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Properties

Label 2520.2.k.a.1889.13
Level $2520$
Weight $2$
Character 2520.1889
Analytic conductor $20.122$
Analytic rank $0$
Dimension $24$
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] = [2520,2,Mod(1889,2520)] 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("2520.1889"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2520, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2520.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(20.1223013094\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1889.13
Character \(\chi\) \(=\) 2520.1889
Dual form 2520.2.k.a.1889.14

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.655755 - 2.13775i) q^{5} +(-2.09441 + 1.61662i) q^{7} +6.16020i q^{11} +0.742412 q^{13} -3.80978i q^{17} -7.08718i q^{19} -5.47817 q^{23} +(-4.13997 - 2.80368i) q^{25} -3.48294i q^{29} +7.72681i q^{31} +(2.08251 + 5.53743i) q^{35} -4.20866i q^{37} +1.52402 q^{41} -12.0150i q^{43} +0.165189i q^{47} +(1.77309 - 6.77172i) q^{49} +1.52623 q^{53} +(13.1690 + 4.03959i) q^{55} -11.2218 q^{59} -7.03535i q^{61} +(0.486841 - 1.58709i) q^{65} +0.383095i q^{67} -5.81423i q^{71} +11.8823 q^{73} +(-9.95870 - 12.9020i) q^{77} +10.5244 q^{79} -10.5797i q^{83} +(-8.14436 - 2.49828i) q^{85} -0.779764 q^{89} +(-1.55491 + 1.20020i) q^{91} +(-15.1506 - 4.64745i) q^{95} -7.92966 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 8 q^{23} - 16 q^{25} + 4 q^{35} - 12 q^{49} + 24 q^{53} + 8 q^{65} + 4 q^{77} + 40 q^{79} + 24 q^{85} - 36 q^{91} - 48 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(1081\) \(1261\) \(2017\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0.655755 − 2.13775i 0.293263 − 0.956032i
\(6\) 0 0
\(7\) −2.09441 + 1.61662i −0.791612 + 0.611025i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 6.16020i 1.85737i 0.370868 + 0.928686i \(0.379060\pi\)
−0.370868 + 0.928686i \(0.620940\pi\)
\(12\) 0 0
\(13\) 0.742412 0.205908 0.102954 − 0.994686i \(-0.467171\pi\)
0.102954 + 0.994686i \(0.467171\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 3.80978i − 0.924007i −0.886878 − 0.462004i \(-0.847131\pi\)
0.886878 − 0.462004i \(-0.152869\pi\)
\(18\) 0 0
\(19\) − 7.08718i − 1.62591i −0.582327 − 0.812955i \(-0.697858\pi\)
0.582327 − 0.812955i \(-0.302142\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.47817 −1.14228 −0.571138 − 0.820854i \(-0.693498\pi\)
−0.571138 + 0.820854i \(0.693498\pi\)
\(24\) 0 0
\(25\) −4.13997 − 2.80368i −0.827994 − 0.560737i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 3.48294i − 0.646766i −0.946268 − 0.323383i \(-0.895180\pi\)
0.946268 − 0.323383i \(-0.104820\pi\)
\(30\) 0 0
\(31\) 7.72681i 1.38778i 0.720083 + 0.693888i \(0.244104\pi\)
−0.720083 + 0.693888i \(0.755896\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.08251 + 5.53743i 0.352009 + 0.935997i
\(36\) 0 0
\(37\) − 4.20866i − 0.691899i −0.938253 − 0.345950i \(-0.887557\pi\)
0.938253 − 0.345950i \(-0.112443\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.52402 0.238012 0.119006 − 0.992894i \(-0.462029\pi\)
0.119006 + 0.992894i \(0.462029\pi\)
\(42\) 0 0
\(43\) − 12.0150i − 1.83227i −0.400868 − 0.916136i \(-0.631292\pi\)
0.400868 − 0.916136i \(-0.368708\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.165189i 0.0240953i 0.999927 + 0.0120477i \(0.00383498\pi\)
−0.999927 + 0.0120477i \(0.996165\pi\)
\(48\) 0 0
\(49\) 1.77309 − 6.77172i 0.253298 − 0.967388i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.52623 0.209643 0.104822 − 0.994491i \(-0.466573\pi\)
0.104822 + 0.994491i \(0.466573\pi\)
\(54\) 0 0
\(55\) 13.1690 + 4.03959i 1.77571 + 0.544698i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −11.2218 −1.46095 −0.730477 − 0.682937i \(-0.760702\pi\)
−0.730477 + 0.682937i \(0.760702\pi\)
\(60\) 0 0
\(61\) − 7.03535i − 0.900785i −0.892831 − 0.450392i \(-0.851284\pi\)
0.892831 − 0.450392i \(-0.148716\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.486841 − 1.58709i 0.0603852 − 0.196855i
\(66\) 0 0
\(67\) 0.383095i 0.0468025i 0.999726 + 0.0234012i \(0.00744952\pi\)
−0.999726 + 0.0234012i \(0.992550\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 5.81423i − 0.690022i −0.938599 − 0.345011i \(-0.887875\pi\)
0.938599 − 0.345011i \(-0.112125\pi\)
\(72\) 0 0
\(73\) 11.8823 1.39072 0.695358 − 0.718663i \(-0.255246\pi\)
0.695358 + 0.718663i \(0.255246\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −9.95870 − 12.9020i −1.13490 − 1.47032i
\(78\) 0 0
\(79\) 10.5244 1.18409 0.592045 − 0.805905i \(-0.298321\pi\)
0.592045 + 0.805905i \(0.298321\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 10.5797i − 1.16127i −0.814164 − 0.580635i \(-0.802804\pi\)
0.814164 − 0.580635i \(-0.197196\pi\)
\(84\) 0 0
\(85\) −8.14436 − 2.49828i −0.883380 − 0.270977i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −0.779764 −0.0826549 −0.0413274 − 0.999146i \(-0.513159\pi\)
−0.0413274 + 0.999146i \(0.513159\pi\)
\(90\) 0 0
\(91\) −1.55491 + 1.20020i −0.162999 + 0.125815i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −15.1506 − 4.64745i −1.55442 − 0.476818i
\(96\) 0 0
\(97\) −7.92966 −0.805135 −0.402568 − 0.915390i \(-0.631882\pi\)
−0.402568 + 0.915390i \(0.631882\pi\)
\(98\) 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 2520.2.k.a.1889.13 yes 24
3.2 odd 2 2520.2.k.b.1889.12 yes 24
4.3 odd 2 5040.2.k.i.1889.13 24
5.4 even 2 2520.2.k.b.1889.14 yes 24
7.6 odd 2 inner 2520.2.k.a.1889.12 yes 24
12.11 even 2 5040.2.k.h.1889.12 24
15.14 odd 2 inner 2520.2.k.a.1889.11 ✓ 24
20.19 odd 2 5040.2.k.h.1889.14 24
21.20 even 2 2520.2.k.b.1889.13 yes 24
28.27 even 2 5040.2.k.i.1889.12 24
35.34 odd 2 2520.2.k.b.1889.11 yes 24
60.59 even 2 5040.2.k.i.1889.11 24
84.83 odd 2 5040.2.k.h.1889.13 24
105.104 even 2 inner 2520.2.k.a.1889.14 yes 24
140.139 even 2 5040.2.k.h.1889.11 24
420.419 odd 2 5040.2.k.i.1889.14 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2520.2.k.a.1889.11 ✓ 24 15.14 odd 2 inner
2520.2.k.a.1889.12 yes 24 7.6 odd 2 inner
2520.2.k.a.1889.13 yes 24 1.1 even 1 trivial
2520.2.k.a.1889.14 yes 24 105.104 even 2 inner
2520.2.k.b.1889.11 yes 24 35.34 odd 2
2520.2.k.b.1889.12 yes 24 3.2 odd 2
2520.2.k.b.1889.13 yes 24 21.20 even 2
2520.2.k.b.1889.14 yes 24 5.4 even 2
5040.2.k.h.1889.11 24 140.139 even 2
5040.2.k.h.1889.12 24 12.11 even 2
5040.2.k.h.1889.13 24 84.83 odd 2
5040.2.k.h.1889.14 24 20.19 odd 2
5040.2.k.i.1889.11 24 60.59 even 2
5040.2.k.i.1889.12 24 28.27 even 2
5040.2.k.i.1889.13 24 4.3 odd 2
5040.2.k.i.1889.14 24 420.419 odd 2