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Properties

Label 380.3.q.a.11.8
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 11.8
Character \(\chi\) \(=\) 380.11
Dual form 380.3.q.a.311.8

$q$-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+(-1.92602 - 0.538947i) q^{2} +(-4.13473 - 2.38719i) q^{3} +(3.41907 + 2.07604i) q^{4} +(1.11803 - 1.93649i) q^{5} +(6.67698 + 6.82616i) q^{6} -2.68655i q^{7} +(-5.46631 - 5.84119i) q^{8} +(6.89731 + 11.9465i) q^{9} +(-3.19702 + 3.12715i) q^{10} +4.69031i q^{11} +(-9.18103 - 16.7458i) q^{12} +(-0.228760 - 0.396224i) q^{13} +(-1.44791 + 5.17433i) q^{14} +(-9.24553 + 5.33791i) q^{15} +(7.38010 + 14.1963i) q^{16} +(-6.39447 + 11.0756i) q^{17} +(-6.84579 - 26.7264i) q^{18} +(-6.56692 + 17.8291i) q^{19} +(7.84288 - 4.29992i) q^{20} +(-6.41329 + 11.1081i) q^{21} +(2.52783 - 9.03361i) q^{22} +(-14.4549 + 8.34553i) q^{23} +(8.65768 + 37.2008i) q^{24} +(-2.50000 - 4.33013i) q^{25} +(0.227052 + 0.886424i) q^{26} -22.8913i q^{27} +(5.57739 - 9.18550i) q^{28} +(7.50504 + 12.9991i) q^{29} +(20.6839 - 5.29804i) q^{30} -12.5190i q^{31} +(-6.56313 - 31.3197i) q^{32} +(11.1966 - 19.3932i) q^{33} +(18.2850 - 17.8854i) q^{34} +(-5.20248 - 3.00365i) q^{35} +(-1.21903 + 55.1650i) q^{36} +39.3914 q^{37} +(22.2569 - 30.7998i) q^{38} +2.18437i q^{39} +(-17.4229 + 4.05481i) q^{40} +(18.7545 - 32.4838i) q^{41} +(18.3388 - 17.9380i) q^{42} +(64.2597 + 37.1004i) q^{43} +(-9.73729 + 16.0365i) q^{44} +30.8457 q^{45} +(32.3381 - 8.28320i) q^{46} +(-9.62255 + 5.55558i) q^{47} +(3.37446 - 76.3154i) q^{48} +41.7825 q^{49} +(2.48133 + 9.68726i) q^{50} +(52.8788 - 30.5296i) q^{51} +(0.0404310 - 1.82963i) q^{52} +(-27.7604 - 48.0824i) q^{53} +(-12.3372 + 44.0889i) q^{54} +(9.08275 + 5.24393i) q^{55} +(-15.6926 + 14.6855i) q^{56} +(69.7137 - 58.0418i) q^{57} +(-7.44898 - 29.0813i) q^{58} +(40.2939 + 23.2637i) q^{59} +(-42.6928 - 0.943421i) q^{60} +(10.4548 + 18.1083i) q^{61} +(-6.74709 + 24.1118i) q^{62} +(32.0948 - 18.5300i) q^{63} +(-4.23899 + 63.8595i) q^{64} -1.02305 q^{65} +(-32.0168 + 31.3171i) q^{66} +(80.3487 - 46.3893i) q^{67} +(-44.8565 + 24.5929i) q^{68} +79.6893 q^{69} +(8.40124 + 8.58895i) q^{70} +(-59.5305 - 34.3700i) q^{71} +(32.0789 - 105.592i) q^{72} +(-18.6835 + 32.3608i) q^{73} +(-75.8685 - 21.2299i) q^{74} +23.8719i q^{75} +(-59.4667 + 47.3256i) q^{76} +12.6008 q^{77} +(1.17726 - 4.20713i) q^{78} +(-39.1021 - 22.5756i) q^{79} +(35.7422 + 1.58042i) q^{80} +(7.43008 - 12.8693i) q^{81} +(-53.6286 + 52.4566i) q^{82} +96.9916i q^{83} +(-44.9885 + 24.6653i) q^{84} +(14.2985 + 24.7657i) q^{85} +(-103.770 - 106.088i) q^{86} -71.6637i q^{87} +(27.3970 - 25.6387i) q^{88} +(-16.2918 - 28.2183i) q^{89} +(-59.4093 - 16.6242i) q^{90} +(-1.06448 + 0.614575i) q^{91} +(-66.7479 - 1.47499i) q^{92} +(-29.8852 + 51.7627i) q^{93} +(21.5273 - 5.51409i) q^{94} +(27.1838 + 32.6503i) q^{95} +(-47.6292 + 145.166i) q^{96} +(-38.7396 + 67.0990i) q^{97} +(-80.4737 - 22.5185i) q^{98} +(-56.0327 + 32.3505i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.92602 − 0.538947i −0.963008 − 0.269474i
\(3\) −4.13473 − 2.38719i −1.37824 − 0.795728i −0.386294 − 0.922376i \(-0.626245\pi\)
−0.991948 + 0.126647i \(0.959578\pi\)
\(4\) 3.41907 + 2.07604i 0.854768 + 0.519011i
\(5\) 1.11803 − 1.93649i 0.223607 − 0.387298i
\(6\) 6.67698 + 6.82616i 1.11283 + 1.13769i
\(7\) − 2.68655i − 0.383793i −0.981415 − 0.191896i \(-0.938536\pi\)
0.981415 − 0.191896i \(-0.0614637\pi\)
\(8\) −5.46631 − 5.84119i −0.683288 − 0.730149i
\(9\) 6.89731 + 11.9465i 0.766367 + 1.32739i
\(10\) −3.19702 + 3.12715i −0.319702 + 0.312715i
\(11\) 4.69031i 0.426392i 0.977009 + 0.213196i \(0.0683873\pi\)
−0.977009 + 0.213196i \(0.931613\pi\)
\(12\) −9.18103 − 16.7458i −0.765085 − 1.39549i
\(13\) −0.228760 − 0.396224i −0.0175969 − 0.0304788i 0.857093 − 0.515162i \(-0.172268\pi\)
−0.874690 + 0.484683i \(0.838935\pi\)
\(14\) −1.44791 + 5.17433i −0.103422 + 0.369595i
\(15\) −9.24553 + 5.33791i −0.616369 + 0.355861i
\(16\) 7.38010 + 14.1963i 0.461256 + 0.887267i
\(17\) −6.39447 + 11.0756i −0.376146 + 0.651503i −0.990498 − 0.137529i \(-0.956084\pi\)
0.614352 + 0.789032i \(0.289417\pi\)
\(18\) −6.84579 − 26.7264i −0.380322 − 1.48480i
\(19\) −6.56692 + 17.8291i −0.345627 + 0.938372i
\(20\) 7.84288 − 4.29992i 0.392144 − 0.214996i
\(21\) −6.41329 + 11.1081i −0.305395 + 0.528959i
\(22\) 2.52783 − 9.03361i 0.114901 − 0.410619i
\(23\) −14.4549 + 8.34553i −0.628473 + 0.362849i −0.780161 − 0.625579i \(-0.784863\pi\)
0.151687 + 0.988429i \(0.451529\pi\)
\(24\) 8.65768 + 37.2008i 0.360737 + 1.55003i
\(25\) −2.50000 − 4.33013i −0.100000 − 0.173205i
\(26\) 0.227052 + 0.886424i 0.00873275 + 0.0340932i
\(27\) − 22.8913i − 0.847825i
\(28\) 5.57739 − 9.18550i 0.199192 − 0.328054i
\(29\) 7.50504 + 12.9991i 0.258794 + 0.448245i 0.965919 − 0.258844i \(-0.0833414\pi\)
−0.707125 + 0.707089i \(0.750008\pi\)
\(30\) 20.6839 − 5.29804i 0.689463 − 0.176601i
\(31\) − 12.5190i − 0.403839i −0.979402 − 0.201920i \(-0.935282\pi\)
0.979402 − 0.201920i \(-0.0647179\pi\)
\(32\) −6.56313 − 31.3197i −0.205098 − 0.978741i
\(33\) 11.1966 − 19.3932i 0.339292 − 0.587671i
\(34\) 18.2850 − 17.8854i 0.537794 − 0.526041i
\(35\) −5.20248 − 3.00365i −0.148642 − 0.0858187i
\(36\) −1.21903 + 55.1650i −0.0338619 + 1.53236i
\(37\) 39.3914 1.06463 0.532317 − 0.846545i \(-0.321322\pi\)
0.532317 + 0.846545i \(0.321322\pi\)
\(38\) 22.2569 − 30.7998i 0.585708 − 0.810522i
\(39\) 2.18437i 0.0560095i
\(40\) −17.4229 + 4.05481i −0.435573 + 0.101370i
\(41\) 18.7545 − 32.4838i 0.457427 − 0.792288i −0.541397 − 0.840767i \(-0.682104\pi\)
0.998824 + 0.0484796i \(0.0154376\pi\)
\(42\) 18.3388 − 17.9380i 0.436638 − 0.427096i
\(43\) 64.2597 + 37.1004i 1.49441 + 0.862799i 0.999979 − 0.00641663i \(-0.00204249\pi\)
0.494433 + 0.869216i \(0.335376\pi\)
\(44\) −9.73729 + 16.0365i −0.221302 + 0.364466i
\(45\) 30.8457 0.685460
\(46\) 32.3381 − 8.28320i 0.703003 − 0.180070i
\(47\) −9.62255 + 5.55558i −0.204735 + 0.118204i −0.598862 − 0.800852i \(-0.704380\pi\)
0.394127 + 0.919056i \(0.371047\pi\)
\(48\) 3.37446 − 76.3154i 0.0703013 − 1.58990i
\(49\) 41.7825 0.852703
\(50\) 2.48133 + 9.68726i 0.0496266 + 0.193745i
\(51\) 52.8788 − 30.5296i 1.03684 − 0.598619i
\(52\) 0.0404310 − 1.82963i 0.000777519 − 0.0351853i
\(53\) −27.7604 − 48.0824i −0.523780 − 0.907214i −0.999617 − 0.0276804i \(-0.991188\pi\)
0.475836 − 0.879534i \(-0.342145\pi\)
\(54\) −12.3372 + 44.0889i −0.228466 + 0.816462i
\(55\) 9.08275 + 5.24393i 0.165141 + 0.0953442i
\(56\) −15.6926 + 14.6855i −0.280226 + 0.262241i
\(57\) 69.7137 − 58.0418i 1.22305 − 1.01828i
\(58\) −7.44898 − 29.0813i −0.128431 − 0.501402i
\(59\) 40.2939 + 23.2637i 0.682947 + 0.394300i 0.800964 − 0.598712i \(-0.204321\pi\)
−0.118017 + 0.993012i \(0.537654\pi\)
\(60\) −42.6928 − 0.943421i −0.711547 − 0.0157237i
\(61\) 10.4548 + 18.1083i 0.171391 + 0.296858i 0.938906 − 0.344173i \(-0.111841\pi\)
−0.767515 + 0.641030i \(0.778507\pi\)
\(62\) −6.74709 + 24.1118i −0.108824 + 0.388900i
\(63\) 32.0948 − 18.5300i 0.509442 − 0.294126i
\(64\) −4.23899 + 63.8595i −0.0662343 + 0.997804i
\(65\) −1.02305 −0.0157392
\(66\) −32.0168 + 31.3171i −0.485103 + 0.474502i
\(67\) 80.3487 − 46.3893i 1.19923 − 0.692378i 0.238849 − 0.971057i \(-0.423230\pi\)
0.960384 + 0.278679i \(0.0898964\pi\)
\(68\) −44.8565 + 24.5929i −0.659654 + 0.361660i
\(69\) 79.6893 1.15492
\(70\) 8.40124 + 8.58895i 0.120018 + 0.122699i
\(71\) −59.5305 − 34.3700i −0.838458 − 0.484084i 0.0182818 − 0.999833i \(-0.494180\pi\)
−0.856740 + 0.515749i \(0.827514\pi\)
\(72\) 32.0789 − 105.592i 0.445540 − 1.46655i
\(73\) −18.6835 + 32.3608i −0.255938 + 0.443298i −0.965150 − 0.261698i \(-0.915718\pi\)
0.709212 + 0.704996i \(0.249051\pi\)
\(74\) −75.8685 − 21.2299i −1.02525 − 0.286891i
\(75\) 23.8719i 0.318291i
\(76\) −59.4667 + 47.3256i −0.782456 + 0.622706i
\(77\) 12.6008 0.163646
\(78\) 1.17726 − 4.20713i 0.0150931 − 0.0539376i
\(79\) −39.1021 − 22.5756i −0.494964 − 0.285768i 0.231668 − 0.972795i \(-0.425582\pi\)
−0.726631 + 0.687027i \(0.758915\pi\)
\(80\) 35.7422 + 1.58042i 0.446777 + 0.0197553i
\(81\) 7.43008 − 12.8693i 0.0917293 − 0.158880i
\(82\) −53.6286 + 52.4566i −0.654007 + 0.639714i
\(83\) 96.9916i 1.16857i 0.811547 + 0.584287i \(0.198626\pi\)
−0.811547 + 0.584287i \(0.801374\pi\)
\(84\) −44.9885 + 24.6653i −0.535577 + 0.293634i
\(85\) 14.2985 + 24.7657i 0.168217 + 0.291361i
\(86\) −103.770 − 106.088i −1.20663 − 1.23359i
\(87\) − 71.6637i − 0.823720i
\(88\) 27.3970 − 25.6387i 0.311330 − 0.291349i
\(89\) −16.2918 − 28.2183i −0.183054 − 0.317060i 0.759865 − 0.650081i \(-0.225265\pi\)
−0.942919 + 0.333022i \(0.891932\pi\)
\(90\) −59.4093 − 16.6242i −0.660103 − 0.184713i
\(91\) −1.06448 + 0.614575i −0.0116975 + 0.00675357i
\(92\) −66.7479 − 1.47499i −0.725521 − 0.0160325i
\(93\) −29.8852 + 51.7627i −0.321346 + 0.556588i
\(94\) 21.5273 − 5.51409i 0.229014 − 0.0586605i
\(95\) 27.1838 + 32.6503i 0.286145 + 0.343687i
\(96\) −47.6292 + 145.166i −0.496138 + 1.51214i
\(97\) −38.7396 + 67.0990i −0.399378 + 0.691742i −0.993649 − 0.112522i \(-0.964107\pi\)
0.594272 + 0.804264i \(0.297440\pi\)
\(98\) −80.4737 − 22.5185i −0.821160 − 0.229781i
\(99\) −56.0327 + 32.3505i −0.565987 + 0.326773i
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.11.8 ✓ 160
4.3 odd 2 inner 380.3.q.a.11.61 yes 160
19.7 even 3 inner 380.3.q.a.311.61 yes 160
76.7 odd 6 inner 380.3.q.a.311.8 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.q.a.11.8 ✓ 160 1.1 even 1 trivial
380.3.q.a.11.61 yes 160 4.3 odd 2 inner
380.3.q.a.311.8 yes 160 76.7 odd 6 inner
380.3.q.a.311.61 yes 160 19.7 even 3 inner