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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 303.100
Character \(\chi\) \(=\) 380.303
Dual form 380.3.j.a.227.100

$q$-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+(1.77962 + 0.912666i) q^{2} +(-3.66846 - 3.66846i) q^{3} +(2.33408 + 3.24839i) q^{4} +(4.37305 + 2.42414i) q^{5} +(-3.18038 - 9.87655i) q^{6} +(-2.51789 - 2.51789i) q^{7} +(1.18908 + 7.91114i) q^{8} +17.9153i q^{9} +(5.56992 + 8.30518i) q^{10} +14.1899i q^{11} +(3.35412 - 20.4791i) q^{12} +(0.192716 - 0.192716i) q^{13} +(-2.18289 - 6.77886i) q^{14} +(-7.14948 - 24.9352i) q^{15} +(-5.10412 + 15.1640i) q^{16} +(18.2406 - 18.2406i) q^{17} +(-16.3506 + 31.8823i) q^{18} +(17.8506 + 6.50817i) q^{19} +(2.33248 + 19.8635i) q^{20} +18.4735i q^{21} +(-12.9507 + 25.2527i) q^{22} +(-18.5273 + 18.5273i) q^{23} +(24.6596 - 33.3838i) q^{24} +(13.2471 + 21.2018i) q^{25} +(0.518847 - 0.167076i) q^{26} +(32.7053 - 32.7053i) q^{27} +(2.30213 - 14.0560i) q^{28} +31.6301 q^{29} +(10.0342 - 50.9003i) q^{30} +35.7449 q^{31} +(-22.9231 + 22.3278i) q^{32} +(52.0552 - 52.0552i) q^{33} +(49.1088 - 15.8137i) q^{34} +(-4.90711 - 17.1145i) q^{35} +(-58.1958 + 41.8157i) q^{36} +(-17.7701 - 17.7701i) q^{37} +(25.8275 + 27.8737i) q^{38} -1.41395 q^{39} +(-13.9778 + 37.4783i) q^{40} +49.7840i q^{41} +(-16.8602 + 32.8759i) q^{42} +(13.5259 - 13.5259i) q^{43} +(-46.0945 + 33.1205i) q^{44} +(-43.4291 + 78.3442i) q^{45} +(-49.8808 + 16.0623i) q^{46} +(37.4331 + 37.4331i) q^{47} +(74.3530 - 36.9044i) q^{48} -36.3205i q^{49} +(4.22457 + 49.8212i) q^{50} -133.830 q^{51} +(1.07583 + 0.176203i) q^{52} +(-20.7721 + 20.7721i) q^{53} +(88.0519 - 28.3539i) q^{54} +(-34.3984 + 62.0532i) q^{55} +(16.9254 - 22.9133i) q^{56} +(-41.6092 - 89.3593i) q^{57} +(56.2895 + 28.8677i) q^{58} +17.7674i q^{59} +(64.3120 - 81.4252i) q^{60} -85.5197 q^{61} +(63.6124 + 32.6232i) q^{62} +(45.1086 - 45.1086i) q^{63} +(-61.1722 + 18.8139i) q^{64} +(1.30993 - 0.375585i) q^{65} +(140.147 - 45.1294i) q^{66} +(28.1260 - 28.1260i) q^{67} +(101.827 + 16.6775i) q^{68} +135.934 q^{69} +(6.88707 - 34.9359i) q^{70} -62.1745 q^{71} +(-141.730 + 21.3026i) q^{72} +(-24.6006 - 24.6006i) q^{73} +(-15.4058 - 47.8421i) q^{74} +(29.1816 - 126.374i) q^{75} +(20.5236 + 73.1764i) q^{76} +(35.7286 - 35.7286i) q^{77} +(-2.51628 - 1.29046i) q^{78} -116.222i q^{79} +(-59.0803 + 53.9399i) q^{80} -78.7189 q^{81} +(-45.4362 + 88.5965i) q^{82} +(26.7366 - 26.7366i) q^{83} +(-60.0093 + 43.1188i) q^{84} +(123.985 - 35.5491i) q^{85} +(36.4155 - 11.7263i) q^{86} +(-116.034 - 116.034i) q^{87} +(-112.258 + 16.8729i) q^{88} -149.178 q^{89} +(-148.789 + 99.7865i) q^{90} -0.970476 q^{91} +(-103.428 - 16.9397i) q^{92} +(-131.129 - 131.129i) q^{93} +(32.4527 + 100.781i) q^{94} +(62.2847 + 71.7329i) q^{95} +(166.001 + 2.18362i) q^{96} +(9.47531 + 9.47531i) q^{97} +(33.1485 - 64.6366i) q^{98} -254.216 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 \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.77962 + 0.912666i 0.889809 + 0.456333i
\(3\) −3.66846 − 3.66846i −1.22282 − 1.22282i −0.966625 − 0.256196i \(-0.917531\pi\)
−0.256196 − 0.966625i \(-0.582469\pi\)
\(4\) 2.33408 + 3.24839i 0.583521 + 0.812098i
\(5\) 4.37305 + 2.42414i 0.874609 + 0.484829i
\(6\) −3.18038 − 9.87655i −0.530064 − 1.64609i
\(7\) −2.51789 − 2.51789i −0.359698 − 0.359698i 0.504004 − 0.863702i \(-0.331860\pi\)
−0.863702 + 0.504004i \(0.831860\pi\)
\(8\) 1.18908 + 7.91114i 0.148635 + 0.988892i
\(9\) 17.9153i 1.99058i
\(10\) 5.56992 + 8.30518i 0.556992 + 0.830518i
\(11\) 14.1899i 1.28999i 0.764185 + 0.644997i \(0.223141\pi\)
−0.764185 + 0.644997i \(0.776859\pi\)
\(12\) 3.35412 − 20.4791i 0.279510 − 1.70659i
\(13\) 0.192716 − 0.192716i 0.0148243 − 0.0148243i −0.699656 − 0.714480i \(-0.746663\pi\)
0.714480 + 0.699656i \(0.246663\pi\)
\(14\) −2.18289 − 6.77886i −0.155921 − 0.484205i
\(15\) −7.14948 − 24.9352i −0.476632 − 1.66235i
\(16\) −5.10412 + 15.1640i −0.319007 + 0.947752i
\(17\) 18.2406 − 18.2406i 1.07297 − 1.07297i 0.0758551 − 0.997119i \(-0.475831\pi\)
0.997119 − 0.0758551i \(-0.0241686\pi\)
\(18\) −16.3506 + 31.8823i −0.908369 + 1.77124i
\(19\) 17.8506 + 6.50817i 0.939505 + 0.342536i
\(20\) 2.33248 + 19.8635i 0.116624 + 0.993176i
\(21\) 18.4735i 0.879693i
\(22\) −12.9507 + 25.2527i −0.588666 + 1.14785i
\(23\) −18.5273 + 18.5273i −0.805535 + 0.805535i −0.983955 − 0.178419i \(-0.942902\pi\)
0.178419 + 0.983955i \(0.442902\pi\)
\(24\) 24.6596 − 33.3838i 1.02748 − 1.39099i
\(25\) 13.2471 + 21.2018i 0.529882 + 0.848071i
\(26\) 0.518847 − 0.167076i 0.0199557 − 0.00642600i
\(27\) 32.7053 − 32.7053i 1.21131 − 1.21131i
\(28\) 2.30213 − 14.0560i 0.0822190 − 0.502001i
\(29\) 31.6301 1.09069 0.545347 − 0.838211i \(-0.316398\pi\)
0.545347 + 0.838211i \(0.316398\pi\)
\(30\) 10.0342 − 50.9003i 0.334473 − 1.69668i
\(31\) 35.7449 1.15306 0.576531 − 0.817075i \(-0.304406\pi\)
0.576531 + 0.817075i \(0.304406\pi\)
\(32\) −22.9231 + 22.3278i −0.716346 + 0.697745i
\(33\) 52.0552 − 52.0552i 1.57743 − 1.57743i
\(34\) 49.1088 − 15.8137i 1.44438 − 0.465109i
\(35\) −4.90711 − 17.1145i −0.140203 − 0.488987i
\(36\) −58.1958 + 41.8157i −1.61655 + 1.16155i
\(37\) −17.7701 − 17.7701i −0.480272 − 0.480272i 0.424946 − 0.905219i \(-0.360293\pi\)
−0.905219 + 0.424946i \(0.860293\pi\)
\(38\) 25.8275 + 27.8737i 0.679670 + 0.733518i
\(39\) −1.41395 −0.0362550
\(40\) −13.9778 + 37.4783i −0.349446 + 0.936957i
\(41\) 49.7840i 1.21424i 0.794609 + 0.607122i \(0.207676\pi\)
−0.794609 + 0.607122i \(0.792324\pi\)
\(42\) −16.8602 + 32.8759i −0.401433 + 0.782759i
\(43\) 13.5259 − 13.5259i 0.314556 − 0.314556i −0.532116 − 0.846672i \(-0.678603\pi\)
0.846672 + 0.532116i \(0.178603\pi\)
\(44\) −46.0945 + 33.1205i −1.04760 + 0.752738i
\(45\) −43.4291 + 78.3442i −0.965092 + 1.74098i
\(46\) −49.8808 + 16.0623i −1.08436 + 0.349180i
\(47\) 37.4331 + 37.4331i 0.796449 + 0.796449i 0.982534 − 0.186084i \(-0.0595798\pi\)
−0.186084 + 0.982534i \(0.559580\pi\)
\(48\) 74.3530 − 36.9044i 1.54902 − 0.768842i
\(49\) − 36.3205i − 0.741235i
\(50\) 4.22457 + 49.8212i 0.0844915 + 0.996424i
\(51\) −133.830 −2.62411
\(52\) 1.07583 + 0.176203i 0.0206891 + 0.00338851i
\(53\) −20.7721 + 20.7721i −0.391926 + 0.391926i −0.875373 − 0.483447i \(-0.839384\pi\)
0.483447 + 0.875373i \(0.339384\pi\)
\(54\) 88.0519 − 28.3539i 1.63059 − 0.525073i
\(55\) −34.3984 + 62.0532i −0.625426 + 1.12824i
\(56\) 16.9254 − 22.9133i 0.302239 − 0.409166i
\(57\) −41.6092 − 89.3593i −0.729987 − 1.56771i
\(58\) 56.2895 + 28.8677i 0.970509 + 0.497719i
\(59\) 17.7674i 0.301143i 0.988599 + 0.150571i \(0.0481114\pi\)
−0.988599 + 0.150571i \(0.951889\pi\)
\(60\) 64.3120 − 81.4252i 1.07187 − 1.35709i
\(61\) −85.5197 −1.40196 −0.700981 − 0.713180i \(-0.747254\pi\)
−0.700981 + 0.713180i \(0.747254\pi\)
\(62\) 63.6124 + 32.6232i 1.02601 + 0.526180i
\(63\) 45.1086 − 45.1086i 0.716009 − 0.716009i
\(64\) −61.1722 + 18.8139i −0.955815 + 0.293968i
\(65\) 1.30993 − 0.375585i 0.0201528 − 0.00577824i
\(66\) 140.147 − 45.1294i 2.12345 − 0.683779i
\(67\) 28.1260 − 28.1260i 0.419791 − 0.419791i −0.465341 − 0.885132i \(-0.654068\pi\)
0.885132 + 0.465341i \(0.154068\pi\)
\(68\) 101.827 + 16.6775i 1.49746 + 0.245258i
\(69\) 135.934 1.97005
\(70\) 6.88707 − 34.9359i 0.0983867 − 0.499084i
\(71\) −62.1745 −0.875697 −0.437848 − 0.899049i \(-0.644259\pi\)
−0.437848 + 0.899049i \(0.644259\pi\)
\(72\) −141.730 + 21.3026i −1.96847 + 0.295870i
\(73\) −24.6006 − 24.6006i −0.336995 − 0.336995i 0.518240 − 0.855235i \(-0.326587\pi\)
−0.855235 + 0.518240i \(0.826587\pi\)
\(74\) −15.4058 − 47.8421i −0.208187 − 0.646515i
\(75\) 29.1816 − 126.374i 0.389088 − 1.68499i
\(76\) 20.5236 + 73.1764i 0.270048 + 0.962847i
\(77\) 35.7286 − 35.7286i 0.464008 − 0.464008i
\(78\) −2.51628 − 1.29046i −0.0322601 − 0.0165444i
\(79\) − 116.222i − 1.47117i −0.677435 − 0.735583i \(-0.736908\pi\)
0.677435 − 0.735583i \(-0.263092\pi\)
\(80\) −59.0803 + 53.9399i −0.738504 + 0.674249i
\(81\) −78.7189 −0.971839
\(82\) −45.4362 + 88.5965i −0.554100 + 1.08045i
\(83\) 26.7366 − 26.7366i 0.322127 − 0.322127i −0.527455 − 0.849583i \(-0.676854\pi\)
0.849583 + 0.527455i \(0.176854\pi\)
\(84\) −60.0093 + 43.1188i −0.714397 + 0.513319i
\(85\) 123.985 − 35.5491i 1.45864 − 0.418224i
\(86\) 36.4155 − 11.7263i 0.423436 − 0.136352i
\(87\) −116.034 − 116.034i −1.33372 − 1.33372i
\(88\) −112.258 + 16.8729i −1.27566 + 0.191738i
\(89\) −149.178 −1.67616 −0.838081 − 0.545546i \(-0.816323\pi\)
−0.838081 + 0.545546i \(0.816323\pi\)
\(90\) −148.789 + 99.7865i −1.65322 + 1.10874i
\(91\) −0.970476 −0.0106646
\(92\) −103.428 − 16.9397i −1.12422 − 0.184127i
\(93\) −131.129 − 131.129i −1.40999 − 1.40999i
\(94\) 32.4527 + 100.781i 0.345242 + 1.07213i
\(95\) 62.2847 + 71.7329i 0.655629 + 0.755084i
\(96\) 166.001 + 2.18362i 1.72918 + 0.0227460i
\(97\) 9.47531 + 9.47531i 0.0976837 + 0.0976837i 0.754260 − 0.656576i \(-0.227996\pi\)
−0.656576 + 0.754260i \(0.727996\pi\)
\(98\) 33.1485 − 64.6366i 0.338250 − 0.659557i
\(99\) −254.216 −2.56784
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.303.100 yes 232
4.3 odd 2 inner 380.3.j.a.303.75 yes 232
5.2 odd 4 inner 380.3.j.a.227.42 yes 232
19.18 odd 2 inner 380.3.j.a.303.17 yes 232
20.7 even 4 inner 380.3.j.a.227.17 ✓ 232
76.75 even 2 inner 380.3.j.a.303.42 yes 232
95.37 even 4 inner 380.3.j.a.227.75 yes 232
380.227 odd 4 inner 380.3.j.a.227.100 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.j.a.227.17 ✓ 232 20.7 even 4 inner
380.3.j.a.227.42 yes 232 5.2 odd 4 inner
380.3.j.a.227.75 yes 232 95.37 even 4 inner
380.3.j.a.227.100 yes 232 380.227 odd 4 inner
380.3.j.a.303.17 yes 232 19.18 odd 2 inner
380.3.j.a.303.42 yes 232 76.75 even 2 inner
380.3.j.a.303.75 yes 232 4.3 odd 2 inner
380.3.j.a.303.100 yes 232 1.1 even 1 trivial