↑

Properties

Label 380.3.j.a.227.20
Level $380$
Weight $3$
Character 380.227
Analytic conductor $10.354$
Analytic rank $0$
Dimension $232$
Inner twists $8$

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(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 227.20
Character \(\chi\) \(=\) 380.227
Dual form 380.3.j.a.303.20

$q$-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+(-1.74129 - 0.983821i) q^{2} +(2.37942 - 2.37942i) q^{3} +(2.06419 + 3.42624i) q^{4} +(-3.94275 + 3.07485i) q^{5} +(-6.48420 + 1.80234i) q^{6} +(-6.50795e-5 + 6.50795e-5i) q^{7} +(-0.223556 - 7.99688i) q^{8} -2.32331i q^{9} +(9.89058 - 1.47524i) q^{10} -4.87427i q^{11} +(13.0641 + 3.24088i) q^{12} +(-5.44907 - 5.44907i) q^{13} +(0.000177349 - 4.92958e-5i) q^{14} +(-2.06512 + 16.6978i) q^{15} +(-7.47822 + 14.1448i) q^{16} +(-2.40653 - 2.40653i) q^{17} +(-2.28572 + 4.04556i) q^{18} +(-17.3725 - 7.69387i) q^{19} +(-18.6738 - 7.16174i) q^{20} +0.000309704i q^{21} +(-4.79541 + 8.48752i) q^{22} +(-29.6081 - 29.6081i) q^{23} +(-19.5599 - 18.4960i) q^{24} +(6.09063 - 24.2467i) q^{25} +(4.12751 + 14.8493i) q^{26} +(15.8867 + 15.8867i) q^{27} +(-0.000357315 - 8.86413e-5i) q^{28} -49.9647 q^{29} +(20.0237 - 27.0441i) q^{30} -13.5435 q^{31} +(26.9377 - 17.2730i) q^{32} +(-11.5979 - 11.5979i) q^{33} +(1.82287 + 6.55805i) q^{34} +(5.64830e-5 - 0.000456702i) q^{35} +(7.96022 - 4.79576i) q^{36} +(28.4811 - 28.4811i) q^{37} +(22.6812 + 30.4887i) q^{38} -25.9313 q^{39} +(25.4706 + 30.8423i) q^{40} +8.33311i q^{41} +(0.000304693 - 0.000539284i) q^{42} +(21.0964 + 21.0964i) q^{43} +(16.7004 - 10.0614i) q^{44} +(7.14383 + 9.16024i) q^{45} +(22.4273 + 80.6855i) q^{46} +(-15.8678 + 15.8678i) q^{47} +(15.8627 + 51.4504i) q^{48} +49.0000i q^{49} +(-34.4600 + 36.2285i) q^{50} -11.4523 q^{51} +(7.42188 - 29.9177i) q^{52} +(-46.5208 - 46.5208i) q^{53} +(-12.0337 - 43.2930i) q^{54} +(14.9876 + 19.2180i) q^{55} +(0.000534982 + 0.000505884i) q^{56} +(-59.6435 + 23.0296i) q^{57} +(87.0032 + 49.1564i) q^{58} -85.3395i q^{59} +(-61.4736 + 27.3920i) q^{60} -66.9339 q^{61} +(23.5832 + 13.3244i) q^{62} +(0.000151200 + 0.000151200i) q^{63} +(-63.9000 + 3.57549i) q^{64} +(38.2394 + 4.72929i) q^{65} +(8.78510 + 31.6057i) q^{66} +(52.7602 + 52.7602i) q^{67} +(3.27780 - 13.2129i) q^{68} -140.901 q^{69} +(-0.000547667 + 0.000739683i) q^{70} +16.4194 q^{71} +(-18.5792 + 0.519389i) q^{72} +(-79.6617 + 79.6617i) q^{73} +(-77.6143 + 21.5736i) q^{74} +(-43.2011 - 72.1854i) q^{75} +(-9.49918 - 75.4040i) q^{76} +(0.000317215 + 0.000317215i) q^{77} +(45.1539 + 25.5117i) q^{78} +9.17360i q^{79} +(-14.0084 - 78.7640i) q^{80} +96.5120 q^{81} +(8.19829 - 14.5104i) q^{82} +(56.4929 + 56.4929i) q^{83} +(-0.00106112 + 0.000639288i) q^{84} +(16.8880 + 2.08864i) q^{85} +(-15.9799 - 57.4901i) q^{86} +(-118.887 + 118.887i) q^{87} +(-38.9789 + 1.08967i) q^{88} -101.957 q^{89} +(-3.42744 - 22.9789i) q^{90} +0.000709246 q^{91} +(40.3276 - 162.561i) q^{92} +(-32.2258 + 32.2258i) q^{93} +(43.2414 - 12.0194i) q^{94} +(92.1531 - 23.0828i) q^{95} +(22.9964 - 105.196i) q^{96} +(61.5781 - 61.5781i) q^{97} +(48.2072 - 85.3233i) q^{98} -11.3244 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.74129 − 0.983821i −0.870646 − 0.491911i
\(3\) 2.37942 − 2.37942i 0.793141 − 0.793141i −0.188862 − 0.982004i \(-0.560480\pi\)
0.982004 + 0.188862i \(0.0604800\pi\)
\(4\) 2.06419 + 3.42624i 0.516048 + 0.856560i
\(5\) −3.94275 + 3.07485i −0.788551 + 0.614969i
\(6\) −6.48420 + 1.80234i −1.08070 + 0.300391i
\(7\) −6.50795e−5 0 6.50795e-5i −9.29708e−6 0 9.29708e-6i −0.707111 − 0.707102i \(-0.750002\pi\)
0.707102 + 0.707111i \(0.250002\pi\)
\(8\) −0.223556 − 7.99688i −0.0279444 − 0.999609i
\(9\) − 2.32331i − 0.258146i
\(10\) 9.89058 − 1.47524i 0.989058 − 0.147524i
\(11\) − 4.87427i − 0.443115i −0.975147 − 0.221558i \(-0.928886\pi\)
0.975147 − 0.221558i \(-0.0711141\pi\)
\(12\) 13.0641 + 3.24088i 1.08867 + 0.270074i
\(13\) −5.44907 − 5.44907i −0.419159 − 0.419159i 0.465755 − 0.884914i \(-0.345783\pi\)
−0.884914 + 0.465755i \(0.845783\pi\)
\(14\) 0.000177349 0 4.92958e-5i 1.26678e−5 0 3.52113e-6i
\(15\) −2.06512 + 16.6978i −0.137675 + 1.11319i
\(16\) −7.47822 + 14.1448i −0.467389 + 0.884052i
\(17\) −2.40653 − 2.40653i −0.141560 − 0.141560i 0.632775 − 0.774336i \(-0.281916\pi\)
−0.774336 + 0.632775i \(0.781916\pi\)
\(18\) −2.28572 + 4.04556i −0.126985 + 0.224753i
\(19\) −17.3725 − 7.69387i −0.914343 − 0.404941i
\(20\) −18.6738 − 7.16174i −0.933688 − 0.358087i
\(21\) 0 0.000309704i 0 1.47478e-5i
\(22\) −4.79541 + 8.48752i −0.217973 + 0.385796i
\(23\) −29.6081 − 29.6081i −1.28731 − 1.28731i −0.936415 − 0.350895i \(-0.885877\pi\)
−0.350895 − 0.936415i \(-0.614123\pi\)
\(24\) −19.5599 − 18.4960i −0.814995 − 0.770667i
\(25\) 6.09063 − 24.2467i 0.243625 − 0.969869i
\(26\) 4.12751 + 14.8493i 0.158750 + 0.571128i
\(27\) 15.8867 + 15.8867i 0.588395 + 0.588395i
\(28\) −0.000357315 0 8.86413e-5i −1.27612e−5 0 3.16576e-6i
\(29\) −49.9647 −1.72292 −0.861461 − 0.507824i \(-0.830450\pi\)
−0.861461 + 0.507824i \(0.830450\pi\)
\(30\) 20.0237 − 27.0441i 0.667456 − 0.901470i
\(31\) −13.5435 −0.436888 −0.218444 − 0.975850i \(-0.570098\pi\)
−0.218444 + 0.975850i \(0.570098\pi\)
\(32\) 26.9377 − 17.2730i 0.841804 − 0.539783i
\(33\) −11.5979 − 11.5979i −0.351453 − 0.351453i
\(34\) 1.82287 + 6.55805i 0.0536139 + 0.192884i
\(35\) 5.64830e−5 0 0.000456702i 1.61380e−6 0 1.30486e-5i
\(36\) 7.96022 − 4.79576i 0.221117 − 0.133216i
\(37\) 28.4811 − 28.4811i 0.769761 − 0.769761i −0.208304 − 0.978064i \(-0.566794\pi\)
0.978064 + 0.208304i \(0.0667942\pi\)
\(38\) 22.6812 + 30.4887i 0.596874 + 0.802335i
\(39\) −25.9313 −0.664905
\(40\) 25.4706 + 30.8423i 0.636765 + 0.771058i
\(41\) 8.33311i 0.203247i 0.994823 + 0.101623i \(0.0324037\pi\)
−0.994823 + 0.101623i \(0.967596\pi\)
\(42\) 0.000304693 0 0.000539284i 7.25459e−6 0 1.28401e-5i
\(43\) 21.0964 + 21.0964i 0.490615 + 0.490615i 0.908500 − 0.417885i \(-0.137229\pi\)
−0.417885 + 0.908500i \(0.637229\pi\)
\(44\) 16.7004 − 10.0614i 0.379555 − 0.228669i
\(45\) 7.14383 + 9.16024i 0.158752 + 0.203561i
\(46\) 22.4273 + 80.6855i 0.487550 + 1.75403i
\(47\) −15.8678 + 15.8678i −0.337612 + 0.337612i −0.855468 − 0.517856i \(-0.826730\pi\)
0.517856 + 0.855468i \(0.326730\pi\)
\(48\) 15.8627 + 51.4504i 0.330473 + 1.07188i
\(49\) 49.0000i 1.00000i
\(50\) −34.4600 + 36.2285i −0.689200 + 0.724571i
\(51\) −11.4523 −0.224555
\(52\) 7.42188 − 29.9177i 0.142728 − 0.575341i
\(53\) −46.5208 − 46.5208i −0.877751 − 0.877751i 0.115550 − 0.993302i \(-0.463137\pi\)
−0.993302 + 0.115550i \(0.963137\pi\)
\(54\) −12.0337 − 43.2930i −0.222846 − 0.801722i
\(55\) 14.9876 + 19.2180i 0.272502 + 0.349419i
\(56\) 0.000534982 0 0.000505884i 9.55325e−6 0 9.03365e-6i
\(57\) −59.6435 + 23.0296i −1.04638 + 0.404028i
\(58\) 87.0032 + 49.1564i 1.50005 + 0.847523i
\(59\) − 85.3395i − 1.44643i −0.690622 − 0.723216i \(-0.742663\pi\)
0.690622 − 0.723216i \(-0.257337\pi\)
\(60\) −61.4736 + 27.3920i −1.02456 + 0.456533i
\(61\) −66.9339 −1.09728 −0.548638 − 0.836060i \(-0.684854\pi\)
−0.548638 + 0.836060i \(0.684854\pi\)
\(62\) 23.5832 + 13.3244i 0.380374 + 0.214910i
\(63\) 0.000151200 0 0.000151200i 2.40000e−6 0 2.40000e-6i
\(64\) −63.9000 + 3.57549i −0.998438 + 0.0558671i
\(65\) 38.2394 + 4.72929i 0.588298 + 0.0727583i
\(66\) 8.78510 + 31.6057i 0.133108 + 0.478874i
\(67\) 52.7602 + 52.7602i 0.787466 + 0.787466i 0.981078 − 0.193612i \(-0.0620204\pi\)
−0.193612 + 0.981078i \(0.562020\pi\)
\(68\) 3.27780 − 13.2129i 0.0482029 − 0.194307i
\(69\) −140.901 −2.04204
\(70\) −0.000547667 0 0.000739683i −7.82381e−6 0 1.05669e-5i
\(71\) 16.4194 0.231259 0.115629 − 0.993292i \(-0.463111\pi\)
0.115629 + 0.993292i \(0.463111\pi\)
\(72\) −18.5792 + 0.519389i −0.258045 + 0.00721374i
\(73\) −79.6617 + 79.6617i −1.09126 + 1.09126i −0.0958610 + 0.995395i \(0.530560\pi\)
−0.995395 + 0.0958610i \(0.969440\pi\)
\(74\) −77.6143 + 21.5736i −1.04884 + 0.291535i
\(75\) −43.2011 − 72.1854i −0.576014 − 0.962473i
\(76\) −9.49918 − 75.4040i −0.124989 − 0.992158i
\(77\) 0.000317215 0 0.000317215i 4.11968e−6 0 4.11968e-6i
\(78\) 45.1539 + 25.5117i 0.578896 + 0.327074i
\(79\) 9.17360i 0.116122i 0.998313 + 0.0580608i \(0.0184917\pi\)
−0.998313 + 0.0580608i \(0.981508\pi\)
\(80\) −14.0084 − 78.7640i −0.175105 − 0.984550i
\(81\) 96.5120 1.19151
\(82\) 8.19829 − 14.5104i 0.0999792 − 0.176956i
\(83\) 56.4929 + 56.4929i 0.680638 + 0.680638i 0.960144 − 0.279506i \(-0.0901708\pi\)
−0.279506 + 0.960144i \(0.590171\pi\)
\(84\) −0.00106112 0.000639288i −1.26324e−5 7.61057e-6i
\(85\) 16.8880 + 2.08864i 0.198683 + 0.0245723i
\(86\) −15.9799 − 57.4901i −0.185813 − 0.668490i
\(87\) −118.887 + 118.887i −1.36652 + 1.36652i
\(88\) −38.9789 + 1.08967i −0.442942 + 0.0123826i
\(89\) −101.957 −1.14558 −0.572790 − 0.819702i \(-0.694139\pi\)
−0.572790 + 0.819702i \(0.694139\pi\)
\(90\) −3.42744 − 22.9789i −0.0380827 − 0.255321i
\(91\) 0.000709246 0 7.79391e−6 0
\(92\) 40.3276 − 162.561i 0.438344 − 1.76697i
\(93\) −32.2258 + 32.2258i −0.346513 + 0.346513i
\(94\) 43.2414 − 12.0194i 0.460015 − 0.127865i
\(95\) 92.1531 − 23.0828i 0.970032 − 0.242977i
\(96\) 22.9964 − 105.196i 0.239546 − 1.09579i
\(97\) 61.5781 − 61.5781i 0.634826 − 0.634826i −0.314449 − 0.949274i \(-0.601820\pi\)
0.949274 + 0.314449i \(0.101820\pi\)
\(98\) 48.2072 − 85.3233i 0.491911 − 0.870646i
\(99\) −11.3244 −0.114388
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.227.20 ✓ 232
4.3 odd 2 inner 380.3.j.a.227.78 yes 232
5.3 odd 4 inner 380.3.j.a.303.39 yes 232
19.18 odd 2 inner 380.3.j.a.227.97 yes 232
20.3 even 4 inner 380.3.j.a.303.97 yes 232
76.75 even 2 inner 380.3.j.a.227.39 yes 232
95.18 even 4 inner 380.3.j.a.303.78 yes 232
380.303 odd 4 inner 380.3.j.a.303.20 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.j.a.227.20 ✓ 232 1.1 even 1 trivial
380.3.j.a.227.39 yes 232 76.75 even 2 inner
380.3.j.a.227.78 yes 232 4.3 odd 2 inner
380.3.j.a.227.97 yes 232 19.18 odd 2 inner
380.3.j.a.303.20 yes 232 380.303 odd 4 inner
380.3.j.a.303.39 yes 232 5.3 odd 4 inner
380.3.j.a.303.78 yes 232 95.18 even 4 inner
380.3.j.a.303.97 yes 232 20.3 even 4 inner