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Properties

Label 380.2.r.a.49.7
Level $380$
Weight $2$
Character 380.49
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 49.7
Root \(1.08802 - 0.628167i\) of defining polynomial
Character \(\chi\) \(=\) 380.49
Dual form 380.2.r.a.349.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.08802 + 0.628167i) q^{3} +(-1.99256 - 1.01474i) q^{5} -4.97100i q^{7} +(-0.710812 - 1.23116i) q^{9} -3.85491 q^{11} +(2.31178 - 1.33470i) q^{13} +(-1.53051 - 2.35572i) q^{15} +(-2.24337 - 1.29521i) q^{17} +(1.24479 + 4.17738i) q^{19} +(3.12262 - 5.40854i) q^{21} +(1.88243 - 1.08682i) q^{23} +(2.94060 + 4.04387i) q^{25} -5.55504i q^{27} +(1.29432 + 2.24183i) q^{29} +7.76610 q^{31} +(-4.19421 - 2.42153i) q^{33} +(-5.04429 + 9.90503i) q^{35} +2.75768i q^{37} +3.35367 q^{39} +(3.66243 - 6.34351i) q^{41} +(1.55033 + 0.895083i) q^{43} +(0.167024 + 3.17445i) q^{45} +(1.47942 - 0.854141i) q^{47} -17.7109 q^{49} +(-1.62722 - 2.81842i) q^{51} +(-6.89738 + 3.98220i) q^{53} +(7.68113 + 3.91173i) q^{55} +(-1.26974 + 5.32700i) q^{57} +(-0.127300 + 0.220490i) q^{59} +(-1.66702 - 2.88737i) q^{61} +(-6.12011 + 3.53345i) q^{63} +(-5.96073 + 0.313624i) q^{65} +(11.4356 - 6.60237i) q^{67} +2.73082 q^{69} +(3.85760 - 6.68156i) q^{71} +(-3.90558 - 2.25489i) q^{73} +(0.659195 + 6.24699i) q^{75} +19.1628i q^{77} +(5.52715 - 9.57330i) q^{79} +(1.35706 - 2.35050i) q^{81} -3.04360i q^{83} +(3.15575 + 4.85723i) q^{85} +3.25221i q^{87} +(4.76212 + 8.24824i) q^{89} +(-6.63482 - 11.4918i) q^{91} +(8.44966 + 4.87841i) q^{93} +(1.75865 - 9.58682i) q^{95} +(-9.72851 - 5.61676i) q^{97} +(2.74011 + 4.74601i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + q^{5} + 10 q^{9} - 5 q^{15} + 14 q^{19} - 8 q^{21} + 9 q^{25} - 16 q^{29} + 8 q^{31} - 2 q^{35} - 8 q^{39} + 26 q^{41} - 32 q^{45} - 44 q^{49} + 26 q^{51} - 12 q^{55} + 4 q^{59} + 2 q^{61} - 18 q^{65}+ \cdots - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(1\)

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
\(3\) 1.08802 + 0.628167i 0.628167 + 0.362673i 0.780042 − 0.625727i \(-0.215198\pi\)
−0.151875 + 0.988400i \(0.548531\pi\)
\(4\) 0 0
\(5\) −1.99256 − 1.01474i −0.891100 − 0.453806i
\(6\) 0 0
\(7\) − 4.97100i − 1.87886i −0.342736 − 0.939432i \(-0.611354\pi\)
0.342736 − 0.939432i \(-0.388646\pi\)
\(8\) 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
\(13\) 2.31178 − 1.33470i 0.641171 − 0.370180i −0.143894 − 0.989593i \(-0.545963\pi\)
0.785066 + 0.619413i \(0.212629\pi\)
\(14\) 0 0
\(15\) −1.53051 − 2.35572i −0.395177 − 0.608244i
\(16\) 0 0
\(17\) −2.24337 − 1.29521i −0.544097 − 0.314135i 0.202641 − 0.979253i \(-0.435048\pi\)
−0.746738 + 0.665118i \(0.768381\pi\)
\(18\) 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
\(23\) 1.88243 − 1.08682i 0.392514 − 0.226618i −0.290735 − 0.956804i \(-0.593900\pi\)
0.683249 + 0.730186i \(0.260566\pi\)
\(24\) 0 0
\(25\) 2.94060 + 4.04387i 0.588119 + 0.808774i
\(26\) 0 0
\(27\) − 5.55504i − 1.06907i
\(28\) 0 0
\(29\) 1.29432 + 2.24183i 0.240350 + 0.416298i 0.960814 − 0.277194i \(-0.0894045\pi\)
−0.720464 + 0.693492i \(0.756071\pi\)
\(30\) 0 0
\(31\) 7.76610 1.39483 0.697417 − 0.716666i \(-0.254333\pi\)
0.697417 + 0.716666i \(0.254333\pi\)
\(32\) 0 0
\(33\) −4.19421 − 2.42153i −0.730117 − 0.421533i
\(34\) 0 0
\(35\) −5.04429 + 9.90503i −0.852640 + 1.67426i
\(36\) 0 0
\(37\) 2.75768i 0.453359i 0.973969 + 0.226680i \(0.0727870\pi\)
−0.973969 + 0.226680i \(0.927213\pi\)
\(38\) 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
\(43\) 1.55033 + 0.895083i 0.236423 + 0.136499i 0.613532 − 0.789670i \(-0.289748\pi\)
−0.377109 + 0.926169i \(0.623082\pi\)
\(44\) 0 0
\(45\) 0.167024 + 3.17445i 0.0248984 + 0.473220i
\(46\) 0 0
\(47\) 1.47942 − 0.854141i 0.215795 − 0.124589i −0.388207 − 0.921572i \(-0.626905\pi\)
0.604002 + 0.796983i \(0.293572\pi\)
\(48\) 0 0
\(49\) −17.7109 −2.53013
\(50\) 0 0
\(51\) −1.62722 − 2.81842i −0.227856 − 0.394658i
\(52\) 0 0
\(53\) −6.89738 + 3.98220i −0.947428 + 0.546998i −0.892281 − 0.451480i \(-0.850896\pi\)
−0.0551469 + 0.998478i \(0.517563\pi\)
\(54\) 0 0
\(55\) 7.68113 + 3.91173i 1.03572 + 0.527458i
\(56\) 0 0
\(57\) −1.26974 + 5.32700i −0.168182 + 0.705578i
\(58\) 0 0
\(59\) −0.127300 + 0.220490i −0.0165730 + 0.0287053i −0.874193 − 0.485579i \(-0.838609\pi\)
0.857620 + 0.514284i \(0.171942\pi\)
\(60\) 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
\(63\) −6.12011 + 3.53345i −0.771062 + 0.445173i
\(64\) 0 0
\(65\) −5.96073 + 0.313624i −0.739338 + 0.0389002i
\(66\) 0 0
\(67\) 11.4356 − 6.60237i 1.39708 − 0.806607i 0.402999 − 0.915201i \(-0.367968\pi\)
0.994086 + 0.108593i \(0.0346346\pi\)
\(68\) 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
\(73\) −3.90558 − 2.25489i −0.457114 − 0.263915i 0.253716 − 0.967279i \(-0.418347\pi\)
−0.710830 + 0.703364i \(0.751680\pi\)
\(74\) 0 0
\(75\) 0.659195 + 6.24699i 0.0761173 + 0.721340i
\(76\) 0 0
\(77\) 19.1628i 2.18380i
\(78\) 0 0
\(79\) 5.52715 − 9.57330i 0.621852 − 1.07708i −0.367288 − 0.930107i \(-0.619714\pi\)
0.989141 − 0.146973i \(-0.0469530\pi\)
\(80\) 0 0
\(81\) 1.35706 − 2.35050i 0.150784 − 0.261166i
\(82\) 0 0
\(83\) − 3.04360i − 0.334079i −0.985950 − 0.167040i \(-0.946579\pi\)
0.985950 − 0.167040i \(-0.0534207\pi\)
\(84\) 0 0
\(85\) 3.15575 + 4.85723i 0.342289 + 0.526840i
\(86\) 0 0
\(87\) 3.25221i 0.348673i
\(88\) 0 0
\(89\) 4.76212 + 8.24824i 0.504784 + 0.874311i 0.999985 + 0.00553277i \(0.00176114\pi\)
−0.495201 + 0.868779i \(0.664906\pi\)
\(90\) 0 0
\(91\) −6.63482 − 11.4918i −0.695518 − 1.20467i
\(92\) 0 0
\(93\) 8.44966 + 4.87841i 0.876189 + 0.505868i
\(94\) 0 0
\(95\) 1.75865 − 9.58682i 0.180433 − 0.983587i
\(96\) 0 0
\(97\) −9.72851 − 5.61676i −0.987780 − 0.570295i −0.0831703 − 0.996535i \(-0.526505\pi\)
−0.904610 + 0.426240i \(0.859838\pi\)
\(98\) 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 380.2.r.a.49.7 yes 20
3.2 odd 2 3420.2.bj.c.1189.9 20
5.2 odd 4 1900.2.i.g.201.7 20
5.3 odd 4 1900.2.i.g.201.4 20
5.4 even 2 inner 380.2.r.a.49.4 ✓ 20
15.14 odd 2 3420.2.bj.c.1189.2 20
19.7 even 3 inner 380.2.r.a.349.4 yes 20
57.26 odd 6 3420.2.bj.c.2629.2 20
95.7 odd 12 1900.2.i.g.501.7 20
95.64 even 6 inner 380.2.r.a.349.7 yes 20
95.83 odd 12 1900.2.i.g.501.4 20
285.254 odd 6 3420.2.bj.c.2629.9 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.r.a.49.4 ✓ 20 5.4 even 2 inner
380.2.r.a.49.7 yes 20 1.1 even 1 trivial
380.2.r.a.349.4 yes 20 19.7 even 3 inner
380.2.r.a.349.7 yes 20 95.64 even 6 inner
1900.2.i.g.201.4 20 5.3 odd 4
1900.2.i.g.201.7 20 5.2 odd 4
1900.2.i.g.501.4 20 95.83 odd 12
1900.2.i.g.501.7 20 95.7 odd 12
3420.2.bj.c.1189.2 20 15.14 odd 2
3420.2.bj.c.1189.9 20 3.2 odd 2
3420.2.bj.c.2629.2 20 57.26 odd 6
3420.2.bj.c.2629.9 20 285.254 odd 6