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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,3,Mod(39,380)] 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("380.39"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.h (of order \(2\), degree \(1\), 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: \(108\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 39.100
Character \(\chi\) \(=\) 380.39
Dual form 380.3.h.a.39.99

$q$-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+(1.91201 + 0.586697i) q^{2} +5.34961 q^{3} +(3.31157 + 2.24354i) q^{4} +(-3.58581 - 3.48454i) q^{5} +(10.2285 + 3.13860i) q^{6} -0.878631 q^{7} +(5.01548 + 6.23257i) q^{8} +19.6184 q^{9} +(-4.81173 - 8.76625i) q^{10} -9.57856i q^{11} +(17.7156 + 12.0021i) q^{12} -6.75472i q^{13} +(-1.67995 - 0.515490i) q^{14} +(-19.1827 - 18.6409i) q^{15} +(5.93303 + 14.8593i) q^{16} +24.8661i q^{17} +(37.5105 + 11.5100i) q^{18} -4.35890i q^{19} +(-4.05695 - 19.5842i) q^{20} -4.70033 q^{21} +(5.61972 - 18.3143i) q^{22} -19.9805 q^{23} +(26.8309 + 33.3418i) q^{24} +(0.716019 + 24.9897i) q^{25} +(3.96297 - 12.9151i) q^{26} +56.8042 q^{27} +(-2.90965 - 1.97125i) q^{28} -13.7220 q^{29} +(-25.7409 - 46.8961i) q^{30} +37.3799i q^{31} +(2.62610 + 31.8921i) q^{32} -51.2416i q^{33} +(-14.5889 + 47.5442i) q^{34} +(3.15060 + 3.06162i) q^{35} +(64.9677 + 44.0147i) q^{36} -35.9721i q^{37} +(2.55735 - 8.33426i) q^{38} -36.1351i q^{39} +(3.73306 - 39.8254i) q^{40} -7.57460 q^{41} +(-8.98709 - 2.75767i) q^{42} -27.7577 q^{43} +(21.4899 - 31.7201i) q^{44} +(-70.3477 - 68.3609i) q^{45} +(-38.2029 - 11.7225i) q^{46} -62.6391 q^{47} +(31.7394 + 79.4916i) q^{48} -48.2280 q^{49} +(-13.2924 + 48.2008i) q^{50} +133.024i q^{51} +(15.1545 - 22.3687i) q^{52} -60.3963i q^{53} +(108.610 + 33.3269i) q^{54} +(-33.3768 + 34.3469i) q^{55} +(-4.40676 - 5.47613i) q^{56} -23.3184i q^{57} +(-26.2365 - 8.05063i) q^{58} -18.4875i q^{59} +(-21.7031 - 104.768i) q^{60} -5.91165 q^{61} +(-21.9307 + 71.4709i) q^{62} -17.2373 q^{63} +(-13.6898 + 62.5187i) q^{64} +(-23.5371 + 24.2211i) q^{65} +(30.0633 - 97.9745i) q^{66} +108.010 q^{67} +(-55.7882 + 82.3459i) q^{68} -106.888 q^{69} +(4.22774 + 7.70230i) q^{70} +30.6097i q^{71} +(98.3956 + 122.273i) q^{72} -90.3567i q^{73} +(21.1047 - 68.7790i) q^{74} +(3.83043 + 133.685i) q^{75} +(9.77938 - 14.4348i) q^{76} +8.41602i q^{77} +(21.2004 - 69.0908i) q^{78} -32.0898i q^{79} +(30.5031 - 73.9565i) q^{80} +127.315 q^{81} +(-14.4827 - 4.44400i) q^{82} +151.434 q^{83} +(-15.5655 - 10.5454i) q^{84} +(86.6468 - 89.1650i) q^{85} +(-53.0729 - 16.2853i) q^{86} -73.4072 q^{87} +(59.6991 - 48.0411i) q^{88} -24.5129 q^{89} +(-94.3984 - 171.980i) q^{90} +5.93490i q^{91} +(-66.1668 - 44.8271i) q^{92} +199.968i q^{93} +(-119.767 - 36.7502i) q^{94} +(-15.1887 + 15.6302i) q^{95} +(14.0486 + 170.610i) q^{96} +32.0221i q^{97} +(-92.2125 - 28.2952i) q^{98} -187.916i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 108 q + 4 q^{5} + 324 q^{9} - 8 q^{10} + 8 q^{14} - 104 q^{16} - 16 q^{21} - 8 q^{24} - 76 q^{25} + 80 q^{26} - 88 q^{29} - 140 q^{30} - 88 q^{34} - 256 q^{36} + 44 q^{40} - 200 q^{41} - 8 q^{44} + 108 q^{45}+ \cdots + 720 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.91201 + 0.586697i 0.956006 + 0.293349i
\(3\) 5.34961 1.78320 0.891602 − 0.452819i \(-0.149582\pi\)
0.891602 + 0.452819i \(0.149582\pi\)
\(4\) 3.31157 + 2.24354i 0.827893 + 0.560886i
\(5\) −3.58581 − 3.48454i −0.717161 − 0.696907i
\(6\) 10.2285 + 3.13860i 1.70475 + 0.523101i
\(7\) −0.878631 −0.125519 −0.0627593 − 0.998029i \(-0.519990\pi\)
−0.0627593 + 0.998029i \(0.519990\pi\)
\(8\) 5.01548 + 6.23257i 0.626935 + 0.779071i
\(9\) 19.6184 2.17982
\(10\) −4.81173 − 8.76625i −0.481173 − 0.876625i
\(11\) − 9.57856i − 0.870779i −0.900242 − 0.435389i \(-0.856611\pi\)
0.900242 − 0.435389i \(-0.143389\pi\)
\(12\) 17.7156 + 12.0021i 1.47630 + 1.00017i
\(13\) − 6.75472i − 0.519594i −0.965663 − 0.259797i \(-0.916344\pi\)
0.965663 − 0.259797i \(-0.0836556\pi\)
\(14\) −1.67995 − 0.515490i −0.119997 − 0.0368207i
\(15\) −19.1827 − 18.6409i −1.27885 − 1.24273i
\(16\) 5.93303 + 14.8593i 0.370814 + 0.928707i
\(17\) 24.8661i 1.46271i 0.681996 + 0.731356i \(0.261112\pi\)
−0.681996 + 0.731356i \(0.738888\pi\)
\(18\) 37.5105 + 11.5100i 2.08392 + 0.639447i
\(19\) − 4.35890i − 0.229416i
\(20\) −4.05695 − 19.5842i −0.202848 − 0.979210i
\(21\) −4.70033 −0.223825
\(22\) 5.61972 − 18.3143i 0.255442 − 0.832469i
\(23\) −19.9805 −0.868716 −0.434358 − 0.900740i \(-0.643025\pi\)
−0.434358 + 0.900740i \(0.643025\pi\)
\(24\) 26.8309 + 33.3418i 1.11795 + 1.38924i
\(25\) 0.716019 + 24.9897i 0.0286408 + 0.999590i
\(26\) 3.96297 − 12.9151i 0.152422 − 0.496735i
\(27\) 56.8042 2.10386
\(28\) −2.90965 − 1.97125i −0.103916 − 0.0704016i
\(29\) −13.7220 −0.473171 −0.236585 − 0.971611i \(-0.576028\pi\)
−0.236585 + 0.971611i \(0.576028\pi\)
\(30\) −25.7409 − 46.8961i −0.858031 − 1.56320i
\(31\) 37.3799i 1.20580i 0.797815 + 0.602902i \(0.205989\pi\)
−0.797815 + 0.602902i \(0.794011\pi\)
\(32\) 2.62610 + 31.8921i 0.0820657 + 0.996627i
\(33\) − 51.2416i − 1.55278i
\(34\) −14.5889 + 47.5442i −0.429084 + 1.39836i
\(35\) 3.15060 + 3.06162i 0.0900171 + 0.0874748i
\(36\) 64.9677 + 44.0147i 1.80466 + 1.22263i
\(37\) − 35.9721i − 0.972218i −0.873898 − 0.486109i \(-0.838416\pi\)
0.873898 − 0.486109i \(-0.161584\pi\)
\(38\) 2.55735 − 8.33426i 0.0672988 − 0.219323i
\(39\) − 36.1351i − 0.926542i
\(40\) 3.73306 − 39.8254i 0.0933264 − 0.995636i
\(41\) −7.57460 −0.184746 −0.0923731 − 0.995724i \(-0.529445\pi\)
−0.0923731 + 0.995724i \(0.529445\pi\)
\(42\) −8.98709 − 2.75767i −0.213978 − 0.0656589i
\(43\) −27.7577 −0.645527 −0.322763 − 0.946480i \(-0.604612\pi\)
−0.322763 + 0.946480i \(0.604612\pi\)
\(44\) 21.4899 − 31.7201i 0.488407 − 0.720912i
\(45\) −70.3477 − 68.3609i −1.56328 − 1.51913i
\(46\) −38.2029 − 11.7225i −0.830498 − 0.254837i
\(47\) −62.6391 −1.33275 −0.666373 − 0.745619i \(-0.732154\pi\)
−0.666373 + 0.745619i \(0.732154\pi\)
\(48\) 31.7394 + 79.4916i 0.661238 + 1.65607i
\(49\) −48.2280 −0.984245
\(50\) −13.2924 + 48.2008i −0.265848 + 0.964015i
\(51\) 133.024i 2.60831i
\(52\) 15.1545 − 22.3687i 0.291433 − 0.430168i
\(53\) − 60.3963i − 1.13955i −0.821800 − 0.569777i \(-0.807030\pi\)
0.821800 − 0.569777i \(-0.192970\pi\)
\(54\) 108.610 + 33.3269i 2.01130 + 0.617164i
\(55\) −33.3768 + 34.3469i −0.606852 + 0.624489i
\(56\) −4.40676 − 5.47613i −0.0786921 − 0.0977880i
\(57\) − 23.3184i − 0.409095i
\(58\) −26.2365 − 8.05063i −0.452354 − 0.138804i
\(59\) − 18.4875i − 0.313348i −0.987650 − 0.156674i \(-0.949923\pi\)
0.987650 − 0.156674i \(-0.0500772\pi\)
\(60\) −21.7031 − 104.768i −0.361719 − 1.74613i
\(61\) −5.91165 −0.0969123 −0.0484562 − 0.998825i \(-0.515430\pi\)
−0.0484562 + 0.998825i \(0.515430\pi\)
\(62\) −21.9307 + 71.4709i −0.353721 + 1.15276i
\(63\) −17.2373 −0.273608
\(64\) −13.6898 + 62.5187i −0.213904 + 0.976855i
\(65\) −23.5371 + 24.2211i −0.362109 + 0.372633i
\(66\) 30.0633 − 97.9745i 0.455505 − 1.48446i
\(67\) 108.010 1.61208 0.806042 − 0.591858i \(-0.201605\pi\)
0.806042 + 0.591858i \(0.201605\pi\)
\(68\) −55.7882 + 82.3459i −0.820414 + 1.21097i
\(69\) −106.888 −1.54910
\(70\) 4.22774 + 7.70230i 0.0603962 + 0.110033i
\(71\) 30.6097i 0.431122i 0.976490 + 0.215561i \(0.0691581\pi\)
−0.976490 + 0.215561i \(0.930842\pi\)
\(72\) 98.3956 + 122.273i 1.36661 + 1.69823i
\(73\) − 90.3567i − 1.23776i −0.785484 − 0.618881i \(-0.787586\pi\)
0.785484 − 0.618881i \(-0.212414\pi\)
\(74\) 21.1047 − 68.7790i 0.285199 − 0.929446i
\(75\) 3.83043 + 133.685i 0.0510724 + 1.78247i
\(76\) 9.77938 − 14.4348i 0.128676 − 0.189932i
\(77\) 8.41602i 0.109299i
\(78\) 21.2004 − 69.0908i 0.271800 − 0.885779i
\(79\) − 32.0898i − 0.406200i −0.979158 − 0.203100i \(-0.934898\pi\)
0.979158 − 0.203100i \(-0.0651017\pi\)
\(80\) 30.5031 − 73.9565i 0.381289 − 0.924456i
\(81\) 127.315 1.57179
\(82\) −14.4827 − 4.44400i −0.176618 − 0.0541951i
\(83\) 151.434 1.82450 0.912251 − 0.409631i \(-0.134343\pi\)
0.912251 + 0.409631i \(0.134343\pi\)
\(84\) −15.5655 − 10.5454i −0.185304 − 0.125541i
\(85\) 86.6468 − 89.1650i 1.01937 − 1.04900i
\(86\) −53.0729 − 16.2853i −0.617127 − 0.189364i
\(87\) −73.4072 −0.843760
\(88\) 59.6991 − 48.0411i 0.678398 − 0.545922i
\(89\) −24.5129 −0.275426 −0.137713 − 0.990472i \(-0.543975\pi\)
−0.137713 + 0.990472i \(0.543975\pi\)
\(90\) −94.3984 − 171.980i −1.04887 − 1.91088i
\(91\) 5.93490i 0.0652187i
\(92\) −66.1668 − 44.8271i −0.719204 − 0.487251i
\(93\) 199.968i 2.15020i
\(94\) −119.767 − 36.7502i −1.27411 − 0.390959i
\(95\) −15.1887 + 15.6302i −0.159881 + 0.164528i
\(96\) 14.0486 + 170.610i 0.146340 + 1.77719i
\(97\) 32.0221i 0.330125i 0.986283 + 0.165062i \(0.0527825\pi\)
−0.986283 + 0.165062i \(0.947217\pi\)
\(98\) −92.2125 − 28.2952i −0.940944 − 0.288727i
\(99\) − 187.916i − 1.89814i
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.h.a.39.100 yes 108
4.3 odd 2 inner 380.3.h.a.39.10 yes 108
5.4 even 2 inner 380.3.h.a.39.9 ✓ 108
20.19 odd 2 inner 380.3.h.a.39.99 yes 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.h.a.39.9 ✓ 108 5.4 even 2 inner
380.3.h.a.39.10 yes 108 4.3 odd 2 inner
380.3.h.a.39.99 yes 108 20.19 odd 2 inner
380.3.h.a.39.100 yes 108 1.1 even 1 trivial