↑

Properties

Label 735.2.u.a
Level $735$
Weight $2$
Character orbit 735.u
Analytic conductor $5.869$
Analytic rank $0$
Dimension $48$
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(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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - q^{2} - 8 q^{3} - 9 q^{4} + 8 q^{5} - q^{6} + 7 q^{7} - 3 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - q^{2} - 8 q^{3} - 9 q^{4} + 8 q^{5} - q^{6} + 7 q^{7} - 3 q^{8} - 8 q^{9} + q^{10} - 2 q^{11} - 9 q^{12} - 21 q^{14} + 8 q^{15} + 25 q^{16} + 5 q^{17} + 6 q^{18} + 30 q^{19} + 9 q^{20} + 7 q^{21} + 49 q^{22} - 24 q^{24} - 8 q^{25} + 42 q^{26} - 8 q^{27} - 21 q^{28} + 12 q^{29} - 6 q^{30} + 86 q^{31} + 35 q^{32} - 9 q^{33} - 12 q^{34} - 7 q^{35} - 9 q^{36} - 37 q^{37} - 57 q^{38} - 28 q^{39} + 24 q^{40} + 28 q^{41} - 14 q^{42} + 2 q^{43} - 31 q^{44} + 8 q^{45} + 4 q^{46} - 2 q^{47} - 10 q^{48} + 21 q^{49} + 6 q^{50} - 23 q^{51} - 76 q^{52} + 5 q^{53} - q^{54} + 9 q^{55} + 133 q^{56} - 5 q^{57} + 20 q^{58} + 3 q^{59} + 9 q^{60} - 55 q^{61} + 26 q^{62} - 21 q^{63} - 69 q^{64} + 28 q^{65} + 49 q^{66} + 42 q^{67} + 78 q^{68} + 14 q^{70} - 40 q^{71} + 18 q^{72} - 28 q^{73} + 36 q^{74} - 8 q^{75} + 54 q^{76} + 42 q^{78} - 42 q^{79} + 10 q^{80} - 8 q^{81} + 9 q^{82} - 31 q^{83} + 23 q^{85} - 41 q^{86} - 9 q^{87} - 95 q^{88} + 37 q^{89} + q^{90} - 49 q^{91} + 37 q^{92} + 2 q^{93} - 108 q^{94} + 5 q^{95} - 49 q^{96} - 14 q^{97} + 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
106.1 −1.38677 − 1.73895i −0.900969 + 0.433884i −0.655788 + 2.87319i 0.900969 − 0.433884i 2.00394 + 0.965046i −0.736272 − 2.54124i 1.89790 − 0.913979i 0.623490 − 0.781831i −2.00394 − 0.965046i
106.2 −0.902006 − 1.13108i −0.900969 + 0.433884i −0.0206845 + 0.0906246i 0.900969 − 0.433884i 1.30344 + 0.627702i −2.26542 + 1.36670i −2.48571 + 1.19706i 0.623490 − 0.781831i −1.30344 − 0.627702i
106.3 −0.610424 − 0.765447i −0.900969 + 0.433884i 0.231750 − 1.01536i 0.900969 − 0.433884i 0.882088 + 0.424791i 2.61254 − 0.417910i −2.68285 + 1.29199i 0.623490 − 0.781831i −0.882088 − 0.424791i
106.4 −0.0334506 − 0.0419457i −0.900969 + 0.433884i 0.444401 − 1.94705i 0.900969 − 0.433884i 0.0483375 + 0.0232781i −1.44505 − 2.21627i −0.193211 + 0.0930455i 0.623490 − 0.781831i −0.0483375 − 0.0232781i
106.5 0.420027 + 0.526697i −0.900969 + 0.433884i 0.344055 − 1.50740i 0.900969 − 0.433884i −0.606956 − 0.292295i −2.60802 + 0.445248i 2.15237 − 1.03653i 0.623490 − 0.781831i 0.606956 + 0.292295i
106.6 0.695126 + 0.871661i −0.900969 + 0.433884i 0.168450 − 0.738027i 0.900969 − 0.433884i −1.00449 − 0.483735i 2.61823 − 0.380587i 2.76938 − 1.33366i 0.623490 − 0.781831i 1.00449 + 0.483735i
106.7 1.13908 + 1.42836i −0.900969 + 0.433884i −0.297670 + 1.30418i 0.900969 − 0.433884i −1.64602 − 0.792680i −0.346679 + 2.62294i 1.09013 − 0.524979i 0.623490 − 0.781831i 1.64602 + 0.792680i
106.8 1.30191 + 1.63254i −0.900969 + 0.433884i −0.525180 + 2.30097i 0.900969 − 0.433884i −1.88131 − 0.905990i 1.02271 − 2.44009i −0.677533 + 0.326282i 0.623490 − 0.781831i 1.88131 + 0.905990i
211.1 −0.577572 + 2.53051i 0.623490 − 0.781831i −4.26794 − 2.05533i −0.623490 + 0.781831i 1.61832 + 2.02931i 2.36933 + 1.17740i 4.42944 − 5.55434i −0.222521 − 0.974928i −1.61832 − 2.02931i
211.2 −0.397682 + 1.74236i 0.623490 − 0.781831i −1.07572 − 0.518040i −0.623490 + 0.781831i 1.11428 + 1.39726i −1.21817 − 2.34863i −0.898152 + 1.12625i −0.222521 − 0.974928i −1.11428 − 1.39726i
211.3 −0.276771 + 1.21261i 0.623490 − 0.781831i 0.408112 + 0.196536i −0.623490 + 0.781831i 0.775495 + 0.972439i 2.64213 + 0.138409i −1.90226 + 2.38536i −0.222521 − 0.974928i −0.775495 − 0.972439i
211.4 −0.243538 + 1.06701i 0.623490 − 0.781831i 0.722737 + 0.348052i −0.623490 + 0.781831i 0.682379 + 0.855676i −2.43609 + 1.03221i −1.91215 + 2.39776i −0.222521 − 0.974928i −0.682379 − 0.855676i
211.5 0.109674 − 0.480514i 0.623490 − 0.781831i 1.58307 + 0.762368i −0.623490 + 0.781831i −0.307300 − 0.385342i 0.338117 − 2.62406i 1.15455 − 1.44776i −0.222521 − 0.974928i 0.307300 + 0.385342i
211.6 0.190151 − 0.833107i 0.623490 − 0.781831i 1.14403 + 0.550935i −0.623490 + 0.781831i −0.532792 − 0.668100i 1.06866 + 2.42032i 1.74211 − 2.18454i −0.222521 − 0.974928i 0.532792 + 0.668100i
211.7 0.403234 − 1.76668i 0.623490 − 0.781831i −1.15664 − 0.557007i −0.623490 + 0.781831i −1.12984 − 1.41677i 1.77371 − 1.96315i 0.809224 − 1.01473i −0.222521 − 0.974928i 1.12984 + 1.41677i
211.8 0.569982 − 2.49726i 0.623490 − 0.781831i −4.10947 − 1.97902i −0.623490 + 0.781831i −1.59706 − 2.00264i −2.46916 − 0.950388i −4.09033 + 5.12911i −0.222521 − 0.974928i 1.59706 + 2.00264i
316.1 −2.43785 + 1.17401i −0.222521 + 0.974928i 3.31784 − 4.16043i 0.222521 − 0.974928i −0.602099 − 2.63797i −1.04880 − 2.42899i −1.99981 + 8.76173i −0.900969 − 0.433884i 0.602099 + 2.63797i
316.2 −1.75766 + 0.846443i −0.222521 + 0.974928i 1.12591 − 1.41185i 0.222521 − 0.974928i −0.434105 − 1.90194i 2.64347 − 0.109737i 0.0842924 − 0.369309i −0.900969 − 0.433884i 0.434105 + 1.90194i
316.3 −0.980330 + 0.472102i −0.222521 + 0.974928i −0.508813 + 0.638031i 0.222521 − 0.974928i −0.242122 − 1.06080i −0.163706 + 2.64068i 0.681832 − 2.98730i −0.900969 − 0.433884i 0.242122 + 1.06080i
316.4 −0.226674 + 0.109161i −0.222521 + 0.974928i −1.20751 + 1.51418i 0.222521 − 0.974928i −0.0559839 − 0.245282i 2.64550 + 0.0366989i 0.220392 − 0.965600i −0.900969 − 0.433884i 0.0559839 + 0.245282i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 106.8
Significant digits:
Format:

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} + T_{2}^{47} + 13 T_{2}^{46} + 15 T_{2}^{45} + 89 T_{2}^{44} + 81 T_{2}^{43} + 542 T_{2}^{42} + \cdots + 49 \) acting on \(S_{2}^{\mathrm{new}}(735, [\chi])\). Copy content Toggle raw display