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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [168,2,Mod(5,168)] 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("168.5"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(168, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 3, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 168.ba (of order \(6\), degree \(2\), 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: \(1.34148675396\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 5.17
Character \(\chi\) \(=\) 168.5
Dual form 168.2.ba.c.101.17

$q$-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+(0.968446 + 1.03059i) q^{2} +(-1.70874 - 0.283220i) q^{3} +(-0.124224 + 1.99614i) q^{4} +(2.24840 + 1.29811i) q^{5} +(-1.36294 - 2.03529i) q^{6} +(-2.53678 + 0.751482i) q^{7} +(-2.17750 + 1.80513i) q^{8} +(2.83957 + 0.967897i) q^{9} +(0.839632 + 3.57432i) q^{10} +(1.63808 + 2.83724i) q^{11} +(0.777612 - 3.37570i) q^{12} +0.912635 q^{13} +(-3.23121 - 1.88661i) q^{14} +(-3.47427 - 2.85492i) q^{15} +(-3.96914 - 0.495937i) q^{16} +(-2.39448 - 4.14736i) q^{17} +(1.75247 + 3.86379i) q^{18} +(2.66693 - 4.61926i) q^{19} +(-2.87052 + 4.32685i) q^{20} +(4.54754 - 0.565618i) q^{21} +(-1.33763 + 4.43590i) q^{22} +(4.45569 + 2.57249i) q^{23} +(4.23203 - 2.46778i) q^{24} +(0.870190 + 1.50721i) q^{25} +(0.883838 + 0.940551i) q^{26} +(-4.57796 - 2.45811i) q^{27} +(-1.18493 - 5.15712i) q^{28} +1.35950 q^{29} +(-0.422391 - 6.34538i) q^{30} +(8.18469 - 4.72543i) q^{31} +(-3.33279 - 4.57083i) q^{32} +(-1.99549 - 5.31204i) q^{33} +(1.95529 - 6.48421i) q^{34} +(-6.67920 - 1.60340i) q^{35} +(-2.28480 + 5.54794i) q^{36} +(1.59746 + 0.922292i) q^{37} +(7.34333 - 1.72500i) q^{38} +(-1.55945 - 0.258476i) q^{39} +(-7.23914 + 1.23200i) q^{40} -5.91833 q^{41} +(4.98696 + 4.13887i) q^{42} -8.00125i q^{43} +(-5.86701 + 2.91738i) q^{44} +(5.12805 + 5.86230i) q^{45} +(1.66391 + 7.08330i) q^{46} +(-3.29243 + 5.70266i) q^{47} +(6.64176 + 1.97156i) q^{48} +(5.87055 - 3.81269i) q^{49} +(-0.710584 + 2.35646i) q^{50} +(2.91692 + 7.76491i) q^{51} +(-0.113371 + 1.82175i) q^{52} +(0.841712 + 1.45789i) q^{53} +(-1.90021 - 7.09853i) q^{54} +8.50565i q^{55} +(4.16733 - 6.21557i) q^{56} +(-5.86535 + 7.13777i) q^{57} +(1.31661 + 1.40109i) q^{58} +(1.50266 - 0.867561i) q^{59} +(6.13041 - 6.58047i) q^{60} +(-4.72347 + 8.18130i) q^{61} +(12.7964 + 3.85872i) q^{62} +(-7.93074 - 0.321460i) q^{63} +(1.48302 - 7.86134i) q^{64} +(2.05197 + 1.18470i) q^{65} +(3.54200 - 7.20095i) q^{66} +(-10.8634 + 6.27198i) q^{67} +(8.57615 - 4.26451i) q^{68} +(-6.88502 - 5.65766i) q^{69} +(-4.81600 - 8.43632i) q^{70} +0.603989i q^{71} +(-7.93035 + 3.01820i) q^{72} +(-1.29220 + 0.746053i) q^{73} +(0.596548 + 2.53951i) q^{74} +(-1.06005 - 2.82189i) q^{75} +(8.88938 + 5.89738i) q^{76} +(-6.28759 - 5.96647i) q^{77} +(-1.24387 - 1.85748i) q^{78} +(-0.0625152 + 0.108280i) q^{79} +(-8.28041 - 6.26745i) q^{80} +(7.12635 + 5.49683i) q^{81} +(-5.73158 - 6.09936i) q^{82} +0.246431i q^{83} +(0.564138 + 9.14777i) q^{84} -12.4332i q^{85} +(8.24600 - 7.74878i) q^{86} +(-2.32303 - 0.385038i) q^{87} +(-8.68850 - 3.22114i) q^{88} +(-1.80671 + 3.12931i) q^{89} +(-1.07538 + 10.9622i) q^{90} +(-2.31516 + 0.685829i) q^{91} +(-5.68856 + 8.57461i) q^{92} +(-15.3238 + 5.75646i) q^{93} +(-9.06563 + 2.12958i) q^{94} +(11.9926 - 6.92395i) q^{95} +(4.40031 + 8.75427i) q^{96} +5.10324i q^{97} +(9.61463 + 2.35773i) q^{98} +(1.90529 + 9.64204i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 6 q^{4} - 4 q^{7} - 14 q^{9} - 30 q^{10} - 18 q^{12} + 4 q^{15} + 6 q^{16} - 36 q^{22} - 12 q^{24} - 8 q^{25} - 10 q^{28} + 22 q^{30} + 48 q^{31} - 42 q^{33} + 52 q^{36} - 8 q^{39} - 18 q^{40} + 12 q^{42}+ \cdots + 90 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(85\) \(113\) \(127\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.968446 + 1.03059i 0.684795 + 0.728736i
\(3\) −1.70874 − 0.283220i −0.986541 − 0.163517i
\(4\) −0.124224 + 1.99614i −0.0621120 + 0.998069i
\(5\) 2.24840 + 1.29811i 1.00551 + 0.580533i 0.909875 − 0.414883i \(-0.136177\pi\)
0.0956384 + 0.995416i \(0.469511\pi\)
\(6\) −1.36294 − 2.03529i −0.556417 − 0.830903i
\(7\) −2.53678 + 0.751482i −0.958814 + 0.284033i
\(8\) −2.17750 + 1.80513i −0.769863 + 0.638209i
\(9\) 2.83957 + 0.967897i 0.946524 + 0.322632i
\(10\) 0.839632 + 3.57432i 0.265515 + 1.13030i
\(11\) 1.63808 + 2.83724i 0.493900 + 0.855460i 0.999975 − 0.00702958i \(-0.00223760\pi\)
−0.506075 + 0.862489i \(0.668904\pi\)
\(12\) 0.777612 − 3.37570i 0.224477 − 0.974479i
\(13\) 0.912635 0.253119 0.126560 − 0.991959i \(-0.459606\pi\)
0.126560 + 0.991959i \(0.459606\pi\)
\(14\) −3.23121 − 1.88661i −0.863577 − 0.504218i
\(15\) −3.47427 − 2.85492i −0.897052 − 0.737138i
\(16\) −3.96914 − 0.495937i −0.992284 − 0.123984i
\(17\) −2.39448 − 4.14736i −0.580746 − 1.00588i −0.995391 − 0.0958985i \(-0.969428\pi\)
0.414645 − 0.909983i \(-0.363906\pi\)
\(18\) 1.75247 + 3.86379i 0.413061 + 0.910703i
\(19\) 2.66693 − 4.61926i 0.611836 − 1.05973i −0.379095 − 0.925358i \(-0.623765\pi\)
0.990931 − 0.134373i \(-0.0429019\pi\)
\(20\) −2.87052 + 4.32685i −0.641867 + 0.967514i
\(21\) 4.54754 − 0.565618i 0.992354 − 0.123428i
\(22\) −1.33763 + 4.43590i −0.285184 + 0.945737i
\(23\) 4.45569 + 2.57249i 0.929075 + 0.536402i 0.886519 − 0.462692i \(-0.153116\pi\)
0.0425563 + 0.999094i \(0.486450\pi\)
\(24\) 4.23203 − 2.46778i 0.863859 − 0.503734i
\(25\) 0.870190 + 1.50721i 0.174038 + 0.301443i
\(26\) 0.883838 + 0.940551i 0.173335 + 0.184457i
\(27\) −4.57796 − 2.45811i −0.881029 − 0.473063i
\(28\) −1.18493 − 5.15712i −0.223931 − 0.974605i
\(29\) 1.35950 0.252453 0.126227 − 0.992001i \(-0.459713\pi\)
0.126227 + 0.992001i \(0.459713\pi\)
\(30\) −0.422391 − 6.34538i −0.0771178 − 1.15850i
\(31\) 8.18469 − 4.72543i 1.47001 − 0.848713i 0.470580 − 0.882357i \(-0.344045\pi\)
0.999434 + 0.0336443i \(0.0107113\pi\)
\(32\) −3.33279 − 4.57083i −0.589159 − 0.808017i
\(33\) −1.99549 − 5.31204i −0.347370 − 0.924707i
\(34\) 1.95529 − 6.48421i 0.335330 − 1.11203i
\(35\) −6.67920 − 1.60340i −1.12899 − 0.271024i
\(36\) −2.28480 + 5.54794i −0.380800 + 0.924657i
\(37\) 1.59746 + 0.922292i 0.262620 + 0.151624i 0.625529 − 0.780201i \(-0.284883\pi\)
−0.362909 + 0.931825i \(0.618216\pi\)
\(38\) 7.34333 − 1.72500i 1.19125 − 0.279831i
\(39\) −1.55945 − 0.258476i −0.249713 − 0.0413894i
\(40\) −7.23914 + 1.23200i −1.14461 + 0.194797i
\(41\) −5.91833 −0.924288 −0.462144 − 0.886805i \(-0.652920\pi\)
−0.462144 + 0.886805i \(0.652920\pi\)
\(42\) 4.98696 + 4.13887i 0.769505 + 0.638641i
\(43\) − 8.00125i − 1.22018i −0.792332 − 0.610090i \(-0.791133\pi\)
0.792332 − 0.610090i \(-0.208867\pi\)
\(44\) −5.86701 + 2.91738i −0.884485 + 0.439812i
\(45\) 5.12805 + 5.86230i 0.764444 + 0.873900i
\(46\) 1.66391 + 7.08330i 0.245331 + 1.04438i
\(47\) −3.29243 + 5.70266i −0.480250 + 0.831818i −0.999743 − 0.0226567i \(-0.992788\pi\)
0.519493 + 0.854475i \(0.326121\pi\)
\(48\) 6.64176 + 1.97156i 0.958655 + 0.284571i
\(49\) 5.87055 − 3.81269i 0.838650 − 0.544671i
\(50\) −0.710584 + 2.35646i −0.100492 + 0.333254i
\(51\) 2.91692 + 7.76491i 0.408451 + 1.08730i
\(52\) −0.113371 + 1.82175i −0.0157218 + 0.252631i
\(53\) 0.841712 + 1.45789i 0.115618 + 0.200256i 0.918027 − 0.396519i \(-0.129782\pi\)
−0.802409 + 0.596775i \(0.796448\pi\)
\(54\) −1.90021 − 7.09853i −0.258586 − 0.965988i
\(55\) 8.50565i 1.14690i
\(56\) 4.16733 − 6.21557i 0.556883 − 0.830591i
\(57\) −5.86535 + 7.13777i −0.776885 + 0.945421i
\(58\) 1.31661 + 1.40109i 0.172879 + 0.183972i
\(59\) 1.50266 − 0.867561i 0.195630 − 0.112947i −0.398986 − 0.916957i \(-0.630638\pi\)
0.594615 + 0.804010i \(0.297304\pi\)
\(60\) 6.13041 − 6.58047i 0.791433 − 0.849535i
\(61\) −4.72347 + 8.18130i −0.604779 + 1.04751i 0.387308 + 0.921951i \(0.373405\pi\)
−0.992086 + 0.125557i \(0.959928\pi\)
\(62\) 12.7964 + 3.85872i 1.62515 + 0.490058i
\(63\) −7.93074 − 0.321460i −0.999180 − 0.0405001i
\(64\) 1.48302 − 7.86134i 0.185378 − 0.982667i
\(65\) 2.05197 + 1.18470i 0.254515 + 0.146944i
\(66\) 3.54200 − 7.20095i 0.435990 − 0.886375i
\(67\) −10.8634 + 6.27198i −1.32717 + 0.766245i −0.984862 − 0.173341i \(-0.944544\pi\)
−0.342313 + 0.939586i \(0.611210\pi\)
\(68\) 8.57615 − 4.26451i 1.04001 − 0.517147i
\(69\) −6.88502 − 5.65766i −0.828860 − 0.681102i
\(70\) −4.81600 − 8.43632i −0.575622 − 1.00833i
\(71\) 0.603989i 0.0716803i 0.999358 + 0.0358401i \(0.0114107\pi\)
−0.999358 + 0.0358401i \(0.988589\pi\)
\(72\) −7.93035 + 3.01820i −0.934601 + 0.355698i
\(73\) −1.29220 + 0.746053i −0.151241 + 0.0873189i −0.573711 − 0.819058i \(-0.694497\pi\)
0.422470 + 0.906377i \(0.361163\pi\)
\(74\) 0.596548 + 2.53951i 0.0693473 + 0.295212i
\(75\) −1.06005 − 2.82189i −0.122404 − 0.325844i
\(76\) 8.88938 + 5.89738i 1.01968 + 0.676476i
\(77\) −6.28759 − 5.96647i −0.716537 − 0.679943i
\(78\) −1.24387 − 1.85748i −0.140840 − 0.210318i
\(79\) −0.0625152 + 0.108280i −0.00703351 + 0.0121824i −0.869521 − 0.493896i \(-0.835572\pi\)
0.862487 + 0.506079i \(0.168906\pi\)
\(80\) −8.28041 − 6.26745i −0.925778 − 0.700722i
\(81\) 7.12635 + 5.49683i 0.791817 + 0.610759i
\(82\) −5.73158 − 6.09936i −0.632947 − 0.673562i
\(83\) 0.246431i 0.0270493i 0.999909 + 0.0135247i \(0.00430516\pi\)
−0.999909 + 0.0135247i \(0.995695\pi\)
\(84\) 0.564138 + 9.14777i 0.0615525 + 0.998104i
\(85\) − 12.4332i − 1.34857i
\(86\) 8.24600 − 7.74878i 0.889189 − 0.835573i
\(87\) −2.32303 − 0.385038i −0.249055 − 0.0412804i
\(88\) −8.68850 − 3.22114i −0.926198 − 0.343375i
\(89\) −1.80671 + 3.12931i −0.191511 + 0.331706i −0.945751 − 0.324892i \(-0.894672\pi\)
0.754241 + 0.656598i \(0.228005\pi\)
\(90\) −1.07538 + 10.9622i −0.113355 + 1.15552i
\(91\) −2.31516 + 0.685829i −0.242695 + 0.0718944i
\(92\) −5.68856 + 8.57461i −0.593073 + 0.893964i
\(93\) −15.3238 + 5.75646i −1.58901 + 0.596917i
\(94\) −9.06563 + 2.12958i −0.935049 + 0.219649i
\(95\) 11.9926 − 6.92395i 1.23042 − 0.710382i
\(96\) 4.40031 + 8.75427i 0.449105 + 0.893479i
\(97\) 5.10324i 0.518155i 0.965857 + 0.259078i \(0.0834185\pi\)
−0.965857 + 0.259078i \(0.916581\pi\)
\(98\) 9.61463 + 2.35773i 0.971224 + 0.238167i
\(99\) 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 168.2.ba.c.5.17 yes 48
3.2 odd 2 inner 168.2.ba.c.5.8 yes 48
4.3 odd 2 672.2.bi.c.593.24 48
7.3 odd 6 inner 168.2.ba.c.101.2 yes 48
8.3 odd 2 672.2.bi.c.593.1 48
8.5 even 2 inner 168.2.ba.c.5.23 yes 48
12.11 even 2 672.2.bi.c.593.9 48
21.17 even 6 inner 168.2.ba.c.101.23 yes 48
24.5 odd 2 inner 168.2.ba.c.5.2 ✓ 48
24.11 even 2 672.2.bi.c.593.16 48
28.3 even 6 672.2.bi.c.17.16 48
56.3 even 6 672.2.bi.c.17.9 48
56.45 odd 6 inner 168.2.ba.c.101.8 yes 48
84.59 odd 6 672.2.bi.c.17.1 48
168.59 odd 6 672.2.bi.c.17.24 48
168.101 even 6 inner 168.2.ba.c.101.17 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.ba.c.5.2 ✓ 48 24.5 odd 2 inner
168.2.ba.c.5.8 yes 48 3.2 odd 2 inner
168.2.ba.c.5.17 yes 48 1.1 even 1 trivial
168.2.ba.c.5.23 yes 48 8.5 even 2 inner
168.2.ba.c.101.2 yes 48 7.3 odd 6 inner
168.2.ba.c.101.8 yes 48 56.45 odd 6 inner
168.2.ba.c.101.17 yes 48 168.101 even 6 inner
168.2.ba.c.101.23 yes 48 21.17 even 6 inner
672.2.bi.c.17.1 48 84.59 odd 6
672.2.bi.c.17.9 48 56.3 even 6
672.2.bi.c.17.16 48 28.3 even 6
672.2.bi.c.17.24 48 168.59 odd 6
672.2.bi.c.593.1 48 8.3 odd 2
672.2.bi.c.593.9 48 12.11 even 2
672.2.bi.c.593.16 48 24.11 even 2
672.2.bi.c.593.24 48 4.3 odd 2