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Properties

Label 385.2.n.e
Level $385$
Weight $2$
Character orbit 385.n
Analytic conductor $3.074$
Analytic rank $0$
Dimension $28$
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] = [385,2,Mod(36,385)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(385, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("385.36");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 385 = 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 385.n (of order \(5\), degree \(4\), 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: \(3.07424047782\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(7\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 3 q^{2} - 3 q^{4} + 7 q^{5} + 7 q^{6} - 7 q^{7} - q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 3 q^{2} - 3 q^{4} + 7 q^{5} + 7 q^{6} - 7 q^{7} - q^{8} - q^{9} + 2 q^{10} - 3 q^{11} + 30 q^{12} - 5 q^{13} - 2 q^{14} - 33 q^{16} + 7 q^{17} - 17 q^{18} - 13 q^{19} + 18 q^{20} + 10 q^{21} + 3 q^{22} + 16 q^{23} + 39 q^{24} - 7 q^{25} + 44 q^{26} - 18 q^{27} - 18 q^{28} + 7 q^{29} + 8 q^{30} + 7 q^{31} - 84 q^{32} + 4 q^{33} + 32 q^{34} + 7 q^{35} + 6 q^{36} - 24 q^{37} - 15 q^{38} - 18 q^{39} + q^{40} - 4 q^{41} + 7 q^{42} + q^{44} - 34 q^{45} + 3 q^{46} - 33 q^{47} + 83 q^{48} - 7 q^{49} + 3 q^{50} + 14 q^{51} + 24 q^{52} - 8 q^{53} - 114 q^{54} - 17 q^{55} - 6 q^{56} + 57 q^{57} - 16 q^{58} - 3 q^{59} + 15 q^{60} - 21 q^{61} - 19 q^{62} - 16 q^{63} - 19 q^{64} - 20 q^{65} - 126 q^{66} + 90 q^{67} - 7 q^{68} - 55 q^{69} + 2 q^{70} - 37 q^{71} + 117 q^{72} + 17 q^{73} + 49 q^{74} - 5 q^{75} - 10 q^{76} - 3 q^{77} + 104 q^{78} - 45 q^{79} - 2 q^{80} + 8 q^{81} - 48 q^{82} + q^{83} + 18 q^{85} + 134 q^{86} + 46 q^{87} + 74 q^{88} - 30 q^{89} - 38 q^{90} - 5 q^{91} + 18 q^{92} - 57 q^{93} + 43 q^{94} + 13 q^{95} - 119 q^{96} - 82 q^{97} - 2 q^{98} - 72 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
36.1 −2.09818 + 1.52442i 0.519461 + 1.59874i 1.46048 − 4.49491i 0.809017 + 0.587785i −3.52707 − 2.56257i 0.309017 − 0.951057i 2.18489 + 6.72441i 0.140929 − 0.102391i −2.59350
36.2 −1.79013 + 1.30061i −0.342550 − 1.05426i 0.894962 − 2.75441i 0.809017 + 0.587785i 1.98439 + 1.44174i 0.309017 − 0.951057i 0.612766 + 1.88590i 1.43293 − 1.04108i −2.21273
36.3 −0.538699 + 0.391388i −0.176441 − 0.543029i −0.481022 + 1.48043i 0.809017 + 0.587785i 0.307584 + 0.223473i 0.309017 − 0.951057i −0.731827 − 2.25233i 2.16330 − 1.57173i −0.665869
36.4 0.285045 − 0.207098i 0.872686 + 2.68585i −0.579673 + 1.78405i 0.809017 + 0.587785i 0.804989 + 0.584858i 0.309017 − 0.951057i 0.421995 + 1.29877i −4.02517 + 2.92446i 0.352336
36.5 1.42900 − 1.03823i 0.0755236 + 0.232438i 0.346092 − 1.06516i 0.809017 + 0.587785i 0.349248 + 0.253743i 0.309017 − 0.951057i 0.480344 + 1.47835i 2.37873 − 1.72825i 1.76635
36.6 1.76389 − 1.28154i −0.720548 − 2.21762i 0.850927 − 2.61888i 0.809017 + 0.587785i −4.11294 − 2.98822i 0.309017 − 0.951057i −0.507773 − 1.56276i −1.97159 + 1.43245i 2.18029
36.7 2.25809 − 1.64060i 0.889902 + 2.73884i 1.78938 − 5.50715i 0.809017 + 0.587785i 6.50282 + 4.72457i 0.309017 − 0.951057i −3.26942 − 10.0622i −4.28224 + 3.11123i 2.79116
71.1 −0.822693 − 2.53199i 1.19323 + 0.866936i −4.11611 + 2.99053i −0.309017 + 0.951057i 1.21341 − 3.73448i −0.809017 + 0.587785i 6.65059 + 4.83194i −0.254819 − 0.784253i 2.66229
71.2 −0.422768 − 1.30115i −2.62739 − 1.90891i 0.103787 − 0.0754058i −0.309017 + 0.951057i −1.37299 + 4.22564i −0.809017 + 0.587785i −2.35563 − 1.71147i 2.33219 + 7.17774i 1.36811
71.3 −0.243737 − 0.750146i −0.550648 − 0.400069i 1.11472 − 0.809894i −0.309017 + 0.951057i −0.165897 + 0.510578i −0.809017 + 0.587785i −2.15546 − 1.56603i −0.783893 − 2.41258i 0.788750
71.4 −0.0912842 − 0.280944i 2.12462 + 1.54362i 1.54744 − 1.12428i −0.309017 + 0.951057i 0.239728 − 0.737806i −0.809017 + 0.587785i −0.935086 − 0.679380i 1.20417 + 3.70604i 0.295402
71.5 0.461021 + 1.41888i −1.09297 − 0.794089i −0.182637 + 0.132694i −0.309017 + 0.951057i 0.622832 − 1.91688i −0.809017 + 0.587785i 2.14146 + 1.55586i −0.363046 − 1.11734i −1.49190
71.6 0.543901 + 1.67395i 1.80845 + 1.31392i −0.888261 + 0.645360i −0.309017 + 0.951057i −1.21582 + 3.74190i −0.809017 + 0.587785i 1.28447 + 0.933224i 0.617067 + 1.89914i −1.76010
71.7 0.766543 + 2.35918i −1.97333 − 1.43371i −3.36010 + 2.44125i −0.309017 + 0.951057i 1.86973 − 5.75443i −0.809017 + 0.587785i −4.32133 − 3.13963i 0.911457 + 2.80518i −2.48059
141.1 −0.822693 + 2.53199i 1.19323 − 0.866936i −4.11611 − 2.99053i −0.309017 − 0.951057i 1.21341 + 3.73448i −0.809017 − 0.587785i 6.65059 − 4.83194i −0.254819 + 0.784253i 2.66229
141.2 −0.422768 + 1.30115i −2.62739 + 1.90891i 0.103787 + 0.0754058i −0.309017 − 0.951057i −1.37299 − 4.22564i −0.809017 − 0.587785i −2.35563 + 1.71147i 2.33219 − 7.17774i 1.36811
141.3 −0.243737 + 0.750146i −0.550648 + 0.400069i 1.11472 + 0.809894i −0.309017 − 0.951057i −0.165897 − 0.510578i −0.809017 − 0.587785i −2.15546 + 1.56603i −0.783893 + 2.41258i 0.788750
141.4 −0.0912842 + 0.280944i 2.12462 − 1.54362i 1.54744 + 1.12428i −0.309017 − 0.951057i 0.239728 + 0.737806i −0.809017 − 0.587785i −0.935086 + 0.679380i 1.20417 − 3.70604i 0.295402
141.5 0.461021 − 1.41888i −1.09297 + 0.794089i −0.182637 − 0.132694i −0.309017 − 0.951057i 0.622832 + 1.91688i −0.809017 − 0.587785i 2.14146 − 1.55586i −0.363046 + 1.11734i −1.49190
141.6 0.543901 − 1.67395i 1.80845 − 1.31392i −0.888261 − 0.645360i −0.309017 − 0.951057i −1.21582 − 3.74190i −0.809017 − 0.587785i 1.28447 − 0.933224i 0.617067 − 1.89914i −1.76010
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 36.7
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 385.2.n.e ✓ 28
11.c even 5 1 inner 385.2.n.e ✓ 28
11.c even 5 1 4235.2.a.bm 14
11.d odd 10 1 4235.2.a.bn 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.n.e ✓ 28 1.a even 1 1 trivial
385.2.n.e ✓ 28 11.c even 5 1 inner
4235.2.a.bm 14 11.c even 5 1
4235.2.a.bn 14 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} - 3 T_{2}^{27} + 13 T_{2}^{26} - 27 T_{2}^{25} + 118 T_{2}^{24} - 223 T_{2}^{23} + \cdots + 6400 \) acting on \(S_{2}^{\mathrm{new}}(385, [\chi])\). Copy content Toggle raw display