Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3420,3,Mod(2089,3420)] 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("3420.2089"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3420, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 3420.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(93.1882504112\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-6}, \sqrt{14})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2089.1
Root \(-1.87083 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 3420.2089
Dual form 3420.3.h.c.2089.3

$q$-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)
 
\(f(q)\) \(=\) \(q+5.00000 q^{5} -9.79796i q^{7} -4.00000 q^{11} -11.2250 q^{13} +19.5959i q^{17} +(-5.00000 + 18.3303i) q^{19} -9.79796i q^{23} +25.0000 q^{25} -36.6606i q^{29} +36.6606i q^{31} -48.9898i q^{35} -33.6749 q^{37} +36.6606i q^{41} +68.5857i q^{43} -9.79796i q^{47} -47.0000 q^{49} -56.1249 q^{53} -20.0000 q^{55} +73.3212i q^{59} +100.000 q^{61} -56.1249 q^{65} -11.2250 q^{67} -36.6606i q^{71} +19.5959i q^{73} +39.1918i q^{77} +109.982i q^{79} -29.3939i q^{83} +97.9796i q^{85} +146.642i q^{89} +109.982i q^{91} +(-25.0000 + 91.6515i) q^{95} -123.475 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 20 q^{5} - 16 q^{11} - 20 q^{19} + 100 q^{25} - 188 q^{49} - 80 q^{55} + 400 q^{61} - 100 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times\).

\(n\) \(1711\) \(1901\) \(2737\) \(3061\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 5.00000 1.00000
\(6\) 0 0
\(7\) − 9.79796i − 1.39971i −0.714286 − 0.699854i \(-0.753248\pi\)
0.714286 − 0.699854i \(-0.246752\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.00000 −0.363636 −0.181818 − 0.983332i \(-0.558198\pi\)
−0.181818 + 0.983332i \(0.558198\pi\)
\(12\) 0 0
\(13\) −11.2250 −0.863459 −0.431730 − 0.902003i \(-0.642097\pi\)
−0.431730 + 0.902003i \(0.642097\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 19.5959i 1.15270i 0.817203 + 0.576351i \(0.195524\pi\)
−0.817203 + 0.576351i \(0.804476\pi\)
\(18\) 0 0
\(19\) −5.00000 + 18.3303i −0.263158 + 0.964753i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 9.79796i − 0.425998i −0.977052 − 0.212999i \(-0.931677\pi\)
0.977052 − 0.212999i \(-0.0683231\pi\)
\(24\) 0 0
\(25\) 25.0000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 36.6606i − 1.26416i −0.774904 − 0.632079i \(-0.782202\pi\)
0.774904 − 0.632079i \(-0.217798\pi\)
\(30\) 0 0
\(31\) 36.6606i 1.18260i 0.806452 + 0.591300i \(0.201385\pi\)
−0.806452 + 0.591300i \(0.798615\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 48.9898i − 1.39971i
\(36\) 0 0
\(37\) −33.6749 −0.910133 −0.455066 − 0.890457i \(-0.650384\pi\)
−0.455066 + 0.890457i \(0.650384\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 36.6606i 0.894161i 0.894494 + 0.447081i \(0.147536\pi\)
−0.894494 + 0.447081i \(0.852464\pi\)
\(42\) 0 0
\(43\) 68.5857i 1.59502i 0.603308 + 0.797508i \(0.293849\pi\)
−0.603308 + 0.797508i \(0.706151\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 9.79796i − 0.208467i −0.994553 − 0.104234i \(-0.966761\pi\)
0.994553 − 0.104234i \(-0.0332390\pi\)
\(48\) 0 0
\(49\) −47.0000 −0.959184
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −56.1249 −1.05896 −0.529480 − 0.848323i \(-0.677613\pi\)
−0.529480 + 0.848323i \(0.677613\pi\)
\(54\) 0 0
\(55\) −20.0000 −0.363636
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 73.3212i 1.24273i 0.783520 + 0.621366i \(0.213422\pi\)
−0.783520 + 0.621366i \(0.786578\pi\)
\(60\) 0 0
\(61\) 100.000 1.63934 0.819672 − 0.572833i \(-0.194156\pi\)
0.819672 + 0.572833i \(0.194156\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −56.1249 −0.863459
\(66\) 0 0
\(67\) −11.2250 −0.167537 −0.0837684 − 0.996485i \(-0.526696\pi\)
−0.0837684 + 0.996485i \(0.526696\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 36.6606i − 0.516347i −0.966099 − 0.258173i \(-0.916879\pi\)
0.966099 − 0.258173i \(-0.0831205\pi\)
\(72\) 0 0
\(73\) 19.5959i 0.268437i 0.990952 + 0.134219i \(0.0428524\pi\)
−0.990952 + 0.134219i \(0.957148\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 39.1918i 0.508985i
\(78\) 0 0
\(79\) 109.982i 1.39217i 0.717957 + 0.696087i \(0.245077\pi\)
−0.717957 + 0.696087i \(0.754923\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 29.3939i − 0.354143i −0.984198 − 0.177072i \(-0.943338\pi\)
0.984198 − 0.177072i \(-0.0566624\pi\)
\(84\) 0 0
\(85\) 97.9796i 1.15270i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 146.642i 1.64767i 0.566831 + 0.823834i \(0.308169\pi\)
−0.566831 + 0.823834i \(0.691831\pi\)
\(90\) 0 0
\(91\) 109.982i 1.20859i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −25.0000 + 91.6515i −0.263158 + 0.964753i
\(96\) 0 0
\(97\) −123.475 −1.27293 −0.636467 − 0.771304i \(-0.719605\pi\)
−0.636467 + 0.771304i \(0.719605\pi\)
\(98\) 0 0
\(99\) 0 0
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 3420.3.h.c.2089.1 4
3.2 odd 2 380.3.g.a.189.1 ✓ 4
5.4 even 2 inner 3420.3.h.c.2089.4 4
15.2 even 4 1900.3.e.c.1101.4 4
15.8 even 4 1900.3.e.c.1101.1 4
15.14 odd 2 380.3.g.a.189.4 yes 4
19.18 odd 2 inner 3420.3.h.c.2089.2 4
57.56 even 2 380.3.g.a.189.3 yes 4
95.94 odd 2 inner 3420.3.h.c.2089.3 4
285.113 odd 4 1900.3.e.c.1101.3 4
285.227 odd 4 1900.3.e.c.1101.2 4
285.284 even 2 380.3.g.a.189.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.g.a.189.1 ✓ 4 3.2 odd 2
380.3.g.a.189.2 yes 4 285.284 even 2
380.3.g.a.189.3 yes 4 57.56 even 2
380.3.g.a.189.4 yes 4 15.14 odd 2
1900.3.e.c.1101.1 4 15.8 even 4
1900.3.e.c.1101.2 4 285.227 odd 4
1900.3.e.c.1101.3 4 285.113 odd 4
1900.3.e.c.1101.4 4 15.2 even 4
3420.3.h.c.2089.1 4 1.1 even 1 trivial
3420.3.h.c.2089.2 4 19.18 odd 2 inner
3420.3.h.c.2089.3 4 95.94 odd 2 inner
3420.3.h.c.2089.4 4 5.4 even 2 inner