↑

Properties

Label 155.3.t.b
Level $155$
Weight $3$
Character orbit 155.t
Analytic conductor $4.223$
Analytic rank $0$
Dimension $88$
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] = [155,3,Mod(11,155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(155, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 23]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("155.11");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 155 = 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 155.t (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: \(4.22344409758\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(11\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 88 q + 3 q^{3} - 46 q^{4} - 6 q^{7} - 7 q^{8} - 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q + 3 q^{3} - 46 q^{4} - 6 q^{7} - 7 q^{8} - 46 q^{9} - 5 q^{10} + 18 q^{11} + 9 q^{12} + 42 q^{13} + 2 q^{14} - 110 q^{16} + 190 q^{17} + 177 q^{18} + 47 q^{19} - 95 q^{20} - 86 q^{21} + 156 q^{22} - 150 q^{23} - 170 q^{24} - 220 q^{25} - 438 q^{26} - 195 q^{27} - 96 q^{28} - 250 q^{29} + 106 q^{31} + 72 q^{32} + 115 q^{33} - 112 q^{34} + 30 q^{35} + 279 q^{36} - 12 q^{37} - 92 q^{38} + 249 q^{39} + 50 q^{40} + 224 q^{41} + 798 q^{42} + 82 q^{43} + 218 q^{44} + 190 q^{45} - 20 q^{46} - 72 q^{47} + 627 q^{48} + 215 q^{49} - 75 q^{50} - 320 q^{51} + 71 q^{52} + 89 q^{53} - 330 q^{54} + 35 q^{55} - 179 q^{56} - 1170 q^{57} - 355 q^{58} - 61 q^{59} + 75 q^{60} + 373 q^{62} - 330 q^{63} + 193 q^{64} - 60 q^{65} + 110 q^{66} + 168 q^{67} - 300 q^{68} + 644 q^{69} + 80 q^{70} - 364 q^{71} + 1095 q^{72} + 903 q^{73} + 84 q^{74} - 15 q^{75} - 262 q^{76} + 150 q^{77} - 396 q^{78} + 323 q^{79} + 485 q^{80} - 405 q^{81} - 908 q^{82} + 360 q^{83} - 476 q^{84} + 45 q^{85} - 181 q^{86} - 412 q^{87} - 1584 q^{88} - 90 q^{89} - 75 q^{90} + 575 q^{91} + 253 q^{93} - 574 q^{94} + 100 q^{95} - 428 q^{96} + 648 q^{97} + 291 q^{98} - 717 q^{99}+O(q^{100}) \) 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 −1.13169 − 3.48297i −3.98014 + 3.58373i −7.61433 + 5.53213i 1.11803 + 1.93649i 16.9863 + 9.80705i −0.417411 − 3.97140i 16.0341 + 11.6495i 2.05760 − 19.5768i 5.47949 − 6.08559i
11.2 −0.952708 − 2.93213i 0.0870895 − 0.0784157i −4.45368 + 3.23579i 1.11803 + 1.93649i −0.312896 − 0.180651i 1.34902 + 12.8351i 3.75394 + 2.72740i −0.939321 + 8.93704i 4.61289 − 5.12313i
11.3 −0.683074 − 2.10228i 4.02353 − 3.62280i −0.716941 + 0.520888i 1.11803 + 1.93649i −10.3645 − 5.98396i 0.377370 + 3.59044i −5.56846 − 4.04572i 2.12333 − 20.2022i 3.30736 − 3.67319i
11.4 −0.650713 − 2.00269i −1.32218 + 1.19049i −0.351272 + 0.255214i 1.11803 + 1.93649i 3.24455 + 1.87324i −0.739103 − 7.03209i −6.07467 − 4.41351i −0.609879 + 5.80261i 3.15067 − 3.49918i
11.5 −0.312145 − 0.960682i −2.73199 + 2.45990i 2.41059 − 1.75140i 1.11803 + 1.93649i 3.21596 + 1.85673i 0.288222 + 2.74225i −5.70381 − 4.14406i 0.471935 − 4.49016i 1.51137 − 1.67854i
11.6 −0.0360915 − 0.111078i 2.17496 − 1.95835i 3.22503 − 2.34312i 1.11803 + 1.93649i −0.296028 − 0.170912i −1.43775 − 13.6793i −0.754622 − 0.548265i −0.0454086 + 0.432034i 0.174751 − 0.194080i
11.7 0.161465 + 0.496940i 1.60674 − 1.44672i 3.01519 − 2.19066i 1.11803 + 1.93649i 0.978365 + 0.564859i 0.680140 + 6.47110i 3.26637 + 2.37315i −0.452126 + 4.30169i −0.781796 + 0.868272i
11.8 0.660747 + 2.03357i −0.638017 + 0.574473i −0.462757 + 0.336213i 1.11803 + 1.93649i −1.58980 − 0.917871i −0.0321765 − 0.306139i 5.92996 + 4.30837i −0.863710 + 8.21765i −3.19926 + 3.55313i
11.9 0.923319 + 2.84168i 3.82757 − 3.44636i −3.98659 + 2.89642i 1.11803 + 1.93649i 13.3276 + 7.69466i −0.0321734 − 0.306109i −2.24249 − 1.62926i 1.83215 − 17.4317i −4.47060 + 4.96510i
11.10 0.938673 + 2.88894i −3.58357 + 3.22666i −4.22880 + 3.07240i 1.11803 + 1.93649i −12.6854 − 7.32393i 1.24583 + 11.8533i −3.01552 − 2.19090i 1.48987 − 14.1752i −4.54494 + 5.04767i
11.11 1.22667 + 3.77529i −0.751173 + 0.676359i −9.51206 + 6.91092i 1.11803 + 1.93649i −3.47489 − 2.00623i −0.788286 − 7.50004i −24.9130 − 18.1004i −0.833957 + 7.93457i −5.93937 + 6.59634i
21.1 −3.13379 − 2.27683i −0.240021 + 0.0252272i 3.40062 + 10.4660i −1.11803 − 1.93649i 0.809614 + 0.467431i 7.23831 + 1.53855i 8.38456 − 25.8050i −8.74635 + 1.85910i −0.905383 + 8.61414i
21.2 −2.26467 − 1.64538i −2.89284 + 0.304050i 1.18539 + 3.64826i −1.11803 − 1.93649i 7.05160 + 4.07125i −7.46893 − 1.58757i −0.141856 + 0.436588i −0.527261 + 0.112073i −0.654285 + 6.22511i
21.3 −2.12856 − 1.54649i 3.97825 − 0.418131i 0.903072 + 2.77937i −1.11803 − 1.93649i −9.11458 − 5.26231i −9.44198 − 2.00695i −0.876128 + 2.69644i 6.84831 − 1.45565i −0.614962 + 5.85097i
21.4 −0.642886 − 0.467084i −1.76437 + 0.185442i −1.04093 − 3.20366i −1.11803 − 1.93649i 1.22090 + 0.704889i 2.77148 + 0.589097i −1.80942 + 5.56883i −5.72473 + 1.21683i −0.185736 + 1.76716i
21.5 0.0478693 + 0.0347791i 3.04816 − 0.320374i −1.23499 − 3.80090i −1.11803 − 1.93649i 0.157056 + 0.0906761i 7.35583 + 1.56353i 0.146212 − 0.449993i 0.385290 − 0.0818960i 0.0138299 − 0.131583i
21.6 0.981020 + 0.712753i −0.802856 + 0.0843835i −0.781684 − 2.40578i −1.11803 − 1.93649i −0.847762 − 0.489456i −8.35307 − 1.77550i 2.44674 − 7.53029i −8.16587 + 1.73571i 0.283426 − 2.69662i
21.7 1.08798 + 0.790465i 5.50837 − 0.578953i −0.677199 − 2.08420i −1.11803 − 1.93649i 6.45064 + 3.72428i −8.96753 − 1.90611i 2.57300 − 7.91887i 21.2036 − 4.50696i 0.314328 − 2.99063i
21.8 1.50037 + 1.09008i −4.44956 + 0.467668i −0.173235 − 0.533162i −1.11803 − 1.93649i −7.18579 − 4.14872i 13.0068 + 2.76468i 2.61364 − 8.04396i 10.7766 − 2.29063i 0.433472 − 4.12421i
21.9 2.36719 + 1.71987i 2.98209 − 0.313430i 1.40960 + 4.33829i −1.11803 − 1.93649i 7.59823 + 4.38684i 7.02263 + 1.49271i −0.507746 + 1.56268i −0.00873257 + 0.00185617i 0.683905 − 6.50692i
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.h odd 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 155.3.t.b ✓ 88
31.h odd 30 1 inner 155.3.t.b ✓ 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.3.t.b ✓ 88 1.a even 1 1 trivial
155.3.t.b ✓ 88 31.h odd 30 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{88} + 67 T_{2}^{86} - 11 T_{2}^{85} + 2571 T_{2}^{84} - 777 T_{2}^{83} + 74555 T_{2}^{82} + \cdots + 48\!\cdots\!61 \) acting on \(S_{3}^{\mathrm{new}}(155, [\chi])\). Copy content Toggle raw display