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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 13.75
Character \(\chi\) \(=\) 80.13
Dual form 80.11.i.a.37.75

$q$-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+(12.9122 + 29.2793i) q^{2} -227.008i q^{3} +(-690.550 + 756.119i) q^{4} +(-1223.61 - 2875.48i) q^{5} +(6646.63 - 2931.18i) q^{6} +(11637.9 + 11637.9i) q^{7} +(-31055.1 - 10455.6i) q^{8} +7516.24 q^{9} +(68392.4 - 72955.3i) q^{10} +(67431.3 - 67431.3i) q^{11} +(171645. + 156761. i) q^{12} +136243. i q^{13} +(-190478. + 491018. i) q^{14} +(-652758. + 277770. i) q^{15} +(-94856.3 - 1.04428e6i) q^{16} +(-716000. + 716000. i) q^{17} +(97051.2 + 220070. i) q^{18} +(1.93473e6 - 1.93473e6i) q^{19} +(3.01917e6 + 1.06047e6i) q^{20} +(2.64189e6 - 2.64189e6i) q^{21} +(2.84502e6 + 1.10365e6i) 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} -1.51109e7i q^{27} +(-1.68361e7 + 763080. i) q^{28} +(2.24558e7 - 2.24558e7i) q^{29} +(-1.65615e7 - 1.55257e7i) q^{30} +6.83661e6 q^{31} +(2.93508e7 - 1.62612e7i) q^{32} +(-1.53075e7 - 1.53075e7i) q^{33} +(-3.02091e7 - 1.17188e7i) q^{34} +(1.92242e7 - 4.77047e7i) q^{35} +(-5.19035e6 + 5.68318e6i) q^{36} -8.63666e7i q^{37} +(8.16289e7 + 3.16658e7i) q^{38} +3.09282e7 q^{39} +(7.93446e6 + 1.02092e8i) q^{40} -2.13292e8i q^{41} +(1.11465e8 + 4.32400e7i) q^{42} -1.88731e8 q^{43} +(4.42138e6 + 9.75508e7i) q^{44} +(-9.19698e6 - 2.16128e7i) q^{45} +(-3.30329e8 - 1.28142e8i) q^{46} +(-4.06607e7 + 4.06607e7i) q^{47} +(-2.37059e8 + 2.15332e7i) q^{48} -1.15956e7i q^{49} +(-2.93468e8 - 1.07392e8i) q^{50} +(1.62538e8 + 1.62538e8i) q^{51} +(-1.03016e8 - 9.40825e7i) q^{52} +9.52654e7 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} +(9.47442e8 + 3.67535e8i) q^{58} +(-3.32846e8 - 3.32846e8i) q^{59} +(2.40735e8 - 6.85377e8i) q^{60} +(-7.28302e8 - 7.28302e8i) q^{61} +(8.82756e7 + 2.00171e8i) 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} +1.33046e9 q^{67} +(-4.69472e7 - 1.03582e9i) q^{68} +(1.77731e9 + 1.77731e9i) q^{69} +(1.64499e9 - 5.31018e7i) q^{70} -5.02194e8i q^{71} +(-2.33418e8 - 7.85872e7i) q^{72} +(9.57995e8 - 9.57995e8i) q^{73} +(2.52875e9 - 1.11518e9i) q^{74} +(1.59745e9 + 1.53711e9i) q^{75} +(1.26857e8 + 2.79891e9i) q^{76} +1.56951e9 q^{77} +(3.99351e8 + 9.05556e8i) q^{78} -8.06323e8i q^{79} +(-2.88673e9 + 1.55055e9i) q^{80} -2.98646e9 q^{81} +(6.24504e9 - 2.75407e9i) q^{82} -6.43594e9i q^{83} +(1.73225e8 + 3.82194e9i) q^{84} +(2.93495e9 + 1.18274e9i) q^{85} +(-2.43694e9 - 5.52592e9i) q^{86} +(-5.09765e9 - 5.09765e9i) q^{87} +(-2.79912e9 + 1.38905e9i) q^{88} +5.67633e9 q^{89} +(5.14054e8 - 5.48350e8i) q^{90} +(-1.58557e9 + 1.58557e9i) q^{91} +(-5.13356e8 - 1.13264e10i) q^{92} -1.55197e9i 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} +(3.39510e8 - 1.49724e8i) q^{98} +(5.06830e8 - 5.06830e8i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 12.9122 + 29.2793i 0.403506 + 0.914977i
\(3\) − 227.008i − 0.934190i −0.884207 − 0.467095i \(-0.845301\pi\)
0.884207 − 0.467095i \(-0.154699\pi\)
\(4\) −690.550 + 756.119i −0.674366 + 0.738398i
\(5\) −1223.61 − 2875.48i −0.391556 − 0.920154i
\(6\) 6646.63 − 2931.18i 0.854763 − 0.376952i
\(7\) 11637.9 + 11637.9i 0.692441 + 0.692441i 0.962769 − 0.270327i \(-0.0871319\pi\)
−0.270327 + 0.962769i \(0.587132\pi\)
\(8\) −31055.1 − 10455.6i −0.947727 − 0.319081i
\(9\) 7516.24 0.127288
\(10\) 68392.4 − 72955.3i 0.683924 − 0.729553i
\(11\) 67431.3 − 67431.3i 0.418695 − 0.418695i −0.466059 − 0.884754i \(-0.654326\pi\)
0.884754 + 0.466059i \(0.154326\pi\)
\(12\) 171645. + 156761.i 0.689804 + 0.629986i
\(13\) 136243.i 0.366941i 0.983025 + 0.183471i \(0.0587332\pi\)
−0.983025 + 0.183471i \(0.941267\pi\)
\(14\) −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 \(0.366056\pi\)
\(18\) 97051.2 + 220070.i 0.0513616 + 0.116466i
\(19\) 1.93473e6 − 1.93473e6i 0.781360 − 0.781360i −0.198700 − 0.980060i \(-0.563672\pi\)
0.980060 + 0.198700i \(0.0636720\pi\)
\(20\) 3.01917e6 + 1.06047e6i 0.943492 + 0.331396i
\(21\) 2.64189e6 − 2.64189e6i 0.646872 − 0.646872i
\(22\) 2.84502e6 + 1.10365e6i 0.552042 + 0.214150i
\(23\) −7.82927e6 + 7.82927e6i −1.21642 + 1.21642i −0.247539 + 0.968878i \(0.579622\pi\)
−0.968878 + 0.247539i \(0.920378\pi\)
\(24\) −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
\(27\) − 1.51109e7i − 1.05310i
\(28\) −1.68361e7 + 763080.i −0.978256 + 0.0443384i
\(29\) 2.24558e7 − 2.24558e7i 1.09481 − 1.09481i 0.0998010 − 0.995007i \(-0.468179\pi\)
0.995007 − 0.0998010i \(-0.0318206\pi\)
\(30\) −1.65615e7 − 1.55257e7i −0.681541 − 0.638916i
\(31\) 6.83661e6 0.238799 0.119399 − 0.992846i \(-0.461903\pi\)
0.119399 + 0.992846i \(0.461903\pi\)
\(32\) 2.93508e7 − 1.62612e7i 0.874723 − 0.484622i
\(33\) −1.53075e7 − 1.53075e7i −0.391141 − 0.391141i
\(34\) −3.02091e7 − 1.17188e7i −0.664880 − 0.257923i
\(35\) 1.92242e7 − 4.77047e7i 0.366023 − 0.908283i
\(36\) −5.19035e6 + 5.68318e6i −0.0858388 + 0.0939893i
\(37\) − 8.63666e7i − 1.24548i −0.782428 − 0.622741i \(-0.786019\pi\)
0.782428 − 0.622741i \(-0.213981\pi\)
\(38\) 8.16289e7 + 3.16658e7i 1.03021 + 0.399643i
\(39\) 3.09282e7 0.342793
\(40\) 7.93446e6 + 1.02092e8i 0.0774850 + 0.996994i
\(41\) − 2.13292e8i − 1.84101i −0.390731 − 0.920505i \(-0.627778\pi\)
0.390731 − 0.920505i \(-0.372222\pi\)
\(42\) 1.11465e8 + 4.32400e7i 0.852890 + 0.330856i
\(43\) −1.88731e8 −1.28381 −0.641907 − 0.766783i \(-0.721856\pi\)
−0.641907 + 0.766783i \(0.721856\pi\)
\(44\) 4.42138e6 + 9.75508e7i 0.0268099 + 0.591517i
\(45\) −9.19698e6 − 2.16128e7i −0.0498405 − 0.117125i
\(46\) −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 \(0.290010\pi\)
\(48\) −2.37059e8 + 2.15332e7i −0.930360 + 0.0845087i
\(49\) − 1.15956e7i − 0.0410499i
\(50\) −2.93468e8 − 1.07392e8i −0.939096 − 0.343655i
\(51\) 1.62538e8 + 1.62538e8i 0.471090 + 0.471090i
\(52\) −1.03016e8 − 9.40825e7i −0.270949 − 0.247453i
\(53\) 9.52654e7 0.227801 0.113901 − 0.993492i \(-0.463665\pi\)
0.113901 + 0.993492i \(0.463665\pi\)
\(54\) 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
\(58\) 9.47442e8 + 3.67535e8i 1.44349 + 0.559963i
\(59\) −3.32846e8 − 3.32846e8i −0.465568 − 0.465568i 0.434907 − 0.900475i \(-0.356781\pi\)
−0.900475 + 0.434907i \(0.856781\pi\)
\(60\) 2.40735e8 − 6.85377e8i 0.309587 − 0.881401i
\(61\) −7.28302e8 − 7.28302e8i −0.862308 − 0.862308i 0.129298 − 0.991606i \(-0.458728\pi\)
−0.991606 + 0.129298i \(0.958728\pi\)
\(62\) 8.82756e7 + 2.00171e8i 0.0963568 + 0.218495i
\(63\) 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
\(67\) 1.33046e9 0.985436 0.492718 − 0.870189i \(-0.336003\pi\)
0.492718 + 0.870189i \(0.336003\pi\)
\(68\) −4.69472e7 − 1.03582e9i −0.0322898 − 0.712423i
\(69\) 1.77731e9 + 1.77731e9i 1.13636 + 1.13636i
\(70\) 1.64499e9 − 5.31018e7i 0.978750 − 0.0315950i
\(71\) − 5.02194e8i − 0.278342i −0.990268 − 0.139171i \(-0.955556\pi\)
0.990268 − 0.139171i \(-0.0444438\pi\)
\(72\) −2.33418e8 − 7.85872e7i −0.120635 − 0.0406153i
\(73\) 9.57995e8 − 9.57995e8i 0.462114 − 0.462114i −0.437234 − 0.899348i \(-0.644042\pi\)
0.899348 + 0.437234i \(0.144042\pi\)
\(74\) 2.52875e9 − 1.11518e9i 1.13959 − 0.502559i
\(75\) 1.59745e9 + 1.53711e9i 0.673163 + 0.647737i
\(76\) 1.26857e8 + 2.79891e9i 0.0500320 + 1.10388i
\(77\) 1.56951e9 0.579844
\(78\) 3.99351e8 + 9.05556e8i 0.138319 + 0.313648i
\(79\) − 8.06323e8i − 0.262044i −0.991379 − 0.131022i \(-0.958174\pi\)
0.991379 − 0.131022i \(-0.0418258\pi\)
\(80\) −2.88673e9 + 1.55055e9i −0.880960 + 0.473190i
\(81\) −2.98646e9 −0.856509
\(82\) 6.24504e9 − 2.75407e9i 1.68448 − 0.742859i
\(83\) − 6.43594e9i − 1.63389i −0.576719 − 0.816943i \(-0.695667\pi\)
0.576719 − 0.816943i \(-0.304333\pi\)
\(84\) 1.73225e8 + 3.82194e9i 0.0414205 + 0.913877i
\(85\) 2.93495e9 + 1.18274e9i 0.661464 + 0.266559i
\(86\) −2.43694e9 − 5.52592e9i −0.518027 − 1.17466i
\(87\) −5.09765e9 − 5.09765e9i −1.02276 − 1.02276i
\(88\) −2.79912e9 + 1.38905e9i −0.530406 + 0.263211i
\(89\) 5.67633e9 1.01652 0.508262 − 0.861202i \(-0.330288\pi\)
0.508262 + 0.861202i \(0.330288\pi\)
\(90\) 5.14054e8 − 5.48350e8i 0.0870555 − 0.0928635i
\(91\) −1.58557e9 + 1.58557e9i −0.254085 + 0.254085i
\(92\) −5.13356e8 − 1.13264e10i −0.0778895 − 1.71851i
\(93\) − 1.55197e9i − 0.223084i
\(94\) −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 \(0.954581\pi\)
\(98\) 3.39510e8 − 1.49724e8i 0.0375597 − 0.0165639i
\(99\) 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.i.a.13.75 ✓ 236
5.2 odd 4 80.11.t.a.77.16 yes 236
16.5 even 4 80.11.t.a.53.16 yes 236
80.37 odd 4 inner 80.11.i.a.37.75 yes 236
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.11.i.a.13.75 ✓ 236 1.1 even 1 trivial
80.11.i.a.37.75 yes 236 80.37 odd 4 inner
80.11.t.a.53.16 yes 236 16.5 even 4
80.11.t.a.77.16 yes 236 5.2 odd 4