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Properties

Label 5040.2.k.i.1889.9
Level $5040$
Weight $2$
Character 5040.1889
Analytic conductor $40.245$
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] = [5040,2,Mod(1889,5040)] 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("5040.1889"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5040, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5040.k (of order \(2\), degree \(1\), not 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: \(40.2446026187\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 2520)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1889.9
Character \(\chi\) \(=\) 5040.1889
Dual form 5040.2.k.i.1889.10

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.727958 - 2.11426i) q^{5} +(1.21521 + 2.35016i) q^{7} +2.91501i q^{11} -0.380654 q^{13} +7.43045i q^{17} +4.51415i q^{19} -2.63030 q^{23} +(-3.94016 + 3.07818i) q^{25} -8.44541i q^{29} +5.79961i q^{31} +(4.08422 - 4.28008i) q^{35} -9.46775i q^{37} +0.570545 q^{41} +2.65721i q^{43} -8.79311i q^{47} +(-4.04653 + 5.71188i) q^{49} -14.0197 q^{53} +(6.16307 - 2.12200i) q^{55} -6.55468 q^{59} +2.66361i q^{61} +(0.277100 + 0.804799i) q^{65} -8.87911i q^{67} -13.6005i q^{71} -6.55607 q^{73} +(-6.85074 + 3.54234i) q^{77} -6.74215 q^{79} -15.5747i q^{83} +(15.7099 - 5.40905i) q^{85} -17.4408 q^{89} +(-0.462574 - 0.894598i) q^{91} +(9.54407 - 3.28611i) q^{95} -6.88869 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 8 q^{23} - 16 q^{25} - 4 q^{35} - 12 q^{49} + 24 q^{53} + 8 q^{65} + 4 q^{77} - 40 q^{79} + 24 q^{85} + 36 q^{91} + 48 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5040\mathbb{Z}\right)^\times\).

\(n\) \(2017\) \(2801\) \(3151\) \(3601\) \(3781\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\) \(1\)

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
\(5\) −0.727958 − 2.11426i −0.325553 − 0.945524i
\(6\) 0 0
\(7\) 1.21521 + 2.35016i 0.459306 + 0.888278i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.91501i 0.878908i 0.898265 + 0.439454i \(0.144828\pi\)
−0.898265 + 0.439454i \(0.855172\pi\)
\(12\) 0 0
\(13\) −0.380654 −0.105574 −0.0527872 − 0.998606i \(-0.516810\pi\)
−0.0527872 + 0.998606i \(0.516810\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.43045i 1.80215i 0.433666 + 0.901074i \(0.357220\pi\)
−0.433666 + 0.901074i \(0.642780\pi\)
\(18\) 0 0
\(19\) 4.51415i 1.03562i 0.855496 + 0.517809i \(0.173252\pi\)
−0.855496 + 0.517809i \(0.826748\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.63030 −0.548456 −0.274228 − 0.961665i \(-0.588422\pi\)
−0.274228 + 0.961665i \(0.588422\pi\)
\(24\) 0 0
\(25\) −3.94016 + 3.07818i −0.788031 + 0.615636i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 8.44541i − 1.56827i −0.620588 − 0.784137i \(-0.713106\pi\)
0.620588 − 0.784137i \(-0.286894\pi\)
\(30\) 0 0
\(31\) 5.79961i 1.04164i 0.853667 + 0.520820i \(0.174374\pi\)
−0.853667 + 0.520820i \(0.825626\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.08422 − 4.28008i 0.690360 − 0.723466i
\(36\) 0 0
\(37\) − 9.46775i − 1.55649i −0.627962 − 0.778244i \(-0.716111\pi\)
0.627962 − 0.778244i \(-0.283889\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.570545 0.0891042 0.0445521 − 0.999007i \(-0.485814\pi\)
0.0445521 + 0.999007i \(0.485814\pi\)
\(42\) 0 0
\(43\) 2.65721i 0.405221i 0.979259 + 0.202611i \(0.0649426\pi\)
−0.979259 + 0.202611i \(0.935057\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 8.79311i − 1.28261i −0.767287 − 0.641303i \(-0.778394\pi\)
0.767287 − 0.641303i \(-0.221606\pi\)
\(48\) 0 0
\(49\) −4.04653 + 5.71188i −0.578076 + 0.815983i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −14.0197 −1.92576 −0.962878 − 0.269936i \(-0.912997\pi\)
−0.962878 + 0.269936i \(0.912997\pi\)
\(54\) 0 0
\(55\) 6.16307 − 2.12200i 0.831028 − 0.286131i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.55468 −0.853347 −0.426673 − 0.904406i \(-0.640315\pi\)
−0.426673 + 0.904406i \(0.640315\pi\)
\(60\) 0 0
\(61\) 2.66361i 0.341040i 0.985354 + 0.170520i \(0.0545448\pi\)
−0.985354 + 0.170520i \(0.945455\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.277100 + 0.804799i 0.0343700 + 0.0998231i
\(66\) 0 0
\(67\) − 8.87911i − 1.08476i −0.840135 − 0.542378i \(-0.817524\pi\)
0.840135 − 0.542378i \(-0.182476\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 13.6005i − 1.61408i −0.590497 − 0.807040i \(-0.701068\pi\)
0.590497 − 0.807040i \(-0.298932\pi\)
\(72\) 0 0
\(73\) −6.55607 −0.767330 −0.383665 − 0.923472i \(-0.625338\pi\)
−0.383665 + 0.923472i \(0.625338\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.85074 + 3.54234i −0.780715 + 0.403688i
\(78\) 0 0
\(79\) −6.74215 −0.758551 −0.379276 − 0.925284i \(-0.623827\pi\)
−0.379276 + 0.925284i \(0.623827\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 15.5747i − 1.70954i −0.519005 − 0.854771i \(-0.673698\pi\)
0.519005 − 0.854771i \(-0.326302\pi\)
\(84\) 0 0
\(85\) 15.7099 − 5.40905i 1.70397 − 0.586694i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −17.4408 −1.84873 −0.924363 − 0.381514i \(-0.875403\pi\)
−0.924363 + 0.381514i \(0.875403\pi\)
\(90\) 0 0
\(91\) −0.462574 − 0.894598i −0.0484909 − 0.0937794i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 9.54407 − 3.28611i 0.979201 − 0.337148i
\(96\) 0 0
\(97\) −6.88869 −0.699441 −0.349720 − 0.936854i \(-0.613723\pi\)
−0.349720 + 0.936854i \(0.613723\pi\)
\(98\) 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 5040.2.k.i.1889.9 24
3.2 odd 2 5040.2.k.h.1889.16 24
4.3 odd 2 2520.2.k.a.1889.9 ✓ 24
5.4 even 2 5040.2.k.h.1889.10 24
7.6 odd 2 inner 5040.2.k.i.1889.16 24
12.11 even 2 2520.2.k.b.1889.16 yes 24
15.14 odd 2 inner 5040.2.k.i.1889.15 24
20.19 odd 2 2520.2.k.b.1889.10 yes 24
21.20 even 2 5040.2.k.h.1889.9 24
28.27 even 2 2520.2.k.a.1889.16 yes 24
35.34 odd 2 5040.2.k.h.1889.15 24
60.59 even 2 2520.2.k.a.1889.15 yes 24
84.83 odd 2 2520.2.k.b.1889.9 yes 24
105.104 even 2 inner 5040.2.k.i.1889.10 24
140.139 even 2 2520.2.k.b.1889.15 yes 24
420.419 odd 2 2520.2.k.a.1889.10 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2520.2.k.a.1889.9 ✓ 24 4.3 odd 2
2520.2.k.a.1889.10 yes 24 420.419 odd 2
2520.2.k.a.1889.15 yes 24 60.59 even 2
2520.2.k.a.1889.16 yes 24 28.27 even 2
2520.2.k.b.1889.9 yes 24 84.83 odd 2
2520.2.k.b.1889.10 yes 24 20.19 odd 2
2520.2.k.b.1889.15 yes 24 140.139 even 2
2520.2.k.b.1889.16 yes 24 12.11 even 2
5040.2.k.h.1889.9 24 21.20 even 2
5040.2.k.h.1889.10 24 5.4 even 2
5040.2.k.h.1889.15 24 35.34 odd 2
5040.2.k.h.1889.16 24 3.2 odd 2
5040.2.k.i.1889.9 24 1.1 even 1 trivial
5040.2.k.i.1889.10 24 105.104 even 2 inner
5040.2.k.i.1889.15 24 15.14 odd 2 inner
5040.2.k.i.1889.16 24 7.6 odd 2 inner