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Properties

Label 380.2.s.a.179.9
Level $380$
Weight $2$
Character 380.179
Analytic conductor $3.034$
Analytic rank $0$
Dimension $112$
Inner twists $8$

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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] = [380,2,Mod(179,380)] 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("380.179"); S:= CuspForms(chi, 2); 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, 3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.s (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: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(56\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 179.9
Character \(\chi\) \(=\) 380.179
Dual form 380.2.s.a.259.9

$q$-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+(-1.31410 + 0.522619i) q^{2} +(1.49300 + 0.861986i) q^{3} +(1.45374 - 1.37355i) q^{4} +(2.15570 + 0.594110i) q^{5} +(-2.41245 - 0.352467i) q^{6} -2.20094 q^{7} +(-1.19252 + 2.56474i) q^{8} +(-0.0139597 - 0.0241789i) q^{9} +(-3.14330 + 0.345886i) q^{10} +4.26641i q^{11} +(3.35442 - 0.797614i) q^{12} +(2.52454 + 4.37263i) q^{13} +(2.89227 - 1.15025i) q^{14} +(2.70635 + 2.74519i) q^{15} +(0.226714 - 3.99357i) q^{16} +(2.47016 + 1.42615i) q^{17} +(0.0309809 + 0.0244780i) q^{18} +(-3.03460 - 3.12909i) q^{19} +(3.94986 - 2.09728i) q^{20} +(-3.28601 - 1.89718i) q^{21} +(-2.22971 - 5.60651i) q^{22} +(-0.151127 - 0.261759i) q^{23} +(-3.99121 + 2.80123i) q^{24} +(4.29407 + 2.56144i) q^{25} +(-5.60272 - 4.42672i) q^{26} -5.22005i q^{27} +(-3.19959 + 3.02310i) q^{28} +(5.15709 - 2.97745i) q^{29} +(-4.99111 - 2.19308i) q^{30} +5.96280 q^{31} +(1.78919 + 5.36645i) q^{32} +(-3.67759 + 6.36977i) q^{33} +(-3.99138 - 0.583154i) q^{34} +(-4.74456 - 1.30760i) q^{35} +(-0.0535048 - 0.0159755i) q^{36} -5.24968 q^{37} +(5.62310 + 2.52600i) q^{38} +8.70447i q^{39} +(-4.09445 + 4.82032i) q^{40} +(8.43819 + 4.87179i) q^{41} +(5.30967 + 0.775760i) q^{42} +(-5.03659 + 8.72364i) q^{43} +(5.86014 + 6.20225i) q^{44} +(-0.0157280 - 0.0604161i) q^{45} +(0.335397 + 0.264997i) q^{46} +(-3.14434 - 5.44615i) q^{47} +(3.78089 - 5.76699i) q^{48} -2.15586 q^{49} +(-6.98151 - 1.12184i) q^{50} +(2.45864 + 4.25849i) q^{51} +(9.67605 + 2.88908i) q^{52} +(-6.08414 - 10.5380i) q^{53} +(2.72810 + 6.85969i) q^{54} +(-2.53472 + 9.19710i) q^{55} +(2.62467 - 5.64484i) q^{56} +(-1.83345 - 7.28752i) q^{57} +(-5.22089 + 6.60787i) q^{58} +(-0.530432 + 0.918736i) q^{59} +(7.70499 + 0.273479i) q^{60} +(-6.60128 - 11.4337i) q^{61} +(-7.83574 + 3.11627i) q^{62} +(0.0307245 + 0.0532164i) q^{63} +(-5.15579 - 6.11701i) q^{64} +(2.84432 + 10.9259i) q^{65} +(1.50377 - 10.2925i) q^{66} +(3.69169 - 2.13140i) q^{67} +(5.54985 - 1.31964i) q^{68} -0.521077i q^{69} +(6.91823 - 0.761275i) q^{70} +(2.49831 - 4.32720i) q^{71} +(0.0786599 - 0.00696917i) q^{72} +(-3.80804 - 2.19857i) q^{73} +(6.89862 - 2.74358i) q^{74} +(4.20313 + 7.52567i) q^{75} +(-8.70948 - 0.380690i) q^{76} -9.39012i q^{77} +(-4.54912 - 11.4386i) q^{78} +(-3.31453 + 5.74093i) q^{79} +(2.86135 - 8.47424i) q^{80} +(4.45773 - 7.72102i) q^{81} +(-13.6348 - 1.99208i) q^{82} +5.63889 q^{83} +(-7.38288 + 1.75550i) q^{84} +(4.47763 + 4.54189i) q^{85} +(2.05947 - 14.0960i) q^{86} +10.2661 q^{87} +(-10.9422 - 5.08778i) q^{88} +(10.0778 - 5.81843i) q^{89} +(0.0522428 + 0.0711733i) q^{90} +(-5.55636 - 9.62389i) q^{91} +(-0.579239 - 0.172949i) q^{92} +(8.90248 + 5.13985i) q^{93} +(6.97825 + 5.51352i) q^{94} +(-4.68267 - 8.54825i) q^{95} +(-1.95454 + 9.55439i) q^{96} +(-4.77238 + 8.26600i) q^{97} +(2.83302 - 1.12669i) q^{98} +(0.103157 - 0.0595579i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 112 q - 2 q^{5} - 8 q^{6} + 44 q^{9} + 6 q^{10} - 36 q^{14} - 4 q^{16} + 44 q^{20} - 48 q^{21} + 2 q^{24} - 2 q^{25} - 36 q^{26} - 12 q^{29} - 32 q^{30} - 30 q^{34} + 20 q^{36} - 24 q^{40} - 24 q^{41} - 14 q^{44}+ \cdots - 84 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.31410 + 0.522619i −0.929212 + 0.369547i
\(3\) 1.49300 + 0.861986i 0.861986 + 0.497668i 0.864677 − 0.502328i \(-0.167523\pi\)
−0.00269074 + 0.999996i \(0.500856\pi\)
\(4\) 1.45374 − 1.37355i 0.726869 − 0.686776i
\(5\) 2.15570 + 0.594110i 0.964057 + 0.265694i
\(6\) −2.41245 − 0.352467i −0.984880 − 0.143894i
\(7\) −2.20094 −0.831877 −0.415939 − 0.909393i \(-0.636547\pi\)
−0.415939 + 0.909393i \(0.636547\pi\)
\(8\) −1.19252 + 2.56474i −0.421620 + 0.906773i
\(9\) −0.0139597 − 0.0241789i −0.00465324 − 0.00805964i
\(10\) −3.14330 + 0.345886i −0.994000 + 0.109379i
\(11\) 4.26641i 1.28637i 0.765710 + 0.643186i \(0.222388\pi\)
−0.765710 + 0.643186i \(0.777612\pi\)
\(12\) 3.35442 − 0.797614i 0.968338 − 0.230251i
\(13\) 2.52454 + 4.37263i 0.700181 + 1.21275i 0.968403 + 0.249392i \(0.0802306\pi\)
−0.268222 + 0.963357i \(0.586436\pi\)
\(14\) 2.89227 − 1.15025i 0.772990 − 0.307418i
\(15\) 2.70635 + 2.74519i 0.698777 + 0.708805i
\(16\) 0.226714 − 3.99357i 0.0566785 − 0.998392i
\(17\) 2.47016 + 1.42615i 0.599102 + 0.345892i 0.768688 − 0.639624i \(-0.220910\pi\)
−0.169586 + 0.985515i \(0.554243\pi\)
\(18\) 0.0309809 + 0.0244780i 0.00730226 + 0.00576952i
\(19\) −3.03460 − 3.12909i −0.696186 − 0.717862i
\(20\) 3.94986 − 2.09728i 0.883216 − 0.468966i
\(21\) −3.28601 − 1.89718i −0.717067 − 0.413999i
\(22\) −2.22971 − 5.60651i −0.475375 − 1.19531i
\(23\) −0.151127 − 0.261759i −0.0315121 − 0.0545806i 0.849839 − 0.527042i \(-0.176699\pi\)
−0.881351 + 0.472461i \(0.843366\pi\)
\(24\) −3.99121 + 2.80123i −0.814702 + 0.571799i
\(25\) 4.29407 + 2.56144i 0.858813 + 0.512289i
\(26\) −5.60272 − 4.42672i −1.09878 − 0.868150i
\(27\) − 5.22005i − 1.00460i
\(28\) −3.19959 + 3.02310i −0.604666 + 0.571313i
\(29\) 5.15709 − 2.97745i 0.957648 − 0.552898i 0.0621997 − 0.998064i \(-0.480188\pi\)
0.895448 + 0.445165i \(0.146855\pi\)
\(30\) −4.99111 − 2.19308i −0.911249 − 0.400399i
\(31\) 5.96280 1.07095 0.535475 − 0.844551i \(-0.320133\pi\)
0.535475 + 0.844551i \(0.320133\pi\)
\(32\) 1.78919 + 5.36645i 0.316287 + 0.948664i
\(33\) −3.67759 + 6.36977i −0.640186 + 1.10883i
\(34\) −3.99138 − 0.583154i −0.684516 − 0.100010i
\(35\) −4.74456 − 1.30760i −0.801978 − 0.221025i
\(36\) −0.0535048 − 0.0159755i −0.00891746 − 0.00266258i
\(37\) −5.24968 −0.863041 −0.431521 − 0.902103i \(-0.642023\pi\)
−0.431521 + 0.902103i \(0.642023\pi\)
\(38\) 5.62310 + 2.52600i 0.912188 + 0.409772i
\(39\) 8.70447i 1.39383i
\(40\) −4.09445 + 4.82032i −0.647390 + 0.762159i
\(41\) 8.43819 + 4.87179i 1.31782 + 0.760846i 0.983378 − 0.181569i \(-0.0581175\pi\)
0.334446 + 0.942415i \(0.391451\pi\)
\(42\) 5.30967 + 0.775760i 0.819299 + 0.119702i
\(43\) −5.03659 + 8.72364i −0.768073 + 1.33034i 0.170533 + 0.985352i \(0.445451\pi\)
−0.938606 + 0.344990i \(0.887882\pi\)
\(44\) 5.86014 + 6.20225i 0.883449 + 0.935024i
\(45\) −0.0157280 − 0.0604161i −0.00234459 − 0.00900629i
\(46\) 0.335397 + 0.264997i 0.0494516 + 0.0390717i
\(47\) −3.14434 − 5.44615i −0.458649 − 0.794403i 0.540241 − 0.841510i \(-0.318333\pi\)
−0.998890 + 0.0471076i \(0.985000\pi\)
\(48\) 3.78089 − 5.76699i 0.545724 − 0.832393i
\(49\) −2.15586 −0.307980
\(50\) −6.98151 − 1.12184i −0.987334 − 0.158652i
\(51\) 2.45864 + 4.25849i 0.344278 + 0.596307i
\(52\) 9.67605 + 2.88908i 1.34183 + 0.400643i
\(53\) −6.08414 − 10.5380i −0.835720 − 1.44751i −0.893443 − 0.449177i \(-0.851717\pi\)
0.0577223 − 0.998333i \(-0.481616\pi\)
\(54\) 2.72810 + 6.85969i 0.371247 + 0.933485i
\(55\) −2.53472 + 9.19710i −0.341781 + 1.24014i
\(56\) 2.62467 − 5.64484i 0.350736 − 0.754324i
\(57\) −1.83345 − 7.28752i −0.242846 − 0.965256i
\(58\) −5.22089 + 6.60787i −0.685536 + 0.867656i
\(59\) −0.530432 + 0.918736i −0.0690564 + 0.119609i −0.898486 − 0.439002i \(-0.855332\pi\)
0.829430 + 0.558611i \(0.188666\pi\)
\(60\) 7.70499 + 0.273479i 0.994710 + 0.0353059i
\(61\) −6.60128 − 11.4337i −0.845207 − 1.46394i −0.885441 − 0.464751i \(-0.846144\pi\)
0.0402343 − 0.999190i \(-0.487190\pi\)
\(62\) −7.83574 + 3.11627i −0.995139 + 0.395767i
\(63\) 0.0307245 + 0.0532164i 0.00387092 + 0.00670463i
\(64\) −5.15579 − 6.11701i −0.644474 − 0.764626i
\(65\) 2.84432 + 10.9259i 0.352794 + 1.35519i
\(66\) 1.50377 − 10.2925i 0.185101 − 1.26692i
\(67\) 3.69169 − 2.13140i 0.451011 − 0.260391i −0.257246 − 0.966346i \(-0.582815\pi\)
0.708257 + 0.705954i \(0.249482\pi\)
\(68\) 5.54985 − 1.31964i 0.673019 − 0.160030i
\(69\) − 0.521077i − 0.0627303i
\(70\) 6.91823 − 0.761275i 0.826886 − 0.0909898i
\(71\) 2.49831 − 4.32720i 0.296495 − 0.513544i −0.678837 − 0.734289i \(-0.737516\pi\)
0.975332 + 0.220745i \(0.0708489\pi\)
\(72\) 0.0786599 − 0.00696917i 0.00927016 − 0.000821325i
\(73\) −3.80804 − 2.19857i −0.445697 − 0.257323i 0.260314 − 0.965524i \(-0.416174\pi\)
−0.706011 + 0.708201i \(0.749507\pi\)
\(74\) 6.89862 − 2.74358i 0.801948 − 0.318935i
\(75\) 4.20313 + 7.52567i 0.485336 + 0.868990i
\(76\) −8.70948 − 0.380690i −0.999046 − 0.0436682i
\(77\) − 9.39012i − 1.07010i
\(78\) −4.54912 − 11.4386i −0.515086 − 1.29516i
\(79\) −3.31453 + 5.74093i −0.372914 + 0.645905i −0.990012 − 0.140980i \(-0.954975\pi\)
0.617099 + 0.786886i \(0.288308\pi\)
\(80\) 2.86135 − 8.47424i 0.319908 − 0.947449i
\(81\) 4.45773 − 7.72102i 0.495303 − 0.857891i
\(82\) −13.6348 − 1.99208i −1.50571 − 0.219989i
\(83\) 5.63889 0.618949 0.309475 − 0.950908i \(-0.399847\pi\)
0.309475 + 0.950908i \(0.399847\pi\)
\(84\) −7.38288 + 1.75550i −0.805538 + 0.191541i
\(85\) 4.47763 + 4.54189i 0.485667 + 0.492637i
\(86\) 2.05947 − 14.0960i 0.222079 − 1.52001i
\(87\) 10.2661 1.10064
\(88\) −10.9422 − 5.08778i −1.16645 − 0.542360i
\(89\) 10.0778 − 5.81843i 1.06825 − 0.616753i 0.140545 − 0.990074i \(-0.455115\pi\)
0.927702 + 0.373322i \(0.121781\pi\)
\(90\) 0.0522428 + 0.0711733i 0.00550687 + 0.00750232i
\(91\) −5.55636 − 9.62389i −0.582465 − 1.00886i
\(92\) −0.579239 − 0.172949i −0.0603898 − 0.0180312i
\(93\) 8.90248 + 5.13985i 0.923144 + 0.532977i
\(94\) 6.97825 + 5.51352i 0.719751 + 0.568676i
\(95\) −4.68267 − 8.54825i −0.480432 − 0.877032i
\(96\) −1.95454 + 9.55439i −0.199484 + 0.975141i
\(97\) −4.77238 + 8.26600i −0.484562 + 0.839286i −0.999843 − 0.0177358i \(-0.994354\pi\)
0.515281 + 0.857021i \(0.327688\pi\)
\(98\) 2.83302 − 1.12669i 0.286179 − 0.113813i
\(99\) 0.103157 − 0.0595579i 0.0103677 − 0.00598579i
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 380.2.s.a.179.9 ✓ 112
4.3 odd 2 inner 380.2.s.a.179.30 yes 112
5.4 even 2 inner 380.2.s.a.179.48 yes 112
19.12 odd 6 inner 380.2.s.a.259.27 yes 112
20.19 odd 2 inner 380.2.s.a.179.27 yes 112
76.31 even 6 inner 380.2.s.a.259.48 yes 112
95.69 odd 6 inner 380.2.s.a.259.30 yes 112
380.259 even 6 inner 380.2.s.a.259.9 yes 112
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.s.a.179.9 ✓ 112 1.1 even 1 trivial
380.2.s.a.179.27 yes 112 20.19 odd 2 inner
380.2.s.a.179.30 yes 112 4.3 odd 2 inner
380.2.s.a.179.48 yes 112 5.4 even 2 inner
380.2.s.a.259.9 yes 112 380.259 even 6 inner
380.2.s.a.259.27 yes 112 19.12 odd 6 inner
380.2.s.a.259.30 yes 112 95.69 odd 6 inner
380.2.s.a.259.48 yes 112 76.31 even 6 inner