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Properties

Label 380.2.r.a.349.8
Level $380$
Weight $2$
Character 380.349
Analytic conductor $3.034$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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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 349.8
Root \(1.74361 + 1.00667i\) of defining polynomial
Character \(\chi\) \(=\) 380.349
Dual form 380.2.r.a.49.8

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.74361 - 1.00667i) q^{3} +(0.0408382 - 2.23570i) q^{5} +1.34403i q^{7} +(0.526784 - 0.912416i) q^{9} +5.25594 q^{11} +(-2.10918 - 1.21773i) q^{13} +(-2.17941 - 3.93929i) q^{15} +(1.17765 - 0.679914i) q^{17} +(2.89815 - 3.25587i) q^{19} +(1.35300 + 2.34346i) q^{21} +(-7.05514 - 4.07329i) q^{23} +(-4.99666 - 0.182603i) q^{25} +3.91884i q^{27} +(-1.03597 + 1.79435i) q^{29} -0.513207 q^{31} +(9.16431 - 5.29102i) q^{33} +(3.00483 + 0.0548876i) q^{35} +5.57175i q^{37} -4.90344 q^{39} +(2.70353 + 4.68265i) q^{41} +(-11.0197 + 6.36221i) q^{43} +(-2.01837 - 1.21499i) q^{45} +(2.82785 + 1.63266i) q^{47} +5.19359 q^{49} +(1.36890 - 2.37101i) q^{51} +(10.1892 + 5.88276i) q^{53} +(0.214643 - 11.7507i) q^{55} +(1.77564 - 8.59447i) q^{57} +(0.0175979 + 0.0304805i) q^{59} +(0.518372 - 0.897846i) q^{61} +(1.22631 + 0.708011i) q^{63} +(-2.80861 + 4.66574i) q^{65} +(0.664028 + 0.383377i) q^{67} -16.4019 q^{69} +(5.68450 + 9.84583i) q^{71} +(-1.86429 + 1.07635i) q^{73} +(-8.89606 + 4.71162i) q^{75} +7.06413i q^{77} +(-6.48576 - 11.2337i) q^{79} +(5.52535 + 9.57019i) q^{81} +4.20304i q^{83} +(-1.47199 - 2.66062i) q^{85} +4.17153i q^{87} +(-3.65426 + 6.32937i) q^{89} +(1.63667 - 2.83479i) q^{91} +(-0.894833 + 0.516632i) q^{93} +(-7.16079 - 6.61235i) q^{95} +(0.721716 - 0.416683i) q^{97} +(2.76875 - 4.79561i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + q^{5} + 10 q^{9} - 5 q^{15} + 14 q^{19} - 8 q^{21} + 9 q^{25} - 16 q^{29} + 8 q^{31} - 2 q^{35} - 8 q^{39} + 26 q^{41} - 32 q^{45} - 44 q^{49} + 26 q^{51} - 12 q^{55} + 4 q^{59} + 2 q^{61} - 18 q^{65}+ \cdots - 20 q^{99}+O(q^{100}) \) 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
\(3\) 1.74361 − 1.00667i 1.00667 − 0.581203i 0.0964577 − 0.995337i \(-0.469249\pi\)
0.910216 + 0.414134i \(0.135915\pi\)
\(4\) 0 0
\(5\) 0.0408382 − 2.23570i 0.0182634 − 0.999833i
\(6\) 0 0
\(7\) 1.34403i 0.507994i 0.967205 + 0.253997i \(0.0817454\pi\)
−0.967205 + 0.253997i \(0.918255\pi\)
\(8\) 0 0
\(9\) 0.526784 − 0.912416i 0.175595 − 0.304139i
\(10\) 0 0
\(11\) 5.25594 1.58473 0.792363 − 0.610050i \(-0.208851\pi\)
0.792363 + 0.610050i \(0.208851\pi\)
\(12\) 0 0
\(13\) −2.10918 − 1.21773i −0.584980 − 0.337738i 0.178130 − 0.984007i \(-0.442995\pi\)
−0.763110 + 0.646269i \(0.776329\pi\)
\(14\) 0 0
\(15\) −2.17941 − 3.93929i −0.562721 − 1.01712i
\(16\) 0 0
\(17\) 1.17765 − 0.679914i 0.285621 − 0.164903i −0.350344 − 0.936621i \(-0.613935\pi\)
0.635965 + 0.771718i \(0.280602\pi\)
\(18\) 0 0
\(19\) 2.89815 − 3.25587i 0.664882 − 0.746949i
\(20\) 0 0
\(21\) 1.35300 + 2.34346i 0.295248 + 0.511384i
\(22\) 0 0
\(23\) −7.05514 − 4.07329i −1.47110 − 0.849339i −0.471625 − 0.881799i \(-0.656332\pi\)
−0.999473 + 0.0324603i \(0.989666\pi\)
\(24\) 0 0
\(25\) −4.99666 − 0.182603i −0.999333 − 0.0365207i
\(26\) 0 0
\(27\) 3.91884i 0.754182i
\(28\) 0 0
\(29\) −1.03597 + 1.79435i −0.192375 + 0.333203i −0.946037 − 0.324059i \(-0.894952\pi\)
0.753662 + 0.657262i \(0.228286\pi\)
\(30\) 0 0
\(31\) −0.513207 −0.0921747 −0.0460873 − 0.998937i \(-0.514675\pi\)
−0.0460873 + 0.998937i \(0.514675\pi\)
\(32\) 0 0
\(33\) 9.16431 − 5.29102i 1.59530 − 0.921048i
\(34\) 0 0
\(35\) 3.00483 + 0.0548876i 0.507910 + 0.00927770i
\(36\) 0 0
\(37\) 5.57175i 0.915991i 0.888955 + 0.457995i \(0.151432\pi\)
−0.888955 + 0.457995i \(0.848568\pi\)
\(38\) 0 0
\(39\) −4.90344 −0.785179
\(40\) 0 0
\(41\) 2.70353 + 4.68265i 0.422220 + 0.731307i 0.996156 − 0.0875933i \(-0.0279176\pi\)
−0.573936 + 0.818900i \(0.694584\pi\)
\(42\) 0 0
\(43\) −11.0197 + 6.36221i −1.68048 + 0.970228i −0.719146 + 0.694859i \(0.755467\pi\)
−0.961339 + 0.275369i \(0.911200\pi\)
\(44\) 0 0
\(45\) −2.01837 − 1.21499i −0.300881 − 0.181120i
\(46\) 0 0
\(47\) 2.82785 + 1.63266i 0.412485 + 0.238148i 0.691857 − 0.722035i \(-0.256793\pi\)
−0.279372 + 0.960183i \(0.590126\pi\)
\(48\) 0 0
\(49\) 5.19359 0.741942
\(50\) 0 0
\(51\) 1.36890 − 2.37101i 0.191685 − 0.332008i
\(52\) 0 0
\(53\) 10.1892 + 5.88276i 1.39960 + 0.808060i 0.994351 − 0.106146i \(-0.0338512\pi\)
0.405250 + 0.914206i \(0.367185\pi\)
\(54\) 0 0
\(55\) 0.214643 − 11.7507i 0.0289425 − 1.58446i
\(56\) 0 0
\(57\) 1.77564 − 8.59447i 0.235190 − 1.13837i
\(58\) 0 0
\(59\) 0.0175979 + 0.0304805i 0.00229105 + 0.00396822i 0.867169 − 0.498015i \(-0.165937\pi\)
−0.864878 + 0.501983i \(0.832604\pi\)
\(60\) 0 0
\(61\) 0.518372 − 0.897846i 0.0663707 − 0.114957i −0.830930 − 0.556376i \(-0.812191\pi\)
0.897301 + 0.441419i \(0.145525\pi\)
\(62\) 0 0
\(63\) 1.22631 + 0.708011i 0.154501 + 0.0892010i
\(64\) 0 0
\(65\) −2.80861 + 4.66574i −0.348366 + 0.578714i
\(66\) 0 0
\(67\) 0.664028 + 0.383377i 0.0811239 + 0.0468369i 0.540013 − 0.841657i \(-0.318419\pi\)
−0.458889 + 0.888493i \(0.651753\pi\)
\(68\) 0 0
\(69\) −16.4019 −1.97455
\(70\) 0 0
\(71\) 5.68450 + 9.84583i 0.674625 + 1.16849i 0.976578 + 0.215163i \(0.0690282\pi\)
−0.301953 + 0.953323i \(0.597638\pi\)
\(72\) 0 0
\(73\) −1.86429 + 1.07635i −0.218199 + 0.125977i −0.605116 − 0.796137i \(-0.706873\pi\)
0.386917 + 0.922115i \(0.373540\pi\)
\(74\) 0 0
\(75\) −8.89606 + 4.71162i −1.02723 + 0.544051i
\(76\) 0 0
\(77\) 7.06413i 0.805032i
\(78\) 0 0
\(79\) −6.48576 − 11.2337i −0.729705 − 1.26389i −0.957008 − 0.290062i \(-0.906324\pi\)
0.227302 − 0.973824i \(-0.427009\pi\)
\(80\) 0 0
\(81\) 5.52535 + 9.57019i 0.613928 + 1.06335i
\(82\) 0 0
\(83\) 4.20304i 0.461343i 0.973032 + 0.230672i \(0.0740923\pi\)
−0.973032 + 0.230672i \(0.925908\pi\)
\(84\) 0 0
\(85\) −1.47199 − 2.66062i −0.159660 − 0.288585i
\(86\) 0 0
\(87\) 4.17153i 0.447235i
\(88\) 0 0
\(89\) −3.65426 + 6.32937i −0.387351 + 0.670912i −0.992092 − 0.125510i \(-0.959943\pi\)
0.604741 + 0.796422i \(0.293277\pi\)
\(90\) 0 0
\(91\) 1.63667 − 2.83479i 0.171569 − 0.297166i
\(92\) 0 0
\(93\) −0.894833 + 0.516632i −0.0927898 + 0.0535722i
\(94\) 0 0
\(95\) −7.16079 − 6.61235i −0.734681 − 0.678413i
\(96\) 0 0
\(97\) 0.721716 − 0.416683i 0.0732791 − 0.0423077i −0.462913 − 0.886404i \(-0.653196\pi\)
0.536192 + 0.844096i \(0.319862\pi\)
\(98\) 0 0
\(99\) 2.76875 − 4.79561i 0.278269 − 0.481977i
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.349.8 yes 20
3.2 odd 2 3420.2.bj.c.2629.6 20
5.2 odd 4 1900.2.i.g.501.3 20
5.3 odd 4 1900.2.i.g.501.8 20
5.4 even 2 inner 380.2.r.a.349.3 yes 20
15.14 odd 2 3420.2.bj.c.2629.8 20
19.11 even 3 inner 380.2.r.a.49.3 ✓ 20
57.11 odd 6 3420.2.bj.c.1189.8 20
95.49 even 6 inner 380.2.r.a.49.8 yes 20
95.68 odd 12 1900.2.i.g.201.8 20
95.87 odd 12 1900.2.i.g.201.3 20
285.239 odd 6 3420.2.bj.c.1189.6 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.r.a.49.3 ✓ 20 19.11 even 3 inner
380.2.r.a.49.8 yes 20 95.49 even 6 inner
380.2.r.a.349.3 yes 20 5.4 even 2 inner
380.2.r.a.349.8 yes 20 1.1 even 1 trivial
1900.2.i.g.201.3 20 95.87 odd 12
1900.2.i.g.201.8 20 95.68 odd 12
1900.2.i.g.501.3 20 5.2 odd 4
1900.2.i.g.501.8 20 5.3 odd 4
3420.2.bj.c.1189.6 20 285.239 odd 6
3420.2.bj.c.1189.8 20 57.11 odd 6
3420.2.bj.c.2629.6 20 3.2 odd 2
3420.2.bj.c.2629.8 20 15.14 odd 2