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 303.42
Character \(\chi\) \(=\) 380.303
Dual form 380.3.j.a.227.42

$q$-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+(-0.912666 - 1.77962i) 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} +(7.91114 + 1.18908i) q^{8} +17.9153i q^{9} +(0.322920 - 9.99478i) q^{10} -14.1899i q^{11} +(20.4791 - 3.35412i) 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} +(31.8823 - 16.3506i) q^{18} +(-17.8506 + 6.50817i) q^{19} +(-18.0816 + 8.54722i) q^{20} -18.4735i q^{21} +(-25.2527 + 12.9507i) 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} +(-14.0560 + 2.30213i) q^{28} -31.6301 q^{29} +(-37.8501 + 35.4809i) q^{30} +35.7449 q^{31} +(-22.3278 + 22.9231i) 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} +(27.8737 + 25.8275i) q^{38} +1.41395 q^{39} +(31.7133 + 24.3776i) q^{40} -49.7840i q^{41} +(-32.8759 + 16.8602i) 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} +(-36.9044 + 74.3530i) q^{48} -36.3205i q^{49} +(25.6409 - 42.9248i) q^{50} -133.830 q^{51} +(-0.176203 - 1.07583i) 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} +(89.3593 + 41.6092i) q^{57} +(28.8677 + 56.2895i) q^{58} +17.7674i q^{59} +(97.6869 + 34.9766i) q^{60} -85.5197 q^{61} +(-32.6232 - 63.6124i) 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} +(16.6775 + 101.827i) q^{68} -135.934 q^{69} +(25.9788 - 24.3527i) q^{70} -62.1745 q^{71} +(-21.3026 + 141.730i) 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} +(-1.29046 - 2.51628i) q^{78} -116.222i q^{79} +(14.4392 - 78.6861i) q^{80} -78.7189 q^{81} +(-88.5965 + 45.4362i) 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} +(16.8729 - 112.258i) q^{88} +149.178 q^{89} +(179.059 + 5.78519i) q^{90} -0.970476 q^{91} +(16.9397 + 103.428i) q^{92} +(-131.129 - 131.129i) q^{93} +(-32.4527 + 100.781i) q^{94} +(-93.8382 - 14.8118i) q^{95} +(166.001 - 2.18362i) q^{96} +(-9.47531 - 9.47531i) q^{97} +(-64.6366 + 33.1485i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.912666 − 1.77962i −0.456333 − 0.889809i
\(3\) −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 0.863702 − 0.504004i \(-0.168140\pi\)
−0.504004 + 0.863702i \(0.668140\pi\)
\(8\) 7.91114 + 1.18908i 0.988892 + 0.148635i
\(9\) 17.9153i 1.99058i
\(10\) 0.322920 − 9.99478i 0.0322920 − 0.999478i
\(11\) − 14.1899i − 1.28999i −0.764185 − 0.644997i \(-0.776859\pi\)
0.764185 − 0.644997i \(-0.223141\pi\)
\(12\) 20.4791 − 3.35412i 1.70659 − 0.279510i
\(13\) −0.192716 + 0.192716i −0.0148243 + 0.0148243i −0.714480 − 0.699656i \(-0.753337\pi\)
0.699656 + 0.714480i \(0.253337\pi\)
\(14\) 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 \(-0.0241686\pi\)
\(18\) 31.8823 − 16.3506i 1.77124 − 0.908369i
\(19\) −17.8506 + 6.50817i −0.939505 + 0.342536i
\(20\) −18.0816 + 8.54722i −0.904081 + 0.427361i
\(21\) − 18.4735i − 0.879693i
\(22\) −25.2527 + 12.9507i −1.14785 + 0.588666i
\(23\) 18.5273 − 18.5273i 0.805535 − 0.805535i −0.178419 − 0.983955i \(-0.557098\pi\)
0.983955 + 0.178419i \(0.0570983\pi\)
\(24\) −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
\(28\) −14.0560 + 2.30213i −0.502001 + 0.0822190i
\(29\) −31.6301 −1.09069 −0.545347 − 0.838211i \(-0.683602\pi\)
−0.545347 + 0.838211i \(0.683602\pi\)
\(30\) −37.8501 + 35.4809i −1.26167 + 1.18270i
\(31\) 35.7449 1.15306 0.576531 − 0.817075i \(-0.304406\pi\)
0.576531 + 0.817075i \(0.304406\pi\)
\(32\) −22.3278 + 22.9231i −0.697745 + 0.716346i
\(33\) −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 0.905219 − 0.424946i \(-0.139707\pi\)
−0.424946 + 0.905219i \(0.639707\pi\)
\(38\) 27.8737 + 25.8275i 0.733518 + 0.679670i
\(39\) 1.41395 0.0362550
\(40\) 31.7133 + 24.3776i 0.792832 + 0.609441i
\(41\) − 49.7840i − 1.21424i −0.794609 − 0.607122i \(-0.792324\pi\)
0.794609 − 0.607122i \(-0.207676\pi\)
\(42\) −32.8759 + 16.8602i −0.782759 + 0.401433i
\(43\) −13.5259 + 13.5259i −0.314556 + 0.314556i −0.846672 − 0.532116i \(-0.821397\pi\)
0.532116 + 0.846672i \(0.321397\pi\)
\(44\) 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 0.186084 − 0.982534i \(-0.440420\pi\)
−0.982534 + 0.186084i \(0.940420\pi\)
\(48\) −36.9044 + 74.3530i −0.768842 + 1.54902i
\(49\) − 36.3205i − 0.741235i
\(50\) 25.6409 − 42.9248i 0.512819 − 0.858497i
\(51\) −133.830 −2.62411
\(52\) −0.176203 − 1.07583i −0.00338851 − 0.0206891i
\(53\) 20.7721 − 20.7721i 0.391926 − 0.391926i −0.483447 − 0.875373i \(-0.660616\pi\)
0.875373 + 0.483447i \(0.160616\pi\)
\(54\) −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
\(57\) 89.3593 + 41.6092i 1.56771 + 0.729987i
\(58\) 28.8677 + 56.2895i 0.497719 + 0.970509i
\(59\) 17.7674i 0.301143i 0.988599 + 0.150571i \(0.0481114\pi\)
−0.988599 + 0.150571i \(0.951889\pi\)
\(60\) 97.6869 + 34.9766i 1.62812 + 0.582943i
\(61\) −85.5197 −1.40196 −0.700981 − 0.713180i \(-0.747254\pi\)
−0.700981 + 0.713180i \(0.747254\pi\)
\(62\) −32.6232 − 63.6124i −0.526180 − 1.02601i
\(63\) −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 \(0.154068\pi\)
\(68\) 16.6775 + 101.827i 0.245258 + 1.49746i
\(69\) −135.934 −1.97005
\(70\) 25.9788 − 24.3527i 0.371126 − 0.347895i
\(71\) −62.1745 −0.875697 −0.437848 − 0.899049i \(-0.644259\pi\)
−0.437848 + 0.899049i \(0.644259\pi\)
\(72\) −21.3026 + 141.730i −0.295870 + 1.96847i
\(73\) −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
\(78\) −1.29046 − 2.51628i −0.0165444 − 0.0322601i
\(79\) − 116.222i − 1.47117i −0.677435 − 0.735583i \(-0.736908\pi\)
0.677435 − 0.735583i \(-0.263092\pi\)
\(80\) 14.4392 − 78.6861i 0.180491 − 0.983577i
\(81\) −78.7189 −0.971839
\(82\) −88.5965 + 45.4362i −1.08045 + 0.554100i
\(83\) −26.7366 + 26.7366i −0.322127 + 0.322127i −0.849583 − 0.527455i \(-0.823146\pi\)
0.527455 + 0.849583i \(0.323146\pi\)
\(84\) 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
\(88\) 16.8729 − 112.258i 0.191738 − 1.27566i
\(89\) 149.178 1.67616 0.838081 − 0.545546i \(-0.183677\pi\)
0.838081 + 0.545546i \(0.183677\pi\)
\(90\) 179.059 + 5.78519i 1.98955 + 0.0642799i
\(91\) −0.970476 −0.0106646
\(92\) 16.9397 + 103.428i 0.184127 + 1.12422i
\(93\) −131.129 − 131.129i −1.40999 − 1.40999i
\(94\) −32.4527 + 100.781i −0.345242 + 1.07213i
\(95\) −93.8382 − 14.8118i −0.987771 − 0.155914i
\(96\) 166.001 − 2.18362i 1.72918 − 0.0227460i
\(97\) −9.47531 − 9.47531i −0.0976837 − 0.0976837i 0.656576 − 0.754260i \(-0.272004\pi\)
−0.754260 + 0.656576i \(0.772004\pi\)
\(98\) −64.6366 + 33.1485i −0.659557 + 0.338250i
\(99\) 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.303.42 yes 232
4.3 odd 2 inner 380.3.j.a.303.17 yes 232
5.2 odd 4 inner 380.3.j.a.227.100 yes 232
19.18 odd 2 inner 380.3.j.a.303.75 yes 232
20.7 even 4 inner 380.3.j.a.227.75 yes 232
76.75 even 2 inner 380.3.j.a.303.100 yes 232
95.37 even 4 inner 380.3.j.a.227.17 ✓ 232
380.227 odd 4 inner 380.3.j.a.227.42 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.j.a.227.17 ✓ 232 95.37 even 4 inner
380.3.j.a.227.42 yes 232 380.227 odd 4 inner
380.3.j.a.227.75 yes 232 20.7 even 4 inner
380.3.j.a.227.100 yes 232 5.2 odd 4 inner
380.3.j.a.303.17 yes 232 4.3 odd 2 inner
380.3.j.a.303.42 yes 232 1.1 even 1 trivial
380.3.j.a.303.75 yes 232 19.18 odd 2 inner
380.3.j.a.303.100 yes 232 76.75 even 2 inner