Properties

Label 1445.4.a.q.1.1
Level $1445$
Weight $4$
Character 1445.1
Self dual yes
Analytic conductor $85.258$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 \(-0.441210\pi\)
0.183645 + 0.982993i \(0.441210\pi\)
\(4\) 22.1428 2.76785
\(5\) 5.00000 0.447214
\(6\) −10.4781 −0.712945
\(7\) −14.4921 −0.782502 −0.391251 − 0.920284i \(-0.627957\pi\)
−0.391251 + 0.920284i \(0.627957\pi\)
\(8\) −77.6476 −3.43157
\(9\) −23.3576 −0.865098
\(10\) −27.4512 −0.868084
\(11\) −45.9068 −1.25831 −0.629155 − 0.777279i \(-0.716599\pi\)
−0.629155 + 0.777279i \(0.716599\pi\)
\(12\) 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 \(-0.465510\pi\)
0.108141 + 0.994136i \(0.465510\pi\)
\(24\) −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 \(-0.774035\pi\)
−0.758433 + 0.651751i \(0.774035\pi\)
\(30\) −52.3905 −0.318839
\(31\) −38.4845 −0.222969 −0.111484 − 0.993766i \(-0.535560\pi\)
−0.111484 + 0.993766i \(0.535560\pi\)
\(32\) −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 \(-0.539832\pi\)
−0.124808 + 0.992181i \(0.539832\pi\)
\(38\) −902.779 −3.85395
\(39\) 105.388 0.432708
\(40\) −388.238 −1.53465
\(41\) 15.7904 0.0601474 0.0300737 − 0.999548i \(-0.490426\pi\)
0.0300737 + 0.999548i \(0.490426\pi\)
\(42\) 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 \(-0.642867\pi\)
−0.433912 + 0.900955i \(0.642867\pi\)
\(62\) 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 \(-0.825477\pi\)
−0.853422 + 0.521220i \(0.825477\pi\)
\(72\) 1813.67 2.96865
\(73\) −745.320 −1.19497 −0.597487 − 0.801878i \(-0.703834\pi\)
−0.597487 + 0.801878i \(0.703834\pi\)
\(74\) 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 \(-0.691146\pi\)
−0.565057 + 0.825052i \(0.691146\pi\)
\(80\) 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 \(-0.284122\pi\)
0.627392 + 0.778703i \(0.284122\pi\)
\(98\) 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.q.1.1 ✓ 8
17.16 even 2 1445.4.a.r.1.1 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1445.4.a.q.1.1 ✓ 8 1.1 even 1 trivial
1445.4.a.r.1.1 yes 8 17.16 even 2