Properties

Label 93.2.p.b
Level $93$
Weight $2$
Character orbit 93.p
Analytic conductor $0.743$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 93 = 3 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 93.p (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.742608738798\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(8\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64q - 10q^{3} + 12q^{4} - 9q^{6} - 26q^{7} - 8q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 64q - 10q^{3} + 12q^{4} - 9q^{6} - 26q^{7} - 8q^{9} - 36q^{10} + 15q^{12} - 32q^{13} - 20q^{15} - 24q^{16} - 6q^{18} + 5q^{21} - 24q^{22} - 48q^{24} + 38q^{25} + 5q^{27} + 76q^{28} + 30q^{31} - 7q^{33} - 4q^{34} - 5q^{36} + 48q^{37} - 7q^{39} + 8q^{40} + 15q^{42} - 92q^{43} - 63q^{45} - 70q^{46} + 12q^{48} - 2q^{49} + 58q^{51} + 72q^{52} + 100q^{54} + 10q^{55} + 93q^{57} + 50q^{58} + 85q^{60} - 18q^{63} + 46q^{64} + 6q^{66} - 46q^{67} + 110q^{69} - 158q^{70} + 163q^{72} - 30q^{73} + 55q^{75} + 34q^{76} - 11q^{78} + 24q^{79} - 108q^{81} - 116q^{82} - 80q^{84} - 130q^{85} - 9q^{87} - 222q^{88} - 93q^{90} - 20q^{91} - 121q^{93} + 128q^{94} - 122q^{96} + 18q^{97} - 102q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −2.49541 + 0.810809i 0.448212 + 1.67305i 3.95165 2.87104i 2.63849 1.52333i −2.47500 3.81155i 0.249973 + 2.37834i −4.44863 + 6.12302i −2.59821 + 1.49976i −5.34899 + 5.94065i
11.2 −1.72336 + 0.559954i −1.73134 + 0.0496529i 1.03839 0.754435i 0.322909 0.186432i 2.95592 1.05504i −0.413561 3.93477i 0.763119 1.05034i 2.99507 0.171932i −0.452096 + 0.502103i
11.3 −1.25999 + 0.409396i 1.20011 1.24889i −0.198061 + 0.143900i 1.61219 0.930798i −1.00084 + 2.06491i 0.00741560 + 0.0705547i 1.74808 2.40602i −0.119450 2.99762i −1.65028 + 1.83282i
11.4 −1.13134 + 0.367594i 1.09350 + 1.34323i −0.473233 + 0.343824i −3.25867 + 1.88139i −1.73088 1.11768i −0.121266 1.15377i 1.80741 2.48769i −0.608530 + 2.93763i 2.99507 3.32636i
11.5 1.13134 0.367594i −1.22157 1.22791i −0.473233 + 0.343824i 3.25867 1.88139i −1.83338 0.940142i −0.121266 1.15377i −1.80741 + 2.48769i −0.0155363 + 2.99996i 2.99507 3.32636i
11.6 1.25999 0.409396i 1.36749 1.06300i −0.198061 + 0.143900i −1.61219 + 0.930798i 1.28784 1.89921i 0.00741560 + 0.0705547i −1.74808 + 2.40602i 0.740080 2.90728i −1.65028 + 1.83282i
11.7 1.72336 0.559954i −0.230355 + 1.71666i 1.03839 0.754435i −0.322909 + 0.186432i 0.564268 + 3.08742i −0.413561 3.93477i −0.763119 + 1.05034i −2.89387 0.790885i −0.452096 + 0.502103i
11.8 2.49541 0.810809i −1.61704 0.620638i 3.95165 2.87104i −2.63849 + 1.52333i −4.53840 0.237641i 0.249973 + 2.37834i 4.44863 6.12302i 2.22962 + 2.00719i −5.34899 + 5.94065i
17.1 −2.49541 0.810809i 0.448212 1.67305i 3.95165 + 2.87104i 2.63849 + 1.52333i −2.47500 + 3.81155i 0.249973 2.37834i −4.44863 6.12302i −2.59821 1.49976i −5.34899 5.94065i
17.2 −1.72336 0.559954i −1.73134 0.0496529i 1.03839 + 0.754435i 0.322909 + 0.186432i 2.95592 + 1.05504i −0.413561 + 3.93477i 0.763119 + 1.05034i 2.99507 + 0.171932i −0.452096 0.502103i
17.3 −1.25999 0.409396i 1.20011 + 1.24889i −0.198061 0.143900i 1.61219 + 0.930798i −1.00084 2.06491i 0.00741560 0.0705547i 1.74808 + 2.40602i −0.119450 + 2.99762i −1.65028 1.83282i
17.4 −1.13134 0.367594i 1.09350 1.34323i −0.473233 0.343824i −3.25867 1.88139i −1.73088 + 1.11768i −0.121266 + 1.15377i 1.80741 + 2.48769i −0.608530 2.93763i 2.99507 + 3.32636i
17.5 1.13134 + 0.367594i −1.22157 + 1.22791i −0.473233 0.343824i 3.25867 + 1.88139i −1.83338 + 0.940142i −0.121266 + 1.15377i −1.80741 2.48769i −0.0155363 2.99996i 2.99507 + 3.32636i
17.6 1.25999 + 0.409396i 1.36749 + 1.06300i −0.198061 0.143900i −1.61219 0.930798i 1.28784 + 1.89921i 0.00741560 0.0705547i −1.74808 2.40602i 0.740080 + 2.90728i −1.65028 1.83282i
17.7 1.72336 + 0.559954i −0.230355 1.71666i 1.03839 + 0.754435i −0.322909 0.186432i 0.564268 3.08742i −0.413561 + 3.93477i −0.763119 1.05034i −2.89387 + 0.790885i −0.452096 0.502103i
17.8 2.49541 + 0.810809i −1.61704 + 0.620638i 3.95165 + 2.87104i −2.63849 1.52333i −4.53840 + 0.237641i 0.249973 2.37834i 4.44863 + 6.12302i 2.22962 2.00719i −5.34899 5.94065i
44.1 −1.59081 2.18957i −1.15089 1.29440i −1.64548 + 5.06426i −1.13653 + 0.656178i −1.00332 + 4.57908i −1.89531 2.10496i 8.55821 2.78073i −0.350919 + 2.97941i 3.24476 + 1.44466i
44.2 −1.13925 1.56805i 1.68300 0.409301i −0.542842 + 1.67070i 1.08227 0.624852i −2.55916 2.17272i 0.706282 + 0.784405i −0.448535 + 0.145738i 2.66495 1.37770i −2.21278 0.985195i
44.3 −0.888042 1.22228i −1.27658 + 1.17061i −0.0873279 + 0.268768i 3.07140 1.77328i 2.56448 + 0.520794i −2.50044 2.77703i −2.46770 + 0.801805i 0.259328 2.98877i −4.89498 2.17939i
44.4 −0.399926 0.550451i 0.304138 1.70514i 0.474979 1.46183i −1.60778 + 0.928255i −1.06023 + 0.514516i 0.402313 + 0.446814i −2.28881 + 0.743680i −2.81500 1.03720i 1.15395 + 0.513773i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 86.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
31.h odd 30 1 inner
93.p even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 93.2.p.b 64
3.b odd 2 1 inner 93.2.p.b 64
31.h odd 30 1 inner 93.2.p.b 64
93.p even 30 1 inner 93.2.p.b 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
93.2.p.b 64 1.a even 1 1 trivial
93.2.p.b 64 3.b odd 2 1 inner
93.2.p.b 64 31.h odd 30 1 inner
93.2.p.b 64 93.p even 30 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(12\!\cdots\!79\)\( T_{2}^{34} + \)\(31\!\cdots\!35\)\( T_{2}^{32} - \)\(76\!\cdots\!81\)\( T_{2}^{30} + \)\(16\!\cdots\!29\)\( T_{2}^{28} - \)\(32\!\cdots\!14\)\( T_{2}^{26} + \)\(56\!\cdots\!68\)\( T_{2}^{24} - \)\(85\!\cdots\!53\)\( T_{2}^{22} + \)\(11\!\cdots\!23\)\( T_{2}^{20} - \)\(12\!\cdots\!54\)\( T_{2}^{18} + \)\(10\!\cdots\!71\)\( T_{2}^{16} - \)\(68\!\cdots\!01\)\( T_{2}^{14} + \)\(29\!\cdots\!59\)\( T_{2}^{12} - \)\(11\!\cdots\!77\)\( T_{2}^{10} + \)\(47\!\cdots\!57\)\( T_{2}^{8} - 484368202445 T_{2}^{6} + 57419844963 T_{2}^{4} - 6212328222 T_{2}^{2} + 492884401 \)">\(T_{2}^{64} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(93, [\chi])\).