Properties

Label 85.4.a.e.1.1
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 1.1
Root \(3.40405\) 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)
 
\(f(q)\) \(=\) \(q-5.40405 q^{2} -8.99158 q^{3} +21.2037 q^{4} -5.00000 q^{5} +48.5909 q^{6} +27.3417 q^{7} -71.3535 q^{8} +53.8486 q^{9} +27.0202 q^{10} -12.9511 q^{11} -190.655 q^{12} -6.73396 q^{13} -147.756 q^{14} +44.9579 q^{15} +215.968 q^{16} +17.0000 q^{17} -291.000 q^{18} -95.8959 q^{19} -106.019 q^{20} -245.845 q^{21} +69.9884 q^{22} +41.6582 q^{23} +641.581 q^{24} +25.0000 q^{25} +36.3906 q^{26} -241.411 q^{27} +579.746 q^{28} +165.529 q^{29} -242.955 q^{30} -197.049 q^{31} -596.273 q^{32} +116.451 q^{33} -91.8688 q^{34} -136.709 q^{35} +1141.79 q^{36} -187.515 q^{37} +518.226 q^{38} +60.5489 q^{39} +356.768 q^{40} -291.596 q^{41} +1328.56 q^{42} -139.562 q^{43} -274.612 q^{44} -269.243 q^{45} -225.123 q^{46} -373.384 q^{47} -1941.90 q^{48} +404.570 q^{49} -135.101 q^{50} -152.857 q^{51} -142.785 q^{52} +76.4382 q^{53} +1304.60 q^{54} +64.7556 q^{55} -1950.93 q^{56} +862.256 q^{57} -894.526 q^{58} -467.311 q^{59} +953.275 q^{60} -466.021 q^{61} +1064.86 q^{62} +1472.31 q^{63} +1494.55 q^{64} +33.6698 q^{65} -629.307 q^{66} -206.925 q^{67} +360.463 q^{68} -374.573 q^{69} +738.780 q^{70} +378.892 q^{71} -3842.29 q^{72} +345.797 q^{73} +1013.34 q^{74} -224.790 q^{75} -2033.35 q^{76} -354.106 q^{77} -327.209 q^{78} -194.264 q^{79} -1079.84 q^{80} +716.757 q^{81} +1575.80 q^{82} -947.568 q^{83} -5212.84 q^{84} -85.0000 q^{85} +754.199 q^{86} -1488.37 q^{87} +924.108 q^{88} +108.202 q^{89} +1455.00 q^{90} -184.118 q^{91} +883.309 q^{92} +1771.78 q^{93} +2017.79 q^{94} +479.480 q^{95} +5361.44 q^{96} +1696.33 q^{97} -2186.32 q^{98} -697.399 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 6 q^{2} - 4 q^{3} + 10 q^{4} - 15 q^{5} + 36 q^{6} + 8 q^{7} - 66 q^{8} + 39 q^{9} + 30 q^{10} - 118 q^{11} - 212 q^{12} - 10 q^{13} - 68 q^{14} + 20 q^{15} + 242 q^{16} + 51 q^{17} - 342 q^{18} - 160 q^{19}+ \cdots + 770 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.40405 −1.91062 −0.955310 − 0.295607i \(-0.904478\pi\)
−0.955310 + 0.295607i \(0.904478\pi\)
\(3\) −8.99158 −1.73043 −0.865215 − 0.501400i \(-0.832818\pi\)
−0.865215 + 0.501400i \(0.832818\pi\)
\(4\) 21.2037 2.65047
\(5\) −5.00000 −0.447214
\(6\) 48.5909 3.30619
\(7\) 27.3417 1.47631 0.738157 − 0.674629i \(-0.235696\pi\)
0.738157 + 0.674629i \(0.235696\pi\)
\(8\) −71.3535 −3.15341
\(9\) 53.8486 1.99439
\(10\) 27.0202 0.854455
\(11\) −12.9511 −0.354992 −0.177496 − 0.984122i \(-0.556800\pi\)
−0.177496 + 0.984122i \(0.556800\pi\)
\(12\) −190.655 −4.58645
\(13\) −6.73396 −0.143666 −0.0718332 − 0.997417i \(-0.522885\pi\)
−0.0718332 + 0.997417i \(0.522885\pi\)
\(14\) −147.756 −2.82067
\(15\) 44.9579 0.773872
\(16\) 215.968 3.37450
\(17\) 17.0000 0.242536
\(18\) −291.000 −3.81052
\(19\) −95.8959 −1.15790 −0.578948 − 0.815364i \(-0.696537\pi\)
−0.578948 + 0.815364i \(0.696537\pi\)
\(20\) −106.019 −1.18532
\(21\) −245.845 −2.55466
\(22\) 69.9884 0.678254
\(23\) 41.6582 0.377667 0.188833 − 0.982009i \(-0.439529\pi\)
0.188833 + 0.982009i \(0.439529\pi\)
\(24\) 641.581 5.45676
\(25\) 25.0000 0.200000
\(26\) 36.3906 0.274492
\(27\) −241.411 −1.72073
\(28\) 579.746 3.91292
\(29\) 165.529 1.05993 0.529965 − 0.848020i \(-0.322205\pi\)
0.529965 + 0.848020i \(0.322205\pi\)
\(30\) −242.955 −1.47858
\(31\) −197.049 −1.14165 −0.570823 − 0.821073i \(-0.693376\pi\)
−0.570823 + 0.821073i \(0.693376\pi\)
\(32\) −596.273 −3.29398
\(33\) 116.451 0.614288
\(34\) −91.8688 −0.463393
\(35\) −136.709 −0.660228
\(36\) 1141.79 5.28607
\(37\) −187.515 −0.833170 −0.416585 − 0.909097i \(-0.636773\pi\)
−0.416585 + 0.909097i \(0.636773\pi\)
\(38\) 518.226 2.21230
\(39\) 60.5489 0.248605
\(40\) 356.768 1.41025
\(41\) −291.596 −1.11072 −0.555362 − 0.831609i \(-0.687420\pi\)
−0.555362 + 0.831609i \(0.687420\pi\)
\(42\) 1328.56 4.88098
\(43\) −139.562 −0.494954 −0.247477 − 0.968894i \(-0.579601\pi\)
−0.247477 + 0.968894i \(0.579601\pi\)
\(44\) −274.612 −0.940893
\(45\) −269.243 −0.891919
\(46\) −225.123 −0.721577
\(47\) −373.384 −1.15880 −0.579401 − 0.815043i \(-0.696713\pi\)
−0.579401 + 0.815043i \(0.696713\pi\)
\(48\) −1941.90 −5.83934
\(49\) 404.570 1.17950
\(50\) −135.101 −0.382124
\(51\) −152.857 −0.419691
\(52\) −142.785 −0.380783
\(53\) 76.4382 0.198106 0.0990528 − 0.995082i \(-0.468419\pi\)
0.0990528 + 0.995082i \(0.468419\pi\)
\(54\) 1304.60 3.28765
\(55\) 64.7556 0.158757
\(56\) −1950.93 −4.65543
\(57\) 862.256 2.00366
\(58\) −894.526 −2.02512
\(59\) −467.311 −1.03117 −0.515583 − 0.856840i \(-0.672424\pi\)
−0.515583 + 0.856840i \(0.672424\pi\)
\(60\) 953.275 2.05112
\(61\) −466.021 −0.978161 −0.489081 − 0.872239i \(-0.662668\pi\)
−0.489081 + 0.872239i \(0.662668\pi\)
\(62\) 1064.86 2.18125
\(63\) 1472.31 2.94435
\(64\) 1494.55 2.91903
\(65\) 33.6698 0.0642496
\(66\) −629.307 −1.17367
\(67\) −206.925 −0.377312 −0.188656 − 0.982043i \(-0.560413\pi\)
−0.188656 + 0.982043i \(0.560413\pi\)
\(68\) 360.463 0.642832
\(69\) −374.573 −0.653526
\(70\) 738.780 1.26144
\(71\) 378.892 0.633327 0.316663 − 0.948538i \(-0.397437\pi\)
0.316663 + 0.948538i \(0.397437\pi\)
\(72\) −3842.29 −6.28914
\(73\) 345.797 0.554417 0.277209 − 0.960810i \(-0.410591\pi\)
0.277209 + 0.960810i \(0.410591\pi\)
\(74\) 1013.34 1.59187
\(75\) −224.790 −0.346086
\(76\) −2033.35 −3.06896
\(77\) −354.106 −0.524079
\(78\) −327.209 −0.474989
\(79\) −194.264 −0.276664 −0.138332 − 0.990386i \(-0.544174\pi\)
−0.138332 + 0.990386i \(0.544174\pi\)
\(80\) −1079.84 −1.50912
\(81\) 716.757 0.983205
\(82\) 1575.80 2.12217
\(83\) −947.568 −1.25312 −0.626561 − 0.779372i \(-0.715538\pi\)
−0.626561 + 0.779372i \(0.715538\pi\)
\(84\) −5212.84 −6.77104
\(85\) −85.0000 −0.108465
\(86\) 754.199 0.945668
\(87\) −1488.37 −1.83413
\(88\) 924.108 1.11943
\(89\) 108.202 0.128869 0.0644345 − 0.997922i \(-0.479476\pi\)
0.0644345 + 0.997922i \(0.479476\pi\)
\(90\) 1455.00 1.70412
\(91\) −184.118 −0.212097
\(92\) 883.309 1.00099
\(93\) 1771.78 1.97554
\(94\) 2017.79 2.21403
\(95\) 479.480 0.517827
\(96\) 5361.44 5.70000
\(97\) 1696.33 1.77563 0.887815 − 0.460200i \(-0.152222\pi\)
0.887815 + 0.460200i \(0.152222\pi\)
\(98\) −2186.32 −2.25358
\(99\) −697.399 −0.707992
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.1.1 ✓ 3
3.2 odd 2 765.4.a.l.1.3 3
4.3 odd 2 1360.4.a.s.1.3 3
5.2 odd 4 425.4.b.g.324.1 6
5.3 odd 4 425.4.b.g.324.6 6
5.4 even 2 425.4.a.h.1.3 3
17.16 even 2 1445.4.a.j.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.e.1.1 ✓ 3 1.1 even 1 trivial
425.4.a.h.1.3 3 5.4 even 2
425.4.b.g.324.1 6 5.2 odd 4
425.4.b.g.324.6 6 5.3 odd 4
765.4.a.l.1.3 3 3.2 odd 2
1360.4.a.s.1.3 3 4.3 odd 2
1445.4.a.j.1.1 3 17.16 even 2