Properties

Label 85.4.a.g.1.3
Level $85$
Weight $4$
Character 85.1
Self dual yes
Analytic conductor $5.015$
Analytic rank $0$
Dimension $5$
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] = [85,4,Mod(1,85)] 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("85.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(85, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 85 = 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 85.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,2,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(5.01516235049\)
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: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(4.05155\) of defining polynomial
Character \(\chi\) \(=\) 85.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.290753 q^{2} -7.70584 q^{3} -7.91546 q^{4} +5.00000 q^{5} -2.24049 q^{6} +22.4661 q^{7} -4.62746 q^{8} +32.3800 q^{9} +1.45376 q^{10} +39.6109 q^{11} +60.9953 q^{12} -6.70527 q^{13} +6.53208 q^{14} -38.5292 q^{15} +61.9783 q^{16} +17.0000 q^{17} +9.41457 q^{18} +42.9490 q^{19} -39.5773 q^{20} -173.120 q^{21} +11.5170 q^{22} -115.726 q^{23} +35.6585 q^{24} +25.0000 q^{25} -1.94958 q^{26} -41.4574 q^{27} -177.830 q^{28} +184.009 q^{29} -11.2025 q^{30} +201.848 q^{31} +55.0401 q^{32} -305.235 q^{33} +4.94280 q^{34} +112.330 q^{35} -256.303 q^{36} -189.885 q^{37} +12.4876 q^{38} +51.6697 q^{39} -23.1373 q^{40} +297.987 q^{41} -50.3352 q^{42} +428.676 q^{43} -313.539 q^{44} +161.900 q^{45} -33.6476 q^{46} -311.134 q^{47} -477.595 q^{48} +161.725 q^{49} +7.26882 q^{50} -130.999 q^{51} +53.0753 q^{52} -307.329 q^{53} -12.0538 q^{54} +198.055 q^{55} -103.961 q^{56} -330.959 q^{57} +53.5011 q^{58} +704.985 q^{59} +304.977 q^{60} -929.140 q^{61} +58.6879 q^{62} +727.452 q^{63} -479.823 q^{64} -33.5263 q^{65} -88.7480 q^{66} +587.978 q^{67} -134.563 q^{68} +891.764 q^{69} +32.6604 q^{70} +507.339 q^{71} -149.837 q^{72} -13.3237 q^{73} -55.2094 q^{74} -192.646 q^{75} -339.962 q^{76} +889.903 q^{77} +15.0231 q^{78} -143.573 q^{79} +309.891 q^{80} -554.796 q^{81} +86.6407 q^{82} +1017.25 q^{83} +1370.33 q^{84} +85.0000 q^{85} +124.639 q^{86} -1417.94 q^{87} -183.298 q^{88} +772.337 q^{89} +47.0729 q^{90} -150.641 q^{91} +916.023 q^{92} -1555.41 q^{93} -90.4630 q^{94} +214.745 q^{95} -424.130 q^{96} -1700.56 q^{97} +47.0221 q^{98} +1282.60 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{2} - q^{3} + 34 q^{4} + 25 q^{5} - 5 q^{6} + 10 q^{7} + 30 q^{8} - 30 q^{9} + 10 q^{10} + 126 q^{11} + 15 q^{12} + 83 q^{13} + 90 q^{14} - 5 q^{15} + 322 q^{16} + 85 q^{17} - 97 q^{18} + 55 q^{19}+ \cdots - 858 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.290753 0.102797 0.0513983 − 0.998678i \(-0.483632\pi\)
0.0513983 + 0.998678i \(0.483632\pi\)
\(3\) −7.70584 −1.48299 −0.741495 − 0.670958i \(-0.765883\pi\)
−0.741495 + 0.670958i \(0.765883\pi\)
\(4\) −7.91546 −0.989433
\(5\) 5.00000 0.447214
\(6\) −2.24049 −0.152446
\(7\) 22.4661 1.21306 0.606528 − 0.795062i \(-0.292562\pi\)
0.606528 + 0.795062i \(0.292562\pi\)
\(8\) −4.62746 −0.204507
\(9\) 32.3800 1.19926
\(10\) 1.45376 0.0459720
\(11\) 39.6109 1.08574 0.542870 − 0.839817i \(-0.317338\pi\)
0.542870 + 0.839817i \(0.317338\pi\)
\(12\) 60.9953 1.46732
\(13\) −6.70527 −0.143054 −0.0715272 − 0.997439i \(-0.522787\pi\)
−0.0715272 + 0.997439i \(0.522787\pi\)
\(14\) 6.53208 0.124698
\(15\) −38.5292 −0.663213
\(16\) 61.9783 0.968410
\(17\) 17.0000 0.242536
\(18\) 9.41457 0.123280
\(19\) 42.9490 0.518589 0.259294 − 0.965798i \(-0.416510\pi\)
0.259294 + 0.965798i \(0.416510\pi\)
\(20\) −39.5773 −0.442488
\(21\) −173.120 −1.79895
\(22\) 11.5170 0.111610
\(23\) −115.726 −1.04915 −0.524576 − 0.851364i \(-0.675776\pi\)
−0.524576 + 0.851364i \(0.675776\pi\)
\(24\) 35.6585 0.303282
\(25\) 25.0000 0.200000
\(26\) −1.94958 −0.0147055
\(27\) −41.4574 −0.295499
\(28\) −177.830 −1.20024
\(29\) 184.009 1.17826 0.589131 − 0.808037i \(-0.299470\pi\)
0.589131 + 0.808037i \(0.299470\pi\)
\(30\) −11.2025 −0.0681761
\(31\) 201.848 1.16945 0.584726 − 0.811231i \(-0.301202\pi\)
0.584726 + 0.811231i \(0.301202\pi\)
\(32\) 55.0401 0.304056
\(33\) −305.235 −1.61014
\(34\) 4.94280 0.0249318
\(35\) 112.330 0.542495
\(36\) −256.303 −1.18659
\(37\) −189.885 −0.843698 −0.421849 − 0.906666i \(-0.638619\pi\)
−0.421849 + 0.906666i \(0.638619\pi\)
\(38\) 12.4876 0.0533092
\(39\) 51.6697 0.212148
\(40\) −23.1373 −0.0914583
\(41\) 297.987 1.13507 0.567534 − 0.823350i \(-0.307897\pi\)
0.567534 + 0.823350i \(0.307897\pi\)
\(42\) −50.3352 −0.184926
\(43\) 428.676 1.52029 0.760144 − 0.649754i \(-0.225128\pi\)
0.760144 + 0.649754i \(0.225128\pi\)
\(44\) −313.539 −1.07427
\(45\) 161.900 0.536325
\(46\) −33.6476 −0.107849
\(47\) −311.134 −0.965607 −0.482803 − 0.875729i \(-0.660381\pi\)
−0.482803 + 0.875729i \(0.660381\pi\)
\(48\) −477.595 −1.43614
\(49\) 161.725 0.471503
\(50\) 7.26882 0.0205593
\(51\) −130.999 −0.359678
\(52\) 53.0753 0.141543
\(53\) −307.329 −0.796507 −0.398253 − 0.917275i \(-0.630383\pi\)
−0.398253 + 0.917275i \(0.630383\pi\)
\(54\) −12.0538 −0.0303763
\(55\) 198.055 0.485558
\(56\) −103.961 −0.248078
\(57\) −330.959 −0.769062
\(58\) 53.5011 0.121121
\(59\) 704.985 1.55561 0.777807 − 0.628503i \(-0.216332\pi\)
0.777807 + 0.628503i \(0.216332\pi\)
\(60\) 304.977 0.656205
\(61\) −929.140 −1.95023 −0.975117 − 0.221693i \(-0.928842\pi\)
−0.975117 + 0.221693i \(0.928842\pi\)
\(62\) 58.6879 0.120216
\(63\) 727.452 1.45477
\(64\) −479.823 −0.937154
\(65\) −33.5263 −0.0639759
\(66\) −88.7480 −0.165517
\(67\) 587.978 1.07213 0.536067 − 0.844175i \(-0.319909\pi\)
0.536067 + 0.844175i \(0.319909\pi\)
\(68\) −134.563 −0.239973
\(69\) 891.764 1.55588
\(70\) 32.6604 0.0557666
\(71\) 507.339 0.848030 0.424015 − 0.905655i \(-0.360620\pi\)
0.424015 + 0.905655i \(0.360620\pi\)
\(72\) −149.837 −0.245257
\(73\) −13.3237 −0.0213619 −0.0106810 − 0.999943i \(-0.503400\pi\)
−0.0106810 + 0.999943i \(0.503400\pi\)
\(74\) −55.2094 −0.0867293
\(75\) −192.646 −0.296598
\(76\) −339.962 −0.513109
\(77\) 889.903 1.31706
\(78\) 15.0231 0.0218081
\(79\) −143.573 −0.204472 −0.102236 − 0.994760i \(-0.532600\pi\)
−0.102236 + 0.994760i \(0.532600\pi\)
\(80\) 309.891 0.433086
\(81\) −554.796 −0.761037
\(82\) 86.6407 0.116681
\(83\) 1017.25 1.34528 0.672638 − 0.739972i \(-0.265161\pi\)
0.672638 + 0.739972i \(0.265161\pi\)
\(84\) 1370.33 1.77994
\(85\) 85.0000 0.108465
\(86\) 124.639 0.156280
\(87\) −1417.94 −1.74735
\(88\) −183.298 −0.222041
\(89\) 772.337 0.919860 0.459930 − 0.887955i \(-0.347874\pi\)
0.459930 + 0.887955i \(0.347874\pi\)
\(90\) 47.0729 0.0551324
\(91\) −150.641 −0.173533
\(92\) 916.023 1.03806
\(93\) −1555.41 −1.73429
\(94\) −90.4630 −0.0992611
\(95\) 214.745 0.231920
\(96\) −424.130 −0.450912
\(97\) −1700.56 −1.78006 −0.890031 − 0.455899i \(-0.849318\pi\)
−0.890031 + 0.455899i \(0.849318\pi\)
\(98\) 47.0221 0.0484689
\(99\) 1282.60 1.30208
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 85.4.a.g.1.3 ✓ 5
3.2 odd 2 765.4.a.m.1.3 5
4.3 odd 2 1360.4.a.w.1.5 5
5.2 odd 4 425.4.b.i.324.6 10
5.3 odd 4 425.4.b.i.324.5 10
5.4 even 2 425.4.a.i.1.3 5
17.16 even 2 1445.4.a.l.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.g.1.3 ✓ 5 1.1 even 1 trivial
425.4.a.i.1.3 5 5.4 even 2
425.4.b.i.324.5 10 5.3 odd 4
425.4.b.i.324.6 10 5.2 odd 4
765.4.a.m.1.3 5 3.2 odd 2
1360.4.a.w.1.5 5 4.3 odd 2
1445.4.a.l.1.3 5 17.16 even 2