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Properties

Label 403.2.f.c
Level $403$
Weight $2$
Character orbit 403.f
Analytic conductor $3.218$
Analytic rank $0$
Dimension $36$
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: \(36\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 5 q^{2} - 19 q^{4} + 12 q^{5} - 6 q^{6} - 4 q^{7} + 30 q^{8} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 5 q^{2} - 19 q^{4} + 12 q^{5} - 6 q^{6} - 4 q^{7} + 30 q^{8} - 22 q^{9} - 3 q^{10} - 17 q^{11} + 8 q^{12} + 8 q^{13} + 4 q^{14} - 12 q^{15} - 17 q^{16} + 7 q^{17} - 14 q^{18} - 4 q^{19} - 15 q^{20} + 28 q^{21} - 30 q^{22} + 4 q^{23} - 48 q^{24} + 24 q^{25} + 25 q^{26} - 6 q^{27} - q^{28} + 15 q^{29} + 35 q^{30} + 36 q^{31} - 35 q^{32} - 17 q^{33} - 6 q^{34} - 17 q^{35} - 35 q^{36} - 3 q^{37} + 14 q^{38} + 3 q^{39} - 2 q^{40} - 44 q^{41} + 57 q^{42} + 4 q^{43} + 64 q^{44} + 5 q^{45} - 13 q^{46} + 24 q^{47} - 89 q^{48} - 44 q^{49} - 84 q^{50} - 28 q^{51} + 50 q^{52} - 28 q^{53} - 21 q^{54} + 29 q^{55} + 11 q^{56} + 32 q^{57} + 49 q^{58} - 11 q^{59} + 54 q^{60} - 4 q^{61} - 5 q^{62} - 9 q^{63} + 34 q^{64} - 5 q^{65} + 52 q^{66} - 16 q^{67} + 53 q^{68} + 4 q^{69} - 44 q^{70} - 5 q^{71} - 27 q^{72} + 64 q^{73} + q^{74} - 98 q^{75} - 42 q^{76} - 22 q^{77} + 143 q^{78} - 6 q^{79} - 2 q^{80} + 10 q^{81} + 22 q^{82} + 36 q^{83} + 38 q^{84} + 2 q^{85} + 84 q^{86} - 34 q^{87} - 69 q^{88} - 54 q^{89} - 32 q^{90} - 43 q^{91} - 86 q^{92} + 44 q^{94} - 2 q^{95} + 170 q^{96} - 28 q^{97} - 29 q^{98} + 154 q^{99}+O(q^{100}) \) 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 −1.39316 − 2.41302i −1.13229 − 1.96118i −2.88179 + 4.99141i −3.38130 −3.15492 + 5.46447i −0.923334 + 1.59926i 10.4866 −1.06415 + 1.84316i 4.71069 + 8.15916i
94.2 −1.33157 − 2.30635i −1.12120 − 1.94198i −2.54616 + 4.41008i 4.27464 −2.98592 + 5.17177i 1.44001 − 2.49418i 8.23530 −1.01419 + 1.75663i −5.69198 − 9.85880i
94.3 −1.19633 − 2.07211i 1.01324 + 1.75498i −1.86241 + 3.22579i 3.38808 2.42434 − 4.19908i −2.00851 + 3.47885i 4.12693 −0.553314 + 0.958368i −4.05326 − 7.02046i
94.4 −1.00222 − 1.73589i −0.398982 − 0.691057i −1.00888 + 1.74743i −2.51167 −0.799734 + 1.38518i 2.36824 − 4.10191i 0.0356005 1.18163 − 2.04664i 2.51724 + 4.35999i
94.5 −0.944860 − 1.63655i 0.990948 + 1.71637i −0.785520 + 1.36056i −1.89350 1.87261 − 3.24346i −0.456547 + 0.790763i −0.810613 −0.463958 + 0.803598i 1.78910 + 3.09881i
94.6 −0.748359 − 1.29620i 0.472887 + 0.819064i −0.120081 + 0.207987i 2.74524 0.707778 − 1.22591i 1.47907 − 2.56182i −2.63398 1.05276 − 1.82343i −2.05442 − 3.55837i
94.7 −0.685864 − 1.18795i −0.770575 − 1.33468i 0.0591818 − 0.102506i 3.15415 −1.05702 + 1.83081i −1.71049 + 2.96266i −2.90582 0.312428 − 0.541141i −2.16332 − 3.74698i
94.8 −0.683087 − 1.18314i 1.43249 + 2.48115i 0.0667843 − 0.115674i −1.11791 1.95703 − 3.38968i −1.05496 + 1.82724i −2.91483 −2.60407 + 4.51038i 0.763627 + 1.32264i
94.9 −0.0573543 − 0.0993406i −1.57377 − 2.72585i 0.993421 − 1.72066i −1.85062 −0.180525 + 0.312678i −1.95903 + 3.39313i −0.457325 −3.45350 + 5.98164i 0.106141 + 0.183841i
94.10 −0.0496225 − 0.0859486i 0.816280 + 1.41384i 0.995075 − 1.72352i −0.352370 0.0810117 − 0.140316i 2.05703 − 3.56288i −0.396002 0.167373 − 0.289899i 0.0174854 + 0.0302857i
94.11 0.105154 + 0.182132i 0.0693408 + 0.120102i 0.977885 − 1.69375i 1.89472 −0.0145830 + 0.0252584i −2.28636 + 3.96010i 0.831932 1.49038 − 2.58142i 0.199238 + 0.345090i
94.12 0.137984 + 0.238996i −0.732250 − 1.26829i 0.961921 − 1.66610i −1.37475 0.202078 − 0.350009i 0.239780 − 0.415311i 1.08286 0.427621 − 0.740662i −0.189694 − 0.328559i
94.13 0.497248 + 0.861259i 1.12055 + 1.94084i 0.505488 − 0.875531i 0.441402 −1.11438 + 1.93016i 0.363263 − 0.629189i 2.99441 −1.01125 + 1.75153i 0.219486 + 0.380161i
94.14 0.646873 + 1.12042i −1.49698 − 2.59285i 0.163110 − 0.282515i 2.87150 1.93671 − 3.35449i 0.577201 − 0.999742i 3.00954 −2.98190 + 5.16480i 1.85749 + 3.21727i
94.15 0.809863 + 1.40272i −0.263517 − 0.456425i −0.311757 + 0.539980i −1.44443 0.426825 − 0.739283i 1.00072 − 1.73330i 2.22953 1.36112 − 2.35752i −1.16979 − 2.02614i
94.16 1.06256 + 1.84040i 0.793339 + 1.37410i −1.25806 + 2.17902i −1.24766 −1.68594 + 2.92013i −1.62133 + 2.80822i −1.09681 0.241228 − 0.417818i −1.32571 − 2.29620i
94.17 1.12604 + 1.95037i 1.66294 + 2.88030i −1.53595 + 2.66034i −0.820579 −3.74509 + 6.48669i 1.96872 − 3.40992i −2.41402 −4.03076 + 6.98148i −0.924008 − 1.60043i
94.18 1.20670 + 2.09007i −0.882453 − 1.52845i −1.91225 + 3.31212i 3.22507 2.12971 − 3.68877i −1.47346 + 2.55211i −4.40325 −0.0574470 + 0.0995011i 3.89169 + 6.74061i
373.1 −1.39316 + 2.41302i −1.13229 + 1.96118i −2.88179 − 4.99141i −3.38130 −3.15492 − 5.46447i −0.923334 − 1.59926i 10.4866 −1.06415 − 1.84316i 4.71069 − 8.15916i
373.2 −1.33157 + 2.30635i −1.12120 + 1.94198i −2.54616 − 4.41008i 4.27464 −2.98592 − 5.17177i 1.44001 + 2.49418i 8.23530 −1.01419 − 1.75663i −5.69198 + 9.85880i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 94.18
Significant digits:
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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.c ✓ 36
13.c even 3 1 inner 403.2.f.c ✓ 36
13.c even 3 1 5239.2.a.p 18
13.e even 6 1 5239.2.a.o 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.f.c ✓ 36 1.a even 1 1 trivial
403.2.f.c ✓ 36 13.c even 3 1 inner
5239.2.a.o 18 13.e even 6 1
5239.2.a.p 18 13.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} + 5 T_{2}^{35} + 40 T_{2}^{34} + 135 T_{2}^{33} + 700 T_{2}^{32} + 1951 T_{2}^{31} + 7983 T_{2}^{30} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(403, [\chi])\). Copy content Toggle raw display