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Properties

Label 380.3.q.a.11.5
Level $380$
Weight $3$
Character 380.11
Analytic conductor $10.354$
Analytic rank $0$
Dimension $160$
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] = [380,3,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 11.5
Character \(\chi\) \(=\) 380.11
Dual form 380.3.q.a.311.5

$q$-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+(-1.98163 - 0.270469i) q^{2} +(-0.144974 - 0.0837008i) q^{3} +(3.85369 + 1.07194i) q^{4} +(1.11803 - 1.93649i) q^{5} +(0.264646 + 0.205075i) q^{6} +9.41366i q^{7} +(-7.34666 - 3.16648i) q^{8} +(-4.48599 - 7.76996i) q^{9} +(-2.73929 + 3.53501i) q^{10} +10.2613i q^{11} +(-0.468963 - 0.477960i) q^{12} +(-11.3778 - 19.7070i) q^{13} +(2.54610 - 18.6544i) q^{14} +(-0.324172 + 0.187161i) q^{15} +(13.7019 + 8.26184i) q^{16} +(-4.16952 + 7.22181i) q^{17} +(6.78802 + 16.6105i) q^{18} +(-2.69940 - 18.8073i) q^{19} +(6.38436 - 6.26418i) q^{20} +(0.787931 - 1.36474i) q^{21} +(2.77535 - 20.3340i) q^{22} +(2.10884 - 1.21754i) q^{23} +(0.800037 + 1.07398i) q^{24} +(-2.50000 - 4.33013i) q^{25} +(17.2165 + 42.1292i) q^{26} +3.00854i q^{27} +(-10.0909 + 36.2774i) q^{28} +(8.50162 + 14.7252i) q^{29} +(0.693009 - 0.283204i) q^{30} -26.6804i q^{31} +(-24.9175 - 20.0778i) q^{32} +(0.858876 - 1.48762i) q^{33} +(10.2157 - 13.1832i) q^{34} +(18.2295 + 10.5248i) q^{35} +(-8.95871 - 34.7517i) q^{36} -35.6396 q^{37} +(0.262426 + 37.9991i) q^{38} +3.80934i q^{39} +(-14.3457 + 10.6865i) q^{40} +(30.0433 - 52.0365i) q^{41} +(-1.93050 + 2.49129i) q^{42} +(-64.6213 - 37.3091i) q^{43} +(-10.9994 + 39.5438i) q^{44} -20.0619 q^{45} +(-4.50824 + 1.84233i) q^{46} +(-26.0728 + 15.0532i) q^{47} +(-1.29490 - 2.34461i) q^{48} -39.6170 q^{49} +(3.78290 + 9.25687i) q^{50} +(1.20894 - 0.697984i) q^{51} +(-22.7220 - 88.1410i) q^{52} +(-2.71658 - 4.70525i) q^{53} +(0.813716 - 5.96180i) q^{54} +(19.8709 + 11.4724i) q^{55} +(29.8082 - 69.1589i) q^{56} +(-1.18284 + 2.95251i) q^{57} +(-12.8643 - 31.4793i) q^{58} +(-53.5235 - 30.9018i) q^{59} +(-1.44988 + 0.373768i) q^{60} +(-55.3656 - 95.8960i) q^{61} +(-7.21621 + 52.8705i) q^{62} +(73.1438 - 42.2296i) q^{63} +(43.9467 + 46.5262i) q^{64} -50.8832 q^{65} +(-2.10433 + 2.71560i) q^{66} +(-41.9987 + 24.2479i) q^{67} +(-23.8094 + 23.3612i) q^{68} -0.407636 q^{69} +(-33.2774 - 25.7867i) q^{70} +(22.2399 + 12.8402i) q^{71} +(8.35356 + 71.2880i) q^{72} +(-30.1843 + 52.2808i) q^{73} +(70.6243 + 9.63940i) q^{74} +0.837008i q^{75} +(9.75755 - 75.3710i) q^{76} -96.5961 q^{77} +(1.03031 - 7.54868i) q^{78} +(-70.3212 - 40.5999i) q^{79} +(31.3182 - 17.2966i) q^{80} +(-40.1221 + 69.4935i) q^{81} +(-73.6088 + 94.9912i) q^{82} +13.8217i q^{83} +(4.49936 - 4.41466i) q^{84} +(9.32332 + 16.1485i) q^{85} +(117.964 + 91.4108i) q^{86} -2.84637i q^{87} +(32.4921 - 75.3860i) q^{88} +(25.5101 + 44.1848i) q^{89} +(39.7553 + 5.42614i) q^{90} +(185.515 - 107.107i) q^{91} +(9.43194 - 2.43148i) q^{92} +(-2.23317 + 3.86796i) q^{93} +(55.7381 - 22.7779i) q^{94} +(-39.4381 - 15.7998i) q^{95} +(1.93186 + 4.99638i) q^{96} +(73.1975 - 126.782i) q^{97} +(78.5062 + 10.7152i) q^{98} +(79.7296 - 46.0319i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.98163 − 0.270469i −0.990814 − 0.135234i
\(3\) −0.144974 − 0.0837008i −0.0483247 − 0.0279003i 0.475643 − 0.879638i \(-0.342215\pi\)
−0.523968 + 0.851738i \(0.675549\pi\)
\(4\) 3.85369 + 1.07194i 0.963423 + 0.267984i
\(5\) 1.11803 − 1.93649i 0.223607 − 0.387298i
\(6\) 0.264646 + 0.205075i 0.0441077 + 0.0341791i
\(7\) 9.41366i 1.34481i 0.740184 + 0.672404i \(0.234738\pi\)
−0.740184 + 0.672404i \(0.765262\pi\)
\(8\) −7.34666 − 3.16648i −0.918332 − 0.395811i
\(9\) −4.48599 − 7.76996i −0.498443 − 0.863329i
\(10\) −2.73929 + 3.53501i −0.273929 + 0.353501i
\(11\) 10.2613i 0.932842i 0.884563 + 0.466421i \(0.154457\pi\)
−0.884563 + 0.466421i \(0.845543\pi\)
\(12\) −0.468963 − 0.477960i −0.0390803 − 0.0398300i
\(13\) −11.3778 − 19.7070i −0.875218 − 1.51592i −0.856530 − 0.516097i \(-0.827385\pi\)
−0.0186875 − 0.999825i \(-0.505949\pi\)
\(14\) 2.54610 − 18.6544i 0.181865 − 1.33245i
\(15\) −0.324172 + 0.187161i −0.0216115 + 0.0124774i
\(16\) 13.7019 + 8.26184i 0.856369 + 0.516365i
\(17\) −4.16952 + 7.22181i −0.245266 + 0.424813i −0.962206 − 0.272322i \(-0.912209\pi\)
0.716941 + 0.697134i \(0.245542\pi\)
\(18\) 6.78802 + 16.6105i 0.377112 + 0.922805i
\(19\) −2.69940 − 18.8073i −0.142074 − 0.989856i
\(20\) 6.38436 − 6.26418i 0.319218 − 0.313209i
\(21\) 0.787931 − 1.36474i 0.0375205 − 0.0649875i
\(22\) 2.77535 − 20.3340i 0.126152 − 0.924273i
\(23\) 2.10884 − 1.21754i 0.0916886 − 0.0529365i −0.453455 − 0.891279i \(-0.649809\pi\)
0.545143 + 0.838343i \(0.316475\pi\)
\(24\) 0.800037 + 1.07398i 0.0333349 + 0.0447491i
\(25\) −2.50000 − 4.33013i −0.100000 − 0.173205i
\(26\) 17.2165 + 42.1292i 0.662173 + 1.62036i
\(27\) 3.00854i 0.111427i
\(28\) −10.0909 + 36.2774i −0.360388 + 1.29562i
\(29\) 8.50162 + 14.7252i 0.293159 + 0.507767i 0.974555 − 0.224148i \(-0.0719601\pi\)
−0.681396 + 0.731915i \(0.738627\pi\)
\(30\) 0.693009 − 0.283204i 0.0231003 − 0.00944014i
\(31\) − 26.6804i − 0.860657i −0.902672 − 0.430328i \(-0.858398\pi\)
0.902672 − 0.430328i \(-0.141602\pi\)
\(32\) −24.9175 − 20.0778i −0.778672 − 0.627432i
\(33\) 0.858876 − 1.48762i 0.0260265 − 0.0450793i
\(34\) 10.2157 − 13.1832i 0.300462 − 0.387742i
\(35\) 18.2295 + 10.5248i 0.520842 + 0.300708i
\(36\) −8.95871 − 34.7517i −0.248853 − 0.965326i
\(37\) −35.6396 −0.963232 −0.481616 − 0.876382i \(-0.659950\pi\)
−0.481616 + 0.876382i \(0.659950\pi\)
\(38\) 0.262426 + 37.9991i 0.00690596 + 0.999976i
\(39\) 3.80934i 0.0976753i
\(40\) −14.3457 + 10.6865i −0.358642 + 0.267163i
\(41\) 30.0433 − 52.0365i 0.732763 − 1.26918i −0.222935 − 0.974833i \(-0.571564\pi\)
0.955698 − 0.294349i \(-0.0951029\pi\)
\(42\) −1.93050 + 2.49129i −0.0459644 + 0.0593164i
\(43\) −64.6213 − 37.3091i −1.50282 − 0.867654i −0.999995 − 0.00326587i \(-0.998960\pi\)
−0.502826 − 0.864388i \(-0.667706\pi\)
\(44\) −10.9994 + 39.5438i −0.249987 + 0.898722i
\(45\) −20.0619 −0.445821
\(46\) −4.50824 + 1.84233i −0.0980052 + 0.0400507i
\(47\) −26.0728 + 15.0532i −0.554741 + 0.320280i −0.751032 − 0.660266i \(-0.770444\pi\)
0.196291 + 0.980546i \(0.437110\pi\)
\(48\) −1.29490 − 2.34461i −0.0269770 − 0.0488461i
\(49\) −39.6170 −0.808510
\(50\) 3.78290 + 9.25687i 0.0756581 + 0.185137i
\(51\) 1.20894 − 0.697984i 0.0237048 − 0.0136860i
\(52\) −22.7220 − 88.1410i −0.436962 − 1.69502i
\(53\) −2.71658 − 4.70525i −0.0512562 − 0.0887783i 0.839259 − 0.543732i \(-0.182989\pi\)
−0.890515 + 0.454954i \(0.849656\pi\)
\(54\) 0.813716 − 5.96180i 0.0150688 − 0.110404i
\(55\) 19.8709 + 11.4724i 0.361288 + 0.208590i
\(56\) 29.8082 − 69.1589i 0.532290 − 1.23498i
\(57\) −1.18284 + 2.95251i −0.0207516 + 0.0517984i
\(58\) −12.8643 − 31.4793i −0.221799 − 0.542747i
\(59\) −53.5235 − 30.9018i −0.907179 − 0.523760i −0.0276564 − 0.999617i \(-0.508804\pi\)
−0.879522 + 0.475858i \(0.842138\pi\)
\(60\) −1.44988 + 0.373768i −0.0241647 + 0.00622947i
\(61\) −55.3656 − 95.8960i −0.907632 − 1.57206i −0.817345 − 0.576149i \(-0.804555\pi\)
−0.0902874 − 0.995916i \(-0.528779\pi\)
\(62\) −7.21621 + 52.8705i −0.116390 + 0.852750i
\(63\) 73.1438 − 42.2296i 1.16101 − 0.670311i
\(64\) 43.9467 + 46.5262i 0.686668 + 0.726971i
\(65\) −50.8832 −0.782819
\(66\) −2.10433 + 2.71560i −0.0318837 + 0.0411455i
\(67\) −41.9987 + 24.2479i −0.626846 + 0.361910i −0.779529 − 0.626366i \(-0.784542\pi\)
0.152684 + 0.988275i \(0.451208\pi\)
\(68\) −23.8094 + 23.3612i −0.350138 + 0.343547i
\(69\) −0.407636 −0.00590777
\(70\) −33.2774 − 25.7867i −0.475391 − 0.368382i
\(71\) 22.2399 + 12.8402i 0.313239 + 0.180848i 0.648375 − 0.761321i \(-0.275449\pi\)
−0.335136 + 0.942170i \(0.608782\pi\)
\(72\) 8.35356 + 71.2880i 0.116022 + 0.990112i
\(73\) −30.1843 + 52.2808i −0.413484 + 0.716175i −0.995268 − 0.0971684i \(-0.969021\pi\)
0.581784 + 0.813343i \(0.302355\pi\)
\(74\) 70.6243 + 9.63940i 0.954383 + 0.130262i
\(75\) 0.837008i 0.0111601i
\(76\) 9.75755 − 75.3710i 0.128389 − 0.991724i
\(77\) −96.5961 −1.25449
\(78\) 1.03031 − 7.54868i 0.0132091 − 0.0967780i
\(79\) −70.3212 − 40.5999i −0.890141 − 0.513923i −0.0161526 − 0.999870i \(-0.505142\pi\)
−0.873989 + 0.485946i \(0.838475\pi\)
\(80\) 31.3182 − 17.2966i 0.391477 − 0.216208i
\(81\) −40.1221 + 69.4935i −0.495334 + 0.857944i
\(82\) −73.6088 + 94.9912i −0.897669 + 1.15843i
\(83\) 13.8217i 0.166527i 0.996528 + 0.0832634i \(0.0265343\pi\)
−0.996528 + 0.0832634i \(0.973466\pi\)
\(84\) 4.49936 − 4.41466i 0.0535638 − 0.0525555i
\(85\) 9.32332 + 16.1485i 0.109686 + 0.189982i
\(86\) 117.964 + 91.4108i 1.37168 + 1.06292i
\(87\) − 2.84637i − 0.0327169i
\(88\) 32.4921 − 75.3860i 0.369229 − 0.856659i
\(89\) 25.5101 + 44.1848i 0.286631 + 0.496459i 0.973003 − 0.230791i \(-0.0741314\pi\)
−0.686373 + 0.727250i \(0.740798\pi\)
\(90\) 39.7553 + 5.42614i 0.441726 + 0.0602904i
\(91\) 185.515 − 107.107i 2.03862 − 1.17700i
\(92\) 9.43194 − 2.43148i 0.102521 − 0.0264291i
\(93\) −2.23317 + 3.86796i −0.0240125 + 0.0415910i
\(94\) 55.7381 − 22.7779i 0.592958 − 0.242318i
\(95\) −39.4381 − 15.7998i −0.415138 − 0.166314i
\(96\) 1.93186 + 4.99638i 0.0201235 + 0.0520456i
\(97\) 73.1975 − 126.782i 0.754614 − 1.30703i −0.190952 − 0.981599i \(-0.561158\pi\)
0.945566 − 0.325430i \(-0.105509\pi\)
\(98\) 78.5062 + 10.7152i 0.801083 + 0.109339i
\(99\) 79.7296 − 46.0319i 0.805350 − 0.464969i
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.11.5 ✓ 160
4.3 odd 2 inner 380.3.q.a.11.57 yes 160
19.7 even 3 inner 380.3.q.a.311.57 yes 160
76.7 odd 6 inner 380.3.q.a.311.5 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.q.a.11.5 ✓ 160 1.1 even 1 trivial
380.3.q.a.11.57 yes 160 4.3 odd 2 inner
380.3.q.a.311.5 yes 160 76.7 odd 6 inner
380.3.q.a.311.57 yes 160 19.7 even 3 inner