Show commands: Magma / Pari/GP / SageMath

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 16.17
Character \(\chi\) \(=\) 137.16
Dual form 137.4.e.a.60.17

$q$-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+(-0.488805 + 0.0913735i) q^{2} +(5.41372 + 7.16892i) q^{3} +(-7.22920 + 2.80061i) q^{4} +(-7.46078 - 1.39466i) q^{5} +(-3.30130 - 3.00953i) q^{6} +(-8.95977 - 17.9936i) q^{7} +(6.66008 - 4.12375i) q^{8} +(-14.6962 + 51.6517i) q^{9} +3.77430 q^{10} +(-28.9764 + 11.2255i) q^{11} +(-59.2141 - 36.6638i) q^{12} +(3.58277 + 7.19518i) q^{13} +(6.02372 + 7.97669i) q^{14} +(-30.3923 - 61.0360i) q^{15} +(42.9560 - 39.1595i) q^{16} +(-56.0117 - 34.6810i) q^{17} +(2.46397 - 26.5904i) q^{18} +(-4.00257 + 43.1946i) q^{19} +(57.8413 - 10.8124i) q^{20} +(80.4893 - 161.644i) q^{21} +(13.1381 - 8.13475i) q^{22} +(44.2423 - 40.3322i) q^{23} +(65.6186 + 25.4208i) q^{24} +(-62.8409 - 24.3447i) q^{25} +(-2.40872 - 3.18967i) q^{26} +(-223.675 + 86.6521i) q^{27} +(115.165 + 104.987i) q^{28} +(-49.2699 + 44.9154i) q^{29} +(20.4330 + 27.0577i) q^{30} +(16.8698 + 182.054i) q^{31} +(-55.1843 + 73.0758i) q^{32} +(-237.345 - 146.958i) q^{33} +(30.5477 + 11.8342i) q^{34} +(41.7518 + 146.742i) q^{35} +(-38.4145 - 414.559i) q^{36} -345.092 q^{37} +(-1.99037 - 21.4795i) q^{38} +(-32.1855 + 64.6373i) q^{39} +(-55.4406 + 21.4778i) q^{40} +13.4548 q^{41} +(-24.5736 + 86.3671i) q^{42} +(-11.6619 + 125.852i) q^{43} +(178.038 - 162.303i) q^{44} +(181.682 - 364.866i) q^{45} +(-17.9406 + 23.7571i) q^{46} +(-121.022 + 425.348i) q^{47} +(513.283 + 95.9492i) q^{48} +(-36.7898 + 48.7176i) q^{49} +(32.9414 + 6.15781i) q^{50} +(-54.6064 - 589.296i) q^{51} +(-46.0514 - 41.9814i) q^{52} +(-64.6155 + 697.312i) q^{53} +(101.416 - 62.7939i) q^{54} +(231.842 - 43.3388i) q^{55} +(-133.874 - 82.8912i) q^{56} +(-331.327 + 205.149i) q^{57} +(19.9793 - 26.4568i) q^{58} +(72.3752 - 254.372i) q^{59} +(390.650 + 356.125i) q^{60} +(140.530 + 493.910i) q^{61} +(-24.8810 - 87.4475i) q^{62} +(1061.08 - 198.350i) q^{63} +(-186.976 + 375.499i) q^{64} +(-16.6954 - 58.6784i) q^{65} +(129.443 + 50.1465i) q^{66} +(-1.85912 - 3.73362i) q^{67} +(502.047 + 93.8489i) q^{68} +(528.654 + 98.8225i) q^{69} +(-33.8168 - 67.9134i) q^{70} +(325.338 + 126.037i) q^{71} +(115.121 + 404.608i) q^{72} +(528.515 - 1061.40i) q^{73} +(168.683 - 31.5323i) q^{74} +(-165.678 - 582.296i) q^{75} +(-92.0357 - 323.472i) q^{76} +(461.609 + 420.812i) q^{77} +(9.82631 - 34.5359i) q^{78} +(593.091 - 785.380i) q^{79} +(-375.099 + 232.252i) q^{80} +(-599.343 - 371.098i) q^{81} +(-6.57675 + 1.22941i) q^{82} +(-854.161 + 528.874i) q^{83} +(-129.171 + 1393.98i) q^{84} +(369.523 + 336.865i) q^{85} +(-5.79913 - 62.5826i) q^{86} +(-588.728 - 110.052i) q^{87} +(-146.694 + 194.254i) q^{88} +(-1508.67 - 282.019i) q^{89} +(-55.4678 + 194.949i) q^{90} +(97.3666 - 128.934i) q^{91} +(-206.882 + 415.475i) q^{92} +(-1213.80 + 1106.53i) q^{93} +(20.2906 - 218.970i) q^{94} +(90.1042 - 316.683i) q^{95} -822.627 q^{96} +(89.3893 - 34.6296i) q^{97} +(13.5315 - 27.1750i) q^{98} +(-153.975 - 1661.65i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.488805 + 0.0913735i −0.172819 + 0.0323054i −0.269447 − 0.963015i \(-0.586841\pi\)
0.0966289 + 0.995320i \(0.469194\pi\)
\(3\) 5.41372 + 7.16892i 1.04187 + 1.37966i 0.922142 + 0.386853i \(0.126438\pi\)
0.119729 + 0.992807i \(0.461798\pi\)
\(4\) −7.22920 + 2.80061i −0.903650 + 0.350076i
\(5\) −7.46078 − 1.39466i −0.667312 − 0.124742i −0.160816 − 0.986984i \(-0.551413\pi\)
−0.506496 + 0.862242i \(0.669060\pi\)
\(6\) −3.30130 − 3.00953i −0.224625 − 0.204773i
\(7\) −8.95977 − 17.9936i −0.483782 − 0.971565i −0.993777 − 0.111388i \(-0.964470\pi\)
0.509995 − 0.860177i \(-0.329647\pi\)
\(8\) 6.66008 − 4.12375i 0.294337 − 0.182246i
\(9\) −14.6962 + 51.6517i −0.544303 + 1.91303i
\(10\) 3.77430 0.119354
\(11\) −28.9764 + 11.2255i −0.794246 + 0.307692i −0.723967 − 0.689835i \(-0.757683\pi\)
−0.0702789 + 0.997527i \(0.522389\pi\)
\(12\) −59.2141 − 36.6638i −1.42447 − 0.881995i
\(13\) 3.58277 + 7.19518i 0.0764371 + 0.153506i 0.930085 − 0.367344i \(-0.119733\pi\)
−0.853648 + 0.520850i \(0.825615\pi\)
\(14\) 6.02372 + 7.97669i 0.114993 + 0.152276i
\(15\) −30.3923 − 61.0360i −0.523151 − 1.05063i
\(16\) 42.9560 − 39.1595i 0.671187 − 0.611868i
\(17\) −56.0117 − 34.6810i −0.799108 − 0.494787i 0.0651259 − 0.997877i \(-0.479255\pi\)
−0.864234 + 0.503090i \(0.832196\pi\)
\(18\) 2.46397 − 26.5904i 0.0322646 − 0.348190i
\(19\) −4.00257 + 43.1946i −0.0483291 + 0.521554i 0.936815 + 0.349825i \(0.113759\pi\)
−0.985144 + 0.171729i \(0.945065\pi\)
\(20\) 57.8413 − 10.8124i 0.646686 − 0.120887i
\(21\) 80.4893 − 161.644i 0.836391 − 1.67970i
\(22\) 13.1381 − 8.13475i 0.127320 − 0.0788334i
\(23\) 44.2423 − 40.3322i 0.401094 − 0.365645i −0.447868 − 0.894100i \(-0.647816\pi\)
0.848962 + 0.528454i \(0.177228\pi\)
\(24\) 65.6186 + 25.4208i 0.558097 + 0.216208i
\(25\) −62.8409 − 24.3447i −0.502727 − 0.194757i
\(26\) −2.40872 − 3.18967i −0.0181688 − 0.0240594i
\(27\) −223.675 + 86.6521i −1.59430 + 0.617637i
\(28\) 115.165 + 104.987i 0.777291 + 0.708594i
\(29\) −49.2699 + 44.9154i −0.315489 + 0.287606i −0.815914 − 0.578174i \(-0.803766\pi\)
0.500424 + 0.865780i \(0.333177\pi\)
\(30\) 20.4330 + 27.0577i 0.124351 + 0.164668i
\(31\) 16.8698 + 182.054i 0.0977389 + 1.05477i 0.893471 + 0.449122i \(0.148263\pi\)
−0.795732 + 0.605649i \(0.792913\pi\)
\(32\) −55.1843 + 73.0758i −0.304853 + 0.403691i
\(33\) −237.345 − 146.958i −1.25201 − 0.775213i
\(34\) 30.5477 + 11.8342i 0.154085 + 0.0596929i
\(35\) 41.7518 + 146.742i 0.201638 + 0.708685i
\(36\) −38.4145 − 414.559i −0.177845 − 1.91925i
\(37\) −345.092 −1.53332 −0.766660 − 0.642054i \(-0.778083\pi\)
−0.766660 + 0.642054i \(0.778083\pi\)
\(38\) −1.99037 − 21.4795i −0.00849684 − 0.0916955i
\(39\) −32.1855 + 64.6373i −0.132149 + 0.265391i
\(40\) −55.4406 + 21.4778i −0.219148 + 0.0848985i
\(41\) 13.4548 0.0512507 0.0256254 − 0.999672i \(-0.491842\pi\)
0.0256254 + 0.999672i \(0.491842\pi\)
\(42\) −24.5736 + 86.3671i −0.0902805 + 0.317303i
\(43\) −11.6619 + 125.852i −0.0413586 + 0.446331i 0.949725 + 0.313087i \(0.101363\pi\)
−0.991083 + 0.133244i \(0.957461\pi\)
\(44\) 178.038 − 162.303i 0.610004 − 0.556092i
\(45\) 181.682 − 364.866i 0.601855 − 1.20869i
\(46\) −17.9406 + 23.7571i −0.0575042 + 0.0761478i
\(47\) −121.022 + 425.348i −0.375593 + 1.32007i 0.511819 + 0.859093i \(0.328972\pi\)
−0.887412 + 0.460978i \(0.847499\pi\)
\(48\) 513.283 + 95.9492i 1.54346 + 0.288522i
\(49\) −36.7898 + 48.7176i −0.107259 + 0.142034i
\(50\) 32.9414 + 6.15781i 0.0931723 + 0.0174169i
\(51\) −54.6064 − 589.296i −0.149930 − 1.61800i
\(52\) −46.0514 − 41.9814i −0.122811 − 0.111957i
\(53\) −64.6155 + 697.312i −0.167465 + 1.80723i 0.334557 + 0.942375i \(0.391413\pi\)
−0.502022 + 0.864855i \(0.667410\pi\)
\(54\) 101.416 − 62.7939i 0.255572 − 0.158244i
\(55\) 231.842 − 43.3388i 0.568392 − 0.106251i
\(56\) −133.874 − 82.8912i −0.319458 − 0.197800i
\(57\) −331.327 + 205.149i −0.769919 + 0.476714i
\(58\) 19.9793 − 26.4568i 0.0452312 − 0.0598957i
\(59\) 72.3752 − 254.372i 0.159702 − 0.561296i −0.840119 − 0.542402i \(-0.817515\pi\)
0.999821 − 0.0188939i \(-0.00601448\pi\)
\(60\) 390.650 + 356.125i 0.840545 + 0.766258i
\(61\) 140.530 + 493.910i 0.294967 + 1.03670i 0.957682 + 0.287829i \(0.0929334\pi\)
−0.662715 + 0.748871i \(0.730596\pi\)
\(62\) −24.8810 − 87.4475i −0.0509659 − 0.179127i
\(63\) 1061.08 − 198.350i 2.12195 − 0.396662i
\(64\) −186.976 + 375.499i −0.365188 + 0.733397i
\(65\) −16.6954 − 58.6784i −0.0318587 − 0.111972i
\(66\) 129.443 + 50.1465i 0.241414 + 0.0935244i
\(67\) −1.85912 − 3.73362i −0.00338996 − 0.00680797i 0.893462 − 0.449139i \(-0.148269\pi\)
−0.896852 + 0.442331i \(0.854152\pi\)
\(68\) 502.047 + 93.8489i 0.895326 + 0.167366i
\(69\) 528.654 + 98.8225i 0.922354 + 0.172418i
\(70\) −33.8168 − 67.9134i −0.0577412 − 0.115960i
\(71\) 325.338 + 126.037i 0.543811 + 0.210673i 0.617448 − 0.786612i \(-0.288167\pi\)
−0.0736372 + 0.997285i \(0.523461\pi\)
\(72\) 115.121 + 404.608i 0.188432 + 0.662271i
\(73\) 528.515 − 1061.40i 0.847371 − 1.70175i 0.145604 − 0.989343i \(-0.453488\pi\)
0.701767 − 0.712406i \(-0.252395\pi\)
\(74\) 168.683 − 31.5323i 0.264986 − 0.0495345i
\(75\) −165.678 − 582.296i −0.255077 − 0.896504i
\(76\) −92.0357 − 323.472i −0.138911 − 0.488221i
\(77\) 461.609 + 420.812i 0.683185 + 0.622805i
\(78\) 9.82631 − 34.5359i 0.0142642 − 0.0501336i
\(79\) 593.091 − 785.380i 0.844658 − 1.11851i −0.146940 − 0.989145i \(-0.546942\pi\)
0.991597 − 0.129362i \(-0.0412930\pi\)
\(80\) −375.099 + 232.252i −0.524217 + 0.324582i
\(81\) −599.343 − 371.098i −0.822145 − 0.509050i
\(82\) −6.57675 + 1.22941i −0.00885708 + 0.00165568i
\(83\) −854.161 + 528.874i −1.12959 + 0.699415i −0.958288 − 0.285803i \(-0.907740\pi\)
−0.171306 + 0.985218i \(0.554799\pi\)
\(84\) −129.171 + 1393.98i −0.167782 + 1.81066i
\(85\) 369.523 + 336.865i 0.471534 + 0.429860i
\(86\) −5.79913 − 62.5826i −0.00727135 − 0.0784704i
\(87\) −588.728 − 110.052i −0.725497 − 0.135619i
\(88\) −146.694 + 194.254i −0.177700 + 0.235313i
\(89\) −1508.67 − 282.019i −1.79684 − 0.335887i −0.823862 − 0.566790i \(-0.808185\pi\)
−0.972977 + 0.230902i \(0.925832\pi\)
\(90\) −55.4678 + 194.949i −0.0649647 + 0.228327i
\(91\) 97.3666 − 128.934i 0.112163 − 0.148527i
\(92\) −206.882 + 415.475i −0.234445 + 0.470828i
\(93\) −1213.80 + 1106.53i −1.35339 + 1.23378i
\(94\) 20.2906 − 218.970i 0.0222640 − 0.240266i
\(95\) 90.1042 − 316.683i 0.0973104 − 0.342011i
\(96\) −822.627 −0.874573
\(97\) 89.3893 − 34.6296i 0.0935682 − 0.0362485i −0.314003 − 0.949422i \(-0.601670\pi\)
0.407571 + 0.913174i \(0.366376\pi\)
\(98\) 13.5315 − 27.1750i 0.0139479 − 0.0280111i
\(99\) −153.975 − 1661.65i −0.156313 − 1.68689i
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.16.17 ✓ 544
137.60 even 17 inner 137.4.e.a.60.17 yes 544
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
137.4.e.a.16.17 ✓ 544 1.1 even 1 trivial
137.4.e.a.60.17 yes 544 137.60 even 17 inner