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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [441,2,Mod(17,441)] 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("441.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(441, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([21, 25])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.bg (of order \(42\), degree \(12\), 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: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(216\)
Relative dimension: \(18\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label 395.9
Character \(\chi\) \(=\) 441.395
Dual form 441.2.bg.a.278.9

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.132489 - 0.0519980i) q^{2} +(-1.45125 - 1.34657i) q^{4} +(-1.27700 - 0.870646i) q^{5} +(-1.61234 - 2.09770i) q^{7} +(0.245763 + 0.510332i) q^{8} +(0.123917 + 0.181752i) q^{10} +(0.796043 + 5.28140i) q^{11} +(-4.42567 + 3.52936i) q^{13} +(0.104541 + 0.361760i) q^{14} +(0.289868 + 3.86802i) q^{16} +(2.27068 + 0.700411i) q^{17} +(2.04813 - 1.18249i) q^{19} +(0.680873 + 2.98310i) q^{20} +(0.169155 - 0.741118i) q^{22} +(1.72793 + 5.60180i) q^{23} +(-0.953993 - 2.43073i) q^{25} +(0.769871 - 0.237474i) q^{26} +(-0.484776 + 5.21543i) q^{28} +(-9.35903 + 2.13614i) q^{29} +(-6.48566 - 3.74450i) q^{31} +(0.496639 - 1.61006i) q^{32} +(-0.264419 - 0.210867i) q^{34} +(0.232612 + 4.08255i) q^{35} +(1.55935 - 1.44686i) q^{37} +(-0.332840 + 0.0501676i) q^{38} +(0.130478 - 0.865668i) q^{40} +(0.180363 - 0.0868585i) q^{41} +(-5.56553 - 2.68022i) q^{43} +(5.95650 - 8.73658i) q^{44} +(0.0623516 - 0.832024i) q^{46} +(4.15730 - 10.5926i) q^{47} +(-1.80070 + 6.76443i) q^{49} +0.371651i q^{50} +(11.1753 + 0.837473i) q^{52} +(-2.89757 + 3.12284i) q^{53} +(3.58168 - 7.43743i) q^{55} +(0.674270 - 1.33837i) q^{56} +(1.35104 + 0.203637i) q^{58} +(-10.3645 + 7.06637i) q^{59} +(3.33764 + 3.59712i) q^{61} +(0.664571 + 0.833345i) q^{62} +(4.68735 - 5.87775i) q^{64} +(8.72442 - 0.653805i) q^{65} +(-4.48870 + 7.77466i) q^{67} +(-2.35218 - 4.07409i) q^{68} +(0.181466 - 0.552987i) q^{70} +(-12.2266 - 2.79063i) q^{71} +(5.02779 - 1.97326i) q^{73} +(-0.281830 + 0.110610i) q^{74} +(-4.56465 - 1.04185i) q^{76} +(9.79530 - 10.1853i) q^{77} +(1.88456 + 3.26415i) q^{79} +(2.99751 - 5.19185i) q^{80} +(-0.0284126 + 0.00212923i) q^{82} +(-5.58349 + 7.00147i) q^{83} +(-2.28985 - 2.87138i) q^{85} +(0.598003 + 0.644494i) q^{86} +(-2.49963 + 1.70422i) q^{88} +(-1.95325 - 0.294405i) q^{89} +(14.5392 + 3.59321i) q^{91} +(5.03554 - 10.4564i) q^{92} +(-1.10159 + 1.18723i) q^{94} +(-3.64499 - 0.273154i) q^{95} +8.13308i q^{97} +(0.590309 - 0.802577i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 216 q - 16 q^{4} + 2 q^{7} + 12 q^{10} + 12 q^{16} - 6 q^{19} + 44 q^{22} + 26 q^{25} + 84 q^{28} - 6 q^{31} - 112 q^{34} + 60 q^{37} - 304 q^{40} + 20 q^{43} - 20 q^{46} - 86 q^{49} - 168 q^{52} - 84 q^{55}+ \cdots + 52 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(e\left(\frac{1}{42}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.132489 − 0.0519980i −0.0936836 − 0.0367681i 0.318036 − 0.948079i \(-0.396977\pi\)
−0.411720 + 0.911310i \(0.635072\pi\)
\(3\) 0 0
\(4\) −1.45125 − 1.34657i −0.725627 − 0.673284i
\(5\) −1.27700 − 0.870646i −0.571093 − 0.389365i 0.243063 − 0.970010i \(-0.421848\pi\)
−0.814156 + 0.580646i \(0.802800\pi\)
\(6\) 0 0
\(7\) −1.61234 − 2.09770i −0.609408 − 0.792857i
\(8\) 0.245763 + 0.510332i 0.0868903 + 0.180430i
\(9\) 0 0
\(10\) 0.123917 + 0.181752i 0.0391859 + 0.0574751i
\(11\) 0.796043 + 5.28140i 0.240016 + 1.59240i 0.706396 + 0.707817i \(0.250320\pi\)
−0.466380 + 0.884584i \(0.654442\pi\)
\(12\) 0 0
\(13\) −4.42567 + 3.52936i −1.22746 + 0.978868i −0.227474 + 0.973784i \(0.573047\pi\)
−0.999987 + 0.00508348i \(0.998382\pi\)
\(14\) 0.104541 + 0.361760i 0.0279398 + 0.0966845i
\(15\) 0 0
\(16\) 0.289868 + 3.86802i 0.0724670 + 0.967006i
\(17\) 2.27068 + 0.700411i 0.550720 + 0.169875i 0.557612 − 0.830102i \(-0.311718\pi\)
−0.00689217 + 0.999976i \(0.502194\pi\)
\(18\) 0 0
\(19\) 2.04813 − 1.18249i 0.469872 − 0.271281i −0.246314 − 0.969190i \(-0.579219\pi\)
0.716186 + 0.697909i \(0.245886\pi\)
\(20\) 0.680873 + 2.98310i 0.152248 + 0.667041i
\(21\) 0 0
\(22\) 0.169155 − 0.741118i 0.0360640 − 0.158007i
\(23\) 1.72793 + 5.60180i 0.360298 + 1.16806i 0.936476 + 0.350732i \(0.114067\pi\)
−0.576178 + 0.817324i \(0.695457\pi\)
\(24\) 0 0
\(25\) −0.953993 − 2.43073i −0.190799 − 0.486147i
\(26\) 0.769871 − 0.237474i 0.150984 − 0.0465724i
\(27\) 0 0
\(28\) −0.484776 + 5.21543i −0.0916140 + 0.985623i
\(29\) −9.35903 + 2.13614i −1.73793 + 0.396671i −0.969854 − 0.243688i \(-0.921643\pi\)
−0.768076 + 0.640359i \(0.778785\pi\)
\(30\) 0 0
\(31\) −6.48566 − 3.74450i −1.16486 − 0.672532i −0.212396 − 0.977184i \(-0.568127\pi\)
−0.952464 + 0.304652i \(0.901460\pi\)
\(32\) 0.496639 − 1.61006i 0.0877942 − 0.284622i
\(33\) 0 0
\(34\) −0.264419 − 0.210867i −0.0453475 − 0.0361634i
\(35\) 0.232612 + 4.08255i 0.0393186 + 0.690077i
\(36\) 0 0
\(37\) 1.55935 − 1.44686i 0.256355 − 0.237863i −0.541517 − 0.840690i \(-0.682150\pi\)
0.797872 + 0.602827i \(0.205959\pi\)
\(38\) −0.332840 + 0.0501676i −0.0539938 + 0.00813826i
\(39\) 0 0
\(40\) 0.130478 − 0.865668i 0.0206305 − 0.136874i
\(41\) 0.180363 − 0.0868585i 0.0281680 − 0.0135650i −0.419747 − 0.907641i \(-0.637881\pi\)
0.447915 + 0.894076i \(0.352167\pi\)
\(42\) 0 0
\(43\) −5.56553 − 2.68022i −0.848735 − 0.408729i −0.0416270 − 0.999133i \(-0.513254\pi\)
−0.807108 + 0.590404i \(0.798968\pi\)
\(44\) 5.95650 − 8.73658i 0.897976 − 1.31709i
\(45\) 0 0
\(46\) 0.0623516 − 0.832024i 0.00919323 − 0.122675i
\(47\) 4.15730 − 10.5926i 0.606405 − 1.54509i −0.216121 − 0.976367i \(-0.569340\pi\)
0.822526 − 0.568728i \(-0.192564\pi\)
\(48\) 0 0
\(49\) −1.80070 + 6.76443i −0.257243 + 0.966347i
\(50\) 0.371651i 0.0525593i
\(51\) 0 0
\(52\) 11.1753 + 0.837473i 1.54973 + 0.116137i
\(53\) −2.89757 + 3.12284i −0.398012 + 0.428955i −0.899980 − 0.435931i \(-0.856419\pi\)
0.501968 + 0.864886i \(0.332610\pi\)
\(54\) 0 0
\(55\) 3.58168 − 7.43743i 0.482953 − 1.00286i
\(56\) 0.674270 − 1.33837i 0.0901031 − 0.178847i
\(57\) 0 0
\(58\) 1.35104 + 0.203637i 0.177400 + 0.0267388i
\(59\) −10.3645 + 7.06637i −1.34934 + 0.919962i −0.999856 − 0.0169808i \(-0.994595\pi\)
−0.349481 + 0.936943i \(0.613642\pi\)
\(60\) 0 0
\(61\) 3.33764 + 3.59712i 0.427341 + 0.460564i 0.909612 − 0.415458i \(-0.136379\pi\)
−0.482272 + 0.876022i \(0.660188\pi\)
\(62\) 0.664571 + 0.833345i 0.0844005 + 0.105835i
\(63\) 0 0
\(64\) 4.68735 − 5.87775i 0.585919 − 0.734719i
\(65\) 8.72442 − 0.653805i 1.08213 − 0.0810945i
\(66\) 0 0
\(67\) −4.48870 + 7.77466i −0.548383 + 0.949826i 0.450003 + 0.893027i \(0.351423\pi\)
−0.998386 + 0.0567994i \(0.981910\pi\)
\(68\) −2.35218 − 4.07409i −0.285244 − 0.494056i
\(69\) 0 0
\(70\) 0.181466 − 0.552987i 0.0216893 − 0.0660946i
\(71\) −12.2266 − 2.79063i −1.45103 − 0.331187i −0.576871 − 0.816835i \(-0.695727\pi\)
−0.874154 + 0.485648i \(0.838584\pi\)
\(72\) 0 0
\(73\) 5.02779 − 1.97326i 0.588458 − 0.230953i −0.0523820 − 0.998627i \(-0.516681\pi\)
0.640840 + 0.767674i \(0.278586\pi\)
\(74\) −0.281830 + 0.110610i −0.0327621 + 0.0128582i
\(75\) 0 0
\(76\) −4.56465 − 1.04185i −0.523601 − 0.119509i
\(77\) 9.79530 − 10.1853i 1.11628 − 1.16072i
\(78\) 0 0
\(79\) 1.88456 + 3.26415i 0.212029 + 0.367245i 0.952349 − 0.305009i \(-0.0986595\pi\)
−0.740320 + 0.672254i \(0.765326\pi\)
\(80\) 2.99751 − 5.19185i 0.335132 − 0.580466i
\(81\) 0 0
\(82\) −0.0284126 + 0.00212923i −0.00313765 + 0.000235134i
\(83\) −5.58349 + 7.00147i −0.612867 + 0.768511i −0.987321 − 0.158734i \(-0.949259\pi\)
0.374454 + 0.927245i \(0.377830\pi\)
\(84\) 0 0
\(85\) −2.28985 − 2.87138i −0.248369 − 0.311445i
\(86\) 0.598003 + 0.644494i 0.0644844 + 0.0694976i
\(87\) 0 0
\(88\) −2.49963 + 1.70422i −0.266461 + 0.181670i
\(89\) −1.95325 − 0.294405i −0.207044 − 0.0312068i 0.0447013 − 0.999000i \(-0.485766\pi\)
−0.251745 + 0.967794i \(0.581004\pi\)
\(90\) 0 0
\(91\) 14.5392 + 3.59321i 1.52413 + 0.376670i
\(92\) 5.03554 − 10.4564i 0.524991 − 1.09016i
\(93\) 0 0
\(94\) −1.10159 + 1.18723i −0.113620 + 0.122454i
\(95\) −3.64499 − 0.273154i −0.373968 − 0.0280250i
\(96\) 0 0
\(97\) 8.13308i 0.825789i 0.910779 + 0.412894i \(0.135482\pi\)
−0.910779 + 0.412894i \(0.864518\pi\)
\(98\) 0.590309 − 0.802577i 0.0596302 − 0.0810726i
\(99\) 0 0
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 441.2.bg.a.395.9 yes 216
3.2 odd 2 inner 441.2.bg.a.395.10 yes 216
49.33 odd 42 inner 441.2.bg.a.278.10 yes 216
147.131 even 42 inner 441.2.bg.a.278.9 ✓ 216
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
441.2.bg.a.278.9 ✓ 216 147.131 even 42 inner
441.2.bg.a.278.10 yes 216 49.33 odd 42 inner
441.2.bg.a.395.9 yes 216 1.1 even 1 trivial
441.2.bg.a.395.10 yes 216 3.2 odd 2 inner