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(32,465)] 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("465.32"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(465, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [60,0,0,0,0,-4,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(30\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 32.2
Character \(\chi\) \(=\) 465.32
Dual form 465.2.k.b.218.2

$q$-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+(-1.79199 + 1.79199i) q^{2} +(-0.946283 + 1.45071i) q^{3} -4.42248i q^{4} +(-1.54734 - 1.61423i) q^{5} +(-0.903922 - 4.29539i) q^{6} +(1.64971 + 1.64971i) q^{7} +(4.34106 + 4.34106i) q^{8} +(-1.20910 - 2.74556i) q^{9} +(5.66551 + 0.119858i) q^{10} +0.702890i q^{11} +(6.41571 + 4.18492i) q^{12} +(3.87532 - 3.87532i) q^{13} -5.91254 q^{14} +(3.80599 - 0.717222i) q^{15} -6.71335 q^{16} +(-0.621687 + 0.621687i) q^{17} +(7.08671 + 2.75333i) q^{18} +3.66579i q^{19} +(-7.13889 + 6.84309i) q^{20} +(-3.95434 + 0.832152i) q^{21} +(-1.25957 - 1.25957i) q^{22} +(4.17587 + 4.17587i) q^{23} +(-10.4055 + 2.18973i) q^{24} +(-0.211462 + 4.99553i) q^{25} +13.8891i q^{26} +(5.12714 + 0.844033i) q^{27} +(7.29581 - 7.29581i) q^{28} +1.06976 q^{29} +(-5.53506 + 8.10557i) q^{30} -1.00000 q^{31} +(3.34816 - 3.34816i) q^{32} +(-1.01969 - 0.665133i) q^{33} -2.22812i q^{34} +(0.110341 - 5.21568i) q^{35} +(-12.1422 + 5.34720i) q^{36} +(-1.37372 - 1.37372i) q^{37} +(-6.56908 - 6.56908i) q^{38} +(1.95480 + 9.28910i) q^{39} +(0.290353 - 13.7246i) q^{40} -10.9953i q^{41} +(5.59494 - 8.57736i) q^{42} +(-6.40788 + 6.40788i) q^{43} +3.10852 q^{44} +(-2.56107 + 6.20007i) q^{45} -14.9663 q^{46} +(-2.39962 + 2.39962i) q^{47} +(6.35273 - 9.73910i) q^{48} -1.55690i q^{49} +(-8.57301 - 9.33089i) q^{50} +(-0.313593 - 1.49018i) q^{51} +(-17.1385 - 17.1385i) q^{52} +(7.84103 + 7.84103i) q^{53} +(-10.7003 + 7.67530i) q^{54} +(1.13462 - 1.08761i) q^{55} +14.3230i q^{56} +(-5.31799 - 3.46888i) q^{57} +(-1.91700 + 1.91700i) q^{58} +8.48651 q^{59} +(-3.17190 - 16.8319i) q^{60} +10.2786 q^{61} +(1.79199 - 1.79199i) q^{62} +(2.53472 - 6.52404i) q^{63} -1.42696i q^{64} +(-12.2521 - 0.259202i) q^{65} +(3.01919 - 0.635358i) q^{66} +(7.06743 + 7.06743i) q^{67} +(2.74940 + 2.74940i) q^{68} +(-10.0095 + 2.10640i) q^{69} +(9.14873 + 9.54419i) q^{70} -0.732223i q^{71} +(6.66988 - 17.1674i) q^{72} +(-8.32802 + 8.32802i) q^{73} +4.92339 q^{74} +(-7.04694 - 5.03395i) q^{75} +16.2119 q^{76} +(-1.15957 + 1.15957i) q^{77} +(-20.1490 - 13.1430i) q^{78} +14.1455i q^{79} +(10.3879 + 10.8369i) q^{80} +(-6.07618 + 6.63928i) q^{81} +(19.7036 + 19.7036i) q^{82} +(8.24844 + 8.24844i) q^{83} +(3.68017 + 17.4880i) q^{84} +(1.96551 + 0.0415816i) q^{85} -22.9658i q^{86} +(-1.01229 + 1.55190i) q^{87} +(-3.05129 + 3.05129i) q^{88} +1.63716 q^{89} +(-6.52107 - 15.6999i) q^{90} +12.7863 q^{91} +(18.4677 - 18.4677i) q^{92} +(0.946283 - 1.45071i) q^{93} -8.60021i q^{94} +(5.91742 - 5.67224i) q^{95} +(1.68889 + 8.02549i) q^{96} +(-1.46919 - 1.46919i) q^{97} +(2.78996 + 2.78996i) q^{98} +(1.92983 - 0.849862i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 60 q - 4 q^{6} + 16 q^{10} - 12 q^{13} - 8 q^{15} - 60 q^{16} + 34 q^{18} - 4 q^{21} + 8 q^{22} + 8 q^{25} - 6 q^{27} - 80 q^{28} - 54 q^{30} - 60 q^{31} + 10 q^{33} + 28 q^{36} - 12 q^{37} + 60 q^{40}+ \cdots + 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(187\) \(311\) \(406\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.79199 + 1.79199i −1.26713 + 1.26713i −0.319567 + 0.947564i \(0.603537\pi\)
−0.947564 + 0.319567i \(0.896463\pi\)
\(3\) −0.946283 + 1.45071i −0.546337 + 0.837565i
\(4\) − 4.42248i − 2.21124i
\(5\) −1.54734 − 1.61423i −0.691993 − 0.721905i
\(6\) −0.903922 − 4.29539i −0.369025 − 1.75358i
\(7\) 1.64971 + 1.64971i 0.623533 + 0.623533i 0.946433 − 0.322900i \(-0.104658\pi\)
−0.322900 + 0.946433i \(0.604658\pi\)
\(8\) 4.34106 + 4.34106i 1.53480 + 1.53480i
\(9\) −1.20910 − 2.74556i −0.403032 − 0.915186i
\(10\) 5.66551 + 0.119858i 1.79159 + 0.0379024i
\(11\) 0.702890i 0.211929i 0.994370 + 0.105965i \(0.0337931\pi\)
−0.994370 + 0.105965i \(0.966207\pi\)
\(12\) 6.41571 + 4.18492i 1.85206 + 1.20808i
\(13\) 3.87532 − 3.87532i 1.07482 − 1.07482i 0.0778557 − 0.996965i \(-0.475193\pi\)
0.996965 − 0.0778557i \(-0.0248073\pi\)
\(14\) −5.91254 −1.58019
\(15\) 3.80599 − 0.717222i 0.982703 − 0.185186i
\(16\) −6.71335 −1.67834
\(17\) −0.621687 + 0.621687i −0.150781 + 0.150781i −0.778467 − 0.627686i \(-0.784002\pi\)
0.627686 + 0.778467i \(0.284002\pi\)
\(18\) 7.08671 + 2.75333i 1.67035 + 0.648966i
\(19\) 3.66579i 0.840991i 0.907295 + 0.420495i \(0.138144\pi\)
−0.907295 + 0.420495i \(0.861856\pi\)
\(20\) −7.13889 + 6.84309i −1.59630 + 1.53016i
\(21\) −3.95434 + 0.832152i −0.862908 + 0.181590i
\(22\) −1.25957 − 1.25957i −0.268542 − 0.268542i
\(23\) 4.17587 + 4.17587i 0.870729 + 0.870729i 0.992552 − 0.121823i \(-0.0388740\pi\)
−0.121823 + 0.992552i \(0.538874\pi\)
\(24\) −10.4055 + 2.18973i −2.12401 + 0.446977i
\(25\) −0.211462 + 4.99553i −0.0422924 + 0.999105i
\(26\) 13.8891i 2.72387i
\(27\) 5.12714 + 0.844033i 0.986719 + 0.162434i
\(28\) 7.29581 − 7.29581i 1.37878 − 1.37878i
\(29\) 1.06976 0.198649 0.0993245 − 0.995055i \(-0.468332\pi\)
0.0993245 + 0.995055i \(0.468332\pi\)
\(30\) −5.53506 + 8.10557i −1.01056 + 1.47987i
\(31\) −1.00000 −0.179605
\(32\) 3.34816 − 3.34816i 0.591876 − 0.591876i
\(33\) −1.01969 − 0.665133i −0.177505 − 0.115785i
\(34\) − 2.22812i − 0.382119i
\(35\) 0.110341 − 5.21568i 0.0186511 − 0.881611i
\(36\) −12.1422 + 5.34720i −2.02369 + 0.891200i
\(37\) −1.37372 − 1.37372i −0.225838 − 0.225838i 0.585113 − 0.810952i \(-0.301050\pi\)
−0.810952 + 0.585113i \(0.801050\pi\)
\(38\) −6.56908 − 6.56908i −1.06564 − 1.06564i
\(39\) 1.95480 + 9.28910i 0.313018 + 1.48745i
\(40\) 0.290353 − 13.7246i 0.0459088 − 2.17005i
\(41\) − 10.9953i − 1.71718i −0.512661 − 0.858591i \(-0.671340\pi\)
0.512661 − 0.858591i \(-0.328660\pi\)
\(42\) 5.59494 − 8.57736i 0.863318 − 1.32352i
\(43\) −6.40788 + 6.40788i −0.977193 + 0.977193i −0.999746 − 0.0225528i \(-0.992821\pi\)
0.0225528 + 0.999746i \(0.492821\pi\)
\(44\) 3.10852 0.468626
\(45\) −2.56107 + 6.20007i −0.381782 + 0.924252i
\(46\) −14.9663 −2.20665
\(47\) −2.39962 + 2.39962i −0.350021 + 0.350021i −0.860117 − 0.510096i \(-0.829610\pi\)
0.510096 + 0.860117i \(0.329610\pi\)
\(48\) 6.35273 − 9.73910i 0.916938 − 1.40572i
\(49\) − 1.55690i − 0.222414i
\(50\) −8.57301 − 9.33089i −1.21241 − 1.31959i
\(51\) −0.313593 − 1.49018i −0.0439118 − 0.208666i
\(52\) −17.1385 − 17.1385i −2.37668 − 2.37668i
\(53\) 7.84103 + 7.84103i 1.07705 + 1.07705i 0.996773 + 0.0802754i \(0.0255800\pi\)
0.0802754 + 0.996773i \(0.474420\pi\)
\(54\) −10.7003 + 7.67530i −1.45613 + 1.04448i
\(55\) 1.13462 − 1.08761i 0.152993 − 0.146654i
\(56\) 14.3230i 1.91399i
\(57\) −5.31799 − 3.46888i −0.704385 − 0.459464i
\(58\) −1.91700 + 1.91700i −0.251714 + 0.251714i
\(59\) 8.48651 1.10485 0.552425 − 0.833563i \(-0.313703\pi\)
0.552425 + 0.833563i \(0.313703\pi\)
\(60\) −3.17190 − 16.8319i −0.409490 − 2.17299i
\(61\) 10.2786 1.31604 0.658019 − 0.753001i \(-0.271394\pi\)
0.658019 + 0.753001i \(0.271394\pi\)
\(62\) 1.79199 − 1.79199i 0.227583 − 0.227583i
\(63\) 2.53472 − 6.52404i 0.319345 − 0.821952i
\(64\) − 1.42696i − 0.178371i
\(65\) −12.2521 − 0.259202i −1.51969 − 0.0321500i
\(66\) 3.01919 − 0.635358i 0.371636 − 0.0782071i
\(67\) 7.06743 + 7.06743i 0.863424 + 0.863424i 0.991734 − 0.128311i \(-0.0409554\pi\)
−0.128311 + 0.991734i \(0.540955\pi\)
\(68\) 2.74940 + 2.74940i 0.333413 + 0.333413i
\(69\) −10.0095 + 2.10640i −1.20500 + 0.253581i
\(70\) 9.14873 + 9.54419i 1.09348 + 1.14075i
\(71\) − 0.732223i − 0.0868988i −0.999056 − 0.0434494i \(-0.986165\pi\)
0.999056 − 0.0434494i \(-0.0138347\pi\)
\(72\) 6.66988 − 17.1674i 0.786053 − 2.02320i
\(73\) −8.32802 + 8.32802i −0.974721 + 0.974721i −0.999688 − 0.0249672i \(-0.992052\pi\)
0.0249672 + 0.999688i \(0.492052\pi\)
\(74\) 4.92339 0.572333
\(75\) −7.04694 − 5.03395i −0.813710 − 0.581271i
\(76\) 16.2119 1.85963
\(77\) −1.15957 + 1.15957i −0.132145 + 0.132145i
\(78\) −20.1490 − 13.1430i −2.28142 − 1.48815i
\(79\) 14.1455i 1.59150i 0.605627 + 0.795749i \(0.292922\pi\)
−0.605627 + 0.795749i \(0.707078\pi\)
\(80\) 10.3879 + 10.8369i 1.16140 + 1.21160i
\(81\) −6.07618 + 6.63928i −0.675131 + 0.737698i
\(82\) 19.7036 + 19.7036i 2.17589 + 2.17589i
\(83\) 8.24844 + 8.24844i 0.905384 + 0.905384i 0.995895 − 0.0905119i \(-0.0288503\pi\)
−0.0905119 + 0.995895i \(0.528850\pi\)
\(84\) 3.68017 + 17.4880i 0.401540 + 1.90810i
\(85\) 1.96551 + 0.0415816i 0.213189 + 0.00451016i
\(86\) − 22.9658i − 2.47646i
\(87\) −1.01229 + 1.55190i −0.108529 + 0.166382i
\(88\) −3.05129 + 3.05129i −0.325269 + 0.325269i
\(89\) 1.63716 0.173539 0.0867694 − 0.996228i \(-0.472346\pi\)
0.0867694 + 0.996228i \(0.472346\pi\)
\(90\) −6.52107 − 15.6999i −0.687381 − 1.65492i
\(91\) 12.7863 1.34037
\(92\) 18.4677 − 18.4677i 1.92539 − 1.92539i
\(93\) 0.946283 − 1.45071i 0.0981250 − 0.150431i
\(94\) − 8.60021i − 0.887044i
\(95\) 5.91742 − 5.67224i 0.607115 − 0.581959i
\(96\) 1.68889 + 8.02549i 0.172371 + 0.819098i
\(97\) −1.46919 − 1.46919i −0.149173 − 0.149173i 0.628575 − 0.777749i \(-0.283638\pi\)
−0.777749 + 0.628575i \(0.783638\pi\)
\(98\) 2.78996 + 2.78996i 0.281828 + 0.281828i
\(99\) 1.92983 − 0.849862i 0.193955 − 0.0854143i
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.k.b.32.2 ✓ 60
3.2 odd 2 inner 465.2.k.b.32.29 yes 60
5.3 odd 4 inner 465.2.k.b.218.29 yes 60
15.8 even 4 inner 465.2.k.b.218.2 yes 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.k.b.32.2 ✓ 60 1.1 even 1 trivial
465.2.k.b.32.29 yes 60 3.2 odd 2 inner
465.2.k.b.218.2 yes 60 15.8 even 4 inner
465.2.k.b.218.29 yes 60 5.3 odd 4 inner