Properties

 Label 2.28.a.b Level $2$ Weight $28$ Character orbit 2.a Self dual yes Analytic conductor $9.237$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2,28,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, 28, names="a")

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

chi := DirichletCharacter("2.1");

S:= CuspForms(chi, 28);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2$$ Weight: $$k$$ $$=$$ $$28$$ 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: $$9.23711149676$$ Analytic rank: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + 8192 q^{2} - 1016388 q^{3} + 67108864 q^{4} - 3341197410 q^{5} - 8326250496 q^{6} - 51021361384 q^{7} + 549755813888 q^{8} - 6592552918443 q^{9}+O(q^{10})$$ q + 8192 * q^2 - 1016388 * q^3 + 67108864 * q^4 - 3341197410 * q^5 - 8326250496 * q^6 - 51021361384 * q^7 + 549755813888 * q^8 - 6592552918443 * q^9 $$q + 8192 q^{2} - 1016388 q^{3} + 67108864 q^{4} - 3341197410 q^{5} - 8326250496 q^{6} - 51021361384 q^{7} + 549755813888 q^{8} - 6592552918443 q^{9} - 27371089182720 q^{10} - 177413845094508 q^{11} - 68208644063232 q^{12} - 264386643393418 q^{13} - 417966992457728 q^{14} + 33\!\cdots\!80 q^{15}+ \cdots + 11\!\cdots\!44 q^{99}+O(q^{100})$$ q + 8192 * q^2 - 1016388 * q^3 + 67108864 * q^4 - 3341197410 * q^5 - 8326250496 * q^6 - 51021361384 * q^7 + 549755813888 * q^8 - 6592552918443 * q^9 - 27371089182720 * q^10 - 177413845094508 * q^11 - 68208644063232 * q^12 - 264386643393418 * q^13 - 417966992457728 * q^14 + 3395952953155080 * q^15 + 4503599627370496 * q^16 + 76811888571465906 * q^17 - 54006193507885056 * q^18 - 147764402234885140 * q^19 - 224223962584842240 * q^20 + 51857499454360992 * q^21 - 1453374219014209536 * q^22 + 3583628816727527112 * q^23 - 558765212165996544 * q^24 + 3713019535666879975 * q^25 - 2165855382678880256 * q^26 + 14451157452241410840 * q^27 - 3423985602213707776 * q^28 - 75208679254256666970 * q^29 + 27819646592246415360 * q^30 - 131071363974700812448 * q^31 + 36893488147419103232 * q^32 + 180321303187916797104 * q^33 + 629242991177448701952 * q^34 + 170472440510894815440 * q^35 - 442418737216594378752 * q^36 - 662288183049353372674 * q^37 - 1210485983108179066880 * q^38 + 268719411705349334184 * q^39 - 1836842701495027630080 * q^40 + 638680458671096629242 * q^41 + 424816635530125246464 * q^42 + 6917115499338769700852 * q^43 - 11906041602164404518912 * q^44 + 22027020736389692832630 * q^45 + 29357087266631902101504 * q^46 - 30908908867583003138544 * q^47 - 4577404618063843688448 * q^48 - 63109183046057553744087 * q^49 + 30417056036183080755200 * q^50 - 78070681801375089267528 * q^51 - 17742687294905387057152 * q^52 - 72917877175183391659218 * q^53 + 118383881848761637601280 * q^54 + 592774679727911334824280 * q^55 - 28049290053334694100992 * q^56 + 150185965258710437674320 * q^57 - 616109500450870615818240 * q^58 - 428694162599071555282140 * q^59 + 227898544883682634629120 * q^60 - 308035309306328670786298 * q^61 - 1073736613680749055574016 * q^62 + 336361024895024181605112 * q^63 + 302231454903657293676544 * q^64 + 883367968144681832647380 * q^65 + 1477192115715414401875968 * q^66 - 2019363317538472735190404 * q^67 + 5154758583725659766390784 * q^68 - 3642357325776057826311456 * q^69 + 1396510232665250328084480 * q^70 + 12815663453552821957376472 * q^71 - 3624294295278341150736384 * q^72 - 22881210928165048507710118 * q^73 - 5425464795540302828945408 * q^74 - 3773868499817388804030300 * q^75 - 9916301173622202915880960 * q^76 + 9051895905091888301679072 * q^77 + 2201349420690221745635328 * q^78 + 53028364881871099261766000 * q^79 - 15047415410647266345615360 * q^80 + 35584171933953904647069321 * q^81 + 5232070317433623586750464 * q^82 + 52517451913599957686712492 * q^83 + 3480097878262786019033088 * q^84 - 256643683152190485030503460 * q^85 + 56665010170583201389379584 * q^86 + 76441199089875425228304360 * q^87 - 97534292804930801818927104 * q^88 - 215234163567542209160339190 * q^89 + 180445353872504363684904960 * q^90 + 13489366477678315864970512 * q^91 + 240493258888248542015520768 * q^92 + 133219361487518209362397824 * q^93 - 253205781443239961710952448 * q^94 + 493710038037396441415487400 * q^95 - 37498098631179007495766016 * q^96 - 356303023225402052577970654 * q^97 - 516990427513303480271560704 * q^98 + 1169610162249993034551211044 * 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
 0
8192.00 −1.01639e6 6.71089e7 −3.34120e9 −8.32625e9 −5.10214e10 5.49756e11 −6.59255e12 −2.73711e13
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

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 2.28.a.b 1
3.b odd 2 1 18.28.a.c 1
4.b odd 2 1 16.28.a.b 1
5.b even 2 1 50.28.a.a 1
5.c odd 4 2 50.28.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.28.a.b 1 1.a even 1 1 trivial
16.28.a.b 1 4.b odd 2 1
18.28.a.c 1 3.b odd 2 1
50.28.a.a 1 5.b even 2 1
50.28.b.b 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 1016388$$ acting on $$S_{28}^{\mathrm{new}}(\Gamma_0(2))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 8192$$
$3$ $$T + 1016388$$
$5$ $$T + 3341197410$$
$7$ $$T + 51021361384$$
$11$ $$T + 177413845094508$$
$13$ $$T + 264386643393418$$
$17$ $$T - 76\!\cdots\!06$$
$19$ $$T + 14\!\cdots\!40$$
$23$ $$T - 35\!\cdots\!12$$
$29$ $$T + 75\!\cdots\!70$$
$31$ $$T + 13\!\cdots\!48$$
$37$ $$T + 66\!\cdots\!74$$
$41$ $$T - 63\!\cdots\!42$$
$43$ $$T - 69\!\cdots\!52$$
$47$ $$T + 30\!\cdots\!44$$
$53$ $$T + 72\!\cdots\!18$$
$59$ $$T + 42\!\cdots\!40$$
$61$ $$T + 30\!\cdots\!98$$
$67$ $$T + 20\!\cdots\!04$$
$71$ $$T - 12\!\cdots\!72$$
$73$ $$T + 22\!\cdots\!18$$
$79$ $$T - 53\!\cdots\!00$$
$83$ $$T - 52\!\cdots\!92$$
$89$ $$T + 21\!\cdots\!90$$
$97$ $$T + 35\!\cdots\!54$$