Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [507,4,Mod(1,507)] 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("507.1"); 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, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,30,60] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
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: \( 2^{3}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(2.04224\) of defining polynomial
Character \(\chi\) \(=\) 507.1

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.04224 0.722042 0.361021 − 0.932558i \(-0.382428\pi\)
0.361021 + 0.932558i \(0.382428\pi\)
\(3\) 3.00000 0.577350
\(4\) −3.82924 −0.478656
\(5\) 12.0825 1.08069 0.540344 − 0.841444i \(-0.318294\pi\)
0.540344 + 0.841444i \(0.318294\pi\)
\(6\) 6.12673 0.416871
\(7\) 29.7373 1.60566 0.802832 − 0.596206i \(-0.203326\pi\)
0.802832 + 0.596206i \(0.203326\pi\)
\(8\) −24.1582 −1.06765
\(9\) 9.00000 0.333333
\(10\) 24.6753 0.780302
\(11\) 28.0636 0.769226 0.384613 − 0.923078i \(-0.374335\pi\)
0.384613 + 0.923078i \(0.374335\pi\)
\(12\) −11.4877 −0.276352
\(13\) 0 0
\(14\) 60.7308 1.15936
\(15\) 36.2474 0.623935
\(16\) −18.7029 −0.292233
\(17\) −50.6556 −0.722693 −0.361347 − 0.932432i \(-0.617683\pi\)
−0.361347 + 0.932432i \(0.617683\pi\)
\(18\) 18.3802 0.240681
\(19\) 105.148 1.26962 0.634808 − 0.772670i \(-0.281079\pi\)
0.634808 + 0.772670i \(0.281079\pi\)
\(20\) −46.2667 −0.517277
\(21\) 89.2119 0.927030
\(22\) 57.3126 0.555413
\(23\) −160.592 −1.45590 −0.727951 − 0.685629i \(-0.759527\pi\)
−0.727951 + 0.685629i \(0.759527\pi\)
\(24\) −72.4746 −0.616409
\(25\) 20.9857 0.167886
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) −113.871 −0.768560
\(29\) 140.105 0.897132 0.448566 − 0.893750i \(-0.351935\pi\)
0.448566 + 0.893750i \(0.351935\pi\)
\(30\) 74.0259 0.450507
\(31\) 223.593 1.29544 0.647718 − 0.761880i \(-0.275724\pi\)
0.647718 + 0.761880i \(0.275724\pi\)
\(32\) 155.070 0.856647
\(33\) 84.1907 0.444113
\(34\) −103.451 −0.521815
\(35\) 359.300 1.73522
\(36\) −34.4632 −0.159552
\(37\) −228.352 −1.01462 −0.507308 − 0.861765i \(-0.669359\pi\)
−0.507308 + 0.861765i \(0.669359\pi\)
\(38\) 214.739 0.916716
\(39\) 0 0
\(40\) −291.890 −1.15380
\(41\) −295.902 −1.12713 −0.563563 − 0.826073i \(-0.690570\pi\)
−0.563563 + 0.826073i \(0.690570\pi\)
\(42\) 182.192 0.669354
\(43\) 192.103 0.681291 0.340645 − 0.940192i \(-0.389354\pi\)
0.340645 + 0.940192i \(0.389354\pi\)
\(44\) −107.462 −0.368194
\(45\) 108.742 0.360229
\(46\) −327.968 −1.05122
\(47\) 36.9300 0.114613 0.0573063 − 0.998357i \(-0.481749\pi\)
0.0573063 + 0.998357i \(0.481749\pi\)
\(48\) −56.1088 −0.168721
\(49\) 541.307 1.57815
\(50\) 42.8579 0.121220
\(51\) −151.967 −0.417247
\(52\) 0 0
\(53\) 149.102 0.386429 0.193214 − 0.981157i \(-0.438109\pi\)
0.193214 + 0.981157i \(0.438109\pi\)
\(54\) 55.1406 0.138957
\(55\) 339.077 0.831293
\(56\) −718.399 −1.71429
\(57\) 315.445 0.733013
\(58\) 286.128 0.647767
\(59\) 438.867 0.968400 0.484200 − 0.874957i \(-0.339111\pi\)
0.484200 + 0.874957i \(0.339111\pi\)
\(60\) −138.800 −0.298650
\(61\) 286.146 0.600610 0.300305 − 0.953843i \(-0.402912\pi\)
0.300305 + 0.953843i \(0.402912\pi\)
\(62\) 456.631 0.935358
\(63\) 267.636 0.535221
\(64\) 466.313 0.910768
\(65\) 0 0
\(66\) 171.938 0.320668
\(67\) −537.128 −0.979412 −0.489706 − 0.871888i \(-0.662896\pi\)
−0.489706 + 0.871888i \(0.662896\pi\)
\(68\) 193.973 0.345921
\(69\) −481.776 −0.840566
\(70\) 733.777 1.25290
\(71\) 102.729 0.171713 0.0858567 − 0.996307i \(-0.472637\pi\)
0.0858567 + 0.996307i \(0.472637\pi\)
\(72\) −217.424 −0.355884
\(73\) −75.5209 −0.121083 −0.0605414 − 0.998166i \(-0.519283\pi\)
−0.0605414 + 0.998166i \(0.519283\pi\)
\(74\) −466.350 −0.732596
\(75\) 62.9571 0.0969288
\(76\) −402.639 −0.607709
\(77\) 834.535 1.23512
\(78\) 0 0
\(79\) 17.5526 0.0249978 0.0124989 − 0.999922i \(-0.496021\pi\)
0.0124989 + 0.999922i \(0.496021\pi\)
\(80\) −225.977 −0.315813
\(81\) 81.0000 0.111111
\(82\) −604.304 −0.813831
\(83\) 1463.08 1.93487 0.967434 − 0.253122i \(-0.0814573\pi\)
0.967434 + 0.253122i \(0.0814573\pi\)
\(84\) −341.614 −0.443728
\(85\) −612.044 −0.781005
\(86\) 392.322 0.491920
\(87\) 420.315 0.517960
\(88\) −677.965 −0.821265
\(89\) −334.905 −0.398875 −0.199438 − 0.979911i \(-0.563911\pi\)
−0.199438 + 0.979911i \(0.563911\pi\)
\(90\) 222.078 0.260101
\(91\) 0 0
\(92\) 614.946 0.696876
\(93\) 670.779 0.747920
\(94\) 75.4200 0.0827551
\(95\) 1270.45 1.37206
\(96\) 465.209 0.494585
\(97\) −748.756 −0.783760 −0.391880 − 0.920016i \(-0.628175\pi\)
−0.391880 + 0.920016i \(0.628175\pi\)
\(98\) 1105.48 1.13949
\(99\) 252.572 0.256409
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.a.r.1.7 10
3.2 odd 2 1521.4.a.bk.1.4 10
13.5 odd 4 507.4.b.i.337.4 10
13.6 odd 12 39.4.j.c.10.2 yes 10
13.8 odd 4 507.4.b.i.337.7 10
13.11 odd 12 39.4.j.c.4.2 ✓ 10
13.12 even 2 inner 507.4.a.r.1.4 10
39.11 even 12 117.4.q.e.82.4 10
39.32 even 12 117.4.q.e.10.4 10
39.38 odd 2 1521.4.a.bk.1.7 10
52.11 even 12 624.4.bv.h.433.4 10
52.19 even 12 624.4.bv.h.49.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.c.4.2 ✓ 10 13.11 odd 12
39.4.j.c.10.2 yes 10 13.6 odd 12
117.4.q.e.10.4 10 39.32 even 12
117.4.q.e.82.4 10 39.11 even 12
507.4.a.r.1.4 10 13.12 even 2 inner
507.4.a.r.1.7 10 1.1 even 1 trivial
507.4.b.i.337.4 10 13.5 odd 4
507.4.b.i.337.7 10 13.8 odd 4
624.4.bv.h.49.2 10 52.19 even 12
624.4.bv.h.433.4 10 52.11 even 12
1521.4.a.bk.1.4 10 3.2 odd 2
1521.4.a.bk.1.7 10 39.38 odd 2