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Properties

Label 380.2.bh.a.33.9
Level $380$
Weight $2$
Character 380.33
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 33.9
Character \(\chi\) \(=\) 380.33
Dual form 380.2.bh.a.357.9

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.60791 + 2.29633i) q^{3} +(1.74139 - 1.40270i) q^{5} +(-0.546238 - 0.146364i) q^{7} +(-1.66171 + 4.56550i) q^{9} +(0.973552 - 1.68624i) q^{11} +(4.44432 + 3.11195i) q^{13} +(6.02105 + 1.74340i) q^{15} +(0.656923 - 1.40878i) q^{17} +(-4.30338 + 0.693479i) q^{19} +(-0.542200 - 1.48968i) q^{21} +(-8.31664 - 0.727612i) q^{23} +(1.06488 - 4.88529i) q^{25} +(-5.03242 + 1.34843i) q^{27} +(1.88234 + 0.685117i) q^{29} +(-6.30203 + 3.63848i) q^{31} +(5.43755 - 0.475724i) q^{33} +(-1.15652 + 0.511330i) q^{35} +(4.89676 + 4.89676i) q^{37} +15.2094i q^{39} +(-5.71247 - 1.00726i) q^{41} +(-0.926985 - 10.5955i) q^{43} +(3.51034 + 10.2812i) q^{45} +(5.99419 - 2.79514i) q^{47} +(-5.78522 - 3.34010i) q^{49} +(4.29129 - 0.756670i) q^{51} +(0.198390 - 2.26761i) q^{53} +(-0.669954 - 4.30200i) q^{55} +(-8.51190 - 8.76694i) q^{57} +(3.62555 - 1.31959i) q^{59} +(3.63452 - 3.04972i) q^{61} +(1.57591 - 2.25063i) q^{63} +(12.1044 - 0.814927i) q^{65} +(0.894831 + 1.91897i) q^{67} +(-11.7016 - 20.2677i) q^{69} +(1.35617 - 1.61623i) q^{71} +(-0.728519 + 0.510115i) q^{73} +(12.9305 - 5.40978i) q^{75} +(-0.778595 + 0.778595i) q^{77} +(-2.16887 + 12.3003i) q^{79} +(-0.0226334 - 0.0189917i) q^{81} +(0.674124 - 2.51586i) q^{83} +(-0.832128 - 3.37469i) q^{85} +(1.45338 + 5.42409i) q^{87} +(-0.660318 - 3.74485i) q^{89} +(-1.97218 - 2.35035i) q^{91} +(-18.4882 - 8.62121i) q^{93} +(-6.52112 + 7.24396i) q^{95} +(-14.7372 - 6.87208i) q^{97} +(6.08078 + 7.24679i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 6 q^{7} + 18 q^{15} - 18 q^{17} + 48 q^{21} - 36 q^{23} - 24 q^{25} - 60 q^{33} - 18 q^{35} - 12 q^{41} - 36 q^{43} + 18 q^{45} - 24 q^{47} + 96 q^{51} - 18 q^{53} + 72 q^{55} - 6 q^{57} - 24 q^{61}+ \cdots - 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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
\(3\) 1.60791 + 2.29633i 0.928326 + 1.32579i 0.945393 + 0.325933i \(0.105678\pi\)
−0.0170667 + 0.999854i \(0.505433\pi\)
\(4\) 0 0
\(5\) 1.74139 − 1.40270i 0.778773 − 0.627306i
\(6\) 0 0
\(7\) −0.546238 − 0.146364i −0.206458 − 0.0553204i 0.154108 − 0.988054i \(-0.450750\pi\)
−0.360566 + 0.932734i \(0.617416\pi\)
\(8\) 0 0
\(9\) −1.66171 + 4.56550i −0.553902 + 1.52183i
\(10\) 0 0
\(11\) 0.973552 − 1.68624i 0.293537 − 0.508421i −0.681107 − 0.732184i \(-0.738501\pi\)
0.974643 + 0.223764i \(0.0718343\pi\)
\(12\) 0 0
\(13\) 4.44432 + 3.11195i 1.23263 + 0.863098i 0.994146 − 0.108047i \(-0.0344598\pi\)
0.238487 + 0.971146i \(0.423349\pi\)
\(14\) 0 0
\(15\) 6.02105 + 1.74340i 1.55463 + 0.450143i
\(16\) 0 0
\(17\) 0.656923 − 1.40878i 0.159327 − 0.341678i −0.810397 − 0.585881i \(-0.800748\pi\)
0.969724 + 0.244203i \(0.0785263\pi\)
\(18\) 0 0
\(19\) −4.30338 + 0.693479i −0.987263 + 0.159095i
\(20\) 0 0
\(21\) −0.542200 − 1.48968i −0.118318 − 0.325075i
\(22\) 0 0
\(23\) −8.31664 − 0.727612i −1.73414 − 0.151718i −0.824157 − 0.566362i \(-0.808350\pi\)
−0.909983 + 0.414645i \(0.863906\pi\)
\(24\) 0 0
\(25\) 1.06488 − 4.88529i 0.212975 − 0.977058i
\(26\) 0 0
\(27\) −5.03242 + 1.34843i −0.968490 + 0.259506i
\(28\) 0 0
\(29\) 1.88234 + 0.685117i 0.349542 + 0.127223i 0.510824 − 0.859685i \(-0.329340\pi\)
−0.161281 + 0.986908i \(0.551563\pi\)
\(30\) 0 0
\(31\) −6.30203 + 3.63848i −1.13188 + 0.653490i −0.944407 − 0.328780i \(-0.893363\pi\)
−0.187472 + 0.982270i \(0.560029\pi\)
\(32\) 0 0
\(33\) 5.43755 − 0.475724i 0.946556 − 0.0828129i
\(34\) 0 0
\(35\) −1.15652 + 0.511330i −0.195487 + 0.0864305i
\(36\) 0 0
\(37\) 4.89676 + 4.89676i 0.805023 + 0.805023i 0.983876 − 0.178853i \(-0.0572386\pi\)
−0.178853 + 0.983876i \(0.557239\pi\)
\(38\) 0 0
\(39\) 15.2094i 2.43545i
\(40\) 0 0
\(41\) −5.71247 − 1.00726i −0.892138 − 0.157308i −0.291255 − 0.956645i \(-0.594073\pi\)
−0.600883 + 0.799337i \(0.705184\pi\)
\(42\) 0 0
\(43\) −0.926985 − 10.5955i −0.141364 − 1.61580i −0.653523 − 0.756906i \(-0.726710\pi\)
0.512160 − 0.858890i \(-0.328846\pi\)
\(44\) 0 0
\(45\) 3.51034 + 10.2812i 0.523290 + 1.53263i
\(46\) 0 0
\(47\) 5.99419 − 2.79514i 0.874342 − 0.407712i 0.0669699 − 0.997755i \(-0.478667\pi\)
0.807372 + 0.590043i \(0.200889\pi\)
\(48\) 0 0
\(49\) −5.78522 − 3.34010i −0.826461 − 0.477157i
\(50\) 0 0
\(51\) 4.29129 − 0.756670i 0.600901 − 0.105955i
\(52\) 0 0
\(53\) 0.198390 − 2.26761i 0.0272510 − 0.311480i −0.970320 − 0.241824i \(-0.922254\pi\)
0.997571 − 0.0696561i \(-0.0221902\pi\)
\(54\) 0 0
\(55\) −0.669954 − 4.30200i −0.0903365 − 0.580082i
\(56\) 0 0
\(57\) −8.51190 − 8.76694i −1.12743 − 1.16121i
\(58\) 0 0
\(59\) 3.62555 − 1.31959i 0.472006 − 0.171796i −0.0950548 − 0.995472i \(-0.530303\pi\)
0.567061 + 0.823676i \(0.308080\pi\)
\(60\) 0 0
\(61\) 3.63452 − 3.04972i 0.465353 − 0.390477i −0.379743 − 0.925092i \(-0.623988\pi\)
0.845096 + 0.534615i \(0.179543\pi\)
\(62\) 0 0
\(63\) 1.57591 − 2.25063i 0.198546 − 0.283553i
\(64\) 0 0
\(65\) 12.1044 − 0.814927i 1.50137 − 0.101079i
\(66\) 0 0
\(67\) 0.894831 + 1.91897i 0.109321 + 0.234440i 0.953356 − 0.301849i \(-0.0976037\pi\)
−0.844035 + 0.536288i \(0.819826\pi\)
\(68\) 0 0
\(69\) −11.7016 − 20.2677i −1.40870 − 2.43994i
\(70\) 0 0
\(71\) 1.35617 − 1.61623i 0.160948 − 0.191811i −0.679543 − 0.733636i \(-0.737822\pi\)
0.840491 + 0.541825i \(0.182266\pi\)
\(72\) 0 0
\(73\) −0.728519 + 0.510115i −0.0852668 + 0.0597044i −0.615432 − 0.788190i \(-0.711019\pi\)
0.530166 + 0.847894i \(0.322130\pi\)
\(74\) 0 0
\(75\) 12.9305 − 5.40978i 1.49308 − 0.624668i
\(76\) 0 0
\(77\) −0.778595 + 0.778595i −0.0887292 + 0.0887292i
\(78\) 0 0
\(79\) −2.16887 + 12.3003i −0.244017 + 1.38389i 0.578749 + 0.815506i \(0.303541\pi\)
−0.822765 + 0.568381i \(0.807570\pi\)
\(80\) 0 0
\(81\) −0.0226334 − 0.0189917i −0.00251482 − 0.00211019i
\(82\) 0 0
\(83\) 0.674124 − 2.51586i 0.0739947 − 0.276152i −0.919009 − 0.394237i \(-0.871009\pi\)
0.993003 + 0.118085i \(0.0376756\pi\)
\(84\) 0 0
\(85\) −0.832128 − 3.37469i −0.0902570 − 0.366037i
\(86\) 0 0
\(87\) 1.45338 + 5.42409i 0.155819 + 0.581523i
\(88\) 0 0
\(89\) −0.660318 − 3.74485i −0.0699935 − 0.396953i −0.999597 − 0.0283915i \(-0.990961\pi\)
0.929603 − 0.368562i \(-0.120150\pi\)
\(90\) 0 0
\(91\) −1.97218 − 2.35035i −0.206740 − 0.246384i
\(92\) 0 0
\(93\) −18.4882 − 8.62121i −1.91714 − 0.893978i
\(94\) 0 0
\(95\) −6.52112 + 7.24396i −0.669053 + 0.743215i
\(96\) 0 0
\(97\) −14.7372 − 6.87208i −1.49634 − 0.697754i −0.509651 − 0.860381i \(-0.670225\pi\)
−0.986688 + 0.162627i \(0.948003\pi\)
\(98\) 0 0
\(99\) 6.08078 + 7.24679i 0.611141 + 0.728329i
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.33.9 ✓ 120
5.2 odd 4 inner 380.2.bh.a.337.9 yes 120
19.15 odd 18 inner 380.2.bh.a.53.9 yes 120
95.72 even 36 inner 380.2.bh.a.357.9 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.bh.a.33.9 ✓ 120 1.1 even 1 trivial
380.2.bh.a.53.9 yes 120 19.15 odd 18 inner
380.2.bh.a.337.9 yes 120 5.2 odd 4 inner
380.2.bh.a.357.9 yes 120 95.72 even 36 inner