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Properties

Label 3229.2.a.d
Level $3229$
Weight $2$
Character orbit 3229.a
Self dual yes
Analytic conductor $25.784$
Analytic rank $0$
Dimension $142$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3229,2,Mod(1,3229)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3229, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3229.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3229 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3229.a (trivial)

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: yes
Analytic conductor: \(25.7836948127\)
Analytic rank: \(0\)
Dimension: \(142\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 142 q + 11 q^{2} + 28 q^{3} + 153 q^{4} + 40 q^{5} + 19 q^{6} + 15 q^{7} + 27 q^{8} + 168 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 142 q + 11 q^{2} + 28 q^{3} + 153 q^{4} + 40 q^{5} + 19 q^{6} + 15 q^{7} + 27 q^{8} + 168 q^{9} + 10 q^{10} + 107 q^{11} + 48 q^{12} + 7 q^{13} + 47 q^{14} + 46 q^{15} + 171 q^{16} + 48 q^{17} + 20 q^{18} + 83 q^{19} + 114 q^{20} + 31 q^{21} + 5 q^{22} + 40 q^{23} + 59 q^{24} + 158 q^{25} + 104 q^{26} + 94 q^{27} + 10 q^{28} + 34 q^{29} + q^{30} + 45 q^{31} + 50 q^{32} + 34 q^{33} + 44 q^{34} + 100 q^{35} + 190 q^{36} + 13 q^{37} + 48 q^{38} + 41 q^{39} - q^{40} + 74 q^{41} - 11 q^{42} + 48 q^{43} + 177 q^{44} + 92 q^{45} - 4 q^{46} + 135 q^{47} + 51 q^{48} + 183 q^{49} + 35 q^{50} + 75 q^{51} - 20 q^{52} + 53 q^{53} + 42 q^{54} + 61 q^{55} + 95 q^{56} + 11 q^{57} - 10 q^{58} + 311 q^{59} + 48 q^{60} + 23 q^{61} + 21 q^{62} + 57 q^{63} + 169 q^{64} + 42 q^{65} + 29 q^{66} + 45 q^{67} + 113 q^{68} + 44 q^{69} - 50 q^{70} + 147 q^{71} - 17 q^{72} + 22 q^{73} + 54 q^{74} + 173 q^{75} + 104 q^{76} + 68 q^{77} - q^{78} + 30 q^{79} + 209 q^{80} + 226 q^{81} - 25 q^{82} + 259 q^{83} + 22 q^{84} - 24 q^{85} + 131 q^{86} + 48 q^{87} - 17 q^{88} + 205 q^{89} + 4 q^{90} + 115 q^{91} + 33 q^{92} - 14 q^{93} - 8 q^{94} + 78 q^{95} + 85 q^{96} + 10 q^{97} + 14 q^{98} + 272 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.80557 −1.91231 5.87123 −1.40753 5.36511 −3.47665 −10.8610 0.656916 3.94893
1.2 −2.79320 1.21125 5.80199 3.01940 −3.38328 4.82705 −10.6197 −1.53287 −8.43381
1.3 −2.76974 0.117337 5.67148 3.30782 −0.324993 −3.65409 −10.1691 −2.98623 −9.16181
1.4 −2.64138 −3.31018 4.97689 2.46856 8.74343 1.79722 −7.86309 7.95727 −6.52039
1.5 −2.60818 2.05747 4.80261 −3.06960 −5.36626 −2.38584 −7.30970 1.23319 8.00606
1.6 −2.59427 2.51111 4.73025 1.07629 −6.51450 0.965766 −7.08300 3.30566 −2.79219
1.7 −2.58777 3.34924 4.69655 4.28161 −8.66707 −1.80920 −6.97806 8.21743 −11.0798
1.8 −2.57241 −2.32463 4.61728 1.65817 5.97989 −3.21269 −6.73273 2.40388 −4.26548
1.9 −2.56265 −2.47906 4.56715 −0.296995 6.35296 2.24177 −6.57871 3.14576 0.761093
1.10 −2.51559 −0.841388 4.32822 0.739321 2.11659 −0.183704 −5.85685 −2.29207 −1.85983
1.11 −2.45540 0.921824 4.02898 −2.64428 −2.26345 0.327263 −4.98196 −2.15024 6.49277
1.12 −2.44230 0.407055 3.96483 1.52022 −0.994152 −3.30449 −4.79872 −2.83431 −3.71284
1.13 −2.40724 0.0392651 3.79478 −1.12689 −0.0945202 −2.18915 −4.32047 −2.99846 2.71270
1.14 −2.34388 2.00001 3.49377 −1.38011 −4.68778 0.225012 −3.50122 1.00003 3.23481
1.15 −2.30132 1.45170 3.29608 4.02369 −3.34082 0.643991 −2.98270 −0.892576 −9.25980
1.16 −2.30101 3.33318 3.29463 −1.88388 −7.66967 −3.89658 −2.97895 8.11008 4.33481
1.17 −2.28023 −3.17199 3.19944 −2.75291 7.23286 −2.42052 −2.73499 7.06152 6.27726
1.18 −2.27439 −1.43888 3.17287 0.657732 3.27258 5.28608 −2.66757 −0.929628 −1.49594
1.19 −2.22495 3.25186 2.95042 1.96415 −7.23525 3.64746 −2.11464 7.57462 −4.37015
1.20 −2.21202 2.64340 2.89303 −1.11191 −5.84725 3.01444 −1.97539 3.98757 2.45956
See next 80 embeddings (of 142 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.142
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3229\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3229.2.a.d ✓ 142
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3229.2.a.d ✓ 142 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{142} - 11 T_{2}^{141} - 158 T_{2}^{140} + 2158 T_{2}^{139} + 10966 T_{2}^{138} + \cdots + 20544503888 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3229))\). Copy content Toggle raw display