Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [880,6,Mod(1,880)] 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("880.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(880, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 880.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,-100,0,90] 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: \(141.137761435\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 33x^{2} - 8x + 116 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-3.95665\) of defining polynomial
Character \(\chi\) \(=\) 880.1

$q$-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)
 
\(f(q)\) \(=\) \(q+16.3044 q^{3} -25.0000 q^{5} +125.436 q^{7} +22.8321 q^{9} +121.000 q^{11} +532.300 q^{13} -407.609 q^{15} -1373.09 q^{17} +554.639 q^{19} +2045.15 q^{21} -4250.72 q^{23} +625.000 q^{25} -3589.70 q^{27} -6973.40 q^{29} -3130.03 q^{31} +1972.83 q^{33} -3135.89 q^{35} +1384.70 q^{37} +8678.81 q^{39} +679.385 q^{41} +1721.06 q^{43} -570.803 q^{45} +15143.3 q^{47} -1072.91 q^{49} -22387.4 q^{51} -9544.44 q^{53} -3025.00 q^{55} +9043.04 q^{57} -27582.7 q^{59} -40527.5 q^{61} +2863.96 q^{63} -13307.5 q^{65} +58726.5 q^{67} -69305.2 q^{69} +42527.8 q^{71} +23753.0 q^{73} +10190.2 q^{75} +15177.7 q^{77} +78690.1 q^{79} -64075.9 q^{81} -52252.1 q^{83} +34327.3 q^{85} -113697. q^{87} +8156.09 q^{89} +66769.3 q^{91} -51033.2 q^{93} -13866.0 q^{95} +79010.1 q^{97} +2762.69 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 100 q^{5} + 90 q^{7} + 22 q^{9} + 484 q^{11} + 820 q^{13} - 3800 q^{17} + 3394 q^{19} - 4708 q^{21} + 3020 q^{23} + 2500 q^{25} - 5400 q^{27} - 5248 q^{29} - 4732 q^{31} - 2250 q^{35} + 10210 q^{37}+ \cdots + 2662 q^{99}+O(q^{100}) \) 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
\(3\) 16.3044 1.04593 0.522963 − 0.852356i \(-0.324827\pi\)
0.522963 + 0.852356i \(0.324827\pi\)
\(4\) 0 0
\(5\) −25.0000 −0.447214
\(6\) 0 0
\(7\) 125.436 0.967555 0.483778 − 0.875191i \(-0.339264\pi\)
0.483778 + 0.875191i \(0.339264\pi\)
\(8\) 0 0
\(9\) 22.8321 0.0939594
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) 0 0
\(13\) 532.300 0.873570 0.436785 − 0.899566i \(-0.356117\pi\)
0.436785 + 0.899566i \(0.356117\pi\)
\(14\) 0 0
\(15\) −407.609 −0.467752
\(16\) 0 0
\(17\) −1373.09 −1.15233 −0.576166 − 0.817333i \(-0.695452\pi\)
−0.576166 + 0.817333i \(0.695452\pi\)
\(18\) 0 0
\(19\) 554.639 0.352474 0.176237 − 0.984348i \(-0.443608\pi\)
0.176237 + 0.984348i \(0.443608\pi\)
\(20\) 0 0
\(21\) 2045.15 1.01199
\(22\) 0 0
\(23\) −4250.72 −1.67549 −0.837746 − 0.546060i \(-0.816127\pi\)
−0.837746 + 0.546060i \(0.816127\pi\)
\(24\) 0 0
\(25\) 625.000 0.200000
\(26\) 0 0
\(27\) −3589.70 −0.947651
\(28\) 0 0
\(29\) −6973.40 −1.53975 −0.769874 − 0.638196i \(-0.779681\pi\)
−0.769874 + 0.638196i \(0.779681\pi\)
\(30\) 0 0
\(31\) −3130.03 −0.584985 −0.292493 − 0.956268i \(-0.594485\pi\)
−0.292493 + 0.956268i \(0.594485\pi\)
\(32\) 0 0
\(33\) 1972.83 0.315358
\(34\) 0 0
\(35\) −3135.89 −0.432704
\(36\) 0 0
\(37\) 1384.70 0.166284 0.0831422 − 0.996538i \(-0.473504\pi\)
0.0831422 + 0.996538i \(0.473504\pi\)
\(38\) 0 0
\(39\) 8678.81 0.913689
\(40\) 0 0
\(41\) 679.385 0.0631185 0.0315592 − 0.999502i \(-0.489953\pi\)
0.0315592 + 0.999502i \(0.489953\pi\)
\(42\) 0 0
\(43\) 1721.06 0.141946 0.0709732 − 0.997478i \(-0.477390\pi\)
0.0709732 + 0.997478i \(0.477390\pi\)
\(44\) 0 0
\(45\) −570.803 −0.0420199
\(46\) 0 0
\(47\) 15143.3 0.999945 0.499972 − 0.866041i \(-0.333344\pi\)
0.499972 + 0.866041i \(0.333344\pi\)
\(48\) 0 0
\(49\) −1072.91 −0.0638372
\(50\) 0 0
\(51\) −22387.4 −1.20525
\(52\) 0 0
\(53\) −9544.44 −0.466724 −0.233362 − 0.972390i \(-0.574973\pi\)
−0.233362 + 0.972390i \(0.574973\pi\)
\(54\) 0 0
\(55\) −3025.00 −0.134840
\(56\) 0 0
\(57\) 9043.04 0.368661
\(58\) 0 0
\(59\) −27582.7 −1.03159 −0.515794 − 0.856713i \(-0.672503\pi\)
−0.515794 + 0.856713i \(0.672503\pi\)
\(60\) 0 0
\(61\) −40527.5 −1.39452 −0.697261 − 0.716817i \(-0.745598\pi\)
−0.697261 + 0.716817i \(0.745598\pi\)
\(62\) 0 0
\(63\) 2863.96 0.0909109
\(64\) 0 0
\(65\) −13307.5 −0.390672
\(66\) 0 0
\(67\) 58726.5 1.59826 0.799130 − 0.601159i \(-0.205294\pi\)
0.799130 + 0.601159i \(0.205294\pi\)
\(68\) 0 0
\(69\) −69305.2 −1.75244
\(70\) 0 0
\(71\) 42527.8 1.00121 0.500607 − 0.865675i \(-0.333110\pi\)
0.500607 + 0.865675i \(0.333110\pi\)
\(72\) 0 0
\(73\) 23753.0 0.521690 0.260845 − 0.965381i \(-0.415999\pi\)
0.260845 + 0.965381i \(0.415999\pi\)
\(74\) 0 0
\(75\) 10190.2 0.209185
\(76\) 0 0
\(77\) 15177.7 0.291729
\(78\) 0 0
\(79\) 78690.1 1.41857 0.709287 − 0.704919i \(-0.249017\pi\)
0.709287 + 0.704919i \(0.249017\pi\)
\(80\) 0 0
\(81\) −64075.9 −1.08513
\(82\) 0 0
\(83\) −52252.1 −0.832546 −0.416273 − 0.909240i \(-0.636664\pi\)
−0.416273 + 0.909240i \(0.636664\pi\)
\(84\) 0 0
\(85\) 34327.3 0.515338
\(86\) 0 0
\(87\) −113697. −1.61046
\(88\) 0 0
\(89\) 8156.09 0.109146 0.0545729 − 0.998510i \(-0.482620\pi\)
0.0545729 + 0.998510i \(0.482620\pi\)
\(90\) 0 0
\(91\) 66769.3 0.845227
\(92\) 0 0
\(93\) −51033.2 −0.611851
\(94\) 0 0
\(95\) −13866.0 −0.157631
\(96\) 0 0
\(97\) 79010.1 0.852615 0.426308 − 0.904578i \(-0.359814\pi\)
0.426308 + 0.904578i \(0.359814\pi\)
\(98\) 0 0
\(99\) 2762.69 0.0283298
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 880.6.a.n.1.4 4
4.3 odd 2 55.6.a.b.1.4 ✓ 4
12.11 even 2 495.6.a.g.1.1 4
20.3 even 4 275.6.b.d.199.2 8
20.7 even 4 275.6.b.d.199.7 8
20.19 odd 2 275.6.a.d.1.1 4
44.43 even 2 605.6.a.c.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.6.a.b.1.4 ✓ 4 4.3 odd 2
275.6.a.d.1.1 4 20.19 odd 2
275.6.b.d.199.2 8 20.3 even 4
275.6.b.d.199.7 8 20.7 even 4
495.6.a.g.1.1 4 12.11 even 2
605.6.a.c.1.1 4 44.43 even 2
880.6.a.n.1.4 4 1.1 even 1 trivial