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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 259.46
Character \(\chi\) \(=\) 380.259
Dual form 380.2.s.a.179.46

$q$-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} +(2.18710 + 0.465374i) q^{5} +(-0.217274 + 0.563105i) q^{6} +3.44511 q^{7} +(-1.27098 - 2.52678i) q^{8} +(-1.40893 + 2.44033i) q^{9} +(2.82040 - 1.43016i) q^{10} -2.55436i q^{11} +(0.260958 + 0.812706i) q^{12} +(-3.08972 + 5.35155i) q^{13} +(3.79165 - 3.05957i) q^{14} +(-0.907681 + 0.294708i) 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} +(1.83399 - 4.07879i) 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} +(4.56685 + 2.03564i) q^{25} +(1.35215 + 8.63380i) q^{26} -2.48299i q^{27} +(1.45588 - 6.73466i) q^{28} +(-3.80617 - 2.19750i) q^{29} +(-0.737256 + 1.13045i) q^{30} -3.36388 q^{31} +(-5.47657 + 1.41677i) q^{32} +(0.545084 + 0.944113i) q^{33} +(2.76523 - 7.16657i) q^{34} +(7.53483 + 1.60327i) q^{35} +(4.17507 + 3.78550i) q^{36} +2.93703 q^{37} +(-5.79128 + 2.11212i) q^{38} -2.63731i q^{39} +(-1.60386 - 6.11781i) 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} +(6.83406 - 1.81537i) 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} +(1.18873 - 5.58665i) 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} +(0.192530 + 1.89892i) 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} +(9.71659 - 4.92707i) 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} +(-7.19836 - 5.30882i) q^{80} +(-3.69692 - 6.40326i) q^{81} +(-2.84354 + 7.36954i) q^{82} +1.04506 q^{83} +(0.899030 + 2.79987i) q^{84} +(11.5520 - 3.75072i) q^{85} +(-7.04067 - 2.71665i) q^{86} +1.87573 q^{87} +(-6.45430 + 3.24653i) q^{88} +(7.94252 + 4.58561i) q^{89} +(-0.483667 + 8.89770i) 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} +(-8.36548 - 5.00187i) 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 \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(-1\)

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 \(-0.705983\pi\)
0.389494 + 0.921029i \(0.372650\pi\)
\(4\) 0.422592 − 1.95484i 0.211296 − 0.977422i
\(5\) 2.18710 + 0.465374i 0.978103 + 0.208122i
\(6\) −0.217274 + 0.563105i −0.0887019 + 0.229887i
\(7\) 3.44511 1.30213 0.651065 − 0.759022i \(-0.274322\pi\)
0.651065 + 0.759022i \(0.274322\pi\)
\(8\) −1.27098 − 2.52678i −0.449359 − 0.893351i
\(9\) −1.40893 + 2.44033i −0.469642 + 0.813444i
\(10\) 2.82040 − 1.43016i 0.891888 − 0.452256i
\(11\) − 2.55436i − 0.770168i −0.922882 − 0.385084i \(-0.874173\pi\)
0.922882 − 0.385084i \(-0.125827\pi\)
\(12\) 0.260958 + 0.812706i 0.0753320 + 0.234608i
\(13\) −3.08972 + 5.35155i −0.856934 + 1.48425i 0.0179060 + 0.999840i \(0.494300\pi\)
−0.874840 + 0.484413i \(0.839033\pi\)
\(14\) 3.79165 − 3.05957i 1.01336 − 0.817705i
\(15\) −0.907681 + 0.294708i −0.234362 + 0.0760933i
\(16\) −3.64283 − 1.65220i −0.910708 − 0.413051i
\(17\) 4.70397 − 2.71584i 1.14088 − 0.658688i 0.194232 − 0.980956i \(-0.437779\pi\)
0.946648 + 0.322268i \(0.104445\pi\)
\(18\) 0.616586 + 3.93706i 0.145331 + 0.927973i
\(19\) −4.12479 − 1.40931i −0.946291 − 0.323317i
\(20\) 1.83399 − 4.07879i 0.410092 − 0.912044i
\(21\) −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 \(-0.714371\pi\)
0.988791 + 0.149306i \(0.0477039\pi\)
\(24\) 1.00896 + 0.662701i 0.205954 + 0.135273i
\(25\) 4.56685 + 2.03564i 0.913371 + 0.407129i
\(26\) 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 0.103082 − 0.994673i \(-0.467130\pi\)
−0.809871 + 0.586608i \(0.800463\pi\)
\(30\) −0.737256 + 1.13045i −0.134604 + 0.206392i
\(31\) −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
\(35\) 7.53483 + 1.60327i 1.27362 + 0.271002i
\(36\) 4.17507 + 3.78550i 0.695845 + 0.630916i
\(37\) 2.93703 0.482845 0.241422 − 0.970420i \(-0.422386\pi\)
0.241422 + 0.970420i \(0.422386\pi\)
\(38\) −5.79128 + 2.11212i −0.939470 + 0.342630i
\(39\) − 2.63731i − 0.422307i
\(40\) −1.60386 − 6.11781i −0.253593 − 0.967311i
\(41\) −4.83719 + 2.79275i −0.755442 + 0.436155i −0.827657 − 0.561234i \(-0.810327\pi\)
0.0722146 + 0.997389i \(0.476993\pi\)
\(42\) −0.748535 + 1.93996i −0.115502 + 0.299342i
\(43\) −2.66813 − 4.62133i −0.406885 − 0.704746i 0.587653 − 0.809113i \(-0.300052\pi\)
−0.994539 + 0.104366i \(0.966719\pi\)
\(44\) −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 0.140284 + 0.990111i \(0.455199\pi\)
−0.927603 + 0.373566i \(0.878135\pi\)
\(48\) 1.69899 − 0.166689i 0.245228 − 0.0240595i
\(49\) 4.86881 0.695545
\(50\) 6.83406 − 1.81537i 0.966483 − 0.256732i
\(51\) −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 \(-0.917531\pi\)
0.705184 + 0.709024i \(0.250864\pi\)
\(54\) −2.20512 − 2.73275i −0.300078 − 0.371880i
\(55\) 1.18873 − 5.58665i 0.160289 − 0.753303i
\(56\) −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 0.923802 + 0.382871i \(0.125065\pi\)
−0.130325 + 0.991471i \(0.541602\pi\)
\(60\) 0.192530 + 1.89892i 0.0248555 + 0.245149i
\(61\) −4.11002 + 7.11877i −0.526234 + 0.911465i 0.473299 + 0.880902i \(0.343063\pi\)
−0.999533 + 0.0305625i \(0.990270\pi\)
\(62\) −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 \(-0.346535\pi\)
−0.535477 + 0.844550i \(0.679868\pi\)
\(68\) −3.32118 − 10.3432i −0.402752 − 1.25430i
\(69\) 1.49455i 0.179922i
\(70\) 9.71659 − 4.92707i 1.16135 − 0.588897i
\(71\) 3.36699 + 5.83180i 0.399588 + 0.692107i 0.993675 − 0.112294i \(-0.0358197\pi\)
−0.594087 + 0.804401i \(0.702486\pi\)
\(72\) 7.95689 + 0.458438i 0.937729 + 0.0540274i
\(73\) 3.37686 − 1.94963i 0.395232 − 0.228187i −0.289193 − 0.957271i \(-0.593387\pi\)
0.684425 + 0.729084i \(0.260053\pi\)
\(74\) 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 0.538780 − 0.842446i \(-0.318885\pi\)
−0.998970 + 0.0453742i \(0.985552\pi\)
\(80\) −7.19836 − 5.30882i −0.804801 − 0.593544i
\(81\) −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 \(-0.481733\pi\)
0.0573552 + 0.998354i \(0.481733\pi\)
\(84\) 0.899030 + 2.79987i 0.0980922 + 0.305490i
\(85\) 11.5520 − 3.75072i 1.25299 − 0.406822i
\(86\) −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 0.857911 − 0.513798i \(-0.171762\pi\)
−0.0160063 + 0.999872i \(0.505095\pi\)
\(90\) −0.483667 + 8.89770i −0.0509829 + 0.937899i
\(91\) −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
\(95\) −8.36548 − 5.00187i −0.858281 − 0.513181i
\(96\) 1.72186 − 1.69231i 0.175736 − 0.172721i
\(97\) −4.94059 − 8.55736i −0.501641 − 0.868868i −0.999998 − 0.00189613i \(-0.999396\pi\)
0.498357 − 0.866972i \(-0.333937\pi\)
\(98\) 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.259.46 yes 112
4.3 odd 2 inner 380.2.s.a.259.8 yes 112
5.4 even 2 inner 380.2.s.a.259.11 yes 112
19.8 odd 6 inner 380.2.s.a.179.49 yes 112
20.19 odd 2 inner 380.2.s.a.259.49 yes 112
76.27 even 6 inner 380.2.s.a.179.11 yes 112
95.84 odd 6 inner 380.2.s.a.179.8 ✓ 112
380.179 even 6 inner 380.2.s.a.179.46 yes 112
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.s.a.179.8 ✓ 112 95.84 odd 6 inner
380.2.s.a.179.11 yes 112 76.27 even 6 inner
380.2.s.a.179.46 yes 112 380.179 even 6 inner
380.2.s.a.179.49 yes 112 19.8 odd 6 inner
380.2.s.a.259.8 yes 112 4.3 odd 2 inner
380.2.s.a.259.11 yes 112 5.4 even 2 inner
380.2.s.a.259.46 yes 112 1.1 even 1 trivial
380.2.s.a.259.49 yes 112 20.19 odd 2 inner