↑

Properties

Label 441.4.e.q.226.2
Level $441$
Weight $4$
Character 441.226
Analytic conductor $26.020$
Analytic rank $0$
Dimension $4$
Inner twists $2$

Related objects

Downloads

Learn more

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(226,441)] 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("441.226"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(441, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-3,0,-17,6,0,0,174,0,66,-6,0,32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(26.0198423125\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 226.2
Root \(-1.63746 + 1.52274i\) of defining polynomial
Character \(\chi\) \(=\) 441.226
Dual form 441.4.e.q.361.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.13746 - 1.97014i) q^{2} +(1.41238 + 2.44631i) q^{4} +(-2.27492 + 3.94027i) q^{5} +24.6254 q^{8} +(5.17525 + 8.96379i) q^{10} +(-20.3746 - 35.2898i) q^{11} +53.2990 q^{13} +(16.7114 - 28.9450i) q^{16} +(2.27492 + 3.94027i) q^{17} +(-61.2990 + 106.173i) q^{19} -12.8522 q^{20} -92.7010 q^{22} +(65.6736 - 113.750i) q^{23} +(52.1495 + 90.3256i) q^{25} +(60.6254 - 105.006i) q^{26} +216.598 q^{29} +(125.897 + 218.060i) q^{31} +(60.4846 + 104.762i) q^{32} +10.3505 q^{34} +(-5.94851 + 10.3031i) q^{37} +(139.450 + 241.535i) q^{38} +(-56.0208 + 97.0308i) q^{40} +111.752 q^{41} +369.196 q^{43} +(57.5531 - 99.6850i) q^{44} +(-149.402 - 258.772i) q^{46} +(-131.347 + 227.500i) q^{47} +237.272 q^{50} +(75.2782 + 130.386i) q^{52} +(-283.550 - 491.123i) q^{53} +185.402 q^{55} +(246.371 - 426.728i) q^{58} +(419.945 + 727.366i) q^{59} +(242.897 - 420.710i) q^{61} +572.811 q^{62} +542.577 q^{64} +(-121.251 + 210.013i) q^{65} +(166.846 + 288.985i) q^{67} +(-6.42608 + 11.1303i) q^{68} -590.248 q^{71} +(-245.350 - 424.960i) q^{73} +(13.5324 + 23.4387i) q^{74} -346.309 q^{76} +(-60.8455 + 105.388i) q^{79} +(76.0340 + 131.695i) q^{80} +(127.114 - 220.168i) q^{82} -609.608 q^{83} -20.7010 q^{85} +(419.945 - 727.366i) q^{86} +(-501.733 - 869.026i) q^{88} +(359.519 - 622.705i) q^{89} +371.023 q^{92} +(298.804 + 517.544i) q^{94} +(-278.900 - 483.070i) q^{95} -637.877 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{2} - 17 q^{4} + 6 q^{5} + 174 q^{8} + 66 q^{10} - 6 q^{11} + 32 q^{13} - 137 q^{16} - 6 q^{17} - 64 q^{19} - 444 q^{20} - 552 q^{22} + 6 q^{23} + 118 q^{25} + 318 q^{26} + 504 q^{29} - 40 q^{31}+ \cdots + 1616 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.13746 − 1.97014i 0.402152 − 0.696548i −0.591833 − 0.806061i \(-0.701596\pi\)
0.993985 + 0.109512i \(0.0349289\pi\)
\(3\) 0 0
\(4\) 1.41238 + 2.44631i 0.176547 + 0.305788i
\(5\) −2.27492 + 3.94027i −0.203475 + 0.352429i −0.949646 − 0.313326i \(-0.898557\pi\)
0.746171 + 0.665754i \(0.231890\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 24.6254 1.08830
\(9\) 0 0
\(10\) 5.17525 + 8.96379i 0.163656 + 0.283460i
\(11\) −20.3746 − 35.2898i −0.558470 − 0.967298i −0.997624 − 0.0688867i \(-0.978055\pi\)
0.439155 − 0.898412i \(-0.355278\pi\)
\(12\) 0 0
\(13\) 53.2990 1.13711 0.568557 − 0.822644i \(-0.307502\pi\)
0.568557 + 0.822644i \(0.307502\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 16.7114 − 28.9450i 0.261115 − 0.452265i
\(17\) 2.27492 + 3.94027i 0.0324558 + 0.0562151i 0.881797 − 0.471629i \(-0.156334\pi\)
−0.849341 + 0.527844i \(0.823001\pi\)
\(18\) 0 0
\(19\) −61.2990 + 106.173i −0.740156 + 1.28199i 0.212269 + 0.977211i \(0.431915\pi\)
−0.952424 + 0.304776i \(0.901418\pi\)
\(20\) −12.8522 −0.143691
\(21\) 0 0
\(22\) −92.7010 −0.898360
\(23\) 65.6736 − 113.750i 0.595387 − 1.03124i −0.398106 − 0.917340i \(-0.630332\pi\)
0.993492 − 0.113900i \(-0.0363344\pi\)
\(24\) 0 0
\(25\) 52.1495 + 90.3256i 0.417196 + 0.722605i
\(26\) 60.6254 − 105.006i 0.457293 − 0.792055i
\(27\) 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
\(31\) 125.897 + 218.060i 0.729412 + 1.26338i 0.957132 + 0.289652i \(0.0935396\pi\)
−0.227720 + 0.973727i \(0.573127\pi\)
\(32\) 60.4846 + 104.762i 0.334134 + 0.578736i
\(33\) 0 0
\(34\) 10.3505 0.0522087
\(35\) 0 0
\(36\) 0 0
\(37\) −5.94851 + 10.3031i −0.0264305 + 0.0457790i −0.878938 − 0.476936i \(-0.841747\pi\)
0.852508 + 0.522715i \(0.175081\pi\)
\(38\) 139.450 + 241.535i 0.595311 + 1.03111i
\(39\) 0 0
\(40\) −56.0208 + 97.0308i −0.221442 + 0.383548i
\(41\) 111.752 0.425678 0.212839 − 0.977087i \(-0.431729\pi\)
0.212839 + 0.977087i \(0.431729\pi\)
\(42\) 0 0
\(43\) 369.196 1.30935 0.654673 − 0.755912i \(-0.272806\pi\)
0.654673 + 0.755912i \(0.272806\pi\)
\(44\) 57.5531 − 99.6850i 0.197192 − 0.341547i
\(45\) 0 0
\(46\) −149.402 − 258.772i −0.478872 − 0.829431i
\(47\) −131.347 + 227.500i −0.407637 + 0.706049i −0.994624 − 0.103548i \(-0.966981\pi\)
0.586987 + 0.809596i \(0.300314\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 237.272 0.671105
\(51\) 0 0
\(52\) 75.2782 + 130.386i 0.200754 + 0.347716i
\(53\) −283.550 − 491.123i −0.734879 − 1.27285i −0.954777 − 0.297324i \(-0.903906\pi\)
0.219898 − 0.975523i \(-0.429428\pi\)
\(54\) 0 0
\(55\) 185.402 0.454538
\(56\) 0 0
\(57\) 0 0
\(58\) 246.371 − 426.728i 0.557761 − 0.966070i
\(59\) 419.945 + 727.366i 0.926648 + 1.60500i 0.788890 + 0.614535i \(0.210656\pi\)
0.137758 + 0.990466i \(0.456010\pi\)
\(60\) 0 0
\(61\) 242.897 − 420.710i 0.509832 − 0.883056i −0.490103 − 0.871665i \(-0.663041\pi\)
0.999935 − 0.0113909i \(-0.00362593\pi\)
\(62\) 572.811 1.17334
\(63\) 0 0
\(64\) 542.577 1.05972
\(65\) −121.251 + 210.013i −0.231374 + 0.400752i
\(66\) 0 0
\(67\) 166.846 + 288.985i 0.304230 + 0.526942i 0.977090 − 0.212828i \(-0.0682675\pi\)
−0.672859 + 0.739770i \(0.734934\pi\)
\(68\) −6.42608 + 11.1303i −0.0114599 + 0.0198492i
\(69\) 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
\(73\) −245.350 − 424.960i −0.393371 − 0.681339i 0.599521 − 0.800359i \(-0.295358\pi\)
−0.992892 + 0.119020i \(0.962025\pi\)
\(74\) 13.5324 + 23.4387i 0.0212582 + 0.0368203i
\(75\) 0 0
\(76\) −346.309 −0.522689
\(77\) 0 0
\(78\) 0 0
\(79\) −60.8455 + 105.388i −0.0866539 + 0.150089i −0.906095 − 0.423075i \(-0.860951\pi\)
0.819441 + 0.573164i \(0.194284\pi\)
\(80\) 76.0340 + 131.695i 0.106261 + 0.184049i
\(81\) 0 0
\(82\) 127.114 − 220.168i 0.171187 − 0.296505i
\(83\) −609.608 −0.806183 −0.403091 − 0.915160i \(-0.632064\pi\)
−0.403091 + 0.915160i \(0.632064\pi\)
\(84\) 0 0
\(85\) −20.7010 −0.0264157
\(86\) 419.945 − 727.366i 0.526556 − 0.912023i
\(87\) 0 0
\(88\) −501.733 − 869.026i −0.607783 − 1.05271i
\(89\) 359.519 − 622.705i 0.428190 − 0.741648i −0.568522 − 0.822668i \(-0.692485\pi\)
0.996712 + 0.0810204i \(0.0258179\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 371.023 0.420455
\(93\) 0 0
\(94\) 298.804 + 517.544i 0.327865 + 0.567878i
\(95\) −278.900 − 483.070i −0.301206 − 0.521704i
\(96\) 0 0
\(97\) −637.877 −0.667697 −0.333849 − 0.942627i \(-0.608347\pi\)
−0.333849 + 0.942627i \(0.608347\pi\)
\(98\) 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.e.q.226.2 4
3.2 odd 2 147.4.e.l.79.1 4
7.2 even 3 63.4.a.e.1.1 2
7.3 odd 6 441.4.e.p.361.2 4
7.4 even 3 inner 441.4.e.q.361.2 4
7.5 odd 6 441.4.a.r.1.1 2
7.6 odd 2 441.4.e.p.226.2 4
21.2 odd 6 21.4.a.c.1.2 ✓ 2
21.5 even 6 147.4.a.i.1.2 2
21.11 odd 6 147.4.e.l.67.1 4
21.17 even 6 147.4.e.m.67.1 4
21.20 even 2 147.4.e.m.79.1 4
28.23 odd 6 1008.4.a.ba.1.2 2
35.9 even 6 1575.4.a.p.1.2 2
84.23 even 6 336.4.a.m.1.1 2
84.47 odd 6 2352.4.a.bz.1.2 2
105.2 even 12 525.4.d.g.274.3 4
105.23 even 12 525.4.d.g.274.2 4
105.44 odd 6 525.4.a.n.1.1 2
168.107 even 6 1344.4.a.bo.1.2 2
168.149 odd 6 1344.4.a.bg.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.c.1.2 ✓ 2 21.2 odd 6
63.4.a.e.1.1 2 7.2 even 3
147.4.a.i.1.2 2 21.5 even 6
147.4.e.l.67.1 4 21.11 odd 6
147.4.e.l.79.1 4 3.2 odd 2
147.4.e.m.67.1 4 21.17 even 6
147.4.e.m.79.1 4 21.20 even 2
336.4.a.m.1.1 2 84.23 even 6
441.4.a.r.1.1 2 7.5 odd 6
441.4.e.p.226.2 4 7.6 odd 2
441.4.e.p.361.2 4 7.3 odd 6
441.4.e.q.226.2 4 1.1 even 1 trivial
441.4.e.q.361.2 4 7.4 even 3 inner
525.4.a.n.1.1 2 105.44 odd 6
525.4.d.g.274.2 4 105.23 even 12
525.4.d.g.274.3 4 105.2 even 12
1008.4.a.ba.1.2 2 28.23 odd 6
1344.4.a.bg.1.2 2 168.149 odd 6
1344.4.a.bo.1.2 2 168.107 even 6
1575.4.a.p.1.2 2 35.9 even 6
2352.4.a.bz.1.2 2 84.47 odd 6