# 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 $8$

# Related objects

## Newspace parameters

 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

 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.

 $$\operatorname{Tr}(f)(q) =$$ $$144 q - 12 q^{4} - 12 q^{5} - 80 q^{9}+O(q^{10})$$ 144 * q - 12 * q^4 - 12 * q^5 - 80 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$144 q - 12 q^{4} - 12 q^{5} - 80 q^{9} - 12 q^{10} - 32 q^{14} - 36 q^{16} + 18 q^{20} + 8 q^{21} - 28 q^{24} + 28 q^{26} - 6 q^{30} + 16 q^{34} + 24 q^{36} - 88 q^{40} - 72 q^{41} + 68 q^{45} + 64 q^{46} - 24 q^{49} - 34 q^{50} + 80 q^{54} + 96 q^{56} - 48 q^{60} - 8 q^{61} + 40 q^{66} - 24 q^{69} - 80 q^{70} - 24 q^{74} - 28 q^{76} - 92 q^{80} - 8 q^{81} + 84 q^{84} + 20 q^{85} - 52 q^{86} - 104 q^{89} - 52 q^{94} - 124 q^{96}+O(q^{100})$$ 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

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
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 219.36 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.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 $$S_{2}^{\mathrm{new}}(260, [\chi])$$:

 $$T_{3}^{72} + 74 T_{3}^{70} + 2991 T_{3}^{68} + 83530 T_{3}^{66} + 1781091 T_{3}^{64} + 30507056 T_{3}^{62} + 433199814 T_{3}^{60} + 5204629396 T_{3}^{58} + 53662585809 T_{3}^{56} + \cdots + 812773572345856$$ T3^72 + 74*T3^70 + 2991*T3^68 + 83530*T3^66 + 1781091*T3^64 + 30507056*T3^62 + 433199814*T3^60 + 5204629396*T3^58 + 53662585809*T3^56 + 479559876910*T3^54 + 3741234611413*T3^52 + 25608093041614*T3^50 + 154337448126341*T3^48 + 820932679344560*T3^46 + 3859349813810230*T3^44 + 16047144864076396*T3^42 + 59026527177601487*T3^40 + 192029048795587082*T3^38 + 552275082977384223*T3^36 + 1403159659090291970*T3^34 + 3146735682338990177*T3^32 + 6222403160538960172*T3^30 + 10835821602534213664*T3^28 + 16590529765186972808*T3^26 + 22286786561478866732*T3^24 + 26193033610694087008*T3^22 + 26832325040015860672*T3^20 + 23837564237511637120*T3^18 + 18243914974626413328*T3^16 + 11920397582138587264*T3^14 + 6570289895160172224*T3^12 + 3003577099831769600*T3^10 + 1112794661449711872*T3^8 + 322342512501501952*T3^6 + 69311219702611968*T3^4 + 9966533389058048*T3^2 + 812773572345856 $$T_{17}^{72} + 280 T_{17}^{70} + 43546 T_{17}^{68} + 4647000 T_{17}^{66} + 375765649 T_{17}^{64} + 24159965420 T_{17}^{62} + 1274590868726 T_{17}^{60} + 56283658647300 T_{17}^{58} + \cdots + 55\!\cdots\!36$$ T17^72 + 280*T17^70 + 43546*T17^68 + 4647000*T17^66 + 375765649*T17^64 + 24159965420*T17^62 + 1274590868726*T17^60 + 56283658647300*T17^58 + 2110311987036806*T17^56 + 67845131916605764*T17^54 + 1883648518828734394*T17^52 + 45378977669980650672*T17^50 + 951612216531262643029*T17^48 + 17398439788813580413004*T17^46 + 277467684496044422828426*T17^44 + 3856937476245651869958252*T17^42 + 46655061624107723137910486*T17^40 + 489823462163740652812828732*T17^38 + 4449190683949894967894790006*T17^36 + 34829316118211757769721451776*T17^34 + 234101365875390444736260684381*T17^32 + 1345790526396410336901208713476*T17^30 + 6597717576735167478536387222770*T17^28 + 27486252509782494596340717066292*T17^26 + 96971392014135587243645444301309*T17^24 + 288093842877986743904393962444900*T17^22 + 715803688233257642537372819349896*T17^20 + 1470135418548849782309934014975008*T17^18 + 2458414488501811958372422278696716*T17^16 + 3260098510642465705550133904488736*T17^14 + 3309141044223875032738473903263584*T17^12 + 2387779985231277010996850501894496*T17^10 + 1102680063875697200116408151563024*T17^8 + 202883062794732735282057377927168*T17^6 + 26558472576125120486039664656384*T17^4 + 1413033620855362766198959767552*T17^2 + 55331397307632199773020225536