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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [137,4,Mod(16,137)] 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("137.16"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(137, base_ring=CyclotomicField(34)) chi = DirichletCharacter(H, H._module([10])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 137 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 137.e (of order \(17\), degree \(16\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.08326167079\)
Analytic rank: \(0\)
Dimension: \(544\)
Relative dimension: \(34\) over \(\Q(\zeta_{17})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{17}]$

Embedding invariants

Embedding label 16.12
Character \(\chi\) \(=\) 137.16
Dual form 137.4.e.a.60.12

$q$-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+(-1.78477 + 0.333632i) q^{2} +(-4.20790 - 5.57217i) q^{3} +(-4.38567 + 1.69902i) q^{4} +(-14.0262 - 2.62195i) q^{5} +(9.36920 + 8.54116i) q^{6} +(-12.9047 - 25.9162i) q^{7} +(19.6104 - 12.1423i) q^{8} +(-5.95367 + 20.9250i) q^{9} +25.9083 q^{10} +(-48.3889 + 18.7460i) q^{11} +(27.9217 + 17.2884i) q^{12} +(-22.1277 - 44.4384i) q^{13} +(31.6785 + 41.9491i) q^{14} +(44.4109 + 89.1890i) q^{15} +(-3.14299 + 2.86521i) q^{16} +(45.3635 + 28.0879i) q^{17} +(3.64471 - 39.3327i) q^{18} +(-6.66326 + 71.9080i) q^{19} +(65.9689 - 12.3317i) q^{20} +(-90.1074 + 180.960i) q^{21} +(80.1089 - 49.6013i) q^{22} +(155.868 - 142.093i) q^{23} +(-150.177 - 58.1790i) q^{24} +(73.2997 + 28.3965i) q^{25} +(54.3189 + 71.9299i) q^{26} +(-34.1470 + 13.2286i) q^{27} +(100.628 + 91.7346i) q^{28} +(30.4332 - 27.7435i) q^{29} +(-109.020 - 144.365i) q^{30} +(-16.7319 - 180.566i) q^{31} +(-106.545 + 141.089i) q^{32} +(308.071 + 190.750i) q^{33} +(-90.3346 - 34.9958i) q^{34} +(113.053 + 397.340i) q^{35} +(-9.44109 - 101.886i) q^{36} -159.842 q^{37} +(-12.0984 - 130.562i) q^{38} +(-154.507 + 310.292i) q^{39} +(-306.895 + 118.892i) q^{40} -327.311 q^{41} +(100.447 - 353.035i) q^{42} +(-7.27433 + 78.5025i) q^{43} +(180.368 - 164.427i) q^{44} +(138.371 - 277.887i) q^{45} +(-230.783 + 305.605i) q^{46} +(79.2916 - 278.681i) q^{47} +(29.1908 + 5.45671i) q^{48} +(-298.413 + 395.163i) q^{49} +(-140.297 - 26.2261i) q^{50} +(-34.3749 - 370.964i) q^{51} +(172.546 + 157.297i) q^{52} +(39.4290 - 425.507i) q^{53} +(56.5312 - 35.0026i) q^{54} +(727.861 - 136.061i) q^{55} +(-567.748 - 351.535i) q^{56} +(428.722 - 265.453i) q^{57} +(-45.0602 + 59.6693i) q^{58} +(88.9376 - 312.583i) q^{59} +(-346.305 - 315.699i) q^{60} +(59.7273 + 209.920i) q^{61} +(90.1053 + 316.687i) q^{62} +(619.126 - 115.735i) q^{63} +(158.253 - 317.816i) q^{64} +(193.852 + 681.318i) q^{65} +(-613.477 - 237.662i) q^{66} +(-305.581 - 613.689i) q^{67} +(-246.672 - 46.1109i) q^{68} +(-1447.64 - 270.611i) q^{69} +(-334.339 - 671.444i) q^{70} +(229.013 + 88.7201i) q^{71} +(137.323 + 482.639i) q^{72} +(-423.321 + 850.143i) q^{73} +(285.282 - 53.3285i) q^{74} +(-150.208 - 527.928i) q^{75} +(-92.9502 - 326.686i) q^{76} +(1110.27 + 1012.14i) q^{77} +(172.236 - 605.348i) q^{78} +(111.927 - 148.216i) q^{79} +(51.5965 - 31.9472i) q^{80} +(716.815 + 443.833i) q^{81} +(584.175 - 109.201i) q^{82} +(294.921 - 182.607i) q^{83} +(87.7271 - 946.726i) q^{84} +(-562.632 - 512.906i) q^{85} +(-13.2079 - 142.536i) q^{86} +(-282.651 - 52.8367i) q^{87} +(-721.308 + 955.166i) q^{88} +(1198.46 + 224.032i) q^{89} +(-154.249 + 542.130i) q^{90} +(-866.121 + 1146.93i) q^{91} +(-442.169 + 887.995i) q^{92} +(-935.738 + 853.038i) q^{93} +(-48.5406 + 523.836i) q^{94} +(281.999 - 991.123i) q^{95} +1234.50 q^{96} +(-1031.81 + 399.725i) q^{97} +(400.760 - 804.835i) q^{98} +(-104.167 - 1124.14i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 544 q - 17 q^{2} - 17 q^{3} - 51 q^{4} - 7 q^{5} + 33 q^{6} - 59 q^{7} + 19 q^{8} - 319 q^{9} - 32 q^{10} + 39 q^{11} - 213 q^{12} + 69 q^{13} + 15 q^{14} - 117 q^{15} - 1627 q^{16} + 377 q^{17} + 917 q^{18}+ \cdots + 2609 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/137\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{5}{17}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.78477 + 0.333632i −0.631012 + 0.117957i −0.489533 − 0.871985i \(-0.662833\pi\)
−0.141479 + 0.989941i \(0.545186\pi\)
\(3\) −4.20790 − 5.57217i −0.809812 − 1.07236i −0.995956 − 0.0898414i \(-0.971364\pi\)
0.186145 − 0.982522i \(-0.440401\pi\)
\(4\) −4.38567 + 1.69902i −0.548209 + 0.212377i
\(5\) −14.0262 − 2.62195i −1.25454 − 0.234514i −0.485796 − 0.874072i \(-0.661470\pi\)
−0.768743 + 0.639558i \(0.779117\pi\)
\(6\) 9.36920 + 8.54116i 0.637494 + 0.581152i
\(7\) −12.9047 − 25.9162i −0.696790 − 1.39934i −0.906765 − 0.421636i \(-0.861456\pi\)
0.209976 − 0.977707i \(-0.432662\pi\)
\(8\) 19.6104 − 12.1423i 0.866666 − 0.536617i
\(9\) −5.95367 + 20.9250i −0.220506 + 0.774999i
\(10\) 25.9083 0.819292
\(11\) −48.3889 + 18.7460i −1.32635 + 0.513829i −0.917122 − 0.398606i \(-0.869494\pi\)
−0.409223 + 0.912435i \(0.634200\pi\)
\(12\) 27.9217 + 17.2884i 0.671692 + 0.415894i
\(13\) −22.1277 − 44.4384i −0.472086 − 0.948076i −0.995388 − 0.0959290i \(-0.969418\pi\)
0.523302 − 0.852147i \(-0.324700\pi\)
\(14\) 31.6785 + 41.9491i 0.604745 + 0.800811i
\(15\) 44.4109 + 89.1890i 0.764456 + 1.53523i
\(16\) −3.14299 + 2.86521i −0.0491092 + 0.0447689i
\(17\) 45.3635 + 28.0879i 0.647193 + 0.400725i 0.810408 − 0.585866i \(-0.199245\pi\)
−0.163216 + 0.986590i \(0.552187\pi\)
\(18\) 3.64471 − 39.3327i 0.0477259 − 0.515044i
\(19\) −6.66326 + 71.9080i −0.0804556 + 0.868254i 0.855886 + 0.517164i \(0.173012\pi\)
−0.936342 + 0.351090i \(0.885811\pi\)
\(20\) 65.9689 − 12.3317i 0.737555 − 0.137873i
\(21\) −90.1074 + 180.960i −0.936336 + 1.88042i
\(22\) 80.1089 − 49.6013i 0.776331 − 0.480684i
\(23\) 155.868 − 142.093i 1.41308 − 1.28819i 0.509033 − 0.860747i \(-0.330003\pi\)
0.904043 − 0.427442i \(-0.140585\pi\)
\(24\) −150.177 − 58.1790i −1.27728 − 0.494823i
\(25\) 73.2997 + 28.3965i 0.586398 + 0.227172i
\(26\) 54.3189 + 71.9299i 0.409724 + 0.542562i
\(27\) −34.1470 + 13.2286i −0.243393 + 0.0942908i
\(28\) 100.628 + 91.7346i 0.679175 + 0.619150i
\(29\) 30.4332 − 27.7435i 0.194872 − 0.177650i −0.570378 − 0.821382i \(-0.693203\pi\)
0.765251 + 0.643733i \(0.222615\pi\)
\(30\) −109.020 − 144.365i −0.663472 − 0.878579i
\(31\) −16.7319 − 180.566i −0.0969400 − 1.04615i −0.895748 − 0.444563i \(-0.853359\pi\)
0.798808 − 0.601587i \(-0.205465\pi\)
\(32\) −106.545 + 141.089i −0.588586 + 0.779414i
\(33\) 308.071 + 190.750i 1.62510 + 1.00622i
\(34\) −90.3346 − 34.9958i −0.455655 − 0.176522i
\(35\) 113.053 + 397.340i 0.545984 + 1.91894i
\(36\) −9.44109 − 101.886i −0.0437087 − 0.471692i
\(37\) −159.842 −0.710214 −0.355107 − 0.934826i \(-0.615556\pi\)
−0.355107 + 0.934826i \(0.615556\pi\)
\(38\) −12.0984 − 130.562i −0.0516479 − 0.557369i
\(39\) −154.507 + 310.292i −0.634382 + 1.27401i
\(40\) −306.895 + 118.892i −1.21311 + 0.469961i
\(41\) −327.311 −1.24676 −0.623382 − 0.781917i \(-0.714242\pi\)
−0.623382 + 0.781917i \(0.714242\pi\)
\(42\) 100.447 − 353.035i 0.369032 − 1.29701i
\(43\) −7.27433 + 78.5025i −0.0257983 + 0.278408i 0.973057 + 0.230566i \(0.0740578\pi\)
−0.998855 + 0.0478415i \(0.984766\pi\)
\(44\) 180.368 − 164.427i 0.617989 − 0.563372i
\(45\) 138.371 − 277.887i 0.458382 − 0.920555i
\(46\) −230.783 + 305.605i −0.739718 + 0.979545i
\(47\) 79.2916 − 278.681i 0.246082 − 0.864890i −0.736086 − 0.676888i \(-0.763328\pi\)
0.982169 − 0.188002i \(-0.0602011\pi\)
\(48\) 29.1908 + 5.45671i 0.0877778 + 0.0164085i
\(49\) −298.413 + 395.163i −0.870009 + 1.15208i
\(50\) −140.297 − 26.2261i −0.396821 − 0.0741787i
\(51\) −34.3749 − 370.964i −0.0943814 − 1.01854i
\(52\) 172.546 + 157.297i 0.460152 + 0.419484i
\(53\) 39.4290 − 425.507i 0.102189 − 1.10279i −0.778032 − 0.628225i \(-0.783782\pi\)
0.880220 − 0.474565i \(-0.157395\pi\)
\(54\) 56.5312 − 35.0026i 0.142462 − 0.0882084i
\(55\) 727.861 − 136.061i 1.78445 − 0.333572i
\(56\) −567.748 − 351.535i −1.35480 − 0.838854i
\(57\) 428.722 − 265.453i 0.996238 − 0.616844i
\(58\) −45.0602 + 59.6693i −0.102012 + 0.135086i
\(59\) 88.9376 − 312.583i 0.196249 − 0.689743i −0.799746 − 0.600338i \(-0.795033\pi\)
0.995995 − 0.0894054i \(-0.0284967\pi\)
\(60\) −346.305 − 315.699i −0.745131 − 0.679276i
\(61\) 59.7273 + 209.920i 0.125365 + 0.440614i 0.998938 − 0.0460806i \(-0.0146731\pi\)
−0.873572 + 0.486694i \(0.838203\pi\)
\(62\) 90.1053 + 316.687i 0.184571 + 0.648699i
\(63\) 619.126 − 115.735i 1.23814 − 0.231448i
\(64\) 158.253 − 317.816i 0.309089 − 0.620734i
\(65\) 193.852 + 681.318i 0.369913 + 1.30011i
\(66\) −613.477 − 237.662i −1.14415 − 0.443246i
\(67\) −305.581 − 613.689i −0.557203 − 1.11901i −0.977364 − 0.211565i \(-0.932144\pi\)
0.420161 − 0.907450i \(-0.361974\pi\)
\(68\) −246.672 − 46.1109i −0.439902 − 0.0822319i
\(69\) −1447.64 − 270.611i −2.52573 − 0.472141i
\(70\) −334.339 − 671.444i −0.570874 − 1.14647i
\(71\) 229.013 + 88.7201i 0.382801 + 0.148298i 0.544950 − 0.838468i \(-0.316549\pi\)
−0.162150 + 0.986766i \(0.551843\pi\)
\(72\) 137.323 + 482.639i 0.224772 + 0.789993i
\(73\) −423.321 + 850.143i −0.678712 + 1.36304i 0.241032 + 0.970517i \(0.422514\pi\)
−0.919744 + 0.392520i \(0.871603\pi\)
\(74\) 285.282 − 53.3285i 0.448154 − 0.0837745i
\(75\) −150.208 − 527.928i −0.231261 − 0.812798i
\(76\) −92.9502 − 326.686i −0.140291 − 0.493072i
\(77\) 1110.27 + 1012.14i 1.64321 + 1.49798i
\(78\) 172.236 − 605.348i 0.250025 − 0.878746i
\(79\) 111.927 − 148.216i 0.159403 − 0.211083i −0.711306 − 0.702883i \(-0.751896\pi\)
0.870709 + 0.491799i \(0.163661\pi\)
\(80\) 51.5965 − 31.9472i 0.0721083 − 0.0446476i
\(81\) 716.815 + 443.833i 0.983286 + 0.608825i
\(82\) 584.175 − 109.201i 0.786724 − 0.147064i
\(83\) 294.921 − 182.607i 0.390021 − 0.241491i −0.317387 − 0.948296i \(-0.602805\pi\)
0.707408 + 0.706805i \(0.249864\pi\)
\(84\) 87.7271 − 946.726i 0.113950 − 1.22972i
\(85\) −562.632 − 512.906i −0.717952 − 0.654500i
\(86\) −13.2079 − 142.536i −0.0165610 − 0.178722i
\(87\) −282.651 − 52.8367i −0.348315 − 0.0651114i
\(88\) −721.308 + 955.166i −0.873769 + 1.15706i
\(89\) 1198.46 + 224.032i 1.42738 + 0.266824i 0.840043 − 0.542519i \(-0.182529\pi\)
0.587337 + 0.809343i \(0.300176\pi\)
\(90\) −154.249 + 542.130i −0.180659 + 0.634951i
\(91\) −866.121 + 1146.93i −0.997738 + 1.32122i
\(92\) −442.169 + 887.995i −0.501079 + 1.00630i
\(93\) −935.738 + 853.038i −1.04335 + 0.951139i
\(94\) −48.5406 + 523.836i −0.0532615 + 0.574783i
\(95\) 281.999 − 991.123i 0.304552 − 1.07039i
\(96\) 1234.50 1.31246
\(97\) −1031.81 + 399.725i −1.08004 + 0.418412i −0.834519 − 0.550979i \(-0.814255\pi\)
−0.245525 + 0.969390i \(0.578961\pi\)
\(98\) 400.760 − 804.835i 0.413091 − 0.829599i
\(99\) −104.167 − 1124.14i −0.105750 − 1.14122i
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 137.4.e.a.16.12 ✓ 544
137.60 even 17 inner 137.4.e.a.60.12 yes 544
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
137.4.e.a.16.12 ✓ 544 1.1 even 1 trivial
137.4.e.a.60.12 yes 544 137.60 even 17 inner