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Properties

Label 1008.2.t.h.193.3
Level $1008$
Weight $2$
Character 1008.193
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(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 = [6,0,-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: \(6\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 193.3
Root \(0.500000 - 1.41036i\) of defining polynomial
Character \(\chi\) \(=\) 1008.193
Dual form 1008.2.t.h.961.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.09097 + 1.34528i) q^{3} -3.18194 q^{5} +(-0.710533 + 2.54856i) q^{7} +(-0.619562 + 2.93533i) q^{9} -3.18194 q^{11} +(2.85185 - 4.93955i) q^{13} +(-3.47141 - 4.28061i) q^{15} +(-0.760877 + 1.31788i) q^{17} +(0.641315 + 1.11079i) q^{19} +(-4.20370 + 1.82454i) q^{21} -2.23912 q^{23} +5.12476 q^{25} +(-4.62476 + 2.36887i) q^{27} +(-3.54063 - 6.13255i) q^{29} +(-4.71053 - 8.15888i) q^{31} +(-3.47141 - 4.28061i) q^{33} +(2.26088 - 8.10936i) q^{35} +(0.500000 + 0.866025i) q^{37} +(9.75636 - 1.55237i) q^{39} +(-2.80150 + 4.85235i) q^{41} +(-3.41423 - 5.91362i) q^{43} +(1.97141 - 9.34004i) q^{45} +(-2.91423 + 5.04759i) q^{47} +(-5.99028 - 3.62167i) q^{49} +(-2.60301 + 0.414174i) q^{51} +(1.02859 - 1.78157i) q^{53} +10.1248 q^{55} +(-0.794668 + 2.07459i) q^{57} +(-0.562382 - 0.974074i) q^{59} +(-1.56238 + 2.70612i) q^{61} +(-7.04063 - 3.66464i) q^{63} +(-9.07442 + 15.7174i) q^{65} +(5.48345 + 9.49761i) q^{67} +(-2.44282 - 3.01225i) q^{69} -8.69002 q^{71} +(-2.48345 + 4.30146i) q^{73} +(5.59097 + 6.89425i) q^{75} +(2.26088 - 8.10936i) q^{77} +(-2.06922 + 3.58399i) q^{79} +(-8.23229 - 3.63723i) q^{81} +(4.03379 + 6.98673i) q^{83} +(2.42107 - 4.19341i) q^{85} +(4.38727 - 11.4536i) q^{87} +(0.112725 + 0.195246i) q^{89} +(10.5624 + 10.7778i) q^{91} +(5.83693 - 15.2381i) q^{93} +(-2.04063 - 3.53447i) q^{95} +(7.42107 + 12.8537i) q^{97} +(1.97141 - 9.34004i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} - 2 q^{5} + 4 q^{7} - 4 q^{9} - 2 q^{11} + 8 q^{13} - 12 q^{15} - 4 q^{17} + 3 q^{19} - 7 q^{21} - 14 q^{23} - 4 q^{25} + 7 q^{27} - 5 q^{29} - 20 q^{31} - 12 q^{33} + 13 q^{35} + 3 q^{37}+ \cdots + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 1.09097 + 1.34528i 0.629873 + 0.776698i
\(4\) 0 0
\(5\) −3.18194 −1.42301 −0.711504 − 0.702682i \(-0.751986\pi\)
−0.711504 + 0.702682i \(0.751986\pi\)
\(6\) 0 0
\(7\) −0.710533 + 2.54856i −0.268556 + 0.963264i
\(8\) 0 0
\(9\) −0.619562 + 2.93533i −0.206521 + 0.978442i
\(10\) 0 0
\(11\) −3.18194 −0.959392 −0.479696 − 0.877435i \(-0.659253\pi\)
−0.479696 + 0.877435i \(0.659253\pi\)
\(12\) 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 −0.943421 − 0.331596i \(-0.892413\pi\)
0.758882 + 0.651229i \(0.225746\pi\)
\(18\) 0 0
\(19\) 0.641315 + 1.11079i 0.147128 + 0.254833i 0.930165 − 0.367142i \(-0.119664\pi\)
−0.783037 + 0.621975i \(0.786330\pi\)
\(20\) 0 0
\(21\) −4.20370 + 1.82454i −0.917322 + 0.398147i
\(22\) 0 0
\(23\) −2.23912 −0.466889 −0.233445 − 0.972370i \(-0.575000\pi\)
−0.233445 + 0.972370i \(0.575000\pi\)
\(24\) 0 0
\(25\) 5.12476 1.02495
\(26\) 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
\(31\) −4.71053 − 8.15888i −0.846037 − 1.46538i −0.884718 − 0.466127i \(-0.845649\pi\)
0.0386810 − 0.999252i \(-0.487684\pi\)
\(32\) 0 0
\(33\) −3.47141 − 4.28061i −0.604295 − 0.745158i
\(34\) 0 0
\(35\) 2.26088 − 8.10936i 0.382158 − 1.37073i
\(36\) 0 0
\(37\) 0.500000 + 0.866025i 0.0821995 + 0.142374i 0.904194 − 0.427121i \(-0.140472\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 9.75636 − 1.55237i 1.56227 − 0.248578i
\(40\) 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
\(45\) 1.97141 − 9.34004i 0.293880 − 1.39233i
\(46\) 0 0
\(47\) −2.91423 + 5.04759i −0.425084 + 0.736267i −0.996428 − 0.0844432i \(-0.973089\pi\)
0.571344 + 0.820711i \(0.306422\pi\)
\(48\) 0 0
\(49\) −5.99028 − 3.62167i −0.855755 − 0.517381i
\(50\) 0 0
\(51\) −2.60301 + 0.414174i −0.364494 + 0.0579959i
\(52\) 0 0
\(53\) 1.02859 − 1.78157i 0.141288 − 0.244717i −0.786694 − 0.617343i \(-0.788209\pi\)
0.927982 + 0.372626i \(0.121542\pi\)
\(54\) 0 0
\(55\) 10.1248 1.36522
\(56\) 0 0
\(57\) −0.794668 + 2.07459i −0.105256 + 0.274786i
\(58\) 0 0
\(59\) −0.562382 − 0.974074i −0.0732159 − 0.126814i 0.827093 − 0.562065i \(-0.189993\pi\)
−0.900309 + 0.435251i \(0.856660\pi\)
\(60\) 0 0
\(61\) −1.56238 + 2.70612i −0.200042 + 0.346484i −0.948542 − 0.316652i \(-0.897441\pi\)
0.748499 + 0.663135i \(0.230775\pi\)
\(62\) 0 0
\(63\) −7.04063 − 3.66464i −0.887036 − 0.461701i
\(64\) 0 0
\(65\) −9.07442 + 15.7174i −1.12554 + 1.94950i
\(66\) 0 0
\(67\) 5.48345 + 9.49761i 0.669910 + 1.16032i 0.977929 + 0.208938i \(0.0670006\pi\)
−0.308019 + 0.951380i \(0.599666\pi\)
\(68\) 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 −0.973967 − 0.226689i \(-0.927210\pi\)
0.683302 + 0.730136i \(0.260543\pi\)
\(74\) 0 0
\(75\) 5.59097 + 6.89425i 0.645590 + 0.796079i
\(76\) 0 0
\(77\) 2.26088 − 8.10936i 0.257651 − 0.924148i
\(78\) 0 0
\(79\) −2.06922 + 3.58399i −0.232805 + 0.403231i −0.958633 − 0.284646i \(-0.908124\pi\)
0.725827 + 0.687877i \(0.241457\pi\)
\(80\) 0 0
\(81\) −8.23229 − 3.63723i −0.914699 − 0.404137i
\(82\) 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
\(87\) 4.38727 − 11.4536i 0.470365 − 1.22795i
\(88\) 0 0
\(89\) 0.112725 + 0.195246i 0.0119488 + 0.0206960i 0.871938 − 0.489616i \(-0.162863\pi\)
−0.859989 + 0.510312i \(0.829530\pi\)
\(90\) 0 0
\(91\) 10.5624 + 10.7778i 1.10724 + 1.12982i
\(92\) 0 0
\(93\) 5.83693 − 15.2381i 0.605262 − 1.58012i
\(94\) 0 0
\(95\) −2.04063 − 3.53447i −0.209364 − 0.362629i
\(96\) 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.t.h.193.3 6
3.2 odd 2 3024.2.t.h.1873.3 6
4.3 odd 2 126.2.h.d.67.1 yes 6
7.2 even 3 1008.2.q.g.625.1 6
9.2 odd 6 3024.2.q.g.2881.1 6
9.7 even 3 1008.2.q.g.529.1 6
12.11 even 2 378.2.h.c.361.3 6
21.2 odd 6 3024.2.q.g.2305.1 6
28.3 even 6 882.2.f.o.589.2 6
28.11 odd 6 882.2.f.n.589.2 6
28.19 even 6 882.2.e.o.373.1 6
28.23 odd 6 126.2.e.c.121.3 yes 6
28.27 even 2 882.2.h.p.67.3 6
36.7 odd 6 126.2.e.c.25.3 ✓ 6
36.11 even 6 378.2.e.d.235.1 6
36.23 even 6 1134.2.g.l.487.1 6
36.31 odd 6 1134.2.g.m.487.3 6
63.2 odd 6 3024.2.t.h.289.3 6
63.16 even 3 inner 1008.2.t.h.961.3 6
84.11 even 6 2646.2.f.l.1765.1 6
84.23 even 6 378.2.e.d.37.1 6
84.47 odd 6 2646.2.e.p.1549.3 6
84.59 odd 6 2646.2.f.m.1765.3 6
84.83 odd 2 2646.2.h.o.361.1 6
252.11 even 6 2646.2.f.l.883.1 6
252.23 even 6 1134.2.g.l.163.1 6
252.31 even 6 7938.2.a.bw.1.3 3
252.47 odd 6 2646.2.h.o.667.1 6
252.59 odd 6 7938.2.a.bz.1.1 3
252.67 odd 6 7938.2.a.bv.1.1 3
252.79 odd 6 126.2.h.d.79.1 yes 6
252.83 odd 6 2646.2.e.p.2125.3 6
252.95 even 6 7938.2.a.ca.1.3 3
252.115 even 6 882.2.f.o.295.2 6
252.151 odd 6 882.2.f.n.295.2 6
252.187 even 6 882.2.h.p.79.3 6
252.191 even 6 378.2.h.c.289.3 6
252.223 even 6 882.2.e.o.655.1 6
252.227 odd 6 2646.2.f.m.883.3 6
252.247 odd 6 1134.2.g.m.163.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.e.c.25.3 ✓ 6 36.7 odd 6
126.2.e.c.121.3 yes 6 28.23 odd 6
126.2.h.d.67.1 yes 6 4.3 odd 2
126.2.h.d.79.1 yes 6 252.79 odd 6
378.2.e.d.37.1 6 84.23 even 6
378.2.e.d.235.1 6 36.11 even 6
378.2.h.c.289.3 6 252.191 even 6
378.2.h.c.361.3 6 12.11 even 2
882.2.e.o.373.1 6 28.19 even 6
882.2.e.o.655.1 6 252.223 even 6
882.2.f.n.295.2 6 252.151 odd 6
882.2.f.n.589.2 6 28.11 odd 6
882.2.f.o.295.2 6 252.115 even 6
882.2.f.o.589.2 6 28.3 even 6
882.2.h.p.67.3 6 28.27 even 2
882.2.h.p.79.3 6 252.187 even 6
1008.2.q.g.529.1 6 9.7 even 3
1008.2.q.g.625.1 6 7.2 even 3
1008.2.t.h.193.3 6 1.1 even 1 trivial
1008.2.t.h.961.3 6 63.16 even 3 inner
1134.2.g.l.163.1 6 252.23 even 6
1134.2.g.l.487.1 6 36.23 even 6
1134.2.g.m.163.3 6 252.247 odd 6
1134.2.g.m.487.3 6 36.31 odd 6
2646.2.e.p.1549.3 6 84.47 odd 6
2646.2.e.p.2125.3 6 252.83 odd 6
2646.2.f.l.883.1 6 252.11 even 6
2646.2.f.l.1765.1 6 84.11 even 6
2646.2.f.m.883.3 6 252.227 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.h.o.667.1 6 252.47 odd 6
3024.2.q.g.2305.1 6 21.2 odd 6
3024.2.q.g.2881.1 6 9.2 odd 6
3024.2.t.h.289.3 6 63.2 odd 6
3024.2.t.h.1873.3 6 3.2 odd 2
7938.2.a.bv.1.1 3 252.67 odd 6
7938.2.a.bw.1.3 3 252.31 even 6
7938.2.a.bz.1.1 3 252.59 odd 6
7938.2.a.ca.1.3 3 252.95 even 6