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 = [5,0,-4,0,-25,0,3] 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: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 115)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-3.93900\) 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)
 
\(f(q)\) \(=\) \(q+3.85751 q^{3} -5.00000 q^{5} +23.5932 q^{7} -12.1196 q^{9} +58.2387 q^{11} +68.5380 q^{13} -19.2876 q^{15} +101.409 q^{17} +7.02478 q^{19} +91.0112 q^{21} +23.0000 q^{23} +25.0000 q^{25} -150.904 q^{27} +206.062 q^{29} -54.8201 q^{31} +224.657 q^{33} -117.966 q^{35} -241.694 q^{37} +264.386 q^{39} -122.510 q^{41} +320.358 q^{43} +60.5979 q^{45} -107.032 q^{47} +213.641 q^{49} +391.186 q^{51} +127.122 q^{53} -291.194 q^{55} +27.0982 q^{57} -693.766 q^{59} -899.524 q^{61} -285.940 q^{63} -342.690 q^{65} -110.435 q^{67} +88.7228 q^{69} -225.992 q^{71} +746.207 q^{73} +96.4378 q^{75} +1374.04 q^{77} +1094.02 q^{79} -254.887 q^{81} -1289.81 q^{83} -507.045 q^{85} +794.887 q^{87} +1208.07 q^{89} +1617.03 q^{91} -211.469 q^{93} -35.1239 q^{95} +903.864 q^{97} -705.829 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 4 q^{3} - 25 q^{5} + 3 q^{7} + 77 q^{9} - 23 q^{11} + 132 q^{13} + 20 q^{15} + 23 q^{17} + 161 q^{19} - 60 q^{21} + 115 q^{23} + 125 q^{25} - 577 q^{27} + 401 q^{29} - 32 q^{31} + 189 q^{33} - 15 q^{35}+ \cdots - 2021 q^{99}+O(q^{100}) \) 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
\(3\) 3.85751 0.742379 0.371189 − 0.928557i \(-0.378950\pi\)
0.371189 + 0.928557i \(0.378950\pi\)
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 23.5932 1.27392 0.636958 − 0.770899i \(-0.280193\pi\)
0.636958 + 0.770899i \(0.280193\pi\)
\(8\) 0 0
\(9\) −12.1196 −0.448874
\(10\) 0 0
\(11\) 58.2387 1.59633 0.798165 − 0.602439i \(-0.205804\pi\)
0.798165 + 0.602439i \(0.205804\pi\)
\(12\) 0 0
\(13\) 68.5380 1.46223 0.731116 − 0.682253i \(-0.239000\pi\)
0.731116 + 0.682253i \(0.239000\pi\)
\(14\) 0 0
\(15\) −19.2876 −0.332002
\(16\) 0 0
\(17\) 101.409 1.44678 0.723391 − 0.690439i \(-0.242583\pi\)
0.723391 + 0.690439i \(0.242583\pi\)
\(18\) 0 0
\(19\) 7.02478 0.0848208 0.0424104 − 0.999100i \(-0.486496\pi\)
0.0424104 + 0.999100i \(0.486496\pi\)
\(20\) 0 0
\(21\) 91.0112 0.945728
\(22\) 0 0
\(23\) 23.0000 0.208514
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −150.904 −1.07561
\(28\) 0 0
\(29\) 206.062 1.31947 0.659737 − 0.751497i \(-0.270668\pi\)
0.659737 + 0.751497i \(0.270668\pi\)
\(30\) 0 0
\(31\) −54.8201 −0.317612 −0.158806 − 0.987310i \(-0.550764\pi\)
−0.158806 + 0.987310i \(0.550764\pi\)
\(32\) 0 0
\(33\) 224.657 1.18508
\(34\) 0 0
\(35\) −117.966 −0.569712
\(36\) 0 0
\(37\) −241.694 −1.07390 −0.536948 − 0.843615i \(-0.680423\pi\)
−0.536948 + 0.843615i \(0.680423\pi\)
\(38\) 0 0
\(39\) 264.386 1.08553
\(40\) 0 0
\(41\) −122.510 −0.466653 −0.233327 − 0.972398i \(-0.574961\pi\)
−0.233327 + 0.972398i \(0.574961\pi\)
\(42\) 0 0
\(43\) 320.358 1.13614 0.568071 − 0.822979i \(-0.307690\pi\)
0.568071 + 0.822979i \(0.307690\pi\)
\(44\) 0 0
\(45\) 60.5979 0.200742
\(46\) 0 0
\(47\) −107.032 −0.332175 −0.166088 − 0.986111i \(-0.553113\pi\)
−0.166088 + 0.986111i \(0.553113\pi\)
\(48\) 0 0
\(49\) 213.641 0.622860
\(50\) 0 0
\(51\) 391.186 1.07406
\(52\) 0 0
\(53\) 127.122 0.329463 0.164732 − 0.986338i \(-0.447324\pi\)
0.164732 + 0.986338i \(0.447324\pi\)
\(54\) 0 0
\(55\) −291.194 −0.713901
\(56\) 0 0
\(57\) 27.0982 0.0629692
\(58\) 0 0
\(59\) −693.766 −1.53086 −0.765429 − 0.643521i \(-0.777473\pi\)
−0.765429 + 0.643521i \(0.777473\pi\)
\(60\) 0 0
\(61\) −899.524 −1.88807 −0.944034 − 0.329847i \(-0.893003\pi\)
−0.944034 + 0.329847i \(0.893003\pi\)
\(62\) 0 0
\(63\) −285.940 −0.571827
\(64\) 0 0
\(65\) −342.690 −0.653930
\(66\) 0 0
\(67\) −110.435 −0.201369 −0.100685 − 0.994918i \(-0.532103\pi\)
−0.100685 + 0.994918i \(0.532103\pi\)
\(68\) 0 0
\(69\) 88.7228 0.154797
\(70\) 0 0
\(71\) −225.992 −0.377752 −0.188876 − 0.982001i \(-0.560484\pi\)
−0.188876 + 0.982001i \(0.560484\pi\)
\(72\) 0 0
\(73\) 746.207 1.19640 0.598198 − 0.801348i \(-0.295884\pi\)
0.598198 + 0.801348i \(0.295884\pi\)
\(74\) 0 0
\(75\) 96.4378 0.148476
\(76\) 0 0
\(77\) 1374.04 2.03359
\(78\) 0 0
\(79\) 1094.02 1.55807 0.779033 − 0.626983i \(-0.215710\pi\)
0.779033 + 0.626983i \(0.215710\pi\)
\(80\) 0 0
\(81\) −254.887 −0.349639
\(82\) 0 0
\(83\) −1289.81 −1.70572 −0.852862 − 0.522137i \(-0.825135\pi\)
−0.852862 + 0.522137i \(0.825135\pi\)
\(84\) 0 0
\(85\) −507.045 −0.647020
\(86\) 0 0
\(87\) 794.887 0.979550
\(88\) 0 0
\(89\) 1208.07 1.43883 0.719413 − 0.694582i \(-0.244411\pi\)
0.719413 + 0.694582i \(0.244411\pi\)
\(90\) 0 0
\(91\) 1617.03 1.86276
\(92\) 0 0
\(93\) −211.469 −0.235788
\(94\) 0 0
\(95\) −35.1239 −0.0379330
\(96\) 0 0
\(97\) 903.864 0.946119 0.473059 − 0.881031i \(-0.343150\pi\)
0.473059 + 0.881031i \(0.343150\pi\)
\(98\) 0 0
\(99\) −705.829 −0.716551
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.n.1.4 5
4.3 odd 2 115.4.a.e.1.1 ✓ 5
12.11 even 2 1035.4.a.k.1.5 5
20.3 even 4 575.4.b.i.24.8 10
20.7 even 4 575.4.b.i.24.3 10
20.19 odd 2 575.4.a.j.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.e.1.1 ✓ 5 4.3 odd 2
575.4.a.j.1.5 5 20.19 odd 2
575.4.b.i.24.3 10 20.7 even 4
575.4.b.i.24.8 10 20.3 even 4
1035.4.a.k.1.5 5 12.11 even 2
1840.4.a.n.1.4 5 1.1 even 1 trivial