Properties

Label 381.2.q.a
Level $381$
Weight $2$
Character orbit 381.q
Analytic conductor $3.042$
Analytic rank $0$
Dimension $120$
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] = [381,2,Mod(25,381)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(381, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("381.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 381 = 3 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 381.q (of order \(21\), degree \(12\), 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: \(3.04230031701\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(10\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 3 q^{2} - 10 q^{3} - 15 q^{4} + 2 q^{5} + 2 q^{6} + 11 q^{7} + 20 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q - 3 q^{2} - 10 q^{3} - 15 q^{4} + 2 q^{5} + 2 q^{6} + 11 q^{7} + 20 q^{8} + 10 q^{9} + 4 q^{10} - 36 q^{11} - 18 q^{12} + q^{13} + 4 q^{14} + q^{15} - 15 q^{16} - 5 q^{17} - 2 q^{18} - 54 q^{19} + 18 q^{20} - 11 q^{21} + 19 q^{22} - 22 q^{23} - 18 q^{24} - 26 q^{25} + 54 q^{26} + 20 q^{27} - 19 q^{28} + 31 q^{29} - 19 q^{30} - 3 q^{31} - q^{32} + 33 q^{33} + 23 q^{34} + 70 q^{35} + 4 q^{36} + 36 q^{37} - 55 q^{38} - q^{39} - 80 q^{40} + 8 q^{41} + 10 q^{42} - 41 q^{43} - 118 q^{44} - q^{45} + 46 q^{46} + 15 q^{47} + 52 q^{48} - 11 q^{49} - 26 q^{50} - 10 q^{51} + 72 q^{52} + 30 q^{53} + 3 q^{54} - 87 q^{55} - 103 q^{56} + 8 q^{57} + 36 q^{58} - 16 q^{59} - 96 q^{60} + 43 q^{61} + 6 q^{63} - 28 q^{64} - 33 q^{65} + 3 q^{66} - 10 q^{67} + 67 q^{68} - 6 q^{69} - 72 q^{70} + 55 q^{71} - 10 q^{72} + 47 q^{73} + 101 q^{74} + 29 q^{75} - 59 q^{76} - 15 q^{77} + 2 q^{78} - 148 q^{79} + 27 q^{80} + 10 q^{81} + 106 q^{82} - 4 q^{83} + 12 q^{84} - 47 q^{85} - 157 q^{86} - 8 q^{87} + 85 q^{88} + 16 q^{89} + 5 q^{90} - 111 q^{91} - 157 q^{92} - 25 q^{93} - 18 q^{94} - 24 q^{95} + 10 q^{96} + 52 q^{97} - 4 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −0.529775 2.32110i −0.365341 0.930874i −3.30488 + 1.59155i 0.773044 + 0.372278i −1.96710 + 1.34114i −1.87928 + 1.28127i 2.47619 + 3.10504i −0.733052 + 0.680173i 0.454554 1.99153i
25.2 −0.376211 1.64829i −0.365341 0.930874i −0.773378 + 0.372439i 3.15783 + 1.52073i −1.39690 + 0.952392i −0.607278 + 0.414035i −1.20340 1.50901i −0.733052 + 0.680173i 1.31859 5.77712i
25.3 −0.370028 1.62120i −0.365341 0.930874i −0.689423 + 0.332009i −0.478317 0.230345i −1.37394 + 0.936739i 4.17508 2.84652i −1.28023 1.60536i −0.733052 + 0.680173i −0.196445 + 0.860681i
25.4 −0.209591 0.918278i −0.365341 0.930874i 1.00263 0.482842i −2.49467 1.20137i −0.778228 + 0.530587i 0.155612 0.106095i −1.82805 2.29230i −0.733052 + 0.680173i −0.580331 + 2.54260i
25.5 0.0597923 + 0.261967i −0.365341 0.930874i 1.73689 0.836440i 3.54884 + 1.70903i 0.222014 0.151366i 3.49359 2.38189i 0.658041 + 0.825157i −0.733052 + 0.680173i −0.235517 + 1.03187i
25.6 0.203474 + 0.891478i −0.365341 0.930874i 1.04861 0.504982i −0.0127261 0.00612854i 0.755516 0.515102i 0.396252 0.270160i 1.80379 + 2.26188i −0.733052 + 0.680173i 0.00287404 0.0125920i
25.7 0.241228 + 1.05689i −0.365341 0.930874i 0.743115 0.357865i −3.37877 1.62713i 0.895699 0.610678i −4.21964 + 2.87691i 1.90930 + 2.39418i −0.733052 + 0.680173i 0.904641 3.96349i
25.8 0.484696 + 2.12359i −0.365341 0.930874i −2.47277 + 1.19082i 2.10785 + 1.01509i 1.79972 1.22703i 0.505716 0.344791i −1.01119 1.26799i −0.733052 + 0.680173i −1.13396 + 4.96821i
25.9 0.515942 + 2.26049i −0.365341 0.930874i −3.04168 + 1.46479i −3.58612 1.72698i 1.91574 1.30613i 3.85246 2.62656i −1.98921 2.49439i −0.733052 + 0.680173i 2.05360 8.99740i
25.10 0.529232 + 2.31872i −0.365341 0.930874i −3.29442 + 1.58651i −0.295284 0.142201i 1.96508 1.33977i −2.06860 + 1.41035i −2.45643 3.08027i −0.733052 + 0.680173i 0.173451 0.759937i
61.1 −0.529775 + 2.32110i −0.365341 + 0.930874i −3.30488 1.59155i 0.773044 0.372278i −1.96710 1.34114i −1.87928 1.28127i 2.47619 3.10504i −0.733052 0.680173i 0.454554 + 1.99153i
61.2 −0.376211 + 1.64829i −0.365341 + 0.930874i −0.773378 0.372439i 3.15783 1.52073i −1.39690 0.952392i −0.607278 0.414035i −1.20340 + 1.50901i −0.733052 0.680173i 1.31859 + 5.77712i
61.3 −0.370028 + 1.62120i −0.365341 + 0.930874i −0.689423 0.332009i −0.478317 + 0.230345i −1.37394 0.936739i 4.17508 + 2.84652i −1.28023 + 1.60536i −0.733052 0.680173i −0.196445 0.860681i
61.4 −0.209591 + 0.918278i −0.365341 + 0.930874i 1.00263 + 0.482842i −2.49467 + 1.20137i −0.778228 0.530587i 0.155612 + 0.106095i −1.82805 + 2.29230i −0.733052 0.680173i −0.580331 2.54260i
61.5 0.0597923 0.261967i −0.365341 + 0.930874i 1.73689 + 0.836440i 3.54884 1.70903i 0.222014 + 0.151366i 3.49359 + 2.38189i 0.658041 0.825157i −0.733052 0.680173i −0.235517 1.03187i
61.6 0.203474 0.891478i −0.365341 + 0.930874i 1.04861 + 0.504982i −0.0127261 + 0.00612854i 0.755516 + 0.515102i 0.396252 + 0.270160i 1.80379 2.26188i −0.733052 0.680173i 0.00287404 + 0.0125920i
61.7 0.241228 1.05689i −0.365341 + 0.930874i 0.743115 + 0.357865i −3.37877 + 1.62713i 0.895699 + 0.610678i −4.21964 2.87691i 1.90930 2.39418i −0.733052 0.680173i 0.904641 + 3.96349i
61.8 0.484696 2.12359i −0.365341 + 0.930874i −2.47277 1.19082i 2.10785 1.01509i 1.79972 + 1.22703i 0.505716 + 0.344791i −1.01119 + 1.26799i −0.733052 0.680173i −1.13396 4.96821i
61.9 0.515942 2.26049i −0.365341 + 0.930874i −3.04168 1.46479i −3.58612 + 1.72698i 1.91574 + 1.30613i 3.85246 + 2.62656i −1.98921 + 2.49439i −0.733052 0.680173i 2.05360 + 8.99740i
61.10 0.529232 2.31872i −0.365341 + 0.930874i −3.29442 1.58651i −0.295284 + 0.142201i 1.96508 + 1.33977i −2.06860 1.41035i −2.45643 + 3.08027i −0.733052 0.680173i 0.173451 + 0.759937i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
127.i even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 381.2.q.a 120
127.i even 21 1 inner 381.2.q.a 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
381.2.q.a 120 1.a even 1 1 trivial
381.2.q.a 120 127.i even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{120} + 3 T_{2}^{119} + 32 T_{2}^{118} + 75 T_{2}^{117} + 525 T_{2}^{116} + 1018 T_{2}^{115} + 6458 T_{2}^{114} + 11527 T_{2}^{113} + 72154 T_{2}^{112} + 126963 T_{2}^{111} + 757046 T_{2}^{110} + 1297295 T_{2}^{109} + \cdots + 531625249 \) acting on \(S_{2}^{\mathrm{new}}(381, [\chi])\). Copy content Toggle raw display