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Properties

Label 465.2.ba.a.4.8
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 4.8
Character \(\chi\) \(=\) 465.4
Dual form 465.2.ba.a.349.8

$q$-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+(-1.71190 + 0.556229i) q^{2} +(0.951057 + 0.309017i) q^{3} +(1.00316 - 0.728840i) q^{4} +(-1.35623 - 1.77782i) q^{5} -1.79999 q^{6} +(-2.23429 - 3.07523i) q^{7} +(0.804112 - 1.10677i) q^{8} +(0.809017 + 0.587785i) q^{9} +(3.31060 + 2.28907i) q^{10} +(-3.35309 + 2.43616i) q^{11} +(1.17929 - 0.383174i) q^{12} +(2.54667 + 0.827464i) q^{13} +(5.53539 + 4.02170i) q^{14} +(-0.740473 - 2.10991i) q^{15} +(-1.52729 + 4.70051i) q^{16} +(-4.27966 + 5.89045i) q^{17} +(-1.71190 - 0.556229i) q^{18} +(2.37198 + 7.30021i) q^{19} +(-2.65626 - 0.794968i) q^{20} +(-1.17463 - 3.61515i) q^{21} +(4.38507 - 6.03553i) q^{22} +(-2.34599 + 3.22898i) q^{23} +(1.10677 - 0.804112i) q^{24} +(-1.32129 + 4.82226i) q^{25} -4.81990 q^{26} +(0.587785 + 0.809017i) q^{27} +(-4.48270 - 1.45652i) q^{28} +(-0.555054 - 1.70828i) q^{29} +(2.44120 + 3.20006i) q^{30} +(5.42397 - 1.25722i) q^{31} -6.16023i q^{32} +(-3.94179 + 1.28076i) q^{33} +(4.04990 - 12.4643i) q^{34} +(-2.43700 + 8.14287i) q^{35} +1.23998 q^{36} -5.87822i q^{37} +(-8.12117 - 11.1778i) q^{38} +(2.16633 + 1.57393i) q^{39} +(-3.05819 + 0.0714597i) q^{40} +(0.819508 + 2.52219i) q^{41} +(4.02170 + 5.53539i) q^{42} +(-1.47548 + 0.479414i) q^{43} +(-1.58812 + 4.88773i) q^{44} +(-0.0522352 - 2.23546i) q^{45} +(2.22004 - 6.83259i) q^{46} +(-0.335158 - 0.108900i) q^{47} +(-2.90508 + 3.99849i) q^{48} +(-2.30189 + 7.08448i) q^{49} +(-0.420369 - 8.99014i) q^{50} +(-5.89045 + 4.27966i) q^{51} +(3.15781 - 1.02604i) q^{52} +(-0.521920 + 0.718361i) q^{53} +(-1.45623 - 1.05801i) q^{54} +(8.87860 + 2.65720i) q^{55} -5.20018 q^{56} +7.67590i q^{57} +(1.90039 + 2.61566i) q^{58} +(0.620031 - 1.90826i) q^{59} +(-2.28060 - 1.57689i) q^{60} -11.3748 q^{61} +(-8.58596 + 5.16919i) q^{62} -3.80119i q^{63} +(0.371919 + 1.14465i) q^{64} +(-1.98279 - 5.64976i) q^{65} +(6.03553 - 4.38507i) q^{66} +14.2570i q^{67} +9.02826i q^{68} +(-3.22898 + 2.34599i) q^{69} +(-0.357400 - 15.2953i) q^{70} +(-5.56772 - 4.04519i) q^{71} +(1.30108 - 0.422747i) q^{72} +(-4.99304 - 6.87234i) q^{73} +(3.26963 + 10.0629i) q^{74} +(-2.74678 + 4.17794i) q^{75} +(7.70017 + 5.59450i) q^{76} +(14.9835 + 4.86843i) q^{77} +(-4.58400 - 1.48943i) q^{78} +(9.30430 + 6.75997i) q^{79} +(10.4280 - 3.65972i) q^{80} +(0.309017 + 0.951057i) q^{81} +(-2.80582 - 3.86189i) q^{82} +(-6.15134 + 1.99869i) q^{83} +(-3.81321 - 2.77046i) q^{84} +(16.2764 - 0.380324i) q^{85} +(2.25921 - 1.64141i) q^{86} -1.79619i q^{87} +5.67003i q^{88} +(-2.50174 + 1.81762i) q^{89} +(1.33285 + 3.79782i) q^{90} +(-3.14535 - 9.68040i) q^{91} +4.94905i q^{92} +(5.54700 + 0.480411i) q^{93} +0.634329 q^{94} +(9.76151 - 14.1177i) q^{95} +(1.90362 - 5.85873i) q^{96} +(-9.05917 - 12.4689i) q^{97} -13.4083i q^{98} -4.14464 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.71190 + 0.556229i −1.21049 + 0.393313i −0.843611 − 0.536955i \(-0.819575\pi\)
−0.366882 + 0.930268i \(0.619575\pi\)
\(3\) 0.951057 + 0.309017i 0.549093 + 0.178411i
\(4\) 1.00316 − 0.728840i 0.501581 − 0.364420i
\(5\) −1.35623 − 1.77782i −0.606524 − 0.795065i
\(6\) −1.79999 −0.734844
\(7\) −2.23429 − 3.07523i −0.844480 − 1.16233i −0.985052 − 0.172257i \(-0.944894\pi\)
0.140572 − 0.990071i \(-0.455106\pi\)
\(8\) 0.804112 − 1.10677i 0.284297 − 0.391301i
\(9\) 0.809017 + 0.587785i 0.269672 + 0.195928i
\(10\) 3.31060 + 2.28907i 1.04690 + 0.723867i
\(11\) −3.35309 + 2.43616i −1.01099 + 0.734530i −0.964417 − 0.264385i \(-0.914831\pi\)
−0.0465763 + 0.998915i \(0.514831\pi\)
\(12\) 1.17929 − 0.383174i 0.340431 − 0.110613i
\(13\) 2.54667 + 0.827464i 0.706320 + 0.229497i 0.640082 − 0.768307i \(-0.278900\pi\)
0.0662381 + 0.997804i \(0.478900\pi\)
\(14\) 5.53539 + 4.02170i 1.47940 + 1.07484i
\(15\) −0.740473 − 2.10991i −0.191189 − 0.544775i
\(16\) −1.52729 + 4.70051i −0.381822 + 1.17513i
\(17\) −4.27966 + 5.89045i −1.03797 + 1.42864i −0.139178 + 0.990267i \(0.544446\pi\)
−0.898792 + 0.438376i \(0.855554\pi\)
\(18\) −1.71190 − 0.556229i −0.403498 − 0.131104i
\(19\) 2.37198 + 7.30021i 0.544170 + 1.67478i 0.722955 + 0.690895i \(0.242783\pi\)
−0.178785 + 0.983888i \(0.557217\pi\)
\(20\) −2.65626 − 0.794968i −0.593958 − 0.177760i
\(21\) −1.17463 − 3.61515i −0.256326 − 0.788890i
\(22\) 4.38507 − 6.03553i 0.934900 − 1.28678i
\(23\) −2.34599 + 3.22898i −0.489174 + 0.673290i −0.980235 − 0.197835i \(-0.936609\pi\)
0.491062 + 0.871125i \(0.336609\pi\)
\(24\) 1.10677 − 0.804112i 0.225918 − 0.164139i
\(25\) −1.32129 + 4.82226i −0.264258 + 0.964452i
\(26\) −4.81990 −0.945260
\(27\) 0.587785 + 0.809017i 0.113119 + 0.155695i
\(28\) −4.48270 − 1.45652i −0.847151 − 0.275256i
\(29\) −0.555054 − 1.70828i −0.103071 − 0.317220i 0.886202 − 0.463299i \(-0.153335\pi\)
−0.989273 + 0.146079i \(0.953335\pi\)
\(30\) 2.44120 + 3.20006i 0.445700 + 0.584249i
\(31\) 5.42397 − 1.25722i 0.974173 − 0.225803i
\(32\) − 6.16023i − 1.08899i
\(33\) −3.94179 + 1.28076i −0.686177 + 0.222953i
\(34\) 4.04990 − 12.4643i 0.694551 − 2.13761i
\(35\) −2.43700 + 8.14287i −0.411929 + 1.37640i
\(36\) 1.23998 0.206663
\(37\) − 5.87822i − 0.966374i −0.875517 − 0.483187i \(-0.839479\pi\)
0.875517 − 0.483187i \(-0.160521\pi\)
\(38\) −8.12117 − 11.1778i −1.31743 − 1.81328i
\(39\) 2.16633 + 1.57393i 0.346890 + 0.252031i
\(40\) −3.05819 + 0.0714597i −0.483542 + 0.0112988i
\(41\) 0.819508 + 2.52219i 0.127986 + 0.393899i 0.994433 − 0.105369i \(-0.0336023\pi\)
−0.866448 + 0.499268i \(0.833602\pi\)
\(42\) 4.02170 + 5.53539i 0.620562 + 0.854130i
\(43\) −1.47548 + 0.479414i −0.225009 + 0.0731100i −0.419352 − 0.907824i \(-0.637743\pi\)
0.194343 + 0.980934i \(0.437743\pi\)
\(44\) −1.58812 + 4.88773i −0.239418 + 0.736852i
\(45\) −0.0522352 − 2.23546i −0.00778676 − 0.333242i
\(46\) 2.22004 − 6.83259i 0.327328 − 1.00741i
\(47\) −0.335158 − 0.108900i −0.0488879 − 0.0158846i 0.284471 − 0.958685i \(-0.408182\pi\)
−0.333359 + 0.942800i \(0.608182\pi\)
\(48\) −2.90508 + 3.99849i −0.419312 + 0.577133i
\(49\) −2.30189 + 7.08448i −0.328841 + 1.01207i
\(50\) −0.420369 − 8.99014i −0.0594491 − 1.27140i
\(51\) −5.89045 + 4.27966i −0.824828 + 0.599272i
\(52\) 3.15781 − 1.02604i 0.437910 − 0.142286i
\(53\) −0.521920 + 0.718361i −0.0716912 + 0.0986745i −0.843357 − 0.537354i \(-0.819424\pi\)
0.771666 + 0.636028i \(0.219424\pi\)
\(54\) −1.45623 − 1.05801i −0.198167 − 0.143977i
\(55\) 8.87860 + 2.65720i 1.19719 + 0.358296i
\(56\) −5.20018 −0.694903
\(57\) 7.67590i 1.01670i
\(58\) 1.90039 + 2.61566i 0.249533 + 0.343453i
\(59\) 0.620031 − 1.90826i 0.0807212 − 0.248434i −0.902549 − 0.430587i \(-0.858307\pi\)
0.983270 + 0.182153i \(0.0583065\pi\)
\(60\) −2.28060 − 1.57689i −0.294424 − 0.203576i
\(61\) −11.3748 −1.45639 −0.728194 − 0.685371i \(-0.759640\pi\)
−0.728194 + 0.685371i \(0.759640\pi\)
\(62\) −8.58596 + 5.16919i −1.09042 + 0.656488i
\(63\) − 3.80119i − 0.478905i
\(64\) 0.371919 + 1.14465i 0.0464899 + 0.143081i
\(65\) −1.98279 − 5.64976i −0.245935 − 0.700766i
\(66\) 6.03553 − 4.38507i 0.742923 − 0.539765i
\(67\) 14.2570i 1.74177i 0.491491 + 0.870883i \(0.336452\pi\)
−0.491491 + 0.870883i \(0.663548\pi\)
\(68\) 9.02826i 1.09484i
\(69\) −3.22898 + 2.34599i −0.388724 + 0.282424i
\(70\) −0.357400 − 15.2953i −0.0427174 − 1.82814i
\(71\) −5.56772 − 4.04519i −0.660767 − 0.480075i 0.206155 − 0.978519i \(-0.433905\pi\)
−0.866922 + 0.498444i \(0.833905\pi\)
\(72\) 1.30108 − 0.422747i 0.153334 − 0.0498212i
\(73\) −4.99304 − 6.87234i −0.584392 − 0.804346i 0.409777 − 0.912186i \(-0.365607\pi\)
−0.994168 + 0.107840i \(0.965607\pi\)
\(74\) 3.26963 + 10.0629i 0.380087 + 1.16979i
\(75\) −2.74678 + 4.17794i −0.317171 + 0.482427i
\(76\) 7.70017 + 5.59450i 0.883270 + 0.641733i
\(77\) 14.9835 + 4.86843i 1.70753 + 0.554810i
\(78\) −4.58400 − 1.48943i −0.519035 − 0.168645i
\(79\) 9.30430 + 6.75997i 1.04682 + 0.760556i 0.971604 − 0.236611i \(-0.0760368\pi\)
0.0752116 + 0.997168i \(0.476037\pi\)
\(80\) 10.4280 − 3.65972i 1.16589 − 0.409169i
\(81\) 0.309017 + 0.951057i 0.0343352 + 0.105673i
\(82\) −2.80582 − 3.86189i −0.309851 − 0.426474i
\(83\) −6.15134 + 1.99869i −0.675197 + 0.219385i −0.626491 − 0.779428i \(-0.715510\pi\)
−0.0487058 + 0.998813i \(0.515510\pi\)
\(84\) −3.81321 − 2.77046i −0.416056 − 0.302282i
\(85\) 16.2764 − 0.380324i 1.76542 − 0.0412519i
\(86\) 2.25921 − 1.64141i 0.243617 − 0.176998i
\(87\) − 1.79619i − 0.192572i
\(88\) 5.67003i 0.604427i
\(89\) −2.50174 + 1.81762i −0.265184 + 0.192668i −0.712429 − 0.701744i \(-0.752405\pi\)
0.447245 + 0.894411i \(0.352405\pi\)
\(90\) 1.33285 + 3.79782i 0.140494 + 0.400325i
\(91\) −3.14535 − 9.68040i −0.329722 − 1.01478i
\(92\) 4.94905i 0.515974i
\(93\) 5.54700 + 0.480411i 0.575197 + 0.0498163i
\(94\) 0.634329 0.0654260
\(95\) 9.76151 − 14.1177i 1.00151 − 1.44845i
\(96\) 1.90362 − 5.85873i 0.194287 − 0.597954i
\(97\) −9.05917 − 12.4689i −0.919819 − 1.26602i −0.963700 − 0.266986i \(-0.913972\pi\)
0.0438810 − 0.999037i \(-0.486028\pi\)
\(98\) − 13.4083i − 1.35444i
\(99\) −4.14464 −0.416552
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.4.8 ✓ 128
5.4 even 2 inner 465.2.ba.a.4.25 yes 128
31.8 even 5 inner 465.2.ba.a.349.25 yes 128
155.39 even 10 inner 465.2.ba.a.349.8 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.ba.a.4.8 ✓ 128 1.1 even 1 trivial
465.2.ba.a.4.25 yes 128 5.4 even 2 inner
465.2.ba.a.349.8 yes 128 155.39 even 10 inner
465.2.ba.a.349.25 yes 128 31.8 even 5 inner