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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 311.5
Character \(\chi\) \(=\) 380.311
Dual form 380.3.q.a.11.5

$q$-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 \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

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

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 −0.523968 − 0.851738i \(-0.675549\pi\)
0.475643 + 0.879638i \(0.342215\pi\)
\(4\) 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 \(-0.765262\pi\)
0.740184 − 0.672404i \(-0.234738\pi\)
\(8\) −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 \(-0.845543\pi\)
0.884563 − 0.466421i \(-0.154457\pi\)
\(12\) −0.468963 + 0.477960i −0.0390803 + 0.0398300i
\(13\) −11.3778 + 19.7070i −0.875218 + 1.51592i −0.0186875 + 0.999825i \(0.505949\pi\)
−0.856530 + 0.516097i \(0.827385\pi\)
\(14\) 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 0.716941 − 0.697134i \(-0.245542\pi\)
−0.962206 + 0.272322i \(0.912209\pi\)
\(18\) 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 0.545143 − 0.838343i \(-0.316475\pi\)
−0.453455 + 0.891279i \(0.649809\pi\)
\(24\) 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 −0.681396 − 0.731915i \(-0.738627\pi\)
0.974555 + 0.224148i \(0.0719601\pi\)
\(30\) 0.693009 + 0.283204i 0.0231003 + 0.00944014i
\(31\) 26.6804i 0.860657i 0.902672 + 0.430328i \(0.141602\pi\)
−0.902672 + 0.430328i \(0.858398\pi\)
\(32\) −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 0.955698 + 0.294349i \(0.0951029\pi\)
−0.222935 + 0.974833i \(0.571564\pi\)
\(42\) −1.93050 − 2.49129i −0.0459644 − 0.0593164i
\(43\) −64.6213 + 37.3091i −1.50282 + 0.867654i −0.502826 + 0.864388i \(0.667706\pi\)
−0.999995 + 0.00326587i \(0.998960\pi\)
\(44\) −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 0.196291 − 0.980546i \(-0.437110\pi\)
−0.751032 + 0.660266i \(0.770444\pi\)
\(48\) −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 −0.890515 − 0.454954i \(-0.849656\pi\)
0.839259 + 0.543732i \(0.182989\pi\)
\(54\) 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 −0.879522 − 0.475858i \(-0.842138\pi\)
−0.0276564 + 0.999617i \(0.508804\pi\)
\(60\) −1.44988 − 0.373768i −0.0241647 − 0.00622947i
\(61\) −55.3656 + 95.8960i −0.907632 + 1.57206i −0.0902874 + 0.995916i \(0.528779\pi\)
−0.817345 + 0.576149i \(0.804555\pi\)
\(62\) −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 0.152684 − 0.988275i \(-0.451208\pi\)
−0.779529 + 0.626366i \(0.784542\pi\)
\(68\) −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 −0.335136 − 0.942170i \(-0.608782\pi\)
0.648375 + 0.761321i \(0.275449\pi\)
\(72\) 8.35356 − 71.2880i 0.116022 − 0.990112i
\(73\) −30.1843 − 52.2808i −0.413484 − 0.716175i 0.581784 − 0.813343i \(-0.302355\pi\)
−0.995268 + 0.0971684i \(0.969021\pi\)
\(74\) 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 −0.873989 − 0.485946i \(-0.838475\pi\)
−0.0161526 + 0.999870i \(0.505142\pi\)
\(80\) 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 \(-0.973466\pi\)
0.996528 − 0.0832634i \(-0.0265343\pi\)
\(84\) 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 −0.686373 − 0.727250i \(-0.740798\pi\)
0.973003 + 0.230791i \(0.0741314\pi\)
\(90\) 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 0.945566 + 0.325430i \(0.105509\pi\)
−0.190952 + 0.981599i \(0.561158\pi\)
\(98\) 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.311.5 yes 160
4.3 odd 2 inner 380.3.q.a.311.57 yes 160
19.11 even 3 inner 380.3.q.a.11.57 yes 160
76.11 odd 6 inner 380.3.q.a.11.5 ✓ 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.q.a.11.5 ✓ 160 76.11 odd 6 inner
380.3.q.a.11.57 yes 160 19.11 even 3 inner
380.3.q.a.311.5 yes 160 1.1 even 1 trivial
380.3.q.a.311.57 yes 160 4.3 odd 2 inner