Properties

Label 154.2.a.d.1.1
Level $154$
Weight $2$
Character 154.1
Self dual yes
Analytic conductor $1.230$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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] = [154,2,Mod(1,154)] 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("154.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(154, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 154 = 2 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 154.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2,-2] 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: \(1.22969619113\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 154.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -3.23607 q^{3} +1.00000 q^{4} +3.23607 q^{5} -3.23607 q^{6} +1.00000 q^{7} +1.00000 q^{8} +7.47214 q^{9} +3.23607 q^{10} +1.00000 q^{11} -3.23607 q^{12} +1.23607 q^{13} +1.00000 q^{14} -10.4721 q^{15} +1.00000 q^{16} -6.47214 q^{17} +7.47214 q^{18} -2.76393 q^{19} +3.23607 q^{20} -3.23607 q^{21} +1.00000 q^{22} +4.00000 q^{23} -3.23607 q^{24} +5.47214 q^{25} +1.23607 q^{26} -14.4721 q^{27} +1.00000 q^{28} -4.47214 q^{29} -10.4721 q^{30} +2.00000 q^{31} +1.00000 q^{32} -3.23607 q^{33} -6.47214 q^{34} +3.23607 q^{35} +7.47214 q^{36} -10.9443 q^{37} -2.76393 q^{38} -4.00000 q^{39} +3.23607 q^{40} +6.47214 q^{41} -3.23607 q^{42} -1.52786 q^{43} +1.00000 q^{44} +24.1803 q^{45} +4.00000 q^{46} -2.00000 q^{47} -3.23607 q^{48} +1.00000 q^{49} +5.47214 q^{50} +20.9443 q^{51} +1.23607 q^{52} -0.472136 q^{53} -14.4721 q^{54} +3.23607 q^{55} +1.00000 q^{56} +8.94427 q^{57} -4.47214 q^{58} +7.23607 q^{59} -10.4721 q^{60} -5.23607 q^{61} +2.00000 q^{62} +7.47214 q^{63} +1.00000 q^{64} +4.00000 q^{65} -3.23607 q^{66} -15.4164 q^{67} -6.47214 q^{68} -12.9443 q^{69} +3.23607 q^{70} -2.47214 q^{71} +7.47214 q^{72} -4.94427 q^{73} -10.9443 q^{74} -17.7082 q^{75} -2.76393 q^{76} +1.00000 q^{77} -4.00000 q^{78} +3.23607 q^{80} +24.4164 q^{81} +6.47214 q^{82} +10.1803 q^{83} -3.23607 q^{84} -20.9443 q^{85} -1.52786 q^{86} +14.4721 q^{87} +1.00000 q^{88} +10.0000 q^{89} +24.1803 q^{90} +1.23607 q^{91} +4.00000 q^{92} -6.47214 q^{93} -2.00000 q^{94} -8.94427 q^{95} -3.23607 q^{96} +3.52786 q^{97} +1.00000 q^{98} +7.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} + 6 q^{9} + 2 q^{10} + 2 q^{11} - 2 q^{12} - 2 q^{13} + 2 q^{14} - 12 q^{15} + 2 q^{16} - 4 q^{17} + 6 q^{18} - 10 q^{19} + 2 q^{20}+ \cdots + 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −3.23607 −1.86834 −0.934172 − 0.356822i \(-0.883860\pi\)
−0.934172 + 0.356822i \(0.883860\pi\)
\(4\) 1.00000 0.500000
\(5\) 3.23607 1.44721 0.723607 − 0.690212i \(-0.242483\pi\)
0.723607 + 0.690212i \(0.242483\pi\)
\(6\) −3.23607 −1.32112
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 7.47214 2.49071
\(10\) 3.23607 1.02333
\(11\) 1.00000 0.301511
\(12\) −3.23607 −0.934172
\(13\) 1.23607 0.342824 0.171412 − 0.985199i \(-0.445167\pi\)
0.171412 + 0.985199i \(0.445167\pi\)
\(14\) 1.00000 0.267261
\(15\) −10.4721 −2.70389
\(16\) 1.00000 0.250000
\(17\) −6.47214 −1.56972 −0.784862 − 0.619671i \(-0.787266\pi\)
−0.784862 + 0.619671i \(0.787266\pi\)
\(18\) 7.47214 1.76120
\(19\) −2.76393 −0.634089 −0.317045 − 0.948411i \(-0.602691\pi\)
−0.317045 + 0.948411i \(0.602691\pi\)
\(20\) 3.23607 0.723607
\(21\) −3.23607 −0.706168
\(22\) 1.00000 0.213201
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) −3.23607 −0.660560
\(25\) 5.47214 1.09443
\(26\) 1.23607 0.242413
\(27\) −14.4721 −2.78516
\(28\) 1.00000 0.188982
\(29\) −4.47214 −0.830455 −0.415227 − 0.909718i \(-0.636298\pi\)
−0.415227 + 0.909718i \(0.636298\pi\)
\(30\) −10.4721 −1.91194
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.23607 −0.563327
\(34\) −6.47214 −1.10996
\(35\) 3.23607 0.546995
\(36\) 7.47214 1.24536
\(37\) −10.9443 −1.79923 −0.899614 − 0.436687i \(-0.856152\pi\)
−0.899614 + 0.436687i \(0.856152\pi\)
\(38\) −2.76393 −0.448369
\(39\) −4.00000 −0.640513
\(40\) 3.23607 0.511667
\(41\) 6.47214 1.01078 0.505389 − 0.862892i \(-0.331349\pi\)
0.505389 + 0.862892i \(0.331349\pi\)
\(42\) −3.23607 −0.499336
\(43\) −1.52786 −0.232997 −0.116499 − 0.993191i \(-0.537167\pi\)
−0.116499 + 0.993191i \(0.537167\pi\)
\(44\) 1.00000 0.150756
\(45\) 24.1803 3.60459
\(46\) 4.00000 0.589768
\(47\) −2.00000 −0.291730 −0.145865 − 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) −3.23607 −0.467086
\(49\) 1.00000 0.142857
\(50\) 5.47214 0.773877
\(51\) 20.9443 2.93278
\(52\) 1.23607 0.171412
\(53\) −0.472136 −0.0648529 −0.0324264 − 0.999474i \(-0.510323\pi\)
−0.0324264 + 0.999474i \(0.510323\pi\)
\(54\) −14.4721 −1.96941
\(55\) 3.23607 0.436351
\(56\) 1.00000 0.133631
\(57\) 8.94427 1.18470
\(58\) −4.47214 −0.587220
\(59\) 7.23607 0.942056 0.471028 − 0.882118i \(-0.343883\pi\)
0.471028 + 0.882118i \(0.343883\pi\)
\(60\) −10.4721 −1.35195
\(61\) −5.23607 −0.670410 −0.335205 − 0.942145i \(-0.608806\pi\)
−0.335205 + 0.942145i \(0.608806\pi\)
\(62\) 2.00000 0.254000
\(63\) 7.47214 0.941401
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) −3.23607 −0.398332
\(67\) −15.4164 −1.88341 −0.941707 − 0.336434i \(-0.890779\pi\)
−0.941707 + 0.336434i \(0.890779\pi\)
\(68\) −6.47214 −0.784862
\(69\) −12.9443 −1.55831
\(70\) 3.23607 0.386784
\(71\) −2.47214 −0.293389 −0.146694 − 0.989182i \(-0.546863\pi\)
−0.146694 + 0.989182i \(0.546863\pi\)
\(72\) 7.47214 0.880600
\(73\) −4.94427 −0.578683 −0.289342 − 0.957226i \(-0.593436\pi\)
−0.289342 + 0.957226i \(0.593436\pi\)
\(74\) −10.9443 −1.27225
\(75\) −17.7082 −2.04477
\(76\) −2.76393 −0.317045
\(77\) 1.00000 0.113961
\(78\) −4.00000 −0.452911
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 3.23607 0.361803
\(81\) 24.4164 2.71293
\(82\) 6.47214 0.714728
\(83\) 10.1803 1.11744 0.558719 − 0.829357i \(-0.311293\pi\)
0.558719 + 0.829357i \(0.311293\pi\)
\(84\) −3.23607 −0.353084
\(85\) −20.9443 −2.27173
\(86\) −1.52786 −0.164754
\(87\) 14.4721 1.55158
\(88\) 1.00000 0.106600
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 24.1803 2.54883
\(91\) 1.23607 0.129575
\(92\) 4.00000 0.417029
\(93\) −6.47214 −0.671129
\(94\) −2.00000 −0.206284
\(95\) −8.94427 −0.917663
\(96\) −3.23607 −0.330280
\(97\) 3.52786 0.358200 0.179100 − 0.983831i \(-0.442681\pi\)
0.179100 + 0.983831i \(0.442681\pi\)
\(98\) 1.00000 0.101015
\(99\) 7.47214 0.750978
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 154.2.a.d.1.1 ✓ 2
3.2 odd 2 1386.2.a.m.1.1 2
4.3 odd 2 1232.2.a.p.1.2 2
5.2 odd 4 3850.2.c.q.1849.4 4
5.3 odd 4 3850.2.c.q.1849.1 4
5.4 even 2 3850.2.a.bj.1.2 2
7.2 even 3 1078.2.e.q.67.2 4
7.3 odd 6 1078.2.e.n.177.1 4
7.4 even 3 1078.2.e.q.177.2 4
7.5 odd 6 1078.2.e.n.67.1 4
7.6 odd 2 1078.2.a.w.1.2 2
8.3 odd 2 4928.2.a.bk.1.1 2
8.5 even 2 4928.2.a.bt.1.2 2
11.10 odd 2 1694.2.a.l.1.1 2
21.20 even 2 9702.2.a.cu.1.2 2
28.27 even 2 8624.2.a.bf.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.a.d.1.1 ✓ 2 1.1 even 1 trivial
1078.2.a.w.1.2 2 7.6 odd 2
1078.2.e.n.67.1 4 7.5 odd 6
1078.2.e.n.177.1 4 7.3 odd 6
1078.2.e.q.67.2 4 7.2 even 3
1078.2.e.q.177.2 4 7.4 even 3
1232.2.a.p.1.2 2 4.3 odd 2
1386.2.a.m.1.1 2 3.2 odd 2
1694.2.a.l.1.1 2 11.10 odd 2
3850.2.a.bj.1.2 2 5.4 even 2
3850.2.c.q.1849.1 4 5.3 odd 4
3850.2.c.q.1849.4 4 5.2 odd 4
4928.2.a.bk.1.1 2 8.3 odd 2
4928.2.a.bt.1.2 2 8.5 even 2
8624.2.a.bf.1.1 2 28.27 even 2
9702.2.a.cu.1.2 2 21.20 even 2