Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7406,2,Mod(1,7406)] 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("7406.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7406, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7406 = 2 \cdot 7 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7406.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,1,2,1,-1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(59.1372077370\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) 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 \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 7406.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.618034 q^{3} +1.00000 q^{4} +1.61803 q^{5} +0.618034 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.61803 q^{9} -1.61803 q^{10} -4.85410 q^{11} -0.618034 q^{12} -4.61803 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -3.47214 q^{17} +2.61803 q^{18} +1.47214 q^{19} +1.61803 q^{20} +0.618034 q^{21} +4.85410 q^{22} +0.618034 q^{24} -2.38197 q^{25} +4.61803 q^{26} +3.47214 q^{27} -1.00000 q^{28} -5.70820 q^{29} +1.00000 q^{30} +9.94427 q^{31} -1.00000 q^{32} +3.00000 q^{33} +3.47214 q^{34} -1.61803 q^{35} -2.61803 q^{36} -3.00000 q^{37} -1.47214 q^{38} +2.85410 q^{39} -1.61803 q^{40} +0.145898 q^{41} -0.618034 q^{42} +5.47214 q^{43} -4.85410 q^{44} -4.23607 q^{45} -7.09017 q^{47} -0.618034 q^{48} +1.00000 q^{49} +2.38197 q^{50} +2.14590 q^{51} -4.61803 q^{52} -3.70820 q^{53} -3.47214 q^{54} -7.85410 q^{55} +1.00000 q^{56} -0.909830 q^{57} +5.70820 q^{58} -11.2361 q^{59} -1.00000 q^{60} -7.70820 q^{61} -9.94427 q^{62} +2.61803 q^{63} +1.00000 q^{64} -7.47214 q^{65} -3.00000 q^{66} -15.4164 q^{67} -3.47214 q^{68} +1.61803 q^{70} -2.09017 q^{71} +2.61803 q^{72} -0.708204 q^{73} +3.00000 q^{74} +1.47214 q^{75} +1.47214 q^{76} +4.85410 q^{77} -2.85410 q^{78} +7.47214 q^{79} +1.61803 q^{80} +5.70820 q^{81} -0.145898 q^{82} -3.61803 q^{83} +0.618034 q^{84} -5.61803 q^{85} -5.47214 q^{86} +3.52786 q^{87} +4.85410 q^{88} -7.94427 q^{89} +4.23607 q^{90} +4.61803 q^{91} -6.14590 q^{93} +7.09017 q^{94} +2.38197 q^{95} +0.618034 q^{96} +4.52786 q^{97} -1.00000 q^{98} +12.7082 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + q^{3} + 2 q^{4} + q^{5} - q^{6} - 2 q^{7} - 2 q^{8} - 3 q^{9} - q^{10} - 3 q^{11} + q^{12} - 7 q^{13} + 2 q^{14} - 2 q^{15} + 2 q^{16} + 2 q^{17} + 3 q^{18} - 6 q^{19} + q^{20} - q^{21}+ \cdots + 12 q^{99}+O(q^{100}) \) 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
\(3\) −0.618034 −0.356822 −0.178411 − 0.983956i \(-0.557096\pi\)
−0.178411 + 0.983956i \(0.557096\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.61803 0.723607 0.361803 − 0.932254i \(-0.382161\pi\)
0.361803 + 0.932254i \(0.382161\pi\)
\(6\) 0.618034 0.252311
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) −2.61803 −0.872678
\(10\) −1.61803 −0.511667
\(11\) −4.85410 −1.46357 −0.731783 − 0.681537i \(-0.761312\pi\)
−0.731783 + 0.681537i \(0.761312\pi\)
\(12\) −0.618034 −0.178411
\(13\) −4.61803 −1.28081 −0.640406 − 0.768036i \(-0.721234\pi\)
−0.640406 + 0.768036i \(0.721234\pi\)
\(14\) 1.00000 0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −3.47214 −0.842117 −0.421058 − 0.907034i \(-0.638341\pi\)
−0.421058 + 0.907034i \(0.638341\pi\)
\(18\) 2.61803 0.617077
\(19\) 1.47214 0.337731 0.168866 − 0.985639i \(-0.445990\pi\)
0.168866 + 0.985639i \(0.445990\pi\)
\(20\) 1.61803 0.361803
\(21\) 0.618034 0.134866
\(22\) 4.85410 1.03490
\(23\) 0 0
\(24\) 0.618034 0.126156
\(25\) −2.38197 −0.476393
\(26\) 4.61803 0.905671
\(27\) 3.47214 0.668213
\(28\) −1.00000 −0.188982
\(29\) −5.70820 −1.05999 −0.529993 − 0.848002i \(-0.677806\pi\)
−0.529993 + 0.848002i \(0.677806\pi\)
\(30\) 1.00000 0.182574
\(31\) 9.94427 1.78604 0.893022 − 0.450013i \(-0.148581\pi\)
0.893022 + 0.450013i \(0.148581\pi\)
\(32\) −1.00000 −0.176777
\(33\) 3.00000 0.522233
\(34\) 3.47214 0.595466
\(35\) −1.61803 −0.273498
\(36\) −2.61803 −0.436339
\(37\) −3.00000 −0.493197 −0.246598 − 0.969118i \(-0.579313\pi\)
−0.246598 + 0.969118i \(0.579313\pi\)
\(38\) −1.47214 −0.238812
\(39\) 2.85410 0.457022
\(40\) −1.61803 −0.255834
\(41\) 0.145898 0.0227854 0.0113927 − 0.999935i \(-0.496374\pi\)
0.0113927 + 0.999935i \(0.496374\pi\)
\(42\) −0.618034 −0.0953647
\(43\) 5.47214 0.834493 0.417246 − 0.908793i \(-0.362995\pi\)
0.417246 + 0.908793i \(0.362995\pi\)
\(44\) −4.85410 −0.731783
\(45\) −4.23607 −0.631476
\(46\) 0 0
\(47\) −7.09017 −1.03421 −0.517104 − 0.855923i \(-0.672990\pi\)
−0.517104 + 0.855923i \(0.672990\pi\)
\(48\) −0.618034 −0.0892055
\(49\) 1.00000 0.142857
\(50\) 2.38197 0.336861
\(51\) 2.14590 0.300486
\(52\) −4.61803 −0.640406
\(53\) −3.70820 −0.509361 −0.254680 − 0.967025i \(-0.581970\pi\)
−0.254680 + 0.967025i \(0.581970\pi\)
\(54\) −3.47214 −0.472498
\(55\) −7.85410 −1.05905
\(56\) 1.00000 0.133631
\(57\) −0.909830 −0.120510
\(58\) 5.70820 0.749524
\(59\) −11.2361 −1.46281 −0.731406 − 0.681943i \(-0.761135\pi\)
−0.731406 + 0.681943i \(0.761135\pi\)
\(60\) −1.00000 −0.129099
\(61\) −7.70820 −0.986934 −0.493467 − 0.869764i \(-0.664271\pi\)
−0.493467 + 0.869764i \(0.664271\pi\)
\(62\) −9.94427 −1.26292
\(63\) 2.61803 0.329841
\(64\) 1.00000 0.125000
\(65\) −7.47214 −0.926804
\(66\) −3.00000 −0.369274
\(67\) −15.4164 −1.88341 −0.941707 − 0.336434i \(-0.890779\pi\)
−0.941707 + 0.336434i \(0.890779\pi\)
\(68\) −3.47214 −0.421058
\(69\) 0 0
\(70\) 1.61803 0.193392
\(71\) −2.09017 −0.248058 −0.124029 − 0.992279i \(-0.539582\pi\)
−0.124029 + 0.992279i \(0.539582\pi\)
\(72\) 2.61803 0.308538
\(73\) −0.708204 −0.0828890 −0.0414445 − 0.999141i \(-0.513196\pi\)
−0.0414445 + 0.999141i \(0.513196\pi\)
\(74\) 3.00000 0.348743
\(75\) 1.47214 0.169988
\(76\) 1.47214 0.168866
\(77\) 4.85410 0.553176
\(78\) −2.85410 −0.323163
\(79\) 7.47214 0.840681 0.420340 − 0.907366i \(-0.361911\pi\)
0.420340 + 0.907366i \(0.361911\pi\)
\(80\) 1.61803 0.180902
\(81\) 5.70820 0.634245
\(82\) −0.145898 −0.0161117
\(83\) −3.61803 −0.397131 −0.198565 − 0.980088i \(-0.563628\pi\)
−0.198565 + 0.980088i \(0.563628\pi\)
\(84\) 0.618034 0.0674330
\(85\) −5.61803 −0.609361
\(86\) −5.47214 −0.590076
\(87\) 3.52786 0.378227
\(88\) 4.85410 0.517449
\(89\) −7.94427 −0.842091 −0.421046 − 0.907039i \(-0.638337\pi\)
−0.421046 + 0.907039i \(0.638337\pi\)
\(90\) 4.23607 0.446521
\(91\) 4.61803 0.484102
\(92\) 0 0
\(93\) −6.14590 −0.637300
\(94\) 7.09017 0.731295
\(95\) 2.38197 0.244385
\(96\) 0.618034 0.0630778
\(97\) 4.52786 0.459735 0.229867 − 0.973222i \(-0.426171\pi\)
0.229867 + 0.973222i \(0.426171\pi\)
\(98\) −1.00000 −0.101015
\(99\) 12.7082 1.27722
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 7406.2.a.p.1.1 yes 2
23.22 odd 2 7406.2.a.o.1.1 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7406.2.a.o.1.1 ✓ 2 23.22 odd 2
7406.2.a.p.1.1 yes 2 1.1 even 1 trivial