# Properties

 Label 16.28.e.a Level $16$ Weight $28$ Character orbit 16.e Analytic conductor $73.897$ Analytic rank $0$ Dimension $106$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,28,Mod(5,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 28, names="a")

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

chi := DirichletCharacter("16.5");

S:= CuspForms(chi, 28);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$16 = 2^{4}$$ Weight: $$k$$ $$=$$ $$28$$ Character orbit: $$[\chi]$$ $$=$$ 16.e (of order $$4$$, degree $$2$$, 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: $$73.8968919741$$ Analytic rank: $$0$$ Dimension: $$106$$ Relative dimension: $$53$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$106 q - 2 q^{2} - 2 q^{3} + 125095688 q^{4} - 2 q^{5} - 59499919472 q^{6} - 287180933804 q^{8}+O(q^{10})$$ 106 * q - 2 * q^2 - 2 * q^3 + 125095688 * q^4 - 2 * q^5 - 59499919472 * q^6 - 287180933804 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$106 q - 2 q^{2} - 2 q^{3} + 125095688 q^{4} - 2 q^{5} - 59499919472 q^{6} - 287180933804 q^{8} + 8247210235372 q^{10} - 34039821810222 q^{11} + 17\!\cdots\!80 q^{12}+ \cdots + 16\!\cdots\!66 q^{99}+O(q^{100})$$ 106 * q - 2 * q^2 - 2 * q^3 + 125095688 * q^4 - 2 * q^5 - 59499919472 * q^6 - 287180933804 * q^8 + 8247210235372 * q^10 - 34039821810222 * q^11 + 1748989708884580 * q^12 - 2 * q^13 - 7155218316814660 * q^14 - 15569560546875004 * q^15 - 11277270898466024 * q^16 - 4 * q^17 - 142926567667036386 * q^18 - 361273435637780378 * q^19 + 442470438134323004 * q^20 + 15251194969972 * q^21 - 1169169681057307068 * q^22 + 10984755246449911984 * q^24 + 17037255844415508360 * q^26 - 16968259081125930056 * q^27 - 60190300783057535720 * q^28 - 27121773564864273802 * q^29 - 676957577616991436364 * q^30 + 293010555569340511088 * q^31 - 1166901104384201893432 * q^32 - 4 * q^33 - 290590654320714186380 * q^34 - 1483431258098321642404 * q^35 + 2272823118258680645548 * q^36 - 470486119578222882250 * q^37 - 7667804020978176078752 * q^38 - 13361029860556314821176 * q^40 - 46808273599641292573640 * q^42 + 30832447383975033909242 * q^43 + 78151525061974584923236 * q^44 + 14901145942652686274 * q^45 - 30741189488837489213204 * q^46 - 109219994122411663546544 * q^47 - 189351174993879449193752 * q^48 - 882423151738888904731010 * q^49 + 548328807383867462419206 * q^50 - 115559869623218432212572 * q^51 - 870962460180776269413356 * q^52 - 5971340109916911897018 * q^53 + 1136059014139929162305056 * q^54 + 1200724087407057566405656 * q^56 - 2828993593282901350448872 * q^58 - 34394779779101149275670 * q^59 + 7255680995179605171006528 * q^60 - 2475467954128860415560434 * q^61 - 3746979776437258967395280 * q^62 + 5188426743337460248956228 * q^63 - 998638142638832585325568 * q^64 + 2911521903927484929681108 * q^65 + 19080130794608880088211092 * q^66 + 19927279436319849143450902 * q^67 - 8373829219145321982498160 * q^68 - 11740592568211854067772156 * q^69 - 5132370883366275255574400 * q^70 + 19086224804727807421919172 * q^72 + 42828123668533070044258924 * q^74 - 81164500905568543837973834 * q^75 - 105896749199037309084347596 * q^76 + 43639762623313421635882292 * q^77 - 98604570863977418199548932 * q^78 - 27720061020991987308873792 * q^79 - 62182770892870708184304728 * q^80 - 581497370030400596903901694 * q^81 - 341107120874158685100150464 * q^82 + 26201266111333778096099838 * q^83 + 507935364828252675231522472 * q^84 + 224109171520266208496093748 * q^85 - 704077428371873697577134268 * q^86 - 681166827660725136677389832 * q^88 + 482826168478152776777239896 * q^90 + 67827511015091818004015876 * q^91 - 83646902794933560160672648 * q^92 + 886600140227681407873186288 * q^93 - 168289477479668477956229936 * q^94 + 239338208955746404979764980 * q^95 - 752456785864472227742519632 * q^96 - 4 * q^97 - 1043862744854197068838248646 * q^98 + 1641925710538879681648895966 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −11581.2 + 305.932i −2.33243e6 + 2.33243e6i 1.34031e8 7.08611e6i −3.25506e9 3.25506e9i 2.62988e10 2.77259e10i 4.57721e11i −1.55007e12 + 1.23070e11i 3.25487e12i 3.86933e13 + 3.67017e13i
5.2 −11578.9 382.145i −2.39173e6 + 2.39173e6i 1.33926e8 + 8.84966e6i 1.10058e8 + 1.10058e8i 2.86077e10 2.67797e10i 2.84560e11i −1.54733e12 1.53649e11i 3.81516e12i −1.23230e12 1.31641e12i
5.3 −11565.4 677.061i 663110. 663110.i 1.33301e8 + 1.56610e7i 2.21940e8 + 2.21940e8i −8.11812e9 + 7.22019e9i 2.02008e11i −1.53108e12 2.71379e11i 6.74617e12i −2.41657e12 2.71710e12i
5.4 −11441.8 + 1817.09i 153675. 153675.i 1.27614e8 4.15818e7i 2.87142e9 + 2.87142e9i −1.47908e9 + 2.03757e9i 4.55239e11i −1.38458e12 + 7.07660e11i 7.57837e12i −3.80720e13 2.76367e13i
5.5 −11221.9 + 2878.54i 3.13170e6 3.13170e6i 1.17646e8 6.46055e7i 2.99171e9 + 2.99171e9i −2.61290e10 + 4.41585e10i 3.77678e11i −1.13424e12 + 1.06365e12i 1.19895e13i −4.21845e13 2.49610e13i
5.6 −10946.6 + 3793.44i 2.44645e6 2.44645e6i 1.05437e8 8.30503e7i −2.13207e9 2.13207e9i −1.74998e10 + 3.60607e10i 9.04455e10i −8.39133e11 + 1.30909e12i 4.34460e12i 3.14268e13 + 1.52510e13i
5.7 −10852.0 4056.00i 3.04255e6 3.04255e6i 1.01315e8 + 8.80317e7i −1.59485e9 1.59485e9i −4.53584e10 + 2.06772e10i 2.58969e10i −7.42421e11 1.36626e12i 1.08886e13i 1.08387e13 + 2.37761e13i
5.8 −10534.4 4821.21i 923601. 923601.i 8.77296e7 + 1.01577e8i −5.91123e8 5.91123e8i −1.41825e10 + 5.27671e9i 1.68267e11i −4.34455e11 1.49302e12i 5.91952e12i 3.37720e12 + 9.07706e12i
5.9 −9759.73 + 6242.23i −2.84375e6 + 2.84375e6i 5.62868e7 1.21845e8i 1.41523e9 + 1.41523e9i 1.00029e10 4.55056e10i 4.92833e10i 2.11241e11 + 1.54053e12i 8.54826e12i −2.26465e13 4.97807e12i
5.10 −9666.34 6385.89i −1.23946e6 + 1.23946e6i 5.26586e7 + 1.23456e8i 3.17355e9 + 3.17355e9i 1.98961e10 4.06599e9i 1.31495e11i 2.79363e11 1.52964e12i 4.55307e12i −1.04107e13 5.09425e13i
5.11 −9440.77 + 6714.88i −474715. + 474715.i 4.40386e7 1.26787e8i −2.37654e9 2.37654e9i 1.29402e9 7.66932e9i 2.04420e11i 4.35602e11 + 1.49268e12i 7.17489e12i 3.83945e13 + 6.47820e12i
5.12 −9065.18 7213.90i −3.30251e6 + 3.30251e6i 3.01372e7 + 1.30790e8i 3.04061e8 + 3.04061e8i 5.37618e10 6.11388e9i 2.05985e11i 6.70310e11 1.40305e12i 1.41875e13i −5.62903e11 4.94983e12i
5.13 −8692.02 7659.41i −956364. + 956364.i 1.68847e7 + 1.33151e8i −3.72252e9 3.72252e9i 1.56379e10 9.87553e8i 2.96868e11i 8.73099e11 1.28668e12i 5.79634e12i 3.84392e12 + 6.08685e13i
5.14 −8651.50 7705.14i 3.24050e6 3.24050e6i 1.54792e7 + 1.33322e8i 2.55225e9 + 2.55225e9i −5.30037e10 + 3.06667e9i 2.23686e11i 8.93348e11 1.27271e12i 1.33761e13i −2.41534e12 4.17462e13i
5.15 −8488.66 + 7884.18i 356479. 356479.i 9.89700e6 1.33852e8i 8.81914e8 + 8.81914e8i −2.15483e8 + 5.83658e9i 2.95444e10i 9.71304e11 + 1.21426e12i 7.37144e12i −1.44394e13 5.33097e11i
5.16 −6847.52 + 9345.01i 2.95774e6 2.95774e6i −4.04408e7 1.27980e8i 6.75988e8 + 6.75988e8i 7.38694e9 + 4.78933e10i 3.29800e11i 1.47290e12 + 4.98428e11i 9.87086e12i −1.09460e13 + 1.68828e12i
5.17 −5798.80 10029.5i 1.92086e6 1.92086e6i −6.69655e7 + 1.16319e8i 4.80331e8 + 4.80331e8i −3.04040e10 8.12664e9i 4.11420e11i 1.55494e12 2.87564e9i 2.46204e11i 2.03215e12 7.60284e12i
5.18 −5041.21 10430.9i −329571. + 329571.i −8.33902e7 + 1.05169e8i −2.69387e8 2.69387e8i 5.09916e9 + 1.77629e9i 2.71695e11i 1.51739e12 + 3.39659e11i 7.40836e12i −1.45192e12 + 4.16799e12i
5.19 −4635.48 + 10617.4i 631535. 631535.i −9.12424e7 9.84339e7i 2.52651e9 + 2.52651e9i 3.77782e9 + 9.63275e9i 3.51372e11i 1.46807e12 5.12473e11i 6.82793e12i −3.85367e13 + 1.51135e13i
5.20 −4567.90 + 10646.7i −1.19622e6 + 1.19622e6i −9.24863e7 9.72661e7i −1.71680e9 1.71680e9i −7.27159e9 1.82001e10i 4.18265e11i 1.45803e12 5.40371e11i 4.76369e12i 2.61205e13 1.04361e13i
See next 80 embeddings (of 106 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.53 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 16.28.e.a 106
16.e even 4 1 inner 16.28.e.a 106

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.28.e.a 106 1.a even 1 1 trivial
16.28.e.a 106 16.e even 4 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{28}^{\mathrm{new}}(16, [\chi])$$.