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Properties

Label 441.4.a.r.1.1
Level $441$
Weight $4$
Character 441.1
Self dual yes
Analytic conductor $26.020$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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] = [441,4,Mod(1,441)] 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("441.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(441, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,3,0,17,6,0,0,87,0,66] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(26.0198423125\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.27492\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.27492 q^{2} -2.82475 q^{4} -4.54983 q^{5} +24.6254 q^{8} +10.3505 q^{10} +40.7492 q^{11} -53.2990 q^{13} -33.4228 q^{16} +4.54983 q^{17} -122.598 q^{19} +12.8522 q^{20} -92.7010 q^{22} -131.347 q^{23} -104.299 q^{25} +121.251 q^{26} +216.598 q^{29} +251.794 q^{31} -120.969 q^{32} -10.3505 q^{34} +11.8970 q^{37} +278.900 q^{38} -112.042 q^{40} -111.752 q^{41} +369.196 q^{43} -115.106 q^{44} +298.804 q^{46} -262.694 q^{47} +237.272 q^{50} +150.556 q^{52} +567.100 q^{53} -185.402 q^{55} -492.743 q^{58} +839.890 q^{59} +485.794 q^{61} -572.811 q^{62} +542.577 q^{64} +242.502 q^{65} -333.691 q^{67} -12.8522 q^{68} -590.248 q^{71} -490.701 q^{73} -27.0647 q^{74} +346.309 q^{76} +121.691 q^{79} +152.068 q^{80} +254.228 q^{82} +609.608 q^{83} -20.7010 q^{85} -839.890 q^{86} +1003.47 q^{88} +719.038 q^{89} +371.023 q^{92} +597.608 q^{94} +557.801 q^{95} +637.877 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + 17 q^{4} + 6 q^{5} + 87 q^{8} + 66 q^{10} + 6 q^{11} - 16 q^{13} + 137 q^{16} - 6 q^{17} - 64 q^{19} + 222 q^{20} - 276 q^{22} - 6 q^{23} - 118 q^{25} + 318 q^{26} + 252 q^{29} - 40 q^{31}+ \cdots - 808 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.27492 −0.804305 −0.402152 − 0.915573i \(-0.631738\pi\)
−0.402152 + 0.915573i \(0.631738\pi\)
\(3\) 0 0
\(4\) −2.82475 −0.353094
\(5\) −4.54983 −0.406950 −0.203475 − 0.979080i \(-0.565223\pi\)
−0.203475 + 0.979080i \(0.565223\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 24.6254 1.08830
\(9\) 0 0
\(10\) 10.3505 0.327311
\(11\) 40.7492 1.11694 0.558470 − 0.829525i \(-0.311389\pi\)
0.558470 + 0.829525i \(0.311389\pi\)
\(12\) 0 0
\(13\) −53.2990 −1.13711 −0.568557 − 0.822644i \(-0.692498\pi\)
−0.568557 + 0.822644i \(0.692498\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −33.4228 −0.522231
\(17\) 4.54983 0.0649116 0.0324558 − 0.999473i \(-0.489667\pi\)
0.0324558 + 0.999473i \(0.489667\pi\)
\(18\) 0 0
\(19\) −122.598 −1.48031 −0.740156 − 0.672436i \(-0.765248\pi\)
−0.740156 + 0.672436i \(0.765248\pi\)
\(20\) 12.8522 0.143691
\(21\) 0 0
\(22\) −92.7010 −0.898360
\(23\) −131.347 −1.19077 −0.595387 − 0.803439i \(-0.703001\pi\)
−0.595387 + 0.803439i \(0.703001\pi\)
\(24\) 0 0
\(25\) −104.299 −0.834392
\(26\) 121.251 0.914586
\(27\) 0 0
\(28\) 0 0
\(29\) 216.598 1.38694 0.693470 − 0.720486i \(-0.256081\pi\)
0.693470 + 0.720486i \(0.256081\pi\)
\(30\) 0 0
\(31\) 251.794 1.45882 0.729412 − 0.684075i \(-0.239794\pi\)
0.729412 + 0.684075i \(0.239794\pi\)
\(32\) −120.969 −0.668267
\(33\) 0 0
\(34\) −10.3505 −0.0522087
\(35\) 0 0
\(36\) 0 0
\(37\) 11.8970 0.0528610 0.0264305 − 0.999651i \(-0.491586\pi\)
0.0264305 + 0.999651i \(0.491586\pi\)
\(38\) 278.900 1.19062
\(39\) 0 0
\(40\) −112.042 −0.442883
\(41\) −111.752 −0.425678 −0.212839 − 0.977087i \(-0.568271\pi\)
−0.212839 + 0.977087i \(0.568271\pi\)
\(42\) 0 0
\(43\) 369.196 1.30935 0.654673 − 0.755912i \(-0.272806\pi\)
0.654673 + 0.755912i \(0.272806\pi\)
\(44\) −115.106 −0.394385
\(45\) 0 0
\(46\) 298.804 0.957744
\(47\) −262.694 −0.815275 −0.407637 − 0.913144i \(-0.633647\pi\)
−0.407637 + 0.913144i \(0.633647\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 237.272 0.671105
\(51\) 0 0
\(52\) 150.556 0.401508
\(53\) 567.100 1.46976 0.734879 − 0.678199i \(-0.237239\pi\)
0.734879 + 0.678199i \(0.237239\pi\)
\(54\) 0 0
\(55\) −185.402 −0.454538
\(56\) 0 0
\(57\) 0 0
\(58\) −492.743 −1.11552
\(59\) 839.890 1.85330 0.926648 − 0.375931i \(-0.122677\pi\)
0.926648 + 0.375931i \(0.122677\pi\)
\(60\) 0 0
\(61\) 485.794 1.01966 0.509832 − 0.860274i \(-0.329707\pi\)
0.509832 + 0.860274i \(0.329707\pi\)
\(62\) −572.811 −1.17334
\(63\) 0 0
\(64\) 542.577 1.05972
\(65\) 242.502 0.462748
\(66\) 0 0
\(67\) −333.691 −0.608460 −0.304230 − 0.952599i \(-0.598399\pi\)
−0.304230 + 0.952599i \(0.598399\pi\)
\(68\) −12.8522 −0.0229199
\(69\) 0 0
\(70\) 0 0
\(71\) −590.248 −0.986613 −0.493306 − 0.869856i \(-0.664212\pi\)
−0.493306 + 0.869856i \(0.664212\pi\)
\(72\) 0 0
\(73\) −490.701 −0.786743 −0.393371 − 0.919380i \(-0.628691\pi\)
−0.393371 + 0.919380i \(0.628691\pi\)
\(74\) −27.0647 −0.0425164
\(75\) 0 0
\(76\) 346.309 0.522689
\(77\) 0 0
\(78\) 0 0
\(79\) 121.691 0.173308 0.0866539 − 0.996238i \(-0.472383\pi\)
0.0866539 + 0.996238i \(0.472383\pi\)
\(80\) 152.068 0.212522
\(81\) 0 0
\(82\) 254.228 0.342375
\(83\) 609.608 0.806183 0.403091 − 0.915160i \(-0.367936\pi\)
0.403091 + 0.915160i \(0.367936\pi\)
\(84\) 0 0
\(85\) −20.7010 −0.0264157
\(86\) −839.890 −1.05311
\(87\) 0 0
\(88\) 1003.47 1.21557
\(89\) 719.038 0.856381 0.428190 − 0.903689i \(-0.359151\pi\)
0.428190 + 0.903689i \(0.359151\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 371.023 0.420455
\(93\) 0 0
\(94\) 597.608 0.655729
\(95\) 557.801 0.602412
\(96\) 0 0
\(97\) 637.877 0.667697 0.333849 − 0.942627i \(-0.391653\pi\)
0.333849 + 0.942627i \(0.391653\pi\)
\(98\) 0 0
\(99\) 0 0
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 441.4.a.r.1.1 2
3.2 odd 2 147.4.a.i.1.2 2
7.2 even 3 441.4.e.p.361.2 4
7.3 odd 6 441.4.e.q.226.2 4
7.4 even 3 441.4.e.p.226.2 4
7.5 odd 6 441.4.e.q.361.2 4
7.6 odd 2 63.4.a.e.1.1 2
12.11 even 2 2352.4.a.bz.1.2 2
21.2 odd 6 147.4.e.m.67.1 4
21.5 even 6 147.4.e.l.67.1 4
21.11 odd 6 147.4.e.m.79.1 4
21.17 even 6 147.4.e.l.79.1 4
21.20 even 2 21.4.a.c.1.2 ✓ 2
28.27 even 2 1008.4.a.ba.1.2 2
35.34 odd 2 1575.4.a.p.1.2 2
84.83 odd 2 336.4.a.m.1.1 2
105.62 odd 4 525.4.d.g.274.3 4
105.83 odd 4 525.4.d.g.274.2 4
105.104 even 2 525.4.a.n.1.1 2
168.83 odd 2 1344.4.a.bo.1.2 2
168.125 even 2 1344.4.a.bg.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.c.1.2 ✓ 2 21.20 even 2
63.4.a.e.1.1 2 7.6 odd 2
147.4.a.i.1.2 2 3.2 odd 2
147.4.e.l.67.1 4 21.5 even 6
147.4.e.l.79.1 4 21.17 even 6
147.4.e.m.67.1 4 21.2 odd 6
147.4.e.m.79.1 4 21.11 odd 6
336.4.a.m.1.1 2 84.83 odd 2
441.4.a.r.1.1 2 1.1 even 1 trivial
441.4.e.p.226.2 4 7.4 even 3
441.4.e.p.361.2 4 7.2 even 3
441.4.e.q.226.2 4 7.3 odd 6
441.4.e.q.361.2 4 7.5 odd 6
525.4.a.n.1.1 2 105.104 even 2
525.4.d.g.274.2 4 105.83 odd 4
525.4.d.g.274.3 4 105.62 odd 4
1008.4.a.ba.1.2 2 28.27 even 2
1344.4.a.bg.1.2 2 168.125 even 2
1344.4.a.bo.1.2 2 168.83 odd 2
1575.4.a.p.1.2 2 35.34 odd 2
2352.4.a.bz.1.2 2 12.11 even 2