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Properties

Label 1248.4.a.i.1.4
Level $1248$
Weight $4$
Character 1248.1
Self dual yes
Analytic conductor $73.634$
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] = [1248,4,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, 4); 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, 4, names="a")
 
Level: \( N \) \(=\) \( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1248.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,0,-15,0,8,0,-38] 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: \(73.6343836872\)
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-4.13319\) 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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} +10.2664 q^{5} +24.9904 q^{7} +9.00000 q^{9} -15.9360 q^{11} +13.0000 q^{13} -30.7991 q^{15} -22.8280 q^{17} -74.8039 q^{19} -74.9712 q^{21} -123.962 q^{23} -19.6014 q^{25} -27.0000 q^{27} -149.421 q^{29} -154.413 q^{31} +47.8081 q^{33} +256.561 q^{35} -321.805 q^{37} -39.0000 q^{39} +271.139 q^{41} -346.908 q^{43} +92.3974 q^{45} -80.7794 q^{47} +281.519 q^{49} +68.4840 q^{51} -38.9816 q^{53} -163.605 q^{55} +224.412 q^{57} +47.9740 q^{59} +320.699 q^{61} +224.913 q^{63} +133.463 q^{65} -580.693 q^{67} +371.887 q^{69} -82.0910 q^{71} +547.936 q^{73} +58.8043 q^{75} -398.248 q^{77} -580.035 q^{79} +81.0000 q^{81} +99.3174 q^{83} -234.361 q^{85} +448.262 q^{87} -318.211 q^{89} +324.875 q^{91} +463.239 q^{93} -767.965 q^{95} +1782.81 q^{97} -143.424 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 15 q^{3} + 8 q^{5} - 38 q^{7} + 45 q^{9} - 32 q^{11} + 65 q^{13} - 24 q^{15} + 34 q^{17} - 42 q^{19} + 114 q^{21} - 144 q^{23} + 143 q^{25} - 135 q^{27} + 262 q^{29} - 6 q^{31} + 96 q^{33} - 72 q^{35}+ \cdots - 288 q^{99}+O(q^{100}) \) 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
\(5\) 10.2664 0.918253 0.459126 − 0.888371i \(-0.348162\pi\)
0.459126 + 0.888371i \(0.348162\pi\)
\(6\) 0 0
\(7\) 24.9904 1.34935 0.674677 − 0.738113i \(-0.264283\pi\)
0.674677 + 0.738113i \(0.264283\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −15.9360 −0.436809 −0.218404 − 0.975858i \(-0.570085\pi\)
−0.218404 + 0.975858i \(0.570085\pi\)
\(12\) 0 0
\(13\) 13.0000 0.277350
\(14\) 0 0
\(15\) −30.7991 −0.530154
\(16\) 0 0
\(17\) −22.8280 −0.325682 −0.162841 − 0.986652i \(-0.552066\pi\)
−0.162841 + 0.986652i \(0.552066\pi\)
\(18\) 0 0
\(19\) −74.8039 −0.903220 −0.451610 − 0.892215i \(-0.649150\pi\)
−0.451610 + 0.892215i \(0.649150\pi\)
\(20\) 0 0
\(21\) −74.9712 −0.779050
\(22\) 0 0
\(23\) −123.962 −1.12382 −0.561911 − 0.827198i \(-0.689934\pi\)
−0.561911 + 0.827198i \(0.689934\pi\)
\(24\) 0 0
\(25\) −19.6014 −0.156812
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −149.421 −0.956783 −0.478391 − 0.878147i \(-0.658780\pi\)
−0.478391 + 0.878147i \(0.658780\pi\)
\(30\) 0 0
\(31\) −154.413 −0.894625 −0.447312 − 0.894378i \(-0.647619\pi\)
−0.447312 + 0.894378i \(0.647619\pi\)
\(32\) 0 0
\(33\) 47.8081 0.252192
\(34\) 0 0
\(35\) 256.561 1.23905
\(36\) 0 0
\(37\) −321.805 −1.42985 −0.714924 − 0.699202i \(-0.753539\pi\)
−0.714924 + 0.699202i \(0.753539\pi\)
\(38\) 0 0
\(39\) −39.0000 −0.160128
\(40\) 0 0
\(41\) 271.139 1.03280 0.516399 − 0.856348i \(-0.327272\pi\)
0.516399 + 0.856348i \(0.327272\pi\)
\(42\) 0 0
\(43\) −346.908 −1.23030 −0.615151 − 0.788409i \(-0.710905\pi\)
−0.615151 + 0.788409i \(0.710905\pi\)
\(44\) 0 0
\(45\) 92.3974 0.306084
\(46\) 0 0
\(47\) −80.7794 −0.250700 −0.125350 − 0.992113i \(-0.540005\pi\)
−0.125350 + 0.992113i \(0.540005\pi\)
\(48\) 0 0
\(49\) 281.519 0.820757
\(50\) 0 0
\(51\) 68.4840 0.188033
\(52\) 0 0
\(53\) −38.9816 −0.101029 −0.0505145 − 0.998723i \(-0.516086\pi\)
−0.0505145 + 0.998723i \(0.516086\pi\)
\(54\) 0 0
\(55\) −163.605 −0.401101
\(56\) 0 0
\(57\) 224.412 0.521474
\(58\) 0 0
\(59\) 47.9740 0.105859 0.0529296 − 0.998598i \(-0.483144\pi\)
0.0529296 + 0.998598i \(0.483144\pi\)
\(60\) 0 0
\(61\) 320.699 0.673136 0.336568 − 0.941659i \(-0.390734\pi\)
0.336568 + 0.941659i \(0.390734\pi\)
\(62\) 0 0
\(63\) 224.913 0.449785
\(64\) 0 0
\(65\) 133.463 0.254678
\(66\) 0 0
\(67\) −580.693 −1.05885 −0.529425 − 0.848357i \(-0.677592\pi\)
−0.529425 + 0.848357i \(0.677592\pi\)
\(68\) 0 0
\(69\) 371.887 0.648839
\(70\) 0 0
\(71\) −82.0910 −0.137217 −0.0686085 − 0.997644i \(-0.521856\pi\)
−0.0686085 + 0.997644i \(0.521856\pi\)
\(72\) 0 0
\(73\) 547.936 0.878508 0.439254 − 0.898363i \(-0.355243\pi\)
0.439254 + 0.898363i \(0.355243\pi\)
\(74\) 0 0
\(75\) 58.8043 0.0905352
\(76\) 0 0
\(77\) −398.248 −0.589410
\(78\) 0 0
\(79\) −580.035 −0.826064 −0.413032 − 0.910716i \(-0.635530\pi\)
−0.413032 + 0.910716i \(0.635530\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 99.3174 0.131343 0.0656717 − 0.997841i \(-0.479081\pi\)
0.0656717 + 0.997841i \(0.479081\pi\)
\(84\) 0 0
\(85\) −234.361 −0.299059
\(86\) 0 0
\(87\) 448.262 0.552399
\(88\) 0 0
\(89\) −318.211 −0.378992 −0.189496 − 0.981881i \(-0.560685\pi\)
−0.189496 + 0.981881i \(0.560685\pi\)
\(90\) 0 0
\(91\) 324.875 0.374243
\(92\) 0 0
\(93\) 463.239 0.516512
\(94\) 0 0
\(95\) −767.965 −0.829385
\(96\) 0 0
\(97\) 1782.81 1.86615 0.933076 − 0.359679i \(-0.117114\pi\)
0.933076 + 0.359679i \(0.117114\pi\)
\(98\) 0 0
\(99\) −143.424 −0.145603
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 1248.4.a.i.1.4 ✓ 5
4.3 odd 2 1248.4.a.n.1.4 yes 5
8.3 odd 2 2496.4.a.by.1.2 5
8.5 even 2 2496.4.a.cd.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1248.4.a.i.1.4 ✓ 5 1.1 even 1 trivial
1248.4.a.n.1.4 yes 5 4.3 odd 2
2496.4.a.by.1.2 5 8.3 odd 2
2496.4.a.cd.1.2 5 8.5 even 2