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 $8$

Related objects

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.

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

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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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 $$S_{2}^{\mathrm{new}}(260, [\chi])$$:

 $$T_{3}^{144} - 478 T_{3}^{140} + 131231 T_{3}^{136} - 24290390 T_{3}^{132} + 3362718895 T_{3}^{128} - 360758553092 T_{3}^{124} + 30930126607498 T_{3}^{120} + \cdots + 43\!\cdots\!96$$ T3^144 - 478*T3^140 + 131231*T3^136 - 24290390*T3^132 + 3362718895*T3^128 - 360758553092*T3^124 + 30930126607498*T3^120 - 2148591424851964*T3^116 + 122660326634845709*T3^112 - 5800313077844589562*T3^108 + 229109634498088950897*T3^104 - 7596302234620343068794*T3^100 + 212384166066291425332149*T3^96 - 5016021757495165571281708*T3^92 + 100185699945283227862273850*T3^88 - 1689767747691181029711472564*T3^84 + 24022972703665976984568420311*T3^80 - 286643430293751552383697758134*T3^76 + 2859335911296101792070943235503*T3^72 - 23685097338747637678450969031422*T3^68 + 162031089979483627719423878281241*T3^64 - 907191667443920647920580789308192*T3^60 + 4134550706347385021223573900296936*T3^56 - 15149967382814563829297815880416240*T3^52 + 44268535769379925859740314351502960*T3^48 - 100737300441391671072616216744036128*T3^44 + 175213174404867233273402808388386144*T3^40 - 219368144306860451623971437959223808*T3^36 + 189610966201625033634197220295668352*T3^32 - 90294141650787440006079957006642944*T3^28 + 30221491395672331906138950362662912*T3^24 - 5685348080440477475623852023080960*T3^20 + 740642714932888643713684290797824*T3^16 - 35385964521257517729782714025984*T3^12 + 1240184689693823082338713939968*T3^8 - 7902730308709362646821666816*T3^4 + 43096964632701710851178496 $$T_{17}^{72} + 12 T_{17}^{71} + 72 T_{17}^{70} + 520 T_{17}^{69} + 474 T_{17}^{68} - 21916 T_{17}^{67} - 161920 T_{17}^{66} - 1233888 T_{17}^{65} - 3127227 T_{17}^{64} + 28630684 T_{17}^{63} + 244012400 T_{17}^{62} + \cdots + 27\!\cdots\!00$$ T17^72 + 12*T17^71 + 72*T17^70 + 520*T17^69 + 474*T17^68 - 21916*T17^67 - 161920*T17^66 - 1233888*T17^65 - 3127227*T17^64 + 28630684*T17^63 + 244012400*T17^62 + 1914607704*T17^61 + 7447475830*T17^60 - 13941817248*T17^59 - 196176806368*T17^58 - 1652759838560*T17^57 - 7348544270550*T17^56 + 2175062145060*T17^55 + 110721504683896*T17^54 + 1006669860225920*T17^53 + 5048102689774378*T17^52 + 4335486108852972*T17^51 - 35743935610912896*T17^50 - 396977993855988560*T17^49 - 2211556878925592771*T17^48 - 3199448567155085180*T17^47 + 7090352622094736688*T17^46 + 112877764890580122216*T17^45 + 708832523122505816482*T17^44 + 1386646833813475798992*T17^43 - 10065374555313870976*T17^42 - 20585070243418127074608*T17^41 - 154174325186289792625638*T17^40 - 354111495042736941004228*T17^39 - 279427543564348115251032*T17^38 + 2635608569219556552890080*T17^37 + 24851256229107182743294918*T17^36 + 64483796077804425516882564*T17^35 + 82489347080058284732294976*T17^34 - 183405244011421568906116128*T17^33 - 2793202813649030151828166043*T17^32 - 7948155195604835951049413628*T17^31 - 11848969276941350286041076000*T17^30 + 5581157943786310147784258424*T17^29 + 236064283462326815705362029306*T17^28 + 718864050133035767870909632728*T17^27 + 1166033692635622758676230537312*T17^26 + 572440459677325302999460558320*T17^25 - 14033342434347953341112582624655*T17^24 - 45890382873926387079811940184936*T17^23 - 76550787016931998566623113263072*T17^22 - 72996493148290866974990733521208*T17^21 + 603853163030102791202649435604104*T17^20 + 2112892665570980136034967068154064*T17^19 + 3625180898152476070263056783690016*T17^18 + 5173641708559915013571222793117872*T17^17 - 14649274786602426806134633113251712*T17^16 - 63513171798819236947623106693026864*T17^15 - 113588248398857214989560580109149568*T17^14 - 202003709290754895731935661034772224*T17^13 + 167870653324326657588405893577203304*T17^12 + 1265043097265600556255129010914529440*T17^11 + 2397424541929295814881427767519203200*T17^10 + 5283303862867270394945199622594596000*T17^9 + 3978353155946749521273512441962290000*T17^8 - 7522884152151822674080945537669200000*T17^7 - 15828706542393340307506277467182000000*T17^6 - 38364822700375518859268507630535000000*T17^5 - 38198167230651441813684379223212500000*T17^4 + 39002039645050788447931242519375000000*T17^3 + 91787788023314523852212704080000000000*T17^2 + 225551053650402080474096711025000000000*T17 + 277124435060383572862529405003906250000