Properties

Label 185.3.ba.a
Level $185$
Weight $3$
Character orbit 185.ba
Analytic conductor $5.041$
Analytic rank $0$
Dimension $432$
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] = [185,3,Mod(19,185)]
 
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([18, 35]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("185.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 185 = 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 185.ba (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: \(5.04088489067\)
Analytic rank: \(0\)
Dimension: \(432\)
Relative dimension: \(36\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 432 q - 24 q^{4} - 30 q^{5} - 24 q^{6} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 432 q - 24 q^{4} - 30 q^{5} - 24 q^{6} + 36 q^{9} - 6 q^{10} - 36 q^{11} - 72 q^{14} + 30 q^{15} - 24 q^{16} + 48 q^{19} + 48 q^{20} - 24 q^{21} + 396 q^{24} - 138 q^{25} - 72 q^{26} - 24 q^{29} + 516 q^{30} - 300 q^{31} - 108 q^{34} + 240 q^{35} - 240 q^{39} - 276 q^{40} + 168 q^{41} - 216 q^{44} - 528 q^{45} + 336 q^{46} + 240 q^{49} - 288 q^{50} - 24 q^{51} - 504 q^{54} - 108 q^{55} - 24 q^{56} - 840 q^{59} - 672 q^{60} + 624 q^{61} - 1980 q^{64} - 354 q^{65} + 96 q^{66} - 192 q^{69} - 1038 q^{70} + 72 q^{71} + 1404 q^{74} + 120 q^{75} - 648 q^{76} + 264 q^{79} - 192 q^{80} + 276 q^{81} + 2352 q^{84} + 954 q^{85} - 1944 q^{86} + 1476 q^{89} + 2286 q^{90} + 312 q^{91} - 1080 q^{94} + 342 q^{95} + 612 q^{96} - 1956 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −0.341572 3.90419i −0.780895 + 0.655248i −11.1868 + 1.97253i −0.126406 + 4.99840i 2.82494 + 2.82494i 2.14262 5.88680i 7.46486 + 27.8592i −1.38239 + 7.83991i 19.5579 1.21380i
19.2 −0.320071 3.65843i 0.00857152 0.00719236i −9.34241 + 1.64732i 0.828602 4.93086i −0.0290562 0.0290562i −3.95176 + 10.8574i 5.21487 + 19.4622i −1.56281 + 8.86315i −18.3044 1.45315i
19.3 −0.307827 3.51848i 3.26477 2.73947i −8.34573 + 1.47158i −2.40766 4.38214i −10.6438 10.6438i 2.41283 6.62919i 4.09024 + 15.2650i 1.59121 9.02419i −14.6773 + 9.82025i
19.4 −0.296650 3.39073i −4.04044 + 3.39033i −7.46981 + 1.31713i −4.76055 1.52877i 12.6943 + 12.6943i −0.415189 + 1.14072i 3.15820 + 11.7866i 3.26798 18.5336i −3.77141 + 16.5953i
19.5 −0.274090 3.13286i 3.27219 2.74569i −5.80047 + 1.02278i 4.75259 + 1.55335i −9.49874 9.49874i −0.0635093 + 0.174490i 1.53831 + 5.74106i 1.60555 9.10555i 3.56378 15.3150i
19.6 −0.263106 3.00731i −2.52465 + 2.11843i −5.03546 + 0.887888i 4.15063 2.78788i 7.03502 + 7.03502i 2.92086 8.02500i 0.869722 + 3.24585i 0.323259 1.83329i −9.47607 11.7487i
19.7 −0.234327 2.67837i −0.0818783 + 0.0687040i −3.17955 + 0.560640i −4.91109 + 0.938739i 0.203201 + 0.203201i −3.08645 + 8.47996i −0.536790 2.00333i −1.56085 + 8.85202i 3.66510 + 12.9338i
19.8 −0.230294 2.63227i −0.796749 + 0.668552i −2.93657 + 0.517797i 3.20664 + 3.83634i 1.94329 + 1.94329i −2.03603 + 5.59395i −0.696280 2.59855i −1.37499 + 7.79793i 9.35980 9.32421i
19.9 −0.223778 2.55780i 1.45401 1.22006i −2.55301 + 0.450165i −4.75203 + 1.55507i −3.44604 3.44604i 2.93621 8.06716i −0.935402 3.49097i −0.937232 + 5.31531i 5.04094 + 11.8067i
19.10 −0.179984 2.05723i −3.68274 + 3.09019i −0.260569 + 0.0459453i 0.359327 + 4.98707i 7.02006 + 7.02006i −0.0988642 + 0.271627i −1.99652 7.45111i 2.45049 13.8974i 10.1949 1.63681i
19.11 −0.160446 1.83391i 1.95538 1.64076i 0.601757 0.106106i 4.02587 2.96519i −3.32274 3.32274i −0.170800 + 0.469270i −2.19699 8.19928i −0.431406 + 2.44662i −6.08382 6.90732i
19.12 −0.122167 1.39637i 4.30022 3.60832i 2.00431 0.353413i −3.99166 3.01108i −5.56389 5.56389i −4.03440 + 11.0844i −2.18951 8.17135i 3.90914 22.1699i −3.71694 + 5.94169i
19.13 −0.110522 1.26327i 0.575333 0.482762i 2.35559 0.415353i −1.04246 4.89012i −0.673447 0.673447i 1.57851 4.33691i −2.09788 7.82940i −1.46488 + 8.30777i −6.06234 + 1.85738i
19.14 −0.102965 1.17690i 3.28226 2.75414i 2.56475 0.452234i 0.167379 + 4.99720i −3.57930 3.57930i 0.859163 2.36053i −2.01938 7.53643i 1.62509 9.21633i 5.86396 0.711526i
19.15 −0.0974015 1.11330i −3.29672 + 2.76628i 2.70927 0.477718i 3.61437 3.45490i 3.40082 + 3.40082i −2.85674 + 7.84883i −1.95271 7.28762i 1.65325 9.37604i −4.19840 3.68739i
19.16 −0.0841671 0.962034i −2.51465 + 2.11004i 3.02081 0.532650i −3.73675 3.32216i 2.24159 + 2.24159i −1.33590 + 3.67037i −1.76645 6.59250i 0.308359 1.74879i −2.88152 + 3.87450i
19.17 −0.0581376 0.664516i −2.60569 + 2.18644i 3.50103 0.617326i −4.46481 + 2.25067i 1.60441 + 1.60441i 4.17366 11.4670i −1.30435 4.86790i 0.446303 2.53111i 1.75518 + 2.83609i
19.18 −0.00858096 0.0980809i −0.373760 + 0.313622i 3.92968 0.692909i 4.80438 + 1.38489i 0.0339675 + 0.0339675i 2.76903 7.60786i −0.203610 0.759884i −1.52150 + 8.62883i 0.0946053 0.483101i
19.19 0.00858096 + 0.0980809i 0.373760 0.313622i 3.92968 0.692909i −2.02731 + 4.57056i 0.0339675 + 0.0339675i −2.76903 + 7.60786i 0.203610 + 0.759884i −1.52150 + 8.62883i −0.465681 0.159620i
19.20 0.0581376 + 0.664516i 2.60569 2.18644i 3.50103 0.617326i 4.59403 1.97354i 1.60441 + 1.60441i −4.17366 + 11.4670i 1.30435 + 4.86790i 0.446303 2.53111i 1.57853 + 2.93807i
See next 80 embeddings (of 432 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
37.i odd 36 1 inner
185.ba odd 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 185.3.ba.a 432
5.b even 2 1 inner 185.3.ba.a 432
37.i odd 36 1 inner 185.3.ba.a 432
185.ba odd 36 1 inner 185.3.ba.a 432
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.3.ba.a 432 1.a even 1 1 trivial
185.3.ba.a 432 5.b even 2 1 inner
185.3.ba.a 432 37.i odd 36 1 inner
185.3.ba.a 432 185.ba odd 36 1 inner

Hecke kernels

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