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Properties

Label 731.2.f.d
Level $731$
Weight $2$
Character orbit 731.f
Analytic conductor $5.837$
Analytic rank $0$
Dimension $68$
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] = [731,2,Mod(259,731)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(731, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("731.259");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 731 = 17 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 731.f (of order \(4\), degree \(2\), 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: \(5.83706438776\)
Analytic rank: \(0\)
Dimension: \(68\)
Relative dimension: \(34\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 76 q^{4} - 4 q^{5} + 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q - 76 q^{4} - 4 q^{5} + 4 q^{6} + 2 q^{10} - 6 q^{11} - 10 q^{12} - 24 q^{13} - 22 q^{14} + 84 q^{16} - 2 q^{17} + 28 q^{18} + 10 q^{20} - 36 q^{21} + 8 q^{22} + 14 q^{23} - 62 q^{24} - 12 q^{27} - 58 q^{28} + 2 q^{29} + 160 q^{30} - 26 q^{31} + 44 q^{33} + 16 q^{34} + 56 q^{35} - 6 q^{37} - 56 q^{38} - 24 q^{39} + 70 q^{40} + 6 q^{41} + 14 q^{44} + 10 q^{45} + 2 q^{46} - 68 q^{47} - 58 q^{48} + 40 q^{50} + 16 q^{51} + 4 q^{52} + 26 q^{54} - 16 q^{55} + 50 q^{56} + 18 q^{57} - 94 q^{58} + 22 q^{61} - 48 q^{62} + 16 q^{63} + 60 q^{64} - 22 q^{65} + 24 q^{67} + 20 q^{68} + 8 q^{69} - 14 q^{71} - 84 q^{72} + 34 q^{73} + 26 q^{74} - 102 q^{75} + 40 q^{78} + 4 q^{79} - 30 q^{80} - 92 q^{81} - 76 q^{82} + 108 q^{84} + 8 q^{85} + 8 q^{86} + 16 q^{88} - 72 q^{89} + 132 q^{90} + 12 q^{91} - 174 q^{92} + 50 q^{95} + 10 q^{96} - 16 q^{97} - 28 q^{98} - 86 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
259.1 − 2.72425i −1.14824 + 1.14824i −5.42152 2.76466 − 2.76466i 3.12810 + 3.12810i 1.65149 + 1.65149i 9.32106i 0.363073i −7.53160 − 7.53160i
259.2 − 2.38680i −1.32556 + 1.32556i −3.69681 −0.620851 + 0.620851i 3.16384 + 3.16384i −1.89151 − 1.89151i 4.04996i − 0.514207i 1.48185 + 1.48185i
259.3 − 2.33650i −0.692341 + 0.692341i −3.45924 0.878052 − 0.878052i 1.61766 + 1.61766i −2.34961 − 2.34961i 3.40953i 2.04133i −2.05157 − 2.05157i
259.4 − 2.33024i 2.15102 − 2.15102i −3.43001 −2.49284 + 2.49284i −5.01238 − 5.01238i 0.968194 + 0.968194i 3.33226i − 6.25374i 5.80892 + 5.80892i
259.5 − 2.24230i 0.751257 − 0.751257i −3.02793 −1.86200 + 1.86200i −1.68455 − 1.68455i 2.49628 + 2.49628i 2.30493i 1.87123i 4.17518 + 4.17518i
259.6 − 2.17355i −2.09748 + 2.09748i −2.72432 0.426175 − 0.426175i 4.55898 + 4.55898i 2.68508 + 2.68508i 1.57435i − 5.79886i −0.926312 − 0.926312i
259.7 − 2.06339i 0.856325 − 0.856325i −2.25758 −2.54838 + 2.54838i −1.76693 − 1.76693i −3.18040 − 3.18040i 0.531478i 1.53342i 5.25831 + 5.25831i
259.8 − 1.84807i 1.37079 − 1.37079i −1.41535 2.26460 − 2.26460i −2.53331 − 2.53331i −1.84433 − 1.84433i − 1.08047i − 0.758116i −4.18514 − 4.18514i
259.9 − 1.48494i 1.38481 − 1.38481i −0.205050 −0.478281 + 0.478281i −2.05636 − 2.05636i −1.44950 − 1.44950i − 2.66539i − 0.835396i 0.710219 + 0.710219i
259.10 − 1.19080i −1.82510 + 1.82510i 0.581989 0.840782 − 0.840782i 2.17334 + 2.17334i 1.07906 + 1.07906i − 3.07464i − 3.66201i −1.00121 − 1.00121i
259.11 − 1.12754i 1.76939 − 1.76939i 0.728657 −0.189570 + 0.189570i −1.99505 − 1.99505i 3.45806 + 3.45806i − 3.07667i − 3.26148i 0.213748 + 0.213748i
259.12 − 0.810360i −0.0502309 + 0.0502309i 1.34332 0.925022 − 0.925022i 0.0407051 + 0.0407051i −0.994972 − 0.994972i − 2.70929i 2.99495i −0.749601 − 0.749601i
259.13 − 0.660947i −1.41031 + 1.41031i 1.56315 −0.660156 + 0.660156i 0.932141 + 0.932141i −2.22740 − 2.22740i − 2.35505i − 0.977951i 0.436328 + 0.436328i
259.14 − 0.436020i 0.705146 − 0.705146i 1.80989 2.58895 − 2.58895i −0.307458 − 0.307458i 0.364609 + 0.364609i − 1.66119i 2.00554i −1.12884 − 1.12884i
259.15 − 0.236577i 2.31349 − 2.31349i 1.94403 −2.11402 + 2.11402i −0.547317 − 0.547317i −3.33868 − 3.33868i − 0.933066i − 7.70443i 0.500128 + 0.500128i
259.16 − 0.166310i −1.09373 + 1.09373i 1.97234 −0.776668 + 0.776668i 0.181899 + 0.181899i −0.631548 − 0.631548i − 0.660639i 0.607490i 0.129167 + 0.129167i
259.17 − 0.0920143i −0.256358 + 0.256358i 1.99153 −2.26856 + 2.26856i 0.0235886 + 0.0235886i 3.14718 + 3.14718i − 0.367278i 2.86856i 0.208740 + 0.208740i
259.18 0.323065i 1.32352 − 1.32352i 1.89563 −1.60497 + 1.60497i 0.427583 + 0.427583i 1.08526 + 1.08526i 1.25854i − 0.503415i −0.518509 − 0.518509i
259.19 0.374564i −0.409333 + 0.409333i 1.85970 2.01640 − 2.01640i −0.153322 − 0.153322i 1.28867 + 1.28867i 1.44571i 2.66489i 0.755271 + 0.755271i
259.20 0.488618i −2.30085 + 2.30085i 1.76125 −0.0210259 + 0.0210259i −1.12424 − 1.12424i −2.78195 − 2.78195i 1.83782i − 7.58778i −0.0102736 − 0.0102736i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 259.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 731.2.f.d ✓ 68
17.c even 4 1 inner 731.2.f.d ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
731.2.f.d ✓ 68 1.a even 1 1 trivial
731.2.f.d ✓ 68 17.c even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(731, [\chi])\):

\( T_{2}^{68} + 106 T_{2}^{66} + 5347 T_{2}^{64} + 170816 T_{2}^{62} + 3879979 T_{2}^{60} + 66694494 T_{2}^{58} + \cdots + 82944 \) Copy content Toggle raw display
\( T_{3}^{68} + 4 T_{3}^{65} + 509 T_{3}^{64} + 10 T_{3}^{63} + 8 T_{3}^{62} + 2070 T_{3}^{61} + \cdots + 1150566400 \) Copy content Toggle raw display