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Properties

Label 465.2.t.d.119.10
Level $465$
Weight $2$
Character 465.119
Analytic conductor $3.713$
Analytic rank $0$
Dimension $104$
Inner twists $8$

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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] = [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 119.10
Character \(\chi\) \(=\) 465.119
Dual form 465.2.t.d.254.10

$q$-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} +(0.359264 - 2.20702i) q^{5} +(1.73209 + 2.66276i) q^{6} +(0.777962 + 0.449157i) q^{7} +1.16737 q^{8} +(-1.21606 + 2.74248i) q^{9} +(-0.658883 + 4.04762i) 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} +(-3.54369 + 1.56278i) q^{15} -4.86788 q^{16} +(0.297429 + 0.171721i) q^{17} +(2.23023 - 5.02965i) q^{18} +(3.30496 - 5.72436i) q^{19} +(0.489849 - 3.00922i) q^{20} +(-0.0826063 - 1.55373i) q^{21} +(-3.76316 - 6.51798i) q^{22} +7.30236i q^{23} +(-1.10251 - 1.69491i) q^{24} +(-4.74186 - 1.58580i) 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} +(-1.65807 + 3.73931i) q^{36} +(-2.52152 + 4.36741i) q^{37} +(-6.06122 + 10.4983i) q^{38} +(4.09098 - 8.04404i) q^{39} +(0.419394 - 2.57640i) 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} +(5.61582 + 3.66914i) q^{45} -13.3924i q^{46} +4.20502 q^{47} +(4.59743 + 7.06771i) q^{48} +(-3.09652 - 5.36332i) q^{49} +(8.69647 + 2.90833i) 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} +(8.58095 - 3.25177i) q^{55} +(0.908168 + 0.524331i) q^{56} +(-11.4326 + 0.607829i) 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} +(10.8946 - 4.12855i) 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} +(2.17597 + 8.38243i) q^{75} +(4.50624 - 7.80503i) q^{76} +3.68652i q^{77} +(-7.50277 + 14.7526i) q^{78} +(-3.85293 - 2.22449i) q^{79} +(-1.74886 + 10.7435i) 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} +(-10.2993 - 6.72913i) 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 \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).

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

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 \(0.724565\pi\)
\(3\) −0.944441 − 1.45191i −0.545273 − 0.838258i
\(4\) 1.36348 0.681739
\(5\) 0.359264 − 2.20702i 0.160668 − 0.987009i
\(6\) 1.73209 + 2.66276i 0.707121 + 1.08707i
\(7\) 0.777962 + 0.449157i 0.294042 + 0.169765i 0.639763 − 0.768572i \(-0.279032\pi\)
−0.345721 + 0.938337i \(0.612366\pi\)
\(8\) 1.16737 0.412727
\(9\) −1.21606 + 2.74248i −0.405354 + 0.914160i
\(10\) −0.658883 + 4.04762i −0.208357 + 1.27997i
\(11\) 2.05191 + 3.55401i 0.618674 + 1.07158i 0.989728 + 0.142964i \(0.0456632\pi\)
−0.371054 + 0.928611i \(0.621003\pi\)
\(12\) −1.28772 − 1.97964i −0.371734 − 0.571473i
\(13\) 2.60516 + 4.51228i 0.722543 + 1.25148i 0.959977 + 0.280077i \(0.0903601\pi\)
−0.237435 + 0.971403i \(0.576307\pi\)
\(14\) −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 0.535635 − 0.844450i \(-0.320072\pi\)
−0.463498 + 0.886098i \(0.653406\pi\)
\(18\) 2.23023 − 5.02965i 0.525670 − 1.18550i
\(19\) 3.30496 − 5.72436i 0.758209 − 1.31326i −0.185553 − 0.982634i \(-0.559408\pi\)
0.943763 − 0.330623i \(-0.107259\pi\)
\(20\) 0.489849 − 3.00922i 0.109533 − 0.672882i
\(21\) −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
\(25\) −4.74186 − 1.58580i −0.948372 − 0.317161i
\(26\) −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 \(0.318927\pi\)
\(30\) 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
\(36\) −1.65807 + 3.73931i −0.276345 + 0.623218i
\(37\) −2.52152 + 4.36741i −0.414536 + 0.717997i −0.995380 − 0.0960177i \(-0.969389\pi\)
0.580844 + 0.814015i \(0.302723\pi\)
\(38\) −6.06122 + 10.4983i −0.983260 + 1.70306i
\(39\) 4.09098 − 8.04404i 0.655081 − 1.28808i
\(40\) 0.419394 − 2.57640i 0.0663119 − 0.407365i
\(41\) 2.21605 − 1.27944i 0.346089 − 0.199815i −0.316872 − 0.948468i \(-0.602633\pi\)
0.662962 + 0.748653i \(0.269299\pi\)
\(42\) 0.151498 + 2.84951i 0.0233767 + 0.439689i
\(43\) 6.48540 − 11.2330i 0.989014 − 1.71302i 0.366486 − 0.930423i \(-0.380561\pi\)
0.622527 − 0.782598i \(-0.286106\pi\)
\(44\) 2.79773 + 4.84582i 0.421774 + 0.730534i
\(45\) 5.61582 + 3.66914i 0.837156 + 0.546964i
\(46\) − 13.3924i − 1.97460i
\(47\) 4.20502 0.613365 0.306683 − 0.951812i \(-0.400781\pi\)
0.306683 + 0.951812i \(0.400781\pi\)
\(48\) 4.59743 + 7.06771i 0.663582 + 1.02014i
\(49\) −3.09652 − 5.36332i −0.442360 − 0.766189i
\(50\) 8.69647 + 2.90833i 1.22987 + 0.411300i
\(51\) −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 \(-0.665455\pi\)
0.503294 + 0.864115i \(0.332121\pi\)
\(54\) −9.40890 + 1.51212i −1.28039 + 0.205774i
\(55\) 8.58095 − 3.25177i 1.15705 − 0.438469i
\(56\) 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 0.159921 − 0.987130i \(-0.448876\pi\)
−0.774919 + 0.632061i \(0.782209\pi\)
\(60\) −4.83174 + 2.13082i −0.623775 + 0.275087i
\(61\) 10.5816i 1.35484i 0.735597 + 0.677419i \(0.236902\pi\)
−0.735597 + 0.677419i \(0.763098\pi\)
\(62\) 1.09640 − 10.1521i 0.139242 − 1.28932i
\(63\) −2.17785 + 1.58734i −0.274384 + 0.199986i
\(64\) −2.35539 −0.294424
\(65\) 10.8946 − 4.12855i 1.35131 − 0.512083i
\(66\) −5.90941 + 11.6196i −0.727399 + 1.43027i
\(67\) 1.94833 − 1.12487i 0.238027 − 0.137425i −0.376243 − 0.926521i \(-0.622784\pi\)
0.614270 + 0.789096i \(0.289451\pi\)
\(68\) 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 0.193289 − 0.981142i \(-0.438085\pi\)
0.946338 + 0.323178i \(0.104751\pi\)
\(72\) −1.41959 + 3.20148i −0.167300 + 0.377299i
\(73\) 1.89237 + 3.27769i 0.221485 + 0.383624i 0.955259 − 0.295770i \(-0.0955761\pi\)
−0.733774 + 0.679394i \(0.762243\pi\)
\(74\) 4.62442 − 8.00973i 0.537578 − 0.931113i
\(75\) 2.17597 + 8.38243i 0.251259 + 0.967920i
\(76\) 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 0.267343 − 0.963601i \(-0.413854\pi\)
−0.700832 + 0.713327i \(0.747188\pi\)
\(80\) −1.74886 + 10.7435i −0.195528 + 1.20116i
\(81\) −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 \(-0.727705\pi\)
0.325784 + 0.945444i \(0.394372\pi\)
\(84\) −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 \(0.825495\pi\)
\(90\) −10.2993 − 6.72913i −1.08564 − 0.709313i
\(91\) 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.119.10 yes 104
3.2 odd 2 inner 465.2.t.d.119.44 yes 104
5.4 even 2 inner 465.2.t.d.119.43 yes 104
15.14 odd 2 inner 465.2.t.d.119.9 ✓ 104
31.6 odd 6 inner 465.2.t.d.254.9 yes 104
93.68 even 6 inner 465.2.t.d.254.43 yes 104
155.99 odd 6 inner 465.2.t.d.254.44 yes 104
465.254 even 6 inner 465.2.t.d.254.10 yes 104
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.t.d.119.9 ✓ 104 15.14 odd 2 inner
465.2.t.d.119.10 yes 104 1.1 even 1 trivial
465.2.t.d.119.43 yes 104 5.4 even 2 inner
465.2.t.d.119.44 yes 104 3.2 odd 2 inner
465.2.t.d.254.9 yes 104 31.6 odd 6 inner
465.2.t.d.254.10 yes 104 465.254 even 6 inner
465.2.t.d.254.43 yes 104 93.68 even 6 inner
465.2.t.d.254.44 yes 104 155.99 odd 6 inner