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Properties

Label 507.4.b.i.337.9
Level $507$
Weight $4$
Character 507.337
Analytic conductor $29.914$
Analytic rank $0$
Dimension $10$
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] = [507,4,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 337.9
Root \(5.04537i\) of defining polynomial
Character \(\chi\) \(=\) 507.337
Dual form 507.4.b.i.337.2

$q$-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)
 
\(f(q)\) \(=\) \(q+5.04537i q^{2} +3.00000 q^{3} -17.4557 q^{4} +20.1174i q^{5} +15.1361i q^{6} +15.4279i q^{7} -47.7076i q^{8} +9.00000 q^{9} -101.500 q^{10} +26.9372i q^{11} -52.3672 q^{12} -77.8394 q^{14} +60.3522i q^{15} +101.057 q^{16} -23.2334 q^{17} +45.4083i q^{18} +45.0794i q^{19} -351.164i q^{20} +46.2837i q^{21} -135.908 q^{22} +142.010 q^{23} -143.123i q^{24} -279.710 q^{25} +27.0000 q^{27} -269.305i q^{28} +2.29068 q^{29} -304.499 q^{30} +37.7740i q^{31} +128.207i q^{32} +80.8116i q^{33} -117.221i q^{34} -310.369 q^{35} -157.102 q^{36} +313.840i q^{37} -227.442 q^{38} +959.753 q^{40} +5.86820i q^{41} -233.518 q^{42} +360.898 q^{43} -470.209i q^{44} +181.057i q^{45} +716.493i q^{46} -209.748i q^{47} +303.170 q^{48} +104.980 q^{49} -1411.24i q^{50} -69.7003 q^{51} +276.886 q^{53} +136.225i q^{54} -541.906 q^{55} +736.028 q^{56} +135.238i q^{57} +11.5573i q^{58} -543.189i q^{59} -1053.49i q^{60} +205.788 q^{61} -190.583 q^{62} +138.851i q^{63} +161.602 q^{64} -407.724 q^{66} -492.578i q^{67} +405.557 q^{68} +426.030 q^{69} -1565.93i q^{70} -826.859i q^{71} -429.369i q^{72} +66.1205i q^{73} -1583.44 q^{74} -839.129 q^{75} -786.894i q^{76} -415.584 q^{77} +317.642 q^{79} +2033.00i q^{80} +81.0000 q^{81} -29.6072 q^{82} -141.450i q^{83} -807.915i q^{84} -467.396i q^{85} +1820.86i q^{86} +6.87204 q^{87} +1285.11 q^{88} +641.320i q^{89} -913.497 q^{90} -2478.89 q^{92} +113.322i q^{93} +1058.26 q^{94} -906.880 q^{95} +384.621i q^{96} +1114.92i q^{97} +529.663i q^{98} +242.435i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 30 q^{3} - 60 q^{4} + 90 q^{9} - 80 q^{10} - 180 q^{12} - 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} - 960 q^{25} + 270 q^{27} + 990 q^{29} - 240 q^{30} - 120 q^{35} - 540 q^{36}+ \cdots + 2760 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.04537i 1.78381i 0.452226 + 0.891903i \(0.350630\pi\)
−0.452226 + 0.891903i \(0.649370\pi\)
\(3\) 3.00000 0.577350
\(4\) −17.4557 −2.18197
\(5\) 20.1174i 1.79935i 0.436556 + 0.899677i \(0.356198\pi\)
−0.436556 + 0.899677i \(0.643802\pi\)
\(6\) 15.1361i 1.02988i
\(7\) 15.4279i 0.833028i 0.909129 + 0.416514i \(0.136748\pi\)
−0.909129 + 0.416514i \(0.863252\pi\)
\(8\) − 47.7076i − 2.10840i
\(9\) 9.00000 0.333333
\(10\) −101.500 −3.20970
\(11\) 26.9372i 0.738352i 0.929359 + 0.369176i \(0.120360\pi\)
−0.929359 + 0.369176i \(0.879640\pi\)
\(12\) −52.3672 −1.25976
\(13\) 0 0
\(14\) −77.8394 −1.48596
\(15\) 60.3522i 1.03886i
\(16\) 101.057 1.57901
\(17\) −23.2334 −0.331467 −0.165733 − 0.986171i \(-0.552999\pi\)
−0.165733 + 0.986171i \(0.552999\pi\)
\(18\) 45.4083i 0.594602i
\(19\) 45.0794i 0.544312i 0.962253 + 0.272156i \(0.0877366\pi\)
−0.962253 + 0.272156i \(0.912263\pi\)
\(20\) − 351.164i − 3.92613i
\(21\) 46.2837i 0.480949i
\(22\) −135.908 −1.31708
\(23\) 142.010 1.28744 0.643720 − 0.765261i \(-0.277390\pi\)
0.643720 + 0.765261i \(0.277390\pi\)
\(24\) − 143.123i − 1.21729i
\(25\) −279.710 −2.23768
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) − 269.305i − 1.81764i
\(29\) 2.29068 0.0146679 0.00733394 − 0.999973i \(-0.497666\pi\)
0.00733394 + 0.999973i \(0.497666\pi\)
\(30\) −304.499 −1.85312
\(31\) 37.7740i 0.218852i 0.993995 + 0.109426i \(0.0349012\pi\)
−0.993995 + 0.109426i \(0.965099\pi\)
\(32\) 128.207i 0.708251i
\(33\) 80.8116i 0.426288i
\(34\) − 117.221i − 0.591273i
\(35\) −310.369 −1.49891
\(36\) −157.102 −0.727322
\(37\) 313.840i 1.39446i 0.716849 + 0.697228i \(0.245584\pi\)
−0.716849 + 0.697228i \(0.754416\pi\)
\(38\) −227.442 −0.970947
\(39\) 0 0
\(40\) 959.753 3.79376
\(41\) 5.86820i 0.0223527i 0.999938 + 0.0111763i \(0.00355761\pi\)
−0.999938 + 0.0111763i \(0.996442\pi\)
\(42\) −233.518 −0.857920
\(43\) 360.898 1.27992 0.639958 − 0.768410i \(-0.278952\pi\)
0.639958 + 0.768410i \(0.278952\pi\)
\(44\) − 470.209i − 1.61106i
\(45\) 181.057i 0.599785i
\(46\) 716.493i 2.29655i
\(47\) − 209.748i − 0.650956i −0.945550 − 0.325478i \(-0.894475\pi\)
0.945550 − 0.325478i \(-0.105525\pi\)
\(48\) 303.170 0.911642
\(49\) 104.980 0.306064
\(50\) − 1411.24i − 3.99158i
\(51\) −69.7003 −0.191372
\(52\) 0 0
\(53\) 276.886 0.717609 0.358804 − 0.933413i \(-0.383185\pi\)
0.358804 + 0.933413i \(0.383185\pi\)
\(54\) 136.225i 0.343294i
\(55\) −541.906 −1.32856
\(56\) 736.028 1.75636
\(57\) 135.238i 0.314259i
\(58\) 11.5573i 0.0261647i
\(59\) − 543.189i − 1.19860i −0.800526 − 0.599298i \(-0.795447\pi\)
0.800526 − 0.599298i \(-0.204553\pi\)
\(60\) − 1053.49i − 2.26675i
\(61\) 205.788 0.431942 0.215971 − 0.976400i \(-0.430708\pi\)
0.215971 + 0.976400i \(0.430708\pi\)
\(62\) −190.583 −0.390389
\(63\) 138.851i 0.277676i
\(64\) 161.602 0.315629
\(65\) 0 0
\(66\) −407.724 −0.760415
\(67\) − 492.578i − 0.898179i −0.893487 − 0.449090i \(-0.851748\pi\)
0.893487 − 0.449090i \(-0.148252\pi\)
\(68\) 405.557 0.723249
\(69\) 426.030 0.743304
\(70\) − 1565.93i − 2.67377i
\(71\) − 826.859i − 1.38211i −0.722800 − 0.691057i \(-0.757145\pi\)
0.722800 − 0.691057i \(-0.242855\pi\)
\(72\) − 429.369i − 0.702800i
\(73\) 66.1205i 0.106011i 0.998594 + 0.0530056i \(0.0168801\pi\)
−0.998594 + 0.0530056i \(0.983120\pi\)
\(74\) −1583.44 −2.48744
\(75\) −839.129 −1.29192
\(76\) − 786.894i − 1.18767i
\(77\) −415.584 −0.615068
\(78\) 0 0
\(79\) 317.642 0.452374 0.226187 − 0.974084i \(-0.427374\pi\)
0.226187 + 0.974084i \(0.427374\pi\)
\(80\) 2033.00i 2.84120i
\(81\) 81.0000 0.111111
\(82\) −29.6072 −0.0398728
\(83\) − 141.450i − 0.187063i −0.995616 − 0.0935313i \(-0.970184\pi\)
0.995616 − 0.0935313i \(-0.0298155\pi\)
\(84\) − 807.915i − 1.04941i
\(85\) − 467.396i − 0.596426i
\(86\) 1820.86i 2.28312i
\(87\) 6.87204 0.00846851
\(88\) 1285.11 1.55674
\(89\) 641.320i 0.763818i 0.924200 + 0.381909i \(0.124733\pi\)
−0.924200 + 0.381909i \(0.875267\pi\)
\(90\) −913.497 −1.06990
\(91\) 0 0
\(92\) −2478.89 −2.80915
\(93\) 113.322i 0.126354i
\(94\) 1058.26 1.16118
\(95\) −906.880 −0.979410
\(96\) 384.621i 0.408909i
\(97\) 1114.92i 1.16704i 0.812097 + 0.583522i \(0.198326\pi\)
−0.812097 + 0.583522i \(0.801674\pi\)
\(98\) 529.663i 0.545960i
\(99\) 242.435i 0.246117i
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.337.9 10
13.5 odd 4 507.4.a.r.1.9 10
13.8 odd 4 507.4.a.r.1.2 10
13.9 even 3 39.4.j.c.10.5 yes 10
13.10 even 6 39.4.j.c.4.5 ✓ 10
13.12 even 2 inner 507.4.b.i.337.2 10
39.5 even 4 1521.4.a.bk.1.2 10
39.8 even 4 1521.4.a.bk.1.9 10
39.23 odd 6 117.4.q.e.82.1 10
39.35 odd 6 117.4.q.e.10.1 10
52.23 odd 6 624.4.bv.h.433.1 10
52.35 odd 6 624.4.bv.h.49.5 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.c.4.5 ✓ 10 13.10 even 6
39.4.j.c.10.5 yes 10 13.9 even 3
117.4.q.e.10.1 10 39.35 odd 6
117.4.q.e.82.1 10 39.23 odd 6
507.4.a.r.1.2 10 13.8 odd 4
507.4.a.r.1.9 10 13.5 odd 4
507.4.b.i.337.2 10 13.12 even 2 inner
507.4.b.i.337.9 10 1.1 even 1 trivial
624.4.bv.h.49.5 10 52.35 odd 6
624.4.bv.h.433.1 10 52.23 odd 6
1521.4.a.bk.1.2 10 39.5 even 4
1521.4.a.bk.1.9 10 39.8 even 4