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Properties

Label 441.2.h.d.373.1
Level $441$
Weight $2$
Character 441.373
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 373.1
Root \(0.939693 + 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 441.373
Dual form 441.2.h.d.214.1

$q$-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 \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.879385 −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 0.675158 − 0.737673i \(-0.264075\pi\)
−0.976423 + 0.215867i \(0.930742\pi\)
\(6\) −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 −0.963291 − 0.268458i \(-0.913486\pi\)
0.714137 + 0.700006i \(0.246819\pi\)
\(12\) −2.09240 + 0.368946i −0.604023 + 0.106506i
\(13\) 1.68479 − 2.91815i 0.467277 − 0.809348i −0.532024 − 0.846729i \(-0.678568\pi\)
0.999301 + 0.0373813i \(0.0119016\pi\)
\(14\) 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 0.836259 − 0.548335i \(-0.184738\pi\)
−0.893001 + 0.450054i \(0.851405\pi\)
\(18\) −2.47906 + 0.902302i −0.584319 + 0.212675i
\(19\) 1.61334 − 2.79439i 0.370126 − 0.641077i −0.619459 − 0.785029i \(-0.712648\pi\)
0.989585 + 0.143953i \(0.0459813\pi\)
\(20\) 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 −0.779152 − 0.626835i \(-0.784350\pi\)
−0.153279 − 0.988183i \(-0.548983\pi\)
\(24\) 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 −0.995239 − 0.0974595i \(-0.968928\pi\)
0.413217 − 0.910632i \(-0.364405\pi\)
\(30\) 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 0.184263 + 0.982877i \(0.441010\pi\)
−0.943328 + 0.331862i \(0.892323\pi\)
\(38\) −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 −0.967927 − 0.251231i \(-0.919165\pi\)
0.701536 + 0.712634i \(0.252498\pi\)
\(42\) 0 0
\(43\) 2.20574 + 3.82045i 0.336372 + 0.582613i 0.983747 − 0.179558i \(-0.0574668\pi\)
−0.647376 + 0.762171i \(0.724133\pi\)
\(44\) 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 0.885063 − 0.465472i \(-0.154115\pi\)
−0.845642 + 0.533751i \(0.820782\pi\)
\(54\) −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 0.799800 − 0.600266i \(-0.204939\pi\)
−0.919746 + 0.392514i \(0.871605\pi\)
\(74\) 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 0.901187 + 0.433431i \(0.142697\pi\)
−0.0752309 + 0.997166i \(0.523969\pi\)
\(84\) 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 −0.999777 − 0.0211385i \(-0.993271\pi\)
0.518195 + 0.855263i \(0.326604\pi\)
\(90\) 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 0.910195 − 0.414181i \(-0.135932\pi\)
−0.813788 + 0.581161i \(0.802598\pi\)
\(98\) 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.373.1 6
3.2 odd 2 1323.2.h.c.226.3 6
7.2 even 3 63.2.f.a.22.3 ✓ 6
7.3 odd 6 441.2.g.b.67.3 6
7.4 even 3 441.2.g.c.67.3 6
7.5 odd 6 441.2.f.c.148.3 6
7.6 odd 2 441.2.h.e.373.1 6
9.2 odd 6 1323.2.g.d.667.1 6
9.7 even 3 441.2.g.c.79.3 6
21.2 odd 6 189.2.f.b.64.1 6
21.5 even 6 1323.2.f.d.442.1 6
21.11 odd 6 1323.2.g.d.361.1 6
21.17 even 6 1323.2.g.e.361.1 6
21.20 even 2 1323.2.h.b.226.3 6
28.23 odd 6 1008.2.r.h.337.2 6
63.2 odd 6 189.2.f.b.127.1 6
63.5 even 6 3969.2.a.l.1.3 3
63.11 odd 6 1323.2.h.c.802.3 6
63.16 even 3 63.2.f.a.43.3 yes 6
63.20 even 6 1323.2.g.e.667.1 6
63.23 odd 6 567.2.a.c.1.3 3
63.25 even 3 inner 441.2.h.d.214.1 6
63.34 odd 6 441.2.g.b.79.3 6
63.38 even 6 1323.2.h.b.802.3 6
63.40 odd 6 3969.2.a.q.1.1 3
63.47 even 6 1323.2.f.d.883.1 6
63.52 odd 6 441.2.h.e.214.1 6
63.58 even 3 567.2.a.h.1.1 3
63.61 odd 6 441.2.f.c.295.3 6
84.23 even 6 3024.2.r.k.1009.2 6
252.23 even 6 9072.2.a.bs.1.2 3
252.79 odd 6 1008.2.r.h.673.2 6
252.191 even 6 3024.2.r.k.2017.2 6
252.247 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 7.2 even 3
63.2.f.a.43.3 yes 6 63.16 even 3
189.2.f.b.64.1 6 21.2 odd 6
189.2.f.b.127.1 6 63.2 odd 6
441.2.f.c.148.3 6 7.5 odd 6
441.2.f.c.295.3 6 63.61 odd 6
441.2.g.b.67.3 6 7.3 odd 6
441.2.g.b.79.3 6 63.34 odd 6
441.2.g.c.67.3 6 7.4 even 3
441.2.g.c.79.3 6 9.7 even 3
441.2.h.d.214.1 6 63.25 even 3 inner
441.2.h.d.373.1 6 1.1 even 1 trivial
441.2.h.e.214.1 6 63.52 odd 6
441.2.h.e.373.1 6 7.6 odd 2
567.2.a.c.1.3 3 63.23 odd 6
567.2.a.h.1.1 3 63.58 even 3
1008.2.r.h.337.2 6 28.23 odd 6
1008.2.r.h.673.2 6 252.79 odd 6
1323.2.f.d.442.1 6 21.5 even 6
1323.2.f.d.883.1 6 63.47 even 6
1323.2.g.d.361.1 6 21.11 odd 6
1323.2.g.d.667.1 6 9.2 odd 6
1323.2.g.e.361.1 6 21.17 even 6
1323.2.g.e.667.1 6 63.20 even 6
1323.2.h.b.226.3 6 21.20 even 2
1323.2.h.b.802.3 6 63.38 even 6
1323.2.h.c.226.3 6 3.2 odd 2
1323.2.h.c.802.3 6 63.11 odd 6
3024.2.r.k.1009.2 6 84.23 even 6
3024.2.r.k.2017.2 6 252.191 even 6
3969.2.a.l.1.3 3 63.5 even 6
3969.2.a.q.1.1 3 63.40 odd 6
9072.2.a.bs.1.2 3 252.23 even 6
9072.2.a.ca.1.2 3 252.247 odd 6