Properties

Label 1248.2.a.o.1.3
Level $1248$
Weight $2$
Character 1248.1
Self dual yes
Analytic conductor $9.965$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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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] = [1248,2,Mod(1,1248)] 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("1248.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1248, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1248.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-3,0,2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(9.96533017226\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 1248.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +4.34017 q^{5} +1.07838 q^{7} +1.00000 q^{9} +3.41855 q^{11} +1.00000 q^{13} -4.34017 q^{15} +2.00000 q^{17} -1.07838 q^{19} -1.07838 q^{21} -2.15676 q^{23} +13.8371 q^{25} -1.00000 q^{27} +2.00000 q^{29} -5.75872 q^{31} -3.41855 q^{33} +4.68035 q^{35} -6.68035 q^{37} -1.00000 q^{39} -0.340173 q^{41} -10.8371 q^{43} +4.34017 q^{45} +7.41855 q^{47} -5.83710 q^{49} -2.00000 q^{51} -2.68035 q^{53} +14.8371 q^{55} +1.07838 q^{57} -9.26180 q^{59} -4.52359 q^{61} +1.07838 q^{63} +4.34017 q^{65} +15.9155 q^{67} +2.15676 q^{69} +5.26180 q^{71} +14.6803 q^{73} -13.8371 q^{75} +3.68649 q^{77} +12.0000 q^{79} +1.00000 q^{81} +1.26180 q^{83} +8.68035 q^{85} -2.00000 q^{87} -13.0205 q^{89} +1.07838 q^{91} +5.75872 q^{93} -4.68035 q^{95} +6.68035 q^{97} +3.41855 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 2 q^{5} + 3 q^{9} - 4 q^{11} + 3 q^{13} - 2 q^{15} + 6 q^{17} + 13 q^{25} - 3 q^{27} + 6 q^{29} + 8 q^{31} + 4 q^{33} - 8 q^{35} + 2 q^{37} - 3 q^{39} + 10 q^{41} - 4 q^{43} + 2 q^{45} + 8 q^{47}+ \cdots - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 4.34017 1.94098 0.970492 − 0.241133i \(-0.0775189\pi\)
0.970492 + 0.241133i \(0.0775189\pi\)
\(6\) 0 0
\(7\) 1.07838 0.407588 0.203794 − 0.979014i \(-0.434673\pi\)
0.203794 + 0.979014i \(0.434673\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.41855 1.03073 0.515366 − 0.856970i \(-0.327656\pi\)
0.515366 + 0.856970i \(0.327656\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −4.34017 −1.12063
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −1.07838 −0.247397 −0.123698 − 0.992320i \(-0.539476\pi\)
−0.123698 + 0.992320i \(0.539476\pi\)
\(20\) 0 0
\(21\) −1.07838 −0.235321
\(22\) 0 0
\(23\) −2.15676 −0.449715 −0.224857 − 0.974392i \(-0.572192\pi\)
−0.224857 + 0.974392i \(0.572192\pi\)
\(24\) 0 0
\(25\) 13.8371 2.76742
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −5.75872 −1.03430 −0.517149 − 0.855896i \(-0.673007\pi\)
−0.517149 + 0.855896i \(0.673007\pi\)
\(32\) 0 0
\(33\) −3.41855 −0.595093
\(34\) 0 0
\(35\) 4.68035 0.791123
\(36\) 0 0
\(37\) −6.68035 −1.09824 −0.549121 − 0.835743i \(-0.685037\pi\)
−0.549121 + 0.835743i \(0.685037\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −0.340173 −0.0531261 −0.0265630 − 0.999647i \(-0.508456\pi\)
−0.0265630 + 0.999647i \(0.508456\pi\)
\(42\) 0 0
\(43\) −10.8371 −1.65264 −0.826321 − 0.563199i \(-0.809570\pi\)
−0.826321 + 0.563199i \(0.809570\pi\)
\(44\) 0 0
\(45\) 4.34017 0.646995
\(46\) 0 0
\(47\) 7.41855 1.08211 0.541053 − 0.840988i \(-0.318026\pi\)
0.541053 + 0.840988i \(0.318026\pi\)
\(48\) 0 0
\(49\) −5.83710 −0.833872
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) −2.68035 −0.368174 −0.184087 − 0.982910i \(-0.558933\pi\)
−0.184087 + 0.982910i \(0.558933\pi\)
\(54\) 0 0
\(55\) 14.8371 2.00063
\(56\) 0 0
\(57\) 1.07838 0.142835
\(58\) 0 0
\(59\) −9.26180 −1.20578 −0.602892 − 0.797823i \(-0.705985\pi\)
−0.602892 + 0.797823i \(0.705985\pi\)
\(60\) 0 0
\(61\) −4.52359 −0.579186 −0.289593 − 0.957150i \(-0.593520\pi\)
−0.289593 + 0.957150i \(0.593520\pi\)
\(62\) 0 0
\(63\) 1.07838 0.135863
\(64\) 0 0
\(65\) 4.34017 0.538332
\(66\) 0 0
\(67\) 15.9155 1.94439 0.972193 − 0.234183i \(-0.0752414\pi\)
0.972193 + 0.234183i \(0.0752414\pi\)
\(68\) 0 0
\(69\) 2.15676 0.259643
\(70\) 0 0
\(71\) 5.26180 0.624460 0.312230 − 0.950007i \(-0.398924\pi\)
0.312230 + 0.950007i \(0.398924\pi\)
\(72\) 0 0
\(73\) 14.6803 1.71820 0.859102 − 0.511804i \(-0.171023\pi\)
0.859102 + 0.511804i \(0.171023\pi\)
\(74\) 0 0
\(75\) −13.8371 −1.59777
\(76\) 0 0
\(77\) 3.68649 0.420114
\(78\) 0 0
\(79\) 12.0000 1.35011 0.675053 − 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 1.26180 0.138500 0.0692500 − 0.997599i \(-0.477939\pi\)
0.0692500 + 0.997599i \(0.477939\pi\)
\(84\) 0 0
\(85\) 8.68035 0.941516
\(86\) 0 0
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) −13.0205 −1.38017 −0.690086 − 0.723727i \(-0.742427\pi\)
−0.690086 + 0.723727i \(0.742427\pi\)
\(90\) 0 0
\(91\) 1.07838 0.113045
\(92\) 0 0
\(93\) 5.75872 0.597152
\(94\) 0 0
\(95\) −4.68035 −0.480193
\(96\) 0 0
\(97\) 6.68035 0.678286 0.339143 − 0.940735i \(-0.389863\pi\)
0.339143 + 0.940735i \(0.389863\pi\)
\(98\) 0 0
\(99\) 3.41855 0.343577
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 1248.2.a.o.1.3 ✓ 3
3.2 odd 2 3744.2.a.ba.1.1 3
4.3 odd 2 1248.2.a.p.1.3 yes 3
8.3 odd 2 2496.2.a.bk.1.1 3
8.5 even 2 2496.2.a.bl.1.1 3
12.11 even 2 3744.2.a.z.1.1 3
24.5 odd 2 7488.2.a.cx.1.3 3
24.11 even 2 7488.2.a.cy.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1248.2.a.o.1.3 ✓ 3 1.1 even 1 trivial
1248.2.a.p.1.3 yes 3 4.3 odd 2
2496.2.a.bk.1.1 3 8.3 odd 2
2496.2.a.bl.1.1 3 8.5 even 2
3744.2.a.z.1.1 3 12.11 even 2
3744.2.a.ba.1.1 3 3.2 odd 2
7488.2.a.cx.1.3 3 24.5 odd 2
7488.2.a.cy.1.3 3 24.11 even 2