↑

Properties

Label 603.2.h.c
Level $603$
Weight $2$
Character orbit 603.h
Analytic conductor $4.815$
Analytic rank $0$
Dimension $128$
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] = [603,2,Mod(364,603)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(603, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("603.364");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 603 = 3^{2} \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 603.h (of order \(3\), degree \(2\), 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: \(4.81497924188\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(64\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q - 2 q^{2} - 3 q^{3} + 130 q^{4} + 5 q^{5} - 6 q^{6} + 4 q^{7} - 36 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q - 2 q^{2} - 3 q^{3} + 130 q^{4} + 5 q^{5} - 6 q^{6} + 4 q^{7} - 36 q^{8} + 5 q^{9} - 2 q^{10} + 15 q^{11} - 7 q^{12} - 6 q^{13} - 8 q^{14} + 12 q^{15} + 118 q^{16} - q^{17} - 16 q^{18} + 8 q^{19} + 3 q^{20} - 3 q^{21} - 20 q^{22} + 9 q^{23} - 39 q^{24} - 57 q^{25} + 8 q^{26} - 24 q^{27} - 16 q^{28} + q^{29} - 28 q^{30} - 32 q^{31} - 54 q^{32} - 9 q^{33} + 2 q^{34} + 11 q^{35} - 24 q^{36} + 2 q^{37} + 2 q^{38} - 7 q^{39} - 6 q^{40} + 14 q^{41} + 18 q^{42} - 11 q^{43} - 18 q^{44} + 3 q^{45} - 4 q^{46} + 12 q^{47} - 28 q^{48} - 52 q^{49} - 22 q^{50} - 12 q^{51} - 6 q^{52} - 60 q^{53} - 15 q^{54} + 10 q^{55} + 14 q^{56} + 35 q^{57} + 6 q^{59} + 23 q^{60} + 12 q^{61} - 22 q^{62} + 17 q^{63} + 56 q^{64} + 52 q^{65} + 54 q^{66} + 9 q^{67} - 19 q^{68} - 34 q^{69} + 25 q^{70} + 8 q^{71} + 11 q^{72} + 22 q^{73} + 21 q^{74} - 44 q^{75} - 5 q^{76} + 35 q^{77} - 30 q^{78} + 15 q^{79} + 14 q^{80} - 63 q^{81} - 24 q^{82} + 6 q^{83} - 41 q^{84} - 12 q^{85} - 40 q^{87} + 3 q^{88} + 116 q^{89} - 103 q^{90} - 10 q^{91} + 4 q^{92} - q^{93} + 4 q^{94} + 90 q^{95} - 17 q^{96} - 38 q^{97} - 49 q^{98} + 46 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
364.1 −2.74578 1.27896 + 1.16802i 5.53929 1.71242 − 2.96599i −3.51174 − 3.20711i −0.122480 + 0.212142i −9.71811 0.271480 + 2.98769i −4.70192 + 8.14396i
364.2 −2.74325 −0.742705 − 1.56473i 5.52545 −1.51879 + 2.63061i 2.03743 + 4.29246i −0.0861787 + 0.149266i −9.67119 −1.89678 + 2.32427i 4.16641 − 7.21644i
364.3 −2.72229 1.25787 − 1.19070i 5.41089 0.0741525 − 0.128436i −3.42429 + 3.24143i −0.766055 + 1.32685i −9.28543 0.164473 − 2.99549i −0.201865 + 0.349640i
364.4 −2.60414 −1.43418 − 0.971150i 4.78155 1.32219 − 2.29010i 3.73480 + 2.52901i 1.03598 − 1.79437i −7.24356 1.11374 + 2.78560i −3.44317 + 5.96375i
364.5 −2.56776 −0.454764 + 1.67128i 4.59340 −0.696697 + 1.20671i 1.16773 − 4.29146i 1.03214 − 1.78773i −6.65922 −2.58638 − 1.52008i 1.78895 − 3.09855i
364.6 −2.49433 −1.55203 + 0.768896i 4.22167 0.591175 − 1.02395i 3.87127 − 1.91788i −2.58327 + 4.47435i −5.54157 1.81760 − 2.38670i −1.47458 + 2.55405i
364.7 −2.38827 1.28549 + 1.16082i 3.70385 −1.50924 + 2.61408i −3.07011 − 2.77236i −1.72843 + 2.99373i −4.06926 0.304990 + 2.98446i 3.60448 − 6.24315i
364.8 −2.25835 1.48974 − 0.883560i 3.10015 0.268915 − 0.465775i −3.36435 + 1.99539i 2.07986 − 3.60242i −2.48451 1.43864 − 2.63255i −0.607305 + 1.05188i
364.9 −2.24620 −1.70592 + 0.299744i 3.04541 −1.53025 + 2.65047i 3.83183 − 0.673285i 0.754542 − 1.30690i −2.34821 2.82031 − 1.02268i 3.43725 − 5.95348i
364.10 −2.13782 0.547814 − 1.64314i 2.57029 −0.157720 + 0.273179i −1.17113 + 3.51274i −1.18479 + 2.05211i −1.21919 −2.39980 − 1.80027i 0.337177 − 0.584008i
364.11 −2.12128 0.865200 + 1.50048i 2.49984 0.171940 − 0.297809i −1.83533 − 3.18293i 1.17964 − 2.04320i −1.06030 −1.50286 + 2.59642i −0.364734 + 0.631738i
364.12 −2.05440 −0.358844 − 1.69447i 2.22058 1.67810 − 2.90655i 0.737211 + 3.48113i −0.853884 + 1.47897i −0.453151 −2.74246 + 1.21610i −3.44749 + 5.97123i
364.13 −1.86122 −1.56516 + 0.741797i 1.46414 1.74609 − 3.02432i 2.91311 − 1.38065i 2.17448 − 3.76631i 0.997360 1.89947 − 2.32207i −3.24986 + 5.62892i
364.14 −1.83465 1.73038 − 0.0761224i 1.36596 −0.599169 + 1.03779i −3.17464 + 0.139658i −1.23241 + 2.13459i 1.16325 2.98841 − 0.263441i 1.09927 − 1.90399i
364.15 −1.82043 1.67206 + 0.451896i 1.31398 1.27150 − 2.20230i −3.04388 − 0.822647i 0.793425 − 1.37425i 1.24885 2.59158 + 1.51120i −2.31468 + 4.00914i
364.16 −1.79110 −0.317785 + 1.70265i 1.20805 0.413615 − 0.716402i 0.569186 − 3.04962i −0.748964 + 1.29724i 1.41846 −2.79803 − 1.08215i −0.740827 + 1.28315i
364.17 −1.78552 −0.479463 − 1.66437i 1.18807 −1.08852 + 1.88538i 0.856089 + 2.97175i 2.43299 − 4.21407i 1.44972 −2.54023 + 1.59600i 1.94358 − 3.36637i
364.18 −1.61104 1.22627 − 1.22322i 0.595457 −2.00413 + 3.47125i −1.97557 + 1.97066i −0.145010 + 0.251165i 2.26278 0.00747014 − 2.99999i 3.22874 − 5.59234i
364.19 −1.58681 −1.63018 − 0.585242i 0.517950 0.588622 − 1.01952i 2.58678 + 0.928665i −0.174819 + 0.302795i 2.35172 2.31498 + 1.90810i −0.934029 + 1.61779i
364.20 −1.15690 −1.29015 + 1.15564i −0.661573 0.201329 − 0.348711i 1.49258 − 1.33697i −0.136234 + 0.235964i 3.07918 0.328975 − 2.98191i −0.232918 + 0.403426i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 364.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
603.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 603.2.h.c yes 128
9.c even 3 1 603.2.f.c ✓ 128
67.c even 3 1 603.2.f.c ✓ 128
603.h even 3 1 inner 603.2.h.c yes 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
603.2.f.c ✓ 128 9.c even 3 1
603.2.f.c ✓ 128 67.c even 3 1
603.2.h.c yes 128 1.a even 1 1 trivial
603.2.h.c yes 128 603.h even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(603, [\chi])\):

\( T_{2}^{64} + T_{2}^{63} - 96 T_{2}^{62} - 89 T_{2}^{61} + 4375 T_{2}^{60} + 3734 T_{2}^{59} + \cdots - 2673 \) Copy content Toggle raw display
\( T_{5}^{128} - 5 T_{5}^{127} + 201 T_{5}^{126} - 884 T_{5}^{125} + 20749 T_{5}^{124} + \cdots + 69\!\cdots\!49 \) Copy content Toggle raw display