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(16,85)] 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("85.16"); 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, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 85 = 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 85.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.01516235049\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 16.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 85.16
Dual form 85.4.d.a.16.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +8.00000i q^{3} -7.00000 q^{4} +5.00000i q^{5} +8.00000i q^{6} -14.0000i q^{7} -15.0000 q^{8} -37.0000 q^{9} +5.00000i q^{10} +20.0000i q^{11} -56.0000i q^{12} -58.0000 q^{13} -14.0000i q^{14} -40.0000 q^{15} +41.0000 q^{16} +(-17.0000 + 68.0000i) q^{17} -37.0000 q^{18} +80.0000 q^{19} -35.0000i q^{20} +112.000 q^{21} +20.0000i q^{22} +118.000i q^{23} -120.000i q^{24} -25.0000 q^{25} -58.0000 q^{26} -80.0000i q^{27} +98.0000i q^{28} +126.000i q^{29} -40.0000 q^{30} +70.0000i q^{31} +161.000 q^{32} -160.000 q^{33} +(-17.0000 + 68.0000i) q^{34} +70.0000 q^{35} +259.000 q^{36} -134.000i q^{37} +80.0000 q^{38} -464.000i q^{39} -75.0000i q^{40} -100.000i q^{41} +112.000 q^{42} +272.000 q^{43} -140.000i q^{44} -185.000i q^{45} +118.000i q^{46} -464.000 q^{47} +328.000i q^{48} +147.000 q^{49} -25.0000 q^{50} +(-544.000 - 136.000i) q^{51} +406.000 q^{52} +642.000 q^{53} -80.0000i q^{54} -100.000 q^{55} +210.000i q^{56} +640.000i q^{57} +126.000i q^{58} -180.000 q^{59} +280.000 q^{60} +110.000i q^{61} +70.0000i q^{62} +518.000i q^{63} -167.000 q^{64} -290.000i q^{65} -160.000 q^{66} -924.000 q^{67} +(119.000 - 476.000i) q^{68} -944.000 q^{69} +70.0000 q^{70} +90.0000i q^{71} +555.000 q^{72} +828.000i q^{73} -134.000i q^{74} -200.000i q^{75} -560.000 q^{76} +280.000 q^{77} -464.000i q^{78} -1334.00i q^{79} +205.000i q^{80} -359.000 q^{81} -100.000i q^{82} +552.000 q^{83} -784.000 q^{84} +(-340.000 - 85.0000i) q^{85} +272.000 q^{86} -1008.00 q^{87} -300.000i q^{88} +1490.00 q^{89} -185.000i q^{90} +812.000i q^{91} -826.000i q^{92} -560.000 q^{93} -464.000 q^{94} +400.000i q^{95} +1288.00i q^{96} +1376.00i q^{97} +147.000 q^{98} -740.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 14 q^{4} - 30 q^{8} - 74 q^{9} - 116 q^{13} - 80 q^{15} + 82 q^{16} - 34 q^{17} - 74 q^{18} + 160 q^{19} + 224 q^{21} - 50 q^{25} - 116 q^{26} - 80 q^{30} + 322 q^{32} - 320 q^{33} - 34 q^{34}+ \cdots + 294 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/85\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(71\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.353553 0.176777 − 0.984251i \(-0.443433\pi\)
0.176777 + 0.984251i \(0.443433\pi\)
\(3\) 8.00000i 1.53960i 0.638285 + 0.769800i \(0.279644\pi\)
−0.638285 + 0.769800i \(0.720356\pi\)
\(4\) −7.00000 −0.875000
\(5\) 5.00000i 0.447214i
\(6\) 8.00000i 0.544331i
\(7\) − 14.0000i − 0.755929i −0.925820 − 0.377964i \(-0.876624\pi\)
0.925820 − 0.377964i \(-0.123376\pi\)
\(8\) −15.0000 −0.662913
\(9\) −37.0000 −1.37037
\(10\) 5.00000i 0.158114i
\(11\) 20.0000i 0.548202i 0.961701 + 0.274101i \(0.0883803\pi\)
−0.961701 + 0.274101i \(0.911620\pi\)
\(12\) − 56.0000i − 1.34715i
\(13\) −58.0000 −1.23741 −0.618704 − 0.785624i \(-0.712342\pi\)
−0.618704 + 0.785624i \(0.712342\pi\)
\(14\) − 14.0000i − 0.267261i
\(15\) −40.0000 −0.688530
\(16\) 41.0000 0.640625
\(17\) −17.0000 + 68.0000i −0.242536 + 0.970143i
\(18\) −37.0000 −0.484499
\(19\) 80.0000 0.965961 0.482980 − 0.875631i \(-0.339554\pi\)
0.482980 + 0.875631i \(0.339554\pi\)
\(20\) − 35.0000i − 0.391312i
\(21\) 112.000 1.16383
\(22\) 20.0000i 0.193819i
\(23\) 118.000i 1.06977i 0.844925 + 0.534885i \(0.179645\pi\)
−0.844925 + 0.534885i \(0.820355\pi\)
\(24\) − 120.000i − 1.02062i
\(25\) −25.0000 −0.200000
\(26\) −58.0000 −0.437490
\(27\) − 80.0000i − 0.570222i
\(28\) 98.0000i 0.661438i
\(29\) 126.000i 0.806814i 0.915021 + 0.403407i \(0.132174\pi\)
−0.915021 + 0.403407i \(0.867826\pi\)
\(30\) −40.0000 −0.243432
\(31\) 70.0000i 0.405560i 0.979224 + 0.202780i \(0.0649977\pi\)
−0.979224 + 0.202780i \(0.935002\pi\)
\(32\) 161.000 0.889408
\(33\) −160.000 −0.844013
\(34\) −17.0000 + 68.0000i −0.0857493 + 0.342997i
\(35\) 70.0000 0.338062
\(36\) 259.000 1.19907
\(37\) − 134.000i − 0.595391i −0.954661 − 0.297695i \(-0.903782\pi\)
0.954661 − 0.297695i \(-0.0962180\pi\)
\(38\) 80.0000 0.341519
\(39\) − 464.000i − 1.90511i
\(40\) − 75.0000i − 0.296464i
\(41\) − 100.000i − 0.380912i −0.981696 − 0.190456i \(-0.939003\pi\)
0.981696 − 0.190456i \(-0.0609966\pi\)
\(42\) 112.000 0.411476
\(43\) 272.000 0.964642 0.482321 − 0.875995i \(-0.339794\pi\)
0.482321 + 0.875995i \(0.339794\pi\)
\(44\) − 140.000i − 0.479677i
\(45\) − 185.000i − 0.612848i
\(46\) 118.000i 0.378221i
\(47\) −464.000 −1.44003 −0.720014 − 0.693959i \(-0.755865\pi\)
−0.720014 + 0.693959i \(0.755865\pi\)
\(48\) 328.000i 0.986307i
\(49\) 147.000 0.428571
\(50\) −25.0000 −0.0707107
\(51\) −544.000 − 136.000i −1.49363 − 0.373408i
\(52\) 406.000 1.08273
\(53\) 642.000 1.66388 0.831939 − 0.554868i \(-0.187231\pi\)
0.831939 + 0.554868i \(0.187231\pi\)
\(54\) − 80.0000i − 0.201604i
\(55\) −100.000 −0.245164
\(56\) 210.000i 0.501115i
\(57\) 640.000i 1.48719i
\(58\) 126.000i 0.285252i
\(59\) −180.000 −0.397187 −0.198593 − 0.980082i \(-0.563637\pi\)
−0.198593 + 0.980082i \(0.563637\pi\)
\(60\) 280.000 0.602464
\(61\) 110.000i 0.230886i 0.993314 + 0.115443i \(0.0368288\pi\)
−0.993314 + 0.115443i \(0.963171\pi\)
\(62\) 70.0000i 0.143387i
\(63\) 518.000i 1.03590i
\(64\) −167.000 −0.326172
\(65\) − 290.000i − 0.553386i
\(66\) −160.000 −0.298404
\(67\) −924.000 −1.68484 −0.842422 − 0.538818i \(-0.818871\pi\)
−0.842422 + 0.538818i \(0.818871\pi\)
\(68\) 119.000 − 476.000i 0.212219 − 0.848875i
\(69\) −944.000 −1.64702
\(70\) 70.0000 0.119523
\(71\) 90.0000i 0.150437i 0.997167 + 0.0752186i \(0.0239654\pi\)
−0.997167 + 0.0752186i \(0.976035\pi\)
\(72\) 555.000 0.908436
\(73\) 828.000i 1.32754i 0.747939 + 0.663768i \(0.231044\pi\)
−0.747939 + 0.663768i \(0.768956\pi\)
\(74\) − 134.000i − 0.210502i
\(75\) − 200.000i − 0.307920i
\(76\) −560.000 −0.845216
\(77\) 280.000 0.414402
\(78\) − 464.000i − 0.673560i
\(79\) − 1334.00i − 1.89983i −0.312505 − 0.949916i \(-0.601168\pi\)
0.312505 − 0.949916i \(-0.398832\pi\)
\(80\) 205.000i 0.286496i
\(81\) −359.000 −0.492455
\(82\) − 100.000i − 0.134673i
\(83\) 552.000 0.729998 0.364999 − 0.931008i \(-0.381069\pi\)
0.364999 + 0.931008i \(0.381069\pi\)
\(84\) −784.000 −1.01835
\(85\) −340.000 − 85.0000i −0.433861 − 0.108465i
\(86\) 272.000 0.341052
\(87\) −1008.00 −1.24217
\(88\) − 300.000i − 0.363410i
\(89\) 1490.00 1.77460 0.887302 − 0.461190i \(-0.152577\pi\)
0.887302 + 0.461190i \(0.152577\pi\)
\(90\) − 185.000i − 0.216675i
\(91\) 812.000i 0.935393i
\(92\) − 826.000i − 0.936048i
\(93\) −560.000 −0.624401
\(94\) −464.000 −0.509127
\(95\) 400.000i 0.431991i
\(96\) 1288.00i 1.36933i
\(97\) 1376.00i 1.44033i 0.693805 + 0.720163i \(0.255933\pi\)
−0.693805 + 0.720163i \(0.744067\pi\)
\(98\) 147.000 0.151523
\(99\) − 740.000i − 0.751240i
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.d.a.16.2 yes 2
3.2 odd 2 765.4.g.a.271.1 2
5.2 odd 4 425.4.c.b.424.2 2
5.3 odd 4 425.4.c.a.424.1 2
5.4 even 2 425.4.d.a.101.1 2
17.4 even 4 1445.4.a.e.1.1 1
17.13 even 4 1445.4.a.d.1.1 1
17.16 even 2 inner 85.4.d.a.16.1 ✓ 2
51.50 odd 2 765.4.g.a.271.2 2
85.33 odd 4 425.4.c.b.424.1 2
85.67 odd 4 425.4.c.a.424.2 2
85.84 even 2 425.4.d.a.101.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.d.a.16.1 ✓ 2 17.16 even 2 inner
85.4.d.a.16.2 yes 2 1.1 even 1 trivial
425.4.c.a.424.1 2 5.3 odd 4
425.4.c.a.424.2 2 85.67 odd 4
425.4.c.b.424.1 2 85.33 odd 4
425.4.c.b.424.2 2 5.2 odd 4
425.4.d.a.101.1 2 5.4 even 2
425.4.d.a.101.2 2 85.84 even 2
765.4.g.a.271.1 2 3.2 odd 2
765.4.g.a.271.2 2 51.50 odd 2
1445.4.a.d.1.1 1 17.13 even 4
1445.4.a.e.1.1 1 17.4 even 4