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Properties

Label 507.4.a.r.1.9
Level $507$
Weight $4$
Character 507.1
Self dual yes
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(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 1.9
Root \(5.04537\) 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)
 
\(f(q)\) \(=\) \(q+5.04537 q^{2} +3.00000 q^{3} +17.4557 q^{4} +20.1174 q^{5} +15.1361 q^{6} -15.4279 q^{7} +47.7076 q^{8} +9.00000 q^{9} +101.500 q^{10} -26.9372 q^{11} +52.3672 q^{12} -77.8394 q^{14} +60.3522 q^{15} +101.057 q^{16} +23.2334 q^{17} +45.4083 q^{18} +45.0794 q^{19} +351.164 q^{20} -46.2837 q^{21} -135.908 q^{22} -142.010 q^{23} +143.123 q^{24} +279.710 q^{25} +27.0000 q^{27} -269.305 q^{28} +2.29068 q^{29} +304.499 q^{30} +37.7740 q^{31} +128.207 q^{32} -80.8116 q^{33} +117.221 q^{34} -310.369 q^{35} +157.102 q^{36} -313.840 q^{37} +227.442 q^{38} +959.753 q^{40} +5.86820 q^{41} -233.518 q^{42} -360.898 q^{43} -470.209 q^{44} +181.057 q^{45} -716.493 q^{46} +209.748 q^{47} +303.170 q^{48} -104.980 q^{49} +1411.24 q^{50} +69.7003 q^{51} +276.886 q^{53} +136.225 q^{54} -541.906 q^{55} -736.028 q^{56} +135.238 q^{57} +11.5573 q^{58} +543.189 q^{59} +1053.49 q^{60} +205.788 q^{61} +190.583 q^{62} -138.851 q^{63} -161.602 q^{64} -407.724 q^{66} -492.578 q^{67} +405.557 q^{68} -426.030 q^{69} -1565.93 q^{70} -826.859 q^{71} +429.369 q^{72} -66.1205 q^{73} -1583.44 q^{74} +839.129 q^{75} +786.894 q^{76} +415.584 q^{77} +317.642 q^{79} +2033.00 q^{80} +81.0000 q^{81} +29.6072 q^{82} -141.450 q^{83} -807.915 q^{84} +467.396 q^{85} -1820.86 q^{86} +6.87204 q^{87} -1285.11 q^{88} -641.320 q^{89} +913.497 q^{90} -2478.89 q^{92} +113.322 q^{93} +1058.26 q^{94} +906.880 q^{95} +384.621 q^{96} +1114.92 q^{97} -529.663 q^{98} -242.435 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 30 q^{3} + 60 q^{4} + 90 q^{9} + 80 q^{10} + 180 q^{12} - 60 q^{14} + 500 q^{16} + 210 q^{17} + 580 q^{22} - 120 q^{23} + 960 q^{25} + 270 q^{27} + 990 q^{29} + 240 q^{30} - 120 q^{35} + 540 q^{36}+ \cdots - 2760 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.04537 1.78381 0.891903 − 0.452226i \(-0.149370\pi\)
0.891903 + 0.452226i \(0.149370\pi\)
\(3\) 3.00000 0.577350
\(4\) 17.4557 2.18197
\(5\) 20.1174 1.79935 0.899677 − 0.436556i \(-0.143802\pi\)
0.899677 + 0.436556i \(0.143802\pi\)
\(6\) 15.1361 1.02988
\(7\) −15.4279 −0.833028 −0.416514 − 0.909129i \(-0.636748\pi\)
−0.416514 + 0.909129i \(0.636748\pi\)
\(8\) 47.7076 2.10840
\(9\) 9.00000 0.333333
\(10\) 101.500 3.20970
\(11\) −26.9372 −0.738352 −0.369176 − 0.929359i \(-0.620360\pi\)
−0.369176 + 0.929359i \(0.620360\pi\)
\(12\) 52.3672 1.25976
\(13\) 0 0
\(14\) −77.8394 −1.48596
\(15\) 60.3522 1.03886
\(16\) 101.057 1.57901
\(17\) 23.2334 0.331467 0.165733 − 0.986171i \(-0.447001\pi\)
0.165733 + 0.986171i \(0.447001\pi\)
\(18\) 45.4083 0.594602
\(19\) 45.0794 0.544312 0.272156 − 0.962253i \(-0.412263\pi\)
0.272156 + 0.962253i \(0.412263\pi\)
\(20\) 351.164 3.92613
\(21\) −46.2837 −0.480949
\(22\) −135.908 −1.31708
\(23\) −142.010 −1.28744 −0.643720 − 0.765261i \(-0.722610\pi\)
−0.643720 + 0.765261i \(0.722610\pi\)
\(24\) 143.123 1.21729
\(25\) 279.710 2.23768
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) −269.305 −1.81764
\(29\) 2.29068 0.0146679 0.00733394 − 0.999973i \(-0.497666\pi\)
0.00733394 + 0.999973i \(0.497666\pi\)
\(30\) 304.499 1.85312
\(31\) 37.7740 0.218852 0.109426 − 0.993995i \(-0.465099\pi\)
0.109426 + 0.993995i \(0.465099\pi\)
\(32\) 128.207 0.708251
\(33\) −80.8116 −0.426288
\(34\) 117.221 0.591273
\(35\) −310.369 −1.49891
\(36\) 157.102 0.727322
\(37\) −313.840 −1.39446 −0.697228 − 0.716849i \(-0.745584\pi\)
−0.697228 + 0.716849i \(0.745584\pi\)
\(38\) 227.442 0.970947
\(39\) 0 0
\(40\) 959.753 3.79376
\(41\) 5.86820 0.0223527 0.0111763 − 0.999938i \(-0.496442\pi\)
0.0111763 + 0.999938i \(0.496442\pi\)
\(42\) −233.518 −0.857920
\(43\) −360.898 −1.27992 −0.639958 − 0.768410i \(-0.721048\pi\)
−0.639958 + 0.768410i \(0.721048\pi\)
\(44\) −470.209 −1.61106
\(45\) 181.057 0.599785
\(46\) −716.493 −2.29655
\(47\) 209.748 0.650956 0.325478 − 0.945550i \(-0.394475\pi\)
0.325478 + 0.945550i \(0.394475\pi\)
\(48\) 303.170 0.911642
\(49\) −104.980 −0.306064
\(50\) 1411.24 3.99158
\(51\) 69.7003 0.191372
\(52\) 0 0
\(53\) 276.886 0.717609 0.358804 − 0.933413i \(-0.383185\pi\)
0.358804 + 0.933413i \(0.383185\pi\)
\(54\) 136.225 0.343294
\(55\) −541.906 −1.32856
\(56\) −736.028 −1.75636
\(57\) 135.238 0.314259
\(58\) 11.5573 0.0261647
\(59\) 543.189 1.19860 0.599298 − 0.800526i \(-0.295447\pi\)
0.599298 + 0.800526i \(0.295447\pi\)
\(60\) 1053.49 2.26675
\(61\) 205.788 0.431942 0.215971 − 0.976400i \(-0.430708\pi\)
0.215971 + 0.976400i \(0.430708\pi\)
\(62\) 190.583 0.390389
\(63\) −138.851 −0.277676
\(64\) −161.602 −0.315629
\(65\) 0 0
\(66\) −407.724 −0.760415
\(67\) −492.578 −0.898179 −0.449090 − 0.893487i \(-0.648252\pi\)
−0.449090 + 0.893487i \(0.648252\pi\)
\(68\) 405.557 0.723249
\(69\) −426.030 −0.743304
\(70\) −1565.93 −2.67377
\(71\) −826.859 −1.38211 −0.691057 − 0.722800i \(-0.742855\pi\)
−0.691057 + 0.722800i \(0.742855\pi\)
\(72\) 429.369 0.702800
\(73\) −66.1205 −0.106011 −0.0530056 − 0.998594i \(-0.516880\pi\)
−0.0530056 + 0.998594i \(0.516880\pi\)
\(74\) −1583.44 −2.48744
\(75\) 839.129 1.29192
\(76\) 786.894 1.18767
\(77\) 415.584 0.615068
\(78\) 0 0
\(79\) 317.642 0.452374 0.226187 − 0.974084i \(-0.427374\pi\)
0.226187 + 0.974084i \(0.427374\pi\)
\(80\) 2033.00 2.84120
\(81\) 81.0000 0.111111
\(82\) 29.6072 0.0398728
\(83\) −141.450 −0.187063 −0.0935313 − 0.995616i \(-0.529816\pi\)
−0.0935313 + 0.995616i \(0.529816\pi\)
\(84\) −807.915 −1.04941
\(85\) 467.396 0.596426
\(86\) −1820.86 −2.28312
\(87\) 6.87204 0.00846851
\(88\) −1285.11 −1.55674
\(89\) −641.320 −0.763818 −0.381909 − 0.924200i \(-0.624733\pi\)
−0.381909 + 0.924200i \(0.624733\pi\)
\(90\) 913.497 1.06990
\(91\) 0 0
\(92\) −2478.89 −2.80915
\(93\) 113.322 0.126354
\(94\) 1058.26 1.16118
\(95\) 906.880 0.979410
\(96\) 384.621 0.408909
\(97\) 1114.92 1.16704 0.583522 − 0.812097i \(-0.301674\pi\)
0.583522 + 0.812097i \(0.301674\pi\)
\(98\) −529.663 −0.545960
\(99\) −242.435 −0.246117
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.1.9 10
3.2 odd 2 1521.4.a.bk.1.2 10
13.2 odd 12 39.4.j.c.4.5 ✓ 10
13.5 odd 4 507.4.b.i.337.2 10
13.7 odd 12 39.4.j.c.10.5 yes 10
13.8 odd 4 507.4.b.i.337.9 10
13.12 even 2 inner 507.4.a.r.1.2 10
39.2 even 12 117.4.q.e.82.1 10
39.20 even 12 117.4.q.e.10.1 10
39.38 odd 2 1521.4.a.bk.1.9 10
52.7 even 12 624.4.bv.h.49.5 10
52.15 even 12 624.4.bv.h.433.1 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.c.4.5 ✓ 10 13.2 odd 12
39.4.j.c.10.5 yes 10 13.7 odd 12
117.4.q.e.10.1 10 39.20 even 12
117.4.q.e.82.1 10 39.2 even 12
507.4.a.r.1.2 10 13.12 even 2 inner
507.4.a.r.1.9 10 1.1 even 1 trivial
507.4.b.i.337.2 10 13.5 odd 4
507.4.b.i.337.9 10 13.8 odd 4
624.4.bv.h.49.5 10 52.7 even 12
624.4.bv.h.433.1 10 52.15 even 12
1521.4.a.bk.1.2 10 3.2 odd 2
1521.4.a.bk.1.9 10 39.38 odd 2