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Properties

Label 380.3.j.a.227.10
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 227.10
Character \(\chi\) \(=\) 380.227
Dual form 380.3.j.a.303.10

$q$-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+(-1.92510 + 0.542224i) q^{2} +(2.55276 - 2.55276i) q^{3} +(3.41199 - 2.08766i) q^{4} +(-3.19089 - 3.84944i) q^{5} +(-3.53014 + 6.29847i) q^{6} +(-3.72891 + 3.72891i) q^{7} +(-5.43642 + 5.86901i) q^{8} -4.03317i q^{9} +(8.23003 + 5.68036i) q^{10} -2.65424i q^{11} +(3.38068 - 14.0393i) q^{12} +(4.58062 + 4.58062i) q^{13} +(5.15661 - 9.20041i) q^{14} +(-17.9723 - 1.68111i) q^{15} +(7.28331 - 14.2462i) q^{16} +(-21.0399 - 21.0399i) q^{17} +(2.18688 + 7.76423i) q^{18} +(-17.7254 - 6.84173i) q^{19} +(-18.9236 - 6.47272i) q^{20} +19.0380i q^{21} +(1.43919 + 5.10968i) q^{22} +(-17.1122 - 17.1122i) q^{23} +(1.10431 + 28.8601i) q^{24} +(-4.63639 + 24.5663i) q^{25} +(-11.3019 - 6.33441i) q^{26} +(12.6791 + 12.6791i) q^{27} +(-4.93828 + 20.5077i) q^{28} +22.2079 q^{29} +(35.5099 - 6.50870i) q^{30} -26.4651 q^{31} +(-6.29646 + 31.3744i) q^{32} +(-6.77565 - 6.77565i) q^{33} +(51.9122 + 29.0955i) q^{34} +(26.2528 + 2.45566i) q^{35} +(-8.41990 - 13.7611i) q^{36} +(-49.2364 + 49.2364i) q^{37} +(37.8329 + 3.55984i) q^{38} +23.3864 q^{39} +(39.9395 + 2.19977i) q^{40} -19.1540i q^{41} +(-10.3229 - 36.6500i) q^{42} +(-4.26491 - 4.26491i) q^{43} +(-5.54117 - 9.05625i) q^{44} +(-15.5254 + 12.8694i) q^{45} +(42.2212 + 23.6640i) q^{46} +(-38.0474 + 38.0474i) q^{47} +(-17.7745 - 54.9596i) q^{48} +21.1904i q^{49} +(-4.39495 - 49.8065i) q^{50} -107.420 q^{51} +(25.1918 + 6.06622i) q^{52} +(32.2413 + 32.2413i) q^{53} +(-31.2835 - 17.5336i) q^{54} +(-10.2174 + 8.46941i) q^{55} +(-1.61311 - 42.1570i) q^{56} +(-62.7141 + 27.7835i) q^{57} +(-42.7523 + 12.0417i) q^{58} -32.4154i q^{59} +(-64.8308 + 31.7842i) q^{60} +4.87709 q^{61} +(50.9479 - 14.3500i) q^{62} +(15.0393 + 15.0393i) q^{63} +(-4.89067 - 63.8129i) q^{64} +(3.01655 - 32.2491i) q^{65} +(16.7177 + 9.36986i) q^{66} +(-44.6039 - 44.6039i) q^{67} +(-115.712 - 27.8637i) q^{68} -87.3666 q^{69} +(-51.8706 + 9.50750i) q^{70} -32.8771 q^{71} +(23.6707 + 21.9260i) q^{72} +(27.9288 - 27.9288i) q^{73} +(68.0876 - 121.482i) q^{74} +(50.8763 + 74.5475i) q^{75} +(-74.7622 + 13.6609i) q^{76} +(9.89744 + 9.89744i) q^{77} +(-45.0211 + 12.6807i) q^{78} -134.863i q^{79} +(-78.0801 + 17.4214i) q^{80} +101.032 q^{81} +(10.3858 + 36.8734i) q^{82} +(-66.7197 - 66.7197i) q^{83} +(39.7450 + 64.9575i) q^{84} +(-13.8558 + 148.128i) q^{85} +(10.5229 + 5.89783i) q^{86} +(56.6914 - 56.6914i) q^{87} +(15.5778 + 14.4296i) q^{88} +121.097 q^{89} +(22.9099 - 33.1931i) q^{90} -34.1614 q^{91} +(-94.1110 - 22.6620i) q^{92} +(-67.5591 + 67.5591i) q^{93} +(52.6147 - 93.8751i) q^{94} +(30.2231 + 90.0642i) q^{95} +(64.0180 + 96.1647i) q^{96} +(-27.3745 + 27.3745i) q^{97} +(-11.4900 - 40.7936i) q^{98} -10.7050 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.92510 + 0.542224i −0.962548 + 0.271112i
\(3\) 2.55276 − 2.55276i 0.850920 − 0.850920i −0.139327 − 0.990246i \(-0.544494\pi\)
0.990246 + 0.139327i \(0.0444938\pi\)
\(4\) 3.41199 − 2.08766i 0.852997 − 0.521916i
\(5\) −3.19089 − 3.84944i −0.638179 − 0.769888i
\(6\) −3.53014 + 6.29847i −0.588357 + 1.04975i
\(7\) −3.72891 + 3.72891i −0.532702 + 0.532702i −0.921375 − 0.388674i \(-0.872933\pi\)
0.388674 + 0.921375i \(0.372933\pi\)
\(8\) −5.43642 + 5.86901i −0.679553 + 0.733627i
\(9\) − 4.03317i − 0.448130i
\(10\) 8.23003 + 5.68036i 0.823003 + 0.568036i
\(11\) − 2.65424i − 0.241295i −0.992695 − 0.120647i \(-0.961503\pi\)
0.992695 − 0.120647i \(-0.0384971\pi\)
\(12\) 3.38068 − 14.0393i 0.281723 − 1.16994i
\(13\) 4.58062 + 4.58062i 0.352355 + 0.352355i 0.860985 − 0.508630i \(-0.169848\pi\)
−0.508630 + 0.860985i \(0.669848\pi\)
\(14\) 5.15661 − 9.20041i 0.368329 − 0.657172i
\(15\) −17.9723 − 1.68111i −1.19815 − 0.112074i
\(16\) 7.28331 − 14.2462i 0.455207 − 0.890386i
\(17\) −21.0399 − 21.0399i −1.23764 − 1.23764i −0.960962 − 0.276682i \(-0.910765\pi\)
−0.276682 − 0.960962i \(-0.589235\pi\)
\(18\) 2.18688 + 7.76423i 0.121493 + 0.431346i
\(19\) −17.7254 − 6.84173i −0.932917 − 0.360091i
\(20\) −18.9236 − 6.47272i −0.946182 − 0.323636i
\(21\) 19.0380i 0.906573i
\(22\) 1.43919 + 5.10968i 0.0654179 + 0.232258i
\(23\) −17.1122 − 17.1122i −0.744008 − 0.744008i 0.229339 − 0.973347i \(-0.426344\pi\)
−0.973347 + 0.229339i \(0.926344\pi\)
\(24\) 1.10431 + 28.8601i 0.0460129 + 1.20250i
\(25\) −4.63639 + 24.5663i −0.185455 + 0.982653i
\(26\) −11.3019 − 6.33441i −0.434687 − 0.243631i
\(27\) 12.6791 + 12.6791i 0.469598 + 0.469598i
\(28\) −4.93828 + 20.5077i −0.176367 + 0.732418i
\(29\) 22.2079 0.765790 0.382895 − 0.923792i \(-0.374927\pi\)
0.382895 + 0.923792i \(0.374927\pi\)
\(30\) 35.5099 − 6.50870i 1.18366 − 0.216957i
\(31\) −26.4651 −0.853713 −0.426857 − 0.904319i \(-0.640379\pi\)
−0.426857 + 0.904319i \(0.640379\pi\)
\(32\) −6.29646 + 31.3744i −0.196764 + 0.980451i
\(33\) −6.77565 − 6.77565i −0.205323 − 0.205323i
\(34\) 51.9122 + 29.0955i 1.52683 + 0.855751i
\(35\) 26.2528 + 2.45566i 0.750080 + 0.0701617i
\(36\) −8.41990 − 13.7611i −0.233886 − 0.382253i
\(37\) −49.2364 + 49.2364i −1.33071 + 1.33071i −0.425982 + 0.904732i \(0.640071\pi\)
−0.904732 + 0.425982i \(0.859929\pi\)
\(38\) 37.8329 + 3.55984i 0.995602 + 0.0936800i
\(39\) 23.3864 0.599652
\(40\) 39.9395 + 2.19977i 0.998487 + 0.0549942i
\(41\) − 19.1540i − 0.467172i −0.972336 − 0.233586i \(-0.924954\pi\)
0.972336 − 0.233586i \(-0.0750460\pi\)
\(42\) −10.3229 − 36.6500i −0.245783 − 0.872620i
\(43\) −4.26491 − 4.26491i −0.0991840 − 0.0991840i 0.655774 − 0.754958i \(-0.272343\pi\)
−0.754958 + 0.655774i \(0.772343\pi\)
\(44\) −5.54117 − 9.05625i −0.125936 − 0.205824i
\(45\) −15.5254 + 12.8694i −0.345010 + 0.285987i
\(46\) 42.2212 + 23.6640i 0.917852 + 0.514434i
\(47\) −38.0474 + 38.0474i −0.809519 + 0.809519i −0.984561 − 0.175042i \(-0.943994\pi\)
0.175042 + 0.984561i \(0.443994\pi\)
\(48\) −17.7745 − 54.9596i −0.370302 − 1.14499i
\(49\) 21.1904i 0.432458i
\(50\) −4.39495 − 49.8065i −0.0878991 − 0.996129i
\(51\) −107.420 −2.10627
\(52\) 25.1918 + 6.06622i 0.484458 + 0.116658i
\(53\) 32.2413 + 32.2413i 0.608326 + 0.608326i 0.942508 − 0.334182i \(-0.108460\pi\)
−0.334182 + 0.942508i \(0.608460\pi\)
\(54\) −31.2835 − 17.5336i −0.579324 − 0.324697i
\(55\) −10.2174 + 8.46941i −0.185770 + 0.153989i
\(56\) −1.61311 − 42.1570i −0.0288055 − 0.752803i
\(57\) −62.7141 + 27.7835i −1.10025 + 0.487429i
\(58\) −42.7523 + 12.0417i −0.737109 + 0.207615i
\(59\) − 32.4154i − 0.549414i −0.961528 − 0.274707i \(-0.911419\pi\)
0.961528 − 0.274707i \(-0.0885809\pi\)
\(60\) −64.8308 + 31.7842i −1.08051 + 0.529736i
\(61\) 4.87709 0.0799523 0.0399761 − 0.999201i \(-0.487272\pi\)
0.0399761 + 0.999201i \(0.487272\pi\)
\(62\) 50.9479 − 14.3500i 0.821740 − 0.231452i
\(63\) 15.0393 + 15.0393i 0.238719 + 0.238719i
\(64\) −4.89067 − 63.8129i −0.0764167 − 0.997076i
\(65\) 3.01655 − 32.2491i 0.0464084 − 0.496140i
\(66\) 16.7177 + 9.36986i 0.253298 + 0.141968i
\(67\) −44.6039 − 44.6039i −0.665730 − 0.665730i 0.290994 − 0.956725i \(-0.406014\pi\)
−0.956725 + 0.290994i \(0.906014\pi\)
\(68\) −115.712 − 27.8637i −1.70165 − 0.409760i
\(69\) −87.3666 −1.26618
\(70\) −51.8706 + 9.50750i −0.741009 + 0.135821i
\(71\) −32.8771 −0.463058 −0.231529 − 0.972828i \(-0.574373\pi\)
−0.231529 + 0.972828i \(0.574373\pi\)
\(72\) 23.6707 + 21.9260i 0.328760 + 0.304528i
\(73\) 27.9288 − 27.9288i 0.382586 − 0.382586i −0.489447 − 0.872033i \(-0.662801\pi\)
0.872033 + 0.489447i \(0.162801\pi\)
\(74\) 68.0876 − 121.482i 0.920103 − 1.64165i
\(75\) 50.8763 + 74.5475i 0.678351 + 0.993967i
\(76\) −74.7622 + 13.6609i −0.983713 + 0.179748i
\(77\) 9.89744 + 9.89744i 0.128538 + 0.128538i
\(78\) −45.0211 + 12.6807i −0.577194 + 0.162573i
\(79\) − 134.863i − 1.70712i −0.520991 − 0.853562i \(-0.674437\pi\)
0.520991 − 0.853562i \(-0.325563\pi\)
\(80\) −78.0801 + 17.4214i −0.976001 + 0.217767i
\(81\) 101.032 1.24731
\(82\) 10.3858 + 36.8734i 0.126656 + 0.449675i
\(83\) −66.7197 − 66.7197i −0.803852 − 0.803852i 0.179843 − 0.983695i \(-0.442441\pi\)
−0.983695 + 0.179843i \(0.942441\pi\)
\(84\) 39.7450 + 64.9575i 0.473155 + 0.773304i
\(85\) −13.8558 + 148.128i −0.163009 + 1.74268i
\(86\) 10.5229 + 5.89783i 0.122359 + 0.0685794i
\(87\) 56.6914 − 56.6914i 0.651626 − 0.651626i
\(88\) 15.5778 + 14.4296i 0.177020 + 0.163973i
\(89\) 121.097 1.36064 0.680320 − 0.732916i \(-0.261841\pi\)
0.680320 + 0.732916i \(0.261841\pi\)
\(90\) 22.9099 − 33.1931i 0.254554 − 0.368812i
\(91\) −34.1614 −0.375400
\(92\) −94.1110 − 22.6620i −1.02295 − 0.246326i
\(93\) −67.5591 + 67.5591i −0.726442 + 0.726442i
\(94\) 52.6147 − 93.8751i 0.559731 − 0.998671i
\(95\) 30.2231 + 90.0642i 0.318138 + 0.948044i
\(96\) 64.0180 + 96.1647i 0.666854 + 1.00172i
\(97\) −27.3745 + 27.3745i −0.282211 + 0.282211i −0.833990 − 0.551779i \(-0.813949\pi\)
0.551779 + 0.833990i \(0.313949\pi\)
\(98\) −11.4900 − 40.7936i −0.117244 − 0.416262i
\(99\) −10.7050 −0.108131
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.227.10 ✓ 232
4.3 odd 2 inner 380.3.j.a.227.49 yes 232
5.3 odd 4 inner 380.3.j.a.303.68 yes 232
19.18 odd 2 inner 380.3.j.a.227.107 yes 232
20.3 even 4 inner 380.3.j.a.303.107 yes 232
76.75 even 2 inner 380.3.j.a.227.68 yes 232
95.18 even 4 inner 380.3.j.a.303.49 yes 232
380.303 odd 4 inner 380.3.j.a.303.10 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.j.a.227.10 ✓ 232 1.1 even 1 trivial
380.3.j.a.227.49 yes 232 4.3 odd 2 inner
380.3.j.a.227.68 yes 232 76.75 even 2 inner
380.3.j.a.227.107 yes 232 19.18 odd 2 inner
380.3.j.a.303.10 yes 232 380.303 odd 4 inner
380.3.j.a.303.49 yes 232 95.18 even 4 inner
380.3.j.a.303.68 yes 232 5.3 odd 4 inner
380.3.j.a.303.107 yes 232 20.3 even 4 inner