LMFDB - Newform orbit 690.2.r.b

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} + 8 q^{5} - 12 q^{6} + 12 q^{9}+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 120 q + 12 q^{4} + 8 q^{5} - 12 q^{6} + 12 q^{9} - 18 q^{10} - 8 q^{11} - 4 q^{15} - 12 q^{16} + 16 q^{19} + 14 q^{20} - 22 q^{21} - 120 q^{24} + 28 q^{25} + 8 q^{26} + 8 q^{29} + 8 q^{30} + 8 q^{31} + 44 q^{34} + 58 q^{35} - 12 q^{36} + 14 q^{39} - 4 q^{40} + 8 q^{44} - 8 q^{45} + 12 q^{49} - 4 q^{50} + 12 q^{54} + 92 q^{55} - 94 q^{59} + 4 q^{60} - 60 q^{61} + 12 q^{64} - 44 q^{65} - 8 q^{66} + 16 q^{70} - 16 q^{74} + 4 q^{75} - 16 q^{76} + 172 q^{79} + 8 q^{80} - 12 q^{81} - 32 q^{85} - 40 q^{86} + 48 q^{89} - 4 q^{90} + 288 q^{91} + 24 q^{94} - 78 q^{95} - 12 q^{96} - 36 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	−2.09099	−	0.792321i	−0.959493	−	0.281733i	−0.378014	−	0.588202i	−0.909632	+	0.415415i	0.654861	+	0.755750i	2.18246	+	0.486678i
49.2	−0.989821	+	0.142315i	0.909632	+	0.415415i	0.959493	−	0.281733i	−1.21290	+	1.87853i	−0.959493	−	0.281733i	0.707520	+	1.10092i	−0.909632	+	0.415415i	0.654861	+	0.755750i	0.933214	−	2.03202i
49.3	−0.989821	+	0.142315i	0.909632	+	0.415415i	0.959493	−	0.281733i	−0.0689747	−	2.23500i	−0.959493	−	0.281733i	−1.33127	−	2.07150i	−0.909632	+	0.415415i	0.654861	+	0.755750i	0.386347	+	2.20244i
49.4	−0.989821	+	0.142315i	0.909632	+	0.415415i	0.959493	−	0.281733i	0.934585	+	2.03139i	−0.959493	−	0.281733i	−2.47408	−	3.84974i	−0.909632	+	0.415415i	0.654861	+	0.755750i	−1.21417	−	1.87771i
49.5	−0.989821	+	0.142315i	0.909632	+	0.415415i	0.959493	−	0.281733i	0.981376	+	2.00920i	−0.959493	−	0.281733i	2.34067	+	3.64216i	−0.909632	+	0.415415i	0.654861	+	0.755750i	−1.25733	−	1.84909i
49.6	−0.989821	+	0.142315i	0.909632	+	0.415415i	0.959493	−	0.281733i	2.01587	−	0.967617i	−0.959493	−	0.281733i	−0.0999940	−	0.155594i	−0.909632	+	0.415415i	0.654861	+	0.755750i	−1.85764	+	1.24466i
49.7	0.989821	−	0.142315i	−0.909632	−	0.415415i	0.959493	−	0.281733i	−2.14372	−	0.635975i	−0.959493	−	0.281733i	2.47408	+	3.84974i	0.909632	−	0.415415i	0.654861	+	0.755750i	−2.21241	−	0.324418i
49.8	0.989821	−	0.142315i	−0.909632	−	0.415415i	0.959493	−	0.281733i	−2.12842	−	0.685448i	−0.959493	−	0.281733i	−2.34067	−	3.64216i	0.909632	−	0.415415i	0.654861	+	0.755750i	−2.20430	−	0.375565i
49.9	0.989821	−	0.142315i	−0.909632	−	0.415415i	0.959493	−	0.281733i	−1.68679	+	1.46790i	−0.959493	−	0.281733i	−0.707520	−	1.10092i	0.909632	−	0.415415i	0.654861	+	0.755750i	−1.46072	+	1.69301i
49.10	0.989821	−	0.142315i	−0.909632	−	0.415415i	0.959493	−	0.281733i	0.670880	−	2.13305i	−0.959493	−	0.281733i	0.0999940	+	0.155594i	0.909632	−	0.415415i	0.654861	+	0.755750i	0.360487	−	2.20682i
49.11	0.989821	−	0.142315i	−0.909632	−	0.415415i	0.959493	−	0.281733i	1.08184	+	1.95694i	−0.959493	−	0.281733i	0.378014	+	0.588202i	0.909632	−	0.415415i	0.654861	+	0.755750i	1.34933	+	1.78306i
49.12	0.989821	−	0.142315i	−0.909632	−	0.415415i	0.959493	−	0.281733i	2.22207	−	0.249802i	−0.959493	−	0.281733i	1.33127	+	2.07150i	0.909632	−	0.415415i	0.654861	+	0.755750i	2.16390	−	0.563493i
169.1	−0.989821	−	0.142315i	0.909632	−	0.415415i	0.959493	+	0.281733i	−2.09099	+	0.792321i	−0.959493	+	0.281733i	−0.378014	+	0.588202i	−0.909632	−	0.415415i	0.654861	−	0.755750i	2.18246	−	0.486678i
169.2	−0.989821	−	0.142315i	0.909632	−	0.415415i	0.959493	+	0.281733i	−1.21290	−	1.87853i	−0.959493	+	0.281733i	0.707520	−	1.10092i	−0.909632	−	0.415415i	0.654861	−	0.755750i	0.933214	+	2.03202i
169.3	−0.989821	−	0.142315i	0.909632	−	0.415415i	0.959493	+	0.281733i	−0.0689747	+	2.23500i	−0.959493	+	0.281733i	−1.33127	+	2.07150i	−0.909632	−	0.415415i	0.654861	−	0.755750i	0.386347	−	2.20244i
169.4	−0.989821	−	0.142315i	0.909632	−	0.415415i	0.959493	+	0.281733i	0.934585	−	2.03139i	−0.959493	+	0.281733i	−2.47408	+	3.84974i	−0.909632	−	0.415415i	0.654861	−	0.755750i	−1.21417	+	1.87771i
169.5	−0.989821	−	0.142315i	0.909632	−	0.415415i	0.959493	+	0.281733i	0.981376	−	2.00920i	−0.959493	+	0.281733i	2.34067	−	3.64216i	−0.909632	−	0.415415i	0.654861	−	0.755750i	−1.25733	+	1.84909i
169.6	−0.989821	−	0.142315i	0.909632	−	0.415415i	0.959493	+	0.281733i	2.01587	+	0.967617i	−0.959493	+	0.281733i	−0.0999940	+	0.155594i	−0.909632	−	0.415415i	0.654861	−	0.755750i	−1.85764	−	1.24466i
169.7	0.989821	+	0.142315i	−0.909632	+	0.415415i	0.959493	+	0.281733i	−2.14372	+	0.635975i	−0.959493	+	0.281733i	2.47408	−	3.84974i	0.909632	+	0.415415i	0.654861	−	0.755750i	−2.21241	+	0.324418i
169.8	0.989821	+	0.142315i	−0.909632	+	0.415415i	0.959493	+	0.281733i	−2.12842	+	0.685448i	−0.959493	+	0.281733i	−2.34067	+	3.64216i	0.909632	+	0.415415i	0.654861	−	0.755750i	−2.20430	+	0.375565i

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.b	✓	120
5.b	even	2	1	inner	690.2.r.b	✓	120
23.c	even	11	1	inner	690.2.r.b	✓	120
115.j	even	22	1	inner	690.2.r.b	✓	120

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
690.2.r.b	✓	120	1.a	even	1	1	trivial
690.2.r.b	✓	120	5.b	even	2	1	inner
690.2.r.b	✓	120	23.c	even	11	1	inner
690.2.r.b	✓	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} - 48 T_{7}^{118} + 1594 T_{7}^{116} - 44301 T_{7}^{114} + 1226263 T_{7}^{112} + \cdots + 23\!\cdots\!16 $ T7^120 - 48*T7^118 + 1594*T7^116 - 44301*T7^114 + 1226263*T7^112 - 34187082*T7^110 + 953331133*T7^108 - 26347597400*T7^106 + 678345434347*T7^104 - 16792501821682*T7^102 + 393643712936124*T7^100 - 8132418127569282*T7^98 + 155832246048834891*T7^96 - 2896309720132643948*T7^94 + 51505703426979728495*T7^92 - 886260401607072725640*T7^90 + 14980719521744668058090*T7^88 - 247892032488475984049399*T7^86 + 3946628496715701065164713*T7^84 - 57988909794363929312067137*T7^82 + 754382610913209550844880779*T7^80 - 8452486073982684721717185620*T7^78 + 80425787627691009199620196142*T7^76 - 647066683227076033194291059688*T7^74 + 4443327464444175831672989068315*T7^72 - 26979236449877539698441454080338*T7^70 + 153101861286237253974439150715483*T7^68 - 836541188562494085780649199307927*T7^66 + 4389688183086873279600715142716419*T7^64 - 22777431673283910078502415659675896*T7^62 + 122767260767165289884983673961471506*T7^60 - 650053888437682262560843063430496426*T7^58 + 2991153912865519349832668360479138022*T7^56 - 11278078749938697931571829725995644298*T7^54 + 32996250151308792499562444437508916797*T7^52 - 62473455237567515033775982191076690525*T7^50 + 106121525470014016023030919689092115649*T7^48 - 223057712366961242750394890628288223510*T7^46 + 380960973903808956940696849295810176524*T7^44 - 771897637331021730576132209237365543627*T7^42 + 1650128816258424427682502156710320817951*T7^40 - 2216432553567449395191587270821611933086*T7^38 + 4649806862929323959395619940994092013419*T7^36 - 5741390181169240664966821176927041943328*T7^34 + 6320790065539686480702814555542503357666*T7^32 - 4349018939951193626490593527758029378031*T7^30 + 2576632958939855187177398669177693833880*T7^28 - 1761072447848415945484133883748523236075*T7^26 + 2826585769025169059977431407027917499933*T7^24 - 3676501383783981590759948257532834622682*T7^22 + 2449435731280426956345527363522496228521*T7^20 - 1009491207879582773998961581273310260844*T7^18 + 509596838687927673527965242735471836096*T7^16 - 215021286497288580792552958615734792448*T7^14 + 67495024850474702470885174464158642176*T7^12 + 7679568795306293235340698691836382208*T7^10 + 255155483920504952121943880348078080*T7^8 - 4288969233449107263618798202650624*T7^6 + 1678715124034562092127551860113408*T7^4 + 43553548021242911643152634871808*T7^2 + 2334093950909254219259645526016 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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