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Properties

Label 720.2.x.a.127.1
Level $720$
Weight $2$
Character 720.127
Analytic conductor $5.749$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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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] = [720,2,Mod(127,720)] 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("720.127"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.x (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 127.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 720.127
Dual form 720.2.x.a.703.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.00000 - 1.00000i) q^{5} +(-5.00000 + 5.00000i) q^{13} +(5.00000 + 5.00000i) q^{17} +(3.00000 + 4.00000i) q^{25} +4.00000i q^{29} +(5.00000 + 5.00000i) q^{37} -8.00000 q^{41} +7.00000i q^{49} +(-5.00000 + 5.00000i) q^{53} -12.0000 q^{61} +(15.0000 - 5.00000i) q^{65} +(5.00000 - 5.00000i) q^{73} +(-5.00000 - 15.0000i) q^{85} -16.0000i q^{89} +(5.00000 + 5.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} - 10 q^{13} + 10 q^{17} + 6 q^{25} + 10 q^{37} - 16 q^{41} - 10 q^{53} - 24 q^{61} + 30 q^{65} + 10 q^{73} - 10 q^{85} + 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −2.00000 − 1.00000i −0.894427 − 0.447214i
\(6\) 0 0
\(7\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −5.00000 + 5.00000i −1.38675 + 1.38675i −0.554700 + 0.832050i \(0.687167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.00000 + 5.00000i 1.21268 + 1.21268i 0.970143 + 0.242536i \(0.0779791\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(24\) 0 0
\(25\) 3.00000 + 4.00000i 0.600000 + 0.800000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.00000i 0.742781i 0.928477 + 0.371391i \(0.121119\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.00000 + 5.00000i 0.821995 + 0.821995i 0.986394 − 0.164399i \(-0.0525685\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 − 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(48\) 0 0
\(49\) 7.00000i 1.00000i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.00000 + 5.00000i −0.686803 + 0.686803i −0.961524 − 0.274721i \(-0.911414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −12.0000 −1.53644 −0.768221 − 0.640184i \(-0.778858\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 15.0000 − 5.00000i 1.86052 − 0.620174i
\(66\) 0 0
\(67\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 5.00000 − 5.00000i 0.585206 − 0.585206i −0.351123 − 0.936329i \(-0.614200\pi\)
0.936329 + 0.351123i \(0.114200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) 0 0
\(85\) −5.00000 − 15.0000i −0.542326 − 1.62698i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 16.0000i − 1.69600i −0.529999 − 0.847998i \(-0.677808\pi\)
0.529999 − 0.847998i \(-0.322192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 5.00000 + 5.00000i 0.507673 + 0.507673i 0.913812 − 0.406138i \(-0.133125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 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 720.2.x.a.127.1 2
3.2 odd 2 80.2.n.a.47.1 ✓ 2
4.3 odd 2 CM 720.2.x.a.127.1 2
5.2 odd 4 3600.2.x.c.2143.1 2
5.3 odd 4 inner 720.2.x.a.703.1 2
5.4 even 2 3600.2.x.c.3007.1 2
12.11 even 2 80.2.n.a.47.1 ✓ 2
15.2 even 4 400.2.n.a.143.1 2
15.8 even 4 80.2.n.a.63.1 yes 2
15.14 odd 2 400.2.n.a.207.1 2
20.3 even 4 inner 720.2.x.a.703.1 2
20.7 even 4 3600.2.x.c.2143.1 2
20.19 odd 2 3600.2.x.c.3007.1 2
24.5 odd 2 320.2.n.d.127.1 2
24.11 even 2 320.2.n.d.127.1 2
48.5 odd 4 1280.2.o.h.127.1 2
48.11 even 4 1280.2.o.h.127.1 2
48.29 odd 4 1280.2.o.i.127.1 2
48.35 even 4 1280.2.o.i.127.1 2
60.23 odd 4 80.2.n.a.63.1 yes 2
60.47 odd 4 400.2.n.a.143.1 2
60.59 even 2 400.2.n.a.207.1 2
120.29 odd 2 1600.2.n.g.1407.1 2
120.53 even 4 320.2.n.d.63.1 2
120.59 even 2 1600.2.n.g.1407.1 2
120.77 even 4 1600.2.n.g.1343.1 2
120.83 odd 4 320.2.n.d.63.1 2
120.107 odd 4 1600.2.n.g.1343.1 2
240.53 even 4 1280.2.o.i.383.1 2
240.83 odd 4 1280.2.o.h.383.1 2
240.173 even 4 1280.2.o.h.383.1 2
240.203 odd 4 1280.2.o.i.383.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.n.a.47.1 ✓ 2 3.2 odd 2
80.2.n.a.47.1 ✓ 2 12.11 even 2
80.2.n.a.63.1 yes 2 15.8 even 4
80.2.n.a.63.1 yes 2 60.23 odd 4
320.2.n.d.63.1 2 120.53 even 4
320.2.n.d.63.1 2 120.83 odd 4
320.2.n.d.127.1 2 24.5 odd 2
320.2.n.d.127.1 2 24.11 even 2
400.2.n.a.143.1 2 15.2 even 4
400.2.n.a.143.1 2 60.47 odd 4
400.2.n.a.207.1 2 15.14 odd 2
400.2.n.a.207.1 2 60.59 even 2
720.2.x.a.127.1 2 1.1 even 1 trivial
720.2.x.a.127.1 2 4.3 odd 2 CM
720.2.x.a.703.1 2 5.3 odd 4 inner
720.2.x.a.703.1 2 20.3 even 4 inner
1280.2.o.h.127.1 2 48.5 odd 4
1280.2.o.h.127.1 2 48.11 even 4
1280.2.o.h.383.1 2 240.83 odd 4
1280.2.o.h.383.1 2 240.173 even 4
1280.2.o.i.127.1 2 48.29 odd 4
1280.2.o.i.127.1 2 48.35 even 4
1280.2.o.i.383.1 2 240.53 even 4
1280.2.o.i.383.1 2 240.203 odd 4
1600.2.n.g.1343.1 2 120.77 even 4
1600.2.n.g.1343.1 2 120.107 odd 4
1600.2.n.g.1407.1 2 120.29 odd 2
1600.2.n.g.1407.1 2 120.59 even 2
3600.2.x.c.2143.1 2 5.2 odd 4
3600.2.x.c.2143.1 2 20.7 even 4
3600.2.x.c.3007.1 2 5.4 even 2
3600.2.x.c.3007.1 2 20.19 odd 2