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Properties

Label 507.2.m.b.40.11
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 40.11
Character \(\chi\) \(=\) 507.40
Dual form 507.2.m.b.469.11

$q$-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+(0.781588 + 0.192644i) q^{2} +(0.120537 - 0.992709i) q^{3} +(-1.19714 - 0.628310i) q^{4} +(-1.69825 + 2.46034i) q^{5} +(0.285450 - 0.752669i) q^{6} +(1.79485 + 1.59010i) q^{7} +(-2.01970 - 1.78930i) q^{8} +(-0.970942 - 0.239316i) q^{9} +(-1.80130 + 1.59582i) q^{10} +(5.73460 - 1.41345i) q^{11} +(-0.768028 + 1.11268i) q^{12} +(3.30660 - 1.43750i) q^{13} +(1.09651 + 1.58857i) q^{14} +(2.23770 + 1.98243i) q^{15} +(0.302178 + 0.437781i) q^{16} +(4.38340 + 3.88336i) q^{17} +(-0.712774 - 0.374093i) q^{18} +7.56977 q^{19} +(3.57891 - 1.87836i) q^{20} +(1.79485 - 1.59010i) q^{21} +4.75439 q^{22} -8.81892 q^{23} +(-2.01970 + 1.78930i) q^{24} +(-1.39620 - 3.68147i) q^{25} +(2.86132 - 0.486535i) q^{26} +(-0.354605 + 0.935016i) q^{27} +(-1.14962 - 3.03130i) q^{28} +(-3.11947 - 0.768880i) q^{29} +(1.36706 + 1.98052i) q^{30} +(-2.69919 + 7.11717i) q^{31} +(2.06550 + 5.44628i) q^{32} +(-0.711916 - 5.86316i) q^{33} +(2.67791 + 3.87962i) q^{34} +(-6.96029 + 1.71556i) q^{35} +(1.01199 + 0.896547i) q^{36} +(2.27202 - 5.99081i) q^{37} +(5.91644 + 1.45827i) q^{38} +(-1.02845 - 3.45576i) q^{39} +(7.83226 - 1.93048i) q^{40} +(-0.0958837 + 0.789673i) q^{41} +(1.70916 - 0.897035i) q^{42} +(-0.595068 - 1.56906i) q^{43} +(-7.75322 - 1.91100i) q^{44} +(2.23770 - 1.98243i) q^{45} +(-6.89276 - 1.69891i) q^{46} +(6.55556 - 3.44062i) q^{47} +(0.471012 - 0.247206i) q^{48} +(-0.150682 - 1.24098i) q^{49} +(-0.382038 - 3.14637i) q^{50} +(4.38340 - 3.88336i) q^{51} +(-4.86167 - 0.356675i) q^{52} +(-2.50886 - 2.22266i) q^{53} +(-0.457280 + 0.662485i) q^{54} +(-6.26121 + 16.5095i) q^{55} +(-0.779901 - 6.42306i) q^{56} +(0.912434 - 7.51457i) q^{57} +(-2.29002 - 1.20190i) q^{58} +(-5.51294 + 7.98688i) q^{59} +(-1.43327 - 3.77922i) q^{60} +(3.45553 - 3.06134i) q^{61} +(-3.48073 + 5.04271i) q^{62} +(-1.36216 - 1.97343i) q^{63} +(0.436940 + 3.59853i) q^{64} +(-2.07870 + 10.5766i) q^{65} +(0.573078 - 4.71972i) q^{66} +(-2.10923 + 1.10701i) q^{67} +(-2.80762 - 7.40307i) q^{68} +(-1.06300 + 8.75462i) q^{69} -5.77058 q^{70} +(-0.623533 + 5.13526i) q^{71} +(1.53281 + 2.22066i) q^{72} +(2.74130 - 0.675671i) q^{73} +(2.92988 - 4.24466i) q^{74} +(-3.82293 + 0.942267i) q^{75} +(-9.06210 - 4.75616i) q^{76} +(12.5403 + 6.58164i) q^{77} +(-0.138093 - 2.89911i) q^{78} +(1.10300 - 0.578898i) q^{79} -1.59026 q^{80} +(0.885456 + 0.464723i) q^{81} +(-0.227067 + 0.598728i) q^{82} +(-0.263030 - 2.16624i) q^{83} +(-3.14777 + 0.775856i) q^{84} +(-16.9985 + 4.18976i) q^{85} +(-0.162827 - 1.34100i) q^{86} +(-1.13928 + 3.00405i) q^{87} +(-14.1113 - 7.40617i) q^{88} -0.866479 q^{89} +(2.13086 - 1.11836i) q^{90} +(8.22062 + 2.67773i) q^{91} +(10.5575 + 5.54101i) q^{92} +(6.73993 + 3.53739i) q^{93} +(5.78657 - 1.42626i) q^{94} +(-12.8554 + 18.6242i) q^{95} +(5.65554 - 1.39397i) q^{96} +(-8.64326 - 12.5219i) q^{97} +(0.121296 - 0.998963i) q^{98} -5.90622 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.781588 + 0.192644i 0.552666 + 0.136220i 0.505752 − 0.862679i \(-0.331215\pi\)
0.0469142 + 0.998899i \(0.485061\pi\)
\(3\) 0.120537 − 0.992709i 0.0695919 − 0.573141i
\(4\) −1.19714 − 0.628310i −0.598572 − 0.314155i
\(5\) −1.69825 + 2.46034i −0.759481 + 1.10030i 0.232337 + 0.972635i \(0.425363\pi\)
−0.991818 + 0.127663i \(0.959252\pi\)
\(6\) 0.285450 − 0.752669i 0.116534 − 0.307276i
\(7\) 1.79485 + 1.59010i 0.678390 + 0.601001i 0.930322 − 0.366744i \(-0.119527\pi\)
−0.251932 + 0.967745i \(0.581066\pi\)
\(8\) −2.01970 − 1.78930i −0.714073 − 0.632614i
\(9\) −0.970942 − 0.239316i −0.323647 − 0.0797719i
\(10\) −1.80130 + 1.59582i −0.569622 + 0.504641i
\(11\) 5.73460 − 1.41345i 1.72905 − 0.426172i 0.755590 − 0.655045i \(-0.227350\pi\)
0.973456 + 0.228874i \(0.0735042\pi\)
\(12\) −0.768028 + 1.11268i −0.221711 + 0.321203i
\(13\) 3.30660 − 1.43750i 0.917086 − 0.398690i
\(14\) 1.09651 + 1.58857i 0.293055 + 0.424563i
\(15\) 2.23770 + 1.98243i 0.577772 + 0.511861i
\(16\) 0.302178 + 0.437781i 0.0755446 + 0.109445i
\(17\) 4.38340 + 3.88336i 1.06313 + 0.941853i 0.998428 − 0.0560520i \(-0.0178512\pi\)
0.0647039 + 0.997905i \(0.479390\pi\)
\(18\) −0.712774 − 0.374093i −0.168002 − 0.0881745i
\(19\) 7.56977 1.73662 0.868312 − 0.496019i \(-0.165205\pi\)
0.868312 + 0.496019i \(0.165205\pi\)
\(20\) 3.57891 − 1.87836i 0.800268 − 0.420013i
\(21\) 1.79485 − 1.59010i 0.391669 − 0.346988i
\(22\) 4.75439 1.01364
\(23\) −8.81892 −1.83887 −0.919435 − 0.393241i \(-0.871354\pi\)
−0.919435 + 0.393241i \(0.871354\pi\)
\(24\) −2.01970 + 1.78930i −0.412270 + 0.365240i
\(25\) −1.39620 − 3.68147i −0.279240 − 0.736295i
\(26\) 2.86132 − 0.486535i 0.561152 − 0.0954173i
\(27\) −0.354605 + 0.935016i −0.0682437 + 0.179944i
\(28\) −1.14962 − 3.03130i −0.217258 − 0.572862i
\(29\) −3.11947 − 0.768880i −0.579271 − 0.142778i −0.0612157 − 0.998125i \(-0.519498\pi\)
−0.518055 + 0.855347i \(0.673344\pi\)
\(30\) 1.36706 + 1.98052i 0.249589 + 0.361593i
\(31\) −2.69919 + 7.11717i −0.484788 + 1.27828i 0.440068 + 0.897964i \(0.354954\pi\)
−0.924856 + 0.380317i \(0.875815\pi\)
\(32\) 2.06550 + 5.44628i 0.365133 + 0.962775i
\(33\) −0.711916 − 5.86316i −0.123929 − 1.02064i
\(34\) 2.67791 + 3.87962i 0.459258 + 0.665350i
\(35\) −6.96029 + 1.71556i −1.17650 + 0.289982i
\(36\) 1.01199 + 0.896547i 0.168665 + 0.149425i
\(37\) 2.27202 − 5.99081i 0.373517 − 0.984884i −0.608073 − 0.793881i \(-0.708057\pi\)
0.981590 − 0.191002i \(-0.0611737\pi\)
\(38\) 5.91644 + 1.45827i 0.959773 + 0.236563i
\(39\) −1.02845 − 3.45576i −0.164684 − 0.553365i
\(40\) 7.83226 − 1.93048i 1.23839 − 0.305235i
\(41\) −0.0958837 + 0.789673i −0.0149745 + 0.123326i −0.998279 − 0.0586408i \(-0.981323\pi\)
0.983305 + 0.181967i \(0.0582464\pi\)
\(42\) 1.70916 − 0.897035i 0.263729 − 0.138416i
\(43\) −0.595068 − 1.56906i −0.0907470 − 0.239280i 0.881957 − 0.471330i \(-0.156226\pi\)
−0.972704 + 0.232050i \(0.925457\pi\)
\(44\) −7.75322 − 1.91100i −1.16884 − 0.288094i
\(45\) 2.23770 − 1.98243i 0.333577 − 0.295523i
\(46\) −6.89276 − 1.69891i −1.01628 − 0.250491i
\(47\) 6.55556 − 3.44062i 0.956227 − 0.501867i 0.0869613 − 0.996212i \(-0.472284\pi\)
0.869266 + 0.494345i \(0.164592\pi\)
\(48\) 0.471012 − 0.247206i 0.0679848 − 0.0356812i
\(49\) −0.150682 − 1.24098i −0.0215260 − 0.177283i
\(50\) −0.382038 − 3.14637i −0.0540283 − 0.444963i
\(51\) 4.38340 − 3.88336i 0.613799 − 0.543779i
\(52\) −4.86167 − 0.356675i −0.674192 − 0.0494620i
\(53\) −2.50886 − 2.22266i −0.344618 − 0.305305i 0.473004 − 0.881060i \(-0.343170\pi\)
−0.817622 + 0.575755i \(0.804708\pi\)
\(54\) −0.457280 + 0.662485i −0.0622280 + 0.0901528i
\(55\) −6.26121 + 16.5095i −0.844262 + 2.22614i
\(56\) −0.779901 − 6.42306i −0.104219 − 0.858318i
\(57\) 0.912434 − 7.51457i 0.120855 − 0.995330i
\(58\) −2.29002 − 1.20190i −0.300694 − 0.157817i
\(59\) −5.51294 + 7.98688i −0.717724 + 1.03980i 0.279107 + 0.960260i \(0.409961\pi\)
−0.996831 + 0.0795430i \(0.974654\pi\)
\(60\) −1.43327 − 3.77922i −0.185034 − 0.487896i
\(61\) 3.45553 − 3.06134i 0.442436 − 0.391964i −0.412283 − 0.911056i \(-0.635268\pi\)
0.854718 + 0.519092i \(0.173730\pi\)
\(62\) −3.48073 + 5.04271i −0.442054 + 0.640425i
\(63\) −1.36216 − 1.97343i −0.171616 − 0.248629i
\(64\) 0.436940 + 3.59853i 0.0546175 + 0.449816i
\(65\) −2.07870 + 10.5766i −0.257831 + 1.31187i
\(66\) 0.573078 − 4.71972i 0.0705410 − 0.580957i
\(67\) −2.10923 + 1.10701i −0.257683 + 0.135243i −0.588619 − 0.808411i \(-0.700328\pi\)
0.330936 + 0.943653i \(0.392636\pi\)
\(68\) −2.80762 − 7.40307i −0.340473 − 0.897754i
\(69\) −1.06300 + 8.75462i −0.127970 + 1.05393i
\(70\) −5.77058 −0.689716
\(71\) −0.623533 + 5.13526i −0.0739998 + 0.609443i 0.907784 + 0.419439i \(0.137773\pi\)
−0.981783 + 0.190004i \(0.939150\pi\)
\(72\) 1.53281 + 2.22066i 0.180643 + 0.261707i
\(73\) 2.74130 − 0.675671i 0.320845 − 0.0790813i −0.0756039 − 0.997138i \(-0.524088\pi\)
0.396449 + 0.918057i \(0.370242\pi\)
\(74\) 2.92988 − 4.24466i 0.340591 − 0.493431i
\(75\) −3.82293 + 0.942267i −0.441433 + 0.108804i
\(76\) −9.06210 − 4.75616i −1.03949 − 0.545569i
\(77\) 12.5403 + 6.58164i 1.42910 + 0.750048i
\(78\) −0.138093 − 2.89911i −0.0156359 − 0.328259i
\(79\) 1.10300 − 0.578898i 0.124097 − 0.0651311i −0.401532 − 0.915845i \(-0.631522\pi\)
0.525629 + 0.850714i \(0.323830\pi\)
\(80\) −1.59026 −0.177797
\(81\) 0.885456 + 0.464723i 0.0983840 + 0.0516359i
\(82\) −0.227067 + 0.598728i −0.0250754 + 0.0661184i
\(83\) −0.263030 − 2.16624i −0.0288713 − 0.237776i 0.971128 − 0.238557i \(-0.0766745\pi\)
−1.00000 0.000781041i \(0.999751\pi\)
\(84\) −3.14777 + 0.775856i −0.343450 + 0.0846528i
\(85\) −16.9985 + 4.18976i −1.84375 + 0.454443i
\(86\) −0.162827 − 1.34100i −0.0175581 − 0.144604i
\(87\) −1.13928 + 3.00405i −0.122144 + 0.322068i
\(88\) −14.1113 − 7.40617i −1.50427 − 0.789501i
\(89\) −0.866479 −0.0918466 −0.0459233 − 0.998945i \(-0.514623\pi\)
−0.0459233 + 0.998945i \(0.514623\pi\)
\(90\) 2.13086 − 1.11836i 0.224613 − 0.117886i
\(91\) 8.22062 + 2.67773i 0.861755 + 0.280702i
\(92\) 10.5575 + 5.54101i 1.10070 + 0.577690i
\(93\) 6.73993 + 3.53739i 0.698898 + 0.366810i
\(94\) 5.78657 − 1.42626i 0.596839 − 0.147108i
\(95\) −12.8554 + 18.6242i −1.31893 + 1.91080i
\(96\) 5.65554 − 1.39397i 0.577216 − 0.142271i
\(97\) −8.64326 − 12.5219i −0.877590 − 1.27141i −0.961464 − 0.274932i \(-0.911345\pi\)
0.0838737 − 0.996476i \(-0.473271\pi\)
\(98\) 0.121296 − 0.998963i 0.0122528 − 0.100911i
\(99\) −5.90622 −0.593597
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.40.11 ✓ 204
169.131 even 13 inner 507.2.m.b.469.11 yes 204
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.m.b.40.11 ✓ 204 1.1 even 1 trivial
507.2.m.b.469.11 yes 204 169.131 even 13 inner