↑

Properties

Label 380.2.s.a.179.4
Level $380$
Weight $2$
Character 380.179
Analytic conductor $3.034$
Analytic rank $0$
Dimension $112$
Inner twists $8$

Related objects

Downloads

Learn more

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 179.4
Character \(\chi\) \(=\) 380.179
Dual form 380.2.s.a.259.4

$q$-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+(-1.37100 + 0.346927i) q^{2} +(-1.86434 - 1.07638i) q^{3} +(1.75928 - 0.951274i) q^{4} +(1.15834 + 1.91265i) q^{5} +(2.92944 + 0.828924i) q^{6} -1.21448 q^{7} +(-2.08195 + 1.91454i) q^{8} +(0.817179 + 1.41540i) q^{9} +(-2.25164 - 2.22038i) q^{10} -0.196505i q^{11} +(-4.30384 - 0.120153i) q^{12} +(-1.26045 - 2.18315i) q^{13} +(1.66505 - 0.421337i) q^{14} +(-0.100814 - 4.81265i) q^{15} +(2.19015 - 3.34712i) q^{16} +(0.977929 + 0.564608i) q^{17} +(-1.61139 - 1.65701i) q^{18} +(-2.27203 + 3.71993i) q^{19} +(3.85731 + 2.26299i) q^{20} +(2.26421 + 1.30724i) q^{21} +(0.0681731 + 0.269409i) q^{22} +(1.98038 + 3.43012i) q^{23} +(5.94224 - 1.32839i) q^{24} +(-2.31647 + 4.43102i) q^{25} +(2.48547 + 2.55582i) q^{26} +2.93989i q^{27} +(-2.13662 + 1.15531i) q^{28} +(-8.63983 + 4.98821i) q^{29} +(1.80786 + 6.56317i) q^{30} -6.48089 q^{31} +(-1.84149 + 5.34873i) q^{32} +(-0.211514 + 0.366353i) q^{33} +(-1.53662 - 0.434807i) q^{34} +(-1.40679 - 2.32288i) q^{35} +(2.78408 + 1.71272i) q^{36} +0.366677 q^{37} +(1.82441 - 5.88825i) q^{38} +5.42686i q^{39} +(-6.07347 - 1.76436i) q^{40} +(4.32289 + 2.49582i) q^{41} +(-3.55775 - 1.00671i) q^{42} +(-0.770759 + 1.33499i) q^{43} +(-0.186931 - 0.345709i) q^{44} +(-1.76058 + 3.20250i) q^{45} +(-3.90510 - 4.01565i) q^{46} +(4.22465 + 7.31730i) q^{47} +(-7.68596 + 3.88275i) q^{48} -5.52503 q^{49} +(1.63865 - 6.87858i) q^{50} +(-1.21546 - 2.10524i) q^{51} +(-4.29426 - 2.64176i) q^{52} +(2.10738 + 3.65009i) q^{53} +(-1.01993 - 4.03059i) q^{54} +(0.375847 - 0.227621i) q^{55} +(2.52850 - 2.32517i) q^{56} +(8.23989 - 4.48965i) q^{57} +(10.1147 - 9.83622i) q^{58} +(1.12383 - 1.94654i) q^{59} +(-4.75551 - 8.37092i) q^{60} +(5.25927 + 9.10933i) q^{61} +(8.88530 - 2.24840i) q^{62} +(-0.992449 - 1.71897i) q^{63} +(0.669068 - 7.97197i) q^{64} +(2.71559 - 4.93964i) q^{65} +(0.162888 - 0.575650i) q^{66} +(-7.42296 + 4.28565i) q^{67} +(2.25755 + 0.0630256i) q^{68} -8.52655i q^{69} +(2.73458 + 2.69662i) q^{70} +(4.08269 - 7.07143i) q^{71} +(-4.41116 - 1.38227i) q^{72} +(-12.0471 - 6.95541i) q^{73} +(-0.502714 + 0.127210i) q^{74} +(9.08815 - 5.76753i) q^{75} +(-0.458474 + 8.70573i) q^{76} +0.238652i q^{77} +(-1.88273 - 7.44023i) q^{78} +(-0.888103 + 1.53824i) q^{79} +(8.93883 + 0.311880i) q^{80} +(5.61597 - 9.72715i) q^{81} +(-6.79256 - 1.92205i) q^{82} -11.2495 q^{83} +(5.22693 + 0.145924i) q^{84} +(0.0528813 + 2.52445i) q^{85} +(0.593565 - 2.09767i) q^{86} +21.4768 q^{87} +(0.376218 + 0.409115i) q^{88} +(-0.585245 + 0.337891i) q^{89} +(1.30273 - 5.00142i) q^{90} +(1.53079 + 2.65140i) q^{91} +(6.74703 + 4.15066i) q^{92} +(12.0826 + 6.97589i) q^{93} +(-8.33056 - 8.56638i) q^{94} +(-9.74673 - 0.0366462i) q^{95} +(9.19043 - 7.98971i) q^{96} +(6.45572 - 11.1816i) q^{97} +(7.57482 - 1.91678i) q^{98} +(0.278133 - 0.160580i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.37100 + 0.346927i −0.969444 + 0.245315i
\(3\) −1.86434 − 1.07638i −1.07638 − 0.621447i −0.146461 − 0.989216i \(-0.546788\pi\)
−0.929917 + 0.367769i \(0.880122\pi\)
\(4\) 1.75928 − 0.951274i 0.879642 − 0.475637i
\(5\) 1.15834 + 1.91265i 0.518028 + 0.855364i
\(6\) 2.92944 + 0.828924i 1.19594 + 0.338407i
\(7\) −1.21448 −0.459031 −0.229515 − 0.973305i \(-0.573714\pi\)
−0.229515 + 0.973305i \(0.573714\pi\)
\(8\) −2.08195 + 1.91454i −0.736082 + 0.676892i
\(9\) 0.817179 + 1.41540i 0.272393 + 0.471799i
\(10\) −2.25164 − 2.22038i −0.712032 − 0.702147i
\(11\) − 0.196505i − 0.0592486i −0.999561 − 0.0296243i \(-0.990569\pi\)
0.999561 − 0.0296243i \(-0.00943109\pi\)
\(12\) −4.30384 − 0.120153i −1.24241 − 0.0346852i
\(13\) −1.26045 − 2.18315i −0.349585 − 0.605498i 0.636591 − 0.771202i \(-0.280344\pi\)
−0.986176 + 0.165703i \(0.947011\pi\)
\(14\) 1.66505 − 0.421337i 0.445004 − 0.112607i
\(15\) −0.100814 − 4.81265i −0.0260300 − 1.24262i
\(16\) 2.19015 − 3.34712i 0.547538 − 0.836781i
\(17\) 0.977929 + 0.564608i 0.237183 + 0.136937i 0.613881 − 0.789398i \(-0.289607\pi\)
−0.376699 + 0.926336i \(0.622941\pi\)
\(18\) −1.61139 − 1.65701i −0.379809 − 0.390560i
\(19\) −2.27203 + 3.71993i −0.521240 + 0.853410i
\(20\) 3.85731 + 2.26299i 0.862522 + 0.506020i
\(21\) 2.26421 + 1.30724i 0.494091 + 0.285263i
\(22\) 0.0681731 + 0.269409i 0.0145346 + 0.0574382i
\(23\) 1.98038 + 3.43012i 0.412938 + 0.715229i 0.995210 − 0.0977650i \(-0.0311694\pi\)
−0.582272 + 0.812994i \(0.697836\pi\)
\(24\) 5.94224 − 1.32839i 1.21296 − 0.271156i
\(25\) −2.31647 + 4.43102i −0.463295 + 0.886204i
\(26\) 2.48547 + 2.55582i 0.487440 + 0.501238i
\(27\) 2.93989i 0.565783i
\(28\) −2.13662 + 1.15531i −0.403783 + 0.218332i
\(29\) −8.63983 + 4.98821i −1.60438 + 0.926287i −0.613778 + 0.789479i \(0.710351\pi\)
−0.990598 + 0.136808i \(0.956316\pi\)
\(30\) 1.80786 + 6.56317i 0.330068 + 1.19827i
\(31\) −6.48089 −1.16400 −0.582001 − 0.813188i \(-0.697730\pi\)
−0.582001 + 0.813188i \(0.697730\pi\)
\(32\) −1.84149 + 5.34873i −0.325533 + 0.945531i
\(33\) −0.211514 + 0.366353i −0.0368199 + 0.0637739i
\(34\) −1.53662 − 0.434807i −0.263528 − 0.0745688i
\(35\) −1.40679 − 2.32288i −0.237791 − 0.392638i
\(36\) 2.78408 + 1.71272i 0.464013 + 0.285453i
\(37\) 0.366677 0.0602813 0.0301406 − 0.999546i \(-0.490404\pi\)
0.0301406 + 0.999546i \(0.490404\pi\)
\(38\) 1.82441 − 5.88825i 0.295959 − 0.955201i
\(39\) 5.42686i 0.868993i
\(40\) −6.07347 − 1.76436i −0.960300 − 0.278969i
\(41\) 4.32289 + 2.49582i 0.675123 + 0.389782i 0.798015 − 0.602638i \(-0.205884\pi\)
−0.122892 + 0.992420i \(0.539217\pi\)
\(42\) −3.55775 − 1.00671i −0.548972 − 0.155339i
\(43\) −0.770759 + 1.33499i −0.117540 + 0.203585i −0.918792 − 0.394742i \(-0.870834\pi\)
0.801252 + 0.598326i \(0.204167\pi\)
\(44\) −0.186931 − 0.345709i −0.0281809 − 0.0521176i
\(45\) −1.76058 + 3.20250i −0.262452 + 0.477400i
\(46\) −3.90510 − 4.01565i −0.575776 − 0.592075i
\(47\) 4.22465 + 7.31730i 0.616228 + 1.06734i 0.990168 + 0.139885i \(0.0446733\pi\)
−0.373940 + 0.927453i \(0.621993\pi\)
\(48\) −7.68596 + 3.88275i −1.10937 + 0.560426i
\(49\) −5.52503 −0.789291
\(50\) 1.63865 − 6.87858i 0.231739 − 0.972778i
\(51\) −1.21546 − 2.10524i −0.170199 − 0.294793i
\(52\) −4.29426 − 2.64176i −0.595507 − 0.366346i
\(53\) 2.10738 + 3.65009i 0.289471 + 0.501379i 0.973684 − 0.227904i \(-0.0731872\pi\)
−0.684212 + 0.729283i \(0.739854\pi\)
\(54\) −1.01993 − 4.03059i −0.138795 − 0.548494i
\(55\) 0.375847 − 0.227621i 0.0506791 − 0.0306924i
\(56\) 2.52850 − 2.32517i 0.337884 − 0.310714i
\(57\) 8.23989 − 4.48965i 1.09140 − 0.594669i
\(58\) 10.1147 − 9.83622i 1.32812 − 1.29156i
\(59\) 1.12383 − 1.94654i 0.146311 − 0.253417i −0.783551 − 0.621328i \(-0.786593\pi\)
0.929861 + 0.367911i \(0.119927\pi\)
\(60\) −4.75551 − 8.37092i −0.613934 − 1.08068i
\(61\) 5.25927 + 9.10933i 0.673381 + 1.16633i 0.976939 + 0.213517i \(0.0684919\pi\)
−0.303558 + 0.952813i \(0.598175\pi\)
\(62\) 8.88530 − 2.24840i 1.12843 − 0.285547i
\(63\) −0.992449 − 1.71897i −0.125037 − 0.216570i
\(64\) 0.669068 − 7.97197i 0.0836335 − 0.996497i
\(65\) 2.71559 − 4.93964i 0.336827 − 0.612687i
\(66\) 0.162888 − 0.575650i 0.0200501 − 0.0708577i
\(67\) −7.42296 + 4.28565i −0.906858 + 0.523575i −0.879419 − 0.476049i \(-0.842069\pi\)
−0.0274394 + 0.999623i \(0.508735\pi\)
\(68\) 2.25755 + 0.0630256i 0.273768 + 0.00764298i
\(69\) − 8.52655i − 1.02648i
\(70\) 2.73458 + 2.69662i 0.326845 + 0.322307i
\(71\) 4.08269 − 7.07143i 0.484526 − 0.839224i −0.515316 − 0.857000i \(-0.672325\pi\)
0.999842 + 0.0177762i \(0.00565863\pi\)
\(72\) −4.41116 − 1.38227i −0.519861 − 0.162902i
\(73\) −12.0471 − 6.95541i −1.41001 − 0.814069i −0.414621 − 0.909994i \(-0.636086\pi\)
−0.995389 + 0.0959247i \(0.969419\pi\)
\(74\) −0.502714 + 0.127210i −0.0584393 + 0.0147879i
\(75\) 9.08815 − 5.76753i 1.04941 − 0.665977i
\(76\) −0.458474 + 8.70573i −0.0525906 + 0.998616i
\(77\) 0.238652i 0.0271970i
\(78\) −1.88273 − 7.44023i −0.213177 − 0.842440i
\(79\) −0.888103 + 1.53824i −0.0999194 + 0.173065i −0.911651 − 0.410965i \(-0.865192\pi\)
0.811732 + 0.584030i \(0.198525\pi\)
\(80\) 8.93883 + 0.311880i 0.999392 + 0.0348692i
\(81\) 5.61597 − 9.72715i 0.623997 − 1.08079i
\(82\) −6.79256 − 1.92205i −0.750112 − 0.212254i
\(83\) −11.2495 −1.23479 −0.617396 − 0.786652i \(-0.711813\pi\)
−0.617396 + 0.786652i \(0.711813\pi\)
\(84\) 5.22693 + 0.145924i 0.570305 + 0.0159216i
\(85\) 0.0528813 + 2.52445i 0.00573579 + 0.273815i
\(86\) 0.593565 − 2.09767i 0.0640058 − 0.226198i
\(87\) 21.4768 2.30255
\(88\) 0.376218 + 0.409115i 0.0401049 + 0.0436119i
\(89\) −0.585245 + 0.337891i −0.0620359 + 0.0358164i −0.530697 − 0.847562i \(-0.678070\pi\)
0.468661 + 0.883378i \(0.344736\pi\)
\(90\) 1.30273 − 5.00142i 0.137320 − 0.527196i
\(91\) 1.53079 + 2.65140i 0.160470 + 0.277942i
\(92\) 6.74703 + 4.15066i 0.703427 + 0.432737i
\(93\) 12.0826 + 6.97589i 1.25291 + 0.723366i
\(94\) −8.33056 − 8.56638i −0.859232 − 0.883554i
\(95\) −9.74673 − 0.0366462i −0.999993 − 0.00375983i
\(96\) 9.19043 − 7.98971i 0.937994 − 0.815447i
\(97\) 6.45572 − 11.1816i 0.655479 − 1.13532i −0.326294 − 0.945268i \(-0.605800\pi\)
0.981773 − 0.190055i \(-0.0608667\pi\)
\(98\) 7.57482 − 1.91678i 0.765173 − 0.193625i
\(99\) 0.278133 − 0.160580i 0.0279534 − 0.0161389i
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.179.4 ✓ 112
4.3 odd 2 inner 380.2.s.a.179.34 yes 112
5.4 even 2 inner 380.2.s.a.179.53 yes 112
19.12 odd 6 inner 380.2.s.a.259.23 yes 112
20.19 odd 2 inner 380.2.s.a.179.23 yes 112
76.31 even 6 inner 380.2.s.a.259.53 yes 112
95.69 odd 6 inner 380.2.s.a.259.34 yes 112
380.259 even 6 inner 380.2.s.a.259.4 yes 112
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.s.a.179.4 ✓ 112 1.1 even 1 trivial
380.2.s.a.179.23 yes 112 20.19 odd 2 inner
380.2.s.a.179.34 yes 112 4.3 odd 2 inner
380.2.s.a.179.53 yes 112 5.4 even 2 inner
380.2.s.a.259.4 yes 112 380.259 even 6 inner
380.2.s.a.259.23 yes 112 19.12 odd 6 inner
380.2.s.a.259.34 yes 112 95.69 odd 6 inner
380.2.s.a.259.53 yes 112 76.31 even 6 inner