Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1840,4,Mod(1,1840)] 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("1840.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1840.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,1,0,-50,0,-28] 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: \(108.563514411\)
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: \( x^{10} - x^{9} - 204 x^{8} + 42 x^{7} + 12958 x^{6} + 5872 x^{5} - 259871 x^{4} - 149461 x^{3} + \cdots - 43712 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(4.78870\) of defining polynomial
Character \(\chi\) \(=\) 1840.1

$q$-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)
 
\(f(q)\) \(=\) \(q+4.78870 q^{3} -5.00000 q^{5} -0.156416 q^{7} -4.06831 q^{9} +57.4097 q^{11} -17.3244 q^{13} -23.9435 q^{15} -37.0472 q^{17} +121.841 q^{19} -0.749032 q^{21} -23.0000 q^{23} +25.0000 q^{25} -148.777 q^{27} +89.1452 q^{29} +76.6700 q^{31} +274.918 q^{33} +0.782082 q^{35} -183.023 q^{37} -82.9616 q^{39} -237.272 q^{41} +342.074 q^{43} +20.3415 q^{45} +245.602 q^{47} -342.976 q^{49} -177.408 q^{51} +197.588 q^{53} -287.048 q^{55} +583.460 q^{57} +311.314 q^{59} +130.256 q^{61} +0.636350 q^{63} +86.6222 q^{65} +293.742 q^{67} -110.140 q^{69} -765.950 q^{71} +699.714 q^{73} +119.718 q^{75} -8.97981 q^{77} -691.925 q^{79} -602.605 q^{81} +952.614 q^{83} +185.236 q^{85} +426.890 q^{87} -20.9162 q^{89} +2.70983 q^{91} +367.150 q^{93} -609.204 q^{95} +455.261 q^{97} -233.560 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{3} - 50 q^{5} - 28 q^{7} + 139 q^{9} + 14 q^{11} + 11 q^{13} - 5 q^{15} + 68 q^{17} - 114 q^{19} - 232 q^{21} - 230 q^{23} + 250 q^{25} + 433 q^{27} - 273 q^{29} + 129 q^{31} + 98 q^{33} + 140 q^{35}+ \cdots - 4356 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 4.78870 0.921587 0.460793 − 0.887507i \(-0.347565\pi\)
0.460793 + 0.887507i \(0.347565\pi\)
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) −0.156416 −0.00844569 −0.00422284 − 0.999991i \(-0.501344\pi\)
−0.00422284 + 0.999991i \(0.501344\pi\)
\(8\) 0 0
\(9\) −4.06831 −0.150678
\(10\) 0 0
\(11\) 57.4097 1.57361 0.786803 − 0.617204i \(-0.211735\pi\)
0.786803 + 0.617204i \(0.211735\pi\)
\(12\) 0 0
\(13\) −17.3244 −0.369610 −0.184805 − 0.982775i \(-0.559165\pi\)
−0.184805 + 0.982775i \(0.559165\pi\)
\(14\) 0 0
\(15\) −23.9435 −0.412146
\(16\) 0 0
\(17\) −37.0472 −0.528545 −0.264272 − 0.964448i \(-0.585132\pi\)
−0.264272 + 0.964448i \(0.585132\pi\)
\(18\) 0 0
\(19\) 121.841 1.47117 0.735584 − 0.677433i \(-0.236908\pi\)
0.735584 + 0.677433i \(0.236908\pi\)
\(20\) 0 0
\(21\) −0.749032 −0.00778343
\(22\) 0 0
\(23\) −23.0000 −0.208514
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −148.777 −1.06045
\(28\) 0 0
\(29\) 89.1452 0.570822 0.285411 − 0.958405i \(-0.407870\pi\)
0.285411 + 0.958405i \(0.407870\pi\)
\(30\) 0 0
\(31\) 76.6700 0.444205 0.222102 − 0.975023i \(-0.428708\pi\)
0.222102 + 0.975023i \(0.428708\pi\)
\(32\) 0 0
\(33\) 274.918 1.45021
\(34\) 0 0
\(35\) 0.782082 0.00377703
\(36\) 0 0
\(37\) −183.023 −0.813212 −0.406606 − 0.913604i \(-0.633288\pi\)
−0.406606 + 0.913604i \(0.633288\pi\)
\(38\) 0 0
\(39\) −82.9616 −0.340628
\(40\) 0 0
\(41\) −237.272 −0.903796 −0.451898 − 0.892070i \(-0.649253\pi\)
−0.451898 + 0.892070i \(0.649253\pi\)
\(42\) 0 0
\(43\) 342.074 1.21316 0.606578 − 0.795024i \(-0.292542\pi\)
0.606578 + 0.795024i \(0.292542\pi\)
\(44\) 0 0
\(45\) 20.3415 0.0673853
\(46\) 0 0
\(47\) 245.602 0.762229 0.381114 − 0.924528i \(-0.375540\pi\)
0.381114 + 0.924528i \(0.375540\pi\)
\(48\) 0 0
\(49\) −342.976 −0.999929
\(50\) 0 0
\(51\) −177.408 −0.487100
\(52\) 0 0
\(53\) 197.588 0.512091 0.256045 − 0.966665i \(-0.417580\pi\)
0.256045 + 0.966665i \(0.417580\pi\)
\(54\) 0 0
\(55\) −287.048 −0.703738
\(56\) 0 0
\(57\) 583.460 1.35581
\(58\) 0 0
\(59\) 311.314 0.686943 0.343472 − 0.939163i \(-0.388397\pi\)
0.343472 + 0.939163i \(0.388397\pi\)
\(60\) 0 0
\(61\) 130.256 0.273402 0.136701 − 0.990612i \(-0.456350\pi\)
0.136701 + 0.990612i \(0.456350\pi\)
\(62\) 0 0
\(63\) 0.636350 0.00127258
\(64\) 0 0
\(65\) 86.6222 0.165295
\(66\) 0 0
\(67\) 293.742 0.535616 0.267808 − 0.963472i \(-0.413701\pi\)
0.267808 + 0.963472i \(0.413701\pi\)
\(68\) 0 0
\(69\) −110.140 −0.192164
\(70\) 0 0
\(71\) −765.950 −1.28030 −0.640152 − 0.768248i \(-0.721129\pi\)
−0.640152 + 0.768248i \(0.721129\pi\)
\(72\) 0 0
\(73\) 699.714 1.12185 0.560927 − 0.827865i \(-0.310445\pi\)
0.560927 + 0.827865i \(0.310445\pi\)
\(74\) 0 0
\(75\) 119.718 0.184317
\(76\) 0 0
\(77\) −8.97981 −0.0132902
\(78\) 0 0
\(79\) −691.925 −0.985414 −0.492707 − 0.870195i \(-0.663992\pi\)
−0.492707 + 0.870195i \(0.663992\pi\)
\(80\) 0 0
\(81\) −602.605 −0.826618
\(82\) 0 0
\(83\) 952.614 1.25979 0.629897 − 0.776679i \(-0.283097\pi\)
0.629897 + 0.776679i \(0.283097\pi\)
\(84\) 0 0
\(85\) 185.236 0.236372
\(86\) 0 0
\(87\) 426.890 0.526062
\(88\) 0 0
\(89\) −20.9162 −0.0249114 −0.0124557 − 0.999922i \(-0.503965\pi\)
−0.0124557 + 0.999922i \(0.503965\pi\)
\(90\) 0 0
\(91\) 2.70983 0.00312161
\(92\) 0 0
\(93\) 367.150 0.409373
\(94\) 0 0
\(95\) −609.204 −0.657927
\(96\) 0 0
\(97\) 455.261 0.476544 0.238272 − 0.971199i \(-0.423419\pi\)
0.238272 + 0.971199i \(0.423419\pi\)
\(98\) 0 0
\(99\) −233.560 −0.237108
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 1840.4.a.bb.1.8 10
4.3 odd 2 920.4.a.g.1.3 ✓ 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.4.a.g.1.3 ✓ 10 4.3 odd 2
1840.4.a.bb.1.8 10 1.1 even 1 trivial