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Properties

Label 161.4.i.b
Level $161$
Weight $4$
Character orbit 161.i
Analytic conductor $9.499$
Analytic rank $0$
Dimension $190$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [161,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("161.8");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 161 = 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(9.49930751092\)
Analytic rank: \(0\)
Dimension: \(190\)
Relative dimension: \(19\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 190 q + 2 q^{2} - 2 q^{3} - 58 q^{4} - 26 q^{5} - 9 q^{6} + 133 q^{7} - 27 q^{8} - 589 q^{9} + 12 q^{10} + 122 q^{11} - 117 q^{12} + 64 q^{13} - 14 q^{14} + 582 q^{15} - 626 q^{16} + 104 q^{17} + 33 q^{18}+ \cdots + 1926 q^{99}+O(q^{100}) \) 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 −0.784639 + 5.45728i 3.79863 + 8.31783i −21.4903 − 6.31013i −3.66136 − 4.22543i −48.3733 + 14.2037i −5.88877 − 3.78449i 32.9755 − 72.2063i −37.0756 + 42.7875i 25.9322 − 16.6656i
8.2 −0.725440 + 5.04554i −0.932678 − 2.04228i −17.2553 − 5.06661i 4.25990 + 4.91618i 10.9810 − 3.22432i −5.88877 − 3.78449i 21.1411 − 46.2926i 14.3802 − 16.5957i −27.8951 + 17.9271i
8.3 −0.716011 + 4.97996i −4.21093 − 9.22064i −16.6114 − 4.87755i −12.9523 − 14.9478i 48.9335 − 14.3682i −5.88877 − 3.78449i 19.4638 − 42.6197i −49.6071 + 57.2496i 83.7134 − 53.7993i
8.4 −0.597539 + 4.15598i 1.67107 + 3.65913i −9.23916 − 2.71286i 12.3462 + 14.2482i −16.2058 + 4.75846i −5.88877 − 3.78449i 2.84168 − 6.22241i 7.08445 − 8.17589i −66.5926 + 42.7965i
8.5 −0.500080 + 3.47813i −2.21254 − 4.84479i −4.17136 − 1.22482i 1.39913 + 1.61468i 17.9573 − 5.27273i −5.88877 − 3.78449i −5.33170 + 11.6748i −0.895414 + 1.03336i −6.31573 + 4.05887i
8.6 −0.493587 + 3.43297i 2.21824 + 4.85727i −3.86571 − 1.13508i −13.0052 − 15.0088i −17.7697 + 5.21767i −5.88877 − 3.78449i −5.72144 + 12.5282i −0.991220 + 1.14393i 57.9439 − 37.2383i
8.7 −0.395830 + 2.75306i −0.928144 − 2.03235i 0.253281 + 0.0743700i −5.55132 − 6.40657i 5.96258 − 1.75077i −5.88877 − 3.78449i −9.54839 + 20.9081i 14.4122 − 16.6326i 19.8351 − 12.7472i
8.8 −0.241487 + 1.67958i 4.17788 + 9.14828i 4.91326 + 1.44266i 6.99290 + 8.07023i −16.3742 + 4.80790i −5.88877 − 3.78449i −9.24875 + 20.2519i −48.5551 + 56.0356i −15.2433 + 9.79628i
8.9 −0.232852 + 1.61952i 2.64230 + 5.78584i 5.10731 + 1.49964i −4.13270 − 4.76939i −9.98557 + 2.93203i −5.88877 − 3.78449i −9.05549 + 19.8288i −8.81293 + 10.1707i 8.68644 − 5.58244i
8.10 −0.196624 + 1.36755i −3.87809 − 8.49182i 5.84442 + 1.71608i 12.3707 + 14.2765i 12.3755 − 3.63377i −5.88877 − 3.78449i −8.08750 + 17.7092i −39.3903 + 45.4588i −21.9562 + 14.1104i
8.11 −0.0500600 + 0.348175i −2.15442 − 4.71752i 7.55722 + 2.21900i −3.24150 − 3.74089i 1.75037 − 0.513956i −5.88877 − 3.78449i −2.31991 + 5.07990i 0.0677575 − 0.0781963i 1.46476 − 0.941341i
8.12 0.189790 − 1.32002i 1.98779 + 4.35265i 5.96952 + 1.75281i 9.00589 + 10.3933i 6.12284 − 1.79783i −5.88877 − 3.78449i 7.87864 − 17.2518i 2.68697 − 3.10093i 15.4286 − 9.91537i
8.13 0.192108 − 1.33614i 0.975006 + 2.13496i 5.92757 + 1.74049i −3.18903 − 3.68033i 3.03992 − 0.892602i −5.88877 − 3.78449i 7.95037 − 17.4089i 14.0738 − 16.2420i −5.53009 + 3.55397i
8.14 0.342734 − 2.38377i −0.715276 − 1.56623i 2.11105 + 0.619862i −13.4984 − 15.5780i −3.97869 + 1.16825i −5.88877 − 3.78449i 10.2046 − 22.3450i 15.7398 − 18.1647i −41.7606 + 26.8379i
8.15 0.391154 − 2.72053i −3.50684 − 7.67891i 0.427647 + 0.125569i −2.98289 − 3.44243i −22.2624 + 6.53685i −5.88877 − 3.78449i 9.64306 − 21.1154i −28.9865 + 33.4522i −10.5320 + 6.76852i
8.16 0.425855 − 2.96189i −1.87521 − 4.10613i −0.915470 − 0.268806i 12.1922 + 14.0705i −12.9605 + 3.80554i −5.88877 − 3.78449i 8.75848 − 19.1784i 4.33731 − 5.00552i 46.8674 − 30.1199i
8.17 0.609616 − 4.23997i 2.05223 + 4.49376i −9.92980 − 2.91565i −2.51236 − 2.89942i 20.3045 − 5.96193i −5.88877 − 3.78449i −4.17997 + 9.15286i 1.69904 − 1.96080i −13.8250 + 8.88480i
8.18 0.655966 − 4.56234i 3.90027 + 8.54039i −12.7087 − 3.73162i 7.70390 + 8.89077i 41.5226 − 12.1921i −5.88877 − 3.78449i −10.0434 + 21.9919i −40.0450 + 46.2143i 45.6162 − 29.3158i
8.19 0.738366 − 5.13545i −1.91468 − 4.19256i −18.1517 − 5.32981i −2.51350 − 2.90073i −22.9444 + 6.73709i −5.88877 − 3.78449i −23.5313 + 51.5264i 3.76965 − 4.35041i −16.7524 + 10.7661i
29.1 −2.16445 + 4.73948i −3.10823 − 0.912658i −12.5389 − 14.4707i −14.7326 + 9.46804i 11.0531 − 12.7560i 0.996204 − 6.92875i 55.7291 − 16.3635i −13.8857 − 8.92381i −12.9857 − 90.3177i
See next 80 embeddings (of 190 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.19
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 161.4.i.b ✓ 190
23.c even 11 1 inner 161.4.i.b ✓ 190
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.4.i.b ✓ 190 1.a even 1 1 trivial
161.4.i.b ✓ 190 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{190} - 2 T_{2}^{189} + 107 T_{2}^{188} - 213 T_{2}^{187} + 7169 T_{2}^{186} + \cdots + 55\!\cdots\!56 \) acting on \(S_{4}^{\mathrm{new}}(161, [\chi])\). Copy content Toggle raw display