# Properties

 Label 224.2.x.b Level $224$ Weight $2$ Character orbit 224.x Analytic conductor $1.789$ Analytic rank $0$ Dimension $112$ CM no Inner twists $4$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [224,2,Mod(27,224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(224, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([4, 1, 4]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("224.27");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 224.x (of order $$8$$, degree $$4$$, 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: $$1.78864900528$$ Analytic rank: $$0$$ Dimension: $$112$$ Relative dimension: $$28$$ over $$\Q(\zeta_{8})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$112 q - 8 q^{2} - 8 q^{4} - 4 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10})$$ 112 * q - 8 * q^2 - 8 * q^4 - 4 * q^7 - 8 * q^8 - 8 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$112 q - 8 q^{2} - 8 q^{4} - 4 q^{7} - 8 q^{8} - 8 q^{9} - 8 q^{11} - 20 q^{14} - 16 q^{15} + 16 q^{16} - 16 q^{18} - 4 q^{21} + 32 q^{22} - 48 q^{23} - 8 q^{25} - 24 q^{28} - 8 q^{29} + 16 q^{30} - 8 q^{32} + 20 q^{35} + 32 q^{36} - 8 q^{37} - 8 q^{39} + 16 q^{42} + 32 q^{43} - 144 q^{44} - 8 q^{46} + 16 q^{50} - 32 q^{51} - 32 q^{53} + 64 q^{56} - 8 q^{57} - 72 q^{58} - 88 q^{60} + 112 q^{64} - 16 q^{65} - 64 q^{67} - 40 q^{70} + 56 q^{71} - 8 q^{72} - 168 q^{74} + 52 q^{77} + 192 q^{78} - 16 q^{79} - 24 q^{84} - 48 q^{85} - 8 q^{86} - 96 q^{88} - 52 q^{91} - 8 q^{92} - 32 q^{93} + 88 q^{98} + 16 q^{99}+O(q^{100})$$ 112 * q - 8 * q^2 - 8 * q^4 - 4 * q^7 - 8 * q^8 - 8 * q^9 - 8 * q^11 - 20 * q^14 - 16 * q^15 + 16 * q^16 - 16 * q^18 - 4 * q^21 + 32 * q^22 - 48 * q^23 - 8 * q^25 - 24 * q^28 - 8 * q^29 + 16 * q^30 - 8 * q^32 + 20 * q^35 + 32 * q^36 - 8 * q^37 - 8 * q^39 + 16 * q^42 + 32 * q^43 - 144 * q^44 - 8 * q^46 + 16 * q^50 - 32 * q^51 - 32 * q^53 + 64 * q^56 - 8 * q^57 - 72 * q^58 - 88 * q^60 + 112 * q^64 - 16 * q^65 - 64 * q^67 - 40 * q^70 + 56 * q^71 - 8 * q^72 - 168 * q^74 + 52 * q^77 + 192 * q^78 - 16 * q^79 - 24 * q^84 - 48 * q^85 - 8 * q^86 - 96 * q^88 - 52 * q^91 - 8 * q^92 - 32 * q^93 + 88 * q^98 + 16 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
27.1 −1.40726 0.140045i −0.311327 0.751610i 1.96078 + 0.394159i −1.37157 + 3.31126i 0.332860 + 1.10131i −2.28614 1.33176i −2.70413 0.829281i 1.65333 1.65333i 2.39388 4.46773i
27.2 −1.40726 0.140045i 0.311327 + 0.751610i 1.96078 + 0.394159i 1.37157 3.31126i −0.332860 1.10131i 1.33176 + 2.28614i −2.70413 0.829281i 1.65333 1.65333i −2.39388 + 4.46773i
27.3 −1.36868 + 0.355955i −1.01638 2.45376i 1.74659 0.974380i 1.11188 2.68430i 2.26454 + 2.99664i 0.0435134 2.64539i −2.04370 + 1.95533i −2.86660 + 2.86660i −0.566314 + 4.06974i
27.4 −1.36868 + 0.355955i 1.01638 + 2.45376i 1.74659 0.974380i −1.11188 + 2.68430i −2.26454 2.99664i 2.64539 0.0435134i −2.04370 + 1.95533i −2.86660 + 2.86660i 0.566314 4.06974i
27.5 −1.23575 0.687692i −0.592616 1.43070i 1.05416 + 1.69963i −0.188106 + 0.454128i −0.251557 + 2.17553i 2.64138 + 0.152070i −0.133852 2.82526i 0.425604 0.425604i 0.544752 0.431830i
27.6 −1.23575 0.687692i 0.592616 + 1.43070i 1.05416 + 1.69963i 0.188106 0.454128i 0.251557 2.17553i −0.152070 2.64138i −0.133852 2.82526i 0.425604 0.425604i −0.544752 + 0.431830i
27.7 −0.783777 + 1.17715i −0.935381 2.25821i −0.771386 1.84525i 0.395947 0.955900i 3.39139 + 0.668846i −0.935627 + 2.47479i 2.77674 + 0.538228i −2.10326 + 2.10326i 0.814908 + 1.21530i
27.8 −0.783777 + 1.17715i 0.935381 + 2.25821i −0.771386 1.84525i −0.395947 + 0.955900i −3.39139 0.668846i −2.47479 + 0.935627i 2.77674 + 0.538228i −2.10326 + 2.10326i −0.814908 1.21530i
27.9 −0.756841 + 1.19465i −0.613776 1.48179i −0.854385 1.80832i −1.25513 + 3.03015i 2.23475 + 0.388228i 1.88664 1.85488i 2.80695 + 0.347919i 0.302349 0.302349i −2.67004 3.79278i
27.10 −0.756841 + 1.19465i 0.613776 + 1.48179i −0.854385 1.80832i 1.25513 3.03015i −2.23475 0.388228i 1.85488 1.88664i 2.80695 + 0.347919i 0.302349 0.302349i 2.67004 + 3.79278i
27.11 −0.522376 1.31420i −0.167131 0.403490i −1.45425 + 1.37301i −0.883342 + 2.13258i −0.442962 + 0.430417i 0.895629 + 2.48955i 2.56408 + 1.19394i 1.98645 1.98645i 3.26407 + 0.0468829i
27.12 −0.522376 1.31420i 0.167131 + 0.403490i −1.45425 + 1.37301i 0.883342 2.13258i 0.442962 0.430417i −2.48955 0.895629i 2.56408 + 1.19394i 1.98645 1.98645i −3.26407 0.0468829i
27.13 −0.242331 1.39330i −1.25862 3.03857i −1.88255 + 0.675278i −0.824079 + 1.98950i −3.92863 + 2.48997i −0.0837494 2.64443i 1.39706 + 2.45931i −5.52747 + 5.52747i 2.97167 + 0.666069i
27.14 −0.242331 1.39330i 1.25862 + 3.03857i −1.88255 + 0.675278i 0.824079 1.98950i 3.92863 2.48997i 2.64443 + 0.0837494i 1.39706 + 2.45931i −5.52747 + 5.52747i −2.97167 0.666069i
27.15 0.207749 + 1.39887i −0.517660 1.24974i −1.91368 + 0.581228i 0.818546 1.97615i 1.64068 0.983771i −1.50177 2.17823i −1.21063 2.55624i 0.827440 0.827440i 2.93442 + 0.734499i
27.16 0.207749 + 1.39887i 0.517660 + 1.24974i −1.91368 + 0.581228i −0.818546 + 1.97615i −1.64068 + 0.983771i 2.17823 + 1.50177i −1.21063 2.55624i 0.827440 0.827440i −2.93442 0.734499i
27.17 0.329116 1.37538i −0.884481 2.13533i −1.78336 0.905323i 1.60624 3.87780i −3.22799 + 0.513731i 0.663558 + 2.56119i −1.83210 + 2.15486i −1.65599 + 1.65599i −4.80483 3.48544i
27.18 0.329116 1.37538i 0.884481 + 2.13533i −1.78336 0.905323i −1.60624 + 3.87780i 3.22799 0.513731i −2.56119 0.663558i −1.83210 + 2.15486i −1.65599 + 1.65599i 4.80483 + 3.48544i
27.19 0.589446 + 1.28552i −1.08865 2.62824i −1.30511 + 1.51548i −1.22393 + 2.95482i 2.73695 2.94869i −1.81057 + 1.92921i −2.71747 0.784442i −3.60116 + 3.60116i −4.51992 + 0.168328i
27.20 0.589446 + 1.28552i 1.08865 + 2.62824i −1.30511 + 1.51548i 1.22393 2.95482i −2.73695 + 2.94869i −1.92921 + 1.81057i −2.71747 0.784442i −3.60116 + 3.60116i 4.51992 0.168328i
See next 80 embeddings (of 112 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 195.28 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
32.h odd 8 1 inner
224.x even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.2.x.b 112
4.b odd 2 1 896.2.x.b 112
7.b odd 2 1 inner 224.2.x.b 112
28.d even 2 1 896.2.x.b 112
32.g even 8 1 896.2.x.b 112
32.h odd 8 1 inner 224.2.x.b 112
224.v odd 8 1 896.2.x.b 112
224.x even 8 1 inner 224.2.x.b 112

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.x.b 112 1.a even 1 1 trivial
224.2.x.b 112 7.b odd 2 1 inner
224.2.x.b 112 32.h odd 8 1 inner
224.2.x.b 112 224.x even 8 1 inner
896.2.x.b 112 4.b odd 2 1
896.2.x.b 112 28.d even 2 1
896.2.x.b 112 32.g even 8 1
896.2.x.b 112 224.v odd 8 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{112} + 4 T_{3}^{110} + 8 T_{3}^{108} - 168 T_{3}^{106} + 27036 T_{3}^{104} + 96192 T_{3}^{102} + 182592 T_{3}^{100} - 3442432 T_{3}^{98} + 258527648 T_{3}^{96} + 826311552 T_{3}^{94} + \cdots + 50\!\cdots\!76$$ acting on $$S_{2}^{\mathrm{new}}(224, [\chi])$$.