# Properties

 Label 176.2.j.a Level $176$ Weight $2$ Character orbit 176.j Analytic conductor $1.405$ Analytic rank $0$ Dimension $40$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [176,2,Mod(45,176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(176, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 0]))

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

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

chi := DirichletCharacter("176.45");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$176 = 2^{4} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 176.j (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: $$1.40536707557$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$20$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40 q - 12 q^{6} - 12 q^{8}+O(q^{10})$$ 40 * q - 12 * q^6 - 12 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$40 q - 12 q^{6} - 12 q^{8} - 4 q^{10} - 12 q^{12} + 8 q^{14} - 16 q^{15} - 8 q^{16} - 20 q^{18} - 16 q^{19} - 20 q^{20} + 28 q^{24} + 24 q^{27} + 20 q^{28} + 12 q^{30} + 24 q^{31} - 20 q^{32} - 40 q^{34} + 24 q^{35} + 20 q^{36} + 48 q^{38} - 8 q^{40} - 8 q^{44} - 52 q^{46} - 40 q^{47} - 40 q^{49} + 64 q^{50} - 24 q^{51} - 36 q^{52} + 32 q^{54} - 8 q^{56} + 40 q^{58} - 8 q^{59} + 12 q^{60} + 32 q^{61} - 24 q^{64} - 16 q^{65} - 20 q^{66} - 24 q^{67} + 4 q^{68} + 32 q^{69} - 8 q^{70} - 20 q^{72} - 56 q^{74} - 56 q^{75} + 4 q^{76} + 64 q^{78} + 32 q^{79} + 32 q^{80} - 40 q^{81} + 24 q^{82} + 40 q^{83} + 76 q^{84} - 32 q^{85} - 16 q^{86} + 20 q^{90} - 16 q^{91} + 12 q^{92} - 48 q^{93} + 16 q^{94} - 48 q^{95} + 36 q^{96} + 72 q^{98}+O(q^{100})$$ 40 * q - 12 * q^6 - 12 * q^8 - 4 * q^10 - 12 * q^12 + 8 * q^14 - 16 * q^15 - 8 * q^16 - 20 * q^18 - 16 * q^19 - 20 * q^20 + 28 * q^24 + 24 * q^27 + 20 * q^28 + 12 * q^30 + 24 * q^31 - 20 * q^32 - 40 * q^34 + 24 * q^35 + 20 * q^36 + 48 * q^38 - 8 * q^40 - 8 * q^44 - 52 * q^46 - 40 * q^47 - 40 * q^49 + 64 * q^50 - 24 * q^51 - 36 * q^52 + 32 * q^54 - 8 * q^56 + 40 * q^58 - 8 * q^59 + 12 * q^60 + 32 * q^61 - 24 * q^64 - 16 * q^65 - 20 * q^66 - 24 * q^67 + 4 * q^68 + 32 * q^69 - 8 * q^70 - 20 * q^72 - 56 * q^74 - 56 * q^75 + 4 * q^76 + 64 * q^78 + 32 * q^79 + 32 * q^80 - 40 * q^81 + 24 * q^82 + 40 * q^83 + 76 * q^84 - 32 * q^85 - 16 * q^86 + 20 * q^90 - 16 * q^91 + 12 * q^92 - 48 * q^93 + 16 * q^94 - 48 * q^95 + 36 * q^96 + 72 * q^98

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
45.1 −1.41184 + 0.0819819i 0.529792 + 0.529792i 1.98656 0.231490i 1.79526 1.79526i −0.791413 0.704546i 1.13150i −2.78571 + 0.489687i 2.43864i −2.38743 + 2.68179i
45.2 −1.39935 + 0.204478i −2.27066 2.27066i 1.91638 0.572275i 1.24014 1.24014i 3.64176 + 2.71316i 3.02382i −2.56467 + 1.19267i 7.31182i −1.48181 + 1.98898i
45.3 −1.25745 0.647159i −0.273979 0.273979i 1.16237 + 1.62754i −2.48024 + 2.48024i 0.167207 + 0.521823i 4.65464i −0.408346 2.79880i 2.84987i 4.72389 1.51367i
45.4 −1.21415 0.725142i −1.27582 1.27582i 0.948339 + 1.76087i −0.795196 + 0.795196i 0.623892 + 2.47419i 4.67475i 0.125448 2.82564i 0.255441i 1.54212 0.388861i
45.5 −1.15225 0.819948i 2.04787 + 2.04787i 0.655370 + 1.88957i 0.814963 0.814963i −0.680516 4.03881i 2.42498i 0.794202 2.71464i 5.38755i −1.60727 + 0.270815i
45.6 −1.15173 + 0.820676i 2.16054 + 2.16054i 0.652980 1.89040i −1.53482 + 1.53482i −4.26148 0.715263i 0.310169i 0.799348 + 2.71312i 6.33589i 0.508112 3.02729i
45.7 −0.526471 1.31257i −0.151436 0.151436i −1.44566 + 1.38205i 1.85628 1.85628i −0.119043 + 0.278497i 0.615059i 2.57513 + 1.16991i 2.95413i −3.41376 1.45921i
45.8 −0.403781 1.35535i 1.28225 + 1.28225i −1.67392 + 1.09452i −2.06176 + 2.06176i 1.22014 2.25564i 3.54041i 2.15936 + 1.82679i 0.288318i 3.62690 + 1.96190i
45.9 −0.398671 + 1.35686i 1.34498 + 1.34498i −1.68212 1.08188i 2.40794 2.40794i −2.36114 + 1.28874i 4.88654i 2.13857 1.85109i 0.617917i 2.30726 + 4.22721i
45.10 −0.0986166 + 1.41077i −1.15526 1.15526i −1.98055 0.278251i 0.731846 0.731846i 1.74374 1.51588i 2.61186i 0.587863 2.76666i 0.330747i 0.960295 + 1.10464i
45.11 0.0241205 + 1.41401i 0.791470 + 0.791470i −1.99884 + 0.0682132i −2.78615 + 2.78615i −1.10005 + 1.13823i 0.0391973i −0.144667 2.82473i 1.74715i −4.00683 3.87243i
45.12 0.494240 1.32504i 0.190424 + 0.190424i −1.51145 1.30977i 0.291418 0.291418i 0.346435 0.158204i 2.64639i −2.48252 + 1.35539i 2.92748i −0.242109 0.530170i
45.13 0.686639 + 1.23634i 1.72812 + 1.72812i −1.05705 + 1.69783i 0.893733 0.893733i −0.949940 + 3.32312i 2.98840i −2.82491 0.141074i 2.97277i 1.71863 + 0.491282i
45.14 0.783980 1.17702i −1.97112 1.97112i −0.770751 1.84552i 2.29075 2.29075i −3.86537 + 0.774730i 3.79929i −2.77647 0.539662i 4.77065i −0.900354 4.49215i
45.15 0.947542 + 1.04984i −2.24228 2.24228i −0.204327 + 1.98954i −2.71114 + 2.71114i 0.229379 4.47868i 1.46050i −2.28230 + 1.67066i 7.05562i −5.41518 0.277342i
45.16 0.988599 1.01127i 1.60387 + 1.60387i −0.0453448 1.99949i −0.215519 + 0.215519i 3.20753 0.0363659i 1.18021i −2.06685 1.93083i 2.14479i 0.00488664 + 0.431010i
45.17 1.10124 + 0.887278i −1.15765 1.15765i 0.425474 + 1.95422i 2.69017 2.69017i −0.247696 2.30200i 0.0889173i −1.26539 + 2.52958i 0.319711i 5.34946 0.575603i
45.18 1.28704 + 0.586108i 0.586507 + 0.586507i 1.31295 + 1.50869i −1.32116 + 1.32116i 0.411103 + 1.09862i 1.42100i 0.805572 + 2.71128i 2.31202i −2.47473 + 0.926044i
45.19 1.28990 0.579802i −1.51913 1.51913i 1.32766 1.49577i −1.06305 + 1.06305i −2.84031 1.07872i 3.12852i 0.845294 2.69916i 1.61549i −0.754864 + 1.98758i
45.20 1.41102 0.0950426i −0.248483 0.248483i 1.98193 0.268213i −0.0434775 + 0.0434775i −0.374231 0.326998i 3.27975i 2.77105 0.566821i 2.87651i −0.0572153 + 0.0654797i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 45.20 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 176.2.j.a 40
4.b odd 2 1 704.2.j.a 40
8.b even 2 1 1408.2.j.a 40
8.d odd 2 1 1408.2.j.b 40
16.e even 4 1 inner 176.2.j.a 40
16.e even 4 1 1408.2.j.a 40
16.f odd 4 1 704.2.j.a 40
16.f odd 4 1 1408.2.j.b 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
176.2.j.a 40 1.a even 1 1 trivial
176.2.j.a 40 16.e even 4 1 inner
704.2.j.a 40 4.b odd 2 1
704.2.j.a 40 16.f odd 4 1
1408.2.j.a 40 8.b even 2 1
1408.2.j.a 40 16.e even 4 1
1408.2.j.b 40 8.d odd 2 1
1408.2.j.b 40 16.f odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(176, [\chi])$$.