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

$q$-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 \(-0.794928\pi\)
0.799549 − 0.600601i \(-0.205072\pi\)
\(6\) 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 \(-0.714647\pi\)
0.624376 − 0.781124i \(-0.285353\pi\)
\(12\) −183.933 + 221.614i −0.368728 + 0.444267i
\(13\) 186.748i 0.306477i 0.988189 + 0.153238i \(0.0489703\pi\)
−0.988189 + 0.153238i \(0.951030\pi\)
\(14\) −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 \(-0.994748\pi\)
0.999864 − 0.0164974i \(-0.00525152\pi\)
\(20\) −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 \(-0.955006\pi\)
0.990026 − 0.140884i \(-0.0449944\pi\)
\(30\) 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 \(0.346752\pi\)
−0.463057 + 0.886328i \(0.653248\pi\)
\(38\) 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 \(-0.934944\pi\)
0.979187 − 0.202958i \(-0.0650555\pi\)
\(44\) −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 \(-0.791907\pi\)
0.793813 − 0.608162i \(-0.208093\pi\)
\(54\) −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 \(0.0234533\pi\)
−0.997287 + 0.0736140i \(0.976547\pi\)
\(60\) 14881.2 + 12350.9i 0.533654 + 0.442917i
\(61\) − 9930.43i − 0.341699i −0.985297 − 0.170849i \(-0.945349\pi\)
0.985297 − 0.170849i \(-0.0546512\pi\)
\(62\) −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 \(-0.763254\pi\)
0.735928 − 0.677060i \(-0.236746\pi\)
\(68\) 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 \(0.185612\pi\)
−0.834750 + 0.550629i \(0.814388\pi\)
\(84\) −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.13.4 yes 10
3.2 odd 2 72.6.d.d.37.7 10
4.3 odd 2 96.6.d.a.49.6 10
8.3 odd 2 96.6.d.a.49.5 10
8.5 even 2 inner 24.6.d.a.13.3 ✓ 10
12.11 even 2 288.6.d.d.145.9 10
16.3 odd 4 768.6.a.bb.1.1 5
16.5 even 4 768.6.a.ba.1.5 5
16.11 odd 4 768.6.a.bc.1.5 5
16.13 even 4 768.6.a.bd.1.1 5
24.5 odd 2 72.6.d.d.37.8 10
24.11 even 2 288.6.d.d.145.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.6.d.a.13.3 ✓ 10 8.5 even 2 inner
24.6.d.a.13.4 yes 10 1.1 even 1 trivial
72.6.d.d.37.7 10 3.2 odd 2
72.6.d.d.37.8 10 24.5 odd 2
96.6.d.a.49.5 10 8.3 odd 2
96.6.d.a.49.6 10 4.3 odd 2
288.6.d.d.145.2 10 24.11 even 2
288.6.d.d.145.9 10 12.11 even 2
768.6.a.ba.1.5 5 16.5 even 4
768.6.a.bb.1.1 5 16.3 odd 4
768.6.a.bc.1.5 5 16.11 odd 4
768.6.a.bd.1.1 5 16.13 even 4