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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [507,4,Mod(337,507)] 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("507.337"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(507, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,30,-60,0,0,0,0,90,-80,0,-180,0,-60,0,500,-210] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(29.9139683729\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 70x^{8} + 1645x^{6} + 14700x^{4} + 44100x^{2} + 27648 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.4
Root \(-2.04224i\) of defining polynomial
Character \(\chi\) \(=\) 507.337
Dual form 507.4.b.i.337.7

$q$-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)
 
\(f(q)\) \(=\) \(q-2.04224i q^{2} +3.00000 q^{3} +3.82924 q^{4} -12.0825i q^{5} -6.12673i q^{6} +29.7373i q^{7} -24.1582i q^{8} +9.00000 q^{9} -24.6753 q^{10} +28.0636i q^{11} +11.4877 q^{12} +60.7308 q^{14} -36.2474i q^{15} -18.7029 q^{16} +50.6556 q^{17} -18.3802i q^{18} -105.148i q^{19} -46.2667i q^{20} +89.2119i q^{21} +57.3126 q^{22} +160.592 q^{23} -72.4746i q^{24} -20.9857 q^{25} +27.0000 q^{27} +113.871i q^{28} +140.105 q^{29} -74.0259 q^{30} -223.593i q^{31} -155.070i q^{32} +84.1907i q^{33} -103.451i q^{34} +359.300 q^{35} +34.4632 q^{36} -228.352i q^{37} -214.739 q^{38} -291.890 q^{40} +295.902i q^{41} +182.192 q^{42} -192.103 q^{43} +107.462i q^{44} -108.742i q^{45} -327.968i q^{46} +36.9300i q^{47} -56.1088 q^{48} -541.307 q^{49} +42.8579i q^{50} +151.967 q^{51} +149.102 q^{53} -55.1406i q^{54} +339.077 q^{55} +718.399 q^{56} -315.445i q^{57} -286.128i q^{58} +438.867i q^{59} -138.800i q^{60} +286.146 q^{61} -456.631 q^{62} +267.636i q^{63} -466.313 q^{64} +171.938 q^{66} +537.128i q^{67} +193.973 q^{68} +481.776 q^{69} -733.777i q^{70} -102.729i q^{71} -217.424i q^{72} -75.5209i q^{73} -466.350 q^{74} -62.9571 q^{75} -402.639i q^{76} -834.535 q^{77} +17.5526 q^{79} +225.977i q^{80} +81.0000 q^{81} +604.304 q^{82} -1463.08i q^{83} +341.614i q^{84} -612.044i q^{85} +392.322i q^{86} +420.315 q^{87} +677.965 q^{88} -334.905i q^{89} -222.078 q^{90} +614.946 q^{92} -670.779i q^{93} +75.4200 q^{94} -1270.45 q^{95} -465.209i q^{96} +748.756i q^{97} +1105.48i q^{98} +252.572i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 30 q^{3} - 60 q^{4} + 90 q^{9} - 80 q^{10} - 180 q^{12} - 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} - 960 q^{25} + 270 q^{27} + 990 q^{29} - 240 q^{30} - 120 q^{35} - 540 q^{36}+ \cdots + 2760 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.04224i − 0.722042i −0.932558 − 0.361021i \(-0.882428\pi\)
0.932558 − 0.361021i \(-0.117572\pi\)
\(3\) 3.00000 0.577350
\(4\) 3.82924 0.478656
\(5\) − 12.0825i − 1.08069i −0.841444 − 0.540344i \(-0.818294\pi\)
0.841444 − 0.540344i \(-0.181706\pi\)
\(6\) − 6.12673i − 0.416871i
\(7\) 29.7373i 1.60566i 0.596206 + 0.802832i \(0.296674\pi\)
−0.596206 + 0.802832i \(0.703326\pi\)
\(8\) − 24.1582i − 1.06765i
\(9\) 9.00000 0.333333
\(10\) −24.6753 −0.780302
\(11\) 28.0636i 0.769226i 0.923078 + 0.384613i \(0.125665\pi\)
−0.923078 + 0.384613i \(0.874335\pi\)
\(12\) 11.4877 0.276352
\(13\) 0 0
\(14\) 60.7308 1.15936
\(15\) − 36.2474i − 0.623935i
\(16\) −18.7029 −0.292233
\(17\) 50.6556 0.722693 0.361347 − 0.932432i \(-0.382317\pi\)
0.361347 + 0.932432i \(0.382317\pi\)
\(18\) − 18.3802i − 0.240681i
\(19\) − 105.148i − 1.26962i −0.772670 − 0.634808i \(-0.781079\pi\)
0.772670 − 0.634808i \(-0.218921\pi\)
\(20\) − 46.2667i − 0.517277i
\(21\) 89.2119i 0.927030i
\(22\) 57.3126 0.555413
\(23\) 160.592 1.45590 0.727951 − 0.685629i \(-0.240473\pi\)
0.727951 + 0.685629i \(0.240473\pi\)
\(24\) − 72.4746i − 0.616409i
\(25\) −20.9857 −0.167886
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 113.871i 0.768560i
\(29\) 140.105 0.897132 0.448566 − 0.893750i \(-0.351935\pi\)
0.448566 + 0.893750i \(0.351935\pi\)
\(30\) −74.0259 −0.450507
\(31\) − 223.593i − 1.29544i −0.761880 − 0.647718i \(-0.775724\pi\)
0.761880 − 0.647718i \(-0.224276\pi\)
\(32\) − 155.070i − 0.856647i
\(33\) 84.1907i 0.444113i
\(34\) − 103.451i − 0.521815i
\(35\) 359.300 1.73522
\(36\) 34.4632 0.159552
\(37\) − 228.352i − 1.01462i −0.861765 − 0.507308i \(-0.830641\pi\)
0.861765 − 0.507308i \(-0.169359\pi\)
\(38\) −214.739 −0.916716
\(39\) 0 0
\(40\) −291.890 −1.15380
\(41\) 295.902i 1.12713i 0.826073 + 0.563563i \(0.190570\pi\)
−0.826073 + 0.563563i \(0.809430\pi\)
\(42\) 182.192 0.669354
\(43\) −192.103 −0.681291 −0.340645 − 0.940192i \(-0.610646\pi\)
−0.340645 + 0.940192i \(0.610646\pi\)
\(44\) 107.462i 0.368194i
\(45\) − 108.742i − 0.360229i
\(46\) − 327.968i − 1.05122i
\(47\) 36.9300i 0.114613i 0.998357 + 0.0573063i \(0.0182512\pi\)
−0.998357 + 0.0573063i \(0.981749\pi\)
\(48\) −56.1088 −0.168721
\(49\) −541.307 −1.57815
\(50\) 42.8579i 0.121220i
\(51\) 151.967 0.417247
\(52\) 0 0
\(53\) 149.102 0.386429 0.193214 − 0.981157i \(-0.438109\pi\)
0.193214 + 0.981157i \(0.438109\pi\)
\(54\) − 55.1406i − 0.138957i
\(55\) 339.077 0.831293
\(56\) 718.399 1.71429
\(57\) − 315.445i − 0.733013i
\(58\) − 286.128i − 0.647767i
\(59\) 438.867i 0.968400i 0.874957 + 0.484200i \(0.160889\pi\)
−0.874957 + 0.484200i \(0.839111\pi\)
\(60\) − 138.800i − 0.298650i
\(61\) 286.146 0.600610 0.300305 − 0.953843i \(-0.402912\pi\)
0.300305 + 0.953843i \(0.402912\pi\)
\(62\) −456.631 −0.935358
\(63\) 267.636i 0.535221i
\(64\) −466.313 −0.910768
\(65\) 0 0
\(66\) 171.938 0.320668
\(67\) 537.128i 0.979412i 0.871888 + 0.489706i \(0.162896\pi\)
−0.871888 + 0.489706i \(0.837104\pi\)
\(68\) 193.973 0.345921
\(69\) 481.776 0.840566
\(70\) − 733.777i − 1.25290i
\(71\) − 102.729i − 0.171713i −0.996307 − 0.0858567i \(-0.972637\pi\)
0.996307 − 0.0858567i \(-0.0273627\pi\)
\(72\) − 217.424i − 0.355884i
\(73\) − 75.5209i − 0.121083i −0.998166 − 0.0605414i \(-0.980717\pi\)
0.998166 − 0.0605414i \(-0.0192827\pi\)
\(74\) −466.350 −0.732596
\(75\) −62.9571 −0.0969288
\(76\) − 402.639i − 0.607709i
\(77\) −834.535 −1.23512
\(78\) 0 0
\(79\) 17.5526 0.0249978 0.0124989 − 0.999922i \(-0.496021\pi\)
0.0124989 + 0.999922i \(0.496021\pi\)
\(80\) 225.977i 0.315813i
\(81\) 81.0000 0.111111
\(82\) 604.304 0.813831
\(83\) − 1463.08i − 1.93487i −0.253122 − 0.967434i \(-0.581457\pi\)
0.253122 − 0.967434i \(-0.418543\pi\)
\(84\) 341.614i 0.443728i
\(85\) − 612.044i − 0.781005i
\(86\) 392.322i 0.491920i
\(87\) 420.315 0.517960
\(88\) 677.965 0.821265
\(89\) − 334.905i − 0.398875i −0.979911 − 0.199438i \(-0.936089\pi\)
0.979911 − 0.199438i \(-0.0639115\pi\)
\(90\) −222.078 −0.260101
\(91\) 0 0
\(92\) 614.946 0.696876
\(93\) − 670.779i − 0.747920i
\(94\) 75.4200 0.0827551
\(95\) −1270.45 −1.37206
\(96\) − 465.209i − 0.494585i
\(97\) 748.756i 0.783760i 0.920016 + 0.391880i \(0.128175\pi\)
−0.920016 + 0.391880i \(0.871825\pi\)
\(98\) 1105.48i 1.13949i
\(99\) 252.572i 0.256409i
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 507.4.b.i.337.4 10
13.5 odd 4 507.4.a.r.1.4 10
13.8 odd 4 507.4.a.r.1.7 10
13.9 even 3 39.4.j.c.10.2 yes 10
13.10 even 6 39.4.j.c.4.2 ✓ 10
13.12 even 2 inner 507.4.b.i.337.7 10
39.5 even 4 1521.4.a.bk.1.7 10
39.8 even 4 1521.4.a.bk.1.4 10
39.23 odd 6 117.4.q.e.82.4 10
39.35 odd 6 117.4.q.e.10.4 10
52.23 odd 6 624.4.bv.h.433.4 10
52.35 odd 6 624.4.bv.h.49.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.c.4.2 ✓ 10 13.10 even 6
39.4.j.c.10.2 yes 10 13.9 even 3
117.4.q.e.10.4 10 39.35 odd 6
117.4.q.e.82.4 10 39.23 odd 6
507.4.a.r.1.4 10 13.5 odd 4
507.4.a.r.1.7 10 13.8 odd 4
507.4.b.i.337.4 10 1.1 even 1 trivial
507.4.b.i.337.7 10 13.12 even 2 inner
624.4.bv.h.49.2 10 52.35 odd 6
624.4.bv.h.433.4 10 52.23 odd 6
1521.4.a.bk.1.4 10 39.8 even 4
1521.4.a.bk.1.7 10 39.5 even 4