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Properties

Label 880.2.bd.c.417.1
Level $880$
Weight $2$
Character 880.417
Analytic conductor $7.027$
Analytic rank $0$
Dimension $4$
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(417,880)] 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("880.417"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(880, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 880.bd (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-4,0,0,0,0,0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.02683537787\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 110)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 417.1
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 880.417
Dual form 880.2.bd.c.593.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.70711 + 1.70711i) q^{3} +(-0.707107 + 2.12132i) q^{5} +(-2.12132 + 2.12132i) q^{7} -2.82843i q^{9} +(1.41421 + 3.00000i) q^{11} +(3.00000 + 3.00000i) q^{13} +(-2.41421 - 4.82843i) q^{15} +(-5.12132 + 5.12132i) q^{17} +3.00000 q^{19} -7.24264i q^{21} +(0.171573 - 0.171573i) q^{23} +(-4.00000 - 3.00000i) q^{25} +(-0.292893 - 0.292893i) q^{27} -1.24264 q^{29} +7.24264 q^{31} +(-7.53553 - 2.70711i) q^{33} +(-3.00000 - 6.00000i) q^{35} +(0.121320 + 0.121320i) q^{37} -10.2426 q^{39} -1.75736i q^{41} +(-1.24264 - 1.24264i) q^{43} +(6.00000 + 2.00000i) q^{45} +(-4.41421 - 4.41421i) q^{47} -2.00000i q^{49} -17.4853i q^{51} +(9.53553 - 9.53553i) q^{53} +(-7.36396 + 0.878680i) q^{55} +(-5.12132 + 5.12132i) q^{57} -1.41421i q^{59} -7.24264i q^{61} +(6.00000 + 6.00000i) q^{63} +(-8.48528 + 4.24264i) q^{65} +(4.00000 + 4.00000i) q^{67} +0.585786i q^{69} +1.24264 q^{71} +(6.00000 + 6.00000i) q^{73} +(11.9497 - 1.70711i) q^{75} +(-9.36396 - 3.36396i) q^{77} -10.2426 q^{79} +9.48528 q^{81} +(7.24264 + 7.24264i) q^{83} +(-7.24264 - 14.4853i) q^{85} +(2.12132 - 2.12132i) q^{87} +5.48528i q^{89} -12.7279 q^{91} +(-12.3640 + 12.3640i) q^{93} +(-2.12132 + 6.36396i) q^{95} +(-2.24264 - 2.24264i) q^{97} +(8.48528 - 4.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 12 q^{13} - 4 q^{15} - 12 q^{17} + 12 q^{19} + 12 q^{23} - 16 q^{25} - 4 q^{27} + 12 q^{29} + 12 q^{31} - 16 q^{33} - 12 q^{35} - 8 q^{37} - 24 q^{39} + 12 q^{43} + 24 q^{45} - 12 q^{47}+ \cdots + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(111\) \(177\) \(321\) \(661\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(1\)

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
\(3\) −1.70711 + 1.70711i −0.985599 + 0.985599i −0.999898 − 0.0142992i \(-0.995448\pi\)
0.0142992 + 0.999898i \(0.495448\pi\)
\(4\) 0 0
\(5\) −0.707107 + 2.12132i −0.316228 + 0.948683i
\(6\) 0 0
\(7\) −2.12132 + 2.12132i −0.801784 + 0.801784i −0.983374 − 0.181591i \(-0.941875\pi\)
0.181591 + 0.983374i \(0.441875\pi\)
\(8\) 0 0
\(9\) − 2.82843i − 0.942809i
\(10\) 0 0
\(11\) 1.41421 + 3.00000i 0.426401 + 0.904534i
\(12\) 0 0
\(13\) 3.00000 + 3.00000i 0.832050 + 0.832050i 0.987797 − 0.155747i \(-0.0497784\pi\)
−0.155747 + 0.987797i \(0.549778\pi\)
\(14\) 0 0
\(15\) −2.41421 − 4.82843i −0.623347 − 1.24669i
\(16\) 0 0
\(17\) −5.12132 + 5.12132i −1.24210 + 1.24210i −0.282975 + 0.959127i \(0.591322\pi\)
−0.959127 + 0.282975i \(0.908678\pi\)
\(18\) 0 0
\(19\) 3.00000 0.688247 0.344124 − 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) 0 0
\(21\) − 7.24264i − 1.58047i
\(22\) 0 0
\(23\) 0.171573 − 0.171573i 0.0357754 − 0.0357754i −0.688993 − 0.724768i \(-0.741947\pi\)
0.724768 + 0.688993i \(0.241947\pi\)
\(24\) 0 0
\(25\) −4.00000 − 3.00000i −0.800000 − 0.600000i
\(26\) 0 0
\(27\) −0.292893 − 0.292893i −0.0563673 − 0.0563673i
\(28\) 0 0
\(29\) −1.24264 −0.230753 −0.115376 − 0.993322i \(-0.536807\pi\)
−0.115376 + 0.993322i \(0.536807\pi\)
\(30\) 0 0
\(31\) 7.24264 1.30082 0.650408 − 0.759585i \(-0.274598\pi\)
0.650408 + 0.759585i \(0.274598\pi\)
\(32\) 0 0
\(33\) −7.53553 − 2.70711i −1.31177 − 0.471247i
\(34\) 0 0
\(35\) −3.00000 − 6.00000i −0.507093 − 1.01419i
\(36\) 0 0
\(37\) 0.121320 + 0.121320i 0.0199449 + 0.0199449i 0.717009 − 0.697064i \(-0.245511\pi\)
−0.697064 + 0.717009i \(0.745511\pi\)
\(38\) 0 0
\(39\) −10.2426 −1.64014
\(40\) 0 0
\(41\) − 1.75736i − 0.274453i −0.990540 − 0.137227i \(-0.956181\pi\)
0.990540 − 0.137227i \(-0.0438189\pi\)
\(42\) 0 0
\(43\) −1.24264 − 1.24264i −0.189501 − 0.189501i 0.605979 − 0.795480i \(-0.292781\pi\)
−0.795480 + 0.605979i \(0.792781\pi\)
\(44\) 0 0
\(45\) 6.00000 + 2.00000i 0.894427 + 0.298142i
\(46\) 0 0
\(47\) −4.41421 − 4.41421i −0.643879 − 0.643879i 0.307628 − 0.951507i \(-0.400465\pi\)
−0.951507 + 0.307628i \(0.900465\pi\)
\(48\) 0 0
\(49\) − 2.00000i − 0.285714i
\(50\) 0 0
\(51\) − 17.4853i − 2.44843i
\(52\) 0 0
\(53\) 9.53553 − 9.53553i 1.30981 − 1.30981i 0.388254 − 0.921552i \(-0.373078\pi\)
0.921552 − 0.388254i \(-0.126922\pi\)
\(54\) 0 0
\(55\) −7.36396 + 0.878680i −0.992956 + 0.118481i
\(56\) 0 0
\(57\) −5.12132 + 5.12132i −0.678335 + 0.678335i
\(58\) 0 0
\(59\) − 1.41421i − 0.184115i −0.995754 − 0.0920575i \(-0.970656\pi\)
0.995754 − 0.0920575i \(-0.0293443\pi\)
\(60\) 0 0
\(61\) − 7.24264i − 0.927325i −0.886012 − 0.463663i \(-0.846535\pi\)
0.886012 − 0.463663i \(-0.153465\pi\)
\(62\) 0 0
\(63\) 6.00000 + 6.00000i 0.755929 + 0.755929i
\(64\) 0 0
\(65\) −8.48528 + 4.24264i −1.05247 + 0.526235i
\(66\) 0 0
\(67\) 4.00000 + 4.00000i 0.488678 + 0.488678i 0.907889 − 0.419211i \(-0.137693\pi\)
−0.419211 + 0.907889i \(0.637693\pi\)
\(68\) 0 0
\(69\) 0.585786i 0.0705204i
\(70\) 0 0
\(71\) 1.24264 0.147474 0.0737372 − 0.997278i \(-0.476507\pi\)
0.0737372 + 0.997278i \(0.476507\pi\)
\(72\) 0 0
\(73\) 6.00000 + 6.00000i 0.702247 + 0.702247i 0.964892 − 0.262646i \(-0.0845950\pi\)
−0.262646 + 0.964892i \(0.584595\pi\)
\(74\) 0 0
\(75\) 11.9497 − 1.70711i 1.37984 − 0.197120i
\(76\) 0 0
\(77\) −9.36396 − 3.36396i −1.06712 − 0.383359i
\(78\) 0 0
\(79\) −10.2426 −1.15239 −0.576194 − 0.817313i \(-0.695463\pi\)
−0.576194 + 0.817313i \(0.695463\pi\)
\(80\) 0 0
\(81\) 9.48528 1.05392
\(82\) 0 0
\(83\) 7.24264 + 7.24264i 0.794983 + 0.794983i 0.982300 − 0.187317i \(-0.0599790\pi\)
−0.187317 + 0.982300i \(0.559979\pi\)
\(84\) 0 0
\(85\) −7.24264 − 14.4853i −0.785575 − 1.57115i
\(86\) 0 0
\(87\) 2.12132 − 2.12132i 0.227429 − 0.227429i
\(88\) 0 0
\(89\) 5.48528i 0.581439i 0.956808 + 0.290719i \(0.0938946\pi\)
−0.956808 + 0.290719i \(0.906105\pi\)
\(90\) 0 0
\(91\) −12.7279 −1.33425
\(92\) 0 0
\(93\) −12.3640 + 12.3640i −1.28208 + 1.28208i
\(94\) 0 0
\(95\) −2.12132 + 6.36396i −0.217643 + 0.652929i
\(96\) 0 0
\(97\) −2.24264 − 2.24264i −0.227706 − 0.227706i 0.584028 − 0.811734i \(-0.301476\pi\)
−0.811734 + 0.584028i \(0.801476\pi\)
\(98\) 0 0
\(99\) 8.48528 − 4.00000i 0.852803 − 0.402015i
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.bd.c.417.1 4
4.3 odd 2 110.2.f.b.87.1 yes 4
5.3 odd 4 880.2.bd.b.593.1 4
11.10 odd 2 880.2.bd.b.417.1 4
12.11 even 2 990.2.m.c.307.2 4
20.3 even 4 110.2.f.c.43.2 yes 4
20.7 even 4 550.2.f.b.43.1 4
20.19 odd 2 550.2.f.a.307.2 4
44.43 even 2 110.2.f.c.87.2 yes 4
55.43 even 4 inner 880.2.bd.c.593.1 4
60.23 odd 4 990.2.m.d.703.1 4
132.131 odd 2 990.2.m.d.307.1 4
220.43 odd 4 110.2.f.b.43.1 ✓ 4
220.87 odd 4 550.2.f.a.43.2 4
220.219 even 2 550.2.f.b.307.1 4
660.263 even 4 990.2.m.c.703.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.2.f.b.43.1 ✓ 4 220.43 odd 4
110.2.f.b.87.1 yes 4 4.3 odd 2
110.2.f.c.43.2 yes 4 20.3 even 4
110.2.f.c.87.2 yes 4 44.43 even 2
550.2.f.a.43.2 4 220.87 odd 4
550.2.f.a.307.2 4 20.19 odd 2
550.2.f.b.43.1 4 20.7 even 4
550.2.f.b.307.1 4 220.219 even 2
880.2.bd.b.417.1 4 11.10 odd 2
880.2.bd.b.593.1 4 5.3 odd 4
880.2.bd.c.417.1 4 1.1 even 1 trivial
880.2.bd.c.593.1 4 55.43 even 4 inner
990.2.m.c.307.2 4 12.11 even 2
990.2.m.c.703.2 4 660.263 even 4
990.2.m.d.307.1 4 132.131 odd 2
990.2.m.d.703.1 4 60.23 odd 4