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Properties

Label 80.11.t.a.53.20
Level $80$
Weight $11$
Character 80.53
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(53,80)] 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("80.53"); 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, 1, 3])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 80.t (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 53.20
Character \(\chi\) \(=\) 80.53
Dual form 80.11.t.a.77.20

$q$-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+(-27.7151 - 15.9960i) q^{2} +275.712 q^{3} +(512.253 + 886.664i) q^{4} +(673.877 - 3051.48i) q^{5} +(-7641.37 - 4410.29i) q^{6} +(14140.7 + 14140.7i) q^{7} +(-14.0368 - 32768.0i) q^{8} +16967.8 q^{9} +(-67488.1 + 73792.6i) q^{10} +(-171153. - 171153. i) q^{11} +(141234. + 244463. i) q^{12} +224586. q^{13} +(-165716. - 618108. i) q^{14} +(185796. - 841327. i) q^{15} +(-523769. + 908393. i) q^{16} +(-1.87256e6 + 1.87256e6i) q^{17} +(-470265. - 271418. i) q^{18} +(-2.84791e6 - 2.84791e6i) q^{19} +(3.05083e6 - 965627. i) q^{20} +(3.89877e6 + 3.89877e6i) q^{21} +(2.00575e6 + 7.48130e6i) q^{22} +(-2.66227e6 + 2.66227e6i) q^{23} +(-3870.10 - 9.03451e6i) q^{24} +(-8.85740e6 - 4.11264e6i) q^{25} +(-6.22443e6 - 3.59249e6i) q^{26} -1.16023e7 q^{27} +(-5.29444e6 + 1.97817e7i) q^{28} +(-7.26064e6 - 7.26064e6i) q^{29} +(-1.86073e7 + 2.03455e7i) q^{30} +1.72157e7 q^{31} +(2.90470e7 - 1.67980e7i) q^{32} +(-4.71889e7 - 4.71889e7i) q^{33} +(8.18517e7 - 2.19446e7i) q^{34} +(5.26793e7 - 3.36210e7i) q^{35} +(8.69183e6 + 1.50448e7i) q^{36} +6.33819e7 q^{37} +(3.33748e7 + 1.24485e8i) q^{38} +6.19210e7 q^{39} +(-1.00000e8 - 2.20388e7i) q^{40} -2.11897e8i q^{41} +(-4.56899e7 - 1.70420e8i) q^{42} -4.83978e7i q^{43} +(6.40816e7 - 2.39429e8i) q^{44} +(1.14342e7 - 5.17770e7i) q^{45} +(1.16371e8 - 3.11992e7i) q^{46} +(-2.81322e7 + 2.81322e7i) q^{47} +(-1.44409e8 + 2.50454e8i) q^{48} +1.17446e8i q^{49} +(1.79698e8 + 2.55666e8i) q^{50} +(-5.16286e8 + 5.16286e8i) q^{51} +(1.15045e8 + 1.99132e8i) q^{52} -2.26989e8i q^{53} +(3.21558e8 + 1.85590e8i) q^{54} +(-6.37607e8 + 4.06934e8i) q^{55} +(4.63165e8 - 4.63562e8i) q^{56} +(-7.85201e8 - 7.85201e8i) q^{57} +(8.50878e7 + 3.17371e8i) q^{58} +(-2.67198e8 + 2.67198e8i) q^{59} +(8.41149e8 - 2.66234e8i) q^{60} +(-8.28111e8 + 8.28111e8i) q^{61} +(-4.77136e8 - 2.75384e8i) q^{62} +(2.39938e8 + 2.39938e8i) q^{63} +(-1.07374e9 + 919913. i) q^{64} +(1.51344e8 - 6.85320e8i) q^{65} +(5.53010e8 + 2.06268e9i) q^{66} +2.13326e9i q^{67} +(-2.61956e9 - 7.01106e8i) q^{68} +(-7.34017e8 + 7.34017e8i) q^{69} +(-1.99782e9 + 8.91502e7i) q^{70} -2.82443e9i q^{71} +(-238173. - 5.56002e8i) q^{72} +(-1.81160e9 + 1.81160e9i) q^{73} +(-1.75664e9 - 1.01386e9i) q^{74} +(-2.44209e9 - 1.13390e9i) q^{75} +(1.06629e9 - 3.98399e9i) q^{76} -4.84047e9i q^{77} +(-1.71615e9 - 9.90491e8i) q^{78} -1.75836e9i q^{79} +(2.41898e9 + 2.21042e9i) q^{80} -4.20081e9 q^{81} +(-3.38952e9 + 5.87276e9i) q^{82} -1.20090e9 q^{83} +(-1.45974e9 + 5.45405e9i) q^{84} +(4.45220e9 + 6.97595e9i) q^{85} +(-7.74173e8 + 1.34135e9i) q^{86} +(-2.00184e9 - 2.00184e9i) q^{87} +(-5.60595e9 + 5.61075e9i) q^{88} -1.20611e9 q^{89} +(-1.14513e9 + 1.25210e9i) q^{90} +(3.17582e9 + 3.17582e9i) q^{91} +(-3.72429e9 - 9.96780e8i) q^{92} +4.74658e9 q^{93} +(1.22969e9 - 3.29682e8i) q^{94} +(-1.06095e10 + 6.77119e9i) q^{95} +(8.00859e9 - 4.63139e9i) q^{96} +(1.02975e10 - 1.02975e10i) q^{97} +(1.87867e9 - 3.25503e9i) q^{98} +(-2.90410e9 - 2.90410e9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 236 q - 2 q^{2} - 4 q^{3} - 1220 q^{4} - 2 q^{5} - 4 q^{6} - 69128 q^{8} + 4487724 q^{9} - 2050 q^{10} - 4 q^{11} + 502036 q^{12} - 4 q^{13} - 4 q^{15} - 2041064 q^{16} - 4 q^{17} - 5928762 q^{18} + 5107040 q^{19}+ \cdots + 28071956444 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −27.7151 − 15.9960i −0.866097 − 0.499876i
\(3\) 275.712 1.13462 0.567308 − 0.823506i \(-0.307985\pi\)
0.567308 + 0.823506i \(0.307985\pi\)
\(4\) 512.253 + 886.664i 0.500247 + 0.865883i
\(5\) 673.877 − 3051.48i 0.215641 − 0.976473i
\(6\) −7641.37 − 4410.29i −0.982687 − 0.567167i
\(7\) 14140.7 + 14140.7i 0.841360 + 0.841360i 0.989036 − 0.147675i \(-0.0471791\pi\)
−0.147675 + 0.989036i \(0.547179\pi\)
\(8\) −14.0368 − 32768.0i −0.000428368 − 1.00000i
\(9\) 16967.8 0.287352
\(10\) −67488.1 + 73792.6i −0.674881 + 0.737926i
\(11\) −171153. − 171153.i −1.06273 − 1.06273i −0.997896 − 0.0648308i \(-0.979349\pi\)
−0.0648308 − 0.997896i \(-0.520651\pi\)
\(12\) 141234. + 244463.i 0.567588 + 0.982444i
\(13\) 224586. 0.604876 0.302438 − 0.953169i \(-0.402199\pi\)
0.302438 + 0.953169i \(0.402199\pi\)
\(14\) −165716. − 618108.i −0.308123 − 1.14928i
\(15\) 185796. − 841327.i 0.244669 − 1.10792i
\(16\) −523769. + 908393.i −0.499505 + 0.866311i
\(17\) −1.87256e6 + 1.87256e6i −1.31884 + 1.31884i −0.404139 + 0.914697i \(0.632429\pi\)
−0.914697 + 0.404139i \(0.867571\pi\)
\(18\) −470265. − 271418.i −0.248875 − 0.143640i
\(19\) −2.84791e6 − 2.84791e6i −1.15016 − 1.15016i −0.986520 − 0.163639i \(-0.947677\pi\)
−0.163639 − 0.986520i \(-0.552323\pi\)
\(20\) 3.05083e6 − 965627.i 0.953384 − 0.301758i
\(21\) 3.89877e6 + 3.89877e6i 0.954620 + 0.954620i
\(22\) 2.00575e6 + 7.48130e6i 0.389192 + 1.45166i
\(23\) −2.66227e6 + 2.66227e6i −0.413630 + 0.413630i −0.883001 − 0.469371i \(-0.844481\pi\)
0.469371 + 0.883001i \(0.344481\pi\)
\(24\) −3870.10 − 9.03451e6i −0.000486033 − 1.13462i
\(25\) −8.85740e6 − 4.11264e6i −0.906998 − 0.421135i
\(26\) −6.22443e6 − 3.59249e6i −0.523881 − 0.302363i
\(27\) −1.16023e7 −0.808581
\(28\) −5.29444e6 + 1.97817e7i −0.307631 + 1.14941i
\(29\) −7.26064e6 − 7.26064e6i −0.353985 − 0.353985i 0.507605 − 0.861590i \(-0.330531\pi\)
−0.861590 + 0.507605i \(0.830531\pi\)
\(30\) −1.86073e7 + 2.03455e7i −0.765731 + 0.837262i
\(31\) 1.72157e7 0.601336 0.300668 − 0.953729i \(-0.402790\pi\)
0.300668 + 0.953729i \(0.402790\pi\)
\(32\) 2.90470e7 − 1.67980e7i 0.865668 − 0.500618i
\(33\) −4.71889e7 − 4.71889e7i −1.20579 − 1.20579i
\(34\) 8.18517e7 − 2.19446e7i 1.80150 − 0.482985i
\(35\) 5.26793e7 − 3.36210e7i 1.00300 − 0.640134i
\(36\) 8.69183e6 + 1.50448e7i 0.143747 + 0.248813i
\(37\) 6.33819e7 0.914022 0.457011 − 0.889461i \(-0.348920\pi\)
0.457011 + 0.889461i \(0.348920\pi\)
\(38\) 3.33748e7 + 1.24485e8i 0.421212 + 1.57109i
\(39\) 6.19210e7 0.686302
\(40\) −1.00000e8 − 2.20388e7i −0.976565 − 0.215222i
\(41\) − 2.11897e8i − 1.82897i −0.404620 − 0.914485i \(-0.632596\pi\)
0.404620 − 0.914485i \(-0.367404\pi\)
\(42\) −4.56899e7 − 1.70420e8i −0.349602 − 1.30399i
\(43\) − 4.83978e7i − 0.329218i −0.986359 − 0.164609i \(-0.947364\pi\)
0.986359 − 0.164609i \(-0.0526362\pi\)
\(44\) 6.40816e7 − 2.39429e8i 0.388571 − 1.45182i
\(45\) 1.14342e7 − 5.17770e7i 0.0619648 − 0.280591i
\(46\) 1.16371e8 − 3.11992e7i 0.565008 − 0.151480i
\(47\) −2.81322e7 + 2.81322e7i −0.122663 + 0.122663i −0.765773 − 0.643110i \(-0.777644\pi\)
0.643110 + 0.765773i \(0.277644\pi\)
\(48\) −1.44409e8 + 2.50454e8i −0.566746 + 0.982930i
\(49\) 1.17446e8i 0.415775i
\(50\) 1.79698e8 + 2.55666e8i 0.575033 + 0.818130i
\(51\) −5.16286e8 + 5.16286e8i −1.49637 + 1.49637i
\(52\) 1.15045e8 + 1.99132e8i 0.302588 + 0.523752i
\(53\) − 2.26989e8i − 0.542782i −0.962469 − 0.271391i \(-0.912516\pi\)
0.962469 − 0.271391i \(-0.0874836\pi\)
\(54\) 3.21558e8 + 1.85590e8i 0.700310 + 0.404191i
\(55\) −6.37607e8 + 4.06934e8i −1.26689 + 0.808557i
\(56\) 4.63165e8 − 4.63562e8i 0.841000 − 0.841721i
\(57\) −7.85201e8 − 7.85201e8i −1.30499 − 1.30499i
\(58\) 8.50878e7 + 3.17371e8i 0.129637 + 0.483534i
\(59\) −2.67198e8 + 2.67198e8i −0.373744 + 0.373744i −0.868839 − 0.495095i \(-0.835133\pi\)
0.495095 + 0.868839i \(0.335133\pi\)
\(60\) 8.41149e8 − 2.66234e8i 1.08172 − 0.342380i
\(61\) −8.28111e8 + 8.28111e8i −0.980482 + 0.980482i −0.999813 − 0.0193312i \(-0.993846\pi\)
0.0193312 + 0.999813i \(0.493846\pi\)
\(62\) −4.77136e8 − 2.75384e8i −0.520815 − 0.300594i
\(63\) 2.39938e8 + 2.39938e8i 0.241767 + 0.241767i
\(64\) −1.07374e9 + 919913.i −1.00000 + 0.000856735i
\(65\) 1.51344e8 − 6.85320e8i 0.130436 − 0.590645i
\(66\) 5.53010e8 + 2.06268e9i 0.441584 + 1.64707i
\(67\) 2.13326e9i 1.58004i 0.613079 + 0.790022i \(0.289931\pi\)
−0.613079 + 0.790022i \(0.710069\pi\)
\(68\) −2.61956e9 − 7.01106e8i −1.80170 − 0.482213i
\(69\) −7.34017e8 + 7.34017e8i −0.469311 + 0.469311i
\(70\) −1.99782e9 + 8.91502e7i −1.18868 + 0.0530435i
\(71\) − 2.82443e9i − 1.56545i −0.622367 − 0.782725i \(-0.713829\pi\)
0.622367 − 0.782725i \(-0.286171\pi\)
\(72\) −238173. − 5.56002e8i −0.000123092 − 0.287352i
\(73\) −1.81160e9 + 1.81160e9i −0.873873 + 0.873873i −0.992892 − 0.119019i \(-0.962025\pi\)
0.119019 + 0.992892i \(0.462025\pi\)
\(74\) −1.75664e9 − 1.01386e9i −0.791631 − 0.456898i
\(75\) −2.44209e9 − 1.13390e9i −1.02909 − 0.477826i
\(76\) 1.06629e9 − 3.98399e9i 0.420539 − 1.57127i
\(77\) − 4.84047e9i − 1.78827i
\(78\) −1.71615e9 − 9.90491e8i −0.594404 − 0.343066i
\(79\) − 1.75836e9i − 0.571443i −0.958313 − 0.285721i \(-0.907767\pi\)
0.958313 − 0.285721i \(-0.0922332\pi\)
\(80\) 2.41898e9 + 2.21042e9i 0.738215 + 0.674565i
\(81\) −4.20081e9 −1.20478
\(82\) −3.38952e9 + 5.87276e9i −0.914259 + 1.58406i
\(83\) −1.20090e9 −0.304872 −0.152436 − 0.988313i \(-0.548712\pi\)
−0.152436 + 0.988313i \(0.548712\pi\)
\(84\) −1.45974e9 + 5.45405e9i −0.349043 + 1.30414i
\(85\) 4.45220e9 + 6.97595e9i 1.00341 + 1.57220i
\(86\) −7.74173e8 + 1.34135e9i −0.164568 + 0.285134i
\(87\) −2.00184e9 − 2.00184e9i −0.401637 − 0.401637i
\(88\) −5.60595e9 + 5.61075e9i −1.06227 + 1.06318i
\(89\) −1.20611e9 −0.215991 −0.107995 − 0.994151i \(-0.534443\pi\)
−0.107995 + 0.994151i \(0.534443\pi\)
\(90\) −1.14513e9 + 1.25210e9i −0.193928 + 0.212045i
\(91\) 3.17582e9 + 3.17582e9i 0.508919 + 0.508919i
\(92\) −3.72429e9 − 9.96780e8i −0.565073 − 0.151238i
\(93\) 4.74658e9 0.682285
\(94\) 1.22969e9 − 3.29682e8i 0.167554 − 0.0449217i
\(95\) −1.06095e10 + 6.77119e9i −1.37112 + 0.875078i
\(96\) 8.00859e9 − 4.63139e9i 0.982200 − 0.568009i
\(97\) 1.02975e10 − 1.02975e10i 1.19915 − 1.19915i 0.224723 − 0.974423i \(-0.427852\pi\)
0.974423 − 0.224723i \(-0.0721478\pi\)
\(98\) 1.87867e9 − 3.25503e9i 0.207836 − 0.360101i
\(99\) −2.90410e9 − 2.90410e9i −0.305377 − 0.305377i
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.t.a.53.20 yes 236
5.2 odd 4 80.11.i.a.37.81 yes 236
16.13 even 4 80.11.i.a.13.81 ✓ 236
80.77 odd 4 inner 80.11.t.a.77.20 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.81 ✓ 236 16.13 even 4
80.11.i.a.37.81 yes 236 5.2 odd 4
80.11.t.a.53.20 yes 236 1.1 even 1 trivial
80.11.t.a.77.20 yes 236 80.77 odd 4 inner