Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [24,6,Mod(13,24)] 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("24.13"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(24, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 24.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.84921167551\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4 x^{9} + 20 x^{8} - 16 x^{7} + 290 x^{6} + 896 x^{5} + 416 x^{4} + 13952 x^{3} + 60997 x^{2} + \cdots + 750564 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{17}\cdot 3^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 13.3
Root \(-0.700222 + 3.62067i\) of defining polynomial
Character \(\chi\) \(=\) 24.13
Dual form 24.6.d.a.13.4

$q$-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+(-1.92045 - 5.32089i) q^{2} +9.00000i q^{3} +(-24.6238 + 20.4370i) q^{4} +67.1492i q^{5} +(47.8880 - 17.2840i) q^{6} +37.4101 q^{7} +(156.032 + 91.7722i) q^{8} -81.0000 q^{9} +(357.294 - 128.957i) q^{10} +626.948i q^{11} +(-183.933 - 221.614i) q^{12} -186.748i q^{13} +(-71.8442 - 199.055i) q^{14} -604.343 q^{15} +(188.659 - 1006.47i) q^{16} -1941.78 q^{17} +(155.556 + 430.992i) q^{18} +51.9193i q^{19} +(-1372.33 - 1653.47i) q^{20} +336.691i q^{21} +(3335.92 - 1204.02i) q^{22} +4048.84 q^{23} +(-825.950 + 1404.28i) q^{24} -1384.02 q^{25} +(-993.666 + 358.640i) q^{26} -729.000i q^{27} +(-921.178 + 764.550i) q^{28} +1276.10i q^{29} +(1160.61 + 3215.64i) q^{30} +5782.69 q^{31} +(-5717.63 + 929.040i) q^{32} -5642.53 q^{33} +(3729.08 + 10332.0i) q^{34} +2512.06i q^{35} +(1994.52 - 1655.40i) q^{36} -14761.4i q^{37} +(276.257 - 99.7083i) q^{38} +1680.73 q^{39} +(-6162.43 + 10477.4i) q^{40} +11535.0 q^{41} +(1791.50 - 646.598i) q^{42} +4921.61i q^{43} +(-12812.9 - 15437.8i) q^{44} -5439.09i q^{45} +(-7775.59 - 21543.4i) q^{46} +13463.0 q^{47} +(9058.24 + 1697.93i) q^{48} -15407.5 q^{49} +(2657.93 + 7364.20i) q^{50} -17476.0i q^{51} +(3816.57 + 4598.44i) q^{52} +24873.6i q^{53} +(-3878.93 + 1400.01i) q^{54} -42099.1 q^{55} +(5837.16 + 3433.21i) q^{56} -467.274 q^{57} +(6790.01 - 2450.69i) q^{58} -3936.59i q^{59} +(14881.2 - 12350.9i) q^{60} +9930.43i q^{61} +(-11105.4 - 30769.1i) q^{62} -3030.22 q^{63} +(15923.7 + 28638.7i) q^{64} +12540.0 q^{65} +(10836.2 + 30023.3i) q^{66} +49755.9i q^{67} +(47813.8 - 39684.0i) q^{68} +36439.6i q^{69} +(13366.4 - 4824.28i) q^{70} -23133.9 q^{71} +(-12638.6 - 7433.55i) q^{72} +72460.8 q^{73} +(-78544.1 + 28348.6i) q^{74} -12456.1i q^{75} +(-1061.07 - 1278.45i) q^{76} +23454.2i q^{77} +(-3227.76 - 8943.00i) q^{78} -32406.1 q^{79} +(67583.7 + 12668.3i) q^{80} +6561.00 q^{81} +(-22152.4 - 61376.5i) q^{82} -69116.9i q^{83} +(-6880.95 - 8290.60i) q^{84} -130389. i q^{85} +(26187.4 - 9451.70i) q^{86} -11484.9 q^{87} +(-57536.4 + 97823.7i) q^{88} -65109.9 q^{89} +(-28940.8 + 10445.5i) q^{90} -6986.27i q^{91} +(-99697.7 + 82746.1i) q^{92} +52044.2i q^{93} +(-25854.9 - 71635.0i) q^{94} -3486.34 q^{95} +(-8361.36 - 51458.7i) q^{96} +112052. q^{97} +(29589.3 + 81981.5i) q^{98} -50782.8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{2} - 32 q^{4} + 90 q^{6} - 196 q^{7} + 68 q^{8} - 810 q^{9} - 868 q^{10} - 396 q^{12} + 3692 q^{14} + 900 q^{15} + 3064 q^{16} - 404 q^{17} - 162 q^{18} - 6376 q^{20} + 5888 q^{22} + 1672 q^{23}+ \cdots - 459846 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/24\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(13\) \(17\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.92045 − 5.32089i −0.339490 − 0.940610i
\(3\) 9.00000i 0.577350i
\(4\) −24.6238 + 20.4370i −0.769493 + 0.638656i
\(5\) 67.1492i 1.20120i 0.799549 + 0.600601i \(0.205072\pi\)
−0.799549 + 0.600601i \(0.794928\pi\)
\(6\) 47.8880 − 17.2840i 0.543061 − 0.196005i
\(7\) 37.4101 0.288565 0.144283 − 0.989537i \(-0.453913\pi\)
0.144283 + 0.989537i \(0.453913\pi\)
\(8\) 156.032 + 91.7722i 0.861961 + 0.506974i
\(9\) −81.0000 −0.333333
\(10\) 357.294 − 128.957i 1.12986 − 0.407796i
\(11\) 626.948i 1.56225i 0.624376 + 0.781124i \(0.285353\pi\)
−0.624376 + 0.781124i \(0.714647\pi\)
\(12\) −183.933 − 221.614i −0.368728 − 0.444267i
\(13\) − 186.748i − 0.306477i −0.988189 − 0.153238i \(-0.951030\pi\)
0.988189 − 0.153238i \(-0.0489703\pi\)
\(14\) −71.8442 − 199.055i −0.0979651 − 0.271427i
\(15\) −604.343 −0.693514
\(16\) 188.659 − 1006.47i 0.184237 − 0.982882i
\(17\) −1941.78 −1.62958 −0.814792 − 0.579753i \(-0.803149\pi\)
−0.814792 + 0.579753i \(0.803149\pi\)
\(18\) 155.556 + 430.992i 0.113163 + 0.313537i
\(19\) 51.9193i 0.0329948i 0.999864 + 0.0164974i \(0.00525152\pi\)
−0.999864 + 0.0164974i \(0.994748\pi\)
\(20\) −1372.33 − 1653.47i −0.767154 − 0.924316i
\(21\) 336.691i 0.166603i
\(22\) 3335.92 − 1204.02i 1.46947 − 0.530368i
\(23\) 4048.84 1.59592 0.797960 − 0.602710i \(-0.205912\pi\)
0.797960 + 0.602710i \(0.205912\pi\)
\(24\) −825.950 + 1404.28i −0.292702 + 0.497653i
\(25\) −1384.02 −0.442885
\(26\) −993.666 + 358.640i −0.288275 + 0.104046i
\(27\) − 729.000i − 0.192450i
\(28\) −921.178 + 764.550i −0.222049 + 0.184294i
\(29\) 1276.10i 0.281768i 0.990026 + 0.140884i \(0.0449944\pi\)
−0.990026 + 0.140884i \(0.955006\pi\)
\(30\) 1160.61 + 3215.64i 0.235441 + 0.652326i
\(31\) 5782.69 1.08075 0.540376 − 0.841424i \(-0.318282\pi\)
0.540376 + 0.841424i \(0.318282\pi\)
\(32\) −5717.63 + 929.040i −0.987055 + 0.160383i
\(33\) −5642.53 −0.901964
\(34\) 3729.08 + 10332.0i 0.553228 + 1.53280i
\(35\) 2512.06i 0.346625i
\(36\) 1994.52 − 1655.40i 0.256498 − 0.212885i
\(37\) − 14761.4i − 1.77266i −0.463057 − 0.886328i \(-0.653248\pi\)
0.463057 − 0.886328i \(-0.346752\pi\)
\(38\) 276.257 − 99.7083i 0.0310352 − 0.0112014i
\(39\) 1680.73 0.176945
\(40\) −6162.43 + 10477.4i −0.608979 + 1.03539i
\(41\) 11535.0 1.07166 0.535831 − 0.844325i \(-0.319998\pi\)
0.535831 + 0.844325i \(0.319998\pi\)
\(42\) 1791.50 − 646.598i 0.156709 − 0.0565602i
\(43\) 4921.61i 0.405916i 0.979187 + 0.202958i \(0.0650555\pi\)
−0.979187 + 0.202958i \(0.934944\pi\)
\(44\) −12812.9 − 15437.8i −0.997739 − 1.20214i
\(45\) − 5439.09i − 0.400401i
\(46\) −7775.59 − 21543.4i −0.541800 − 1.50114i
\(47\) 13463.0 0.888989 0.444494 − 0.895782i \(-0.353383\pi\)
0.444494 + 0.895782i \(0.353383\pi\)
\(48\) 9058.24 + 1697.93i 0.567467 + 0.106370i
\(49\) −15407.5 −0.916730
\(50\) 2657.93 + 7364.20i 0.150355 + 0.416582i
\(51\) − 17476.0i − 0.940841i
\(52\) 3816.57 + 4598.44i 0.195733 + 0.235832i
\(53\) 24873.6i 1.21632i 0.793813 + 0.608162i \(0.208093\pi\)
−0.793813 + 0.608162i \(0.791907\pi\)
\(54\) −3878.93 + 1400.01i −0.181020 + 0.0653350i
\(55\) −42099.1 −1.87657
\(56\) 5837.16 + 3433.21i 0.248732 + 0.146295i
\(57\) −467.274 −0.0190495
\(58\) 6790.01 − 2450.69i 0.265033 − 0.0956574i
\(59\) − 3936.59i − 0.147228i −0.997287 − 0.0736140i \(-0.976547\pi\)
0.997287 − 0.0736140i \(-0.0234533\pi\)
\(60\) 14881.2 − 12350.9i 0.533654 − 0.442917i
\(61\) 9930.43i 0.341699i 0.985297 + 0.170849i \(0.0546512\pi\)
−0.985297 + 0.170849i \(0.945349\pi\)
\(62\) −11105.4 − 30769.1i −0.366905 − 1.01657i
\(63\) −3030.22 −0.0961884
\(64\) 15923.7 + 28638.7i 0.485954 + 0.873985i
\(65\) 12540.0 0.368141
\(66\) 10836.2 + 30023.3i 0.306208 + 0.848396i
\(67\) 49755.9i 1.35412i 0.735928 + 0.677060i \(0.236746\pi\)
−0.735928 + 0.677060i \(0.763254\pi\)
\(68\) 47813.8 − 39684.0i 1.25395 − 1.04074i
\(69\) 36439.6i 0.921405i
\(70\) 13366.4 − 4824.28i 0.326039 − 0.117676i
\(71\) −23133.9 −0.544632 −0.272316 − 0.962208i \(-0.587790\pi\)
−0.272316 + 0.962208i \(0.587790\pi\)
\(72\) −12638.6 − 7433.55i −0.287320 − 0.168991i
\(73\) 72460.8 1.59146 0.795731 − 0.605650i \(-0.207087\pi\)
0.795731 + 0.605650i \(0.207087\pi\)
\(74\) −78544.1 + 28348.6i −1.66738 + 0.601800i
\(75\) − 12456.1i − 0.255700i
\(76\) −1061.07 − 1278.45i −0.0210723 − 0.0253892i
\(77\) 23454.2i 0.450810i
\(78\) −3227.76 − 8943.00i −0.0600710 − 0.166436i
\(79\) −32406.1 −0.584198 −0.292099 − 0.956388i \(-0.594354\pi\)
−0.292099 + 0.956388i \(0.594354\pi\)
\(80\) 67583.7 + 12668.3i 1.18064 + 0.221306i
\(81\) 6561.00 0.111111
\(82\) −22152.4 − 61376.5i −0.363819 − 1.00802i
\(83\) − 69116.9i − 1.10126i −0.834750 − 0.550629i \(-0.814388\pi\)
0.834750 − 0.550629i \(-0.185612\pi\)
\(84\) −6880.95 − 8290.60i −0.106402 − 0.128200i
\(85\) − 130389.i − 1.95746i
\(86\) 26187.4 − 9451.70i 0.381809 − 0.137805i
\(87\) −11484.9 −0.162679
\(88\) −57536.4 + 97823.7i −0.792020 + 1.34660i
\(89\) −65109.9 −0.871308 −0.435654 − 0.900114i \(-0.643483\pi\)
−0.435654 + 0.900114i \(0.643483\pi\)
\(90\) −28940.8 + 10445.5i −0.376621 + 0.135932i
\(91\) − 6986.27i − 0.0884386i
\(92\) −99697.7 + 82746.1i −1.22805 + 1.01924i
\(93\) 52044.2i 0.623972i
\(94\) −25854.9 − 71635.0i −0.301803 − 0.836191i
\(95\) −3486.34 −0.0396334
\(96\) −8361.36 − 51458.7i −0.0925974 − 0.569876i
\(97\) 112052. 1.20918 0.604589 − 0.796538i \(-0.293337\pi\)
0.604589 + 0.796538i \(0.293337\pi\)
\(98\) 29589.3 + 81981.5i 0.311221 + 0.862285i
\(99\) − 50782.8i − 0.520749i
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 24.6.d.a.13.3 ✓ 10
3.2 odd 2 72.6.d.d.37.8 10
4.3 odd 2 96.6.d.a.49.5 10
8.3 odd 2 96.6.d.a.49.6 10
8.5 even 2 inner 24.6.d.a.13.4 yes 10
12.11 even 2 288.6.d.d.145.2 10
16.3 odd 4 768.6.a.bc.1.5 5
16.5 even 4 768.6.a.bd.1.1 5
16.11 odd 4 768.6.a.bb.1.1 5
16.13 even 4 768.6.a.ba.1.5 5
24.5 odd 2 72.6.d.d.37.7 10
24.11 even 2 288.6.d.d.145.9 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.6.d.a.13.3 ✓ 10 1.1 even 1 trivial
24.6.d.a.13.4 yes 10 8.5 even 2 inner
72.6.d.d.37.7 10 24.5 odd 2
72.6.d.d.37.8 10 3.2 odd 2
96.6.d.a.49.5 10 4.3 odd 2
96.6.d.a.49.6 10 8.3 odd 2
288.6.d.d.145.2 10 12.11 even 2
288.6.d.d.145.9 10 24.11 even 2
768.6.a.ba.1.5 5 16.13 even 4
768.6.a.bb.1.1 5 16.11 odd 4
768.6.a.bc.1.5 5 16.3 odd 4
768.6.a.bd.1.1 5 16.5 even 4