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Properties

Label 185.2.z.a
Level $185$
Weight $2$
Character orbit 185.z
Analytic conductor $1.477$
Analytic rank $0$
Dimension $204$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [185,2,Mod(17,185)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(185, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([9, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("185.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 185 = 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 185.z (of order \(36\), degree \(12\), 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: \(1.47723243739\)
Analytic rank: \(0\)
Dimension: \(204\)
Relative dimension: \(17\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 204 q - 12 q^{2} - 6 q^{3} - 12 q^{5} - 24 q^{6} - 12 q^{7} - 24 q^{8} - 6 q^{10} - 36 q^{11} + 36 q^{12} - 12 q^{13} - 24 q^{14} - 12 q^{15} - 24 q^{16} - 30 q^{17} - 90 q^{18} + 42 q^{20} - 24 q^{21}+ \cdots + 192 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −0.469713 − 2.66388i −1.96091 + 1.37304i −4.99622 + 1.81848i −0.175517 − 2.22917i 4.57868 + 4.57868i 0.315692 + 3.60837i 4.48602 + 7.77001i 0.933852 − 2.56574i −5.85579 + 1.51463i
17.2 −0.433198 − 2.45679i 1.24169 − 0.869438i −3.96876 + 1.44451i 2.20276 − 0.384486i −2.67392 − 2.67392i −0.255866 − 2.92456i 2.77342 + 4.80370i −0.240199 + 0.659941i −1.89883 − 5.24516i
17.3 −0.321176 − 1.82148i −2.27797 + 1.59505i −1.33525 + 0.485992i 1.54034 + 1.62091i 3.63698 + 3.63698i −0.309940 − 3.54263i −0.535504 − 0.927519i 1.61890 − 4.44788i 2.45773 − 3.32630i
17.4 −0.301161 − 1.70797i 2.59563 − 1.81748i −0.947076 + 0.344707i −0.204668 + 2.22668i −3.88590 − 3.88590i 0.191282 + 2.18636i −0.860345 − 1.49016i 2.40801 − 6.61594i 3.86474 − 0.321023i
17.5 −0.284025 − 1.61078i 1.15722 − 0.810292i −0.634568 + 0.230964i −1.17293 − 1.90374i −1.63388 − 1.63388i 0.0979711 + 1.11982i −1.08337 − 1.87645i −0.343483 + 0.943711i −2.73337 + 2.43005i
17.6 −0.264015 − 1.49730i −1.51227 + 1.05890i −0.292825 + 0.106580i −2.21300 − 0.320340i 1.98476 + 1.98476i −0.177838 − 2.03269i −1.28351 − 2.22310i 0.139628 − 0.383625i 0.104619 + 3.39811i
17.7 −0.176891 − 1.00320i −0.0575614 + 0.0403049i 0.904265 − 0.329126i 2.12084 − 0.708554i 0.0506160 + 0.0506160i 0.159768 + 1.82615i −1.50881 − 2.61334i −1.02437 + 2.81444i −1.08598 − 2.00229i
17.8 −0.0435497 − 0.246983i 0.942704 − 0.660089i 1.82028 − 0.662528i 1.09022 + 1.95228i −0.204085 − 0.204085i −0.194050 − 2.21800i −0.493699 − 0.855111i −0.573086 + 1.57454i 0.434701 − 0.354288i
17.9 −0.0371751 − 0.210830i −1.08293 + 0.758277i 1.83632 − 0.668365i −1.35134 + 1.78154i 0.200126 + 0.200126i 0.262998 + 3.00608i −0.423260 − 0.733108i −0.428304 + 1.17675i 0.425839 + 0.218675i
17.10 0.0705497 + 0.400107i −0.765944 + 0.536320i 1.72428 − 0.627585i 0.310406 − 2.21442i −0.268623 − 0.268623i −0.428931 − 4.90270i 0.779029 + 1.34932i −0.727029 + 1.99750i 0.907904 − 0.0320308i
17.11 0.125030 + 0.709080i 1.92078 − 1.34494i 1.39222 − 0.506728i −2.19797 − 0.411008i 1.19383 + 1.19383i 0.00129848 + 0.0148417i 1.25340 + 2.17095i 0.854452 − 2.34759i 0.0166256 − 1.60993i
17.12 0.129279 + 0.733177i −2.44076 + 1.70904i 1.35855 − 0.494472i 2.21527 + 0.304255i −1.56857 − 1.56857i 0.170584 + 1.94978i 1.28265 + 2.22162i 2.01044 − 5.52363i 0.0633153 + 1.66352i
17.13 0.268354 + 1.52191i −0.402831 + 0.282066i −0.364808 + 0.132779i −0.00459685 − 2.23606i −0.537379 − 0.537379i 0.387671 + 4.43110i 1.24541 + 2.15712i −0.943348 + 2.59183i 3.40185 − 0.607052i
17.14 0.324760 + 1.84181i 0.372392 − 0.260752i −1.40740 + 0.512251i −0.590023 + 2.15682i 0.601193 + 0.601193i −0.113451 − 1.29675i 0.469686 + 0.813520i −0.955376 + 2.62487i −4.16406 − 0.386258i
17.15 0.337973 + 1.91674i −2.14225 + 1.50002i −1.68029 + 0.611575i −2.23000 − 0.164667i −3.59918 − 3.59918i −0.0824997 − 0.942976i 0.206187 + 0.357126i 1.31312 − 3.60778i −0.438056 − 4.32998i
17.16 0.382399 + 2.16869i 1.71987 − 1.20426i −2.67761 + 0.974572i 1.60324 − 1.55873i 3.26935 + 3.26935i −0.142598 − 1.62991i −0.935317 − 1.62002i 0.481630 − 1.32327i 3.99348 + 2.88087i
17.17 0.463880 + 2.63079i −1.36341 + 0.954671i −4.82650 + 1.75670i 1.95555 + 1.08435i −3.14400 − 3.14400i −0.0709307 − 0.810742i −4.18905 − 7.25565i −0.0785653 + 0.215857i −1.94555 + 5.64766i
18.1 −1.98709 − 1.66737i 0.135184 − 1.54516i 0.821120 + 4.65681i 1.41577 + 1.73078i −2.84498 + 2.84498i 3.63042 − 1.69289i 3.53900 − 6.12973i 0.585164 + 0.103180i 0.0725704 − 5.79982i
18.2 −1.95495 − 1.64040i −0.0617260 + 0.705532i 0.783627 + 4.44417i −1.89600 − 1.18540i 1.27802 − 1.27802i −1.75899 + 0.820228i 3.20625 − 5.55338i 2.46046 + 0.433845i 1.76206 + 5.42760i
18.3 −1.39978 − 1.17455i −0.0319999 + 0.365761i 0.232506 + 1.31861i 0.831127 − 2.07587i 0.474398 − 0.474398i 1.06911 − 0.498533i −0.603963 + 1.04610i 2.82167 + 0.497536i −3.60161 + 1.92955i
See next 80 embeddings (of 204 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.17
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.z even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 185.2.z.a ✓ 204
5.b even 2 1 925.2.bn.b 204
5.c odd 4 1 185.2.bc.a yes 204
5.c odd 4 1 925.2.bq.b 204
37.i odd 36 1 185.2.bc.a yes 204
185.z even 36 1 inner 185.2.z.a ✓ 204
185.ba odd 36 1 925.2.bq.b 204
185.bc even 36 1 925.2.bn.b 204
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.z.a ✓ 204 1.a even 1 1 trivial
185.2.z.a ✓ 204 185.z even 36 1 inner
185.2.bc.a yes 204 5.c odd 4 1
185.2.bc.a yes 204 37.i odd 36 1
925.2.bn.b 204 5.b even 2 1
925.2.bn.b 204 185.bc even 36 1
925.2.bq.b 204 5.c odd 4 1
925.2.bq.b 204 185.ba odd 36 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(185, [\chi])\).