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Properties

Label 380.2.s.a.179.10
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 179.10
Character \(\chi\) \(=\) 380.179
Dual form 380.2.s.a.259.10

$q$-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+(-1.21903 + 0.716913i) q^{2} +(-2.56289 - 1.47969i) q^{3} +(0.972072 - 1.74788i) q^{4} +(-2.23484 + 0.0740448i) q^{5} +(4.18505 - 0.0335865i) q^{6} +4.78381 q^{7} +(0.0680894 + 2.82761i) q^{8} +(2.87894 + 4.98647i) q^{9} +(2.67126 - 1.69245i) q^{10} -2.47729i q^{11} +(-5.07763 + 3.04126i) q^{12} +(0.900209 + 1.55921i) q^{13} +(-5.83161 + 3.42957i) q^{14} +(5.83722 + 3.11710i) q^{15} +(-2.11015 - 3.39813i) q^{16} +(-1.50351 - 0.868052i) q^{17} +(-7.08438 - 4.01471i) q^{18} +(0.00235679 - 4.35890i) q^{19} +(-2.04301 + 3.97821i) q^{20} +(-12.2604 - 7.07853i) q^{21} +(1.77600 + 3.01989i) q^{22} +(-2.29937 - 3.98263i) q^{23} +(4.00947 - 7.34760i) q^{24} +(4.98903 - 0.330957i) q^{25} +(-2.21520 - 1.25535i) q^{26} -8.16160i q^{27} +(4.65021 - 8.36151i) q^{28} +(1.43155 - 0.826504i) q^{29} +(-9.35044 + 0.384942i) q^{30} -10.2012 q^{31} +(5.00850 + 2.62963i) q^{32} +(-3.66561 + 6.34902i) q^{33} +(2.45514 - 0.0197034i) q^{34} +(-10.6911 + 0.354216i) q^{35} +(11.5143 - 0.184824i) q^{36} -1.78457 q^{37} +(3.12208 + 5.31532i) q^{38} -5.32810i q^{39} +(-0.361539 - 6.31421i) q^{40} +(-0.233836 - 0.135005i) q^{41} +(20.0205 - 0.160672i) q^{42} +(0.436260 - 0.755625i) q^{43} +(-4.33000 - 2.40810i) q^{44} +(-6.80320 - 10.9308i) q^{45} +(5.65820 + 3.20650i) q^{46} +(-1.12676 - 1.95161i) q^{47} +(0.379927 + 11.8314i) q^{48} +15.8848 q^{49} +(-5.84452 + 3.98015i) q^{50} +(2.56889 + 4.44945i) q^{51} +(3.60037 - 0.0577923i) q^{52} +(-3.52269 - 6.10148i) q^{53} +(5.85116 + 9.94924i) q^{54} +(0.183430 + 5.53635i) q^{55} +(0.325727 + 13.5267i) q^{56} +(-6.45584 + 11.1679i) q^{57} +(-1.15257 + 2.03383i) q^{58} +(0.407141 - 0.705190i) q^{59} +(11.1225 - 7.17270i) q^{60} +(-5.56896 - 9.64573i) q^{61} +(12.4356 - 7.31337i) q^{62} +(13.7723 + 23.8543i) q^{63} +(-7.99073 + 0.385060i) q^{64} +(-2.12727 - 3.41792i) q^{65} +(-0.0832036 - 10.3676i) q^{66} +(-0.641213 + 0.370205i) q^{67} +(-2.97877 + 1.78414i) q^{68} +13.6094i q^{69} +(12.7788 - 8.09635i) q^{70} +(6.93656 - 12.0145i) q^{71} +(-13.9038 + 8.48004i) q^{72} +(-6.88596 - 3.97561i) q^{73} +(2.17544 - 1.27938i) q^{74} +(-13.2761 - 6.53400i) q^{75} +(-7.61653 - 4.24128i) q^{76} -11.8509i q^{77} +(3.81979 + 6.49512i) q^{78} +(-6.02065 + 10.4281i) q^{79} +(4.96747 + 7.43803i) q^{80} +(-3.43978 + 5.95788i) q^{81} +(0.381840 - 0.00306441i) q^{82} +1.31945 q^{83} +(-24.2904 + 14.5488i) q^{84} +(3.42438 + 1.82863i) q^{85} +(0.00990242 + 1.23389i) q^{86} -4.89186 q^{87} +(7.00480 - 0.168677i) q^{88} +(-4.85070 + 2.80055i) q^{89} +(16.1297 + 8.44769i) q^{90} +(4.30642 + 7.45895i) q^{91} +(-9.19630 + 0.147617i) q^{92} +(26.1446 + 15.0946i) q^{93} +(2.77269 + 1.57128i) q^{94} +(0.317487 + 9.74162i) q^{95} +(-8.94522 - 14.1505i) q^{96} +(1.12429 - 1.94732i) q^{97} +(-19.3641 + 11.3880i) q^{98} +(12.3529 - 7.13197i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.21903 + 0.716913i −0.861985 + 0.506934i
\(3\) −2.56289 − 1.47969i −1.47969 − 0.854297i −0.479951 − 0.877296i \(-0.659345\pi\)
−0.999736 + 0.0229984i \(0.992679\pi\)
\(4\) 0.972072 − 1.74788i 0.486036 − 0.873939i
\(5\) −2.23484 + 0.0740448i −0.999452 + 0.0331138i
\(6\) 4.18505 − 0.0335865i 1.70854 − 0.0137116i
\(7\) 4.78381 1.80811 0.904055 − 0.427417i \(-0.140576\pi\)
0.904055 + 0.427417i \(0.140576\pi\)
\(8\) 0.0680894 + 2.82761i 0.0240732 + 0.999710i
\(9\) 2.87894 + 4.98647i 0.959647 + 1.66216i
\(10\) 2.67126 − 1.69245i 0.844726 − 0.535199i
\(11\) − 2.47729i − 0.746931i −0.927644 − 0.373465i \(-0.878169\pi\)
0.927644 − 0.373465i \(-0.121831\pi\)
\(12\) −5.07763 + 3.04126i −1.46578 + 0.877936i
\(13\) 0.900209 + 1.55921i 0.249673 + 0.432446i 0.963435 − 0.267942i \(-0.0863436\pi\)
−0.713762 + 0.700388i \(0.753010\pi\)
\(14\) −5.83161 + 3.42957i −1.55856 + 0.916592i
\(15\) 5.83722 + 3.11710i 1.50716 + 0.804831i
\(16\) −2.11015 − 3.39813i −0.527538 − 0.849532i
\(17\) −1.50351 − 0.868052i −0.364655 − 0.210534i 0.306466 − 0.951882i \(-0.400853\pi\)
−0.671121 + 0.741348i \(0.734187\pi\)
\(18\) −7.08438 − 4.01471i −1.66981 − 0.946277i
\(19\) 0.00235679 − 4.35890i 0.000540684 − 1.00000i
\(20\) −2.04301 + 3.97821i −0.456830 + 0.889554i
\(21\) −12.2604 − 7.07853i −2.67543 − 1.54466i
\(22\) 1.77600 + 3.01989i 0.378645 + 0.643843i
\(23\) −2.29937 − 3.98263i −0.479452 − 0.830435i 0.520270 − 0.854002i \(-0.325831\pi\)
−0.999722 + 0.0235664i \(0.992498\pi\)
\(24\) 4.00947 − 7.34760i 0.818429 − 1.49982i
\(25\) 4.98903 − 0.330957i 0.997807 − 0.0661913i
\(26\) −2.21520 − 1.25535i −0.434436 − 0.246194i
\(27\) − 8.16160i − 1.57070i
\(28\) 4.65021 − 8.36151i 0.878807 − 1.58018i
\(29\) 1.43155 − 0.826504i 0.265832 − 0.153478i −0.361160 − 0.932504i \(-0.617619\pi\)
0.626992 + 0.779026i \(0.284286\pi\)
\(30\) −9.35044 + 0.384942i −1.70715 + 0.0702804i
\(31\) −10.2012 −1.83219 −0.916094 − 0.400963i \(-0.868676\pi\)
−0.916094 + 0.400963i \(0.868676\pi\)
\(32\) 5.00850 + 2.62963i 0.885386 + 0.464857i
\(33\) −3.66561 + 6.34902i −0.638101 + 1.10522i
\(34\) 2.45514 − 0.0197034i 0.421054 − 0.00337911i
\(35\) −10.6911 + 0.354216i −1.80712 + 0.0598734i
\(36\) 11.5143 − 0.184824i 1.91905 − 0.0308041i
\(37\) −1.78457 −0.293381 −0.146691 − 0.989182i \(-0.546862\pi\)
−0.146691 + 0.989182i \(0.546862\pi\)
\(38\) 3.12208 + 5.31532i 0.506468 + 0.862259i
\(39\) − 5.32810i − 0.853179i
\(40\) −0.361539 − 6.31421i −0.0571643 − 0.998365i
\(41\) −0.233836 − 0.135005i −0.0365191 − 0.0210843i 0.481629 − 0.876375i \(-0.340045\pi\)
−0.518148 + 0.855291i \(0.673379\pi\)
\(42\) 20.0205 − 0.160672i 3.08923 − 0.0247922i
\(43\) 0.436260 − 0.755625i 0.0665291 − 0.115232i −0.830842 − 0.556508i \(-0.812141\pi\)
0.897371 + 0.441276i \(0.145474\pi\)
\(44\) −4.33000 − 2.40810i −0.652772 − 0.363035i
\(45\) −6.80320 − 10.9308i −1.01416 − 1.62947i
\(46\) 5.65820 + 3.20650i 0.834256 + 0.472772i
\(47\) −1.12676 − 1.95161i −0.164355 − 0.284671i 0.772071 − 0.635536i \(-0.219221\pi\)
−0.936426 + 0.350865i \(0.885888\pi\)
\(48\) 0.379927 + 11.8314i 0.0548377 + 1.70771i
\(49\) 15.8848 2.26926
\(50\) −5.84452 + 3.98015i −0.826540 + 0.562878i
\(51\) 2.56889 + 4.44945i 0.359717 + 0.623047i
\(52\) 3.60037 − 0.0577923i 0.499282 − 0.00801435i
\(53\) −3.52269 − 6.10148i −0.483879 − 0.838103i 0.515950 − 0.856619i \(-0.327439\pi\)
−0.999829 + 0.0185160i \(0.994106\pi\)
\(54\) 5.85116 + 9.94924i 0.796241 + 1.35392i
\(55\) 0.183430 + 5.53635i 0.0247337 + 0.746521i
\(56\) 0.325727 + 13.5267i 0.0435271 + 1.80759i
\(57\) −6.45584 + 11.1679i −0.855097 + 1.47922i
\(58\) −1.15257 + 2.03383i −0.151340 + 0.267055i
\(59\) 0.407141 − 0.705190i 0.0530053 − 0.0918079i −0.838305 − 0.545201i \(-0.816453\pi\)
0.891311 + 0.453393i \(0.149787\pi\)
\(60\) 11.1225 − 7.17270i 1.43591 − 0.925992i
\(61\) −5.56896 − 9.64573i −0.713033 − 1.23501i −0.963713 − 0.266939i \(-0.913988\pi\)
0.250681 − 0.968070i \(-0.419346\pi\)
\(62\) 12.4356 − 7.31337i 1.57932 − 0.928798i
\(63\) 13.7723 + 23.8543i 1.73515 + 3.00536i
\(64\) −7.99073 + 0.385060i −0.998841 + 0.0481325i
\(65\) −2.12727 − 3.41792i −0.263856 − 0.423941i
\(66\) −0.0832036 − 10.3676i −0.0102417 − 1.27616i
\(67\) −0.641213 + 0.370205i −0.0783366 + 0.0452277i −0.538657 − 0.842525i \(-0.681068\pi\)
0.460320 + 0.887753i \(0.347735\pi\)
\(68\) −2.97877 + 1.78414i −0.361229 + 0.216359i
\(69\) 13.6094i 1.63838i
\(70\) 12.7788 − 8.09635i 1.52736 − 0.967699i
\(71\) 6.93656 − 12.0145i 0.823219 − 1.42586i −0.0800547 − 0.996790i \(-0.525509\pi\)
0.903273 − 0.429066i \(-0.141157\pi\)
\(72\) −13.9038 + 8.48004i −1.63857 + 0.999383i
\(73\) −6.88596 − 3.97561i −0.805941 − 0.465310i 0.0396032 − 0.999215i \(-0.487391\pi\)
−0.845544 + 0.533905i \(0.820724\pi\)
\(74\) 2.17544 − 1.27938i 0.252890 − 0.148725i
\(75\) −13.2761 − 6.53400i −1.53299 − 0.754481i
\(76\) −7.61653 − 4.24128i −0.873676 − 0.486509i
\(77\) − 11.8509i − 1.35053i
\(78\) 3.81979 + 6.49512i 0.432506 + 0.735428i
\(79\) −6.02065 + 10.4281i −0.677376 + 1.17325i 0.298393 + 0.954443i \(0.403549\pi\)
−0.975768 + 0.218806i \(0.929784\pi\)
\(80\) 4.96747 + 7.43803i 0.555380 + 0.831597i
\(81\) −3.43978 + 5.95788i −0.382198 + 0.661987i
\(82\) 0.381840 − 0.00306441i 0.0421672 − 0.000338407i
\(83\) 1.31945 0.144828 0.0724142 − 0.997375i \(-0.476930\pi\)
0.0724142 + 0.997375i \(0.476930\pi\)
\(84\) −24.2904 + 14.5488i −2.65030 + 1.58740i
\(85\) 3.42438 + 1.82863i 0.371426 + 0.198343i
\(86\) 0.00990242 + 1.23389i 0.00106781 + 0.133054i
\(87\) −4.89186 −0.524463
\(88\) 7.00480 − 0.168677i 0.746715 − 0.0179810i
\(89\) −4.85070 + 2.80055i −0.514173 + 0.296858i −0.734547 − 0.678558i \(-0.762605\pi\)
0.220374 + 0.975415i \(0.429272\pi\)
\(90\) 16.1297 + 8.44769i 1.70022 + 0.890465i
\(91\) 4.30642 + 7.45895i 0.451436 + 0.781910i
\(92\) −9.19630 + 0.147617i −0.958781 + 0.0153901i
\(93\) 26.1446 + 15.0946i 2.71106 + 1.56523i
\(94\) 2.77269 + 1.57128i 0.285981 + 0.162065i
\(95\) 0.317487 + 9.74162i 0.0325734 + 0.999469i
\(96\) −8.94522 − 14.1505i −0.912967 − 1.44422i
\(97\) 1.12429 − 1.94732i 0.114154 − 0.197721i −0.803287 − 0.595592i \(-0.796918\pi\)
0.917441 + 0.397871i \(0.130251\pi\)
\(98\) −19.3641 + 11.3880i −1.95607 + 1.15036i
\(99\) 12.3529 − 7.13197i 1.24152 − 0.716790i
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.179.10 ✓ 112
4.3 odd 2 inner 380.2.s.a.179.28 yes 112
5.4 even 2 inner 380.2.s.a.179.47 yes 112
19.12 odd 6 inner 380.2.s.a.259.29 yes 112
20.19 odd 2 inner 380.2.s.a.179.29 yes 112
76.31 even 6 inner 380.2.s.a.259.47 yes 112
95.69 odd 6 inner 380.2.s.a.259.28 yes 112
380.259 even 6 inner 380.2.s.a.259.10 yes 112
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.s.a.179.10 ✓ 112 1.1 even 1 trivial
380.2.s.a.179.28 yes 112 4.3 odd 2 inner
380.2.s.a.179.29 yes 112 20.19 odd 2 inner
380.2.s.a.179.47 yes 112 5.4 even 2 inner
380.2.s.a.259.10 yes 112 380.259 even 6 inner
380.2.s.a.259.28 yes 112 95.69 odd 6 inner
380.2.s.a.259.29 yes 112 19.12 odd 6 inner
380.2.s.a.259.47 yes 112 76.31 even 6 inner