# Properties

 Label 68.2.i.b Level $68$ Weight $2$ Character orbit 68.i Analytic conductor $0.543$ Analytic rank $0$ Dimension $48$ CM no Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$68 = 2^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 68.i (of order $$16$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.542982733745$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$6$$ over $$\Q(\zeta_{16})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$48 q - 8 q^{2} - 8 q^{4} - 16 q^{5} - 8 q^{6} - 8 q^{8} - 16 q^{9}+O(q^{10})$$ 48 * q - 8 * q^2 - 8 * q^4 - 16 * q^5 - 8 * q^6 - 8 * q^8 - 16 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$48 q - 8 q^{2} - 8 q^{4} - 16 q^{5} - 8 q^{6} - 8 q^{8} - 16 q^{9} + 16 q^{10} - 8 q^{12} - 16 q^{13} - 8 q^{14} - 16 q^{17} - 16 q^{18} - 24 q^{20} - 16 q^{21} - 8 q^{22} + 8 q^{24} + 16 q^{25} - 16 q^{26} + 40 q^{28} + 56 q^{30} + 32 q^{32} + 56 q^{34} + 56 q^{36} - 16 q^{37} + 32 q^{38} + 56 q^{40} - 48 q^{41} + 40 q^{42} + 24 q^{44} - 64 q^{45} + 8 q^{46} - 32 q^{48} - 16 q^{49} - 16 q^{52} + 48 q^{53} - 24 q^{54} - 48 q^{56} + 64 q^{57} - 64 q^{58} - 112 q^{60} + 16 q^{61} - 64 q^{62} - 56 q^{64} + 96 q^{65} - 96 q^{66} - 32 q^{68} + 32 q^{69} - 80 q^{70} - 64 q^{72} + 64 q^{73} - 16 q^{74} - 64 q^{76} + 16 q^{77} - 112 q^{78} - 24 q^{80} + 64 q^{81} - 40 q^{82} - 80 q^{85} + 64 q^{86} + 56 q^{88} - 16 q^{89} + 48 q^{90} + 104 q^{92} - 16 q^{93} + 88 q^{94} + 144 q^{96} - 16 q^{97} + 72 q^{98}+O(q^{100})$$ 48 * q - 8 * q^2 - 8 * q^4 - 16 * q^5 - 8 * q^6 - 8 * q^8 - 16 * q^9 + 16 * q^10 - 8 * q^12 - 16 * q^13 - 8 * q^14 - 16 * q^17 - 16 * q^18 - 24 * q^20 - 16 * q^21 - 8 * q^22 + 8 * q^24 + 16 * q^25 - 16 * q^26 + 40 * q^28 + 56 * q^30 + 32 * q^32 + 56 * q^34 + 56 * q^36 - 16 * q^37 + 32 * q^38 + 56 * q^40 - 48 * q^41 + 40 * q^42 + 24 * q^44 - 64 * q^45 + 8 * q^46 - 32 * q^48 - 16 * q^49 - 16 * q^52 + 48 * q^53 - 24 * q^54 - 48 * q^56 + 64 * q^57 - 64 * q^58 - 112 * q^60 + 16 * q^61 - 64 * q^62 - 56 * q^64 + 96 * q^65 - 96 * q^66 - 32 * q^68 + 32 * q^69 - 80 * q^70 - 64 * q^72 + 64 * q^73 - 16 * q^74 - 64 * q^76 + 16 * q^77 - 112 * q^78 - 24 * q^80 + 64 * q^81 - 40 * q^82 - 80 * q^85 + 64 * q^86 + 56 * q^88 - 16 * q^89 + 48 * q^90 + 104 * q^92 - 16 * q^93 + 88 * q^94 + 144 * q^96 - 16 * q^97 + 72 * q^98

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 −1.17748 0.783289i 0.418777 2.10534i 0.772915 + 1.84461i −0.561585 0.840472i −2.14219 + 2.15097i −1.73093 1.15657i 0.534775 2.77741i −1.48543 0.615285i 0.00292235 + 1.42952i
3.2 −0.705238 1.22582i −0.563072 + 2.83076i −1.00528 + 1.72899i 0.991762 + 1.48428i 3.86710 1.30613i −1.11612 0.745767i 2.82840 + 0.0129412i −4.92449 2.03979i 1.12003 2.26249i
3.3 −0.278734 + 1.38647i −0.418777 + 2.10534i −1.84461 0.772915i −0.561585 0.840472i −2.80226 1.16745i 1.73093 + 1.15657i 1.58578 2.34207i −1.48543 0.615285i 1.32182 0.544354i
3.4 0.238486 1.39396i 0.0741201 0.372627i −1.88625 0.664880i −1.26427 1.89211i −0.501751 0.192187i 3.23087 + 2.15880i −1.37666 + 2.47079i 2.63828 + 1.09281i −2.93903 + 1.31109i
3.5 0.368108 + 1.36547i 0.563072 2.83076i −1.72899 + 1.00528i 0.991762 + 1.48428i 4.07257 0.273169i 1.11612 + 0.745767i −2.00913 1.99083i −4.92449 2.03979i −1.66165 + 1.90059i
3.6 1.15431 + 0.817044i −0.0741201 + 0.372627i 0.664880 + 1.88625i −1.26427 1.89211i −0.390010 + 0.369569i −3.23087 2.15880i −0.773668 + 2.72056i 2.63828 + 1.09281i 0.0865750 3.21705i
7.1 −1.26407 0.634144i 0.968913 + 0.647407i 1.19572 + 1.60320i 0.0404181 + 0.203195i −0.814220 1.43280i 2.58907 + 0.514998i −0.494813 2.78481i −0.628394 1.51708i 0.0777640 0.282483i
7.2 −1.19689 + 0.753296i −1.51958 1.01535i 0.865091 1.80322i −0.618394 3.10888i 2.58363 + 0.0705700i 1.63813 + 0.325845i 0.322943 + 2.80993i 0.130134 + 0.314170i 3.08205 + 3.25515i
7.3 0.226060 1.39603i 1.80045 + 1.20302i −1.89779 0.631174i 0.177432 + 0.892011i 2.08647 2.24153i −3.16204 0.628968i −1.31015 + 2.50669i 0.646311 + 1.56033i 1.28538 0.0460519i
7.4 0.313668 + 1.37899i 1.51958 + 1.01535i −1.80322 + 0.865091i −0.618394 3.10888i −0.923513 + 2.41397i −1.63813 0.325845i −1.75857 2.21528i 0.130134 + 0.314170i 4.09314 1.82791i
7.5 0.827293 1.14699i −1.80045 1.20302i −0.631174 1.89779i 0.177432 + 0.892011i −2.86936 + 1.06985i 3.16204 + 0.628968i −2.69892 0.846081i 0.646311 + 1.56033i 1.16992 + 0.534441i
7.6 1.34224 + 0.445422i −0.968913 0.647407i 1.60320 + 1.19572i 0.0404181 + 0.203195i −1.01214 1.30055i −2.58907 0.514998i 1.61927 + 2.31904i −0.628394 1.51708i −0.0362570 + 0.290739i
11.1 −1.41399 0.0249511i −1.51338 + 0.301030i 1.99875 + 0.0705615i 1.90890 + 1.27549i 2.14742 0.387894i 2.53332 + 3.79138i −2.82447 0.149645i −0.571942 + 0.236906i −2.66735 1.85116i
11.2 −1.28454 + 0.591578i 2.70865 0.538783i 1.30007 1.51981i −2.42403 1.61969i −3.16062 + 2.29446i 0.747003 + 1.11797i −0.770902 + 2.72134i 4.27484 1.77070i 4.07193 + 0.646542i
11.3 −1.01749 0.982201i 1.51338 0.301030i 0.0705615 + 1.99875i 1.90890 + 1.27549i −1.83552 1.18015i −2.53332 3.79138i 1.89138 2.10301i −0.571942 + 0.236906i −0.689499 3.17272i
11.4 −0.489996 1.32661i −2.70865 + 0.538783i −1.51981 + 1.30007i −2.42403 1.61969i 2.04198 + 3.32933i −0.747003 1.11797i 2.46939 + 1.37917i 4.27484 1.77070i −0.960932 + 4.00939i
11.5 0.114599 + 1.40956i 0.935416 0.186066i −1.97373 + 0.323068i 0.763429 + 0.510107i 0.369469 + 1.29720i −0.225807 0.337944i −0.681573 2.74508i −1.93126 + 0.799952i −0.631540 + 1.13456i
11.6 1.07774 0.915678i −0.935416 + 0.186066i 0.323068 1.97373i 0.763429 + 0.510107i −0.837764 + 1.05707i 0.225807 + 0.337944i −1.45912 2.42301i −1.93126 + 0.799952i 1.28988 0.149290i
23.1 −1.17748 + 0.783289i 0.418777 + 2.10534i 0.772915 1.84461i −0.561585 + 0.840472i −2.14219 2.15097i −1.73093 + 1.15657i 0.534775 + 2.77741i −1.48543 + 0.615285i 0.00292235 1.42952i
23.2 −0.705238 + 1.22582i −0.563072 2.83076i −1.00528 1.72899i 0.991762 1.48428i 3.86710 + 1.30613i −1.11612 + 0.745767i 2.82840 0.0129412i −4.92449 + 2.03979i 1.12003 + 2.26249i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 63.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
17.e odd 16 1 inner
68.i even 16 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 68.2.i.b 48
3.b odd 2 1 612.2.bd.d 48
4.b odd 2 1 inner 68.2.i.b 48
12.b even 2 1 612.2.bd.d 48
17.e odd 16 1 inner 68.2.i.b 48
51.i even 16 1 612.2.bd.d 48
68.i even 16 1 inner 68.2.i.b 48
204.t odd 16 1 612.2.bd.d 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.2.i.b 48 1.a even 1 1 trivial
68.2.i.b 48 4.b odd 2 1 inner
68.2.i.b 48 17.e odd 16 1 inner
68.2.i.b 48 68.i even 16 1 inner
612.2.bd.d 48 3.b odd 2 1
612.2.bd.d 48 12.b even 2 1
612.2.bd.d 48 51.i even 16 1
612.2.bd.d 48 204.t odd 16 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{48} + 8 T_{3}^{46} + 16 T_{3}^{44} - 384 T_{3}^{42} - 2944 T_{3}^{40} + 13600 T_{3}^{38} + 221488 T_{3}^{36} - 86208 T_{3}^{34} + 6461256 T_{3}^{32} + 25939456 T_{3}^{30} + 35736928 T_{3}^{28} + \cdots + 17866064896$$ acting on $$S_{2}^{\mathrm{new}}(68, [\chi])$$.