Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2940,2,Mod(949,2940)] 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("2940.949"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2940, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2940.bb (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,4,0,0,0,2,0,8] 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: \(23.4760181943\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1549.2
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 2940.1549
Dual form 2940.2.bb.h.949.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.866025 - 0.500000i) q^{3} +(1.86603 - 1.23205i) q^{5} +(0.500000 - 0.866025i) q^{9} +(2.00000 + 3.46410i) q^{11} +6.00000i q^{13} +(1.00000 - 2.00000i) q^{15} +(1.73205 - 1.00000i) q^{17} +(3.00000 - 5.19615i) q^{19} +(1.73205 + 1.00000i) q^{23} +(1.96410 - 4.59808i) q^{25} -1.00000i q^{27} -6.00000 q^{29} +(1.00000 + 1.73205i) q^{31} +(3.46410 + 2.00000i) q^{33} +(3.46410 + 2.00000i) q^{37} +(3.00000 + 5.19615i) q^{39} +8.00000 q^{41} +4.00000i q^{43} +(-0.133975 - 2.23205i) q^{45} +(-3.46410 - 2.00000i) q^{47} +(1.00000 - 1.73205i) q^{51} +(-5.19615 + 3.00000i) q^{53} +(8.00000 + 4.00000i) q^{55} -6.00000i q^{57} +(2.00000 + 3.46410i) q^{59} +(-7.00000 + 12.1244i) q^{61} +(7.39230 + 11.1962i) q^{65} +(3.46410 - 2.00000i) q^{67} +2.00000 q^{69} +(8.66025 - 5.00000i) q^{73} +(-0.598076 - 4.96410i) q^{75} +(-0.500000 - 0.866025i) q^{81} +16.0000i q^{83} +(2.00000 - 4.00000i) q^{85} +(-5.19615 + 3.00000i) q^{87} +(4.00000 - 6.92820i) q^{89} +(1.73205 + 1.00000i) q^{93} +(-0.803848 - 13.3923i) q^{95} +10.0000i q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} + 2 q^{9} + 8 q^{11} + 4 q^{15} + 12 q^{19} - 6 q^{25} - 24 q^{29} + 4 q^{31} + 12 q^{39} + 32 q^{41} - 4 q^{45} + 4 q^{51} + 32 q^{55} + 8 q^{59} - 28 q^{61} - 12 q^{65} + 8 q^{69} + 8 q^{75}+ \cdots + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2940\mathbb{Z}\right)^\times\).

\(n\) \(1081\) \(1177\) \(1471\) \(1961\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(1\) \(1\)

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
\(3\) 0.866025 − 0.500000i 0.500000 − 0.288675i
\(4\) 0 0
\(5\) 1.86603 − 1.23205i 0.834512 − 0.550990i
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0.500000 − 0.866025i 0.166667 − 0.288675i
\(10\) 0 0
\(11\) 2.00000 + 3.46410i 0.603023 + 1.04447i 0.992361 + 0.123371i \(0.0393705\pi\)
−0.389338 + 0.921095i \(0.627296\pi\)
\(12\) 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
\(17\) 1.73205 − 1.00000i 0.420084 − 0.242536i −0.275029 − 0.961436i \(-0.588688\pi\)
0.695113 + 0.718900i \(0.255354\pi\)
\(18\) 0 0
\(19\) 3.00000 − 5.19615i 0.688247 − 1.19208i −0.284157 − 0.958778i \(-0.591714\pi\)
0.972404 − 0.233301i \(-0.0749529\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.73205 + 1.00000i 0.361158 + 0.208514i 0.669588 − 0.742732i \(-0.266471\pi\)
−0.308431 + 0.951247i \(0.599804\pi\)
\(24\) 0 0
\(25\) 1.96410 − 4.59808i 0.392820 − 0.919615i
\(26\) 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
\(31\) 1.00000 + 1.73205i 0.179605 + 0.311086i 0.941745 − 0.336327i \(-0.109185\pi\)
−0.762140 + 0.647412i \(0.775851\pi\)
\(32\) 0 0
\(33\) 3.46410 + 2.00000i 0.603023 + 0.348155i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.46410 + 2.00000i 0.569495 + 0.328798i 0.756948 − 0.653476i \(-0.226690\pi\)
−0.187453 + 0.982274i \(0.560023\pi\)
\(38\) 0 0
\(39\) 3.00000 + 5.19615i 0.480384 + 0.832050i
\(40\) 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
\(45\) −0.133975 − 2.23205i −0.0199718 − 0.332734i
\(46\) 0 0
\(47\) −3.46410 − 2.00000i −0.505291 − 0.291730i 0.225605 − 0.974219i \(-0.427564\pi\)
−0.730896 + 0.682489i \(0.760898\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 1.00000 − 1.73205i 0.140028 − 0.242536i
\(52\) 0 0
\(53\) −5.19615 + 3.00000i −0.713746 + 0.412082i −0.812447 − 0.583036i \(-0.801865\pi\)
0.0987002 + 0.995117i \(0.468532\pi\)
\(54\) 0 0
\(55\) 8.00000 + 4.00000i 1.07872 + 0.539360i
\(56\) 0 0
\(57\) − 6.00000i − 0.794719i
\(58\) 0 0
\(59\) 2.00000 + 3.46410i 0.260378 + 0.450988i 0.966342 − 0.257260i \(-0.0828195\pi\)
−0.705965 + 0.708247i \(0.749486\pi\)
\(60\) 0 0
\(61\) −7.00000 + 12.1244i −0.896258 + 1.55236i −0.0640184 + 0.997949i \(0.520392\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 7.39230 + 11.1962i 0.916903 + 1.38871i
\(66\) 0 0
\(67\) 3.46410 − 2.00000i 0.423207 − 0.244339i −0.273241 − 0.961946i \(-0.588096\pi\)
0.696449 + 0.717607i \(0.254762\pi\)
\(68\) 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
\(73\) 8.66025 − 5.00000i 1.01361 − 0.585206i 0.101361 − 0.994850i \(-0.467680\pi\)
0.912245 + 0.409644i \(0.134347\pi\)
\(74\) 0 0
\(75\) −0.598076 − 4.96410i −0.0690599 − 0.573205i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(80\) 0 0
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 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
\(87\) −5.19615 + 3.00000i −0.557086 + 0.321634i
\(88\) 0 0
\(89\) 4.00000 − 6.92820i 0.423999 − 0.734388i −0.572327 − 0.820025i \(-0.693959\pi\)
0.996326 + 0.0856373i \(0.0272926\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 1.73205 + 1.00000i 0.179605 + 0.103695i
\(94\) 0 0
\(95\) −0.803848 − 13.3923i −0.0824730 − 1.37402i
\(96\) 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 2940.2.bb.h.1549.2 4
5.4 even 2 inner 2940.2.bb.h.1549.1 4
7.2 even 3 420.2.k.a.169.2 yes 2
7.3 odd 6 2940.2.bb.c.949.2 4
7.4 even 3 inner 2940.2.bb.h.949.1 4
7.5 odd 6 2940.2.k.d.589.1 2
7.6 odd 2 2940.2.bb.c.1549.1 4
21.2 odd 6 1260.2.k.d.1009.2 2
28.23 odd 6 1680.2.t.a.1009.1 2
35.2 odd 12 2100.2.a.j.1.1 1
35.4 even 6 inner 2940.2.bb.h.949.2 4
35.9 even 6 420.2.k.a.169.1 ✓ 2
35.19 odd 6 2940.2.k.d.589.2 2
35.23 odd 12 2100.2.a.e.1.1 1
35.24 odd 6 2940.2.bb.c.949.1 4
35.34 odd 2 2940.2.bb.c.1549.2 4
84.23 even 6 5040.2.t.o.1009.2 2
105.2 even 12 6300.2.a.n.1.1 1
105.23 even 12 6300.2.a.bc.1.1 1
105.44 odd 6 1260.2.k.d.1009.1 2
140.23 even 12 8400.2.a.cd.1.1 1
140.79 odd 6 1680.2.t.a.1009.2 2
140.107 even 12 8400.2.a.bh.1.1 1
420.359 even 6 5040.2.t.o.1009.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.k.a.169.1 ✓ 2 35.9 even 6
420.2.k.a.169.2 yes 2 7.2 even 3
1260.2.k.d.1009.1 2 105.44 odd 6
1260.2.k.d.1009.2 2 21.2 odd 6
1680.2.t.a.1009.1 2 28.23 odd 6
1680.2.t.a.1009.2 2 140.79 odd 6
2100.2.a.e.1.1 1 35.23 odd 12
2100.2.a.j.1.1 1 35.2 odd 12
2940.2.k.d.589.1 2 7.5 odd 6
2940.2.k.d.589.2 2 35.19 odd 6
2940.2.bb.c.949.1 4 35.24 odd 6
2940.2.bb.c.949.2 4 7.3 odd 6
2940.2.bb.c.1549.1 4 7.6 odd 2
2940.2.bb.c.1549.2 4 35.34 odd 2
2940.2.bb.h.949.1 4 7.4 even 3 inner
2940.2.bb.h.949.2 4 35.4 even 6 inner
2940.2.bb.h.1549.1 4 5.4 even 2 inner
2940.2.bb.h.1549.2 4 1.1 even 1 trivial
5040.2.t.o.1009.1 2 420.359 even 6
5040.2.t.o.1009.2 2 84.23 even 6
6300.2.a.n.1.1 1 105.2 even 12
6300.2.a.bc.1.1 1 105.23 even 12
8400.2.a.bh.1.1 1 140.107 even 12
8400.2.a.cd.1.1 1 140.23 even 12