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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1665,2,Mod(334,1665)] 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("1665.334"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1665, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1665 = 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1665.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [26,0,0,-32,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.2950919365\)
Analytic rank: \(0\)
Dimension: \(26\)
Twist minimal: no (minimal twist has level 555)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 334.12
Character \(\chi\) \(=\) 1665.334
Dual form 1665.2.c.f.334.15

$q$-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)
 
\(f(q)\) \(=\) \(q-0.214651i q^{2} +1.95393 q^{4} +(2.20564 - 0.367653i) q^{5} -4.41925i q^{7} -0.848712i q^{8} +(-0.0789169 - 0.473441i) q^{10} -5.57516 q^{11} +0.780716i q^{13} -0.948594 q^{14} +3.72567 q^{16} +4.10713i q^{17} +4.94347 q^{19} +(4.30965 - 0.718366i) q^{20} +1.19671i q^{22} -7.70902i q^{23} +(4.72966 - 1.62182i) q^{25} +0.167581 q^{26} -8.63488i q^{28} -1.53583 q^{29} -2.52604 q^{31} -2.49714i q^{32} +0.881597 q^{34} +(-1.62475 - 9.74725i) q^{35} -1.00000i q^{37} -1.06112i q^{38} +(-0.312032 - 1.87195i) q^{40} +11.2120 q^{41} -2.78517i q^{43} -10.8934 q^{44} -1.65475 q^{46} -2.53935i q^{47} -12.5297 q^{49} +(-0.348124 - 1.01522i) q^{50} +1.52546i q^{52} -1.49186i q^{53} +(-12.2968 + 2.04972i) q^{55} -3.75067 q^{56} +0.329667i q^{58} -3.84326 q^{59} +5.82961 q^{61} +0.542216i q^{62} +6.91533 q^{64} +(0.287033 + 1.72198i) q^{65} -1.94661i q^{67} +8.02502i q^{68} +(-2.09225 + 0.348753i) q^{70} -9.36234 q^{71} -11.9612i q^{73} -0.214651 q^{74} +9.65917 q^{76} +24.6380i q^{77} -8.12132 q^{79} +(8.21748 - 1.36975i) q^{80} -2.40667i q^{82} +15.7827i q^{83} +(1.51000 + 9.05883i) q^{85} -0.597839 q^{86} +4.73170i q^{88} +0.343059 q^{89} +3.45018 q^{91} -15.0629i q^{92} -0.545073 q^{94} +(10.9035 - 1.81748i) q^{95} +12.1360i q^{97} +2.68952i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 26 q - 32 q^{4} + 2 q^{5} - 20 q^{11} + 16 q^{14} + 44 q^{16} - 28 q^{19} - 4 q^{20} + 10 q^{25} - 32 q^{26} + 46 q^{29} + 32 q^{31} + 20 q^{34} + 12 q^{35} - 36 q^{40} - 56 q^{41} + 20 q^{44} + 36 q^{46}+ \cdots + 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1665\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.214651i − 0.151781i −0.997116 − 0.0758904i \(-0.975820\pi\)
0.997116 − 0.0758904i \(-0.0241799\pi\)
\(3\) 0 0
\(4\) 1.95393 0.976963
\(5\) 2.20564 − 0.367653i 0.986391 − 0.164419i
\(6\) 0 0
\(7\) − 4.41925i − 1.67032i −0.550008 − 0.835159i \(-0.685375\pi\)
0.550008 − 0.835159i \(-0.314625\pi\)
\(8\) − 0.848712i − 0.300065i
\(9\) 0 0
\(10\) −0.0789169 − 0.473441i −0.0249557 − 0.149715i
\(11\) −5.57516 −1.68097 −0.840486 − 0.541833i \(-0.817731\pi\)
−0.840486 + 0.541833i \(0.817731\pi\)
\(12\) 0 0
\(13\) 0.780716i 0.216532i 0.994122 + 0.108266i \(0.0345298\pi\)
−0.994122 + 0.108266i \(0.965470\pi\)
\(14\) −0.948594 −0.253522
\(15\) 0 0
\(16\) 3.72567 0.931418
\(17\) 4.10713i 0.996125i 0.867141 + 0.498062i \(0.165955\pi\)
−0.867141 + 0.498062i \(0.834045\pi\)
\(18\) 0 0
\(19\) 4.94347 1.13411 0.567055 − 0.823680i \(-0.308083\pi\)
0.567055 + 0.823680i \(0.308083\pi\)
\(20\) 4.30965 − 0.718366i 0.963667 − 0.160632i
\(21\) 0 0
\(22\) 1.19671i 0.255140i
\(23\) − 7.70902i − 1.60744i −0.595006 − 0.803721i \(-0.702850\pi\)
0.595006 − 0.803721i \(-0.297150\pi\)
\(24\) 0 0
\(25\) 4.72966 − 1.62182i 0.945933 − 0.324363i
\(26\) 0.167581 0.0328654
\(27\) 0 0
\(28\) − 8.63488i − 1.63184i
\(29\) −1.53583 −0.285197 −0.142598 − 0.989781i \(-0.545546\pi\)
−0.142598 + 0.989781i \(0.545546\pi\)
\(30\) 0 0
\(31\) −2.52604 −0.453690 −0.226845 − 0.973931i \(-0.572841\pi\)
−0.226845 + 0.973931i \(0.572841\pi\)
\(32\) − 2.49714i − 0.441437i
\(33\) 0 0
\(34\) 0.881597 0.151193
\(35\) −1.62475 − 9.74725i −0.274633 − 1.64759i
\(36\) 0 0
\(37\) − 1.00000i − 0.164399i
\(38\) − 1.06112i − 0.172136i
\(39\) 0 0
\(40\) −0.312032 − 1.87195i −0.0493365 − 0.295981i
\(41\) 11.2120 1.75102 0.875512 − 0.483197i \(-0.160524\pi\)
0.875512 + 0.483197i \(0.160524\pi\)
\(42\) 0 0
\(43\) − 2.78517i − 0.424735i −0.977190 − 0.212367i \(-0.931883\pi\)
0.977190 − 0.212367i \(-0.0681174\pi\)
\(44\) −10.8934 −1.64225
\(45\) 0 0
\(46\) −1.65475 −0.243979
\(47\) − 2.53935i − 0.370402i −0.982701 − 0.185201i \(-0.940706\pi\)
0.982701 − 0.185201i \(-0.0592936\pi\)
\(48\) 0 0
\(49\) −12.5297 −1.78996
\(50\) −0.348124 − 1.01522i −0.0492322 − 0.143574i
\(51\) 0 0
\(52\) 1.52546i 0.211543i
\(53\) − 1.49186i − 0.204922i −0.994737 − 0.102461i \(-0.967328\pi\)
0.994737 − 0.102461i \(-0.0326717\pi\)
\(54\) 0 0
\(55\) −12.2968 + 2.04972i −1.65810 + 0.276385i
\(56\) −3.75067 −0.501204
\(57\) 0 0
\(58\) 0.329667i 0.0432874i
\(59\) −3.84326 −0.500349 −0.250175 − 0.968201i \(-0.580488\pi\)
−0.250175 + 0.968201i \(0.580488\pi\)
\(60\) 0 0
\(61\) 5.82961 0.746405 0.373202 − 0.927750i \(-0.378260\pi\)
0.373202 + 0.927750i \(0.378260\pi\)
\(62\) 0.542216i 0.0688615i
\(63\) 0 0
\(64\) 6.91533 0.864417
\(65\) 0.287033 + 1.72198i 0.0356020 + 0.213585i
\(66\) 0 0
\(67\) − 1.94661i − 0.237816i −0.992905 − 0.118908i \(-0.962061\pi\)
0.992905 − 0.118908i \(-0.0379394\pi\)
\(68\) 8.02502i 0.973176i
\(69\) 0 0
\(70\) −2.09225 + 0.348753i −0.250072 + 0.0416840i
\(71\) −9.36234 −1.11110 −0.555552 − 0.831481i \(-0.687493\pi\)
−0.555552 + 0.831481i \(0.687493\pi\)
\(72\) 0 0
\(73\) − 11.9612i − 1.39995i −0.714168 − 0.699975i \(-0.753195\pi\)
0.714168 − 0.699975i \(-0.246805\pi\)
\(74\) −0.214651 −0.0249526
\(75\) 0 0
\(76\) 9.65917 1.10798
\(77\) 24.6380i 2.80776i
\(78\) 0 0
\(79\) −8.12132 −0.913720 −0.456860 − 0.889539i \(-0.651026\pi\)
−0.456860 + 0.889539i \(0.651026\pi\)
\(80\) 8.21748 − 1.36975i 0.918742 − 0.153143i
\(81\) 0 0
\(82\) − 2.40667i − 0.265772i
\(83\) 15.7827i 1.73238i 0.499718 + 0.866188i \(0.333437\pi\)
−0.499718 + 0.866188i \(0.666563\pi\)
\(84\) 0 0
\(85\) 1.51000 + 9.05883i 0.163782 + 0.982568i
\(86\) −0.597839 −0.0644666
\(87\) 0 0
\(88\) 4.73170i 0.504401i
\(89\) 0.343059 0.0363642 0.0181821 − 0.999835i \(-0.494212\pi\)
0.0181821 + 0.999835i \(0.494212\pi\)
\(90\) 0 0
\(91\) 3.45018 0.361677
\(92\) − 15.0629i − 1.57041i
\(93\) 0 0
\(94\) −0.545073 −0.0562199
\(95\) 10.9035 − 1.81748i 1.11868 − 0.186470i
\(96\) 0 0
\(97\) 12.1360i 1.23222i 0.787659 + 0.616111i \(0.211293\pi\)
−0.787659 + 0.616111i \(0.788707\pi\)
\(98\) 2.68952i 0.271682i
\(99\) 0 0
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 1665.2.c.f.334.12 26
3.2 odd 2 555.2.c.c.334.15 yes 26
5.2 odd 4 8325.2.a.cv.1.8 13
5.3 odd 4 8325.2.a.cu.1.6 13
5.4 even 2 inner 1665.2.c.f.334.15 26
15.2 even 4 2775.2.a.bh.1.6 13
15.8 even 4 2775.2.a.bg.1.8 13
15.14 odd 2 555.2.c.c.334.12 ✓ 26
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
555.2.c.c.334.12 ✓ 26 15.14 odd 2
555.2.c.c.334.15 yes 26 3.2 odd 2
1665.2.c.f.334.12 26 1.1 even 1 trivial
1665.2.c.f.334.15 26 5.4 even 2 inner
2775.2.a.bg.1.8 13 15.8 even 4
2775.2.a.bh.1.6 13 15.2 even 4
8325.2.a.cu.1.6 13 5.3 odd 4
8325.2.a.cv.1.8 13 5.2 odd 4