Properties

Label 85.4.a.e.1.2
Level $85$
Weight $4$
Character 85.1
Self dual yes
Analytic conductor $5.015$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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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 = [3,-6,-4] 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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1304.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 11x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.182370\) 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)
 
\(f(q)\) \(=\) \(q-1.81763 q^{2} +6.14911 q^{3} -4.69622 q^{4} -5.00000 q^{5} -11.1768 q^{6} -34.0161 q^{7} +23.0770 q^{8} +10.8116 q^{9} +9.08815 q^{10} -33.6746 q^{11} -28.8776 q^{12} +25.1375 q^{13} +61.8287 q^{14} -30.7456 q^{15} -4.37576 q^{16} +17.0000 q^{17} -19.6514 q^{18} -150.063 q^{19} +23.4811 q^{20} -209.169 q^{21} +61.2079 q^{22} -126.479 q^{23} +141.903 q^{24} +25.0000 q^{25} -45.6907 q^{26} -99.5445 q^{27} +159.747 q^{28} +235.632 q^{29} +55.8841 q^{30} +282.666 q^{31} -176.663 q^{32} -207.069 q^{33} -30.8997 q^{34} +170.080 q^{35} -50.7735 q^{36} +8.51881 q^{37} +272.760 q^{38} +154.573 q^{39} -115.385 q^{40} -23.8338 q^{41} +380.191 q^{42} +105.057 q^{43} +158.143 q^{44} -54.0578 q^{45} +229.893 q^{46} -74.5279 q^{47} -26.9071 q^{48} +814.093 q^{49} -45.4408 q^{50} +104.535 q^{51} -118.051 q^{52} -680.632 q^{53} +180.935 q^{54} +168.373 q^{55} -784.990 q^{56} -922.756 q^{57} -428.292 q^{58} -435.334 q^{59} +144.388 q^{60} +365.958 q^{61} -513.782 q^{62} -367.767 q^{63} +356.114 q^{64} -125.687 q^{65} +376.374 q^{66} +338.847 q^{67} -79.8357 q^{68} -777.736 q^{69} -309.143 q^{70} -69.0565 q^{71} +249.499 q^{72} -768.149 q^{73} -15.4840 q^{74} +153.728 q^{75} +704.730 q^{76} +1145.48 q^{77} -280.957 q^{78} +344.423 q^{79} +21.8788 q^{80} -904.022 q^{81} +43.3210 q^{82} -231.238 q^{83} +982.302 q^{84} -85.0000 q^{85} -190.954 q^{86} +1448.93 q^{87} -777.110 q^{88} -905.395 q^{89} +98.2571 q^{90} -855.079 q^{91} +593.975 q^{92} +1738.14 q^{93} +135.464 q^{94} +750.316 q^{95} -1086.32 q^{96} -439.268 q^{97} -1479.72 q^{98} -364.075 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 6 q^{2} - 4 q^{3} + 10 q^{4} - 15 q^{5} + 36 q^{6} + 8 q^{7} - 66 q^{8} + 39 q^{9} + 30 q^{10} - 118 q^{11} - 212 q^{12} - 10 q^{13} - 68 q^{14} + 20 q^{15} + 242 q^{16} + 51 q^{17} - 342 q^{18} - 160 q^{19}+ \cdots + 770 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.81763 −0.642629 −0.321315 − 0.946972i \(-0.604125\pi\)
−0.321315 + 0.946972i \(0.604125\pi\)
\(3\) 6.14911 1.18340 0.591698 − 0.806159i \(-0.298458\pi\)
0.591698 + 0.806159i \(0.298458\pi\)
\(4\) −4.69622 −0.587027
\(5\) −5.00000 −0.447214
\(6\) −11.1768 −0.760486
\(7\) −34.0161 −1.83670 −0.918348 − 0.395774i \(-0.870476\pi\)
−0.918348 + 0.395774i \(0.870476\pi\)
\(8\) 23.0770 1.01987
\(9\) 10.8116 0.400428
\(10\) 9.08815 0.287393
\(11\) −33.6746 −0.923024 −0.461512 − 0.887134i \(-0.652693\pi\)
−0.461512 + 0.887134i \(0.652693\pi\)
\(12\) −28.8776 −0.694687
\(13\) 25.1375 0.536299 0.268149 − 0.963377i \(-0.413588\pi\)
0.268149 + 0.963377i \(0.413588\pi\)
\(14\) 61.8287 1.18031
\(15\) −30.7456 −0.529231
\(16\) −4.37576 −0.0683713
\(17\) 17.0000 0.242536
\(18\) −19.6514 −0.257327
\(19\) −150.063 −1.81194 −0.905971 − 0.423341i \(-0.860857\pi\)
−0.905971 + 0.423341i \(0.860857\pi\)
\(20\) 23.4811 0.262527
\(21\) −209.169 −2.17354
\(22\) 61.2079 0.593163
\(23\) −126.479 −1.14664 −0.573322 − 0.819330i \(-0.694345\pi\)
−0.573322 + 0.819330i \(0.694345\pi\)
\(24\) 141.903 1.20691
\(25\) 25.0000 0.200000
\(26\) −45.6907 −0.344641
\(27\) −99.5445 −0.709531
\(28\) 159.747 1.07819
\(29\) 235.632 1.50882 0.754410 − 0.656404i \(-0.227923\pi\)
0.754410 + 0.656404i \(0.227923\pi\)
\(30\) 55.8841 0.340100
\(31\) 282.666 1.63769 0.818843 − 0.574018i \(-0.194616\pi\)
0.818843 + 0.574018i \(0.194616\pi\)
\(32\) −176.663 −0.975933
\(33\) −207.069 −1.09230
\(34\) −30.8997 −0.155861
\(35\) 170.080 0.821395
\(36\) −50.7735 −0.235062
\(37\) 8.51881 0.0378509 0.0189255 − 0.999821i \(-0.493975\pi\)
0.0189255 + 0.999821i \(0.493975\pi\)
\(38\) 272.760 1.16441
\(39\) 154.573 0.634654
\(40\) −115.385 −0.456100
\(41\) −23.8338 −0.0907857 −0.0453929 − 0.998969i \(-0.514454\pi\)
−0.0453929 + 0.998969i \(0.514454\pi\)
\(42\) 380.191 1.39678
\(43\) 105.057 0.372581 0.186291 − 0.982495i \(-0.440353\pi\)
0.186291 + 0.982495i \(0.440353\pi\)
\(44\) 158.143 0.541841
\(45\) −54.0578 −0.179077
\(46\) 229.893 0.736867
\(47\) −74.5279 −0.231298 −0.115649 − 0.993290i \(-0.536895\pi\)
−0.115649 + 0.993290i \(0.536895\pi\)
\(48\) −26.9071 −0.0809104
\(49\) 814.093 2.37345
\(50\) −45.4408 −0.128526
\(51\) 104.535 0.287016
\(52\) −118.051 −0.314822
\(53\) −680.632 −1.76400 −0.881999 − 0.471250i \(-0.843803\pi\)
−0.881999 + 0.471250i \(0.843803\pi\)
\(54\) 180.935 0.455966
\(55\) 168.373 0.412789
\(56\) −784.990 −1.87319
\(57\) −922.756 −2.14425
\(58\) −428.292 −0.969612
\(59\) −435.334 −0.960605 −0.480302 − 0.877103i \(-0.659473\pi\)
−0.480302 + 0.877103i \(0.659473\pi\)
\(60\) 144.388 0.310673
\(61\) 365.958 0.768132 0.384066 − 0.923306i \(-0.374523\pi\)
0.384066 + 0.923306i \(0.374523\pi\)
\(62\) −513.782 −1.05243
\(63\) −367.767 −0.735465
\(64\) 356.114 0.695535
\(65\) −125.687 −0.239840
\(66\) 376.374 0.701947
\(67\) 338.847 0.617863 0.308931 − 0.951084i \(-0.400029\pi\)
0.308931 + 0.951084i \(0.400029\pi\)
\(68\) −79.8357 −0.142375
\(69\) −777.736 −1.35693
\(70\) −309.143 −0.527853
\(71\) −69.0565 −0.115430 −0.0577148 − 0.998333i \(-0.518381\pi\)
−0.0577148 + 0.998333i \(0.518381\pi\)
\(72\) 249.499 0.408385
\(73\) −768.149 −1.23158 −0.615788 − 0.787912i \(-0.711162\pi\)
−0.615788 + 0.787912i \(0.711162\pi\)
\(74\) −15.4840 −0.0243241
\(75\) 153.728 0.236679
\(76\) 704.730 1.06366
\(77\) 1145.48 1.69531
\(78\) −280.957 −0.407848
\(79\) 344.423 0.490514 0.245257 − 0.969458i \(-0.421128\pi\)
0.245257 + 0.969458i \(0.421128\pi\)
\(80\) 21.8788 0.0305766
\(81\) −904.022 −1.24009
\(82\) 43.3210 0.0583416
\(83\) −231.238 −0.305803 −0.152902 − 0.988241i \(-0.548862\pi\)
−0.152902 + 0.988241i \(0.548862\pi\)
\(84\) 982.302 1.27593
\(85\) −85.0000 −0.108465
\(86\) −190.954 −0.239432
\(87\) 1448.93 1.78553
\(88\) −777.110 −0.941365
\(89\) −905.395 −1.07833 −0.539167 − 0.842199i \(-0.681261\pi\)
−0.539167 + 0.842199i \(0.681261\pi\)
\(90\) 98.2571 0.115080
\(91\) −855.079 −0.985018
\(92\) 593.975 0.673111
\(93\) 1738.14 1.93803
\(94\) 135.464 0.148639
\(95\) 750.316 0.810325
\(96\) −1086.32 −1.15492
\(97\) −439.268 −0.459803 −0.229902 − 0.973214i \(-0.573840\pi\)
−0.229902 + 0.973214i \(0.573840\pi\)
\(98\) −1479.72 −1.52525
\(99\) −364.075 −0.369605
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.e.1.2 ✓ 3
3.2 odd 2 765.4.a.l.1.2 3
4.3 odd 2 1360.4.a.s.1.1 3
5.2 odd 4 425.4.b.g.324.2 6
5.3 odd 4 425.4.b.g.324.5 6
5.4 even 2 425.4.a.h.1.2 3
17.16 even 2 1445.4.a.j.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.e.1.2 ✓ 3 1.1 even 1 trivial
425.4.a.h.1.2 3 5.4 even 2
425.4.b.g.324.2 6 5.2 odd 4
425.4.b.g.324.5 6 5.3 odd 4
765.4.a.l.1.2 3 3.2 odd 2
1360.4.a.s.1.1 3 4.3 odd 2
1445.4.a.j.1.2 3 17.16 even 2