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Properties

Label 380.3.h.a.39.9
Level $380$
Weight $3$
Character 380.39
Analytic conductor $10.354$
Analytic rank $0$
Dimension $108$
Inner twists $4$

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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] = [380,3,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 39.9
Character \(\chi\) \(=\) 380.39
Dual form 380.3.h.a.39.10

$q$-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} +(8.90047 - 4.55869i) 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} +(-19.6924 + 3.49438i) 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} +(-47.6141 + 24.3872i) 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} +(39.7022 + 4.87216i) 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} +(-16.0304 + 47.3606i) 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} +(105.347 - 18.6936i) 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} +(7.82022 - 4.00540i) 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} +(-73.0525 - 32.6088i) 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} +(174.613 - 89.4341i) 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 \(-0.850418\pi\)
−0.891602 + 0.452819i \(0.850418\pi\)
\(4\) 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 \(-0.480010\pi\)
0.0627593 + 0.998029i \(0.480010\pi\)
\(8\) −5.01548 − 6.23257i −0.626935 − 0.779071i
\(9\) 19.6184 2.17982
\(10\) 8.90047 − 4.55869i 0.890047 − 0.455869i
\(11\) − 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 \(0.0836556\pi\)
−0.965663 + 0.259797i \(0.916344\pi\)
\(14\) −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 \(-0.738888\pi\)
0.681996 − 0.731356i \(-0.261112\pi\)
\(18\) −37.5105 − 11.5100i −2.08392 − 0.639447i
\(19\) − 4.35890i − 0.229416i
\(20\) −19.6924 + 3.49438i −0.984618 + 0.174719i
\(21\) −4.70033 −0.223825
\(22\) −5.61972 + 18.3143i −0.255442 + 0.832469i
\(23\) 19.9805 0.868716 0.434358 − 0.900740i \(-0.356975\pi\)
0.434358 + 0.900740i \(0.356975\pi\)
\(24\) 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 \(0.576028\pi\)
\(30\) −47.6141 + 24.3872i −1.58714 + 0.812908i
\(31\) 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 \(0.161584\pi\)
−0.873898 + 0.486109i \(0.838416\pi\)
\(38\) −2.55735 + 8.33426i −0.0672988 + 0.219323i
\(39\) − 36.1351i − 0.926542i
\(40\) 39.7022 + 4.87216i 0.992554 + 0.121804i
\(41\) −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 \(-0.395388\pi\)
0.322763 + 0.946480i \(0.395388\pi\)
\(44\) 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 \(-0.267846\pi\)
0.666373 + 0.745619i \(0.267846\pi\)
\(48\) −31.7394 − 79.4916i −0.661238 − 1.65607i
\(49\) −48.2280 −0.984245
\(50\) −16.0304 + 47.3606i −0.320609 + 0.947212i
\(51\) 133.024i 2.60831i
\(52\) −15.1545 + 22.3687i −0.291433 + 0.430168i
\(53\) 60.3963i 1.13955i 0.821800 + 0.569777i \(0.192970\pi\)
−0.821800 + 0.569777i \(0.807030\pi\)
\(54\) 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 \(-0.0500772\pi\)
\(60\) 105.347 − 18.6936i 1.75578 − 0.311560i
\(61\) −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 \(-0.798395\pi\)
−0.806042 + 0.591858i \(0.798395\pi\)
\(68\) 55.7882 − 82.3459i 0.820414 − 1.21097i
\(69\) −106.888 −1.54910
\(70\) 7.82022 − 4.00540i 0.111717 − 0.0572200i
\(71\) 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 \(0.212414\pi\)
−0.785484 + 0.618881i \(0.787586\pi\)
\(74\) 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 \(-0.0651017\pi\)
\(80\) −73.0525 − 32.6088i −0.913156 − 0.407610i
\(81\) 127.315 1.57179
\(82\) 14.4827 + 4.44400i 0.176618 + 0.0541951i
\(83\) −151.434 −1.82450 −0.912251 − 0.409631i \(-0.865657\pi\)
−0.912251 + 0.409631i \(0.865657\pi\)
\(84\) −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 \(0.543975\pi\)
\(90\) 174.613 − 89.4341i 1.94014 − 0.993712i
\(91\) 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 \(-0.947217\pi\)
0.986283 − 0.165062i \(-0.0527825\pi\)
\(98\) 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.39.9 ✓ 108
4.3 odd 2 inner 380.3.h.a.39.99 yes 108
5.4 even 2 inner 380.3.h.a.39.100 yes 108
20.19 odd 2 inner 380.3.h.a.39.10 yes 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.h.a.39.9 ✓ 108 1.1 even 1 trivial
380.3.h.a.39.10 yes 108 20.19 odd 2 inner
380.3.h.a.39.99 yes 108 4.3 odd 2 inner
380.3.h.a.39.100 yes 108 5.4 even 2 inner