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Properties

Label 385.2.j.d
Level $385$
Weight $2$
Character orbit 385.j
Analytic conductor $3.074$
Analytic rank $0$
Dimension $40$
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] = [385,2,Mod(188,385)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(385, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("385.188");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 385 = 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 385.j (of order \(4\), degree \(2\), 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: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 24 q^{8} - 40 q^{11} - 28 q^{15} - 32 q^{16} - 52 q^{18} + 4 q^{21} + 4 q^{23} + 16 q^{25} + 12 q^{28} + 44 q^{30} + 28 q^{32} - 42 q^{35} + 24 q^{36} - 8 q^{37} + 44 q^{42} - 4 q^{43} - 32 q^{46} - 120 q^{50} + 64 q^{51} - 20 q^{53} - 84 q^{56} - 36 q^{57} + 28 q^{58} + 164 q^{60} + 16 q^{63} + 16 q^{67} - 4 q^{70} + 32 q^{71} - 8 q^{72} + 176 q^{78} - 104 q^{81} - 84 q^{85} - 24 q^{86} - 24 q^{88} - 96 q^{91} - 36 q^{92} - 84 q^{93} + 20 q^{95} + 76 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
188.1 −1.88561 + 1.88561i −1.27229 + 1.27229i − 5.11105i 2.19274 + 0.438071i − 4.79808i −1.82521 + 1.91536i 5.86622 + 5.86622i − 0.237444i −4.96068 + 3.30861i
188.2 −1.88561 + 1.88561i 1.27229 − 1.27229i − 5.11105i −2.19274 − 0.438071i 4.79808i −1.91536 + 1.82521i 5.86622 + 5.86622i − 0.237444i 4.96068 − 3.30861i
188.3 −1.54584 + 1.54584i −0.253262 + 0.253262i − 2.77923i 1.85982 + 1.24140i − 0.783004i 0.988478 − 2.45416i 1.20457 + 1.20457i 2.87172i −4.79398 + 0.955982i
188.4 −1.54584 + 1.54584i 0.253262 − 0.253262i − 2.77923i −1.85982 − 1.24140i 0.783004i 2.45416 − 0.988478i 1.20457 + 1.20457i 2.87172i 4.79398 − 0.955982i
188.5 −1.18335 + 1.18335i −1.20606 + 1.20606i − 0.800639i −0.578411 + 2.15996i − 2.85439i 1.07188 + 2.41890i −1.41926 − 1.41926i 0.0908366i −1.87153 − 3.24046i
188.6 −1.18335 + 1.18335i 1.20606 − 1.20606i − 0.800639i 0.578411 − 2.15996i 2.85439i −2.41890 − 1.07188i −1.41926 − 1.41926i 0.0908366i 1.87153 + 3.24046i
188.7 −0.351766 + 0.351766i −1.60375 + 1.60375i 1.75252i −1.93332 − 1.12350i − 1.12829i −0.235274 − 2.63527i −1.32001 − 1.32001i − 2.14404i 1.07529 − 0.284867i
188.8 −0.351766 + 0.351766i 1.60375 − 1.60375i 1.75252i 1.93332 + 1.12350i 1.12829i 2.63527 + 0.235274i −1.32001 − 1.32001i − 2.14404i −1.07529 + 0.284867i
188.9 −0.222820 + 0.222820i −0.132131 + 0.132131i 1.90070i 0.771580 + 2.09873i − 0.0588826i −2.49744 − 0.873369i −0.869153 − 0.869153i 2.96508i −0.639561 − 0.295715i
188.10 −0.222820 + 0.222820i 0.132131 − 0.132131i 1.90070i −0.771580 − 2.09873i 0.0588826i 0.873369 + 2.49744i −0.869153 − 0.869153i 2.96508i 0.639561 + 0.295715i
188.11 0.484215 − 0.484215i −2.31786 + 2.31786i 1.53107i −1.60547 + 1.55642i 2.24469i 0.0359100 + 2.64551i 1.70980 + 1.70980i − 7.74495i −0.0237495 + 1.53104i
188.12 0.484215 − 0.484215i 2.31786 − 2.31786i 1.53107i 1.60547 − 1.55642i − 2.24469i −2.64551 − 0.0359100i 1.70980 + 1.70980i − 7.74495i 0.0237495 − 1.53104i
188.13 0.625287 − 0.625287i −1.82490 + 1.82490i 1.21803i 2.17139 + 0.533912i 2.28218i 2.11261 − 1.59276i 2.01219 + 2.01219i − 3.66053i 1.69159 − 1.02389i
188.14 0.625287 − 0.625287i 1.82490 − 1.82490i 1.21803i −2.17139 − 0.533912i − 2.28218i 1.59276 − 2.11261i 2.01219 + 2.01219i − 3.66053i −1.69159 + 1.02389i
188.15 0.924360 − 0.924360i −0.244165 + 0.244165i 0.291117i −2.09828 + 0.772794i 0.451393i −2.64408 + 0.0940746i 2.11782 + 2.11782i 2.88077i −1.22523 + 2.65391i
188.16 0.924360 − 0.924360i 0.244165 − 0.244165i 0.291117i 2.09828 − 0.772794i − 0.451393i −0.0940746 + 2.64408i 2.11782 + 2.11782i 2.88077i 1.22523 − 2.65391i
188.17 1.47145 − 1.47145i −0.503694 + 0.503694i − 2.33031i 0.0653050 + 2.23511i 1.48232i 2.64113 − 0.156238i −0.486040 − 0.486040i 2.49258i 3.38495 + 3.19276i
188.18 1.47145 − 1.47145i 0.503694 − 0.503694i − 2.33031i −0.0653050 − 2.23511i − 1.48232i 0.156238 − 2.64113i −0.486040 − 0.486040i 2.49258i −3.38495 − 3.19276i
188.19 1.68407 − 1.68407i −2.06325 + 2.06325i − 3.67221i 1.53812 − 1.62302i 6.94934i −2.00834 − 1.72237i −2.81613 − 2.81613i − 5.51401i −0.142985 − 5.32359i
188.20 1.68407 − 1.68407i 2.06325 − 2.06325i − 3.67221i −1.53812 + 1.62302i − 6.94934i 1.72237 + 2.00834i −2.81613 − 2.81613i − 5.51401i 0.142985 + 5.32359i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 188.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 385.2.j.d ✓ 40
5.c odd 4 1 inner 385.2.j.d ✓ 40
7.b odd 2 1 inner 385.2.j.d ✓ 40
35.f even 4 1 inner 385.2.j.d ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.j.d ✓ 40 1.a even 1 1 trivial
385.2.j.d ✓ 40 5.c odd 4 1 inner
385.2.j.d ✓ 40 7.b odd 2 1 inner
385.2.j.d ✓ 40 35.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(385, [\chi])\):

\( T_{2}^{20} - 4 T_{2}^{17} + 68 T_{2}^{16} - 22 T_{2}^{15} + 8 T_{2}^{14} - 146 T_{2}^{13} + 1266 T_{2}^{12} + \cdots + 36 \) Copy content Toggle raw display
\( T_{3}^{40} + 278 T_{3}^{36} + 27925 T_{3}^{32} + 1277316 T_{3}^{28} + 27627508 T_{3}^{24} + 266161816 T_{3}^{20} + \cdots + 64 \) Copy content Toggle raw display