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Properties

Label 380.3.q.a.11.11
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 11.11
Character \(\chi\) \(=\) 380.11
Dual form 380.3.q.a.311.11

$q$-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+(-1.86315 + 0.727102i) q^{2} +(4.59307 + 2.65181i) q^{3} +(2.94265 - 2.70940i) q^{4} +(-1.11803 + 1.93649i) q^{5} +(-10.4857 - 1.60109i) q^{6} +10.5847i q^{7} +(-3.51258 + 7.18761i) q^{8} +(9.56418 + 16.5656i) q^{9} +(0.675037 - 4.42090i) q^{10} +2.97000i q^{11} +(20.7006 - 4.64112i) q^{12} +(-6.49943 - 11.2573i) q^{13} +(-7.69616 - 19.7209i) q^{14} +(-10.2704 + 5.92962i) q^{15} +(1.31832 - 15.9456i) q^{16} +(-7.98860 + 13.8367i) q^{17} +(-29.8644 - 23.9101i) q^{18} +(-16.6781 - 9.10175i) q^{19} +(1.95675 + 8.72761i) q^{20} +(-28.0686 + 48.6163i) q^{21} +(-2.15949 - 5.53355i) q^{22} +(29.7422 - 17.1717i) q^{23} +(-35.1937 + 23.6985i) q^{24} +(-2.50000 - 4.33013i) q^{25} +(20.2946 + 16.2484i) q^{26} +53.7169i q^{27} +(28.6782 + 31.1470i) q^{28} +(-3.01300 - 5.21868i) q^{29} +(14.8239 - 18.5154i) q^{30} -13.9012i q^{31} +(9.13784 + 30.6676i) q^{32} +(-7.87586 + 13.6414i) q^{33} +(4.82328 - 31.5883i) q^{34} +(-20.4972 - 11.8341i) q^{35} +(73.0269 + 22.8336i) q^{36} +37.0840 q^{37} +(37.6916 + 4.83125i) q^{38} -68.9410i q^{39} +(-9.99158 - 14.8381i) q^{40} +(-22.8717 + 39.6150i) q^{41} +(16.9470 - 110.988i) q^{42} +(0.00326943 + 0.00188761i) q^{43} +(8.04690 + 8.73965i) q^{44} -42.7723 q^{45} +(-42.9286 + 53.6190i) q^{46} +(59.0534 - 34.0945i) q^{47} +(48.3398 - 69.7433i) q^{48} -63.0360 q^{49} +(7.80632 + 6.24992i) q^{50} +(-73.3843 + 42.3685i) q^{51} +(-49.6261 - 15.5168i) q^{52} +(40.6943 + 70.4846i) q^{53} +(-39.0577 - 100.083i) q^{54} +(-5.75137 - 3.32056i) q^{55} +(-76.0788 - 37.1796i) q^{56} +(-52.4674 - 86.0320i) q^{57} +(9.40818 + 7.53241i) q^{58} +(-13.2228 - 7.63421i) q^{59} +(-14.1565 + 45.2754i) q^{60} +(38.2744 + 66.2932i) q^{61} +(10.1076 + 25.9001i) q^{62} +(-175.342 + 101.234i) q^{63} +(-39.3236 - 50.4941i) q^{64} +29.0663 q^{65} +(4.75522 - 31.1425i) q^{66} +(-102.669 + 59.2761i) q^{67} +(13.9814 + 62.3606i) q^{68} +182.144 q^{69} +(46.7939 + 7.14506i) q^{70} +(44.3202 + 25.5883i) q^{71} +(-152.662 + 10.5555i) q^{72} +(46.0213 - 79.7113i) q^{73} +(-69.0930 + 26.9638i) q^{74} -26.5181i q^{75} +(-73.7379 + 18.4043i) q^{76} -31.4365 q^{77} +(50.1271 + 128.447i) q^{78} +(-79.1663 - 45.7067i) q^{79} +(29.4046 + 20.3806i) q^{80} +(-56.3694 + 97.6347i) q^{81} +(13.8093 - 90.4387i) q^{82} +105.899i q^{83} +(49.1248 + 219.110i) q^{84} +(-17.8630 - 30.9397i) q^{85} +(-0.00746393 - 0.00113968i) q^{86} -31.9596i q^{87} +(-21.3472 - 10.4323i) q^{88} +(38.9887 + 67.5303i) q^{89} +(79.6912 - 31.0998i) q^{90} +(119.156 - 68.7946i) q^{91} +(40.9959 - 131.114i) q^{92} +(36.8634 - 63.8493i) q^{93} +(-85.2351 + 106.461i) q^{94} +(36.2721 - 22.1209i) q^{95} +(-39.3538 + 165.090i) q^{96} +(-2.55639 + 4.42780i) q^{97} +(117.445 - 45.8336i) q^{98} +(-49.1999 + 28.4056i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.86315 + 0.727102i −0.931574 + 0.363551i
\(3\) 4.59307 + 2.65181i 1.53102 + 0.883936i 0.999315 + 0.0370082i \(0.0117828\pi\)
0.531708 + 0.846928i \(0.321551\pi\)
\(4\) 2.94265 − 2.70940i 0.735661 − 0.677350i
\(5\) −1.11803 + 1.93649i −0.223607 + 0.387298i
\(6\) −10.4857 − 1.60109i −1.74762 − 0.266848i
\(7\) 10.5847i 1.51210i 0.654513 + 0.756051i \(0.272874\pi\)
−0.654513 + 0.756051i \(0.727126\pi\)
\(8\) −3.51258 + 7.18761i −0.439072 + 0.898452i
\(9\) 9.56418 + 16.5656i 1.06269 + 1.84063i
\(10\) 0.675037 − 4.42090i 0.0675037 − 0.442090i
\(11\) 2.97000i 0.270000i 0.990846 + 0.135000i \(0.0431034\pi\)
−0.990846 + 0.135000i \(0.956897\pi\)
\(12\) 20.7006 − 4.64112i 1.72505 − 0.386760i
\(13\) −6.49943 − 11.2573i −0.499956 − 0.865950i 0.500044 − 0.866000i \(-0.333317\pi\)
−1.00000 5.05630e-5i \(0.999984\pi\)
\(14\) −7.69616 − 19.7209i −0.549726 − 1.40863i
\(15\) −10.2704 + 5.92962i −0.684694 + 0.395308i
\(16\) 1.31832 − 15.9456i 0.0823953 − 0.996600i
\(17\) −7.98860 + 13.8367i −0.469917 + 0.813921i −0.999408 − 0.0343948i \(-0.989050\pi\)
0.529491 + 0.848316i \(0.322383\pi\)
\(18\) −29.8644 − 23.9101i −1.65913 − 1.32834i
\(19\) −16.6781 − 9.10175i −0.877793 − 0.479039i
\(20\) 1.95675 + 8.72761i 0.0978375 + 0.436380i
\(21\) −28.0686 + 48.6163i −1.33660 + 2.31506i
\(22\) −2.15949 − 5.53355i −0.0981587 − 0.251525i
\(23\) 29.7422 − 17.1717i 1.29314 − 0.746594i 0.313930 − 0.949446i \(-0.398354\pi\)
0.979209 + 0.202852i \(0.0650211\pi\)
\(24\) −35.1937 + 23.6985i −1.46640 + 0.987438i
\(25\) −2.50000 − 4.33013i −0.100000 − 0.173205i
\(26\) 20.2946 + 16.2484i 0.780563 + 0.624937i
\(27\) 53.7169i 1.98952i
\(28\) 28.6782 + 31.1470i 1.02422 + 1.11239i
\(29\) −3.01300 − 5.21868i −0.103897 − 0.179954i 0.809390 − 0.587271i \(-0.199798\pi\)
−0.913287 + 0.407317i \(0.866464\pi\)
\(30\) 14.8239 − 18.5154i 0.494129 − 0.617180i
\(31\) − 13.9012i − 0.448427i −0.974540 − 0.224214i \(-0.928019\pi\)
0.974540 − 0.224214i \(-0.0719813\pi\)
\(32\) 9.13784 + 30.6676i 0.285558 + 0.958362i
\(33\) −7.87586 + 13.6414i −0.238663 + 0.413376i
\(34\) 4.82328 − 31.5883i 0.141861 − 0.929067i
\(35\) −20.4972 − 11.8341i −0.585634 − 0.338116i
\(36\) 73.0269 + 22.8336i 2.02853 + 0.634268i
\(37\) 37.0840 1.00227 0.501135 − 0.865369i \(-0.332916\pi\)
0.501135 + 0.865369i \(0.332916\pi\)
\(38\) 37.6916 + 4.83125i 0.991885 + 0.127138i
\(39\) − 68.9410i − 1.76772i
\(40\) −9.99158 − 14.8381i −0.249789 − 0.370952i
\(41\) −22.8717 + 39.6150i −0.557847 + 0.966220i 0.439829 + 0.898082i \(0.355039\pi\)
−0.997676 + 0.0681380i \(0.978294\pi\)
\(42\) 16.9470 − 110.988i 0.403500 − 2.64257i
\(43\) 0.00326943 + 0.00188761i 7.60334e−5 + 4.38979e-5i 0.500038 − 0.866003i \(-0.333319\pi\)
−0.499962 + 0.866047i \(0.666653\pi\)
\(44\) 8.04690 + 8.73965i 0.182884 + 0.198628i
\(45\) −42.7723 −0.950496
\(46\) −42.9286 + 53.6190i −0.933230 + 1.16563i
\(47\) 59.0534 − 34.0945i 1.25646 − 0.725415i 0.284072 − 0.958803i \(-0.408314\pi\)
0.972384 + 0.233388i \(0.0749811\pi\)
\(48\) 48.3398 − 69.7433i 1.00708 − 1.45298i
\(49\) −63.0360 −1.28645
\(50\) 7.80632 + 6.24992i 0.156126 + 0.124998i
\(51\) −73.3843 + 42.3685i −1.43891 + 0.830754i
\(52\) −49.6261 − 15.5168i −0.954349 − 0.298401i
\(53\) 40.6943 + 70.4846i 0.767817 + 1.32990i 0.938744 + 0.344614i \(0.111990\pi\)
−0.170927 + 0.985284i \(0.554676\pi\)
\(54\) −39.0577 − 100.083i −0.723291 − 1.85338i
\(55\) −5.75137 − 3.32056i −0.104570 − 0.0603738i
\(56\) −76.0788 − 37.1796i −1.35855 − 0.663921i
\(57\) −52.4674 − 86.0320i −0.920481 − 1.50933i
\(58\) 9.40818 + 7.53241i 0.162210 + 0.129869i
\(59\) −13.2228 − 7.63421i −0.224116 − 0.129393i 0.383739 − 0.923442i \(-0.374636\pi\)
−0.607855 + 0.794048i \(0.707970\pi\)
\(60\) −14.1565 + 45.2754i −0.235941 + 0.754590i
\(61\) 38.2744 + 66.2932i 0.627449 + 1.08677i 0.988062 + 0.154058i \(0.0492342\pi\)
−0.360613 + 0.932716i \(0.617432\pi\)
\(62\) 10.1076 + 25.9001i 0.163026 + 0.417743i
\(63\) −175.342 + 101.234i −2.78321 + 1.60689i
\(64\) −39.3236 − 50.4941i −0.614431 − 0.788970i
\(65\) 29.0663 0.447174
\(66\) 4.75522 − 31.1425i 0.0720488 − 0.471856i
\(67\) −102.669 + 59.2761i −1.53238 + 0.884717i −0.533124 + 0.846037i \(0.678982\pi\)
−0.999252 + 0.0386803i \(0.987685\pi\)
\(68\) 13.9814 + 62.3606i 0.205609 + 0.917068i
\(69\) 182.144 2.63977
\(70\) 46.7939 + 7.14506i 0.668484 + 0.102072i
\(71\) 44.3202 + 25.5883i 0.624228 + 0.360398i 0.778513 − 0.627628i \(-0.215974\pi\)
−0.154286 + 0.988026i \(0.549308\pi\)
\(72\) −152.662 + 10.5555i −2.12031 + 0.146605i
\(73\) 46.0213 − 79.7113i 0.630429 − 1.09194i −0.357035 − 0.934091i \(-0.616212\pi\)
0.987464 − 0.157844i \(-0.0504544\pi\)
\(74\) −69.0930 + 26.9638i −0.933689 + 0.364376i
\(75\) − 26.5181i − 0.353574i
\(76\) −73.7379 + 18.4043i −0.970236 + 0.242162i
\(77\) −31.4365 −0.408267
\(78\) 50.1271 + 128.447i 0.642656 + 1.64676i
\(79\) −79.1663 − 45.7067i −1.00211 − 0.578566i −0.0932348 − 0.995644i \(-0.529721\pi\)
−0.908871 + 0.417078i \(0.863054\pi\)
\(80\) 29.4046 + 20.3806i 0.367557 + 0.254758i
\(81\) −56.3694 + 97.6347i −0.695919 + 1.20537i
\(82\) 13.8093 − 90.4387i 0.168406 − 1.10291i
\(83\) 105.899i 1.27589i 0.770081 + 0.637946i \(0.220216\pi\)
−0.770081 + 0.637946i \(0.779784\pi\)
\(84\) 49.1248 + 219.110i 0.584820 + 2.60845i
\(85\) −17.8630 − 30.9397i −0.210153 − 0.363996i
\(86\) −0.00746393 − 0.00113968i −8.67898e−5 − 1.32521e-5i
\(87\) − 31.9596i − 0.367352i
\(88\) −21.3472 − 10.4323i −0.242582 − 0.118549i
\(89\) 38.9887 + 67.5303i 0.438075 + 0.758768i 0.997541 − 0.0700853i \(-0.0223271\pi\)
−0.559466 + 0.828853i \(0.688994\pi\)
\(90\) 79.6912 − 31.0998i 0.885457 − 0.345554i
\(91\) 119.156 − 68.7946i 1.30940 − 0.755984i
\(92\) 40.9959 − 131.114i 0.445607 − 1.42515i
\(93\) 36.8634 − 63.8493i 0.396381 − 0.686552i
\(94\) −85.2351 + 106.461i −0.906757 + 1.13256i
\(95\) 36.2721 − 22.1209i 0.381812 − 0.232851i
\(96\) −39.3538 + 165.090i −0.409936 + 1.71969i
\(97\) −2.55639 + 4.42780i −0.0263546 + 0.0456474i −0.878902 − 0.477003i \(-0.841723\pi\)
0.852547 + 0.522650i \(0.175057\pi\)
\(98\) 117.445 − 45.8336i 1.19842 − 0.467690i
\(99\) −49.1999 + 28.4056i −0.496969 + 0.286925i
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.11.11 ✓ 160
4.3 odd 2 inner 380.3.q.a.11.43 yes 160
19.7 even 3 inner 380.3.q.a.311.43 yes 160
76.7 odd 6 inner 380.3.q.a.311.11 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.q.a.11.11 ✓ 160 1.1 even 1 trivial
380.3.q.a.11.43 yes 160 4.3 odd 2 inner
380.3.q.a.311.11 yes 160 76.7 odd 6 inner
380.3.q.a.311.43 yes 160 19.7 even 3 inner