Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1445,4,Mod(1,1445)] 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("1445.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1445, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1445 = 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1445.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,2,1,34,-25] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(85.2577599583\)
Analytic rank: \(0\)
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: \(\Z[a_1, a_2, a_3]\)
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 1.4
Root \(1.68729\) of defining polynomial
Character \(\chi\) \(=\) 1445.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.58234 q^{2} -2.57070 q^{3} +4.83317 q^{4} -5.00000 q^{5} -9.20914 q^{6} -25.2782 q^{7} -11.3447 q^{8} -20.3915 q^{9} -17.9117 q^{10} -11.7931 q^{11} -12.4247 q^{12} +21.3725 q^{13} -90.5551 q^{14} +12.8535 q^{15} -79.3058 q^{16} -73.0493 q^{18} -148.641 q^{19} -24.1659 q^{20} +64.9827 q^{21} -42.2468 q^{22} +7.58989 q^{23} +29.1637 q^{24} +25.0000 q^{25} +76.5636 q^{26} +121.829 q^{27} -122.174 q^{28} +5.09866 q^{29} +46.0457 q^{30} +157.775 q^{31} -193.343 q^{32} +30.3165 q^{33} +126.391 q^{35} -98.5556 q^{36} -425.395 q^{37} -532.484 q^{38} -54.9423 q^{39} +56.7233 q^{40} +381.427 q^{41} +232.790 q^{42} -146.952 q^{43} -56.9980 q^{44} +101.957 q^{45} +27.1896 q^{46} +500.876 q^{47} +203.872 q^{48} +295.986 q^{49} +89.5586 q^{50} +103.297 q^{52} -234.490 q^{53} +436.435 q^{54} +58.9654 q^{55} +286.772 q^{56} +382.113 q^{57} +18.2651 q^{58} +425.699 q^{59} +62.1233 q^{60} -309.393 q^{61} +565.205 q^{62} +515.460 q^{63} -58.1754 q^{64} -106.862 q^{65} +108.604 q^{66} -191.944 q^{67} -19.5114 q^{69} +452.775 q^{70} -768.770 q^{71} +231.334 q^{72} +447.674 q^{73} -1523.91 q^{74} -64.2676 q^{75} -718.410 q^{76} +298.108 q^{77} -196.822 q^{78} -1344.69 q^{79} +396.529 q^{80} +237.382 q^{81} +1366.40 q^{82} +1046.10 q^{83} +314.073 q^{84} -526.432 q^{86} -13.1071 q^{87} +133.788 q^{88} -313.947 q^{89} +365.246 q^{90} -540.258 q^{91} +36.6833 q^{92} -405.594 q^{93} +1794.31 q^{94} +743.207 q^{95} +497.029 q^{96} -276.758 q^{97} +1060.32 q^{98} +240.478 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{2} + q^{3} + 34 q^{4} - 25 q^{5} + 5 q^{6} - 10 q^{7} + 30 q^{8} - 30 q^{9} - 10 q^{10} - 126 q^{11} - 15 q^{12} + 83 q^{13} - 90 q^{14} - 5 q^{15} + 322 q^{16} - 97 q^{18} + 55 q^{19} - 170 q^{20}+ \cdots + 858 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.58234 1.26655 0.633275 − 0.773927i \(-0.281710\pi\)
0.633275 + 0.773927i \(0.281710\pi\)
\(3\) −2.57070 −0.494732 −0.247366 − 0.968922i \(-0.579565\pi\)
−0.247366 + 0.968922i \(0.579565\pi\)
\(4\) 4.83317 0.604147
\(5\) −5.00000 −0.447214
\(6\) −9.20914 −0.626603
\(7\) −25.2782 −1.36489 −0.682447 − 0.730935i \(-0.739084\pi\)
−0.682447 + 0.730935i \(0.739084\pi\)
\(8\) −11.3447 −0.501368
\(9\) −20.3915 −0.755240
\(10\) −17.9117 −0.566418
\(11\) −11.7931 −0.323250 −0.161625 − 0.986852i \(-0.551673\pi\)
−0.161625 + 0.986852i \(0.551673\pi\)
\(12\) −12.4247 −0.298891
\(13\) 21.3725 0.455974 0.227987 − 0.973664i \(-0.426786\pi\)
0.227987 + 0.973664i \(0.426786\pi\)
\(14\) −90.5551 −1.72870
\(15\) 12.8535 0.221251
\(16\) −79.3058 −1.23915
\(17\) 0 0
\(18\) −73.0493 −0.956549
\(19\) −148.641 −1.79477 −0.897386 − 0.441246i \(-0.854537\pi\)
−0.897386 + 0.441246i \(0.854537\pi\)
\(20\) −24.1659 −0.270183
\(21\) 64.9827 0.675257
\(22\) −42.2468 −0.409412
\(23\) 7.58989 0.0688087 0.0344044 − 0.999408i \(-0.489047\pi\)
0.0344044 + 0.999408i \(0.489047\pi\)
\(24\) 29.1637 0.248043
\(25\) 25.0000 0.200000
\(26\) 76.5636 0.577513
\(27\) 121.829 0.868374
\(28\) −122.174 −0.824596
\(29\) 5.09866 0.0326482 0.0163241 − 0.999867i \(-0.494804\pi\)
0.0163241 + 0.999867i \(0.494804\pi\)
\(30\) 46.0457 0.280225
\(31\) 157.775 0.914106 0.457053 − 0.889439i \(-0.348905\pi\)
0.457053 + 0.889439i \(0.348905\pi\)
\(32\) −193.343 −1.06808
\(33\) 30.3165 0.159922
\(34\) 0 0
\(35\) 126.391 0.610399
\(36\) −98.5556 −0.456276
\(37\) −425.395 −1.89012 −0.945062 − 0.326892i \(-0.893999\pi\)
−0.945062 + 0.326892i \(0.893999\pi\)
\(38\) −532.484 −2.27317
\(39\) −54.9423 −0.225585
\(40\) 56.7233 0.224218
\(41\) 381.427 1.45290 0.726450 − 0.687220i \(-0.241169\pi\)
0.726450 + 0.687220i \(0.241169\pi\)
\(42\) 232.790 0.855246
\(43\) −146.952 −0.521162 −0.260581 − 0.965452i \(-0.583914\pi\)
−0.260581 + 0.965452i \(0.583914\pi\)
\(44\) −56.9980 −0.195290
\(45\) 101.957 0.337754
\(46\) 27.1896 0.0871497
\(47\) 500.876 1.55447 0.777237 − 0.629208i \(-0.216621\pi\)
0.777237 + 0.629208i \(0.216621\pi\)
\(48\) 203.872 0.613049
\(49\) 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 \(-0.598277\pi\)
−0.303865 + 0.952715i \(0.598277\pi\)
\(54\) 436.435 1.09984
\(55\) 58.9654 0.144562
\(56\) 286.772 0.684313
\(57\) 382.113 0.887932
\(58\) 18.2651 0.0413505
\(59\) 425.699 0.939344 0.469672 − 0.882841i \(-0.344372\pi\)
0.469672 + 0.882841i \(0.344372\pi\)
\(60\) 62.1233 0.133668
\(61\) −309.393 −0.649405 −0.324703 − 0.945816i \(-0.605264\pi\)
−0.324703 + 0.945816i \(0.605264\pi\)
\(62\) 565.205 1.15776
\(63\) 515.460 1.03082
\(64\) −58.1754 −0.113624
\(65\) −106.862 −0.203918
\(66\) 108.604 0.202549
\(67\) −191.944 −0.349996 −0.174998 − 0.984569i \(-0.555992\pi\)
−0.174998 + 0.984569i \(0.555992\pi\)
\(68\) 0 0
\(69\) −19.5114 −0.0340419
\(70\) 452.775 0.773100
\(71\) −768.770 −1.28502 −0.642509 − 0.766278i \(-0.722106\pi\)
−0.642509 + 0.766278i \(0.722106\pi\)
\(72\) 231.334 0.378653
\(73\) 447.674 0.717758 0.358879 − 0.933384i \(-0.383159\pi\)
0.358879 + 0.933384i \(0.383159\pi\)
\(74\) −1523.91 −2.39393
\(75\) −64.2676 −0.0989464
\(76\) −718.410 −1.08431
\(77\) 298.108 0.441201
\(78\) −196.822 −0.285715
\(79\) −1344.69 −1.91506 −0.957529 − 0.288337i \(-0.906898\pi\)
−0.957529 + 0.288337i \(0.906898\pi\)
\(80\) 396.529 0.554166
\(81\) 237.382 0.325628
\(82\) 1366.40 1.84017
\(83\) 1046.10 1.38343 0.691715 − 0.722171i \(-0.256856\pi\)
0.691715 + 0.722171i \(0.256856\pi\)
\(84\) 314.073 0.407954
\(85\) 0 0
\(86\) −526.432 −0.660077
\(87\) −13.1071 −0.0161521
\(88\) 133.788 0.162067
\(89\) −313.947 −0.373914 −0.186957 − 0.982368i \(-0.559863\pi\)
−0.186957 + 0.982368i \(0.559863\pi\)
\(90\) 365.246 0.427782
\(91\) −540.258 −0.622356
\(92\) 36.6833 0.0415706
\(93\) −405.594 −0.452238
\(94\) 1794.31 1.96882
\(95\) 743.207 0.802647
\(96\) 497.029 0.528414
\(97\) −276.758 −0.289696 −0.144848 − 0.989454i \(-0.546269\pi\)
−0.144848 + 0.989454i \(0.546269\pi\)
\(98\) 1060.32 1.09295
\(99\) 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 1445.4.a.l.1.4 5
17.16 even 2 85.4.a.g.1.4 ✓ 5
51.50 odd 2 765.4.a.m.1.2 5
68.67 odd 2 1360.4.a.w.1.2 5
85.33 odd 4 425.4.b.i.324.3 10
85.67 odd 4 425.4.b.i.324.8 10
85.84 even 2 425.4.a.i.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.g.1.4 ✓ 5 17.16 even 2
425.4.a.i.1.2 5 85.84 even 2
425.4.b.i.324.3 10 85.33 odd 4
425.4.b.i.324.8 10 85.67 odd 4
765.4.a.m.1.2 5 51.50 odd 2
1360.4.a.w.1.2 5 68.67 odd 2
1445.4.a.l.1.4 5 1.1 even 1 trivial