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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 13.2
Character \(\chi\) \(=\) 80.13
Dual form 80.11.i.a.37.2

$q$-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+(-31.9536 - 1.72183i) q^{2} +388.128i q^{3} +(1018.07 + 110.037i) q^{4} +(3121.22 + 153.749i) q^{5} +(668.289 - 12402.1i) q^{6} +(1274.43 + 1274.43i) q^{7} +(-32341.6 - 5269.03i) q^{8} -91594.5 q^{9} +(-99469.5 - 10287.0i) q^{10} +(194203. - 194203. i) q^{11} +(-42708.6 + 395142. i) q^{12} +503179. i q^{13} +(-38528.2 - 42916.9i) q^{14} +(-59674.4 + 1.21143e6i) q^{15} +(1.02436e6 + 224051. i) q^{16} +(1.59786e6 - 1.59786e6i) q^{17} +(2.92678e6 + 157710. i) q^{18} +(-1.54562e6 + 1.54562e6i) q^{19} +(3.16070e6 + 499978. i) q^{20} +(-494641. + 494641. i) q^{21} +(-6.53988e6 + 5.87111e6i) q^{22} +(6.34526e6 - 6.34526e6i) q^{23} +(2.04506e6 - 1.25527e7i) q^{24} +(9.71835e6 + 959768. i) q^{25} +(866387. - 1.60784e7i) q^{26} -1.26318e7i q^{27} +(1.15722e6 + 1.43769e6i) q^{28} +(2.37105e7 - 2.37105e7i) q^{29} +(3.99269e6 - 3.86069e7i) q^{30} -1.55756e7 q^{31} +(-3.23462e7 - 8.92303e6i) q^{32} +(7.53757e7 + 7.53757e7i) q^{33} +(-5.38088e7 + 4.83064e7i) q^{34} +(3.78182e6 + 4.17370e6i) q^{35} +(-9.32496e7 - 1.00788e7i) q^{36} -1.55083e7i q^{37} +(5.20494e7 - 4.67269e7i) q^{38} -1.95298e8 q^{39} +(-1.00135e8 - 2.14183e7i) q^{40} -8.23434e7i q^{41} +(1.66573e7 - 1.49539e7i) q^{42} +2.51380e8 q^{43} +(2.19082e8 - 1.76343e8i) q^{44} +(-2.85886e8 - 1.40826e7i) q^{45} +(-2.13680e8 + 1.91829e8i) q^{46} +(7.44739e7 - 7.44739e7i) q^{47} +(-8.69607e7 + 3.97583e8i) q^{48} -2.79227e8i q^{49} +(-3.08884e8 - 4.74014e7i) q^{50} +(6.20176e8 + 6.20176e8i) q^{51} +(-5.53684e7 + 5.12272e8i) q^{52} +1.65009e8 q^{53} +(-2.17498e7 + 4.03632e8i) q^{54} +(6.36009e8 - 5.76291e8i) q^{55} +(-3.45020e7 - 4.79320e7i) q^{56} +(-5.99898e8 - 5.99898e8i) q^{57} +(-7.98462e8 + 7.16812e8i) q^{58} +(5.36087e8 + 5.36087e8i) q^{59} +(-1.94055e8 + 1.22676e9i) q^{60} +(-1.78865e8 - 1.78865e8i) q^{61} +(4.97699e8 + 2.68186e7i) q^{62} +(-1.16730e8 - 1.16730e8i) q^{63} +(1.01822e9 + 3.40818e8i) q^{64} +(-7.73633e7 + 1.57053e9i) q^{65} +(-2.27874e9 - 2.53831e9i) q^{66} +1.12235e9 q^{67} +(1.80256e9 - 1.45091e9i) q^{68} +(2.46277e9 + 2.46277e9i) q^{69} +(-1.13656e8 - 1.39877e8i) q^{70} -1.76021e8i q^{71} +(2.96231e9 + 4.82614e8i) q^{72} +(-9.74289e8 + 9.74289e8i) q^{73} +(-2.67025e7 + 4.95545e8i) q^{74} +(-3.72513e8 + 3.77196e9i) q^{75} +(-1.74362e9 + 1.40347e9i) q^{76} +4.94995e8 q^{77} +(6.24048e9 + 3.36269e8i) q^{78} -9.87003e7i q^{79} +(3.16280e9 + 8.56807e8i) q^{80} -5.05801e8 q^{81} +(-1.41781e8 + 2.63117e9i) q^{82} -3.21307e8i q^{83} +(-5.58008e8 + 4.49150e8i) q^{84} +(5.23295e9 - 4.74161e9i) q^{85} +(-8.03252e9 - 4.32834e8i) q^{86} +(9.20271e9 + 9.20271e9i) q^{87} +(-7.30410e9 + 5.25758e9i) q^{88} -1.05052e10 q^{89} +(9.11085e9 + 9.42236e8i) q^{90} +(-6.41264e8 + 6.41264e8i) q^{91} +(7.15814e9 - 5.76171e9i) q^{92} -6.04535e9i q^{93} +(-2.50794e9 + 2.25148e9i) q^{94} +(-5.06185e9 + 4.58657e9i) q^{95} +(3.46328e9 - 1.25545e10i) q^{96} +(1.02017e9 - 1.02017e9i) q^{97} +(-4.80780e8 + 8.92232e9i) q^{98} +(-1.77879e10 + 1.77879e10i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −31.9536 − 1.72183i −0.998551 − 0.0538071i
\(3\) 388.128i 1.59724i 0.601839 + 0.798618i \(0.294435\pi\)
−0.601839 + 0.798618i \(0.705565\pi\)
\(4\) 1018.07 + 110.037i 0.994210 + 0.107458i
\(5\) 3121.22 + 153.749i 0.998789 + 0.0491997i
\(6\) 668.289 − 12402.1i 0.0859426 − 1.59492i
\(7\) 1274.43 + 1274.43i 0.0758271 + 0.0758271i 0.744003 − 0.668176i \(-0.232925\pi\)
−0.668176 + 0.744003i \(0.732925\pi\)
\(8\) −32341.6 − 5269.03i −0.986987 − 0.160798i
\(9\) −91594.5 −1.55116
\(10\) −99469.5 − 10287.0i −0.994695 − 0.102870i
\(11\) 194203. − 194203.i 1.20585 − 1.20585i 0.233490 − 0.972359i \(-0.424986\pi\)
0.972359 − 0.233490i \(-0.0750145\pi\)
\(12\) −42708.6 + 395142.i −0.171636 + 1.58799i
\(13\) 503179.i 1.35521i 0.735427 + 0.677604i \(0.236981\pi\)
−0.735427 + 0.677604i \(0.763019\pi\)
\(14\) −38528.2 − 42916.9i −0.0716372 − 0.0797973i
\(15\) −59674.4 + 1.21143e6i −0.0785835 + 1.59530i
\(16\) 1.02436e6 + 224051.i 0.976905 + 0.213672i
\(17\) 1.59786e6 − 1.59786e6i 1.12537 − 1.12537i 0.134450 − 0.990920i \(-0.457073\pi\)
0.990920 − 0.134450i \(-0.0429267\pi\)
\(18\) 2.92678e6 + 157710.i 1.54891 + 0.0834634i
\(19\) −1.54562e6 + 1.54562e6i −0.624215 + 0.624215i −0.946606 − 0.322391i \(-0.895513\pi\)
0.322391 + 0.946606i \(0.395513\pi\)
\(20\) 3.16070e6 + 499978.i 0.987719 + 0.156243i
\(21\) −494641. + 494641.i −0.121114 + 0.121114i
\(22\) −6.53988e6 + 5.87111e6i −1.26899 + 1.13922i
\(23\) 6.34526e6 − 6.34526e6i 0.985849 − 0.985849i −0.0140524 − 0.999901i \(-0.504473\pi\)
0.999901 + 0.0140524i \(0.00447316\pi\)
\(24\) 2.04506e6 − 1.25527e7i 0.256832 − 1.57645i
\(25\) 9.71835e6 + 959768.i 0.995159 + 0.0982803i
\(26\) 866387. − 1.60784e7i 0.0729198 − 1.35324i
\(27\) − 1.26318e7i − 0.880332i
\(28\) 1.15722e6 + 1.43769e6i 0.0672398 + 0.0835363i
\(29\) 2.37105e7 − 2.37105e7i 1.15598 − 1.15598i 0.170649 − 0.985332i \(-0.445413\pi\)
0.985332 − 0.170649i \(-0.0545866\pi\)
\(30\) 3.99269e6 − 3.86069e7i 0.164308 − 1.58876i
\(31\) −1.55756e7 −0.544049 −0.272024 − 0.962290i \(-0.587693\pi\)
−0.272024 + 0.962290i \(0.587693\pi\)
\(32\) −3.23462e7 − 8.92303e6i −0.963993 − 0.265927i
\(33\) 7.53757e7 + 7.53757e7i 1.92602 + 1.92602i
\(34\) −5.38088e7 + 4.83064e7i −1.18429 + 1.06319i
\(35\) 3.78182e6 + 4.17370e6i 0.0720046 + 0.0794659i
\(36\) −9.32496e7 − 1.00788e7i −1.54218 − 0.166685i
\(37\) − 1.55083e7i − 0.223643i −0.993728 − 0.111821i \(-0.964332\pi\)
0.993728 − 0.111821i \(-0.0356684\pi\)
\(38\) 5.20494e7 − 4.67269e7i 0.656898 − 0.589724i
\(39\) −1.95298e8 −2.16458
\(40\) −1.00135e8 − 2.14183e7i −0.977881 − 0.209163i
\(41\) − 8.23434e7i − 0.710738i −0.934726 − 0.355369i \(-0.884355\pi\)
0.934726 − 0.355369i \(-0.115645\pi\)
\(42\) 1.66573e7 − 1.49539e7i 0.127455 − 0.114421i
\(43\) 2.51380e8 1.70997 0.854987 − 0.518650i \(-0.173565\pi\)
0.854987 + 0.518650i \(0.173565\pi\)
\(44\) 2.19082e8 − 1.76343e8i 1.32844 − 1.06929i
\(45\) −2.85886e8 − 1.40826e7i −1.54928 − 0.0763166i
\(46\) −2.13680e8 + 1.91829e8i −1.03747 + 0.931375i
\(47\) 7.44739e7 − 7.44739e7i 0.324724 − 0.324724i −0.525852 − 0.850576i \(-0.676253\pi\)
0.850576 + 0.525852i \(0.176253\pi\)
\(48\) −8.69607e7 + 3.97583e8i −0.341285 + 1.56035i
\(49\) − 2.79227e8i − 0.988501i
\(50\) −3.08884e8 − 4.74014e7i −0.988429 − 0.151684i
\(51\) 6.20176e8 + 6.20176e8i 1.79748 + 1.79748i
\(52\) −5.53684e7 + 5.12272e8i −0.145628 + 1.34736i
\(53\) 1.65009e8 0.394575 0.197287 − 0.980346i \(-0.436787\pi\)
0.197287 + 0.980346i \(0.436787\pi\)
\(54\) −2.17498e7 + 4.03632e8i −0.0473681 + 0.879057i
\(55\) 6.36009e8 − 5.76291e8i 1.26372 − 1.14506i
\(56\) −3.45020e7 − 4.79320e7i −0.0626475 − 0.0870332i
\(57\) −5.99898e8 − 5.99898e8i −0.997019 − 0.997019i
\(58\) −7.98462e8 + 7.16812e8i −1.21651 + 1.09211i
\(59\) 5.36087e8 + 5.36087e8i 0.749851 + 0.749851i 0.974451 − 0.224600i \(-0.0721076\pi\)
−0.224600 + 0.974451i \(0.572108\pi\)
\(60\) −1.94055e8 + 1.22676e9i −0.249557 + 1.57762i
\(61\) −1.78865e8 − 1.78865e8i −0.211776 − 0.211776i 0.593246 − 0.805021i \(-0.297846\pi\)
−0.805021 + 0.593246i \(0.797846\pi\)
\(62\) 4.97699e8 + 2.68186e7i 0.543260 + 0.0292737i
\(63\) −1.16730e8 − 1.16730e8i −0.117620 − 0.117620i
\(64\) 1.01822e9 + 3.40818e8i 0.948288 + 0.317411i
\(65\) −7.73633e7 + 1.57053e9i −0.0666758 + 1.35357i
\(66\) −2.27874e9 − 2.53831e9i −1.81960 − 2.02687i
\(67\) 1.12235e9 0.831291 0.415645 − 0.909527i \(-0.363556\pi\)
0.415645 + 0.909527i \(0.363556\pi\)
\(68\) 1.80256e9 − 1.45091e9i 1.23978 − 0.997923i
\(69\) 2.46277e9 + 2.46277e9i 1.57463 + 1.57463i
\(70\) −1.13656e8 − 1.39877e8i −0.0676245 − 0.0832252i
\(71\) − 1.76021e8i − 0.0975600i −0.998810 − 0.0487800i \(-0.984467\pi\)
0.998810 − 0.0487800i \(-0.0155333\pi\)
\(72\) 2.96231e9 + 4.82614e8i 1.53098 + 0.249424i
\(73\) −9.74289e8 + 9.74289e8i −0.469974 + 0.469974i −0.901906 − 0.431932i \(-0.857832\pi\)
0.431932 + 0.901906i \(0.357832\pi\)
\(74\) −2.67025e7 + 4.95545e8i −0.0120336 + 0.223319i
\(75\) −3.72513e8 + 3.77196e9i −0.156977 + 1.58950i
\(76\) −1.74362e9 + 1.40347e9i −0.687678 + 0.553524i
\(77\) 4.94995e8 0.182872
\(78\) 6.24048e9 + 3.36269e8i 2.16145 + 0.116470i
\(79\) − 9.87003e7i − 0.0320762i −0.999871 − 0.0160381i \(-0.994895\pi\)
0.999871 − 0.0160381i \(-0.00510530\pi\)
\(80\) 3.16280e9 + 8.56807e8i 0.965210 + 0.261477i
\(81\) −5.05801e8 −0.145062
\(82\) −1.41781e8 + 2.63117e9i −0.0382428 + 0.709709i
\(83\) − 3.21307e8i − 0.0815698i −0.999168 − 0.0407849i \(-0.987014\pi\)
0.999168 − 0.0407849i \(-0.0129858\pi\)
\(84\) −5.58008e8 + 4.49150e8i −0.133427 + 0.107398i
\(85\) 5.23295e9 − 4.74161e9i 1.17938 − 1.06864i
\(86\) −8.03252e9 − 4.32834e8i −1.70750 − 0.0920087i
\(87\) 9.20271e9 + 9.20271e9i 1.84637 + 1.84637i
\(88\) −7.30410e9 + 5.25758e9i −1.38406 + 0.996259i
\(89\) −1.05052e10 −1.88129 −0.940643 − 0.339398i \(-0.889777\pi\)
−0.940643 + 0.339398i \(0.889777\pi\)
\(90\) 9.11085e9 + 9.42236e8i 1.54293 + 0.159568i
\(91\) −6.41264e8 + 6.41264e8i −0.102761 + 0.102761i
\(92\) 7.15814e9 − 5.76171e9i 1.08608 − 0.874203i
\(93\) − 6.04535e9i − 0.868974i
\(94\) −2.50794e9 + 2.25148e9i −0.341726 + 0.306781i
\(95\) −5.06185e9 + 4.58657e9i −0.654170 + 0.592748i
\(96\) 3.46328e9 − 1.25545e10i 0.424748 − 1.53972i
\(97\) 1.02017e9 − 1.02017e9i 0.118800 − 0.118800i −0.645208 − 0.764007i \(-0.723229\pi\)
0.764007 + 0.645208i \(0.223229\pi\)
\(98\) −4.80780e8 + 8.92232e9i −0.0531883 + 0.987069i
\(99\) −1.77879e10 + 1.77879e10i −1.87046 + 1.87046i
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.13.2 ✓ 236
5.2 odd 4 80.11.t.a.77.61 yes 236
16.5 even 4 80.11.t.a.53.61 yes 236
80.37 odd 4 inner 80.11.i.a.37.2 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.2 ✓ 236 1.1 even 1 trivial
80.11.i.a.37.2 yes 236 80.37 odd 4 inner
80.11.t.a.53.61 yes 236 16.5 even 4
80.11.t.a.77.61 yes 236 5.2 odd 4