Properties

Label 169.2.h.a
Level $169$
Weight $2$
Character orbit 169.h
Analytic conductor $1.349$
Analytic rank $0$
Dimension $168$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 169.h (of order \(26\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.34947179416\)
Analytic rank: \(0\)
Dimension: \(168\)
Relative dimension: \(14\) over \(\Q(\zeta_{26})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{26}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 168q - 13q^{2} - 13q^{3} + q^{4} - 13q^{5} - 13q^{6} - 13q^{7} - 13q^{8} - 27q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 168q - 13q^{2} - 13q^{3} + q^{4} - 13q^{5} - 13q^{6} - 13q^{7} - 13q^{8} - 27q^{9} - 9q^{10} - 13q^{11} - 9q^{12} + 52q^{13} - 15q^{14} + 39q^{15} - 31q^{16} - 11q^{17} + 26q^{18} - 13q^{20} - 13q^{21} - 6q^{22} - 102q^{23} - 91q^{24} + 3q^{25} - 13q^{26} - 19q^{27} - 13q^{28} - 17q^{29} + 23q^{30} + 39q^{31} - 13q^{33} + 52q^{34} - 21q^{35} + 11q^{36} - 13q^{37} - 40q^{38} - 65q^{39} + 53q^{40} - 13q^{41} + 101q^{42} - 3q^{43} - 39q^{44} - 78q^{45} - 39q^{46} - 39q^{47} + 8q^{48} + 85q^{49} - 13q^{50} + 62q^{51} - 78q^{52} + 18q^{53} + 65q^{54} - 103q^{55} + 3q^{56} + 65q^{57} + 39q^{58} + 91q^{59} - 117q^{60} - 23q^{61} - 64q^{62} - 78q^{63} + 15q^{64} - 13q^{65} + 134q^{66} - 65q^{67} + 40q^{68} - 40q^{69} + 26q^{71} + 156q^{72} - 13q^{73} - 53q^{74} + 35q^{75} + 221q^{76} + 3q^{77} - 143q^{78} - 23q^{79} - 43q^{81} - 129q^{82} + 117q^{83} + 234q^{84} + 26q^{85} - 91q^{86} + 79q^{87} + 95q^{88} + 17q^{90} + 39q^{91} - 89q^{92} + 52q^{93} - 61q^{94} + 122q^{95} - 182q^{96} - 91q^{97} - 13q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
12.1 −2.11580 + 1.46043i 0.527540 + 0.276874i 1.63454 4.30994i 0.267437 + 0.301873i −1.52053 + 0.184625i −1.09769 4.45349i 1.60549 + 6.51374i −1.50256 2.17683i −1.00671 0.248131i
12.2 −2.08715 + 1.44065i −1.12364 0.589730i 1.57149 4.14367i −2.16703 2.44607i 3.19479 0.387918i 1.08188 + 4.38937i 1.47582 + 5.98765i −0.789415 1.14367i 8.04684 + 1.98337i
12.3 −1.71806 + 1.18589i 2.94692 + 1.54666i 0.836178 2.20482i −0.781452 0.882078i −6.89716 + 0.837466i 0.664295 + 2.69515i 0.178883 + 0.725755i 4.58799 + 6.64685i 2.38863 + 0.588744i
12.4 −1.24935 + 0.862366i −1.71162 0.898326i 0.107998 0.284768i 0.371060 + 0.418840i 2.91310 0.353715i −0.356102 1.44476i −0.615953 2.49902i 0.418452 + 0.606232i −0.824778 0.203290i
12.5 −1.19961 + 0.828028i 0.960584 + 0.504153i 0.0442133 0.116581i 2.00541 + 2.26364i −1.56977 + 0.190605i 0.228271 + 0.926132i −0.654173 2.65409i −1.03564 1.50039i −4.28006 1.05494i
12.6 −0.705316 + 0.486844i −1.15919 0.608391i −0.448757 + 1.18327i −0.816038 0.921117i 1.11379 0.135238i 0.328944 + 1.33458i −0.669753 2.71730i −0.730609 1.05847i 1.02401 + 0.252395i
12.7 0.156578 0.108078i 2.47375 + 1.29833i −0.696374 + 1.83619i 0.268145 + 0.302673i 0.527655 0.0640690i −0.983215 3.98906i 0.180477 + 0.732225i 2.72961 + 3.95452i 0.0746977 + 0.0184113i
12.8 0.197515 0.136335i 1.40158 + 0.735608i −0.688785 + 1.81618i −0.369290 0.416842i 0.377124 0.0457911i 0.397115 + 1.61116i 0.226434 + 0.918678i −0.280874 0.406917i −0.129771 0.0319856i
12.9 0.323126 0.223038i −2.84614 1.49377i −0.654545 + 1.72589i 1.31816 + 1.48790i −1.25283 + 0.152121i 0.0500939 + 0.203239i 0.361363 + 1.46611i 4.16496 + 6.03399i 0.757790 + 0.186779i
12.10 0.651938 0.450000i −1.14686 0.601918i −0.486687 + 1.28329i −2.65258 2.99414i −1.01854 + 0.123674i −0.815660 3.30926i 0.639345 + 2.59392i −0.751216 1.08832i −3.07668 0.758333i
12.11 1.05649 0.729245i −0.277577 0.145684i −0.124830 + 0.329149i 2.67487 + 3.01931i −0.399497 + 0.0485078i 0.147281 + 0.597543i 0.722584 + 2.93164i −1.64837 2.38807i 5.02780 + 1.23924i
12.12 1.62113 1.11899i 0.214934 + 0.112806i 0.666728 1.75802i 0.457146 + 0.516011i 0.474664 0.0576347i −0.520249 2.11073i 0.0564755 + 0.229130i −1.67072 2.42046i 1.31850 + 0.324982i
12.13 1.63772 1.13043i 1.71106 + 0.898032i 0.695024 1.83263i −2.41202 2.72261i 3.81739 0.463516i 0.903389 + 3.66519i 0.0190509 + 0.0772924i 0.417062 + 0.604219i −7.02794 1.73223i
12.14 2.07618 1.43308i −1.76851 0.928185i 1.54757 4.08062i −0.134767 0.152121i −5.00190 + 0.607340i 0.524392 + 2.12754i −1.42736 5.79101i 0.561902 + 0.814056i −0.497802 0.122697i
25.1 −2.49582 0.946540i −0.730291 1.05801i 3.83616 + 3.39854i −0.453685 0.0550874i 0.821227 + 3.33185i −1.38308 2.63524i −3.87657 7.38618i 0.477757 1.25974i 1.08017 + 0.566920i
25.2 −2.16507 0.821103i 1.73033 + 2.50681i 2.51629 + 2.22924i 3.34407 + 0.406044i −1.68793 6.84821i −0.858950 1.63659i −1.46534 2.79198i −2.22627 + 5.87018i −6.90674 3.62494i
25.3 −1.68720 0.639871i 0.0407127 + 0.0589826i 0.940192 + 0.832938i 0.654244 + 0.0794397i −0.0309493 0.125566i −0.153151 0.291806i 0.623830 + 1.18861i 1.06199 2.80025i −1.05301 0.552663i
25.4 −1.52590 0.578696i −1.45490 2.10779i 0.496446 + 0.439813i −3.46589 0.420835i 1.00026 + 4.05821i 0.131788 + 0.251102i 1.01380 + 1.93163i −1.26222 + 3.32819i 5.04505 + 2.64785i
25.5 −1.49527 0.567082i 1.18578 + 1.71790i 0.417236 + 0.369639i −2.96326 0.359805i −0.798874 3.24116i 1.24562 + 2.37332i 1.07210 + 2.04271i −0.481287 + 1.26905i 4.22684 + 2.21842i
25.6 −0.622921 0.236243i −1.59376 2.30896i −1.16480 1.03192i 3.47190 + 0.421565i 0.447311 + 1.81481i −1.63679 3.11863i 1.10100 + 2.09779i −1.72740 + 4.55479i −2.06313 1.08281i
See next 80 embeddings (of 168 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 155.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
169.h even 26 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.2.h.a 168
169.h even 26 1 inner 169.2.h.a 168
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.2.h.a 168 1.a even 1 1 trivial
169.2.h.a 168 169.h even 26 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(169, [\chi])\).