Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,2,Mod(49,380)] 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("380.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.r (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: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 20 x^{18} + 261 x^{16} - 1994 x^{14} + 11074 x^{12} - 39211 x^{10} + 99376 x^{8} - 134299 x^{6} + \cdots + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 49.9
Root \(2.10552 - 1.21562i\) of defining polynomial
Character \(\chi\) \(=\) 380.49
Dual form 380.2.r.a.349.9

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.10552 + 1.21562i) q^{3} +(2.22225 - 0.248224i) q^{5} +0.663818i q^{7} +(1.45548 + 2.52097i) q^{9} -1.80905 q^{11} +(1.99526 - 1.15197i) q^{13} +(4.98074 + 2.17878i) q^{15} +(-3.77643 - 2.18033i) q^{17} +(-4.21168 + 1.12329i) q^{19} +(-0.806953 + 1.39768i) q^{21} +(1.81374 - 1.04716i) q^{23} +(4.87677 - 1.10323i) q^{25} -0.216466i q^{27} +(0.974621 + 1.68809i) q^{29} -9.52527 q^{31} +(-3.80900 - 2.19913i) q^{33} +(0.164775 + 1.47517i) q^{35} +2.97461i q^{37} +5.60143 q^{39} +(-0.247657 + 0.428954i) q^{41} +(6.81715 + 3.93588i) q^{43} +(3.86021 + 5.24093i) q^{45} +(-5.69449 + 3.28772i) q^{47} +6.55935 q^{49} +(-5.30091 - 9.18145i) q^{51} +(1.99575 - 1.15225i) q^{53} +(-4.02016 + 0.449050i) q^{55} +(-10.2333 - 2.75471i) q^{57} +(3.88559 - 6.73003i) q^{59} +(-5.36021 - 9.28415i) q^{61} +(-1.67347 + 0.966176i) q^{63} +(4.14802 - 3.05522i) q^{65} +(3.96984 - 2.29199i) q^{67} +5.09182 q^{69} +(-2.95914 + 5.12538i) q^{71} +(-4.86313 - 2.80773i) q^{73} +(11.6093 + 3.60545i) q^{75} -1.20088i q^{77} +(-2.99810 + 5.19286i) q^{79} +(4.62959 - 8.01868i) q^{81} +6.20090i q^{83} +(-8.93338 - 3.90782i) q^{85} +4.73909i q^{87} +(-6.65028 - 11.5186i) q^{89} +(0.764696 + 1.32449i) q^{91} +(-20.0557 - 11.5791i) q^{93} +(-9.08057 + 3.54166i) q^{95} +(8.80695 + 5.08470i) q^{97} +(-2.63305 - 4.56057i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + q^{5} + 10 q^{9} - 5 q^{15} + 14 q^{19} - 8 q^{21} + 9 q^{25} - 16 q^{29} + 8 q^{31} - 2 q^{35} - 8 q^{39} + 26 q^{41} - 32 q^{45} - 44 q^{49} + 26 q^{51} - 12 q^{55} + 4 q^{59} + 2 q^{61} - 18 q^{65}+ \cdots - 20 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.10552 + 1.21562i 1.21562 + 0.701841i 0.963979 − 0.265979i \(-0.0856951\pi\)
0.251645 + 0.967820i \(0.419028\pi\)
\(4\) 0 0
\(5\) 2.22225 − 0.248224i 0.993819 − 0.111009i
\(6\) 0 0
\(7\) 0.663818i 0.250900i 0.992100 + 0.125450i \(0.0400374\pi\)
−0.992100 + 0.125450i \(0.959963\pi\)
\(8\) 0 0
\(9\) 1.45548 + 2.52097i 0.485161 + 0.840323i
\(10\) 0 0
\(11\) −1.80905 −0.545450 −0.272725 − 0.962092i \(-0.587925\pi\)
−0.272725 + 0.962092i \(0.587925\pi\)
\(12\) 0 0
\(13\) 1.99526 − 1.15197i 0.553386 − 0.319498i −0.197100 − 0.980383i \(-0.563153\pi\)
0.750487 + 0.660886i \(0.229819\pi\)
\(14\) 0 0
\(15\) 4.98074 + 2.17878i 1.28602 + 0.562558i
\(16\) 0 0
\(17\) −3.77643 − 2.18033i −0.915920 − 0.528807i −0.0335887 − 0.999436i \(-0.510694\pi\)
−0.882331 + 0.470629i \(0.844027\pi\)
\(18\) 0 0
\(19\) −4.21168 + 1.12329i −0.966225 + 0.257699i
\(20\) 0 0
\(21\) −0.806953 + 1.39768i −0.176092 + 0.305000i
\(22\) 0 0
\(23\) 1.81374 − 1.04716i 0.378191 − 0.218349i −0.298840 − 0.954303i \(-0.596600\pi\)
0.677031 + 0.735955i \(0.263266\pi\)
\(24\) 0 0
\(25\) 4.87677 − 1.10323i 0.975354 − 0.220646i
\(26\) 0 0
\(27\) − 0.216466i − 0.0416588i
\(28\) 0 0
\(29\) 0.974621 + 1.68809i 0.180983 + 0.313471i 0.942215 − 0.335008i \(-0.108739\pi\)
−0.761233 + 0.648479i \(0.775406\pi\)
\(30\) 0 0
\(31\) −9.52527 −1.71079 −0.855394 − 0.517977i \(-0.826685\pi\)
−0.855394 + 0.517977i \(0.826685\pi\)
\(32\) 0 0
\(33\) −3.80900 − 2.19913i −0.663062 − 0.382819i
\(34\) 0 0
\(35\) 0.164775 + 1.47517i 0.0278521 + 0.249349i
\(36\) 0 0
\(37\) 2.97461i 0.489023i 0.969646 + 0.244511i \(0.0786276\pi\)
−0.969646 + 0.244511i \(0.921372\pi\)
\(38\) 0 0
\(39\) 5.60143 0.896946
\(40\) 0 0
\(41\) −0.247657 + 0.428954i −0.0386775 + 0.0669914i −0.884716 − 0.466130i \(-0.845648\pi\)
0.846039 + 0.533122i \(0.178981\pi\)
\(42\) 0 0
\(43\) 6.81715 + 3.93588i 1.03960 + 0.600216i 0.919721 − 0.392572i \(-0.128415\pi\)
0.119884 + 0.992788i \(0.461748\pi\)
\(44\) 0 0
\(45\) 3.86021 + 5.24093i 0.575446 + 0.781272i
\(46\) 0 0
\(47\) −5.69449 + 3.28772i −0.830627 + 0.479563i −0.854067 − 0.520163i \(-0.825871\pi\)
0.0234403 + 0.999725i \(0.492538\pi\)
\(48\) 0 0
\(49\) 6.55935 0.937049
\(50\) 0 0
\(51\) −5.30091 − 9.18145i −0.742276 − 1.28566i
\(52\) 0 0
\(53\) 1.99575 − 1.15225i 0.274137 − 0.158273i −0.356629 − 0.934246i \(-0.616074\pi\)
0.630766 + 0.775973i \(0.282741\pi\)
\(54\) 0 0
\(55\) −4.02016 + 0.449050i −0.542079 + 0.0605498i
\(56\) 0 0
\(57\) −10.2333 − 2.75471i −1.35543 − 0.364871i
\(58\) 0 0
\(59\) 3.88559 − 6.73003i 0.505860 − 0.876176i −0.494117 − 0.869396i \(-0.664508\pi\)
0.999977 − 0.00678007i \(-0.00215818\pi\)
\(60\) 0 0
\(61\) −5.36021 − 9.28415i −0.686304 − 1.18871i −0.973025 − 0.230700i \(-0.925899\pi\)
0.286721 − 0.958014i \(-0.407435\pi\)
\(62\) 0 0
\(63\) −1.67347 + 0.966176i −0.210837 + 0.121727i
\(64\) 0 0
\(65\) 4.14802 − 3.05522i 0.514499 − 0.378954i
\(66\) 0 0
\(67\) 3.96984 − 2.29199i 0.484993 − 0.280011i −0.237502 − 0.971387i \(-0.576329\pi\)
0.722495 + 0.691376i \(0.242995\pi\)
\(68\) 0 0
\(69\) 5.09182 0.612984
\(70\) 0 0
\(71\) −2.95914 + 5.12538i −0.351185 + 0.608270i −0.986457 − 0.164018i \(-0.947554\pi\)
0.635272 + 0.772288i \(0.280888\pi\)
\(72\) 0 0
\(73\) −4.86313 − 2.80773i −0.569187 − 0.328620i 0.187638 − 0.982238i \(-0.439917\pi\)
−0.756824 + 0.653618i \(0.773250\pi\)
\(74\) 0 0
\(75\) 11.6093 + 3.60545i 1.34052 + 0.416321i
\(76\) 0 0
\(77\) − 1.20088i − 0.136853i
\(78\) 0 0
\(79\) −2.99810 + 5.19286i −0.337312 + 0.584242i −0.983926 − 0.178575i \(-0.942851\pi\)
0.646614 + 0.762817i \(0.276185\pi\)
\(80\) 0 0
\(81\) 4.62959 − 8.01868i 0.514399 − 0.890965i
\(82\) 0 0
\(83\) 6.20090i 0.680638i 0.940310 + 0.340319i \(0.110535\pi\)
−0.940310 + 0.340319i \(0.889465\pi\)
\(84\) 0 0
\(85\) −8.93338 − 3.90782i −0.968961 − 0.423863i
\(86\) 0 0
\(87\) 4.73909i 0.508084i
\(88\) 0 0
\(89\) −6.65028 − 11.5186i −0.704928 − 1.22097i −0.966717 − 0.255847i \(-0.917646\pi\)
0.261789 − 0.965125i \(-0.415688\pi\)
\(90\) 0 0
\(91\) 0.764696 + 1.32449i 0.0801619 + 0.138844i
\(92\) 0 0
\(93\) −20.0557 − 11.5791i −2.07968 − 1.20070i
\(94\) 0 0
\(95\) −9.08057 + 3.54166i −0.931646 + 0.363366i
\(96\) 0 0
\(97\) 8.80695 + 5.08470i 0.894211 + 0.516273i 0.875317 − 0.483549i \(-0.160652\pi\)
0.0188932 + 0.999822i \(0.493986\pi\)
\(98\) 0 0
\(99\) −2.63305 − 4.56057i −0.264631 − 0.458354i
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 380.2.r.a.49.9 yes 20
3.2 odd 2 3420.2.bj.c.1189.1 20
5.2 odd 4 1900.2.i.g.201.9 20
5.3 odd 4 1900.2.i.g.201.2 20
5.4 even 2 inner 380.2.r.a.49.2 ✓ 20
15.14 odd 2 3420.2.bj.c.1189.7 20
19.7 even 3 inner 380.2.r.a.349.2 yes 20
57.26 odd 6 3420.2.bj.c.2629.7 20
95.7 odd 12 1900.2.i.g.501.9 20
95.64 even 6 inner 380.2.r.a.349.9 yes 20
95.83 odd 12 1900.2.i.g.501.2 20
285.254 odd 6 3420.2.bj.c.2629.1 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.r.a.49.2 ✓ 20 5.4 even 2 inner
380.2.r.a.49.9 yes 20 1.1 even 1 trivial
380.2.r.a.349.2 yes 20 19.7 even 3 inner
380.2.r.a.349.9 yes 20 95.64 even 6 inner
1900.2.i.g.201.2 20 5.3 odd 4
1900.2.i.g.201.9 20 5.2 odd 4
1900.2.i.g.501.2 20 95.83 odd 12
1900.2.i.g.501.9 20 95.7 odd 12
3420.2.bj.c.1189.1 20 3.2 odd 2
3420.2.bj.c.1189.7 20 15.14 odd 2
3420.2.bj.c.2629.1 20 285.254 odd 6
3420.2.bj.c.2629.7 20 57.26 odd 6