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,-4,0,2,0,0,6,0,2] 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: \(\Q(i, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) 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)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.4
Root \(0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.2.b.i.599.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +0.618034i q^{3} -1.00000 q^{4} -0.618034 q^{6} +1.61803i q^{7} -1.00000i q^{8} +2.61803 q^{9} +3.85410 q^{11} -0.618034i q^{12} -4.09017i q^{13} -1.61803 q^{14} +1.00000 q^{16} -5.09017i q^{17} +2.61803i q^{18} +4.85410 q^{19} -1.00000 q^{21} +3.85410i q^{22} -1.00000i q^{23} +0.618034 q^{24} +4.09017 q^{26} +3.47214i q^{27} -1.61803i q^{28} +4.76393 q^{29} -2.09017 q^{31} +1.00000i q^{32} +2.38197i q^{33} +5.09017 q^{34} -2.61803 q^{36} -2.47214i q^{37} +4.85410i q^{38} +2.52786 q^{39} -12.3262 q^{41} -1.00000i q^{42} -3.85410 q^{44} +1.00000 q^{46} +9.70820i q^{47} +0.618034i q^{48} +4.38197 q^{49} +3.14590 q^{51} +4.09017i q^{52} +8.47214i q^{53} -3.47214 q^{54} +1.61803 q^{56} +3.00000i q^{57} +4.76393i q^{58} +11.7082 q^{59} +6.32624 q^{61} -2.09017i q^{62} +4.23607i q^{63} -1.00000 q^{64} -2.38197 q^{66} +5.52786i q^{67} +5.09017i q^{68} +0.618034 q^{69} +7.09017 q^{71} -2.61803i q^{72} +1.23607i q^{73} +2.47214 q^{74} -4.85410 q^{76} +6.23607i q^{77} +2.52786i q^{78} -10.4721 q^{79} +5.70820 q^{81} -12.3262i q^{82} -10.9443i q^{83} +1.00000 q^{84} +2.94427i q^{87} -3.85410i q^{88} +1.52786 q^{89} +6.61803 q^{91} +1.00000i q^{92} -1.29180i q^{93} -9.70820 q^{94} -0.618034 q^{96} +14.6180i q^{97} +4.38197i q^{98} +10.0902 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 2 q^{6} + 6 q^{9} + 2 q^{11} - 2 q^{14} + 4 q^{16} + 6 q^{19} - 4 q^{21} - 2 q^{24} - 6 q^{26} + 28 q^{29} + 14 q^{31} - 2 q^{34} - 6 q^{36} + 28 q^{39} - 18 q^{41} - 2 q^{44} + 4 q^{46}+ \cdots + 18 q^{99}+O(q^{100}) \) 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
\(3\) 0.618034i 0.356822i 0.983956 + 0.178411i \(0.0570957\pi\)
−0.983956 + 0.178411i \(0.942904\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −0.618034 −0.252311
\(7\) 1.61803i 0.611559i 0.952102 + 0.305780i \(0.0989171\pi\)
−0.952102 + 0.305780i \(0.901083\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 2.61803 0.872678
\(10\) 0 0
\(11\) 3.85410 1.16206 0.581028 − 0.813884i \(-0.302651\pi\)
0.581028 + 0.813884i \(0.302651\pi\)
\(12\) − 0.618034i − 0.178411i
\(13\) − 4.09017i − 1.13441i −0.823577 − 0.567205i \(-0.808025\pi\)
0.823577 − 0.567205i \(-0.191975\pi\)
\(14\) −1.61803 −0.432438
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 5.09017i − 1.23455i −0.786748 − 0.617274i \(-0.788237\pi\)
0.786748 − 0.617274i \(-0.211763\pi\)
\(18\) 2.61803i 0.617077i
\(19\) 4.85410 1.11361 0.556804 − 0.830644i \(-0.312028\pi\)
0.556804 + 0.830644i \(0.312028\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 3.85410i 0.821697i
\(23\) − 1.00000i − 0.208514i
\(24\) 0.618034 0.126156
\(25\) 0 0
\(26\) 4.09017 0.802148
\(27\) 3.47214i 0.668213i
\(28\) − 1.61803i − 0.305780i
\(29\) 4.76393 0.884640 0.442320 − 0.896857i \(-0.354156\pi\)
0.442320 + 0.896857i \(0.354156\pi\)
\(30\) 0 0
\(31\) −2.09017 −0.375406 −0.187703 − 0.982226i \(-0.560104\pi\)
−0.187703 + 0.982226i \(0.560104\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 2.38197i 0.414647i
\(34\) 5.09017 0.872957
\(35\) 0 0
\(36\) −2.61803 −0.436339
\(37\) − 2.47214i − 0.406417i −0.979136 − 0.203208i \(-0.934863\pi\)
0.979136 − 0.203208i \(-0.0651369\pi\)
\(38\) 4.85410i 0.787439i
\(39\) 2.52786 0.404782
\(40\) 0 0
\(41\) −12.3262 −1.92503 −0.962517 − 0.271220i \(-0.912573\pi\)
−0.962517 + 0.271220i \(0.912573\pi\)
\(42\) − 1.00000i − 0.154303i
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) −3.85410 −0.581028
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 9.70820i 1.41609i 0.706169 + 0.708044i \(0.250422\pi\)
−0.706169 + 0.708044i \(0.749578\pi\)
\(48\) 0.618034i 0.0892055i
\(49\) 4.38197 0.625995
\(50\) 0 0
\(51\) 3.14590 0.440514
\(52\) 4.09017i 0.567205i
\(53\) 8.47214i 1.16374i 0.813283 + 0.581869i \(0.197678\pi\)
−0.813283 + 0.581869i \(0.802322\pi\)
\(54\) −3.47214 −0.472498
\(55\) 0 0
\(56\) 1.61803 0.216219
\(57\) 3.00000i 0.397360i
\(58\) 4.76393i 0.625535i
\(59\) 11.7082 1.52428 0.762139 − 0.647413i \(-0.224149\pi\)
0.762139 + 0.647413i \(0.224149\pi\)
\(60\) 0 0
\(61\) 6.32624 0.809992 0.404996 − 0.914319i \(-0.367273\pi\)
0.404996 + 0.914319i \(0.367273\pi\)
\(62\) − 2.09017i − 0.265452i
\(63\) 4.23607i 0.533694i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −2.38197 −0.293200
\(67\) 5.52786i 0.675336i 0.941265 + 0.337668i \(0.109638\pi\)
−0.941265 + 0.337668i \(0.890362\pi\)
\(68\) 5.09017i 0.617274i
\(69\) 0.618034 0.0744025
\(70\) 0 0
\(71\) 7.09017 0.841448 0.420724 − 0.907189i \(-0.361776\pi\)
0.420724 + 0.907189i \(0.361776\pi\)
\(72\) − 2.61803i − 0.308538i
\(73\) 1.23607i 0.144671i 0.997380 + 0.0723354i \(0.0230452\pi\)
−0.997380 + 0.0723354i \(0.976955\pi\)
\(74\) 2.47214 0.287380
\(75\) 0 0
\(76\) −4.85410 −0.556804
\(77\) 6.23607i 0.710666i
\(78\) 2.52786i 0.286224i
\(79\) −10.4721 −1.17821 −0.589104 − 0.808057i \(-0.700519\pi\)
−0.589104 + 0.808057i \(0.700519\pi\)
\(80\) 0 0
\(81\) 5.70820 0.634245
\(82\) − 12.3262i − 1.36121i
\(83\) − 10.9443i − 1.20129i −0.799516 − 0.600645i \(-0.794911\pi\)
0.799516 − 0.600645i \(-0.205089\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) 0 0
\(87\) 2.94427i 0.315659i
\(88\) − 3.85410i − 0.410849i
\(89\) 1.52786 0.161953 0.0809766 − 0.996716i \(-0.474196\pi\)
0.0809766 + 0.996716i \(0.474196\pi\)
\(90\) 0 0
\(91\) 6.61803 0.693758
\(92\) 1.00000i 0.104257i
\(93\) − 1.29180i − 0.133953i
\(94\) −9.70820 −1.00132
\(95\) 0 0
\(96\) −0.618034 −0.0630778
\(97\) 14.6180i 1.48424i 0.670269 + 0.742118i \(0.266179\pi\)
−0.670269 + 0.742118i \(0.733821\pi\)
\(98\) 4.38197i 0.442645i
\(99\) 10.0902 1.01410
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.i.599.4 4
5.2 odd 4 1150.2.a.j.1.2 2
5.3 odd 4 230.2.a.c.1.1 ✓ 2
5.4 even 2 inner 1150.2.b.i.599.1 4
15.8 even 4 2070.2.a.u.1.2 2
20.3 even 4 1840.2.a.l.1.2 2
20.7 even 4 9200.2.a.bu.1.1 2
40.3 even 4 7360.2.a.bn.1.1 2
40.13 odd 4 7360.2.a.bh.1.2 2
115.68 even 4 5290.2.a.o.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.c.1.1 ✓ 2 5.3 odd 4
1150.2.a.j.1.2 2 5.2 odd 4
1150.2.b.i.599.1 4 5.4 even 2 inner
1150.2.b.i.599.4 4 1.1 even 1 trivial
1840.2.a.l.1.2 2 20.3 even 4
2070.2.a.u.1.2 2 15.8 even 4
5290.2.a.o.1.1 2 115.68 even 4
7360.2.a.bh.1.2 2 40.13 odd 4
7360.2.a.bn.1.1 2 40.3 even 4
9200.2.a.bu.1.1 2 20.7 even 4