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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 227.14
Character \(\chi\) \(=\) 380.227
Dual form 380.3.j.a.303.14

$q$-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+(-1.88290 + 0.674294i) q^{2} +(-2.71125 + 2.71125i) q^{3} +(3.09066 - 2.53926i) q^{4} +(2.06771 - 4.55243i) q^{5} +(3.27685 - 6.93322i) q^{6} +(-4.53921 + 4.53921i) q^{7} +(-4.10720 + 6.86519i) q^{8} -5.70181i q^{9} +(-0.823628 + 9.96602i) q^{10} +1.12715i q^{11} +(-1.49498 + 15.2641i) q^{12} +(-4.27635 - 4.27635i) q^{13} +(5.48614 - 11.6077i) q^{14} +(6.73670 + 17.9489i) q^{15} +(3.10431 - 15.6960i) q^{16} +(-7.95995 - 7.95995i) q^{17} +(3.84469 + 10.7360i) q^{18} +(10.5162 - 15.8243i) q^{19} +(-5.16921 - 19.3204i) q^{20} -24.6139i q^{21} +(-0.760027 - 2.12231i) q^{22} +(10.9472 + 10.9472i) q^{23} +(-7.47762 - 29.7490i) q^{24} +(-16.4492 - 18.8262i) q^{25} +(10.9355 + 5.16844i) q^{26} +(-8.94224 - 8.94224i) q^{27} +(-2.50290 + 25.5554i) q^{28} +28.5585 q^{29} +(-24.7874 - 29.2535i) q^{30} -1.29403 q^{31} +(4.73856 + 31.6472i) q^{32} +(-3.05598 - 3.05598i) q^{33} +(20.3552 + 9.62048i) q^{34} +(11.2787 + 30.0502i) q^{35} +(-14.4784 - 17.6223i) q^{36} +(27.1892 - 27.1892i) q^{37} +(-9.13075 + 36.8867i) q^{38} +23.1885 q^{39} +(22.7608 + 32.8930i) q^{40} -1.39696i q^{41} +(16.5970 + 46.3457i) q^{42} +(43.4298 + 43.4298i) q^{43} +(2.86211 + 3.48362i) q^{44} +(-25.9571 - 11.7897i) q^{45} +(-27.9942 - 13.2309i) q^{46} +(13.0820 - 13.0820i) q^{47} +(34.1392 + 50.9723i) q^{48} +7.79111i q^{49} +(43.6666 + 24.3564i) q^{50} +43.1629 q^{51} +(-24.0755 - 2.35796i) q^{52} +(52.6857 + 52.6857i) q^{53} +(22.8671 + 10.8077i) q^{54} +(5.13124 + 2.33061i) q^{55} +(-12.5191 - 49.8060i) q^{56} +(14.3917 + 71.4159i) q^{57} +(-53.7730 + 19.2568i) q^{58} +17.9143i q^{59} +(66.3977 + 38.3676i) q^{60} +91.6996 q^{61} +(2.43654 - 0.872559i) q^{62} +(25.8817 + 25.8817i) q^{63} +(-30.2618 - 56.3935i) q^{64} +(-28.3100 + 10.6255i) q^{65} +(7.81474 + 3.69349i) q^{66} +(-11.3949 - 11.3949i) q^{67} +(-44.8139 - 4.38909i) q^{68} -59.3615 q^{69} +(-41.4993 - 48.9765i) q^{70} +24.9183 q^{71} +(39.1440 + 23.4185i) q^{72} +(83.9292 - 83.9292i) q^{73} +(-32.8611 + 69.5281i) q^{74} +(95.6404 + 6.44475i) q^{75} +(-7.68014 - 75.6109i) q^{76} +(-5.11635 - 5.11635i) q^{77} +(-43.6618 + 15.6359i) q^{78} -52.2198i q^{79} +(-65.0359 - 46.5869i) q^{80} +99.8057 q^{81} +(0.941958 + 2.63033i) q^{82} +(-64.2544 - 64.2544i) q^{83} +(-62.5012 - 76.0732i) q^{84} +(-52.6960 + 19.7782i) q^{85} +(-111.058 - 52.4897i) q^{86} +(-77.4294 + 77.4294i) q^{87} +(-7.73807 - 4.62941i) q^{88} -52.2732 q^{89} +(56.8243 + 4.69617i) q^{90} +38.8225 q^{91} +(61.6320 + 6.03626i) q^{92} +(3.50846 - 3.50846i) q^{93} +(-15.8110 + 33.4532i) q^{94} +(-50.2946 - 80.5943i) q^{95} +(-98.6511 - 72.9562i) q^{96} +(97.5489 - 97.5489i) q^{97} +(-5.25350 - 14.6699i) q^{98} +6.42676 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.88290 + 0.674294i −0.941452 + 0.337147i
\(3\) −2.71125 + 2.71125i −0.903752 + 0.903752i −0.995758 − 0.0920067i \(-0.970672\pi\)
0.0920067 + 0.995758i \(0.470672\pi\)
\(4\) 3.09066 − 2.53926i 0.772664 − 0.634815i
\(5\) 2.06771 − 4.55243i 0.413542 − 0.910485i
\(6\) 3.27685 − 6.93322i 0.546142 − 1.15554i
\(7\) −4.53921 + 4.53921i −0.648459 + 0.648459i −0.952620 − 0.304162i \(-0.901624\pi\)
0.304162 + 0.952620i \(0.401624\pi\)
\(8\) −4.10720 + 6.86519i −0.513400 + 0.858149i
\(9\) − 5.70181i − 0.633534i
\(10\) −0.823628 + 9.96602i −0.0823628 + 0.996602i
\(11\) 1.12715i 0.102468i 0.998687 + 0.0512339i \(0.0163154\pi\)
−0.998687 + 0.0512339i \(0.983685\pi\)
\(12\) −1.49498 + 15.2641i −0.124581 + 1.27201i
\(13\) −4.27635 − 4.27635i −0.328950 − 0.328950i 0.523237 − 0.852187i \(-0.324724\pi\)
−0.852187 + 0.523237i \(0.824724\pi\)
\(14\) 5.48614 − 11.6077i 0.391867 − 0.829119i
\(15\) 6.73670 + 17.9489i 0.449113 + 1.19659i
\(16\) 3.10431 − 15.6960i 0.194020 − 0.980998i
\(17\) −7.95995 − 7.95995i −0.468233 − 0.468233i 0.433109 − 0.901342i \(-0.357417\pi\)
−0.901342 + 0.433109i \(0.857417\pi\)
\(18\) 3.84469 + 10.7360i 0.213594 + 0.596442i
\(19\) 10.5162 − 15.8243i 0.553484 − 0.832860i
\(20\) −5.16921 − 19.3204i −0.258461 − 0.966022i
\(21\) − 24.6139i − 1.17209i
\(22\) −0.760027 − 2.12231i −0.0345467 − 0.0964685i
\(23\) 10.9472 + 10.9472i 0.475966 + 0.475966i 0.903839 − 0.427873i \(-0.140737\pi\)
−0.427873 + 0.903839i \(0.640737\pi\)
\(24\) −7.47762 − 29.7490i −0.311567 − 1.23954i
\(25\) −16.4492 − 18.8262i −0.657966 − 0.753048i
\(26\) 10.9355 + 5.16844i 0.420595 + 0.198786i
\(27\) −8.94224 − 8.94224i −0.331194 − 0.331194i
\(28\) −2.50290 + 25.5554i −0.0893894 + 0.912692i
\(29\) 28.5585 0.984777 0.492388 − 0.870376i \(-0.336124\pi\)
0.492388 + 0.870376i \(0.336124\pi\)
\(30\) −24.7874 − 29.2535i −0.826246 − 0.975117i
\(31\) −1.29403 −0.0417430 −0.0208715 − 0.999782i \(-0.506644\pi\)
−0.0208715 + 0.999782i \(0.506644\pi\)
\(32\) 4.73856 + 31.6472i 0.148080 + 0.988975i
\(33\) −3.05598 − 3.05598i −0.0926054 − 0.0926054i
\(34\) 20.3552 + 9.62048i 0.598682 + 0.282955i
\(35\) 11.2787 + 30.0502i 0.322247 + 0.858577i
\(36\) −14.4784 − 17.6223i −0.402177 − 0.489509i
\(37\) 27.1892 − 27.1892i 0.734842 − 0.734842i −0.236732 − 0.971575i \(-0.576076\pi\)
0.971575 + 0.236732i \(0.0760765\pi\)
\(38\) −9.13075 + 36.8867i −0.240283 + 0.970703i
\(39\) 23.1885 0.594578
\(40\) 22.7608 + 32.8930i 0.569019 + 0.822324i
\(41\) − 1.39696i − 0.0340721i −0.999855 − 0.0170360i \(-0.994577\pi\)
0.999855 − 0.0170360i \(-0.00542300\pi\)
\(42\) 16.5970 + 46.3457i 0.395167 + 1.10347i
\(43\) 43.4298 + 43.4298i 1.00999 + 1.00999i 0.999950 + 0.0100448i \(0.00319742\pi\)
0.0100448 + 0.999950i \(0.496803\pi\)
\(44\) 2.86211 + 3.48362i 0.0650481 + 0.0791731i
\(45\) −25.9571 − 11.7897i −0.576823 − 0.261993i
\(46\) −27.9942 − 13.2309i −0.608570 − 0.287629i
\(47\) 13.0820 − 13.0820i 0.278340 − 0.278340i −0.554106 − 0.832446i \(-0.686940\pi\)
0.832446 + 0.554106i \(0.186940\pi\)
\(48\) 34.1392 + 50.9723i 0.711233 + 1.06192i
\(49\) 7.79111i 0.159002i
\(50\) 43.6666 + 24.3564i 0.873331 + 0.487127i
\(51\) 43.1629 0.846332
\(52\) −24.0755 − 2.35796i −0.462990 − 0.0453454i
\(53\) 52.6857 + 52.6857i 0.994070 + 0.994070i 0.999983 − 0.00591245i \(-0.00188200\pi\)
−0.00591245 + 0.999983i \(0.501882\pi\)
\(54\) 22.8671 + 10.8077i 0.423464 + 0.200142i
\(55\) 5.13124 + 2.33061i 0.0932953 + 0.0423747i
\(56\) −12.5191 − 49.8060i −0.223555 − 0.889393i
\(57\) 14.3917 + 71.4159i 0.252486 + 1.25291i
\(58\) −53.7730 + 19.2568i −0.927120 + 0.332014i
\(59\) 17.9143i 0.303631i 0.988409 + 0.151816i \(0.0485120\pi\)
−0.988409 + 0.151816i \(0.951488\pi\)
\(60\) 66.3977 + 38.3676i 1.10663 + 0.639460i
\(61\) 91.6996 1.50327 0.751636 − 0.659578i \(-0.229265\pi\)
0.751636 + 0.659578i \(0.229265\pi\)
\(62\) 2.43654 − 0.872559i 0.0392991 − 0.0140735i
\(63\) 25.8817 + 25.8817i 0.410821 + 0.410821i
\(64\) −30.2618 − 56.3935i −0.472840 − 0.881148i
\(65\) −28.3100 + 10.6255i −0.435538 + 0.163469i
\(66\) 7.81474 + 3.69349i 0.118405 + 0.0559619i
\(67\) −11.3949 − 11.3949i −0.170073 − 0.170073i 0.616939 − 0.787011i \(-0.288373\pi\)
−0.787011 + 0.616939i \(0.788373\pi\)
\(68\) −44.8139 − 4.38909i −0.659028 − 0.0645454i
\(69\) −59.3615 −0.860311
\(70\) −41.4993 − 48.9765i −0.592847 − 0.699665i
\(71\) 24.9183 0.350961 0.175481 − 0.984483i \(-0.443852\pi\)
0.175481 + 0.984483i \(0.443852\pi\)
\(72\) 39.1440 + 23.4185i 0.543667 + 0.325257i
\(73\) 83.9292 − 83.9292i 1.14972 − 1.14972i 0.163107 − 0.986608i \(-0.447848\pi\)
0.986608 − 0.163107i \(-0.0521516\pi\)
\(74\) −32.8611 + 69.5281i −0.444069 + 0.939569i
\(75\) 95.6404 + 6.44475i 1.27521 + 0.0859300i
\(76\) −7.68014 − 75.6109i −0.101054 − 0.994881i
\(77\) −5.11635 − 5.11635i −0.0664461 − 0.0664461i
\(78\) −43.6618 + 15.6359i −0.559767 + 0.200460i
\(79\) − 52.2198i − 0.661011i −0.943804 − 0.330505i \(-0.892781\pi\)
0.943804 − 0.330505i \(-0.107219\pi\)
\(80\) −65.0359 − 46.5869i −0.812948 − 0.582336i
\(81\) 99.8057 1.23217
\(82\) 0.941958 + 2.63033i 0.0114873 + 0.0320772i
\(83\) −64.2544 − 64.2544i −0.774149 − 0.774149i 0.204680 − 0.978829i \(-0.434385\pi\)
−0.978829 + 0.204680i \(0.934385\pi\)
\(84\) −62.5012 − 76.0732i −0.744061 − 0.905633i
\(85\) −52.6960 + 19.7782i −0.619953 + 0.232685i
\(86\) −111.058 − 52.4897i −1.29138 − 0.610345i
\(87\) −77.4294 + 77.4294i −0.889994 + 0.889994i
\(88\) −7.73807 − 4.62941i −0.0879326 − 0.0526070i
\(89\) −52.2732 −0.587339 −0.293669 − 0.955907i \(-0.594876\pi\)
−0.293669 + 0.955907i \(0.594876\pi\)
\(90\) 56.8243 + 4.69617i 0.631382 + 0.0521797i
\(91\) 38.8225 0.426621
\(92\) 61.6320 + 6.03626i 0.669913 + 0.0656115i
\(93\) 3.50846 − 3.50846i 0.0377253 − 0.0377253i
\(94\) −15.8110 + 33.4532i −0.168203 + 0.355886i
\(95\) −50.2946 − 80.5943i −0.529417 − 0.848362i
\(96\) −98.6511 − 72.9562i −1.02762 − 0.759961i
\(97\) 97.5489 − 97.5489i 1.00566 − 1.00566i 0.00567483 − 0.999984i \(-0.498194\pi\)
0.999984 − 0.00567483i \(-0.00180636\pi\)
\(98\) −5.25350 − 14.6699i −0.0536071 − 0.149693i
\(99\) 6.42676 0.0649168
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.227.14 ✓ 232
4.3 odd 2 inner 380.3.j.a.227.45 yes 232
5.3 odd 4 inner 380.3.j.a.303.72 yes 232
19.18 odd 2 inner 380.3.j.a.227.103 yes 232
20.3 even 4 inner 380.3.j.a.303.103 yes 232
76.75 even 2 inner 380.3.j.a.227.72 yes 232
95.18 even 4 inner 380.3.j.a.303.45 yes 232
380.303 odd 4 inner 380.3.j.a.303.14 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.j.a.227.14 ✓ 232 1.1 even 1 trivial
380.3.j.a.227.45 yes 232 4.3 odd 2 inner
380.3.j.a.227.72 yes 232 76.75 even 2 inner
380.3.j.a.227.103 yes 232 19.18 odd 2 inner
380.3.j.a.303.14 yes 232 380.303 odd 4 inner
380.3.j.a.303.45 yes 232 95.18 even 4 inner
380.3.j.a.303.72 yes 232 5.3 odd 4 inner
380.3.j.a.303.103 yes 232 20.3 even 4 inner