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Properties

Label 507.2.m.b.40.17
Level $507$
Weight $2$
Character 507.40
Analytic conductor $4.048$
Analytic rank $0$
Dimension $204$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [507,2,Mod(40,507)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("507.40"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(507, base_ring=CyclotomicField(26)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.m (of order \(13\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [204] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(204\)
Relative dimension: \(17\) over \(\Q(\zeta_{13})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{13}]$

Embedding invariants

Embedding label 40.17
Character \(\chi\) \(=\) 507.40
Dual form 507.2.m.b.469.17

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.66897 + 0.657842i) q^{2} +(0.120537 - 0.992709i) q^{3} +(4.91973 + 2.58207i) q^{4} +(0.113107 - 0.163863i) q^{5} +(0.974754 - 2.57022i) q^{6} +(-1.95577 - 1.73266i) q^{7} +(7.31692 + 6.48223i) q^{8} +(-0.970942 - 0.239316i) q^{9} +(0.409674 - 0.362939i) q^{10} +(2.78789 - 0.687153i) q^{11} +(3.15625 - 4.57262i) q^{12} +(-3.28223 + 1.49231i) q^{13} +(-4.08008 - 5.91102i) q^{14} +(-0.149035 - 0.132033i) q^{15} +(8.95186 + 12.9690i) q^{16} +(-2.02855 - 1.79714i) q^{17} +(-2.43398 - 1.27745i) q^{18} -1.21614 q^{19} +(0.979560 - 0.514113i) q^{20} +(-1.95577 + 1.73266i) q^{21} +7.89283 q^{22} -6.60947 q^{23} +(7.31692 - 6.48223i) q^{24} +(1.75897 + 4.63801i) q^{25} +(-9.74186 + 1.82375i) q^{26} +(-0.354605 + 0.935016i) q^{27} +(-5.14801 - 13.5742i) q^{28} +(4.36264 + 1.07529i) q^{29} +(-0.310912 - 0.450434i) q^{30} +(-2.66381 + 7.02388i) q^{31} +(8.42793 + 22.2226i) q^{32} +(-0.346100 - 2.85039i) q^{33} +(-4.23190 - 6.13096i) q^{34} +(-0.505130 + 0.124503i) q^{35} +(-4.15884 - 3.68441i) q^{36} +(3.75089 - 9.89028i) q^{37} +(-3.24585 - 0.800030i) q^{38} +(1.08580 + 3.43817i) q^{39} +(1.88979 - 0.465792i) q^{40} +(0.662454 - 5.45580i) q^{41} +(-6.35972 + 3.33784i) q^{42} +(-2.31766 - 6.11117i) q^{43} +(15.4899 + 3.81792i) q^{44} +(-0.149035 + 0.132033i) q^{45} +(-17.6405 - 4.34798i) q^{46} +(-8.48580 + 4.45369i) q^{47} +(13.9535 - 7.32335i) q^{48} +(-0.0208321 - 0.171568i) q^{49} +(1.64355 + 13.5358i) q^{50} +(-2.02855 + 1.79714i) q^{51} +(-20.0009 - 1.13318i) q^{52} +(0.727118 + 0.644170i) q^{53} +(-1.56152 + 2.26226i) q^{54} +(0.202729 - 0.534554i) q^{55} +(-3.07872 - 25.3555i) q^{56} +(-0.146590 + 1.20728i) q^{57} +(10.9364 + 5.73986i) q^{58} +(0.0165082 - 0.0239163i) q^{59} +(-0.392291 - 1.03439i) q^{60} +(-1.85348 + 1.64204i) q^{61} +(-11.7302 + 16.9942i) q^{62} +(1.48429 + 2.15036i) q^{63} +(4.07595 + 33.5685i) q^{64} +(-0.126707 + 0.706626i) q^{65} +(0.951375 - 7.83528i) q^{66} +(-0.551770 + 0.289591i) q^{67} +(-5.33956 - 14.0793i) q^{68} +(-0.796683 + 6.56128i) q^{69} -1.43008 q^{70} +(-1.67773 + 13.8174i) q^{71} +(-5.55301 - 8.04492i) q^{72} +(7.77596 - 1.91660i) q^{73} +(16.5172 - 23.9294i) q^{74} +(4.81622 - 1.18709i) q^{75} +(-5.98309 - 3.14017i) q^{76} +(-6.64308 - 3.48656i) q^{77} +(0.636197 + 9.89066i) q^{78} +(9.43685 - 4.95284i) q^{79} +3.13766 q^{80} +(0.885456 + 0.464723i) q^{81} +(5.35712 - 14.1256i) q^{82} +(-1.56174 - 12.8621i) q^{83} +(-14.0957 + 3.47429i) q^{84} +(-0.523926 + 0.129136i) q^{85} +(-2.16558 - 17.8352i) q^{86} +(1.59331 - 4.20122i) q^{87} +(24.8531 + 13.0439i) q^{88} +3.85120 q^{89} +(-0.484627 + 0.254352i) q^{90} +(9.00496 + 2.76837i) q^{91} +(-32.5168 - 17.0661i) q^{92} +(6.65158 + 3.49102i) q^{93} +(-25.5782 + 6.30445i) q^{94} +(-0.137554 + 0.199281i) q^{95} +(23.0765 - 5.68784i) q^{96} +(6.19553 + 8.97577i) q^{97} +(0.0572643 - 0.471614i) q^{98} -2.87132 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 204 q - q^{2} - 17 q^{3} - 21 q^{4} - 6 q^{5} - q^{6} - 8 q^{7} - 9 q^{8} - 17 q^{9} - 6 q^{10} - 8 q^{11} - 21 q^{12} + 54 q^{13} - 30 q^{14} - 6 q^{15} - 45 q^{16} - 18 q^{17} - q^{18} - 20 q^{19}+ \cdots + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{13}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.66897 + 0.657842i 1.88725 + 0.465164i 0.999992 + 0.00388245i \(0.00123583\pi\)
0.887254 + 0.461282i \(0.152610\pi\)
\(3\) 0.120537 − 0.992709i 0.0695919 − 0.573141i
\(4\) 4.91973 + 2.58207i 2.45986 + 1.29104i
\(5\) 0.113107 − 0.163863i 0.0505828 − 0.0732818i −0.796876 − 0.604142i \(-0.793516\pi\)
0.847459 + 0.530861i \(0.178131\pi\)
\(6\) 0.974754 − 2.57022i 0.397942 − 1.04929i
\(7\) −1.95577 − 1.73266i −0.739213 − 0.654885i 0.207033 − 0.978334i \(-0.433619\pi\)
−0.946246 + 0.323449i \(0.895158\pi\)
\(8\) 7.31692 + 6.48223i 2.58692 + 2.29181i
\(9\) −0.970942 − 0.239316i −0.323647 − 0.0797719i
\(10\) 0.409674 − 0.362939i 0.129550 − 0.114772i
\(11\) 2.78789 − 0.687153i 0.840580 − 0.207184i 0.204557 − 0.978855i \(-0.434425\pi\)
0.636023 + 0.771670i \(0.280578\pi\)
\(12\) 3.15625 − 4.57262i 0.911132 − 1.32000i
\(13\) −3.28223 + 1.49231i −0.910326 + 0.413893i
\(14\) −4.08008 − 5.91102i −1.09045 − 1.57979i
\(15\) −0.149035 − 0.132033i −0.0384806 − 0.0340909i
\(16\) 8.95186 + 12.9690i 2.23796 + 3.24225i
\(17\) −2.02855 − 1.79714i −0.491995 − 0.435870i 0.380290 − 0.924867i \(-0.375824\pi\)
−0.872285 + 0.488998i \(0.837363\pi\)
\(18\) −2.43398 − 1.27745i −0.573695 − 0.301098i
\(19\) −1.21614 −0.279002 −0.139501 − 0.990222i \(-0.544550\pi\)
−0.139501 + 0.990222i \(0.544550\pi\)
\(20\) 0.979560 − 0.514113i 0.219036 − 0.114959i
\(21\) −1.95577 + 1.73266i −0.426785 + 0.378098i
\(22\) 7.89283 1.68276
\(23\) −6.60947 −1.37817 −0.689085 − 0.724681i \(-0.741987\pi\)
−0.689085 + 0.724681i \(0.741987\pi\)
\(24\) 7.31692 − 6.48223i 1.49356 − 1.32318i
\(25\) 1.75897 + 4.63801i 0.351793 + 0.927603i
\(26\) −9.74186 + 1.82375i −1.91054 + 0.357666i
\(27\) −0.354605 + 0.935016i −0.0682437 + 0.179944i
\(28\) −5.14801 − 13.5742i −0.972882 − 2.56528i
\(29\) 4.36264 + 1.07529i 0.810122 + 0.199677i 0.622557 − 0.782575i \(-0.286094\pi\)
0.187566 + 0.982252i \(0.439940\pi\)
\(30\) −0.310912 − 0.450434i −0.0567646 − 0.0822377i
\(31\) −2.66381 + 7.02388i −0.478434 + 1.26153i 0.450996 + 0.892526i \(0.351069\pi\)
−0.929429 + 0.369000i \(0.879700\pi\)
\(32\) 8.42793 + 22.2226i 1.48986 + 3.92844i
\(33\) −0.346100 − 2.85039i −0.0602482 − 0.496189i
\(34\) −4.23190 − 6.13096i −0.725765 − 1.05145i
\(35\) −0.505130 + 0.124503i −0.0853826 + 0.0210449i
\(36\) −4.15884 − 3.68441i −0.693140 − 0.614068i
\(37\) 3.75089 − 9.89028i 0.616642 − 1.62595i −0.153550 − 0.988141i \(-0.549071\pi\)
0.770193 − 0.637811i \(-0.220160\pi\)
\(38\) −3.24585 − 0.800030i −0.526546 − 0.129782i
\(39\) 1.08580 + 3.43817i 0.173867 + 0.550548i
\(40\) 1.88979 − 0.465792i 0.298802 − 0.0736481i
\(41\) 0.662454 − 5.45580i 0.103458 − 0.852053i −0.844916 − 0.534899i \(-0.820350\pi\)
0.948374 − 0.317154i \(-0.102727\pi\)
\(42\) −6.35972 + 3.33784i −0.981326 + 0.515039i
\(43\) −2.31766 − 6.11117i −0.353440 − 0.931945i −0.987353 − 0.158540i \(-0.949321\pi\)
0.633913 − 0.773405i \(-0.281448\pi\)
\(44\) 15.4899 + 3.81792i 2.33519 + 0.575574i
\(45\) −0.149035 + 0.132033i −0.0222168 + 0.0196824i
\(46\) −17.6405 − 4.34798i −2.60095 − 0.641075i
\(47\) −8.48580 + 4.45369i −1.23778 + 0.649638i −0.951246 − 0.308432i \(-0.900196\pi\)
−0.286535 + 0.958070i \(0.592503\pi\)
\(48\) 13.9535 − 7.32335i 2.01401 − 1.05703i
\(49\) −0.0208321 − 0.171568i −0.00297602 − 0.0245097i
\(50\) 1.64355 + 13.5358i 0.232433 + 1.91426i
\(51\) −2.02855 + 1.79714i −0.284053 + 0.251649i
\(52\) −20.0009 − 1.13318i −2.77363 − 0.157144i
\(53\) 0.727118 + 0.644170i 0.0998773 + 0.0884835i 0.711584 − 0.702601i \(-0.247978\pi\)
−0.611707 + 0.791085i \(0.709517\pi\)
\(54\) −1.56152 + 2.26226i −0.212496 + 0.307854i
\(55\) 0.202729 − 0.534554i 0.0273360 − 0.0720792i
\(56\) −3.07872 − 25.3555i −0.411411 − 3.38828i
\(57\) −0.146590 + 1.20728i −0.0194163 + 0.159908i
\(58\) 10.9364 + 5.73986i 1.43602 + 0.753680i
\(59\) 0.0165082 − 0.0239163i 0.00214919 − 0.00311364i −0.821908 − 0.569621i \(-0.807090\pi\)
0.824057 + 0.566507i \(0.191706\pi\)
\(60\) −0.392291 − 1.03439i −0.0506446 − 0.133539i
\(61\) −1.85348 + 1.64204i −0.237313 + 0.210241i −0.773394 − 0.633925i \(-0.781443\pi\)
0.536081 + 0.844167i \(0.319904\pi\)
\(62\) −11.7302 + 16.9942i −1.48974 + 2.15826i
\(63\) 1.48429 + 2.15036i 0.187003 + 0.270920i
\(64\) 4.07595 + 33.5685i 0.509494 + 4.19606i
\(65\) −0.126707 + 0.706626i −0.0157160 + 0.0876462i
\(66\) 0.951375 − 7.83528i 0.117106 − 0.964456i
\(67\) −0.551770 + 0.289591i −0.0674094 + 0.0353792i −0.498091 − 0.867125i \(-0.665965\pi\)
0.430682 + 0.902504i \(0.358273\pi\)
\(68\) −5.33956 − 14.0793i −0.647517 − 1.70736i
\(69\) −0.796683 + 6.56128i −0.0959094 + 0.789885i
\(70\) −1.43008 −0.170927
\(71\) −1.67773 + 13.8174i −0.199110 + 1.63982i 0.460105 + 0.887865i \(0.347812\pi\)
−0.659215 + 0.751955i \(0.729111\pi\)
\(72\) −5.55301 − 8.04492i −0.654428 − 0.948103i
\(73\) 7.77596 − 1.91660i 0.910107 − 0.224321i 0.243649 − 0.969864i \(-0.421656\pi\)
0.666458 + 0.745542i \(0.267809\pi\)
\(74\) 16.5172 − 23.9294i 1.92009 − 2.78173i
\(75\) 4.81622 − 1.18709i 0.556129 − 0.137073i
\(76\) −5.98309 − 3.14017i −0.686308 − 0.360202i
\(77\) −6.64308 − 3.48656i −0.757049 − 0.397330i
\(78\) 0.636197 + 9.89066i 0.0720351 + 1.11990i
\(79\) 9.43685 − 4.95284i 1.06173 − 0.557238i 0.158925 − 0.987291i \(-0.449197\pi\)
0.902804 + 0.430053i \(0.141505\pi\)
\(80\) 3.13766 0.350801
\(81\) 0.885456 + 0.464723i 0.0983840 + 0.0516359i
\(82\) 5.35712 − 14.1256i 0.591595 − 1.55991i
\(83\) −1.56174 − 12.8621i −0.171423 − 1.41180i −0.783693 − 0.621148i \(-0.786666\pi\)
0.612270 − 0.790649i \(-0.290257\pi\)
\(84\) −14.0957 + 3.47429i −1.53797 + 0.379076i
\(85\) −0.523926 + 0.129136i −0.0568278 + 0.0140068i
\(86\) −2.16558 − 17.8352i −0.233521 − 1.92322i
\(87\) 1.59331 − 4.20122i 0.170821 − 0.450418i
\(88\) 24.8531 + 13.0439i 2.64934 + 1.39048i
\(89\) 3.85120 0.408227 0.204113 − 0.978947i \(-0.434569\pi\)
0.204113 + 0.978947i \(0.434569\pi\)
\(90\) −0.484627 + 0.254352i −0.0510841 + 0.0268110i
\(91\) 9.00496 + 2.76837i 0.943977 + 0.290204i
\(92\) −32.5168 − 17.0661i −3.39011 − 1.77927i
\(93\) 6.65158 + 3.49102i 0.689737 + 0.362002i
\(94\) −25.5782 + 6.30445i −2.63819 + 0.650254i
\(95\) −0.137554 + 0.199281i −0.0141127 + 0.0204458i
\(96\) 23.0765 − 5.68784i 2.35523 − 0.580513i
\(97\) 6.19553 + 8.97577i 0.629061 + 0.911352i 0.999849 − 0.0173657i \(-0.00552794\pi\)
−0.370789 + 0.928717i \(0.620913\pi\)
\(98\) 0.0572643 − 0.471614i 0.00578456 − 0.0476402i
\(99\) −2.87132 −0.288579
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 507.2.m.b.40.17 ✓ 204
169.131 even 13 inner 507.2.m.b.469.17 yes 204
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.m.b.40.17 ✓ 204 1.1 even 1 trivial
507.2.m.b.469.17 yes 204 169.131 even 13 inner