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Properties

Label 475.2.h.a
Level $475$
Weight $2$
Character orbit 475.h
Analytic conductor $3.793$
Analytic rank $0$
Dimension $84$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [475,2,Mod(96,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([6, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.96");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.h (of order \(5\), degree \(4\), 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: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(84\)
Relative dimension: \(21\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 84 q - q^{2} - 2 q^{3} - 19 q^{4} + 2 q^{5} + 2 q^{6} - 3 q^{8} - 17 q^{9} - 41 q^{10} - 35 q^{12} - 8 q^{13} + 10 q^{14} - 2 q^{15} - 3 q^{16} - 17 q^{17} + 20 q^{18} + 21 q^{19} + 24 q^{20} + 16 q^{21}+ \cdots - 82 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
96.1 −0.798184 + 2.45656i 1.57386 + 1.14348i −3.77954 − 2.74600i 0.794270 + 2.09025i −4.06524 + 2.95357i 2.92921 5.58312 − 4.05637i 0.242447 + 0.746174i −5.76878 + 0.282769i
96.2 −0.792539 + 2.43918i −1.31286 − 0.953851i −3.70346 − 2.69072i −1.80442 + 1.32064i 3.36711 − 2.44635i −4.04488 5.34852 − 3.88593i −0.113274 − 0.348620i −1.79120 − 5.44796i
96.3 −0.673159 + 2.07177i 1.35439 + 0.984022i −2.22105 − 1.61369i 1.06862 − 1.96419i −2.95038 + 2.14358i 3.71610 1.31361 − 0.954392i −0.0609786 − 0.187673i 3.35000 + 3.53615i
96.4 −0.624361 + 1.92158i 0.195933 + 0.142353i −1.68463 − 1.22395i −2.22797 − 0.190171i −0.395877 + 0.287621i 0.918631 0.134550 − 0.0977560i −0.908926 − 2.79739i 1.75648 − 4.16249i
96.5 −0.486653 + 1.49777i −2.72464 − 1.97956i −0.388435 − 0.282215i 2.16402 − 0.563050i 4.29088 − 3.11751i −3.30777 −1.93643 + 1.40690i 2.57792 + 7.93402i −0.209810 + 3.51520i
96.6 −0.480220 + 1.47796i 0.183137 + 0.133057i −0.335734 − 0.243925i 1.08074 + 1.95755i −0.284600 + 0.206774i −4.09551 −1.99272 + 1.44780i −0.911216 − 2.80443i −3.41218 + 0.657242i
96.7 −0.348213 + 1.07169i −1.39728 − 1.01518i 0.590766 + 0.429217i 2.15112 − 0.610485i 1.57451 − 1.14395i 5.08951 −2.48897 + 1.80834i −0.00525905 − 0.0161857i −0.0947964 + 2.51791i
96.8 −0.207777 + 0.639473i 1.91077 + 1.38826i 1.25228 + 0.909834i −0.623025 + 2.14752i −1.28477 + 0.933439i −1.46029 −1.92995 + 1.40219i 0.796743 + 2.45212i −1.24383 − 0.844614i
96.9 −0.151456 + 0.466135i 1.61572 + 1.17389i 1.42369 + 1.03437i 1.59666 − 1.56547i −0.791900 + 0.575349i −0.642137 −1.49082 + 1.08314i 0.305477 + 0.940161i 0.487895 + 0.981358i
96.10 −0.104113 + 0.320426i −0.352507 − 0.256111i 1.52620 + 1.10885i −1.42960 + 1.71938i 0.118765 − 0.0862879i 4.00453 −1.05934 + 0.769656i −0.868383 − 2.67261i −0.402093 − 0.637088i
96.11 −0.0370819 + 0.114126i −1.94518 − 1.41325i 1.60638 + 1.16711i −0.396172 − 2.20069i 0.233420 − 0.169590i −2.00436 −0.386929 + 0.281120i 0.859377 + 2.64489i 0.265848 + 0.0363922i
96.12 0.124567 − 0.383378i −0.404433 − 0.293838i 1.48657 + 1.08006i 2.23318 + 0.113670i −0.163030 + 0.118448i −1.36982 1.25149 − 0.909260i −0.849825 − 2.61549i 0.321759 − 0.841991i
96.13 0.192824 − 0.593450i −0.0659841 − 0.0479403i 1.30303 + 0.946708i −1.97016 − 1.05757i −0.0411735 + 0.0299143i 0.0995699 1.82272 − 1.32428i −0.924995 − 2.84684i −1.00751 + 0.965270i
96.14 0.279723 − 0.860900i 2.37071 + 1.72242i 0.955131 + 0.693943i −1.82014 + 1.29888i 2.14597 − 1.55914i −1.13919 2.32924 − 1.69229i 1.72647 + 5.31353i 0.609067 + 1.93029i
96.15 0.409150 − 1.25924i −2.56971 − 1.86700i 0.199764 + 0.145137i −2.08226 + 0.814974i −3.40239 + 2.47198i −2.53256 2.40683 − 1.74867i 2.19065 + 6.74213i 0.174286 + 2.95551i
96.16 0.551726 − 1.69804i −1.61783 − 1.17542i −0.960894 − 0.698131i 1.36964 + 1.76751i −2.88850 + 2.09862i 2.50181 1.17327 − 0.852432i 0.308700 + 0.950081i 3.75696 − 1.35052i
96.17 0.573163 − 1.76401i −1.32920 − 0.965723i −1.16519 − 0.846563i 0.188362 − 2.22812i −2.46540 + 1.79122i −1.03768 0.839925 − 0.610241i −0.0928897 − 0.285885i −3.82247 − 1.60935i
96.18 0.595943 − 1.83413i 1.94502 + 1.41314i −1.39083 − 1.01050i 2.17273 + 0.528449i 3.75101 − 2.72526i −1.88033 0.438159 − 0.318341i 0.859095 + 2.64402i 2.26406 − 3.67013i
96.19 0.643199 − 1.97956i 1.30761 + 0.950033i −1.88693 − 1.37094i −1.59256 − 1.56963i 2.72170 − 1.97743i 3.67802 −0.559699 + 0.406645i −0.119774 − 0.368626i −4.13152 + 2.14299i
96.20 0.801327 − 2.46623i −0.562359 − 0.408578i −3.82214 − 2.77695i −1.56090 + 1.60112i −1.45828 + 1.05950i −0.170579 −5.71557 + 4.15261i −0.777739 − 2.39363i 2.69794 + 5.13257i
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 96.21
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.h.a ✓ 84
25.d even 5 1 inner 475.2.h.a ✓ 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.h.a ✓ 84 1.a even 1 1 trivial
475.2.h.a ✓ 84 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{84} + T_{2}^{83} + 31 T_{2}^{82} + 33 T_{2}^{81} + 554 T_{2}^{80} + 571 T_{2}^{79} + \cdots + 958441 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\). Copy content Toggle raw display