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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [984,2,Mod(493,984)] 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("984.493"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(984, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 984 = 2^{3} \cdot 3 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 984.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40,2,0,0,0,2,4,2,-40,-2,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.85727955889\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 493.13
Character \(\chi\) \(=\) 984.493
Dual form 984.2.f.b.493.14

$q$-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+(-0.589421 - 1.28553i) q^{2} +1.00000i q^{3} +(-1.30516 + 1.51544i) q^{4} -0.836655i q^{5} +(1.28553 - 0.589421i) q^{6} +0.923115 q^{7} +(2.71743 + 0.784595i) q^{8} -1.00000 q^{9} +(-1.07554 + 0.493142i) q^{10} +1.70474i q^{11} +(-1.51544 - 1.30516i) q^{12} +3.71831i q^{13} +(-0.544104 - 1.18669i) q^{14} +0.836655 q^{15} +(-0.593090 - 3.95579i) q^{16} -3.25205 q^{17} +(0.589421 + 1.28553i) q^{18} +2.88887i q^{19} +(1.26790 + 1.09197i) q^{20} +0.923115i q^{21} +(2.19148 - 1.00481i) q^{22} -6.37693 q^{23} +(-0.784595 + 2.71743i) q^{24} +4.30001 q^{25} +(4.77999 - 2.19165i) q^{26} -1.00000i q^{27} +(-1.20482 + 1.39892i) q^{28} -4.62442i q^{29} +(-0.493142 - 1.07554i) q^{30} -4.41278 q^{31} +(-4.73569 + 3.09406i) q^{32} -1.70474 q^{33} +(1.91683 + 4.18060i) q^{34} -0.772328i q^{35} +(1.30516 - 1.51544i) q^{36} +1.94774i q^{37} +(3.71372 - 1.70276i) q^{38} -3.71831 q^{39} +(0.656435 - 2.27355i) q^{40} -1.00000 q^{41} +(1.18669 - 0.544104i) q^{42} +10.2546i q^{43} +(-2.58342 - 2.22496i) q^{44} +0.836655i q^{45} +(3.75870 + 8.19772i) q^{46} +8.05403 q^{47} +(3.95579 - 0.593090i) q^{48} -6.14786 q^{49} +(-2.53452 - 5.52778i) q^{50} -3.25205i q^{51} +(-5.63486 - 4.85300i) q^{52} +0.801403i q^{53} +(-1.28553 + 0.589421i) q^{54} +1.42627 q^{55} +(2.50850 + 0.724272i) q^{56} -2.88887 q^{57} +(-5.94482 + 2.72573i) q^{58} +14.9454i q^{59} +(-1.09197 + 1.26790i) q^{60} +9.67578i q^{61} +(2.60098 + 5.67275i) q^{62} -0.923115 q^{63} +(6.76882 + 4.26416i) q^{64} +3.11094 q^{65} +(1.00481 + 2.19148i) q^{66} +11.1605i q^{67} +(4.24446 - 4.92827i) q^{68} -6.37693i q^{69} +(-0.992850 + 0.455227i) q^{70} +11.4027 q^{71} +(-2.71743 - 0.784595i) q^{72} -3.15575 q^{73} +(2.50388 - 1.14804i) q^{74} +4.30001i q^{75} +(-4.37789 - 3.77045i) q^{76} +1.57367i q^{77} +(2.19165 + 4.77999i) q^{78} -0.775978 q^{79} +(-3.30963 + 0.496212i) q^{80} +1.00000 q^{81} +(0.589421 + 1.28553i) q^{82} +8.64172i q^{83} +(-1.39892 - 1.20482i) q^{84} +2.72084i q^{85} +(13.1826 - 6.04429i) q^{86} +4.62442 q^{87} +(-1.33753 + 4.63249i) q^{88} -18.4191 q^{89} +(1.07554 - 0.493142i) q^{90} +3.43243i q^{91} +(8.32294 - 9.66382i) q^{92} -4.41278i q^{93} +(-4.74722 - 10.3537i) q^{94} +2.41698 q^{95} +(-3.09406 - 4.73569i) q^{96} -12.2790 q^{97} +(3.62368 + 7.90324i) q^{98} -1.70474i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q + 2 q^{2} + 2 q^{6} + 4 q^{7} + 2 q^{8} - 40 q^{9} - 2 q^{10} - 24 q^{15} + 20 q^{16} - 2 q^{18} - 16 q^{20} + 28 q^{22} + 24 q^{23} - 2 q^{24} - 40 q^{25} - 2 q^{26} - 32 q^{28} + 8 q^{30} + 8 q^{31}+ \cdots + 42 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/984\mathbb{Z}\right)^\times\).

\(n\) \(247\) \(329\) \(457\) \(493\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.589421 − 1.28553i −0.416784 − 0.909006i
\(3\) 1.00000i 0.577350i
\(4\) −1.30516 + 1.51544i −0.652582 + 0.757718i
\(5\) − 0.836655i − 0.374163i −0.982344 − 0.187082i \(-0.940097\pi\)
0.982344 − 0.187082i \(-0.0599029\pi\)
\(6\) 1.28553 − 0.589421i 0.524815 − 0.240630i
\(7\) 0.923115 0.348905 0.174452 − 0.984666i \(-0.444185\pi\)
0.174452 + 0.984666i \(0.444185\pi\)
\(8\) 2.71743 + 0.784595i 0.960756 + 0.277396i
\(9\) −1.00000 −0.333333
\(10\) −1.07554 + 0.493142i −0.340117 + 0.155945i
\(11\) 1.70474i 0.513997i 0.966412 + 0.256998i \(0.0827335\pi\)
−0.966412 + 0.256998i \(0.917266\pi\)
\(12\) −1.51544 − 1.30516i −0.437469 − 0.376769i
\(13\) 3.71831i 1.03127i 0.856807 + 0.515637i \(0.172445\pi\)
−0.856807 + 0.515637i \(0.827555\pi\)
\(14\) −0.544104 − 1.18669i −0.145418 − 0.317156i
\(15\) 0.836655 0.216023
\(16\) −0.593090 − 3.95579i −0.148273 − 0.988947i
\(17\) −3.25205 −0.788737 −0.394369 − 0.918952i \(-0.629037\pi\)
−0.394369 + 0.918952i \(0.629037\pi\)
\(18\) 0.589421 + 1.28553i 0.138928 + 0.303002i
\(19\) 2.88887i 0.662752i 0.943499 + 0.331376i \(0.107513\pi\)
−0.943499 + 0.331376i \(0.892487\pi\)
\(20\) 1.26790 + 1.09197i 0.283510 + 0.244172i
\(21\) 0.923115i 0.201440i
\(22\) 2.19148 − 1.00481i 0.467226 − 0.214226i
\(23\) −6.37693 −1.32968 −0.664841 − 0.746985i \(-0.731501\pi\)
−0.664841 + 0.746985i \(0.731501\pi\)
\(24\) −0.784595 + 2.71743i −0.160155 + 0.554692i
\(25\) 4.30001 0.860002
\(26\) 4.77999 − 2.19165i 0.937433 − 0.429818i
\(27\) − 1.00000i − 0.192450i
\(28\) −1.20482 + 1.39892i −0.227689 + 0.264371i
\(29\) − 4.62442i − 0.858733i −0.903130 − 0.429367i \(-0.858737\pi\)
0.903130 − 0.429367i \(-0.141263\pi\)
\(30\) −0.493142 − 1.07554i −0.0900350 − 0.196366i
\(31\) −4.41278 −0.792558 −0.396279 − 0.918130i \(-0.629699\pi\)
−0.396279 + 0.918130i \(0.629699\pi\)
\(32\) −4.73569 + 3.09406i −0.837160 + 0.546958i
\(33\) −1.70474 −0.296756
\(34\) 1.91683 + 4.18060i 0.328733 + 0.716967i
\(35\) − 0.772328i − 0.130547i
\(36\) 1.30516 − 1.51544i 0.217527 − 0.252573i
\(37\) 1.94774i 0.320207i 0.987100 + 0.160103i \(0.0511828\pi\)
−0.987100 + 0.160103i \(0.948817\pi\)
\(38\) 3.71372 − 1.70276i 0.602445 − 0.276224i
\(39\) −3.71831 −0.595406
\(40\) 0.656435 − 2.27355i 0.103792 − 0.359479i
\(41\) −1.00000 −0.156174
\(42\) 1.18669 − 0.544104i 0.183110 − 0.0839570i
\(43\) 10.2546i 1.56381i 0.623395 + 0.781907i \(0.285753\pi\)
−0.623395 + 0.781907i \(0.714247\pi\)
\(44\) −2.58342 − 2.22496i −0.389465 − 0.335425i
\(45\) 0.836655i 0.124721i
\(46\) 3.75870 + 8.19772i 0.554190 + 1.20869i
\(47\) 8.05403 1.17480 0.587400 − 0.809296i \(-0.300151\pi\)
0.587400 + 0.809296i \(0.300151\pi\)
\(48\) 3.95579 − 0.593090i 0.570969 − 0.0856052i
\(49\) −6.14786 −0.878266
\(50\) −2.53452 − 5.52778i −0.358435 − 0.781746i
\(51\) − 3.25205i − 0.455378i
\(52\) −5.63486 − 4.85300i −0.781414 − 0.672991i
\(53\) 0.801403i 0.110081i 0.998484 + 0.0550406i \(0.0175288\pi\)
−0.998484 + 0.0550406i \(0.982471\pi\)
\(54\) −1.28553 + 0.589421i −0.174938 + 0.0802101i
\(55\) 1.42627 0.192319
\(56\) 2.50850 + 0.724272i 0.335212 + 0.0967849i
\(57\) −2.88887 −0.382640
\(58\) −5.94482 + 2.72573i −0.780594 + 0.357906i
\(59\) 14.9454i 1.94572i 0.231390 + 0.972861i \(0.425673\pi\)
−0.231390 + 0.972861i \(0.574327\pi\)
\(60\) −1.09197 + 1.26790i −0.140973 + 0.163685i
\(61\) 9.67578i 1.23886i 0.785053 + 0.619428i \(0.212635\pi\)
−0.785053 + 0.619428i \(0.787365\pi\)
\(62\) 2.60098 + 5.67275i 0.330325 + 0.720439i
\(63\) −0.923115 −0.116302
\(64\) 6.76882 + 4.26416i 0.846102 + 0.533020i
\(65\) 3.11094 0.385865
\(66\) 1.00481 + 2.19148i 0.123683 + 0.269753i
\(67\) 11.1605i 1.36348i 0.731596 + 0.681738i \(0.238776\pi\)
−0.731596 + 0.681738i \(0.761224\pi\)
\(68\) 4.24446 − 4.92827i 0.514716 − 0.597640i
\(69\) − 6.37693i − 0.767692i
\(70\) −0.992850 + 0.455227i −0.118668 + 0.0544100i
\(71\) 11.4027 1.35325 0.676627 − 0.736326i \(-0.263441\pi\)
0.676627 + 0.736326i \(0.263441\pi\)
\(72\) −2.71743 − 0.784595i −0.320252 − 0.0924655i
\(73\) −3.15575 −0.369352 −0.184676 − 0.982799i \(-0.559124\pi\)
−0.184676 + 0.982799i \(0.559124\pi\)
\(74\) 2.50388 − 1.14804i 0.291070 − 0.133457i
\(75\) 4.30001i 0.496522i
\(76\) −4.37789 − 3.77045i −0.502179 − 0.432500i
\(77\) 1.57367i 0.179336i
\(78\) 2.19165 + 4.77999i 0.248156 + 0.541227i
\(79\) −0.775978 −0.0873044 −0.0436522 − 0.999047i \(-0.513899\pi\)
−0.0436522 + 0.999047i \(0.513899\pi\)
\(80\) −3.30963 + 0.496212i −0.370027 + 0.0554782i
\(81\) 1.00000 0.111111
\(82\) 0.589421 + 1.28553i 0.0650907 + 0.141963i
\(83\) 8.64172i 0.948552i 0.880376 + 0.474276i \(0.157290\pi\)
−0.880376 + 0.474276i \(0.842710\pi\)
\(84\) −1.39892 − 1.20482i −0.152635 − 0.131456i
\(85\) 2.72084i 0.295117i
\(86\) 13.1826 − 6.04429i 1.42152 − 0.651773i
\(87\) 4.62442 0.495790
\(88\) −1.33753 + 4.63249i −0.142581 + 0.493825i
\(89\) −18.4191 −1.95242 −0.976211 − 0.216821i \(-0.930431\pi\)
−0.976211 + 0.216821i \(0.930431\pi\)
\(90\) 1.07554 − 0.493142i 0.113372 − 0.0519817i
\(91\) 3.43243i 0.359816i
\(92\) 8.32294 − 9.66382i 0.867727 − 1.00752i
\(93\) − 4.41278i − 0.457583i
\(94\) −4.74722 − 10.3537i −0.489638 − 1.06790i
\(95\) 2.41698 0.247977
\(96\) −3.09406 − 4.73569i −0.315786 − 0.483335i
\(97\) −12.2790 −1.24675 −0.623374 − 0.781924i \(-0.714238\pi\)
−0.623374 + 0.781924i \(0.714238\pi\)
\(98\) 3.62368 + 7.90324i 0.366047 + 0.798348i
\(99\) − 1.70474i − 0.171332i
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 984.2.f.b.493.13 ✓ 40
4.3 odd 2 3936.2.f.b.1969.7 40
8.3 odd 2 3936.2.f.b.1969.34 40
8.5 even 2 inner 984.2.f.b.493.14 yes 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
984.2.f.b.493.13 ✓ 40 1.1 even 1 trivial
984.2.f.b.493.14 yes 40 8.5 even 2 inner
3936.2.f.b.1969.7 40 4.3 odd 2
3936.2.f.b.1969.34 40 8.3 odd 2