↑

Properties

Label 735.2.bm.b
Level $735$
Weight $2$
Character orbit 735.bm
Analytic conductor $5.869$
Analytic rank $0$
Dimension $444$
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] = [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.

\(\operatorname{Tr}(f)(q) = \) \( 444 q - 36 q^{4} + 37 q^{5} + 2 q^{6} + 3 q^{7} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 444 q - 36 q^{4} + 37 q^{5} + 2 q^{6} + 3 q^{7} + 20 q^{9} + 6 q^{11} - 13 q^{12} - 18 q^{14} + 34 q^{16} + 163 q^{18} + 3 q^{19} + 72 q^{20} + 16 q^{21} + 12 q^{22} + 24 q^{23} - 6 q^{24} + 37 q^{25} - 16 q^{26} - 51 q^{27} + 40 q^{28} + 42 q^{29} - 4 q^{30} + 27 q^{31} + 30 q^{32} - 77 q^{33} - 3 q^{35} + 14 q^{36} + 4 q^{37} - 8 q^{38} - 24 q^{39} - 46 q^{41} - 139 q^{42} - 10 q^{43} + 6 q^{44} + 10 q^{45} - 78 q^{46} - 32 q^{47} + 10 q^{48} - 35 q^{49} - 11 q^{51} - 52 q^{52} + 24 q^{53} - 8 q^{54} - 24 q^{56} - 10 q^{57} - 52 q^{58} - 8 q^{59} + 41 q^{60} + 71 q^{61} + 4 q^{62} + 117 q^{63} + 60 q^{64} - 25 q^{65} - 5 q^{66} - 5 q^{67} - 14 q^{68} - 52 q^{69} - 28 q^{71} - 18 q^{72} - 21 q^{73} - 54 q^{77} - 230 q^{78} + 3 q^{79} - 204 q^{80} - 136 q^{81} - 36 q^{82} - 324 q^{84} + 208 q^{86} - 84 q^{87} - 8 q^{89} + 33 q^{90} + 25 q^{91} - 10 q^{93} + 52 q^{94} - 3 q^{95} - 107 q^{96} + 26 q^{98} + 82 q^{99}+O(q^{100}) \) 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 −2.64236 − 0.198018i 0.883699 − 1.48966i 4.96520 + 0.748383i −0.733052 − 0.680173i −2.63003 + 3.76122i −0.638989 + 2.56743i −7.80498 − 1.78144i −1.43815 − 2.63282i 1.80230 + 1.94242i
26.2 −2.62309 − 0.196573i −1.26915 − 1.17866i 4.86430 + 0.733176i −0.733052 − 0.680173i 3.09741 + 3.34123i 0.382655 − 2.61793i −7.48639 − 1.70872i 0.221499 + 2.99181i 1.78916 + 1.92825i
26.3 −2.39157 − 0.179224i 0.531511 + 1.64848i 3.70984 + 0.559168i −0.733052 − 0.680173i −0.975700 − 4.03773i 2.45607 − 0.983735i −4.09583 − 0.934846i −2.43499 + 1.75237i 1.63124 + 1.75806i
26.4 −2.36604 − 0.177310i 1.73201 + 0.0121178i 3.58903 + 0.540959i −0.733052 − 0.680173i −4.09585 − 0.335773i 1.75969 − 1.97573i −3.76948 − 0.860360i 2.99971 + 0.0419763i 1.61383 + 1.73929i
26.5 −2.36173 − 0.176987i −1.03875 + 1.38600i 3.56878 + 0.537907i −0.733052 − 0.680173i 2.69856 − 3.08951i −2.50569 − 0.849426i −3.71534 − 0.848002i −0.841981 − 2.87942i 1.61089 + 1.73612i
26.6 −2.35662 − 0.176604i −1.32365 − 1.11712i 3.54480 + 0.534292i −0.733052 − 0.680173i 2.92204 + 2.86639i −0.172364 + 2.64013i −3.65143 − 0.833414i 0.504088 + 2.95735i 1.60740 + 1.73237i
26.7 −2.00398 − 0.150178i 1.55007 + 0.772836i 2.01573 + 0.303822i −0.733052 − 0.680173i −2.99025 − 1.78154i −0.475024 + 2.60276i −0.0754238 − 0.0172150i 1.80545 + 2.39590i 1.36688 + 1.47314i
26.8 −1.88511 − 0.141270i −1.63551 + 0.570171i 1.55604 + 0.234535i −0.733052 − 0.680173i 3.16368 − 0.843790i 1.85530 + 1.88623i 0.785829 + 0.179360i 2.34981 − 1.86505i 1.28580 + 1.38576i
26.9 −1.55536 − 0.116558i −0.181064 − 1.72256i 0.427893 + 0.0644944i −0.733052 − 0.680173i 0.0808415 + 2.70030i 2.20809 − 1.45751i 2.38322 + 0.543954i −2.93443 + 0.623788i 1.06088 + 1.14336i
26.10 −1.40710 − 0.105448i −0.0610268 + 1.73098i −0.00884992 − 0.00133391i −0.733052 − 0.680173i 0.268398 − 2.42922i −2.12383 + 1.57777i 2.76365 + 0.630785i −2.99255 − 0.211272i 0.959755 + 1.03437i
26.11 −1.35954 − 0.101884i 0.938837 − 1.45554i −0.139683 − 0.0210538i −0.733052 − 0.680173i −1.42469 + 1.88321i −1.06142 − 2.42351i 2.84611 + 0.649605i −1.23717 − 2.73302i 0.927318 + 0.999411i
26.12 −1.33002 − 0.0996712i −1.72963 − 0.0914612i −0.218645 − 0.0329554i −0.733052 − 0.680173i 2.29133 + 0.294040i 0.225270 − 2.63614i 2.88813 + 0.659198i 2.98327 + 0.316389i 0.907180 + 0.977707i
26.13 −1.07013 − 0.0801949i 1.43451 + 0.970659i −0.838923 − 0.126447i −0.733052 − 0.680173i −1.45727 − 1.15377i −2.07432 − 1.64231i 2.98006 + 0.680178i 1.11564 + 2.78484i 0.729912 + 0.786658i
26.14 −0.661397 − 0.0495648i −0.807138 − 1.53249i −1.54267 − 0.232520i −0.733052 − 0.680173i 0.457881 + 1.05359i 1.39149 + 2.25028i 2.30204 + 0.525425i −1.69706 + 2.47386i 0.451126 + 0.486198i
26.15 −0.656203 − 0.0491756i 1.72891 + 0.104207i −1.54948 − 0.233546i −0.733052 − 0.680173i −1.12939 − 0.153401i 2.46879 − 0.951345i 2.28838 + 0.522307i 2.97828 + 0.360330i 0.447583 + 0.482379i
26.16 −0.630874 − 0.0472775i 0.854508 − 1.50659i −1.58189 − 0.238432i −0.733052 − 0.680173i −0.610314 + 0.910070i −1.88437 + 1.85719i 2.22027 + 0.506761i −1.53963 − 2.57479i 0.430306 + 0.463760i
26.17 −0.587200 − 0.0440045i −1.03735 + 1.38705i −1.63479 − 0.246406i −0.733052 − 0.680173i 0.670169 − 0.768826i 2.52046 + 0.804532i 2.09727 + 0.478689i −0.847804 − 2.87771i 0.400517 + 0.431655i
26.18 −0.132025 − 0.00989392i −1.17059 − 1.27661i −1.96033 − 0.295472i −0.733052 − 0.680173i 0.141916 + 0.180126i −2.56321 − 0.655707i 0.514042 + 0.117327i −0.259452 + 2.98876i 0.0900518 + 0.0970527i
26.19 −0.0993298 − 0.00744374i −0.139389 + 1.72643i −1.96785 − 0.296606i −0.733052 − 0.680173i 0.0266966 − 0.170449i −1.04822 + 2.42925i 0.387480 + 0.0884398i −2.96114 − 0.481291i 0.0677509 + 0.0730181i
26.20 0.241156 + 0.0180721i 0.675880 + 1.59474i −1.91983 − 0.289368i −0.733052 − 0.680173i 0.134172 + 0.396795i 0.932196 − 2.47609i −0.929287 − 0.212104i −2.08637 + 2.15570i −0.164488 − 0.177275i
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.b yes 444
3.b odd 2 1 735.2.bm.a ✓ 444
49.h odd 42 1 735.2.bm.a ✓ 444
147.o even 42 1 inner 735.2.bm.b yes 444
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
735.2.bm.a ✓ 444 3.b odd 2 1
735.2.bm.a ✓ 444 49.h odd 42 1
735.2.bm.b yes 444 1.a even 1 1 trivial
735.2.bm.b yes 444 147.o even 42 1 inner

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