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Properties

Label 880.2.bo.i.81.1
Level $880$
Weight $2$
Character 880.81
Analytic conductor $7.027$
Analytic rank $0$
Dimension $12$
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] = [880,2,Mod(81,880)] 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("880.81"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(880, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 880.bo (of order \(5\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,1,0,3,0,1] 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: no
Analytic conductor: \(7.02683537787\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 5 x^{10} + 4 x^{9} + 28 x^{8} - 81 x^{7} + 335 x^{6} - 235 x^{5} + 782 x^{4} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 81.1
Root \(-2.19470 + 1.59454i\) of defining polynomial
Character \(\chi\) \(=\) 880.81
Dual form 880.2.bo.i.641.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.529284 + 1.62897i) q^{3} +(0.809017 - 0.587785i) q^{5} +(-1.14492 - 3.52372i) q^{7} +(0.0536500 + 0.0389790i) q^{9} +(-2.68927 + 1.94109i) q^{11} +(0.952617 + 0.692117i) q^{13} +(0.529284 + 1.62897i) q^{15} +(4.36891 - 3.17420i) q^{17} +(1.18895 - 3.65921i) q^{19} +6.34602 q^{21} +8.68237 q^{23} +(0.309017 - 0.951057i) q^{25} +(-4.24895 + 3.08704i) q^{27} +(-2.12479 - 6.53944i) q^{29} +(7.08327 + 5.14630i) q^{31} +(-1.73859 - 5.40813i) q^{33} +(-2.99745 - 2.17778i) q^{35} +(0.696735 + 2.14433i) q^{37} +(-1.63164 + 1.18546i) q^{39} +(0.493602 - 1.51915i) q^{41} -4.11979 q^{43} +0.0663151 q^{45} +(3.91833 - 12.0594i) q^{47} +(-5.44260 + 3.95428i) q^{49} +(2.85828 + 8.79688i) q^{51} +(10.0152 + 7.27650i) q^{53} +(-1.03472 + 3.15109i) q^{55} +(5.33145 + 3.87353i) q^{57} +(0.121753 + 0.374716i) q^{59} +(-1.45692 + 1.05851i) q^{61} +(0.0759258 - 0.233676i) q^{63} +1.17750 q^{65} +14.3809 q^{67} +(-4.59544 + 14.1433i) q^{69} +(-5.54603 + 4.02943i) q^{71} +(3.10719 + 9.56296i) q^{73} +(1.38568 + 1.00676i) q^{75} +(9.91887 + 7.25381i) q^{77} +(-0.901110 - 0.654695i) q^{79} +(-2.71832 - 8.36612i) q^{81} +(0.0140132 - 0.0101812i) q^{83} +(1.66878 - 5.13596i) q^{85} +11.7772 q^{87} -8.49434 q^{89} +(1.34815 - 4.14917i) q^{91} +(-12.1322 + 8.81458i) q^{93} +(-1.18895 - 3.65921i) q^{95} +(-8.50719 - 6.18083i) q^{97} +(-0.219941 - 0.000685401i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + q^{3} + 3 q^{5} + q^{7} - 10 q^{9} - 4 q^{11} + 18 q^{13} - q^{15} + 3 q^{17} - 4 q^{19} - 28 q^{21} + 18 q^{23} - 3 q^{25} - 23 q^{27} + 15 q^{29} + 8 q^{31} + 4 q^{33} - 6 q^{35} + 6 q^{37}+ \cdots - 79 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(111\) \(177\) \(321\) \(661\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{5}\right)\) \(1\)

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
\(3\) −0.529284 + 1.62897i −0.305582 + 0.940486i 0.673877 + 0.738844i \(0.264628\pi\)
−0.979459 + 0.201642i \(0.935372\pi\)
\(4\) 0 0
\(5\) 0.809017 − 0.587785i 0.361803 − 0.262866i
\(6\) 0 0
\(7\) −1.14492 − 3.52372i −0.432741 − 1.33184i −0.895384 − 0.445295i \(-0.853099\pi\)
0.462643 − 0.886545i \(-0.346901\pi\)
\(8\) 0 0
\(9\) 0.0536500 + 0.0389790i 0.0178833 + 0.0129930i
\(10\) 0 0
\(11\) −2.68927 + 1.94109i −0.810845 + 0.585261i
\(12\) 0 0
\(13\) 0.952617 + 0.692117i 0.264208 + 0.191959i 0.712000 − 0.702179i \(-0.247789\pi\)
−0.447792 + 0.894138i \(0.647789\pi\)
\(14\) 0 0
\(15\) 0.529284 + 1.62897i 0.136661 + 0.420598i
\(16\) 0 0
\(17\) 4.36891 − 3.17420i 1.05962 − 0.769856i 0.0855991 − 0.996330i \(-0.472720\pi\)
0.974017 + 0.226473i \(0.0727196\pi\)
\(18\) 0 0
\(19\) 1.18895 − 3.65921i 0.272764 − 0.839480i −0.717039 − 0.697033i \(-0.754503\pi\)
0.989802 − 0.142447i \(-0.0454971\pi\)
\(20\) 0 0
\(21\) 6.34602 1.38481
\(22\) 0 0
\(23\) 8.68237 1.81040 0.905200 − 0.424986i \(-0.139721\pi\)
0.905200 + 0.424986i \(0.139721\pi\)
\(24\) 0 0
\(25\) 0.309017 − 0.951057i 0.0618034 − 0.190211i
\(26\) 0 0
\(27\) −4.24895 + 3.08704i −0.817710 + 0.594101i
\(28\) 0 0
\(29\) −2.12479 − 6.53944i −0.394564 − 1.21434i −0.929301 − 0.369324i \(-0.879589\pi\)
0.534736 − 0.845019i \(-0.320411\pi\)
\(30\) 0 0
\(31\) 7.08327 + 5.14630i 1.27219 + 0.924302i 0.999288 − 0.0377385i \(-0.0120154\pi\)
0.272905 + 0.962041i \(0.412015\pi\)
\(32\) 0 0
\(33\) −1.73859 − 5.40813i −0.302650 − 0.941434i
\(34\) 0 0
\(35\) −2.99745 − 2.17778i −0.506662 − 0.368111i
\(36\) 0 0
\(37\) 0.696735 + 2.14433i 0.114543 + 0.352526i 0.991851 − 0.127401i \(-0.0406634\pi\)
−0.877309 + 0.479926i \(0.840663\pi\)
\(38\) 0 0
\(39\) −1.63164 + 1.18546i −0.261272 + 0.189825i
\(40\) 0 0
\(41\) 0.493602 − 1.51915i 0.0770877 − 0.237251i −0.905086 − 0.425229i \(-0.860193\pi\)
0.982173 + 0.187978i \(0.0601934\pi\)
\(42\) 0 0
\(43\) −4.11979 −0.628262 −0.314131 − 0.949380i \(-0.601713\pi\)
−0.314131 + 0.949380i \(0.601713\pi\)
\(44\) 0 0
\(45\) 0.0663151 0.00988567
\(46\) 0 0
\(47\) 3.91833 − 12.0594i 0.571547 − 1.75904i −0.0761009 − 0.997100i \(-0.524247\pi\)
0.647648 − 0.761940i \(-0.275753\pi\)
\(48\) 0 0
\(49\) −5.44260 + 3.95428i −0.777514 + 0.564897i
\(50\) 0 0
\(51\) 2.85828 + 8.79688i 0.400239 + 1.23181i
\(52\) 0 0
\(53\) 10.0152 + 7.27650i 1.37570 + 0.999504i 0.997268 + 0.0738747i \(0.0235365\pi\)
0.378432 + 0.925629i \(0.376464\pi\)
\(54\) 0 0
\(55\) −1.03472 + 3.15109i −0.139521 + 0.424893i
\(56\) 0 0
\(57\) 5.33145 + 3.87353i 0.706168 + 0.513061i
\(58\) 0 0
\(59\) 0.121753 + 0.374716i 0.0158508 + 0.0487838i 0.958669 − 0.284523i \(-0.0918354\pi\)
−0.942818 + 0.333307i \(0.891835\pi\)
\(60\) 0 0
\(61\) −1.45692 + 1.05851i −0.186539 + 0.135529i −0.677136 − 0.735858i \(-0.736779\pi\)
0.490596 + 0.871387i \(0.336779\pi\)
\(62\) 0 0
\(63\) 0.0759258 − 0.233676i 0.00956575 − 0.0294404i
\(64\) 0 0
\(65\) 1.17750 0.146051
\(66\) 0 0
\(67\) 14.3809 1.75691 0.878455 − 0.477826i \(-0.158575\pi\)
0.878455 + 0.477826i \(0.158575\pi\)
\(68\) 0 0
\(69\) −4.59544 + 14.1433i −0.553227 + 1.70266i
\(70\) 0 0
\(71\) −5.54603 + 4.02943i −0.658192 + 0.478205i −0.866052 − 0.499954i \(-0.833350\pi\)
0.207860 + 0.978159i \(0.433350\pi\)
\(72\) 0 0
\(73\) 3.10719 + 9.56296i 0.363670 + 1.11926i 0.950810 + 0.309775i \(0.100254\pi\)
−0.587140 + 0.809485i \(0.699746\pi\)
\(74\) 0 0
\(75\) 1.38568 + 1.00676i 0.160005 + 0.116250i
\(76\) 0 0
\(77\) 9.91887 + 7.25381i 1.13036 + 0.826649i
\(78\) 0 0
\(79\) −0.901110 − 0.654695i −0.101383 − 0.0736589i 0.535939 − 0.844257i \(-0.319958\pi\)
−0.637322 + 0.770598i \(0.719958\pi\)
\(80\) 0 0
\(81\) −2.71832 − 8.36612i −0.302035 − 0.929569i
\(82\) 0 0
\(83\) 0.0140132 − 0.0101812i 0.00153814 − 0.00111753i −0.587016 − 0.809575i \(-0.699697\pi\)
0.588554 + 0.808458i \(0.299697\pi\)
\(84\) 0 0
\(85\) 1.66878 − 5.13596i 0.181004 − 0.557073i
\(86\) 0 0
\(87\) 11.7772 1.26265
\(88\) 0 0
\(89\) −8.49434 −0.900399 −0.450199 − 0.892928i \(-0.648647\pi\)
−0.450199 + 0.892928i \(0.648647\pi\)
\(90\) 0 0
\(91\) 1.34815 − 4.14917i 0.141324 − 0.434951i
\(92\) 0 0
\(93\) −12.1322 + 8.81458i −1.25805 + 0.914029i
\(94\) 0 0
\(95\) −1.18895 − 3.65921i −0.121984 − 0.375427i
\(96\) 0 0
\(97\) −8.50719 − 6.18083i −0.863774 − 0.627569i 0.0651351 − 0.997876i \(-0.479252\pi\)
−0.928909 + 0.370308i \(0.879252\pi\)
\(98\) 0 0
\(99\) −0.219941 0.000685401i −0.0221049 6.88853e-5i
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 880.2.bo.i.81.1 12
4.3 odd 2 440.2.y.c.81.3 ✓ 12
11.3 even 5 inner 880.2.bo.i.641.1 12
11.5 even 5 9680.2.a.dc.1.1 6
11.6 odd 10 9680.2.a.dd.1.1 6
44.3 odd 10 440.2.y.c.201.3 yes 12
44.27 odd 10 4840.2.a.bb.1.6 6
44.39 even 10 4840.2.a.ba.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.c.81.3 ✓ 12 4.3 odd 2
440.2.y.c.201.3 yes 12 44.3 odd 10
880.2.bo.i.81.1 12 1.1 even 1 trivial
880.2.bo.i.641.1 12 11.3 even 5 inner
4840.2.a.ba.1.6 6 44.39 even 10
4840.2.a.bb.1.6 6 44.27 odd 10
9680.2.a.dc.1.1 6 11.5 even 5
9680.2.a.dd.1.1 6 11.6 odd 10