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 1.2
Root \(1.68729\) 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)
 
\(f(q)\) \(=\) \(q-2.57070 q^{3} +5.00000 q^{5} -25.2782 q^{7} -20.3915 q^{9} -11.7931 q^{11} +21.3725 q^{13} -12.8535 q^{15} +17.0000 q^{17} +148.641 q^{19} +64.9827 q^{21} +7.58989 q^{23} +25.0000 q^{25} +121.829 q^{27} -5.09866 q^{29} +157.775 q^{31} +30.3165 q^{33} -126.391 q^{35} +425.395 q^{37} -54.9423 q^{39} -381.427 q^{41} +146.952 q^{43} -101.957 q^{45} -500.876 q^{47} +295.986 q^{49} -43.7020 q^{51} -234.490 q^{53} -58.9654 q^{55} -382.113 q^{57} -425.699 q^{59} +309.393 q^{61} +515.460 q^{63} +106.862 q^{65} +191.944 q^{67} -19.5114 q^{69} -768.770 q^{71} -447.674 q^{73} -64.2676 q^{75} +298.108 q^{77} -1344.69 q^{79} +237.382 q^{81} -1046.10 q^{83} +85.0000 q^{85} +13.1071 q^{87} -313.947 q^{89} -540.258 q^{91} -405.594 q^{93} +743.207 q^{95} +276.758 q^{97} +240.478 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{3} + 25 q^{5} - 10 q^{7} - 30 q^{9} - 126 q^{11} + 83 q^{13} + 5 q^{15} + 85 q^{17} - 55 q^{19} + 6 q^{21} + 2 q^{23} + 125 q^{25} + 163 q^{27} + 195 q^{29} - 97 q^{31} - 352 q^{33} - 50 q^{35}+ \cdots + 858 q^{99}+O(q^{100}) \) 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
\(3\) −2.57070 −0.494732 −0.247366 − 0.968922i \(-0.579565\pi\)
−0.247366 + 0.968922i \(0.579565\pi\)
\(4\) 0 0
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) −25.2782 −1.36489 −0.682447 − 0.730935i \(-0.739084\pi\)
−0.682447 + 0.730935i \(0.739084\pi\)
\(8\) 0 0
\(9\) −20.3915 −0.755240
\(10\) 0 0
\(11\) −11.7931 −0.323250 −0.161625 − 0.986852i \(-0.551673\pi\)
−0.161625 + 0.986852i \(0.551673\pi\)
\(12\) 0 0
\(13\) 21.3725 0.455974 0.227987 − 0.973664i \(-0.426786\pi\)
0.227987 + 0.973664i \(0.426786\pi\)
\(14\) 0 0
\(15\) −12.8535 −0.221251
\(16\) 0 0
\(17\) 17.0000 0.242536
\(18\) 0 0
\(19\) 148.641 1.79477 0.897386 − 0.441246i \(-0.145463\pi\)
0.897386 + 0.441246i \(0.145463\pi\)
\(20\) 0 0
\(21\) 64.9827 0.675257
\(22\) 0 0
\(23\) 7.58989 0.0688087 0.0344044 − 0.999408i \(-0.489047\pi\)
0.0344044 + 0.999408i \(0.489047\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 121.829 0.868374
\(28\) 0 0
\(29\) −5.09866 −0.0326482 −0.0163241 − 0.999867i \(-0.505196\pi\)
−0.0163241 + 0.999867i \(0.505196\pi\)
\(30\) 0 0
\(31\) 157.775 0.914106 0.457053 − 0.889439i \(-0.348905\pi\)
0.457053 + 0.889439i \(0.348905\pi\)
\(32\) 0 0
\(33\) 30.3165 0.159922
\(34\) 0 0
\(35\) −126.391 −0.610399
\(36\) 0 0
\(37\) 425.395 1.89012 0.945062 − 0.326892i \(-0.106001\pi\)
0.945062 + 0.326892i \(0.106001\pi\)
\(38\) 0 0
\(39\) −54.9423 −0.225585
\(40\) 0 0
\(41\) −381.427 −1.45290 −0.726450 − 0.687220i \(-0.758831\pi\)
−0.726450 + 0.687220i \(0.758831\pi\)
\(42\) 0 0
\(43\) 146.952 0.521162 0.260581 − 0.965452i \(-0.416086\pi\)
0.260581 + 0.965452i \(0.416086\pi\)
\(44\) 0 0
\(45\) −101.957 −0.337754
\(46\) 0 0
\(47\) −500.876 −1.55447 −0.777237 − 0.629208i \(-0.783379\pi\)
−0.777237 + 0.629208i \(0.783379\pi\)
\(48\) 0 0
\(49\) 295.986 0.862934
\(50\) 0 0
\(51\) −43.7020 −0.119990
\(52\) 0 0
\(53\) −234.490 −0.607730 −0.303865 − 0.952715i \(-0.598277\pi\)
−0.303865 + 0.952715i \(0.598277\pi\)
\(54\) 0 0
\(55\) −58.9654 −0.144562
\(56\) 0 0
\(57\) −382.113 −0.887932
\(58\) 0 0
\(59\) −425.699 −0.939344 −0.469672 − 0.882841i \(-0.655628\pi\)
−0.469672 + 0.882841i \(0.655628\pi\)
\(60\) 0 0
\(61\) 309.393 0.649405 0.324703 − 0.945816i \(-0.394736\pi\)
0.324703 + 0.945816i \(0.394736\pi\)
\(62\) 0 0
\(63\) 515.460 1.03082
\(64\) 0 0
\(65\) 106.862 0.203918
\(66\) 0 0
\(67\) 191.944 0.349996 0.174998 − 0.984569i \(-0.444008\pi\)
0.174998 + 0.984569i \(0.444008\pi\)
\(68\) 0 0
\(69\) −19.5114 −0.0340419
\(70\) 0 0
\(71\) −768.770 −1.28502 −0.642509 − 0.766278i \(-0.722106\pi\)
−0.642509 + 0.766278i \(0.722106\pi\)
\(72\) 0 0
\(73\) −447.674 −0.717758 −0.358879 − 0.933384i \(-0.616841\pi\)
−0.358879 + 0.933384i \(0.616841\pi\)
\(74\) 0 0
\(75\) −64.2676 −0.0989464
\(76\) 0 0
\(77\) 298.108 0.441201
\(78\) 0 0
\(79\) −1344.69 −1.91506 −0.957529 − 0.288337i \(-0.906898\pi\)
−0.957529 + 0.288337i \(0.906898\pi\)
\(80\) 0 0
\(81\) 237.382 0.325628
\(82\) 0 0
\(83\) −1046.10 −1.38343 −0.691715 − 0.722171i \(-0.743144\pi\)
−0.691715 + 0.722171i \(0.743144\pi\)
\(84\) 0 0
\(85\) 85.0000 0.108465
\(86\) 0 0
\(87\) 13.1071 0.0161521
\(88\) 0 0
\(89\) −313.947 −0.373914 −0.186957 − 0.982368i \(-0.559863\pi\)
−0.186957 + 0.982368i \(0.559863\pi\)
\(90\) 0 0
\(91\) −540.258 −0.622356
\(92\) 0 0
\(93\) −405.594 −0.452238
\(94\) 0 0
\(95\) 743.207 0.802647
\(96\) 0 0
\(97\) 276.758 0.289696 0.144848 − 0.989454i \(-0.453731\pi\)
0.144848 + 0.989454i \(0.453731\pi\)
\(98\) 0 0
\(99\) 240.478 0.244131
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.1.2 5
4.3 odd 2 85.4.a.g.1.4 ✓ 5
12.11 even 2 765.4.a.m.1.2 5
20.3 even 4 425.4.b.i.324.3 10
20.7 even 4 425.4.b.i.324.8 10
20.19 odd 2 425.4.a.i.1.2 5
68.67 odd 2 1445.4.a.l.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.g.1.4 ✓ 5 4.3 odd 2
425.4.a.i.1.2 5 20.19 odd 2
425.4.b.i.324.3 10 20.3 even 4
425.4.b.i.324.8 10 20.7 even 4
765.4.a.m.1.2 5 12.11 even 2
1360.4.a.w.1.2 5 1.1 even 1 trivial
1445.4.a.l.1.4 5 68.67 odd 2