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Properties

Label 380.2.s.a.179.16
Level $380$
Weight $2$
Character 380.179
Analytic conductor $3.034$
Analytic rank $0$
Dimension $112$
Inner twists $8$

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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] = [380,2,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 179.16
Character \(\chi\) \(=\) 380.179
Dual form 380.2.s.a.259.16

$q$-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+(-0.982532 - 1.01717i) q^{2} +(-1.99347 - 1.15093i) q^{3} +(-0.0692634 + 1.99880i) q^{4} +(-0.174859 - 2.22922i) q^{5} +(0.787956 + 3.15852i) q^{6} +2.46740 q^{7} +(2.10117 - 1.89343i) q^{8} +(1.14928 + 1.99061i) q^{9} +(-2.09569 + 2.36814i) q^{10} -5.51285i q^{11} +(2.43855 - 3.90483i) q^{12} +(-2.19483 - 3.80155i) q^{13} +(-2.42430 - 2.50976i) q^{14} +(-2.21710 + 4.64513i) q^{15} +(-3.99041 - 0.276888i) q^{16} +(1.55609 + 0.898409i) q^{17} +(0.895582 - 3.12484i) q^{18} +(-2.41756 + 3.62704i) q^{19} +(4.46788 - 0.195104i) q^{20} +(-4.91869 - 2.83981i) q^{21} +(-5.60750 + 5.41655i) q^{22} +(2.71762 + 4.70705i) q^{23} +(-6.36782 + 1.35620i) q^{24} +(-4.93885 + 0.779598i) q^{25} +(-1.71033 + 5.96765i) q^{26} +1.61462i q^{27} +(-0.170901 + 4.93185i) q^{28} +(2.31508 - 1.33661i) q^{29} +(6.90325 - 2.30882i) q^{30} -0.664627 q^{31} +(3.63906 + 4.33097i) q^{32} +(-6.34491 + 10.9897i) q^{33} +(-0.615074 - 2.46552i) q^{34} +(-0.431447 - 5.50039i) q^{35} +(-4.05843 + 2.15930i) q^{36} -7.71655 q^{37} +(6.06464 - 1.10461i) q^{38} +10.1044i q^{39} +(-4.58829 - 4.35289i) q^{40} +(-6.69145 - 3.86331i) q^{41} +(1.94421 + 7.79334i) q^{42} +(4.60729 - 7.98005i) q^{43} +(11.0191 + 0.381839i) q^{44} +(4.23654 - 2.91007i) q^{45} +(2.11772 - 7.38910i) q^{46} +(-2.82660 - 4.89582i) q^{47} +(7.63607 + 5.14464i) q^{48} -0.911921 q^{49} +(5.64556 + 4.25766i) q^{50} +(-2.06801 - 3.58190i) q^{51} +(7.75056 - 4.12371i) q^{52} +(-1.36441 - 2.36322i) q^{53} +(1.64235 - 1.58642i) q^{54} +(-12.2894 + 0.963971i) q^{55} +(5.18443 - 4.67186i) q^{56} +(8.99380 - 4.44794i) q^{57} +(-3.63420 - 1.04156i) q^{58} +(-5.14567 + 8.91256i) q^{59} +(-9.13113 - 4.75328i) q^{60} +(5.81823 + 10.0775i) q^{61} +(0.653017 + 0.676038i) q^{62} +(2.83573 + 4.91163i) q^{63} +(0.829832 - 7.95684i) q^{64} +(-8.09071 + 5.55748i) q^{65} +(17.4124 - 4.34389i) q^{66} +(-7.45986 + 4.30695i) q^{67} +(-1.90352 + 3.04809i) q^{68} -12.5111i q^{69} +(-5.17091 + 5.84316i) q^{70} +(6.06770 - 10.5096i) q^{71} +(6.18391 + 2.00653i) q^{72} +(8.71390 + 5.03097i) q^{73} +(7.58175 + 7.84903i) q^{74} +(10.7427 + 4.13016i) q^{75} +(-7.08227 - 5.08345i) q^{76} -13.6024i q^{77} +(10.2778 - 9.92785i) q^{78} +(0.959826 - 1.66247i) q^{79} +(0.0805140 + 8.94391i) q^{80} +(5.30615 - 9.19053i) q^{81} +(2.64492 + 10.6022i) q^{82} +0.520746 q^{83} +(6.01689 - 9.63479i) q^{84} +(1.73066 - 3.62596i) q^{85} +(-12.6439 + 3.15427i) q^{86} -6.15339 q^{87} +(-10.4382 - 11.5834i) q^{88} +(3.56937 - 2.06078i) q^{89} +(-7.12257 - 1.45004i) q^{90} +(-5.41552 - 9.37996i) q^{91} +(-9.59668 + 5.10594i) q^{92} +(1.32491 + 0.764939i) q^{93} +(-2.20265 + 7.68543i) q^{94} +(8.50819 + 4.75506i) q^{95} +(-2.26971 - 12.8219i) q^{96} +(1.48746 - 2.57636i) q^{97} +(0.895991 + 0.927577i) q^{98} +(10.9739 - 6.33580i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.982532 − 1.01717i −0.694755 − 0.719247i
\(3\) −1.99347 − 1.15093i −1.15093 − 0.664490i −0.201816 − 0.979423i \(-0.564684\pi\)
−0.949114 + 0.314934i \(0.898018\pi\)
\(4\) −0.0692634 + 1.99880i −0.0346317 + 0.999400i
\(5\) −0.174859 − 2.22922i −0.0781992 − 0.996938i
\(6\) 0.787956 + 3.15852i 0.321682 + 1.28946i
\(7\) 2.46740 0.932591 0.466295 − 0.884629i \(-0.345588\pi\)
0.466295 + 0.884629i \(0.345588\pi\)
\(8\) 2.10117 − 1.89343i 0.742876 − 0.669429i
\(9\) 1.14928 + 1.99061i 0.383093 + 0.663536i
\(10\) −2.09569 + 2.36814i −0.662715 + 0.748872i
\(11\) − 5.51285i − 1.66219i −0.556132 − 0.831094i \(-0.687715\pi\)
0.556132 − 0.831094i \(-0.312285\pi\)
\(12\) 2.43855 − 3.90483i 0.703950 − 1.12723i
\(13\) −2.19483 − 3.80155i −0.608735 − 1.05436i −0.991449 − 0.130493i \(-0.958344\pi\)
0.382714 − 0.923867i \(-0.374989\pi\)
\(14\) −2.42430 − 2.50976i −0.647922 − 0.670763i
\(15\) −2.21710 + 4.64513i −0.572453 + 1.19937i
\(16\) −3.99041 − 0.276888i −0.997601 − 0.0692219i
\(17\) 1.55609 + 0.898409i 0.377407 + 0.217896i 0.676690 − 0.736268i \(-0.263414\pi\)
−0.299282 + 0.954165i \(0.596747\pi\)
\(18\) 0.895582 − 3.12484i 0.211091 − 0.736533i
\(19\) −2.41756 + 3.62704i −0.554627 + 0.832099i
\(20\) 4.46788 − 0.195104i 0.999048 − 0.0436267i
\(21\) −4.91869 − 2.83981i −1.07335 − 0.619697i
\(22\) −5.60750 + 5.41655i −1.19552 + 1.15481i
\(23\) 2.71762 + 4.70705i 0.566662 + 0.981487i 0.996893 + 0.0787684i \(0.0250988\pi\)
−0.430231 + 0.902719i \(0.641568\pi\)
\(24\) −6.36782 + 1.35620i −1.29983 + 0.276833i
\(25\) −4.93885 + 0.779598i −0.987770 + 0.155920i
\(26\) −1.71033 + 5.96765i −0.335423 + 1.17035i
\(27\) 1.61462i 0.310735i
\(28\) −0.170901 + 4.93185i −0.0322972 + 0.932031i
\(29\) 2.31508 − 1.33661i 0.429900 − 0.248203i −0.269404 − 0.963027i \(-0.586827\pi\)
0.699304 + 0.714824i \(0.253493\pi\)
\(30\) 6.90325 − 2.30882i 1.26036 − 0.421532i
\(31\) −0.664627 −0.119371 −0.0596853 − 0.998217i \(-0.519010\pi\)
−0.0596853 + 0.998217i \(0.519010\pi\)
\(32\) 3.63906 + 4.33097i 0.643301 + 0.765614i
\(33\) −6.34491 + 10.9897i −1.10451 + 1.91306i
\(34\) −0.615074 − 2.46552i −0.105484 − 0.422833i
\(35\) −0.431447 − 5.50039i −0.0729279 − 0.929735i
\(36\) −4.05843 + 2.15930i −0.676405 + 0.359884i
\(37\) −7.71655 −1.26859 −0.634296 − 0.773090i \(-0.718710\pi\)
−0.634296 + 0.773090i \(0.718710\pi\)
\(38\) 6.06464 − 1.10461i 0.983814 − 0.179191i
\(39\) 10.1044i 1.61799i
\(40\) −4.58829 − 4.35289i −0.725472 − 0.688252i
\(41\) −6.69145 − 3.86331i −1.04503 − 0.603348i −0.123776 − 0.992310i \(-0.539500\pi\)
−0.921254 + 0.388962i \(0.872834\pi\)
\(42\) 1.94421 + 7.79334i 0.299998 + 1.20254i
\(43\) 4.60729 − 7.98005i 0.702605 − 1.21695i −0.264944 − 0.964264i \(-0.585354\pi\)
0.967549 − 0.252683i \(-0.0813130\pi\)
\(44\) 11.0191 + 0.381839i 1.66119 + 0.0575644i
\(45\) 4.23654 − 2.91007i 0.631547 − 0.433808i
\(46\) 2.11772 − 7.38910i 0.312240 − 1.08946i
\(47\) −2.82660 − 4.89582i −0.412303 − 0.714129i 0.582838 − 0.812588i \(-0.301942\pi\)
−0.995141 + 0.0984589i \(0.968609\pi\)
\(48\) 7.63607 + 5.14464i 1.10217 + 0.742565i
\(49\) −0.911921 −0.130274
\(50\) 5.64556 + 4.25766i 0.798402 + 0.602124i
\(51\) −2.06801 − 3.58190i −0.289579 − 0.501566i
\(52\) 7.75056 − 4.12371i 1.07481 − 0.571856i
\(53\) −1.36441 − 2.36322i −0.187416 − 0.324614i 0.756972 − 0.653447i \(-0.226678\pi\)
−0.944388 + 0.328834i \(0.893344\pi\)
\(54\) 1.64235 − 1.58642i 0.223495 − 0.215884i
\(55\) −12.2894 + 0.963971i −1.65710 + 0.129982i
\(56\) 5.18443 − 4.67186i 0.692799 − 0.624304i
\(57\) 8.99380 − 4.44794i 1.19126 − 0.589143i
\(58\) −3.63420 − 1.04156i −0.477194 − 0.136764i
\(59\) −5.14567 + 8.91256i −0.669909 + 1.16032i 0.308020 + 0.951380i \(0.400334\pi\)
−0.977929 + 0.208937i \(0.933000\pi\)
\(60\) −9.13113 − 4.75328i −1.17882 − 0.613646i
\(61\) 5.81823 + 10.0775i 0.744948 + 1.29029i 0.950219 + 0.311583i \(0.100859\pi\)
−0.205271 + 0.978705i \(0.565808\pi\)
\(62\) 0.653017 + 0.676038i 0.0829333 + 0.0858569i
\(63\) 2.83573 + 4.91163i 0.357269 + 0.618808i
\(64\) 0.829832 − 7.95684i 0.103729 − 0.994606i
\(65\) −8.09071 + 5.55748i −1.00353 + 0.689321i
\(66\) 17.4124 − 4.34389i 2.14332 − 0.534696i
\(67\) −7.45986 + 4.30695i −0.911367 + 0.526178i −0.880871 − 0.473357i \(-0.843042\pi\)
−0.0304960 + 0.999535i \(0.509709\pi\)
\(68\) −1.90352 + 3.04809i −0.230836 + 0.369635i
\(69\) − 12.5111i − 1.50616i
\(70\) −5.17091 + 5.84316i −0.618042 + 0.698391i
\(71\) 6.06770 − 10.5096i 0.720103 − 1.24726i −0.240855 − 0.970561i \(-0.577428\pi\)
0.960958 − 0.276694i \(-0.0892389\pi\)
\(72\) 6.18391 + 2.00653i 0.728781 + 0.236471i
\(73\) 8.71390 + 5.03097i 1.01988 + 0.588831i 0.914071 − 0.405555i \(-0.132922\pi\)
0.105814 + 0.994386i \(0.466255\pi\)
\(74\) 7.58175 + 7.84903i 0.881361 + 0.912431i
\(75\) 10.7427 + 4.13016i 1.24046 + 0.476910i
\(76\) −7.08227 − 5.08345i −0.812392 − 0.583111i
\(77\) − 13.6024i − 1.55014i
\(78\) 10.2778 − 9.92785i 1.16374 − 1.12411i
\(79\) 0.959826 − 1.66247i 0.107989 − 0.187042i −0.806967 − 0.590597i \(-0.798892\pi\)
0.914955 + 0.403555i \(0.132226\pi\)
\(80\) 0.0805140 + 8.94391i 0.00900174 + 0.999959i
\(81\) 5.30615 − 9.19053i 0.589573 − 1.02117i
\(82\) 2.64492 + 10.6022i 0.292083 + 1.17081i
\(83\) 0.520746 0.0571593 0.0285796 − 0.999592i \(-0.490902\pi\)
0.0285796 + 0.999592i \(0.490902\pi\)
\(84\) 6.01689 − 9.63479i 0.656497 − 1.05124i
\(85\) 1.73066 − 3.62596i 0.187716 − 0.393291i
\(86\) −12.6439 + 3.15427i −1.36342 + 0.340134i
\(87\) −6.15339 −0.659713
\(88\) −10.4382 − 11.5834i −1.11272 − 1.23480i
\(89\) 3.56937 − 2.06078i 0.378352 − 0.218442i −0.298749 − 0.954332i \(-0.596569\pi\)
0.677101 + 0.735890i \(0.263236\pi\)
\(90\) −7.12257 − 1.45004i −0.750785 − 0.152848i
\(91\) −5.41552 − 9.37996i −0.567701 − 0.983287i
\(92\) −9.59668 + 5.10594i −1.00052 + 0.532331i
\(93\) 1.32491 + 0.764939i 0.137387 + 0.0793205i
\(94\) −2.20265 + 7.68543i −0.227186 + 0.792692i
\(95\) 8.50819 + 4.75506i 0.872922 + 0.487859i
\(96\) −2.26971 − 12.8219i −0.231651 − 1.30863i
\(97\) 1.48746 − 2.57636i 0.151029 − 0.261590i −0.780577 − 0.625060i \(-0.785075\pi\)
0.931606 + 0.363470i \(0.118408\pi\)
\(98\) 0.895991 + 0.927577i 0.0905087 + 0.0936994i
\(99\) 10.9739 − 6.33580i 1.10292 − 0.636772i
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.179.16 yes 112
4.3 odd 2 inner 380.2.s.a.179.54 yes 112
5.4 even 2 inner 380.2.s.a.179.41 yes 112
19.12 odd 6 inner 380.2.s.a.259.3 yes 112
20.19 odd 2 inner 380.2.s.a.179.3 ✓ 112
76.31 even 6 inner 380.2.s.a.259.41 yes 112
95.69 odd 6 inner 380.2.s.a.259.54 yes 112
380.259 even 6 inner 380.2.s.a.259.16 yes 112
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.s.a.179.3 ✓ 112 20.19 odd 2 inner
380.2.s.a.179.16 yes 112 1.1 even 1 trivial
380.2.s.a.179.41 yes 112 5.4 even 2 inner
380.2.s.a.179.54 yes 112 4.3 odd 2 inner
380.2.s.a.259.3 yes 112 19.12 odd 6 inner
380.2.s.a.259.16 yes 112 380.259 even 6 inner
380.2.s.a.259.41 yes 112 76.31 even 6 inner
380.2.s.a.259.54 yes 112 95.69 odd 6 inner