↑

Properties

Label 1008.2.r.h.337.1
Level $1008$
Weight $2$
Character 1008.337
Analytic conductor $8.049$
Analytic rank $0$
Dimension $6$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1008,2,Mod(337,1008)] 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("1008.337"); 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, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.r (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,-3,0,3] 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: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 337.1
Root \(-0.766044 + 0.642788i\) of defining polynomial
Character \(\chi\) \(=\) 1008.337
Dual form 1008.2.r.h.673.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.70574 + 0.300767i) q^{3} +(-1.26604 + 2.19285i) q^{5} +(0.500000 + 0.866025i) q^{7} +(2.81908 - 1.02606i) q^{9} +(0.233956 + 0.405223i) q^{11} +(-2.91147 + 5.04282i) q^{13} +(1.50000 - 4.12122i) q^{15} +3.87939 q^{17} +2.18479 q^{19} +(-1.11334 - 1.32683i) q^{21} +(-0.0530334 + 0.0918566i) q^{23} +(-0.705737 - 1.22237i) q^{25} +(-4.50000 + 2.59808i) q^{27} +(-4.39053 - 7.60462i) q^{29} +(-3.84002 + 6.65111i) q^{31} +(-0.520945 - 0.620838i) q^{33} -2.53209 q^{35} -7.68004 q^{37} +(3.44949 - 9.47740i) q^{39} +(1.11334 - 1.92836i) q^{41} +(0.613341 + 1.06234i) q^{43} +(-1.31908 + 7.48086i) q^{45} +(-2.66637 - 4.61830i) q^{47} +(-0.500000 + 0.866025i) q^{49} +(-6.61721 + 1.16679i) q^{51} -0.716881 q^{53} -1.18479 q^{55} +(-3.72668 + 0.657115i) q^{57} +(0.368241 - 0.637812i) q^{59} +(-0.479055 - 0.829748i) q^{61} +(2.29813 + 1.92836i) q^{63} +(-7.37211 - 12.7689i) q^{65} +(-4.81908 + 8.34689i) q^{67} +(0.0628336 - 0.172634i) q^{69} -13.2344 q^{71} -10.2686 q^{73} +(1.57145 + 1.87278i) q^{75} +(-0.233956 + 0.405223i) q^{77} +(-6.31908 - 10.9450i) q^{79} +(6.89440 - 5.78509i) q^{81} +(-1.36571 - 2.36549i) q^{83} +(-4.91147 + 8.50692i) q^{85} +(9.77631 + 11.6510i) q^{87} -8.11381 q^{89} -5.82295 q^{91} +(4.54963 - 12.5000i) q^{93} +(-2.76604 + 4.79093i) q^{95} +(6.80200 + 11.7814i) q^{97} +(1.07532 + 0.902302i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{5} + 3 q^{7} + 6 q^{11} + 3 q^{13} + 9 q^{15} + 12 q^{17} + 6 q^{19} + 12 q^{23} + 6 q^{25} - 27 q^{27} - 9 q^{29} - 3 q^{31} - 6 q^{35} - 6 q^{37} + 18 q^{39} - 3 q^{43} + 9 q^{45} + 3 q^{47}+ \cdots - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) −1.70574 + 0.300767i −0.984808 + 0.173648i
\(4\) 0 0
\(5\) −1.26604 + 2.19285i −0.566192 + 0.980674i 0.430745 + 0.902473i \(0.358251\pi\)
−0.996938 + 0.0782003i \(0.975083\pi\)
\(6\) 0 0
\(7\) 0.500000 + 0.866025i 0.188982 + 0.327327i
\(8\) 0 0
\(9\) 2.81908 − 1.02606i 0.939693 − 0.342020i
\(10\) 0 0
\(11\) 0.233956 + 0.405223i 0.0705403 + 0.122179i 0.899138 − 0.437665i \(-0.144194\pi\)
−0.828598 + 0.559844i \(0.810861\pi\)
\(12\) 0 0
\(13\) −2.91147 + 5.04282i −0.807498 + 1.39863i 0.107094 + 0.994249i \(0.465845\pi\)
−0.914592 + 0.404378i \(0.867488\pi\)
\(14\) 0 0
\(15\) 1.50000 − 4.12122i 0.387298 − 1.06409i
\(16\) 0 0
\(17\) 3.87939 0.940889 0.470445 − 0.882430i \(-0.344094\pi\)
0.470445 + 0.882430i \(0.344094\pi\)
\(18\) 0 0
\(19\) 2.18479 0.501226 0.250613 − 0.968087i \(-0.419368\pi\)
0.250613 + 0.968087i \(0.419368\pi\)
\(20\) 0 0
\(21\) −1.11334 − 1.32683i −0.242951 − 0.289538i
\(22\) 0 0
\(23\) −0.0530334 + 0.0918566i −0.0110582 + 0.0191534i −0.871502 − 0.490393i \(-0.836853\pi\)
0.860443 + 0.509546i \(0.170187\pi\)
\(24\) 0 0
\(25\) −0.705737 − 1.22237i −0.141147 − 0.244474i
\(26\) 0 0
\(27\) −4.50000 + 2.59808i −0.866025 + 0.500000i
\(28\) 0 0
\(29\) −4.39053 − 7.60462i −0.815301 − 1.41214i −0.909112 − 0.416552i \(-0.863238\pi\)
0.0938108 − 0.995590i \(-0.470095\pi\)
\(30\) 0 0
\(31\) −3.84002 + 6.65111i −0.689688 + 1.19458i 0.282250 + 0.959341i \(0.408919\pi\)
−0.971939 + 0.235235i \(0.924414\pi\)
\(32\) 0 0
\(33\) −0.520945 − 0.620838i −0.0906848 − 0.108074i
\(34\) 0 0
\(35\) −2.53209 −0.428001
\(36\) 0 0
\(37\) −7.68004 −1.26259 −0.631296 − 0.775542i \(-0.717477\pi\)
−0.631296 + 0.775542i \(0.717477\pi\)
\(38\) 0 0
\(39\) 3.44949 − 9.47740i 0.552361 − 1.51760i
\(40\) 0 0
\(41\) 1.11334 − 1.92836i 0.173875 − 0.301160i −0.765897 − 0.642964i \(-0.777705\pi\)
0.939771 + 0.341804i \(0.111038\pi\)
\(42\) 0 0
\(43\) 0.613341 + 1.06234i 0.0935336 + 0.162005i 0.908996 − 0.416806i \(-0.136850\pi\)
−0.815462 + 0.578811i \(0.803517\pi\)
\(44\) 0 0
\(45\) −1.31908 + 7.48086i −0.196637 + 1.11518i
\(46\) 0 0
\(47\) −2.66637 − 4.61830i −0.388931 − 0.673648i 0.603375 − 0.797457i \(-0.293822\pi\)
−0.992306 + 0.123810i \(0.960489\pi\)
\(48\) 0 0
\(49\) −0.500000 + 0.866025i −0.0714286 + 0.123718i
\(50\) 0 0
\(51\) −6.61721 + 1.16679i −0.926595 + 0.163384i
\(52\) 0 0
\(53\) −0.716881 −0.0984712 −0.0492356 − 0.998787i \(-0.515679\pi\)
−0.0492356 + 0.998787i \(0.515679\pi\)
\(54\) 0 0
\(55\) −1.18479 −0.159757
\(56\) 0 0
\(57\) −3.72668 + 0.657115i −0.493611 + 0.0870369i
\(58\) 0 0
\(59\) 0.368241 − 0.637812i 0.0479409 − 0.0830360i −0.841059 − 0.540943i \(-0.818067\pi\)
0.889000 + 0.457907i \(0.151401\pi\)
\(60\) 0 0
\(61\) −0.479055 − 0.829748i −0.0613368 − 0.106238i 0.833726 − 0.552178i \(-0.186203\pi\)
−0.895063 + 0.445939i \(0.852870\pi\)
\(62\) 0 0
\(63\) 2.29813 + 1.92836i 0.289538 + 0.242951i
\(64\) 0 0
\(65\) −7.37211 − 12.7689i −0.914398 − 1.58378i
\(66\) 0 0
\(67\) −4.81908 + 8.34689i −0.588744 + 1.01973i 0.405653 + 0.914027i \(0.367044\pi\)
−0.994397 + 0.105708i \(0.966289\pi\)
\(68\) 0 0
\(69\) 0.0628336 − 0.172634i 0.00756428 − 0.0207827i
\(70\) 0 0
\(71\) −13.2344 −1.57064 −0.785318 − 0.619092i \(-0.787501\pi\)
−0.785318 + 0.619092i \(0.787501\pi\)
\(72\) 0 0
\(73\) −10.2686 −1.20185 −0.600923 − 0.799307i \(-0.705200\pi\)
−0.600923 + 0.799307i \(0.705200\pi\)
\(74\) 0 0
\(75\) 1.57145 + 1.87278i 0.181456 + 0.216250i
\(76\) 0 0
\(77\) −0.233956 + 0.405223i −0.0266617 + 0.0461794i
\(78\) 0 0
\(79\) −6.31908 − 10.9450i −0.710952 − 1.23140i −0.964500 − 0.264082i \(-0.914931\pi\)
0.253548 − 0.967323i \(-0.418402\pi\)
\(80\) 0 0
\(81\) 6.89440 − 5.78509i 0.766044 − 0.642788i
\(82\) 0 0
\(83\) −1.36571 − 2.36549i −0.149907 − 0.259646i 0.781286 − 0.624173i \(-0.214564\pi\)
−0.931193 + 0.364527i \(0.881231\pi\)
\(84\) 0 0
\(85\) −4.91147 + 8.50692i −0.532724 + 0.922705i
\(86\) 0 0
\(87\) 9.77631 + 11.6510i 1.04813 + 1.24911i
\(88\) 0 0
\(89\) −8.11381 −0.860062 −0.430031 − 0.902814i \(-0.641497\pi\)
−0.430031 + 0.902814i \(0.641497\pi\)
\(90\) 0 0
\(91\) −5.82295 −0.610411
\(92\) 0 0
\(93\) 4.54963 − 12.5000i 0.471775 − 1.29619i
\(94\) 0 0
\(95\) −2.76604 + 4.79093i −0.283790 + 0.491539i
\(96\) 0 0
\(97\) 6.80200 + 11.7814i 0.690639 + 1.19622i 0.971629 + 0.236511i \(0.0760039\pi\)
−0.280990 + 0.959711i \(0.590663\pi\)
\(98\) 0 0
\(99\) 1.07532 + 0.902302i 0.108074 + 0.0906848i
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.r.h.337.1 6
3.2 odd 2 3024.2.r.k.1009.3 6
4.3 odd 2 63.2.f.a.22.2 ✓ 6
9.2 odd 6 3024.2.r.k.2017.3 6
9.4 even 3 9072.2.a.ca.1.3 3
9.5 odd 6 9072.2.a.bs.1.1 3
9.7 even 3 inner 1008.2.r.h.673.1 6
12.11 even 2 189.2.f.b.64.2 6
28.3 even 6 441.2.h.e.373.2 6
28.11 odd 6 441.2.h.d.373.2 6
28.19 even 6 441.2.g.b.67.2 6
28.23 odd 6 441.2.g.c.67.2 6
28.27 even 2 441.2.f.c.148.2 6
36.7 odd 6 63.2.f.a.43.2 yes 6
36.11 even 6 189.2.f.b.127.2 6
36.23 even 6 567.2.a.c.1.2 3
36.31 odd 6 567.2.a.h.1.2 3
84.11 even 6 1323.2.h.c.226.2 6
84.23 even 6 1323.2.g.d.361.2 6
84.47 odd 6 1323.2.g.e.361.2 6
84.59 odd 6 1323.2.h.b.226.2 6
84.83 odd 2 1323.2.f.d.442.2 6
252.11 even 6 1323.2.g.d.667.2 6
252.47 odd 6 1323.2.h.b.802.2 6
252.79 odd 6 441.2.h.d.214.2 6
252.83 odd 6 1323.2.f.d.883.2 6
252.115 even 6 441.2.g.b.79.2 6
252.139 even 6 3969.2.a.q.1.2 3
252.151 odd 6 441.2.g.c.79.2 6
252.167 odd 6 3969.2.a.l.1.2 3
252.187 even 6 441.2.h.e.214.2 6
252.191 even 6 1323.2.h.c.802.2 6
252.223 even 6 441.2.f.c.295.2 6
252.227 odd 6 1323.2.g.e.667.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.a.22.2 ✓ 6 4.3 odd 2
63.2.f.a.43.2 yes 6 36.7 odd 6
189.2.f.b.64.2 6 12.11 even 2
189.2.f.b.127.2 6 36.11 even 6
441.2.f.c.148.2 6 28.27 even 2
441.2.f.c.295.2 6 252.223 even 6
441.2.g.b.67.2 6 28.19 even 6
441.2.g.b.79.2 6 252.115 even 6
441.2.g.c.67.2 6 28.23 odd 6
441.2.g.c.79.2 6 252.151 odd 6
441.2.h.d.214.2 6 252.79 odd 6
441.2.h.d.373.2 6 28.11 odd 6
441.2.h.e.214.2 6 252.187 even 6
441.2.h.e.373.2 6 28.3 even 6
567.2.a.c.1.2 3 36.23 even 6
567.2.a.h.1.2 3 36.31 odd 6
1008.2.r.h.337.1 6 1.1 even 1 trivial
1008.2.r.h.673.1 6 9.7 even 3 inner
1323.2.f.d.442.2 6 84.83 odd 2
1323.2.f.d.883.2 6 252.83 odd 6
1323.2.g.d.361.2 6 84.23 even 6
1323.2.g.d.667.2 6 252.11 even 6
1323.2.g.e.361.2 6 84.47 odd 6
1323.2.g.e.667.2 6 252.227 odd 6
1323.2.h.b.226.2 6 84.59 odd 6
1323.2.h.b.802.2 6 252.47 odd 6
1323.2.h.c.226.2 6 84.11 even 6
1323.2.h.c.802.2 6 252.191 even 6
3024.2.r.k.1009.3 6 3.2 odd 2
3024.2.r.k.2017.3 6 9.2 odd 6
3969.2.a.l.1.2 3 252.167 odd 6
3969.2.a.q.1.2 3 252.139 even 6
9072.2.a.bs.1.1 3 9.5 odd 6
9072.2.a.ca.1.3 3 9.4 even 3