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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1008,2,Mod(193,1008)] 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("1008.193"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.t (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [22,0,2,0,-2] 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: \(8.04892052375\)
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 961.11
Character \(\chi\) \(=\) 1008.961
Dual form 1008.2.t.l.193.11

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.72692 + 0.133195i) q^{3} -4.22296 q^{5} +(2.37802 + 1.15974i) q^{7} +(2.96452 + 0.460034i) q^{9} -1.92915 q^{11} +(-0.291529 - 0.504943i) q^{13} +(-7.29273 - 0.562477i) q^{15} +(3.61082 + 6.25412i) q^{17} +(-2.10268 + 3.64194i) q^{19} +(3.95219 + 2.31953i) q^{21} -1.27988 q^{23} +12.8334 q^{25} +(5.05822 + 1.18930i) q^{27} +(-4.20305 + 7.27990i) q^{29} +(-0.476061 + 0.824561i) q^{31} +(-3.33149 - 0.256953i) q^{33} +(-10.0423 - 4.89755i) q^{35} +(3.03329 - 5.25381i) q^{37} +(-0.436192 - 0.910828i) q^{39} +(1.31299 + 2.27416i) q^{41} +(-0.442349 + 0.766171i) q^{43} +(-12.5191 - 1.94271i) q^{45} +(2.88201 + 4.99178i) q^{47} +(4.30999 + 5.51579i) q^{49} +(5.40259 + 11.2813i) q^{51} +(-0.962456 - 1.66702i) q^{53} +8.14673 q^{55} +(-4.11625 + 6.00929i) q^{57} +(-2.27614 + 3.94240i) q^{59} +(5.29008 + 9.16268i) q^{61} +(6.51617 + 4.53205i) q^{63} +(1.23112 + 2.13236i) q^{65} +(-2.43191 + 4.21220i) q^{67} +(-2.21025 - 0.170473i) q^{69} -11.5443 q^{71} +(0.446138 + 0.772734i) q^{73} +(22.1623 + 1.70934i) q^{75} +(-4.58756 - 2.23732i) q^{77} +(-5.93520 - 10.2801i) q^{79} +(8.57674 + 2.72756i) q^{81} +(5.24250 - 9.08028i) q^{83} +(-15.2484 - 26.4109i) q^{85} +(-8.22798 + 12.0120i) q^{87} +(3.87906 - 6.71874i) q^{89} +(-0.107659 - 1.53887i) q^{91} +(-0.931947 + 1.36054i) q^{93} +(8.87953 - 15.3798i) q^{95} +(-1.98651 + 3.44073i) q^{97} +(-5.71900 - 0.887474i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q + 2 q^{3} - 2 q^{5} + q^{7} + 6 q^{11} + 7 q^{13} + q^{15} - q^{17} - 13 q^{19} + 33 q^{21} + 44 q^{25} + 2 q^{27} - 7 q^{29} - 6 q^{31} + 9 q^{33} - 2 q^{35} + 6 q^{37} + 4 q^{39} + 4 q^{41} - 2 q^{43}+ \cdots - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) 1.72692 + 0.133195i 0.997039 + 0.0769000i
\(4\) 0 0
\(5\) −4.22296 −1.88857 −0.944283 − 0.329134i \(-0.893243\pi\)
−0.944283 + 0.329134i \(0.893243\pi\)
\(6\) 0 0
\(7\) 2.37802 + 1.15974i 0.898809 + 0.438341i
\(8\) 0 0
\(9\) 2.96452 + 0.460034i 0.988173 + 0.153345i
\(10\) 0 0
\(11\) −1.92915 −0.581660 −0.290830 − 0.956775i \(-0.593931\pi\)
−0.290830 + 0.956775i \(0.593931\pi\)
\(12\) 0 0
\(13\) −0.291529 − 0.504943i −0.0808557 − 0.140046i 0.822762 − 0.568386i \(-0.192432\pi\)
−0.903618 + 0.428340i \(0.859099\pi\)
\(14\) 0 0
\(15\) −7.29273 − 0.562477i −1.88297 − 0.145231i
\(16\) 0 0
\(17\) 3.61082 + 6.25412i 0.875753 + 1.51685i 0.855959 + 0.517044i \(0.172968\pi\)
0.0197936 + 0.999804i \(0.493699\pi\)
\(18\) 0 0
\(19\) −2.10268 + 3.64194i −0.482387 + 0.835519i −0.999796 − 0.0202194i \(-0.993564\pi\)
0.517408 + 0.855739i \(0.326897\pi\)
\(20\) 0 0
\(21\) 3.95219 + 2.31953i 0.862438 + 0.506162i
\(22\) 0 0
\(23\) −1.27988 −0.266873 −0.133437 − 0.991057i \(-0.542601\pi\)
−0.133437 + 0.991057i \(0.542601\pi\)
\(24\) 0 0
\(25\) 12.8334 2.56668
\(26\) 0 0
\(27\) 5.05822 + 1.18930i 0.973454 + 0.228881i
\(28\) 0 0
\(29\) −4.20305 + 7.27990i −0.780487 + 1.35184i 0.151171 + 0.988508i \(0.451695\pi\)
−0.931658 + 0.363335i \(0.881638\pi\)
\(30\) 0 0
\(31\) −0.476061 + 0.824561i −0.0855030 + 0.148096i −0.905605 − 0.424121i \(-0.860583\pi\)
0.820102 + 0.572217i \(0.193916\pi\)
\(32\) 0 0
\(33\) −3.33149 − 0.256953i −0.579938 − 0.0447297i
\(34\) 0 0
\(35\) −10.0423 − 4.89755i −1.69746 − 0.827837i
\(36\) 0 0
\(37\) 3.03329 − 5.25381i 0.498669 − 0.863721i −0.501330 − 0.865256i \(-0.667156\pi\)
0.999999 + 0.00153588i \(0.000488885\pi\)
\(38\) 0 0
\(39\) −0.436192 − 0.910828i −0.0698467 − 0.145849i
\(40\) 0 0
\(41\) 1.31299 + 2.27416i 0.205054 + 0.355164i 0.950150 − 0.311794i \(-0.100930\pi\)
−0.745096 + 0.666957i \(0.767596\pi\)
\(42\) 0 0
\(43\) −0.442349 + 0.766171i −0.0674576 + 0.116840i −0.897782 − 0.440441i \(-0.854822\pi\)
0.830324 + 0.557281i \(0.188155\pi\)
\(44\) 0 0
\(45\) −12.5191 − 1.94271i −1.86623 − 0.289602i
\(46\) 0 0
\(47\) 2.88201 + 4.99178i 0.420384 + 0.728126i 0.995977 − 0.0896103i \(-0.0285622\pi\)
−0.575593 + 0.817736i \(0.695229\pi\)
\(48\) 0 0
\(49\) 4.30999 + 5.51579i 0.615713 + 0.787970i
\(50\) 0 0
\(51\) 5.40259 + 11.2813i 0.756514 + 1.57970i
\(52\) 0 0
\(53\) −0.962456 − 1.66702i −0.132204 − 0.228983i 0.792322 − 0.610103i \(-0.208872\pi\)
−0.924526 + 0.381120i \(0.875539\pi\)
\(54\) 0 0
\(55\) 8.14673 1.09850
\(56\) 0 0
\(57\) −4.11625 + 6.00929i −0.545210 + 0.795950i
\(58\) 0 0
\(59\) −2.27614 + 3.94240i −0.296329 + 0.513256i −0.975293 − 0.220915i \(-0.929096\pi\)
0.678964 + 0.734171i \(0.262429\pi\)
\(60\) 0 0
\(61\) 5.29008 + 9.16268i 0.677325 + 1.17316i 0.975783 + 0.218739i \(0.0701943\pi\)
−0.298458 + 0.954423i \(0.596472\pi\)
\(62\) 0 0
\(63\) 6.51617 + 4.53205i 0.820961 + 0.570985i
\(64\) 0 0
\(65\) 1.23112 + 2.13236i 0.152701 + 0.264486i
\(66\) 0 0
\(67\) −2.43191 + 4.21220i −0.297106 + 0.514602i −0.975473 − 0.220121i \(-0.929355\pi\)
0.678367 + 0.734723i \(0.262688\pi\)
\(68\) 0 0
\(69\) −2.21025 − 0.170473i −0.266083 − 0.0205226i
\(70\) 0 0
\(71\) −11.5443 −1.37005 −0.685027 − 0.728518i \(-0.740209\pi\)
−0.685027 + 0.728518i \(0.740209\pi\)
\(72\) 0 0
\(73\) 0.446138 + 0.772734i 0.0522165 + 0.0904417i 0.890952 − 0.454097i \(-0.150038\pi\)
−0.838736 + 0.544539i \(0.816705\pi\)
\(74\) 0 0
\(75\) 22.1623 + 1.70934i 2.55908 + 0.197378i
\(76\) 0 0
\(77\) −4.58756 − 2.23732i −0.522801 − 0.254966i
\(78\) 0 0
\(79\) −5.93520 − 10.2801i −0.667763 − 1.15660i −0.978528 − 0.206112i \(-0.933919\pi\)
0.310766 − 0.950487i \(-0.399415\pi\)
\(80\) 0 0
\(81\) 8.57674 + 2.72756i 0.952971 + 0.303062i
\(82\) 0 0
\(83\) 5.24250 − 9.08028i 0.575439 − 0.996690i −0.420555 − 0.907267i \(-0.638164\pi\)
0.995994 − 0.0894227i \(-0.0285022\pi\)
\(84\) 0 0
\(85\) −15.2484 − 26.4109i −1.65392 − 2.86467i
\(86\) 0 0
\(87\) −8.22798 + 12.0120i −0.882133 + 1.28782i
\(88\) 0 0
\(89\) 3.87906 − 6.71874i 0.411180 − 0.712185i −0.583839 − 0.811869i \(-0.698450\pi\)
0.995019 + 0.0996849i \(0.0317835\pi\)
\(90\) 0 0
\(91\) −0.107659 − 1.53887i −0.0112857 − 0.161317i
\(92\) 0 0
\(93\) −0.931947 + 1.36054i −0.0966384 + 0.141082i
\(94\) 0 0
\(95\) 8.87953 − 15.3798i 0.911020 − 1.57793i
\(96\) 0 0
\(97\) −1.98651 + 3.44073i −0.201699 + 0.349353i −0.949076 − 0.315047i \(-0.897980\pi\)
0.747377 + 0.664400i \(0.231313\pi\)
\(98\) 0 0
\(99\) −5.71900 − 0.887474i −0.574781 − 0.0891945i
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 1008.2.t.l.961.11 22
3.2 odd 2 3024.2.t.k.289.11 22
4.3 odd 2 504.2.t.c.457.1 yes 22
7.4 even 3 1008.2.q.l.529.5 22
9.4 even 3 1008.2.q.l.625.5 22
9.5 odd 6 3024.2.q.l.2305.1 22
12.11 even 2 1512.2.t.c.289.11 22
21.11 odd 6 3024.2.q.l.2881.1 22
28.11 odd 6 504.2.q.c.25.7 ✓ 22
36.23 even 6 1512.2.q.d.793.1 22
36.31 odd 6 504.2.q.c.121.7 yes 22
63.4 even 3 inner 1008.2.t.l.193.11 22
63.32 odd 6 3024.2.t.k.1873.11 22
84.11 even 6 1512.2.q.d.1369.1 22
252.67 odd 6 504.2.t.c.193.1 yes 22
252.95 even 6 1512.2.t.c.361.11 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.q.c.25.7 ✓ 22 28.11 odd 6
504.2.q.c.121.7 yes 22 36.31 odd 6
504.2.t.c.193.1 yes 22 252.67 odd 6
504.2.t.c.457.1 yes 22 4.3 odd 2
1008.2.q.l.529.5 22 7.4 even 3
1008.2.q.l.625.5 22 9.4 even 3
1008.2.t.l.193.11 22 63.4 even 3 inner
1008.2.t.l.961.11 22 1.1 even 1 trivial
1512.2.q.d.793.1 22 36.23 even 6
1512.2.q.d.1369.1 22 84.11 even 6
1512.2.t.c.289.11 22 12.11 even 2
1512.2.t.c.361.11 22 252.95 even 6
3024.2.q.l.2305.1 22 9.5 odd 6
3024.2.q.l.2881.1 22 21.11 odd 6
3024.2.t.k.289.11 22 3.2 odd 2
3024.2.t.k.1873.11 22 63.32 odd 6