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Properties

Label 507.4.a.r.1.6
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 1.6
Root \(0.917374\) 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)
 
\(f(q)\) \(=\) \(q+0.917374 q^{2} +3.00000 q^{3} -7.15843 q^{4} -15.4704 q^{5} +2.75212 q^{6} -20.5833 q^{7} -13.9059 q^{8} +9.00000 q^{9} -14.1922 q^{10} -65.8420 q^{11} -21.4753 q^{12} -18.8826 q^{14} -46.4112 q^{15} +44.5105 q^{16} +44.2956 q^{17} +8.25636 q^{18} +147.053 q^{19} +110.744 q^{20} -61.7500 q^{21} -60.4017 q^{22} +53.1586 q^{23} -41.7178 q^{24} +114.334 q^{25} +27.0000 q^{27} +147.344 q^{28} -38.6257 q^{29} -42.5765 q^{30} +88.3894 q^{31} +152.080 q^{32} -197.526 q^{33} +40.6357 q^{34} +318.433 q^{35} -64.4258 q^{36} +78.9587 q^{37} +134.903 q^{38} +215.131 q^{40} -354.966 q^{41} -56.6478 q^{42} -407.846 q^{43} +471.325 q^{44} -139.234 q^{45} +48.7663 q^{46} -67.9674 q^{47} +133.531 q^{48} +80.6738 q^{49} +104.887 q^{50} +132.887 q^{51} +226.572 q^{53} +24.7691 q^{54} +1018.60 q^{55} +286.231 q^{56} +441.160 q^{57} -35.4342 q^{58} +142.031 q^{59} +332.231 q^{60} +266.831 q^{61} +81.0862 q^{62} -185.250 q^{63} -216.569 q^{64} -181.205 q^{66} -411.187 q^{67} -317.087 q^{68} +159.476 q^{69} +292.122 q^{70} -91.5052 q^{71} -125.153 q^{72} +63.1328 q^{73} +72.4347 q^{74} +343.001 q^{75} -1052.67 q^{76} +1355.25 q^{77} -287.115 q^{79} -688.595 q^{80} +81.0000 q^{81} -325.637 q^{82} -373.812 q^{83} +442.033 q^{84} -685.272 q^{85} -374.147 q^{86} -115.877 q^{87} +915.595 q^{88} +119.403 q^{89} -127.729 q^{90} -380.532 q^{92} +265.168 q^{93} -62.3515 q^{94} -2274.97 q^{95} +456.241 q^{96} +554.650 q^{97} +74.0080 q^{98} -592.578 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 30 q^{3} + 60 q^{4} + 90 q^{9} + 80 q^{10} + 180 q^{12} - 60 q^{14} + 500 q^{16} + 210 q^{17} + 580 q^{22} - 120 q^{23} + 960 q^{25} + 270 q^{27} + 990 q^{29} + 240 q^{30} - 120 q^{35} + 540 q^{36}+ \cdots - 2760 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.917374 0.324341 0.162170 − 0.986763i \(-0.448151\pi\)
0.162170 + 0.986763i \(0.448151\pi\)
\(3\) 3.00000 0.577350
\(4\) −7.15843 −0.894803
\(5\) −15.4704 −1.38372 −0.691858 − 0.722034i \(-0.743208\pi\)
−0.691858 + 0.722034i \(0.743208\pi\)
\(6\) 2.75212 0.187258
\(7\) −20.5833 −1.11140 −0.555698 − 0.831384i \(-0.687549\pi\)
−0.555698 + 0.831384i \(0.687549\pi\)
\(8\) −13.9059 −0.614562
\(9\) 9.00000 0.333333
\(10\) −14.1922 −0.448795
\(11\) −65.8420 −1.80474 −0.902369 − 0.430964i \(-0.858173\pi\)
−0.902369 + 0.430964i \(0.858173\pi\)
\(12\) −21.4753 −0.516615
\(13\) 0 0
\(14\) −18.8826 −0.360471
\(15\) −46.4112 −0.798889
\(16\) 44.5105 0.695476
\(17\) 44.2956 0.631957 0.315979 − 0.948766i \(-0.397667\pi\)
0.315979 + 0.948766i \(0.397667\pi\)
\(18\) 8.25636 0.108114
\(19\) 147.053 1.77560 0.887798 − 0.460234i \(-0.152234\pi\)
0.887798 + 0.460234i \(0.152234\pi\)
\(20\) 110.744 1.23815
\(21\) −61.7500 −0.641665
\(22\) −60.4017 −0.585350
\(23\) 53.1586 0.481928 0.240964 − 0.970534i \(-0.422536\pi\)
0.240964 + 0.970534i \(0.422536\pi\)
\(24\) −41.7178 −0.354817
\(25\) 114.334 0.914669
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 147.344 0.994480
\(29\) −38.6257 −0.247331 −0.123666 − 0.992324i \(-0.539465\pi\)
−0.123666 + 0.992324i \(0.539465\pi\)
\(30\) −42.5765 −0.259112
\(31\) 88.3894 0.512104 0.256052 − 0.966663i \(-0.417578\pi\)
0.256052 + 0.966663i \(0.417578\pi\)
\(32\) 152.080 0.840133
\(33\) −197.526 −1.04197
\(34\) 40.6357 0.204969
\(35\) 318.433 1.53786
\(36\) −64.4258 −0.298268
\(37\) 78.9587 0.350831 0.175415 − 0.984495i \(-0.443873\pi\)
0.175415 + 0.984495i \(0.443873\pi\)
\(38\) 134.903 0.575898
\(39\) 0 0
\(40\) 215.131 0.850379
\(41\) −354.966 −1.35211 −0.676054 − 0.736852i \(-0.736311\pi\)
−0.676054 + 0.736852i \(0.736311\pi\)
\(42\) −56.6478 −0.208118
\(43\) −407.846 −1.44642 −0.723208 − 0.690630i \(-0.757333\pi\)
−0.723208 + 0.690630i \(0.757333\pi\)
\(44\) 471.325 1.61489
\(45\) −139.234 −0.461239
\(46\) 48.7663 0.156309
\(47\) −67.9674 −0.210938 −0.105469 − 0.994423i \(-0.533634\pi\)
−0.105469 + 0.994423i \(0.533634\pi\)
\(48\) 133.531 0.401533
\(49\) 80.6738 0.235201
\(50\) 104.887 0.296664
\(51\) 132.887 0.364861
\(52\) 0 0
\(53\) 226.572 0.587209 0.293604 − 0.955927i \(-0.405145\pi\)
0.293604 + 0.955927i \(0.405145\pi\)
\(54\) 24.7691 0.0624194
\(55\) 1018.60 2.49724
\(56\) 286.231 0.683021
\(57\) 441.160 1.02514
\(58\) −35.4342 −0.0802196
\(59\) 142.031 0.313404 0.156702 − 0.987646i \(-0.449914\pi\)
0.156702 + 0.987646i \(0.449914\pi\)
\(60\) 332.231 0.714848
\(61\) 266.831 0.560069 0.280035 − 0.959990i \(-0.409654\pi\)
0.280035 + 0.959990i \(0.409654\pi\)
\(62\) 81.0862 0.166096
\(63\) −185.250 −0.370465
\(64\) −216.569 −0.422987
\(65\) 0 0
\(66\) −181.205 −0.337952
\(67\) −411.187 −0.749768 −0.374884 − 0.927072i \(-0.622317\pi\)
−0.374884 + 0.927072i \(0.622317\pi\)
\(68\) −317.087 −0.565477
\(69\) 159.476 0.278241
\(70\) 292.122 0.498789
\(71\) −91.5052 −0.152953 −0.0764765 − 0.997071i \(-0.524367\pi\)
−0.0764765 + 0.997071i \(0.524367\pi\)
\(72\) −125.153 −0.204854
\(73\) 63.1328 0.101221 0.0506105 − 0.998718i \(-0.483883\pi\)
0.0506105 + 0.998718i \(0.483883\pi\)
\(74\) 72.4347 0.113789
\(75\) 343.001 0.528084
\(76\) −1052.67 −1.58881
\(77\) 1355.25 2.00578
\(78\) 0 0
\(79\) −287.115 −0.408899 −0.204449 − 0.978877i \(-0.565540\pi\)
−0.204449 + 0.978877i \(0.565540\pi\)
\(80\) −688.595 −0.962341
\(81\) 81.0000 0.111111
\(82\) −325.637 −0.438543
\(83\) −373.812 −0.494352 −0.247176 − 0.968971i \(-0.579503\pi\)
−0.247176 + 0.968971i \(0.579503\pi\)
\(84\) 442.033 0.574164
\(85\) −685.272 −0.874449
\(86\) −374.147 −0.469132
\(87\) −115.877 −0.142797
\(88\) 915.595 1.10912
\(89\) 119.403 0.142209 0.0711047 − 0.997469i \(-0.477348\pi\)
0.0711047 + 0.997469i \(0.477348\pi\)
\(90\) −127.729 −0.149598
\(91\) 0 0
\(92\) −380.532 −0.431231
\(93\) 265.168 0.295663
\(94\) −62.3515 −0.0684157
\(95\) −2274.97 −2.45692
\(96\) 456.241 0.485051
\(97\) 554.650 0.580579 0.290290 − 0.956939i \(-0.406248\pi\)
0.290290 + 0.956939i \(0.406248\pi\)
\(98\) 74.0080 0.0762851
\(99\) −592.578 −0.601579
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.1.6 10
3.2 odd 2 1521.4.a.bk.1.5 10
13.5 odd 4 507.4.b.i.337.5 10
13.6 odd 12 39.4.j.c.10.3 yes 10
13.8 odd 4 507.4.b.i.337.6 10
13.11 odd 12 39.4.j.c.4.3 ✓ 10
13.12 even 2 inner 507.4.a.r.1.5 10
39.11 even 12 117.4.q.e.82.3 10
39.32 even 12 117.4.q.e.10.3 10
39.38 odd 2 1521.4.a.bk.1.6 10
52.11 even 12 624.4.bv.h.433.2 10
52.19 even 12 624.4.bv.h.49.4 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.c.4.3 ✓ 10 13.11 odd 12
39.4.j.c.10.3 yes 10 13.6 odd 12
117.4.q.e.10.3 10 39.32 even 12
117.4.q.e.82.3 10 39.11 even 12
507.4.a.r.1.5 10 13.12 even 2 inner
507.4.a.r.1.6 10 1.1 even 1 trivial
507.4.b.i.337.5 10 13.5 odd 4
507.4.b.i.337.6 10 13.8 odd 4
624.4.bv.h.49.4 10 52.19 even 12
624.4.bv.h.433.2 10 52.11 even 12
1521.4.a.bk.1.5 10 3.2 odd 2
1521.4.a.bk.1.6 10 39.38 odd 2