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Properties

Label 1008.2.q.g.625.1
Level $1008$
Weight $2$
Character 1008.625
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(529,1008)] 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("1008.529"); 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, 4, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,-2,0,1,0,-2] 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: 6.0.309123.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 625.1
Root \(0.500000 - 1.41036i\) of defining polynomial
Character \(\chi\) \(=\) 1008.625
Dual form 1008.2.q.g.529.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.71053 + 0.272169i) q^{3} +(1.59097 + 2.75564i) q^{5} +(2.56238 + 0.658939i) q^{7} +(2.85185 - 0.931107i) q^{9} +(1.59097 - 2.75564i) q^{11} +(2.85185 - 4.93955i) q^{13} +(-3.47141 - 4.28061i) q^{15} +(-0.760877 - 1.31788i) q^{17} +(0.641315 - 1.11079i) q^{19} +(-4.56238 - 0.429736i) q^{21} +(1.11956 + 1.93914i) q^{23} +(-2.56238 + 4.43818i) q^{25} +(-4.62476 + 2.36887i) q^{27} +(-3.54063 - 6.13255i) q^{29} +9.42107 q^{31} +(-1.97141 + 5.14663i) q^{33} +(2.26088 + 8.10936i) q^{35} +(0.500000 - 0.866025i) q^{37} +(-3.53379 + 9.22544i) q^{39} +(-2.80150 + 4.85235i) q^{41} +(-3.41423 - 5.91362i) q^{43} +(7.10301 + 6.37731i) q^{45} +5.82846 q^{47} +(6.13160 + 3.37690i) q^{49} +(1.66019 + 2.04719i) q^{51} +(1.02859 + 1.78157i) q^{53} +10.1248 q^{55} +(-0.794668 + 2.07459i) q^{57} +1.12476 q^{59} +3.12476 q^{61} +(7.92107 - 0.506659i) q^{63} +18.1488 q^{65} -10.9669 q^{67} +(-2.44282 - 3.01225i) q^{69} -8.69002 q^{71} +(-2.48345 - 4.30146i) q^{73} +(3.17511 - 8.28905i) q^{75} +(5.89248 - 6.01266i) q^{77} +4.13844 q^{79} +(7.26608 - 5.31075i) q^{81} +(4.03379 + 6.98673i) q^{83} +(2.42107 - 4.19341i) q^{85} +(7.72545 + 9.52628i) q^{87} +(0.112725 - 0.195246i) q^{89} +(10.5624 - 10.7778i) q^{91} +(-16.1150 + 2.56412i) q^{93} +4.08126 q^{95} +(7.42107 + 12.8537i) q^{97} +(1.97141 - 9.34004i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} + q^{5} - 2 q^{7} + 8 q^{9} + q^{11} + 8 q^{13} - 12 q^{15} - 4 q^{17} + 3 q^{19} - 10 q^{21} + 7 q^{23} + 2 q^{25} + 7 q^{27} - 5 q^{29} + 40 q^{31} - 3 q^{33} + 13 q^{35} + 3 q^{37} + 5 q^{39}+ \cdots + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) −1.71053 + 0.272169i −0.987577 + 0.157137i
\(4\) 0 0
\(5\) 1.59097 + 2.75564i 0.711504 + 1.23236i 0.964292 + 0.264840i \(0.0853191\pi\)
−0.252788 + 0.967522i \(0.581348\pi\)
\(6\) 0 0
\(7\) 2.56238 + 0.658939i 0.968489 + 0.249055i
\(8\) 0 0
\(9\) 2.85185 − 0.931107i 0.950616 − 0.310369i
\(10\) 0 0
\(11\) 1.59097 − 2.75564i 0.479696 − 0.830858i −0.520033 − 0.854146i \(-0.674080\pi\)
0.999729 + 0.0232884i \(0.00741361\pi\)
\(12\) 0 0
\(13\) 2.85185 − 4.93955i 0.790960 − 1.36998i −0.134412 − 0.990925i \(-0.542915\pi\)
0.925373 − 0.379058i \(-0.123752\pi\)
\(14\) 0 0
\(15\) −3.47141 − 4.28061i −0.896314 − 1.10525i
\(16\) 0 0
\(17\) −0.760877 − 1.31788i −0.184540 − 0.319632i 0.758882 − 0.651229i \(-0.225746\pi\)
−0.943421 + 0.331596i \(0.892413\pi\)
\(18\) 0 0
\(19\) 0.641315 − 1.11079i 0.147128 − 0.254833i −0.783037 − 0.621975i \(-0.786330\pi\)
0.930165 + 0.367142i \(0.119664\pi\)
\(20\) 0 0
\(21\) −4.56238 − 0.429736i −0.995593 − 0.0937761i
\(22\) 0 0
\(23\) 1.11956 + 1.93914i 0.233445 + 0.404338i 0.958820 − 0.284016i \(-0.0916669\pi\)
−0.725375 + 0.688354i \(0.758334\pi\)
\(24\) 0 0
\(25\) −2.56238 + 4.43818i −0.512476 + 0.887635i
\(26\) 0 0
\(27\) −4.62476 + 2.36887i −0.890036 + 0.455890i
\(28\) 0 0
\(29\) −3.54063 − 6.13255i −0.657478 − 1.13879i −0.981266 − 0.192656i \(-0.938290\pi\)
0.323788 − 0.946130i \(-0.395043\pi\)
\(30\) 0 0
\(31\) 9.42107 1.69207 0.846037 − 0.533125i \(-0.178982\pi\)
0.846037 + 0.533125i \(0.178982\pi\)
\(32\) 0 0
\(33\) −1.97141 + 5.14663i −0.343178 + 0.895914i
\(34\) 0 0
\(35\) 2.26088 + 8.10936i 0.382158 + 1.37073i
\(36\) 0 0
\(37\) 0.500000 − 0.866025i 0.0821995 − 0.142374i −0.821995 − 0.569495i \(-0.807139\pi\)
0.904194 + 0.427121i \(0.140472\pi\)
\(38\) 0 0
\(39\) −3.53379 + 9.22544i −0.565860 + 1.47725i
\(40\) 0 0
\(41\) −2.80150 + 4.85235i −0.437522 + 0.757810i −0.997498 − 0.0706992i \(-0.977477\pi\)
0.559976 + 0.828509i \(0.310810\pi\)
\(42\) 0 0
\(43\) −3.41423 − 5.91362i −0.520665 − 0.901819i −0.999711 − 0.0240288i \(-0.992351\pi\)
0.479046 − 0.877790i \(-0.340983\pi\)
\(44\) 0 0
\(45\) 7.10301 + 6.37731i 1.05885 + 0.950674i
\(46\) 0 0
\(47\) 5.82846 0.850168 0.425084 − 0.905154i \(-0.360245\pi\)
0.425084 + 0.905154i \(0.360245\pi\)
\(48\) 0 0
\(49\) 6.13160 + 3.37690i 0.875943 + 0.482415i
\(50\) 0 0
\(51\) 1.66019 + 2.04719i 0.232473 + 0.286663i
\(52\) 0 0
\(53\) 1.02859 + 1.78157i 0.141288 + 0.244717i 0.927982 − 0.372626i \(-0.121542\pi\)
−0.786694 + 0.617343i \(0.788209\pi\)
\(54\) 0 0
\(55\) 10.1248 1.36522
\(56\) 0 0
\(57\) −0.794668 + 2.07459i −0.105256 + 0.274786i
\(58\) 0 0
\(59\) 1.12476 0.146432 0.0732159 − 0.997316i \(-0.476674\pi\)
0.0732159 + 0.997316i \(0.476674\pi\)
\(60\) 0 0
\(61\) 3.12476 0.400085 0.200042 − 0.979787i \(-0.435892\pi\)
0.200042 + 0.979787i \(0.435892\pi\)
\(62\) 0 0
\(63\) 7.92107 − 0.506659i 0.997961 − 0.0638331i
\(64\) 0 0
\(65\) 18.1488 2.25109
\(66\) 0 0
\(67\) −10.9669 −1.33982 −0.669910 − 0.742442i \(-0.733667\pi\)
−0.669910 + 0.742442i \(0.733667\pi\)
\(68\) 0 0
\(69\) −2.44282 − 3.01225i −0.294081 − 0.362632i
\(70\) 0 0
\(71\) −8.69002 −1.03132 −0.515658 − 0.856794i \(-0.672452\pi\)
−0.515658 + 0.856794i \(0.672452\pi\)
\(72\) 0 0
\(73\) −2.48345 − 4.30146i −0.290666 − 0.503448i 0.683302 − 0.730136i \(-0.260543\pi\)
−0.973967 + 0.226689i \(0.927210\pi\)
\(74\) 0 0
\(75\) 3.17511 − 8.28905i 0.366630 − 0.957137i
\(76\) 0 0
\(77\) 5.89248 − 6.01266i 0.671510 − 0.685206i
\(78\) 0 0
\(79\) 4.13844 0.465610 0.232805 − 0.972523i \(-0.425210\pi\)
0.232805 + 0.972523i \(0.425210\pi\)
\(80\) 0 0
\(81\) 7.26608 − 5.31075i 0.807342 − 0.590084i
\(82\) 0 0
\(83\) 4.03379 + 6.98673i 0.442766 + 0.766893i 0.997894 − 0.0648718i \(-0.0206639\pi\)
−0.555127 + 0.831765i \(0.687331\pi\)
\(84\) 0 0
\(85\) 2.42107 − 4.19341i 0.262602 − 0.454839i
\(86\) 0 0
\(87\) 7.72545 + 9.52628i 0.828255 + 1.02132i
\(88\) 0 0
\(89\) 0.112725 − 0.195246i 0.0119488 − 0.0206960i −0.859989 − 0.510312i \(-0.829530\pi\)
0.871938 + 0.489616i \(0.162863\pi\)
\(90\) 0 0
\(91\) 10.5624 − 10.7778i 1.10724 − 1.12982i
\(92\) 0 0
\(93\) −16.1150 + 2.56412i −1.67105 + 0.265887i
\(94\) 0 0
\(95\) 4.08126 0.418728
\(96\) 0 0
\(97\) 7.42107 + 12.8537i 0.753495 + 1.30509i 0.946119 + 0.323819i \(0.104967\pi\)
−0.192624 + 0.981273i \(0.561700\pi\)
\(98\) 0 0
\(99\) 1.97141 − 9.34004i 0.198134 − 0.938710i
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.q.g.625.1 6
3.2 odd 2 3024.2.q.g.2305.1 6
4.3 odd 2 126.2.e.c.121.3 yes 6
7.4 even 3 1008.2.t.h.193.3 6
9.2 odd 6 3024.2.t.h.289.3 6
9.7 even 3 1008.2.t.h.961.3 6
12.11 even 2 378.2.e.d.37.1 6
21.11 odd 6 3024.2.t.h.1873.3 6
28.3 even 6 882.2.h.p.67.3 6
28.11 odd 6 126.2.h.d.67.1 yes 6
28.19 even 6 882.2.f.o.589.2 6
28.23 odd 6 882.2.f.n.589.2 6
28.27 even 2 882.2.e.o.373.1 6
36.7 odd 6 126.2.h.d.79.1 yes 6
36.11 even 6 378.2.h.c.289.3 6
36.23 even 6 1134.2.g.l.163.1 6
36.31 odd 6 1134.2.g.m.163.3 6
63.11 odd 6 3024.2.q.g.2881.1 6
63.25 even 3 inner 1008.2.q.g.529.1 6
84.11 even 6 378.2.h.c.361.3 6
84.23 even 6 2646.2.f.l.1765.1 6
84.47 odd 6 2646.2.f.m.1765.3 6
84.59 odd 6 2646.2.h.o.361.1 6
84.83 odd 2 2646.2.e.p.1549.3 6
252.11 even 6 378.2.e.d.235.1 6
252.23 even 6 7938.2.a.ca.1.3 3
252.47 odd 6 2646.2.f.m.883.3 6
252.67 odd 6 1134.2.g.m.487.3 6
252.79 odd 6 882.2.f.n.295.2 6
252.83 odd 6 2646.2.h.o.667.1 6
252.95 even 6 1134.2.g.l.487.1 6
252.103 even 6 7938.2.a.bw.1.3 3
252.115 even 6 882.2.e.o.655.1 6
252.131 odd 6 7938.2.a.bz.1.1 3
252.151 odd 6 126.2.e.c.25.3 ✓ 6
252.187 even 6 882.2.f.o.295.2 6
252.191 even 6 2646.2.f.l.883.1 6
252.223 even 6 882.2.h.p.79.3 6
252.227 odd 6 2646.2.e.p.2125.3 6
252.247 odd 6 7938.2.a.bv.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.e.c.25.3 ✓ 6 252.151 odd 6
126.2.e.c.121.3 yes 6 4.3 odd 2
126.2.h.d.67.1 yes 6 28.11 odd 6
126.2.h.d.79.1 yes 6 36.7 odd 6
378.2.e.d.37.1 6 12.11 even 2
378.2.e.d.235.1 6 252.11 even 6
378.2.h.c.289.3 6 36.11 even 6
378.2.h.c.361.3 6 84.11 even 6
882.2.e.o.373.1 6 28.27 even 2
882.2.e.o.655.1 6 252.115 even 6
882.2.f.n.295.2 6 252.79 odd 6
882.2.f.n.589.2 6 28.23 odd 6
882.2.f.o.295.2 6 252.187 even 6
882.2.f.o.589.2 6 28.19 even 6
882.2.h.p.67.3 6 28.3 even 6
882.2.h.p.79.3 6 252.223 even 6
1008.2.q.g.529.1 6 63.25 even 3 inner
1008.2.q.g.625.1 6 1.1 even 1 trivial
1008.2.t.h.193.3 6 7.4 even 3
1008.2.t.h.961.3 6 9.7 even 3
1134.2.g.l.163.1 6 36.23 even 6
1134.2.g.l.487.1 6 252.95 even 6
1134.2.g.m.163.3 6 36.31 odd 6
1134.2.g.m.487.3 6 252.67 odd 6
2646.2.e.p.1549.3 6 84.83 odd 2
2646.2.e.p.2125.3 6 252.227 odd 6
2646.2.f.l.883.1 6 252.191 even 6
2646.2.f.l.1765.1 6 84.23 even 6
2646.2.f.m.883.3 6 252.47 odd 6
2646.2.f.m.1765.3 6 84.47 odd 6
2646.2.h.o.361.1 6 84.59 odd 6
2646.2.h.o.667.1 6 252.83 odd 6
3024.2.q.g.2305.1 6 3.2 odd 2
3024.2.q.g.2881.1 6 63.11 odd 6
3024.2.t.h.289.3 6 9.2 odd 6
3024.2.t.h.1873.3 6 21.11 odd 6
7938.2.a.bv.1.1 3 252.247 odd 6
7938.2.a.bw.1.3 3 252.103 even 6
7938.2.a.bz.1.1 3 252.131 odd 6
7938.2.a.ca.1.3 3 252.23 even 6