Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [765,4,Mod(1,765)] 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("765.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(765, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 765 = 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 765.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,-2,0,34,-25,0,10,-30,0,10,-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: \(45.1364611544\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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 \(\chi\) \(=\) 765.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.58234 q^{2} +4.83317 q^{4} -5.00000 q^{5} +25.2782 q^{7} +11.3447 q^{8} +17.9117 q^{10} -11.7931 q^{11} +21.3725 q^{13} -90.5551 q^{14} -79.3058 q^{16} -17.0000 q^{17} -148.641 q^{19} -24.1659 q^{20} +42.2468 q^{22} +7.58989 q^{23} +25.0000 q^{25} -76.5636 q^{26} +122.174 q^{28} +5.09866 q^{29} -157.775 q^{31} +193.343 q^{32} +60.8998 q^{34} -126.391 q^{35} +425.395 q^{37} +532.484 q^{38} -56.7233 q^{40} +381.427 q^{41} -146.952 q^{43} -56.9980 q^{44} -27.1896 q^{46} -500.876 q^{47} +295.986 q^{49} -89.5586 q^{50} +103.297 q^{52} +234.490 q^{53} +58.9654 q^{55} +286.772 q^{56} -18.2651 q^{58} -425.699 q^{59} +309.393 q^{61} +565.205 q^{62} -58.1754 q^{64} -106.862 q^{65} -191.944 q^{67} -82.1640 q^{68} +452.775 q^{70} -768.770 q^{71} -447.674 q^{73} -1523.91 q^{74} -718.410 q^{76} -298.108 q^{77} +1344.69 q^{79} +396.529 q^{80} -1366.40 q^{82} -1046.10 q^{83} +85.0000 q^{85} +526.432 q^{86} -133.788 q^{88} +313.947 q^{89} +540.258 q^{91} +36.6833 q^{92} +1794.31 q^{94} +743.207 q^{95} +276.758 q^{97} -1060.32 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} + 34 q^{4} - 25 q^{5} + 10 q^{7} - 30 q^{8} + 10 q^{10} - 126 q^{11} + 83 q^{13} - 90 q^{14} + 322 q^{16} - 85 q^{17} + 55 q^{19} - 170 q^{20} - 240 q^{22} + 2 q^{23} + 125 q^{25} + 395 q^{26}+ \cdots - 502 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.58234 −1.26655 −0.633275 − 0.773927i \(-0.718290\pi\)
−0.633275 + 0.773927i \(0.718290\pi\)
\(3\) 0 0
\(4\) 4.83317 0.604147
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 25.2782 1.36489 0.682447 − 0.730935i \(-0.260916\pi\)
0.682447 + 0.730935i \(0.260916\pi\)
\(8\) 11.3447 0.501368
\(9\) 0 0
\(10\) 17.9117 0.566418
\(11\) −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 \(0.426786\pi\)
\(14\) −90.5551 −1.72870
\(15\) 0 0
\(16\) −79.3058 −1.23915
\(17\) −17.0000 −0.242536
\(18\) 0 0
\(19\) −148.641 −1.79477 −0.897386 − 0.441246i \(-0.854537\pi\)
−0.897386 + 0.441246i \(0.854537\pi\)
\(20\) −24.1659 −0.270183
\(21\) 0 0
\(22\) 42.2468 0.409412
\(23\) 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
\(26\) −76.5636 −0.577513
\(27\) 0 0
\(28\) 122.174 0.824596
\(29\) 5.09866 0.0326482 0.0163241 − 0.999867i \(-0.494804\pi\)
0.0163241 + 0.999867i \(0.494804\pi\)
\(30\) 0 0
\(31\) −157.775 −0.914106 −0.457053 − 0.889439i \(-0.651095\pi\)
−0.457053 + 0.889439i \(0.651095\pi\)
\(32\) 193.343 1.06808
\(33\) 0 0
\(34\) 60.8998 0.307183
\(35\) −126.391 −0.610399
\(36\) 0 0
\(37\) 425.395 1.89012 0.945062 − 0.326892i \(-0.106001\pi\)
0.945062 + 0.326892i \(0.106001\pi\)
\(38\) 532.484 2.27317
\(39\) 0 0
\(40\) −56.7233 −0.224218
\(41\) 381.427 1.45290 0.726450 − 0.687220i \(-0.241169\pi\)
0.726450 + 0.687220i \(0.241169\pi\)
\(42\) 0 0
\(43\) −146.952 −0.521162 −0.260581 − 0.965452i \(-0.583914\pi\)
−0.260581 + 0.965452i \(0.583914\pi\)
\(44\) −56.9980 −0.195290
\(45\) 0 0
\(46\) −27.1896 −0.0871497
\(47\) −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
\(50\) −89.5586 −0.253310
\(51\) 0 0
\(52\) 103.297 0.275475
\(53\) 234.490 0.607730 0.303865 − 0.952715i \(-0.401723\pi\)
0.303865 + 0.952715i \(0.401723\pi\)
\(54\) 0 0
\(55\) 58.9654 0.144562
\(56\) 286.772 0.684313
\(57\) 0 0
\(58\) −18.2651 −0.0413505
\(59\) −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 \(0.394736\pi\)
\(62\) 565.205 1.15776
\(63\) 0 0
\(64\) −58.1754 −0.113624
\(65\) −106.862 −0.203918
\(66\) 0 0
\(67\) −191.944 −0.349996 −0.174998 − 0.984569i \(-0.555992\pi\)
−0.174998 + 0.984569i \(0.555992\pi\)
\(68\) −82.1640 −0.146527
\(69\) 0 0
\(70\) 452.775 0.773100
\(71\) −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 \(0.616841\pi\)
\(74\) −1523.91 −2.39393
\(75\) 0 0
\(76\) −718.410 −1.08431
\(77\) −298.108 −0.441201
\(78\) 0 0
\(79\) 1344.69 1.91506 0.957529 − 0.288337i \(-0.0931024\pi\)
0.957529 + 0.288337i \(0.0931024\pi\)
\(80\) 396.529 0.554166
\(81\) 0 0
\(82\) −1366.40 −1.84017
\(83\) −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
\(86\) 526.432 0.660077
\(87\) 0 0
\(88\) −133.788 −0.162067
\(89\) 313.947 0.373914 0.186957 − 0.982368i \(-0.440137\pi\)
0.186957 + 0.982368i \(0.440137\pi\)
\(90\) 0 0
\(91\) 540.258 0.622356
\(92\) 36.6833 0.0415706
\(93\) 0 0
\(94\) 1794.31 1.96882
\(95\) 743.207 0.802647
\(96\) 0 0
\(97\) 276.758 0.289696 0.144848 − 0.989454i \(-0.453731\pi\)
0.144848 + 0.989454i \(0.453731\pi\)
\(98\) −1060.32 −1.09295
\(99\) 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 765.4.a.m.1.2 5
3.2 odd 2 85.4.a.g.1.4 ✓ 5
12.11 even 2 1360.4.a.w.1.2 5
15.2 even 4 425.4.b.i.324.8 10
15.8 even 4 425.4.b.i.324.3 10
15.14 odd 2 425.4.a.i.1.2 5
51.50 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 3.2 odd 2
425.4.a.i.1.2 5 15.14 odd 2
425.4.b.i.324.3 10 15.8 even 4
425.4.b.i.324.8 10 15.2 even 4
765.4.a.m.1.2 5 1.1 even 1 trivial
1360.4.a.w.1.2 5 12.11 even 2
1445.4.a.l.1.4 5 51.50 odd 2