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Properties

Label 2496.4.a.by.1.1
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 1.1
Root \(-6.84895\) 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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -15.6979 q^{5} +20.4412 q^{7} +9.00000 q^{9} -44.7496 q^{11} -13.0000 q^{13} +47.0937 q^{15} +53.4502 q^{17} +31.9767 q^{19} -61.3235 q^{21} -82.4786 q^{23} +121.424 q^{25} -27.0000 q^{27} -100.775 q^{29} +263.622 q^{31} +134.249 q^{33} -320.884 q^{35} -104.742 q^{37} +39.0000 q^{39} -398.232 q^{41} +180.308 q^{43} -141.281 q^{45} +118.222 q^{47} +74.8419 q^{49} -160.351 q^{51} +526.934 q^{53} +702.475 q^{55} -95.9302 q^{57} +224.738 q^{59} +256.401 q^{61} +183.971 q^{63} +204.073 q^{65} +494.496 q^{67} +247.436 q^{69} +146.727 q^{71} +164.841 q^{73} -364.273 q^{75} -914.734 q^{77} -305.708 q^{79} +81.0000 q^{81} -208.668 q^{83} -839.056 q^{85} +302.325 q^{87} +284.988 q^{89} -265.735 q^{91} -790.865 q^{93} -501.968 q^{95} -104.131 q^{97} -402.746 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 15 q^{3} - 8 q^{5} + 38 q^{7} + 45 q^{9} - 32 q^{11} - 65 q^{13} + 24 q^{15} + 34 q^{17} - 42 q^{19} - 114 q^{21} + 144 q^{23} + 143 q^{25} - 135 q^{27} - 262 q^{29} + 6 q^{31} + 96 q^{33} - 72 q^{35}+ \cdots - 288 q^{99}+O(q^{100}) \) 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
\(5\) −15.6979 −1.40406 −0.702032 − 0.712146i \(-0.747724\pi\)
−0.702032 + 0.712146i \(0.747724\pi\)
\(6\) 0 0
\(7\) 20.4412 1.10372 0.551860 − 0.833937i \(-0.313918\pi\)
0.551860 + 0.833937i \(0.313918\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −44.7496 −1.22659 −0.613296 − 0.789853i \(-0.710157\pi\)
−0.613296 + 0.789853i \(0.710157\pi\)
\(12\) 0 0
\(13\) −13.0000 −0.277350
\(14\) 0 0
\(15\) 47.0937 0.810636
\(16\) 0 0
\(17\) 53.4502 0.762563 0.381282 − 0.924459i \(-0.375483\pi\)
0.381282 + 0.924459i \(0.375483\pi\)
\(18\) 0 0
\(19\) 31.9767 0.386103 0.193052 − 0.981189i \(-0.438161\pi\)
0.193052 + 0.981189i \(0.438161\pi\)
\(20\) 0 0
\(21\) −61.3235 −0.637233
\(22\) 0 0
\(23\) −82.4786 −0.747738 −0.373869 − 0.927482i \(-0.621969\pi\)
−0.373869 + 0.927482i \(0.621969\pi\)
\(24\) 0 0
\(25\) 121.424 0.971394
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −100.775 −0.645292 −0.322646 − 0.946520i \(-0.604572\pi\)
−0.322646 + 0.946520i \(0.604572\pi\)
\(30\) 0 0
\(31\) 263.622 1.52735 0.763675 − 0.645601i \(-0.223393\pi\)
0.763675 + 0.645601i \(0.223393\pi\)
\(32\) 0 0
\(33\) 134.249 0.708173
\(34\) 0 0
\(35\) −320.884 −1.54969
\(36\) 0 0
\(37\) −104.742 −0.465391 −0.232695 − 0.972550i \(-0.574755\pi\)
−0.232695 + 0.972550i \(0.574755\pi\)
\(38\) 0 0
\(39\) 39.0000 0.160128
\(40\) 0 0
\(41\) −398.232 −1.51691 −0.758456 − 0.651725i \(-0.774046\pi\)
−0.758456 + 0.651725i \(0.774046\pi\)
\(42\) 0 0
\(43\) 180.308 0.639459 0.319729 − 0.947509i \(-0.396408\pi\)
0.319729 + 0.947509i \(0.396408\pi\)
\(44\) 0 0
\(45\) −141.281 −0.468021
\(46\) 0 0
\(47\) 118.222 0.366903 0.183451 − 0.983029i \(-0.441273\pi\)
0.183451 + 0.983029i \(0.441273\pi\)
\(48\) 0 0
\(49\) 74.8419 0.218198
\(50\) 0 0
\(51\) −160.351 −0.440266
\(52\) 0 0
\(53\) 526.934 1.36566 0.682830 − 0.730577i \(-0.260749\pi\)
0.682830 + 0.730577i \(0.260749\pi\)
\(54\) 0 0
\(55\) 702.475 1.72221
\(56\) 0 0
\(57\) −95.9302 −0.222917
\(58\) 0 0
\(59\) 224.738 0.495906 0.247953 − 0.968772i \(-0.420242\pi\)
0.247953 + 0.968772i \(0.420242\pi\)
\(60\) 0 0
\(61\) 256.401 0.538177 0.269089 − 0.963115i \(-0.413278\pi\)
0.269089 + 0.963115i \(0.413278\pi\)
\(62\) 0 0
\(63\) 183.971 0.367907
\(64\) 0 0
\(65\) 204.073 0.389417
\(66\) 0 0
\(67\) 494.496 0.901676 0.450838 − 0.892606i \(-0.351125\pi\)
0.450838 + 0.892606i \(0.351125\pi\)
\(68\) 0 0
\(69\) 247.436 0.431707
\(70\) 0 0
\(71\) 146.727 0.245257 0.122629 − 0.992453i \(-0.460868\pi\)
0.122629 + 0.992453i \(0.460868\pi\)
\(72\) 0 0
\(73\) 164.841 0.264291 0.132145 − 0.991230i \(-0.457813\pi\)
0.132145 + 0.991230i \(0.457813\pi\)
\(74\) 0 0
\(75\) −364.273 −0.560834
\(76\) 0 0
\(77\) −914.734 −1.35381
\(78\) 0 0
\(79\) −305.708 −0.435378 −0.217689 − 0.976018i \(-0.569852\pi\)
−0.217689 + 0.976018i \(0.569852\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −208.668 −0.275955 −0.137978 − 0.990435i \(-0.544060\pi\)
−0.137978 + 0.990435i \(0.544060\pi\)
\(84\) 0 0
\(85\) −839.056 −1.07069
\(86\) 0 0
\(87\) 302.325 0.372560
\(88\) 0 0
\(89\) 284.988 0.339423 0.169711 − 0.985494i \(-0.445716\pi\)
0.169711 + 0.985494i \(0.445716\pi\)
\(90\) 0 0
\(91\) −265.735 −0.306117
\(92\) 0 0
\(93\) −790.865 −0.881816
\(94\) 0 0
\(95\) −501.968 −0.542114
\(96\) 0 0
\(97\) −104.131 −0.108999 −0.0544994 − 0.998514i \(-0.517356\pi\)
−0.0544994 + 0.998514i \(0.517356\pi\)
\(98\) 0 0
\(99\) −402.746 −0.408864
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.1.1 5
4.3 odd 2 2496.4.a.cd.1.1 5
8.3 odd 2 1248.4.a.i.1.5 ✓ 5
8.5 even 2 1248.4.a.n.1.5 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1248.4.a.i.1.5 ✓ 5 8.3 odd 2
1248.4.a.n.1.5 yes 5 8.5 even 2
2496.4.a.by.1.1 5 1.1 even 1 trivial
2496.4.a.cd.1.1 5 4.3 odd 2