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Properties

Label 1900.2.i.g.201.7
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 201.7
Root \(-0.628167 - 1.08802i\) of defining polynomial
Character \(\chi\) \(=\) 1900.201
Dual form 1900.2.i.g.501.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.628167 - 1.08802i) q^{3} +4.97100 q^{7} +(0.710812 + 1.23116i) q^{9} -3.85491 q^{11} +(-1.33470 - 2.31178i) q^{13} +(1.29521 - 2.24337i) q^{17} +(-1.24479 - 4.17738i) q^{19} +(3.12262 - 5.40854i) q^{21} +(-1.08682 - 1.88243i) q^{23} +5.55504 q^{27} +(-1.29432 - 2.24183i) q^{29} +7.76610 q^{31} +(-2.42153 + 4.19421i) q^{33} -2.75768 q^{37} -3.35367 q^{39} +(3.66243 - 6.34351i) q^{41} +(0.895083 - 1.55033i) q^{43} +(0.854141 + 1.47942i) q^{47} +17.7109 q^{49} +(-1.62722 - 2.81842i) q^{51} +(3.98220 + 6.89738i) q^{53} +(-5.32700 - 1.26974i) q^{57} +(0.127300 - 0.220490i) q^{59} +(-1.66702 - 2.88737i) q^{61} +(3.53345 + 6.12011i) q^{63} +(6.60237 + 11.4356i) q^{67} -2.73082 q^{69} +(3.85760 - 6.68156i) q^{71} +(-2.25489 + 3.90558i) q^{73} -19.1628 q^{77} +(-5.52715 + 9.57330i) q^{79} +(1.35706 - 2.35050i) q^{81} -3.04360 q^{83} -3.25221 q^{87} +(-4.76212 - 8.24824i) q^{89} +(-6.63482 - 11.4918i) q^{91} +(4.87841 - 8.44966i) q^{93} +(5.61676 - 9.72851i) q^{97} +(-2.74011 - 4.74601i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 10 q^{9} - 14 q^{19} - 8 q^{21} + 16 q^{29} + 8 q^{31} + 8 q^{39} + 26 q^{41} + 44 q^{49} + 26 q^{51} - 4 q^{59} + 2 q^{61} - 48 q^{69} - 2 q^{71} + 16 q^{79} + 26 q^{81} + 40 q^{89} - 4 q^{91} + 20 q^{99}+O(q^{100}) \) 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
\(3\) 0.628167 − 1.08802i 0.362673 − 0.628167i −0.625727 − 0.780042i \(-0.715198\pi\)
0.988400 + 0.151875i \(0.0485310\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.97100 1.87886 0.939432 − 0.342736i \(-0.111354\pi\)
0.939432 + 0.342736i \(0.111354\pi\)
\(8\) 0 0
\(9\) 0.710812 + 1.23116i 0.236937 + 0.410387i
\(10\) 0 0
\(11\) −3.85491 −1.16230 −0.581149 − 0.813797i \(-0.697397\pi\)
−0.581149 + 0.813797i \(0.697397\pi\)
\(12\) 0 0
\(13\) −1.33470 − 2.31178i −0.370180 − 0.641171i 0.619413 − 0.785066i \(-0.287371\pi\)
−0.989593 + 0.143894i \(0.954037\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.29521 − 2.24337i 0.314135 − 0.544097i −0.665118 − 0.746738i \(-0.731619\pi\)
0.979253 + 0.202641i \(0.0649523\pi\)
\(18\) 0 0
\(19\) −1.24479 − 4.17738i −0.285574 − 0.958357i
\(20\) 0 0
\(21\) 3.12262 − 5.40854i 0.681412 − 1.18024i
\(22\) 0 0
\(23\) −1.08682 − 1.88243i −0.226618 − 0.392514i 0.730186 − 0.683249i \(-0.239434\pi\)
−0.956804 + 0.290735i \(0.906100\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.55504 1.06907
\(28\) 0 0
\(29\) −1.29432 − 2.24183i −0.240350 − 0.416298i 0.720464 − 0.693492i \(-0.243929\pi\)
−0.960814 + 0.277194i \(0.910595\pi\)
\(30\) 0 0
\(31\) 7.76610 1.39483 0.697417 − 0.716666i \(-0.254333\pi\)
0.697417 + 0.716666i \(0.254333\pi\)
\(32\) 0 0
\(33\) −2.42153 + 4.19421i −0.421533 + 0.730117i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.75768 −0.453359 −0.226680 − 0.973969i \(-0.572787\pi\)
−0.226680 + 0.973969i \(0.572787\pi\)
\(38\) 0 0
\(39\) −3.35367 −0.537017
\(40\) 0 0
\(41\) 3.66243 − 6.34351i 0.571975 − 0.990690i −0.424388 − 0.905481i \(-0.639511\pi\)
0.996363 − 0.0852097i \(-0.0271560\pi\)
\(42\) 0 0
\(43\) 0.895083 − 1.55033i 0.136499 − 0.236423i −0.789670 − 0.613532i \(-0.789748\pi\)
0.926169 + 0.377109i \(0.123082\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.854141 + 1.47942i 0.124589 + 0.215795i 0.921572 − 0.388207i \(-0.126905\pi\)
−0.796983 + 0.604002i \(0.793572\pi\)
\(48\) 0 0
\(49\) 17.7109 2.53013
\(50\) 0 0
\(51\) −1.62722 − 2.81842i −0.227856 − 0.394658i
\(52\) 0 0
\(53\) 3.98220 + 6.89738i 0.546998 + 0.947428i 0.998478 + 0.0551469i \(0.0175627\pi\)
−0.451480 + 0.892281i \(0.649104\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −5.32700 − 1.26974i −0.705578 − 0.168182i
\(58\) 0 0
\(59\) 0.127300 − 0.220490i 0.0165730 − 0.0287053i −0.857620 − 0.514284i \(-0.828058\pi\)
0.874193 + 0.485579i \(0.161391\pi\)
\(60\) 0 0
\(61\) −1.66702 − 2.88737i −0.213441 − 0.369690i 0.739349 − 0.673323i \(-0.235134\pi\)
−0.952789 + 0.303633i \(0.901800\pi\)
\(62\) 0 0
\(63\) 3.53345 + 6.12011i 0.445173 + 0.771062i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.60237 + 11.4356i 0.806607 + 1.39708i 0.915201 + 0.402999i \(0.132032\pi\)
−0.108593 + 0.994086i \(0.534635\pi\)
\(68\) 0 0
\(69\) −2.73082 −0.328752
\(70\) 0 0
\(71\) 3.85760 − 6.68156i 0.457813 − 0.792955i −0.541032 − 0.841002i \(-0.681966\pi\)
0.998845 + 0.0480468i \(0.0152997\pi\)
\(72\) 0 0
\(73\) −2.25489 + 3.90558i −0.263915 + 0.457114i −0.967279 − 0.253716i \(-0.918347\pi\)
0.703364 + 0.710830i \(0.251680\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −19.1628 −2.18380
\(78\) 0 0
\(79\) −5.52715 + 9.57330i −0.621852 + 1.07708i 0.367288 + 0.930107i \(0.380286\pi\)
−0.989141 + 0.146973i \(0.953047\pi\)
\(80\) 0 0
\(81\) 1.35706 − 2.35050i 0.150784 − 0.261166i
\(82\) 0 0
\(83\) −3.04360 −0.334079 −0.167040 − 0.985950i \(-0.553421\pi\)
−0.167040 + 0.985950i \(0.553421\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −3.25221 −0.348673
\(88\) 0 0
\(89\) −4.76212 − 8.24824i −0.504784 − 0.874311i −0.999985 − 0.00553277i \(-0.998239\pi\)
0.495201 − 0.868779i \(-0.335094\pi\)
\(90\) 0 0
\(91\) −6.63482 − 11.4918i −0.695518 − 1.20467i
\(92\) 0 0
\(93\) 4.87841 − 8.44966i 0.505868 − 0.876189i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 5.61676 − 9.72851i 0.570295 − 0.987780i −0.426240 − 0.904610i \(-0.640162\pi\)
0.996535 − 0.0831703i \(-0.0265045\pi\)
\(98\) 0 0
\(99\) −2.74011 − 4.74601i −0.275392 − 0.476992i
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.201.7 20
5.2 odd 4 380.2.r.a.49.4 ✓ 20
5.3 odd 4 380.2.r.a.49.7 yes 20
5.4 even 2 inner 1900.2.i.g.201.4 20
15.2 even 4 3420.2.bj.c.1189.2 20
15.8 even 4 3420.2.bj.c.1189.9 20
19.7 even 3 inner 1900.2.i.g.501.7 20
95.7 odd 12 380.2.r.a.349.7 yes 20
95.64 even 6 inner 1900.2.i.g.501.4 20
95.83 odd 12 380.2.r.a.349.4 yes 20
285.83 even 12 3420.2.bj.c.2629.2 20
285.197 even 12 3420.2.bj.c.2629.9 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.r.a.49.4 ✓ 20 5.2 odd 4
380.2.r.a.49.7 yes 20 5.3 odd 4
380.2.r.a.349.4 yes 20 95.83 odd 12
380.2.r.a.349.7 yes 20 95.7 odd 12
1900.2.i.g.201.4 20 5.4 even 2 inner
1900.2.i.g.201.7 20 1.1 even 1 trivial
1900.2.i.g.501.4 20 95.64 even 6 inner
1900.2.i.g.501.7 20 19.7 even 3 inner
3420.2.bj.c.1189.2 20 15.2 even 4
3420.2.bj.c.1189.9 20 15.8 even 4
3420.2.bj.c.2629.2 20 285.83 even 12
3420.2.bj.c.2629.9 20 285.197 even 12