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Properties

Label 80.11.i.a.13.5
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 13.5
Character \(\chi\) \(=\) 80.13
Dual form 80.11.i.a.37.5

$q$-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+(-31.8587 - 3.00404i) q^{2} -26.1884i q^{3} +(1005.95 + 191.409i) q^{4} +(629.600 + 3060.92i) q^{5} +(-78.6710 + 834.328i) q^{6} +(8782.74 + 8782.74i) q^{7} +(-31473.3 - 9119.97i) q^{8} +58363.2 q^{9} +(-10863.1 - 99408.2i) q^{10} +(4757.29 - 4757.29i) q^{11} +(5012.71 - 26344.3i) q^{12} -466998. i q^{13} +(-253423. - 306190. i) q^{14} +(80160.6 - 16488.2i) q^{15} +(975301. + 385097. i) q^{16} +(565396. - 565396. i) q^{17} +(-1.85937e6 - 175325. i) q^{18} +(1.38283e6 - 1.38283e6i) q^{19} +(47458.1 + 3.19965e6i) q^{20} +(230006. - 230006. i) q^{21} +(-165852. + 137270. i) q^{22} +(6.86584e6 - 6.86584e6i) q^{23} +(-238837. + 824235. i) q^{24} +(-8.97283e6 + 3.85431e6i) q^{25} +(-1.40288e6 + 1.48780e7i) q^{26} -3.07484e6i q^{27} +(7.15391e6 + 1.05161e7i) q^{28} +(-6.87132e6 + 6.87132e6i) q^{29} +(-2.60334e6 + 284487. i) q^{30} -2.75723e7 q^{31} +(-2.99150e7 - 1.51985e7i) q^{32} +(-124586. - 124586. i) q^{33} +(-1.97112e7 + 1.63143e7i) q^{34} +(-2.13537e7 + 3.24129e7i) q^{35} +(5.87105e7 + 1.11713e7i) q^{36} -1.12774e7i q^{37} +(-4.82091e7 + 3.99010e7i) q^{38} -1.22299e7 q^{39} +(8.09991e6 - 1.02079e8i) q^{40} -5.42469e7i q^{41} +(-8.01864e6 + 6.63674e6i) q^{42} -8.81188e7 q^{43} +(5.69619e6 - 3.87501e6i) q^{44} +(3.67454e7 + 1.78645e8i) q^{45} +(-2.39362e8 + 1.98111e8i) q^{46} +(8.37641e7 - 8.37641e7i) q^{47} +(1.00851e7 - 2.55416e7i) q^{48} -1.28202e8i q^{49} +(2.97441e8 - 9.58385e7i) q^{50} +(-1.48068e7 - 1.48068e7i) q^{51} +(8.93879e7 - 4.69778e8i) q^{52} +5.57728e8 q^{53} +(-9.23693e6 + 9.79602e7i) q^{54} +(1.75569e7 + 1.15665e7i) q^{55} +(-1.96323e8 - 3.56520e8i) q^{56} +(-3.62140e7 - 3.62140e7i) q^{57} +(2.39553e8 - 1.98269e8i) q^{58} +(4.88884e8 + 4.88884e8i) q^{59} +(8.37937e7 - 1.24285e6i) q^{60} +(-4.07500e6 - 4.07500e6i) q^{61} +(8.78417e8 + 8.28282e7i) q^{62} +(5.12589e8 + 5.12589e8i) q^{63} +(9.07394e8 + 5.74071e8i) q^{64} +(1.42944e9 - 2.94022e8i) q^{65} +(3.59488e6 + 4.34340e6i) q^{66} -3.07255e8 q^{67} +(6.76983e8 - 4.60539e8i) q^{68} +(-1.79805e8 - 1.79805e8i) q^{69} +(7.77669e8 - 9.68485e8i) q^{70} -1.49616e9i q^{71} +(-1.83688e9 - 5.32270e8i) q^{72} +(1.26872e9 - 1.26872e9i) q^{73} +(-3.38778e7 + 3.59284e8i) q^{74} +(1.00938e8 + 2.34984e8i) q^{75} +(1.65574e9 - 1.12637e9i) q^{76} +8.35641e7 q^{77} +(3.89630e8 + 3.67392e7i) q^{78} -9.19313e8i q^{79} +(-5.64702e8 + 3.22777e9i) q^{80} +3.36576e9 q^{81} +(-1.62960e8 + 1.72823e9i) q^{82} -3.85139e9i q^{83} +(2.75400e8 - 1.87350e8i) q^{84} +(2.08660e9 + 1.37466e9i) q^{85} +(2.80735e9 + 2.64712e8i) q^{86} +(1.79949e8 + 1.79949e8i) q^{87} +(-1.93114e8 + 1.06341e8i) q^{88} +4.02424e9 q^{89} +(-6.34005e8 - 5.80178e9i) q^{90} +(4.10153e9 - 4.10153e9i) q^{91} +(8.22089e9 - 5.59252e9i) q^{92} +7.22074e8i q^{93} +(-2.92024e9 + 2.41698e9i) q^{94} +(5.10335e9 + 3.36210e9i) q^{95} +(-3.98025e8 + 7.83425e8i) q^{96} +(5.59860e9 - 5.59860e9i) q^{97} +(-3.85124e8 + 4.08435e9i) q^{98} +(2.77650e8 - 2.77650e8i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −31.8587 − 3.00404i −0.995584 − 0.0938762i
\(3\) − 26.1884i − 0.107771i −0.998547 − 0.0538856i \(-0.982839\pi\)
0.998547 − 0.0538856i \(-0.0171606\pi\)
\(4\) 1005.95 + 191.409i 0.982375 + 0.186923i
\(5\) 629.600 + 3060.92i 0.201472 + 0.979494i
\(6\) −78.6710 + 834.328i −0.0101172 + 0.107295i
\(7\) 8782.74 + 8782.74i 0.522565 + 0.522565i 0.918345 − 0.395781i \(-0.129526\pi\)
−0.395781 + 0.918345i \(0.629526\pi\)
\(8\) −31473.3 − 9119.97i −0.960489 − 0.278319i
\(9\) 58363.2 0.988385
\(10\) −10863.1 − 99408.2i −0.108631 − 0.994082i
\(11\) 4757.29 − 4757.29i 0.0295390 − 0.0295390i −0.692183 − 0.721722i \(-0.743351\pi\)
0.721722 + 0.692183i \(0.243351\pi\)
\(12\) 5012.71 − 26344.3i 0.0201449 − 0.105872i
\(13\) − 466998.i − 1.25776i −0.777501 − 0.628881i \(-0.783513\pi\)
0.777501 − 0.628881i \(-0.216487\pi\)
\(14\) −253423. − 306190.i −0.471201 − 0.569313i
\(15\) 80160.6 − 16488.2i 0.105561 − 0.0217129i
\(16\) 975301. + 385097.i 0.930119 + 0.367257i
\(17\) 565396. − 565396.i 0.398206 − 0.398206i −0.479394 − 0.877600i \(-0.659143\pi\)
0.877600 + 0.479394i \(0.159143\pi\)
\(18\) −1.85937e6 − 175325.i −0.984021 − 0.0927859i
\(19\) 1.38283e6 − 1.38283e6i 0.558470 − 0.558470i −0.370402 − 0.928872i \(-0.620780\pi\)
0.928872 + 0.370402i \(0.120780\pi\)
\(20\) 47458.1 + 3.19965e6i 0.0148307 + 0.999890i
\(21\) 230006. − 230006.i 0.0563174 − 0.0563174i
\(22\) −165852. + 137270.i −0.0321816 + 0.0266356i
\(23\) 6.86584e6 − 6.86584e6i 1.06673 − 1.06673i 0.0691224 − 0.997608i \(-0.477980\pi\)
0.997608 − 0.0691224i \(-0.0220199\pi\)
\(24\) −238837. + 824235.i −0.0299948 + 0.103513i
\(25\) −8.97283e6 + 3.85431e6i −0.918818 + 0.394681i
\(26\) −1.40288e6 + 1.48780e7i −0.118074 + 1.25221i
\(27\) − 3.07484e6i − 0.214291i
\(28\) 7.15391e6 + 1.05161e7i 0.415675 + 0.611034i
\(29\) −6.87132e6 + 6.87132e6i −0.335004 + 0.335004i −0.854483 − 0.519479i \(-0.826126\pi\)
0.519479 + 0.854483i \(0.326126\pi\)
\(30\) −2.60334e6 + 284487.i −0.107133 + 0.0117073i
\(31\) −2.75723e7 −0.963084 −0.481542 − 0.876423i \(-0.659923\pi\)
−0.481542 + 0.876423i \(0.659923\pi\)
\(32\) −2.99150e7 − 1.51985e7i −0.891535 − 0.452952i
\(33\) −124586. − 124586.i −0.00318345 − 0.00318345i
\(34\) −1.97112e7 + 1.63143e7i −0.433830 + 0.359065i
\(35\) −2.13537e7 + 3.24129e7i −0.406567 + 0.617131i
\(36\) 5.87105e7 + 1.11713e7i 0.970965 + 0.184752i
\(37\) − 1.12774e7i − 0.162630i −0.996688 − 0.0813151i \(-0.974088\pi\)
0.996688 − 0.0813151i \(-0.0259120\pi\)
\(38\) −4.82091e7 + 3.99010e7i −0.608431 + 0.503577i
\(39\) −1.22299e7 −0.135551
\(40\) 8.09991e6 − 1.02079e8i 0.0791007 − 0.996867i
\(41\) − 5.42469e7i − 0.468226i −0.972209 − 0.234113i \(-0.924781\pi\)
0.972209 − 0.234113i \(-0.0752186\pi\)
\(42\) −8.01864e6 + 6.63674e6i −0.0613556 + 0.0507818i
\(43\) −8.81188e7 −0.599414 −0.299707 − 0.954031i \(-0.596889\pi\)
−0.299707 + 0.954031i \(0.596889\pi\)
\(44\) 5.69619e6 − 3.87501e6i 0.0345399 − 0.0234968i
\(45\) 3.67454e7 + 1.78645e8i 0.199132 + 0.968118i
\(46\) −2.39362e8 + 1.98111e8i −1.16216 + 0.961879i
\(47\) 8.37641e7 − 8.37641e7i 0.365232 − 0.365232i −0.500503 − 0.865735i \(-0.666852\pi\)
0.865735 + 0.500503i \(0.166852\pi\)
\(48\) 1.00851e7 − 2.55416e7i 0.0395798 − 0.100240i
\(49\) − 1.28202e8i − 0.453852i
\(50\) 2.97441e8 − 9.58385e7i 0.951812 − 0.306683i
\(51\) −1.48068e7 − 1.48068e7i −0.0429151 − 0.0429151i
\(52\) 8.93879e7 − 4.69778e8i 0.235105 − 1.23559i
\(53\) 5.57728e8 1.33365 0.666827 − 0.745212i \(-0.267652\pi\)
0.666827 + 0.745212i \(0.267652\pi\)
\(54\) −9.23693e6 + 9.79602e7i −0.0201168 + 0.213344i
\(55\) 1.75569e7 + 1.15665e7i 0.0348846 + 0.0229820i
\(56\) −1.96323e8 − 3.56520e8i −0.356477 − 0.647357i
\(57\) −3.62140e7 − 3.62140e7i −0.0601870 − 0.0601870i
\(58\) 2.39553e8 − 1.98269e8i 0.364973 − 0.302076i
\(59\) 4.88884e8 + 4.88884e8i 0.683826 + 0.683826i 0.960860 − 0.277034i \(-0.0893514\pi\)
−0.277034 + 0.960860i \(0.589351\pi\)
\(60\) 8.37937e7 − 1.24285e6i 0.107759 − 0.00159832i
\(61\) −4.07500e6 − 4.07500e6i −0.00482479 − 0.00482479i 0.704690 − 0.709515i \(-0.251086\pi\)
−0.709515 + 0.704690i \(0.751086\pi\)
\(62\) 8.78417e8 + 8.28282e7i 0.958831 + 0.0904107i
\(63\) 5.12589e8 + 5.12589e8i 0.516495 + 0.516495i
\(64\) 9.07394e8 + 5.74071e8i 0.845077 + 0.534645i
\(65\) 1.42944e9 − 2.94022e8i 1.23197 − 0.253404i
\(66\) 3.59488e6 + 4.34340e6i 0.00287055 + 0.00346825i
\(67\) −3.07255e8 −0.227575 −0.113788 − 0.993505i \(-0.536298\pi\)
−0.113788 + 0.993505i \(0.536298\pi\)
\(68\) 6.76983e8 − 4.60539e8i 0.465621 − 0.316754i
\(69\) −1.79805e8 − 1.79805e8i −0.114963 − 0.114963i
\(70\) 7.77669e8 − 9.68485e8i 0.462705 − 0.576239i
\(71\) − 1.49616e9i − 0.829253i −0.909992 − 0.414627i \(-0.863912\pi\)
0.909992 − 0.414627i \(-0.136088\pi\)
\(72\) −1.83688e9 − 5.32270e8i −0.949333 − 0.275087i
\(73\) 1.26872e9 − 1.26872e9i 0.611998 − 0.611998i −0.331468 − 0.943466i \(-0.607544\pi\)
0.943466 + 0.331468i \(0.107544\pi\)
\(74\) −3.38778e7 + 3.59284e8i −0.0152671 + 0.161912i
\(75\) 1.00938e8 + 2.34984e8i 0.0425353 + 0.0990221i
\(76\) 1.65574e9 − 1.12637e9i 0.653018 − 0.444236i
\(77\) 8.35641e7 0.0308721
\(78\) 3.89630e8 + 3.67392e7i 0.134952 + 0.0127250i
\(79\) − 9.19313e8i − 0.298764i −0.988780 − 0.149382i \(-0.952272\pi\)
0.988780 − 0.149382i \(-0.0477284\pi\)
\(80\) −5.64702e8 + 3.22777e9i −0.172333 + 0.985039i
\(81\) 3.36576e9 0.965291
\(82\) −1.62960e8 + 1.72823e9i −0.0439553 + 0.466158i
\(83\) − 3.85139e9i − 0.977747i −0.872355 − 0.488874i \(-0.837408\pi\)
0.872355 − 0.488874i \(-0.162592\pi\)
\(84\) 2.75400e8 − 1.87350e8i 0.0658518 − 0.0447978i
\(85\) 2.08660e9 + 1.37466e9i 0.470268 + 0.309813i
\(86\) 2.80735e9 + 2.64712e8i 0.596766 + 0.0562707i
\(87\) 1.79949e8 + 1.79949e8i 0.0361038 + 0.0361038i
\(88\) −1.93114e8 + 1.06341e8i −0.0365932 + 0.0201506i
\(89\) 4.02424e9 0.720665 0.360333 − 0.932824i \(-0.382663\pi\)
0.360333 + 0.932824i \(0.382663\pi\)
\(90\) −6.34005e8 − 5.80178e9i −0.107369 − 0.982536i
\(91\) 4.10153e9 − 4.10153e9i 0.657262 − 0.657262i
\(92\) 8.22089e9 − 5.59252e9i 1.24733 − 0.848532i
\(93\) 7.22074e8i 0.103793i
\(94\) −2.92024e9 + 2.41698e9i −0.397906 + 0.329332i
\(95\) 5.10335e9 + 3.36210e9i 0.659535 + 0.434502i
\(96\) −3.98025e8 + 7.83425e8i −0.0488151 + 0.0960818i
\(97\) 5.59860e9 − 5.59860e9i 0.651960 − 0.651960i −0.301505 − 0.953465i \(-0.597489\pi\)
0.953465 + 0.301505i \(0.0974889\pi\)
\(98\) −3.85124e8 + 4.08435e9i −0.0426059 + 0.451848i
\(99\) 2.77650e8 − 2.77650e8i 0.0291959 − 0.0291959i
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.13.5 ✓ 236
5.2 odd 4 80.11.t.a.77.63 yes 236
16.5 even 4 80.11.t.a.53.63 yes 236
80.37 odd 4 inner 80.11.i.a.37.5 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.5 ✓ 236 1.1 even 1 trivial
80.11.i.a.37.5 yes 236 80.37 odd 4 inner
80.11.t.a.53.63 yes 236 16.5 even 4
80.11.t.a.77.63 yes 236 5.2 odd 4