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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 179.11
Character \(\chi\) \(=\) 380.179
Dual form 380.2.s.a.259.11

$q$-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+(-1.10059 - 0.888090i) q^{2} +(0.369609 + 0.213394i) q^{3} +(0.422592 + 1.95484i) q^{4} +(-0.690526 - 2.12678i) q^{5} +(-0.217274 - 0.563105i) q^{6} -3.44511 q^{7} +(1.27098 - 2.52678i) q^{8} +(-1.40893 - 2.44033i) q^{9} +(-1.12878 + 2.95395i) q^{10} +2.55436i q^{11} +(-0.260958 + 0.812706i) q^{12} +(3.08972 + 5.35155i) q^{13} +(3.79165 + 3.05957i) q^{14} +(0.198616 - 0.933429i) q^{15} +(-3.64283 + 1.65220i) q^{16} +(-4.70397 - 2.71584i) q^{17} +(-0.616586 + 3.93706i) q^{18} +(-4.12479 + 1.40931i) q^{19} +(3.86570 - 2.24863i) q^{20} +(-1.27334 - 0.735166i) q^{21} +(2.26850 - 2.81130i) q^{22} +(-1.75092 - 3.03269i) q^{23} +(1.00896 - 0.662701i) q^{24} +(-4.04635 + 2.93719i) q^{25} +(1.35215 - 8.63380i) q^{26} -2.48299i q^{27} +(-1.45588 - 6.73466i) q^{28} +(-3.80617 + 2.19750i) q^{29} +(-1.04756 + 0.850933i) q^{30} -3.36388 q^{31} +(5.47657 + 1.41677i) q^{32} +(-0.545084 + 0.944113i) q^{33} +(2.76523 + 7.16657i) q^{34} +(2.37894 + 7.32698i) q^{35} +(4.17507 - 3.78550i) q^{36} -2.93703 q^{37} +(5.79128 + 2.11212i) q^{38} +2.63731i q^{39} +(-6.25154 - 0.958277i) q^{40} +(-4.83719 - 2.79275i) q^{41} +(0.748535 + 1.93996i) q^{42} +(2.66813 - 4.62133i) q^{43} +(-4.99337 + 1.07945i) q^{44} +(-4.21714 + 4.68158i) q^{45} +(-0.766254 + 4.89272i) q^{46} +(5.39759 + 9.34890i) q^{47} +(-1.69899 - 0.166689i) q^{48} +4.86881 q^{49} +(7.06185 + 0.360884i) q^{50} +(-1.15909 - 2.00760i) q^{51} +(-9.15575 + 8.30144i) q^{52} +(1.90332 + 3.29665i) q^{53} +(-2.20512 + 2.73275i) q^{54} +(5.43254 - 1.76385i) q^{55} +(-4.37866 + 8.70504i) q^{56} +(-1.82529 - 0.359312i) q^{57} +(6.14061 + 0.961687i) q^{58} +(6.09481 - 10.5565i) q^{59} +(1.90864 - 0.00619615i) q^{60} +(-4.11002 - 7.11877i) q^{61} +(3.70224 + 2.98742i) q^{62} +(4.85391 + 8.40722i) q^{63} +(-4.76923 - 6.42296i) q^{64} +(9.24801 - 10.2665i) q^{65} +(1.43837 - 0.554997i) q^{66} +(0.587816 - 0.339376i) q^{67} +(3.32118 - 10.3432i) q^{68} -1.49455i q^{69} +(3.88879 - 10.1767i) q^{70} +(3.36699 - 5.83180i) q^{71} +(-7.95689 + 0.458438i) q^{72} +(-3.37686 - 1.94963i) q^{73} +(3.23246 + 2.60835i) q^{74} +(-2.12234 + 0.222146i) q^{75} +(-4.49807 - 7.46775i) q^{76} -8.80005i q^{77} +(2.34217 - 2.90259i) q^{78} +(-4.09026 + 7.08453i) q^{79} +(6.02933 + 6.60660i) q^{80} +(-3.69692 + 6.40326i) q^{81} +(2.84354 + 7.36954i) q^{82} -1.04506 q^{83} +(0.899030 - 2.79987i) q^{84} +(-2.52776 + 11.8796i) q^{85} +(-7.04067 + 2.71665i) q^{86} -1.87573 q^{87} +(6.45430 + 3.24653i) q^{88} +(7.94252 - 4.58561i) q^{89} +(8.79900 - 1.40730i) q^{90} +(-10.6444 - 18.4367i) q^{91} +(5.18851 - 4.70437i) q^{92} +(-1.24332 - 0.717830i) q^{93} +(2.36214 - 15.0828i) q^{94} +(5.84555 + 7.79933i) q^{95} +(1.72186 + 1.69231i) q^{96} +(4.94059 - 8.55736i) q^{97} +(-5.35856 - 4.32395i) q^{98} +(6.23348 - 3.59890i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.10059 − 0.888090i −0.778234 − 0.627975i
\(3\) 0.369609 + 0.213394i 0.213394 + 0.123203i 0.602888 − 0.797826i \(-0.294017\pi\)
−0.389494 + 0.921029i \(0.627350\pi\)
\(4\) 0.422592 + 1.95484i 0.211296 + 0.977422i
\(5\) −0.690526 − 2.12678i −0.308813 − 0.951123i
\(6\) −0.217274 − 0.563105i −0.0887019 − 0.229887i
\(7\) −3.44511 −1.30213 −0.651065 − 0.759022i \(-0.725678\pi\)
−0.651065 + 0.759022i \(0.725678\pi\)
\(8\) 1.27098 − 2.52678i 0.449359 − 0.893351i
\(9\) −1.40893 − 2.44033i −0.469642 − 0.813444i
\(10\) −1.12878 + 2.95395i −0.356952 + 0.934123i
\(11\) 2.55436i 0.770168i 0.922882 + 0.385084i \(0.125827\pi\)
−0.922882 + 0.385084i \(0.874173\pi\)
\(12\) −0.260958 + 0.812706i −0.0753320 + 0.234608i
\(13\) 3.08972 + 5.35155i 0.856934 + 1.48425i 0.874840 + 0.484413i \(0.160967\pi\)
−0.0179060 + 0.999840i \(0.505700\pi\)
\(14\) 3.79165 + 3.05957i 1.01336 + 0.817705i
\(15\) 0.198616 − 0.933429i 0.0512824 − 0.241010i
\(16\) −3.64283 + 1.65220i −0.910708 + 0.413051i
\(17\) −4.70397 − 2.71584i −1.14088 − 0.658688i −0.194232 − 0.980956i \(-0.562221\pi\)
−0.946648 + 0.322268i \(0.895555\pi\)
\(18\) −0.616586 + 3.93706i −0.145331 + 0.927973i
\(19\) −4.12479 + 1.40931i −0.946291 + 0.323317i
\(20\) 3.86570 − 2.24863i 0.864398 − 0.502809i
\(21\) −1.27334 − 0.735166i −0.277867 − 0.160426i
\(22\) 2.26850 − 2.81130i 0.483646 − 0.599371i
\(23\) −1.75092 − 3.03269i −0.365093 − 0.632360i 0.623698 − 0.781665i \(-0.285629\pi\)
−0.988791 + 0.149306i \(0.952296\pi\)
\(24\) 1.00896 − 0.662701i 0.205954 − 0.135273i
\(25\) −4.04635 + 2.93719i −0.809269 + 0.587438i
\(26\) 1.35215 − 8.63380i 0.265178 − 1.69323i
\(27\) − 2.48299i − 0.477851i
\(28\) −1.45588 − 6.73466i −0.275135 − 1.27273i
\(29\) −3.80617 + 2.19750i −0.706789 + 0.408065i −0.809871 − 0.586608i \(-0.800463\pi\)
0.103082 + 0.994673i \(0.467130\pi\)
\(30\) −1.04756 + 0.850933i −0.191258 + 0.155358i
\(31\) −3.36388 −0.604170 −0.302085 − 0.953281i \(-0.597683\pi\)
−0.302085 + 0.953281i \(0.597683\pi\)
\(32\) 5.47657 + 1.41677i 0.968129 + 0.250452i
\(33\) −0.545084 + 0.944113i −0.0948869 + 0.164349i
\(34\) 2.76523 + 7.16657i 0.474233 + 1.22906i
\(35\) 2.37894 + 7.32698i 0.402115 + 1.23849i
\(36\) 4.17507 − 3.78550i 0.695845 − 0.630916i
\(37\) −2.93703 −0.482845 −0.241422 − 0.970420i \(-0.577614\pi\)
−0.241422 + 0.970420i \(0.577614\pi\)
\(38\) 5.79128 + 2.11212i 0.939470 + 0.342630i
\(39\) 2.63731i 0.422307i
\(40\) −6.25154 − 0.958277i −0.988455 − 0.151517i
\(41\) −4.83719 − 2.79275i −0.755442 − 0.436155i 0.0722146 − 0.997389i \(-0.476993\pi\)
−0.827657 + 0.561234i \(0.810327\pi\)
\(42\) 0.748535 + 1.93996i 0.115502 + 0.299342i
\(43\) 2.66813 − 4.62133i 0.406885 − 0.704746i −0.587653 − 0.809113i \(-0.699948\pi\)
0.994539 + 0.104366i \(0.0332815\pi\)
\(44\) −4.99337 + 1.07945i −0.752779 + 0.162733i
\(45\) −4.21714 + 4.68158i −0.628654 + 0.697889i
\(46\) −0.766254 + 4.89272i −0.112978 + 0.721393i
\(47\) 5.39759 + 9.34890i 0.787320 + 1.36368i 0.927603 + 0.373566i \(0.121865\pi\)
−0.140284 + 0.990111i \(0.544801\pi\)
\(48\) −1.69899 − 0.166689i −0.245228 − 0.0240595i
\(49\) 4.86881 0.695545
\(50\) 7.06185 + 0.360884i 0.998697 + 0.0510366i
\(51\) −1.15909 − 2.00760i −0.162304 − 0.281120i
\(52\) −9.15575 + 8.30144i −1.26967 + 1.15120i
\(53\) 1.90332 + 3.29665i 0.261441 + 0.452830i 0.966625 − 0.256195i \(-0.0824689\pi\)
−0.705184 + 0.709024i \(0.749136\pi\)
\(54\) −2.20512 + 2.73275i −0.300078 + 0.371880i
\(55\) 5.43254 − 1.76385i 0.732524 − 0.237838i
\(56\) −4.37866 + 8.70504i −0.585124 + 1.16326i
\(57\) −1.82529 − 0.359312i −0.241766 − 0.0475920i
\(58\) 6.14061 + 0.961687i 0.806301 + 0.126276i
\(59\) 6.09481 − 10.5565i 0.793477 − 1.37434i −0.130325 − 0.991471i \(-0.541602\pi\)
0.923802 − 0.382871i \(-0.125065\pi\)
\(60\) 1.90864 − 0.00619615i 0.246405 − 0.000799920i
\(61\) −4.11002 − 7.11877i −0.526234 − 0.911465i −0.999533 − 0.0305625i \(-0.990270\pi\)
0.473299 − 0.880902i \(-0.343063\pi\)
\(62\) 3.70224 + 2.98742i 0.470185 + 0.379403i
\(63\) 4.85391 + 8.40722i 0.611535 + 1.05921i
\(64\) −4.76923 − 6.42296i −0.596154 − 0.802870i
\(65\) 9.24801 − 10.2665i 1.14707 − 1.27341i
\(66\) 1.43837 − 0.554997i 0.177051 − 0.0683154i
\(67\) 0.587816 − 0.339376i 0.0718131 − 0.0414613i −0.463664 − 0.886011i \(-0.653465\pi\)
0.535477 + 0.844550i \(0.320132\pi\)
\(68\) 3.32118 − 10.3432i 0.402752 − 1.25430i
\(69\) − 1.49455i − 0.179922i
\(70\) 3.88879 − 10.1767i 0.464799 − 1.21635i
\(71\) 3.36699 − 5.83180i 0.399588 − 0.692107i −0.594087 − 0.804401i \(-0.702486\pi\)
0.993675 + 0.112294i \(0.0358197\pi\)
\(72\) −7.95689 + 0.458438i −0.937729 + 0.0540274i
\(73\) −3.37686 − 1.94963i −0.395232 − 0.228187i 0.289193 − 0.957271i \(-0.406613\pi\)
−0.684425 + 0.729084i \(0.739947\pi\)
\(74\) 3.23246 + 2.60835i 0.375766 + 0.303214i
\(75\) −2.12234 + 0.222146i −0.245067 + 0.0256512i
\(76\) −4.49807 − 7.46775i −0.515964 − 0.856610i
\(77\) − 8.80005i − 1.00286i
\(78\) 2.34217 − 2.90259i 0.265198 − 0.328654i
\(79\) −4.09026 + 7.08453i −0.460190 + 0.797072i −0.998970 − 0.0453742i \(-0.985552\pi\)
0.538780 + 0.842446i \(0.318885\pi\)
\(80\) 6.02933 + 6.60660i 0.674100 + 0.738640i
\(81\) −3.69692 + 6.40326i −0.410769 + 0.711474i
\(82\) 2.84354 + 7.36954i 0.314017 + 0.813829i
\(83\) −1.04506 −0.114710 −0.0573552 − 0.998354i \(-0.518267\pi\)
−0.0573552 + 0.998354i \(0.518267\pi\)
\(84\) 0.899030 − 2.79987i 0.0980922 − 0.305490i
\(85\) −2.52776 + 11.8796i −0.274174 + 1.28853i
\(86\) −7.04067 + 2.71665i −0.759215 + 0.292944i
\(87\) −1.87573 −0.201099
\(88\) 6.45430 + 3.24653i 0.688030 + 0.346081i
\(89\) 7.94252 − 4.58561i 0.841905 − 0.486074i −0.0160063 − 0.999872i \(-0.505095\pi\)
0.857911 + 0.513798i \(0.171762\pi\)
\(90\) 8.79900 − 1.40730i 0.927496 − 0.148342i
\(91\) −10.6444 − 18.4367i −1.11584 − 1.93269i
\(92\) 5.18851 − 4.70437i 0.540940 − 0.490465i
\(93\) −1.24332 − 0.717830i −0.128926 − 0.0744355i
\(94\) 2.36214 − 15.0828i 0.243636 − 1.55568i
\(95\) 5.84555 + 7.79933i 0.599741 + 0.800194i
\(96\) 1.72186 + 1.69231i 0.175736 + 0.172721i
\(97\) 4.94059 − 8.55736i 0.501641 − 0.868868i −0.498357 − 0.866972i \(-0.666063\pi\)
0.999998 − 0.00189613i \(-0.000603558\pi\)
\(98\) −5.35856 − 4.32395i −0.541296 − 0.436784i
\(99\) 6.23348 − 3.59890i 0.626488 − 0.361703i
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.179.11 yes 112
4.3 odd 2 inner 380.2.s.a.179.49 yes 112
5.4 even 2 inner 380.2.s.a.179.46 yes 112
19.12 odd 6 inner 380.2.s.a.259.8 yes 112
20.19 odd 2 inner 380.2.s.a.179.8 ✓ 112
76.31 even 6 inner 380.2.s.a.259.46 yes 112
95.69 odd 6 inner 380.2.s.a.259.49 yes 112
380.259 even 6 inner 380.2.s.a.259.11 yes 112
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.s.a.179.8 ✓ 112 20.19 odd 2 inner
380.2.s.a.179.11 yes 112 1.1 even 1 trivial
380.2.s.a.179.46 yes 112 5.4 even 2 inner
380.2.s.a.179.49 yes 112 4.3 odd 2 inner
380.2.s.a.259.8 yes 112 19.12 odd 6 inner
380.2.s.a.259.11 yes 112 380.259 even 6 inner
380.2.s.a.259.46 yes 112 76.31 even 6 inner
380.2.s.a.259.49 yes 112 95.69 odd 6 inner