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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [465,2,Mod(119,465)] 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("465.119"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(465, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [104,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(104\)
Relative dimension: \(52\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 254.43
Character \(\chi\) \(=\) 465.254
Dual form 465.2.t.d.119.43

$q$-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+1.83398 q^{2} +(0.944441 - 1.45191i) q^{3} +1.36348 q^{4} +(2.09097 - 0.792377i) q^{5} +(1.73209 - 2.66276i) q^{6} +(-0.777962 + 0.449157i) q^{7} -1.16737 q^{8} +(-1.21606 - 2.74248i) q^{9} +(3.83479 - 1.45320i) q^{10} +(2.05191 - 3.55401i) q^{11} +(1.28772 - 1.97964i) q^{12} +(-2.60516 + 4.51228i) q^{13} +(-1.42677 + 0.823744i) q^{14} +(0.824337 - 3.78424i) q^{15} -4.86788 q^{16} +(-0.297429 + 0.171721i) q^{17} +(-2.23023 - 5.02965i) q^{18} +(3.30496 + 5.72436i) q^{19} +(2.85099 - 1.08039i) q^{20} +(-0.0826063 + 1.55373i) q^{21} +(3.76316 - 6.51798i) q^{22} +7.30236i q^{23} +(-1.10251 + 1.69491i) q^{24} +(3.74428 - 3.31367i) q^{25} +(-4.77782 + 8.27542i) q^{26} +(-5.13032 - 0.824504i) q^{27} +(-1.06073 + 0.612415i) q^{28} +5.80166 q^{29} +(1.51182 - 6.94021i) q^{30} +(-0.597823 - 5.53558i) q^{31} -6.59286 q^{32} +(-3.22218 - 6.33574i) q^{33} +(-0.545479 + 0.314932i) q^{34} +(-1.27079 + 1.55561i) q^{35} +(-1.65807 - 3.73931i) q^{36} +(2.52152 + 4.36741i) q^{37} +(6.06122 + 10.4983i) q^{38} +(4.09098 + 8.04404i) q^{39} +(-2.44093 + 0.924996i) q^{40} +(2.21605 + 1.27944i) q^{41} +(-0.151498 + 2.84951i) q^{42} +(-6.48540 - 11.2330i) q^{43} +(2.79773 - 4.84582i) q^{44} +(-4.71582 - 4.77085i) q^{45} +13.3924i q^{46} -4.20502 q^{47} +(-4.59743 + 7.06771i) q^{48} +(-3.09652 + 5.36332i) q^{49} +(6.86692 - 6.07720i) q^{50} +(-0.0315819 + 0.594019i) q^{51} +(-3.55208 + 6.15239i) q^{52} +(-0.0480085 - 0.0277177i) q^{53} +(-9.40890 - 1.51212i) q^{54} +(1.47436 - 9.05721i) q^{55} +(0.908168 - 0.524331i) q^{56} +(11.4326 + 0.607829i) q^{57} +10.6401 q^{58} +(-4.72388 + 2.72734i) q^{59} +(1.12397 - 5.15973i) q^{60} -10.5816i q^{61} +(-1.09640 - 10.1521i) q^{62} +(2.17785 + 1.58734i) q^{63} -2.35539 q^{64} +(-1.87188 + 11.4993i) q^{65} +(-5.90941 - 11.6196i) q^{66} +(-1.94833 - 1.12487i) q^{67} +(-0.405538 + 0.234138i) q^{68} +(10.6023 + 6.89665i) q^{69} +(-2.33060 + 2.85296i) q^{70} +(9.60267 + 5.54410i) q^{71} +(1.41959 + 3.20148i) q^{72} +(-1.89237 + 3.27769i) q^{73} +(4.62442 + 8.00973i) q^{74} +(-1.27488 - 8.56590i) q^{75} +(4.50624 + 7.80503i) q^{76} +3.68652i q^{77} +(7.50277 + 14.7526i) q^{78} +(-3.85293 + 2.22449i) q^{79} +(-10.1786 + 3.85720i) q^{80} +(-6.04239 + 6.67005i) q^{81} +(4.06420 + 2.34646i) q^{82} +(3.00738 + 1.73631i) q^{83} +(-0.112632 + 2.11848i) q^{84} +(-0.485847 + 0.594739i) q^{85} +(-11.8941 - 20.6011i) q^{86} +(5.47933 - 8.42346i) q^{87} +(-2.39533 + 4.14884i) q^{88} -16.1029 q^{89} +(-8.64872 - 8.74964i) q^{90} -4.68051i q^{91} +9.95660i q^{92} +(-8.60175 - 4.36004i) q^{93} -7.71192 q^{94} +(11.4464 + 9.35066i) q^{95} +(-6.22657 + 9.57221i) q^{96} +7.72813i q^{97} +(-5.67895 + 9.83622i) q^{98} +(-12.2421 - 1.30542i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 104 q + 56 q^{4} - 12 q^{6} - 2 q^{9} - 20 q^{10} - 24 q^{16} + 12 q^{19} - 6 q^{21} - 48 q^{24} + 8 q^{25} - 40 q^{31} + 108 q^{34} + 36 q^{36} + 52 q^{39} - 76 q^{40} - 4 q^{45} + 80 q^{49} - 20 q^{51}+ \cdots + 66 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(406\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.83398 1.29682 0.648409 − 0.761292i \(-0.275435\pi\)
0.648409 + 0.761292i \(0.275435\pi\)
\(3\) 0.944441 − 1.45191i 0.545273 − 0.838258i
\(4\) 1.36348 0.681739
\(5\) 2.09097 − 0.792377i 0.935108 − 0.354362i
\(6\) 1.73209 − 2.66276i 0.707121 − 1.08707i
\(7\) −0.777962 + 0.449157i −0.294042 + 0.169765i −0.639763 − 0.768572i \(-0.720968\pi\)
0.345721 + 0.938337i \(0.387634\pi\)
\(8\) −1.16737 −0.412727
\(9\) −1.21606 − 2.74248i −0.405354 − 0.914160i
\(10\) 3.83479 − 1.45320i 1.21267 − 0.459543i
\(11\) 2.05191 − 3.55401i 0.618674 − 1.07158i −0.371054 − 0.928611i \(-0.621003\pi\)
0.989728 − 0.142964i \(-0.0456632\pi\)
\(12\) 1.28772 − 1.97964i 0.371734 − 0.571473i
\(13\) −2.60516 + 4.51228i −0.722543 + 1.25148i 0.237435 + 0.971403i \(0.423693\pi\)
−0.959977 + 0.280077i \(0.909640\pi\)
\(14\) −1.42677 + 0.823744i −0.381319 + 0.220155i
\(15\) 0.824337 − 3.78424i 0.212843 − 0.977086i
\(16\) −4.86788 −1.21697
\(17\) −0.297429 + 0.171721i −0.0721372 + 0.0416484i −0.535635 − 0.844450i \(-0.679928\pi\)
0.463498 + 0.886098i \(0.346594\pi\)
\(18\) −2.23023 − 5.02965i −0.525670 − 1.18550i
\(19\) 3.30496 + 5.72436i 0.758209 + 1.31326i 0.943763 + 0.330623i \(0.107259\pi\)
−0.185553 + 0.982634i \(0.559408\pi\)
\(20\) 2.85099 − 1.08039i 0.637500 − 0.241582i
\(21\) −0.0826063 + 1.55373i −0.0180262 + 0.339052i
\(22\) 3.76316 − 6.51798i 0.802308 − 1.38964i
\(23\) 7.30236i 1.52265i 0.648372 + 0.761323i \(0.275450\pi\)
−0.648372 + 0.761323i \(0.724550\pi\)
\(24\) −1.10251 + 1.69491i −0.225049 + 0.345972i
\(25\) 3.74428 − 3.31367i 0.748855 − 0.662734i
\(26\) −4.77782 + 8.27542i −0.937007 + 1.62294i
\(27\) −5.13032 − 0.824504i −0.987331 − 0.158676i
\(28\) −1.06073 + 0.612415i −0.200460 + 0.115736i
\(29\) 5.80166 1.07734 0.538671 − 0.842516i \(-0.318927\pi\)
0.538671 + 0.842516i \(0.318927\pi\)
\(30\) 1.51182 − 6.94021i 0.276019 − 1.26710i
\(31\) −0.597823 − 5.53558i −0.107372 − 0.994219i
\(32\) −6.59286 −1.16546
\(33\) −3.22218 − 6.33574i −0.560910 − 1.10291i
\(34\) −0.545479 + 0.314932i −0.0935488 + 0.0540105i
\(35\) −1.27079 + 1.55561i −0.214803 + 0.262946i
\(36\) −1.65807 − 3.73931i −0.276345 − 0.623218i
\(37\) 2.52152 + 4.36741i 0.414536 + 0.717997i 0.995380 − 0.0960177i \(-0.0306106\pi\)
−0.580844 + 0.814015i \(0.697277\pi\)
\(38\) 6.06122 + 10.4983i 0.983260 + 1.70306i
\(39\) 4.09098 + 8.04404i 0.655081 + 1.28808i
\(40\) −2.44093 + 0.924996i −0.385945 + 0.146255i
\(41\) 2.21605 + 1.27944i 0.346089 + 0.199815i 0.662962 − 0.748653i \(-0.269299\pi\)
−0.316872 + 0.948468i \(0.602633\pi\)
\(42\) −0.151498 + 2.84951i −0.0233767 + 0.439689i
\(43\) −6.48540 − 11.2330i −0.989014 − 1.71302i −0.622527 − 0.782598i \(-0.713894\pi\)
−0.366486 − 0.930423i \(-0.619439\pi\)
\(44\) 2.79773 − 4.84582i 0.421774 − 0.730534i
\(45\) −4.71582 − 4.77085i −0.702993 − 0.711197i
\(46\) 13.3924i 1.97460i
\(47\) −4.20502 −0.613365 −0.306683 − 0.951812i \(-0.599219\pi\)
−0.306683 + 0.951812i \(0.599219\pi\)
\(48\) −4.59743 + 7.06771i −0.663582 + 1.02014i
\(49\) −3.09652 + 5.36332i −0.442360 + 0.766189i
\(50\) 6.86692 − 6.07720i 0.971130 − 0.859445i
\(51\) −0.0315819 + 0.594019i −0.00442235 + 0.0831794i
\(52\) −3.55208 + 6.15239i −0.492585 + 0.853183i
\(53\) −0.0480085 − 0.0277177i −0.00659448 − 0.00380732i 0.496699 − 0.867923i \(-0.334545\pi\)
−0.503294 + 0.864115i \(0.667879\pi\)
\(54\) −9.40890 − 1.51212i −1.28039 − 0.205774i
\(55\) 1.47436 − 9.05721i 0.198802 − 1.22127i
\(56\) 0.908168 − 0.524331i 0.121359 − 0.0700667i
\(57\) 11.4326 + 0.607829i 1.51428 + 0.0805089i
\(58\) 10.6401 1.39712
\(59\) −4.72388 + 2.72734i −0.614997 + 0.355069i −0.774919 − 0.632061i \(-0.782209\pi\)
0.159921 + 0.987130i \(0.448876\pi\)
\(60\) 1.12397 − 5.15973i 0.145103 − 0.666118i
\(61\) − 10.5816i − 1.35484i −0.735597 − 0.677419i \(-0.763098\pi\)
0.735597 − 0.677419i \(-0.236902\pi\)
\(62\) −1.09640 − 10.1521i −0.139242 − 1.28932i
\(63\) 2.17785 + 1.58734i 0.274384 + 0.199986i
\(64\) −2.35539 −0.294424
\(65\) −1.87188 + 11.4993i −0.232179 + 1.42631i
\(66\) −5.90941 − 11.6196i −0.727399 − 1.43027i
\(67\) −1.94833 − 1.12487i −0.238027 − 0.137425i 0.376243 − 0.926521i \(-0.377216\pi\)
−0.614270 + 0.789096i \(0.710549\pi\)
\(68\) −0.405538 + 0.234138i −0.0491787 + 0.0283933i
\(69\) 10.6023 + 6.89665i 1.27637 + 0.830259i
\(70\) −2.33060 + 2.85296i −0.278560 + 0.340994i
\(71\) 9.60267 + 5.54410i 1.13963 + 0.657964i 0.946338 − 0.323178i \(-0.104751\pi\)
0.193289 + 0.981142i \(0.438085\pi\)
\(72\) 1.41959 + 3.20148i 0.167300 + 0.377299i
\(73\) −1.89237 + 3.27769i −0.221485 + 0.383624i −0.955259 − 0.295770i \(-0.904424\pi\)
0.733774 + 0.679394i \(0.237757\pi\)
\(74\) 4.62442 + 8.00973i 0.537578 + 0.931113i
\(75\) −1.27488 − 8.56590i −0.147211 − 0.989105i
\(76\) 4.50624 + 7.80503i 0.516901 + 0.895299i
\(77\) 3.68652i 0.420118i
\(78\) 7.50277 + 14.7526i 0.849521 + 1.67040i
\(79\) −3.85293 + 2.22449i −0.433488 + 0.250275i −0.700832 − 0.713327i \(-0.747188\pi\)
0.267343 + 0.963601i \(0.413854\pi\)
\(80\) −10.1786 + 3.85720i −1.13800 + 0.431248i
\(81\) −6.04239 + 6.67005i −0.671377 + 0.741116i
\(82\) 4.06420 + 2.34646i 0.448815 + 0.259124i
\(83\) 3.00738 + 1.73631i 0.330103 + 0.190585i 0.655887 − 0.754859i \(-0.272295\pi\)
−0.325784 + 0.945444i \(0.605628\pi\)
\(84\) −0.112632 + 2.11848i −0.0122891 + 0.231145i
\(85\) −0.485847 + 0.594739i −0.0526975 + 0.0645085i
\(86\) −11.8941 − 20.6011i −1.28257 − 2.22148i
\(87\) 5.47933 − 8.42346i 0.587445 − 0.903090i
\(88\) −2.39533 + 4.14884i −0.255344 + 0.442268i
\(89\) −16.1029 −1.70690 −0.853451 − 0.521173i \(-0.825495\pi\)
−0.853451 + 0.521173i \(0.825495\pi\)
\(90\) −8.64872 − 8.74964i −0.911655 − 0.922293i
\(91\) − 4.68051i − 0.490651i
\(92\) 9.95660i 1.03805i
\(93\) −8.60175 − 4.36004i −0.891959 − 0.452115i
\(94\) −7.71192 −0.795424
\(95\) 11.4464 + 9.35066i 1.17438 + 0.959357i
\(96\) −6.22657 + 9.57221i −0.635496 + 0.976960i
\(97\) 7.72813i 0.784672i 0.919822 + 0.392336i \(0.128333\pi\)
−0.919822 + 0.392336i \(0.871667\pi\)
\(98\) −5.67895 + 9.83622i −0.573660 + 0.993608i
\(99\) −12.2421 − 1.30542i −1.23037 − 0.131200i
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 465.2.t.d.254.43 yes 104
3.2 odd 2 inner 465.2.t.d.254.9 yes 104
5.4 even 2 inner 465.2.t.d.254.10 yes 104
15.14 odd 2 inner 465.2.t.d.254.44 yes 104
31.26 odd 6 inner 465.2.t.d.119.44 yes 104
93.26 even 6 inner 465.2.t.d.119.10 yes 104
155.119 odd 6 inner 465.2.t.d.119.9 ✓ 104
465.119 even 6 inner 465.2.t.d.119.43 yes 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.d.119.9 ✓ 104 155.119 odd 6 inner
465.2.t.d.119.10 yes 104 93.26 even 6 inner
465.2.t.d.119.43 yes 104 465.119 even 6 inner
465.2.t.d.119.44 yes 104 31.26 odd 6 inner
465.2.t.d.254.9 yes 104 3.2 odd 2 inner
465.2.t.d.254.10 yes 104 5.4 even 2 inner
465.2.t.d.254.43 yes 104 1.1 even 1 trivial
465.2.t.d.254.44 yes 104 15.14 odd 2 inner