# Properties

 Label 55.6.e.b Level $55$ Weight $6$ Character orbit 55.e Analytic conductor $8.821$ Analytic rank $0$ Dimension $52$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [55,6,Mod(32,55)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(55, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([1, 2]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("55.32");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$55 = 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 55.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: $$8.82111008971$$ Analytic rank: $$0$$ Dimension: $$52$$ Relative dimension: $$26$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$52 q + 56 q^{3} - 48 q^{5}+O(q^{10})$$ 52 * q + 56 * q^3 - 48 * q^5 $$\operatorname{Tr}(f)(q) =$$ $$52 q + 56 q^{3} - 48 q^{5} - 364 q^{11} + 4792 q^{12} + 2564 q^{15} - 6416 q^{16} + 18272 q^{20} + 2680 q^{22} - 144 q^{23} - 12248 q^{25} + 2432 q^{26} + 3644 q^{27} + 9672 q^{31} + 30576 q^{33} - 148776 q^{36} + 24152 q^{37} + 4400 q^{38} - 43480 q^{42} - 4984 q^{45} - 44148 q^{47} + 26416 q^{48} + 75596 q^{53} + 59092 q^{55} + 304264 q^{56} - 220200 q^{58} - 240016 q^{60} + 207000 q^{66} - 17888 q^{67} - 216880 q^{70} + 285704 q^{71} + 278404 q^{75} - 341160 q^{77} + 15160 q^{78} + 616192 q^{80} - 397204 q^{81} + 349720 q^{82} - 94768 q^{86} + 539000 q^{88} - 864768 q^{91} + 242072 q^{92} + 15772 q^{93} - 633128 q^{97}+O(q^{100})$$ 52 * q + 56 * q^3 - 48 * q^5 - 364 * q^11 + 4792 * q^12 + 2564 * q^15 - 6416 * q^16 + 18272 * q^20 + 2680 * q^22 - 144 * q^23 - 12248 * q^25 + 2432 * q^26 + 3644 * q^27 + 9672 * q^31 + 30576 * q^33 - 148776 * q^36 + 24152 * q^37 + 4400 * q^38 - 43480 * q^42 - 4984 * q^45 - 44148 * q^47 + 26416 * q^48 + 75596 * q^53 + 59092 * q^55 + 304264 * q^56 - 220200 * q^58 - 240016 * q^60 + 207000 * q^66 - 17888 * q^67 - 216880 * q^70 + 285704 * q^71 + 278404 * q^75 - 341160 * q^77 + 15160 * q^78 + 616192 * q^80 - 397204 * q^81 + 349720 * q^82 - 94768 * q^86 + 539000 * q^88 - 864768 * q^91 + 242072 * q^92 + 15772 * q^93 - 633128 * q^97

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
32.1 −7.70910 + 7.70910i −16.8901 + 16.8901i 86.8603i −34.5372 + 43.9566i 260.415i 97.1832 97.1832i 422.923 + 422.923i 327.553i −72.6145 605.116i
32.2 −7.05963 + 7.05963i 1.68959 1.68959i 67.6769i 55.5275 6.45743i 23.8558i −73.3168 + 73.3168i 251.866 + 251.866i 237.291i −346.417 + 437.591i
32.3 −6.82111 + 6.82111i 5.46634 5.46634i 61.0552i −38.9249 40.1230i 74.5730i 17.4759 17.4759i 198.189 + 198.189i 183.238i 539.194 + 8.17222i
32.4 −6.66565 + 6.66565i 19.1413 19.1413i 56.8618i −13.1183 + 54.3407i 255.179i 17.8098 17.8098i 165.720 + 165.720i 489.780i −274.774 449.658i
32.5 −5.22937 + 5.22937i −7.05670 + 7.05670i 22.6926i 7.96459 + 55.3314i 73.8041i −142.717 + 142.717i −48.6720 48.6720i 143.406i −330.998 247.699i
32.6 −4.99853 + 4.99853i −11.9326 + 11.9326i 17.9706i 53.2298 17.0759i 119.291i 159.303 159.303i −70.1263 70.1263i 41.7749i −180.716 + 351.425i
32.7 −4.80775 + 4.80775i −19.8987 + 19.8987i 14.2290i −6.50942 55.5214i 191.336i −113.651 + 113.651i −85.4386 85.4386i 548.917i 298.229 + 235.638i
32.8 −4.43002 + 4.43002i 16.3501 16.3501i 7.25019i 27.0484 48.9222i 144.862i 30.3495 30.3495i −109.642 109.642i 291.649i 96.9014 + 336.552i
32.9 −4.04500 + 4.04500i −4.18256 + 4.18256i 0.723976i −55.4681 6.94934i 33.8369i 23.8782 23.8782i −126.511 126.511i 208.012i 252.478 196.258i
32.10 −2.88475 + 2.88475i 5.55436 5.55436i 15.3564i −11.0950 + 54.7896i 32.0459i 103.743 103.743i −136.612 136.612i 181.298i −126.048 190.061i
32.11 −2.33566 + 2.33566i 11.4271 11.4271i 21.0894i 53.4426 + 16.3977i 53.3797i −37.7988 + 37.7988i −123.999 123.999i 18.1587i −163.123 + 86.5243i
32.12 −1.37494 + 1.37494i 16.5868 16.5868i 28.2191i −55.3028 8.16074i 45.6116i −178.839 + 178.839i −82.7976 82.7976i 307.243i 87.2585 64.8175i
32.13 −0.819850 + 0.819850i −2.25487 + 2.25487i 30.6557i 5.74279 55.6059i 3.69731i −88.0749 + 88.0749i −51.3683 51.3683i 232.831i 40.8803 + 50.2967i
32.14 0.819850 0.819850i −2.25487 + 2.25487i 30.6557i 5.74279 55.6059i 3.69731i 88.0749 88.0749i 51.3683 + 51.3683i 232.831i −40.8803 50.2967i
32.15 1.37494 1.37494i 16.5868 16.5868i 28.2191i −55.3028 8.16074i 45.6116i 178.839 178.839i 82.7976 + 82.7976i 307.243i −87.2585 + 64.8175i
32.16 2.33566 2.33566i 11.4271 11.4271i 21.0894i 53.4426 + 16.3977i 53.3797i 37.7988 37.7988i 123.999 + 123.999i 18.1587i 163.123 86.5243i
32.17 2.88475 2.88475i 5.55436 5.55436i 15.3564i −11.0950 + 54.7896i 32.0459i −103.743 + 103.743i 136.612 + 136.612i 181.298i 126.048 + 190.061i
32.18 4.04500 4.04500i −4.18256 + 4.18256i 0.723976i −55.4681 6.94934i 33.8369i −23.8782 + 23.8782i 126.511 + 126.511i 208.012i −252.478 + 196.258i
32.19 4.43002 4.43002i 16.3501 16.3501i 7.25019i 27.0484 48.9222i 144.862i −30.3495 + 30.3495i 109.642 + 109.642i 291.649i −96.9014 336.552i
32.20 4.80775 4.80775i −19.8987 + 19.8987i 14.2290i −6.50942 55.5214i 191.336i 113.651 113.651i 85.4386 + 85.4386i 548.917i −298.229 235.638i
See all 52 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 32.26 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
11.b odd 2 1 inner
55.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 55.6.e.b 52
5.c odd 4 1 inner 55.6.e.b 52
11.b odd 2 1 inner 55.6.e.b 52
55.e even 4 1 inner 55.6.e.b 52

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.6.e.b 52 1.a even 1 1 trivial
55.6.e.b 52 5.c odd 4 1 inner
55.6.e.b 52 11.b odd 2 1 inner
55.6.e.b 52 55.e even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{52} + 51268 T_{2}^{48} + 1084701006 T_{2}^{44} + 12359132998700 T_{2}^{40} + \cdots + 21\!\cdots\!00$$ acting on $$S_{6}^{\mathrm{new}}(55, [\chi])$$.