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 1.2
Root \(1.74666\) 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)
 
\(f(q)\) \(=\) \(q-6.66534 q^{3} -25.0000 q^{5} +12.8802 q^{7} -198.573 q^{9} +121.000 q^{11} -485.167 q^{13} +166.634 q^{15} +266.661 q^{17} +149.702 q^{19} -85.8507 q^{21} +3213.11 q^{23} +625.000 q^{25} +2943.24 q^{27} +2948.81 q^{29} -2145.87 q^{31} -806.507 q^{33} -322.004 q^{35} -808.357 q^{37} +3233.80 q^{39} +10105.2 q^{41} -2763.15 q^{43} +4964.33 q^{45} -9973.36 q^{47} -16641.1 q^{49} -1777.39 q^{51} +7126.92 q^{53} -3025.00 q^{55} -997.814 q^{57} +33337.2 q^{59} -11871.1 q^{61} -2557.66 q^{63} +12129.2 q^{65} -4500.58 q^{67} -21416.5 q^{69} +45977.8 q^{71} -62039.1 q^{73} -4165.84 q^{75} +1558.50 q^{77} +57486.6 q^{79} +28635.6 q^{81} +90511.7 q^{83} -6666.53 q^{85} -19654.8 q^{87} -127861. q^{89} -6249.03 q^{91} +14303.0 q^{93} -3742.54 q^{95} +132338. q^{97} -24027.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 100 q^{5} + 90 q^{7} + 22 q^{9} + 484 q^{11} + 820 q^{13} - 3800 q^{17} + 3394 q^{19} - 4708 q^{21} + 3020 q^{23} + 2500 q^{25} - 5400 q^{27} - 5248 q^{29} - 4732 q^{31} - 2250 q^{35} + 10210 q^{37}+ \cdots + 2662 q^{99}+O(q^{100}) \) 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
\(3\) −6.66534 −0.427582 −0.213791 − 0.976879i \(-0.568581\pi\)
−0.213791 + 0.976879i \(0.568581\pi\)
\(4\) 0 0
\(5\) −25.0000 −0.447214
\(6\) 0 0
\(7\) 12.8802 0.0993520 0.0496760 − 0.998765i \(-0.484181\pi\)
0.0496760 + 0.998765i \(0.484181\pi\)
\(8\) 0 0
\(9\) −198.573 −0.817174
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) 0 0
\(13\) −485.167 −0.796219 −0.398110 − 0.917338i \(-0.630334\pi\)
−0.398110 + 0.917338i \(0.630334\pi\)
\(14\) 0 0
\(15\) 166.634 0.191220
\(16\) 0 0
\(17\) 266.661 0.223788 0.111894 − 0.993720i \(-0.464308\pi\)
0.111894 + 0.993720i \(0.464308\pi\)
\(18\) 0 0
\(19\) 149.702 0.0951356 0.0475678 − 0.998868i \(-0.484853\pi\)
0.0475678 + 0.998868i \(0.484853\pi\)
\(20\) 0 0
\(21\) −85.8507 −0.0424811
\(22\) 0 0
\(23\) 3213.11 1.26650 0.633251 − 0.773946i \(-0.281720\pi\)
0.633251 + 0.773946i \(0.281720\pi\)
\(24\) 0 0
\(25\) 625.000 0.200000
\(26\) 0 0
\(27\) 2943.24 0.776991
\(28\) 0 0
\(29\) 2948.81 0.651106 0.325553 − 0.945524i \(-0.394450\pi\)
0.325553 + 0.945524i \(0.394450\pi\)
\(30\) 0 0
\(31\) −2145.87 −0.401051 −0.200525 − 0.979689i \(-0.564265\pi\)
−0.200525 + 0.979689i \(0.564265\pi\)
\(32\) 0 0
\(33\) −806.507 −0.128921
\(34\) 0 0
\(35\) −322.004 −0.0444315
\(36\) 0 0
\(37\) −808.357 −0.0970731 −0.0485366 − 0.998821i \(-0.515456\pi\)
−0.0485366 + 0.998821i \(0.515456\pi\)
\(38\) 0 0
\(39\) 3233.80 0.340449
\(40\) 0 0
\(41\) 10105.2 0.938829 0.469414 − 0.882978i \(-0.344465\pi\)
0.469414 + 0.882978i \(0.344465\pi\)
\(42\) 0 0
\(43\) −2763.15 −0.227894 −0.113947 − 0.993487i \(-0.536349\pi\)
−0.113947 + 0.993487i \(0.536349\pi\)
\(44\) 0 0
\(45\) 4964.33 0.365451
\(46\) 0 0
\(47\) −9973.36 −0.658562 −0.329281 − 0.944232i \(-0.606806\pi\)
−0.329281 + 0.944232i \(0.606806\pi\)
\(48\) 0 0
\(49\) −16641.1 −0.990129
\(50\) 0 0
\(51\) −1777.39 −0.0956878
\(52\) 0 0
\(53\) 7126.92 0.348508 0.174254 − 0.984701i \(-0.444249\pi\)
0.174254 + 0.984701i \(0.444249\pi\)
\(54\) 0 0
\(55\) −3025.00 −0.134840
\(56\) 0 0
\(57\) −997.814 −0.0406783
\(58\) 0 0
\(59\) 33337.2 1.24681 0.623403 − 0.781901i \(-0.285750\pi\)
0.623403 + 0.781901i \(0.285750\pi\)
\(60\) 0 0
\(61\) −11871.1 −0.408476 −0.204238 − 0.978921i \(-0.565472\pi\)
−0.204238 + 0.978921i \(0.565472\pi\)
\(62\) 0 0
\(63\) −2557.66 −0.0811878
\(64\) 0 0
\(65\) 12129.2 0.356080
\(66\) 0 0
\(67\) −4500.58 −0.122485 −0.0612423 − 0.998123i \(-0.519506\pi\)
−0.0612423 + 0.998123i \(0.519506\pi\)
\(68\) 0 0
\(69\) −21416.5 −0.541533
\(70\) 0 0
\(71\) 45977.8 1.08244 0.541218 − 0.840882i \(-0.317963\pi\)
0.541218 + 0.840882i \(0.317963\pi\)
\(72\) 0 0
\(73\) −62039.1 −1.36257 −0.681284 − 0.732019i \(-0.738578\pi\)
−0.681284 + 0.732019i \(0.738578\pi\)
\(74\) 0 0
\(75\) −4165.84 −0.0855164
\(76\) 0 0
\(77\) 1558.50 0.0299557
\(78\) 0 0
\(79\) 57486.6 1.03633 0.518166 − 0.855280i \(-0.326615\pi\)
0.518166 + 0.855280i \(0.326615\pi\)
\(80\) 0 0
\(81\) 28635.6 0.484946
\(82\) 0 0
\(83\) 90511.7 1.44215 0.721074 − 0.692858i \(-0.243649\pi\)
0.721074 + 0.692858i \(0.243649\pi\)
\(84\) 0 0
\(85\) −6666.53 −0.100081
\(86\) 0 0
\(87\) −19654.8 −0.278401
\(88\) 0 0
\(89\) −127861. −1.71105 −0.855524 − 0.517764i \(-0.826765\pi\)
−0.855524 + 0.517764i \(0.826765\pi\)
\(90\) 0 0
\(91\) −6249.03 −0.0791059
\(92\) 0 0
\(93\) 14303.0 0.171482
\(94\) 0 0
\(95\) −3742.54 −0.0425459
\(96\) 0 0
\(97\) 132338. 1.42809 0.714046 − 0.700099i \(-0.246861\pi\)
0.714046 + 0.700099i \(0.246861\pi\)
\(98\) 0 0
\(99\) −24027.4 −0.246387
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.1.2 4
4.3 odd 2 55.6.a.b.1.3 ✓ 4
12.11 even 2 495.6.a.g.1.2 4
20.3 even 4 275.6.b.d.199.4 8
20.7 even 4 275.6.b.d.199.5 8
20.19 odd 2 275.6.a.d.1.2 4
44.43 even 2 605.6.a.c.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.6.a.b.1.3 ✓ 4 4.3 odd 2
275.6.a.d.1.2 4 20.19 odd 2
275.6.b.d.199.4 8 20.3 even 4
275.6.b.d.199.5 8 20.7 even 4
495.6.a.g.1.2 4 12.11 even 2
605.6.a.c.1.2 4 44.43 even 2
880.6.a.n.1.2 4 1.1 even 1 trivial