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Properties

Label 806.2.cc.a
Level $806$
Weight $2$
Character orbit 806.cc
Analytic conductor $6.436$
Analytic rank $0$
Dimension $576$
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] = [806,2,Mod(21,806)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(806, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([15, 58]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("806.21");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 806 = 2 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 806.cc (of order \(60\), degree \(16\), 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: \(6.43594240292\)
Analytic rank: \(0\)
Dimension: \(576\)
Relative dimension: \(36\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 576 q + 4 q^{7} - 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 576 q + 4 q^{7} - 64 q^{9} + 8 q^{11} + 20 q^{13} - 24 q^{14} + 144 q^{16} - 16 q^{18} - 8 q^{21} + 12 q^{22} + 48 q^{26} + 60 q^{27} - 36 q^{28} + 36 q^{31} - 52 q^{33} - 56 q^{34} - 8 q^{35} + 60 q^{37} - 8 q^{39} - 40 q^{41} - 36 q^{42} - 12 q^{44} - 36 q^{45} - 40 q^{46} + 8 q^{47} + 52 q^{53} + 144 q^{57} - 4 q^{59} - 328 q^{63} - 76 q^{65} - 32 q^{66} - 48 q^{70} - 20 q^{71} + 16 q^{72} - 48 q^{73} + 48 q^{74} + 60 q^{76} + 40 q^{78} + 16 q^{79} + 216 q^{81} - 284 q^{83} + 32 q^{84} - 80 q^{85} + 156 q^{86} - 32 q^{87} - 220 q^{89} - 60 q^{91} + 44 q^{93} - 16 q^{94} + 248 q^{97} - 72 q^{98} + 240 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1 −0.156434 − 0.987688i −3.37142 + 0.354351i −0.951057 + 0.309017i −0.720130 + 2.68756i 0.877395 + 3.27448i −1.31254 + 0.852375i 0.453990 + 0.891007i 8.30650 − 1.76560i 2.76713 + 0.290837i
21.2 −0.156434 − 0.987688i −2.82310 + 0.296720i −0.951057 + 0.309017i −0.111558 + 0.416339i 0.734697 + 2.74193i 0.909083 − 0.590365i 0.453990 + 0.891007i 4.94742 − 1.05161i 0.428665 + 0.0450545i
21.3 −0.156434 − 0.987688i −2.52074 + 0.264940i −0.951057 + 0.309017i 0.516545 − 1.92777i 0.656009 + 2.44826i −0.0172225 + 0.0111844i 0.453990 + 0.891007i 3.34949 − 0.711955i −1.98484 − 0.208615i
21.4 −0.156434 − 0.987688i −2.10283 + 0.221017i −0.951057 + 0.309017i −0.362316 + 1.35218i 0.547251 + 2.04237i 3.63638 − 2.36150i 0.453990 + 0.891007i 1.43861 − 0.305787i 1.39221 + 0.146327i
21.5 −0.156434 − 0.987688i −1.42035 + 0.149285i −0.951057 + 0.309017i −0.809137 + 3.01974i 0.369638 + 1.37951i −3.93108 + 2.55287i 0.453990 + 0.891007i −0.939341 + 0.199663i 3.10914 + 0.326784i
21.6 −0.156434 − 0.987688i −1.05871 + 0.111275i −0.951057 + 0.309017i 0.886879 − 3.30988i 0.275523 + 1.02827i −3.07333 + 1.99584i 0.453990 + 0.891007i −1.82596 + 0.388121i −3.40787 − 0.358181i
21.7 −0.156434 − 0.987688i −1.04329 + 0.109654i −0.951057 + 0.309017i 0.778449 − 2.90521i 0.271511 + 1.01329i 2.22770 − 1.44669i 0.453990 + 0.891007i −1.85801 + 0.394932i −2.99122 − 0.314390i
21.8 −0.156434 − 0.987688i −0.941015 + 0.0989046i −0.951057 + 0.309017i −0.138671 + 0.517528i 0.244894 + 0.913957i −0.644986 + 0.418859i 0.453990 + 0.891007i −2.05872 + 0.437594i 0.532849 + 0.0560047i
21.9 −0.156434 − 0.987688i −0.0465632 + 0.00489399i −0.951057 + 0.309017i −0.0189851 + 0.0708534i 0.0121178 + 0.0452243i 4.05729 − 2.63484i 0.453990 + 0.891007i −2.93230 + 0.623279i 0.0729510 + 0.00766746i
21.10 −0.156434 − 0.987688i 0.240211 − 0.0252472i −0.951057 + 0.309017i 1.08515 − 4.04985i −0.0625136 − 0.233304i 1.33120 − 0.864489i 0.453990 + 0.891007i −2.87738 + 0.611606i −4.16975 − 0.438258i
21.11 −0.156434 − 0.987688i 0.625094 − 0.0657000i −0.951057 + 0.309017i −0.261455 + 0.975763i −0.162677 − 0.607120i 0.0763943 − 0.0496110i 0.453990 + 0.891007i −2.54802 + 0.541598i 1.00465 + 0.105593i
21.12 −0.156434 − 0.987688i 0.673006 − 0.0707358i −0.951057 + 0.309017i −0.900464 + 3.36058i −0.175146 − 0.653655i 0.178409 − 0.115860i 0.453990 + 0.891007i −2.48651 + 0.528524i 3.46007 + 0.363668i
21.13 −0.156434 − 0.987688i 1.18167 − 0.124198i −0.951057 + 0.309017i 0.534192 − 1.99363i −0.307523 − 1.14769i −0.497158 + 0.322858i 0.453990 + 0.891007i −1.55353 + 0.330213i −2.05265 − 0.215742i
21.14 −0.156434 − 0.987688i 1.52546 − 0.160332i −0.951057 + 0.309017i 0.0611000 − 0.228028i −0.396993 − 1.48160i −3.83494 + 2.49044i 0.453990 + 0.891007i −0.633120 + 0.134574i −0.234779 − 0.0246763i
21.15 −0.156434 − 0.987688i 2.42197 − 0.254559i −0.951057 + 0.309017i 0.434015 − 1.61977i −0.630304 − 2.35233i 2.69387 − 1.74942i 0.453990 + 0.891007i 2.86668 − 0.609332i −1.66772 − 0.175284i
21.16 −0.156434 − 0.987688i 2.48784 − 0.261482i −0.951057 + 0.309017i −0.716889 + 2.67546i −0.647446 − 2.41630i 3.17915 − 2.06457i 0.453990 + 0.891007i 3.18651 − 0.677314i 2.75467 + 0.289528i
21.17 −0.156434 − 0.987688i 3.03293 − 0.318773i −0.951057 + 0.309017i 0.568925 − 2.12326i −0.789303 − 2.94572i −1.43763 + 0.933608i 0.453990 + 0.891007i 6.16258 − 1.30990i −2.18611 − 0.229770i
21.18 −0.156434 − 0.987688i 3.13985 − 0.330012i −0.951057 + 0.309017i −0.825654 + 3.08138i −0.817130 − 3.04957i −2.14787 + 1.39484i 0.453990 + 0.891007i 6.81532 − 1.44864i 3.17261 + 0.333455i
21.19 0.156434 + 0.987688i −2.94499 + 0.309531i −0.951057 + 0.309017i 1.14728 − 4.28171i −0.766418 − 2.86031i −1.12508 + 0.730632i −0.453990 − 0.891007i 5.64271 − 1.19940i 4.40847 + 0.463349i
21.20 0.156434 + 0.987688i −2.67408 + 0.281057i −0.951057 + 0.309017i −0.624492 + 2.33064i −0.695915 − 2.59719i −1.95381 + 1.26882i −0.453990 − 0.891007i 4.13726 − 0.879402i −2.39964 − 0.252212i
See next 80 embeddings (of 576 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner
31.h odd 30 1 inner
403.cd even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 806.2.cc.a ✓ 576
13.d odd 4 1 inner 806.2.cc.a ✓ 576
31.h odd 30 1 inner 806.2.cc.a ✓ 576
403.cd even 60 1 inner 806.2.cc.a ✓ 576
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
806.2.cc.a ✓ 576 1.a even 1 1 trivial
806.2.cc.a ✓ 576 13.d odd 4 1 inner
806.2.cc.a ✓ 576 31.h odd 30 1 inner
806.2.cc.a ✓ 576 403.cd even 60 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(806, [\chi])\).