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 1549.1
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 2940.1549
Dual form 2940.2.bb.h.949.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.866025 + 0.500000i) q^{3} +(0.133975 - 2.23205i) q^{5} +(0.500000 - 0.866025i) q^{9} +(2.00000 + 3.46410i) q^{11} -6.00000i q^{13} +(1.00000 + 2.00000i) q^{15} +(-1.73205 + 1.00000i) q^{17} +(3.00000 - 5.19615i) q^{19} +(-1.73205 - 1.00000i) q^{23} +(-4.96410 - 0.598076i) q^{25} +1.00000i q^{27} -6.00000 q^{29} +(1.00000 + 1.73205i) q^{31} +(-3.46410 - 2.00000i) q^{33} +(-3.46410 - 2.00000i) q^{37} +(3.00000 + 5.19615i) q^{39} +8.00000 q^{41} -4.00000i q^{43} +(-1.86603 - 1.23205i) q^{45} +(3.46410 + 2.00000i) q^{47} +(1.00000 - 1.73205i) q^{51} +(5.19615 - 3.00000i) q^{53} +(8.00000 - 4.00000i) q^{55} +6.00000i q^{57} +(2.00000 + 3.46410i) q^{59} +(-7.00000 + 12.1244i) q^{61} +(-13.3923 - 0.803848i) q^{65} +(-3.46410 + 2.00000i) q^{67} +2.00000 q^{69} +(-8.66025 + 5.00000i) q^{73} +(4.59808 - 1.96410i) q^{75} +(-0.500000 - 0.866025i) q^{81} -16.0000i q^{83} +(2.00000 + 4.00000i) q^{85} +(5.19615 - 3.00000i) q^{87} +(4.00000 - 6.92820i) q^{89} +(-1.73205 - 1.00000i) q^{93} +(-11.1962 - 7.39230i) q^{95} -10.0000i q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} + 2 q^{9} + 8 q^{11} + 4 q^{15} + 12 q^{19} - 6 q^{25} - 24 q^{29} + 4 q^{31} + 12 q^{39} + 32 q^{41} - 4 q^{45} + 4 q^{51} + 32 q^{55} + 8 q^{59} - 28 q^{61} - 12 q^{65} + 8 q^{69} + 8 q^{75}+ \cdots + 16 q^{99}+O(q^{100}) \) 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
\(5\) 0.133975 − 2.23205i 0.0599153 − 0.998203i
\(6\) 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 \(-0.687167\pi\)
0.554700 − 0.832050i \(-0.312833\pi\)
\(14\) 0 0
\(15\) 1.00000 + 2.00000i 0.258199 + 0.516398i
\(16\) 0 0
\(17\) −1.73205 + 1.00000i −0.420084 + 0.242536i −0.695113 − 0.718900i \(-0.744646\pi\)
0.275029 + 0.961436i \(0.411312\pi\)
\(18\) 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 0.308431 − 0.951247i \(-0.400196\pi\)
−0.669588 + 0.742732i \(0.733529\pi\)
\(24\) 0 0
\(25\) −4.96410 − 0.598076i −0.992820 − 0.119615i
\(26\) 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 0.187453 − 0.982274i \(-0.439977\pi\)
−0.756948 + 0.653476i \(0.773310\pi\)
\(38\) 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 \(-0.901344\pi\)
0.952353 − 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) −1.86603 − 1.23205i −0.278171 − 0.183663i
\(46\) 0 0
\(47\) 3.46410 + 2.00000i 0.505291 + 0.291730i 0.730896 − 0.682489i \(-0.239102\pi\)
−0.225605 + 0.974219i \(0.572436\pi\)
\(48\) 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 −0.0987002 − 0.995117i \(-0.531468\pi\)
0.812447 + 0.583036i \(0.198135\pi\)
\(54\) 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
\(65\) −13.3923 − 0.803848i −1.66111 − 0.0997050i
\(66\) 0 0
\(67\) −3.46410 + 2.00000i −0.423207 + 0.244339i −0.696449 − 0.717607i \(-0.745238\pi\)
0.273241 + 0.961946i \(0.411904\pi\)
\(68\) 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 −0.912245 − 0.409644i \(-0.865653\pi\)
−0.101361 + 0.994850i \(0.532320\pi\)
\(74\) 0 0
\(75\) 4.59808 − 1.96410i 0.530940 − 0.226795i
\(76\) 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 \(-0.658802\pi\)
0.478451 − 0.878114i \(-0.341198\pi\)
\(84\) 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
\(95\) −11.1962 − 7.39230i −1.14870 − 0.758434i
\(96\) 0 0
\(97\) − 10.0000i − 1.01535i −0.861550 − 0.507673i \(-0.830506\pi\)
0.861550 − 0.507673i \(-0.169494\pi\)
\(98\) 0 0
\(99\) 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.1549.1 4
5.4 even 2 inner 2940.2.bb.h.1549.2 4
7.2 even 3 420.2.k.a.169.1 ✓ 2
7.3 odd 6 2940.2.bb.c.949.1 4
7.4 even 3 inner 2940.2.bb.h.949.2 4
7.5 odd 6 2940.2.k.d.589.2 2
7.6 odd 2 2940.2.bb.c.1549.2 4
21.2 odd 6 1260.2.k.d.1009.1 2
28.23 odd 6 1680.2.t.a.1009.2 2
35.2 odd 12 2100.2.a.e.1.1 1
35.4 even 6 inner 2940.2.bb.h.949.1 4
35.9 even 6 420.2.k.a.169.2 yes 2
35.19 odd 6 2940.2.k.d.589.1 2
35.23 odd 12 2100.2.a.j.1.1 1
35.24 odd 6 2940.2.bb.c.949.2 4
35.34 odd 2 2940.2.bb.c.1549.1 4
84.23 even 6 5040.2.t.o.1009.1 2
105.2 even 12 6300.2.a.bc.1.1 1
105.23 even 12 6300.2.a.n.1.1 1
105.44 odd 6 1260.2.k.d.1009.2 2
140.23 even 12 8400.2.a.bh.1.1 1
140.79 odd 6 1680.2.t.a.1009.1 2
140.107 even 12 8400.2.a.cd.1.1 1
420.359 even 6 5040.2.t.o.1009.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.k.a.169.1 ✓ 2 7.2 even 3
420.2.k.a.169.2 yes 2 35.9 even 6
1260.2.k.d.1009.1 2 21.2 odd 6
1260.2.k.d.1009.2 2 105.44 odd 6
1680.2.t.a.1009.1 2 140.79 odd 6
1680.2.t.a.1009.2 2 28.23 odd 6
2100.2.a.e.1.1 1 35.2 odd 12
2100.2.a.j.1.1 1 35.23 odd 12
2940.2.k.d.589.1 2 35.19 odd 6
2940.2.k.d.589.2 2 7.5 odd 6
2940.2.bb.c.949.1 4 7.3 odd 6
2940.2.bb.c.949.2 4 35.24 odd 6
2940.2.bb.c.1549.1 4 35.34 odd 2
2940.2.bb.c.1549.2 4 7.6 odd 2
2940.2.bb.h.949.1 4 35.4 even 6 inner
2940.2.bb.h.949.2 4 7.4 even 3 inner
2940.2.bb.h.1549.1 4 1.1 even 1 trivial
2940.2.bb.h.1549.2 4 5.4 even 2 inner
5040.2.t.o.1009.1 2 84.23 even 6
5040.2.t.o.1009.2 2 420.359 even 6
6300.2.a.n.1.1 1 105.23 even 12
6300.2.a.bc.1.1 1 105.2 even 12
8400.2.a.bh.1.1 1 140.23 even 12
8400.2.a.cd.1.1 1 140.107 even 12