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(599,1150)] 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("1150.599"); 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([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1150.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-4,0,2,0,0,-6,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.18279623245\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.1
Root \(-1.56155i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.2.b.h.599.4

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.56155i q^{3} -1.00000 q^{4} -1.56155 q^{6} -2.56155i q^{7} +1.00000i q^{8} +0.561553 q^{9} -1.00000 q^{11} +1.56155i q^{12} -0.561553i q^{13} -2.56155 q^{14} +1.00000 q^{16} -5.56155i q^{17} -0.561553i q^{18} -3.00000 q^{19} -4.00000 q^{21} +1.00000i q^{22} +1.00000i q^{23} +1.56155 q^{24} -0.561553 q^{26} -5.56155i q^{27} +2.56155i q^{28} -1.43845 q^{29} -5.12311 q^{31} -1.00000i q^{32} +1.56155i q^{33} -5.56155 q^{34} -0.561553 q^{36} -3.12311i q^{37} +3.00000i q^{38} -0.876894 q^{39} -1.87689 q^{41} +4.00000i q^{42} +7.68466i q^{43} +1.00000 q^{44} +1.00000 q^{46} -6.00000i q^{47} -1.56155i q^{48} +0.438447 q^{49} -8.68466 q^{51} +0.561553i q^{52} +9.12311i q^{53} -5.56155 q^{54} +2.56155 q^{56} +4.68466i q^{57} +1.43845i q^{58} -4.00000 q^{59} +2.24621 q^{61} +5.12311i q^{62} -1.43845i q^{63} -1.00000 q^{64} +1.56155 q^{66} +5.56155i q^{67} +5.56155i q^{68} +1.56155 q^{69} +1.12311 q^{71} +0.561553i q^{72} -6.12311i q^{73} -3.12311 q^{74} +3.00000 q^{76} +2.56155i q^{77} +0.876894i q^{78} -15.9309 q^{79} -7.00000 q^{81} +1.87689i q^{82} +6.12311i q^{83} +4.00000 q^{84} +7.68466 q^{86} +2.24621i q^{87} -1.00000i q^{88} +8.43845 q^{89} -1.43845 q^{91} -1.00000i q^{92} +8.00000i q^{93} -6.00000 q^{94} -1.56155 q^{96} -8.24621i q^{97} -0.438447i q^{98} -0.561553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 2 q^{6} - 6 q^{9} - 4 q^{11} - 2 q^{14} + 4 q^{16} - 12 q^{19} - 16 q^{21} - 2 q^{24} + 6 q^{26} - 14 q^{29} - 4 q^{31} - 14 q^{34} + 6 q^{36} - 20 q^{39} - 24 q^{41} + 4 q^{44} + 4 q^{46}+ \cdots + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) − 1.56155i − 0.901563i −0.892634 − 0.450781i \(-0.851145\pi\)
0.892634 − 0.450781i \(-0.148855\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.56155 −0.637501
\(7\) − 2.56155i − 0.968176i −0.875019 − 0.484088i \(-0.839151\pi\)
0.875019 − 0.484088i \(-0.160849\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0.561553 0.187184
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 − 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 1.56155i 0.450781i
\(13\) − 0.561553i − 0.155747i −0.996963 − 0.0778734i \(-0.975187\pi\)
0.996963 − 0.0778734i \(-0.0248130\pi\)
\(14\) −2.56155 −0.684604
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 5.56155i − 1.34887i −0.738332 − 0.674437i \(-0.764386\pi\)
0.738332 − 0.674437i \(-0.235614\pi\)
\(18\) − 0.561553i − 0.132359i
\(19\) −3.00000 −0.688247 −0.344124 − 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 1.00000i 0.213201i
\(23\) 1.00000i 0.208514i
\(24\) 1.56155 0.318751
\(25\) 0 0
\(26\) −0.561553 −0.110130
\(27\) − 5.56155i − 1.07032i
\(28\) 2.56155i 0.484088i
\(29\) −1.43845 −0.267113 −0.133556 − 0.991041i \(-0.542640\pi\)
−0.133556 + 0.991041i \(0.542640\pi\)
\(30\) 0 0
\(31\) −5.12311 −0.920137 −0.460068 − 0.887883i \(-0.652175\pi\)
−0.460068 + 0.887883i \(0.652175\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 1.56155i 0.271831i
\(34\) −5.56155 −0.953798
\(35\) 0 0
\(36\) −0.561553 −0.0935921
\(37\) − 3.12311i − 0.513435i −0.966486 − 0.256718i \(-0.917359\pi\)
0.966486 − 0.256718i \(-0.0826411\pi\)
\(38\) 3.00000i 0.486664i
\(39\) −0.876894 −0.140415
\(40\) 0 0
\(41\) −1.87689 −0.293122 −0.146561 − 0.989202i \(-0.546820\pi\)
−0.146561 + 0.989202i \(0.546820\pi\)
\(42\) 4.00000i 0.617213i
\(43\) 7.68466i 1.17190i 0.810347 + 0.585950i \(0.199278\pi\)
−0.810347 + 0.585950i \(0.800722\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) − 6.00000i − 0.875190i −0.899172 − 0.437595i \(-0.855830\pi\)
0.899172 − 0.437595i \(-0.144170\pi\)
\(48\) − 1.56155i − 0.225391i
\(49\) 0.438447 0.0626353
\(50\) 0 0
\(51\) −8.68466 −1.21610
\(52\) 0.561553i 0.0778734i
\(53\) 9.12311i 1.25315i 0.779359 + 0.626577i \(0.215545\pi\)
−0.779359 + 0.626577i \(0.784455\pi\)
\(54\) −5.56155 −0.756831
\(55\) 0 0
\(56\) 2.56155 0.342302
\(57\) 4.68466i 0.620498i
\(58\) 1.43845i 0.188877i
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 2.24621 0.287598 0.143799 − 0.989607i \(-0.454068\pi\)
0.143799 + 0.989607i \(0.454068\pi\)
\(62\) 5.12311i 0.650635i
\(63\) − 1.43845i − 0.181227i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 1.56155 0.192214
\(67\) 5.56155i 0.679452i 0.940524 + 0.339726i \(0.110334\pi\)
−0.940524 + 0.339726i \(0.889666\pi\)
\(68\) 5.56155i 0.674437i
\(69\) 1.56155 0.187989
\(70\) 0 0
\(71\) 1.12311 0.133288 0.0666441 − 0.997777i \(-0.478771\pi\)
0.0666441 + 0.997777i \(0.478771\pi\)
\(72\) 0.561553i 0.0661796i
\(73\) − 6.12311i − 0.716655i −0.933596 − 0.358328i \(-0.883347\pi\)
0.933596 − 0.358328i \(-0.116653\pi\)
\(74\) −3.12311 −0.363054
\(75\) 0 0
\(76\) 3.00000 0.344124
\(77\) 2.56155i 0.291916i
\(78\) 0.876894i 0.0992887i
\(79\) −15.9309 −1.79236 −0.896181 − 0.443688i \(-0.853670\pi\)
−0.896181 + 0.443688i \(0.853670\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 1.87689i 0.207268i
\(83\) 6.12311i 0.672098i 0.941844 + 0.336049i \(0.109091\pi\)
−0.941844 + 0.336049i \(0.890909\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) 7.68466 0.828658
\(87\) 2.24621i 0.240819i
\(88\) − 1.00000i − 0.106600i
\(89\) 8.43845 0.894474 0.447237 − 0.894416i \(-0.352408\pi\)
0.447237 + 0.894416i \(0.352408\pi\)
\(90\) 0 0
\(91\) −1.43845 −0.150790
\(92\) − 1.00000i − 0.104257i
\(93\) 8.00000i 0.829561i
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) −1.56155 −0.159375
\(97\) − 8.24621i − 0.837276i −0.908153 − 0.418638i \(-0.862508\pi\)
0.908153 − 0.418638i \(-0.137492\pi\)
\(98\) − 0.438447i − 0.0442899i
\(99\) −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.b.h.599.1 4
5.2 odd 4 1150.2.a.p.1.1 yes 2
5.3 odd 4 1150.2.a.k.1.2 ✓ 2
5.4 even 2 inner 1150.2.b.h.599.4 4
20.3 even 4 9200.2.a.bw.1.1 2
20.7 even 4 9200.2.a.bp.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1150.2.a.k.1.2 ✓ 2 5.3 odd 4
1150.2.a.p.1.1 yes 2 5.2 odd 4
1150.2.b.h.599.1 4 1.1 even 1 trivial
1150.2.b.h.599.4 4 5.4 even 2 inner
9200.2.a.bp.1.2 2 20.7 even 4
9200.2.a.bw.1.1 2 20.3 even 4