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Properties

Label 80.11.i.a.13.1
Level $80$
Weight $11$
Character 80.13
Analytic conductor $50.829$
Analytic rank $0$
Dimension $236$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [80,11,Mod(13,80)] 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("80.13"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 3])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 80.i (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: \(50.8285802139\)
Analytic rank: \(0\)
Dimension: \(236\)
Relative dimension: \(118\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 13.1
Character \(\chi\) \(=\) 80.13
Dual form 80.11.i.a.37.1

$q$-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+(-31.9615 - 1.56973i) q^{2} -214.607i q^{3} +(1019.07 + 100.341i) q^{4} +(2623.04 + 1698.61i) q^{5} +(-336.874 + 6859.15i) q^{6} +(-21161.1 - 21161.1i) q^{7} +(-32413.5 - 4806.72i) q^{8} +12992.9 q^{9} +(-81170.0 - 58407.5i) q^{10} +(-46152.0 + 46152.0i) q^{11} +(21534.0 - 218700. i) q^{12} +480366. i q^{13} +(643123. + 709558. i) q^{14} +(364533. - 562923. i) q^{15} +(1.02844e6 + 204510. i) q^{16} +(1.33380e6 - 1.33380e6i) q^{17} +(-415273. - 20395.3i) q^{18} +(429208. - 429208. i) q^{19} +(2.50263e6 + 1.99420e6i) q^{20} +(-4.54132e6 + 4.54132e6i) q^{21} +(1.54753e6 - 1.40264e6i) q^{22} +(-7.93820e6 + 7.93820e6i) q^{23} +(-1.03156e6 + 6.95617e6i) q^{24} +(3.99509e6 + 8.91104e6i) q^{25} +(754042. - 1.53532e7i) q^{26} -1.54607e7i q^{27} +(-1.94414e7 - 2.36880e7i) q^{28} +(-1.91931e7 + 1.91931e7i) q^{29} +(-1.25346e7 + 1.74196e7i) q^{30} -1.48717e7 q^{31} +(-3.25494e7 - 8.15082e6i) q^{32} +(9.90454e6 + 9.90454e6i) q^{33} +(-4.47241e7 + 4.05367e7i) q^{34} +(-1.95621e7 - 9.14509e7i) q^{35} +(1.32407e7 + 1.30373e6i) q^{36} -5.04314e7i q^{37} +(-1.43919e7 + 1.30444e7i) q^{38} +1.03090e8 q^{39} +(-7.68574e7 - 6.76661e7i) q^{40} +1.12715e8i q^{41} +(1.52276e8 - 1.38019e8i) q^{42} +1.16916e8 q^{43} +(-5.16632e7 + 4.24013e7i) q^{44} +(3.40810e7 + 2.20699e7i) q^{45} +(2.66177e8 - 2.41256e8i) q^{46} +(-1.75161e7 + 1.75161e7i) q^{47} +(4.38893e7 - 2.20710e8i) q^{48} +6.13110e8i q^{49} +(-1.13701e8 - 2.91081e8i) q^{50} +(-2.86244e8 - 2.86244e8i) q^{51} +(-4.82006e7 + 4.89527e8i) q^{52} -1.93359e8 q^{53} +(-2.42690e7 + 4.94146e8i) q^{54} +(-1.99453e8 + 4.26646e7i) q^{55} +(5.84191e8 + 7.87622e8i) q^{56} +(-9.21110e7 - 9.21110e7i) q^{57} +(6.43568e8 - 5.83312e8i) q^{58} +(-1.52100e8 - 1.52100e8i) q^{59} +(4.27970e8 - 5.37081e8i) q^{60} +(6.27227e8 + 6.27227e8i) q^{61} +(4.75320e8 + 2.33444e7i) q^{62} +(-2.74945e8 - 2.74945e8i) q^{63} +(1.02753e9 + 3.11606e8i) q^{64} +(-8.15953e8 + 1.26002e9i) q^{65} +(-3.01016e8 - 3.32111e8i) q^{66} +2.78495e8 q^{67} +(1.49308e9 - 1.22541e9i) q^{68} +(1.70359e9 + 1.70359e9i) q^{69} +(4.81680e8 + 2.95361e9i) q^{70} +1.55224e9i q^{71} +(-4.21146e8 - 6.24534e7i) q^{72} +(1.08626e9 - 1.08626e9i) q^{73} +(-7.91634e7 + 1.61186e9i) q^{74} +(1.91237e9 - 8.57373e8i) q^{75} +(4.80461e8 - 3.94326e8i) q^{76} +1.95326e9 q^{77} +(-3.29490e9 - 1.61823e8i) q^{78} +7.56780e8i q^{79} +(2.35026e9 + 2.28335e9i) q^{80} -2.55075e9 q^{81} +(1.76932e8 - 3.60254e9i) q^{82} +5.62752e9i q^{83} +(-5.08361e9 + 4.17225e9i) q^{84} +(5.76424e9 - 1.23302e9i) q^{85} +(-3.73682e9 - 1.83527e8i) q^{86} +(4.11897e9 + 4.11897e9i) q^{87} +(1.71779e9 - 1.27411e9i) q^{88} +7.35487e9 q^{89} +(-1.05464e9 - 7.58884e8i) q^{90} +(1.01651e10 - 1.01651e10i) q^{91} +(-8.88613e9 + 7.29307e9i) q^{92} +3.19156e9i q^{93} +(5.87337e8 - 5.32346e8i) q^{94} +(1.85489e9 - 3.96775e8i) q^{95} +(-1.74922e9 + 6.98532e9i) q^{96} +(1.10052e9 - 1.10052e9i) q^{97} +(9.62414e8 - 1.95959e10i) q^{98} +(-5.99649e8 + 5.99649e8i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 236 q - 2 q^{2} + 1220 q^{4} - 2 q^{5} - 4 q^{6} - 69128 q^{8} - 4487724 q^{9} + 2046 q^{10} - 4 q^{11} + 738232 q^{12} - 4 q^{15} - 2041064 q^{16} - 4 q^{17} + 5924662 q^{18} - 5107040 q^{19} + 2913160 q^{20}+ \cdots - 28071956444 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −31.9615 − 1.56973i −0.998796 − 0.0490539i
\(3\) − 214.607i − 0.883156i −0.897223 − 0.441578i \(-0.854419\pi\)
0.897223 − 0.441578i \(-0.145581\pi\)
\(4\) 1019.07 + 100.341i 0.995187 + 0.0979897i
\(5\) 2623.04 + 1698.61i 0.839374 + 0.543554i
\(6\) −336.874 + 6859.15i −0.0433222 + 0.882092i
\(7\) −21161.1 − 21161.1i −1.25907 − 1.25907i −0.951540 − 0.307526i \(-0.900499\pi\)
−0.307526 − 0.951540i \(-0.599501\pi\)
\(8\) −32413.5 − 4806.72i −0.989183 − 0.146690i
\(9\) 12992.9 0.220036
\(10\) −81170.0 − 58407.5i −0.811700 − 0.584075i
\(11\) −46152.0 + 46152.0i −0.286568 + 0.286568i −0.835721 − 0.549154i \(-0.814950\pi\)
0.549154 + 0.835721i \(0.314950\pi\)
\(12\) 21534.0 − 218700.i 0.0865402 − 0.878905i
\(13\) 480366.i 1.29376i 0.762590 + 0.646882i \(0.223927\pi\)
−0.762590 + 0.646882i \(0.776073\pi\)
\(14\) 643123. + 709558.i 1.19579 + 1.31931i
\(15\) 364533. − 562923.i 0.480043 − 0.741298i
\(16\) 1.02844e6 + 204510.i 0.980796 + 0.195036i
\(17\) 1.33380e6 − 1.33380e6i 0.939394 − 0.939394i −0.0588718 − 0.998266i \(-0.518750\pi\)
0.998266 + 0.0588718i \(0.0187503\pi\)
\(18\) −415273. − 20395.3i −0.219771 − 0.0107936i
\(19\) 429208. − 429208.i 0.173340 − 0.173340i −0.615105 − 0.788445i \(-0.710886\pi\)
0.788445 + 0.615105i \(0.210886\pi\)
\(20\) 2.50263e6 + 1.99420e6i 0.782072 + 0.623189i
\(21\) −4.54132e6 + 4.54132e6i −1.11195 + 1.11195i
\(22\) 1.54753e6 − 1.40264e6i 0.300280 − 0.272165i
\(23\) −7.93820e6 + 7.93820e6i −1.23334 + 1.23334i −0.270668 + 0.962673i \(0.587244\pi\)
−0.962673 + 0.270668i \(0.912756\pi\)
\(24\) −1.03156e6 + 6.95617e6i −0.129550 + 0.873602i
\(25\) 3.99509e6 + 8.91104e6i 0.409097 + 0.912491i
\(26\) 754042. − 1.53532e7i 0.0634642 − 1.29221i
\(27\) − 1.54607e7i − 1.07748i
\(28\) −1.94414e7 − 2.36880e7i −1.12963 − 1.37638i
\(29\) −1.91931e7 + 1.91931e7i −0.935740 + 0.935740i −0.998056 − 0.0623160i \(-0.980151\pi\)
0.0623160 + 0.998056i \(0.480151\pi\)
\(30\) −1.25346e7 + 1.74196e7i −0.515829 + 0.716857i
\(31\) −1.48717e7 −0.519459 −0.259729 − 0.965681i \(-0.583633\pi\)
−0.259729 + 0.965681i \(0.583633\pi\)
\(32\) −3.25494e7 − 8.15082e6i −0.970048 − 0.242913i
\(33\) 9.90454e6 + 9.90454e6i 0.253084 + 0.253084i
\(34\) −4.47241e7 + 4.05367e7i −0.984344 + 0.892182i
\(35\) −1.95621e7 − 9.14509e7i −0.372456 − 1.74120i
\(36\) 1.32407e7 + 1.30373e6i 0.218977 + 0.0215613i
\(37\) − 5.04314e7i − 0.727264i −0.931543 − 0.363632i \(-0.881537\pi\)
0.931543 − 0.363632i \(-0.118463\pi\)
\(38\) −1.43919e7 + 1.30444e7i −0.181635 + 0.164629i
\(39\) 1.03090e8 1.14259
\(40\) −7.68574e7 − 6.76661e7i −0.750560 − 0.660802i
\(41\) 1.12715e8i 0.972887i 0.873712 + 0.486444i \(0.161706\pi\)
−0.873712 + 0.486444i \(0.838294\pi\)
\(42\) 1.52276e8 − 1.38019e8i 1.16516 − 1.05607i
\(43\) 1.16916e8 0.795304 0.397652 − 0.917536i \(-0.369825\pi\)
0.397652 + 0.917536i \(0.369825\pi\)
\(44\) −5.16632e7 + 4.24013e7i −0.313269 + 0.257108i
\(45\) 3.40810e7 + 2.20699e7i 0.184693 + 0.119602i
\(46\) 2.66177e8 − 2.41256e8i 1.29236 − 1.17136i
\(47\) −1.75161e7 + 1.75161e7i −0.0763745 + 0.0763745i −0.744262 − 0.667888i \(-0.767199\pi\)
0.667888 + 0.744262i \(0.267199\pi\)
\(48\) 4.38893e7 − 2.20710e8i 0.172247 − 0.866195i
\(49\) 6.13110e8i 2.17049i
\(50\) −1.13701e8 − 2.91081e8i −0.363843 − 0.931460i
\(51\) −2.86244e8 − 2.86244e8i −0.829631 − 0.829631i
\(52\) −4.82006e7 + 4.89527e8i −0.126776 + 1.28754i
\(53\) −1.93359e8 −0.462365 −0.231182 − 0.972910i \(-0.574259\pi\)
−0.231182 + 0.972910i \(0.574259\pi\)
\(54\) −2.42690e7 + 4.94146e8i −0.0528547 + 1.07618i
\(55\) −1.99453e8 + 4.26646e7i −0.396303 + 0.0847723i
\(56\) 5.84191e8 + 7.87622e8i 1.06075 + 1.43014i
\(57\) −9.21110e7 − 9.21110e7i −0.153087 − 0.153087i
\(58\) 6.43568e8 − 5.83312e8i 0.980516 − 0.888712i
\(59\) −1.52100e8 − 1.52100e8i −0.212750 − 0.212750i 0.592685 − 0.805435i \(-0.298068\pi\)
−0.805435 + 0.592685i \(0.798068\pi\)
\(60\) 4.27970e8 − 5.37081e8i 0.550372 − 0.690691i
\(61\) 6.27227e8 + 6.27227e8i 0.742635 + 0.742635i 0.973084 − 0.230449i \(-0.0740195\pi\)
−0.230449 + 0.973084i \(0.574020\pi\)
\(62\) 4.75320e8 + 2.33444e7i 0.518833 + 0.0254815i
\(63\) −2.74945e8 − 2.74945e8i −0.277040 − 0.277040i
\(64\) 1.02753e9 + 3.11606e8i 0.956964 + 0.290206i
\(65\) −8.15953e8 + 1.26002e9i −0.703231 + 1.08595i
\(66\) −3.01016e8 − 3.32111e8i −0.240364 − 0.265194i
\(67\) 2.78495e8 0.206273 0.103137 − 0.994667i \(-0.467112\pi\)
0.103137 + 0.994667i \(0.467112\pi\)
\(68\) 1.49308e9 − 1.22541e9i 1.02692 − 0.842822i
\(69\) 1.70359e9 + 1.70359e9i 1.08923 + 1.08923i
\(70\) 4.81680e8 + 2.95361e9i 0.286595 + 1.75737i
\(71\) 1.55224e9i 0.860334i 0.902749 + 0.430167i \(0.141545\pi\)
−0.902749 + 0.430167i \(0.858455\pi\)
\(72\) −4.21146e8 − 6.24534e7i −0.217656 − 0.0322770i
\(73\) 1.08626e9 − 1.08626e9i 0.523984 − 0.523984i −0.394788 − 0.918772i \(-0.629182\pi\)
0.918772 + 0.394788i \(0.129182\pi\)
\(74\) −7.91634e7 + 1.61186e9i −0.0356751 + 0.726388i
\(75\) 1.91237e9 − 8.57373e8i 0.805871 − 0.361296i
\(76\) 4.80461e8 − 3.94326e8i 0.189492 − 0.155521i
\(77\) 1.95326e9 0.721615
\(78\) −3.29490e9 − 1.61823e8i −1.14122 − 0.0560487i
\(79\) 7.56780e8i 0.245943i 0.992410 + 0.122971i \(0.0392424\pi\)
−0.992410 + 0.122971i \(0.960758\pi\)
\(80\) 2.35026e9 + 2.28335e9i 0.717242 + 0.696824i
\(81\) −2.55075e9 −0.731548
\(82\) 1.76932e8 − 3.60254e9i 0.0477239 − 0.971716i
\(83\) 5.62752e9i 1.42865i 0.699813 + 0.714326i \(0.253267\pi\)
−0.699813 + 0.714326i \(0.746733\pi\)
\(84\) −5.08361e9 + 4.17225e9i −1.21556 + 0.997640i
\(85\) 5.76424e9 − 1.23302e9i 1.29911 − 0.277891i
\(86\) −3.73682e9 − 1.83527e8i −0.794347 − 0.0390128i
\(87\) 4.11897e9 + 4.11897e9i 0.826404 + 0.826404i
\(88\) 1.71779e9 − 1.27411e9i 0.325504 − 0.241431i
\(89\) 7.35487e9 1.31712 0.658559 − 0.752529i \(-0.271166\pi\)
0.658559 + 0.752529i \(0.271166\pi\)
\(90\) −1.05464e9 − 7.58884e8i −0.178603 − 0.128518i
\(91\) 1.01651e10 − 1.01651e10i 1.62893 − 1.62893i
\(92\) −8.88613e9 + 7.29307e9i −1.34826 + 1.10655i
\(93\) 3.19156e9i 0.458763i
\(94\) 5.87337e8 − 5.32346e8i 0.0800291 − 0.0725361i
\(95\) 1.85489e9 − 3.96775e8i 0.239717 − 0.0512775i
\(96\) −1.74922e9 + 6.98532e9i −0.214530 + 0.856703i
\(97\) 1.10052e9 − 1.10052e9i 0.128156 − 0.128156i −0.640120 − 0.768275i \(-0.721115\pi\)
0.768275 + 0.640120i \(0.221115\pi\)
\(98\) 9.62414e8 − 1.95959e10i 0.106471 − 2.16788i
\(99\) −5.99649e8 + 5.99649e8i −0.0630553 + 0.0630553i
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 80.11.i.a.13.1 ✓ 236
5.2 odd 4 80.11.t.a.77.60 yes 236
16.5 even 4 80.11.t.a.53.60 yes 236
80.37 odd 4 inner 80.11.i.a.37.1 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.1 ✓ 236 1.1 even 1 trivial
80.11.i.a.37.1 yes 236 80.37 odd 4 inner
80.11.t.a.53.60 yes 236 16.5 even 4
80.11.t.a.77.60 yes 236 5.2 odd 4