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Properties

Label 165.4.m.a
Level $165$
Weight $4$
Character orbit 165.m
Analytic conductor $9.735$
Analytic rank $0$
Dimension $24$
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] = [165,4,Mod(16,165)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(165, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("165.16");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 165.m (of order \(5\), degree \(4\), 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: \(9.73531515095\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{2} - 18 q^{3} - 12 q^{4} - 30 q^{5} - 18 q^{6} - 51 q^{7} - 83 q^{8} - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{2} - 18 q^{3} - 12 q^{4} - 30 q^{5} - 18 q^{6} - 51 q^{7} - 83 q^{8} - 54 q^{9} + 70 q^{10} + 42 q^{11} + 294 q^{12} - 131 q^{13} - 145 q^{14} - 90 q^{15} - 88 q^{16} - 301 q^{17} - 9 q^{18} - 56 q^{19} - 185 q^{20} + 462 q^{21} + 407 q^{22} + 420 q^{23} - 84 q^{24} - 150 q^{25} + 581 q^{26} - 162 q^{27} - 241 q^{28} - 957 q^{29} - 15 q^{30} + 234 q^{31} + 820 q^{32} + 21 q^{33} - 156 q^{34} - 130 q^{35} - 108 q^{36} - 598 q^{37} + 329 q^{38} - 393 q^{39} - 415 q^{40} - 197 q^{41} + 525 q^{42} + 1550 q^{43} + 1475 q^{44} + 1080 q^{45} - 907 q^{46} + 549 q^{47} - 219 q^{48} - 1255 q^{49} - 150 q^{50} - 48 q^{51} - 1333 q^{52} + 40 q^{53} + 378 q^{54} - 265 q^{55} - 838 q^{56} - 543 q^{57} + 1779 q^{58} - 728 q^{59} - 555 q^{60} + 50 q^{61} - 34 q^{62} - 459 q^{63} + 2677 q^{64} + 1720 q^{65} - 564 q^{66} - 370 q^{67} - 2736 q^{68} + 15 q^{69} - 725 q^{70} - 1354 q^{71} - 252 q^{72} - 2250 q^{73} + 519 q^{74} - 450 q^{75} + 3188 q^{76} + 3672 q^{77} + 48 q^{78} - 1467 q^{79} - 365 q^{80} - 486 q^{81} - 2395 q^{82} - 150 q^{83} - 903 q^{84} - 80 q^{85} + 495 q^{86} + 4554 q^{87} - 6342 q^{88} - 946 q^{89} - 270 q^{90} + 2941 q^{91} - 521 q^{92} + 702 q^{93} + 7990 q^{94} - 280 q^{95} - 750 q^{96} - 2254 q^{97} - 388 q^{98} - 252 q^{99}+O(q^{100}) \) 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 −1.32271 − 4.07087i −2.42705 − 1.76336i −8.35030 + 6.06685i 1.54508 − 4.75528i −3.96812 + 12.2126i −7.93869 + 5.76779i 8.03924 + 5.84085i 2.78115 + 8.55951i −21.4018
16.2 −0.641263 − 1.97361i −2.42705 − 1.76336i 2.98824 − 2.17108i 1.54508 − 4.75528i −1.92379 + 5.92082i 19.6118 − 14.2488i −19.6319 − 14.2634i 2.78115 + 8.55951i −10.3759
16.3 −0.419280 − 1.29041i −2.42705 − 1.76336i 4.98277 − 3.62020i 1.54508 − 4.75528i −1.25784 + 3.87123i −29.6695 + 21.5561i −15.5422 − 11.2921i 2.78115 + 8.55951i −6.78409
16.4 0.358909 + 1.10461i −2.42705 − 1.76336i 5.38079 − 3.90938i 1.54508 − 4.75528i 1.07673 − 3.31382i −3.24932 + 2.36077i 13.7666 + 10.0020i 2.78115 + 8.55951i 5.80727
16.5 1.14145 + 3.51301i −2.42705 − 1.76336i −4.56618 + 3.31753i 1.54508 − 4.75528i 3.42434 − 10.5390i −0.0327150 + 0.0237688i 7.04019 + 5.11500i 2.78115 + 8.55951i 18.4690
16.6 1.61896 + 4.98266i −2.42705 − 1.76336i −15.7337 + 11.4312i 1.54508 − 4.75528i 4.85689 − 14.9480i −14.3913 + 10.4559i −48.5220 − 35.2533i 2.78115 + 8.55951i 26.1954
31.1 −1.32271 + 4.07087i −2.42705 + 1.76336i −8.35030 − 6.06685i 1.54508 + 4.75528i −3.96812 − 12.2126i −7.93869 − 5.76779i 8.03924 − 5.84085i 2.78115 − 8.55951i −21.4018
31.2 −0.641263 + 1.97361i −2.42705 + 1.76336i 2.98824 + 2.17108i 1.54508 + 4.75528i −1.92379 − 5.92082i 19.6118 + 14.2488i −19.6319 + 14.2634i 2.78115 − 8.55951i −10.3759
31.3 −0.419280 + 1.29041i −2.42705 + 1.76336i 4.98277 + 3.62020i 1.54508 + 4.75528i −1.25784 − 3.87123i −29.6695 − 21.5561i −15.5422 + 11.2921i 2.78115 − 8.55951i −6.78409
31.4 0.358909 − 1.10461i −2.42705 + 1.76336i 5.38079 + 3.90938i 1.54508 + 4.75528i 1.07673 + 3.31382i −3.24932 − 2.36077i 13.7666 − 10.0020i 2.78115 − 8.55951i 5.80727
31.5 1.14145 − 3.51301i −2.42705 + 1.76336i −4.56618 − 3.31753i 1.54508 + 4.75528i 3.42434 + 10.5390i −0.0327150 − 0.0237688i 7.04019 − 5.11500i 2.78115 − 8.55951i 18.4690
31.6 1.61896 − 4.98266i −2.42705 + 1.76336i −15.7337 − 11.4312i 1.54508 + 4.75528i 4.85689 + 14.9480i −14.3913 − 10.4559i −48.5220 + 35.2533i 2.78115 − 8.55951i 26.1954
91.1 −3.92348 + 2.85057i 0.927051 + 2.85317i 4.79577 − 14.7599i −4.04508 − 2.93893i −11.7704 − 8.55172i 8.06283 − 24.8148i 11.2689 + 34.6821i −7.28115 + 5.29007i 24.2484
91.2 −3.85910 + 2.80380i 0.927051 + 2.85317i 4.55921 − 14.0318i −4.04508 − 2.93893i −11.5773 − 8.41139i −9.45935 + 29.1129i 9.95555 + 30.6400i −7.28115 + 5.29007i 23.8505
91.3 −1.63136 + 1.18525i 0.927051 + 2.85317i −1.21562 + 3.74129i −4.04508 − 2.93893i −4.89409 − 3.55576i 5.30904 − 16.3396i −7.43627 − 22.8865i −7.28115 + 5.29007i 10.0824
91.4 −0.126865 + 0.0921727i 0.927051 + 2.85317i −2.46454 + 7.58507i −4.04508 − 2.93893i −0.380595 − 0.276518i 1.05648 − 3.25151i −0.774138 − 2.38255i −7.28115 + 5.29007i 0.784068
91.5 2.09630 − 1.52305i 0.927051 + 2.85317i −0.397342 + 1.22289i −4.04508 − 2.93893i 6.28891 + 4.56916i −3.89358 + 11.9832i 7.43531 + 22.8835i −7.28115 + 5.29007i −12.9559
91.6 3.70843 − 2.69433i 0.927051 + 2.85317i 4.02089 − 12.3750i −4.04508 − 2.93893i 11.1253 + 8.08300i 9.09427 − 27.9893i −7.09931 − 21.8494i −7.28115 + 5.29007i −22.9194
136.1 −3.92348 − 2.85057i 0.927051 − 2.85317i 4.79577 + 14.7599i −4.04508 + 2.93893i −11.7704 + 8.55172i 8.06283 + 24.8148i 11.2689 − 34.6821i −7.28115 − 5.29007i 24.2484
136.2 −3.85910 − 2.80380i 0.927051 − 2.85317i 4.55921 + 14.0318i −4.04508 + 2.93893i −11.5773 + 8.41139i −9.45935 − 29.1129i 9.95555 − 30.6400i −7.28115 − 5.29007i 23.8505
See all 24 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
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.4.m.a ✓ 24
11.c even 5 1 inner 165.4.m.a ✓ 24
11.c even 5 1 1815.4.a.bn 12
11.d odd 10 1 1815.4.a.bh 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.m.a ✓ 24 1.a even 1 1 trivial
165.4.m.a ✓ 24 11.c even 5 1 inner
1815.4.a.bh 12 11.d odd 10 1
1815.4.a.bn 12 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 6 T_{2}^{23} + 48 T_{2}^{22} + 249 T_{2}^{21} + 1678 T_{2}^{20} + 5564 T_{2}^{19} + \cdots + 553943296 \) acting on \(S_{4}^{\mathrm{new}}(165, [\chi])\). Copy content Toggle raw display