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Properties

Label 1900.2.i.g.201.3
Level $1900$
Weight $2$
Character 1900.201
Analytic conductor $15.172$
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] = [1900,2,Mod(201,1900)] 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("1900.201"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1900, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1900.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.1715763840\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 201.3
Root \(1.00667 + 1.74361i\) of defining polynomial
Character \(\chi\) \(=\) 1900.201
Dual form 1900.2.i.g.501.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.00667 + 1.74361i) q^{3} -1.34403 q^{7} +(-0.526784 - 0.912416i) q^{9} +5.25594 q^{11} +(-1.21773 - 2.10918i) q^{13} +(0.679914 - 1.17765i) q^{17} +(-2.89815 - 3.25587i) q^{19} +(1.35300 - 2.34346i) q^{21} +(-4.07329 - 7.05514i) q^{23} -3.91884 q^{27} +(1.03597 + 1.79435i) q^{29} -0.513207 q^{31} +(-5.29102 + 9.16431i) q^{33} -5.57175 q^{37} +4.90344 q^{39} +(2.70353 - 4.68265i) q^{41} +(6.36221 - 11.0197i) q^{43} +(-1.63266 - 2.82785i) q^{47} -5.19359 q^{49} +(1.36890 + 2.37101i) q^{51} +(5.88276 + 10.1892i) q^{53} +(8.59447 - 1.77564i) q^{57} +(-0.0175979 + 0.0304805i) q^{59} +(0.518372 + 0.897846i) q^{61} +(0.708011 + 1.22631i) q^{63} +(-0.383377 - 0.664028i) q^{67} +16.4019 q^{69} +(5.68450 - 9.84583i) q^{71} +(1.07635 - 1.86429i) q^{73} -7.06413 q^{77} +(6.48576 - 11.2337i) q^{79} +(5.52535 - 9.57019i) q^{81} +4.20304 q^{83} -4.17153 q^{87} +(3.65426 + 6.32937i) q^{89} +(1.63667 + 2.83479i) q^{91} +(0.516632 - 0.894833i) q^{93} +(0.416683 - 0.721716i) q^{97} +(-2.76875 - 4.79561i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 10 q^{9} - 14 q^{19} - 8 q^{21} + 16 q^{29} + 8 q^{31} + 8 q^{39} + 26 q^{41} + 44 q^{49} + 26 q^{51} - 4 q^{59} + 2 q^{61} - 48 q^{69} - 2 q^{71} + 16 q^{79} + 26 q^{81} + 40 q^{89} - 4 q^{91} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00667 + 1.74361i −0.581203 + 1.00667i 0.414134 + 0.910216i \(0.364085\pi\)
−0.995337 + 0.0964577i \(0.969249\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.34403 −0.507994 −0.253997 − 0.967205i \(-0.581745\pi\)
−0.253997 + 0.967205i \(0.581745\pi\)
\(8\) 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
\(13\) −1.21773 − 2.10918i −0.337738 − 0.584980i 0.646269 − 0.763110i \(-0.276329\pi\)
−0.984007 + 0.178130i \(0.942995\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.679914 − 1.17765i 0.164903 − 0.285621i −0.771718 − 0.635965i \(-0.780602\pi\)
0.936621 + 0.350344i \(0.113935\pi\)
\(18\) 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
\(23\) −4.07329 − 7.05514i −0.849339 − 1.47110i −0.881799 − 0.471625i \(-0.843668\pi\)
0.0324603 − 0.999473i \(-0.489666\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −3.91884 −0.754182
\(28\) 0 0
\(29\) 1.03597 + 1.79435i 0.192375 + 0.333203i 0.946037 − 0.324059i \(-0.105048\pi\)
−0.753662 + 0.657262i \(0.771714\pi\)
\(30\) 0 0
\(31\) −0.513207 −0.0921747 −0.0460873 − 0.998937i \(-0.514675\pi\)
−0.0460873 + 0.998937i \(0.514675\pi\)
\(32\) 0 0
\(33\) −5.29102 + 9.16431i −0.921048 + 1.59530i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.57175 −0.915991 −0.457995 − 0.888955i \(-0.651432\pi\)
−0.457995 + 0.888955i \(0.651432\pi\)
\(38\) 0 0
\(39\) 4.90344 0.785179
\(40\) 0 0
\(41\) 2.70353 − 4.68265i 0.422220 − 0.731307i −0.573936 − 0.818900i \(-0.694584\pi\)
0.996156 + 0.0875933i \(0.0279176\pi\)
\(42\) 0 0
\(43\) 6.36221 − 11.0197i 0.970228 − 1.68048i 0.275369 − 0.961339i \(-0.411200\pi\)
0.694859 − 0.719146i \(-0.255467\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.63266 − 2.82785i −0.238148 − 0.412485i 0.722035 − 0.691857i \(-0.243207\pi\)
−0.960183 + 0.279372i \(0.909874\pi\)
\(48\) 0 0
\(49\) −5.19359 −0.741942
\(50\) 0 0
\(51\) 1.36890 + 2.37101i 0.191685 + 0.332008i
\(52\) 0 0
\(53\) 5.88276 + 10.1892i 0.808060 + 1.39960i 0.914206 + 0.405250i \(0.132815\pi\)
−0.106146 + 0.994351i \(0.533851\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 8.59447 − 1.77564i 1.13837 − 0.235190i
\(58\) 0 0
\(59\) −0.0175979 + 0.0304805i −0.00229105 + 0.00396822i −0.867169 − 0.498015i \(-0.834063\pi\)
0.864878 + 0.501983i \(0.167396\pi\)
\(60\) 0 0
\(61\) 0.518372 + 0.897846i 0.0663707 + 0.114957i 0.897301 − 0.441419i \(-0.145525\pi\)
−0.830930 + 0.556376i \(0.812191\pi\)
\(62\) 0 0
\(63\) 0.708011 + 1.22631i 0.0892010 + 0.154501i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −0.383377 − 0.664028i −0.0468369 − 0.0811239i 0.841657 − 0.540013i \(-0.181581\pi\)
−0.888493 + 0.458889i \(0.848247\pi\)
\(68\) 0 0
\(69\) 16.4019 1.97455
\(70\) 0 0
\(71\) 5.68450 − 9.84583i 0.674625 − 1.16849i −0.301953 − 0.953323i \(-0.597638\pi\)
0.976578 − 0.215163i \(-0.0690282\pi\)
\(72\) 0 0
\(73\) 1.07635 − 1.86429i 0.125977 − 0.218199i −0.796137 − 0.605116i \(-0.793127\pi\)
0.922115 + 0.386917i \(0.126460\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −7.06413 −0.805032
\(78\) 0 0
\(79\) 6.48576 − 11.2337i 0.729705 − 1.26389i −0.227302 − 0.973824i \(-0.572991\pi\)
0.957008 − 0.290062i \(-0.0936761\pi\)
\(80\) 0 0
\(81\) 5.52535 − 9.57019i 0.613928 − 1.06335i
\(82\) 0 0
\(83\) 4.20304 0.461343 0.230672 − 0.973032i \(-0.425908\pi\)
0.230672 + 0.973032i \(0.425908\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −4.17153 −0.447235
\(88\) 0 0
\(89\) 3.65426 + 6.32937i 0.387351 + 0.670912i 0.992092 − 0.125510i \(-0.0400568\pi\)
−0.604741 + 0.796422i \(0.706723\pi\)
\(90\) 0 0
\(91\) 1.63667 + 2.83479i 0.171569 + 0.297166i
\(92\) 0 0
\(93\) 0.516632 − 0.894833i 0.0535722 − 0.0927898i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.416683 − 0.721716i 0.0423077 − 0.0732791i −0.844096 − 0.536192i \(-0.819862\pi\)
0.886404 + 0.462913i \(0.153196\pi\)
\(98\) 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 1900.2.i.g.201.3 20
5.2 odd 4 380.2.r.a.49.8 yes 20
5.3 odd 4 380.2.r.a.49.3 ✓ 20
5.4 even 2 inner 1900.2.i.g.201.8 20
15.2 even 4 3420.2.bj.c.1189.6 20
15.8 even 4 3420.2.bj.c.1189.8 20
19.7 even 3 inner 1900.2.i.g.501.3 20
95.7 odd 12 380.2.r.a.349.3 yes 20
95.64 even 6 inner 1900.2.i.g.501.8 20
95.83 odd 12 380.2.r.a.349.8 yes 20
285.83 even 12 3420.2.bj.c.2629.6 20
285.197 even 12 3420.2.bj.c.2629.8 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.r.a.49.3 ✓ 20 5.3 odd 4
380.2.r.a.49.8 yes 20 5.2 odd 4
380.2.r.a.349.3 yes 20 95.7 odd 12
380.2.r.a.349.8 yes 20 95.83 odd 12
1900.2.i.g.201.3 20 1.1 even 1 trivial
1900.2.i.g.201.8 20 5.4 even 2 inner
1900.2.i.g.501.3 20 19.7 even 3 inner
1900.2.i.g.501.8 20 95.64 even 6 inner
3420.2.bj.c.1189.6 20 15.2 even 4
3420.2.bj.c.1189.8 20 15.8 even 4
3420.2.bj.c.2629.6 20 285.83 even 12
3420.2.bj.c.2629.8 20 285.197 even 12