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Properties

Label 441.2.h.d.214.1
Level $441$
Weight $2$
Character 441.214
Analytic conductor $3.521$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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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(214,441)] 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("441.214"); 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([4, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,6,0,6,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == 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: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 214.1
Root \(0.939693 - 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 441.214
Dual form 441.2.h.d.373.1

$q$-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-0.879385 q^{2} +(1.70574 + 0.300767i) q^{3} -1.22668 q^{4} +(-0.673648 + 1.16679i) q^{5} +(-1.50000 - 0.264490i) q^{6} +2.83750 q^{8} +(2.81908 + 1.02606i) q^{9} +(0.592396 - 1.02606i) q^{10} +(-0.826352 - 1.43128i) q^{11} +(-2.09240 - 0.368946i) q^{12} +(1.68479 + 2.91815i) q^{13} +(-1.50000 + 1.78763i) q^{15} -0.0418891 q^{16} +(-0.233956 + 0.405223i) q^{17} +(-2.47906 - 0.902302i) q^{18} +(1.61334 + 2.79439i) q^{19} +(0.826352 - 1.43128i) q^{20} +(0.726682 + 1.25865i) q^{22} +(-4.47178 + 7.74535i) q^{23} +(4.84002 + 0.853427i) q^{24} +(1.59240 + 2.75811i) q^{25} +(-1.48158 - 2.56617i) q^{26} +(4.50000 + 2.59808i) q^{27} +(-3.13429 + 5.42874i) q^{29} +(1.31908 - 1.57202i) q^{30} +9.23442 q^{31} -5.63816 q^{32} +(-0.979055 - 2.68993i) q^{33} +(0.205737 - 0.356347i) q^{34} +(-3.45811 - 1.25865i) q^{36} +(-4.61721 - 7.99724i) q^{37} +(-1.41875 - 2.45734i) q^{38} +(1.99613 + 5.48432i) q^{39} +(-1.91147 + 3.31077i) q^{40} +(-1.70574 - 2.95442i) q^{41} +(2.20574 - 3.82045i) q^{43} +(1.01367 + 1.75573i) q^{44} +(-3.09627 + 2.59808i) q^{45} +(3.93242 - 6.81115i) q^{46} +9.35504 q^{47} +(-0.0714517 - 0.0125989i) q^{48} +(-1.40033 - 2.42544i) q^{50} +(-0.520945 + 0.620838i) q^{51} +(-2.06670 - 3.57964i) q^{52} +(0.286989 - 0.497079i) q^{53} +(-3.95723 - 2.28471i) q^{54} +2.22668 q^{55} +(1.91147 + 5.25173i) q^{57} +(2.75624 - 4.77396i) q^{58} -10.3969 q^{59} +(1.84002 - 2.19285i) q^{60} +7.63816 q^{61} -8.12061 q^{62} +5.04189 q^{64} -4.53983 q^{65} +(0.860967 + 2.36549i) q^{66} +0.596267 q^{67} +(0.286989 - 0.497079i) q^{68} +(-9.95723 + 11.8666i) q^{69} -0.554378 q^{71} +(7.99912 + 2.91144i) q^{72} +(-1.02481 + 1.77503i) q^{73} +(4.06031 + 7.03266i) q^{74} +(1.88666 + 5.18355i) q^{75} +(-1.97906 - 3.42782i) q^{76} +(-1.75537 - 4.82283i) q^{78} -2.40373 q^{79} +(0.0282185 - 0.0488759i) q^{80} +(6.89440 + 5.78509i) q^{81} +(1.50000 + 2.59808i) q^{82} +(7.52481 - 13.0334i) q^{83} +(-0.315207 - 0.545955i) q^{85} +(-1.93969 + 3.35965i) q^{86} +(-6.97906 + 8.31731i) q^{87} +(-2.34477 - 4.06126i) q^{88} +(-4.54323 - 7.86911i) q^{89} +(2.72281 - 2.28471i) q^{90} +(5.48545 - 9.50108i) q^{92} +(15.7515 + 2.77741i) q^{93} -8.22668 q^{94} -4.34730 q^{95} +(-9.61721 - 1.69577i) q^{96} +(0.949493 - 1.64457i) q^{97} +(-0.860967 - 4.88279i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 6 q^{4} - 3 q^{5} - 9 q^{6} + 12 q^{8} - 6 q^{11} - 9 q^{12} + 3 q^{13} - 9 q^{15} + 6 q^{16} - 6 q^{17} - 18 q^{18} + 3 q^{19} + 6 q^{20} - 9 q^{22} - 12 q^{23} + 9 q^{24} + 6 q^{25} + 3 q^{26}+ \cdots + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(199\) \(344\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.879385 −0.621819 −0.310910 − 0.950439i \(-0.600634\pi\)
−0.310910 + 0.950439i \(0.600634\pi\)
\(3\) 1.70574 + 0.300767i 0.984808 + 0.173648i
\(4\) −1.22668 −0.613341
\(5\) −0.673648 + 1.16679i −0.301265 + 0.521806i −0.976423 − 0.215867i \(-0.930742\pi\)
0.675158 + 0.737673i \(0.264075\pi\)
\(6\) −1.50000 − 0.264490i −0.612372 − 0.107978i
\(7\) 0 0
\(8\) 2.83750 1.00321
\(9\) 2.81908 + 1.02606i 0.939693 + 0.342020i
\(10\) 0.592396 − 1.02606i 0.187332 − 0.324469i
\(11\) −0.826352 − 1.43128i −0.249154 − 0.431548i 0.714137 − 0.700006i \(-0.246819\pi\)
−0.963291 + 0.268458i \(0.913486\pi\)
\(12\) −2.09240 − 0.368946i −0.604023 − 0.106506i
\(13\) 1.68479 + 2.91815i 0.467277 + 0.809348i 0.999301 − 0.0373813i \(-0.0119016\pi\)
−0.532024 + 0.846729i \(0.678568\pi\)
\(14\) 0 0
\(15\) −1.50000 + 1.78763i −0.387298 + 0.461564i
\(16\) −0.0418891 −0.0104723
\(17\) −0.233956 + 0.405223i −0.0567426 + 0.0982810i −0.893001 − 0.450054i \(-0.851405\pi\)
0.836259 + 0.548335i \(0.184738\pi\)
\(18\) −2.47906 − 0.902302i −0.584319 − 0.212675i
\(19\) 1.61334 + 2.79439i 0.370126 + 0.641077i 0.989585 − 0.143953i \(-0.0459813\pi\)
−0.619459 + 0.785029i \(0.712648\pi\)
\(20\) 0.826352 − 1.43128i 0.184778 − 0.320045i
\(21\) 0 0
\(22\) 0.726682 + 1.25865i 0.154929 + 0.268345i
\(23\) −4.47178 + 7.74535i −0.932431 + 1.61502i −0.153279 + 0.988183i \(0.548983\pi\)
−0.779152 + 0.626835i \(0.784350\pi\)
\(24\) 4.84002 + 0.853427i 0.987965 + 0.174205i
\(25\) 1.59240 + 2.75811i 0.318479 + 0.551622i
\(26\) −1.48158 − 2.56617i −0.290562 − 0.503268i
\(27\) 4.50000 + 2.59808i 0.866025 + 0.500000i
\(28\) 0 0
\(29\) −3.13429 + 5.42874i −0.582022 + 1.00809i 0.413217 + 0.910632i \(0.364405\pi\)
−0.995239 + 0.0974595i \(0.968928\pi\)
\(30\) 1.31908 − 1.57202i 0.240830 − 0.287010i
\(31\) 9.23442 1.65855 0.829276 − 0.558840i \(-0.188753\pi\)
0.829276 + 0.558840i \(0.188753\pi\)
\(32\) −5.63816 −0.996695
\(33\) −0.979055 − 2.68993i −0.170432 − 0.468257i
\(34\) 0.205737 − 0.356347i 0.0352836 − 0.0611130i
\(35\) 0 0
\(36\) −3.45811 − 1.25865i −0.576352 − 0.209775i
\(37\) −4.61721 − 7.99724i −0.759065 − 1.31474i −0.943328 − 0.331862i \(-0.892323\pi\)
0.184263 − 0.982877i \(-0.441010\pi\)
\(38\) −1.41875 − 2.45734i −0.230151 − 0.398634i
\(39\) 1.99613 + 5.48432i 0.319637 + 0.878194i
\(40\) −1.91147 + 3.31077i −0.302231 + 0.523479i
\(41\) −1.70574 − 2.95442i −0.266391 − 0.461403i 0.701536 − 0.712634i \(-0.252498\pi\)
−0.967927 + 0.251231i \(0.919165\pi\)
\(42\) 0 0
\(43\) 2.20574 − 3.82045i 0.336372 − 0.582613i −0.647376 − 0.762171i \(-0.724133\pi\)
0.983747 + 0.179558i \(0.0574668\pi\)
\(44\) 1.01367 + 1.75573i 0.152817 + 0.264686i
\(45\) −3.09627 + 2.59808i −0.461564 + 0.387298i
\(46\) 3.93242 − 6.81115i 0.579803 − 1.00425i
\(47\) 9.35504 1.36457 0.682286 − 0.731085i \(-0.260986\pi\)
0.682286 + 0.731085i \(0.260986\pi\)
\(48\) −0.0714517 − 0.0125989i −0.0103132 − 0.00181849i
\(49\) 0 0
\(50\) −1.40033 − 2.42544i −0.198037 − 0.343009i
\(51\) −0.520945 + 0.620838i −0.0729468 + 0.0869346i
\(52\) −2.06670 − 3.57964i −0.286600 − 0.496406i
\(53\) 0.286989 − 0.497079i 0.0394210 − 0.0682791i −0.845642 − 0.533751i \(-0.820782\pi\)
0.885063 + 0.465472i \(0.154115\pi\)
\(54\) −3.95723 − 2.28471i −0.538511 − 0.310910i
\(55\) 2.22668 0.300246
\(56\) 0 0
\(57\) 1.91147 + 5.25173i 0.253181 + 0.695609i
\(58\) 2.75624 − 4.77396i 0.361913 − 0.626851i
\(59\) −10.3969 −1.35356 −0.676782 − 0.736183i \(-0.736626\pi\)
−0.676782 + 0.736183i \(0.736626\pi\)
\(60\) 1.84002 − 2.19285i 0.237546 − 0.283096i
\(61\) 7.63816 0.977966 0.488983 − 0.872293i \(-0.337368\pi\)
0.488983 + 0.872293i \(0.337368\pi\)
\(62\) −8.12061 −1.03132
\(63\) 0 0
\(64\) 5.04189 0.630236
\(65\) −4.53983 −0.563097
\(66\) 0.860967 + 2.36549i 0.105978 + 0.291171i
\(67\) 0.596267 0.0728456 0.0364228 − 0.999336i \(-0.488404\pi\)
0.0364228 + 0.999336i \(0.488404\pi\)
\(68\) 0.286989 − 0.497079i 0.0348025 − 0.0602797i
\(69\) −9.95723 + 11.8666i −1.19871 + 1.42857i
\(70\) 0 0
\(71\) −0.554378 −0.0657925 −0.0328963 − 0.999459i \(-0.510473\pi\)
−0.0328963 + 0.999459i \(0.510473\pi\)
\(72\) 7.99912 + 2.91144i 0.942706 + 0.343117i
\(73\) −1.02481 + 1.77503i −0.119946 + 0.207752i −0.919746 − 0.392514i \(-0.871605\pi\)
0.799800 + 0.600266i \(0.204939\pi\)
\(74\) 4.06031 + 7.03266i 0.472001 + 0.817530i
\(75\) 1.88666 + 5.18355i 0.217853 + 0.598545i
\(76\) −1.97906 − 3.42782i −0.227013 − 0.393198i
\(77\) 0 0
\(78\) −1.75537 − 4.82283i −0.198756 − 0.546078i
\(79\) −2.40373 −0.270441 −0.135221 − 0.990816i \(-0.543174\pi\)
−0.135221 + 0.990816i \(0.543174\pi\)
\(80\) 0.0282185 − 0.0488759i 0.00315492 − 0.00546449i
\(81\) 6.89440 + 5.78509i 0.766044 + 0.642788i
\(82\) 1.50000 + 2.59808i 0.165647 + 0.286910i
\(83\) 7.52481 − 13.0334i 0.825956 − 1.43060i −0.0752309 − 0.997166i \(-0.523969\pi\)
0.901187 − 0.433431i \(-0.142697\pi\)
\(84\) 0 0
\(85\) −0.315207 − 0.545955i −0.0341891 − 0.0592172i
\(86\) −1.93969 + 3.35965i −0.209162 + 0.362280i
\(87\) −6.97906 + 8.31731i −0.748233 + 0.891710i
\(88\) −2.34477 − 4.06126i −0.249953 − 0.432932i
\(89\) −4.54323 − 7.86911i −0.481582 − 0.834124i 0.518195 − 0.855263i \(-0.326604\pi\)
−0.999777 + 0.0211385i \(0.993271\pi\)
\(90\) 2.72281 − 2.28471i 0.287010 − 0.240830i
\(91\) 0 0
\(92\) 5.48545 − 9.50108i 0.571898 − 0.990556i
\(93\) 15.7515 + 2.77741i 1.63335 + 0.288004i
\(94\) −8.22668 −0.848517
\(95\) −4.34730 −0.446023
\(96\) −9.61721 − 1.69577i −0.981553 − 0.173074i
\(97\) 0.949493 − 1.64457i 0.0964064 − 0.166981i −0.813788 − 0.581161i \(-0.802598\pi\)
0.910195 + 0.414181i \(0.135932\pi\)
\(98\) 0 0
\(99\) −0.860967 − 4.88279i −0.0865304 − 0.490738i
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.h.d.214.1 6
3.2 odd 2 1323.2.h.c.802.3 6
7.2 even 3 441.2.g.c.79.3 6
7.3 odd 6 441.2.f.c.295.3 6
7.4 even 3 63.2.f.a.43.3 yes 6
7.5 odd 6 441.2.g.b.79.3 6
7.6 odd 2 441.2.h.e.214.1 6
9.4 even 3 441.2.g.c.67.3 6
9.5 odd 6 1323.2.g.d.361.1 6
21.2 odd 6 1323.2.g.d.667.1 6
21.5 even 6 1323.2.g.e.667.1 6
21.11 odd 6 189.2.f.b.127.1 6
21.17 even 6 1323.2.f.d.883.1 6
21.20 even 2 1323.2.h.b.802.3 6
28.11 odd 6 1008.2.r.h.673.2 6
63.4 even 3 63.2.f.a.22.3 ✓ 6
63.5 even 6 1323.2.h.b.226.3 6
63.11 odd 6 567.2.a.c.1.3 3
63.13 odd 6 441.2.g.b.67.3 6
63.23 odd 6 1323.2.h.c.226.3 6
63.25 even 3 567.2.a.h.1.1 3
63.31 odd 6 441.2.f.c.148.3 6
63.32 odd 6 189.2.f.b.64.1 6
63.38 even 6 3969.2.a.l.1.3 3
63.40 odd 6 441.2.h.e.373.1 6
63.41 even 6 1323.2.g.e.361.1 6
63.52 odd 6 3969.2.a.q.1.1 3
63.58 even 3 inner 441.2.h.d.373.1 6
63.59 even 6 1323.2.f.d.442.1 6
84.11 even 6 3024.2.r.k.2017.2 6
252.11 even 6 9072.2.a.bs.1.2 3
252.67 odd 6 1008.2.r.h.337.2 6
252.95 even 6 3024.2.r.k.1009.2 6
252.151 odd 6 9072.2.a.ca.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.a.22.3 ✓ 6 63.4 even 3
63.2.f.a.43.3 yes 6 7.4 even 3
189.2.f.b.64.1 6 63.32 odd 6
189.2.f.b.127.1 6 21.11 odd 6
441.2.f.c.148.3 6 63.31 odd 6
441.2.f.c.295.3 6 7.3 odd 6
441.2.g.b.67.3 6 63.13 odd 6
441.2.g.b.79.3 6 7.5 odd 6
441.2.g.c.67.3 6 9.4 even 3
441.2.g.c.79.3 6 7.2 even 3
441.2.h.d.214.1 6 1.1 even 1 trivial
441.2.h.d.373.1 6 63.58 even 3 inner
441.2.h.e.214.1 6 7.6 odd 2
441.2.h.e.373.1 6 63.40 odd 6
567.2.a.c.1.3 3 63.11 odd 6
567.2.a.h.1.1 3 63.25 even 3
1008.2.r.h.337.2 6 252.67 odd 6
1008.2.r.h.673.2 6 28.11 odd 6
1323.2.f.d.442.1 6 63.59 even 6
1323.2.f.d.883.1 6 21.17 even 6
1323.2.g.d.361.1 6 9.5 odd 6
1323.2.g.d.667.1 6 21.2 odd 6
1323.2.g.e.361.1 6 63.41 even 6
1323.2.g.e.667.1 6 21.5 even 6
1323.2.h.b.226.3 6 63.5 even 6
1323.2.h.b.802.3 6 21.20 even 2
1323.2.h.c.226.3 6 63.23 odd 6
1323.2.h.c.802.3 6 3.2 odd 2
3024.2.r.k.1009.2 6 252.95 even 6
3024.2.r.k.2017.2 6 84.11 even 6
3969.2.a.l.1.3 3 63.38 even 6
3969.2.a.q.1.1 3 63.52 odd 6
9072.2.a.bs.1.2 3 252.11 even 6
9072.2.a.ca.1.2 3 252.151 odd 6