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Properties

Label 1000.2.q.c.49.1
Level $1000$
Weight $2$
Character 1000.49
Analytic conductor $7.985$
Analytic rank $0$
Dimension $32$
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] = [1000,2,Mod(49,1000)] 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("1000.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1000, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1000.q (of order \(10\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,0,0,0,0,0,10] 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: \(7.98504020213\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 200)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label 49.1
Character \(\chi\) \(=\) 1000.49
Dual form 1000.2.q.c.449.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.79539 + 0.908276i) q^{3} +3.13612i q^{7} +(4.56217 - 3.31461i) q^{9} +(2.75289 + 2.00009i) q^{11} +(-1.49708 - 2.06055i) q^{13} +(4.78108 + 1.55347i) q^{17} +(-1.49043 + 4.58707i) q^{19} +(-2.84847 - 8.76668i) q^{21} +(4.65112 - 6.40172i) q^{23} +(-4.55951 + 6.27563i) q^{27} +(-0.109546 - 0.337149i) q^{29} +(-0.453770 + 1.39656i) q^{31} +(-9.51203 - 3.09064i) q^{33} +(3.48437 + 4.79583i) q^{37} +(6.05645 + 4.40027i) q^{39} +(-6.18135 + 4.49101i) q^{41} +9.20779i q^{43} +(-2.87129 + 0.932938i) q^{47} -2.83527 q^{49} -14.7760 q^{51} +(-8.05366 + 2.61679i) q^{53} -14.1764i q^{57} +(-3.49540 + 2.53956i) q^{59} +(-6.86607 - 4.98849i) q^{61} +(10.3950 + 14.3075i) q^{63} +(-8.19301 - 2.66207i) q^{67} +(-7.18715 + 22.1198i) q^{69} +(2.28174 + 7.02248i) q^{71} +(-1.38366 + 1.90444i) q^{73} +(-6.27253 + 8.63340i) q^{77} +(-0.0627911 - 0.193251i) q^{79} +(1.81781 - 5.59466i) q^{81} +(-8.33641 - 2.70866i) q^{83} +(0.612449 + 0.842963i) q^{87} +(-2.41967 - 1.75799i) q^{89} +(6.46213 - 4.69501i) q^{91} -4.31607i q^{93} +(9.72444 - 3.15966i) q^{97} +19.1887 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 10 q^{9} + 6 q^{11} - 6 q^{19} - 4 q^{21} + 30 q^{23} - 2 q^{29} + 6 q^{31} + 40 q^{37} - 12 q^{39} + 20 q^{47} - 60 q^{49} - 60 q^{51} + 30 q^{53} - 30 q^{59} + 14 q^{61} + 20 q^{63} - 4 q^{69}+ \cdots + 92 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1000\mathbb{Z}\right)^\times\).

\(n\) \(377\) \(501\) \(751\)
\(\chi(n)\) \(e\left(\frac{7}{10}\right)\) \(1\) \(1\)

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
\(3\) −2.79539 + 0.908276i −1.61392 + 0.524393i −0.970496 − 0.241119i \(-0.922486\pi\)
−0.643421 + 0.765512i \(0.722486\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.13612i 1.18534i 0.805444 + 0.592671i \(0.201927\pi\)
−0.805444 + 0.592671i \(0.798073\pi\)
\(8\) 0 0
\(9\) 4.56217 − 3.31461i 1.52072 − 1.10487i
\(10\) 0 0
\(11\) 2.75289 + 2.00009i 0.830028 + 0.603050i 0.919567 − 0.392933i \(-0.128540\pi\)
−0.0895396 + 0.995983i \(0.528540\pi\)
\(12\) 0 0
\(13\) −1.49708 − 2.06055i −0.415214 − 0.571493i 0.549266 − 0.835647i \(-0.314907\pi\)
−0.964480 + 0.264154i \(0.914907\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.78108 + 1.55347i 1.15958 + 0.376771i 0.824745 − 0.565504i \(-0.191318\pi\)
0.334838 + 0.942276i \(0.391318\pi\)
\(18\) 0 0
\(19\) −1.49043 + 4.58707i −0.341928 + 1.05235i 0.621280 + 0.783589i \(0.286613\pi\)
−0.963208 + 0.268757i \(0.913387\pi\)
\(20\) 0 0
\(21\) −2.84847 − 8.76668i −0.621586 − 1.91305i
\(22\) 0 0
\(23\) 4.65112 − 6.40172i 0.969825 − 1.33485i 0.0276903 − 0.999617i \(-0.491185\pi\)
0.942135 − 0.335234i \(-0.108815\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −4.55951 + 6.27563i −0.877479 + 1.20775i
\(28\) 0 0
\(29\) −0.109546 − 0.337149i −0.0203422 − 0.0626070i 0.940370 − 0.340153i \(-0.110479\pi\)
−0.960712 + 0.277546i \(0.910479\pi\)
\(30\) 0 0
\(31\) −0.453770 + 1.39656i −0.0814994 + 0.250829i −0.983501 − 0.180904i \(-0.942098\pi\)
0.902001 + 0.431733i \(0.142098\pi\)
\(32\) 0 0
\(33\) −9.51203 − 3.09064i −1.65583 − 0.538012i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.48437 + 4.79583i 0.572828 + 0.788430i 0.992886 − 0.119067i \(-0.0379905\pi\)
−0.420059 + 0.907497i \(0.637990\pi\)
\(38\) 0 0
\(39\) 6.05645 + 4.40027i 0.969808 + 0.704607i
\(40\) 0 0
\(41\) −6.18135 + 4.49101i −0.965365 + 0.701379i −0.954390 − 0.298561i \(-0.903493\pi\)
−0.0109742 + 0.999940i \(0.503493\pi\)
\(42\) 0 0
\(43\) 9.20779i 1.40418i 0.712090 + 0.702088i \(0.247749\pi\)
−0.712090 + 0.702088i \(0.752251\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.87129 + 0.932938i −0.418821 + 0.136083i −0.510843 − 0.859674i \(-0.670667\pi\)
0.0920227 + 0.995757i \(0.470667\pi\)
\(48\) 0 0
\(49\) −2.83527 −0.405038
\(50\) 0 0
\(51\) −14.7760 −2.06905
\(52\) 0 0
\(53\) −8.05366 + 2.61679i −1.10625 + 0.359444i −0.804507 − 0.593944i \(-0.797570\pi\)
−0.301748 + 0.953388i \(0.597570\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 14.1764i − 1.87770i
\(58\) 0 0
\(59\) −3.49540 + 2.53956i −0.455062 + 0.330622i −0.791591 − 0.611051i \(-0.790747\pi\)
0.336529 + 0.941673i \(0.390747\pi\)
\(60\) 0 0
\(61\) −6.86607 − 4.98849i −0.879110 − 0.638711i 0.0539061 − 0.998546i \(-0.482833\pi\)
−0.933016 + 0.359835i \(0.882833\pi\)
\(62\) 0 0
\(63\) 10.3950 + 14.3075i 1.30965 + 1.80258i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −8.19301 − 2.66207i −1.00094 − 0.325223i −0.237696 − 0.971340i \(-0.576392\pi\)
−0.763239 + 0.646116i \(0.776392\pi\)
\(68\) 0 0
\(69\) −7.18715 + 22.1198i −0.865231 + 2.66291i
\(70\) 0 0
\(71\) 2.28174 + 7.02248i 0.270793 + 0.833415i 0.990302 + 0.138932i \(0.0443669\pi\)
−0.719509 + 0.694483i \(0.755633\pi\)
\(72\) 0 0
\(73\) −1.38366 + 1.90444i −0.161945 + 0.222898i −0.882276 − 0.470732i \(-0.843990\pi\)
0.720331 + 0.693630i \(0.243990\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.27253 + 8.63340i −0.714821 + 0.983867i
\(78\) 0 0
\(79\) −0.0627911 − 0.193251i −0.00706455 − 0.0217424i 0.947462 − 0.319868i \(-0.103639\pi\)
−0.954527 + 0.298126i \(0.903639\pi\)
\(80\) 0 0
\(81\) 1.81781 − 5.59466i 0.201979 − 0.621629i
\(82\) 0 0
\(83\) −8.33641 − 2.70866i −0.915040 − 0.297314i −0.186609 − 0.982434i \(-0.559750\pi\)
−0.728430 + 0.685120i \(0.759750\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0.612449 + 0.842963i 0.0656614 + 0.0903752i
\(88\) 0 0
\(89\) −2.41967 − 1.75799i −0.256484 − 0.186347i 0.452112 − 0.891961i \(-0.350671\pi\)
−0.708596 + 0.705615i \(0.750671\pi\)
\(90\) 0 0
\(91\) 6.46213 − 4.69501i 0.677415 − 0.492171i
\(92\) 0 0
\(93\) − 4.31607i − 0.447556i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9.72444 − 3.15966i 0.987367 − 0.320815i 0.229561 − 0.973294i \(-0.426271\pi\)
0.757807 + 0.652479i \(0.226271\pi\)
\(98\) 0 0
\(99\) 19.1887 1.92853
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 1000.2.q.c.49.1 32
5.2 odd 4 1000.2.m.d.201.7 32
5.3 odd 4 1000.2.m.e.201.2 32
5.4 even 2 200.2.q.a.9.8 ✓ 32
20.19 odd 2 400.2.y.d.209.1 32
25.2 odd 20 1000.2.m.d.801.7 32
25.8 odd 20 5000.2.a.q.1.3 16
25.11 even 5 200.2.q.a.89.8 yes 32
25.14 even 10 inner 1000.2.q.c.449.1 32
25.17 odd 20 5000.2.a.r.1.14 16
25.23 odd 20 1000.2.m.e.801.2 32
100.11 odd 10 400.2.y.d.289.1 32
100.67 even 20 10000.2.a.bq.1.3 16
100.83 even 20 10000.2.a.br.1.14 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.2.q.a.9.8 ✓ 32 5.4 even 2
200.2.q.a.89.8 yes 32 25.11 even 5
400.2.y.d.209.1 32 20.19 odd 2
400.2.y.d.289.1 32 100.11 odd 10
1000.2.m.d.201.7 32 5.2 odd 4
1000.2.m.d.801.7 32 25.2 odd 20
1000.2.m.e.201.2 32 5.3 odd 4
1000.2.m.e.801.2 32 25.23 odd 20
1000.2.q.c.49.1 32 1.1 even 1 trivial
1000.2.q.c.449.1 32 25.14 even 10 inner
5000.2.a.q.1.3 16 25.8 odd 20
5000.2.a.r.1.14 16 25.17 odd 20
10000.2.a.bq.1.3 16 100.67 even 20
10000.2.a.br.1.14 16 100.83 even 20