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Properties

Label 260.2.bc.c
Level $260$
Weight $2$
Character orbit 260.bc
Analytic conductor $2.076$
Analytic rank $0$
Dimension $144$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [260,2,Mod(19,260)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(260, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 6, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("260.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 260 = 2^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 260.bc (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: \(2.07611045255\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(36\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 12 q^{4} - 12 q^{5} - 80 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q - 12 q^{4} - 12 q^{5} - 80 q^{9} - 12 q^{10} - 32 q^{14} - 36 q^{16} + 18 q^{20} + 8 q^{21} - 28 q^{24} + 28 q^{26} - 6 q^{30} + 16 q^{34} + 24 q^{36} - 88 q^{40} - 72 q^{41} + 68 q^{45} + 64 q^{46} - 24 q^{49} - 34 q^{50} + 80 q^{54} + 96 q^{56} - 48 q^{60} - 8 q^{61} + 40 q^{66} - 24 q^{69} - 80 q^{70} - 24 q^{74} - 28 q^{76} - 92 q^{80} - 8 q^{81} + 84 q^{84} + 20 q^{85} - 52 q^{86} - 104 q^{89} - 52 q^{94} - 124 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.39033 + 0.258789i −0.828585 + 1.43515i 1.86606 − 0.719606i 2.18883 + 0.457193i 0.780609 − 2.20977i 1.23703 − 4.61665i −2.40822 + 1.48341i 0.126892 + 0.219784i −3.16152 − 0.0692064i
19.2 −1.36326 − 0.376194i 1.63494 − 2.83181i 1.71696 + 1.02570i −2.20931 − 0.344891i −3.29416 + 3.24543i −0.576390 + 2.15112i −1.95479 − 2.04421i −3.84608 − 6.66161i 2.88212 + 1.30131i
19.3 −1.35671 − 0.399163i −1.20070 + 2.07967i 1.68134 + 1.08310i −1.03858 + 1.98024i 2.45913 − 2.34224i −0.925353 + 3.45346i −1.84876 − 2.14058i −1.38334 − 2.39602i 2.19949 − 2.27205i
19.4 −1.33346 + 0.471049i 0.828585 − 1.43515i 1.55623 − 1.25625i 2.18883 + 0.457193i −0.428858 + 2.30402i −1.23703 + 4.61665i −1.48341 + 2.40822i 0.126892 + 0.219784i −3.13407 + 0.421397i
19.5 −1.32112 − 0.504618i 0.414936 − 0.718690i 1.49072 + 1.33332i 0.439941 + 2.19236i −0.910845 + 0.740092i 0.345572 − 1.28969i −1.29660 − 2.51373i 1.15566 + 2.00166i 0.525091 − 3.11838i
19.6 −1.23293 − 0.692736i 0.540386 − 0.935976i 1.04023 + 1.70819i 1.28074 − 1.83295i −1.31464 + 0.779649i 0.116775 − 0.435812i −0.0992103 − 2.82669i 0.915965 + 1.58650i −2.84881 + 1.37268i
19.7 −0.992521 + 1.00742i −1.63494 + 2.83181i −0.0298050 − 1.99978i −2.20931 − 0.344891i −1.23011 − 4.45771i 0.576390 − 2.15112i 2.04421 + 1.95479i −3.84608 − 6.66161i 2.54024 − 1.88340i
19.8 −0.975366 + 1.02404i 1.20070 − 2.07967i −0.0973226 − 1.99763i −1.03858 + 1.98024i 0.958547 + 3.25800i 0.925353 − 3.45346i 2.14058 + 1.84876i −1.38334 − 2.39602i −1.01485 − 2.99501i
19.9 −0.944171 − 1.05287i −1.41558 + 2.45186i −0.217082 + 1.98818i 2.22381 − 0.233813i 3.91805 − 0.824548i 0.196708 − 0.734126i 2.29827 − 1.64863i −2.50775 − 4.34355i −2.34583 − 2.12063i
19.10 −0.891815 + 1.09757i −0.414936 + 0.718690i −0.409332 − 1.95766i 0.439941 + 2.19236i −0.418769 − 1.09636i −0.345572 + 1.28969i 2.51373 + 1.29660i 1.15566 + 2.00166i −2.79862 − 1.47231i
19.11 −0.796363 − 1.16868i −0.640517 + 1.10941i −0.731612 + 1.86138i −2.15621 + 0.592253i 1.80662 − 0.134934i 0.819289 − 3.05763i 2.75798 − 0.627317i 0.679475 + 1.17689i 2.40928 + 2.04826i
19.12 −0.721381 + 1.21639i −0.540386 + 0.935976i −0.959219 − 1.75496i 1.28074 − 1.83295i −0.748690 − 1.33252i −0.116775 + 0.435812i 2.82669 + 0.0992103i 0.915965 + 1.58650i 1.30568 + 2.88014i
19.13 −0.691873 − 1.23341i −0.267386 + 0.463127i −1.04262 + 1.70673i −1.00857 − 1.99569i 0.756225 + 0.00937347i −1.03041 + 3.84553i 2.82647 + 0.105146i 1.35701 + 2.35041i −1.76371 + 2.62475i
19.14 −0.578520 − 1.29047i 1.22430 − 2.12055i −1.33063 + 1.49313i 1.79435 + 1.33428i −3.44479 − 0.353143i 0.498659 − 1.86102i 2.69663 + 0.853333i −1.49783 − 2.59432i 0.683776 − 3.08747i
19.15 −0.291240 + 1.38390i 1.41558 − 2.45186i −1.83036 − 0.806094i 2.22381 − 0.233813i 2.98086 + 2.67310i −0.196708 + 0.734126i 1.64863 − 2.29827i −2.50775 − 4.34355i −0.324088 + 3.14563i
19.16 −0.144222 − 1.40684i 1.22430 − 2.12055i −1.95840 + 0.405795i −1.33428 − 1.79435i −3.15985 − 1.41657i 0.498659 − 1.86102i 0.853333 + 2.69663i −1.49783 − 2.59432i −2.33194 + 2.13590i
19.17 −0.105332 + 1.41029i 0.640517 − 1.10941i −1.97781 − 0.297096i −2.15621 + 0.592253i 1.49712 + 1.02017i −0.819289 + 3.05763i 0.627317 − 2.75798i 0.679475 + 1.17689i −0.608128 − 3.10325i
19.18 −0.0175279 − 1.41410i −0.267386 + 0.463127i −1.99939 + 0.0495727i 1.99569 + 1.00857i 0.659597 + 0.369995i −1.03041 + 3.84553i 0.105146 + 2.82647i 1.35701 + 2.35041i 1.39124 − 2.83980i
19.19 0.0175279 + 1.41410i 0.267386 − 0.463127i −1.99939 + 0.0495727i −1.00857 − 1.99569i 0.659597 + 0.369995i 1.03041 − 3.84553i −0.105146 − 2.82647i 1.35701 + 2.35041i 2.80444 − 1.46120i
19.20 0.105332 − 1.41029i −0.640517 + 1.10941i −1.97781 − 0.297096i −0.592253 + 2.15621i 1.49712 + 1.02017i 0.819289 − 3.05763i −0.627317 + 2.75798i 0.679475 + 1.17689i 2.97849 + 1.06236i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
13.f odd 12 1 inner
20.d odd 2 1 inner
52.l even 12 1 inner
65.s odd 12 1 inner
260.bc even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 260.2.bc.c ✓ 144
4.b odd 2 1 inner 260.2.bc.c ✓ 144
5.b even 2 1 inner 260.2.bc.c ✓ 144
13.f odd 12 1 inner 260.2.bc.c ✓ 144
20.d odd 2 1 inner 260.2.bc.c ✓ 144
52.l even 12 1 inner 260.2.bc.c ✓ 144
65.s odd 12 1 inner 260.2.bc.c ✓ 144
260.bc even 12 1 inner 260.2.bc.c ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.bc.c ✓ 144 1.a even 1 1 trivial
260.2.bc.c ✓ 144 4.b odd 2 1 inner
260.2.bc.c ✓ 144 5.b even 2 1 inner
260.2.bc.c ✓ 144 13.f odd 12 1 inner
260.2.bc.c ✓ 144 20.d odd 2 1 inner
260.2.bc.c ✓ 144 52.l even 12 1 inner
260.2.bc.c ✓ 144 65.s odd 12 1 inner
260.2.bc.c ✓ 144 260.bc even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(260, [\chi])\):

\( T_{3}^{72} + 74 T_{3}^{70} + 2991 T_{3}^{68} + 83530 T_{3}^{66} + 1781091 T_{3}^{64} + \cdots + 812773572345856 \) Copy content Toggle raw display
\( T_{17}^{72} + 280 T_{17}^{70} + 43546 T_{17}^{68} + 4647000 T_{17}^{66} + 375765649 T_{17}^{64} + \cdots + 55\!\cdots\!36 \) Copy content Toggle raw display