# Properties

 Label 260.2.bj.c Level $260$ Weight $2$ Character orbit 260.bj 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.bj (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.

 $$\operatorname{Tr}(f)(q) =$$ $$144 q - 6 q^{2} - 24 q^{5} - 4 q^{6} - 24 q^{8}+O(q^{10})$$ 144 * q - 6 * q^2 - 24 * q^5 - 4 * q^6 - 24 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$144 q - 6 q^{2} - 24 q^{5} - 4 q^{6} - 24 q^{8} - 16 q^{10} + 20 q^{12} - 12 q^{13} - 28 q^{16} - 4 q^{18} + 30 q^{20} - 32 q^{21} - 28 q^{22} - 24 q^{25} - 12 q^{26} + 14 q^{28} - 4 q^{30} + 4 q^{32} - 28 q^{33} + 4 q^{36} + 20 q^{40} + 24 q^{41} - 56 q^{42} - 4 q^{46} + 12 q^{48} + 20 q^{50} - 2 q^{52} + 24 q^{53} - 20 q^{56} - 24 q^{57} - 42 q^{58} + 88 q^{60} - 32 q^{61} - 128 q^{66} - 32 q^{68} + 108 q^{70} + 2 q^{72} - 8 q^{73} + 60 q^{76} - 72 q^{77} - 120 q^{78} - 64 q^{80} - 32 q^{81} - 42 q^{82} - 48 q^{85} - 24 q^{86} - 42 q^{88} - 56 q^{90} - 84 q^{92} + 8 q^{93} + 160 q^{96} + 68 q^{97} + 98 q^{98}+O(q^{100})$$ 144 * q - 6 * q^2 - 24 * q^5 - 4 * q^6 - 24 * q^8 - 16 * q^10 + 20 * q^12 - 12 * q^13 - 28 * q^16 - 4 * q^18 + 30 * q^20 - 32 * q^21 - 28 * q^22 - 24 * q^25 - 12 * q^26 + 14 * q^28 - 4 * q^30 + 4 * q^32 - 28 * q^33 + 4 * q^36 + 20 * q^40 + 24 * q^41 - 56 * q^42 - 4 * q^46 + 12 * q^48 + 20 * q^50 - 2 * q^52 + 24 * q^53 - 20 * q^56 - 24 * q^57 - 42 * q^58 + 88 * q^60 - 32 * q^61 - 128 * q^66 - 32 * q^68 + 108 * q^70 + 2 * q^72 - 8 * q^73 + 60 * q^76 - 72 * q^77 - 120 * q^78 - 64 * q^80 - 32 * q^81 - 42 * q^82 - 48 * q^85 - 24 * q^86 - 42 * q^88 - 56 * q^90 - 84 * q^92 + 8 * q^93 + 160 * q^96 + 68 * q^97 + 98 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 −1.41378 + 0.0350218i −1.66173 0.445260i 1.99755 0.0990263i 1.78102 + 1.35202i 2.36492 + 0.571303i −1.76644 + 0.473317i −2.82062 + 0.209959i −0.0349771 0.0201940i −2.56533 1.84908i
3.2 −1.40734 + 0.139220i 0.160249 + 0.0429387i 1.96124 0.391860i 1.36197 1.77343i −0.231504 0.0381197i 3.90057 1.04516i −2.70558 + 0.824524i −2.57424 1.48624i −1.66986 + 2.68544i
3.3 −1.38499 0.285996i 2.68406 + 0.719191i 1.83641 + 0.792206i −2.23362 0.104689i −3.51171 1.76370i 1.57219 0.421268i −2.31685 1.62241i 4.08885 + 2.36070i 3.06360 + 0.783800i
3.4 −1.34244 0.444817i −2.68406 0.719191i 1.60428 + 1.19428i −2.23362 0.104689i 3.28327 + 2.15938i −1.57219 + 0.421268i −1.62241 2.31685i 4.08885 + 2.36070i 2.95192 + 1.13409i
3.5 −1.34050 + 0.450613i −0.804148 0.215471i 1.59390 1.20810i −2.02083 + 0.957199i 1.17506 0.0735202i 1.98521 0.531935i −1.59224 + 2.33769i −1.99785 1.15346i 2.27761 2.19374i
3.6 −1.29282 + 0.573250i 3.02314 + 0.810048i 1.34277 1.48222i 2.23524 0.0606684i −4.37274 + 0.685770i −2.22941 + 0.597369i −0.886276 + 2.68599i 5.88512 + 3.39778i −2.85499 + 1.35979i
3.7 −1.20686 0.737220i 1.66173 + 0.445260i 0.913014 + 1.77944i 1.78102 + 1.35202i −1.67722 1.76243i 1.76644 0.473317i 0.209959 2.82062i −0.0349771 0.0201940i −1.15271 2.94470i
3.8 −1.14919 0.824240i −0.160249 0.0429387i 0.641258 + 1.89441i 1.36197 1.77343i 0.148765 + 0.181428i −3.90057 + 1.04516i 0.824524 2.70558i −2.57424 1.48624i −3.02688 + 0.915413i
3.9 −1.11075 + 0.875350i −2.57196 0.689155i 0.467524 1.94459i 0.222626 2.22496i 3.46005 1.48589i −1.04514 + 0.280044i 1.18289 + 2.56919i 3.54197 + 2.04496i 1.70034 + 2.66624i
3.10 −1.10569 + 0.881731i 0.840154 + 0.225119i 0.445099 1.94984i −0.905596 + 2.04448i −1.12744 + 0.491879i −2.80535 + 0.751691i 1.22710 + 2.54838i −1.94290 1.12173i −0.801374 3.05905i
3.11 −0.935603 1.06049i 0.804148 + 0.215471i −0.249293 + 1.98440i −2.02083 + 0.957199i −0.523858 1.05439i −1.98521 + 0.531935i 2.33769 1.59224i −1.99785 1.15346i 2.90580 + 1.24752i
3.12 −0.924149 + 1.07049i 1.38963 + 0.372351i −0.291896 1.97858i −0.609763 2.15132i −1.68282 + 1.14348i 0.0945931 0.0253462i 2.38781 + 1.51604i −0.805646 0.465140i 2.86648 + 1.33540i
3.13 −0.832990 1.14286i −3.02314 0.810048i −0.612255 + 1.90398i 2.23524 0.0606684i 1.59247 + 4.12979i 2.22941 0.597369i 2.68599 0.886276i 5.88512 + 3.39778i −1.93127 2.50403i
3.14 −0.594250 + 1.28330i 0.960123 + 0.257264i −1.29373 1.52521i 1.38665 + 1.75420i −0.900701 + 1.07925i 4.50882 1.20813i 2.72610 0.753900i −1.74242 1.00599i −3.07518 + 0.737057i
3.15 −0.524261 1.31345i 2.57196 + 0.689155i −1.45030 + 1.37718i 0.222626 2.22496i −0.443209 3.73944i 1.04514 0.280044i 2.56919 + 1.18289i 3.54197 + 2.04496i −3.03908 + 0.874051i
3.16 −0.516690 1.31645i −0.840154 0.225119i −1.46606 + 1.36039i −0.905596 + 2.04448i 0.137742 + 1.22233i 2.80535 0.751691i 2.54838 + 1.22710i −1.94290 1.12173i 3.15936 + 0.135807i
3.17 −0.358012 + 1.36815i −2.61303 0.700158i −1.74365 0.979628i −0.297936 + 2.21613i 1.89342 3.32434i −3.10919 + 0.833104i 1.96452 2.03486i 3.73961 + 2.15907i −2.92533 1.20102i
3.18 −0.265092 1.38915i −1.38963 0.372351i −1.85945 + 0.736503i −0.609763 2.15132i −0.148869 + 2.02911i −0.0945931 + 0.0253462i 1.51604 + 2.38781i −0.805646 0.465140i −2.82686 + 1.41735i
3.19 −0.0733540 + 1.41231i 2.99344 + 0.802089i −1.98924 0.207197i −1.68089 + 1.47466i −1.35238 + 4.16882i −0.986808 + 0.264414i 0.438545 2.79422i 5.71924 + 3.30200i −1.95938 2.48210i
3.20 −0.0703644 + 1.41246i −0.124346 0.0333185i −1.99010 0.198774i −1.85457 1.24923i 0.0558107 0.173290i −0.462161 + 0.123836i 0.420793 2.79695i −2.58372 1.49171i 1.89498 2.53161i
See next 80 embeddings (of 144 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 243.36 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.c even 3 1 inner
20.e even 4 1 inner
52.j odd 6 1 inner
65.q odd 12 1 inner
260.bj 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.bj.c 144
4.b odd 2 1 inner 260.2.bj.c 144
5.c odd 4 1 inner 260.2.bj.c 144
13.c even 3 1 inner 260.2.bj.c 144
20.e even 4 1 inner 260.2.bj.c 144
52.j odd 6 1 inner 260.2.bj.c 144
65.q odd 12 1 inner 260.2.bj.c 144
260.bj even 12 1 inner 260.2.bj.c 144

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.bj.c 144 1.a even 1 1 trivial
260.2.bj.c 144 4.b odd 2 1 inner
260.2.bj.c 144 5.c odd 4 1 inner
260.2.bj.c 144 13.c even 3 1 inner
260.2.bj.c 144 20.e even 4 1 inner
260.2.bj.c 144 52.j odd 6 1 inner
260.2.bj.c 144 65.q odd 12 1 inner
260.2.bj.c 144 260.bj 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 $$S_{2}^{\mathrm{new}}(260, [\chi])$$:

 $$T_{3}^{144} - 478 T_{3}^{140} + 129567 T_{3}^{136} - 23966422 T_{3}^{132} + 3345451151 T_{3}^{128} - 367242535236 T_{3}^{124} + 32658104919274 T_{3}^{120} + \cdots + 13\!\cdots\!56$$ T3^144 - 478*T3^140 + 129567*T3^136 - 23966422*T3^132 + 3345451151*T3^128 - 367242535236*T3^124 + 32658104919274*T3^120 - 2388026427116988*T3^116 + 145206272532439885*T3^112 - 7375875949896315322*T3^108 + 313978792022979359345*T3^104 - 11190225651621918888442*T3^100 + 333708641158558611928853*T3^96 - 8301024328371789692396844*T3^92 + 172165315930833608495972314*T3^88 - 2968914069442635923540438964*T3^84 + 42528085241497168807284456663*T3^80 - 503670314152407003504899596022*T3^76 + 4914061375634132148853976865135*T3^72 - 39138596454348125641567150011070*T3^68 + 252581663937352413597838019704249*T3^64 - 1298911370262562886475198759332320*T3^60 + 5253672509237834207000491907279112*T3^56 - 16192723490404967697298762280851184*T3^52 + 37452569634625873385269447113050608*T3^48 - 61492106883549313376325116678388384*T3^44 + 75781516365696536552294706910636896*T3^40 - 69064262726792607355321259624032768*T3^36 + 47953356752807460390247631537021056*T3^32 - 24374521354616630898007656576677632*T3^28 + 9121772343527504207361777923108864*T3^24 - 2204452304049162811713267303264256*T3^20 + 323663409818490753478353119109376*T3^16 - 333617262610071662339692070912*T3^12 + 276016265732641584272125952*T3^8 - 69112212375712500842496*T3^4 + 13911327357083385856 $$T_{17}^{72} - 80 T_{17}^{69} - 3662 T_{17}^{68} + 472 T_{17}^{67} + 3200 T_{17}^{66} + 297116 T_{17}^{65} + 8540021 T_{17}^{64} - 2948256 T_{17}^{63} - 11939488 T_{17}^{62} - 728819188 T_{17}^{61} + \cdots + 14\!\cdots\!00$$ T17^72 - 80*T17^69 - 3662*T17^68 + 472*T17^67 + 3200*T17^66 + 297116*T17^65 + 8540021*T17^64 - 2948256*T17^63 - 11939488*T17^62 - 728819188*T17^61 - 11854522578*T17^60 + 7302560072*T17^59 + 30002444680*T17^58 + 1003816242780*T17^57 + 11770283454874*T17^56 - 9977520793152*T17^55 - 39289323292448*T17^54 - 980236849305664*T17^53 - 7534599635881622*T17^52 + 8329389260923264*T17^51 + 36717466133923376*T17^50 + 549253535340872748*T17^49 + 3486674698327450189*T17^48 - 3242476664891909360*T17^47 - 16475882701560397120*T17^46 - 208027232864402071756*T17^45 - 1188509862456217162022*T17^44 + 688837764902468951880*T17^43 + 5211391873186110567896*T17^42 + 50839649178111499716020*T17^41 + 297615074165915037496458*T17^40 + 64780325675061578876448*T17^39 - 681713714511798692151616*T17^38 - 7497015323710181772376848*T17^37 - 48846725867227123896091130*T17^36 - 40145987076591649508785152*T17^35 + 44512501872841522157881200*T17^34 + 581028083977348785137619140*T17^33 + 5296370795981732210937835637*T17^32 + 10212182358059630897372257968*T17^31 + 12640157942255021081776576928*T17^30 + 15784233961134826885274588876*T17^29 - 197265547429845594295929642766*T17^28 - 541234060960403059339652985592*T17^27 - 734686579201993438565005375672*T17^26 - 1683547906729352254560171996396*T17^25 + 5060200356648724659774332259201*T17^24 + 22730747112692701391782138122496*T17^23 + 38878330039603340868011879313152*T17^22 + 114254481484069047205216024659944*T17^21 + 119023193098016299076577926444336*T17^20 - 127789571032054531662083209280792*T17^19 - 164992097224797785487167617716640*T17^18 - 772948574891504516911312215804048*T17^17 - 1344952947324349230206562227964288*T17^16 + 820666866492755707371925831894992*T17^15 + 1762411486617809943595620638258080*T17^14 + 4825331465423323380188377923979920*T17^13 + 4377091932135283491759941541130408*T17^12 - 7653803359039182918536016347181888*T17^11 - 3059385023641954279069289372050304*T17^10 - 8506790310135690286754206506042784*T17^9 + 4987284291339938772877847774043760*T17^8 + 5994874947873477741882068355184544*T17^7 + 2151928674140674032814522070621312*T17^6 + 912718378670205760149914890793856*T17^5 + 344049273966941445170195852160864*T17^4 + 64211902849921680858877614596800*T17^3 + 9498544678357549604515278080000*T17^2 + 1666374282446381929949396000000*T17 + 146169931459376363311506250000