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Properties

Label 855.2.cg.d
Level $855$
Weight $2$
Character orbit 855.cg
Analytic conductor $6.827$
Analytic rank $0$
Dimension $216$
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] = [855,2,Mod(77,855)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(855, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([10, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("855.77");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 855.cg (of order \(12\), 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: \(6.82720937282\)
Analytic rank: \(0\)
Dimension: \(216\)
Relative dimension: \(54\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 216 q + 2 q^{3} - 16 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 216 q + 2 q^{3} - 16 q^{6} - 12 q^{11} + 18 q^{12} + 8 q^{15} + 84 q^{16} - 24 q^{20} - 8 q^{21} - 6 q^{25} - 64 q^{27} - 64 q^{30} - 60 q^{32} - 26 q^{33} + 208 q^{36} - 12 q^{37} - 12 q^{41} + 40 q^{42} - 28 q^{45} - 24 q^{46} - 24 q^{47} + 118 q^{48} - 78 q^{50} - 32 q^{51} - 48 q^{52} - 12 q^{55} - 48 q^{56} + 2 q^{57} + 12 q^{58} + 2 q^{60} - 12 q^{61} + 22 q^{63} + 54 q^{65} - 116 q^{66} - 6 q^{67} + 132 q^{68} - 146 q^{72} + 48 q^{73} - 44 q^{75} - 108 q^{76} - 18 q^{78} + 80 q^{81} - 48 q^{82} - 12 q^{83} - 24 q^{85} + 4 q^{87} - 24 q^{88} + 38 q^{90} + 24 q^{91} + 114 q^{92} - 12 q^{93} - 6 q^{95} - 80 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
77.1 −2.64102 − 0.707660i 1.17070 − 1.27650i 4.74217 + 2.73789i 0.409873 + 2.19818i −3.99518 + 2.54281i −0.267962 + 1.00005i −6.71995 − 6.71995i −0.258905 − 2.98881i 0.473082 − 6.09550i
77.2 −2.59002 − 0.693995i 1.55964 − 0.753337i 4.49454 + 2.59493i 1.13334 − 1.92758i −4.56232 + 0.868778i 0.346103 − 1.29167i −6.04805 − 6.04805i 1.86497 − 2.34987i −4.27309 + 4.20594i
77.3 −2.47168 − 0.662285i −1.65582 − 0.508201i 3.93854 + 2.27392i 0.983960 + 2.00794i 3.75608 + 2.35273i −1.02267 + 3.81664i −4.61004 − 4.61004i 2.48346 + 1.68297i −1.10221 − 5.61465i
77.4 −2.44481 − 0.655085i −1.73119 − 0.0544764i 3.81592 + 2.20312i −2.23378 − 0.101193i 4.19676 + 1.26726i 0.419789 − 1.56667i −4.30652 − 4.30652i 2.99406 + 0.188619i 5.39488 + 1.71071i
77.5 −2.43296 − 0.651911i 1.06794 + 1.36364i 3.76228 + 2.17215i −1.23569 − 1.86362i −1.70929 − 4.01388i −0.715538 + 2.67042i −4.17533 − 4.17533i −0.719003 + 2.91256i 1.79147 + 5.33968i
77.6 −2.29703 − 0.615488i 1.56849 + 0.734732i 3.16548 + 1.82759i 2.09266 + 0.787904i −3.15066 − 2.65309i 0.467645 − 1.74528i −2.78325 − 2.78325i 1.92034 + 2.30484i −4.32195 − 3.09784i
77.7 −2.11738 − 0.567349i −0.0390834 − 1.73161i 2.42935 + 1.40259i 2.22906 − 0.176837i −0.899674 + 3.68864i −0.276757 + 1.03287i −1.24804 − 1.24804i −2.99694 + 0.135354i −4.82010 − 0.890227i
77.8 −2.09712 − 0.561921i −0.564996 + 1.63731i 2.35009 + 1.35683i −0.843480 − 2.07088i 2.10490 − 3.11614i 0.666941 − 2.48906i −1.09560 − 1.09560i −2.36156 − 1.85015i 0.605205 + 4.81685i
77.9 −2.05611 − 0.550934i 0.277870 − 1.70962i 2.19202 + 1.26556i −2.14472 + 0.632602i −1.51322 + 3.36208i 1.10633 − 4.12887i −0.799441 − 0.799441i −2.84558 − 0.950102i 4.75830 − 0.119103i
77.10 −2.04033 − 0.546705i −1.68155 + 0.415189i 2.13202 + 1.23092i 1.43984 − 1.71080i 3.65791 + 0.0721900i 0.705599 − 2.63333i −0.689816 − 0.689816i 2.65524 − 1.39632i −3.87306 + 2.70344i
77.11 −1.97489 − 0.529171i 1.65843 + 0.499621i 1.88813 + 1.09011i −1.08514 + 1.95511i −3.01083 − 1.86429i 0.222194 − 0.829241i −0.260562 − 0.260562i 2.50076 + 1.65717i 3.17763 − 3.28692i
77.12 −1.65047 − 0.442243i −0.602717 + 1.62380i 0.796437 + 0.459823i −2.13299 + 0.671089i 1.71288 − 2.41350i −0.426638 + 1.59223i 1.30532 + 1.30532i −2.27347 − 1.95739i 3.81723 − 0.164316i
77.13 −1.64766 − 0.441490i −0.942188 − 1.45337i 0.787834 + 0.454856i −1.96831 + 1.06101i 0.910762 + 2.81063i −1.15771 + 4.32064i 1.31508 + 1.31508i −1.22456 + 2.73869i 3.71155 − 0.879201i
77.14 −1.45039 − 0.388632i 1.70559 − 0.301610i 0.220554 + 0.127337i −0.905405 − 2.04456i −2.59099 − 0.225392i −0.188852 + 0.704807i 1.85312 + 1.85312i 2.81806 − 1.02885i 0.518611 + 3.31729i
77.15 −1.29047 − 0.345781i 1.49735 − 0.870597i −0.186298 − 0.107559i −1.78443 + 1.34751i −2.23333 + 0.605725i 0.523616 − 1.95416i 2.09260 + 2.09260i 1.48412 − 2.60718i 2.76871 − 1.12191i
77.16 −1.20364 − 0.322513i −1.17588 + 1.27174i −0.387325 − 0.223622i 1.88620 + 1.20094i 1.82548 − 1.15147i −1.15446 + 4.30850i 2.15632 + 2.15632i −0.234623 − 2.99081i −1.88298 − 2.05381i
77.17 −1.11785 − 0.299526i −0.451940 + 1.67205i −0.572185 − 0.330351i 1.64321 − 1.51653i 1.00602 − 1.73373i −0.404226 + 1.50859i 2.17731 + 2.17731i −2.59150 − 1.51133i −2.29110 + 1.20307i
77.18 −1.05439 − 0.282523i −1.66826 − 0.465744i −0.700128 − 0.404219i 1.43036 + 1.71874i 1.62741 + 0.962399i 0.127927 − 0.477431i 2.16774 + 2.16774i 2.56616 + 1.55396i −1.02257 − 2.21634i
77.19 −1.03952 − 0.278539i 1.40632 + 1.01107i −0.729029 − 0.420905i 1.13869 − 1.92442i −1.18028 − 1.44274i −0.670261 + 2.50145i 2.16257 + 2.16257i 0.955479 + 2.84378i −1.71972 + 1.68330i
77.20 −0.999654 − 0.267856i 0.173873 + 1.72330i −0.804490 − 0.464473i −0.273731 + 2.21925i 0.287784 − 1.76928i 1.17867 − 4.39885i 2.14339 + 2.14339i −2.93954 + 0.599272i 0.868077 − 2.14516i
See next 80 embeddings (of 216 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 77.54
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
9.d odd 6 1 inner
45.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 855.2.cg.d ✓ 216
5.c odd 4 1 inner 855.2.cg.d ✓ 216
9.d odd 6 1 inner 855.2.cg.d ✓ 216
45.l even 12 1 inner 855.2.cg.d ✓ 216
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
855.2.cg.d ✓ 216 1.a even 1 1 trivial
855.2.cg.d ✓ 216 5.c odd 4 1 inner
855.2.cg.d ✓ 216 9.d odd 6 1 inner
855.2.cg.d ✓ 216 45.l even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{216} - 399 T_{2}^{212} + 12 T_{2}^{211} - 84 T_{2}^{209} + 86592 T_{2}^{208} + \cdots + 31\!\cdots\!21 \) acting on \(S_{2}^{\mathrm{new}}(855, [\chi])\). Copy content Toggle raw display