Properties

Label 507.2.m.a
Level $507$
Weight $2$
Character orbit 507.m
Analytic conductor $4.048$
Analytic rank $0$
Dimension $180$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(180\)
Relative dimension: \(15\) over \(\Q(\zeta_{13})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{13}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 180q - q^{2} + 15q^{3} - 15q^{4} - 2q^{5} + q^{6} + 4q^{7} + 3q^{8} - 15q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 180q - q^{2} + 15q^{3} - 15q^{4} - 2q^{5} + q^{6} + 4q^{7} + 3q^{8} - 15q^{9} - 2q^{10} - 4q^{11} + 15q^{12} - 14q^{13} + 6q^{14} + 2q^{15} - 15q^{16} - 2q^{17} - q^{18} + 2q^{20} - 4q^{21} - 28q^{22} - 52q^{23} - 3q^{24} - 67q^{25} - 40q^{26} + 15q^{27} - 4q^{28} - 27q^{29} + 2q^{30} + 22q^{31} - 5q^{32} - 9q^{33} + 63q^{34} - 31q^{35} - 15q^{36} + 2q^{37} + 65q^{38} + q^{39} + 45q^{40} - 6q^{41} + 59q^{42} - 60q^{43} - 35q^{44} - 2q^{45} - 156q^{46} + 15q^{48} + 59q^{49} - 51q^{50} + 2q^{51} + 66q^{52} + 50q^{53} + q^{54} + 55q^{55} - 14q^{56} - 13q^{57} + 36q^{58} + 92q^{59} - 15q^{60} + 6q^{61} + 61q^{62} + 4q^{63} - 203q^{64} - 54q^{65} + 54q^{66} + 86q^{67} + 32q^{68} + 112q^{70} + 39q^{71} + 3q^{72} - 158q^{73} - 80q^{74} + 15q^{75} + 130q^{76} - 64q^{77} + 66q^{78} - 10q^{79} - 310q^{80} - 15q^{81} + 59q^{82} - 82q^{83} + 4q^{84} + 22q^{85} - q^{86} + 40q^{87} + 10q^{88} + 2q^{89} - 2q^{90} - 100q^{91} - 54q^{92} + 43q^{93} + 65q^{94} + 58q^{95} - 60q^{96} + 16q^{97} - 113q^{98} - 30q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1 −2.58082 0.636114i −0.120537 + 0.992709i 4.48506 + 2.35394i 0.134864 0.195384i 0.942559 2.48533i 1.30603 + 1.15704i −6.09859 5.40288i −0.970942 0.239316i −0.472345 + 0.418461i
40.2 −2.31903 0.571589i −0.120537 + 0.992709i 3.28027 + 1.72162i −2.03486 + 2.94800i 0.846950 2.23322i −0.933245 0.826783i −3.04745 2.69981i −0.970942 0.239316i 6.40394 5.67340i
40.3 −1.76615 0.435317i −0.120537 + 0.992709i 1.15887 + 0.608224i 1.67811 2.43117i 0.645029 1.70080i 3.62124 + 3.20814i 0.941117 + 0.833757i −0.970942 0.239316i −4.02213 + 3.56330i
40.4 −1.73397 0.427384i −0.120537 + 0.992709i 1.05307 + 0.552694i 0.457873 0.663343i 0.633275 1.66981i −0.852902 0.755605i 1.08370 + 0.960071i −0.970942 0.239316i −1.07744 + 0.954527i
40.5 −1.56270 0.385172i −0.120537 + 0.992709i 0.522773 + 0.274373i −0.878504 + 1.27273i 0.570727 1.50488i 0.273931 + 0.242682i 1.69816 + 1.50443i −0.970942 0.239316i 1.86306 1.65053i
40.6 −0.926138 0.228272i −0.120537 + 0.992709i −0.965289 0.506623i −0.278592 + 0.403610i 0.338242 0.891870i −3.10155 2.74774i 2.20628 + 1.95460i −0.970942 0.239316i 0.350148 0.310204i
40.7 −0.584736 0.144124i −0.120537 + 0.992709i −1.44977 0.760897i −2.25739 + 3.27039i 0.213556 0.563100i 2.17116 + 1.92348i 1.63963 + 1.45258i −0.970942 0.239316i 1.79132 1.58697i
40.8 −0.0301922 0.00744170i −0.120537 + 0.992709i −1.77006 0.928997i 1.67342 2.42437i 0.0110267 0.0290750i −1.02495 0.908024i 0.0930795 + 0.0824612i −0.970942 0.239316i −0.0685655 + 0.0607437i
40.9 −0.00839886 0.00207013i −0.120537 + 0.992709i −1.77085 0.929412i 0.646404 0.936478i 0.00306741 0.00808810i 2.26833 + 2.00956i 0.0258987 + 0.0229442i −0.970942 0.239316i −0.00736769 + 0.00652721i
40.10 0.869211 + 0.214241i −0.120537 + 0.992709i −1.06128 0.557005i −1.13901 + 1.65015i −0.317451 + 0.837049i −1.32969 1.17800i −2.14332 1.89881i −0.970942 0.239316i −1.34357 + 1.19030i
40.11 1.36051 + 0.335337i −0.120537 + 0.992709i −0.0323644 0.0169861i 1.40752 2.03915i −0.496883 + 1.31017i −3.59937 3.18876i −2.13601 1.89234i −0.970942 0.239316i 2.59876 2.30230i
40.12 1.56583 + 0.385942i −0.120537 + 0.992709i 0.531960 + 0.279194i 1.72727 2.50239i −0.571868 + 1.50789i 1.75934 + 1.55864i −1.68903 1.49635i −0.970942 0.239316i 3.67039 3.25168i
40.13 1.74042 + 0.428975i −0.120537 + 0.992709i 1.07413 + 0.563748i −1.82535 + 2.64447i −0.635632 + 1.67602i 2.27352 + 2.01416i −1.05581 0.935370i −0.970942 0.239316i −4.31129 + 3.81947i
40.14 2.46984 + 0.608762i −0.120537 + 0.992709i 3.95863 + 2.07765i 0.951102 1.37791i −0.902030 + 2.37846i 0.576934 + 0.511119i 4.70433 + 4.16767i −0.970942 0.239316i 3.18789 2.82423i
40.15 2.53537 + 0.624913i −0.120537 + 0.992709i 4.26668 + 2.23932i −2.06622 + 2.99344i −0.925962 + 2.44156i −3.04788 2.70019i 5.50913 + 4.88066i −0.970942 0.239316i −7.10928 + 6.29828i
79.1 −2.26213 1.18726i 0.970942 + 0.239316i 2.57152 + 3.72549i −1.11131 2.93028i −1.91227 1.69412i 0.198155 + 1.63195i −0.778110 6.40831i 0.885456 + 0.464723i −0.965071 + 7.94807i
79.2 −2.25517 1.18361i 0.970942 + 0.239316i 2.54876 + 3.69251i 1.29246 + 3.40793i −1.90639 1.68891i 0.394148 + 3.24610i −0.763417 6.28731i 0.885456 + 0.464723i 1.11893 9.21525i
79.3 −1.85232 0.972175i 0.970942 + 0.239316i 1.34985 + 1.95560i 0.359355 + 0.947542i −1.56584 1.38722i −0.194000 1.59773i −0.0948689 0.781316i 0.885456 + 0.464723i 0.255534 2.10451i
79.4 −1.32045 0.693027i 0.970942 + 0.239316i 0.127180 + 0.184253i −0.0244689 0.0645191i −1.11623 0.988894i 0.0316924 + 0.261010i 0.319262 + 2.62936i 0.885456 + 0.464723i −0.0124035 + 0.102152i
79.5 −0.890994 0.467630i 0.970942 + 0.239316i −0.560937 0.812658i −1.54798 4.08168i −0.753192 0.667270i −0.324090 2.66912i 0.362350 + 2.98422i 0.885456 + 0.464723i −0.529477 + 4.36063i
See next 80 embeddings (of 180 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 469.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
169.g even 13 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.m.a 180
169.g even 13 1 inner 507.2.m.a 180
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.m.a 180 1.a even 1 1 trivial
507.2.m.a 180 169.g even 13 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(37\!\cdots\!11\)\( T_{2}^{154} + \)\(37\!\cdots\!12\)\( T_{2}^{153} + \)\(25\!\cdots\!50\)\( T_{2}^{152} + \)\(24\!\cdots\!05\)\( T_{2}^{151} + \)\(16\!\cdots\!02\)\( T_{2}^{150} + \)\(15\!\cdots\!42\)\( T_{2}^{149} + \)\(10\!\cdots\!78\)\( T_{2}^{148} + \)\(92\!\cdots\!41\)\( T_{2}^{147} + \)\(60\!\cdots\!37\)\( T_{2}^{146} + \)\(58\!\cdots\!21\)\( T_{2}^{145} + \)\(36\!\cdots\!03\)\( T_{2}^{144} + \)\(37\!\cdots\!15\)\( T_{2}^{143} + \)\(21\!\cdots\!89\)\( T_{2}^{142} + \)\(24\!\cdots\!05\)\( T_{2}^{141} + \)\(12\!\cdots\!49\)\( T_{2}^{140} + \)\(15\!\cdots\!83\)\( T_{2}^{139} + \)\(73\!\cdots\!13\)\( T_{2}^{138} + \)\(87\!\cdots\!72\)\( T_{2}^{137} + \)\(39\!\cdots\!17\)\( T_{2}^{136} + \)\(46\!\cdots\!70\)\( T_{2}^{135} + \)\(20\!\cdots\!61\)\( T_{2}^{134} + \)\(23\!\cdots\!65\)\( T_{2}^{133} + \)\(10\!\cdots\!33\)\( T_{2}^{132} + \)\(11\!\cdots\!56\)\( T_{2}^{131} + \)\(49\!\cdots\!30\)\( T_{2}^{130} + \)\(52\!\cdots\!74\)\( T_{2}^{129} + \)\(21\!\cdots\!05\)\( T_{2}^{128} + \)\(21\!\cdots\!91\)\( T_{2}^{127} + \)\(89\!\cdots\!45\)\( T_{2}^{126} + \)\(89\!\cdots\!62\)\( T_{2}^{125} + \)\(35\!\cdots\!56\)\( T_{2}^{124} + \)\(34\!\cdots\!41\)\( T_{2}^{123} + \)\(13\!\cdots\!09\)\( T_{2}^{122} + \)\(12\!\cdots\!01\)\( T_{2}^{121} + \)\(48\!\cdots\!51\)\( T_{2}^{120} + \)\(43\!\cdots\!06\)\( T_{2}^{119} + \)\(16\!\cdots\!32\)\( T_{2}^{118} + \)\(14\!\cdots\!95\)\( T_{2}^{117} + \)\(56\!\cdots\!70\)\( T_{2}^{116} + \)\(47\!\cdots\!69\)\( T_{2}^{115} + \)\(19\!\cdots\!79\)\( T_{2}^{114} + \)\(15\!\cdots\!10\)\( T_{2}^{113} + \)\(65\!\cdots\!80\)\( T_{2}^{112} + \)\(44\!\cdots\!84\)\( T_{2}^{111} + \)\(21\!\cdots\!77\)\( T_{2}^{110} + \)\(12\!\cdots\!78\)\( T_{2}^{109} + \)\(70\!\cdots\!78\)\( T_{2}^{108} + \)\(39\!\cdots\!87\)\( T_{2}^{107} + \)\(20\!\cdots\!55\)\( T_{2}^{106} + \)\(12\!\cdots\!95\)\( T_{2}^{105} + \)\(58\!\cdots\!53\)\( T_{2}^{104} + \)\(38\!\cdots\!58\)\( T_{2}^{103} + \)\(16\!\cdots\!24\)\( T_{2}^{102} + \)\(95\!\cdots\!44\)\( T_{2}^{101} + \)\(49\!\cdots\!18\)\( T_{2}^{100} + \)\(18\!\cdots\!90\)\( T_{2}^{99} + \)\(14\!\cdots\!63\)\( T_{2}^{98} + \)\(28\!\cdots\!87\)\( T_{2}^{97} + \)\(38\!\cdots\!94\)\( T_{2}^{96} + \)\(40\!\cdots\!10\)\( T_{2}^{95} + \)\(98\!\cdots\!10\)\( T_{2}^{94} + \)\(92\!\cdots\!53\)\( T_{2}^{93} + \)\(25\!\cdots\!74\)\( T_{2}^{92} + \)\(26\!\cdots\!17\)\( T_{2}^{91} + \)\(63\!\cdots\!35\)\( T_{2}^{90} + \)\(88\!\cdots\!25\)\( T_{2}^{89} + \)\(15\!\cdots\!58\)\( T_{2}^{88} + \)\(12\!\cdots\!49\)\( T_{2}^{87} + \)\(36\!\cdots\!49\)\( T_{2}^{86} - \)\(11\!\cdots\!21\)\( T_{2}^{85} + \)\(85\!\cdots\!61\)\( T_{2}^{84} - \)\(96\!\cdots\!99\)\( T_{2}^{83} + \)\(19\!\cdots\!22\)\( T_{2}^{82} - \)\(40\!\cdots\!87\)\( T_{2}^{81} + \)\(42\!\cdots\!89\)\( T_{2}^{80} - \)\(14\!\cdots\!98\)\( T_{2}^{79} + \)\(90\!\cdots\!03\)\( T_{2}^{78} - \)\(38\!\cdots\!69\)\( T_{2}^{77} + \)\(18\!\cdots\!88\)\( T_{2}^{76} - \)\(82\!\cdots\!94\)\( T_{2}^{75} + \)\(34\!\cdots\!72\)\( T_{2}^{74} - \)\(14\!\cdots\!59\)\( T_{2}^{73} + \)\(59\!\cdots\!38\)\( T_{2}^{72} - \)\(19\!\cdots\!41\)\( T_{2}^{71} + \)\(97\!\cdots\!97\)\( T_{2}^{70} - \)\(12\!\cdots\!29\)\( T_{2}^{69} + \)\(15\!\cdots\!35\)\( T_{2}^{68} + \)\(16\!\cdots\!09\)\( T_{2}^{67} + \)\(21\!\cdots\!57\)\( T_{2}^{66} + \)\(59\!\cdots\!25\)\( T_{2}^{65} + \)\(26\!\cdots\!32\)\( T_{2}^{64} + \)\(11\!\cdots\!12\)\( T_{2}^{63} + \)\(32\!\cdots\!18\)\( T_{2}^{62} + \)\(20\!\cdots\!21\)\( T_{2}^{61} + \)\(41\!\cdots\!68\)\( T_{2}^{60} + \)\(29\!\cdots\!61\)\( T_{2}^{59} + \)\(43\!\cdots\!53\)\( T_{2}^{58} + \)\(28\!\cdots\!48\)\( T_{2}^{57} + \)\(41\!\cdots\!43\)\( T_{2}^{56} + \)\(27\!\cdots\!81\)\( T_{2}^{55} + \)\(35\!\cdots\!96\)\( T_{2}^{54} + \)\(20\!\cdots\!46\)\( T_{2}^{53} + \)\(23\!\cdots\!11\)\( T_{2}^{52} + \)\(80\!\cdots\!08\)\( T_{2}^{51} + \)\(13\!\cdots\!70\)\( T_{2}^{50} + \)\(47\!\cdots\!51\)\( T_{2}^{49} + \)\(13\!\cdots\!53\)\( T_{2}^{48} + \)\(43\!\cdots\!48\)\( T_{2}^{47} + \)\(11\!\cdots\!62\)\( T_{2}^{46} + \)\(71\!\cdots\!36\)\( T_{2}^{45} + \)\(11\!\cdots\!08\)\( T_{2}^{44} + \)\(71\!\cdots\!89\)\( T_{2}^{43} + \)\(73\!\cdots\!45\)\( T_{2}^{42} + \)\(58\!\cdots\!47\)\( T_{2}^{41} + \)\(53\!\cdots\!01\)\( T_{2}^{40} + \)\(26\!\cdots\!58\)\( T_{2}^{39} + \)\(25\!\cdots\!55\)\( T_{2}^{38} + \)\(13\!\cdots\!77\)\( T_{2}^{37} + \)\(91\!\cdots\!25\)\( T_{2}^{36} + \)\(48\!\cdots\!81\)\( T_{2}^{35} + \)\(18\!\cdots\!24\)\( T_{2}^{34} + \)\(12\!\cdots\!87\)\( T_{2}^{33} + \)\(68\!\cdots\!81\)\( T_{2}^{32} + \)\(16\!\cdots\!79\)\( T_{2}^{31} + \)\(18\!\cdots\!28\)\( T_{2}^{30} + \)\(41\!\cdots\!10\)\( T_{2}^{29} + \)\(32\!\cdots\!26\)\( T_{2}^{28} + \)\(72\!\cdots\!99\)\( T_{2}^{27} + \)\(73\!\cdots\!19\)\( T_{2}^{26} - \)\(27\!\cdots\!65\)\( T_{2}^{25} + \)\(14\!\cdots\!54\)\( T_{2}^{24} - \)\(11\!\cdots\!74\)\( T_{2}^{23} + \)\(16\!\cdots\!88\)\( T_{2}^{22} - \)\(15\!\cdots\!80\)\( T_{2}^{21} + \)\(30\!\cdots\!80\)\( T_{2}^{20} - \)\(81\!\cdots\!43\)\( T_{2}^{19} + \)\(42\!\cdots\!34\)\( T_{2}^{18} - \)\(10\!\cdots\!80\)\( T_{2}^{17} + \)\(29\!\cdots\!55\)\( T_{2}^{16} - \)\(54\!\cdots\!59\)\( T_{2}^{15} + \)\(95\!\cdots\!93\)\( T_{2}^{14} - \)\(12\!\cdots\!97\)\( T_{2}^{13} + \)\(12\!\cdots\!21\)\( T_{2}^{12} - \)\(81\!\cdots\!84\)\( T_{2}^{11} + \)\(30\!\cdots\!51\)\( T_{2}^{10} - \)\(11\!\cdots\!15\)\( T_{2}^{9} - \)\(37\!\cdots\!29\)\( T_{2}^{8} + \)\(16\!\cdots\!26\)\( T_{2}^{7} - \)\(78\!\cdots\!72\)\( T_{2}^{6} - \)\(55\!\cdots\!90\)\( T_{2}^{5} + 212730439324 T_{2}^{4} - 7505345263 T_{2}^{3} + 204668126 T_{2}^{2} + 5540834 T_{2} + 28561 \)">\(T_{2}^{180} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\).