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Properties

Label 2496.4.a.by.1.4
Level $2496$
Weight $4$
Character 2496.1
Self dual yes
Analytic conductor $147.269$
Analytic rank $1$
Dimension $5$
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] = [2496,4,Mod(1,2496)] 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("2496.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2496, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2496.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,0,-15,0,-8,0,38,0,45,0,-32] 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: yes
Analytic conductor: \(147.268767374\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 94x^{3} - 92x^{2} + 1858x + 4112 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 1248)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(5.50703\) of defining polynomial
Character \(\chi\) \(=\) 2496.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} +9.01405 q^{5} +7.78722 q^{7} +9.00000 q^{9} -15.2727 q^{11} -13.0000 q^{13} -27.0422 q^{15} +116.941 q^{17} -79.0055 q^{19} -23.3617 q^{21} -30.2219 q^{23} -43.7468 q^{25} -27.0000 q^{27} +217.877 q^{29} -250.570 q^{31} +45.8181 q^{33} +70.1944 q^{35} -360.793 q^{37} +39.0000 q^{39} +185.409 q^{41} +407.250 q^{43} +81.1265 q^{45} -34.8091 q^{47} -282.359 q^{49} -350.822 q^{51} -196.423 q^{53} -137.669 q^{55} +237.016 q^{57} -229.759 q^{59} -385.294 q^{61} +70.0850 q^{63} -117.183 q^{65} -1060.50 q^{67} +90.6656 q^{69} +311.085 q^{71} -288.906 q^{73} +131.241 q^{75} -118.932 q^{77} +1168.67 q^{79} +81.0000 q^{81} -160.282 q^{83} +1054.11 q^{85} -653.630 q^{87} +1089.97 q^{89} -101.234 q^{91} +751.711 q^{93} -712.160 q^{95} -12.2706 q^{97} -137.454 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 15 q^{3} - 8 q^{5} + 38 q^{7} + 45 q^{9} - 32 q^{11} - 65 q^{13} + 24 q^{15} + 34 q^{17} - 42 q^{19} - 114 q^{21} + 144 q^{23} + 143 q^{25} - 135 q^{27} - 262 q^{29} + 6 q^{31} + 96 q^{33} - 72 q^{35}+ \cdots - 288 q^{99}+O(q^{100}) \) 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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) 9.01405 0.806241 0.403121 − 0.915147i \(-0.367925\pi\)
0.403121 + 0.915147i \(0.367925\pi\)
\(6\) 0 0
\(7\) 7.78722 0.420470 0.210235 − 0.977651i \(-0.432577\pi\)
0.210235 + 0.977651i \(0.432577\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −15.2727 −0.418626 −0.209313 − 0.977849i \(-0.567123\pi\)
−0.209313 + 0.977849i \(0.567123\pi\)
\(12\) 0 0
\(13\) −13.0000 −0.277350
\(14\) 0 0
\(15\) −27.0422 −0.465484
\(16\) 0 0
\(17\) 116.941 1.66837 0.834184 − 0.551486i \(-0.185939\pi\)
0.834184 + 0.551486i \(0.185939\pi\)
\(18\) 0 0
\(19\) −79.0055 −0.953953 −0.476976 − 0.878916i \(-0.658267\pi\)
−0.476976 + 0.878916i \(0.658267\pi\)
\(20\) 0 0
\(21\) −23.3617 −0.242759
\(22\) 0 0
\(23\) −30.2219 −0.273987 −0.136993 − 0.990572i \(-0.543744\pi\)
−0.136993 + 0.990572i \(0.543744\pi\)
\(24\) 0 0
\(25\) −43.7468 −0.349975
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 217.877 1.39513 0.697563 − 0.716523i \(-0.254268\pi\)
0.697563 + 0.716523i \(0.254268\pi\)
\(30\) 0 0
\(31\) −250.570 −1.45173 −0.725867 − 0.687835i \(-0.758561\pi\)
−0.725867 + 0.687835i \(0.758561\pi\)
\(32\) 0 0
\(33\) 45.8181 0.241694
\(34\) 0 0
\(35\) 70.1944 0.339001
\(36\) 0 0
\(37\) −360.793 −1.60308 −0.801542 − 0.597939i \(-0.795986\pi\)
−0.801542 + 0.597939i \(0.795986\pi\)
\(38\) 0 0
\(39\) 39.0000 0.160128
\(40\) 0 0
\(41\) 185.409 0.706246 0.353123 − 0.935577i \(-0.385120\pi\)
0.353123 + 0.935577i \(0.385120\pi\)
\(42\) 0 0
\(43\) 407.250 1.44430 0.722152 − 0.691734i \(-0.243153\pi\)
0.722152 + 0.691734i \(0.243153\pi\)
\(44\) 0 0
\(45\) 81.1265 0.268747
\(46\) 0 0
\(47\) −34.8091 −0.108030 −0.0540152 − 0.998540i \(-0.517202\pi\)
−0.0540152 + 0.998540i \(0.517202\pi\)
\(48\) 0 0
\(49\) −282.359 −0.823205
\(50\) 0 0
\(51\) −350.822 −0.963233
\(52\) 0 0
\(53\) −196.423 −0.509071 −0.254536 − 0.967063i \(-0.581923\pi\)
−0.254536 + 0.967063i \(0.581923\pi\)
\(54\) 0 0
\(55\) −137.669 −0.337514
\(56\) 0 0
\(57\) 237.016 0.550765
\(58\) 0 0
\(59\) −229.759 −0.506984 −0.253492 − 0.967338i \(-0.581579\pi\)
−0.253492 + 0.967338i \(0.581579\pi\)
\(60\) 0 0
\(61\) −385.294 −0.808718 −0.404359 − 0.914600i \(-0.632505\pi\)
−0.404359 + 0.914600i \(0.632505\pi\)
\(62\) 0 0
\(63\) 70.0850 0.140157
\(64\) 0 0
\(65\) −117.183 −0.223611
\(66\) 0 0
\(67\) −1060.50 −1.93375 −0.966874 − 0.255254i \(-0.917841\pi\)
−0.966874 + 0.255254i \(0.917841\pi\)
\(68\) 0 0
\(69\) 90.6656 0.158186
\(70\) 0 0
\(71\) 311.085 0.519986 0.259993 − 0.965611i \(-0.416280\pi\)
0.259993 + 0.965611i \(0.416280\pi\)
\(72\) 0 0
\(73\) −288.906 −0.463204 −0.231602 − 0.972811i \(-0.574397\pi\)
−0.231602 + 0.972811i \(0.574397\pi\)
\(74\) 0 0
\(75\) 131.241 0.202058
\(76\) 0 0
\(77\) −118.932 −0.176020
\(78\) 0 0
\(79\) 1168.67 1.66438 0.832190 − 0.554491i \(-0.187087\pi\)
0.832190 + 0.554491i \(0.187087\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −160.282 −0.211967 −0.105983 − 0.994368i \(-0.533799\pi\)
−0.105983 + 0.994368i \(0.533799\pi\)
\(84\) 0 0
\(85\) 1054.11 1.34511
\(86\) 0 0
\(87\) −653.630 −0.805477
\(88\) 0 0
\(89\) 1089.97 1.29816 0.649081 − 0.760719i \(-0.275154\pi\)
0.649081 + 0.760719i \(0.275154\pi\)
\(90\) 0 0
\(91\) −101.234 −0.116617
\(92\) 0 0
\(93\) 751.711 0.838159
\(94\) 0 0
\(95\) −712.160 −0.769116
\(96\) 0 0
\(97\) −12.2706 −0.0128442 −0.00642212 − 0.999979i \(-0.502044\pi\)
−0.00642212 + 0.999979i \(0.502044\pi\)
\(98\) 0 0
\(99\) −137.454 −0.139542
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 2496.4.a.by.1.4 5
4.3 odd 2 2496.4.a.cd.1.4 5
8.3 odd 2 1248.4.a.i.1.2 ✓ 5
8.5 even 2 1248.4.a.n.1.2 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1248.4.a.i.1.2 ✓ 5 8.3 odd 2
1248.4.a.n.1.2 yes 5 8.5 even 2
2496.4.a.by.1.4 5 1.1 even 1 trivial
2496.4.a.cd.1.4 5 4.3 odd 2