↑

Properties

Label 5040.2.t.bb.1009.3
Level $5040$
Weight $2$
Character 5040.1009
Analytic conductor $40.245$
Analytic rank $0$
Dimension $8$
Inner twists $4$

Related objects

Downloads

Learn more

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(1009,5040)] 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("5040.1009"); 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, 0, 1, 0])) 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.t (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(40.2446026187\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(i, \sqrt{3}, \sqrt{7})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 1260)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1009.3
Root \(-0.228425 - 1.39564i\) of defining polynomial
Character \(\chi\) \(=\) 5040.1009
Dual form 5040.2.t.bb.1009.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.456850 - 2.18890i) q^{5} +1.00000i q^{7} +3.46410 q^{11} -3.58258i q^{13} +2.55040i q^{17} -3.58258 q^{19} +0.913701i q^{23} +(-4.58258 + 2.00000i) q^{25} -8.75560 q^{29} +3.58258 q^{31} +(2.18890 - 0.456850i) q^{35} -4.00000i q^{37} -9.66930 q^{41} -11.1652i q^{43} +1.82740i q^{47} -1.00000 q^{49} -9.66930i q^{53} +(-1.58258 - 7.58258i) q^{55} +1.82740 q^{59} +9.16515 q^{61} +(-7.84190 + 1.63670i) q^{65} +8.00000i q^{67} -3.46410 q^{71} +3.58258i q^{73} +3.46410i q^{77} +11.1652 q^{79} -6.92820i q^{83} +(5.58258 - 1.16515i) q^{85} -12.9427 q^{89} +3.58258 q^{91} +(1.63670 + 7.84190i) q^{95} -4.41742i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{19} - 8 q^{31} - 8 q^{49} + 24 q^{55} + 16 q^{79} + 8 q^{85} - 8 q^{91}+O(q^{100}) \) 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
\(5\) −0.456850 − 2.18890i −0.204310 − 0.978906i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.46410 1.04447 0.522233 − 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) 0 0
\(13\) − 3.58258i − 0.993628i −0.867857 − 0.496814i \(-0.834503\pi\)
0.867857 − 0.496814i \(-0.165497\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.55040i 0.618563i 0.950971 + 0.309282i \(0.100089\pi\)
−0.950971 + 0.309282i \(0.899911\pi\)
\(18\) 0 0
\(19\) −3.58258 −0.821899 −0.410950 − 0.911658i \(-0.634803\pi\)
−0.410950 + 0.911658i \(0.634803\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.913701i 0.190520i 0.995452 + 0.0952599i \(0.0303682\pi\)
−0.995452 + 0.0952599i \(0.969632\pi\)
\(24\) 0 0
\(25\) −4.58258 + 2.00000i −0.916515 + 0.400000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −8.75560 −1.62587 −0.812937 − 0.582351i \(-0.802133\pi\)
−0.812937 + 0.582351i \(0.802133\pi\)
\(30\) 0 0
\(31\) 3.58258 0.643450 0.321725 − 0.946833i \(-0.395737\pi\)
0.321725 + 0.946833i \(0.395737\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.18890 − 0.456850i 0.369992 − 0.0772218i
\(36\) 0 0
\(37\) − 4.00000i − 0.657596i −0.944400 − 0.328798i \(-0.893356\pi\)
0.944400 − 0.328798i \(-0.106644\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −9.66930 −1.51009 −0.755046 − 0.655672i \(-0.772385\pi\)
−0.755046 + 0.655672i \(0.772385\pi\)
\(42\) 0 0
\(43\) − 11.1652i − 1.70267i −0.524623 − 0.851335i \(-0.675794\pi\)
0.524623 − 0.851335i \(-0.324206\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.82740i 0.266554i 0.991079 + 0.133277i \(0.0425500\pi\)
−0.991079 + 0.133277i \(0.957450\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 9.66930i − 1.32818i −0.747652 − 0.664091i \(-0.768819\pi\)
0.747652 − 0.664091i \(-0.231181\pi\)
\(54\) 0 0
\(55\) −1.58258 − 7.58258i −0.213394 − 1.02243i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.82740 0.237907 0.118954 − 0.992900i \(-0.462046\pi\)
0.118954 + 0.992900i \(0.462046\pi\)
\(60\) 0 0
\(61\) 9.16515 1.17348 0.586739 − 0.809776i \(-0.300412\pi\)
0.586739 + 0.809776i \(0.300412\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −7.84190 + 1.63670i −0.972668 + 0.203008i
\(66\) 0 0
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.46410 −0.411113 −0.205557 − 0.978645i \(-0.565900\pi\)
−0.205557 + 0.978645i \(0.565900\pi\)
\(72\) 0 0
\(73\) 3.58258i 0.419309i 0.977776 + 0.209654i \(0.0672339\pi\)
−0.977776 + 0.209654i \(0.932766\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.46410i 0.394771i
\(78\) 0 0
\(79\) 11.1652 1.25618 0.628089 − 0.778142i \(-0.283837\pi\)
0.628089 + 0.778142i \(0.283837\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 6.92820i − 0.760469i −0.924890 − 0.380235i \(-0.875843\pi\)
0.924890 − 0.380235i \(-0.124157\pi\)
\(84\) 0 0
\(85\) 5.58258 − 1.16515i 0.605515 − 0.126378i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −12.9427 −1.37192 −0.685962 − 0.727637i \(-0.740618\pi\)
−0.685962 + 0.727637i \(0.740618\pi\)
\(90\) 0 0
\(91\) 3.58258 0.375556
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.63670 + 7.84190i 0.167922 + 0.804562i
\(96\) 0 0
\(97\) − 4.41742i − 0.448521i −0.974529 − 0.224261i \(-0.928003\pi\)
0.974529 − 0.224261i \(-0.0719967\pi\)
\(98\) 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.t.bb.1009.3 8
3.2 odd 2 inner 5040.2.t.bb.1009.6 8
4.3 odd 2 1260.2.k.e.1009.3 ✓ 8
5.4 even 2 inner 5040.2.t.bb.1009.4 8
12.11 even 2 1260.2.k.e.1009.6 yes 8
15.14 odd 2 inner 5040.2.t.bb.1009.5 8
20.3 even 4 6300.2.a.bk.1.2 4
20.7 even 4 6300.2.a.bl.1.1 4
20.19 odd 2 1260.2.k.e.1009.4 yes 8
60.23 odd 4 6300.2.a.bk.1.4 4
60.47 odd 4 6300.2.a.bl.1.3 4
60.59 even 2 1260.2.k.e.1009.5 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1260.2.k.e.1009.3 ✓ 8 4.3 odd 2
1260.2.k.e.1009.4 yes 8 20.19 odd 2
1260.2.k.e.1009.5 yes 8 60.59 even 2
1260.2.k.e.1009.6 yes 8 12.11 even 2
5040.2.t.bb.1009.3 8 1.1 even 1 trivial
5040.2.t.bb.1009.4 8 5.4 even 2 inner
5040.2.t.bb.1009.5 8 15.14 odd 2 inner
5040.2.t.bb.1009.6 8 3.2 odd 2 inner
6300.2.a.bk.1.2 4 20.3 even 4
6300.2.a.bk.1.4 4 60.23 odd 4
6300.2.a.bl.1.1 4 20.7 even 4
6300.2.a.bl.1.3 4 60.47 odd 4