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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1260,2,Mod(1009,1260)] 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("1260.1009"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1260, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1260.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,4,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.0611506547\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1009.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1260.1009
Dual form 1260.2.k.d.1009.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.00000 - 1.00000i) q^{5} -1.00000i q^{7} +4.00000 q^{11} -6.00000i q^{13} +2.00000i q^{17} -6.00000 q^{19} -2.00000i q^{23} +(3.00000 - 4.00000i) q^{25} +6.00000 q^{29} -2.00000 q^{31} +(-1.00000 - 2.00000i) q^{35} +4.00000i q^{37} -8.00000 q^{41} -4.00000i q^{43} +4.00000i q^{47} -1.00000 q^{49} -6.00000i q^{53} +(8.00000 - 4.00000i) q^{55} +4.00000 q^{59} +14.0000 q^{61} +(-6.00000 - 12.0000i) q^{65} -4.00000i q^{67} -10.0000i q^{73} -4.00000i q^{77} +16.0000i q^{83} +(2.00000 + 4.00000i) q^{85} +8.00000 q^{89} -6.00000 q^{91} +(-12.0000 + 6.00000i) q^{95} -10.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5} + 8 q^{11} - 12 q^{19} + 6 q^{25} + 12 q^{29} - 4 q^{31} - 2 q^{35} - 16 q^{41} - 2 q^{49} + 16 q^{55} + 8 q^{59} + 28 q^{61} - 12 q^{65} + 4 q^{85} + 16 q^{89} - 12 q^{91} - 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 2.00000 − 1.00000i 0.894427 − 0.447214i
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) − 6.00000i − 1.66410i −0.554700 − 0.832050i \(-0.687167\pi\)
0.554700 − 0.832050i \(-0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 − 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 2.00000i − 0.417029i −0.978019 − 0.208514i \(-0.933137\pi\)
0.978019 − 0.208514i \(-0.0668628\pi\)
\(24\) 0 0
\(25\) 3.00000 − 4.00000i 0.600000 − 0.800000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.00000 − 2.00000i −0.169031 − 0.338062i
\(36\) 0 0
\(37\) 4.00000i 0.657596i 0.944400 + 0.328798i \(0.106644\pi\)
−0.944400 + 0.328798i \(0.893356\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 − 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 − 0.304997i \(-0.901344\pi\)
0.952353 − 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.00000i 0.583460i 0.956501 + 0.291730i \(0.0942309\pi\)
−0.956501 + 0.291730i \(0.905769\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 6.00000i − 0.824163i −0.911147 − 0.412082i \(-0.864802\pi\)
0.911147 − 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 8.00000 − 4.00000i 1.07872 − 0.539360i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 − 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.00000 − 12.0000i −0.744208 − 1.48842i
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 − 0.244339i \(-0.921429\pi\)
0.969690 − 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) − 10.0000i − 1.17041i −0.810885 − 0.585206i \(-0.801014\pi\)
0.810885 − 0.585206i \(-0.198986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 4.00000i − 0.455842i
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 16.0000i 1.75623i 0.478451 + 0.878114i \(0.341198\pi\)
−0.478451 + 0.878114i \(0.658802\pi\)
\(84\) 0 0
\(85\) 2.00000 + 4.00000i 0.216930 + 0.433861i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 8.00000 0.847998 0.423999 − 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −12.0000 + 6.00000i −1.23117 + 0.615587i
\(96\) 0 0
\(97\) − 10.0000i − 1.01535i −0.861550 − 0.507673i \(-0.830506\pi\)
0.861550 − 0.507673i \(-0.169494\pi\)
\(98\) 0 0
\(99\) 0 0
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 1260.2.k.d.1009.1 2
3.2 odd 2 420.2.k.a.169.1 ✓ 2
4.3 odd 2 5040.2.t.o.1009.1 2
5.2 odd 4 6300.2.a.bc.1.1 1
5.3 odd 4 6300.2.a.n.1.1 1
5.4 even 2 inner 1260.2.k.d.1009.2 2
12.11 even 2 1680.2.t.a.1009.2 2
15.2 even 4 2100.2.a.e.1.1 1
15.8 even 4 2100.2.a.j.1.1 1
15.14 odd 2 420.2.k.a.169.2 yes 2
20.19 odd 2 5040.2.t.o.1009.2 2
21.2 odd 6 2940.2.bb.h.949.2 4
21.5 even 6 2940.2.bb.c.949.1 4
21.11 odd 6 2940.2.bb.h.1549.1 4
21.17 even 6 2940.2.bb.c.1549.2 4
21.20 even 2 2940.2.k.d.589.2 2
60.23 odd 4 8400.2.a.bh.1.1 1
60.47 odd 4 8400.2.a.cd.1.1 1
60.59 even 2 1680.2.t.a.1009.1 2
105.44 odd 6 2940.2.bb.h.949.1 4
105.59 even 6 2940.2.bb.c.1549.1 4
105.74 odd 6 2940.2.bb.h.1549.2 4
105.89 even 6 2940.2.bb.c.949.2 4
105.104 even 2 2940.2.k.d.589.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.k.a.169.1 ✓ 2 3.2 odd 2
420.2.k.a.169.2 yes 2 15.14 odd 2
1260.2.k.d.1009.1 2 1.1 even 1 trivial
1260.2.k.d.1009.2 2 5.4 even 2 inner
1680.2.t.a.1009.1 2 60.59 even 2
1680.2.t.a.1009.2 2 12.11 even 2
2100.2.a.e.1.1 1 15.2 even 4
2100.2.a.j.1.1 1 15.8 even 4
2940.2.k.d.589.1 2 105.104 even 2
2940.2.k.d.589.2 2 21.20 even 2
2940.2.bb.c.949.1 4 21.5 even 6
2940.2.bb.c.949.2 4 105.89 even 6
2940.2.bb.c.1549.1 4 105.59 even 6
2940.2.bb.c.1549.2 4 21.17 even 6
2940.2.bb.h.949.1 4 105.44 odd 6
2940.2.bb.h.949.2 4 21.2 odd 6
2940.2.bb.h.1549.1 4 21.11 odd 6
2940.2.bb.h.1549.2 4 105.74 odd 6
5040.2.t.o.1009.1 2 4.3 odd 2
5040.2.t.o.1009.2 2 20.19 odd 2
6300.2.a.n.1.1 1 5.3 odd 4
6300.2.a.bc.1.1 1 5.2 odd 4
8400.2.a.bh.1.1 1 60.23 odd 4
8400.2.a.cd.1.1 1 60.47 odd 4