Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [441,2,Mod(148,441)] 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("441.148"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(441, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,1,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 148.2
Root \(0.500000 + 1.41036i\) of defining polynomial
Character \(\chi\) \(=\) 441.148
Dual form 441.2.f.d.295.2

$q$-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+(0.119562 + 0.207087i) q^{2} +(0.619562 + 1.61745i) q^{3} +(0.971410 - 1.68253i) q^{4} +(0.590972 - 1.02359i) q^{5} +(-0.260877 + 0.321688i) q^{6} +0.942820 q^{8} +(-2.23229 + 2.00422i) q^{9} +0.282630 q^{10} +(1.85185 + 3.20750i) q^{11} +(3.32326 + 0.528775i) q^{12} +(0.500000 - 0.866025i) q^{13} +(2.02175 + 0.321688i) q^{15} +(-1.83009 - 3.16982i) q^{16} +6.94282 q^{17} +(-0.681943 - 0.222649i) q^{18} -1.94282 q^{19} +(-1.14815 - 1.98866i) q^{20} +(-0.442820 + 0.766987i) q^{22} +(2.80150 - 4.85235i) q^{23} +(0.584135 + 1.52496i) q^{24} +(1.80150 + 3.12030i) q^{25} +0.239123 q^{26} +(-4.62476 - 2.36887i) q^{27} +(-0.119562 - 0.207087i) q^{29} +(0.175107 + 0.457140i) q^{30} +(0.830095 - 1.43777i) q^{31} +(1.38044 - 2.39099i) q^{32} +(-4.04063 + 4.98251i) q^{33} +(0.830095 + 1.43777i) q^{34} +(1.20370 + 5.70281i) q^{36} -9.54583 q^{37} +(-0.232287 - 0.402332i) q^{38} +(1.71053 + 0.272169i) q^{39} +(0.557180 - 0.965064i) q^{40} +(-5.09097 + 8.81782i) q^{41} +(-1.11273 - 1.92730i) q^{43} +7.19562 q^{44} +(0.732287 + 3.46939i) q^{45} +1.33981 q^{46} +(2.91423 + 5.04759i) q^{47} +(3.99316 - 4.92398i) q^{48} +(-0.430782 + 0.746136i) q^{50} +(4.30150 + 11.2297i) q^{51} +(-0.971410 - 1.68253i) q^{52} -11.6030 q^{53} +(-0.0623817 - 1.24095i) q^{54} +4.37756 q^{55} +(-1.20370 - 3.14241i) q^{57} +(0.0285900 - 0.0495193i) q^{58} +(1.30150 - 2.25427i) q^{59} +(2.50520 - 3.08917i) q^{60} +(-3.80150 - 6.58440i) q^{61} +0.396990 q^{62} -6.66019 q^{64} +(-0.590972 - 1.02359i) q^{65} +(-1.51492 - 0.241044i) q^{66} +(-1.75404 + 3.03809i) q^{67} +(6.74433 - 11.6815i) q^{68} +(9.58414 + 1.52496i) q^{69} +8.60301 q^{71} +(-2.10464 + 1.88962i) q^{72} -15.1488 q^{73} +(-1.14132 - 1.97682i) q^{74} +(-3.93078 + 4.84706i) q^{75} +(-1.88727 + 3.26886i) q^{76} +(0.148152 + 0.386770i) q^{78} +(-3.68878 - 6.38915i) q^{79} -4.32614 q^{80} +(0.966208 - 8.94799i) q^{81} -2.43474 q^{82} +(-3.47141 - 6.01266i) q^{83} +(4.10301 - 7.10662i) q^{85} +(0.266078 - 0.460861i) q^{86} +(0.260877 - 0.321688i) q^{87} +(1.74596 + 3.02409i) q^{88} -2.74720 q^{89} +(-0.630912 + 0.566453i) q^{90} +(-5.44282 - 9.42724i) q^{92} +(2.83981 + 0.451852i) q^{93} +(-0.696860 + 1.20700i) q^{94} +(-1.14815 + 1.98866i) q^{95} +(4.72257 + 0.751424i) q^{96} +(3.58414 + 6.20790i) q^{97} +(-10.5624 - 3.44854i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} + 4 q^{3} - 3 q^{4} - 5 q^{5} - q^{6} - 12 q^{8} - 4 q^{9} + 2 q^{11} + 2 q^{12} + 3 q^{13} + 11 q^{15} - 3 q^{16} + 24 q^{17} + 13 q^{18} + 6 q^{19} - 16 q^{20} + 15 q^{22} - 15 q^{24} - 6 q^{25}+ \cdots - 46 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(199\) \(344\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.119562 + 0.207087i 0.0845428 + 0.146433i 0.905196 − 0.424994i \(-0.139724\pi\)
−0.820653 + 0.571426i \(0.806390\pi\)
\(3\) 0.619562 + 1.61745i 0.357704 + 0.933835i
\(4\) 0.971410 − 1.68253i 0.485705 − 0.841266i
\(5\) 0.590972 − 1.02359i 0.264291 − 0.457765i −0.703087 − 0.711104i \(-0.748196\pi\)
0.967378 + 0.253339i \(0.0815289\pi\)
\(6\) −0.260877 + 0.321688i −0.106502 + 0.131329i
\(7\) 0 0
\(8\) 0.942820 0.333337
\(9\) −2.23229 + 2.00422i −0.744096 + 0.668073i
\(10\) 0.282630 0.0893755
\(11\) 1.85185 + 3.20750i 0.558353 + 0.967096i 0.997634 + 0.0687465i \(0.0219000\pi\)
−0.439281 + 0.898350i \(0.644767\pi\)
\(12\) 3.32326 + 0.528775i 0.959342 + 0.152644i
\(13\) 0.500000 − 0.866025i 0.138675 − 0.240192i −0.788320 − 0.615265i \(-0.789049\pi\)
0.926995 + 0.375073i \(0.122382\pi\)
\(14\) 0 0
\(15\) 2.02175 + 0.321688i 0.522014 + 0.0830595i
\(16\) −1.83009 − 3.16982i −0.457524 − 0.792454i
\(17\) 6.94282 1.68388 0.841941 − 0.539570i \(-0.181413\pi\)
0.841941 + 0.539570i \(0.181413\pi\)
\(18\) −0.681943 − 0.222649i −0.160736 − 0.0524790i
\(19\) −1.94282 −0.445713 −0.222857 − 0.974851i \(-0.571538\pi\)
−0.222857 + 0.974851i \(0.571538\pi\)
\(20\) −1.14815 − 1.98866i −0.256735 − 0.444677i
\(21\) 0 0
\(22\) −0.442820 + 0.766987i −0.0944096 + 0.163522i
\(23\) 2.80150 − 4.85235i 0.584154 − 1.01178i −0.410826 − 0.911714i \(-0.634760\pi\)
0.994980 − 0.100071i \(-0.0319070\pi\)
\(24\) 0.584135 + 1.52496i 0.119236 + 0.311282i
\(25\) 1.80150 + 3.12030i 0.360301 + 0.624060i
\(26\) 0.239123 0.0468959
\(27\) −4.62476 − 2.36887i −0.890036 − 0.455890i
\(28\) 0 0
\(29\) −0.119562 − 0.207087i −0.0222020 − 0.0384551i 0.854711 − 0.519104i \(-0.173734\pi\)
−0.876913 + 0.480649i \(0.840401\pi\)
\(30\) 0.175107 + 0.457140i 0.0319700 + 0.0834620i
\(31\) 0.830095 − 1.43777i 0.149089 − 0.258231i −0.781802 − 0.623527i \(-0.785699\pi\)
0.930891 + 0.365297i \(0.119032\pi\)
\(32\) 1.38044 − 2.39099i 0.244029 − 0.422671i
\(33\) −4.04063 + 4.98251i −0.703383 + 0.867344i
\(34\) 0.830095 + 1.43777i 0.142360 + 0.246575i
\(35\) 0 0
\(36\) 1.20370 + 5.70281i 0.200616 + 0.950469i
\(37\) −9.54583 −1.56932 −0.784662 − 0.619923i \(-0.787164\pi\)
−0.784662 + 0.619923i \(0.787164\pi\)
\(38\) −0.232287 − 0.402332i −0.0376819 − 0.0652669i
\(39\) 1.71053 + 0.272169i 0.273905 + 0.0435819i
\(40\) 0.557180 − 0.965064i 0.0880979 − 0.152590i
\(41\) −5.09097 + 8.81782i −0.795076 + 1.37711i 0.127715 + 0.991811i \(0.459236\pi\)
−0.922791 + 0.385301i \(0.874097\pi\)
\(42\) 0 0
\(43\) −1.11273 − 1.92730i −0.169689 − 0.293910i 0.768622 − 0.639704i \(-0.220943\pi\)
−0.938311 + 0.345794i \(0.887610\pi\)
\(44\) 7.19562 1.08478
\(45\) 0.732287 + 3.46939i 0.109163 + 0.517186i
\(46\) 1.33981 0.197544
\(47\) 2.91423 + 5.04759i 0.425084 + 0.736267i 0.996428 − 0.0844432i \(-0.0269112\pi\)
−0.571344 + 0.820711i \(0.693578\pi\)
\(48\) 3.99316 − 4.92398i 0.576364 − 0.710716i
\(49\) 0 0
\(50\) −0.430782 + 0.746136i −0.0609217 + 0.105520i
\(51\) 4.30150 + 11.2297i 0.602331 + 1.57247i
\(52\) −0.971410 − 1.68253i −0.134710 − 0.233325i
\(53\) −11.6030 −1.59380 −0.796898 − 0.604114i \(-0.793527\pi\)
−0.796898 + 0.604114i \(0.793527\pi\)
\(54\) −0.0623817 − 1.24095i −0.00848907 − 0.168872i
\(55\) 4.37756 0.590270
\(56\) 0 0
\(57\) −1.20370 − 3.14241i −0.159434 − 0.416223i
\(58\) 0.0285900 − 0.0495193i 0.00375405 − 0.00650220i
\(59\) 1.30150 − 2.25427i 0.169442 − 0.293481i −0.768782 − 0.639511i \(-0.779137\pi\)
0.938224 + 0.346029i \(0.112470\pi\)
\(60\) 2.50520 − 3.08917i 0.323420 − 0.398811i
\(61\) −3.80150 − 6.58440i −0.486733 − 0.843046i 0.513151 − 0.858298i \(-0.328478\pi\)
−0.999884 + 0.0152524i \(0.995145\pi\)
\(62\) 0.396990 0.0504178
\(63\) 0 0
\(64\) −6.66019 −0.832524
\(65\) −0.590972 − 1.02359i −0.0733010 − 0.126961i
\(66\) −1.51492 − 0.241044i −0.186473 − 0.0296704i
\(67\) −1.75404 + 3.03809i −0.214290 + 0.371161i −0.953053 − 0.302804i \(-0.902077\pi\)
0.738763 + 0.673966i \(0.235410\pi\)
\(68\) 6.74433 − 11.6815i 0.817870 − 1.41659i
\(69\) 9.58414 + 1.52496i 1.15379 + 0.183584i
\(70\) 0 0
\(71\) 8.60301 1.02099 0.510495 − 0.859881i \(-0.329462\pi\)
0.510495 + 0.859881i \(0.329462\pi\)
\(72\) −2.10464 + 1.88962i −0.248035 + 0.222694i
\(73\) −15.1488 −1.77304 −0.886519 − 0.462693i \(-0.846883\pi\)
−0.886519 + 0.462693i \(0.846883\pi\)
\(74\) −1.14132 − 1.97682i −0.132675 − 0.229800i
\(75\) −3.93078 + 4.84706i −0.453888 + 0.559690i
\(76\) −1.88727 + 3.26886i −0.216485 + 0.374963i
\(77\) 0 0
\(78\) 0.148152 + 0.386770i 0.0167749 + 0.0437931i
\(79\) −3.68878 − 6.38915i −0.415020 − 0.718836i 0.580410 − 0.814324i \(-0.302892\pi\)
−0.995431 + 0.0954881i \(0.969559\pi\)
\(80\) −4.32614 −0.483677
\(81\) 0.966208 − 8.94799i 0.107356 − 0.994221i
\(82\) −2.43474 −0.268872
\(83\) −3.47141 − 6.01266i −0.381037 − 0.659975i 0.610174 − 0.792267i \(-0.291100\pi\)
−0.991211 + 0.132292i \(0.957766\pi\)
\(84\) 0 0
\(85\) 4.10301 − 7.10662i 0.445034 − 0.770821i
\(86\) 0.266078 − 0.460861i 0.0286920 − 0.0496960i
\(87\) 0.260877 − 0.321688i 0.0279689 − 0.0344886i
\(88\) 1.74596 + 3.02409i 0.186120 + 0.322369i
\(89\) −2.74720 −0.291203 −0.145602 − 0.989343i \(-0.546512\pi\)
−0.145602 + 0.989343i \(0.546512\pi\)
\(90\) −0.630912 + 0.566453i −0.0665039 + 0.0597094i
\(91\) 0 0
\(92\) −5.44282 − 9.42724i −0.567453 − 0.982858i
\(93\) 2.83981 + 0.451852i 0.294475 + 0.0468548i
\(94\) −0.696860 + 1.20700i −0.0718756 + 0.124492i
\(95\) −1.14815 + 1.98866i −0.117798 + 0.204032i
\(96\) 4.72257 + 0.751424i 0.481995 + 0.0766919i
\(97\) 3.58414 + 6.20790i 0.363914 + 0.630317i 0.988601 − 0.150558i \(-0.0481069\pi\)
−0.624687 + 0.780875i \(0.714774\pi\)
\(98\) 0 0
\(99\) −10.5624 − 3.44854i −1.06156 − 0.346591i
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.f.d.148.2 6
3.2 odd 2 1323.2.f.c.442.2 6
7.2 even 3 441.2.g.d.67.2 6
7.3 odd 6 441.2.h.c.373.2 6
7.4 even 3 441.2.h.b.373.2 6
7.5 odd 6 441.2.g.e.67.2 6
7.6 odd 2 63.2.f.b.22.2 ✓ 6
9.2 odd 6 1323.2.f.c.883.2 6
9.4 even 3 3969.2.a.m.1.2 3
9.5 odd 6 3969.2.a.p.1.2 3
9.7 even 3 inner 441.2.f.d.295.2 6
21.2 odd 6 1323.2.g.b.361.2 6
21.5 even 6 1323.2.g.c.361.2 6
21.11 odd 6 1323.2.h.e.226.2 6
21.17 even 6 1323.2.h.d.226.2 6
21.20 even 2 189.2.f.a.64.2 6
28.27 even 2 1008.2.r.k.337.2 6
63.2 odd 6 1323.2.h.e.802.2 6
63.11 odd 6 1323.2.g.b.667.2 6
63.13 odd 6 567.2.a.d.1.2 3
63.16 even 3 441.2.h.b.214.2 6
63.20 even 6 189.2.f.a.127.2 6
63.25 even 3 441.2.g.d.79.2 6
63.34 odd 6 63.2.f.b.43.2 yes 6
63.38 even 6 1323.2.g.c.667.2 6
63.41 even 6 567.2.a.g.1.2 3
63.47 even 6 1323.2.h.d.802.2 6
63.52 odd 6 441.2.g.e.79.2 6
63.61 odd 6 441.2.h.c.214.2 6
84.83 odd 2 3024.2.r.g.1009.3 6
252.83 odd 6 3024.2.r.g.2017.3 6
252.139 even 6 9072.2.a.bq.1.3 3
252.167 odd 6 9072.2.a.cd.1.1 3
252.223 even 6 1008.2.r.k.673.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.b.22.2 ✓ 6 7.6 odd 2
63.2.f.b.43.2 yes 6 63.34 odd 6
189.2.f.a.64.2 6 21.20 even 2
189.2.f.a.127.2 6 63.20 even 6
441.2.f.d.148.2 6 1.1 even 1 trivial
441.2.f.d.295.2 6 9.7 even 3 inner
441.2.g.d.67.2 6 7.2 even 3
441.2.g.d.79.2 6 63.25 even 3
441.2.g.e.67.2 6 7.5 odd 6
441.2.g.e.79.2 6 63.52 odd 6
441.2.h.b.214.2 6 63.16 even 3
441.2.h.b.373.2 6 7.4 even 3
441.2.h.c.214.2 6 63.61 odd 6
441.2.h.c.373.2 6 7.3 odd 6
567.2.a.d.1.2 3 63.13 odd 6
567.2.a.g.1.2 3 63.41 even 6
1008.2.r.k.337.2 6 28.27 even 2
1008.2.r.k.673.2 6 252.223 even 6
1323.2.f.c.442.2 6 3.2 odd 2
1323.2.f.c.883.2 6 9.2 odd 6
1323.2.g.b.361.2 6 21.2 odd 6
1323.2.g.b.667.2 6 63.11 odd 6
1323.2.g.c.361.2 6 21.5 even 6
1323.2.g.c.667.2 6 63.38 even 6
1323.2.h.d.226.2 6 21.17 even 6
1323.2.h.d.802.2 6 63.47 even 6
1323.2.h.e.226.2 6 21.11 odd 6
1323.2.h.e.802.2 6 63.2 odd 6
3024.2.r.g.1009.3 6 84.83 odd 2
3024.2.r.g.2017.3 6 252.83 odd 6
3969.2.a.m.1.2 3 9.4 even 3
3969.2.a.p.1.2 3 9.5 odd 6
9072.2.a.bq.1.3 3 252.139 even 6
9072.2.a.cd.1.1 3 252.167 odd 6