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Properties

Label 80.11.t.a.53.16
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 53.16
Character \(\chi\) \(=\) 80.53
Dual form 80.11.t.a.77.16

$q$-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+(-29.2793 - 12.9122i) q^{2} -227.008 q^{3} +(690.550 + 756.119i) q^{4} +(2875.48 - 1223.61i) q^{5} +(6646.63 + 2931.18i) q^{6} +(-11637.9 - 11637.9i) q^{7} +(-10455.6 - 31055.1i) q^{8} -7516.24 q^{9} +(-99991.5 - 1302.28i) q^{10} +(67431.3 + 67431.3i) q^{11} +(-156761. - 171645. i) q^{12} +136243. q^{13} +(190478. + 491018. i) q^{14} +(-652758. + 277770. i) q^{15} +(-94856.3 + 1.04428e6i) q^{16} +(-716000. + 716000. i) q^{17} +(220070. + 97051.2i) q^{18} +(-1.93473e6 - 1.93473e6i) q^{19} +(2.91086e6 + 1.32924e6i) q^{20} +(2.64189e6 + 2.64189e6i) q^{21} +(-1.10365e6 - 2.84502e6i) q^{22} +(7.82927e6 - 7.82927e6i) q^{23} +(2.37352e6 + 7.04977e6i) q^{24} +(6.77116e6 - 7.03696e6i) q^{25} +(-3.98909e6 - 1.75919e6i) q^{26} +1.51109e7 q^{27} +(763080. - 1.68361e7i) q^{28} +(-2.24558e7 - 2.24558e7i) q^{29} +(2.26989e7 + 295627. i) q^{30} +6.83661e6 q^{31} +(1.62612e7 - 2.93508e7i) q^{32} +(-1.53075e7 - 1.53075e7i) q^{33} +(3.02091e7 - 1.17188e7i) q^{34} +(-4.77047e7 - 1.92242e7i) q^{35} +(-5.19035e6 - 5.68318e6i) q^{36} +8.63666e7 q^{37} +(3.16658e7 + 8.16289e7i) q^{38} -3.09282e7 q^{39} +(-6.80645e7 - 7.65048e7i) q^{40} +2.13292e8i q^{41} +(-4.32400e7 - 1.11465e8i) q^{42} -1.88731e8i q^{43} +(-4.42138e6 + 9.75508e7i) q^{44} +(-2.16128e7 + 9.19698e6i) q^{45} +(-3.30329e8 + 1.28142e8i) q^{46} +(-4.06607e7 + 4.06607e7i) q^{47} +(2.15332e7 - 2.37059e8i) q^{48} -1.15956e7i q^{49} +(-2.89117e8 + 1.18606e8i) q^{50} +(1.62538e8 - 1.62538e8i) q^{51} +(9.40825e7 + 1.03016e8i) q^{52} +9.52654e7i q^{53} +(-4.42435e8 - 1.95114e8i) q^{54} +(2.76407e8 + 1.11388e8i) q^{55} +(-2.39734e8 + 4.83097e8i) q^{56} +(4.39199e8 + 4.39199e8i) q^{57} +(3.67535e8 + 9.47442e8i) q^{58} +(3.32846e8 - 3.32846e8i) q^{59} +(-6.60790e8 - 3.01748e8i) q^{60} +(-7.28302e8 + 7.28302e8i) q^{61} +(-2.00171e8 - 8.82756e7i) q^{62} +(8.74730e7 + 8.74730e7i) q^{63} +(-8.55101e8 + 6.49403e8i) q^{64} +(3.91763e8 - 1.66708e8i) q^{65} +(2.50538e8 + 6.45844e8i) q^{66} -1.33046e9i q^{67} +(-1.03582e9 - 4.69472e7i) q^{68} +(-1.77731e9 + 1.77731e9i) q^{69} +(1.14853e9 + 1.17884e9i) q^{70} +5.02194e8i q^{71} +(7.85872e7 + 2.33418e8i) q^{72} +(-9.57995e8 + 9.57995e8i) q^{73} +(-2.52875e9 - 1.11518e9i) q^{74} +(-1.53711e9 + 1.59745e9i) q^{75} +(1.26857e8 - 2.79891e9i) q^{76} -1.56951e9i q^{77} +(9.05556e8 + 3.99351e8i) q^{78} -8.06323e8i q^{79} +(1.00503e9 + 3.11887e9i) q^{80} -2.98646e9 q^{81} +(2.75407e9 - 6.24504e9i) q^{82} -6.43594e9 q^{83} +(-1.73225e8 + 3.82194e9i) q^{84} +(-1.18274e9 + 2.93495e9i) q^{85} +(-2.43694e9 + 5.52592e9i) q^{86} +(5.09765e9 + 5.09765e9i) q^{87} +(1.38905e9 - 2.79912e9i) q^{88} -5.67633e9 q^{89} +(7.51561e8 + 9.78822e6i) q^{90} +(-1.58557e9 - 1.58557e9i) q^{91} +(1.13264e10 + 5.13356e8i) q^{92} -1.55197e9 q^{93} +(1.71554e9 - 6.65497e8i) q^{94} +(-7.93062e9 - 3.19591e9i) q^{95} +(-3.69143e9 + 6.66289e9i) q^{96} +(-9.72123e9 + 9.72123e9i) q^{97} +(-1.49724e8 + 3.39510e8i) q^{98} +(-5.06830e8 - 5.06830e8i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −29.2793 − 12.9122i −0.914977 − 0.403506i
\(3\) −227.008 −0.934190 −0.467095 − 0.884207i \(-0.654699\pi\)
−0.467095 + 0.884207i \(0.654699\pi\)
\(4\) 690.550 + 756.119i 0.674366 + 0.738398i
\(5\) 2875.48 − 1223.61i 0.920154 − 0.391556i
\(6\) 6646.63 + 2931.18i 0.854763 + 0.376952i
\(7\) −11637.9 − 11637.9i −0.692441 − 0.692441i 0.270327 − 0.962769i \(-0.412868\pi\)
−0.962769 + 0.270327i \(0.912868\pi\)
\(8\) −10455.6 − 31055.1i −0.319081 − 0.947727i
\(9\) −7516.24 −0.127288
\(10\) −99991.5 − 1302.28i −0.999915 − 0.0130228i
\(11\) 67431.3 + 67431.3i 0.418695 + 0.418695i 0.884754 − 0.466059i \(-0.154326\pi\)
−0.466059 + 0.884754i \(0.654326\pi\)
\(12\) −156761. − 171645.i −0.629986 − 0.689804i
\(13\) 136243. 0.366941 0.183471 − 0.983025i \(-0.441267\pi\)
0.183471 + 0.983025i \(0.441267\pi\)
\(14\) 190478. + 491018.i 0.354164 + 0.912972i
\(15\) −652758. + 277770.i −0.859599 + 0.365788i
\(16\) −94856.3 + 1.04428e6i −0.0904620 + 0.995900i
\(17\) −716000. + 716000.i −0.504276 + 0.504276i −0.912764 − 0.408488i \(-0.866056\pi\)
0.408488 + 0.912764i \(0.366056\pi\)
\(18\) 220070. + 97051.2i 0.116466 + 0.0513616i
\(19\) −1.93473e6 − 1.93473e6i −0.781360 − 0.781360i 0.198700 − 0.980060i \(-0.436328\pi\)
−0.980060 + 0.198700i \(0.936328\pi\)
\(20\) 2.91086e6 + 1.32924e6i 0.909645 + 0.415387i
\(21\) 2.64189e6 + 2.64189e6i 0.646872 + 0.646872i
\(22\) −1.10365e6 − 2.84502e6i −0.214150 − 0.552042i
\(23\) 7.82927e6 − 7.82927e6i 1.21642 − 1.21642i 0.247539 − 0.968878i \(-0.420378\pi\)
0.968878 − 0.247539i \(-0.0796217\pi\)
\(24\) 2.37352e6 + 7.04977e6i 0.298082 + 0.885358i
\(25\) 6.77116e6 − 7.03696e6i 0.693367 − 0.720584i
\(26\) −3.98909e6 − 1.75919e6i −0.335743 − 0.148063i
\(27\) 1.51109e7 1.05310
\(28\) 763080. − 1.68361e7i 0.0443384 − 0.978256i
\(29\) −2.24558e7 − 2.24558e7i −1.09481 − 1.09481i −0.995007 − 0.0998010i \(-0.968179\pi\)
−0.0998010 − 0.995007i \(-0.531821\pi\)
\(30\) 2.26989e7 + 295627.i 0.934111 + 0.0121657i
\(31\) 6.83661e6 0.238799 0.119399 − 0.992846i \(-0.461903\pi\)
0.119399 + 0.992846i \(0.461903\pi\)
\(32\) 1.62612e7 − 2.93508e7i 0.484622 − 0.874723i
\(33\) −1.53075e7 − 1.53075e7i −0.391141 − 0.391141i
\(34\) 3.02091e7 − 1.17188e7i 0.664880 − 0.257923i
\(35\) −4.77047e7 − 1.92242e7i −0.908283 − 0.366023i
\(36\) −5.19035e6 − 5.68318e6i −0.0858388 − 0.0939893i
\(37\) 8.63666e7 1.24548 0.622741 − 0.782428i \(-0.286019\pi\)
0.622741 + 0.782428i \(0.286019\pi\)
\(38\) 3.16658e7 + 8.16289e7i 0.399643 + 1.03021i
\(39\) −3.09282e7 −0.342793
\(40\) −6.80645e7 − 7.65048e7i −0.664692 − 0.747117i
\(41\) 2.13292e8i 1.84101i 0.390731 + 0.920505i \(0.372222\pi\)
−0.390731 + 0.920505i \(0.627778\pi\)
\(42\) −4.32400e7 − 1.11465e8i −0.330856 − 0.852890i
\(43\) − 1.88731e8i − 1.28381i −0.766783 − 0.641907i \(-0.778144\pi\)
0.766783 − 0.641907i \(-0.221856\pi\)
\(44\) −4.42138e6 + 9.75508e7i −0.0268099 + 0.591517i
\(45\) −2.16128e7 + 9.19698e6i −0.117125 + 0.0498405i
\(46\) −3.30329e8 + 1.28142e8i −1.60382 + 0.622162i
\(47\) −4.06607e7 + 4.06607e7i −0.177291 + 0.177291i −0.790174 − 0.612883i \(-0.790010\pi\)
0.612883 + 0.790174i \(0.290010\pi\)
\(48\) 2.15332e7 − 2.37059e8i 0.0845087 − 0.930360i
\(49\) − 1.15956e7i − 0.0410499i
\(50\) −2.89117e8 + 1.18606e8i −0.925175 + 0.379540i
\(51\) 1.62538e8 − 1.62538e8i 0.471090 − 0.471090i
\(52\) 9.40825e7 + 1.03016e8i 0.247453 + 0.270949i
\(53\) 9.52654e7i 0.227801i 0.993492 + 0.113901i \(0.0363345\pi\)
−0.993492 + 0.113901i \(0.963665\pi\)
\(54\) −4.42435e8 − 1.95114e8i −0.963564 − 0.424933i
\(55\) 2.76407e8 + 1.11388e8i 0.549207 + 0.221321i
\(56\) −2.39734e8 + 4.83097e8i −0.435301 + 0.877191i
\(57\) 4.39199e8 + 4.39199e8i 0.729939 + 0.729939i
\(58\) 3.67535e8 + 9.47442e8i 0.559963 + 1.44349i
\(59\) 3.32846e8 − 3.32846e8i 0.465568 − 0.465568i −0.434907 − 0.900475i \(-0.643219\pi\)
0.900475 + 0.434907i \(0.143219\pi\)
\(60\) −6.60790e8 − 3.01748e8i −0.849781 − 0.388051i
\(61\) −7.28302e8 + 7.28302e8i −0.862308 + 0.862308i −0.991606 − 0.129298i \(-0.958728\pi\)
0.129298 + 0.991606i \(0.458728\pi\)
\(62\) −2.00171e8 − 8.82756e7i −0.218495 − 0.0963568i
\(63\) 8.74730e7 + 8.74730e7i 0.0881397 + 0.0881397i
\(64\) −8.55101e8 + 6.49403e8i −0.796375 + 0.604804i
\(65\) 3.91763e8 − 1.66708e8i 0.337642 − 0.143678i
\(66\) 2.50538e8 + 6.45844e8i 0.200057 + 0.515713i
\(67\) − 1.33046e9i − 0.985436i −0.870189 − 0.492718i \(-0.836003\pi\)
0.870189 − 0.492718i \(-0.163997\pi\)
\(68\) −1.03582e9 − 4.69472e7i −0.712423 − 0.0322898i
\(69\) −1.77731e9 + 1.77731e9i −1.13636 + 1.13636i
\(70\) 1.14853e9 + 1.17884e9i 0.683365 + 0.701400i
\(71\) 5.02194e8i 0.278342i 0.990268 + 0.139171i \(0.0444438\pi\)
−0.990268 + 0.139171i \(0.955556\pi\)
\(72\) 7.85872e7 + 2.33418e8i 0.0406153 + 0.120635i
\(73\) −9.57995e8 + 9.57995e8i −0.462114 + 0.462114i −0.899348 − 0.437234i \(-0.855958\pi\)
0.437234 + 0.899348i \(0.355958\pi\)
\(74\) −2.52875e9 − 1.11518e9i −1.13959 − 0.502559i
\(75\) −1.53711e9 + 1.59745e9i −0.647737 + 0.673163i
\(76\) 1.26857e8 − 2.79891e9i 0.0500320 − 1.10388i
\(77\) − 1.56951e9i − 0.579844i
\(78\) 9.05556e8 + 3.99351e8i 0.313648 + 0.138319i
\(79\) − 8.06323e8i − 0.262044i −0.991379 − 0.131022i \(-0.958174\pi\)
0.991379 − 0.131022i \(-0.0418258\pi\)
\(80\) 1.00503e9 + 3.11887e9i 0.306712 + 0.951802i
\(81\) −2.98646e9 −0.856509
\(82\) 2.75407e9 − 6.24504e9i 0.742859 − 1.68448i
\(83\) −6.43594e9 −1.63389 −0.816943 − 0.576719i \(-0.804333\pi\)
−0.816943 + 0.576719i \(0.804333\pi\)
\(84\) −1.73225e8 + 3.82194e9i −0.0414205 + 0.913877i
\(85\) −1.18274e9 + 2.93495e9i −0.266559 + 0.661464i
\(86\) −2.43694e9 + 5.52592e9i −0.518027 + 1.17466i
\(87\) 5.09765e9 + 5.09765e9i 1.02276 + 1.02276i
\(88\) 1.38905e9 − 2.79912e9i 0.263211 − 0.530406i
\(89\) −5.67633e9 −1.01652 −0.508262 − 0.861202i \(-0.669712\pi\)
−0.508262 + 0.861202i \(0.669712\pi\)
\(90\) 7.51561e8 + 9.78822e6i 0.127277 + 0.00165764i
\(91\) −1.58557e9 − 1.58557e9i −0.254085 − 0.254085i
\(92\) 1.13264e10 + 5.13356e8i 1.71851 + 0.0778895i
\(93\) −1.55197e9 −0.223084
\(94\) 1.71554e9 − 6.65497e8i 0.233755 − 0.0906790i
\(95\) −7.93062e9 − 3.19591e9i −1.02492 − 0.413025i
\(96\) −3.69143e9 + 6.66289e9i −0.452730 + 0.817158i
\(97\) −9.72123e9 + 9.72123e9i −1.13204 + 1.13204i −0.142205 + 0.989837i \(0.545419\pi\)
−0.989837 + 0.142205i \(0.954581\pi\)
\(98\) −1.49724e8 + 3.39510e8i −0.0165639 + 0.0375597i
\(99\) −5.06830e8 − 5.06830e8i −0.0532950 − 0.0532950i
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.53.16 yes 236
5.2 odd 4 80.11.i.a.37.75 yes 236
16.13 even 4 80.11.i.a.13.75 ✓ 236
80.77 odd 4 inner 80.11.t.a.77.16 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.75 ✓ 236 16.13 even 4
80.11.i.a.37.75 yes 236 5.2 odd 4
80.11.t.a.53.16 yes 236 1.1 even 1 trivial
80.11.t.a.77.16 yes 236 80.77 odd 4 inner