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Properties

Label 216.4.t.a
Level $216$
Weight $4$
Character orbit 216.t
Analytic conductor $12.744$
Analytic rank $0$
Dimension $636$
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] = [216,4,Mod(13,216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(216, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 9, 8]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("216.13");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 216.t (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: \(12.7444125612\)
Analytic rank: \(0\)
Dimension: \(636\)
Relative dimension: \(106\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 636 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 3 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 636 q - 6 q^{2} - 6 q^{4} - 6 q^{6} - 12 q^{7} - 3 q^{8} - 12 q^{9} - 3 q^{10} + 111 q^{12} + 69 q^{14} - 12 q^{15} - 6 q^{16} - 6 q^{17} + 351 q^{18} - 237 q^{20} - 6 q^{22} - 12 q^{23} + 288 q^{24} - 12 q^{25} - 1002 q^{26} - 12 q^{28} + 33 q^{30} - 12 q^{31} + 684 q^{32} - 360 q^{33} + 18 q^{34} + 1044 q^{36} + 840 q^{38} - 12 q^{39} - 381 q^{40} + 48 q^{41} - 2586 q^{42} + 885 q^{44} - 3 q^{46} - 12 q^{47} - 2613 q^{48} - 12 q^{49} + 2817 q^{50} + 1371 q^{52} + 4446 q^{54} - 24 q^{55} - 2925 q^{56} - 174 q^{57} - 897 q^{58} + 966 q^{60} - 630 q^{62} - 72 q^{63} - 3 q^{64} - 12 q^{65} - 6795 q^{66} + 2595 q^{68} + 999 q^{70} + 5958 q^{71} - 1452 q^{72} - 6 q^{73} - 5931 q^{74} - 198 q^{76} - 2652 q^{78} - 12 q^{79} + 78 q^{80} - 12 q^{81} - 12 q^{82} - 402 q^{84} + 8322 q^{86} - 6096 q^{87} - 1542 q^{88} - 6 q^{89} + 15612 q^{90} + 3243 q^{92} - 2031 q^{94} - 762 q^{95} - 594 q^{96} - 12 q^{97} - 972 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −2.82833 − 0.0232434i −2.45150 + 4.58150i 7.99892 + 0.131480i 4.66407 − 5.55842i 7.04015 − 12.9010i −8.39129 − 3.05418i −22.6205 − 0.557791i −14.9803 − 22.4631i −13.3207 + 15.6126i
13.2 −2.82365 + 0.164344i −5.14582 − 0.721501i 7.94598 − 0.928101i 8.72330 − 10.3960i 14.6486 + 1.19158i 17.1856 + 6.25504i −22.2841 + 3.92651i 25.9589 + 7.42543i −22.9230 + 30.7884i
13.3 −2.81394 − 0.285939i 4.53072 + 2.54412i 7.83648 + 1.60923i 1.57172 − 1.87311i −12.0217 − 8.45450i −10.2681 − 3.73728i −21.5912 − 6.76902i 14.0549 + 23.0534i −4.95832 + 4.82138i
13.4 −2.81348 − 0.290395i 4.52308 − 2.55769i 7.83134 + 1.63404i 3.38486 − 4.03392i −13.4683 + 5.88252i −1.55456 − 0.565814i −21.5588 − 6.87152i 13.9165 − 23.1372i −10.6947 + 10.3664i
13.5 −2.80577 − 0.357309i −4.19036 − 3.07261i 7.74466 + 2.00505i −8.24274 + 9.82331i 10.6593 + 10.1183i 4.23380 + 1.54098i −21.0133 − 8.39294i 8.11818 + 25.7506i 26.6372 − 24.6167i
13.6 −2.80472 + 0.365460i −1.05246 − 5.08845i 7.73288 − 2.05003i 10.5704 − 12.5973i 4.81147 + 13.8870i −27.5268 − 10.0189i −20.9393 + 8.57580i −24.7847 + 10.7107i −25.0432 + 39.1950i
13.7 −2.75786 + 0.627866i −0.282432 + 5.18847i 7.21157 − 3.46313i −9.09467 + 10.8386i −2.47876 − 14.4864i 2.29934 + 0.836890i −17.7141 + 14.0787i −26.8405 − 2.93078i 18.2766 − 35.6016i
13.8 −2.72249 + 0.766860i 1.70736 − 4.90764i 6.82385 − 4.17553i −4.49049 + 5.35155i −0.884793 + 14.6703i 27.5724 + 10.0355i −15.3758 + 16.6008i −21.1698 − 16.7582i 8.12139 − 18.0131i
13.9 −2.72103 − 0.772003i 5.16538 + 0.564659i 6.80802 + 4.20129i −12.8989 + 15.3723i −13.6192 − 5.52414i 27.3207 + 9.94391i −15.2814 − 16.6876i 26.3623 + 5.83336i 46.9657 − 31.8705i
13.10 −2.66739 + 0.940767i 2.90472 + 4.30843i 6.22991 − 5.01878i 5.68364 − 6.77350i −11.8013 − 8.75959i 28.6996 + 10.4458i −11.8961 + 19.2479i −10.1252 + 25.0296i −8.78819 + 23.4145i
13.11 −2.65820 + 0.966427i 4.16005 − 3.11352i 6.13204 − 5.13791i −12.0804 + 14.3968i −8.04925 + 12.2967i −28.0138 − 10.1962i −11.3348 + 19.5837i 7.61202 − 25.9048i 18.1985 − 49.9444i
13.12 −2.62974 + 1.04138i −5.19014 + 0.249981i 5.83107 − 5.47710i −2.09232 + 2.49353i 13.3884 − 6.06227i −22.9522 − 8.35393i −9.63050 + 20.4757i 26.8750 − 2.59487i 2.90556 − 8.73622i
13.13 −2.58549 + 1.14685i −0.255611 − 5.18986i 5.36949 − 5.93031i −3.27299 + 3.90060i 6.61285 + 13.1252i 8.29649 + 3.01968i −7.08159 + 21.4907i −26.8693 + 2.65317i 3.98889 − 13.8386i
13.14 −2.57154 − 1.17778i −0.223972 − 5.19132i 5.22567 + 6.05742i −2.96305 + 3.53122i −5.53828 + 13.6135i −17.0976 − 6.22302i −6.30372 − 21.7316i −26.8997 + 2.32542i 11.7786 − 5.59087i
13.15 −2.57122 − 1.17847i 1.15411 + 5.06636i 5.22239 + 6.06025i 12.6273 − 15.0486i 3.00310 − 14.3868i −7.37857 − 2.68558i −6.28610 − 21.7367i −24.3360 + 11.6943i −50.2021 + 23.8125i
13.16 −2.56771 − 1.18612i −4.42428 + 2.72502i 5.18626 + 6.09120i −6.30914 + 7.51894i 14.5925 − 1.74936i −26.3482 − 9.58995i −6.09194 − 21.7919i 12.1485 − 24.1125i 25.1184 − 11.8231i
13.17 −2.55888 − 1.20504i −3.77922 + 3.56616i 5.09577 + 6.16710i −1.08437 + 1.29230i 13.9679 − 4.57129i 30.2565 + 11.0125i −5.60788 − 21.9215i 1.56497 − 26.9546i 4.33205 − 2.00014i
13.18 −2.49507 − 1.33215i 2.52814 − 4.53966i 4.45076 + 6.64761i 5.88418 − 7.01250i −12.3554 + 7.95892i 18.9317 + 6.89058i −2.24935 − 22.5153i −14.2171 − 22.9538i −24.0231 + 9.65807i
13.19 −2.45348 + 1.40729i 4.76236 − 2.07845i 4.03910 − 6.90548i 13.2895 − 15.8378i −8.75937 + 11.8014i 3.91737 + 1.42580i −0.191839 + 22.6266i 18.3601 − 19.7966i −10.3171 + 57.5597i
13.20 −2.36371 + 1.55335i 4.86625 + 1.82198i 3.17422 − 7.34332i 0.520651 − 0.620488i −14.3326 + 3.25236i −17.5070 − 6.37202i 3.90381 + 22.2881i 20.3608 + 17.7324i −0.266833 + 2.27540i
See next 80 embeddings (of 636 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.106
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
27.e even 9 1 inner
216.t even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.4.t.a ✓ 636
8.b even 2 1 inner 216.4.t.a ✓ 636
27.e even 9 1 inner 216.4.t.a ✓ 636
216.t even 18 1 inner 216.4.t.a ✓ 636
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.4.t.a ✓ 636 1.a even 1 1 trivial
216.4.t.a ✓ 636 8.b even 2 1 inner
216.4.t.a ✓ 636 27.e even 9 1 inner
216.4.t.a ✓ 636 216.t even 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(216, [\chi])\).