Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [85,4,Mod(26,85)] 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("85.26"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(85, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 85 = 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 85.l (of order \(8\), degree \(4\), 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: \(5.01516235049\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 66.8
Character \(\chi\) \(=\) 85.66
Dual form 85.4.l.a.76.8

$q$-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+(-0.550473 - 0.550473i) q^{2} +(-2.40197 - 5.79887i) q^{3} -7.39396i q^{4} +(-4.61940 + 1.91342i) q^{5} +(-1.86990 + 4.51434i) q^{6} +(-3.51664 - 1.45664i) q^{7} +(-8.47397 + 8.47397i) q^{8} +(-8.76551 + 8.76551i) q^{9} +(3.59614 + 1.48957i) q^{10} +(2.86778 - 6.92343i) q^{11} +(-42.8766 + 17.7601i) q^{12} +91.2737i q^{13} +(1.13398 + 2.73766i) q^{14} +(22.1913 + 22.1913i) q^{15} -49.8223 q^{16} +(-6.69881 - 69.7720i) q^{17} +9.65036 q^{18} +(-108.873 - 108.873i) q^{19} +(14.1477 + 34.1556i) q^{20} +23.8913i q^{21} +(-5.38980 + 2.23253i) q^{22} +(-5.98565 + 14.4506i) q^{23} +(69.4936 + 28.7852i) q^{24} +(17.6777 - 17.6777i) q^{25} +(50.2438 - 50.2438i) q^{26} +(-84.6849 - 35.0776i) q^{27} +(-10.7703 + 26.0019i) q^{28} +(214.315 - 88.7722i) q^{29} -24.4314i q^{30} +(-40.5491 - 97.8941i) q^{31} +(95.2176 + 95.2176i) q^{32} -47.0364 q^{33} +(-34.7201 + 42.0951i) q^{34} +19.0319 q^{35} +(64.8118 + 64.8118i) q^{36} +(79.6668 + 192.333i) q^{37} +119.864i q^{38} +(529.284 - 219.237i) q^{39} +(22.9304 - 55.3588i) q^{40} +(-155.811 - 64.5389i) q^{41} +(13.1515 - 13.1515i) q^{42} +(-41.2824 + 41.2824i) q^{43} +(-51.1916 - 21.2042i) q^{44} +(23.7193 - 57.2635i) q^{45} +(11.2496 - 4.65975i) q^{46} -380.480i q^{47} +(119.672 + 288.913i) q^{48} +(-232.293 - 232.293i) q^{49} -19.4622 q^{50} +(-388.508 + 206.436i) q^{51} +674.874 q^{52} +(-412.252 - 412.252i) q^{53} +(27.3075 + 65.9261i) q^{54} +37.4694i q^{55} +(42.1434 - 17.4564i) q^{56} +(-369.831 + 892.851i) q^{57} +(-166.841 - 69.1080i) q^{58} +(381.150 - 381.150i) q^{59} +(164.082 - 164.082i) q^{60} +(445.712 + 184.620i) q^{61} +(-31.5669 + 76.2093i) q^{62} +(43.5933 - 18.0570i) q^{63} +293.749i q^{64} +(-174.645 - 421.630i) q^{65} +(25.8923 + 25.8923i) q^{66} -629.171 q^{67} +(-515.891 + 49.5307i) q^{68} +98.1746 q^{69} +(-10.4766 - 10.4766i) q^{70} +(47.6691 + 115.083i) q^{71} -148.557i q^{72} +(-389.824 + 161.471i) q^{73} +(62.0196 - 149.728i) q^{74} +(-144.972 - 60.0492i) q^{75} +(-805.003 + 805.003i) q^{76} +(-20.1699 + 20.1699i) q^{77} +(-412.041 - 170.673i) q^{78} +(134.629 - 325.024i) q^{79} +(230.149 - 95.3308i) q^{80} +910.032i q^{81} +(50.2427 + 121.297i) q^{82} +(363.357 + 363.357i) q^{83} +176.651 q^{84} +(164.447 + 309.487i) q^{85} +45.4497 q^{86} +(-1029.56 - 1029.56i) q^{87} +(34.3675 + 82.9704i) q^{88} -213.950i q^{89} +(-44.5789 + 18.4652i) q^{90} +(132.953 - 320.977i) q^{91} +(106.847 + 44.2576i) q^{92} +(-470.277 + 470.277i) q^{93} +(-209.444 + 209.444i) q^{94} +(711.248 + 294.609i) q^{95} +(323.444 - 780.864i) q^{96} +(173.373 - 71.8133i) q^{97} +255.742i q^{98} +(35.5499 + 85.8250i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q - 24 q^{6} + 120 q^{9} - 200 q^{11} + 96 q^{12} + 48 q^{14} - 80 q^{15} - 1152 q^{16} - 112 q^{17} - 16 q^{18} - 32 q^{19} + 1232 q^{22} + 544 q^{23} + 416 q^{24} + 368 q^{26} + 216 q^{27} - 144 q^{28}+ \cdots - 12032 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/85\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(71\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{8}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.550473 − 0.550473i −0.194622 − 0.194622i 0.603068 − 0.797690i \(-0.293945\pi\)
−0.797690 + 0.603068i \(0.793945\pi\)
\(3\) −2.40197 − 5.79887i −0.462259 − 1.11599i −0.967468 − 0.252995i \(-0.918584\pi\)
0.505208 − 0.862997i \(-0.331416\pi\)
\(4\) − 7.39396i − 0.924245i
\(5\) −4.61940 + 1.91342i −0.413171 + 0.171141i
\(6\) −1.86990 + 4.51434i −0.127231 + 0.307162i
\(7\) −3.51664 − 1.45664i −0.189881 − 0.0786512i 0.285717 − 0.958314i \(-0.407768\pi\)
−0.475598 + 0.879663i \(0.657768\pi\)
\(8\) −8.47397 + 8.47397i −0.374500 + 0.374500i
\(9\) −8.76551 + 8.76551i −0.324649 + 0.324649i
\(10\) 3.59614 + 1.48957i 0.113720 + 0.0471044i
\(11\) 2.86778 − 6.92343i 0.0786062 − 0.189772i −0.879691 − 0.475546i \(-0.842251\pi\)
0.958297 + 0.285774i \(0.0922506\pi\)
\(12\) −42.8766 + 17.7601i −1.03145 + 0.427241i
\(13\) 91.2737i 1.94729i 0.228067 + 0.973645i \(0.426760\pi\)
−0.228067 + 0.973645i \(0.573240\pi\)
\(14\) 1.13398 + 2.73766i 0.0216477 + 0.0522621i
\(15\) 22.1913 + 22.1913i 0.381985 + 0.381985i
\(16\) −49.8223 −0.778473
\(17\) −6.69881 − 69.7720i −0.0955706 − 0.995423i
\(18\) 9.65036 0.126367
\(19\) −108.873 − 108.873i −1.31459 − 1.31459i −0.917993 − 0.396598i \(-0.870191\pi\)
−0.396598 − 0.917993i \(-0.629809\pi\)
\(20\) 14.1477 + 34.1556i 0.158176 + 0.381872i
\(21\) 23.8913i 0.248263i
\(22\) −5.38980 + 2.23253i −0.0522323 + 0.0216353i
\(23\) −5.98565 + 14.4506i −0.0542649 + 0.131007i −0.948687 − 0.316217i \(-0.897587\pi\)
0.894422 + 0.447224i \(0.147587\pi\)
\(24\) 69.4936 + 28.7852i 0.591055 + 0.244823i
\(25\) 17.6777 − 17.6777i 0.141421 − 0.141421i
\(26\) 50.2438 − 50.2438i 0.378985 − 0.378985i
\(27\) −84.6849 − 35.0776i −0.603615 − 0.250026i
\(28\) −10.7703 + 26.0019i −0.0726929 + 0.175496i
\(29\) 214.315 − 88.7722i 1.37232 − 0.568434i 0.429904 − 0.902875i \(-0.358548\pi\)
0.942417 + 0.334441i \(0.108548\pi\)
\(30\) − 24.4314i − 0.148685i
\(31\) −40.5491 − 97.8941i −0.234930 − 0.567171i 0.761815 − 0.647795i \(-0.224309\pi\)
−0.996745 + 0.0806239i \(0.974309\pi\)
\(32\) 95.2176 + 95.2176i 0.526008 + 0.526008i
\(33\) −47.0364 −0.248121
\(34\) −34.7201 + 42.0951i −0.175131 + 0.212331i
\(35\) 19.0319 0.0919137
\(36\) 64.8118 + 64.8118i 0.300055 + 0.300055i
\(37\) 79.6668 + 192.333i 0.353977 + 0.854575i 0.996121 + 0.0879917i \(0.0280449\pi\)
−0.642145 + 0.766584i \(0.721955\pi\)
\(38\) 119.864i 0.511696i
\(39\) 529.284 − 219.237i 2.17316 − 0.900153i
\(40\) 22.9304 − 55.3588i 0.0906403 − 0.218825i
\(41\) −155.811 − 64.5389i −0.593501 − 0.245836i 0.0656551 − 0.997842i \(-0.479086\pi\)
−0.659156 + 0.752006i \(0.729086\pi\)
\(42\) 13.1515 − 13.1515i 0.0483173 − 0.0483173i
\(43\) −41.2824 + 41.2824i −0.146407 + 0.146407i −0.776511 − 0.630104i \(-0.783012\pi\)
0.630104 + 0.776511i \(0.283012\pi\)
\(44\) −51.1916 − 21.2042i −0.175396 − 0.0726514i
\(45\) 23.7193 − 57.2635i 0.0785748 − 0.189696i
\(46\) 11.2496 − 4.65975i 0.0360580 − 0.0149357i
\(47\) − 380.480i − 1.18082i −0.807102 − 0.590412i \(-0.798965\pi\)
0.807102 − 0.590412i \(-0.201035\pi\)
\(48\) 119.672 + 288.913i 0.359856 + 0.868770i
\(49\) −232.293 − 232.293i −0.677238 − 0.677238i
\(50\) −19.4622 −0.0550473
\(51\) −388.508 + 206.436i −1.06671 + 0.566799i
\(52\) 674.874 1.79977
\(53\) −412.252 − 412.252i −1.06844 − 1.06844i −0.997479 − 0.0709590i \(-0.977394\pi\)
−0.0709590 − 0.997479i \(-0.522606\pi\)
\(54\) 27.3075 + 65.9261i 0.0688162 + 0.166137i
\(55\) 37.4694i 0.0918612i
\(56\) 42.1434 − 17.4564i 0.100565 − 0.0416555i
\(57\) −369.831 + 892.851i −0.859391 + 2.07475i
\(58\) −166.841 − 69.1080i −0.377713 − 0.156454i
\(59\) 381.150 − 381.150i 0.841042 − 0.841042i −0.147953 − 0.988994i \(-0.547268\pi\)
0.988994 + 0.147953i \(0.0472683\pi\)
\(60\) 164.082 − 164.082i 0.353047 − 0.353047i
\(61\) 445.712 + 184.620i 0.935535 + 0.387511i 0.797775 − 0.602955i \(-0.206010\pi\)
0.137759 + 0.990466i \(0.456010\pi\)
\(62\) −31.5669 + 76.2093i −0.0646614 + 0.156106i
\(63\) 43.5933 − 18.0570i 0.0871785 − 0.0361105i
\(64\) 293.749i 0.573728i
\(65\) −174.645 − 421.630i −0.333262 − 0.804565i
\(66\) 25.8923 + 25.8923i 0.0482897 + 0.0482897i
\(67\) −629.171 −1.14725 −0.573623 − 0.819119i \(-0.694463\pi\)
−0.573623 + 0.819119i \(0.694463\pi\)
\(68\) −515.891 + 49.5307i −0.920014 + 0.0883306i
\(69\) 98.1746 0.171287
\(70\) −10.4766 − 10.4766i −0.0178884 − 0.0178884i
\(71\) 47.6691 + 115.083i 0.0796799 + 0.192364i 0.958699 − 0.284422i \(-0.0918016\pi\)
−0.879019 + 0.476786i \(0.841802\pi\)
\(72\) − 148.557i − 0.243162i
\(73\) −389.824 + 161.471i −0.625007 + 0.258886i −0.672630 − 0.739979i \(-0.734835\pi\)
0.0476231 + 0.998865i \(0.484835\pi\)
\(74\) 62.0196 − 149.728i 0.0974274 − 0.235211i
\(75\) −144.972 − 60.0492i −0.223198 − 0.0924518i
\(76\) −805.003 + 805.003i −1.21500 + 1.21500i
\(77\) −20.1699 + 20.1699i −0.0298516 + 0.0298516i
\(78\) −412.041 − 170.673i −0.598134 − 0.247755i
\(79\) 134.629 − 325.024i 0.191734 − 0.462887i −0.798553 − 0.601924i \(-0.794401\pi\)
0.990287 + 0.139037i \(0.0444008\pi\)
\(80\) 230.149 − 95.3308i 0.321643 − 0.133229i
\(81\) 910.032i 1.24833i
\(82\) 50.2427 + 121.297i 0.0676631 + 0.163353i
\(83\) 363.357 + 363.357i 0.480525 + 0.480525i 0.905299 − 0.424774i \(-0.139646\pi\)
−0.424774 + 0.905299i \(0.639646\pi\)
\(84\) 176.651 0.229455
\(85\) 164.447 + 309.487i 0.209845 + 0.394924i
\(86\) 45.4497 0.0569880
\(87\) −1029.56 − 1029.56i −1.26874 − 1.26874i
\(88\) 34.3675 + 82.9704i 0.0416316 + 0.100508i
\(89\) − 213.950i − 0.254816i −0.991850 − 0.127408i \(-0.959334\pi\)
0.991850 − 0.127408i \(-0.0406658\pi\)
\(90\) −44.5789 + 18.4652i −0.0522114 + 0.0216267i
\(91\) 132.953 − 320.977i 0.153157 − 0.369753i
\(92\) 106.847 + 44.2576i 0.121083 + 0.0501541i
\(93\) −470.277 + 470.277i −0.524360 + 0.524360i
\(94\) −209.444 + 209.444i −0.229814 + 0.229814i
\(95\) 711.248 + 294.609i 0.768132 + 0.318171i
\(96\) 323.444 − 780.864i 0.343869 − 0.830172i
\(97\) 173.373 − 71.8133i 0.181477 − 0.0751704i −0.290095 − 0.956998i \(-0.593687\pi\)
0.471572 + 0.881828i \(0.343687\pi\)
\(98\) 255.742i 0.263611i
\(99\) 35.5499 + 85.8250i 0.0360899 + 0.0871287i
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 85.4.l.a.66.8 ✓ 72
17.5 odd 16 1445.4.a.z.1.20 36
17.8 even 8 inner 85.4.l.a.76.8 yes 72
17.12 odd 16 1445.4.a.y.1.20 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.l.a.66.8 ✓ 72 1.1 even 1 trivial
85.4.l.a.76.8 yes 72 17.8 even 8 inner
1445.4.a.y.1.20 36 17.12 odd 16
1445.4.a.z.1.20 36 17.5 odd 16