↑

Properties

Label 380.3.q.a.11.3
Level $380$
Weight $3$
Character 380.11
Analytic conductor $10.354$
Analytic rank $0$
Dimension $160$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

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 11.3
Character \(\chi\) \(=\) 380.11
Dual form 380.3.q.a.311.3

$q$-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+(-1.99167 - 0.182322i) q^{2} +(2.87776 + 1.66148i) q^{3} +(3.93352 + 0.726252i) q^{4} +(1.11803 - 1.93649i) q^{5} +(-5.42864 - 3.83380i) q^{6} +8.40212i q^{7} +(-7.70187 - 2.16362i) q^{8} +(1.02102 + 1.76846i) q^{9} +(-2.57982 + 3.65301i) q^{10} -15.0813i q^{11} +(10.1131 + 8.62544i) q^{12} +(12.4542 + 21.5712i) q^{13} +(1.53189 - 16.7343i) q^{14} +(6.43488 - 3.71518i) q^{15} +(14.9451 + 5.71345i) q^{16} +(-12.9899 + 22.4991i) q^{17} +(-1.71111 - 3.70835i) q^{18} +(10.9898 + 15.4992i) q^{19} +(5.80419 - 6.80525i) q^{20} +(-13.9599 + 24.1793i) q^{21} +(-2.74965 + 30.0369i) q^{22} +(20.3638 - 11.7570i) q^{23} +(-18.5693 - 19.0229i) q^{24} +(-2.50000 - 4.33013i) q^{25} +(-20.8717 - 45.2335i) q^{26} -23.1210i q^{27} +(-6.10205 + 33.0499i) q^{28} +(2.36143 + 4.09012i) q^{29} +(-13.4935 + 6.22620i) q^{30} +7.07389i q^{31} +(-28.7241 - 14.1041i) q^{32} +(25.0572 - 43.4003i) q^{33} +(29.9737 - 42.4426i) q^{34} +(16.2706 + 9.39385i) q^{35} +(2.73185 + 7.69778i) q^{36} +11.2361 q^{37} +(-19.0623 - 32.8729i) q^{38} +82.7693i q^{39} +(-12.8008 + 12.4956i) q^{40} +(-26.9163 + 46.6204i) q^{41} +(32.2120 - 45.6121i) q^{42} +(-21.1892 - 12.2336i) q^{43} +(10.9528 - 59.3224i) q^{44} +4.56614 q^{45} +(-42.7016 + 19.7034i) q^{46} +(2.66879 - 1.54083i) q^{47} +(33.5158 + 41.2729i) q^{48} -21.5956 q^{49} +(4.18970 + 9.08000i) q^{50} +(-74.7637 + 43.1648i) q^{51} +(33.3225 + 93.8957i) q^{52} +(25.8957 + 44.8527i) q^{53} +(-4.21547 + 46.0494i) q^{54} +(-29.2047 - 16.8614i) q^{55} +(18.1790 - 64.7120i) q^{56} +(5.87461 + 62.8623i) q^{57} +(-3.95748 - 8.57672i) q^{58} +(79.7845 + 46.0636i) q^{59} +(28.0099 - 9.94038i) q^{60} +(-4.86547 - 8.42725i) q^{61} +(1.28973 - 14.0889i) q^{62} +(-14.8588 + 8.57873i) q^{63} +(54.6375 + 33.3279i) q^{64} +55.6967 q^{65} +(-57.8185 + 81.8707i) q^{66} +(73.6716 - 42.5343i) q^{67} +(-67.4360 + 79.0669i) q^{68} +78.1362 q^{69} +(-30.6931 - 21.6760i) q^{70} +(-51.8723 - 29.9485i) q^{71} +(-4.03748 - 15.8295i) q^{72} +(-54.2612 + 93.9832i) q^{73} +(-22.3787 - 2.04860i) q^{74} -16.6148i q^{75} +(31.9724 + 68.9476i) q^{76} +126.714 q^{77} +(15.0907 - 164.849i) q^{78} +(-50.0890 - 28.9189i) q^{79} +(27.7732 - 22.5533i) q^{80} +(47.6042 - 82.4529i) q^{81} +(62.1084 - 87.9451i) q^{82} -5.00562i q^{83} +(-72.4719 + 84.9713i) q^{84} +(29.0463 + 50.3096i) q^{85} +(39.9715 + 28.2286i) q^{86} +15.6939i q^{87} +(-32.6301 + 116.154i) q^{88} +(-26.0954 - 45.1985i) q^{89} +(-9.09426 - 0.832509i) q^{90} +(-181.244 + 104.641i) q^{91} +(88.6399 - 31.4573i) q^{92} +(-11.7531 + 20.3570i) q^{93} +(-5.59629 + 2.58225i) q^{94} +(42.3010 - 3.95311i) q^{95} +(-59.2274 - 88.3129i) q^{96} +(58.0139 - 100.483i) q^{97} +(43.0113 + 3.93735i) q^{98} +(26.6706 - 15.3983i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.99167 − 0.182322i −0.995836 − 0.0911611i
\(3\) 2.87776 + 1.66148i 0.959255 + 0.553826i 0.895944 − 0.444168i \(-0.146501\pi\)
0.0633113 + 0.997994i \(0.479834\pi\)
\(4\) 3.93352 + 0.726252i 0.983379 + 0.181563i
\(5\) 1.11803 − 1.93649i 0.223607 − 0.387298i
\(6\) −5.42864 − 3.83380i −0.904773 − 0.638967i
\(7\) 8.40212i 1.20030i 0.799887 + 0.600151i \(0.204893\pi\)
−0.799887 + 0.600151i \(0.795107\pi\)
\(8\) −7.70187 − 2.16362i −0.962733 − 0.270453i
\(9\) 1.02102 + 1.76846i 0.113447 + 0.196495i
\(10\) −2.57982 + 3.65301i −0.257982 + 0.365301i
\(11\) − 15.0813i − 1.37102i −0.728062 − 0.685511i \(-0.759579\pi\)
0.728062 − 0.685511i \(-0.240421\pi\)
\(12\) 10.1131 + 8.62544i 0.842757 + 0.718786i
\(13\) 12.4542 + 21.5712i 0.958013 + 1.65933i 0.727318 + 0.686300i \(0.240766\pi\)
0.230695 + 0.973026i \(0.425900\pi\)
\(14\) 1.53189 − 16.7343i 0.109421 − 1.19530i
\(15\) 6.43488 − 3.71518i 0.428992 − 0.247679i
\(16\) 14.9451 + 5.71345i 0.934070 + 0.357091i
\(17\) −12.9899 + 22.4991i −0.764111 + 1.32348i 0.176604 + 0.984282i \(0.443489\pi\)
−0.940715 + 0.339197i \(0.889845\pi\)
\(18\) −1.71111 − 3.70835i −0.0950616 − 0.206019i
\(19\) 10.9898 + 15.4992i 0.578412 + 0.815745i
\(20\) 5.80419 − 6.80525i 0.290209 − 0.340262i
\(21\) −13.9599 + 24.1793i −0.664759 + 1.15140i
\(22\) −2.74965 + 30.0369i −0.124984 + 1.36531i
\(23\) 20.3638 − 11.7570i 0.885382 − 0.511175i 0.0129528 − 0.999916i \(-0.495877\pi\)
0.872429 + 0.488741i \(0.162544\pi\)
\(24\) −18.5693 − 19.0229i −0.773723 − 0.792620i
\(25\) −2.50000 − 4.33013i −0.100000 − 0.173205i
\(26\) −20.8717 − 45.2335i −0.802758 − 1.73975i
\(27\) − 23.1210i − 0.856333i
\(28\) −6.10205 + 33.0499i −0.217930 + 1.18035i
\(29\) 2.36143 + 4.09012i 0.0814287 + 0.141039i 0.903864 − 0.427820i \(-0.140718\pi\)
−0.822435 + 0.568859i \(0.807385\pi\)
\(30\) −13.4935 + 6.22620i −0.449784 + 0.207540i
\(31\) 7.07389i 0.228190i 0.993470 + 0.114095i \(0.0363968\pi\)
−0.993470 + 0.114095i \(0.963603\pi\)
\(32\) −28.7241 − 14.1041i −0.897628 − 0.440754i
\(33\) 25.0572 − 43.4003i 0.759308 − 1.31516i
\(34\) 29.9737 − 42.4426i 0.881579 − 1.24831i
\(35\) 16.2706 + 9.39385i 0.464875 + 0.268396i
\(36\) 2.73185 + 7.69778i 0.0758848 + 0.213827i
\(37\) 11.2361 0.303679 0.151840 − 0.988405i \(-0.451480\pi\)
0.151840 + 0.988405i \(0.451480\pi\)
\(38\) −19.0623 − 32.8729i −0.501639 − 0.865077i
\(39\) 82.7693i 2.12229i
\(40\) −12.8008 + 12.4956i −0.320020 + 0.312390i
\(41\) −26.9163 + 46.6204i −0.656495 + 1.13708i 0.325022 + 0.945706i \(0.394628\pi\)
−0.981517 + 0.191376i \(0.938705\pi\)
\(42\) 32.2120 − 45.6121i 0.766953 − 1.08600i
\(43\) −21.1892 − 12.2336i −0.492773 − 0.284502i 0.232951 − 0.972488i \(-0.425162\pi\)
−0.725724 + 0.687986i \(0.758495\pi\)
\(44\) 10.9528 − 59.3224i 0.248927 − 1.34824i
\(45\) 4.56614 0.101470
\(46\) −42.7016 + 19.7034i −0.928295 + 0.428335i
\(47\) 2.66879 − 1.54083i 0.0567829 − 0.0327836i −0.471340 − 0.881952i \(-0.656229\pi\)
0.528123 + 0.849168i \(0.322896\pi\)
\(48\) 33.5158 + 41.2729i 0.698245 + 0.859853i
\(49\) −21.5956 −0.440726
\(50\) 4.18970 + 9.08000i 0.0837941 + 0.181600i
\(51\) −74.7637 + 43.1648i −1.46595 + 0.846369i
\(52\) 33.3225 + 93.8957i 0.640818 + 1.80569i
\(53\) 25.8957 + 44.8527i 0.488598 + 0.846277i 0.999914 − 0.0131162i \(-0.00417513\pi\)
−0.511316 + 0.859393i \(0.670842\pi\)
\(54\) −4.21547 + 46.0494i −0.0780642 + 0.852768i
\(55\) −29.2047 − 16.8614i −0.530995 − 0.306570i
\(56\) 18.1790 − 64.7120i 0.324625 − 1.15557i
\(57\) 5.87461 + 62.8623i 0.103063 + 1.10285i
\(58\) −3.95748 − 8.57672i −0.0682324 − 0.147875i
\(59\) 79.7845 + 46.0636i 1.35228 + 0.780739i 0.988568 − 0.150773i \(-0.0481764\pi\)
0.363710 + 0.931512i \(0.381510\pi\)
\(60\) 28.0099 − 9.94038i 0.466831 − 0.165673i
\(61\) −4.86547 − 8.42725i −0.0797619 − 0.138152i 0.823385 − 0.567483i \(-0.192083\pi\)
−0.903147 + 0.429331i \(0.858749\pi\)
\(62\) 1.28973 − 14.0889i 0.0208020 − 0.227240i
\(63\) −14.8588 + 8.57873i −0.235854 + 0.136170i
\(64\) 54.6375 + 33.3279i 0.853710 + 0.520748i
\(65\) 55.6967 0.856873
\(66\) −57.8185 + 81.8707i −0.876038 + 1.24046i
\(67\) 73.6716 − 42.5343i 1.09958 − 0.634841i 0.163466 − 0.986549i \(-0.447732\pi\)
0.936110 + 0.351708i \(0.114399\pi\)
\(68\) −67.4360 + 79.0669i −0.991706 + 1.16275i
\(69\) 78.1362 1.13241
\(70\) −30.6931 − 21.6760i −0.438472 − 0.309657i
\(71\) −51.8723 − 29.9485i −0.730595 − 0.421809i 0.0880447 − 0.996117i \(-0.471938\pi\)
−0.818640 + 0.574307i \(0.805271\pi\)
\(72\) −4.03748 − 15.8295i −0.0560761 − 0.219855i
\(73\) −54.2612 + 93.9832i −0.743304 + 1.28744i 0.207678 + 0.978197i \(0.433409\pi\)
−0.950983 + 0.309244i \(0.899924\pi\)
\(74\) −22.3787 − 2.04860i −0.302415 − 0.0276837i
\(75\) − 16.6148i − 0.221530i
\(76\) 31.9724 + 68.9476i 0.420689 + 0.907205i
\(77\) 126.714 1.64564
\(78\) 15.0907 − 164.849i 0.193470 − 2.11345i
\(79\) −50.0890 − 28.9189i −0.634038 − 0.366062i 0.148276 − 0.988946i \(-0.452627\pi\)
−0.782314 + 0.622884i \(0.785961\pi\)
\(80\) 27.7732 − 22.5533i 0.347165 − 0.281916i
\(81\) 47.6042 − 82.4529i 0.587706 − 1.01794i
\(82\) 62.1084 − 87.9451i 0.757419 − 1.07250i
\(83\) − 5.00562i − 0.0603087i −0.999545 − 0.0301544i \(-0.990400\pi\)
0.999545 − 0.0301544i \(-0.00959989\pi\)
\(84\) −72.4719 + 84.9713i −0.862761 + 1.01156i
\(85\) 29.0463 + 50.3096i 0.341721 + 0.591878i
\(86\) 39.9715 + 28.2286i 0.464785 + 0.328239i
\(87\) 15.6939i 0.180389i
\(88\) −32.6301 + 116.154i −0.370797 + 1.31993i
\(89\) −26.0954 − 45.1985i −0.293206 − 0.507848i 0.681360 − 0.731949i \(-0.261389\pi\)
−0.974566 + 0.224100i \(0.928056\pi\)
\(90\) −9.09426 − 0.832509i −0.101047 − 0.00925010i
\(91\) −181.244 + 104.641i −1.99169 + 1.14990i
\(92\) 88.6399 − 31.4573i 0.963477 − 0.341927i
\(93\) −11.7531 + 20.3570i −0.126378 + 0.218892i
\(94\) −5.59629 + 2.58225i −0.0595350 + 0.0274707i
\(95\) 42.3010 − 3.95311i 0.445273 − 0.0416117i
\(96\) −59.2274 − 88.3129i −0.616952 − 0.919926i
\(97\) 58.0139 − 100.483i 0.598082 − 1.03591i −0.395022 − 0.918672i \(-0.629263\pi\)
0.993104 − 0.117236i \(-0.0374035\pi\)
\(98\) 43.0113 + 3.93735i 0.438891 + 0.0401770i
\(99\) 26.6706 − 15.3983i 0.269400 − 0.155538i
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.11.3 ✓ 160
4.3 odd 2 inner 380.3.q.a.11.56 yes 160
19.7 even 3 inner 380.3.q.a.311.56 yes 160
76.7 odd 6 inner 380.3.q.a.311.3 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.q.a.11.3 ✓ 160 1.1 even 1 trivial
380.3.q.a.11.56 yes 160 4.3 odd 2 inner
380.3.q.a.311.3 yes 160 76.7 odd 6 inner
380.3.q.a.311.56 yes 160 19.7 even 3 inner