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 = [2,2,1,2,0,1,1,2,3] 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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.56155 q^{3} +1.00000 q^{4} -1.56155 q^{6} +2.56155 q^{7} +1.00000 q^{8} -0.561553 q^{9} -1.00000 q^{11} -1.56155 q^{12} -0.561553 q^{13} +2.56155 q^{14} +1.00000 q^{16} +5.56155 q^{17} -0.561553 q^{18} +3.00000 q^{19} -4.00000 q^{21} -1.00000 q^{22} +1.00000 q^{23} -1.56155 q^{24} -0.561553 q^{26} +5.56155 q^{27} +2.56155 q^{28} +1.43845 q^{29} -5.12311 q^{31} +1.00000 q^{32} +1.56155 q^{33} +5.56155 q^{34} -0.561553 q^{36} +3.12311 q^{37} +3.00000 q^{38} +0.876894 q^{39} -1.87689 q^{41} -4.00000 q^{42} +7.68466 q^{43} -1.00000 q^{44} +1.00000 q^{46} +6.00000 q^{47} -1.56155 q^{48} -0.438447 q^{49} -8.68466 q^{51} -0.561553 q^{52} +9.12311 q^{53} +5.56155 q^{54} +2.56155 q^{56} -4.68466 q^{57} +1.43845 q^{58} +4.00000 q^{59} +2.24621 q^{61} -5.12311 q^{62} -1.43845 q^{63} +1.00000 q^{64} +1.56155 q^{66} -5.56155 q^{67} +5.56155 q^{68} -1.56155 q^{69} +1.12311 q^{71} -0.561553 q^{72} -6.12311 q^{73} +3.12311 q^{74} +3.00000 q^{76} -2.56155 q^{77} +0.876894 q^{78} +15.9309 q^{79} -7.00000 q^{81} -1.87689 q^{82} +6.12311 q^{83} -4.00000 q^{84} +7.68466 q^{86} -2.24621 q^{87} -1.00000 q^{88} -8.43845 q^{89} -1.43845 q^{91} +1.00000 q^{92} +8.00000 q^{93} +6.00000 q^{94} -1.56155 q^{96} +8.24621 q^{97} -0.438447 q^{98} +0.561553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + q^{3} + 2 q^{4} + q^{6} + q^{7} + 2 q^{8} + 3 q^{9} - 2 q^{11} + q^{12} + 3 q^{13} + q^{14} + 2 q^{16} + 7 q^{17} + 3 q^{18} + 6 q^{19} - 8 q^{21} - 2 q^{22} + 2 q^{23} + q^{24} + 3 q^{26}+ \cdots - 3 q^{99}+O(q^{100}) \) 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
\(3\) −1.56155 −0.901563 −0.450781 − 0.892634i \(-0.648855\pi\)
−0.450781 + 0.892634i \(0.648855\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.56155 −0.637501
\(7\) 2.56155 0.968176 0.484088 − 0.875019i \(-0.339151\pi\)
0.484088 + 0.875019i \(0.339151\pi\)
\(8\) 1.00000 0.353553
\(9\) −0.561553 −0.187184
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 − 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) −1.56155 −0.450781
\(13\) −0.561553 −0.155747 −0.0778734 − 0.996963i \(-0.524813\pi\)
−0.0778734 + 0.996963i \(0.524813\pi\)
\(14\) 2.56155 0.684604
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.56155 1.34887 0.674437 − 0.738332i \(-0.264386\pi\)
0.674437 + 0.738332i \(0.264386\pi\)
\(18\) −0.561553 −0.132359
\(19\) 3.00000 0.688247 0.344124 − 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) −1.00000 −0.213201
\(23\) 1.00000 0.208514
\(24\) −1.56155 −0.318751
\(25\) 0 0
\(26\) −0.561553 −0.110130
\(27\) 5.56155 1.07032
\(28\) 2.56155 0.484088
\(29\) 1.43845 0.267113 0.133556 − 0.991041i \(-0.457360\pi\)
0.133556 + 0.991041i \(0.457360\pi\)
\(30\) 0 0
\(31\) −5.12311 −0.920137 −0.460068 − 0.887883i \(-0.652175\pi\)
−0.460068 + 0.887883i \(0.652175\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.56155 0.271831
\(34\) 5.56155 0.953798
\(35\) 0 0
\(36\) −0.561553 −0.0935921
\(37\) 3.12311 0.513435 0.256718 − 0.966486i \(-0.417359\pi\)
0.256718 + 0.966486i \(0.417359\pi\)
\(38\) 3.00000 0.486664
\(39\) 0.876894 0.140415
\(40\) 0 0
\(41\) −1.87689 −0.293122 −0.146561 − 0.989202i \(-0.546820\pi\)
−0.146561 + 0.989202i \(0.546820\pi\)
\(42\) −4.00000 −0.617213
\(43\) 7.68466 1.17190 0.585950 − 0.810347i \(-0.300722\pi\)
0.585950 + 0.810347i \(0.300722\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) −1.56155 −0.225391
\(49\) −0.438447 −0.0626353
\(50\) 0 0
\(51\) −8.68466 −1.21610
\(52\) −0.561553 −0.0778734
\(53\) 9.12311 1.25315 0.626577 − 0.779359i \(-0.284455\pi\)
0.626577 + 0.779359i \(0.284455\pi\)
\(54\) 5.56155 0.756831
\(55\) 0 0
\(56\) 2.56155 0.342302
\(57\) −4.68466 −0.620498
\(58\) 1.43845 0.188877
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 2.24621 0.287598 0.143799 − 0.989607i \(-0.454068\pi\)
0.143799 + 0.989607i \(0.454068\pi\)
\(62\) −5.12311 −0.650635
\(63\) −1.43845 −0.181227
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.56155 0.192214
\(67\) −5.56155 −0.679452 −0.339726 − 0.940524i \(-0.610334\pi\)
−0.339726 + 0.940524i \(0.610334\pi\)
\(68\) 5.56155 0.674437
\(69\) −1.56155 −0.187989
\(70\) 0 0
\(71\) 1.12311 0.133288 0.0666441 − 0.997777i \(-0.478771\pi\)
0.0666441 + 0.997777i \(0.478771\pi\)
\(72\) −0.561553 −0.0661796
\(73\) −6.12311 −0.716655 −0.358328 − 0.933596i \(-0.616653\pi\)
−0.358328 + 0.933596i \(0.616653\pi\)
\(74\) 3.12311 0.363054
\(75\) 0 0
\(76\) 3.00000 0.344124
\(77\) −2.56155 −0.291916
\(78\) 0.876894 0.0992887
\(79\) 15.9309 1.79236 0.896181 − 0.443688i \(-0.146330\pi\)
0.896181 + 0.443688i \(0.146330\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) −1.87689 −0.207268
\(83\) 6.12311 0.672098 0.336049 − 0.941844i \(-0.390909\pi\)
0.336049 + 0.941844i \(0.390909\pi\)
\(84\) −4.00000 −0.436436
\(85\) 0 0
\(86\) 7.68466 0.828658
\(87\) −2.24621 −0.240819
\(88\) −1.00000 −0.106600
\(89\) −8.43845 −0.894474 −0.447237 − 0.894416i \(-0.647592\pi\)
−0.447237 + 0.894416i \(0.647592\pi\)
\(90\) 0 0
\(91\) −1.43845 −0.150790
\(92\) 1.00000 0.104257
\(93\) 8.00000 0.829561
\(94\) 6.00000 0.618853
\(95\) 0 0
\(96\) −1.56155 −0.159375
\(97\) 8.24621 0.837276 0.418638 − 0.908153i \(-0.362508\pi\)
0.418638 + 0.908153i \(0.362508\pi\)
\(98\) −0.438447 −0.0442899
\(99\) 0.561553 0.0564382
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.p.1.1 yes 2
4.3 odd 2 9200.2.a.bp.1.2 2
5.2 odd 4 1150.2.b.h.599.4 4
5.3 odd 4 1150.2.b.h.599.1 4
5.4 even 2 1150.2.a.k.1.2 ✓ 2
20.19 odd 2 9200.2.a.bw.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1150.2.a.k.1.2 ✓ 2 5.4 even 2
1150.2.a.p.1.1 yes 2 1.1 even 1 trivial
1150.2.b.h.599.1 4 5.3 odd 4
1150.2.b.h.599.4 4 5.2 odd 4
9200.2.a.bp.1.2 2 4.3 odd 2
9200.2.a.bw.1.1 2 20.19 odd 2