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Properties

Label 300.3.u.a
Level $300$
Weight $3$
Character orbit 300.u
Analytic conductor $8.174$
Analytic rank $0$
Dimension $928$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [300,3,Mod(23,300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(300, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 10, 11]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.23");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 300.u (of order \(20\), degree \(8\), 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: \(8.17440793081\)
Analytic rank: \(0\)
Dimension: \(928\)
Relative dimension: \(116\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 928 q - 20 q^{4} - 6 q^{6} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 928 q - 20 q^{4} - 6 q^{6} - 20 q^{9} - 8 q^{10} + 10 q^{12} - 32 q^{13} - 12 q^{16} + 14 q^{18} - 12 q^{21} + 56 q^{22} - 32 q^{25} + 64 q^{28} - 78 q^{30} + 20 q^{33} - 20 q^{34} - 70 q^{36} - 124 q^{40} + 454 q^{42} + 84 q^{45} - 12 q^{46} - 76 q^{48} - 324 q^{52} - 660 q^{54} + 52 q^{57} - 200 q^{58} - 826 q^{60} - 24 q^{61} - 20 q^{64} + 138 q^{66} - 20 q^{69} + 352 q^{70} + 590 q^{72} - 144 q^{73} + 96 q^{76} + 308 q^{78} - 12 q^{81} + 20 q^{82} - 10 q^{84} + 864 q^{85} - 760 q^{88} - 538 q^{90} - 388 q^{93} - 1420 q^{94} - 6 q^{96} + 288 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −1.99999 + 0.00504968i −2.92237 + 0.678037i 3.99995 − 0.0201986i −2.32110 − 4.42860i 5.84130 − 1.37083i −3.36739 − 3.36739i −7.99977 + 0.0605956i 8.08053 − 3.96295i 4.66454 + 8.84546i
23.2 −1.99964 − 0.0379676i 1.14158 − 2.77431i 3.99712 + 0.151843i 4.27257 − 2.59714i −2.38809 + 5.50427i −4.07277 − 4.07277i −7.98703 − 0.455393i −6.39357 − 6.33422i −8.64221 + 5.03113i
23.3 −1.99617 + 0.123641i −1.53625 + 2.57681i 3.96943 − 0.493620i 3.39000 + 3.67531i 2.74802 − 5.33370i −3.39236 − 3.39236i −7.86263 + 1.47614i −4.27987 − 7.91724i −7.22144 − 6.91742i
23.4 −1.99422 + 0.151996i −1.59435 − 2.54127i 3.95379 − 0.606225i −0.179543 − 4.99678i 3.56574 + 4.82550i 8.22741 + 8.22741i −7.79258 + 1.80990i −3.91609 + 8.10335i 1.11754 + 9.93736i
23.5 −1.99345 − 0.161712i −2.33835 − 1.87939i 3.94770 + 0.644731i −2.39489 + 4.38913i 4.35747 + 4.12462i −4.78889 − 4.78889i −7.76528 − 1.92363i 1.93578 + 8.78936i 5.48388 − 8.36224i
23.6 −1.98821 + 0.216842i 2.97041 + 0.420345i 3.90596 − 0.862255i 3.62538 + 3.44334i −5.99694 − 0.191625i −8.40935 − 8.40935i −7.57889 + 2.56132i 8.64662 + 2.49719i −7.95468 − 6.05995i
23.7 −1.96224 + 0.386823i 0.991266 − 2.83150i 3.70074 − 1.51807i −4.31389 + 2.52791i −0.849810 + 5.93951i 3.94176 + 3.94176i −6.67449 + 4.41035i −7.03478 − 5.61354i 7.48702 − 6.62907i
23.8 −1.95646 + 0.415062i 1.78111 + 2.41406i 3.65545 − 1.62410i 4.46722 − 2.24588i −4.48664 − 3.98372i 4.75879 + 4.75879i −6.47762 + 4.69473i −2.65532 + 8.59937i −7.80774 + 6.24814i
23.9 −1.94586 − 0.462213i 2.33835 + 1.87939i 3.57272 + 1.79880i −2.39489 + 4.38913i −3.68142 − 4.73784i 4.78889 + 4.78889i −6.12057 − 5.15156i 1.93578 + 8.78936i 6.68883 − 7.43368i
23.10 −1.91350 − 0.581813i −1.14158 + 2.77431i 3.32299 + 2.22660i 4.27257 − 2.59714i 3.79855 − 4.64446i 4.07277 + 4.07277i −5.06308 − 6.19397i −6.39357 − 6.33422i −9.68663 + 2.48380i
23.11 −1.90055 − 0.622835i 2.92237 − 0.678037i 3.22415 + 2.36745i −2.32110 − 4.42860i −5.97641 − 0.531515i 3.36739 + 3.36739i −4.65312 − 6.50757i 8.08053 − 3.96295i 1.65306 + 9.86242i
23.12 −1.88157 + 0.678006i 2.98267 − 0.322016i 3.08061 − 2.55143i −4.99748 + 0.158790i −5.39377 + 2.62816i −3.65865 − 3.65865i −4.06650 + 6.88938i 8.79261 − 1.92093i 9.29545 − 3.68710i
23.13 −1.87761 + 0.688911i 0.443817 + 2.96699i 3.05080 − 2.58701i −4.57873 − 2.00878i −2.87731 − 5.26508i −0.467335 − 0.467335i −3.94598 + 6.95911i −8.60605 + 2.63360i 9.98093 + 0.617358i
23.14 −1.86027 − 0.734442i 1.53625 − 2.57681i 2.92119 + 2.73252i 3.39000 + 3.67531i −4.75035 + 3.66526i 3.39236 + 3.39236i −3.42732 − 7.22866i −4.27987 − 7.91724i −3.60699 − 9.32682i
23.15 −1.84964 − 0.760803i 1.59435 + 2.54127i 2.84236 + 2.81443i −0.179543 − 4.99678i −1.01557 − 5.91343i −8.22741 − 8.22741i −3.11612 − 7.36816i −3.91609 + 8.10335i −3.46947 + 9.37885i
23.16 −1.82389 − 0.820620i −2.97041 − 0.420345i 2.65317 + 2.99344i 3.62538 + 3.44334i 5.07276 + 3.20424i 8.40935 + 8.40935i −2.38261 − 7.63696i 8.64662 + 2.49719i −3.78664 − 9.25534i
23.17 −1.81868 + 0.832105i −2.64320 + 1.41898i 2.61520 − 3.02667i −1.23074 + 4.84616i 3.62640 − 4.78009i 7.10365 + 7.10365i −2.23771 + 7.68067i 4.97301 − 7.50128i −1.79419 − 9.83773i
23.18 −1.74666 − 0.974254i −0.991266 + 2.83150i 2.10166 + 3.40339i −4.31389 + 2.52791i 4.49001 − 3.97993i −3.94176 − 3.94176i −0.355122 − 7.99211i −7.03478 − 5.61354i 9.99774 − 0.212576i
23.19 −1.74162 + 0.983243i −2.95108 − 0.539548i 2.06647 − 3.42487i 4.89865 − 1.00161i 5.67017 − 1.96194i −1.88925 − 1.88925i −0.231522 + 7.99665i 8.41778 + 3.18450i −7.54675 + 6.56099i
23.20 −1.73244 − 0.999326i −1.78111 − 2.41406i 2.00269 + 3.46254i 4.46722 − 2.24588i 0.673230 + 5.96211i −4.75879 − 4.75879i −0.00933631 − 7.99999i −2.65532 + 8.59937i −9.98355 − 0.573357i
See next 80 embeddings (of 928 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.116
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
25.f odd 20 1 inner
75.l even 20 1 inner
100.l even 20 1 inner
300.u odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.3.u.a ✓ 928
3.b odd 2 1 inner 300.3.u.a ✓ 928
4.b odd 2 1 inner 300.3.u.a ✓ 928
12.b even 2 1 inner 300.3.u.a ✓ 928
25.f odd 20 1 inner 300.3.u.a ✓ 928
75.l even 20 1 inner 300.3.u.a ✓ 928
100.l even 20 1 inner 300.3.u.a ✓ 928
300.u odd 20 1 inner 300.3.u.a ✓ 928
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.3.u.a ✓ 928 1.a even 1 1 trivial
300.3.u.a ✓ 928 3.b odd 2 1 inner
300.3.u.a ✓ 928 4.b odd 2 1 inner
300.3.u.a ✓ 928 12.b even 2 1 inner
300.3.u.a ✓ 928 25.f odd 20 1 inner
300.3.u.a ✓ 928 75.l even 20 1 inner
300.3.u.a ✓ 928 100.l even 20 1 inner
300.3.u.a ✓ 928 300.u odd 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(300, [\chi])\).