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Properties

Label 880.3.j.c.241.4
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 241.4
Root \(1.51954 - 1.14412i\) of defining polynomial
Character \(\chi\) \(=\) 880.241
Dual form 880.3.j.c.241.3

$q$-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)
 
\(f(q)\) \(=\) \(q+1.35781 q^{3} +2.23607 q^{5} +12.4389i q^{7} -7.15636 q^{9} +(7.23901 + 8.28232i) q^{11} -1.28012i q^{13} +3.03615 q^{15} +3.33028i q^{17} +1.88632i q^{19} +16.8896i q^{21} -32.3565 q^{23} +5.00000 q^{25} -21.9372 q^{27} -27.9138i q^{29} -16.5112 q^{31} +(9.82918 + 11.2458i) q^{33} +27.8142i q^{35} -22.4573 q^{37} -1.73816i q^{39} +52.3967i q^{41} -15.7171i q^{43} -16.0021 q^{45} -87.6835 q^{47} -105.726 q^{49} +4.52188i q^{51} +74.2161 q^{53} +(16.1869 + 18.5198i) q^{55} +2.56125i q^{57} +26.8351 q^{59} +47.4483i q^{61} -89.0171i q^{63} -2.86244i q^{65} +79.2475 q^{67} -43.9339 q^{69} +74.9405 q^{71} -64.0267i q^{73} +6.78904 q^{75} +(-103.023 + 90.0451i) q^{77} +151.980i q^{79} +34.6207 q^{81} +151.469i q^{83} +7.44674i q^{85} -37.9016i q^{87} +127.627 q^{89} +15.9233 q^{91} -22.4190 q^{93} +4.21793i q^{95} -88.1768 q^{97} +(-51.8050 - 59.2712i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} + 40 q^{9} + 40 q^{15} + 136 q^{23} + 40 q^{25} - 64 q^{27} + 64 q^{31} + 88 q^{33} - 48 q^{37} - 152 q^{47} - 232 q^{49} + 352 q^{53} - 80 q^{59} - 24 q^{67} + 112 q^{69} + 256 q^{71} + 40 q^{75}+ \cdots - 704 q^{99}+O(q^{100}) \) 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
\(3\) 1.35781 0.452602 0.226301 − 0.974057i \(-0.427337\pi\)
0.226301 + 0.974057i \(0.427337\pi\)
\(4\) 0 0
\(5\) 2.23607 0.447214
\(6\) 0 0
\(7\) 12.4389i 1.77698i 0.458894 + 0.888491i \(0.348246\pi\)
−0.458894 + 0.888491i \(0.651754\pi\)
\(8\) 0 0
\(9\) −7.15636 −0.795151
\(10\) 0 0
\(11\) 7.23901 + 8.28232i 0.658092 + 0.752938i
\(12\) 0 0
\(13\) − 1.28012i − 0.0984708i −0.998787 − 0.0492354i \(-0.984322\pi\)
0.998787 − 0.0492354i \(-0.0156785\pi\)
\(14\) 0 0
\(15\) 3.03615 0.202410
\(16\) 0 0
\(17\) 3.33028i 0.195899i 0.995191 + 0.0979495i \(0.0312284\pi\)
−0.995191 + 0.0979495i \(0.968772\pi\)
\(18\) 0 0
\(19\) 1.88632i 0.0992798i 0.998767 + 0.0496399i \(0.0158074\pi\)
−0.998767 + 0.0496399i \(0.984193\pi\)
\(20\) 0 0
\(21\) 16.8896i 0.804266i
\(22\) 0 0
\(23\) −32.3565 −1.40680 −0.703402 − 0.710792i \(-0.748337\pi\)
−0.703402 + 0.710792i \(0.748337\pi\)
\(24\) 0 0
\(25\) 5.00000 0.200000
\(26\) 0 0
\(27\) −21.9372 −0.812490
\(28\) 0 0
\(29\) − 27.9138i − 0.962546i −0.876571 − 0.481273i \(-0.840175\pi\)
0.876571 − 0.481273i \(-0.159825\pi\)
\(30\) 0 0
\(31\) −16.5112 −0.532618 −0.266309 − 0.963888i \(-0.585804\pi\)
−0.266309 + 0.963888i \(0.585804\pi\)
\(32\) 0 0
\(33\) 9.82918 + 11.2458i 0.297854 + 0.340781i
\(34\) 0 0
\(35\) 27.8142i 0.794690i
\(36\) 0 0
\(37\) −22.4573 −0.606954 −0.303477 − 0.952839i \(-0.598148\pi\)
−0.303477 + 0.952839i \(0.598148\pi\)
\(38\) 0 0
\(39\) − 1.73816i − 0.0445681i
\(40\) 0 0
\(41\) 52.3967i 1.27797i 0.769220 + 0.638984i \(0.220645\pi\)
−0.769220 + 0.638984i \(0.779355\pi\)
\(42\) 0 0
\(43\) − 15.7171i − 0.365513i −0.983158 − 0.182756i \(-0.941498\pi\)
0.983158 − 0.182756i \(-0.0585020\pi\)
\(44\) 0 0
\(45\) −16.0021 −0.355602
\(46\) 0 0
\(47\) −87.6835 −1.86561 −0.932804 − 0.360385i \(-0.882645\pi\)
−0.932804 + 0.360385i \(0.882645\pi\)
\(48\) 0 0
\(49\) −105.726 −2.15766
\(50\) 0 0
\(51\) 4.52188i 0.0886644i
\(52\) 0 0
\(53\) 74.2161 1.40030 0.700152 − 0.713994i \(-0.253116\pi\)
0.700152 + 0.713994i \(0.253116\pi\)
\(54\) 0 0
\(55\) 16.1869 + 18.5198i 0.294308 + 0.336724i
\(56\) 0 0
\(57\) 2.56125i 0.0449343i
\(58\) 0 0
\(59\) 26.8351 0.454832 0.227416 − 0.973798i \(-0.426972\pi\)
0.227416 + 0.973798i \(0.426972\pi\)
\(60\) 0 0
\(61\) 47.4483i 0.777841i 0.921271 + 0.388921i \(0.127152\pi\)
−0.921271 + 0.388921i \(0.872848\pi\)
\(62\) 0 0
\(63\) − 89.0171i − 1.41297i
\(64\) 0 0
\(65\) − 2.86244i − 0.0440375i
\(66\) 0 0
\(67\) 79.2475 1.18280 0.591399 − 0.806379i \(-0.298576\pi\)
0.591399 + 0.806379i \(0.298576\pi\)
\(68\) 0 0
\(69\) −43.9339 −0.636723
\(70\) 0 0
\(71\) 74.9405 1.05550 0.527750 − 0.849400i \(-0.323036\pi\)
0.527750 + 0.849400i \(0.323036\pi\)
\(72\) 0 0
\(73\) − 64.0267i − 0.877078i −0.898712 − 0.438539i \(-0.855496\pi\)
0.898712 − 0.438539i \(-0.144504\pi\)
\(74\) 0 0
\(75\) 6.78904 0.0905205
\(76\) 0 0
\(77\) −103.023 + 90.0451i −1.33796 + 1.16942i
\(78\) 0 0
\(79\) 151.980i 1.92380i 0.273400 + 0.961901i \(0.411852\pi\)
−0.273400 + 0.961901i \(0.588148\pi\)
\(80\) 0 0
\(81\) 34.6207 0.427416
\(82\) 0 0
\(83\) 151.469i 1.82492i 0.409161 + 0.912462i \(0.365821\pi\)
−0.409161 + 0.912462i \(0.634179\pi\)
\(84\) 0 0
\(85\) 7.44674i 0.0876087i
\(86\) 0 0
\(87\) − 37.9016i − 0.435651i
\(88\) 0 0
\(89\) 127.627 1.43401 0.717005 − 0.697068i \(-0.245513\pi\)
0.717005 + 0.697068i \(0.245513\pi\)
\(90\) 0 0
\(91\) 15.9233 0.174981
\(92\) 0 0
\(93\) −22.4190 −0.241064
\(94\) 0 0
\(95\) 4.21793i 0.0443993i
\(96\) 0 0
\(97\) −88.1768 −0.909040 −0.454520 − 0.890737i \(-0.650189\pi\)
−0.454520 + 0.890737i \(0.650189\pi\)
\(98\) 0 0
\(99\) −51.8050 − 59.2712i −0.523282 − 0.598699i
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.241.4 8
4.3 odd 2 110.3.d.a.21.3 ✓ 8
11.10 odd 2 inner 880.3.j.c.241.3 8
12.11 even 2 990.3.b.b.901.5 8
20.3 even 4 550.3.c.b.549.4 16
20.7 even 4 550.3.c.b.549.13 16
20.19 odd 2 550.3.d.f.351.6 8
44.43 even 2 110.3.d.a.21.7 yes 8
132.131 odd 2 990.3.b.b.901.2 8
220.43 odd 4 550.3.c.b.549.12 16
220.87 odd 4 550.3.c.b.549.5 16
220.219 even 2 550.3.d.f.351.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.3.d.a.21.3 ✓ 8 4.3 odd 2
110.3.d.a.21.7 yes 8 44.43 even 2
550.3.c.b.549.4 16 20.3 even 4
550.3.c.b.549.5 16 220.87 odd 4
550.3.c.b.549.12 16 220.43 odd 4
550.3.c.b.549.13 16 20.7 even 4
550.3.d.f.351.2 8 220.219 even 2
550.3.d.f.351.6 8 20.19 odd 2
880.3.j.c.241.3 8 11.10 odd 2 inner
880.3.j.c.241.4 8 1.1 even 1 trivial
990.3.b.b.901.2 8 132.131 odd 2
990.3.b.b.901.5 8 12.11 even 2