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Properties

Label 380.2.s.a.179.13
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 179.13
Character \(\chi\) \(=\) 380.179
Dual form 380.2.s.a.259.13

$q$-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+(-1.05024 - 0.947103i) q^{2} +(1.43647 + 0.829346i) q^{3} +(0.205992 + 1.98936i) q^{4} +(-2.15560 - 0.594467i) q^{5} +(-0.723156 - 2.23149i) q^{6} +4.79169 q^{7} +(1.66779 - 2.28440i) q^{8} +(-0.124369 - 0.215414i) q^{9} +(1.70087 + 2.66591i) q^{10} +5.02999i q^{11} +(-1.35397 + 3.02850i) q^{12} +(-1.27269 - 2.20437i) q^{13} +(-5.03240 - 4.53822i) q^{14} +(-2.60343 - 2.64167i) q^{15} +(-3.91513 + 0.819585i) q^{16} +(3.06401 + 1.76901i) q^{17} +(-0.0734021 + 0.344026i) q^{18} +(4.35677 + 0.136247i) q^{19} +(0.738576 - 4.41073i) q^{20} +(6.88312 + 3.97397i) q^{21} +(4.76392 - 5.28267i) q^{22} +(0.613547 + 1.06269i) q^{23} +(4.29029 - 1.89829i) q^{24} +(4.29322 + 2.56287i) q^{25} +(-0.751137 + 3.52048i) q^{26} -5.38866i q^{27} +(0.987048 + 9.53241i) q^{28} +(-1.98471 + 1.14588i) q^{29} +(0.232285 + 5.24010i) q^{30} +5.54876 q^{31} +(4.88805 + 2.84728i) q^{32} +(-4.17160 + 7.22543i) q^{33} +(-1.54250 - 4.75981i) q^{34} +(-10.3290 - 2.84850i) q^{35} +(0.402918 - 0.291789i) q^{36} +2.52683 q^{37} +(-4.44660 - 4.26940i) q^{38} -4.22202i q^{39} +(-4.95309 + 3.93280i) q^{40} +(-0.902159 - 0.520862i) q^{41} +(-3.46514 - 10.6926i) q^{42} +(-2.77704 + 4.80998i) q^{43} +(-10.0065 + 1.03614i) q^{44} +(0.140034 + 0.538280i) q^{45} +(0.362112 - 1.69717i) q^{46} +(-0.492773 - 0.853508i) q^{47} +(-6.30369 - 2.06969i) q^{48} +15.9603 q^{49} +(-2.08159 - 6.75773i) q^{50} +(2.93424 + 5.08225i) q^{51} +(4.12313 - 2.98594i) q^{52} +(-4.98156 - 8.62832i) q^{53} +(-5.10362 + 5.65936i) q^{54} +(2.99016 - 10.8426i) q^{55} +(7.99154 - 10.9461i) q^{56} +(6.14537 + 3.80899i) q^{57} +(3.16968 + 0.676289i) q^{58} +(-0.000131383 + 0.000227563i) q^{59} +(4.71896 - 5.72334i) q^{60} +(-2.99940 - 5.19512i) q^{61} +(-5.82751 - 5.25525i) q^{62} +(-0.595939 - 1.03220i) q^{63} +(-2.43694 - 7.61980i) q^{64} +(1.43299 + 5.50832i) q^{65} +(11.2244 - 3.63747i) q^{66} +(-11.8116 + 6.81943i) q^{67} +(-2.88804 + 6.45983i) q^{68} +2.03537i q^{69} +(8.15002 + 12.7742i) q^{70} +(3.41092 - 5.90788i) q^{71} +(-0.699513 - 0.0751568i) q^{72} +(-12.2071 - 7.04778i) q^{73} +(-2.65376 - 2.39316i) q^{74} +(4.04157 + 7.24204i) q^{75} +(0.626413 + 8.69526i) q^{76} +24.1021i q^{77} +(-3.99869 + 4.43412i) q^{78} +(-4.23485 + 7.33498i) q^{79} +(8.92668 + 0.560722i) q^{80} +(4.09596 - 7.09441i) q^{81} +(0.454170 + 1.40147i) q^{82} -7.78986 q^{83} +(-6.48780 + 14.5116i) q^{84} +(-5.55316 - 5.63472i) q^{85} +(7.47210 - 2.42147i) q^{86} -3.80131 q^{87} +(11.4905 + 8.38897i) q^{88} +(-10.6815 + 6.16699i) q^{89} +(0.362738 - 0.697947i) q^{90} +(-6.09836 - 10.5627i) q^{91} +(-1.98770 + 1.43947i) q^{92} +(7.97063 + 4.60185i) q^{93} +(-0.290832 + 1.36309i) q^{94} +(-9.31045 - 2.88365i) q^{95} +(4.66015 + 8.14391i) q^{96} +(2.71921 - 4.70981i) q^{97} +(-16.7621 - 15.1160i) q^{98} +(1.08353 - 0.625576i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.05024 − 0.947103i −0.742629 − 0.669703i
\(3\) 1.43647 + 0.829346i 0.829346 + 0.478823i 0.853629 − 0.520882i \(-0.174397\pi\)
−0.0242825 + 0.999705i \(0.507730\pi\)
\(4\) 0.205992 + 1.98936i 0.102996 + 0.994682i
\(5\) −2.15560 − 0.594467i −0.964013 − 0.265854i
\(6\) −0.723156 − 2.23149i −0.295227 − 0.911004i
\(7\) 4.79169 1.81109 0.905544 − 0.424253i \(-0.139463\pi\)
0.905544 + 0.424253i \(0.139463\pi\)
\(8\) 1.66779 − 2.28440i 0.589654 − 0.807656i
\(9\) −0.124369 − 0.215414i −0.0414564 − 0.0718046i
\(10\) 1.70087 + 2.66591i 0.537861 + 0.843033i
\(11\) 5.02999i 1.51660i 0.651907 + 0.758299i \(0.273969\pi\)
−0.651907 + 0.758299i \(0.726031\pi\)
\(12\) −1.35397 + 3.02850i −0.390858 + 0.874253i
\(13\) −1.27269 − 2.20437i −0.352982 − 0.611383i 0.633788 − 0.773506i \(-0.281499\pi\)
−0.986770 + 0.162124i \(0.948166\pi\)
\(14\) −5.03240 − 4.53822i −1.34497 − 1.21289i
\(15\) −2.60343 − 2.64167i −0.672204 − 0.682077i
\(16\) −3.91513 + 0.819585i −0.978784 + 0.204896i
\(17\) 3.06401 + 1.76901i 0.743131 + 0.429047i 0.823207 − 0.567742i \(-0.192183\pi\)
−0.0800754 + 0.996789i \(0.525516\pi\)
\(18\) −0.0734021 + 0.344026i −0.0173010 + 0.0810877i
\(19\) 4.35677 + 0.136247i 0.999511 + 0.0312573i
\(20\) 0.738576 − 4.41073i 0.165151 − 0.986268i
\(21\) 6.88312 + 3.97397i 1.50202 + 0.867191i
\(22\) 4.76392 − 5.28267i 1.01567 − 1.12627i
\(23\) 0.613547 + 1.06269i 0.127933 + 0.221587i 0.922876 − 0.385098i \(-0.125832\pi\)
−0.794942 + 0.606685i \(0.792499\pi\)
\(24\) 4.29029 − 1.89829i 0.875752 − 0.387487i
\(25\) 4.29322 + 2.56287i 0.858643 + 0.512573i
\(26\) −0.751137 + 3.52048i −0.147310 + 0.690424i
\(27\) − 5.38866i − 1.03705i
\(28\) 0.987048 + 9.53241i 0.186535 + 1.80146i
\(29\) −1.98471 + 1.14588i −0.368552 + 0.212784i −0.672826 − 0.739801i \(-0.734920\pi\)
0.304274 + 0.952585i \(0.401586\pi\)
\(30\) 0.232285 + 5.24010i 0.0424092 + 0.956707i
\(31\) 5.54876 0.996587 0.498294 − 0.867008i \(-0.333960\pi\)
0.498294 + 0.867008i \(0.333960\pi\)
\(32\) 4.88805 + 2.84728i 0.864093 + 0.503332i
\(33\) −4.17160 + 7.22543i −0.726183 + 1.25779i
\(34\) −1.54250 − 4.75981i −0.264537 − 0.816300i
\(35\) −10.3290 − 2.84850i −1.74591 − 0.481485i
\(36\) 0.402918 − 0.291789i 0.0671529 − 0.0486315i
\(37\) 2.52683 0.415408 0.207704 − 0.978192i \(-0.433401\pi\)
0.207704 + 0.978192i \(0.433401\pi\)
\(38\) −4.44660 − 4.26940i −0.721333 − 0.692588i
\(39\) − 4.22202i − 0.676064i
\(40\) −4.95309 + 3.93280i −0.783152 + 0.621830i
\(41\) −0.902159 − 0.520862i −0.140894 − 0.0813450i 0.427896 − 0.903828i \(-0.359255\pi\)
−0.568790 + 0.822483i \(0.692588\pi\)
\(42\) −3.46514 − 10.6926i −0.534683 − 1.64991i
\(43\) −2.77704 + 4.80998i −0.423495 + 0.733515i −0.996279 − 0.0861917i \(-0.972530\pi\)
0.572783 + 0.819707i \(0.305864\pi\)
\(44\) −10.0065 + 1.03614i −1.50853 + 0.156203i
\(45\) 0.140034 + 0.538280i 0.0208750 + 0.0802420i
\(46\) 0.362112 − 1.69717i 0.0533905 − 0.250234i
\(47\) −0.492773 − 0.853508i −0.0718784 − 0.124497i 0.827846 − 0.560955i \(-0.189566\pi\)
−0.899725 + 0.436458i \(0.856233\pi\)
\(48\) −6.30369 − 2.06969i −0.909860 − 0.298734i
\(49\) 15.9603 2.28004
\(50\) −2.08159 − 6.75773i −0.294382 − 0.955688i
\(51\) 2.93424 + 5.08225i 0.410876 + 0.711657i
\(52\) 4.12313 − 2.98594i 0.571776 − 0.414075i
\(53\) −4.98156 − 8.62832i −0.684270 − 1.18519i −0.973665 − 0.227981i \(-0.926787\pi\)
0.289395 − 0.957210i \(-0.406546\pi\)
\(54\) −5.10362 + 5.65936i −0.694514 + 0.770142i
\(55\) 2.99016 − 10.8426i 0.403193 − 1.46202i
\(56\) 7.99154 − 10.9461i 1.06791 − 1.46274i
\(57\) 6.14537 + 3.80899i 0.813974 + 0.504513i
\(58\) 3.16968 + 0.676289i 0.416200 + 0.0888011i
\(59\) −0.000131383 0 0.000227563i −1.71047e−5 0 2.96262e-5i −0.866034 − 0.499985i \(-0.833339\pi\)
0.866017 + 0.500015i \(0.166672\pi\)
\(60\) 4.71896 − 5.72334i 0.609215 − 0.738880i
\(61\) −2.99940 − 5.19512i −0.384034 − 0.665167i 0.607601 − 0.794243i \(-0.292132\pi\)
−0.991635 + 0.129076i \(0.958799\pi\)
\(62\) −5.82751 − 5.25525i −0.740095 − 0.667417i
\(63\) −0.595939 − 1.03220i −0.0750812 − 0.130045i
\(64\) −2.43694 − 7.61980i −0.304617 − 0.952475i
\(65\) 1.43299 + 5.50832i 0.177741 + 0.683223i
\(66\) 11.2244 − 3.63747i 1.38163 − 0.447741i
\(67\) −11.8116 + 6.81943i −1.44302 + 0.833126i −0.998050 − 0.0624211i \(-0.980118\pi\)
−0.444967 + 0.895547i \(0.646784\pi\)
\(68\) −2.88804 + 6.45983i −0.350226 + 0.783369i
\(69\) 2.03537i 0.245030i
\(70\) 8.15002 + 12.7742i 0.974114 + 1.52681i
\(71\) 3.41092 − 5.90788i 0.404801 − 0.701136i −0.589497 − 0.807771i \(-0.700674\pi\)
0.994298 + 0.106634i \(0.0340074\pi\)
\(72\) −0.699513 − 0.0751568i −0.0824384 − 0.00885732i
\(73\) −12.2071 − 7.04778i −1.42873 − 0.824880i −0.431713 − 0.902011i \(-0.642091\pi\)
−0.997021 + 0.0771308i \(0.975424\pi\)
\(74\) −2.65376 − 2.39316i −0.308494 − 0.278200i
\(75\) 4.04157 + 7.24204i 0.466681 + 0.836239i
\(76\) 0.626413 + 8.69526i 0.0718545 + 0.997415i
\(77\) 24.1021i 2.74669i
\(78\) −3.99869 + 4.43412i −0.452762 + 0.502065i
\(79\) −4.23485 + 7.33498i −0.476458 + 0.825249i −0.999636 − 0.0269740i \(-0.991413\pi\)
0.523178 + 0.852223i \(0.324746\pi\)
\(80\) 8.92668 + 0.560722i 0.998033 + 0.0626906i
\(81\) 4.09596 − 7.09441i 0.455106 − 0.788267i
\(82\) 0.454170 + 1.40147i 0.0501547 + 0.154766i
\(83\) −7.78986 −0.855048 −0.427524 − 0.904004i \(-0.640614\pi\)
−0.427524 + 0.904004i \(0.640614\pi\)
\(84\) −6.48780 + 14.5116i −0.707877 + 1.58335i
\(85\) −5.55316 − 5.63472i −0.602325 − 0.611171i
\(86\) 7.47210 − 2.42147i 0.805737 − 0.261114i
\(87\) −3.80131 −0.407543
\(88\) 11.4905 + 8.38897i 1.22489 + 0.894268i
\(89\) −10.6815 + 6.16699i −1.13224 + 0.653699i −0.944497 − 0.328519i \(-0.893451\pi\)
−0.187743 + 0.982218i \(0.560117\pi\)
\(90\) 0.362738 − 0.697947i 0.0382359 − 0.0735701i
\(91\) −6.09836 − 10.5627i −0.639281 − 1.10727i
\(92\) −1.98770 + 1.43947i −0.207232 + 0.150075i
\(93\) 7.97063 + 4.60185i 0.826516 + 0.477189i
\(94\) −0.290832 + 1.36309i −0.0299970 + 0.140592i
\(95\) −9.31045 − 2.88365i −0.955232 − 0.295856i
\(96\) 4.66015 + 8.14391i 0.475625 + 0.831185i
\(97\) 2.71921 − 4.70981i 0.276094 − 0.478209i −0.694316 − 0.719670i \(-0.744293\pi\)
0.970411 + 0.241461i \(0.0776265\pi\)
\(98\) −16.7621 − 15.1160i −1.69322 − 1.52695i
\(99\) 1.08353 − 0.625576i 0.108899 − 0.0628728i
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.179.13 yes 112
4.3 odd 2 inner 380.2.s.a.179.51 yes 112
5.4 even 2 inner 380.2.s.a.179.44 yes 112
19.12 odd 6 inner 380.2.s.a.259.6 yes 112
20.19 odd 2 inner 380.2.s.a.179.6 ✓ 112
76.31 even 6 inner 380.2.s.a.259.44 yes 112
95.69 odd 6 inner 380.2.s.a.259.51 yes 112
380.259 even 6 inner 380.2.s.a.259.13 yes 112
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.s.a.179.6 ✓ 112 20.19 odd 2 inner
380.2.s.a.179.13 yes 112 1.1 even 1 trivial
380.2.s.a.179.44 yes 112 5.4 even 2 inner
380.2.s.a.179.51 yes 112 4.3 odd 2 inner
380.2.s.a.259.6 yes 112 19.12 odd 6 inner
380.2.s.a.259.13 yes 112 380.259 even 6 inner
380.2.s.a.259.44 yes 112 76.31 even 6 inner
380.2.s.a.259.51 yes 112 95.69 odd 6 inner