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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [440,2,Mod(219,440)] 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("440.219"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 440.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.51341768894\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.599695360000.19
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 219.2
Root \(-1.35246 + 0.413333i\) of defining polynomial
Character \(\chi\) \(=\) 440.219
Dual form 440.2.c.b.219.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.35246 + 0.413333i) q^{2} +(1.65831 - 1.11803i) q^{4} +2.23607i q^{5} +5.22173i q^{7} +(-1.78069 + 2.19753i) q^{8} +3.00000 q^{9} +(-0.924240 - 3.02420i) q^{10} -3.31662 q^{11} -3.56840i q^{13} +(-2.15831 - 7.06220i) q^{14} +(1.50000 - 3.70810i) q^{16} -4.55341 q^{17} +(-4.05739 + 1.24000i) q^{18} +(2.50000 + 3.70810i) q^{20} +(4.48561 - 1.37087i) q^{22} -5.00000 q^{25} +(1.47494 + 4.82613i) q^{26} +(5.83808 + 8.65927i) q^{28} +8.94427i q^{31} +(-0.496016 + 5.63507i) q^{32} +(6.15831 - 1.88207i) q^{34} -11.6762 q^{35} +(4.97494 - 3.35410i) q^{36} +(-4.91384 - 3.98174i) q^{40} +9.96326 q^{43} +(-5.50000 + 3.70810i) q^{44} +6.70820i q^{45} -20.2665 q^{49} +(6.76232 - 2.06666i) q^{50} +(-3.98960 - 5.91753i) q^{52} -7.41620i q^{55} +(-11.4749 - 9.29827i) q^{56} -4.00000 q^{59} +(-3.69696 - 12.0968i) q^{62} +15.6652i q^{63} +(-1.65831 - 7.82624i) q^{64} +7.97919 q^{65} +(-7.55097 + 5.09086i) q^{68} +(15.7916 - 4.82613i) q^{70} +14.8324i q^{71} +(-5.34206 + 6.59260i) q^{72} +17.0860 q^{73} -17.3185i q^{77} +(8.29156 + 3.35410i) q^{80} +9.00000 q^{81} -13.3890 q^{83} -10.1817i q^{85} +(-13.4749 + 4.11814i) q^{86} +(5.90587 - 7.28840i) q^{88} +13.2665 q^{89} +(-2.77272 - 9.07260i) q^{90} +18.6332 q^{91} +(27.4097 - 8.37680i) q^{98} -9.94987 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{9} - 4 q^{14} + 12 q^{16} + 20 q^{20} - 40 q^{25} - 28 q^{26} + 36 q^{34} - 44 q^{44} - 56 q^{49} - 52 q^{56} - 32 q^{59} + 60 q^{70} + 72 q^{81} - 68 q^{86} + 96 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/440\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(177\) \(221\) \(321\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.35246 + 0.413333i −0.956336 + 0.292270i
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 1.65831 − 1.11803i 0.829156 − 0.559017i
\(5\) 2.23607i 1.00000i
\(6\) 0 0
\(7\) 5.22173i 1.97363i 0.161853 + 0.986815i \(0.448253\pi\)
−0.161853 + 0.986815i \(0.551747\pi\)
\(8\) −1.78069 + 2.19753i −0.629568 + 0.776946i
\(9\) 3.00000 1.00000
\(10\) −0.924240 − 3.02420i −0.292270 − 0.956336i
\(11\) −3.31662 −1.00000
\(12\) 0 0
\(13\) − 3.56840i − 0.989697i −0.868979 − 0.494848i \(-0.835224\pi\)
0.868979 − 0.494848i \(-0.164776\pi\)
\(14\) −2.15831 − 7.06220i −0.576833 − 1.88745i
\(15\) 0 0
\(16\) 1.50000 − 3.70810i 0.375000 − 0.927025i
\(17\) −4.55341 −1.10436 −0.552182 − 0.833724i \(-0.686204\pi\)
−0.552182 + 0.833724i \(0.686204\pi\)
\(18\) −4.05739 + 1.24000i −0.956336 + 0.292270i
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 2.50000 + 3.70810i 0.559017 + 0.829156i
\(21\) 0 0
\(22\) 4.48561 − 1.37087i 0.956336 − 0.292270i
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 1.47494 + 4.82613i 0.289259 + 0.946483i
\(27\) 0 0
\(28\) 5.83808 + 8.65927i 1.10329 + 1.63645i
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 8.94427i 1.60644i 0.595683 + 0.803219i \(0.296881\pi\)
−0.595683 + 0.803219i \(0.703119\pi\)
\(32\) −0.496016 + 5.63507i −0.0876841 + 0.996148i
\(33\) 0 0
\(34\) 6.15831 − 1.88207i 1.05614 − 0.322772i
\(35\) −11.6762 −1.97363
\(36\) 4.97494 − 3.35410i 0.829156 − 0.559017i
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −4.91384 − 3.98174i −0.776946 − 0.629568i
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 9.96326 1.51938 0.759691 − 0.650284i \(-0.225350\pi\)
0.759691 + 0.650284i \(0.225350\pi\)
\(44\) −5.50000 + 3.70810i −0.829156 + 0.559017i
\(45\) 6.70820i 1.00000i
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −20.2665 −2.89521
\(50\) 6.76232 − 2.06666i 0.956336 − 0.292270i
\(51\) 0 0
\(52\) −3.98960 − 5.91753i −0.553257 − 0.820613i
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) − 7.41620i − 1.00000i
\(56\) −11.4749 − 9.29827i −1.53340 − 1.24253i
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) −3.69696 − 12.0968i −0.469514 − 1.53629i
\(63\) 15.6652i 1.97363i
\(64\) −1.65831 − 7.82624i −0.207289 − 0.978280i
\(65\) 7.97919 0.989697
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −7.55097 + 5.09086i −0.915689 + 0.617358i
\(69\) 0 0
\(70\) 15.7916 − 4.82613i 1.88745 − 0.576833i
\(71\) 14.8324i 1.76028i 0.474713 + 0.880141i \(0.342552\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −5.34206 + 6.59260i −0.629568 + 0.776946i
\(73\) 17.0860 1.99977 0.999883 − 0.0153173i \(-0.00487585\pi\)
0.999883 + 0.0153173i \(0.00487585\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 17.3185i − 1.97363i
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 8.29156 + 3.35410i 0.927025 + 0.375000i
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −13.3890 −1.46964 −0.734819 − 0.678263i \(-0.762733\pi\)
−0.734819 + 0.678263i \(0.762733\pi\)
\(84\) 0 0
\(85\) − 10.1817i − 1.10436i
\(86\) −13.4749 + 4.11814i −1.45304 + 0.444070i
\(87\) 0 0
\(88\) 5.90587 − 7.28840i 0.629568 − 0.776946i
\(89\) 13.2665 1.40625 0.703123 − 0.711068i \(-0.251788\pi\)
0.703123 + 0.711068i \(0.251788\pi\)
\(90\) −2.77272 − 9.07260i −0.292270 − 0.956336i
\(91\) 18.6332 1.95330
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 27.4097 − 8.37680i 2.76880 − 0.846185i
\(99\) −9.94987 −1.00000
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 440.2.c.b.219.2 yes 8
4.3 odd 2 1760.2.c.b.879.5 8
5.4 even 2 inner 440.2.c.b.219.7 yes 8
8.3 odd 2 inner 440.2.c.b.219.1 ✓ 8
8.5 even 2 1760.2.c.b.879.4 8
11.10 odd 2 inner 440.2.c.b.219.7 yes 8
20.19 odd 2 1760.2.c.b.879.8 8
40.19 odd 2 inner 440.2.c.b.219.8 yes 8
40.29 even 2 1760.2.c.b.879.1 8
44.43 even 2 1760.2.c.b.879.8 8
55.54 odd 2 CM 440.2.c.b.219.2 yes 8
88.21 odd 2 1760.2.c.b.879.1 8
88.43 even 2 inner 440.2.c.b.219.8 yes 8
220.219 even 2 1760.2.c.b.879.5 8
440.109 odd 2 1760.2.c.b.879.4 8
440.219 even 2 inner 440.2.c.b.219.1 ✓ 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.c.b.219.1 ✓ 8 8.3 odd 2 inner
440.2.c.b.219.1 ✓ 8 440.219 even 2 inner
440.2.c.b.219.2 yes 8 1.1 even 1 trivial
440.2.c.b.219.2 yes 8 55.54 odd 2 CM
440.2.c.b.219.7 yes 8 5.4 even 2 inner
440.2.c.b.219.7 yes 8 11.10 odd 2 inner
440.2.c.b.219.8 yes 8 40.19 odd 2 inner
440.2.c.b.219.8 yes 8 88.43 even 2 inner
1760.2.c.b.879.1 8 40.29 even 2
1760.2.c.b.879.1 8 88.21 odd 2
1760.2.c.b.879.4 8 8.5 even 2
1760.2.c.b.879.4 8 440.109 odd 2
1760.2.c.b.879.5 8 4.3 odd 2
1760.2.c.b.879.5 8 220.219 even 2
1760.2.c.b.879.8 8 20.19 odd 2
1760.2.c.b.879.8 8 44.43 even 2