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Properties

Label 415.2.j.a
Level $415$
Weight $2$
Character orbit 415.j
Analytic conductor $3.314$
Analytic rank $0$
Dimension $1600$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [415,2,Mod(4,415)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(415, base_ring=CyclotomicField(82))
 
chi = DirichletCharacter(H, H._module([41, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("415.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 415 = 5 \cdot 83 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 415.j (of order \(82\), degree \(40\), 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: \(3.31379168388\)
Analytic rank: \(0\)
Dimension: \(1600\)
Relative dimension: \(40\) over \(\Q(\zeta_{82})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{82}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1600 q - 44 q^{4} - 37 q^{5} - 82 q^{6} - 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1600 q - 44 q^{4} - 37 q^{5} - 82 q^{6} - 50 q^{9} - 31 q^{10} - 78 q^{11} - 90 q^{14} - 49 q^{15} - 116 q^{16} - 78 q^{19} - 47 q^{20} - 70 q^{21} - 66 q^{24} - 53 q^{25} - 98 q^{26} - 90 q^{29} - 53 q^{30} - 74 q^{31} - 106 q^{34} - 61 q^{35} - 68 q^{36} - 86 q^{39} - 35 q^{40} - 74 q^{41} - 78 q^{44} - 35 q^{45} - 102 q^{46} - 22 q^{49} - 41 q^{50} - 50 q^{51} - 114 q^{54} - 59 q^{55} - 86 q^{56} - 102 q^{59} + 9 q^{60} - 70 q^{61} - 92 q^{64} - 33 q^{65} - 118 q^{66} - 102 q^{69} + 111 q^{70} - 86 q^{71} - 10 q^{74} + 262 q^{75} - 34 q^{76} - 22 q^{79} + 367 q^{80} - 130 q^{81} - 172 q^{84} + 78 q^{85} - 26 q^{86} - 78 q^{89} + 267 q^{90} - 138 q^{91} - 130 q^{94} + 82 q^{95} - 118 q^{96} - 18 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −0.211870 + 2.75964i −0.597117 − 0.620444i −5.59414 − 0.864068i 1.85457 + 1.24923i 1.83871 − 1.51637i 2.11408 − 3.00558i 2.30845 − 9.86481i 0.0865068 − 2.25684i −3.84035 + 4.85326i
4.2 −0.199236 + 2.59508i −2.00963 − 2.08814i −4.71819 − 0.728769i −0.0931389 − 2.23413i 5.81927 − 4.79912i −0.601039 + 0.854496i 1.64517 − 7.03037i −0.206793 + 5.39496i 5.81630 + 0.203416i
4.3 −0.186256 + 2.42601i 0.797129 + 0.828269i −3.87426 − 0.598417i 2.14810 − 0.621021i −2.15786 + 1.77957i −2.37191 + 3.37215i 1.06456 − 4.54924i 0.0642932 − 1.67732i 1.10651 + 5.32698i
4.4 −0.183399 + 2.38880i −0.328764 − 0.341607i −3.69618 − 0.570909i −2.07464 + 0.834190i 0.876326 − 0.722701i −1.30911 + 1.86115i 0.949867 − 4.05911i 0.106298 − 2.77318i −1.61223 − 5.10889i
4.5 −0.166560 + 2.16947i 0.287568 + 0.298802i −2.70231 − 0.417398i −0.0587919 − 2.23529i −0.696141 + 0.574103i 1.99119 − 2.83087i 0.364078 − 1.55583i 0.108321 − 2.82595i 4.85921 + 0.244764i
4.6 −0.162793 + 2.12040i 2.27629 + 2.36522i −2.49304 − 0.385074i 2.07168 − 0.841524i −5.38578 + 4.44162i 1.54302 − 2.19371i 0.253235 − 1.08216i −0.297833 + 7.77006i 1.44711 + 4.52978i
4.7 −0.161832 + 2.10789i 1.47255 + 1.53008i −2.44045 − 0.376951i −1.10235 + 1.94546i −3.46354 + 2.85636i 1.25209 − 1.78009i 0.226107 − 0.966232i −0.0578195 + 1.50843i −3.92242 − 2.63848i
4.8 −0.143973 + 1.87527i −2.21332 − 2.29979i −1.51936 − 0.234680i −0.692460 + 2.12615i 4.63139 − 3.81948i 1.71738 − 2.44159i −0.198254 + 0.847205i −0.275313 + 7.18255i −3.88741 − 1.60466i
4.9 −0.137275 + 1.78803i −0.444862 − 0.462241i −1.20166 − 0.185607i 1.23255 + 1.86569i 0.887570 − 0.731974i 0.0317759 − 0.0451758i −0.320388 + 1.36912i 0.0991440 − 2.58653i −3.50512 + 1.94773i
4.10 −0.127680 + 1.66306i −0.995328 − 1.03421i −0.772898 − 0.119381i −1.10780 − 1.94237i 1.84704 − 1.52324i −1.01916 + 1.44894i −0.462875 + 1.97802i 0.0359942 − 0.939040i 3.37171 − 1.59433i
4.11 −0.124964 + 1.62768i 1.92362 + 1.99876i −0.657161 − 0.101505i −1.96980 − 1.05824i −3.49372 + 2.88125i −2.20302 + 3.13203i −0.496589 + 2.12209i −0.179844 + 4.69190i 1.96864 − 3.07396i
4.12 −0.123418 + 1.60753i −1.59422 − 1.65650i −0.592373 − 0.0914976i 2.15888 − 0.582430i 2.85963 − 2.35832i −0.364655 + 0.518429i −0.514526 + 2.19874i −0.0875416 + 2.28384i 0.669831 + 3.54236i
4.13 −0.0929248 + 1.21036i 1.13600 + 1.18038i 0.520225 + 0.0803537i 1.44196 + 1.70902i −1.53424 + 1.26528i −1.46089 + 2.07695i −0.698791 + 2.98617i 0.0121119 − 0.315984i −2.20252 + 1.58648i
4.14 −0.0873216 + 1.13738i 0.0205021 + 0.0213030i 0.690560 + 0.106664i −2.09140 − 0.791242i −0.0260198 + 0.0214584i 2.70042 − 3.83918i −0.701453 + 2.99755i 0.114875 − 2.99693i 1.08256 − 2.30961i
4.15 −0.0714958 + 0.931244i −1.54575 − 1.60613i 1.11446 + 0.172138i −2.23606 + 0.00603673i 1.60621 − 1.32463i −0.191435 + 0.272163i −0.665605 + 2.84436i −0.0754174 + 1.96754i 0.154247 − 2.08275i
4.16 −0.0642865 + 0.837341i 1.17171 + 1.21749i 1.27955 + 0.197639i 0.811973 − 2.08343i −1.09478 + 0.902856i −0.159314 + 0.226496i −0.630455 + 2.69415i 0.00554677 − 0.144708i 1.69235 + 0.813835i
4.17 −0.0364948 + 0.475350i −0.133720 − 0.138944i 1.75194 + 0.270603i 2.20145 − 0.391952i 0.0709272 − 0.0584933i 0.725353 − 1.03123i −0.409825 + 1.75132i 0.113484 − 2.96064i 0.105973 + 1.06076i
4.18 −0.0280598 + 0.365484i 1.48347 + 1.54142i 1.84377 + 0.284788i −0.918358 + 2.03878i −0.604991 + 0.498933i 0.972204 − 1.38218i −0.322865 + 1.37971i −0.0603909 + 1.57552i −0.719372 − 0.392853i
4.19 −0.00544261 + 0.0708908i −2.18996 − 2.27551i 1.97156 + 0.304527i 2.05687 + 0.877094i 0.173232 − 0.142863i −2.37036 + 3.36994i −0.0647191 + 0.276567i −0.267118 + 6.96875i −0.0733726 + 0.141039i
4.20 −0.00195611 + 0.0254786i −0.717388 − 0.745413i 1.97592 + 0.305199i 0.332047 + 2.21128i 0.0203954 − 0.0168199i 2.17856 − 3.09725i −0.0232861 + 0.0995093i 0.0739133 − 1.92830i −0.0569897 + 0.00413458i
See next 80 embeddings (of 1600 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
83.c even 41 1 inner
415.j even 82 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 415.2.j.a ✓ 1600
5.b even 2 1 inner 415.2.j.a ✓ 1600
83.c even 41 1 inner 415.2.j.a ✓ 1600
415.j even 82 1 inner 415.2.j.a ✓ 1600
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
415.2.j.a ✓ 1600 1.a even 1 1 trivial
415.2.j.a ✓ 1600 5.b even 2 1 inner
415.2.j.a ✓ 1600 83.c even 41 1 inner
415.2.j.a ✓ 1600 415.j even 82 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(415, [\chi])\).