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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 = [18,0,-54,-64,0,0,0,0,162,-396,0,192,0,196,0,64,268] 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: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 97 x^{16} + 3906 x^{14} + 84743 x^{12} + 1077128 x^{10} + 8187552 x^{8} + 36483705 x^{6} + \cdots + 26460736 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 13^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.15
Root \(2.37739i\) of defining polynomial
Character \(\chi\) \(=\) 507.337
Dual form 507.4.b.j.337.4

$q$-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)
 
\(f(q)\) \(=\) \(q+3.37739i q^{2} -3.00000 q^{3} -3.40677 q^{4} -15.7127i q^{5} -10.1322i q^{6} +17.1681i q^{7} +15.5131i q^{8} +9.00000 q^{9} +53.0679 q^{10} -52.8187i q^{11} +10.2203 q^{12} -57.9835 q^{14} +47.1380i q^{15} -79.6481 q^{16} +71.0654 q^{17} +30.3965i q^{18} +92.6916i q^{19} +53.5295i q^{20} -51.5044i q^{21} +178.390 q^{22} -190.712 q^{23} -46.5394i q^{24} -121.888 q^{25} -27.0000 q^{27} -58.4878i q^{28} -128.204 q^{29} -159.204 q^{30} +3.29674i q^{31} -144.898i q^{32} +158.456i q^{33} +240.016i q^{34} +269.757 q^{35} -30.6609 q^{36} -241.546i q^{37} -313.056 q^{38} +243.753 q^{40} -97.1824i q^{41} +173.950 q^{42} -376.151 q^{43} +179.941i q^{44} -141.414i q^{45} -644.109i q^{46} -577.354i q^{47} +238.944 q^{48} +48.2555 q^{49} -411.664i q^{50} -213.196 q^{51} -307.686 q^{53} -91.1896i q^{54} -829.924 q^{55} -266.331 q^{56} -278.075i q^{57} -432.995i q^{58} -349.914i q^{59} -160.588i q^{60} +127.467 q^{61} -11.1344 q^{62} +154.513i q^{63} -147.809 q^{64} -535.169 q^{66} -903.564i q^{67} -242.104 q^{68} +572.136 q^{69} +911.076i q^{70} +826.106i q^{71} +139.618i q^{72} +131.760i q^{73} +815.796 q^{74} +365.665 q^{75} -315.779i q^{76} +906.799 q^{77} -556.244 q^{79} +1251.48i q^{80} +81.0000 q^{81} +328.223 q^{82} -254.664i q^{83} +175.464i q^{84} -1116.63i q^{85} -1270.41i q^{86} +384.612 q^{87} +819.384 q^{88} -183.410i q^{89} +477.611 q^{90} +649.712 q^{92} -9.89023i q^{93} +1949.95 q^{94} +1456.43 q^{95} +434.693i q^{96} -780.498i q^{97} +162.978i q^{98} -475.369i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 54 q^{3} - 64 q^{4} + 162 q^{9} - 396 q^{10} + 192 q^{12} + 196 q^{14} + 64 q^{16} + 268 q^{17} + 548 q^{22} - 452 q^{23} - 1224 q^{25} - 486 q^{27} - 1094 q^{29} + 1188 q^{30} + 276 q^{35} - 576 q^{36}+ \cdots + 444 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.37739i 1.19409i 0.802208 + 0.597044i \(0.203658\pi\)
−0.802208 + 0.597044i \(0.796342\pi\)
\(3\) −3.00000 −0.577350
\(4\) −3.40677 −0.425846
\(5\) − 15.7127i − 1.40538i −0.711494 − 0.702692i \(-0.751981\pi\)
0.711494 − 0.702692i \(-0.248019\pi\)
\(6\) − 10.1322i − 0.689407i
\(7\) 17.1681i 0.926992i 0.886099 + 0.463496i \(0.153405\pi\)
−0.886099 + 0.463496i \(0.846595\pi\)
\(8\) 15.5131i 0.685590i
\(9\) 9.00000 0.333333
\(10\) 53.0679 1.67815
\(11\) − 52.8187i − 1.44777i −0.689922 − 0.723884i \(-0.742355\pi\)
0.689922 − 0.723884i \(-0.257645\pi\)
\(12\) 10.2203 0.245862
\(13\) 0 0
\(14\) −57.9835 −1.10691
\(15\) 47.1380i 0.811399i
\(16\) −79.6481 −1.24450
\(17\) 71.0654 1.01388 0.506938 − 0.861982i \(-0.330777\pi\)
0.506938 + 0.861982i \(0.330777\pi\)
\(18\) 30.3965i 0.398029i
\(19\) 92.6916i 1.11921i 0.828761 + 0.559603i \(0.189046\pi\)
−0.828761 + 0.559603i \(0.810954\pi\)
\(20\) 53.5295i 0.598478i
\(21\) − 51.5044i − 0.535199i
\(22\) 178.390 1.72876
\(23\) −190.712 −1.72896 −0.864482 − 0.502663i \(-0.832354\pi\)
−0.864482 + 0.502663i \(0.832354\pi\)
\(24\) − 46.5394i − 0.395826i
\(25\) −121.888 −0.975106
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) − 58.4878i − 0.394756i
\(29\) −128.204 −0.820927 −0.410463 − 0.911877i \(-0.634633\pi\)
−0.410463 + 0.911877i \(0.634633\pi\)
\(30\) −159.204 −0.968882
\(31\) 3.29674i 0.0191004i 0.999954 + 0.00955020i \(0.00303997\pi\)
−0.999954 + 0.00955020i \(0.996960\pi\)
\(32\) − 144.898i − 0.800454i
\(33\) 158.456i 0.835869i
\(34\) 240.016i 1.21066i
\(35\) 269.757 1.30278
\(36\) −30.6609 −0.141949
\(37\) − 241.546i − 1.07324i −0.843823 − 0.536621i \(-0.819700\pi\)
0.843823 − 0.536621i \(-0.180300\pi\)
\(38\) −313.056 −1.33643
\(39\) 0 0
\(40\) 243.753 0.963518
\(41\) − 97.1824i − 0.370179i −0.982722 − 0.185090i \(-0.940742\pi\)
0.982722 − 0.185090i \(-0.0592575\pi\)
\(42\) 173.950 0.639075
\(43\) −376.151 −1.33401 −0.667006 − 0.745052i \(-0.732425\pi\)
−0.667006 + 0.745052i \(0.732425\pi\)
\(44\) 179.941i 0.616527i
\(45\) − 141.414i − 0.468462i
\(46\) − 644.109i − 2.06454i
\(47\) − 577.354i − 1.79182i −0.444231 − 0.895912i \(-0.646523\pi\)
0.444231 − 0.895912i \(-0.353477\pi\)
\(48\) 238.944 0.718513
\(49\) 48.2555 0.140686
\(50\) − 411.664i − 1.16436i
\(51\) −213.196 −0.585362
\(52\) 0 0
\(53\) −307.686 −0.797433 −0.398716 − 0.917074i \(-0.630544\pi\)
−0.398716 + 0.917074i \(0.630544\pi\)
\(54\) − 91.1896i − 0.229802i
\(55\) −829.924 −2.03467
\(56\) −266.331 −0.635536
\(57\) − 278.075i − 0.646174i
\(58\) − 432.995i − 0.980259i
\(59\) − 349.914i − 0.772117i −0.922474 − 0.386059i \(-0.873836\pi\)
0.922474 − 0.386059i \(-0.126164\pi\)
\(60\) − 160.588i − 0.345531i
\(61\) 127.467 0.267548 0.133774 − 0.991012i \(-0.457290\pi\)
0.133774 + 0.991012i \(0.457290\pi\)
\(62\) −11.1344 −0.0228076
\(63\) 154.513i 0.308997i
\(64\) −147.809 −0.288689
\(65\) 0 0
\(66\) −535.169 −0.998102
\(67\) − 903.564i − 1.64758i −0.566894 − 0.823791i \(-0.691855\pi\)
0.566894 − 0.823791i \(-0.308145\pi\)
\(68\) −242.104 −0.431755
\(69\) 572.136 0.998218
\(70\) 911.076i 1.55563i
\(71\) 826.106i 1.38086i 0.723401 + 0.690428i \(0.242578\pi\)
−0.723401 + 0.690428i \(0.757422\pi\)
\(72\) 139.618i 0.228530i
\(73\) 131.760i 0.211252i 0.994406 + 0.105626i \(0.0336846\pi\)
−0.994406 + 0.105626i \(0.966315\pi\)
\(74\) 815.796 1.28155
\(75\) 365.665 0.562978
\(76\) − 315.779i − 0.476610i
\(77\) 906.799 1.34207
\(78\) 0 0
\(79\) −556.244 −0.792182 −0.396091 − 0.918211i \(-0.629633\pi\)
−0.396091 + 0.918211i \(0.629633\pi\)
\(80\) 1251.48i 1.74900i
\(81\) 81.0000 0.111111
\(82\) 328.223 0.442026
\(83\) − 254.664i − 0.336783i −0.985720 − 0.168391i \(-0.946143\pi\)
0.985720 − 0.168391i \(-0.0538573\pi\)
\(84\) 175.464i 0.227912i
\(85\) − 1116.63i − 1.42489i
\(86\) − 1270.41i − 1.59293i
\(87\) 384.612 0.473962
\(88\) 819.384 0.992576
\(89\) − 183.410i − 0.218443i −0.994017 − 0.109222i \(-0.965164\pi\)
0.994017 − 0.109222i \(-0.0348358\pi\)
\(90\) 477.611 0.559384
\(91\) 0 0
\(92\) 649.712 0.736273
\(93\) − 9.89023i − 0.0110276i
\(94\) 1949.95 2.13960
\(95\) 1456.43 1.57292
\(96\) 434.693i 0.462142i
\(97\) − 780.498i − 0.816986i −0.912762 − 0.408493i \(-0.866054\pi\)
0.912762 − 0.408493i \(-0.133946\pi\)
\(98\) 162.978i 0.167992i
\(99\) − 475.369i − 0.482589i
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.j.337.15 18
13.5 odd 4 507.4.a.q.1.7 yes 9
13.8 odd 4 507.4.a.n.1.3 ✓ 9
13.12 even 2 inner 507.4.b.j.337.4 18
39.5 even 4 1521.4.a.be.1.3 9
39.8 even 4 1521.4.a.bj.1.7 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.4.a.n.1.3 ✓ 9 13.8 odd 4
507.4.a.q.1.7 yes 9 13.5 odd 4
507.4.b.j.337.4 18 13.12 even 2 inner
507.4.b.j.337.15 18 1.1 even 1 trivial
1521.4.a.be.1.3 9 39.5 even 4
1521.4.a.bj.1.7 9 39.8 even 4