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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 16.18
Character \(\chi\) \(=\) 137.16
Dual form 137.4.e.a.60.18

$q$-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+(-0.410351 + 0.0767078i) q^{2} +(-2.48305 - 3.28809i) q^{3} +(-7.29727 + 2.82698i) q^{4} +(-16.0067 - 2.99218i) q^{5} +(1.27114 + 1.15880i) q^{6} +(4.27155 + 8.57843i) q^{7} +(5.61703 - 3.47792i) q^{8} +(2.74291 - 9.64034i) q^{9} +6.79790 q^{10} +(20.9511 - 8.11650i) q^{11} +(27.4148 + 16.9745i) q^{12} +(35.9486 + 72.1945i) q^{13} +(-2.41087 - 3.19250i) q^{14} +(29.9070 + 60.0613i) q^{15} +(44.2281 - 40.3192i) q^{16} +(-16.3599 - 10.1296i) q^{17} +(-0.386067 + 4.16632i) q^{18} +(2.51013 - 27.0886i) q^{19} +(125.264 - 23.4160i) q^{20} +(17.6002 - 35.3459i) q^{21} +(-7.97470 + 4.93772i) q^{22} +(11.0192 - 10.0454i) q^{23} +(-25.3831 - 9.83345i) q^{24} +(130.704 + 50.6349i) q^{25} +(-20.2894 - 26.8675i) q^{26} +(-142.245 + 55.1061i) q^{27} +(-55.4217 - 50.5236i) q^{28} +(32.7213 - 29.8294i) q^{29} +(-16.8795 - 22.3521i) q^{30} +(24.4409 + 263.759i) q^{31} +(-46.9071 + 62.1150i) q^{32} +(-78.7103 - 48.7354i) q^{33} +(7.49030 + 2.90176i) q^{34} +(-42.7054 - 150.094i) q^{35} +(7.23725 + 78.1024i) q^{36} +214.116 q^{37} +(1.04787 + 11.3084i) q^{38} +(148.120 - 297.464i) q^{39} +(-100.317 + 38.8630i) q^{40} +263.565 q^{41} +(-4.51093 + 15.8543i) q^{42} +(-19.0034 + 205.079i) q^{43} +(-129.941 + 118.457i) q^{44} +(-72.7507 + 146.103i) q^{45} +(-3.75119 + 4.96738i) q^{46} +(19.5783 - 68.8105i) q^{47} +(-242.394 - 45.3112i) q^{48} +(151.360 - 200.433i) q^{49} +(-57.5185 - 10.7521i) q^{50} +(7.31532 + 78.9449i) q^{51} +(-466.419 - 425.197i) q^{52} +(-50.4433 + 544.369i) q^{53} +(54.1434 - 33.5242i) q^{54} +(-359.645 + 67.2293i) q^{55} +(53.8285 + 33.3292i) q^{56} +(-95.3024 + 59.0087i) q^{57} +(-11.1391 + 14.7505i) q^{58} +(16.4723 - 57.8940i) q^{59} +(-388.031 - 353.737i) q^{60} +(-85.9495 - 302.081i) q^{61} +(-30.2617 - 106.359i) q^{62} +(94.4155 - 17.6493i) q^{63} +(-198.928 + 399.502i) q^{64} +(-359.401 - 1263.16i) q^{65} +(36.0372 + 13.9609i) q^{66} +(133.124 + 267.349i) q^{67} +(148.019 + 27.6695i) q^{68} +(-60.3913 - 11.2891i) q^{69} +(29.0376 + 58.3153i) q^{70} +(114.247 + 44.2596i) q^{71} +(-18.1213 - 63.6897i) q^{72} +(113.644 - 228.227i) q^{73} +(-87.8627 + 16.4244i) q^{74} +(-158.052 - 555.494i) q^{75} +(58.2618 + 204.769i) q^{76} +(159.121 + 145.058i) q^{77} +(-37.9631 + 133.427i) q^{78} +(19.3805 - 25.6640i) q^{79} +(-828.590 + 513.041i) q^{80} +(304.310 + 188.421i) q^{81} +(-108.154 + 20.2175i) q^{82} +(1157.85 - 716.913i) q^{83} +(-28.5111 + 307.684i) q^{84} +(231.559 + 211.094i) q^{85} +(-7.93312 - 85.6120i) q^{86} +(-179.330 - 33.5227i) q^{87} +(89.4545 - 118.457i) q^{88} +(-177.484 - 33.1774i) q^{89} +(18.6461 - 65.5341i) q^{90} +(-465.759 + 616.765i) q^{91} +(-52.0123 + 104.455i) q^{92} +(806.575 - 735.290i) q^{93} +(-2.75565 + 29.7382i) q^{94} +(-121.233 + 426.089i) q^{95} +320.712 q^{96} +(-1211.73 + 469.427i) q^{97} +(-46.7360 + 93.8585i) q^{98} +(-20.7788 - 224.239i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.410351 + 0.0767078i −0.145081 + 0.0271203i −0.255789 − 0.966733i \(-0.582335\pi\)
0.110708 + 0.993853i \(0.464688\pi\)
\(3\) −2.48305 − 3.28809i −0.477863 − 0.632793i 0.493684 − 0.869642i \(-0.335650\pi\)
−0.971546 + 0.236849i \(0.923885\pi\)
\(4\) −7.29727 + 2.82698i −0.912159 + 0.353372i
\(5\) −16.0067 − 2.99218i −1.43169 − 0.267629i −0.589917 − 0.807464i \(-0.700839\pi\)
−0.841771 + 0.539835i \(0.818486\pi\)
\(6\) 1.27114 + 1.15880i 0.0864903 + 0.0788463i
\(7\) 4.27155 + 8.57843i 0.230642 + 0.463192i 0.980006 − 0.198966i \(-0.0637584\pi\)
−0.749364 + 0.662158i \(0.769641\pi\)
\(8\) 5.61703 − 3.47792i 0.248240 − 0.153704i
\(9\) 2.74291 − 9.64034i 0.101589 − 0.357050i
\(10\) 6.79790 0.214969
\(11\) 20.9511 − 8.11650i 0.574272 − 0.222474i −0.0565525 − 0.998400i \(-0.518011\pi\)
0.630825 + 0.775925i \(0.282717\pi\)
\(12\) 27.4148 + 16.9745i 0.659498 + 0.408344i
\(13\) 35.9486 + 72.1945i 0.766949 + 1.54024i 0.839798 + 0.542899i \(0.182673\pi\)
−0.0728490 + 0.997343i \(0.523209\pi\)
\(14\) −2.41087 − 3.19250i −0.0460237 − 0.0609452i
\(15\) 29.9070 + 60.0613i 0.514797 + 1.03385i
\(16\) 44.2281 − 40.3192i 0.691064 − 0.629988i
\(17\) −16.3599 − 10.1296i −0.233403 − 0.144517i 0.404748 − 0.914428i \(-0.367359\pi\)
−0.638151 + 0.769911i \(0.720300\pi\)
\(18\) −0.386067 + 4.16632i −0.00505538 + 0.0545562i
\(19\) 2.51013 − 27.0886i 0.0303086 − 0.327081i −0.967058 − 0.254556i \(-0.918071\pi\)
0.997367 − 0.0725253i \(-0.0231058\pi\)
\(20\) 125.264 − 23.4160i 1.40050 − 0.261799i
\(21\) 17.6002 − 35.3459i 0.182889 − 0.367291i
\(22\) −7.97470 + 4.93772i −0.0772823 + 0.0478512i
\(23\) 11.0192 − 10.0454i 0.0998986 − 0.0910696i −0.622206 − 0.782854i \(-0.713763\pi\)
0.722104 + 0.691784i \(0.243175\pi\)
\(24\) −25.3831 − 9.83345i −0.215887 − 0.0836352i
\(25\) 130.704 + 50.6349i 1.04563 + 0.405079i
\(26\) −20.2894 − 26.8675i −0.153041 − 0.202660i
\(27\) −142.245 + 55.1061i −1.01389 + 0.392784i
\(28\) −55.4217 − 50.5236i −0.374062 − 0.341002i
\(29\) 32.7213 − 29.8294i 0.209524 − 0.191006i −0.562130 − 0.827049i \(-0.690018\pi\)
0.771654 + 0.636042i \(0.219430\pi\)
\(30\) −16.8795 − 22.3521i −0.102725 − 0.136030i
\(31\) 24.4409 + 263.759i 0.141604 + 1.52815i 0.709299 + 0.704908i \(0.249012\pi\)
−0.567696 + 0.823239i \(0.692165\pi\)
\(32\) −46.9071 + 62.1150i −0.259127 + 0.343140i
\(33\) −78.7103 − 48.7354i −0.415203 − 0.257083i
\(34\) 7.49030 + 2.90176i 0.0377816 + 0.0146367i
\(35\) −42.7054 − 150.094i −0.206244 − 0.724872i
\(36\) 7.23725 + 78.1024i 0.0335058 + 0.361585i
\(37\) 214.116 0.951364 0.475682 − 0.879617i \(-0.342201\pi\)
0.475682 + 0.879617i \(0.342201\pi\)
\(38\) 1.04787 + 11.3084i 0.00447336 + 0.0482752i
\(39\) 148.120 − 297.464i 0.608157 − 1.22134i
\(40\) −100.317 + 38.8630i −0.396538 + 0.153620i
\(41\) 263.565 1.00395 0.501974 − 0.864883i \(-0.332607\pi\)
0.501974 + 0.864883i \(0.332607\pi\)
\(42\) −4.51093 + 15.8543i −0.0165727 + 0.0582469i
\(43\) −19.0034 + 205.079i −0.0673951 + 0.727308i 0.893256 + 0.449549i \(0.148415\pi\)
−0.960651 + 0.277759i \(0.910408\pi\)
\(44\) −129.941 + 118.457i −0.445211 + 0.405864i
\(45\) −72.7507 + 146.103i −0.241001 + 0.483995i
\(46\) −3.75119 + 4.96738i −0.0120235 + 0.0159217i
\(47\) 19.5783 − 68.8105i 0.0607614 − 0.213554i −0.925500 − 0.378748i \(-0.876355\pi\)
0.986261 + 0.165194i \(0.0528251\pi\)
\(48\) −242.394 − 45.3112i −0.728886 − 0.136252i
\(49\) 151.360 − 200.433i 0.441284 − 0.584354i
\(50\) −57.5185 − 10.7521i −0.162687 − 0.0304114i
\(51\) 7.31532 + 78.9449i 0.0200853 + 0.216755i
\(52\) −466.419 − 425.197i −1.24386 − 1.13393i
\(53\) −50.4433 + 544.369i −0.130734 + 1.41085i 0.637388 + 0.770543i \(0.280015\pi\)
−0.768122 + 0.640304i \(0.778808\pi\)
\(54\) 54.1434 − 33.5242i 0.136444 − 0.0844826i
\(55\) −359.645 + 67.2293i −0.881719 + 0.164822i
\(56\) 53.8285 + 33.3292i 0.128449 + 0.0795322i
\(57\) −95.3024 + 59.0087i −0.221458 + 0.137121i
\(58\) −11.1391 + 14.7505i −0.0252178 + 0.0333937i
\(59\) 16.4723 − 57.8940i 0.0363475 − 0.127748i −0.941529 − 0.336933i \(-0.890610\pi\)
0.977876 + 0.209185i \(0.0670810\pi\)
\(60\) −388.031 − 353.737i −0.834911 − 0.761122i
\(61\) −85.9495 − 302.081i −0.180405 − 0.634058i −0.998278 − 0.0586593i \(-0.981317\pi\)
0.817873 − 0.575399i \(-0.195153\pi\)
\(62\) −30.2617 − 106.359i −0.0619878 − 0.217864i
\(63\) 94.4155 − 17.6493i 0.188813 − 0.0352953i
\(64\) −198.928 + 399.502i −0.388532 + 0.780277i
\(65\) −359.401 − 1263.16i −0.685818 − 2.41040i
\(66\) 36.0372 + 13.9609i 0.0672102 + 0.0260374i
\(67\) 133.124 + 267.349i 0.242742 + 0.487492i 0.982710 − 0.185150i \(-0.0592772\pi\)
−0.739968 + 0.672642i \(0.765160\pi\)
\(68\) 148.019 + 27.6695i 0.263969 + 0.0493443i
\(69\) −60.3913 − 11.2891i −0.105366 − 0.0196963i
\(70\) 29.0376 + 58.3153i 0.0495808 + 0.0995717i
\(71\) 114.247 + 44.2596i 0.190967 + 0.0739809i 0.454826 − 0.890580i \(-0.349701\pi\)
−0.263859 + 0.964561i \(0.584995\pi\)
\(72\) −18.1213 − 63.6897i −0.0296613 − 0.104249i
\(73\) 113.644 − 228.227i 0.182205 − 0.365918i −0.785164 − 0.619288i \(-0.787421\pi\)
0.967369 + 0.253370i \(0.0815390\pi\)
\(74\) −87.8627 + 16.4244i −0.138025 + 0.0258013i
\(75\) −158.052 − 555.494i −0.243337 − 0.855240i
\(76\) 58.2618 + 204.769i 0.0879353 + 0.309061i
\(77\) 159.121 + 145.058i 0.235500 + 0.214686i
\(78\) −37.9631 + 133.427i −0.0551087 + 0.193687i
\(79\) 19.3805 − 25.6640i 0.0276010 − 0.0365497i −0.784007 − 0.620751i \(-0.786828\pi\)
0.811608 + 0.584202i \(0.198592\pi\)
\(80\) −828.590 + 513.041i −1.15799 + 0.716997i
\(81\) 304.310 + 188.421i 0.417435 + 0.258465i
\(82\) −108.154 + 20.2175i −0.145654 + 0.0272274i
\(83\) 1157.85 − 716.913i 1.53122 − 0.948090i 0.536672 − 0.843791i \(-0.319681\pi\)
0.994546 − 0.104299i \(-0.0332598\pi\)
\(84\) −28.5111 + 307.684i −0.0370335 + 0.399656i
\(85\) 231.559 + 211.094i 0.295483 + 0.269368i
\(86\) −7.93312 − 85.6120i −0.00994710 − 0.107346i
\(87\) −179.330 − 33.5227i −0.220991 − 0.0413104i
\(88\) 89.4545 − 118.457i 0.108362 − 0.143495i
\(89\) −177.484 − 33.1774i −0.211385 − 0.0395146i 0.0769920 − 0.997032i \(-0.475468\pi\)
−0.288377 + 0.957517i \(0.593115\pi\)
\(90\) 18.6461 − 65.5341i 0.0218385 − 0.0767544i
\(91\) −465.759 + 616.765i −0.536536 + 0.710489i
\(92\) −52.0123 + 104.455i −0.0589420 + 0.118371i
\(93\) 806.575 − 735.290i 0.899332 − 0.819850i
\(94\) −2.75565 + 29.7382i −0.00302366 + 0.0326305i
\(95\) −121.233 + 426.089i −0.130929 + 0.460167i
\(96\) 320.712 0.340964
\(97\) −1211.73 + 469.427i −1.26838 + 0.491372i −0.899017 − 0.437914i \(-0.855718\pi\)
−0.369361 + 0.929286i \(0.620423\pi\)
\(98\) −46.7360 + 93.8585i −0.0481740 + 0.0967463i
\(99\) −20.7788 − 224.239i −0.0210944 − 0.227645i
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.16.18 ✓ 544
137.60 even 17 inner 137.4.e.a.60.18 yes 544
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
137.4.e.a.16.18 ✓ 544 1.1 even 1 trivial
137.4.e.a.60.18 yes 544 137.60 even 17 inner