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Properties

Label 130.3.r.a
Level $130$
Weight $3$
Character orbit 130.r
Analytic conductor $3.542$
Analytic rank $0$
Dimension $28$
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] = [130,3,Mod(17,130)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(130, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("130.17");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 130 = 2 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 130.r (of order \(12\), degree \(4\), 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: \(3.54224343668\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(7\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 14 q^{2} + 8 q^{5} + 4 q^{7} - 56 q^{8} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 14 q^{2} + 8 q^{5} + 4 q^{7} - 56 q^{8} - 48 q^{9} - 12 q^{10} - 12 q^{11} - 24 q^{13} + 10 q^{15} + 56 q^{16} + 50 q^{17} - 92 q^{18} - 36 q^{19} + 8 q^{20} + 12 q^{22} + 68 q^{23} - 6 q^{25} + 42 q^{26} - 24 q^{27} - 8 q^{28} + 186 q^{29} + 12 q^{30} + 56 q^{32} - 164 q^{33} - 20 q^{34} - 18 q^{35} + 92 q^{36} + 24 q^{37} + 72 q^{38} + 156 q^{39} - 20 q^{40} + 48 q^{41} - 204 q^{42} + 80 q^{43} + 26 q^{45} - 96 q^{46} + 104 q^{47} + 96 q^{49} + 18 q^{50} - 336 q^{51} - 96 q^{52} + 126 q^{53} + 24 q^{54} - 502 q^{55} - 16 q^{56} + 160 q^{57} - 130 q^{58} + 80 q^{59} + 136 q^{60} - 336 q^{61} - 108 q^{62} + 396 q^{63} + 160 q^{65} - 64 q^{66} - 32 q^{67} + 100 q^{68} + 264 q^{69} + 200 q^{70} - 60 q^{71} + 188 q^{72} - 286 q^{73} + 6 q^{74} - 684 q^{75} + 24 q^{76} + 424 q^{77} + 128 q^{78} + 40 q^{80} + 86 q^{81} + 42 q^{82} - 64 q^{83} - 96 q^{84} + 356 q^{85} + 384 q^{87} + 24 q^{88} - 348 q^{89} - 294 q^{90} + 192 q^{91} + 80 q^{92} - 336 q^{93} + 432 q^{94} - 446 q^{95} - 86 q^{97} - 82 q^{98} + 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0.366025 + 1.36603i −1.48198 − 5.53084i −1.73205 + 1.00000i 4.80922 − 1.36799i 7.01282 − 4.04885i −9.14020 − 2.44911i −2.00000 − 2.00000i −20.5996 + 11.8932i 3.62900 + 6.06880i
17.2 0.366025 + 1.36603i −0.768948 − 2.86975i −1.73205 + 1.00000i 0.194205 + 4.99623i 3.63870 − 2.10081i 7.21829 + 1.93414i −2.00000 − 2.00000i 0.150026 − 0.0866177i −6.75389 + 2.09403i
17.3 0.366025 + 1.36603i −0.575387 − 2.14737i −1.73205 + 1.00000i −4.99982 + 0.0418782i 2.72276 − 1.57199i −6.51256 − 1.74504i −2.00000 − 2.00000i 3.51408 − 2.02886i −1.88727 − 6.81456i
17.4 0.366025 + 1.36603i −0.307866 − 1.14897i −1.73205 + 1.00000i 0.471217 − 4.97775i 1.45684 − 0.841105i 6.02734 + 1.61502i −2.00000 − 2.00000i 6.56888 − 3.79254i 6.97220 − 1.17829i
17.5 0.366025 + 1.36603i 0.780467 + 2.91274i −1.73205 + 1.00000i 0.179287 + 4.99678i −3.69321 + 2.13227i −6.82767 − 1.82947i −2.00000 − 2.00000i −0.0807020 + 0.0465933i −6.76011 + 2.07386i
17.6 0.366025 + 1.36603i 0.957104 + 3.57196i −1.73205 + 1.00000i 4.99095 − 0.300686i −4.52907 + 2.61486i 9.12999 + 2.44637i −2.00000 − 2.00000i −4.04863 + 2.33748i 2.23756 + 6.70771i
17.7 0.366025 + 1.36603i 1.39661 + 5.21223i −1.73205 + 1.00000i −4.51108 − 2.15642i −6.60884 + 3.81562i 2.83686 + 0.760135i −2.00000 − 2.00000i −17.4226 + 10.0589i 1.29455 − 6.95156i
23.1 0.366025 − 1.36603i −1.48198 + 5.53084i −1.73205 − 1.00000i 4.80922 + 1.36799i 7.01282 + 4.04885i −9.14020 + 2.44911i −2.00000 + 2.00000i −20.5996 − 11.8932i 3.62900 − 6.06880i
23.2 0.366025 − 1.36603i −0.768948 + 2.86975i −1.73205 − 1.00000i 0.194205 − 4.99623i 3.63870 + 2.10081i 7.21829 − 1.93414i −2.00000 + 2.00000i 0.150026 + 0.0866177i −6.75389 − 2.09403i
23.3 0.366025 − 1.36603i −0.575387 + 2.14737i −1.73205 − 1.00000i −4.99982 − 0.0418782i 2.72276 + 1.57199i −6.51256 + 1.74504i −2.00000 + 2.00000i 3.51408 + 2.02886i −1.88727 + 6.81456i
23.4 0.366025 − 1.36603i −0.307866 + 1.14897i −1.73205 − 1.00000i 0.471217 + 4.97775i 1.45684 + 0.841105i 6.02734 − 1.61502i −2.00000 + 2.00000i 6.56888 + 3.79254i 6.97220 + 1.17829i
23.5 0.366025 − 1.36603i 0.780467 − 2.91274i −1.73205 − 1.00000i 0.179287 − 4.99678i −3.69321 − 2.13227i −6.82767 + 1.82947i −2.00000 + 2.00000i −0.0807020 − 0.0465933i −6.76011 − 2.07386i
23.6 0.366025 − 1.36603i 0.957104 − 3.57196i −1.73205 − 1.00000i 4.99095 + 0.300686i −4.52907 − 2.61486i 9.12999 − 2.44637i −2.00000 + 2.00000i −4.04863 − 2.33748i 2.23756 − 6.70771i
23.7 0.366025 − 1.36603i 1.39661 − 5.21223i −1.73205 − 1.00000i −4.51108 + 2.15642i −6.60884 − 3.81562i 2.83686 − 0.760135i −2.00000 + 2.00000i −17.4226 − 10.0589i 1.29455 + 6.95156i
43.1 −1.36603 + 0.366025i −4.18824 + 1.12224i 1.73205 − 1.00000i −2.95422 − 4.03393i 5.31047 − 3.06600i 0.174925 − 0.652828i −2.00000 + 2.00000i 8.48770 − 4.90038i 5.51206 + 4.42912i
43.2 −1.36603 + 0.366025i −4.08481 + 1.09452i 1.73205 − 1.00000i 4.55836 + 2.05460i 5.17934 − 2.99029i −1.49577 + 5.58230i −2.00000 + 2.00000i 7.69349 − 4.44184i −6.97887 − 1.13817i
43.3 −1.36603 + 0.366025i −0.217304 + 0.0582264i 1.73205 − 1.00000i 3.61019 + 3.45927i 0.275530 − 0.159078i 2.62366 − 9.79163i −2.00000 + 2.00000i −7.75040 + 4.47469i −6.19779 − 3.40403i
43.4 −1.36603 + 0.366025i 0.172010 − 0.0460900i 1.73205 − 1.00000i −2.29260 − 4.44342i −0.218100 + 0.125920i −0.0464055 + 0.173188i −2.00000 + 2.00000i −7.76677 + 4.48414i 4.75816 + 5.23067i
43.5 −1.36603 + 0.366025i 0.201090 − 0.0538820i 1.73205 − 1.00000i −3.92337 + 3.09954i −0.254972 + 0.147208i 0.627223 − 2.34083i −2.00000 + 2.00000i −7.75669 + 4.47833i 4.22491 − 5.67011i
43.6 −1.36603 + 0.366025i 3.95449 − 1.05960i 1.73205 − 1.00000i −0.205548 + 4.99577i −5.01410 + 2.89489i −3.38404 + 12.6294i −2.00000 + 2.00000i 6.72102 − 3.88038i −1.54780 − 6.89959i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.7
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.r odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 130.3.r.a ✓ 28
5.c odd 4 1 130.3.r.b yes 28
13.e even 6 1 130.3.r.b yes 28
65.r odd 12 1 inner 130.3.r.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.3.r.a ✓ 28 1.a even 1 1 trivial
130.3.r.a ✓ 28 65.r odd 12 1 inner
130.3.r.b yes 28 5.c odd 4 1
130.3.r.b yes 28 13.e even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{28} + 24 T_{3}^{26} + 8 T_{3}^{25} - 611 T_{3}^{24} + 576 T_{3}^{23} - 19240 T_{3}^{22} + \cdots + 53348416 \) acting on \(S_{3}^{\mathrm{new}}(130, [\chi])\). Copy content Toggle raw display