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Properties

Label 441.2.g.c.67.3
Level $441$
Weight $2$
Character 441.67
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 67.3
Root \(0.939693 + 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 441.67
Dual form 441.2.g.c.79.3

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.439693 − 0.761570i 0.310910 − 0.538511i −0.667650 − 0.744475i \(-0.732700\pi\)
0.978560 + 0.205964i \(0.0660330\pi\)
\(3\) −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 \(-0.402591\pi\)
0.301265 + 0.953541i \(0.402591\pi\)
\(6\) −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 \(-0.678568\pi\)
0.999301 + 0.0373813i \(0.0119016\pi\)
\(14\) 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 \(-0.851405\pi\)
0.836259 + 0.548335i \(0.184738\pi\)
\(18\) 2.02094 + 1.69577i 0.476341 + 0.399698i
\(19\) 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
\(23\) 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 −0.995239 − 0.0974595i \(-0.968928\pi\)
0.413217 − 0.910632i \(-0.364405\pi\)
\(30\) −2.02094 + 0.356347i −0.368972 + 0.0650598i
\(31\) −4.61721 − 7.99724i −0.829276 − 1.43635i −0.898607 − 0.438754i \(-0.855420\pi\)
0.0693317 − 0.997594i \(-0.477913\pi\)
\(32\) 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 −0.943328 − 0.331862i \(-0.892323\pi\)
0.184263 − 0.982877i \(-0.441010\pi\)
\(38\) 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 \(-0.919165\pi\)
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
\(45\) −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 0.291995 + 0.956420i \(0.405681\pi\)
−0.974281 + 0.225335i \(0.927652\pi\)
\(48\) −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 −0.845642 − 0.533751i \(-0.820782\pi\)
0.885063 + 0.465472i \(0.154115\pi\)
\(54\) − 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 0.975945 + 0.218019i \(0.0699595\pi\)
−0.299162 + 0.954202i \(0.596707\pi\)
\(60\) 0.979055 − 2.68993i 0.126396 − 0.347269i
\(61\) −3.81908 + 6.61484i −0.488983 + 0.846943i −0.999920 − 0.0126752i \(-0.995965\pi\)
0.510937 + 0.859618i \(0.329299\pi\)
\(62\) −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 0.847239 − 0.531211i \(-0.178263\pi\)
−0.883662 + 0.468125i \(0.844930\pi\)
\(68\) −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 \(-0.871605\pi\)
0.799800 + 0.600266i \(0.204939\pi\)
\(74\) −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 −0.790461 − 0.612512i \(-0.790159\pi\)
0.925682 + 0.378303i \(0.123492\pi\)
\(80\) 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 0.901187 + 0.433431i \(0.142697\pi\)
−0.0752309 + 0.997166i \(0.523969\pi\)
\(84\) 0 0
\(85\) −0.315207 + 0.545955i −0.0341891 + 0.0592172i
\(86\) 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 \(-0.326604\pi\)
−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
\(93\) −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 \(-0.135932\pi\)
−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.g.c.67.3 6
3.2 odd 2 1323.2.g.d.361.1 6
7.2 even 3 441.2.h.d.373.1 6
7.3 odd 6 441.2.f.c.148.3 6
7.4 even 3 63.2.f.a.22.3 ✓ 6
7.5 odd 6 441.2.h.e.373.1 6
7.6 odd 2 441.2.g.b.67.3 6
9.2 odd 6 1323.2.h.c.802.3 6
9.7 even 3 441.2.h.d.214.1 6
21.2 odd 6 1323.2.h.c.226.3 6
21.5 even 6 1323.2.h.b.226.3 6
21.11 odd 6 189.2.f.b.64.1 6
21.17 even 6 1323.2.f.d.442.1 6
21.20 even 2 1323.2.g.e.361.1 6
28.11 odd 6 1008.2.r.h.337.2 6
63.2 odd 6 1323.2.g.d.667.1 6
63.4 even 3 567.2.a.h.1.1 3
63.11 odd 6 189.2.f.b.127.1 6
63.16 even 3 inner 441.2.g.c.79.3 6
63.20 even 6 1323.2.h.b.802.3 6
63.25 even 3 63.2.f.a.43.3 yes 6
63.31 odd 6 3969.2.a.q.1.1 3
63.32 odd 6 567.2.a.c.1.3 3
63.34 odd 6 441.2.h.e.214.1 6
63.38 even 6 1323.2.f.d.883.1 6
63.47 even 6 1323.2.g.e.667.1 6
63.52 odd 6 441.2.f.c.295.3 6
63.59 even 6 3969.2.a.l.1.3 3
63.61 odd 6 441.2.g.b.79.3 6
84.11 even 6 3024.2.r.k.1009.2 6
252.11 even 6 3024.2.r.k.2017.2 6
252.67 odd 6 9072.2.a.ca.1.2 3
252.95 even 6 9072.2.a.bs.1.2 3
252.151 odd 6 1008.2.r.h.673.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.a.22.3 ✓ 6 7.4 even 3
63.2.f.a.43.3 yes 6 63.25 even 3
189.2.f.b.64.1 6 21.11 odd 6
189.2.f.b.127.1 6 63.11 odd 6
441.2.f.c.148.3 6 7.3 odd 6
441.2.f.c.295.3 6 63.52 odd 6
441.2.g.b.67.3 6 7.6 odd 2
441.2.g.b.79.3 6 63.61 odd 6
441.2.g.c.67.3 6 1.1 even 1 trivial
441.2.g.c.79.3 6 63.16 even 3 inner
441.2.h.d.214.1 6 9.7 even 3
441.2.h.d.373.1 6 7.2 even 3
441.2.h.e.214.1 6 63.34 odd 6
441.2.h.e.373.1 6 7.5 odd 6
567.2.a.c.1.3 3 63.32 odd 6
567.2.a.h.1.1 3 63.4 even 3
1008.2.r.h.337.2 6 28.11 odd 6
1008.2.r.h.673.2 6 252.151 odd 6
1323.2.f.d.442.1 6 21.17 even 6
1323.2.f.d.883.1 6 63.38 even 6
1323.2.g.d.361.1 6 3.2 odd 2
1323.2.g.d.667.1 6 63.2 odd 6
1323.2.g.e.361.1 6 21.20 even 2
1323.2.g.e.667.1 6 63.47 even 6
1323.2.h.b.226.3 6 21.5 even 6
1323.2.h.b.802.3 6 63.20 even 6
1323.2.h.c.226.3 6 21.2 odd 6
1323.2.h.c.802.3 6 9.2 odd 6
3024.2.r.k.1009.2 6 84.11 even 6
3024.2.r.k.2017.2 6 252.11 even 6
3969.2.a.l.1.3 3 63.59 even 6
3969.2.a.q.1.1 3 63.31 odd 6
9072.2.a.bs.1.2 3 252.95 even 6
9072.2.a.ca.1.2 3 252.67 odd 6