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Properties

Label 507.4.b.i.337.5
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 337.5
Root \(-0.917374i\) of defining polynomial
Character \(\chi\) \(=\) 507.337
Dual form 507.4.b.i.337.6

$q$-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)
 
\(f(q)\) \(=\) \(q-0.917374i q^{2} +3.00000 q^{3} +7.15843 q^{4} +15.4704i q^{5} -2.75212i q^{6} -20.5833i q^{7} -13.9059i q^{8} +9.00000 q^{9} +14.1922 q^{10} -65.8420i q^{11} +21.4753 q^{12} -18.8826 q^{14} +46.4112i q^{15} +44.5105 q^{16} -44.2956 q^{17} -8.25636i q^{18} -147.053i q^{19} +110.744i q^{20} -61.7500i q^{21} -60.4017 q^{22} -53.1586 q^{23} -41.7178i q^{24} -114.334 q^{25} +27.0000 q^{27} -147.344i q^{28} -38.6257 q^{29} +42.5765 q^{30} -88.3894i q^{31} -152.080i q^{32} -197.526i q^{33} +40.6357i q^{34} +318.433 q^{35} +64.4258 q^{36} +78.9587i q^{37} -134.903 q^{38} +215.131 q^{40} +354.966i q^{41} -56.6478 q^{42} +407.846 q^{43} -471.325i q^{44} +139.234i q^{45} +48.7663i q^{46} -67.9674i q^{47} +133.531 q^{48} -80.6738 q^{49} +104.887i q^{50} -132.887 q^{51} +226.572 q^{53} -24.7691i q^{54} +1018.60 q^{55} -286.231 q^{56} -441.160i q^{57} +35.4342i q^{58} +142.031i q^{59} +332.231i q^{60} +266.831 q^{61} -81.0862 q^{62} -185.250i q^{63} +216.569 q^{64} -181.205 q^{66} +411.187i q^{67} -317.087 q^{68} -159.476 q^{69} -292.122i q^{70} +91.5052i q^{71} -125.153i q^{72} +63.1328i q^{73} +72.4347 q^{74} -343.001 q^{75} -1052.67i q^{76} -1355.25 q^{77} -287.115 q^{79} +688.595i q^{80} +81.0000 q^{81} +325.637 q^{82} +373.812i q^{83} -442.033i q^{84} -685.272i q^{85} -374.147i q^{86} -115.877 q^{87} -915.595 q^{88} +119.403i q^{89} +127.729 q^{90} -380.532 q^{92} -265.168i q^{93} -62.3515 q^{94} +2274.97 q^{95} -456.241i q^{96} -554.650i q^{97} +74.0080i q^{98} -592.578i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 30 q^{3} - 60 q^{4} + 90 q^{9} - 80 q^{10} - 180 q^{12} - 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} - 960 q^{25} + 270 q^{27} + 990 q^{29} - 240 q^{30} - 120 q^{35} - 540 q^{36}+ \cdots + 2760 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.917374i − 0.324341i −0.986763 − 0.162170i \(-0.948151\pi\)
0.986763 − 0.162170i \(-0.0518494\pi\)
\(3\) 3.00000 0.577350
\(4\) 7.15843 0.894803
\(5\) 15.4704i 1.38372i 0.722034 + 0.691858i \(0.243208\pi\)
−0.722034 + 0.691858i \(0.756792\pi\)
\(6\) − 2.75212i − 0.187258i
\(7\) − 20.5833i − 1.11140i −0.831384 − 0.555698i \(-0.812451\pi\)
0.831384 − 0.555698i \(-0.187549\pi\)
\(8\) − 13.9059i − 0.614562i
\(9\) 9.00000 0.333333
\(10\) 14.1922 0.448795
\(11\) − 65.8420i − 1.80474i −0.430964 − 0.902369i \(-0.641827\pi\)
0.430964 − 0.902369i \(-0.358173\pi\)
\(12\) 21.4753 0.516615
\(13\) 0 0
\(14\) −18.8826 −0.360471
\(15\) 46.4112i 0.798889i
\(16\) 44.5105 0.695476
\(17\) −44.2956 −0.631957 −0.315979 − 0.948766i \(-0.602333\pi\)
−0.315979 + 0.948766i \(0.602333\pi\)
\(18\) − 8.25636i − 0.108114i
\(19\) − 147.053i − 1.77560i −0.460234 − 0.887798i \(-0.652234\pi\)
0.460234 − 0.887798i \(-0.347766\pi\)
\(20\) 110.744i 1.23815i
\(21\) − 61.7500i − 0.641665i
\(22\) −60.4017 −0.585350
\(23\) −53.1586 −0.481928 −0.240964 − 0.970534i \(-0.577464\pi\)
−0.240964 + 0.970534i \(0.577464\pi\)
\(24\) − 41.7178i − 0.354817i
\(25\) −114.334 −0.914669
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) − 147.344i − 0.994480i
\(29\) −38.6257 −0.247331 −0.123666 − 0.992324i \(-0.539465\pi\)
−0.123666 + 0.992324i \(0.539465\pi\)
\(30\) 42.5765 0.259112
\(31\) − 88.3894i − 0.512104i −0.966663 − 0.256052i \(-0.917578\pi\)
0.966663 − 0.256052i \(-0.0824218\pi\)
\(32\) − 152.080i − 0.840133i
\(33\) − 197.526i − 1.04197i
\(34\) 40.6357i 0.204969i
\(35\) 318.433 1.53786
\(36\) 64.4258 0.298268
\(37\) 78.9587i 0.350831i 0.984495 + 0.175415i \(0.0561268\pi\)
−0.984495 + 0.175415i \(0.943873\pi\)
\(38\) −134.903 −0.575898
\(39\) 0 0
\(40\) 215.131 0.850379
\(41\) 354.966i 1.35211i 0.736852 + 0.676054i \(0.236311\pi\)
−0.736852 + 0.676054i \(0.763689\pi\)
\(42\) −56.6478 −0.208118
\(43\) 407.846 1.44642 0.723208 − 0.690630i \(-0.242667\pi\)
0.723208 + 0.690630i \(0.242667\pi\)
\(44\) − 471.325i − 1.61489i
\(45\) 139.234i 0.461239i
\(46\) 48.7663i 0.156309i
\(47\) − 67.9674i − 0.210938i −0.994423 − 0.105469i \(-0.966366\pi\)
0.994423 − 0.105469i \(-0.0336343\pi\)
\(48\) 133.531 0.401533
\(49\) −80.6738 −0.235201
\(50\) 104.887i 0.296664i
\(51\) −132.887 −0.364861
\(52\) 0 0
\(53\) 226.572 0.587209 0.293604 − 0.955927i \(-0.405145\pi\)
0.293604 + 0.955927i \(0.405145\pi\)
\(54\) − 24.7691i − 0.0624194i
\(55\) 1018.60 2.49724
\(56\) −286.231 −0.683021
\(57\) − 441.160i − 1.02514i
\(58\) 35.4342i 0.0802196i
\(59\) 142.031i 0.313404i 0.987646 + 0.156702i \(0.0500862\pi\)
−0.987646 + 0.156702i \(0.949914\pi\)
\(60\) 332.231i 0.714848i
\(61\) 266.831 0.560069 0.280035 − 0.959990i \(-0.409654\pi\)
0.280035 + 0.959990i \(0.409654\pi\)
\(62\) −81.0862 −0.166096
\(63\) − 185.250i − 0.370465i
\(64\) 216.569 0.422987
\(65\) 0 0
\(66\) −181.205 −0.337952
\(67\) 411.187i 0.749768i 0.927072 + 0.374884i \(0.122317\pi\)
−0.927072 + 0.374884i \(0.877683\pi\)
\(68\) −317.087 −0.565477
\(69\) −159.476 −0.278241
\(70\) − 292.122i − 0.498789i
\(71\) 91.5052i 0.152953i 0.997071 + 0.0764765i \(0.0243670\pi\)
−0.997071 + 0.0764765i \(0.975633\pi\)
\(72\) − 125.153i − 0.204854i
\(73\) 63.1328i 0.101221i 0.998718 + 0.0506105i \(0.0161167\pi\)
−0.998718 + 0.0506105i \(0.983883\pi\)
\(74\) 72.4347 0.113789
\(75\) −343.001 −0.528084
\(76\) − 1052.67i − 1.58881i
\(77\) −1355.25 −2.00578
\(78\) 0 0
\(79\) −287.115 −0.408899 −0.204449 − 0.978877i \(-0.565540\pi\)
−0.204449 + 0.978877i \(0.565540\pi\)
\(80\) 688.595i 0.962341i
\(81\) 81.0000 0.111111
\(82\) 325.637 0.438543
\(83\) 373.812i 0.494352i 0.968971 + 0.247176i \(0.0795026\pi\)
−0.968971 + 0.247176i \(0.920497\pi\)
\(84\) − 442.033i − 0.574164i
\(85\) − 685.272i − 0.874449i
\(86\) − 374.147i − 0.469132i
\(87\) −115.877 −0.142797
\(88\) −915.595 −1.10912
\(89\) 119.403i 0.142209i 0.997469 + 0.0711047i \(0.0226525\pi\)
−0.997469 + 0.0711047i \(0.977348\pi\)
\(90\) 127.729 0.149598
\(91\) 0 0
\(92\) −380.532 −0.431231
\(93\) − 265.168i − 0.295663i
\(94\) −62.3515 −0.0684157
\(95\) 2274.97 2.45692
\(96\) − 456.241i − 0.485051i
\(97\) − 554.650i − 0.580579i −0.956939 − 0.290290i \(-0.906248\pi\)
0.956939 − 0.290290i \(-0.0937516\pi\)
\(98\) 74.0080i 0.0762851i
\(99\) − 592.578i − 0.601579i
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.337.5 10
13.5 odd 4 507.4.a.r.1.5 10
13.8 odd 4 507.4.a.r.1.6 10
13.9 even 3 39.4.j.c.10.3 yes 10
13.10 even 6 39.4.j.c.4.3 ✓ 10
13.12 even 2 inner 507.4.b.i.337.6 10
39.5 even 4 1521.4.a.bk.1.6 10
39.8 even 4 1521.4.a.bk.1.5 10
39.23 odd 6 117.4.q.e.82.3 10
39.35 odd 6 117.4.q.e.10.3 10
52.23 odd 6 624.4.bv.h.433.2 10
52.35 odd 6 624.4.bv.h.49.4 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.c.4.3 ✓ 10 13.10 even 6
39.4.j.c.10.3 yes 10 13.9 even 3
117.4.q.e.10.3 10 39.35 odd 6
117.4.q.e.82.3 10 39.23 odd 6
507.4.a.r.1.5 10 13.5 odd 4
507.4.a.r.1.6 10 13.8 odd 4
507.4.b.i.337.5 10 1.1 even 1 trivial
507.4.b.i.337.6 10 13.12 even 2 inner
624.4.bv.h.49.4 10 52.35 odd 6
624.4.bv.h.433.2 10 52.23 odd 6
1521.4.a.bk.1.5 10 39.8 even 4
1521.4.a.bk.1.6 10 39.5 even 4