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Properties

Label 152.5.u.b
Level $152$
Weight $5$
Character orbit 152.u
Analytic conductor $15.712$
Analytic rank $0$
Dimension $456$
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] = [152,5,Mod(35,152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(152, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 9, 4]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("152.35");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 152 = 2^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 152.u (of order \(18\), degree \(6\), 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: \(15.7122343887\)
Analytic rank: \(0\)
Dimension: \(456\)
Relative dimension: \(76\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 456 q - 6 q^{2} - 54 q^{3} + 36 q^{4} - 120 q^{6} + 381 q^{8} + 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 456 q - 6 q^{2} - 54 q^{3} + 36 q^{4} - 120 q^{6} + 381 q^{8} + 90 q^{9} - 453 q^{10} - 6 q^{11} - 3 q^{12} - 189 q^{14} - 720 q^{16} - 12 q^{17} - 3900 q^{18} - 12 q^{19} + 1158 q^{20} + 1146 q^{22} - 36 q^{24} - 12 q^{25} - 3099 q^{26} - 1980 q^{27} + 3276 q^{28} - 2586 q^{30} - 4281 q^{32} - 2592 q^{33} + 3696 q^{34} - 3762 q^{35} + 2145 q^{36} - 5238 q^{38} + 7200 q^{40} - 7062 q^{41} - 24333 q^{42} - 12 q^{43} + 21189 q^{44} - 1488 q^{46} + 15399 q^{48} + 82314 q^{49} + 23070 q^{50} - 20706 q^{51} - 6381 q^{52} - 21849 q^{54} - 14418 q^{56} - 12 q^{57} - 8292 q^{58} - 726 q^{59} - 7890 q^{60} - 27504 q^{62} + 19995 q^{64} - 6 q^{65} + 10020 q^{66} - 15414 q^{67} + 30348 q^{68} + 11259 q^{70} + 29670 q^{72} - 68088 q^{73} - 34623 q^{74} + 90024 q^{75} - 26838 q^{76} + 3243 q^{78} - 39633 q^{80} - 57660 q^{81} - 115083 q^{82} - 6 q^{83} - 42597 q^{84} - 19212 q^{86} + 22389 q^{88} - 12 q^{89} + 38094 q^{90} - 14418 q^{91} + 64128 q^{92} + 102186 q^{94} - 98646 q^{96} - 29958 q^{97} + 52635 q^{98} + 145764 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1 −3.99975 + 0.0444998i 10.8842 − 3.96151i 15.9960 − 0.355976i 3.17076 + 0.559090i −43.3577 + 16.3294i 72.3599 − 41.7770i −63.9644 + 2.13564i 40.7218 − 34.1697i −12.7071 − 2.09512i
35.2 −3.99962 − 0.0549938i −0.616405 + 0.224353i 15.9940 + 0.439909i −26.1998 − 4.61974i 2.47772 − 0.863429i 40.3917 − 23.3202i −63.9456 − 2.63904i −61.7200 + 51.7892i 104.535 + 19.9180i
35.3 −3.97945 − 0.404893i −0.765197 + 0.278509i 15.6721 + 3.22251i 29.4454 + 5.19202i 3.15783 − 0.798490i −42.1970 + 24.3625i −61.0617 − 19.1694i −61.5416 + 51.6396i −115.074 − 32.5837i
35.4 −3.95717 + 0.583809i −15.6108 + 5.68185i 15.3183 − 4.62046i −2.80257 − 0.494168i 58.4573 − 31.5977i −59.5580 + 34.3858i −57.9197 + 27.2269i 149.363 − 125.330i 11.3787 + 0.319342i
35.5 −3.91027 − 0.842475i 14.1257 − 5.14132i 14.5805 + 6.58862i −24.4715 − 4.31498i −59.5667 + 8.20345i −19.6127 + 11.3234i −51.4629 − 38.0470i 111.052 − 93.1834i 92.0549 + 37.4894i
35.6 −3.89271 + 0.920206i −2.24871 + 0.818463i 14.3064 − 7.16419i −48.1940 − 8.49791i 8.00042 − 5.25531i −18.6011 + 10.7393i −49.0984 + 41.0530i −57.6628 + 48.3848i 195.425 − 11.2685i
35.7 −3.84485 + 1.10325i −12.2677 + 4.46509i 13.5657 − 8.48363i −7.32085 − 1.29086i 42.2414 − 30.7019i 40.9739 − 23.6563i −42.7985 + 47.5846i 68.5105 − 57.4871i 29.5717 − 3.11353i
35.8 −3.81734 − 1.19495i −6.61922 + 2.40920i 13.1442 + 9.12308i −3.91723 − 0.690713i 28.1467 − 1.28708i −30.7324 + 17.7434i −39.2741 − 50.5326i −24.0398 + 20.1717i 14.1280 + 7.31759i
35.9 −3.75677 − 1.37357i 14.5384 − 5.29155i 12.2266 + 10.3204i 38.1337 + 6.72399i −61.8857 − 0.0904226i −44.2961 + 25.5743i −31.7567 − 55.5654i 121.315 − 101.796i −134.023 − 77.6398i
35.10 −3.73957 + 1.41973i −8.82341 + 3.21146i 11.9687 − 10.6184i 42.1571 + 7.43344i 28.4363 − 24.5363i 49.9405 − 28.8332i −29.6827 + 56.7004i 5.48945 − 4.60619i −168.203 + 32.0539i
35.11 −3.67454 + 1.58043i 11.7666 − 4.28268i 11.0045 − 11.6147i −14.1235 − 2.49035i −36.4682 + 34.3331i −36.6011 + 21.1316i −22.0800 + 60.0706i 58.0610 − 48.7189i 55.8330 − 13.1703i
35.12 −3.66507 + 1.60226i 5.27830 − 1.92115i 10.8655 − 11.7448i 22.0587 + 3.88955i −16.2672 + 15.4984i −11.1721 + 6.45024i −21.0047 + 60.4550i −37.8799 + 31.7850i −87.0789 + 21.0884i
35.13 −3.64933 − 1.63781i −8.93763 + 3.25303i 10.6352 + 11.9538i 21.4291 + 3.77853i 37.9442 + 2.76673i 53.6708 − 30.9868i −19.2333 − 61.0416i 7.24944 − 6.08300i −72.0134 − 48.8859i
35.14 −3.46764 − 1.99387i 5.95008 − 2.16565i 8.04899 + 13.8280i −14.4398 − 2.54613i −24.9507 − 4.35397i 25.0127 − 14.4411i −0.339773 − 63.9991i −31.3362 + 26.2942i 44.9954 + 37.6201i
35.15 −3.37734 − 2.14326i −14.0634 + 5.11866i 6.81288 + 14.4770i −39.2895 − 6.92781i 58.4676 + 12.8541i 36.4164 − 21.0250i 8.01862 − 63.4957i 109.529 − 91.9060i 117.846 + 107.605i
35.16 −3.15786 − 2.45518i 5.02700 − 1.82968i 3.94418 + 15.5062i −31.9084 − 5.62631i −20.3668 − 6.56433i −57.5713 + 33.2388i 25.6155 − 58.6502i −40.1266 + 33.6702i 86.9487 + 96.1080i
35.17 −2.96660 + 2.68315i −8.71221 + 3.17099i 1.60141 − 15.9197i −12.2422 − 2.15864i 17.3374 − 32.7832i 18.9232 − 10.9253i 37.9641 + 51.5241i 3.79786 − 3.18678i 42.1097 − 26.4439i
35.18 −2.96624 − 2.68355i 5.02700 − 1.82968i 1.59713 + 15.9201i 31.9084 + 5.62631i −19.8213 − 8.06294i 57.5713 − 33.2388i 37.9849 − 51.5087i −40.1266 + 33.6702i −79.5494 − 102.317i
35.19 −2.95239 + 2.69878i −4.97990 + 1.81253i 1.43319 − 15.9357i −7.68438 − 1.35496i 9.81096 − 18.7909i −75.3156 + 43.4835i 38.7756 + 50.9162i −40.5355 + 34.0133i 26.3440 − 16.7381i
35.20 −2.71839 + 2.93434i 1.54745 − 0.563227i −1.22071 − 15.9534i −10.4049 − 1.83466i −2.55388 + 6.07183i 62.3309 − 35.9867i 50.1310 + 39.7855i −59.9722 + 50.3227i 33.6680 − 25.5441i
See next 80 embeddings (of 456 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.76
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
19.e even 9 1 inner
152.u odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 152.5.u.b ✓ 456
8.d odd 2 1 inner 152.5.u.b ✓ 456
19.e even 9 1 inner 152.5.u.b ✓ 456
152.u odd 18 1 inner 152.5.u.b ✓ 456
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.5.u.b ✓ 456 1.a even 1 1 trivial
152.5.u.b ✓ 456 8.d odd 2 1 inner
152.5.u.b ✓ 456 19.e even 9 1 inner
152.5.u.b ✓ 456 152.u odd 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{228} + 27 T_{3}^{227} + 342 T_{3}^{226} + 3003 T_{3}^{225} + 36327 T_{3}^{224} + \cdots + 75\!\cdots\!21 \) acting on \(S_{5}^{\mathrm{new}}(152, [\chi])\). Copy content Toggle raw display