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 = [8,4,2,48,-40] 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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 48x^{6} + 112x^{5} + 767x^{4} - 496x^{3} - 3404x^{2} + 576x + 4128 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(6.49025\) 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)
 
\(f(q)\) \(=\) \(q-5.49025 q^{2} -1.90849 q^{3} +22.1428 q^{4} -5.00000 q^{5} +10.4781 q^{6} +14.4921 q^{7} -77.6476 q^{8} -23.3576 q^{9} +27.4512 q^{10} +45.9068 q^{11} -42.2595 q^{12} +55.2206 q^{13} -79.5654 q^{14} +9.54247 q^{15} +249.162 q^{16} +128.239 q^{18} +164.433 q^{19} -110.714 q^{20} -27.6582 q^{21} -252.040 q^{22} -23.8569 q^{23} +148.190 q^{24} +25.0000 q^{25} -303.175 q^{26} +96.1073 q^{27} +320.897 q^{28} +236.889 q^{29} -52.3905 q^{30} +38.4845 q^{31} -746.781 q^{32} -87.6129 q^{33} -72.4607 q^{35} -517.204 q^{36} +56.1791 q^{37} -902.779 q^{38} -105.388 q^{39} +388.238 q^{40} -15.7904 q^{41} +151.850 q^{42} +373.408 q^{43} +1016.51 q^{44} +116.788 q^{45} +130.980 q^{46} +74.3781 q^{47} -475.525 q^{48} -132.978 q^{49} -137.256 q^{50} +1222.74 q^{52} +119.118 q^{53} -527.653 q^{54} -229.534 q^{55} -1125.28 q^{56} -313.820 q^{57} -1300.58 q^{58} +46.3762 q^{59} +211.297 q^{60} +413.453 q^{61} -211.290 q^{62} -338.502 q^{63} +2106.72 q^{64} -276.103 q^{65} +481.016 q^{66} -25.5720 q^{67} +45.5307 q^{69} +397.827 q^{70} +1021.13 q^{71} +1813.67 q^{72} +745.320 q^{73} -308.437 q^{74} -47.7124 q^{75} +3641.01 q^{76} +665.287 q^{77} +578.608 q^{78} +793.529 q^{79} -1245.81 q^{80} +447.236 q^{81} +86.6931 q^{82} -1236.22 q^{83} -612.430 q^{84} -2050.10 q^{86} -452.101 q^{87} -3564.55 q^{88} +734.112 q^{89} -641.196 q^{90} +800.264 q^{91} -528.259 q^{92} -73.4475 q^{93} -408.354 q^{94} -822.165 q^{95} +1425.23 q^{96} -1198.74 q^{97} +730.083 q^{98} -1072.27 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} + 2 q^{3} + 48 q^{4} - 40 q^{5} - 44 q^{6} - 32 q^{7} - 36 q^{8} + 162 q^{9} - 20 q^{10} + 44 q^{11} + 36 q^{12} + 86 q^{13} - 146 q^{14} - 10 q^{15} + 348 q^{16} - 8 q^{18} + 288 q^{19} - 240 q^{20}+ \cdots + 5258 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.49025 −1.94110 −0.970548 − 0.240908i \(-0.922555\pi\)
−0.970548 + 0.240908i \(0.922555\pi\)
\(3\) −1.90849 −0.367290 −0.183645 − 0.982993i \(-0.558790\pi\)
−0.183645 + 0.982993i \(0.558790\pi\)
\(4\) 22.1428 2.76785
\(5\) −5.00000 −0.447214
\(6\) 10.4781 0.712945
\(7\) 14.4921 0.782502 0.391251 − 0.920284i \(-0.372043\pi\)
0.391251 + 0.920284i \(0.372043\pi\)
\(8\) −77.6476 −3.43157
\(9\) −23.3576 −0.865098
\(10\) 27.4512 0.868084
\(11\) 45.9068 1.25831 0.629155 − 0.777279i \(-0.283401\pi\)
0.629155 + 0.777279i \(0.283401\pi\)
\(12\) −42.2595 −1.01660
\(13\) 55.2206 1.17811 0.589055 − 0.808093i \(-0.299500\pi\)
0.589055 + 0.808093i \(0.299500\pi\)
\(14\) −79.5654 −1.51891
\(15\) 9.54247 0.164257
\(16\) 249.162 3.89316
\(17\) 0 0
\(18\) 128.239 1.67924
\(19\) 164.433 1.98545 0.992725 − 0.120406i \(-0.0384197\pi\)
0.992725 + 0.120406i \(0.0384197\pi\)
\(20\) −110.714 −1.23782
\(21\) −27.6582 −0.287405
\(22\) −252.040 −2.44250
\(23\) −23.8569 −0.216283 −0.108141 − 0.994136i \(-0.534490\pi\)
−0.108141 + 0.994136i \(0.534490\pi\)
\(24\) 148.190 1.26038
\(25\) 25.0000 0.200000
\(26\) −303.175 −2.28683
\(27\) 96.1073 0.685032
\(28\) 320.897 2.16585
\(29\) 236.889 1.51687 0.758433 − 0.651751i \(-0.225965\pi\)
0.758433 + 0.651751i \(0.225965\pi\)
\(30\) −52.3905 −0.318839
\(31\) 38.4845 0.222969 0.111484 − 0.993766i \(-0.464440\pi\)
0.111484 + 0.993766i \(0.464440\pi\)
\(32\) −746.781 −4.12542
\(33\) −87.6129 −0.462165
\(34\) 0 0
\(35\) −72.4607 −0.349945
\(36\) −517.204 −2.39446
\(37\) 56.1791 0.249616 0.124808 − 0.992181i \(-0.460168\pi\)
0.124808 + 0.992181i \(0.460168\pi\)
\(38\) −902.779 −3.85395
\(39\) −105.388 −0.432708
\(40\) 388.238 1.53465
\(41\) −15.7904 −0.0601474 −0.0300737 − 0.999548i \(-0.509574\pi\)
−0.0300737 + 0.999548i \(0.509574\pi\)
\(42\) 151.850 0.557881
\(43\) 373.408 1.32428 0.662142 − 0.749378i \(-0.269648\pi\)
0.662142 + 0.749378i \(0.269648\pi\)
\(44\) 1016.51 3.48282
\(45\) 116.788 0.386884
\(46\) 130.980 0.419825
\(47\) 74.3781 0.230833 0.115417 − 0.993317i \(-0.463180\pi\)
0.115417 + 0.993317i \(0.463180\pi\)
\(48\) −475.525 −1.42992
\(49\) −132.978 −0.387691
\(50\) −137.256 −0.388219
\(51\) 0 0
\(52\) 1222.74 3.26084
\(53\) 119.118 0.308720 0.154360 − 0.988015i \(-0.450668\pi\)
0.154360 + 0.988015i \(0.450668\pi\)
\(54\) −527.653 −1.32971
\(55\) −229.534 −0.562734
\(56\) −1125.28 −2.68521
\(57\) −313.820 −0.729236
\(58\) −1300.58 −2.94438
\(59\) 46.3762 0.102333 0.0511666 − 0.998690i \(-0.483706\pi\)
0.0511666 + 0.998690i \(0.483706\pi\)
\(60\) 211.297 0.454639
\(61\) 413.453 0.867824 0.433912 − 0.900955i \(-0.357133\pi\)
0.433912 + 0.900955i \(0.357133\pi\)
\(62\) −211.290 −0.432804
\(63\) −338.502 −0.676941
\(64\) 2106.72 4.11468
\(65\) −276.103 −0.526867
\(66\) 481.016 0.897107
\(67\) −25.5720 −0.0466286 −0.0233143 − 0.999728i \(-0.507422\pi\)
−0.0233143 + 0.999728i \(0.507422\pi\)
\(68\) 0 0
\(69\) 45.5307 0.0794385
\(70\) 397.827 0.679277
\(71\) 1021.13 1.70684 0.853422 − 0.521220i \(-0.174523\pi\)
0.853422 + 0.521220i \(0.174523\pi\)
\(72\) 1813.67 2.96865
\(73\) 745.320 1.19497 0.597487 − 0.801878i \(-0.296166\pi\)
0.597487 + 0.801878i \(0.296166\pi\)
\(74\) −308.437 −0.484529
\(75\) −47.7124 −0.0734580
\(76\) 3641.01 5.49543
\(77\) 665.287 0.984630
\(78\) 578.608 0.839928
\(79\) 793.529 1.13011 0.565057 − 0.825052i \(-0.308854\pi\)
0.565057 + 0.825052i \(0.308854\pi\)
\(80\) −1245.81 −1.74107
\(81\) 447.236 0.613493
\(82\) 86.6931 0.116752
\(83\) −1236.22 −1.63485 −0.817424 − 0.576036i \(-0.804599\pi\)
−0.817424 + 0.576036i \(0.804599\pi\)
\(84\) −612.430 −0.795495
\(85\) 0 0
\(86\) −2050.10 −2.57056
\(87\) −452.101 −0.557129
\(88\) −3564.55 −4.31799
\(89\) 734.112 0.874334 0.437167 − 0.899380i \(-0.355982\pi\)
0.437167 + 0.899380i \(0.355982\pi\)
\(90\) −641.196 −0.750978
\(91\) 800.264 0.921874
\(92\) −528.259 −0.598639
\(93\) −73.4475 −0.0818942
\(94\) −408.354 −0.448069
\(95\) −822.165 −0.887920
\(96\) 1425.23 1.51523
\(97\) −1198.74 −1.25478 −0.627392 − 0.778703i \(-0.715878\pi\)
−0.627392 + 0.778703i \(0.715878\pi\)
\(98\) 730.083 0.752546
\(99\) −1072.27 −1.08856
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.r.1.1 yes 8
17.16 even 2 1445.4.a.q.1.1 ✓ 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1445.4.a.q.1.1 ✓ 8 17.16 even 2
1445.4.a.r.1.1 yes 8 1.1 even 1 trivial