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Properties

Label 152.3.u.b
Level $152$
Weight $3$
Character orbit 152.u
Analytic conductor $4.142$
Analytic rank $0$
Dimension $216$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [152,3,Mod(35,152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(152, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 9, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("152.35");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 152 = 2^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 152.u (of order \(18\), degree \(6\), 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: \(4.14170001828\)
Analytic rank: \(0\)
Dimension: \(216\)
Relative dimension: \(36\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 216 q - 6 q^{2} - 18 q^{3} - 12 q^{4} - 12 q^{6} - 51 q^{8} - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 216 q - 6 q^{2} - 18 q^{3} - 12 q^{4} - 12 q^{6} - 51 q^{8} - 54 q^{9} - 9 q^{10} - 6 q^{11} - 3 q^{12} + 27 q^{14} - 48 q^{16} - 12 q^{17} + 204 q^{18} - 12 q^{19} - 42 q^{20} + 174 q^{22} - 36 q^{24} - 12 q^{25} + 69 q^{26} - 144 q^{27} + 108 q^{28} - 30 q^{30} + 39 q^{32} + 144 q^{33} - 72 q^{34} - 162 q^{35} - 159 q^{36} + 246 q^{38} - 72 q^{40} - 78 q^{41} + 219 q^{42} - 12 q^{43} - 603 q^{44} + 132 q^{46} - 369 q^{48} + 834 q^{49} - 186 q^{50} - 1158 q^{51} - 21 q^{52} + 75 q^{54} - 306 q^{56} - 12 q^{57} + 60 q^{58} - 258 q^{59} - 42 q^{60} - 468 q^{62} + 171 q^{64} - 6 q^{65} - 204 q^{66} + 174 q^{67} - 84 q^{68} - 261 q^{70} - 714 q^{72} + 1080 q^{73} - 219 q^{74} - 696 q^{75} + 42 q^{76} + 291 q^{78} - 9 q^{80} + 732 q^{81} + 765 q^{82} - 6 q^{83} + 27 q^{84} + 444 q^{86} + 693 q^{88} - 12 q^{89} + 1014 q^{90} - 306 q^{91} + 336 q^{92} + 1074 q^{94} - 726 q^{96} - 294 q^{97} + 831 q^{98} + 864 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1 −1.99954 − 0.0430175i 2.02536 − 0.737170i 3.99630 + 0.172030i −4.07739 − 0.718954i −4.08149 + 1.38687i −8.60687 + 4.96918i −7.98335 − 0.515891i −3.33574 + 2.79902i 8.12197 + 1.61297i
35.2 −1.99807 − 0.0878964i 2.58324 − 0.940221i 3.98455 + 0.351246i 7.82333 + 1.37946i −5.24412 + 1.65157i 6.21534 − 3.58843i −7.93052 − 1.05204i −1.10530 + 0.927460i −15.5103 − 3.44390i
35.3 −1.99072 + 0.192441i −1.59398 + 0.580162i 3.92593 − 0.766193i −1.87589 − 0.330770i 3.06153 − 1.46169i 3.46445 − 2.00020i −7.66799 + 2.28079i −4.69021 + 3.93555i 3.79802 + 0.297472i
35.4 −1.96267 − 0.384622i −4.92092 + 1.79107i 3.70413 + 1.50977i 4.49450 + 0.792502i 10.3470 − 1.62258i 1.71971 − 0.992873i −6.68929 − 4.38787i 14.1131 − 11.8423i −8.51640 − 3.28410i
35.5 −1.75857 + 0.952584i 5.23909 − 1.90687i 2.18517 − 3.35038i −5.63828 − 0.994182i −7.39687 + 8.34405i 6.18084 − 3.56851i −0.651263 + 7.97345i 16.9175 − 14.1955i 10.8624 − 3.62260i
35.6 −1.66838 + 1.10295i −2.25066 + 0.819173i 1.56701 − 3.68028i 3.20953 + 0.565927i 2.85146 − 3.84905i −2.85189 + 1.64654i 1.44477 + 7.86846i −2.49998 + 2.09773i −5.97892 + 2.59576i
35.7 −1.63424 − 1.15293i 0.0347648 − 0.0126533i 1.34148 + 3.76834i −7.45238 − 1.31406i −0.0714024 − 0.0194029i 9.90562 − 5.71901i 2.15235 − 7.70502i −6.89335 + 5.78421i 10.6640 + 10.7396i
35.8 −1.62842 + 1.16114i −4.29285 + 1.56247i 1.30350 − 3.78165i −9.59691 − 1.69219i 5.17631 − 7.52895i −1.13692 + 0.656399i 2.26838 + 7.67167i 9.09282 − 7.62978i 17.5927 − 8.38777i
35.9 −1.44111 + 1.38679i 2.98937 − 1.08804i 0.153618 − 3.99705i 6.13731 + 1.08217i −2.79913 + 5.71362i −3.52794 + 2.03686i 5.32169 + 5.97324i 0.858086 − 0.720019i −10.3453 + 6.95164i
35.10 −1.41920 − 1.40921i 0.0347648 − 0.0126533i 0.0282673 + 3.99990i 7.45238 + 1.31406i −0.0671694 − 0.0310331i −9.90562 + 5.71901i 5.59657 − 5.71650i −6.89335 + 5.78421i −8.72465 − 12.3669i
35.11 −1.01813 + 1.72146i 1.80552 − 0.657154i −1.92682 − 3.50534i −2.68636 − 0.473679i −0.706991 + 3.77719i −5.17357 + 2.98696i 7.99603 + 0.251963i −4.06636 + 3.41208i 3.55049 − 4.14219i
35.12 −0.823242 + 1.82271i −2.48769 + 0.905444i −2.64454 − 3.00106i 4.79912 + 0.846215i 0.397608 − 5.27973i 11.0833 − 6.39897i 7.64717 − 2.34963i −1.52564 + 1.28017i −5.49325 + 8.05077i
35.13 −0.719592 − 1.86606i −4.92092 + 1.79107i −2.96437 + 2.68561i −4.49450 − 0.792502i 6.88330 + 7.89390i −1.71971 + 0.992873i 7.14465 + 3.59916i 14.1131 − 11.8423i 1.75535 + 8.95729i
35.14 −0.484715 + 1.94037i 0.839443 − 0.305532i −3.53010 − 1.88106i −7.22003 − 1.27309i 0.185957 + 1.77693i 5.32261 − 3.07301i 5.36104 − 5.93795i −6.28309 + 5.27214i 5.96992 − 13.3925i
35.15 −0.433522 − 1.95245i 2.58324 − 0.940221i −3.62412 + 1.69286i −7.82333 − 1.37946i −2.95562 − 4.63603i −6.21534 + 3.58843i 4.87636 + 6.34202i −1.10530 + 0.927460i 0.698250 + 15.8727i
35.16 −0.389580 − 1.96169i 2.02536 − 0.737170i −3.69645 + 1.52847i 4.07739 + 0.718954i −2.23514 − 3.68594i 8.60687 − 4.96918i 4.43845 + 6.65584i −3.33574 + 2.79902i −0.178105 − 8.27867i
35.17 −0.315856 + 1.97490i −4.34914 + 1.58296i −3.80047 − 1.24757i 2.48427 + 0.438045i −1.75248 − 9.08910i −10.5409 + 6.08579i 3.66423 − 7.11150i 9.51484 − 7.98390i −1.64977 + 4.76784i
35.18 −0.156167 − 1.99389i −1.59398 + 0.580162i −3.95122 + 0.622762i 1.87589 + 0.330770i 1.40571 + 3.08763i −3.46445 + 2.00020i 1.85877 + 7.78106i −4.69021 + 3.93555i 0.366567 − 3.79198i
35.19 0.122239 + 1.99626i 4.11766 − 1.49871i −3.97012 + 0.488041i 4.24720 + 0.748896i 3.49514 + 8.03672i 3.17866 − 1.83520i −1.45956 − 7.86573i 7.81459 − 6.55722i −0.975820 + 8.57007i
35.20 0.632739 − 1.89727i 5.23909 − 1.90687i −3.19928 − 2.40096i 5.63828 + 0.994182i −0.302881 − 11.1465i −6.18084 + 3.56851i −6.57958 + 4.55073i 16.9175 − 14.1955i 5.45379 − 10.0683i
See next 80 embeddings (of 216 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
19.e even 9 1 inner
152.u odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 152.3.u.b ✓ 216
8.d odd 2 1 inner 152.3.u.b ✓ 216
19.e even 9 1 inner 152.3.u.b ✓ 216
152.u odd 18 1 inner 152.3.u.b ✓ 216
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.3.u.b ✓ 216 1.a even 1 1 trivial
152.3.u.b ✓ 216 8.d odd 2 1 inner
152.3.u.b ✓ 216 19.e even 9 1 inner
152.3.u.b ✓ 216 152.u odd 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{108} + 9 T_{3}^{107} + 54 T_{3}^{106} + 267 T_{3}^{105} + 732 T_{3}^{104} + 597 T_{3}^{103} + \cdots + 88\!\cdots\!25 \) acting on \(S_{3}^{\mathrm{new}}(152, [\chi])\). Copy content Toggle raw display