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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 227.8
Character \(\chi\) \(=\) 380.227
Dual form 380.3.j.a.303.8

$q$-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+(-1.93420 - 0.508790i) q^{2} +(-3.89344 + 3.89344i) q^{3} +(3.48227 + 1.96820i) q^{4} +(3.99170 + 3.01103i) q^{5} +(9.51163 - 5.54975i) q^{6} +(-5.88298 + 5.88298i) q^{7} +(-5.73400 - 5.57864i) q^{8} -21.3177i q^{9} +(-6.18877 - 7.85488i) q^{10} -8.21055i q^{11} +(-21.2211 + 5.89491i) q^{12} +(-1.11629 - 1.11629i) q^{13} +(14.3721 - 8.38567i) q^{14} +(-27.2647 + 3.81818i) q^{15} +(8.25236 + 13.7076i) q^{16} +(-6.22724 - 6.22724i) q^{17} +(-10.8462 + 41.2327i) q^{18} +(-17.9570 - 6.20865i) q^{19} +(7.97385 + 18.3417i) q^{20} -45.8100i q^{21} +(-4.17744 + 15.8809i) q^{22} +(-1.57158 - 1.57158i) q^{23} +(44.0451 - 0.604889i) q^{24} +(6.86737 + 24.0383i) q^{25} +(1.59117 + 2.72708i) q^{26} +(47.9582 + 47.9582i) q^{27} +(-32.0650 + 8.90721i) q^{28} -29.0250 q^{29} +(54.6781 + 6.48688i) q^{30} +24.3985 q^{31} +(-8.98743 - 30.7120i) q^{32} +(31.9673 + 31.9673i) q^{33} +(8.87638 + 15.2131i) q^{34} +(-41.1970 + 5.76926i) q^{35} +(41.9576 - 74.2339i) q^{36} +(21.2310 - 21.2310i) q^{37} +(31.5735 + 21.1451i) q^{38} +8.69238 q^{39} +(-6.09096 - 39.5335i) q^{40} -29.4182i q^{41} +(-23.3077 + 88.6058i) q^{42} +(-54.6774 - 54.6774i) q^{43} +(16.1600 - 28.5913i) q^{44} +(64.1883 - 85.0939i) q^{45} +(2.24015 + 3.83937i) q^{46} +(54.7418 - 54.7418i) q^{47} +(-85.4998 - 21.2397i) q^{48} -20.2189i q^{49} +(-1.05245 - 49.9889i) q^{50} +48.4908 q^{51} +(-1.69013 - 6.08428i) q^{52} +(32.2812 + 32.2812i) q^{53} +(-68.3602 - 117.161i) q^{54} +(24.7222 - 32.7741i) q^{55} +(66.5521 - 0.913987i) q^{56} +(94.0873 - 45.7413i) q^{57} +(56.1402 + 14.7676i) q^{58} +81.4243i q^{59} +(-102.458 - 40.3666i) q^{60} +13.9818 q^{61} +(-47.1917 - 12.4137i) q^{62} +(125.412 + 125.412i) q^{63} +(1.75755 + 63.9759i) q^{64} +(-1.09471 - 7.81705i) q^{65} +(-45.5665 - 78.0958i) q^{66} +(-71.9988 - 71.9988i) q^{67} +(-9.42844 - 33.9414i) q^{68} +12.2377 q^{69} +(82.6185 + 9.80167i) q^{70} +88.5150 q^{71} +(-118.924 + 122.236i) q^{72} +(-69.8114 + 69.8114i) q^{73} +(-51.8671 + 30.2629i) q^{74} +(-120.329 - 66.8539i) q^{75} +(-50.3111 - 56.9631i) q^{76} +(48.3025 + 48.3025i) q^{77} +(-16.8128 - 4.42259i) q^{78} -71.3866i q^{79} +(-8.33311 + 79.5648i) q^{80} -181.585 q^{81} +(-14.9677 + 56.9007i) q^{82} +(72.2575 + 72.2575i) q^{83} +(90.1634 - 159.523i) q^{84} +(-6.10687 - 43.6077i) q^{85} +(77.9377 + 133.576i) q^{86} +(113.007 - 113.007i) q^{87} +(-45.8037 + 47.0793i) q^{88} +25.0477 q^{89} +(-167.448 + 131.930i) q^{90} +13.1342 q^{91} +(-2.37948 - 8.56587i) q^{92} +(-94.9941 + 94.9941i) q^{93} +(-133.734 + 78.0295i) q^{94} +(-52.9844 - 78.8521i) q^{95} +(154.567 + 84.5832i) q^{96} +(-16.3527 + 16.3527i) q^{97} +(-10.2872 + 39.1075i) q^{98} -175.030 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.93420 − 0.508790i −0.967100 − 0.254395i
\(3\) −3.89344 + 3.89344i −1.29781 + 1.29781i −0.367978 + 0.929835i \(0.619950\pi\)
−0.929835 + 0.367978i \(0.880050\pi\)
\(4\) 3.48227 + 1.96820i 0.870567 + 0.492051i
\(5\) 3.99170 + 3.01103i 0.798340 + 0.602206i
\(6\) 9.51163 − 5.54975i 1.58527 − 0.924958i
\(7\) −5.88298 + 5.88298i −0.840426 + 0.840426i −0.988914 − 0.148488i \(-0.952559\pi\)
0.148488 + 0.988914i \(0.452559\pi\)
\(8\) −5.73400 − 5.57864i −0.716750 − 0.697330i
\(9\) − 21.3177i − 2.36863i
\(10\) −6.18877 − 7.85488i −0.618877 − 0.785488i
\(11\) − 8.21055i − 0.746414i −0.927748 − 0.373207i \(-0.878258\pi\)
0.927748 − 0.373207i \(-0.121742\pi\)
\(12\) −21.2211 + 5.89491i −1.76842 + 0.491243i
\(13\) −1.11629 − 1.11629i −0.0858681 − 0.0858681i 0.662868 − 0.748736i \(-0.269339\pi\)
−0.748736 + 0.662868i \(0.769339\pi\)
\(14\) 14.3721 − 8.38567i 1.02658 − 0.598976i
\(15\) −27.2647 + 3.81818i −1.81765 + 0.254545i
\(16\) 8.25236 + 13.7076i 0.515772 + 0.856726i
\(17\) −6.22724 − 6.22724i −0.366308 − 0.366308i 0.499821 − 0.866129i \(-0.333399\pi\)
−0.866129 + 0.499821i \(0.833399\pi\)
\(18\) −10.8462 + 41.2327i −0.602568 + 2.29071i
\(19\) −17.9570 − 6.20865i −0.945104 − 0.326771i
\(20\) 7.97385 + 18.3417i 0.398692 + 0.917085i
\(21\) − 45.8100i − 2.18143i
\(22\) −4.17744 + 15.8809i −0.189884 + 0.721857i
\(23\) −1.57158 − 1.57158i −0.0683297 − 0.0683297i 0.672116 − 0.740446i \(-0.265386\pi\)
−0.740446 + 0.672116i \(0.765386\pi\)
\(24\) 44.0451 − 0.604889i 1.83521 − 0.0252037i
\(25\) 6.86737 + 24.0383i 0.274695 + 0.961531i
\(26\) 1.59117 + 2.72708i 0.0611987 + 0.104888i
\(27\) 47.9582 + 47.9582i 1.77623 + 1.77623i
\(28\) −32.0650 + 8.90721i −1.14518 + 0.318115i
\(29\) −29.0250 −1.00086 −0.500431 − 0.865776i \(-0.666825\pi\)
−0.500431 + 0.865776i \(0.666825\pi\)
\(30\) 54.6781 + 6.48688i 1.82260 + 0.216229i
\(31\) 24.3985 0.787049 0.393525 − 0.919314i \(-0.371256\pi\)
0.393525 + 0.919314i \(0.371256\pi\)
\(32\) −8.98743 − 30.7120i −0.280857 − 0.959750i
\(33\) 31.9673 + 31.9673i 0.968705 + 0.968705i
\(34\) 8.87638 + 15.2131i 0.261070 + 0.447444i
\(35\) −41.1970 + 5.76926i −1.17706 + 0.164836i
\(36\) 41.9576 − 74.2339i 1.16549 − 2.06205i
\(37\) 21.2310 − 21.2310i 0.573810 − 0.573810i −0.359381 − 0.933191i \(-0.617012\pi\)
0.933191 + 0.359381i \(0.117012\pi\)
\(38\) 31.5735 + 21.1451i 0.830881 + 0.556450i
\(39\) 8.69238 0.222881
\(40\) −6.09096 − 39.5335i −0.152274 − 0.988338i
\(41\) − 29.4182i − 0.717517i −0.933430 − 0.358759i \(-0.883200\pi\)
0.933430 − 0.358759i \(-0.116800\pi\)
\(42\) −23.3077 + 88.6058i −0.554944 + 2.10966i
\(43\) −54.6774 − 54.6774i −1.27157 − 1.27157i −0.945266 − 0.326301i \(-0.894198\pi\)
−0.326301 − 0.945266i \(-0.605802\pi\)
\(44\) 16.1600 − 28.5913i 0.367274 − 0.649803i
\(45\) 64.1883 − 85.0939i 1.42641 − 1.89098i
\(46\) 2.24015 + 3.83937i 0.0486990 + 0.0834645i
\(47\) 54.7418 − 54.7418i 1.16472 − 1.16472i 0.181289 − 0.983430i \(-0.441973\pi\)
0.983430 − 0.181289i \(-0.0580268\pi\)
\(48\) −85.4998 − 21.2397i −1.78124 − 0.442494i
\(49\) − 20.2189i − 0.412631i
\(50\) −1.05245 − 49.9889i −0.0210489 − 0.999778i
\(51\) 48.4908 0.950799
\(52\) −1.69013 − 6.08428i −0.0325025 − 0.117005i
\(53\) 32.2812 + 32.2812i 0.609080 + 0.609080i 0.942706 − 0.333626i \(-0.108272\pi\)
−0.333626 + 0.942706i \(0.608272\pi\)
\(54\) −68.3602 − 117.161i −1.26593 − 2.16966i
\(55\) 24.7222 − 32.7741i 0.449495 − 0.595893i
\(56\) 66.5521 − 0.913987i 1.18843 − 0.0163212i
\(57\) 94.0873 − 45.7413i 1.65065 − 0.802480i
\(58\) 56.1402 + 14.7676i 0.967935 + 0.254614i
\(59\) 81.4243i 1.38007i 0.723775 + 0.690036i \(0.242405\pi\)
−0.723775 + 0.690036i \(0.757595\pi\)
\(60\) −102.458 − 40.3666i −1.70763 − 0.672776i
\(61\) 13.9818 0.229210 0.114605 − 0.993411i \(-0.463440\pi\)
0.114605 + 0.993411i \(0.463440\pi\)
\(62\) −47.1917 − 12.4137i −0.761156 − 0.200221i
\(63\) 125.412 + 125.412i 1.99066 + 1.99066i
\(64\) 1.75755 + 63.9759i 0.0274617 + 0.999623i
\(65\) −1.09471 − 7.81705i −0.0168417 − 0.120262i
\(66\) −45.5665 − 78.0958i −0.690402 − 1.18327i
\(67\) −71.9988 − 71.9988i −1.07461 − 1.07461i −0.996983 − 0.0776266i \(-0.975266\pi\)
−0.0776266 − 0.996983i \(-0.524734\pi\)
\(68\) −9.42844 − 33.9414i −0.138654 − 0.499138i
\(69\) 12.2377 0.177358
\(70\) 82.6185 + 9.80167i 1.18026 + 0.140024i
\(71\) 88.5150 1.24669 0.623345 − 0.781947i \(-0.285773\pi\)
0.623345 + 0.781947i \(0.285773\pi\)
\(72\) −118.924 + 122.236i −1.65172 + 1.69772i
\(73\) −69.8114 + 69.8114i −0.956321 + 0.956321i −0.999085 − 0.0427641i \(-0.986384\pi\)
0.0427641 + 0.999085i \(0.486384\pi\)
\(74\) −51.8671 + 30.2629i −0.700906 + 0.408958i
\(75\) −120.329 − 66.8539i −1.60439 − 0.891385i
\(76\) −50.3111 − 56.9631i −0.661988 − 0.749515i
\(77\) 48.3025 + 48.3025i 0.627306 + 0.627306i
\(78\) −16.8128 − 4.42259i −0.215549 − 0.0566999i
\(79\) − 71.3866i − 0.903628i −0.892112 − 0.451814i \(-0.850777\pi\)
0.892112 − 0.451814i \(-0.149223\pi\)
\(80\) −8.33311 + 79.5648i −0.104164 + 0.994560i
\(81\) −181.585 −2.24179
\(82\) −14.9677 + 56.9007i −0.182533 + 0.693911i
\(83\) 72.2575 + 72.2575i 0.870572 + 0.870572i 0.992535 − 0.121962i \(-0.0389187\pi\)
−0.121962 + 0.992535i \(0.538919\pi\)
\(84\) 90.1634 − 159.523i 1.07337 − 1.89908i
\(85\) −6.10687 − 43.6077i −0.0718456 − 0.513032i
\(86\) 77.9377 + 133.576i 0.906253 + 1.55321i
\(87\) 113.007 − 113.007i 1.29893 − 1.29893i
\(88\) −45.8037 + 47.0793i −0.520497 + 0.534992i
\(89\) 25.0477 0.281435 0.140718 − 0.990050i \(-0.455059\pi\)
0.140718 + 0.990050i \(0.455059\pi\)
\(90\) −167.448 + 131.930i −1.86053 + 1.46589i
\(91\) 13.1342 0.144332
\(92\) −2.37948 − 8.56587i −0.0258639 − 0.0931073i
\(93\) −94.9941 + 94.9941i −1.02144 + 1.02144i
\(94\) −133.734 + 78.0295i −1.42270 + 0.830102i
\(95\) −52.9844 − 78.8521i −0.557731 − 0.830022i
\(96\) 154.567 + 84.5832i 1.61007 + 0.881075i
\(97\) −16.3527 + 16.3527i −0.168584 + 0.168584i −0.786357 − 0.617773i \(-0.788035\pi\)
0.617773 + 0.786357i \(0.288035\pi\)
\(98\) −10.2872 + 39.1075i −0.104971 + 0.399056i
\(99\) −175.030 −1.76798
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.227.8 ✓ 232
4.3 odd 2 inner 380.3.j.a.227.66 yes 232
5.3 odd 4 inner 380.3.j.a.303.51 yes 232
19.18 odd 2 inner 380.3.j.a.227.109 yes 232
20.3 even 4 inner 380.3.j.a.303.109 yes 232
76.75 even 2 inner 380.3.j.a.227.51 yes 232
95.18 even 4 inner 380.3.j.a.303.66 yes 232
380.303 odd 4 inner 380.3.j.a.303.8 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.j.a.227.8 ✓ 232 1.1 even 1 trivial
380.3.j.a.227.51 yes 232 76.75 even 2 inner
380.3.j.a.227.66 yes 232 4.3 odd 2 inner
380.3.j.a.227.109 yes 232 19.18 odd 2 inner
380.3.j.a.303.8 yes 232 380.303 odd 4 inner
380.3.j.a.303.51 yes 232 5.3 odd 4 inner
380.3.j.a.303.66 yes 232 95.18 even 4 inner
380.3.j.a.303.109 yes 232 20.3 even 4 inner