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Properties

Label 775.2.bf.e
Level $775$
Weight $2$
Character orbit 775.bf
Analytic conductor $6.188$
Analytic rank $0$
Dimension $72$
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] = [775,2,Mod(349,775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("775.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.bf (of order \(10\), degree \(4\), not 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: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 20 q^{4} - 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 20 q^{4} - 12 q^{6} - 2 q^{11} + 30 q^{14} + 4 q^{16} - 22 q^{19} + 14 q^{21} - 20 q^{24} - 56 q^{26} - 50 q^{29} - 18 q^{31} - 36 q^{34} - 48 q^{36} - 14 q^{39} + 2 q^{41} - 22 q^{44} + 72 q^{46} - 14 q^{49} - 68 q^{51} + 4 q^{54} - 72 q^{56} + 50 q^{59} + 108 q^{61} + 120 q^{64} - 16 q^{66} + 88 q^{69} + 64 q^{71} + 112 q^{74} - 30 q^{76} - 28 q^{79} - 92 q^{81} - 54 q^{84} + 50 q^{86} + 12 q^{89} - 128 q^{91} - 44 q^{94} + 72 q^{96} + 92 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1 −2.32935 − 0.756851i −1.70597 + 0.554304i 3.23501 + 2.35037i 0 4.39333 −1.45441 + 2.00183i −2.87734 − 3.96032i 0.176040 − 0.127900i 0
349.2 −2.22226 − 0.722056i 2.36255 − 0.767639i 2.79904 + 2.03362i 0 −5.80447 −3.06349 + 4.21653i −2.00493 − 2.75955i 2.56532 − 1.86381i 0
349.3 −2.09193 − 0.679709i 0.393934 − 0.127997i 2.29613 + 1.66823i 0 −0.911081 1.78335 − 2.45457i −1.08366 − 1.49152i −2.28825 + 1.66251i 0
349.4 −2.04473 − 0.664374i 1.02137 − 0.331864i 2.12151 + 1.54137i 0 −2.30891 2.58487 − 3.55777i −0.786446 − 1.08245i −1.49398 + 1.08544i 0
349.5 −1.82648 − 0.593461i 2.72397 − 0.885070i 1.36582 + 0.992323i 0 −5.50054 −0.217697 + 0.299634i 0.351923 + 0.484381i 4.20959 − 3.05845i 0
349.6 −0.809091 − 0.262890i 1.36007 − 0.441915i −1.03252 − 0.750167i 0 −1.21660 0.649960 − 0.894593i 1.63828 + 2.25490i −0.772536 + 0.561280i 0
349.7 −0.730690 − 0.237416i −0.621004 + 0.201777i −1.14049 − 0.828616i 0 0.501667 −2.68327 + 3.69321i 1.53980 + 2.11936i −2.08212 + 1.51275i 0
349.8 −0.666437 − 0.216539i −2.76572 + 0.898636i −1.22078 − 0.886952i 0 2.03777 −0.325102 + 0.447464i 1.44528 + 1.98926i 4.41459 − 3.20739i 0
349.9 −0.431417 − 0.140176i 1.08477 − 0.352463i −1.45156 − 1.05462i 0 −0.517395 −1.20825 + 1.66302i 1.01166 + 1.39243i −1.37455 + 0.998672i 0
349.10 0.431417 + 0.140176i −1.08477 + 0.352463i −1.45156 − 1.05462i 0 −0.517395 1.20825 − 1.66302i −1.01166 − 1.39243i −1.37455 + 0.998672i 0
349.11 0.666437 + 0.216539i 2.76572 − 0.898636i −1.22078 − 0.886952i 0 2.03777 0.325102 − 0.447464i −1.44528 − 1.98926i 4.41459 − 3.20739i 0
349.12 0.730690 + 0.237416i 0.621004 − 0.201777i −1.14049 − 0.828616i 0 0.501667 2.68327 − 3.69321i −1.53980 − 2.11936i −2.08212 + 1.51275i 0
349.13 0.809091 + 0.262890i −1.36007 + 0.441915i −1.03252 − 0.750167i 0 −1.21660 −0.649960 + 0.894593i −1.63828 − 2.25490i −0.772536 + 0.561280i 0
349.14 1.82648 + 0.593461i −2.72397 + 0.885070i 1.36582 + 0.992323i 0 −5.50054 0.217697 − 0.299634i −0.351923 − 0.484381i 4.20959 − 3.05845i 0
349.15 2.04473 + 0.664374i −1.02137 + 0.331864i 2.12151 + 1.54137i 0 −2.30891 −2.58487 + 3.55777i 0.786446 + 1.08245i −1.49398 + 1.08544i 0
349.16 2.09193 + 0.679709i −0.393934 + 0.127997i 2.29613 + 1.66823i 0 −0.911081 −1.78335 + 2.45457i 1.08366 + 1.49152i −2.28825 + 1.66251i 0
349.17 2.22226 + 0.722056i −2.36255 + 0.767639i 2.79904 + 2.03362i 0 −5.80447 3.06349 − 4.21653i 2.00493 + 2.75955i 2.56532 − 1.86381i 0
349.18 2.32935 + 0.756851i 1.70597 − 0.554304i 3.23501 + 2.35037i 0 4.39333 1.45441 − 2.00183i 2.87734 + 3.96032i 0.176040 − 0.127900i 0
374.1 −1.53032 + 2.10630i −1.04725 − 1.44142i −1.47660 − 4.54452i 0 4.63870 −1.74246 + 0.566161i 6.87960 + 2.23532i −0.0539012 + 0.165891i 0
374.2 −1.43706 + 1.97795i 0.393939 + 0.542210i −1.22909 − 3.78276i 0 −1.63858 −1.26835 + 0.412110i 4.59795 + 1.49396i 0.788247 − 2.42597i 0
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
31.d even 5 1 inner
155.n even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.bf.e 72
5.b even 2 1 inner 775.2.bf.e 72
5.c odd 4 1 775.2.k.f ✓ 36
5.c odd 4 1 775.2.k.g yes 36
31.d even 5 1 inner 775.2.bf.e 72
155.n even 10 1 inner 775.2.bf.e 72
155.s odd 20 1 775.2.k.f ✓ 36
155.s odd 20 1 775.2.k.g yes 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
775.2.k.f ✓ 36 5.c odd 4 1
775.2.k.f ✓ 36 155.s odd 20 1
775.2.k.g yes 36 5.c odd 4 1
775.2.k.g yes 36 155.s odd 20 1
775.2.bf.e 72 1.a even 1 1 trivial
775.2.bf.e 72 5.b even 2 1 inner
775.2.bf.e 72 31.d even 5 1 inner
775.2.bf.e 72 155.n even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{72} - 28 T_{2}^{70} + 442 T_{2}^{68} - 5252 T_{2}^{66} + 52679 T_{2}^{64} - 453058 T_{2}^{62} + \cdots + 492884401 \) acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\). Copy content Toggle raw display