# Properties

 Label 154.2.n.a Level $154$ Weight $2$ Character orbit 154.n Analytic conductor $1.230$ Analytic rank $0$ Dimension $64$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [154,2,Mod(17,154)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(154, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([5, 27]))

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

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

chi := DirichletCharacter("154.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$154 = 2 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 154.n (of order $$30$$, degree $$8$$, 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: $$1.22969619113$$ Analytic rank: $$0$$ Dimension: $$64$$ Relative dimension: $$8$$ over $$\Q(\zeta_{30})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$64 q - 8 q^{4} + 12 q^{5} - 10 q^{7} + 4 q^{9}+O(q^{10})$$ 64 * q - 8 * q^4 + 12 * q^5 - 10 * q^7 + 4 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$64 q - 8 q^{4} + 12 q^{5} - 10 q^{7} + 4 q^{9} - 8 q^{11} - 2 q^{14} - 12 q^{15} + 8 q^{16} - 30 q^{17} + 8 q^{22} - 16 q^{23} - 20 q^{25} - 24 q^{26} + 10 q^{28} + 20 q^{29} - 18 q^{31} - 126 q^{33} + 30 q^{35} - 32 q^{36} + 16 q^{37} - 12 q^{38} + 20 q^{39} - 30 q^{40} - 2 q^{42} - 2 q^{44} + 108 q^{45} - 24 q^{47} - 78 q^{49} - 60 q^{51} + 8 q^{53} + 4 q^{56} - 80 q^{57} - 28 q^{58} - 60 q^{59} + 4 q^{60} + 30 q^{61} + 50 q^{63} + 16 q^{64} + 72 q^{66} - 32 q^{67} + 30 q^{68} - 10 q^{70} - 8 q^{71} + 20 q^{72} + 90 q^{73} + 20 q^{74} + 180 q^{75} + 46 q^{77} + 96 q^{78} + 30 q^{79} + 18 q^{80} + 48 q^{81} + 60 q^{82} + 40 q^{84} + 140 q^{85} + 10 q^{86} + 14 q^{88} - 36 q^{89} + 106 q^{91} + 28 q^{92} + 94 q^{93} - 120 q^{94} - 70 q^{95} + 104 q^{99}+O(q^{100})$$ 64 * q - 8 * q^4 + 12 * q^5 - 10 * q^7 + 4 * q^9 - 8 * q^11 - 2 * q^14 - 12 * q^15 + 8 * q^16 - 30 * q^17 + 8 * q^22 - 16 * q^23 - 20 * q^25 - 24 * q^26 + 10 * q^28 + 20 * q^29 - 18 * q^31 - 126 * q^33 + 30 * q^35 - 32 * q^36 + 16 * q^37 - 12 * q^38 + 20 * q^39 - 30 * q^40 - 2 * q^42 - 2 * q^44 + 108 * q^45 - 24 * q^47 - 78 * q^49 - 60 * q^51 + 8 * q^53 + 4 * q^56 - 80 * q^57 - 28 * q^58 - 60 * q^59 + 4 * q^60 + 30 * q^61 + 50 * q^63 + 16 * q^64 + 72 * q^66 - 32 * q^67 + 30 * q^68 - 10 * q^70 - 8 * q^71 + 20 * q^72 + 90 * q^73 + 20 * q^74 + 180 * q^75 + 46 * q^77 + 96 * q^78 + 30 * q^79 + 18 * q^80 + 48 * q^81 + 60 * q^82 + 40 * q^84 + 140 * q^85 + 10 * q^86 + 14 * q^88 - 36 * q^89 + 106 * q^91 + 28 * q^92 + 94 * q^93 - 120 * q^94 - 70 * q^95 + 104 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1 −0.743145 0.669131i −1.26932 2.85094i 0.104528 + 0.994522i −0.660681 3.10826i −0.964361 + 2.96800i 2.06657 + 1.65206i 0.587785 0.809017i −4.50928 + 5.00806i −1.58885 + 2.75197i
17.2 −0.743145 0.669131i −0.288057 0.646987i 0.104528 + 0.994522i 0.507623 + 2.38818i −0.218851 + 0.673553i 2.40050 + 1.11248i 0.587785 0.809017i 1.67178 1.85670i 1.22077 2.11443i
17.3 −0.743145 0.669131i 0.242993 + 0.545771i 0.104528 + 0.994522i −0.828137 3.89608i 0.184613 0.568181i −2.64015 + 0.172111i 0.587785 0.809017i 1.76857 1.96420i −1.99156 + 3.44948i
17.4 −0.743145 0.669131i 1.31438 + 2.95215i 0.104528 + 0.994522i −0.0248203 0.116770i 0.998599 3.07337i 1.24260 2.33580i 0.587785 0.809017i −4.98021 + 5.53108i −0.0596895 + 0.103385i
17.5 0.743145 + 0.669131i −0.795049 1.78571i 0.104528 + 0.994522i −0.578901 2.72352i 0.604036 1.85903i −0.540264 2.59000i −0.587785 + 0.809017i −0.549265 + 0.610021i 1.39218 2.41133i
17.6 0.743145 + 0.669131i −0.655191 1.47158i 0.104528 + 0.994522i 0.737883 + 3.47147i 0.497779 1.53201i 2.53479 0.758189i −0.587785 + 0.809017i 0.271110 0.301098i −1.77451 + 3.07354i
17.7 0.743145 + 0.669131i 0.491544 + 1.10403i 0.104528 + 0.994522i −0.307168 1.44511i −0.373449 + 1.14936i 1.23185 + 2.34148i −0.587785 + 0.809017i 1.03013 1.14408i 0.738698 1.27946i
17.8 0.743145 + 0.669131i 0.958696 + 2.15327i 0.104528 + 0.994522i −0.0111506 0.0524597i −0.728367 + 2.24168i −1.55622 2.13967i −0.587785 + 0.809017i −1.71007 + 1.89922i 0.0268158 0.0464464i
19.1 −0.994522 + 0.104528i −1.54204 1.38846i 0.978148 0.207912i 1.40155 + 3.14793i 1.67872 + 1.21966i 0.662121 2.56156i −0.951057 + 0.309017i 0.136481 + 1.29853i −1.72292 2.98419i
19.2 −0.994522 + 0.104528i −1.17132 1.05466i 0.978148 0.207912i −0.450544 1.01194i 1.27514 + 0.926445i −2.33469 + 1.24468i −0.951057 + 0.309017i −0.0539063 0.512885i 0.553852 + 0.959299i
19.3 −0.994522 + 0.104528i 0.679730 + 0.612031i 0.978148 0.207912i 0.418703 + 0.940423i −0.739981 0.537628i 0.184148 + 2.63934i −0.951057 + 0.309017i −0.226135 2.15153i −0.514710 0.891504i
19.4 −0.994522 + 0.104528i 2.03362 + 1.83108i 0.978148 0.207912i 0.405705 + 0.911227i −2.21388 1.60848i 1.62068 2.09127i −0.951057 + 0.309017i 0.469175 + 4.46390i −0.498731 0.863828i
19.5 0.994522 0.104528i −1.40184 1.26222i 0.978148 0.207912i −1.61021 3.61659i −1.52610 1.10878i −1.72212 + 2.00856i 0.951057 0.309017i 0.0583646 + 0.555302i −1.97943 3.42847i
19.6 0.994522 0.104528i −0.572295 0.515297i 0.978148 0.207912i 0.136527 + 0.306644i −0.623023 0.452653i 2.46505 0.960996i 0.951057 0.309017i −0.251594 2.39376i 0.167832 + 0.290693i
19.7 0.994522 0.104528i 0.118353 + 0.106566i 0.978148 0.207912i 1.34550 + 3.02205i 0.128844 + 0.0936108i −2.10827 + 1.59850i 0.951057 0.309017i −0.310934 2.95834i 1.65402 + 2.86485i
19.8 0.994522 0.104528i 1.85578 + 1.67095i 0.978148 0.207912i −0.776435 1.74390i 2.02028 + 1.46782i −2.08284 1.63149i 0.951057 0.309017i 0.338255 + 3.21829i −0.954469 1.65319i
61.1 −0.207912 + 0.978148i −3.10364 0.326206i −0.913545 0.406737i 2.36149 2.12630i 0.964361 2.96800i 0.932594 + 2.47594i 0.587785 0.809017i 6.59174 + 1.40112i 1.58885 + 2.75197i
61.2 −0.207912 + 0.978148i −0.704336 0.0740287i −0.913545 0.406737i −1.81441 + 1.63370i 0.218851 0.673553i 0.316242 + 2.62678i 0.587785 0.809017i −2.44383 0.519453i −1.22077 2.11443i
61.3 −0.207912 + 0.978148i 0.594148 + 0.0624475i −0.913545 0.406737i 2.96003 2.66523i −0.184613 + 0.568181i 0.979538 2.45774i 0.587785 0.809017i −2.58533 0.549529i 1.99156 + 3.44948i
61.4 −0.207912 + 0.978148i 3.21383 + 0.337787i −0.913545 0.406737i 0.0887159 0.0798801i −0.998599 + 3.07337i −2.60546 + 0.459985i 0.587785 0.809017i 7.28016 + 1.54745i 0.0596895 + 0.103385i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 17.8 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.d odd 6 1 inner
11.d odd 10 1 inner
77.n even 30 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 154.2.n.a 64
7.d odd 6 1 inner 154.2.n.a 64
11.d odd 10 1 inner 154.2.n.a 64
77.n even 30 1 inner 154.2.n.a 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.n.a 64 1.a even 1 1 trivial
154.2.n.a 64 7.d odd 6 1 inner
154.2.n.a 64 11.d odd 10 1 inner
154.2.n.a 64 77.n even 30 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(154, [\chi])$$.