Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [147,2,Mod(5,147)] 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("147.5"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(147, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([21, 29])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 147.o (of order \(42\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [192] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.17380090971\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(16\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label 101.7
Character \(\chi\) \(=\) 147.101
Dual form 147.2.o.b.131.7

$q$-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+(-0.742405 - 0.291372i) q^{2} +(0.610083 + 1.62105i) q^{3} +(-0.999837 - 0.927713i) q^{4} +(-3.49434 - 2.38240i) q^{5} +(0.0194000 - 1.38124i) q^{6} +(-2.63459 - 0.242725i) q^{7} +(1.16405 + 2.41717i) q^{8} +(-2.25560 + 1.97795i) q^{9} +(1.90005 + 2.78686i) q^{10} +(-0.481626 - 3.19538i) q^{11} +(0.893884 - 2.18677i) q^{12} +(1.47939 - 1.17977i) q^{13} +(1.88521 + 0.947848i) q^{14} +(1.73015 - 7.11795i) q^{15} +(0.0439563 + 0.586557i) q^{16} +(0.330642 + 0.101990i) q^{17} +(2.25088 - 0.811220i) q^{18} +(-3.79474 + 2.19089i) q^{19} +(1.28358 + 5.62375i) q^{20} +(-1.21385 - 4.41889i) q^{21} +(-0.573484 + 2.51260i) q^{22} +(-0.617691 - 2.00251i) q^{23} +(-3.20819 + 3.36166i) q^{24} +(4.70786 + 11.9954i) q^{25} +(-1.44206 + 0.444815i) q^{26} +(-4.58245 - 2.44972i) q^{27} +(2.40899 + 2.68683i) q^{28} +(3.02682 - 0.690852i) q^{29} +(-3.35844 + 4.78028i) q^{30} +(-6.61413 - 3.81867i) q^{31} +(1.71985 - 5.57560i) q^{32} +(4.88603 - 2.73019i) q^{33} +(-0.215753 - 0.172058i) q^{34} +(8.62789 + 7.12481i) q^{35} +(4.09020 + 0.114920i) q^{36} +(4.19171 - 3.88934i) q^{37} +(3.45560 - 0.520848i) q^{38} +(2.81502 + 1.67840i) q^{39} +(1.69109 - 11.2196i) q^{40} +(-5.35893 + 2.58073i) q^{41} +(-0.386372 + 3.63428i) q^{42} +(-1.13376 - 0.545989i) q^{43} +(-2.48285 + 3.64167i) q^{44} +(12.5941 - 1.53789i) q^{45} +(-0.124898 + 1.66665i) q^{46} +(0.205644 - 0.523971i) q^{47} +(-0.924020 + 0.429104i) q^{48} +(6.88217 + 1.27896i) q^{49} -10.2772i q^{50} +(0.0363892 + 0.598209i) q^{51} +(-2.57363 - 0.192867i) q^{52} +(0.0176983 - 0.0190742i) q^{53} +(2.68825 + 3.15388i) q^{54} +(-5.92970 + 12.3132i) q^{55} +(-2.48009 - 6.65081i) q^{56} +(-5.86666 - 4.81483i) q^{57} +(-2.44842 - 0.369040i) q^{58} +(-2.74909 + 1.87430i) q^{59} +(-8.33328 + 5.51171i) q^{60} +(-4.20943 - 4.53669i) q^{61} +(3.79771 + 4.76217i) q^{62} +(6.42268 - 4.66360i) q^{63} +(-2.16792 + 2.71849i) q^{64} +(-7.98016 + 0.598030i) q^{65} +(-4.42291 + 0.603248i) q^{66} +(6.45002 - 11.1718i) q^{67} +(-0.235971 - 0.408714i) q^{68} +(2.86932 - 2.22300i) q^{69} +(-4.32941 - 7.80342i) q^{70} +(-7.01155 - 1.60034i) q^{71} +(-7.40667 - 3.14974i) q^{72} +(-2.47732 + 0.972276i) q^{73} +(-4.24519 + 1.66612i) q^{74} +(-16.5730 + 14.9499i) q^{75} +(5.82665 + 1.32989i) q^{76} +(0.493290 + 8.53543i) q^{77} +(-1.60084 - 2.06627i) q^{78} +(-1.49210 - 2.58439i) q^{79} +(1.24381 - 2.15435i) q^{80} +(1.17543 - 8.92291i) q^{81} +(4.73045 - 0.354498i) q^{82} +(-5.31648 + 6.66666i) q^{83} +(-2.88581 + 5.54427i) q^{84} +(-0.912395 - 1.14411i) q^{85} +(0.682621 + 0.735691i) q^{86} +(2.96652 + 4.48514i) q^{87} +(7.16315 - 4.88375i) q^{88} +(1.88700 + 0.284420i) q^{89} +(-9.79800 - 2.52783i) q^{90} +(-4.18394 + 2.74913i) q^{91} +(-1.24016 + 2.57522i) q^{92} +(2.15508 - 13.0515i) q^{93} +(-0.305341 + 0.329080i) q^{94} +(18.4797 + 1.38486i) q^{95} +(10.0876 - 0.613629i) q^{96} -11.3787i q^{97} +(-4.73670 - 2.95478i) q^{98} +(7.40665 + 6.25486i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 192 q - 14 q^{3} - 44 q^{4} - 28 q^{6} - 28 q^{7} - 26 q^{9} - 28 q^{10} + 21 q^{12} - 28 q^{13} - 10 q^{15} - 20 q^{16} - 3 q^{18} - 42 q^{19} - 14 q^{21} - 34 q^{22} - 14 q^{24} - 16 q^{25} + 7 q^{27}+ \cdots - 136 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\) \(50\) \(52\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{42}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.742405 − 0.291372i −0.524959 − 0.206031i 0.0880369 − 0.996117i \(-0.471941\pi\)
−0.612996 + 0.790086i \(0.710036\pi\)
\(3\) 0.610083 + 1.62105i 0.352232 + 0.935913i
\(4\) −0.999837 − 0.927713i −0.499919 − 0.463857i
\(5\) −3.49434 − 2.38240i −1.56271 − 1.06544i −0.964492 − 0.264110i \(-0.914922\pi\)
−0.598222 − 0.801330i \(-0.704126\pi\)
\(6\) 0.0194000 − 1.38124i 0.00792001 − 0.563887i
\(7\) −2.63459 − 0.242725i −0.995783 − 0.0917415i
\(8\) 1.16405 + 2.41717i 0.411553 + 0.854599i
\(9\) −2.25560 + 1.97795i −0.751866 + 0.659317i
\(10\) 1.90005 + 2.78686i 0.600847 + 0.881281i
\(11\) −0.481626 − 3.19538i −0.145216 − 0.963443i −0.936165 − 0.351561i \(-0.885651\pi\)
0.790949 − 0.611882i \(-0.209587\pi\)
\(12\) 0.893884 − 2.18677i 0.258042 − 0.631265i
\(13\) 1.47939 − 1.17977i 0.410308 − 0.327210i −0.396488 − 0.918040i \(-0.629771\pi\)
0.806796 + 0.590830i \(0.201200\pi\)
\(14\) 1.88521 + 0.947848i 0.503844 + 0.253323i
\(15\) 1.73015 − 7.11795i 0.446722 − 1.83785i
\(16\) 0.0439563 + 0.586557i 0.0109891 + 0.146639i
\(17\) 0.330642 + 0.101990i 0.0801925 + 0.0247361i 0.334592 − 0.942363i \(-0.391402\pi\)
−0.254399 + 0.967099i \(0.581878\pi\)
\(18\) 2.25088 − 0.811220i 0.530539 − 0.191206i
\(19\) −3.79474 + 2.19089i −0.870573 + 0.502626i −0.867539 − 0.497370i \(-0.834299\pi\)
−0.00303461 + 0.999995i \(0.500966\pi\)
\(20\) 1.28358 + 5.62375i 0.287018 + 1.25751i
\(21\) −1.21385 − 4.41889i −0.264884 − 0.964280i
\(22\) −0.573484 + 2.51260i −0.122267 + 0.535687i
\(23\) −0.617691 − 2.00251i −0.128798 − 0.417551i 0.868088 − 0.496410i \(-0.165349\pi\)
−0.996886 + 0.0788585i \(0.974872\pi\)
\(24\) −3.20819 + 3.36166i −0.654868 + 0.686195i
\(25\) 4.70786 + 11.9954i 0.941572 + 2.39909i
\(26\) −1.44206 + 0.444815i −0.282810 + 0.0872355i
\(27\) −4.58245 − 2.44972i −0.881894 − 0.471448i
\(28\) 2.40899 + 2.68683i 0.455255 + 0.507764i
\(29\) 3.02682 − 0.690852i 0.562066 − 0.128288i 0.0679577 − 0.997688i \(-0.478352\pi\)
0.494108 + 0.869400i \(0.335495\pi\)
\(30\) −3.35844 + 4.78028i −0.613165 + 0.872756i
\(31\) −6.61413 − 3.81867i −1.18793 − 0.685853i −0.230097 − 0.973168i \(-0.573904\pi\)
−0.957836 + 0.287314i \(0.907238\pi\)
\(32\) 1.71985 − 5.57560i 0.304029 − 0.985637i
\(33\) 4.88603 − 2.73019i 0.850549 − 0.475265i
\(34\) −0.215753 − 0.172058i −0.0370014 − 0.0295076i
\(35\) 8.62789 + 7.12481i 1.45838 + 1.20431i
\(36\) 4.09020 + 0.114920i 0.681700 + 0.0191533i
\(37\) 4.19171 − 3.88934i 0.689113 − 0.639404i −0.255798 − 0.966730i \(-0.582338\pi\)
0.944911 + 0.327327i \(0.106148\pi\)
\(38\) 3.45560 − 0.520848i 0.560572 − 0.0844927i
\(39\) 2.81502 + 1.67840i 0.450763 + 0.268759i
\(40\) 1.69109 − 11.2196i 0.267385 − 1.77398i
\(41\) −5.35893 + 2.58073i −0.836925 + 0.403042i −0.802708 − 0.596373i \(-0.796608\pi\)
−0.0342172 + 0.999414i \(0.510894\pi\)
\(42\) −0.386372 + 3.63428i −0.0596184 + 0.560782i
\(43\) −1.13376 − 0.545989i −0.172896 − 0.0832626i 0.345434 − 0.938443i \(-0.387732\pi\)
−0.518330 + 0.855181i \(0.673446\pi\)
\(44\) −2.48285 + 3.64167i −0.374303 + 0.549002i
\(45\) 12.5941 − 1.53789i 1.87741 − 0.229255i
\(46\) −0.124898 + 1.66665i −0.0184152 + 0.245734i
\(47\) 0.205644 − 0.523971i 0.0299962 − 0.0764290i −0.915091 − 0.403248i \(-0.867881\pi\)
0.945087 + 0.326819i \(0.105977\pi\)
\(48\) −0.924020 + 0.429104i −0.133371 + 0.0619358i
\(49\) 6.88217 + 1.27896i 0.983167 + 0.182709i
\(50\) − 10.2772i − 1.45342i
\(51\) 0.0363892 + 0.598209i 0.00509551 + 0.0837661i
\(52\) −2.57363 − 0.192867i −0.356899 − 0.0267459i
\(53\) 0.0176983 − 0.0190742i 0.00243104 − 0.00262004i −0.731835 − 0.681482i \(-0.761336\pi\)
0.734266 + 0.678862i \(0.237526\pi\)
\(54\) 2.68825 + 3.15388i 0.365825 + 0.429189i
\(55\) −5.92970 + 12.3132i −0.799561 + 1.66031i
\(56\) −2.48009 − 6.65081i −0.331416 − 0.888752i
\(57\) −5.86666 − 4.81483i −0.777057 − 0.637740i
\(58\) −2.44842 − 0.369040i −0.321493 − 0.0484573i
\(59\) −2.74909 + 1.87430i −0.357901 + 0.244013i −0.728899 − 0.684621i \(-0.759968\pi\)
0.370999 + 0.928633i \(0.379015\pi\)
\(60\) −8.33328 + 5.51171i −1.07582 + 0.711559i
\(61\) −4.20943 − 4.53669i −0.538963 − 0.580864i 0.403582 − 0.914943i \(-0.367765\pi\)
−0.942545 + 0.334080i \(0.891575\pi\)
\(62\) 3.79771 + 4.76217i 0.482309 + 0.604797i
\(63\) 6.42268 − 4.66360i 0.809181 − 0.587559i
\(64\) −2.16792 + 2.71849i −0.270990 + 0.339811i
\(65\) −7.98016 + 0.598030i −0.989817 + 0.0741765i
\(66\) −4.42291 + 0.603248i −0.544423 + 0.0742547i
\(67\) 6.45002 − 11.1718i 0.787995 − 1.36485i −0.139198 − 0.990265i \(-0.544452\pi\)
0.927193 − 0.374583i \(-0.122214\pi\)
\(68\) −0.235971 − 0.408714i −0.0286157 − 0.0495639i
\(69\) 2.86932 − 2.22300i 0.345425 − 0.267618i
\(70\) −4.32941 − 7.80342i −0.517463 − 0.932687i
\(71\) −7.01155 − 1.60034i −0.832118 − 0.189925i −0.214818 − 0.976654i \(-0.568916\pi\)
−0.617299 + 0.786729i \(0.711773\pi\)
\(72\) −7.40667 − 3.14974i −0.872884 − 0.371200i
\(73\) −2.47732 + 0.972276i −0.289948 + 0.113796i −0.505857 − 0.862617i \(-0.668824\pi\)
0.215909 + 0.976413i \(0.430729\pi\)
\(74\) −4.24519 + 1.66612i −0.493494 + 0.193682i
\(75\) −16.5730 + 14.9499i −1.91368 + 1.72626i
\(76\) 5.82665 + 1.32989i 0.668362 + 0.152549i
\(77\) 0.493290 + 8.53543i 0.0562156 + 0.972703i
\(78\) −1.60084 − 2.06627i −0.181260 − 0.233959i
\(79\) −1.49210 − 2.58439i −0.167874 − 0.290767i 0.769798 − 0.638288i \(-0.220357\pi\)
−0.937672 + 0.347521i \(0.887024\pi\)
\(80\) 1.24381 − 2.15435i 0.139063 − 0.240863i
\(81\) 1.17543 − 8.92291i 0.130603 − 0.991435i
\(82\) 4.73045 − 0.354498i 0.522391 − 0.0391478i
\(83\) −5.31648 + 6.66666i −0.583560 + 0.731761i −0.982716 − 0.185122i \(-0.940732\pi\)
0.399156 + 0.916883i \(0.369303\pi\)
\(84\) −2.88581 + 5.54427i −0.314867 + 0.604930i
\(85\) −0.912395 − 1.14411i −0.0989632 − 0.124096i
\(86\) 0.682621 + 0.735691i 0.0736089 + 0.0793316i
\(87\) 2.96652 + 4.48514i 0.318044 + 0.480858i
\(88\) 7.16315 − 4.88375i 0.763594 − 0.520610i
\(89\) 1.88700 + 0.284420i 0.200022 + 0.0301484i 0.248289 − 0.968686i \(-0.420132\pi\)
−0.0482672 + 0.998834i \(0.515370\pi\)
\(90\) −9.79800 − 2.52783i −1.03280 − 0.266456i
\(91\) −4.18394 + 2.74913i −0.438596 + 0.288188i
\(92\) −1.24016 + 2.57522i −0.129296 + 0.268485i
\(93\) 2.15508 − 13.0515i 0.223471 − 1.35338i
\(94\) −0.305341 + 0.329080i −0.0314936 + 0.0339420i
\(95\) 18.4797 + 1.38486i 1.89598 + 0.142084i
\(96\) 10.0876 − 0.613629i 1.02956 − 0.0626283i
\(97\) − 11.3787i − 1.15533i −0.816275 − 0.577664i \(-0.803965\pi\)
0.816275 − 0.577664i \(-0.196035\pi\)
\(98\) −4.73670 − 2.95478i −0.478479 − 0.298478i
\(99\) 7.40665 + 6.25486i 0.744397 + 0.628637i
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 147.2.o.b.101.7 ✓ 192
3.2 odd 2 inner 147.2.o.b.101.10 yes 192
49.33 odd 42 inner 147.2.o.b.131.10 yes 192
147.131 even 42 inner 147.2.o.b.131.7 yes 192
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.2.o.b.101.7 ✓ 192 1.1 even 1 trivial
147.2.o.b.101.10 yes 192 3.2 odd 2 inner
147.2.o.b.131.7 yes 192 147.131 even 42 inner
147.2.o.b.131.10 yes 192 49.33 odd 42 inner