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Properties

Label 380.3.q.a.11.10
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 11.10
Character \(\chi\) \(=\) 380.11
Dual form 380.3.q.a.311.10

$q$-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+(-1.87590 + 0.693529i) q^{2} +(-0.511454 - 0.295288i) q^{3} +(3.03804 - 2.60199i) q^{4} +(1.11803 - 1.93649i) q^{5} +(1.16423 + 0.199224i) q^{6} -0.545604i q^{7} +(-3.89451 + 6.98805i) q^{8} +(-4.32561 - 7.49218i) q^{9} +(-0.754312 + 4.40806i) q^{10} +18.8150i q^{11} +(-2.32215 + 0.433702i) q^{12} +(9.81776 + 17.0049i) q^{13} +(0.378392 + 1.02350i) q^{14} +(-1.14365 + 0.660285i) q^{15} +(2.45932 - 15.8099i) q^{16} +(12.1867 - 21.1079i) q^{17} +(13.3105 + 11.0547i) q^{18} +(-17.5493 + 7.28157i) q^{19} +(-1.64210 - 8.79224i) q^{20} +(-0.161111 + 0.279052i) q^{21} +(-13.0487 - 35.2951i) q^{22} +(35.3476 - 20.4080i) q^{23} +(4.05535 - 2.42406i) q^{24} +(-2.50000 - 4.33013i) q^{25} +(-30.2105 - 25.0906i) q^{26} +10.4244i q^{27} +(-1.41966 - 1.65757i) q^{28} +(-3.34688 - 5.79697i) q^{29} +(1.68745 - 2.03178i) q^{30} +26.2260i q^{31} +(6.35115 + 31.3634i) q^{32} +(5.55585 - 9.62301i) q^{33} +(-8.22207 + 48.0483i) q^{34} +(-1.05656 - 0.610004i) q^{35} +(-32.6359 - 11.5063i) q^{36} +41.8955 q^{37} +(27.8709 - 25.8305i) q^{38} -11.5963i q^{39} +(9.17810 + 15.3546i) q^{40} +(26.3527 - 45.6442i) q^{41} +(0.108698 - 0.635209i) q^{42} +(7.63053 + 4.40549i) q^{43} +(48.9564 + 57.1606i) q^{44} -19.3447 q^{45} +(-52.1552 + 62.7980i) q^{46} +(42.3108 - 24.4282i) q^{47} +(-5.92630 + 7.35982i) q^{48} +48.7023 q^{49} +(7.69283 + 6.38908i) q^{50} +(-12.4659 + 7.19717i) q^{51} +(74.0731 + 26.1157i) q^{52} +(6.36711 + 11.0282i) q^{53} +(-7.22962 - 19.5552i) q^{54} +(36.4351 + 21.0358i) q^{55} +(3.81271 + 2.12486i) q^{56} +(11.1258 + 1.45792i) q^{57} +(10.2988 + 8.55341i) q^{58} +(32.1613 + 18.5683i) q^{59} +(-1.75639 + 4.98172i) q^{60} +(31.1468 + 53.9478i) q^{61} +(-18.1885 - 49.1975i) q^{62} +(-4.08776 + 2.36007i) q^{63} +(-33.6656 - 54.4300i) q^{64} +43.9063 q^{65} +(-3.74840 + 21.9050i) q^{66} +(-3.68328 + 2.12654i) q^{67} +(-17.8991 - 95.8363i) q^{68} -24.1049 q^{69} +(2.40506 + 0.411556i) q^{70} +(9.89842 + 5.71485i) q^{71} +(69.2018 - 1.04920i) q^{72} +(-17.6540 + 30.5776i) q^{73} +(-78.5919 + 29.0557i) q^{74} +2.95288i q^{75} +(-34.3689 + 67.7848i) q^{76} +10.2655 q^{77} +(8.04235 + 21.7535i) q^{78} +(50.9514 + 29.4168i) q^{79} +(-27.8661 - 22.4384i) q^{80} +(-35.8523 + 62.0980i) q^{81} +(-17.7796 + 103.901i) q^{82} +100.068i q^{83} +(0.236630 + 1.26698i) q^{84} +(-27.2502 - 47.1988i) q^{85} +(-17.3695 - 2.97228i) q^{86} +3.95318i q^{87} +(-131.480 - 73.2751i) q^{88} +(-25.8120 - 44.7076i) q^{89} +(36.2888 - 13.4161i) q^{90} +(9.27792 - 5.35661i) q^{91} +(54.2861 - 153.974i) q^{92} +(7.74424 - 13.4134i) q^{93} +(-62.4294 + 75.1687i) q^{94} +(-5.52005 + 42.1252i) q^{95} +(6.01292 - 17.9164i) q^{96} +(49.8152 - 86.2825i) q^{97} +(-91.3609 + 33.7765i) q^{98} +(140.965 - 81.3863i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.87590 + 0.693529i −0.937952 + 0.346764i
\(3\) −0.511454 − 0.295288i −0.170485 − 0.0984295i 0.412330 − 0.911035i \(-0.364715\pi\)
−0.582814 + 0.812605i \(0.698049\pi\)
\(4\) 3.03804 − 2.60199i 0.759509 − 0.650497i
\(5\) 1.11803 − 1.93649i 0.223607 − 0.387298i
\(6\) 1.16423 + 0.199224i 0.194038 + 0.0332041i
\(7\) − 0.545604i − 0.0779435i −0.999240 − 0.0389717i \(-0.987592\pi\)
0.999240 − 0.0389717i \(-0.0124082\pi\)
\(8\) −3.89451 + 6.98805i −0.486814 + 0.873506i
\(9\) −4.32561 − 7.49218i −0.480623 − 0.832464i
\(10\) −0.754312 + 4.40806i −0.0754312 + 0.440806i
\(11\) 18.8150i 1.71045i 0.518255 + 0.855226i \(0.326582\pi\)
−0.518255 + 0.855226i \(0.673418\pi\)
\(12\) −2.32215 + 0.433702i −0.193513 + 0.0361418i
\(13\) 9.81776 + 17.0049i 0.755212 + 1.30807i 0.945269 + 0.326293i \(0.105799\pi\)
−0.190057 + 0.981773i \(0.560867\pi\)
\(14\) 0.378392 + 1.02350i 0.0270280 + 0.0731073i
\(15\) −1.14365 + 0.660285i −0.0762431 + 0.0440190i
\(16\) 2.45932 − 15.8099i 0.153707 − 0.988116i
\(17\) 12.1867 − 21.1079i 0.716863 − 1.24164i −0.245373 − 0.969429i \(-0.578910\pi\)
0.962236 − 0.272215i \(-0.0877562\pi\)
\(18\) 13.3105 + 11.0547i 0.739471 + 0.614148i
\(19\) −17.5493 + 7.28157i −0.923649 + 0.383240i
\(20\) −1.64210 − 8.79224i −0.0821050 − 0.439612i
\(21\) −0.161111 + 0.279052i −0.00767193 + 0.0132882i
\(22\) −13.0487 − 35.2951i −0.593124 − 1.60432i
\(23\) 35.3476 − 20.4080i 1.53685 − 0.887302i 0.537832 − 0.843052i \(-0.319243\pi\)
0.999020 − 0.0442506i \(-0.0140900\pi\)
\(24\) 4.05535 − 2.42406i 0.168973 − 0.101003i
\(25\) −2.50000 − 4.33013i −0.100000 − 0.173205i
\(26\) −30.2105 − 25.0906i −1.16194 − 0.965023i
\(27\) 10.4244i 0.386089i
\(28\) −1.41966 − 1.65757i −0.0507020 − 0.0591988i
\(29\) −3.34688 − 5.79697i −0.115410 − 0.199896i 0.802534 − 0.596607i \(-0.203485\pi\)
−0.917944 + 0.396711i \(0.870151\pi\)
\(30\) 1.68745 − 2.03178i 0.0562482 − 0.0677261i
\(31\) 26.2260i 0.846000i 0.906130 + 0.423000i \(0.139023\pi\)
−0.906130 + 0.423000i \(0.860977\pi\)
\(32\) 6.35115 + 31.3634i 0.198473 + 0.980106i
\(33\) 5.55585 − 9.62301i 0.168359 − 0.291606i
\(34\) −8.22207 + 48.0483i −0.241826 + 1.41319i
\(35\) −1.05656 − 0.610004i −0.0301874 − 0.0174287i
\(36\) −32.6359 − 11.5063i −0.906553 − 0.319620i
\(37\) 41.8955 1.13231 0.566155 − 0.824299i \(-0.308430\pi\)
0.566155 + 0.824299i \(0.308430\pi\)
\(38\) 27.8709 − 25.8305i 0.733444 − 0.679750i
\(39\) − 11.5963i − 0.297340i
\(40\) 9.17810 + 15.3546i 0.229452 + 0.383864i
\(41\) 26.3527 − 45.6442i 0.642749 − 1.11327i −0.342068 − 0.939675i \(-0.611127\pi\)
0.984817 − 0.173598i \(-0.0555393\pi\)
\(42\) 0.108698 − 0.635209i 0.00258804 − 0.0151240i
\(43\) 7.63053 + 4.40549i 0.177454 + 0.102453i 0.586096 − 0.810242i \(-0.300664\pi\)
−0.408642 + 0.912695i \(0.633997\pi\)
\(44\) 48.9564 + 57.1606i 1.11264 + 1.29910i
\(45\) −19.3447 −0.429883
\(46\) −52.1552 + 62.7980i −1.13381 + 1.36517i
\(47\) 42.3108 − 24.4282i 0.900230 − 0.519748i 0.0229554 − 0.999736i \(-0.492692\pi\)
0.877275 + 0.479988i \(0.159359\pi\)
\(48\) −5.92630 + 7.35982i −0.123465 + 0.153330i
\(49\) 48.7023 0.993925
\(50\) 7.69283 + 6.38908i 0.153857 + 0.127782i
\(51\) −12.4659 + 7.19717i −0.244429 + 0.141121i
\(52\) 74.0731 + 26.1157i 1.42448 + 0.502224i
\(53\) 6.36711 + 11.0282i 0.120134 + 0.208079i 0.919820 − 0.392340i \(-0.128334\pi\)
−0.799686 + 0.600418i \(0.795001\pi\)
\(54\) −7.22962 − 19.5552i −0.133882 − 0.362133i
\(55\) 36.4351 + 21.0358i 0.662456 + 0.382469i
\(56\) 3.81271 + 2.12486i 0.0680841 + 0.0379440i
\(57\) 11.1258 + 1.45792i 0.195190 + 0.0255776i
\(58\) 10.2988 + 8.55341i 0.177566 + 0.147473i
\(59\) 32.1613 + 18.5683i 0.545107 + 0.314717i 0.747146 − 0.664660i \(-0.231424\pi\)
−0.202039 + 0.979377i \(0.564757\pi\)
\(60\) −1.75639 + 4.98172i −0.0292731 + 0.0830287i
\(61\) 31.1468 + 53.9478i 0.510603 + 0.884390i 0.999925 + 0.0122869i \(0.00391115\pi\)
−0.489321 + 0.872103i \(0.662756\pi\)
\(62\) −18.1885 − 49.1975i −0.293363 − 0.793508i
\(63\) −4.08776 + 2.36007i −0.0648851 + 0.0374614i
\(64\) −33.6656 − 54.4300i −0.526025 − 0.850469i
\(65\) 43.9063 0.675482
\(66\) −3.74840 + 21.9050i −0.0567940 + 0.331894i
\(67\) −3.68328 + 2.12654i −0.0549743 + 0.0317394i −0.527235 − 0.849719i \(-0.676771\pi\)
0.472261 + 0.881459i \(0.343438\pi\)
\(68\) −17.8991 − 95.8363i −0.263221 − 1.40936i
\(69\) −24.1049 −0.349347
\(70\) 2.40506 + 0.411556i 0.0343580 + 0.00587937i
\(71\) 9.89842 + 5.71485i 0.139414 + 0.0804909i 0.568085 − 0.822970i \(-0.307685\pi\)
−0.428671 + 0.903461i \(0.641018\pi\)
\(72\) 69.2018 − 1.04920i 0.961136 − 0.0145722i
\(73\) −17.6540 + 30.5776i −0.241835 + 0.418871i −0.961237 − 0.275723i \(-0.911083\pi\)
0.719402 + 0.694594i \(0.244416\pi\)
\(74\) −78.5919 + 29.0557i −1.06205 + 0.392645i
\(75\) 2.95288i 0.0393718i
\(76\) −34.3689 + 67.7848i −0.452223 + 0.891905i
\(77\) 10.2655 0.133319
\(78\) 8.04235 + 21.7535i 0.103107 + 0.278891i
\(79\) 50.9514 + 29.4168i 0.644955 + 0.372365i 0.786521 − 0.617564i \(-0.211880\pi\)
−0.141566 + 0.989929i \(0.545214\pi\)
\(80\) −27.8661 − 22.4384i −0.348326 − 0.280480i
\(81\) −35.8523 + 62.0980i −0.442621 + 0.766642i
\(82\) −17.7796 + 103.901i −0.216824 + 1.26708i
\(83\) 100.068i 1.20563i 0.797879 + 0.602817i \(0.205955\pi\)
−0.797879 + 0.602817i \(0.794045\pi\)
\(84\) 0.236630 + 1.26698i 0.00281702 + 0.0150831i
\(85\) −27.2502 − 47.1988i −0.320591 − 0.555280i
\(86\) −17.3695 − 2.97228i −0.201971 − 0.0345614i
\(87\) 3.95318i 0.0454389i
\(88\) −131.480 − 73.2751i −1.49409 − 0.832672i
\(89\) −25.8120 − 44.7076i −0.290022 − 0.502333i 0.683793 − 0.729676i \(-0.260329\pi\)
−0.973815 + 0.227343i \(0.926996\pi\)
\(90\) 36.2888 − 13.4161i 0.403209 − 0.149068i
\(91\) 9.27792 − 5.35661i 0.101955 − 0.0588639i
\(92\) 54.2861 − 153.974i 0.590066 − 1.67363i
\(93\) 7.74424 − 13.4134i 0.0832714 − 0.144230i
\(94\) −62.4294 + 75.1687i −0.664143 + 0.799667i
\(95\) −5.52005 + 42.1252i −0.0581058 + 0.443423i
\(96\) 6.01292 − 17.9164i 0.0626346 − 0.186629i
\(97\) 49.8152 − 86.2825i 0.513559 − 0.889511i −0.486317 − 0.873782i \(-0.661660\pi\)
0.999876 − 0.0157282i \(-0.00500664\pi\)
\(98\) −91.3609 + 33.7765i −0.932254 + 0.344658i
\(99\) 140.965 − 81.3863i 1.42389 − 0.822083i
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.11.10 ✓ 160
4.3 odd 2 inner 380.3.q.a.11.44 yes 160
19.7 even 3 inner 380.3.q.a.311.44 yes 160
76.7 odd 6 inner 380.3.q.a.311.10 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.q.a.11.10 ✓ 160 1.1 even 1 trivial
380.3.q.a.11.44 yes 160 4.3 odd 2 inner
380.3.q.a.311.10 yes 160 76.7 odd 6 inner
380.3.q.a.311.44 yes 160 19.7 even 3 inner