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Properties

Label 802.2.e.b
Level $802$
Weight $2$
Character orbit 802.e
Analytic conductor $6.404$
Analytic rank $0$
Dimension $68$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [802,2,Mod(45,802)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(802, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("802.45");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 802 = 2 \cdot 401 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 802.e (of order \(8\), 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: \(6.40400224211\)
Analytic rank: \(0\)
Dimension: \(68\)
Relative dimension: \(17\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 68 q + 4 q^{6} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q + 4 q^{6} + 20 q^{9} - 4 q^{10} + 12 q^{11} - 4 q^{12} - 8 q^{13} + 8 q^{14} - 4 q^{15} - 68 q^{16} + 4 q^{17} + 20 q^{19} - 40 q^{21} - 16 q^{23} + 20 q^{25} - 8 q^{26} - 12 q^{27} + 16 q^{29} - 4 q^{30} - 8 q^{31} + 8 q^{33} + 4 q^{34} + 16 q^{35} - 20 q^{36} + 12 q^{37} - 20 q^{38} + 40 q^{39} + 4 q^{40} + 24 q^{41} - 40 q^{42} + 40 q^{43} - 12 q^{44} + 28 q^{45} + 16 q^{46} + 4 q^{47} - 8 q^{50} + 24 q^{51} + 4 q^{52} + 12 q^{53} + 12 q^{54} - 24 q^{55} - 4 q^{58} + 4 q^{59} + 4 q^{60} - 20 q^{61} + 8 q^{62} - 16 q^{63} + 36 q^{65} - 8 q^{66} + 48 q^{67} + 8 q^{68} - 16 q^{71} - 68 q^{72} + 40 q^{73} - 8 q^{74} - 96 q^{75} - 16 q^{77} + 32 q^{78} + 28 q^{79} - 16 q^{82} + 16 q^{83} - 4 q^{84} + 16 q^{85} - 32 q^{86} + 8 q^{87} + 8 q^{88} - 20 q^{89} - 4 q^{91} + 8 q^{92} - 24 q^{95} - 4 q^{96} + 24 q^{97} + 24 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1 0.707107 + 0.707107i −3.13632 − 1.29911i 1.00000i 3.57036 −1.29911 − 3.13632i 3.40854 − 3.40854i −0.707107 + 0.707107i 6.02752 + 6.02752i 2.52463 + 2.52463i
45.2 0.707107 + 0.707107i −2.60306 − 1.07822i 1.00000i 0.573607 −1.07822 − 2.60306i −2.49076 + 2.49076i −0.707107 + 0.707107i 3.49205 + 3.49205i 0.405601 + 0.405601i
45.3 0.707107 + 0.707107i −2.43681 − 1.00936i 1.00000i −3.88924 −1.00936 − 2.43681i −0.0905001 + 0.0905001i −0.707107 + 0.707107i 2.79793 + 2.79793i −2.75011 − 2.75011i
45.4 0.707107 + 0.707107i −2.25577 − 0.934372i 1.00000i −0.971927 −0.934372 − 2.25577i 0.630025 − 0.630025i −0.707107 + 0.707107i 2.09414 + 2.09414i −0.687256 − 0.687256i
45.5 0.707107 + 0.707107i −1.62035 − 0.671173i 1.00000i 2.02594 −0.671173 − 1.62035i −1.35018 + 1.35018i −0.707107 + 0.707107i 0.0537565 + 0.0537565i 1.43256 + 1.43256i
45.6 0.707107 + 0.707107i −1.04368 − 0.432308i 1.00000i −1.91544 −0.432308 − 1.04368i 2.26723 − 2.26723i −0.707107 + 0.707107i −1.21893 − 1.21893i −1.35442 − 1.35442i
45.7 0.707107 + 0.707107i −0.865572 − 0.358532i 1.00000i 1.42002 −0.358532 − 0.865572i 2.24756 − 2.24756i −0.707107 + 0.707107i −1.50065 − 1.50065i 1.00411 + 1.00411i
45.8 0.707107 + 0.707107i 0.0377837 + 0.0156505i 1.00000i −2.89099 0.0156505 + 0.0377837i 1.93372 − 1.93372i −0.707107 + 0.707107i −2.12014 − 2.12014i −2.04424 − 2.04424i
45.9 0.707107 + 0.707107i 0.101711 + 0.0421303i 1.00000i −2.13856 0.0421303 + 0.101711i −1.65263 + 1.65263i −0.707107 + 0.707107i −2.11275 − 2.11275i −1.51219 − 1.51219i
45.10 0.707107 + 0.707107i 0.373451 + 0.154688i 1.00000i −0.903310 0.154688 + 0.373451i −0.943205 + 0.943205i −0.707107 + 0.707107i −2.00578 − 2.00578i −0.638737 − 0.638737i
45.11 0.707107 + 0.707107i 0.868578 + 0.359777i 1.00000i 3.24260 0.359777 + 0.868578i 1.17950 − 1.17950i −0.707107 + 0.707107i −1.49633 − 1.49633i 2.29286 + 2.29286i
45.12 0.707107 + 0.707107i 1.29760 + 0.537484i 1.00000i 2.57737 0.537484 + 1.29760i −3.54527 + 3.54527i −0.707107 + 0.707107i −0.726439 − 0.726439i 1.82248 + 1.82248i
45.13 0.707107 + 0.707107i 1.99059 + 0.824530i 1.00000i 0.448980 0.824530 + 1.99059i −1.76081 + 1.76081i −0.707107 + 0.707107i 1.16128 + 1.16128i 0.317476 + 0.317476i
45.14 0.707107 + 0.707107i 2.08581 + 0.863972i 1.00000i 0.323043 0.863972 + 2.08581i 3.25849 − 3.25849i −0.707107 + 0.707107i 1.48285 + 1.48285i 0.228426 + 0.228426i
45.15 0.707107 + 0.707107i 2.28476 + 0.946378i 1.00000i −4.02154 0.946378 + 2.28476i −1.21285 + 1.21285i −0.707107 + 0.707107i 2.20317 + 2.20317i −2.84366 − 2.84366i
45.16 0.707107 + 0.707107i 2.73669 + 1.13357i 1.00000i 0.245522 1.13357 + 2.73669i 1.24807 − 1.24807i −0.707107 + 0.707107i 4.08315 + 4.08315i 0.173610 + 0.173610i
45.17 0.707107 + 0.707107i 2.89171 + 1.19779i 1.00000i 0.889344 1.19779 + 2.89171i −1.71273 + 1.71273i −0.707107 + 0.707107i 4.80599 + 4.80599i 0.628861 + 0.628861i
303.1 0.707107 − 0.707107i −3.13632 + 1.29911i − 1.00000i 3.57036 −1.29911 + 3.13632i 3.40854 + 3.40854i −0.707107 − 0.707107i 6.02752 − 6.02752i 2.52463 − 2.52463i
303.2 0.707107 − 0.707107i −2.60306 + 1.07822i − 1.00000i 0.573607 −1.07822 + 2.60306i −2.49076 − 2.49076i −0.707107 − 0.707107i 3.49205 − 3.49205i 0.405601 − 0.405601i
303.3 0.707107 − 0.707107i −2.43681 + 1.00936i − 1.00000i −3.88924 −1.00936 + 2.43681i −0.0905001 − 0.0905001i −0.707107 − 0.707107i 2.79793 − 2.79793i −2.75011 + 2.75011i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 45.17
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
401.e even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 802.2.e.b ✓ 68
401.e even 8 1 inner 802.2.e.b ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
802.2.e.b ✓ 68 1.a even 1 1 trivial
802.2.e.b ✓ 68 401.e even 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{68} - 10 T_{3}^{66} - 4 T_{3}^{65} + 50 T_{3}^{64} + 28 T_{3}^{63} + 66 T_{3}^{62} + \cdots + 207368 \) acting on \(S_{2}^{\mathrm{new}}(802, [\chi])\). Copy content Toggle raw display