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Properties

Label 153.4.e.a
Level $153$
Weight $4$
Character orbit 153.e
Analytic conductor $9.027$
Analytic rank $0$
Dimension $44$
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] = [153,4,Mod(52,153)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(153, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("153.52");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 153 = 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 153.e (of order \(3\), degree \(2\), 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: \(9.02729223088\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 2 q^{3} - 72 q^{4} + 14 q^{5} - 48 q^{6} + 76 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q + 2 q^{3} - 72 q^{4} + 14 q^{5} - 48 q^{6} + 76 q^{7} - 6 q^{8} + 2 q^{9} - 120 q^{10} + 26 q^{11} + 65 q^{12} + 92 q^{13} + 100 q^{14} - 36 q^{15} - 160 q^{16} + 748 q^{17} - 423 q^{18} - 244 q^{19} + 62 q^{20} + 592 q^{21} + 308 q^{22} + 74 q^{23} - 336 q^{24} - 322 q^{25} + 388 q^{26} - 160 q^{27} - 1312 q^{28} + 156 q^{29} + 732 q^{30} + 600 q^{31} - 91 q^{32} + 6 q^{33} + 260 q^{35} - 1552 q^{36} - 1236 q^{37} - 526 q^{38} + 358 q^{39} + 1285 q^{40} + 534 q^{41} + 153 q^{42} + 1168 q^{43} + 1650 q^{44} + 198 q^{45} - 2526 q^{46} + 4 q^{47} + 242 q^{48} - 184 q^{49} - 46 q^{50} + 34 q^{51} + 2011 q^{52} - 4 q^{53} - 3651 q^{54} - 5656 q^{55} + 42 q^{56} + 1504 q^{57} + 2846 q^{58} + 426 q^{59} + 1551 q^{60} + 1600 q^{61} - 1134 q^{62} - 1508 q^{63} - 4770 q^{64} + 1654 q^{65} + 3492 q^{66} + 3260 q^{67} - 1224 q^{68} + 516 q^{69} + 4106 q^{70} - 844 q^{71} - 3960 q^{72} - 3472 q^{73} - 242 q^{74} - 2488 q^{75} + 2734 q^{76} + 876 q^{77} + 6480 q^{78} + 3312 q^{79} - 1190 q^{80} - 1474 q^{81} - 8188 q^{82} - 192 q^{83} - 3590 q^{84} + 238 q^{85} - 1643 q^{86} + 6852 q^{87} + 6235 q^{88} + 764 q^{89} - 4509 q^{90} - 3992 q^{91} + 3002 q^{92} - 458 q^{93} + 5625 q^{94} - 2900 q^{95} + 4902 q^{96} + 3424 q^{97} - 5174 q^{98} - 1632 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
52.1 −2.66446 + 4.61498i −5.19308 − 0.178654i −10.1987 − 17.6647i 3.95101 + 6.84336i 14.6612 − 23.4900i 0.950227 − 1.64584i 66.0648 26.9362 + 1.85553i −42.1093
52.2 −2.35178 + 4.07339i 4.69081 + 2.23523i −7.06170 − 12.2312i −3.68481 − 6.38227i −20.1367 + 13.8508i 17.1481 − 29.7014i 28.8017 17.0075 + 20.9701i 34.6633
52.3 −2.19297 + 3.79833i −1.92857 + 4.82500i −5.61821 − 9.73103i 9.78266 + 16.9441i −14.0977 − 17.9064i 1.58733 − 2.74934i 14.1947 −19.5612 − 18.6107i −85.8122
52.4 −2.15367 + 3.73027i 1.05798 + 5.08731i −5.27662 − 9.13937i −2.00751 − 3.47710i −21.2556 − 7.00984i 4.38852 − 7.60115i 10.9977 −24.7614 + 10.7645i 17.2940
52.5 −1.90975 + 3.30778i −1.73336 − 4.89852i −3.29425 − 5.70582i 0.804239 + 1.39298i 19.5135 + 3.62137i −6.79079 + 11.7620i −5.39118 −20.9909 + 16.9818i −6.14357
52.6 −1.36283 + 2.36049i −4.89773 − 1.73558i 0.285392 + 0.494313i −10.1093 − 17.5098i 10.7716 − 9.19574i 2.60573 − 4.51325i −23.3610 20.9755 + 17.0008i 55.1091
52.7 −1.29547 + 2.24383i 5.19605 + 0.0326894i 0.643498 + 1.11457i 4.95635 + 8.58465i −6.80469 + 11.6167i −5.21298 + 9.02915i −24.0621 26.9979 + 0.339711i −25.6833
52.8 −0.973828 + 1.68672i 2.00226 − 4.79489i 2.10332 + 3.64305i −2.95528 − 5.11869i 6.13777 + 8.04665i 12.6609 − 21.9293i −23.7743 −18.9819 − 19.2012i 11.5117
52.9 −0.836273 + 1.44847i −2.89984 + 4.31172i 2.60130 + 4.50558i −2.51476 − 4.35570i −3.82033 − 7.80609i −10.4385 + 18.0800i −22.0819 −10.1819 − 25.0066i 8.41211
52.10 −0.293084 + 0.507636i −5.15984 + 0.613200i 3.82820 + 6.63064i 9.16437 + 15.8732i 1.20098 − 2.79904i −3.24542 + 5.62123i −9.17728 26.2480 − 6.32804i −10.7437
52.11 −0.00128968 + 0.00223379i −3.06541 − 4.19563i 4.00000 + 6.92820i 1.80027 + 3.11816i 0.0133255 − 0.00143646i 6.63938 − 11.4997i −0.0412697 −8.20656 + 25.7226i −0.00928709
52.12 0.149336 − 0.258658i 4.09427 + 3.19953i 3.95540 + 6.85095i 8.49086 + 14.7066i 1.43901 − 0.581208i 16.4608 − 28.5110i 4.75211 6.52602 + 26.1994i 5.07197
52.13 0.362340 − 0.627592i 4.39980 + 2.76438i 3.73742 + 6.47340i −3.39995 − 5.88888i 3.32913 − 1.75963i −11.7437 + 20.3407i 11.2143 11.7164 + 24.3254i −4.92776
52.14 0.433002 − 0.749982i −0.112814 + 5.19493i 3.62502 + 6.27872i −10.0510 − 17.4089i 3.84725 + 2.33402i 12.3949 − 21.4686i 13.2066 −26.9745 − 1.17213i −17.4084
52.15 1.15898 − 2.00741i 3.50823 − 3.83306i 1.31353 + 2.27510i −5.63510 − 9.76029i −3.62856 − 11.4849i −1.68531 + 2.91905i 24.6331 −2.38464 − 26.8945i −26.1239
52.16 1.17924 − 2.04250i 1.35221 − 5.01712i 1.21879 + 2.11101i 6.33653 + 10.9752i −8.65290 − 8.67828i 3.42994 − 5.94084i 24.6168 −23.3430 − 13.5684i 29.8891
52.17 1.48068 − 2.56462i −3.19883 − 4.09481i −0.384845 − 0.666572i 3.51129 + 6.08173i −15.2381 + 2.14068i −16.4970 + 28.5736i 21.4116 −6.53492 + 26.1972i 20.7964
52.18 1.87858 − 3.25380i 2.09269 + 4.75612i −3.05816 − 5.29689i 3.97043 + 6.87698i 19.4068 + 2.12555i −1.70690 + 2.95644i 7.07732 −18.2413 + 19.9062i 29.8351
52.19 2.01219 − 3.48522i −3.31466 + 4.00163i −4.09783 − 7.09765i 0.262829 + 0.455234i 7.27684 + 19.6043i 6.48383 − 11.2303i −0.787399 −5.02612 − 26.5281i 2.11545
52.20 2.44029 − 4.22671i 4.88630 − 1.76752i −7.91006 − 13.7006i −7.07984 − 12.2626i 4.45321 − 24.9662i −7.81538 + 13.5366i −38.1667 20.7518 − 17.2732i −69.1075
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 52.22
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 153.4.e.a ✓ 44
3.b odd 2 1 459.4.e.a 44
9.c even 3 1 inner 153.4.e.a ✓ 44
9.c even 3 1 1377.4.a.e 22
9.d odd 6 1 459.4.e.a 44
9.d odd 6 1 1377.4.a.f 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
153.4.e.a ✓ 44 1.a even 1 1 trivial
153.4.e.a ✓ 44 9.c even 3 1 inner
459.4.e.a 44 3.b odd 2 1
459.4.e.a 44 9.d odd 6 1
1377.4.a.e 22 9.c even 3 1
1377.4.a.f 22 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{44} + 124 T_{2}^{42} + 2 T_{2}^{41} + 8864 T_{2}^{40} + 247 T_{2}^{39} + 429661 T_{2}^{38} + \cdots + 118247952384 \) acting on \(S_{4}^{\mathrm{new}}(153, [\chi])\). Copy content Toggle raw display