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Properties

Label 380.3.j.a.227.3
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 227.3
Character \(\chi\) \(=\) 380.227
Dual form 380.3.j.a.303.3

$q$-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+(-1.99610 + 0.124883i) q^{2} +(-0.756575 + 0.756575i) q^{3} +(3.96881 - 0.498556i) q^{4} +(4.99668 - 0.182084i) q^{5} +(1.41571 - 1.60468i) q^{6} +(-0.187105 + 0.187105i) q^{7} +(-7.85987 + 1.49080i) q^{8} +7.85519i q^{9} +(-9.95113 + 0.987456i) q^{10} +3.97709i q^{11} +(-2.62550 + 3.37989i) q^{12} +(1.06644 + 1.06644i) q^{13} +(0.350114 - 0.396847i) q^{14} +(-3.64260 + 3.91812i) q^{15} +(15.5029 - 3.95734i) q^{16} +(8.43174 + 8.43174i) q^{17} +(-0.980977 - 15.6797i) q^{18} +(-12.1595 + 14.5995i) q^{19} +(19.7401 - 3.21378i) q^{20} -0.283118i q^{21} +(-0.496669 - 7.93865i) q^{22} +(-24.4679 - 24.4679i) q^{23} +(4.81867 - 7.07448i) q^{24} +(24.9337 - 1.81963i) q^{25} +(-2.26190 - 1.99554i) q^{26} +(-12.7522 - 12.7522i) q^{27} +(-0.649303 + 0.835868i) q^{28} +24.5863 q^{29} +(6.78169 - 8.27585i) q^{30} +26.2886 q^{31} +(-30.4511 + 9.83529i) q^{32} +(-3.00896 - 3.00896i) q^{33} +(-17.8835 - 15.7776i) q^{34} +(-0.900838 + 0.968976i) q^{35} +(3.91625 + 31.1757i) q^{36} +(-18.8143 + 18.8143i) q^{37} +(22.4483 - 30.6606i) q^{38} -1.61369 q^{39} +(-39.0018 + 8.88021i) q^{40} +61.5706i q^{41} +(0.0353566 + 0.565132i) q^{42} +(13.1019 + 13.1019i) q^{43} +(1.98280 + 15.7843i) q^{44} +(1.43030 + 39.2499i) q^{45} +(51.8960 + 45.7847i) q^{46} +(-40.2257 + 40.2257i) q^{47} +(-8.73506 + 14.7231i) q^{48} +48.9300i q^{49} +(-49.5428 + 6.74594i) q^{50} -12.7585 q^{51} +(4.76418 + 3.70082i) q^{52} +(-24.9633 - 24.9633i) q^{53} +(27.0472 + 23.8621i) q^{54} +(0.724162 + 19.8722i) q^{55} +(1.19169 - 1.74956i) q^{56} +(-1.84606 - 20.2452i) q^{57} +(-49.0766 + 3.07040i) q^{58} +47.0834i q^{59} +(-12.5034 + 17.3663i) q^{60} +30.6701 q^{61} +(-52.4746 + 3.28299i) q^{62} +(-1.46975 - 1.46975i) q^{63} +(59.5550 - 23.4350i) q^{64} +(5.52285 + 5.13449i) q^{65} +(6.38195 + 5.63041i) q^{66} +(75.7961 + 75.7961i) q^{67} +(37.6677 + 29.2603i) q^{68} +37.0236 q^{69} +(1.67715 - 2.04667i) q^{70} -26.3120 q^{71} +(-11.7105 - 61.7408i) q^{72} +(-24.9341 + 24.9341i) q^{73} +(35.2057 - 39.9049i) q^{74} +(-17.4875 + 20.2409i) q^{75} +(-40.9801 + 64.0049i) q^{76} +(-0.744134 - 0.744134i) q^{77} +(3.22107 - 0.201521i) q^{78} -66.1805i q^{79} +(76.7424 - 22.5964i) q^{80} -51.4007 q^{81} +(-7.68910 - 122.901i) q^{82} +(13.6610 + 13.6610i) q^{83} +(-0.141150 - 1.12364i) q^{84} +(43.6660 + 40.5955i) q^{85} +(-27.7888 - 24.5164i) q^{86} +(-18.6013 + 18.6013i) q^{87} +(-5.92904 - 31.2594i) q^{88} +59.3165 q^{89} +(-7.75665 - 78.1680i) q^{90} -0.399074 q^{91} +(-109.307 - 84.9099i) q^{92} +(-19.8893 + 19.8893i) q^{93} +(75.2708 - 85.3178i) q^{94} +(-58.0989 + 75.1633i) q^{95} +(15.5974 - 30.4796i) q^{96} +(115.923 - 115.923i) q^{97} +(-6.11051 - 97.6690i) q^{98} -31.2408 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.99610 + 0.124883i −0.998049 + 0.0624413i
\(3\) −0.756575 + 0.756575i −0.252192 + 0.252192i −0.821869 − 0.569677i \(-0.807068\pi\)
0.569677 + 0.821869i \(0.307068\pi\)
\(4\) 3.96881 − 0.498556i 0.992202 − 0.124639i
\(5\) 4.99668 − 0.182084i 0.999337 − 0.0364167i
\(6\) 1.41571 − 1.60468i 0.235952 − 0.267447i
\(7\) −0.187105 + 0.187105i −0.0267294 + 0.0267294i −0.720345 − 0.693616i \(-0.756017\pi\)
0.693616 + 0.720345i \(0.256017\pi\)
\(8\) −7.85987 + 1.49080i −0.982483 + 0.186350i
\(9\) 7.85519i 0.872799i
\(10\) −9.95113 + 0.987456i −0.995113 + 0.0987456i
\(11\) 3.97709i 0.361553i 0.983524 + 0.180777i \(0.0578611\pi\)
−0.983524 + 0.180777i \(0.942139\pi\)
\(12\) −2.62550 + 3.37989i −0.218792 + 0.281658i
\(13\) 1.06644 + 1.06644i 0.0820340 + 0.0820340i 0.746933 − 0.664899i \(-0.231525\pi\)
−0.664899 + 0.746933i \(0.731525\pi\)
\(14\) 0.350114 − 0.396847i 0.0250082 − 0.0283462i
\(15\) −3.64260 + 3.91812i −0.242840 + 0.261208i
\(16\) 15.5029 − 3.95734i 0.968930 − 0.247334i
\(17\) 8.43174 + 8.43174i 0.495985 + 0.495985i 0.910186 − 0.414201i \(-0.135939\pi\)
−0.414201 + 0.910186i \(0.635939\pi\)
\(18\) −0.980977 − 15.6797i −0.0544987 − 0.871096i
\(19\) −12.1595 + 14.5995i −0.639974 + 0.768396i
\(20\) 19.7401 − 3.21378i 0.987005 − 0.160689i
\(21\) − 0.283118i − 0.0134818i
\(22\) −0.496669 − 7.93865i −0.0225759 − 0.360848i
\(23\) −24.4679 − 24.4679i −1.06382 − 1.06382i −0.997819 − 0.0660035i \(-0.978975\pi\)
−0.0660035 − 0.997819i \(-0.521025\pi\)
\(24\) 4.81867 − 7.07448i 0.200778 − 0.294770i
\(25\) 24.9337 − 1.81963i 0.997348 − 0.0727851i
\(26\) −2.26190 − 1.99554i −0.0869962 − 0.0767516i
\(27\) −12.7522 − 12.7522i −0.472304 − 0.472304i
\(28\) −0.649303 + 0.835868i −0.0231894 + 0.0298524i
\(29\) 24.5863 0.847802 0.423901 − 0.905708i \(-0.360660\pi\)
0.423901 + 0.905708i \(0.360660\pi\)
\(30\) 6.78169 − 8.27585i 0.226056 − 0.275862i
\(31\) 26.2886 0.848020 0.424010 − 0.905658i \(-0.360622\pi\)
0.424010 + 0.905658i \(0.360622\pi\)
\(32\) −30.4511 + 9.83529i −0.951596 + 0.307353i
\(33\) −3.00896 − 3.00896i −0.0911807 − 0.0911807i
\(34\) −17.8835 − 15.7776i −0.525987 − 0.464047i
\(35\) −0.900838 + 0.968976i −0.0257382 + 0.0276850i
\(36\) 3.91625 + 31.1757i 0.108785 + 0.865993i
\(37\) −18.8143 + 18.8143i −0.508496 + 0.508496i −0.914065 − 0.405569i \(-0.867074\pi\)
0.405569 + 0.914065i \(0.367074\pi\)
\(38\) 22.4483 − 30.6606i 0.590746 − 0.806858i
\(39\) −1.61369 −0.0413765
\(40\) −39.0018 + 8.88021i −0.975045 + 0.222005i
\(41\) 61.5706i 1.50172i 0.660461 + 0.750861i \(0.270361\pi\)
−0.660461 + 0.750861i \(0.729639\pi\)
\(42\) 0.0353566 + 0.565132i 0.000841823 + 0.0134555i
\(43\) 13.1019 + 13.1019i 0.304694 + 0.304694i 0.842847 − 0.538153i \(-0.180878\pi\)
−0.538153 + 0.842847i \(0.680878\pi\)
\(44\) 1.98280 + 15.7843i 0.0450636 + 0.358734i
\(45\) 1.43030 + 39.2499i 0.0317845 + 0.872220i
\(46\) 51.8960 + 45.7847i 1.12817 + 0.995320i
\(47\) −40.2257 + 40.2257i −0.855865 + 0.855865i −0.990848 − 0.134983i \(-0.956902\pi\)
0.134983 + 0.990848i \(0.456902\pi\)
\(48\) −8.73506 + 14.7231i −0.181980 + 0.306732i
\(49\) 48.9300i 0.998571i
\(50\) −49.5428 + 6.74594i −0.990857 + 0.134919i
\(51\) −12.7585 −0.250166
\(52\) 4.76418 + 3.70082i 0.0916189 + 0.0711697i
\(53\) −24.9633 − 24.9633i −0.471005 − 0.471005i 0.431235 − 0.902240i \(-0.358078\pi\)
−0.902240 + 0.431235i \(0.858078\pi\)
\(54\) 27.0472 + 23.8621i 0.500874 + 0.441891i
\(55\) 0.724162 + 19.8722i 0.0131666 + 0.361313i
\(56\) 1.19169 − 1.74956i 0.0212801 − 0.0312422i
\(57\) −1.84606 − 20.2452i −0.0323869 − 0.355179i
\(58\) −49.0766 + 3.07040i −0.846148 + 0.0529379i
\(59\) 47.0834i 0.798024i 0.916946 + 0.399012i \(0.130647\pi\)
−0.916946 + 0.399012i \(0.869353\pi\)
\(60\) −12.5034 + 17.3663i −0.208390 + 0.289439i
\(61\) 30.6701 0.502788 0.251394 − 0.967885i \(-0.419111\pi\)
0.251394 + 0.967885i \(0.419111\pi\)
\(62\) −52.4746 + 3.28299i −0.846365 + 0.0529515i
\(63\) −1.46975 − 1.46975i −0.0233293 − 0.0233293i
\(64\) 59.5550 − 23.4350i 0.930547 − 0.366172i
\(65\) 5.52285 + 5.13449i 0.0849670 + 0.0789922i
\(66\) 6.38195 + 5.63041i 0.0966962 + 0.0853093i
\(67\) 75.7961 + 75.7961i 1.13128 + 1.13128i 0.989964 + 0.141321i \(0.0451350\pi\)
0.141321 + 0.989964i \(0.454865\pi\)
\(68\) 37.6677 + 29.2603i 0.553936 + 0.430298i
\(69\) 37.0236 0.536574
\(70\) 1.67715 − 2.04667i 0.0239593 − 0.0292381i
\(71\) −26.3120 −0.370591 −0.185295 − 0.982683i \(-0.559324\pi\)
−0.185295 + 0.982683i \(0.559324\pi\)
\(72\) −11.7105 − 61.7408i −0.162646 − 0.857510i
\(73\) −24.9341 + 24.9341i −0.341562 + 0.341562i −0.856954 − 0.515392i \(-0.827646\pi\)
0.515392 + 0.856954i \(0.327646\pi\)
\(74\) 35.2057 − 39.9049i 0.475752 − 0.539255i
\(75\) −17.4875 + 20.2409i −0.233167 + 0.269878i
\(76\) −40.9801 + 64.0049i −0.539212 + 0.842170i
\(77\) −0.744134 − 0.744134i −0.00966408 − 0.00966408i
\(78\) 3.22107 − 0.201521i 0.0412958 − 0.00258361i
\(79\) − 66.1805i − 0.837728i −0.908049 − 0.418864i \(-0.862428\pi\)
0.908049 − 0.418864i \(-0.137572\pi\)
\(80\) 76.7424 − 22.5964i 0.959280 − 0.282455i
\(81\) −51.4007 −0.634577
\(82\) −7.68910 − 122.901i −0.0937695 − 1.49879i
\(83\) 13.6610 + 13.6610i 0.164590 + 0.164590i 0.784597 − 0.620007i \(-0.212870\pi\)
−0.620007 + 0.784597i \(0.712870\pi\)
\(84\) −0.141150 − 1.12364i −0.00168036 − 0.0133767i
\(85\) 43.6660 + 40.5955i 0.513718 + 0.477594i
\(86\) −27.7888 − 24.5164i −0.323125 − 0.285074i
\(87\) −18.6013 + 18.6013i −0.213809 + 0.213809i
\(88\) −5.92904 − 31.2594i −0.0673755 − 0.355220i
\(89\) 59.3165 0.666477 0.333239 − 0.942842i \(-0.391859\pi\)
0.333239 + 0.942842i \(0.391859\pi\)
\(90\) −7.75665 − 78.1680i −0.0861850 − 0.868533i
\(91\) −0.399074 −0.00438543
\(92\) −109.307 − 84.9099i −1.18812 − 0.922934i
\(93\) −19.8893 + 19.8893i −0.213863 + 0.213863i
\(94\) 75.2708 − 85.3178i 0.800754 − 0.907636i
\(95\) −58.0989 + 75.1633i −0.611567 + 0.791192i
\(96\) 15.5974 − 30.4796i 0.162473 − 0.317496i
\(97\) 115.923 − 115.923i 1.19508 − 1.19508i 0.219455 − 0.975623i \(-0.429572\pi\)
0.975623 − 0.219455i \(-0.0704279\pi\)
\(98\) −6.11051 − 97.6690i −0.0623521 − 0.996623i
\(99\) −31.2408 −0.315563
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.227.3 ✓ 232
4.3 odd 2 inner 380.3.j.a.227.56 yes 232
5.3 odd 4 inner 380.3.j.a.303.61 yes 232
19.18 odd 2 inner 380.3.j.a.227.114 yes 232
20.3 even 4 inner 380.3.j.a.303.114 yes 232
76.75 even 2 inner 380.3.j.a.227.61 yes 232
95.18 even 4 inner 380.3.j.a.303.56 yes 232
380.303 odd 4 inner 380.3.j.a.303.3 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.j.a.227.3 ✓ 232 1.1 even 1 trivial
380.3.j.a.227.56 yes 232 4.3 odd 2 inner
380.3.j.a.227.61 yes 232 76.75 even 2 inner
380.3.j.a.227.114 yes 232 19.18 odd 2 inner
380.3.j.a.303.3 yes 232 380.303 odd 4 inner
380.3.j.a.303.56 yes 232 95.18 even 4 inner
380.3.j.a.303.61 yes 232 5.3 odd 4 inner
380.3.j.a.303.114 yes 232 20.3 even 4 inner