Properties

Label 138.4.a.d.1.2
Level $138$
Weight $4$
Character 138.1
Self dual yes
Analytic conductor $8.142$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [138,4,Mod(1,138)] 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("138.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(138, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 138.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-4,6] 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: yes
Analytic conductor: \(8.14226358079\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{277}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 69 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-7.82166\) of defining polynomial
Character \(\chi\) \(=\) 138.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +17.6433 q^{5} -6.00000 q^{6} +14.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -35.2866 q^{10} -17.6433 q^{11} +12.0000 q^{12} -9.28663 q^{13} -28.0000 q^{14} +52.9300 q^{15} +16.0000 q^{16} -35.2866 q^{17} -18.0000 q^{18} +55.6433 q^{19} +70.5733 q^{20} +42.0000 q^{21} +35.2866 q^{22} +23.0000 q^{23} -24.0000 q^{24} +186.287 q^{25} +18.5733 q^{26} +27.0000 q^{27} +56.0000 q^{28} -74.4332 q^{29} -105.860 q^{30} +187.287 q^{31} -32.0000 q^{32} -52.9300 q^{33} +70.5733 q^{34} +247.006 q^{35} +36.0000 q^{36} +234.930 q^{37} -111.287 q^{38} -27.8599 q^{39} -141.147 q^{40} -393.440 q^{41} -84.0000 q^{42} +209.503 q^{43} -70.5733 q^{44} +158.790 q^{45} -46.0000 q^{46} -64.2802 q^{47} +48.0000 q^{48} -147.000 q^{49} -372.573 q^{50} -105.860 q^{51} -37.1465 q^{52} -379.783 q^{53} -54.0000 q^{54} -311.287 q^{55} -112.000 q^{56} +166.930 q^{57} +148.866 q^{58} -396.000 q^{59} +211.720 q^{60} +594.803 q^{61} -374.573 q^{62} +126.000 q^{63} +64.0000 q^{64} -163.847 q^{65} +105.860 q^{66} -1065.81 q^{67} -141.147 q^{68} +69.0000 q^{69} -494.013 q^{70} +911.159 q^{71} -72.0000 q^{72} -941.593 q^{73} -469.860 q^{74} +558.860 q^{75} +222.573 q^{76} -247.006 q^{77} +55.7198 q^{78} -520.293 q^{79} +282.293 q^{80} +81.0000 q^{81} +786.879 q^{82} -418.382 q^{83} +168.000 q^{84} -622.573 q^{85} -419.006 q^{86} -223.300 q^{87} +141.147 q^{88} +599.159 q^{89} -317.580 q^{90} -130.013 q^{91} +92.0000 q^{92} +561.860 q^{93} +128.560 q^{94} +981.733 q^{95} -96.0000 q^{96} +40.7262 q^{97} +294.000 q^{98} -158.790 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 6 q^{3} + 8 q^{4} + 2 q^{5} - 12 q^{6} + 28 q^{7} - 16 q^{8} + 18 q^{9} - 4 q^{10} - 2 q^{11} + 24 q^{12} + 48 q^{13} - 56 q^{14} + 6 q^{15} + 32 q^{16} - 4 q^{17} - 36 q^{18} + 78 q^{19}+ \cdots - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 3.00000 0.577350
\(4\) 4.00000 0.500000
\(5\) 17.6433 1.57807 0.789033 − 0.614351i \(-0.210582\pi\)
0.789033 + 0.614351i \(0.210582\pi\)
\(6\) −6.00000 −0.408248
\(7\) 14.0000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −8.00000 −0.353553
\(9\) 9.00000 0.333333
\(10\) −35.2866 −1.11586
\(11\) −17.6433 −0.483605 −0.241803 − 0.970325i \(-0.577739\pi\)
−0.241803 + 0.970325i \(0.577739\pi\)
\(12\) 12.0000 0.288675
\(13\) −9.28663 −0.198127 −0.0990634 − 0.995081i \(-0.531585\pi\)
−0.0990634 + 0.995081i \(0.531585\pi\)
\(14\) −28.0000 −0.534522
\(15\) 52.9300 0.911097
\(16\) 16.0000 0.250000
\(17\) −35.2866 −0.503427 −0.251714 − 0.967802i \(-0.580994\pi\)
−0.251714 + 0.967802i \(0.580994\pi\)
\(18\) −18.0000 −0.235702
\(19\) 55.6433 0.671866 0.335933 − 0.941886i \(-0.390948\pi\)
0.335933 + 0.941886i \(0.390948\pi\)
\(20\) 70.5733 0.789033
\(21\) 42.0000 0.436436
\(22\) 35.2866 0.341961
\(23\) 23.0000 0.208514
\(24\) −24.0000 −0.204124
\(25\) 186.287 1.49029
\(26\) 18.5733 0.140097
\(27\) 27.0000 0.192450
\(28\) 56.0000 0.377964
\(29\) −74.4332 −0.476617 −0.238308 − 0.971190i \(-0.576593\pi\)
−0.238308 + 0.971190i \(0.576593\pi\)
\(30\) −105.860 −0.644243
\(31\) 187.287 1.08509 0.542543 − 0.840028i \(-0.317462\pi\)
0.542543 + 0.840028i \(0.317462\pi\)
\(32\) −32.0000 −0.176777
\(33\) −52.9300 −0.279210
\(34\) 70.5733 0.355977
\(35\) 247.006 1.19291
\(36\) 36.0000 0.166667
\(37\) 234.930 1.04384 0.521922 − 0.852993i \(-0.325215\pi\)
0.521922 + 0.852993i \(0.325215\pi\)
\(38\) −111.287 −0.475081
\(39\) −27.8599 −0.114389
\(40\) −141.147 −0.557931
\(41\) −393.440 −1.49866 −0.749329 − 0.662198i \(-0.769624\pi\)
−0.749329 + 0.662198i \(0.769624\pi\)
\(42\) −84.0000 −0.308607
\(43\) 209.503 0.742999 0.371499 − 0.928433i \(-0.378844\pi\)
0.371499 + 0.928433i \(0.378844\pi\)
\(44\) −70.5733 −0.241803
\(45\) 158.790 0.526022
\(46\) −46.0000 −0.147442
\(47\) −64.2802 −0.199494 −0.0997471 − 0.995013i \(-0.531803\pi\)
−0.0997471 + 0.995013i \(0.531803\pi\)
\(48\) 48.0000 0.144338
\(49\) −147.000 −0.428571
\(50\) −372.573 −1.05380
\(51\) −105.860 −0.290654
\(52\) −37.1465 −0.0990634
\(53\) −379.783 −0.984288 −0.492144 − 0.870514i \(-0.663787\pi\)
−0.492144 + 0.870514i \(0.663787\pi\)
\(54\) −54.0000 −0.136083
\(55\) −311.287 −0.763161
\(56\) −112.000 −0.267261
\(57\) 166.930 0.387902
\(58\) 148.866 0.337019
\(59\) −396.000 −0.873810 −0.436905 − 0.899508i \(-0.643925\pi\)
−0.436905 + 0.899508i \(0.643925\pi\)
\(60\) 211.720 0.455548
\(61\) 594.803 1.24847 0.624235 − 0.781237i \(-0.285411\pi\)
0.624235 + 0.781237i \(0.285411\pi\)
\(62\) −374.573 −0.767272
\(63\) 126.000 0.251976
\(64\) 64.0000 0.125000
\(65\) −163.847 −0.312657
\(66\) 105.860 0.197431
\(67\) −1065.81 −1.94342 −0.971711 − 0.236172i \(-0.924107\pi\)
−0.971711 + 0.236172i \(0.924107\pi\)
\(68\) −141.147 −0.251714
\(69\) 69.0000 0.120386
\(70\) −494.013 −0.843512
\(71\) 911.159 1.52302 0.761512 − 0.648151i \(-0.224457\pi\)
0.761512 + 0.648151i \(0.224457\pi\)
\(72\) −72.0000 −0.117851
\(73\) −941.593 −1.50966 −0.754829 − 0.655921i \(-0.772280\pi\)
−0.754829 + 0.655921i \(0.772280\pi\)
\(74\) −469.860 −0.738110
\(75\) 558.860 0.860421
\(76\) 222.573 0.335933
\(77\) −247.006 −0.365571
\(78\) 55.7198 0.0808849
\(79\) −520.293 −0.740982 −0.370491 − 0.928836i \(-0.620810\pi\)
−0.370491 + 0.928836i \(0.620810\pi\)
\(80\) 282.293 0.394517
\(81\) 81.0000 0.111111
\(82\) 786.879 1.05971
\(83\) −418.382 −0.553294 −0.276647 − 0.960972i \(-0.589223\pi\)
−0.276647 + 0.960972i \(0.589223\pi\)
\(84\) 168.000 0.218218
\(85\) −622.573 −0.794442
\(86\) −419.006 −0.525379
\(87\) −223.300 −0.275175
\(88\) 141.147 0.170980
\(89\) 599.159 0.713604 0.356802 − 0.934180i \(-0.383867\pi\)
0.356802 + 0.934180i \(0.383867\pi\)
\(90\) −317.580 −0.371954
\(91\) −130.013 −0.149770
\(92\) 92.0000 0.104257
\(93\) 561.860 0.626475
\(94\) 128.560 0.141064
\(95\) 981.733 1.06025
\(96\) −96.0000 −0.102062
\(97\) 40.7262 0.0426301 0.0213151 − 0.999773i \(-0.493215\pi\)
0.0213151 + 0.999773i \(0.493215\pi\)
\(98\) 294.000 0.303046
\(99\) −158.790 −0.161202
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 138.4.a.d.1.2 ✓ 2
3.2 odd 2 414.4.a.k.1.1 2
4.3 odd 2 1104.4.a.k.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.4.a.d.1.2 ✓ 2 1.1 even 1 trivial
414.4.a.k.1.1 2 3.2 odd 2
1104.4.a.k.1.2 2 4.3 odd 2