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Properties

Label 230.2.g.d
Level $230$
Weight $2$
Character orbit 230.g
Analytic conductor $1.837$
Analytic rank $0$
Dimension $30$
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] = [230,2,Mod(31,230)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(230, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("230.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 230.g (of order \(11\), degree \(10\), 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: \(1.83655924649\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(3\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 3 q^{2} - q^{3} - 3 q^{4} + 3 q^{5} - q^{6} + 8 q^{7} - 3 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 3 q^{2} - q^{3} - 3 q^{4} + 3 q^{5} - q^{6} + 8 q^{7} - 3 q^{8} + 12 q^{9} + 3 q^{10} - 3 q^{11} - q^{12} + 12 q^{13} - 3 q^{14} + q^{15} - 3 q^{16} - 26 q^{17} + 12 q^{18} - 25 q^{19} + 3 q^{20} - 11 q^{21} - 14 q^{22} + 14 q^{23} + 10 q^{24} - 3 q^{25} - 10 q^{26} - 52 q^{27} - 3 q^{28} + 4 q^{29} + q^{30} + 5 q^{31} - 3 q^{32} + 9 q^{33} + 7 q^{34} + 3 q^{35} + 12 q^{36} - 20 q^{37} + 19 q^{38} + 41 q^{39} + 3 q^{40} + 32 q^{41} + 22 q^{42} - 46 q^{43} - 3 q^{44} - 56 q^{45} + 3 q^{46} + 80 q^{47} - q^{48} - 41 q^{49} - 3 q^{50} - 34 q^{51} - 21 q^{52} + 29 q^{53} + 47 q^{54} + 3 q^{55} + 8 q^{56} - 33 q^{57} - 7 q^{58} - 3 q^{59} + q^{60} - q^{61} - 28 q^{62} - 8 q^{63} - 3 q^{64} + 21 q^{65} - 46 q^{66} - 63 q^{67} - 4 q^{68} + 78 q^{69} - 8 q^{70} + 55 q^{71} - 32 q^{72} - 36 q^{73} - 20 q^{74} - q^{75} - 3 q^{76} - 9 q^{77} - 80 q^{78} - 29 q^{79} + 3 q^{80} + 15 q^{81} - 23 q^{82} - 63 q^{83} + 4 q^{85} + 31 q^{86} + 63 q^{87} - 3 q^{88} - 95 q^{89} + 10 q^{90} - 12 q^{91} + 14 q^{92} + 126 q^{93} + 3 q^{94} - 19 q^{95} - q^{96} + 11 q^{97} + 58 q^{98} + 97 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −0.142315 + 0.989821i −0.844960 − 1.85020i −0.959493 − 0.281733i 0.654861 + 0.755750i 1.95162 − 0.573048i 1.31877 + 0.847525i 0.415415 − 0.909632i −0.744718 + 0.859450i −0.841254 + 0.540641i
31.2 −0.142315 + 0.989821i −0.0488649 − 0.106999i −0.959493 − 0.281733i 0.654861 + 0.755750i 0.112864 − 0.0331400i 1.43095 + 0.919619i 0.415415 − 0.909632i 1.95552 − 2.25679i −0.841254 + 0.540641i
31.3 −0.142315 + 0.989821i 1.30924 + 2.86684i −0.959493 − 0.281733i 0.654861 + 0.755750i −3.02398 + 0.887921i −0.493060 − 0.316870i 0.415415 − 0.909632i −4.54006 + 5.23951i −0.841254 + 0.540641i
41.1 −0.959493 − 0.281733i −1.80420 + 2.08215i 0.841254 + 0.540641i 0.142315 − 0.989821i 2.31772 − 1.48951i 1.84799 + 4.04654i −0.654861 − 0.755750i −0.653295 − 4.54377i −0.415415 + 0.909632i
41.2 −0.959493 − 0.281733i −0.858654 + 0.990939i 0.841254 + 0.540641i 0.142315 − 0.989821i 1.10305 − 0.708888i −1.50220 − 3.28936i −0.654861 − 0.755750i 0.182270 + 1.26772i −0.415415 + 0.909632i
41.3 −0.959493 − 0.281733i 2.00799 − 2.31734i 0.841254 + 0.540641i 0.142315 − 0.989821i −2.57952 + 1.65776i 0.414762 + 0.908202i −0.654861 − 0.755750i −0.911115 − 6.33694i −0.415415 + 0.909632i
71.1 0.841254 + 0.540641i −0.369448 − 2.56957i 0.415415 + 0.909632i 0.959493 + 0.281733i 1.07841 − 2.36139i 1.04688 − 1.20816i −0.142315 + 0.989821i −3.58770 + 1.05344i 0.654861 + 0.755750i
71.2 0.841254 + 0.540641i 0.00806642 + 0.0561032i 0.415415 + 0.909632i 0.959493 + 0.281733i −0.0235458 + 0.0515580i −3.27862 + 3.78373i −0.142315 + 0.989821i 2.87540 − 0.844293i 0.654861 + 0.755750i
71.3 0.841254 + 0.540641i 0.219067 + 1.52364i 0.415415 + 0.909632i 0.959493 + 0.281733i −0.639452 + 1.40020i 2.43457 − 2.80964i −0.142315 + 0.989821i 0.604987 − 0.177640i 0.654861 + 0.755750i
81.1 0.841254 − 0.540641i −0.369448 + 2.56957i 0.415415 − 0.909632i 0.959493 − 0.281733i 1.07841 + 2.36139i 1.04688 + 1.20816i −0.142315 − 0.989821i −3.58770 − 1.05344i 0.654861 − 0.755750i
81.2 0.841254 − 0.540641i 0.00806642 − 0.0561032i 0.415415 − 0.909632i 0.959493 − 0.281733i −0.0235458 − 0.0515580i −3.27862 − 3.78373i −0.142315 − 0.989821i 2.87540 + 0.844293i 0.654861 − 0.755750i
81.3 0.841254 − 0.540641i 0.219067 − 1.52364i 0.415415 − 0.909632i 0.959493 − 0.281733i −0.639452 − 1.40020i 2.43457 + 2.80964i −0.142315 − 0.989821i 0.604987 + 0.177640i 0.654861 − 0.755750i
101.1 −0.959493 + 0.281733i −1.80420 − 2.08215i 0.841254 − 0.540641i 0.142315 + 0.989821i 2.31772 + 1.48951i 1.84799 − 4.04654i −0.654861 + 0.755750i −0.653295 + 4.54377i −0.415415 − 0.909632i
101.2 −0.959493 + 0.281733i −0.858654 − 0.990939i 0.841254 − 0.540641i 0.142315 + 0.989821i 1.10305 + 0.708888i −1.50220 + 3.28936i −0.654861 + 0.755750i 0.182270 − 1.26772i −0.415415 − 0.909632i
101.3 −0.959493 + 0.281733i 2.00799 + 2.31734i 0.841254 − 0.540641i 0.142315 + 0.989821i −2.57952 − 1.65776i 0.414762 − 0.908202i −0.654861 + 0.755750i −0.911115 + 6.33694i −0.415415 − 0.909632i
121.1 0.415415 − 0.909632i −3.22429 − 0.946737i −0.654861 − 0.755750i −0.841254 + 0.540641i −2.20060 + 2.53963i −0.324589 + 2.25757i −0.959493 + 0.281733i 6.97598 + 4.48319i 0.142315 + 0.989821i
121.2 0.415415 − 0.909632i −0.732525 − 0.215089i −0.654861 − 0.755750i −0.841254 + 0.540641i −0.499953 + 0.576977i 0.575516 − 4.00280i −0.959493 + 0.281733i −2.03343 − 1.30681i 0.142315 + 0.989821i
121.3 0.415415 − 0.909632i 2.99732 + 0.880093i −0.654861 − 0.755750i −0.841254 + 0.540641i 2.04569 − 2.36086i −0.352734 + 2.45332i −0.959493 + 0.281733i 5.68562 + 3.65393i 0.142315 + 0.989821i
131.1 −0.654861 + 0.755750i −0.925739 + 0.594936i −0.142315 − 0.989821i −0.415415 − 0.909632i 0.156607 − 1.08923i 3.86758 − 1.13562i 0.841254 + 0.540641i −0.743202 + 1.62739i 0.959493 + 0.281733i
131.2 −0.654861 + 0.755750i −0.578669 + 0.371888i −0.142315 − 0.989821i −0.415415 − 0.909632i 0.0978934 − 0.680863i −2.88525 + 0.847187i 0.841254 + 0.540641i −1.04969 + 2.29850i 0.959493 + 0.281733i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.2.g.d ✓ 30
23.c even 11 1 inner 230.2.g.d ✓ 30
23.c even 11 1 5290.2.a.bk 15
23.d odd 22 1 5290.2.a.bl 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.g.d ✓ 30 1.a even 1 1 trivial
230.2.g.d ✓ 30 23.c even 11 1 inner
5290.2.a.bk 15 23.c even 11 1
5290.2.a.bl 15 23.d odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{30} + T_{3}^{29} - T_{3}^{28} + 18 T_{3}^{27} + 52 T_{3}^{26} + 39 T_{3}^{25} + 16 T_{3}^{24} + \cdots + 1024 \) acting on \(S_{2}^{\mathrm{new}}(230, [\chi])\). Copy content Toggle raw display