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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [137,5,Mod(10,137)] 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("137.10"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(137, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([5])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 137 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 137.d (of order \(8\), degree \(4\), 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: \(14.1616849425\)
Analytic rank: \(0\)
Dimension: \(180\)
Relative dimension: \(45\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 10.8
Character \(\chi\) \(=\) 137.10
Dual form 137.5.d.a.96.8

$q$-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+(-4.30796 - 4.30796i) q^{2} +(3.22032 - 7.77454i) q^{3} +21.1170i q^{4} +(-32.9153 + 13.6339i) q^{5} +(-47.3654 + 19.6194i) q^{6} +(58.9245 + 58.9245i) q^{7} +(22.0439 - 22.0439i) q^{8} +(7.20270 + 7.20270i) q^{9} +(200.532 + 83.0631i) q^{10} +(-93.5130 - 93.5130i) q^{11} +(164.175 + 68.0035i) q^{12} +(95.8765 - 231.466i) q^{13} -507.689i q^{14} +299.806i q^{15} +147.944 q^{16} +(179.339 - 179.339i) q^{17} -62.0579i q^{18} +(-340.949 + 340.949i) q^{19} +(-287.908 - 695.072i) q^{20} +(647.867 - 268.355i) q^{21} +805.700i q^{22} +(265.299 + 109.891i) q^{23} +(-100.393 - 242.369i) q^{24} +(455.588 - 455.588i) q^{25} +(-1410.18 + 584.116i) q^{26} +(708.930 - 293.648i) q^{27} +(-1244.31 + 1244.31i) q^{28} +(794.394 - 329.049i) q^{29} +(1291.55 - 1291.55i) q^{30} +(-659.050 + 1591.09i) q^{31} +(-990.039 - 990.039i) q^{32} +(-1028.16 + 425.878i) q^{33} -1545.17 q^{34} +(-2742.89 - 1136.14i) q^{35} +(-152.100 + 152.100i) q^{36} +(1278.93 - 1278.93i) q^{37} +2937.59 q^{38} +(-1490.79 - 1490.79i) q^{39} +(-425.035 + 1026.12i) q^{40} +(-497.729 - 1201.62i) q^{41} +(-3947.04 - 1634.92i) q^{42} +(590.107 - 1424.64i) q^{43} +(1974.71 - 1974.71i) q^{44} +(-335.280 - 138.878i) q^{45} +(-669.494 - 1616.30i) q^{46} +(4003.41 - 1658.27i) q^{47} +(476.427 - 1150.20i) q^{48} +4543.20i q^{49} -3925.31 q^{50} +(-816.750 - 1971.81i) q^{51} +(4887.88 + 2024.63i) q^{52} +(-2518.78 + 1043.31i) q^{53} +(-4319.07 - 1789.02i) q^{54} +(4352.95 + 1803.05i) q^{55} +2597.85 q^{56} +(1552.76 + 3748.69i) q^{57} +(-4839.75 - 2004.69i) q^{58} -613.019 q^{59} -6331.02 q^{60} +(3006.36 - 3006.36i) q^{61} +(9693.49 - 4015.18i) q^{62} +848.832i q^{63} +6162.99i q^{64} +8925.95i q^{65} +(6263.94 + 2594.61i) q^{66} +(2967.33 - 1229.11i) q^{67} +(3787.11 + 3787.11i) q^{68} +(1708.70 - 1708.70i) q^{69} +(6921.80 + 16710.7i) q^{70} +(139.351 - 336.423i) q^{71} +317.551 q^{72} +4484.76 q^{73} -11019.2 q^{74} +(-2074.85 - 5009.12i) q^{75} +(-7199.83 - 7199.83i) q^{76} -11020.4i q^{77} +12844.5i q^{78} +(4004.43 - 9667.55i) q^{79} +(-4869.61 + 2017.06i) q^{80} -5632.16i q^{81} +(-3032.35 + 7320.74i) q^{82} +(-3536.16 + 8537.06i) q^{83} +(5666.86 + 13681.0i) q^{84} +(-3457.90 + 8348.10i) q^{85} +(-8679.46 + 3595.15i) q^{86} -7235.69i q^{87} -4122.78 q^{88} +(1836.23 + 760.592i) q^{89} +(846.094 + 2042.65i) q^{90} +(19288.5 - 7989.57i) q^{91} +(-2320.56 + 5602.33i) q^{92} +(10247.6 + 10247.6i) q^{93} +(-24390.3 - 10102.8i) q^{94} +(6573.95 - 15870.9i) q^{95} +(-10885.3 + 4508.85i) q^{96} +(-92.5425 - 38.3323i) q^{97} +(19571.9 - 19571.9i) q^{98} -1347.09i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 180 q - 16 q^{2} - 64 q^{5} - 68 q^{6} - 4 q^{7} + 72 q^{8} - 200 q^{9} + 388 q^{10} + 56 q^{11} + 320 q^{12} + 356 q^{13} - 9520 q^{16} + 272 q^{17} + 812 q^{19} + 3324 q^{20} + 1016 q^{21} + 1928 q^{23}+ \cdots - 9068 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{5}{8}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.30796 − 4.30796i −1.07699 − 1.07699i −0.996778 − 0.0802118i \(-0.974440\pi\)
−0.0802118 − 0.996778i \(-0.525560\pi\)
\(3\) 3.22032 − 7.77454i 0.357813 − 0.863837i −0.637794 − 0.770207i \(-0.720153\pi\)
0.995607 − 0.0936302i \(-0.0298471\pi\)
\(4\) 21.1170i 1.31981i
\(5\) −32.9153 + 13.6339i −1.31661 + 0.545358i −0.926806 − 0.375540i \(-0.877457\pi\)
−0.389804 + 0.920898i \(0.627457\pi\)
\(6\) −47.3654 + 19.6194i −1.31570 + 0.544983i
\(7\) 58.9245 + 58.9245i 1.20254 + 1.20254i 0.973391 + 0.229150i \(0.0735948\pi\)
0.229150 + 0.973391i \(0.426405\pi\)
\(8\) 22.0439 − 22.0439i 0.344436 − 0.344436i
\(9\) 7.20270 + 7.20270i 0.0889222 + 0.0889222i
\(10\) 200.532 + 83.0631i 2.00532 + 0.830631i
\(11\) −93.5130 − 93.5130i −0.772834 − 0.772834i 0.205767 − 0.978601i \(-0.434031\pi\)
−0.978601 + 0.205767i \(0.934031\pi\)
\(12\) 164.175 + 68.0035i 1.14010 + 0.472246i
\(13\) 95.8765 − 231.466i 0.567317 − 1.36962i −0.336492 − 0.941686i \(-0.609240\pi\)
0.903808 − 0.427937i \(-0.140760\pi\)
\(14\) − 507.689i − 2.59025i
\(15\) 299.806i 1.33247i
\(16\) 147.944 0.577906
\(17\) 179.339 − 179.339i 0.620551 − 0.620551i −0.325121 − 0.945672i \(-0.605405\pi\)
0.945672 + 0.325121i \(0.105405\pi\)
\(18\) − 62.0579i − 0.191537i
\(19\) −340.949 + 340.949i −0.944458 + 0.944458i −0.998537 − 0.0540790i \(-0.982778\pi\)
0.0540790 + 0.998537i \(0.482778\pi\)
\(20\) −287.908 − 695.072i −0.719771 − 1.73768i
\(21\) 647.867 − 268.355i 1.46909 − 0.608515i
\(22\) 805.700i 1.66467i
\(23\) 265.299 + 109.891i 0.501511 + 0.207733i 0.619074 − 0.785333i \(-0.287508\pi\)
−0.117563 + 0.993065i \(0.537508\pi\)
\(24\) −100.393 − 242.369i −0.174293 − 0.420780i
\(25\) 455.588 − 455.588i 0.728941 − 0.728941i
\(26\) −1410.18 + 584.116i −2.08606 + 0.864076i
\(27\) 708.930 − 293.648i 0.972469 − 0.402810i
\(28\) −1244.31 + 1244.31i −1.58713 + 1.58713i
\(29\) 794.394 − 329.049i 0.944583 − 0.391259i 0.143391 − 0.989666i \(-0.454199\pi\)
0.801192 + 0.598407i \(0.204199\pi\)
\(30\) 1291.55 − 1291.55i 1.43506 − 1.43506i
\(31\) −659.050 + 1591.09i −0.685796 + 1.65566i 0.0672888 + 0.997734i \(0.478565\pi\)
−0.753085 + 0.657924i \(0.771435\pi\)
\(32\) −990.039 − 990.039i −0.966835 − 0.966835i
\(33\) −1028.16 + 425.878i −0.944133 + 0.391073i
\(34\) −1545.17 −1.33665
\(35\) −2742.89 − 1136.14i −2.23909 − 0.927463i
\(36\) −152.100 + 152.100i −0.117361 + 0.117361i
\(37\) 1278.93 − 1278.93i 0.934210 − 0.934210i −0.0637558 − 0.997966i \(-0.520308\pi\)
0.997966 + 0.0637558i \(0.0203079\pi\)
\(38\) 2937.59 2.03434
\(39\) −1490.79 − 1490.79i −0.980138 − 0.980138i
\(40\) −425.035 + 1026.12i −0.265647 + 0.641328i
\(41\) −497.729 − 1201.62i −0.296091 − 0.714827i −0.999990 − 0.00452660i \(-0.998559\pi\)
0.703899 − 0.710300i \(-0.251441\pi\)
\(42\) −3947.04 − 1634.92i −2.23755 − 0.926825i
\(43\) 590.107 − 1424.64i 0.319149 − 0.770494i −0.680150 − 0.733073i \(-0.738086\pi\)
0.999300 − 0.0374217i \(-0.0119145\pi\)
\(44\) 1974.71 − 1974.71i 1.02000 − 1.02000i
\(45\) −335.280 − 138.878i −0.165570 − 0.0685815i
\(46\) −669.494 − 1616.30i −0.316396 − 0.763848i
\(47\) 4003.41 − 1658.27i 1.81232 − 0.750687i 0.831568 − 0.555423i \(-0.187444\pi\)
0.980751 − 0.195264i \(-0.0625564\pi\)
\(48\) 476.427 − 1150.20i 0.206782 − 0.499217i
\(49\) 4543.20i 1.89221i
\(50\) −3925.31 −1.57012
\(51\) −816.750 − 1971.81i −0.314014 − 0.758097i
\(52\) 4887.88 + 2024.63i 1.80765 + 0.748752i
\(53\) −2518.78 + 1043.31i −0.896683 + 0.371418i −0.782944 − 0.622092i \(-0.786283\pi\)
−0.113739 + 0.993511i \(0.536283\pi\)
\(54\) −4319.07 − 1789.02i −1.48116 − 0.613517i
\(55\) 4352.95 + 1803.05i 1.43899 + 0.596050i
\(56\) 2597.85 0.828396
\(57\) 1552.76 + 3748.69i 0.477918 + 1.15380i
\(58\) −4839.75 − 2004.69i −1.43869 − 0.595924i
\(59\) −613.019 −0.176104 −0.0880521 − 0.996116i \(-0.528064\pi\)
−0.0880521 + 0.996116i \(0.528064\pi\)
\(60\) −6331.02 −1.75862
\(61\) 3006.36 − 3006.36i 0.807943 − 0.807943i −0.176379 − 0.984322i \(-0.556439\pi\)
0.984322 + 0.176379i \(0.0564386\pi\)
\(62\) 9693.49 − 4015.18i 2.52172 − 1.04453i
\(63\) 848.832i 0.213865i
\(64\) 6162.99i 1.50464i
\(65\) 8925.95i 2.11265i
\(66\) 6263.94 + 2594.61i 1.43800 + 0.595641i
\(67\) 2967.33 − 1229.11i 0.661023 − 0.273805i −0.0268461 − 0.999640i \(-0.508546\pi\)
0.687869 + 0.725835i \(0.258546\pi\)
\(68\) 3787.11 + 3787.11i 0.819012 + 0.819012i
\(69\) 1708.70 − 1708.70i 0.358894 − 0.358894i
\(70\) 6921.80 + 16710.7i 1.41261 + 3.41035i
\(71\) 139.351 − 336.423i 0.0276435 − 0.0667373i −0.909455 − 0.415802i \(-0.863501\pi\)
0.937098 + 0.349065i \(0.113501\pi\)
\(72\) 317.551 0.0612560
\(73\) 4484.76 0.841576 0.420788 − 0.907159i \(-0.361754\pi\)
0.420788 + 0.907159i \(0.361754\pi\)
\(74\) −11019.2 −2.01227
\(75\) −2074.85 − 5009.12i −0.368862 − 0.890511i
\(76\) −7199.83 − 7199.83i −1.24651 − 1.24651i
\(77\) − 11020.4i − 1.85873i
\(78\) 12844.5i 2.11120i
\(79\) 4004.43 − 9667.55i 0.641632 − 1.54904i −0.182844 − 0.983142i \(-0.558530\pi\)
0.824477 − 0.565896i \(-0.191470\pi\)
\(80\) −4869.61 + 2017.06i −0.760877 + 0.315166i
\(81\) − 5632.16i − 0.858431i
\(82\) −3032.35 + 7320.74i −0.450974 + 1.08875i
\(83\) −3536.16 + 8537.06i −0.513306 + 1.23923i 0.428643 + 0.903474i \(0.358992\pi\)
−0.941949 + 0.335756i \(0.891008\pi\)
\(84\) 5666.86 + 13681.0i 0.803126 + 1.93892i
\(85\) −3457.90 + 8348.10i −0.478602 + 1.15545i
\(86\) −8679.46 + 3595.15i −1.17353 + 0.486094i
\(87\) − 7235.69i − 0.955964i
\(88\) −4122.78 −0.532383
\(89\) 1836.23 + 760.592i 0.231818 + 0.0960223i 0.495569 − 0.868569i \(-0.334960\pi\)
−0.263751 + 0.964591i \(0.584960\pi\)
\(90\) 846.094 + 2042.65i 0.104456 + 0.252179i
\(91\) 19288.5 − 7989.57i 2.32925 − 0.964807i
\(92\) −2320.56 + 5602.33i −0.274168 + 0.661900i
\(93\) 10247.6 + 10247.6i 1.18483 + 1.18483i
\(94\) −24390.3 − 10102.8i −2.76033 − 1.14337i
\(95\) 6573.95 − 15870.9i 0.728415 − 1.75855i
\(96\) −10885.3 + 4508.85i −1.18113 + 0.489242i
\(97\) −92.5425 − 38.3323i −0.00983553 − 0.00407401i 0.377760 − 0.925903i \(-0.376694\pi\)
−0.387596 + 0.921829i \(0.626694\pi\)
\(98\) 19571.9 − 19571.9i 2.03789 − 2.03789i
\(99\) − 1347.09i − 0.137444i
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 137.5.d.a.10.8 ✓ 180
137.96 odd 8 inner 137.5.d.a.96.8 yes 180
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
137.5.d.a.10.8 ✓ 180 1.1 even 1 trivial
137.5.d.a.96.8 yes 180 137.96 odd 8 inner