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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 11.6
Character \(\chi\) \(=\) 380.11
Dual form 380.3.q.a.311.6

$q$-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+(-1.95337 + 0.429344i) q^{2} +(0.161149 + 0.0930393i) q^{3} +(3.63133 - 1.67734i) q^{4} +(-1.11803 + 1.93649i) q^{5} +(-0.354730 - 0.112552i) q^{6} +4.44882i q^{7} +(-6.37318 + 4.83555i) q^{8} +(-4.48269 - 7.76424i) q^{9} +(1.35252 - 4.26271i) q^{10} -5.33286i q^{11} +(0.741243 + 0.0675554i) q^{12} +(-2.99276 - 5.18362i) q^{13} +(-1.91007 - 8.69020i) q^{14} +(-0.360340 + 0.208042i) q^{15} +(10.3731 - 12.1819i) 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} +(2.28963 + 10.4171i) q^{22} +(-11.5933 + 6.69342i) q^{23} +(-1.47693 + 0.186287i) q^{24} +(-2.50000 - 4.33013i) q^{25} +(8.07154 + 8.84061i) q^{26} -3.34297i q^{27} +(7.46217 + 16.1551i) q^{28} +(-19.5200 - 33.8097i) q^{29} +(0.614556 - 0.561094i) q^{30} -37.4045i q^{31} +(-15.0323 + 28.2494i) q^{32} +(0.496166 - 0.859385i) q^{33} +(-6.61843 + 20.8592i) q^{34} +(-8.61510 - 4.97393i) q^{35} +(-29.3014 - 20.6755i) q^{36} +15.4100 q^{37} +(-33.0731 - 18.7128i) q^{38} -1.11378i q^{39} +(-2.23857 - 17.7479i) q^{40} +(18.2956 - 31.6888i) q^{41} +(0.500724 - 1.57813i) q^{42} +(30.8670 + 17.8211i) q^{43} +(-8.94501 - 19.3654i) q^{44} +20.0472 q^{45} +(19.7723 - 18.0523i) q^{46} +(39.7406 - 22.9442i) q^{47} +(2.80501 - 0.997998i) q^{48} +29.2080 q^{49} +(6.74254 + 7.38499i) q^{50} +(1.76329 - 1.01804i) q^{51} +(-19.5624 - 13.8035i) q^{52} +(-25.0258 - 43.3460i) q^{53} +(1.43529 + 6.53007i) 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} +(-0.959555 + 1.35988i) q^{60} +(-52.2132 - 90.4360i) q^{61} +(16.0594 + 73.0648i) q^{62} +(34.5417 - 19.9427i) q^{63} +(17.2349 - 61.6357i) q^{64} +13.3840 q^{65} +(-0.600226 + 1.89172i) q^{66} +(89.2934 - 51.5536i) q^{67} +(3.97247 - 43.5874i) q^{68} -2.49100 q^{69} +(18.9640 + 6.01710i) q^{70} +(-101.073 - 58.3543i) q^{71} +(66.1134 + 27.8067i) q^{72} +(19.9994 - 34.6400i) q^{73} +(-30.1015 + 6.61620i) q^{74} -0.930393i q^{75} +(72.6384 + 22.3533i) q^{76} +23.7249 q^{77} +(0.478194 + 2.17562i) q^{78} +(45.6586 + 26.3610i) q^{79} +(11.9927 + 33.7072i) q^{80} +(-40.0332 + 69.3395i) q^{81} +(-22.1326 + 69.7552i) q^{82} -22.6330i q^{83} +(-0.300542 + 3.29765i) q^{84} +(12.2336 + 21.1891i) q^{85} +(-67.9461 - 21.5586i) q^{86} -7.26452i q^{87} +(25.7873 + 33.9873i) q^{88} +(-39.7688 - 68.8816i) q^{89} +(-39.1596 + 8.60714i) q^{90} +(23.0610 - 13.3143i) q^{91} +(-30.8721 + 43.7519i) 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} +(-57.0541 + 12.5403i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.95337 + 0.429344i −0.976686 + 0.214672i
\(3\) 0.161149 + 0.0930393i 0.0537163 + 0.0310131i 0.526618 − 0.850102i \(-0.323460\pi\)
−0.472901 + 0.881115i \(0.656793\pi\)
\(4\) 3.63133 − 1.67734i 0.907832 − 0.419334i
\(5\) −1.11803 + 1.93649i −0.223607 + 0.387298i
\(6\) −0.354730 − 0.112552i −0.0591216 − 0.0187587i
\(7\) 4.44882i 0.635545i 0.948167 + 0.317773i \(0.102935\pi\)
−0.948167 + 0.317773i \(0.897065\pi\)
\(8\) −6.37318 + 4.83555i −0.796648 + 0.604444i
\(9\) −4.48269 − 7.76424i −0.498076 − 0.862694i
\(10\) 1.35252 − 4.26271i 0.135252 − 0.426271i
\(11\) − 5.33286i − 0.484806i −0.970176 − 0.242403i \(-0.922064\pi\)
0.970176 − 0.242403i \(-0.0779356\pi\)
\(12\) 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 \(0.907275\pi\)
\(14\) −1.91007 − 8.69020i −0.136434 − 0.620728i
\(15\) −0.360340 + 0.208042i −0.0240227 + 0.0138695i
\(16\) 10.3731 − 12.1819i 0.648318 − 0.761370i
\(17\) 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
\(22\) 2.28963 + 10.4171i 0.104074 + 0.473503i
\(23\) −11.5933 + 6.69342i −0.504058 + 0.291018i −0.730388 − 0.683033i \(-0.760661\pi\)
0.226330 + 0.974051i \(0.427327\pi\)
\(24\) −1.47693 + 0.186287i −0.0615386 + 0.00776197i
\(25\) −2.50000 − 4.33013i −0.100000 − 0.173205i
\(26\) 8.07154 + 8.84061i 0.310444 + 0.340024i
\(27\) − 3.34297i − 0.123814i
\(28\) 7.46217 + 16.1551i 0.266506 + 0.576968i
\(29\) −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 \(-0.793853\pi\)
0.797516 − 0.603298i \(-0.206147\pi\)
\(32\) −15.0323 + 28.2494i −0.469758 + 0.882795i
\(33\) 0.496166 − 0.859385i 0.0150353 − 0.0260420i
\(34\) −6.61843 + 20.8592i −0.194660 + 0.613507i
\(35\) −8.61510 − 4.97393i −0.246146 − 0.142112i
\(36\) −29.3014 − 20.6755i −0.813927 − 0.574320i
\(37\) 15.4100 0.416487 0.208244 − 0.978077i \(-0.433225\pi\)
0.208244 + 0.978077i \(0.433225\pi\)
\(38\) −33.0731 − 18.7128i −0.870345 − 0.492442i
\(39\) − 1.11378i − 0.0285584i
\(40\) −2.23857 − 17.7479i −0.0559643 − 0.443698i
\(41\) 18.2956 − 31.6888i 0.446233 − 0.772899i −0.551904 − 0.833908i \(-0.686099\pi\)
0.998137 + 0.0610091i \(0.0194319\pi\)
\(42\) 0.500724 − 1.57813i 0.0119220 − 0.0375745i
\(43\) 30.8670 + 17.8211i 0.717837 + 0.414443i 0.813956 − 0.580926i \(-0.197310\pi\)
−0.0961190 + 0.995370i \(0.530643\pi\)
\(44\) −8.94501 − 19.3654i −0.203296 − 0.440122i
\(45\) 20.0472 0.445493
\(46\) 19.7723 − 18.0523i 0.429833 − 0.392441i
\(47\) 39.7406 − 22.9442i 0.845544 − 0.488175i −0.0136010 − 0.999908i \(-0.504329\pi\)
0.859145 + 0.511733i \(0.170996\pi\)
\(48\) 2.80501 − 0.997998i 0.0584377 − 0.0207916i
\(49\) 29.2080 0.596082
\(50\) 6.74254 + 7.38499i 0.134851 + 0.147700i
\(51\) 1.76329 − 1.01804i 0.0345744 − 0.0199615i
\(52\) −19.5624 − 13.8035i −0.376200 − 0.265453i
\(53\) −25.0258 − 43.3460i −0.472185 − 0.817848i 0.527308 − 0.849674i \(-0.323201\pi\)
−0.999493 + 0.0318256i \(0.989868\pi\)
\(54\) 1.43529 + 6.53007i 0.0265794 + 0.120927i
\(55\) 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 0.628622 − 0.777711i \(-0.283619\pi\)
−0.359206 + 0.933258i \(0.616952\pi\)
\(60\) −0.959555 + 1.35988i −0.0159926 + 0.0226647i
\(61\) −52.2132 − 90.4360i −0.855955 − 1.48256i −0.875756 − 0.482753i \(-0.839637\pi\)
0.0198015 − 0.999804i \(-0.493697\pi\)
\(62\) 16.0594 + 73.0648i 0.259022 + 1.17847i
\(63\) 34.5417 − 19.9427i 0.548281 − 0.316550i
\(64\) 17.2349 − 61.6357i 0.269295 − 0.963058i
\(65\) 13.3840 0.205908
\(66\) −0.600226 + 1.89172i −0.00909433 + 0.0286625i
\(67\) 89.2934 − 51.5536i 1.33274 − 0.769456i 0.347019 − 0.937858i \(-0.387194\pi\)
0.985718 + 0.168402i \(0.0538607\pi\)
\(68\) 3.97247 − 43.5874i 0.0584187 − 0.640992i
\(69\) −2.49100 −0.0361015
\(70\) 18.9640 + 6.01710i 0.270915 + 0.0859585i
\(71\) −101.073 − 58.3543i −1.42356 − 0.821891i −0.426956 − 0.904272i \(-0.640414\pi\)
−0.996601 + 0.0823814i \(0.973747\pi\)
\(72\) 66.1134 + 27.8067i 0.918241 + 0.386203i
\(73\) 19.9994 − 34.6400i 0.273965 − 0.474521i −0.695909 − 0.718130i \(-0.744998\pi\)
0.969873 + 0.243610i \(0.0783315\pi\)
\(74\) −30.1015 + 6.61620i −0.406777 + 0.0894082i
\(75\) − 0.930393i − 0.0124052i
\(76\) 72.6384 + 22.3533i 0.955768 + 0.294122i
\(77\) 23.7249 0.308116
\(78\) 0.478194 + 2.17562i 0.00613069 + 0.0278926i
\(79\) 45.6586 + 26.3610i 0.577957 + 0.333684i 0.760321 − 0.649547i \(-0.225042\pi\)
−0.182364 + 0.983231i \(0.558375\pi\)
\(80\) 11.9927 + 33.7072i 0.149909 + 0.421340i
\(81\) −40.0332 + 69.3395i −0.494237 + 0.856043i
\(82\) −22.1326 + 69.7552i −0.269910 + 0.850673i
\(83\) − 22.6330i − 0.272687i −0.990662 − 0.136344i \(-0.956465\pi\)
0.990662 − 0.136344i \(-0.0435351\pi\)
\(84\) −0.300542 + 3.29765i −0.00357788 + 0.0392578i
\(85\) 12.2336 + 21.1891i 0.143924 + 0.249284i
\(86\) −67.9461 − 21.5586i −0.790071 − 0.250682i
\(87\) − 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 \(0.980784\pi\)
\(90\) −39.1596 + 8.60714i −0.435107 + 0.0956349i
\(91\) 23.0610 − 13.3143i 0.253417 − 0.146311i
\(92\) −30.8721 + 43.7519i −0.335566 + 0.475565i
\(93\) 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 \(0.906072\pi\)
\(98\) −57.0541 + 12.5403i −0.582185 + 0.127962i
\(99\) −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.11.6 ✓ 160
4.3 odd 2 inner 380.3.q.a.11.49 yes 160
19.7 even 3 inner 380.3.q.a.311.49 yes 160
76.7 odd 6 inner 380.3.q.a.311.6 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.q.a.11.6 ✓ 160 1.1 even 1 trivial
380.3.q.a.11.49 yes 160 4.3 odd 2 inner
380.3.q.a.311.6 yes 160 76.7 odd 6 inner
380.3.q.a.311.49 yes 160 19.7 even 3 inner