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(16,465)] 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("465.16"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(465, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.n (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,4] 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: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6 x^{15} + 18 x^{14} - 7 x^{13} + 168 x^{12} - 290 x^{11} + 2849 x^{10} - 4031 x^{9} + \cdots + 259081 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 376.3
Root \(-0.681323 - 2.09690i\) of defining polynomial
Character \(\chi\) \(=\) 465.376
Dual form 465.2.n.d.256.3

$q$-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+(0.647481 + 1.99274i) q^{2} +(0.309017 - 0.951057i) q^{3} +(-1.93376 + 1.40496i) q^{4} -1.00000 q^{5} +2.09529 q^{6} +(-3.54727 + 2.57724i) q^{7} +(-0.661536 - 0.480634i) q^{8} +(-0.809017 - 0.587785i) q^{9} +(-0.647481 - 1.99274i) q^{10} +(-3.50499 + 2.54652i) q^{11} +(0.738630 + 2.27327i) q^{12} +(0.387184 - 1.19163i) q^{13} +(-7.43258 - 5.40008i) q^{14} +(-0.309017 + 0.951057i) q^{15} +(-0.947813 + 2.91707i) q^{16} +(1.71747 + 1.24781i) q^{17} +(0.647481 - 1.99274i) q^{18} +(1.52455 + 4.69209i) q^{19} +(1.93376 - 1.40496i) q^{20} +(1.35494 + 4.17007i) q^{21} +(-7.34398 - 5.33571i) q^{22} +(-2.57536 - 1.87111i) q^{23} +(-0.661536 + 0.480634i) q^{24} +1.00000 q^{25} +2.62531 q^{26} +(-0.809017 + 0.587785i) q^{27} +(3.23865 - 9.96754i) q^{28} +(-1.89588 - 5.83493i) q^{29} -2.09529 q^{30} +(0.898526 + 5.49478i) q^{31} -8.06206 q^{32} +(1.33879 + 4.12036i) q^{33} +(-1.37454 + 4.23040i) q^{34} +(3.54727 - 2.57724i) q^{35} +2.39026 q^{36} +6.37108 q^{37} +(-8.36300 + 6.07608i) q^{38} +(-1.01366 - 0.736468i) q^{39} +(0.661536 + 0.480634i) q^{40} +(1.64666 + 5.06789i) q^{41} +(-7.43258 + 5.40008i) q^{42} +(-1.08653 - 3.34400i) q^{43} +(3.20004 - 9.84872i) q^{44} +(0.809017 + 0.587785i) q^{45} +(2.06114 - 6.34353i) q^{46} +(1.46520 - 4.50941i) q^{47} +(2.48141 + 1.80285i) q^{48} +(3.77783 - 11.6270i) q^{49} +(0.647481 + 1.99274i) q^{50} +(1.71747 - 1.24781i) q^{51} +(0.925469 + 2.84830i) q^{52} +(2.27057 + 1.64967i) q^{53} +(-1.69513 - 1.23158i) q^{54} +(3.50499 - 2.54652i) q^{55} +3.58536 q^{56} +4.93355 q^{57} +(10.4000 - 7.55602i) q^{58} +(-2.94936 + 9.07721i) q^{59} +(-0.738630 - 2.27327i) q^{60} +8.17845 q^{61} +(-10.3679 + 5.34830i) q^{62} +4.38467 q^{63} +(-3.32441 - 10.2315i) q^{64} +(-0.387184 + 1.19163i) q^{65} +(-7.34398 + 5.33571i) q^{66} +5.14459 q^{67} -5.07429 q^{68} +(-2.57536 + 1.87111i) q^{69} +(7.43258 + 5.40008i) q^{70} +(-1.20556 - 0.875891i) q^{71} +(0.252684 + 0.777682i) q^{72} +(8.91303 - 6.47570i) q^{73} +(4.12516 + 12.6959i) q^{74} +(0.309017 - 0.951057i) q^{75} +(-9.54030 - 6.93143i) q^{76} +(5.87014 - 18.0664i) q^{77} +(0.811264 - 2.49681i) q^{78} +(12.7438 + 9.25890i) q^{79} +(0.947813 - 2.91707i) q^{80} +(0.309017 + 0.951057i) q^{81} +(-9.03282 + 6.56273i) q^{82} +(-0.171116 - 0.526641i) q^{83} +(-8.47889 - 6.16028i) q^{84} +(-1.71747 - 1.24781i) q^{85} +(5.96023 - 4.33036i) q^{86} -6.13521 q^{87} +3.54262 q^{88} +(-9.61689 + 6.98708i) q^{89} +(-0.647481 + 1.99274i) q^{90} +(1.69767 + 5.22490i) q^{91} +7.60894 q^{92} +(5.50351 + 0.843432i) q^{93} +9.93478 q^{94} +(-1.52455 - 4.69209i) q^{95} +(-2.49131 + 7.66748i) q^{96} +(-7.42038 + 5.39122i) q^{97} +25.6157 q^{98} +4.33240 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} - 4 q^{3} - 4 q^{4} - 16 q^{5} + 4 q^{6} + 7 q^{7} - 8 q^{8} - 4 q^{9} - 4 q^{10} + 4 q^{11} + 6 q^{12} + 8 q^{13} - q^{14} + 4 q^{15} + 8 q^{16} + 4 q^{17} + 4 q^{18} + 17 q^{19} + 4 q^{20}+ \cdots + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(187\) \(311\) \(406\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.647481 + 1.99274i 0.457839 + 1.40908i 0.867770 + 0.496966i \(0.165553\pi\)
−0.409932 + 0.912116i \(0.634447\pi\)
\(3\) 0.309017 − 0.951057i 0.178411 − 0.549093i
\(4\) −1.93376 + 1.40496i −0.966879 + 0.702479i
\(5\) −1.00000 −0.447214
\(6\) 2.09529 0.855400
\(7\) −3.54727 + 2.57724i −1.34074 + 0.974107i −0.341327 + 0.939945i \(0.610876\pi\)
−0.999416 + 0.0341620i \(0.989124\pi\)
\(8\) −0.661536 − 0.480634i −0.233888 − 0.169930i
\(9\) −0.809017 − 0.587785i −0.269672 − 0.195928i
\(10\) −0.647481 − 1.99274i −0.204752 − 0.630161i
\(11\) −3.50499 + 2.54652i −1.05679 + 0.767806i −0.973493 − 0.228719i \(-0.926546\pi\)
−0.0833011 + 0.996524i \(0.526546\pi\)
\(12\) 0.738630 + 2.27327i 0.213224 + 0.656236i
\(13\) 0.387184 − 1.19163i 0.107386 − 0.330499i −0.882897 − 0.469566i \(-0.844410\pi\)
0.990283 + 0.139067i \(0.0444104\pi\)
\(14\) −7.43258 − 5.40008i −1.98644 − 1.44323i
\(15\) −0.309017 + 0.951057i −0.0797878 + 0.245562i
\(16\) −0.947813 + 2.91707i −0.236953 + 0.729267i
\(17\) 1.71747 + 1.24781i 0.416547 + 0.302639i 0.776247 − 0.630429i \(-0.217121\pi\)
−0.359700 + 0.933068i \(0.617121\pi\)
\(18\) 0.647481 − 1.99274i 0.152613 − 0.469694i
\(19\) 1.52455 + 4.69209i 0.349756 + 1.07644i 0.958988 + 0.283446i \(0.0914778\pi\)
−0.609232 + 0.792992i \(0.708522\pi\)
\(20\) 1.93376 − 1.40496i 0.432402 − 0.314158i
\(21\) 1.35494 + 4.17007i 0.295672 + 0.909984i
\(22\) −7.34398 − 5.33571i −1.56574 − 1.13758i
\(23\) −2.57536 − 1.87111i −0.536999 − 0.390153i 0.285970 − 0.958238i \(-0.407684\pi\)
−0.822969 + 0.568086i \(0.807684\pi\)
\(24\) −0.661536 + 0.480634i −0.135035 + 0.0981089i
\(25\) 1.00000 0.200000
\(26\) 2.62531 0.514865
\(27\) −0.809017 + 0.587785i −0.155695 + 0.113119i
\(28\) 3.23865 − 9.96754i 0.612047 − 1.88369i
\(29\) −1.89588 − 5.83493i −0.352057 − 1.08352i −0.957697 − 0.287780i \(-0.907083\pi\)
0.605640 − 0.795739i \(-0.292917\pi\)
\(30\) −2.09529 −0.382547
\(31\) 0.898526 + 5.49478i 0.161380 + 0.986892i
\(32\) −8.06206 −1.42518
\(33\) 1.33879 + 4.12036i 0.233053 + 0.717263i
\(34\) −1.37454 + 4.23040i −0.235732 + 0.725508i
\(35\) 3.54727 − 2.57724i 0.599598 − 0.435634i
\(36\) 2.39026 0.398376
\(37\) 6.37108 1.04740 0.523700 − 0.851903i \(-0.324551\pi\)
0.523700 + 0.851903i \(0.324551\pi\)
\(38\) −8.36300 + 6.07608i −1.35666 + 0.985670i
\(39\) −1.01366 − 0.736468i −0.162316 − 0.117929i
\(40\) 0.661536 + 0.480634i 0.104598 + 0.0759949i
\(41\) 1.64666 + 5.06789i 0.257165 + 0.791472i 0.993395 + 0.114741i \(0.0366038\pi\)
−0.736231 + 0.676731i \(0.763396\pi\)
\(42\) −7.43258 + 5.40008i −1.14687 + 0.833251i
\(43\) −1.08653 − 3.34400i −0.165695 − 0.509956i 0.833392 − 0.552682i \(-0.186396\pi\)
−0.999087 + 0.0427264i \(0.986396\pi\)
\(44\) 3.20004 − 9.84872i 0.482425 − 1.48475i
\(45\) 0.809017 + 0.587785i 0.120601 + 0.0876219i
\(46\) 2.06114 − 6.34353i 0.303898 − 0.935302i
\(47\) 1.46520 − 4.50941i 0.213721 − 0.657765i −0.785521 − 0.618835i \(-0.787605\pi\)
0.999242 − 0.0389301i \(-0.0123950\pi\)
\(48\) 2.48141 + 1.80285i 0.358160 + 0.260218i
\(49\) 3.77783 − 11.6270i 0.539691 − 1.66100i
\(50\) 0.647481 + 1.99274i 0.0915677 + 0.281816i
\(51\) 1.71747 − 1.24781i 0.240493 − 0.174729i
\(52\) 0.925469 + 2.84830i 0.128339 + 0.394988i
\(53\) 2.27057 + 1.64967i 0.311887 + 0.226599i 0.732706 − 0.680545i \(-0.238257\pi\)
−0.420819 + 0.907145i \(0.638257\pi\)
\(54\) −1.69513 − 1.23158i −0.230678 − 0.167597i
\(55\) 3.50499 − 2.54652i 0.472612 − 0.343373i
\(56\) 3.58536 0.479113
\(57\) 4.93355 0.653465
\(58\) 10.4000 − 7.55602i 1.36558 − 0.992153i
\(59\) −2.94936 + 9.07721i −0.383974 + 1.18175i 0.553247 + 0.833017i \(0.313388\pi\)
−0.937222 + 0.348734i \(0.886612\pi\)
\(60\) −0.738630 − 2.27327i −0.0953567 − 0.293478i
\(61\) 8.17845 1.04714 0.523571 − 0.851982i \(-0.324599\pi\)
0.523571 + 0.851982i \(0.324599\pi\)
\(62\) −10.3679 + 5.34830i −1.31673 + 0.679235i
\(63\) 4.38467 0.552416
\(64\) −3.32441 − 10.2315i −0.415551 − 1.27894i
\(65\) −0.387184 + 1.19163i −0.0480243 + 0.147803i
\(66\) −7.34398 + 5.33571i −0.903981 + 0.656781i
\(67\) 5.14459 0.628512 0.314256 − 0.949338i \(-0.398245\pi\)
0.314256 + 0.949338i \(0.398245\pi\)
\(68\) −5.07429 −0.615348
\(69\) −2.57536 + 1.87111i −0.310036 + 0.225255i
\(70\) 7.43258 + 5.40008i 0.888363 + 0.645434i
\(71\) −1.20556 − 0.875891i −0.143074 − 0.103949i 0.513946 − 0.857823i \(-0.328183\pi\)
−0.657020 + 0.753873i \(0.728183\pi\)
\(72\) 0.252684 + 0.777682i 0.0297791 + 0.0916507i
\(73\) 8.91303 − 6.47570i 1.04319 − 0.757923i 0.0722857 − 0.997384i \(-0.476971\pi\)
0.970906 + 0.239461i \(0.0769707\pi\)
\(74\) 4.12516 + 12.6959i 0.479540 + 1.47587i
\(75\) 0.309017 − 0.951057i 0.0356822 − 0.109819i
\(76\) −9.54030 − 6.93143i −1.09435 − 0.795090i
\(77\) 5.87014 − 18.0664i 0.668964 − 2.05886i
\(78\) 0.811264 − 2.49681i 0.0918576 − 0.282709i
\(79\) 12.7438 + 9.25890i 1.43379 + 1.04171i 0.989296 + 0.145924i \(0.0466157\pi\)
0.444492 + 0.895783i \(0.353384\pi\)
\(80\) 0.947813 − 2.91707i 0.105969 − 0.326138i
\(81\) 0.309017 + 0.951057i 0.0343352 + 0.105673i
\(82\) −9.03282 + 6.56273i −0.997508 + 0.724732i
\(83\) −0.171116 − 0.526641i −0.0187824 − 0.0578063i 0.941226 − 0.337777i \(-0.109675\pi\)
−0.960008 + 0.279971i \(0.909675\pi\)
\(84\) −8.47889 − 6.16028i −0.925123 − 0.672141i
\(85\) −1.71747 − 1.24781i −0.186285 − 0.135344i
\(86\) 5.96023 − 4.33036i 0.642708 − 0.466955i
\(87\) −6.13521 −0.657763
\(88\) 3.54262 0.377644
\(89\) −9.61689 + 6.98708i −1.01939 + 0.740629i −0.966157 − 0.257954i \(-0.916952\pi\)
−0.0532306 + 0.998582i \(0.516952\pi\)
\(90\) −0.647481 + 1.99274i −0.0682505 + 0.210054i
\(91\) 1.69767 + 5.22490i 0.177965 + 0.547719i
\(92\) 7.60894 0.793287
\(93\) 5.50351 + 0.843432i 0.570687 + 0.0874599i
\(94\) 9.93478 1.02469
\(95\) −1.52455 − 4.69209i −0.156416 − 0.481398i
\(96\) −2.49131 + 7.66748i −0.254269 + 0.782559i
\(97\) −7.42038 + 5.39122i −0.753425 + 0.547395i −0.896887 − 0.442261i \(-0.854177\pi\)
0.143462 + 0.989656i \(0.454177\pi\)
\(98\) 25.6157 2.58757
\(99\) 4.33240 0.435423
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.n.d.376.3 yes 16
31.8 even 5 inner 465.2.n.d.256.3 ✓ 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.n.d.256.3 ✓ 16 31.8 even 5 inner
465.2.n.d.376.3 yes 16 1.1 even 1 trivial