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Properties

Label 185.3.g.a
Level $185$
Weight $3$
Character orbit 185.g
Analytic conductor $5.041$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [185,3,Mod(6,185)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(185, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("185.6");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 185 = 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 185.g (of order \(4\), degree \(2\), 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: \(5.04088489067\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{2} - 36 q^{8} - 120 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{2} - 36 q^{8} - 120 q^{9} + 44 q^{13} + 104 q^{14} - 56 q^{16} + 4 q^{17} - 160 q^{18} - 16 q^{19} - 64 q^{22} - 80 q^{23} + 124 q^{24} + 64 q^{26} - 100 q^{29} - 60 q^{31} - 260 q^{32} - 48 q^{33} - 128 q^{34} + 20 q^{35} + 56 q^{37} + 216 q^{38} - 4 q^{39} - 28 q^{42} - 48 q^{43} + 248 q^{44} - 40 q^{45} - 24 q^{46} + 376 q^{47} + 216 q^{49} + 20 q^{50} + 444 q^{51} + 76 q^{52} - 64 q^{53} + 88 q^{54} + 40 q^{55} - 308 q^{56} - 276 q^{57} - 64 q^{59} + 140 q^{60} + 120 q^{61} + 472 q^{63} - 336 q^{66} - 428 q^{68} - 276 q^{69} - 32 q^{71} + 624 q^{72} + 544 q^{74} + 220 q^{76} - 200 q^{79} - 320 q^{80} + 400 q^{81} + 80 q^{82} + 16 q^{83} + 1192 q^{84} - 656 q^{86} + 568 q^{87} + 760 q^{88} - 704 q^{89} + 472 q^{91} - 744 q^{92} - 396 q^{93} - 172 q^{94} - 1344 q^{96} + 544 q^{97} - 504 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1 −2.50948 + 2.50948i − 2.82579i − 8.59502i 1.58114 + 1.58114i 7.09128 + 7.09128i −3.69611 11.5311 + 11.5311i 1.01490 −7.93568
6.2 −2.26990 + 2.26990i 0.985369i − 6.30493i 1.58114 + 1.58114i −2.23669 − 2.23669i 6.61646 5.23197 + 5.23197i 8.02905 −7.17807
6.3 −2.18819 + 2.18819i − 2.57121i − 5.57637i −1.58114 − 1.58114i 5.62630 + 5.62630i −11.4681 3.44940 + 3.44940i 2.38888 6.91967
6.4 −2.11848 + 2.11848i 2.16491i − 4.97595i −1.58114 − 1.58114i −4.58633 − 4.58633i −3.12931 2.06754 + 2.06754i 4.31316 6.69924
6.5 −1.96847 + 1.96847i 1.20273i − 3.74975i −1.58114 − 1.58114i −2.36755 − 2.36755i 12.6099 −0.492604 − 0.492604i 7.55343 6.22485
6.6 −1.51033 + 1.51033i 3.68848i − 0.562165i 1.58114 + 1.58114i −5.57081 − 5.57081i −7.63317 −5.19225 − 5.19225i −4.60490 −4.77607
6.7 −1.19745 + 1.19745i − 2.29969i 1.13224i 1.58114 + 1.58114i 2.75376 + 2.75376i 4.25447 −6.14559 − 6.14559i 3.71144 −3.78666
6.8 −1.18485 + 1.18485i 5.55093i 1.19224i −1.58114 − 1.58114i −6.57704 − 6.57704i −1.18863 −6.15205 − 6.15205i −21.8128 3.74684
6.9 −0.996874 + 0.996874i − 3.34646i 2.01249i −1.58114 − 1.58114i 3.33599 + 3.33599i 1.34281 −5.99369 − 5.99369i −2.19876 3.15239
6.10 −0.552361 + 0.552361i − 5.66185i 3.38980i 1.58114 + 1.58114i 3.12738 + 3.12738i −12.7437 −4.08183 − 4.08183i −23.0566 −1.74672
6.11 −0.299984 + 0.299984i 2.75633i 3.82002i 1.58114 + 1.58114i −0.826854 − 0.826854i 5.28767 −2.34588 − 2.34588i 1.40264 −0.948631
6.12 0.0405140 − 0.0405140i − 1.16830i 3.99672i −1.58114 − 1.58114i −0.0473325 − 0.0473325i 1.62484 0.323979 + 0.323979i 7.63507 −0.128117
6.13 0.110683 − 0.110683i 2.37237i 3.97550i −1.58114 − 1.58114i 0.262580 + 0.262580i −12.1493 0.882749 + 0.882749i 3.37187 −0.350009
6.14 0.560858 − 0.560858i − 4.05378i 3.37088i 1.58114 + 1.58114i −2.27359 − 2.27359i 10.8478 4.13401 + 4.13401i −7.43310 1.77359
6.15 0.847479 − 0.847479i 5.14280i 2.56356i 1.58114 + 1.58114i 4.35842 + 4.35842i −2.18183 5.56248 + 5.56248i −17.4484 2.67996
6.16 0.972726 − 0.972726i 0.115433i 2.10761i 1.58114 + 1.58114i 0.112285 + 0.112285i −6.49332 5.94103 + 5.94103i 8.98668 3.07603
6.17 1.37547 − 1.37547i − 4.58692i 0.216145i −1.58114 − 1.58114i −6.30919 − 6.30919i −8.83445 5.79920 + 5.79920i −12.0398 −4.34963
6.18 1.45780 − 1.45780i 0.478408i − 0.250335i −1.58114 − 1.58114i 0.697421 + 0.697421i 6.29110 5.46624 + 5.46624i 8.77113 −4.60995
6.19 1.66895 − 1.66895i 4.86789i − 1.57079i −1.58114 − 1.58114i 8.12427 + 8.12427i 3.16344 4.05423 + 4.05423i −14.6964 −5.27768
6.20 1.90054 − 1.90054i 1.10267i − 3.22411i 1.58114 + 1.58114i 2.09566 + 2.09566i 4.33787 1.47461 + 1.47461i 7.78413 6.01004
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.24
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 185.3.g.a ✓ 48
37.d odd 4 1 inner 185.3.g.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.3.g.a ✓ 48 1.a even 1 1 trivial
185.3.g.a ✓ 48 37.d odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(185, [\chi])\).