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Properties

Label 672.2.bi.c.17.16
Level $672$
Weight $2$
Character 672.17
Analytic conductor $5.366$
Analytic rank $0$
Dimension $48$
Inner twists $8$

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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] = [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 17.16
Character \(\chi\) \(=\) 672.17
Dual form 672.2.bi.c.593.16

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.609093 + 1.62142i) q^{3} +(2.24840 - 1.29811i) q^{5} +(2.53678 + 0.751482i) q^{7} +(-2.25801 + 1.97519i) q^{9} +(-1.63808 + 2.83724i) q^{11} -0.912635 q^{13} +(3.47427 + 2.85492i) q^{15} +(2.39448 - 4.14736i) q^{17} +(2.66693 + 4.61926i) q^{19} +(0.326671 + 4.57092i) q^{21} +(4.45569 - 2.57249i) q^{23} +(0.870190 - 1.50721i) q^{25} +(-4.57796 - 2.45811i) q^{27} +1.35950 q^{29} +(-8.18469 - 4.72543i) q^{31} +(-5.59810 - 0.927874i) q^{33} +(6.67920 - 1.60340i) q^{35} +(-1.59746 + 0.922292i) q^{37} +(-0.555880 - 1.47977i) q^{39} +5.91833 q^{41} +8.00125i q^{43} +(-2.51288 + 7.37217i) q^{45} +(-3.29243 - 5.70266i) q^{47} +(5.87055 + 3.81269i) q^{49} +(8.18307 + 1.35633i) q^{51} +(0.841712 - 1.45789i) q^{53} +8.50565i q^{55} +(-5.86535 + 7.13777i) q^{57} +(-1.50266 - 0.867561i) q^{59} +(4.72347 + 8.18130i) q^{61} +(-7.21241 + 3.31379i) q^{63} +(-2.05197 + 1.18470i) q^{65} +(-10.8634 - 6.27198i) q^{67} +(6.88502 + 5.65766i) q^{69} -0.603989i q^{71} +(-1.29220 - 0.746053i) q^{73} +(2.97385 + 0.492910i) q^{75} +(-6.28759 + 5.96647i) q^{77} +(0.0625152 + 0.108280i) q^{79} +(1.19722 - 8.92001i) q^{81} +0.246431i q^{83} -12.4332i q^{85} +(0.828064 + 2.20433i) q^{87} +(1.80671 + 3.12931i) q^{89} +(-2.31516 - 0.685829i) q^{91} +(2.67667 - 16.1491i) q^{93} +(11.9926 + 6.92395i) q^{95} -5.10324i q^{97} +(-1.90529 - 9.64204i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 4 q^{7} - 14 q^{9} - 4 q^{15} - 8 q^{25} - 48 q^{31} - 42 q^{33} + 8 q^{39} - 36 q^{49} + 4 q^{57} + 6 q^{63} - 36 q^{73} + 56 q^{79} + 42 q^{81} + 132 q^{87}+O(q^{100}) \) 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
\(3\) 0.609093 + 1.62142i 0.351660 + 0.936128i
\(4\) 0 0
\(5\) 2.24840 − 1.29811i 1.00551 − 0.580533i 0.0956384 − 0.995416i \(-0.469511\pi\)
0.909875 + 0.414883i \(0.136177\pi\)
\(6\) 0 0
\(7\) 2.53678 + 0.751482i 0.958814 + 0.284033i
\(8\) 0 0
\(9\) −2.25801 + 1.97519i −0.752670 + 0.658398i
\(10\) 0 0
\(11\) −1.63808 + 2.83724i −0.493900 + 0.855460i −0.999975 − 0.00702958i \(-0.997762\pi\)
0.506075 + 0.862489i \(0.331096\pi\)
\(12\) 0 0
\(13\) −0.912635 −0.253119 −0.126560 − 0.991959i \(-0.540394\pi\)
−0.126560 + 0.991959i \(0.540394\pi\)
\(14\) 0 0
\(15\) 3.47427 + 2.85492i 0.897052 + 0.737138i
\(16\) 0 0
\(17\) 2.39448 − 4.14736i 0.580746 − 1.00588i −0.414645 − 0.909983i \(-0.636094\pi\)
0.995391 − 0.0958985i \(-0.0305724\pi\)
\(18\) 0 0
\(19\) 2.66693 + 4.61926i 0.611836 + 1.05973i 0.990931 + 0.134373i \(0.0429019\pi\)
−0.379095 + 0.925358i \(0.623765\pi\)
\(20\) 0 0
\(21\) 0.326671 + 4.57092i 0.0712854 + 0.997456i
\(22\) 0 0
\(23\) 4.45569 − 2.57249i 0.929075 − 0.536402i 0.0425563 − 0.999094i \(-0.486450\pi\)
0.886519 + 0.462692i \(0.153116\pi\)
\(24\) 0 0
\(25\) 0.870190 − 1.50721i 0.174038 − 0.301443i
\(26\) 0 0
\(27\) −4.57796 − 2.45811i −0.881029 − 0.473063i
\(28\) 0 0
\(29\) 1.35950 0.252453 0.126227 − 0.992001i \(-0.459713\pi\)
0.126227 + 0.992001i \(0.459713\pi\)
\(30\) 0 0
\(31\) −8.18469 − 4.72543i −1.47001 − 0.848713i −0.470580 − 0.882357i \(-0.655955\pi\)
−0.999434 + 0.0336443i \(0.989289\pi\)
\(32\) 0 0
\(33\) −5.59810 − 0.927874i −0.974504 − 0.161522i
\(34\) 0 0
\(35\) 6.67920 − 1.60340i 1.12899 − 0.271024i
\(36\) 0 0
\(37\) −1.59746 + 0.922292i −0.262620 + 0.151624i −0.625529 − 0.780201i \(-0.715117\pi\)
0.362909 + 0.931825i \(0.381784\pi\)
\(38\) 0 0
\(39\) −0.555880 − 1.47977i −0.0890121 − 0.236952i
\(40\) 0 0
\(41\) 5.91833 0.924288 0.462144 − 0.886805i \(-0.347080\pi\)
0.462144 + 0.886805i \(0.347080\pi\)
\(42\) 0 0
\(43\) 8.00125i 1.22018i 0.792332 + 0.610090i \(0.208867\pi\)
−0.792332 + 0.610090i \(0.791133\pi\)
\(44\) 0 0
\(45\) −2.51288 + 7.37217i −0.374598 + 1.09898i
\(46\) 0 0
\(47\) −3.29243 − 5.70266i −0.480250 − 0.831818i 0.519493 − 0.854475i \(-0.326121\pi\)
−0.999743 + 0.0226567i \(0.992788\pi\)
\(48\) 0 0
\(49\) 5.87055 + 3.81269i 0.838650 + 0.544671i
\(50\) 0 0
\(51\) 8.18307 + 1.35633i 1.14586 + 0.189924i
\(52\) 0 0
\(53\) 0.841712 − 1.45789i 0.115618 − 0.200256i −0.802409 − 0.596775i \(-0.796448\pi\)
0.918027 + 0.396519i \(0.129782\pi\)
\(54\) 0 0
\(55\) 8.50565i 1.14690i
\(56\) 0 0
\(57\) −5.86535 + 7.13777i −0.776885 + 0.945421i
\(58\) 0 0
\(59\) −1.50266 − 0.867561i −0.195630 − 0.112947i 0.398986 − 0.916957i \(-0.369362\pi\)
−0.594615 + 0.804010i \(0.702696\pi\)
\(60\) 0 0
\(61\) 4.72347 + 8.18130i 0.604779 + 1.04751i 0.992086 + 0.125557i \(0.0400718\pi\)
−0.387308 + 0.921951i \(0.626595\pi\)
\(62\) 0 0
\(63\) −7.21241 + 3.31379i −0.908678 + 0.417498i
\(64\) 0 0
\(65\) −2.05197 + 1.18470i −0.254515 + 0.146944i
\(66\) 0 0
\(67\) −10.8634 − 6.27198i −1.32717 − 0.766245i −0.342313 − 0.939586i \(-0.611210\pi\)
−0.984862 + 0.173341i \(0.944544\pi\)
\(68\) 0 0
\(69\) 6.88502 + 5.65766i 0.828860 + 0.681102i
\(70\) 0 0
\(71\) − 0.603989i − 0.0716803i −0.999358 − 0.0358401i \(-0.988589\pi\)
0.999358 − 0.0358401i \(-0.0114107\pi\)
\(72\) 0 0
\(73\) −1.29220 − 0.746053i −0.151241 − 0.0873189i 0.422470 − 0.906377i \(-0.361163\pi\)
−0.573711 + 0.819058i \(0.694497\pi\)
\(74\) 0 0
\(75\) 2.97385 + 0.492910i 0.343391 + 0.0569164i
\(76\) 0 0
\(77\) −6.28759 + 5.96647i −0.716537 + 0.679943i
\(78\) 0 0
\(79\) 0.0625152 + 0.108280i 0.00703351 + 0.0121824i 0.869521 − 0.493896i \(-0.164428\pi\)
−0.862487 + 0.506079i \(0.831094\pi\)
\(80\) 0 0
\(81\) 1.19722 − 8.92001i 0.133025 − 0.991113i
\(82\) 0 0
\(83\) 0.246431i 0.0270493i 0.999909 + 0.0135247i \(0.00430516\pi\)
−0.999909 + 0.0135247i \(0.995695\pi\)
\(84\) 0 0
\(85\) − 12.4332i − 1.34857i
\(86\) 0 0
\(87\) 0.828064 + 2.20433i 0.0887778 + 0.236329i
\(88\) 0 0
\(89\) 1.80671 + 3.12931i 0.191511 + 0.331706i 0.945751 − 0.324892i \(-0.105328\pi\)
−0.754241 + 0.656598i \(0.771995\pi\)
\(90\) 0 0
\(91\) −2.31516 − 0.685829i −0.242695 − 0.0718944i
\(92\) 0 0
\(93\) 2.67667 − 16.1491i 0.277558 − 1.67458i
\(94\) 0 0
\(95\) 11.9926 + 6.92395i 1.23042 + 0.710382i
\(96\) 0 0
\(97\) − 5.10324i − 0.518155i −0.965857 − 0.259078i \(-0.916581\pi\)
0.965857 − 0.259078i \(-0.0834185\pi\)
\(98\) 0 0
\(99\) −1.90529 − 9.64204i −0.191489 − 0.969061i
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.17.16 48
3.2 odd 2 inner 672.2.bi.c.17.1 48
4.3 odd 2 168.2.ba.c.101.2 yes 48
7.5 odd 6 inner 672.2.bi.c.593.24 48
8.3 odd 2 168.2.ba.c.101.8 yes 48
8.5 even 2 inner 672.2.bi.c.17.9 48
12.11 even 2 168.2.ba.c.101.23 yes 48
21.5 even 6 inner 672.2.bi.c.593.9 48
24.5 odd 2 inner 672.2.bi.c.17.24 48
24.11 even 2 168.2.ba.c.101.17 yes 48
28.19 even 6 168.2.ba.c.5.17 yes 48
56.5 odd 6 inner 672.2.bi.c.593.1 48
56.19 even 6 168.2.ba.c.5.23 yes 48
84.47 odd 6 168.2.ba.c.5.8 yes 48
168.5 even 6 inner 672.2.bi.c.593.16 48
168.131 odd 6 168.2.ba.c.5.2 ✓ 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.ba.c.5.2 ✓ 48 168.131 odd 6
168.2.ba.c.5.8 yes 48 84.47 odd 6
168.2.ba.c.5.17 yes 48 28.19 even 6
168.2.ba.c.5.23 yes 48 56.19 even 6
168.2.ba.c.101.2 yes 48 4.3 odd 2
168.2.ba.c.101.8 yes 48 8.3 odd 2
168.2.ba.c.101.17 yes 48 24.11 even 2
168.2.ba.c.101.23 yes 48 12.11 even 2
672.2.bi.c.17.1 48 3.2 odd 2 inner
672.2.bi.c.17.9 48 8.5 even 2 inner
672.2.bi.c.17.16 48 1.1 even 1 trivial
672.2.bi.c.17.24 48 24.5 odd 2 inner
672.2.bi.c.593.1 48 56.5 odd 6 inner
672.2.bi.c.593.9 48 21.5 even 6 inner
672.2.bi.c.593.16 48 168.5 even 6 inner
672.2.bi.c.593.24 48 7.5 odd 6 inner