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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,3,Mod(39,380)] 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("380.39"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.h (of order \(2\), degree \(1\), 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: \(108\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 39.3
Character \(\chi\) \(=\) 380.39
Dual form 380.3.h.a.39.4

$q$-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+(-1.99455 - 0.147607i) q^{2} -0.578687 q^{3} +(3.95642 + 0.588817i) q^{4} +(-3.52665 - 3.54440i) q^{5} +(1.15422 + 0.0854181i) q^{6} +11.0384 q^{7} +(-7.80436 - 1.75842i) q^{8} -8.66512 q^{9} +(6.51089 + 7.59002i) q^{10} +19.3643i q^{11} +(-2.28953 - 0.340740i) q^{12} +4.94590i q^{13} +(-22.0165 - 1.62934i) q^{14} +(2.04082 + 2.05110i) q^{15} +(15.3066 + 4.65922i) q^{16} -20.8304i q^{17} +(17.2830 + 1.27903i) q^{18} -4.35890i q^{19} +(-11.8659 - 16.0997i) q^{20} -6.38775 q^{21} +(2.85831 - 38.6230i) q^{22} -7.14491 q^{23} +(4.51628 + 1.01757i) q^{24} +(-0.125495 + 24.9997i) q^{25} +(0.730047 - 9.86481i) q^{26} +10.2226 q^{27} +(43.6724 + 6.49957i) q^{28} +34.0400 q^{29} +(-3.76776 - 4.39224i) q^{30} +23.8062i q^{31} +(-29.8420 - 11.5524i) q^{32} -11.2059i q^{33} +(-3.07471 + 41.5472i) q^{34} +(-38.9284 - 39.1243i) q^{35} +(-34.2829 - 5.10217i) q^{36} +67.7450i q^{37} +(-0.643403 + 8.69402i) q^{38} -2.86212i q^{39} +(21.2907 + 33.8631i) q^{40} +20.9967 q^{41} +(12.7407 + 0.942875i) q^{42} +47.7631 q^{43} +(-11.4020 + 76.6135i) q^{44} +(30.5588 + 30.7126i) q^{45} +(14.2509 + 1.05464i) q^{46} +23.3276 q^{47} +(-8.85772 - 2.69623i) q^{48} +72.8454 q^{49} +(3.94043 - 49.8445i) q^{50} +12.0543i q^{51} +(-2.91223 + 19.5681i) q^{52} -12.8768i q^{53} +(-20.3894 - 1.50892i) q^{54} +(68.6349 - 68.2912i) q^{55} +(-86.1473 - 19.4100i) q^{56} +2.52244i q^{57} +(-67.8944 - 5.02453i) q^{58} +76.9352i q^{59} +(6.86665 + 9.31667i) q^{60} +55.2257 q^{61} +(3.51396 - 47.4826i) q^{62} -95.6487 q^{63} +(57.8159 + 27.4466i) q^{64} +(17.5302 - 17.4424i) q^{65} +(-1.65406 + 22.3506i) q^{66} +51.0598 q^{67} +(12.2653 - 82.4140i) q^{68} +4.13467 q^{69} +(71.8695 + 83.7813i) q^{70} +129.596i q^{71} +(67.6257 + 15.2369i) q^{72} -64.4246i q^{73} +(9.99961 - 135.120i) q^{74} +(0.0726220 - 14.4670i) q^{75} +(2.56659 - 17.2457i) q^{76} +213.750i q^{77} +(-0.422469 + 5.70864i) q^{78} -27.9745i q^{79} +(-37.4669 - 70.6841i) q^{80} +72.0704 q^{81} +(-41.8789 - 3.09925i) q^{82} +82.5330 q^{83} +(-25.2727 - 3.76122i) q^{84} +(-73.8313 + 73.4616i) q^{85} +(-95.2657 - 7.05016i) q^{86} -19.6985 q^{87} +(34.0506 - 151.126i) q^{88} -104.970 q^{89} +(-56.4176 - 65.7684i) q^{90} +54.5946i q^{91} +(-28.2683 - 4.20705i) q^{92} -13.7763i q^{93} +(-46.5280 - 3.44331i) q^{94} +(-15.4497 + 15.3723i) q^{95} +(17.2691 + 6.68521i) q^{96} +14.4820i q^{97} +(-145.293 - 10.7525i) q^{98} -167.794i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 108 q + 4 q^{5} + 324 q^{9} - 8 q^{10} + 8 q^{14} - 104 q^{16} - 16 q^{21} - 8 q^{24} - 76 q^{25} + 80 q^{26} - 88 q^{29} - 140 q^{30} - 88 q^{34} - 256 q^{36} + 44 q^{40} - 200 q^{41} - 8 q^{44} + 108 q^{45}+ \cdots + 720 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.99455 − 0.147607i −0.997273 − 0.0738034i
\(3\) −0.578687 −0.192896 −0.0964478 − 0.995338i \(-0.530748\pi\)
−0.0964478 + 0.995338i \(0.530748\pi\)
\(4\) 3.95642 + 0.588817i 0.989106 + 0.147204i
\(5\) −3.52665 − 3.54440i −0.705330 − 0.708879i
\(6\) 1.15422 + 0.0854181i 0.192369 + 0.0142363i
\(7\) 11.0384 1.57691 0.788454 − 0.615093i \(-0.210882\pi\)
0.788454 + 0.615093i \(0.210882\pi\)
\(8\) −7.80436 − 1.75842i −0.975544 − 0.219802i
\(9\) −8.66512 −0.962791
\(10\) 6.51089 + 7.59002i 0.651089 + 0.759002i
\(11\) 19.3643i 1.76039i 0.474609 + 0.880197i \(0.342589\pi\)
−0.474609 + 0.880197i \(0.657411\pi\)
\(12\) −2.28953 − 0.340740i −0.190794 − 0.0283950i
\(13\) 4.94590i 0.380453i 0.981740 + 0.190227i \(0.0609223\pi\)
−0.981740 + 0.190227i \(0.939078\pi\)
\(14\) −22.0165 − 1.62934i −1.57261 − 0.116381i
\(15\) 2.04082 + 2.05110i 0.136055 + 0.136740i
\(16\) 15.3066 + 4.65922i 0.956662 + 0.291201i
\(17\) − 20.8304i − 1.22532i −0.790347 − 0.612660i \(-0.790100\pi\)
0.790347 − 0.612660i \(-0.209900\pi\)
\(18\) 17.2830 + 1.27903i 0.960166 + 0.0710572i
\(19\) − 4.35890i − 0.229416i
\(20\) −11.8659 − 16.0997i −0.593296 − 0.804984i
\(21\) −6.38775 −0.304179
\(22\) 2.85831 − 38.6230i 0.129923 − 1.75559i
\(23\) −7.14491 −0.310648 −0.155324 − 0.987864i \(-0.549642\pi\)
−0.155324 + 0.987864i \(0.549642\pi\)
\(24\) 4.51628 + 1.01757i 0.188178 + 0.0423988i
\(25\) −0.125495 + 24.9997i −0.00501978 + 0.999987i
\(26\) 0.730047 − 9.86481i 0.0280787 − 0.379416i
\(27\) 10.2226 0.378614
\(28\) 43.6724 + 6.49957i 1.55973 + 0.232128i
\(29\) 34.0400 1.17379 0.586897 − 0.809662i \(-0.300349\pi\)
0.586897 + 0.809662i \(0.300349\pi\)
\(30\) −3.76776 − 4.39224i −0.125592 − 0.146408i
\(31\) 23.8062i 0.767942i 0.923345 + 0.383971i \(0.125444\pi\)
−0.923345 + 0.383971i \(0.874556\pi\)
\(32\) −29.8420 − 11.5524i −0.932561 − 0.361012i
\(33\) − 11.2059i − 0.339572i
\(34\) −3.07471 + 41.5472i −0.0904327 + 1.22198i
\(35\) −38.9284 − 39.1243i −1.11224 − 1.11784i
\(36\) −34.2829 − 5.10217i −0.952303 − 0.141727i
\(37\) 67.7450i 1.83095i 0.402380 + 0.915473i \(0.368183\pi\)
−0.402380 + 0.915473i \(0.631817\pi\)
\(38\) −0.643403 + 8.69402i −0.0169317 + 0.228790i
\(39\) − 2.86212i − 0.0733878i
\(40\) 21.2907 + 33.8631i 0.532267 + 0.846576i
\(41\) 20.9967 0.512115 0.256057 − 0.966662i \(-0.417576\pi\)
0.256057 + 0.966662i \(0.417576\pi\)
\(42\) 12.7407 + 0.942875i 0.303349 + 0.0224494i
\(43\) 47.7631 1.11077 0.555385 − 0.831593i \(-0.312571\pi\)
0.555385 + 0.831593i \(0.312571\pi\)
\(44\) −11.4020 + 76.6135i −0.259137 + 1.74122i
\(45\) 30.5588 + 30.7126i 0.679085 + 0.682503i
\(46\) 14.2509 + 1.05464i 0.309801 + 0.0229269i
\(47\) 23.3276 0.496332 0.248166 − 0.968718i \(-0.420172\pi\)
0.248166 + 0.968718i \(0.420172\pi\)
\(48\) −8.85772 − 2.69623i −0.184536 − 0.0561714i
\(49\) 72.8454 1.48664
\(50\) 3.94043 − 49.8445i 0.0788085 − 0.996890i
\(51\) 12.0543i 0.236359i
\(52\) −2.91223 + 19.5681i −0.0560043 + 0.376309i
\(53\) − 12.8768i − 0.242958i −0.992594 − 0.121479i \(-0.961236\pi\)
0.992594 − 0.121479i \(-0.0387638\pi\)
\(54\) −20.3894 − 1.50892i −0.377581 − 0.0279430i
\(55\) 68.6349 − 68.2912i 1.24791 − 1.24166i
\(56\) −86.1473 − 19.4100i −1.53834 − 0.346608i
\(57\) 2.52244i 0.0442533i
\(58\) −67.8944 − 5.02453i −1.17059 − 0.0866299i
\(59\) 76.9352i 1.30399i 0.758225 + 0.651993i \(0.226067\pi\)
−0.758225 + 0.651993i \(0.773933\pi\)
\(60\) 6.86665 + 9.31667i 0.114444 + 0.155278i
\(61\) 55.2257 0.905339 0.452670 − 0.891678i \(-0.350472\pi\)
0.452670 + 0.891678i \(0.350472\pi\)
\(62\) 3.51396 − 47.4826i 0.0566767 − 0.765848i
\(63\) −95.6487 −1.51823
\(64\) 57.8159 + 27.4466i 0.903374 + 0.428853i
\(65\) 17.5302 − 17.4424i 0.269696 − 0.268345i
\(66\) −1.65406 + 22.3506i −0.0250616 + 0.338646i
\(67\) 51.0598 0.762087 0.381044 − 0.924557i \(-0.375565\pi\)
0.381044 + 0.924557i \(0.375565\pi\)
\(68\) 12.2653 − 82.4140i 0.180372 − 1.21197i
\(69\) 4.13467 0.0599227
\(70\) 71.8695 + 83.7813i 1.02671 + 1.19688i
\(71\) 129.596i 1.82529i 0.408754 + 0.912644i \(0.365963\pi\)
−0.408754 + 0.912644i \(0.634037\pi\)
\(72\) 67.6257 + 15.2369i 0.939246 + 0.211624i
\(73\) − 64.4246i − 0.882529i −0.897377 − 0.441265i \(-0.854530\pi\)
0.897377 − 0.441265i \(-0.145470\pi\)
\(74\) 9.99961 − 135.120i 0.135130 − 1.82595i
\(75\) 0.0726220 − 14.4670i 0.000968294 − 0.192893i
\(76\) 2.56659 − 17.2457i 0.0337710 − 0.226917i
\(77\) 213.750i 2.77598i
\(78\) −0.422469 + 5.70864i −0.00541627 + 0.0731876i
\(79\) − 27.9745i − 0.354108i −0.984201 − 0.177054i \(-0.943343\pi\)
0.984201 − 0.177054i \(-0.0566567\pi\)
\(80\) −37.4669 − 70.6841i −0.468336 − 0.883551i
\(81\) 72.0704 0.889758
\(82\) −41.8789 − 3.09925i −0.510718 − 0.0377958i
\(83\) 82.5330 0.994373 0.497187 − 0.867644i \(-0.334366\pi\)
0.497187 + 0.867644i \(0.334366\pi\)
\(84\) −25.2727 − 3.76122i −0.300865 − 0.0447764i
\(85\) −73.8313 + 73.4616i −0.868603 + 0.864254i
\(86\) −95.2657 − 7.05016i −1.10774 − 0.0819786i
\(87\) −19.6985 −0.226420
\(88\) 34.0506 − 151.126i 0.386938 − 1.71734i
\(89\) −104.970 −1.17943 −0.589717 − 0.807610i \(-0.700761\pi\)
−0.589717 + 0.807610i \(0.700761\pi\)
\(90\) −56.4176 − 65.7684i −0.626862 − 0.730760i
\(91\) 54.5946i 0.599940i
\(92\) −28.2683 − 4.20705i −0.307264 − 0.0457288i
\(93\) − 13.7763i − 0.148133i
\(94\) −46.5280 − 3.44331i −0.494978 − 0.0366310i
\(95\) −15.4497 + 15.3723i −0.162628 + 0.161814i
\(96\) 17.2691 + 6.68521i 0.179887 + 0.0696376i
\(97\) 14.4820i 0.149299i 0.997210 + 0.0746494i \(0.0237838\pi\)
−0.997210 + 0.0746494i \(0.976216\pi\)
\(98\) −145.293 − 10.7525i −1.48259 − 0.109719i
\(99\) − 167.794i − 1.69489i
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.h.a.39.3 ✓ 108
4.3 odd 2 inner 380.3.h.a.39.105 yes 108
5.4 even 2 inner 380.3.h.a.39.106 yes 108
20.19 odd 2 inner 380.3.h.a.39.4 yes 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.h.a.39.3 ✓ 108 1.1 even 1 trivial
380.3.h.a.39.4 yes 108 20.19 odd 2 inner
380.3.h.a.39.105 yes 108 4.3 odd 2 inner
380.3.h.a.39.106 yes 108 5.4 even 2 inner