↑

Properties

Label 380.3.q.a.11.16
Level $380$
Weight $3$
Character 380.11
Analytic conductor $10.354$
Analytic rank $0$
Dimension $160$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,3,Mod(11,380)] 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("380.11"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 4])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.q (of order \(6\), degree \(2\), 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: \(10.3542500457\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(80\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 11.16
Character \(\chi\) \(=\) 380.11
Dual form 380.3.q.a.311.16

$q$-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+(-1.58052 - 1.22554i) q^{2} +(2.60636 + 1.50478i) q^{3} +(0.996097 + 3.87399i) q^{4} +(-1.11803 + 1.93649i) q^{5} +(-2.27523 - 5.57254i) q^{6} -3.75845i q^{7} +(3.17338 - 7.34368i) q^{8} +(0.0287371 + 0.0497742i) q^{9} +(4.14033 - 1.69047i) q^{10} -10.2592i q^{11} +(-3.23332 + 11.5959i) q^{12} +(-6.68139 - 11.5725i) q^{13} +(-4.60614 + 5.94032i) q^{14} +(-5.82800 + 3.36479i) q^{15} +(-14.0156 + 7.71774i) q^{16} +(-3.22495 + 5.58578i) q^{17} +(0.0155806 - 0.113888i) q^{18} +(13.1289 - 13.7343i) q^{19} +(-8.61562 - 2.40232i) q^{20} +(5.65565 - 9.79588i) q^{21} +(-12.5731 + 16.2149i) q^{22} +(31.7753 - 18.3455i) q^{23} +(19.3216 - 14.3650i) q^{24} +(-2.50000 - 4.33013i) q^{25} +(-3.62250 + 26.4789i) q^{26} -26.9131i q^{27} +(14.5602 - 3.74379i) q^{28} +(11.5334 + 19.9764i) q^{29} +(13.3350 + 1.82432i) q^{30} -9.19445i q^{31} +(31.6103 + 4.97861i) q^{32} +(15.4378 - 26.7391i) q^{33} +(11.9427 - 4.87614i) q^{34} +(7.27822 + 4.20208i) q^{35} +(-0.164200 + 0.160907i) q^{36} +54.0057 q^{37} +(-37.5825 + 5.61735i) q^{38} -40.2162i q^{39} +(10.6730 + 14.3557i) q^{40} +(-22.9075 + 39.6769i) q^{41} +(-20.9441 + 8.55136i) q^{42} +(20.7909 + 12.0037i) q^{43} +(39.7440 - 10.2192i) q^{44} -0.128516 q^{45} +(-72.7046 - 9.94650i) q^{46} +(-38.1988 + 22.0541i) q^{47} +(-48.1431 - 0.975190i) q^{48} +34.8740 q^{49} +(-1.35545 + 9.90771i) q^{50} +(-16.8108 + 9.70571i) q^{51} +(38.1765 - 37.4110i) q^{52} +(-27.5614 - 47.7377i) q^{53} +(-32.9831 + 42.5367i) q^{54} +(19.8668 + 11.4701i) q^{55} +(-27.6009 - 11.9270i) q^{56} +(54.8858 - 16.0404i) q^{57} +(6.25314 - 45.7077i) q^{58} +(16.5844 + 9.57499i) q^{59} +(-18.8404 - 19.2259i) q^{60} +(-58.7304 - 101.724i) q^{61} +(-11.2682 + 14.5320i) q^{62} +(0.187074 - 0.108007i) q^{63} +(-43.8593 - 46.6086i) q^{64} +29.8801 q^{65} +(-57.1698 + 23.3421i) q^{66} +(19.6264 - 11.3313i) q^{67} +(-24.8516 - 6.92945i) q^{68} +110.424 q^{69} +(-6.35356 - 15.5612i) q^{70} +(-89.9473 - 51.9311i) q^{71} +(0.456720 - 0.0530840i) q^{72} +(-2.26357 + 3.92062i) q^{73} +(-85.3571 - 66.1862i) q^{74} -15.0478i q^{75} +(66.2843 + 37.1806i) q^{76} -38.5587 q^{77} +(-49.2866 + 63.5625i) q^{78} +(-103.937 - 60.0081i) q^{79} +(0.724554 - 35.7698i) q^{80} +(40.7570 - 70.5932i) q^{81} +(84.8315 - 34.6362i) q^{82} +105.943i q^{83} +(43.5827 + 12.1523i) q^{84} +(-7.21122 - 12.4902i) q^{85} +(-18.1496 - 44.4522i) q^{86} +69.4208i q^{87} +(-75.3402 - 32.5563i) q^{88} +(2.30732 + 3.99639i) q^{89} +(0.203123 + 0.157502i) q^{90} +(-43.4948 + 25.1117i) q^{91} +(102.721 + 104.823i) q^{92} +(13.8356 - 23.9640i) q^{93} +(87.4022 + 11.9572i) q^{94} +(11.9178 + 40.7795i) q^{95} +(74.8961 + 60.5427i) q^{96} +(27.3937 - 47.4472i) q^{97} +(-55.1191 - 42.7395i) q^{98} +(0.510643 - 0.294820i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 160 q + 2 q^{4} + 6 q^{6} + 248 q^{9} - 10 q^{10} - 16 q^{13} - 14 q^{16} + 48 q^{17} + 48 q^{21} - 44 q^{24} - 400 q^{25} + 68 q^{26} + 60 q^{28} - 80 q^{30} + 30 q^{32} - 40 q^{33} - 22 q^{34} + 52 q^{36}+ \cdots - 226 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.58052 − 1.22554i −0.790261 − 0.612771i
\(3\) 2.60636 + 1.50478i 0.868786 + 0.501594i 0.866945 − 0.498404i \(-0.166080\pi\)
0.00184153 + 0.999998i \(0.499414\pi\)
\(4\) 0.996097 + 3.87399i 0.249024 + 0.968497i
\(5\) −1.11803 + 1.93649i −0.223607 + 0.387298i
\(6\) −2.27523 − 5.57254i −0.379206 − 0.928757i
\(7\) − 3.75845i − 0.536922i −0.963291 − 0.268461i \(-0.913485\pi\)
0.963291 − 0.268461i \(-0.0865151\pi\)
\(8\) 3.17338 − 7.34368i 0.396672 − 0.917960i
\(9\) 0.0287371 + 0.0497742i 0.00319302 + 0.00553047i
\(10\) 4.14033 − 1.69047i 0.414033 − 0.169047i
\(11\) − 10.2592i − 0.932654i −0.884612 − 0.466327i \(-0.845577\pi\)
0.884612 − 0.466327i \(-0.154423\pi\)
\(12\) −3.23332 + 11.5959i −0.269443 + 0.966326i
\(13\) −6.68139 − 11.5725i −0.513953 − 0.890193i −0.999869 − 0.0161876i \(-0.994847\pi\)
0.485916 − 0.874006i \(-0.338486\pi\)
\(14\) −4.60614 + 5.94032i −0.329010 + 0.424309i
\(15\) −5.82800 + 3.36479i −0.388533 + 0.224320i
\(16\) −14.0156 + 7.71774i −0.875974 + 0.482359i
\(17\) −3.22495 + 5.58578i −0.189703 + 0.328576i −0.945151 − 0.326633i \(-0.894086\pi\)
0.755448 + 0.655209i \(0.227419\pi\)
\(18\) 0.0155806 − 0.113888i 0.000865592 − 0.00632710i
\(19\) 13.1289 − 13.7343i 0.690996 − 0.722859i
\(20\) −8.61562 − 2.40232i −0.430781 − 0.120116i
\(21\) 5.65565 − 9.79588i 0.269317 − 0.466471i
\(22\) −12.5731 + 16.2149i −0.571503 + 0.737040i
\(23\) 31.7753 − 18.3455i 1.38153 − 0.797628i 0.389192 − 0.921157i \(-0.372754\pi\)
0.992341 + 0.123528i \(0.0394210\pi\)
\(24\) 19.3216 − 14.3650i 0.805067 − 0.598543i
\(25\) −2.50000 − 4.33013i −0.100000 − 0.173205i
\(26\) −3.62250 + 26.4789i −0.139327 + 1.01842i
\(27\) − 26.9131i − 0.996782i
\(28\) 14.5602 − 3.74379i 0.520008 − 0.133707i
\(29\) 11.5334 + 19.9764i 0.397702 + 0.688841i 0.993442 − 0.114337i \(-0.0364743\pi\)
−0.595740 + 0.803178i \(0.703141\pi\)
\(30\) 13.3350 + 1.82432i 0.444499 + 0.0608106i
\(31\) − 9.19445i − 0.296595i −0.988943 − 0.148298i \(-0.952621\pi\)
0.988943 − 0.148298i \(-0.0473793\pi\)
\(32\) 31.6103 + 4.97861i 0.987823 + 0.155582i
\(33\) 15.4378 − 26.7391i 0.467813 − 0.810277i
\(34\) 11.9427 − 4.87614i 0.351256 − 0.143416i
\(35\) 7.27822 + 4.20208i 0.207949 + 0.120059i
\(36\) −0.164200 + 0.160907i −0.00456110 + 0.00446965i
\(37\) 54.0057 1.45961 0.729806 − 0.683654i \(-0.239610\pi\)
0.729806 + 0.683654i \(0.239610\pi\)
\(38\) −37.5825 + 5.61735i −0.989014 + 0.147825i
\(39\) − 40.2162i − 1.03118i
\(40\) 10.6730 + 14.3557i 0.266826 + 0.358893i
\(41\) −22.9075 + 39.6769i −0.558719 + 0.967730i 0.438885 + 0.898543i \(0.355374\pi\)
−0.997604 + 0.0691863i \(0.977960\pi\)
\(42\) −20.9441 + 8.55136i −0.498670 + 0.203604i
\(43\) 20.7909 + 12.0037i 0.483510 + 0.279155i 0.721878 − 0.692020i \(-0.243279\pi\)
−0.238368 + 0.971175i \(0.576612\pi\)
\(44\) 39.7440 − 10.2192i 0.903273 − 0.232253i
\(45\) −0.128516 −0.00285592
\(46\) −72.7046 − 9.94650i −1.58053 − 0.216228i
\(47\) −38.1988 + 22.0541i −0.812740 + 0.469236i −0.847906 − 0.530146i \(-0.822137\pi\)
0.0351665 + 0.999381i \(0.488804\pi\)
\(48\) −48.1431 − 0.975190i −1.00298 − 0.0203165i
\(49\) 34.8740 0.711715
\(50\) −1.35545 + 9.90771i −0.0271089 + 0.198154i
\(51\) −16.8108 + 9.70571i −0.329623 + 0.190308i
\(52\) 38.1765 − 37.4110i 0.734163 − 0.719442i
\(53\) −27.5614 − 47.7377i −0.520026 − 0.900711i −0.999729 − 0.0232804i \(-0.992589\pi\)
0.479703 − 0.877431i \(-0.340744\pi\)
\(54\) −32.9831 + 42.5367i −0.610798 + 0.787717i
\(55\) 19.8668 + 11.4701i 0.361215 + 0.208548i
\(56\) −27.6009 − 11.9270i −0.492873 − 0.212982i
\(57\) 54.8858 − 16.0404i 0.962909 − 0.281410i
\(58\) 6.25314 − 45.7077i 0.107813 − 0.788064i
\(59\) 16.5844 + 9.57499i 0.281091 + 0.162288i 0.633917 − 0.773401i \(-0.281446\pi\)
−0.352826 + 0.935689i \(0.614779\pi\)
\(60\) −18.8404 − 19.2259i −0.314007 − 0.320432i
\(61\) −58.7304 − 101.724i −0.962794 − 1.66761i −0.715429 − 0.698685i \(-0.753769\pi\)
−0.247364 − 0.968923i \(-0.579564\pi\)
\(62\) −11.2682 + 14.5320i −0.181745 + 0.234387i
\(63\) 0.187074 − 0.108007i 0.00296943 − 0.00171440i
\(64\) −43.8593 − 46.6086i −0.685302 − 0.728259i
\(65\) 29.8801 0.459694
\(66\) −57.1698 + 23.3421i −0.866208 + 0.353668i
\(67\) 19.6264 − 11.3313i 0.292931 − 0.169124i −0.346332 − 0.938112i \(-0.612573\pi\)
0.639263 + 0.768988i \(0.279240\pi\)
\(68\) −24.8516 − 6.92945i −0.365465 − 0.101904i
\(69\) 110.424 1.60034
\(70\) −6.35356 − 15.5612i −0.0907651 − 0.222303i
\(71\) −89.9473 − 51.9311i −1.26686 − 0.731424i −0.292470 − 0.956275i \(-0.594477\pi\)
−0.974393 + 0.224850i \(0.927811\pi\)
\(72\) 0.456720 − 0.0530840i 0.00634333 − 0.000737278i
\(73\) −2.26357 + 3.92062i −0.0310078 + 0.0537071i −0.881113 − 0.472906i \(-0.843205\pi\)
0.850105 + 0.526613i \(0.176538\pi\)
\(74\) −85.3571 − 66.1862i −1.15347 − 0.894408i
\(75\) − 15.0478i − 0.200638i
\(76\) 66.2843 + 37.1806i 0.872161 + 0.489218i
\(77\) −38.5587 −0.500762
\(78\) −49.2866 + 63.5625i −0.631879 + 0.814904i
\(79\) −103.937 − 60.0081i −1.31566 − 0.759596i −0.332632 − 0.943057i \(-0.607937\pi\)
−0.983027 + 0.183460i \(0.941270\pi\)
\(80\) 0.724554 − 35.7698i 0.00905693 − 0.447122i
\(81\) 40.7570 − 70.5932i 0.503173 − 0.871521i
\(82\) 84.8315 − 34.6362i 1.03453 − 0.422392i
\(83\) 105.943i 1.27643i 0.769860 + 0.638213i \(0.220326\pi\)
−0.769860 + 0.638213i \(0.779674\pi\)
\(84\) 43.5827 + 12.1523i 0.518842 + 0.144670i
\(85\) −7.21122 − 12.4902i −0.0848378 − 0.146943i
\(86\) −18.1496 − 44.4522i −0.211041 − 0.516886i
\(87\) 69.4208i 0.797940i
\(88\) −75.3402 − 32.5563i −0.856139 − 0.369958i
\(89\) 2.30732 + 3.99639i 0.0259249 + 0.0449033i 0.878697 − 0.477380i \(-0.158414\pi\)
−0.852772 + 0.522284i \(0.825080\pi\)
\(90\) 0.203123 + 0.157502i 0.00225692 + 0.00175002i
\(91\) −43.4948 + 25.1117i −0.477965 + 0.275953i
\(92\) 102.721 + 104.823i 1.11654 + 1.13938i
\(93\) 13.8356 − 23.9640i 0.148770 − 0.257678i
\(94\) 87.4022 + 11.9572i 0.929810 + 0.127205i
\(95\) 11.9178 + 40.7795i 0.125451 + 0.429258i
\(96\) 74.8961 + 60.5427i 0.780168 + 0.630653i
\(97\) 27.3937 − 47.4472i 0.282409 − 0.489146i −0.689569 − 0.724220i \(-0.742200\pi\)
0.971978 + 0.235074i \(0.0755332\pi\)
\(98\) −55.1191 − 42.7395i −0.562440 − 0.436118i
\(99\) 0.510643 − 0.294820i 0.00515801 − 0.00297798i
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 380.3.q.a.11.16 ✓ 160
4.3 odd 2 inner 380.3.q.a.11.71 yes 160
19.7 even 3 inner 380.3.q.a.311.71 yes 160
76.7 odd 6 inner 380.3.q.a.311.16 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.q.a.11.16 ✓ 160 1.1 even 1 trivial
380.3.q.a.11.71 yes 160 4.3 odd 2 inner
380.3.q.a.311.16 yes 160 76.7 odd 6 inner
380.3.q.a.311.71 yes 160 19.7 even 3 inner