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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [483,2,Mod(10,483)] 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("483.10"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(483, base_ring=CyclotomicField(66)) chi = DirichletCharacter(H, H._module([0, 11, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.bf (of order \(66\), degree \(20\), 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: \(3.85677441763\)
Analytic rank: \(0\)
Dimension: \(640\)
Relative dimension: \(32\) over \(\Q(\zeta_{66})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label 10.21
Character \(\chi\) \(=\) 483.10
Dual form 483.2.bf.a.145.21

$q$-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+(0.310216 - 0.896309i) q^{2} +(0.458227 + 0.888835i) q^{3} +(0.864970 + 0.680220i) q^{4} +(2.14536 - 0.204857i) q^{5} +(0.938820 - 0.134982i) q^{6} +(-1.73488 + 1.99754i) q^{7} +(2.47383 - 1.58983i) q^{8} +(-0.580057 + 0.814576i) q^{9} +(0.481908 - 1.98645i) q^{10} +(0.899616 - 0.311360i) q^{11} +(-0.208251 + 1.08051i) q^{12} +(0.327403 + 1.11503i) q^{13} +(1.25223 + 2.17466i) q^{14} +(1.16514 + 1.81300i) q^{15} +(-0.138705 - 0.571750i) q^{16} +(-2.90371 - 1.16247i) q^{17} +(0.550169 + 0.772604i) q^{18} +(0.161704 - 0.0647363i) q^{19} +(1.99502 + 1.28212i) q^{20} +(-2.57046 - 0.626697i) q^{21} -0.902923i q^{22} +(1.08389 - 4.67174i) q^{23} +(2.54667 + 1.47032i) q^{24} +(-0.349047 + 0.0672733i) q^{25} +(1.10098 + 0.0524461i) q^{26} +(-0.989821 - 0.142315i) q^{27} +(-2.85939 + 0.547715i) q^{28} +(1.21531 + 8.45264i) q^{29} +(1.98645 - 0.481908i) q^{30} +(-0.772324 + 0.0367903i) q^{31} +(5.29917 + 0.506009i) q^{32} +(0.688976 + 0.656937i) q^{33} +(-1.94271 + 2.24200i) q^{34} +(-3.31273 + 4.64085i) q^{35} +(-1.05582 + 0.310018i) q^{36} +(-3.63934 - 2.59156i) q^{37} +(-0.00786080 - 0.165019i) q^{38} +(-0.841056 + 0.801945i) q^{39} +(4.98156 - 3.91754i) q^{40} +(-3.22152 - 1.47122i) q^{41} +(-1.35911 + 2.10951i) q^{42} +(7.02243 - 10.9271i) q^{43} +(0.989935 + 0.342620i) q^{44} +(-1.07756 + 1.86639i) q^{45} +(-3.85109 - 2.42075i) q^{46} +(8.45717 - 4.88275i) q^{47} +(0.444633 - 0.385277i) q^{48} +(-0.980370 - 6.93101i) q^{49} +(-0.0479821 + 0.333723i) q^{50} +(-0.297312 - 3.11359i) q^{51} +(-0.475274 + 1.18718i) q^{52} +(5.40166 + 5.66510i) q^{53} +(-0.434616 + 0.843038i) q^{54} +(1.86621 - 0.852272i) q^{55} +(-1.11604 + 7.69976i) q^{56} +(0.131637 + 0.114064i) q^{57} +(7.95319 + 1.53285i) q^{58} +(-6.24761 - 1.51565i) q^{59} +(-0.225424 + 2.36075i) q^{60} +(-8.62256 - 4.44524i) q^{61} +(-0.206611 + 0.703654i) q^{62} +(-0.620822 - 2.57188i) q^{63} +(2.58623 - 5.66306i) q^{64} +(0.930819 + 2.32507i) q^{65} +(0.802550 - 0.413743i) q^{66} +(-1.20037 - 6.22810i) q^{67} +(-1.72089 - 2.98066i) q^{68} +(4.64908 - 1.17732i) q^{69} +(3.13198 + 4.40890i) q^{70} +(-4.69301 - 5.41603i) q^{71} +(-0.139922 + 2.93732i) q^{72} +(-7.64589 + 9.72254i) q^{73} +(-3.45182 + 2.45803i) q^{74} +(-0.219737 - 0.279419i) q^{75} +(0.183904 + 0.0539990i) q^{76} +(-0.938772 + 2.33720i) q^{77} +(0.457882 + 1.00262i) q^{78} +(2.62002 - 2.74780i) q^{79} +(-0.414699 - 1.19819i) q^{80} +(-0.327068 - 0.945001i) q^{81} +(-2.31803 + 2.43108i) q^{82} +(2.15074 + 4.70947i) q^{83} +(-1.79708 - 2.29055i) q^{84} +(-6.46764 - 1.89907i) q^{85} +(-7.61561 - 9.68403i) q^{86} +(-6.95612 + 4.95343i) q^{87} +(1.73049 - 2.20049i) q^{88} +(0.459142 - 9.63857i) q^{89} +(1.33858 + 1.54481i) q^{90} +(-2.79533 - 1.28045i) q^{91} +(4.11534 - 3.30364i) q^{92} +(-0.386600 - 0.669611i) q^{93} +(-1.75291 - 9.09495i) q^{94} +(0.333650 - 0.172009i) q^{95} +(1.97846 + 4.94196i) q^{96} +(-4.50877 + 9.87283i) q^{97} +(-6.51645 - 1.27139i) q^{98} +(-0.268202 + 0.913412i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9} + 4 q^{18} - 28 q^{23} + 56 q^{25} - 84 q^{26} - 176 q^{28} - 24 q^{29} + 12 q^{31} + 36 q^{32} - 76 q^{35} + 28 q^{36} + 44 q^{37} - 110 q^{42} - 88 q^{43}+ \cdots - 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\) \(323\) \(346\) \(442\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{3}{22}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.310216 − 0.896309i 0.219356 − 0.633786i −0.780606 − 0.625023i \(-0.785089\pi\)
0.999962 − 0.00876283i \(-0.00278933\pi\)
\(3\) 0.458227 + 0.888835i 0.264557 + 0.513169i
\(4\) 0.864970 + 0.680220i 0.432485 + 0.340110i
\(5\) 2.14536 − 0.204857i 0.959433 − 0.0916148i 0.396426 − 0.918067i \(-0.370250\pi\)
0.563007 + 0.826452i \(0.309644\pi\)
\(6\) 0.938820 − 0.134982i 0.383272 − 0.0551062i
\(7\) −1.73488 + 1.99754i −0.655724 + 0.755001i
\(8\) 2.47383 − 1.58983i 0.874631 − 0.562091i
\(9\) −0.580057 + 0.814576i −0.193352 + 0.271525i
\(10\) 0.481908 − 1.98645i 0.152393 − 0.628172i
\(11\) 0.899616 − 0.311360i 0.271244 − 0.0938786i −0.188060 − 0.982158i \(-0.560220\pi\)
0.459304 + 0.888279i \(0.348099\pi\)
\(12\) −0.208251 + 1.08051i −0.0601170 + 0.311917i
\(13\) 0.327403 + 1.11503i 0.0908053 + 0.309254i 0.992354 − 0.123423i \(-0.0393871\pi\)
−0.901549 + 0.432677i \(0.857569\pi\)
\(14\) 1.25223 + 2.17466i 0.334673 + 0.581202i
\(15\) 1.16514 + 1.81300i 0.300839 + 0.468115i
\(16\) −0.138705 − 0.571750i −0.0346763 − 0.142937i
\(17\) −2.90371 − 1.16247i −0.704253 − 0.281940i −0.00823643 − 0.999966i \(-0.502622\pi\)
−0.696017 + 0.718026i \(0.745046\pi\)
\(18\) 0.550169 + 0.772604i 0.129676 + 0.182105i
\(19\) 0.161704 − 0.0647363i 0.0370973 − 0.0148515i −0.353040 − 0.935608i \(-0.614852\pi\)
0.390137 + 0.920757i \(0.372428\pi\)
\(20\) 1.99502 + 1.28212i 0.446100 + 0.286691i
\(21\) −2.57046 − 0.626697i −0.560920 − 0.136756i
\(22\) − 0.902923i − 0.192504i
\(23\) 1.08389 − 4.67174i 0.226006 − 0.974126i
\(24\) 2.54667 + 1.47032i 0.519838 + 0.300128i
\(25\) −0.349047 + 0.0672733i −0.0698094 + 0.0134547i
\(26\) 1.10098 + 0.0524461i 0.215920 + 0.0102855i
\(27\) −0.989821 − 0.142315i −0.190491 − 0.0273885i
\(28\) −2.85939 + 0.547715i −0.540374 + 0.103508i
\(29\) 1.21531 + 8.45264i 0.225677 + 1.56962i 0.716016 + 0.698084i \(0.245964\pi\)
−0.490339 + 0.871532i \(0.663127\pi\)
\(30\) 1.98645 − 0.481908i 0.362675 − 0.0879840i
\(31\) −0.772324 + 0.0367903i −0.138714 + 0.00660774i −0.116824 − 0.993153i \(-0.537271\pi\)
−0.0218897 + 0.999760i \(0.506968\pi\)
\(32\) 5.29917 + 0.506009i 0.936770 + 0.0894506i
\(33\) 0.688976 + 0.656937i 0.119935 + 0.114358i
\(34\) −1.94271 + 2.24200i −0.333172 + 0.384501i
\(35\) −3.31273 + 4.64085i −0.559954 + 0.784447i
\(36\) −1.05582 + 0.310018i −0.175970 + 0.0516696i
\(37\) −3.63934 − 2.59156i −0.598304 − 0.426050i 0.240372 − 0.970681i \(-0.422731\pi\)
−0.838676 + 0.544630i \(0.816670\pi\)
\(38\) −0.00786080 − 0.165019i −0.00127519 − 0.0267695i
\(39\) −0.841056 + 0.801945i −0.134677 + 0.128414i
\(40\) 4.98156 − 3.91754i 0.787654 − 0.619418i
\(41\) −3.22152 − 1.47122i −0.503117 − 0.229766i 0.147655 − 0.989039i \(-0.452827\pi\)
−0.650772 + 0.759273i \(0.725555\pi\)
\(42\) −1.35911 + 2.10951i −0.209715 + 0.325505i
\(43\) 7.02243 − 10.9271i 1.07091 − 1.66637i 0.419222 − 0.907884i \(-0.362303\pi\)
0.651689 − 0.758486i \(-0.274061\pi\)
\(44\) 0.989935 + 0.342620i 0.149238 + 0.0516519i
\(45\) −1.07756 + 1.86639i −0.160633 + 0.278224i
\(46\) −3.85109 − 2.42075i −0.567812 − 0.356920i
\(47\) 8.45717 − 4.88275i 1.23361 − 0.712223i 0.265826 − 0.964021i \(-0.414355\pi\)
0.967780 + 0.251798i \(0.0810220\pi\)
\(48\) 0.444633 − 0.385277i 0.0641773 − 0.0556099i
\(49\) −0.980370 − 6.93101i −0.140053 − 0.990144i
\(50\) −0.0479821 + 0.333723i −0.00678570 + 0.0471956i
\(51\) −0.297312 − 3.11359i −0.0416320 − 0.435990i
\(52\) −0.475274 + 1.18718i −0.0659086 + 0.164632i
\(53\) 5.40166 + 5.66510i 0.741975 + 0.778161i 0.981239 − 0.192793i \(-0.0617547\pi\)
−0.239264 + 0.970955i \(0.576906\pi\)
\(54\) −0.434616 + 0.843038i −0.0591438 + 0.114723i
\(55\) 1.86621 − 0.852272i 0.251640 − 0.114920i
\(56\) −1.11604 + 7.69976i −0.149137 + 1.02892i
\(57\) 0.131637 + 0.114064i 0.0174357 + 0.0151081i
\(58\) 7.95319 + 1.53285i 1.04430 + 0.201273i
\(59\) −6.24761 − 1.51565i −0.813370 − 0.197321i −0.192563 − 0.981285i \(-0.561680\pi\)
−0.620807 + 0.783963i \(0.713195\pi\)
\(60\) −0.225424 + 2.36075i −0.0291021 + 0.304771i
\(61\) −8.62256 − 4.44524i −1.10401 − 0.569155i −0.192855 − 0.981227i \(-0.561775\pi\)
−0.911151 + 0.412073i \(0.864805\pi\)
\(62\) −0.206611 + 0.703654i −0.0262397 + 0.0893642i
\(63\) −0.620822 − 2.57188i −0.0782162 − 0.324027i
\(64\) 2.58623 − 5.66306i 0.323279 − 0.707882i
\(65\) 0.930819 + 2.32507i 0.115454 + 0.288390i
\(66\) 0.802550 − 0.413743i 0.0987871 − 0.0509283i
\(67\) −1.20037 − 6.22810i −0.146648 − 0.760883i −0.978558 − 0.205972i \(-0.933964\pi\)
0.831910 − 0.554911i \(-0.187248\pi\)
\(68\) −1.72089 − 2.98066i −0.208688 − 0.361459i
\(69\) 4.64908 − 1.17732i 0.559683 − 0.141732i
\(70\) 3.13198 + 4.40890i 0.374343 + 0.526964i
\(71\) −4.69301 − 5.41603i −0.556958 − 0.642764i 0.405532 − 0.914081i \(-0.367086\pi\)
−0.962490 + 0.271317i \(0.912541\pi\)
\(72\) −0.139922 + 2.93732i −0.0164899 + 0.346166i
\(73\) −7.64589 + 9.72254i −0.894884 + 1.13794i 0.0949176 + 0.995485i \(0.469741\pi\)
−0.989802 + 0.142452i \(0.954501\pi\)
\(74\) −3.45182 + 2.45803i −0.401266 + 0.285740i
\(75\) −0.219737 − 0.279419i −0.0253731 − 0.0322645i
\(76\) 0.183904 + 0.0539990i 0.0210952 + 0.00619411i
\(77\) −0.938772 + 2.33720i −0.106983 + 0.266348i
\(78\) 0.457882 + 1.00262i 0.0518449 + 0.113525i
\(79\) 2.62002 − 2.74780i 0.294776 − 0.309152i −0.559646 − 0.828732i \(-0.689063\pi\)
0.854422 + 0.519580i \(0.173911\pi\)
\(80\) −0.414699 − 1.19819i −0.0463647 − 0.133962i
\(81\) −0.327068 − 0.945001i −0.0363409 − 0.105000i
\(82\) −2.31803 + 2.43108i −0.255984 + 0.268468i
\(83\) 2.15074 + 4.70947i 0.236075 + 0.516932i 0.990176 − 0.139827i \(-0.0446545\pi\)
−0.754101 + 0.656758i \(0.771927\pi\)
\(84\) −1.79708 − 2.29055i −0.196077 − 0.249920i
\(85\) −6.46764 − 1.89907i −0.701514 − 0.205983i
\(86\) −7.61561 − 9.68403i −0.821212 − 1.04426i
\(87\) −6.95612 + 4.95343i −0.745775 + 0.531064i
\(88\) 1.73049 − 2.20049i 0.184470 − 0.234573i
\(89\) 0.459142 − 9.63857i 0.0486689 − 1.02169i −0.834022 − 0.551732i \(-0.813967\pi\)
0.882691 − 0.469955i \(-0.155730\pi\)
\(90\) 1.33858 + 1.54481i 0.141099 + 0.162837i
\(91\) −2.79533 − 1.28045i −0.293031 − 0.134227i
\(92\) 4.11534 − 3.30364i 0.429054 − 0.344428i
\(93\) −0.386600 − 0.669611i −0.0400886 − 0.0694354i
\(94\) −1.75291 − 9.09495i −0.180799 − 0.938072i
\(95\) 0.333650 − 0.172009i 0.0342318 − 0.0176477i
\(96\) 1.97846 + 4.94196i 0.201926 + 0.504386i
\(97\) −4.50877 + 9.87283i −0.457796 + 1.00243i 0.530188 + 0.847880i \(0.322121\pi\)
−0.987984 + 0.154554i \(0.950606\pi\)
\(98\) −6.51645 − 1.27139i −0.658261 − 0.128430i
\(99\) −0.268202 + 0.913412i −0.0269553 + 0.0918014i
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 483.2.bf.a.10.21 ✓ 640
7.5 odd 6 inner 483.2.bf.a.355.12 yes 640
23.7 odd 22 inner 483.2.bf.a.283.12 yes 640
161.145 even 66 inner 483.2.bf.a.145.21 yes 640
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.bf.a.10.21 ✓ 640 1.1 even 1 trivial
483.2.bf.a.145.21 yes 640 161.145 even 66 inner
483.2.bf.a.283.12 yes 640 23.7 odd 22 inner
483.2.bf.a.355.12 yes 640 7.5 odd 6 inner