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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3024,2,Mod(289,3024)] 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("3024.289"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3024, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.t (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [22,0,0,0,2,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1873.11
Character \(\chi\) \(=\) 3024.1873
Dual form 3024.2.t.k.289.11

$q$-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)
 
\(f(q)\) \(=\) \(q+4.22296 q^{5} +(2.37802 - 1.15974i) q^{7} +1.92915 q^{11} +(-0.291529 + 0.504943i) q^{13} +(-3.61082 + 6.25412i) q^{17} +(-2.10268 - 3.64194i) q^{19} +1.27988 q^{23} +12.8334 q^{25} +(4.20305 + 7.27990i) q^{29} +(-0.476061 - 0.824561i) q^{31} +(10.0423 - 4.89755i) q^{35} +(3.03329 + 5.25381i) q^{37} +(-1.31299 + 2.27416i) q^{41} +(-0.442349 - 0.766171i) q^{43} +(-2.88201 + 4.99178i) q^{47} +(4.30999 - 5.51579i) q^{49} +(0.962456 - 1.66702i) q^{53} +8.14673 q^{55} +(2.27614 + 3.94240i) q^{59} +(5.29008 - 9.16268i) q^{61} +(-1.23112 + 2.13236i) q^{65} +(-2.43191 - 4.21220i) q^{67} +11.5443 q^{71} +(0.446138 - 0.772734i) q^{73} +(4.58756 - 2.23732i) q^{77} +(-5.93520 + 10.2801i) q^{79} +(-5.24250 - 9.08028i) q^{83} +(-15.2484 + 26.4109i) q^{85} +(-3.87906 - 6.71874i) q^{89} +(-0.107659 + 1.53887i) q^{91} +(-8.87953 - 15.3798i) q^{95} +(-1.98651 - 3.44073i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q + 2 q^{5} + q^{7} - 6 q^{11} + 7 q^{13} + q^{17} - 13 q^{19} + 44 q^{25} + 7 q^{29} - 6 q^{31} + 2 q^{35} + 6 q^{37} - 4 q^{41} - 2 q^{43} + 17 q^{47} + 29 q^{49} - q^{53} - 2 q^{55} - 21 q^{59}+ \cdots + 19 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(5\) 4.22296 1.88857 0.944283 − 0.329134i \(-0.106757\pi\)
0.944283 + 0.329134i \(0.106757\pi\)
\(6\) 0 0
\(7\) 2.37802 − 1.15974i 0.898809 − 0.438341i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.92915 0.581660 0.290830 − 0.956775i \(-0.406069\pi\)
0.290830 + 0.956775i \(0.406069\pi\)
\(12\) 0 0
\(13\) −0.291529 + 0.504943i −0.0808557 + 0.140046i −0.903618 − 0.428340i \(-0.859099\pi\)
0.822762 + 0.568386i \(0.192432\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.61082 + 6.25412i −0.875753 + 1.51685i −0.0197936 + 0.999804i \(0.506301\pi\)
−0.855959 + 0.517044i \(0.827032\pi\)
\(18\) 0 0
\(19\) −2.10268 − 3.64194i −0.482387 − 0.835519i 0.517408 − 0.855739i \(-0.326897\pi\)
−0.999796 + 0.0202194i \(0.993564\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.27988 0.266873 0.133437 − 0.991057i \(-0.457399\pi\)
0.133437 + 0.991057i \(0.457399\pi\)
\(24\) 0 0
\(25\) 12.8334 2.56668
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.20305 + 7.27990i 0.780487 + 1.35184i 0.931658 + 0.363335i \(0.118362\pi\)
−0.151171 + 0.988508i \(0.548305\pi\)
\(30\) 0 0
\(31\) −0.476061 − 0.824561i −0.0855030 − 0.148096i 0.820102 − 0.572217i \(-0.193916\pi\)
−0.905605 + 0.424121i \(0.860583\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 10.0423 − 4.89755i 1.69746 − 0.827837i
\(36\) 0 0
\(37\) 3.03329 + 5.25381i 0.498669 + 0.863721i 0.999999 − 0.00153588i \(-0.000488885\pi\)
−0.501330 + 0.865256i \(0.667156\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.31299 + 2.27416i −0.205054 + 0.355164i −0.950150 − 0.311794i \(-0.899070\pi\)
0.745096 + 0.666957i \(0.232404\pi\)
\(42\) 0 0
\(43\) −0.442349 − 0.766171i −0.0674576 − 0.116840i 0.830324 − 0.557281i \(-0.188155\pi\)
−0.897782 + 0.440441i \(0.854822\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.88201 + 4.99178i −0.420384 + 0.728126i −0.995977 − 0.0896103i \(-0.971438\pi\)
0.575593 + 0.817736i \(0.304771\pi\)
\(48\) 0 0
\(49\) 4.30999 − 5.51579i 0.615713 − 0.787970i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.962456 − 1.66702i 0.132204 − 0.228983i −0.792322 − 0.610103i \(-0.791128\pi\)
0.924526 + 0.381120i \(0.124461\pi\)
\(54\) 0 0
\(55\) 8.14673 1.09850
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.27614 + 3.94240i 0.296329 + 0.513256i 0.975293 − 0.220915i \(-0.0709043\pi\)
−0.678964 + 0.734171i \(0.737571\pi\)
\(60\) 0 0
\(61\) 5.29008 − 9.16268i 0.677325 − 1.17316i −0.298458 − 0.954423i \(-0.596472\pi\)
0.975783 − 0.218739i \(-0.0701943\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.23112 + 2.13236i −0.152701 + 0.264486i
\(66\) 0 0
\(67\) −2.43191 − 4.21220i −0.297106 − 0.514602i 0.678367 − 0.734723i \(-0.262688\pi\)
−0.975473 + 0.220121i \(0.929355\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 11.5443 1.37005 0.685027 − 0.728518i \(-0.259791\pi\)
0.685027 + 0.728518i \(0.259791\pi\)
\(72\) 0 0
\(73\) 0.446138 − 0.772734i 0.0522165 − 0.0904417i −0.838736 − 0.544539i \(-0.816705\pi\)
0.890952 + 0.454097i \(0.150038\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.58756 − 2.23732i 0.522801 − 0.254966i
\(78\) 0 0
\(79\) −5.93520 + 10.2801i −0.667763 + 1.15660i 0.310766 + 0.950487i \(0.399415\pi\)
−0.978528 + 0.206112i \(0.933919\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −5.24250 − 9.08028i −0.575439 − 0.996690i −0.995994 − 0.0894227i \(-0.971498\pi\)
0.420555 − 0.907267i \(-0.361836\pi\)
\(84\) 0 0
\(85\) −15.2484 + 26.4109i −1.65392 + 2.86467i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.87906 − 6.71874i −0.411180 − 0.712185i 0.583839 − 0.811869i \(-0.301550\pi\)
−0.995019 + 0.0996849i \(0.968217\pi\)
\(90\) 0 0
\(91\) −0.107659 + 1.53887i −0.0112857 + 0.161317i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −8.87953 − 15.3798i −0.911020 − 1.57793i
\(96\) 0 0
\(97\) −1.98651 − 3.44073i −0.201699 − 0.349353i 0.747377 − 0.664400i \(-0.231313\pi\)
−0.949076 + 0.315047i \(0.897980\pi\)
\(98\) 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 3024.2.t.k.1873.11 22
3.2 odd 2 1008.2.t.l.193.11 22
4.3 odd 2 1512.2.t.c.361.11 22
7.2 even 3 3024.2.q.l.2305.1 22
9.2 odd 6 1008.2.q.l.529.5 22
9.7 even 3 3024.2.q.l.2881.1 22
12.11 even 2 504.2.t.c.193.1 yes 22
21.2 odd 6 1008.2.q.l.625.5 22
28.23 odd 6 1512.2.q.d.793.1 22
36.7 odd 6 1512.2.q.d.1369.1 22
36.11 even 6 504.2.q.c.25.7 ✓ 22
63.2 odd 6 1008.2.t.l.961.11 22
63.16 even 3 inner 3024.2.t.k.289.11 22
84.23 even 6 504.2.q.c.121.7 yes 22
252.79 odd 6 1512.2.t.c.289.11 22
252.191 even 6 504.2.t.c.457.1 yes 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.q.c.25.7 ✓ 22 36.11 even 6
504.2.q.c.121.7 yes 22 84.23 even 6
504.2.t.c.193.1 yes 22 12.11 even 2
504.2.t.c.457.1 yes 22 252.191 even 6
1008.2.q.l.529.5 22 9.2 odd 6
1008.2.q.l.625.5 22 21.2 odd 6
1008.2.t.l.193.11 22 3.2 odd 2
1008.2.t.l.961.11 22 63.2 odd 6
1512.2.q.d.793.1 22 28.23 odd 6
1512.2.q.d.1369.1 22 36.7 odd 6
1512.2.t.c.289.11 22 252.79 odd 6
1512.2.t.c.361.11 22 4.3 odd 2
3024.2.q.l.2305.1 22 7.2 even 3
3024.2.q.l.2881.1 22 9.7 even 3
3024.2.t.k.289.11 22 63.16 even 3 inner
3024.2.t.k.1873.11 22 1.1 even 1 trivial