Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1150,2,Mod(1,1150)] 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("1150.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1150.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,1,2,0,-1,3,-2,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(9.18279623245\)
Analytic rank: \(1\)
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: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 1150.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.618034 q^{3} +1.00000 q^{4} +0.618034 q^{6} +4.85410 q^{7} -1.00000 q^{8} -2.61803 q^{9} -3.38197 q^{11} -0.618034 q^{12} +0.381966 q^{13} -4.85410 q^{14} +1.00000 q^{16} -5.85410 q^{17} +2.61803 q^{18} -6.85410 q^{19} -3.00000 q^{21} +3.38197 q^{22} +1.00000 q^{23} +0.618034 q^{24} -0.381966 q^{26} +3.47214 q^{27} +4.85410 q^{28} +3.70820 q^{29} -8.85410 q^{31} -1.00000 q^{32} +2.09017 q^{33} +5.85410 q^{34} -2.61803 q^{36} +3.70820 q^{37} +6.85410 q^{38} -0.236068 q^{39} -3.38197 q^{41} +3.00000 q^{42} +6.76393 q^{43} -3.38197 q^{44} -1.00000 q^{46} -11.7082 q^{47} -0.618034 q^{48} +16.5623 q^{49} +3.61803 q^{51} +0.381966 q^{52} -2.00000 q^{53} -3.47214 q^{54} -4.85410 q^{56} +4.23607 q^{57} -3.70820 q^{58} -6.00000 q^{59} -3.85410 q^{61} +8.85410 q^{62} -12.7082 q^{63} +1.00000 q^{64} -2.09017 q^{66} +0.763932 q^{67} -5.85410 q^{68} -0.618034 q^{69} +2.61803 q^{71} +2.61803 q^{72} -7.52786 q^{73} -3.70820 q^{74} -6.85410 q^{76} -16.4164 q^{77} +0.236068 q^{78} +5.70820 q^{79} +5.70820 q^{81} +3.38197 q^{82} -5.70820 q^{83} -3.00000 q^{84} -6.76393 q^{86} -2.29180 q^{87} +3.38197 q^{88} -9.70820 q^{89} +1.85410 q^{91} +1.00000 q^{92} +5.47214 q^{93} +11.7082 q^{94} +0.618034 q^{96} -16.0344 q^{97} -16.5623 q^{98} +8.85410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + q^{3} + 2 q^{4} - q^{6} + 3 q^{7} - 2 q^{8} - 3 q^{9} - 9 q^{11} + q^{12} + 3 q^{13} - 3 q^{14} + 2 q^{16} - 5 q^{17} + 3 q^{18} - 7 q^{19} - 6 q^{21} + 9 q^{22} + 2 q^{23} - q^{24}+ \cdots + 11 q^{99}+O(q^{100}) \) 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
\(3\) −0.618034 −0.356822 −0.178411 − 0.983956i \(-0.557096\pi\)
−0.178411 + 0.983956i \(0.557096\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0.618034 0.252311
\(7\) 4.85410 1.83468 0.917339 − 0.398107i \(-0.130333\pi\)
0.917339 + 0.398107i \(0.130333\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.61803 −0.872678
\(10\) 0 0
\(11\) −3.38197 −1.01970 −0.509851 − 0.860263i \(-0.670299\pi\)
−0.509851 + 0.860263i \(0.670299\pi\)
\(12\) −0.618034 −0.178411
\(13\) 0.381966 0.105938 0.0529692 − 0.998596i \(-0.483131\pi\)
0.0529692 + 0.998596i \(0.483131\pi\)
\(14\) −4.85410 −1.29731
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −5.85410 −1.41983 −0.709914 − 0.704288i \(-0.751266\pi\)
−0.709914 + 0.704288i \(0.751266\pi\)
\(18\) 2.61803 0.617077
\(19\) −6.85410 −1.57244 −0.786219 − 0.617947i \(-0.787964\pi\)
−0.786219 + 0.617947i \(0.787964\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 3.38197 0.721038
\(23\) 1.00000 0.208514
\(24\) 0.618034 0.126156
\(25\) 0 0
\(26\) −0.381966 −0.0749097
\(27\) 3.47214 0.668213
\(28\) 4.85410 0.917339
\(29\) 3.70820 0.688596 0.344298 − 0.938860i \(-0.388117\pi\)
0.344298 + 0.938860i \(0.388117\pi\)
\(30\) 0 0
\(31\) −8.85410 −1.59024 −0.795122 − 0.606450i \(-0.792593\pi\)
−0.795122 + 0.606450i \(0.792593\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.09017 0.363852
\(34\) 5.85410 1.00397
\(35\) 0 0
\(36\) −2.61803 −0.436339
\(37\) 3.70820 0.609625 0.304812 − 0.952412i \(-0.401406\pi\)
0.304812 + 0.952412i \(0.401406\pi\)
\(38\) 6.85410 1.11188
\(39\) −0.236068 −0.0378011
\(40\) 0 0
\(41\) −3.38197 −0.528174 −0.264087 − 0.964499i \(-0.585071\pi\)
−0.264087 + 0.964499i \(0.585071\pi\)
\(42\) 3.00000 0.462910
\(43\) 6.76393 1.03149 0.515745 − 0.856742i \(-0.327515\pi\)
0.515745 + 0.856742i \(0.327515\pi\)
\(44\) −3.38197 −0.509851
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) −11.7082 −1.70782 −0.853909 − 0.520423i \(-0.825774\pi\)
−0.853909 + 0.520423i \(0.825774\pi\)
\(48\) −0.618034 −0.0892055
\(49\) 16.5623 2.36604
\(50\) 0 0
\(51\) 3.61803 0.506626
\(52\) 0.381966 0.0529692
\(53\) −2.00000 −0.274721 −0.137361 − 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) −3.47214 −0.472498
\(55\) 0 0
\(56\) −4.85410 −0.648657
\(57\) 4.23607 0.561081
\(58\) −3.70820 −0.486911
\(59\) −6.00000 −0.781133 −0.390567 − 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −3.85410 −0.493467 −0.246734 − 0.969083i \(-0.579357\pi\)
−0.246734 + 0.969083i \(0.579357\pi\)
\(62\) 8.85410 1.12447
\(63\) −12.7082 −1.60108
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.09017 −0.257282
\(67\) 0.763932 0.0933292 0.0466646 − 0.998911i \(-0.485141\pi\)
0.0466646 + 0.998911i \(0.485141\pi\)
\(68\) −5.85410 −0.709914
\(69\) −0.618034 −0.0744025
\(70\) 0 0
\(71\) 2.61803 0.310703 0.155352 − 0.987859i \(-0.450349\pi\)
0.155352 + 0.987859i \(0.450349\pi\)
\(72\) 2.61803 0.308538
\(73\) −7.52786 −0.881070 −0.440535 − 0.897735i \(-0.645211\pi\)
−0.440535 + 0.897735i \(0.645211\pi\)
\(74\) −3.70820 −0.431070
\(75\) 0 0
\(76\) −6.85410 −0.786219
\(77\) −16.4164 −1.87082
\(78\) 0.236068 0.0267294
\(79\) 5.70820 0.642223 0.321112 − 0.947041i \(-0.395944\pi\)
0.321112 + 0.947041i \(0.395944\pi\)
\(80\) 0 0
\(81\) 5.70820 0.634245
\(82\) 3.38197 0.373476
\(83\) −5.70820 −0.626557 −0.313278 − 0.949661i \(-0.601427\pi\)
−0.313278 + 0.949661i \(0.601427\pi\)
\(84\) −3.00000 −0.327327
\(85\) 0 0
\(86\) −6.76393 −0.729374
\(87\) −2.29180 −0.245706
\(88\) 3.38197 0.360519
\(89\) −9.70820 −1.02907 −0.514534 − 0.857470i \(-0.672035\pi\)
−0.514534 + 0.857470i \(0.672035\pi\)
\(90\) 0 0
\(91\) 1.85410 0.194363
\(92\) 1.00000 0.104257
\(93\) 5.47214 0.567434
\(94\) 11.7082 1.20761
\(95\) 0 0
\(96\) 0.618034 0.0630778
\(97\) −16.0344 −1.62805 −0.814025 − 0.580829i \(-0.802728\pi\)
−0.814025 + 0.580829i \(0.802728\pi\)
\(98\) −16.5623 −1.67305
\(99\) 8.85410 0.889871
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 1150.2.a.l.1.1 2
4.3 odd 2 9200.2.a.bo.1.2 2
5.2 odd 4 230.2.b.a.139.2 ✓ 4
5.3 odd 4 230.2.b.a.139.3 yes 4
5.4 even 2 1150.2.a.n.1.2 2
15.2 even 4 2070.2.d.c.829.3 4
15.8 even 4 2070.2.d.c.829.1 4
20.3 even 4 1840.2.e.c.369.3 4
20.7 even 4 1840.2.e.c.369.2 4
20.19 odd 2 9200.2.a.by.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.b.a.139.2 ✓ 4 5.2 odd 4
230.2.b.a.139.3 yes 4 5.3 odd 4
1150.2.a.l.1.1 2 1.1 even 1 trivial
1150.2.a.n.1.2 2 5.4 even 2
1840.2.e.c.369.2 4 20.7 even 4
1840.2.e.c.369.3 4 20.3 even 4
2070.2.d.c.829.1 4 15.8 even 4
2070.2.d.c.829.3 4 15.2 even 4
9200.2.a.bo.1.2 2 4.3 odd 2
9200.2.a.by.1.1 2 20.19 odd 2