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Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 260 = 2^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 260.bg (of order \(12\), degree \(4\), minimal)

Newform invariants

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.

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 6 q^{2} - 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q - 6 q^{2} - 12 q^{6} + 12 q^{10} - 12 q^{12} + 4 q^{13} - 12 q^{16} - 24 q^{17} - 42 q^{20} - 12 q^{22} - 24 q^{25} - 36 q^{26} - 6 q^{28} - 36 q^{32} - 12 q^{33} - 76 q^{36} + 48 q^{37} - 16 q^{38} + 76 q^{40} - 72 q^{41} + 40 q^{42} - 120 q^{45} - 12 q^{46} - 40 q^{48} + 120 q^{50} - 10 q^{52} + 8 q^{53} - 20 q^{56} + 126 q^{58} + 16 q^{61} - 44 q^{62} + 32 q^{65} + 96 q^{66} - 64 q^{68} - 54 q^{72} - 12 q^{76} + 40 q^{77} - 100 q^{78} - 24 q^{80} - 32 q^{81} - 78 q^{82} - 24 q^{85} - 10 q^{88} - 8 q^{90} + 100 q^{92} - 48 q^{93} - 12 q^{97} - 162 q^{98}+O(q^{100}) \) 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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −1.39962 − 0.202672i −0.654675 − 0.175420i 1.91785 + 0.567327i 1.99368 + 1.01254i 0.880741 + 0.378205i 0.970879 + 3.62337i −2.56927 − 1.18273i −2.20025 − 1.27031i −2.58517 − 1.82123i
23.2 −1.38580 + 0.282082i 2.11574 + 0.566910i 1.84086 − 0.781815i −0.526400 − 2.17322i −3.09190 − 0.188811i 1.32407 + 4.94149i −2.33052 + 1.60271i 1.55689 + 0.898869i 1.34251 + 2.86316i
23.3 −1.37541 + 0.329025i 1.27282 + 0.341051i 1.78348 − 0.905087i −2.19727 − 0.414735i −1.86286 − 0.0502939i −1.05068 − 3.92118i −2.15522 + 1.83167i −1.09432 − 0.631809i 3.15860 − 0.152528i
23.4 −1.35231 + 0.413846i −2.09023 − 0.560075i 1.65746 − 1.11929i 0.553491 − 2.16648i 3.05841 − 0.107639i −0.371776 − 1.38749i −1.77818 + 2.19956i 1.45729 + 0.841367i 0.148100 + 3.15881i
23.5 −1.33237 − 0.474120i −1.74962 − 0.468810i 1.55042 + 1.26341i −2.23215 + 0.132344i 2.10887 + 1.45416i −0.184185 − 0.687388i −1.46673 − 2.41841i 0.243315 + 0.140478i 3.03679 + 0.881975i
23.6 −1.31533 − 0.519519i 3.04677 + 0.816378i 1.46020 + 1.36668i −0.215330 + 2.22568i −3.58339 − 2.65666i −0.0424015 − 0.158244i −1.21063 − 2.55624i 6.01823 + 3.47463i 1.43951 − 2.81564i
23.7 −1.26807 − 0.626106i 1.34938 + 0.361566i 1.21598 + 1.58789i 1.82758 − 1.28839i −1.48473 − 1.30335i −0.711458 − 2.65520i −0.547758 − 2.77488i −0.907969 − 0.524216i −3.12416 + 0.489502i
23.8 −1.23451 + 0.689918i −2.57063 − 0.688800i 1.04803 − 1.70342i −1.36165 + 1.77367i 3.64869 − 0.923198i 0.948066 + 3.53823i −0.118577 + 2.82594i 3.53564 + 2.04130i 0.457283 − 3.12904i
23.9 −1.08563 + 0.906314i −0.274584 − 0.0735746i 0.357191 − 1.96785i 2.19604 − 0.421205i 0.364779 − 0.168984i −0.487864 − 1.82073i 1.39571 + 2.46008i −2.52809 − 1.45960i −2.00235 + 2.44757i
23.10 −0.941226 + 1.05551i 1.16081 + 0.311038i −0.228185 − 1.98694i −1.13827 + 1.92466i −1.42089 + 0.932484i 0.352983 + 1.31735i 2.31200 + 1.62931i −1.34734 − 0.777889i −0.960121 − 3.01300i
23.11 −0.932110 − 1.06357i −0.450342 − 0.120669i −0.262342 + 1.98272i −0.808965 − 2.08460i 0.291429 + 0.591445i 0.415219 + 1.54962i 2.35328 − 1.56909i −2.40983 − 1.39132i −1.46307 + 2.80347i
23.12 −0.849309 − 1.13078i −0.754665 − 0.202212i −0.557348 + 1.92077i 0.282716 + 2.21812i 0.412286 + 1.02510i −0.814984 − 3.04156i 2.64534 − 1.00109i −2.06945 − 1.19480i 2.26811 − 2.20356i
23.13 −0.633301 − 1.26449i 1.48318 + 0.397416i −1.19786 + 1.60160i −1.88795 + 1.19818i −0.436769 − 2.12715i 0.726248 + 2.71040i 2.78381 + 0.500385i −0.556198 − 0.321121i 2.71072 + 1.62849i
23.14 −0.595380 + 1.28278i −2.12058 − 0.568207i −1.29105 − 1.52748i 1.57238 + 1.58985i 1.99143 − 2.38193i −0.751521 − 2.80471i 2.72808 − 0.746695i 1.57591 + 0.909854i −2.97559 + 1.07045i
23.15 −0.565829 + 1.29609i 2.90282 + 0.777809i −1.35968 − 1.46672i 0.446275 − 2.19108i −2.65061 + 3.32220i −0.874770 − 3.26469i 2.67034 − 0.932342i 5.22332 + 3.01568i 2.58731 + 1.81819i
23.16 −0.300622 − 1.38189i 2.28888 + 0.613303i −1.81925 + 0.830853i 2.18429 − 0.478409i 0.159433 − 3.34735i 0.371096 + 1.38495i 1.69506 + 2.26424i 2.26474 + 1.30755i −1.31776 − 2.87463i
23.17 −0.173304 − 1.40355i −3.13012 − 0.838713i −1.93993 + 0.486484i −2.04631 − 0.901445i −0.634717 + 4.53865i −0.315858 − 1.17880i 1.01900 + 2.63849i 6.49613 + 3.75054i −0.910593 + 3.02834i
23.18 −0.158021 + 1.40536i −2.90282 − 0.777809i −1.95006 − 0.444152i 0.446275 − 2.19108i 1.55181 − 3.95659i 0.874770 + 3.26469i 0.932342 − 2.67034i 5.22332 + 3.01568i 3.00873 + 0.973413i
23.19 −0.125776 + 1.40861i 2.12058 + 0.568207i −1.96836 − 0.354338i 1.57238 + 1.58985i −1.06710 + 2.91560i 0.751521 + 2.80471i 0.746695 − 2.72808i 1.57591 + 0.909854i −2.43725 + 2.01490i
23.20 −0.0281072 − 1.41393i −1.77624 − 0.475943i −1.99842 + 0.0794833i 1.35785 + 1.77658i −0.623027 + 2.52487i 0.755969 + 2.82131i 0.168554 + 2.82340i 0.330446 + 0.190783i 2.47380 − 1.96985i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 147.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
13.e even 6 1 inner
20.e even 4 1 inner
52.i odd 6 1 inner
65.r odd 12 1 inner
260.bg 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.bg.c ✓ 144
4.b odd 2 1 inner 260.2.bg.c ✓ 144
5.c odd 4 1 inner 260.2.bg.c ✓ 144
13.e even 6 1 inner 260.2.bg.c ✓ 144
20.e even 4 1 inner 260.2.bg.c ✓ 144
52.i odd 6 1 inner 260.2.bg.c ✓ 144
65.r odd 12 1 inner 260.2.bg.c ✓ 144
260.bg even 12 1 inner 260.2.bg.c ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.bg.c ✓ 144 1.a even 1 1 trivial
260.2.bg.c ✓ 144 4.b odd 2 1 inner
260.2.bg.c ✓ 144 5.c odd 4 1 inner
260.2.bg.c ✓ 144 13.e even 6 1 inner
260.2.bg.c ✓ 144 20.e even 4 1 inner
260.2.bg.c ✓ 144 52.i odd 6 1 inner
260.2.bg.c ✓ 144 65.r odd 12 1 inner
260.2.bg.c ✓ 144 260.bg 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 \(S_{2}^{\mathrm{new}}(260, [\chi])\):

\( T_{3}^{144} - 478 T_{3}^{140} + 131231 T_{3}^{136} - 24290390 T_{3}^{132} + 3362718895 T_{3}^{128} - 360758553092 T_{3}^{124} + 30930126607498 T_{3}^{120} + \cdots + 43\!\cdots\!96 \) Copy content Toggle raw display
\( T_{17}^{72} + 12 T_{17}^{71} + 72 T_{17}^{70} + 520 T_{17}^{69} + 474 T_{17}^{68} - 21916 T_{17}^{67} - 161920 T_{17}^{66} - 1233888 T_{17}^{65} - 3127227 T_{17}^{64} + 28630684 T_{17}^{63} + 244012400 T_{17}^{62} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display