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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 = [3,-3,-1,3,0,1,-3,-3,10] 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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1101.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 9x + 12 \) 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 1.2
Root \(1.43163\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.43163 q^{3} +1.00000 q^{4} +1.43163 q^{6} -3.08719 q^{7} -1.00000 q^{8} -0.950444 q^{9} -6.46926 q^{11} -1.43163 q^{12} -3.95044 q^{13} +3.08719 q^{14} +1.00000 q^{16} +3.43163 q^{17} +0.950444 q^{18} +3.08719 q^{19} +4.41970 q^{21} +6.46926 q^{22} +1.00000 q^{23} +1.43163 q^{24} +3.95044 q^{26} +5.65556 q^{27} -3.08719 q^{28} +0.863254 q^{29} -5.95044 q^{31} -1.00000 q^{32} +9.26157 q^{33} -3.43163 q^{34} -0.950444 q^{36} +7.03763 q^{37} -3.08719 q^{38} +5.65556 q^{39} +5.60601 q^{41} -4.41970 q^{42} -8.00000 q^{43} -6.46926 q^{44} -1.00000 q^{46} -3.90089 q^{47} -1.43163 q^{48} +2.53074 q^{49} -4.91281 q^{51} -3.95044 q^{52} +6.00000 q^{53} -5.65556 q^{54} +3.08719 q^{56} -4.41970 q^{57} -0.863254 q^{58} +6.86325 q^{59} -13.5069 q^{61} +5.95044 q^{62} +2.93420 q^{63} +1.00000 q^{64} -9.26157 q^{66} +10.0753 q^{67} +3.43163 q^{68} -1.43163 q^{69} +2.56837 q^{71} +0.950444 q^{72} -5.90089 q^{73} -7.03763 q^{74} +3.08719 q^{76} +19.9718 q^{77} -5.65556 q^{78} +15.8018 q^{79} -5.24533 q^{81} -5.60601 q^{82} -9.03763 q^{83} +4.41970 q^{84} +8.00000 q^{86} -1.23586 q^{87} +6.46926 q^{88} +16.7641 q^{89} +12.1958 q^{91} +1.00000 q^{92} +8.51882 q^{93} +3.90089 q^{94} +1.43163 q^{96} +14.2949 q^{97} -2.53074 q^{98} +6.14867 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - q^{3} + 3 q^{4} + q^{6} - 3 q^{7} - 3 q^{8} + 10 q^{9} + 3 q^{11} - q^{12} + q^{13} + 3 q^{14} + 3 q^{16} + 7 q^{17} - 10 q^{18} + 3 q^{19} - 22 q^{21} - 3 q^{22} + 3 q^{23} + q^{24}+ \cdots + 57 q^{99}+O(q^{100}) \) 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
\(3\) −1.43163 −0.826550 −0.413275 − 0.910606i \(-0.635615\pi\)
−0.413275 + 0.910606i \(0.635615\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.43163 0.584459
\(7\) −3.08719 −1.16685 −0.583424 − 0.812168i \(-0.698287\pi\)
−0.583424 + 0.812168i \(0.698287\pi\)
\(8\) −1.00000 −0.353553
\(9\) −0.950444 −0.316815
\(10\) 0 0
\(11\) −6.46926 −1.95056 −0.975278 − 0.220983i \(-0.929074\pi\)
−0.975278 + 0.220983i \(0.929074\pi\)
\(12\) −1.43163 −0.413275
\(13\) −3.95044 −1.09566 −0.547828 − 0.836591i \(-0.684545\pi\)
−0.547828 + 0.836591i \(0.684545\pi\)
\(14\) 3.08719 0.825086
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.43163 0.832292 0.416146 − 0.909298i \(-0.363381\pi\)
0.416146 + 0.909298i \(0.363381\pi\)
\(18\) 0.950444 0.224022
\(19\) 3.08719 0.708250 0.354125 − 0.935198i \(-0.384779\pi\)
0.354125 + 0.935198i \(0.384779\pi\)
\(20\) 0 0
\(21\) 4.41970 0.964459
\(22\) 6.46926 1.37925
\(23\) 1.00000 0.208514
\(24\) 1.43163 0.292230
\(25\) 0 0
\(26\) 3.95044 0.774746
\(27\) 5.65556 1.08841
\(28\) −3.08719 −0.583424
\(29\) 0.863254 0.160302 0.0801511 − 0.996783i \(-0.474460\pi\)
0.0801511 + 0.996783i \(0.474460\pi\)
\(30\) 0 0
\(31\) −5.95044 −1.06873 −0.534366 − 0.845253i \(-0.679449\pi\)
−0.534366 + 0.845253i \(0.679449\pi\)
\(32\) −1.00000 −0.176777
\(33\) 9.26157 1.61223
\(34\) −3.43163 −0.588519
\(35\) 0 0
\(36\) −0.950444 −0.158407
\(37\) 7.03763 1.15698 0.578490 − 0.815690i \(-0.303642\pi\)
0.578490 + 0.815690i \(0.303642\pi\)
\(38\) −3.08719 −0.500808
\(39\) 5.65556 0.905615
\(40\) 0 0
\(41\) 5.60601 0.875511 0.437756 − 0.899094i \(-0.355774\pi\)
0.437756 + 0.899094i \(0.355774\pi\)
\(42\) −4.41970 −0.681975
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −6.46926 −0.975278
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) −3.90089 −0.569003 −0.284501 − 0.958676i \(-0.591828\pi\)
−0.284501 + 0.958676i \(0.591828\pi\)
\(48\) −1.43163 −0.206638
\(49\) 2.53074 0.361534
\(50\) 0 0
\(51\) −4.91281 −0.687931
\(52\) −3.95044 −0.547828
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −5.65556 −0.769625
\(55\) 0 0
\(56\) 3.08719 0.412543
\(57\) −4.41970 −0.585404
\(58\) −0.863254 −0.113351
\(59\) 6.86325 0.893520 0.446760 − 0.894654i \(-0.352578\pi\)
0.446760 + 0.894654i \(0.352578\pi\)
\(60\) 0 0
\(61\) −13.5069 −1.72938 −0.864690 − 0.502305i \(-0.832485\pi\)
−0.864690 + 0.502305i \(0.832485\pi\)
\(62\) 5.95044 0.755707
\(63\) 2.93420 0.369674
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −9.26157 −1.14002
\(67\) 10.0753 1.23089 0.615445 − 0.788180i \(-0.288976\pi\)
0.615445 + 0.788180i \(0.288976\pi\)
\(68\) 3.43163 0.416146
\(69\) −1.43163 −0.172348
\(70\) 0 0
\(71\) 2.56837 0.304810 0.152405 − 0.988318i \(-0.451298\pi\)
0.152405 + 0.988318i \(0.451298\pi\)
\(72\) 0.950444 0.112011
\(73\) −5.90089 −0.690647 −0.345323 − 0.938484i \(-0.612231\pi\)
−0.345323 + 0.938484i \(0.612231\pi\)
\(74\) −7.03763 −0.818108
\(75\) 0 0
\(76\) 3.08719 0.354125
\(77\) 19.9718 2.27600
\(78\) −5.65556 −0.640366
\(79\) 15.8018 1.77784 0.888919 − 0.458064i \(-0.151457\pi\)
0.888919 + 0.458064i \(0.151457\pi\)
\(80\) 0 0
\(81\) −5.24533 −0.582814
\(82\) −5.60601 −0.619080
\(83\) −9.03763 −0.992009 −0.496005 − 0.868320i \(-0.665200\pi\)
−0.496005 + 0.868320i \(0.665200\pi\)
\(84\) 4.41970 0.482229
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) −1.23586 −0.132498
\(88\) 6.46926 0.689625
\(89\) 16.7641 1.77700 0.888498 − 0.458881i \(-0.151750\pi\)
0.888498 + 0.458881i \(0.151750\pi\)
\(90\) 0 0
\(91\) 12.1958 1.27846
\(92\) 1.00000 0.104257
\(93\) 8.51882 0.883360
\(94\) 3.90089 0.402346
\(95\) 0 0
\(96\) 1.43163 0.146115
\(97\) 14.2949 1.45143 0.725713 − 0.687998i \(-0.241510\pi\)
0.725713 + 0.687998i \(0.241510\pi\)
\(98\) −2.53074 −0.255643
\(99\) 6.14867 0.617964
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.q.1.2 3
4.3 odd 2 9200.2.a.cf.1.2 3
5.2 odd 4 1150.2.b.j.599.2 6
5.3 odd 4 1150.2.b.j.599.5 6
5.4 even 2 230.2.a.d.1.2 ✓ 3
15.14 odd 2 2070.2.a.z.1.2 3
20.19 odd 2 1840.2.a.r.1.2 3
40.19 odd 2 7360.2.a.ce.1.2 3
40.29 even 2 7360.2.a.bz.1.2 3
115.114 odd 2 5290.2.a.r.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.d.1.2 ✓ 3 5.4 even 2
1150.2.a.q.1.2 3 1.1 even 1 trivial
1150.2.b.j.599.2 6 5.2 odd 4
1150.2.b.j.599.5 6 5.3 odd 4
1840.2.a.r.1.2 3 20.19 odd 2
2070.2.a.z.1.2 3 15.14 odd 2
5290.2.a.r.1.2 3 115.114 odd 2
7360.2.a.bz.1.2 3 40.29 even 2
7360.2.a.ce.1.2 3 40.19 odd 2
9200.2.a.cf.1.2 3 4.3 odd 2