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Properties

Label 441.2.g.b.79.3
Level $441$
Weight $2$
Character 441.79
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(67,441)] 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("441.67"); 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, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.g (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-3,0,-3,-6] 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 79.3
Root \(0.939693 - 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 441.79
Dual form 441.2.g.b.67.3

$q$-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+(0.439693 + 0.761570i) q^{2} +(1.11334 - 1.32683i) q^{3} +(0.613341 - 1.06234i) q^{4} -1.34730 q^{5} +(1.50000 + 0.264490i) q^{6} +2.83750 q^{8} +(-0.520945 - 2.95442i) q^{9} +(-0.592396 - 1.02606i) q^{10} +1.65270 q^{11} +(-0.726682 - 1.99654i) q^{12} +(-1.68479 - 2.91815i) q^{13} +(-1.50000 + 1.78763i) q^{15} +(0.0209445 + 0.0362770i) q^{16} +(0.233956 + 0.405223i) q^{17} +(2.02094 - 1.69577i) q^{18} +(-1.61334 + 2.79439i) q^{19} +(-0.826352 + 1.43128i) q^{20} +(0.726682 + 1.25865i) q^{22} +8.94356 q^{23} +(3.15910 - 3.76487i) q^{24} -3.18479 q^{25} +(1.48158 - 2.56617i) q^{26} +(-4.50000 - 2.59808i) q^{27} +(-3.13429 + 5.42874i) q^{29} +(-2.02094 - 0.356347i) q^{30} +(4.61721 - 7.99724i) q^{31} +(2.81908 - 4.88279i) q^{32} +(1.84002 - 2.19285i) q^{33} +(-0.205737 + 0.356347i) q^{34} +(-3.45811 - 1.25865i) q^{36} +(-4.61721 + 7.99724i) q^{37} -2.83750 q^{38} +(-5.74763 - 1.01346i) q^{39} -3.82295 q^{40} +(1.70574 + 2.95442i) q^{41} +(2.20574 - 3.82045i) q^{43} +(1.01367 - 1.75573i) q^{44} +(0.701867 + 3.98048i) q^{45} +(3.93242 + 6.81115i) q^{46} +(4.67752 + 8.10170i) q^{47} +(0.0714517 + 0.0125989i) q^{48} +(-1.40033 - 2.42544i) q^{50} +(0.798133 + 0.140732i) q^{51} -4.13341 q^{52} +(0.286989 + 0.497079i) q^{53} -4.56942i q^{54} -2.22668 q^{55} +(1.91147 + 5.25173i) q^{57} -5.51249 q^{58} +(-5.19846 + 9.00400i) q^{59} +(0.979055 + 2.68993i) q^{60} +(3.81908 + 6.61484i) q^{61} +8.12061 q^{62} +5.04189 q^{64} +(2.26991 + 3.93161i) q^{65} +(2.47906 + 0.437124i) q^{66} +(-0.298133 + 0.516382i) q^{67} +0.573978 q^{68} +(9.95723 - 11.8666i) q^{69} -0.554378 q^{71} +(-1.47818 - 8.38316i) q^{72} +(1.02481 + 1.77503i) q^{73} -8.12061 q^{74} +(-3.54576 + 4.22567i) q^{75} +(1.97906 + 3.42782i) q^{76} +(-1.75537 - 4.82283i) q^{78} +(1.20187 + 2.08169i) q^{79} +(-0.0282185 - 0.0488759i) q^{80} +(-8.45723 + 3.07818i) q^{81} +(-1.50000 + 2.59808i) q^{82} +(-7.52481 + 13.0334i) q^{83} +(-0.315207 - 0.545955i) q^{85} +3.87939 q^{86} +(3.71348 + 10.2027i) q^{87} +4.68954 q^{88} +(4.54323 - 7.86911i) q^{89} +(-2.72281 + 2.28471i) q^{90} +(5.48545 - 9.50108i) q^{92} +(-5.47044 - 15.0299i) q^{93} +(-4.11334 + 7.12452i) q^{94} +(2.17365 - 3.76487i) q^{95} +(-3.34002 - 9.17664i) q^{96} +(-0.949493 + 1.64457i) q^{97} +(-0.860967 - 4.88279i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} - 3 q^{4} - 6 q^{5} + 9 q^{6} + 12 q^{8} + 12 q^{11} + 9 q^{12} - 3 q^{13} - 9 q^{15} - 3 q^{16} + 6 q^{17} + 9 q^{18} - 3 q^{19} - 6 q^{20} - 9 q^{22} + 24 q^{23} - 18 q^{24} - 12 q^{25}+ \cdots + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(199\) \(344\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.439693 + 0.761570i 0.310910 + 0.538511i 0.978560 − 0.205964i \(-0.0660330\pi\)
−0.667650 + 0.744475i \(0.732700\pi\)
\(3\) 1.11334 − 1.32683i 0.642788 − 0.766044i
\(4\) 0.613341 − 1.06234i 0.306670 − 0.531169i
\(5\) −1.34730 −0.602529 −0.301265 − 0.953541i \(-0.597409\pi\)
−0.301265 + 0.953541i \(0.597409\pi\)
\(6\) 1.50000 + 0.264490i 0.612372 + 0.107978i
\(7\) 0 0
\(8\) 2.83750 1.00321
\(9\) −0.520945 − 2.95442i −0.173648 − 0.984808i
\(10\) −0.592396 − 1.02606i −0.187332 − 0.324469i
\(11\) 1.65270 0.498309 0.249154 − 0.968464i \(-0.419847\pi\)
0.249154 + 0.968464i \(0.419847\pi\)
\(12\) −0.726682 − 1.99654i −0.209775 − 0.576352i
\(13\) −1.68479 − 2.91815i −0.467277 − 0.809348i 0.532024 − 0.846729i \(-0.321432\pi\)
−0.999301 + 0.0373813i \(0.988098\pi\)
\(14\) 0 0
\(15\) −1.50000 + 1.78763i −0.387298 + 0.461564i
\(16\) 0.0209445 + 0.0362770i 0.00523613 + 0.00906925i
\(17\) 0.233956 + 0.405223i 0.0567426 + 0.0982810i 0.893001 − 0.450054i \(-0.148595\pi\)
−0.836259 + 0.548335i \(0.815262\pi\)
\(18\) 2.02094 − 1.69577i 0.476341 − 0.399698i
\(19\) −1.61334 + 2.79439i −0.370126 + 0.641077i −0.989585 − 0.143953i \(-0.954019\pi\)
0.619459 + 0.785029i \(0.287352\pi\)
\(20\) −0.826352 + 1.43128i −0.184778 + 0.320045i
\(21\) 0 0
\(22\) 0.726682 + 1.25865i 0.154929 + 0.268345i
\(23\) 8.94356 1.86486 0.932431 − 0.361348i \(-0.117683\pi\)
0.932431 + 0.361348i \(0.117683\pi\)
\(24\) 3.15910 − 3.76487i 0.644849 − 0.768501i
\(25\) −3.18479 −0.636959
\(26\) 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 \(0.968928\pi\)
\(30\) −2.02094 − 0.356347i −0.368972 − 0.0650598i
\(31\) 4.61721 − 7.99724i 0.829276 − 1.43635i −0.0693317 − 0.997594i \(-0.522087\pi\)
0.898607 − 0.438754i \(-0.144580\pi\)
\(32\) 2.81908 − 4.88279i 0.498347 − 0.863163i
\(33\) 1.84002 − 2.19285i 0.320307 − 0.381727i
\(34\) −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 \(0.892323\pi\)
\(38\) −2.83750 −0.460303
\(39\) −5.74763 − 1.01346i −0.920357 − 0.162284i
\(40\) −3.82295 −0.604461
\(41\) 1.70574 + 2.95442i 0.266391 + 0.461403i 0.967927 − 0.251231i \(-0.0808353\pi\)
−0.701536 + 0.712634i \(0.747502\pi\)
\(42\) 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
\(45\) 0.701867 + 3.98048i 0.104628 + 0.593375i
\(46\) 3.93242 + 6.81115i 0.579803 + 1.00425i
\(47\) 4.67752 + 8.10170i 0.682286 + 1.18175i 0.974281 + 0.225335i \(0.0723475\pi\)
−0.291995 + 0.956420i \(0.594319\pi\)
\(48\) 0.0714517 + 0.0125989i 0.0103132 + 0.00181849i
\(49\) 0 0
\(50\) −1.40033 − 2.42544i −0.198037 − 0.343009i
\(51\) 0.798133 + 0.140732i 0.111761 + 0.0197065i
\(52\) −4.13341 −0.573201
\(53\) 0.286989 + 0.497079i 0.0394210 + 0.0682791i 0.885063 − 0.465472i \(-0.154115\pi\)
−0.845642 + 0.533751i \(0.820782\pi\)
\(54\) − 4.56942i − 0.621819i
\(55\) −2.22668 −0.300246
\(56\) 0 0
\(57\) 1.91147 + 5.25173i 0.253181 + 0.695609i
\(58\) −5.51249 −0.723825
\(59\) −5.19846 + 9.00400i −0.676782 + 1.17222i 0.299162 + 0.954202i \(0.403293\pi\)
−0.975945 + 0.218019i \(0.930041\pi\)
\(60\) 0.979055 + 2.68993i 0.126396 + 0.347269i
\(61\) 3.81908 + 6.61484i 0.488983 + 0.846943i 0.999920 − 0.0126752i \(-0.00403474\pi\)
−0.510937 + 0.859618i \(0.670701\pi\)
\(62\) 8.12061 1.03132
\(63\) 0 0
\(64\) 5.04189 0.630236
\(65\) 2.26991 + 3.93161i 0.281548 + 0.487656i
\(66\) 2.47906 + 0.437124i 0.305151 + 0.0538063i
\(67\) −0.298133 + 0.516382i −0.0364228 + 0.0630861i −0.883662 − 0.468125i \(-0.844930\pi\)
0.847239 + 0.531211i \(0.178263\pi\)
\(68\) 0.573978 0.0696051
\(69\) 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 \(0.510473\pi\)
\(72\) −1.47818 − 8.38316i −0.174205 − 0.987965i
\(73\) 1.02481 + 1.77503i 0.119946 + 0.207752i 0.919746 − 0.392514i \(-0.128395\pi\)
−0.799800 + 0.600266i \(0.795061\pi\)
\(74\) −8.12061 −0.944002
\(75\) −3.54576 + 4.22567i −0.409429 + 0.487939i
\(76\) 1.97906 + 3.42782i 0.227013 + 0.393198i
\(77\) 0 0
\(78\) −1.75537 − 4.82283i −0.198756 − 0.546078i
\(79\) 1.20187 + 2.08169i 0.135221 + 0.234209i 0.925682 − 0.378303i \(-0.123492\pi\)
−0.790461 + 0.612512i \(0.790159\pi\)
\(80\) −0.0282185 − 0.0488759i −0.00315492 − 0.00546449i
\(81\) −8.45723 + 3.07818i −0.939693 + 0.342020i
\(82\) −1.50000 + 2.59808i −0.165647 + 0.286910i
\(83\) −7.52481 + 13.0334i −0.825956 + 1.43060i 0.0752309 + 0.997166i \(0.476031\pi\)
−0.901187 + 0.433431i \(0.857303\pi\)
\(84\) 0 0
\(85\) −0.315207 − 0.545955i −0.0341891 − 0.0592172i
\(86\) 3.87939 0.418325
\(87\) 3.71348 + 10.2027i 0.398127 + 1.09384i
\(88\) 4.68954 0.499907
\(89\) 4.54323 − 7.86911i 0.481582 − 0.834124i −0.518195 − 0.855263i \(-0.673396\pi\)
0.999777 + 0.0211385i \(0.00672911\pi\)
\(90\) −2.72281 + 2.28471i −0.287010 + 0.240830i
\(91\) 0 0
\(92\) 5.48545 − 9.50108i 0.571898 − 0.990556i
\(93\) −5.47044 − 15.0299i −0.567258 − 1.55853i
\(94\) −4.11334 + 7.12452i −0.424259 + 0.734838i
\(95\) 2.17365 − 3.76487i 0.223012 − 0.386267i
\(96\) −3.34002 − 9.17664i −0.340890 − 0.936587i
\(97\) −0.949493 + 1.64457i −0.0964064 + 0.166981i −0.910195 − 0.414181i \(-0.864068\pi\)
0.813788 + 0.581161i \(0.197402\pi\)
\(98\) 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.g.b.79.3 6
3.2 odd 2 1323.2.g.e.667.1 6
7.2 even 3 441.2.f.c.295.3 6
7.3 odd 6 441.2.h.d.214.1 6
7.4 even 3 441.2.h.e.214.1 6
7.5 odd 6 63.2.f.a.43.3 yes 6
7.6 odd 2 441.2.g.c.79.3 6
9.4 even 3 441.2.h.e.373.1 6
9.5 odd 6 1323.2.h.b.226.3 6
21.2 odd 6 1323.2.f.d.883.1 6
21.5 even 6 189.2.f.b.127.1 6
21.11 odd 6 1323.2.h.b.802.3 6
21.17 even 6 1323.2.h.c.802.3 6
21.20 even 2 1323.2.g.d.667.1 6
28.19 even 6 1008.2.r.h.673.2 6
63.2 odd 6 3969.2.a.l.1.3 3
63.4 even 3 inner 441.2.g.b.67.3 6
63.5 even 6 189.2.f.b.64.1 6
63.13 odd 6 441.2.h.d.373.1 6
63.16 even 3 3969.2.a.q.1.1 3
63.23 odd 6 1323.2.f.d.442.1 6
63.31 odd 6 441.2.g.c.67.3 6
63.32 odd 6 1323.2.g.e.361.1 6
63.40 odd 6 63.2.f.a.22.3 ✓ 6
63.41 even 6 1323.2.h.c.226.3 6
63.47 even 6 567.2.a.c.1.3 3
63.58 even 3 441.2.f.c.148.3 6
63.59 even 6 1323.2.g.d.361.1 6
63.61 odd 6 567.2.a.h.1.1 3
84.47 odd 6 3024.2.r.k.2017.2 6
252.47 odd 6 9072.2.a.bs.1.2 3
252.103 even 6 1008.2.r.h.337.2 6
252.131 odd 6 3024.2.r.k.1009.2 6
252.187 even 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 63.40 odd 6
63.2.f.a.43.3 yes 6 7.5 odd 6
189.2.f.b.64.1 6 63.5 even 6
189.2.f.b.127.1 6 21.5 even 6
441.2.f.c.148.3 6 63.58 even 3
441.2.f.c.295.3 6 7.2 even 3
441.2.g.b.67.3 6 63.4 even 3 inner
441.2.g.b.79.3 6 1.1 even 1 trivial
441.2.g.c.67.3 6 63.31 odd 6
441.2.g.c.79.3 6 7.6 odd 2
441.2.h.d.214.1 6 7.3 odd 6
441.2.h.d.373.1 6 63.13 odd 6
441.2.h.e.214.1 6 7.4 even 3
441.2.h.e.373.1 6 9.4 even 3
567.2.a.c.1.3 3 63.47 even 6
567.2.a.h.1.1 3 63.61 odd 6
1008.2.r.h.337.2 6 252.103 even 6
1008.2.r.h.673.2 6 28.19 even 6
1323.2.f.d.442.1 6 63.23 odd 6
1323.2.f.d.883.1 6 21.2 odd 6
1323.2.g.d.361.1 6 63.59 even 6
1323.2.g.d.667.1 6 21.20 even 2
1323.2.g.e.361.1 6 63.32 odd 6
1323.2.g.e.667.1 6 3.2 odd 2
1323.2.h.b.226.3 6 9.5 odd 6
1323.2.h.b.802.3 6 21.11 odd 6
1323.2.h.c.226.3 6 63.41 even 6
1323.2.h.c.802.3 6 21.17 even 6
3024.2.r.k.1009.2 6 252.131 odd 6
3024.2.r.k.2017.2 6 84.47 odd 6
3969.2.a.l.1.3 3 63.2 odd 6
3969.2.a.q.1.1 3 63.16 even 3
9072.2.a.bs.1.2 3 252.47 odd 6
9072.2.a.ca.1.2 3 252.187 even 6