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Properties

Label 115.3.h.a
Level $115$
Weight $3$
Character orbit 115.h
Analytic conductor $3.134$
Analytic rank $0$
Dimension $160$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [115,3,Mod(11,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 9]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("115.11");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 115.h (of order \(22\), degree \(10\), 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: \(3.13352304014\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(16\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q + 4 q^{2} - 16 q^{4} - 26 q^{6} - 22 q^{8} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q + 4 q^{2} - 16 q^{4} - 26 q^{6} - 22 q^{8} - 32 q^{9} + 30 q^{12} + 12 q^{13} - 256 q^{16} - 110 q^{17} + 70 q^{18} - 66 q^{19} - 66 q^{21} - 34 q^{23} + 180 q^{24} + 80 q^{25} + 238 q^{26} + 234 q^{27} + 128 q^{29} + 188 q^{31} + 496 q^{32} - 242 q^{34} - 170 q^{35} - 736 q^{36} - 770 q^{38} - 188 q^{39} - 440 q^{40} - 234 q^{41} - 176 q^{43} - 22 q^{44} + 80 q^{46} - 224 q^{47} + 754 q^{48} + 518 q^{49} + 90 q^{50} + 528 q^{51} - 82 q^{52} + 352 q^{53} + 510 q^{54} + 400 q^{55} + 418 q^{56} - 726 q^{57} + 376 q^{58} - 62 q^{59} + 330 q^{60} - 308 q^{61} - 662 q^{62} - 550 q^{63} - 206 q^{64} - 176 q^{66} - 44 q^{67} - 280 q^{69} - 120 q^{70} - 18 q^{71} + 1126 q^{72} + 52 q^{73} + 154 q^{74} + 704 q^{76} - 726 q^{77} - 1434 q^{78} - 572 q^{79} + 476 q^{81} + 46 q^{82} + 286 q^{83} - 1100 q^{84} - 130 q^{85} + 396 q^{86} - 1012 q^{87} - 528 q^{88} - 264 q^{89} + 350 q^{92} + 604 q^{93} + 444 q^{94} - 80 q^{95} - 394 q^{96} + 792 q^{97} + 540 q^{98} + 2112 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −3.20529 + 2.05992i −0.0243799 + 0.169566i 4.36899 − 9.56674i 0.629973 + 2.14549i −0.271147 − 0.593730i −1.96278 + 1.70076i 3.53386 + 24.5785i 8.60728 + 2.52732i −6.43878 − 5.57924i
11.2 −2.96935 + 1.90828i 0.852548 − 5.92960i 3.51382 − 7.69420i −0.629973 − 2.14549i 8.78386 + 19.2340i 4.46805 − 3.87159i 2.23966 + 15.5772i −25.7979 − 7.57495i 5.96482 + 5.16855i
11.3 −2.76404 + 1.77634i −0.745160 + 5.18270i 2.82287 − 6.18121i −0.629973 − 2.14549i −7.14658 − 15.6488i −7.41885 + 6.42847i 1.30706 + 9.09078i −17.6697 − 5.18828i 5.55239 + 4.81117i
11.4 −2.44761 + 1.57298i −0.0493905 + 0.343518i 1.85486 − 4.06157i −0.629973 − 2.14549i −0.419460 − 0.918490i 2.57506 − 2.23130i 0.192568 + 1.33934i 8.51987 + 2.50166i 4.91675 + 4.26039i
11.5 −2.09072 + 1.34362i 0.431029 − 2.99787i 0.904125 − 1.97976i 0.629973 + 2.14549i 3.12685 + 6.84685i −1.58303 + 1.37170i −0.644971 − 4.48587i −0.166015 − 0.0487463i −4.19983 − 3.63918i
11.6 −1.36722 + 0.878660i 0.341987 − 2.37857i −0.564407 + 1.23588i −0.629973 − 2.14549i 1.62239 + 3.55253i −8.63354 + 7.48100i −1.23942 − 8.62036i 3.09477 + 0.908708i 2.74647 + 2.37983i
11.7 −1.00847 + 0.648107i −0.522349 + 3.63302i −1.06468 + 2.33133i 0.629973 + 2.14549i −1.82781 − 4.00234i −2.64490 + 2.29182i −1.11966 − 7.78741i −4.29053 − 1.25981i −2.02582 − 1.75538i
11.8 −0.337085 + 0.216631i 0.283787 − 1.97378i −1.59496 + 3.49248i −0.629973 − 2.14549i 0.331922 + 0.726809i 8.28605 − 7.17990i −0.447041 − 3.10924i 4.82015 + 1.41532i 0.677134 + 0.586740i
11.9 0.235723 − 0.151490i 0.0812454 − 0.565074i −1.62904 + 3.56711i 0.629973 + 2.14549i −0.0664516 − 0.145509i −3.04533 + 2.63880i 0.315887 + 2.19704i 8.32273 + 2.44377i 0.473519 + 0.410307i
11.10 0.291588 − 0.187393i −0.754507 + 5.24771i −1.61175 + 3.52924i −0.629973 − 2.14549i 0.763376 + 1.67156i 4.68392 − 4.05864i 0.388698 + 2.70345i −18.3338 − 5.38328i −0.585742 − 0.507548i
11.11 1.35039 − 0.867843i −0.0102654 + 0.0713975i −0.591258 + 1.29468i 0.629973 + 2.14549i 0.0480995 + 0.105323i 9.09465 − 7.88056i 1.23893 + 8.61693i 8.63044 + 2.53413i 2.71266 + 2.35053i
11.12 2.25069 − 1.44643i 0.464668 − 3.23184i 1.31179 − 2.87243i −0.629973 − 2.14549i −3.62881 − 7.94599i −1.48905 + 1.29027i 0.320672 + 2.23032i −1.59344 − 0.467876i −4.52119 − 3.91763i
11.13 2.25259 − 1.44765i −0.677841 + 4.71449i 1.31681 − 2.88342i 0.629973 + 2.14549i 5.29805 + 11.6011i −1.25341 + 1.08608i 0.316337 + 2.20017i −13.1315 − 3.85576i 4.52500 + 3.92094i
11.14 2.28753 − 1.47011i 0.630718 − 4.38674i 1.40992 − 3.08729i 0.629973 + 2.14549i −5.00618 − 10.9620i 2.08243 − 1.80444i 0.234513 + 1.63107i −10.2102 − 2.99799i 4.59518 + 3.98175i
11.15 2.92018 − 1.87668i −0.393934 + 2.73987i 3.34383 − 7.32197i −0.629973 − 2.14549i 3.99152 + 8.74020i 4.90483 − 4.25006i −2.00042 − 13.9132i 1.28373 + 0.376936i −5.86604 − 5.08296i
11.16 3.20322 − 2.05859i 0.0918438 − 0.638788i 4.36119 − 9.54968i 0.629973 + 2.14549i −1.02080 − 2.23525i −8.06410 + 6.98759i −3.52141 − 24.4920i 8.23582 + 2.41826i 6.43462 + 5.57563i
21.1 −3.20529 − 2.05992i −0.0243799 − 0.169566i 4.36899 + 9.56674i 0.629973 − 2.14549i −0.271147 + 0.593730i −1.96278 − 1.70076i 3.53386 − 24.5785i 8.60728 − 2.52732i −6.43878 + 5.57924i
21.2 −2.96935 − 1.90828i 0.852548 + 5.92960i 3.51382 + 7.69420i −0.629973 + 2.14549i 8.78386 − 19.2340i 4.46805 + 3.87159i 2.23966 − 15.5772i −25.7979 + 7.57495i 5.96482 − 5.16855i
21.3 −2.76404 − 1.77634i −0.745160 − 5.18270i 2.82287 + 6.18121i −0.629973 + 2.14549i −7.14658 + 15.6488i −7.41885 − 6.42847i 1.30706 − 9.09078i −17.6697 + 5.18828i 5.55239 − 4.81117i
21.4 −2.44761 − 1.57298i −0.0493905 − 0.343518i 1.85486 + 4.06157i −0.629973 + 2.14549i −0.419460 + 0.918490i 2.57506 + 2.23130i 0.192568 − 1.33934i 8.51987 − 2.50166i 4.91675 − 4.26039i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 111.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.d odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.3.h.a ✓ 160
23.d odd 22 1 inner 115.3.h.a ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.3.h.a ✓ 160 1.a even 1 1 trivial
115.3.h.a ✓ 160 23.d odd 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(115, [\chi])\).