↑

Properties

Label 185.2.o.a
Level $185$
Weight $2$
Character orbit 185.o
Analytic conductor $1.477$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [185,2,Mod(16,185)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(185, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("185.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 185 = 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 185.o (of order \(9\), degree \(6\), 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: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 3 q^{2} - 3 q^{4} - 12 q^{6} + 6 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 3 q^{2} - 3 q^{4} - 12 q^{6} + 6 q^{7} + 12 q^{8} - 3 q^{10} - 6 q^{11} + 6 q^{12} + 24 q^{13} - 9 q^{14} - 33 q^{16} - 6 q^{17} - 3 q^{18} + 9 q^{19} + 3 q^{20} - 12 q^{21} + 15 q^{22} - 6 q^{23} - 15 q^{24} - 12 q^{27} - 39 q^{28} + 21 q^{29} - 12 q^{30} - 66 q^{31} + 36 q^{32} + 18 q^{33} - 21 q^{34} + 3 q^{35} + 54 q^{36} + 24 q^{37} + 48 q^{38} + 12 q^{39} + 6 q^{40} - 9 q^{41} - 72 q^{42} - 48 q^{43} + 6 q^{44} + 21 q^{45} + 39 q^{46} - 3 q^{47} - 54 q^{48} + 18 q^{49} - 3 q^{50} + 42 q^{51} - 36 q^{52} - 15 q^{53} - 135 q^{54} + 3 q^{55} + 30 q^{56} - 33 q^{57} + 39 q^{58} + 45 q^{59} + 6 q^{60} - 18 q^{61} - 42 q^{62} - 3 q^{63} + 6 q^{64} - 24 q^{65} + 63 q^{66} + 33 q^{67} - 66 q^{68} - 21 q^{69} - 27 q^{70} - 30 q^{71} + 33 q^{72} + 72 q^{73} + 63 q^{74} + 6 q^{75} - 6 q^{76} + 6 q^{77} + 54 q^{78} - 84 q^{79} - 18 q^{80} + 12 q^{81} + 18 q^{82} - 15 q^{83} + 132 q^{84} - 9 q^{85} - 30 q^{86} + 63 q^{87} - 9 q^{88} - 60 q^{89} - 42 q^{90} - 90 q^{91} + 108 q^{92} + 15 q^{93} + 51 q^{94} + 18 q^{95} + 93 q^{96} + 117 q^{98} - 150 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −2.08136 − 0.757553i −0.0387037 + 0.0140870i 2.22608 + 1.86791i −0.173648 + 0.984808i 0.0912280 0.444852 − 2.52288i −1.00331 − 1.73778i −2.29683 + 1.92727i 1.10747 − 1.91819i
16.2 −1.04995 − 0.382151i 2.82629 − 1.02868i −0.575729 − 0.483094i −0.173648 + 0.984808i −3.36058 0.418871 − 2.37554i 1.53721 + 2.66252i 4.63157 − 3.88635i 0.558668 − 0.967641i
16.3 −0.503620 − 0.183303i −0.599885 + 0.218340i −1.31206 − 1.10095i −0.173648 + 0.984808i 0.342137 −0.460639 + 2.61241i 0.994913 + 1.72324i −1.98594 + 1.66640i 0.267971 − 0.464139i
16.4 0.451449 + 0.164314i −1.97862 + 0.720158i −1.35528 − 1.13722i −0.173648 + 0.984808i −1.01158 0.642947 − 3.64633i −0.905403 − 1.56820i 1.09816 − 0.921470i −0.240211 + 0.416058i
16.5 1.54344 + 0.561767i 1.56472 − 0.569510i 0.534543 + 0.448535i −0.173648 + 0.984808i 2.73498 −0.133213 + 0.755490i −1.06943 − 1.85231i −0.174136 + 0.146117i −0.821249 + 1.42244i
16.6 2.25338 + 0.820164i −2.71349 + 0.987630i 2.87297 + 2.41071i −0.173648 + 0.984808i −6.92455 −0.331566 + 1.88040i 2.09872 + 3.63509i 4.08949 − 3.43149i −1.19900 + 2.07673i
46.1 −0.419337 − 2.37818i −0.428005 + 2.42734i −3.60050 + 1.31047i −0.766044 − 0.642788i 5.95212 3.16379 + 2.65474i 2.21149 + 3.83042i −2.88970 − 1.05176i −1.20743 + 2.09133i
46.2 −0.235743 − 1.33697i −0.0334057 + 0.189453i 0.147483 − 0.0536795i −0.766044 − 0.642788i 0.261167 −2.49840 − 2.09640i −1.46413 − 2.53594i 2.78430 + 1.01340i −0.678795 + 1.17571i
46.3 −0.123274 − 0.699122i 0.365137 − 2.07079i 1.40581 − 0.511673i −0.766044 − 0.642788i −1.49275 2.22647 + 1.86823i −1.24093 − 2.14935i −1.33578 − 0.486185i −0.354954 + 0.614798i
46.4 0.198087 + 1.12341i 0.00283048 − 0.0160525i 0.656575 − 0.238974i −0.766044 − 0.642788i 0.0185942 0.374673 + 0.314388i 1.53926 + 2.66608i 2.81883 + 1.02597i 0.570370 − 0.987910i
46.5 0.262189 + 1.48695i 0.581059 − 3.29535i −0.262887 + 0.0956832i −0.766044 − 0.642788i 5.05236 −1.39315 − 1.16899i 1.29868 + 2.24939i −7.70262 − 2.80353i 0.754944 − 1.30760i
46.6 0.410474 + 2.32791i −0.313967 + 1.78060i −3.37130 + 1.22705i −0.766044 − 0.642788i −4.27395 1.59840 + 1.34121i −1.87648 − 3.25015i −0.252874 − 0.0920385i 1.18191 − 2.04713i
71.1 −2.08409 − 1.74876i 2.15734 − 1.81023i 0.937973 + 5.31951i 0.939693 + 0.342020i −7.66173 2.64249 + 0.961786i 4.62713 − 8.01442i 0.856266 − 4.85613i −1.36029 − 2.35609i
71.2 −1.65139 − 1.38568i −0.532186 + 0.446557i 0.459683 + 2.60699i 0.939693 + 0.342020i 1.49763 −3.30199 − 1.20182i 0.697607 − 1.20829i −0.437136 + 2.47912i −1.07787 − 1.86692i
71.3 −0.634662 − 0.532544i −2.38372 + 2.00018i −0.228104 − 1.29364i 0.939693 + 0.342020i 2.57804 −0.820213 − 0.298533i −1.37265 + 2.37749i 1.16046 − 6.58132i −0.414246 − 0.717495i
71.4 −0.231115 − 0.193929i 1.70862 − 1.43370i −0.331490 − 1.87998i 0.939693 + 0.342020i −0.672925 −0.211021 − 0.0768055i −0.589669 + 1.02134i 0.342937 − 1.94489i −0.150850 − 0.261280i
71.5 0.708323 + 0.594353i −0.838601 + 0.703670i −0.198831 − 1.12763i 0.939693 + 0.342020i −1.01223 3.72635 + 1.35628i 1.45402 − 2.51844i −0.312844 + 1.77423i 0.462325 + 0.800770i
71.6 1.68720 + 1.41573i 0.654588 − 0.549265i 0.495055 + 2.80760i 0.939693 + 0.342020i 1.88203 −3.08865 − 1.12418i −0.937056 + 1.62303i −0.394150 + 2.23534i 1.10124 + 1.90740i
81.1 −2.08136 + 0.757553i −0.0387037 − 0.0140870i 2.22608 − 1.86791i −0.173648 − 0.984808i 0.0912280 0.444852 + 2.52288i −1.00331 + 1.73778i −2.29683 − 1.92727i 1.10747 + 1.91819i
81.2 −1.04995 + 0.382151i 2.82629 + 1.02868i −0.575729 + 0.483094i −0.173648 − 0.984808i −3.36058 0.418871 + 2.37554i 1.53721 − 2.66252i 4.63157 + 3.88635i 0.558668 + 0.967641i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 185.2.o.a ✓ 36
5.b even 2 1 925.2.p.c 36
5.c odd 4 2 925.2.bc.c 72
37.f even 9 1 inner 185.2.o.a ✓ 36
37.f even 9 1 6845.2.a.q 18
37.h even 18 1 6845.2.a.s 18
185.x even 18 1 925.2.p.c 36
185.bd odd 36 2 925.2.bc.c 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.o.a ✓ 36 1.a even 1 1 trivial
185.2.o.a ✓ 36 37.f even 9 1 inner
925.2.p.c 36 5.b even 2 1
925.2.p.c 36 185.x even 18 1
925.2.bc.c 72 5.c odd 4 2
925.2.bc.c 72 185.bd odd 36 2
6845.2.a.q 18 37.f even 9 1
6845.2.a.s 18 37.h even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} + 3 T_{2}^{35} + 6 T_{2}^{34} + T_{2}^{33} - 48 T_{2}^{31} + 121 T_{2}^{30} + 486 T_{2}^{29} + \cdots + 5041 \) acting on \(S_{2}^{\mathrm{new}}(185, [\chi])\). Copy content Toggle raw display