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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,3,Mod(159,380)] 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("380.159"); S:= CuspForms(chi, 3); 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, 2])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.p (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: \(10.3542500457\)
Analytic rank: \(0\)
Dimension: \(232\)
Relative dimension: \(116\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 159.13
Character \(\chi\) \(=\) 380.159
Dual form 380.3.p.a.239.13

$q$-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+(-1.85215 + 0.754679i) q^{2} +(2.72599 + 4.72155i) q^{3} +(2.86092 - 2.79556i) q^{4} +(-4.08217 - 2.88720i) q^{5} +(-8.61220 - 6.68778i) q^{6} +7.14895 q^{7} +(-3.18910 + 7.33687i) q^{8} +(-10.3621 + 17.9476i) q^{9} +(9.73969 + 2.26680i) q^{10} -2.36056i q^{11} +(20.9982 + 5.88732i) q^{12} +(-9.56121 - 5.52017i) q^{13} +(-13.2409 + 5.39517i) q^{14} +(2.50411 - 27.1446i) q^{15} +(0.369710 - 15.9957i) q^{16} +(-22.2125 + 12.8244i) q^{17} +(5.64739 - 41.0617i) q^{18} +(-5.55509 + 18.1698i) q^{19} +(-19.7501 + 3.15190i) q^{20} +(19.4880 + 33.7542i) q^{21} +(1.78146 + 4.37210i) q^{22} +(-17.9519 + 31.0936i) q^{23} +(-43.3349 + 4.94272i) q^{24} +(8.32818 + 23.5720i) q^{25} +(21.8748 + 3.00853i) q^{26} -63.9196 q^{27} +(20.4526 - 19.9853i) q^{28} +(-4.52769 + 7.84219i) q^{29} +(15.8475 + 52.1658i) q^{30} +5.39342i q^{31} +(11.3869 + 29.9055i) q^{32} +(11.1455 - 6.43485i) q^{33} +(31.4625 - 40.5160i) q^{34} +(-29.1832 - 20.6404i) q^{35} +(20.5286 + 80.3143i) q^{36} -27.8867i q^{37} +(-3.42350 - 37.8455i) q^{38} -60.1917i q^{39} +(34.2014 - 20.7428i) q^{40} +(-16.1347 - 27.9461i) q^{41} +(-61.5682 - 47.8106i) q^{42} +(16.3616 + 28.3392i) q^{43} +(-6.59907 - 6.75336i) q^{44} +(94.1179 - 43.3478i) q^{45} +(9.78391 - 71.1379i) q^{46} +(4.86459 - 8.42572i) q^{47} +(76.5325 - 41.8586i) q^{48} +2.10753 q^{49} +(-33.2144 - 37.3739i) q^{50} +(-121.102 - 69.9183i) q^{51} +(-42.7858 + 10.9362i) q^{52} +(74.3419 + 42.9213i) q^{53} +(118.389 - 48.2388i) q^{54} +(-6.81539 + 9.63618i) q^{55} +(-22.7987 + 52.4509i) q^{56} +(-100.933 + 23.3020i) q^{57} +(2.46762 - 17.9419i) q^{58} +(-26.3916 + 15.2372i) q^{59} +(-68.7204 - 84.6590i) q^{60} +(-10.5562 + 18.2839i) q^{61} +(-4.07031 - 9.98943i) q^{62} +(-74.0778 + 128.307i) q^{63} +(-43.6593 - 46.7960i) q^{64} +(23.0927 + 50.1394i) q^{65} +(-15.7869 + 20.3296i) q^{66} +(28.8165 - 49.9116i) q^{67} +(-27.6968 + 98.7857i) q^{68} -195.747 q^{69} +(69.6286 + 16.2052i) q^{70} +(35.0956 - 20.2624i) q^{71} +(-98.6335 - 133.262i) q^{72} +(78.9017 - 45.5539i) q^{73} +(21.0455 + 51.6503i) q^{74} +(-88.5942 + 103.579i) q^{75} +(34.9020 + 67.5118i) q^{76} -16.8755i q^{77} +(45.4254 + 111.484i) q^{78} +(91.2310 - 52.6722i) q^{79} +(-47.6920 + 64.2298i) q^{80} +(-80.9857 - 140.271i) q^{81} +(50.9742 + 39.5839i) q^{82} -67.1097 q^{83} +(150.115 + 42.0882i) q^{84} +(127.702 + 11.7805i) q^{85} +(-51.6912 - 40.1406i) q^{86} -49.3698 q^{87} +(17.3191 + 7.52805i) q^{88} +(5.46244 - 9.46123i) q^{89} +(-141.607 + 151.315i) q^{90} +(-68.3527 - 39.4634i) q^{91} +(35.5650 + 139.142i) q^{92} +(-25.4653 + 14.7024i) q^{93} +(-2.65124 + 19.2769i) q^{94} +(75.1366 - 58.1334i) q^{95} +(-110.160 + 135.286i) q^{96} +(-148.774 + 85.8949i) q^{97} +(-3.90345 + 1.59051i) q^{98} +(42.3663 + 24.4602i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 232 q - 2 q^{5} + 8 q^{6} - 328 q^{9} + 20 q^{14} + 12 q^{16} + 92 q^{20} - 40 q^{21} - 134 q^{24} - 2 q^{25} + 28 q^{26} - 4 q^{29} + 268 q^{30} - 70 q^{34} + 12 q^{36} - 42 q^{40} - 12 q^{41} + 98 q^{44}+ \cdots - 628 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.85215 + 0.754679i −0.926075 + 0.377340i
\(3\) 2.72599 + 4.72155i 0.908664 + 1.57385i 0.815923 + 0.578161i \(0.196230\pi\)
0.0927410 + 0.995690i \(0.470437\pi\)
\(4\) 2.86092 − 2.79556i 0.715230 − 0.698890i
\(5\) −4.08217 − 2.88720i −0.816433 − 0.577440i
\(6\) −8.61220 − 6.68778i −1.43537 − 1.11463i
\(7\) 7.14895 1.02128 0.510639 − 0.859795i \(-0.329409\pi\)
0.510639 + 0.859795i \(0.329409\pi\)
\(8\) −3.18910 + 7.33687i −0.398638 + 0.917109i
\(9\) −10.3621 + 17.9476i −1.15134 + 1.99418i
\(10\) 9.73969 + 2.26680i 0.973969 + 0.226680i
\(11\) − 2.36056i − 0.214596i −0.994227 − 0.107298i \(-0.965780\pi\)
0.994227 − 0.107298i \(-0.0342199\pi\)
\(12\) 20.9982 + 5.88732i 1.74985 + 0.490610i
\(13\) −9.56121 − 5.52017i −0.735478 − 0.424628i 0.0849449 − 0.996386i \(-0.472929\pi\)
−0.820423 + 0.571757i \(0.806262\pi\)
\(14\) −13.2409 + 5.39517i −0.945781 + 0.385369i
\(15\) 2.50411 − 27.1446i 0.166941 − 1.80964i
\(16\) 0.369710 − 15.9957i 0.0231069 − 0.999733i
\(17\) −22.2125 + 12.8244i −1.30662 + 0.754375i −0.981530 − 0.191309i \(-0.938727\pi\)
−0.325086 + 0.945684i \(0.605393\pi\)
\(18\) 5.64739 − 41.0617i 0.313744 − 2.28120i
\(19\) −5.55509 + 18.1698i −0.292373 + 0.956304i
\(20\) −19.7501 + 3.15190i −0.987504 + 0.157595i
\(21\) 19.4880 + 33.7542i 0.927999 + 1.60734i
\(22\) 1.78146 + 4.37210i 0.0809756 + 0.198732i
\(23\) −17.9519 + 31.0936i −0.780517 + 1.35190i 0.151123 + 0.988515i \(0.451711\pi\)
−0.931641 + 0.363381i \(0.881622\pi\)
\(24\) −43.3349 + 4.94272i −1.80562 + 0.205947i
\(25\) 8.32818 + 23.5720i 0.333127 + 0.942882i
\(26\) 21.8748 + 3.00853i 0.841337 + 0.115713i
\(27\) −63.9196 −2.36739
\(28\) 20.4526 − 19.9853i 0.730449 − 0.713761i
\(29\) −4.52769 + 7.84219i −0.156127 + 0.270420i −0.933469 − 0.358658i \(-0.883234\pi\)
0.777342 + 0.629079i \(0.216568\pi\)
\(30\) 15.8475 + 52.1658i 0.528250 + 1.73886i
\(31\) 5.39342i 0.173981i 0.996209 + 0.0869907i \(0.0277250\pi\)
−0.996209 + 0.0869907i \(0.972275\pi\)
\(32\) 11.3869 + 29.9055i 0.355840 + 0.934547i
\(33\) 11.1455 − 6.43485i 0.337742 − 0.194996i
\(34\) 31.4625 − 40.5160i 0.925369 − 1.19165i
\(35\) −29.1832 − 20.6404i −0.833806 − 0.589727i
\(36\) 20.5286 + 80.3143i 0.570238 + 2.23095i
\(37\) − 27.8867i − 0.753693i −0.926276 − 0.376847i \(-0.877008\pi\)
0.926276 − 0.376847i \(-0.122992\pi\)
\(38\) −3.42350 − 37.8455i −0.0900921 − 0.995933i
\(39\) − 60.1917i − 1.54338i
\(40\) 34.2014 − 20.7428i 0.855036 − 0.518569i
\(41\) −16.1347 − 27.9461i −0.393529 − 0.681613i 0.599383 − 0.800462i \(-0.295413\pi\)
−0.992912 + 0.118850i \(0.962079\pi\)
\(42\) −61.5682 − 47.8106i −1.46591 − 1.13835i
\(43\) 16.3616 + 28.3392i 0.380503 + 0.659051i 0.991134 − 0.132864i \(-0.0424175\pi\)
−0.610631 + 0.791915i \(0.709084\pi\)
\(44\) −6.59907 − 6.75336i −0.149979 − 0.153485i
\(45\) 94.1179 − 43.3478i 2.09151 − 0.963285i
\(46\) 9.78391 − 71.1379i 0.212694 − 1.54648i
\(47\) 4.86459 − 8.42572i 0.103502 − 0.179271i −0.809623 − 0.586950i \(-0.800329\pi\)
0.913125 + 0.407679i \(0.133662\pi\)
\(48\) 76.5325 − 41.8586i 1.59443 − 0.872054i
\(49\) 2.10753 0.0430107
\(50\) −33.2144 − 37.3739i −0.664287 − 0.747477i
\(51\) −121.102 − 69.9183i −2.37455 − 1.37095i
\(52\) −42.7858 + 10.9362i −0.822804 + 0.210311i
\(53\) 74.3419 + 42.9213i 1.40268 + 0.809836i 0.994667 − 0.103141i \(-0.0328892\pi\)
0.408011 + 0.912977i \(0.366223\pi\)
\(54\) 118.389 − 48.2388i 2.19238 − 0.893311i
\(55\) −6.81539 + 9.63618i −0.123916 + 0.175203i
\(56\) −22.7987 + 52.4509i −0.407120 + 0.936624i
\(57\) −100.933 + 23.3020i −1.77075 + 0.408807i
\(58\) 2.46762 − 17.9419i 0.0425452 − 0.309342i
\(59\) −26.3916 + 15.2372i −0.447315 + 0.258257i −0.706695 − 0.707518i \(-0.749815\pi\)
0.259381 + 0.965775i \(0.416482\pi\)
\(60\) −68.7204 − 84.6590i −1.14534 − 1.41098i
\(61\) −10.5562 + 18.2839i −0.173052 + 0.299735i −0.939485 − 0.342589i \(-0.888696\pi\)
0.766433 + 0.642324i \(0.222030\pi\)
\(62\) −4.07031 − 9.98943i −0.0656501 − 0.161120i
\(63\) −74.0778 + 128.307i −1.17584 + 2.03661i
\(64\) −43.6593 − 46.7960i −0.682176 − 0.731188i
\(65\) 23.0927 + 50.1394i 0.355272 + 0.771375i
\(66\) −15.7869 + 20.3296i −0.239195 + 0.308024i
\(67\) 28.8165 − 49.9116i 0.430097 − 0.744949i −0.566784 − 0.823866i \(-0.691813\pi\)
0.996881 + 0.0789167i \(0.0251461\pi\)
\(68\) −27.6968 + 98.7857i −0.407306 + 1.45273i
\(69\) −195.747 −2.83691
\(70\) 69.6286 + 16.2052i 0.994694 + 0.231503i
\(71\) 35.0956 − 20.2624i 0.494304 − 0.285387i −0.232054 − 0.972703i \(-0.574545\pi\)
0.726358 + 0.687316i \(0.241211\pi\)
\(72\) −98.6335 − 133.262i −1.36991 − 1.85086i
\(73\) 78.9017 − 45.5539i 1.08084 − 0.624026i 0.149721 − 0.988728i \(-0.452163\pi\)
0.931124 + 0.364702i \(0.118829\pi\)
\(74\) 21.0455 + 51.6503i 0.284398 + 0.697977i
\(75\) −88.5942 + 103.579i −1.18126 + 1.38106i
\(76\) 34.9020 + 67.5118i 0.459237 + 0.888314i
\(77\) − 16.8755i − 0.219162i
\(78\) 45.4254 + 111.484i 0.582377 + 1.42928i
\(79\) 91.2310 − 52.6722i 1.15482 − 0.666737i 0.204765 − 0.978811i \(-0.434357\pi\)
0.950058 + 0.312074i \(0.101024\pi\)
\(80\) −47.6920 + 64.2298i −0.596151 + 0.802873i
\(81\) −80.9857 − 140.271i −0.999824 − 1.73175i
\(82\) 50.9742 + 39.5839i 0.621637 + 0.482730i
\(83\) −67.1097 −0.808551 −0.404275 − 0.914637i \(-0.632476\pi\)
−0.404275 + 0.914637i \(0.632476\pi\)
\(84\) 150.115 + 42.0882i 1.78709 + 0.501049i
\(85\) 127.702 + 11.7805i 1.50237 + 0.138595i
\(86\) −51.6912 − 40.1406i −0.601060 − 0.466752i
\(87\) −49.3698 −0.567468
\(88\) 17.3191 + 7.52805i 0.196808 + 0.0855460i
\(89\) 5.46244 − 9.46123i 0.0613757 − 0.106306i −0.833705 − 0.552210i \(-0.813785\pi\)
0.895081 + 0.445904i \(0.147118\pi\)
\(90\) −141.607 + 151.315i −1.57341 + 1.68128i
\(91\) −68.3527 − 39.4634i −0.751128 − 0.433664i
\(92\) 35.5650 + 139.142i 0.386577 + 1.51241i
\(93\) −25.4653 + 14.7024i −0.273821 + 0.158091i
\(94\) −2.65124 + 19.2769i −0.0282047 + 0.205074i
\(95\) 75.1366 − 58.1334i 0.790911 − 0.611931i
\(96\) −110.160 + 135.286i −1.14750 + 1.40923i
\(97\) −148.774 + 85.8949i −1.53376 + 0.885515i −0.534573 + 0.845122i \(0.679527\pi\)
−0.999184 + 0.0403924i \(0.987139\pi\)
\(98\) −3.90345 + 1.59051i −0.0398312 + 0.0162297i
\(99\) 42.3663 + 24.4602i 0.427942 + 0.247073i
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.p.a.159.13 ✓ 232
4.3 odd 2 inner 380.3.p.a.159.25 yes 232
5.4 even 2 inner 380.3.p.a.159.104 yes 232
19.11 even 3 inner 380.3.p.a.239.92 yes 232
20.19 odd 2 inner 380.3.p.a.159.92 yes 232
76.11 odd 6 inner 380.3.p.a.239.104 yes 232
95.49 even 6 inner 380.3.p.a.239.25 yes 232
380.239 odd 6 inner 380.3.p.a.239.13 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.13 ✓ 232 1.1 even 1 trivial
380.3.p.a.159.25 yes 232 4.3 odd 2 inner
380.3.p.a.159.92 yes 232 20.19 odd 2 inner
380.3.p.a.159.104 yes 232 5.4 even 2 inner
380.3.p.a.239.13 yes 232 380.239 odd 6 inner
380.3.p.a.239.25 yes 232 95.49 even 6 inner
380.3.p.a.239.92 yes 232 19.11 even 3 inner
380.3.p.a.239.104 yes 232 76.11 odd 6 inner