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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1150,4,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, 4); 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, 4, names="a")
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
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,-16,0,12,0,0,26,0,-24] 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: \(67.8521965066\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{73})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 37x^{2} + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 46)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.1
Root \(-3.77200i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.4.b.j.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)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} -2.77200i q^{3} -4.00000 q^{4} -5.54400 q^{6} -23.0880i q^{7} +8.00000i q^{8} +19.3160 q^{9} -6.00000 q^{11} +11.0880i q^{12} -72.4920i q^{13} -46.1760 q^{14} +16.0000 q^{16} +58.6320i q^{17} -38.6320i q^{18} -77.4400 q^{19} -64.0000 q^{21} +12.0000i q^{22} +23.0000i q^{23} +22.1760 q^{24} -144.984 q^{26} -128.388i q^{27} +92.3520i q^{28} +169.476 q^{29} -303.268 q^{31} -32.0000i q^{32} +16.6320i q^{33} +117.264 q^{34} -77.2640 q^{36} -265.336i q^{37} +154.880i q^{38} -200.948 q^{39} +310.388 q^{41} +128.000i q^{42} -193.304i q^{43} +24.0000 q^{44} +46.0000 q^{46} -322.700i q^{47} -44.3520i q^{48} -190.056 q^{49} +162.528 q^{51} +289.968i q^{52} -137.336i q^{53} -256.776 q^{54} +184.704 q^{56} +214.664i q^{57} -338.952i q^{58} -562.320 q^{59} -283.224 q^{61} +606.536i q^{62} -445.968i q^{63} -64.0000 q^{64} +33.2640 q^{66} +182.768i q^{67} -234.528i q^{68} +63.7560 q^{69} +109.788 q^{71} +154.528i q^{72} -625.684i q^{73} -530.672 q^{74} +309.760 q^{76} +138.528i q^{77} +401.896i q^{78} -172.072 q^{79} +165.640 q^{81} -620.776i q^{82} +994.984i q^{83} +256.000 q^{84} -386.608 q^{86} -469.788i q^{87} -48.0000i q^{88} -1099.90 q^{89} -1673.70 q^{91} -92.0000i q^{92} +840.660i q^{93} -645.400 q^{94} -88.7041 q^{96} +1625.53i q^{97} +380.112i q^{98} -115.896 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{4} + 12 q^{6} + 26 q^{9} - 24 q^{11} - 48 q^{14} + 64 q^{16} + 32 q^{19} - 256 q^{21} - 48 q^{24} - 204 q^{26} + 114 q^{29} - 34 q^{31} + 264 q^{34} - 104 q^{36} - 650 q^{39} + 746 q^{41} + 96 q^{44}+ \cdots - 156 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.00000i − 0.707107i
\(3\) − 2.77200i − 0.533472i −0.963770 − 0.266736i \(-0.914055\pi\)
0.963770 − 0.266736i \(-0.0859452\pi\)
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) −5.54400 −0.377222
\(7\) − 23.0880i − 1.24664i −0.781969 − 0.623318i \(-0.785784\pi\)
0.781969 − 0.623318i \(-0.214216\pi\)
\(8\) 8.00000i 0.353553i
\(9\) 19.3160 0.715408
\(10\) 0 0
\(11\) −6.00000 −0.164461 −0.0822304 − 0.996613i \(-0.526204\pi\)
−0.0822304 + 0.996613i \(0.526204\pi\)
\(12\) 11.0880i 0.266736i
\(13\) − 72.4920i − 1.54659i −0.634046 − 0.773295i \(-0.718607\pi\)
0.634046 − 0.773295i \(-0.281393\pi\)
\(14\) −46.1760 −0.881504
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 58.6320i 0.836491i 0.908334 + 0.418246i \(0.137355\pi\)
−0.908334 + 0.418246i \(0.862645\pi\)
\(18\) − 38.6320i − 0.505870i
\(19\) −77.4400 −0.935051 −0.467525 − 0.883980i \(-0.654854\pi\)
−0.467525 + 0.883980i \(0.654854\pi\)
\(20\) 0 0
\(21\) −64.0000 −0.665045
\(22\) 12.0000i 0.116291i
\(23\) 23.0000i 0.208514i
\(24\) 22.1760 0.188611
\(25\) 0 0
\(26\) −144.984 −1.09360
\(27\) − 128.388i − 0.915122i
\(28\) 92.3520i 0.623318i
\(29\) 169.476 1.08520 0.542602 − 0.839990i \(-0.317439\pi\)
0.542602 + 0.839990i \(0.317439\pi\)
\(30\) 0 0
\(31\) −303.268 −1.75705 −0.878525 − 0.477696i \(-0.841472\pi\)
−0.878525 + 0.477696i \(0.841472\pi\)
\(32\) − 32.0000i − 0.176777i
\(33\) 16.6320i 0.0877352i
\(34\) 117.264 0.591489
\(35\) 0 0
\(36\) −77.2640 −0.357704
\(37\) − 265.336i − 1.17895i −0.807788 − 0.589473i \(-0.799336\pi\)
0.807788 − 0.589473i \(-0.200664\pi\)
\(38\) 154.880i 0.661181i
\(39\) −200.948 −0.825062
\(40\) 0 0
\(41\) 310.388 1.18230 0.591152 − 0.806560i \(-0.298673\pi\)
0.591152 + 0.806560i \(0.298673\pi\)
\(42\) 128.000i 0.470258i
\(43\) − 193.304i − 0.685549i −0.939418 − 0.342775i \(-0.888633\pi\)
0.939418 − 0.342775i \(-0.111367\pi\)
\(44\) 24.0000 0.0822304
\(45\) 0 0
\(46\) 46.0000 0.147442
\(47\) − 322.700i − 1.00150i −0.865591 − 0.500751i \(-0.833057\pi\)
0.865591 − 0.500751i \(-0.166943\pi\)
\(48\) − 44.3520i − 0.133368i
\(49\) −190.056 −0.554099
\(50\) 0 0
\(51\) 162.528 0.446245
\(52\) 289.968i 0.773295i
\(53\) − 137.336i − 0.355935i −0.984036 − 0.177968i \(-0.943048\pi\)
0.984036 − 0.177968i \(-0.0569522\pi\)
\(54\) −256.776 −0.647089
\(55\) 0 0
\(56\) 184.704 0.440752
\(57\) 214.664i 0.498823i
\(58\) − 338.952i − 0.767355i
\(59\) −562.320 −1.24081 −0.620405 − 0.784281i \(-0.713032\pi\)
−0.620405 + 0.784281i \(0.713032\pi\)
\(60\) 0 0
\(61\) −283.224 −0.594477 −0.297239 − 0.954803i \(-0.596066\pi\)
−0.297239 + 0.954803i \(0.596066\pi\)
\(62\) 606.536i 1.24242i
\(63\) − 445.968i − 0.891852i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) 33.2640 0.0620382
\(67\) 182.768i 0.333263i 0.986019 + 0.166632i \(0.0532891\pi\)
−0.986019 + 0.166632i \(0.946711\pi\)
\(68\) − 234.528i − 0.418246i
\(69\) 63.7560 0.111237
\(70\) 0 0
\(71\) 109.788 0.183513 0.0917566 − 0.995781i \(-0.470752\pi\)
0.0917566 + 0.995781i \(0.470752\pi\)
\(72\) 154.528i 0.252935i
\(73\) − 625.684i − 1.00316i −0.865111 − 0.501581i \(-0.832752\pi\)
0.865111 − 0.501581i \(-0.167248\pi\)
\(74\) −530.672 −0.833640
\(75\) 0 0
\(76\) 309.760 0.467525
\(77\) 138.528i 0.205023i
\(78\) 401.896i 0.583407i
\(79\) −172.072 −0.245058 −0.122529 − 0.992465i \(-0.539101\pi\)
−0.122529 + 0.992465i \(0.539101\pi\)
\(80\) 0 0
\(81\) 165.640 0.227216
\(82\) − 620.776i − 0.836015i
\(83\) 994.984i 1.31583i 0.753093 + 0.657914i \(0.228561\pi\)
−0.753093 + 0.657914i \(0.771439\pi\)
\(84\) 256.000 0.332522
\(85\) 0 0
\(86\) −386.608 −0.484756
\(87\) − 469.788i − 0.578926i
\(88\) − 48.0000i − 0.0581456i
\(89\) −1099.90 −1.30999 −0.654993 − 0.755635i \(-0.727328\pi\)
−0.654993 + 0.755635i \(0.727328\pi\)
\(90\) 0 0
\(91\) −1673.70 −1.92803
\(92\) − 92.0000i − 0.104257i
\(93\) 840.660i 0.937337i
\(94\) −645.400 −0.708169
\(95\) 0 0
\(96\) −88.7041 −0.0943054
\(97\) 1625.53i 1.70152i 0.525555 + 0.850760i \(0.323858\pi\)
−0.525555 + 0.850760i \(0.676142\pi\)
\(98\) 380.112i 0.391807i
\(99\) −115.896 −0.117656
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 1150.4.b.j.599.1 4
5.2 odd 4 46.4.a.d.1.1 ✓ 2
5.3 odd 4 1150.4.a.j.1.2 2
5.4 even 2 inner 1150.4.b.j.599.4 4
15.2 even 4 414.4.a.f.1.1 2
20.7 even 4 368.4.a.f.1.2 2
35.27 even 4 2254.4.a.f.1.2 2
40.27 even 4 1472.4.a.n.1.1 2
40.37 odd 4 1472.4.a.k.1.2 2
115.22 even 4 1058.4.a.j.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
46.4.a.d.1.1 ✓ 2 5.2 odd 4
368.4.a.f.1.2 2 20.7 even 4
414.4.a.f.1.1 2 15.2 even 4
1058.4.a.j.1.1 2 115.22 even 4
1150.4.a.j.1.2 2 5.3 odd 4
1150.4.b.j.599.1 4 1.1 even 1 trivial
1150.4.b.j.599.4 4 5.4 even 2 inner
1472.4.a.k.1.2 2 40.37 odd 4
1472.4.a.n.1.1 2 40.27 even 4
2254.4.a.f.1.2 2 35.27 even 4