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Properties

Label 184.5.e.e
Level $184$
Weight $5$
Character orbit 184.e
Analytic conductor $19.020$
Analytic rank $0$
Dimension $84$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [184,5,Mod(45,184)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(184, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("184.45");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 184 = 2^{3} \cdot 23 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 184.e (of order \(2\), degree \(1\), 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: \(19.0200732074\)
Analytic rank: \(0\)
Dimension: \(84\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 84 q + 4 q^{2} - 32 q^{4} - 120 q^{6} - 212 q^{8} - 2272 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q + 4 q^{2} - 32 q^{4} - 120 q^{6} - 212 q^{8} - 2272 q^{9} + 1036 q^{12} - 240 q^{16} - 2392 q^{18} + 2596 q^{23} + 1348 q^{24} + 11996 q^{25} - 1560 q^{26} - 260 q^{31} + 1744 q^{32} - 8600 q^{36} + 18740 q^{39} - 4 q^{41} - 7652 q^{46} - 14732 q^{47} - 10216 q^{48} - 52140 q^{49} - 7264 q^{50} + 18936 q^{52} - 68 q^{54} - 208 q^{55} - 27268 q^{58} - 500 q^{62} - 6536 q^{64} - 16188 q^{70} + 47860 q^{71} + 15584 q^{72} - 4 q^{73} + 37384 q^{78} - 3372 q^{81} + 86300 q^{82} - 34420 q^{87} - 15932 q^{92} - 93956 q^{94} + 109872 q^{95} + 26072 q^{96} + 35848 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1 −3.97302 − 0.463762i 8.59777i 15.5699 + 3.68507i 16.2308 3.98732 − 34.1591i 91.2242i −60.1504 − 21.8616i 7.07839 −64.4853 − 7.52722i
45.2 −3.97302 − 0.463762i 8.59777i 15.5699 + 3.68507i −16.2308 3.98732 − 34.1591i − 91.2242i −60.1504 − 21.8616i 7.07839 64.4853 + 7.52722i
45.3 −3.97302 + 0.463762i − 8.59777i 15.5699 − 3.68507i 16.2308 3.98732 + 34.1591i − 91.2242i −60.1504 + 21.8616i 7.07839 −64.4853 + 7.52722i
45.4 −3.97302 + 0.463762i − 8.59777i 15.5699 − 3.68507i −16.2308 3.98732 + 34.1591i 91.2242i −60.1504 + 21.8616i 7.07839 64.4853 − 7.52722i
45.5 −3.82439 − 1.17220i 12.2734i 13.2519 + 8.96586i 23.2173 14.3868 − 46.9381i − 2.51446i −40.1708 − 49.8228i −69.6354 −88.7919 − 27.2152i
45.6 −3.82439 − 1.17220i 12.2734i 13.2519 + 8.96586i −23.2173 14.3868 − 46.9381i 2.51446i −40.1708 − 49.8228i −69.6354 88.7919 + 27.2152i
45.7 −3.82439 + 1.17220i − 12.2734i 13.2519 − 8.96586i 23.2173 14.3868 + 46.9381i 2.51446i −40.1708 + 49.8228i −69.6354 −88.7919 + 27.2152i
45.8 −3.82439 + 1.17220i − 12.2734i 13.2519 − 8.96586i −23.2173 14.3868 + 46.9381i − 2.51446i −40.1708 + 49.8228i −69.6354 88.7919 − 27.2152i
45.9 −3.78790 − 1.28524i − 14.8841i 12.6963 + 9.73671i −37.2743 −19.1297 + 56.3796i − 29.4469i −35.5783 − 53.1995i −140.538 141.191 + 47.9064i
45.10 −3.78790 − 1.28524i − 14.8841i 12.6963 + 9.73671i 37.2743 −19.1297 + 56.3796i 29.4469i −35.5783 − 53.1995i −140.538 −141.191 − 47.9064i
45.11 −3.78790 + 1.28524i 14.8841i 12.6963 − 9.73671i −37.2743 −19.1297 − 56.3796i 29.4469i −35.5783 + 53.1995i −140.538 141.191 − 47.9064i
45.12 −3.78790 + 1.28524i 14.8841i 12.6963 − 9.73671i 37.2743 −19.1297 − 56.3796i − 29.4469i −35.5783 + 53.1995i −140.538 −141.191 + 47.9064i
45.13 −3.48223 − 1.96827i − 4.47691i 8.25182 + 13.7079i −16.6301 −8.81177 + 15.5896i − 19.2078i −1.75381 − 63.9760i 60.9573 57.9098 + 32.7326i
45.14 −3.48223 − 1.96827i − 4.47691i 8.25182 + 13.7079i 16.6301 −8.81177 + 15.5896i 19.2078i −1.75381 − 63.9760i 60.9573 −57.9098 − 32.7326i
45.15 −3.48223 + 1.96827i 4.47691i 8.25182 − 13.7079i −16.6301 −8.81177 − 15.5896i 19.2078i −1.75381 + 63.9760i 60.9573 57.9098 − 32.7326i
45.16 −3.48223 + 1.96827i 4.47691i 8.25182 − 13.7079i 16.6301 −8.81177 − 15.5896i − 19.2078i −1.75381 + 63.9760i 60.9573 −57.9098 + 32.7326i
45.17 −3.06955 − 2.56473i 3.88102i 2.84432 + 15.7452i −45.6985 9.95376 − 11.9130i 85.4583i 31.6513 − 55.6255i 65.9377 140.274 + 117.204i
45.18 −3.06955 − 2.56473i 3.88102i 2.84432 + 15.7452i 45.6985 9.95376 − 11.9130i − 85.4583i 31.6513 − 55.6255i 65.9377 −140.274 − 117.204i
45.19 −3.06955 + 2.56473i − 3.88102i 2.84432 − 15.7452i −45.6985 9.95376 + 11.9130i − 85.4583i 31.6513 + 55.6255i 65.9377 140.274 − 117.204i
45.20 −3.06955 + 2.56473i − 3.88102i 2.84432 − 15.7452i 45.6985 9.95376 + 11.9130i 85.4583i 31.6513 + 55.6255i 65.9377 −140.274 + 117.204i
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 45.84
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
23.b odd 2 1 inner
184.e odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 184.5.e.e ✓ 84
4.b odd 2 1 736.5.e.e 84
8.b even 2 1 inner 184.5.e.e ✓ 84
8.d odd 2 1 736.5.e.e 84
23.b odd 2 1 inner 184.5.e.e ✓ 84
92.b even 2 1 736.5.e.e 84
184.e odd 2 1 inner 184.5.e.e ✓ 84
184.h even 2 1 736.5.e.e 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
184.5.e.e ✓ 84 1.a even 1 1 trivial
184.5.e.e ✓ 84 8.b even 2 1 inner
184.5.e.e ✓ 84 23.b odd 2 1 inner
184.5.e.e ✓ 84 184.e odd 2 1 inner
736.5.e.e 84 4.b odd 2 1
736.5.e.e 84 8.d odd 2 1
736.5.e.e 84 92.b even 2 1
736.5.e.e 84 184.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(184, [\chi])\):

\( T_{3}^{42} + 2269 T_{3}^{40} + 2367809 T_{3}^{38} + 1507912797 T_{3}^{36} + 655765508021 T_{3}^{34} + \cdots + 31\!\cdots\!76 \) Copy content Toggle raw display
\( T_{5}^{42} - 16124 T_{5}^{40} + 119171112 T_{5}^{38} - 535890864176 T_{5}^{36} + \cdots - 12\!\cdots\!24 \) Copy content Toggle raw display