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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [137,4,Mod(37,137)] 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("137.37"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(137, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([3])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 137 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 137.c (of order \(4\), degree \(2\), 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: \(66\)
Relative dimension: \(33\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 37.6
Character \(\chi\) \(=\) 137.37
Dual form 137.4.c.a.100.28

$q$-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-4.16498i q^{2} +(-4.74102 + 4.74102i) q^{3} -9.34708 q^{4} +(5.85868 + 5.85868i) q^{5} +(19.7463 + 19.7463i) q^{6} -4.01344i q^{7} +5.61056i q^{8} -17.9545i q^{9} +(24.4013 - 24.4013i) q^{10} -39.7172i q^{11} +(44.3147 - 44.3147i) q^{12} +(29.3069 - 29.3069i) q^{13} -16.7159 q^{14} -55.5522 q^{15} -51.4087 q^{16} -18.0473i q^{17} -74.7802 q^{18} -53.3009i q^{19} +(-54.7615 - 54.7615i) q^{20} +(19.0278 + 19.0278i) q^{21} -165.422 q^{22} +(27.1036 - 27.1036i) q^{23} +(-26.5998 - 26.5998i) q^{24} -56.3517i q^{25} +(-122.063 - 122.063i) q^{26} +(-42.8849 - 42.8849i) q^{27} +37.5140i q^{28} +(72.4156 + 72.4156i) q^{29} +231.374i q^{30} +(144.526 - 144.526i) q^{31} +259.001i q^{32} +(188.300 + 188.300i) q^{33} -75.1667 q^{34} +(23.5135 - 23.5135i) q^{35} +167.822i q^{36} +84.3180i q^{37} -221.997 q^{38} +277.889i q^{39} +(-32.8705 + 32.8705i) q^{40} +(-169.648 - 169.648i) q^{41} +(79.2505 - 79.2505i) q^{42} +(137.613 - 137.613i) q^{43} +371.240i q^{44} +(105.190 - 105.190i) q^{45} +(-112.886 - 112.886i) q^{46} +(-324.540 - 324.540i) q^{47} +(243.730 - 243.730i) q^{48} +326.892 q^{49} -234.704 q^{50} +(85.5626 + 85.5626i) q^{51} +(-273.934 + 273.934i) q^{52} +(-49.2169 - 49.2169i) q^{53} +(-178.615 + 178.615i) q^{54} +(232.691 - 232.691i) q^{55} +22.5177 q^{56} +(252.700 + 252.700i) q^{57} +(301.610 - 301.610i) q^{58} +272.817 q^{59} +519.251 q^{60} +274.088i q^{61} +(-601.948 - 601.948i) q^{62} -72.0593 q^{63} +667.465 q^{64} +343.400 q^{65} +(784.267 - 784.267i) q^{66} +(193.598 + 193.598i) q^{67} +168.690i q^{68} +256.997i q^{69} +(-97.9332 - 97.9332i) q^{70} +(39.5283 - 39.5283i) q^{71} +100.735 q^{72} -65.6881 q^{73} +351.183 q^{74} +(267.165 + 267.165i) q^{75} +498.207i q^{76} -159.403 q^{77} +1157.40 q^{78} +(-687.937 + 687.937i) q^{79} +(-301.187 - 301.187i) q^{80} +891.407 q^{81} +(-706.583 + 706.583i) q^{82} +(-54.5033 + 54.5033i) q^{83} +(-177.854 - 177.854i) q^{84} +(105.733 - 105.733i) q^{85} +(-573.157 - 573.157i) q^{86} -686.647 q^{87} +222.836 q^{88} +(-628.905 + 628.905i) q^{89} +(-438.113 - 438.113i) q^{90} +(-117.622 - 117.622i) q^{91} +(-253.339 + 253.339i) q^{92} +1370.40i q^{93} +(-1351.70 + 1351.70i) q^{94} +(312.273 - 312.273i) q^{95} +(-1227.93 - 1227.93i) q^{96} +(-530.326 + 530.326i) q^{97} -1361.50i q^{98} -713.103 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 66 q - 272 q^{4} - 4 q^{5} + 6 q^{6} - 174 q^{10} - 104 q^{12} - 56 q^{13} + 140 q^{14} + 276 q^{15} + 1616 q^{16} + 136 q^{18} - 18 q^{20} - 12 q^{21} + 1124 q^{22} + 170 q^{23} + 132 q^{24} + 68 q^{26}+ \cdots + 11068 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 4.16498i − 1.47254i −0.676686 − 0.736272i \(-0.736584\pi\)
0.676686 − 0.736272i \(-0.263416\pi\)
\(3\) −4.74102 + 4.74102i −0.912409 + 0.912409i −0.996461 − 0.0840521i \(-0.973214\pi\)
0.0840521 + 0.996461i \(0.473214\pi\)
\(4\) −9.34708 −1.16838
\(5\) 5.85868 + 5.85868i 0.524016 + 0.524016i 0.918782 − 0.394766i \(-0.129174\pi\)
−0.394766 + 0.918782i \(0.629174\pi\)
\(6\) 19.7463 + 19.7463i 1.34356 + 1.34356i
\(7\) − 4.01344i − 0.216706i −0.994113 − 0.108353i \(-0.965442\pi\)
0.994113 − 0.108353i \(-0.0345576\pi\)
\(8\) 5.61056i 0.247954i
\(9\) − 17.9545i − 0.664981i
\(10\) 24.4013 − 24.4013i 0.771637 − 0.771637i
\(11\) − 39.7172i − 1.08865i −0.838873 − 0.544327i \(-0.816785\pi\)
0.838873 − 0.544327i \(-0.183215\pi\)
\(12\) 44.3147 − 44.3147i 1.06605 − 1.06605i
\(13\) 29.3069 − 29.3069i 0.625253 − 0.625253i −0.321617 − 0.946870i \(-0.604226\pi\)
0.946870 + 0.321617i \(0.104226\pi\)
\(14\) −16.7159 −0.319108
\(15\) −55.5522 −0.956235
\(16\) −51.4087 −0.803262
\(17\) − 18.0473i − 0.257477i −0.991679 − 0.128739i \(-0.958907\pi\)
0.991679 − 0.128739i \(-0.0410929\pi\)
\(18\) −74.7802 −0.979214
\(19\) − 53.3009i − 0.643582i −0.946811 − 0.321791i \(-0.895715\pi\)
0.946811 − 0.321791i \(-0.104285\pi\)
\(20\) −54.7615 − 54.7615i −0.612253 − 0.612253i
\(21\) 19.0278 + 19.0278i 0.197724 + 0.197724i
\(22\) −165.422 −1.60309
\(23\) 27.1036 − 27.1036i 0.245717 − 0.245717i −0.573493 − 0.819210i \(-0.694412\pi\)
0.819210 + 0.573493i \(0.194412\pi\)
\(24\) −26.5998 − 26.5998i −0.226236 − 0.226236i
\(25\) − 56.3517i − 0.450814i
\(26\) −122.063 − 122.063i −0.920712 − 0.920712i
\(27\) −42.8849 − 42.8849i −0.305674 − 0.305674i
\(28\) 37.5140i 0.253195i
\(29\) 72.4156 + 72.4156i 0.463698 + 0.463698i 0.899865 − 0.436168i \(-0.143665\pi\)
−0.436168 + 0.899865i \(0.643665\pi\)
\(30\) 231.374i 1.40810i
\(31\) 144.526 − 144.526i 0.837343 − 0.837343i −0.151166 − 0.988508i \(-0.548303\pi\)
0.988508 + 0.151166i \(0.0483028\pi\)
\(32\) 259.001i 1.43079i
\(33\) 188.300 + 188.300i 0.993298 + 0.993298i
\(34\) −75.1667 −0.379147
\(35\) 23.5135 − 23.5135i 0.113557 − 0.113557i
\(36\) 167.822i 0.776954i
\(37\) 84.3180i 0.374643i 0.982299 + 0.187322i \(0.0599806\pi\)
−0.982299 + 0.187322i \(0.940019\pi\)
\(38\) −221.997 −0.947703
\(39\) 277.889i 1.14097i
\(40\) −32.8705 + 32.8705i −0.129932 + 0.129932i
\(41\) −169.648 − 169.648i −0.646211 − 0.646211i 0.305864 − 0.952075i \(-0.401055\pi\)
−0.952075 + 0.305864i \(0.901055\pi\)
\(42\) 79.2505 − 79.2505i 0.291157 − 0.291157i
\(43\) 137.613 − 137.613i 0.488042 − 0.488042i −0.419646 − 0.907688i \(-0.637846\pi\)
0.907688 + 0.419646i \(0.137846\pi\)
\(44\) 371.240i 1.27197i
\(45\) 105.190 − 105.190i 0.348461 − 0.348461i
\(46\) −112.886 − 112.886i −0.361829 − 0.361829i
\(47\) −324.540 − 324.540i −1.00721 − 1.00721i −0.999974 − 0.00723809i \(-0.997696\pi\)
−0.00723809 − 0.999974i \(-0.502304\pi\)
\(48\) 243.730 − 243.730i 0.732903 − 0.732903i
\(49\) 326.892 0.953039
\(50\) −234.704 −0.663843
\(51\) 85.5626 + 85.5626i 0.234925 + 0.234925i
\(52\) −273.934 + 273.934i −0.730536 + 0.730536i
\(53\) −49.2169 − 49.2169i −0.127556 − 0.127556i 0.640447 − 0.768003i \(-0.278749\pi\)
−0.768003 + 0.640447i \(0.778749\pi\)
\(54\) −178.615 + 178.615i −0.450119 + 0.450119i
\(55\) 232.691 − 232.691i 0.570473 − 0.570473i
\(56\) 22.5177 0.0537330
\(57\) 252.700 + 252.700i 0.587210 + 0.587210i
\(58\) 301.610 − 301.610i 0.682815 − 0.682815i
\(59\) 272.817 0.601997 0.300998 − 0.953625i \(-0.402680\pi\)
0.300998 + 0.953625i \(0.402680\pi\)
\(60\) 519.251 1.11725
\(61\) 274.088i 0.575301i 0.957735 + 0.287650i \(0.0928741\pi\)
−0.957735 + 0.287650i \(0.907126\pi\)
\(62\) −601.948 − 601.948i −1.23302 − 1.23302i
\(63\) −72.0593 −0.144105
\(64\) 667.465 1.30364
\(65\) 343.400 0.655285
\(66\) 784.267 − 784.267i 1.46268 − 1.46268i
\(67\) 193.598 + 193.598i 0.353012 + 0.353012i 0.861229 − 0.508217i \(-0.169695\pi\)
−0.508217 + 0.861229i \(0.669695\pi\)
\(68\) 168.690i 0.300833i
\(69\) 256.997i 0.448389i
\(70\) −97.9332 − 97.9332i −0.167218 − 0.167218i
\(71\) 39.5283 − 39.5283i 0.0660725 − 0.0660725i −0.673298 − 0.739371i \(-0.735123\pi\)
0.739371 + 0.673298i \(0.235123\pi\)
\(72\) 100.735 0.164885
\(73\) −65.6881 −0.105318 −0.0526590 − 0.998613i \(-0.516770\pi\)
−0.0526590 + 0.998613i \(0.516770\pi\)
\(74\) 351.183 0.551678
\(75\) 267.165 + 267.165i 0.411327 + 0.411327i
\(76\) 498.207i 0.751951i
\(77\) −159.403 −0.235917
\(78\) 1157.40 1.68013
\(79\) −687.937 + 687.937i −0.979734 + 0.979734i −0.999799 − 0.0200651i \(-0.993613\pi\)
0.0200651 + 0.999799i \(0.493613\pi\)
\(80\) −301.187 − 301.187i −0.420922 − 0.420922i
\(81\) 891.407 1.22278
\(82\) −706.583 + 706.583i −0.951574 + 0.951574i
\(83\) −54.5033 + 54.5033i −0.0720785 + 0.0720785i −0.742227 − 0.670149i \(-0.766230\pi\)
0.670149 + 0.742227i \(0.266230\pi\)
\(84\) −177.854 − 177.854i −0.231018 − 0.231018i
\(85\) 105.733 − 105.733i 0.134922 − 0.134922i
\(86\) −573.157 − 573.157i −0.718663 − 0.718663i
\(87\) −686.647 −0.846164
\(88\) 222.836 0.269936
\(89\) −628.905 + 628.905i −0.749031 + 0.749031i −0.974297 − 0.225266i \(-0.927675\pi\)
0.225266 + 0.974297i \(0.427675\pi\)
\(90\) −438.113 − 438.113i −0.513124 − 0.513124i
\(91\) −117.622 − 117.622i −0.135496 − 0.135496i
\(92\) −253.339 + 253.339i −0.287092 + 0.287092i
\(93\) 1370.40i 1.52800i
\(94\) −1351.70 + 1351.70i −1.48316 + 1.48316i
\(95\) 312.273 − 312.273i 0.337247 − 0.337247i
\(96\) −1227.93 − 1227.93i −1.30547 − 1.30547i
\(97\) −530.326 + 530.326i −0.555118 + 0.555118i −0.927914 − 0.372795i \(-0.878399\pi\)
0.372795 + 0.927914i \(0.378399\pi\)
\(98\) − 1361.50i − 1.40339i
\(99\) −713.103 −0.723935
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.c.a.37.6 ✓ 66
137.100 even 4 inner 137.4.c.a.100.28 yes 66
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
137.4.c.a.37.6 ✓ 66 1.1 even 1 trivial
137.4.c.a.100.28 yes 66 137.100 even 4 inner