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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,3,Mod(11,380)] 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("380.11"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 4])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.q (of order \(6\), 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: \(160\)
Relative dimension: \(80\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 11.7
Character \(\chi\) \(=\) 380.11
Dual form 380.3.q.a.311.7

$q$-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+(-1.92805 - 0.531634i) q^{2} +(-1.89857 - 1.09614i) q^{3} +(3.43473 + 2.05003i) q^{4} +(-1.11803 + 1.93649i) q^{5} +(3.07778 + 3.12275i) q^{6} +2.39245i q^{7} +(-5.53245 - 5.77858i) q^{8} +(-2.09696 - 3.63204i) q^{9} +(3.18513 - 3.13926i) q^{10} -11.2116i q^{11} +(-4.27395 - 7.65707i) q^{12} +(6.14180 + 10.6379i) q^{13} +(1.27191 - 4.61276i) q^{14} +(4.24533 - 2.45104i) q^{15} +(7.59474 + 14.0826i) q^{16} +(-8.70675 + 15.0805i) q^{17} +(2.11212 + 8.11756i) q^{18} +(-3.50542 - 18.6738i) q^{19} +(-7.81001 + 4.35932i) q^{20} +(2.62246 - 4.54223i) q^{21} +(-5.96045 + 21.6164i) q^{22} +(-5.15281 + 2.97497i) q^{23} +(4.16961 + 17.0354i) q^{24} +(-2.50000 - 4.33013i) q^{25} +(-6.18620 - 23.7756i) q^{26} +28.9247i q^{27} +(-4.90460 + 8.21742i) q^{28} +(6.81645 + 11.8064i) q^{29} +(-9.48825 + 2.46876i) q^{30} +40.6747i q^{31} +(-7.15620 - 31.1896i) q^{32} +(-12.2894 + 21.2859i) q^{33} +(24.8044 - 24.4472i) q^{34} +(-4.63296 - 2.67484i) q^{35} +(0.243311 - 16.7739i) q^{36} +42.4940 q^{37} +(-3.16904 + 37.8676i) q^{38} -26.9291i q^{39} +(17.3756 - 4.25290i) q^{40} +(34.1823 - 59.2054i) q^{41} +(-7.47103 + 7.36344i) q^{42} +(45.0061 + 25.9843i) q^{43} +(22.9841 - 38.5087i) q^{44} +9.37789 q^{45} +(11.5165 - 2.99648i) q^{46} +(7.94113 - 4.58481i) q^{47} +(1.01738 - 35.0617i) q^{48} +43.2762 q^{49} +(2.51807 + 9.67777i) q^{50} +(33.0607 - 19.0876i) q^{51} +(-0.712635 + 49.1292i) q^{52} +(50.6793 + 87.7791i) q^{53} +(15.3774 - 55.7682i) q^{54} +(21.7111 + 12.5349i) q^{55} +(13.8250 - 13.2361i) q^{56} +(-13.8138 + 39.2960i) q^{57} +(-6.86573 - 26.3872i) q^{58} +(-23.2094 - 13.3999i) q^{59} +(19.6063 + 0.284395i) q^{60} +(37.2040 + 64.4392i) q^{61} +(21.6241 - 78.4228i) q^{62} +(8.68948 - 5.01687i) q^{63} +(-2.78395 + 63.9394i) q^{64} -27.4670 q^{65} +(35.0109 - 34.5068i) q^{66} +(-92.6129 + 53.4701i) q^{67} +(-60.8209 + 33.9485i) q^{68} +13.0439 q^{69} +(7.51053 + 7.62026i) q^{70} +(47.4942 + 27.4208i) q^{71} +(-9.38670 + 32.2115i) q^{72} +(38.5356 - 66.7457i) q^{73} +(-81.9303 - 22.5912i) q^{74} +10.9614i q^{75} +(26.2418 - 71.3258i) q^{76} +26.8231 q^{77} +(-14.3164 + 51.9205i) q^{78} +(85.9331 + 49.6135i) q^{79} +(-35.7620 - 1.03770i) q^{80} +(12.8329 - 22.2272i) q^{81} +(-97.3806 + 95.9784i) q^{82} -121.735i q^{83} +(18.3192 - 10.2252i) q^{84} +(-19.4689 - 33.7211i) q^{85} +(-72.9597 - 74.0257i) q^{86} -29.8871i q^{87} +(-64.7869 + 62.0274i) q^{88} +(39.0705 + 67.6722i) q^{89} +(-18.0810 - 4.98561i) q^{90} +(-25.4507 + 14.6940i) q^{91} +(-23.7973 - 0.345187i) q^{92} +(44.5851 - 77.2237i) q^{93} +(-17.7483 + 4.61796i) q^{94} +(40.0809 + 14.0898i) q^{95} +(-20.6015 + 67.0597i) q^{96} +(-32.9466 + 57.0652i) q^{97} +(-83.4385 - 23.0071i) q^{98} +(-40.7208 + 23.5102i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 160 q + 2 q^{4} + 6 q^{6} + 248 q^{9} - 10 q^{10} - 16 q^{13} - 14 q^{16} + 48 q^{17} + 48 q^{21} - 44 q^{24} - 400 q^{25} + 68 q^{26} + 60 q^{28} - 80 q^{30} + 30 q^{32} - 40 q^{33} - 22 q^{34} + 52 q^{36}+ \cdots - 226 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.92805 − 0.531634i −0.964023 − 0.265817i
\(3\) −1.89857 − 1.09614i −0.632856 − 0.365380i 0.149001 − 0.988837i \(-0.452394\pi\)
−0.781857 + 0.623457i \(0.785727\pi\)
\(4\) 3.43473 + 2.05003i 0.858682 + 0.512508i
\(5\) −1.11803 + 1.93649i −0.223607 + 0.387298i
\(6\) 3.07778 + 3.12275i 0.512964 + 0.520459i
\(7\) 2.39245i 0.341779i 0.985290 + 0.170889i \(0.0546641\pi\)
−0.985290 + 0.170889i \(0.945336\pi\)
\(8\) −5.53245 − 5.77858i −0.691557 − 0.722322i
\(9\) −2.09696 − 3.63204i −0.232995 − 0.403560i
\(10\) 3.18513 − 3.13926i 0.318513 − 0.313926i
\(11\) − 11.2116i − 1.01923i −0.860402 − 0.509616i \(-0.829787\pi\)
0.860402 − 0.509616i \(-0.170213\pi\)
\(12\) −4.27395 − 7.65707i −0.356162 − 0.638089i
\(13\) 6.14180 + 10.6379i 0.472446 + 0.818301i 0.999503 − 0.0315294i \(-0.0100378\pi\)
−0.527057 + 0.849830i \(0.676704\pi\)
\(14\) 1.27191 − 4.61276i 0.0908506 − 0.329483i
\(15\) 4.24533 − 2.45104i 0.283022 − 0.163403i
\(16\) 7.59474 + 14.0826i 0.474671 + 0.880163i
\(17\) −8.70675 + 15.0805i −0.512162 + 0.887091i 0.487739 + 0.872990i \(0.337822\pi\)
−0.999901 + 0.0141008i \(0.995511\pi\)
\(18\) 2.11212 + 8.11756i 0.117340 + 0.450976i
\(19\) −3.50542 − 18.6738i −0.184496 − 0.982833i
\(20\) −7.81001 + 4.35932i −0.390501 + 0.217966i
\(21\) 2.62246 − 4.54223i 0.124879 − 0.216297i
\(22\) −5.96045 + 21.6164i −0.270930 + 0.982564i
\(23\) −5.15281 + 2.97497i −0.224035 + 0.129347i −0.607817 − 0.794077i \(-0.707955\pi\)
0.383782 + 0.923424i \(0.374621\pi\)
\(24\) 4.16961 + 17.0354i 0.173734 + 0.709807i
\(25\) −2.50000 − 4.33013i −0.100000 − 0.173205i
\(26\) −6.18620 − 23.7756i −0.237931 − 0.914445i
\(27\) 28.9247i 1.07129i
\(28\) −4.90460 + 8.21742i −0.175164 + 0.293479i
\(29\) 6.81645 + 11.8064i 0.235050 + 0.407118i 0.959287 − 0.282432i \(-0.0911413\pi\)
−0.724237 + 0.689551i \(0.757808\pi\)
\(30\) −9.48825 + 2.46876i −0.316275 + 0.0822920i
\(31\) 40.6747i 1.31209i 0.754723 + 0.656044i \(0.227771\pi\)
−0.754723 + 0.656044i \(0.772229\pi\)
\(32\) −7.15620 − 31.1896i −0.223631 − 0.974674i
\(33\) −12.2894 + 21.2859i −0.372407 + 0.645028i
\(34\) 24.8044 − 24.4472i 0.729540 − 0.719035i
\(35\) −4.63296 − 2.67484i −0.132370 − 0.0764240i
\(36\) 0.243311 − 16.7739i 0.00675864 − 0.465942i
\(37\) 42.4940 1.14849 0.574243 − 0.818685i \(-0.305297\pi\)
0.574243 + 0.818685i \(0.305297\pi\)
\(38\) −3.16904 + 37.8676i −0.0833959 + 0.996516i
\(39\) − 26.9291i − 0.690489i
\(40\) 17.3756 − 4.25290i 0.434391 − 0.106322i
\(41\) 34.1823 − 59.2054i 0.833714 − 1.44403i −0.0613597 − 0.998116i \(-0.519544\pi\)
0.895073 − 0.445919i \(-0.147123\pi\)
\(42\) −7.47103 + 7.36344i −0.177882 + 0.175320i
\(43\) 45.0061 + 25.9843i 1.04665 + 0.604285i 0.921710 − 0.387879i \(-0.126792\pi\)
0.124943 + 0.992164i \(0.460125\pi\)
\(44\) 22.9841 − 38.5087i 0.522365 − 0.875197i
\(45\) 9.37789 0.208397
\(46\) 11.5165 − 2.99648i 0.250358 − 0.0651409i
\(47\) 7.94113 − 4.58481i 0.168960 − 0.0975492i −0.413135 − 0.910670i \(-0.635566\pi\)
0.582095 + 0.813120i \(0.302233\pi\)
\(48\) 1.01738 − 35.0617i 0.0211953 − 0.730452i
\(49\) 43.2762 0.883187
\(50\) 2.51807 + 9.67777i 0.0503615 + 0.193555i
\(51\) 33.0607 − 19.0876i 0.648250 − 0.374267i
\(52\) −0.712635 + 49.1292i −0.0137045 + 0.944793i
\(53\) 50.6793 + 87.7791i 0.956213 + 1.65621i 0.731568 + 0.681769i \(0.238789\pi\)
0.224645 + 0.974441i \(0.427878\pi\)
\(54\) 15.3774 − 55.7682i 0.284766 − 1.03275i
\(55\) 21.7111 + 12.5349i 0.394747 + 0.227907i
\(56\) 13.8250 − 13.2361i 0.246874 − 0.236359i
\(57\) −13.8138 + 39.2960i −0.242348 + 0.689403i
\(58\) −6.86573 − 26.3872i −0.118375 − 0.454952i
\(59\) −23.2094 − 13.3999i −0.393379 − 0.227118i 0.290244 − 0.956953i \(-0.406264\pi\)
−0.683623 + 0.729835i \(0.739597\pi\)
\(60\) 19.6063 + 0.284395i 0.326771 + 0.00473992i
\(61\) 37.2040 + 64.4392i 0.609901 + 1.05638i 0.991256 + 0.131951i \(0.0421242\pi\)
−0.381355 + 0.924429i \(0.624542\pi\)
\(62\) 21.6241 − 78.4228i 0.348775 − 1.26488i
\(63\) 8.68948 − 5.01687i 0.137928 − 0.0796329i
\(64\) −2.78395 + 63.9394i −0.0434992 + 0.999053i
\(65\) −27.4670 −0.422569
\(66\) 35.0109 − 34.5068i 0.530468 − 0.522830i
\(67\) −92.6129 + 53.4701i −1.38228 + 0.798061i −0.992429 − 0.122816i \(-0.960808\pi\)
−0.389853 + 0.920877i \(0.627474\pi\)
\(68\) −60.8209 + 33.9485i −0.894426 + 0.499242i
\(69\) 13.0439 0.189043
\(70\) 7.51053 + 7.62026i 0.107293 + 0.108861i
\(71\) 47.4942 + 27.4208i 0.668932 + 0.386208i 0.795672 − 0.605728i \(-0.207118\pi\)
−0.126740 + 0.991936i \(0.540451\pi\)
\(72\) −9.38670 + 32.2115i −0.130371 + 0.447382i
\(73\) 38.5356 − 66.7457i 0.527886 − 0.914325i −0.471586 − 0.881820i \(-0.656318\pi\)
0.999472 − 0.0325045i \(-0.0103483\pi\)
\(74\) −81.9303 − 22.5912i −1.10717 − 0.305287i
\(75\) 10.9614i 0.146152i
\(76\) 26.2418 − 71.3258i 0.345287 − 0.938497i
\(77\) 26.8231 0.348352
\(78\) −14.3164 + 51.9205i −0.183544 + 0.665647i
\(79\) 85.9331 + 49.6135i 1.08776 + 0.628019i 0.932979 − 0.359930i \(-0.117199\pi\)
0.154781 + 0.987949i \(0.450533\pi\)
\(80\) −35.7620 − 1.03770i −0.447025 − 0.0129712i
\(81\) 12.8329 − 22.2272i 0.158431 − 0.274410i
\(82\) −97.3806 + 95.9784i −1.18757 + 1.17047i
\(83\) − 121.735i − 1.46669i −0.679858 − 0.733344i \(-0.737959\pi\)
0.679858 − 0.733344i \(-0.262041\pi\)
\(84\) 18.3192 − 10.2252i 0.218085 − 0.121729i
\(85\) −19.4689 − 33.7211i −0.229046 − 0.396719i
\(86\) −72.9597 − 74.0257i −0.848369 − 0.860764i
\(87\) − 29.8871i − 0.343530i
\(88\) −64.7869 + 62.0274i −0.736215 + 0.704857i
\(89\) 39.0705 + 67.6722i 0.438995 + 0.760361i 0.997612 − 0.0690641i \(-0.0220013\pi\)
−0.558617 + 0.829425i \(0.688668\pi\)
\(90\) −18.0810 − 4.98561i −0.200900 − 0.0553956i
\(91\) −25.4507 + 14.6940i −0.279678 + 0.161472i
\(92\) −23.7973 − 0.345187i −0.258666 − 0.00375204i
\(93\) 44.5851 − 77.2237i 0.479410 − 0.830362i
\(94\) −17.7483 + 4.61796i −0.188812 + 0.0491272i
\(95\) 40.0809 + 14.0898i 0.421904 + 0.148313i
\(96\) −20.6015 + 67.0597i −0.214599 + 0.698539i
\(97\) −32.9466 + 57.0652i −0.339656 + 0.588301i −0.984368 − 0.176125i \(-0.943644\pi\)
0.644712 + 0.764425i \(0.276977\pi\)
\(98\) −83.4385 − 23.0071i −0.851413 − 0.234766i
\(99\) −40.7208 + 23.5102i −0.411322 + 0.237477i
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.q.a.11.7 ✓ 160
4.3 odd 2 inner 380.3.q.a.11.60 yes 160
19.7 even 3 inner 380.3.q.a.311.60 yes 160
76.7 odd 6 inner 380.3.q.a.311.7 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.q.a.11.7 ✓ 160 1.1 even 1 trivial
380.3.q.a.11.60 yes 160 4.3 odd 2 inner
380.3.q.a.311.7 yes 160 76.7 odd 6 inner
380.3.q.a.311.60 yes 160 19.7 even 3 inner