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Properties

Label 155.3.s.a
Level $155$
Weight $3$
Character orbit 155.s
Analytic conductor $4.223$
Analytic rank $0$
Dimension $240$
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] = [155,3,Mod(2,155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(155, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([5, 16]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("155.2");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 155 = 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 155.s (of order \(20\), 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: \(240\)
Relative dimension: \(30\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 6 q^{2} - 10 q^{3} - 16 q^{5} - 56 q^{6} + 18 q^{7} - 28 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q - 6 q^{2} - 10 q^{3} - 16 q^{5} - 56 q^{6} + 18 q^{7} - 28 q^{8} - 38 q^{10} - 12 q^{11} - 98 q^{12} - 6 q^{13} - 22 q^{15} + 140 q^{16} - 6 q^{17} - 76 q^{18} + 156 q^{20} + 60 q^{21} - 208 q^{22} - 68 q^{23} - 68 q^{25} + 80 q^{26} + 104 q^{27} + 64 q^{28} - 376 q^{30} - 20 q^{31} + 12 q^{32} - 302 q^{33} + 102 q^{35} + 704 q^{36} + 484 q^{37} + 164 q^{38} - 178 q^{40} - 120 q^{41} - 186 q^{42} - 30 q^{43} + 56 q^{45} - 260 q^{46} + 402 q^{47} - 506 q^{48} - 564 q^{50} + 236 q^{51} - 256 q^{52} - 114 q^{53} - 388 q^{55} - 472 q^{56} - 384 q^{57} + 358 q^{58} + 400 q^{60} - 176 q^{61} - 10 q^{62} - 256 q^{63} + 556 q^{65} - 276 q^{66} + 764 q^{67} - 448 q^{68} + 708 q^{70} + 408 q^{71} - 662 q^{72} + 354 q^{73} + 424 q^{75} - 1208 q^{76} + 226 q^{77} - 636 q^{78} + 818 q^{80} + 160 q^{81} - 392 q^{82} + 348 q^{83} - 150 q^{85} + 292 q^{86} + 344 q^{87} - 836 q^{88} + 68 q^{90} - 548 q^{91} + 288 q^{92} + 790 q^{93} + 720 q^{95} + 1072 q^{96} + 452 q^{97} - 272 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −0.604770 − 3.81837i −0.362894 + 2.29122i −10.4099 + 3.38240i 4.52476 + 2.12757i 8.96819 0.230781 − 0.452934i 12.1904 + 23.9250i 3.44151 + 1.11821i 5.38742 − 18.5639i
2.2 −0.535176 − 3.37897i 0.0748117 − 0.472342i −7.32680 + 2.38062i −4.97406 + 0.508628i −1.63607 −3.98906 + 7.82897i 5.75262 + 11.2901i 8.34200 + 2.71048i 4.38064 + 16.5350i
2.3 −0.517385 − 3.26664i 0.443456 − 2.79987i −6.59901 + 2.14415i −2.18979 − 4.49498i −9.37560 4.31388 − 8.46647i 4.41236 + 8.65974i 0.916885 + 0.297914i −13.5505 + 9.47889i
2.4 −0.516704 − 3.26234i 0.781416 − 4.93367i −6.57166 + 2.13526i 4.84435 − 1.23785i −16.4991 −4.29583 + 8.43104i 4.36343 + 8.56371i −15.1710 − 4.92934i −6.54138 − 15.1643i
2.5 −0.479725 − 3.02887i −0.648200 + 4.09257i −5.13967 + 1.66998i −3.89936 + 3.12969i 12.7068 5.86013 − 11.5012i 1.95491 + 3.83672i −7.76947 − 2.52446i 11.3500 + 10.3092i
2.6 −0.471569 − 2.97737i −0.843294 + 5.32435i −4.83812 + 1.57200i −0.120112 − 4.99856i 16.2502 −2.44971 + 4.80782i 1.48774 + 2.91986i −19.0781 − 6.19884i −14.8259 + 2.71478i
2.7 −0.402379 − 2.54052i −0.0711001 + 0.448908i −2.48813 + 0.808442i 3.91763 − 3.10679i 1.16907 0.0992101 − 0.194711i −1.61597 − 3.17151i 8.36305 + 2.71732i −9.46925 − 8.70273i
2.8 −0.352593 − 2.22618i −0.203850 + 1.28706i −1.02735 + 0.333806i −0.584883 + 4.96567i 2.93710 −2.87878 + 5.64992i −2.98771 − 5.86371i 6.94455 + 2.25642i 11.2607 − 0.448804i
2.9 −0.316922 − 2.00097i 0.397510 − 2.50978i −0.0992023 + 0.0322328i 2.88521 + 4.08357i −5.14797 3.36790 − 6.60988i −3.58304 − 7.03211i 2.41852 + 0.785823i 7.25671 − 7.06738i
2.10 −0.293213 − 1.85127i 0.831496 − 5.24986i 0.462986 − 0.150433i −4.35932 + 2.44875i −9.96273 −1.71446 + 3.36482i −3.81799 − 7.49324i −18.3101 − 5.94932i 5.81151 + 7.35228i
2.11 −0.223623 − 1.41190i −0.385214 + 2.43214i 1.86078 − 0.604604i −2.98394 − 4.01199i 3.52008 0.981003 − 1.92533i −3.86566 − 7.58679i 2.79258 + 0.907363i −4.99724 + 5.11019i
2.12 −0.108226 − 0.683309i 0.388044 − 2.45002i 3.34903 − 1.08817i −4.77747 − 1.47505i −1.71611 0.703897 − 1.38148i −2.36233 − 4.63634i 2.70751 + 0.879723i −0.490873 + 3.42413i
2.13 −0.0998941 − 0.630706i −0.392989 + 2.48124i 3.41641 − 1.11006i 4.38348 − 2.40523i 1.60419 −1.38564 + 2.71948i −2.20102 − 4.31974i 2.55742 + 0.830956i −1.95488 − 2.52442i
2.14 −0.0924042 − 0.583417i −0.743494 + 4.69424i 3.47239 − 1.12825i 4.33671 + 2.48857i 2.80740 5.29387 − 10.3898i −2.05177 − 4.02683i −12.9236 − 4.19912i 1.05114 − 2.76006i
2.15 −0.0905404 − 0.571649i 0.616383 − 3.89169i 3.48564 − 1.13255i 2.62489 − 4.25558i −2.28049 0.0625067 − 0.122676i −2.01405 − 3.95279i −6.20582 − 2.01639i −2.67036 − 1.11522i
2.16 −0.0306570 − 0.193561i −0.703624 + 4.44251i 3.76770 − 1.22420i −4.24404 + 2.64351i 0.881466 −3.26366 + 6.40529i −0.708343 − 1.39020i −10.6813 − 3.47056i 0.641790 + 0.740436i
2.17 0.0264577 + 0.167047i 0.240594 − 1.51905i 3.77702 − 1.22723i 4.12279 + 2.82888i 0.260119 −5.29789 + 10.3977i 0.612069 + 1.20125i 6.30988 + 2.05020i −0.363477 + 0.763547i
2.18 0.114570 + 0.723366i 0.164193 − 1.03667i 3.29409 − 1.07032i −3.07653 + 3.94144i 0.768706 3.69336 − 7.24863i 2.48161 + 4.87044i 7.51178 + 2.44072i −3.20358 − 1.77389i
2.19 0.216084 + 1.36430i −0.106487 + 0.672332i 1.98960 − 0.646461i −3.92111 − 3.10240i −0.940273 −3.98428 + 7.81959i 3.82029 + 7.49774i 8.11882 + 2.63796i 3.38532 − 6.01995i
2.20 0.234488 + 1.48050i −0.242354 + 1.53016i 1.66734 − 0.541750i 0.548196 − 4.96986i −2.32223 4.69696 − 9.21831i 3.91507 + 7.68376i 6.27685 + 2.03947i 7.48641 − 0.353769i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
31.d even 5 1 inner
155.s odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 155.3.s.a ✓ 240
5.c odd 4 1 inner 155.3.s.a ✓ 240
31.d even 5 1 inner 155.3.s.a ✓ 240
155.s odd 20 1 inner 155.3.s.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.3.s.a ✓ 240 1.a even 1 1 trivial
155.3.s.a ✓ 240 5.c odd 4 1 inner
155.3.s.a ✓ 240 31.d even 5 1 inner
155.3.s.a ✓ 240 155.s odd 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(155, [\chi])\).