Properties

Label 161.2.i.a
Level $161$
Weight $2$
Character orbit 161.i
Analytic conductor $1.286$
Analytic rank $0$
Dimension $50$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [161,2,Mod(8,161)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(161, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("161.8");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 161 = 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 161.i (of order \(11\), 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: \(1.28559147254\)
Analytic rank: \(0\)
Dimension: \(50\)
Relative dimension: \(5\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 50 q + 2 q^{2} - 4 q^{4} - 11 q^{5} - 5 q^{6} + 5 q^{7} + 9 q^{8} - 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 50 q + 2 q^{2} - 4 q^{4} - 11 q^{5} - 5 q^{6} + 5 q^{7} + 9 q^{8} - 17 q^{9} - 4 q^{11} + 9 q^{12} + 2 q^{13} - 2 q^{14} + q^{15} - 54 q^{16} + 7 q^{17} + 21 q^{18} + 13 q^{19} - 73 q^{20} + 24 q^{22} + 10 q^{23} - 12 q^{24} + 16 q^{25} + 25 q^{26} - 12 q^{27} - 18 q^{28} + 7 q^{29} + 22 q^{30} - 9 q^{31} - 46 q^{32} + 13 q^{33} + 28 q^{34} - 11 q^{35} - 49 q^{36} + 26 q^{37} - 102 q^{39} + 36 q^{41} + 5 q^{42} - 10 q^{43} + 27 q^{44} - 12 q^{45} + 88 q^{46} - 106 q^{47} + 95 q^{48} - 5 q^{49} - 10 q^{50} - 62 q^{51} - 27 q^{52} - 2 q^{53} + 49 q^{54} - 66 q^{55} - 9 q^{56} + 57 q^{57} - 44 q^{58} - 11 q^{59} + 142 q^{60} - 10 q^{61} + 47 q^{62} - 5 q^{63} + 71 q^{64} + 18 q^{65} - 187 q^{66} - 36 q^{67} - 20 q^{68} + 29 q^{69} + q^{71} - 19 q^{72} - 27 q^{73} - 9 q^{74} + 42 q^{75} + 85 q^{76} + 15 q^{77} - 75 q^{78} + 38 q^{79} + 132 q^{80} + 47 q^{81} - 69 q^{82} + 53 q^{83} + 68 q^{84} - 106 q^{85} - 26 q^{86} + 16 q^{87} + 30 q^{88} + 29 q^{89} + 13 q^{90} + 42 q^{91} - 94 q^{92} - 74 q^{93} - 35 q^{94} + 36 q^{95} + 74 q^{96} + 55 q^{97} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −0.188704 + 1.31247i 0.554762 + 1.21476i 0.232022 + 0.0681278i 0.605110 + 0.698334i −1.69902 + 0.498877i −0.841254 0.540641i −1.23485 + 2.70395i 0.796701 0.919442i −1.03073 + 0.662409i
8.2 −0.0827955 + 0.575856i −0.801904 1.75592i 1.59423 + 0.468109i −2.45011 2.82758i 1.07755 0.316398i −0.841254 0.540641i −0.884916 + 1.93770i −0.475639 + 0.548917i 1.83114 1.17680i
8.3 0.0810785 0.563913i 1.06791 + 2.33839i 1.60756 + 0.472023i −0.715681 0.825940i 1.40523 0.412614i −0.841254 0.540641i 0.869852 1.90471i −2.36306 + 2.72711i −0.523785 + 0.336616i
8.4 0.222230 1.54565i −1.15077 2.51984i −0.420652 0.123514i −0.138046 0.159314i −4.15052 + 1.21870i −0.841254 0.540641i 1.01298 2.21813i −3.06074 + 3.53228i −0.276921 + 0.177967i
8.5 0.252821 1.75841i 0.198117 + 0.433815i −1.10909 0.325658i 2.02744 + 2.33979i 0.812911 0.238692i −0.841254 0.540641i 0.622919 1.36400i 1.81564 2.09536i 4.62688 2.97352i
29.1 −1.13379 + 2.48266i −1.50202 0.441033i −3.56837 4.11812i 1.57859 1.01450i 2.79791 3.22896i 0.142315 0.989821i 9.03219 2.65209i −0.462204 0.297041i 0.728860 + 5.06933i
29.2 −0.813943 + 1.78229i 1.29442 + 0.380077i −1.20432 1.38986i −3.05799 + 1.96525i −1.73099 + 1.99767i 0.142315 0.989821i −0.302592 + 0.0888490i −0.992685 0.637960i −1.01361 7.04982i
29.3 −0.282831 + 0.619314i 0.745240 + 0.218822i 1.00616 + 1.16118i 2.35286 1.51209i −0.346297 + 0.399648i 0.142315 0.989821i −2.31023 + 0.678345i −2.01626 1.29577i 0.270997 + 1.88483i
29.4 0.490936 1.07500i 1.94080 + 0.569869i 0.395116 + 0.455989i −2.35800 + 1.51540i 1.56541 1.80658i 0.142315 0.989821i 2.95201 0.866789i 0.918175 + 0.590075i 0.471422 + 3.27881i
29.5 0.908799 1.98999i −0.286353 0.0840808i −1.82443 2.10551i 0.768064 0.493605i −0.427557 + 0.493427i 0.142315 0.989821i −1.64984 + 0.484437i −2.44883 1.57377i −0.284254 1.97703i
36.1 −1.59835 1.84459i −1.86860 1.20088i −0.563170 + 3.91694i −0.732261 + 1.60343i 0.771545 + 5.36621i 0.959493 + 0.281733i 4.01871 2.58267i 0.803315 + 1.75901i 4.12807 1.21211i
36.2 −0.737191 0.850763i 1.61822 + 1.03996i 0.104281 0.725293i 0.719526 1.57554i −0.308170 2.14337i 0.959493 + 0.281733i −2.58796 + 1.66318i 0.290854 + 0.636882i −1.87084 + 0.549328i
36.3 0.515215 + 0.594590i −1.14233 0.734130i 0.196539 1.36696i 0.0858162 0.187911i −0.152039 1.05745i 0.959493 + 0.281733i 2.23776 1.43812i −0.480277 1.05166i 0.155944 0.0457893i
36.4 1.34132 + 1.54796i 0.455720 + 0.292874i −0.312428 + 2.17298i 0.0741704 0.162411i 0.157908 + 1.09828i 0.959493 + 0.281733i −0.336572 + 0.216302i −1.12434 2.46196i 0.350892 0.103031i
36.5 1.78872 + 2.06430i −2.35987 1.51660i −0.777161 + 5.40527i −1.03647 + 2.26955i −1.09045 7.58425i 0.959493 + 0.281733i −7.95252 + 5.11077i 2.02268 + 4.42905i −6.53898 + 1.92002i
50.1 −1.13379 2.48266i −1.50202 + 0.441033i −3.56837 + 4.11812i 1.57859 + 1.01450i 2.79791 + 3.22896i 0.142315 + 0.989821i 9.03219 + 2.65209i −0.462204 + 0.297041i 0.728860 5.06933i
50.2 −0.813943 1.78229i 1.29442 0.380077i −1.20432 + 1.38986i −3.05799 1.96525i −1.73099 1.99767i 0.142315 + 0.989821i −0.302592 0.0888490i −0.992685 + 0.637960i −1.01361 + 7.04982i
50.3 −0.282831 0.619314i 0.745240 0.218822i 1.00616 1.16118i 2.35286 + 1.51209i −0.346297 0.399648i 0.142315 + 0.989821i −2.31023 0.678345i −2.01626 + 1.29577i 0.270997 1.88483i
50.4 0.490936 + 1.07500i 1.94080 0.569869i 0.395116 0.455989i −2.35800 1.51540i 1.56541 + 1.80658i 0.142315 + 0.989821i 2.95201 + 0.866789i 0.918175 0.590075i 0.471422 3.27881i
50.5 0.908799 + 1.98999i −0.286353 + 0.0840808i −1.82443 + 2.10551i 0.768064 + 0.493605i −0.427557 0.493427i 0.142315 + 0.989821i −1.64984 0.484437i −2.44883 + 1.57377i −0.284254 + 1.97703i
See all 50 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 161.2.i.a 50
23.c even 11 1 inner 161.2.i.a 50
23.c even 11 1 3703.2.a.q 25
23.d odd 22 1 3703.2.a.r 25
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.2.i.a 50 1.a even 1 1 trivial
161.2.i.a 50 23.c even 11 1 inner
3703.2.a.q 25 23.c even 11 1
3703.2.a.r 25 23.d odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{50} - 2 T_{2}^{49} + 9 T_{2}^{48} - 21 T_{2}^{47} + 113 T_{2}^{46} - 195 T_{2}^{45} + 554 T_{2}^{44} - 1316 T_{2}^{43} + 4277 T_{2}^{42} - 7850 T_{2}^{41} + 17228 T_{2}^{40} - 30695 T_{2}^{39} + 89086 T_{2}^{38} - 139147 T_{2}^{37} + \cdots + 279841 \) acting on \(S_{2}^{\mathrm{new}}(161, [\chi])\). Copy content Toggle raw display