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Properties

Label 80.11.i.a.13.14
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 13.14
Character \(\chi\) \(=\) 80.13
Dual form 80.11.i.a.37.14

$q$-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+(-30.7560 - 8.83561i) q^{2} -445.379i q^{3} +(867.864 + 543.496i) q^{4} +(2471.93 + 1911.85i) q^{5} +(-3935.20 + 13698.1i) q^{6} +(15417.6 + 15417.6i) q^{7} +(-21889.9 - 24383.9i) q^{8} -139314. q^{9} +(-59134.5 - 80641.9i) q^{10} +(72783.4 - 72783.4i) q^{11} +(242062. - 386528. i) q^{12} -137857. i q^{13} +(-337959. - 610406. i) q^{14} +(851497. - 1.10095e6i) q^{15} +(457800. + 943362. i) q^{16} +(-410060. + 410060. i) q^{17} +(4.28473e6 + 1.23092e6i) q^{18} +(-2.06510e6 + 2.06510e6i) q^{19} +(1.10622e6 + 3.00271e6i) q^{20} +(6.86666e6 - 6.86666e6i) q^{21} +(-2.88161e6 + 1.59544e6i) q^{22} +(-6.32399e6 + 6.32399e6i) q^{23} +(-1.08601e7 + 9.74931e6i) q^{24} +(2.45530e6 + 9.45193e6i) q^{25} +(-1.21805e6 + 4.23993e6i) q^{26} +3.57481e7i q^{27} +(5.00096e6 + 2.17597e7i) q^{28} +(2.79262e7 - 2.79262e7i) q^{29} +(-3.59162e7 + 2.63373e7i) q^{30} +4.48583e7 q^{31} +(-5.74492e6 - 3.30590e7i) q^{32} +(-3.24162e7 - 3.24162e7i) q^{33} +(1.62350e7 - 8.98869e6i) q^{34} +(8.63517e6 + 6.75872e7i) q^{35} +(-1.20905e8 - 7.57164e7i) q^{36} +1.01461e8i q^{37} +(8.17605e7 - 4.52677e7i) q^{38} -6.13987e7 q^{39} +(-7.49216e6 - 1.02126e8i) q^{40} +1.26196e8i q^{41} +(-2.71862e8 + 1.50520e8i) q^{42} -2.17053e7 q^{43} +(1.02724e8 - 2.36086e7i) q^{44} +(-3.44374e8 - 2.66346e8i) q^{45} +(2.50377e8 - 1.38624e8i) q^{46} +(-1.95932e8 + 1.95932e8i) q^{47} +(4.20153e8 - 2.03894e8i) q^{48} +1.92927e8i q^{49} +(7.99834e6 - 3.12398e8i) q^{50} +(1.82632e8 + 1.82632e8i) q^{51} +(7.49248e7 - 1.19641e8i) q^{52} +5.89050e8 q^{53} +(3.15857e8 - 1.09947e9i) q^{54} +(3.19067e8 - 4.07650e7i) q^{55} +(3.84509e7 - 7.13429e8i) q^{56} +(9.19751e8 + 9.19751e8i) q^{57} +(-1.10564e9 + 6.12152e8i) q^{58} +(594374. + 594374. i) q^{59} +(1.33734e9 - 4.92688e8i) q^{60} +(7.97824e7 + 7.97824e7i) q^{61} +(-1.37966e9 - 3.96350e8i) q^{62} +(-2.14787e9 - 2.14787e9i) q^{63} +(-1.15405e8 + 1.06752e9i) q^{64} +(2.63562e8 - 3.40774e8i) q^{65} +(7.10576e8 + 1.28341e9i) q^{66} +1.75301e9 q^{67} +(-5.78743e8 + 1.33010e8i) q^{68} +(2.81657e9 + 2.81657e9i) q^{69} +(3.31591e8 - 2.15501e9i) q^{70} -2.40024e8i q^{71} +(3.04956e9 + 3.39700e9i) q^{72} +(-1.35240e9 + 1.35240e9i) q^{73} +(8.96473e8 - 3.12055e9i) q^{74} +(4.20969e9 - 1.09354e9i) q^{75} +(-2.91459e9 + 6.69851e8i) q^{76} +2.24428e9 q^{77} +(1.88838e9 + 5.42495e8i) q^{78} +7.00094e8i q^{79} +(-6.71912e8 + 3.20717e9i) q^{80} +7.69515e9 q^{81} +(1.11502e9 - 3.88127e9i) q^{82} +1.07400e9i q^{83} +(9.69133e9 - 2.22732e9i) q^{84} +(-1.79762e9 + 2.29669e8i) q^{85} +(6.67569e8 + 1.91780e8i) q^{86} +(-1.24377e10 - 1.24377e10i) q^{87} +(-3.36796e9 - 1.81520e8i) q^{88} +4.45073e9 q^{89} +(8.23824e9 + 1.12345e10i) q^{90} +(2.12542e9 - 2.12542e9i) q^{91} +(-8.92542e9 + 2.05130e9i) q^{92} -1.99789e10i q^{93} +(7.75727e9 - 4.29491e9i) q^{94} +(-9.05293e9 + 1.15663e9i) q^{95} +(-1.47238e10 + 2.55867e9i) q^{96} +(3.85397e9 - 3.85397e9i) q^{97} +(1.70463e9 - 5.93366e9i) q^{98} +(-1.01397e10 + 1.01397e10i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −30.7560 − 8.83561i −0.961125 − 0.276113i
\(3\) − 445.379i − 1.83284i −0.400223 − 0.916418i \(-0.631067\pi\)
0.400223 − 0.916418i \(-0.368933\pi\)
\(4\) 867.864 + 543.496i 0.847523 + 0.530758i
\(5\) 2471.93 + 1911.85i 0.791019 + 0.611791i
\(6\) −3935.20 + 13698.1i −0.506069 + 1.76158i
\(7\) 15417.6 + 15417.6i 0.917330 + 0.917330i 0.996834 − 0.0795049i \(-0.0253339\pi\)
−0.0795049 + 0.996834i \(0.525334\pi\)
\(8\) −21889.9 − 24383.9i −0.668027 − 0.744137i
\(9\) −139314. −2.35929
\(10\) −59134.5 − 80641.9i −0.591345 − 0.806419i
\(11\) 72783.4 − 72783.4i 0.451928 − 0.451928i −0.444066 − 0.895994i \(-0.646465\pi\)
0.895994 + 0.444066i \(0.146465\pi\)
\(12\) 242062. − 386528.i 0.972792 − 1.55337i
\(13\) − 137857.i − 0.371289i −0.982617 − 0.185645i \(-0.940563\pi\)
0.982617 − 0.185645i \(-0.0594373\pi\)
\(14\) −337959. − 610406.i −0.628382 − 1.13496i
\(15\) 851497. − 1.10095e6i 1.12131 − 1.44981i
\(16\) 457800. + 943362.i 0.436592 + 0.899660i
\(17\) −410060. + 410060.i −0.288804 + 0.288804i −0.836607 − 0.547803i \(-0.815464\pi\)
0.547803 + 0.836607i \(0.315464\pi\)
\(18\) 4.28473e6 + 1.23092e6i 2.26757 + 0.651429i
\(19\) −2.06510e6 + 2.06510e6i −0.834012 + 0.834012i −0.988063 − 0.154051i \(-0.950768\pi\)
0.154051 + 0.988063i \(0.450768\pi\)
\(20\) 1.10622e6 + 3.00271e6i 0.345694 + 0.938347i
\(21\) 6.86666e6 − 6.86666e6i 1.68131 − 1.68131i
\(22\) −2.88161e6 + 1.59544e6i −0.559142 + 0.309576i
\(23\) −6.32399e6 + 6.32399e6i −0.982543 + 0.982543i −0.999850 − 0.0173070i \(-0.994491\pi\)
0.0173070 + 0.999850i \(0.494491\pi\)
\(24\) −1.08601e7 + 9.74931e6i −1.36388 + 1.22438i
\(25\) 2.45530e6 + 9.45193e6i 0.251423 + 0.967877i
\(26\) −1.21805e6 + 4.23993e6i −0.102518 + 0.356855i
\(27\) 3.57481e7i 2.49135i
\(28\) 5.00096e6 + 2.17597e7i 0.290578 + 1.26434i
\(29\) 2.79262e7 − 2.79262e7i 1.36151 − 1.36151i 0.489518 − 0.871993i \(-0.337173\pi\)
0.871993 − 0.489518i \(-0.162827\pi\)
\(30\) −3.59162e7 + 2.63373e7i −1.47803 + 1.08384i
\(31\) 4.48583e7 1.56687 0.783437 − 0.621472i \(-0.213465\pi\)
0.783437 + 0.621472i \(0.213465\pi\)
\(32\) −5.74492e6 − 3.30590e7i −0.171212 − 0.985234i
\(33\) −3.24162e7 − 3.24162e7i −0.828309 − 0.828309i
\(34\) 1.62350e7 − 8.98869e6i 0.357319 − 0.197834i
\(35\) 8.63517e6 + 6.75872e7i 0.164411 + 1.28684i
\(36\) −1.20905e8 − 7.57164e7i −1.99955 − 1.25221i
\(37\) 1.01461e8i 1.46316i 0.681755 + 0.731581i \(0.261217\pi\)
−0.681755 + 0.731581i \(0.738783\pi\)
\(38\) 8.17605e7 − 4.52677e7i 1.03187 − 0.571309i
\(39\) −6.13987e7 −0.680512
\(40\) −7.49216e6 − 1.02126e8i −0.0731657 − 0.997320i
\(41\) 1.26196e8i 1.08924i 0.838682 + 0.544622i \(0.183327\pi\)
−0.838682 + 0.544622i \(0.816673\pi\)
\(42\) −2.71862e8 + 1.50520e8i −2.08019 + 1.15172i
\(43\) −2.17053e7 −0.147647 −0.0738234 − 0.997271i \(-0.523520\pi\)
−0.0738234 + 0.997271i \(0.523520\pi\)
\(44\) 1.02724e8 − 2.36086e7i 0.622883 − 0.143155i
\(45\) −3.44374e8 − 2.66346e8i −1.86624 − 1.44339i
\(46\) 2.50377e8 − 1.38624e8i 1.21564 − 0.673054i
\(47\) −1.95932e8 + 1.95932e8i −0.854312 + 0.854312i −0.990661 − 0.136349i \(-0.956463\pi\)
0.136349 + 0.990661i \(0.456463\pi\)
\(48\) 4.20153e8 − 2.03894e8i 1.64893 − 0.800201i
\(49\) 1.92927e8i 0.682987i
\(50\) 7.99834e6 − 3.12398e8i 0.0255947 − 0.999672i
\(51\) 1.82632e8 + 1.82632e8i 0.529330 + 0.529330i
\(52\) 7.49248e7 − 1.19641e8i 0.197065 − 0.314676i
\(53\) 5.89050e8 1.40855 0.704275 − 0.709927i \(-0.251272\pi\)
0.704275 + 0.709927i \(0.251272\pi\)
\(54\) 3.15857e8 − 1.09947e9i 0.687893 − 2.39450i
\(55\) 3.19067e8 − 4.07650e7i 0.633969 − 0.0809980i
\(56\) 3.84509e7 − 7.13429e8i 0.0698179 − 1.29542i
\(57\) 9.19751e8 + 9.19751e8i 1.52861 + 1.52861i
\(58\) −1.10564e9 + 6.12152e8i −1.68451 + 0.932652i
\(59\) 594374. + 594374.i 0.000831380 + 0.000831380i 0.707522 − 0.706691i \(-0.249813\pi\)
−0.706691 + 0.707522i \(0.749813\pi\)
\(60\) 1.33734e9 − 4.92688e8i 1.71984 − 0.633601i
\(61\) 7.97824e7 + 7.97824e7i 0.0944622 + 0.0944622i 0.752759 − 0.658297i \(-0.228723\pi\)
−0.658297 + 0.752759i \(0.728723\pi\)
\(62\) −1.37966e9 − 3.96350e8i −1.50596 − 0.432634i
\(63\) −2.14787e9 − 2.14787e9i −2.16424 − 2.16424i
\(64\) −1.15405e8 + 1.06752e9i −0.107480 + 0.994207i
\(65\) 2.63562e8 − 3.40774e8i 0.227152 − 0.293697i
\(66\) 7.10576e8 + 1.28341e9i 0.567402 + 1.02482i
\(67\) 1.75301e9 1.29840 0.649202 − 0.760616i \(-0.275103\pi\)
0.649202 + 0.760616i \(0.275103\pi\)
\(68\) −5.78743e8 + 1.33010e8i −0.398053 + 0.0914831i
\(69\) 2.81657e9 + 2.81657e9i 1.80084 + 1.80084i
\(70\) 3.31591e8 − 2.15501e9i 0.197293 − 1.28221i
\(71\) − 2.40024e8i − 0.133034i −0.997785 − 0.0665171i \(-0.978811\pi\)
0.997785 − 0.0665171i \(-0.0211887\pi\)
\(72\) 3.04956e9 + 3.39700e9i 1.57607 + 1.75563i
\(73\) −1.35240e9 + 1.35240e9i −0.652364 + 0.652364i −0.953562 − 0.301198i \(-0.902614\pi\)
0.301198 + 0.953562i \(0.402614\pi\)
\(74\) 8.96473e8 − 3.12055e9i 0.403998 − 1.40628i
\(75\) 4.20969e9 − 1.09354e9i 1.77396 − 0.460816i
\(76\) −2.91459e9 + 6.69851e8i −1.14950 + 0.264186i
\(77\) 2.24428e9 0.829133
\(78\) 1.88838e9 + 5.42495e8i 0.654057 + 0.187898i
\(79\) 7.00094e8i 0.227521i 0.993508 + 0.113760i \(0.0362896\pi\)
−0.993508 + 0.113760i \(0.963710\pi\)
\(80\) −6.71912e8 + 3.20717e9i −0.205051 + 0.978751i
\(81\) 7.69515e9 2.20695
\(82\) 1.11502e9 − 3.88127e9i 0.300754 − 1.04690i
\(83\) 1.07400e9i 0.272656i 0.990664 + 0.136328i \(0.0435301\pi\)
−0.990664 + 0.136328i \(0.956470\pi\)
\(84\) 9.69133e9 − 2.22732e9i 2.31732 − 0.532582i
\(85\) −1.79762e9 + 2.29669e8i −0.405137 + 0.0517617i
\(86\) 6.67569e8 + 1.91780e8i 0.141907 + 0.0407672i
\(87\) −1.24377e10 − 1.24377e10i −2.49543 − 2.49543i
\(88\) −3.36796e9 − 1.81520e8i −0.638196 − 0.0343962i
\(89\) 4.45073e9 0.797042 0.398521 − 0.917159i \(-0.369524\pi\)
0.398521 + 0.917159i \(0.369524\pi\)
\(90\) 8.23824e9 + 1.12345e10i 1.39515 + 1.90257i
\(91\) 2.12542e9 − 2.12542e9i 0.340595 − 0.340595i
\(92\) −8.92542e9 + 2.05130e9i −1.35422 + 0.311236i
\(93\) − 1.99789e10i − 2.87182i
\(94\) 7.75727e9 − 4.29491e9i 1.05699 − 0.585214i
\(95\) −9.05293e9 + 1.15663e9i −1.16996 + 0.149478i
\(96\) −1.47238e10 + 2.55867e9i −1.80577 + 0.313803i
\(97\) 3.85397e9 − 3.85397e9i 0.448796 − 0.448796i −0.446158 − 0.894954i \(-0.647208\pi\)
0.894954 + 0.446158i \(0.147208\pi\)
\(98\) 1.70463e9 − 5.93366e9i 0.188581 − 0.656436i
\(99\) −1.01397e10 + 1.01397e10i −1.06623 + 1.06623i
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.13.14 ✓ 236
5.2 odd 4 80.11.t.a.77.70 yes 236
16.5 even 4 80.11.t.a.53.70 yes 236
80.37 odd 4 inner 80.11.i.a.37.14 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.14 ✓ 236 1.1 even 1 trivial
80.11.i.a.37.14 yes 236 80.37 odd 4 inner
80.11.t.a.53.70 yes 236 16.5 even 4
80.11.t.a.77.70 yes 236 5.2 odd 4