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Properties

Label 735.2.bm.a
Level $735$
Weight $2$
Character orbit 735.bm
Analytic conductor $5.869$
Analytic rank $0$
Dimension $444$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [735,2,Mod(26,735)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([21, 0, 17]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("735.26");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.bm (of order \(42\), degree \(12\), 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: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(444\)
Relative dimension: \(37\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 444 q - 36 q^{4} - 37 q^{5} + 12 q^{6} + 3 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 444 q - 36 q^{4} - 37 q^{5} + 12 q^{6} + 3 q^{7} - 4 q^{9} - 6 q^{11} - 27 q^{12} + 18 q^{14} + 34 q^{16} - 161 q^{18} + 3 q^{19} - 72 q^{20} - 22 q^{21} + 12 q^{22} - 24 q^{23} - 12 q^{24} + 37 q^{25} + 16 q^{26} + 9 q^{27} + 40 q^{28} - 42 q^{29} + 2 q^{30} + 27 q^{31} - 30 q^{32} + 53 q^{33} + 3 q^{35} - 84 q^{36} + 4 q^{37} + 8 q^{38} + 12 q^{39} + 46 q^{41} + 25 q^{42} - 10 q^{43} - 6 q^{44} + 8 q^{45} - 78 q^{46} + 32 q^{47} - 10 q^{48} - 35 q^{49} + 25 q^{51} - 52 q^{52} - 24 q^{53} - 16 q^{54} + 24 q^{56} - 10 q^{57} - 52 q^{58} + 8 q^{59} + 71 q^{60} + 71 q^{61} - 4 q^{62} - 83 q^{63} + 60 q^{64} + 25 q^{65} - 25 q^{66} - 5 q^{67} + 14 q^{68} - 116 q^{69} + 28 q^{71} - 21 q^{73} + 54 q^{77} + 162 q^{78} + 3 q^{79} + 204 q^{80} - 80 q^{81} - 36 q^{82} + 64 q^{84} - 208 q^{86} - 134 q^{87} + 8 q^{89} - 33 q^{90} + 25 q^{91} - 8 q^{93} + 52 q^{94} + 3 q^{95} - 477 q^{96} - 26 q^{98} + 82 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1 −2.77212 − 0.207742i 1.73018 + 0.0805277i 5.66382 + 0.853684i 0.733052 + 0.680173i −4.77953 − 0.582662i 2.03593 + 1.68967i −10.1031 − 2.30596i 2.98703 + 0.278655i −1.89081 − 2.03780i
26.2 −2.64485 − 0.198204i −1.72728 + 0.128453i 4.97828 + 0.750355i 0.733052 + 0.680173i 4.59386 + 0.00261575i −2.38725 + 1.14062i −7.84655 − 1.79092i 2.96700 − 0.443748i −1.80400 − 1.94425i
26.3 −2.62857 − 0.196984i 0.265818 − 1.71153i 4.89290 + 0.737486i 0.733052 + 0.680173i −1.03587 + 4.44651i −1.64760 − 2.07013i −7.57635 − 1.72925i −2.85868 − 0.909913i −1.79289 − 1.93228i
26.4 −2.32482 − 0.174221i −0.189834 + 1.72162i 3.39676 + 0.511979i 0.733052 + 0.680173i 0.741270 − 3.96937i 0.500339 + 2.59801i −3.26188 − 0.744502i −2.92793 − 0.653642i −1.58571 − 1.70899i
26.5 −2.09482 − 0.156985i −1.71868 + 0.214770i 2.38597 + 0.359626i 0.733052 + 0.680173i 3.63405 − 0.180097i 2.50104 − 0.863027i −0.845662 − 0.193017i 2.90775 − 0.738243i −1.42883 − 1.53992i
26.6 −2.08738 − 0.156427i −0.804505 + 1.53387i 2.35501 + 0.354961i 0.733052 + 0.680173i 1.91925 − 3.07593i 1.08520 − 2.41295i −0.778776 − 0.177750i −1.70554 − 2.46802i −1.42376 − 1.53445i
26.7 −1.91733 − 0.143684i 1.58140 − 0.706523i 1.67784 + 0.252894i 0.733052 + 0.680173i −3.13358 + 1.12741i 0.564185 − 2.58490i 0.568355 + 0.129723i 2.00165 − 2.23459i −1.30777 − 1.40944i
26.8 −1.74448 − 0.130730i −1.19941 − 1.24957i 1.04845 + 0.158028i 0.733052 + 0.680173i 1.92898 + 2.33664i −1.29508 − 2.30711i 1.60268 + 0.365802i −0.122835 + 2.99748i −1.18987 − 1.28238i
26.9 −1.69357 − 0.126915i 1.42142 − 0.989728i 0.874403 + 0.131795i 0.733052 + 0.680173i −2.53289 + 1.49577i 1.99166 + 1.74163i 1.84734 + 0.421643i 1.04088 − 2.81364i −1.15515 − 1.24495i
26.10 −1.67501 − 0.125524i 0.163349 − 1.72433i 0.812234 + 0.122425i 0.733052 + 0.680173i −0.490057 + 2.86776i −1.70353 + 2.02435i 1.93005 + 0.440522i −2.94663 − 0.563337i −1.14249 − 1.23131i
26.11 −1.33514 − 0.100055i 0.353481 + 1.69560i −0.205075 − 0.0309101i 0.733052 + 0.680173i −0.302294 − 2.29923i −2.62647 − 0.318819i 2.88134 + 0.657647i −2.75010 + 1.19872i −0.910672 − 0.981471i
26.12 −1.30176 − 0.0975535i 1.31873 + 1.12292i −0.292596 − 0.0441018i 0.733052 + 0.680173i −1.60713 − 1.59042i 2.64011 + 0.172665i 2.92195 + 0.666916i 0.478105 + 2.96166i −0.887905 − 0.956934i
26.13 −1.06838 − 0.0800638i −1.42092 − 0.990451i −0.842641 − 0.127008i 0.733052 + 0.680173i 1.43878 + 1.17194i 1.41530 + 2.23538i 2.97911 + 0.679963i 1.03801 + 2.81470i −0.728719 − 0.785372i
26.14 −1.02438 − 0.0767668i −1.57164 + 0.727965i −0.934198 − 0.140808i 0.733052 + 0.680173i 1.66585 − 0.625063i −2.64478 − 0.0718366i 2.94916 + 0.673127i 1.94013 − 2.28820i −0.698710 − 0.753030i
26.15 −0.640838 − 0.0480242i −1.24825 + 1.20078i −1.56929 − 0.236533i 0.733052 + 0.680173i 0.857592 − 0.709561i −0.0216235 + 2.64566i 2.24735 + 0.512943i 0.116248 − 2.99775i −0.437103 − 0.471085i
26.16 −0.522803 − 0.0391787i 1.73195 − 0.0184969i −1.70587 − 0.257119i 0.733052 + 0.680173i −0.906195 − 0.0581854i −2.16228 − 1.52464i 1.90401 + 0.434578i 2.99932 − 0.0640714i −0.356594 − 0.384317i
26.17 −0.402292 − 0.0301476i 0.961638 − 1.44057i −1.81673 − 0.273828i 0.733052 + 0.680173i −0.430289 + 0.550540i 1.82948 − 1.91128i 1.50921 + 0.344468i −1.15051 − 2.77062i −0.274395 − 0.295728i
26.18 −0.241156 − 0.0180721i 0.339909 + 1.69837i −1.91983 − 0.289368i 0.733052 + 0.680173i −0.0512778 − 0.415715i 0.932196 − 2.47609i 0.929287 + 0.212104i −2.76892 + 1.15458i −0.164488 − 0.177275i
26.19 0.0993298 + 0.00744374i 1.08770 + 1.34793i −1.96785 − 0.296606i 0.733052 + 0.680173i 0.0980077 + 0.141986i −1.04822 + 2.42925i −0.387480 − 0.0884398i −0.633806 + 2.93228i 0.0677509 + 0.0730181i
26.20 0.132025 + 0.00989392i 0.248046 − 1.71420i −1.96033 − 0.295472i 0.733052 + 0.680173i 0.0497085 − 0.223863i −2.56321 − 0.655707i −0.514042 − 0.117327i −2.87695 − 0.850399i 0.0900518 + 0.0970527i
See next 80 embeddings (of 444 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.37
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
147.o even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.bm.a ✓ 444
3.b odd 2 1 735.2.bm.b yes 444
49.h odd 42 1 735.2.bm.b yes 444
147.o even 42 1 inner 735.2.bm.a ✓ 444
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
735.2.bm.a ✓ 444 1.a even 1 1 trivial
735.2.bm.a ✓ 444 147.o even 42 1 inner
735.2.bm.b yes 444 3.b odd 2 1
735.2.bm.b yes 444 49.h odd 42 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{444} + 55 T_{2}^{442} + 1376 T_{2}^{440} + 6 T_{2}^{439} + 19173 T_{2}^{438} + \cdots + 17\!\cdots\!24 \) acting on \(S_{2}^{\mathrm{new}}(735, [\chi])\). Copy content Toggle raw display