Properties

Label 712.2.t.a
Level $712$
Weight $2$
Character orbit 712.t
Analytic conductor $5.685$
Analytic rank $0$
Dimension $880$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [712,2,Mod(85,712)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(712, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 11, 19]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("712.85");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 712 = 2^{3} \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 712.t (of order \(22\), degree \(10\), 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: \(5.68534862392\)
Analytic rank: \(0\)
Dimension: \(880\)
Relative dimension: \(88\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 880 q - 9 q^{2} - 9 q^{4} - 11 q^{6} - 22 q^{7} - 9 q^{8} - 102 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 880 q - 9 q^{2} - 9 q^{4} - 11 q^{6} - 22 q^{7} - 9 q^{8} - 102 q^{9} - 5 q^{10} + 44 q^{14} - 22 q^{15} - 9 q^{16} - 14 q^{17} - 72 q^{18} + 15 q^{20} + 43 q^{22} - 22 q^{23} - 11 q^{24} + 66 q^{25} - 11 q^{26} - 11 q^{28} - 88 q^{30} - 22 q^{31} + 26 q^{32} - 22 q^{33} - 4 q^{34} - 28 q^{36} - 11 q^{38} - 22 q^{39} - 88 q^{40} - 63 q^{42} - 47 q^{44} + 77 q^{46} - 38 q^{47} - 132 q^{48} + 58 q^{49} - 43 q^{50} - 11 q^{54} - 188 q^{55} - 132 q^{56} + 2 q^{57} - 11 q^{58} - 66 q^{60} - 88 q^{62} - 22 q^{63} - 9 q^{64} - 22 q^{65} + 33 q^{66} - 5 q^{68} - 132 q^{70} - 22 q^{71} + 166 q^{72} - 38 q^{73} - 33 q^{74} - 66 q^{76} + 169 q^{78} - 38 q^{79} - 84 q^{80} - 86 q^{81} - 121 q^{82} - 46 q^{84} - 11 q^{86} + 26 q^{87} - 86 q^{88} - 38 q^{89} - 50 q^{90} - 110 q^{92} - 119 q^{94} + 198 q^{95} - 11 q^{96} - 14 q^{97} - 76 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
85.1 −1.41408 + 0.0192097i −0.669586 1.46619i 1.99926 0.0543283i 1.34067 0.192759i 0.975016 + 2.06045i 2.70826 0.389389i −2.82608 + 0.115230i 0.263216 0.303768i −1.89212 + 0.298332i
85.2 −1.41352 0.0442608i 1.27406 + 2.78980i 1.99608 + 0.125127i 3.44160 0.494827i −1.67743 3.99983i 5.17583 0.744172i −2.81597 0.265218i −4.19518 + 4.84150i −4.88668 + 0.547121i
85.3 −1.40519 + 0.159466i 0.485601 + 1.06332i 1.94914 0.448161i 1.57279 0.226133i −0.851926 1.41673i 0.0267271 0.00384277i −2.66746 + 0.940575i 1.06975 1.23455i −2.17402 + 0.568567i
85.4 −1.40063 0.195543i −1.41003 3.08753i 1.92353 + 0.547767i −1.40119 + 0.201460i 1.37118 + 4.60021i −3.81848 + 0.549015i −2.58703 1.14335i −5.58010 + 6.43978i 2.00194 0.00817865i
85.5 −1.39418 0.237183i 1.35569 + 2.96854i 1.88749 + 0.661353i −1.54432 + 0.222039i −1.18599 4.46023i −2.11521 + 0.304121i −2.47464 1.36973i −5.00976 + 5.78157i 2.20572 + 0.0567224i
85.6 −1.38825 + 0.269764i 0.0607019 + 0.132919i 1.85445 0.748998i −1.65789 + 0.238369i −0.120126 0.168149i −3.25832 + 0.468475i −2.37239 + 1.54006i 1.95060 2.25111i 2.23726 0.778154i
85.7 −1.38616 0.280305i 0.658002 + 1.44082i 1.84286 + 0.777093i −3.18682 + 0.458196i −0.508223 2.18165i 3.03966 0.437037i −2.33667 1.59374i 0.321578 0.371121i 4.54587 + 0.258152i
85.8 −1.37553 0.328487i −0.287478 0.629489i 1.78419 + 0.903690i 0.403160 0.0579657i 0.188657 + 0.960317i −1.18553 + 0.170454i −2.15737 1.82914i 1.65097 1.90532i −0.573602 0.0526990i
85.9 −1.37498 + 0.330808i −0.700665 1.53424i 1.78113 0.909708i 4.13654 0.594744i 1.47094 + 1.87777i −2.98673 + 0.429427i −2.14808 + 1.84004i 0.101613 0.117268i −5.49090 + 2.18616i
85.10 −1.31714 + 0.514929i 1.03166 + 2.25903i 1.46970 1.35646i −2.35114 + 0.338043i −2.52208 2.44422i −1.32338 + 0.190273i −1.23731 + 2.54344i −2.07430 + 2.39387i 2.92270 1.65592i
85.11 −1.31126 + 0.529710i −0.859677 1.88243i 1.43881 1.38918i −3.39203 + 0.487700i 2.12440 + 2.01298i 0.0843246 0.0121240i −1.15080 + 2.58373i −0.839913 + 0.969311i 4.18950 2.43630i
85.12 −1.28844 0.583034i 0.819041 + 1.79345i 1.32014 + 1.50241i 3.71925 0.534748i −0.00964071 2.78828i −4.70012 + 0.675775i −0.824967 2.70544i −0.581052 + 0.670570i −5.10380 1.47946i
85.13 −1.28834 + 0.583249i 0.149949 + 0.328342i 1.31964 1.50285i −1.97078 + 0.283355i −0.384690 0.335558i 4.21470 0.605983i −0.823613 + 2.70586i 1.87926 2.16878i 2.37377 1.51451i
85.14 −1.22497 0.706718i −0.860766 1.88481i 1.00110 + 1.73142i −2.43475 + 0.350064i −0.277620 + 2.91716i 2.81093 0.404150i −0.00269505 2.82843i −0.847023 + 0.977517i 3.22989 + 1.29186i
85.15 −1.21317 0.726780i −0.0387383 0.0848251i 0.943582 + 1.76342i 2.49889 0.359286i −0.0146528 + 0.131062i 2.76914 0.398142i 0.136890 2.82511i 1.95889 2.26068i −3.29271 1.38026i
85.16 −1.20889 0.733883i −0.442122 0.968112i 0.922831 + 1.77437i −1.09028 + 0.156758i −0.176004 + 1.49481i −3.31475 + 0.476589i 0.186577 2.82227i 1.22281 1.41120i 1.43306 + 0.610631i
85.17 −1.19267 + 0.759957i 0.872513 + 1.91054i 0.844931 1.81276i 0.846421 0.121697i −2.49255 1.61557i −1.44373 + 0.207577i 0.369894 + 2.80414i −0.924288 + 1.06669i −0.917017 + 0.788388i
85.18 −1.18673 + 0.769195i −0.872513 1.91054i 0.816679 1.82566i −0.846421 + 0.121697i 2.50502 + 1.59617i −1.44373 + 0.207577i 0.435106 + 2.79476i −0.924288 + 1.06669i 0.910868 0.795484i
85.19 −1.15226 0.819936i −1.23417 2.70245i 0.655411 + 1.88956i 4.10263 0.589869i −0.793753 + 4.12587i 1.32836 0.190989i 0.794112 2.71466i −3.81550 + 4.40333i −5.21096 2.68421i
85.20 −1.06574 + 0.929625i −0.149949 0.328342i 0.271594 1.98147i 1.97078 0.283355i 0.465041 + 0.210530i 4.21470 0.605983i 1.55258 + 2.36421i 1.87926 2.16878i −1.83692 + 2.13407i
See next 80 embeddings (of 880 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 85.88
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
89.f even 22 1 inner
712.t even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 712.2.t.a 880
8.b even 2 1 inner 712.2.t.a 880
89.f even 22 1 inner 712.2.t.a 880
712.t even 22 1 inner 712.2.t.a 880
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
712.2.t.a 880 1.a even 1 1 trivial
712.2.t.a 880 8.b even 2 1 inner
712.2.t.a 880 89.f even 22 1 inner
712.2.t.a 880 712.t even 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(712, [\chi])\).