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Properties

Label 2880.2.t.c.721.6
Level $2880$
Weight $2$
Character 2880.721
Analytic conductor $22.997$
Analytic rank $0$
Dimension $16$
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] = [2880,2,Mod(721,2880)] 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("2880.721"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2880, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2880.t (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16] 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: \(22.9969157821\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 721.6
Root \(-0.296075 - 1.38287i\) of defining polynomial
Character \(\chi\) \(=\) 2880.721
Dual form 2880.2.t.c.2161.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.707107 - 0.707107i) q^{5} -2.66881i q^{7} +(-3.49714 + 3.49714i) q^{11} +(2.94072 + 2.94072i) q^{13} -1.85116 q^{17} +(3.44856 + 3.44856i) q^{19} -0.707288i q^{23} -1.00000i q^{25} +(3.49909 + 3.49909i) q^{29} -6.84272 q^{31} +(-1.88714 - 1.88714i) q^{35} +(-0.0975060 + 0.0975060i) q^{37} +10.2052i q^{41} +(-4.43844 + 4.43844i) q^{43} -1.89428 q^{47} -0.122561 q^{49} +(7.43897 - 7.43897i) q^{53} +4.94571i q^{55} +(0.959574 - 0.959574i) q^{59} +(6.49825 + 6.49825i) q^{61} +4.15881 q^{65} +(-3.49691 - 3.49691i) q^{67} +7.86777i q^{71} +15.6564i q^{73} +(9.33322 + 9.33322i) q^{77} +6.70212 q^{79} +(-3.87327 - 3.87327i) q^{83} +(-1.30896 + 1.30896i) q^{85} -10.5055i q^{89} +(7.84824 - 7.84824i) q^{91} +4.87701 q^{95} +4.79937 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{11} + 8 q^{19} + 16 q^{29} - 16 q^{37} - 8 q^{43} - 40 q^{47} - 16 q^{49} - 16 q^{53} - 8 q^{59} + 16 q^{61} - 40 q^{67} - 16 q^{77} - 16 q^{79} + 40 q^{83} - 16 q^{85} - 32 q^{91} + 32 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(2431\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\) \(1\)

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
\(5\) 0.707107 − 0.707107i 0.316228 − 0.316228i
\(6\) 0 0
\(7\) − 2.66881i − 1.00872i −0.863495 − 0.504358i \(-0.831729\pi\)
0.863495 − 0.504358i \(-0.168271\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.49714 + 3.49714i −1.05443 + 1.05443i −0.0559977 + 0.998431i \(0.517834\pi\)
−0.998431 + 0.0559977i \(0.982166\pi\)
\(12\) 0 0
\(13\) 2.94072 + 2.94072i 0.815610 + 0.815610i 0.985468 − 0.169858i \(-0.0543310\pi\)
−0.169858 + 0.985468i \(0.554331\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.85116 −0.448971 −0.224486 − 0.974477i \(-0.572070\pi\)
−0.224486 + 0.974477i \(0.572070\pi\)
\(18\) 0 0
\(19\) 3.44856 + 3.44856i 0.791155 + 0.791155i 0.981682 − 0.190527i \(-0.0610197\pi\)
−0.190527 + 0.981682i \(0.561020\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 0.707288i − 0.147480i −0.997278 − 0.0737399i \(-0.976507\pi\)
0.997278 − 0.0737399i \(-0.0234935\pi\)
\(24\) 0 0
\(25\) − 1.00000i − 0.200000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.49909 + 3.49909i 0.649766 + 0.649766i 0.952936 − 0.303171i \(-0.0980452\pi\)
−0.303171 + 0.952936i \(0.598045\pi\)
\(30\) 0 0
\(31\) −6.84272 −1.22899 −0.614494 − 0.788921i \(-0.710640\pi\)
−0.614494 + 0.788921i \(0.710640\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.88714 − 1.88714i −0.318984 − 0.318984i
\(36\) 0 0
\(37\) −0.0975060 + 0.0975060i −0.0160299 + 0.0160299i −0.715076 − 0.699046i \(-0.753608\pi\)
0.699046 + 0.715076i \(0.253608\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.2052i 1.59379i 0.604117 + 0.796896i \(0.293526\pi\)
−0.604117 + 0.796896i \(0.706474\pi\)
\(42\) 0 0
\(43\) −4.43844 + 4.43844i −0.676855 + 0.676855i −0.959287 − 0.282432i \(-0.908859\pi\)
0.282432 + 0.959287i \(0.408859\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.89428 −0.276310 −0.138155 − 0.990411i \(-0.544117\pi\)
−0.138155 + 0.990411i \(0.544117\pi\)
\(48\) 0 0
\(49\) −0.122561 −0.0175087
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.43897 − 7.43897i 1.02182 − 1.02182i 0.0220650 − 0.999757i \(-0.492976\pi\)
0.999757 − 0.0220650i \(-0.00702407\pi\)
\(54\) 0 0
\(55\) 4.94571i 0.666879i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.959574 − 0.959574i 0.124926 − 0.124926i −0.641880 − 0.766805i \(-0.721845\pi\)
0.766805 + 0.641880i \(0.221845\pi\)
\(60\) 0 0
\(61\) 6.49825 + 6.49825i 0.832015 + 0.832015i 0.987792 − 0.155777i \(-0.0497881\pi\)
−0.155777 + 0.987792i \(0.549788\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.15881 0.515837
\(66\) 0 0
\(67\) −3.49691 − 3.49691i −0.427216 − 0.427216i 0.460463 − 0.887679i \(-0.347683\pi\)
−0.887679 + 0.460463i \(0.847683\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.86777i 0.933733i 0.884328 + 0.466866i \(0.154617\pi\)
−0.884328 + 0.466866i \(0.845383\pi\)
\(72\) 0 0
\(73\) 15.6564i 1.83244i 0.400675 + 0.916220i \(0.368776\pi\)
−0.400675 + 0.916220i \(0.631224\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 9.33322 + 9.33322i 1.06362 + 1.06362i
\(78\) 0 0
\(79\) 6.70212 0.754047 0.377024 − 0.926204i \(-0.376948\pi\)
0.377024 + 0.926204i \(0.376948\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.87327 − 3.87327i −0.425147 − 0.425147i 0.461825 − 0.886971i \(-0.347195\pi\)
−0.886971 + 0.461825i \(0.847195\pi\)
\(84\) 0 0
\(85\) −1.30896 + 1.30896i −0.141977 + 0.141977i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 10.5055i − 1.11358i −0.830653 − 0.556790i \(-0.812033\pi\)
0.830653 − 0.556790i \(-0.187967\pi\)
\(90\) 0 0
\(91\) 7.84824 − 7.84824i 0.822719 − 0.822719i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.87701 0.500370
\(96\) 0 0
\(97\) 4.79937 0.487303 0.243651 − 0.969863i \(-0.421655\pi\)
0.243651 + 0.969863i \(0.421655\pi\)
\(98\) 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 2880.2.t.c.721.6 16
3.2 odd 2 320.2.l.a.81.5 16
4.3 odd 2 720.2.t.c.541.2 16
12.11 even 2 80.2.l.a.61.7 yes 16
15.2 even 4 1600.2.q.g.849.5 16
15.8 even 4 1600.2.q.h.849.4 16
15.14 odd 2 1600.2.l.i.401.4 16
16.5 even 4 inner 2880.2.t.c.2161.7 16
16.11 odd 4 720.2.t.c.181.2 16
24.5 odd 2 640.2.l.a.161.4 16
24.11 even 2 640.2.l.b.161.5 16
48.5 odd 4 320.2.l.a.241.5 16
48.11 even 4 80.2.l.a.21.7 ✓ 16
48.29 odd 4 640.2.l.a.481.4 16
48.35 even 4 640.2.l.b.481.5 16
60.23 odd 4 400.2.q.g.349.3 16
60.47 odd 4 400.2.q.h.349.6 16
60.59 even 2 400.2.l.h.301.2 16
96.5 odd 8 5120.2.a.u.1.4 8
96.11 even 8 5120.2.a.v.1.4 8
96.53 odd 8 5120.2.a.t.1.5 8
96.59 even 8 5120.2.a.s.1.5 8
240.53 even 4 1600.2.q.g.49.5 16
240.59 even 4 400.2.l.h.101.2 16
240.107 odd 4 400.2.q.g.149.3 16
240.149 odd 4 1600.2.l.i.1201.4 16
240.197 even 4 1600.2.q.h.49.4 16
240.203 odd 4 400.2.q.h.149.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.l.a.21.7 ✓ 16 48.11 even 4
80.2.l.a.61.7 yes 16 12.11 even 2
320.2.l.a.81.5 16 3.2 odd 2
320.2.l.a.241.5 16 48.5 odd 4
400.2.l.h.101.2 16 240.59 even 4
400.2.l.h.301.2 16 60.59 even 2
400.2.q.g.149.3 16 240.107 odd 4
400.2.q.g.349.3 16 60.23 odd 4
400.2.q.h.149.6 16 240.203 odd 4
400.2.q.h.349.6 16 60.47 odd 4
640.2.l.a.161.4 16 24.5 odd 2
640.2.l.a.481.4 16 48.29 odd 4
640.2.l.b.161.5 16 24.11 even 2
640.2.l.b.481.5 16 48.35 even 4
720.2.t.c.181.2 16 16.11 odd 4
720.2.t.c.541.2 16 4.3 odd 2
1600.2.l.i.401.4 16 15.14 odd 2
1600.2.l.i.1201.4 16 240.149 odd 4
1600.2.q.g.49.5 16 240.53 even 4
1600.2.q.g.849.5 16 15.2 even 4
1600.2.q.h.49.4 16 240.197 even 4
1600.2.q.h.849.4 16 15.8 even 4
2880.2.t.c.721.6 16 1.1 even 1 trivial
2880.2.t.c.2161.7 16 16.5 even 4 inner
5120.2.a.s.1.5 8 96.59 even 8
5120.2.a.t.1.5 8 96.53 odd 8
5120.2.a.u.1.4 8 96.5 odd 8
5120.2.a.v.1.4 8 96.11 even 8