# Properties

 Label 507.2.p.b Level $507$ Weight $2$ Character orbit 507.p Analytic conductor $4.048$ Analytic rank $0$ Dimension $192$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 507.p (of order $$26$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.04841538248$$ Analytic rank: $$0$$ Dimension: $$192$$ Relative dimension: $$16$$ over $$\Q(\zeta_{26})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{26}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$192q + 16q^{3} + 18q^{4} - 16q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$192q + 16q^{3} + 18q^{4} - 16q^{9} - 18q^{12} - 63q^{13} - 10q^{14} - 6q^{16} + 12q^{17} + 16q^{22} - 52q^{23} + 58q^{25} + 51q^{26} + 16q^{27} - 49q^{29} - 26q^{31} - 13q^{33} - 65q^{34} + 39q^{35} + 18q^{36} + 77q^{38} - 2q^{39} - 55q^{42} - 76q^{43} + 39q^{44} + 6q^{48} - 58q^{49} + 52q^{50} - 12q^{51} + 63q^{52} - 73q^{53} + 37q^{55} - 10q^{56} + 13q^{57} - 26q^{58} - 104q^{59} - 13q^{60} + 8q^{61} + 53q^{62} + 42q^{64} + 52q^{65} - 42q^{66} + 26q^{67} - 34q^{68} - 39q^{71} + 52q^{73} + 59q^{74} - 6q^{75} - 130q^{76} - 52q^{77} + 53q^{78} + 14q^{79} - 16q^{81} + 41q^{82} - 78q^{83} + 91q^{85} + 169q^{86} - 42q^{87} - 270q^{88} + 80q^{91} - 54q^{92} - 91q^{93} + 25q^{94} - 58q^{95} - 65q^{96} + 130q^{97} + 104q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1 −2.52295 0.956827i −0.568065 0.822984i 3.95272 + 3.50180i 4.16354 + 0.505545i 0.645743 + 2.61988i 0.140914 + 0.268489i −4.11396 7.83849i −0.354605 + 0.935016i −10.0207 5.25925i
25.2 −2.42142 0.918325i −0.568065 0.822984i 3.52295 + 3.12106i −2.78849 0.338584i 0.619759 + 2.51446i 0.357250 + 0.680683i −3.25740 6.20646i −0.354605 + 0.935016i 6.44118 + 3.38059i
25.3 −2.13008 0.807834i −0.568065 0.822984i 2.38764 + 2.11526i −1.00892 0.122506i 0.545191 + 2.21193i 1.86515 + 3.55375i −1.25969 2.40014i −0.354605 + 0.935016i 2.05013 + 1.07599i
25.4 −1.50260 0.569861i −0.568065 0.822984i 0.436045 + 0.386302i 2.17889 + 0.264565i 0.384588 + 1.56033i 0.728343 + 1.38774i 1.05859 + 2.01697i −0.354605 + 0.935016i −3.12324 1.63920i
25.5 −1.31279 0.497877i −0.568065 0.822984i −0.0214779 0.0190278i 1.11474 + 0.135354i 0.336007 + 1.36323i −0.369513 0.704048i 1.32369 + 2.52209i −0.354605 + 0.935016i −1.39603 0.732692i
25.6 −1.13089 0.428890i −0.568065 0.822984i −0.402056 0.356191i −4.00621 0.486442i 0.289449 + 1.17434i −2.12665 4.05199i 1.42607 + 2.71715i −0.354605 + 0.935016i 4.32195 + 2.26833i
25.7 −0.468954 0.177851i −0.568065 0.822984i −1.30873 1.15944i −1.41049 0.171264i 0.120028 + 0.486973i 2.16100 + 4.11744i 0.873690 + 1.66468i −0.354605 + 0.935016i 0.630994 + 0.331171i
25.8 −0.103579 0.0392824i −0.568065 0.822984i −1.48784 1.31811i −1.67597 0.203500i 0.0265109 + 0.107559i 0.366680 + 0.698651i 0.205293 + 0.391152i −0.354605 + 0.935016i 0.165602 + 0.0869145i
25.9 0.283929 + 0.107680i −0.568065 0.822984i −1.42800 1.26510i −0.879635 0.106807i −0.0726712 0.294839i −0.935242 1.78196i −0.551463 1.05073i −0.354605 + 0.935016i −0.238253 0.125045i
25.10 0.494430 + 0.187512i −0.568065 0.822984i −1.28772 1.14082i 3.75871 + 0.456390i −0.126548 0.513427i 1.26721 + 2.41446i −0.914254 1.74197i −0.354605 + 0.935016i 1.77284 + 0.930459i
25.11 0.917259 + 0.347870i −0.568065 0.822984i −0.776671 0.688071i 0.986534 + 0.119787i −0.234771 0.952502i −0.506792 0.965611i −1.38484 2.63860i −0.354605 + 0.935016i 0.863237 + 0.453062i
25.12 1.43480 + 0.544146i −0.568065 0.822984i 0.265521 + 0.235231i −3.88465 0.471682i −0.367233 1.48992i 2.17406 + 4.14233i −1.17328 2.23550i −0.354605 + 0.935016i −5.31702 2.79059i
25.13 1.78796 + 0.678085i −0.568065 0.822984i 1.23999 + 1.09854i −2.39960 0.291364i −0.457626 1.85666i −1.52079 2.89763i −0.305152 0.581418i −0.354605 + 0.935016i −4.09283 2.14808i
25.14 1.98795 + 0.753931i −0.568065 0.822984i 1.88652 + 1.67131i 3.81359 + 0.463053i −0.508813 2.06433i −2.20899 4.20888i 0.514149 + 0.979628i −0.354605 + 0.935016i 7.23212 + 3.79571i
25.15 2.12140 + 0.804542i −0.568065 0.822984i 2.35604 + 2.08727i 2.11775 + 0.257141i −0.542969 2.20291i 1.12326 + 2.14019i 1.21005 + 2.30557i −0.354605 + 0.935016i 4.28571 + 2.24932i
25.16 2.56554 + 0.972980i −0.568065 0.822984i 4.13827 + 3.66618i −1.46104 0.177402i −0.656644 2.66411i 0.543789 + 1.03610i 4.49950 + 8.57307i −0.354605 + 0.935016i −3.57574 1.87669i
64.1 −2.62119 0.318270i 0.748511 0.663123i 4.82745 + 1.18986i −0.880115 1.67692i −2.17304 + 1.49994i −2.02452 0.767800i −7.33724 2.78265i 0.120537 0.992709i 1.77323 + 4.67563i
64.2 −2.60940 0.316839i 0.748511 0.663123i 4.76672 + 1.17489i 1.60429 + 3.05673i −2.16327 + 1.49320i 2.76033 + 1.04686i −7.15054 2.71184i 0.120537 0.992709i −3.21776 8.48454i
64.3 −2.10986 0.256183i 0.748511 0.663123i 2.44400 + 0.602391i 1.00545 + 1.91572i −1.74913 + 1.20734i −2.58799 0.981496i −1.02769 0.389750i 0.120537 0.992709i −1.63057 4.29947i
64.4 −1.61341 0.195904i 0.748511 0.663123i 0.622835 + 0.153515i −1.54944 2.95221i −1.33756 + 0.923254i −1.00058 0.379471i 2.06448 + 0.782953i 0.120537 0.992709i 1.92153 + 5.06666i
See next 80 embeddings (of 192 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 493.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
169.h even 26 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.p.b 192
169.h even 26 1 inner 507.2.p.b 192

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.p.b 192 1.a even 1 1 trivial
507.2.p.b 192 169.h even 26 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$11\!\cdots\!44$$$$T_{2}^{168} + 256460285133 T_{2}^{167} -$$$$86\!\cdots\!76$$$$T_{2}^{166} -$$$$27\!\cdots\!84$$$$T_{2}^{165} +$$$$61\!\cdots\!99$$$$T_{2}^{164} +$$$$27\!\cdots\!79$$$$T_{2}^{163} -$$$$42\!\cdots\!49$$$$T_{2}^{162} -$$$$24\!\cdots\!02$$$$T_{2}^{161} +$$$$28\!\cdots\!14$$$$T_{2}^{160} +$$$$20\!\cdots\!81$$$$T_{2}^{159} -$$$$18\!\cdots\!90$$$$T_{2}^{158} -$$$$15\!\cdots\!92$$$$T_{2}^{157} +$$$$11\!\cdots\!02$$$$T_{2}^{156} +$$$$11\!\cdots\!49$$$$T_{2}^{155} -$$$$68\!\cdots\!20$$$$T_{2}^{154} -$$$$74\!\cdots\!34$$$$T_{2}^{153} +$$$$40\!\cdots\!01$$$$T_{2}^{152} +$$$$46\!\cdots\!42$$$$T_{2}^{151} -$$$$23\!\cdots\!36$$$$T_{2}^{150} -$$$$27\!\cdots\!34$$$$T_{2}^{149} +$$$$13\!\cdots\!91$$$$T_{2}^{148} +$$$$16\!\cdots\!05$$$$T_{2}^{147} -$$$$78\!\cdots\!25$$$$T_{2}^{146} -$$$$90\!\cdots\!53$$$$T_{2}^{145} +$$$$43\!\cdots\!09$$$$T_{2}^{144} +$$$$50\!\cdots\!18$$$$T_{2}^{143} -$$$$23\!\cdots\!67$$$$T_{2}^{142} -$$$$27\!\cdots\!02$$$$T_{2}^{141} +$$$$12\!\cdots\!46$$$$T_{2}^{140} +$$$$14\!\cdots\!90$$$$T_{2}^{139} -$$$$64\!\cdots\!26$$$$T_{2}^{138} -$$$$72\!\cdots\!31$$$$T_{2}^{137} +$$$$32\!\cdots\!17$$$$T_{2}^{136} +$$$$35\!\cdots\!07$$$$T_{2}^{135} -$$$$16\!\cdots\!46$$$$T_{2}^{134} -$$$$16\!\cdots\!08$$$$T_{2}^{133} +$$$$80\!\cdots\!90$$$$T_{2}^{132} +$$$$68\!\cdots\!67$$$$T_{2}^{131} -$$$$38\!\cdots\!21$$$$T_{2}^{130} -$$$$26\!\cdots\!14$$$$T_{2}^{129} +$$$$18\!\cdots\!94$$$$T_{2}^{128} +$$$$86\!\cdots\!40$$$$T_{2}^{127} -$$$$82\!\cdots\!36$$$$T_{2}^{126} -$$$$22\!\cdots\!29$$$$T_{2}^{125} +$$$$36\!\cdots\!87$$$$T_{2}^{124} +$$$$26\!\cdots\!05$$$$T_{2}^{123} -$$$$14\!\cdots\!45$$$$T_{2}^{122} +$$$$18\!\cdots\!38$$$$T_{2}^{121} +$$$$57\!\cdots\!21$$$$T_{2}^{120} -$$$$18\!\cdots\!19$$$$T_{2}^{119} -$$$$21\!\cdots\!38$$$$T_{2}^{118} +$$$$10\!\cdots\!41$$$$T_{2}^{117} +$$$$72\!\cdots\!59$$$$T_{2}^{116} -$$$$49\!\cdots\!17$$$$T_{2}^{115} -$$$$23\!\cdots\!16$$$$T_{2}^{114} +$$$$19\!\cdots\!91$$$$T_{2}^{113} +$$$$75\!\cdots\!33$$$$T_{2}^{112} -$$$$70\!\cdots\!25$$$$T_{2}^{111} -$$$$23\!\cdots\!06$$$$T_{2}^{110} +$$$$22\!\cdots\!80$$$$T_{2}^{109} +$$$$72\!\cdots\!22$$$$T_{2}^{108} -$$$$64\!\cdots\!98$$$$T_{2}^{107} -$$$$21\!\cdots\!51$$$$T_{2}^{106} +$$$$16\!\cdots\!61$$$$T_{2}^{105} +$$$$65\!\cdots\!42$$$$T_{2}^{104} -$$$$39\!\cdots\!38$$$$T_{2}^{103} -$$$$18\!\cdots\!71$$$$T_{2}^{102} +$$$$75\!\cdots\!22$$$$T_{2}^{101} +$$$$50\!\cdots\!56$$$$T_{2}^{100} -$$$$10\!\cdots\!69$$$$T_{2}^{99} -$$$$13\!\cdots\!24$$$$T_{2}^{98} -$$$$12\!\cdots\!28$$$$T_{2}^{97} +$$$$31\!\cdots\!98$$$$T_{2}^{96} +$$$$61\!\cdots\!77$$$$T_{2}^{95} -$$$$68\!\cdots\!47$$$$T_{2}^{94} -$$$$26\!\cdots\!78$$$$T_{2}^{93} +$$$$13\!\cdots\!86$$$$T_{2}^{92} +$$$$80\!\cdots\!64$$$$T_{2}^{91} -$$$$25\!\cdots\!66$$$$T_{2}^{90} -$$$$20\!\cdots\!68$$$$T_{2}^{89} +$$$$41\!\cdots\!67$$$$T_{2}^{88} +$$$$44\!\cdots\!41$$$$T_{2}^{87} -$$$$63\!\cdots\!16$$$$T_{2}^{86} -$$$$89\!\cdots\!46$$$$T_{2}^{85} +$$$$93\!\cdots\!79$$$$T_{2}^{84} +$$$$16\!\cdots\!93$$$$T_{2}^{83} -$$$$12\!\cdots\!93$$$$T_{2}^{82} -$$$$26\!\cdots\!84$$$$T_{2}^{81} +$$$$15\!\cdots\!64$$$$T_{2}^{80} +$$$$40\!\cdots\!08$$$$T_{2}^{79} -$$$$17\!\cdots\!27$$$$T_{2}^{78} -$$$$56\!\cdots\!49$$$$T_{2}^{77} +$$$$26\!\cdots\!52$$$$T_{2}^{76} +$$$$82\!\cdots\!56$$$$T_{2}^{75} -$$$$32\!\cdots\!20$$$$T_{2}^{74} -$$$$89\!\cdots\!73$$$$T_{2}^{73} +$$$$72\!\cdots\!06$$$$T_{2}^{72} +$$$$14\!\cdots\!89$$$$T_{2}^{71} -$$$$31\!\cdots\!69$$$$T_{2}^{70} -$$$$13\!\cdots\!86$$$$T_{2}^{69} +$$$$10\!\cdots\!32$$$$T_{2}^{68} +$$$$28\!\cdots\!60$$$$T_{2}^{67} -$$$$22\!\cdots\!99$$$$T_{2}^{66} -$$$$20\!\cdots\!48$$$$T_{2}^{65} +$$$$14\!\cdots\!00$$$$T_{2}^{64} +$$$$36\!\cdots\!98$$$$T_{2}^{63} +$$$$55\!\cdots\!90$$$$T_{2}^{62} -$$$$27\!\cdots\!39$$$$T_{2}^{61} -$$$$41\!\cdots\!29$$$$T_{2}^{60} +$$$$40\!\cdots\!19$$$$T_{2}^{59} +$$$$95\!\cdots\!94$$$$T_{2}^{58} -$$$$44\!\cdots\!97$$$$T_{2}^{57} +$$$$96\!\cdots\!39$$$$T_{2}^{56} +$$$$31\!\cdots\!81$$$$T_{2}^{55} -$$$$94\!\cdots\!31$$$$T_{2}^{54} -$$$$91\!\cdots\!36$$$$T_{2}^{53} -$$$$16\!\cdots\!55$$$$T_{2}^{52} +$$$$12\!\cdots\!60$$$$T_{2}^{51} +$$$$46\!\cdots\!93$$$$T_{2}^{50} -$$$$21\!\cdots\!18$$$$T_{2}^{49} -$$$$49\!\cdots\!82$$$$T_{2}^{48} +$$$$20\!\cdots\!50$$$$T_{2}^{47} -$$$$27\!\cdots\!82$$$$T_{2}^{46} -$$$$23\!\cdots\!23$$$$T_{2}^{45} +$$$$19\!\cdots\!60$$$$T_{2}^{44} +$$$$17\!\cdots\!37$$$$T_{2}^{43} +$$$$11\!\cdots\!14$$$$T_{2}^{42} -$$$$44\!\cdots\!79$$$$T_{2}^{41} +$$$$30\!\cdots\!84$$$$T_{2}^{40} +$$$$16\!\cdots\!06$$$$T_{2}^{39} -$$$$22\!\cdots\!99$$$$T_{2}^{38} +$$$$70\!\cdots\!95$$$$T_{2}^{37} +$$$$32\!\cdots\!82$$$$T_{2}^{36} +$$$$13\!\cdots\!09$$$$T_{2}^{35} -$$$$82\!\cdots\!78$$$$T_{2}^{34} -$$$$83\!\cdots\!48$$$$T_{2}^{33} -$$$$13\!\cdots\!31$$$$T_{2}^{32} +$$$$13\!\cdots\!53$$$$T_{2}^{31} +$$$$86\!\cdots\!08$$$$T_{2}^{30} +$$$$19\!\cdots\!06$$$$T_{2}^{29} -$$$$21\!\cdots\!73$$$$T_{2}^{28} -$$$$44\!\cdots\!52$$$$T_{2}^{27} -$$$$26\!\cdots\!40$$$$T_{2}^{26} -$$$$81\!\cdots\!58$$$$T_{2}^{25} +$$$$23\!\cdots\!34$$$$T_{2}^{24} +$$$$11\!\cdots\!54$$$$T_{2}^{23} +$$$$48\!\cdots\!01$$$$T_{2}^{22} +$$$$11\!\cdots\!76$$$$T_{2}^{21} +$$$$20\!\cdots\!81$$$$T_{2}^{20} +$$$$22\!\cdots\!73$$$$T_{2}^{19} -$$$$13\!\cdots\!44$$$$T_{2}^{18} -$$$$95\!\cdots\!26$$$$T_{2}^{17} -$$$$17\!\cdots\!62$$$$T_{2}^{16} +$$$$47\!\cdots\!12$$$$T_{2}^{15} +$$$$34\!\cdots\!21$$$$T_{2}^{14} +$$$$96\!\cdots\!64$$$$T_{2}^{13} +$$$$16\!\cdots\!30$$$$T_{2}^{12} +$$$$18\!\cdots\!55$$$$T_{2}^{11} +$$$$12\!\cdots\!67$$$$T_{2}^{10} -$$$$38\!\cdots\!82$$$$T_{2}^{9} -$$$$15\!\cdots\!31$$$$T_{2}^{8} -$$$$23\!\cdots\!28$$$$T_{2}^{7} -$$$$18\!\cdots\!60$$$$T_{2}^{6} -$$$$33\!\cdots\!24$$$$T_{2}^{5} +$$$$84\!\cdots\!16$$$$T_{2}^{4} +$$$$92\!\cdots\!98$$$$T_{2}^{3} +$$$$44\!\cdots\!11$$$$T_{2}^{2} +$$$$45\!\cdots\!13$$$$T_{2} +$$$$25\!\cdots\!69$$">$$T_{2}^{192} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(507, [\chi])$$.