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Properties

Label 1008.2.r.h.337.2
Level $1008$
Weight $2$
Character 1008.337
Analytic conductor $8.049$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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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 337.2
Root \(-0.173648 - 0.984808i\) of defining polynomial
Character \(\chi\) \(=\) 1008.337
Dual form 1008.2.r.h.673.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.592396 - 1.62760i) q^{3} +(-0.673648 + 1.16679i) q^{5} +(0.500000 + 0.866025i) q^{7} +(-2.29813 - 1.92836i) q^{9} +(0.826352 + 1.43128i) q^{11} +(1.68479 - 2.91815i) q^{13} +(1.50000 + 1.78763i) q^{15} +0.467911 q^{17} +3.22668 q^{19} +(1.70574 - 0.300767i) q^{21} +(4.47178 - 7.74535i) q^{23} +(1.59240 + 2.75811i) q^{25} +(-4.50000 + 2.59808i) q^{27} +(-3.13429 - 5.42874i) q^{29} +(4.61721 - 7.99724i) q^{31} +(2.81908 - 0.497079i) q^{33} -1.34730 q^{35} +9.23442 q^{37} +(-3.75150 - 4.47086i) q^{39} +(-1.70574 + 2.95442i) q^{41} +(-2.20574 - 3.82045i) q^{43} +(3.79813 - 1.38241i) q^{45} +(4.67752 + 8.10170i) q^{47} +(-0.500000 + 0.866025i) q^{49} +(0.277189 - 0.761570i) q^{51} -0.573978 q^{53} -2.22668 q^{55} +(1.91147 - 5.25173i) q^{57} +(-5.19846 + 9.00400i) q^{59} +(-3.81908 - 6.61484i) q^{61} +(0.520945 - 2.95442i) q^{63} +(2.26991 + 3.93161i) q^{65} +(0.298133 - 0.516382i) q^{67} +(-9.95723 - 11.8666i) q^{69} +0.554378 q^{71} +2.04963 q^{73} +(5.43242 - 0.957882i) q^{75} +(-0.826352 + 1.43128i) q^{77} +(-1.20187 - 2.08169i) q^{79} +(1.56283 + 8.86327i) q^{81} +(-7.52481 - 13.0334i) q^{83} +(-0.315207 + 0.545955i) q^{85} +(-10.6925 + 1.88538i) q^{87} +9.08647 q^{89} +3.36959 q^{91} +(-10.2811 - 12.2525i) q^{93} +(-2.17365 + 3.76487i) q^{95} +(0.949493 + 1.64457i) q^{97} +(0.860967 - 4.88279i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{5} + 3 q^{7} + 6 q^{11} + 3 q^{13} + 9 q^{15} + 12 q^{17} + 6 q^{19} + 12 q^{23} + 6 q^{25} - 27 q^{27} - 9 q^{29} - 3 q^{31} - 6 q^{35} - 6 q^{37} + 18 q^{39} - 3 q^{43} + 9 q^{45} + 3 q^{47}+ \cdots - 18 q^{99}+O(q^{100}) \) 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
\(3\) 0.592396 − 1.62760i 0.342020 − 0.939693i
\(4\) 0 0
\(5\) −0.673648 + 1.16679i −0.301265 + 0.521806i −0.976423 − 0.215867i \(-0.930742\pi\)
0.675158 + 0.737673i \(0.264075\pi\)
\(6\) 0 0
\(7\) 0.500000 + 0.866025i 0.188982 + 0.327327i
\(8\) 0 0
\(9\) −2.29813 − 1.92836i −0.766044 − 0.642788i
\(10\) 0 0
\(11\) 0.826352 + 1.43128i 0.249154 + 0.431548i 0.963291 − 0.268458i \(-0.0865140\pi\)
−0.714137 + 0.700006i \(0.753181\pi\)
\(12\) 0 0
\(13\) 1.68479 − 2.91815i 0.467277 − 0.809348i −0.532024 − 0.846729i \(-0.678568\pi\)
0.999301 + 0.0373813i \(0.0119016\pi\)
\(14\) 0 0
\(15\) 1.50000 + 1.78763i 0.387298 + 0.461564i
\(16\) 0 0
\(17\) 0.467911 0.113485 0.0567426 − 0.998389i \(-0.481929\pi\)
0.0567426 + 0.998389i \(0.481929\pi\)
\(18\) 0 0
\(19\) 3.22668 0.740252 0.370126 − 0.928982i \(-0.379315\pi\)
0.370126 + 0.928982i \(0.379315\pi\)
\(20\) 0 0
\(21\) 1.70574 − 0.300767i 0.372222 − 0.0656328i
\(22\) 0 0
\(23\) 4.47178 − 7.74535i 0.932431 − 1.61502i 0.153279 − 0.988183i \(-0.451017\pi\)
0.779152 − 0.626835i \(-0.215650\pi\)
\(24\) 0 0
\(25\) 1.59240 + 2.75811i 0.318479 + 0.551622i
\(26\) 0 0
\(27\) −4.50000 + 2.59808i −0.866025 + 0.500000i
\(28\) 0 0
\(29\) −3.13429 − 5.42874i −0.582022 − 1.00809i −0.995239 − 0.0974595i \(-0.968928\pi\)
0.413217 − 0.910632i \(-0.364405\pi\)
\(30\) 0 0
\(31\) 4.61721 − 7.99724i 0.829276 − 1.43635i −0.0693317 − 0.997594i \(-0.522087\pi\)
0.898607 − 0.438754i \(-0.144580\pi\)
\(32\) 0 0
\(33\) 2.81908 − 0.497079i 0.490738 − 0.0865304i
\(34\) 0 0
\(35\) −1.34730 −0.227735
\(36\) 0 0
\(37\) 9.23442 1.51813 0.759065 − 0.651015i \(-0.225657\pi\)
0.759065 + 0.651015i \(0.225657\pi\)
\(38\) 0 0
\(39\) −3.75150 − 4.47086i −0.600720 − 0.715910i
\(40\) 0 0
\(41\) −1.70574 + 2.95442i −0.266391 + 0.461403i −0.967927 − 0.251231i \(-0.919165\pi\)
0.701536 + 0.712634i \(0.252498\pi\)
\(42\) 0 0
\(43\) −2.20574 − 3.82045i −0.336372 − 0.582613i 0.647376 − 0.762171i \(-0.275867\pi\)
−0.983747 + 0.179558i \(0.942533\pi\)
\(44\) 0 0
\(45\) 3.79813 − 1.38241i 0.566192 − 0.206077i
\(46\) 0 0
\(47\) 4.67752 + 8.10170i 0.682286 + 1.18175i 0.974281 + 0.225335i \(0.0723475\pi\)
−0.291995 + 0.956420i \(0.594319\pi\)
\(48\) 0 0
\(49\) −0.500000 + 0.866025i −0.0714286 + 0.123718i
\(50\) 0 0
\(51\) 0.277189 − 0.761570i 0.0388142 − 0.106641i
\(52\) 0 0
\(53\) −0.573978 −0.0788419 −0.0394210 − 0.999223i \(-0.512551\pi\)
−0.0394210 + 0.999223i \(0.512551\pi\)
\(54\) 0 0
\(55\) −2.22668 −0.300246
\(56\) 0 0
\(57\) 1.91147 − 5.25173i 0.253181 − 0.695609i
\(58\) 0 0
\(59\) −5.19846 + 9.00400i −0.676782 + 1.17222i 0.299162 + 0.954202i \(0.403293\pi\)
−0.975945 + 0.218019i \(0.930041\pi\)
\(60\) 0 0
\(61\) −3.81908 − 6.61484i −0.488983 − 0.846943i 0.510937 − 0.859618i \(-0.329299\pi\)
−0.999920 + 0.0126752i \(0.995965\pi\)
\(62\) 0 0
\(63\) 0.520945 − 2.95442i 0.0656328 − 0.372222i
\(64\) 0 0
\(65\) 2.26991 + 3.93161i 0.281548 + 0.487656i
\(66\) 0 0
\(67\) 0.298133 − 0.516382i 0.0364228 − 0.0630861i −0.847239 − 0.531211i \(-0.821737\pi\)
0.883662 + 0.468125i \(0.155070\pi\)
\(68\) 0 0
\(69\) −9.95723 − 11.8666i −1.19871 − 1.42857i
\(70\) 0 0
\(71\) 0.554378 0.0657925 0.0328963 − 0.999459i \(-0.489527\pi\)
0.0328963 + 0.999459i \(0.489527\pi\)
\(72\) 0 0
\(73\) 2.04963 0.239891 0.119946 − 0.992780i \(-0.461728\pi\)
0.119946 + 0.992780i \(0.461728\pi\)
\(74\) 0 0
\(75\) 5.43242 − 0.957882i 0.627282 − 0.110607i
\(76\) 0 0
\(77\) −0.826352 + 1.43128i −0.0941715 + 0.163110i
\(78\) 0 0
\(79\) −1.20187 − 2.08169i −0.135221 − 0.234209i 0.790461 − 0.612512i \(-0.209841\pi\)
−0.925682 + 0.378303i \(0.876508\pi\)
\(80\) 0 0
\(81\) 1.56283 + 8.86327i 0.173648 + 0.984808i
\(82\) 0 0
\(83\) −7.52481 − 13.0334i −0.825956 − 1.43060i −0.901187 − 0.433431i \(-0.857303\pi\)
0.0752309 − 0.997166i \(-0.476031\pi\)
\(84\) 0 0
\(85\) −0.315207 + 0.545955i −0.0341891 + 0.0592172i
\(86\) 0 0
\(87\) −10.6925 + 1.88538i −1.14636 + 0.202134i
\(88\) 0 0
\(89\) 9.08647 0.963164 0.481582 − 0.876401i \(-0.340062\pi\)
0.481582 + 0.876401i \(0.340062\pi\)
\(90\) 0 0
\(91\) 3.36959 0.353228
\(92\) 0 0
\(93\) −10.2811 − 12.2525i −1.06610 − 1.27052i
\(94\) 0 0
\(95\) −2.17365 + 3.76487i −0.223012 + 0.386267i
\(96\) 0 0
\(97\) 0.949493 + 1.64457i 0.0964064 + 0.166981i 0.910195 − 0.414181i \(-0.135932\pi\)
−0.813788 + 0.581161i \(0.802598\pi\)
\(98\) 0 0
\(99\) 0.860967 − 4.88279i 0.0865304 − 0.490738i
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.337.2 6
3.2 odd 2 3024.2.r.k.1009.2 6
4.3 odd 2 63.2.f.a.22.3 ✓ 6
9.2 odd 6 3024.2.r.k.2017.2 6
9.4 even 3 9072.2.a.ca.1.2 3
9.5 odd 6 9072.2.a.bs.1.2 3
9.7 even 3 inner 1008.2.r.h.673.2 6
12.11 even 2 189.2.f.b.64.1 6
28.3 even 6 441.2.h.e.373.1 6
28.11 odd 6 441.2.h.d.373.1 6
28.19 even 6 441.2.g.b.67.3 6
28.23 odd 6 441.2.g.c.67.3 6
28.27 even 2 441.2.f.c.148.3 6
36.7 odd 6 63.2.f.a.43.3 yes 6
36.11 even 6 189.2.f.b.127.1 6
36.23 even 6 567.2.a.c.1.3 3
36.31 odd 6 567.2.a.h.1.1 3
84.11 even 6 1323.2.h.c.226.3 6
84.23 even 6 1323.2.g.d.361.1 6
84.47 odd 6 1323.2.g.e.361.1 6
84.59 odd 6 1323.2.h.b.226.3 6
84.83 odd 2 1323.2.f.d.442.1 6
252.11 even 6 1323.2.g.d.667.1 6
252.47 odd 6 1323.2.h.b.802.3 6
252.79 odd 6 441.2.h.d.214.1 6
252.83 odd 6 1323.2.f.d.883.1 6
252.115 even 6 441.2.g.b.79.3 6
252.139 even 6 3969.2.a.q.1.1 3
252.151 odd 6 441.2.g.c.79.3 6
252.167 odd 6 3969.2.a.l.1.3 3
252.187 even 6 441.2.h.e.214.1 6
252.191 even 6 1323.2.h.c.802.3 6
252.223 even 6 441.2.f.c.295.3 6
252.227 odd 6 1323.2.g.e.667.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.a.22.3 ✓ 6 4.3 odd 2
63.2.f.a.43.3 yes 6 36.7 odd 6
189.2.f.b.64.1 6 12.11 even 2
189.2.f.b.127.1 6 36.11 even 6
441.2.f.c.148.3 6 28.27 even 2
441.2.f.c.295.3 6 252.223 even 6
441.2.g.b.67.3 6 28.19 even 6
441.2.g.b.79.3 6 252.115 even 6
441.2.g.c.67.3 6 28.23 odd 6
441.2.g.c.79.3 6 252.151 odd 6
441.2.h.d.214.1 6 252.79 odd 6
441.2.h.d.373.1 6 28.11 odd 6
441.2.h.e.214.1 6 252.187 even 6
441.2.h.e.373.1 6 28.3 even 6
567.2.a.c.1.3 3 36.23 even 6
567.2.a.h.1.1 3 36.31 odd 6
1008.2.r.h.337.2 6 1.1 even 1 trivial
1008.2.r.h.673.2 6 9.7 even 3 inner
1323.2.f.d.442.1 6 84.83 odd 2
1323.2.f.d.883.1 6 252.83 odd 6
1323.2.g.d.361.1 6 84.23 even 6
1323.2.g.d.667.1 6 252.11 even 6
1323.2.g.e.361.1 6 84.47 odd 6
1323.2.g.e.667.1 6 252.227 odd 6
1323.2.h.b.226.3 6 84.59 odd 6
1323.2.h.b.802.3 6 252.47 odd 6
1323.2.h.c.226.3 6 84.11 even 6
1323.2.h.c.802.3 6 252.191 even 6
3024.2.r.k.1009.2 6 3.2 odd 2
3024.2.r.k.2017.2 6 9.2 odd 6
3969.2.a.l.1.3 3 252.167 odd 6
3969.2.a.q.1.1 3 252.139 even 6
9072.2.a.bs.1.2 3 9.5 odd 6
9072.2.a.ca.1.2 3 9.4 even 3