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Properties

Label 154.4.m.b
Level $154$
Weight $4$
Character orbit 154.m
Analytic conductor $9.086$
Analytic rank $0$
Dimension $96$
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] = [154,4,Mod(9,154)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(154, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([10, 18]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("154.9");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 154 = 2 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 154.m (of order \(15\), degree \(8\), 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: \(9.08629414088\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(12\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 24 q^{2} - 6 q^{3} + 48 q^{4} - 8 q^{5} + 24 q^{6} - 16 q^{7} - 192 q^{8} + 114 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 24 q^{2} - 6 q^{3} + 48 q^{4} - 8 q^{5} + 24 q^{6} - 16 q^{7} - 192 q^{8} + 114 q^{9} + 84 q^{10} + 48 q^{11} + 96 q^{12} - 22 q^{13} - 14 q^{14} - 30 q^{15} + 192 q^{16} - 233 q^{17} + 208 q^{18} + 21 q^{19} + 64 q^{20} - 38 q^{21} + 228 q^{22} + 108 q^{23} - 48 q^{24} + 200 q^{25} - 68 q^{26} + 468 q^{27} - 108 q^{28} - 38 q^{29} + 30 q^{30} - 34 q^{31} - 1536 q^{32} - 1617 q^{33} - 1128 q^{34} + 1610 q^{35} - 832 q^{36} - 74 q^{37} - 98 q^{38} + 514 q^{39} - 104 q^{40} - 746 q^{41} - 614 q^{42} - 1556 q^{43} + 52 q^{44} + 3104 q^{45} + 146 q^{46} - 404 q^{47} + 192 q^{48} + 690 q^{49} - 1700 q^{50} + 434 q^{51} - 136 q^{52} + 1683 q^{53} + 1752 q^{54} + 924 q^{55} - 128 q^{56} - 2728 q^{57} + 38 q^{58} - 753 q^{59} - 80 q^{60} + 1495 q^{61} + 2456 q^{62} + 2181 q^{63} - 1536 q^{64} + 3284 q^{65} + 686 q^{66} + 330 q^{67} - 932 q^{68} + 3096 q^{69} - 1128 q^{70} - 2342 q^{71} + 912 q^{72} - 1512 q^{73} - 148 q^{74} - 2027 q^{75} - 448 q^{76} - 3652 q^{77} + 2464 q^{78} - 2414 q^{79} - 208 q^{80} + 1850 q^{81} - 1024 q^{82} - 916 q^{83} - 2920 q^{84} - 304 q^{85} - 1334 q^{86} + 2286 q^{87} - 456 q^{88} + 2860 q^{89} + 8184 q^{90} + 2951 q^{91} + 1016 q^{92} + 431 q^{93} - 268 q^{94} - 66 q^{95} - 192 q^{96} - 430 q^{97} + 608 q^{98} + 4736 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 1.33826 + 1.48629i −8.55412 − 3.80854i −0.418114 + 3.97809i −11.5566 − 2.45644i −5.78705 − 17.8107i 0.142461 − 18.5197i −6.47214 + 4.70228i 40.6014 + 45.0925i −11.8148 − 20.4638i
9.2 1.33826 + 1.48629i −8.08582 − 3.60004i −0.418114 + 3.97809i 14.7700 + 3.13947i −5.47024 − 16.8357i −18.4975 + 0.917520i −6.47214 + 4.70228i 34.3536 + 38.1536i 15.1000 + 26.1540i
9.3 1.33826 + 1.48629i −5.93051 − 2.64043i −0.418114 + 3.97809i 2.85067 + 0.605930i −4.01213 − 12.3481i 17.3644 + 6.44025i −6.47214 + 4.70228i 10.1326 + 11.2534i 2.91436 + 5.04782i
9.4 1.33826 + 1.48629i −4.35525 − 1.93908i −0.418114 + 3.97809i −9.67372 − 2.05621i −2.94643 − 9.06817i 13.5034 + 12.6751i −6.47214 + 4.70228i −2.85834 − 3.17451i −9.88984 − 17.1297i
9.5 1.33826 + 1.48629i −2.35444 − 1.04826i −0.418114 + 3.97809i 6.70170 + 1.42449i −1.59283 − 4.90222i −13.6223 + 12.5472i −6.47214 + 4.70228i −13.6220 − 15.1288i 6.85142 + 11.8670i
9.6 1.33826 + 1.48629i −1.28676 − 0.572901i −0.418114 + 3.97809i −3.97894 − 0.845749i −0.870520 − 2.67919i 0.310616 − 18.5177i −6.47214 + 4.70228i −16.7390 − 18.5905i −4.06783 − 7.04569i
9.7 1.33826 + 1.48629i −0.929365 − 0.413780i −0.418114 + 3.97809i 21.0140 + 4.46665i −0.628736 − 1.93505i 8.13267 − 16.6391i −6.47214 + 4.70228i −17.3740 − 19.2958i 21.4834 + 37.2104i
9.8 1.33826 + 1.48629i 1.66592 + 0.741717i −0.418114 + 3.97809i −11.5631 − 2.45782i 1.12703 + 3.46866i −18.1926 − 3.46833i −6.47214 + 4.70228i −15.8414 − 17.5936i −11.8214 − 20.4753i
9.9 1.33826 + 1.48629i 4.60534 + 2.05043i −0.418114 + 3.97809i −19.1772 − 4.07623i 3.11562 + 9.58889i 6.43761 + 17.3654i −6.47214 + 4.70228i −1.06161 − 1.17903i −19.6056 − 33.9579i
9.10 1.33826 + 1.48629i 5.04372 + 2.24561i −0.418114 + 3.97809i 14.0210 + 2.98026i 3.41219 + 10.5016i −1.74904 + 18.4375i −6.47214 + 4.70228i 2.32980 + 2.58751i 14.3343 + 24.8277i
9.11 1.33826 + 1.48629i 6.55108 + 2.91673i −0.418114 + 3.97809i 4.50513 + 0.957594i 4.43195 + 13.6401i 18.4741 − 1.30610i −6.47214 + 4.70228i 16.3428 + 18.1505i 4.60577 + 7.97743i
9.12 1.33826 + 1.48629i 8.14893 + 3.62814i −0.418114 + 3.97809i 3.45120 + 0.733576i 5.51293 + 16.9671i −16.5803 − 8.25186i −6.47214 + 4.70228i 35.1751 + 39.0659i 3.52831 + 6.11121i
25.1 1.82709 + 0.813473i −8.15361 − 1.73310i 2.67652 + 2.97258i 1.95272 − 18.5789i −13.4876 − 9.79928i −10.0624 + 15.5482i 2.47214 + 7.60845i 38.8120 + 17.2802i 18.6813 − 32.3569i
25.2 1.82709 + 0.813473i −7.75306 − 1.64796i 2.67652 + 2.97258i −0.505217 + 4.80682i −12.8250 − 9.31788i −6.10769 − 17.4842i 2.47214 + 7.60845i 32.7284 + 14.5716i −4.83330 + 8.37152i
25.3 1.82709 + 0.813473i −6.05642 − 1.28733i 2.67652 + 2.97258i −0.691745 + 6.58151i −10.0184 − 7.27880i 17.2657 − 6.70034i 2.47214 + 7.60845i 10.3572 + 4.61134i −6.61777 + 11.4623i
25.4 1.82709 + 0.813473i −2.64175 − 0.561521i 2.67652 + 2.97258i 0.873854 − 8.31417i −4.36993 − 3.17494i 18.4072 + 2.04320i 2.47214 + 7.60845i −18.0022 − 8.01509i 8.35996 − 14.4799i
25.5 1.82709 + 0.813473i −1.96202 − 0.417041i 2.67652 + 2.97258i 1.11173 − 10.5774i −3.24555 − 2.35803i −18.4794 − 1.22987i 2.47214 + 7.60845i −20.9901 − 9.34540i 10.6357 − 18.4215i
25.6 1.82709 + 0.813473i −1.38309 − 0.293984i 2.67652 + 2.97258i −2.05885 + 19.5886i −2.28788 − 1.66224i −17.4091 − 6.31858i 2.47214 + 7.60845i −22.8392 − 10.1687i −19.6965 + 34.1154i
25.7 1.82709 + 0.813473i −0.0518447 − 0.0110199i 2.67652 + 2.97258i −0.554576 + 5.27644i −0.0857605 − 0.0623087i −4.51846 + 17.9606i 2.47214 + 7.60845i −24.6632 − 10.9807i −5.30551 + 9.18941i
25.8 1.82709 + 0.813473i 4.80854 + 1.02209i 2.67652 + 2.97258i −0.997764 + 9.49309i 7.95420 + 5.77907i 14.0005 + 12.1238i 2.47214 + 7.60845i −2.58831 − 1.15239i −9.54538 + 16.5331i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
11.c even 5 1 inner
77.m even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 154.4.m.b ✓ 96
7.c even 3 1 inner 154.4.m.b ✓ 96
11.c even 5 1 inner 154.4.m.b ✓ 96
77.m even 15 1 inner 154.4.m.b ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.4.m.b ✓ 96 1.a even 1 1 trivial
154.4.m.b ✓ 96 7.c even 3 1 inner
154.4.m.b ✓ 96 11.c even 5 1 inner
154.4.m.b ✓ 96 77.m even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{96} + 6 T_{3}^{95} - 201 T_{3}^{94} - 1650 T_{3}^{93} + 10918 T_{3}^{92} + 132080 T_{3}^{91} + \cdots + 26\!\cdots\!25 \) acting on \(S_{4}^{\mathrm{new}}(154, [\chi])\). Copy content Toggle raw display