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Properties

Label 880.3.j.c.241.8
Level $880$
Weight $3$
Character 880.241
Analytic conductor $23.978$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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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 241.8
Root \(-1.51954 + 1.14412i\) of defining polynomial
Character \(\chi\) \(=\) 880.241
Dual form 880.3.j.c.241.7

$q$-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)
 
\(f(q)\) \(=\) \(q+5.11433 q^{3} +2.23607 q^{5} +6.78202i q^{7} +17.1564 q^{9} +(-7.23901 + 8.28232i) q^{11} +9.87597i q^{13} +11.4360 q^{15} -29.1178i q^{17} +27.1845i q^{19} +34.6855i q^{21} +35.0515 q^{23} +5.00000 q^{25} +41.7143 q^{27} +21.3472i q^{29} -21.1545 q^{31} +(-37.0227 + 42.3585i) q^{33} +15.1651i q^{35} -7.43123 q^{37} +50.5089i q^{39} -44.2799i q^{41} +15.2380i q^{43} +38.3628 q^{45} +0.490048 q^{47} +3.00421 q^{49} -148.918i q^{51} +40.6167 q^{53} +(-16.1869 + 18.5198i) q^{55} +139.031i q^{57} +15.7748 q^{59} +36.9599i q^{61} +116.355i q^{63} +22.0833i q^{65} -98.6639 q^{67} +179.265 q^{69} +114.279 q^{71} -106.453i q^{73} +25.5716 q^{75} +(-56.1708 - 49.0951i) q^{77} -35.4008i q^{79} +58.9334 q^{81} -119.115i q^{83} -65.1094i q^{85} +109.177i q^{87} -121.240 q^{89} -66.9790 q^{91} -108.191 q^{93} +60.7865i q^{95} +124.898 q^{97} +(-124.195 + 142.094i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} + 40 q^{9} + 40 q^{15} + 136 q^{23} + 40 q^{25} - 64 q^{27} + 64 q^{31} + 88 q^{33} - 48 q^{37} - 152 q^{47} - 232 q^{49} + 352 q^{53} - 80 q^{59} - 24 q^{67} + 112 q^{69} + 256 q^{71} + 40 q^{75}+ \cdots - 704 q^{99}+O(q^{100}) \) 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
\(3\) 5.11433 1.70478 0.852388 − 0.522910i \(-0.175153\pi\)
0.852388 + 0.522910i \(0.175153\pi\)
\(4\) 0 0
\(5\) 2.23607 0.447214
\(6\) 0 0
\(7\) 6.78202i 0.968860i 0.874830 + 0.484430i \(0.160973\pi\)
−0.874830 + 0.484430i \(0.839027\pi\)
\(8\) 0 0
\(9\) 17.1564 1.90626
\(10\) 0 0
\(11\) −7.23901 + 8.28232i −0.658092 + 0.752938i
\(12\) 0 0
\(13\) 9.87597i 0.759690i 0.925050 + 0.379845i \(0.124023\pi\)
−0.925050 + 0.379845i \(0.875977\pi\)
\(14\) 0 0
\(15\) 11.4360 0.762399
\(16\) 0 0
\(17\) − 29.1178i − 1.71281i −0.516302 − 0.856406i \(-0.672692\pi\)
0.516302 − 0.856406i \(-0.327308\pi\)
\(18\) 0 0
\(19\) 27.1845i 1.43077i 0.698733 + 0.715383i \(0.253748\pi\)
−0.698733 + 0.715383i \(0.746252\pi\)
\(20\) 0 0
\(21\) 34.6855i 1.65169i
\(22\) 0 0
\(23\) 35.0515 1.52398 0.761990 − 0.647589i \(-0.224222\pi\)
0.761990 + 0.647589i \(0.224222\pi\)
\(24\) 0 0
\(25\) 5.00000 0.200000
\(26\) 0 0
\(27\) 41.7143 1.54497
\(28\) 0 0
\(29\) 21.3472i 0.736110i 0.929804 + 0.368055i \(0.119976\pi\)
−0.929804 + 0.368055i \(0.880024\pi\)
\(30\) 0 0
\(31\) −21.1545 −0.682402 −0.341201 − 0.939990i \(-0.610834\pi\)
−0.341201 + 0.939990i \(0.610834\pi\)
\(32\) 0 0
\(33\) −37.0227 + 42.3585i −1.12190 + 1.28359i
\(34\) 0 0
\(35\) 15.1651i 0.433287i
\(36\) 0 0
\(37\) −7.43123 −0.200844 −0.100422 − 0.994945i \(-0.532019\pi\)
−0.100422 + 0.994945i \(0.532019\pi\)
\(38\) 0 0
\(39\) 50.5089i 1.29510i
\(40\) 0 0
\(41\) − 44.2799i − 1.08000i −0.841666 − 0.539999i \(-0.818425\pi\)
0.841666 − 0.539999i \(-0.181575\pi\)
\(42\) 0 0
\(43\) 15.2380i 0.354373i 0.984177 + 0.177186i \(0.0566995\pi\)
−0.984177 + 0.177186i \(0.943300\pi\)
\(44\) 0 0
\(45\) 38.3628 0.852506
\(46\) 0 0
\(47\) 0.490048 0.0104265 0.00521327 − 0.999986i \(-0.498341\pi\)
0.00521327 + 0.999986i \(0.498341\pi\)
\(48\) 0 0
\(49\) 3.00421 0.0613105
\(50\) 0 0
\(51\) − 148.918i − 2.91996i
\(52\) 0 0
\(53\) 40.6167 0.766353 0.383177 − 0.923675i \(-0.374830\pi\)
0.383177 + 0.923675i \(0.374830\pi\)
\(54\) 0 0
\(55\) −16.1869 + 18.5198i −0.294308 + 0.336724i
\(56\) 0 0
\(57\) 139.031i 2.43913i
\(58\) 0 0
\(59\) 15.7748 0.267370 0.133685 − 0.991024i \(-0.457319\pi\)
0.133685 + 0.991024i \(0.457319\pi\)
\(60\) 0 0
\(61\) 36.9599i 0.605901i 0.953006 + 0.302950i \(0.0979716\pi\)
−0.953006 + 0.302950i \(0.902028\pi\)
\(62\) 0 0
\(63\) 116.355i 1.84690i
\(64\) 0 0
\(65\) 22.0833i 0.339744i
\(66\) 0 0
\(67\) −98.6639 −1.47260 −0.736298 − 0.676658i \(-0.763428\pi\)
−0.736298 + 0.676658i \(0.763428\pi\)
\(68\) 0 0
\(69\) 179.265 2.59805
\(70\) 0 0
\(71\) 114.279 1.60957 0.804784 − 0.593568i \(-0.202281\pi\)
0.804784 + 0.593568i \(0.202281\pi\)
\(72\) 0 0
\(73\) − 106.453i − 1.45826i −0.684374 − 0.729131i \(-0.739925\pi\)
0.684374 − 0.729131i \(-0.260075\pi\)
\(74\) 0 0
\(75\) 25.5716 0.340955
\(76\) 0 0
\(77\) −56.1708 − 49.0951i −0.729491 − 0.637599i
\(78\) 0 0
\(79\) − 35.4008i − 0.448112i −0.974576 − 0.224056i \(-0.928070\pi\)
0.974576 − 0.224056i \(-0.0719298\pi\)
\(80\) 0 0
\(81\) 58.9334 0.727573
\(82\) 0 0
\(83\) − 119.115i − 1.43512i −0.696499 − 0.717558i \(-0.745260\pi\)
0.696499 − 0.717558i \(-0.254740\pi\)
\(84\) 0 0
\(85\) − 65.1094i − 0.765993i
\(86\) 0 0
\(87\) 109.177i 1.25490i
\(88\) 0 0
\(89\) −121.240 −1.36225 −0.681123 − 0.732169i \(-0.738508\pi\)
−0.681123 + 0.732169i \(0.738508\pi\)
\(90\) 0 0
\(91\) −66.9790 −0.736033
\(92\) 0 0
\(93\) −108.191 −1.16334
\(94\) 0 0
\(95\) 60.7865i 0.639858i
\(96\) 0 0
\(97\) 124.898 1.28761 0.643805 − 0.765190i \(-0.277355\pi\)
0.643805 + 0.765190i \(0.277355\pi\)
\(98\) 0 0
\(99\) −124.195 + 142.094i −1.25450 + 1.43530i
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.241.8 8
4.3 odd 2 110.3.d.a.21.5 yes 8
11.10 odd 2 inner 880.3.j.c.241.7 8
12.11 even 2 990.3.b.b.901.1 8
20.3 even 4 550.3.c.b.549.9 16
20.7 even 4 550.3.c.b.549.8 16
20.19 odd 2 550.3.d.f.351.4 8
44.43 even 2 110.3.d.a.21.1 ✓ 8
132.131 odd 2 990.3.b.b.901.6 8
220.43 odd 4 550.3.c.b.549.1 16
220.87 odd 4 550.3.c.b.549.16 16
220.219 even 2 550.3.d.f.351.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.3.d.a.21.1 ✓ 8 44.43 even 2
110.3.d.a.21.5 yes 8 4.3 odd 2
550.3.c.b.549.1 16 220.43 odd 4
550.3.c.b.549.8 16 20.7 even 4
550.3.c.b.549.9 16 20.3 even 4
550.3.c.b.549.16 16 220.87 odd 4
550.3.d.f.351.4 8 20.19 odd 2
550.3.d.f.351.8 8 220.219 even 2
880.3.j.c.241.7 8 11.10 odd 2 inner
880.3.j.c.241.8 8 1.1 even 1 trivial
990.3.b.b.901.1 8 12.11 even 2
990.3.b.b.901.6 8 132.131 odd 2