Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [234,2,Mod(25,234)] 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("234.25"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(234, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 234.t (of order \(6\), 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: \(1.86849940730\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 103.8
Character \(\chi\) \(=\) 234.103
Dual form 234.2.t.a.25.8

$q$-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+(0.866025 - 0.500000i) q^{2} +(-1.62009 + 0.612619i) q^{3} +(0.500000 - 0.866025i) q^{4} +(-0.548308 - 0.316566i) q^{5} +(-1.09673 + 1.34059i) q^{6} +(2.15366 - 1.24342i) q^{7} -1.00000i q^{8} +(2.24940 - 1.98500i) q^{9} -0.633132 q^{10} +(4.20891 - 2.43002i) q^{11} +(-0.279503 + 1.70935i) q^{12} +(3.35794 - 1.31310i) q^{13} +(1.24342 - 2.15366i) q^{14} +(1.08224 + 0.176962i) q^{15} +(-0.500000 - 0.866025i) q^{16} -6.27837 q^{17} +(0.955536 - 2.84376i) q^{18} +4.86004i q^{19} +(-0.548308 + 0.316566i) q^{20} +(-2.72739 + 3.33382i) q^{21} +(2.43002 - 4.20891i) q^{22} +(-1.77384 + 3.07238i) q^{23} +(0.612619 + 1.62009i) q^{24} +(-2.29957 - 3.98298i) q^{25} +(2.25152 - 2.81615i) q^{26} +(-2.42818 + 4.59390i) q^{27} -2.48683i q^{28} +(0.415447 + 0.719576i) q^{29} +(1.02573 - 0.387868i) q^{30} +(3.73461 + 2.15618i) q^{31} +(-0.866025 - 0.500000i) q^{32} +(-5.33015 + 6.51531i) q^{33} +(-5.43723 + 3.13918i) q^{34} -1.57449 q^{35} +(-0.594360 - 2.94053i) q^{36} -7.81310i q^{37} +(2.43002 + 4.20891i) q^{38} +(-4.63575 + 4.18448i) q^{39} +(-0.316566 + 0.548308i) q^{40} +(-0.0678657 - 0.0391823i) q^{41} +(-0.695076 + 4.25087i) q^{42} +(4.84370 + 8.38954i) q^{43} -4.86004i q^{44} +(-1.86175 + 0.376308i) q^{45} +3.54768i q^{46} +(-4.30069 + 2.48301i) q^{47} +(1.34059 + 1.09673i) q^{48} +(-0.407830 + 0.706382i) q^{49} +(-3.98298 - 2.29957i) q^{50} +(10.1715 - 3.84625i) q^{51} +(0.541797 - 3.56461i) q^{52} -6.36026 q^{53} +(0.194083 + 5.19253i) q^{54} -3.07704 q^{55} +(-1.24342 - 2.15366i) q^{56} +(-2.97735 - 7.87371i) q^{57} +(0.719576 + 0.415447i) q^{58} +(-7.86994 - 4.54371i) q^{59} +(0.694376 - 0.848770i) q^{60} +(5.28535 + 9.15449i) q^{61} +4.31236 q^{62} +(2.37626 - 7.07195i) q^{63} -1.00000 q^{64} +(-2.25687 - 0.343029i) q^{65} +(-1.35839 + 8.30750i) q^{66} +(6.82824 + 3.94228i) q^{67} +(-3.13918 + 5.43723i) q^{68} +(0.991586 - 6.06423i) q^{69} +(-1.36355 + 0.787247i) q^{70} +12.3085i q^{71} +(-1.98500 - 2.24940i) q^{72} -1.05367i q^{73} +(-3.90655 - 6.76635i) q^{74} +(6.16556 + 5.04403i) q^{75} +(4.20891 + 2.43002i) q^{76} +(6.04305 - 10.4669i) q^{77} +(-1.92244 + 5.94174i) q^{78} +(-1.68838 - 2.92436i) q^{79} +0.633132i q^{80} +(1.11957 - 8.93009i) q^{81} -0.0783645 q^{82} +(13.1755 - 7.60686i) q^{83} +(1.52348 + 4.02890i) q^{84} +(3.44248 + 1.98752i) q^{85} +(8.38954 + 4.84370i) q^{86} +(-1.11389 - 0.911268i) q^{87} +(-2.43002 - 4.20891i) q^{88} +0.595093i q^{89} +(-1.42416 + 1.25677i) q^{90} +(5.59914 - 7.00329i) q^{91} +(1.77384 + 3.07238i) q^{92} +(-7.37133 - 1.20531i) q^{93} +(-2.48301 + 4.30069i) q^{94} +(1.53852 - 2.66480i) q^{95} +(1.70935 + 0.279503i) q^{96} +(-14.7770 + 8.53151i) q^{97} +0.815659i q^{98} +(4.64394 - 13.8208i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q + 14 q^{4} - 16 q^{9} + 2 q^{13} + 8 q^{14} - 14 q^{16} + 16 q^{17} - 8 q^{23} + 14 q^{25} + 8 q^{26} + 18 q^{27} - 16 q^{29} - 8 q^{30} - 68 q^{35} - 8 q^{36} - 8 q^{39} - 10 q^{42} - 4 q^{43} + 10 q^{49}+ \cdots + 8 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/234\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.866025 − 0.500000i 0.612372 − 0.353553i
\(3\) −1.62009 + 0.612619i −0.935361 + 0.353696i
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) −0.548308 − 0.316566i −0.245211 − 0.141573i 0.372358 − 0.928089i \(-0.378549\pi\)
−0.617569 + 0.786516i \(0.711883\pi\)
\(6\) −1.09673 + 1.34059i −0.447739 + 0.547293i
\(7\) 2.15366 − 1.24342i 0.814007 − 0.469967i −0.0343382 − 0.999410i \(-0.510932\pi\)
0.848346 + 0.529443i \(0.177599\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 2.24940 − 1.98500i 0.749799 − 0.661666i
\(10\) −0.633132 −0.200214
\(11\) 4.20891 − 2.43002i 1.26904 − 0.732678i 0.294230 − 0.955735i \(-0.404937\pi\)
0.974805 + 0.223057i \(0.0716035\pi\)
\(12\) −0.279503 + 1.70935i −0.0806855 + 0.493447i
\(13\) 3.35794 − 1.31310i 0.931326 − 0.364187i
\(14\) 1.24342 − 2.15366i 0.332317 − 0.575590i
\(15\) 1.08224 + 0.176962i 0.279434 + 0.0456914i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −6.27837 −1.52273 −0.761364 − 0.648325i \(-0.775470\pi\)
−0.761364 + 0.648325i \(0.775470\pi\)
\(18\) 0.955536 − 2.84376i 0.225222 − 0.670280i
\(19\) 4.86004i 1.11497i 0.830187 + 0.557484i \(0.188233\pi\)
−0.830187 + 0.557484i \(0.811767\pi\)
\(20\) −0.548308 + 0.316566i −0.122605 + 0.0707863i
\(21\) −2.72739 + 3.33382i −0.595165 + 0.727500i
\(22\) 2.43002 − 4.20891i 0.518082 − 0.897344i
\(23\) −1.77384 + 3.07238i −0.369871 + 0.640636i −0.989545 − 0.144224i \(-0.953932\pi\)
0.619674 + 0.784859i \(0.287265\pi\)
\(24\) 0.612619 + 1.62009i 0.125050 + 0.330700i
\(25\) −2.29957 − 3.98298i −0.459914 − 0.796595i
\(26\) 2.25152 − 2.81615i 0.441559 − 0.552292i
\(27\) −2.42818 + 4.59390i −0.467304 + 0.884097i
\(28\) − 2.48683i − 0.469967i
\(29\) 0.415447 + 0.719576i 0.0771466 + 0.133622i 0.902018 − 0.431699i \(-0.142086\pi\)
−0.824871 + 0.565321i \(0.808752\pi\)
\(30\) 1.02573 − 0.387868i 0.187272 − 0.0708148i
\(31\) 3.73461 + 2.15618i 0.670756 + 0.387261i 0.796363 − 0.604819i \(-0.206755\pi\)
−0.125607 + 0.992080i \(0.540088\pi\)
\(32\) −0.866025 − 0.500000i −0.153093 − 0.0883883i
\(33\) −5.33015 + 6.51531i −0.927861 + 1.13417i
\(34\) −5.43723 + 3.13918i −0.932477 + 0.538366i
\(35\) −1.57449 −0.266138
\(36\) −0.594360 − 2.94053i −0.0990600 − 0.490089i
\(37\) − 7.81310i − 1.28447i −0.766509 − 0.642233i \(-0.778008\pi\)
0.766509 − 0.642233i \(-0.221992\pi\)
\(38\) 2.43002 + 4.20891i 0.394201 + 0.682776i
\(39\) −4.63575 + 4.18448i −0.742314 + 0.670052i
\(40\) −0.316566 + 0.548308i −0.0500535 + 0.0866952i
\(41\) −0.0678657 − 0.0391823i −0.0105988 − 0.00611924i 0.494691 − 0.869069i \(-0.335281\pi\)
−0.505290 + 0.862950i \(0.668614\pi\)
\(42\) −0.695076 + 4.25087i −0.107253 + 0.655923i
\(43\) 4.84370 + 8.38954i 0.738658 + 1.27939i 0.953100 + 0.302656i \(0.0978733\pi\)
−0.214442 + 0.976737i \(0.568793\pi\)
\(44\) − 4.86004i − 0.732678i
\(45\) −1.86175 + 0.376308i −0.277533 + 0.0560967i
\(46\) 3.54768i 0.523077i
\(47\) −4.30069 + 2.48301i −0.627320 + 0.362184i −0.779714 − 0.626136i \(-0.784635\pi\)
0.152393 + 0.988320i \(0.451302\pi\)
\(48\) 1.34059 + 1.09673i 0.193497 + 0.158300i
\(49\) −0.407830 + 0.706382i −0.0582614 + 0.100912i
\(50\) −3.98298 − 2.29957i −0.563278 − 0.325209i
\(51\) 10.1715 − 3.84625i 1.42430 − 0.538582i
\(52\) 0.541797 − 3.56461i 0.0751337 − 0.494323i
\(53\) −6.36026 −0.873649 −0.436824 − 0.899547i \(-0.643897\pi\)
−0.436824 + 0.899547i \(0.643897\pi\)
\(54\) 0.194083 + 5.19253i 0.0264113 + 0.706613i
\(55\) −3.07704 −0.414909
\(56\) −1.24342 − 2.15366i −0.166159 − 0.287795i
\(57\) −2.97735 − 7.87371i −0.394360 − 1.04290i
\(58\) 0.719576 + 0.415447i 0.0944849 + 0.0545509i
\(59\) −7.86994 − 4.54371i −1.02458 − 0.591541i −0.109152 − 0.994025i \(-0.534813\pi\)
−0.915427 + 0.402484i \(0.868147\pi\)
\(60\) 0.694376 − 0.848770i 0.0896435 − 0.109576i
\(61\) 5.28535 + 9.15449i 0.676719 + 1.17211i 0.975963 + 0.217935i \(0.0699322\pi\)
−0.299244 + 0.954177i \(0.596734\pi\)
\(62\) 4.31236 0.547670
\(63\) 2.37626 − 7.07195i 0.299380 − 0.890982i
\(64\) −1.00000 −0.125000
\(65\) −2.25687 − 0.343029i −0.279930 − 0.0425475i
\(66\) −1.35839 + 8.30750i −0.167207 + 1.02258i
\(67\) 6.82824 + 3.94228i 0.834202 + 0.481627i 0.855289 − 0.518151i \(-0.173380\pi\)
−0.0210874 + 0.999778i \(0.506713\pi\)
\(68\) −3.13918 + 5.43723i −0.380682 + 0.659361i
\(69\) 0.991586 − 6.06423i 0.119373 − 0.730047i
\(70\) −1.36355 + 0.787247i −0.162976 + 0.0940940i
\(71\) 12.3085i 1.46075i 0.683048 + 0.730374i \(0.260654\pi\)
−0.683048 + 0.730374i \(0.739346\pi\)
\(72\) −1.98500 − 2.24940i −0.233934 − 0.265094i
\(73\) − 1.05367i − 0.123323i −0.998097 − 0.0616615i \(-0.980360\pi\)
0.998097 − 0.0616615i \(-0.0196399\pi\)
\(74\) −3.90655 − 6.76635i −0.454127 − 0.786572i
\(75\) 6.16556 + 5.04403i 0.711938 + 0.582434i
\(76\) 4.20891 + 2.43002i 0.482796 + 0.278742i
\(77\) 6.04305 − 10.4669i 0.688669 − 1.19281i
\(78\) −1.92244 + 5.94174i −0.217673 + 0.672769i
\(79\) −1.68838 − 2.92436i −0.189958 − 0.329017i 0.755278 − 0.655404i \(-0.227502\pi\)
−0.945236 + 0.326388i \(0.894168\pi\)
\(80\) 0.633132i 0.0707863i
\(81\) 1.11957 − 8.93009i 0.124397 − 0.992233i
\(82\) −0.0783645 −0.00865391
\(83\) 13.1755 − 7.60686i 1.44620 − 0.834961i 0.447943 − 0.894062i \(-0.352157\pi\)
0.998252 + 0.0591005i \(0.0188232\pi\)
\(84\) 1.52348 + 4.02890i 0.166225 + 0.439589i
\(85\) 3.44248 + 1.98752i 0.373390 + 0.215577i
\(86\) 8.38954 + 4.84370i 0.904667 + 0.522310i
\(87\) −1.11389 − 0.911268i −0.119421 − 0.0976982i
\(88\) −2.43002 − 4.20891i −0.259041 − 0.448672i
\(89\) 0.595093i 0.0630797i 0.999502 + 0.0315398i \(0.0100411\pi\)
−0.999502 + 0.0315398i \(0.989959\pi\)
\(90\) −1.42416 + 1.25677i −0.150120 + 0.132475i
\(91\) 5.59914 − 7.00329i 0.586950 − 0.734144i
\(92\) 1.77384 + 3.07238i 0.184936 + 0.320318i
\(93\) −7.37133 − 1.20531i −0.764371 − 0.124985i
\(94\) −2.48301 + 4.30069i −0.256103 + 0.443583i
\(95\) 1.53852 − 2.66480i 0.157849 − 0.273403i
\(96\) 1.70935 + 0.279503i 0.174460 + 0.0285266i
\(97\) −14.7770 + 8.53151i −1.50038 + 0.866244i −0.500378 + 0.865807i \(0.666806\pi\)
−1.00000 0.000436560i \(0.999861\pi\)
\(98\) 0.815659i 0.0823940i
\(99\) 4.64394 − 13.8208i 0.466733 − 1.38904i
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 234.2.t.a.103.8 yes 28
3.2 odd 2 702.2.t.a.415.5 28
9.2 odd 6 702.2.t.a.181.10 28
9.4 even 3 2106.2.b.c.649.12 14
9.5 odd 6 2106.2.b.d.649.3 14
9.7 even 3 inner 234.2.t.a.25.1 ✓ 28
13.12 even 2 inner 234.2.t.a.103.1 yes 28
39.38 odd 2 702.2.t.a.415.10 28
117.25 even 6 inner 234.2.t.a.25.8 yes 28
117.38 odd 6 702.2.t.a.181.5 28
117.77 odd 6 2106.2.b.d.649.12 14
117.103 even 6 2106.2.b.c.649.3 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
234.2.t.a.25.1 ✓ 28 9.7 even 3 inner
234.2.t.a.25.8 yes 28 117.25 even 6 inner
234.2.t.a.103.1 yes 28 13.12 even 2 inner
234.2.t.a.103.8 yes 28 1.1 even 1 trivial
702.2.t.a.181.5 28 117.38 odd 6
702.2.t.a.181.10 28 9.2 odd 6
702.2.t.a.415.5 28 3.2 odd 2
702.2.t.a.415.10 28 39.38 odd 2
2106.2.b.c.649.3 14 117.103 even 6
2106.2.b.c.649.12 14 9.4 even 3
2106.2.b.d.649.3 14 9.5 odd 6
2106.2.b.d.649.12 14 117.77 odd 6