 8 q^{51} - 12 q^{54} - 56 q^{55} + 8 q^{56} + 138 q^{59} + 4 q^{60} - 36 q^{61} + 12 q^{64} + 52 q^{65} + 96 q^{70} + 8 q^{71} + 8 q^{74} - 4 q^{75} - 60 q^{79} - 12 q^{81} + 8 q^{84} + 24 q^{85} - 8 q^{86} - 104 q^{89} + 4 q^{90} - 144 q^{91} - 24 q^{94} - 14 q^{95} + 12 q^{96} - 44 q^{99}+O(q^{100}) $

Display 100 coefficients 10 coefficients

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	$ a_{2} $			$ a_{3} $			$ a_{4} $			$ a_{5} $			$ a_{6} $			$ a_{7} $			$ a_{8} $			$ a_{9} $			$ a_{10} $
49.1	−0.989821	+	0.142315i	−0.909632	−	0.415415i	0.959493	−	0.281733i	−2.05037	−	0.892171i	0.959493	+	0.281733i	−0.0262635	−	0.0408667i	−0.909632	+	0.415415i	0.654861	+	0.755750i	2.15647	+	0.591291i
49.2	−0.989821	+	0.142315i	−0.909632	−	0.415415i	0.959493	−	0.281733i	−1.41393	+	1.73228i	0.959493	+	0.281733i	−0.692568	−	1.07766i	−0.909632	+	0.415415i	0.654861	+	0.755750i	1.15301	−	1.91587i
49.3	−0.989821	+	0.142315i	−0.909632	−	0.415415i	0.959493	−	0.281733i	−0.967142	−	2.01609i	0.959493	+	0.281733i	0.684801	+	1.06557i	−0.909632	+	0.415415i	0.654861	+	0.755750i	1.24422	+	1.85793i
49.4	−0.989821	+	0.142315i	−0.909632	−	0.415415i	0.959493	−	0.281733i	1.14908	−	1.91823i	0.959493	+	0.281733i	−1.79620	−	2.79494i	−0.909632	+	0.415415i	0.654861	+	0.755750i	−0.864395	+	2.06224i
49.5	−0.989821	+	0.142315i	−0.909632	−	0.415415i	0.959493	−	0.281733i	1.87948	−	1.21142i	0.959493	+	0.281733i	2.74068	+	4.26458i	−0.909632	+	0.415415i	0.654861	+	0.755750i	−1.68795	+	1.46657i
49.6	−0.989821	+	0.142315i	−0.909632	−	0.415415i	0.959493	−	0.281733i	1.93207	+	1.12565i	0.959493	+	0.281733i	0.0169454	+	0.0263676i	−0.909632	+	0.415415i	0.654861	+	0.755750i	−2.07260	−	0.839235i
49.7	0.989821	−	0.142315i	0.909632	+	0.415415i	0.959493	−	0.281733i	−1.51343	+	1.64607i	0.959493	+	0.281733i	0.692568	+	1.07766i	0.909632	−	0.415415i	0.654861	+	0.755750i	−1.26376	+	1.84470i
49.8	0.989821	−	0.142315i	0.909632	+	0.415415i	0.959493	−	0.281733i	−1.38916	−	1.75221i	0.959493	+	0.281733i	−0.0169454	−	0.0263676i	0.909632	−	0.415415i	0.654861	+	0.755750i	−1.62439	−	1.53668i
49.9	0.989821	−	0.142315i	0.909632	+	0.415415i	0.959493	−	0.281733i	0.931614	−	2.03276i	0.959493	+	0.281733i	−2.74068	−	4.26458i	0.909632	−	0.415415i	0.654861	+	0.755750i	0.632841	−	2.14465i
49.10	0.989821	−	0.142315i	0.909632	+	0.415415i	0.959493	−	0.281733i	1.17489	+	1.90253i	0.959493	+	0.281733i	0.0262635	+	0.0408667i	0.909632	−	0.415415i	0.654861	+	0.755750i	1.43369	+	1.71597i
49.11	0.989821	−	0.142315i	0.909632	+	0.415415i	0.959493	−	0.281733i	1.73517	−	1.41038i	0.959493	+	0.281733i	1.79620	+	2.79494i	0.909632	−	0.415415i	0.654861	+	0.755750i	1.51679	−	1.64297i
49.12	0.989821	−	0.142315i	0.909632	+	0.415415i	0.959493	−	0.281733i	2.13321	+	0.670378i	0.959493	+	0.281733i	−0.684801	−	1.06557i	0.909632	−	0.415415i	0.654861	+	0.755750i	2.20690	+	0.359967i
169.1	−0.989821	−	0.142315i	−0.909632	+	0.415415i	0.959493	+	0.281733i	−2.05037	+	0.892171i	0.959493	−	0.281733i	−0.0262635	+	0.0408667i	−0.909632	−	0.415415i	0.654861	−	0.755750i	2.15647	−	0.591291i
169.2	−0.989821	−	0.142315i	−0.909632	+	0.415415i	0.959493	+	0.281733i	−1.41393	−	1.73228i	0.959493	−	0.281733i	−0.692568	+	1.07766i	−0.909632	−	0.415415i	0.654861	−	0.755750i	1.15301	+	1.91587i
169.3	−0.989821	−	0.142315i	−0.909632	+	0.415415i	0.959493	+	0.281733i	−0.967142	+	2.01609i	0.959493	−	0.281733i	0.684801	−	1.06557i	−0.909632	−	0.415415i	0.654861	−	0.755750i	1.24422	−	1.85793i
169.4	−0.989821	−	0.142315i	−0.909632	+	0.415415i	0.959493	+	0.281733i	1.14908	+	1.91823i	0.959493	−	0.281733i	−1.79620	+	2.79494i	−0.909632	−	0.415415i	0.654861	−	0.755750i	−0.864395	−	2.06224i
169.5	−0.989821	−	0.142315i	−0.909632	+	0.415415i	0.959493	+	0.281733i	1.87948	+	1.21142i	0.959493	−	0.281733i	2.74068	−	4.26458i	−0.909632	−	0.415415i	0.654861	−	0.755750i	−1.68795	−	1.46657i
169.6	−0.989821	−	0.142315i	−0.909632	+	0.415415i	0.959493	+	0.281733i	1.93207	−	1.12565i	0.959493	−	0.281733i	0.0169454	−	0.0263676i	−0.909632	−	0.415415i	0.654861	−	0.755750i	−2.07260	+	0.839235i
169.7	0.989821	+	0.142315i	0.909632	−	0.415415i	0.959493	+	0.281733i	−1.51343	−	1.64607i	0.959493	−	0.281733i	0.692568	−	1.07766i	0.909632	+	0.415415i	0.654861	−	0.755750i	−1.26376	−	1.84470i
169.8	0.989821	+	0.142315i	0.909632	−	0.415415i	0.959493	+	0.281733i	−1.38916	+	1.75221i	0.959493	−	0.281733i	−0.0169454	+	0.0263676i	0.909632	+	0.415415i	0.654861	−	0.755750i	−1.62439	+	1.53668i

See next 80 embeddings (of 120 total)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
23.c	even	11	1	inner
115.j	even	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	690.2.r.a	✓	120
5.b	even	2	1	inner	690.2.r.a	✓	120
23.c	even	11	1	inner	690.2.r.a	✓	120
115.j	even	22	1	inner	690.2.r.a	✓	120

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
690.2.r.a	✓	120	1.a	even	1	1	trivial
690.2.r.a	✓	120	5.b	even	2	1	inner
690.2.r.a	✓	120	23.c	even	11	1	inner
690.2.r.a	✓	120	115.j	even	22	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{120} - 40 T_{7}^{118} + 1194 T_{7}^{116} - 40549 T_{7}^{114} + 995879 T_{7}^{112} + \cdots + 10\!\cdots\!16 $ T7^120 - 40*T7^118 + 1194*T7^116 - 40549*T7^114 + 995879*T7^112 - 20494634*T7^110 + 390330005*T7^108 - 5284328400*T7^106 + 96904725579*T7^104 - 1832104312866*T7^102 + 37937162569700*T7^100 - 626996651153810*T7^98 + 8964863935847147*T7^96 - 112214693483801668*T7^94 + 1472507678603229911*T7^92 - 21990448349933312632*T7^90 + 326452733128837542018*T7^88 - 4417499771818355770631*T7^86 + 53519349306415463840529*T7^84 - 577149476872135012222969*T7^82 + 5539706254913503685355099*T7^80 - 46198988965065179944714132*T7^78 + 325694070151965104542961526*T7^76 - 2028412655153330427750632432*T7^74 + 15058724489999355435642481211*T7^72 - 149540211138568637345738306154*T7^70 + 1332838901961291985376808780219*T7^68 - 8415408707723329904048545467127*T7^66 + 35187796435501819733915688858483*T7^64 - 93575649773321124769881285994696*T7^62 + 162350733330006126885924974224194*T7^60 - 302207024287048223202662721894882*T7^58 + 1009259260250402116792973576601478*T7^56 - 3061515668526788433013784762513730*T7^54 + 6805741192205698845516279999911149*T7^52 - 13025276604669418574303913280808621*T7^50 + 28833871319640706307586009895153729*T7^48 - 47749797610504552142817510241795062*T7^46 + 83914014482303678458845811876549716*T7^44 - 105727830951972296285527688765206019*T7^42 + 204498660457295248864899615774921127*T7^40 - 260179532918795450996449466206148422*T7^38 + 502362562391346156683195539354583275*T7^36 - 600047444117922772738118630958978416*T7^34 + 808004752089314641181741995719834690*T7^32 - 903578612219511539795092732597579551*T7^30 + 1030106478433132516500727573349750168*T7^28 - 522536775512945181969029180857494395*T7^26 - 325237982511377213324077881719594027*T7^24 + 85111043114405014445290588086177206*T7^22 + 328298602351657611094668798055610105*T7^20 - 52329686871308214360719210008561644*T7^18 + 19412977177993353231446718705847360*T7^16 - 1836707175626499854914079030792192*T7^14 + 169074154945617569477239943911424*T7^12 - 8894316125421077886596562768896*T7^10 + 162060409978519848675893399552*T7^8 + 443263033901022209248591872*T7^6 + 1458972661668687578267648*T7^4 + 1151290242967872208896*T7^2 + 1003194354909577216 acting on $S_{2}^{\mathrm{new}}(690, [\chi])$.

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	690.r (of order \(22\), degree \(10\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 49.12
Significant digits:
Format:

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