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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,3,Mod(11,380)] 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("380.11"); 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, 0, 4])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.q (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: \(160\)
Relative dimension: \(80\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 11.49
Character \(\chi\) \(=\) 380.11
Dual form 380.3.q.a.311.49

$q$-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+(0.604863 + 1.90634i) q^{2} +(-0.161149 - 0.0930393i) q^{3} +(-3.26828 + 2.30615i) q^{4} +(-1.11803 + 1.93649i) q^{5} +(0.0798917 - 0.363481i) q^{6} -4.44882i q^{7} +(-6.37318 - 4.83555i) q^{8} +(-4.48269 - 7.76424i) q^{9} +(-4.36787 - 0.960042i) q^{10} +5.33286i q^{11} +(0.741243 - 0.0675554i) q^{12} +(-2.99276 - 5.18362i) q^{13} +(8.48097 - 2.69093i) q^{14} +(0.360340 - 0.208042i) q^{15} +(5.36331 - 15.0743i) q^{16} +(5.47101 - 9.47607i) q^{17} +(12.0899 - 13.2418i) q^{18} +(-14.1425 - 12.6882i) q^{19} +(-0.811798 - 8.90736i) q^{20} +(-0.413915 + 0.716922i) q^{21} +(-10.1663 + 3.22565i) q^{22} +(11.5933 - 6.69342i) q^{23} +(0.577134 + 1.37220i) q^{24} +(-2.50000 - 4.33013i) q^{25} +(8.07154 - 8.84061i) q^{26} +3.34297i q^{27} +(10.2597 + 14.5400i) q^{28} +(-19.5200 - 33.8097i) q^{29} +(0.614556 + 0.561094i) q^{30} +37.4045i q^{31} +(31.9809 + 1.10641i) q^{32} +(0.496166 - 0.859385i) q^{33} +(21.3738 + 4.69789i) q^{34} +(8.61510 + 4.97393i) q^{35} +(32.5562 + 15.0380i) q^{36} +15.4100 q^{37} +(15.6338 - 34.6350i) q^{38} +1.11378i q^{39} +(16.4894 - 6.93530i) q^{40} +(18.2956 - 31.6888i) q^{41} +(-1.61706 - 0.355424i) q^{42} +(-30.8670 - 17.8211i) q^{43} +(-12.2984 - 17.4293i) q^{44} +20.0472 q^{45} +(19.7723 + 18.0523i) q^{46} +(-39.7406 + 22.9442i) q^{47} +(-2.26680 + 1.93021i) q^{48} +29.2080 q^{49} +(6.74254 - 7.38499i) q^{50} +(-1.76329 + 1.01804i) q^{51} +(21.7354 + 10.0397i) q^{52} +(-25.0258 - 43.3460i) q^{53} +(-6.37285 + 2.02204i) q^{54} +(-10.3270 - 5.96232i) q^{55} +(-21.5125 + 28.3531i) q^{56} +(1.09854 + 3.36050i) q^{57} +(52.6458 - 57.6621i) q^{58} +(-15.8956 - 9.17731i) q^{59} +(-0.697914 + 1.51094i) q^{60} +(-52.2132 - 90.4360i) q^{61} +(-71.3057 + 22.6246i) q^{62} +(-34.5417 + 19.9427i) q^{63} +(17.2349 + 61.6357i) q^{64} +13.3840 q^{65} +(1.93839 + 0.426052i) q^{66} +(-89.2934 + 51.5536i) q^{67} +(3.97247 + 43.5874i) q^{68} -2.49100 q^{69} +(-4.27105 + 19.4319i) q^{70} +(101.073 + 58.3543i) q^{71} +(-8.97542 + 71.1592i) q^{72} +(19.9994 - 34.6400i) q^{73} +(9.32097 + 29.3768i) q^{74} +0.930393i q^{75} +(75.4825 + 8.85386i) q^{76} +23.7249 q^{77} +(-2.12324 + 0.673684i) q^{78} +(-45.6586 - 26.3610i) q^{79} +(23.1949 + 27.2396i) q^{80} +(-40.0332 + 69.3395i) q^{81} +(71.4761 + 15.7102i) q^{82} +22.6330i q^{83} +(-0.300542 - 3.29765i) q^{84} +(12.2336 + 21.1891i) q^{85} +(15.3027 - 69.6224i) q^{86} +7.26452i q^{87} +(25.7873 - 33.9873i) q^{88} +(-39.7688 - 68.8816i) q^{89} +(12.1258 + 38.2168i) q^{90} +(-23.0610 + 13.3143i) q^{91} +(-22.4542 + 48.6120i) q^{92} +(3.48009 - 6.02769i) q^{93} +(-67.7771 - 61.8810i) q^{94} +(40.3824 - 13.2010i) q^{95} +(-5.05074 - 3.15377i) q^{96} +(-70.8339 + 122.688i) q^{97} +(17.6669 + 55.6805i) q^{98} +(41.4056 - 23.9056i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 160 q + 2 q^{4} + 6 q^{6} + 248 q^{9} - 10 q^{10} - 16 q^{13} - 14 q^{16} + 48 q^{17} + 48 q^{21} - 44 q^{24} - 400 q^{25} + 68 q^{26} + 60 q^{28} - 80 q^{30} + 30 q^{32} - 40 q^{33} - 22 q^{34} + 52 q^{36}+ \cdots - 226 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.604863 + 1.90634i 0.302432 + 0.953171i
\(3\) −0.161149 − 0.0930393i −0.0537163 − 0.0310131i 0.472901 − 0.881115i \(-0.343207\pi\)
−0.526618 + 0.850102i \(0.676540\pi\)
\(4\) −3.26828 + 2.30615i −0.817070 + 0.576538i
\(5\) −1.11803 + 1.93649i −0.223607 + 0.387298i
\(6\) 0.0798917 − 0.363481i 0.0133153 − 0.0605802i
\(7\) − 4.44882i − 0.635545i −0.948167 − 0.317773i \(-0.897065\pi\)
0.948167 − 0.317773i \(-0.102935\pi\)
\(8\) −6.37318 − 4.83555i −0.796648 − 0.604444i
\(9\) −4.48269 − 7.76424i −0.498076 − 0.862694i
\(10\) −4.36787 − 0.960042i −0.436787 − 0.0960042i
\(11\) 5.33286i 0.484806i 0.970176 + 0.242403i \(0.0779356\pi\)
−0.970176 + 0.242403i \(0.922064\pi\)
\(12\) 0.741243 − 0.0675554i 0.0617702 − 0.00562961i
\(13\) −2.99276 − 5.18362i −0.230213 − 0.398740i 0.727658 − 0.685940i \(-0.240609\pi\)
−0.957871 + 0.287200i \(0.907275\pi\)
\(14\) 8.48097 − 2.69093i 0.605783 − 0.192209i
\(15\) 0.360340 − 0.208042i 0.0240227 − 0.0138695i
\(16\) 5.36331 − 15.0743i 0.335207 − 0.942144i
\(17\) 5.47101 − 9.47607i 0.321824 − 0.557416i −0.659040 − 0.752108i \(-0.729037\pi\)
0.980864 + 0.194692i \(0.0623707\pi\)
\(18\) 12.0899 − 13.2418i 0.671660 − 0.735658i
\(19\) −14.1425 − 12.6882i −0.744341 − 0.667800i
\(20\) −0.811798 − 8.90736i −0.0405899 − 0.445368i
\(21\) −0.413915 + 0.716922i −0.0197102 + 0.0341391i
\(22\) −10.1663 + 3.22565i −0.462103 + 0.146621i
\(23\) 11.5933 − 6.69342i 0.504058 − 0.291018i −0.226330 − 0.974051i \(-0.572673\pi\)
0.730388 + 0.683033i \(0.239339\pi\)
\(24\) 0.577134 + 1.37220i 0.0240473 + 0.0571750i
\(25\) −2.50000 − 4.33013i −0.100000 − 0.173205i
\(26\) 8.07154 − 8.84061i 0.310444 − 0.340024i
\(27\) 3.34297i 0.123814i
\(28\) 10.2597 + 14.5400i 0.366416 + 0.519285i
\(29\) −19.5200 − 33.8097i −0.673104 − 1.16585i −0.977019 − 0.213151i \(-0.931627\pi\)
0.303915 − 0.952699i \(-0.401706\pi\)
\(30\) 0.614556 + 0.561094i 0.0204852 + 0.0187031i
\(31\) 37.4045i 1.20660i 0.797516 + 0.603298i \(0.206147\pi\)
−0.797516 + 0.603298i \(0.793853\pi\)
\(32\) 31.9809 + 1.10641i 0.999402 + 0.0345752i
\(33\) 0.496166 − 0.859385i 0.0150353 − 0.0260420i
\(34\) 21.3738 + 4.69789i 0.628643 + 0.138173i
\(35\) 8.61510 + 4.97393i 0.246146 + 0.142112i
\(36\) 32.5562 + 15.0380i 0.904339 + 0.417721i
\(37\) 15.4100 0.416487 0.208244 − 0.978077i \(-0.433225\pi\)
0.208244 + 0.978077i \(0.433225\pi\)
\(38\) 15.6338 − 34.6350i 0.411415 − 0.911448i
\(39\) 1.11378i 0.0285584i
\(40\) 16.4894 − 6.93530i 0.412236 − 0.173382i
\(41\) 18.2956 − 31.6888i 0.446233 − 0.772899i −0.551904 − 0.833908i \(-0.686099\pi\)
0.998137 + 0.0610091i \(0.0194319\pi\)
\(42\) −1.61706 − 0.355424i −0.0385014 − 0.00846247i
\(43\) −30.8670 − 17.8211i −0.717837 − 0.414443i 0.0961190 − 0.995370i \(-0.469357\pi\)
−0.813956 + 0.580926i \(0.802690\pi\)
\(44\) −12.2984 − 17.4293i −0.279509 − 0.396120i
\(45\) 20.0472 0.445493
\(46\) 19.7723 + 18.0523i 0.429833 + 0.392441i
\(47\) −39.7406 + 22.9442i −0.845544 + 0.488175i −0.859145 − 0.511733i \(-0.829004\pi\)
0.0136010 + 0.999908i \(0.495671\pi\)
\(48\) −2.26680 + 1.93021i −0.0472249 + 0.0402127i
\(49\) 29.2080 0.596082
\(50\) 6.74254 − 7.38499i 0.134851 − 0.147700i
\(51\) −1.76329 + 1.01804i −0.0345744 + 0.0199615i
\(52\) 21.7354 + 10.0397i 0.417989 + 0.193072i
\(53\) −25.0258 − 43.3460i −0.472185 − 0.817848i 0.527308 − 0.849674i \(-0.323201\pi\)
−0.999493 + 0.0318256i \(0.989868\pi\)
\(54\) −6.37285 + 2.02204i −0.118016 + 0.0374452i
\(55\) −10.3270 − 5.96232i −0.187764 − 0.108406i
\(56\) −21.5125 + 28.3531i −0.384152 + 0.506306i
\(57\) 1.09854 + 3.36050i 0.0192727 + 0.0589561i
\(58\) 52.6458 − 57.6621i 0.907687 − 0.994174i
\(59\) −15.8956 − 9.17731i −0.269416 − 0.155548i 0.359206 − 0.933258i \(-0.383048\pi\)
−0.628622 + 0.777711i \(0.716381\pi\)
\(60\) −0.697914 + 1.51094i −0.0116319 + 0.0251823i
\(61\) −52.2132 − 90.4360i −0.855955 − 1.48256i −0.875756 − 0.482753i \(-0.839637\pi\)
0.0198015 − 0.999804i \(-0.493697\pi\)
\(62\) −71.3057 + 22.6246i −1.15009 + 0.364913i
\(63\) −34.5417 + 19.9427i −0.548281 + 0.316550i
\(64\) 17.2349 + 61.6357i 0.269295 + 0.963058i
\(65\) 13.3840 0.205908
\(66\) 1.93839 + 0.426052i 0.0293696 + 0.00645533i
\(67\) −89.2934 + 51.5536i −1.33274 + 0.769456i −0.985718 − 0.168402i \(-0.946139\pi\)
−0.347019 + 0.937858i \(0.612806\pi\)
\(68\) 3.97247 + 43.5874i 0.0584187 + 0.640992i
\(69\) −2.49100 −0.0361015
\(70\) −4.27105 + 19.4319i −0.0610150 + 0.277598i
\(71\) 101.073 + 58.3543i 1.42356 + 0.821891i 0.996601 − 0.0823814i \(-0.0262526\pi\)
0.426956 + 0.904272i \(0.359586\pi\)
\(72\) −8.97542 + 71.1592i −0.124659 + 0.988322i
\(73\) 19.9994 − 34.6400i 0.273965 − 0.474521i −0.695909 − 0.718130i \(-0.744998\pi\)
0.969873 + 0.243610i \(0.0783315\pi\)
\(74\) 9.32097 + 29.3768i 0.125959 + 0.396984i
\(75\) 0.930393i 0.0124052i
\(76\) 75.4825 + 8.85386i 0.993191 + 0.116498i
\(77\) 23.7249 0.308116
\(78\) −2.12324 + 0.673684i −0.0272211 + 0.00863698i
\(79\) −45.6586 − 26.3610i −0.577957 − 0.333684i 0.182364 − 0.983231i \(-0.441625\pi\)
−0.760321 + 0.649547i \(0.774958\pi\)
\(80\) 23.1949 + 27.2396i 0.289936 + 0.340495i
\(81\) −40.0332 + 69.3395i −0.494237 + 0.856043i
\(82\) 71.4761 + 15.7102i 0.871660 + 0.191587i
\(83\) 22.6330i 0.272687i 0.990662 + 0.136344i \(0.0435351\pi\)
−0.990662 + 0.136344i \(0.956465\pi\)
\(84\) −0.300542 − 3.29765i −0.00357788 − 0.0392578i
\(85\) 12.2336 + 21.1891i 0.143924 + 0.249284i
\(86\) 15.3027 − 69.6224i 0.177939 − 0.809562i
\(87\) 7.26452i 0.0835002i
\(88\) 25.7873 − 33.9873i 0.293038 − 0.386219i
\(89\) −39.7688 − 68.8816i −0.446840 − 0.773950i 0.551338 − 0.834282i \(-0.314118\pi\)
−0.998178 + 0.0603317i \(0.980784\pi\)
\(90\) 12.1258 + 38.2168i 0.134731 + 0.424631i
\(91\) −23.0610 + 13.3143i −0.253417 + 0.146311i
\(92\) −22.4542 + 48.6120i −0.244068 + 0.528391i
\(93\) 3.48009 − 6.02769i 0.0374203 − 0.0648138i
\(94\) −67.7771 − 61.8810i −0.721033 − 0.658308i
\(95\) 40.3824 − 13.2010i 0.425077 − 0.138957i
\(96\) −5.05074 − 3.15377i −0.0526119 − 0.0328518i
\(97\) −70.8339 + 122.688i −0.730246 + 1.26482i 0.226532 + 0.974004i \(0.427261\pi\)
−0.956778 + 0.290820i \(0.906072\pi\)
\(98\) 17.6669 + 55.6805i 0.180274 + 0.568168i
\(99\) 41.4056 − 23.9056i 0.418239 − 0.241470i
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.q.a.11.49 yes 160
4.3 odd 2 inner 380.3.q.a.11.6 ✓ 160
19.7 even 3 inner 380.3.q.a.311.6 yes 160
76.7 odd 6 inner 380.3.q.a.311.49 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.q.a.11.6 ✓ 160 4.3 odd 2 inner
380.3.q.a.11.49 yes 160 1.1 even 1 trivial
380.3.q.a.311.6 yes 160 19.7 even 3 inner
380.3.q.a.311.49 yes 160 76.7 odd 6 inner