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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 11.57
Character \(\chi\) \(=\) 380.11
Dual form 380.3.q.a.311.57

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.22505 + 1.58091i 0.612523 + 0.790453i
\(3\) 0.144974 + 0.0837008i 0.0483247 + 0.0279003i 0.523968 − 0.851738i \(-0.324451\pi\)
−0.475643 + 0.879638i \(0.657785\pi\)
\(4\) −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 \(-0.765262\pi\)
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
\(10\) 4.43105 − 0.604787i 0.443105 − 0.0604787i
\(11\) − 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 −0.856530 − 0.516097i \(-0.827385\pi\)
−0.0186875 − 0.999825i \(-0.505949\pi\)
\(14\) 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 −0.962206 − 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
\(22\) 16.2221 − 12.5705i 0.737367 − 0.571388i
\(23\) −2.10884 + 1.21754i −0.0916886 + 0.0529365i −0.545143 − 0.838343i \(-0.683525\pi\)
0.453455 + 0.891279i \(0.350191\pi\)
\(24\) −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 0.974555 − 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 \(0.141602\pi\)
−0.902672 + 0.430328i \(0.858398\pi\)
\(32\) −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 −0.222935 − 0.974833i \(-0.571564\pi\)
0.955698 − 0.294349i \(-0.0951029\pi\)
\(42\) 3.12277 − 0.426222i 0.0743517 − 0.0101481i
\(43\) 64.6213 + 37.3091i 1.50282 + 0.867654i 0.999995 + 0.00326587i \(0.00103956\pi\)
0.502826 + 0.864388i \(0.332294\pi\)
\(44\) 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 −0.196291 − 0.980546i \(-0.562890\pi\)
0.751032 + 0.660266i \(0.229556\pi\)
\(48\) −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 0.839259 − 0.543732i \(-0.182989\pi\)
−0.890515 + 0.454954i \(0.849656\pi\)
\(54\) 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 0.879522 − 0.475858i \(-0.157862\pi\)
0.0276564 + 0.999617i \(0.491196\pi\)
\(60\) 0.401249 + 1.44252i 0.00668749 + 0.0240420i
\(61\) −55.3656 − 95.8960i −0.907632 − 1.57206i −0.817345 − 0.576149i \(-0.804555\pi\)
−0.0902874 − 0.995916i \(-0.528779\pi\)
\(62\) −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 −0.152684 − 0.988275i \(-0.548792\pi\)
0.779529 + 0.626366i \(0.215458\pi\)
\(68\) −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 0.335136 − 0.942170i \(-0.391218\pi\)
−0.648375 + 0.761321i \(0.724551\pi\)
\(72\) 57.5605 + 42.8784i 0.799451 + 0.595534i
\(73\) −30.1843 + 52.2808i −0.413484 + 0.716175i −0.995268 − 0.0971684i \(-0.969021\pi\)
0.581784 + 0.813343i \(0.302355\pi\)
\(74\) −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 0.873989 − 0.485946i \(-0.161525\pi\)
0.0161526 + 0.999870i \(0.494858\pi\)
\(80\) −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 \(-0.973466\pi\)
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
\(86\) 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 0.973003 − 0.230791i \(-0.0741314\pi\)
−0.686373 + 0.727250i \(0.740798\pi\)
\(90\) −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 −0.190952 − 0.981599i \(-0.561158\pi\)
0.945566 − 0.325430i \(-0.105509\pi\)
\(98\) −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.11.57 yes 160
4.3 odd 2 inner 380.3.q.a.11.5 ✓ 160
19.7 even 3 inner 380.3.q.a.311.5 yes 160
76.7 odd 6 inner 380.3.q.a.311.57 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.q.a.11.5 ✓ 160 4.3 odd 2 inner
380.3.q.a.11.57 yes 160 1.1 even 1 trivial
380.3.q.a.311.5 yes 160 19.7 even 3 inner
380.3.q.a.311.57 yes 160 76.7 odd 6 inner