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Properties

Label 380.3.j.a.227.16
Level $380$
Weight $3$
Character 380.227
Analytic conductor $10.354$
Analytic rank $0$
Dimension $232$
Inner twists $8$

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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(227,380)] 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("380.227"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 2])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.j (of order \(4\), 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 \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 227.16
Character \(\chi\) \(=\) 380.227
Dual form 380.3.j.a.303.16

$q$-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+(-1.83249 - 0.801229i) q^{2} +(-3.50213 + 3.50213i) q^{3} +(2.71606 + 2.93649i) q^{4} +(-0.641092 - 4.95873i) q^{5} +(9.22364 - 3.61162i) q^{6} +(2.56617 - 2.56617i) q^{7} +(-2.62436 - 7.55730i) q^{8} -15.5298i q^{9} +(-2.79828 + 9.60050i) q^{10} +18.6927i q^{11} +(-19.7960 - 0.771980i) q^{12} +(1.33481 + 1.33481i) q^{13} +(-6.75857 + 2.64640i) q^{14} +(19.6113 + 15.1209i) q^{15} +(-1.24600 + 15.9514i) q^{16} +(2.61596 + 2.61596i) q^{17} +(-12.4430 + 28.4583i) q^{18} +(10.2483 + 15.9991i) q^{19} +(12.8200 - 15.3508i) q^{20} +17.9741i q^{21} +(14.9771 - 34.2542i) q^{22} +(-22.6151 - 22.6151i) q^{23} +(35.6575 + 17.2758i) q^{24} +(-24.1780 + 6.35801i) q^{25} +(-1.37654 - 3.51553i) q^{26} +(22.8684 + 22.8684i) q^{27} +(14.5054 + 0.565664i) q^{28} +8.57971 q^{29} +(-23.8223 - 43.4222i) q^{30} -13.0055 q^{31} +(15.0640 - 28.2325i) q^{32} +(-65.4642 - 65.4642i) q^{33} +(-2.69775 - 6.88971i) q^{34} +(-14.3701 - 11.0798i) q^{35} +(45.6033 - 42.1800i) q^{36} +(-4.50579 + 4.50579i) q^{37} +(-5.96098 - 37.5295i) q^{38} -9.34938 q^{39} +(-35.7921 + 17.8584i) q^{40} -69.0095i q^{41} +(14.4014 - 32.9374i) q^{42} +(-33.7742 - 33.7742i) q^{43} +(-54.8910 + 50.7705i) q^{44} +(-77.0083 + 9.95607i) q^{45} +(23.3221 + 59.5618i) q^{46} +(-22.4191 + 22.4191i) q^{47} +(-51.5003 - 60.2276i) q^{48} +35.8296i q^{49} +(49.4002 + 7.72111i) q^{50} -18.3229 q^{51} +(-0.294235 + 7.54510i) q^{52} +(-21.5312 - 21.5312i) q^{53} +(-23.5833 - 60.2290i) q^{54} +(92.6920 - 11.9837i) q^{55} +(-26.1278 - 12.6587i) q^{56} +(-91.9220 - 20.1401i) q^{57} +(-15.7223 - 6.87432i) q^{58} -41.7519i q^{59} +(8.86303 + 98.6579i) q^{60} +27.3710 q^{61} +(23.8325 + 10.4204i) q^{62} +(-39.8522 - 39.8522i) q^{63} +(-50.2254 + 39.6662i) q^{64} +(5.76324 - 7.47471i) q^{65} +(67.5109 + 172.415i) q^{66} +(22.2473 + 22.2473i) q^{67} +(-0.576640 + 14.7869i) q^{68} +158.402 q^{69} +(17.4556 + 31.8174i) q^{70} -126.266 q^{71} +(-117.364 + 40.7559i) q^{72} +(-60.8346 + 60.8346i) q^{73} +(11.8670 - 4.64666i) q^{74} +(62.4079 - 106.941i) q^{75} +(-19.1463 + 73.5487i) q^{76} +(47.9686 + 47.9686i) q^{77} +(17.1327 + 7.49099i) q^{78} -77.7490i q^{79} +(79.8975 - 4.04775i) q^{80} -20.4075 q^{81} +(-55.2925 + 126.460i) q^{82} +(-93.8670 - 93.8670i) q^{83} +(-52.7809 + 48.8188i) q^{84} +(11.2948 - 14.6489i) q^{85} +(34.8302 + 88.9520i) q^{86} +(-30.0473 + 30.0473i) q^{87} +(141.266 - 49.0564i) q^{88} -144.779 q^{89} +(149.094 + 43.4569i) q^{90} +6.85071 q^{91} +(4.98507 - 127.833i) q^{92} +(45.5470 - 45.5470i) q^{93} +(59.0456 - 23.1200i) q^{94} +(72.7653 - 61.0755i) q^{95} +(46.1178 + 151.630i) q^{96} +(47.3733 - 47.3733i) q^{97} +(28.7077 - 65.6574i) q^{98} +290.294 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 232 q - 8 q^{5} - 32 q^{6} - 64 q^{16} - 8 q^{17} + 80 q^{20} - 8 q^{25} + 40 q^{26} + 40 q^{28} - 104 q^{30} + 104 q^{36} + 184 q^{38} + 192 q^{42} + 192 q^{45} + 280 q^{58} + 112 q^{61} - 280 q^{62} + 216 q^{66}+ \cdots + 504 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.83249 − 0.801229i −0.916247 − 0.400615i
\(3\) −3.50213 + 3.50213i −1.16738 + 1.16738i −0.184555 + 0.982822i \(0.559084\pi\)
−0.982822 + 0.184555i \(0.940916\pi\)
\(4\) 2.71606 + 2.93649i 0.679016 + 0.734124i
\(5\) −0.641092 − 4.95873i −0.128218 − 0.991746i
\(6\) 9.22364 − 3.61162i 1.53727 − 0.601937i
\(7\) 2.56617 − 2.56617i 0.366595 − 0.366595i −0.499639 − 0.866234i \(-0.666534\pi\)
0.866234 + 0.499639i \(0.166534\pi\)
\(8\) −2.62436 − 7.55730i −0.328045 − 0.944662i
\(9\) − 15.5298i − 1.72554i
\(10\) −2.79828 + 9.60050i −0.279828 + 0.960050i
\(11\) 18.6927i 1.69933i 0.527319 + 0.849667i \(0.323197\pi\)
−0.527319 + 0.849667i \(0.676803\pi\)
\(12\) −19.7960 − 0.771980i −1.64967 − 0.0643316i
\(13\) 1.33481 + 1.33481i 0.102678 + 0.102678i 0.756580 − 0.653902i \(-0.226869\pi\)
−0.653902 + 0.756580i \(0.726869\pi\)
\(14\) −6.75857 + 2.64640i −0.482755 + 0.189028i
\(15\) 19.6113 + 15.1209i 1.30742 + 1.00806i
\(16\) −1.24600 + 15.9514i −0.0778751 + 0.996963i
\(17\) 2.61596 + 2.61596i 0.153880 + 0.153880i 0.779848 − 0.625968i \(-0.215296\pi\)
−0.625968 + 0.779848i \(0.715296\pi\)
\(18\) −12.4430 + 28.4583i −0.691276 + 1.58102i
\(19\) 10.2483 + 15.9991i 0.539385 + 0.842060i
\(20\) 12.8200 − 15.3508i 0.641002 − 0.767539i
\(21\) 17.9741i 0.855910i
\(22\) 14.9771 − 34.2542i 0.680778 − 1.55701i
\(23\) −22.6151 − 22.6151i −0.983263 − 0.983263i 0.0165990 − 0.999862i \(-0.494716\pi\)
−0.999862 + 0.0165990i \(0.994716\pi\)
\(24\) 35.6575 + 17.2758i 1.48573 + 0.719824i
\(25\) −24.1780 + 6.35801i −0.967120 + 0.254320i
\(26\) −1.37654 − 3.51553i −0.0529440 − 0.135213i
\(27\) 22.8684 + 22.8684i 0.846977 + 0.846977i
\(28\) 14.5054 + 0.565664i 0.518050 + 0.0202023i
\(29\) 8.57971 0.295852 0.147926 − 0.988998i \(-0.452740\pi\)
0.147926 + 0.988998i \(0.452740\pi\)
\(30\) −23.8223 − 43.4222i −0.794075 − 1.44741i
\(31\) −13.0055 −0.419533 −0.209766 − 0.977752i \(-0.567270\pi\)
−0.209766 + 0.977752i \(0.567270\pi\)
\(32\) 15.0640 − 28.2325i 0.470751 − 0.882266i
\(33\) −65.4642 − 65.4642i −1.98376 − 1.98376i
\(34\) −2.69775 − 6.88971i −0.0793454 − 0.202639i
\(35\) −14.3701 − 11.0798i −0.410574 − 0.316565i
\(36\) 45.6033 − 42.1800i 1.26676 − 1.17167i
\(37\) −4.50579 + 4.50579i −0.121778 + 0.121778i −0.765369 − 0.643591i \(-0.777444\pi\)
0.643591 + 0.765369i \(0.277444\pi\)
\(38\) −5.96098 − 37.5295i −0.156868 − 0.987620i
\(39\) −9.34938 −0.239728
\(40\) −35.7921 + 17.8584i −0.894803 + 0.446461i
\(41\) − 69.0095i − 1.68316i −0.540133 − 0.841580i \(-0.681626\pi\)
0.540133 − 0.841580i \(-0.318374\pi\)
\(42\) 14.4014 − 32.9374i 0.342890 − 0.784225i
\(43\) −33.7742 − 33.7742i −0.785447 − 0.785447i 0.195297 − 0.980744i \(-0.437433\pi\)
−0.980744 + 0.195297i \(0.937433\pi\)
\(44\) −54.8910 + 50.7705i −1.24752 + 1.15388i
\(45\) −77.0083 + 9.95607i −1.71130 + 0.221246i
\(46\) 23.3221 + 59.5618i 0.507002 + 1.29482i
\(47\) −22.4191 + 22.4191i −0.477001 + 0.477001i −0.904171 − 0.427170i \(-0.859511\pi\)
0.427170 + 0.904171i \(0.359511\pi\)
\(48\) −51.5003 − 60.2276i −1.07292 − 1.25474i
\(49\) 35.8296i 0.731216i
\(50\) 49.4002 + 7.72111i 0.988005 + 0.154422i
\(51\) −18.3229 −0.359272
\(52\) −0.294235 + 7.54510i −0.00565836 + 0.145098i
\(53\) −21.5312 − 21.5312i −0.406250 − 0.406250i 0.474179 − 0.880429i \(-0.342745\pi\)
−0.880429 + 0.474179i \(0.842745\pi\)
\(54\) −23.5833 − 60.2290i −0.436728 − 1.11535i
\(55\) 92.6920 − 11.9837i 1.68531 − 0.217886i
\(56\) −26.1278 − 12.6587i −0.466569 − 0.226049i
\(57\) −91.9220 − 20.1401i −1.61267 − 0.353336i
\(58\) −15.7223 − 6.87432i −0.271074 − 0.118523i
\(59\) − 41.7519i − 0.707660i −0.935310 − 0.353830i \(-0.884879\pi\)
0.935310 − 0.353830i \(-0.115121\pi\)
\(60\) 8.86303 + 98.6579i 0.147717 + 1.64430i
\(61\) 27.3710 0.448705 0.224352 − 0.974508i \(-0.427973\pi\)
0.224352 + 0.974508i \(0.427973\pi\)
\(62\) 23.8325 + 10.4204i 0.384395 + 0.168071i
\(63\) −39.8522 − 39.8522i −0.632574 − 0.632574i
\(64\) −50.2254 + 39.6662i −0.784773 + 0.619784i
\(65\) 5.76324 − 7.47471i 0.0886652 − 0.114996i
\(66\) 67.5109 + 172.415i 1.02289 + 2.61234i
\(67\) 22.2473 + 22.2473i 0.332049 + 0.332049i 0.853364 − 0.521315i \(-0.174558\pi\)
−0.521315 + 0.853364i \(0.674558\pi\)
\(68\) −0.576640 + 14.7869i −0.00848000 + 0.217454i
\(69\) 158.402 2.29568
\(70\) 17.4556 + 31.8174i 0.249366 + 0.454534i
\(71\) −126.266 −1.77839 −0.889194 − 0.457531i \(-0.848734\pi\)
−0.889194 + 0.457531i \(0.848734\pi\)
\(72\) −117.364 + 40.7559i −1.63005 + 0.566055i
\(73\) −60.8346 + 60.8346i −0.833350 + 0.833350i −0.987974 − 0.154623i \(-0.950584\pi\)
0.154623 + 0.987974i \(0.450584\pi\)
\(74\) 11.8670 − 4.64666i 0.160365 − 0.0627927i
\(75\) 62.4079 − 106.941i 0.832106 − 1.42588i
\(76\) −19.1463 + 73.5487i −0.251925 + 0.967747i
\(77\) 47.9686 + 47.9686i 0.622968 + 0.622968i
\(78\) 17.1327 + 7.49099i 0.219650 + 0.0960384i
\(79\) − 77.7490i − 0.984164i −0.870549 − 0.492082i \(-0.836236\pi\)
0.870549 − 0.492082i \(-0.163764\pi\)
\(80\) 79.8975 − 4.04775i 0.998719 − 0.0505968i
\(81\) −20.4075 −0.251944
\(82\) −55.2925 + 126.460i −0.674298 + 1.54219i
\(83\) −93.8670 − 93.8670i −1.13093 − 1.13093i −0.990023 − 0.140905i \(-0.954999\pi\)
−0.140905 − 0.990023i \(-0.545001\pi\)
\(84\) −52.7809 + 48.8188i −0.628344 + 0.581177i
\(85\) 11.2948 − 14.6489i 0.132880 − 0.172340i
\(86\) 34.8302 + 88.9520i 0.405002 + 1.03433i
\(87\) −30.0473 + 30.0473i −0.345371 + 0.345371i
\(88\) 141.266 − 49.0564i 1.60530 − 0.557459i
\(89\) −144.779 −1.62673 −0.813367 − 0.581751i \(-0.802368\pi\)
−0.813367 + 0.581751i \(0.802368\pi\)
\(90\) 149.094 + 43.4569i 1.65660 + 0.482854i
\(91\) 6.85071 0.0752825
\(92\) 4.98507 − 127.833i 0.0541855 − 1.38949i
\(93\) 45.5470 − 45.5470i 0.489753 − 0.489753i
\(94\) 59.0456 − 23.1200i 0.628144 − 0.245957i
\(95\) 72.7653 − 61.0755i 0.765950 − 0.642900i
\(96\) 46.1178 + 151.630i 0.480394 + 1.57948i
\(97\) 47.3733 − 47.3733i 0.488385 − 0.488385i −0.419411 − 0.907796i \(-0.637764\pi\)
0.907796 + 0.419411i \(0.137764\pi\)
\(98\) 28.7077 − 65.6574i 0.292936 − 0.669974i
\(99\) 290.294 2.93227
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.j.a.227.16 ✓ 232
4.3 odd 2 inner 380.3.j.a.227.74 yes 232
5.3 odd 4 inner 380.3.j.a.303.43 yes 232
19.18 odd 2 inner 380.3.j.a.227.101 yes 232
20.3 even 4 inner 380.3.j.a.303.101 yes 232
76.75 even 2 inner 380.3.j.a.227.43 yes 232
95.18 even 4 inner 380.3.j.a.303.74 yes 232
380.303 odd 4 inner 380.3.j.a.303.16 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.j.a.227.16 ✓ 232 1.1 even 1 trivial
380.3.j.a.227.43 yes 232 76.75 even 2 inner
380.3.j.a.227.74 yes 232 4.3 odd 2 inner
380.3.j.a.227.101 yes 232 19.18 odd 2 inner
380.3.j.a.303.16 yes 232 380.303 odd 4 inner
380.3.j.a.303.43 yes 232 5.3 odd 4 inner
380.3.j.a.303.74 yes 232 95.18 even 4 inner
380.3.j.a.303.101 yes 232 20.3 even 4 inner