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Properties

Label 403.2.h.b
Level $403$
Weight $2$
Character orbit 403.h
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(118,403)]
 
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([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("403.118");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 403 = 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 403.h (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.

\(\operatorname{Tr}(f)(q) = \) \( 34 q + 6 q^{2} - 2 q^{3} + 34 q^{4} - 5 q^{5} - 2 q^{7} + 36 q^{8} - 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 34 q + 6 q^{2} - 2 q^{3} + 34 q^{4} - 5 q^{5} - 2 q^{7} + 36 q^{8} - 23 q^{9} - 7 q^{10} - 5 q^{11} - 28 q^{12} - 17 q^{13} - 7 q^{14} + 8 q^{15} + 18 q^{16} - 8 q^{17} + 6 q^{18} + 3 q^{19} - 8 q^{20} + 13 q^{21} + 12 q^{22} - 14 q^{23} - 6 q^{24} - 26 q^{25} - 3 q^{26} + 28 q^{27} - 7 q^{28} - 18 q^{29} - 60 q^{30} - 9 q^{31} + 58 q^{32} - 14 q^{33} - 15 q^{34} + 50 q^{35} - 49 q^{36} - 6 q^{37} + 2 q^{38} + 4 q^{39} - 29 q^{40} - 5 q^{41} + 8 q^{42} - q^{43} - 22 q^{44} + 13 q^{45} + 34 q^{46} + 16 q^{47} - 49 q^{48} + 3 q^{49} - 35 q^{51} - 17 q^{52} + 30 q^{53} - 2 q^{54} + 21 q^{55} - 7 q^{56} + 34 q^{58} - 9 q^{59} - 38 q^{60} - 28 q^{61} - 62 q^{62} + 88 q^{63} + 56 q^{64} - 5 q^{65} + 140 q^{66} - 31 q^{67} - 39 q^{68} + 5 q^{69} + 56 q^{70} + q^{71} - 32 q^{72} - 10 q^{73} - 39 q^{74} - 2 q^{75} - 16 q^{76} + 76 q^{77} - 23 q^{79} - 22 q^{80} - 29 q^{81} - 10 q^{82} + 3 q^{83} + 52 q^{84} - 32 q^{85} + 4 q^{86} + 18 q^{87} - 10 q^{88} + 26 q^{89} + 35 q^{90} + 4 q^{91} - 94 q^{92} - 41 q^{93} + 70 q^{94} + 28 q^{95} - 23 q^{96} + 32 q^{97} - 38 q^{98} - 70 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
118.1 −2.49327 −1.44053 − 2.49508i 4.21641 0.778554 − 1.34849i 3.59165 + 6.22091i −0.277831 − 0.481217i −5.52611 −2.65028 + 4.59042i −1.94115 + 3.36216i
118.2 −2.00723 0.235463 + 0.407833i 2.02898 −0.859670 + 1.48899i −0.472628 − 0.818616i 1.82223 + 3.15619i −0.0581753 1.38911 − 2.40602i 1.72556 − 2.98875i
118.3 −1.85162 0.240159 + 0.415967i 1.42848 0.854909 − 1.48075i −0.444682 − 0.770212i −0.531962 − 0.921385i 1.05824 1.38465 − 2.39828i −1.58296 + 2.74177i
118.4 −1.72349 −1.49143 − 2.58324i 0.970414 −1.96711 + 3.40713i 2.57047 + 4.45218i 0.142448 + 0.246727i 1.77448 −2.94874 + 5.10737i 3.39029 − 5.87216i
118.5 −1.51349 1.68586 + 2.92000i 0.290659 1.24235 − 2.15182i −2.55154 − 4.41940i −1.03520 − 1.79302i 2.58707 −4.18427 + 7.24738i −1.88029 + 3.25676i
118.6 −0.837386 0.883556 + 1.53036i −1.29878 −2.21305 + 3.83311i −0.739877 − 1.28150i −1.99716 − 3.45918i 2.76236 −0.0613409 + 0.106246i 1.85318 − 3.20980i
118.7 −0.708536 −0.837990 − 1.45144i −1.49798 1.10081 − 1.90665i 0.593746 + 1.02840i 1.50089 + 2.59962i 2.47844 0.0955444 − 0.165488i −0.779961 + 1.35093i
118.8 −0.291491 −0.965220 − 1.67181i −1.91503 −0.817652 + 1.41622i 0.281353 + 0.487317i −0.566002 − 0.980344i 1.14120 −0.363299 + 0.629252i 0.238338 − 0.412814i
118.9 −0.272181 0.850363 + 1.47287i −1.92592 1.58639 − 2.74770i −0.231453 − 0.400888i 1.18195 + 2.04720i 1.06856 0.0537640 − 0.0931220i −0.431784 + 0.747872i
118.10 0.680422 0.896149 + 1.55218i −1.53703 −0.204382 + 0.354001i 0.609760 + 1.05613i 1.47430 + 2.55357i −2.40667 −0.106167 + 0.183887i −0.139066 + 0.240870i
118.11 1.28826 1.07551 + 1.86284i −0.340382 −1.69851 + 2.94190i 1.38554 + 2.39982i 0.278041 + 0.481581i −3.01502 −0.813438 + 1.40892i −2.18812 + 3.78994i
118.12 1.35626 0.0933728 + 0.161726i −0.160553 1.08899 − 1.88618i 0.126638 + 0.219343i −1.43912 − 2.49263i −2.93028 1.48256 − 2.56787i 1.47695 − 2.55816i
118.13 1.55799 −1.19850 − 2.07586i 0.427341 −0.908804 + 1.57410i −1.86725 − 3.23417i −1.80502 − 3.12639i −2.45019 −1.37279 + 2.37775i −1.41591 + 2.45243i
118.14 2.14110 −0.301124 − 0.521562i 2.58432 0.249792 − 0.432653i −0.644738 − 1.11672i 2.01488 + 3.48987i 1.25109 1.31865 − 2.28397i 0.534831 − 0.926355i
118.15 2.35132 1.34746 + 2.33387i 3.52869 −0.0458962 + 0.0794945i 3.16831 + 5.48767i −2.00349 − 3.47014i 3.59444 −2.13130 + 3.69152i −0.107916 + 0.186917i
118.16 2.53634 −1.60552 − 2.78084i 4.43304 1.34893 − 2.33641i −4.07215 − 7.05318i 0.661783 + 1.14624i 6.17102 −3.65540 + 6.33133i 3.42134 − 5.92593i
118.17 2.78699 −0.467575 − 0.809863i 5.76734 −2.03564 + 3.52583i −1.30313 − 2.25709i −0.420736 − 0.728737i 10.4996 1.06275 − 1.84073i −5.67332 + 9.82648i
222.1 −2.49327 −1.44053 + 2.49508i 4.21641 0.778554 + 1.34849i 3.59165 − 6.22091i −0.277831 + 0.481217i −5.52611 −2.65028 − 4.59042i −1.94115 − 3.36216i
222.2 −2.00723 0.235463 − 0.407833i 2.02898 −0.859670 − 1.48899i −0.472628 + 0.818616i 1.82223 − 3.15619i −0.0581753 1.38911 + 2.40602i 1.72556 + 2.98875i
222.3 −1.85162 0.240159 − 0.415967i 1.42848 0.854909 + 1.48075i −0.444682 + 0.770212i −0.531962 + 0.921385i 1.05824 1.38465 + 2.39828i −1.58296 − 2.74177i
See all 34 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 118.17
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.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.h.b ✓ 34
31.c even 3 1 inner 403.2.h.b ✓ 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.h.b ✓ 34 1.a even 1 1 trivial
403.2.h.b ✓ 34 31.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{17} - 3 T_{2}^{16} - 21 T_{2}^{15} + 62 T_{2}^{14} + 182 T_{2}^{13} - 514 T_{2}^{12} - 854 T_{2}^{11} + \cdots + 75 \) acting on \(S_{2}^{\mathrm{new}}(403, [\chi])\). Copy content Toggle raw display