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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 10.32
Character \(\chi\) \(=\) 483.10
Dual form 483.2.bf.a.145.32

$q$-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+(0.904830 - 2.61433i) q^{2} +(-0.458227 - 0.888835i) q^{3} +(-4.44392 - 3.49473i) q^{4} +(-2.90497 + 0.277391i) q^{5} +(-2.73833 + 0.393712i) q^{6} +(1.81111 + 1.92870i) q^{7} +(-8.50275 + 5.46439i) q^{8} +(-0.580057 + 0.814576i) q^{9} +(-1.90331 + 7.84556i) q^{10} +(1.83416 - 0.634810i) q^{11} +(-1.06992 + 5.55129i) q^{12} +(-1.56688 - 5.33629i) q^{13} +(6.68101 - 2.98970i) q^{14} +(1.57769 + 2.45493i) q^{15} +(3.92648 + 16.1852i) q^{16} +(-1.82046 - 0.728802i) q^{17} +(1.60472 + 2.25351i) q^{18} +(-3.59927 + 1.44093i) q^{19} +(13.8789 + 8.91940i) q^{20} +(0.884399 - 2.49356i) q^{21} -5.36951i q^{22} +(-3.24008 + 3.53580i) q^{23} +(8.75313 + 5.05362i) q^{24} +(3.45228 - 0.665372i) q^{25} +(-15.3686 - 0.732096i) q^{26} +(0.989821 + 0.142315i) q^{27} +(-1.30812 - 14.9003i) q^{28} +(-0.964793 - 6.71029i) q^{29} +(7.84556 - 1.90331i) q^{30} +(-0.198691 + 0.00946481i) q^{31} +(25.7433 + 2.45819i) q^{32} +(-1.40470 - 1.33938i) q^{33} +(-3.55254 + 4.09985i) q^{34} +(-5.79623 - 5.10043i) q^{35} +(5.42445 - 1.59276i) q^{36} +(5.84350 + 4.16114i) q^{37} +(0.510346 + 10.7135i) q^{38} +(-4.02510 + 3.83792i) q^{39} +(23.1845 - 18.2325i) q^{40} +(-9.74867 - 4.45207i) q^{41} +(-5.71877 - 4.56836i) q^{42} +(3.04441 - 4.73720i) q^{43} +(-10.3694 - 3.58887i) q^{44} +(1.45909 - 2.52722i) q^{45} +(6.31205 + 11.6699i) q^{46} +(-5.51234 + 3.18255i) q^{47} +(12.5867 - 10.9065i) q^{48} +(-0.439767 + 6.98617i) q^{49} +(1.38422 - 9.62745i) q^{50} +(0.186398 + 1.95205i) q^{51} +(-11.6858 + 29.1898i) q^{52} +(-7.68016 - 8.05472i) q^{53} +(1.26768 - 2.45895i) q^{54} +(-5.15210 + 2.35289i) q^{55} +(-25.9386 - 6.50265i) q^{56} +(2.93003 + 2.53889i) q^{57} +(-18.4159 - 3.54937i) q^{58} +(-4.25150 - 1.03140i) q^{59} +(1.56822 - 16.4231i) q^{60} +(3.90196 + 2.01160i) q^{61} +(-0.155037 + 0.528008i) q^{62} +(-2.62162 + 0.356530i) q^{63} +(15.8826 - 34.7781i) q^{64} +(6.03197 + 15.0671i) q^{65} +(-4.77261 + 2.46045i) q^{66} +(-0.685990 - 3.55925i) q^{67} +(5.54300 + 9.60076i) q^{68} +(4.62744 + 1.25970i) q^{69} +(-18.5788 + 10.5382i) q^{70} +(3.36160 + 3.87949i) q^{71} +(0.480922 - 10.0958i) q^{72} +(0.207920 - 0.264392i) q^{73} +(16.1660 - 11.5117i) q^{74} +(-2.17333 - 2.76362i) q^{75} +(21.0305 + 6.17512i) q^{76} +(4.54623 + 2.38784i) q^{77} +(6.39158 + 13.9956i) q^{78} +(3.93917 - 4.13128i) q^{79} +(-15.8959 - 45.9283i) q^{80} +(-0.327068 - 0.945001i) q^{81} +(-20.4601 + 21.4579i) q^{82} +(-1.40584 - 3.07835i) q^{83} +(-12.6445 + 7.99043i) q^{84} +(5.49055 + 1.61217i) q^{85} +(-9.62994 - 12.2455i) q^{86} +(-5.52225 + 3.93237i) q^{87} +(-12.1266 + 15.4202i) q^{88} +(0.569021 - 11.9452i) q^{89} +(-5.28677 - 6.10126i) q^{90} +(7.45432 - 12.6866i) q^{91} +(26.7553 - 4.38959i) q^{92} +(0.0994581 + 0.172266i) q^{93} +(3.33252 + 17.2908i) q^{94} +(10.0561 - 5.18427i) q^{95} +(-9.61134 - 24.0080i) q^{96} +(1.54462 - 3.38225i) q^{97} +(17.8663 + 7.47099i) q^{98} +(-0.546818 + 1.86229i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.904830 − 2.61433i 0.639811 − 1.84861i 0.123222 − 0.992379i \(-0.460677\pi\)
0.516589 − 0.856234i \(-0.327202\pi\)
\(3\) −0.458227 − 0.888835i −0.264557 − 0.513169i
\(4\) −4.44392 − 3.49473i −2.22196 − 1.74737i
\(5\) −2.90497 + 0.277391i −1.29914 + 0.124053i −0.721602 − 0.692308i \(-0.756594\pi\)
−0.577541 + 0.816361i \(0.695988\pi\)
\(6\) −2.73833 + 0.393712i −1.11792 + 0.160732i
\(7\) 1.81111 + 1.92870i 0.684535 + 0.728980i
\(8\) −8.50275 + 5.46439i −3.00618 + 1.93195i
\(9\) −0.580057 + 0.814576i −0.193352 + 0.271525i
\(10\) −1.90331 + 7.84556i −0.601880 + 2.48098i
\(11\) 1.83416 − 0.634810i 0.553021 − 0.191402i −0.0362430 − 0.999343i \(-0.511539\pi\)
0.589264 + 0.807941i \(0.299418\pi\)
\(12\) −1.06992 + 5.55129i −0.308860 + 1.60252i
\(13\) −1.56688 − 5.33629i −0.434573 − 1.48002i −0.828033 − 0.560679i \(-0.810540\pi\)
0.393460 − 0.919342i \(-0.371278\pi\)
\(14\) 6.68101 − 2.98970i 1.78557 − 0.799031i
\(15\) 1.57769 + 2.45493i 0.407358 + 0.633861i
\(16\) 3.92648 + 16.1852i 0.981620 + 4.04629i
\(17\) −1.82046 − 0.728802i −0.441526 − 0.176761i 0.140238 − 0.990118i \(-0.455213\pi\)
−0.581764 + 0.813357i \(0.697637\pi\)
\(18\) 1.60472 + 2.25351i 0.378236 + 0.531158i
\(19\) −3.59927 + 1.44093i −0.825729 + 0.330572i −0.745751 − 0.666225i \(-0.767909\pi\)
−0.0799779 + 0.996797i \(0.525485\pi\)
\(20\) 13.8789 + 8.91940i 3.10341 + 1.99444i
\(21\) 0.884399 − 2.49356i 0.192992 − 0.544139i
\(22\) − 5.36951i − 1.14478i
\(23\) −3.24008 + 3.53580i −0.675603 + 0.737266i
\(24\) 8.75313 + 5.05362i 1.78672 + 1.03157i
\(25\) 3.45228 − 0.665372i 0.690455 − 0.133074i
\(26\) −15.3686 − 0.732096i −3.01403 − 0.143576i
\(27\) 0.989821 + 0.142315i 0.190491 + 0.0273885i
\(28\) −1.30812 − 14.9003i −0.247212 − 2.81590i
\(29\) −0.964793 − 6.71029i −0.179158 − 1.24607i −0.858718 − 0.512449i \(-0.828738\pi\)
0.679560 − 0.733620i \(-0.262171\pi\)
\(30\) 7.84556 − 1.90331i 1.43240 − 0.347496i
\(31\) −0.198691 + 0.00946481i −0.0356859 + 0.00169993i −0.0654173 − 0.997858i \(-0.520838\pi\)
0.0297314 + 0.999558i \(0.490535\pi\)
\(32\) 25.7433 + 2.45819i 4.55081 + 0.434550i
\(33\) −1.40470 − 1.33938i −0.244528 − 0.233157i
\(34\) −3.55254 + 4.09985i −0.609255 + 0.703118i
\(35\) −5.79623 − 5.10043i −0.979741 − 0.862131i
\(36\) 5.42445 − 1.59276i 0.904075 − 0.265460i
\(37\) 5.84350 + 4.16114i 0.960665 + 0.684087i 0.948811 − 0.315843i \(-0.102287\pi\)
0.0118538 + 0.999930i \(0.496227\pi\)
\(38\) 0.510346 + 10.7135i 0.0827891 + 1.73796i
\(39\) −4.02510 + 3.83792i −0.644532 + 0.614560i
\(40\) 23.1845 − 18.2325i 3.66579 − 2.88281i
\(41\) −9.74867 − 4.45207i −1.52249 − 0.695296i −0.533844 − 0.845583i \(-0.679253\pi\)
−0.988643 + 0.150286i \(0.951980\pi\)
\(42\) −5.71877 − 4.56836i −0.882425 − 0.704913i
\(43\) 3.04441 − 4.73720i 0.464268 − 0.722416i −0.527626 − 0.849476i \(-0.676918\pi\)
0.991895 + 0.127061i \(0.0405543\pi\)
\(44\) −10.3694 − 3.58887i −1.56324 − 0.541042i
\(45\) 1.45909 − 2.52722i 0.217509 − 0.376736i
\(46\) 6.31205 + 11.6699i 0.930660 + 1.72064i
\(47\) −5.51234 + 3.18255i −0.804058 + 0.464223i −0.844888 − 0.534943i \(-0.820333\pi\)
0.0408302 + 0.999166i \(0.487000\pi\)
\(48\) 12.5867 − 10.9065i 1.81674 − 1.57421i
\(49\) −0.439767 + 6.98617i −0.0628238 + 0.998025i
\(50\) 1.38422 − 9.62745i 0.195758 − 1.36153i
\(51\) 0.186398 + 1.95205i 0.0261009 + 0.273341i
\(52\) −11.6858 + 29.1898i −1.62054 + 4.04790i
\(53\) −7.68016 − 8.05472i −1.05495 − 1.10640i −0.994123 − 0.108252i \(-0.965475\pi\)
−0.0608275 − 0.998148i \(-0.519374\pi\)
\(54\) 1.26768 − 2.45895i 0.172509 − 0.334621i
\(55\) −5.15210 + 2.35289i −0.694709 + 0.317263i
\(56\) −25.9386 − 6.50265i −3.46619 − 0.868954i
\(57\) 2.93003 + 2.53889i 0.388092 + 0.336284i
\(58\) −18.4159 − 3.54937i −2.41813 − 0.466056i
\(59\) −4.25150 − 1.03140i −0.553498 − 0.134277i −0.0507521 − 0.998711i \(-0.516162\pi\)
−0.502746 + 0.864434i \(0.667677\pi\)
\(60\) 1.56822 − 16.4231i 0.202456 − 2.12022i
\(61\) 3.90196 + 2.01160i 0.499595 + 0.257559i 0.689556 − 0.724233i \(-0.257806\pi\)
−0.189961 + 0.981792i \(0.560836\pi\)
\(62\) −0.155037 + 0.528008i −0.0196897 + 0.0670571i
\(63\) −2.62162 + 0.356530i −0.330293 + 0.0449186i
\(64\) 15.8826 − 34.7781i 1.98533 − 4.34726i
\(65\) 6.03197 + 15.0671i 0.748174 + 1.86885i
\(66\) −4.77261 + 2.46045i −0.587468 + 0.302861i
\(67\) −0.685990 − 3.55925i −0.0838070 − 0.434832i −0.999460 − 0.0328622i \(-0.989538\pi\)
0.915653 − 0.401970i \(-0.131674\pi\)
\(68\) 5.54300 + 9.60076i 0.672188 + 1.16426i
\(69\) 4.62744 + 1.25970i 0.557078 + 0.151650i
\(70\) −18.5788 + 10.5382i −2.22060 + 1.25956i
\(71\) 3.36160 + 3.87949i 0.398948 + 0.460411i 0.919309 − 0.393535i \(-0.128748\pi\)
−0.520361 + 0.853946i \(0.674203\pi\)
\(72\) 0.480922 − 10.0958i 0.0566772 − 1.18980i
\(73\) 0.207920 − 0.264392i 0.0243352 − 0.0309447i −0.773733 − 0.633512i \(-0.781613\pi\)
0.798068 + 0.602567i \(0.205855\pi\)
\(74\) 16.1660 − 11.5117i 1.87926 − 1.33821i
\(75\) −2.17333 − 2.76362i −0.250955 − 0.319115i
\(76\) 21.0305 + 6.17512i 2.41237 + 0.708335i
\(77\) 4.54623 + 2.38784i 0.518091 + 0.272120i
\(78\) 6.39158 + 13.9956i 0.723704 + 1.58469i
\(79\) 3.93917 − 4.13128i 0.443191 − 0.464805i −0.463810 − 0.885935i \(-0.653518\pi\)
0.907000 + 0.421130i \(0.138366\pi\)
\(80\) −15.8959 − 45.9283i −1.77722 − 5.13494i
\(81\) −0.327068 − 0.945001i −0.0363409 − 0.105000i
\(82\) −20.4601 + 21.4579i −2.25944 + 2.36963i
\(83\) −1.40584 − 3.07835i −0.154311 − 0.337893i 0.816650 − 0.577134i \(-0.195829\pi\)
−0.970960 + 0.239240i \(0.923102\pi\)
\(84\) −12.6445 + 7.99043i −1.37963 + 0.871827i
\(85\) 5.49055 + 1.61217i 0.595534 + 0.174865i
\(86\) −9.62994 − 12.2455i −1.03842 − 1.32046i
\(87\) −5.52225 + 3.93237i −0.592047 + 0.421595i
\(88\) −12.1266 + 15.4202i −1.29270 + 1.64380i
\(89\) 0.569021 − 11.9452i 0.0603161 − 1.26619i −0.742452 − 0.669899i \(-0.766337\pi\)
0.802768 − 0.596291i \(-0.203360\pi\)
\(90\) −5.28677 − 6.10126i −0.557275 − 0.643130i
\(91\) 7.45432 − 12.6866i 0.781425 − 1.32992i
\(92\) 26.7553 − 4.38959i 2.78944 − 0.457647i
\(93\) 0.0994581 + 0.172266i 0.0103133 + 0.0178632i
\(94\) 3.33252 + 17.2908i 0.343724 + 1.78341i
\(95\) 10.0561 − 5.18427i 1.03173 − 0.531895i
\(96\) −9.61134 − 24.0080i −0.980953 − 2.45030i
\(97\) 1.54462 − 3.38225i 0.156833 − 0.343416i −0.814862 − 0.579655i \(-0.803187\pi\)
0.971695 + 0.236239i \(0.0759147\pi\)
\(98\) 17.8663 + 7.47099i 1.80477 + 0.754684i
\(99\) −0.546818 + 1.86229i −0.0549573 + 0.187167i
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.10.32 ✓ 640
7.5 odd 6 inner 483.2.bf.a.355.1 yes 640
23.7 odd 22 inner 483.2.bf.a.283.1 yes 640
161.145 even 66 inner 483.2.bf.a.145.32 yes 640
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.bf.a.10.32 ✓ 640 1.1 even 1 trivial
483.2.bf.a.145.32 yes 640 161.145 even 66 inner
483.2.bf.a.283.1 yes 640 23.7 odd 22 inner
483.2.bf.a.355.1 yes 640 7.5 odd 6 inner