Show commands: Magma / Pari/GP / SageMath

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 = [2,0,0,-8,0,-36,0,0,-108,0,-104] 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: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) 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 46)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.4.b.a.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)
 
\(f(q)\) \(=\) \(q+2.00000i q^{2} +9.00000i q^{3} -4.00000 q^{4} -18.0000 q^{6} +2.00000i q^{7} -8.00000i q^{8} -54.0000 q^{9} -52.0000 q^{11} -36.0000i q^{12} -43.0000i q^{13} -4.00000 q^{14} +16.0000 q^{16} -50.0000i q^{17} -108.000i q^{18} +74.0000 q^{19} -18.0000 q^{21} -104.000i q^{22} +23.0000i q^{23} +72.0000 q^{24} +86.0000 q^{26} -243.000i q^{27} -8.00000i q^{28} +7.00000 q^{29} -273.000 q^{31} +32.0000i q^{32} -468.000i q^{33} +100.000 q^{34} +216.000 q^{36} -4.00000i q^{37} +148.000i q^{38} +387.000 q^{39} +123.000 q^{41} -36.0000i q^{42} +152.000i q^{43} +208.000 q^{44} -46.0000 q^{46} +75.0000i q^{47} +144.000i q^{48} +339.000 q^{49} +450.000 q^{51} +172.000i q^{52} -86.0000i q^{53} +486.000 q^{54} +16.0000 q^{56} +666.000i q^{57} +14.0000i q^{58} +444.000 q^{59} +262.000 q^{61} -546.000i q^{62} -108.000i q^{63} -64.0000 q^{64} +936.000 q^{66} +764.000i q^{67} +200.000i q^{68} -207.000 q^{69} -21.0000 q^{71} +432.000i q^{72} -681.000i q^{73} +8.00000 q^{74} -296.000 q^{76} -104.000i q^{77} +774.000i q^{78} -426.000 q^{79} +729.000 q^{81} +246.000i q^{82} -902.000i q^{83} +72.0000 q^{84} -304.000 q^{86} +63.0000i q^{87} +416.000i q^{88} +1272.00 q^{89} +86.0000 q^{91} -92.0000i q^{92} -2457.00i q^{93} -150.000 q^{94} -288.000 q^{96} -342.000i q^{97} +678.000i q^{98} +2808.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} - 36 q^{6} - 108 q^{9} - 104 q^{11} - 8 q^{14} + 32 q^{16} + 148 q^{19} - 36 q^{21} + 144 q^{24} + 172 q^{26} + 14 q^{29} - 546 q^{31} + 200 q^{34} + 432 q^{36} + 774 q^{39} + 246 q^{41} + 416 q^{44}+ \cdots + 5616 q^{99}+O(q^{100}) \) 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
\(3\) 9.00000i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) −18.0000 −1.22474
\(7\) 2.00000i 0.107990i 0.998541 + 0.0539949i \(0.0171955\pi\)
−0.998541 + 0.0539949i \(0.982805\pi\)
\(8\) − 8.00000i − 0.353553i
\(9\) −54.0000 −2.00000
\(10\) 0 0
\(11\) −52.0000 −1.42533 −0.712663 − 0.701506i \(-0.752511\pi\)
−0.712663 + 0.701506i \(0.752511\pi\)
\(12\) − 36.0000i − 0.866025i
\(13\) − 43.0000i − 0.917389i −0.888594 − 0.458694i \(-0.848317\pi\)
0.888594 − 0.458694i \(-0.151683\pi\)
\(14\) −4.00000 −0.0763604
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) − 50.0000i − 0.713340i −0.934230 − 0.356670i \(-0.883912\pi\)
0.934230 − 0.356670i \(-0.116088\pi\)
\(18\) − 108.000i − 1.41421i
\(19\) 74.0000 0.893514 0.446757 − 0.894655i \(-0.352579\pi\)
0.446757 + 0.894655i \(0.352579\pi\)
\(20\) 0 0
\(21\) −18.0000 −0.187044
\(22\) − 104.000i − 1.00786i
\(23\) 23.0000i 0.208514i
\(24\) 72.0000 0.612372
\(25\) 0 0
\(26\) 86.0000 0.648692
\(27\) − 243.000i − 1.73205i
\(28\) − 8.00000i − 0.0539949i
\(29\) 7.00000 0.0448230 0.0224115 − 0.999749i \(-0.492866\pi\)
0.0224115 + 0.999749i \(0.492866\pi\)
\(30\) 0 0
\(31\) −273.000 −1.58169 −0.790843 − 0.612019i \(-0.790357\pi\)
−0.790843 + 0.612019i \(0.790357\pi\)
\(32\) 32.0000i 0.176777i
\(33\) − 468.000i − 2.46874i
\(34\) 100.000 0.504408
\(35\) 0 0
\(36\) 216.000 1.00000
\(37\) − 4.00000i − 0.0177729i −0.999961 − 0.00888643i \(-0.997171\pi\)
0.999961 − 0.00888643i \(-0.00282868\pi\)
\(38\) 148.000i 0.631810i
\(39\) 387.000 1.58896
\(40\) 0 0
\(41\) 123.000 0.468521 0.234261 − 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) − 36.0000i − 0.132260i
\(43\) 152.000i 0.539065i 0.962991 + 0.269532i \(0.0868691\pi\)
−0.962991 + 0.269532i \(0.913131\pi\)
\(44\) 208.000 0.712663
\(45\) 0 0
\(46\) −46.0000 −0.147442
\(47\) 75.0000i 0.232763i 0.993205 + 0.116382i \(0.0371296\pi\)
−0.993205 + 0.116382i \(0.962870\pi\)
\(48\) 144.000i 0.433013i
\(49\) 339.000 0.988338
\(50\) 0 0
\(51\) 450.000 1.23554
\(52\) 172.000i 0.458694i
\(53\) − 86.0000i − 0.222887i −0.993771 − 0.111443i \(-0.964453\pi\)
0.993771 − 0.111443i \(-0.0355474\pi\)
\(54\) 486.000 1.22474
\(55\) 0 0
\(56\) 16.0000 0.0381802
\(57\) 666.000i 1.54761i
\(58\) 14.0000i 0.0316947i
\(59\) 444.000 0.979727 0.489863 − 0.871799i \(-0.337047\pi\)
0.489863 + 0.871799i \(0.337047\pi\)
\(60\) 0 0
\(61\) 262.000 0.549929 0.274964 − 0.961454i \(-0.411334\pi\)
0.274964 + 0.961454i \(0.411334\pi\)
\(62\) − 546.000i − 1.11842i
\(63\) − 108.000i − 0.215980i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) 936.000 1.74566
\(67\) 764.000i 1.39310i 0.717510 + 0.696548i \(0.245282\pi\)
−0.717510 + 0.696548i \(0.754718\pi\)
\(68\) 200.000i 0.356670i
\(69\) −207.000 −0.361158
\(70\) 0 0
\(71\) −21.0000 −0.0351020 −0.0175510 − 0.999846i \(-0.505587\pi\)
−0.0175510 + 0.999846i \(0.505587\pi\)
\(72\) 432.000i 0.707107i
\(73\) − 681.000i − 1.09185i −0.837834 − 0.545925i \(-0.816178\pi\)
0.837834 − 0.545925i \(-0.183822\pi\)
\(74\) 8.00000 0.0125673
\(75\) 0 0
\(76\) −296.000 −0.446757
\(77\) − 104.000i − 0.153921i
\(78\) 774.000i 1.12357i
\(79\) −426.000 −0.606693 −0.303346 − 0.952880i \(-0.598104\pi\)
−0.303346 + 0.952880i \(0.598104\pi\)
\(80\) 0 0
\(81\) 729.000 1.00000
\(82\) 246.000i 0.331295i
\(83\) − 902.000i − 1.19286i −0.802665 − 0.596430i \(-0.796585\pi\)
0.802665 − 0.596430i \(-0.203415\pi\)
\(84\) 72.0000 0.0935220
\(85\) 0 0
\(86\) −304.000 −0.381176
\(87\) 63.0000i 0.0776357i
\(88\) 416.000i 0.503929i
\(89\) 1272.00 1.51496 0.757482 − 0.652856i \(-0.226430\pi\)
0.757482 + 0.652856i \(0.226430\pi\)
\(90\) 0 0
\(91\) 86.0000 0.0990687
\(92\) − 92.0000i − 0.104257i
\(93\) − 2457.00i − 2.73956i
\(94\) −150.000 −0.164588
\(95\) 0 0
\(96\) −288.000 −0.306186
\(97\) − 342.000i − 0.357988i −0.983850 − 0.178994i \(-0.942716\pi\)
0.983850 − 0.178994i \(-0.0572843\pi\)
\(98\) 678.000i 0.698861i
\(99\) 2808.00 2.85065
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.a.599.2 2
5.2 odd 4 1150.4.a.d.1.1 1
5.3 odd 4 46.4.a.b.1.1 ✓ 1
5.4 even 2 inner 1150.4.b.a.599.1 2
15.8 even 4 414.4.a.b.1.1 1
20.3 even 4 368.4.a.e.1.1 1
35.13 even 4 2254.4.a.b.1.1 1
40.3 even 4 1472.4.a.a.1.1 1
40.13 odd 4 1472.4.a.j.1.1 1
115.68 even 4 1058.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
46.4.a.b.1.1 ✓ 1 5.3 odd 4
368.4.a.e.1.1 1 20.3 even 4
414.4.a.b.1.1 1 15.8 even 4
1058.4.a.b.1.1 1 115.68 even 4
1150.4.a.d.1.1 1 5.2 odd 4
1150.4.b.a.599.1 2 5.4 even 2 inner
1150.4.b.a.599.2 2 1.1 even 1 trivial
1472.4.a.a.1.1 1 40.3 even 4
1472.4.a.j.1.1 1 40.13 odd 4
2254.4.a.b.1.1 1 35.13 even 4