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Properties

Label 803.2.bk.a
Level $803$
Weight $2$
Character orbit 803.bk
Analytic conductor $6.412$
Analytic rank $0$
Dimension $1152$
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] = [803,2,Mod(3,803)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(803, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([48, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("803.3");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 803 = 11 \cdot 73 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 803.bk (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: \(6.41198728231\)
Analytic rank: \(0\)
Dimension: \(1152\)
Relative dimension: \(72\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 1152 q - 2 q^{2} + 134 q^{4} - 8 q^{5} - 42 q^{6} - 8 q^{7} - 64 q^{8} + 272 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1152 q - 2 q^{2} + 134 q^{4} - 8 q^{5} - 42 q^{6} - 8 q^{7} - 64 q^{8} + 272 q^{9} - 32 q^{10} - 14 q^{11} - 84 q^{12} - 12 q^{13} - 12 q^{14} - 36 q^{15} + 134 q^{16} - 42 q^{17} - 28 q^{18} - 18 q^{19} - 2 q^{20} - 64 q^{21} - 28 q^{22} - 48 q^{23} + 42 q^{25} - 34 q^{26} - 92 q^{28} - 4 q^{29} + 36 q^{30} - 36 q^{31} - 148 q^{32} - 84 q^{33} + 54 q^{35} - 70 q^{36} - 18 q^{37} + 42 q^{38} + 76 q^{39} + 136 q^{40} - 38 q^{41} + 96 q^{42} - 40 q^{43} - 72 q^{44} - 68 q^{45} + 8 q^{47} - 6 q^{48} - 90 q^{50} - 78 q^{51} - 98 q^{52} + 10 q^{53} - 36 q^{54} + 4 q^{55} + 24 q^{56} - 78 q^{57} + 116 q^{58} + 76 q^{59} - 250 q^{60} + 6 q^{61} - 68 q^{62} + 12 q^{63} - 264 q^{64} - 272 q^{65} - 70 q^{66} - 48 q^{67} - 12 q^{68} + 12 q^{69} + 54 q^{71} - 16 q^{72} - 36 q^{73} - 104 q^{74} - 76 q^{75} + 2 q^{77} + 336 q^{78} - 66 q^{79} - 248 q^{80} - 268 q^{81} - 40 q^{82} - 114 q^{83} - 70 q^{84} - 180 q^{85} - 2 q^{86} - 116 q^{87} - 46 q^{88} - 64 q^{89} - 148 q^{90} - 18 q^{91} + 36 q^{92} - 112 q^{93} + 84 q^{94} + 158 q^{95} - 156 q^{96} - 72 q^{98} + 112 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −0.291495 − 2.77339i −2.91780 + 0.948052i −5.65042 + 1.20103i −2.28706 − 2.82429i 3.47984 + 7.81585i 0.313872 + 0.616008i 3.25452 + 10.0164i 5.18772 − 3.76910i −7.16618 + 7.16618i
3.2 −0.286069 − 2.72176i 3.02878 − 0.984110i −5.36985 + 1.14140i 0.647454 + 0.799539i −3.54495 − 7.96209i 1.68285 + 3.30278i 2.95135 + 9.08332i 5.77798 − 4.19795i 1.99094 − 1.99094i
3.3 −0.282319 − 2.68609i 1.34201 − 0.436047i −5.17906 + 1.10084i −1.94648 − 2.40371i −1.55014 − 3.48166i −0.584678 − 1.14750i 2.74987 + 8.46324i −0.816185 + 0.592993i −5.90703 + 5.90703i
3.4 −0.275556 − 2.62174i 0.878282 − 0.285371i −4.84129 + 1.02905i 1.81600 + 2.24257i −0.990184 − 2.22399i −0.874452 − 1.71621i 2.40269 + 7.39472i −1.73711 + 1.26208i 5.37903 − 5.37903i
3.5 −0.273091 − 2.59829i −0.722200 + 0.234657i −4.72022 + 1.00331i 0.115429 + 0.142543i 0.806932 + 1.81240i 1.57963 + 3.10020i 2.28128 + 7.02104i −1.96054 + 1.42442i 0.338846 − 0.338846i
3.6 −0.254166 − 2.41822i −0.682160 + 0.221647i −3.82691 + 0.813436i −0.444506 − 0.548919i 0.709375 + 1.59328i −1.68537 − 3.30772i 1.43696 + 4.42252i −2.01084 + 1.46096i −1.21443 + 1.21443i
3.7 −0.251413 − 2.39203i −1.79889 + 0.584493i −3.70233 + 0.786954i 0.852503 + 1.05275i 1.85039 + 4.15605i −0.518556 − 1.01772i 1.32673 + 4.08326i 0.467306 − 0.339518i 2.30389 − 2.30389i
3.8 −0.241127 − 2.29417i 0.514021 − 0.167016i −3.24879 + 0.690552i −1.44939 − 1.78985i −0.507107 − 1.13898i 1.61543 + 3.17047i 0.941931 + 2.89896i −2.19073 + 1.59166i −3.75673 + 3.75673i
3.9 −0.232607 − 2.21310i 2.24984 − 0.731017i −2.88743 + 0.613742i 1.38806 + 1.71411i −2.14114 − 4.80909i −1.41222 − 2.77163i 0.654601 + 2.01465i 2.10034 − 1.52599i 3.47063 − 3.47063i
3.10 −0.230224 − 2.19043i 2.94535 − 0.957003i −2.78870 + 0.592758i −1.26904 − 1.56713i −2.77434 − 6.23128i −1.29781 − 2.54710i 0.579203 + 1.78260i 5.33220 − 3.87407i −3.14053 + 3.14053i
3.11 −0.227301 − 2.16263i −2.32672 + 0.755996i −2.66899 + 0.567312i 0.776019 + 0.958303i 2.16380 + 4.85998i 1.01189 + 1.98594i 0.489611 + 1.50687i 2.41503 − 1.75462i 1.89606 − 1.89606i
3.12 −0.212057 − 2.01759i −1.00833 + 0.327626i −2.06940 + 0.439864i −2.11739 − 2.61476i 0.874837 + 1.96492i −1.79665 − 3.52612i 0.0724896 + 0.223100i −1.51766 + 1.10265i −4.82650 + 4.82650i
3.13 −0.208618 − 1.98487i 0.853689 − 0.277380i −1.93989 + 0.412336i 2.45300 + 3.02921i −0.728658 − 1.63659i 1.45379 + 2.85323i −0.0103445 − 0.0318372i −1.77521 + 1.28976i 5.50084 − 5.50084i
3.14 −0.208121 − 1.98014i 1.38126 − 0.448799i −1.92134 + 0.408394i −1.49752 − 1.84928i −1.17615 − 2.64168i −0.231747 − 0.454829i −0.0219852 − 0.0676634i −0.720591 + 0.523540i −3.35016 + 3.35016i
3.15 −0.207085 − 1.97028i −2.78926 + 0.906286i −1.88282 + 0.400206i −0.606443 − 0.748895i 2.36325 + 5.30794i −0.00714023 − 0.0140135i −0.0459863 − 0.141531i 4.53157 − 3.29238i −1.34995 + 1.34995i
3.16 −0.184259 − 1.75311i −1.10662 + 0.359563i −1.08315 + 0.230232i 2.40651 + 2.97179i 0.834260 + 1.87378i 0.587833 + 1.15369i −0.486248 − 1.49652i −1.33173 + 0.967555i 4.76646 − 4.76646i
3.17 −0.172592 − 1.64210i 2.27618 − 0.739574i −0.710408 + 0.151002i 1.04153 + 1.28618i −1.60730 − 3.61006i 1.40675 + 2.76090i −0.649893 − 2.00016i 2.20696 − 1.60345i 1.93228 − 1.93228i
3.18 −0.153529 − 1.46073i 1.53227 − 0.497863i −0.153864 + 0.0327049i −0.726055 − 0.896603i −0.962490 − 2.16179i 0.389717 + 0.764864i −0.836358 − 2.57404i −0.327082 + 0.237639i −1.19822 + 1.19822i
3.19 −0.152546 − 1.45138i −0.608293 + 0.197646i −0.126947 + 0.0269834i 1.75587 + 2.16832i 0.379654 + 0.852716i −1.80158 − 3.53581i −0.843416 − 2.59577i −2.09609 + 1.52290i 2.87921 − 2.87921i
3.20 −0.150468 − 1.43161i −2.03329 + 0.660657i −0.0705665 + 0.0149994i 0.574210 + 0.709090i 1.25175 + 2.81147i −0.494812 − 0.971124i −0.857565 − 2.63931i 1.27076 − 0.923260i 0.928739 − 0.928739i
See next 80 embeddings (of 1152 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.72
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
73.h even 12 1 inner
803.bk even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 803.2.bk.a ✓ 1152
11.c even 5 1 inner 803.2.bk.a ✓ 1152
73.h even 12 1 inner 803.2.bk.a ✓ 1152
803.bk even 60 1 inner 803.2.bk.a ✓ 1152
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
803.2.bk.a ✓ 1152 1.a even 1 1 trivial
803.2.bk.a ✓ 1152 11.c even 5 1 inner
803.2.bk.a ✓ 1152 73.h even 12 1 inner
803.2.bk.a ✓ 1152 803.bk even 60 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(803, [\chi])\).