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 349.9
Root \(2.10552 + 1.21562i\) of defining polynomial
Character \(\chi\) \(=\) 380.349
Dual form 380.2.r.a.49.9

$q$-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 \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

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

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 0.251645 − 0.967820i \(-0.419028\pi\)
0.963979 + 0.265979i \(0.0856951\pi\)
\(4\) 0 0
\(5\) 2.22225 + 0.248224i 0.993819 + 0.111009i
\(6\) 0 0
\(7\) − 0.663818i − 0.250900i −0.992100 − 0.125450i \(-0.959963\pi\)
0.992100 − 0.125450i \(-0.0400374\pi\)
\(8\) 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 0.750487 − 0.660886i \(-0.229819\pi\)
−0.197100 + 0.980383i \(0.563153\pi\)
\(14\) 0 0
\(15\) 4.98074 − 2.17878i 1.28602 − 0.562558i
\(16\) 0 0
\(17\) −3.77643 + 2.18033i −0.915920 + 0.528807i −0.882331 − 0.470629i \(-0.844027\pi\)
−0.0335887 + 0.999436i \(0.510694\pi\)
\(18\) 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 0.677031 − 0.735955i \(-0.263266\pi\)
−0.298840 + 0.954303i \(0.596600\pi\)
\(24\) 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 −0.761233 − 0.648479i \(-0.775406\pi\)
0.942215 + 0.335008i \(0.108739\pi\)
\(30\) 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 \(-0.921372\pi\)
0.969646 − 0.244511i \(-0.0786276\pi\)
\(38\) 0 0
\(39\) 5.60143 0.896946
\(40\) 0 0
\(41\) −0.247657 − 0.428954i −0.0386775 − 0.0669914i 0.846039 − 0.533122i \(-0.178981\pi\)
−0.884716 + 0.466130i \(0.845648\pi\)
\(42\) 0 0
\(43\) 6.81715 − 3.93588i 1.03960 − 0.600216i 0.119884 − 0.992788i \(-0.461748\pi\)
0.919721 + 0.392572i \(0.128415\pi\)
\(44\) 0 0
\(45\) 3.86021 − 5.24093i 0.575446 − 0.781272i
\(46\) 0 0
\(47\) −5.69449 − 3.28772i −0.830627 − 0.479563i 0.0234403 − 0.999725i \(-0.492538\pi\)
−0.854067 + 0.520163i \(0.825871\pi\)
\(48\) 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 0.630766 − 0.775973i \(-0.282741\pi\)
−0.356629 + 0.934246i \(0.616074\pi\)
\(54\) 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 0.999977 + 0.00678007i \(0.00215818\pi\)
−0.494117 + 0.869396i \(0.664508\pi\)
\(60\) 0 0
\(61\) −5.36021 + 9.28415i −0.686304 + 1.18871i 0.286721 + 0.958014i \(0.407435\pi\)
−0.973025 + 0.230700i \(0.925899\pi\)
\(62\) 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 0.722495 − 0.691376i \(-0.242995\pi\)
−0.237502 + 0.971387i \(0.576329\pi\)
\(68\) 0 0
\(69\) 5.09182 0.612984
\(70\) 0 0
\(71\) −2.95914 − 5.12538i −0.351185 − 0.608270i 0.635272 − 0.772288i \(-0.280888\pi\)
−0.986457 + 0.164018i \(0.947554\pi\)
\(72\) 0 0
\(73\) −4.86313 + 2.80773i −0.569187 + 0.328620i −0.756824 − 0.653618i \(-0.773250\pi\)
0.187638 + 0.982238i \(0.439917\pi\)
\(74\) 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 0.646614 − 0.762817i \(-0.276185\pi\)
−0.983926 + 0.178575i \(0.942851\pi\)
\(80\) 0 0
\(81\) 4.62959 + 8.01868i 0.514399 + 0.890965i
\(82\) 0 0
\(83\) − 6.20090i − 0.680638i −0.940310 − 0.340319i \(-0.889465\pi\)
0.940310 − 0.340319i \(-0.110535\pi\)
\(84\) 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 0.261789 + 0.965125i \(0.415688\pi\)
−0.966717 + 0.255847i \(0.917646\pi\)
\(90\) 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 0.0188932 − 0.999822i \(-0.493986\pi\)
0.875317 + 0.483549i \(0.160652\pi\)
\(98\) 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.349.9 yes 20
3.2 odd 2 3420.2.bj.c.2629.1 20
5.2 odd 4 1900.2.i.g.501.2 20
5.3 odd 4 1900.2.i.g.501.9 20
5.4 even 2 inner 380.2.r.a.349.2 yes 20
15.14 odd 2 3420.2.bj.c.2629.7 20
19.11 even 3 inner 380.2.r.a.49.2 ✓ 20
57.11 odd 6 3420.2.bj.c.1189.7 20
95.49 even 6 inner 380.2.r.a.49.9 yes 20
95.68 odd 12 1900.2.i.g.201.9 20
95.87 odd 12 1900.2.i.g.201.2 20
285.239 odd 6 3420.2.bj.c.1189.1 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.r.a.49.2 ✓ 20 19.11 even 3 inner
380.2.r.a.49.9 yes 20 95.49 even 6 inner
380.2.r.a.349.2 yes 20 5.4 even 2 inner
380.2.r.a.349.9 yes 20 1.1 even 1 trivial
1900.2.i.g.201.2 20 95.87 odd 12
1900.2.i.g.201.9 20 95.68 odd 12
1900.2.i.g.501.2 20 5.2 odd 4
1900.2.i.g.501.9 20 5.3 odd 4
3420.2.bj.c.1189.1 20 285.239 odd 6
3420.2.bj.c.1189.7 20 57.11 odd 6
3420.2.bj.c.2629.1 20 3.2 odd 2
3420.2.bj.c.2629.7 20 15.14 odd 2