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Properties

Label 465.2.ba.a.4.3
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 4.3
Character \(\chi\) \(=\) 465.4
Dual form 465.2.ba.a.349.3

$q$-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+(-2.34433 + 0.761718i) q^{2} +(0.951057 + 0.309017i) q^{3} +(3.29762 - 2.39586i) q^{4} +(-2.05846 + 0.873354i) q^{5} -2.46497 q^{6} +(1.42844 + 1.96608i) q^{7} +(-3.00798 + 4.14013i) q^{8} +(0.809017 + 0.587785i) q^{9} +(4.16045 - 3.61539i) q^{10} +(2.88524 - 2.09625i) q^{11} +(3.87658 - 1.25958i) q^{12} +(1.08966 + 0.354052i) q^{13} +(-4.84633 - 3.52107i) q^{14} +(-2.22759 + 0.194510i) q^{15} +(1.37892 - 4.24388i) q^{16} +(-4.25139 + 5.85154i) q^{17} +(-2.34433 - 0.761718i) q^{18} +(-0.363399 - 1.11843i) q^{19} +(-4.69558 + 7.81177i) q^{20} +(0.750976 + 2.31127i) q^{21} +(-5.16719 + 7.11203i) q^{22} +(3.91435 - 5.38764i) q^{23} +(-4.14013 + 3.00798i) q^{24} +(3.47451 - 3.59553i) q^{25} -2.82420 q^{26} +(0.587785 + 0.809017i) q^{27} +(9.42091 + 3.06104i) q^{28} +(2.21118 + 6.80530i) q^{29} +(5.07404 - 2.15279i) q^{30} +(0.404174 + 5.55308i) q^{31} +0.764425i q^{32} +(3.39180 - 1.10206i) q^{33} +(5.50943 - 16.9563i) q^{34} +(-4.65747 - 2.79956i) q^{35} +4.07608 q^{36} +5.08933i q^{37} +(1.70385 + 2.34515i) q^{38} +(0.926919 + 0.673446i) q^{39} +(2.57601 - 11.1493i) q^{40} +(2.43486 + 7.49373i) q^{41} +(-3.52107 - 4.84633i) q^{42} +(-11.3137 + 3.67604i) q^{43} +(4.49210 - 13.8253i) q^{44} +(-2.17867 - 0.503374i) q^{45} +(-5.07265 + 15.6120i) q^{46} +(-1.29032 - 0.419251i) q^{47} +(2.62286 - 3.61006i) q^{48} +(0.338089 - 1.04053i) q^{49} +(-5.40660 + 11.0757i) q^{50} +(-5.85154 + 4.25139i) q^{51} +(4.44154 - 1.44314i) q^{52} +(-3.83753 + 5.28191i) q^{53} +(-1.99420 - 1.44887i) q^{54} +(-4.10838 + 6.83488i) q^{55} -12.4366 q^{56} -1.17598i q^{57} +(-10.3674 - 14.2696i) q^{58} +(-0.384827 + 1.18438i) q^{59} +(-6.87973 + 5.97842i) q^{60} -7.48057 q^{61} +(-5.17739 - 12.7104i) q^{62} +2.43021i q^{63} +(2.17556 + 6.69570i) q^{64} +(-2.55223 + 0.222857i) q^{65} +(-7.11203 + 5.16719i) q^{66} +2.71966i q^{67} +29.4819i q^{68} +(5.38764 - 3.91435i) q^{69} +(13.0511 + 3.01541i) q^{70} +(5.21523 + 3.78909i) q^{71} +(-4.86701 + 1.58139i) q^{72} +(3.85383 + 5.30434i) q^{73} +(-3.87663 - 11.9310i) q^{74} +(4.41553 - 2.34587i) q^{75} +(-3.87794 - 2.81749i) q^{76} +(8.24279 + 2.67825i) q^{77} +(-2.68598 - 0.872727i) q^{78} +(-8.65758 - 6.29010i) q^{79} +(0.867957 + 9.94014i) q^{80} +(0.309017 + 0.951057i) q^{81} +(-11.4162 - 15.7131i) q^{82} +(3.19968 - 1.03964i) q^{83} +(8.01391 + 5.82245i) q^{84} +(3.64086 - 15.7581i) q^{85} +(23.7229 - 17.2357i) q^{86} +7.15552i q^{87} +18.2507i q^{88} +(6.17622 - 4.48729i) q^{89} +(5.49095 - 0.479461i) q^{90} +(0.860420 + 2.64810i) q^{91} -27.1446i q^{92} +(-1.33160 + 5.40618i) q^{93} +3.34429 q^{94} +(1.72482 + 1.98486i) q^{95} +(-0.236220 + 0.727011i) q^{96} +(2.14800 + 2.95647i) q^{97} +2.69687i q^{98} +3.56635 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.34433 + 0.761718i −1.65769 + 0.538616i −0.980386 − 0.197088i \(-0.936852\pi\)
−0.677303 + 0.735704i \(0.736852\pi\)
\(3\) 0.951057 + 0.309017i 0.549093 + 0.178411i
\(4\) 3.29762 − 2.39586i 1.64881 − 1.19793i
\(5\) −2.05846 + 0.873354i −0.920571 + 0.390576i
\(6\) −2.46497 −1.00632
\(7\) 1.42844 + 1.96608i 0.539900 + 0.743109i 0.988599 − 0.150574i \(-0.0481123\pi\)
−0.448698 + 0.893683i \(0.648112\pi\)
\(8\) −3.00798 + 4.14013i −1.06348 + 1.46376i
\(9\) 0.809017 + 0.587785i 0.269672 + 0.195928i
\(10\) 4.16045 − 3.61539i 1.31565 − 1.14329i
\(11\) 2.88524 − 2.09625i 0.869932 − 0.632043i −0.0606363 − 0.998160i \(-0.519313\pi\)
0.930569 + 0.366117i \(0.119313\pi\)
\(12\) 3.87658 − 1.25958i 1.11907 − 0.363609i
\(13\) 1.08966 + 0.354052i 0.302217 + 0.0981962i 0.456200 − 0.889877i \(-0.349210\pi\)
−0.153983 + 0.988073i \(0.549210\pi\)
\(14\) −4.84633 − 3.52107i −1.29524 − 0.941045i
\(15\) −2.22759 + 0.194510i −0.575162 + 0.0502222i
\(16\) 1.37892 − 4.24388i 0.344730 − 1.06097i
\(17\) −4.25139 + 5.85154i −1.03111 + 1.41921i −0.127007 + 0.991902i \(0.540537\pi\)
−0.904107 + 0.427306i \(0.859463\pi\)
\(18\) −2.34433 − 0.761718i −0.552563 − 0.179539i
\(19\) −0.363399 − 1.11843i −0.0833693 − 0.256584i 0.900679 − 0.434485i \(-0.143070\pi\)
−0.984049 + 0.177900i \(0.943070\pi\)
\(20\) −4.69558 + 7.81177i −1.04996 + 1.74676i
\(21\) 0.750976 + 2.31127i 0.163876 + 0.504360i
\(22\) −5.16719 + 7.11203i −1.10165 + 1.51629i
\(23\) 3.91435 − 5.38764i 0.816198 − 1.12340i −0.174139 − 0.984721i \(-0.555714\pi\)
0.990337 − 0.138679i \(-0.0442857\pi\)
\(24\) −4.14013 + 3.00798i −0.845100 + 0.614001i
\(25\) 3.47451 − 3.59553i 0.694901 − 0.719105i
\(26\) −2.82420 −0.553872
\(27\) 0.587785 + 0.809017i 0.113119 + 0.155695i
\(28\) 9.42091 + 3.06104i 1.78039 + 0.578482i
\(29\) 2.21118 + 6.80530i 0.410605 + 1.26371i 0.916123 + 0.400897i \(0.131301\pi\)
−0.505518 + 0.862816i \(0.668699\pi\)
\(30\) 5.07404 − 2.15279i 0.926389 − 0.393044i
\(31\) 0.404174 + 5.55308i 0.0725917 + 0.997362i
\(32\) 0.764425i 0.135132i
\(33\) 3.39180 − 1.10206i 0.590437 − 0.191845i
\(34\) 5.50943 − 16.9563i 0.944860 − 2.90798i
\(35\) −4.65747 − 2.79956i −0.787257 − 0.473213i
\(36\) 4.07608 0.679347
\(37\) 5.08933i 0.836680i 0.908290 + 0.418340i \(0.137388\pi\)
−0.908290 + 0.418340i \(0.862612\pi\)
\(38\) 1.70385 + 2.34515i 0.276401 + 0.380433i
\(39\) 0.926919 + 0.673446i 0.148426 + 0.107838i
\(40\) 2.57601 − 11.1493i 0.407302 − 1.76286i
\(41\) 2.43486 + 7.49373i 0.380261 + 1.17032i 0.939860 + 0.341560i \(0.110955\pi\)
−0.559598 + 0.828764i \(0.689045\pi\)
\(42\) −3.52107 − 4.84633i −0.543312 − 0.747805i
\(43\) −11.3137 + 3.67604i −1.72532 + 0.560590i −0.992760 − 0.120112i \(-0.961675\pi\)
−0.732560 + 0.680703i \(0.761675\pi\)
\(44\) 4.49210 − 13.8253i 0.677210 − 2.08424i
\(45\) −2.17867 − 0.503374i −0.324777 − 0.0750386i
\(46\) −5.07265 + 15.6120i −0.747922 + 2.30187i
\(47\) −1.29032 − 0.419251i −0.188213 − 0.0611541i 0.213394 − 0.976966i \(-0.431548\pi\)
−0.401607 + 0.915812i \(0.631548\pi\)
\(48\) 2.62286 − 3.61006i 0.378577 − 0.521067i
\(49\) 0.338089 − 1.04053i 0.0482984 − 0.148647i
\(50\) −5.40660 + 11.0757i −0.764609 + 1.56634i
\(51\) −5.85154 + 4.25139i −0.819380 + 0.595314i
\(52\) 4.44154 − 1.44314i 0.615930 − 0.200128i
\(53\) −3.83753 + 5.28191i −0.527126 + 0.725526i −0.986689 − 0.162619i \(-0.948006\pi\)
0.459563 + 0.888145i \(0.348006\pi\)
\(54\) −1.99420 − 1.44887i −0.271377 − 0.197167i
\(55\) −4.10838 + 6.83488i −0.553974 + 0.921615i
\(56\) −12.4366 −1.66190
\(57\) − 1.17598i − 0.155763i
\(58\) −10.3674 − 14.2696i −1.36131 − 1.87368i
\(59\) −0.384827 + 1.18438i −0.0501002 + 0.154193i −0.972977 − 0.230904i \(-0.925832\pi\)
0.922876 + 0.385096i \(0.125832\pi\)
\(60\) −6.87973 + 5.97842i −0.888170 + 0.771811i
\(61\) −7.48057 −0.957789 −0.478895 − 0.877872i \(-0.658962\pi\)
−0.478895 + 0.877872i \(0.658962\pi\)
\(62\) −5.17739 − 12.7104i −0.657529 − 1.61422i
\(63\) 2.43021i 0.306178i
\(64\) 2.17556 + 6.69570i 0.271945 + 0.836962i
\(65\) −2.55223 + 0.222857i −0.316565 + 0.0276420i
\(66\) −7.11203 + 5.16719i −0.875431 + 0.636038i
\(67\) 2.71966i 0.332260i 0.986104 + 0.166130i \(0.0531271\pi\)
−0.986104 + 0.166130i \(0.946873\pi\)
\(68\) 29.4819i 3.57521i
\(69\) 5.38764 − 3.91435i 0.648596 − 0.471232i
\(70\) 13.0511 + 3.01541i 1.55991 + 0.360410i
\(71\) 5.21523 + 3.78909i 0.618934 + 0.449682i 0.852549 − 0.522647i \(-0.175056\pi\)
−0.233615 + 0.972329i \(0.575056\pi\)
\(72\) −4.86701 + 1.58139i −0.573583 + 0.186368i
\(73\) 3.85383 + 5.30434i 0.451056 + 0.620826i 0.972624 − 0.232384i \(-0.0746527\pi\)
−0.521568 + 0.853210i \(0.674653\pi\)
\(74\) −3.87663 − 11.9310i −0.450649 − 1.38696i
\(75\) 4.41553 − 2.34587i 0.509862 − 0.270877i
\(76\) −3.87794 − 2.81749i −0.444830 − 0.323188i
\(77\) 8.24279 + 2.67825i 0.939354 + 0.305214i
\(78\) −2.68598 − 0.872727i −0.304127 − 0.0988169i
\(79\) −8.65758 − 6.29010i −0.974054 − 0.707692i −0.0176824 − 0.999844i \(-0.505629\pi\)
−0.956372 + 0.292152i \(0.905629\pi\)
\(80\) 0.867957 + 9.94014i 0.0970405 + 1.11134i
\(81\) 0.309017 + 0.951057i 0.0343352 + 0.105673i
\(82\) −11.4162 − 15.7131i −1.26071 − 1.73522i
\(83\) 3.19968 − 1.03964i 0.351210 − 0.114115i −0.128099 − 0.991761i \(-0.540887\pi\)
0.479309 + 0.877646i \(0.340887\pi\)
\(84\) 8.01391 + 5.82245i 0.874389 + 0.635281i
\(85\) 3.64086 − 15.7581i 0.394906 − 1.70921i
\(86\) 23.7229 − 17.2357i 2.55810 − 1.85857i
\(87\) 7.15552i 0.767152i
\(88\) 18.2507i 1.94554i
\(89\) 6.17622 − 4.48729i 0.654678 − 0.475651i −0.210183 − 0.977662i \(-0.567406\pi\)
0.864862 + 0.502010i \(0.167406\pi\)
\(90\) 5.49095 − 0.479461i 0.578797 − 0.0505396i
\(91\) 0.860420 + 2.64810i 0.0901965 + 0.277596i
\(92\) − 27.1446i − 2.83002i
\(93\) −1.33160 + 5.40618i −0.138081 + 0.560595i
\(94\) 3.34429 0.344937
\(95\) 1.72482 + 1.98486i 0.176963 + 0.203642i
\(96\) −0.236220 + 0.727011i −0.0241091 + 0.0742003i
\(97\) 2.14800 + 2.95647i 0.218096 + 0.300184i 0.904020 − 0.427489i \(-0.140602\pi\)
−0.685924 + 0.727673i \(0.740602\pi\)
\(98\) 2.69687i 0.272425i
\(99\) 3.56635 0.358432
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.4.3 ✓ 128
5.4 even 2 inner 465.2.ba.a.4.30 yes 128
31.8 even 5 inner 465.2.ba.a.349.30 yes 128
155.39 even 10 inner 465.2.ba.a.349.3 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.ba.a.4.3 ✓ 128 1.1 even 1 trivial
465.2.ba.a.4.30 yes 128 5.4 even 2 inner
465.2.ba.a.349.3 yes 128 155.39 even 10 inner
465.2.ba.a.349.30 yes 128 31.8 even 5 inner