Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [420,2,Mod(169,420)] 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("420.169"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(420, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 420.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.35371688489\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 169.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 420.169
Dual form 420.2.k.a.169.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} +(-2.00000 - 1.00000i) q^{5} +1.00000i q^{7} -1.00000 q^{9} -4.00000 q^{11} +6.00000i q^{13} +(1.00000 - 2.00000i) q^{15} +2.00000i q^{17} -6.00000 q^{19} -1.00000 q^{21} -2.00000i q^{23} +(3.00000 + 4.00000i) q^{25} -1.00000i q^{27} -6.00000 q^{29} -2.00000 q^{31} -4.00000i q^{33} +(1.00000 - 2.00000i) q^{35} -4.00000i q^{37} -6.00000 q^{39} +8.00000 q^{41} +4.00000i q^{43} +(2.00000 + 1.00000i) q^{45} +4.00000i q^{47} -1.00000 q^{49} -2.00000 q^{51} -6.00000i q^{53} +(8.00000 + 4.00000i) q^{55} -6.00000i q^{57} -4.00000 q^{59} +14.0000 q^{61} -1.00000i q^{63} +(6.00000 - 12.0000i) q^{65} +4.00000i q^{67} +2.00000 q^{69} +10.0000i q^{73} +(-4.00000 + 3.00000i) q^{75} -4.00000i q^{77} +1.00000 q^{81} +16.0000i q^{83} +(2.00000 - 4.00000i) q^{85} -6.00000i q^{87} -8.00000 q^{89} -6.00000 q^{91} -2.00000i q^{93} +(12.0000 + 6.00000i) q^{95} +10.0000i q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} - 2 q^{9} - 8 q^{11} + 2 q^{15} - 12 q^{19} - 2 q^{21} + 6 q^{25} - 12 q^{29} - 4 q^{31} + 2 q^{35} - 12 q^{39} + 16 q^{41} + 4 q^{45} - 2 q^{49} - 4 q^{51} + 16 q^{55} - 8 q^{59} + 28 q^{61}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/420\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(241\) \(281\) \(337\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) −2.00000 − 1.00000i −0.894427 − 0.447214i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 6.00000i 1.66410i 0.554700 + 0.832050i \(0.312833\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 1.00000 − 2.00000i 0.258199 − 0.516398i
\(16\) 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
\(21\) −1.00000 −0.218218
\(22\) 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
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) − 4.00000i − 0.696311i
\(34\) 0 0
\(35\) 1.00000 − 2.00000i 0.169031 − 0.338062i
\(36\) 0 0
\(37\) − 4.00000i − 0.657596i −0.944400 − 0.328798i \(-0.893356\pi\)
0.944400 − 0.328798i \(-0.106644\pi\)
\(38\) 0 0
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 − 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 2.00000 + 1.00000i 0.298142 + 0.149071i
\(46\) 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
\(51\) −2.00000 −0.280056
\(52\) 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
\(57\) − 6.00000i − 0.794719i
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 − 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) − 1.00000i − 0.125988i
\(64\) 0 0
\(65\) 6.00000 − 12.0000i 0.744208 − 1.48842i
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 0 0
\(69\) 2.00000 0.240772
\(70\) 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 \(0.198986\pi\)
−0.810885 + 0.585206i \(0.801014\pi\)
\(74\) 0 0
\(75\) −4.00000 + 3.00000i −0.461880 + 0.346410i
\(76\) 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
\(81\) 1.00000 0.111111
\(82\) 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
\(87\) − 6.00000i − 0.643268i
\(88\) 0 0
\(89\) −8.00000 −0.847998 −0.423999 − 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 0 0
\(93\) − 2.00000i − 0.207390i
\(94\) 0 0
\(95\) 12.0000 + 6.00000i 1.23117 + 0.615587i
\(96\) 0 0
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
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 420.2.k.a.169.2 yes 2
3.2 odd 2 1260.2.k.d.1009.2 2
4.3 odd 2 1680.2.t.a.1009.1 2
5.2 odd 4 2100.2.a.j.1.1 1
5.3 odd 4 2100.2.a.e.1.1 1
5.4 even 2 inner 420.2.k.a.169.1 ✓ 2
7.2 even 3 2940.2.bb.h.949.1 4
7.3 odd 6 2940.2.bb.c.1549.1 4
7.4 even 3 2940.2.bb.h.1549.2 4
7.5 odd 6 2940.2.bb.c.949.2 4
7.6 odd 2 2940.2.k.d.589.1 2
12.11 even 2 5040.2.t.o.1009.2 2
15.2 even 4 6300.2.a.n.1.1 1
15.8 even 4 6300.2.a.bc.1.1 1
15.14 odd 2 1260.2.k.d.1009.1 2
20.3 even 4 8400.2.a.cd.1.1 1
20.7 even 4 8400.2.a.bh.1.1 1
20.19 odd 2 1680.2.t.a.1009.2 2
35.4 even 6 2940.2.bb.h.1549.1 4
35.9 even 6 2940.2.bb.h.949.2 4
35.19 odd 6 2940.2.bb.c.949.1 4
35.24 odd 6 2940.2.bb.c.1549.2 4
35.34 odd 2 2940.2.k.d.589.2 2
60.59 even 2 5040.2.t.o.1009.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.k.a.169.1 ✓ 2 5.4 even 2 inner
420.2.k.a.169.2 yes 2 1.1 even 1 trivial
1260.2.k.d.1009.1 2 15.14 odd 2
1260.2.k.d.1009.2 2 3.2 odd 2
1680.2.t.a.1009.1 2 4.3 odd 2
1680.2.t.a.1009.2 2 20.19 odd 2
2100.2.a.e.1.1 1 5.3 odd 4
2100.2.a.j.1.1 1 5.2 odd 4
2940.2.k.d.589.1 2 7.6 odd 2
2940.2.k.d.589.2 2 35.34 odd 2
2940.2.bb.c.949.1 4 35.19 odd 6
2940.2.bb.c.949.2 4 7.5 odd 6
2940.2.bb.c.1549.1 4 7.3 odd 6
2940.2.bb.c.1549.2 4 35.24 odd 6
2940.2.bb.h.949.1 4 7.2 even 3
2940.2.bb.h.949.2 4 35.9 even 6
2940.2.bb.h.1549.1 4 35.4 even 6
2940.2.bb.h.1549.2 4 7.4 even 3
5040.2.t.o.1009.1 2 60.59 even 2
5040.2.t.o.1009.2 2 12.11 even 2
6300.2.a.n.1.1 1 15.2 even 4
6300.2.a.bc.1.1 1 15.8 even 4
8400.2.a.bh.1.1 1 20.7 even 4
8400.2.a.cd.1.1 1 20.3 even 4