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Properties

Label 731.2.u.b
Level $731$
Weight $2$
Character orbit 731.u
Analytic conductor $5.837$
Analytic rank $0$
Dimension $348$
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(52,731)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(731, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("731.52");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 731 = 17 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 731.u (of order \(21\), 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: \(5.83706438776\)
Analytic rank: \(0\)
Dimension: \(348\)
Relative dimension: \(29\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 348 q + 6 q^{2} - 3 q^{3} - 54 q^{4} + q^{5} - 12 q^{6} + 49 q^{7} - 2 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 348 q + 6 q^{2} - 3 q^{3} - 54 q^{4} + q^{5} - 12 q^{6} + 49 q^{7} - 2 q^{8} + 8 q^{9} - 4 q^{10} - 16 q^{11} - 12 q^{12} - 9 q^{13} - 3 q^{14} - 14 q^{15} - 30 q^{16} - 29 q^{17} - 204 q^{18} - 8 q^{19} + 61 q^{20} + 26 q^{21} + 85 q^{22} - 79 q^{23} - 12 q^{24} + 22 q^{25} - 14 q^{26} + 12 q^{27} - 15 q^{28} + 54 q^{29} - 46 q^{30} - 52 q^{31} + 33 q^{32} + 39 q^{33} + 3 q^{34} + 32 q^{35} - 161 q^{36} + 68 q^{37} - 29 q^{38} + 61 q^{39} + 107 q^{40} + 63 q^{41} - 56 q^{42} - 45 q^{43} - 74 q^{44} + 144 q^{45} - 22 q^{46} - 70 q^{47} + 168 q^{48} - 119 q^{49} + 97 q^{50} - 6 q^{51} + 27 q^{52} + 10 q^{53} - 34 q^{54} - 78 q^{55} - 44 q^{56} - 40 q^{57} - 38 q^{58} - 47 q^{59} - 72 q^{60} - 76 q^{61} - 41 q^{62} + 63 q^{63} - 112 q^{64} - 31 q^{65} - 598 q^{66} - 39 q^{67} - 20 q^{68} - 17 q^{69} - 72 q^{70} - 37 q^{71} + 245 q^{72} + 14 q^{73} + 77 q^{74} - 121 q^{75} + 152 q^{76} + 59 q^{77} + 220 q^{78} + 4 q^{79} - 42 q^{80} - 109 q^{81} + 148 q^{82} + 59 q^{83} + 390 q^{84} + 2 q^{85} + 12 q^{86} - 78 q^{87} + 344 q^{88} - 103 q^{89} - 75 q^{90} + 148 q^{91} + 28 q^{92} + 140 q^{93} + 24 q^{94} - 302 q^{95} + 22 q^{96} + 57 q^{97} - 72 q^{98} - 128 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
52.1 −1.74998 − 2.19441i 0.963898 + 0.145284i −1.30795 + 5.73049i 0.755186 + 0.514877i −1.36799 − 2.36943i −0.932904 + 1.61584i 9.80631 − 4.72247i −1.95873 − 0.604187i −0.191712 − 2.55821i
52.2 −1.67210 − 2.09675i 2.29142 + 0.345377i −1.15539 + 5.06209i −2.26489 − 1.54418i −3.10732 − 5.38204i 2.23836 − 3.87695i 7.71335 − 3.71455i 2.26462 + 0.698543i 0.549377 + 7.33093i
52.3 −1.50074 − 1.88187i −1.04625 − 0.157697i −0.844170 + 3.69855i −1.03583 − 0.706220i 1.27338 + 2.20557i −1.99658 + 3.45819i 3.88980 − 1.87323i −1.79695 − 0.554286i 0.225505 + 3.00916i
52.4 −1.43554 − 1.80010i −3.00731 − 0.453278i −0.734572 + 3.21837i 1.89413 + 1.29140i 3.50115 + 6.06416i −1.09737 + 1.90070i 2.69909 − 1.29981i 5.97171 + 1.84203i −0.394444 − 5.26349i
52.5 −1.32549 − 1.66211i 2.59933 + 0.391785i −0.560648 + 2.45636i 2.86311 + 1.95203i −2.79419 − 4.83967i 2.53065 − 4.38321i 0.995100 − 0.479215i 3.73628 + 1.15249i −0.550522 − 7.34620i
52.6 −1.15937 − 1.45380i −0.101371 − 0.0152793i −0.324361 + 1.42112i −3.42704 − 2.33652i 0.0953136 + 0.165088i 0.698039 − 1.20904i −0.908594 + 0.437556i −2.85668 − 0.881168i 0.576369 + 7.69111i
52.7 −1.04859 − 1.31489i 2.19207 + 0.330401i −0.184358 + 0.807723i 1.73827 + 1.18513i −1.86414 − 3.22879i −0.383582 + 0.664384i −1.77514 + 0.854860i 1.82928 + 0.564257i −0.264414 − 3.52837i
52.8 −1.02252 − 1.28220i −0.661263 − 0.0996693i −0.153446 + 0.672293i −0.0486121 − 0.0331432i 0.548358 + 0.949785i 0.748574 − 1.29657i −1.93625 + 0.932450i −2.43938 − 0.752450i 0.00721069 + 0.0962200i
52.9 −0.916857 − 1.14970i −1.68959 − 0.254664i −0.0361471 + 0.158371i 2.78996 + 1.90216i 1.25632 + 2.17601i 0.435372 − 0.754087i −2.43457 + 1.17243i −0.0768702 − 0.0237113i −0.371074 − 4.95163i
52.10 −0.636414 − 0.798038i −2.64174 − 0.398179i 0.213200 − 0.934089i −2.05536 − 1.40132i 1.36348 + 2.36162i −0.398801 + 0.690743i −2.72041 + 1.31008i 3.95355 + 1.21951i 0.189753 + 2.53208i
52.11 −0.614029 − 0.769968i 1.49724 + 0.225673i 0.229223 − 1.00429i 0.606327 + 0.413386i −0.745588 − 1.29140i 0.356455 − 0.617398i −2.68862 + 1.29477i −0.675919 − 0.208493i −0.0540078 − 0.720683i
52.12 −0.362567 − 0.454644i 1.29324 + 0.194925i 0.369795 − 1.62018i −2.61272 − 1.78132i −0.380264 − 0.658637i −1.55140 + 2.68710i −1.91853 + 0.923914i −1.23224 − 0.380097i 0.137417 + 1.83370i
52.13 −0.266678 − 0.334404i 3.15596 + 0.475684i 0.404333 − 1.77150i 2.29587 + 1.56530i −0.682553 − 1.18222i −2.60569 + 4.51320i −1.47094 + 0.708369i 6.86706 + 2.11821i −0.0888167 − 1.18518i
52.14 −0.251421 − 0.315272i 1.17665 + 0.177352i 0.408858 − 1.79132i −1.00912 − 0.688006i −0.239921 − 0.415555i 1.75229 − 3.03505i −1.39418 + 0.671401i −1.51366 − 0.466903i 0.0368048 + 0.491126i
52.15 −0.123719 − 0.155139i −1.15042 − 0.173398i 0.436280 − 1.91147i 1.32212 + 0.901405i 0.115429 + 0.199928i −1.26206 + 2.18595i −0.708080 + 0.340993i −1.57331 − 0.485303i −0.0237284 − 0.316634i
52.16 0.176668 + 0.221534i 3.18926 + 0.480704i 0.427176 − 1.87158i −2.99852 − 2.04436i 0.456948 + 0.791457i 0.233016 − 0.403596i 1.00067 − 0.481898i 7.07361 + 2.18192i −0.0768466 − 1.02545i
52.17 0.384419 + 0.482046i −0.444042 − 0.0669286i 0.360451 − 1.57924i 0.0789639 + 0.0538367i −0.138436 − 0.239777i −1.46965 + 2.54550i 2.01084 − 0.968367i −2.67402 − 0.824827i 0.00440346 + 0.0587601i
52.18 0.429469 + 0.538537i −2.21655 − 0.334092i 0.339463 − 1.48729i 2.91429 + 1.98693i −0.772020 − 1.33718i 1.74515 − 3.02268i 2.18795 − 1.05366i 1.93477 + 0.596798i 0.181562 + 2.42278i
52.19 0.443477 + 0.556103i 1.01721 + 0.153320i 0.332464 − 1.45662i 2.99169 + 2.03970i 0.365847 + 0.633666i 0.454177 − 0.786658i 2.23916 − 1.07832i −1.85551 − 0.572349i 0.192464 + 2.56825i
52.20 0.486709 + 0.610314i −1.78702 − 0.269350i 0.309445 − 1.35577i −3.20929 − 2.18806i −0.705372 − 1.22174i 2.22375 − 3.85164i 2.38468 − 1.14840i 0.254185 + 0.0784058i −0.226589 − 3.02362i
See next 80 embeddings (of 348 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 52.29
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 731.2.u.b ✓ 348
43.g even 21 1 inner 731.2.u.b ✓ 348
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
731.2.u.b ✓ 348 1.a even 1 1 trivial
731.2.u.b ✓ 348 43.g even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{348} - 6 T_{2}^{347} + 103 T_{2}^{346} - 544 T_{2}^{345} + 5389 T_{2}^{344} + \cdots + 56\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(731, [\chi])\). Copy content Toggle raw display