↑

Properties

Label 672.2.bi.c.17.23
Level $672$
Weight $2$
Character 672.17
Analytic conductor $5.366$
Analytic rank $0$
Dimension $48$
Inner twists $8$

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] = [672,2,Mod(17,672)] 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("672.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(672, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.bi (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 17.23
Character \(\chi\) \(=\) 672.17
Dual form 672.2.bi.c.593.23

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.70617 + 0.298296i) q^{3} +(-0.337879 + 0.195075i) q^{5} +(-1.39526 + 2.24795i) q^{7} +(2.82204 + 1.01789i) q^{9} +(-0.748582 + 1.29658i) q^{11} +3.28768 q^{13} +(-0.634670 + 0.232043i) q^{15} +(1.68169 - 2.91278i) q^{17} +(2.56203 + 4.43756i) q^{19} +(-3.05110 + 3.41918i) q^{21} +(-4.72764 + 2.72950i) q^{23} +(-2.42389 + 4.19830i) q^{25} +(4.51125 + 2.57850i) q^{27} +4.13801 q^{29} +(3.60237 + 2.07983i) q^{31} +(-1.66397 + 1.98889i) q^{33} +(0.0329110 - 1.03171i) q^{35} +(7.46581 - 4.31038i) q^{37} +(5.60934 + 0.980702i) q^{39} -11.1607 q^{41} -4.79323i q^{43} +(-1.15207 + 0.206585i) q^{45} +(-2.51067 - 4.34861i) q^{47} +(-3.10652 - 6.27292i) q^{49} +(3.73813 - 4.46806i) q^{51} +(0.499243 - 0.864715i) q^{53} -0.584118i q^{55} +(3.04755 + 8.33548i) q^{57} +(1.36034 + 0.785391i) q^{59} +(3.40889 + 5.90437i) q^{61} +(-6.22563 + 4.92357i) q^{63} +(-1.11084 + 0.641343i) q^{65} +(-3.05467 - 1.76361i) q^{67} +(-8.88036 + 3.24676i) q^{69} -14.3360i q^{71} +(2.76107 + 1.59410i) q^{73} +(-5.38791 + 6.43999i) q^{75} +(-1.87018 - 3.49184i) q^{77} +(0.239413 + 0.414676i) q^{79} +(6.92781 + 5.74504i) q^{81} -17.4548i q^{83} +1.31222i q^{85} +(7.06016 + 1.23435i) q^{87} +(-2.54840 - 4.41396i) q^{89} +(-4.58716 + 7.39053i) q^{91} +(5.52586 + 4.62312i) q^{93} +(-1.73131 - 0.999573i) q^{95} -9.00074i q^{97} +(-3.43230 + 2.89703i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 4 q^{7} - 14 q^{9} - 4 q^{15} - 8 q^{25} - 48 q^{31} - 42 q^{33} + 8 q^{39} - 36 q^{49} + 4 q^{57} + 6 q^{63} - 36 q^{73} + 56 q^{79} + 42 q^{81} + 132 q^{87}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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
\(3\) 1.70617 + 0.298296i 0.985058 + 0.172221i
\(4\) 0 0
\(5\) −0.337879 + 0.195075i −0.151104 + 0.0872401i −0.573646 − 0.819103i \(-0.694471\pi\)
0.422542 + 0.906344i \(0.361138\pi\)
\(6\) 0 0
\(7\) −1.39526 + 2.24795i −0.527357 + 0.849644i
\(8\) 0 0
\(9\) 2.82204 + 1.01789i 0.940680 + 0.339296i
\(10\) 0 0
\(11\) −0.748582 + 1.29658i −0.225706 + 0.390934i −0.956531 − 0.291631i \(-0.905802\pi\)
0.730825 + 0.682565i \(0.239136\pi\)
\(12\) 0 0
\(13\) 3.28768 0.911838 0.455919 − 0.890021i \(-0.349311\pi\)
0.455919 + 0.890021i \(0.349311\pi\)
\(14\) 0 0
\(15\) −0.634670 + 0.232043i −0.163871 + 0.0599132i
\(16\) 0 0
\(17\) 1.68169 − 2.91278i 0.407871 − 0.706453i −0.586780 − 0.809746i \(-0.699605\pi\)
0.994651 + 0.103293i \(0.0329381\pi\)
\(18\) 0 0
\(19\) 2.56203 + 4.43756i 0.587769 + 1.01805i 0.994524 + 0.104509i \(0.0333270\pi\)
−0.406755 + 0.913537i \(0.633340\pi\)
\(20\) 0 0
\(21\) −3.05110 + 3.41918i −0.665805 + 0.746126i
\(22\) 0 0
\(23\) −4.72764 + 2.72950i −0.985780 + 0.569141i −0.904010 − 0.427511i \(-0.859391\pi\)
−0.0817700 + 0.996651i \(0.526057\pi\)
\(24\) 0 0
\(25\) −2.42389 + 4.19830i −0.484778 + 0.839661i
\(26\) 0 0
\(27\) 4.51125 + 2.57850i 0.868190 + 0.496232i
\(28\) 0 0
\(29\) 4.13801 0.768410 0.384205 − 0.923248i \(-0.374476\pi\)
0.384205 + 0.923248i \(0.374476\pi\)
\(30\) 0 0
\(31\) 3.60237 + 2.07983i 0.647005 + 0.373549i 0.787308 − 0.616560i \(-0.211474\pi\)
−0.140303 + 0.990109i \(0.544808\pi\)
\(32\) 0 0
\(33\) −1.66397 + 1.98889i −0.289661 + 0.346222i
\(34\) 0 0
\(35\) 0.0329110 − 1.03171i 0.00556298 − 0.174391i
\(36\) 0 0
\(37\) 7.46581 − 4.31038i 1.22737 − 0.708623i 0.260892 − 0.965368i \(-0.415983\pi\)
0.966479 + 0.256745i \(0.0826501\pi\)
\(38\) 0 0
\(39\) 5.60934 + 0.980702i 0.898214 + 0.157038i
\(40\) 0 0
\(41\) −11.1607 −1.74301 −0.871505 − 0.490387i \(-0.836855\pi\)
−0.871505 + 0.490387i \(0.836855\pi\)
\(42\) 0 0
\(43\) − 4.79323i − 0.730961i −0.930819 − 0.365480i \(-0.880905\pi\)
0.930819 − 0.365480i \(-0.119095\pi\)
\(44\) 0 0
\(45\) −1.15207 + 0.206585i −0.171741 + 0.0307959i
\(46\) 0 0
\(47\) −2.51067 − 4.34861i −0.366219 − 0.634310i 0.622752 − 0.782419i \(-0.286015\pi\)
−0.988971 + 0.148109i \(0.952681\pi\)
\(48\) 0 0
\(49\) −3.10652 − 6.27292i −0.443788 − 0.896132i
\(50\) 0 0
\(51\) 3.73813 − 4.46806i 0.523443 − 0.625653i
\(52\) 0 0
\(53\) 0.499243 − 0.864715i 0.0685763 − 0.118778i −0.829699 − 0.558212i \(-0.811488\pi\)
0.898275 + 0.439434i \(0.144821\pi\)
\(54\) 0 0
\(55\) − 0.584118i − 0.0787624i
\(56\) 0 0
\(57\) 3.04755 + 8.33548i 0.403657 + 1.10406i
\(58\) 0 0
\(59\) 1.36034 + 0.785391i 0.177101 + 0.102249i 0.585930 − 0.810362i \(-0.300729\pi\)
−0.408829 + 0.912611i \(0.634063\pi\)
\(60\) 0 0
\(61\) 3.40889 + 5.90437i 0.436464 + 0.755977i 0.997414 − 0.0718723i \(-0.0228974\pi\)
−0.560950 + 0.827850i \(0.689564\pi\)
\(62\) 0 0
\(63\) −6.22563 + 4.92357i −0.784355 + 0.620312i
\(64\) 0 0
\(65\) −1.11084 + 0.641343i −0.137783 + 0.0795488i
\(66\) 0 0
\(67\) −3.05467 − 1.76361i −0.373187 − 0.215460i 0.301663 − 0.953415i \(-0.402458\pi\)
−0.674850 + 0.737955i \(0.735792\pi\)
\(68\) 0 0
\(69\) −8.88036 + 3.24676i −1.06907 + 0.390864i
\(70\) 0 0
\(71\) − 14.3360i − 1.70137i −0.525677 − 0.850684i \(-0.676188\pi\)
0.525677 − 0.850684i \(-0.323812\pi\)
\(72\) 0 0
\(73\) 2.76107 + 1.59410i 0.323159 + 0.186576i 0.652800 − 0.757531i \(-0.273594\pi\)
−0.329641 + 0.944106i \(0.606928\pi\)
\(74\) 0 0
\(75\) −5.38791 + 6.43999i −0.622142 + 0.743626i
\(76\) 0 0
\(77\) −1.87018 − 3.49184i −0.213127 − 0.397932i
\(78\) 0 0
\(79\) 0.239413 + 0.414676i 0.0269361 + 0.0466547i 0.879179 − 0.476491i \(-0.158092\pi\)
−0.852243 + 0.523146i \(0.824758\pi\)
\(80\) 0 0
\(81\) 6.92781 + 5.74504i 0.769756 + 0.638338i
\(82\) 0 0
\(83\) − 17.4548i − 1.91591i −0.286911 − 0.957957i \(-0.592628\pi\)
0.286911 − 0.957957i \(-0.407372\pi\)
\(84\) 0 0
\(85\) 1.31222i 0.142331i
\(86\) 0 0
\(87\) 7.06016 + 1.23435i 0.756928 + 0.132337i
\(88\) 0 0
\(89\) −2.54840 − 4.41396i −0.270130 − 0.467879i 0.698765 − 0.715351i \(-0.253733\pi\)
−0.968895 + 0.247473i \(0.920400\pi\)
\(90\) 0 0
\(91\) −4.58716 + 7.39053i −0.480865 + 0.774738i
\(92\) 0 0
\(93\) 5.52586 + 4.62312i 0.573005 + 0.479395i
\(94\) 0 0
\(95\) −1.73131 − 0.999573i −0.177629 − 0.102554i
\(96\) 0 0
\(97\) − 9.00074i − 0.913887i −0.889496 − 0.456943i \(-0.848944\pi\)
0.889496 − 0.456943i \(-0.151056\pi\)
\(98\) 0 0
\(99\) −3.43230 + 2.89703i −0.344959 + 0.291163i
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 672.2.bi.c.17.23 48
3.2 odd 2 inner 672.2.bi.c.17.6 48
4.3 odd 2 168.2.ba.c.101.9 yes 48
7.5 odd 6 inner 672.2.bi.c.593.19 48
8.3 odd 2 168.2.ba.c.101.1 yes 48
8.5 even 2 inner 672.2.bi.c.17.2 48
12.11 even 2 168.2.ba.c.101.16 yes 48
21.5 even 6 inner 672.2.bi.c.593.2 48
24.5 odd 2 inner 672.2.bi.c.17.19 48
24.11 even 2 168.2.ba.c.101.24 yes 48
28.19 even 6 168.2.ba.c.5.24 yes 48
56.5 odd 6 inner 672.2.bi.c.593.6 48
56.19 even 6 168.2.ba.c.5.16 yes 48
84.47 odd 6 168.2.ba.c.5.1 ✓ 48
168.5 even 6 inner 672.2.bi.c.593.23 48
168.131 odd 6 168.2.ba.c.5.9 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.ba.c.5.1 ✓ 48 84.47 odd 6
168.2.ba.c.5.9 yes 48 168.131 odd 6
168.2.ba.c.5.16 yes 48 56.19 even 6
168.2.ba.c.5.24 yes 48 28.19 even 6
168.2.ba.c.101.1 yes 48 8.3 odd 2
168.2.ba.c.101.9 yes 48 4.3 odd 2
168.2.ba.c.101.16 yes 48 12.11 even 2
168.2.ba.c.101.24 yes 48 24.11 even 2
672.2.bi.c.17.2 48 8.5 even 2 inner
672.2.bi.c.17.6 48 3.2 odd 2 inner
672.2.bi.c.17.19 48 24.5 odd 2 inner
672.2.bi.c.17.23 48 1.1 even 1 trivial
672.2.bi.c.593.2 48 21.5 even 6 inner
672.2.bi.c.593.6 48 56.5 odd 6 inner
672.2.bi.c.593.19 48 7.5 odd 6 inner
672.2.bi.c.593.23 48 168.5 even 6 inner