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Properties

Label 155.3.w.a
Level $155$
Weight $3$
Character orbit 155.w
Analytic conductor $4.223$
Analytic rank $0$
Dimension $480$
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(7,155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(155, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([15, 56]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("155.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 155 = 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 155.w (of order \(60\), degree \(16\), 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: \(480\)
Relative dimension: \(30\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 480 q - 12 q^{2} - 14 q^{3} - 8 q^{5} - 28 q^{6} - 66 q^{7} - 56 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 480 q - 12 q^{2} - 14 q^{3} - 8 q^{5} - 28 q^{6} - 66 q^{7} - 56 q^{8} + 14 q^{10} - 36 q^{11} + 44 q^{12} - 18 q^{13} + 148 q^{15} + 400 q^{16} - 18 q^{17} + 28 q^{18} - 132 q^{20} - 180 q^{21} - 134 q^{22} + 50 q^{23} - 34 q^{25} - 128 q^{26} + 100 q^{27} - 64 q^{28} + 40 q^{30} - 172 q^{31} + 84 q^{32} + 152 q^{33} - 264 q^{35} - 1088 q^{36} + 50 q^{37} - 254 q^{38} - 212 q^{40} + 156 q^{41} + 630 q^{42} - 30 q^{43} - 242 q^{45} + 224 q^{46} - 132 q^{47} + 632 q^{48} - 1362 q^{50} + 4 q^{51} + 322 q^{52} - 270 q^{53} - 182 q^{55} - 80 q^{56} + 72 q^{57} - 424 q^{58} - 880 q^{60} + 80 q^{61} + 130 q^{62} + 208 q^{63} - 76 q^{65} + 1416 q^{66} - 482 q^{67} + 238 q^{68} + 1254 q^{70} - 792 q^{71} - 448 q^{72} - 402 q^{73} + 2282 q^{75} + 152 q^{76} - 244 q^{77} - 942 q^{78} - 194 q^{80} - 592 q^{81} + 104 q^{82} - 1326 q^{83} + 132 q^{85} + 116 q^{86} + 184 q^{87} - 1012 q^{88} + 1450 q^{90} + 1664 q^{91} - 1500 q^{92} - 910 q^{93} + 366 q^{95} - 700 q^{96} - 974 q^{97} + 1226 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −1.70373 + 3.34375i 0.173847 − 0.267700i −5.92685 − 8.15761i 3.35923 − 3.70345i 0.598936 + 1.03739i −0.646950 − 1.68536i 22.5485 − 3.57133i 3.61919 + 8.12883i 6.66022 + 17.5421i
7.2 −1.59914 + 3.13850i −2.90182 + 4.46841i −4.94177 − 6.80176i −4.94246 − 0.756371i −9.38366 − 16.2530i 3.24619 + 8.45662i 15.3337 − 2.42862i −7.88548 − 17.7111i 10.2776 − 14.3024i
7.3 −1.53249 + 3.00767i 1.05949 − 1.63147i −4.34644 − 5.98236i −0.524036 + 4.97246i 3.28327 + 5.68680i 0.897790 + 2.33882i 11.3177 − 1.79255i 2.12145 + 4.76486i −14.1525 − 9.19636i
7.4 −1.50433 + 2.95241i 0.197700 − 0.304432i −4.10259 − 5.64673i −4.87197 − 1.12424i 0.601402 + 1.04166i −1.11665 − 2.90897i 9.75203 − 1.54457i 3.60704 + 8.10154i 10.6483 − 12.6928i
7.5 −1.30088 + 2.55312i 2.73281 − 4.20816i −2.47498 − 3.40651i 4.96907 + 0.555307i 7.18886 + 12.4515i −1.04258 − 2.71602i 0.596265 − 0.0944390i −6.57971 − 14.7783i −7.88191 + 11.9642i
7.6 −1.19111 + 2.33768i −2.33025 + 3.58827i −1.69486 − 2.33278i 2.93087 − 4.05093i −5.61264 − 9.72137i −2.99329 − 7.79778i −2.89330 + 0.458254i −3.78497 − 8.50119i 5.97879 + 11.6765i
7.7 −1.14401 + 2.24525i −1.58109 + 2.43467i −1.38123 − 1.90110i 4.60565 + 1.94627i −3.65764 − 6.33522i 3.88743 + 10.1271i −4.10691 + 0.650471i 0.232878 + 0.523052i −9.63876 + 8.11428i
7.8 −1.05436 + 2.06930i −1.80077 + 2.77294i −0.819189 − 1.12752i −1.68274 + 4.70833i −3.83939 − 6.65002i −4.27815 − 11.1450i −5.97846 + 0.946895i −0.785806 − 1.76495i −7.96875 − 8.44637i
7.9 −0.986241 + 1.93561i 2.33779 − 3.59989i −0.422759 − 0.581878i −3.33625 − 3.72417i 4.66234 + 8.07541i −3.78351 − 9.85637i −7.03931 + 1.11492i −3.83328 − 8.60969i 10.4989 − 2.78475i
7.10 −0.837675 + 1.64403i 0.0435902 − 0.0671230i 0.350006 + 0.481742i −0.245086 − 4.99399i 0.0738378 + 0.127891i 2.15827 + 5.62250i −8.37488 + 1.32645i 3.65802 + 8.21606i 8.41557 + 3.78041i
7.11 −0.711667 + 1.39672i 2.19925 − 3.38655i 0.906771 + 1.24806i −3.87465 + 3.16024i 3.16494 + 5.48184i 2.84100 + 7.40106i −8.58164 + 1.35920i −2.97137 − 6.67380i −1.65652 − 7.66085i
7.12 −0.411744 + 0.808093i −1.20927 + 1.86211i 1.86766 + 2.57061i −4.86237 − 1.16507i −1.00685 − 1.74391i 0.845758 + 2.20327i −6.42941 + 1.01832i 1.65551 + 3.71833i 2.94353 − 3.44954i
7.13 −0.391255 + 0.767881i 0.376748 − 0.580140i 1.91458 + 2.63519i 3.28724 + 3.76750i 0.298074 + 0.516280i −1.19853 − 3.12227i −6.17741 + 0.978406i 3.46601 + 7.78478i −4.17914 + 1.05015i
7.14 −0.0952519 + 0.186942i 2.03448 − 3.13283i 2.32527 + 3.20045i 4.03859 − 2.94784i 0.391871 + 0.678740i 2.14170 + 5.57931i −1.64870 + 0.261128i −2.01487 − 4.52548i 0.166393 + 1.03577i
7.15 −0.0301697 + 0.0592113i −2.88826 + 4.44753i 2.34855 + 3.23250i −1.24450 + 4.84264i −0.176206 − 0.305198i 1.59511 + 4.15541i −0.524800 + 0.0831201i −7.77784 − 17.4693i −0.249193 − 0.219790i
7.16 0.179471 − 0.352233i −2.31955 + 3.57179i 2.25928 + 3.10964i 4.77352 − 1.48779i 0.841808 + 1.45805i −0.0581079 − 0.151376i 3.06260 − 0.485069i −3.71675 − 8.34796i 0.332661 − 1.94840i
7.17 0.213780 − 0.419567i 0.214276 − 0.329956i 2.22081 + 3.05668i 0.929580 − 4.91283i −0.0926307 − 0.160441i −4.51498 − 11.7619i 3.61762 − 0.572975i 3.59767 + 8.08051i −1.86254 − 1.44029i
7.18 0.431126 − 0.846132i −0.392066 + 0.603728i 1.82107 + 2.50649i −4.73644 + 1.60191i 0.341804 + 0.592022i −0.895716 − 2.33342i 6.65771 − 1.05448i 3.44986 + 7.74851i −0.686576 + 4.69828i
7.19 0.546739 − 1.07304i 2.65422 − 4.08714i 1.49866 + 2.06273i −4.15113 − 2.78714i −2.93448 − 5.08267i 0.988700 + 2.57565i 7.79063 − 1.23392i −5.99918 − 13.4744i −5.26028 + 2.93048i
7.20 0.592153 − 1.16217i 2.83226 − 4.36130i 1.35116 + 1.85971i 1.72673 + 4.69238i −3.39142 − 5.87411i −3.62623 − 9.44665i 8.11446 − 1.28520i −7.33859 − 16.4828i 6.47581 + 0.771861i
See next 80 embeddings (of 480 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
31.g even 15 1 inner
155.w odd 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 155.3.w.a ✓ 480
5.c odd 4 1 inner 155.3.w.a ✓ 480
31.g even 15 1 inner 155.3.w.a ✓ 480
155.w odd 60 1 inner 155.3.w.a ✓ 480
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.3.w.a ✓ 480 1.a even 1 1 trivial
155.3.w.a ✓ 480 5.c odd 4 1 inner
155.3.w.a ✓ 480 31.g even 15 1 inner
155.3.w.a ✓ 480 155.w odd 60 1 inner

Hecke kernels

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