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Properties

Label 380.2.bh.a.13.1
Level $380$
Weight $2$
Character 380.13
Analytic conductor $3.034$
Analytic rank $0$
Dimension $120$
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] = [380,2,Mod(13,380)] 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("380.13"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([0, 27, 10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.bh (of order \(36\), degree \(12\), 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: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(10\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label 13.1
Character \(\chi\) \(=\) 380.13
Dual form 380.2.bh.a.117.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.74765 + 1.28125i) q^{3} +(-1.16017 - 1.91154i) q^{5} +(-0.370262 - 1.38184i) q^{7} +(3.97961 - 4.74272i) q^{9} +(2.73107 + 4.73035i) q^{11} +(-0.160884 + 0.345016i) q^{13} +(5.63692 + 3.76578i) q^{15} +(-0.0144528 + 0.165196i) q^{17} +(1.45553 + 4.10870i) q^{19} +(2.78783 + 3.32240i) q^{21} +(0.363161 - 0.254288i) q^{23} +(-2.30799 + 4.43544i) q^{25} +(-2.50399 + 9.34503i) q^{27} +(1.09856 + 0.921802i) q^{29} +(6.67979 + 3.85658i) q^{31} +(-13.5648 - 9.49816i) q^{33} +(-2.21187 + 2.31094i) q^{35} +(6.77341 + 6.77341i) q^{37} -1.15412i q^{39} +(2.40119 - 6.59722i) q^{41} +(6.61211 - 9.44307i) q^{43} +(-13.6830 - 2.10482i) q^{45} +(-3.11441 + 0.272476i) q^{47} +(4.28980 - 2.47672i) q^{49} +(-0.171947 - 0.472419i) q^{51} +(6.56681 + 9.37838i) q^{53} +(5.87375 - 10.7086i) q^{55} +(-9.26357 - 9.42437i) q^{57} +(-8.39207 + 7.04179i) q^{59} +(2.07149 - 11.7480i) q^{61} +(-8.02716 - 3.74313i) q^{63} +(0.846167 - 0.0927428i) q^{65} +(0.773226 + 8.83802i) q^{67} +(-0.672034 + 1.16400i) q^{69} +(-0.325803 + 0.0574478i) q^{71} +(-0.964863 - 2.06915i) q^{73} +(0.658643 - 15.1442i) q^{75} +(5.52535 - 5.52535i) q^{77} +(-2.57267 - 0.936374i) q^{79} +(-1.86796 - 10.5937i) q^{81} +(-11.1195 + 2.97945i) q^{83} +(0.332548 - 0.164029i) q^{85} +(-4.19952 - 1.12526i) q^{87} +(-5.02263 + 1.82809i) q^{89} +(0.536325 + 0.0945686i) q^{91} +(-23.2950 - 2.03804i) q^{93} +(6.16528 - 7.54912i) q^{95} +(0.497197 + 0.0434991i) q^{97} +(33.3033 + 5.87227i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 6 q^{7} + 18 q^{15} - 18 q^{17} + 48 q^{21} - 36 q^{23} - 24 q^{25} - 60 q^{33} - 18 q^{35} - 12 q^{41} - 36 q^{43} + 18 q^{45} - 24 q^{47} + 96 q^{51} - 18 q^{53} + 72 q^{55} - 6 q^{57} - 24 q^{61}+ \cdots - 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.74765 + 1.28125i −1.58636 + 0.739730i −0.997643 − 0.0686122i \(-0.978143\pi\)
−0.588713 + 0.808342i \(0.700365\pi\)
\(4\) 0 0
\(5\) −1.16017 − 1.91154i −0.518846 − 0.854868i
\(6\) 0 0
\(7\) −0.370262 − 1.38184i −0.139946 − 0.522285i −0.999928 − 0.0119616i \(-0.996192\pi\)
0.859983 − 0.510323i \(-0.170474\pi\)
\(8\) 0 0
\(9\) 3.97961 − 4.74272i 1.32654 − 1.58091i
\(10\) 0 0
\(11\) 2.73107 + 4.73035i 0.823448 + 1.42625i 0.903100 + 0.429431i \(0.141286\pi\)
−0.0796521 + 0.996823i \(0.525381\pi\)
\(12\) 0 0
\(13\) −0.160884 + 0.345016i −0.0446211 + 0.0956903i −0.927355 − 0.374184i \(-0.877923\pi\)
0.882733 + 0.469874i \(0.155701\pi\)
\(14\) 0 0
\(15\) 5.63692 + 3.76578i 1.45545 + 0.972320i
\(16\) 0 0
\(17\) −0.0144528 + 0.165196i −0.00350532 + 0.0400660i −0.997746 − 0.0671051i \(-0.978624\pi\)
0.994241 + 0.107171i \(0.0341793\pi\)
\(18\) 0 0
\(19\) 1.45553 + 4.10870i 0.333922 + 0.942601i
\(20\) 0 0
\(21\) 2.78783 + 3.32240i 0.608354 + 0.725008i
\(22\) 0 0
\(23\) 0.363161 − 0.254288i 0.0757244 − 0.0530228i −0.535101 − 0.844788i \(-0.679727\pi\)
0.610826 + 0.791765i \(0.290838\pi\)
\(24\) 0 0
\(25\) −2.30799 + 4.43544i −0.461599 + 0.887089i
\(26\) 0 0
\(27\) −2.50399 + 9.34503i −0.481894 + 1.79845i
\(28\) 0 0
\(29\) 1.09856 + 0.921802i 0.203998 + 0.171174i 0.739063 − 0.673636i \(-0.235269\pi\)
−0.535065 + 0.844811i \(0.679713\pi\)
\(30\) 0 0
\(31\) 6.67979 + 3.85658i 1.19973 + 0.692662i 0.960494 − 0.278301i \(-0.0897712\pi\)
0.239231 + 0.970963i \(0.423105\pi\)
\(32\) 0 0
\(33\) −13.5648 − 9.49816i −2.36132 − 1.65342i
\(34\) 0 0
\(35\) −2.21187 + 2.31094i −0.373874 + 0.390620i
\(36\) 0 0
\(37\) 6.77341 + 6.77341i 1.11354 + 1.11354i 0.992668 + 0.120874i \(0.0385699\pi\)
0.120874 + 0.992668i \(0.461430\pi\)
\(38\) 0 0
\(39\) − 1.15412i − 0.184807i
\(40\) 0 0
\(41\) 2.40119 − 6.59722i 0.375003 − 1.03031i −0.598397 − 0.801200i \(-0.704195\pi\)
0.973400 − 0.229112i \(-0.0735824\pi\)
\(42\) 0 0
\(43\) 6.61211 − 9.44307i 1.00834 − 1.44005i 0.113565 − 0.993531i \(-0.463773\pi\)
0.894772 − 0.446524i \(-0.147338\pi\)
\(44\) 0 0
\(45\) −13.6830 − 2.10482i −2.03973 − 0.313769i
\(46\) 0 0
\(47\) −3.11441 + 0.272476i −0.454284 + 0.0397447i −0.312002 − 0.950081i \(-0.601000\pi\)
−0.142282 + 0.989826i \(0.545444\pi\)
\(48\) 0 0
\(49\) 4.28980 − 2.47672i 0.612829 − 0.353817i
\(50\) 0 0
\(51\) −0.171947 − 0.472419i −0.0240773 − 0.0661519i
\(52\) 0 0
\(53\) 6.56681 + 9.37838i 0.902021 + 1.28822i 0.957106 + 0.289738i \(0.0935681\pi\)
−0.0550851 + 0.998482i \(0.517543\pi\)
\(54\) 0 0
\(55\) 5.87375 − 10.7086i 0.792016 − 1.44394i
\(56\) 0 0
\(57\) −9.26357 − 9.42437i −1.22699 − 1.24829i
\(58\) 0 0
\(59\) −8.39207 + 7.04179i −1.09255 + 0.916762i −0.996902 − 0.0786523i \(-0.974938\pi\)
−0.0956529 + 0.995415i \(0.530494\pi\)
\(60\) 0 0
\(61\) 2.07149 − 11.7480i 0.265227 − 1.50418i −0.503159 − 0.864194i \(-0.667829\pi\)
0.768387 − 0.639986i \(-0.221060\pi\)
\(62\) 0 0
\(63\) −8.02716 − 3.74313i −1.01133 − 0.471589i
\(64\) 0 0
\(65\) 0.846167 − 0.0927428i 0.104954 − 0.0115033i
\(66\) 0 0
\(67\) 0.773226 + 8.83802i 0.0944646 + 1.07974i 0.884458 + 0.466620i \(0.154528\pi\)
−0.789993 + 0.613116i \(0.789916\pi\)
\(68\) 0 0
\(69\) −0.672034 + 1.16400i −0.0809033 + 0.140129i
\(70\) 0 0
\(71\) −0.325803 + 0.0574478i −0.0386657 + 0.00681780i −0.192948 − 0.981209i \(-0.561805\pi\)
0.154282 + 0.988027i \(0.450694\pi\)
\(72\) 0 0
\(73\) −0.964863 − 2.06915i −0.112929 − 0.242176i 0.841718 − 0.539917i \(-0.181545\pi\)
−0.954647 + 0.297741i \(0.903767\pi\)
\(74\) 0 0
\(75\) 0.658643 − 15.1442i 0.0760535 − 1.74870i
\(76\) 0 0
\(77\) 5.52535 − 5.52535i 0.629672 − 0.629672i
\(78\) 0 0
\(79\) −2.57267 − 0.936374i −0.289447 − 0.105350i 0.193216 − 0.981156i \(-0.438108\pi\)
−0.482663 + 0.875806i \(0.660330\pi\)
\(80\) 0 0
\(81\) −1.86796 − 10.5937i −0.207551 − 1.17708i
\(82\) 0 0
\(83\) −11.1195 + 2.97945i −1.22052 + 0.327037i −0.810881 − 0.585211i \(-0.801012\pi\)
−0.409638 + 0.912248i \(0.634345\pi\)
\(84\) 0 0
\(85\) 0.332548 − 0.164029i 0.0360698 − 0.0177915i
\(86\) 0 0
\(87\) −4.19952 − 1.12526i −0.450236 − 0.120640i
\(88\) 0 0
\(89\) −5.02263 + 1.82809i −0.532397 + 0.193777i −0.594208 − 0.804311i \(-0.702535\pi\)
0.0618111 + 0.998088i \(0.480312\pi\)
\(90\) 0 0
\(91\) 0.536325 + 0.0945686i 0.0562222 + 0.00991348i
\(92\) 0 0
\(93\) −23.2950 − 2.03804i −2.41557 − 0.211335i
\(94\) 0 0
\(95\) 6.16528 − 7.54912i 0.632545 − 0.774524i
\(96\) 0 0
\(97\) 0.497197 + 0.0434991i 0.0504827 + 0.00441667i 0.112369 − 0.993667i \(-0.464156\pi\)
−0.0618864 + 0.998083i \(0.519712\pi\)
\(98\) 0 0
\(99\) 33.3033 + 5.87227i 3.34711 + 0.590185i
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 380.2.bh.a.13.1 ✓ 120
5.2 odd 4 inner 380.2.bh.a.317.10 yes 120
19.3 odd 18 inner 380.2.bh.a.193.10 yes 120
95.22 even 36 inner 380.2.bh.a.117.1 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.bh.a.13.1 ✓ 120 1.1 even 1 trivial
380.2.bh.a.117.1 yes 120 95.22 even 36 inner
380.2.bh.a.193.10 yes 120 19.3 odd 18 inner
380.2.bh.a.317.10 yes 120 5.2 odd 4 inner