Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [600,2,Mod(293,600)] 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("600.293"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(600, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 2, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 600.w (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,0,0,-4,0] 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: no
Analytic conductor: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 293.9
Character \(\chi\) \(=\) 600.293
Dual form 600.2.w.j.557.9

$q$-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+(0.0941764 + 1.41107i) q^{2} +(0.416519 + 1.68122i) q^{3} +(-1.98226 + 0.265780i) q^{4} +(-2.33310 + 0.746071i) q^{6} +(0.361989 - 0.361989i) q^{7} +(-0.561717 - 2.77209i) q^{8} +(-2.65302 + 1.40052i) q^{9} -2.63380 q^{11} +(-1.27249 - 3.22192i) q^{12} +(-3.49376 + 3.49376i) q^{13} +(0.544885 + 0.476703i) q^{14} +(3.85872 - 1.05369i) q^{16} +(-3.61339 - 3.61339i) q^{17} +(-2.22609 - 3.61172i) q^{18} -0.672266 q^{19} +(0.759360 + 0.457809i) q^{21} +(-0.248041 - 3.71648i) q^{22} +(-4.31851 + 4.31851i) q^{23} +(4.42653 - 2.09900i) q^{24} +(-5.25899 - 4.60093i) q^{26} +(-3.45963 - 3.87698i) q^{27} +(-0.621348 + 0.813767i) q^{28} +4.76080i q^{29} +3.73793 q^{31} +(1.85024 + 5.34571i) q^{32} +(-1.09703 - 4.42800i) q^{33} +(4.75847 - 5.43906i) q^{34} +(4.88676 - 3.48132i) q^{36} +(2.82150 + 2.82150i) q^{37} +(-0.0633116 - 0.948617i) q^{38} +(-7.32901 - 4.41857i) q^{39} +4.10027i q^{41} +(-0.574489 + 1.11463i) q^{42} +(7.57996 - 7.57996i) q^{43} +(5.22087 - 0.700010i) q^{44} +(-6.50044 - 5.68704i) q^{46} +(-0.987537 - 0.987537i) q^{47} +(3.37872 + 6.04849i) q^{48} +6.73793i q^{49} +(4.56987 - 7.57996i) q^{51} +(5.99698 - 7.85412i) q^{52} +(0.646149 + 0.646149i) q^{53} +(5.14489 - 5.24691i) q^{54} +(-1.20680 - 0.800131i) q^{56} +(-0.280012 - 1.13023i) q^{57} +(-6.71784 + 0.448355i) q^{58} -4.92247i q^{59} +6.07190i q^{61} +(0.352025 + 5.27449i) q^{62} +(-0.453392 + 1.46734i) q^{63} +(-7.36895 + 3.11426i) q^{64} +(6.14492 - 1.96500i) q^{66} +(0.349085 + 0.349085i) q^{67} +(8.12305 + 6.20232i) q^{68} +(-9.05912 - 5.46164i) q^{69} +8.63702i q^{71} +(5.37262 + 6.56772i) q^{72} +(11.3261 + 11.3261i) q^{73} +(-3.71562 + 4.24706i) q^{74} +(1.33261 - 0.178675i) q^{76} +(-0.953406 + 0.953406i) q^{77} +(5.54472 - 10.7579i) q^{78} +4.07707i q^{79} +(5.07707 - 7.43124i) q^{81} +(-5.78579 + 0.386149i) q^{82} +(-8.53893 - 8.53893i) q^{83} +(-1.62693 - 0.705675i) q^{84} +(11.4097 + 9.98203i) q^{86} +(-8.00397 + 1.98296i) q^{87} +(1.47945 + 7.30111i) q^{88} -6.58584 q^{89} +2.52941i q^{91} +(7.41264 - 9.70819i) q^{92} +(1.55692 + 6.28429i) q^{93} +(1.30049 - 1.48649i) q^{94} +(-8.21668 + 5.33725i) q^{96} +(0.660859 - 0.660859i) q^{97} +(-9.50772 + 0.634554i) q^{98} +(6.98752 - 3.68869i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q - 4 q^{6} + 8 q^{12} + 28 q^{16} + 20 q^{18} + 52 q^{22} - 12 q^{28} - 32 q^{31} - 8 q^{33} - 20 q^{36} - 16 q^{42} + 24 q^{46} - 44 q^{48} - 8 q^{52} + 16 q^{57} - 28 q^{58} - 48 q^{63} + 16 q^{66}+ \cdots - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0941764 + 1.41107i 0.0665928 + 0.997780i
\(3\) 0.416519 + 1.68122i 0.240477 + 0.970655i
\(4\) −1.98226 + 0.265780i −0.991131 + 0.132890i
\(5\) 0 0
\(6\) −2.33310 + 0.746071i −0.952486 + 0.304582i
\(7\) 0.361989 − 0.361989i 0.136819 − 0.136819i −0.635380 − 0.772199i \(-0.719157\pi\)
0.772199 + 0.635380i \(0.219157\pi\)
\(8\) −0.561717 − 2.77209i −0.198597 − 0.980081i
\(9\) −2.65302 + 1.40052i −0.884341 + 0.466841i
\(10\) 0 0
\(11\) −2.63380 −0.794119 −0.397060 − 0.917793i \(-0.629969\pi\)
−0.397060 + 0.917793i \(0.629969\pi\)
\(12\) −1.27249 − 3.22192i −0.367335 − 0.930089i
\(13\) −3.49376 + 3.49376i −0.968995 + 0.968995i −0.999534 − 0.0305386i \(-0.990278\pi\)
0.0305386 + 0.999534i \(0.490278\pi\)
\(14\) 0.544885 + 0.476703i 0.145627 + 0.127404i
\(15\) 0 0
\(16\) 3.85872 − 1.05369i 0.964681 − 0.263423i
\(17\) −3.61339 − 3.61339i −0.876376 − 0.876376i 0.116782 − 0.993158i \(-0.462742\pi\)
−0.993158 + 0.116782i \(0.962742\pi\)
\(18\) −2.22609 − 3.61172i −0.524696 − 0.851290i
\(19\) −0.672266 −0.154228 −0.0771142 − 0.997022i \(-0.524571\pi\)
−0.0771142 + 0.997022i \(0.524571\pi\)
\(20\) 0 0
\(21\) 0.759360 + 0.457809i 0.165706 + 0.0999022i
\(22\) −0.248041 − 3.71648i −0.0528826 − 0.792357i
\(23\) −4.31851 + 4.31851i −0.900472 + 0.900472i −0.995477 − 0.0950052i \(-0.969713\pi\)
0.0950052 + 0.995477i \(0.469713\pi\)
\(24\) 4.42653 − 2.09900i 0.903562 − 0.428457i
\(25\) 0 0
\(26\) −5.25899 − 4.60093i −1.03137 − 0.902316i
\(27\) −3.45963 − 3.87698i −0.665806 − 0.746125i
\(28\) −0.621348 + 0.813767i −0.117424 + 0.153787i
\(29\) 4.76080i 0.884058i 0.897001 + 0.442029i \(0.145741\pi\)
−0.897001 + 0.442029i \(0.854259\pi\)
\(30\) 0 0
\(31\) 3.73793 0.671352 0.335676 − 0.941978i \(-0.391035\pi\)
0.335676 + 0.941978i \(0.391035\pi\)
\(32\) 1.85024 + 5.34571i 0.327079 + 0.944997i
\(33\) −1.09703 − 4.42800i −0.190968 − 0.770816i
\(34\) 4.75847 − 5.43906i 0.816070 − 0.932791i
\(35\) 0 0
\(36\) 4.88676 − 3.48132i 0.814459 − 0.580221i
\(37\) 2.82150 + 2.82150i 0.463851 + 0.463851i 0.899915 − 0.436064i \(-0.143628\pi\)
−0.436064 + 0.899915i \(0.643628\pi\)
\(38\) −0.0633116 − 0.948617i −0.0102705 − 0.153886i
\(39\) −7.32901 − 4.41857i −1.17358 − 0.707538i
\(40\) 0 0
\(41\) 4.10027i 0.640355i 0.947358 + 0.320177i \(0.103743\pi\)
−0.947358 + 0.320177i \(0.896257\pi\)
\(42\) −0.574489 + 1.11463i −0.0886456 + 0.171991i
\(43\) 7.57996 − 7.57996i 1.15593 − 1.15593i 0.170591 − 0.985342i \(-0.445432\pi\)
0.985342 − 0.170591i \(-0.0545678\pi\)
\(44\) 5.22087 − 0.700010i 0.787076 − 0.105530i
\(45\) 0 0
\(46\) −6.50044 − 5.68704i −0.958438 − 0.838508i
\(47\) −0.987537 − 0.987537i −0.144047 − 0.144047i 0.631406 − 0.775453i \(-0.282478\pi\)
−0.775453 + 0.631406i \(0.782478\pi\)
\(48\) 3.37872 + 6.04849i 0.487676 + 0.873025i
\(49\) 6.73793i 0.962561i
\(50\) 0 0
\(51\) 4.56987 − 7.57996i 0.639910 − 1.06141i
\(52\) 5.99698 − 7.85412i 0.831631 − 1.08917i
\(53\) 0.646149 + 0.646149i 0.0887554 + 0.0887554i 0.750091 − 0.661335i \(-0.230010\pi\)
−0.661335 + 0.750091i \(0.730010\pi\)
\(54\) 5.14489 − 5.24691i 0.700131 − 0.714014i
\(55\) 0 0
\(56\) −1.20680 − 0.800131i −0.161266 − 0.106922i
\(57\) −0.280012 − 1.13023i −0.0370884 − 0.149702i
\(58\) −6.71784 + 0.448355i −0.882096 + 0.0588719i
\(59\) − 4.92247i − 0.640851i −0.947274 − 0.320425i \(-0.896174\pi\)
0.947274 − 0.320425i \(-0.103826\pi\)
\(60\) 0 0
\(61\) 6.07190i 0.777428i 0.921359 + 0.388714i \(0.127081\pi\)
−0.921359 + 0.388714i \(0.872919\pi\)
\(62\) 0.352025 + 5.27449i 0.0447072 + 0.669861i
\(63\) −0.453392 + 1.46734i −0.0571220 + 0.184868i
\(64\) −7.36895 + 3.11426i −0.921118 + 0.389283i
\(65\) 0 0
\(66\) 6.14492 − 1.96500i 0.756388 − 0.241875i
\(67\) 0.349085 + 0.349085i 0.0426476 + 0.0426476i 0.728109 − 0.685461i \(-0.240399\pi\)
−0.685461 + 0.728109i \(0.740399\pi\)
\(68\) 8.12305 + 6.20232i 0.985064 + 0.752141i
\(69\) −9.05912 − 5.46164i −1.09059 − 0.657504i
\(70\) 0 0
\(71\) 8.63702i 1.02503i 0.858680 + 0.512513i \(0.171285\pi\)
−0.858680 + 0.512513i \(0.828715\pi\)
\(72\) 5.37262 + 6.56772i 0.633170 + 0.774013i
\(73\) 11.3261 + 11.3261i 1.32562 + 1.32562i 0.909152 + 0.416465i \(0.136731\pi\)
0.416465 + 0.909152i \(0.363269\pi\)
\(74\) −3.71562 + 4.24706i −0.431932 + 0.493711i
\(75\) 0 0
\(76\) 1.33261 − 0.178675i 0.152860 − 0.0204954i
\(77\) −0.953406 + 0.953406i −0.108651 + 0.108651i
\(78\) 5.54472 − 10.7579i 0.627816 − 1.21809i
\(79\) 4.07707i 0.458706i 0.973343 + 0.229353i \(0.0736610\pi\)
−0.973343 + 0.229353i \(0.926339\pi\)
\(80\) 0 0
\(81\) 5.07707 − 7.43124i 0.564119 − 0.825694i
\(82\) −5.78579 + 0.386149i −0.638933 + 0.0426430i
\(83\) −8.53893 − 8.53893i −0.937270 − 0.937270i 0.0608758 − 0.998145i \(-0.480611\pi\)
−0.998145 + 0.0608758i \(0.980611\pi\)
\(84\) −1.62693 − 0.705675i −0.177512 − 0.0769955i
\(85\) 0 0
\(86\) 11.4097 + 9.98203i 1.23034 + 1.07639i
\(87\) −8.00397 + 1.98296i −0.858115 + 0.212596i
\(88\) 1.47945 + 7.30111i 0.157710 + 0.778301i
\(89\) −6.58584 −0.698097 −0.349049 − 0.937105i \(-0.613495\pi\)
−0.349049 + 0.937105i \(0.613495\pi\)
\(90\) 0 0
\(91\) 2.52941i 0.265154i
\(92\) 7.41264 − 9.70819i 0.772822 − 1.01215i
\(93\) 1.55692 + 6.28429i 0.161445 + 0.651651i
\(94\) 1.30049 − 1.48649i 0.134135 − 0.153320i
\(95\) 0 0
\(96\) −8.21668 + 5.33725i −0.838611 + 0.544731i
\(97\) 0.660859 − 0.660859i 0.0671001 − 0.0671001i −0.672760 − 0.739860i \(-0.734892\pi\)
0.739860 + 0.672760i \(0.234892\pi\)
\(98\) −9.50772 + 0.634554i −0.960424 + 0.0640996i
\(99\) 6.98752 − 3.68869i 0.702272 − 0.370728i
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 600.2.w.j.293.9 32
3.2 odd 2 inner 600.2.w.j.293.8 32
5.2 odd 4 inner 600.2.w.j.557.1 32
5.3 odd 4 120.2.w.c.77.16 yes 32
5.4 even 2 120.2.w.c.53.8 yes 32
8.5 even 2 inner 600.2.w.j.293.16 32
15.2 even 4 inner 600.2.w.j.557.16 32
15.8 even 4 120.2.w.c.77.1 yes 32
15.14 odd 2 120.2.w.c.53.9 yes 32
20.3 even 4 480.2.bi.c.17.15 32
20.19 odd 2 480.2.bi.c.113.10 32
24.5 odd 2 inner 600.2.w.j.293.1 32
40.3 even 4 480.2.bi.c.17.2 32
40.13 odd 4 120.2.w.c.77.9 yes 32
40.19 odd 2 480.2.bi.c.113.7 32
40.29 even 2 120.2.w.c.53.1 ✓ 32
40.37 odd 4 inner 600.2.w.j.557.8 32
60.23 odd 4 480.2.bi.c.17.7 32
60.59 even 2 480.2.bi.c.113.2 32
120.29 odd 2 120.2.w.c.53.16 yes 32
120.53 even 4 120.2.w.c.77.8 yes 32
120.59 even 2 480.2.bi.c.113.15 32
120.77 even 4 inner 600.2.w.j.557.9 32
120.83 odd 4 480.2.bi.c.17.10 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.w.c.53.1 ✓ 32 40.29 even 2
120.2.w.c.53.8 yes 32 5.4 even 2
120.2.w.c.53.9 yes 32 15.14 odd 2
120.2.w.c.53.16 yes 32 120.29 odd 2
120.2.w.c.77.1 yes 32 15.8 even 4
120.2.w.c.77.8 yes 32 120.53 even 4
120.2.w.c.77.9 yes 32 40.13 odd 4
120.2.w.c.77.16 yes 32 5.3 odd 4
480.2.bi.c.17.2 32 40.3 even 4
480.2.bi.c.17.7 32 60.23 odd 4
480.2.bi.c.17.10 32 120.83 odd 4
480.2.bi.c.17.15 32 20.3 even 4
480.2.bi.c.113.2 32 60.59 even 2
480.2.bi.c.113.7 32 40.19 odd 2
480.2.bi.c.113.10 32 20.19 odd 2
480.2.bi.c.113.15 32 120.59 even 2
600.2.w.j.293.1 32 24.5 odd 2 inner
600.2.w.j.293.8 32 3.2 odd 2 inner
600.2.w.j.293.9 32 1.1 even 1 trivial
600.2.w.j.293.16 32 8.5 even 2 inner
600.2.w.j.557.1 32 5.2 odd 4 inner
600.2.w.j.557.8 32 40.37 odd 4 inner
600.2.w.j.557.9 32 120.77 even 4 inner
600.2.w.j.557.16 32 15.2 even 4 inner