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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 254.9
Character \(\chi\) \(=\) 465.254
Dual form 465.2.t.d.119.9

$q$-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} +(-1.72961 + 0.0919572i) q^{3} +1.36348 q^{4} +(-2.09097 + 0.792377i) q^{5} +(3.17206 - 0.168647i) q^{6} +(-0.777962 + 0.449157i) q^{7} +1.16737 q^{8} +(2.98309 - 0.318100i) q^{9} +(3.83479 - 1.45320i) q^{10} +(-2.05191 + 3.55401i) q^{11} +(-2.35828 + 0.125382i) q^{12} +(-2.60516 + 4.51228i) q^{13} +(1.42677 - 0.823744i) q^{14} +(3.54369 - 1.56278i) q^{15} -4.86788 q^{16} +(0.297429 - 0.171721i) q^{17} +(-5.47092 + 0.583388i) q^{18} +(3.30496 + 5.72436i) q^{19} +(-2.85099 + 1.08039i) q^{20} +(1.30427 - 0.848404i) q^{21} +(3.76316 - 6.51798i) q^{22} -7.30236i q^{23} +(-2.01909 + 0.107348i) 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} +(-6.49905 + 2.86611i) 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} +(4.06737 - 0.433722i) 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} +(-2.39200 + 1.55596i) q^{42} +(-6.48540 - 11.2330i) q^{43} +(-2.79773 + 4.84582i) q^{44} +(-5.98548 + 3.02887i) q^{45} +13.3924i q^{46} +4.20502 q^{47} +(8.41953 - 0.447637i) q^{48} +(-3.09652 + 5.36332i) q^{49} +(-6.86692 + 6.07720i) q^{50} +(-0.498645 + 0.324360i) 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} +(-6.24268 - 9.59698i) q^{57} +10.6401 q^{58} +(4.72388 - 2.72734i) q^{59} +(4.83174 - 2.13082i) 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} +(0.671504 + 12.6302i) q^{69} +(-2.33060 + 2.85296i) q^{70} +(-9.60267 - 5.54410i) q^{71} +(3.48236 - 0.371340i) q^{72} +(-1.89237 + 3.27769i) q^{73} +(-4.62442 - 8.00973i) q^{74} +(-6.17142 + 6.07566i) 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} +(8.79763 - 1.89784i) q^{81} +(4.06420 + 2.34646i) q^{82} +(-3.00738 - 1.73631i) q^{83} +(1.77834 - 1.15678i) q^{84} +(-0.485847 + 0.594739i) q^{85} +(11.8941 + 20.6011i) q^{86} +(10.0346 - 0.533504i) q^{87} +(-2.39533 + 4.14884i) q^{88} +16.1029 q^{89} +(10.9772 - 5.55488i) q^{90} -4.68051i q^{91} -9.95660i q^{92} +(1.54304 + 9.51940i) q^{93} -7.71192 q^{94} +(-11.4464 - 9.35066i) q^{95} +(-11.4031 + 0.606261i) q^{96} +7.72813i q^{97} +(5.67895 - 9.83622i) q^{98} +(-4.99050 + 11.2546i) 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 \(-0.724565\pi\)
−0.648409 + 0.761292i \(0.724565\pi\)
\(3\) −1.72961 + 0.0919572i −0.998590 + 0.0530915i
\(4\) 1.36348 0.681739
\(5\) −2.09097 + 0.792377i −0.935108 + 0.354362i
\(6\) 3.17206 − 0.168647i 1.29499 − 0.0688500i
\(7\) −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
\(9\) 2.98309 − 0.318100i 0.994363 − 0.106033i
\(10\) 3.83479 − 1.45320i 1.21267 − 0.459543i
\(11\) −2.05191 + 3.55401i −0.618674 + 1.07158i 0.371054 + 0.928611i \(0.378997\pi\)
−0.989728 + 0.142964i \(0.954337\pi\)
\(12\) −2.35828 + 0.125382i −0.680777 + 0.0361945i
\(13\) −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
\(15\) 3.54369 − 1.56278i 0.914976 − 0.403508i
\(16\) −4.86788 −1.21697
\(17\) 0.297429 − 0.171721i 0.0721372 − 0.0416484i −0.463498 − 0.886098i \(-0.653406\pi\)
0.535635 + 0.844450i \(0.320072\pi\)
\(18\) −5.47092 + 0.583388i −1.28951 + 0.137506i
\(19\) 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
\(21\) 1.30427 − 0.848404i 0.284614 − 0.185137i
\(22\) 3.76316 − 6.51798i 0.802308 − 1.38964i
\(23\) − 7.30236i − 1.52265i −0.648372 − 0.761323i \(-0.724550\pi\)
0.648372 − 0.761323i \(-0.275450\pi\)
\(24\) −2.01909 + 0.107348i −0.412145 + 0.0219123i
\(25\) 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 \(-0.681073\pi\)
−0.538671 + 0.842516i \(0.681073\pi\)
\(30\) −6.49905 + 2.86611i −1.18656 + 0.523277i
\(31\) −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
\(36\) 4.06737 − 0.433722i 0.677896 − 0.0722870i
\(37\) 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 0.316872 − 0.948468i \(-0.397367\pi\)
−0.662962 + 0.748653i \(0.730701\pi\)
\(42\) −2.39200 + 1.55596i −0.369093 + 0.240089i
\(43\) −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
\(45\) −5.98548 + 3.02887i −0.892263 + 0.451517i
\(46\) 13.3924i 1.97460i
\(47\) 4.20502 0.613365 0.306683 − 0.951812i \(-0.400781\pi\)
0.306683 + 0.951812i \(0.400781\pi\)
\(48\) 8.41953 − 0.447637i 1.21525 − 0.0646108i
\(49\) −3.09652 + 5.36332i −0.442360 + 0.766189i
\(50\) −6.86692 + 6.07720i −0.971130 + 0.859445i
\(51\) −0.498645 + 0.324360i −0.0698243 + 0.0454196i
\(52\) −3.55208 + 6.15239i −0.492585 + 0.853183i
\(53\) 0.0480085 + 0.0277177i 0.00659448 + 0.00380732i 0.503294 − 0.864115i \(-0.332121\pi\)
−0.496699 + 0.867923i \(0.665455\pi\)
\(54\) 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
\(57\) −6.24268 − 9.59698i −0.826863 − 1.27115i
\(58\) 10.6401 1.39712
\(59\) 4.72388 − 2.72734i 0.614997 − 0.355069i −0.159921 − 0.987130i \(-0.551124\pi\)
0.774919 + 0.632061i \(0.217791\pi\)
\(60\) 4.83174 − 2.13082i 0.623775 − 0.275087i
\(61\) − 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
\(69\) 0.671504 + 12.6302i 0.0808396 + 1.52050i
\(70\) −2.33060 + 2.85296i −0.278560 + 0.340994i
\(71\) −9.60267 − 5.54410i −1.13963 − 0.657964i −0.193289 − 0.981142i \(-0.561915\pi\)
−0.946338 + 0.323178i \(0.895249\pi\)
\(72\) 3.48236 − 0.371340i 0.410400 − 0.0437628i
\(73\) −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
\(75\) −6.17142 + 6.07566i −0.712614 + 0.701557i
\(76\) 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
\(81\) 8.79763 − 1.89784i 0.977514 − 0.210871i
\(82\) 4.06420 + 2.34646i 0.448815 + 0.259124i
\(83\) −3.00738 − 1.73631i −0.330103 − 0.190585i 0.325784 − 0.945444i \(-0.394372\pi\)
−0.655887 + 0.754859i \(0.727705\pi\)
\(84\) 1.77834 − 1.15678i 0.194033 − 0.126215i
\(85\) −0.485847 + 0.594739i −0.0526975 + 0.0645085i
\(86\) 11.8941 + 20.6011i 1.28257 + 2.22148i
\(87\) 10.0346 − 0.533504i 1.07582 − 0.0571976i
\(88\) −2.39533 + 4.14884i −0.255344 + 0.442268i
\(89\) 16.1029 1.70690 0.853451 − 0.521173i \(-0.174505\pi\)
0.853451 + 0.521173i \(0.174505\pi\)
\(90\) 10.9772 − 5.55488i 1.15710 − 0.585535i
\(91\) − 4.68051i − 0.490651i
\(92\) − 9.95660i − 1.03805i
\(93\) 1.54304 + 9.51940i 0.160005 + 0.987116i
\(94\) −7.71192 −0.795424
\(95\) −11.4464 − 9.35066i −1.17438 − 0.959357i
\(96\) −11.4031 + 0.606261i −1.16382 + 0.0618762i
\(97\) 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
\(99\) −4.99050 + 11.2546i −0.501564 + 1.13113i
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.254.9 yes 104
3.2 odd 2 inner 465.2.t.d.254.43 yes 104
5.4 even 2 inner 465.2.t.d.254.44 yes 104
15.14 odd 2 inner 465.2.t.d.254.10 yes 104
31.26 odd 6 inner 465.2.t.d.119.10 yes 104
93.26 even 6 inner 465.2.t.d.119.44 yes 104
155.119 odd 6 inner 465.2.t.d.119.43 yes 104
465.119 even 6 inner 465.2.t.d.119.9 ✓ 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.d.119.9 ✓ 104 465.119 even 6 inner
465.2.t.d.119.10 yes 104 31.26 odd 6 inner
465.2.t.d.119.43 yes 104 155.119 odd 6 inner
465.2.t.d.119.44 yes 104 93.26 even 6 inner
465.2.t.d.254.9 yes 104 1.1 even 1 trivial
465.2.t.d.254.10 yes 104 15.14 odd 2 inner
465.2.t.d.254.43 yes 104 3.2 odd 2 inner
465.2.t.d.254.44 yes 104 5.4 even 2 inner