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Properties

Label 380.3.q.a.11.9
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 11.9
Character \(\chi\) \(=\) 380.11
Dual form 380.3.q.a.311.9

$q$-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+(-1.89765 + 0.631592i) q^{2} +(-2.04261 - 1.17930i) q^{3} +(3.20218 - 2.39709i) q^{4} +(1.11803 - 1.93649i) q^{5} +(4.62101 + 0.947812i) q^{6} -9.35011i q^{7} +(-4.56266 + 6.57131i) q^{8} +(-1.71849 - 2.97651i) q^{9} +(-0.898570 + 4.38093i) q^{10} -13.3271i q^{11} +(-9.36771 + 1.11997i) q^{12} +(1.47875 + 2.56128i) q^{13} +(5.90545 + 17.7433i) q^{14} +(-4.56742 + 2.63700i) q^{15} +(4.50796 - 15.3518i) q^{16} +(-6.25822 + 10.8396i) q^{17} +(5.14104 + 4.56300i) q^{18} +(17.6266 - 7.09233i) q^{19} +(-1.06179 - 8.88103i) q^{20} +(-11.0266 + 19.0986i) q^{21} +(8.41727 + 25.2902i) q^{22} +(3.43981 - 1.98597i) q^{23} +(17.0693 - 8.04189i) q^{24} +(-2.50000 - 4.33013i) q^{25} +(-4.42385 - 3.92645i) q^{26} +29.3339i q^{27} +(-22.4130 - 29.9408i) q^{28} +(6.41804 + 11.1164i) q^{29} +(7.00188 - 7.88886i) q^{30} -17.2216i q^{31} +(1.14153 + 31.9796i) q^{32} +(-15.7167 + 27.2220i) q^{33} +(5.02977 - 24.5224i) q^{34} +(-18.1064 - 10.4537i) q^{35} +(-12.6379 - 5.41196i) q^{36} -27.5585 q^{37} +(-28.9698 + 24.5916i) q^{38} -6.97560i q^{39} +(7.62409 + 16.1825i) q^{40} +(-3.04784 + 5.27901i) q^{41} +(8.86214 - 43.2069i) q^{42} +(-30.3090 - 17.4989i) q^{43} +(-31.9461 - 42.6757i) q^{44} -7.68531 q^{45} +(-5.27324 + 5.94124i) q^{46} +(-45.8303 + 26.4602i) q^{47} +(-27.3125 + 26.0416i) q^{48} -38.4245 q^{49} +(7.47901 + 6.63810i) q^{50} +(25.5663 - 14.7607i) q^{51} +(10.8748 + 4.65698i) q^{52} +(-19.9994 - 34.6399i) q^{53} +(-18.5271 - 55.6657i) q^{54} +(-25.8078 - 14.9001i) q^{55} +(61.4425 + 42.6613i) q^{56} +(-44.3684 - 6.30027i) q^{57} +(-19.2002 - 17.0414i) q^{58} +(-72.5161 - 41.8672i) q^{59} +(-8.30460 + 19.3927i) q^{60} +(15.2720 + 26.4519i) q^{61} +(10.8770 + 32.6806i) q^{62} +(-27.8307 + 16.0680i) q^{63} +(-22.3643 - 59.9653i) q^{64} +6.61319 q^{65} +(12.6316 - 61.5845i) q^{66} +(-13.0349 + 7.52571i) q^{67} +(5.94338 + 49.7118i) q^{68} -9.36826 q^{69} +(40.9622 + 8.40172i) q^{70} +(10.6743 + 6.16281i) q^{71} +(27.4004 + 2.28807i) q^{72} +(61.4766 - 106.481i) q^{73} +(52.2966 - 17.4057i) q^{74} +11.7930i q^{75} +(39.4428 - 64.9636i) q^{76} -124.610 q^{77} +(4.40573 + 13.2373i) q^{78} +(-17.2207 - 9.94240i) q^{79} +(-24.6886 - 25.8935i) q^{80} +(19.1272 - 33.1293i) q^{81} +(2.44956 - 11.9427i) q^{82} +103.014i q^{83} +(10.4719 + 87.5891i) q^{84} +(13.9938 + 24.2380i) q^{85} +(68.5681 + 14.0639i) q^{86} -30.2752i q^{87} +(87.5764 + 60.8069i) q^{88} +(71.6196 + 124.049i) q^{89} +(14.5841 - 4.85398i) q^{90} +(23.9482 - 13.8265i) q^{91} +(6.25434 - 14.6050i) q^{92} +(-20.3094 + 35.1770i) q^{93} +(70.2581 - 79.1583i) q^{94} +(5.97295 - 42.0633i) q^{95} +(35.3820 - 66.6682i) q^{96} +(-83.9410 + 145.390i) q^{97} +(72.9164 - 24.2686i) q^{98} +(-39.6681 + 22.9024i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.89765 + 0.631592i −0.948827 + 0.315796i
\(3\) −2.04261 − 1.17930i −0.680871 − 0.393101i 0.119312 − 0.992857i \(-0.461931\pi\)
−0.800183 + 0.599756i \(0.795264\pi\)
\(4\) 3.20218 − 2.39709i 0.800546 − 0.599271i
\(5\) 1.11803 − 1.93649i 0.223607 − 0.387298i
\(6\) 4.62101 + 0.947812i 0.770169 + 0.157969i
\(7\) − 9.35011i − 1.33573i −0.744283 − 0.667865i \(-0.767208\pi\)
0.744283 − 0.667865i \(-0.232792\pi\)
\(8\) −4.56266 + 6.57131i −0.570332 + 0.821414i
\(9\) −1.71849 − 2.97651i −0.190943 − 0.330723i
\(10\) −0.898570 + 4.38093i −0.0898570 + 0.438093i
\(11\) − 13.3271i − 1.21155i −0.795635 − 0.605776i \(-0.792863\pi\)
0.795635 − 0.605776i \(-0.207137\pi\)
\(12\) −9.36771 + 1.11997i −0.780643 + 0.0933311i
\(13\) 1.47875 + 2.56128i 0.113750 + 0.197021i 0.917280 − 0.398244i \(-0.130380\pi\)
−0.803529 + 0.595265i \(0.797047\pi\)
\(14\) 5.90545 + 17.7433i 0.421818 + 1.26738i
\(15\) −4.56742 + 2.63700i −0.304495 + 0.175800i
\(16\) 4.50796 − 15.3518i 0.281747 − 0.959489i
\(17\) −6.25822 + 10.8396i −0.368131 + 0.637621i −0.989273 − 0.146076i \(-0.953335\pi\)
0.621143 + 0.783698i \(0.286669\pi\)
\(18\) 5.14104 + 4.56300i 0.285613 + 0.253500i
\(19\) 17.6266 − 7.09233i 0.927718 − 0.373281i
\(20\) −1.06179 − 8.88103i −0.0530893 − 0.444051i
\(21\) −11.0266 + 19.0986i −0.525077 + 0.909459i
\(22\) 8.41727 + 25.2902i 0.382603 + 1.14955i
\(23\) 3.43981 − 1.98597i 0.149557 − 0.0863466i −0.423354 − 0.905964i \(-0.639147\pi\)
0.572911 + 0.819618i \(0.305814\pi\)
\(24\) 17.0693 − 8.04189i 0.711221 − 0.335079i
\(25\) −2.50000 − 4.33013i −0.100000 − 0.173205i
\(26\) −4.42385 − 3.92645i −0.170148 − 0.151017i
\(27\) 29.3339i 1.08644i
\(28\) −22.4130 − 29.9408i −0.800464 − 1.06931i
\(29\) 6.41804 + 11.1164i 0.221312 + 0.383323i 0.955207 − 0.295940i \(-0.0956329\pi\)
−0.733895 + 0.679263i \(0.762300\pi\)
\(30\) 7.00188 − 7.88886i 0.233396 − 0.262962i
\(31\) − 17.2216i − 0.555534i −0.960648 − 0.277767i \(-0.910406\pi\)
0.960648 − 0.277767i \(-0.0895944\pi\)
\(32\) 1.14153 + 31.9796i 0.0356729 + 0.999364i
\(33\) −15.7167 + 27.2220i −0.476262 + 0.824911i
\(34\) 5.02977 − 24.5224i 0.147934 − 0.721247i
\(35\) −18.1064 − 10.4537i −0.517326 − 0.298678i
\(36\) −12.6379 − 5.41196i −0.351052 − 0.150332i
\(37\) −27.5585 −0.744825 −0.372413 − 0.928067i \(-0.621469\pi\)
−0.372413 + 0.928067i \(0.621469\pi\)
\(38\) −28.9698 + 24.5916i −0.762364 + 0.647149i
\(39\) − 6.97560i − 0.178861i
\(40\) 7.62409 + 16.1825i 0.190602 + 0.404562i
\(41\) −3.04784 + 5.27901i −0.0743375 + 0.128756i −0.900798 − 0.434238i \(-0.857018\pi\)
0.826460 + 0.562995i \(0.190351\pi\)
\(42\) 8.86214 − 43.2069i 0.211003 − 1.02874i
\(43\) −30.3090 − 17.4989i −0.704860 − 0.406951i 0.104295 − 0.994546i \(-0.466741\pi\)
−0.809155 + 0.587595i \(0.800075\pi\)
\(44\) −31.9461 − 42.6757i −0.726048 − 0.969903i
\(45\) −7.68531 −0.170785
\(46\) −5.27324 + 5.94124i −0.114636 + 0.129157i
\(47\) −45.8303 + 26.4602i −0.975114 + 0.562982i −0.900791 − 0.434252i \(-0.857013\pi\)
−0.0743222 + 0.997234i \(0.523679\pi\)
\(48\) −27.3125 + 26.0416i −0.569010 + 0.542533i
\(49\) −38.4245 −0.784173
\(50\) 7.47901 + 6.63810i 0.149580 + 0.132762i
\(51\) 25.5663 − 14.7607i 0.501299 − 0.289425i
\(52\) 10.8748 + 4.65698i 0.209132 + 0.0895573i
\(53\) −19.9994 − 34.6399i −0.377347 − 0.653583i 0.613329 − 0.789828i \(-0.289830\pi\)
−0.990675 + 0.136244i \(0.956497\pi\)
\(54\) −18.5271 − 55.6657i −0.343094 − 1.03085i
\(55\) −25.8078 − 14.9001i −0.469232 − 0.270911i
\(56\) 61.4425 + 42.6613i 1.09719 + 0.761809i
\(57\) −44.3684 − 6.30027i −0.778394 − 0.110531i
\(58\) −19.2002 − 17.0414i −0.331038 − 0.293818i
\(59\) −72.5161 − 41.8672i −1.22909 − 0.709613i −0.262247 − 0.965001i \(-0.584464\pi\)
−0.966839 + 0.255388i \(0.917797\pi\)
\(60\) −8.30460 + 19.3927i −0.138410 + 0.323211i
\(61\) 15.2720 + 26.4519i 0.250361 + 0.433637i 0.963625 − 0.267258i \(-0.0861175\pi\)
−0.713264 + 0.700895i \(0.752784\pi\)
\(62\) 10.8770 + 32.6806i 0.175435 + 0.527106i
\(63\) −27.8307 + 16.0680i −0.441757 + 0.255048i
\(64\) −22.3643 − 59.9653i −0.349442 − 0.936958i
\(65\) 6.61319 0.101741
\(66\) 12.6316 − 61.5845i 0.191387 − 0.933099i
\(67\) −13.0349 + 7.52571i −0.194551 + 0.112324i −0.594111 − 0.804383i \(-0.702496\pi\)
0.399560 + 0.916707i \(0.369163\pi\)
\(68\) 5.94338 + 49.7118i 0.0874026 + 0.731055i
\(69\) −9.36826 −0.135772
\(70\) 40.9622 + 8.40172i 0.585174 + 0.120025i
\(71\) 10.6743 + 6.16281i 0.150342 + 0.0868001i 0.573284 − 0.819357i \(-0.305669\pi\)
−0.422942 + 0.906157i \(0.639003\pi\)
\(72\) 27.4004 + 2.28807i 0.380562 + 0.0317787i
\(73\) 61.4766 − 106.481i 0.842145 − 1.45864i −0.0459322 − 0.998945i \(-0.514626\pi\)
0.888077 − 0.459694i \(-0.152041\pi\)
\(74\) 52.2966 − 17.4057i 0.706710 − 0.235213i
\(75\) 11.7930i 0.157240i
\(76\) 39.4428 − 64.9636i 0.518985 − 0.854784i
\(77\) −124.610 −1.61831
\(78\) 4.40573 + 13.2373i 0.0564837 + 0.169709i
\(79\) −17.2207 − 9.94240i −0.217984 − 0.125853i 0.387032 − 0.922066i \(-0.373500\pi\)
−0.605017 + 0.796213i \(0.706833\pi\)
\(80\) −24.6886 − 25.8935i −0.308608 − 0.323668i
\(81\) 19.1272 − 33.1293i 0.236138 − 0.409004i
\(82\) 2.44956 − 11.9427i 0.0298727 − 0.145643i
\(83\) 103.014i 1.24113i 0.784154 + 0.620566i \(0.213097\pi\)
−0.784154 + 0.620566i \(0.786903\pi\)
\(84\) 10.4719 + 87.5891i 0.124665 + 1.04273i
\(85\) 13.9938 + 24.2380i 0.164633 + 0.285153i
\(86\) 68.5681 + 14.0639i 0.797303 + 0.163534i
\(87\) − 30.2752i − 0.347991i
\(88\) 87.5764 + 60.8069i 0.995186 + 0.690987i
\(89\) 71.6196 + 124.049i 0.804714 + 1.39381i 0.916484 + 0.400072i \(0.131015\pi\)
−0.111769 + 0.993734i \(0.535652\pi\)
\(90\) 14.5841 − 4.85398i 0.162045 − 0.0539331i
\(91\) 23.9482 − 13.8265i 0.263167 − 0.151940i
\(92\) 6.25434 − 14.6050i 0.0679820 − 0.158750i
\(93\) −20.3094 + 35.1770i −0.218381 + 0.378247i
\(94\) 70.2581 − 79.1583i 0.747427 − 0.842109i
\(95\) 5.97295 − 42.0633i 0.0628731 − 0.442772i
\(96\) 35.3820 − 66.6682i 0.368562 − 0.694461i
\(97\) −83.9410 + 145.390i −0.865371 + 1.49887i 0.00130642 + 0.999999i \(0.499584\pi\)
−0.866678 + 0.498868i \(0.833749\pi\)
\(98\) 72.9164 − 24.2686i 0.744044 − 0.247639i
\(99\) −39.6681 + 22.9024i −0.400688 + 0.231338i
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.11.9 ✓ 160
4.3 odd 2 inner 380.3.q.a.11.46 yes 160
19.7 even 3 inner 380.3.q.a.311.46 yes 160
76.7 odd 6 inner 380.3.q.a.311.9 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.q.a.11.9 ✓ 160 1.1 even 1 trivial
380.3.q.a.11.46 yes 160 4.3 odd 2 inner
380.3.q.a.311.9 yes 160 76.7 odd 6 inner
380.3.q.a.311.46 yes 160 19.7 even 3 inner