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Properties

Label 735.2.u.b
Level $735$
Weight $2$
Character orbit 735.u
Analytic conductor $5.869$
Analytic rank $0$
Dimension $48$
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(106,735)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 0, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("735.106");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.u (of order \(7\), degree \(6\), 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: \(48\)
Relative dimension: \(8\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + q^{2} - 8 q^{3} - q^{4} - 8 q^{5} + q^{6} - q^{7} + 15 q^{8} - 8 q^{9} + q^{10} + 10 q^{11} - q^{12} + 20 q^{13} - q^{14} - 8 q^{15} - 15 q^{16} + 7 q^{17} - 6 q^{18} - 2 q^{19} - q^{20}+ \cdots - 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
106.1 −1.50084 − 1.88200i −0.900969 + 0.433884i −0.844346 + 3.69932i −0.900969 + 0.433884i 2.16878 + 1.04443i −2.16503 + 1.52074i 3.89178 − 1.87418i 0.623490 − 0.781831i 2.16878 + 1.04443i
106.2 −1.47354 − 1.84776i −0.900969 + 0.433884i −0.797863 + 3.49567i −0.900969 + 0.433884i 2.12933 + 1.02543i −1.04651 − 2.42998i 3.37619 − 1.62589i 0.623490 − 0.781831i 2.12933 + 1.02543i
106.3 −1.00535 − 1.26067i −0.900969 + 0.433884i −0.133517 + 0.584976i −0.900969 + 0.433884i 1.45277 + 0.699619i 2.49967 + 0.866972i −2.03386 + 0.979453i 0.623490 − 0.781831i 1.45277 + 0.699619i
106.4 −0.0768248 − 0.0963352i −0.900969 + 0.433884i 0.441663 − 1.93505i −0.900969 + 0.433884i 0.111015 + 0.0534620i 1.60567 − 2.10281i −0.442374 + 0.213036i 0.623490 − 0.781831i 0.111015 + 0.0534620i
106.5 0.0311710 + 0.0390871i −0.900969 + 0.433884i 0.444486 − 1.94742i −0.900969 + 0.433884i −0.0450433 − 0.0216917i 1.31746 + 2.29440i 0.180061 − 0.0867127i 0.623490 − 0.781831i −0.0450433 − 0.0216917i
106.6 0.837439 + 1.05012i −0.900969 + 0.433884i 0.0436035 − 0.191039i −0.900969 + 0.433884i −1.21013 − 0.582770i −2.55563 − 0.684649i 2.65740 − 1.27974i 0.623490 − 0.781831i −1.21013 − 0.582770i
106.7 1.02388 + 1.28391i −0.900969 + 0.433884i −0.155040 + 0.679276i −0.900969 + 0.433884i −1.47955 − 0.712514i −0.516929 − 2.59476i 1.92823 − 0.928588i 0.623490 − 0.781831i −1.47955 − 0.712514i
106.8 1.54058 + 1.93183i −0.900969 + 0.433884i −0.913529 + 4.00243i −0.900969 + 0.433884i −2.22621 − 1.07208i −1.73169 + 2.00031i −4.68697 + 2.25713i 0.623490 − 0.781831i −2.22621 − 1.07208i
211.1 −0.489922 + 2.14649i 0.623490 − 0.781831i −2.56545 − 1.23546i 0.623490 − 0.781831i 1.37273 + 1.72135i −2.64571 + 0.0145107i 1.16331 − 1.45874i −0.222521 − 0.974928i 1.37273 + 1.72135i
211.2 −0.430109 + 1.88443i 0.623490 − 0.781831i −1.56415 − 0.753254i 0.623490 − 0.781831i 1.20514 + 1.51120i 1.44043 − 2.21927i −0.318067 + 0.398843i −0.222521 − 0.974928i 1.20514 + 1.51120i
211.3 −0.105053 + 0.460266i 0.623490 − 0.781831i 1.60113 + 0.771063i 0.623490 − 0.781831i 0.294351 + 0.369105i −2.25104 + 1.39025i −1.11180 + 1.39415i −0.222521 − 0.974928i 0.294351 + 0.369105i
211.4 −0.0253319 + 0.110986i 0.623490 − 0.781831i 1.79026 + 0.862145i 0.623490 − 0.781831i 0.0709783 + 0.0890040i 2.44876 − 1.00178i −0.282993 + 0.354862i −0.222521 − 0.974928i 0.0709783 + 0.0890040i
211.5 0.179043 − 0.784438i 0.623490 − 0.781831i 1.21865 + 0.586872i 0.623490 − 0.781831i −0.501667 − 0.629070i 2.64379 + 0.101941i 1.68189 − 2.10902i −0.222521 − 0.974928i −0.501667 − 0.629070i
211.6 0.189678 − 0.831032i 0.623490 − 0.781831i 1.14730 + 0.552511i 0.623490 − 0.781831i −0.531465 − 0.666436i −2.46818 − 0.952935i 1.73970 − 2.18152i −0.222521 − 0.974928i −0.531465 − 0.666436i
211.7 0.414664 − 1.81676i 0.623490 − 0.781831i −1.32673 − 0.638921i 0.623490 − 0.781831i −1.16186 − 1.45693i −1.39768 − 2.24644i 0.612808 − 0.768437i −0.222521 − 0.974928i −1.16186 − 1.45693i
211.8 0.489553 − 2.14487i 0.623490 − 0.781831i −2.55887 − 1.23229i 0.623490 − 0.781831i −1.37170 − 1.72005i 1.49623 + 2.18204i −1.15241 + 1.44507i −0.222521 − 0.974928i −1.37170 − 1.72005i
316.1 −2.19446 + 1.05679i −0.222521 + 0.974928i 2.45184 − 3.07451i −0.222521 + 0.974928i −0.541986 − 2.37459i 2.00777 − 1.72304i −1.04736 + 4.58878i −0.900969 − 0.433884i −0.541986 − 2.37459i
316.2 −1.61143 + 0.776023i −0.222521 + 0.974928i 0.747508 − 0.937345i −0.222521 + 0.974928i −0.397990 − 1.74371i 0.837875 + 2.50957i 0.318826 − 1.39687i −0.900969 − 0.433884i −0.397990 − 1.74371i
316.3 −0.557211 + 0.268339i −0.222521 + 0.974928i −1.00850 + 1.26462i −0.222521 + 0.974928i −0.137620 − 0.602952i −2.57754 + 0.596878i 0.497841 − 2.18118i −0.900969 − 0.433884i −0.137620 − 0.602952i
316.4 −0.441873 + 0.212795i −0.222521 + 0.974928i −1.09701 + 1.37561i −0.222521 + 0.974928i −0.109134 − 0.478146i 1.08258 − 2.41413i 0.410284 − 1.79757i −0.900969 − 0.433884i −0.109134 − 0.478146i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 106.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.u.b ✓ 48
49.e even 7 1 inner 735.2.u.b ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
735.2.u.b ✓ 48 1.a even 1 1 trivial
735.2.u.b ✓ 48 49.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} - T_{2}^{47} + 9 T_{2}^{46} - 15 T_{2}^{45} + 89 T_{2}^{44} - 133 T_{2}^{43} + 630 T_{2}^{42} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(735, [\chi])\). Copy content Toggle raw display