LMFDB - Newform orbit 2.30.a.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2,30,Mod(1,2)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2, base_ring=CyclotomicField(1))

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

N = Newforms(chi, 30, names="a")

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

chi := DirichletCharacter("2.1");

S:= CuspForms(chi, 30);

N := Newforms(S);

Level:	$N$	$=$	$2$
Weight:	$k$	$=$	$30$
Character orbit:	$[\chi]$	$=$	2.a (trivial)

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:	yes
Analytic conductor:	$10.6556084766$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q + 16384 q^{2} + 4782996 q^{3} + 268435456 q^{4} + 6065841750 q^{5} + 78364606464 q^{6} + 904018883432 q^{7} + 4398046511104 q^{8} - 45753326628867 q^{9} + 99382751232000 q^{10} + 23\!\cdots\!32 q^{11}+ \cdots - 10\!\cdots\!44 q^{99}+O(q^{100})

q + 16384 * q^2 + 4782996 * q^3 + 268435456 * q^4 + 6065841750 * q^5 + 78364606464 * q^6 + 904018883432 * q^7 + 4398046511104 * q^8 - 45753326628867 * q^9 + 99382751232000 * q^10 + 2348011244715132 * q^11 + 1283925712306176 * q^12 + 16003222193389886 * q^13 + 14811445386149888 * q^14 + 29012896826883000 * q^15 + 72057594037927936 * q^16 - 535853837930780718 * q^17 - 749622503487356928 * q^18 - 4500449383992496540 * q^19 + 1628286996185088000 * q^20 + 4323918703379722272 * q^21 + 38469816233412722688 * q^22 + 6054145763195418936 * q^23 + 21035838870424387584 * q^24 - 149470078787052640625 * q^25 + 262196792416499892224 * q^26 - 547096798667290275000 * q^27 + 242670721206679764992 * q^28 - 2361414770177543957490 * q^29 + 475347301611651072000 * q^30 + 4580859651056014465952 * q^31 + 1180591620717411303424 * q^32 + 11230528391427497495472 * q^33 - 8779429280657911283712 * q^34 + 5483635485910208886000 * q^35 - 12281815097136855908352 * q^36 - 50376596834828206261258 * q^37 - 73735362707333063311360 * q^38 + 76543347738095051178456 * q^39 + 26677854145496481792000 * q^40 + 157191196896822750812682 * q^41 + 70843084036173369704448 * q^42 - 210053142195379460736484 * q^43 + 630289469168234048520192 * q^44 - 277532438866768203797250 * q^45 + 99191124184193743847424 * q^46 - 885237004192860923541648 * q^47 + 344651184053033166176256 * q^48 - 2402655614211539722738983 * q^49 - 2448917770847070464000000 * q^50 - 2562986763407572451071128 * q^51 + 4295832246951934234198016 * q^52 + 19545079942227702370848966 * q^53 - 8963633949364883865600000 * q^54 + 14242664637662514542361000 * q^55 + 3975917096250241269628928 * q^56 - 21525631401838574980833840 * q^57 - 38689419594588880199516160 * q^58 + 4147355257120207357648620 * q^59 + 7788090189605291163648000 * q^60 - 63157257621786600629654098 * q^61 + 75052804522901741010157568 * q^62 - 41361871252327937999231544 * q^63 + 19342813113834066795298816 * q^64 + 97073013315190944526540500 * q^65 + 184000977165148118965813248 * q^66 + 15782714250030816425172692 * q^67 - 143842169334299218472337408 * q^68 + 28956954968780635989212256 * q^69 + 89843883801152862388224000 * q^70 - 458578117673329990520126808 * q^71 - 201225258551490247202439168 * q^72 + 382939334311990038717870506 * q^73 - 825370162541825331384451072 * q^74 - 714914788958157631898812500 * q^75 - 1208080182596944909293322240 * q^76 + 2122646503733154141554493024 * q^77 + 1254086209340949318507823104 * q^78 + 4055656551380978437159131920 * q^79 + 437089962319814357680128000 * q^80 + 523306272599437657015977561 * q^81 + 2575420569957543949314981888 * q^82 - 12017293371822907068052905564 * q^83 + 1160693088848664489237676032 * q^84 - 3250404582018263289339376500 * q^85 - 3441510681729097084706553856 * q^86 - 11294637400100112038498840040 * q^87 + 10326662662852346650954825728 * q^88 + 10442412205106441731707520890 * q^89 - 4547091478393130251014144000 * q^90 + 14467215058582526712761768752 * q^91 + 1625147378633830299196194816 * q^92 + 21910233387562312966590552192 * q^93 - 14503723076695833371306360832 * q^94 - 27299013767183467199062545000 * q^95 + 5646764999524895394631778304 * q^96 + 108412279645050844119569076962 * q^97 - 39365109583241866817355497472 * q^98 - 107429325407703998959302915444 * q^99

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}

1.1

16384.0

4.78300e6

2.68435e8

6.06584e9

7.83646e10

9.04019e11

4.39805e12

−4.57533e13

9.93828e13

Atkin-Lehner signs

$p$	Sign
$2$	$-1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2.30.a.b	✓	1
3.b	odd	2	1		18.30.a.a		1
4.b	odd	2	1		16.30.a.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2.30.a.b	✓	1	1.a	even	1	1	trivial
16.30.a.a		1	4.b	odd	2	1
18.30.a.a		1	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3} - 4782996$ T3 - 4782996 acting on $S_{30}^{\mathrm{new}}(\Gamma_0(2))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T - 16384$ T - 16384
$3$	$T - 4782996$ T - 4782996
$5$	$T - 6065841750$ T - 6065841750
$7$	$T - 904018883432$ T - 904018883432
$11$	$T - 2348011244715132$ T - 2348011244715132
$13$	$T - 16\!\cdots\!86$ T - 16003222193389886
$17$	$T + 53\!\cdots\!18$ T + 535853837930780718
$19$	$T + 45\!\cdots\!40$ T + 4500449383992496540
$23$	$T - 60\!\cdots\!36$ T - 6054145763195418936
$29$	$T + 23\!\cdots\!90$ T + 2361414770177543957490
$31$	$T - 45\!\cdots\!52$ T - 4580859651056014465952
$37$	$T + 50\!\cdots\!58$ T + 50376596834828206261258
$41$	$T - 15\!\cdots\!82$ T - 157191196896822750812682
$43$	$T + 21\!\cdots\!84$ T + 210053142195379460736484
$47$	$T + 88\!\cdots\!48$ T + 885237004192860923541648
$53$	$T - 19\!\cdots\!66$ T - 19545079942227702370848966
$59$	$T - 41\!\cdots\!20$ T - 4147355257120207357648620
$61$	$T + 63\!\cdots\!98$ T + 63157257621786600629654098
$67$	$T - 15\!\cdots\!92$ T - 15782714250030816425172692
$71$	$T + 45\!\cdots\!08$ T + 458578117673329990520126808
$73$	$T - 38\!\cdots\!06$ T - 382939334311990038717870506
$79$	$T - 40\!\cdots\!20$ T - 4055656551380978437159131920
$83$	$T + 12\!\cdots\!64$ T + 12017293371822907068052905564
$89$	$T - 10\!\cdots\!90$ T - 10442412205106441731707520890
$97$	$T - 10\!\cdots\!62$ T - 108412279645050844119569076962
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