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Properties

Label 1260.2.k.e.1009.8
Level $1260$
Weight $2$
Character 1260.1009
Analytic conductor $10.061$
Analytic rank $0$
Dimension $8$
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] = [1260,2,Mod(1009,1260)] 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("1260.1009"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1260, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1260.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.0611506547\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1009.8
Root \(-1.09445 + 0.895644i\) of defining polynomial
Character \(\chi\) \(=\) 1260.1009
Dual form 1260.2.k.e.1009.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.18890 + 0.456850i) q^{5} -1.00000i q^{7} -3.46410 q^{11} +5.58258i q^{13} +7.84190i q^{17} -5.58258 q^{19} +4.37780i q^{23} +(4.58258 + 2.00000i) q^{25} +1.82740 q^{29} +5.58258 q^{31} +(0.456850 - 2.18890i) q^{35} -4.00000i q^{37} +6.20520 q^{41} -7.16515i q^{43} +8.75560i q^{47} -1.00000 q^{49} +6.20520i q^{53} +(-7.58258 - 1.58258i) q^{55} +8.75560 q^{59} -9.16515 q^{61} +(-2.55040 + 12.2197i) q^{65} -8.00000i q^{67} +3.46410 q^{71} -5.58258i q^{73} +3.46410i q^{77} +7.16515 q^{79} +6.92820i q^{83} +(-3.58258 + 17.1652i) q^{85} -18.2342 q^{89} +5.58258 q^{91} +(-12.2197 - 2.55040i) q^{95} -13.5826i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{19} + 8 q^{31} - 8 q^{49} - 24 q^{55} - 16 q^{79} + 8 q^{85} + 8 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(5\) 2.18890 + 0.456850i 0.978906 + 0.204310i
\(6\) 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 \(-0.674901\pi\)
−0.522233 + 0.852803i \(0.674901\pi\)
\(12\) 0 0
\(13\) 5.58258i 1.54833i 0.632985 + 0.774164i \(0.281829\pi\)
−0.632985 + 0.774164i \(0.718171\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.84190i 1.90194i 0.309282 + 0.950971i \(0.399911\pi\)
−0.309282 + 0.950971i \(0.600089\pi\)
\(18\) 0 0
\(19\) −5.58258 −1.28073 −0.640365 − 0.768070i \(-0.721217\pi\)
−0.640365 + 0.768070i \(0.721217\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.37780i 0.912835i 0.889766 + 0.456417i \(0.150868\pi\)
−0.889766 + 0.456417i \(0.849132\pi\)
\(24\) 0 0
\(25\) 4.58258 + 2.00000i 0.916515 + 0.400000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.82740 0.339340 0.169670 − 0.985501i \(-0.445730\pi\)
0.169670 + 0.985501i \(0.445730\pi\)
\(30\) 0 0
\(31\) 5.58258 1.00266 0.501330 − 0.865256i \(-0.332844\pi\)
0.501330 + 0.865256i \(0.332844\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.456850 − 2.18890i 0.0772218 − 0.369992i
\(36\) 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
\(41\) 6.20520 0.969090 0.484545 − 0.874766i \(-0.338985\pi\)
0.484545 + 0.874766i \(0.338985\pi\)
\(42\) 0 0
\(43\) − 7.16515i − 1.09268i −0.837565 − 0.546338i \(-0.816022\pi\)
0.837565 − 0.546338i \(-0.183978\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.75560i 1.27714i 0.769565 + 0.638568i \(0.220473\pi\)
−0.769565 + 0.638568i \(0.779527\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.20520i 0.852350i 0.904641 + 0.426175i \(0.140139\pi\)
−0.904641 + 0.426175i \(0.859861\pi\)
\(54\) 0 0
\(55\) −7.58258 − 1.58258i −1.02243 − 0.213394i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.75560 1.13988 0.569941 − 0.821685i \(-0.306966\pi\)
0.569941 + 0.821685i \(0.306966\pi\)
\(60\) 0 0
\(61\) −9.16515 −1.17348 −0.586739 − 0.809776i \(-0.699588\pi\)
−0.586739 + 0.809776i \(0.699588\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.55040 + 12.2197i −0.316338 + 1.51567i
\(66\) 0 0
\(67\) − 8.00000i − 0.977356i −0.872464 − 0.488678i \(-0.837479\pi\)
0.872464 − 0.488678i \(-0.162521\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.46410 0.411113 0.205557 − 0.978645i \(-0.434100\pi\)
0.205557 + 0.978645i \(0.434100\pi\)
\(72\) 0 0
\(73\) − 5.58258i − 0.653391i −0.945130 − 0.326696i \(-0.894065\pi\)
0.945130 − 0.326696i \(-0.105935\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.46410i 0.394771i
\(78\) 0 0
\(79\) 7.16515 0.806143 0.403071 − 0.915169i \(-0.367943\pi\)
0.403071 + 0.915169i \(0.367943\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.92820i 0.760469i 0.924890 + 0.380235i \(0.124157\pi\)
−0.924890 + 0.380235i \(0.875843\pi\)
\(84\) 0 0
\(85\) −3.58258 + 17.1652i −0.388585 + 1.86182i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −18.2342 −1.93282 −0.966411 − 0.257001i \(-0.917266\pi\)
−0.966411 + 0.257001i \(0.917266\pi\)
\(90\) 0 0
\(91\) 5.58258 0.585213
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −12.2197 − 2.55040i −1.25372 − 0.261666i
\(96\) 0 0
\(97\) − 13.5826i − 1.37910i −0.724237 − 0.689551i \(-0.757808\pi\)
0.724237 − 0.689551i \(-0.242192\pi\)
\(98\) 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 1260.2.k.e.1009.8 yes 8
3.2 odd 2 inner 1260.2.k.e.1009.1 ✓ 8
4.3 odd 2 5040.2.t.bb.1009.8 8
5.2 odd 4 6300.2.a.bl.1.2 4
5.3 odd 4 6300.2.a.bk.1.1 4
5.4 even 2 inner 1260.2.k.e.1009.7 yes 8
12.11 even 2 5040.2.t.bb.1009.1 8
15.2 even 4 6300.2.a.bl.1.4 4
15.8 even 4 6300.2.a.bk.1.3 4
15.14 odd 2 inner 1260.2.k.e.1009.2 yes 8
20.19 odd 2 5040.2.t.bb.1009.7 8
60.59 even 2 5040.2.t.bb.1009.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1260.2.k.e.1009.1 ✓ 8 3.2 odd 2 inner
1260.2.k.e.1009.2 yes 8 15.14 odd 2 inner
1260.2.k.e.1009.7 yes 8 5.4 even 2 inner
1260.2.k.e.1009.8 yes 8 1.1 even 1 trivial
5040.2.t.bb.1009.1 8 12.11 even 2
5040.2.t.bb.1009.2 8 60.59 even 2
5040.2.t.bb.1009.7 8 20.19 odd 2
5040.2.t.bb.1009.8 8 4.3 odd 2
6300.2.a.bk.1.1 4 5.3 odd 4
6300.2.a.bk.1.3 4 15.8 even 4
6300.2.a.bl.1.2 4 5.2 odd 4
6300.2.a.bl.1.4 4 15.2 even 4