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Properties

Label 465.2.ba.a.4.9
Level $465$
Weight $2$
Character 465.4
Analytic conductor $3.713$
Analytic rank $0$
Dimension $128$
Inner twists $4$

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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(4,465)] 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("465.4"); 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, 5, 6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.ba (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(32\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label 4.9
Character \(\chi\) \(=\) 465.4
Dual form 465.2.ba.a.349.9

$q$-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+(-1.36240 + 0.442671i) q^{2} +(-0.951057 - 0.309017i) q^{3} +(0.0421447 - 0.0306199i) q^{4} +(2.03530 - 0.926033i) q^{5} +1.43251 q^{6} +(-1.00919 - 1.38903i) q^{7} +(1.64016 - 2.25748i) q^{8} +(0.809017 + 0.587785i) q^{9} +(-2.36297 + 2.16260i) q^{10} +(-2.35160 + 1.70854i) q^{11} +(-0.0495441 + 0.0160979i) q^{12} +(0.880226 + 0.286003i) q^{13} +(1.98980 + 1.44567i) q^{14} +(-2.22185 + 0.251766i) q^{15} +(-1.26743 + 3.90073i) q^{16} +(-0.872706 + 1.20118i) q^{17} +(-1.36240 - 0.442671i) q^{18} +(-2.54894 - 7.84482i) q^{19} +(0.0574223 - 0.101348i) q^{20} +(0.530561 + 1.63290i) q^{21} +(2.44750 - 3.36870i) q^{22} +(-4.10960 + 5.65638i) q^{23} +(-2.25748 + 1.64016i) q^{24} +(3.28493 - 3.76952i) q^{25} -1.32583 q^{26} +(-0.587785 - 0.809017i) q^{27} +(-0.0850638 - 0.0276389i) q^{28} +(-1.36200 - 4.19179i) q^{29} +(2.91560 - 1.32655i) q^{30} +(3.31973 - 4.46983i) q^{31} -0.294613i q^{32} +(2.76447 - 0.898231i) q^{33} +(0.657250 - 2.02281i) q^{34} +(-3.34029 - 1.89255i) q^{35} +0.0520938 q^{36} -8.97455i q^{37} +(6.94534 + 9.55945i) q^{38} +(-0.748765 - 0.544010i) q^{39} +(1.24771 - 6.11350i) q^{40} +(-1.55764 - 4.79392i) q^{41} +(-1.44567 - 1.98980i) q^{42} +(-4.01068 + 1.30315i) q^{43} +(-0.0467923 + 0.144012i) q^{44} +(2.19090 + 0.447145i) q^{45} +(3.09501 - 9.52546i) q^{46} +(-8.32489 - 2.70492i) q^{47} +(2.41079 - 3.31816i) q^{48} +(1.25218 - 3.85382i) q^{49} +(-2.80673 + 6.58974i) q^{50} +(1.20118 - 0.872706i) q^{51} +(0.0458543 - 0.0148990i) q^{52} +(4.73117 - 6.51190i) q^{53} +(1.15893 + 0.842010i) q^{54} +(-3.20406 + 5.65505i) q^{55} -4.79093 q^{56} +8.24853i q^{57} +(3.71117 + 5.10799i) q^{58} +(-0.267482 + 0.823226i) q^{59} +(-0.0859302 + 0.0786435i) q^{60} -13.2122 q^{61} +(-2.54414 + 7.55925i) q^{62} -1.71693i q^{63} +(-2.40443 - 7.40009i) q^{64} +(2.05638 - 0.233016i) q^{65} +(-3.36870 + 2.44750i) q^{66} -7.44875i q^{67} +0.0773455i q^{68} +(5.65638 - 4.10960i) q^{69} +(5.38859 + 1.09977i) q^{70} +(10.3705 + 7.53458i) q^{71} +(2.65383 - 0.862281i) q^{72} +(9.02915 + 12.4276i) q^{73} +(3.97277 + 12.2269i) q^{74} +(-4.28899 + 2.56993i) q^{75} +(-0.347632 - 0.252570i) q^{76} +(4.74641 + 1.54220i) q^{77} +(1.26094 + 0.409703i) q^{78} +(-11.7452 - 8.53342i) q^{79} +(1.03261 + 9.11286i) q^{80} +(0.309017 + 0.951057i) q^{81} +(4.24426 + 5.84172i) q^{82} +(0.277881 - 0.0902892i) q^{83} +(0.0723596 + 0.0525723i) q^{84} +(-0.663893 + 3.25292i) q^{85} +(4.88729 - 3.55082i) q^{86} +4.40751i q^{87} +8.11096i q^{88} +(-8.47948 + 6.16070i) q^{89} +(-3.18283 + 0.360658i) q^{90} +(-0.491047 - 1.51129i) q^{91} +0.364223i q^{92} +(-4.53851 + 3.22521i) q^{93} +12.5392 q^{94} +(-12.4524 - 13.6062i) q^{95} +(-0.0910403 + 0.280193i) q^{96} +(1.27022 + 1.74831i) q^{97} +5.80475i q^{98} -2.90674 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q + 36 q^{4} - 20 q^{6} + 32 q^{9} + 2 q^{10} + 4 q^{14} - 4 q^{15} - 40 q^{16} - 4 q^{19} + 36 q^{20} + 20 q^{21} - 20 q^{24} + 12 q^{25} + 24 q^{26} - 28 q^{29} - 8 q^{30} - 12 q^{31} - 44 q^{34}+ \cdots - 20 q^{96}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.36240 + 0.442671i −0.963363 + 0.313016i −0.748134 − 0.663548i \(-0.769050\pi\)
−0.215229 + 0.976564i \(0.569050\pi\)
\(3\) −0.951057 − 0.309017i −0.549093 − 0.178411i
\(4\) 0.0421447 − 0.0306199i 0.0210724 − 0.0153100i
\(5\) 2.03530 − 0.926033i 0.910216 − 0.414135i
\(6\) 1.43251 0.584821
\(7\) −1.00919 − 1.38903i −0.381437 − 0.525003i 0.574528 − 0.818485i \(-0.305186\pi\)
−0.955964 + 0.293482i \(0.905186\pi\)
\(8\) 1.64016 − 2.25748i 0.579883 − 0.798140i
\(9\) 0.809017 + 0.587785i 0.269672 + 0.195928i
\(10\) −2.36297 + 2.16260i −0.747237 + 0.683874i
\(11\) −2.35160 + 1.70854i −0.709034 + 0.515143i −0.882862 − 0.469633i \(-0.844386\pi\)
0.173828 + 0.984776i \(0.444386\pi\)
\(12\) −0.0495441 + 0.0160979i −0.0143022 + 0.00464705i
\(13\) 0.880226 + 0.286003i 0.244131 + 0.0793229i 0.428527 − 0.903529i \(-0.359033\pi\)
−0.184396 + 0.982852i \(0.559033\pi\)
\(14\) 1.98980 + 1.44567i 0.531796 + 0.386372i
\(15\) −2.22185 + 0.251766i −0.573679 + 0.0650058i
\(16\) −1.26743 + 3.90073i −0.316856 + 0.975183i
\(17\) −0.872706 + 1.20118i −0.211662 + 0.291328i −0.901627 − 0.432515i \(-0.857626\pi\)
0.689964 + 0.723843i \(0.257626\pi\)
\(18\) −1.36240 − 0.442671i −0.321121 − 0.104339i
\(19\) −2.54894 − 7.84482i −0.584766 − 1.79972i −0.600208 − 0.799844i \(-0.704916\pi\)
0.0154424 − 0.999881i \(-0.495084\pi\)
\(20\) 0.0574223 − 0.101348i 0.0128400 − 0.0226622i
\(21\) 0.530561 + 1.63290i 0.115778 + 0.356328i
\(22\) 2.44750 − 3.36870i 0.521809 − 0.718209i
\(23\) −4.10960 + 5.65638i −0.856911 + 1.17944i 0.125386 + 0.992108i \(0.459983\pi\)
−0.982297 + 0.187329i \(0.940017\pi\)
\(24\) −2.25748 + 1.64016i −0.460807 + 0.334796i
\(25\) 3.28493 − 3.76952i 0.656985 − 0.753904i
\(26\) −1.32583 −0.260016
\(27\) −0.587785 − 0.809017i −0.113119 − 0.155695i
\(28\) −0.0850638 − 0.0276389i −0.0160756 − 0.00522326i
\(29\) −1.36200 − 4.19179i −0.252916 − 0.778396i −0.994233 − 0.107241i \(-0.965798\pi\)
0.741317 − 0.671155i \(-0.234202\pi\)
\(30\) 2.91560 − 1.32655i 0.532313 − 0.242195i
\(31\) 3.31973 − 4.46983i 0.596241 − 0.802805i
\(32\) − 0.294613i − 0.0520806i
\(33\) 2.76447 − 0.898231i 0.481233 − 0.156362i
\(34\) 0.657250 − 2.02281i 0.112717 − 0.346908i
\(35\) −3.34029 − 1.89255i −0.564611 − 0.319899i
\(36\) 0.0520938 0.00868229
\(37\) − 8.97455i − 1.47541i −0.675125 − 0.737703i \(-0.735910\pi\)
0.675125 − 0.737703i \(-0.264090\pi\)
\(38\) 6.94534 + 9.55945i 1.12668 + 1.55075i
\(39\) −0.748765 − 0.544010i −0.119898 − 0.0871113i
\(40\) 1.24771 − 6.11350i 0.197281 − 0.966629i
\(41\) −1.55764 − 4.79392i −0.243262 − 0.748684i −0.995917 − 0.0902694i \(-0.971227\pi\)
0.752655 − 0.658415i \(-0.228773\pi\)
\(42\) −1.44567 − 1.98980i −0.223072 − 0.307033i
\(43\) −4.01068 + 1.30315i −0.611623 + 0.198728i −0.598418 − 0.801184i \(-0.704204\pi\)
−0.0132057 + 0.999913i \(0.504204\pi\)
\(44\) −0.0467923 + 0.144012i −0.00705420 + 0.0217106i
\(45\) 2.19090 + 0.447145i 0.326601 + 0.0666565i
\(46\) 3.09501 − 9.52546i 0.456334 − 1.40445i
\(47\) −8.32489 − 2.70492i −1.21431 − 0.394553i −0.369303 − 0.929309i \(-0.620404\pi\)
−0.845007 + 0.534756i \(0.820404\pi\)
\(48\) 2.41079 − 3.31816i 0.347967 − 0.478935i
\(49\) 1.25218 − 3.85382i 0.178883 − 0.550546i
\(50\) −2.80673 + 6.58974i −0.396931 + 0.931929i
\(51\) 1.20118 − 0.872706i 0.168198 − 0.122203i
\(52\) 0.0458543 − 0.0148990i 0.00635885 − 0.00206612i
\(53\) 4.73117 − 6.51190i 0.649876 − 0.894478i −0.349218 − 0.937042i \(-0.613553\pi\)
0.999094 + 0.0425640i \(0.0135526\pi\)
\(54\) 1.15893 + 0.842010i 0.157710 + 0.114583i
\(55\) −3.20406 + 5.65505i −0.432035 + 0.762527i
\(56\) −4.79093 −0.640214
\(57\) 8.24853i 1.09254i
\(58\) 3.71117 + 5.10799i 0.487300 + 0.670711i
\(59\) −0.267482 + 0.823226i −0.0348232 + 0.107175i −0.966957 − 0.254939i \(-0.917945\pi\)
0.932134 + 0.362114i \(0.117945\pi\)
\(60\) −0.0859302 + 0.0786435i −0.0110935 + 0.0101528i
\(61\) −13.2122 −1.69165 −0.845826 − 0.533459i \(-0.820892\pi\)
−0.845826 + 0.533459i \(0.820892\pi\)
\(62\) −2.54414 + 7.55925i −0.323106 + 0.960026i
\(63\) − 1.71693i − 0.216313i
\(64\) −2.40443 − 7.40009i −0.300554 − 0.925011i
\(65\) 2.05638 − 0.233016i 0.255062 − 0.0289021i
\(66\) −3.36870 + 2.44750i −0.414658 + 0.301267i
\(67\) − 7.44875i − 0.910009i −0.890489 − 0.455005i \(-0.849638\pi\)
0.890489 − 0.455005i \(-0.150362\pi\)
\(68\) 0.0773455i 0.00937952i
\(69\) 5.65638 − 4.10960i 0.680948 − 0.494738i
\(70\) 5.38859 + 1.09977i 0.644059 + 0.131447i
\(71\) 10.3705 + 7.53458i 1.23075 + 0.894190i 0.996946 − 0.0780950i \(-0.0248838\pi\)
0.233800 + 0.972285i \(0.424884\pi\)
\(72\) 2.65383 − 0.862281i 0.312757 − 0.101621i
\(73\) 9.02915 + 12.4276i 1.05678 + 1.45454i 0.882778 + 0.469791i \(0.155671\pi\)
0.174004 + 0.984745i \(0.444329\pi\)
\(74\) 3.97277 + 12.2269i 0.461825 + 1.42135i
\(75\) −4.28899 + 2.56993i −0.495250 + 0.296750i
\(76\) −0.347632 − 0.252570i −0.0398761 − 0.0289717i
\(77\) 4.74641 + 1.54220i 0.540903 + 0.175750i
\(78\) 1.26094 + 0.409703i 0.142773 + 0.0463897i
\(79\) −11.7452 − 8.53342i −1.32144 − 0.960085i −0.999913 − 0.0131852i \(-0.995803\pi\)
−0.321530 − 0.946899i \(-0.604197\pi\)
\(80\) 1.03261 + 9.11286i 0.115450 + 1.01885i
\(81\) 0.309017 + 0.951057i 0.0343352 + 0.105673i
\(82\) 4.24426 + 5.84172i 0.468700 + 0.645110i
\(83\) 0.277881 − 0.0902892i 0.0305015 − 0.00991052i −0.293727 − 0.955889i \(-0.594896\pi\)
0.324228 + 0.945979i \(0.394896\pi\)
\(84\) 0.0723596 + 0.0525723i 0.00789508 + 0.00573611i
\(85\) −0.663893 + 3.25292i −0.0720093 + 0.352828i
\(86\) 4.88729 − 3.55082i 0.527010 − 0.382895i
\(87\) 4.40751i 0.472535i
\(88\) 8.11096i 0.864632i
\(89\) −8.47948 + 6.16070i −0.898823 + 0.653033i −0.938163 − 0.346193i \(-0.887474\pi\)
0.0393407 + 0.999226i \(0.487474\pi\)
\(90\) −3.18283 + 0.360658i −0.335500 + 0.0380167i
\(91\) −0.491047 − 1.51129i −0.0514757 − 0.158426i
\(92\) 0.364223i 0.0379728i
\(93\) −4.53851 + 3.22521i −0.470621 + 0.334439i
\(94\) 12.5392 1.29332
\(95\) −12.4524 − 13.6062i −1.27759 − 1.39597i
\(96\) −0.0910403 + 0.280193i −0.00929176 + 0.0285971i
\(97\) 1.27022 + 1.74831i 0.128971 + 0.177514i 0.868619 − 0.495480i \(-0.165008\pi\)
−0.739648 + 0.672994i \(0.765008\pi\)
\(98\) 5.80475i 0.586369i
\(99\) −2.90674 −0.292138
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.ba.a.4.9 ✓ 128
5.4 even 2 inner 465.2.ba.a.4.24 yes 128
31.8 even 5 inner 465.2.ba.a.349.24 yes 128
155.39 even 10 inner 465.2.ba.a.349.9 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.ba.a.4.9 ✓ 128 1.1 even 1 trivial
465.2.ba.a.4.24 yes 128 5.4 even 2 inner
465.2.ba.a.349.9 yes 128 155.39 even 10 inner
465.2.ba.a.349.24 yes 128 31.8 even 5 inner