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Properties

Label 380.2.bh.a.33.3
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 33.3
Character \(\chi\) \(=\) 380.33
Dual form 380.2.bh.a.357.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.02333 - 1.46147i) q^{3} +(0.710865 + 2.12006i) q^{5} +(-2.65380 - 0.711084i) q^{7} +(-0.0626300 + 0.172074i) q^{9} +(2.84045 - 4.91980i) q^{11} +(-1.87803 - 1.31501i) q^{13} +(2.37096 - 3.20844i) q^{15} +(1.57974 - 3.38777i) q^{17} +(-3.00247 - 3.15994i) q^{19} +(1.67650 + 4.60614i) q^{21} +(-3.36528 - 0.294424i) q^{23} +(-3.98934 + 3.01416i) q^{25} +(-4.85444 + 1.30074i) q^{27} +(-1.09210 - 0.397492i) q^{29} +(7.63721 - 4.40934i) q^{31} +(-10.0969 + 0.883363i) q^{33} +(-0.378951 - 6.13172i) q^{35} +(3.85089 + 3.85089i) q^{37} +4.09038i q^{39} +(-2.15157 - 0.379380i) q^{41} +(-0.126691 - 1.44809i) q^{43} +(-0.409330 - 0.0104579i) q^{45} +(0.0279851 - 0.0130497i) q^{47} +(0.474851 + 0.274156i) q^{49} +(-6.56773 + 1.15807i) q^{51} +(0.107930 - 1.23364i) q^{53} +(12.4495 + 2.52462i) q^{55} +(-1.54564 + 7.62170i) q^{57} +(6.00753 - 2.18656i) q^{59} +(8.16469 - 6.85099i) q^{61} +(0.288567 - 0.412116i) q^{63} +(1.45288 - 4.91633i) q^{65} +(5.85638 + 12.5590i) q^{67} +(3.01352 + 5.21957i) q^{69} +(-7.74802 + 9.23373i) q^{71} +(10.1192 - 7.08552i) q^{73} +(8.48754 + 2.74583i) q^{75} +(-11.0364 + 11.0364i) q^{77} +(-2.15191 + 12.2041i) q^{79} +(7.28954 + 6.11665i) q^{81} +(3.01260 - 11.2432i) q^{83} +(8.30526 + 0.940909i) q^{85} +(0.536661 + 2.00284i) q^{87} +(1.92434 + 10.9135i) q^{89} +(4.04883 + 4.82520i) q^{91} +(-14.2596 - 6.64934i) q^{93} +(4.56492 - 8.61171i) q^{95} +(-10.5280 - 4.90928i) q^{97} +(0.668674 + 0.796895i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 6 q^{7} + 18 q^{15} - 18 q^{17} + 48 q^{21} - 36 q^{23} - 24 q^{25} - 60 q^{33} - 18 q^{35} - 12 q^{41} - 36 q^{43} + 18 q^{45} - 24 q^{47} + 96 q^{51} - 18 q^{53} + 72 q^{55} - 6 q^{57} - 24 q^{61}+ \cdots - 60 q^{97}+O(q^{100}) \) 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
\(3\) −1.02333 − 1.46147i −0.590823 − 0.843782i 0.406741 − 0.913543i \(-0.366665\pi\)
−0.997564 + 0.0697615i \(0.977776\pi\)
\(4\) 0 0
\(5\) 0.710865 + 2.12006i 0.317908 + 0.948121i
\(6\) 0 0
\(7\) −2.65380 − 0.711084i −1.00304 − 0.268765i −0.280323 − 0.959906i \(-0.590442\pi\)
−0.722720 + 0.691141i \(0.757108\pi\)
\(8\) 0 0
\(9\) −0.0626300 + 0.172074i −0.0208767 + 0.0573581i
\(10\) 0 0
\(11\) 2.84045 − 4.91980i 0.856427 − 1.48337i −0.0188880 − 0.999822i \(-0.506013\pi\)
0.875315 − 0.483553i \(-0.160654\pi\)
\(12\) 0 0
\(13\) −1.87803 − 1.31501i −0.520870 − 0.364717i 0.283380 − 0.959008i \(-0.408544\pi\)
−0.804251 + 0.594290i \(0.797433\pi\)
\(14\) 0 0
\(15\) 2.37096 − 3.20844i 0.612180 − 0.828417i
\(16\) 0 0
\(17\) 1.57974 − 3.38777i 0.383144 − 0.821654i −0.616274 − 0.787532i \(-0.711359\pi\)
0.999418 − 0.0341223i \(-0.0108636\pi\)
\(18\) 0 0
\(19\) −3.00247 − 3.15994i −0.688813 − 0.724939i
\(20\) 0 0
\(21\) 1.67650 + 4.60614i 0.365842 + 1.00514i
\(22\) 0 0
\(23\) −3.36528 − 0.294424i −0.701710 − 0.0613917i −0.269284 − 0.963061i \(-0.586787\pi\)
−0.432426 + 0.901669i \(0.642342\pi\)
\(24\) 0 0
\(25\) −3.98934 + 3.01416i −0.797869 + 0.602832i
\(26\) 0 0
\(27\) −4.85444 + 1.30074i −0.934237 + 0.250328i
\(28\) 0 0
\(29\) −1.09210 − 0.397492i −0.202798 − 0.0738125i 0.238624 − 0.971112i \(-0.423304\pi\)
−0.441422 + 0.897300i \(0.645526\pi\)
\(30\) 0 0
\(31\) 7.63721 − 4.40934i 1.37168 − 0.791942i 0.380543 − 0.924763i \(-0.375737\pi\)
0.991140 + 0.132822i \(0.0424037\pi\)
\(32\) 0 0
\(33\) −10.0969 + 0.883363i −1.75764 + 0.153774i
\(34\) 0 0
\(35\) −0.378951 − 6.13172i −0.0640544 − 1.03645i
\(36\) 0 0
\(37\) 3.85089 + 3.85089i 0.633083 + 0.633083i 0.948840 − 0.315757i \(-0.102258\pi\)
−0.315757 + 0.948840i \(0.602258\pi\)
\(38\) 0 0
\(39\) 4.09038i 0.654984i
\(40\) 0 0
\(41\) −2.15157 − 0.379380i −0.336019 − 0.0592491i 0.00309324 − 0.999995i \(-0.499015\pi\)
−0.339112 + 0.940746i \(0.610127\pi\)
\(42\) 0 0
\(43\) −0.126691 − 1.44809i −0.0193203 − 0.220832i −0.999717 − 0.0237970i \(-0.992424\pi\)
0.980397 − 0.197035i \(-0.0631311\pi\)
\(44\) 0 0
\(45\) −0.409330 − 0.0104579i −0.0610193 − 0.00155897i
\(46\) 0 0
\(47\) 0.0279851 − 0.0130497i 0.00408204 − 0.00190349i −0.420576 − 0.907257i \(-0.638172\pi\)
0.424658 + 0.905354i \(0.360394\pi\)
\(48\) 0 0
\(49\) 0.474851 + 0.274156i 0.0678359 + 0.0391651i
\(50\) 0 0
\(51\) −6.56773 + 1.15807i −0.919667 + 0.162162i
\(52\) 0 0
\(53\) 0.107930 − 1.23364i 0.0148253 − 0.169454i −0.985172 − 0.171567i \(-0.945117\pi\)
0.999998 + 0.00211296i \(0.000672577\pi\)
\(54\) 0 0
\(55\) 12.4495 + 2.52462i 1.67868 + 0.340419i
\(56\) 0 0
\(57\) −1.54564 + 7.62170i −0.204724 + 1.00952i
\(58\) 0 0
\(59\) 6.00753 − 2.18656i 0.782114 − 0.284666i 0.0800600 − 0.996790i \(-0.474489\pi\)
0.702054 + 0.712124i \(0.252267\pi\)
\(60\) 0 0
\(61\) 8.16469 − 6.85099i 1.04538 − 0.877179i 0.0527806 − 0.998606i \(-0.483192\pi\)
0.992600 + 0.121427i \(0.0387472\pi\)
\(62\) 0 0
\(63\) 0.288567 − 0.412116i 0.0363560 − 0.0519218i
\(64\) 0 0
\(65\) 1.45288 − 4.91633i 0.180207 − 0.609795i
\(66\) 0 0
\(67\) 5.85638 + 12.5590i 0.715471 + 1.53433i 0.840281 + 0.542151i \(0.182390\pi\)
−0.124810 + 0.992181i \(0.539832\pi\)
\(68\) 0 0
\(69\) 3.01352 + 5.21957i 0.362785 + 0.628362i
\(70\) 0 0
\(71\) −7.74802 + 9.23373i −0.919521 + 1.09584i 0.0755958 + 0.997139i \(0.475914\pi\)
−0.995117 + 0.0987040i \(0.968530\pi\)
\(72\) 0 0
\(73\) 10.1192 − 7.08552i 1.18436 − 0.829298i 0.195877 − 0.980629i \(-0.437245\pi\)
0.988483 + 0.151331i \(0.0483559\pi\)
\(74\) 0 0
\(75\) 8.48754 + 2.74583i 0.980057 + 0.317061i
\(76\) 0 0
\(77\) −11.0364 + 11.0364i −1.25771 + 1.25771i
\(78\) 0 0
\(79\) −2.15191 + 12.2041i −0.242109 + 1.37307i 0.585002 + 0.811032i \(0.301094\pi\)
−0.827111 + 0.562038i \(0.810017\pi\)
\(80\) 0 0
\(81\) 7.28954 + 6.11665i 0.809949 + 0.679628i
\(82\) 0 0
\(83\) 3.01260 − 11.2432i 0.330676 − 1.23410i −0.577806 − 0.816174i \(-0.696091\pi\)
0.908482 − 0.417925i \(-0.137242\pi\)
\(84\) 0 0
\(85\) 8.30526 + 0.940909i 0.900832 + 0.102056i
\(86\) 0 0
\(87\) 0.536661 + 2.00284i 0.0575361 + 0.214727i
\(88\) 0 0
\(89\) 1.92434 + 10.9135i 0.203980 + 1.15683i 0.899037 + 0.437874i \(0.144268\pi\)
−0.695056 + 0.718955i \(0.744621\pi\)
\(90\) 0 0
\(91\) 4.04883 + 4.82520i 0.424432 + 0.505819i
\(92\) 0 0
\(93\) −14.2596 − 6.64934i −1.47865 − 0.689505i
\(94\) 0 0
\(95\) 4.56492 − 8.61171i 0.468351 − 0.883543i
\(96\) 0 0
\(97\) −10.5280 − 4.90928i −1.06896 − 0.498462i −0.193171 − 0.981165i \(-0.561877\pi\)
−0.875784 + 0.482703i \(0.839655\pi\)
\(98\) 0 0
\(99\) 0.668674 + 0.796895i 0.0672043 + 0.0800910i
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.33.3 ✓ 120
5.2 odd 4 inner 380.2.bh.a.337.3 yes 120
19.15 odd 18 inner 380.2.bh.a.53.3 yes 120
95.72 even 36 inner 380.2.bh.a.357.3 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.bh.a.33.3 ✓ 120 1.1 even 1 trivial
380.2.bh.a.53.3 yes 120 19.15 odd 18 inner
380.2.bh.a.337.3 yes 120 5.2 odd 4 inner
380.2.bh.a.357.3 yes 120 95.72 even 36 inner