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Properties

Label 184.4.b.b
Level $184$
Weight $4$
Character orbit 184.b
Analytic conductor $10.856$
Analytic rank $0$
Dimension $36$
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] = [184,4,Mod(93,184)]
 
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, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("184.93");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 184 = 2^{3} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 184.b (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: \(10.8563514411\)
Analytic rank: \(0\)
Dimension: \(36\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 33 q^{6} + 45 q^{8} - 378 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 33 q^{6} + 45 q^{8} - 378 q^{9} + 124 q^{10} - 53 q^{12} - 168 q^{14} + 48 q^{16} + 264 q^{17} + 175 q^{18} - 186 q^{20} - 28 q^{22} + 828 q^{23} - 2 q^{24} - 1328 q^{25} - 351 q^{26} - 132 q^{28} + 290 q^{30} - 186 q^{31} + 140 q^{32} + 676 q^{33} - 568 q^{34} + 167 q^{36} + 188 q^{38} - 810 q^{39} - 294 q^{40} - 518 q^{41} - 66 q^{42} + 204 q^{44} - 674 q^{47} - 927 q^{48} + 3300 q^{49} - 486 q^{50} + 1721 q^{52} + 161 q^{54} + 880 q^{55} - 1830 q^{56} - 1936 q^{57} - 2233 q^{58} + 4622 q^{60} + 3601 q^{62} - 20 q^{63} - 3417 q^{64} + 428 q^{65} - 4524 q^{66} + 2746 q^{68} + 5422 q^{70} + 546 q^{71} - 4749 q^{72} - 3210 q^{73} - 3844 q^{74} + 4868 q^{76} + 4609 q^{78} - 2454 q^{80} + 5320 q^{81} - 3991 q^{82} + 7020 q^{84} + 6438 q^{86} - 374 q^{87} - 6808 q^{88} - 1712 q^{89} - 10260 q^{90} + 6579 q^{94} + 412 q^{95} - 5391 q^{96} + 3636 q^{97} - 3818 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
93.1 −2.82843 − 0.00210008i 5.68587i 7.99999 + 0.0118798i − 2.36073i 0.0119408 − 16.0821i 29.3009 −22.6274 − 0.0504018i −5.32907 −0.00495771 + 6.67714i
93.2 −2.82843 + 0.00210008i − 5.68587i 7.99999 − 0.0118798i 2.36073i 0.0119408 + 16.0821i 29.3009 −22.6274 + 0.0504018i −5.32907 −0.00495771 − 6.67714i
93.3 −2.67455 − 0.920215i − 1.12231i 6.30641 + 4.92232i 2.80881i −1.03277 + 3.00168i −21.7446 −12.3372 − 18.9682i 25.7404 2.58471 − 7.51230i
93.4 −2.67455 + 0.920215i 1.12231i 6.30641 − 4.92232i − 2.80881i −1.03277 − 3.00168i −21.7446 −12.3372 + 18.9682i 25.7404 2.58471 + 7.51230i
93.5 −2.44914 − 1.41482i 9.89305i 3.99658 + 6.93018i 1.21363i 13.9969 − 24.2295i −23.3303 0.0167437 − 22.6274i −70.8725 1.71707 − 2.97236i
93.6 −2.44914 + 1.41482i − 9.89305i 3.99658 − 6.93018i − 1.21363i 13.9969 + 24.2295i −23.3303 0.0167437 + 22.6274i −70.8725 1.71707 + 2.97236i
93.7 −2.25141 − 1.71206i − 1.03333i 2.13767 + 7.70911i − 16.9109i −1.76912 + 2.32644i 16.1844 8.38572 − 21.0162i 25.9322 −28.9525 + 38.0733i
93.8 −2.25141 + 1.71206i 1.03333i 2.13767 − 7.70911i 16.9109i −1.76912 − 2.32644i 16.1844 8.38572 + 21.0162i 25.9322 −28.9525 − 38.0733i
93.9 −2.15504 − 1.83188i − 9.29414i 1.28840 + 7.89557i 19.3527i −17.0258 + 20.0293i 21.7760 11.6872 − 19.3755i −59.3811 35.4519 − 41.7058i
93.10 −2.15504 + 1.83188i 9.29414i 1.28840 − 7.89557i − 19.3527i −17.0258 − 20.0293i 21.7760 11.6872 + 19.3755i −59.3811 35.4519 + 41.7058i
93.11 −1.50792 − 2.39295i − 3.48289i −3.45238 + 7.21672i − 3.77949i −8.33436 + 5.25191i 10.4826 22.4751 − 2.62087i 14.8695 −9.04410 + 5.69915i
93.12 −1.50792 + 2.39295i 3.48289i −3.45238 − 7.21672i 3.77949i −8.33436 − 5.25191i 10.4826 22.4751 + 2.62087i 14.8695 −9.04410 − 5.69915i
93.13 −1.41614 − 2.44838i 4.72053i −3.98909 + 6.93449i 14.9130i 11.5576 − 6.68494i −24.5143 22.6274 − 0.0534403i 4.71657 36.5126 − 21.1189i
93.14 −1.41614 + 2.44838i − 4.72053i −3.98909 − 6.93449i − 14.9130i 11.5576 + 6.68494i −24.5143 22.6274 + 0.0534403i 4.71657 36.5126 + 21.1189i
93.15 −0.481515 − 2.78714i 8.89267i −7.53629 + 2.68410i 17.6826i 24.7851 − 4.28195i 32.1817 11.1098 + 19.7122i −52.0796 49.2840 − 8.51445i
93.16 −0.481515 + 2.78714i − 8.89267i −7.53629 − 2.68410i − 17.6826i 24.7851 + 4.28195i 32.1817 11.1098 − 19.7122i −52.0796 49.2840 + 8.51445i
93.17 −0.145054 − 2.82471i − 4.06576i −7.95792 + 0.819467i − 21.7202i −11.4846 + 0.589753i −27.4687 3.46908 + 22.3599i 10.4696 −61.3533 + 3.15060i
93.18 −0.145054 + 2.82471i 4.06576i −7.95792 − 0.819467i 21.7202i −11.4846 − 0.589753i −27.4687 3.46908 − 22.3599i 10.4696 −61.3533 − 3.15060i
93.19 0.0101231 − 2.82841i − 9.43952i −7.99980 − 0.0572647i − 4.54371i −26.6988 − 0.0955575i 5.16629 −0.242951 + 22.6261i −62.1046 −12.8515 − 0.0459966i
93.20 0.0101231 + 2.82841i 9.43952i −7.99980 + 0.0572647i 4.54371i −26.6988 + 0.0955575i 5.16629 −0.242951 − 22.6261i −62.1046 −12.8515 + 0.0459966i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 93.36
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 184.4.b.b ✓ 36
4.b odd 2 1 736.4.b.b 36
8.b even 2 1 inner 184.4.b.b ✓ 36
8.d odd 2 1 736.4.b.b 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
184.4.b.b ✓ 36 1.a even 1 1 trivial
184.4.b.b ✓ 36 8.b even 2 1 inner
736.4.b.b 36 4.b odd 2 1
736.4.b.b 36 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{36} + 675 T_{3}^{34} + 205706 T_{3}^{32} + 37488546 T_{3}^{30} + 4563455623 T_{3}^{28} + \cdots + 39\!\cdots\!00 \) acting on \(S_{4}^{\mathrm{new}}(184, [\chi])\). Copy content Toggle raw display