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Properties

Label 441.4.e.m.226.1
Level $441$
Weight $4$
Character 441.226
Analytic conductor $26.020$
Analytic rank $0$
Dimension $2$
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] = [441,4,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 = [2,4,0,-8,-4,0,0,0,0,16,62,0,-124] 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: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 226.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 441.226
Dual form 441.4.e.m.361.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.00000 - 3.46410i) q^{2} +(-4.00000 - 6.92820i) q^{4} +(-2.00000 + 3.46410i) q^{5} +(8.00000 + 13.8564i) q^{10} +(31.0000 + 53.6936i) q^{11} -62.0000 q^{13} +(32.0000 - 55.4256i) q^{16} +(42.0000 + 72.7461i) q^{17} +(-50.0000 + 86.6025i) q^{19} +32.0000 q^{20} +248.000 q^{22} +(-21.0000 + 36.3731i) q^{23} +(54.5000 + 94.3968i) q^{25} +(-124.000 + 214.774i) q^{26} +10.0000 q^{29} +(24.0000 + 41.5692i) q^{31} +(-128.000 - 221.703i) q^{32} +336.000 q^{34} +(123.000 - 213.042i) q^{37} +(200.000 + 346.410i) q^{38} +248.000 q^{41} +68.0000 q^{43} +(248.000 - 429.549i) q^{44} +(84.0000 + 145.492i) q^{46} +(162.000 - 280.592i) q^{47} +436.000 q^{50} +(248.000 + 429.549i) q^{52} +(129.000 + 223.435i) q^{53} -248.000 q^{55} +(20.0000 - 34.6410i) q^{58} +(60.0000 + 103.923i) q^{59} +(-311.000 + 538.668i) q^{61} +192.000 q^{62} -512.000 q^{64} +(124.000 - 214.774i) q^{65} +(-452.000 - 782.887i) q^{67} +(336.000 - 581.969i) q^{68} +678.000 q^{71} +(321.000 + 555.988i) q^{73} +(-492.000 - 852.169i) q^{74} +800.000 q^{76} +(-370.000 + 640.859i) q^{79} +(128.000 + 221.703i) q^{80} +(496.000 - 859.097i) q^{82} -468.000 q^{83} -336.000 q^{85} +(136.000 - 235.559i) q^{86} +(100.000 - 173.205i) q^{89} +336.000 q^{92} +(-648.000 - 1122.37i) q^{94} +(-200.000 - 346.410i) q^{95} -1266.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} - 8 q^{4} - 4 q^{5} + 16 q^{10} + 62 q^{11} - 124 q^{13} + 64 q^{16} + 84 q^{17} - 100 q^{19} + 64 q^{20} + 496 q^{22} - 42 q^{23} + 109 q^{25} - 248 q^{26} + 20 q^{29} + 48 q^{31} - 256 q^{32}+ \cdots - 2532 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 − 3.46410i 0.707107 − 1.22474i −0.258819 − 0.965926i \(-0.583333\pi\)
0.965926 − 0.258819i \(-0.0833333\pi\)
\(3\) 0 0
\(4\) −4.00000 − 6.92820i −0.500000 − 0.866025i
\(5\) −2.00000 + 3.46410i −0.178885 + 0.309839i −0.941499 − 0.337016i \(-0.890582\pi\)
0.762614 + 0.646854i \(0.223916\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 8.00000 + 13.8564i 0.252982 + 0.438178i
\(11\) 31.0000 + 53.6936i 0.849714 + 1.47175i 0.881464 + 0.472252i \(0.156559\pi\)
−0.0317500 + 0.999496i \(0.510108\pi\)
\(12\) 0 0
\(13\) −62.0000 −1.32275 −0.661373 − 0.750057i \(-0.730026\pi\)
−0.661373 + 0.750057i \(0.730026\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 32.0000 − 55.4256i 0.500000 − 0.866025i
\(17\) 42.0000 + 72.7461i 0.599206 + 1.03785i 0.992939 + 0.118630i \(0.0378502\pi\)
−0.393733 + 0.919225i \(0.628817\pi\)
\(18\) 0 0
\(19\) −50.0000 + 86.6025i −0.603726 + 1.04568i 0.388526 + 0.921438i \(0.372984\pi\)
−0.992251 + 0.124246i \(0.960349\pi\)
\(20\) 32.0000 0.357771
\(21\) 0 0
\(22\) 248.000 2.40335
\(23\) −21.0000 + 36.3731i −0.190383 + 0.329753i −0.945377 − 0.325979i \(-0.894306\pi\)
0.754994 + 0.655731i \(0.227640\pi\)
\(24\) 0 0
\(25\) 54.5000 + 94.3968i 0.436000 + 0.755174i
\(26\) −124.000 + 214.774i −0.935323 + 1.62003i
\(27\) 0 0
\(28\) 0 0
\(29\) 10.0000 0.0640329 0.0320164 − 0.999487i \(-0.489807\pi\)
0.0320164 + 0.999487i \(0.489807\pi\)
\(30\) 0 0
\(31\) 24.0000 + 41.5692i 0.139049 + 0.240840i 0.927137 − 0.374723i \(-0.122262\pi\)
−0.788088 + 0.615563i \(0.788929\pi\)
\(32\) −128.000 − 221.703i −0.707107 − 1.22474i
\(33\) 0 0
\(34\) 336.000 1.69481
\(35\) 0 0
\(36\) 0 0
\(37\) 123.000 − 213.042i 0.546516 − 0.946593i −0.451994 − 0.892021i \(-0.649287\pi\)
0.998510 − 0.0545719i \(-0.0173794\pi\)
\(38\) 200.000 + 346.410i 0.853797 + 1.47882i
\(39\) 0 0
\(40\) 0 0
\(41\) 248.000 0.944661 0.472330 − 0.881422i \(-0.343413\pi\)
0.472330 + 0.881422i \(0.343413\pi\)
\(42\) 0 0
\(43\) 68.0000 0.241161 0.120580 − 0.992704i \(-0.461524\pi\)
0.120580 + 0.992704i \(0.461524\pi\)
\(44\) 248.000 − 429.549i 0.849714 − 1.47175i
\(45\) 0 0
\(46\) 84.0000 + 145.492i 0.269242 + 0.466341i
\(47\) 162.000 − 280.592i 0.502769 − 0.870821i −0.497226 − 0.867621i \(-0.665648\pi\)
0.999995 − 0.00319997i \(-0.00101858\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 436.000 1.23319
\(51\) 0 0
\(52\) 248.000 + 429.549i 0.661373 + 1.14553i
\(53\) 129.000 + 223.435i 0.334330 + 0.579077i 0.983356 − 0.181689i \(-0.0581565\pi\)
−0.649026 + 0.760767i \(0.724823\pi\)
\(54\) 0 0
\(55\) −248.000 −0.608006
\(56\) 0 0
\(57\) 0 0
\(58\) 20.0000 − 34.6410i 0.0452781 − 0.0784239i
\(59\) 60.0000 + 103.923i 0.132396 + 0.229316i 0.924600 − 0.380941i \(-0.124400\pi\)
−0.792204 + 0.610256i \(0.791066\pi\)
\(60\) 0 0
\(61\) −311.000 + 538.668i −0.652778 + 1.13064i 0.329668 + 0.944097i \(0.393063\pi\)
−0.982446 + 0.186548i \(0.940270\pi\)
\(62\) 192.000 0.393291
\(63\) 0 0
\(64\) −512.000 −1.00000
\(65\) 124.000 − 214.774i 0.236620 − 0.409838i
\(66\) 0 0
\(67\) −452.000 − 782.887i −0.824188 − 1.42754i −0.902538 − 0.430609i \(-0.858299\pi\)
0.0783505 − 0.996926i \(-0.475035\pi\)
\(68\) 336.000 − 581.969i 0.599206 − 1.03785i
\(69\) 0 0
\(70\) 0 0
\(71\) 678.000 1.13329 0.566646 − 0.823961i \(-0.308241\pi\)
0.566646 + 0.823961i \(0.308241\pi\)
\(72\) 0 0
\(73\) 321.000 + 555.988i 0.514660 + 0.891418i 0.999855 + 0.0170119i \(0.00541532\pi\)
−0.485195 + 0.874406i \(0.661251\pi\)
\(74\) −492.000 − 852.169i −0.772890 − 1.33868i
\(75\) 0 0
\(76\) 800.000 1.20745
\(77\) 0 0
\(78\) 0 0
\(79\) −370.000 + 640.859i −0.526940 + 0.912687i 0.472567 + 0.881295i \(0.343327\pi\)
−0.999507 + 0.0313921i \(0.990006\pi\)
\(80\) 128.000 + 221.703i 0.178885 + 0.309839i
\(81\) 0 0
\(82\) 496.000 − 859.097i 0.667976 − 1.15697i
\(83\) −468.000 −0.618912 −0.309456 − 0.950914i \(-0.600147\pi\)
−0.309456 + 0.950914i \(0.600147\pi\)
\(84\) 0 0
\(85\) −336.000 −0.428757
\(86\) 136.000 − 235.559i 0.170526 − 0.295360i
\(87\) 0 0
\(88\) 0 0
\(89\) 100.000 − 173.205i 0.119101 − 0.206289i −0.800311 − 0.599585i \(-0.795332\pi\)
0.919412 + 0.393297i \(0.128665\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 336.000 0.380765
\(93\) 0 0
\(94\) −648.000 − 1122.37i −0.711022 − 1.23153i
\(95\) −200.000 − 346.410i −0.215995 − 0.374115i
\(96\) 0 0
\(97\) −1266.00 −1.32518 −0.662592 − 0.748981i \(-0.730544\pi\)
−0.662592 + 0.748981i \(0.730544\pi\)
\(98\) 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.m.226.1 2
3.2 odd 2 147.4.e.c.79.1 2
7.2 even 3 63.4.a.a.1.1 1
7.3 odd 6 441.4.e.n.361.1 2
7.4 even 3 inner 441.4.e.m.361.1 2
7.5 odd 6 441.4.a.b.1.1 1
7.6 odd 2 441.4.e.n.226.1 2
21.2 odd 6 21.4.a.b.1.1 ✓ 1
21.5 even 6 147.4.a.g.1.1 1
21.11 odd 6 147.4.e.c.67.1 2
21.17 even 6 147.4.e.b.67.1 2
21.20 even 2 147.4.e.b.79.1 2
28.23 odd 6 1008.4.a.m.1.1 1
35.9 even 6 1575.4.a.k.1.1 1
84.23 even 6 336.4.a.h.1.1 1
84.47 odd 6 2352.4.a.l.1.1 1
105.2 even 12 525.4.d.b.274.2 2
105.23 even 12 525.4.d.b.274.1 2
105.44 odd 6 525.4.a.b.1.1 1
168.107 even 6 1344.4.a.i.1.1 1
168.149 odd 6 1344.4.a.w.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.b.1.1 ✓ 1 21.2 odd 6
63.4.a.a.1.1 1 7.2 even 3
147.4.a.g.1.1 1 21.5 even 6
147.4.e.b.67.1 2 21.17 even 6
147.4.e.b.79.1 2 21.20 even 2
147.4.e.c.67.1 2 21.11 odd 6
147.4.e.c.79.1 2 3.2 odd 2
336.4.a.h.1.1 1 84.23 even 6
441.4.a.b.1.1 1 7.5 odd 6
441.4.e.m.226.1 2 1.1 even 1 trivial
441.4.e.m.361.1 2 7.4 even 3 inner
441.4.e.n.226.1 2 7.6 odd 2
441.4.e.n.361.1 2 7.3 odd 6
525.4.a.b.1.1 1 105.44 odd 6
525.4.d.b.274.1 2 105.23 even 12
525.4.d.b.274.2 2 105.2 even 12
1008.4.a.m.1.1 1 28.23 odd 6
1344.4.a.i.1.1 1 168.107 even 6
1344.4.a.w.1.1 1 168.149 odd 6
1575.4.a.k.1.1 1 35.9 even 6
2352.4.a.l.1.1 1 84.47 odd 6