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Properties

Label 403.2.f.b
Level $403$
Weight $2$
Character orbit 403.f
Analytic conductor $3.218$
Analytic rank $0$
Dimension $34$
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] = [403,2,Mod(94,403)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(403, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("403.94");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 403 = 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 403.f (of order \(3\), degree \(2\), 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: \(3.21797120146\)
Analytic rank: \(0\)
Dimension: \(34\)
Relative dimension: \(17\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 34 q + 4 q^{2} - 20 q^{4} - 14 q^{5} + 6 q^{6} + 6 q^{7} - 12 q^{8} - 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 34 q + 4 q^{2} - 20 q^{4} - 14 q^{5} + 6 q^{6} + 6 q^{7} - 12 q^{8} - 17 q^{9} - 6 q^{10} + 13 q^{11} + 8 q^{12} - 3 q^{13} + 4 q^{15} - 34 q^{16} + 6 q^{17} + 24 q^{18} + 4 q^{19} + 28 q^{20} - 36 q^{21} + 34 q^{22} + 8 q^{23} + 40 q^{24} + 16 q^{25} - 26 q^{26} - 6 q^{27} + 21 q^{28} + 6 q^{29} - 19 q^{30} - 34 q^{31} + 6 q^{32} + 7 q^{33} - 48 q^{34} + 9 q^{35} + 14 q^{37} + 22 q^{38} - 21 q^{39} - 20 q^{40} + 43 q^{41} - 33 q^{42} - 18 q^{43} - 56 q^{44} + 26 q^{45} + 7 q^{46} - 12 q^{47} + 95 q^{48} + q^{49} + 44 q^{50} + 52 q^{51} - 24 q^{52} - 10 q^{53} + 27 q^{54} - 39 q^{55} - 39 q^{56} - 92 q^{57} + 8 q^{58} - q^{59} - 42 q^{60} + 19 q^{61} - 4 q^{62} + 5 q^{63} + 84 q^{64} - 32 q^{65} + 52 q^{66} + 10 q^{67} - 34 q^{68} - 32 q^{69} + 48 q^{70} + 35 q^{71} - 26 q^{72} - 22 q^{73} + 68 q^{74} + 62 q^{75} + 2 q^{76} + 42 q^{77} - 81 q^{78} + 2 q^{79} + 49 q^{80} - 37 q^{81} - 35 q^{82} - 48 q^{83} - 34 q^{84} - 13 q^{85} - 152 q^{86} + 22 q^{87} + 37 q^{88} + 42 q^{89} + 30 q^{90} - 39 q^{91} + 30 q^{92} - 42 q^{94} - 34 q^{95} - 66 q^{96} - 38 q^{97} + 8 q^{98} - 34 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
94.1 −1.36162 − 2.35840i 1.42643 + 2.47066i −2.70804 + 4.69046i −1.48160 3.88453 − 6.72821i 1.05061 − 1.81970i 9.30282 −2.56943 + 4.45038i 2.01739 + 3.49422i
94.2 −1.21866 − 2.11078i 0.165727 + 0.287048i −1.97025 + 3.41258i −0.0487430 0.403929 − 0.699625i 0.449852 − 0.779167i 4.72962 1.44507 − 2.50293i 0.0594010 + 0.102886i
94.3 −1.04303 − 1.80658i −0.534702 − 0.926131i −1.17583 + 2.03660i −1.36190 −1.11542 + 1.93197i −2.24250 + 3.88412i 0.733593 0.928187 − 1.60767i 1.42050 + 2.46038i
94.4 −0.806301 − 1.39655i −1.60737 − 2.78404i −0.300243 + 0.520037i −1.40674 −2.59204 + 4.48955i 1.36764 − 2.36882i −2.25686 −3.66725 + 6.35186i 1.13426 + 1.96459i
94.5 −0.579157 − 1.00313i −0.862835 − 1.49447i 0.329155 − 0.570113i 1.50607 −0.999433 + 1.73107i 0.722094 − 1.25070i −3.07916 0.0110326 − 0.0191090i −0.872252 − 1.51078i
94.6 −0.399878 − 0.692609i 0.607925 + 1.05296i 0.680195 − 1.17813i 2.33311 0.486192 − 0.842109i −0.0878879 + 0.152226i −2.68749 0.760853 − 1.31784i −0.932958 − 1.61593i
94.7 −0.0914916 − 0.158468i 0.877667 + 1.52016i 0.983259 − 1.70305i −3.62872 0.160598 − 0.278164i 1.84724 − 3.19952i −0.725806 −0.0405970 + 0.0703161i 0.331997 + 0.575036i
94.8 −0.0361865 − 0.0626768i 1.64544 + 2.84999i 0.997381 − 1.72751i 2.88575 0.119086 − 0.206262i −0.540773 + 0.936646i −0.289113 −3.91498 + 6.78094i −0.104425 − 0.180869i
94.9 0.0205421 + 0.0355799i −1.04665 − 1.81285i 0.999156 − 1.73059i 1.98920 0.0430008 − 0.0744796i 1.34392 − 2.32775i 0.164267 −0.690959 + 1.19678i 0.0408622 + 0.0707754i
94.10 0.466912 + 0.808715i 1.35923 + 2.35426i 0.563986 − 0.976853i −3.82118 −1.26928 + 2.19846i −2.11839 + 3.66916i 2.92098 −2.19501 + 3.80187i −1.78416 − 3.09025i
94.11 0.578965 + 1.00280i −0.235957 − 0.408690i 0.329599 − 0.570882i −3.25594 0.273222 − 0.473235i 0.0133059 − 0.0230465i 3.07917 1.38865 − 2.40521i −1.88508 − 3.26505i
94.12 0.600575 + 1.04023i 0.0302705 + 0.0524300i 0.278620 − 0.482584i 2.34059 −0.0363594 + 0.0629763i −0.828861 + 1.43563i 3.07163 1.49817 − 2.59490i 1.40570 + 2.43474i
94.13 0.948581 + 1.64299i −1.29227 − 2.23827i −0.799611 + 1.38497i −2.49083 2.45164 − 4.24636i −1.48428 + 2.57085i 0.760339 −1.83991 + 3.18681i −2.36275 − 4.09241i
94.14 1.11298 + 1.92773i −0.337463 − 0.584504i −1.47744 + 2.55900i 1.70140 0.751178 − 1.30108i 2.45364 − 4.24983i −2.12551 1.27224 − 2.20358i 1.89362 + 3.27984i
94.15 1.16646 + 2.02036i 0.786870 + 1.36290i −1.72124 + 2.98128i 2.09117 −1.83570 + 3.17953i −0.532652 + 0.922581i −3.36520 0.261670 − 0.453226i 2.43926 + 4.22492i
94.16 1.23104 + 2.13222i −1.38318 − 2.39574i −2.03090 + 3.51763i −1.87256 3.40549 − 5.89848i 1.18933 − 2.05998i −5.07632 −2.32637 + 4.02939i −2.30518 − 3.99270i
94.17 1.41028 + 2.44268i 0.400852 + 0.694296i −2.97779 + 5.15768i −2.47906 −1.13063 + 1.95831i 0.397709 − 0.688852i −11.1570 1.17864 − 2.04146i −3.49618 − 6.05556i
373.1 −1.36162 + 2.35840i 1.42643 − 2.47066i −2.70804 − 4.69046i −1.48160 3.88453 + 6.72821i 1.05061 + 1.81970i 9.30282 −2.56943 − 4.45038i 2.01739 − 3.49422i
373.2 −1.21866 + 2.11078i 0.165727 − 0.287048i −1.97025 − 3.41258i −0.0487430 0.403929 + 0.699625i 0.449852 + 0.779167i 4.72962 1.44507 + 2.50293i 0.0594010 − 0.102886i
373.3 −1.04303 + 1.80658i −0.534702 + 0.926131i −1.17583 − 2.03660i −1.36190 −1.11542 − 1.93197i −2.24250 − 3.88412i 0.733593 0.928187 + 1.60767i 1.42050 − 2.46038i
See all 34 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 94.17
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.f.b ✓ 34
13.c even 3 1 inner 403.2.f.b ✓ 34
13.c even 3 1 5239.2.a.m 17
13.e even 6 1 5239.2.a.n 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.f.b ✓ 34 1.a even 1 1 trivial
403.2.f.b ✓ 34 13.c even 3 1 inner
5239.2.a.m 17 13.c even 3 1
5239.2.a.n 17 13.e even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{34} - 4 T_{2}^{33} + 35 T_{2}^{32} - 104 T_{2}^{31} + 605 T_{2}^{30} - 1558 T_{2}^{29} + 6908 T_{2}^{28} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(403, [\chi])\). Copy content Toggle raw display