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Properties

Label 507.4.a.f.1.2
Level $507$
Weight $4$
Character 507.1
Self dual yes
Analytic conductor $29.914$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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] = [507,4,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 = [2,-2,-6,14] 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: \(2\)
Coefficient field: \(\Q(\sqrt{14}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.74166\) 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)
 
\(f(q)\) \(=\) \(q+2.74166 q^{2} -3.00000 q^{3} -0.483315 q^{4} -19.4833 q^{5} -8.22497 q^{6} -7.48331 q^{7} -23.2583 q^{8} +9.00000 q^{9} -53.4166 q^{10} -22.8999 q^{11} +1.44994 q^{12} -20.5167 q^{14} +58.4499 q^{15} -59.8999 q^{16} +67.0334 q^{17} +24.6749 q^{18} -16.5167 q^{19} +9.41657 q^{20} +22.4499 q^{21} -62.7836 q^{22} -175.600 q^{23} +69.7750 q^{24} +254.600 q^{25} -27.0000 q^{27} +3.61680 q^{28} +291.800 q^{29} +160.250 q^{30} -117.283 q^{31} +21.8418 q^{32} +68.6997 q^{33} +183.783 q^{34} +145.800 q^{35} -4.34983 q^{36} +154.766 q^{37} -45.2831 q^{38} +453.150 q^{40} +251.716 q^{41} +61.5501 q^{42} -502.566 q^{43} +11.0679 q^{44} -175.350 q^{45} -481.434 q^{46} +281.733 q^{47} +179.700 q^{48} -287.000 q^{49} +698.025 q^{50} -201.100 q^{51} +366.999 q^{53} -74.0247 q^{54} +446.166 q^{55} +174.049 q^{56} +49.5501 q^{57} +800.015 q^{58} +79.6663 q^{59} -28.2497 q^{60} -194.865 q^{61} -321.550 q^{62} -67.3498 q^{63} +539.082 q^{64} +188.351 q^{66} -400.082 q^{67} -32.3982 q^{68} +526.799 q^{69} +399.733 q^{70} -528.299 q^{71} -209.325 q^{72} +734.366 q^{73} +424.316 q^{74} -763.799 q^{75} +7.98276 q^{76} +171.367 q^{77} +113.266 q^{79} +1167.05 q^{80} +81.0000 q^{81} +690.118 q^{82} +933.466 q^{83} -10.8504 q^{84} -1306.03 q^{85} -1377.86 q^{86} -875.399 q^{87} +532.613 q^{88} -1190.91 q^{89} -480.749 q^{90} +84.8699 q^{92} +351.849 q^{93} +772.415 q^{94} +321.800 q^{95} -65.5253 q^{96} -557.165 q^{97} -786.856 q^{98} -206.099 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 6 q^{3} + 14 q^{4} - 24 q^{5} + 6 q^{6} - 54 q^{8} + 18 q^{9} - 32 q^{10} + 44 q^{11} - 42 q^{12} - 56 q^{14} + 72 q^{15} - 30 q^{16} + 164 q^{17} - 18 q^{18} - 48 q^{19} - 56 q^{20} - 380 q^{22}+ \cdots + 396 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.74166 0.969322 0.484661 − 0.874702i \(-0.338943\pi\)
0.484661 + 0.874702i \(0.338943\pi\)
\(3\) −3.00000 −0.577350
\(4\) −0.483315 −0.0604143
\(5\) −19.4833 −1.74264 −0.871320 − 0.490715i \(-0.836736\pi\)
−0.871320 + 0.490715i \(0.836736\pi\)
\(6\) −8.22497 −0.559638
\(7\) −7.48331 −0.404061 −0.202031 − 0.979379i \(-0.564754\pi\)
−0.202031 + 0.979379i \(0.564754\pi\)
\(8\) −23.2583 −1.02788
\(9\) 9.00000 0.333333
\(10\) −53.4166 −1.68918
\(11\) −22.8999 −0.627689 −0.313844 − 0.949474i \(-0.601617\pi\)
−0.313844 + 0.949474i \(0.601617\pi\)
\(12\) 1.44994 0.0348802
\(13\) 0 0
\(14\) −20.5167 −0.391665
\(15\) 58.4499 1.00611
\(16\) −59.8999 −0.935936
\(17\) 67.0334 0.956352 0.478176 − 0.878264i \(-0.341298\pi\)
0.478176 + 0.878264i \(0.341298\pi\)
\(18\) 24.6749 0.323107
\(19\) −16.5167 −0.199431 −0.0997155 − 0.995016i \(-0.531793\pi\)
−0.0997155 + 0.995016i \(0.531793\pi\)
\(20\) 9.41657 0.105280
\(21\) 22.4499 0.233285
\(22\) −62.7836 −0.608433
\(23\) −175.600 −1.59196 −0.795979 − 0.605324i \(-0.793044\pi\)
−0.795979 + 0.605324i \(0.793044\pi\)
\(24\) 69.7750 0.593449
\(25\) 254.600 2.03680
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 3.61680 0.0244111
\(29\) 291.800 1.86848 0.934239 − 0.356648i \(-0.116080\pi\)
0.934239 + 0.356648i \(0.116080\pi\)
\(30\) 160.250 0.975249
\(31\) −117.283 −0.679505 −0.339753 − 0.940515i \(-0.610343\pi\)
−0.339753 + 0.940515i \(0.610343\pi\)
\(32\) 21.8418 0.120660
\(33\) 68.6997 0.362396
\(34\) 183.783 0.927013
\(35\) 145.800 0.704133
\(36\) −4.34983 −0.0201381
\(37\) 154.766 0.687661 0.343830 − 0.939032i \(-0.388276\pi\)
0.343830 + 0.939032i \(0.388276\pi\)
\(38\) −45.2831 −0.193313
\(39\) 0 0
\(40\) 453.150 1.79123
\(41\) 251.716 0.958815 0.479407 − 0.877592i \(-0.340852\pi\)
0.479407 + 0.877592i \(0.340852\pi\)
\(42\) 61.5501 0.226128
\(43\) −502.566 −1.78234 −0.891170 − 0.453669i \(-0.850115\pi\)
−0.891170 + 0.453669i \(0.850115\pi\)
\(44\) 11.0679 0.0379214
\(45\) −175.350 −0.580880
\(46\) −481.434 −1.54312
\(47\) 281.733 0.874361 0.437181 − 0.899374i \(-0.355977\pi\)
0.437181 + 0.899374i \(0.355977\pi\)
\(48\) 179.700 0.540363
\(49\) −287.000 −0.836735
\(50\) 698.025 1.97431
\(51\) −201.100 −0.552150
\(52\) 0 0
\(53\) 366.999 0.951154 0.475577 − 0.879674i \(-0.342239\pi\)
0.475577 + 0.879674i \(0.342239\pi\)
\(54\) −74.0247 −0.186546
\(55\) 446.166 1.09384
\(56\) 174.049 0.415328
\(57\) 49.5501 0.115141
\(58\) 800.015 1.81116
\(59\) 79.6663 0.175791 0.0878955 − 0.996130i \(-0.471986\pi\)
0.0878955 + 0.996130i \(0.471986\pi\)
\(60\) −28.2497 −0.0607837
\(61\) −194.865 −0.409016 −0.204508 − 0.978865i \(-0.565559\pi\)
−0.204508 + 0.978865i \(0.565559\pi\)
\(62\) −321.550 −0.658660
\(63\) −67.3498 −0.134687
\(64\) 539.082 1.05289
\(65\) 0 0
\(66\) 188.351 0.351279
\(67\) −400.082 −0.729519 −0.364759 − 0.931102i \(-0.618849\pi\)
−0.364759 + 0.931102i \(0.618849\pi\)
\(68\) −32.3982 −0.0577774
\(69\) 526.799 0.919117
\(70\) 399.733 0.682532
\(71\) −528.299 −0.883065 −0.441532 − 0.897245i \(-0.645565\pi\)
−0.441532 + 0.897245i \(0.645565\pi\)
\(72\) −209.325 −0.342628
\(73\) 734.366 1.17741 0.588706 − 0.808347i \(-0.299638\pi\)
0.588706 + 0.808347i \(0.299638\pi\)
\(74\) 424.316 0.666565
\(75\) −763.799 −1.17594
\(76\) 7.98276 0.0120485
\(77\) 171.367 0.253625
\(78\) 0 0
\(79\) 113.266 0.161309 0.0806545 − 0.996742i \(-0.474299\pi\)
0.0806545 + 0.996742i \(0.474299\pi\)
\(80\) 1167.05 1.63100
\(81\) 81.0000 0.111111
\(82\) 690.118 0.929400
\(83\) 933.466 1.23447 0.617236 − 0.786778i \(-0.288252\pi\)
0.617236 + 0.786778i \(0.288252\pi\)
\(84\) −10.8504 −0.0140937
\(85\) −1306.03 −1.66658
\(86\) −1377.86 −1.72766
\(87\) −875.399 −1.07877
\(88\) 532.613 0.645191
\(89\) −1190.91 −1.41839 −0.709195 − 0.705012i \(-0.750941\pi\)
−0.709195 + 0.705012i \(0.750941\pi\)
\(90\) −480.749 −0.563060
\(91\) 0 0
\(92\) 84.8699 0.0961771
\(93\) 351.849 0.392313
\(94\) 772.415 0.847538
\(95\) 321.800 0.347536
\(96\) −65.5253 −0.0696630
\(97\) −557.165 −0.583211 −0.291606 − 0.956539i \(-0.594189\pi\)
−0.291606 + 0.956539i \(0.594189\pi\)
\(98\) −786.856 −0.811066
\(99\) −206.099 −0.209230
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.f.1.2 2
3.2 odd 2 1521.4.a.s.1.1 2
13.5 odd 4 507.4.b.f.337.2 4
13.8 odd 4 507.4.b.f.337.3 4
13.12 even 2 39.4.a.b.1.1 ✓ 2
39.38 odd 2 117.4.a.c.1.2 2
52.51 odd 2 624.4.a.r.1.2 2
65.64 even 2 975.4.a.j.1.2 2
91.90 odd 2 1911.4.a.h.1.1 2
104.51 odd 2 2496.4.a.s.1.1 2
104.77 even 2 2496.4.a.bc.1.1 2
156.155 even 2 1872.4.a.t.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.b.1.1 ✓ 2 13.12 even 2
117.4.a.c.1.2 2 39.38 odd 2
507.4.a.f.1.2 2 1.1 even 1 trivial
507.4.b.f.337.2 4 13.5 odd 4
507.4.b.f.337.3 4 13.8 odd 4
624.4.a.r.1.2 2 52.51 odd 2
975.4.a.j.1.2 2 65.64 even 2
1521.4.a.s.1.1 2 3.2 odd 2
1872.4.a.t.1.1 2 156.155 even 2
1911.4.a.h.1.1 2 91.90 odd 2
2496.4.a.s.1.1 2 104.51 odd 2
2496.4.a.bc.1.1 2 104.77 even 2