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Properties

Label 80.11.i.a.13.16
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 13.16
Character \(\chi\) \(=\) 80.13
Dual form 80.11.i.a.37.16

$q$-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+(-29.2822 + 12.9055i) q^{2} -135.733i q^{3} +(690.895 - 755.804i) q^{4} +(-2255.45 + 2163.00i) q^{5} +(1751.71 + 3974.57i) q^{6} +(7878.52 + 7878.52i) q^{7} +(-10476.9 + 31048.0i) q^{8} +40625.4 q^{9} +(38129.9 - 92445.2i) q^{10} +(200812. - 200812. i) q^{11} +(-102588. - 93777.5i) q^{12} +215252. i q^{13} +(-332377. - 129024. i) q^{14} +(293591. + 306140. i) q^{15} +(-93904.5 - 1.04436e6i) q^{16} +(1317.75 - 1317.75i) q^{17} +(-1.18960e6 + 524293. i) q^{18} +(-689836. + 689836. i) q^{19} +(76527.5 + 3.19908e6i) q^{20} +(1.06938e6 - 1.06938e6i) q^{21} +(-3.28864e6 + 8.47181e6i) q^{22} +(-7.89039e6 + 7.89039e6i) q^{23} +(4.21425e6 + 1.42206e6i) q^{24} +(408477. - 9.75708e6i) q^{25} +(-2.77793e6 - 6.30304e6i) q^{26} -1.35291e7i q^{27} +(1.13979e7 - 511391. i) q^{28} +(1.04972e7 - 1.04972e7i) q^{29} +(-1.25479e7 - 5.17549e6i) q^{30} -1.53099e7 q^{31} +(1.62278e7 + 2.93694e7i) q^{32} +(-2.72569e7 - 2.72569e7i) q^{33} +(-21580.4 + 55592.9i) q^{34} +(-3.48109e7 - 728354. i) q^{35} +(2.80679e7 - 3.07049e7i) q^{36} +2.35995e7i q^{37} +(1.12972e7 - 2.91026e7i) q^{38} +2.92168e7 q^{39} +(-4.35268e7 - 9.26886e7i) q^{40} -5.93876e6i q^{41} +(-1.75129e7 + 4.51147e7i) q^{42} +6.26569e7 q^{43} +(-1.30346e7 - 2.90515e8i) q^{44} +(-9.16286e7 + 8.78729e7i) q^{45} +(1.29218e8 - 3.32878e8i) q^{46} +(4.45484e7 - 4.45484e7i) q^{47} +(-1.41755e8 + 1.27460e7i) q^{48} -1.58333e8i q^{49} +(1.13959e8 + 2.90980e8i) q^{50} +(-178863. - 178863. i) q^{51} +(1.62688e8 + 1.48716e8i) q^{52} +3.00939e8 q^{53} +(1.74601e8 + 3.96163e8i) q^{54} +(-1.85647e7 + 8.87279e8i) q^{55} +(-3.27155e8 + 1.62070e8i) q^{56} +(9.36338e7 + 9.36338e7i) q^{57} +(-1.71909e8 + 4.42853e8i) q^{58} +(4.08460e8 + 4.08460e8i) q^{59} +(4.34223e8 - 1.03873e7i) q^{60} +(8.21636e8 + 8.21636e8i) q^{61} +(4.48309e8 - 1.97583e8i) q^{62} +(3.20069e8 + 3.20069e8i) q^{63} +(-8.54212e8 - 6.50571e8i) q^{64} +(-4.65589e8 - 4.85489e8i) q^{65} +(1.14991e9 + 4.46378e8i) q^{66} -3.81041e8 q^{67} +(-85534.6 - 1.90639e6i) q^{68} +(1.07099e9 + 1.07099e9i) q^{69} +(1.02874e9 - 4.27925e8i) q^{70} +2.91240e9i q^{71} +(-4.25628e8 + 1.26134e9i) q^{72} +(1.43041e9 - 1.43041e9i) q^{73} +(-3.04564e8 - 6.91045e8i) q^{74} +(-1.32436e9 - 5.54439e7i) q^{75} +(4.47769e7 + 9.97986e8i) q^{76} +3.16421e9 q^{77} +(-8.55533e8 + 3.77058e8i) q^{78} -2.31206e9i q^{79} +(2.47075e9 + 2.15239e9i) q^{80} +5.62535e8 q^{81} +(7.66428e7 + 1.73900e8i) q^{82} +2.96728e9i q^{83} +(-6.94128e7 - 1.54707e9i) q^{84} +(-121823. + 5.82242e6i) q^{85} +(-1.83473e9 + 8.08620e8i) q^{86} +(-1.42482e9 - 1.42482e9i) q^{87} +(4.13093e9 + 8.33870e9i) q^{88} +5.82148e9 q^{89} +(1.54904e9 - 3.75563e9i) q^{90} +(-1.69586e9 + 1.69586e9i) q^{91} +(5.12162e8 + 1.14150e10i) q^{92} +2.07807e9i q^{93} +(-7.29555e8 + 1.87940e9i) q^{94} +(6.37740e7 - 3.04801e9i) q^{95} +(3.98640e9 - 2.20265e9i) q^{96} +(3.51373e9 - 3.51373e9i) q^{97} +(2.04337e9 + 4.63634e9i) q^{98} +(8.15809e9 - 8.15809e9i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −29.2822 + 12.9055i −0.915069 + 0.403298i
\(3\) − 135.733i − 0.558574i −0.960208 − 0.279287i \(-0.909902\pi\)
0.960208 − 0.279287i \(-0.0900980\pi\)
\(4\) 690.895 − 755.804i 0.674702 − 0.738090i
\(5\) −2255.45 + 2163.00i −0.721744 + 0.692160i
\(6\) 1751.71 + 3974.57i 0.225271 + 0.511133i
\(7\) 7878.52 + 7878.52i 0.468764 + 0.468764i 0.901514 − 0.432750i \(-0.142457\pi\)
−0.432750 + 0.901514i \(0.642457\pi\)
\(8\) −10476.9 + 31048.0i −0.319729 + 0.947509i
\(9\) 40625.4 0.687996
\(10\) 38129.9 − 92445.2i 0.381299 − 0.924452i
\(11\) 200812. − 200812.i 1.24689 − 1.24689i 0.289798 − 0.957088i \(-0.406412\pi\)
0.957088 − 0.289798i \(-0.0935882\pi\)
\(12\) −102588. − 93777.5i −0.412278 − 0.376871i
\(13\) 215252.i 0.579735i 0.957067 + 0.289867i \(0.0936112\pi\)
−0.957067 + 0.289867i \(0.906389\pi\)
\(14\) −332377. − 129024.i −0.618003 − 0.239900i
\(15\) 293591. + 306140.i 0.386622 + 0.403147i
\(16\) −93904.5 − 1.04436e6i −0.0895543 − 0.995982i
\(17\) 1317.75 − 1317.75i 0.000928087 − 0.000928087i −0.706643 − 0.707571i \(-0.749791\pi\)
0.707571 + 0.706643i \(0.249791\pi\)
\(18\) −1.18960e6 + 524293.i −0.629563 + 0.277467i
\(19\) −689836. + 689836.i −0.278598 + 0.278598i −0.832549 − 0.553951i \(-0.813119\pi\)
0.553951 + 0.832549i \(0.313119\pi\)
\(20\) 76527.5 + 3.19908e6i 0.0239148 + 0.999714i
\(21\) 1.06938e6 − 1.06938e6i 0.261839 − 0.261839i
\(22\) −3.28864e6 + 8.47181e6i −0.638120 + 1.64385i
\(23\) −7.89039e6 + 7.89039e6i −1.22591 + 1.22591i −0.260416 + 0.965496i \(0.583860\pi\)
−0.965496 + 0.260416i \(0.916140\pi\)
\(24\) 4.21425e6 + 1.42206e6i 0.529254 + 0.178592i
\(25\) 408477. − 9.75708e6i 0.0418280 − 0.999125i
\(26\) −2.77793e6 − 6.30304e6i −0.233806 − 0.530497i
\(27\) − 1.35291e7i − 0.942870i
\(28\) 1.13979e7 − 511391.i 0.662267 − 0.0297141i
\(29\) 1.04972e7 − 1.04972e7i 0.511780 − 0.511780i −0.403291 − 0.915072i \(-0.632134\pi\)
0.915072 + 0.403291i \(0.132134\pi\)
\(30\) −1.25479e7 − 5.17549e6i −0.516374 − 0.212983i
\(31\) −1.53099e7 −0.534767 −0.267384 − 0.963590i \(-0.586159\pi\)
−0.267384 + 0.963590i \(0.586159\pi\)
\(32\) 1.62278e7 + 2.93694e7i 0.483626 + 0.875275i
\(33\) −2.72569e7 − 2.72569e7i −0.696478 − 0.696478i
\(34\) −21580.4 + 55592.9i −0.000474968 + 0.00122356i
\(35\) −3.48109e7 − 728354.i −0.662788 − 0.0138676i
\(36\) 2.80679e7 − 3.07049e7i 0.464192 − 0.507803i
\(37\) 2.35995e7i 0.340325i 0.985416 + 0.170163i \(0.0544293\pi\)
−0.985416 + 0.170163i \(0.945571\pi\)
\(38\) 1.12972e7 − 2.91026e7i 0.142578 − 0.367294i
\(39\) 2.92168e7 0.323825
\(40\) −4.35268e7 − 9.26886e7i −0.425066 − 0.905162i
\(41\) − 5.93876e6i − 0.0512597i −0.999672 − 0.0256299i \(-0.991841\pi\)
0.999672 − 0.0256299i \(-0.00815913\pi\)
\(42\) −1.75129e7 + 4.51147e7i −0.134002 + 0.345200i
\(43\) 6.26569e7 0.426213 0.213106 − 0.977029i \(-0.431642\pi\)
0.213106 + 0.977029i \(0.431642\pi\)
\(44\) −1.30346e7 − 2.90515e8i −0.0790378 − 1.76159i
\(45\) −9.16286e7 + 8.78729e7i −0.496556 + 0.476203i
\(46\) 1.29218e8 − 3.32878e8i 0.627387 − 1.61620i
\(47\) 4.45484e7 − 4.45484e7i 0.194242 − 0.194242i −0.603284 − 0.797526i \(-0.706142\pi\)
0.797526 + 0.603284i \(0.206142\pi\)
\(48\) −1.41755e8 + 1.27460e7i −0.556329 + 0.0500227i
\(49\) − 1.58333e8i − 0.560520i
\(50\) 1.13959e8 + 2.90980e8i 0.364669 + 0.931137i
\(51\) −178863. − 178863.i −0.000518405 − 0.000518405i
\(52\) 1.62688e8 + 1.48716e8i 0.427897 + 0.391148i
\(53\) 3.00939e8 0.719613 0.359806 − 0.933027i \(-0.382843\pi\)
0.359806 + 0.933027i \(0.382843\pi\)
\(54\) 1.74601e8 + 3.96163e8i 0.380257 + 0.862791i
\(55\) −1.85647e7 + 8.87279e8i −0.0368871 + 1.76298i
\(56\) −3.27155e8 + 1.62070e8i −0.594036 + 0.294281i
\(57\) 9.36338e7 + 9.36338e7i 0.155618 + 0.155618i
\(58\) −1.71909e8 + 4.42853e8i −0.261914 + 0.674714i
\(59\) 4.08460e8 + 4.08460e8i 0.571333 + 0.571333i 0.932501 − 0.361168i \(-0.117622\pi\)
−0.361168 + 0.932501i \(0.617622\pi\)
\(60\) 4.34223e8 − 1.03873e7i 0.558414 − 0.0133582i
\(61\) 8.21636e8 + 8.21636e8i 0.972815 + 0.972815i 0.999640 − 0.0268256i \(-0.00853986\pi\)
−0.0268256 + 0.999640i \(0.508540\pi\)
\(62\) 4.48309e8 − 1.97583e8i 0.489349 − 0.215670i
\(63\) 3.20069e8 + 3.20069e8i 0.322508 + 0.322508i
\(64\) −8.54212e8 − 6.50571e8i −0.795547 − 0.605892i
\(65\) −4.65589e8 − 4.85489e8i −0.401270 − 0.418420i
\(66\) 1.14991e9 + 4.46378e8i 0.918213 + 0.356437i
\(67\) −3.81041e8 −0.282226 −0.141113 − 0.989993i \(-0.545068\pi\)
−0.141113 + 0.989993i \(0.545068\pi\)
\(68\) −85534.6 − 1.90639e6i −5.88298e−5 − 0.00131119i
\(69\) 1.07099e9 + 1.07099e9i 0.684762 + 0.684762i
\(70\) 1.02874e9 − 4.27925e8i 0.612089 − 0.254611i
\(71\) 2.91240e9i 1.61421i 0.590410 + 0.807103i \(0.298966\pi\)
−0.590410 + 0.807103i \(0.701034\pi\)
\(72\) −4.25628e8 + 1.26134e9i −0.219972 + 0.651882i
\(73\) 1.43041e9 − 1.43041e9i 0.689994 − 0.689994i −0.272236 − 0.962230i \(-0.587763\pi\)
0.962230 + 0.272236i \(0.0877632\pi\)
\(74\) −3.04564e8 − 6.91045e8i −0.137252 − 0.311421i
\(75\) −1.32436e9 − 5.54439e7i −0.558085 − 0.0233640i
\(76\) 4.47769e7 + 9.97986e8i 0.0176598 + 0.393601i
\(77\) 3.16421e9 1.16899
\(78\) −8.55533e8 + 3.77058e8i −0.296322 + 0.130598i
\(79\) − 2.31206e9i − 0.751386i −0.926744 − 0.375693i \(-0.877405\pi\)
0.926744 − 0.375693i \(-0.122595\pi\)
\(80\) 2.47075e9 + 2.15239e9i 0.754015 + 0.656858i
\(81\) 5.62535e8 0.161333
\(82\) 7.66428e7 + 1.73900e8i 0.0206729 + 0.0469062i
\(83\) 2.96728e9i 0.753300i 0.926356 + 0.376650i \(0.122924\pi\)
−0.926356 + 0.376650i \(0.877076\pi\)
\(84\) −6.94128e7 − 1.54707e9i −0.0165975 − 0.369925i
\(85\) −121823. + 5.82242e6i −2.74559e−5 + 0.00131223i
\(86\) −1.83473e9 + 8.08620e8i −0.390014 + 0.171891i
\(87\) −1.42482e9 − 1.42482e9i −0.285867 − 0.285867i
\(88\) 4.13093e9 + 8.33870e9i 0.782770 + 1.58010i
\(89\) 5.82148e9 1.04252 0.521259 − 0.853399i \(-0.325463\pi\)
0.521259 + 0.853399i \(0.325463\pi\)
\(90\) 1.54904e9 − 3.75563e9i 0.262332 − 0.636019i
\(91\) −1.69586e9 + 1.69586e9i −0.271759 + 0.271759i
\(92\) 5.12162e8 + 1.14150e10i 0.0777084 + 1.73196i
\(93\) 2.07807e9i 0.298707i
\(94\) −7.29555e8 + 1.87940e9i −0.0994075 + 0.256082i
\(95\) 6.37740e7 − 3.04801e9i 0.00824187 − 0.393911i
\(96\) 3.98640e9 − 2.20265e9i 0.488905 − 0.270140i
\(97\) 3.51373e9 − 3.51373e9i 0.409175 − 0.409175i −0.472276 − 0.881451i \(-0.656567\pi\)
0.881451 + 0.472276i \(0.156567\pi\)
\(98\) 2.04337e9 + 4.63634e9i 0.226056 + 0.512914i
\(99\) 8.15809e9 − 8.15809e9i 0.857852 − 0.857852i
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.13.16 ✓ 236
5.2 odd 4 80.11.t.a.77.44 yes 236
16.5 even 4 80.11.t.a.53.44 yes 236
80.37 odd 4 inner 80.11.i.a.37.16 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.16 ✓ 236 1.1 even 1 trivial
80.11.i.a.37.16 yes 236 80.37 odd 4 inner
80.11.t.a.53.44 yes 236 16.5 even 4
80.11.t.a.77.44 yes 236 5.2 odd 4