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Properties

Label 385.2.br.a
Level $385$
Weight $2$
Character orbit 385.br
Analytic conductor $3.074$
Analytic rank $0$
Dimension $352$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [385,2,Mod(19,385)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(385, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([15, 25, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("385.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 385 = 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 385.br (of order \(30\), 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: \(3.07424047782\)
Analytic rank: \(0\)
Dimension: \(352\)
Relative dimension: \(44\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 352 q + 34 q^{4} - 15 q^{5} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 352 q + 34 q^{4} - 15 q^{5} + 36 q^{9} - 6 q^{11} - 18 q^{14} - 36 q^{15} + 30 q^{16} - 30 q^{19} - 30 q^{24} + 23 q^{25} - 54 q^{26} - 40 q^{29} - 35 q^{30} - 54 q^{31} - 25 q^{35} - 52 q^{36} + 30 q^{39} - 105 q^{40} - 22 q^{44} - 60 q^{45} - 110 q^{46} - 58 q^{49} - 20 q^{50} + 50 q^{51} + 68 q^{56} + 6 q^{59} + 15 q^{60} - 120 q^{64} + 150 q^{66} - 53 q^{70} + 8 q^{71} - 80 q^{74} - 27 q^{75} - 10 q^{79} - 195 q^{80} + 60 q^{81} + 10 q^{84} - 130 q^{85} - 2 q^{86} + 12 q^{89} - 98 q^{91} - 180 q^{94} - 50 q^{95} + 420 q^{96} - 80 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −0.281576 − 2.67902i −1.71723 + 1.90717i −5.14156 + 1.09287i −1.91881 − 1.14812i 5.59288 + 4.06347i 1.06218 + 2.42318i 2.71072 + 8.34275i −0.374857 − 3.56652i −2.53554 + 5.46381i
19.2 −0.273741 − 2.60447i 0.975686 − 1.08361i −4.75202 + 1.01007i −0.724041 − 2.11560i −3.08931 − 2.24451i 0.628667 − 2.56998i 2.31301 + 7.11872i 0.0913401 + 0.869043i −5.31181 + 2.46487i
19.3 −0.269995 − 2.56883i 0.139100 − 0.154486i −4.56969 + 0.971317i −0.524153 + 2.17377i −0.434405 − 0.315613i 2.64489 + 0.0675964i 2.13257 + 6.56338i 0.309068 + 2.94059i 5.72555 + 0.759553i
19.4 −0.261326 − 2.48635i −1.59367 + 1.76995i −4.15737 + 0.883677i 2.22050 − 0.263424i 4.81718 + 3.49989i −0.530566 − 2.59201i 1.73845 + 5.35040i −0.279351 − 2.65785i −1.23524 − 5.45210i
19.5 −0.244493 − 2.32620i −0.466080 + 0.517634i −3.39512 + 0.721656i 0.939104 + 2.02931i 1.31807 + 0.957635i −2.63626 + 0.223935i 1.06321 + 3.27223i 0.262871 + 2.50105i 4.49096 − 2.68069i
19.6 −0.225022 − 2.14094i 1.20476 − 1.33802i −2.57669 + 0.547693i 2.21499 + 0.306285i −3.13571 − 2.27823i 1.87904 + 1.86258i 0.421929 + 1.29856i −0.0252675 − 0.240404i 0.157317 − 4.81109i
19.7 −0.216555 − 2.06038i 0.327349 − 0.363558i −2.24199 + 0.476549i −2.23601 − 0.0157350i −0.819958 − 0.595734i −1.91871 + 1.82169i 0.186987 + 0.575488i 0.288568 + 2.74555i 0.451800 + 4.61045i
19.8 −0.206687 − 1.96650i 2.20344 − 2.44717i −1.86810 + 0.397077i −1.16405 − 1.90918i −5.26777 − 3.82726i 0.781609 + 2.52766i −0.0550939 − 0.169562i −0.819897 − 7.80079i −3.51381 + 2.68371i
19.9 −0.180882 − 1.72098i 1.31570 − 1.46123i −0.972762 + 0.206767i −1.87057 + 1.22514i −2.75274 − 1.99998i −1.06289 − 2.42286i −0.537686 − 1.65483i −0.0905484 − 0.861511i 2.44679 + 2.99761i
19.10 −0.179465 − 1.70750i −0.440487 + 0.489211i −0.927046 + 0.197050i 0.861570 − 2.06342i 0.914378 + 0.664335i 2.34062 − 1.23349i −0.558270 − 1.71818i 0.268287 + 2.55258i −3.67791 − 1.10082i
19.11 −0.176400 − 1.67833i −1.91114 + 2.12253i −0.829396 + 0.176294i −2.21481 + 0.307585i 3.89945 + 2.83312i 0.539829 − 2.59009i −0.600796 − 1.84906i −0.539116 − 5.12935i 0.906924 + 3.66294i
19.12 −0.176072 − 1.67521i −1.32325 + 1.46962i −0.819037 + 0.174092i −0.0695854 − 2.23498i 2.69492 + 1.95797i −2.63511 + 0.237109i −0.605191 − 1.86259i −0.0952042 − 0.905808i −3.73182 + 0.510088i
19.13 −0.159232 − 1.51499i −1.90564 + 2.11643i −0.313542 + 0.0666453i −0.0987093 + 2.23389i 3.50980 + 2.55002i −0.492876 + 2.59944i −0.790579 − 2.43315i −0.534215 − 5.08271i 3.40003 − 0.206162i
19.14 −0.134303 − 1.27781i 1.10860 − 1.23123i 0.341540 − 0.0725965i 1.64716 − 1.51224i −1.72216 − 1.25122i −2.53513 − 0.757050i −0.932713 − 2.87060i 0.0266633 + 0.253684i −2.15357 − 1.90165i
19.15 −0.116832 − 1.11158i −0.722264 + 0.802156i 0.734336 − 0.156088i 2.20850 + 0.350010i 0.976043 + 0.709137i 0.301796 + 2.62848i −0.950076 − 2.92403i 0.191797 + 1.82483i 0.131041 − 2.49582i
19.16 −0.0952599 − 0.906337i 1.71042 − 1.89962i 1.14392 − 0.243148i −0.472530 + 2.18557i −1.88463 − 1.36926i 2.63221 + 0.267330i −0.892577 − 2.74707i −0.369413 − 3.51473i 2.02588 + 0.220074i
19.17 −0.0884049 − 0.841117i −0.259966 + 0.288721i 1.25663 − 0.267106i 1.61698 + 1.54447i 0.265830 + 0.193137i 0.970994 − 2.46113i −0.858462 − 2.64207i 0.297808 + 2.83345i 1.15613 − 1.49661i
19.18 −0.0699306 − 0.665345i −0.0577669 + 0.0641567i 1.51850 − 0.322767i −2.12279 + 0.702695i 0.0467260 + 0.0339484i 1.66476 + 2.05635i −0.734412 − 2.26029i 0.312806 + 2.97615i 0.615982 + 1.36325i
19.19 −0.0405924 − 0.386211i −2.10285 + 2.33545i 1.80878 − 0.384469i 1.25987 − 1.84736i 0.987335 + 0.717341i 2.63025 + 0.285948i −0.461915 − 1.42163i −0.718768 − 6.83862i −0.764610 − 0.411587i
19.20 −0.0361650 − 0.344087i 1.55035 − 1.72184i 1.83921 − 0.390936i 1.45146 + 1.70096i −0.648529 − 0.471184i −2.35174 + 1.21215i −0.414859 − 1.27681i −0.247555 − 2.35533i 0.532785 − 0.560943i
See next 80 embeddings (of 352 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.d odd 6 1 inner
11.d odd 10 1 inner
35.i odd 6 1 inner
55.h odd 10 1 inner
77.n even 30 1 inner
385.br even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 385.2.br.a ✓ 352
5.b even 2 1 inner 385.2.br.a ✓ 352
7.d odd 6 1 inner 385.2.br.a ✓ 352
11.d odd 10 1 inner 385.2.br.a ✓ 352
35.i odd 6 1 inner 385.2.br.a ✓ 352
55.h odd 10 1 inner 385.2.br.a ✓ 352
77.n even 30 1 inner 385.2.br.a ✓ 352
385.br even 30 1 inner 385.2.br.a ✓ 352
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.br.a ✓ 352 1.a even 1 1 trivial
385.2.br.a ✓ 352 5.b even 2 1 inner
385.2.br.a ✓ 352 7.d odd 6 1 inner
385.2.br.a ✓ 352 11.d odd 10 1 inner
385.2.br.a ✓ 352 35.i odd 6 1 inner
385.2.br.a ✓ 352 55.h odd 10 1 inner
385.2.br.a ✓ 352 77.n even 30 1 inner
385.2.br.a ✓ 352 385.br even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(385, [\chi])\).