Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4840,2,Mod(1,4840)] 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("4840.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4840.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,-3,0,-6,0,7,0,5,0,0,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(38.6475945783\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.25903625.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 7x^{4} + 17x^{3} + 16x^{2} - 20x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.03795\) of defining polynomial
Character \(\chi\) \(=\) 4840.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.03795 q^{3} -1.00000 q^{5} -2.60210 q^{7} -1.92266 q^{9} -2.87756 q^{13} +1.03795 q^{15} +0.810293 q^{17} +5.15069 q^{19} +2.70085 q^{21} +4.98262 q^{23} +1.00000 q^{25} +5.10948 q^{27} +6.72933 q^{29} -4.14084 q^{31} +2.60210 q^{35} +4.05133 q^{37} +2.98677 q^{39} +9.76082 q^{41} -5.92912 q^{43} +1.92266 q^{45} -0.967904 q^{47} -0.229090 q^{49} -0.841046 q^{51} +4.47445 q^{53} -5.34617 q^{57} +6.80756 q^{59} -2.61307 q^{61} +5.00294 q^{63} +2.87756 q^{65} -15.8408 q^{67} -5.17172 q^{69} -1.41346 q^{71} -13.1429 q^{73} -1.03795 q^{75} +5.69123 q^{79} +0.464568 q^{81} +7.96967 q^{83} -0.810293 q^{85} -6.98472 q^{87} -14.5196 q^{89} +7.48768 q^{91} +4.29799 q^{93} -5.15069 q^{95} -2.99325 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} - 6 q^{5} + 7 q^{7} + 5 q^{9} - q^{13} + 3 q^{15} - 6 q^{17} + 7 q^{19} - 14 q^{21} - 9 q^{23} + 6 q^{25} - 21 q^{27} - 10 q^{29} + q^{31} - 7 q^{35} + 3 q^{37} + q^{39} + 6 q^{41} - 18 q^{43}+ \cdots - 33 q^{97}+O(q^{100}) \) 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
\(3\) −1.03795 −0.599262 −0.299631 − 0.954055i \(-0.596864\pi\)
−0.299631 + 0.954055i \(0.596864\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.60210 −0.983500 −0.491750 − 0.870736i \(-0.663643\pi\)
−0.491750 + 0.870736i \(0.663643\pi\)
\(8\) 0 0
\(9\) −1.92266 −0.640885
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −2.87756 −0.798091 −0.399045 − 0.916931i \(-0.630658\pi\)
−0.399045 + 0.916931i \(0.630658\pi\)
\(14\) 0 0
\(15\) 1.03795 0.267998
\(16\) 0 0
\(17\) 0.810293 0.196525 0.0982625 − 0.995161i \(-0.468672\pi\)
0.0982625 + 0.995161i \(0.468672\pi\)
\(18\) 0 0
\(19\) 5.15069 1.18165 0.590824 − 0.806800i \(-0.298803\pi\)
0.590824 + 0.806800i \(0.298803\pi\)
\(20\) 0 0
\(21\) 2.70085 0.589374
\(22\) 0 0
\(23\) 4.98262 1.03895 0.519474 − 0.854486i \(-0.326128\pi\)
0.519474 + 0.854486i \(0.326128\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.10948 0.983320
\(28\) 0 0
\(29\) 6.72933 1.24961 0.624803 − 0.780783i \(-0.285179\pi\)
0.624803 + 0.780783i \(0.285179\pi\)
\(30\) 0 0
\(31\) −4.14084 −0.743717 −0.371858 − 0.928290i \(-0.621279\pi\)
−0.371858 + 0.928290i \(0.621279\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.60210 0.439835
\(36\) 0 0
\(37\) 4.05133 0.666034 0.333017 − 0.942921i \(-0.391933\pi\)
0.333017 + 0.942921i \(0.391933\pi\)
\(38\) 0 0
\(39\) 2.98677 0.478266
\(40\) 0 0
\(41\) 9.76082 1.52438 0.762192 − 0.647351i \(-0.224123\pi\)
0.762192 + 0.647351i \(0.224123\pi\)
\(42\) 0 0
\(43\) −5.92912 −0.904182 −0.452091 − 0.891972i \(-0.649322\pi\)
−0.452091 + 0.891972i \(0.649322\pi\)
\(44\) 0 0
\(45\) 1.92266 0.286613
\(46\) 0 0
\(47\) −0.967904 −0.141183 −0.0705917 − 0.997505i \(-0.522489\pi\)
−0.0705917 + 0.997505i \(0.522489\pi\)
\(48\) 0 0
\(49\) −0.229090 −0.0327272
\(50\) 0 0
\(51\) −0.841046 −0.117770
\(52\) 0 0
\(53\) 4.47445 0.614613 0.307307 − 0.951611i \(-0.400572\pi\)
0.307307 + 0.951611i \(0.400572\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −5.34617 −0.708117
\(58\) 0 0
\(59\) 6.80756 0.886269 0.443135 − 0.896455i \(-0.353866\pi\)
0.443135 + 0.896455i \(0.353866\pi\)
\(60\) 0 0
\(61\) −2.61307 −0.334569 −0.167285 − 0.985909i \(-0.553500\pi\)
−0.167285 + 0.985909i \(0.553500\pi\)
\(62\) 0 0
\(63\) 5.00294 0.630311
\(64\) 0 0
\(65\) 2.87756 0.356917
\(66\) 0 0
\(67\) −15.8408 −1.93526 −0.967629 − 0.252378i \(-0.918788\pi\)
−0.967629 + 0.252378i \(0.918788\pi\)
\(68\) 0 0
\(69\) −5.17172 −0.622602
\(70\) 0 0
\(71\) −1.41346 −0.167746 −0.0838732 − 0.996476i \(-0.526729\pi\)
−0.0838732 + 0.996476i \(0.526729\pi\)
\(72\) 0 0
\(73\) −13.1429 −1.53826 −0.769131 − 0.639091i \(-0.779311\pi\)
−0.769131 + 0.639091i \(0.779311\pi\)
\(74\) 0 0
\(75\) −1.03795 −0.119852
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 5.69123 0.640314 0.320157 − 0.947365i \(-0.396264\pi\)
0.320157 + 0.947365i \(0.396264\pi\)
\(80\) 0 0
\(81\) 0.464568 0.0516187
\(82\) 0 0
\(83\) 7.96967 0.874785 0.437392 − 0.899271i \(-0.355902\pi\)
0.437392 + 0.899271i \(0.355902\pi\)
\(84\) 0 0
\(85\) −0.810293 −0.0878887
\(86\) 0 0
\(87\) −6.98472 −0.748841
\(88\) 0 0
\(89\) −14.5196 −1.53907 −0.769535 − 0.638605i \(-0.779512\pi\)
−0.769535 + 0.638605i \(0.779512\pi\)
\(90\) 0 0
\(91\) 7.48768 0.784923
\(92\) 0 0
\(93\) 4.29799 0.445681
\(94\) 0 0
\(95\) −5.15069 −0.528449
\(96\) 0 0
\(97\) −2.99325 −0.303919 −0.151959 − 0.988387i \(-0.548558\pi\)
−0.151959 + 0.988387i \(0.548558\pi\)
\(98\) 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 4840.2.a.bb.1.3 6
4.3 odd 2 9680.2.a.dc.1.4 6
11.5 even 5 440.2.y.c.201.2 yes 12
11.9 even 5 440.2.y.c.81.2 ✓ 12
11.10 odd 2 4840.2.a.ba.1.3 6
44.27 odd 10 880.2.bo.i.641.2 12
44.31 odd 10 880.2.bo.i.81.2 12
44.43 even 2 9680.2.a.dd.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.c.81.2 ✓ 12 11.9 even 5
440.2.y.c.201.2 yes 12 11.5 even 5
880.2.bo.i.81.2 12 44.31 odd 10
880.2.bo.i.641.2 12 44.27 odd 10
4840.2.a.ba.1.3 6 11.10 odd 2
4840.2.a.bb.1.3 6 1.1 even 1 trivial
9680.2.a.dc.1.4 6 4.3 odd 2
9680.2.a.dd.1.4 6 44.43 even 2