Properties

Label 85.4.a.e.1.3
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 1.3
Root \(-3.22168\) 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)
 
\(f(q)\) \(=\) \(q+1.22168 q^{2} -1.15753 q^{3} -6.50750 q^{4} -5.00000 q^{5} -1.41413 q^{6} +14.6743 q^{7} -17.7235 q^{8} -25.6601 q^{9} -6.10839 q^{10} -71.3743 q^{11} +7.53262 q^{12} -28.4035 q^{13} +17.9273 q^{14} +5.78764 q^{15} +30.4076 q^{16} +17.0000 q^{17} -31.3484 q^{18} +85.9592 q^{19} +32.5375 q^{20} -16.9860 q^{21} -87.1964 q^{22} -7.17872 q^{23} +20.5154 q^{24} +25.0000 q^{25} -34.7000 q^{26} +60.9556 q^{27} -95.4934 q^{28} -27.1610 q^{29} +7.07063 q^{30} -15.6170 q^{31} +178.936 q^{32} +82.6178 q^{33} +20.7685 q^{34} -73.3717 q^{35} +166.983 q^{36} -67.0036 q^{37} +105.014 q^{38} +32.8779 q^{39} +88.6175 q^{40} -38.5701 q^{41} -20.7514 q^{42} -251.495 q^{43} +464.469 q^{44} +128.301 q^{45} -8.77008 q^{46} +57.9121 q^{47} -35.1977 q^{48} -127.663 q^{49} +30.5419 q^{50} -19.6780 q^{51} +184.836 q^{52} -677.807 q^{53} +74.4680 q^{54} +356.872 q^{55} -260.081 q^{56} -99.5002 q^{57} -33.1819 q^{58} +598.645 q^{59} -37.6631 q^{60} +346.063 q^{61} -19.0789 q^{62} -376.546 q^{63} -24.6588 q^{64} +142.018 q^{65} +100.932 q^{66} -849.923 q^{67} -110.628 q^{68} +8.30957 q^{69} -89.6366 q^{70} -911.836 q^{71} +454.787 q^{72} +704.352 q^{73} -81.8568 q^{74} -28.9382 q^{75} -559.380 q^{76} -1047.37 q^{77} +40.1662 q^{78} +55.8414 q^{79} -152.038 q^{80} +622.266 q^{81} -47.1202 q^{82} -1235.19 q^{83} +110.536 q^{84} -85.0000 q^{85} -307.245 q^{86} +31.4396 q^{87} +1265.00 q^{88} -72.8069 q^{89} +156.742 q^{90} -416.803 q^{91} +46.7155 q^{92} +18.0771 q^{93} +70.7499 q^{94} -429.796 q^{95} -207.124 q^{96} +972.939 q^{97} -155.964 q^{98} +1831.47 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 6 q^{2} - 4 q^{3} + 10 q^{4} - 15 q^{5} + 36 q^{6} + 8 q^{7} - 66 q^{8} + 39 q^{9} + 30 q^{10} - 118 q^{11} - 212 q^{12} - 10 q^{13} - 68 q^{14} + 20 q^{15} + 242 q^{16} + 51 q^{17} - 342 q^{18} - 160 q^{19}+ \cdots + 770 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.22168 0.431928 0.215964 − 0.976401i \(-0.430711\pi\)
0.215964 + 0.976401i \(0.430711\pi\)
\(3\) −1.15753 −0.222766 −0.111383 − 0.993778i \(-0.535528\pi\)
−0.111383 + 0.993778i \(0.535528\pi\)
\(4\) −6.50750 −0.813438
\(5\) −5.00000 −0.447214
\(6\) −1.41413 −0.0962191
\(7\) 14.6743 0.792340 0.396170 − 0.918177i \(-0.370339\pi\)
0.396170 + 0.918177i \(0.370339\pi\)
\(8\) −17.7235 −0.783275
\(9\) −25.6601 −0.950375
\(10\) −6.10839 −0.193164
\(11\) −71.3743 −1.95638 −0.978189 − 0.207716i \(-0.933397\pi\)
−0.978189 + 0.207716i \(0.933397\pi\)
\(12\) 7.53262 0.181207
\(13\) −28.4035 −0.605979 −0.302989 − 0.952994i \(-0.597985\pi\)
−0.302989 + 0.952994i \(0.597985\pi\)
\(14\) 17.9273 0.342234
\(15\) 5.78764 0.0996241
\(16\) 30.4076 0.475119
\(17\) 17.0000 0.242536
\(18\) −31.3484 −0.410494
\(19\) 85.9592 1.03792 0.518958 − 0.854800i \(-0.326320\pi\)
0.518958 + 0.854800i \(0.326320\pi\)
\(20\) 32.5375 0.363781
\(21\) −16.9860 −0.176507
\(22\) −87.1964 −0.845015
\(23\) −7.17872 −0.0650811 −0.0325406 − 0.999470i \(-0.510360\pi\)
−0.0325406 + 0.999470i \(0.510360\pi\)
\(24\) 20.5154 0.174487
\(25\) 25.0000 0.200000
\(26\) −34.7000 −0.261739
\(27\) 60.9556 0.434478
\(28\) −95.4934 −0.644520
\(29\) −27.1610 −0.173919 −0.0869597 − 0.996212i \(-0.527715\pi\)
−0.0869597 + 0.996212i \(0.527715\pi\)
\(30\) 7.07063 0.0430305
\(31\) −15.6170 −0.0904803 −0.0452401 − 0.998976i \(-0.514405\pi\)
−0.0452401 + 0.998976i \(0.514405\pi\)
\(32\) 178.936 0.988493
\(33\) 82.6178 0.435815
\(34\) 20.7685 0.104758
\(35\) −73.3717 −0.354345
\(36\) 166.983 0.773071
\(37\) −67.0036 −0.297712 −0.148856 − 0.988859i \(-0.547559\pi\)
−0.148856 + 0.988859i \(0.547559\pi\)
\(38\) 105.014 0.448305
\(39\) 32.8779 0.134992
\(40\) 88.6175 0.350291
\(41\) −38.5701 −0.146918 −0.0734590 − 0.997298i \(-0.523404\pi\)
−0.0734590 + 0.997298i \(0.523404\pi\)
\(42\) −20.7514 −0.0762383
\(43\) −251.495 −0.891920 −0.445960 − 0.895053i \(-0.647138\pi\)
−0.445960 + 0.895053i \(0.647138\pi\)
\(44\) 464.469 1.59139
\(45\) 128.301 0.425021
\(46\) −8.77008 −0.0281104
\(47\) 57.9121 0.179731 0.0898654 − 0.995954i \(-0.471356\pi\)
0.0898654 + 0.995954i \(0.471356\pi\)
\(48\) −35.1977 −0.105841
\(49\) −127.663 −0.372197
\(50\) 30.5419 0.0863856
\(51\) −19.6780 −0.0540288
\(52\) 184.836 0.492926
\(53\) −677.807 −1.75668 −0.878339 − 0.478039i \(-0.841348\pi\)
−0.878339 + 0.478039i \(0.841348\pi\)
\(54\) 74.4680 0.187663
\(55\) 356.872 0.874919
\(56\) −260.081 −0.620620
\(57\) −99.5002 −0.231213
\(58\) −33.1819 −0.0751207
\(59\) 598.645 1.32097 0.660483 − 0.750841i \(-0.270352\pi\)
0.660483 + 0.750841i \(0.270352\pi\)
\(60\) −37.6631 −0.0810381
\(61\) 346.063 0.726375 0.363187 − 0.931716i \(-0.381689\pi\)
0.363187 + 0.931716i \(0.381689\pi\)
\(62\) −19.0789 −0.0390810
\(63\) −376.546 −0.753021
\(64\) −24.6588 −0.0481617
\(65\) 142.018 0.271002
\(66\) 100.932 0.188241
\(67\) −849.923 −1.54977 −0.774885 − 0.632102i \(-0.782192\pi\)
−0.774885 + 0.632102i \(0.782192\pi\)
\(68\) −110.628 −0.197288
\(69\) 8.30957 0.0144979
\(70\) −89.6366 −0.153052
\(71\) −911.836 −1.52415 −0.762077 − 0.647486i \(-0.775820\pi\)
−0.762077 + 0.647486i \(0.775820\pi\)
\(72\) 454.787 0.744405
\(73\) 704.352 1.12929 0.564645 − 0.825334i \(-0.309013\pi\)
0.564645 + 0.825334i \(0.309013\pi\)
\(74\) −81.8568 −0.128590
\(75\) −28.9382 −0.0445533
\(76\) −559.380 −0.844280
\(77\) −1047.37 −1.55012
\(78\) 40.1662 0.0583067
\(79\) 55.8414 0.0795272 0.0397636 − 0.999209i \(-0.487340\pi\)
0.0397636 + 0.999209i \(0.487340\pi\)
\(80\) −152.038 −0.212480
\(81\) 622.266 0.853588
\(82\) −47.1202 −0.0634580
\(83\) −1235.19 −1.63350 −0.816748 − 0.576995i \(-0.804225\pi\)
−0.816748 + 0.576995i \(0.804225\pi\)
\(84\) 110.536 0.143577
\(85\) −85.0000 −0.108465
\(86\) −307.245 −0.385246
\(87\) 31.4396 0.0387434
\(88\) 1265.00 1.53238
\(89\) −72.8069 −0.0867136 −0.0433568 − 0.999060i \(-0.513805\pi\)
−0.0433568 + 0.999060i \(0.513805\pi\)
\(90\) 156.742 0.183578
\(91\) −416.803 −0.480141
\(92\) 46.7155 0.0529395
\(93\) 18.0771 0.0201560
\(94\) 70.7499 0.0776308
\(95\) −429.796 −0.464170
\(96\) −207.124 −0.220203
\(97\) 972.939 1.01842 0.509211 − 0.860642i \(-0.329937\pi\)
0.509211 + 0.860642i \(0.329937\pi\)
\(98\) −155.964 −0.160762
\(99\) 1831.47 1.85929
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.1.3 ✓ 3
3.2 odd 2 765.4.a.l.1.1 3
4.3 odd 2 1360.4.a.s.1.2 3
5.2 odd 4 425.4.b.g.324.4 6
5.3 odd 4 425.4.b.g.324.3 6
5.4 even 2 425.4.a.h.1.1 3
17.16 even 2 1445.4.a.j.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.e.1.3 ✓ 3 1.1 even 1 trivial
425.4.a.h.1.1 3 5.4 even 2
425.4.b.g.324.3 6 5.3 odd 4
425.4.b.g.324.4 6 5.2 odd 4
765.4.a.l.1.1 3 3.2 odd 2
1360.4.a.s.1.2 3 4.3 odd 2
1445.4.a.j.1.3 3 17.16 even 2