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 376.4
Root \(0.872306 + 2.68468i\) of defining polynomial
Character \(\chi\) \(=\) 465.376
Dual form 465.2.n.d.256.4

$q$-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} +(3.66116 - 2.65999i) q^{7} +(-0.661536 - 0.480634i) q^{8} +(-0.809017 - 0.587785i) q^{9} +(-0.647481 - 1.99274i) q^{10} +(1.76221 - 1.28032i) q^{11} +(0.738630 + 2.27327i) q^{12} +(0.147632 - 0.454364i) q^{13} +(7.67121 + 5.57346i) q^{14} +(-0.309017 + 0.951057i) q^{15} +(-0.947813 + 2.91707i) q^{16} +(4.23129 + 3.07421i) q^{17} +(0.647481 - 1.99274i) q^{18} +(-0.247788 - 0.762613i) q^{19} +(1.93376 - 1.40496i) q^{20} +(-1.39844 - 4.30395i) q^{21} +(3.69235 + 2.68265i) q^{22} +(-3.88943 - 2.82584i) q^{23} +(-0.661536 + 0.480634i) q^{24} +1.00000 q^{25} +1.00102 q^{26} +(-0.809017 + 0.587785i) q^{27} +(-3.34263 + 10.2876i) q^{28} +(2.98371 + 9.18293i) q^{29} -2.09529 q^{30} +(-4.87061 - 2.69763i) q^{31} -8.06206 q^{32} +(-0.673105 - 2.07161i) q^{33} +(-3.38643 + 10.4224i) q^{34} +(-3.66116 + 2.65999i) q^{35} +2.39026 q^{36} +3.49248 q^{37} +(1.35925 - 0.987555i) q^{38} +(-0.386505 - 0.280812i) q^{39} +(0.661536 + 0.480634i) q^{40} +(0.332581 + 1.02358i) q^{41} +(7.67121 - 5.57346i) q^{42} +(-2.28628 - 7.03645i) q^{43} +(-1.60889 + 4.95167i) q^{44} +(0.809017 + 0.587785i) q^{45} +(3.11283 - 9.58032i) q^{46} +(-2.60226 + 8.00892i) q^{47} +(2.48141 + 1.80285i) q^{48} +(4.16544 - 12.8199i) q^{49} +(0.647481 + 1.99274i) q^{50} +(4.23129 - 3.07421i) q^{51} +(0.352878 + 1.08605i) q^{52} +(-0.357578 - 0.259796i) q^{53} +(-1.69513 - 1.23158i) q^{54} +(-1.76221 + 1.28032i) q^{55} -3.70047 q^{56} -0.801859 q^{57} +(-16.3673 + 11.8915i) q^{58} +(1.18875 - 3.65859i) q^{59} +(-0.738630 - 2.27327i) q^{60} +2.03780 q^{61} +(2.22205 - 11.4525i) q^{62} -4.52544 q^{63} +(-3.32441 - 10.2315i) q^{64} +(-0.147632 + 0.454364i) q^{65} +(3.69235 - 2.68265i) q^{66} -1.98645 q^{67} -12.5014 q^{68} +(-3.88943 + 2.82584i) q^{69} +(-7.67121 - 5.57346i) q^{70} +(-10.3007 - 7.48387i) q^{71} +(0.252684 + 0.777682i) q^{72} +(-4.24952 + 3.08746i) q^{73} +(2.26132 + 6.95962i) q^{74} +(0.309017 - 0.951057i) q^{75} +(1.55060 + 1.12658i) q^{76} +(3.04610 - 9.37493i) q^{77} +(0.309332 - 0.952025i) q^{78} +(-8.94128 - 6.49622i) q^{79} +(0.947813 - 2.91707i) q^{80} +(0.309017 + 0.951057i) q^{81} +(-1.82439 + 1.32550i) q^{82} +(3.06335 + 9.42803i) q^{83} +(8.75111 + 6.35806i) q^{84} +(-4.23129 - 3.07421i) q^{85} +(12.5415 - 9.11194i) q^{86} +9.65550 q^{87} -1.78113 q^{88} +(3.78522 - 2.75012i) q^{89} +(-0.647481 + 1.99274i) q^{90} +(-0.668099 - 2.05620i) q^{91} +11.4914 q^{92} +(-4.07070 + 3.79861i) q^{93} -17.6446 q^{94} +(0.247788 + 0.762613i) q^{95} +(-2.49131 + 7.66748i) q^{96} +(-9.12206 + 6.62756i) q^{97} +28.2438 q^{98} -2.17821 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
\(7\) 3.66116 − 2.65999i 1.38379 − 1.00538i 0.387274 − 0.921965i \(-0.373417\pi\)
0.996515 − 0.0834167i \(-0.0265832\pi\)
\(8\) −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
\(11\) 1.76221 − 1.28032i 0.531327 − 0.386032i −0.289527 − 0.957170i \(-0.593498\pi\)
0.820854 + 0.571138i \(0.193498\pi\)
\(12\) 0.738630 + 2.27327i 0.213224 + 0.656236i
\(13\) 0.147632 − 0.454364i 0.0409457 − 0.126018i −0.928494 − 0.371347i \(-0.878896\pi\)
0.969440 + 0.245329i \(0.0788960\pi\)
\(14\) 7.67121 + 5.57346i 2.05022 + 1.48957i
\(15\) −0.309017 + 0.951057i −0.0797878 + 0.245562i
\(16\) −0.947813 + 2.91707i −0.236953 + 0.729267i
\(17\) 4.23129 + 3.07421i 1.02624 + 0.745606i 0.967552 − 0.252670i \(-0.0813088\pi\)
0.0586863 + 0.998276i \(0.481309\pi\)
\(18\) 0.647481 − 1.99274i 0.152613 − 0.469694i
\(19\) −0.247788 − 0.762613i −0.0568465 − 0.174955i 0.918602 − 0.395185i \(-0.129319\pi\)
−0.975448 + 0.220229i \(0.929319\pi\)
\(20\) 1.93376 − 1.40496i 0.432402 − 0.314158i
\(21\) −1.39844 − 4.30395i −0.305164 − 0.939199i
\(22\) 3.69235 + 2.68265i 0.787212 + 0.571943i
\(23\) −3.88943 − 2.82584i −0.811003 − 0.589228i 0.103118 − 0.994669i \(-0.467118\pi\)
−0.914121 + 0.405441i \(0.867118\pi\)
\(24\) −0.661536 + 0.480634i −0.135035 + 0.0981089i
\(25\) 1.00000 0.200000
\(26\) 1.00102 0.196316
\(27\) −0.809017 + 0.587785i −0.155695 + 0.113119i
\(28\) −3.34263 + 10.2876i −0.631697 + 1.94416i
\(29\) 2.98371 + 9.18293i 0.554062 + 1.70523i 0.698408 + 0.715700i \(0.253892\pi\)
−0.144347 + 0.989527i \(0.546108\pi\)
\(30\) −2.09529 −0.382547
\(31\) −4.87061 − 2.69763i −0.874787 − 0.484508i
\(32\) −8.06206 −1.42518
\(33\) −0.673105 − 2.07161i −0.117173 − 0.360620i
\(34\) −3.38643 + 10.4224i −0.580769 + 1.78742i
\(35\) −3.66116 + 2.65999i −0.618849 + 0.449620i
\(36\) 2.39026 0.398376
\(37\) 3.49248 0.574160 0.287080 − 0.957907i \(-0.407315\pi\)
0.287080 + 0.957907i \(0.407315\pi\)
\(38\) 1.35925 − 0.987555i 0.220500 − 0.160203i
\(39\) −0.386505 − 0.280812i −0.0618903 − 0.0449659i
\(40\) 0.661536 + 0.480634i 0.104598 + 0.0759949i
\(41\) 0.332581 + 1.02358i 0.0519404 + 0.159856i 0.973662 − 0.227996i \(-0.0732172\pi\)
−0.921722 + 0.387852i \(0.873217\pi\)
\(42\) 7.67121 − 5.57346i 1.18369 − 0.860003i
\(43\) −2.28628 − 7.03645i −0.348654 − 1.07305i −0.959598 − 0.281374i \(-0.909210\pi\)
0.610944 − 0.791674i \(-0.290790\pi\)
\(44\) −1.60889 + 4.95167i −0.242550 + 0.746492i
\(45\) 0.809017 + 0.587785i 0.120601 + 0.0876219i
\(46\) 3.11283 − 9.58032i 0.458962 − 1.41254i
\(47\) −2.60226 + 8.00892i −0.379578 + 1.16822i 0.560759 + 0.827979i \(0.310509\pi\)
−0.940338 + 0.340243i \(0.889491\pi\)
\(48\) 2.48141 + 1.80285i 0.358160 + 0.260218i
\(49\) 4.16544 − 12.8199i 0.595063 − 1.83141i
\(50\) 0.647481 + 1.99274i 0.0915677 + 0.281816i
\(51\) 4.23129 − 3.07421i 0.592499 − 0.430476i
\(52\) 0.352878 + 1.08605i 0.0489353 + 0.150607i
\(53\) −0.357578 − 0.259796i −0.0491171 − 0.0356857i 0.562956 − 0.826487i \(-0.309664\pi\)
−0.612073 + 0.790801i \(0.709664\pi\)
\(54\) −1.69513 − 1.23158i −0.230678 − 0.167597i
\(55\) −1.76221 + 1.28032i −0.237617 + 0.172639i
\(56\) −3.70047 −0.494496
\(57\) −0.801859 −0.106209
\(58\) −16.3673 + 11.8915i −2.14913 + 1.56144i
\(59\) 1.18875 − 3.65859i 0.154762 − 0.476308i −0.843375 − 0.537326i \(-0.819435\pi\)
0.998137 + 0.0610176i \(0.0194346\pi\)
\(60\) −0.738630 − 2.27327i −0.0953567 − 0.293478i
\(61\) 2.03780 0.260914 0.130457 − 0.991454i \(-0.458356\pi\)
0.130457 + 0.991454i \(0.458356\pi\)
\(62\) 2.22205 − 11.4525i 0.282201 − 1.45447i
\(63\) −4.52544 −0.570152
\(64\) −3.32441 − 10.2315i −0.415551 − 1.27894i
\(65\) −0.147632 + 0.454364i −0.0183115 + 0.0563569i
\(66\) 3.69235 − 2.68265i 0.454497 − 0.330212i
\(67\) −1.98645 −0.242683 −0.121342 − 0.992611i \(-0.538720\pi\)
−0.121342 + 0.992611i \(0.538720\pi\)
\(68\) −12.5014 −1.51602
\(69\) −3.88943 + 2.82584i −0.468233 + 0.340191i
\(70\) −7.67121 − 5.57346i −0.916885 − 0.666156i
\(71\) −10.3007 − 7.48387i −1.22246 − 0.888172i −0.226161 − 0.974090i \(-0.572618\pi\)
−0.996302 + 0.0859180i \(0.972618\pi\)
\(72\) 0.252684 + 0.777682i 0.0297791 + 0.0916507i
\(73\) −4.24952 + 3.08746i −0.497369 + 0.361360i −0.808011 − 0.589167i \(-0.799456\pi\)
0.310642 + 0.950527i \(0.399456\pi\)
\(74\) 2.26132 + 6.95962i 0.262873 + 0.809039i
\(75\) 0.309017 − 0.951057i 0.0356822 − 0.109819i
\(76\) 1.55060 + 1.12658i 0.177866 + 0.129227i
\(77\) 3.04610 − 9.37493i 0.347135 − 1.06837i
\(78\) 0.309332 − 0.952025i 0.0350249 − 0.107796i
\(79\) −8.94128 − 6.49622i −1.00597 − 0.730882i −0.0426123 − 0.999092i \(-0.513568\pi\)
−0.963361 + 0.268210i \(0.913568\pi\)
\(80\) 0.947813 − 2.91707i 0.105969 − 0.326138i
\(81\) 0.309017 + 0.951057i 0.0343352 + 0.105673i
\(82\) −1.82439 + 1.32550i −0.201470 + 0.146377i
\(83\) 3.06335 + 9.42803i 0.336246 + 1.03486i 0.966105 + 0.258150i \(0.0831129\pi\)
−0.629858 + 0.776710i \(0.716887\pi\)
\(84\) 8.75111 + 6.35806i 0.954825 + 0.693721i
\(85\) −4.23129 − 3.07421i −0.458948 − 0.333445i
\(86\) 12.5415 − 9.11194i 1.35239 − 0.982566i
\(87\) 9.65550 1.03518
\(88\) −1.78113 −0.189869
\(89\) 3.78522 − 2.75012i 0.401232 − 0.291512i −0.368810 − 0.929505i \(-0.620235\pi\)
0.770043 + 0.637992i \(0.220235\pi\)
\(90\) −0.647481 + 1.99274i −0.0682505 + 0.210054i
\(91\) −0.668099 − 2.05620i −0.0700358 − 0.215548i
\(92\) 11.4914 1.19806
\(93\) −4.07070 + 3.79861i −0.422111 + 0.393897i
\(94\) −17.6446 −1.81991
\(95\) 0.247788 + 0.762613i 0.0254225 + 0.0782424i
\(96\) −2.49131 + 7.66748i −0.254269 + 0.782559i
\(97\) −9.12206 + 6.62756i −0.926205 + 0.672927i −0.945061 − 0.326895i \(-0.893998\pi\)
0.0188561 + 0.999822i \(0.493998\pi\)
\(98\) 28.2438 2.85306
\(99\) −2.17821 −0.218919
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.376.4 yes 16
31.8 even 5 inner 465.2.n.d.256.4 ✓ 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.n.d.256.4 ✓ 16 31.8 even 5 inner
465.2.n.d.376.4 yes 16 1.1 even 1 trivial