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Properties

Label 161.4.i.a
Level $161$
Weight $4$
Character orbit 161.i
Analytic conductor $9.499$
Analytic rank $0$
Dimension $170$
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] = [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: \(170\)
Relative dimension: \(17\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 170 q + 10 q^{3} - 72 q^{4} + 10 q^{5} + 15 q^{6} - 119 q^{7} + 3 q^{8} + 201 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 170 q + 10 q^{3} - 72 q^{4} + 10 q^{5} + 15 q^{6} - 119 q^{7} + 3 q^{8} + 201 q^{9} - 132 q^{10} - 14 q^{11} + 167 q^{12} + 48 q^{13} + 270 q^{15} + 488 q^{16} + 104 q^{17} - 201 q^{18} - 262 q^{19} - 821 q^{20} + 70 q^{21} - 344 q^{22} - 403 q^{23} - 2964 q^{24} - 227 q^{25} - 697 q^{26} + 370 q^{27} - 966 q^{28} + 474 q^{29} + 396 q^{30} + 2276 q^{31} + 544 q^{32} + 648 q^{33} + 1616 q^{34} + 224 q^{35} + 1671 q^{36} - 210 q^{37} + 112 q^{38} - 1680 q^{39} + 1918 q^{40} - 480 q^{41} + 105 q^{42} - 2116 q^{43} - 199 q^{44} + 868 q^{45} - 4144 q^{46} - 8708 q^{47} + 1087 q^{48} - 833 q^{49} + 4798 q^{50} - 4638 q^{51} + 1587 q^{52} - 250 q^{53} + 7701 q^{54} + 458 q^{55} + 21 q^{56} + 4430 q^{57} + 5422 q^{58} + 960 q^{59} + 4700 q^{60} + 1780 q^{61} - 4489 q^{62} - 1519 q^{63} - 8307 q^{64} + 488 q^{65} - 16657 q^{66} + 2086 q^{67} + 3836 q^{68} + 10 q^{69} - 924 q^{70} + 8538 q^{71} + 893 q^{72} + 5336 q^{73} - 5487 q^{74} + 6274 q^{75} + 1757 q^{76} + 2905 q^{77} - 823 q^{78} + 928 q^{79} + 11406 q^{80} + 2433 q^{81} - 475 q^{82} + 5826 q^{83} - 1526 q^{84} - 2012 q^{85} - 6572 q^{86} - 3066 q^{87} + 3976 q^{88} - 9400 q^{89} - 1797 q^{90} - 3668 q^{91} + 12424 q^{92} - 30400 q^{93} + 10175 q^{94} - 5716 q^{95} + 11860 q^{96} - 674 q^{97} + 1078 q^{98} - 1410 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −0.737150 + 5.12699i −0.125880 − 0.275638i −18.0667 − 5.30487i −7.28034 − 8.40197i 1.50598 − 0.442197i 5.88877 + 3.78449i 23.3021 − 51.0244i 17.6211 − 20.3358i 48.4435 − 31.1328i
8.2 −0.715412 + 4.97580i −2.68964 − 5.88948i −16.5708 − 4.86564i 9.41642 + 10.8671i 31.2291 − 9.16969i 5.88877 + 3.78449i 19.3592 − 42.3908i −9.77063 + 11.2759i −60.8093 + 39.0798i
8.3 −0.574680 + 3.99698i 3.02553 + 6.62499i −7.96968 − 2.34011i 3.43706 + 3.96657i −28.2187 + 8.28575i 5.88877 + 3.78449i 0.513540 − 1.12449i −17.0554 + 19.6830i −17.8295 + 11.4584i
8.4 −0.451060 + 3.13719i −3.30014 − 7.22629i −1.96257 − 0.576263i 2.42710 + 2.80102i 24.1588 − 7.09367i 5.88877 + 3.78449i −7.84002 + 17.1672i −23.6471 + 27.2902i −9.88211 + 6.35085i
8.5 −0.396462 + 2.75745i −2.07534 − 4.54435i 0.229576 + 0.0674095i −10.3862 − 11.9863i 13.3536 − 3.92098i 5.88877 + 3.78449i −9.53503 + 20.8788i 1.33713 − 1.54313i 37.1693 − 23.8873i
8.6 −0.306215 + 2.12977i 0.0717938 + 0.157207i 3.23379 + 0.949528i 10.5233 + 12.1446i −0.356798 + 0.104765i 5.88877 + 3.78449i −10.1632 + 22.2543i 17.6617 − 20.3827i −29.0876 + 18.6934i
8.7 −0.153334 + 1.06646i 2.98984 + 6.54684i 6.56211 + 1.92681i −2.87833 − 3.32177i −7.44041 + 2.18470i 5.88877 + 3.78449i −6.64172 + 14.5433i −16.2407 + 18.7428i 3.98389 − 2.56029i
8.8 −0.0438387 + 0.304905i 0.644088 + 1.41036i 7.58490 + 2.22713i −10.0039 − 11.5452i −0.458260 + 0.134557i 5.88877 + 3.78449i −2.03529 + 4.45666i 16.1070 − 18.5884i 3.95873 − 2.54412i
8.9 0.0878020 − 0.610676i −1.60558 − 3.51573i 7.31073 + 2.14662i 4.99783 + 5.76781i −2.28795 + 0.671802i 5.88877 + 3.78449i 4.00313 − 8.76563i 7.89875 − 9.11565i 3.96108 − 2.54563i
8.10 0.276791 − 1.92513i 2.95212 + 6.46424i 4.04645 + 1.18814i 8.61098 + 9.93760i 13.2616 − 3.89396i 5.88877 + 3.78449i 9.87095 − 21.6144i −15.3901 + 17.7612i 21.5146 − 13.8266i
8.11 0.311541 − 2.16681i −2.14455 − 4.69591i 3.07792 + 0.903759i 4.28379 + 4.94376i −10.8433 + 3.18388i 5.88877 + 3.78449i 10.1922 − 22.3179i 0.228735 − 0.263974i 12.0468 − 7.74200i
8.12 0.490376 − 3.41064i −0.157809 − 0.345553i −3.71606 − 1.09113i −4.52713 − 5.22459i −1.25594 + 0.368778i 5.88877 + 3.78449i 5.90748 − 12.9356i 17.5867 − 20.2962i −20.0392 + 12.8784i
8.13 0.518332 − 3.60508i 2.07192 + 4.53688i −5.05199 − 1.48340i −2.22701 − 2.57011i 17.4298 − 5.11784i 5.88877 + 3.78449i 4.13766 − 9.06020i 1.39082 − 1.60510i −10.4198 + 6.69638i
8.14 0.601190 − 4.18137i −2.72343 − 5.96349i −9.44645 − 2.77373i −13.1781 − 15.2084i −26.5728 + 7.80248i 5.88877 + 3.78449i −3.23816 + 7.09058i −10.4648 + 12.0771i −71.5143 + 45.9595i
8.15 0.665917 − 4.63155i −4.19188 − 9.17894i −13.3319 − 3.91459i 7.09842 + 8.19201i −45.3042 + 13.3025i 5.88877 + 3.78449i −11.4582 + 25.0899i −48.9999 + 56.5489i 42.6687 − 27.4215i
8.16 0.742766 − 5.16605i 0.827379 + 1.81171i −18.4605 − 5.42048i 13.4138 + 15.4804i 9.97392 − 2.92861i 5.88877 + 3.78449i −24.3693 + 53.3613i 15.0835 − 17.4073i 89.9357 − 57.7981i
8.17 0.798896 − 5.55645i 2.54120 + 5.56446i −22.5599 − 6.62419i −11.2668 − 13.0025i 32.9488 − 9.67464i 5.88877 + 3.78449i −36.1743 + 79.2106i −6.82425 + 7.87561i −81.2490 + 52.2155i
29.1 −2.22452 + 4.87102i 2.01444 + 0.591494i −13.5395 − 15.6254i 11.4391 − 7.35145i −7.36235 + 8.49661i −0.996204 + 6.92875i 65.1262 − 19.1228i −19.0057 − 12.2142i 10.3626 + 72.0734i
29.2 −2.15599 + 4.72096i −9.21984 − 2.70719i −12.4003 − 14.3107i 3.70437 − 2.38065i 32.6584 − 37.6898i −0.996204 + 6.92875i 54.4573 − 15.9901i 54.9627 + 35.3224i 3.25239 + 22.6208i
29.3 −1.86961 + 4.09388i 0.194794 + 0.0571967i −8.02551 − 9.26194i −5.56403 + 3.57578i −0.598346 + 0.690528i −0.996204 + 6.92875i 18.3756 − 5.39556i −22.6792 − 14.5750i −4.23625 − 29.4638i
See next 80 embeddings (of 170 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.17
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.a ✓ 170
23.c even 11 1 inner 161.4.i.a ✓ 170
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.4.i.a ✓ 170 1.a even 1 1 trivial
161.4.i.a ✓ 170 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{170} + 104 T_{2}^{168} - T_{2}^{167} + 5602 T_{2}^{166} - 584 T_{2}^{165} + 235673 T_{2}^{164} + \cdots + 55\!\cdots\!84 \) acting on \(S_{4}^{\mathrm{new}}(161, [\chi])\). Copy content Toggle raw display