# Properties

 Label 6.28.a.c Level $6$ Weight $28$ Character orbit 6.a Self dual yes Analytic conductor $27.711$ Analytic rank $0$ 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] = [6,28,Mod(1,6)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6.1");

S:= CuspForms(chi, 28);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6 = 2 \cdot 3$$ Weight: $$k$$ $$=$$ $$28$$ Character orbit: $$[\chi]$$ $$=$$ 6.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: $$27.7113344903$$ 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)

 $$f(q)$$ $$=$$ $$q + 8192 q^{2} + 1594323 q^{3} + 67108864 q^{4} + 1220703150 q^{5} + 13060694016 q^{6} + 96889207016 q^{7} + 549755813888 q^{8} + 2541865828329 q^{9}+O(q^{10})$$ q + 8192 * q^2 + 1594323 * q^3 + 67108864 * q^4 + 1220703150 * q^5 + 13060694016 * q^6 + 96889207016 * q^7 + 549755813888 * q^8 + 2541865828329 * q^9 $$q + 8192 q^{2} + 1594323 q^{3} + 67108864 q^{4} + 1220703150 q^{5} + 13060694016 q^{6} + 96889207016 q^{7} + 549755813888 q^{8} + 2541865828329 q^{9} + 10000000204800 q^{10} + 34495064342052 q^{11} + 106993205379072 q^{12} + 300892562137622 q^{13} + 793716383875072 q^{14} + 19\!\cdots\!50 q^{15}+ \cdots + 87\!\cdots\!08 q^{99}+O(q^{100})$$ q + 8192 * q^2 + 1594323 * q^3 + 67108864 * q^4 + 1220703150 * q^5 + 13060694016 * q^6 + 96889207016 * q^7 + 549755813888 * q^8 + 2541865828329 * q^9 + 10000000204800 * q^10 + 34495064342052 * q^11 + 106993205379072 * q^12 + 300892562137622 * q^13 + 793716383875072 * q^14 + 1946195108217450 * q^15 + 4503599627370496 * q^16 + 11406510312331986 * q^17 + 20822964865671168 * q^18 + 62694436994411420 * q^19 + 81920001677721600 * q^20 + 154472691197370168 * q^21 + 282583567090089984 * q^22 - 894750379460289528 * q^23 + 876488338465357824 * q^24 - 5960464416503905625 * q^25 + 2464911869031399424 * q^26 + 4052555153018976267 * q^27 + 6502124616704589824 * q^28 + 104977877616797619030 * q^29 + 15943230326517350400 * q^30 + 242604665669598327632 * q^31 + 36893488147419103232 * q^32 + 54996274467013370796 * q^33 + 93442132478623629312 * q^34 + 118272960205433300400 * q^35 + 170581728179578208256 * q^36 + 1615296776273185563326 * q^37 + 513592827858218352640 * q^38 + 479719932344939919906 * q^39 + 671088653743895347200 * q^40 - 3845227056141271336998 * q^41 + 1265440286288856416256 * q^42 - 457279560936215164348 * q^43 + 2314924581602017148928 * q^44 + 3102863623518569536350 * q^45 - 7329795108538691813376 * q^46 - 48040279177582596731664 * q^47 + 7180192468708211294208 * q^48 - 56324843927344976515287 * q^49 - 48828124499999994880000 * q^50 + 18185661740688068915478 * q^51 + 20192558031105224081408 * q^52 - 277436637507624408709218 * q^53 + 33198531813531453579264 * q^54 + 42108233701795553863800 * q^55 + 53265404860043999838208 * q^56 + 99955182872240998368660 * q^57 + 859978773436806095093760 * q^58 + 257780373920421815739540 * q^59 + 130606942834830134476800 * q^60 + 143521812506920157597702 * q^61 + 1987417421165349499961344 * q^62 + 246279364447864798356264 * q^63 + 302231454903657293676544 * q^64 + 367300498412965908909300 * q^65 + 450529480433773533560832 * q^66 - 5989037992702629692139124 * q^67 + 765477949264884771323904 * q^68 - 1426521109232267181149544 * q^69 + 968892090002909596876800 * q^70 - 5167007853435876621826728 * q^71 + 1397405517247104682033152 * q^72 - 12558020215909057855312678 * q^73 + 13232511191229936134766592 * q^74 - 9502905509913756327766875 * q^75 + 4207352445814524744826880 * q^76 + 3342199430067316062236832 * q^77 + 3929865685769747823869952 * q^78 - 60783355718993851022367520 * q^79 + 5497558251469990684262400 * q^80 + 6461081889226673298932241 * q^81 - 31500100043909294792687616 * q^82 + 3971159179443692816062812 * q^83 + 10366486825278311761969152 * q^84 + 13923963068771139155955900 * q^85 - 3746034163189474626338816 * q^86 + 167368644775645630364766690 * q^87 + 18963862172483724484018176 * q^88 + 312460635134078620023752010 * q^89 + 25418658803864121641779200 * q^90 + 29153241742526701439955952 * q^91 - 60045681529148963335176192 * q^92 + 386790198384351014505233136 * q^93 - 393545967022756632425791488 * q^94 + 76531296726554552789973000 * q^95 + 58820136703657666922151936 * q^96 - 548671865528572742983955614 * q^97 - 461413121452810047613231104 * q^98 + 87681825297072158367591108 * 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 1.59432e6 6.71089e7 1.22070e9 1.30607e10 9.68892e10 5.49756e11 2.54187e12 1.00000e13
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$

## 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 6.28.a.c 1
3.b odd 2 1 18.28.a.b 1
4.b odd 2 1 48.28.a.a 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 1220703150$$ acting on $$S_{28}^{\mathrm{new}}(\Gamma_0(6))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 8192$$
$3$ $$T - 1594323$$
$5$ $$T - 1220703150$$
$7$ $$T - 96889207016$$
$11$ $$T - 34495064342052$$
$13$ $$T - 300892562137622$$
$17$ $$T - 11\!\cdots\!86$$
$19$ $$T - 62\!\cdots\!20$$
$23$ $$T + 89\!\cdots\!28$$
$29$ $$T - 10\!\cdots\!30$$
$31$ $$T - 24\!\cdots\!32$$
$37$ $$T - 16\!\cdots\!26$$
$41$ $$T + 38\!\cdots\!98$$
$43$ $$T + 45\!\cdots\!48$$
$47$ $$T + 48\!\cdots\!64$$
$53$ $$T + 27\!\cdots\!18$$
$59$ $$T - 25\!\cdots\!40$$
$61$ $$T - 14\!\cdots\!02$$
$67$ $$T + 59\!\cdots\!24$$
$71$ $$T + 51\!\cdots\!28$$
$73$ $$T + 12\!\cdots\!78$$
$79$ $$T + 60\!\cdots\!20$$
$83$ $$T - 39\!\cdots\!12$$
$89$ $$T - 31\!\cdots\!10$$
$97$ $$T + 54\!\cdots\!14$$