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Properties

Label 1900.2.i.g.201.2
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 201.2
Root \(1.21562 + 2.10552i\) of defining polynomial
Character \(\chi\) \(=\) 1900.201
Dual form 1900.2.i.g.501.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.21562 + 2.10552i) q^{3} +0.663818 q^{7} +(-1.45548 - 2.52097i) q^{9} -1.80905 q^{11} +(1.15197 + 1.99526i) q^{13} +(-2.18033 + 3.77643i) q^{17} +(4.21168 - 1.12329i) q^{19} +(-0.806953 + 1.39768i) q^{21} +(1.04716 + 1.81374i) q^{23} -0.216466 q^{27} +(-0.974621 - 1.68809i) q^{29} -9.52527 q^{31} +(2.19913 - 3.80900i) q^{33} +2.97461 q^{37} -5.60143 q^{39} +(-0.247657 + 0.428954i) q^{41} +(-3.93588 + 6.81715i) q^{43} +(3.28772 + 5.69449i) q^{47} -6.55935 q^{49} +(-5.30091 - 9.18145i) q^{51} +(1.15225 + 1.99575i) q^{53} +(-2.75471 + 10.2333i) q^{57} +(-3.88559 + 6.73003i) q^{59} +(-5.36021 - 9.28415i) q^{61} +(-0.966176 - 1.67347i) q^{63} +(-2.29199 - 3.96984i) q^{67} -5.09182 q^{69} +(-2.95914 + 5.12538i) q^{71} +(2.80773 - 4.86313i) q^{73} -1.20088 q^{77} +(2.99810 - 5.19286i) q^{79} +(4.62959 - 8.01868i) q^{81} -6.20090 q^{83} +4.73909 q^{87} +(6.65028 + 11.5186i) q^{89} +(0.764696 + 1.32449i) q^{91} +(11.5791 - 20.0557i) q^{93} +(5.08470 - 8.80695i) q^{97} +(2.63305 + 4.56057i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 10 q^{9} - 14 q^{19} - 8 q^{21} + 16 q^{29} + 8 q^{31} + 8 q^{39} + 26 q^{41} + 44 q^{49} + 26 q^{51} - 4 q^{59} + 2 q^{61} - 48 q^{69} - 2 q^{71} + 16 q^{79} + 26 q^{81} + 40 q^{89} - 4 q^{91} + 20 q^{99}+O(q^{100}) \) 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
\(3\) −1.21562 + 2.10552i −0.701841 + 1.21562i 0.265979 + 0.963979i \(0.414305\pi\)
−0.967820 + 0.251645i \(0.919028\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.663818 0.250900 0.125450 − 0.992100i \(-0.459963\pi\)
0.125450 + 0.992100i \(0.459963\pi\)
\(8\) 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
\(13\) 1.15197 + 1.99526i 0.319498 + 0.553386i 0.980383 − 0.197100i \(-0.0631525\pi\)
−0.660886 + 0.750487i \(0.729819\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.18033 + 3.77643i −0.528807 + 0.915920i 0.470629 + 0.882331i \(0.344027\pi\)
−0.999436 + 0.0335887i \(0.989306\pi\)
\(18\) 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
\(23\) 1.04716 + 1.81374i 0.218349 + 0.378191i 0.954303 − 0.298840i \(-0.0965997\pi\)
−0.735955 + 0.677031i \(0.763266\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −0.216466 −0.0416588
\(28\) 0 0
\(29\) −0.974621 − 1.68809i −0.180983 − 0.313471i 0.761233 − 0.648479i \(-0.224594\pi\)
−0.942215 + 0.335008i \(0.891261\pi\)
\(30\) 0 0
\(31\) −9.52527 −1.71079 −0.855394 − 0.517977i \(-0.826685\pi\)
−0.855394 + 0.517977i \(0.826685\pi\)
\(32\) 0 0
\(33\) 2.19913 − 3.80900i 0.382819 − 0.663062i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.97461 0.489023 0.244511 − 0.969646i \(-0.421372\pi\)
0.244511 + 0.969646i \(0.421372\pi\)
\(38\) 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
\(43\) −3.93588 + 6.81715i −0.600216 + 1.03960i 0.392572 + 0.919721i \(0.371585\pi\)
−0.992788 + 0.119884i \(0.961748\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.28772 + 5.69449i 0.479563 + 0.830627i 0.999725 − 0.0234403i \(-0.00746198\pi\)
−0.520163 + 0.854067i \(0.674129\pi\)
\(48\) 0 0
\(49\) −6.55935 −0.937049
\(50\) 0 0
\(51\) −5.30091 − 9.18145i −0.742276 − 1.28566i
\(52\) 0 0
\(53\) 1.15225 + 1.99575i 0.158273 + 0.274137i 0.934246 − 0.356629i \(-0.116074\pi\)
−0.775973 + 0.630766i \(0.782741\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.75471 + 10.2333i −0.364871 + 1.35543i
\(58\) 0 0
\(59\) −3.88559 + 6.73003i −0.505860 + 0.876176i 0.494117 + 0.869396i \(0.335492\pi\)
−0.999977 + 0.00678007i \(0.997842\pi\)
\(60\) 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
\(63\) −0.966176 − 1.67347i −0.121727 − 0.210837i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −2.29199 − 3.96984i −0.280011 − 0.484993i 0.691376 − 0.722495i \(-0.257005\pi\)
−0.971387 + 0.237502i \(0.923671\pi\)
\(68\) 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
\(73\) 2.80773 − 4.86313i 0.328620 − 0.569187i −0.653618 − 0.756824i \(-0.726750\pi\)
0.982238 + 0.187638i \(0.0600831\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.20088 −0.136853
\(78\) 0 0
\(79\) 2.99810 − 5.19286i 0.337312 − 0.584242i −0.646614 − 0.762817i \(-0.723815\pi\)
0.983926 + 0.178575i \(0.0571488\pi\)
\(80\) 0 0
\(81\) 4.62959 − 8.01868i 0.514399 − 0.890965i
\(82\) 0 0
\(83\) −6.20090 −0.680638 −0.340319 − 0.940310i \(-0.610535\pi\)
−0.340319 + 0.940310i \(0.610535\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.73909 0.508084
\(88\) 0 0
\(89\) 6.65028 + 11.5186i 0.704928 + 1.22097i 0.966717 + 0.255847i \(0.0823542\pi\)
−0.261789 + 0.965125i \(0.584312\pi\)
\(90\) 0 0
\(91\) 0.764696 + 1.32449i 0.0801619 + 0.138844i
\(92\) 0 0
\(93\) 11.5791 − 20.0557i 1.20070 − 2.07968i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 5.08470 − 8.80695i 0.516273 − 0.894211i −0.483549 − 0.875317i \(-0.660652\pi\)
0.999822 − 0.0188932i \(-0.00601425\pi\)
\(98\) 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 1900.2.i.g.201.2 20
5.2 odd 4 380.2.r.a.49.9 yes 20
5.3 odd 4 380.2.r.a.49.2 ✓ 20
5.4 even 2 inner 1900.2.i.g.201.9 20
15.2 even 4 3420.2.bj.c.1189.1 20
15.8 even 4 3420.2.bj.c.1189.7 20
19.7 even 3 inner 1900.2.i.g.501.2 20
95.7 odd 12 380.2.r.a.349.2 yes 20
95.64 even 6 inner 1900.2.i.g.501.9 20
95.83 odd 12 380.2.r.a.349.9 yes 20
285.83 even 12 3420.2.bj.c.2629.1 20
285.197 even 12 3420.2.bj.c.2629.7 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.r.a.49.2 ✓ 20 5.3 odd 4
380.2.r.a.49.9 yes 20 5.2 odd 4
380.2.r.a.349.2 yes 20 95.7 odd 12
380.2.r.a.349.9 yes 20 95.83 odd 12
1900.2.i.g.201.2 20 1.1 even 1 trivial
1900.2.i.g.201.9 20 5.4 even 2 inner
1900.2.i.g.501.2 20 19.7 even 3 inner
1900.2.i.g.501.9 20 95.64 even 6 inner
3420.2.bj.c.1189.1 20 15.2 even 4
3420.2.bj.c.1189.7 20 15.8 even 4
3420.2.bj.c.2629.1 20 285.83 even 12
3420.2.bj.c.2629.7 20 285.197 even 12