↑

Properties

Label 880.3.j.c.241.1
Level $880$
Weight $3$
Character 880.241
Analytic conductor $23.978$
Analytic rank $0$
Dimension $8$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [880,3,Mod(241,880)] 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("880.241"); S:= CuspForms(chi, 3); 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, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 880.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,8,0,0,0,0,0,40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(23.9782632637\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.4956160000.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 19x^{4} - 30x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 110)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 241.1
Root \(-1.09132 - 0.437016i\) of defining polynomial
Character \(\chi\) \(=\) 880.241
Dual form 880.3.j.c.241.2

$q$-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)
 
\(f(q)\) \(=\) \(q-4.76766 q^{3} -2.23607 q^{5} -9.73055i q^{7} +13.7306 q^{9} +(-10.0795 - 4.40491i) q^{11} -16.6335i q^{13} +10.6608 q^{15} -12.8351i q^{17} +4.69221i q^{19} +46.3920i q^{21} +32.4557 q^{23} +5.00000 q^{25} -22.5538 q^{27} -29.4791i q^{29} +23.4043 q^{31} +(48.0557 + 21.0011i) q^{33} +21.7582i q^{35} -11.1821 q^{37} +79.3026i q^{39} -69.0178i q^{41} -65.2097i q^{43} -30.7025 q^{45} -45.7097 q^{47} -45.6836 q^{49} +61.1934i q^{51} -1.00392 q^{53} +(22.5385 + 9.84968i) q^{55} -22.3709i q^{57} -94.0820 q^{59} +113.061i q^{61} -133.606i q^{63} +37.1935i q^{65} +13.0292 q^{67} -154.738 q^{69} -36.0057 q^{71} +83.2194i q^{73} -23.8383 q^{75} +(-42.8622 + 98.0793i) q^{77} -0.104581i q^{79} -16.0465 q^{81} +41.9202i q^{83} +28.7002i q^{85} +140.546i q^{87} -145.516 q^{89} -161.853 q^{91} -111.584 q^{93} -10.4921i q^{95} +89.1978 q^{97} +(-138.398 - 60.4820i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} + 40 q^{9} + 40 q^{15} + 136 q^{23} + 40 q^{25} - 64 q^{27} + 64 q^{31} + 88 q^{33} - 48 q^{37} - 152 q^{47} - 232 q^{49} + 352 q^{53} - 80 q^{59} - 24 q^{67} + 112 q^{69} + 256 q^{71} + 40 q^{75}+ \cdots - 704 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/880\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(177\) \(321\) \(661\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(3\) −4.76766 −1.58922 −0.794610 − 0.607120i \(-0.792325\pi\)
−0.794610 + 0.607120i \(0.792325\pi\)
\(4\) 0 0
\(5\) −2.23607 −0.447214
\(6\) 0 0
\(7\) − 9.73055i − 1.39008i −0.718972 − 0.695039i \(-0.755387\pi\)
0.718972 − 0.695039i \(-0.244613\pi\)
\(8\) 0 0
\(9\) 13.7306 1.52562
\(10\) 0 0
\(11\) −10.0795 − 4.40491i −0.916320 − 0.400447i
\(12\) 0 0
\(13\) − 16.6335i − 1.27950i −0.768585 − 0.639748i \(-0.779039\pi\)
0.768585 − 0.639748i \(-0.220961\pi\)
\(14\) 0 0
\(15\) 10.6608 0.710721
\(16\) 0 0
\(17\) − 12.8351i − 0.755007i −0.926008 − 0.377503i \(-0.876783\pi\)
0.926008 − 0.377503i \(-0.123217\pi\)
\(18\) 0 0
\(19\) 4.69221i 0.246958i 0.992347 + 0.123479i \(0.0394052\pi\)
−0.992347 + 0.123479i \(0.960595\pi\)
\(20\) 0 0
\(21\) 46.3920i 2.20914i
\(22\) 0 0
\(23\) 32.4557 1.41112 0.705558 − 0.708652i \(-0.250696\pi\)
0.705558 + 0.708652i \(0.250696\pi\)
\(24\) 0 0
\(25\) 5.00000 0.200000
\(26\) 0 0
\(27\) −22.5538 −0.835325
\(28\) 0 0
\(29\) − 29.4791i − 1.01652i −0.861203 − 0.508260i \(-0.830289\pi\)
0.861203 − 0.508260i \(-0.169711\pi\)
\(30\) 0 0
\(31\) 23.4043 0.754979 0.377489 − 0.926014i \(-0.376787\pi\)
0.377489 + 0.926014i \(0.376787\pi\)
\(32\) 0 0
\(33\) 48.0557 + 21.0011i 1.45623 + 0.636398i
\(34\) 0 0
\(35\) 21.7582i 0.621662i
\(36\) 0 0
\(37\) −11.1821 −0.302219 −0.151109 − 0.988517i \(-0.548285\pi\)
−0.151109 + 0.988517i \(0.548285\pi\)
\(38\) 0 0
\(39\) 79.3026i 2.03340i
\(40\) 0 0
\(41\) − 69.0178i − 1.68336i −0.539976 − 0.841681i \(-0.681567\pi\)
0.539976 − 0.841681i \(-0.318433\pi\)
\(42\) 0 0
\(43\) − 65.2097i − 1.51650i −0.651961 − 0.758252i \(-0.726054\pi\)
0.651961 − 0.758252i \(-0.273946\pi\)
\(44\) 0 0
\(45\) −30.7025 −0.682278
\(46\) 0 0
\(47\) −45.7097 −0.972547 −0.486274 − 0.873807i \(-0.661644\pi\)
−0.486274 + 0.873807i \(0.661644\pi\)
\(48\) 0 0
\(49\) −45.6836 −0.932319
\(50\) 0 0
\(51\) 61.1934i 1.19987i
\(52\) 0 0
\(53\) −1.00392 −0.0189419 −0.00947094 − 0.999955i \(-0.503015\pi\)
−0.00947094 + 0.999955i \(0.503015\pi\)
\(54\) 0 0
\(55\) 22.5385 + 9.84968i 0.409791 + 0.179085i
\(56\) 0 0
\(57\) − 22.3709i − 0.392471i
\(58\) 0 0
\(59\) −94.0820 −1.59461 −0.797305 − 0.603576i \(-0.793742\pi\)
−0.797305 + 0.603576i \(0.793742\pi\)
\(60\) 0 0
\(61\) 113.061i 1.85346i 0.375723 + 0.926732i \(0.377394\pi\)
−0.375723 + 0.926732i \(0.622606\pi\)
\(62\) 0 0
\(63\) − 133.606i − 2.12073i
\(64\) 0 0
\(65\) 37.1935i 0.572208i
\(66\) 0 0
\(67\) 13.0292 0.194466 0.0972329 − 0.995262i \(-0.469001\pi\)
0.0972329 + 0.995262i \(0.469001\pi\)
\(68\) 0 0
\(69\) −154.738 −2.24257
\(70\) 0 0
\(71\) −36.0057 −0.507122 −0.253561 − 0.967319i \(-0.581602\pi\)
−0.253561 + 0.967319i \(0.581602\pi\)
\(72\) 0 0
\(73\) 83.2194i 1.13999i 0.821648 + 0.569996i \(0.193055\pi\)
−0.821648 + 0.569996i \(0.806945\pi\)
\(74\) 0 0
\(75\) −23.8383 −0.317844
\(76\) 0 0
\(77\) −42.8622 + 98.0793i −0.556652 + 1.27376i
\(78\) 0 0
\(79\) − 0.104581i − 0.00132381i −1.00000 0.000661904i \(-0.999789\pi\)
1.00000 0.000661904i \(-0.000210690\pi\)
\(80\) 0 0
\(81\) −16.0465 −0.198105
\(82\) 0 0
\(83\) 41.9202i 0.505062i 0.967589 + 0.252531i \(0.0812630\pi\)
−0.967589 + 0.252531i \(0.918737\pi\)
\(84\) 0 0
\(85\) 28.7002i 0.337649i
\(86\) 0 0
\(87\) 140.546i 1.61548i
\(88\) 0 0
\(89\) −145.516 −1.63501 −0.817505 − 0.575921i \(-0.804643\pi\)
−0.817505 + 0.575921i \(0.804643\pi\)
\(90\) 0 0
\(91\) −161.853 −1.77860
\(92\) 0 0
\(93\) −111.584 −1.19983
\(94\) 0 0
\(95\) − 10.4921i − 0.110443i
\(96\) 0 0
\(97\) 89.1978 0.919565 0.459782 − 0.888032i \(-0.347927\pi\)
0.459782 + 0.888032i \(0.347927\pi\)
\(98\) 0 0
\(99\) −138.398 − 60.4820i −1.39796 − 0.610929i
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 880.3.j.c.241.1 8
4.3 odd 2 110.3.d.a.21.8 yes 8
11.10 odd 2 inner 880.3.j.c.241.2 8
12.11 even 2 990.3.b.b.901.4 8
20.3 even 4 550.3.c.b.549.15 16
20.7 even 4 550.3.c.b.549.2 16
20.19 odd 2 550.3.d.f.351.1 8
44.43 even 2 110.3.d.a.21.4 ✓ 8
132.131 odd 2 990.3.b.b.901.7 8
220.43 odd 4 550.3.c.b.549.7 16
220.87 odd 4 550.3.c.b.549.10 16
220.219 even 2 550.3.d.f.351.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.3.d.a.21.4 ✓ 8 44.43 even 2
110.3.d.a.21.8 yes 8 4.3 odd 2
550.3.c.b.549.2 16 20.7 even 4
550.3.c.b.549.7 16 220.43 odd 4
550.3.c.b.549.10 16 220.87 odd 4
550.3.c.b.549.15 16 20.3 even 4
550.3.d.f.351.1 8 20.19 odd 2
550.3.d.f.351.5 8 220.219 even 2
880.3.j.c.241.1 8 1.1 even 1 trivial
880.3.j.c.241.2 8 11.10 odd 2 inner
990.3.b.b.901.4 8 12.11 even 2
990.3.b.b.901.7 8 132.131 odd 2