Properties

Label 668.2.h.a
Level $668$
Weight $2$
Character orbit 668.h
Analytic conductor $5.334$
Analytic rank $0$
Dimension $6724$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 668 = 2^{2} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 668.h (of order \(166\), degree \(82\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.33400685502\)
Analytic rank: \(0\)
Dimension: \(6724\)
Relative dimension: \(82\) over \(\Q(\zeta_{166})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{166}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 6724q - 81q^{2} - 85q^{4} - 166q^{5} - 75q^{6} - 75q^{8} - 84q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 6724q - 81q^{2} - 85q^{4} - 166q^{5} - 75q^{6} - 75q^{8} - 84q^{9} - 83q^{10} - 101q^{12} - 166q^{13} - 93q^{14} - 93q^{16} - 166q^{17} - 63q^{18} - 83q^{20} - 166q^{21} - 103q^{22} - 93q^{24} - 88q^{25} - 83q^{26} - 83q^{28} - 170q^{29} - 83q^{30} - 101q^{32} - 174q^{33} - 83q^{34} - 113q^{36} - 166q^{37} - 47q^{38} - 83q^{40} - 166q^{41} - 86q^{42} - 78q^{44} - 166q^{45} - 83q^{46} - 110q^{48} - 84q^{49} - 43q^{50} - 83q^{52} - 166q^{53} - 86q^{54} - 133q^{56} - 174q^{57} - 61q^{58} - 83q^{60} - 202q^{61} - 88q^{62} - 91q^{64} - 190q^{65} - 107q^{66} - 83q^{68} - 166q^{69} - 83q^{70} - 52q^{72} - 166q^{73} - 83q^{74} - 47q^{76} - 166q^{77} - 83q^{78} - 83q^{80} - 280q^{81} - 83q^{82} - 116q^{84} - 150q^{85} - 83q^{86} - 63q^{88} - 194q^{89} - 83q^{90} - 83q^{92} - 238q^{93} - 173q^{94} - 85q^{96} - 162q^{97} - 134q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1 −1.41330 0.0508355i −1.89908 2.53233i 1.99483 + 0.143692i 0.0776745 0.336123i 2.55524 + 3.67549i 1.39922 + 0.132799i −2.81199 0.304488i −1.96595 + 6.73829i −0.126864 + 0.471094i
15.2 −1.41324 + 0.0524176i 0.519888 + 0.693245i 1.99450 0.148157i 0.730855 3.16265i −0.771066 0.952472i 3.90748 + 0.370857i −2.81095 + 0.313930i 0.629939 2.15911i −0.867096 + 4.50789i
15.3 −1.40266 0.180378i −1.32817 1.77105i 1.93493 + 0.506019i 0.679950 2.94236i 1.54351 + 2.72375i −1.65201 0.156792i −2.62278 1.05879i −0.532332 + 1.82456i −1.48448 + 4.00449i
15.4 −1.40204 0.185136i 0.267796 + 0.357093i 1.93145 + 0.519139i −0.750457 + 3.24747i −0.309351 0.550239i −1.50739 0.143066i −2.61186 1.08544i 0.784443 2.68867i 1.65340 4.41416i
15.5 −1.39758 + 0.216291i 1.40535 + 1.87397i 1.90644 0.604567i −0.352669 + 1.52611i −2.36941 2.31505i −3.78061 0.358817i −2.53363 + 1.25727i −0.696501 + 2.38725i 0.162797 2.20914i
15.6 −1.37664 0.323827i 1.92126 + 2.56191i 1.79027 + 0.891587i 0.303580 1.31369i −1.81527 4.14898i 1.06823 + 0.101385i −2.17584 1.80713i −2.03189 + 6.96427i −0.843329 + 1.71017i
15.7 −1.36880 0.355518i −0.199932 0.266599i 1.74721 + 0.973265i −0.435572 + 1.88486i 0.178885 + 0.436000i 2.28482 + 0.216852i −2.04557 1.95337i 0.809142 2.77333i 1.26631 2.42514i
15.8 −1.35899 + 0.391340i 1.57222 + 2.09648i 1.69371 1.06365i −0.776138 + 3.35860i −2.95707 2.23382i 4.43983 + 0.421383i −1.88548 + 2.10831i −1.08311 + 3.71235i −0.259592 4.86804i
15.9 −1.35747 0.396595i 1.06894 + 1.42538i 1.68542 + 1.07673i 0.441016 1.90842i −0.885753 2.35885i −3.35265 0.318199i −1.86088 2.13005i −0.0488343 + 0.167379i −1.35553 + 2.41571i
15.10 −1.35512 + 0.404546i −0.666088 0.888195i 1.67269 1.09641i 0.185473 0.802603i 1.26194 + 0.934146i 0.603158 + 0.0572456i −1.82314 + 2.16245i 0.495026 1.69670i 0.0733515 + 1.16265i
15.11 −1.30737 + 0.539237i 0.666088 + 0.888195i 1.41845 1.40997i 0.185473 0.802603i −1.34977 0.802023i −0.603158 0.0572456i −1.09413 + 2.60823i 0.495026 1.69670i 0.190310 + 1.14932i
15.12 −1.30206 + 0.551933i −1.57222 2.09648i 1.39074 1.43730i −0.776138 + 3.35860i 3.20425 + 1.86199i −4.43983 0.421383i −1.01754 + 2.63906i −1.08311 + 3.71235i −0.843142 4.80149i
15.13 −1.22179 + 0.712204i −1.40535 1.87397i 0.985531 1.74032i −0.352669 + 1.52611i 3.05169 + 1.28870i 3.78061 + 0.358817i 0.0353553 + 2.82821i −0.696501 + 2.38725i −0.656017 2.11576i
15.14 −1.21190 0.728895i −1.02052 1.36081i 0.937425 + 1.76670i −0.0807078 + 0.349249i 0.244883 + 2.39303i −3.47450 0.329764i 0.151670 2.82436i 0.0298911 0.102452i 0.352376 0.364429i
15.15 −1.16508 0.801617i −1.64821 2.19780i 0.714821 + 1.86789i −0.943687 + 4.08364i 0.158496 + 3.88184i 0.269428 + 0.0255713i 0.664513 2.74926i −1.27351 + 4.36494i 4.37298 4.00129i
15.16 −1.13075 + 0.849359i −0.519888 0.693245i 0.557178 1.92082i 0.730855 3.16265i 1.17668 + 0.342313i −3.90748 0.370857i 1.00144 + 2.64521i 0.629939 2.15911i 1.85981 + 4.19691i
15.17 −1.12956 0.850944i −0.939846 1.25324i 0.551790 + 1.92238i 0.224144 0.969945i −0.00482749 + 2.21536i 3.52889 + 0.334926i 1.01256 2.64097i 0.152947 0.524225i −1.07855 + 0.904872i
15.18 −1.06576 + 0.929597i 1.89908 + 2.53233i 0.271697 1.98146i 0.0776745 0.336123i −4.37802 0.933483i −1.39922 0.132799i 1.55239 + 2.36433i −1.96595 + 6.73829i 0.229676 + 0.430433i
15.19 −1.03337 0.965479i 0.777226 + 1.03639i 0.135700 + 1.99539i 0.786467 3.40330i 0.197455 1.82137i −2.18353 0.207238i 1.78628 2.19299i 0.370215 1.26891i −4.09852 + 2.75754i
15.20 −1.02853 0.970630i 0.994607 + 1.32626i 0.115755 + 1.99665i −0.166755 + 0.721602i 0.264323 2.32950i 4.08470 + 0.387677i 1.81895 2.16597i 0.0705222 0.241714i 0.871922 0.580334i
See next 80 embeddings (of 6724 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 659.82
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
167.d odd 166 1 inner
668.h even 166 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 668.2.h.a 6724
4.b odd 2 1 inner 668.2.h.a 6724
167.d odd 166 1 inner 668.2.h.a 6724
668.h even 166 1 inner 668.2.h.a 6724
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
668.2.h.a 6724 1.a even 1 1 trivial
668.2.h.a 6724 4.b odd 2 1 inner
668.2.h.a 6724 167.d odd 166 1 inner
668.2.h.a 6724 668.h even 166 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(668, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database