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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 311.57
Character \(\chi\) \(=\) 380.311
Dual form 380.3.q.a.11.57

$q$-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+(1.22505 - 1.58091i) q^{2} +(0.144974 - 0.0837008i) q^{3} +(-0.998521 - 3.87336i) q^{4} +(1.11803 + 1.93649i) q^{5} +(0.0452769 - 0.331728i) q^{6} +9.41366i q^{7} +(-7.34666 - 3.16648i) q^{8} +(-4.48599 + 7.76996i) q^{9} +(4.43105 + 0.604787i) q^{10} +10.2613i q^{11} +(-0.468963 - 0.477960i) q^{12} +(-11.3778 + 19.7070i) q^{13} +(14.8821 + 11.5322i) q^{14} +(0.324172 + 0.187161i) q^{15} +(-14.0059 + 7.73528i) 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} +(16.2221 + 12.5705i) q^{22} +(-2.10884 - 1.21754i) q^{23} +(-1.33011 + 0.155863i) q^{24} +(-2.50000 + 4.33013i) q^{25} +(17.2165 + 42.1292i) q^{26} +3.00854i q^{27} +(36.4625 - 9.39974i) q^{28} +(8.50162 - 14.7252i) q^{29} +(0.693009 - 0.283204i) q^{30} -26.6804i q^{31} +(-4.92916 + 31.6181i) q^{32} +(0.858876 + 1.48762i) q^{33} +(-16.5249 - 2.25545i) q^{34} +(-18.2295 + 10.5248i) q^{35} +(34.5752 + 9.61740i) q^{36} -35.6396 q^{37} +(-26.4256 - 27.3073i) q^{38} +3.80934i q^{39} +(-2.08194 - 17.7670i) q^{40} +(30.0433 + 52.0365i) q^{41} +(3.12277 + 0.426222i) q^{42} +(64.6213 - 37.3091i) q^{43} +(39.7456 - 10.2461i) q^{44} -20.0619 q^{45} +(-4.50824 + 1.84233i) q^{46} +(26.0728 + 15.0532i) q^{47} +(-1.38304 + 2.29372i) q^{48} -39.6170 q^{49} +(3.78290 + 9.25687i) q^{50} +(-1.20894 - 0.697984i) q^{51} +(87.6934 + 24.3926i) q^{52} +(-2.71658 + 4.70525i) q^{53} +(4.75621 + 3.68560i) 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} +(0.401249 - 1.44252i) q^{60} +(-55.3656 + 95.8960i) q^{61} +(-42.1791 - 32.6847i) q^{62} +(-73.1438 - 42.2296i) q^{63} +(43.9467 + 46.5262i) q^{64} -50.8832 q^{65} +(3.40394 + 0.464599i) q^{66} +(41.9987 + 24.2479i) q^{67} +(-23.8094 + 23.3612i) q^{68} -0.407636 q^{69} +(-5.69326 + 41.7124i) q^{70} +(-22.2399 + 12.8402i) q^{71} +(57.5605 - 42.8784i) q^{72} +(-30.1843 - 52.2808i) q^{73} +(-43.6601 + 56.3428i) q^{74} +0.837008i q^{75} +(-75.5428 + 8.32369i) q^{76} -96.5961 q^{77} +(6.02220 + 4.66661i) q^{78} +(70.3212 - 40.5999i) q^{79} +(-30.6384 - 18.4740i) q^{80} +(-40.1221 - 69.4935i) q^{81} +(119.069 + 16.2516i) q^{82} +13.8217i q^{83} +(4.49936 - 4.41466i) q^{84} +(9.32332 - 16.1485i) q^{85} +(20.1819 - 147.865i) q^{86} -2.84637i q^{87} +(32.4921 - 75.3860i) q^{88} +(25.5101 - 44.1848i) q^{89} +(-24.5768 + 31.7160i) q^{90} +(-185.515 - 107.107i) q^{91} +(-2.61025 + 9.38404i) 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} +(-48.5327 + 62.6307i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.22505 − 1.58091i 0.612523 − 0.790453i
\(3\) 0.144974 − 0.0837008i 0.0483247 − 0.0279003i −0.475643 − 0.879638i \(-0.657785\pi\)
0.523968 + 0.851738i \(0.324451\pi\)
\(4\) −0.998521 − 3.87336i −0.249630 − 0.968341i
\(5\) 1.11803 + 1.93649i 0.223607 + 0.387298i
\(6\) 0.0452769 − 0.331728i 0.00754616 − 0.0552879i
\(7\) 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
\(10\) 4.43105 + 0.604787i 0.443105 + 0.0604787i
\(11\) 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 −0.0186875 + 0.999825i \(0.505949\pi\)
−0.856530 + 0.516097i \(0.827385\pi\)
\(14\) 14.8821 + 11.5322i 1.06301 + 0.823727i
\(15\) 0.324172 + 0.187161i 0.0216115 + 0.0124774i
\(16\) −14.0059 + 7.73528i −0.875369 + 0.483455i
\(17\) −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
\(22\) 16.2221 + 12.5705i 0.737367 + 0.571388i
\(23\) −2.10884 − 1.21754i −0.0916886 − 0.0529365i 0.453455 − 0.891279i \(-0.350191\pi\)
−0.545143 + 0.838343i \(0.683525\pi\)
\(24\) −1.33011 + 0.155863i −0.0554213 + 0.00649429i
\(25\) −2.50000 + 4.33013i −0.100000 + 0.173205i
\(26\) 17.2165 + 42.1292i 0.662173 + 1.62036i
\(27\) 3.00854i 0.111427i
\(28\) 36.4625 − 9.39974i 1.30223 − 0.335705i
\(29\) 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 \(-0.858398\pi\)
0.902672 − 0.430328i \(-0.141602\pi\)
\(32\) −4.92916 + 31.6181i −0.154036 + 0.988065i
\(33\) 0.858876 + 1.48762i 0.0260265 + 0.0450793i
\(34\) −16.5249 − 2.25545i −0.486025 − 0.0663368i
\(35\) −18.2295 + 10.5248i −0.520842 + 0.300708i
\(36\) 34.5752 + 9.61740i 0.960423 + 0.267150i
\(37\) −35.6396 −0.963232 −0.481616 − 0.876382i \(-0.659950\pi\)
−0.481616 + 0.876382i \(0.659950\pi\)
\(38\) −26.4256 − 27.3073i −0.695411 − 0.718613i
\(39\) 3.80934i 0.0976753i
\(40\) −2.08194 − 17.7670i −0.0520485 − 0.444174i
\(41\) 30.0433 + 52.0365i 0.732763 + 1.26918i 0.955698 + 0.294349i \(0.0951029\pi\)
−0.222935 + 0.974833i \(0.571564\pi\)
\(42\) 3.12277 + 0.426222i 0.0743517 + 0.0101481i
\(43\) 64.6213 − 37.3091i 1.50282 − 0.867654i 0.502826 − 0.864388i \(-0.332294\pi\)
0.999995 − 0.00326587i \(-0.00103956\pi\)
\(44\) 39.7456 − 10.2461i 0.903309 − 0.232866i
\(45\) −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 \(-0.229556\pi\)
−0.196291 + 0.980546i \(0.562890\pi\)
\(48\) −1.38304 + 2.29372i −0.0288134 + 0.0477858i
\(49\) −39.6170 −0.808510
\(50\) 3.78290 + 9.25687i 0.0756581 + 0.185137i
\(51\) −1.20894 − 0.697984i −0.0237048 − 0.0136860i
\(52\) 87.6934 + 24.3926i 1.68641 + 0.469089i
\(53\) −2.71658 + 4.70525i −0.0512562 + 0.0887783i −0.890515 − 0.454954i \(-0.849656\pi\)
0.839259 + 0.543732i \(0.182989\pi\)
\(54\) 4.75621 + 3.68560i 0.0880780 + 0.0682518i
\(55\) −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 \(-0.491196\pi\)
0.879522 + 0.475858i \(0.157862\pi\)
\(60\) 0.401249 − 1.44252i 0.00668749 − 0.0240420i
\(61\) −55.3656 + 95.8960i −0.907632 + 1.57206i −0.0902874 + 0.995916i \(0.528779\pi\)
−0.817345 + 0.576149i \(0.804555\pi\)
\(62\) −42.1791 − 32.6847i −0.680308 − 0.527172i
\(63\) −73.1438 − 42.2296i −1.16101 − 0.670311i
\(64\) 43.9467 + 46.5262i 0.686668 + 0.726971i
\(65\) −50.8832 −0.782819
\(66\) 3.40394 + 0.464599i 0.0515749 + 0.00703937i
\(67\) 41.9987 + 24.2479i 0.626846 + 0.361910i 0.779529 − 0.626366i \(-0.215458\pi\)
−0.152684 + 0.988275i \(0.548792\pi\)
\(68\) −23.8094 + 23.3612i −0.350138 + 0.343547i
\(69\) −0.407636 −0.00590777
\(70\) −5.69326 + 41.7124i −0.0813323 + 0.595892i
\(71\) −22.2399 + 12.8402i −0.313239 + 0.180848i −0.648375 − 0.761321i \(-0.724551\pi\)
0.335136 + 0.942170i \(0.391218\pi\)
\(72\) 57.5605 − 42.8784i 0.799451 − 0.595534i
\(73\) −30.1843 − 52.2808i −0.413484 − 0.716175i 0.581784 − 0.813343i \(-0.302355\pi\)
−0.995268 + 0.0971684i \(0.969021\pi\)
\(74\) −43.6601 + 56.3428i −0.590002 + 0.761389i
\(75\) 0.837008i 0.0111601i
\(76\) −75.5428 + 8.32369i −0.993984 + 0.109522i
\(77\) −96.5961 −1.25449
\(78\) 6.02220 + 4.66661i 0.0772077 + 0.0598284i
\(79\) 70.3212 − 40.5999i 0.890141 − 0.513923i 0.0161526 − 0.999870i \(-0.494858\pi\)
0.873989 + 0.485946i \(0.161525\pi\)
\(80\) −30.6384 − 18.4740i −0.382980 − 0.230925i
\(81\) −40.1221 − 69.4935i −0.495334 − 0.857944i
\(82\) 119.069 + 16.2516i 1.45206 + 0.198190i
\(83\) 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
\(86\) 20.1819 − 147.865i 0.234673 − 1.71937i
\(87\) − 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 \(0.0741314\pi\)
\(90\) −24.5768 + 31.7160i −0.273076 + 0.352400i
\(91\) −185.515 − 107.107i −2.03862 − 1.17700i
\(92\) −2.61025 + 9.38404i −0.0283723 + 0.102000i
\(93\) −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 \(0.561158\pi\)
\(98\) −48.5327 + 62.6307i −0.495232 + 0.639089i
\(99\) −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.311.57 yes 160
4.3 odd 2 inner 380.3.q.a.311.5 yes 160
19.11 even 3 inner 380.3.q.a.11.5 ✓ 160
76.11 odd 6 inner 380.3.q.a.11.57 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.q.a.11.5 ✓ 160 19.11 even 3 inner
380.3.q.a.11.57 yes 160 76.11 odd 6 inner
380.3.q.a.311.5 yes 160 4.3 odd 2 inner
380.3.q.a.311.57 yes 160 1.1 even 1 trivial