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Properties

Label 440.2.c.a.219.3
Level $440$
Weight $2$
Character 440.219
Analytic conductor $3.513$
Analytic rank $0$
Dimension $4$
CM discriminant -40
Inner twists $8$

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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] = [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 = [4] 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: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 219.3
Root \(-2.28825i\) of defining polynomial
Character \(\chi\) \(=\) 440.219
Dual form 440.2.c.a.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)
 
\(f(q)\) \(=\) \(q+1.41421i q^{2} -2.00000 q^{4} -2.23607 q^{5} -2.82843i q^{7} -2.82843i q^{8} +3.00000 q^{9} -3.16228i q^{10} +(-1.00000 + 3.16228i) q^{11} -5.65685i q^{13} +4.00000 q^{14} +4.00000 q^{16} +4.24264i q^{18} -6.32456i q^{19} +4.47214 q^{20} +(-4.47214 - 1.41421i) q^{22} +4.47214 q^{23} +5.00000 q^{25} +8.00000 q^{26} +5.65685i q^{28} +5.65685i q^{32} +6.32456i q^{35} -6.00000 q^{36} -4.47214 q^{37} +8.94427 q^{38} +6.32456i q^{40} -12.6491i q^{41} +(2.00000 - 6.32456i) q^{44} -6.70820 q^{45} +6.32456i q^{46} -13.4164 q^{47} -1.00000 q^{49} +7.07107i q^{50} +11.3137i q^{52} +13.4164 q^{53} +(2.23607 - 7.07107i) q^{55} -8.00000 q^{56} -14.0000 q^{59} -8.48528i q^{63} -8.00000 q^{64} +12.6491i q^{65} -8.94427 q^{70} -8.48528i q^{72} -6.32456i q^{74} +12.6491i q^{76} +(8.94427 + 2.82843i) q^{77} -8.94427 q^{80} +9.00000 q^{81} +17.8885 q^{82} +(8.94427 + 2.82843i) q^{88} +14.0000 q^{89} -9.48683i q^{90} -16.0000 q^{91} -8.94427 q^{92} -18.9737i q^{94} +14.1421i q^{95} -1.41421i q^{98} +(-3.00000 + 9.48683i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + 12 q^{9} - 4 q^{11} + 16 q^{14} + 16 q^{16} + 20 q^{25} + 32 q^{26} - 24 q^{36} + 8 q^{44} - 4 q^{49} - 32 q^{56} - 56 q^{59} - 32 q^{64} + 36 q^{81} + 56 q^{89} - 64 q^{91} - 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 1.00000i
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) −2.00000 −1.00000
\(5\) −2.23607 −1.00000
\(6\) 0 0
\(7\) − 2.82843i − 1.06904i −0.845154 − 0.534522i \(-0.820491\pi\)
0.845154 − 0.534522i \(-0.179509\pi\)
\(8\) − 2.82843i − 1.00000i
\(9\) 3.00000 1.00000
\(10\) − 3.16228i − 1.00000i
\(11\) −1.00000 + 3.16228i −0.301511 + 0.953463i
\(12\) 0 0
\(13\) − 5.65685i − 1.56893i −0.620174 − 0.784465i \(-0.712938\pi\)
0.620174 − 0.784465i \(-0.287062\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 4.24264i 1.00000i
\(19\) − 6.32456i − 1.45095i −0.688247 − 0.725476i \(-0.741620\pi\)
0.688247 − 0.725476i \(-0.258380\pi\)
\(20\) 4.47214 1.00000
\(21\) 0 0
\(22\) −4.47214 − 1.41421i −0.953463 − 0.301511i
\(23\) 4.47214 0.932505 0.466252 − 0.884652i \(-0.345604\pi\)
0.466252 + 0.884652i \(0.345604\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 8.00000 1.56893
\(27\) 0 0
\(28\) 5.65685i 1.06904i
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 5.65685i 1.00000i
\(33\) 0 0
\(34\) 0 0
\(35\) 6.32456i 1.06904i
\(36\) −6.00000 −1.00000
\(37\) −4.47214 −0.735215 −0.367607 − 0.929981i \(-0.619823\pi\)
−0.367607 + 0.929981i \(0.619823\pi\)
\(38\) 8.94427 1.45095
\(39\) 0 0
\(40\) 6.32456i 1.00000i
\(41\) − 12.6491i − 1.97546i −0.156174 − 0.987730i \(-0.549916\pi\)
0.156174 − 0.987730i \(-0.450084\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 2.00000 − 6.32456i 0.301511 − 0.953463i
\(45\) −6.70820 −1.00000
\(46\) 6.32456i 0.932505i
\(47\) −13.4164 −1.95698 −0.978492 − 0.206284i \(-0.933863\pi\)
−0.978492 + 0.206284i \(0.933863\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 7.07107i 1.00000i
\(51\) 0 0
\(52\) 11.3137i 1.56893i
\(53\) 13.4164 1.84289 0.921443 − 0.388514i \(-0.127012\pi\)
0.921443 + 0.388514i \(0.127012\pi\)
\(54\) 0 0
\(55\) 2.23607 − 7.07107i 0.301511 − 0.953463i
\(56\) −8.00000 −1.06904
\(57\) 0 0
\(58\) 0 0
\(59\) −14.0000 −1.82264 −0.911322 − 0.411693i \(-0.864937\pi\)
−0.911322 + 0.411693i \(0.864937\pi\)
\(60\) 0 0
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) − 8.48528i − 1.06904i
\(64\) −8.00000 −1.00000
\(65\) 12.6491i 1.56893i
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −8.94427 −1.06904
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) − 8.48528i − 1.00000i
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) − 6.32456i − 0.735215i
\(75\) 0 0
\(76\) 12.6491i 1.45095i
\(77\) 8.94427 + 2.82843i 1.01929 + 0.322329i
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −8.94427 −1.00000
\(81\) 9.00000 1.00000
\(82\) 17.8885 1.97546
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 8.94427 + 2.82843i 0.953463 + 0.301511i
\(89\) 14.0000 1.48400 0.741999 − 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) − 9.48683i − 1.00000i
\(91\) −16.0000 −1.67726
\(92\) −8.94427 −0.932505
\(93\) 0 0
\(94\) − 18.9737i − 1.95698i
\(95\) 14.1421i 1.45095i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) − 1.41421i − 0.142857i
\(99\) −3.00000 + 9.48683i −0.301511 + 0.953463i
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.a.219.3 yes 4
4.3 odd 2 1760.2.c.a.879.2 4
5.4 even 2 inner 440.2.c.a.219.2 yes 4
8.3 odd 2 inner 440.2.c.a.219.2 yes 4
8.5 even 2 1760.2.c.a.879.3 4
11.10 odd 2 inner 440.2.c.a.219.1 ✓ 4
20.19 odd 2 1760.2.c.a.879.3 4
40.19 odd 2 CM 440.2.c.a.219.3 yes 4
40.29 even 2 1760.2.c.a.879.2 4
44.43 even 2 1760.2.c.a.879.1 4
55.54 odd 2 inner 440.2.c.a.219.4 yes 4
88.21 odd 2 1760.2.c.a.879.4 4
88.43 even 2 inner 440.2.c.a.219.4 yes 4
220.219 even 2 1760.2.c.a.879.4 4
440.109 odd 2 1760.2.c.a.879.1 4
440.219 even 2 inner 440.2.c.a.219.1 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.c.a.219.1 ✓ 4 11.10 odd 2 inner
440.2.c.a.219.1 ✓ 4 440.219 even 2 inner
440.2.c.a.219.2 yes 4 5.4 even 2 inner
440.2.c.a.219.2 yes 4 8.3 odd 2 inner
440.2.c.a.219.3 yes 4 1.1 even 1 trivial
440.2.c.a.219.3 yes 4 40.19 odd 2 CM
440.2.c.a.219.4 yes 4 55.54 odd 2 inner
440.2.c.a.219.4 yes 4 88.43 even 2 inner
1760.2.c.a.879.1 4 44.43 even 2
1760.2.c.a.879.1 4 440.109 odd 2
1760.2.c.a.879.2 4 4.3 odd 2
1760.2.c.a.879.2 4 40.29 even 2
1760.2.c.a.879.3 4 8.5 even 2
1760.2.c.a.879.3 4 20.19 odd 2
1760.2.c.a.879.4 4 88.21 odd 2
1760.2.c.a.879.4 4 220.219 even 2