Properties

Label 740.2.be.a
Level $740$
Weight $2$
Character orbit 740.be
Analytic conductor $5.909$
Analytic rank $0$
Dimension $304$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [740,2,Mod(51,740)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(740, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 0, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("740.51");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 740.be (of order \(12\), degree \(4\), 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: \(5.90892974957\)
Analytic rank: \(0\)
Dimension: \(304\)
Relative dimension: \(76\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 304 q + 4 q^{2} - 12 q^{4} - 8 q^{8} - 152 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 304 q + 4 q^{2} - 12 q^{4} - 8 q^{8} - 152 q^{9} + 16 q^{13} - 24 q^{14} + 20 q^{16} - 60 q^{18} - 16 q^{22} - 24 q^{24} - 32 q^{29} - 56 q^{32} - 32 q^{37} + 40 q^{38} + 20 q^{42} + 12 q^{44} - 8 q^{46} + 104 q^{49} + 4 q^{50} - 40 q^{52} + 40 q^{54} + 4 q^{56} - 48 q^{57} - 12 q^{58} - 28 q^{60} - 16 q^{61} - 60 q^{62} + 24 q^{65} - 52 q^{66} - 200 q^{68} - 196 q^{72} + 20 q^{74} - 40 q^{76} + 96 q^{77} - 48 q^{78} - 152 q^{81} + 32 q^{82} - 112 q^{84} + 12 q^{88} - 8 q^{89} + 140 q^{92} - 16 q^{93} - 28 q^{94} - 132 q^{96} + 48 q^{97} - 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1 −1.41421 0.000708265i −1.00731 1.74471i 2.00000 + 0.00200328i 0.965926 + 0.258819i 1.42331 + 2.46810i −2.33505 + 1.34814i −2.82842 0.00424959i −0.529335 + 0.916835i −1.36584 0.366709i
51.2 −1.41335 + 0.0493513i 0.328872 + 0.569623i 1.99513 0.139502i 0.965926 + 0.258819i −0.492923 0.788847i −2.40655 + 1.38942i −2.81294 + 0.295627i 1.28369 2.22341i −1.37797 0.318133i
51.3 −1.41162 + 0.0855720i 0.839608 + 1.45424i 1.98535 0.241591i −0.965926 0.258819i −1.30965 1.98100i −0.586292 + 0.338496i −2.78190 + 0.510926i 0.0901155 0.156085i 1.38567 + 0.282699i
51.4 −1.40586 + 0.153475i −1.10372 1.91170i 1.95289 0.431528i −0.965926 0.258819i 1.84508 + 2.51819i −0.0394290 + 0.0227643i −2.67927 + 0.906387i −0.936403 + 1.62190i 1.39768 + 0.215619i
51.5 −1.39498 0.232435i −0.140643 0.243602i 1.89195 + 0.648485i −0.965926 0.258819i 0.139574 + 0.372510i 3.36739 1.94416i −2.48850 1.34438i 1.46044 2.52955i 1.28729 + 0.585563i
51.6 −1.38367 + 0.292354i −1.60937 2.78750i 1.82906 0.809039i 0.965926 + 0.258819i 3.04176 + 3.38647i 4.09048 2.36164i −2.29428 + 1.65417i −3.68012 + 6.37416i −1.41218 0.0757269i
51.7 −1.34883 0.425031i −1.46817 2.54294i 1.63870 + 1.14659i −0.965926 0.258819i 0.899483 + 4.05402i −2.48215 + 1.43307i −1.72299 2.24306i −2.81103 + 4.86885i 1.19287 + 0.759652i
51.8 −1.34447 + 0.438647i 1.60937 + 2.78750i 1.61518 1.17949i 0.965926 + 0.258819i −3.38647 3.04176i −4.09048 + 2.36164i −1.65417 + 2.29428i −3.68012 + 6.37416i −1.41218 + 0.0757269i
51.9 −1.31917 0.509705i 0.769115 + 1.33215i 1.48040 + 1.34477i 0.965926 + 0.258819i −0.335590 2.14935i 1.23671 0.714015i −1.26746 2.52855i 0.316923 0.548926i −1.14230 0.833762i
51.10 −1.29425 + 0.570018i 1.10372 + 1.91170i 1.35016 1.47549i −0.965926 0.258819i −2.51819 1.84508i 0.0394290 0.0227643i −0.906387 + 2.67927i −0.936403 + 1.62190i 1.39768 0.215619i
51.11 −1.26529 + 0.631704i −0.839608 1.45424i 1.20190 1.59857i −0.965926 0.258819i 1.98100 + 1.30965i 0.586292 0.338496i −0.510926 + 2.78190i 0.0901155 0.156085i 1.38567 0.282699i
51.12 −1.24867 + 0.663937i −0.328872 0.569623i 1.11838 1.65808i 0.965926 + 0.258819i 0.788847 + 0.492923i 2.40655 1.38942i −0.295627 + 2.81294i 1.28369 2.22341i −1.37797 + 0.318133i
51.13 −1.22439 + 0.707720i 1.00731 + 1.74471i 0.998265 1.73305i 0.965926 + 0.258819i −2.46810 1.42331i 2.33505 1.34814i 0.00424959 + 2.82842i −0.529335 + 0.916835i −1.36584 + 0.366709i
51.14 −1.16908 0.795771i −0.450967 0.781098i 0.733498 + 1.86064i 0.965926 + 0.258819i −0.0943580 + 1.27203i 2.84609 1.64319i 0.623124 2.75893i 1.09326 1.89358i −0.923284 1.07124i
51.15 −1.16119 0.807234i 1.22756 + 2.12620i 0.696746 + 1.87471i 0.965926 + 0.258819i 0.290903 3.45986i −1.51707 + 0.875879i 0.704274 2.73934i −1.51381 + 2.62200i −0.912700 1.08027i
51.16 −1.13569 0.842742i 0.902444 + 1.56308i 0.579572 + 1.91418i −0.965926 0.258819i 0.292378 2.53570i −0.124486 + 0.0718719i 0.954949 2.66234i −0.128811 + 0.223107i 0.878872 + 1.10796i
51.17 −1.09187 + 0.898785i 0.140643 + 0.243602i 0.384370 1.96272i −0.965926 0.258819i −0.372510 0.139574i −3.36739 + 1.94416i 1.34438 + 2.48850i 1.46044 2.52955i 1.28729 0.585563i
51.18 −1.06502 0.930442i −1.41047 2.44300i 0.268554 + 1.98189i −0.965926 0.258819i −0.770890 + 3.91421i 2.56631 1.48166i 1.55802 2.36063i −2.47883 + 4.29347i 0.787918 + 1.17439i
51.19 −0.994365 1.00560i −0.766676 1.32792i −0.0224764 + 1.99987i −0.965926 0.258819i −0.573007 + 2.09141i −3.13458 + 1.80975i 2.03343 1.96600i 0.324416 0.561906i 0.700214 + 1.22870i
51.20 −0.955608 + 1.04250i 1.46817 + 2.54294i −0.173628 1.99245i −0.965926 0.258819i −4.05402 0.899483i 2.48215 1.43307i 2.24306 + 1.72299i −2.81103 + 4.86885i 1.19287 0.759652i
See next 80 embeddings (of 304 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.76
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
37.g odd 12 1 inner
148.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.be.a 304
4.b odd 2 1 inner 740.2.be.a 304
37.g odd 12 1 inner 740.2.be.a 304
148.l even 12 1 inner 740.2.be.a 304
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.be.a 304 1.a even 1 1 trivial
740.2.be.a 304 4.b odd 2 1 inner
740.2.be.a 304 37.g odd 12 1 inner
740.2.be.a 304 148.l even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(740, [\chi])\).