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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 179.15
Character \(\chi\) \(=\) 380.179
Dual form 380.2.s.a.259.15

$q$-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+(-0.993958 + 1.00601i) q^{2} +(-0.753647 - 0.435118i) q^{3} +(-0.0240957 - 1.99985i) q^{4} +(-1.67009 - 1.48688i) q^{5} +(1.18682 - 0.325684i) q^{6} -1.85484 q^{7} +(2.03582 + 1.96353i) q^{8} +(-1.12134 - 1.94223i) q^{9} +(3.15580 - 0.202225i) q^{10} +5.50475i q^{11} +(-0.852013 + 1.51767i) q^{12} +(0.160712 + 0.278361i) q^{13} +(1.84363 - 1.86598i) q^{14} +(0.611689 + 1.84727i) q^{15} +(-3.99884 + 0.0963757i) q^{16} +(5.09650 + 2.94246i) q^{17} +(3.06846 + 0.802411i) q^{18} +(3.44208 + 2.67434i) q^{19} +(-2.93330 + 3.37576i) q^{20} +(1.39790 + 0.807075i) q^{21} +(-5.53781 - 5.47149i) q^{22} +(1.76944 + 3.06477i) q^{23} +(-0.679918 - 2.36563i) q^{24} +(0.578391 + 4.96643i) q^{25} +(-0.439774 - 0.115002i) q^{26} +4.56238i q^{27} +(0.0446936 + 3.70941i) q^{28} +(1.85199 - 1.06924i) q^{29} +(-2.46635 - 1.22074i) q^{30} -6.78126 q^{31} +(3.87772 - 4.11865i) q^{32} +(2.39522 - 4.14864i) q^{33} +(-8.02584 + 2.20242i) q^{34} +(3.09775 + 2.75792i) q^{35} +(-3.85715 + 2.28933i) q^{36} -2.63450 q^{37} +(-6.11168 + 0.804565i) q^{38} -0.279715i q^{39} +(-0.480463 - 6.30628i) q^{40} +(-1.69627 - 0.979341i) q^{41} +(-2.20137 + 0.604092i) q^{42} +(1.16121 - 2.01128i) q^{43} +(11.0087 - 0.132641i) q^{44} +(-1.01511 + 4.91099i) q^{45} +(-4.84193 - 1.26618i) q^{46} +(-0.698825 - 1.21040i) q^{47} +(3.05565 + 1.66733i) q^{48} -3.55956 q^{49} +(-5.57116 - 4.35456i) q^{50} +(-2.56064 - 4.43515i) q^{51} +(0.552810 - 0.328108i) q^{52} +(1.32919 + 2.30223i) q^{53} +(-4.58978 - 4.53481i) q^{54} +(8.18489 - 9.19342i) q^{55} +(-3.77612 - 3.64204i) q^{56} +(-1.43045 - 3.51322i) q^{57} +(-0.765130 + 2.92589i) q^{58} +(-6.28447 + 10.8850i) q^{59} +(3.67952 - 1.26780i) q^{60} +(4.28278 + 7.41799i) q^{61} +(6.74029 - 6.82199i) q^{62} +(2.07992 + 3.60252i) q^{63} +(0.289092 + 7.99477i) q^{64} +(0.145486 - 0.703847i) q^{65} +(1.79281 + 6.53317i) q^{66} +(-2.28252 + 1.31781i) q^{67} +(5.76170 - 10.2632i) q^{68} -3.07967i q^{69} +(-5.85352 + 0.375096i) q^{70} +(-2.03427 + 3.52346i) q^{71} +(1.53077 - 6.15581i) q^{72} +(-4.16372 - 2.40392i) q^{73} +(2.61858 - 2.65032i) q^{74} +(1.72508 - 3.99460i) q^{75} +(5.26536 - 6.94809i) q^{76} -10.2104i q^{77} +(0.281395 + 0.278025i) q^{78} +(2.39261 - 4.14413i) q^{79} +(6.82171 + 5.78483i) q^{80} +(-1.37886 + 2.38826i) q^{81} +(2.67124 - 0.733032i) q^{82} +11.9539 q^{83} +(1.58035 - 2.81503i) q^{84} +(-4.13652 - 12.4920i) q^{85} +(0.869162 + 3.16731i) q^{86} -1.86099 q^{87} +(-10.8087 + 11.2067i) q^{88} +(12.4618 - 7.19480i) q^{89} +(-3.93151 - 5.90252i) q^{90} +(-0.298095 - 0.516316i) q^{91} +(6.08645 - 3.61248i) q^{92} +(5.11068 + 2.95065i) q^{93} +(1.91227 + 0.500065i) q^{94} +(-1.77215 - 9.58433i) q^{95} +(-4.71453 + 1.41674i) q^{96} +(-7.46897 + 12.9366i) q^{97} +(3.53806 - 3.58094i) q^{98} +(10.6915 - 6.17272i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.993958 + 1.00601i −0.702834 + 0.711354i
\(3\) −0.753647 − 0.435118i −0.435118 − 0.251216i 0.266407 − 0.963861i \(-0.414164\pi\)
−0.701525 + 0.712645i \(0.747497\pi\)
\(4\) −0.0240957 − 1.99985i −0.0120478 − 0.999927i
\(5\) −1.67009 − 1.48688i −0.746886 − 0.664952i
\(6\) 1.18682 − 0.325684i 0.484519 − 0.132960i
\(7\) −1.85484 −0.701064 −0.350532 − 0.936551i \(-0.613999\pi\)
−0.350532 + 0.936551i \(0.613999\pi\)
\(8\) 2.03582 + 1.96353i 0.719770 + 0.694213i
\(9\) −1.12134 − 1.94223i −0.373781 − 0.647409i
\(10\) 3.15580 − 0.202225i 0.997953 − 0.0639493i
\(11\) 5.50475i 1.65974i 0.557953 + 0.829872i \(0.311587\pi\)
−0.557953 + 0.829872i \(0.688413\pi\)
\(12\) −0.852013 + 1.51767i −0.245955 + 0.438113i
\(13\) 0.160712 + 0.278361i 0.0445735 + 0.0772035i 0.887451 − 0.460901i \(-0.152474\pi\)
−0.842878 + 0.538105i \(0.819140\pi\)
\(14\) 1.84363 − 1.86598i 0.492732 − 0.498705i
\(15\) 0.611689 + 1.84727i 0.157938 + 0.476962i
\(16\) −3.99884 + 0.0963757i −0.999710 + 0.0240939i
\(17\) 5.09650 + 2.94246i 1.23608 + 0.713652i 0.968291 − 0.249825i \(-0.0803731\pi\)
0.267791 + 0.963477i \(0.413706\pi\)
\(18\) 3.06846 + 0.802411i 0.723243 + 0.189130i
\(19\) 3.44208 + 2.67434i 0.789666 + 0.613536i
\(20\) −2.93330 + 3.37576i −0.655905 + 0.754843i
\(21\) 1.39790 + 0.807075i 0.305046 + 0.176118i
\(22\) −5.53781 − 5.47149i −1.18067 − 1.16653i
\(23\) 1.76944 + 3.06477i 0.368955 + 0.639048i 0.989402 − 0.145200i \(-0.0463825\pi\)
−0.620448 + 0.784248i \(0.713049\pi\)
\(24\) −0.679918 − 2.36563i −0.138788 − 0.482882i
\(25\) 0.578391 + 4.96643i 0.115678 + 0.993287i
\(26\) −0.439774 − 0.115002i −0.0862468 − 0.0225538i
\(27\) 4.56238i 0.878030i
\(28\) 0.0446936 + 3.70941i 0.00844631 + 0.701013i
\(29\) 1.85199 − 1.06924i 0.343905 − 0.198554i −0.318092 − 0.948060i \(-0.603042\pi\)
0.661998 + 0.749506i \(0.269709\pi\)
\(30\) −2.46635 − 1.22074i −0.450293 − 0.222876i
\(31\) −6.78126 −1.21795 −0.608976 − 0.793189i \(-0.708419\pi\)
−0.608976 + 0.793189i \(0.708419\pi\)
\(32\) 3.87772 − 4.11865i 0.685491 − 0.728081i
\(33\) 2.39522 − 4.14864i 0.416954 − 0.722185i
\(34\) −8.02584 + 2.20242i −1.37642 + 0.377712i
\(35\) 3.09775 + 2.75792i 0.523615 + 0.466174i
\(36\) −3.85715 + 2.28933i −0.642858 + 0.381554i
\(37\) −2.63450 −0.433109 −0.216554 − 0.976271i \(-0.569482\pi\)
−0.216554 + 0.976271i \(0.569482\pi\)
\(38\) −6.11168 + 0.804565i −0.991446 + 0.130518i
\(39\) − 0.279715i − 0.0447902i
\(40\) −0.480463 − 6.30628i −0.0759678 − 0.997110i
\(41\) −1.69627 − 0.979341i −0.264913 − 0.152947i 0.361661 − 0.932310i \(-0.382210\pi\)
−0.626573 + 0.779362i \(0.715543\pi\)
\(42\) −2.20137 + 0.604092i −0.339679 + 0.0932134i
\(43\) 1.16121 − 2.01128i 0.177083 − 0.306717i −0.763797 − 0.645456i \(-0.776667\pi\)
0.940880 + 0.338739i \(0.110001\pi\)
\(44\) 11.0087 − 0.132641i 1.65962 − 0.0199963i
\(45\) −1.01511 + 4.91099i −0.151323 + 0.732087i
\(46\) −4.84193 − 1.26618i −0.713903 − 0.186688i
\(47\) −0.698825 − 1.21040i −0.101934 − 0.176555i 0.810547 − 0.585673i \(-0.199170\pi\)
−0.912481 + 0.409118i \(0.865836\pi\)
\(48\) 3.05565 + 1.66733i 0.441045 + 0.240659i
\(49\) −3.55956 −0.508509
\(50\) −5.57116 − 4.35456i −0.787881 − 0.615828i
\(51\) −2.56064 − 4.43515i −0.358561 − 0.621046i
\(52\) 0.552810 − 0.328108i 0.0766609 − 0.0455004i
\(53\) 1.32919 + 2.30223i 0.182578 + 0.316235i 0.942758 − 0.333478i \(-0.108222\pi\)
−0.760179 + 0.649713i \(0.774889\pi\)
\(54\) −4.58978 − 4.53481i −0.624590 − 0.617110i
\(55\) 8.18489 − 9.19342i 1.10365 − 1.23964i
\(56\) −3.77612 − 3.64204i −0.504605 − 0.486688i
\(57\) −1.43045 − 3.51322i −0.189468 − 0.465337i
\(58\) −0.765130 + 2.92589i −0.100466 + 0.384189i
\(59\) −6.28447 + 10.8850i −0.818168 + 1.41711i 0.0888618 + 0.996044i \(0.471677\pi\)
−0.907030 + 0.421065i \(0.861656\pi\)
\(60\) 3.67952 − 1.26780i 0.475025 − 0.163672i
\(61\) 4.28278 + 7.41799i 0.548353 + 0.949776i 0.998388 + 0.0567646i \(0.0180784\pi\)
−0.450034 + 0.893011i \(0.648588\pi\)
\(62\) 6.74029 − 6.82199i 0.856018 − 0.866394i
\(63\) 2.07992 + 3.60252i 0.262045 + 0.453875i
\(64\) 0.289092 + 7.99477i 0.0361365 + 0.999347i
\(65\) 0.145486 − 0.703847i 0.0180453 − 0.0873015i
\(66\) 1.79281 + 6.53317i 0.220680 + 0.804178i
\(67\) −2.28252 + 1.31781i −0.278854 + 0.160997i −0.632905 − 0.774230i \(-0.718138\pi\)
0.354050 + 0.935226i \(0.384804\pi\)
\(68\) 5.76170 − 10.2632i 0.698708 − 1.24459i
\(69\) − 3.07967i − 0.370748i
\(70\) −5.85352 + 0.375096i −0.699629 + 0.0448326i
\(71\) −2.03427 + 3.52346i −0.241424 + 0.418158i −0.961120 − 0.276131i \(-0.910948\pi\)
0.719696 + 0.694289i \(0.244281\pi\)
\(72\) 1.53077 − 6.15581i 0.180403 − 0.725469i
\(73\) −4.16372 − 2.40392i −0.487327 − 0.281358i 0.236138 − 0.971719i \(-0.424118\pi\)
−0.723465 + 0.690361i \(0.757452\pi\)
\(74\) 2.61858 − 2.65032i 0.304404 − 0.308093i
\(75\) 1.72508 − 3.99460i 0.199195 − 0.461257i
\(76\) 5.26536 − 6.94809i 0.603978 − 0.797001i
\(77\) − 10.2104i − 1.16359i
\(78\) 0.281395 + 0.278025i 0.0318617 + 0.0314801i
\(79\) 2.39261 − 4.14413i 0.269190 − 0.466251i −0.699463 − 0.714669i \(-0.746577\pi\)
0.968653 + 0.248418i \(0.0799107\pi\)
\(80\) 6.82171 + 5.78483i 0.762691 + 0.646763i
\(81\) −1.37886 + 2.38826i −0.153207 + 0.265362i
\(82\) 2.67124 − 0.733032i 0.294989 − 0.0809499i
\(83\) 11.9539 1.31211 0.656054 − 0.754714i \(-0.272224\pi\)
0.656054 + 0.754714i \(0.272224\pi\)
\(84\) 1.58035 − 2.81503i 0.172430 − 0.307145i
\(85\) −4.13652 − 12.4920i −0.448668 − 1.35495i
\(86\) 0.869162 + 3.16731i 0.0937241 + 0.341540i
\(87\) −1.86099 −0.199519
\(88\) −10.8087 + 11.2067i −1.15222 + 1.19463i
\(89\) 12.4618 − 7.19480i 1.32094 − 0.762647i 0.337064 − 0.941482i \(-0.390566\pi\)
0.983879 + 0.178835i \(0.0572327\pi\)
\(90\) −3.93151 − 5.90252i −0.414418 − 0.622180i
\(91\) −0.298095 − 0.516316i −0.0312489 − 0.0541246i
\(92\) 6.08645 − 3.61248i 0.634557 − 0.376627i
\(93\) 5.11068 + 2.95065i 0.529953 + 0.305968i
\(94\) 1.91227 + 0.500065i 0.197236 + 0.0515778i
\(95\) −1.77215 − 9.58433i −0.181819 − 0.983332i
\(96\) −4.71453 + 1.41674i −0.481175 + 0.144595i
\(97\) −7.46897 + 12.9366i −0.758359 + 1.31352i 0.185328 + 0.982677i \(0.440665\pi\)
−0.943687 + 0.330839i \(0.892668\pi\)
\(98\) 3.53806 − 3.58094i 0.357398 − 0.361730i
\(99\) 10.6915 − 6.17272i 1.07453 − 0.620382i
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.179.15 ✓ 112
4.3 odd 2 inner 380.2.s.a.179.24 yes 112
5.4 even 2 inner 380.2.s.a.179.42 yes 112
19.12 odd 6 inner 380.2.s.a.259.33 yes 112
20.19 odd 2 inner 380.2.s.a.179.33 yes 112
76.31 even 6 inner 380.2.s.a.259.42 yes 112
95.69 odd 6 inner 380.2.s.a.259.24 yes 112
380.259 even 6 inner 380.2.s.a.259.15 yes 112
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.s.a.179.15 ✓ 112 1.1 even 1 trivial
380.2.s.a.179.24 yes 112 4.3 odd 2 inner
380.2.s.a.179.33 yes 112 20.19 odd 2 inner
380.2.s.a.179.42 yes 112 5.4 even 2 inner
380.2.s.a.259.15 yes 112 380.259 even 6 inner
380.2.s.a.259.24 yes 112 95.69 odd 6 inner
380.2.s.a.259.33 yes 112 19.12 odd 6 inner
380.2.s.a.259.42 yes 112 76.31 even 6 inner