Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1360,4,Mod(1,1360)] 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("1360.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1360 = 2^{4} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1360.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,0,1,0,25,0,-10,0,-30,0,-126] 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: \(80.2425976078\)
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: \( x^{5} - 2x^{4} - 20x^{3} + 38x^{2} + 69x - 126 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 85)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(4.05155\) of defining polynomial
Character \(\chi\) \(=\) 1360.1

$q$-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)
 
\(f(q)\) \(=\) \(q+7.70584 q^{3} +5.00000 q^{5} -22.4661 q^{7} +32.3800 q^{9} -39.6109 q^{11} -6.70527 q^{13} +38.5292 q^{15} +17.0000 q^{17} -42.9490 q^{19} -173.120 q^{21} +115.726 q^{23} +25.0000 q^{25} +41.4574 q^{27} +184.009 q^{29} -201.848 q^{31} -305.235 q^{33} -112.330 q^{35} -189.885 q^{37} -51.6697 q^{39} +297.987 q^{41} -428.676 q^{43} +161.900 q^{45} +311.134 q^{47} +161.725 q^{49} +130.999 q^{51} -307.329 q^{53} -198.055 q^{55} -330.959 q^{57} -704.985 q^{59} -929.140 q^{61} -727.452 q^{63} -33.5263 q^{65} -587.978 q^{67} +891.764 q^{69} -507.339 q^{71} -13.3237 q^{73} +192.646 q^{75} +889.903 q^{77} +143.573 q^{79} -554.796 q^{81} -1017.25 q^{83} +85.0000 q^{85} +1417.94 q^{87} +772.337 q^{89} +150.641 q^{91} -1555.41 q^{93} -214.745 q^{95} -1700.56 q^{97} -1282.60 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{3} + 25 q^{5} - 10 q^{7} - 30 q^{9} - 126 q^{11} + 83 q^{13} + 5 q^{15} + 85 q^{17} - 55 q^{19} + 6 q^{21} + 2 q^{23} + 125 q^{25} + 163 q^{27} + 195 q^{29} - 97 q^{31} - 352 q^{33} - 50 q^{35}+ \cdots + 858 q^{99}+O(q^{100}) \) 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
\(3\) 7.70584 1.48299 0.741495 − 0.670958i \(-0.234117\pi\)
0.741495 + 0.670958i \(0.234117\pi\)
\(4\) 0 0
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) −22.4661 −1.21306 −0.606528 − 0.795062i \(-0.707438\pi\)
−0.606528 + 0.795062i \(0.707438\pi\)
\(8\) 0 0
\(9\) 32.3800 1.19926
\(10\) 0 0
\(11\) −39.6109 −1.08574 −0.542870 − 0.839817i \(-0.682662\pi\)
−0.542870 + 0.839817i \(0.682662\pi\)
\(12\) 0 0
\(13\) −6.70527 −0.143054 −0.0715272 − 0.997439i \(-0.522787\pi\)
−0.0715272 + 0.997439i \(0.522787\pi\)
\(14\) 0 0
\(15\) 38.5292 0.663213
\(16\) 0 0
\(17\) 17.0000 0.242536
\(18\) 0 0
\(19\) −42.9490 −0.518589 −0.259294 − 0.965798i \(-0.583490\pi\)
−0.259294 + 0.965798i \(0.583490\pi\)
\(20\) 0 0
\(21\) −173.120 −1.79895
\(22\) 0 0
\(23\) 115.726 1.04915 0.524576 − 0.851364i \(-0.324224\pi\)
0.524576 + 0.851364i \(0.324224\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 41.4574 0.295499
\(28\) 0 0
\(29\) 184.009 1.17826 0.589131 − 0.808037i \(-0.299470\pi\)
0.589131 + 0.808037i \(0.299470\pi\)
\(30\) 0 0
\(31\) −201.848 −1.16945 −0.584726 − 0.811231i \(-0.698798\pi\)
−0.584726 + 0.811231i \(0.698798\pi\)
\(32\) 0 0
\(33\) −305.235 −1.61014
\(34\) 0 0
\(35\) −112.330 −0.542495
\(36\) 0 0
\(37\) −189.885 −0.843698 −0.421849 − 0.906666i \(-0.638619\pi\)
−0.421849 + 0.906666i \(0.638619\pi\)
\(38\) 0 0
\(39\) −51.6697 −0.212148
\(40\) 0 0
\(41\) 297.987 1.13507 0.567534 − 0.823350i \(-0.307897\pi\)
0.567534 + 0.823350i \(0.307897\pi\)
\(42\) 0 0
\(43\) −428.676 −1.52029 −0.760144 − 0.649754i \(-0.774872\pi\)
−0.760144 + 0.649754i \(0.774872\pi\)
\(44\) 0 0
\(45\) 161.900 0.536325
\(46\) 0 0
\(47\) 311.134 0.965607 0.482803 − 0.875729i \(-0.339619\pi\)
0.482803 + 0.875729i \(0.339619\pi\)
\(48\) 0 0
\(49\) 161.725 0.471503
\(50\) 0 0
\(51\) 130.999 0.359678
\(52\) 0 0
\(53\) −307.329 −0.796507 −0.398253 − 0.917275i \(-0.630383\pi\)
−0.398253 + 0.917275i \(0.630383\pi\)
\(54\) 0 0
\(55\) −198.055 −0.485558
\(56\) 0 0
\(57\) −330.959 −0.769062
\(58\) 0 0
\(59\) −704.985 −1.55561 −0.777807 − 0.628503i \(-0.783668\pi\)
−0.777807 + 0.628503i \(0.783668\pi\)
\(60\) 0 0
\(61\) −929.140 −1.95023 −0.975117 − 0.221693i \(-0.928842\pi\)
−0.975117 + 0.221693i \(0.928842\pi\)
\(62\) 0 0
\(63\) −727.452 −1.45477
\(64\) 0 0
\(65\) −33.5263 −0.0639759
\(66\) 0 0
\(67\) −587.978 −1.07213 −0.536067 − 0.844175i \(-0.680091\pi\)
−0.536067 + 0.844175i \(0.680091\pi\)
\(68\) 0 0
\(69\) 891.764 1.55588
\(70\) 0 0
\(71\) −507.339 −0.848030 −0.424015 − 0.905655i \(-0.639380\pi\)
−0.424015 + 0.905655i \(0.639380\pi\)
\(72\) 0 0
\(73\) −13.3237 −0.0213619 −0.0106810 − 0.999943i \(-0.503400\pi\)
−0.0106810 + 0.999943i \(0.503400\pi\)
\(74\) 0 0
\(75\) 192.646 0.296598
\(76\) 0 0
\(77\) 889.903 1.31706
\(78\) 0 0
\(79\) 143.573 0.204472 0.102236 − 0.994760i \(-0.467400\pi\)
0.102236 + 0.994760i \(0.467400\pi\)
\(80\) 0 0
\(81\) −554.796 −0.761037
\(82\) 0 0
\(83\) −1017.25 −1.34528 −0.672638 − 0.739972i \(-0.734839\pi\)
−0.672638 + 0.739972i \(0.734839\pi\)
\(84\) 0 0
\(85\) 85.0000 0.108465
\(86\) 0 0
\(87\) 1417.94 1.74735
\(88\) 0 0
\(89\) 772.337 0.919860 0.459930 − 0.887955i \(-0.347874\pi\)
0.459930 + 0.887955i \(0.347874\pi\)
\(90\) 0 0
\(91\) 150.641 0.173533
\(92\) 0 0
\(93\) −1555.41 −1.73429
\(94\) 0 0
\(95\) −214.745 −0.231920
\(96\) 0 0
\(97\) −1700.56 −1.78006 −0.890031 − 0.455899i \(-0.849318\pi\)
−0.890031 + 0.455899i \(0.849318\pi\)
\(98\) 0 0
\(99\) −1282.60 −1.30208
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 1360.4.a.w.1.5 5
4.3 odd 2 85.4.a.g.1.3 ✓ 5
12.11 even 2 765.4.a.m.1.3 5
20.3 even 4 425.4.b.i.324.5 10
20.7 even 4 425.4.b.i.324.6 10
20.19 odd 2 425.4.a.i.1.3 5
68.67 odd 2 1445.4.a.l.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.g.1.3 ✓ 5 4.3 odd 2
425.4.a.i.1.3 5 20.19 odd 2
425.4.b.i.324.5 10 20.3 even 4
425.4.b.i.324.6 10 20.7 even 4
765.4.a.m.1.3 5 12.11 even 2
1360.4.a.w.1.5 5 1.1 even 1 trivial
1445.4.a.l.1.3 5 68.67 odd 2